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UNIVERSITÀ DEGLI STUDI DI PAVIA

DOTTORATO DI RICERCA IN FISICA – XXI CICLO

The structure of the nucleon by electromagnetic

probes in Chiral Effective Field Theory

Marina Dorati

Tesi per il conseguimento del titolo

DOTTORATO DI RICERCA IN FISICA – XXI CICLO

THE STRUCTURE OF THE NUCLEON BY

ELECTROMAGNETIC PROBES IN CHIRAL EFFECTIVE

FIELD THEORY

dissertation submitted by

Marina Dorati

to obtain the degree of

DOTTORE DI RICERCA IN FISICA

Supervisors: Prof. Sigfrido Boffi (Università degli Studi di Pavia)

Dr. Thomas R. Hemmert (TU München -Universität Regensburg)

Prof. Ulf-G. Meißner (HISKP Universität Bonn)

Referee: Prof. Jiunn-Wei Chen (National Taiwan University)

Università degliStudi di Pavia

Dipartimento di FisicaNucleare e Teorica

Istituto Nazionale di Fisica Nucleare

Cover:

The structure of the nucleon by electromagnetic probes in Chiral Effective Field TheoryMarina Dorati

PhD thesis – University of PaviaPrinted in Pavia, Italy, November 2008ISBN 978-88-95767-17-8

Quark and gluon structure of a nucleon,

Jean-Francois COLONNA (CMAP/ECOLE POLYTECHNIQUE, www.lactamme.polytechnique.fr)

Contents

Introduction 1

1 Notions of Chiral Perturbation Theory 5

1.1 Effective Lagrangian and chiral symmetry . . . . . . . . . . . . 5

1.1.1 The mesonic sector . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 Chiral Perturbation Theory with baryons . . . . . . . . . 8

1.2 Regularization schemes in covariant baryon ChPT . . . . . . . . 11

1.3 Including the ∆(1232) as an explicit degree of freedom . . . . . 14

2 The Generalized Form Factors of the Nucleon 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Extracting the First Moments of GPDs . . . . . . . . . . . . . . 21

2.2.1 The generalized form factors of the nucleon . . . . . . . . 21

2.2.2 The generalized form factors of the pion . . . . . . . . . 24

2.3 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Leading order nucleon Lagrangian . . . . . . . . . . . . . 24

2.3.2 Consequences for the meson Lagrangian . . . . . . . . . 26

2.3.3 Power-counting in BChPT with tensor fields . . . . . . . 27

2.3.4 Next-to-leading order nucleon Lagrangian . . . . . . . . 28

2.4 The Generalized Isovector Form Factors in O(p2) BChPT . . . . 30

2.4.1 Moments of the isovector GPDs at t = 0 . . . . . . . . . 30

2.4.2 The slopes of the generalized isovector form factors . . . 36

2.4.3 The generalized isovector form factors of the nucleon . . 38

2.5 The Generalized Isoscalar Form Factors in O(p2) BChPT . . . . 40

2.5.1 Moments of the isoscalar GPDs at t = 0 . . . . . . . . . 40

2.5.2 The contribution of u and d quarks to the spin of thenucleon . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5.3 A first glance at the generalized isoscalar form factors ofthe nucleon . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6 The First Moments of axial GPDs . . . . . . . . . . . . . . . . . 51

2.6.1 The Generalized Axial Form Factors in O(p2) BChPT . . 53

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

I

CONTENTS

3 Doubly Virtual Compton Scattering with ∆’s 633.1 Introduction and summary . . . . . . . . . . . . . . . . . . . . . 633.2 Compton scattering and nucleon polarizabilities . . . . . . . . . 67

3.2.1 Real Compton scattering . . . . . . . . . . . . . . . . . . 673.2.2 Doubly virtual Compton scattering . . . . . . . . . . . . 70

3.3 The spin polarizabilities in covariant ChPT . . . . . . . . . . . . 743.4 Effective Field Theory with ∆’s. . . . . . . . . . . . . . . . . . . 76

3.4.1 Power-counting . . . . . . . . . . . . . . . . . . . . . . . 773.5 Covariant O(ε3) results for the VVCS amplitudes. . . . . . . . . 80

Summary and future perspectives 85

A Appendix to GPDs 87A.1 Feynman Rules for GPDs . . . . . . . . . . . . . . . . . . . . . 87A.2 Basic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90A.3 Regulator Functions . . . . . . . . . . . . . . . . . . . . . . . . 92A.4 Isovector vector Amplitudes in O(p2) BChPT . . . . . . . . . . 94A.5 BChPT Results in the Isoscalar vector Channel . . . . . . . . . 96

A.5.1 Isoscalar vector Amplitudes in O(p2) BChPT . . . . . . 96A.5.2 Estimate of O(p3) contributions . . . . . . . . . . . . . . 96

A.6 Moments of axial GPDs in O(p2) BChPT . . . . . . . . . . . . 99A.6.1 Axial Amplitudes in O(p2) BChPT . . . . . . . . . . . . 99

A.7 O(p3) calculation of the generalized form factors: preliminaryresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100A.7.1 Isovector vector channel . . . . . . . . . . . . . . . . . . 100A.7.2 Isoscalar vector channel . . . . . . . . . . . . . . . . . . 102

B Appendix to VVCS 103B.1 Feynman rules for VVCS . . . . . . . . . . . . . . . . . . . . . . 103B.2 Loop integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105B.3 The VVCS amplitudes in covariant SSE at O(ε3) . . . . . . . . 106

Bibliography 153

II

Introduction

In the last decades much effort has been made among physicists in order toimprove our knowledge about the structure of the nucleon. The magneticmoment of the proton, first measured in 1933 by Frisch and Stern, was theearliest experimental evidence for the internal structure of the nucleon. Begin-ning from that discovery, a series of successful experiments had followed andthanks to the surprising progress made in experimental technology, many ofthe structure parameters of proton and neutron are now known to incredibleprecision. Although the many goals achieved by nuclear physics on both theo-retical and experimental sides, a lot of opened questions around the structureof the nucleon still need to be solved, a lot of interesting issues still have to bediscussed.

Quantum Chromo Dynamics (QCD) is the quantum field theory of stronginteractions within the Standard Model. The nucleon is the lightest baryon inthe hadron spectrum, it is a complex system of interacting quarks and gluonsgoverned by strong interactions. In this work we are going to explore the lowenergy properties of this system. Because of the impossibility of QCD to allowfor the description of low energy properties and processes, we will make useof an effective field theory (EFT) called Chiral Perturbation Theory (ChPT),which represents the low energy limit of QCD.

The first aim of the present work is the analysis of the so-called general-ized form factors of the nucleon within the framework of Chiral PerturbationTheory. These objects are defined as the first moments of the so-called Gener-alized Parton Distributions (GPDs), a quite new class of functions which cane.g. be accessed in Deeply Virtual Compton Scattering (DVCS) off the nucleon.By performing the calculation up to the leading one loop order in covariantbaryon ChPT (second order in the chiral expansion), we will derive the mo-mentum transfer and quark-mass dependence of the generalized form factors.In particular, we will focus on the forward limit of these expressions, wherethe GPDs reduce to the well known Parton Distribution functions (PDFs) anda connection to phenomenology becomes possible. The ChPT results will bealso compared to available lattice QCD data, paving the way for chiral pre-dictions based on chiral extrapolation studies. Furthermore, an interestingaspect related to the generalized form factors is the possibility to gain fromtheir analysis information about the contribution of the quarks to the spin of

1

INTRODUCTION

the nucleon.

The investigation of the spin structure of the nucleon is another centraltopic of present nuclear physics activities. To this purpose a very specialrole is played by the process of forward Doubly Virtual Compton Scattering(VVCS) off nucleons, a particular case of Compton scattering which involvestwo virtual photons. The matrix element of this process can be related tocertain sum rules which have validity at all energy scales and are written interms of the nucleon spin structure functions g1 and g2. Among these sum rulesa really interesting task is represented by the so-called spin polarizabilities,linear combinations of moments of g1 and g2. Despite of the several theoreticaland experimental investigations of this issue carried out in the past years,further improvement in the results is still demanded. In particular one has tobetter understand and estimate the contributions of the nucleon resonances tothe values of the VVCS matrix element. The ∆ resonance plays an importantrole in the nucleon spin sector and thus its inclusion in the calculation ina systematic manner is called for. A second aim of the present work is toextend the chiral analysis of the process of Doubly Virtual Compton Scatteringpresented in reference [BHM03b] including the ∆ resonance as explicit degree offreedom in the calculation. We will present a leading-one-loop order calculationof the spin amplitude of the process of Doubly Virtual Compton Scattering inthe small scale expansion scheme of covariant Chiral Perturbation Theory.

This work is organized as follows: chapter 1 will deal with essentials ofChiral Perturbation Theory. The formalism needed to perform calculationsis provided, with special regards to the regularization schemes in covariantbaryon ChPT and to the general properties of a chiral EFT for spin-3/2 fields.In particular, technical details about the recently introduced IR regularizationscheme are given, with emphasis on the advantages that the adoption of sucha scheme can bring about.Chapter 2 will be dedicated to the discussion of the first moments of the parity-even Generalized Parton Distributions (GPDs) in a nucleon, corresponding tosix (generalized) vector form factors. We explore the chiral analysis of thethree (generalized) isovector and (generalized) isoscalar form factors, whichare currently under investigation in lattice QCD. We study the limit of vanish-ing four-momentum transfer and we first concentrate on the isovector moment〈x〉u−d; our BChPT calculation predicts a new mechanism for chiral curvature,connecting the high values for this moment typically found in lattice QCD stud-ies for large quark masses with the smaller value known from phenomenology.Likewise, we analyse the quark-mass dependence of the isoscalar moments inthe forward limit and extract the contribution of quarks to the total spin ofthe nucleon. We also proceed with a first glance at the momentum dependenceof the isoscalar C-form factor of the nucleon. We close with the analysis ofthe generalized axial form factors of the nucleon, presenting promising resultsfrom a combined fit of different GPDs-moments.In chapter 3 we will first discuss the generalization of the Gerasimov-Hearnsum rule, the Burkhardt-Cottingham sum rule, the nucleon spin polarizabil-ities and other moments of the nucleon spin structure functions. Afterwards

2

INTRODUCTION

we will describe in detail the calculation of the matrix element of the processof Doubly Virtual Compton Scattering (VVCS) up to leading one loop order(third order in the chiral expansion) in covariant baryon Chiral PerturbationTheory, providing the required technicalities for the inclusion of the ∆ reso-nance as explicit degree of freedom in the calculation. The results for the spinamplitudes corresponding to the various diagrams contributing to the processare finally summarized, while for a complete and detailed discussion of theresults presented in this chapter we postpone to a later communication.

3

Chapter 1Notions of Chiral PerturbationTheory

Quantum Chromo Dynamics (QCD) is the fundamental quantum field theoryof strong interactions. Thanks to asymptotic freedom high energy regions areprobed by means of perturbartive techniques. At low momenta and energies(Q < 1 GeV) the runnning coupling αs(Q) is of order one so that an expansionin powers of αs is no longer practical. Moreover, QCD is written in terms of thewrong degress of freedom (quarks and gluons), while low energy experimentsare performed with hadrons.QCD can be mapped on a (chiral) effective field theory (EFT) at low energies.This effective field theory is called Chiral Perturbation Theory (ChPT) and itallows the description of low energy properties and processes within a pertur-bative approach.In this chapter we are going to explore the basic concepts underlying the the-ory of Chiral Perturbation Theory; we provide the standard ingredients forthe formulation of the theory, referring to the specific chapters for pertinenttechnical details.For further information about effective field theories and ChPT we refer toreference [Wei76] and to the recent reviews [Ber07, BM06, Sch03].

1.1 Effective Lagrangian and chiral symmetry

The large value of the strong coupling constant αs forbids a perturbative de-velopment of QCD in the low energy domain, making the testing of QCD inthis region a very investigated task for theoretical physics.In the seventies the concept of effective field theory (EFT) has emerged [Wei76],leading to the development of Chiral Perturbation Theory (ChPT) as a model-independent way of solving QCD [GL84]. The main characteristic of EFTs isthe presence of a specific energy scale Λ. The value of this scale represents athreshold, it selects the relevant degrees of freedom when compared to the par-ticular energy of working. In the case of ChPT, the scale Λ is determined bythe spontaneous breaking of chiral symmetry, with consequent identification

5

1. Notions of Chiral Perturbation Theory

of the meson pions (in the particular case of two light flavors) as responsiblefor the dynamics of a strongly interacting system. The main point of this ap-proach is that the effective Lagrangian has to preserve all the symmetries of theoriginal QCD Lagrangian: the effective Lagrangian has to contain all possibleterms compatible with the assumed symmetry principles of the fundamentaltheory, in this particular case all symmetry properties of QCD.

The strong interaction as described by the Standard Model involves Nf = 6different flavors of quarks described by the fermion spinor q and the gluons,vector gauge bosons that mediate strong color charge interactions and aredescribed by the non-abelian field strength tensor Gµν . The ordinary QCDLagrangian reads:

LQCD =∑Nf

q(iγµDµ −mq)q − 1

4Ga

µνGµν a, (1.1)

where mq denotes the mass of the specific quark q, γµ are Dirac matrices andDµ is the gauge covariant derivative.According to a typical hadron scale of 1 GeV, quarks are separeted into light(u,d,s) and heavy (c,t,b). Since we are interested in processes at low energy,we choose to restrict ourselves to the light quark sector and in particular tothe lightest u and d quarks. The QCD Lagrangian has the property to beinvariant under parity P, charge conjugation C and time reversal T transfor-mations. In the limit mq → 0, the corresponding Lagrangian L0

QCD is invariantunder independent left and right handed rotations of the quark fields in flavorspace. Therefore the Lagrangian L0

QCD is symmetric under the flavor groupSU(2)L × SU(2)R

1, which takes the name of chiral symmetry group. Thereal world reveals that chiral symmetry is a symmetry of L0

QCD but not ofthe ground state: the chiral symmetry group SU(2)L × SU(2)R is sponta-neously broken down to its vectorial subgroup SU(2)V . Low energy hadronphenomenology shows evidence of the spontaneous symmetry breaking via thenon-existence of degenerate parity doublets in the hadron spectrum, i.e statesof equal spin, baryon number and strangeness but opposite parity. In termsof charge operators, this evidence means that the vacuum (the ground state)is not annihilated by both vector and axial charge operators, i.e. the condi-tion QV

a |0〉 = QAa |0〉 = 0 is not fulfilled. Assuming QV

a |0〉 = 0, one has toconclude that QA

a |0〉 6= 0, i.e. the ground state of QCD is not invariant underaxial transformations. According to the Goldstone’s theorem [Gol61], chiralsymmetry and its spontaneous symmetry breaking thus imply the existence ofN2

f − 1 = 3 massless pseudoscalar (QAa is an axial charge!) bosons. Because of

the non-zero quark mass terms, responsible for the so-called explicit breakingof chiral symmetry, such 0− massless bosons do not exist. The best candi-dates to be identified with them are are the pseudoscalar mesons π, which are

1The Lagrangian L0QCD is invariant under the global transformations of the group

SU(2)L × SU(2)R × U(1)V × U(1)A. In this work we are not concerned with the vecto-rial U(1)V and the axial U(1)A groups. The vector one is an exact symmetry, it results inbaryonic number conservation and leads to a classification of hadrons into mesons (B=0)and baryons (B=1). The axial one is exact in the classical theory but violated in the fullquantum field theory; this phenomenon is called the axial U(1) anomaly.

6

1.1. Effective Lagrangian and chiral symmetry

the lightest particles in the hadron spectrum according to the Particle PhysicsBooklet.

Because of the small value of the quark mass, the explicit breaking ofchiral symmetry represents a small effect when compared to the mechanismof spontaneous chiral symmetry breaking and may be therefore treated asa perturbation. Given that the interaction between the pions is weak andthat the quark mass is very small, an expansion in the external momentump and in the mass mq is possible, as long as one stays below the scale Λχ ∼1 GeV, with p,mq ¿ Λχ. Chiral Perturbation Theory takes its origin as theeffective field theory of QCD with pions as the explicit degrees of freedom andwith validity up to the scale Λχ. Concluding, in order to be the low energylimit of LQCD the effective Lagrangian LChPT has to be the most generalLagrangian invariant under the discrete parity (P), charge conjugation (C)and time reversal (T) transformations, in the chiral limit mq → 0 has to beinvariant under SU(2)L × SU(2)R, while the ground state has to be invariantonly under the subgroup SU(2)V .

1.1.1 The mesonic sector

We now proceed with the construction of the effective Lagrangian in themesonic sector. We denote the dynamical variables representing the pion field

by the SU(2) matrix field U = exp(

iτ iπi

Fπ

), where τ i with i = 1, 2, 3 are the

Pauli matrices in the isospin space, πi are the Klein-Gordon pion field oper-ators and Fπ is the pion decay constant. ChPT is characterized by countingrules: since the field U counts as a quantity O(1), ∂µU counts as O(p). Atlow energy, an expansion in powers of meson momenta is equivalent to an ex-pansion of Leff in powers of the derivative ∂µU . Lorentz invariance restrictsthe form of the Lagrangian to only terms with an even number of derivatives,thus the general expression Leff = L(2)

eff + L(4)eff + . . ., where the superscripts

(n = 2, 4, . . .) mark the low-energy dimension.Let us concentrate on the leading part of the Lagrangian, given for n = 2, onlypurely meson term used in the present work. One has:

L(2)ππ =

F 2π

4Tr

[∇µU†∇µU + χ†U + χU †] . (1.2)

The covariant derivative is defined as

∇µU = ∂µU − i (vµ + aµ) U + iU (vµ − aµ) , (1.3)

and contains the couplings to arbitrary vector vµ and axial aµ fields, like forexample photons and W bosons, in the order. As the partial derivative ∂µ, thefields vµ, aµ count as O(p).The field χ = 2B(s + ip) contains the free parameter B and is a linear combi-nation of scalar s and pseudoscalar p fields. The constant B is related to thespontaneous symmetry breaking of the theory, measuring the strength of the

7


chiral condensate via 〈0|uu|0〉 = −F 2πB + O(M), where M = diag(mu,md),

with mu and md the mass of the up and down quark.It is straightforward to see that the symmetry breaking part of the effective

Lagrangian L(2)SB = F 2

π

4Tr

[χ†U + χU †] is proportional to the sum of the quark

masses. Let us set all external sources to zero except for s = M, , so thatχ = 2B diag(mu,md). Expanding in powers of the pion fields one finds:

L(2)ππ = −F 2

πB(mu + md) (1.4)

+1

2

[∂µπ

i∂µπi −B(mu + md)πiπi

]+O(π4).

The first term is thus related to the vacuum expectation value of the mass term−(mu〈0|uu|0〉+md〈0|uu|0〉), while the contribution at order π2 corresponds tothe Klein-Gordon equation for a free pion of mass m2

π = B(mu + md). Thanksto this relation one can always translate the quark-mass dependence of physicalobservables into a pion-mass dependence up to a certain order. According tothe standard ChPT scenario, we assume that B counts as O(1) and mq isO(p2), so that mπ has to be counted as a quantity of order p. Beside thesechiral counting rules a systematic method to estimate the importance of loopdiagrams is also demanded. We make use of Weinberg power-counting scheme[Wei79], according to which a one to one correspondence between loop andchiral order exists, that is diagrams with NL meson loops are suppresed bypowers of (p2)L. The chiral dimension D of a loop diagram in the mesonicsector can then be calculated as D = 2+

∑d Nd(d− 2)+2NL, where Nd is the

number of vertices with chiral dimension d, i.e. generated by a Lagrangian ofchiral dimension d, and NL is the number of pion loops.

1.1.2 Chiral Perturbation Theory with baryons

Chiral Perturbation Theory can be extended to include baryons. We constructa theory able to describe the dynamics of baryons, their interaction with pionsand external fields at low energy. In particular, we are interested in the pion-nucleon (πN) system.We still makes use of the matrix field operator U(x) representing the pionand we denotes by ΨN the isospin doublet nucleon field. We introduce thecovariant derivative Dµ acting on the nucleon field

DµΨN =(∂µ + Γµ − iv(s)

µ

)ΨN , (1.5)

where

Γµ =1

2

[u†, ∂µu

]− i

2u† (vµ + aµ) u− i

2u (vµ − aµ) u† (1.6)

is the so-called chiral connection expressed in terms of the external vectorand axial background fields vµ, aµ, v

(s)µ defines an isosinglet vector field and

u2 = U . Together with the covariant derivative, another building block with

8

1.1. Effective Lagrangian and chiral symmetry

one derivative is the so-called vielbein uµ ≡ i(u†∇µu− u∇µu

†), which underparity transforms as an axial vector.In order to write the leading order Lagrangian we fix some counting rulesassigning chiral dimension p0 to the the elements {M0, Ψ, Ψ, DµΨ, ΨΨ, ΨγµΨ,Ψγµγ5Ψ, ΨσµνΨ, Ψσµνγ5Ψ} and chiral dimension p1 to the objects {(i /D −M0)Ψ,Ψγ5Ψ}, where M0 defines the nucleon mass in the chiral limit.The leading order pion-nucleon Lagrangian thus takes the form [GSS88]:

L(1)πN = ΨN

[iγµDµ −M0 +

gA

2γµγ5uµ

]ΨN , (1.7)

where gA is the chiral limit of the axial coupling constant.For explicit expressions of higher order up to O(p4) πN Lagrangian we

refer to reference [FMMS00]. Their general form can be written in terms of

field monomials O(n)i (the superscript n denotes as usual the chiral order) as

following:

L(2)πN =

7∑i=1

ciΨNO(2)i ΨN , L(3)

πN =23∑i=1

diΨNO(3)i ΨN , L(4)

πN =118∑i=1

eiΨNO(4)i ΨN .

(1.8)

The coefficient of the monomials are low energy constants (LECs) of the the-ory. At higher order the Lagrangians becomes very long and the number of theLECs explodes rapidly. Their values are not constrained by symmetries andmost of them show a scale dependence. Indeed, calculation involving one loopgraphs yield divergent integrals thus the need for renormalization. Since weare working within the framework of an effective field theory, all possible struc-tures conforming to the symmetries of the system are already contained in theLagrangians. The LECs are therefore the best suited objects to act as countert-erms for the renormalization of the theory. As will be discussed later on, loopdiagrams in the nucleon sector start to appear at chiral dimension D = 3. Thisimplies that the low energy constants appearing in lower order Lagrangians arenot affected by loop effects and are therefore scale-independent. In particular,the constants ci belonging to L(2)

πN have been determined by some accuracy fromlow-energy hadron phenomenology. In order to develop a regularized theory,the dimension-three and -four LECs are decomposed into a finite and infinitepart [Eck94, FMS98, MMS00]:

di ≡ dri (λ) +

βi

F 2π

L, (1.9)

where

L =λd−4

16π2

[1

d− 4+

1

2(γE − 1− ln(4π))

], (1.10)

with λ the renormalization scale, d denotes the space-time dimension andγE the Euler-Mascheroni constant. The βi appearing in the infinite part ofthe decomposition denotes the β-function associated with the corresponding

9


couterterm and depends on coupling constants of the theory. The value of therenormalized LEC dr

i (λ) can be finally fixed from phenomenology. The dri (λ)

at two different scales λ1, λ2 are connected by the following relation:

dri (λ2) = dr

i (λ1)− βilogλ2

λ1

. (1.11)

The appearance of a new energy scale, namely the nucleon mass M0, pre-vents the extension of the power-counting scheme introduced in the mesonicsector to include the pion-nucleon system. Because of the zeroth component,one can not say that a derivative acting on the baryon field results in a smallfour-momentum. Since the nucleon mass M0 is of about the same size asthe scale 4πFπ appearing in the calculation of pion-loop contribution, termscoming from higher order diagrams are no more suppressed and a consistenthierarchy of loop diagrams is no more ensured. This issue represents an his-torical problem of ChPT, first noted by the authors of reference [GSS88] whendealing with chiral loops within the scheme of dimensional regularization (MS).Many physicists have faced the task during the years and several interestingalternative solutions have been suggested. The problem was first cleared up byJenkins and Manohar [JM91a], who proposed to apply to the pion-nucleon sec-tor the heavy quark effective field theory methods used in heavy quark physics.The idea is to consider the baryons as extremely heavy sources surrounded bya cloud of light particles (the pions) and to split up their momentum into alarge piece of the order of their mass and a small residual component. One haspµ = M0v

µ + rµ, where vµ is a velocity vector satisfying v2 = 1 and v · r ¿ M0.At this point we introduce the velocity projector operators P±

v ≡ 1±/v

2and

we apply them to the nucleon field defining the two velocity-dependent fieldsNv ≡ e−iM0v·xP+

v ΨN and hv ≡ e−iM0v·xP−v ΨN . The nucleon field can now be

written as ΨN = e−iM0v·x(Nv +hv), where we note that the dependence on thenucleon mass is factorized out through the exponential term. Inserting thisexpression of the nucleon field into the leading order Lagrangian of eq.(1.7)one finds that the heavy component hv is formally suppressed by the powerof 1/M0 relative to the light component Nv. The non-relativist limit of L(1)

πN

finally readsL(1)

πN = N [iv ·D + gAS · u] N, (1.12)

with Sµ = i2γ5σµνv

ν the so-called Pauli-Lubanski spin vector. A systematicformulation of this approach was developed in ref.[BKKM92].Since it is based on the idea of treating baryons as extremely heavy, thisframework is called Heavy Baryon Chiral Perturbation Theory (HBChPT).This construction of the effective Lagrangian guarantees that terms of the kindM0

4πFπ∼ 1 will not appear in the calculation, allowing for the formulation of a

power-counting scheme analogous to the mesonic sector. In the non-relativisticapproach of HBChPT the one-to-one correspondence between the loop and thesmall momentum expansion is newly restored. The chiral dimension D of aFeynman diagram can be then derived via

D = 2NL + 1 +∑

d

(d− 2)NMd +

∑

d

(d− 1)NMBd , (1.13)

10

1.2. Regularization schemes in covariant baryon ChPT

power analyticity non-relativistic ultravioletcounting limit regularization

IR X X X Xstandard IR X - X (X)

MS - X - XEOMS [FGJS03] X X - X

Table 1.1: Overview over several renormalization schemes of BChPT frequentlydiscussed in literature and how they behave with respect to the four conditionslisted in the text [Gai07]. The IR scheme is designed in such a way to fulfillall of the four conditions.

where NL again denotes the number of loops in the diagram, NMd is the number

of vertices generated by a meson Lagrangian of order d and NMBd the number

of vertices from a meson-baryon Lagrangian of order d. The power-countingis very similar to the mesonic one. We note that D ≥ 1 and the loop startscontributing at D = 3. This implies that the second order LECs from L(2)

πN

are scale-independent, they do not contribute to the renormalization of theinfinities from one-loop calculations2.However, the HBChPT prescription still shows some deficiencies: under certaincircumstances the heavy baryon approach may generate Green functions whichdo not satisfy the analytic properties which characterize a fully relativistic fieldtheory [BKM96]. The search for a method which encloses the advantages ofthe heavy baryon approach (presence of a consistent power-counting scheme)as well as of a relativistic theory (absence of non-analyticities) has led to thedevelopment of the so-called Infrared regularization scheme (IR) , which willbe the topic of the next section. As we will see, even this technique has to bemodified in order to develop a consistent renormalization scheme for covari-ant baryon ChPT (BChPT). Essentials of this recently introduced innovativescheme will also be given.

1.2 Regularization schemes in covariant baryon

ChPT

In this section basic aspects of renormalization in covariant BChPT will bediscussed, with particular emphasis on the infrared (IR) regularization schemeand a new scheme called IR recently introduced by the authors of reference[GH06] and described in detail in reference [Gai07]. See these references forfurther information about this modified version of the infrared regularizationprescription.The infrared regularization scheme (IR) was first introduced by Becher and

2We underline that these conditions are fulfilled for strongly interacting systems in thepresence of vector (v), axial (a), scalar (s) and pseudoscalar (p) fields. As we will see inchapter 2, the presence of additional fields nullifies those conditions.

11


Leutwyler [BL99] as solution to the known deficiency of a rigorous power-counting in the dimensional MS scheme of covariant BChPT and in order toavoid the production of non-analytic structures as can happen in the heavybaryon formulation of ChPT. The basic idea underlying infrared regulariza-tion consists of splitting the relativistic one-loop integrals into ”soft”and ”hard”parts. The ”soft”part is regulated by the power-counting scheme as in HBChPT,while the terms contributing to the ”hard” part can be absorbed in the LECsof the theory. Strictly speaking this means that by a particular choice of Feyn-man parametrization any dimensionally regularized one-loop integral H canbe separated into an infrared singular part I and a regular part R. In thelow energy region H and I have the same analytic properties while the con-tribution of R, which corresponds to an infinite sum of terms analytic in thequark mass, can be taken into account just by a proper modification of the lowenergy constants appearing in the general effective Lagrangian. A calculationwhich adopts the IR regularization scheme is obtained by replacing the generalintegral H by its infrared singular part I and by dropping the regular part R.The method of infrared regularization produces a consistent power-counting,to be specific it is characterized by a power-counting scheme which follows thesame formula of eq.(1.13) as the heavy baryon approach; the chiral dimensionD of a Feynman diagram in the IR regularization scheme is then obtained via

D = 2NL + 1 +∑

d

(d− 2)NMd +

∑

d

(d− 1)NMBd . (1.14)

Besides the existence of a meaningful power-counting prescription, the IRscheme also owns the property that an expansion in 1/M0 of the covariantresult exactly reproduces the corresponding non-relativistic heavy baryon ex-pression. This implies that all coupling constants are defined in both schemes(IR and HBChPT) at the same way and their finite parts do have the samenumerical values.However, in addition to power-counting and non-relativistic limit, a consistentrenormalization scheme in BChPT has to fulfill two further conditions: it hasto provide a satisfying ultraviolet regularization and to guarantee the absenceof unphysical non-analyticities. Unfortunately, the IR prescription misses bothrequirements. Indeed, a loop integral calculated in IR can contain divergencesand scale dependent logarithms beyond the order of the calculation. Since ata given order only counterterms from the effective Lagrangian correspondingto the order of the calculation can contribute to the result, the higher orderdivergences and scale-dependent logarithm can not be properly absorbed intothe LECs. The authors of reference [BL99] have argued that the matching be-tween the HBCHPT approach and the relativistic theory requires λ = M0 andhave suggested to remove by hand the divergences which can not be absorbedby the LECs since they represent higher order corrections. This procedure canin principle bring to acceptable numerical results up to the physical pion massbut is off course constrained by the choice of the scale λ and definitely notcompletely satisfactory.Furthermore, one finds out that the infrared regular parts of the loop integrals

12

1.2. Regularization schemes in covariant baryon ChPT

How to calculate loop diagrams in IR- a practitioner’s guide -

1.Calculate all diagrams in MS as usual.

2.The infrared regular part of this result isfound if all integrals H11 appearing in the

MS result are replaced by R11.

3.The IR result is found, if the regular part

expanded in the pion-mass up to the powerat which a counter-term is available is

added to the MS result.

Table 1.2: The three steps which have to be performed in an IR calculation inpractice [Gai07].

can be non-analytic, with bad consequences on analysis of chiral extrapolation.An example for an analyticity problem in IR can be seen in the investigationof the quark-mass dependence of the nucleon mass. Indeed, when looking atthe O(p3) covariant expression for the nucleon mass in the IR scheme given in[PHW04], one can easily observe that for mπ → 2M0 the arccos structure di-verges and becomes complex afterwards. This singularity becomes numericallysignificant for values of the pion mass already outside the range of applicabilityof the theory (above 700 MeV), but the quark-mass dependence of some quan-tities - e.g the isovector magnetic moment κv [GH07, Gai07]- can be anywaystrongly influenced for values of the pion mass not far away from the physicalone.As alternative to the procedure dictated by infrared regularization, the au-thors of [GH06] has developed a new scheme which fulfills all four conditionsdiscussed above. The new scheme ensures that only the terms of the regularpart which can really be absorbed by the proper counterterms (that is by thecounterterms corresponding to the order of working) will appear in the calcu-lation. This is achieved by introducing a new -modified- regular part obtainedby expanding the standard regular part of the loop integral up to the powerof the pion mass at which a corresponding counterterm is available. The finalresult in the IR regularization scheme can be then calculated as the sum ofthe MS result plus the new regular part as defined above. Table 1.2 summa-rizes the steps one has to follow in order to reproduce a calculation in the IRprescription, while table 1.1 shows in a schematic way how the regularizationschemes known from the literature fulfill the four basic conditions which setup a consistent regularization scheme.An example of calculation and relative results within the IR scheme will be

13


provided in chapter 2.We conclude listing the main properties and advantages of this new renormal-ization scheme:

¦ within this framework all ultraviolet divergences cancel and the result isindependent of the renormalization scale;

¦ a rigorous power-counting scheme exists, allowing for a well defined hier-archy of the Feynman diagrams. As happens for the IR scheme, even inthis case the power-counting formula which provides the chiral dimensionof a Feynman diagram is exactly the same as in HBChPT (see eq.(1.13));

¦ the 1/M0 expansion of the IR result reproduces exactly the non-relativisticlimit of HBChPT ;

¦ the LECs are defined in the same way in IR and IR as well as in HBChPT,so that new information about the couplings has validity in both schemes;

¦ in contrast to the standard infrared scheme, IR results show no unphys-ical cuts, singularities or imaginary parts.

1.3 Including the ∆(1232) as an explicit de-

gree of freedom

In section 1.1.2 we have seen how to include spin-12

baryons in the framework ofa chiral effective field theory. Pions and nucleons are the dynamical degrees offreedom in Baryon Chiral Perturbation Theory. Resonant baryons are assumedto be very heavy compared to the nucleon, so that they can be integrated outand their contribution appears in the form of counterterms. This approachcan be considered reasonable for heavier resonances such as the Roper andhigher states but looses its validity when applied to the case of the ∆(1232)resonance. Indeed, the spin-3/2 isospin-3/2 ∆(1232) resonance lies only about300 MeV above the nucleon mass and couples very strongly to the π − Nsector. Phenomenology indicates that the inclusion of the ∆ effects via a meremodification of the low energy constants of the theory is not always sufficientin order to estimate the real contribution provided by the ∆(1232) in manylow- and medium-energy processes. Thus the need to include the ∆(1232)as an explicit degree of freedom in the chiral EFT, as was first suggested byJenkins and Manohar in reference [JM91b].

In the second half of the nineties the authors of [HHK98] have intro-duced the so-called small scale expansion scheme (SSE), a proper power-counting scheme based on the essentials of HBChPT but developed in sucha way that nucleon and ∆ degrees of freedom appear simultaneously in theeffective Lagrangian. The main idea underlying the SSE scheme is to in-troduce an additional small parameter, that is the ∆-nucleon mass splitting∆ ≡ m∆ −M0 = 294 MeV and to set up a low energy expansion in power of

14

1.3. Including the ∆(1232) as an explicit degree of freedom

the parameter ε with

ε ∈{

p

Λχ

,mπ

Λχ

,∆

Λχ

}, (1.15)

where Λχ ' 1 GeV denotes as usual the scale of chiral symmetry breaking.The mass splitting ∆ is therefore treated as a small parameter together withexternal momenta and pion mass. We also observe that it is a dimensionfulparameter of the theory which stays finite in the chiral limit. Because it isbased on the main concepts of the heavy baryon approach and it consists in aphenomenological expansion in the small parameter ε, the SSE scheme can beregarded as a phenomenological extension of HBChPT.Alternative approaches to the inclusion of spin-3/2 particles in chiral EFT havebeen presented in references [DJM94, PP03].

Following reference [HHK98], we now proceed to the development of a chiralEFT for spin-3/2 fields. We adopt the Rarita Schwinger formalism [RS41] andwe represent the ∆ field as a vector-spinor field Ψµ(x). The standard generalform for the relativistic Lagrangian for a spin-3/2 field of mass m∆ is:

L3/2 = Ψα 3

2 Λ(A)αβΨβ, (1.16)

where α,β are Lorentz indeces and32 Λ(A) denotes a matrix depending on a

free unphysical parameter A (A 6= −1/2). This dependence is introducedin order to ensure that the Lagrangian is invariant under the so-called pointtransformation

Ψµ(x) → Ψµ(x) + aγµγνΨν(x),

A → A− 2a

1 + 4a. (1.17)

This transformation means that the presence of a quantity a (a 6= −1/4) of”off-shell”spin-1/2 components γνΨ

ν(x), which always appear in the relativisticspin-3/2 field Ψµ(x), can be compensated by a corresponding change in theparameter A, so that the Lagrangian remains invariant. One can think oftransferring the A-dependence of the matrix

32 Λ(A) to another matrix OA

µν

defined as

OAαµ = gαµ +

1

2Aγαγβ, (1.18)

so that the Lagrangian takes the form:

L3/2 = ΨαO

A 32

αµ ΛµνOAνβΨβ. (1.19)

If we now redefine the spin-3/2 field as

ψµ(x) = OAµνΨ

ν(x), (1.20)

15


we can finally write

L3/2 = ψ32

µ Λµνψν , (1.21)

and we can thus work with an A-independent Lagrangian written in terms ofthe ψµ fields.

The next step is to find a way to match spin-3/2 dynamics and the isospinproperties of the ∆ resonance. The ∆(1232) carries isospin I = 3/2. Thefour physical states of the ∆ can be reproduced if we treat the spin-3/2 fieldψµ(x) as an isospin-doublet and we attach to it an additional isovector indexi = 1, 2, 3. Because of this double index, the ∆ field is a vector and spinor inboth spin and isospin spaces. This construction has to be constrained by thecondition

τ iψiµ(x) = 0, i = 1, 2, 3 (1.22)

with τ i a 2-component Pauli matrix in the isospin space, in order to eliminatethe two extra degrees of freedom generated by such a notation. A representa-tion for the three isospin doublets is for example:

ψ1µ =

1√2

[∆++ − 1√

3∆0

1√3∆+ −∆−

]

µ

ψ2µ =

i√2

[∆++ + 1√

3∆0

1√3∆+ −∆−

]

µ

ψ3µ = −

√2

3

[∆+

∆0

]

µ

(1.23)

Finally, the general form of the relativistic Lagrangian for a spin-3/2, isospin-3/2 field reads:

LS=3/2,I=3/2 = ψµ 3

2i Λ(0)ij

µν ψνj . (1.24)

In addition to this Lagrangian which will be used to describe the ∆ system,the general effective Lagrangian relevant for the study of a general interactingsystem made up of pions, nucleons and ∆ resonances will off course includethe following structures:

Leff = LπN + Lππ + LπN∆ + Lπ∆ + L∆. (1.25)

Explicit expressions of some terms of the effective Lagrangian will be given inchapter 3, where the formalism introduced in this section will be applied tothe description of a particular low energy process.Adopting the choice A = −1, the propagator for a spin-3/2 isospin-3/2 particle

16

1.3. Including the ∆(1232) as an explicit degree of freedom

within the Rarita-Schwinger formalism in d space-time dimensions has thegeneral form:

Gµν(p) = Gijµν(p)ξij

3/2, (1.26)

where

Gijµν(p) = −i

/p + m∆

p2 −m2∆ + iε

{gµν − 1

d− 1γµγν − (d− 2)

(d− 1)

pµpν

m2∆

+pµγν − pνγµ

(d− 1)m∆

},

(1.27)

and ξij3/2 is an isospin projection operator, which together with the correspon-

dent isospin-1/2 term has the following properties:

ξ3/2ij + ξ

1/2ij = δij, (1.28)

ξIijξ

Jjk = δIJξJ

ik, (1.29)

with

ξ3/2ij = δij − 1

3τ iτ j =

2

3δij − i

3εijkτ

k, (1.30)

ξ1/2ij =

1

3τ iτ j =

1

3δij +

i

3εijkτ

k. (1.31)

The ∆ propagator can be rewritten via the projection operators P3/2µν and P

1/2µν

into its spin-3/2 and 1/2 components [BHM03a]:

Gµν(p) = − /p + m∆

p2 −m2∆

P 3/2µν − 1√

d− 1m∆

((P

1/212

)µν

+(P

1/221

)µν

)(1.32)

+d− 2

(d− 1)m2∆

(/p + m∆)(P

1/222

)µν

.

We observe that the spin-1/2 pieces do not propagate and thus in principle canbe absorbed in purely polynomial terms with consequent redefinition of theLECs of the theory. This has been shown for example in reference [BHM03a,TE96] for the nucleon mass. That calculation has demonstrated how the spin-1/2 contribution to the nucleon self-energy can be completely absorbed intopolynomial terms appearing in the chiral expansion of the nucleon mass beyondleading-one-loop order. This reference represents one of the first attempts todevelop a consistent extension of the infrared regularization method in thepresence of spin-3/2 fields, that is a covariant formulation of the small scaleexpansion, up to that time mainly applied in its heavy baryon approach. Weremark that the fact that spin-1/2 terms simply redefine counterterms is nota characteristic of the covariant formalism but the same happens also in thenon-relativistic SSE scheme [HHK98]. The calculation of the nucleon mass incovariant SSE has been later performed at the next-to-leading one loop orderin reference [BHM05], where results for the ∆ mass are also provided.

17


Another example of calculation in the covariant SSE scheme will be presentedin chapter 3, where further technical details about chiral EFT with inclusionof the ∆ resonance will be given.

An effective field theory based on the chiral symmetry properties of QCD inthe zero-mass limit has been introduced. Working in a relativistic frameworkwe have written the main Lagrangians describing the nucleon-system and itsinteraction with pions. The advantages/disadvantages of different renormaliza-tion schemes have been discussed and a systematic manner to include spin-3/2fields as explicit degree of freedom in the calculation has been explored. Weare now ready to start our chiral analysis of the structure of the nucleon byelectromagnetic probes; further pertinent technical details necessary for theunderstanding of the discussed calculations can be found in the specific chap-ters.

18

Chapter 2The Generalized Form Factorsof the Nucleon

2.1 Introduction

About a decade ago the concept of Generalized Parton Distributions (GPDs)has emerged among theorists, constituting a universal framework bringing ahost of seemingly disparate nucleon structure observables like form factors,moments of parton distribution functions, etc. under one theoretical roof. Forreviews of this very active field of research we refer to references [Ji98, Die03,BR05, BP07].Working in twist-two approximation, the parity-even part of the structureof the nucleon is encoded in two Generalized Parton Distribution functionsHq(x, ξ, t) and Eq(x, ξ, t). For a process where the incoming (outgoing) nucleoncarries the four-momentum pµ

1 (pµ2) we define two new momentum variables

qµ = pµ2 − pµ

1 ; p = (pµ1 + pµ

2)/2. (2.1)

The GPDs can e.g. be accessed in Deeply Virtual Compton Scattering (DVCS)off the nucleon where – in the parton model of the nucleon – the GPD variablex can be interpreted as the fraction of the total momentum of the nucleoncarried by the probed quark q. t = q2 denotes the total four-momentumtransfer squared to the nucleon, whereas the “skewdness” variable ξ = −n · q/2with n · p = 1 interpolates between the t- and the x dependence of the GPDs,for details see the reviews [Ji98, Die03, BR05, BP07].The three-dimensional parameter space of GPDs is vast and rich in informationabout nucleon structure. The experimental program for their determination isonly at the beginning at laboratories like CERN, Desy, JLAB, etc. [Ji98, Die03,BR05, BP07]. However, moments of GPDs can be interpreted much easierand are connected to well established hadron structure observables. E.g. thezeroth order Mellin moments in the variable x correspond to the contributionof quark q to the well known Dirac- and Pauli form factors F1(t) and F2(t) of

19

2. The Generalized Form Factors of the Nucleon

the nucleon:∫ 1

−1

dxHq(x, ξ, t) = F q1 (t), (2.2)

∫ 1

−1

dxEq(x, ξ, t) = F q2 (t), (2.3)

We recall that the Dirac- and Pauli form factors appear when writing the mostgeneral form of the nucleon matrix element of the vector current Vµ = qγµq:

〈p2|Vµ|p1〉 = u(p2)

[F q

1 (t)γµ + F q2 (t)

iσµαqα

2MN

]u(p1) (2.4)

where u (u) is a Dirac spinor of the incoming (outgoing) nucleon of mass MN

for which the quark matrix element is evaluated.The form factors play a central role in the understanding the structure of thenucleon. For the case of 2 light flavors the isoscalar and isovector Dirac andPauli form factors of the nucleon have been studied at low values of t at the one-loop level in Chiral Effective Field Theory, both in non-relativistic [BFHM98]and in covariant [GSS88, KM01, SGS05] schemes. The chiral extrapolationof these form factors for lattice QCD data with the help of ChEFT has beendiscussed in refs.[HW02, G+05, AKNT06, PPV06], whereas the status of theexperimental situation is reviewed in ref.[dJ06].

We want to focus on the first moments in x of these nucleon GPDs

∫ 1

−1

dx xHq(x, ξ, t) = Aq2,0(t) + (−2ξ)2Cq

2,0(t), (2.5)

∫ 1

−1

dx xEq(x, ξ, t) = Bq2,0(t)− (−2ξ)2Cq

2,0(t), (2.6)

where one encounters three generalized form factors Aq2,0(t), Bq

2,0(t) and Cq2,0(t)

of the nucleon for each quark flavor q. For the case of two light flavors the gen-eralized isoscalar and isovector form factors have been analysed in a series ofpapers at leading one loop order in the nonrelativistic framework of HBChPT,starting with the pioneering analyses of Chen and Ji as well as Belitsky andJi [CJ02, BJ02, ACK06, DMS06, DMS07]. Our aim is to provide the firstanalysis of these generalized form factors utilizing the methods of covariantBaryon Chiral Perturbation Theory (BChPT) for two light flavors pioneeredin reference [GSS88]. The leading one loop order calculation presented hererelies on the IR renormalization prescription introduced in section 1.2 whichwe consider to be inevitable for a consistent analysis of quark mass depen-dencies in BChPT. We note that (at t = 0) a covariant BChPT calculationdiffers from a nonrelativistic one – provided both are performed at the samechiral order D – by an infinite series of terms ∼ (mπ/M0)

i where M0 denotesthe mass of the nucleon in the chiral limit, estimated to be around 890 MeV[PHW04, AK+04, BHM05, PMW+06]. Such terms quickly become relevantonce the pion mass mπ takes on values larger than 140 MeV, as it typically oc-curs in present day lattice QCD simulations of generalized form factors. Aside

20

2.2. Extracting the First Moments of GPDs

from this resummation property in (1/M0)i, we emphasize again, that the

power-counting analysis determining possible operators and allowed topolo-gies for loop diagrams at a particular chiral order (see section 2.3.3) is identi-cal between covariant and nonrelativistic frameworks. Both schemes organizea perturbative calculation as a power series in (1/(4πFπ))D. Finally we notethat the first moments of nucleon GPDs have also been studied in constituentquark models (e.g. see ref.[BPT03]) and chiral quark soliton models (e.g. seeref.[G+07]) of the nucleon which – in contrast to ChEFT – can also providedynamical insights into the short-distance structure present in the generalizedform factors.This chapter is organized as follows: In the next section we specify the oper-ators with which we are going to obtain information on the three generalizedisoscalar and the three generalized isovector form factors of the nucleon. Insection 2.3 we portray the effective chiral Lagrangian required for the calcu-lation, immediately followed by the sections containing our leading one looporder results of the generalized form factors in the isovector (section 2.4) andin the isoscalar (section 2.5) matrix elements. Finally, a last section (section2.6) extends the discussion to the generalized axial form factors. A summaryof the main results concludes this chapter, while a few technical details regard-ing the calculation of the amplitudes in covariant BChPT are relegated to theappendix section A. We have published the main results of this chapter in refer-ence [DGH08]. After this publication has appeared the results of this analysishave been applied successfully to new sets of lattice data in refs.[H+08, B+07b]

2.2 Extracting the First Moments of GPDs

2.2.1 The generalized form factors of the nucleon

In eqs.(2.5,2.6) of the introduction of this chapter it was shown that the firstmoments of nucleon GPDs are connected to three generalized form factors.In lattice QCD one can directly access the contribution of quark flavor q tothese generalized form factors of the nucleon by evaluating the matrix element[Ji98, Die03, BR05]

i〈p2|qγ{µ←→D ν} q|p1〉 =

u(p2)

[Aq

2,0(q2)γ{µpν} −

Bq2,0(q

2)

2MN

qαiσα{µpν} +Cq

2,0(q2)

MN

q{µqν}

]u(p1).

(2.7)

The brackets {. . .} denote the completely symmetrized and traceless combina-tion of all indices: a{µbν} = aµbν + aνbµ − 2

dgµνa · b. In eq.(2.7) the general-

ized form factors A(q2), B(q2) and C(q2) are real functions of the momentumtransfer squared. In ChEFT we employ the same philosophy and also extractinformation about the first moments of nucleon GPDs of eq.(2.5,2.6) via a cal-culation of the generalized form factors according to eq.(2.7).Studying a strongly interacting system with two light flavors in the nonpertur-

21


bative regime of QCD with the methods of ChEFT one works in the basis ofsinglet (s) and triplet (v) contributions of the quarks to the three form factors:

i〈p2|qγ{µ←→D ν} q|p1〉u+d =

u(p2)

[As

2,0(q2)γ{µpν} −

Bs2,0(q

2)

2MN

qαiσα{µpν} +Cs

2,0(q2)

MN

q{µqν}

]1

2u(p1),

(2.8)

i〈p2|qγ{µ←→D ν} q|p1〉u−d =

u(p2)

[Av

2,0(q2)γ{µpν} −

Bv2,0(q

2)

2MN

qαiσα{µpν} +Cv

2,0(q2)

MN

q{µqν}

]τa

2u(p1).

(2.9)

Note that the 2×2 unit matrix 1 and the Pauli matrices τa with a = 1, 2, 3 onthe right hand sides of eqs.(2.8,2.9) operate in the space of a (proton,neutron)doublet field.At present not much is known yet experimentally about the momentum depen-dence of these 6 form factors. The main source of information at the momentis provided by lattice QCD studies of these objects (e.g. see refs.[G+04, H+03,E+06a, H+08, B+07b]). Given that present day lattice simulations work withquark masses much larger than those realized for u and d quarks in the stan-dard model, one also needs to know the quark mass dependence of all 6 formfactors in order to extrapolate the lattice QCD results down to the real worldof light u and d quarks. This information is also encoded in the ChEFT results,typically expressed in form of a pion mass dependence of the observables understudy. (The connection between the explicit breaking of chiral symmetry dueto non-zero quark-masses and the resulting effective pion-mass is addressed insection 2.3.2). The analysis of the quark-mass dependence of the generalizedform factors proceeds as follow: we first extract numerical values for presentlyunknown ChPT coupling constants by fitting the pion mass dependent BChPTresults to lattice data at different quark masses. Thanks to the extrapolationfunction provided by the EFT framework we are then able to bridge the gapbetween the domain of large quark-masses used in present day lattice simula-tions and the physical world of small quark masses. The information revealedby this crosstalk between lattice QCD and ChPT allows us to give predictionsfor the physical observables under study.The need for a chiral extrapolation of lattice QCD results for the general-ized form factors of the nucleon leads to a further complication in the anal-ysis: One needs to be aware that it is common practice in current latticeQCD analyses that the mass parameter MN in Eqs.(2.8,2.9) does not cor-respond to the physical mass of a nucleon, instead, it represents a (larger)nucleon mass consistent with the values of the quark-masses employed in thesimulation. The quark-mass dependence of MN has been studied in detailin ChEFT, both in non-relativistic [BHM04] and in covariant frameworks[PHW04, BHM05, PMW+06]. The next-to-leading one loop chiral formulaeof ref.[PHW04] provide a stable extrapolation function up to effective pion-masses ∼ 600 MeV. We therefore utilize the O(p4) IR renormalized BChPT

22

2.2. Extracting the First Moments of GPDs

gA 1.2Fπ [GeV] 0.0924M0 [GeV] 0.889c1 [GeV−1] −0.817c2 [GeV−1] 3.2c3[GeV−1] −3.4

er1(1GeV) [GeV−3] 1.44

Table 2.1: Input values used in this chapter for the numerical analysis of thechiral extrapolation functions [Gai07].

result [Gai07]

MN(mπ) = M0 − 4c1m2π

+3g2

Am3π

8π2F 2π

√4− m2

π

M20

(−1 +

m2π

4M20

+ c1m4

π

M30

)arccos

(mπ

2M0

)

+4er1(λ)m4

π −3m4

π

128π2F 2π

[(6g2

A

M0

− c2

)

+4

(g2

A

M0

− 8c1 + c2 + 4c3

)log

(mπ

λ

)]

− 3c1g2Am6

π

8π2F 2πM2

0

log

(mπ

M0

)+O(p5), (2.10)

in order to correct for the mass effects of eqs.(2.8,2.9). The coupling con-stants occurring in this formula are described in detail in reference [PHW04,AK+04, BHM05, PMW+06]. Possible effects of higher orders can be estimated

as O(p5) ∼ δMm5

π

(4πFπ)4where δM could be varied within natural size estimates

−3 < δM < +3 [Gai07].We note that the trivial, purely kinematical effect of MN = MN(mπ) ineqs.(2.8,2.9) could induce quite a strong quark mass dependence into the formfactors Bs,v

2,0(t) and Cs,v2,0(t) and might even be able to mask any“intrinsic”quark

mass dependence in these form factors. We are reminded of the analysis of thePauli form factors F s,v

2 (t) in ref.[G+05] where the absorption of the analogouseffect into a “normalized” magneton even led to a different slope (!) for theisovector anomalous magnetic moment κv = F v

2 (t = 0) when compared to thequark mass dependence of the “unnormalized” lattice data. We therefore urgethe readers that this effect should be taken into account in any quantitative(future) analysis of the quark mass dependence of the generalized form factorsBs,v

2,0(t) and Cs,v2,0(t) as well. For convenience we give the numerical values for

the appearing coupling constants in table 2.1 and emphasize that we use thesame values for all parameters throughout this work.

Finally we note that in the forward limit t → 0 the generalized form factorsAs,v

2,0(t = 0) can be understood as moments of the ordinary Parton Distribution

23


Functions (PDFs) q(x) and q(x) [Ji98, Die03, BR05]:

〈x〉u±d = As,v2,0(t = 0) =

∫ 1

0

dx x (q(x) + q(x))u±d . (2.11)

Experimental results exist for 〈x〉 in proton- and“neutron” targets, from whichone can estimate the isoscalar and isovector quark contributions at the physicalpoint [xpr] at a regularization scale µ. In this work we choose µ = 2 GeVfor our comparisons with phenomenology1. In section 2.4.1 we attempt toconnect the physical value for 〈x〉u−d with recent lattice QCD results from theLHPC collaboration [H+08], whereas in section 2.5.1 we analyse the quark massdependence of 〈x〉u+d with (quenched) lattice QCD results from the QCDSFcollaboration [G+04].

2.2.2 The generalized form factors of the pion

The first moment of a generalized parton distribution function in a pion Hqπ(x, ξ, t)

can be defined analogously to the case of the nucleon discussed above. Oneobtains [Ji98, Die03, BR05]

∫ 1

−1

dx xHqπ(x, ξ, t) = Aq

π(t) + (−2ξ)2Cqπ(t). (2.12)

The two functions Aqπ(t) and Cq

π(t) are the generalized form factors of the pion,generated by contributions of quark flavor q. In the forward limit one recoversthe first moment of the ordinary parton distribution functions in a pion:

〈x〉π = Aqπ(t = 0) =

∫ 1

0

dx x (q(x) + q(x)) . (2.13)

In the analysis of the isoscalar GPD moments of a nucleon we encounter tensorfields directly interacting with the pion-cloud of the nucleon. One thereforeneeds to understand the relevant pion-tensor couplings in terms of the two gen-eralized form factors Aπ(t) and Cπ(t). We note that the two generalized formfactors of the pion have been analysed at one loop level already in ref.[DL91]for the total sum of quark and gluon contributions, whereas the quark con-tribution to the form factors as defined in eq.(2.12) has been the focus of themore recent works [KP02, DMS05].

2.3 Formalism

2.3.1 Leading order nucleon Lagrangian

In this section we apply the basic concepts of ChEFT introduced in chapter1 to the present discussion, adapting them to the particular case of interest.

1Note that this µ dependence is not part of the ChEFT framework, as it clearly involvesshort-distance physics. However, all chiral tensor couplings specified in section 2.3 carry animplicit µ dependence (which we do not indicate) as soon as they are fitted to lattice QCDdata or phenomenological values which do depend on this scale.

24

2.3. Formalism

We briefly introduce the relevant parts of the chiral Lagrangian which arenecessary for an evaluation of the tensor currents eqs.(2.8,2.9) at leading oneloop level.We first extend the Lagrangian of eq.(1.7) to the interaction between externaltensor fields and a strongly interacting system at low energies. In this workwe focus on symmetric, traceless tensor fields with positive parity in order tocalculate the generalized form factors of the nucleon. In particular, we utilizethe chiral tensor structures

V ±µν =

1

2

(gµαgνβ + gµβgνα − 2

dgµνgαβ

)×

(u†V αβ

R u± uV αβL u†

),

V 0µν =

1

2

(gµαgνβ + gµβgνα − 2

dgµνgαβ

)vαβ

(s)

1

2. (2.14)

The right- and left handed fields V(R,L)αβ are related to the symmetric isovector

tensor fields of definite parity viαβ and ai

αβ with i = 1, 2, 3 via

V Rαβ =

(vi

αβ + aiαβ

)× τ i

2,

V Lαβ =

(vi

αβ − aiαβ

)× τ i

2, (2.15)

while v(s)αβ denotes the symmetric isoscalar tensor field of positive parity. In

order to study possible interactions with external tensor fields originating fromthe leading order Lagrangian eq.(1.7), we rewrite it into the equivalent form

LtπN = ΨN

[1

2

(iγµgµν

−→D ν − i

←−D ν gµνγ

µ)−M0 + . . .

]ΨN , (2.16)

where we have introduced

gµν = gµν + as2,0V

0µν +

av2,0

2V +

µν . (2.17)

The coupling as2,0 (av

2,0) has been defined such that it corresponds to the chirallimit value of (〈x〉u+d) (〈x〉u−d) defined in eq.(2.11). We note that the couplingas

2,0 is allowed to be different from unity, as we only sum over the u + d quarkcontributions in the isoscalar moments but neglect the contributions from glu-ons. While this separation between quark- and gluon contributions does notoccur in nature where one obtains the sum rule for the total angular momen-tum Aq+g

2,0 (0) = 1, it can be implemented in lattice QCD analyses at a fixedrenormalization scale (e.g. see refs.[G+04, H+03, E+06a, H+08]).Although the construction of the parity-even tensor interactions with a stronglyinteracting system started from the O(p1) BChPT Lagrangian, an inspectionof the resulting Lagrangian eq.(2.16) reveals that the leading interactions ac-tually start out at O(p0). This appears as a consequence of the fact that we donot assign a nonzero chiral dimension pn, n ≥ 1 to any of the tensor fields. Fur-thermore, symmetries allow the addition of the parity-odd tensor interaction

25


∼ V −µν . We finally obtain [DGH08]

L(0)tπN =

1

2ΨN

[iγµ

(as

2,0V0µν +

av2,0

2V +

µν +∆av

2,0

2V −

µνγ5

)−→D ν

−i←−D νγµ

(as

2,0V0µν +

av2,0

2V +

µν +∆av

2,0

2V −

µνγ5

)]ΨN , (2.18)

with the coupling ∆av2,0 corresponding to the chiral limit value of the axial

quantity 〈∆x〉u−d (see section 2.6). The O(p1) part of the leading order pion-nucleon Lagrangian in the presence of external symmetric, traceless tensorfields with positive parity then reads [DGH08]

L(1)tπN = ΨN

{iγµDµ −M0 +

gA

2γµγ5uµ +

bv2,0

8M0

(iσαµ

[−→Dα, V µν

+

]−→D ν + h.c.

)

+bs2,0

4M0

(iσαµ

[−→∇α, V µν0

]−→D ν + h.c.

)+ . . .

}ΨN , (2.19)

where we have introduced ∇α = ∂α− ivα(s). The two new couplings bv

2,0 and bs2,0

can be interpreted as the chiral limit values of isovector- and isoscalar anoma-lous gravitomagnetic moments Bv

2,0(0) and Bs2,0(0). No further structures enter

our calculation at this order2. Finally we note that the coupling bs2,0 is only

allowed to exist because we do not sum over the quark- and gluon contributionsin the isoscalar moments, otherwise the anomalous gravitomagnetic momentis bound to vanish in the forward limit Bq+g

2,0 (t = 0) = 0 [Ter99].

2.3.2 Consequences for the meson Lagrangian

The choice of assigning the chiral power p0 to the symmetric tensor fieldsV L,R,0

µν also has the consequence that the well known leading order chiral La-grangian for two light flavors in the meson sector [GL84](see eq.(1.2)) is mod-ified [DGH08]:

L(2)tππ =

F 2π

4Tr

[∇µU† (gµν + 4x0

πV µν0

)∇νU + χ†U + χU †] . (2.20)

We note that the new coupling x0π has been defined such that it corresponds

to the chiral limit value of 〈x〉π of eq.(2.13). It is allowed to differ from unitybecause we only sum over the quark-distribution functions in the isoscalarchannel and neglect the contributions from gluons.The explicit breaking of chiral symmetry via the finite quark masses is encodedvia

χ = 2 B0 (s + ip) , (2.21)

if one switches off the external pseudoscalar background field p and identifiesthe two flavor quark mass matrix M = diag(mu,md) with the scalar back-ground field s. To the order we are working here we obtain the resulting pion

2We only show those terms where the tensor fields couple at tree level without simultane-ous emission of pions, photons, etc., as these are the relevant terms for our O(p2) calculationof the form factors according to the power-counting analysis of subsection 2.3.3.

26

2.3. Formalism

mass mπ viam2

π = 2 B0 m +O(mq2), (2.22)

where m = (mu + md)/2 and B0 is connected to the value of the chiral con-densate. The other free parameter at this order Fπ can be identified with thevalue of the pion-decay constant (in the chiral limit).

2.3.3 Power-counting in BChPT with tensor fields

We start from the general power-counting formula of Baryon ChPT:

D = 2NL + 1 +∑

d

(d− 2)NMd +

∑

d

(d− 1)NMBd . (2.23)

Here D denotes the chiral dimension pD of a particular Feynman diagram,NL counts the number of loops in the diagram, whereas the variables NM, MB

d

count the number of vertices of chiral dimension d from the pion (M) andpion-nucleon (MB) Lagrangians. To leading order3 D = 0 we only havethe tree level contributions from the order p0 Lagrangian of eq.(2.18) withNL = 0, NM

2 = 0 and NMB0 = 1. At next-to-leading order D = 1 we find

additional tree level contributions from the order p1 Lagrangian of eq.(2.19)with NL = 0, NM

2 = 0 and NMB1 = 1. The first loop contributions enter at

D = 2 with NL = 1, NM2 = 0 and NMB

0 = 1 plus possible contributions fromNMB

1 . The corresponding diagrams are shown in fig.2.1. Diagram (e) in thatfigure represents loop corrections from the nucleon Z-factor (given in appendixA.4) which at this order only renormalizes the tree level tensor couplings of theorder p0 Lagrangian. Note that there is an additional possibility of obtainingD = 2 contributions via NL = 0, NM

2 = 0 and NMB2 = 1, corresponding to

further tree level contributions discussed in the next subsection.In this work we stop with our analysis of the generalized form factors at theD = 2, i.e. O(p2) level, corresponding to a leading one loop order calcula-tion. Preliminary results for the next-to-leading one loop effects of D = 3 aregiven in appendix A.7 but their analysis is postponed to a later communica-tion. The Feynman rules pertinent to the calculation are listed in AppendixA.1. The (perhaps) surprising finding of this power-counting analysis is theobservation that the tensor coupling to the pion field controlled by the cou-pling x0

π in eq.(2.20) does not contribute at leading one loop order! Here itonly starts to enter at D = 3 via NL = 1, NM

2 = 1 and NMB1 = 1 or 2. (The

corresponding diagram for NMB1 = 2 is shown in fig.2.2 while the not-shown

diagram for NMB1 = 1 is expected to sum to zero due to isospin-symmetry.)

We therefore note that the generalized form factors of the nucleon behave quitedifferent from the standard Dirac- and Pauli form factors of the nucleon wherethe pion-cloud interactions with the external source a la fig.2.2 are part of theleading one loop order result and play a prominent role in the final result. Wediscuss the impact of this particular D = 3 contribution further in section 2.5.3

3Note that in contrast to the analyses presented in the previous chapters where theleading order contributions were of chiral dimension D = 1, the calculation in this chapterstarts at D = 0 due to the presence of external fields with zero chiral dimension.

27


a) b)

c) d)

e)

Figure 2.1: The loop diagrams contributing to the first moments of the GPDsof a nucleon at leading one loop order in BChPT. The solid- and dashed linesrepresent nucleon and pion propagators, respectively. The solid dot denotes acoupling to a tensor field from the O(p0) Lagrangian of eq.(2.18).

when we try to estimate the possible size of higher order corrections to ourO(p2) analysis.We point out that the chiral dimension we are talking about is the one of thecontributing loop diagrams and thus of the currents in eqs.(2.9),(2.8), it is notnecessarily the chiral dimension of the respective contributions to the gener-alized form factors. Indeed, because of the presence of one power of a smallparameter in front of B2,0(q

2) and two powers of a small parameter in front ofC2,0(q

2) in eqs.(2.9),(2.8), if the currents are evaluated at a chiral dimensionD, the corresponding contributions to B2,0(q

2) are of chiral dimension D − 1,the contributions to C2,0(q

2) are of chiral dimension D − 2. The form factorA2,0(q

2) does not carry such a small prefactor and is therefore not affected bythis consideration. Consequently at the D = 2 level systematic uncertaintiesdue to possible higher order effects are expected to be already moderate forA2,0(q

2) but still large for C2,0(q2). We now move on to a discussion of the ten-

sor interactions in the O(p2) Lagrangian which contribute to the generalizedform factors at D = 2 according to our power-counting analysis.

2.3.4 Next-to-leading order nucleon Lagrangian

At next-to-leading order the covariant BChPT Lagrangian for two flavor QCDcontains seven independent terms in the presence of general scalar, pseu-doscalar, vector- and axial vector background fields, governed by the couplings

28

2.3. Formalism

Figure 2.2: (Isoscalar) tensor field coupling to the pion-cloud of the nucleon.This process only starts to contribute at next-to-leading one loop order inBChPT.

c1, . . . , c7 [BKM95]. Extending this scenario to symmetric and traceless tensorbackground fields with positive parity, symmetries allow the construction ofsix additional, independent terms which describe the coupling of a tensor fieldto the nucleon at next-to-leading order tree level [DGH08]:

L(2)tπN =

c8

4M20

ΨN

{Tr(χ+)V +

µνγµi−→D ν + h.c.

}ΨN

+c9

2M20

ΨN

{Tr(χ+)γµi

−→D ν + h.c.

}ΨNV 0

µν

+cv2,0

2M0

ΨN

{[−→Dµ, [

−→D ν , V +

µν ]]}

ΨN

+cs2,0

M0

ΨN

{[−→∇µ, [−→∇ν , V 0

µν ]]}

ΨN

+c12

4M20

ΨN

{[−→Dα, [

−→Dα, V +

µν ]]γµi−→D ν + h.c.

}ΨN

+c13

2M20

ΨN

{γµi−→D ν + h.c.

}ΨN

[−→∇α, [−→∇α, V 0

µν ]]

+ . . .

(2.24)

with χ+ = u†χu† + uχ†u. The physics behind these couplings ci where i =8, . . . 13 with respect to the generalized form factors of the nucleon is quitesimple: c8 and c9 govern the leading quark mass insertion in 〈x〉u−d and 〈x〉u+d,respectively, whereas the couplings c10 and c11 give the values of the generalizedform factors Cv

2,0(0) and Cs2,0(0) in the double limit t → 0 and mπ → 0. We

can therefore denote them as cv2,0 and cs

2,0. Finally the couplings c12 and c13

parametrize the contributions of short-distance physics to the slopes of thegeneralized form factors Av

2,0(t) and As2,0(t) in the chiral limit. Note that the

operator controlled by the coupling c9 is not allowed to exist when we add thegluon-contributions on the left hand side of eq.(2.8) [H+].After laying down the necessary effective Lagrangians for our calculation, weare now proceeding to the results of our calculation.

29


2.4 The Generalized Isovector Form Factors

in O(p2) BChPT

2.4.1 Moments of the isovector GPDs at t = 0

In this subsection we present our results for the generalized isovector formfactors of the nucleon at t = 0. For the PDF-moment Av

2,0(t = 0) we obtain to

O(p2) in IR renormalized BChPT

Av2,0(0) = 〈x〉u−d

= av2,0 +

av2,0m

2π

(4πFπ)2

[− (3g2

A + 1) logm2

π

λ2− 2g2

A

+g2A

m2π

M20

(1 + 3 log

m2π

M20

)− 1

2g2

A

m4π

M40

logm2

π

M20

+g2A

mπ√4M2

0 −m2π

(14− 8

m2π

M20

+m4

π

M40

)arccos

(mπ

2M0

)]

+∆av

2,0gAm2π

3(4πFπ)2

[2m2

π

M20

(1 + 3 log

m2π

M20

)− m4

π

M40

logm2

π

M20

+2mπ(4M2

0 −m2π)

32

M40

arccos

(mπ

2M0

)]

+4m2π

c(r)8 (λ)

M20

+O(p3). (2.25)

Many of the parameters in this expression are well known from analyses ofchiral extrapolation functions. Numerical estimates for their chiral limit valuescan be found in table 2.1. Furthermore, in a first fit to lattice data we constrainthe coupling ∆av

2,0 from the phenomenological value of 〈∆x〉phen.u−d ≈ 0.21 via

(see section 2.6)〈∆x〉u−d = ∆av

2,0 +O(m2π) (2.26)

and perform a fit with two parameters: The couplings av2,0 and c

(r)8 (1GeV) at

the regularization scale λ = 1 GeV. We fit to the LHPC data for this quantityas given in ref.[H+08], including lattice data up to effective pion masses ofmπ ≈ 600 MeV. The resulting values for the fit parameters together withtheir statistical errors are given in table 2.2. The resulting chiral extrapolationfunction is shown as the solid line in figure 2.3. We note that the extrapolationcurve tends towards smaller values for small quark masses but does not quitereach the phenomenological value at the physical point which was not includedin the fit. We therefore again estimate possible corrections to the solid curvearising from higher orders. From dimensional analysis we know that the leadingchiral contribution to 〈x〉u−d beyond our calculation takes the form O(p3) ∼δA

m3π

(4πFπ)2M0+ .... Repeating the fit with values for δA between4 −1, . . . , +1 and

4The natural scale of all couplings in the observables considered here is below one, as allcoupling estimates in this section refer to a moment of a parton distribution itself normalized

30

2.4. The Generalized Isovector Form Factors in O(p2) BChPT

0 0.1 0.2 0.3 0.4 0.5 0.6mπ [GeV]

0

0.1

0.2

0.3

<x>

u-d

Figure 2.3: The quark mass dependence of the isovector parton distribution in anucleon. The solid line displays the best fit curve of theO(p2) result of eq.(2.25)to the LHPC lattice data of ref.[H+08]. The corresponding parameters aregiven in Table 2.2. Note that the phenomenological value at the physical pionmass was not included in the fit. The grey band indicates the size of possibleO(p3) corrections as discussed in the text.

accounting for statistical errors one ends up with an array of curves coveringthe grey shaded area of figure 2.3. Reassuringly, the phenomenological valuefor 〈x〉u−d lies well within that band of uncertainties due to possible next-ordercorrections, giving us no indication that something may be inconsistent withthe large values for 〈x〉u−d typically found in lattice QCD simulations at large

quark masses. The resulting values for the couplings av2,0 and c

(r)8 (1 GeV) of

Fit I are also well within expectations. We can conclude that thanks to thecombined (leading one loop order) BChPT plus lattice analysis the physicalvalue of this observable can predicted with a precision of approximately 30%.

At this point some remarks are needed. First of all, in our analysis wehave compared the results of a SU(2) theory with the phenomenology of asix flavor world. There might therefore be slight differences between our re-sults and phenomenology, which are anyway for sure beyond the precision ofour calculation. Secondly, we have made use of lattice results calculated with2 + 1 flavors. The validity of our SU(2) ChPT calculation can be extendedto a world with a large, fixed strange quark-mass, the effects of such inclusionbeing encoded in the coupling constants of the theory. However, the values wewould find for those coupling constants might differ from the ones in a puretwo flavor scenario.We note that the mechanism of the downward-bending at small quark masses inAv

2,0(0) found in eq.(2.25), see figure 2.4, is quite different from what has been

to unity. This expectation is confirmed by the fit values of tables 2.2 and 2.4 found for theinvestigated coupling constants.

31


Fit I (4 points - 2 parameter) Fit II (6+1 points - 3 parameter)av

2,0 0.157± 0.006 0.141± 0.0057∆av

2,0 0.210 (fixed) 0.144± 0.034cr8(1GeV) −0.283± 0.011 −0.213± 0.03

Table 2.2: The values of the coupling constants resulting from the two fits ofthe O(p2) BChPT result given in eq.(2.25) to the LHPC lattice data for 〈x〉u−d.The errors shown are of only statistical origin and do neither include uncer-tainties from possible higher order corrections in ChEFT nor from systematicuncertainties connected with the lattice simulation.

discussed in literature so far within the nonrelativistic HBChPT framework(e.g. see ref.[DMN+01]). In order to demonstrate this we truncate eq.(2.25) atleading order in 1/M0 to obtain the exact O(p2) HBChPT limit of our results,agreeing with the findings of references [AS02, CJ01]:

Av2,0(0)|p2

HBChPT = av2,0

{1− m2

π

(4πFπ)2

(2g2

A + (3g2A + 1) log

m2π

λ2

)}

+4m2π

c(r)8 (λ)

M20

+O(

m3π

16π2F 2πM0

). (2.27)

As we already stated in chapter 1 and in the Introduction, the covariantBChPT scheme used in this work is able to exactly reproduce the correspondingnonrelativistic HBChPT result at the same order by the appropriate truncationof the 1/M0 expansion. At leading one loop order no recoil effects are includedin HBChPT loop results and our covariant results have thus to be truncated

at order(

1M0

)0

in order to obtain the according nonrelativistic limit. In the

following we therefore denote this truncation by 1/M0 → 0.All differences between the HBChPT limits presented in this chapter and

the findings of previous HBChPT studies [AS02, CJ01, ACK06, DMS06, DMS07]are due to the inclusion of selected terms of higher order, i.e. terms with D = 3(and even larger values of D) according to the counting formula eq.(2.23) inthose references.Fit I is certainly constricted by the assumption that we use the physical valueof 〈∆x〉phen.

u−d ≈ 0.21 for the coupling ∆av2,0 which presumably takes a value in

the chiral limit which is a bit smaller than the phenomenological value at thephysical point [DH]. Furthermore, in order to also numerically compare theO(p2) HBChPT result of eq.(2.27) with the O(p2) covariant BChPT result ofeq.(2.25) we perform a second fit: We fit the covariant expression for 〈x〉u−d

of eq.(2.25) again to the LHPC lattice data, this time however, we constrainthe coupling ∆av

2,0 in such a way that the resulting chiral extrapolation curve

reproduces the phenomenological value of 〈x〉phen.u−d = 0.160 ± 0.006 [xpr] ex-

actly for physical quark masses. The parameter values for this Fit II are againgiven in table 2.2, whereas the resulting chiral extrapolation curve of the co-variant O(p2) expression of eq.(2.25) is shown as the solid line in fig.2.4. First,

32


0 0.2 0.4 0.6 0.8mπ [GeV]

0

0.1

0.2

0.3

<x>

u-d

Figure 2.4: “Fit II”of the O(p2) BChPT result of eq.(2.25) to the LHPC latticedata of ref.[H+08] and to the physical point (solid line). The correspondingfit-parameters are given in table 2.2. The dashed curve shown corresponds tothe O(p2) result in the HBChPT truncation (see eq.(2.27)). The shaded areaindicates the region where one does not expect that ChEFT can provide atrustworthy chiral extrapolation function due to the large pion masses involved.As explained in the text, the difference between the two curves gives indicationof the size of the higher order corrections provided by the BChPT result butdoes not allow us to make strong statements about the radius of convergenceof HBChPT.

we emphasize that the curve looks very reasonable, connecting the physicalpoint with the lattice data of the LHPC collaboration in a smooth fashion.Second, we note that the resulting values for the coupling constants av

2,0 and∆av

2,0 underlying this curve are very reassuring, indicating that both 〈x〉u−d and〈∆x〉u−d are slightly smaller in the chiral limit than at the physical point! Like-

wise, the unknown quark mass insertion c(r)8 (λ) contributes in a strength just

as expected from natural size estimates. For the comparison with HBChPT wenow utilize the very same values5 for av

2,0 and c(r)8 (λ) of Fit II as given in table

2.2. The resulting curve based on the O(p2) HBChPT formula of eq.(2.27)is shown as the dashed curve in fig.2.4. One observes that this leading oneloop HBChPT expression agrees with the covariant result between the chirallimit and the physical point but is not able to extrapolate on towards the lat-tice data. Analogous behaviour has been found for the corresponding leadingone loop HBChPT expressions for the axial coupling constant of the nucleon[HPW03a] and for the anomalous magnetic moments of the nucleon [HW02].These examples show the limited usefulness of leading HBChPT extrapolation

5We emphasize again that due to the definition of the IR renormalization scheme thenumerical values of the ChPT coupling constants have to be the same in HBChPT and IRrenormalized covariant BChPT.

33


〈x〉u−d

Phenomenology 0.16± 0.006[xpr]Extrapolated LHPC 0.157± 0.006 [H+08]values at mphys

π QCDSF/UKQCD 0.198 ± 0.008 [B+07b]

Table 2.3: The phenomenological values of the observables 〈x〉u−d, togetherwith the values obtained from chiral extrapolation studies of lattice data ofrefs.[H+08] and [B+07b] based on our formula of eq.2.25.

formulae, which seems to successfully describe the quark mass dependence onlybetween the chiral limit and the physical point.However, we note that all our numerical comparisons with HBChPT resultsshown in the figures of section 2.4 and 2.5 are based on the assumption that theD = 2 fit values found in tables 2.2 and 2.4 are already reliable estimates of thetrue, correct values of these couplings in low energy QCD. Clearly, the shownHBChPT curves might have to be revised if future D = 3 analyses [DHH] leadto substantially different numerical values for these couplings. The true rangeof applicability of HBChPT versus covariant BChPT can only be determinedonce the stability of the employed couplings is guaranteed. A study of higherorder effects is therefore essential also in this respect. Ideally we would liketo reanalyse the results of Fit II by first fixing the couplings from a fit ofthe HBChPT formula of Eq.(2.27) and then study the resulting O(p2) BChPTcurve for this observable. However, a fit to the present set of lattice data shownin Fig.2.3 leads to a curve that is not compatible with the 2 lightest latticepoints and lies significantly below the grey band shown in Fig.2.3—even downto the chiral limit. We will study this issue further in ref.[DHH].At this point we conclude that the smooth extrapolation behaviour of the co-variant O(p2) BChPT expression for 〈x〉u−d of eq.(2.25) between the chirallimit and the region of present lattice QCD data is due to an infinite tower

of(

mπ

M0

)i

terms. According to our analysis the chiral curvature resulting from

the logarithm of eq.(2.27) governing the leading nonanalytic quark mass be-haviour of this moment is not responsible for the rising behaviour of the chiralextrapolation function as has been hypothesized in ref.[DMN+01].Subsequent to the publication of these results for the momentum fraction〈x〉u−d in ref.[DGH08], the LHPC and QCDSF/UKQCD collaborations uti-lized the formula of eq.(2.25) to perform chiral extrapolation studies on newdynamical lattice data of Au−d

2,0 (0). The corresponding extrapolated values arereported in table 2.3. Before we proceed to the discussion of these values, wehave to mention an important point, that we will have to keep in mind alsofor future analysis of the other form factors: because of the serious differencesin normalization between LHPC and QCDSF/UKQCD data, the comparisonbetween the fits performed by the two collaborations should be done with duecaution. Both collaborations make use of our formula to extrapolate down tothe physical pion mass, but as clearly shown in the table the obtained valuesfor 〈x〉u−d are not consistent with each other. In particular, the LHPC value

34


0 0.1 0.2 0.3 0.4 0.5 0.6mπ [GeV]

-0.5

0

0.5

1

B20

v (0)

b2,0

v = 0

b2,0

v = 0.5

0 0.1 0.2 0.3mπ[GeV]

-0.05

-0.025

0

0.025

0.05

C20

v (0)

c2,0

v = 0

Figure 2.5: Quark mass dependence of the isovector moments Bv2,0(t = 0) and

Cv2,0(t = 0). In Bv

2,0(t = 0) we have varied the (unknown) chiral limit value bv2,0

between 0 and +0.5, as lattice analyses [G+04, H+03, H+08, B+07b] suggestthat this moment has a large positive value. For the chiral limit value cv

2,0 ofCv

2,0(t = 0) we have chosen the value zero, as preliminary lattice QCD analysessuggest that this moment is consistent with zero [H+08]. The grey bands shownindicate the size of possible higher order corrections to these O(p2) results.

agrees very well with both our BChPT value and phenomenology, while theQCDSF/UKQCD value lies higher than the phenomenological point. We willdiscuss once again the extrapolated results for this observable in section 2.6.1,where the possibility to gain new information from combined fits of differentobservables will be explored.

Finally we are discussing the O(p2) BChPT results at t = 0 for the remain-ing two generalized isovector form factors of the nucleon at twist-two level.One obtains

Bv2,0(t = 0) = bv

2,0

MN(mπ)

M0

+av

2,0 g2Am2

π

(4πFπ)2

[(3 + log

m2π

M20

)

−m2π

M20

(2 + 3 log

m2π

M20

)+

m4π

M40

logm2

π

M20

− 2mπ√4M2

0 −m2π

(5− 5

m2π

M20

+m4

π

M40

)arccos

(mπ

2M0

)]

+O(p3), (2.28)

Cv2,0(t = 0) = cv

2,0

MN(mπ)

M0

+av

2,0g2Am2

π

12(4πFπ)2

[− 1 + 2

m2π

M20

(1 + log

m2π

M20

)

−m4π

M40

logm2

π

M20

+2mπ√

4M20 −m2

π

(2− 4

m2π

M20

+m4

π

M40

)

× arccos

(mπ

2M0

)]+O(p3). (2.29)

35


As one can easily observe, one encounters plenty of nonanalytic terms andeven chiral logarithms in these covariant O(p2) BChPT results. However, wenote that in the HBChPT limit to the same chiral order O(p2) one wouldonly obtain the chiral limit values bv

2,0 and cv2,0, nothing else [Dor05]. E.g. the

chiral logarithms calculated in ref.[BJ02] for Bv2,0(t = 0) and Cv

2,0(t = 0) wouldonly show up in a full O(p3), respectively O(p4) BChPT calculation of thegeneralized form factors. From the viewpoint of power-counting in BChPTthey are to be considered part of higher order corrections (i.e. D > 2) to thefull O(p2) results given in eqs.(2.28,2.29).Regrettably, at this point no information from experiments exists for these twostructure quantities of the nucleon. From phenomenology one would expectthat Bv

2,0(t = 0) has a“large”positive value at mπ = 140 MeV, as it correspondsto the next-higher moment of the isovector Pauli form factor F v

2 (t = 0) =κv = 3.71 n.m. (Compare eq.(2.6) and eq.(2.3) at ξ = 0). Lattice QCDanalyses seem to support this expectation [G+04, H+03, H+08, B+07b]. Incontrast, the value of Cv

2,0(t = 0) cannot be estimated from information knownabout nucleon form factors. State-of-the-art lattice QCD analyses (e.g. seeref.[H+08]) suggest that it is consistent with zero. In fig.2.5 we have indicatedhow the corresponding extrapolation curves based upon this information mightlook like. As a caveat we note that in both form factors the intrinsic quarkmass dependence is small and we would see a dominant influence of the quarkmass dependence stemming from the kinematical factor MN(mπ) in eqs.(2.28)and (2.29), if the corresponding chiral limit values bv

2,0 and cv2,0 are nonzero.

A further observation is that the uncertainties connected with possible higherorder corrections from O(p3) to Bv

2,0(t = 0) and Cv2,0(t = 0) could already

become substantial for pion masses around 300 MeV. For both quantities they

can be estimated via O(p3) ∼ δB,Cm2

πMN (mπ)(4πFπ)2M0

where −1 < δB,C < 1. In order to

ultimately test the stability of the results in eqs.(2.28,2.29) it will be very usefulto extend this analysis to next-to-leading one loop order. The systematic errorsof our calculation of Bv

2,0(t = 0) and Cv2,0(t = 0) already become large at the

lowest pion masses at which lattice data are available: we therefore prefer toomit a study of those two form factors based on fits to lattice data. Subsequentto the publication of the results for these two form factors in [DGH08], theLHPC collaboration utilized our results to analyse the form factors Bv

2,0(t) andCv

2,0(t) in ref.[H+08].

2.4.2 The slopes of the generalized isovector form fac-tors

In order to discuss the generalized isovector form factors Av2,0(t), Bv

2,0(t) andCv

2,0(t) at nonzero values of t, we first analyse their slopes ρX , defined via

Xv2,0(t) = Xv

2,0(0) + ρvX t +O(t2); X = A,B,C. (2.30)

36


To O(p2) in BChPT we find

ρvA =

c12

M20

− av2,0g

2A

6(4πFπ)2

m2π

(4M20 −m2

π)

{26 + 8 log

m2π

M20

− m2π

M20

(30 + 32 log

m2π

M20

)

+m4

π

M40

(6 +

39

2log

m2π

M20

)− 3

m6π

M60

logm2

π

M20

− mπ√4M2

0 −m2π

(90− 130

m2π

M20

+ 51m4

π

M40

− 6m6

π

M60

)arccos

(mπ

2M0

)}

+mπδt

A

(4πFπ)2M0

, (2.31)

ρvB =

av2,0g

2A

18(4πFπ)2

1

(4M20 −m2

π)

{4M2

0 + m2π

(83 + 24 log

m2π

M20

)

−114m4

π

M20

(1 + log

m2π

M20

)+

m6π

M40

(24 + 75 log

m2π

M20

)− 12

m8π

M60

logm2

π

M20

− 6m3π√

4M20 −m2

π

(50− 80

m2π

M20

+ 33m4

π

M40

− 4m6

π

M60

)arccos

(mπ

2M0

)}

+δtB

MN(mπ)

(4πFπ)2M0

, (2.32)

ρvC =

av2,0g

2A

180(4πFπ)2

1

(4M20 −m2

π)

{4M2

0 − 13m2π + 12

m4π

M20

(4 + 3 log

m2π

M20

)

−3m6

π

M40

(4 + 11 log

m2π

M20

)+ 6

m8π

M60

logm2

π

M20

+6m3

π√4M2

0 −m2π

(10− 30

m2π

M20

+ 15m4

π

M40

− 2m6

π

M60

)arccos

(mπ

2M0

)}

+δtC

MN(mπ)

(4πFπ)2M0

. (2.33)

The parameters δtA, δt

B and δtC are not part of the covariant O(p2) result. They

are only given to indicate the size of possible higher order corrections fromO(p3). A numerical analysis of the formulae given above suggests that thesize of pion-cloud contributions to the slopes of the generalized isovector formfactors is very small! The physics governing the size of these objects seemsto be hidden in the counter-term contributions c12, δt

B and δtC which dominate

numerically when assuming natural size estimates −1 < c12, δtB, δt

C < +1.We note that this situation reminds us of the isoscalar Dirac- and Pauli formfactors of the nucleon F s

1 (t) and F s2 (t), where the t dependence in SU(2) ChEFT

calculations is also dominated by counter-terms (e.g. see the discussion inreference [BFHM98]).Finally truncating the covariant results of eqs.(2.31-2.33) at leading order in

37


1/M0 we obtain the (trivial) HBChPT expressions to O(p2):

ρvA =

c12

M20

+O(p3), (2.34)

ρvB = 0 +O(p3), (2.35)

ρvC = 0 +O(p3). (2.36)

As already mentioned, the nonzero slope results found in the HBChPT calcu-lations of refs.[ACK06, DMS06, DMS07] are of higher order from the point ofview of our power-counting. Most of them can already be added systematicallyto our covariant O(p2) results of eqs.(2.31-2.33) at O(p3).

2.4.3 The generalized isovector form factors of the nu-cleon

In this subsection we present the full t dependence of the generalized isovectorform factors of a nucleon to O(p2) in BChPT. We note that for all threegeneralized form factors at this order only the amplitude of diagram c) offig.2.1 depends on t. The resulting expressions at this order are therefore quitesimple:

Av2,0(t) = Av

2,0(0) +av

2,0g2A

192π2F 2π

F v2,0(t) +

c12

M20

t +O(p3), (2.37)

with Av2,0(0) given above and

F v2,0(t) = t− 2m3

π (50M40 − 43m2

πM20 + 8m4

π)

M40

√4M2

0 −m2π

arccos

(mπ

2M0

)

+

∫ 12

− 12

du

{2m3

πM20

M8

√4M2 −m2

π

[−10M6 +

(17m2

π + 60M20

)M

−4m2π

(m2

π + 15M20

)M2 + 12m4

πM20

]arccos

(mπ

2M

)

+2(M2

0 − M2)

M20 M6

[12m4

πM40 + 2m2

π

(4m2

π − 9M20

)M2

0 M2

+(m2

π − 2M20

) (8m2

π + M20

)M4

]

+2M2

0

M8

[2M8 + 3m2

π

(3m2

π + 4M20

)M4 − 4m4

π

(m2

π + 9M20

)M2

+12m6πM2

0

]log

(M

M0

)+

2m2π

M40 M8

[m4

π

(−12M8

0 + 4M2M60 + 8M8

)

+9m2πM2

(4M8

0 .− M2M60 − 3M6M2

0

)+ 12M4

0 M4(M4 −M4

0

)]

× log

(mπ

M0

)},

(2.38)

38


where we have introduced M2 = M20 +

(u2 − 1

4

)t. Note that F v

2,0(t = 0) = 0by construction. A conservative estimate for the size of possible higher order

corrections indicated in eq.(2.37) can be obtained via O(p3) ∼ δAm3

π

(4πFπ)2M0+

δtA

mπ

(4πFπ)2M0t, in complete analogy to the discussion in the two previous sub-

sections. For the remaining two generalized isovector form factors we obtain

Bv2,0(t) = bv

2,0

MN(mπ)

M0

+av

2,0g2AM2

0

48π2F 2π

∫ 12

− 12

du

M8

{ (M2

0 − M2)

M6 + 9m2πM2

0 M4

−6m4πM2

0 M2 + 6m2πM2

0

(m4

π − 3m2πM2 + M4

)log

mπ

M

− 6m3πM2

0√4M2 −m2

π

[m4

π − 5m2πM2 + 5M4

]arccos

(mπ

2M

)}

+δBm2

πMN(mπ)

(4πFπ)2M0

+ δtB

MN(mπ)

(4πFπ)2M0

t, (2.39)

Cv2,0(t) = cv

2,0

MN(mπ)

M0

+av

2,0g2AM2

0

48π2F 2π

∫ 12

− 12

du u2

M8

{2(M2

0 − M2)

M6

−3m2πM2

0 M4 + 6m4πM2

0 M2 − 6m4πM2

0

(m2

π − 2M2)

logmπ

M

+6m3

πM20√

4M2 −m2π

[m4

π − 4m2πM2 + 2M4

]arccos

(mπ

2M

)}

+δCm2

πMN(mπ)

(4πFπ)2M0

+ δtC

MN(mπ)

(4πFπ)2M0

t. (2.40)

The parameters δB, δC , δtB and δt

C have again been inserted to study the possi-ble corrections of higher orders to our O(p2) results. Varying these parametersbetween -1 and +1, we conclude that a full O(p3) calculation is required be-fore one wants to make any strong claims regarding the t dependence of Bv

2,0(t)and Cv

2,0(t) beyond the linear t dependence discussed in the previous subsec-tion. On the other hand, the predicted t dependence of the form factor Av

2,0(t)of eq.(2.37) appears to be more reliable at this order, as possible higher or-der contributions only affect terms beyond the leading linear dependence ont. However, due to the unsettled situation in the t dependence of those formfactors at leading one loop level, we only present fits to dipole-Q2-extrapolatedlattice data throughout this chapter and do not dare to fit to lattice data di-rectly before the next-to-leading one loop order effects are calculated. To geta rough idea about such direct fits in the sector of generalized nucleon formfactors the reader is again pointed to reference [H+08].

39


2.5 The Generalized Isoscalar Form Factors in

O(p2) BChPT

At the time of the publication of these results in reference [DGH08], onlyquenched lattice QCD data were available for the moments of the isoscalarGPDs [G+04]. Subsequent to the publication of our results both LHPC [H+08]and QCDSF [B+07b] collaborations utilized our chiral extrapolation formulaefor the analysis of their new lattice data obtained with dynamical fermionsimulations.

2.5.1 Moments of the isoscalar GPDs at t = 0

To O(p2) in two flavor covariant BChPT the only nonzero loop contributionsto the isoscalar moment As

2,0(t = 0) (see eq.(2.11)) arise from diagrams c) ande) in fig.2.1. One obtains [DGH08]

As2,0(0) = 〈x〉u+d

= as2,0 + 4m2

π

c9

M20

− 3as2,0g

2Am2

π

16π2F 2π

[m2

π

M20

+m2

π

M20

(2− m2

π

M20

)log

(mπ

M0

)

+mπ√

4M20 −m2

π

(2− 4

m2π

M20

+m4

π

M40

)arccos

(mπ

2M0

)]

+O(p3). (2.41)

Eq.(2.41) should provide a similarly successful chiral extrapolation functionfor 〈x〉u+d as the covariant O(p2) BChPT result of eq.(2.25) did for the LHPClattice data for 〈x〉u−d in section 2.4.1. The uncertainty arising from higher

orders can be estimated to scale as O(p3) ∼ δ0A

m3π

(4πFπ)2M0where δ0

A shouldagain be a number between −1, . . . , +1, according to natural size estimates.We note that the coupling ∆av

2,0 which played an essential role in the chiralextrapolation function of 〈x〉u−d is not present in the quark mass dependenceof the isoscalar moment 〈x〉u+d. The resulting chiral extrapolation functionis therefore presumably quite different from the one in the isovector channel.The absence of a chiral logarithm ∼ m2

π log mπ in 〈x〉u+d (compare eq.(2.27)and eq.(2.42)) presumably only leads to a difference in the chiral extrapolationfunctions between the isovector- and the isoscalar moment for mπ < 140 MeV.Note that from eq.(2.41) in the limit 1/M0 → 0 we reproduce the leadingHBChPT result for 〈x〉u+d of ref.[AS02] which found a complete cancelation ofthe nonanalytic quark mass dependent terms in this channel:

As2,0(0)|p2

HBChPT = as2,0 + 4m2

π

c9

M20

+O(

m3π

16π2F 2πM0

). (2.42)

The coupling c9 is therefore scale independent (in dimensional regularization)and constitutes the leading correction to the chiral limit value as

2,0 of 〈x〉u+d.At t = 0 we also find nontrivial results for the two other generalized isoscalar

40

2.5. The Generalized Isoscalar Form Factors in O(p2) BChPT

form factors of the nucleon. To order p2 in the covariant calculation they read[DGH08]

Bs2,0(0) = bs

2,0

MN(mπ)

M0

− 3 as2,0 g2

Am2π

(4πFπ)2

[(3 + log

m2π

M20

)− m2

π

M20

(2 + 3 log

m2π

M20

)

+m4

π

M40

logm2

π

M20

− 2mπ√4M2

0 −m2π

(5− 5

m2π

M20

+m4

π

M40

)arccos

(mπ

2M0

)]

+δ0B

m2πMN(mπ)

(4πFπ)2M0

, (2.43)

Cs2,0(0) = cs

2,0

MN(mπ)

M0

− as2,0g

2Am2

π

4(4πFπ)2

[− 1 + 2

m2π

M20

(1 + log

m2π

M20

)

−m4π

M40

logm2

π

M20

+2mπ√

4M20 −m2

π

(2− 4

m2π

M20

+m4

π

M40

)arccos

(mπ

2M0

)]

+δ0C

mπMN(mπ)

(4πFπ)2. (2.44)

The parameters δ0B and δ0

C have been added “by hand” to these results in orderto indicate possible effects of higher order (i.e. O(p3)) corrections. Presumablythey take on values −1, . . . , +1. Note that our O(p2) BChPT prediction forCs

2,0(0) is strongly affected by possible corrections from higher orders. This isdue to the fact that one only receives a nonzero result for this form factor start-ing at O(p2) – the order we are working at. Eq.(2.44) should therefore onlybe considered to provide a rough estimate for the quark mass dependence ofthis form factor at t = 0. For a true, quantitative analysis of its chiral extrap-olation behaviour the complete6 O(p3) corrections should first be added. Inorder to obtain a rough estimate of the chiral extrapolation functions resultingfrom eqs.(2.41,2.43) we utilize the quenched7 data of the QCDSF collaboration[G+04] as input.However, lattice simulations (especially in the isoscalar channel) are charac-terized by several sources of systematic errors. Values for the generalized formfactors of the nucleon at different lattice spacings and volumes at small pionmasses are not available yet. Further simulations are needed in order to con-trol possible finite size and discretization effects. Moreover, because of largecalculational expenses, lattice simulations in the isoscalar sector often neglectdisconnected diagrams, that is processes where the incoming external field isnot directly coupled to a valence-quark. Consequently, present lattice resultsfor the isoscalar form factors contain an unknown systematic error.

6The most prominent correction from O(p3) arises from the triangle diagram of fig.2.2and is given in appendix A.5.2 via ∆Cs

h.o.(t = 0, mπ). We note, however, that this is notthe only next-order correction.

7The quark masses employed in ref.[G+04] are so large that one does not expect to finddifferences between quenched and dynamical simulations, see e.g. the discussion in reference[HW02]. Note that we are utilizing the lattice data of reference [G+04] with the scale setby r0, as we consider the alternative way of scale-setting (via a linear extrapolation to thephysical mass of the nucleon) also discussed in ref.[G+04] to be obsolete in the light of thedetailed chiral extrapolation studies of refs.[PHW04, AK+04, BHM05, PMW+06].

41


0 0.2 0.4 0.6 0.8 1mπ [GeV]

0.45

0.5

0.55

0.6

0.65

<x>

u+d

0 0.2 0.4 0.6 0.8 1mπ [GeV]

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

B20

s (0)

Figure 2.6: Solid lines: The quark mass dependence of the O(p2) BChPTresult for 〈x〉u+d of eq.(2.41) and Bs

2,0(0) of eq.(2.43). The values for the threepreviously unknown parameters resulting from a combined fit to the shownQCDSF data of ref.[G+04] and to the phenomenological value for 〈x〉u+d [xpr]are given in table 2.4. At the physical point we obtain Bs

2,0 = −0.056± 0.016,with the shown error only being statistical. Due to the poor data situation withits unknown systematic errors, however, these results should only be consideredas a rough estimate of the true quark mass dependence. The dashed linescorrespond to the respective HBChPT results at this order which are againfound to only be applicable for very low pion masses. We stress once again thatthe analysis of higher order terms is demanded in order to draw conclusionsabout the radius of convergence of HBChPT.

Performing a combined fit of eqs.(2.41,2.43) to the lattice data shown in fig.2.6and including the phenomenological value of 〈x〉u+d ∼ 0.54 [xpr], we obtainthe two solid curves shown in fig.2.6. The resulting parameters of the fit aregiven in table 2.4. Interestingly, despite the large quark masses and the hugeerror bars in the data of ref.[G+04] we obtain reasonable chiral extrapolationcurves with natural size coupling constants. The analysis of the QCDSF datain combination with the physical value for 〈x〉u+d suggests that the chiral limitvalue of this isoscalar PDF-moment is smaller than the value at the physi-cal point, leading to a monotonically rising chiral extrapolation function asshown in the left panel of fig.2.6. As a second observation we note that thevalue for the generalized form factor Bs

2,0(t = 0) could take on a small negativevalue at the physical point according to the right panel of fig.2.6, albeit witha large uncertainty due to the poor data situation and large systematic er-rors of the BChPT calculation at this order. Because of the small (negative!)value of the isoscalar Pauli form factor F s

2 (t = 0) = κs = −0.12 n.m., it issomewhat expected that the next higher moment yields a value close to zero.However, fig.2.6 now opens the possibility that Bs

2,0(t = 0) ≈ −0.06 might beas large as 50% of its F s

2 (t = 0) analogue. It will be very interesting to observewhether this feature can be reproduced when the new data of QCDSF [LQ] atsmall pion masses and a next-to-leading one loop order BChPT calculation ofthe generalized form factors become available. Fig.2.6 also demonstrates thatthe corresponding O(p2) HBChPT results are again not sufficient for a chiralextrapolation at this order. However, we note again that the true range of ap-

42


as2,0 bs

2,0 c9 〈x〉phen.u+d (µ = 2GeV)

0.527± 0.007 −0.103± 0.016 0.147± 0.002 0.538± 0.012 (fixed)

Table 2.4: The values for the three tensor coupling constants entering As2,0(t)

and Bs2,0(t) at order p2 as extracted from a combined fit to the lattice results

for As2,0(0) and Bs

2,0(0) [G+04] shown in fig.2.6 and to the physical point of

As2,0(t = 0,mπ = 0.14 GeV) = 〈x〉phen.

u+d . Note that we have obtained a smallnegative value for bs

2,0. The indicated uncertainties are the statistical errorsarising in the fit of eqs.(2.41) and (2.43) to the data of references [G+04, xpr]and do not reflect the much larger systematic uncertainties from both thelattice simulations (which e.g. neglect all contributions from disconnecteddiagrams) and possible higher order contributions to our chiral analysis.

Bs2,0(t = 0,mphys

π )

Extrapolated BChPT −0.056± 0.016[DGH08]values LHPC −0.094± 0.050 [H+08]

at mphysπ QCDSF/UKQCD −0.120± 0.023 [B+07b]

Table 2.5: Extrapolated values for the isoscalar generalized form factor Bs2,0(t =

0) at the physical pion mass, as obtained from the combined fit showed infigure 2.6 and from chiral extrapolations to the lattice data of refs.[H+08] and[B+07b].

plicability of HBChPT versus covariant BChPT can only be determined oncethe stability of the employed couplings is guaranteed, see the similar discussionfor 〈x〉u−d in section 2.4.1.The ChPT results for the form factors provided in this section have been sub-sequently used by LHPC [H+08] and QCDSF/UKQCD [B+07b] collaborationsto analyse new dynamical lattice QCD data. The combination of lattice cal-culations in the chiral regime (with pion mass values as low as 350 MeV) andour formulae obtained in chiral perturbation theory had allowed significantprogress in the chiral extrapolation of the generalized form factors of the nu-cleon. In table 2.5.1 we show the values obtained by the two collaborationsfor the forward limit of the form factor Bs

2,0 at the physical pion mass. Al-though these results have to be taken with due caution because of the reasonsdiscussed above, it seems that the results obtained from chiral extrapolationsstudies on more recent dynamical lattice data confirm that the form factorBs

2,0(t = 0) takes on a small negative value at the physical pion mass.

2.5.2 The contribution of u and d quarks to the spin ofthe nucleon

In the past few years a lot of interest in generalized isoscalar form factors ofthe nucleon has focused on the values of As

2,0 and Bs2,0 at the point t = 0 since

one can determine the contribution of quarks to the total spin of the nucleon

43


Ju+d(t = 0,mπ = 0.14GeV)Extrapolated BChPT 0.24± 0.05[DGH08]

values LHPC 0.213± 0.026[H+08]at mphys

π QCDSF/UKQCD 0.226± 0.013[B+07b]

Table 2.6: Values for the contribution of u and d quarks to the spin of anucleon, as obtained from chiral extrapolations in references [DGH08], [H+08]and [B+07b]. All three analysis are based on the same formulae given inref.[DGH08].

via these two structures [Ji97]:

Ju+d =1

2

[As

2,0(t = 0) + Bs2,0(t = 0)

]. (2.45)

To O(p2) in BChPT we find [DGH08]

Ju+d =1

2

{as

2,0 + bs2,0

MN(mπ)

M0

+as

2,0m2π

(4πFπ)2

[3g2

Amπ√4M2

0 −m2π

(8− 6

m2π

M20

+m4

π

M40

)

× arccosmπ

2M0

−3g2A

(3− m2

π

M20

+

(2− 4

m2π

M20

+m4

π

M40

)log

mπ

M0

)]

+4m2π

c9

M20

}+O(p3). (2.46)

Note that despite the plethora of nonanalytic quark mass dependent termscontained in the O(p2) BChPT result of eq.(2.46), the two chiral logarithmscalculated in ref.[CJ02] within the HBChPT framework are not yet containedin our result. Both terms (∼ a(q)π, b(q)N in the notation of ref.[CJ02]) arepart of the complete O(p3) result according to our power-counting and willappear in the calculation of the next order8. We further note that the twologarithms of ref.[CJ02] are UV-divergent and are accompanied by a counter-term, whereas the O(p2) BChPT result of eq.(2.46) happens to be UV-finiteto the order we are working. In ref.[CJ02] the authors also reported that thetwo chiral logarithms (of O(p3)) which they describe presumably are cancelednumerically by pion-cloud contributions around an intermediate ∆(1232) state.We can confirm that this possibility exists, as the described ∆ contributionsalso start at O(p3), assuming a power-counting where the nucleon-∆ massdifference in the chiral limit is counted as a small parameter of chiral dimensionone ∼ p1.Utilizing the O(p2) BChPT result of eq.(2.46) and the fit parameters of table2.4, we obtain a first estimate for the contribution of u and d quarks to thespin of a nucleon [DGH08]:

Ju+d(t = 0, mπ = 0.14 GeV) ≈ 0.24± 0.05, (2.47)

8The contribution ∼ a(q)π is already contained in the function ∆Bh.o.(t = 0, mπ) dis-cussed in subsection 2.5.3.

44


0 0.2 0.4 0.6 0.8 1mπ [GeV]

0

0.1

0.2

0.3

0.4

0.5

Jq

Figure 2.7: The contribution of u+d quarks to the spin of a nucleon as functionof the effective pion mass. The O(p2) BChPT result shown as the solid lineis a prediction of eq.(2.46) which utilizes the fit-parameters of table 2.4. Forcomparison we have also plotted simulation data from QCDSF [G+04] in thefigure. At the physical point one can read off Ju+d ≈ 0.24. (The error barshown at the physical point is only statistical, i.e. arises due to the errorsattached to the parameters in table 2.4, and does not reflect any systematicuncertainties.)

which is only about half of the total spin of the nucleon! We emphasize thatthis number is just a rough estimate, as we are assuming that the true error isdominated by systematic errors from both the lattice input to our analysis andthe possible higher order corrections to our chiral calculation9. Anyway, the au-thors of references [H+08]-LHPC collaboration- and [B+07b]-QCDSF/UKQCDcollaboration- have showed that the same chiral analysis performed on a dif-ferent set of lattice data leads to a result which is consistent with the valuegiven here, even within statistical errors only. The values for this quantity aregiven in table 2.6; one can observe that the results obtained from the threechiral extrapolations are consistent with each other.Based on the same input, we can also predict the quark mass dependence ofJu+d. The result is displayed as the solid line in fig.2.7.Note that in contrast to the analysis given in ref.[G+04], we do not obtaina flat chiral extrapolation function between the lattice data and the physicalpoint. The O(p2) BChPT analysis suggests that the value at the physical pointlies lower than the values obtained in the QCDSF simulation at large quark

9The size of the statistical error read off from the fit of table 2.4 is ±0.01 and thereforenegligible. The small value of this statistical error is of course heavily influenced by the errorassigned to the phenomenological value of 〈x〉u+d given in table 2.4. Note, however, thatthe size of statistical errors will increase once we extend our analysis to next-to-leading oneloop order due to the presence of several so far unknown counter-terms at this order.

45


masses. Following the above discussion, this curve obviously can only be a firstestimate of the true result.

2.5.3 A first glance at the generalized isoscalar formfactors of the nucleon

In this section we present the results for the momentum- and quark massdependence of the generalized form factors of the nucleon for the isoscalarflavor combination u + d at O(p2) in BChPT. We note that at this order theonly nonzero loop contributions to the three isoscalar form factors arise fromdiagrams c) and e) of figure 2.1, as the coupling of the isoscalar tensor field tothe nucleon is not affected by chiral rotations. One obtains

As2,0(t) = As

2,0(0)− as2,0g

2A

64π2F 2π

F s2,0(t) +

c13

M20

t +O(p3) , (2.48)

with As2,0(0) given in eq.(2.41) and F s

2,0(t = 0) = 0 by construction. Interest-ingly, to the order we are working here, the t dependence of this isoscalar formfactor As

2,0(t) is given by the same function

F s2,0(t) = F v

2,0(t) +O(p3), (2.49)

that controls its isovector analogue eq.(2.38) albeit with larger numerical pref-actors (compare eq.(2.48) to eq.(2.37)). We note that F s

2,0(t) does not dependon the scale λ of dimensional regularization for the loop diagrams. The chiralcoupling c13 is therefore also scale-independent10, parametrizing the quark massindependent short-distance contributions to the slope of As

2,0(t). The unknowncontributions from higher orders in the chiral expansion can be estimated froma calculation of the triangle diagram displayed in figure 2.2. Due to the cou-pling of the tensor field to the (long-range) pion-cloud of the nucleon, amongall the contributions at the next chiral order this diagram should give the mostimportant t dependent correction to the covariant O(p2) result of eq.(2.48),resulting in the estimate O(p3) ∼ ∆As

h.o.(t,mπ). The explicit expression forthe function ∆As

h.o.(t,mπ) can be found in appendix A.5.2. For completeness,we also note that in the limit 1/M0 → 0 we obtain the corresponding O(p2)HBChPT result

As2,0(t)|p

2

HBChPT = as2,0 + 4m2

π

c9

M20

+c13

M20

t +O(1/(16π2F 2πM0)) , (2.50)

10We note again that this scale-independence refers to the UV-scales of the ChEFT cal-culation, not to be confused with the scale- and scheme dependence of the quark-operatorson the left hand side of eq.(2.7) which is completely outside the framework of ChEFT.

46


which is just a string of tree level couplings.To order O(p2) in covariant BChPT the two other isoscalar form factors read

Bs2,0(t) = bs

2,0

MN(mπ)

M0

− as2,0g

2AM2

0

16π2F 2π

∫ 12

− 12

du

M8

{ (M2

0 − M2)

M6 + 9m2πM2

0 M4

−6m4πM2

0 M2 + 6m2πM2

0

(m4

π − 3m2πM2 + M4

)log

mπ

M

− 6m3πM2

0√4M2 −m2

π

[m4

π − 5m2πM2 + 5M4

]arccos

(mπ

2M

)}

+∆Bsh.o.(t,mπ), (2.51)

Cs2,0(t) = cs

2,0

MN(mπ)

M0

− as2,0g

2AM2

0

16π2F 2π

∫ 12

− 12

du u2

M8

{2(M2

0 − M2)

M6 − 3m2πM2

0 M4

+6m4πM2

0 M2 − 6m4πM2

0

(m2

π − 2M2)

logmπ

M

+6m3

πM20√

4M2 −m2π

[m4

π − 4m2πM2 + 2M4

]arccos

(mπ

2M

)}

+∆Csh.o.(t,mπ), (2.52)

with M defined in appendix A.4 (c.f. eq.(A.37)). MN(mπ) again denotes the(quark mass dependent) mass function of the nucleon eq.(2.10), introduced viaeq.(2.8). In the limit 1/M0 → 0 we obtain the corresponding O(p2) HBChPTresults for Bs

2,0(t) and Cs2,0(t) which at this order only consist of the tree level

coupling constants bs2,0 and cs

2,0. As in the case of As2,0(t) we have estimated

the contributions from higher orders via O(p3) ∼ ∆Bsh.o.(t, mπ), ∆Cs

h.o.(t,mπ),assuming that the dominant t dependent higher order corrections to our covari-ant O(p2) BChPT results of eqs.(2.51,2.52) originate from the O(p3) trianglediagram displayed in fig.2.2. Explicit expressions are given in appendix A.5.2.We note that the nonanalytic quark mass dependent terms in As

2,0(t), Bs2,0(t)

and Cs2,0(t) calculated in reference [BJ02] with the help of the HBChPT for-

malism11 correspond to the leading terms in a 1/M0 expansion of the O(p3)BChPT corrections ∆Xh.o.(t,mπ) where X = A,B,C of appendix A.5.2.At this point we refrain from a detailed numerical analysis of the t depen-dence of the generalized isoscalar form factors As

2,0(t) and Bs2,0(t). On the one

hand very few lattice data for this flavor combination have been published sofar for pion masses below 600 MeV. Moreover, available lattice data neglectcontributions from disconnected diagrams and are therefore accompanied by

11In refs.[ACK06, DMS06, DMS07] additional terms have been calculated within theHBChPT approach. While some terms correspond to O(p3) and O(p4) contributions accord-ing to our power-counting, expanding the covariant O(p2) result of eq.(2.51) to the order

1(4πFπ)2M1

0one can e.g. also recognize a term ∼ as

2,0 m2π present in ref.[ACK06]. However, as

far as we can see, neither ref.[ACK06] nor ref.[DMS06, DMS07] presents a complete O(p3)HBChPT calculation of the matrix element eq.(2.8).

47


an unknown systematic uncertainty which is very hard to estimate. On theother hand, in the t dependence both of As

2,0(t) and of Bs2,0(t) we encounter

chiral couplings (c13 at O(p2) in eq.(2.48) and B34 at O(p3) in ∆Bsh.o.(t,mπ)

of eq.(A.45)) connected with presently unknown short-distance physics andsystematic uncertainties can therefore not be guaranteed to be small at theorder we are working here. We are therefore postponing this discussion untilfurther information is available, in particular from a calculation of the effectsof next-to-leading one loop order. Despite of these considerations, the authorsof reference [H+08] have afterwards attempted a study of the t dependence ofthe O(p2) BChPT results for the isoscalar generalized form factors and havereported quite satisfactory findings. In the meantime we are preparing a full(next-to-leading one loop) O(p3) BChPT analysis of the isoscalar moments ofthe GPDs which (in addition to several other diagrams!) also contains thecontributions from the triangle diagram shown in fig.2.2, already presented inappendix A.5.2.Before finally proceeding to the summary of this chapter, we take a look atthe third generalized isoscalar form factor Cs

2,0(t) of eq.(2.52). According toour power-counting, short distance contributions to the radius of this form fac-tor are suppressed and only start to enter at O(p4), both in HBChPT and inBChPT. After adding the O(p3) estimate ∆Cs

h.o.(t,mπ) to the O(p2) BChPTresult of eq.(2.52), we can hope to catch a first glance of the t dependenceof this elusive nucleon structure. Utilizing as

2,0 of table 2.4 and assumingx0

π ≈ 〈x〉sπ ≈ 0.5 at a renormalization scale µ2 = 4 GeV2 [GRS99] we candetermine its slope

ρsC =

dCs2,0(t)

d t|t=0

=g2

A

640π2F 2πM6

0

{4as

2,0m4π

(2m2

π − 3M20

)log

mπ

M0

+ as2,0

(m6

π − 8m4πM2

0

+2m2πM4

0 −2

3M6

0

)+ x0

πM0MN(mπ)

(7m4

π − 27m2πM2

0 +20

3M4

0

)

−2x0π

MN(mπ)

M0

(4m6

π − 21m4πM2

0 + 20m2πM4

0 + 5M60

)log

mπ

M0

− 1

m3π − 4mπM2

0

[as

2,0

(m8

π − 4M20 m6

π + 2M40 m4

π

)

−x0πM0MN(mπ)

(m6

π − 7m4πM2

0 + 9m2πM4

0 + 8M60

)

− 1√4M2

0 −m2π

(4as

2,0m4π

(−2m6π + 15m4

πM20 − 30m2

πM40 + 10M6

0

)(2.53)

+2x0π

MN(mπ)

M0

(4m10

π − 45M20 m8

π + 170m6πM4

0 − 225m4πM6

0

+30m2πM8

0 + 32M100

))]

arccos

(mπ

2M0

)}.

(2.54)

48


At the physical point this would give us ρsC(mπ = 0.14GeV) = −0.77 GeV−2.

Truncating eq.(2.54) in 1/M0 we reproduce the chiral singularity ∼ m−1π found

in ref.[BJ02]12

ρsC = −g2

Ax0πMN(mπ)

160πF 2πmπ

− g2A

960π2F 2π

[as

2,0 + x0π

MN(mπ)

M0

(−13 + 15 log

mπ

M0

)]+...

(2.55)This amounts to a slope of ρs

C |χ = −1.12 GeV−2 which is 45% larger than theBChPT estimate of eq.(2.54). Interestingly, among the terms ∼ m0

π shown ineq.(2.55) it is the 1/M0 suppressed corrections to the leading HBChPT resultof ref.[BJ02] that dominate numerically. This gives a strong indication that acovariant calculation of ∆Cs

h.o.(t,mπ) as given in appendix A.5.2 is advisable,automatically containing all associated (1/M0)

i corrections.The quark mass dependence of the slope function ρs

C of eq.(2.54) already sug-gests that one obtains an interesting variation of the t dependence of thisform factor as a function of the quark mass! We therefore close this discus-sion with a look at fig.2.8. There we have fixed the only unknown parametercs2,0 = −0.41 ± 0.1 such that the BChPT result coincides with the dipole

parametrization of the QCDSF collaboration at t = 0 [G+04] for the lightestpion mass in the simulation, i.e. mπ = 640 MeV. We note explicitly that thiscoupling only affects the overall normalization of this form factor but does notimpact its momentum dependence. It is therefore quite remarkable to observethat the resulting t dependence of this form factor according to this ChEFTestimate agrees quite well with the phenomenological dipole-parametrizationof the QCDSF data at this large quark mass, even over quite a long range infour-momentum transfer. The result is rather close to a straight line which is aconsequence of the fact that in the renormalization scheme introduces and ap-plied in this work, pion-cloud effects are switched off at large pion masses (seefig.2.8). We remind the reader that the value of this form factor at t = 0 andmπ = 0.14 GeV determines the strength of the so-called D-term of the nucleon,playing a decisive role in the analysis of DVCS experiments [Ji98, Die03, BR05].Utilizing the extracted value of cs

2,0, we can now study the C-form factor alsoat the physical point, with the result also shown in fig.2.8. At this low value ofthe pion mass one can suddenly observe a nonlinear t dependence for low val-ues of four-momentum transfer due to the pion-cloud of the nucleon. This is avery interesting observation because such a mechanism would allow for a morenegative value of Cs

2,0(t = 0) at the physical point than previously extractedfrom lattice QCD analyses via dipole extrapolations (e.g. see ref.[G+04]). Weobtain [DGH08]

Cs2,0 (t = 0,mπ = 140 MeV) ≈ −0.36± 0.1, (2.56)

The assigned error corresponds to the fit error of Cs2,0(mπ = 0.64 GeV, t =

0)QCDSF given in ref.[G+04], as it directly influences our unknown coupling

12Note that due to a different definition of the covariant derivative in the quark-operatoron the left hand side of eq.(2.7) our definition of the third isoscalar form factor differs fromref.[BJ02] by a factor of 4: CBJ

2 (t) = 4Cs2,0(t).

49


0 0.2 0.4 0.6 0.8 1

-t [GeV2]

-0.4

-0.3

-0.2

-0.1

0

C20

s (t)

mπ = 640 MeV

mπ = 140 MeV

Figure 2.8: Momentum dependence of the form factor Cs2,0(t) [DGH08]. The

quasi linear t dependence of the O(p2) BChPT result at mπ = 640 MeV (solidline) has been normalized to the dipole parametrization of the QCDSF data ofref.[G+04] (dashed line), the shown data point gives the result of this referenceat t = 0 and mπ = 640 MeV. The resulting nonlinear t dependence in thisform factor for smaller values of mπ is then due to the coupling of the tensorfield to the pion-cloud, providing an interesting mechanism to obtain “large”negative values at t = 0 and mπ = 140 MeV.

cs2,0. However, we note that the (unknown) systematic uncertainties, both

from the quenched simulation results of ref.[G+04] and possible further O(p3)contributions (beyond ∆Cs

h.o.(t,mπ)) to our BChPT results are not accountedfor in this error bar.Despite of these considerations we observe that the value of eq.(2.56) is con-sistent with the value Cs

2,0 (t = 0,mπ = 140 MeV) = −0.267± 0.036 derived inthe subsequently published ref.[H+08], where our formula of eq.(2.52) has beenfitted to dynamical lattice data from the LHPC collaboration.

With the value of eq.(2.56) we can finally obtain the first estimate for theradius of this elusive form factor [DGH08]:

(rsC)2 =

6

Cs2,0(t = 0, mπ = 140 MeV)

ρsC(mπ = 140 MeV)

≈ (0.5± 0.1) fm2. (2.57)

We compare this result with the radii of the isovector Dirac- and Pauli formfactors of the nucleon which are also dominated by pion-cloud effects. Inter-estingly, with (rv

1)2 = 0.585 fm2 and (rv

2)2 = 0.80 fm2 [MMD96] the estimated

value for (rsC)2 seems to lie in the same order of magnitude! We note, however,

that our numerical estimate for the slope ρsC of eq.(2.54) given above is signif-

icantly smaller than the corresponding slopes of the isovector Pauli- and Diracform factors, as expected from general arguments and as already observed in

50

2.6. The First Moments of axial GPDs

lattice QCD simulations with dynamical fermions [L+04] (at very large quarkmasses).However, before we can go into a more detailed numerical analysis of these in-teresting new form factors of the nucleon, one should first complete the O(p3)calculation of the generalized isoscalar form factors, as there are additionaldiagrams next to fig.2.2 possibly also affecting the t dependence, albeit pre-sumably in a weaker fashion [DHH].

2.6 The First Moments of axial GPDs

In full analogy to the vector case, we now extend our analysis to the firstmoments of the axial Generalized Parton Distributions [DGH07, DH]. AxialGPDs have been first introduced together with the vector ones and are usuallyreferred to as Hq(x, ξ, t) and Eq(x, ξ, t) [Ji98, Die03, BP07].In eq.(2.3) of the introduction of this chapter we have seen how x-integrals ofvector GPDs are related to the Dirac and Pauli form factors. Integrating thefunctions Hq(x, ξ, t) and Eq(x, ξ, t) over the momentum fraction x we find

∫ 1

−1

dx Hq(x, ξ, t) = Gq

A(t), (2.58)

∫ 1

−1

dx Eq(x, ξ, t) = Gq

P (t), (2.59)

where GqA(t), Gq

P (t) are the axial and the pseudoscalar form factors of thenucleon, respectively. These definitions come clear if we consider the matrixelement of the axial current between two nucleons. Lorentz invariance andparity conservation require this matrix element to take the form

〈p2|qγµγ5q|p1〉 = u(p2)

[Gq

A(t)γµγ5 + GqT (t)

iσµνγ5qν

2MN

+ GqP (t)

γ5qµ

2MN

]u(p1),

(2.60)

where the three form factors GqA(t), Gq

T (t) and GqP (t) reflect the structure of

the nucleon when probed by an external axial field.The axial tensor term, Gq

T (t), is often omitted because it has the oppositebehaviour under charge symmetry from the other two terms. The case GT 6= 0would imply the existence of the so called second-class currents [Wei58], whichare experimentally excluded to a high precision [Wil00]. We therefore chooseto work in absence of second-class terms, assuming GT = 0.Furthermore, the axial-vector form factor GA(t) admits the following expansionfor small momentum transfer

GA(t) = gA

(1 +

1

6〈r2

A〉t +O(t2)

), (2.61)

with 〈r2A〉 the nucleon axial radius and gA the axial-vector coupling constant,

which defines the limit for vanishing momentum transfer of the nucleon axial

51


form factor GA(t). We have already encountered the chiral limit of the axial-vector coupling constant gA in section 1.1.2, when we have introduced theleading-order pion-nucleon Lagrangian L(1)

πN . A calculation of the quark-massdependence of gA in Chiral Perturbation Theory is obtained via studying theresponse of the nucleon to the presence of an external axial field.

An analysis of the axial form factors of the nucleon GqA(t), Gq

P (t) in theframework of HBChPT can be found in references [BKM92a, BKLM94, FLMS97,BFHM98], while a calculation in the covariant scheme has been presented inreference [SFGS07]. Further discussion on data and theoretical prediction canbe found in refs. [BEM02, GF04]. In the last years a lot of chiral extrapolationstudies of lattice data of the coupling constant gA have been also performed,for details we refer to [HPW03b, E+06b, PMHW07, K+06].As for the vector case we focus on the first moments in x of the axial GPDs

∫ 1

−1

dx x Hq(x, ξ, t) = Aq2,0(t), (2.62)

∫ 1

−1

dx x Eq(x, ξ, t) = Bq2,0(t). (2.63)

A study of the leading one loop order contributions to the first moments ofaxial GPDs in the framework of Heavy Baryon ChPT has been presented inreferences [CJ01, ACK06, DMS06, DMS07], while no calculation in covariantBChPT can be found in literature sofar.For each quark q one has two generalized form factors Aq

2,0(t) and Bq2,0(t),

defined by the form factor decomposition of the matrix element [Ji98, Die03,BP07]

i〈p2| qγ{µγ5

←→D ν}q|p1〉 =

Aq

2,0(q2) u(p2)γ{µγ5 pν}u(p1) +

Bq

2,0(q2)

2MN

u(p2)γ5 q{µ pν}u(p1), (2.64)

where we make use of the same notation of eq.(2.7). Like in the vector case,the form factors A

q

2,0(q2), B

q

2,0(q2) are real functions of the momentum transfer

squared. Notice that no Cq(q2) form factor appears in the matrix element ofeq.(2.64); time reversal invariance indeed constraints the current u(p2)γ5u(p1)to come with odd powers of qα, allowing for a two-form factors decompositionof the matrix element.

Working within the methods of ChPT, we concentrate on the triplet (v)contribution of the quarks to the two axial generalized form factors:

i〈p2|qγ{µγ5

←→D ν}q|p1〉u−d =

u(p2)

[A

v

2,0(q2) γ{µγ5 pν} +

Bv

2,0(q2)

2MN

γ5 q{µ pν}]]u(p)× η†

τa

2η. (2.65)

In analogy to the vector case, in the limit of vanishing four-momentum trans-fer the isovector form factor Av

2,0(t → 0) is directly connected to the spin

52


dependent analogue of the mean momentum fraction 〈∆x〉u−d.

Av2,0(0) = 〈∆x〉u−d =

∫ 1

0dx x (q↓(x)− q↑(x))

∣∣u−d

(2.66)

= ∆av2,0 +O(p2),

where q↓(↑)(x) is the distribution with quark spin parallel (antiparallel) to thespin of the nucleon and ∆av

2,0 corresponds to the chiral limit value of 〈∆x〉u−d.Because of this helicity dependence, which becomes explicit in the forwardlimit case, axial GPDs are sometimes referred to as ”polarized” generalizedparton distributions, where the adjective off course refers to the spin of theparton and not of the target.

2.6.1 The Generalized Axial Form Factors in O(p2) BChPT

Like in the vector case, we stop our analysis of the generalized axial form factorsto the leading one loop order in BChPT, i.e to the O(p2) level according tothe power counting scheme presented in section 2.3.3.The loop diagrams contributing to the first moments of the axial GPDs areagain the ones discussed for the vector GPDs (see fig.2.1), off course afterappropriate interchange of the couplings.We remark that the results presented in this section still have to be published.

The relevant parts of the chiral Lagrangian we need for the evaluation ofthe tensor current of eq.(2.65) have been given in section 2.3. Note that sincewe are now dealing with an external axial source (ai

αβ), the leading term to the

form factor A2,0(q2), i.e the coupling ∆av

2,0, appears in the order p0 Lagrangian

of eq.(2.18), while the leading term to the form factor B2,0(q2), i.e ∆bv

2,0, now

belongs to the next-to-leading order Lagrangian L(2)tπN of eq.(2.24), which has

to be extended as follows:

L(2)tπN = . . .

+ψN

{− ∆bv

2,0

8M0

(iγ5

[−→Dµ, V

µν−

]−→D ν + h.c.

)}ψN

+c14

4M20

ψN

{Tr(χ+)V −

µνγµγ5i

−→D ν + h.c.

}ψN

+c16

4M20

ψN

{[−→Dα, [

−→Dα, V −

µν ]]γµγ5i

−→D ν + h.c.

}ψN

+ . . . (2.67)

The new coupling c14 governs the leading quark-mass insertion in 〈∆xu−d〉,while c16 parametrizes the contributions of short-distance physics to the slopesof the generalized form factor Av

2,0(t).In the following we present the O(p2) BChPT expressions for both the

generalized form factors, obtained again in the IR regularization scheme. TheO(p2) amplitudes corresponding to the diagrams of fig.2.1 can be found inappendix A.6.1.

53


We first start providing results for the form factors Av2,0(t) at t = 0. In this

limit, the PDF-moment reads

Av2,0(0) = 〈∆x〉u−d

∆av2,0 +

∆av2,0m

2π

3(4πFπ)2

{− 3(2g2

A + 1) logm2

π

λ2− 3g2

A

+g2A

m2π

M20

(1 + 9 log

mπ

M0

)− g2

A

m4π

M40

logmπ

M0

+g2

Amπ√4M2

0 −m2π

(22− 11

m2π

M20

+m4

π

M40

)arccos

(mπ

2M0

)}

+av

2,0 gA m2π

3(4πFπ)2

{2m2

π

M20

(1 + 3 log

m2π

M20

)− m4

π

M40

logm2

π

M20

+2mπ(4M2

0 −m2π)

32

M40

arccos

(mπ

2M0

)}

+4m2π

c(r)14 (λ)

M20

+O(p3). (2.68)

The truncation of this expression at leading order in 1/M0 gives the exactO(p2) HBChPT limit of our result, in agreement with the findings of references[CJ01, ACK06, DMS06, DMS07]:

Av2,0(0)|p2

HBChPT = ∆av2,0

{1− m2

π

(4πFπ)2

(g2

A + (2g2A + 1) log

m2π

λ2

)}

+4m2π

c(r)14 (λ)

M20

+O(p3). (2.69)

Looking at the covariant expression for 〈∆x〉u−d of eq.(2.68) and at the resultfor 〈x〉u−d given in eq.(2.25), one can easily observe that each isovector moment(〈x〉u−d and 〈∆x〉u−d) depends on 3 unknown parameters: 2 couplings (av

2,0,∆av

2,0) and one counterterm (c8 and c14, respectively). As the same couplingscontribute in both moments, it is hoped that a simultaneous fit of our BChPTresults to the lattice data of ref.[H+08] can considerably reduce the statisticalerrors. As one can see from Fig.2.9, the results of this procedure are prettyoutstanding, given that the values at the physical pion mass were not includedin the fit! The chiral curvature in both observables naturally bends down to thephenomenological value for lighter quark masses, leading to a very satisfactoryextrapolation curve.

The input values used in the fit and the resulting values for the fit param-eters, together with their statistical errors are given in table 2.7. We make thechoice to adopt the same input values used throughout the present work andin ref.[DGH08], i.e. we choose to set the nucleon axial-vector coupling and thepion decay constant equal to their physical values gA = 1.2 and Fπ = 0.0924MeV. Strictly speaking, these quantities should be taken in the chiral limit(gA = 1.267, Fπ = 0.0862[CD04]). The difference resulting from this choice is

54


0 0.2 0.4 0.6 0.8mΠ @GeVD

0

0.05

0.1

0.15

0.2

0.25

0.3

<x>

u-d

0 0.2 0.4 0.6 0.8mΠ @GeVD

0

0.05

0.1

0.15

0.2

0.25

0.3

<x>

u-d

0 0.2 0.4 0.6 0.8mΠ @GeVD

0

0.05

0.1

0.15

0.2

0.25

0.3

<D

x>u-

d

0 0.2 0.4 0.6 0.8mΠ @GeVD

0

0.05

0.1

0.15

0.2

0.25

0.3

<D

x>u-

d

Figure 2.9: Combined FIT of the O(p2) results given in eq.(2.25) and eq.(2.68)to the LHPC data of reference [H+08] for 〈x〉u−d and 〈∆x〉u−d respectively. Thefit is a 4-parameter fit; only data points for mπ < 600 MeV have been includedin the fit, for a total of 8 data points. Note that the phenomenological valuesat physical pion mass were not included in the fit. The bands shown indicateestimate of higher order possible corrections.

Input values Values from the fit

gA 1,2 av2,0 0.14± 0.01

M0 [GeV] 0.889 ∆av2,0 0.16 ± 0.01

Fπ [GeV] 0.0924 cr8(1 GeV) -0.23 ± 0.06

cr14(1 GeV) -0.16 ± 0.05

Table 2.7: The input values and the values of the coupling constants resultingfrom the combined fit of the O(p2) BChPT results of eqs.(2.25) and (2.68) tothe LHPC data of reference [H+08] for 〈x〉u−d and 〈∆x〉u−d respectively. Thecorresponding plot is showed in fig.2.9.

not significant, because the ratio gA/Fπ is very insensitive to this change. Thechoice of the physical values instead of the chiral ones does not lead to anysignificant changes in the final results.The extrapolated values of the two observables are reported in table 2.8, wherethe corresponding phenomenological values 〈x〉phen

u−d and 〈∆x〉phenu−d are also given.

As one can easily see, the obtained ChPT values from this O(p2) leading-one-loop analysis are clearly consistent with the experimental ones! The tableshows also the extrapolated values for 〈x〉u−d at the physical pion mass, asobtained by LHPC and QCDSF/UKQCD collaborations from chiral extrapo-lation studies on recent lattice data. Both analysis make use of the covariantchiral expression for 〈x〉u−d given in ref.[DGH08](i.e. eq.(2.25)). Differentlyto our case, where all couplings are free-parameters in the fit, the analysisperformed by both lattice QCD collaborations implies an input value for thecoupling constant ∆av

2,0. The LHPC collaboration uses the value ∆av2,0 = 0.17

obtained from a chiral fit to own lattice results for 〈∆x〉u−d [E+06a], while theQCDSF/UKQCD collaboration follows ref.[DGH08] and constraints the cou-pling ∆av

2,0 from the phenomenological value of 〈∆x〉u−d = 0.21. The LHPC

55


〈x〉u−d 〈∆x〉u−d

Phenomenology 0.16± 0.006[xpr] 0.21± 0.025 [BB02]Extrapolated BChPT 0.16± 0.01 0.19± 0.01

values LHPC 0.157± 0.006 [H+08]at mphys

π QCDSF/UKQCD 0.198 ± 0.008 [B+07b]

Table 2.8: The phenomenological values of the observables 〈x〉u−d and 〈∆x〉u−d,together with the extrapolated values obtained from the combined fit showedin fig.2.9 and from the chiral extrapolation studies on lattice data from LHPC[H+08] and QCDSF/UKQCD [B+07b] collaborations. All three analysis makeuse of the chiral results of ref.[DGH08].

result agrees very well with both phenomenology and the value obtained fromour combined fit. On the contrary, the QCDSF/UKQCD value is definitely toohigh, it is not consistent with the experimental result. We want to stress onceagain that such comparisons between the results from the different collabora-tions have to be done with due caution, reminding of the serious differenceson normalization between the two sets of lattice data. Therefore, before firmconclusions can be drawn, the combined fit described in this section will haveto be newly performed once QCDSF/UKQCD lattice data for 〈∆x〉u−d willalso be available.

We would like to stress that this efficient cross-talk between the ChPTresults for 〈x〉u−d and 〈∆x〉u−d occurs only in the covariant framework, whileas clearly showed by eqs.(2.27),(2.69)in the non-relativistic approach both ob-servables are completely independent at this order.

We now turn to the full t-dependence of Av2,0(t), which can be written as

Av2,0(t) = Av

2,0(0) +∆av

2,0 g2A

96π2F 2π

F v2,0(t) +

c16

M20

t +O(p3), (2.70)

with Av2,0(0) given above and

F v2,0(t) = −1

6t− m3

π(4m4π − 17m2

πM20 + 10M4

0 )

M40

√4M2

0 −m2π

arccos

(mπ

2M0

)

+m4

π

M40

(4m2π − 9M2

0 ) logmπ

M0

+

∫ 12

− 12

du

{m3

πM20 (4m4

π − 17m2πM2 + 10M4)

M6

√4M2 −m2

π

arccos

(mπ

2M

)

+3m2

π

M2(M2 −M2

0 ) +4m4

π

M20 M4

(M40 − M4) (2.71)

+2M20 log

M0

M+

m4πM2

0

M6(9M2 − 4m2

π) logmπ

M

}, (2.72)

56


where we use the same notation introduced for the vector channel in section2.4.3.Furthermore, we provide the expression of the slope of Av

2,0(t):

ρvA

=c16

M20

− ∆av2,0 g2

A

6(4πFπ)2

m2π

(4M20 −m2

π)

{− 2 + 4

m2π

M20

(2 + 3 log

mπ

M0

)(2.73)

−m4π

M40

(2 + 11 log

mπ

M0

)+ 2

m6π

M60

logmπ

M0

+mπ√

4M20 −m2

π

(10− 30

m2π

M20

+ 15m4

π

M40

− 2m6

π

M60

)arccos

(mπ

2M0

)}

+mπδt

A

(4πFπ)2M0

, (2.74)

where we have introduced the termmπδt

A

(4πFπ)2M0to indicate the size of possible

higher order corrections coming from O(p3), in complete analogy to the vectorcase (see section 2.4.2). A first numerical analysis of the formula given abovereveals a very small contribution of the pion-cloud to the slope of the axialgeneralized form factor, suggesting again that the physics which governs thesize of this object is hidden in the counter-term contribution c16.

Finally, we present the O(p2) BChPT results for Bv2,0(t):

Bv2,0(t) = ∆bv

2,0

MN

M0

+∆av

2,0 g2AM2

0

24π2F 2π

∫ 12

− 12

du

M8u2

{M6(M2 −M2

0 )

+3m2πM2

0 M4

(4 + log

m2π

M2

)− 3m4

πM20 M2

(4 + 5 log

m2π

M2

)

+6m6πM2

0 logm2

π

M2− 6m3

πM20√

4M2 −m2π

(7M4 − 9m2πM2 + 2m4

π)

× arccosmπ

2M

}+O(p3), (2.75)

which in the limit t → 0 reduces to

Bv2,0(0) = ∆bv

2,0

MN

M0

+∆av

2,0 g2A m2

π

6(4πFπ)2

{4 + log

m2π

M20

− m2π

M20

(4 + 5 log

m2π

M20

)

+2m4

π

M40

logm2

π

M20

− 2mπ√4M2

0 −m2π

(7− 9

m2π

M20

+ 2m4

π

M40

)arccos

(mπ

2M0

)}

+O(p3). (2.76)

Furthermore, the relative slope reads:

57


ρvB

=∆av

2,0 g2A

180(4πFπ)2

1

(4M20 −m2

π)

{− 4M2

0 + m2π

(109 + 24 log

m2π

M20

)

−6m4

π

M20

(35 + 31 log

m2π

M20

)+ 3

m6π

M40

(16 + 47 log

m2π

M20

)

−24m8

π

M60

logm2

π

M20

+6m3

π√4M2

0 −m2π

(− 70 + 140

m2π

M20

− 63m4

π

M40

+ 8m6

π

M60

)

× arccos

(mπ

2M0

)}+ δB

MN

(4πFπ)2M0

. (2.77)

Again, δB is not part of the O(p2) result but it is given to indicate the size ofpossible higher order corrections from O(p3).Since the axial generalized form factor Bv

2,0(t) is not directly connected toany phenomenological observables, unfortunately a numerical analysis of theexpressions given above does not provide us with new interesting physicalinformation. Anyway, lattice data for this quantity are available and Bv

2,0(t)shows dependence on the coupling constant ∆av

2,0 which appears in severalquantities discussed in this chapter. Future lattice simulations, together withhigher order calculations from ChPT will then allow to perform simultaneousfits of many observables with consequent minimization of the statistical errors,paving the way to even more reliable chiral predictions.

58

2.7. Summary

2.7 Summary

The pertinent results of this chapter can be summarized as follows:

1. We have constructed the effective chiral Lagrangian for symmetric, trace-less tensor fields of positive parity up to O(p2) in the covariant frameworkof Baryon Chiral Perturbation Theory for two light quark flavors.

2. Within this covariant framework we have calculated the generalized isovec-tor and isoscalar form factors of the nucleon Av,s

2,0(t, m2π), Bv,s

2,0(t,m2π) and

Cv,s2,0(t,m

2π) up to O(p2) which corresponds to leading one loop order.

Our results were IR renormalized and we can thus exactly reproduce thecorresponding nonrelativistic O(p2) results previously obtained in HeavyBaryon ChPT by taking the limit 1/M0 → 0. Several HBChPT resultspublished recently could not yet be reproduced, as they correspond topartial, nonrelativistic results from the higher orders O(p3, p4, p5).

3. According to our numerical analysis of the quark mass dependence of thegeneralized form factors, we have noted that for Bs,v

2,0(t) and Cs,v2,0(t) the

observable quark mass dependencies could be dominated by the (wellknown) quark mass dependence of the mass of the nucleon MN(mπ).This mass function appears in several places in the chiral results dueto kinematical factors in the matrix element used in the definition ofthe generalized form factors. Such a “trivial” but numerically significanteffect is already known from the analysis of lattice QCD data for thePauli form factors of the nucleon.

4. The pion-cloud contributions to all three generalized isovector form fac-tors only show a very weak dependence on t. The momentum trans-fer dependence of these structures seems to be dominated by presentlyunknown short distance contributions. The situation in this isovectorchannel reminds us of an analogous role played by chiral dynamics inthe isoscalar Dirac- and Pauli form factors of the nucleon. At this pointwe are therefore not able to give predictions for the numerical size ofthe slopes of these interesting nucleon structure quantities. It is hopedthat a global fit to new lattice QCD data at small pion masses and smallvalues of t – extrapolated to the physical point with the help of the for-mulae presented in this work – will lead to first insights into this newfield of baryon structure physics. A first step on this way has alreadybeen achieved in reference [H+08, B+07b].

5. The leading one loop order covariant BChPT results for the generalizedisoscalar form factors As

2,0(t) and Bs2,0(t) are quite surprising. As far

as the topology of possible Feynman diagrams is concerned, one is re-minded of the isovector Dirac- and Pauli form factors of the nucleon.A power-counting analysis, however, told us that those diagrams (e.gsee fig.2.2) which one would naively expect to strongly depend on boththe momentum transfer and the quark mass only start to contribute at

59


next-to-leading one loop order. Our analysis therefore suggests that themomentum dependence at low values of t is dominated by short-distancephysics.

6. It is the value of 〈x〉π of a pion in the chiral limit that controls the magni-tude of those long-distance pion-cloud effects in the generalized isoscalarform factors of the nucleon, pointing to the need of a simultaneous anal-ysis of pion- and nucleon structure on the lattice and in ChEFT.

7. In the forward limit, the isovector form factor Av2,0(t → 0) reduces to

〈x〉u−d. Our covariant O(p2) BChPT result for this isovector momentprovides a smooth chiral extrapolation function between the high valuesat large quark masses from the LHPC collaboration and the lower valueknown from phenomenology. The required chiral curvature accordingto this new analysis does not originate from the chiral logarithm of theleading nonanalytic quark mass dependence of this moment – as hadbeen speculated in the literature for the past few years – but is due to aninfinite tower of terms (mπ/M0)

i with well constrained coefficients (seeeq.(2.25)). The well known leading one loop HBChPT result for 〈x〉u−d ofeq.(2.27) was found to be not applicable for chiral extrapolations abovethe physical pion mass, as expected.

8. Judging from the (quenched) lattice data of the QCDSF collaboration,our O(p2) BChPT result of eq.(2.41) for As

2,0(t = 0) = 〈x〉u+d also pro-vides a very stable chiral extrapolation function out to quite large valuesof effective pion masses.

9. A study of the forward limit in the isoscalar sector has led to a firstestimate of the contribution of the u and d quarks to the total spin ofa nucleon Ju+d ≈ 0.24. This low value compared to previous determi-nations arises from the possibility of a small negative contribution ofBs

2,0(t = 0) ≈ −0.06 at the physical point, driven by pion-cloud effects.However, at the moment the uncertainty in such a determination is ratherlarge.

10. In a first glance at the third generalized isoscalar form factor Cs2,0(t)

the quark mass dependence was found to be qualitatively different fromAs

2,0(t) and Bs2,0(t). Its slope contains a chiral singularity ∼ m−1

π and theinfluence of short distance contributions is suppressed. A first numericalestimate of its slope gives ρs

C ≈ −0.75 GeV2 which is much smaller thanthe slopes of the corresponding Dirac- or Pauli form factors. At low t wehave also observed significant changes in the momentum dependence ofthis form factor as a function of the quark mass, resulting in the estimateCs

2,0(t = 0) ≈ −0.35 at the physical point.

11. The extension of our analysis to the axial generalized form factors isstraightforward. The covariant O(p2) results of eq.(2.27) for Av

2,0(t =

0) = 〈x〉u−d and of eq.(2.69) for Av2,0(t = 0) = 〈∆x〉u−d reveals the cross-

talk of coupling constants between the two observables, suggesting the

60

2.7. Summary

idea of a simultaneous fit to the available lattice data. The result ofthe procedure is extremely convincing and we conclude that combinedfits of several observables characterized by a common subset of ChEFTcouplings are the winning strategy towards the most reliable chiral ex-trapolations of lattice QCD results.

12. Throughout this chapter we have indicated how to estimate possible cor-rections of higher orders to our leading one loop order BChPT results.The associated theoretical uncertainties of our O(p2) calculation havebeen discussed in detail. Ultimately, in order to judge the stability ofour results it is mandatory that we analyse the complete next-to-leadingone loop order. In appendix A.7 we provide preliminary results of thiscalculation, referring to a later communication for a more complete anddetailed discussion.

Finally we refer that the BChPT results for the generalized isovector andisoscalar vector form factors presented in this chapter have already been ap-plied in a detailed analysis of recent lattice data in references [H+08, B+07b].

61

Chapter 3Doubly Virtual ComptonScattering with ∆’s

3.1 Introduction and summary

In the last decades much effort has been done in theoretical as well as in exper-imental physics in order to better understand the spin structure of the nucleon[FJ01]. A lot of interesting results have been obtained but at the same timenew exciting challenges have arisen. Nowadays facilities make possible to workwith polarized beams and polarized targets, paving the way to the study ofthe spin structure of the nucleon encoded in the spin structure functions g1

and g2. In particular, the high luminosity available at Jefferson Laboratoryhas allowed to extract the spin structure functions and their moments withan unprecedented precision over a wide kinematic range [Che08]. Results forQ2 values as low as 0.1 GeV2 have been reported and the plan for a future 12GeV upgrade at Jefferson Lab has already been proposed. On the theoreticalside, such a low energy region can be investigated in Chiral Perturbation The-ory, while the region of intermediate momentum transfer can be analysed bymeans of quark or resonance models or using dispersion relations. Dispersionrelations represent the link which can bridge the gap between the high energyand the low energy domain, between perturbative and non-perturbative QCD.To this purpose, a very interesting topic is represented by certain sum rules,such as the Gerasimov-Drell-Hearn (GDH) sum rule and its generalization tofinite photon virtuality or the Burkhardt-Cottingham (BC) sum rule. Thesesum rules have validity at all energy scales and can be written in terms ofthe spin structure functions g1 and g2. They are very special tools becausethanks to dispersion relations they can be related to the spin-dependent struc-ture functions S1,2(ν, Q

2) which appear in the spin amplitude of the processof doubly virtual Compton scattering (VVCS), where ν is the energy transferand Q2 the negative of the photon virtuality. A set of sum rules in terms of thestructure functions S1,2(ν, Q

2) can be derived: in addition to the BC sum ruleand the GDH sum rule at finite virtuality, of particular interest are also thesum rules for the forward (γ0(Q

2)) and longitudinal-transverse (δ0(Q2)) gen-

63

3. Doubly Virtual Compton Scattering with ∆’s

Q2(GeV2)

Γ 1p (n

o el

asti

c)CLAS EG1aSLAC E143

HERMESGDH slope

Burkert-IoffeSoffer-Teryaev (2004)

Bernard et al, Χpt

Ji et al, Χpt

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.5 1 1.5 2 2.5 3Q2(GeV2)

Γ 1n (n

o el

asti

c)

JLab Hall A E94-010CLAS EG1aSLAC E143

CLAS EG1b preliminary

HERMESGDH slopeBurkert-IoffeSoffer-

Teryaev (2004) Bernardet al, Χpt

Ji et al, Χpt

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 0.5 1 1.5 2 2.5 3

Q2(GeV2)

Γ 1p-n

EG1bJLab Hall A E94010/CLAS EG1aCLAS EG1aHERMESE143 E155

GDH slope

pQCD leading twist

Burkert-Ioffe

Soffer-Teryaev (2004)

0

0.05

0.1

0.15

0.2

10-1

1

Q2(GeV2)

Γ 1p (n

o el

asti

c)

CLAS EG1aSLAC E143

CLAS EG1b (preliminary)

HERMESGDH slope

Burkert-IoffeSoffer-Teryaev (2004)

Bernard et al, Χpt

Ji et al, Χpt

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.5 1 1.5 2 2.5 3Q2(GeV2)

Γ 1d (n

o el

asti

c) CLAS EG1a

SLAC E143CLAS EG1b

HERMES

GDH slopeBurkert-Ioffe

Soffer-Teryaev (2004)

ΧpT Bernard et al

Ji et al, Χpt

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 0.5 1 1.5 2 2.5 3

Figure 3.1: The left and right panels show the first moments Γ1(Q2) for proton

and neutron, in the order [Che08]. Experimental results are compared tomodel predictions (full and dashed lines). In the insets the region at low Q2 isselected, where the next-to-leading order ChPT calculations of refs. [JKO00]and [BHM03b] gain validity.

eralized spin polarizabilities. Chiral Perturbation Theory represents the mostsuitable framework to study the process of doubly virtual Compton scatteringin the low energy domain. Indeed, a determination of the two building blocksS1,2(ν = 0, Q2) in ChPT allows to make predictions for all desired spin-relatedsum rules and for all moments of the nucleon’s spin structure functions forQ2 . 0.3 GeV2. Such a calculation has been done at the next-to-leading oneloop order (O(p4)) within HBChPT [KSV03, Bur01, JKO00] and in infraredregularization [BHM02, BHM03b]. At that order of the chiral expansion, nounknown low-energy constants appear and thus the calculation owns the bigadvantage to allow parameter-free predictions for the spin structure functionsS1,2(ν, Q

2). Starting from there one can derive all the desired results for thesum rules and the spin polarizabilities.The calculation has gained relevance because of new recent experimental re-sults. In particular, new data are available for the first moments Γ1(Q

2) ofthe spin structure function g1. They are shown for both proton and neutronin fig.3.1, where experimental data from JLAB Hall A [A+02, M+05], CLASEG1A [F+03, Y+03] and EG1B [D+06, P+08, B+07a], SLAC [A+03] and HER-MES [A+07] are compared to model predictions and to the ChPT results ofreference [JKO00] and [BHM03b]. As one can observe, the calculations arein reasonable agreements with the data at the lowest Q2 values up to ∼ 0.07GeV2, that is within the range of applicability of the theory. In particular, abetter convergence is achieved by the covariant calculation of Bernard et al.[BHM03b], thanks to higher order terms which appear in the covariant butnot in the heavy baryon calculation. The authors of ref.[BHM03b] have alsomade a first attempt to estimate the influence of the ∆(1232) resonance on

64

3.1. Introduction and summary

0 0.1 0.2 0.3Q

2 (GeV

2)

0.0

1.0

2.0

3.0

δ LT (

10−

4 fm4 )

E94010 MAID estimate Kao et al. O(p

3)+O(p

4)

Bernard et al. Bernard et al. (VM + ∆)

−5.0

0.0

γ 0 (1

0−4 fm

4 )

Figure 3.2: Experimental results for the neutron generalized spin polarizabil-ities γ0 and δ0 [Che08]. The squares represent the results with statistical un-certainties, while systematic uncertainties are shown by the light bands. Theresults are compared to the heavy baryon predictions of ref.[KSV03] and tothe relativistic baryon ChPT calculation of ref.[BHM03b], without (dot-dashedcurves) and with (dark bands) ∆ and vector meson resonances contributions.

the values of the VVCS matrix element. The ∆ plays a very important rolein the spin sector of the nucleon and it is reasonable to think that its inclu-sion on the calculation can bring about a considerable change of value to thespin structure fucntions. To this purpose, the ∆ relativistic Born graphs havebeen evaluated. Furthermore, in order to consider the fact that the N → Nγtransition occurs at finite Q2 and in order to get an input from phenomenol-ogy, a transition form factor has been introduced and its value has been fixedfrom pion electroproduction in the ∆(1232) resonance region. Finally, a mod-elling of the less important vector mesons resonances has also been included.Going back to the analysis of the first moment Γ1, we observe that the calcu-lations are in reasonable agreement with the data at the lowest Q2 settings of0.05− 0.1 GeV2. Anyway, the evaluation of the only Born graphs is definitelynot enough for a reliable estimation of the effect of the ∆ resonance. Thusour challenge consists in performing the leading one loop order calculation ofthe structure function S1,2(ν,Q

2) in covariant small scale expansion. We areconfident that the O(ε3) calculation presented in this work will throw lightupon the role played by the ∆ resonance within this sector of nuclear physics.

Besides the case of the first moments Γ1, another very interesting topic isrepresented by the generalized spin polarizabilities γ0 and δ0. They are de-

65


fined in such a way that they represent the ideal objects to test ChPT at lowQ2. Thus the need for their investigation towards a better understanding ofthe dynamics of QCD in the low energy domain of chiral EFT. From ChPT itturns out that γ0 is sensitive to the nucleon resonances, while δ0 is not muchaffected by the ∆ resonance. Figure 3.2 shows the first results for the neu-tron spin polarizabilities obtained at JLAB Hall A [A+02, M+05], togetherwith predictions from the next-to-leading one loop order (O(p4)) calculationin HBChPT [KSV03] and from the corresponding relativistic calculation ofthe already cited reference [BHM03b], with and without a first rough estiman-tion of the vector meson and ∆ resonance contributions. We observe how atlow Q2 the relativistic calculation including the resonance contributions arein very good agreement with the experimental data, while the heavy baryoncurve shows big discrepancies even at Q2 = 0.1GeV2. This can be explainedby either a bad behaviour of the non-relativistic approximation or the signifi-cant role played by the resonances. A calculation of the same quantities withinclusion of the ∆ as explicit degree of freedom will help us solve out the issue.Since the δ0 polarizability is not influenced by the nucleon resonances, we couldthink that it is more suitable to the comparison to the experimental results.On the contrary, even at the lowest Q2 = 0.1GeV2 the ChPT predictions arein significant disagreement with the experimental data, thus the name of ”δ0

puzzle” to this problem of chiral EFT. The reason of this disagreement can berelated to problems generated by the Infrared regularization (IR) scheme: asdiscussed in section 1.2, this scheme suffers from the deficiency of presentingunphysical cuts in the infrared regular part R of the loop integrals. This cutsfirst appears at values of Q2 already outside the range of applicability of thechiral theory but they can influence in a significant way the behaviour of thecurve at lower values of Q2. A possible solution to the ”δ0 puzzle” could bethe use of the modified infrared (IR) instead of the infrared regularization (IR)scheme. Finally, the calculation of this quantity in the covariant SSE schemewill confirm or disprove the statement that the δ0 is not sensitive to the ∆resonance.

Concluding, our aim is to present a leading one loop order calculation of thespin amplitude of the process of doubly virtual Compton scattering (VVCS)off nucleons in the SSE scheme of covariant Chiral Perturbation Theory.

The chapter is organized as follows: In the next section we firstly intro-duced the formalism needed to describe the process of doubly virtual Comptonscattering, we derive the pertinent sum rules and we define the nucleon polar-izabilities. In section 3.3 we explain how to extract the spin structure func-tions in ChPT, while section 3.4 is devoted to the description of the effectiveLagrangians required for the calculation and to the application of the power-counting scheme for the identification of the Feynman diagrams contributingto the VVCS matrix element. Finally, section 3.5 collects the main resultsof our O(ε3) calculation. For a more detailed and complete analysis of thematerial portrayed in this work we refer to a later communication [BDKM].

66

3.2. Compton scattering and nucleon polarizabilities

3.2 Compton scattering and nucleon polariz-

abilities

In this section we provide an overview on spin dependent doubly virtual Comp-ton scattering, focusing in particular on its connection to inelastic electropro-duction and on the derivation of the corresponding sum rules.We first discuss the forward scattering of a real photon by a nucleon andafterwards we generalize the formalism to the case of finite virtuality. For amore detailed introduction to the subject we refer to references [DPV03, DT04,CDM05]

3.2.1 Real Compton scattering

Working in the laboratory frame, we consider the photon scattering process

γ(q, ε) + N(p) → γ(q′, ε′) + N(p′) (3.1)

on a nucleon with initial and final momentum p and p′. The incident photonhas four-momentum q = (ν, ~q) and polarization four-vector ε = (0, ~ε), withq2 = 0 (real photon) and ~ε · ~q = 0 (transerve polarization). The correspondingprimed quantities characterize the kinematics of the scattered photon. Forforward scattering with ~q = ~q ′ the Compton amplitude reduces to

T (ν, θ = 0) = ~ε′ · ~ε f(ν) + i ~σ · (~ε′ × ~ε) g(ν), (3.2)

where θ is the nucleon scattering angle and ν has been defined above and cor-responds to the photon laboratory energy. The two functions g(ν),f(ν) are thespin-flip and non-flip amplitude, respectively. Because of crossing symmetry,the amplitude T of eq.(3.2) has to be invariant under exchange of the in- andoutgoing photons, that is under the transformation ε′ ↔ ε and ν ↔ −ν. Thisimplies that the two amplitudes f(ν) and g(ν) have opposite parity:

f(ν) = f(−ν) , g(ν) = −g(−ν). (3.3)

The amplitudes f and g can be separated by measurement of scattering pro-cesses of circularly polarized photons off nucleons polarized parallel or an-tiparallel with respect to the photon momentum. Let us take for example aright-handed photon (helicity λ = 1): one can have 1/2 or 3/2 helicity inter-mediate states, whose corresponding amplitudes are linear combination of fand g :

f 12

= f + g f 32

= f − g. (3.4)

Similarly, we define the total absorption cross section as the spin average overthe two helicity cross sections,

σT =1

2(σ3/2 + σ1/2), (3.5)

67


and the transverse-transverse interference term by the helicity difference,

σ′TT =1

2(σ3/2 − σ1/2). (3.6)

By the optical theorem, we are able to relate the just defined absorption crosssections to the imaginary part of the respective forward scattering amplitudes:

Im f(ν) =ν

8π(σ1/2(ν) + σ3/2(ν)) =

ν

4πσT (ν), (3.7)

Im g(ν) =ν

8π(σ1/2(ν)− σ3/2(ν)) = − ν

4πσ′TT (ν). (3.8)

Unitarity, crossing symmetry and the analytic properties of the forward Comp-ton scattering amplitude allow us to set up dispersion relations for f(ν) andg(ν). Replacing the imaginary part by the cross sections, as given in eq.(3.7)and (3.8), the dispersion integrals then read

Re f(ν) = f(0) +ν2

2π2P

∫ ∞

ν0

σT (ν ′)ν ′2 − ν2

dν ′, (3.9)

Re g(ν) =ν

4π2P

∫ ∞

ν0

σ1/2(ν′)− σ3/2(ν

′)ν ′2 − ν2

ν ′dν ′, (3.10)

where P denotes the principal value integral and ν0 is the threshold energyfor pion production. We point out that for the function f(ν) a subtractionat ν = 0 has been made in order to ensure the convergence of the integralalso in case of a divergent cross section. If we subtract at ν = 0, i.e., weconsider the difference f(ν)− f(0), we also remove the nucleon pole terms atν = 0. For the function g(ν) we are able to write an unsubtracted dispersionrelation, since we assume that the difference of cross sections σ1/2(ν

′)−σ3/2(ν′)

decreases sufficiently fast at large ν ′ to make the integral converge even withoutsubtraction.At this point we expand the relations of eq.(3.9) and (3.10) in powers of ν

Re f(ν) = f(0) +∑n=1

(1

2π2

∫ ∞

ν0

σT (ν ′)ν ′2n

dν ′)

ν2n, (3.11)

Re g(ν) =∑n=1

(1

4π2

∫σ1/2(ν

′)− σ3/2(ν′)

(ν ′)2n−1dν ′

)ν2n−1, (3.12)

and we recall the historical low energy theorem (LET) of Low [Low54], Gell-Mann and Goldberger [GMG54]. According to this theorem the leading andnext-to-leading terms of the expansion are determined by the global propertiesof the system:

f(ν) = − e2 e2N

4πM0

+ (α + β) ν2 +O(ν4), (3.13)

g(ν) = − e2κ2N

8πM20

ν + γ0ν3 +O(ν5). (3.14)

68


Here e eN is the charge (ep = 1,en = 0), M0 the nucleon mass and (e/2M0)κN

the anomalous magnetic moment. Moreover, f(0) = − e2 e2N

4πM0= −αQED/M0

is the well-known Thomson limit and it indicates that, if the photon wave-length is very large with respect to any dimension of the proton, the scatteringamplitude can only involve the fine structure constant αQED and the protonmass. The spin-flip amplitude g(ν) vanishes at ν = 0 and because of crossingsymmetry f(ν) has no O(ν) term. The terms of order ν2 introduce the electric(α) and magnetic (β) polarizabilities ; they yield information on the internalnucleon structure providing a measure of the global resistance of the nucleon’sinternal degrees of freedom against displacement in an external electric or mag-netic field. The term O(ν3) provide information on the spin structure throughthe forward spin polarizability γ0.By matching eq.(3.11),(3.12) and the just discussed low energy expansions ofeq.(3.13),(3.14) we can write a set of sum rules; in particular we obtain theso-called Baldin’s sum rule for the electric and magnetic polarizabilities[Bal60]

α + β =1

2π2

∫ ∞

ν0

σT (ν ′)ν ′2

dν ′, (3.15)

the sum rule of Gerasimov[Ger66], Drell and Hearn[DH66] (GDH sum rule)

πe2κ2N

2M20

=

∫ ∞

ν0

σ3/2(ν′)− σ1/2(ν

′)ν ′

dν ′ ≡ I, (3.16)

and the forward spin polarizability can be expressed via[GMG54]

γ0 = − 1

4π2

∫ ∞

ν0

σ3/2(ν′)− σ1/2(ν

′)ν ′3

dν ′. (3.17)

Several experiment of Compton scattering off protons have been performedover the last decades (for example at MAMI, ELSA, GRAAL, CEBAF andSpring-8). A weighted average over various measurements give the followingvalues for the static electric and magnetic polarizabilities of the proton[Sch05]:

αp = (12.0± 0.6) · 10−4 fm3,

βp = (1.9∓ 0.6) · 10−4 fm3. (3.18)

Taking into account the typical size of a proton (∼ 1fm3), these findings indi-cate that the protons are rather stiff objects.Since stable single-nucleon targets do not exist, corresponding values for theneutron are much harder to access; a weighted average over several experimentgives[Sch05]:

αn = (12.5± 1.7) · 10−4 fm3,

βn = (2.7∓ 1.8) · 10−4 fm3. (3.19)

On the theory side, reference [BKM92b] provides a calculation of the elec-tric and magnetic nucleon polarizabilities at the leading one loop order in

69


HBChPT, while an extension to the next-to-leading order has been presentedin references [BKSM93] and [BKMS94]. The authors of reference [HGHP04]have proposed a multipole analysis of the so called dynamical polarizabilitieswith inclusion of the ∆(1232) as explicit degree of freedom and have success-fully compared their results to a dispersion relations analysis. A calculation ofthe spin polarizability in HBChPT can be found in reference [BKKM92], whilean analysis in the SSE scheme has been presented in reference [HGHP05]. Fi-nally, for a study in Chiral Effective Field Theory of Compton scattering fromdeuteron as an alternative way to extract the neutron polarizabilities we referto [HGH04].

3.2.2 Doubly virtual Compton scattering

In analogy to the case of real Compton scattering, we now proceed to dis-cuss the process of doubly virtual Compton scattering (VVCS); we derive therelative sum rules and we generalize the concept of polarizabilities to virtualphotons. As already stated in the introduction of this chapter, of particularinterest are the sum rules for the spin-dependent VVCS amplitudes, which willbe the object of our chiral analysis.Let us consider the following reaction:

γ?(q, ε) + N(p, s) → γ?(q, ε′) + N(p, s′), (3.20)

describing a process of doubly virtual Compton scattering off nucleons in theforward direction. The virtual photon is now characterized by a negative pho-ton virtuality, that is q2 = q2

0 − ~q 2 = −Q2 < 0, and s(s′) defines the nucleonspin polarization four-vector of the incoming (outgoing) nucleon.The absorption of a virtual photon on a nucleon is related to inelastic elec-troproduction, e + N → e′ + anything, where e and e′ are the electrons inthe initial and final states, respectively. The differential cross section for thisprocess can be written in terms of four virtual photoabsorption cross sections:σT (ν,Q2), σL(ν,Q2) of eq.(3.5) and (3.6), σ′L(ν, Q2) and σ′LT (ν,Q2). Becauseof the longitudinal polarization of the virtual photon, in addition to the crosssections introduced for the real Compton scattering in section 3.2.1 one has thelongitudinal cross section σ′L and a longitudinal-transverse interference σ′LT .The four virtual photoabsorption cross sections are related to the quark struc-tures functions measured in (un)polarized deep inelastic electron scatteringvia[DKT01]

σT (ν, Q2) =4π2αQED

mKF1(ν, Q

2), (3.21)

σL(ν, Q2) =4π2αQED

K

[F2(ν, Q2)

ν(1 + γ2)− F1(ν, Q

2)

M0

], (3.22)

σ′LT (ν, Q2) = −4π2αQED

Kγ ν

[(G1(ν, Q

2) +ν

M0

G2(ν, Q2))

], (3.23)

σ′TT (ν, Q2) = −4π2αQED

Kν

[(G1(ν, Q

2)− ν

M0

γ2G2(ν, Q2))

], (3.24)

70


where ν = E − E ′ is the virtual photon energy in the lab frame, αQED =e2/4π = 1/137.036 is the fine structure constant, γ = Q/ν and K is a fluxfactor.The VVCS amplitude for forward scattering reads:

T (ν, Q2, θ = 0) = ~ε ′ · ~ε fT (ν, Q2) + fL(ν, Q2)

+ i ~σ · (~ε ′ × ~ε) gTT (ν, Q2)

− i ~σ · [(~ε ′ − ~ε)× q ] gLT (ν, Q2). (3.25)

We note that in the limit Q2 = 0 the amplitude reduces to the one of eq.(3.2)for the real scattering, with f(ν) = fT (ν, Q2 = 0) and g(ν) = gTT (ν, Q2 = 0),while the remaining two structures vanish because they are related to thelongitudinal polarization of the photon.We focus our attention on the spin dependent VVCS amplitudes gTT (ν, Q2),gLT (ν, Q2) and we choose to work on eq.(3.25) in its covariant form,

T [µν](p, q, s) = −iM0

Nεµναβqα

{sβ S1(ν,Q

2)

+ [p · q sβ − s · q pβ]S2(ν, Q

2)

M20

}, (3.26)

where sν is a spin-polarization four-vector satisfying s · p = 0, s2 = −1 andεµναβ is the totally antisymmetric Levi-Civita tensor with ε0123 = 1. Again,ν = p · q/M0 defines the energy transfer and Q2 = −q2 is the negative of thephoton virtuality. The normalization factor N depends on the convention used.As suggested in references [BHM02, BHM03b] we set S1(0, 0) = −e2κ2

N/M20

and as usual in baryon ChPT we use the spinor normalization uu = 1, so thatN = 2M0. We note that crossing symmetry (ν → −ν) implies that S1(ν, Q

2)and S2(ν, Q

2) have opposite parity, even and odd respectively.The optical theorem relates the imaginary parts of the spin amplitudes to thestructure functions Gi(ν, Q

2) introduced in eq.(3.23),(3.24):

Im Si(ν, Q2) = 2π Gi(ν, Q

2), (i = 1, 2) . (3.27)

Assuming an appropriate behaviour at high energy, the functions S1(ν, Q2)

and S2(ν, Q2) satisfy the following dispersion relations [IKL, JO01]:

S1(ν, Q2) = 4e2

∫ ∞

Q2/2M0

dν ′ν ′G1(ν′, Q2)

ν ′2 − ν2,

νS2(ν, Q2) = 4e2

∫ ∞

Q2/2M0

dν ′ν2G2(ν′, Q2)

ν ′2 − ν2. (3.28)

Note that from now on we will refer to Si(ν,Q2) as to the real part of the

relative functions.Before we proceed with the derivation of the sum rules we have to make animportant remark. As we will better see in section 3.5, the contribution of the

71


corresponding diagrams to the spin amplitudes can be separated into elasticand inelastic parts. The elastic part is given by the single nucleon exchange(pole) terms and does not contribute to the nucleon spin structure. In particu-lar, the sum rules we are talking about involve uniquely the contributions frominelastic processes for ν > ν0. In what follows we will therefore consider a sub-tracted version of the Compton amplitudes, i.e. we subtract the contributionfrom the elastic intermediate state:

Si(ν, Q2) = Si(ν, Q

2)− Seli (ν,Q2). (3.29)

We next perform a low energy expansion around ν = 0 for the subtractedstructure functions [JO01, BHM03b],

S1(ν,Q2) = S

(0)1 (0, Q2) + ν2 S

(2)1 (0, Q2) + ν4 S

(4)1 (0, Q2) + . . . ,

νS2(ν,Q2) = S

(−1)2 (0, Q2) + ν2 S

(1)2 (0, Q2) + ν4 S

(3)2 (0, Q2)

+ν5 S(5)2 (0, Q2) + . . . , (3.30)

where the ellipses indicate terms of higher order in ν. Let us first concentrateon the expansion of the structure function S1(ν, Q

2); recalling eq.(3.28) onegets the following sum rules for S1(0, Q

2):

S(0)1 (0, Q2) = 4e2

∫ ∞

ν0

dν ′G1(ν′, Q2)

ν ′=

4e2

M20

I1(Q2) , (3.31)

S(2i)1 (0, Q2) = 4e2

∫ ∞

ν0

dν ′G1(ν′, Q2)

ν ′(2i+1)(i = 1, 2, 3, . . . ), (3.32)

where

I1(Q2) =

2M20

Q2

∫ x0

0

g1(x,Q2)dx (3.33)

is the generalization of the GDH sum rule to finite Q2[AIL89]. Here, x =Q2/2νM0 is the standard scaling variable, x0 corresponds to the pion pro-duction threshold and the spin structure function g1 is related to G1 viaG1 = g1/(M0ν). At zero virtuality I1(0) = −κ2

N/4 and one recovers the GDHsum rule of eq.(3.16), while in the limiting case Q2 → ∞ the often used first

moment Γ1(Q2) ≡ ∫ 1

0dx g1(x,Q2) appears:

I1(Q2) →

{−1

4κ2

N for Q2 → 0,2M2

0

Q2 Γ1 +O(Q−4) for Q2 →∞.(3.34)

From the second line of eq.(3.30) we obtain the sum rules for the second spindependent amplitude S2(0, Q

2):

S(−1)2 (0, Q2) = 4e2

∫ ∞

ν0

dν ′G2(ν′, Q2) =

4

M20

I2(Q2) , (3.35)

S(i)2 (0, Q2) = 4e2

∫ ∞

ν0

dν ′G2(ν′, Q2)

ν ′(i+1)(i = 1, 3, 5, . . .), (3.36)

72


where I2(Q2) is defined by the sum rule for the spin structure function g2,

known as Burkhardt-Cottingham sum rule[BC70]:

I2(Q2) =

2M20

Q2

∫ x0

0

g2(x,Q2)dx (3.37)

=1

4

GM(Q2)−GE(Q2)

(1 + Q2

4M20)

GM(Q2)

=1

4F2(Q

2)(F1(Q

2) + F2(Q2)

). (3.38)

Here, GE(Q2) and GM(Q2) are the electric and magnetic nucleon Sachs formfactors, in order, while F1(Q

2) and F2(Q2) are the Dirac and Pauli nucleon

form factors already encountered in the introduction of chapter 2. Moreover,G2 = g2/ν

2.The Burkhardt-Cottingham sum rule has the limiting cases

I2(Q2) →

{14µNκN for Q2 → 0,O(Q−10) for Q2 →∞.

(3.39)

In addition to the GDH and the BC sum rules, the following sum rules havebeen introduced in the past year[DKT01]:

IA (Q2) =M2

0

8π2α

∫ ∞

ν0

(1− x) (σ1/2 − σ3/2)dν

ν

=2M2

0

Q2

∫ x0

0

(g1 − γ2 g2) dx , (3.40)

=2M2

0

Q2

∫ x0

0

1√1 + γ2

(g1 − γ2 g2) dx , (3.41)

IC (Q2) =M2

0

8π2α

∫ ∞

ν0

(σ1/2 − σ3/2)dν

ν

=2M2

0

Q2

∫ x0

0

1

1− x(g1 − γ2 g2) dx. (3.42)

While IA(Q2) can be easily expressed in terms of Si(ν,Q2) via

IA(Q2) =M2

0

4

(S

(0)1 (0, Q2)−Q2S

(1)2 (0, Q2)

), (3.43)

in order to calculate IC(Q2) we have to introduce the new quantity SR(ν, Q2)defined as:

SR(ν, Q2) = 4e2

∫ ∞

ν0

dν ′G1(ν

′, Q2)−Q2G2(ν′, Q2)/ν

ν ′ − ν. (3.44)

This leads to the relation

73


IC(Q2) =M2

0

4SR(Q2/2M0, Q

2). (3.45)

We now turn to the forward spin polarizability defined in eq.(3.17) and wewrite it as

γ0 =1

4π2

∫dν

ν3(x− 1)σ′TT (3.46)

=1

4π2

∫dν

ν3(1− x)(σ1/2(ν, 0)− σ3/2(ν, 0)).

We observe that in case of zero virtuality, i.e. Q2 = 0, we have that x vanishesand we go back to the dispersion integral of eq.(3.17).Finally, considering the longitudinal-transverse interference cross section σLT

we can define the so called longitudinal-transverse spin polarizability δ0:

δ0 = − 1

2π2

∫dν

ν3lim

Q2→0

(ν

Qσ′LT

), (3.47)

We note that the definitions of the spin polarizabilities present an extra 1/ν2

weighting compared to the GDH and BC sum rules. As a consequence theintegrals defining the generalized spin polarizabilities receive less contributionsfrom the large-ν region and converge much faster, representing the most suit-able object to test ChPT in the low energy domain.The generalization to finite Q2 can be performed also for these quantities,yielding to the following relations between γ0(Q

2), δ0(Q2) and the Si(ν, Q

2):

γ0(Q2) =

1

8π

(S

(2)1 (0, Q2)− Q2

M0

S(3)2 (0, Q2)

), (3.48)

δ0(Q2) =

1

8π

((S

(2)1 (0, Q2) + S

(1)2 (0, Q2)

). (3.49)

3.3 The spin polarizabilities in covariant ChPT

We now fix the most suitable kinematical and gauge conditions for the evalu-ation of the VVCS forward matrix element of eq.(3.26). Working in the restframe of the nucleon, i.e. p = (M0,~0) and adopting the Coulomb gauge condi-tion ε0 = ε′0 = 0, the matrix element T [µν] takes the form:

T[ij]rest(ν, Q

2) =i

2εijkχ

†{

ν σkS1(ν, Q2)

+[ν2σk − σ · q qk

] S2(ν, Q2)

M0

}χ, (3.50)

74

3.3. The spin polarizabilities in covariant ChPT

where ν now corresponds to the photon energy in the laboratory system,σk(k = 1, 2, 3) denote the Pauli spin matrices and χ is a two component spinor.Making use of the following identity

~σ · ~q ~ε ′ · (~ε× ~q) =(ν2 + Q2

)[~σ · (~ε ′ × ~ε)

− (~σ · (~ε ′ × q) ~ε · q − ~σ · (~ε× q) ~ε ′ · q)] (3.51)

we obtain

ε′ · T · ε|rest =1

2M0

χ†{i~σ · (~ε ′ × ~ε)

[M0νS1(ν, Q

2)−Q2S2(ν, Q2)

]

+i [~σ · (~ε ′ × q) ~ε · q − ~σ · (~ε× q) ~ε ′ · q] (ν2 + Q2)S2(ν,Q

2)}

χ.

(3.52)

The VVCS forward matrix element is now written in the most convenient wayfor the calculation in the framework of relativistic Baryon Chiral PerturbationTheory.We recall that the amplitude T has to fulfill crossing symmetry. Calling s =(p+ q)2 the standard Mandelstam variable, one can express the photon energyby ν = (s −M2

0 + Q2)/2M0. Therefore, for each diagram, the correspondingcrossed one can be derived very easily by operating the transformations ε ↔ ε′

and s → (2M20 − 2Q2 − s). These considerations enable us to write eq.(3.52)

via two invariant functions A(ν,Q2) and B(ν,Q2):

ε′ · T · ε|rest = χ†{i~σ · (~ε ′ × ~ε)

[A(s,Q2)− A(2M2

0 − 2Q2 − s,Q2)]

+ i [~σ · (~ε ′ × q) ~ε · q − ~σ · (~ε× q) ~ε ′ · q] |~q |2 [B(s,Q2)

−B(2M20 − 2Q2 − s,Q2)

]}χ.

(3.53)

In this expression both contributions from a diagram and its crossed partner aretherefore taken into account. As we have seen in section 3.2.2, we are mainlyinterested in quantities evaluated at ν = 0, i.e. at s = M2

0 + q2 = M20 − Q2.

Let us for example concentrate on the term proportional to i~σ · (~ε ′ × ~ε) ineq.(3.53): as required by crossing symmetry, if we write the equation in termsof ν the quantity [A(M2

0 −Q2 + 2M0ν, Q2)−A(M2

0 −Q2− 2M0ν,Q2)] appear,

that is the difference of a quantity taken at ν with the same quantity taken at−ν. As a consequence, comparing eq.(3.53) with eq.(3.52) we deduce that theSi(0, Q

2) we need in order to evaluate the sum rules can be simply obtainedas derivatives of order n ≥ 1 of the functions A and B.

75


For example:

S1(0, Q2)− Q2

M0

limν→0

S2(0, Q2)

ν= 4

dA(ν, Q2)

dν

∣∣∣∣ν=0

, (3.54)

Q2 limν→0

S2(0, Q2)

ν= 4

dB(ν, Q2)

dν

∣∣∣∣ν=0

, (3.55)

where νS2(ν, Q2) ≡ νS2(ν, Q

2) − νS2(ν, Q2)

∣∣ν=0

and we recall that the con-

tribution from the elastic intermediate state is subtracted, i.e. Si(ν,Q2) ≡

Si(ν, Q2)− Sel

i (ν, Q2).

3.4 Effective Field Theory with ∆’s.

In section 1.3 we have summarized the basic concepts of a Chiral EffectiveField Theory with inclusion of the ∆ resonance as explicit degree of freedom.Here we give the Lagrangians we need in order to perform a leading one looporder calculation of the Compton amplitudes within the covariant frameworkof SSE.The full relativistic Lagrangian we require for a system of nucleon, ∆ and piondegrees of freedom consists of four terms:

Leff = LπN + Lππ + LπN∆ + Lπ∆. (3.56)

Concerning the first two pieces of Lagrangians listed, the ones relevant for ourstudy are the well known L(1)

πN of eq.(1.7) and the meson L(2)ππ of eq.(1.2). These

terms are responsible for the interaction of a photon with nucleons and pions,respectively.The leading chiral pion-∆-nucleon Lagrangian is given by [HHK98]:

L(1)πN∆ = gπN∆ ψ

µ

i Θµν (Z) wν

i ψN + h.c., (3.57)

with

wνi = 〈τ iuν〉/2, (3.58)

Θµν(Z) = gµν + (Z − 1/2)γµγν , (3.59)

where τ i,i = 1, 2, 3 denote as usual the Pauli matrices in isospin space and Z isthe so-called off-shell parameter. Since this parameter does not appear in thespin 3/2 contributions, the physical observables we are interested in do not de-pend on it. In our calculation we choose to put Z = 1/2. Here, the LEC gπN∆

represents the leading axial-vector N∆ coupling constant, frequently called cA

in the literature. We recall that ψµi denotes the spin-3/2 field in the Rarita-

Schwinger notation.The coupling of the ∆ resonance to the photon is described through the fol-lowing Lagrangian[HHK98],

76

3.4. Effective Field Theory with ∆’s.

L(1)π,∆ = −ψ

µ

i

[(i/Dij −m0

∆δij)gµν − 1

4γµγ

λ(i/Dij −m0

∆δij)γλγν

]ψν

j , (3.60)

where we have listed only the terms pertinent to the calculation. Here, theminimally coupled covariant derivative reads:

Dijµ = Dµδ

ij − iεijkTr[τ kDµ], (3.61)

with Dµ the standard derivative defined in eq.(1.5), whereas m0∆ refers to the

∆(1232) mass in the chiral limit.Finally, we introduce the relativistic counterterm Lagrangian[HHK97]

L(2)γN∆ = i

b1

2MN

ψµ

i gµνγργ51

2Tr[fρν

+ ] ψN + h.c., (3.62)

which defines the γN∆ transition at order ε2 and will be employed for theevaluation of the Born graphs contributing to the process at order O(ε3). Forthe numerical estimate of the magnitude of the finite O(ε2) counterterm b1

we follow reference [HHK97], where the authors employ the phenomenologicalfindings of reference [DMW91] to get b1 = −(2.5 ± 0.35). The chiral tensorfield appearing above is defined as[BKM95]

f+µν = u†FR

µνu± uFLµνu

†,

with

FL,Rµν = ∂µF

L,Rν − ∂νF

L,Rµ − i[FL,R

µ , FL,Rν ]

and FLµ = vµ − aµ, FR

µ = vµ + aµ, where vµ and aµ denote external vector andaxial vector sources, in the order.

We are now provided with the building blocks of the theory underlying theprocess of VVCS at the one-loop level in relativistic SSE. Finally, we remindthat our choice for the ∆ propagator corresponds to the following general formin the Rarita-Schwinger formalism (see section 1.3 for further details):

Gijµν(p) = −i

/p + m∆

p2 −m2∆

{gµν − 1

d− 1γµγν − (d− 2)

(d− 1)

pµpν

m2∆


(d− 1)m∆

}ξij3/2.

(3.63)

3.4.1 Power-counting

In this section we draw the Feynman diagrams contributing to the VVCS pro-cess up to the leading one loop order. Similarly to the study of the generalized

77


Figure 3.3: Born graphs at order O(ε3).

a) b) c) d)

g)f)e)

n)m)l)k)

j)i)h)

Figure 3.4: Loop diagrams contributing to the spin structure functions at thirdorder. Solid, dashed and wiggly lines denote nucleons, pions and photons,respectively. A double solid line denotes the ∆ resonance. Crossed diagramsare not shown.

form factors (see section 2.3.3), we start from the general power-counting for-mula of Baryon ChPT:

D = 2NL + 1 +∑

d

(d− 2)NMd +

∑

d

(d− 1)NMBd . (3.64)

Now the variable NMBd counts the number of vertices of chiral dimension d

from pion-nucleons but also from pion-∆ and pion-nucleon-∆ Lagrangians.The chiral order D = 1 is given by a zero number of loops NL = 0 andan arbitrary number of insertions from the leading meson and meson-baryonLagrangian. Since the leading γN∆ coupling is of chiral dimension d = 2 (its

generated by the L(2)γN∆ of eq.(3.62)), there is no D = 1 diagram contribution

to the spin amplitudes.The first contribution with NL = 0 occurs at the chiral order D = 3 and it

corresponds to the Born graph depicted in figure 3.3, where the crossed part-ner is also shown. We observe that no two-photon seagull term contributesto the spin-dependent structure at this order. Indeed, in order to constructa two-photon observable, one needs two factors of the electromagnetic fieldstrength tensor Fµν , that is a term of fourth order. Since such terms alwaysappear together with structures of the type ε′ · ε, i.e. they just contribute to

78

3.4. Effective Field Theory with ∆’s.

s)

r)

p)

q)

o)

Figure 3.5: Loop diagrams giving a zero contribution to the inelastic structureof the spin dependent VVCS amplitudes.

the spin-independent amplitudes, the terms of our interest will firstly appearat the fifth order.In addition to the Born graphs, at the chiral order D = 3 starts the NL = 1topology. One has the two possibilities NMB

1 = 2,NM2 = 0 and NMB

1 =1,NM

2 = 1, corresponding to the diagrams of figures 3.4 and 3.5. Only thediagrams depicted in figure 3.4 actually give non zero contribution to the spinamplitudes. The diagrams o),p),q),r) do provide no inelastic structure: aftersubtraction of the elastic part they reduce to constant terms, which conse-quently vanish once the corresponding crossed diagrams are added. Finally,diagrams s) provides neither elastic nor inelastic contributions, it simply doesnot contribute to the spin dependent amplitude of the process. Explicit expres-sions for the diagrams in terms of the pertinent loop functions will be discussedin the next section.The Feynman rules pertinent to the calculation can be found in appendix B.

As already stated in the introduction of this chapter, a very importantfeature of this calculation is the possibility to make parameter free predictions.For the observables considered here no contributions from L(3)

πN exist, so that noLECs will appear in the results for the structure functions S1,2(ν, Q

2), whereagain the bar reminds us of the subtraction of the elastic intermediate states.

79


3.5 Covariant O(ε3) results for the VVCS am-

plitudes.

We apply the methods of Chiral EFT in the scheme of small scale expansionto derive the leading one loop order contributions to the functions A(s,Q2)and B(s,Q2) as defined in eq.(3.53). Strictly speaking, we make use of theFeynman rules listed in appendix B to calculate the diagrams of chiral orderD = 3, according to the power-counting scheme presented in section 3.4.1.As already mentioned in the last section, the only tree level contributions aregiven by the Born graphs of fig.3.3, which have to be summed up to the loopdiagrams of figs.3.4,3.5.We are interested exclusively in the inelastic contributions and we thereforesubtract the terms from the elastic intermediate states. In practise, this impliesthe subtraction of the contributions from the single nucleon exchange, thenucleon poles of the kind 1/(s − M2

0 ) generated by the nucleon propagatorswhich appear in the diagrams.In what follows we provide the expressions for A(s,Q2) and B(s,Q2) aftersubtraction of the nucleon pole terms.

We first start our discussion with the Born graphs, the sum of the both(direct plus crossed terms) is given by:

A(s,Q2) =

(e b1

2M0

)21

(s−m2∆)

(3.65)

{M2

0 −Q2 − s

6M0m2∆

[M4

0 + s2 + 5Q4 + Q2(6m2∆ + 2s)

−2M20 (5Q2 + 2s)

] }

B(s,Q2) =

(e b1

2M0

)21

(s−m2∆)

{−2M0 −m∆

3m2∆

(M20 −Q2 − s)

}.(3.66)

Unlike the case with the nucleon as only baryonic degree of freedom [BHM03b],the Born graphs now obviously show no elastic contributions, so that the wholeamplitude survives the subtraction from the elastic intermediate states. Werecall that the value of the LEC b1 coming from the Lagrangian L

(2)γN∆ of

eq.(3.62) can be fixed by fitting resonant multipoles and its values is about−(2.5± 0.35).

We now proceed with the analysis of the diagrams which give a non-zerocontribution to the spin amplitudes. The crossed partners are easily ob-tained by taking the negative of the direct ones once the substitution s →(2M2

0 − 2Q2 − s) has been made. In particular, in this section we providethe expressions for the diagrams involving only one ∆ propagator, while themore complicated ones, involving two or three ∆ propagators are reported inappendix B.3 for practical reasons. Those longer and more difficult diagramsare calculated in an automatic way by using Form and Mathematica. The pro-gram, realized by Hermann Krebs, provides results in terms of the general loop

80

3.5. Covariant O(ε3) results for the VVCS amplitudes.

function II[d + n][{{p1,m1}, k1}, . . . {{pN ,mN}, kN}] of eq.(B.10). As shownin reference [Dav91], such integrals can be rewritten by reduction in terms ofthe scalar loop functions {J0,γ0,Γ0} defined in appendix B.2.The following results still show the dependence on the dimension d and areobtained assuming we make use of either the infrared (IR) or the modified-infrared (IR) regularization scheme. This means that when the graphs exclu-sively involve nucleon or ∆ propagators, no infrared singularities appear andthe integrals can be set equal to zero. The choice of the regularization proce-dure will be of crucial importance for the analysis of the results, in particularto check the vanishing of the divergences, essential requirement of the calcula-tion at the order we are working. The expression of the scalar loop functionsappearing in the results will therefore depend on the scheme of regularizationadopted.The results are given in terms of the two variables s and Q2, where s = (p +q)2 = 2M0ν+M2

0 −Q2 is the standard Mandelstam variable and Q2 = −q2 ≥ 0is the negative of the photon virtuality. In order to extract the spin structurefunctions S1,2(0, Q

2) we will have to make a change of variable and to operatevia derivative with respect to the energy transfer ν, as shown in eq.(3.55).An expansion around ν = 0 will then provide us with the expressions for the

S(i)

1,2(0, Q2), allowing for numerical estimations for the sum rules and the spin

polarizabilities.Diagram a) is the simplest diagram contributing to the spin structures, it ispurely inelastic and gives the following amplitudes:

A(s,Q2) =1

3(d− 1)2sm∆M0

(e gπN∆

Fπ

)2

(3.67)

×{[−2(s + m2

π)M0 + 2m2∆M0 + (d− 1)m∆(Q2 + s + M2

0 )]∆π

− [2(m4

π + (s−m2∆)2 − 2m2

π(s + m2∆)

)M0

−(d− 1)m∆(s + m2π −m2

∆)(Q2 + s + M20 )

+2(d− 1)sm∆(Q2 + s + 2m∆M0 + M20 )

]J0[s]

},

B(s,Q2) = − 1

3(d− 1)2s2 m∆

(e gπN∆

Fπ

)2

(3.68)

×{[

d(−m2π + m2

∆ + 3s)− 4s]∆π

+[−4sm4

∆ + d(−m2π + m2

∆ + s)2]J0[s]

}.

Diagram h) contributes exclusively to the structure A(s,Q2) and it is char-acterized by two nucleon poles. The subtraction of the elastic intermediatestates is made by subtracting the power series expansion for the function abouts = M2

0 up to the power (s − M20 )−1. Besides the scalar two-point function

J0[s], the result thus presents also the scalar loop function evaluated at s = M20

and the function J ′0[M20 ], where the prime denotes differentiation with respect

81


to s. One has:

A(s,Q2) =(d− 2)

16(d− 1)F 2πm2

∆M0

(e gπN∆

Fπ

)2

(τ3 + 1) (3.69)

×{− 1

d sM20

[4m2

πM20 s + d

(m4

π(M20 + Q2)−M2

0 s(Q2 + s)

+(M20 + Q2)m4

∆ + 2m2π(2M2

0 s− (M20 + Q2)m2

∆))]

∆π

+Q2

M20 (M2

0 − s)2(−m2

π + (M0 + m∆)2)[m4

π(s− 3M20 )

+(M0 −m∆)(M0 + m∆)(3M40 − 2M2

0 m∆ + 2M0s m∆ − sm2∆

+M20 (−5s + 3m2

∆)) + m2π(−2M3

0 m∆ + 2M0s m∆ − 2sm2∆

+M20 (4s + 6m2

∆))]J0[M

20 ]

+1

s(M20 − s2)2

[((s−m2

π)(M40 + M2

0 (Q2 − 2s) + s(Q2 + s))

+4M0Q2sm∆ + (M4

0 + M20 (Q2 − 2s) + s(Q2 + s))m2

∆

) (m4

π

+(s−m2∆)2 − 2m2

π(s + m2∆)

]J0[s]

+2Q2

(s−M20 )

((M0 −m∆)2 −m2

π

)(m2

π − (M0 + m∆)2)2

J ′0[M20 ]

},

B(s,Q2) = 0. (3.70)

Diagram i)+j). Whenever appropriate, the contributions from different dia-grams are added together in order to reproduce the proper structures appearingin eq.(3.53). The calculation of these two diagrams give:

A(s,Q2) =(2− d)

24(d− 1)2m2∆M0

(e gπN∆

Fπ

)2

(τ3 + 1) (3.71)

×{

1

d sM20

[dQ2(m2

π −m2∆)2 + 2dQ2m∆(−m2

π + m2∆)M0

+(d s(Q2 + s)− d(m4

π + m2∆) + m2

π(−4(d− 1)s + 2d m2∆)

)M2

0

−2dm∆(s + m2π −m2

∆)M30

]∆π

+

[Q2((m∆ −M0)

2 −m2π)(m2

π − (m∆ + M0)2)2

M20 (M2

0 − s2)

]J0[M

20 ]

+1

s(M20 − s)

[(m4

π + (s−m2∆)2 − 2m2

π(s + m2∆)

) ((s−m2

π)

(Q2 + s−M20 ) + m∆M0(Q

2 − s−M20 )

)]J0[s]

},

82

3.5. Covariant O(ε3) results for the VVCS amplitudes.

B(s,Q2) =(d− 2)

24(d− 1)2s2 m2∆M3

0

(e gπN∆

Fπ

)2

(τ3 + 1) (3.72)

×{[

dm4π(s + M2

0 ) + dm2π(2sM0 + m∆(s + M2

0 ))− 2m2π(d s m∆M0

+2(d− 1)sM20 + dm4

π(s + M20 ))

]∆π

+d s2

(s−M20 )

[((m∆ −M0)

2 −m2π

) (m∆ + M0)

2 + m2π

)2]J0[M

20 ]

− dM30

(s−M20 )

[(m4

π + (s−m2∆)2 −m2

π(s + m2∆)

)(2sm∆

+M0(s−m2π + m2

∆))]

J0[s]

}.

In Diagram d)+e) the interaction of a photon with the pion loop occurs.This implies the appearance of the three-point functions γi[s] in addition tothe two-point function J0[s] already encountered in the previous diagrams:

A(s,Q2) =2

3(d− 1)2sm∆

(e gπN∆

Fπ

)2

(3.73)

×{− (

3s + m2π −m2

∆

)∆π + s(Q2 + 4m2

π)Jππ0 [Q2]

−[m4

π + (s−m2∆)2 − 2m2

π(s + m2∆)

]J0[s]

+4s[(m∆ + M0)

2 −m2π

]γ3[s]

},

B(s,Q2) =1

3(d− 1)2s2m∆

(e gπN∆

Fπ

)2 {s2(Q2 + 4m2

π)Jππ0 [Q2] (3.74)

−[s(s−m2

π − 3m2∆) + d(m2

π −m2∆)(−s + m2

π −m2∆)

]J0[s]

+[s(3Q2 − 4s) + d(2s2 + Q2m2

π −Q2(2s + m2∆))

]∆π

+2s2[(m∆ + M0)

2 −m2π

](γ1[s]− 2γ4[s]

)}.

We remark that the remaining scalar loop function, namely the three-pointfunction Γ0, is defined according to the diagrams of fig.3.4 where the photoninteracts with the ∆ propagator and thus a two-∆’s, 1-π system is created.

We now turn to the diagrams of fig.3.5, which do not give contribution tothe spin structures of the nucleon.Diagram o)+p) show no dependence on the Mandelstam variable s, so that

83


the addition of the crossed partner make the result vanish:

A(s,Q2) =1

24(d− 1)2m2∆M3

0

(e gπN∆

Fπ

)2

(τ3 + 1) (3.75)

×{

(d− 2)[(

(M0 −m∆)2 −m2π

) ((M0 + m∆)2 −m2

π

)2]J0[M

20 ]

−1

d

[dm4

π + d(m∆ −M0)(m∆ + M0)3 − 2m2

π

(dm2

∆

+dm∆M0 − 2(d− 1)M20

)]∆π

}

B(s,Q2) = 0 (3.76)

The same happens for Diagram q)+r), whose contributions read:

A(s,Q2) =(d− 2)

4(d− 1)m2∆M0(M2

0 − p2)

(e gπN∆

Fπ

)2

(τ3 + 1) (3.77)

×{[(

m2π − (m∆ −M0)

2) (

m2π − (m∆ + M0)

2)2

]J0[M

20 ]

+1

d

[4m2

πM20 + d

(m4

π − (M20 −m2

∆)(M0 + m∆)2

+m2π(−2m2

∆ + 6m∆M0 + 4M20 )

)]∆π

},

B(s,Q2) = 0. (3.78)

Finally, Diagram s)’s contribution to the spin dependent terms of the matrixelement of the process is equal to zero:

A(s,Q2) = 0, (3.79)

B(s,Q2) = 0. (3.80)

The final expressions for the functions A(ν, Q2) and B(ν, Q2) are then ob-tained via addition of all the contributions coming from the different diagrams,the ones reported in this section and the ones given in the appendix. The nextstep will be the analysis of the divergences; given that at O(ε3) no coutert-erms are involved, the vanishing of the divergences represent a very importantcheck for the calculation. Afterwards numerical parameter-free prediction forthe generalized sum rules and the spin polarizabilities will be available, givingindication about the role of the ∆ resonance for these quantities. For a morecomplete and detailed discussion of the topic we refer to a future communica-tion [BDKM].

84

Summary and futureperspectives

The aim of the present work was to provide a deeper understanding about thestructure of the nucleon by electromagnetic probes. When limited to the lowenergy region, the most suitable framework for this study is Chiral Pertur-bation Theory. After an introduction on the formalism, we have applied theacquired knowledge in Baryon Chiral Perturbation Theory to a leading oneloop order calculation of the generalized isovector and isoscalar form factorsof the nucleon Av,s

2,0(t,m2π), Bv,s

2,0(t,m2π) and Cv,s

2,0(t,m2π) in the IR renormaliza-

tion scheme. To this purpose, we had to construct the effective chiral La-grangian necessary to describe the interaction between external tensor fieldsand a strongly interacting system. The calculation has provided us with boththe quark-mass and momentum transfer dependence of the six generalized formfactors. These objects are under investigation also in lattice QCD; thanks torecent simulations, lattice data at values of the pion mass as low as 350 MeVare nowadays available, paving the way to reliable studies of chiral extrapo-lation. Of particular interest was the chiral extrapolation of the lattice datafor the moment of the parton distribution function Av

2,0(0) = 〈x〉u−d. Ourcovariant O(p2) BChPT result for this isovector moment provides a smoothchiral extrapolation function between the high values at large quark massesfrom lattice QCD and the lower value known from phenomenology. A fist at-tempt to use our results to make chiral extrapolations of lattice data of thespin-dependent quantity As

2,0(0) = 〈∆x〉u−d and of the isoscalar form factorBs

2,0(0) has also been done, together with a first estimation of the contributionof u and d quarks to the total spin of the nucleon Ju+d. Despite of the surpris-ingly nice results provided by these analysis, the values given in the chapterhave to be taken with due caution and to be considered as a first rough es-timate. Indeed, the lattice data used as input for the chiral extrapolation ofthe isoscalar generalized form factors correspond to very high values of thepion mass, they are characterized by unknown systematic uncertainties sincedisconnected diagrams are not included in the simulations. As pointed outthroughout chapter 2, in order to judge the stability of our results an analysisof the complete next-to-leading one loop order calculation is definitely calledfor. Despite of this considerations, a first comparison to the values obtained

85

SUMMARY AND FUTURE PERSPECTIVES

by chiral extrapolation studies of lattice data subsequent to our publicationand based on our formulae show that our BChPT calculation is definitely im-proving the quality of chiral extrapolation.In the last section of chapter 2, the possibility to perform simultaneous fits ofseveral observables characterized by a common subset of ChEFT couplings hasbeen explored, with particular emphasis on the advantages which such a pro-cedure can bring about in the studies of chiral extrapolations of lattice QCDdata.The second part of this work was dedicated to the possibility to extract newinformation about the spin structure of the nucleon from the analysis of theprocess of doubly virtual Compton scattering (VVCS) off nucleons. Startingfrom the connection between the spin structure functions in VVCS and theones probed in inelastic electroproduction experiments and making use of theoptical theorem and dispersion relations, we have written a set of sum rulesin terms of the spin-dependent structure functions S1,2(ν, Q

2) appearing inthe spin amplitude of the process. A calculation of these structure functionsin Chiral Perturbation Theory allows to make predictions for all spin-relatedsum rules in the low energy region. Although the several calculations on thetopic present in the literature, the issue still owns big relevance in theoreticalas well as experimental physics research. Indeed, given the key role playedby the ∆ resonance in the spin sector of the nucleon and the very active ex-perimental activity at many laboratories e.g. at Jefferson Lab, a calculationof the derived sum rules with inclusion of the ∆ resonance as explicit degreeof freedom is definitely called for, in order to improve the agreement betweentheoretical predictions and experimental data. For this purpose we have per-formed a leading one loop order calculation of the matrix element of VVCS inthe small scale expansion scheme of covariant Chiral Perturbation Theory andwe have provided the results for all the Feynman diagrams contributing to theprocess at this order of working. The technicalities about the calculation havebeen given, together with first indications regarding how to derive numericalresults for the sum rules and the spin polarizabilities starting from the pro-vided ChPT expressions. The analysis of such results is still on the way. Wepostpone the complete and definitive discussion of the obtained results to alater communication [BDKM].

86

Appendix AAppendix to GPDs

A.1 Feynman Rules for GPDs

Isovector vector channel

In this section we collect the expressions of the vertices we need in order tocalculate the O(p2) amplitudes in the isovector channel corresponding to thefive diagrams of fig.2.1. In particular for each diagram we give the pertinentFeynman rule generated by the O(p0) Lagrangian of eq.(2.18).

We make use of the following notation:lµ pion four-momentumpµ nucleon four-momentumτ i Pauli matricesa, b, c, . . . isospin indices

From L(1)πN of eq.(1.7)

¦ nucleon propagator

i

/p−M0 + iε(A.1)

¦ N in, π out, N out:

1

2

gA

Fπ

/l γ5 τa (A.2)

¦ N in, π in, N out:

− 1

2

gA

Fπ

/l γ5 τa (A.3)

87

A. Appendix to GPDs

From L(2)ππ of eq.(1.2)

¦ pion propagator

iδab

l2 −m2π + iε

(A.4)

From L(0)tπN of eq.(2.18)

¦ diagram a)

N in (momentum p), tensor field (isospin a), π out (momentum l, isospinc ), N out (momentum (p′ − l)):

i∆av

2,0

4Fπ

εcadτ d γ5γ{µ(l − 2p)ν} (A.5)

¦ diagram b)

N in (momentum (p − l)), π in (momentum l, isospin c), tensor field(isospin a), N out (momentum p′):

i∆av

2,0

4Fπ

εcadτ d γ5γ{µ(l − 2p)ν} (A.6)

¦ diagram c):

N in (momentum (p − l)), tensor field (isospin a), N out (momentum(p′ − l)):

iav

2,0

2τa γ{µ(p− l)ν} (A.7)

¦ diagram d):

π in (isospin c), tensor field (isospin a), π out (isospin d):

iav

2,0

4F 2π

(δcaτ d + δadτ c − 2δcdτa) γ{µpν} (A.8)

Isoscalar vector channel

As reported in section 2.5, to O(p2) the only nonzero loop contributions tothe isoscalar channel arise from diagram c) and e) in fig.2.1. The pertinentFeynman rule, generated by the O(p0) Lagrangian of eq.2.18, reads:

88

A.1. Feynman Rules for GPDs

¦ diagram c):

N in (momentum (p − l)), tensor field (isospin a), N out (momentum(p′ − l)):

ias

2,0

2τa γ{µ(p− l)ν} (A.9)

89

A. Appendix to GPDs

A.2 Basic Integrals

The integrals required for one-loop calculations in BChPT can be reduced totwo basis integrals in d-dimensions:

∆π (m) ≡ 1

i

∫ddl

(2π)d

1

m2 − l2 − iε, (A.10)

H11

(M2,m2, p2

) ≡ 1

i

∫ddl

(2π)d

1

(m2 − l2 − iε) (M2 − (l − p)2 − iε), (A.11)

where m (M) is a mass function involving the mass of the Goldstone Boson(of the Baryon) and pµ denotes a four-momentum determined by the kine-matics.The propagators are shifted into the complex energy-plane by a smallamount ε to ensure causality. Utilizing the MS-renormalization scheme ofref.[GSS88] one obtains the dimensionally-regularized results

∆π = 2m2

(L +

1

16π2ln

m

λ

)+O(d− 4), (A.12)

H11(M2,m2, p2) = −2L− 1

16π2

[− 1 + log

M2

λ2+

p2 −M2 + m2

p2log

m

M

+2mM

p2

√1−

(p2 −M2 −m2

2mM

)2

arccos

(m2 + M2 − p2

2mM

)]

+O(d− 4), (A.13)

with

L =λd−4

16π2

[1

d− 4+

1

2(γE − 1− ln 4π)

](A.14)

which is divergent as d → 4. λ is the scale parameter introduced in dimensionalregularization and γE = 0.577215 . . . the Euler-Mascheroni constant.

More complicated integral expressions needed during the calculation aredefined via

1

i

∫ddl

(2π)d

{lµ, lµlν}(m2 − l2 − iε)(M2 − (l − p)2 − iε)

=

{pµH(1)11 , gµνH

(2)11 + pµpνH

(3)11 },

(A.15)

1

i

∫ddl

(2π)d

lµlνlα

(m2 − l2 − iε)(M2 − (l − p)2 − iε)=

(pµgµα + pνgµα + pαgµν) H(4)11 + pµpνpα H

(5)11 .

(A.16)

The integrals H(i)11 are related to the three basis integrals of eqs.(A.10-A.11)

90

A.2. Basic Integrals

via the tensor-identities

H(1)11 (M, p, m) =

1

2p2

[∆π −∆N +

(m2 + p2 −M2

)H11(M, p, m)

],

(A.17)

H(2)11 (M, p, m) =

1

2(d− 1)

[−∆N + 2m2H11(M, p, m)

− (m2 + p2 −M2

)H

(1)11

], (A.18)

H(3)11 (M, p, m) =

1

p2(d− 1)

[(1− d

2

)∆N −m2H11(M, p,m)

+d

2

(m2 + p2 −M2

)H

(1)11 (M, p,m)

], (A.19)

H(4)11 (M, p, m) =

1

2p2

[(m2 + p2 −M2

)H

(2)11 (M, p, m)

+1

d

(m2∆π −M2∆N

)], (A.20)

H(5)11 (M, p, m) =

1

2p2

[(m2 + p2 −M2

)H

(3)11 (M, p, m)

−4H(4)11 (M, p,m)−∆N

](A.21)

Finally, we note that the integrals involving more than one baryon propagatorcan be related to the ones defined above via

H(i)′11 (M2,m2, p2) ≡ ∂

∂M2H

(i)11 (M2,m2, p2). (A.22)

91

A. Appendix to GPDs

A.3 Regulator Functions

The basis regulator function is defined as

R11

(M2,m2, p2

) ≡∫ ∞

x=1

dx

∫ddl

(2π)d

[xM2 + (x2 − x)p2 + (1− x)m2 − l2

]−2.

(A.23)

More complicated regulator functions R(i)11 , i = 1 . . . 5 can be defined in analogy

to Eqs.(A.15, A.16). We note that for our O(p2) calculation of the momentsof the GPDs we only need to know these functions up to the power1 of m2

π, tin order to obtain a properly renormalized, scale-independent result, which atthe same time is also consistent with the requirements of power-counting. Theregulator functions needed for our O(p2) BChPT calculation (see AppendixA.4) read

R11(M20 ,m2

π, p2) =

(2− m2

π

M20

)L +

1

16π2

[2 log

M0

λ

−1− 1

2

m2π

M20

(2 log

M0

λ+ 3

)]+ ..., (A.24)

R(1)11 (M2

0 ,m2π, p2) =

(1 +

m2π

M20

)L +

1

16π2

[log

M0

λ+

1

2

m2π

M20

(2 log

M0

λ− 1

) ]

+..., (A.25)

R(2)11 (M2

0 ,m2π, p2) =

(1

3M2

0 +1

2m2

π

)L +

1

48π2

[M2

0

3

(3 log

M0

λ− 1

)

+3

2m2

π

(log

M0

λ− 1

) ]+ ..., (A.26)

R(3)11 (M2

0 ,m2π, p2) =

2

3L +

1

48π2

[2 log

M0

λ+

1

3+

3

2

m2π

M20

]+ ..., (A.27)

1Strictly speaking we need to know the regulator terms contributing to the generalizedform factor As,v

2,0(t) up to power (m2π, t)1, whereas for Bs,v

2,0(t), Cs,v2,0 (t) only the leading terms

(m2π, t)0 are required [Gai07].

92

A.3. Regulator Functions

The derivatives of the regulator functions needed for the calculation (see Ap-pendix A.4) read

R(2)11

′(M2,m2

π, p2) =1

2

(1 +

m2π

M20

)L +

1

32π2

[log

M0

λ+

m2π

2M20

(2 log

M0

λ− 1

) ]

− t

384π2M20

+ ..., (A.28)

R(3)11

′(M2,m2

π, p2) = −m2π

M40

L +1

16π2M20

[1

2+

m2π

M20

(− log

M0

λ+ 1

) ]

+t

192π2M40

+ ..., (A.29)

R(4)11

′(M2,m2

π, p2) =1

3L +

1

16π2

[1

3log

M0

λ+

1

18+

m2π

4M20

]

− t

576π2M20

+ ..., (A.30)

R(5)11

′(M2,m2

π, p2) =1

16π2M20

[1

3− m2

π

2M20

]+

t

288π2M40

+ ... (A.31)

One can clearly observe that all contributions are polynomial in m2π (and there-

fore polynomial in the quark-mass [BL99]) or polynomial in t [Gai07], as ex-pected. Their addition to the MS-results therefore just amounts to a shiftin the coupling constants [BL99, Gai07] of the effective field theory and doesnot affect the non-analytic quark-mass dependencies, which are the scheme-independent signatures of chiral dynamics.

93

A. Appendix to GPDs

A.4 Isovector vector Amplitudes in O(p2) BChPT

The five O(p2) amplitudes in the isovector channel corresponding to the fivediagrams of Fig.2.1 written in terms of the basic integrals

I(i)11

(M2,m2, p2

)= H

(i)11

(M2,m2, p2

)+ R

(i)11

(M2,m2, p2

), i = 0 . . . 5,

(A.32)of Appendices A.2 and A.3 read

Ampa+b =− i∆av

2,0 gA

F 2π

η†τa

2η u(p′) γ{µpν} u(p)

[2m2

πI11(M20 ,m2

π, p2)

(A.33)

−m2πI

(1)11 (M2

0 ,m2π, p2) + 2I

(2)11 (M2

0 ,m2π, p2)

],

Ampc = iav

2,0g2A

4F 2π

η†τa

2η u(p′)

∫ 12

− 12

du (A.34)

×{

γ{µpν}

[−∆π + 4M2

0

(I

(1)11 (M2

0 ,m2π, p2)− I

(3)11 (M2

0 ,m2π, p2)

+ (2M20 − M2)

(I

(3)′11 (M2,m2

π, p2)− I(5)′11 (M2,m2

π, p2))

+ (d− 2)(I

(4)′11 (M2,m2

π, p2)− I(2)′11 (M2,m2

π, p2)) )]

+ i ∆ασα{µpν} 4M30

(I

(3)′11 (M2,m2

π, p2)− I(5)′11 (M2,m2

π, p2)

)

−∆{µ∆ν}

(8M3

0 u2 I(5)′11 (M2,m2

π, p2)

)}u(p),

Ampd =− iav

2,0

F 2π

η†τa

2η u(p′) γ{µpν} u(p) ∆π, (A.35)

Ampe =i av2,0 η†

τa

2η u(p′) γ{µpν} u(p) ZN . (A.36)

Note that the various couplings and parameters are defined in section 2.3. ηdenotes the isospin doublet of proton and neutron. The variables in the integralfunctions are given as

p2 = M2 ≡ M20 +

(u2 − 1

4

)t, (A.37)

where t = ∆2 corresponds to the momentum transfer by the tensor fields.ZN denotes the Z-factor of the nucleon, calculated to the required O(p3)

accuracy in BChPT. It is obtained from the self-energy ΣN at this order viathe prescription

ZN = 1 +∂ΣN

∂/p

∣∣∣∣/p=M0

+O(p4), (A.38)

94

A.4. Isovector vector Amplitudes in O(p2) BChPT

with

ΣN =3g2

A

4F 2π

(M0+/p)[m2

πI11(M20 ,m2

π, p2)+(M0−/p)/pI(1)11 (M2

0 , m2π, p2)−∆N

]+O(p4).

(A.39)

95

A. Appendix to GPDs

A.5 BChPT Results in the Isoscalar vector

Channel

A.5.1 Isoscalar vector Amplitudes in O(p2) BChPT

To O(p2) in BChPT the results in the isoscalar channel are quite simple. Theamplitudes corresponding to the Feynman diagrams of Fig.2.1 can be simplyexpressed in terms of results already obtained in the isovector channel discussedin the previous section A.4. They read

Ampa+b = 0 +O(p3), (A.40)

Ampc = −3as

2,0η†1η

av2,0η

†τaηAmpc +O(p3), (A.41)

Ampd = 0 +O(p3), (A.42)

Ampe =as

2,0η†1η

av2,0η

†τaηAmpe +O(p3). (A.43)

Note that the various couplings and parameters are defined in section 2.3.

A.5.2 Estimate of O(p3) contributions

The contributions from the (higher order) Feynman diagram shown in figure2.2 to the generalized isoscalar form factors read

∆Ash.o.(t,mπ) =

g2Ax0

π

32π2Fπ2

∫ 12

− 12

du

p8

{2p8

(3m2

π + M20

)− tp8

+4M20

(2M2

0 + m2)p6 − 2M2

0

(11M4

0 − 7m2M20 + 2m4

)p4

+12M40

(M2

0 − m2)2

p2 + 2M20

[− p8 + 3p4

(3M4

0 − m4)

−2p2(M2

0 − m2) (

7M40 − 2m2M2

0 + m4)

+ 6M20

(M2

0 − m2)3

]

× logm

M0

+2M2

0√2M2

0 (p2 + m2)− (p2 − m2)2 −M40

×[6M10

0 − 4M80

(6m2 + 5p2

)+ M6

0

(36m4 + 38m2p2 + 23p4

)

−M40

(24m6 + 18p2m4 + 11p4m2 + 9p6

)

+M20

(6m8 + 2p2m6 − 5p4m4 − 2p6m2 − p8

)

+p2(p2 − m2

)3 (p2 + 2m2

) ]arccos

(M2

0 + m2 − p2

2M0m

)},

(A.44)

96

A.5. BChPT Results in the Isoscalar vector Channel

∆Bsh.o.(t, mπ) =

g2Ax0

π

96π2F 2π

MN(mπ)

M0

∫ 12

− 12

du

{− 6M2

0 + 18m2π

(2 log

M0

λ− 3

)

+t

(11− 6 log

M0

λ

)− 36M4

0

p8

√2 (m2 + p2) M2

0 − (p2 − m2)2 −M40

×[M8

0 −(4m2 + 3p2

)M6

0 +(6m4 + 5p2m2 + 3p4

)M4

0

− (4m6 + p2m4 + p4m2 + p6

)M2

0 + m8 − m6p2

]

× arccos

(M2

0 + m2 − p2

2M0m

)

−6M40

p8

[2p6 − 3

(3M2

0 − m2)p4 + 6

(M2

0 − m2)2

p2

+6(M2

0 p4 − 2M20

(M2

0 − m2)p2 +

(M2

0 − m2)3

)log

m

M0

]}

+Br33(λ)

MN(mπ)

M0

4m2π

Λ2χ

+ Br34(λ)

MN(mπ)

M0

t

Λ2χ

, (A.45)

∆Csh.o.(t,mπ) =

g2Ax0

πM20

96π2F 2π

MN(mπ)

M0

∫ 12

− 12

du

p8

{5p8 − 9M2

0 p6 − 6u2p2

[3p6 − 2M2

0 p4

−3M20

(M2

0 − 3m2)p2 − 6

(M3

0 −M0m2)2

]

−9(4u2 − 1

)p8 log

m

M0

+ 9M20

[4u2

(−m2p4 + 2m2p2(m2 −M2

0

)

+(M2

0 − m2)3

)− p4

(M2

0 − m2) ]

logm

M0

+9M2

0√2M2

0 (m2 + p2)− (p2 − m2)2 −M40

[4u2M8

0

−4(4m2 + p2

)u2M6

0 −(−24u2m4 + 4u2p2m2 + p4

)M4

0

+(−16u2m6 + 20u2p2m4 + 2

(1− 2u2

)m2p4 + p6

)M2

0

+m2(m2 − p2

) (4u2m4 − 8u2m2p2 +

(4u2 − 1

)p4

) ]

× arccos

(M2

0 + m2 − p2

2M0m

)},

(A.46)

with the new variable

m2 = m2π +

(u2 − 1

4

)t. (A.47)

97

A. Appendix to GPDs

We note that the contributions of ∆As2,0(t,mπ), ∆Cs

2,0(t,mπ) are finite atO(p3)in BChPT, while ∆Bs

2,0(t,mπ) contains two new counterterms Br33(λ), Br

34(λ)at this order [DHH].

98

A.6. Moments of axial GPDs in O(p2) BChPT

A.6 Moments of axial GPDs in O(p2) BChPT

A.6.1 Axial Amplitudes in O(p2) BChPT

To O(p2) the five amplitudes in the axial sector corresponding to the fivediagrams of fig. 2.1 read

Ampa+b = iav

2,0 gA

F 2π

η†τa

2η u(p′) γ{µγ5 pν} u(p)

[∆N − 2m2

πI11(M20 ,m2

π, p2)(A.48)

+m2πI

(1)11 (M2

0 ,m2π, p2)− 2I

(2)11 (M2

0 ,m2π, p2)

]

Ampc = i∆av

2,0g2A

4F 2π

η†τa

2η u(p′)

∫ 12

− 12

du (A.49)

×{

γ{µγ5 pν}

[−∆π + 4M2

0

(I

(1)11 (M2

0 ,m2π, p2)− I

(3)11 (M2

0 ,m2π, p2)

+(d− 2)(I

(2)′11 (M2,m2

π, p2)− I(4)′11 (M2, m2

π, p2))

+M2

(I

(3)′11 (M2,m2

π, p2)− I(5)′11 (M2,m2

π, p2

)]

+ γ5 p{µqν}

[16M3

0 u2

(2I

(5)′11 (M2,m2

π, p2)− I(3)′11 (M2,m2

π, p2)

)]}u(p)

Ampd = − i∆av

2,0

F 2π

η†τa

2η u(p′) γ{µγ5 pν} u(p) ∆π (A.50)

Ampe = i∆av2,0 η†

τa

2η u(p′) γ{µγ5 pν} u(p) ZN (A.51)

(A.52)

where we use the notation introduced in section A.4.

99

A. Appendix to GPDs

b3)a3)

c3) d3)

e3)

(3) (3)(4) (4)

g3)f3)

Figure A.1: The loop diagrams contributing to first moments of GPDs of anucleon at next-to-leading one loop order in BChPT. The solid- and dashedlines represent nucleon and pion propagators, respectively. The solid dot andthe solid square denote a coupling to a tensor field from the O(p0) Lagrangianof eq.(2.18) and from the O(p1) Lagrangian of eq.(2.19), respectively.

A.7 O(p3) calculation of the generalized form

factors: preliminary results

In this section we provide preliminary results of the first moments of GPDsat next-to-leading one loop order in BChPT. For an accurate analysis of theresults we refer to a future communication [DHH].Figure A.1 shows the diagrams contributing at the chiral order D = 3. Thenew couplings come from the O(p1) Lagrangian of eq.(2.19) and are depictedas solid squares in order to distinguish them from the ones involved in theleading order calculation. We specify that the diagrams f 3) and g3) of figureA.1 contribute only to the isoscalar generalized form factors, in particular anon-zero contribution comes only from diagram f 3).We observe that for a complete O(p3) evaluation one has to consider also thecontributions from the O(p2) Feynman diagrams of figure 2.1 after appropriatep2 nucleon-mass insertions.

A.7.1 Isovector vector channel

To O(p3) the isovector vector amplitudes corresponding to the five diagramsof figure A.1 read:

100

A.7. O(p3) calculation of the generalized form factors: preliminary results

Ampa3+b3 = i4 hv

A gA

F 2π

η†τa

2η u(p′) γ{µpν} u(p)

[2I

(2)11 (M2

0 ,m2π, p2) (A.53)

−m2πI

(1)11 (M2

0 ,m2π, p2) + ∆N

],

Ampc3 = ibv2,0g

2A

8F 2π

η†τa

2η u(p′)

∫ 12

− 12

du (A.54)

×{

γ{µpν}

[4M3

0 q2(I

(3)′11 (M2,m2

π, p2)− I(5)′11 (M2, m2

π, p2)) ]

+ i ∆ασα{µpν}

[4M2

0

(I

(1)11 (M2

0 ,m2π, p2)− I

(3)11 (M2

0 ,m2π, p2)

)

+ m2π

(I ′11(M

2,m2π, p2) + I

(1)′11 (M2,m2

π, p2))

− 4I(2)′11 (M2,m2

π, p2) + 6I(4)′11 (M2,m2

π, p2)

− 2(M2 −M2

0

)(I

(3)′11 (M2,m2

π, p2)− I(5)′11 (M2,m2

π, p2))−∆π

]

+ ∆{µ∆ν}

(4M2

0 I(4)′11 (M2,m2

π, p2)

)}u(p),

Ampd3 =− bv2,0

2M0F 2π

η†τa

2η u(p′) ∆ασα{µpν} u(p) ∆π, (A.55)

Ampe3 = i η†τa

2η u(p′)

[−i

bv2,0

2M0

∆ασα{µpν} ZN (3) (A.56)

+ av2,0 γ{µpν} ZN (4)

]u(p).

We note the appearance of a new coupling called hvA in diagrams a3+b3. This

coupling belongs to the first order Lagrangian L(1)tπN of eq.(2.19). Let us con-

centrate on the third term of that Lagrangian, namely ψN [gA

2γµγ5uµ]ψN =

ψN [gA

2γµγ5g

µνuν ]ψN ; in analogy to what has been done in section 2.3 (see

eq.(2.17)), we introduce the structure gµν = gµν +hv

A

2V +

µν +h0AV 0

µν and we auto-matically get the new couplings hv

A,h0A appearing in the vertices of the diagrams

a3,b3 of the isovector (hvA) and isoscalar (h0

A) vector channels.Again, ZN denotes the Z-factor of the nucleon. Being a O(p3) calculation,power counting indicates two different contributions to diagram e3: the p1

coupling b2,0 associated to the third order ZN (3) of eq.A.39, and the p0 cou-pling a2,0 associated to the fourth order ZN (4). The latter is obtained via:

ZN (4) = 1 +∂Σ(3)

∂/p

∣∣∣∣/p=M0−4c1m2

π

+∂Σ(4)

∂/p

∣∣∣∣/p=M0

. (A.57)

The fourth order self energy Σ(4) involves the two graphs of figure A.2, which

101

A. Appendix to GPDs

Figure A.2: The loop diagrams contributing to the O(p4) nucleon self energy.

The cross denotes a vertex from L(2)πN .

include vertices generated by the Lagrangian L(2)πN [FMMS00]. One has:

Σ(4) = Σ(4)tadpole + Σ(4)

c1, (A.58)

with

Σ(4)tadpole =

3m2π

F 2π

∆π

(2c1 − p2

4M20

c2 − c3

), (A.59)

Σ(4)c1

= − 3g2A

4F 2π

(4m2πc1)

{m2

πI11(M20 ,m2

π, p2) + 2M0/pI(1)11 (M2

0 ,m2π, p2)(A.60)

+∆N + 2M0

[(/p + M0)

∂

∂M20

(m2

πI11(M20 ,m2

π, p2)−∆N

)

+(M20 − p2)/p

∂

∂M20

I(1)11 (M2

0 ,m2π, p2)

]}.

A.7.2 Isoscalar vector channel

As far as the isoscalar vector channel is concerned, the main contribution to thegeneralized form factors, namely the one due to the Feynman diagram shown infigure 2.2 and called f3) in figure A.1, has already been given in section A.5.2.Analogously to the leading order, the amplitudes of the remaining diagramsdepicted in figure A.1 can be simply expressed in terms of the results obtainedin the isovector channel discussed in the previous subsection. They read:

Ampa3+b3 =3

2

h0A η†1η

hvA η†τaη

Ampa3+b3, (A.61)

Ampc3 = −3bs2,0 η†1η

bv2,0 η†τaη

Ampc3, (A.62)

Ampd3 = 0, (A.63)

Ampe3 = i η†τa

2η u(p′)

[− i

bs2,0

2M0

∆ασα{µpν} ZN (3) (A.64)

+ as2,0 γ{µpν} ZN (4)

]u(p).

102

Appendix BAppendix to VVCS

B.1 Feynman rules for VVCS

In this section we collect the expressions of the vertices used for the calculationof the O(ε3) amplitudes corresponding to the diagrams of figs. 3.3, 3.5 and 3.4.

We make use of the following notation:lµ: pion four-momentumε: photon polarization vectorα, β Dirac indexa, b, i, j isospin indices

From L(1)πN of eq.(1.7)

¦ photon:

ie

2(1 + τ 3)/ε (B.1)

From L(2)ππ of eq.(1.2)

¦ π in (momentum l,isospin i), photon, π out (momentum l′, isospin j):

− e ε3ijε · (l + l′) (B.2)

From L(1)πN∆ of eq.(3.57)

¦ N in, ∆ out (isospin i), π out (isospin a):

gπN∆

Fπ

lµδia (B.3)

103

B. Appendix to VVCS

¦ N out, ∆ in (isospin i), π in (isospin a):

− gπN∆

Fπ

lµδia (B.4)

¦ N in, photon, ∆ out (isospin i), π out (isospin a):

igπN∆ e

Fπ

εαεa3i (B.5)

¦ photon, ∆ in (isospin i), π in (isospin a), N out:

igπN∆ e

Fπ

εαεa3i (B.6)

From L(1)π∆ of eq.(3.60)

¦ ∆ propagator

Gijµν(p) = −i

/p + m∆

p2 −m2∆

{gµν − 1

d− 1γµγν − (d− 2)

(d− 1)

pµpν

m2∆


(d− 1)m∆

}ξij

(B.7)

¦ ∆ in (Dirac index β, isospin j), photon, ∆ out (Dirac index α, isospini):

− ie(1

2(1 + τ 3)δij − iεij3

)(/εgαβ +

d− 2

4γα/εγβ

)(B.8)

From L(2)γN∆ of eq.(3.62)

¦ photon, ∆ (isospin i), N

ieb1

2M0

(/qεµ − qµ/ε)γ5δ3i (B.9)

104

B.2. Loop integrals

B.2 Loop integrals

In this section we define the pertinent loop functions appearing in the ampli-tudes given in section 3.5 and B.3.We make use of the general expression of a loop diagram with generic dimen-sion d and arbitrary number of propagators:

II[d + n][{{p1,m1}, k1}, . . . {{pN ,mN}, kN}] = (B.10)∫dd+nl

(2π)d+n

1

[(l + p1)2 −m21 + iε]k1 . . . [(l + pN)2 −m2

N + iε]kN

As shown in reference [Dav91], each integral of the kind of eq.(B.10) can bewritten in terms of scalar loop functions. Following references [BKM92b][BHM03b],we define the following two- and three-point scalar loop functions:

∆π =1

i

∫ddl

(2π)d

1

[m2π − l2 − iε]

, (B.11)

J0[s] =1

i

∫ddl

(2π)d

1

[m2π − l2 − iε][m2

∆ − (p + q − l)2 − iε],

Jππ0 [Q2] =

1

i

∫ddl

(2π)d

1

[m2π − l2 − iε][m2

π − (l + q)2 − iε],

γ0[s] =1

i

∫ddl

(2π)d

1

[m2π − l2 − iε][m2

π − (l + q)2 − iε][m2∆ − (p− l)2 − iε]

,

Γ0[s] =1

i

∫ddl

(2π)d

1

[m2π − l2 − iε][m2

∆ − (p + q − l)2 − iε][m2∆ − (p− l)2 − iε]

,

with s = (p + q)2, p2 = M20 and Q2 = −q2.

According to diagrams e)+d) of fig.3.4 we define further loop functions,which by means of Passarino-Feldmann reduction can also be written in termsof the only scalar loop functions defined above:

1

i

∫ddl

(2π)d

lµ[m2

π − l2 − iε][m2π − (l + q)2 − iε][m2

∆ − (p− l)2 − iε]=

{qµγ1(s) + pµγ2(s)},1

i

∫ddl

(2π)d

lµlν[m2

π − l2 − iε][m2π − (l + q)2 − iε][m2

∆ − (p− l)2 − iε]=

gµνγ3(s) + qµqνγ4(s) + (pµqν + pνqµ)γ5(s) + pµpνγ6(s).

(B.12)

In what follows we provide the basic correspondence between the general basisdefined by eq.(B.10) and the scalar basis of eq.(B.11):

− i ∆π = II[d][{{0,mπ}, 1}] , (B.13)

i J0[s] = II[d][{{0,mπ}, 1}, {{q + p,m∆}, 1}] ,i Jππ

0 [Q2] = II[d][{{0,mπ}, 1}, {{q, mπ}, 1}] ,−i Γ0[s] = II[d][{{0,mπ}, 1}, {{q + p,m∆}, 1}, {{p,m∆}, 1}] ,−i γ0[s] = II[d][{{0,mπ}, 1}, {{q, mπ}, 1}, {{q + p, m∆}, 1}] .

105

B. Appendix to VVCS

B.3 The VVCS amplitudes in covariant SSE

at O(ε3)

In this section we provide the O(ε3) amplitudes corresponding to the diagramsb)+c), f), g), k)+l) and m)+n) depicted in fig.3.4, written in terms ofthe basic integral of eq.(B.10).The various parameters and couplings are defined in chapter 1 and in section3.4.Note that in what follows the kinematical condition s = M2

0 in the loop func-tion is expressed via the introduction of a new momentum pr 6= p such thatthe equation q2 + 2M0ν = 0 is fulfilled.

106

B.3. The VVCS amplitudes in covariant SSE at O(ε3)

Dia

gra

mb)+

c)

A(ν

,Q2)

=1

96(−

1+

d)3

F2M

0

( ie2gπN

∆2(5

+τ[3

])(

1ds2m

4 ∆M

2 0

((16d4s2

m2 πm

∆M

3 0+

16sm

2 πM

2 0

( −( Q

2+

s)(Q

2+

3s) +

2( Q

2+

9s) m

∆M

0+

2sM

2 0+

2m

∆M

3 0+

M4 0

) +4d

( s2M

2 0

( −Q

2−

s+

M2 0

)(−Q

2+

s+

3M

2 0

) −2sm

∆M

3 0

( 2( Q

4+

5Q

2s

+s2

) +( 3

Q2−

7s) M

2 0+

M4 0

) +2m

5 ∆M

0

( −4s2

+( Q

2+

9s) M

2 0+

M4 0

) −sm

2 ∆M

2 0

((Q

2−

8s)(

Q2

+s) +

( Q2

+4s) M

2 0+

4M

4 0

) −2sm

3 ∆M

0

( −2s

( Q2

+s) −

( Q2−

15s) M

2 0+

5M

4 0

) +m

4 ∆

( 2s2

( Q2

+s) −

( Q4

+12Q

2s

+21s2

) M2 0

+6sM

4 0+

M6 0

) +

m4 π

( 2s2

( Q2

+s) −

8s2

m∆

M0

+( −

Q4

+2Q

2s

+s2

) M2 0

+2

( Q2−

s) m∆

M3 0−

4sM

4 0+

2m

∆M

5 0+

M6 0

) +2m

2 π

( −sM

2 0

( −Q

2−

s+

M2 0

)(4Q

2+

13s

+3M

2 0

) −m

2 ∆

( −Q

2−

s+

M2 0

)(−2

s2+

( Q2

+3s) M

2 0+

M4 0

) −2m

3 ∆M

0

( −4s2

+( Q

2+

4s) M

2 0+

M4 0

) −2sm

∆M

0

( s( Q

2+

s) +( 6

Q2

+40s) M

2 0+

3M

4 0

))) +

d3

( s2M

2 0

( −Q

2−

s+

M2 0

)(−Q

2+

s+

3M

2 0

) +m

4 π

( −Q

2−

s+

M2 0

)(−2

s2+

( Q2−

3s) M

2 0+

M4 0

) −2sm

∆M

3 0

( 3Q

4+

8Q

2s

+s2

+( 4

Q2−

2s) M

2 0+

M4 0

) −2sm

2 ∆M

2 0

((Q

2+

s)(2Q

2+

s) +( 3

Q2

+4s) M

2 0+

3M

4 0

) −2m

3 ∆M

0

( −2s2

( Q2

+s) +

( Q4

+s2

) M2 0

+2Q

2M

4 0+

M6 0

) +m

4 ∆

( 2s2

( Q2

+s) +

( −Q

4+

2Q

2s

+9s2

) M2 0

+4sM

4 0+

M6 0

)

+2m

2 π

( −sM

2 0

( −Q

2−

s+

M2 0

)(2Q

2+

7s

+M

2 0

) +m

∆M

0

( −2s2

( Q2

+s) +

( Q4−

4Q

2s−

75s2

) M2 0

+2

( Q2−

2s) M

4 0+

M6 0

) −m

2 ∆

( 2s2

( Q2

+s) −

( Q4−

7s2

) M2 0−

2sM

4 0+

M6 0

))−

4d2

( s2M

2 0

( −Q

2−

s+

M2 0

)(−Q

2+

s+

3M

2 0

) +m

5 ∆M

0

( −4s2

+( Q

2+

3s) M

2 0+

M4 0

) −sm

∆M

3 0

( 5Q

4+

18Q

2s

+3s2

+( 7

Q2−

9s) M

2 0+

2M

4 0

) −sm

2 ∆M

2 0

( 2( Q

2−

s)(Q

2+

s) +3

( Q2

+3s) M

2 0+

5M

4 0

) +m

4 ∆

( 2s2

( Q2

+s) −

( Q2

+s)(

Q2

+5s) M

2 0+

2sM

4 0+

M6 0

) −m

3 ∆M

0

( −4s2

( Q2

+s) +

( Q4

+2Q

2s

+15s2

) M2 0

+2

( Q2

+4s) M

4 0+

M6 0

) +m

4 π

( 2s2

( Q2

+s) −

4s2

m∆

M0

+( −

Q4

+2Q

2s

+s2

) M2 0+

( Q2−

s) m∆

M3 0−

4sM

4 0+

m∆

M5 0

+M

6 0

) +m

2 π

( −sM

2 0

( −Q

2−

s+

M2 0

)(5Q

2+

17s

+3M

2 0

) −2m

3 ∆M

0

( −4s2

+( Q

2+

s) M2 0

+M

4 0

) +

m∆

M0

( −4s2

( Q2

+s) +

( Q4−

14Q

2s−

121s2

) M2 0

+2

( Q2−

4s) M

4 0+

M6 0

) −m

2 ∆

( 4s2

( Q2

+s) +

( −2Q

4−

3Q

2s

+7s2

) M2 0−

sM4 0

+2M

6 0

))))

II[d

][{{

0,m

π},

1}]

−1

m4 ∆

( 2

((−

2+

d)m

∆(−

mπ−

m∆

+M

0)(

mπ−

m∆

+M

0)(−

mπ+

m∆

+M

0)(

mπ+

m∆

+M

0) (

m2 π−

m2 ∆

+M

2 0)

M0

+

4(−

1+

d)m

∆

( −(−

9+

2d)m

4 ∆M

0+

(−2

+d)M

0

( −2Q

2−

s+

M2 0

)(−m

2 π+

M2 0

) +(−

1+

d)m

3 ∆

( −Q

2−

s+

3M

2 0

) +

m2 ∆

M0

( (−2

+d)s

+(−

9+

2d)m

2 π+

(13−

3d)M

2 0

) +m

∆(m

π−

M0)(m

π+

M0)( (−

1+

d)( Q

2+

s) +3(−

3+

d)M

2 0

))−

1M

2 0

( (mπ−

m∆−

M0)(m

π+

m∆−

M0)(m

π−

m∆

+M

0)(m

π+

m∆

+M

0)( 7

(−2

+d)m

3 ∆M

0+

(−2

+d)2

(mπ−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) +(−

2+

d)

m∆

M0

( −2

( Q2

+s) −

7m

2 π+

11M

2 0

) −m

2 ∆

( −(−

2+

d)2

( Q2

+s) +

(24

+(−

8+

d)d

)M2 0

))) −

1M

0

( 2(−

1+

d)m

∆

( m2 π−

m2 ∆

+M

2 0

)( −

2(−

1+

d)m

3 ∆M

0+

2(−

5+

2d)m

∆(m

π−

M0)M

0(m

π+

M0)+

(−2

+d)(−

mπ

+M

0)(m

π+

M0)( −

Q2−

s+

M2 0

) +m

2 ∆

( 2( Q

2+

s) +d

( −Q

2−

s+

M2 0

))))

)

II[d

][{{

0,m

π},

1},{{

p,m

∆},

1}])−

1s2m

4 ∆

((2(−

2+

d)s

2m

∆M

0

( d( Q

2−

s) 2+

12Q

2s−

2( (−

3+

2d)Q

2+

s) M2 0−

(−2

+d)M

4 0

) −m

6 ∆

( −( Q

2+

s)((−

2+

d)2

Q2

+(4

4−

d(2

0+

3d))

s) +4(6

+(−

2+

d)d

)sM

2 0+

(−2

+d)2

M4 0

) +

s2m

2 ∆

((Q

2+

s)(44s

+d

( (−4

+3d)Q

2+

7(−

4+

d)s

))+

4( (1

1+

5(−

3+

d)d

)Q2

+(−

6+

d)s

) M2 0−

(−2

+d)(−1

0+

7d)M

4 0

) +

sm4 ∆

((Q

2+

s)(−1

00s

+d

( (−4

+3d)Q

2+

(52

+5d)s

))+

2( (2

+3(−

2+

d)d

)Q2

+80s

+6(−

11

+d)d

s) M2 0

+(−

2+

d)(−1

0+

7d)M

4 0

) +

(−2

+d)2

s3( −s

2+

( −Q

2+

M2 0

) 2)

+4m

7 ∆M

0

( 3(−

6+

d)s

+(−

2+

d)( Q

2+

M2 0

))+

(−2

+d)m

6 π

( (−2

+d)( Q

2−

3s

+M

2 0

)(−Q

2−

s+

M2 0

) −4m

∆M

0

( Q2−

s+

M2 0

))+

2m

5 ∆M

0

( (128

+d(−

76

+7d))

s2+

(−2

+d)d

( Q2

+M

2 0

) 2−

6s

( dQ

2+

(−4

+3d)M

2 0

))+

4sm

3 ∆M

0

( 2(−

3+

2(−

3+

d)d

)s2

+(−

2+

d)( (−

2+

d)Q

2−

M2 0

)(Q

2+

M2 0

) +3s

( (14

+d(−

14

+3d))

Q2

+(−

4+

3d)M

2 0

))+

m4 π

( −(−

2+

d)2

s( 3

Q2−

7s

+M

2 0

)(−Q

2−

s+

M2 0

) +4m

3 ∆M

0

( (−14

+d)s

+3(−

2+

d)( Q

2+

M2 0

))−

m2 ∆

( −Q

2−

s+

M2 0

)((1

2+

(8−

5d)d

)s+

3(−

2+

d)2

( Q2

+M

2 0

))+

2(−

2+

d)m

∆M

0

( d( Q

2−

3s

+M

2 0

)(Q

2−

s+

M2 0

) +6s

( 3Q

2−

s+

M2 0

))) +

107

B. Appendix to VVCS

m2 π

( −(−

2+

d)2

s2( −

Q2−

s+

M2 0

)(−3

Q2

+5s

+M

2 0

) −4(−

2+

d)s

m∆

M0

( dQ

4−

(−16

+5d)Q

2s

+2(−

1+

d)s

2+

( Q2

+dQ

2+

s−

2ds) M

2 0+

M4 0

) +

m4 ∆

( −(−

2+

d)( Q

2+

s)(3(−

2+

d)Q

2−

(34

+5d)s

) −4(−

10

+d)s

M2 0

+3(−

2+

d)2

M4 0

) −2sm

2 ∆

((Q

2+

s)((6

+d(−

8+

3d))

Q2

+(5

2+

d(−

38

+9d))

s) +2

( (−1

+(−

1+

d)d

)Q2−

6(5

+(−

4+

d)d

)s) M

2 0+

(−2

+d)(−4

+3d)M

4 0

) −4m

5 ∆M

0

( (−34

+5d)s

+3(−

2+

d)( Q

2+

M2 0

))−

4m

3 ∆M

0

( (−42

+d(4

+5d))

s2+

(−2

+d)d

( Q2

+M

2 0

) 2−

2s

( (9+

(−5

+d)d

)Q2

+( −

3+

d+

d2) M

2 0

))))

II[d

][{{

0,m

π},

1},{{

p+

q,m

∆},

1}]) −

1m

3 ∆

( 8(−

1+

d)( (−

2+

d)Q

4(m

π−

M0)M

0(m

π+

M0)−

2(−

5+

d)m

4 ∆M

0

( −Q

2−

2s

+2M

2 0

) +Q

2m

2 ∆M

0

( (−2

+d)( Q

2+

2s) +

2(−

5+

d)m

2 π+

(14−

4d)M

2 0

) +

m5 ∆

( −(−

4+

d)( Q

2+

s) +(−

8+

d)M

2 0

) +Q

2m

∆(m

π−

M0)(m

π+

M0)( (−

1+

d)( Q

2+

s) +(−

5+

3d)M

2 0

) +

m3 ∆

((Q

2+

s)((−

1+

d)( Q

2+

2s) −

(−4

+d)m

2 π

) +( (−

7+

2d)Q

2−

(8+

d)s

+(−

8+

d)m

2 π

) M2 0−

(−14

+d)M

4 0

))II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}]

+1

m2 ∆

(32(−

1+

d)π

( (Q

2−

(−6

+d)m

2 ∆

)(( Q

2+

s) 2+

2( Q

2−

s) M2 0

+M

4 0

)−

( Q2

+s

+M

2 0

)(2m

4 ∆+

(−3

+2d)Q

2(m

π−

M0)(m

π+

M0)−

m2 ∆

( 5Q

2+

4s−

ds

+2m

2 π+

(−6

+d)M

2 0

)))

II[2

+d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])

+1

m4 ∆

( 16(−

1+

d)π

( −8(−

1+

d)m

5 ∆M

0−

2(−

2+

d)Q

2m

∆M

0

( Q2

+s

+2m

2 π−

3M

2 0

) +(−

2+

d)2

Q2(m

π−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) −8m

4 ∆

( (−3

+d)( Q

2+

s) +2M

2 0

) +4m

3 ∆M

0

( 9Q

2−

2dQ

2+

11s−

3ds

+2(−

1+

d)m

2 π+

(−9

+d)M

2 0

) −m

2 ∆

( (−2

+d)( Q

2+

s)(dQ

2+

2(−

1+

d)s

) +( −

(−8

+d)(−2

+d)Q

2−

2(1

2+

(−5

+d)d

)s) M

2 0−

4(−

5+

d)M

4 0+

2(−

8+

(−1

+d)d

)m2 π

( −Q

2−

s+

M2 0

)))

II[2

+d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

m2 ∆

( 64(−

1+

d)π

M2 0

( −2m

4 ∆+

(−3

+2d)Q

2(−

mπ

+M

0)(m

π+

M0)+

m2 ∆

( 5Q

2+

4s−

ds

+2m

2 π+

(−6

+d)M

2 0

))

II[2

+d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])

+256(−

1+

d)π

2(Q

2−

(−6+

d)m

2 ∆)( (Q

2+

s)2

+2(Q

2−

s)M

2 0+

M4 0

) II[4

+d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

3}]

m2 ∆

−128(−

4+

d)(− 1

+d)π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

) II[4

+d][{{

0,m

π} ,

1},{{

p+

q,m

∆} ,

2},{{

p,m

∆} ,

2}]))

B(ν

,Q2)

=1

96(−

1+

d)3

F2m

4 ∆M

0

ie

2gπN

∆2(5

+τ[3

])

2(−

2+

d)(−

1+

d)(−

4s

m2 π

d+

(s+

m2 π−

m2 ∆

)2

−1+

d

)M

0(−

8m

∆+

(−2+

d)M

0)

s2

−4(−

2+

d) (

s+

m2 π−

m2 ∆

)M0((−

3+

d)m

∆+

(−2+

d)M

0)

s+

16(−

2+

d)(−1

+d)M

0((−1

+d)m

∆+

(−2

+d)M

0)−

4(−

2+

d)2

(−1

+d)( −

m2 π

+m

2 ∆+

M2 0

) +

(−2+

d)( 8

sm

2 π+

d2(s

+m

2 π−

m2 ∆

)2+

2d( m

4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)))

M0(−

2m

∆(Q

2−

3s+

M2 0)+

(−2+

d)M

0(Q

2−

s+

M2 0))

ds3

−1 s2

( 2d

( s+

m2 π−

m2 ∆

) M0

( −6(−

4+

d)m

3 ∆+

(20

+(−

10

+d)d

)m2 ∆

M0

+(−

2+

d)2

M0

( 3Q

2−

3s

+m

2 π+

2M

2 0

) +(−

2+

d)m

∆

( (−6

+d)Q

2−

(−12

+d)s

+2m

2 π+

(−2

+d)M

2 0

))) +

1 s

( 4(−

1+

d)M

0

( (32

+(−

15

+d)d

)m3 ∆

+4(1

0+

(−5

+d)d

)m2 ∆

M0

+(−

2+

d)2

M0

( 3Q

2−

3s

+2m

2 π+

M2 0

) +

(−2

+d)m

∆

( −5Q

2+

7s

+(5

+d)m

2 π+

d( 2

Q2−

2s

+M

2 0

))))

) II[d

][{{

0,m

π},

1}]

+

4(−

2+

d)(−1

+d)(m

π−

m∆−

M0)(m

π+

m∆

+M

0)( (−

2+

d)m

2 ∆+

2m

∆M

0+

(−2

+d)(−

mπ

+M

0)(m

π+

M0)) II

[d][{{

0,m

π},

1},{{

p,m

∆},

1}]

+

M0

( −2(−

2+

d) (

s+

m2 π−

m2 ∆

)( m4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)) (−

8m

∆+

(−2+

d)M

0)

s2

+4(−

2+

d)( m

4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)) ((

−3+

d)m

∆+

(−2+

d)M

0)

s+

8(−

1+

d)( (−

2+

d)( (−

1+

d)( Q

2−

s) +(3

+d)m

2 π

) m∆

+3(−

2+

d)(−1

+d)m

3 ∆+

(−2

+d)2

( Q2−

s+

m2 π

) M0

+(2

0+

d(−

10

+3d))

m2 ∆

M0

) +

(−2+

d) (

s+

m2 π−

m2 ∆

)( (2+

d)m

4 π+

(2+

d) (

s−

m2 ∆

)2+

2m

2 π((−

4+

d)s−

(2+

d)m

2 ∆)) (2

m∆

(Q2−

3s+

M2 0)−

(−2+

d)M

0(Q

2−

s+

M2 0))

s3

+

108


1 s2

( 2( −4

sm2 π

+d

( s+

m2 π−

m2 ∆

) 2) (−6

(−4

+d)m

3 ∆+

(20

+(−

10

+d)d

)m2 ∆

M0

+(−

2+

d)2

M0

( 3Q

2−

3s

+m

2 π+

2M

2 0

) +

(−2

+d)m

∆

( (−6

+d)Q

2−

(−12

+d)s

+2m

2 π+

(−2

+d)M

2 0

))) −

1 s

( 4(−

1+

d)( s

+m

2 π−

m2 ∆

)( (3

2+

(−15

+d)d

)m3 ∆

+4(1

0+

(−5

+d)d

)m2 ∆

M0

+(−

2+

d)2

M0

( 3Q

2−

3s

+2m

2 π+

M2 0

) +(−

2+

d)m

∆

( −5Q

2+

7s

+(5

+d)m

2 π+

d( 2

Q2−

2s

+M

2 0

))))

)

II[d

][{{

0,m

π},

1},{{

p+

q,m

∆},

1}]

+8(−

1+

d)M

0

( m2 π

( (−2

+d)(−1

+d)Q

2m

∆+

(−2

+d)2

Q2M

0+

12m

2 ∆M

0

) −(m

∆+

M0)( −

(−2

+d)(−1

+d)( Q

2+

2s) m

2 ∆+

(−2

+d)Q

2m

∆M

0−

12m

3 ∆M

0+

( (−2

+d)2

Q2

+2(8

+(−

3+

d)d

)m2 ∆

) M2 0

))

II[d

][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}]−

32(−

1+

d)π

M0

( −2(−

1+

(−4

+d)d

)m5 ∆−

2(−

7+

d)d

m4 ∆

M0

+2(−

2+

d)2

Q2M

0

( −m

2 π+

M2 0

) +

(−2

+d)Q

2m

∆

( Q2

+s−

(−3

+d)m

2 π+

(−4

+d)M

2 0

) −m

3 ∆

( (−3

+d)( (−

6+

d)Q

2+

2(−

4+

d)s

) +(2−

2(−

4+

d)d

)m2 π

+(−

26

+6d)M

2 0

) +

2m

2 ∆M

0

( −(2

+(−

5+

d)d

)Q2−

2(−

2+

d)(−1

+d)s

+(−

14

+(−

1+

d)d

)m2 π

+(1

8+

(−5

+d)d

)M2 0

))

II[2

+d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}]−

64(−

1+

d)π

m∆

M0

( (−2

+d)m

2 ∆−

2m

∆M

0+

(−2

+d)(−

mπ

+M

0)(m

π+

M0))

II[2

+d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}]−

32(−

1+

d)π

M2 0

( 12m

4 ∆+

(−2

+d)2

Q2(−

mπ

+M

0)(m

π+

M0)−

m2 ∆

( (−2

+d)( d

Q2

+2(−

1+

d)s

) +12m

2 π−

2(8

+(−

3+

d)d

)M2 0

))II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}]−

256(−

1+

d)π

2M

0

( 4(−

1+

d)m

5 ∆−

2(−

4+

(−5

+d)d

)m4 ∆

M0

+(−

2+

d)Q

2m

∆

( Q2

+s

+2m

2 π−

3M

2 0

) +(−

2+

d)2

Q2M

0

( −m

2 π+

M2 0

) −2m

3 ∆

( 9Q

2−

2dQ

2+

11s−

3ds

+2(−

1+

d)m

2 π+

(−9

+d)M

2 0

) −m

2 ∆M

0

( (−6

+d)d

Q2

+2(−

2+

d)(−1

+d)s−

2(−

8+

(−1

+d)d

)m2 π

+4(−

5+

d)M

2 0

))II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

3}]−

128(−

1+

d)π

2M

2 0( −

2(−

8+

(−1

+d)d

)m4 ∆

+(−

2+

d)2

Q2(−

mπ

+M

0)(m

π+

M0)+

m2 ∆

( −(−

2+

d)( d

Q2

+2(−

1+

d)s

) +2(−

8+

(−1

+d)d

)m2 π−

4(−

5+

d)M

2 0

))II

[4+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}])

)

Dia

gra

mf)

A(ν

,Q2)

=−

196(−

1+

d)3

F2m

4 ∆M

0( ie

2gπN

∆2

( −4(−

2+

d)2

Q2

( Q2

+s−

M2 0

) +48(−

2+

d)2

(−1+

d)m

2 π(−

Q2−

s+

M2 0)

d−

8(−

2+

d)( −

Q2−

s+

M2 0

)(−3

(−2

+d)m

2 π−

2(1

+d)m

2 ∆+

(−10

+d)m

∆M

0+

(−2

+d)M

2 0

) −4(−

2+

d) (−

Q2−

s+

M2 0)(

m2 π−

m2 ∆

+M

2 0)(

(−3+

2d)m

2 ∆−

m∆

M0−

(−2+

d)(

mπ−

M0)(

mπ+

M0) )

M2 0

+

1ds2

( (−2

+d)( −

4sm

2 π+

d( s2

−m

4 π−

m4 ∆

+2m

2 π

( 2s

+m

2 ∆

)))(−(−2

+d)Q

2( Q

2+

s−

M2 0

) −2m

∆M

0

( Q2−

s+

M2 0

))) −

1ds2

( (−2

+d)( 4

sm2 π

+d

( m4 π

+( s−

m2 ∆

) 2−

2m

2 π

( s+

m2 ∆

))) (

(−2

+d)( −

s+

M2 0

)(−Q

2−

s+

M2 0

) +m

∆M

0

( Q2

+7s

+M

2 0

)))

+

2(−

2+

d)(−

1+

d)m

∆(−

Q2−

s+

M2 0)(−

4m

2 πM

2 0d

+(m

2 π−

m2 ∆

+M

2 0)2

−1+

d

)

M3 0

−1

−2Q

2−

s+

2M

2 0

( 2( −

2Q

2−

s+

m2 π−

m2 ∆

+2M

2 0

)(−2

(−21

+5d)m

3 ∆M

0+

(−2

+d)m

∆M

0

( −7Q

2−

s+

M2 0

) +(−

2+

d)2

( −Q

2−

s+

M2 0

)(−2

Q2−

s+

2M

2 0

) +

(−2

+d)m

2 π

( (−2

+d)( Q

2+

s) +8m

∆M

0−

(−2

+d)M

2 0

) −m

2 ∆

( −(2

+d)( Q

2+

s) +9(−

2+

d)M

2 0

))) −

1 s

( 2( s

+m

2 π−

m2 ∆

)(2(−

21

+5d)m

3 ∆M

0+

(−2

+d)2

( −s

+m

2 π

)(Q

2+

s−

M2 0

) +(−

2+

d)m

∆M

0

( 5Q

2−

s−

8m

2 π+

M2 0

) +m

2 ∆

( (2+

d)( Q

2+

s) +(−

22

+7d)M

2 0

))) +

1

d(2

Q2+

s−

2M

2 0)2

( (−2

+d)( (−

2+

d)( −

2Q

2−

s+

M2 0

)(−Q

2−

s+

M2 0

) +m

∆M

0

( −13Q

2−

7s

+15M

2 0

))( 4

m2 π

( −2Q

2−

s+

2M

2 0

) +d

( m4 π

+( 2

Q2

+s

+m

2 ∆−

2M

2 0

) 2−

2m

2 π

( −2Q

2−

s+

m2 ∆

+2M

2 0

))))

+1

d(2

Q2+

s−

2M

2 0)2

109

B. Appendix to VVCS

( (−2

+d)( −

(−2

+d)Q

2( Q

2+

s−

M2 0

) −2m

∆M

0

( −3Q

2−

s+

M2 0

))( 4

m2 π

( 2Q

2+

s−

2M

2 0

) −d

( m4 π

+m

4 ∆−

( 2Q

2+

s−

2M

2 0

) 2−

2m

2 π

( −4Q

2−

2s

+m

2 ∆+

4M

2 0

))))

)

(5+

τ[3

])II

[d][{{

0,m

π},

1}])−

148(−

1+

d)3

F2m

4 ∆M

4 0

( i(−2

+d)e

2gπN

∆2(m

π−

m∆−

M0)(m

π+

m∆−

M0)(m

π−

m∆

+M

0)(m

π+

m∆

+M

0)

( −Q

2−

s+

M2 0

)(m

3 ∆+

2(−

3+

2d)m

2 ∆M

0+

2(−

2+

d)M

0

( −m

2 π+

M2 0

) −m

∆

( m2 π

+3M

2 0

))(5

+τ[3

])II

[d][{{

0,m

π},

1},{{

p,m

∆},

1}]

+

196(−

1+

d)3

F2m

4 ∆M

0

( ie2gπN

∆2

( m4 π

+( 2

Q2

+s

+m

2 ∆−

2M

2 0

) 2−

2m

2 π

( −2Q

2−

s+

m2 ∆

+2M

2 0

))(

(−2+

d) (

2Q

2+

s+

m2 π−

m2 ∆−

2M

2 0)(

(−2+

d)Q

2(Q

2+

s−

M2 0)+

2m

∆M

0(−

3Q

2−

s+

M2 0))

(2Q

2+

s−

2M

2 0)2

+

1

(2Q

2+

s−

2M

2 0)2

( (−2

+d)( −

2Q

2−

s+

m2 π−

m2 ∆

+2M

2 0

)((−

2+

d)( −

2Q

2−

s+

M2 0

)(−Q

2−

s+

M2 0

) +m

∆M

0

( −13Q

2−

7s

+15M

2 0

))) +

1−

2Q

2−

s+

2M

2 0

( 2( 2

(−21

+5d)m

3 ∆M

0−

(−2

+d)m

∆M

0

( −7Q

2−

s+

M2 0

) −(−

2+

d)2

( −Q

2−

s+

M2 0

)(−2

Q2−

s+

2M

2 0

) +m

2 ∆( −

(2+

d)( Q

2+

s) +9(−

2+

d)M

2 0

) +(−

2+

d)m

2 π

( −8m

∆M

0+

(−2

+d)( −

Q2−

s+

M2 0

))))

) (5+

τ[3

])II

[d][{{

0,m

π},

1},{{

p−

q,m

∆},

1}]) +

148(−

1+

d)3

F2m

4 ∆M

0

( i(−2

+d)e

2gπN

∆2

( Q2

+4m

2 π

)(Q

2+

s−

M2 0

)(4(1

+d)m

2 ∆−

2(−

10

+d)m

∆M

0+

(−2

+d)( Q

2+

6m

2 π−

2M

2 0

))

(5+

τ[3

])II

[d][{{

0,m

π},

1},{{

q,m

π},

1}])−

196(−

1+

d)3

F2s2m

4 ∆M

0

( ie2gπN

∆2

( m4 π

+( s−

m2 ∆

) 2−

2m

2 π

( s+

m2 ∆

))( (−

2+

d)s

m∆

M0

( 13Q

2+

3s

+5M

2 0

) +(−

2+

d)2

s( −s

2+

( −Q

2+

M2 0

) 2) +

m3 ∆

M0

( 11(−

6+

d)s

+(−

2+

d)( Q

2+

M2 0

))+

(−2

+d)m

2 π

( (−2

+d)( Q

2−

3s

+M

2 0

)(−Q

2−

s+

M2 0

) −m

∆M

0

( Q2

+7s

+M

2 0

))−

m2 ∆

( (−6

+d)d

s2−

2s

( (2+

d)Q

2+

(−3

+d)(

6+

d)M

2 0

) +(−

2+

d)2

( −Q

4+

M4 0

))) (5

+τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1}]) +

16(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2π

((Q

2+

s) m2 ∆

( (−2

+d)( (−

4+

3d)Q

2+

2(−

1+

d)s

) −2(2

+(−

4+

d)d

)m2 ∆

) −2

( Q2

+s) m

∆

( (−2

+d)Q

2+

2(−

5+

d)m

2 ∆

) M0+

( (−2

+d)2

Q2

( Q2

+s) −

( (20

+d(−

30

+7d))

Q2

+4s

+6(−

3+

d)d

s) m2 ∆

+2(−

2+

(−4

+d)d

)m4 ∆

) M2 0

+

2m

∆

( (−2

+d)Q

2+

2(−

5+

d)m

2 ∆

) M3 0−

( (−2

+d)2

Q2−

4(−

3+

d)d

m2 ∆

) M4 0

+m

2 π

( (−2

+d)2

Q2−

2(−

2+

(−3

+d)d

)m2 ∆

)(−Q

2−

s+

M2 0

))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p−

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2π

( −2(−

23

+6d)m

5 ∆M

0+

(−2

+d)2

M2 0

( −Q

2−

s+

M2 0

)(−2

Q2−

s+

2M

2 0

) +2m

3 ∆M

0

( −5(−

3+

d)Q

2+

11s−

3ds

+(6

+d)M

2 0

) +

m4 ∆

( −(−

4+

d)(

2+

d)( Q

2+

s) +(2

6+

(−10

+d)d

)M2 0

) −(−

2+

d)m

∆M

0

( 3( Q

2+

s)(2Q

2+

s) −( 1

1Q

2+

10s) M

2 0+

7M

4 0

) −m

2 ∆

( (−2

+d)( Q

2+

s)(2Q

2+

(−1

+2d)s

) +( (−

16

+(1

9−

4d)d

)Q2−

6(−

2+

d)(−1

+d)s

) M2 0

+(4

+d(−

13

+4d))

M4 0

) +

(−2

+d)m

4 π

( −8m

∆M

0+

(−2

+d)( −

Q2−

s+

M2 0

))+

m2 π

( 2(−

31

+10d)m

3 ∆M

0−

(−2

+d)2

( −Q

2−

s+

M2 0

)(−2

Q2−

s+

3M

2 0

) +2(−

2+

d)m

∆M

0

( 3Q

2+

4M

2 0

) −2m

2 ∆

( −(2

+(−

4+

d)d

)( Q

2+

s) +(2

3+

(−12

+d)d

)M2 0

))) (5

+τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

1}]) +

16(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2π

( 2(−

23

+6d)m

5 ∆M

0+

(−2

+d)2

sM2 0

( −Q

2−

s+

M2 0

) +2m

3 ∆M

0

( 7Q

2+

11s−

d( Q

2+

3s) +

(−28

+5d)M

2 0

) +

m4 ∆

( −(−

4+

d)(

2+

d)( Q

2+

s) +(−

42

+d(6

+d))

M2 0

) −(−

2+

d)m

∆M

0

( −3s

( Q2

+s) +

( −3Q

2+

2s) M

2 0+

M4 0

) +

m2 ∆

( (−2

+d)( Q

2+

s)(4(−

1+

d)Q

2+

(−1

+2d)s

) +( (−

12

+(1

7−

4d)d

)Q2

+2

( 2+

d−

d2) s) M

2 0+

3(−

4+

d)M

4 0

) +

(−2

+d)m

4 π

( 8m

∆M

0+

(−2

+d)( −

Q2−

s+

M2 0

))+

m2 π

( −2(−

31

+10d)m

3 ∆M

0−

(−2

+d)2

( −Q

2−

s+

M2 0

)(s

+M

2 0

) −2(−

2+

d)m

∆M

0

( 3Q

2+

4M

2 0

) −2m

2 ∆

( −(2

+(−

4+

d)d

)( Q

2+

s) +(−

19

+d(4

+d))

M2 0

)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2π

( 2(−

2+

d)Q

2m

∆M

0

( −Q

2−

s+

M2 0

) +4(−

5+

d)m

3 ∆M

0

( −Q

2−

s+

M2 0

) +

(−2

+d)2

Q2(m

π−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) +2m

4 ∆

( −(2

+(−

4+

d)d

)( Q

2+

s) +(6

+(−

4+

d)d

)M2 0

) −m

2 ∆

( (−2

+d)( Q

2+

s)(dQ

2+

2(−

1+

d)s

) +( −

(20

+(−

10

+d)d

)Q2−

2(6

+(−

3+

d)d

)s) M

2 0+

8M

4 0+

2(−

2+

(−3

+d)d

)m2 π

( −Q

2−

s+

M2 0

)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2π

( −Q

2−

s+

M2 0

)(−2

(−3

+d)(−2

+d)m

6 ∆−

8(−

5+

d)m

5 ∆M

0+

4(−

2+

d)Q

2m

∆(m

π−

M0)M

0(m

π+

M0)+

110


(−2

+d)2

Q2

( m2 π−

M2 0

) 2+

m4 ∆

( (−2

+d)2

Q2

+4(2

+(−

4+

d)d

)m2 π

+4(1

2+

(−6

+d)d

)M2 0

) −4m

3 ∆M

0

( (−2

+d)Q

2−

2(−

5+

d)(m

π−

M0)(m

π+

M0)) −

2m

2 ∆(m

π−

M0)(m

π+

M0)( (−

2+

d)2

Q2

+(−

2+

(−3

+d)d

)(m

π−

M0)(m

π+

M0)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 8i(−2

+d)e

2gπN

∆2π

2M

0

( −2Q

2−

s+

M2 0

) (5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

1}]) −

16(−

1+

d)2

F2m

4 ∆M

0

( 4ie

2gπN

∆2π

2( (−

2+

d)2

(mπ−

M0)(m

π+

M0)( −

2Q

2−

s+

M2 0

)(−Q

2−

s+

M2 0

) −2m

3 ∆M

0

( (21−

9d)Q

2+

3(5−

2d)s

+(−

37

+14d)M

2 0

) +

m2 ∆

( (−2

+d)( Q

2+

s)(2Q

2+

ds) −

( (38

+d(−

32

+5d))

Q2

+2(−

3+

d)(−5

+3d)s

) M2 0

+(6

6+

d(−

42

+5d))

M4 0

) −(−

2+

d)m

∆M

0

( −5

( Q2

+s)(

2Q

2+

s) +( 1

7Q

2+

10s) M

2 0+

3M

4 0+

m2 π

( 14Q

2+

8s−

16M

2 0

)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

2}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 8i(−2

+d)e

2gπN

∆2π

2M

0

( −s

+M

2 0

) (5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1}]) +

16(−

1+

d)2

F2m

4 ∆M

0

( 4ie

2gπN

∆2π

2( (−

2+

d)2

(mπ−

M0)(m

π+

M0)( −

s+

M2 0

)(−Q

2−

s+

M2 0

) −2m

3 ∆M

0

( 3(−

3+

d)Q

2+

3(−

5+

2d)s

+(−

7+

2d)M

2 0

) −(−

2+

d)m

∆M

0

( −5s

( Q2

+s) −

2( Q

2+

4s) m

2 π+

( 7Q

2+

10s) M

2 0+

3M

4 0

) −m

2 ∆

( −(−

2+

d)( Q

2+

s)(2(−

1+

d)Q

2+

ds) +

( −(1

4+

(−8

+d)d

)Q2−

2(1

5+

(−10

+d)d

)s) M

2 0+

3(−

2+

(−2

+d)d

)M4 0

))(5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( 4ie

2gπN

∆2π

2( −

(−2

+d)2

Q2(m

π−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) −2(−

2+

d)m

3 ∆M

0

( 3Q

2−

2s

+2M

2 0

) −m

2 ∆

( −(−

2+

d)( Q

2+

s)(Q

2+

s−

ds) +

( −(1

0+

(−8

+d)d

)Q2

+2

( 3+

d−

d2) s) M

2 0+

( −8

+d

+d2) M

4 0

) +

(−2

+d)m

∆M

0

( 5Q

4+

2Q

2s−

s2−

2( Q

2−

2s) M

2 0−

3M

4 0+

2m

2 π

( 2Q

2−

s+

M2 0

))) (5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

1}]) +

16(−

1+

d)2

F2m

4 ∆M

0

( 4ie

2gπN

∆2π

2( −

(−2

+d)2

Q2(m

π−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) −2(−

2+

d)m

3 ∆M

0

( −7Q

2−

2s

+2M

2 0

) +

m2 ∆

( (−2

+d)( Q

2+

s)((−

1+

2d)Q

2+

(−1

+d)s

) +( (−

30

+(2

0−

3d)d

)Q2−

2(7

+(−

5+

d)d

)s) M

2 0+

(−4

+d)(−3

+d)M

4 0

) +

(−2

+d)m

∆M

0

( 3Q

4+

6Q

2s

+s2−

2Q

2M

2 0−

M4 0

+2m

2 π

( −4Q

2−

s+

M2 0

))) (5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{

p,m

∆},

1}]) −

16(−

1+

d)2

F2m

4 ∆M

0

( 4ie

2gπN

∆2π

2( −

(−2

+d)2

Q2(m

π−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) −2m

3 ∆M

0

( (13−

5d)Q

2+

(7−

2d)s

+(−

7+

2d)M

2 0

) +

m2 ∆

( (−2

+d)( Q

2+

s)(Q

2+

(−1

+d)s

) +( −

(20

+(−

14

+d)d

)Q2−

2(−

3+

d)(−2

+d)s

) M2 0

+(−

5+

d)(−2

+d)M

4 0

) +

(−2

+d)m

∆M

0

((Q

2+

s)(7Q

2+

s) −4Q

2M

2 0−

M4 0

+2m

2 π

( −4Q

2−

s+

M2 0

))) (5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

1}]) +

16(−

1+

d)2

F2m

4 ∆M

0

( 4ie

2gπN

∆2π

2( (−

2+

d)2

Q2(m

π−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) +2m

3 ∆M

0

( Q2

+dQ

2+

7s−

2ds

+(−

7+

2d)M

2 0

) +

m2 ∆

( (−2

+d)( Q

2+

s)((−

3+

2d)Q

2+

(−1

+d)s

) +( (−

12

+(1

4−

3d)d

)Q2

+2

( 2+

d−

d2) s) M

2 0+

( −6

+d

+d2) M

4 0

) +

(−2

+d)m

∆M

0

( −5Q

4+

s2+

4Q

2M

2 0+

3M

4 0−

4s

( Q2

+M

2 0

) −2m

2 π

( 2Q

2−

s+

M2 0

)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{

p+

q,m

∆},

1}])

+1

6(−

1+

d)2

F2M

0( 8

i(−4

+d)e

2gπN

∆2π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

) (5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])

−1

6(−

1+

d)2

F2M

0

( 8i(−4

+d)e

2gπN

∆2π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

) (5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])

−1

6(−

1+

d)2

F2M

0

( 8i(−4

+d)e

2gπN

∆2π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

) (5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

1},{{−p

+q,m

∆},

1}])

+1

6(−

1+

d)2

F2M

0( 8

i(−4

+d)e

2gπN

∆2π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

) (5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

−1

6(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3M

0

( −2Q

2−

s+

M2 0

)(−Q

2−

s+

3M

2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

3}]) −

16(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3M

0

( −s

+M

2 0

)(Q

2+

s+

M2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

3}]) −

111

B. Appendix to VVCS

16(−

1+

d)2

F2m

3 ∆M

0

( 16i(−2

+d)e

2gπN

∆2π

3M

0

( −( 3

Q2−

s)(Q

2+

s) −2

( Q2

+s) M

2 0+

M4 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

2}]) +

16(−

1+

d)2

F2m

3 ∆M

0( 1

6i(−2

+d)e

2gπN

∆2π

3M

0

((Q

2+

s)(5Q

2+

s) −2

( 5Q

2+

s) M2 0

+M

4 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{

p,m

∆},

2}]) −

16(−

1+

d)2

F2m

3 ∆M

0

( 16i(−2

+d)e

2gπN

∆2π

3M

0

((Q

2+

s)(5Q

2+

s) −2

( 5Q

2+

s) M2 0

+M

4 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

2}]) +

16(−

1+

d)2

F2m

3 ∆M

0( 1

6i(−2

+d)e

2gπN

∆2π

3M

0

( −( 3

Q2−

s)(Q

2+

s) −2

( Q2

+s) M

2 0+

M4 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{

p+

q,m

∆},

2}]) +

16(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3Q

2M

0

( Q2−

s+

M2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

,m∆},

1}]) +

16(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3Q

2M

0

( −3Q

2−

s+

M2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{

p,m

∆},

1}]) +

16(−

1+

d)2

F2m

3 ∆M

0

( 64i (− 2

+d)e

2gπN

∆2π

3Q

2M

0

( −Q

2−

s+

M2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π} ,

1},{{

q,m

π} ,

3},{{−p

+q,m

∆} ,

1}]) +

16(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3Q

2M

0

( −Q

2−

s+

M2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{

p+

q,m

∆},

1}])

B(ν

,Q2)

=1

192(−

1+

d)3

F2m

4 ∆M

5 0

( ie2gπN

∆2

( −24(−

2+

d)2

m∆

M4 0

+8(−

2+

d)M

4 0(−

6m

∆+

(−2

+d)M

0)+

2(−

2+

d)(−

1+

d)(−

4s

m2 π

d+

(s+

m2 π−

m2 ∆

)2

−1+

d

)M

4 0(−

4m

∆+

(−2+

d)M

0)

s2

−4(−

2+

d) (

s+

m2 π−

m2 ∆

)M4 0((−

3+

d)m

∆+

(−2+

d)M

0)

s+

8(−

2+

d)(−1

+d)M

4 0((

2+

d)m

∆+

(−2

+d)M

0)−

8(−

2+

d)M

4 0((−4

+d)m

∆+

2(−

2+

d)M

0)+

(−2+

d)2

d(s

+m

2 π−

m2 ∆

)M5 0(−

Q2−

s+

M2 0)

s2

+

1ds3

( (−2

+d)2

( 8sm

2 π+

d2

( −s2

+(m

π−

m∆

)2(m

π+

m∆

)2)

+2d

( m4 π

+( s−

m2 ∆

) 2−

2m

2 π

( s+

m2 ∆

))) M

5 0

( −Q

2−

s+

M2 0

))−

1ds3

( 2(−

2+

d)m

∆

( 8sm

2 π+

2d

( −5s2

+m

4 π−

2sm

2 ∆+

m4 ∆−

2m

2 π

( s+

m2 ∆

))+

d2

( 7s2

+m

4 π+

4sm

2 ∆+

m4 ∆−

2m

2 π

( 2s

+m

2 ∆

))) M

4 0

( Q2−

s+

M2 0

))−

2(−

2+

d)m

∆M

2 0

( m2 π−

m2 ∆

+M

2 0

) −2(−

2+

d)M

4 0((−

1+

2d)m

∆+

2(−

2+

d)M

0) (

2Q

2+

s−

m2 π+

m2 ∆−

2M

2 0)

−2Q

2−

s+

2M

2 0+

1ds3

( 2(−

2+

d)( 8

sm2 π

+d2

( s−

m2 π

+m

2 ∆

) 2+

2d

( −s2

+m

4 π−

2sm

2 ∆+

m4 ∆−

2m

2 π

( s+

m2 ∆

))) M

4 0

( (−2

+d)Q

2M

0+

m∆

( Q2

+s

+M

2 0

)))

+

2( 2

M2 0

+d

( m2 π−

m2 ∆−

M2 0

))( 8

(−2

+d)m

3 ∆+

(−16

+d(4

+d))

m2 ∆

M0−

2(−

2+

d)m

∆

( Q2

+s

+4m

2 π−

5M

2 0

) +(−

2+

d)2

M0

( −m

2 π+

M2 0

))−

2d

( m2 π−

m2 ∆

+M

2 0

)(4(−

2+

d)m

3 ∆+

( −6

+d2) m

2 ∆M

0−

(−2

+d)m

∆

( Q2

+s

+4m

2 π−

5M

2 0

) +(−

2+

d)2

M0

( −m

2 π+

M2 0

))+

1 s

( 2(−

1+

d)M

4 0

( −(4

+(−

2+

d)d

)m2 ∆

M0−

(−2

+d)d

m∆

( −Q

2−

s+

M2 0

) −(−

2+

d)2

M0

( −2

( Q2

+s) +

m2 π

+M

2 0

))) +

1 s2

( 4( −

4s

+d

( 3s−

m2 π

+m

2 ∆

))M

4 0

( −2(−

4+

d)m

3 ∆+

(−2

+d)2

Q2M

0−

(−5

+2d)m

2 ∆M

0+

(−2

+d)m

∆

( dQ

2−

s+

2m

2 π+

M2 0

))) +

1 s

( 2(−

1+

d)M

4 0( 2

(−4

+d)(

4+

d)m

3 ∆+

2(−

2+

d)2

( −2Q

2−

s+

m2 π

) M0

+2(−

1+

d)(

3+

d)m

2 ∆M

0+

(−2

+d)m

∆

( 3( Q

2+

s) −2d

( 2Q

2+

s) +2(−

4+

d)m

2 π+

M2 0

))) +

(−2

+d)(−1

+d)M

4 0

( m∆

( 11Q

2+

5s−

9M

2 0

) +(−

2+

d)M

0

( −3Q

2−

s+

M2 0

))( −

8m

2 π

d(2

Q2+

s−

2M

2 0)2

+(2

Q2+

s−

m2 π+

m2 ∆−

2M

2 0)2

(−2Q

2−

s+

2M

2 0)3

+

3(2

Q2+

s−

m2 π+

m2 ∆−

2M

2 0)2

(−1+

d) (−

2Q

2−

s+

2M

2 0)3

)−

2m

∆

( −4(−

2+

d)m

2 ∆−

2(−

5+

2d)m

∆M

0+

(−2

+d)( Q

2+

s+

4m

2 π−

5M

2 0

))( −

4M

2 0+

d( −

m2 π

+m

2 ∆+

3M

2 0

))+

1−

2Q

2−

s+

2M

2 0

( 2(−

1+

d)M

4 0

( 2(−

27

+d(5

+d))

m3 ∆

+(−

26

+d(−

4+

3d))

m2 ∆

M0

+(−

2+

d)2

M0

( −4

( 3Q

2+

s) −3m

2 π+

7M

2 0

) −(−

2+

d)m

∆

( −37Q

2+

9dQ

2−

13s

+3ds

+2(4

+d)m

2 π+

(21−

5d)M

2 0

))) +

1Q

2

( 8(−

2+

d)M

4 0

( −3m

3 ∆+

m2 ∆

M0

+(−

2+

d)2

M0

( −2Q

2−

s+

M2 0

) +(−

2+

d)m

∆

( 4Q

2+

3s−

d( 2

Q2

+s) +

(−3

+d)M

2 0

))) +

1 s2

( 2( −

2s

+d

( s−

m2 π

+m

2 ∆

))M

4 0

( (4+

(−2

+d)d

)m2 ∆

M0

+(−

2+

d)2

M0

( −3Q

2−

2s

+m

2 π+

M2 0

) +(−

2+

d)m

∆

( −(1

+d)( Q

2+

s) +(−

1+

d)M

2 0

))) −

112


1

(2Q

2+

s−

2M

2 0)2

( dM

4 0

( −2Q

2−

s+

m2 π−

m2 ∆

+2M

2 0

)( 4

(−13

+3d)m

3 ∆+

2( −

10

+d2) m

2 ∆M

0+

(−2

+d)2

M0

( −5

( 3Q

2+

s) −2m

2 π+

7M

2 0

) +(−

2+

d)m

∆

( 49Q

2+

19s−

2d

( 3Q

2+

s) −8m

2 π+

(−31

+2d)M

2 0

))) −

2(−

1+

d)M

2 0

( 2(−

2+

d)(

4+

d)m

3 ∆+

2(−

15

+d(4

+d))

m2 ∆

M0

+2(−

2+

d)2

M0

( −m

2 π+

M2 0

) −(−

2+

d)m

∆

( 3( Q

2+

s) +2(4

+d)m

2 π−

(11

+2d)M

2 0

))−

2(−

1+

d)M

2 0

( −2(−

2+

d)(

4+

d)m

3 ∆−

6(−

3+

(−1

+d)d

)m2 ∆

M0

+4(−

2+

d)2

(mπ−

M0)M

0(m

π+

M0)+

(−2

+d)m

∆

( 3( Q

2+

s) +2(4

+d)m

2 π−

(11

+2d)M

2 0

))−

1

d(2

Q2+

s−

2M

2 0)2

( 2(−

2+

d)M

4 0(−

2m

∆+

(−2

+d)M

0)( 4

m2 π

( −2Q

2−

s+

2M

2 0

) +d

( m4 π

+( 2

Q2

+s

+m

2 ∆−

2M

2 0

) 2−

2m

2 π

( −2Q

2−

s+

m2 ∆

+2M

2 0

))))

−1

d(2

Q2+

s−

2M

2 0)2

( 8(−

2+

d)m

∆M

4 0

( 4m

2 π

( 2Q

2+

s−

2M

2 0

) −d

( m4 π

+m

4 ∆−

( 2Q

2+

s−

2M

2 0

) 2−

2m

2 π

( −4Q

2−

2s

+m

2 ∆+

4M

2 0

))))

)

(5+

τ[3

])II

[d][{{

0,m

π},

1}])

+1

48(−

1+

d)3

F2m

3 ∆M

2 0

( i(−2

+d)e

2gπN

∆2

( −m

2 π+

m2 ∆

+2(−

2+

d)d

m∆

M0

+M

2 0

)(−m

2 π+

(m∆

+M

0)2

) (5+

τ[3

])

II[d

][{{

0,m

π},

1},{{

p,m

∆},

1}])−

196(−

1+

d)3

F2m

4 ∆M

0

( ie2gπN

∆2M

0

(1

(2Q

2+

s−

2M

2 0)2

( 8(−

2+

d)m

∆

( 2Q

2+

s+

m2 π−

m2 ∆−

2M

2 0

)(m

4 π+

( 2Q

2+

s+

m2 ∆−

2M

2 0

) 2−

2m

2 π

( −2Q

2−

s+

m2 ∆

+2M

2 0

))) −

1

(2Q

2+

s−

2M

2 0)2

( 2(−

2+

d)(−

2m

∆+

(−2

+d)M

0)( −

2Q

2−

s+

m2 π−

m2 ∆

+2M

2 0

)(m

4 π+

( 2Q

2+

s+

m2 ∆−

2M

2 0

) 2−

2m

2 π

( −2Q

2−

s+

m2 ∆

+2M

2 0

)))

+

1−

2(2

Q2+

s)+

4M

2 0

( 4(−

2+

d)((−1

+2d)m

∆+

2(−

2+

d)M

0)( m

4 π+

( 2Q

2+

s+

m2 ∆−

2M

2 0

) 2−

2m

2 π

( −2Q

2−

s+

m2 ∆

+2M

2 0

)))−

1−

2Q

2−

s+

2M

2 0

( 2(−

1+

d)( −

2Q

2−

s+

m2 π−

m2 ∆

+2M

2 0

)(−2

(−27

+d(5

+d))

m3 ∆−

(−26

+d(−

4+

3d))

m2 ∆

M0+

(−2

+d)2

M0

( 4( 3

Q2

+s) +

3m

2 π−

7M

2 0

) +(−

2+

d)m

∆

( −37Q

2+

9dQ

2−

13s

+3ds

+2(4

+d)m

2 π+

(21−

5d)M

2 0

))) +

4(−

1+

d)( −

(−14

+d(2

+d))

m3 ∆−

(−4

+d)(

2+

d)m

2 ∆M

0+

(−2

+d)2

M0

( 3Q

2+

s+

m2 π−

2M

2 0

) +(−

2+

d)m

∆

( (−3

+d)( 3

Q2

+s) +

(2+

d)m

2 π+

(5−

2d)M

2 0

))+

1

(2Q

2+

s−

2M

2 0)2

((4m

2 π

( 2Q

2+

s−

2M

2 0

) +d

( 2Q

2+

s−

m2 π

+m

2 ∆−

2M

2 0

) 2)

( −4(−

13

+3d)m

3 ∆−

2( −

10

+d2) m

2 ∆M

0+

(−2

+d)2

M0

( 5( 3

Q2

+s) +

2m

2 π−

7M

2 0

) +(−

2+

d)m

∆

( −49Q

2+

6dQ

2−

19s

+2ds

+8m

2 π+

(31−

2d)M

2 0

))) +

1

(−2Q

2−

s+

2M

2 0)3

( (−2

+d)( −

2Q

2−

s+

m2 π−

m2 ∆

+2M

2 0

)(m

∆

( 11Q

2+

5s−

9M

2 0

) +(−

2+

d)M

0

( −3Q

2−

s+

M2 0

))( (2

+d)m

4 π+

(2+

d)( 2

Q2

+s

+m

2 ∆−

2M

2 0

) 2+

2m

2 π

( −(2

+d)m

2 ∆+

(−4

+d)( −

2Q

2−

s+

2M

2 0

))))

)

(5+

τ[3

])II

[d][{{

0,m

π},

1},{{

p−

q,m

∆},

1}])−

ie2gπN

∆2

24(−

1+

d)3

F2Q

2m

4 ∆M

0

((Q

2+

4m

2 π

) M0

( −3m

3 ∆+

m2 ∆

M0

+(−

3+

d)(−2

+d)m

∆

( −Q

2−

s+

M2 0

) +(−

2+

d)2

M0

( −Q

2−

s+

M2 0

))

(5+

τ[3

])II

[d][{{

0,m

π},

1},{{

q,m

π},

1}])

+1

96(−

1+

d)3

F2s3m

4 ∆M

0

( ie2gπN

∆2M

0

( 2sm

∆

( (−2

+d)s

2( (−

3+

d)Q

2+

(−1

+d)s

) +s

( (−2

+d)(

7+

(−4

+d)d

)Q2

+10s−

(−1

+d)2

ds) m

2 ∆+

( (−3

+d)(−2

+d)d

Q2

+(−

5+

d)( 4

+d2) s) m

4 ∆−

2(−

6+

d)d

m6 ∆

) +( sm

2 ∆

( −(−

28

+d(1

6+

(−6

+d)d

))s2

+2(−

30

+d(1

2+

d))

sm2 ∆

+d(2

4+

(−8

+d)d

)m4 ∆

) +

(−2

+d)2

Q2

( 2s3

+(−

2+

d)s

2m

2 ∆+

2(−

3+

d)s

m4 ∆

+(2

+d)m

6 ∆

))M

0+

2(−

2+

d)(−1

+d)s

m∆

( s−

m2 ∆

)(s−

dm

2 ∆

) M2 0

+

(−2

+d)2

m2 ∆

( −s

+m

2 ∆

)((−

4+

d)s

+(2

+d)m

2 ∆

) M3 0−

(−2

+d)m

6 π

( −4dsm

∆−

3(−

2+

d)d

sM0

+( −

4+

d2) M

0

( Q2

+M

2 0

))+

m2 π

( 4d(−

14

+3d)s

m5 ∆

+4s

( −(−

2+

d)2

dQ

2+

( −19

+(−

4+

d)2

d) s) m

2 ∆M

0+

(−2

+d)2

s2M

0

( −(6

+d)Q

2+

(2+

d)s

+(−

4+

d)M

2 0

) +

2(−

2+

d)s

2m

∆

( 5( Q

2+

s) +d

( −dQ

2+

(−4

+d)s

) +(−

3+

d)(−1

+d)M

2 0

) −m

4 ∆M

0

( −d(−

36

+d(4

+d))

s+

3(−

2+

d)2

(2+

d)( Q

2+

M2 0

))+

4sm

3 ∆

( (−12

+d(6

+(−

3+

d)d

))s−

(−2

+d)d

( (−3

+d)Q

2+

(−1

+d)M

2 0

))) +

m4 π

( −4d(−

10

+3d)s

m3 ∆−

2(−

2+

d)2

sM0

( −(3

+d)Q

2+

s+

2ds−

3M

2 0

) +m

2 ∆M

0

( −d2(−

16

+5d)s

+3(−

2+

d)2

(2+

d)( Q

2+

M2 0

))−

2(−

2+

d)s

m∆

( (6+

d(−

5+

3d))

s+

d( 3

Q2

+M

2 0−

d( Q

2+

M2 0

))))

) (5+

τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1}]) +

124(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2M

0

( −2(−

2+

(−3

+d)d

)m5 ∆−

2(8

+(−

5+

d)d

)m4 ∆

M0

+(−

2+

d)2

Q2m

∆M

2 0+

(−2

+d)2

Q2M

3 0−

113

B. Appendix to VVCS

m2 π

( (−2

+d)2

Q2−

2(−

2+

(−3

+d)d

)m2 ∆

) (m∆

+M

0)−

m3 ∆

( −(−

4+

d)d

( 3Q

2+

2s) −

2( 5

Q2

+3s) +

2(1

+d(−

7+

2d))

M2 0

) −m

2 ∆M

0

( −4

( Q2

+s) +

d( 8

Q2−

3dQ

2+

6s−

2ds

+4(−

3+

d)M

2 0

))) (5

+τ[3

])II

[d][{{

0,m

π},

1},{{

p−

q,m

∆},

1},{{

p,m

∆},

1}]) −

124(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2M

0

( (−14

+d(2

+d))

m5 ∆

+(−

6+

d(4

+d))

m4 ∆

M0

+(−

2+

d)2

(mπ−

M0)M

0(m

π+

M0)( 3

Q2

+s

+m

2 π−

2M

2 0

) −m

3 ∆

( 26Q

2+

14s−

d( (8

+d)Q

2+

(−2

+3d)s

) +2

( −9

+d

+d2) m

2 π+

(−2

+d)(

2+

5d)M

2 0

) −m

2 ∆M

0

( 10Q

2−

2s−

d( (6

+d)Q

2+

3(−

2+

d)s

) +2

( −9

+d2) m

2 π+

(12

+5(−

2+

d)d

)M2 0

) +

(−2

+d)m

∆

( 6Q

4+

8Q

2s

+2s2

+(2

+d)m

4 π−

6Q

2M

2 0−

4sM

2 0+

dM

2 0

( −3Q

2−

s+

2M

2 0

) +m

2 π

( (−8

+3d)Q

2+

(−2

+d)s

+(2−

3d)M

2 0

)))

(5+

τ[3

])II

[d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

1}])

+1

24(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2M

0

( −(1

2+

(−4

+d)d

)m5 ∆−

(10

+(−

4+

d)d

)m4 ∆

M0

+(−

2+

d)2

( −Q

2−

s+

m2 π

) M0

( −m

2 π+

M2 0

) +

(−2

+d)m

∆

( −m

2 π+

M2 0

)(Q

2+

s−

d( Q

2+

s) +(−

2+

d)m

2 π+

M2 0

) +m

3 ∆

( (−1

+d)( −

4Q

2+

3dQ

2+

ds) +

2(8

+(−

4+

d)d

)m2 π

+(−

4+

d)(

1+

d)M

2 0

) +

m2 ∆

M0

( (−2

+d)( (−

4+

3d)Q

2+

(−2

+d)s

) +2(1

+(−

4+

d)d

)m2 π

+( 2

+d2) M

2 0

))(5

+τ[3

])II

[d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1}]−

124(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2M

0

( m∆

( (−2

+d)Q

4+

(−2

+d)Q

2s

+(−

4+

d)(

2+

d)Q

2m

2 ∆+

2(−

2+

(−3

+d)d

)m2 ∆

( s+

m2 ∆

))+

m2 ∆

((−2

+d2) Q

2+

2(−

2+

d)( (−

1+

d)s

+(−

3+

d)m

2 ∆

))M

0−

(−2

+d)(−1

+d)Q

2m

∆M

2 0−

( (−2

+d)2

Q2

+8m

2 ∆

) M3 0

+m

2 π

( (−2

+d)2

Q2−

2(−

2+

(−3

+d)d

)m2 ∆

) (m∆

+M

0))

(5+

τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])−

124(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2M

0

( −2(−

2+

(−3

+d)d

)m7 ∆−

2(−

3+

d)d

m6 ∆

M0

+(−

2+

d)2

Q2M

0

( m2 π−

M2 0

) 2+

(−2

+d)Q

2m

∆(m

π−

M0)(m

π+

M0)( Q

2+

s+

(−2

+d)m

2 π−

(−1

+d)M

2 0

) +

m4 ∆

M0

( (18

+(−

12

+d)d

)Q2−

4(−

2+

d)s

+4(7

+(−

5+

d)d

)m2 π

+4(−

3+

d)(

1+

d)M

2 0

) +

m5 ∆

( (14

+(−

6+

d)d

)Q2−

2(−

5+

d)s

+4(−

2+

(−3

+d)d

)m2 π

+2(−

9+

d(−

5+

2d))

M2 0

) −2m

2 ∆(m

π−

M0)M

0(m

π+

M0)

( (−3

+d)(−1

+d)Q

2+

(−2

+(−

3+

d)d

)(m

π−

M0)(m

π+

M0)) −

m3 ∆

((Q

2+

s)((−

4+

3d)Q

2+

2(−

1+

d)s

) +2(−

2+

(−3

+d)d

)m4 π−

( (8+

d(−

5+

2d))

Q2

+2(3

+d)s

) M2 0

+2(−

2+

d)(−1

+d)M

4 0−

2m

2 π

( −(9

+(−

5+

d)d

)Q2

+(−

5+

d)s

+(1

+d(−

7+

2d))

M2 0

)))

(5+

τ[3

])II

[d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2πM

0

( −4(−

2+

(−3

+d)d

)m5 ∆−

2(2

0+

d(−

13

+3d))

m4 ∆

M0

+3(−

2+

d)2

Q2M

0

( −m

2 π+

M2 0

) −(−

2+

d)Q

2m

∆

( Q2

+s

+2(−

2+

d)m

2 π+

(3−

2d)M

2 0

) +

2m

3 ∆

( (15

+d(−

13

+3d))

Q2

+(1

1+

d(−

9+

2d))

s+

2(−

2+

(−3

+d)d

)m2 π

+(−

7+

(15−

4d)d

)M2 0

) +

m2 ∆

M0

( 2( 5

Q2

+6s) +

6(−

2+

(−3

+d)d

)m2 π

+d

( (−22

+9d)Q

2+

6(−

3+

d)s−

12(−

3+

d)M

2 0

)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p−

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2πM

2 0

( −2(2

+(−

5+

d)d

)m4 ∆−

(−2

+d)2

Q2(m

π−

M0)(m

π+

M0)+

m2 ∆

( −(−

2+

d)( d

Q2

+2(−

1+

d)s

) +2(−

2+

(−3

+d)d

)m2 π

+8M

2 0

))(5

+τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 4ie

2gπN

∆2πM

0

( 2(−

2+

d)m

2 ∆+

(−5

+d)m

∆M

0−

2(−

2+

d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[2

+d][{{

0,m

π},

1},{{

p−

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2πM

2 0

( m2 π

( −(−

2+

d)2

Q2

+2(−

2+

(−3

+d)d

)m2 ∆

) +(−

2+

d)m

2 ∆

( (−4

+3d)Q

2+

2(−

1+

d)( s−

m2 ∆

))+

( (−2

+d)2

Q2−

4(−

3+

d)d

m2 ∆

) M2 0

)(5

+τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p−

q,m

∆},

1},{{

p,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2πM

0

( (−2

+d)(−1

+2d)m

3 ∆+

2(−

7+

d)m

2 ∆M

0+

2(−

2+

d)2

M0

( −m

2 π+

M2 0

) +(−

2+

d)m

∆

( −7

( Q2

+s) +

(1−

2d)m

2 π+

2(6

+d)M

2 0

))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2πM

0

( 2(−

27

+d(5

+d))

m5 ∆

+3(−

2+

d)(

6+

d)m

4 ∆M

0+

(−2

+d)2

(mπ−

M0)M

0(m

π+

M0)( 4

( 3Q

2+

s) +3m

2 π−

7M

2 0

) −2m

2 ∆M

0

( 4( 8

Q2

+s) −

d( (2

0+

d)Q

2+

(−6

+5d)s

) +( −

26

+3d2) m

2 π+

2(3

+d(−

7+

5d))

M2 0

) −

114


m3 ∆

( 110Q

2+

62s−

d( (4

0+

d)Q

2+

(4+

7d)s

) +2(−

35

+d(7

+2d))

m2 π

+(−

48

+d(−

14

+15d))

M2 0

) +

(−2

+d)m

∆

( 9( Q

2+

s)(3Q

2+

s) +2(4

+d)m

4 π−

( (32

+9d)Q

2+

3(6

+d)s

) M2 0

+(1

+5d)M

4 0+

m2 π

( −34Q

2+

9dQ

2−

10s

+3ds

+(1

0−

7d)M

2 0

)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

2}])−

16(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2πM

0

( −(−

2+

d)(−1

+2d)m

3 ∆+

2(4

+d)m

2 ∆M

0+

2(−

2+

d)2

M0

( −m

2 π+

M2 0

) −(−

2+

d)m

∆

( 3Q

2+

2s

+(−

5+

2d)m

2 π+

(5−

2d)M

2 0

))(5

+τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2πM

0

( −(−

8+

(−4

+d)d

)m4 ∆

M0

+(−

2+

d)d

m∆

( −Q

2−

s+

M2 0

)(−m

2 π+

M2 0

) +(−

2+

d)2

M0

( −m

2 π+

M2 0

)(−2

( Q2

+s) +

m2 π

+M

2 0

) +

(−2

+d)m

3 ∆

( (−2

+d)( Q

2+

s) +(−

2+

3d)M

2 0

) +2m

2 ∆M

0

( (−2

+d)(−1

+d)( 2

Q2

+s) +

(2+

(−4

+d)d

)m2 π

+(4

+d(−

5+

2d))

M2 0

))(5

+τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2πM

0

( −2(2

2+

(−7

+d)d

)m5 ∆−

2(1

9+

(−6

+d)d

)m4 ∆

M0

+2(−

2+

d)2

( −2Q

2−

s+

m2 π

) M0

( −m

2 π+

M2 0

) +

m3 ∆

( (20

+d(−

19

+6d))

Q2

+3ds

+( 6

0−

26d

+4d2) m

2 π+

(−4

+d)(

7+

4d)M

2 0

) +

2m

2 ∆M

0

( (−2

+d)(−5

+3d)Q

2+

s−

ds

+(1

3+

2(−

6+

d)d

)m2 π

+(4

+d(−

3+

2d))

M2 0

) +

(−2

+d)m

∆

((Q

2+

s)(2Q

2+

s) −2(−

4+

d)m

4 π+

( 2Q

2−

4dQ

2+

3s−

2ds) M

2 0+

m2 π

( −3

( Q2

+s) +

2d

( 2Q

2+

s) +(−

9+

2d)M

2 0

)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2πM

0

( 2(−

27

+d(5

+d))

m5 ∆

+2(−

14

+d(6

+d))

m4 ∆

M0

+2(−

2+

d)2

(mπ−

M0)M

0(m

π+

M0)( 4

Q2

+s

+m

2 π−

2M

2 0

) −2m

2 ∆M

0

( 3( 9

Q2

+s) −

d( (1

7+

d)Q

2+

(−5

+4d)s

) +2(−

14

+d(2

+d))

m2 π

+( 9−

9d

+6d2) M

2 0

) +

(−2

+d)m

∆

((Q

2+

s)(25Q

2+

7s) +

2(4

+d)m

4 π−

( 23Q

2+

8dQ

2+

13s

+2ds) M

2 0+

(2+

4d)M

4 0+

m2 π

( −31Q

2+

8dQ

2−

7s

+2ds

+(3−

6d)M

2 0

))−

m3 ∆

( 104Q

2+

56s

+2(−

35

+d(7

+2d))

m2 π−

22M

2 0+

d( −

35Q

2+

s−

2d

( Q2

+4s) +

3(−

5+

4d)M

2 0

)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

1}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 4ie

2gπN

∆2πM

0

( 2(−

2+

d)m

2 ∆+

(−7

+3d)m

∆M

0−

2(−

2+

d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[2

+d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2πM

0

( −4(−

2+

(−3

+d)d

)m5 ∆−

6(−

4+

d)(−1

+d)m

4 ∆M

0+

3(−

2+

d)2

Q2M

3 0−

m2 π

( (−2

+d)2

Q2−

2(−

2+

(−3

+d)d

)m2 ∆

) (2m

∆+

3M

0)+

m2 ∆

M0

( (10−

d(2

+3d))

Q2−

6(−

2+

d)(−1

+d)s

+24M

2 0

) +2m

3 ∆

( 13Q

2+

9s

+d

( Q2−

dQ

2+

(5−

2d)s

) +(−

5+

d)M

2 0

) +

(−2

+d)Q

2m

∆

( −3

( Q2

+s) +

(−1

+2d)M

2 0

))(5

+τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}]) +

16(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2πM

0

( −4(−

2+

(−3

+d)d

)m5 ∆−

4(5

+(−

5+

d)d

)m4 ∆

M0

+2(−

2+

d)2

Q2M

3 0−

2m

2 π

( (−2

+d)2

Q2−

2(−

2+

(−3

+d)d

)m2 ∆

) (m∆

+M

0)+

2m

2 ∆M

0

( −(−

5+

d(2

+d))

Q2−

2(−

2+

d)(−1

+d)s

+8M

2 0

) +

2m

3 ∆

( 13Q

2+

9s

+d

( Q2−

dQ

2+

(5−

2d)s

) +(−

5+

d)M

2 0

) +(−

2+

d)Q

2m

∆

( −3

( Q2

+s) +

(−1

+2d)M

2 0

))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

2},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0( ie

2gπN

∆2πM

2 0

( −2(−

3+

d)(−2

+d)m

6 ∆+

(−2

+d)2

Q2

( m2 π−

M2 0

) 2+

m4 ∆

( (4+

(−8

+d)d

)Q2−

4ds

+4(2

+(−

4+

d)d

)m2 π

+4(4

+(−

3+

d)d

)M2 0

) −2m

2 ∆(m

π−

M0)(m

π+

M0)( (−

2+

d)2

Q2

+(−

2+

(−3

+d)d

)(m

π−

M0)(m

π+

M0)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

3 ∆M

0( 4

ie2gπN

∆2πM

0

( −2(−

2+

d)m

4 ∆+

(−4

+d)2

m3 ∆

M0

+4(−

3+

d)m

∆(m

π−

M0)M

0(m

π+

M0)−

2(−

2+

d)( m

2 π−

M2 0

) 2+

4m

2 ∆

( (−2

+d)m

2 π−

(−4

+d)M

2 0

))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0

( ie2gπN

∆2πM

0

( −4(−

2+

(−3

+d)d

)m7 ∆−

2(−

3+

d)(−2

+3d)m

6 ∆M

0+

3(−

2+

d)2

Q2M

0

( m2 π−

M2 0

) 2+

2(−

2+

d)Q

2m

∆(m

π−

M0)

115

B. Appendix to VVCS

(mπ

+M

0)( 2

( Q2

+s) +

(−2

+d)m

2 π−

dM

2 0

) +2m

5 ∆

( (24

+(−

8+

d)d

)Q2−

4(−

5+

d)s

+4(−

2+

(−3

+d)d

)m2 π

+4(−

7+

(−2

+d)d

)M2 0

) +

m4 ∆

M0

( (56

+d(−

40

+3d))

Q2−

16(−

2+

d)s

+4(−

2+

d)(−8

+3d)m

2 π+

12(−

2+

(−2

+d)d

)M2 0

) −2m

2 ∆(m

π−

M0)M

0(m

π+

M0)( (1

0+

3(−

4+

d)d

)Q2

+3(−

2+

(−3

+d)d

)(m

π−

M0)(m

π+

M0)) −

4m

3 ∆

((Q

2+

s)((−

3+

2d)Q

2+

(−1

+d)s

) +

(−2

+(−

3+

d)d

)m4 π−

( (8+

(−3

+d)d

)Q2

+8s) M

2 0+

(7+

(−4

+d)d

)M4 0

+m

2 π

( (14

+(−

6+

d)d

)Q2−

2(−

5+

d)s−

2(−

3+

d)(−1

+d)M

2 0

)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])−

16(−

1+

d)2

F2m

4 ∆M

0( 2

ie2gπN

∆2πM

0

( −2(−

2+

(−3

+d)d

)m7 ∆−

2(−

3+

d)d

m6 ∆

M0

+(−

2+

d)2

Q2M

0

( m2 π−

M2 0

) 2+

(−2

+d)Q

2m

∆(m

π−

M0)(m

π+

M0)( 2

( Q2

+s) +

(−2

+d)m

2 π−

dM

2 0

) +

m5 ∆

( (24

+(−

8+

d)d

)Q2−

4(−

5+

d)s

+4(−

2+

(−3

+d)d

)m2 π

+4(−

7+

(−2

+d)d

)M2 0

) +

m4 ∆

M0

( (26

+(−

16

+d)d

)Q2

+16s−

6ds

+4(7

+(−

5+

d)d

)m2 π

+( −

20−

6d

+4d2) M

2 0

) −2m

2 ∆(m

π−

M0)M

0(m

π+

M0)( (−

3+

d)(−1

+d)Q

2+

(−2

+(−

3+

d)d

)(m

π−

M0)(m

π+

M0)) −

2m

3 ∆

((Q

2+

s)((−

3+

2d)Q

2+

(−1

+d)s

) +

(−2

+(−

3+

d)d

)m4 π−

( (8+

(−3

+d)d

)Q2

+8s) M

2 0+

(7+

(−4

+d)d

)M4 0

+m

2 π

( (14

+(−

6+

d)d

)Q2−

2(−

5+

d)s−

2(−

3+

d)(−1

+d)M

2 0

)))

(5+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

1},{{−p

+q,m

∆},

1}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

2 ∆+

(−3

+d)m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[4

+d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p−

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( 16ie

2gπN

∆2π

2M

0

( −2(4

+(−

3+

d)d

)m4 ∆

M0−

(−2

+d)Q

2m

∆

( Q2

+s−

M2 0

) +2(−

5+

d)m

3 ∆

( −Q

2−

s+

M2 0

) +

(−2

+d)2

Q2M

0

( −m

2 π+

M2 0

) +m

2 ∆M

0

( 2( Q

2+

2s) +

2(−

2+

(−3

+d)d

)m2 π

+d

( 3(−

2+

d)Q

2+

2(−

3+

d)s−

4(−

3+

d)M

2 0

)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p−

q,m

∆},

3}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

2 ∆+

(−3

+d)m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[4

+d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

2 ∆+

(−3

+d)m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[4

+d][{{

0,m

π},

1},{{

p−

q,m

∆},

1},{{

p,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0( 8

ie2gπN

∆2π

2M

2 0

( m2 π

( −(−

2+

d)2

Q2

+2(−

2+

(−3

+d)d

)m2 ∆

) +(−

2+

d)m

2 ∆

( (−4

+3d)Q

2+

2(−

1+

d)( s−

m2 ∆

))+

( (−2

+d)2

Q2−

4(−

3+

d)d

m2 ∆

) M2 0

)(5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p−

q,m

∆},

2},{{

p,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( 8ie

2gπN

∆2π

2M

0

( −2(−

2+

d)m

3 ∆−

d(2

+d)m

2 ∆M

0+

(−2

+d)2

M0

( −m

2 π+

M2 0

) +(−

2+

d)m

∆

( −7

( Q2

+s) +

2m

2 π+

12M

2 0

))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

2}])−

16(−

1+

d)2

F2m

4 ∆M

0

( 8ie

2gπN

∆2π

2M

0

( 4(−

13

+3d)m

5 ∆+

2(−

24

+d(4

+d))

m4 ∆

M0

+(−

2+

d)2

(mπ−

M0)M

0(m

π+

M0)( 5

( 3Q

2+

s) +2m

2 π−

7M

2 0

) −m

3 ∆

( (148

+d(−

61

+2d))

Q2

+88s−

d(2

3+

2d)s

+4(−

17

+5d)m

2 π+

(−128

+d(1

9+

10d))

M2 0

) −m

2 ∆M

0

( (122

+(−

78

+d)d

)Q2

+38s−

3d(2

+3d)s

+4

( −8

+d2) m

2 π+

(−46

+d(−

14

+25d))

M2 0

) +

(−2

+d)m

∆

( 14

( Q2

+s)(

3Q

2+

s) +8m

4 π−

( 63Q

2+

6dQ

2+

29s

+2ds) M

2 0+

(3+

2d)M

4 0+

m2 π

( −47Q

2+

6dQ

2−

17s

+2ds

+(2

1−

2d)M

2 0

)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

3}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 16i(−2

+d)e

2gπN

∆2π

2M

0(5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1}]) −

16(−

1+

d)2

F2m

4 ∆M

0( 8

ie2gπN

∆2π

2M

0

( −4(−

2+

d)m

3 ∆−

(−12

+d(2

+d))

m2 ∆

M0

+(−

2+

d)2

M0

( −m

2 π+

M2 0

) −(−

2+

d)m

∆

( Q2

+s−

4m

2 π+

4M

2 0

))(5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( 8i(−2

+d)e

2gπN

∆2π

2M

2 0

( −(−

2+

d)(m

π−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) +m

2 ∆

( d( Q

2+

s) +(−

4+

3d)M

2 0

))

116


(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

3}])−

16(−

1+

d)2

F2m

3 ∆M

0

( 8ie

2gπN

∆2π

2M

0

( 6(−

2+

d)m

2 ∆+

2(−

1+

d)d

m∆

M0−

(−2

+d)( 8

Q2

+3s

+2m

2 π−

M2 0

))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( 4ie

2gπN

∆2π

2M

0

( −2(−

8+

(−4

+d)d

)m4 ∆

M0

+2(−

2+

d)2

M0

( −m

2 π+

M2 0

)(−3

Q2−

2s

+m

2 π+

M2 0

) +

(−2

+d)m

3 ∆

( (−9

+2d)( Q

2+

s) +(−

7+

6d)M

2 0

) +2m

2 ∆M

0

( (−2

+d)( (−

5+

4d)Q

2+

(−3

+d)s

) +2(2

+(−

4+

d)d

)m2 π

+(1

6+

5(−

3+

d)d

)M2 0

) +

(−2

+d)m

∆

((Q

2+

s)(3Q

2+

2s) −

( 2(2

+d)Q

2+

(3+

2d)s

) M2 0

+(−

3+

2d)M

4 0+

m2 π

( (3+

2d)( Q

2+

s) +(1−

2d)M

2 0

)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

2}])

+1

6(−

1+

d)2

F2m

3 ∆M

0( 3

2ie

2gπN

∆2π

2M

0

( (−2

+d)m

2 ∆+

(−3

+d)m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{

p,m

∆},

1}]) +

16(−

1+

d)2

F2m

3 ∆M

0

( 8ie

2gπN

∆2π

2M

0

( −2(−

2+

d)m

2 ∆−

2(2

+(−

1+

d)d

)m∆

M0

+(−

2+

d)( −

7( Q

2+

s) +2m

2 π+

9M

2 0

))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0

( 4ie

2gπN

∆2π

2M

0

( 8(−

13

+3d)m

5 ∆+

2(−

40

+d(8

+d))

m4 ∆

M0

+2(−

2+

d)2

(mπ−

M0)M

0(m

π+

M0)( 9

Q2

+2s

+m

2 π−

3M

2 0

) −2m

2 ∆M

0

( 103Q

2+

31s−

d( 6

5Q

2+

(5+

7d)s

) +2(−

18

+d(4

+d))

m2 π

+( −

35

+d

+13d2) M

2 0

) −m

3 ∆

( (278

+d(−

109

+2d))

Q2

+158s−

3d(1

1+

2d)s

+8(−

17

+5d)m

2 π+

(−170

+d(2

5+

14d))

M2 0

) +

(−2

+d)m

∆

( 7( Q

2+

s)(11Q

2+

3s) +

16m

4 π−

( 89Q

2+

10dQ

2+

37s

+2ds) M

2 0+

2(2

+d)M

4 0+

m2 π

( −85Q

2+

10dQ

2−

25s

+2ds

+(2

1−

2d)M

2 0

)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

2 ∆+

(−3

+d)m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[4

+d][{{

0,m

π},

1},{{

q,m

π},

2},{{

p+

q,m

∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0

( 8ie

2gπN

∆2π

2M

0

( −4(−

10

+3d)m

5 ∆−

4(−

9+

2d)m

4 ∆M

0+

4(−

2+

d)2

Q2M

0

( −m

2 π+

M2 0

) +

2m

3 ∆

( 2(−

9+

4d)Q

2+

(4+

d(−

7+

2d))

s+

2(−

14

+5d)m

2 π+

(16

+(5−

2d)d

)M2 0

) +

2m

2 ∆M

0

( (−7

+4d)Q

2+

(11

+2(−

4+

d)d

)s+

(−22

+8d)m

2 π+

(−9−

2(−

4+

d)d

)M2 0

) −(−

2+

d)m

∆

( 9Q

4+

8Q

2s

+s2

+8m

4 π−

2( Q

2+

2dQ

2−

3s) M

2 0−

7M

4 0+

2m

2 π

( Q2

+2dQ

2−

2s−

2M

2 0

)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

,m∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( 8ie

2gπN

∆2π

2M

0( −

4(−

13

+3d)m

5 ∆−

8(−

4+

d)m

4 ∆M

0+

4(−

2+

d)2

Q2M

0

( −m

2 π+

M2 0

) +4m

2 ∆M

0

( 22Q

2+

7s−

d( 1

4Q

2+

(2+

d)s

) +2(−

5+

2d)m

2 π+

(−5

+d(2

+d))

M2 0

) +

2m

3 ∆

( 65Q

2+

35s−

d( 2

4Q

2+

(5+

2d)s

) +2(−

17

+5d)m

2 π+

(−21

+d(3

+2d))

M2 0

) +

(−2

+d)m

∆

( −7

( Q2

+s)(

5Q

2+

s) +( (3

8−

4d)Q

2+

8s) m

2 π−

8m

4 π+

2( 2

(7+

d)Q

2+

5s) M

2 0−

3M

4 0

))(5

+τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

+q,m

∆},

1}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

2 ∆+

(−3

+d)m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[4

+d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( 16ie

2gπN

∆2π

2M

0

( −2(−

5+

d)d

m4 ∆

M0−

(−2

+d)Q

2m

∆

( Q2

+s−

M2 0

) +2(−

5+

d)m

3 ∆

( −Q

2−

s+

M2 0

) +

(−2

+d)2

Q2M

0

( −m

2 π+

M2 0

) +m

2 ∆M

0

( −(−

6+

d(2

+d))

Q2−

2(−

2+

d)(−1

+d)s

+2(−

2+

(−3

+d)d

)m2 π

+8M

2 0

))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

3}])

+1

6(−

1+

d)2

F2m

4 ∆M

0( 8

ie2gπN

∆2π

2M

2 0

( −2(2

+(−

5+

d)d

)m4 ∆−

(−2

+d)2

Q2(m

π−

M0)(m

π+

M0)+

m2 ∆

( −(−

2+

d)( d

Q2

+2(−

1+

d)s

) +2(−

2+

(−3

+d)d

)m2 π

+8M

2 0

))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}])

+1

6(−

1+

d)2

F2m

4 ∆M

0( 8

ie2gπN

∆2π

2M

2 0

( −2(2

+(−

5+

d)d

)m4 ∆−

(−2

+d)2

Q2(m

π−

M0)(m

π+

M0)+

m2 ∆

( −(−

2+

d)( d

Q2

+2(−

1+

d)s

) +2(−

2+

(−3

+d)d

)m2 π

+8M

2 0

))

117

B. Appendix to VVCS

(5+

τ[3

])II

[4+

d][{{

0,m

π},

2},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

2 ∆+

(−3

+d)m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[4

+d][{{

0,m

π},

2},{{

p−

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

2 ∆+

(−3

+d)m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0)) (5

+τ[3

])

II[4

+d][{{

0,m

π},

2},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

4 ∆M

0

( 8ie

2gπN

∆2π

2M

0

( −2(−

2+

(−5

+d)d

)m4 ∆

M0−

2(−

2+

d)Q

2m

∆

( Q2

+s−

M2 0

) +4(−

5+

d)m

3 ∆

( −Q

2−

s+

M2 0

) +

(−2

+d)2

Q2M

0

( −m

2 π+

M2 0

) +m

2 ∆M

0

( −(−

12

+d(6

+d))

Q2−

2(−

2+

d)(−1

+d)s

+2(−

2+

(−3

+d)d

)m2 π

+8M

2 0

))(5

+τ[3

])II

[4+

d][{{

0,m

π},

2},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 16ie

2gπN

∆2π

2M

0

( −2(−

3+

2d)Q

2m

∆M

0+

4m

3 ∆M

0−

(−2

+d)Q

2( Q

2+

s−

M2 0

) −2(−

5+

d)m

2 ∆

( Q2

+s−

M2 0

))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

3},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])−

16(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0(−

mπ

+m

∆+

M0)(m

π+

m∆

+M

0)( (−

2+

d)m

2 ∆−

2m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])−

16(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

4 ∆−

(−5

+d)(−2

+d)m

3 ∆M

0+

(−2

+d)( m

2 π−

M2 0

) 2+

2(−

3+

d)m

∆M

0

( −m

2 π+

M2 0

) −2m

2 ∆

( (−2

+d)m

2 π−

(−4

+d)M

2 0

))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])−

16(−

1+

d)2

F2m

4 ∆M

0

( 16ie

2gπN

∆2π

2M

0

( −2(−

3+

d)(−2

+d)m

6 ∆M

0+

(−2

+d)2

Q2M

0

( m2 π−

M2 0

) 2+

4(−

5+

d)m

5 ∆

( −Q

2−

s+

M2 0

) +

2(−

2+

d)Q

2m

∆

( −Q

2−

s+

M2 0

)(−m

2 π+

M2 0

) +2m

3 ∆

( −Q

2−

s+

M2 0

)((−

2+

d)Q

2−

2(−

5+

d)(m

π−

M0)(m

π+

M0)) −

2m

2 ∆M

0

( −m

2 π+

M2 0

)(−(−2

+d)2

Q2−

(−2

+(−

3+

d)d

)(m

π−

M0)(m

π+

M0)) +

m4 ∆

M0

( 4( 5

Q2

+4s) +

4(2

+(−

4+

d)d

)m2 π−

8d

( 2Q

2+

s+

M2 0

) +d2

( Q2

+4M

2 0

)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

3}])−

16(−

1+

d)2

F2m

4 ∆M

0( 8

ie2gπN

∆2π

2M

2 0

( −2(−

3+

d)(−2

+d)m

6 ∆+

(−2

+d)2

Q2

( m2 π−

M2 0

) 2+

m4 ∆

( (4+

(−8

+d)d

)Q2−

4ds

+4(2

+(−

4+

d)d

)m2 π

+4(4

+(−

3+

d)d

)M2 0

) −2m

2 ∆(m

π−

M0)(m

π+

M0)( (−

2+

d)2

Q2

+(−

2+

(−3

+d)d

)(m

π−

M0)(m

π+

M0)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}])−

16(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( (−2

+d)m

4 ∆−

(−5

+d)(−2

+d)m

3 ∆M

0+

(−2

+d)( m

2 π−

M2 0

) 2+

2(−

3+

d)m

∆M

0

( −m

2 π+

M2 0

) −2m

2 ∆

( (−2

+d)m

2 π−

(−4

+d)M

2 0

))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

1},{{−p

+q,m

∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0

( 8ie

2gπN

∆2π

2M

0

( −2(−

3+

d)(−2

+d)m

6 ∆M

0+

(−2

+d)2

Q2M

0

( m2 π−

M2 0

) 2+

8(−

5+

d)m

5 ∆

( −Q

2−

s+

M2 0

) +

4(−

2+

d)Q

2m

∆

( −Q

2−

s+

M2 0

)(−m

2 π+

M2 0

) +m

4 ∆M

0

( (36

+(−

24

+d)d

)Q2−

4(−

8+

3d)s

+4(2

+(−

4+

d)d

)m2 π

+4(−

4+

(−1

+d)d

)M2 0

) +

4m

3 ∆

( −Q

2−

s+

M2 0

)((−

2+

d)Q

2−

2(−

5+

d)(m

π−

M0)(m

π+

M0)) −

2m

2 ∆M

0

( −m

2 π+

M2 0

)(−(−2

+d)2

Q2−

(−2

+(−

3+

d)d

)(m

π−

M0)(m

π+

M0)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

1},{{−p

+q,m

∆},

2}])−

16(−

1+

d)2

F2m

4 ∆M

0( 8

ie2gπN

∆2π

2M

2 0

( −2(−

3+

d)(−2

+d)m

6 ∆+

(−2

+d)2

Q2

( m2 π−

M2 0

) 2+

m4 ∆

( (4+

(−8

+d)d

)Q2−

4ds

+4(2

+(−

4+

d)d

)m2 π

+4(4

+(−

3+

d)d

)M2 0

) −2m

2 ∆(m

π−

M0)(m

π+

M0)( (−

2+

d)2

Q2

+(−

2+

(−3

+d)d

)(m

π−

M0)(m

π+

M0)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

1},{{−p

,m∆},

2}])−

16(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0(−

mπ

+m

∆+

M0)(m

π+

m∆

+M

0)( (−

2+

d)m

2 ∆−

2m

∆M

0−

(−2

+d)(m

π−

M0)(m

π+

M0))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 32ie

2gπN

∆2π

2M

0

( −2(−

5+

d)m

4 ∆

( −Q

2−

s+

M2 0

) +(−

2+

d)Q

2(m

π−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) −

118


2m

3 ∆M

0

( 4( Q

2+

s) −d

( 2Q

2+

s) +(−

4+

d)M

2 0

) +m

2 ∆

( −Q

2−

s+

M2 0

)(−(−2

+d)Q

2+

2(−

5+

d)(m

π−

M0)(m

π+

M0)))

(5+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

,m∆},

1},{{−p

+q,m

∆},

1}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3M

0

( −3

( Q2

+s) +

7M

2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

3}]) −

12(−

1+

d)2

F2m

4 ∆M

0

( 32ie

2gπN

∆2π

3M

0

( 2(−

2+

d)2

M0

( −3Q

2−

s+

M2 0

)(−m

2 π+

M2 0

) −2m

3 ∆

( (32−

13d)Q

2+

(20−

7d)s

+3(−

16

+5d)M

2 0

) −2m

2 ∆M

0

( (34

+(−

22

+d)d

)Q2

+14s−

d(6

+d)s

+(−

22

+d(2

+5d))

M2 0

) +(−

2+

d)m

∆( 9

( Q2

+s)(

3Q

2+

s) −2

( 28Q

2+

11s) M

2 0+

5M

4 0+

2m

2 π

( −11Q

2−

5s

+9M

2 0

))) (5

+τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

4}]) +

16(−

1+

d)2

F2m

3 ∆M

0

( 128i(−2

+d)e

2gπN

∆2π

3( Q

2+

s) M0(5

+τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

2}]) +

16(−

1+

d)2

F2m

4 ∆M

0( 3

2i(−2

+d)e

2gπN

∆2π

3M

0

( 2(−

2+

d)M

0

( −Q

2−

s+

M2 0

)(−m

2 π+

M2 0

) +2m

2 ∆M

0

( d( Q

2+

s) +(−

4+

3d)M

2 0

) +m

∆

( (Q

2+

s) 2−

M4 0

))

(5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

3}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3M

0

( −3

( Q2

+s) +

5M

2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

2}]) −

16(−

1+

d)2

F2m

4 ∆M

0

( 64ie

2gπN

∆2π

3M

0

( (−2

+d)2

M0

( −5Q

2−

s+

M2 0

)(−m

2 π+

M2 0

) −2m

3 ∆

( −9

( (−5

+2d)Q

2+

(−3

+d)s

) +(−

55

+17d)M

2 0

) −m

2 ∆M

0

( (84

+(−

52

+d)d

)Q2

+(3

2−

3d(4

+d))

s+

(−56

+d(1

6+

7d))

M2 0

) +

(−2

+d)m

∆

((Q

2+

s)(37Q

2+

10s) −

( 60Q

2+

19s) M

2 0+

M4 0

+2m

2 π

( −15Q

2−

6s

+10M

2 0

)))

(5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

3}])

+1

6(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3M

0

( 7Q

2−

3s

+3M

2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

,m∆},

1}]) +

16(−

1+

d)2

F2m

4 ∆M

0( 3

2ie

2gπN

∆2π

3M

0

( 4(−

2+

d)2

Q2(m

π−

M0)M

0(m

π+

M0)−

2(−

2+

d)m

3 ∆

( 5( Q

2+

s) +3M

2 0

) +4m

2 ∆M

0

( −(−

2+

d)( Q

2+

s+

ds) +

(8+

(−5

+d)d

)M2 0

) +(−

2+

d)

m∆

( 3( Q

2+

s)(3Q

2+

s) −2

( 3Q

2+

2s) M

2 0−

7M

4 0+

2m

2 π

( 3( Q

2+

s) +M

2 0

))) (5

+τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

,m∆},

2}]) +

12(−

1+

d)2

F2m

3 ∆M

0

( 64i(−2

+d)e

2gπN

∆2π

3M

0

( −Q

2−

s+

M2 0

) (5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

+q,m

∆},

1}]) +

16(−

1+

d)2

F2m

4 ∆M

0

( 32ie

2gπN

∆2π

3M

0

( 4(−

2+

d)2

Q2M

0

( −m

2 π+

M2 0

) +2m

3 ∆

( (84−

33d)Q

2+

3(1

6−

5d)s

+(−

76

+23d)M

2 0

) +4m

2 ∆M

0( −

7(−

5+

3d)Q

2+

13s−

d(5

+d)s

+(−

21

+d(9

+d))

M2 0

) +(−

2+

d)m

∆

( −( Q

2+

s)(67Q

2+

13s) +

2( 3

9Q

2+

8s) M

2 0+

5M

4 0+

m2 π

( 18

( 3Q

2+

s) −26M

2 0

)))

(5+

τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

+q,m

∆},

2}])−

12(−

1+

d)2

F2m

3 ∆M

0( 6

4ie

2gπN

∆2π

3M

0

( −2(−

2+

d)m

2 π

( 2Q

2−

s+

M2 0

) +2(−

2+

d)m

2 ∆

( 3Q

2−

2s

+2M

2 0

) +2m

∆M

0

( −3Q

2+

2dQ

2+

5s−

2ds

+(−

5+

2d)M

2 0

) +

(−2

+d)( −

6Q

4−

Q2s

+s2

+( Q

2−

4s) M

2 0+

3M

4 0

))(5

+τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

4},{{−p

,m∆},

1}]) +

12(−

1+

d)2

F2m

3 ∆M

0( 6

4ie

2gπN

∆2π

3M

0

( 2(−

2+

d)m

2 π

( 4Q

2+

s−

M2 0

) +4m

∆M

0

( 5Q

2+

2s−

d( 3

Q2

+s) +

(−2

+d)M

2 0

) +2m

2 ∆

( 13Q

2−

5dQ

2+

7s−

2ds

+(−

7+

2d)M

2 0

) +

(−2

+d)( −

( Q2

+s)(

10Q

2+

s) +7Q

2M

2 0+

M4 0

))(5

+τ[3

])II

[6+

d][{{

0,m

π},

1},{{

q,m

π},

4},{{−p

+q,m

∆},

1}]) −

12(−

1+

d)2

F2m

3 ∆M

0

( 1024i(−2

+d)e

2gπN

∆2π

4M

0

( −3Q

2−

s+

M2 0

)(−Q

2−

s+

3M

2 0

) (5+

τ[3

])II

[8+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p,m

∆},

5}]) −

12(−

1+

d)2

F2m

3 ∆M

0

( 256i(−2

+d)e

2gπN

∆2π

4M

0

((Q

2+

s)(11Q

2+

3s) −

10

( 3Q

2+

s) M2 0

+7M

4 0

) (5+

τ[3

])II

[8+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

+q,m

∆},

4}]) −

16(−

1+

d)2

F2m

3 ∆M

0

( 512i(−2

+d)e

2gπN

∆2π

4M

0

( −( Q

2+

s) 2+

M4 0

) (5+

τ[3

])II

[8+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

,m∆},

3}])

−1

6(−

1+

d)2

F2m

3 ∆M

0

( 512i(−2

+d)e

2gπN

∆2π

4M

0

( 3( Q

2+

s)(5Q

2+

s) −8

( 4Q

2+

s) M2 0

+5M

4 0

) (5+

τ[3

])II

[8+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

+q,m

∆},

3}]) −

119

B. Appendix to VVCS

12(−

1+

d)2

F2m

3 ∆M

0

( 256i(−2

+d)e

2gπN

∆2π

4M

0

( −( 3

Q2−

s)(Q

2+

s) −2

( Q2

+s) M

2 0+

M4 0

) (5+

τ[3

])

II[8

+d][{{

0,m

π},

1},{{

q,m

π},

4},{{−p

,m∆},

2}])−

12(−

1+

d)2

F2m

3 ∆M

0( 2

56i(−2

+d)e

2gπN

∆2π

4M

0

((Q

2+

s)(9Q

2+

s) −2

( 7Q

2+

s) M2 0

+M

4 0

) (5+

τ[3

])II

[8+

d][{{

0,m

π},

1},{{

q,m

π},

4},{{−p

+q,m

∆},

2}]) +

12(−

1+

d)2

F2m

3 ∆M

0

( 2048i(−2

+d)e

2gπN

∆2π

4Q

2M

0

( Q2−

s+

M2 0

) (5+

τ[3

])II

[8+

d][{{

0,m

π},

1},{{

q,m

π},

5},{{−p

,m∆},

1}]) −

12(−

1+

d)2

F2m

3 ∆M

0( 2

048i(−2

+d)e

2gπN

∆2π

4Q

2M

0

( Q2

+s−

M2 0

) (5+

τ[3

])II

[8+

d][{{

0,m

π},

1},{{

q,m

π},

5},{{−p

+q,m

∆},

1}])

Dia

gra

mg)

A(ν

,Q2)

=−

( ie2gπN

∆2

( 2d8s3

M4 0

( 18m

2 πm

6 ∆M

2 0+

4m

8 π

( Q2

+s

+2m

∆M

0+

M2 0

) −8m

6 π

( m2 ∆

+M

2 0

)(Q

2+

s+

2m

∆M

0+

M2 0

) +

3m

6 ∆

( m4 ∆

+4m

2 ∆M

2 0−

5M

4 0

) +m

4 π

( −3m

6 ∆+

8m

5 ∆M

0−

16m

3 ∆M

3 0+

8m

∆M

5 0+

4m

4 ∆

( Q2

+s

+M

2 0

) −8m

2 ∆M

2 0

( Q2

+s

+M

2 0

) +4M

4 0

( Q2

+s

+M

2 0

))) +

64sm

2 π(m

π−

m∆−

M0)(m

π+

m∆−

M0)M

2 0(m

π−

m∆

+M

0)(m

π+

m∆

+M

0)( −1

2sm

3 ∆M

0

( −s

( Q2

+s) +

( Q2

+6s) M

2 0+

M4 0

) +4s2

m∆

M0

( −( Q

2+

s) 2−

2sM

2 0+

15M

4 0

) +

2m

2 π

( 2s2

( Q2

+s) 2

−6s2

( Q2

+s) m

∆M

0+

( −Q

6+

Q4s

+33Q

2s2

+35s3

) M2 0

+2

( Q2−

16s) sm

∆M

3 0−

( Q4−

39s2

) M4 0

+2sm

∆M

5 0+

( Q2−

5s) M

6 0+

M8 0

)−

2m

2 ∆

( 2s2

( Q2

+s) 2

−( Q

6+

4Q

4s

+33Q

2s2

+26s3

) M2 0−

( Q4−

4Q

2s

+s2

) M4 0

+( Q

2+

4s) M

6 0+

M8 0

)+

M2 0

( s4−

M2 0

( Q2

+M

2 0

) 3+

s3( 1

1Q

2+

10M

2 0

) +s

( Q2

+M

2 0

)(3Q

4−

5Q

2M

2 0+

6M

4 0

) +s2

( 13Q

4+

15Q

2M

2 0+

56M

4 0

)))

+

d7s2

( −2sm

10

π

( (Q

2+

s) 2−

4( Q

2+

s) M2 0

+3M

4 0

)+

2sm

8 π

( 5( Q

2+

s) 2m

2 ∆+

2( Q

2+

s) 2m

∆M

0+

( Q2

+s)(

7Q

2+

5s−

16m

2 ∆

) M2 0−

8( Q

2+

s) m∆

M3 0+

( −66Q

2−

64s

+11m

2 ∆

) M4 0−

138m

∆M

5 0−

69M

6 0

) +m

6 π

( −20s

( Q2

+s) 2

m4 ∆−

16s

( Q2

+s) 2

m3 ∆

M0−

8s

( Q2

+s) m

2 ∆

( 5Q

2+

3s−

6m

2 ∆

) M2 0−

8s

( Q2

+s) m

∆

( 3Q

2+

2s−

6m

2 ∆

) M3 0

+( −

4s

( 11Q

4+

14Q

2s

+5s2

) +16s

( 16Q

2+

15s) m

2 ∆−

9( Q

2+

7s) m

4 ∆

) M4 0

+

8sm

∆

( 3Q

2+

2s

+68m

2 ∆

) M5 0

+( 8

s( 3

1Q

2+

29s) +

296sm

2 ∆−

9m

4 ∆

) M6 0

+592sm

∆M

7 0+

316sM

8 0

) −(m

∆+

M0)( 2

sm∆

M8 0

( −7Q

4−

4Q

2s−

s2+

2Q

2M

2 0+

M4 0

) +2sM

9 0

( −7Q

4−

4Q

2s−

s2+

2Q

2M

2 0+

M4 0

) −4sm

3 ∆M

6 0

( −3Q

4−

9Q

2s−

2s2

+5Q

2M

2 0+

2M

4 0

) −4sm

2 ∆M

7 0

( −3Q

4−

9Q

2s−

2s2

+5Q

2M

2 0+

2M

4 0

) +m

5 ∆M

4 0

( −4s2

( Q2

+3s) −

20Q

2sM

2 0+

( 9Q

2−

17s) M

4 0+

9M

6 0

) +2m

9 ∆( −s

( Q2

+s) 2

+9

( Q2

+4s) M

4 0+

9M

6 0

)−

m4 ∆

M5 0

( 4s2

( Q2

+3s) +

20Q

2sM

2 0+

( 9Q

2+

23s) M

4 0+

9M

6 0

) −2m

8 ∆M

0

( s( Q

2+

s) 2+

( 9Q

2+

66s) M

4 0+

9M

6 0

)−

m7 ∆

M2 0

( −4s

( Q2

+s)(

Q2

+2s) −

4Q

2sM

2 0+

( 27Q

2−

287s) M

4 0+

27M

6 0

) +m

6 ∆M

3 0

( 4s

( Q2

+s)(

Q2

+2s) +

4Q

2sM

2 0+

( 27Q

2−

175s) M

4 0+

27M

6 0

))+

m4 π

( −8sm

5 ∆M

0

( −3( Q

2+

s) 2+

6( Q

2+

s) M2 0

+25M

4 0

)−

8sm

∆M

5 0

( −8Q

4−

9Q

2s−

3s2

+( Q

2−

2s) M

2 0+

45M

4 0

) −4sM

6 0

( −17Q

4−

16Q

2s−

5s2

+( 3

4Q

2+

28s) M

2 0+

49M

4 0

) +4sm

6 ∆

( 5( Q

2+

s) 2−

8( Q

2+

s) M2 0

+51M

4 0

)+

8sm

3 ∆M

3 0

((Q

2+

s)(5Q

2+

2s) +

( Q2

+4s) M

2 0+

62M

4 0

) +

4sm

2 ∆M

4 0

( 13Q

4+

10Q

2s

+3s2

+( 4

8Q

2+

52s) M

2 0+

65M

4 0

) +m

4 ∆M

2 0

( 12s

( Q2

+s)(

3Q

2+

s) −24s

( 5Q

2+

4s) M

2 0+

( 27Q

2−

67s) M

4 0+

27M

6 0

))+

m2 π

( −8sm

2 ∆M

6 0

( 3Q

4−

4Q

2s−

s2+

2Q

2M

2 0+

5M

4 0

) +8sm

∆M

7 0

( −9Q

4−

7Q

2s−

2s2

+( Q

2−

2s) M

2 0+

6M

4 0

) −16sm

3 ∆M

5 0( −

s( 3

Q2

+s) −

( 2Q

2+

3s) M

2 0+

6M

4 0

) −16sm

7 ∆M

0

( (Q

2+

s) 2−

( Q2

+s) M

2 0+

8M

4 0

)−

8sm

5 ∆M

3 0

((Q

2−

2s)(

Q2

+s) +

3( Q

2+

2s) M

2 0+

10M

4 0

) +

2sM

8 0

( −25Q

4−

18Q

2s−

5s2

+8Q

2M

2 0+

13M

4 0

) +4m

6 ∆M

2 0

( −2

( Q2−

s) s( Q

2+

s) −4s2

M2 0

+( 9

Q2−

95s) M

4 0+

9M

6 0

) +

m8 ∆

( −10s

( Q2

+s) 2

+8s

( Q2

+s) M

2 0+

( 27Q

2−

85s) M

4 0+

27M

6 0

)−

m4 ∆

M4 0

( −4s

( −Q

4+

6Q

2s

+s2

) +8

( Q2−

s) sM2 0

+( 2

7Q

2+

101s) M

4 0+

27M

6 0

)))−

120


8d2

( 8sm

11

∆M

0

( s2( Q

2+

s) −4s2

M2 0

+( Q

2+

6s) M

4 0+

M6 0

) −2s2

M8 0

( s( Q

2+

s)(3Q

4+

2s2

) +( −

6Q

6+

19Q

4s

+7Q

2s2

+3s3

) M2 0

+( −

2Q

4+

34Q

2s

+17s2

)

M4 0

+( 5

Q2−

23s) M

6 0+

M8 0

) −4sm

9 ∆M

0

( 4s2

( Q2

+s) 2

+2s2

( 26Q

2+

27s) M

2 0−

( 3Q

4+

3Q

2s

+38s2

) M4 0

+( −

2Q

2+

33s) M

6 0+

M8 0

)+

4sm

5 ∆M

5 0

( s( 5

Q6

+9Q

4s−

17Q

2s2−

35s3

) −( 2

Q6

+12Q

4s

+59Q

2s2−

15s3

) M2 0−

( Q4

+10Q

2s

+42s2

) M4 0

+( 4

Q2−

s) M6 0

+3M

8 0

) −4sm

7 ∆M

3 0

( −4s2

( Q2

+s)(

Q2

+5s) −

( Q6

+8Q

4s

+111Q

2s2

+164s3

) M2 0

+4

( Q4

+28s2

) M4 0

+( 9

Q2−

26s) M

6 0+

4M

8 0

) +

4m

10

π

( 3s3

( Q2

+s) 2

−2s3

( Q2

+s) m

∆M

0−

s3( 2

3Q

2+

19s) M

2 0−

( Q6

+Q

4s

+3Q

2s2−

13s3

) M4 0−

( Q2−

3s)(

Q2

+2s) M

6 0+

( Q2−

4s) M

8 0+

M10

0

)+

2sm

2 ∆M

6 0

( 2s2

( Q2

+s)(

3Q

4+

2s2

) −s

( 5Q

6+

41Q

4s−

12Q

2s2

+3s3

) M2 0+

( 3Q

6+

23Q

4s−

44Q

2s2

+21s3

) M4 0

+( 7

Q4

+34Q

2s−

11s2

) M6 0

+5

( Q2−

2s) M

8 0+

M10

0

) −2m

10

∆

( 6s3

( Q2

+s) 2

+2s3

( 23Q

2+

27s) M

2 0−

( 2Q

6+

5Q

4s

+17Q

2s2

+143s3

) M4 0

+( −

2Q

4+

2Q

2s−

17s2

) M6 0

+( 2

Q2−

5s) M

8 0+

2M

10

0

)−

2m

6 ∆M

4 0

( s2( −

7Q

6+

17Q

4s

+56Q

2s2

+49s3

) −s

( 9Q

6+

36Q

4s

+71Q

2s2

+25s3

) M2 0−

2( Q

6+

7Q

4s

+22Q

2s2−

104s3

) M4 0+

( −2Q

4+

25Q

2s−

49s2

) M6 0

+2

( Q2

+9s) M

8 0+

2M

10

0

) +m

8 ∆M

2 0

( 2s3

( Q2

+s)(

8Q

2+

29s) +

s( −

6Q

6−

9Q

4s

+67Q

2s2

+166s3

) M2 0−

( 8Q

6+

22Q

4s

+71Q

2s2−

96s3

) M4 0−

4( 2

Q4−

9Q

2s

+28s2

) M6 0

+4

( 2Q

2+

s) M8 0

+8M

10

0

) +sm

4 ∆M

4 0

( −2s2

( Q2

+s)(

3Q

4+

2s2

) +

2s

( −8Q

6+

75Q

4s

+122Q

2s2

+32s3

) M2 0−

( 18Q

6+

113Q

4s

+153Q

2s2−

84s3

) M4 0−

( 30Q

4+

109Q

2s

+190s2

) M6 0

+2

( 4Q

2+

s) M8 0

+20M

10

0

) +

2m

8 π

( −30s3

( Q2

+s) 2

m2 ∆

+4s3

( Q2

+s) m

∆

( −2

( Q2

+s) +

5m

2 ∆

) M0

+s3

( −52Q

4−

83Q

2s−

31s2

+2

( 69Q

2+

49s) m

2 ∆

) M2 0+

2s3

m∆

( 31Q

2+

29s−

8m

2 ∆

) M3 0

+( s

( 11Q

6+

27Q

4s

+304Q

2s2

+221s3

) +( 1

0Q

6+

13Q

4s

+33Q

2s2−

38s3

) m2 ∆

) M4 0

+

2sm

∆

( Q4

+2Q

2s

+214s2

+2

( Q2−

2s) m

2 ∆

) M5 0

+( 4

Q6

+9Q

4s

+11Q

2s2

+173s3

+( 1

0Q

4−

10Q

2s−

59s2

) m2 ∆

) M6 0

+

4sm

∆

( Q2

+s

+m

2 ∆

) M7 0

+( 4

Q4−

10Q

2( 2

s+

m2 ∆

) +s

( 3s

+37m

2 ∆

))M

8 0+

2sm

∆M

9 0−

2( 2

Q2−

3s

+5m

2 ∆

) M10

0−

4M

12

0

) +

4s2

m∆

M9 0

( −7s3−

Q2

( −5Q

2+

M2 0

)(Q

2+

M2 0

) −3s2

( 7Q

2+

5M

2 0

) +s

( −23Q

4−

25Q

2M

2 0+

20M

4 0

))−

4sm

3 ∆M

7 0

( −26s4−

Q2M

2 0

( Q2

+M

2 0

) 2−

6s3

( 13Q

2+

4M

2 0

) +s2

( −14Q

4+

33Q

2M

2 0+

32M

4 0

) +2s

( 5Q

6−

4Q

2M

4 0+

2M

6 0

))+

2m

6 π

( −8sm

5 ∆M

0

( 5s2

( Q2

+s) −

8s2

M2 0

+2Q

2M

4 0+

2M

6 0

) −2sm

∆M

3 0

( −2s2

( Q2

+s)(

17Q

2+

9s) +

( Q2−

2s)(

Q4−

4Q

2s−

15s2

) M2 0

+2

( 2Q

4+

Q2s

+262s2

) M4 0

+( 5

Q2

+6s) M

6 0+

2M

8 0

) −2sm

3 ∆M

0

( −16s2

( Q2

+s) 2

+s2

( 41Q

2+

33s) M

2 0+

( 6Q

4+

21Q

2s

+536s2

) M4 0

+( 1

6Q

2+

s) M6 0

+10M

8 0

)+

2M

4 0

( s2( 6

Q6

+93Q

4s

+116Q

2s2

+43s3

) −s

( 13Q

6+

7Q

4s

+217Q

2s2

+179s3

) M2 0−

( Q6

+11Q

4s−

7Q

2s2

+260s3

) M4 0−

( Q4−

13Q

2s

+18s2

) M6 0

+( Q

2+

5s) M

8 0+

M10

0

) +

4m

4 ∆

( 15s3

( Q2

+s) 2

−s3

( 23Q

2+

3s) M

2 0−

( 5Q

6+

8Q

4s

+11Q

2s2

+12s3

) M4 0

+( −

5Q

4+

5Q

2s

+31s2

) M6 0

+( 5

Q2−

17s) M

8 0+

5M

10

0

)+

m2 ∆

M2 0

( 4s3

( Q2

+s)(

37Q

2+

16s) +

s( −

30Q

6−

67Q

4s−

103Q

2s2

+34s3

) M2 0−

( 8Q

6+

28Q

4s

+35Q

2s2

+734s3

) M4 0+

( −8Q

4+

42Q

2s−

32s2

) M6 0

+8

( Q2−

s) M8 0

+8M

10

0

))+

m4 π

( 16sm

7 ∆M

0

( 5s2

( Q2

+s) −

12s2

M2 0

+3

( Q2

+2s) M

4 0+

3M

6 0

) +

sm4 ∆

M2 0

( −12s2

( Q2

+s)(

22Q

2+

s) +( 4

8Q

6+

97Q

4s−

1249Q

2s2−

1180s3

) M2 0

+( 3

6Q

4−

77Q

2s

+1298s2

) M4 0

+( −

12Q

2+

78s) M

6 0

) +

4sm

∆M

5 0

( −s

( −5Q

6+

79Q

4s

+101Q

2s2

+31s3

) +2

( Q6−

4Q

4s−

62Q

2s2−

17s3

) M2 0

+( 5

Q4−

3Q

2s

+426s2

) M4 0

+( 4

Q2

+6s) M

6 0+

M8 0

) +

4sm

3 ∆M

3 0

( −16s2

( Q2

+s)(

4Q

2+

s) +( 3

Q6−

4Q

4s−

271Q

2s2−

144s3

) M2 0

+4

( 3Q

4+

8Q

2s

+3s2

) M4 0

+( 1

7Q

2+

2s) M

6 0+

8M

8 0

) +

4sm

5 ∆M

0

( −24s2

( Q2

+s) 2

−3s2

( 21Q

2+

25s) M

2 0+

3( 4

Q4

+13Q

2s

+156s2

) M4 0

+( 2

8Q

2+

27s) M

6 0+

16M

8 0

)−

2m

2 ∆M

4 0

( s2( 1

7Q

6+

229Q

4s

+216Q

2s2

+77s3

) +s

( −5Q

6+

81Q

4s

+1204Q

2s2

+805s3

) M2 0−

( 6Q

6+

34Q

4s

+25Q

2s2

+199s3

) M4 0

+( −

6Q

4−

43Q

2s

+65s2

) M6 0

+( 6

Q2−

50s) M

8 0+

6M

10

0

) −2sM

4 0

( s2( Q

2+

s)(3Q

4+

2s2

) +

s( 1

8Q

6+

259Q

4s

+270Q

2s2

+106s3

) M2 0−

( 19Q

6−

51Q

4s

+106Q

2s2

+188s3

) M4 0

+( −

25Q

4+

42Q

2s−

440s2

) M6 0−

14sM

8 0+

6M

10

0

) −2m

6 ∆

( 60s3

( Q2

+s) 2

+4s3

( 23Q

2+

43s) M

2 0−

( 20Q

6+

38Q

4s

+30Q

2s2−

103s3

) M4 0

+( −

20Q

4+

20Q

2s

+102s2

) M6 0

+( 2

0Q

2−

62s) M

8 0+

20M

10

0

))+

121

B. Appendix to VVCS

2m

2 π

( −4sm

9 ∆M

0

( 5s2

( Q2

+s) −

16s2

M2 0

+4

( Q2

+4s) M

4 0+

4M

6 0

) −2sm

5 ∆M

3 0

( −2

( 13Q

2−

11s) s2

( Q2

+s) +

( 3Q

6+

10Q

4s−

153Q

2s2−

10s3

) M2 0

+2

( 2Q

4−

9Q

2s−

88s2

) M4 0

+( 3

Q2−

26s) M

6 0+

2M

8 0

) −2sm

3 ∆M

5 0

( 2s

( 5Q

6+

9Q

4s

+29Q

2s2

+11s3

) +( 9

Q2−

23s) s

( 4Q

2+

s) M2 0

+( 4

Q4

+31Q

2s−

12s2

) M4 0

+( 8

Q2−

7s) M

6 0+

4M

8 0

) −2sm

7 ∆M

0

( −16s2

( Q2

+s) 2

−s2

( 125Q

2+

133s) M

2 0+

( 10Q

4+

23Q

2s

+184s2

) M4 0

+( 1

6Q

2−

5s) M

6 0+

6M

8 0

)−

sm2 ∆

M4 0

( −2s2

( Q2

+s)(

3Q

4+

2s2

) +2s

( 25Q

6+

18Q

4s

+70Q

2s2

+29s3

) M2 0

+( 6

Q6

+151Q

4s−

541Q

2s2−

412s3

) M4 0+

( −2Q

4+

81Q

2s

+128s2

) M6 0

+( −

8Q

2+

10s) M

8 0

) +2sM

6 0

( s2( Q

2+

s)(3Q

4+

2s2

) +s2

( 69Q

4+

58Q

2s

+24s2

) M2 0+

2( −

Q6

+9Q

4s

+26Q

2s2

+s3

) M4 0

+( −

5Q

4+

14Q

2s−

71s2

) M6 0−

4( Q

2−

s) M8 0−

M10

0

) −2m

6 ∆M

2 0

( −14

( Q2−

2s) s3

( Q2

+s) +

s( Q

6+

2Q

4s−

195Q

2s2−

126s3

) M2 0−

( 4Q

6+

6Q

4s−

23Q

2s2

+158s3

) M4 0

+( −

4Q

4+

17Q

2s

+49s2

) M6 0

+4Q

2M

8 0+

4M

10

0

) +

2m

8 ∆

( 15s3

( Q2

+s) 2

+s3

( 69Q

2+

89s) M

2 0−

( 5Q

6+

11Q

4s

+15Q

2s2−

10s3

) M4 0

+( −

5Q

4+

5Q

2s

+4s2

) M6 0

+( 5

Q2−

14s) M

8 0+

5M

10

0

)−

2sm

∆M

7 0

( −24s4

+Q

2M

2 0

( Q2

+M

2 0

) 2−

26s3

( 3Q

2+

2M

2 0

) +s2

( −72Q

4−

113Q

2M

2 0+

136M

4 0

) +2s

( 5Q

6+

Q4M

2 0−

2Q

2M

4 0+

M6 0

))+

m4 ∆

M4 0

( 40s5

+6M

4 0

( −Q

2+

M2 0

)(Q

2+

M2 0

) 2−

8s4

( −5Q

2+

73M

2 0

) −2sM

2 0

( −2Q

2+

7M

2 0

)(3Q

4+

4Q

2M

2 0+

3M

4 0

) +

s3( 6

0Q

4−

187Q

2M

2 0+

292M

4 0

) +s2

( −2Q

6+

21Q

4M

2 0−

9Q

2M

4 0+

164M

6 0

))))

+

16d

( 4s2

m∆

M9 0

( 2Q

6−

Q4s−

3s3

+( 2

Q4−

11Q

2s−

15s2

) M2 0

+12sM

4 0

) −4s2

m3 ∆

M7 0

((Q

2+

s)(4Q

4+

5Q

2s−

8s2

) −s

( 23Q

2+

56s) M

2 0+

( −5Q

2+

79s) M

4 0+

M6 0

) −4sm

11

∆M

0

( −3s2

( Q2

+s) +

s2M

2 0+

3( Q

2+

3s) M

4 0+

3M

6 0

) +

4s2

m5 ∆

M5 0

( 2( Q

6+

7Q

4s

+5Q

2s2−

4s3

) −2

( 3Q

4+

52Q

2s

+47s2

) M2 0

+( −

11Q

2+

149s) M

4 0+

21M

6 0

) −s2

M8 0

( s( Q

2+

s)(3Q

4−

2Q

2s

+s2

) −( 5

Q6

+6Q

4s

+11Q

2s2−

4s3

) M2 0−

( Q2−

18s)(

3Q

2+

s) M4 0

+3

( Q2−

8s) M

6 0+

M8 0

) −4sm

7 ∆M

3 0

((3Q

2−

4s) s2

( Q2

+s) −

s( 4

Q4

+136Q

2s

+143s2

) M2 0

+s

( −7Q

2+

170s) M

4 0+

3( Q

2+

16s) M

6 0+

3M

8 0

) +

4sm

9 ∆M

0

( −s2

( Q2

+s) 2

−s2

( 59Q

2+

53s) M

2 0+

s( −

Q2

+49s) M

4 0+

( 6Q

2+

37s) M

6 0+

6M

8 0

)+

m6 ∆

M4 0

( −s2

( −5Q

6+

31Q

4s

+62Q

2s2

+36s3

) +s

( 4Q

6−

36Q

4s

+389Q

2s2

+166s3

) M2 0

+( 4

Q6

+26Q

4s

+239Q

2s2−

142s3

) M4 0+

4( 2

Q4−

3Q

2s−

8s2

) M6 0

+( 4

Q2−

26s) M

8 0

) +2m

10

π

( 2s3

( Q2

+s) 2

−6s3

( Q2

+s) m

∆M

0−

4s3

( 5Q

2+

4s) M

2 0−

2s3

m∆

M3 0

+( −

Q6

+Q

4s−

3Q

2s2

+9s3

) M4 0

+2

( Q2−

3s) sm

∆M

5 0−

( Q4−

9s2

) M6 0

+2sm

∆M

7 0+

( Q2−

5s) M

8 0+

M10

0

) −2m

10

∆

( 2s3

( Q2

+s) 2

+2s3

( 21Q

2+

23s) M

2 0−

( Q6

+4Q

4s

+53Q

2s2

+107s3

) M4 0−

( Q4−

4Q

2s

+15s2

) M6 0

+( Q

2+

4s) M

8 0+

M10

0

)+

sm2 ∆

M6 0

( 2s2

( Q2

+s)(

3Q

4−

2Q

2s

+s2

) −s

( 5Q

6+

87Q

4s

+64Q

2s2

+4s3

) M2 0+

( 2Q

6+

16Q

4s

+165Q

2s2

+60s3

) M4 0

+( 6

Q4

+45Q

2s−

136s2

) M6 0

+( 6

Q2−

8s) M

8 0+

2M

10

0

) +m

8 ∆M

2 0( 1

8s4

( Q2

+s) +

s( −

Q6

+29Q

4s

+21Q

2s2

+99s3

) M2 0−

( 5Q

6+

22Q

4s

+259Q

2s2

+214s3

) M4 0

+( −

7Q

4+

21Q

2s−

34s2

) M6 0

+( Q

2+

26s) M

8 0+

3M

10

0

) −m

4 ∆M

4 0

( s3( Q

2+

s)(3Q

4−

2Q

2s

+s2

) −s2

( −5Q

6+

116Q

4s

+145Q

2s2

+30s3

) M2 0

+s

( 5Q

6+

12Q

4s

+478Q

2s2

+103s3

) M4 0+

( Q6

+18Q

4s

+128Q

2s2−

142s3

) M6 0

+( 3

Q4

+7Q

2s−

45s2

) M8 0

+3

( Q2−

2s) M

10

0+

M12

0

) +

m8 π

( −4sm

3 ∆M

0

( −15s2

( Q2

+s) −

3s2

M2 0

+( 7

Q2−

15s) M

4 0+

7M

6 0

) −4sm

∆M

0

( s2( Q

2+

s) 2−

s2( 3

5Q

2+

41s) M

2 0+

2( 6

Q2−

79s) sM

4 0+

2( Q

2−

s) M6 0

+2M

8 0

) +M

2 0

( −2s3

( Q2

+s)(

32Q

2+

23s) −

s( −

19Q

6+

11Q

4s

+245Q

2s2

+325s3

) M2 0+

( 3Q

6+

10Q

4s

+29Q

2s2−

424s3

) M4 0

+( Q

4−

15Q

2s

+22s2

) M6 0

+( −

7Q

2+

10s) M

8 0−

5M

10

0

) −2m

2 ∆

( 10s3

( Q2

+s) 2

−2s3

( 19Q

2+

9s) M

2 0−

5( Q

6+

2Q

2s2

+7s3

) M4 0−

( Q2−

2s)(

5Q

2+

6s) M

6 0+

( 5Q

2−

16s) M

8 0+

5M

10

0

))+

m2 π

( −4s2

m∆

M7 0

( 4Q

6−

18Q

4s−

25Q

2s2−

15s3−

3s

( 12Q

2+

17s) M

2 0−

( Q2−

151s) M

4 0+

M6 0

) +4s2

m3 ∆

M5 0

( −2

( 2Q

6+

12Q

4s

+23Q

2s2

+7s3

) −2

( 10Q

4−

5Q

2s

+26s2

) M2 0

+3

( 5Q

2+

131s) M

4 0+

3M

6 0

) +4sm

9 ∆M

0

( −15s2

( Q2

+s) +

3s2

M2 0

+( 1

3Q

2+

21s) M

4 0+

13M

6 0

) +

122


4sm

5 ∆M

3 0

( 3s2

( Q2

+s)(

8Q

2+

s) +s

( −4Q

4+

29Q

2s

+10s2

) M2 0−

s( 6

1Q

2+

196s) M

4 0+

( 7Q

2−

32s) M

6 0+

7M

8 0

) −4sm

7 ∆M

0

( −4s2

( Q2

+s) 2

−2s2

( 71Q

2+

59s) M

2 0+

s( −

29Q

2+

41s) M

4 0+

( 8Q

2+

27s) M

6 0+

8M

8 0

)+

2sM

6 0

( s2( Q

2+

s)(3Q

4−

2Q

2s

+s2

) +s

( −12Q

6−

29Q

4s−

26Q

2s2

+13s3

) M2 0

+( 2

Q6

+20Q

4s

+35Q

2s2−

15s3

) M4 0+

( 6Q

4+

4Q

2s−

253s2

) M6 0

+6

( Q2−

2s) M

8 0+

2M

10

0

) +2m

4 ∆M

4 0

( −s2

( 12Q

6+

34Q

4s

+51Q

2s2

+11s3

) +

s( 1

9Q

6+

29Q

4s

+297Q

2s2

+190s3

) M2 0−

( 4Q

6−

7Q

4s

+117Q

2s2

+119s3

) M4 0−

( 6Q

4+

7Q

2s

+56s2

) M6 0−

7sM

8 0+

2M

10

0

) +

2m

8 ∆

( 10s3

( Q2

+s) 2

+4s3

( 37Q

2+

42s) M

2 0−

( 5Q

6+

15Q

4s

+125Q

2s2

+234s3

) M4 0

+( −

5Q

4+

16Q

2s−

25s2

) M6 0

+( 5

Q2

+11s) M

8 0+

5M

10

0

)+

m2 ∆

M4 0

( 2s3

( Q2

+s)(

3Q

4−

2Q

2s

+s2

) +2s2

( 4Q

6+

127Q

4s

+107Q

2s2

+12s3

) M2 0−

s( 3

0Q

6+

199Q

4s

+67Q

2s2

+14s3

) M4 0+

( Q2−

20s)(

2Q

4−

4Q

2s−

49s2

) M6 0

+2

( 3Q

4−

13Q

2s

+53s2

) M8 0

+6

( Q2−

2s) M

10

0+

2M

12

0

) −m

6 ∆M

2 0

( 8s5

+4M

4 0

( −3Q

2+

M2 0

)(Q

2+

M2 0

) 2+

4s4

( −14Q

2+

33M

2 0

) +5s2

M2 0

( 23Q

4−

13Q

2M

2 0+

20M

4 0

) −s3

( 64Q

4+

47Q

2M

2 0+

360M

4 0

) +16sM

2 0

( Q6−

Q4M

2 0+

Q2M

4 0+

M6 0

))) +

m6 π

( 8sm

5 ∆M

0

( −15s2

( Q2

+s) −

s2M

2 0+

9( Q

2−

s) M4 0

+9M

6 0

) +4sm

∆M

3 0

( s2( Q

2+

s)(18Q

2+

11s) +

s( 4

Q4−

47Q

2s−

52s2

) M2 0+

5( 5

Q2−

88s) sM

4 0+

( Q2

+4s) M

6 0+

M8 0

) +4sm

3 ∆M

0

( 4s2

( Q2

+s) 2

−2s2

( 23Q

2+

35s) M

2 0+

( 51Q

2−

259s) sM

4 0+

( 8Q

2−

5s) M

6 0+

8M

8 0

)+

2m

4 ∆

( 20s3

( Q2

+s) 2

+8s3

( 6Q

2+

11s) M

2 0−

( 10Q

6+

10Q

4s

+30Q

2s2

+197s3

) M4 0

+2

( −5Q

4+

8Q

2s

+s2

) M6 0

+2

( 5Q

2−

7s) M

8 0+

10M

10

0

)+

m2 ∆

M2 0

( 24s3

( Q2

+s)(

8Q

2+

5s) +

s( −

56Q

6−

35Q

4s

+611Q

2s2

+764s3

) M2 0

+( −

4Q

6−

48Q

4s

+69Q

2s2

+464s3

) M4 0+

4( Q

4+

14Q

2s−

36s2

) M6 0

+4

( 5Q

2+

4s) M

8 0+

12M

10

0

) +2M

4 0

( 49s5

+2M

6 0

( Q2

+M

2 0

) 2+

s4( 8

3Q

2+

340M

2 0

) +

7s3

( 5Q

4+

41Q

2M

2 0+

24M

4 0

) +sM

2 0

( −17Q

6−

7Q

4M

2 0+

21Q

2M

4 0+

7M

6 0

) −s2

( 7Q

6−

34Q

4M

2 0+

19Q

2M

4 0+

62M

6 0

))) +

m4 π

( −8s2

m∆

M5 0

( −Q

6+

17Q

4s

+26Q

2s2

+11s3

+( 3

Q4

+5Q

2s

+11s2

) M2 0

+( 7

Q2−

211s) M

4 0+

M6 0

) −8sm

7 ∆M

0( −

15s2

( Q2

+s) +

s2M

2 0+

( 11Q

2+

3s) M

4 0+

11M

6 0

) −4sm

5 ∆M

0

( 6s2

( Q2

+s) 2

+36s2

( 2Q

2+

s) M2 0

+( 6

7Q

2−

93s) sM

4 0+

( 4Q

2+

7s) M

6 0+

4M

8 0

)−

4sm

3 ∆M

3 0

( 3s2

( Q2

+s)(

13Q

2+

6s) +

s( 4

Q4

+118Q

2s

+101s2

) M2 0

+s

( 27Q

2+

218s) M

4 0+

5( Q

2−

4s) M

6 0+

5M

8 0

) −m

2 ∆M

4 0

( s2( −

33Q

6−

29Q

4s

+2Q

2s2

+40s3

) −s

( −8Q

6+

66Q

4s

+1031Q

2s2

+1174s3

) M2 0+

( −4Q

6+

26Q

4s

+205Q

2s2−

1838s3

) M4 0

+2s

( 8Q

2+

63s) M

6 0+

2( 6

Q2−

13s) M

8 0+

8M

10

0

) −2m

6 ∆

( 20s3

( Q2

+s) 2

+4s3

( 43Q

2+

53s) M

2 0−

5( 2

Q6

+4Q

4s

+19Q

2s2

+56s3

) M4 0

+( −

10Q

4+

24Q

2s−

11s2

) M6 0

+2

( 5Q

2+

2s) M

8 0+

10M

10

0

)−

2m

4 ∆M

2 0

( 42s5

+3M

4 0

( Q2

+M

2 0

) 3+

s4( 1

38Q

2+

203M

2 0

) +2s2

M2 0

( −33Q

4+

46Q

2M

2 0+

6M

4 0

) +sM

2 0

( Q2

+M

2 0

)(−2

7Q

4+

5Q

2M

2 0+

18M

4 0

) +

s3( 9

6Q

4+

217Q

2M

2 0+

65M

4 0

))+

M4 0

( −s6

+s5

( Q2−

78M

2 0

) −M

6 0

( Q2

+M

2 0

) 3−

sM4 0

( Q2

+M

2 0

)(−1

1Q

4+

21Q

2M

2 0+

18M

4 0

) −s4

( Q4

+23Q

2M

2 0+

275M

4 0

) +s2

M2 0

( 33Q

6−

100Q

4M

2 0+

10Q

2M

4 0+

109M

6 0

) +3s3

( −Q

6+

14Q

4M

2 0−

102Q

2M

4 0+

184M

6 0

))))

+

2d4

( 4s2

m∆

M9 0

( 19Q

6−

57Q

4s−

55Q

2s2−

25s3

+( 1

6Q

4−

101Q

2s−

75s2

) M2 0

+( −

3Q

2+

82s) M

4 0

) −8sm

11

∆M

0

( −6s2

( Q2

+s) +

11s2

M2 0

+( Q

2+

s) M4 0

+M

6 0

) −2s2

M8 0

( s( Q

2+

s)(6Q

4−

14Q

2s

+s2

) +( −

23Q

6+

31Q

4s

+14Q

2s2

+29s3

) M2 0

+( −

9Q

4+

165Q

2s

+51s2

) M4 0

+( 1

8Q

2−

85s) M

6 0+

4M

8 0

) +

4sm

5 ∆M

5 0

( s( 1

9Q

6+

51Q

4s−

35Q

2s2−

113s3

) −2

( 3Q

6+

22Q

4s

+212Q

2s2

+81s3

) M2 0

+( −

5Q

4−

67Q

2s

+280s2

) M4 0

+8

( Q2

+4s) M

6 0+

7M

8 0

) +

4sm

9 ∆M

0

( −12s2

( Q2

+s) 2

−s2

( 277Q

2+

271s) M

2 0+

( 7Q

4−

8Q

2s

+298s2

) M4 0

+6

( 3Q

2+

2s) M

6 0+

11M

8 0

)−

4sm

7 ∆M

3 0

( −2s2

( Q2

+s)(

2Q

2+

31s) −

( 3Q

6+

30Q

4s

+641Q

2s2

+810s3

) M2 0

+( 8

Q4−

44Q

2s

+886s2

) M4 0

+3

( 9Q

2+

10s) M

6 0+

16M

8 0

) +

2m

6 ∆M

4 0

( −s2

( −7Q

6+

145Q

4s

+301Q

2s2

+201s3

) +s

( −5Q

6−

24Q

4s

+551Q

2s2

+248s3

) M2 0+

( 4Q

6+

19Q

4s

+441Q

2s2−

592s3

) M4 0

+( 8

Q4−

31Q

2s−

56s2

) M6 0

+( 4

Q2−

59s) M

8 0

) +

123

B. Appendix to VVCS

4m

10

π

( 10s3

( Q2

+s) 2

−12s3

( Q2

+s) m

∆M

0−

s3( 8

5Q

2+

69s) M

2 0−

2s3

m∆

M3 0−

( Q6−

7Q

4s

+3Q

2s2−

48s3

) M4 0+

2( Q

2−

3s) sm

∆M

5 0−

( Q2−

3s)(

Q2

+6s) M

6 0+

2sm

∆M

7 0+

( Q2−

8s) M

8 0+

M10

0

) −2m

10

∆

( 20s3

( Q2

+s) 2

+32s3

( 7Q

2+

8s) M

2 0−

( 2Q

6+

6Q

4s

+99Q

2s2

+649s3

) M4 0

+( −

2Q

4+

2Q

2s−

25s2

) M6 0

+2

( Q2

+6s) M

8 0+

2M

10

0

)+

2sm

2 ∆M

6 0

( 2s2

( Q2

+s)(

6Q

4−

14Q

2s

+s2

) −s

( 39Q

6+

275Q

4s

+87Q

2s2

+s3

) M2 0

+( 1

1Q

6+

66Q

4s

+115Q

2s2

+146s3

) M4 0+

( 27Q

4+

167Q

2s−

214s2

) M6 0

+3

( 7Q

2−

6s) M

8 0+

5M

10

0

) +m

8 ∆M

2 0

( 2s3

( Q2

+s)(

24Q

2+

101s) +

2s

( 8Q

6+

54Q

4s

+175Q

2s2

+372s3

) M2 0−

( 10Q

6+

10Q

4s

+607Q

2s2

+623s3

) M4 0

+( −

14Q

4+

58Q

2s−

45s2

) M6 0

+2

( Q2

+50s) M

8 0+

6M

10

0

) +

2m

8 π

( −100s3

( Q2

+s) 2

m2 ∆−

24s3

( Q2

+s) m

∆

( Q2

+s−

5m

2 ∆

) M0

+s3

( −11

( Q2

+s)(

22Q

2+

15s) +

8( 5

7Q

2+

37s) m

2 ∆

) M2 0+

2s3

m∆

( 209Q

2+

215s−

14m

2 ∆

) M3 0

+( −

18s3

( 25s

+3m

2 ∆

) +2Q

6( 8

s+

5m

2 ∆

) −10Q

4s

( 11s

+5m

2 ∆

) +Q

2s2

( −197s

+26m

2 ∆

))M

4 0+

2sm

∆

( 3Q

4−

2Q

2( 1

8s

+5m

2 ∆

) +2s

( 566s

+11m

2 ∆

))M

5 0+

( 3Q

6−

35Q

4s

+134Q

2s2−

571s3

+2

( 5Q

4+

11Q

2s−

52s2

) m2 ∆

) M6 0

+

4sm

∆

( Q2−

4s−

5m

2 ∆

) M7 0

+( Q

4−

5Q

2( 3

s+

2m

2 ∆

) +s

( −5s

+52m

2 ∆

))M

8 0−

2sm

∆M

9 0+

( −7Q

2+

28s−

10m

2 ∆

) M10

0−

5M

12

0

) −m

4 ∆M

4 0

( 2s3

( Q2

+s)(

6Q

4−

14Q

2s

+s2

) −2s2

( 9Q

6+

447Q

4s

+581Q

2s2

+150s3

) M2 0

+2s

( 14Q

6+

105Q

4s

+716Q

2s2

+51s3

) M4 0+

( 2Q

6+

94Q

4s

+771Q

2s2

+49s3

) M6 0

+( 6

Q4

+34Q

2s−

151s2

) M8 0

+( 6

Q2−

32s) M

10

0+

2M

12

0

) +

m4 π

( −400s3

( Q2

+s) 2

m6 ∆

+96s3

( Q2

+s) m

5 ∆

( −3

( Q2

+s) +

5m

2 ∆

) M0−

4s3

m4 ∆

( 327Q

4+

423Q

2s

+96s2

+332Q

2m

2 ∆+

492sm

2 ∆

) M2 0−

8s3

m3 ∆

( 142Q

4+

197Q

2s

+55s2

+6

( 17Q

2+

14s) m

2 ∆+

62m

4 ∆

) M3 0

+2

( −s3

( Q2

+s)(

6Q

4−

14Q

2s

+s2

) −s2

( −125Q

6+

99Q

4s

+135Q

2s2

+147s3

) m2 ∆

+2s

( 36Q

6+

38Q

4s−

1021Q

2s2−

997s3

) m4 ∆

+2

( 10Q

6−

10Q

4s

+65Q

2s2

+327s3

) m6 ∆

) M4 0

+

4sm

∆

( −s

( −19Q

6+

309Q

4s

+443Q

2s2

+161s3

) +( 9

Q6−

22Q

4s−

1029Q

2s2−

682s3

) m2 ∆

+2

( 15Q

4−

82Q

2s

+627s2

) m4 ∆

+4

( −5Q

2+

3s) m

6 ∆

)

M5 0

+( −

2s2

( −141Q

6+

139Q

4s

+331Q

2s2

+304s3

) +2s

( 25Q

6+

176Q

4s

+237Q

2s2

+1116s3

) m2 ∆

+( −

12Q

6+

80Q

4s−

325Q

2s2

+3437s3

) m4 ∆

+

4( 1

0Q

4+

6Q

2s−

73s2

) m6 ∆

) M6 0

+8sm

∆

( 3( Q

6−

6Q

4s−

57Q

2s2−

26s3

) +( 1

6Q

4−

18Q

2s−

435s2

) m2 ∆

+6

( 5Q

2−

7s) m

4 ∆−

10m

6 ∆

) M7 0

+( −

2s

( −14Q

6+

361Q

4s

+808Q

2s2

+609s3

) +2

( 4Q

6+

Q4s−

469Q

2s2

+5224s3

) m2 ∆−

( 36Q

4+

44Q

2s

+473s2

) m4 ∆−

8( 5

Q2

+2s) m

6 ∆

) M8 0

+

4sm

∆

( 15Q

4−

55Q

2s

+2794s2

+( 3

1Q

2−

2s) m

2 ∆+

30m

4 ∆

) M9 0−

2( Q

6+

25Q

4s

+98Q

2s2−

2486s3−

3( 1

7Q

2−

144s) sm

2 ∆+

2( 9

Q2−

5s) m

4 ∆+

20m

6 ∆

)

M10

0−

16sm

∆

( −3Q

2+

s−

2m

2 ∆

) M11

0−

2( 3

Q4

+57Q

2s−

79s2

+3

( 4Q

2−

21s) m

2 ∆+

6m

4 ∆

) M12

0+

12sm

∆M

13

0−

2( 3

Q2

+18s

+8m

2 ∆

) M14

0−

2M

16

0

) −4sm

3 ∆M

7 0

( −88s4−

3Q

2M

2 0

( Q2

+M

2 0

) 2−

2s3

( 104Q

2+

99M

2 0

) +s2

( −14Q

4+

11Q

2M

2 0+

264M

4 0

) +2s

( 19Q

6+

Q4M

2 0−

17Q

2M

4 0+

6M

6 0

))+

m6 π

( 16sm

5 ∆M

0

( −30s2

( Q2

+s) +

19s2

M2 0

+( 5

Q2−

7s) M

4 0+

5M

6 0

) −8sm

3 ∆M

0

( −24s2

( Q2

+s) 2

+s2

( 175Q

2+

187s) M

2 0+

2( 4

Q4−

33Q

2s

+544s2

) M4 0

+6

( 2Q

2−

5s) M

6 0+

4M

8 0

)−

4sm

∆M

3 0

( −2s2

( Q2

+s)(

72Q

2+

43s) +

( 3Q

6−

26Q

4s

+169Q

2s2

+258s3

) M2 0

+2

( 6Q

4−

40Q

2s

+1525s2

) M4 0

+13

( Q2−

2s) M

6 0+

4M

8 0

) +

2M

4 0

( s2( −

59Q

6+

377Q

4s

+689Q

2s2

+345s3

) +s

( −31Q

6+

350Q

4s

+957Q

2s2

+1230s3

) M2 0

+s

( 37Q

4−

149Q

2s−

536s2

) M4 0+

( 4Q

4+

63Q

2s−

106s2

) M6 0

+( 8

Q2−

s) M8 0

+4M

10

0

) +4m

2 ∆M

2 0

( s3( Q

2+

s)(351Q

2+

197s) +

s( −

28Q

6+

77Q

4s

+749Q

2s2

+986s3

) M2 0−

( 2Q

6−

15Q

4s

+88Q

2s2

+253s3

) M4 0

+( 2

Q4

+24Q

2s−

7s2

) M6 0

+( 1

0Q

2−

11s) M

8 0+

6M

10

0

) +m

4 ∆

( 400s3

( Q2

+s) 2

−8

( 31Q

2−

49s) s3

M2 0−

( 40Q

6−

120Q

4s

+85Q

2s2

+1339s3

) M4 0

+( −

40Q

4−

56Q

2s

+339s2

) M6 0

+8

( 5Q

2−

12s) M

8 0+

40M

10

0

))+

m2 π

( 8sm

9 ∆M

0

( −30s2

( Q2

+s) +

43s2

M2 0

+( 5

Q2

+s) M

4 0+

5M

6 0

) +4sm

5 ∆M

3 0

( 2( 6

8Q

2−

19s) s2

( Q2

+s) +

( −9Q

6−

34Q

4s

+557Q

2s2

+130s3

) M2 0−

2( 6

Q4

+76Q

2s

+441s2

) M4 0

+( 9

Q2−

58s) M

6 0+

12M

8 0

) −8sm

7 ∆M

0

( −24s2

( Q2

+s) 2

−s2

( 311Q

2+

299s) M

2 0+

2( 6

Q4−

19Q

2s

+127s2

) M4 0

+2

( 14Q

2−

5s) M

6 0+

16M

8 0

) −4m

6 ∆M

2 0

( −( 8

5Q

2−

69s) s3

( Q2

+s) +

s( 2

0Q

6+

87Q

4s−

283Q

2s2−

50s3

) M2 0−

( 6Q

6−

15Q

4s

+57Q

2s2

+560s3

) M4 0

+( −

10Q

4+

20Q

2s

+58s2

) M6 0

+( −

2Q

2+

33s) M

8 0+

2M

10

0

) +2sM

6 0

( 2s2

( Q2

+s)(

6Q

4−

14Q

2s

+s2

) +

s( −

105Q

6+

15Q

4s

+23Q

2s2

+133s3

) M2 0

+( Q

6+

112Q

4s

+383Q

2s2

+18s3

) M4 0

+( 9

Q4

+137Q

2s−

1560s2

) M6 0

+15Q

2M

8 0+

7M

10

0

) +m

8 ∆

124


( 200s3

( Q2

+s) 2

+4s3

( 363Q

2+

443s) M

2 0−

( 20Q

6+

20Q

4s

+413Q

2s2

+1351s3

) M4 0

+( −

20Q

4+

4Q

2s

+39s2

) M6 0

+4

( 5Q

2+

16s) M

8 0+

20M

10

0

)+

4m

2 ∆M

4 0

( s3( Q

2+

s)(6Q

4−

14Q

2s

+s2

) +s2

( 29Q

6+

406Q

4s

+275Q

2s2

+45s3

) M2 0

+s

( −24Q

6−

256Q

4s

+485Q

2s2

+233s3

) M4 0+

( Q6−

36Q

4s−

172Q

2s2

+951s3

) M6 0

+( 3

Q2−

17s)(

Q2−

6s) M

8 0+

( 3Q

2−

23s) M

10

0+

M12

0

) −8sm

3 ∆M

5 0

( 63s4

+5M

4 0

( Q2

+M

2 0

) 2+

s3( 1

61Q

2+

81M

2 0

) −s2

( −71Q

4+

95Q

2M

2 0+

830M

4 0

) +s

( 19Q

6+

72Q

4M

2 0+

3Q

2M

4 0−

34M

6 0

))−

4sm

∆M

7 0

( −112s4

+3Q

2M

2 0

( Q2

+M

2 0

) 2−

2s3

( 146Q

2+

143M

2 0

) +s2

( −234Q

4−

415Q

2M

2 0+

956M

4 0

) +2s

( 19Q

6+

3Q

4M

2 0−

7Q

2M

4 0+

4M

6 0

))+

m4 ∆

M4 0

( 6s5

+8M

4 0

( −2Q

2+

M2 0

)(Q

2+

M2 0

) 2−

2s4

( 253Q

2+

542M

2 0

) +s3

( −266Q

4+

1638Q

2M

2 0+

391M

4 0

) −2sM

2 0

( −11Q

6+

57Q

4M

2 0+

83Q

2M

4 0+

3M

6 0

) −s2

( 146Q

6+

28Q

4M

2 0+

253Q

2M

4 0+

395M

6 0

))))

+

d5

( 16s3

m11

∆M

0

( −Q

2−

s+

2M

2 0

) −4s2

m∆

M9 0

( 19Q

6−

133Q

4s−

105Q

2s2−

29s3

+2

( 7Q

4−

40Q

2s−

30s2

) M2 0−

5( Q

2−

17s) M

4 0

) −4sm

9 ∆M

0

( −23s2

( Q2

+s) 2

−4s2

( 37Q

2+

41s) M

2 0+

( 2Q

4+

69s2

) M4 0

+4Q

2M

6 0+

2M

8 0

)+

4sm

7 ∆M

3 0

( −2s2

( Q2

+s)(

21Q

2+

58s) −

( 5Q

6+

18Q

4s

+483Q

2s2

+588s3

) M2 0−

6( Q

4+

3Q

2s−

64s2

) M4 0

+( 3

Q2

+4s) M

6 0+

4M

8 0

) +

s2M

8 0

((3Q

2−

5s)(

3Q

2−

s) s( Q

2+

s) +( −

45Q

6+

280Q

4s

+175Q

2s2

+66s3

) M2 0

+( −

11Q

4+

169Q

2s

+80s2

) M4 0

+( 4

1Q

2−

158s) M

6 0+

7M

8 0

) +

2m

10

π

( −30s3

( Q2

+s) 2

+8s3

( Q2

+s) m

∆M

0+

2s3

( 97Q

2+

83s) M

2 0+

( Q6−

5Q

4s

+3Q

2s2−

127s3

) M4 0

+( Q

2−

3s)(

Q2

+5s) M

6 0−

( Q2−

7s) M

8 0−

M10

0

)+

2m

10

∆

( 30s3

( Q2

+s) 2

+2s3

( 69Q

2+

83s) M

2 0−

( Q6

+3Q

4s

+152Q

2s2

+920s3

) M4 0−

( Q4

+2Q

2s

+118s2

) M6 0

+( Q

2+

s) M8 0

+M

10

0

)−

2sm

2 ∆M

6 0

((3Q

2−

5s)(

3Q

2−

s) s2( Q

2+

s) +s

( −36Q

6−

129Q

4s

+256Q

2s2

+57s3

) M2 0

+( 1

2Q

6+

68Q

4s−

193Q

2s2

+25s3

) M4 0+

( 26Q

4+

132Q

2s−

37s2

) M6 0

+4

( 4Q

2−

s) M8 0

+2M

10

0

) −m

8 ∆M

2 0

( 20s3

( Q2

+s)(

5Q

2+

14s) +

s( 9

Q6

+19Q

4s

+443Q

2s2

+661s3

) M2 0−

( Q6

+760Q

2s2

+1115s3

) M4 0

+( 5

Q4

+17Q

2s−

535s2

) M6 0

+13

( Q2

+2s) M

8 0+

7M

10

0

) +

m4 π

( 600s3

( Q2

+s) 2

m6 ∆

+8s3

( Q2

+s) m

5 ∆

( 69

( Q2

+s) −

20m

2 ∆

) M0

+8s3

m4 ∆

( 9( Q

2+

s)(19Q

2+

4s) +

( 13Q

2+

83s) m

2 ∆

) M2 0+

8s3

m3 ∆

( 173Q

4+

235Q

2s

+62s2−

6( 1

3Q

2+

5s) m

2 ∆+

24m

4 ∆

) M3 0

+((

3Q

2−

5s)(

3Q

2−

s) s3( Q

2+

s) +

2s2

( −63Q

6+

581Q

4s

+469Q

2s2

+197s3

) m2 ∆

+6s

( −13Q

6−

7Q

4s

+321Q

2s2

+447s3

) m4 ∆

+4

( −5Q

6+

Q4s−

36Q

2s2

+362s3

) m6 ∆

) M4 0

+

4sm

∆

( s( −

19Q

6+

452Q

4s

+595Q

2s2

+200s3

) +( −

15Q

6+

42Q

4s

+943Q

2s2

+660s3

) m2 ∆−

6( 2

Q4−

4Q

2s

+311s2

) m4 ∆

) M5 0

+( s2

( −153Q

6+

1672Q

4s

+1887Q

2s2

+838s3

) +2s

( −37Q

6−

4Q

4s

+1825Q

2s2

+1162s3

) m2 ∆−

( 18Q

6−

16Q

4s−

79Q

2s2

+5857s3

) m4 ∆−

4( 5

Q4

+10Q

2s−

54s2

) m6 ∆

) M6 0

+8sm

∆

( −5Q

6+

23Q

4s

+253Q

2s2

+130s3−

( 17Q

4−

13Q

2s−

580s2

) m2 ∆−

12

( Q2−

s) m4 ∆

) M7 0

+( −

s( 4

9Q

6−

534Q

4s

+458Q

2s2

+585s3

) −2

( 6Q

6+

59Q

4s−

235Q

2s2

+2536s3

) m2 ∆

+( −

54Q

4+

162Q

2s

+209s2

) m4 ∆

+4

( 5Q

2−

11s) m

6 ∆

) M8 0−

4sm

∆

( 22Q

4−

57Q

2s

+2978s2

+( 2

3Q

2+

4s) m

2 ∆+

12m

4 ∆

) M9 0−

( 3Q

6+

28Q

4s−

378Q

2s2

+5832s3

+2

( 12Q

4+

55Q

2s−

202s2

) m2 ∆

+2

( 27Q

2−

34s) m

4 ∆−

20m

6 ∆

) M10

0−

8sm

∆

( 7Q

2+

2m

2 ∆

) M11

0−

( 9Q

4−

63Q

2s

+65s2

+6

( 2Q

2+

11s) m

2 ∆+

18m

4 ∆

) M12

0−

8sm

∆M

13

0+

( −9Q

2+

42s) M

14

0−

3M

16

0

) +

4sm

3 ∆M

7 0

( −128s4−

5Q

2M

2 0

( Q2

+M

2 0

) 2−

4s3

( 113Q

2+

34M

2 0

) +s2

( −86Q

4+

73Q

2M

2 0+

184M

4 0

) +2s

( 19Q

6+

5Q

4M

2 0−

14Q

2M

4 0+

2M

6 0

))−

4sm

5 ∆M

5 0

( −192s4

+2M

2 0

( −5Q

2+

M2 0

)(Q

2+

M2 0

) 2+

s3( −

107Q

2+

8M

2 0

) −2s2

( −14Q

4+

269Q

2M

2 0+

45M

4 0

) +s

( 19Q

6−

22Q

4M

2 0−

41Q

2M

4 0+

8M

6 0

))+

2m

6 ∆M

4 0

( 199s5

+2M

4 0

( Q2

+M

2 0

) 2( Q

2+

2M

2 0

) +s4

( 199Q

2+

226M

2 0

) +s3

( 83Q

4+

167Q

2M

2 0+

4M

4 0

) +

sM2 0

( Q2

+M

2 0

)(−3

Q4−

14Q

2M

2 0+

21M

4 0

) −s2

( 9Q

6+

60Q

4M

2 0+

395Q

2M

4 0+

170M

6 0

))−

m4 ∆

M4 0

( −5s6

+3M

6 0

( Q2

+M

2 0

) 3+

13s5

( Q2

+10M

2 0

) +sM

4 0

( Q2

+M

2 0

)(−3

9Q

4−

53Q

2M

2 0+

14M

4 0

) +s4

( 9Q

4+

1049Q

2M

2 0+

441M

4 0

) −s2

M2 0

( −9Q

6+

286Q

4M

2 0+

557Q

2M

4 0+

26M

6 0

) −s3

( 9Q

6−

664Q

4M

2 0+

526Q

2M

4 0+

1089M

6 0

))+

125

B. Appendix to VVCS

m8 π

( 16s3

m3 ∆

M0

( −5

( Q2

+s) +

2M

2 0

) −4sm

∆M

0

( −23s2

( Q2

+s) 2

+8s2

( 25Q

2+

23s) M

2 0+

( 2Q

4−

8Q

2s

+1465s2

) M4 0

+4

( Q2−

2s) M

6 0+

2M

8 0

) +

M2 0

( 4s3

( Q2

+s)(

139Q

2+

94s) −

s( 1

7Q

6−

93Q

4s

+1595Q

2s2

+1053s3

) M2 0−

( 7Q

6−

48Q

4s

+167Q

2s2

+1050s3

) M4 0

+( −

13Q

4+

71Q

2s−

72s2

) M6 0

+( −

5Q

2+

6s) M

8 0+

M10

0

) +

2m

2 ∆

( 150s3

( Q2

+s) 2

−2s3

( 319Q

2+

249s) M

2 0+

( −5Q

6+

17Q

4s−

15Q

2s2

+359s3

) M4 0

+( −

5Q

4−

10Q

2s

+59s2

) M6 0

+( 5

Q2−

27s) M

8 0+

5M

10

0

))+

m6 π

( −32s3

m5 ∆

M0

( −5

( Q2

+s) +

4M

2 0

) +16sm

3 ∆M

0

( −23s2

( Q2

+s) 2

+s2

( 113Q

2+

97s) M

2 0+

2( Q

4−

3Q

2s

+387s2

) M4 0

+( 4

Q2−

6s) M

6 0+

2M

8 0

) +

4sm

∆M

3 0

( −2s2

( Q2

+s)(

97Q

2+

60s) +

( 5Q

6−

30Q

4s

+79Q

2s2

+168s3

) M2 0

+2

( 7Q

4−

19Q

2s

+1822s2

) M4 0

+( 1

3Q

2−

12s) M

6 0+

4M

8 0

) +

4m

2 ∆M

2 0

( −4s3

( Q2

+s)(

98Q

2+

53s) +

s( 1

5Q

6−

32Q

4s

+266Q

2s2−

61s3

) M2 0

+( 5

Q6−

20Q

4s

+100Q

2s2

+1306s3

) M4 0+

( 11Q

4−

49Q

2s

+52s2

) M6 0

+7

( Q2−

2s) M

8 0+

M10

0

) −m

4 ∆

( 600s3

( Q2

+s) 2

−8s3

( 153Q

2+

83s) M

2 0+

( −20Q

6+

36Q

4s

+47Q

2s2

+373s3

) M4 0+

( −20Q

4−

40Q

2s

+343s2

) M6 0

+4

( 5Q

2−

19s) M

8 0+

20M

10

0

) +2M

4 0

( −415s5

+2M

4 0

( Q2

+M

2 0

) 2( 2

Q2

+M

2 0

) +s4

( −1007Q

2+

624M

2 0

) −sM

2 0

( Q2

+M

2 0

)(−2

5Q

4+

44Q

2M

2 0+

29M

4 0

) +5s3

( −141Q

4+

193Q

2M

2 0+

492M

4 0

) +s2

( 27Q

6−

206Q

4M

2 0+

35Q

2M

4 0+

102M

6 0

))) +

m2 π

( −16s3

m9 ∆

M0

( −5

( Q2

+s) +

8M

2 0

) +8sm

3 ∆M

5 0

( s( 1

9Q

6+

76Q

4s

+225Q

2s2

+92s3

) +2s

( 31Q

4+

55Q

2s

+77s2

) M2 0

+( 2

Q4

+31Q

2s−

552s2

) M4 0+

4( Q

2−

3s) M

6 0+

2M

8 0

) +16sm

7 ∆M

0

( −23s2

( Q2

+s) 2

−s2

( 61Q

2+

77s) M

2 0+

2( Q

4−

Q2s

+38s2

) M4 0

+( 4

Q2−

2s) M

6 0+

2M

8 0

)−

4sm

5 ∆M

3 0

( 2( 5

5Q

2−

56s) s2

( Q2

+s) +

( −15Q

6−

6Q

4s

+539Q

2s2

+240s3

) M2 0−

2( 1

3Q

4+

15Q

2s

+78s2

) M4 0−

( 7Q

2+

12s) M

6 0+

4M

8 0

) +

2sM

6 0

( −( 3

Q2−

5s)(

3Q

2−

s) s2( Q

2+

s) −s

( −72Q

6+

519Q

4s

+430Q

2s2

+195s3

) M2 0

+( 8

Q6−

102Q

4s−

217Q

2s2−

11s3

) M4 0+

( 14Q

4−

164Q

2s

+1187s2

) M6 0

+( 4

Q2−

22s) M

8 0−

2M

10

0

) +4m

6 ∆M

2 0

( −4

( 16Q

2−

29s) s3

( Q2

+s) +

s( 1

1Q

6+

24Q

4s−

238Q

2s2−

181s3

) M2 0

+( Q

6+

4Q

4s

+137Q

2s2−

707s3

) M4 0

+( 7

Q4−

5Q

2s

+185s2

) M6 0

+( 1

1Q

2+

2s) M

8 0+

5M

10

0

) −m

8 ∆

( 300s3

( Q2

+s) 2

+4s3

( 179Q

2+

249s) M

2 0−

( 10Q

6+

14Q

4s

+519Q

2s2

+301s3

) M4 0−

5( 2

Q4

+4Q

2s

+55s2

) M6 0

+2

( 5Q

2−

3s) M

8 0+

10M

10

0

)+

4sm

∆M

7 0

( −132s4

+5Q

2M

2 0

( Q2

+M

2 0

) 2−

4s3

( 108Q

2+

77M

2 0

) +s2

( −414Q

4−

469Q

2M

2 0+

884M

4 0

) +2s

( 19Q

6−

Q4M

2 0−

16Q

2M

4 0+

2M

6 0

))−

m4 ∆

M4 0

( −38s5

+12M

6 0

( Q2

+M

2 0

) 2−

6s4

( 113Q

2+

372M

2 0

) −2sM

2 0

( Q2

+M

2 0

)(15Q

4+

80Q

2M

2 0+

41M

4 0

) +s3

( −82Q

4+

170Q

2M

2 0+

2723M

4 0

) +

s2( −

90Q

6−

12Q

4M

2 0+

407Q

2M

4 0+

397M

6 0

))+

2m

2 ∆M

4 0

( −5s6

+3M

6 0

( Q2

+M

2 0

) 3+

s5( 1

3Q

2+

126M

2 0

) +sM

4 0

( Q2

+M

2 0

)(21Q

4−

5Q

2M

2 0+

2M

4 0

) −s4

( −9Q

4−

221Q

2M

2 0+

839M

4 0

) +s2

M2 0

( 9Q

6+

424Q

4M

2 0+

420Q

2M

4 0+

M6 0

) −s3

( 9Q

6+

432Q

4M

2 0+

1196Q

2M

4 0+

648M

6 0

))))

+

d6

( 2s3

m10

π

( 9( Q

2+

s) 2−

2( 2

3Q

2+

21s) M

2 0+

33M

4 0

)−

8s3

m9 ∆

M0

( 4( Q

2+

s) 2+

( 7Q

2+

9s) M

2 0+

35M

4 0

) +

4s2

m∆

M9 0

( 3Q

6−

50Q

4s−

33Q

2s2−

8s3

+2

( Q4−

2Q

2s−

5s2

) M2 0−

( Q2−

18s) M

4 0

) −2s2

m10

∆

( 9s

( Q2

+s) 2

+2s

( 7Q

2+

9s) M

2 0−

2( 3

1Q

2+

121s) M

4 0−

62M

6 0

)+

4sm

3 ∆M

7 0

( 6s

( −Q

6+

6Q

4s

+25Q

2s2

+6s3

) +( Q

6−

2Q

4s−

43Q

2s2

+24s3

) M2 0

+2

( Q4

+2Q

2s−

22s2

) M4 0

+Q

2M

6 0

) +

4sm

7 ∆M

3 0

( 4s2

( Q2

+s)(

4Q

2+

9s) +

( Q2

+s)(

Q4

+Q

2s

+72s2

) M2 0

+2

( Q4

+Q

2s

+50s2

) M4 0

+Q

2M

6 0

) −s2

M8 0

( (Q

2−

s) 2s

( Q2

+s) +

( −7Q

6+

112Q

4s

+69Q

2s2

+18s3

) M2 0−

( Q2−

2s)(

Q2

+5s) M

4 0+

( 7Q

2−

30s) M

6 0+

M8 0

)+

2sm

2 ∆M

6 0( (

Q2−

s) 2s2

( Q2

+s) +

s( −

5Q

6−

7Q

4s

+125Q

2s2

+27s3

) M2 0

+( 2

Q6

+11Q

4s−

79Q

2s2−

12s3

) M4 0

+( 4

Q4

+17Q

2s−

s2) M

6 0+

( 2Q

2+

s) M8 0

)−

4sm

5 ∆M

5 0

( 56s4

+2Q

2M

2 0

( Q2

+M

2 0

) 2+

s3( 2

5Q

2+

4M

2 0

) +Q

2s

( Q2

+M

2 0

)(−3

Q2

+5M

2 0

) +2s2

( −3Q

4+

70Q

2M

2 0+

34M

4 0

))+

m8 ∆

M2 0

( 70s5

+Q

2sM

2 0

( Q2

+M

2 0

) 2+

M4 0

( Q2

+M

2 0

) 3+

s4( 9

6Q

2+

101M

2 0

) −s2

M2 0

( Q2

+M

2 0

)(−3

Q2

+292M

2 0

) +s3

( 26Q

4+

91Q

2M

2 0+

64M

4 0

))−

126


2m

6 ∆M

4 0

( 34s5−

Q2sM

2 0

( Q2

+M

2 0

) 2+

M4 0

( Q2

+M

2 0

) 3+

s4( 2

Q2

+93M

2 0

) +s3

( −2Q

4+

117Q

2M

2 0+

75M

4 0

) −s2

( Q2

+M

2 0

)(2Q

4+

7Q

2M

2 0+

106M

4 0

))+

m4 ∆

M4 0

( −s6

+s5

( Q2−

20M

2 0

) −7Q

2sM

4 0

( Q2

+M

2 0

) 2+

M6 0

( Q2

+M

2 0

) 3−

s2M

2 0

( Q2

+M

2 0

)(Q

4+

43Q

2M

2 0+

45M

4 0

) +

s4( Q

4+

243Q

2M

2 0+

123M

4 0

) −s3

( Q6−

114Q

4M

2 0+

154Q

2M

4 0+

394M

6 0

))+

m8 π

( −2s3

m2 ∆

( 45

( Q2

+s) 2

−10

( 17Q

2+

15s) M

2 0+

121M

4 0

)+

8s3

m∆

M0

( −4( Q

2+

s) 2+

( 23Q

2+

21s) M

2 0+

229M

4 0

)+

M2 0

( −98s5

+

M4 0

( Q2

+M

2 0

) 3−

sM2 0

( Q2

+M

2 0

) 2( −

Q2

+8M

2 0

) +s2

M2 0

( Q2

+M

2 0

)(−5

Q2

+22M

2 0

) +s4

( −240Q

2+

677M

2 0

) +s3

( −142Q

4+

779Q

2M

2 0+

734M

4 0

)))

+

2m

6 π

( −8s3

m3 ∆

M0

( −8( Q

2+

s) 2+

( 31Q

2+

27s) M

2 0+

238M

4 0

)+

s2m

4 ∆

( 90s

( Q2

+s) 2

−20s

( 11Q

2+

9s) M

2 0+

( 29Q

2+

229s) M

4 0+

29M

6 0

) −2sm

∆M

3 0

( −4s2

( Q2

+s)(

14Q

2+

9s) +

( Q2

+s)(

Q4−

7Q

2s

+32s2

) M2 0

+2

( Q4−

3Q

2s

+530s2

) M4 0

+Q

2M

6 0

) −2m

2 ∆M

2 0

( −56s5

+M

4 0

( Q2

+M

2 0

) 3−

sM2 0

( Q2

+M

2 0

) 2( −

Q2

+6M

2 0

) +s2

M2 0

( Q2

+M

2 0

)(−3

Q2

+16M

2 0

) +s4

( −156Q

2+

233M

2 0

) +

s3( −

100Q

4+

305Q

2M

2 0+

504M

4 0

))+

M4 0

( 106s5−

M4 0

( Q2

+M

2 0

) 3+

sM2 0

( Q2

+M

2 0

) 2( −

3Q

2+

8M

2 0

) −3s4

( −94Q

2+

187M

2 0

) −s2

( Q2

+M

2 0

)(2Q

4−

23Q

2M

2 0+

18M

4 0

) −s3

( −214Q

4+

649Q

2M

2 0+

1006M

4 0

))) +

2m

2 π

( 8s3

m7 ∆

M0

( 8( Q

2+

s) 2−

( Q2−

3s) M

2 0+

26M

4 0

)+

4s2

m3 ∆

M5 0

( −3Q

6−

14Q

4s−

63Q

2s2−

24s3−

2( 5

Q4

+30Q

2s

+29s2

) M2 0

+( −

7Q

2+

116s) M

4 0

) +

s2m

8 ∆

( 45s

( Q2

+s) 2

+10s

( Q2

+3s) M

2 0+

( −95Q

2+

188s) M

4 0−

95M

6 0

)−

2sm

∆M

7 0

( −2s

( −3Q

6+

70Q

4s

+63Q

2s2

+18s3

) +( Q

2+

s)(Q

4−

3Q

2s−

72s2

) M2 0

+2

( Q4−

4Q

2s

+90s2

) M4 0

+Q

2M

6 0

) −2sm

5 ∆M

3 0

( −12

( 2Q

2−

3s) s2

( Q2

+s) +

( Q2

+s)(

3Q

4−

5Q

2s−

96s2

) M2 0

+2

( 3Q

4−

Q2s−

58s2

) M4 0

+3Q

2M

6 0

) +sM

6 0

( (Q

2−

s) 2s2

( Q2

+s) +

s( −

9Q

6+

211Q

4s

+165Q

2s2

+53s3

) M2 0

+Q

2( −

2Q

4+

15Q

2s−

11s2

) M4 0

+( −

4Q

4+

29Q

2s−

203s2

) M6 0

+( −

2Q

2+

5s) M

8 0

) −2m

6 ∆M

2 0

( 28s5

+M

4 0

( Q2

+M

2 0

) 3−

sM2 0

( Q2

+M

2 0

) 2( −

Q2

+2M

2 0

) −3s4

( −4Q

2+

17M

2 0

) +s2

M2 0

( Q2

+M

2 0

)(Q

2+

74M

2 0

) −s3

( 16Q

4+

35Q

2M

2 0+

353M

4 0

))+

m4 ∆

M4 0

( −18s5

+M

4 0

( Q2

+M

2 0

) 3−

sM2 0

( Q2

+M

2 0

) 2( 5

Q2

+8M

2 0

) −s4

( 114Q

2+

199M

2 0

) +s2

( Q2

+M

2 0

)(−6

Q4−

7Q

2M

2 0+

97M

4 0

) +

s3( 2

Q4−

39Q

2M

2 0+

535M

4 0

))−

m2 ∆

M4 0

( −s6

+M

6 0

( Q2

+M

2 0

) 3−

sM4 0

( Q2

+M

2 0

) 2( −

Q2

+4M

2 0

) +s5

( Q2

+56M

2 0

) +

s2M

2 0

( Q2

+M

2 0

)(7Q

4+

49Q

2M

2 0+

17M

4 0

) −s4

( −Q

4−

163Q

2M

2 0+

165M

4 0

) −s3

( Q6

+74Q

4M

2 0+

214Q

2M

4 0+

72M

6 0

))) +

m4 π

( 16s3

m5 ∆

M0

( −12

( Q2

+s) 2

+6

( 4Q

2+

3s) M

2 0+

115M

4 0

)+

2s2

m6 ∆

( −90s

( Q2

+s) 2

+20s

( 5Q

2+

3s) M

2 0+

( 4Q

2−

571s) M

4 0+

4M

6 0

) +

4sm

3 ∆M

3 0

( −12s2

( Q2

+s)(

8Q

2+

3s) +

( Q2

+s)(

3Q

4−

13Q

2s−

136s2

) M2 0

+2

( 3Q

4−

5Q

2s−

282s2

) M4 0

+3Q

2M

6 0

) +2m

2 ∆M

4 0( −5

4s5

+7Q

2sM

2 0

( Q2

+M

2 0

) 2+

M4 0

( Q2

+M

2 0

) 3−

s4( 1

66Q

2+

627M

2 0

) −s2

( Q2

+M

2 0

)(−6

Q4

+23Q

2M

2 0+

10M

4 0

) −s3

( 218Q

4+

675Q

2M

2 0+

174M

4 0

))+

4sm

∆M

5 0

( −56s4

+Q

2s

( 3Q

2−

13M

2 0

)(Q

2+

M2 0

) +2Q

2M

2 0

( Q2

+M

2 0

) 2−

s3( 1

69Q

2+

72M

2 0

) +2s2

( −69Q

4−

46Q

2M

2 0+

382M

4 0

))+

2m

4 ∆M

2 0

( −42s5

+3M

4 0

( Q2

+M

2 0

) 3−

3sM

2 0

( Q2

+M

2 0

) 2( −

Q2

+4M

2 0

) −s4

( 216Q

2+

25M

2 0

) −s2

M2 0

( Q2

+M

2 0

)(3Q

2+

49M

2 0

) +

s3( −

174Q

4+

105Q

2M

2 0+

589M

4 0

))+

M4 0

( −s6

+s5

( Q2−

220M

2 0

) +M

6 0

( Q2

+M

2 0

) 3−

sM4 0

( Q2

+M

2 0

) 2( −

9Q

2+

8M

2 0

) +

s2M

2 0

( Q2

+M

2 0

)(15Q

4−

83Q

2M

2 0+

5M

4 0

) +s4

( Q4−

621Q

2M

2 0+

539M

4 0

) +s3

( −Q

6−

614Q

4M

2 0+

630Q

2M

4 0+

1588M

6 0

))))

+

4d3M

0

( −4s2

m∆

M8 0

( 3Q

6+

26Q

4s

+21Q

2s2−

4s3

+( Q

2−

7s)(

4Q

2+

5s) M

2 0+

( Q2

+21s) M

4 0

) +4sm

11

∆

( −9s2

( Q2

+s) +

7s2

M2 0

+( 7

Q2

+11s) M

4 0+

7M

6 0

) +

sm10

∆M

0

( 4s2

( 55Q

2+

59s) −

( 4Q

4+

241Q

2s

+165s2

) M2 0

+( 1

2Q

2−

109s) M

4 0+

40M

6 0

) +

4sm

3 ∆M

6 0

( s( 6

Q6

+47Q

4s

+91Q

2s2

+2s3

) +( Q

6−

4Q

4s−

63Q

2s2−

133s3

) M2 0

+( 2

Q4−

11Q

2s

+190s2

) M4 0

+( Q

2+

5s) M

6 0

) +

127

B. Appendix to VVCS

4sm

10

π

( 9s2

( Q2

+s) m

∆+

s2( 1

3Q

2+

9s) M

0+

3s2

m∆

M2 0−

3( 2

Q4

+s2

) M3 0−

3( Q

2−

3s) m

∆M

4 0+

3( Q

2−

3s) M

5 0−

3m

∆M

6 0+

3M

7 0

) +

2s2

M7 0

( s( Q

2+

s)(Q

4−

8Q

2s−

2s2

) −( 4

Q6

+47Q

4s

+33Q

2s2−

9s3

) M2 0

+( −

4Q

4+

78Q

2s

+13s2

) M4 0

+( Q

2−

21s) M

6 0+

M8 0

) +

4sm

7 ∆M

2 0

( 7s2

( Q2

+s)(

3Q

2+

2s) +

( Q6−

12Q

4s−

306Q

2s2−

330s3

) M2 0

+( 6

Q4−

Q2s

+397s2

) M4 0

+4

( 4Q

2+

31s) M

6 0+

11M

8 0

) −4sm

9 ∆

( 2s2

( Q2

+s) 2

−s2

( 161Q

2+

139s) M

2 0+

( 2Q

4+

6Q

2s

+127s2

) M4 0

+2

( 9Q

2+

38s) M

6 0+

16M

8 0

)+

m8 ∆

M0

( 2s3

( Q2

+s)(

12Q

2+

s) −2s

( 7Q

6+

80Q

4s

+15Q

2s2

+103s3

) M2 0+

( 4Q

6+

6Q

4s

+601Q

2s2

+229s3

) M4 0

+( 1

2Q

4−

28Q

2s

+79s2

) M6 0

+12

( Q2−

8s) M

8 0+

4M

10

0

) −sm

2 ∆M

5 0

( 4s2

( Q2

+s)(

Q4−

8Q

2s−

2s2

) −s

( 21Q

6+

317Q

4s

+386Q

2s2

+30s3

) M2 0

+( 2

Q6

+10Q

4s

+617Q

2s2

+212s3

) M4 0+

( 10Q

4+

115Q

2s−

448s2

) M6 0

+2

( 7Q

2−

6s) M

8 0+

6M

10

0

) +

2m

8 π

( −2s3

( Q2

+s) m

∆

( 2( Q

2+

s) +45m

2 ∆

) +s3

( 54Q

4+

97Q

2s

+43s2

+6Q

2m

2 ∆+

46sm

2 ∆

) M0−

2s3

m∆

( 83Q

2+

105s

+5m

2 ∆

) M2 0+

s( −

5Q

6+

99Q

4s

+808Q

2s2

+837s3

+2

( 23Q

4+

6Q

2s−

90s2

) m2 ∆

) M3 0

+2sm

∆

((40Q

2−

251s) s

+( 1

9Q

2−

41s) m

2 ∆

) M4 0

+( 2

Q6

+23Q

4s−

71Q

2s2

+913s3−

2s

( 9Q

2+

s) m2 ∆

) M5 0

+2sm

∆

( 6Q

2−

2s

+19m

2 ∆

) M6 0

+( 6

Q4−

20Q

2s

+21s2−

4sm

2 ∆

) M7 0

+12sm

∆M

8 0+

6( Q

2−

4s) M

9 0+

2M

11

0

) +

m4 ∆

M3 0

( 2s3

( Q2

+s)(

Q4−

8Q

2s−

2s2

) −2s2

( 9Q

6+

188Q

4s

+186Q

2s2

+56s3

) M2 0

+2s

( −5Q

6−

58Q

4s

+688Q

2s2

+248s3

) M4 0+

( 4Q

6+

18Q

4s

+343Q

2s2−

699s3

) M6 0

+( 1

2Q

4+

24Q

2s−

169s2

) M8 0

+4

( 3Q

2−

s) M10

0+

4M

12

0

) +

m4 π

( −24s3

( Q2

+s) m

5 ∆

( 2( Q

2+

s) +15m

2 ∆

) +4s3

m4 ∆

( 99Q

4+

165Q

2s

+66s2

+278Q

2m

2 ∆+

318sm

2 ∆

) M0+

4s3

m3 ∆

( 61Q

4+

101Q

2s

+40s2

+6

( 39Q

2+

17s) m

2 ∆+

30m

4 ∆

) M2 0

+s

( 2s2

( Q2

+s)(

Q4−

8Q

2s−

2s2

) −s

( 203Q

6+

735Q

4s

+550Q

2s2

+42s3

) m2 ∆−

4( 1

8Q

6+

33Q

4s−

367Q

2s2−

258s3

) m4 ∆

+( 7

2Q

4−

307Q

2s−

1739s2

) m6 ∆

) M3 0

+

4sm

∆

( s( −

3Q

6+

20Q

4s

+57Q

2s2

+36s3

) +( 3

Q6

+226Q

2s2

+214s3

) m2 ∆

+2

( −3Q

4+

107Q

2s

+117s2

) m4 ∆−

18

( −3Q

2+

s) m6 ∆

) M4 0

+( −

2s2

( 108Q

6+

477Q

4s

+364Q

2s2

+4s3

) −s

( −10Q

6+

472Q

4s

+4915Q

2s2

+5008s3

) m2 ∆

+( 2

4Q

6−

68Q

4s

+445Q

2s2

+1875s3

) m4 ∆

+( 2

4Q

2−

283s) sm

6 ∆

) M5 0

+4sm

∆

( 2( Q

6+

4Q

4s−

40Q

2s2

+5s3

) +( 2

Q4

+81Q

2s

+557s2

) m2 ∆−

2( 2

Q2−

19s) m

4 ∆+

54m

6 ∆

) M6 0

+( 1

0s

( 3Q

6+

31Q

4s

+174Q

2s2

+166s3

) +( 8

Q6

+142Q

4s

+827Q

2s2−

6488s3

) m2 ∆

+( 7

2Q

4−

12Q

2s

+445s2

) m4 ∆

+192sm

6 ∆

) M7 0

+

4sm

∆

( 4Q

4+

33Q

2s−

920s2

+4

( 2Q

2−

5s) m

2 ∆+

2m

4 ∆

) M8 0

+2

( 2Q

6+

55Q

4s−

58Q

2s2−

600s3

+( 1

2Q

4+

35Q

2s

+224s2

) m2 ∆

+4

( 9Q

2−

2s) m

4 ∆

)

M9 0

+4sm

∆

( 2Q

2+

10s

+9m

2 ∆

) M10

0+

2( 6

Q4

+44Q

2s−

70s2

+12Q

2m

2 ∆+

5sm

2 ∆+

12m

4 ∆

) M11

0+

4( 3

Q2

+2

( s+

m2 ∆

))M

13

0+

4M

15

0

) −4sm

5 ∆M

4 0

( 18s4

+2M

2 0

( Q2

+M

2 0

) 3−

19s3

( −3Q

2+

14M

2 0

) +2s2

( 20Q

4−

75Q

2M

2 0+

207M

4 0

) +s

( 3Q

6−

20Q

4M

2 0−

19Q

2M

4 0+

64M

6 0

))−

m6 ∆

M3 0

( −62s5

+8M

4 0

( Q2

+M

2 0

) 3+

2s4

( −97Q

2+

358M

2 0

) −s3

( 129Q

4−

1273Q

2M

2 0+

605M

4 0

) −2sM

2 0

( 13Q

6−

5Q

4M

2 0+

3Q

2M

4 0+

33M

6 0

) −s2

( 5Q

6+

294Q

4M

2 0−

590Q

2M

4 0+

185M

6 0

))+

m6 π

( 8sm

5 ∆

( 45s2

( Q2

+s) −

5s2

M2 0

+( −

23Q

2+

29s) M

4 0−

23M

6 0

) −sm

4 ∆M

0

( 8s2

( 71Q

2+

91s) +

( 128Q

4+

3Q

2s−

1187s2

) M2 0−

3( 8

Q2

+55s) M

4 0+

88M

6 0

) +8sm

3 ∆

( 4s2

( Q2

+s) 2

+44s2

( Q2

+2s) M

2 0+

( Q4−

85Q

2s

+98s2

) M4 0−

( 8Q

2+

11s) M

6 0−

9M

8 0

)−

4sm

∆M

2 0

( s2( Q

2+

s)(20Q

2+

13s) +

( Q6

+6Q

4s−

158Q

2s2−

152s3

) M2 0

+( 2

Q4

+77Q

2s−

820s2

) M4 0

+4

( Q2

+5s) M

6 0+

3M

8 0

) −m

2 ∆M

0

( 4s3

( Q2

+s)(

87Q

2+

65s) +

s( −

44Q

6+

311Q

4s

+3201Q

2s2

+3188s3

) M2 0

+( 1

6Q

6+

32Q

4s

+67Q

2s2

+2860s3

) M4 0+

( 48Q

4−

52Q

2s−

4s2

) M6 0

+16

( 3Q

2−

2s) M

8 0+

16M

10

0

) +4M

3 0

( −29s5−

2M

4 0

( Q2

+M

2 0

) 3−

10s4

( −Q

2+

85M

2 0

) +

2s2

( −Q

2+

M2 0

)(−1

3Q

4+

44Q

2M

2 0+

10M

4 0

) +sM

4 0

( −19Q

4−

2Q

2M

2 0+

11M

4 0

) −s3

( −67Q

4+

830Q

2M

2 0+

534M

4 0

))) +

m2 π

( −4sm

9 ∆

( −45s2

( Q2

+s) +

25s2

M2 0

+( 3

1Q

2+

19s) M

4 0+

31M

6 0

) −sm

8 ∆M

0

( 4s2

( 207Q

2+

227s) +

( 8Q

4−

527Q

2s−

1089s2

) M2 0

+3

( 12Q

2−

89s) M

4 0+

148M

6 0

) +8sm

3 ∆M

4 0

( s( 3

Q6

+24Q

4s

+28Q

2s2

+5s3

) +s

( 22Q

4−

75Q

2s

+42s2

) M2 0

+( Q

4−

34Q

2s−

597s2

) M4 0

+( 2

Q2−

17s) M

6 0+

M8 0

) +

8sm

7 ∆

( 4s2

( Q2

+s) 2

−4s2

( 50Q

2+

39s) M

2 0+

( 3Q

4−

39Q

2s−

26s2

) M4 0

+( 1

6Q

2+

31s) M

6 0+

13M

8 0

)−

128


4sm

5 ∆M

2 0

( s2( Q

2+

s)(62Q

2+

41s) +

3( Q

6−

6Q

4s

+26Q

2s2

+12s3

) M2 0

+( 6

Q4−

221Q

2s−

802s2

) M4 0

+20

( Q2−

7s) M

6 0+

17M

8 0

) +

4sM

5 0

( s2( Q

2+

s)(−Q

4+

8Q

2s

+2s2

) +s

( 30Q

6+

168Q

4s

+140Q

2s2

+5s3

) M2 0

+( −

5Q

6−

11Q

4s−

61Q

2s2

+s3

) M4 0+

( −14Q

4−

4Q

2s

+391s2

) M6 0−

13

( Q2−

s) M8 0−

4M

10

0

) −m

6 ∆M

0

( 4s3

( Q2

+s)(

45Q

2+

23s) −

s( 5

2Q

6+

405Q

4s

+147Q

2s2

+688s3

) M2 0+

( 16Q

6−

48Q

4s

+149Q

2s2

+334s3

) M4 0

+2

( 24Q

4−

14Q

2s−

59s2

) M6 0

+16

( 3Q

2−

8s) M

8 0+

16M

10

0

) +

4sm

∆M

6 0

( −25s4

+s3

( Q2−

101M

2 0

) −Q

2M

2 0

( Q2

+M

2 0

) 2+

s2( 2

8Q

4−

27Q

2M

2 0+

369M

4 0

) +s

( 6Q

6+

2Q

4M

2 0+

5Q

2M

4 0+

3M

6 0

))+

m4 ∆

M3 0

( 96s5

+8M

4 0

( Q2

+M

2 0

) 3−

4s4

( −79Q

2+

442M

2 0

) +s3

( 338Q

4−

2008Q

2M

2 0+

1137M

4 0

) −4sM

2 0

( 9Q

6+

14Q

4M

2 0+

17Q

2M

4 0+

30M

6 0

) +

s2( 9

4Q

6+

142Q

4M

2 0+

623Q

2M

4 0+

911M

6 0

))−

m2 ∆

M3 0

( −8s6

+8M

6 0

( Q2

+M

2 0

) 3+

8s5

( −5Q

2+

46M

2 0

) −4sM

4 0

( Q2

+M

2 0

)(17Q

4+

17Q

2M

2 0+

21M

4 0

) −4s4

( 7Q

4−

363Q

2M

2 0+

215M

4 0

) +s2

M2 0

( 202Q

6−

273Q

4M

2 0+

61Q

2M

4 0+

428M

6 0

) +s3

( 4Q

6+

1226Q

4M

2 0−

737Q

2M

4 0+

3652M

6 0

))))

)

(9+

5τ[3

])II

[d][{{

0,m

π},

1}])

/( 7

68(−

1+

d)4

d(1

+d)F

2s3

M0

m6 ∆

(mπ−

m∆−

M0)(m

π+

m∆−

M0)M

4 0(m

π−

m∆

+M

0)(m

π+

m∆

+M

0)) −

( ie2gπN

∆2

( −2(−

2+

d)m

10

π

( 3(−

2+

d)4

( Q2

+s) 2

m2 ∆

+( Q

2+

s) m∆

( (−2

+d)3

(−1

+d)( Q

2+

s) −24(−

3+

d)(−2

+d)m

2 ∆

) M0+

( (−2

+d)2

( Q2

+s)(

(18

+d(−

14

+3d))

Q2

+(9

+2(−

4+

d)d

)s) −

2( (1

16

+d(−

242

+d(1

69

+d(−

49

+5d))

))Q

2+

(68

+d(−

170

+d(1

33

+d(−

43

+5d))

))s) m

2 ∆

) M2 0

+

2m

∆

( −(−

2+

d)( (−

56

+d(4

7+

d(−

17

+2d))

)Q2

+(−

4+

d)(

17

+d(−

7+

2d))

s) +4(−

4+

(−2

+d)d

)m2 ∆

) M3 0

+( −

(−2

+d)2

( (125

+d(−

78

+11d))

Q2

+2(4

5+

d(−

29

+5d))

s) +(−

24

+d(−

188

+7d(3

0+

(−10

+d)d

)))m

2 ∆

) M4 0

+

(−2

+d)(−1

+d)(

36

+d(−

22

+3d))

m∆

M5 0

+(−

2+

d)2

( 81−

50d

+8d2) M

6 0

) +

m8 π

( 15(−

2+

d)5

( Q2

+s) 2

m4 ∆

+10(−

2+

d)2

( Q2

+s) m

3 ∆

( (−2

+d)2

(−1

+d)( Q

2+

s) −12(−

3+

d)m

2 ∆

) M0+

2(−

2+

d)m

2 ∆

( (280

+d(−

480

+d(3

12

+d(−

94

+11d))

))Q

4+

(380

+d(−

660

+d(4

39

+d(−

138

+17d))

))Q

2s−

5(−

8+

(−4

+d)(−2

+d)d

(−11

+4d))

Q2m

2 ∆+

s((

100

+d

( −180

+d

( 127−

44d

+6d2))

) s+

5(5

6+

d(1

6+

d(−

62

+(2

9−

4d)d

)))m

2 ∆

))M

2 0+

2m

∆

( (−2

+d)2

( Q2

+s)(

(−36

+d(5

2+

d(−

29

+5d))

)Q2

+(−

1+

d)(

8+

3(−

4+

d)d

)s) −

2( (−

400

+d(9

36

+d(−

876

+d(4

09

+d(−

93

+8d))

)))Q

2+

(−640

+d(1

136

+d(−

856

+d(3

59

+d(−

83

+8d))

)))s

) m2 ∆

+

20(−

2+

d)(−2

+d(−

7+

2d))

m4 ∆

) M3 0

+( (−

2+

d)3

( 5(2

4+

d(−

16

+3d))

Q4

+2(6

7+

8(−

5+

d)d

)Q2s

+5(6

+(−

4+

d)d

)s2) −

2(−

2+

d)( (4

24

+d(−

1222

+d(9

61

+29(−

10

+d)d

)))Q

2+

2(9

6+

d(−

362

+d(3

02

+d(−

101

+12d))

))s) m

2 ∆+

(2992

+d(−

2576

+d(−

132

+d(9

46

+d(−

359

+38d))

)))m

4 ∆

) M4 0

+

2m

∆

( −2(−

2+

d)2

( (−208

+3d(7

3+

d(−

28

+3d))

)Q2

+(−

238

+d(1

75

+d(−

65

+8d))

)s) +

(368

+d(4

80

+d(−

1088

+d(6

06

+d(−

137

+11d))

)))m

2 ∆

) M5 0

+

2( −

(−2

+d)3

( (317

+d(−

185

+23d))

Q2

+5(4

2+

d(−

25

+4d))

s) +(−

2+

d)(−4

68

+d(−

264

+d(5

57

+18(−

11

+d)d

)))m

2 ∆

) M6 0

+

2(−

2+

d)2

(−228

+d(2

70

+d(−

115

+13d))

)m∆

M7 0

+5(−

2+

d)3

(78

+d(−

46

+7d))

M8 0

) +

m6 π

( −20(−

2+

d)5

( Q2

+s) 2

m6 ∆

+20(−

2+

d)2

( Q2

+s) m

5 ∆

( −(−

2+

d)2

(−1

+d)( Q

2+

s) +8(−

3+

d)m

2 ∆

) M0−

4(−

2+

d)m

4 ∆

( (200

+d(−

320

+d(1

94

+d(−

58

+7d))

))Q

4+

( 220

+d

( −340

+d

( 203−

66d

+9d2))

) Q2s−

10(−

44

+d(2

2+

d(9

+(−

7+

d)d

)))Q

2m

2 ∆+

s( (2

0+

d(−

20

+d(9

+2(−

4+

d)d

)))s−

10(−

60

+d(4

6+

d(−

3+

(−5

+d)d

)))m

2 ∆

))M

2 0−

8m

3 ∆

( (−2

+d)( Q

2+

s)((5

8+

d(−

99

+d(7

0+

3(−

8+

d)d

)))Q

2+

(−1

+d)(−2

+d(5

+(−

5+

d)d

))s) −

( (320

+d(−

188

+d(−

192

+d(2

03

+d(−

61

+6d))

)))Q

2+

(80

+d(1

2+

d(−

172

+3d(5

1+

d(−

17

+2d))

)))s

) m2 ∆

+20(−

4+

d)(−2

+d)d

m4 ∆

)

M3 0

+m

2 ∆

( −4(−

2+

d)( (2

34

+d(−

378

+d(2

30

+d(−

66

+7d))

))Q

4+

2(9

1+

2d(−

62

+d(3

2+

(−9

+d)d

)))Q

2s

+( 6

+(−

4+

d)d

3) s2

) +

129

B. Appendix to VVCS

4( (1

432

+d(−

1536

+d(1

89

+d(2

71

+d(−

105

+11d))

)))Q

2+

2(5

12

+d(−

621

+d(2

18

+(−

6+

d)d

(−1

+3d))

))s) m

2 ∆+

(−5792

+d(5

000

+d(−

156

+d(−

1906

+d(9

07

+3(−

44

+d)d

))))

)m4 ∆

) M4 0−

4m

∆

( (−2

+d)2

( (−52

+(−

4+

d)d

(−17

+5d))

Q4

+2(−

16

+d(1

9+

d(−

13

+2d))

)Q2s

+( 4

+(−

5+

d)d

2) s2

) −4

( (88

+d(4

2+

d(−

193

+d(1

35

+d(−

35

+3d))

)))Q

2+

(−48

+d(1

36

+d(−

135

+d(7

3+

2(−

10

+d)d

))))

s) m2 ∆

+

(1216

+d(−

816

+d(−

272

+d(4

10

+d(−

137

+15d))

)))m

4 ∆

) M5 0−

4( (−

2+

d)3

( (44

+d(−

22

+3d))

Q4

+(3

9+

(−10

+d)d

)Q2s

+5s2

) −2(−

2+

d)( (−

115

+d(−

82

+d(1

39

+d(−

47

+4d))

))Q

2+

(−102

+d(6

+d(4

5+

d(−

23

+3d))

))s) m

2 ∆+

( 1984

+d

( −2054

+d

( 598

+d

( 81−

62d

+6d2))

))m

4 ∆

) M6 0

+8m

∆

( (−2

+d)2

(−188

+7(−

6+

d)(−5

+d)d

)Q2+

(−2

+d)2

(−196

+d(1

45

+d(−

55

+6d))

)s+

(−596

+d(3

36

+d(2

39

+d(−

212−

3(−

16

+d)d

))))

m2 ∆

) M7 0

+

4(−

2+

d)( 1

1(−

2+

d)2

(19

+(−

10

+d)d

)Q2

+10(−

2+

d)2

(13

+(−

7+

d)d

)s+

(770−

d(2

92

+d(1

70

+d(−

82

+5d))

))m

2 ∆

) M8 0−

4(−

2+

d)2

(−236

+(−

5+

d)d

(−50

+11d))

m∆

M9 0−

20(−

2+

d)3

(25

+2(−

7+

d)d

)M10

0

) +

m4 π

( 15(−

2+

d)5

( Q2

+s) 2

m8 ∆

+20(−

2+

d)2

( Q2

+s) m

7 ∆

( (−2

+d)2

(−1

+d)( Q

2+

s) −6(−

3+

d)m

2 ∆

) M0+

2(−

2+

d)m

6 ∆

( 2( Q

2+

s)((1

20

+d(−

160

+d(7

6+

d(−

22

+3d))

))Q

2+

(−60

+d(1

40

+d(−

109−

2(−

14

+d)d

)))s

) −5

( (−256

+d(2

20

+d(−

44

+d(−

7+

2d))

))Q

2+

(−304

+d(2

92

+d(−

80

+d(−

1+

2d))

))s) m

2 ∆

) M2 0

+

4m

5 ∆

( 3(−

2+

d)( Q

2+

s)((4

4+

d(−

58

+d(3

0+

(−9

+d)d

)))Q

2−

(−1

+d)(−1

2+

d(2

2+

(−8

+d)d

))s) −

2( (1

040

+d(−

1312

+d(4

92

+d(−

3+

d(−

29

+4d))

)))Q

2+

(800

+d(−

1112

+d(5

12

+d(−

53

+d(−

19

+4d))

)))s

) m2 ∆

+

10(−

2+

d)(

2+

d(−

17

+4d))

m4 ∆

) M3 0−

m4 ∆

( 2(−

2+

d)( (−

236

+d(3

64

+d(−

200

+(5

6−

5d)d

)))Q

4+

2(−

6+

d(−

64

+d(8

9+

4(−

8+

d)d

)))Q

2s+

(68

+d(−

144

+d(1

02

+(−

24

+d)d

)))s

2) +

4( (1

904

+d(−

2752

+d(1

261

+d(−

223

+d(1

1+

d))

)))Q

2+

720s−

2d

(569

+d(−

339

+d(1

19

+2(−

12

+d)d

)))s

)m2 ∆

+(−

5024

+d(3

800

+d(2

8+

d(−

1934

+d(1

019

+d(−

161

+3d))

))))

m4 ∆

) M4 0

+

4m

3 ∆

( (−2

+d)( (1

16

+d(−

190

+d(1

22

+d(−

35

+3d))

))Q

4−

2d(−

3+

2d)(

14

+(−

7+

d)d

)Q2s−

(−1

+d)(−2

0+

d(2

6+

(−8

+d)d

))s2

) +

2( (−

896

+d(1

408

+d(−

798

+d(1

95

+(−

22

+d)d

))))

Q2

+(−

440

+d(6

76

+d(−

480

+d(1

95

+d(−

43

+4d))

)))s

) m2 ∆

+

(1824

+d(−

2008

+d(6

08

+(−

2+

d)d

(−131

+27d))

))m

4 ∆

) M5 0

+

m2 ∆

( 4(−

2+

d)(

138

+d(−

266

+d(1

90

+d(−

58

+5d))

))Q

4−

4( −

36

+d

( 98

+d

( −66

+9d

+d3))

) Q2s−

12(−

2+

d)(

14

+d(−

20

+7d))

s2+

4(−

1748

+d(2

702

+d(−

1457

+d(3

73

+d(−

58

+5d))

)))Q

2m

2 ∆+

4(−

616

+d(9

42

+d(−

584

+d(1

96

+d(−

33

+2d))

)))s

m2 ∆

+(1

184

+d(1

76

+d(−

1012

+d(1

278

+d(−

695

+3(5

6−

5d)d

))))

)m4 ∆

) M6 0

+

4m

∆

( (−2

+d)2

( (−44

+d(4

4+

(−25

+d)d

))Q

4−

2(2

+d(1

1+

(−2

+d)d

))Q

2s−

(−1

+d)(

16

+(−

4+

d)d

)s2) +

2( (−

976

+d(1

500

+d(−

992

+d(3

44

+5(−

13

+d)d

))))

Q2

+(−

4+

d)(

184

+d(−

234

+d(1

35

+d(−

37

+4d))

))s) m

2 ∆+

(1144

+d(−

928

+d(1

26

+d(6

2+

3(−

9+

d)d

))))

m4 ∆

) M7 0

+( (−

2+

d)3

( (108−

d(4

+9d))

Q4−

2(−

42

+d(−

20

+7d))

Q2s−

5(−

4+

d)d

s2) +

4(−

2+

d)( (7

06

+d(−

518

+d(1

59

+5(−

6+

d)d

)))Q

2+

2(2

54

+d(−

254

+d(1

06

+d(−

19

+2d))

))s) m

2 ∆−

2(−

2440

+d(2

632

+d(−

1360

+d(7

16

+d(−

241

+30d))

)))m

4 ∆

) M8 0−

4(−

2+

d)m

∆( 2

(−2

+d)( (−

175

+d(1

93

+4(−

16

+d)d

))Q

2+

(−169

+d(1

20

+d(−

45

+4d))

)s) +

(1308

+d(−

818

+d(1

44

+d(−

33

+7d))

))m

2 ∆

) M9 0−

2(−

2+

d)( (−

2+

d)2

( (302

+d(−

135

+8d))

Q2

+5(−

4+

d)(−9

+2d)s

) +2

( 1026

+d

( −728

+d

( 111

+2d

+4d2))

) m2 ∆

) M10

0+

4(−

2+

d)2

(−234

+d(2

30

+d(−

95

+9d))

)m∆

M11

0+

5(−

4+

d)(−2

+d)3

(−18

+5d)M

12

0

) +

(−m

∆+

M0)(m

∆+

M0)( −(

−2+

d)5

( Q2

+s) 2

m10

∆+

2(−

2+

d)2

( Q2

+s) m

9 ∆

( −(−

2+

d)2

(−1

+d)( Q

2+

s) +4(−

3+

d)m

2 ∆

) M0+

(−2

+d)m

8 ∆

( 72s

( Q2

+s) +

d4

( Q2

+s)(

Q2

+3s) −

16

( 21Q

2+

23s) m

2 ∆−

2d3

( 6Q

4+

22Q

2s

+16s2−

7Q

2m

2 ∆−

9sm

2 ∆

) −8d

( 4Q

4+

23Q

2s

+19s2−

2( 2

2Q

2+

25s) m

2 ∆

) +2d2

( 20Q

4+

77Q

2s

+57s2−

2( 3

1Q

2+

37s) m

2 ∆

))M

2 0+

130


4m

7 ∆

( (−2

+d)( Q

2+

s)((−

12

+d(−

2+

d(1

6+

(−7

+d)d

)))Q

2+

16s

+2(−

3+

d)d

(8+

(−5

+d)d

)s) +

( (472

+d(−

680

+d(3

58

+d(−

81

+7d))

))Q

2+

(424

+d(−

640

+d(3

62

+d(−

91

+9d))

))s) m

2 ∆−

2(−

2+

d)(

2+

d(−

9+

2d))

m4 ∆

) M3 0

+

m6 ∆

( 2(−

2+

d)( (−

20

+d(3

6+

d(−

10

+(−

4+

d)d

)))Q

4+

2(−

26

+d(3

6+

d(−

5+

(−7

+d)d

)))Q

2s−

(52

+d(−

80

+d(3

6+

(−4

+d)d

)))s

2) +

2( (−

128

+d(2

28

+(−

10

+d)d

(27

+(−

9+

d)d

)))Q

2+

(−560

+d(8

52

+d(−

572

+(1

76−

21d)d

)))s

) m2 ∆−

(912

+d(−

1944

+d(2

088

+d(−

980

+3d(7

0+

(−10

+d)d

))))

)m4 ∆

) M4 0−

2m

5 ∆

( 2(−

2+

d)( 2

(−16

+d(1

0+

(−1

+d)d

))Q

4+

(−36

+d(1

4+

d(1

2+

(−3

+d)d

)))Q

2s

+(−

1+

d)(−2

8+

d(5

4+

d(−

22

+3d))

)s2) −

2( (−

936

+d(1

292

+d(−

762

+d(2

22

+(−

29

+d)d

))))

Q2−

4( 2

92

+d

( −469

+d

( 298−

84d

+9d2))

) s) m2 ∆

+

(448

+d(−

736

+5d(9

2+

d(6

+d(−

17

+3d))

)))m

4 ∆

) M5 0

+m

4 ∆( 2

(−2

+d)( (1

20

+d(−

176

+d(7

4+

d(−

16

+3d))

))Q

4+

2(8

4+

d(−

136

+d(7

4+

d(−

21

+4d))

))Q

2s−

(−32

+d(8

+d(3

8+

(−20

+d)d

)))s

2) −

4( (5

12

+d(−

566

+d(−

11

+2d(7

2+

(−20

+d)d

))))

Q2

+(2

64−

d(3

44

+d(5

4+

d(−

124

+27d))

))s) m

2 ∆−

(−1056

+d(6

56

+d(−

472

+d(1

36

+d(1

79

+2d(−

47

+6d))

))))

m4 ∆

) M6 0

+

4m

3 ∆

( (−2

+d)( (1

2+

d(−

62

+d(5

2+

d(−

17

+3d))

))Q

4+

(60

+d(−

202

+d(1

86

+d(−

65

+9d))

))Q

2s

+2(−

1+

d)2

(12

+(−

6+

d)d

)s2) −

( (−632

+d(8

44

+d(−

694

+d(3

30

+d(−

73

+5d))

)))Q

2−

2(2

96

+d(−

206

+d(2

+d(1

1+

d))

))s) m

2 ∆+

2(5

84

+d(−

926

+d(5

74

+d(−

73

+d(−

32

+7d))

)))m

4 ∆

) M7 0

+m

2 ∆( (−

2+

d)( −

(152

+d(−

152

+d(2

+d)(

6+

d))

)Q4

+2(−

12

+d(−

104

+d(1

62

+d(−

66

+7d))

))Q

2s

+(−

8+

d(−

48

+(−

4+

d)d

(−20

+3d))

)s2) −

4( (−

540

+d(6

16

+d(−

200

+d(3

7+

d(−

19

+5d))

)))Q

2+

(−112

+d(−

6+

d(2

00

+d(−

122

+21d))

))s) m

2 ∆+

(800

+d(−

2576

+d(1

456

+d(−

32

+d(8

0+

3d(−

31

+5d))

))))

m4 ∆

) M8 0−

2m

∆

( (−2

+d)2

( (−28

+d(6

0+

d(−

31

+7d))

)Q4

+4(−

5+

d(1

1+

(−5

+d)d

))Q

2s

+(−

1+

d)(

8+

(−4

+d)d

)s2) +

2( (1

12

+d(4

4+

d(−

124

+d(8

0+

d(−

29

+5d))

)))Q

2+

2(2

04

+d(−

262

+d(1

45

+d(−

43

+6d))

))s) m

2 ∆+

2(9

52

+d(−

860

+d(1

58

+d(9

2+

d(−

39

+5d))

)))m

4 ∆

) M9 0−

( (−2

+d)3

( (8+

d(−

24

+7d))

Q4

+2(1

+2(−

4+

d)d

)Q2s

+(2

+(−

4+

d)d

)s2) +

2(−

2+

d)(

(−80

+d(−

118

+d(1

26

+d(−

41

+4d))

))

Q2−

(128

+d(−

104

+d(1

4+

5d))

)s) m

2 ∆+

(368

+d(1

68

+d(−

1004

+d(9

30

+d(−

297

+29d))

)))m

4 ∆

) M10

0+

4m

∆

( (−2

+d)2

(28

+d(−

25

+d(4

+d))

)Q2

+(−

2+

d)2

(26

+5(−

3+

d)d

)s−

2(−

284

+d(3

58

+d(−

183

+d(5

7+

(−12

+d)d

))))

m2 ∆

) M11

0+

(−2

+d)( 2

(−2

+d)2

( (−17

+d(3

+d))

Q2

+(−

10

+3d)s

) −(4

72

+d(−

368

+d(7

6+

3(−

2+

d)d

)))m

2 ∆

) M12

0+

2(−

2+

d)2

(−44

+d(3

8+

(−15

+d)d

))m

∆M

13

0+

(−2

+d)3

(22

+(−

10

+d)d

)M14

0

) +

m2 π

( −6(−

2+

d)5

( Q2

+s) 2

m10

∆+

2(−

2+

d)2

( Q2

+s) m

9 ∆

( −5(−

2+

d)2

(−1

+d)( Q

2+

s) +24(−

3+

d)m

2 ∆

) M0+

2(−

2+

d)m

8 ∆

((−4

0+

d2(4

2+

(−14

+d)d

)) Q4

+(1

00

+d(−

300

+d(2

69

+d(−

78

+7d))

))Q

2s

+2(−

380

+d(3

74

+d(−

115

+d(7

+d))

))Q

2m

2 ∆+

s((

140

+d

( −300

+d

( 227−

64d

+6d2))

) s+

2(−

428

+d(4

46

+d(−

151

+d(1

3+

d))

))m

2 ∆

))M

2 0+

4m

7 ∆

( 2(−

2+

d)( Q

2+

s)((−

30

+d(1

+d)(

17

+(−

7+

d)d

))Q

2+

26s

+(−

3+

d)d

(25

+3(−

5+

d)d

)s) +

( (1760

+d(−

2436

+d(1

176

+d(1

1+

d)(−1

9+

2d))

))Q

2+

(1520

+d(−

2236

+d(1

196

+d(−

259

+d(1

3+

2d))

)))s

) m2 ∆−

4(−

2+

d)(

4+

d(−

22

+5d))

m4 ∆

) M3 0

+

m6 ∆

( 4(−

2+

d)( (−

14

+d(3

0+

d(−

10

+(−

2+

d)d

)))Q

4+

2(−

1+

2d(−

4+

d(1

1+

(−7

+d)d

)))Q

2s−

(34

+d(−

56

+d(2

8+

(−4

+d)d

)))s

2) −

2( (−

632

+d(9

40

+d(−

204

+d(−

21

+(−

2+

d)d

))))

Q2

+2(6

00

+d(−

870

+d(5

26

+d(−

109

+d(−

5+

3d))

)))s

) m2 ∆−

(2816

+d(−

3544

+d(2

580

+d(−

534

+d(−

129

+d(2

6+

3d))

))))

m4 ∆

) M4 0

+

2m

5 ∆

( 2(−

2+

d)( (3

2+

d(−

16

+(−

6+

d)(−1

+d)d

))Q

4+

2(4

8+

d(−

78

+d(5

7+

d(−

21

+2d))

))Q

2s−

(−1

+d)(−3

2+

d(5

6+

d(−

22

+3d))

)s2) −

8( (−

4+

d)(−5

0+

d(3

1+

d(−

3+

(−3

+d)d

)))Q

2+

(368

+d(−

568

+d(3

13

+d(−

53

+2(−

3+

d)d

))))

s) m2 ∆−

131

B. Appendix to VVCS

(2192

+d(−

3096

+d(1

552

+d(1

62

+d(−

281

+47d))

)))m

4 ∆

) M5 0

+

2m

4 ∆

( 2(−

2+

d)( −

( 4+

(−2

+d)d

( 4+

d2))

Q4

+(4

4+

d(−

152

+d(1

59

+d(−

58

+5d))

))Q

2s

+(−

24

+d(2

0+

(9−

8d)d

))s2

) −4

( (278

+d(−

275

+d(−

17

+d(9

1+

2(−

14

+d)d

))))

Q2

+404s−

d(6

46

+d(−

256

+d(9

+d(3

+d))

))s) m

2 ∆+

(4056

+d(−

7640

+d(6

234

+d(−

2527

+d(5

55

+d(−

62

+3d))

))))

m4 ∆

) M6 0

+

8m

3 ∆

( (−2

+d)( (−

6+

d(5

7+

d(−

62

+(2

2−

3d)d

)))Q

4−

2d(−

15

+d(1

5+

(−5

+d)d

))Q

2s−

(−1

+d)(−1

4+

d(1

9+

(−7

+d)d

))s2

) +( (−

528

+d(5

12

+d(−

112

+(−

2+

d)(−1

+d)d

)))Q

2+

(−640

+d(1

044

+d(−

800

+d(3

39

+d(−

73

+6d))

)))s

) m2 ∆

+

(948

+d(−

1480

+d(8

29

+d(−

80

+d(−

46

+9d))

)))m

4 ∆

) M7 0

+

m2 ∆

( 2(−

2+

d)( (2

0+

d(1

48

+d(−

220−

7(−

12

+d)d

)))Q

4−

2(−

62

+d(−

16

+d(1

08

+d(−

52

+5d))

))Q

2s

+( 4

4−

(−8

+d)d

2(−

8+

3d)) s2

) −4

( (184

+d(3

92

+d(−

695

+d(3

47

+3(−

23

+d)d

))))

Q2

+368s−

2d

( 201

+d

( −34

+d

( −14

+d

+d2))

) s) m2 ∆−

(−4528

+d(3

696

+d(1

268

+d(−

946

+d(5

9+

d(4

+3d))

))))

m4 ∆

) M8 0

+

2m

∆

( (−2

+d)2

( (−12

+d(6

4+

d(−

31

+11d))

)Q4

+2(−

20

+d(5

4+

d(−

23

+5d))

)Q2s

+(−

1+

d)(

28

+3(−

4+

d)d

)s2) −

8( (−

392

+d(6

48

+d(−

496

+d(1

97

+d(−

37

+2d))

)))Q

2+

2(−

210

+d(3

16

+d(−

214

+d(7

9+

(−15

+d)d

))))

s) m2 ∆−

2(−

1872

+d(2

608

+d(−

1436

+d(3

34

+(−

35

+d)d

))))

m4 ∆

) M9 0

+

2( (−

2+

d)3

( (−6

+d(−

26

+9d))

Q4

+(−

7+

d(−

26

+7d))

Q2s

+(3

+2(−

4+

d)d

)s2) −

2(−

2+

d)(

(454

+d(−

294

+d(7

5+

(−7

+d)d

)))

Q2

+(3

60

+d(−

382

+d(1

51

+d(−

25

+3d))

))s) m

2 ∆+

2(3

44

+d(−

354

+d(−

170

+d(4

13

+2d(−

82

+9d))

)))m

4 ∆

) M10

0+

4m

∆

( (−2

+d)2

( −5

( 32−

33d

+9d2) Q

2+

(−148

+d(9

7+

d(−

35

+2d))

)s) +

2(−

1116

+d(1

456

+d(−

775

+d(2

50

+d(−

54

+5d))

)))m

2 ∆

) M11

0+

2( −

(−2

+d)3

( (−113

+d(3

8+

d))

Q2−

2(3

3+

(−13

+d)d

)s) +

(−2

+d)( 1

236

+d

( −1044

+d

( 286−

42d

+9d2))

) m2 ∆

) M12

0−

2(−

2+

d)2

(−228

+d(2

10

+d(−

85

+7d))

)m∆

M13

0−

2(−

2+

d)3

( 69−

34d

+4d2) M

14

0

) +(−

2+

d)2

m12

π( 8

m∆

M0

( −(−

3+

d)( Q

2+

s) +M

2 0

) +(−

2+

d)( (−

2+

d)2

( Q2

+s) 2

−2

( (−4

+d)(−5

+2d)Q

2+

(16

+d(−

11

+2d))

s) M2 0

+(2

8+

3(−

6+

d)d

)M4 0

)))

(9+

5τ[3

])II

[d][{{

0,m

π},

1},{{

p,m

∆},

1}])

/(384(−

1+

d)4

F2m

6 ∆M

0(m

π−

m∆−

M0)(m

π+

m∆−

M0)M

4 0(m

π−

m∆

+M

0)(m

π+

m∆

+M

0)) −

1768(−

1+

d)4

(1+

d)F

2s3m

6 ∆M

0

( ie2gπN

∆2

((Q

2+

s)((−

2+

d)3

s4( (−

6+

(−3

+d)d

)Q4−

2(−

2+

(−3

+d)d

)Q2s

+( −

2+

d+

d2) s2

) −(−

2+

d)s

3( (−

2+

d)(

56

+d(−

20

+13(−

3+

d)d

))Q

4+

2(8

0+

d(6

0+

d(−

82

+5d(3

+d))

))Q

2s

+(4

0−

d(6

0+

d(7

4+

d(−

83

+15d))

))s2

) m2 ∆

+

s2( (−

2+

d)(−4

8+

d(5

6+

d(−

10

+3(−

1+

d)d

)))Q

4−

2(6

88

+d(−

316

+d(−

660

+d(5

28

+d(−

129

+11d))

)))Q

2s−

(−2

+d)(

88

+d(5

16

+d(2

34

+d(−

237

+65d))

))s2

) m4 ∆

+s

( (−2

+d)3

(6+

(−1

+d)d

)Q4+

2(1

92

+d(3

6+

d(−

284

+d(8

4+

(−3

+d)d

))))

Q2s

+(−

2160

+d(−

976

+d(1

564

+d(4

76

+d(−

145

+d(−

19

+6d))

))))

s2) m

6 ∆−

2( (−

2+

d)3

(2+

d)Q

4+

2(−

2+

d)( 1

2+

(−2

+d)d

2) Q

2s

+(−

800

+d(1

12

+d(4

74

+d(−

190

+d(1

50

+d(−

62

+9d))

))))

s2) m

8 ∆

) +

4(1

+d)s

m∆

( (−2

+d)2

s2( (−

2+

d)d

Q6−

2(8

+(−

3+

d)d

)Q4s

+(−

20

+d(2

+d))

Q2s2−

2(−

2+

d)s

3) +

2(−

2+

d)s

( (−2

+d)3

Q6

+(−

80

+d(1

26

+d(−

65

+12d))

)Q4s

+(−

48

+d(1

10

+7(−

8+

d)d

))Q

2s2

+(8

+(4−

5d)d

)s3) m

2 ∆+

( (−2

+d)3

dQ

6+

2(−

2+

d)2

(8+

(−5

+d)d

)Q4s

+(8

00

+d(−

1208

+d(6

20

+d(−

146

+17d))

))Q

2s2−

2(−

128

+d(2

92

+d(−

144

+19d))

)s3) m

4 ∆−

2( (−

6+

d)(−2

+d)d

Q4

+2(4

+d)(−2

+3d)Q

2s

+(−

240

+d(2

00

+d(−

62

+7d))

)s2) m

6 ∆−

4(−

6+

d)( (−

2+

d)Q

2+

(−6

+d)s

) m8 ∆

) M0

+( (−

2+

d)3

(1+

d)Q

6( (−

14

+17d)s

3+

3(−

2+

d)s

2m

2 ∆+

3(−

2+

d)s

m4 ∆

+(2

+d)m

6 ∆

) +

2(−

2+

d)Q

4( (−

2+

d)2

(10

+d(1

2+

d))

s4+

(−400

+d(2

52

+d(3

52

+d(−

277

+59d))

))s3

m2 ∆

+

(−112

+d(5

6+

d(5

2+

d(−

51

+11d))

))s2

m4 ∆

+(3

2+

d(4

+d(4

+(−

3+

d)d

)))s

m6 ∆−

(−2

+d)2

(2+

d)m

8 ∆

) +

Q2s

( −(−

2+

d)3

(−26

+d(−

19

+3d))

s4+

(−2

+d)(

32

+d(7

92

+7d(6

+d(−

65

+17d))

))s3

m2 ∆−

132


(−5472

+d(2

616

+d(4

160

+d(−

3358

+d(1

070

+d(−

145

+3d))

))))

s2m

4 ∆+

(−624

+d(−

8+

d(4

28

+d(−

414

+d(1

56

+d(−

41

+9d))

))))

sm6 ∆−

4(−

2+

d)2

(2+

d)(

4+

d)m

8 ∆

) +

s2( −

2(−

2+

d)3

(2+

5d)s

4+

8(−

2+

d)(

2+

d(4

+d(−

17

+8d))

)s3m

2 ∆−

(−960

+d(1

680

+d(7

72

+d(−

1842

+d(8

45

+3(−

50

+d)d

))))

)s2m

4 ∆+

(1120

+d(−

1408

+d(−

256

+d(1

430

+d(−

271

+d(−

62

+15d))

))))

sm6 ∆−

2(−

240

+d(1

36

+d(2

18

+d(−

50

+d(1

18

+d(−

62

+9d))

))))

m8 ∆

))

M2 0

+8(1

+d)s

m∆

( 4(−

2+

d)2

s4−

2(−

16

+d(2

+d))

s3m

2 ∆−

2(1

64

+d(−

132

+5d))

s2m

4 ∆+

2(8

8+

d(−

66

+5d))

sm6 ∆−

2(−

6+

d)(−2

+d)m

8 ∆+

(−2

+d)3

Q4

( (6−

5d)s

2−

2sm

2 ∆+

dm

4 ∆

) +

(−2

+d)Q

2( (−

2+

d)( −

6+

d2) s3

−2(−

6+

d)2

(−1

+d)s

2m

2 ∆+

(−20

+d(1

0+

(−6

+d)d

))sm

4 ∆−

2(−

6+

d)d

m6 ∆

))M

3 0+

( −(−

2+

d)3

(1+

d)Q

4( (−

46

+41d)s

3+

(−6

+7d)s

2m

2 ∆−

(−18

+5d)s

m4 ∆−

3(2

+d)m

6 ∆

) +

(−2

+d)Q

2( (−

2+

d)2

(−38

+5(−

7+

d)d

)s4−

( −1024

+d

( 280

+d

( 890−

429d

+57d2))

) s3m

2 ∆+

(−272

+(−

4+

d)d

(14

+d(−

43

+19d))

)s2m

4 ∆+

(−56

+d(1

00

+d(5

8+

(−25

+d)d

)))s

m6 ∆

+2(−

2+

d)2

(2+

d)m

8 ∆

) +

s( −

2(−

2+

d)3

(−2

+(−

4+

d)d

)s4−

2(−

2+

d)(

64

+d(1

40

+d(−

56

+d(−

17

+7d))

))s3

m2 ∆−

(1120

+d(1

12

+d(−

1012

+d(6

42

+d(−

289

+d(4

0+

3d))

))))

s2m

4 ∆+

(544

+d(4

32

+d(−

716

+d(1

58

+d(6

1+

d(−

44

+9d))

))))

sm6 ∆

+2(−

2+

d)(

32

+d(3

6+

(−22

+d)d

))m

8 ∆

))M

4 0+

4(−

2+

d)(

1+

d)s

m∆

( s−

m2 ∆

) 2( (−

2+

d)2

( dQ

2+

2s) −

2(−

6+

d)d

m2 ∆

) M5 0

+(−

2+

d)

( s−

m2 ∆

) 2( 2

(−2

+d)2

(2+

d)s

2−

4(−

4+

5d)(−4

+(−

2+

d)d

)sm

2 ∆+

2(−

2+

d)2

(2+

d)m

4 ∆+

(−2

+d)2

(1+

d)Q

2( (−

6+

7d)s

+3(2

+d)m

2 ∆

))

M6 0

+(−

2+

d)3

(1+

d)( s−

m2 ∆

) 2( (−

2+

d)s

+(2

+d)m

2 ∆

) M8 0−

(−2

+d)

m6 π

( (−2

+d)2

s( Q

2+

s)((−

18

+(−

9+

d)d

)Q4

+2(2

6+

(31−

3d)d

)Q2s

+(−

58

+d(−

41

+9d))

s2) −

8( (−

2+

d)2

(2+

d)Q

6−

(−2

+d(1

0+

3(−

4+

d)d

))Q

4s

+(1

4+

d(2

+d)(−1

0+

3d))

Q2s2

+(2

0+

d(−

14

+d(−

18

+7d))

)s3) m

2 ∆+

8(1

+d)s

m∆

( −(−

2+

d)( Q

2−

s)(d

( Q2−

3s) +

22s) −

8(−

3+

d)( Q

2−

2s) m

2 ∆

) M0

+( (−

2+

d)2

( (1+

d)(

2+

d)Q

6−

2(4

+3d(1

+d))

Q4s

+(−

50

+d(−

49

+17d))

Q2s2

+8(1

1+

(5−

3d)d

)s3) −

8( (−

2+

d)2

(2+

d)Q

4+

2(−

2+

d)( 2

+d2) Q

2s

+(−

6+

d(4

4+

(12−

11d)d

))s2

) m2 ∆

)

M2 0

+16(1

+d)s

m∆

( −(−

2+

d)( d

( Q2−

2s) +

3s) −

4(−

3+

d)m

2 ∆

) M3 0

+( (−

2+

d)2

( 3(1

+d)(

2+

d)Q

4−

(−10

+d(1

3+

15d))

Q2s

+2(−

16

+d(7

+11d))

s2) +

8( (−

2+

d)2

(2+

d)Q

2+

(−22

+d(1

8+

(8−

5d)d

))s) m

2 ∆

)

M4 0−

8(−

2+

d)d

(1+

d)s

m∆

M5 0

+(−

2+

d)2

(2+

d)( 3

(1+

d)Q

2−

8ds

+8m

2 ∆

) M6 0

+(−

2+

d)2

(1+

d)(

2+

d)M

8 0

) −m

4 π

( −( Q

2+

s)((−

2+

d)3

s2( 3

(−10

+(−

5+

d)d

)Q4−

2(−

54

+d(−

47

+19d))

Q2s

+3(−

3+

d)(

6+

5d)s

2) +

(−2

+d)s

( (−2

+d)2

(−30

+d(−

19

+3d))

Q4−

2(−

4+

d(6

+d(5

8+

d(−

19

+5d))

))Q

2s

+(1

68

+d(3

64

+d(−

102

+d(−

175

+51d))

))s2

) m2 ∆

+( −

12(−

2+

d)3

(2+

d)Q

4+

24(−

2+

d)(−2

+d(2

+(−

4+

d)d

))Q

2s

+(6

40

+d(−

128

+d(−

324

+d(2

34

+d(4

7+

d(−

58

+9d))

))))

s2) m

4 ∆

) +

4(1

+d)s

m∆

( (−2

+d)2

( −d2Q

2( Q

2−

3s) 2

+2d

( Q2−

7s)(

Q2−

s)(Q

2+

s) +8s

( 3Q

4+

11Q

2s−

7s2

))+

2( (−

2+

d)d

(−10

+3d)Q

4−

2(−

2+

d)(

76

+d(−

41

+4d))

Q2s

+(−

144

+d(1

76

+d(−

82

+13d))

)s2) m

2 ∆+

8( 3

(−4

+d)(−2

+d)Q

2+

(−12

+(1

6−

3d)d

)s) m

4 ∆

) M0

+( −

3(−

2+

d)3

(1+

d)(

2+

d)Q

6( s

+m

2 ∆

) +2Q

4( (−

2+

d)3

(−16

+d(−

9+

13d))

s2+

(−3

+d)(−2

+d)d

( 4+

5d2) sm

2 ∆+

6(−

2+

d)3

(2+

d)m

4 ∆

) +

(−2

+d)Q

2s

( −(−

2+

d)2

(−118

+d(−

67

+75d))

s2−

(528

+d(3

60

+d(−

198

+d(−

139

+35d))

))sm

2 ∆+

24

( −8

+d3) m

4 ∆

) +

s2( 4

(−2

+d)3

(−12

+d(5

+8d))

s2+

8(−

2+

d)(

134

+d(1

08

+d(−

87

+2d(−

12

+5d))

))s

133

B. Appendix to VVCS

m2 ∆

+(−

32

+d(−

1232

+d(5

00

+d(4

06

+d(−

255

+(5

8−

9d)d

))))

)m4 ∆

))M

2 0+

8(−

2+

d)(

1+

d)s

m∆

( −(−

2+

d)( (−

2+

d)d

Q4−

(10

+d(−

4+

3d))

Q2s

+2(−

1+

4d)s

2) +

2( −

4s

+d

( (−10

+3d)Q

2+

(17−

4d)s

))m

2 ∆+

12(−

4+

d)m

4 ∆

) M3 0

+

(−2

+d)( (−

2+

d)2

s( −

(1+

d)(

18

+5d)Q

4+

(26

+d(6

7+

29d))

Q2s−

6(4

+d)(

1+

3d)s

2) −

( 9(−

2+

d)2

(1+

d)(

2+

d)Q

4−

(−24

+d(8

4+

d(−

62

+d(−

69

+29d))

))Q

2s

+2(1

52

+d(8

4+

d(−

208

+d(−

11

+21d))

))s2

) m2 ∆

+

12

( −(−

2+

d)2

(2+

d)Q

2+

(4+

d(−

24

+d(2

+3d))

)s) m

4 ∆

) M4 0−

4(−

2+

d)(

1+

d)s

m∆

( 8s

+d

( (−2

+d)2

Q2

+20m

2 ∆−

2d

( s+

3m

2 ∆

))) M

5 0+

(−2

+d)( 1

2(−

2+

d)2

(2+

3d)s

2+

8( 1

2+

d(−

1+

2d)( −

8+

d2))

sm2 ∆−

12(−

2+

d)2

(2+

d)m

4 ∆−

(−2

+d)2

(1+

d)Q

2( (1

8+

d)s

+9(2

+d)m

2 ∆

))M

6 0+

(−2

+d)3

(1+

d)( (−

6+

d)s−

3(2

+d)m

2 ∆

) M8 0

) +

m2 π

( −( Q

2+

s)((−

2+

d)3

s3( (−

38

+d(−

11

+19d))

Q4−

2(−

26

+d(−

25

+9d))

Q2s

+(−

18

+d(−

3+

7d))

s2) +

2(−

2+

d)s

2( (−

20

+d(−

6+

d(3

2+

3(−

5+

d)d

)))Q

4+

2(−

258

+d(5

1+

d(2

06

+d(−

122

+17d))

))Q

2s

+(−

8+

d(1

64

+d(4

4+

d(−

133

+31d))

))s2

) m2 ∆

+

s( (−

2+

d)3

(−6

+d(−

11

+3d))

Q4−

2(−

392

+d(1

2+

d(4

14

+d(−

184

+d(2

3+

d))

)))Q

2

s+

(2064

+d(3

52

+d(−

1260

+d(4

18−

(−9

+d)d

(−22

+3d))

)))s

2) m

4 ∆−

( 8(−

2+

d)3

(2+

d)Q

4+

16(−

2+

d)(

7+

(−1

+d)d

)Q2s

+( −

2080

+d

( 64

+d

( 1192

+d

( −446

+d

( 215−

66d

+9d2))

))) s2

) m6 ∆

) −8(1

+d)s

m∆

( (−2

+d)2

s( (−

2+

d)d

Q6−

(28

+d(−

23

+8d))

Q4s

+(−

38

+d(4

+3d))

Q2s2−

(−14

+5d)s

3) +

( (−2

+d)3

dQ

6−

2(−

3+

d)(−2

+d)( 4

+d2) Q

4s

+(−

2+

d)(

96

+d(−

16

+d(−

24

+5d))

)Q2s2−

2(1

6+

d(−

8+

d(−

12

+5d))

)s3) m

2 ∆−

( (−2

+d)d

(−14

+3d)Q

4−

2(−

100

+d(9

6+

d(−

33

+2d))

)Q2s

+(2

16

+d(2

0+

d(−

110

+17d))

)s2) m

4 ∆−

8( (−

5+

d)(−2

+d)Q

2−

2(−

6+

d)s

) m6 ∆

) M0

+( −

3(−

2+

d)3

(1+

d)Q

6( (2

+d)s

2+

2dsm

2 ∆+

(2+

d)m

4 ∆

) +2(−

2+

d)Q

4( (−

2+

d)2

(−60

+d(−

23

+41d))

s3+

2

(−72

+d(−

6+

d(3

4+

(−13

+d)d

)))s

2m

2 ∆+

(2+

d)(−2

4+

(−3

+d)(−2

+d)d

)sm

4 ∆+

4(−

2+

d)2

(2+

d)m

6 ∆

) +

Q2s

( −(−

2+

d)3

(6+

d(−

3+

7d))

s3+

2(−

2+

d)(−1

376

+d(−

168

+d(1

028

+d(−

247

+5d))

))s2

m2 ∆−

(−2080

+d(8

+d(1

608

+d(−

782

+d(9

8+

d(−

23

+9d))

))))

sm4 ∆

+16(−

2+

d)2

(6+

d(4

+d))

m6 ∆

) +

s2( 8

(−2

+d)3

(1+

d(5

+d))

s3+

8(−

2+

d)(

18

+d(6

4+

d(2

0+

d(−

45

+8d))

))s2

m2 ∆−

2(−

880+

d(−

368

+d(5

80

+d(−

66

+d(1

41

+d(−

46

+3d))

))))

sm4 ∆

+(−

320

+d(−

400

+d(6

32

+d(1

78

+3d(1

3+

d(−

22

+3d))

))))

m6 ∆

))M

2 0+

16(1

+d)s

m∆

( −(−

2+

d)2

s( (−

2+

d)Q

4+

(−4

+d)(

1+

d)Q

2s

+(3

+2d)s

2) +

( −(−

2+

d)3

dQ

4+

(−2

+d)(−1

6+

d(4

+(−

2+

d)d

))

Q2s

+2(−

4+

d)(

12

+d(−

9+

2d))

s2) m

2 ∆+

( −92s

+d

( (−2

+d)(−1

4+

3d)Q

2+

(48

+(9−

2d)d

)s))

m4 ∆

+4(−

5+

d)(−2

+d)m

6 ∆

) M3 0

+( (−

2+

d)3

s2( (1

+d)(−2

6+

7d)Q

4+

( 118

+87d−

39d2) Q

2s

+2(−

20

+d(−

29

+3d))

s2) −

2(−

2+

d)s

( 5(−

2+

d)2

d(1

+d)Q

4−

(−4

+d(−

50

+d(−

12

+d(3

+5d))

))Q

2s

+2(−

176

+d(−

64

+d(1

30

+(−

35

+d)d

)))s

2) m

2 ∆−

( 9(−

2+

d)3

(1+

d)(

2+

d)Q

4−

(−2

+d)(

2+

d)(

36

+d(−

72

+d(−

23

+13d))

)Q2s

+(−

320+

d(8

96

+d(1

00

+d(−

362

+d(7

3+

3d(−

8+

3d))

))))

s2) m

4 ∆+

8(−

2+

d)( −

(−2

+d)2

(2+

d)Q

2+

(−14

+d(−

30

+d(1

2+

d))

)s) m

6 ∆

) M4 0

+

8(−

2+

d)(

1+

d)s

m∆

( (−2

+d)s

( (−2

+d)2

Q2−

(−4

+d)s

) −(−

2+

d)( (−

2+

d)d

Q2

+2(−

6+

d)s

) m2 ∆

+d(−

14

+3d)m

4 ∆

) M5 0

+

(−2

+d)( −

8(−

2+

d)2

(−2

+(−

2+

d)d

)s3−

8(2

4+

d(8

+5(−

4+

d)d

))s2

m2 ∆

+8(2

4+

d(−

8+

d(−

20

+d(4

+d))

))sm

4 ∆−

8(−

2+

d)2

(2+

d)m

6 ∆+

(−2

+d)2

(1+

d)Q

2( 1

1(−

2+

d)s

2−

2dsm

2 ∆−

9(2

+d)m

4 ∆

))M

6 0+

(−2

+d)3

(1+

d)( (−

2+

d)s

2+

2dsm

2 ∆−

3(2

+d)m

4 ∆

) M8 0

) +

2(−

2+

d)2

m8 π

( −8(1

+d)s

m∆

M0

( Q2−

3s

+M

2 0

) +(−

2+

d)( −

Q2−

s+

M2 0

)((1

0+

9d)s

2+

(2+

d)( Q

2+

M2 0

) 2−

2s

( (2+

3d)Q

2+

(4+

3d)M

2 0

))))

(9+

5τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1}])−

1384(−

1+

d)3

F2m

6 ∆M

0

( ie2gπN

∆2

( 4(−

2+

d)3

Q4M

4 0

( −5Q

2−

s+

5M

2 0

) +

32m

7 ∆M

0

( (−44

+d(1

7+

(−6

+d)d

))Q

2−

4(−

5+

d)(−2

+d)s

+4(−

5+

d)(−2

+d)M

2 0

) +

m8 ∆

( −(3

68

+d(−

1064

+d(4

20

+d(−

86

+9d))

))( Q

2+

s) +(7

20

+d(3

28

+d(2

0+

d(−

130

+27d))

))M

2 0

) +

m4 π

((Q

2+

s)(−4

(−2

+d)3

Q4

+4(−

2+

d)(−1

8+

7d)Q

2m

2 ∆+

(80

+d(1

12

+d(−

4+

(2−

3d)d

)))m

4 ∆

) +

134


16(−

1+

d)Q

2m

∆

( (−2

+d)2

Q2−

2(−

4+

d)(−1

+d)m

2 ∆

) M0

+( 2

0(−

2+

d)3

Q4−

4(5

2+

d(8

+d(−

33

+8d))

)Q2m

2 ∆+

(464

+d(−

368

+d(1

64

+d(−

82

+15d))

))m

4 ∆

) M2 0

) −16m

5 ∆M

0

( −(−

2+

d)(

28

+3(−

5+

d)d

)Q4−

2(−

50

+d(6

4+

d(−

23

+3d))

)Q2s

+8(−

4+

d)(−1

+d)s

2+

2( (−

3+

d)(

30

+d(−

14

+3d))

Q2−

4(8

+(−

5+

d)d

)s) M

2 0+

32M

4 0

) −16Q

2m

3 ∆M

0

( (−2

+d)( −

(−2

+d)2

Q4−

(8+

d(−

9+

2d))

Q2s

+2(−

1+

d)s

2) +

( (−2

+d)(

38

+3d(−

9+

2d))

Q2

+2(−

22

+d(2

4+

(−9

+d)d

))s) M

2 0+

4(−

4+

d)(−2

+d)M

4 0

) +

4(−

2+

d)Q

2m

2 ∆M

2 0

((28−

28d

+6d2) Q

4+

2(2

4+

d(−

27

+7d))

Q2s

+4(−

2+

d)(−1

+d)s

2+

( −5(3

8+

3d(−

9+

2d))

Q2

+(−

70

+(6

5−

16d)d

)s) M

2 0+

(70

+d(−

29

+4d))

M4 0

) −2m

6 ∆

( −2

( Q2

+s)(

(76

+d(−

76

+13d))

Q2

+4(2

2+

d(−

22

+3d))

s) +

2( (3

88−

d(1

8+

(−3

+d)d

(−5

+3d))

)Q2

+2(3

16

+d(−

173

+d(5

4+

d(−

20

+3d))

))s) M

2 0+

(−640

+d(1

084

+3d(−

104

+d(6

+d))

))M

4 0

) +

m4 ∆

( −4(−

2+

d)( Q

2+

s)((8

+(−

8+

d)d

)Q4

+4(−

3+

d)(−1

+d)Q

2s

+4(−

1+

d)2

s2) +

4( (−

368

+d(4

00

+17(−

8+

d)d

))Q

4+

2(−

276

+d(3

35

+d(−

129

+20d))

)Q2s

+2

( −32

+d

( 65−

33d

+6d2))

s2) M

2 0+

( (3408−

d(2

744

+3d(−

324

+d(4

2+

d))

))Q

2+

(1008

+d(−

1024

+d(5

80

+3d(−

54

+5d))

))s) M

4 0+

(−240

+d(3

28

+d(−

220

+(5

0−

3d)d

)))M

6 0

) +16(−

2+

d)2

Q4m

∆M

3 0

( s+

(−2

+d)( −

Q2

+M

2 0

))+

2m

2 π

( −64(−

5+

d)d

m5 ∆

M0

( −s

+M

2 0

) −4(−

2+

d)3

Q4M

2 0

( −3Q

2−

s+

5M

2 0

) −8(−

2+

d)2

Q4m

∆M

0

( −(−

2+

d)Q

2+

s+

(−3

+2d)M

2 0

) +

16Q

2m

3 ∆M

0

( (−2

+d)(

11

+d(−

9+

2d))

Q2

+(−

3+

d)(

6+

(−4

+d)d

)s+

( 10

+(−

5+

d)d

2) M

2 0

) +

m6 ∆

( −2(1

36

+d(−

278

+d(1

70

+d(−

38

+3d))

))( Q

2+

s) +(7

20

+d(1

2+

d(2

20

+3d(−

46

+7d))

))M

2 0

) +4(−

2+

d)Q

2m

2 ∆( −

( Q2

+s)(

(4+

(−6

+d)d

)Q2

+2(−

2+

d)(−1

+d)s

) +( (8

0+

d(−

56

+11d))

Q2

+4(1

1+

d(−

9+

2d))

s) M2 0

+2(−

24

+d(3

+d))

M4 0

) −m

4 ∆

( −4

( Q2

+s)(

2(1

6+

d(−

27

+5d))

Q2

+(2

0+

d(−

27

+5d))

s) +( (5

92

+d(−

364

+d(1

08

+(2−

3d)d

)))Q

2+

2(2

48

+d(−

136

+d(8

0+

d(−

28

+3d))

))s) M

2 0+

2(6

4+

d(−

46

+d(1

0+

3(−

4+

d)d

)))M

4 0

)))

(9+

5τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])−

1384(−

1+

d)3

F2m

6 ∆M

0

(ie2

gπN

∆2(3

2( −

14

+d−

4d2

+d3) Q

2m

9 ∆M

0+

8(−

2+

d)3

Q6m

∆M

0

( m2 π−

M2 0

) 2+

m10

∆

( −(3

20

+(−

8+

d)d

(68

+3(−

6+

d)d

))( Q

2+

s) +(5

76

+d(−

304

+3d(6

8+

5(−

6+

d)d

)))M

2 0

) +8(−

2+

d)3

Q6

( −m

2 πM

0+

M3 0

) 2−

32m

7 ∆M

0( (−

5+

d)( (−

2+

d)Q

4+

4(−

1+

d)Q

2s

+4(−

1+

d)s

2) +

2( −

14

+d−

4d2

+d3) Q

2m

2 π+

2( (−

24

+d(1

3+

(−6

+d)d

))Q

2−

4(−

5+

d)(−1

+d)s

) M2 0

+4(−

5+

d)(−1

+d)M

4 0

) −8(−

2+

d)Q

4m

2 ∆

( m4 π

( Q2

+s−

d( Q

2+

s) +(−

17

+4d)M

2 0

) +M

4 0

( (−3

+2d)( (−

3+

d)Q

2+

(−5

+2d)s

) +(−

31

+(1

9−

4d)d

)M2 0

) +

m2 πM

2 0

( −2(3

+(−

5+

d)d

)Q2

+(−

16

+(1

7−

4d)d

)s+

(48

+d(−

23

+4d))

M2 0

))+

8Q

2m

5 ∆M

0

( (−2

+d)( (−

2+

d)2

Q4

+4(−

3+

d)(−1

+d)Q

2s

+4(−

3+

d)(−1

+d)s

2) +

4( −

14

+d−

4d2

+d3) m

4 π+

4( −

(−2

+d)(

13

+(−

6+

d)d

)Q2−

2(−

1+

d)(

16

+(−

7+

d)d

)s) M

2 0+

8(−

20

+d(1

8+

(−7

+d)d

))M

4 0−

8m

2 π

( (−5

+d)( d

Q2

+2(−

1+

d)s

) +(−

24

+d(1

3+

(−6

+d)d

))M

2 0

))+

2m

8 ∆

( 2( Q

2+

s)((1

6+

(−16

+d)d

)Q2

+2(2

4+

(−17

+d)d

)s) +

( (96

+d(−

380

+d(1

10

+d(−

26

+3d))

))Q

2−

2(1

12

+d(−

88

+d(3

4+

3(−

4+

d)d

)))s

) M2 0−

2(6

4+

3d(5

8+

(−6

+d)d

(2+

d))

)M4 0

+m

2 π

( −(3

20

+(−

8+

d)d

(68

+3(−

6+

d)d

))( Q

2+

s) +(5

76

+d(−

304

+3d(6

8+

5(−

6+

d)d

)))M

2 0

))+

m6 ∆

( 8(−

2+

d)(−1

+d)Q

2( Q

2+

s)(Q

2+

2s) +

8( (−

54

+(4

7−

7d)d

)Q4−

(140

+d(−

107

+11d))

Q2s

+(−

80

+d(3

9+

d))

s2) M

2 0+

( (1152−

d(5

20

+d(−

244

+d(2

2+

3d))

))Q

2+

(1664

+d(−

1248

+d(3

16

+15(−

6+

d)d

)))s

) M4 0

+

135

B. Appendix to VVCS

(−768

+d(1

176

+d(−

332−

3(−

14

+d)d

)))M

6 0+

m4 π

( −(3

20

+(−

8+

d)d

(68

+3(−

6+

d)d

))( Q

2+

s) +(5

76

+d(−

304

+3d(6

8+

5(−

6+

d)d

)))M

2 0

) +

2m

2 π

( 4( Q

2+

s)((1

6+

(−16

+d)d

)Q2

+(2

4+

(−17

+d)d

)s) +

( (864

+d(−

348

+d(2

68

+d(−

74

+3d))

))Q

2−

2(1

12

+d(−

88

+d(3

4+

3(−

4+

d)d

)))s

) M2 0−

2(6

4+

3d(5

8+

(−6

+d)d

(2+

d))

)M4 0

))+

4Q

2m

4 ∆

( m4 π

( (16

+(−

16

+d)d

)( Q

2+

s) +(−

64

+d(4

8+

d(−

49

+8d))

)M2 0

) +2m

2 π

( 2(−

2+

d)(−1

+d)( Q

2+

s) 2−

( (8+

d(−

56

+11d))

Q2

+2(5

8+

5(−

9+

d)d

)s) M

2 0−

4(−

40

+d(2

9+

2(−

7+

d)d

))M

4 0

) +

M2 0

( 2(−

2+

d)( (6

+(−

6+

d)d

)Q4

+(1

8+

d(−

19

+4d))

Q2s

+4(−

3+

d)(−1

+d)s

2) +

( (216

+d(−

216

+(6

7−

8d)d

))Q

2+

(296

+d(−

316

+(1

07−

16d)d

))s) M

2 0+

(−296

+d(2

60

+d(−

107

+16d))

)M4 0

))−

16(−

2+

d)Q

4m

3 ∆M

0

( −2(−

3+

d)(−1

+d)s

(mπ−

M0)(m

π+

M0)−

Q2

( (2+

(−4

+d)d

)m2 π−

(−2

+d)2

M2 0

) +

2(m

π−

M0)(m

π+

M0)( 8

M2 0

+(−

5+

d)( m

2 π+

dM

2 0

))))

(9+

5τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])−

196(−

1+

d)3

F2m

6 ∆M

0(i

e2gπN

∆2π(−

(944

+d(−

272

+d(2

20

+d(−

130

+21d))

))m

8 ∆

( Q2

+s

+M

2 0

) +8

(−2

+d)3

Q4(m

π−

M0)(m

π+

M0)( Q

2+

m2 π−

M2 0

)(Q

2+

s+

M2 0

) +

m6 ∆

( 2( Q

2+

s)((−

272

+d(−

232

+d(1

26

+d(−

22

+3d))

))Q

2+

8(−

34

+d(−

7+

3d))

s) +( (1

440

+d(−

1760

+d(1

048

+d(−

334

+45d))

))Q

2−

( 992

+d2(2

92

+d(−

206

+33d))

) s) M2 0

+

(1984

+d(−

1296

+d(7

96

+d(−

290

+39d))

))M

4 0−

(224

+d(−

704

+3d(9

2+

(−14

+d)d

)))m

2 π

( Q2

+s

+M

2 0

))−

4(−

2+

d)Q

2m

2 ∆

( −2Q

2( Q

2+

s)((5

+(−

5+

d)d

)Q2

+2(−

2+

d)2

s) +( (6

6+

d(−

41

+10d))

Q4

+(6

8+

d(−

59

+12d))

Q2s

+4(−

4+

d)(−1

+d)s

2) M

2 0+

( (52

+d(−

49

+12d))

Q2

+2d(−

3+

2d)s

) M4 0

+2(−

12

+d)M

6 0+

4(−

1+

(−3

+d)d

)m4 π

( Q2

+s

+M

2 0

) +

m2 π

( −( Q

2+

s)((4

8+

d(−

37

+8d))

Q2

+4(−

4+

d)(−1

+d)s

) +( (−

20

+(4

7−

12d)d

)Q2

+2(2

+(9−

4d)d

)s) M

2 0+

(28

+2(5−

2d)d

)M4 0

))+

m4 ∆

( 8( Q

2+

s)((−

68

+d(7

3+

3(−

8+

d)d

))Q

4+

2(−

49

+d(5

6+

3(−

7+

d)d

))Q

2s−

4(−

3+

d)(−1

+d)s

2) +

( (784−

d(−

80

+3d(1

6+

d(2

+d))

))Q

4−

(−288

+d(2

24

+d(−

40

+3(−

6+

d)d

)))Q

2s

+8(3

6+

d(−

33

+5d))

s2) M

2 0+

( (1184−

d(3

04

+d(4

8+

d(−

46

+15d))

))Q

2+

4( 1

24

+d

( −80

+d

( 86−

39d

+6d2))

) s) M4 0−

4(3

6+

d(−

50

+d(4

8+

d(−

19

+3d))

))M

6 0+

2(1

36

+d(−

64

+d(4

0+

d(−

20

+3d))

))m

4 π

( Q2

+s

+M

2 0

) +

m2 π

((Q

2+

s)((−

96

+d(−

152

+d(8

0+

3(−

6+

d)d

)))Q

2+

8d(−

7+

3d)s

) +((−1

28

+3d(−

96

+d(6

4+

3(−

6+

d)d

)))

Q2−

2(5

28

+d(−

352

+d(2

28

+d(−

98

+15d))

))s) M

2 0+

2(−

16

+d(−

68

+d(5

6+

3(−

6+

d)d

)))M

4 0

)))

(9+

5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])−

196(−

1+

d)3

F2m

6 ∆M

0(i

e2gπN

∆2π(−

64(−

6+

d)(−1

+d)m

7 ∆M

0+

16(−

2+

d)3

Q2(m

π−

M0)(m

π+

M0)( Q

2+

m2 π−

M2 0

)(−Q

2−

s+

M2 0

) −32m

5 ∆M

0

( (86

+d(−

44

+7d))

Q2

+2(2

9+

d(−

15

+2d))

s−

4(−

4+

d)(−1

+d)m

2 π+

2(−

17

+d)M

2 0

) +

2m

6 ∆

( −(2

32

+d(3

40

+d(−

468

+d(1

82

+3(−

11

+d)d

))))

( Q2

+s) +

(−712

+d(1

196

+d(−

392

+d(2

58

+d(−

109

+15d))

)))M

2 0

) +

32(−

2+

d)2

Q2m

∆M

0

( −Q

2( Q

2+

s) +3m

4 π+

2sM

2 0+

M4 0

+m

2 π

( Q2−

2s−

4M

2 0

))−

32m

3 ∆M

0

( (36

+d(−

30

+7d))

Q4

+2(1

4+

3(−

4+

d)d

)Q2s

+2(−

2+

d)(−1

+d)s

2+

2(−

2+

d)(−1

+d)m

4 π−

2( (1

3+

(−10

+d)d

)Q2

+(−

1+

d)s

) M2 0

+2(−

1+

d)M

4 0−

2m

2 π

( (13

+2(−

4+

d)d

)Q2

+(−

1+

d)(−5

+2d)s

+(−

1+

d)M

2 0

))+

m4 ∆

( −8

( Q2

+s)(

2( 1

9+

d( −

10

+d

+d2))

Q2

+(−

1+

d)(

64

+d(−

45

+11d))

s) +( (−

3424

+d(3

048

+d(−

980

+d(1

42

+3(−

7+

d)d

))))

Q2

+(−

1904

+d(2

192

+d(−

1028

+3d(8

2+

(−7

+d)d

))))

s) M2 0

+(1

552

+d(−

1608

+d(8

68

+d(−

398

+(1

13−

15d)d

))))

M4 0

+m

2 π

( −(−

688

+d(2

72

+d(−

164

+d(8

6+

3(−

7+

d)d

))))

( Q2

+s) +

(−848

+d(5

60

+d(−

452

+d(3

26

+d(−

113

+15d))

)))M

2 0

))+

4(−

2+

d)m

2 ∆

( −2Q

2( Q

2+

s)((−

1+

(−2

+d)d

)Q2

+4(−

2+

d)(−1

+d)s

) +( (1

72

+d(−

129

+25d))

Q4

+(1

18

+d(−

131

+37d))

Q2s

+8(−

2+

d)(−1

+d)s

2) M

2 0+

136


( (−238

+(1

39−

21d)d

)Q2−

8(−

2+

d)(−1

+d)s

) M4 0

+8(−

2+

d)(−1

+d)m

4 π

( Q2

+s−

M2 0

) +

m2 π

( −( Q

2+

s)((8

6+

d(−

59

+13d))

Q2

+8(−

2+

d)(−1

+d)s

) +(2

22

+d(−

91

+5d))

Q2M

2 0+

8(−

2+

d)(−1

+d)M

4 0

)))

(9+

5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])−

196(−

1+

d)3

F2m

6 ∆M

0(i

e2gπN

∆2π(3

2(−

2+

d)2

Q4m

∆M

3 0

( Q2

+s

+M

2 0

) +8(−

2+

d)3

Q4M

4 0

( −3Q

2−

s+

3M

2 0

)

+128m

7 ∆M

0

( Q2−

5s−

d( (−

4+

d)Q

2+

(−6

+d)s

) +(−

5+

d)(−1

+d)M

2 0

) +

m8 ∆

( −(−

1+

d)(

256

+d(−

640

+d(2

44

+3(−

14

+d)d

)))( Q

2+

s) +(−

2400

+d(1

456

+d(−

1076

+d(5

86

+3d(−

49

+5d))

)))M

2 0

) +

+4m

4 π

( 2( Q

2+

s)(−(−2

+d)3

Q4

+4(−

3+

d)(−2

+d)Q

2m

2 ∆−

4( 2

+d−

4d2

+d3) m

4 ∆

) +16Q

2m

∆

( (−2

+d)2

Q2−

4(−

3+

d)m

2 ∆

) M0+

( 6(−

2+

d)3

Q4−

8( 8

+(−

4+

d)d

2) Q

2m

2 ∆+

(152

+d(−

56

+d(8

+3(−

4+

d)d

)))m

4 ∆

) M2 0

) −32Q

2m

3 ∆M

0

( (−2

+d)( Q

2+

s)(dQ

2+

2(−

1+

d)s

) +( −

(44

+5(−

6+

d)d

)Q2−

2(8

+(−

5+

d)d

)s) M

2 0+

4(−

3+

d)M

4 0

) +8(−

2+

d)Q

2m

2 ∆M

2 0( (7

+d(−

11

+3d))

Q4

+(2

1+

d(−

29

+9d))

Q2s

+4(−

2+

d)(−1

+d)s

2−

( (143

+2d(−

44

+9d))

Q2

+(4

6+

3d(−

15

+4d))

s) M2 0

+(4

2+

d(−

21

+4d))

M4 0

) +

m6 ∆

( 4( Q

2+

s)((−

48

+d(8

8+

(−25

+d)d

))Q

2+

2(−

1+

d)(

64

+(−

25

+d)d

)s) +

( (−768

+d(8

24

+3d(−

288

+d(1

34

+(−

23

+d)d

))))

Q2

+(−

2944

+d(1

392

+d(−

876

+d(4

22

+3(−

27

+d)d

))))

s) M2 0

+

(4160

+d(−

3528

+d(1

388

+d(−

554

+3(4

9−

5d)d

))))

M4 0

) −2m

4 ∆

( 4(−

2+

d)( Q

2+

s)((−

1+

2d)Q

4+

2( −

1+

d2) Q

2s

+4(−

1+

d)2

s2) +

2( (1

48

+d(−

246

+(1

01−

17d)d

))Q

4−

(−512

+d(6

88

+3d(−

97

+17d))

)Q2s−

8(−

16

+d(2

9+

3(−

5+

d)d

))s2

) M2 0

+( −

2(1

528

+d(−

896

+d(2

77

+3(−

17

+d)d

)))Q

2+

(−640

+d(8

72

+d(−

476

+(1

22−

9d)d

)))s

) M4 0

+

(144

+d(−

360

+d(2

20

+d(−

50

+3d))

))M

6 0

) +

m2 π

( −16(−

2+

d)3

Q4M

2 0

( −2Q

2−

s+

3M

2 0

) −32(−

2+

d)2

Q4m

∆M

0

( Q2

+s

+3M

2 0

) −128m

5 ∆M

0

( 7Q

2−

5s−

d( (−

2+

d)Q

2+

(−6

+d)s

) +(−

5+

d)(−1

+d)M

2 0

) +

m6 ∆

( −(−

1+

d)(

448

+d(−

352

+d(1

48

+3(−

14

+d)d

)))( Q

2+

s) +(−

1152

+d(2

224

+d(−

804

+d(3

14

+3d(−

37

+5d))

)))M

2 0

) +

8(−

2+

d)Q

2m

2 ∆

( −( Q

2+

s)((−

1+

(−2

+d)d

)Q2

+4(−

2+

d)(−1

+d)s

) +( 7

(19

+d(−

11

+2d))

Q2

+(5

8+

d(−

49

+12d))

s) M2 0

+(−

58

+13d)M

4 0

) −2m

4 ∆

( −2

( Q2

+s)(

(−32

+(−

16

+d)(−1

+d)d

)Q2

+16(−

5+

d)(−1

+d)s

) +( 2

(896

+d(−

300

+d(5

7+

d(−

19

+3d))

))Q

2+

(512

+d(−

584

+d(3

16

+9(−

10

+d)d

)))s

) M2 0

+(1

92

+d(1

36

+d(−

124

+d(1

0+

3d))

))M

4 0

) −64Q

2m

3 ∆M

0

( −(−

4+

(−1

+d)d

)s+

(−4

+d)( (−

3+

d)Q

2+

(−5

+d)M

2 0

))) +

64m

5 ∆M

0

( −2(−

5+

d)(−1

+d)s

2+

Q2

( (−2

+(9−

2d)d

)Q2

+(6

2+

3(−

7+

d)d

)M2 0

) +s

( −16Q

2+

(−5

+d)( −

5dQ

2+

2(−

1+

d)M

2 0

))))

(9+

5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])

+1

24(−

1+

d)3

F2m

6 ∆M

0(i

e2gπN

∆2πM

2 0(2

(64

+d(6

0+

d(−

2+

3(−

4+

d)d

)))m

10

∆−

4(−

2+

d)3

Q6

( m2 π−

M2 0

) 2+

m8 ∆

( 2(2

40

+d(7

6+

d(1

4+

d(−

20

+3d))

))Q

2+

(544

+3d(−

88

+d(4

8+

d(−

22

+3d))

))s

+(−

544

+3d(8

8+

d(−

48

+(2

2−

3d)d

)))M

2 0

) +

4(−

2+

d)Q

4m

2 ∆

( (−16

+3d)m

4 π+

M2 0

( (12

+d(−

9+

2d))

Q2

+(−

2+

d)(−7

+4d)s−

2(1

5+

d(−

9+

2d))

M2 0

) +

m2 π

( (−8

+(9−

2d)d

)Q2−

(−2

+d)(−7

+4d)s

+(4

6+

d(−

21

+4d))

M2 0

))−

4Q

2m

4 ∆

( (−2

+d)( (8

+(−

5+

d)d

)Q4

+(1

6+

d(−

17

+4d))

Q2s

+2(−

1+

d)(−5

+2d)s

2) +

4(−

6+

d(4

+(−

6+

d)d

))m

4 π+

( (144

+d(−

94

+(2

5−

4d)d

))Q

2+

(132

+d(−

138

+(4

9−

8d)d

))s) M

2 0+

(−136

+d(1

16

+d(−

51

+8d))

)M4 0

+

m2 π

( 4(−

8+

7d)Q

2+

(−92

+(6

2−

5d)d

)s+

(140

+d(−

94

+(5

3−

8d)d

))M

2 0

))−

m6 ∆

( 4(1

0+

d)(−8

+3d)Q

4+

4(−

124

+d(3

4+

7d))

Q2s

+8(−

28

+d(1

1+

d))

s2+

2(6

4+

d(6

0+

d(−

2+

3(−

4+

d)d

)))m

4 π+

( 2(5

36

+d(−

24

+d(4

8+

d(−

28

+3d))

))Q

2+

( 800

+d

( −760

+d

( 256−

66d

+9d2))

) s) M2 0

+(−

448

+d(7

92

+d(−

268−

3(−

14

+d)d

)))M

4 0−

m2 π

( 2(−

160

+d(7

6+

d(−

66

+d(4

+3d))

))Q

2+

( 352

+d

( −584

+d

( 272−

66d

+9d2))

) s+

(−96

+d(8

24

+d(−

280

+3d(6

+d))

))M

2 0

)))

(9+

5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

3}])−

196(−

1+

d)3

F2m

6 ∆M

0(i

e2gπN

∆2π(−

2(6

4+

d(6

0+

d(−

2+

3(−

4+

d)d

)))m

10

∆

( Q2

+s

+M

2 0

) +

2m

4 π

( 2(−

2+

d)3

Q6−

2(−

2+

d)(−1

6+

3d)Q

4m

2 ∆+

8(−

6+

d(4

+(−

6+

d)d

))Q

2m

4 ∆+

(64

+d(6

0+

d(−

2+

3(−

4+

d)d

)))m

6 ∆

)(Q

2+

s+

M2 0

) +4(−

2+

d)3

Q6M

4 0

137

B. Appendix to VVCS

( Q2

+s

+M

2 0

) +

4(−

2+

d)Q

4m

2 ∆M

2 0

( −2

( Q2

+s)(

(5+

(−5

+d)d

)Q2

+2(−

3+

d)(−1

+d)s

) +( (2

2+

d(−

7+

2d))

Q2

+12s−

5ds) M

2 0+

(32

+d(−

17

+4d))

M4 0

) +m

8 ∆((

Q2

+s)(

(−32

+d(−

576

+d(1

80

+d(−

26

+3d))

))Q

2+

32(−

3+

d)(

1+

2d)s

) +( (9

60

+d(−

1264

+d(5

32

+d(−

158

+21d))

))Q

2−

2(−

8+

3d)(−5

6+

d(3

2+

d(−

14

+3d))

)s) M

2 0+

2( 4

96

+d

( −344

+d

( 176−

66d

+9d2))

) M4 0

) +

4Q

2m

4 ∆

( (−2

+d)( Q

2+

s)((6

+(−

6+

d)d

)Q4

+4(−

4+

d)(−1

+d)Q

2s

+4(−

3+

d)(−1

+d)s

2) +

( (124

+d(−

110−

3(−

9+

d)d

))Q

4−

(−204

+d(2

38

+d(−

69

+8d))

)Q2s−

2(−

1+

d)(

54

+d(−

19

+2d))

s2) M

2 0+

( (−12

+d(−

46

+d(−

5+

4d))

)Q2

+2(8−

13(−

2+

d)d

)s) M

4 0+

2(−

74

+d(4

1+

4(−

5+

d)d

))M

6 0

) +

m6 ∆

( −8

( Q2

+s)(

(32

+d(−

25

+3d))

Q4

+2(3

3+

d(−

34

+5d))

Q2s

+8(−

5+

d)(−1

+d)s

2) +

( (528

+d(3

92

+d(−

100

+(1

0−

3d)d

)))Q

4−

(−352

+d(4

8+

d(−

36

+d(−

10

+3d))

))Q

2s

+16(2

6+

d(−

71

+17d))

s2) M

2 0+

( (−16

+d(1

112

+d(−

480

+(1

18−

15d)d

)))Q

2+

4( 2

40

+d

( 18

+d

( 47−

39d

+6d2))

) s) M4 0−

4(2

00

+d(−

230

+d(1

01

+3(−

9+

d)d

)))M

6 0

) +m

2 π

( −8(−

2+

d)3

Q6M

2 0

( Q2

+s

+M

2 0

) −8(−

2+

d)Q

4m

2 ∆

( −( Q

2+

s)((3

+(−

5+

d)d

)Q2

+2(−

3+

d)(−1

+d)s

) +( (2

1+

(−5

+d)d

)Q2

+14s−

4ds) M

2 0+

2(1

2+

(−5

+d)d

)M4 0

) −4Q

2m

4 ∆

((Q

2+

s)((1

6+

d(−

56

+9d))

Q2

+2(−

1+

d)(−3

8+

7d)s

) +( (−

140

+d(1

0+

d(−

35

+8d))

)Q2

+2(−

8+

d(4

4+

d(−

33

+4d))

)s) M

2 0+

2( −

78

+d

( 33−

22d

+4d2))

M4 0

) +

m6 ∆

((Q

2+

s)((7

68

+d(−

576

+d(3

40

+d(−

74

+3d))

))Q

2−

32(−

3+

d)(

1+

2d)s

) +((

1312

+d(−

1824

+d(8

28

+d(−

158

+9d))

))

Q2−

2(5

76

+d(−

304

+3d(6

8+

5(−

6+

d)d

)))s

) M2 0

+2(2

72

+d(−

624

+d(2

44

+3(−

14

+d)d

)))M

4 0

)))

(9+

5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}])−

112(−

1+

d)3

F2m

4 ∆M

0(i

e2gπN

∆2π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

) (2(3

04

+d(−

176

+d(1

42

+d(−

62

+9d))

))m

4 ∆+

m2 ∆

( −8(−

8+

d(−

19

+7d))

Q2−

16(6

+(−

6+

d)d

)s+

(256

+d(−

224

+d(1

48

+d(−

58

+9d))

))m

2 π+

(−160

+d(1

28

+d(−

132

+(5

8−

9d)d

)))M

2 0

) −4(−

2+

d)Q

2( (2

+d)Q

2+

(2+

3d)(m

π−

M0)(m

π+

M0)))

(9+

5τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

3}])

+1

12(−

1+

d)3

F2m

4 ∆M

0(i

e2gπN

∆2π

2( (

Q2

+s) 2

+

2( Q

2−

s) M2 0

+M

4 0

)(−(−8

+3d)(−5

6+

d(3

2+

d(−

14

+3d))

)m6 ∆

+4

( −4

+d2) Q

4(m

π−

M0)(m

π+

M0)+

4Q

2m

2 ∆

( (−2

+d)( (2

+d)Q

2+

2(−

1+

d)s

) +(−

16

+d(−

28

+9d))

m2 π

+(1

2+

(34−

11d)d

)M2 0

) −m

4 ∆( 6

4Q

2−

96s

+4d

( (28−

9d)Q

2+

2(3

7−

9d)s

) +(−

8+

3d)(−5

6+

d(3

2+

d(−

14

+3d))

)m2 π

+( −

352

+d

( 128

+d

( −136

+66d−

9d2))

) M2 0

))

(9+

5τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

2},{{

p+

q,m

∆},

3}])

+1

24(−

1+

d)3

F2m

4 ∆M

0(i

e2gπN

∆2π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

)( −

2(5

6+

d(−

252

+d(1

36

+3(−

10

+d)d

)))m

4 ∆+

m2 ∆

( 16(−

1+

d)(−7

+2d)Q

2+

8(−

3+

d)d

s+

(16

+d(9

6+

d(−

68−

3(−

6+

d)d

)))m

2 π+

(−16

+3(−

2+

d)d

(12

+(−

4+

d)d

))M

2 0

) +

4(−

2+

d)Q

2( 2

(−1

+d)Q

2+

(−2

+3d)(m

π−

M0)(m

π+

M0)))

(9+

5τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}])

+1

12(−

1+

d)3

F2m

4 ∆M

0(i

e2gπN

∆2π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

)( −

(320

+(−

8+

d)d

(68

+3(−

6+

d)d

))m

6 ∆+

8(−

2+

d)(−1

+d)Q

4(m

π−

M0)(m

π+

M0)+

4Q

2m

2 ∆( 2

(−2

+d)(−1

+d)( Q

2+

2s) +

(16

+(−

16

+d)d

)m2 π

+(−

24

+(2

8−

5d)d

)M2 0

) −m

4 ∆

( −4(1

6+

(−16

+d)d

)Q2−

8(2

4+

(−17

+d)d

)s+

(320

+(−

8+

d)d

(68

+3(−

6+

d)d

))m

2 π+

(−128−

3d(−

136

+d(6

8+

(−14

+d)d

)))M

2 0

))(9

+5τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

3}])

138


B(ν

,Q2)

=1

384(−

1+

d)4

F2m

6 ∆M

0

( ie2gπN

∆2

( −4(−

2+

d)3

(s+

m2 π−

m2 ∆

)( 4sm

2 π+

d( m

4 π+(s−

m2 ∆

)2−

2m

2 π(3

s+

m2 ∆

)))M

2 0

d(1

+d)s

2+

64(−

2+

d)2

(−1

+d)m

2 ∆M

0(m

∆+

M0)−

2(−

2+

d)3

(s+

m2 π−

m2 ∆

)( 8sm

2 π+

2d( m

4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)) +

d2( m

4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)))

M2 0(−

Q2−

3s+

M2 0)

d(1

+d)s

3−

32(−

2+

d)2

(−1

+d)m

2 ∆

( m2 π−

m2 ∆

+M

2 0

) −32(−

2+

d)(−

1+

d)M

0(2

dm

∆M

0(−

m∆

+(−

2+

d)M

0)+

(−2+

d)m

2 π((

2−

5d)m

∆+

(−2+

d)M

0) )

d+

(−2+

d)3

(2+

d) (

s+

m2 π−

m2 ∆

)( 16sm

2 π+

d2(s

+m

2 π−

m2 ∆

)2+

4d(s

2−

3sm

2 π+

m4 π−

2(s

+m

2 π)m

2 ∆+

m4 ∆

)) M2 0

( −s2+(Q

2+

M2 0)2

)

d(1

+d)s

4+

16(−

2+

d) (

m2 π−

m2 ∆

+M

2 0)(

(−6+

d+

d2)m

3 ∆+(−

10+

d2)m

2 ∆M

0+(−

6+

d+

d2)m

∆(−

m2 π+

M2 0)+

(−2+

d)2

M0(−

m2 π+

M2 0))

M0

−8(−

2+

d)3

(−1+

d) (−

m2 π+

m2 ∆

+M

2 0)(−

4m

2 πM

2 0d

+(m

2 π−

m2 ∆

+M

2 0)2

−1+

d

)

M2 0

+1

ds2

( 4( 4

sm2 π

+d

( m4 π

+( s−

m2 ∆

) 2−

2m

2 π

( s+

m2 ∆

))) M

0

( −4(1

0+

d(−

14

+5d))

m3 ∆−

(−2

+d)(

40

+d(−

28

+5d))

m2 ∆

M0−

2(−

2+

d)2

m∆

( (−1

+d)Q

2+

(−9

+d)s

+2m

2 π+

M2 0

) +(−

2+

d)3

M0

( −2Q

2−

3s−

3m

2 π+

2M

2 0

))) −

1 s

( 8( s

+m

2 π−

m2 ∆

) M0

( −(3

6+

d(2

+d)(−1

0+

3d))

m3 ∆−

(−2

+d)(

4+

d(−

4+

3d))

m2 ∆

M0

+(−

2+

d)3

M0

( −Q

2−

3m

2 π+

M2 0

) +

(−2

+d)2

m∆

( −dQ

2+

6s

+(1

6−

3d)m

2 π+

(−10

+d)M

2 0

))) −

4d(m

2 π−

m2 ∆

+M

2 0)( (−

2+

d)3

m4 ∆

+(−

2+

d)3

(m2 π−

M2 0)2−

2(−

2+

d)m

2 ∆((−

2+

d)2

m2 π−

(2+

(−4+

d)d

)M2 0))

M2 0

+

1M

0

( 8(−

2+

d)(−1

+d)( (−4

+d2) m

5 ∆+

( −16

+d2) m

4 ∆M

0+

( −4

+d2) m

∆

( m2 π−

M2 0

) 2+

(−2

+d)2

M0

( m2 π−

M2 0

) 2+

2m

2 ∆M

0

( (6−

(−2

+d)d

)m2 π

+(−

4+

d)(

2+

d)M

2 0

) +2m

3 ∆

( −( −

4+

d2) m

2 π+

( −14

+d2) M

2 0

))) −

1ds3

( 2( 8

sm2 π

+d2

( s+

m2 π−

m2 ∆

) 2+

2d

( m4 π

+( s−

m2 ∆

) 2−

2m

2 π

( s+

m2 ∆

))) M

0

( (−2

+d)3

M0

( Q2−

s+

M2 0

)(3Q

2+

2s

+3m

2 π+

M2 0

) +

(−2

+d)m

2 ∆M

0

( −(3

2+

d(−

24

+5d))

s+

(−4

+d)2

( Q2

+M

2 0

))+

(−2

+d)2

m∆

( Q2−

s+

M2 0

)((−

8+

d)s

+2m

2 π+

(−2

+d)( Q

2+

M2 0

))−

2m

3 ∆

( (32

+d(−

36

+11d))

s+

(−6

+d)(−2

+d)( Q

2+

M2 0

))))−

1 s

( 8(−

1+

d)M

0

( (56

+d(2

4+

d(−

24

+7d))

)m5 ∆

+(−

112

+d(1

16

+d(−

46

+7d))

)m4 ∆

M0−

(−2

+d)m

2 ∆M

0( −

36Q

2+

20s−

d( 7

(−4

+d)Q

2+

ds) +

4(7

+(−

3+

d)d

)m2 π

+(−

2+

d)2

M2 0

) −m

3 ∆

( −(1

6+

d(2

0+

d(−

26

+7d))

)Q2

+44s−

d(4

8+

(−20

+d)d

)s+

4(1

4+

d(−

2+

(−2

+d)d

))m

2 π+

( −4

+(−

2+

d)2

d) M

2 0

) +(−

2+

d)3

M0

( Q2

( Q2−

s) +m

4 π+

( −2Q

2+

s) M2 0

+m

2 π

( 5Q

2−

3s

+M

2 0

))+

(−2

+d)2

m∆

((dQ

2−

4s)(

Q2−

s) +(−

4+

d)m

4 π+

( −2(−

1+

d)Q

2+

(−8

+d)s

) M2 0

+m

2 π

( −14Q

2+

5dQ

2+

16s−

3ds

+(−

8+

d)M

2 0

))))

+1 s2

( 4d

( s+

m2 π−

m2 ∆

) M0

( 2(−

2+

d(−

12

+5d))

m5 ∆

+(−

104

+d(1

04

+7(−

6+

d)d

))m

4 ∆M

0−

4(−

2+

d)m

2 ∆M

0( −

2(7

+(−

5+

d)d

)Q2

+(1

4+

(−6

+d)d

)s+

(−2

+(−

2+

d)d

)m2 π

+M

2 0

) +(−

2+

d)3

M0

((Q

2−

s)(3Q

2+

s) +m

4 π+

2sM

2 0+

m2 π

( 8Q

2−

6s

+4M

2 0

))+

(−2

+d)2

m∆

( −(−

12

+d)s

2+

2m

4 π+

2(−

2+

d)Q

2( Q

2+

M2 0

) +s

( −(1

2+

d)Q

2+

(−10

+d)M

2 0

) +m

2 π

( −4Q

2+

3dQ

2+

6s−

3ds

+(−

10

+3d)M

2 0

))+

m3 ∆

( −(1

00

+d(−

72

+5d(2

+d))

)s−

4(1

3+

3(−

4+

d)d

)m2 π

+(−

2+

d)( (2

0+

(−6

+d)d

)Q2

+( −

8+

d2) M

2 0

))))

)

(9+

5τ[3

])II

[d][{{

0,m

π},

1}])

+1

96(−

1+

d)4

F2m

6 ∆M

3 0(i

(−2

+d)e

2gπN

∆2

( m2 π−

(m∆

+M

0)2

)( −(

−2+

d)3

m6 ∆−

6(−

4+

d)(−2

+d)m

5 ∆M

0+

(−2

+d)2

(mπ−

M0)2

(mπ

+M

0)2

( (−2

+d)m

2 π+

dM

2 0

) +

2(−

2+

d)m

∆M

0

( −m

2 π+

M2 0

)(3(−

4+

d)m

2 π+

(8+

d)M

2 0

) −4m

3 ∆M

0

( −3(−

4+

d)(−2

+d)m

2 π+

( −10

+d2) M

2 0

) +

m4 ∆

( 3(−

2+

d)3

m2 π−

( 16−

16d

+d3) M

2 0

) −m

2 ∆

( 3(−

2+

d)3

m4 π−

2d(2

+(−

4+

d)d

)m2 πM

2 0−

(48

+d(−

56

+d(6

+d))

)M4 0

))

(9+

5τ[3

])II

[d][{{

0,m

π},

1},{{

p,m

∆},

1}])

+1

384(−

1+

d)4

F2m

6 ∆M

0(i

e2gπN

∆2M

0

(4(−

2+

d)3

( m4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)) 2

M0

(1+

d)s

2+

139

B. Appendix to VVCS

2(−

2+

d)3

( m4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆))(

(s+

m2 π−

m2 ∆

)2+

m4 π

+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)

1+

d

)M

0(−

Q2−

3s+

M2 0)

s3

−(−

2+

d)3

(−1+

d)

3( m

4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)) 2

−1+

d2

+(s

+m

2 π−

m2 ∆

)2((s

+m

2 π−

m2 ∆

)2+

6( m

4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆))

−1+

d

)

M0

( −s2+(Q

2+

M2 0)2

)

s4

+

1 s2

( 4( s

+m

2 π−

m2 ∆

)(m

4 π+

( s−

m2 ∆

) 2−

2m

2 π

( s+

m2 ∆

))( 4

(10

+d(−

14

+5d))

m3 ∆

+(−

2+

d)(

40

+d(−

28

+5d))

m2 ∆

M0+

(−2

+d)3

M0

( 2Q

2+

3s

+3m

2 π−

2M

2 0

) +2(−

2+

d)2

m∆

( (−1

+d)Q

2+

(−9

+d)s

+2m

2 π+

M2 0

))) +

1 s

( 8( m

4 π+

( s−

m2 ∆

) 2−

2m

2 π

( s+

m2 ∆

))( −

(36

+d(2

+d)(−1

0+

3d))

m3 ∆−

(−2

+d)(

4+

d(−

4+

3d))

m2 ∆

M0

+(−

2+

d)3

M0

( −Q

2−

3m

2 π+

M2 0

) +

(−2

+d)2

m∆

( −dQ

2+

6s

+(1

6−

3d)m

2 π+

(−10

+d)M

2 0

))) +

1 s3

( 2( s

+m

2 π−

m2 ∆

)((2

+d)m

4 π+

(2+

d)( s−

m2 ∆

) 2+

2m

2 π

( (−4

+d)s−

(2+

d)m

2 ∆

))( (−

2+

d)3

M0

( Q2−

s+

M2 0

)(3Q

2+

2s

+3m

2 π+

M2 0

) +(−

2+

d)m

2 ∆M

0

( −(3

2+

d(−

24

+5d))

s+

(−4

+d)2

( Q2

+M

2 0

))+

(−2

+d)2

m∆

( Q2−

s+

M2 0

)((−

8+

d)s

+2m

2 π+

(−2

+d)( Q

2+

M2 0

))−

2m

3 ∆

( (32

+d(−

36

+11d))

s+

(−6

+d)(−2

+d)( Q

2+

M2 0

))))

+

32(−

1+

d)m

∆

( (−10

+(−

2+

d)d

)m4 ∆

+4(−

6+

d)m

3 ∆M

0+

2(−

2+

d)m

∆M

0

( Q2

+6m

2 π−

2M

2 0

) +

m2 ∆

( (−2

+d)d

( Q2

+s) +

6(5

+(−

3+

d)d

)m2 π

+(−

10

+(8−

3d)d

)M2 0

) +(−

2+

d)2

( m4 π

+( −

Q2

+s) M

2 0+

m2 π

( 3Q

2−

3s

+M

2 0

))) +

1 s

( 8(−

1+

d)( s

+m

2 π−

m2 ∆

)((5

6+

d(2

4+

d(−

24

+7d))

)m5 ∆

+(−

112

+d(1

16

+d(−

46

+7d))

)m4 ∆

M0−

(−2

+d)m

2 ∆M

0( −

36Q

2+

20s−

d( 7

(−4

+d)Q

2+

ds) +

4(7

+(−

3+

d)d

)m2 π

+(−

2+

d)2

M2 0

) −m

3 ∆

( −(1

6+

d(2

0+

d(−

26

+7d))

)Q2

+44s−

d(4

8+

(−20

+d)d

)s+

4(1

4+

d(−

2+

(−2

+d)d

))m

2 π+

( −4

+(−

2+

d)2

d) M

2 0

) +(−

2+

d)3

M0

( Q2

( Q2−

s) +m

4 π+

( −2Q

2+

s) M2 0

+m

2 π

( 5Q

2−

3s

+M

2 0

))+

(−2

+d)2

m∆

((dQ

2−

4s)(

Q2−

s) +(−

4+

d)m

4 π+

( −2(−

1+

d)Q

2+

(−8

+d)s

) M2 0

+m

2 π

( −14Q

2+

5dQ

2+

16s−

3ds

+(−

8+

d)M

2 0

))))−

1 s2

( 4( −4

sm2 π

+d

( s+

m2 π−

m2 ∆

) 2) (

2(−

2+

d(−

12

+5d))

m5 ∆

+(−

104

+d(1

04

+7(−

6+

d)d

))m

4 ∆M

0−

4(−

2+

d)m

2 ∆M

0( −

2(7

+(−

5+

d)d

)Q2

+(1

4+

(−6

+d)d

)s+

(−2

+(−

2+

d)d

)m2 π

+M

2 0

) +(−

2+

d)3

M0

((Q

2−

s)(3Q

2+

s) +m

4 π+

2sM

2 0+

m2 π

( 8Q

2−

6s

+4M

2 0

))+

(−2

+d)2

m∆

( −(−

12

+d)s

2+

2m

4 π+

2(−

2+

d)Q

2( Q

2+

M2 0

) +s

( −(1

2+

d)Q

2+

(−10

+d)M

2 0

) +m

2 π

( −4Q

2+

3dQ

2+

6s−

3ds

+(−

10

+3d)M

2 0

))+

m3 ∆

( −(1

00

+d(−

72

+5d(2

+d))

)s−

4(1

3+

3(−

4+

d)d

)m2 π

+(−

2+

d)( (2

0+

(−6

+d)d

)Q2

+( −

8+

d2) M

2 0

))))

)

(9+

5τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1}])

+1

12(−

1+

d)3

F2m

5 ∆M

0(i

e2gπN

∆2M

0

( m4 π

( (−2

+d)2

Q2

+2m

∆(m

∆+

dm

∆+

2(−

2+

d)M

0)) +

(−m

∆+

M0)(m

∆+

M0)

( −2(−

15

+d)m

4 ∆−

4(−

10

+d)m

3 ∆M

0+

(−2

+d)2

Q2M

2 0+

4(−

2+

d)m

∆M

0

( −Q

2+

M2 0

) +m

2 ∆

( −(−

2+

d)( (2

+d)Q

2+

2(−

1+

d)s

) +2(3

+(−

2+

d)d

)M2 0

))−

2m

2 π

( 2(−

7+

d)m

4 ∆+

12m

3 ∆M

0+

(−2

+d)2

Q2M

2 0+

2(−

2+

d)m

∆M

0

( −Q

2+

2M

2 0

) +m

2 ∆

( −(−

2+

d)( (−

2+

d)Q

2+

(−1

+d)s

) +(4

+(−

1+

d)d

)M2 0

)))

(9+

5τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)3

F2m

4 ∆M

0(i

e2gπN

∆2

( (−2

+d)Q

2−

8m

2 ∆

) (mπ−

m∆−

M0)(m

π+

m∆−

M0)M

0(m

π−

m∆

+M

0)

(m∆

+M

0)(m

π+

m∆

+M

0)(9

+5τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])

−1

6(−

1+

d)3

F2m

6 ∆M

0(2

ie2gπN

∆2πM

0

( (−2

+d)Q

2( (2

+(−

1+

d)d

)Q2

+2(−

1+

d)d

s) m3 ∆

+( (−

22

+(−

2+

d)d

(−4

+3d))

Q2

+2(−

9+

d(1

4+

3(−

4+

d)d

))s) m

5 ∆+

2( 5

+(−

4+

d)d

2) m

7 ∆+

m2 ∆

( (−2

+d)Q

2( (8

+(−

4+

d)d

)Q2

+2(−

2+

d)(−1

+d)s

) +( (−

84

+d(4

4+

d(−

16

+3d))

)Q2

+2(−

2+

d)(

14

+3(−

3+

d)d

)s) m

2 ∆+

2(−

16

+(−

4+

d)d

(1+

d))

m4 ∆

) M0−

m∆

( (−2

+d)2

(1+

d)Q

4+

2( (3

+d(1

6+

d(−

13

+3d))

)Q2

+s

+(−

4+

d)(−2

+d)d

s) m2 ∆

+2(1

5+

d(2

0+

d(−

14

+3d))

)m4 ∆

) M2 0

+( −

(−2

+d)3

Q4−

2(−

2+

d)( (1

6+

3(−

4+

d)d

)Q2

+(−

2+

d)(−1

+d)s

) m2 ∆−

2(−

2+

3d)(

8+

(−5

+d)d

)m4 ∆

) M3 0

+

m∆

( (−2

+d)3

Q2−

4dm

2 ∆

) M4 0

+(−

2+

d)( (−

2+

d)2

Q2−

4m

2 ∆

) M5 0

+

m4 π

( (−2

+d)3

Q2m

∆−

2(1

+d(1

0+

(−6

+d)d

))m

3 ∆+

(−2

+d)3

Q2M

0−

2(−

2+

d)(

4+

(−3

+d)d

)m2 ∆

M0

) +

m2 π

( −4(2

+(−

5+

d)d

)m5 ∆−

4(4

+(−

7+

d)d

)m4 ∆

M0−

(−2

+d)3

Q2M

0

( −Q

2+

2M

2 0

) +(−

2+

d)2

Q2m

∆

( Q2

+dQ

2−

2(−

2+

d)M

2 0

) +

140


2(−

2+

d)m

2 ∆M

0

( (12

+d(−

9+

2d))

Q2

+(−

2+

d)(−1

+d)s

+(6

+(−

3+

d)d

)M2 0

) +

2m

3 ∆

( 7Q

2+

s+

(−2

+d)d

( 2(−

3+

d)Q

2+

(−4

+d)s

) +(1

+d(1

2+

(−6

+d)d

))M

2 0

)))

(9+

5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])

+1

6(−

1+

d)3

F2m

5 ∆M

0(2

ie2gπN

∆2πM

0

( −(2

0+

3(−

4+

d)d

)m4 ∆−

2(1

8+

d(−

11

+3d))

m3 ∆

M0−

2(−

3+

d)(−2

+d)

m∆

M0

( Q2

+m

2 π−

M2 0

) +(−

2+

d)2

(mπ−

M0)(m

π+

M0)( 2

Q2

+m

2 π−

M2 0

) +2m

2 ∆

( (−2

+d)( Q

2+

(−1

+d)s

) +(8

+(−

4+

d)d

)m2 π

+(−

6+

(5−

2d)d

)M2 0

))(9

+5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

6(−

1+

d)3

F2m

5 ∆M

0

( 2ie

2gπN

∆2πM

0

( −4(−

3+

d)m

6 ∆−

4(−

9+

2d)m

5 ∆M

0+

(−2

+d)2

Q2

( m2 π−

M2 0

) 2+

4(−

2+

d)m

∆(m

π−

M0)M

0(m

π+

M0)( Q

2+

m2 π−

M2 0

) +

4m

3 ∆M

0

( −(−

2+

d)Q

2+

(−7

+d)m

2 π+

(13−

3d)M

2 0

) +m

4 ∆

( −(−

2+

d)( d

Q2

+2(−

1+

d)s

) +8(−

3+

d)m

2 π+

2(3

0+

(−7

+d)d

)M2 0

) −2m

2 ∆

( 2(−

3+

d)m

4 π+

m2 π

( −(−

2+

d)( Q

2+

(−1

+d)s

) +(1

4+

(−7

+d)d

)M2 0

) +M

2 0

( (−2

+d)( 3

Q2

+(−

1+

d)s

) −(8

+(−

5+

d)d

)M2 0

)))

(9+

5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])

+1

6(−

1+

d)3

F2m

4 ∆M

0

( 8ie

2gπN

∆2π

( (−2

+d)Q

2−

8m

2 ∆

) (mπ−

m∆−

M0)(m

π+

m∆−

M0)

M2 0(m

π−

m∆

+M

0)(m

π+

m∆

+M

0)(9

+5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

3}]−

16(−

1+

d)3

F2m

6 ∆M

0( ie

2gπN

∆2πM

0

( m4 π

( (−2

+d)2

dQ

4m

∆−

4(−

3+

d)d

Q2m

3 ∆+

4( 2

+d−

4d2

+d3) m

5 ∆+

( (−2

+d)3

Q4−

4(−

2+

d)2

Q2m

2 ∆+

4( 1

0+

d−

4d2

+d3) m

4 ∆

) M0

) +

(m∆

+M

0)( m

4 ∆

( (−2

+d)( (8

+(−

2+

d)d

)Q4

+4(3

+(−

2+

d)d

)Q2s

+4(−

1+

d)2

s2) +

4( (−

10

+(−

5+

d)d

)Q2

+2(−

5+

d)(

1+

d)s

) m2 ∆

+

4( 2

+d−

4d2

+d3) m

4 ∆

) −2(−

2+

d)Q

2( (−

2+

d)Q

2+

2(−

3+

d)s

) m3 ∆

M0

+4

( (−4

+d)Q

2+

4(−

5+

d)s

) m5 ∆

M0

+32m

7 ∆M

0−

2m

2 ∆

( (−2

+d)Q

2( (6

+(−

3+

d)d

)Q2

+2(−

2+

d)(−1

+d)s

) +2

( (−20

+d(6

+(−

4+

d)d

))Q

2+

2(−

1+

d)(

7+

(−4

+d)d

)s) m

2 ∆+

4(1

+d)

(1+

(−4

+d)d

)m4 ∆

) M2 0

+2m

∆

( Q2

+2m

2 ∆

)((−

2+

d)2

Q2−

4(−

3+

d)m

2 ∆

) M3 0

+( (−

2+

d)3

Q4

+4(−

2+

d)3

Q2m

2 ∆+

8(−

1+

d(9

+(−

5+

d)d

))m

4 ∆

) M4 0

) −2m

2 π

( 4( 2

+d−

4d2

+d3) m

7 ∆+

4( 1

0+

d−

4d2

+d3) m

6 ∆M

0+

(−2

+d)2

dQ

4m

∆M

2 0+

(−2

+d)3

Q4M

3 0+

(−2

+d)Q

2m

2 ∆M

0

( −(6

+(−

4+

d)d

)Q2−

2(−

2+

d)(−1

+d)s

+2(−

3+

d)(−2

+d)M

2 0

) −Q

2m

3 ∆

( d(2

+(−

4+

d)d

)Q2

+2(−

3+

d)(−1

+d)d

s+

4( Q

2+

s) −2

( 2+

(−3

+d)2

d) M

2 0

) +

4m

5 ∆

( −(5

+d)Q

2+

(−5

+d)(

1+

d)s

+(7

+d(5

+(−

5+

d)d

))M

2 0

) +4m

4 ∆M

0

( (11−

4d)Q

2+

(−5

+d)(−1

+d)s

+(5

+d(7

+(−

5+

d)d

))M

2 0

)))

(9+

5τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}])−

16(−

1+

d)3

F2m

6 ∆M

0

( 16ie

2gπN

∆2π

2M

0

( −2(−

6+

d)(−1

+d)m

7 ∆+

2(4

+d(1

8+

(−7

+d)d

))m

6 ∆M

0+

(−2

+d)m

4 π

( 3(−

2+

d)Q

2m

∆−

2(−

1+

d)m

3 ∆+

(−2

+d)( (−

2+

d)Q

2−

2(−

1+

d)m

2 ∆

) M0

) +

(−2

+d)3

Q2M

3 0

( −Q

2+

M2 0

) −m

5 ∆

( (86

+d(−

44

+7d))

Q2

+2(2

9+

d(−

15

+2d))

s+

2(−

17

+d)M

2 0

) −m

4 ∆M

0

( −(−

2+

d)(

32

+3(−

6+

d)d

)Q2−

2(−

1+

d)(

26

+d(−

14

+3d))

s+

(−68

+6d(1

0+

(−5

+d)d

))M

2 0

) −(−

2+

d)m

2 ∆M

0

( −Q

2( (6

+(−

6+

d)d

)Q2

+2(−

3+

d)d

s) +2

( (22

+d(−

14

+3d))

Q2

+(−

2+

d)(−1

+d)s

) M2 0

) +(−

2+

d)2

Q2m

∆

( −Q

2( Q

2+

s) +2sM

2 0+

M4 0

) −m

3 ∆

( (36

+d(−

30

+7d))

Q4

+2(1

4+

3(−

4+

d)d

)Q2s

+2(−

2+

d)(−1

+d)s

2−

2( (1

3+

(−10

+d)d

)Q2

+(−

1+

d)s

) M2 0

+2(−

1+

d)M

4 0

) +

m2 π

( 4(−

4+

d)(−1

+d)m

5 ∆+

4(−

8+

(−1

+d)d

)m4 ∆

M0

+(−

2+

d)2

Q2m

∆

( Q2−

2s−

4M

2 0

) −(−

2+

d)3

Q2M

0

( −Q

2+

2M

2 0

) +2m

3 ∆( (1

3+

2(−

4+

d)d

)Q2

+(−

1+

d)(−5

+2d)s

+(−

1+

d)M

2 0

) +2(−

2+

d)m

2 ∆M

0

( (20

+d(−

11

+2d))

Q2

+(−

2+

d)(−1

+d)s

+(−

2+

d)(−1

+d)M

2 0

)))

(9+

5τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

3}])−

16(−

1+

d)3

F2m

6 ∆M

0( 8

ie2gπN

∆2π

2M

0

( 4( 2

+d−

4d2

+d3) m

8 ∆M

0+

(−2

+d)3

Q4M

0

( m2 π−

M2 0

) 2+

2(−

2+

d)2

Q4m

∆(m

π−

M0)(m

π+

M0)( −

Q2−

s+

2m

2 π−

M2 0

) +

8m

7 ∆

( Q2−

5s−

d( (−

4+

d)Q

2+

(−6

+d)s

) +(−

5+

d)(−1

+d)M

2 0

) −4m

6 ∆M

0

( 3(−

8+

(−3

+d)d

)Q2

+2(−

5+

d)(−1

+d)s

+2

( 2+

d−

4d2

+d3) m

2 π+

2(−

3+

d)(−1

+d)2

M2 0

) −4m

5 ∆

( (2+

d(−

9+

2d))

Q4

+(1

6+

5(−

5+

d)d

)Q2s

+2(−

5+

d)(−1

+d)s

2+

( −(6

2+

3(−

7+

d)d

)Q2−

2(−

5+

d)(−1

+d)s

) M2 0+

2m

2 π

( 7Q

2−

5s−

d( (−

2+

d)Q

2+

(−6

+d)s

) +(−

5+

d)(−1

+d)M

2 0

))+

m4 ∆

M0

( (40

+(−

8+

d)(−2

+d)d

)Q4

+4(−

2+

d(1

6+

(−7

+d)d

))Q

2s+

4(−

2+

d)(−1

+d)2

s2+

4( 2

+d−

4d2

+d3) m

4 π+

4( −

(−44

+d(2

2+

(−7

+d)d

))Q

2−

2(−

1+

d)(

7+

(−4

+d)d

)s) M

2 0+

8(−

1+

d)(

5+

(−4

+d)d

)M4 0−

8m

2 π

( (19−

5d)Q

2+

5s

+(−

6+

d)d

s+

(−3

+d)(−1

+d)2

M2 0

))−

2(−

2+

d)Q

2m

2 ∆M

0

( 2Q

2( Q

2+

s) +2(−

3+

d)m

4 π+

( (12

+(−

6+

d)d

)Q2

+2(−

2+

d)(−1

+d)s

) M2 0−

2(5

+(−

4+

d)d

)M4 0+

141

B. Appendix to VVCS

m2 π

( −(1

2+

(−6

+d)d

)Q2−

2(−

2+

d)(−1

+d)s

+2(8

+(−

5+

d)d

)M2 0

))−

2Q

2m

3 ∆

( (−2

+d)( Q

2+

s)(dQ

2+

2(−

1+

d)s

) +8(−

3+

d)m

4 π+

( −(4

4+

5(−

6+

d)d

)Q2−

2(8

+(−

5+

d)d

)s) M

2 0+

4(−

3+

d)M

4 0+

2m

2 π

( −(−

4+

(−1

+d)d

)s+

(−4

+d)( (−

3+

d)Q

2+

(−5

+d)M

2 0

))))

(9+

5τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

2},{{

p+

q,m

∆},

3}])−

16(−

1+

d)3

F2m

6 ∆M

0( 8

ie2gπN

∆2π

2M

2 0

( 2(1

8+

d(2

+(−

5+

d)d

))m

6 ∆+

(−2

+d)3

Q2(m

π−

M0)(m

π+

M0)( Q

2+

m2 π−

M2 0

) +

m4 ∆

( (12

+d(6

+d(−

10

+3d))

)Q2

+2(−

1+

d)(

16

+3(−

4+

d)d

)s+

4(−

11

+3d)m

2 π+

(60−

2d(2

6+

3(−

5+

d)d

))M

2 0

) +

(−2

+d)m

2 ∆

( −2(−

2+

d)(−1

+d)m

4 π+

2(−

2+

d)(−1

+d)s

( Q2−

M2 0

) +

2m

2 π

( (11

+2(−

4+

d)d

)Q2

+(−

2+

d)(−1

+d)s

+(−

2+

d)(−1

+d)M

2 0

) +Q

2( (−

2+

d)d

Q2

+( −

30

+22d−

6d2) M

2 0

)))

(9+

5τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}])−

16(−

1+

d)3

F2m

6 ∆M

0

( 8ie

2gπN

∆2π

2M

2 0

( 4( 2

+d−

4d2

+d3) m

8 ∆+

(−2

+d)3

Q4

( m2 π−

M2 0

) 2−

4m

6 ∆

( −(4

+(−

7+

d)d

)Q2−

2(−

5+

d)(−1

+d)s

+2

( 2+

d−

4d2

+d3) m

2 π+

2(−

1+

d)(−7

+(−

2+

d)d

)M2 0

) +

m4 ∆

( (−2

+d)( d

Q2

+2(−

1+

d)s

) 2+

4( 2

+d−

4d2

+d3) m

4 π+

4((

2−

(−3

+d)d

2) Q

2−

2(−

1+

d)(

7+

(−4

+d)d

)s) M

2 0+

8(−

1+

d)(

5+

(−4

+d)d

)M4 0−

8m

2 π

( −(1

+d)Q

2+

(−5

+d)(−1

+d)s

+(−

3+

d)(−1

+d)2

M2 0

))−

2(−

2+

d)Q

2m

2 ∆( 2

(−3

+d)m

4 π+

m2 π

( −(−

2+

d)( d

Q2

+2(−

1+

d)s

) +2(8

+(−

5+

d)d

)M2 0

) +M

2 0

( (2+

(−2

+d)d

)Q2

+2(−

2+

d)(−1

+d)s−

2(5

+(−

4+

d)d

)M2 0

)))

(9+

5τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

3}])

Dia

gra

mk)+

l)

A(ν

,Q2)

=−

196(−

1+

d)2

F2m

2 ∆M

2 0( e2

gπN

∆2(1

+τ[3

])

(1

dsM

2 0

( 2( −2

4sm

2 πM

2 0+

d2

( (Q

2−

s)(m

2 π−

m2 ∆

) 2−

( −Q

2s−

2s2

+m

4 π+

2sm

2 ∆+

m4 ∆

+2m

2 π

( 9s−

m2 ∆

))M

2 0+

2sM

4 0

)−

2d

( (Q

2−

s)(m

2 π−

m2 ∆

) 2+

( Q2−

s) m∆

( −m

2 π+

m2 ∆

) M0−

( −Q

2s−

2s2

+m

4 π+

2sm

2 ∆+

m4 ∆

+m

2 π

( 24s−

2m

2 ∆

))M

2 0+

( −m

2 πm

∆+

m3 ∆

) M3 0

+2sM

4 0

))II

[d][{{

0,m

π},

1}]) −

1−

sM

2 0+

M4 0

( 2( Q

2−

s+

M2 0

)(−(−2

+d)( m

2 π−

m2 ∆

) −2m

∆M

0+

(−2

+d)M

2 0

)(( m

2 π−

m2 ∆

) 2−

2( m

2 π+

m2 ∆

) M2 0

+M

4 0

) II[d

][{{

0,m

π},

1},{{

p,m

∆},

1}])

+

(−2

+d)( Q

2+

4m

2 π

)(s−

6m

2 π−

2m

2 ∆+

M2 0

) II[d

][{{

0,m

π},

1},{{

q,m

π},

1}]

+1

s(−

s+

M2 0)

( 2( m

4 π+

( s−

m2 ∆

) 2−

2m

2 π

( s+

m2 ∆

))( −

(−2

+d)( −

s+

m2 π

)(Q

2+

s−

M2 0

) +(−

2+

d)m

2 ∆

( Q2

+s−

M2 0

) −2m

∆M

0

( Q2−

s+

M2 0

))II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1}]) +

1−

s+

M2 0

( 16(−

1+

d)π

( 4Q

2m

3 ∆M

0+

(−2

+d)m

4 ∆

( −Q

2−

s+

M2 0

) +(−

2+

d)( −

s+

m2 π

) (mπ−

M0)(m

π+

M0)( −

Q2−

s+

M2 0

) +2m

∆M

0( s

( Q2

+s) −

2Q

2m

2 π+

( Q2−

2s) M

2 0+

M4 0

) −m

2 ∆

( −( Q

2+

s)(2(−

1+

d)Q

2+

ds

+2(−

2+

d)m

2 π

) +( (−

6+

d)Q

2+

4s

+2(−

2+

d)m

2 π

) M2 0

+(−

4+

d)M

4 0

))

II[2

+d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1}])−

1−

s+

M2 0

( 16(−

1+

d)π

Q2

( −(−

2+

d)m

4 π+

m2 ∆

( 2(−

1+

d)Q

2−

(−2

+d)m

2 ∆

) +4m

3 ∆M

0+

2(2

+d)m

2 ∆M

2 0+

4m

∆M

3 0−

(−2

+d)M

4 0+

2m

2 π

( (−2

+d)m

2 ∆−

2m

∆M

0+

(−2

+d)M

2 0

))II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

pr+

q,m

∆},

1}]))

)

B(ν

,Q2)

=1

24(−

1+

d)2

F2m

2 ∆M

3 0

( ie2gπN

∆2(1

+τ[3

])(

1Q

2s2

((2sM

0

( Q2m

∆

( m2 π−

m2 ∆

) +3Q

2( m

2 π−

m2 ∆

) M0

+4sm

∆M

2 0+

4sM

3 0

) +d

( 2Q

2s

( m2 π−

m2 ∆

) 2+

6Q

2sm

∆

( −m

2 π+

m2 ∆

) M0

+Q

2( −

9s

+2m

2 π−

2m

2 ∆

)(m

2 π−

m2 ∆

) M2 0−

8s2

142


m∆

M3 0−

8s2

M4 0

) +d2

( −Q2s

( m2 π−

m2 ∆

) 2+

2Q

2sm

∆

( m2 π−

m2 ∆

) M0−

Q2

( m2 π−

m2 ∆

)(−3

s+

m2 π−

m2 ∆

) M2 0

+2s2

m∆

M3 0

+2s2

M4 0

))II

[d][{{

0,m

π},

1}])

+

1−

s+

M2 0

( (−m

2 π+

m2 ∆

+2m

∆M

0+

M2 0

)((−

2+

d)d

( m2 π−

m2 ∆

) 2+

2(−

1+

d)m

∆

( m2 π−

m2 ∆

) M0

+((

2+

d−

d2) m

2 π+

( 2−

d+

d2) m

2 ∆

) M2 0−

2(−

1+

d)m

∆M

3 0+

(−2

+d)M

4 0

)

II[d

][{{

0,m

π},

1},{{

p,m

∆},

1}]

[{}])−

(−2+

d) (

Q2+

4m

2 π)M

3 0(m

∆+

M0)I

I[d][{{

0,m

π},

1},{{

q,m

π},

1}]

Q2

−1

s2(−

s+

M2 0)

( M3 0

( 2s

( −s

+m

2 π

)(s

+(1

+(−

3+

d)d

)m2 π

) m∆

+2s

( (4+

(−3

+d)d

)s−

2(1

+(−

3+

d)d

)m2 π

) m3 ∆

+

2(1

+(−

3+

d)d

)sm

5 ∆−

(−2

+d)( −

s+

m2 π

) 2( −

s+

dm

2 π

) M0

+(−

2+

d)( −

s+

m2 π

)(2s−

ds

+3dm

2 π

) m2 ∆

M0

+

(−2

+d)( −

3s

+2ds−

3dm

2 π

) m4 ∆

M0

+(−

2+

d)d

m6 ∆

M0

) II[d

][{{

0,m

π},

1},{{

p+

q,m

∆},

1}]) +

1−

s+

M2 0

( 2(−

1+

d)M

3 0

( m4 π

(m∆−

dm

∆−

(−2

+d)M

0)+

m2 π

( (−1

+d)m

∆

( s+

2m

2 ∆

) +(−

2+

d)s

M0

+2(−

1+

d)m

2 ∆M

0+

(−3

+d)m

∆M

2 0+

(−2

+d)M

3 0

) +

(m∆

+M

0)( (−

1+

d)m

2 ∆

( 2Q

2+

s−

m2 ∆

) −m

∆

( s+

m2 ∆

) M0

+(−

2+

d)( −

s+

m2 ∆

) M2 0

+2m

∆M

3 0

))II

[d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

1}]) +

1−

s+

M2 0

( 2(−

1+

d)M

3 0

( m4 π

((−1

+d)m

∆+

(−2

+d)M

0)−

2m

2 π(m

∆+

M0)( (−

1+

d)m

2 ∆+

(−2

+d)M

2 0

) +

(m∆

+M

0)( (−

1+

d)m

2 ∆

( −2Q

2+

m2 ∆

) +m

3 ∆M

0+

(3−

2d)m

2 ∆M

2 0−

m∆

M3 0

+(−

2+

d)M

4 0

))II

[d][{{

0,m

π},

1},{{

q,m

π},

1},{{

pr+

q,m

∆},

1}]) +

1−

s+

M2 0

( 8(−

1+

d)π

M4 0

( −(−

2+

d)m

4 ∆+

2m

3 ∆M

0+

(−2

+d)( −

s+

m2 π

) (−m

π+

M0)(m

π+

M0)+

2m

∆M

0

( −m

2 π+

M2 0

) +m

2 ∆

( −2Q

2+

2(−

2+

d)m

2 π+

d( 2

Q2

+s

+M

2 0

)))

II[2

+d][{{

0,m

π},

1},{{

q,m

π},

1},{{

p+

q,m

∆},

2}])

+1

−s+

M2 0

( 8(−

1+

d)π

M4 0

( (−2

+d)m

4 π+

m2 ∆

( −2(−

1+

d)Q

2+

(−2

+d)m

2 ∆

) −2m

3 ∆M

0−

2dm

2 ∆M

2 0−

2m

∆M

3 0+

(−2

+d)M

4 0+

m2 π

( −2(−

2+

d)m

2 ∆+

2m

∆M

0−

2(−

2+

d)M

2 0

))II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

1},{{

pr+

q,m

∆},

2}])

+1

−s+

M2 0

( 8(−

1+

d)π

M3 0

( −2(−

1+

d)m

5 ∆−

2dm

4 ∆M

0+

2(−

2+

d)( −

s+

m2 π

) M0

( −m

2 π+

M2 0

) +2m

2 ∆M

0

( (−1

+2d)Q

2+

(−3

+d)s

+2(−

1+

d)m

2 π+

(1+

d)M

2 0

) +

m3 ∆

( −3

( Q2

+s) +

2d

( 2Q

2+

s) +4(−

1+

d)m

2 π+

(−5

+2d)M

2 0

) +m

∆

( s( Q

2+

s) −( Q

2+

s−

2ds) m

2 π−

2(−

1+

d)m

4 π+

( −(1

+2d)s

+(−

7+

2d)m

2 π

) M2 0

+6M

4 0

))II

[2+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

1}]) +

1−

s+

M2 0

( 8(−

1+

d)π

M3 0

( 2m

4 π((−1

+d)m

∆+

(−2

+d)M

0)+

m2 π

( m∆

( Q2−

4(−

1+

d)m

2 ∆

) −4(−

1+

d)m

2 ∆M

0−

4(−

2+

d)m

∆M

2 0−

4(−

2+

d)M

3 0

) +

(m∆

+M

0)( m

2 ∆

( (3−

4d)Q

2+

2(−

1+

d)m

2 ∆

) +( −

Q2m

∆+

2m

3 ∆

) M0

+2(3−

2d)m

2 ∆M

2 0−

2m

∆M

3 0+

2(−

2+

d)M

4 0

))

II[2

+d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

r,m

∆},

1}])

+1

−s+

M2 0

( 64(−

1+

d)π

2M

4 0

( −(−

2+

d)m

4 ∆+

2m

3 ∆M

0+

(−2

+d)( −

s+

m2 π

) (−m

π+

M0)(m

π+

M0)+

2m

∆M

0

( −m

2 π+

M2 0

) +m

2 ∆

( −2Q

2+

2(−

2+

d)m

2 π+

d( 2

Q2

+s

+M

2 0

)))

II[4

+d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

,m∆},

2}])

+1

−s+

M2 0

( 64(−

1+

d)π

2M

4 0

( (−2

+d)m

4 π+

m2 ∆

( −2(−

1+

d)Q

2+

(−2

+d)m

2 ∆

) −2m

3 ∆M

0−

2dm

2 ∆M

2 0−

2m

∆M

3 0+

(−2

+d)M

4 0+

m2 π

( −2(−

2+

d)m

2 ∆+

2m

∆M

0−

2(−

2+

d)M

2 0

))II

[4+

d][{{

0,m

π},

1},{{

q,m

π},

2},{{−p

r,m

∆},

2}])

+1

−s+

M2 0

( 128(−

1+

d)π

2m

∆M

3 0

( Q2s

+s2−

3sM

2 0+

2M

4 0−

m2 π

( Q2−

s+

M2 0

) +m

2 ∆

( Q2−

s+

M2 0

) +2m

∆M

0

( Q2−

s+

M2 0

))

II[4

+d][{{

0,m

π},

1},{{

q,m

π},

3},{{−p

,m∆},

1}])−

128(−

1+

d)π

2Q

2m

∆M

3 0(−

m2 π+

(m∆

+M

0)2

)II[4+

d][{{

0,m

π},

1},{{

q,m

π},

3},{{−

pr,

m∆},

1}]

−s+

M2 0

))

Dia

gra

mm

)+n)

A(ν

,Q2)

=1

48(−

1+

d)3

dF

2s2m

4 ∆M

5 0

( ie2gπN

∆2

( 16d5s2

m4 πM

4 0−

32sm

2 πM

2 0

((Q

4−

2Q

2s−

s2) m

2 ∆+

( Q4−

2Q

2s−

s2) m

∆M

0+

2s

( −4Q

2−

3s

+m

2 ∆

) M2 0

+2Q

2m

∆M

3 0−

( 10s

+m

2 ∆

) M4 0

+m

∆M

5 0+

m2 π

( −Q

4+

2Q

2s

+s2

+10sM

2 0+

M4 0

))+

143

B. Appendix to VVCS

d4M

2 0

( Q2sm

4 ∆

( 3Q

2−

4s

+2m

2 ∆

) +8Q

2( Q

2−

s) sm3 ∆

M0

+( Q

2sm

2 ∆

( −11s

+m

2 ∆

) +Q

4( 2

s2+

sm2 ∆

+2m

4 ∆

) +2s

( −s3−

3s2

m2 ∆

+4sm

4 ∆+

m6 ∆

))M

2 0+

8Q

2sm

∆

( −2s

+m

2 ∆

) M3 0−

( 2sm

2 ∆

( s+

m2 ∆

) +Q

2( 7

s2+

sm2 ∆−

4m

4 ∆

))M

4 0+

2( s2

−sm

2 ∆+

m4 ∆

) M6 0

+

m4 π

( Q2s

( 3Q

2−

4s

+2m

2 ∆

) +( 2

Q4−

5Q

2s

+2s

( −85s

+m

2 ∆

))M

2 0+

4( Q

2−

2s) M

4 0+

2M

6 0

) −m

2 π

( 2Q

2sm

2 ∆

( 5Q

2−

4s

+2m

2 ∆

) +8Q

2( Q

2−

s) sm∆

M0+

( Q2s

( −23s

+4m

2 ∆

) +Q

4( 7

s+

4m

2 ∆

) −2s

( 10s2

+19sm

2 ∆−

2m

4 ∆

))M

2 0+

8Q

2sm

∆M

3 0+

( 11Q

2s

+8Q

2m

2 ∆−

6sm

2 ∆

) M4 0

+4

( s+

m2 ∆

) M6 0

))−

4d

( −2s

( −Q

4+

2Q

2s

+s2

) m6 ∆−

2s

( −Q

4+

2Q

2s

+s2

) m5 ∆

M0

+m

4 ∆

( 6s2

( s−

m2 ∆

) +2Q

4( s

+m

2 ∆

) +Q

2s

( 4s

+17m

2 ∆

))M

2 0+

m3 ∆

( 2s2

( 2s−

11m

2 ∆

) +Q

4( 1

1s

+2m

2 ∆

) +Q

2s

( −4s

+49m

2 ∆

))M

3 0+

s( −

2s3

+19s2

m2 ∆−

15sm

4 ∆+

3m

6 ∆+

2Q

4( 3

s+

m2 ∆

) +Q

2( 4

s2+

16sm

2 ∆+

15m

4 ∆

))M

4 0+

m∆

( −Q

4s

+Q

2( −

2s2

+sm

2 ∆+

4m

4 ∆

) +s

( −8s2

+11sm

2 ∆+

39m

4 ∆

))M

5 0−

( 8s3−

5s2

m2 ∆−

29sm

4 ∆+

2m

6 ∆+

Q2s

( 7s

+m

2 ∆

))M

6 0+

( −Q

2sm

∆+

2( 4

s2m

∆−

5sm

3 ∆+

m5 ∆

))M

7 0+

s( 2

s−

3m

2 ∆

) M8 0

+2m

6 π

( s+

M2 0

)(−Q

4+

2Q

2s

+s2−

4sM

2 0+

M4 0

) +

m4 π

( −6s

( −Q

4+

2Q

2s

+s2

) m2 ∆−

2s

( −Q

4+

2Q

2s

+s2

) m∆

M0

+( −

Q2s

( 36s

+m

2 ∆

) +3Q

4( 7

s+

2m

2 ∆

) +2s2

( −7s

+3m

2 ∆

))M

2 0+

2( Q

4+

6Q

2s−

s2) m

∆M

3 0+

s( −

13Q

2−

248s

+5m

2 ∆

) M4 0

+2

( 2Q

2+

s) m∆

M5 0−

6( 3

s+

m2 ∆

) M6 0

+2m

∆M

7 0

) −m

2 π

( −6s

( −Q

4+

2Q

2s

+s2

) m4 ∆−

4s

( −Q

4+

2Q

2s

+s2

) m3 ∆

M0

+2m

2 ∆

( −s2

( 4s

+3m

2 ∆

) +Q

4( 1

3s

+3m

2 ∆

) +Q

2s

( −16s

+9m

2 ∆

))M

2 0+

m∆

( −8s2

( s+

3m

2 ∆

) +Q

4( 1

7s

+4m

2 ∆

) +Q

2s

( −28s

+27m

2 ∆

))M

3 0+

s( 7

Q4

+Q

2( −

151s

+m

2 ∆

) −s

( 120s

+19m

2 ∆

))M

4 0+

( 25Q

2sm

∆+

8Q

2m

3 ∆+

7sm

3 ∆

) M5 0

+( 1

1Q

2s−

164s2

+7sm

2 ∆−

6m

4 ∆

) M6 0

+4

( 2sm

∆+

m3 ∆

) M7 0

+4sM

8 0

))+

2d2

( −4s

( −Q

4+

2Q

2s

+s2

) m6 ∆−

2s

( −Q

4+

2Q

2s

+s2

) m5 ∆

M0

+m

4 ∆

( 8s2

( 2s−

m2 ∆

) +Q

4( 7

s+

4m

2 ∆

) +2Q

2s

( 4s

+19m

2 ∆

))M

2 0+

m3 ∆

( 2s2

( 2s−

17m

2 ∆

) +Q

4( 4

1s

+2m

2 ∆

) +4Q

2s

( −7s

+22m

2 ∆

))M

3 0+

( Q4

( 16s2

+7sm

2 ∆+

4m

4 ∆

) +Q

2s

( 8s2−

11sm

2 ∆+

31m

4 ∆

) +s

( −8s3

+7s2

m2 ∆

+4sm

4 ∆+

18m

6 ∆

))M

4 0+

m∆

( −3Q

4s

+Q

2( −

38s2

+23sm

2 ∆+

4m

4 ∆

) +2s

( −8s2

+17sm

2 ∆+

39m

4 ∆

))M

5 0−

2( 8

s3−

3s2

m2 ∆−

28sm

4 ∆+

2m

6 ∆+

2Q

2( 7

s2+

sm2 ∆−

2m

4 ∆

))M

6 0+

( −3Q

2sm

∆+

2( 8

s2m

∆−

9sm

3 ∆+

m5 ∆

))M

7 0+

( 8s2−

11sm

2 ∆+

4m

4 ∆

) M8 0

+4m

6 π

( s+

M2 0

)(−Q

4+

2Q

2s

+s2−

4sM

2 0+

M4 0

) +

2m

4 π

( −6s

( −Q

4+

2Q

2s

+s2

) m2 ∆

+s

( Q4−

2Q

2s−

s2) m

∆M

0+

( 8s2

( −s

+m

2 ∆

) +6Q

4( 3

s+

m2 ∆

) −Q

2s

( 28s

+m

2 ∆

))M

2 0+

( Q4

+10Q

2s−

5s2

) m∆

M3 0

+( 2

Q4−

18Q

2s

+3s

( −98s

+3m

2 ∆

))M

4 0+

( 2Q

2+

5s) m

∆M

5 0+

( 4Q

2−

20s−

6m

2 ∆

) M6 0

+m

∆M

7 0+

2M

8 0

) −m

2 π

( −12s

( −Q

4+

2Q

2s

+s2

) m4 ∆−

4s

( −Q

4+

2Q

2s

+s2

) m3 ∆

M0

+m

2 ∆

( −4s2

m2 ∆

+16Q

2s

( −3s

+2m

2 ∆

) +Q

4( 5

5s

+12m

2 ∆

))M

2 0+

m∆

( Q4

( 31s

+4m

2 ∆

) +4Q

2s

( −11s

+5m

2 ∆

) −4s2

( s+

11m

2 ∆

))M

3 0+

( 4Q

4( 7

s+

2m

2 ∆

) +Q

2s

( −252s

+5m

2 ∆

) −2s

( 104s2

+57sm

2 ∆−

6m

4 ∆

))M

4 0+

Q2m

∆

( 35s

+8m

2 ∆

) M5 0

+2

( 22Q

2s−

104s2

+8Q

2m

2 ∆+

7sm

2 ∆−

6m

4 ∆

) M6 0

+4m

∆

( s+

m2 ∆

) M7 0

+8

( 2s

+m

2 ∆

) M8 0

))−

d3

( −2s

( −Q

4+

2Q

2s

+s2

) m6 ∆

+m

4 ∆

( 2s2

( 5s−

m2 ∆

) +Q

4( 1

1s

+2m

2 ∆

) +Q

2( −

4s2

+22sm

2 ∆

))M

2 0+

2sm

3 ∆

( 23Q

4−

20Q

2s

+14Q

2m

2 ∆−

4sm

2 ∆

) M3 0+

( Q4

( 14s2

+7sm

2 ∆+

8m

4 ∆

) +Q

2s

( 4s2−

49sm

2 ∆+

11m

4 ∆

) +2s

( −5s3−

12s2

m2 ∆

+15sm

4 ∆+

8m

6 ∆

))M

4 0−

2sm

∆

( Q4

+34Q

2s

+4s2−

21Q

2m

2 ∆−

12sm

2 ∆−

14m

4 ∆

) M5 0

+( −

35Q

2s2−

8s3−

5Q

2sm

2 ∆+

2s2

m2 ∆

+16Q

2m

4 ∆+

16sm

4 ∆−

2m

6 ∆

) M6 0−

2sm

∆

( Q2−

4s

+2m

2 ∆

) M7 0

+2

( 5s2−

6sm

2 ∆+

4m

4 ∆

) M8 0

+2m

6 π

( s+

M2 0

)(−Q

4+

2Q

2s

+s2−

4sM

2 0+

M4 0

) +

m4 π

( −6s

( −Q

4+

2Q

2s

+s2

) m2 ∆

+( 6

Q2s

( −6s

+m

2 ∆

) +2s2

( −3s

+5m

2 ∆

) +Q

4( 2

5s

+6m

2 ∆

))M

2 0+

8( Q

2−

s) sm∆

M3 0

+( 8

Q4−

33Q

2s

+4s

( −164s

+5m

2 ∆

))M

4 0+

8sm

∆M

5 0+

2( 8

Q2−

3( 7

s+

m2 ∆

))M

6 0+

8M

8 0

) −m

2 π

( −6s

( −Q

4+

2Q

2s

+s2

) m4 ∆

+2m

2 ∆

( s2( 2

s+

m2 ∆

) +4Q

2s

( −5s

+2m

2 ∆

) +Q

4( 2

6s

+3m

2 ∆

))M

2 0−

2sm

∆

( −17Q

4+

8sm

2 ∆+

2Q

2( 1

0s

+m

2 ∆

))M

3 0+

( Q2s

( −179s

+10m

2 ∆

) +Q

4( 3

5s

+16m

2 ∆

) −2s

( 76s2

+85sm

2 ∆−

7m

4 ∆

))M

4 0+

( 34Q

2sm

∆−

4sm

3 ∆

) M5 0

+( 5

5Q

2s−

84s2

+32Q

2m

2 ∆−

10sm

2 ∆−

6m

4 ∆

) M6 0

+4

( 5s

+4m

2 ∆

) M8 0

))) (1

+τ[3

])II

[d][{{

0,m

π},

1}]) −

124(−

1+

d)3

F2m

4 ∆M

5 0

( ie2gπN

∆2

( −(−

2+

d)2

( 2Q

2+

s) m8 ∆

+2(−

2+

d)( 2

Q2

+s) m

7 ∆M

0+

(−2

+d)m

6 ∆

( 2( −

4+

2d

+d2) Q

2+

(−8

+6d)s−

(−6

+d)m

2 ∆

) M2 0+

2m

5 ∆

( 2d3Q

2+

6s−

22m

2 ∆−

2d2

( 5Q

2+

m2 ∆

) +d

( 12Q

2−

3s

+17m

2 ∆

))M

3 0+

2m

4 ∆

((4−

12d

+13d2−

4d3) Q

2−

2( 2−

d−

2d2

+d3) s

+( 1

0−

6d−

7d2

+2d3) m

2 ∆

) M4 0

+

2m

3 ∆

((28−

62d

+40d2−

8d3) Q

2+

3(−

2+

d)s

+( 7

8−

77d

+12d2) m

2 ∆

) M5 0

+2m

2 ∆

( −(−

2+

d)2

(−3

+4d)Q

2+

34m

2 ∆+

d2

( s+

4m

2 ∆

) −2d

( s+

14m

2 ∆

))M

6 0−

2m

∆

( 2d3Q

2+

d( 4

Q2

+s−

17m

2 ∆

) −2

( s−

7m

2 ∆

) +d2

( −6Q

2+

8m

2 ∆

))M

7 0+

( −2(−

2+

d)2

(−1

+d)Q

2+

(−2

+d)2

s−

2( 1

2−

6d

+d2) m

2 ∆

) M8 0−

2( 6−

7d

+2d2) m

∆M

9 0+

(−2

+d)2

M10

0+

(−2

+d)2

m8 π

( −2Q

2−

s+

3M

2 0

) −

144


2(−

2+

d)m

6 π

( −2(−

2+

d)( 2

Q2

+s) m

2 ∆+

( 2Q

2+

s) m∆

M0

+((

8−

6d

+d2) Q

2−

(−2

+d)s

+2(−

3+

2d)m

2 ∆

) M2 0

+(m

∆−

2dm

∆)M

3 0+

5(−

2+

d)M

4 0

) +

2m

4 π

( −3(−

2+

d)2

( 2Q

2+

s) m4 ∆

+3(−

2+

d)( 2

Q2

+s) m

3 ∆M

0+

(−2

+d)m

2 ∆

((12−

10d

+3d2) Q

2+

d( s

+3m

2 ∆

))M

2 0+

m∆

( 2( −

8+

10d−

5d2

+d3) Q

2+

(−2

+d)s

+( −

26

+27d−

6d2) m

2 ∆

) M3 0

+

(−2

+d)((

10−

9d

+2d2) Q

2+

( 4+

5d−

2d2) m

2 ∆

) M4 0

+( −

10

+17d−

6d2) m

∆M

5 0+

6(−

2+

d)2

M6 0

) −2m

2 π

( −2(−

2+

d)2

( 2Q

2+

s) m6 ∆

+3(−

2+

d)( 2

Q2

+s) m

5 ∆M

0+

(−2

+d)m

4 ∆

( −2dQ

2+

3d2Q

2−

6s

+5ds

+6m

2 ∆

) M2 0+

m3 ∆

( 4(−

2+

d)2

(−1

+d)Q

2−

2(−

2+

d)s

+( −

46

+39d−

6d2) m

2 ∆

) M3 0

+

m2 ∆

( −2

( −8

+16d−

8d2

+d3) Q

2−

2( −

6+

11d−

6d2

+d3) s

+( −

18

+8d−

3d2) m

2 ∆

) M4 0

+m

∆

( 2(−

2+

d)Q

2−

(−2

+d)s

+6

( −2−

2d

+d2) m

2 ∆

) M5 0

+( −

(−2

+d)2

Q2

+(−

2+

d)2

s−

4(−

3+

d)2

m2 ∆

) M6 0

+( −

14

+19d−

6d2) m

∆M

7 0+

3(−

2+

d)2

M8 0

))

(1+

τ[3

])II

[d][{{

0,m

π},

1},{{

p,m

∆},

1}])−

148(−

1+

d)3

F2m

4 ∆M

5 0(−

s+

M2 0)(i

e2gπN

∆2

( 2(−

2+

d)2

Q2m

8 π

( −Q

2+

2M

2 0

) +

m6 π

( 8(−

2+

d)2

Q4m

2 ∆−

4(−

2+

d)Q

4m

∆M

0+

Q2

( −(−

2+

d)2

(−11

+3d)Q

2+

2( −

10

+10d

+d2−

d3) m

2 ∆

) M2 0+

8( 6−

5d

+d2) Q

2m

∆M

3 0+

2( (−

2+

d)2

(−13

+5d)Q

2−

2( 2

+2d−

5d2

+d3) m

2 ∆

) M4 0

) +

m4 π

( −12(−

2+

d)2

Q4m

4 ∆+

12(−

2+

d)Q

4m

3 ∆M

0+

Q2m

2 ∆

((−1

00

+142d−

71d2

+13d3) Q

2+

2( −

34

+30d−

11d2

+3d3) m

2 ∆

) M2 0

+2Q

2m

∆((−2

2+

29d−

17d2

+4d3) Q

2−

2( 3

9−

20d

+d2) m

2 ∆

) M3 0

+( (−

2+

d)2

(−23

+11d)Q

4−

4( 2

9−

10d−

4d2

+d3) Q

2m

2 ∆+

4( −

10

+18d−

11d2

+3d3) m

4 ∆

)

M4 0−

4m

∆

((14

+11d−

17d2

+4d3) Q

2−

2( 1

7−

22d

+5d2) m

2 ∆

) M5 0

+( −

2(−

2+

d)2

(−29

+17d)Q

2+

4( 1

0+

6d−

21d2

+5d3) m

2 ∆

) M6 0

) +( −

m2 ∆

+M

2 0

)(2(−

2+

d)2

Q4m

6 ∆−

4(−

2+

d)Q

4m

5 ∆M

0+

Q2m

4 ∆

((8−

14d

+11d2−

3d3) Q

2−

2( −

34

+38d−

11d2

+d3) m

2 ∆

) M2 0+

2Q

2m

3 ∆

((22−

41d

+23d2−

4d3) Q

2+

2( 4

9−

44d

+7d2) m

2 ∆

) M3 0−

2m

2 ∆

((−1

0+

19d−

9d2

+d3) Q

4+

( −76

+62d−

9d2

+d3) Q

2m

2 ∆+

2( −

18

+26d−

9d2

+d3) m

4 ∆

) M4 0

+

2m

∆

((−1

8+

31d−

19d2

+4d3) Q

4−

2( 3

+21d−

20d2

+4d3) Q

2m

2 ∆+

4( 1

7−

22d

+5d2) m

4 ∆

) M5 0

+( (−

2+

d)2

(−7

+5d)Q

4−

2( 7

0−

20d−

27d2

+7d3) Q

2m

2 ∆+

8( −

2+

2d

+d2−

d3) m

4 ∆

) M6 0−

4m

∆

((−6

+25d−

19d2

+4d3) Q

2+

6( −

1+

d2) m

2 ∆

) M7 0

+2

( −(−

2+

d)2

(−9

+7d)Q

2+

2( 6

+2d−

11d2

+3d3) m

2 ∆

) M8 0

) +

m2 π

( 8(−

2+

d)2

Q4m

6 ∆−

12(−

2+

d)Q

4m

5 ∆M

0+

Q2m

4 ∆

((56−

92d

+57d2−

13d3) Q

2+

2( 7

0−

70d

+19d2−

3d3) m

2 ∆

) M2 0−

8Q

2m

3 ∆

((−1

0+

17d−

10d2

+2d3) Q

2+

( −38

+27d−

3d2) m

2 ∆

) M3 0−

2m

2 ∆

((−1

0+

12d−

9d2

+3d3) Q

4+

( −188

+160d−

31d2

+3d3) Q

2m

2 ∆+

6( −

10

+14d−

5d2

+d3) m

4 ∆

) M4 0−

4Q

2m

∆((−1

8+

29d−

18d2

+4d3) Q

2−

2( 1

8−

7d

+d2) m

2 ∆

) M5 0

+( −

(−2

+d)2

(−21

+13d)Q

4+

2( 1

58−

98d

+d2

+3d3) Q

2m

2 ∆+

8( 2−

10d

+11d2−

3d3) m

4 ∆

)

M6 0

+8m

∆

((−2

+23d−

19d2

+4d3) Q

2−

2( 1

0−

11d

+d2) m

2 ∆

) M7 0

+2

( (−2

+d)2

(−27

+19d)Q

2−

2( 1

4+

6d−

27d2

+7d3) m

2 ∆

) M8 0

))

(1+

τ[3

])II

[d][{{

0,m

π},

1},{{

p,m

∆},

1}])

+1

48(−

1+

d)3

F2s2m

4 ∆M

0(−

s+

M2 0)(i

e2gπN

∆2

((Q

2+

s)(2(−

2+

d)2

s4( (−

3+

d)Q

2+

s−

ds) +

(−2

+d)s

3+

( (−22

+(2

3−

3d)d

)Q2

+54s

+4(−

4+

d)d

s) m2 ∆

2s2

( (−2

+(9−

4d)d

)Q2

+(9

2+

d(−

43

+d(−

19

+7d))

)s) m

4 ∆−

(−2

+d)s

( −d(1

+d)Q

2+

2(−

6+

(−6

+d)d

)s) m

6 ∆−

2( (−

2+

d)2

Q2

+30s−

d(3

4+

(−10

+d)d

)s) m

8 ∆

) +2m

∆

( (−2

+d)Q

4( s−

m2 ∆

)(2(4

+d(−

5+

2d))

s2−

(−1

+d)(−1

1+

4d)s

m2 ∆−

2m

4 ∆

) +

2s2

( (−2

+d)(−3

+2d)s

3−

(21

+2d(−

19

+6d))

s2m

2 ∆+

(−30

+d(1

2+

5d))

sm4 ∆

+(5

+(5−

3d)d

)m6 ∆

) +

Q2s

( −2(2

+(−

3+

d)d

(−3

+2d))

s3+

(−52

+d(1

03

+d(−

71

+16d))

)s2m

2 ∆+

(98

+d(−

65

+d(−

5+

4d))

)sm

4 ∆−

2(4

5+

7(−

6+

d)d

)m6 ∆

))

M0

+( (−

2+

d)2

s3( (−

1+

3d)Q

4+

(11−

7d)Q

2s

+2(2

+d)s

2) −

s2( 2

(−2

+d)(

1+

2(−

2+

d)d

)Q4−

(−156

+d(1

86

+d(−

57

+7d))

)Q2s

+2(−

68

+d(5

3+

d(−

13

+3d))

)s2) m

2 ∆−

s( (−

2+

d)( −

4+

d+

d2) Q

4−

(108

+d(−

72

+d(−

11

+3d))

)Q2s

+2(2

32

+d(−

165

+d(−

31

+19d))

)s2) m

4 ∆+

( 2(−

2+

d)2

dQ

4−

(60

+d(−

46

+d(−

5+

3d))

)Q2s

+2(1

18

+d(−

114

+d(1

5+

4d))

)s2) m

6 ∆+

2(−

5+

d)(

2+

(−4

+d)d

)sm

8 ∆

) M2 0

+

2m

∆

( −(−

2+

d)s

2( (−

1+

d)Q

4+

(−1

+d)(−9

+4d)Q

2s

+4(−

2+

d)s

2) +

s( (−

2+

d)(−1

+d)Q

4−

4(−

8+

d(2

3+

2d(−

9+

2d))

)Q2s+

2(3

9+

d(−

48

+7d))

s2) m

2 ∆+

s( (6

+d(1

9+

d(−

21

+4d))

)Q2

+4(2

3+

d(−

23

+5d))

s) m4 ∆

+2

( 2(−

2+

d)Q

2+

(−37

+(3

8−

7d)d

)s) m

6 ∆

) M3 0

+

145

B. Appendix to VVCS

( (−2

+d)2

s3( (1

1−

7d)Q

2+

2(−

5+

d)s

) +2s2

( −2(−

2+

d)(

1+

2(−

2+

d)d

)Q2

+(1

2+

d(−

9+

d(−

11

+5d))

)s) m

2 ∆+

s( −

(−2

+d)2

(−1

+5d)Q

2+

2(1

14

+d(−

103

+4d(1

+d))

)s) m

4 ∆+

2( 2

(−2

+d)2

dQ

2−

(62

+d(−

56

+d(5

+2d))

)s) m

6 ∆+

2(−

2+

d)2

m8 ∆

) M4 0−

2m

∆

( s−

m2 ∆

)((−

2+

d)s

( (−1

+d)Q

2+

2s) +

2(1

0+

(−9

+d)d

)sm

2 ∆+

2(−

2+

d)m

4 ∆

) M5 0−

2(−

2+

d)( s−

m2 ∆

)((−

2+

d)2

s2−

(−3

+d)(−1

+d)s

m2 ∆

+(−

2+

d)d

m4 ∆

) M6 0

+

2(−

2+

d)2

m8 π

( Q2−

3s

+M

2 0

)(−Q

2−

s+

M2 0

) +

m4 π

((Q

2+

s)(2(−

2+

d)2

s2( 3

(−4

+d)Q

2+

(16−

7d)s

) −s

( (100

+d(−

118

+(4

7−

7d)d

))Q

2+

2(3

2+

d(8

+5(−

4+

d)d

))s) m

2 ∆−

2( 6

(−2

+d)2

Q2−

(−5

+3d)( 2

+d2) s) m

4 ∆

) +2(−

2+

d)s

( (11

+d(−

9+

4d))

Q4

+(1

+3(9−

4d)d

)Q2s

+2(−

5+

6d)s

2) m

∆M

0+

4( 3

(−2

+d)Q

4−

(27

+(−

14

+d)d

)Q2s

+(4

3+

d(−

49

+11d))

s2) m

3 ∆M

0+

( (−2

+d)2

s( (1

+5d)Q

4+

(45−

33d)Q

2s

+6(−

6+

5d)s

2) +

( 6(−

2+

d)2

dQ

4−

(−48

+d(1

10

+d(−

71

+13d))

)Q2s

+2(7

4+

d(2

6+

d(−

85

+22d))

)s2) m

2 ∆+

2(−

1+

d)(

34

+d(−

20

+3d))

sm4 ∆

) M2 0

+

2m

∆

( (−2

+d)s

( (13

+d(−

9+

4d))

Q2

+4(2−

3d)s

) −2

( 3s

+(−

2+

d)( −

6Q

2+

ds))

m2 ∆

) M3 0

+( (−

2+

d)2

s( (5

+7d)Q

2−

18ds) +

2( 6

(−2

+d)2

dQ

2−

(22

+d(1

8+

5d(−

7+

2d))

)s) m

2 ∆+

12(−

2+

d)2

m4 ∆

) M4 0

+

4(−

2+

d)m

∆

( s+

3m

2 ∆

) M5 0

+2(−

2+

d)2

( (2+

d)s

+3dm

2 ∆

) M6 0

) +

m2 π

( −( Q

2+

s)((−

2+

d)2

s3( 7

(−3

+d)Q

2+

2(8−

5d)s

) +2s2

((−8

+d

+d3) Q

2−

(−3

+d)(−4

4+

d(1

7+

2d))

s) m2 ∆−

s( (5

6+

d(−

68

+(3

3−

7d)d

))Q

2+

2(1

28

+d(−

90

+d(4

+3d))

)s) m

4 ∆−

2( 4

(−2

+d)2

Q2

+54s−

3d(1

8+

(−5

+d)d

)s) m

6 ∆

) +

2m

∆

( (−2

+d)s

2( (−

19

+(2

1−

8d)d

)Q4

+(1

1+

d(−

47

+16d))

Q2s

+2(7−

6d)s

2) −

2s

( 2(−

2+

d)(

5+

2(−

3+

d)d

)Q4−

(26

+d(1

1+

d(−

17

+4d))

)Q2s

+(6

6+

d(−

53

+5d))

s2) m

2 ∆−

2( 3

(−2

+d)Q

4−

2(3

2+

3(−

8+

d)d

)Q2s

+( 4

6−

39d

+6d2) s2

) m4 ∆

) M0

+( −

(−2

+d)2

s2( d

( 2Q

2−

9s)(

3Q

2−

2s) +

( 43Q

2−

8s) s) −

2s

( (−2

+d)(

1+

d(−

5+

2d))

Q4−

(30

+d(−

10

+d(−

11

+3d))

)Q2s

+(2

48

+d(−

195

+d(9

+8d))

)s2) m

2 ∆−

( 6(−

2+

d)2

dQ

4−

(64

+d(−

24

+d(−

27

+7d))

)Q2s

+2(2

20

+d(−

136

+d(−

34

+19d))

)s2) m

4 ∆−

2(−

22

+d(4

6+

d(−

19

+3d))

)sm

6 ∆

) M2 0

+

2m

∆

( (−2

+d)s

( −(−

1+

d)Q

4+

4(−

2+

d)Q

2s

+4(−

4+

3d)s

2) +

2s

( −(−

1+

d)(

10

+d(−

15

+4d))

Q2

+2(−

6+

d)(−3

+d)s

) m2 ∆

+

4( −

3(−

2+

d)Q

2+

(20

+3(−

6+

d)d

)s) m

4 ∆

) M3 0

+2

( (−2

+d)2

(8+

3d)s

3−

( −102

+d

( 131−

60d

+9d2))

s2m

2 ∆+

(100

+d(−

74

+d(−

6+

7d))

)sm

4 ∆−

4(−

2+

d)2

m6 ∆

+(−

2+

d)2

Q2

( 2(−

4+

d)s

2−

(3+

d)s

m2 ∆−

6dm

4 ∆

))M

4 0+

2m

∆

( −(−

2+

d)( (−

1+

d)Q

2−

2s) s−

2(1

0+

(−9

+d)d

)sm

2 ∆−

6(−

2+

d)m

4 ∆

) M5 0

+2(−

2+

d)( (−

4+

d)(−2

+d)s

2+

(7+

(−6

+d)d

)sm

2 ∆−

3(−

2+

d)d

m4 ∆

) M6 0

) −m

6 π

( 4s

( −( 1

1Q

2−

24s)(

Q2

+s) +

( 13Q

2−

32s) M

2 0+

8M

4 0

) +4m

2 ∆

( −( 8

Q2−

13s)(

Q2

+s) −

19sM

2 0+

8M

4 0

) +

d2

( −( Q

2+

s)(−6

s( 8

s+

m2 ∆

) +Q

2( 1

5s

+8m

2 ∆

))−

8( Q

2−

s) sm∆

M0−

( 8Q

4−

49Q

2s

+88s2

+34sm

2 ∆

) M2 0−

8sm

∆M

3 0+

8( −

2Q

2+

6s

+m

2 ∆

) M4 0−

8M

6 0

) +

d3

( s( Q

2+

s)(Q

2−

6s

+2m

2 ∆

) +( 2

Q4−

9Q

2s

+2s

( 7s

+m

2 ∆

))M

2 0+

2( 2

Q2−

5s) M

4 0+

2M

6 0

) +

4d

((Q

2+

s)(6

( 2Q

2−

5s) s

+( 8

Q2−

13s) m

2 ∆

) +( Q

4+

8Q

2s−

5s2

) m∆

M0

+( 2

Q4−

22Q

2s

+23s

( 2s

+m

2 ∆

))M

2 0+

2( Q

2+

2s) m

∆M

3 0+

2( 2

Q2−

9s−

4m

2 ∆

) M4 0

+m

∆M

5 0+

2M

6 0

) −8m

∆M

0

( 4Q

2s−

s2+

( Q2

+M

2 0

) 2))

)

(1+

τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1}])−

112(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)(i

e2gπN

∆2

( m6 π

( (−2

+d)2

Q2−

2(−

2+

(−2

+d)d

)m2 ∆

) M2 0+

2(−

2+

d)Q

2( (−

3+

2d)Q

2−

s) sm∆

M3 0

+(−

2+

d)2

Q2

( 2Q

2−

s) sM4 0−

2m

7 ∆M

0

( (2+

d(−

13

+4d))

Q2

+(−

10

+3d)s

+(−

4+

5d)M

2 0

) +

2m

8 ∆

( −(1

+2(−

3+

d)d

)( Q

2+

s) +(1

3+

(−8

+d)d

)M2 0

) +

2m

5 ∆M

0

( (−2

+d)(−3

+2d)Q

4+

2(2

+d(−

7+

2d))

Q2s

+(−

30

+7d)s

2+

( (2+

d(−

13

+4d))

Q2

+4(6

+d)s

) M2 0

+(−

22

+5d)M

4 0

) −m

6 ∆

( −( Q

2+

s)((−

2+

d)(−3

+2d)Q

2+

2(−

1+

d)(−5

+2d)s

) +( (−

5+

d)d

Q2−

4( −

9+

d+

d2) s) M

2 0+

2(1

3+

(−7

+d)d

)M4 0

) −2m

3 ∆M

0( (−

2+

d)Q

2s

( (−3

+2d)Q

2+

(−7

+4d)s

) +( (−

2+

d)(−3

+2d)Q

4−

2(1

0+

d(−

7+

2d))

Q2s

+(−

12

+7d)s

2) M

2 0+

(−2

+d)( (−

6+

4d)Q

2+

s) M4 0

) −m

2 ∆M

2 0

( (−2

+d)s

( −2Q

4+

(−4

+d)Q

2s−

2(−

1+

d)s

2) +

( (−2

+d)(−3

+2d)Q

4−

(42

+d(−

21

+4d))

Q2s

+2(6

+(−

4+

d)d

)s2) M

2 0+

146


(−2

+d)( (−

7+

4d)Q

2−

2(−

1+

d)s

) M4 0

) +m

4 ∆

( −2(−

2+

d)(−1

+d)s

( Q2

+s)(

Q2

+2s) +

2s

( (−5

+d(−

7+

3d))

Q2

+(−

9+

2(−

2+

d)d

)s) M

2 0+

( (−8

+(−

4+

d)d

)Q2

+86s−

8d2s) M

4 0+

2( −

14

+d2) M

6 0

) +

m4 π

( 2(−

2+

d)Q

2( (−

3+

2d)Q

2−

s) m∆

M0−

(−2

+d)2

Q2M

2 0

( −2Q

2+

2s

+M

2 0

) −2m

3 ∆M

0

( (2+

d(−

13

+4d))

Q2

+(−

10

+3d)s

+(−

4+

5d)M

2 0

) +

2m

4 ∆

( −(−

17

+2(−

3+

d)d

)( Q

2+

s) +3(−

3+

(−4

+d)d

)M2 0

) +

m2 ∆

( (8+

d(−

7+

2d))

Q2

( Q2

+s) +

( (4+

(7−

3d)d

)Q2

+4(−

2+

(−2

+d)d

)s) M

2 0+

2(−

2+

(−2

+d)d

)M4 0

))+

m2 π

( −4m

5 ∆M

0

( (18

+(1

1−

4d)d

)Q2−

5(−

6+

d)s

+(−

44

+13d)M

2 0

) −2m

6 ∆

( −4(−

3+

d)d

( Q2

+s) +

(−18

+d(−

2+

3d))

M2 0

) −(−

2+

d)2

Q2M

2 0

((2Q

2−

s) s+

2( Q

2−

s) M2 0

) −2(−

2+

d)Q

2m

∆M

0

(((−

5+

2d)Q

2−

s) s+

( (−1

+2d)Q

2−

s) M2 0

) +

2m

3 ∆M

0

( 2(2

+d(−

7+

2d))

Q4

+(−

30

+d(−

3+

4d))

Q2s

+(−

10

+3d)s

2+

( (38

+d(−

25

+4d))

Q2

+4(−

4+

3d)s

) M2 0

+(−

2+

d)M

4 0

) −m

4 ∆

( −2

( Q2

+s)(

(−3

+d)(−1

+2d)Q

2+

(−15

+d(−

3+

2d))

s) +( (2

0+

(40−

11d)d

)Q2

+4(1

8−

7d)s

) M2 0

+2(−

27

+7d)M

4 0

) −2m

2 ∆

( (−3

+d)(−2

+d)Q

2s

( Q2

+s) +

( −( 3−

6d

+d2) Q

4+

3(5−

2d)Q

2s

+d(−

5+

2d)s

2) M

2 0+

( −(−

5+

d)(−1

+2d)Q

2+

2(−

4+

d)s

) M4 0

+(−

2+

d)(−1

+d)M

6 0

)))

(1+

τ[3

])II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

12(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)(i

e2gπN

∆2

( m6 π

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M2 0+

m4 π

( Q2m

2 ∆

((8−

7d

+2d2) Q

2+

2( 1

7+

6d−

2d2) m

2 ∆

) +2Q

2m

∆

((6−

7d

+2d2) Q

2+

( −2

+13d−

4d2) m

2 ∆

) M0+

( 2(−

2+

d)2

Q4−

( −12

+d2) Q

2m

2 ∆+

2( 8−

6d

+d2) m

4 ∆

) M2 0−

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M3 0−

3( (−

2+

d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M4 0

) +( −

m2 ∆

+M

2 0

) 2( Q

2m

2 ∆

((6−

7d

+2d2) Q

2−

2( 1−

6d

+2d2) m

2 ∆

) +2Q

2m

∆

((6−

7d

+2d2) Q

2+

( −2

+13d−

4d2) m

2 ∆

) M0+

( 2(−

2+

d)2

Q4

+( 1

2+

8d−

3d2) Q

2m

2 ∆−

2( −

12

+2d

+d2) m

4 ∆

) M2 0−

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M3 0−

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M4 0

) +

m2 π

(m∆

+M

0)( 2

(−3

+d)Q

2m

3 ∆

( (−1

+2d)Q

2+

4dm

2 ∆

) +2Q

2m

2 ∆

((1−

7d

+2d2) Q

2+

2( −

18−

5d

+2d2) m

2 ∆

) M0+

m∆

( −4

( 2−

3d

+d2) Q

4+

( 28−

40d

+11d2) Q

2m

2 ∆+

2( 1

8−

10d

+d2) m

4 ∆

) M2 0−

( 4(−

2+

d)2

Q4

+( 1

2+

16d−

5d2) Q

2m

2 ∆+

2( −

10

+6d

+d2) m

4 ∆

) M3 0

+

m∆

((−2

0+

16d−

3d2) Q

2+

2( −

34

+10d

+3d2) m

2 ∆

) M4 0

+3

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M5 0

))

(1+

τ[3

])II

[d][{{

0,m

π},

1},{{

pr+

q,m

∆},

1},{{

pr,

m∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)(i

e2gπN

∆2π

( 4M

2 0

( Q2

+(Q

2+

s−

M2 0)2

4M

2 0

)( 2

(18

+(−

8+

d)d

)m6 ∆−

4(−

17

+5d)m

5 ∆M

0−

(−2

+d)2

Q2

( −s

+m

2 π

) (mπ−

M0)(m

π+

M0)+

2(−

2+

d)Q

2m

∆M

0

( 2s−

3m

2 π+

M2 0

) +

2m

3 ∆M

0

( 6Q

2−

3dQ

2−

22s

+8ds

+(3

4−

10d)m

2 π+

2(−

6+

d)M

2 0

) −m

4 ∆

( 44s

+2(−

10

+d)d

s+

4(6

+(−

2+

d)d

)m2 π−

12M

2 0+

(−2

+d)d

( Q2

+2M

2 0

))+

m2 ∆

( (−2

+d)s

( (2+

d)Q

2+

2(−

1+

d)s

) +2(−

2+

(−4

+d)d

)m4 π

+( (−

4+

d)(−2

+d)Q

2−

2(6

+(−

2+

d)d

)s) M

2 0+

2(−

2+

d)(−1

+d)M

4 0−

2m

2 π

( (4+

(−3

+d)d

)Q2

+(−

2+

(−4

+d)d

)s+

(−2

+(−

4+

d)d

)M2 0

))) −

( Q2

+s

+M

2 0

)((−

2+

d)s

( (−2

+d)Q

4+

(−8

+d)Q

2s−

2(−

1+

d)s

2) m

2 ∆−

( (−2

+d)2

Q4

+2(6

+(−

9+

d)d

)Q2s

+2(−

32

+5d)s

2) m

4 ∆+

( (20

+d(−

20

+3d))

Q2

+4(−

3+

2d)s

) m6 ∆−

2(−

12

+d(2

+d))

m8 ∆

+m

6 π

( −(−

2+

d)2

Q2

+2(−

2+

(−2

+d)d

)m2 ∆

) +

2m

3 ∆

( −(−

2+

d)Q

4+

(−2

+4d)s

2+

4(1

3−

2d)s

m2 ∆

+2(7−

4d)m

4 ∆+

Q2

( (8+

d)s

+(1

6−

9d)m

2 ∆

))M

0+

( (−2

+d)2

Q2s

( −Q

2+

s) +( (−

2+

d)2

Q4−

2(1

2+

(−5

+d)d

)Q2s

+2(6

+(−

4+

d)d

)s2) m

2 ∆+

( (−4

+(1

0−

3d)d

)Q2

+4

( −17

+d

+d2) s) m

4 ∆+

2(4

+(−

6+

d)d

)m6 ∆

) M2 0

+2m

∆

( (−2

+d)Q

2( Q

2+

s) +( (−

22

+7d)Q

2+

4(−

3+

d)s

) m2 ∆

+4(−

13

+2d)m

4 ∆

) M3 0

+

2m

2 ∆

( (−2

+d)( d

( Q2−

s) +s) −

(10

+(−

5+

d)d

)m2 ∆

) M4 0

+m

4 π

( −2(4

+d(−

10

+3d))

m4 ∆

+2(−

2+

d)Q

2m

∆M

0+

4(−

7+

4d)m

3 ∆M

0−

(−2

+d)2

Q2

( Q2−

2s−

M2 0

) +m

2 ∆

( 8s

+d

( (−4

+d)Q

2−

4(−

2+

d)s

) +(4−

2(−

2+

d)d

)M2 0

))+

m2 π

( 6(−

2+

(−2

+d)d

)m6 ∆−

2(−

2+

d)Q

2m

∆M

0

( Q2

+s

+M

2 0

) −4m

3 ∆M

0

( −7Q

2+

4dQ

2−

8s

+6ds

+2(−

3+

d)M

2 0

) +

m4 ∆

( (24

+(4−

3d)d

)Q2

+4(3

+(−

8+

d)d

)s+

4(5

+2d)M

2 0

) +(−

2+

d)2

Q2

((Q

2−

s) s+

( Q2−

2s) M

2 0

) +

2m

2 ∆

( −(−

2+

d)2

Q4−

3(−

2+

d)Q

2s

+d(−

5+

2d)s

2+

( −(2

+(−

7+

d)d

)Q2

+2(−

4+

d)s

) M2 0

+(−

2+

d)(−1

+d)M

4 0

))))

(1+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}])−

4iπ

147

B. Appendix to VVCS

( −1

12(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)

( e2gπN

∆2M

2 0

( −m

6 π

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) −m

4 π

( (−2

+d)2

Q4−

(−4

+d)d

Q2m

2 ∆+

2( 4−

10d

+3d2) m

4 ∆−

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M0−

3( (−

2+

d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M2 0

) −( −

m2 ∆

+M

2 0

)(−m

2 ∆

( (−2

+d)2

Q4

+( −

20

+20d−

3d2) Q

2m

2 ∆+

2( −

12

+2d

+d2) m

4 ∆

) −2m

∆

( Q2

+m

2 ∆

)((−

2+

d)Q

2+

2(−

7+

4d)m

2 ∆

) M0+

( (−2

+d)2

Q4−

2( −

2−

4d

+d2) Q

2m

2 ∆+

4(5−

2d)m

4 ∆

) M2 0−

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M3 0−

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M4 0

) −m

2 π

( m2 ∆

( 2(−

2+

d)2

Q4

+( −

24−

4d

+3d2) Q

2m

2 ∆−

6( −

2−

2d

+d2) m

4 ∆

) +2Q

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M0−

2( (−

2+

d)2

Q4+

( 4+

4d−

d2) Q

2m

2 ∆+

2( 8−

6d

+d2) m

4 ∆

) M2 0

+4m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M3 0

+3

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M4 0

))(1

+τ[3

])) +

124(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)

( e2gπN

∆2

( −m

6 π

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

)(Q

2+

2M

2 0

) +m

4 π

( Q2m

2 ∆

((4

+4d−

d2) Q

2−

2( 4−

10d

+3d2) m

2 ∆

) +

2Q

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M0

+( 5

(−2

+d)2

Q4−

4( −

7−

9d

+3d2) Q

2m

2 ∆−

4( 4−

10d

+3d2) m

4 ∆

) M2 0

+

4m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M3 0

+6

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M4 0

) +

m2 π

( Q2m

2 ∆

( 48Q

2m

2 ∆−

12m

4 ∆+

d2m

2 ∆

( Q2

+6m

2 ∆

) +2d

( Q4−

2Q

2m

2 ∆−

6m

4 ∆

))+

4Q

4m

∆

( (−2

+d)Q

2+

(−10

+d)m

2 ∆

) M0+

2m

2 ∆

((8−

8d

+3d2) Q

4+

( 88−

24d

+7d2) Q

2m

2 ∆+

6( −

2−

2d

+d2) m

4 ∆

) M2 0

+16Q

2m

∆

( (−2

+d)Q

2+

(−10

+d)m

2 ∆

) M3 0

+( −

7(−

2+

d)2

Q4

+2

( −14−

30d

+9d2) Q

2m

2 ∆+

8( 8−

6d

+d2) m

4 ∆

) M4 0−

8m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M5 0−

6( (−

2+

d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M6 0

) +( −

m2 ∆

+M

2 0

)(Q

2m

2 ∆

( −2(−

2+

d)Q

4+

( 16

+4d−

d2) Q

2m

2 ∆+

2( −

12

+2d

+d2) m

4 ∆

) −2Q

2m

∆

( 2(−

2+

d)Q

4+

(−18

+d)Q

2m

2 ∆+

2(7−

4d)m

4 ∆

) M0+

2m

2 ∆

( −( −

4+

d2) Q

4+

( 42−

8d

+d2) Q

2m

2 ∆+

2( −

12

+2d

+d2) m

4 ∆

) M2 0−

2m

∆

( 9(−

2+

d)Q

4+

2(−

45

+7d)Q

2m

2 ∆+

4(7−

4d)m

4 ∆

) M3 0

+( 3

(−2

+d)2

Q4+

2( 6

+10d−

3d2) Q

2m

2 ∆+

8(−

5+

2d)m

4 ∆

) M4 0

+4m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M5 0

+2

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M6 0

))(1

+τ[3

])))

II[2

+d][{{

0,m

π},

1},{{

pr,

m∆},

1},{{

pr+

q,m

∆},

2}]−

16(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)(2

ie2gπN

∆2π

( (−2

+d)2

Q2sM

2 0

( (−3

+d)( Q

2+

s) +2M

2 0

) −2m

5 ∆M

0

( −(−

11

+d)d

( Q2

+s) −

2( Q

2+

5s) +

3(−

1+

d)(−8

+3d)M

2 0

) −2(−

1+

d)m

6 ∆

( (7+

(−5

+d)d

)( Q

2+

s) +(−

19

+7d)M

2 0

) +

2(−

2+

d)Q

2m

∆M

0

( 2s

( Q2

+s) +

( −(−

3+

d)Q

2+

(3+

d)s

) M2 0

+2M

4 0

) +

2m

3 ∆M

0

( (2+

(−7

+d)d

)Q4

+2

( −13

+d2) Q

2s

+(−

20

+d(3

+d))

s2+

( −2(2

1+

d(−

11

+2d))

Q2

+(5

2+

d(−

27

+7d))

s) M2 0

+2(−

9+

d)M

4 0

) +

m4 ∆

((Q

2+

s)((−

2+

(−5

+d)(−1

+d)d

)Q2

+2(−

9+

d(5

+(−

5+

d)d

))s) +

( (−66

+d(3

1+

d(−

13

+2d))

)Q2

+2(3

7+

d(−

25

+d(3

+d))

)s) M

2 0+

4(−

7+

3d)M

4 0

) −m

2 ∆

( (−2

+d)s

( Q2

+s)(

(2+

(−5

+d)d

)Q2

+2(−

3+

d)(−1

+d)s

) +((

6+

(−3

+d)2

d) Q

4−

(−74

+d(3

7+

(−4

+d)d

))Q

2s−

2(−

2+

d)(

13

+(−

6+

d)d

)s2) M

2 0+

((26

+d−

7d2

+2d3) Q

2+

2(−

34

+d(2

4+

(−7

+d)d

))s) M

4 0+

4(8

+(−

5+

d)d

)M6 0

) +

m4 π

( 16(−

2+

d)Q

2m

∆M

0+

(−2

+d)2

Q2

( (−3

+d)( Q

2+

s) +2M

2 0

) −2(−

1+

d)m

2 ∆

( (2+

(−5

+d)d

)( Q

2+

s) +(−

4+

3d)M

2 0

))+

m2 π

( −(−

2+

d)2

Q2

( s+

M2 0

)((−

3+

d)( Q

2+

s) +2M

2 0

) +2(−

2+

d)Q

2m

∆M

0

( (−5

+d)Q

2+

(−9

+d)s−

2(3

+d)M

2 0

) +

2(−

1+

d)m

4 ∆

( (17

+2(−

5+

d)d

)( Q

2+

s) +(−

11

+2d)M

2 0

) −2m

3 ∆M

0

( −(−

17

+d)(−2

+d)Q

2−

(−10

+d)(−1

+d)s

+3(−

1+

d)(−8

+3d)M

2 0

) +

m2 ∆

((Q

2+

s)((−

4+

d(1

9+

d(−

13

+2d))

)Q2

+2(−

5+

d)(−2

+d)(

1+

d)s

) +( (3

+d)(

24

+d(−

15

+2d))

Q2

+2(−

22

+d(8

+(−

3+

d)d

))s) M

2 0+

( 32−

22d

+6d2) M

4 0

)))

(1+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}])

+1

12(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)

( e2gπN

∆2M

2 0

( (−2

+d)s

( (−2

+d)Q

4+

(−8

+d)Q

2s−

2(−

1+

d)s

2) m

2 ∆−

( (−2

+d)2

Q4

+2(6

+(−

9+

d)d

)Q2s

+2(−

32

+5d)s

2) m

4 ∆+

( (20

+d(−

20

+3d))

Q2

+4(−

3+

2d)s

) m6 ∆−

2(−

12

+d(2

+d))

m8 ∆

+m

6 π

( −(−

2+

d)2

Q2

+2(−

2+

(−2

+d)d

)m2 ∆

) +

2m

3 ∆

( −(−

2+

d)Q

4+

(−2

+4d)s

2+

4(1

3−

2d)s

m2 ∆

+2(7−

4d)m

4 ∆+

Q2

( (8+

d)s

+(1

6−

9d)m

2 ∆

))M

0+

( (−2

+d)2

Q2s

( −Q

2+

s) +( (−

2+

d)2

Q4−

2(1

2+

(−5

+d)d

)Q2s

+2(6

+(−

4+

d)d

)s2) m

2 ∆+

( (−4

+(1

0−

3d)d

)Q2

+4

( −17

+d

+d2) s) m

4 ∆+

2(4

+(−

6+

d)d

)m6 ∆

) M2 0

+

2m

∆

( (−2

+d)Q

2( Q

2+

s) +( (−

22

+7d)Q

2+

4(−

3+

d)s

) m2 ∆

+4(−

13

+2d)m

4 ∆

) M3 0

+2m

2 ∆

( (−2

+d)( d

( Q2−

s) +s) −

(10

+(−

5+

d)d

)m2 ∆

) M4 0

+

m4 π

( −2(4

+d(−

10

+3d))

m4 ∆

+2(−

2+

d)Q

2m

∆M

0+

4(−

7+

4d)m

3 ∆M

0−

(−2

+d)2

Q2

( Q2−

2s−

M2 0

) +m

2 ∆

( 8s

+d

( (−4

+d)Q

2−

4(−

2+

d)s

) +(4−

2(−

2+

d)d

)M2 0

))+

m2 π

( 6(−

2+

(−2

+d)d

)m6 ∆−

2(−

2+

d)Q

2m

∆M

0

( Q2

+s

+M

2 0

) −4m

3 ∆M

0

( −7Q

2+

4dQ

2−

8s

+6ds

+2(−

3+

d)M

2 0

) +

m4 ∆

( (24

+(4−

3d)d

)Q2

+4(3

+(−

8+

d)d

)s+

4(5

+2d)M

2 0

) +(−

2+

d)2

Q2

((Q

2−

s) s+

( Q2−

2s) M

2 0

) +

148


2m

2 ∆

( −(−

2+

d)2

Q4−

3(−

2+

d)Q

2s

+d(−

5+

2d)s

2+

( −(2

+(−

7+

d)d

)Q2

+2(−

4+

d)s

) M2 0

+(−

2+

d)(−1

+d)M

4 0

))) (1

+τ[3

])(−

4iπ

II[2

+d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}]

+4i(π

II[2

+d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}]

+πII

[2+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}])

))+

16(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)

( 2ie

2gπN

∆2π

( m4 π

( Q2

( (−3

+d)(−2

+d)2

Q2−

2( −

2+

7d−

6d2

+d3) m

2 ∆

) +16(−

2+

d)Q

2m

∆M

0+

(−1

+d)( (−

2+

d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M2 0

) +

m2 π

( Q2m

2 ∆

((−4

+19d−

13d2

+2d3) Q

2+

2( −

17

+27d−

12d2

+2d3) m

2 ∆

) +2(−

2+

d)Q

2m

∆

( (−5

+d)Q

2+

(−17

+d)m

2 ∆

) M0+

( −2(−

3+

d)(−2

+d)2

Q4

+2

( 44

+2d−

17d2

+3d3) Q

2m

2 ∆+

4(−

3+

d)(−1

+d)2

m4 ∆

) M2 0−

2m

∆

((−3

0+

13d

+d2) Q

2+

2( 7−

11d

+4d2) m

2 ∆

)

M3 0−

2(−

1+

d)( (−

2+

d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M4 0

) −(m

∆+

M0)( Q

2m

3 ∆

( −( −

2+

5d−

6d2

+d3) Q

2+

2( −

7+

12d−

6d2

+d3) m

2 ∆

) +Q

2m

2 ∆

((−6

+19d−

8d2

+d3) Q

2−

2( −

5+

d−

5d2

+d3) m

2 ∆

) M0+

m∆

((8

+2d−

5d2

+d3) Q

4+

( 76−

44d

+19d2−

3d3) Q

2m

2 ∆+

2( 1

2−

14d

+d2

+d3) m

4 ∆

) M2 0+

( −(−

3+

d)(−2

+d)2

Q4

+3

( 20−

5d2

+d3) Q

2m

2 ∆−

2( −

2+

8d−

7d2

+d3) m

4 ∆

) M3 0+

(−2

+d)m

∆

((−1

2−

5d

+d2) Q

2−

2( −

8+

7d

+d2) m

2 ∆

) M4 0−

(−1

+d)( (−

2+

d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M5 0

))

(1+

τ[3

])II

[2+

d][{{

0,m

π},

1},{{

pr+

q,m

∆},

1},{{

pr,

m∆},

1}])−

16(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)(2

ie2gπN

∆2M

2 0

( −m

6 π

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) −m

4 π

( (−2

+d)2

Q4−

(−4

+d)d

Q2m

2 ∆+

2( 4−

10d

+3d2) m

4 ∆−

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M0−

3( (−

2+

d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M2 0

) −( −

m2 ∆

+M

2 0

)(−m

2 ∆

( (−2

+d)2

Q4

+( −

20

+20d−

3d2) Q

2m

2 ∆+

2( −

12

+2d

+d2) m

4 ∆

) −2m

∆

( Q2

+m

2 ∆

)((−

2+

d)Q

2+

2(−

7+

4d)m

2 ∆

) M0+

( (−2

+d)2

Q4−

2( −

2−

4d

+d2) Q

2m

2 ∆+

4(5−

2d)m

4 ∆

) M2 0−

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M3 0−

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M4 0

) −m

2 π

( m2 ∆

( 2(−

2+

d)2

Q4

+( −

24−

4d

+3d2) Q

2m

2 ∆−

6( −

2−

2d

+d2) m

4 ∆

) +2Q

2m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M0−

2( (−

2+

d)2

Q4

+( 4

+4d−

d2) Q

2m

2 ∆+

2( 8−

6d

+d2) m

4 ∆

) M2 0

+4m

∆

( (−2

+d)Q

2+

2(−

7+

4d)m

2 ∆

) M3 0

+3

( (−2

+d)2

Q2−

2( −

2−

2d

+d2) m

2 ∆

) M4 0

))(1

+τ[3

])(π

II[2

+d][{{

0,m

π},

1},{{

pr,

m∆},

1},{{

pr+

q,m

∆},

2}]

+πII

[2+

d][{{

0,m

π},

1},{{

pr+

q,m

∆},

1},{{

pr,

m∆},

2}])

)−

16(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)(8

ie2gπN

∆2π

2( (

Q2

+s) 2

+2

( Q2−

s) M2 0

+M

4 0

) (2(1

8+

(−8

+d)d

)m6 ∆−

4(−

17

+5d)m

5 ∆M

0−

(−2

+d)2

Q2

( −s

+m

2 π

) (mπ−

M0)(m

π+

M0)+

2(−

2+

d)Q

2m

∆M

0

( 2s−

3m

2 π+

M2 0

) +2m

3 ∆M

0

( 6Q

2−

3dQ

2−

22s

+8ds

+(3

4−

10d)m

2 π+

2(−

6+

d)M

2 0

) −m

4 ∆

( 44s

+2(−

10

+d)d

s+

4(6

+(−

2+

d)d

)m2 π−

12M

2 0+

(−2

+d)d

( Q2

+2M

2 0

))+

m2 ∆

( (−2

+d)s

( (2+

d)Q

2+

2(−

1+

d)s

) +

+2(−

2+

(−4

+d)d

)m4 π

+( (−

4+

d)(−2

+d)Q

2−

2(6

+(−

2+

d)d

)s) M

2 02(−

2+

d)(−1

+d)M

4 0−

2m

2 π

( (4+

(−3

+d)d

)Q2

+(−

2+

(−4

+d)d

)s+

(−2

+(−

4+

d)d

)M2 0

)))

(1+

τ[3

])II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

3}])

+32iπ

2

( −1

24(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)

( e2gπN

∆2Q

2( Q

2+

4M

2 0

)(m

4 π

( (−2

+d)2

Q2−

2( −

2−

4d

+d2) m

2 ∆

) −2m

2 π

( −m

2 ∆

((4−

3d

+d2) Q

2+

2( 6−

2d

+d2) m

2 ∆

) +( −

3(−

2+

d)Q

2m

∆+

2(1

7−

5d)m

3 ∆

) M0

+( (−

2+

d)2

Q2−

2( −

2−

4d

+d2) m

2 ∆

) M2 0

) +( −

m2 ∆

+M

2 0

)( m

2 ∆

( 36m

2 ∆+

2d

( Q2−

8m

2 ∆

) −d2

( Q2−

2m

2 ∆

))+

( −6(−

2+

d)Q

2m

∆+

4(1

7−

5d)m

3 ∆

) M0

+( (−

2+

d)2

Q2−

2( −

2−

4d

+d2) m

2 ∆

) M2 0

))(1

+τ[3

])) +

112(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)

( e2gπN

∆2Q

2( Q

2+

4M

2 0

)(m

4 π

( (−2

+d)2

Q2−

2( −

2−

3d

+d2) m

2 ∆

) +

m2 π

( m2 ∆

((4−

7d

+2d2) Q

2+

2( 9−

6d

+2d2) m

2 ∆

) +2m

∆

( (−2

+d)Q

2+

(−10

+d)m

2 ∆

) M0−

2( (−

2+

d)2

Q2−

2( −

2−

3d

+d2) m

2 ∆

) M2 0

) +( −

m2 ∆

+M

2 0

)(m

2 ∆

( −( 2−

3d

+d2) Q

2+

2( 3−

3d

+d2) m

2 ∆

) −2m

∆

( (−2

+d)Q

2+

(−10

+d)m

2 ∆

) M0

+( (−

2+

d)2

Q2−

2( −

2−

3d

+d2) m

2 ∆

) M2 0

))(1

+τ[3

])))

II[4

+d][{{

0,m

π},

1},{{

pr,

m∆},

1},{{

pr+

q,m

∆},

3}]

+1

12(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)

( e2gπN

∆2

( (Q

2+

s) 2+

2( Q

2−

s) M2 0

+M

4 0

)( 2

(3+

(−3

+d)d

)m6 ∆−

2(−

10

+d)m

5 ∆M

0−

(−2

+d)2

Q2

( −s

+m

2 π

) (mπ−

M0)(m

π+

M0)+

2(−

2+

d)Q

2m

∆M

0

( −m

2 π+

M2 0

) −2m

3 ∆M

0

( (−2

+d)Q

2+

ds

+(−

10

+d)m

2 π−

−2(−

5+

d)M

2 0

) m4 ∆

( (−2

+d)(−1

+d)Q

2+

2(−

3+

d)(

1+

d)s

+2(9

+2(−

3+

d)d

)m2 π

+2(−

2+

d)2

M2 0

) +

m2 ∆

( (−2

+d)s

( (−2

+d)Q

2+

2(−

1+

d)s

) +2(−

2+

(−3

+d)d

)m4 π

+( (−

2+

d)(−1

+d)Q

2−

2(6

+(−

3+

d)d

)s) M

2 0+

2(−

2+

d)(−1

+d)M

4 0+

m2 π

( (−4

+(7−

2d)d

)Q2

+4s−

2(−

3+

d)d

s+

(4−

2(−

3+

d)d

)M2 0

)))

(1+

τ[3

])( 3

2iπ

2II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

3}]−

149

B. Appendix to VVCS

16i( 2

π2II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

3}]

+π

2II

[4+

d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}]))

) −1

6(−

1+

d)2

F2m

4 ∆M

0(−

s+

M2 0)

( 8ie

2gπN

∆2Q

2( Q

2+

4M

2 0

)(m

4 π

( (−2

+d)2

Q2−

2( −

2−

3d

+d2) m

2 ∆

) +

m2 π

( m2 ∆

((4−

7d

+2d2) Q

2+

2( 9−

6d

+2d2) m

2 ∆

) +2m

∆

( (−2

+d)Q

2+

(−10

+d)m

2 ∆

) M0−

2( (−

2+

d)2

Q2−

2( −

2−

3d

+d2) m

2 ∆

) M2 0

) +( −

m2 ∆

+M

2 0

)(m

2 ∆

( −( 2−

3d

+d2) Q

2+

2( 3−

3d

+d2) m

2 ∆

) −2m

∆

( (−2

+d)Q

2+

(−10

+d)m

2 ∆

) M0

+( (−

2+

d)2

Q2−

2( −

2−

3d

+d2) m

2 ∆

) M2 0

))(1

+τ[3

])( 2

π2II

[4+

d][{{

0,m

π},

1},{{

pr,

m∆},

1},{{

pr+

q,m

∆},

3}]

+π

2II

[4+

d][{{

0,m

π},

1},{{

pr+

q,m

∆},

2},{{

pr,

m∆},

2}]))

B(ν

,Q2)

=1

24(−

1+

d)3

F2m

4 ∆M

5 0

( ie2gπN

∆2(1

+τ[3

])(

1ds3

((−3

2sm

2 πM

2 0

( Q2s

( m2 π−

m2 ∆

) −Q

2sm

∆M

0+

((Q

2+

s) m2 π−

sm2 ∆−

Q2

( s+

m2 ∆

))M

2 0+

( −s

+m

2 π−

m2 ∆

) M4 0

) +d4

( −Q2s2

( m2 π−

m2 ∆

) 3+

2Q

2s2

m∆

( m2 π−

m2 ∆

) 2M

0+

s( m

2 π−

m2 ∆

)(−

( Q2−

2s) m

4 π−

2sm

4 ∆+

Q2m

2 ∆

( s−

m2 ∆

) +Q

2m

2 π

( 3s

+2m

2 ∆

))M

2 0+

2sm

∆

((Q

2−

2s) m

4 π+

2sm

4 ∆−

2Q

2m

2 π

( s+

m2 ∆

) +Q

2( −

2sm

2 ∆+

m4 ∆

))M

3 0−

((Q

2−

2s) m

6 π−

2sm

4 ∆

( s+

m2 ∆

) +Q

2m

2 ∆

( s2+

sm2 ∆−

m4 ∆

) +m

4 π

( −3Q

2( s

+m

2 ∆

) +2s

( 3s

+m

2 ∆

))+

m2 π

( 2sm

2 ∆

( −2s

+m

2 ∆

) +Q

2( 3

s2+

2sm

2 ∆+

3m

4 ∆

))) M

4 0+

2sm

∆

( s2+

m4 π−

4sm

2 ∆−

2m

2 πm

2 ∆+

m4 ∆

) M5 0

+( s3

−m

6 π−

5s2

m2 ∆−

3sm

4 ∆+

m6 ∆

+m

4 π

( s+

3m

2 ∆

) +m

2 π

( s2+

2sm

2 ∆−

3m

4 ∆

))M

6 0

) +

2d3

( Q2s2

( m2 π−

m2 ∆

) 3−

3Q

2s2

m∆

( m2 π−

m2 ∆

) 2M

0+

s( m

2 π−

m2 ∆

)(( Q

2−

5s) m

4 π−

Q2sm

2 ∆+

Q2m

4 ∆+

5sm

4 ∆−

Q2m

2 π

( 3s

+2m

2 ∆

))M

2 0+

sm∆

((−3

Q2

+16s) m

4 π+

6Q

2sm

2 ∆−

3Q

2m

4 ∆−

8sm

4 ∆+

m2 π

( −8sm

2 ∆+

6Q

2( s

+m

2 ∆

))) M

3 0+

((Q

2−

5s) m

6 π+

3Q

2s2

m2 ∆

+Q

2sm

4 ∆+

7s2

m4 ∆−

Q2m

6 ∆−

5sm

6 ∆+

m4 π

( −3Q

2( s

+m

2 ∆

) +s

( 19s

+5m

2 ∆

))+

m2 π

( 5sm

2 ∆

( −2s

+m

2 ∆

) +Q

2( 5

s2+

2sm

2 ∆+

3m

4 ∆

)))

M4 0

+sm

∆

( −3s2−

5m

4 π+

24sm

2 ∆−

5m

4 ∆+

2m

2 π

( 2s

+5m

2 ∆

))M

5 0+

( −2s3

+m

6 π+

13s2

m2 ∆

+5sm

4 ∆−

m6 ∆

+m

4 π

( s−

3m

2 ∆

) +m

2 π

( −5s2−

6sm

2 ∆+

3m

4 ∆

))M

6 0

) −8d

( Q2s2

( m2 π−

m2 ∆

) 3−

Q2s2

m∆

( m2 π−

m2 ∆

) 2M

0+

s( m

2 π−

m2 ∆

)(( Q

2+

s) m4 π−

Q2sm

2 ∆+

Q2m

4 ∆+

sm4 ∆−

m2 π

( 7Q

2s

+2Q

2m

2 ∆+

2sm

2 ∆

))M

2 0−

sm∆

((Q

2+

7s) m

4 π+

( Q2

+6s) m

4 ∆−

m2 π

( 5Q

2s

+2Q

2m

2 ∆+

13sm

2 ∆

))M

3 0+

((Q

2+

s) m6 π−

m4 π

( 7Q

2s

+11s2

+3Q

2m

2 ∆+

3sm

2 ∆

) −m

2 ∆

( sm2 ∆

( 11s

+m

2 ∆

) +Q

2( 2

s2−

sm2 ∆

+m

4 ∆

))+

m2 π

( sm2 ∆

( 14s

+3m

2 ∆

) +Q

2( 5

s2+

6sm

2 ∆+

3m

4 ∆

)))

M4 0−

s2m

∆

( s+

2m

2 π+

5m

2 ∆

) M5 0

+( m

6 π−

4s2

m2 ∆

+sm

4 ∆−

m6 ∆−

m4 π

( 7s

+3m

2 ∆

) +m

2 π

( 7s2

+6sm

2 ∆+

3m

4 ∆

))M

6 0

) +

4d2

( Q2s2

( m2 π−

m2 ∆

) 3+

s( m

2 π−

m2 ∆

)(( Q

2+

4s) m

4 π−

2sm

4 ∆−

m2 π

( 5Q

2s

+2Q

2m

2 ∆+

2sm

2 ∆

) +Q

2( −

sm2 ∆

+m

4 ∆

))M

2 0+

s2m

∆

( −19m

4 π−

3m

2 ∆

( Q2

+m

2 ∆

) +m

2 π

( Q2

+22m

2 ∆

))M

3 0+

((Q

2+

4s) m

6 π+

2sm

4 ∆

( −11s

+m

2 ∆

) −m

4 π

( 5Q

2s

+22s2

+3Q

2m

2 ∆+

6sm

2 ∆

) −Q

2( 5

s2m

2 ∆−

sm4 ∆

+m

6 ∆

) +m

2 π

( 20s2

m2 ∆

+Q

2( s2

+4sm

2 ∆+

3m

4 ∆

))) M

4 0+

sm∆

( 3m

4 π−

20sm

2 ∆+

3m

4 ∆−

6m

2 π

( s+

m2 ∆

))M

5 0+

( s3+

m6 π−

13s2

m2 ∆−

sm4 ∆−

m6 ∆−

m4 π

( 7s

+3m

2 ∆

) +m

2 π

( 9s2

+8sm

2 ∆+

3m

4 ∆

))M

6 0

))II

[d][{{

0,m

π},

1}]) +

1s−

M2 0

( M5 0

(1

M5 0

((m

2 π−

m2 ∆

+M

2 0

)((−

2+

d)2

Q2

( m2 π−

m2 ∆

) +2m

∆

( (−2

+d)Q

2−

2(−

2+

d)m

2 π−

2m

2 ∆

) M0−

( (−2

+d)2

Q2

+4

( 5−

4d

+d2) m

2 ∆

)

M2 0

+4(−

2+

d)m

∆M

3 0

)((2

+d)( m

2 π−

m2 ∆

) 2+

2( (−

4+

d)m

2 π−

(2+

d)m

2 ∆

) M2 0

+(2

+d)M

4 0

))+

2(m

2 π−

m2 ∆

+M

2 0)( (m

2 π−

m2 ∆

)2−

2(m

2 π+

m2 ∆

)M2 0+

M4 0

) (4m

3 ∆+

(16+

3(−

4+

d)d

)m2 ∆

M0+

(−2+

d)2

M0(m

2 π−

M2 0)−

2(−

2+

d)m

∆(−

2m

2 π+

M2 0))

M4 0

+

8(−

1+

d)m

∆

( 3(−

2+

d)m

4 π−

(−m

∆+

M0)(m

∆+

M0)2

(dm

∆+

(−2

+d)M

0)+

m2 π

( 2(−

1+

2d)m

2 ∆+

2(3

+d)m

∆M

0−

2(−

2+

d)M

2 0

))−

4( (m

2 π−

m2 ∆

)2−

2(m

2 π+

m2 ∆

)M2 0+

M4 0

) ((−

2+

d)m

2 π((

2+

d)m

∆+

(−2+

d)M

0)−

(m∆

+M

0) (−

(4−

2d+

d2)m

2 ∆+

2(−

2+

d)m

∆M

0+

(−2+

d)2

M2 0))

M2 0

−1

M4 0

( 2( −4

m2 πM

2 0+

d( m

2 π−

m2 ∆

+M

2 0

) 2) (

2m

5 ∆+

(2+

(−2

+d)d

)m4 ∆

M0

+(−

2+

d)2

M0

( m2 π−

M2 0

)(2Q

2+

m2 π−

M2 0

) +m

3 ∆

( −(−

2+

d)d

Q2

+2(−

3+

d)m

2 π−

150


2( 1

3−

7d

+2d2) M

2 0

) −2m

2 ∆M

0

((3−

3d

+d2) m

2 π+

( 25−

15d

+3d2) M

2 0

) +(−

2+

d)m

∆

( dQ

2m

2 π−

2m

4 π−

( (−4

+d)Q

2+

8m

2 π

) M2 0

+10M

4 0

))) +

1M

2 0

( 4(−

1+

d)( m

2 π−

m2 ∆

+M

2 0

)((−

2+

d)m

4 π((−8

+d)m

∆+

(−2

+d)M

0)−

m2 π

( m∆

((2

+d−

d2) Q

2+

2( 1

5−

7d

+d2) m

2 ∆

) +( −

(−2

+d)2

Q2

+2

( 10−

4d

+d2) m

2 ∆

) M0

+2(−

2+

d)2

m∆

M2 0

+2(−

2+

d)2

M3 0

) +(m

∆+

M0)

( m2 ∆

((4−

3d

+d2) Q

2+

( 14−

4d

+d2) m

2 ∆

) +( −

(−2

+d)Q

2m

∆+

2(9−

4d)m

3 ∆

) M0−

( (−2

+d)2

Q2

+2

( 6−

3d

+d2) m

2 ∆

) M2 0+

6(−

2+

d)m

∆M

3 0+

(−2

+d)2

M4 0

))))

II[d

][{{

0,m

π},

1},{{

p,m

∆},

1}]) −

1−

s+

M2 0

( M5 0

(4( m

4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)) ((

−2+

d) (−

ds+

(2+

d)m

2 π)m

∆+

(4+

(−2+

d)d

)m3 ∆

+(−

2+

d)2

(−s+

m2 π)M

0+

(8+

(−4+

d)d

)m2 ∆

M0)

s−

2(s

+m

2 π−

m2 ∆

)( m4 π+(s−

m2 ∆

)2−

2m

2 π(s

+m

2 ∆)) (4

m3 ∆

+(−

2+

d)2

(−s+

m2 π)M

0+

(16+

3(−

4+

d)d

)m2 ∆

M0−

2(−

2+

d)m

∆(−

2m

2 π+

M2 0))

s2

+

1 s3

( (s

+m

2 π−

m2 ∆

)((2

+d)m

4 π+

(2+

d)( s−

m2 ∆

) 2+

2m

2 π

( (−4

+d)s−

(2+

d)m

2 ∆

))( 4

sm3 ∆

+(−

2+

d)2

( s−

m2 π

) M0

( Q2−

s+

M2 0

) −2(−

2+

d)s

m∆

( Q2

+s−

2m

2 π+

M2 0

) +m

2 ∆M

0

( (16

+3(−

4+

d)d

)s+

(−2

+d)2

( Q2

+M

2 0

))))−

8(−

1+

d)m

∆

( 3(−

2+

d)m

4 π−

(m∆

+M

0)2

((−2

+d)s

+m

∆(−

dm

∆+

2M

0))

+m

2 π

( 2(−

1+

2d)m

2 ∆+

2(3

+d)m

∆M

0+

(−2

+d)( −

3s

+M

2 0

)))−

1 s

( 4(−

1+

d)( s

+m

2 π−

m2 ∆

)((1

4+

(−4

+d)d

)m5 ∆

+(−

2+

d)2

( −s

+m

2 π

)(Q

2−

s+

m2 π

) M0

+(−

8+

d)(−4

+d)m

4 ∆M

0+

m3 ∆

( (4+

(−3

+d)d

)Q2−

4(−

1+

d)2

s−

2(1

5+

(−7

+d)d

)m2 π

+2(5

+(−

5+

d)d

)M2 0

) +m

2 ∆M

0

( 6Q

2−

34s−

4(−

5+

d)d

s−

2(1

0+

(−4

+d)d

)m2 π+

10M

2 0+

(−4

+d)d

( Q2

+2M

2 0

))+

(−2

+d)m

∆

( (−8

+d)m

4 π+

m2 π

( Q2

+dQ

2+

6s−

2ds−

2M

2 0

) +s

( Q2−

dQ

2+

(2+

d)s

+2M

2 0

))))

+1 s2

( 2( −4

sm2 π

+d

( s+

m2 π−

m2 ∆

) 2) (

2m

5 ∆+

(2+

(−2

+d)d

)m4 ∆

M0

+(−

2+

d)2

( −s

+m

2 π

) M0

( 2Q

2−

2s

+m

2 π+

M2 0

) −m

3 ∆( (−

2+

d)d

Q2

+(2

2+

d(−

10

+3d))

s−

2(−

3+

d)m

2 π+

(−2

+d)2

M2 0

) +m

2 ∆M

0

( (−54

+(3

4−

7d)d

)s−

2(3

+(−

3+

d)d

)m2 π

+(−

2+

d)2

M2 0

) +

(−2

+d)m

∆

( −2m

4 π+

m2 π

( dQ

2−

(6+

d)s

+(−

2+

d)M

2 0

) +s

( (6+

d)s−

(−4

+d)( Q

2+

M2 0

))))

))II

[d][{{

0,m

π},

1},{{

p+

q,m

∆},

1}]) +

1−

s+

M2 0

( 8(−

1+

d)M

5 0

( m∆

( m2 π−

(m∆

+M

0)2

) ((−

2+

d)m

4 π−

(−m

∆+

M0)(m

∆+

M0)((−2

+d)s

+m

∆(−

dm

∆+

2M

0))

+m

2 π

( 2m

2 ∆+

6m

∆M

0+

(−2

+d)( −

s+

M2 0

)))

II[d

][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

1}]−

m∆

( −m2 π

+(m

∆+

M0)2

)( −(

−2+

d)m

4 π−

2m

2 πm

∆(m

∆+

3M

0)+

(−m

∆+

M0)2

(m∆

+M

0)(d

m∆

+(−

2+

d)M

0)) II

[d][{{

0,m

π},

1},{{

pr+

q,m

∆},

1},{{

pr,

m∆},

1}]

+

4π

( 2(−

2+

d)m

6 πm

∆+

m4 π

( 2(7

+(−

5+

d)d

)m3 ∆−

(−2

+d)2

Q2M

0+

2(1

0+

(−5

+d)d

)m2 ∆

M0−

(−2

+d)m

∆

( dQ

2+

s−

M2 0

))+

m2 π

( −2(8

+d(−

7+

2d))

m5 ∆−

4(9

+(−

4+

d)d

)m4 ∆

M0

+(−

2+

d)2

Q2M

0

( s+

M2 0

) −2m

3 ∆

( (−2

+d)d

Q2

+(−

5+

(−1

+d)d

)s+

(19

+(−

7+

d)d

)M2 0

) −2m

2 ∆M

0

( (4+

(−3

+d)d

)Q2

+(−

4+

(−1

+d)d

)s+

(6+

(−3

+d)d

)M2 0

) +(−

2+

d)m

∆

(((−

2+

d)Q

2−

s) s+

( dQ

2−

2s) M

2 0−

3M

4 0

))+

(m∆

+M

0)( 2

(3+

(−3

+d)d

)m6 ∆

+10m

5 ∆M

0−

(−2

+d)2

Q2sM

2 0+

(−2

+d)s

m∆

M0

( s+

3M

2 0

) −m

3 ∆M

0

( (4+

d)s

+(−

2+

3d)M

2 0

) −m

4 ∆

( (4+

(−2

+d)d

)Q2

+(1

6+

d(−

7+

2d))

s+

(−18

+d(−

1+

2d))

M2 0

) +m

2 ∆

( (−2

+d)s

( −s

+d

( Q2

+2s))

+( (8

+(−

4+

d)d

)Q2+

(−22

+(1

1−

2d)d

)s)M

2 0+

2(6

+(−

4+

d)d

)M4 0

))) II

[2+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

2}]−

4π

( 2(−

2+

d)m

6 πm

∆+

m4 π

( 14m

3 ∆+

2dm

∆

( Q2−

5m

2 ∆

) +d2

( −Q

2m

∆+

2m

3 ∆

) +( −

(−2

+d)2

Q2

+2

( 10−

5d

+d2) m

2 ∆

) M0

) +

(−m

∆+

M0)(m

∆+

M0)2

( m2 ∆

((4−

2d

+d2) Q

2−

2( 3−

3d

+d2) m

2 ∆

) −10m

3 ∆M

0−

( (−2

+d)2

Q2

+2

( 4+

d−

d2) m

2 ∆

) M2 0

+4(−

2+

d)m

∆M

3 0

) −2m

2 π

( m3 ∆

( 8m

2 ∆+

d2

( Q2

+2m

2 ∆

) −d

( 2Q

2+

7m

2 ∆

))+

m2 ∆

((4−

3d

+d2) Q

2+

2( 9−

4d

+d2) m

2 ∆

) M0+

m∆

( −( 2−

3d

+d2) Q

2+

2( 7−

4d

+d2) m

2 ∆

) M2 0

+( −

(−2

+d)2

Q2

+2(−

1+

d)2

m2 ∆

) M3 0

+3(−

2+

d)m

∆M

4 0

))II

[2+

d][{{

0,m

π},

1},{{

pr,

m∆},

1},{{

pr+

q,m

∆},

2}]−

8πm

∆M

0(−

mπ

+m

∆+

M0)(m

π+

m∆

+M

0)

( 2m

∆

( m2 π−

m2 ∆

) +( 2

s−

ds

+(−

2+

d)m

2 π+

dm

2 ∆

) M0

) II[2

+d][{{

0,m

π},

1},{{

p+

q,m

∆},

1},{{

p,m

∆},

2}]−

8πm

∆M

0(−

mπ

+m

∆+

M0)

(mπ

+m

∆+

M0)( −

2m

2 πm

∆+

2m

3 ∆−

( (−2

+d)m

2 π+

dm

2 ∆

) M0

+(−

2+

d)M

3 0

) II[2

+d][{{

0,m

π},

1},{{

pr+

q,m

∆},

1},{{

pr,

m∆},

2}]

+

151

B. Appendix to VVCS

32π

2( 2

(−2

+d)(−1

+d)m

6 ∆M

0−

2m

5 ∆

( −Q

2+

s+

M2 0

) +(−

2+

d)2

Q2

( −s

+m

2 π

) M0

( −m

2 π+

M2 0

) −m

4 ∆M

0

((−8

+d2) Q

2+

2(1

0+

(−3

+d)d

)s+

4(−

2+

d)

(−1

+d)m

2 π+

2(−

2+

d)(−1

+d)M

2 0

) +(−

2+

d)m

∆

( 2m

4 π

( −Q

2+

s+

M2 0

) +s

( Q2

+s

+M

2 0

)(−Q

2+

2M

2 0

) −m

2 π

( −Q

2+

s+

M2 0

)(Q

2+

2s

+2M

2 0

))−

m3 ∆

( 2(−

4+

d)s

2+

( −Q

2+

M2 0

)(dQ

2+

2(−

3+

d)m

2 π+

2M

2 0

) +s

( 3(−

4+

d)Q

2+

2(−

3+

d)m

2 π+

(34−

6d)M

2 0

))+

m2 ∆

M0

( 2Q

4+

(−2

+d)2

Q2s

+2(−

1+

(−2

+d)d

)s2

+2(−

2+

d)(−1

+d)m

4 π+

( (8+

(−4

+d)d

)Q2−

2(1

4+

(−7

+d)d

)s) M

2 0+

2(5

+(−

4+

d)d

)

M4 0−

2m

2 π

( (6+

(−4

+d)d

)Q2

+( −

10

+d

+d2) s

+(−

2+

d)(−1

+d)M

2 0

))) II

[4+

d][{{

0,m

π},

1},{{

p,m

∆},

1},{{

p+

q,m

∆},

3}]−

32π

2( (−

2+

d)m

4 π

( −2Q

2m

∆+

( −(−

2+

d)Q

2+

2(−

1+

d)m

2 ∆

) M0

+4m

∆M

2 0

) +m

2 π

( Q2m

∆

( (−2

+d)Q

2+

2(−

3+

d)m

2 ∆

) −2m

2 ∆

((6−

4d

+d2) Q

2+

2( 2−

3d

+d2) m

2 ∆

) M0

+2m

∆

( (−2

+d)Q

2−

2(−

3+

d)m

2 ∆

) M2 0

+2

( (−2

+d)2

Q2

+2

( 4+

d−

d2) m

2 ∆

) M3 0−

8(−

2+

d)m

∆M

4 0

) +

(m∆

+M

0)( Q

2m

2 ∆

( dQ

2+

2m

2 ∆

) +( −

(−2

+d)Q

4m

∆−

( −6

+d2) Q

2m

3 ∆+

2( 2−

3d

+d2) m

5 ∆

) M0

+m

2 ∆

((8−

4d

+d2) Q

2−

2( 4−

3d

+d2) m

2 ∆

) M2 0+

m∆

( (−2

+d)2

Q2−

2( 8−

3d

+d2) m

2 ∆

) M3 0−

( (−2

+d)2

Q2

+2

( 6+

d−

d2) m

2 ∆

) M4 0

+4(−

2+

d)m

∆M

5 0

))

II[4

+d][{{

0,m

π},

1},{{

pr,

m∆},

1},{{

pr+

q,m

∆},

3}]

+16π

2M

0

( 2(−

2+

d)(−1

+d)m

6 ∆−

4m

5 ∆M

0+

2(−

2+

d)

( −Q

2−

2s

+2m

2 π

) m∆

(mπ−

M0)M

0(m

π+

M0)+

(−2

+d)2

Q2

( −s

+m

2 π

) (−m

π+

M0)(m

π+

M0)−

2m

3 ∆M

0

( dQ

2+

2s

+2(−

3+

d)m

2 π−

2(−

6+

d)M

2 0

) −m

4 ∆

( d2Q

2+

2(6

+(−

3+

d)d

)s+

4(−

2+

d)(−1

+d)m

2 π+

2(6

+(−

3+

d)d

)M2 0

) +m

2 ∆

( (−2

+d)s

( dQ

2+

2(−

1+

d)s

) +2(−

2+

d)(−1

+d)m

4 π+

( (−2

+d)d

Q2−

2(1

4+

(−7

+d)d

)s) M

2 0+

2(−

2+

d)(−1

+d)M

4 0+

m2 π

( 8s−

2d2

( Q2

+s) +

2d

( 2Q

2+

s) +(8−

2(−

1+

d)d

)M2 0

)))

II[4

+d][{{

0,m

π},

1},{{

p+

q,m

∆},

2},{{

p,m

∆},

2}]−

16π

2M

0

( −(−

2+

d)m

4 π

( (−2

+d)Q

2+

2m

∆(m

∆−

dm

∆−

2M

0)) −

2m

2 π

( (−2

+d)m

2 ∆

( −2m

2 ∆+

d( Q

2+

2m

2 ∆

))+

m∆

( (−2

+d)Q

2+

2(−

3+

d)m

2 ∆

) M0−

( (−2

+d)2

Q2

+2

( 4+

d−

d2) m

2 ∆

) M2 0

+4(−

2+

d)m

∆M

3 0

) +

(m∆

+M

0)( 4

m5 ∆−

6dm

5 ∆+

d2

( −Q

2m

3 ∆+

2m

5 ∆

) +m

2 ∆

( −8m

2 ∆−

2d

( Q2−

3m

2 ∆

) +d2

( Q2−

2m

2 ∆

))M

0+

m∆

( −16m

2 ∆−

2d

( Q2−

3m

2 ∆

) +d2

( Q2−

2m

2 ∆

))M

2 0−

( (−2

+d)2

Q2

+2

( 6+

d−

d2) m

2 ∆

) M3 0

+4(−

2+

d)m

∆M

4 0

))II

[4+

d][{{

0,m

π},

1},{{

pr+

q,m

∆},

2},{{

pr,

m∆},

2}]))

))

152

Bibliography

[A+02] M. Amarian et al., The Q2 evolution of the generalized Gerasimov-Drell- Hearn integral for the neutron using a He-3 target, Phys.Rev. Lett. 89 (2002), 242301.

[A+03] P. L. Anthony et al., Precision measurement of the proton anddeuteron spin structure functions g2 and asymmetries A(2), Phys.Lett. B553 (2003), 18–24.

[A+07] A. Airapetian et al., Precise determination of the spin structurefunction g(1) of the proton, deuteron and neutron, Phys. Rev. D75(2007), 012007.

[ACK06] S. Ando, J.-W. Chen, and C.-W. Kao, Leading chiral correctionsto the nucleon generalized parton distributions, Phys. Rev. D74(2006), 094013.

[AIL89] M. Anselmino, B. L. Ioffe, and E. Leader, On Possible Resolutionsof the Spin Crisis in the Parton Model, Sov. J. Nucl. Phys. 49(1989), 136–154.

[AK+04] A. Ali Khan et al., The nucleon mass in N(f) = 2 lattice QCD: Fi-nite size effects from chiral perturbation theory, Nucl. Phys. B689(2004), 175–194.

[AKNT06] C. Alexandrou, G. Koutsou, J. W. Negele, and A. Tsapalis, Thenucleon electromagnetic form factors from lattice QCD, Phys. Rev.D74 (2006), 034508.

[AS02] D. Arndt and M. J. Savage, Chiral Corrections to Matrix Elementsof Twist-2 Operators, Nucl. Phys. A697 (2002), 429–439.

[B+07a] P. E. Bosted et al., Quark-hadron duality in spin structure func-tions g1(p) and g1(d), Phys. Rev. C75 (2007), 035203.

[B+07b] D. Brommel et al., Moments of generalized parton distributionsand quark angular momentum of the nucleon, PoS LAT2007(2007), 158.

153

BIBLIOGRAPHY

[Bal60] A. M. Baldin, Nucl. Phys. 18 (1960), 310.

[BB02] J. Bluemlein and H. Bottcher, QCD analysis of polarized deep in-elastic scattering data and parton distributions, Nucl. Phys. B636(2002), 225–263.

[BC70] H. Burkhardt and W. N. Cottingham, Sum rules for forward vir-tual Compton scattering, Annals Phys. 56 (1970), 453–463.

[BDKM] V. Bernard, M. Dorati, H. Krebs, and U.-G. Meißner, The spin po-larizabilities of the nucleon in the covariant SSE scheme, in prepa-ration.

[BEM02] V. Bernard, L. Elouadrhiri, and U.-G. Meißner, Axial structure ofthe nucleon, J. Phys. G28 (2002), R1–R35.

[Ber07] V. Bernard, Chiral Perturbation Theory and Baryon Properties,arXiv:0706.0312 [hep-ph].

[BFHM98] V. Bernard, H. W. Fearing, T. R. Hemmert, and U.-G. Meißner,The form factors of the nucleon at small momentum transfer, Nucl.Phys. A635 (1998), 121–145.

[BHM02] V. Bernard, T. R. Hemmert, and U.-G. Meißner, Novel analysis ofchiral loop effects in the generalized Gerasimov-Drell-Hearn sumrule, Phys. Lett. B545 (2002), 105–111.

[BHM03a] V. Bernard, T. R. Hemmert, and U.-G. Meißner, Infrared regular-ization with spin-3/2 fields, Phys. Lett. B565 (2003), 137–145.

[BHM03b] V. Bernard, T. R. Hemmert, and U.-G. Meißner, Spin structure ofthe nucleon at low energies, Phys. Rev. D67 (2003), 076008.

[BHM04] V. Bernard, T. R. Hemmert, and U.-G. Meißner, Cutoff schemesin chiral perturbation theory and the quark mass expansion of thenucleon mass, Nucl. Phys. A732 (2004), 149–170.

[BHM05] V. Bernard, T. R. Hemmert, and U.-G. Meißner, Chiral extrapo-lations and the covariant small scale expansion, Phys. Lett. B622(2005), 141–150.

[BJ02] A. V. Belitsky and X. Ji, Chiral structure of nucleon gravitationalform factors, Phys. Lett. B538 (2002), 289–297.

[BKKM92] V. Bernard, N. Kaiser, J. Kambor, and U.-G. Meißner, Chiralstructure of the nucleon, Nucl. Phys. B388 (1992), 315–345.

[BKLM94] V. Bernard, N. Kaiser, T. S. H. Lee, and U.-G. Meißner, Thresholdpion electroproduction in chiral perturbation theory, Phys. Rept.246 (1994), 315–363.

154

BIBLIOGRAPHY

[BKM92a] V. Bernard, N. Kaiser, and U.-G. Meißner, Measuring the axialradius of the nucleon in pion electroproduction, Phys. Rev. Lett.69 (1992), 1877–1879.

[BKM92b] V. Bernard, N. Kaiser, and U.-G. Meißner, Nucleons with chiralloops: Electromagnetic polarizabilities, Nucl. Phys. B373 (1992),346–370.

[BKM95] V. Bernard, N. Kaiser, and U.-G. Meißner, Chiral dynamics innucleons and nuclei, Int. J. Mod. Phys. E4 (1995), 193–346.

[BKM96] V. Bernard, N. Kaiser, and U.-G. Meißner, Nucleon electroweakform factors: Analysis of their spectral functions, Nucl. Phys.A611 (1996), 429–441.

[BKMS94] V. Bernard, N. Kaiser, U.-G. Meißner, and A. Schmidt, Aspects ofnucleon Compton scattering, Z. Phys. A348 (1994), 317–341.

[BKSM93] V. Bernard, N. Kaiser, A. Schmidt, and U.-G. Meißner, Consistentcalculation of the nucleon electromagnetic polarizabilities in chi-ral perturbation theory beyond next- to-leading order, Phys. Lett.B319 (1993), 269–275.

[BL99] T. Becher and H. Leutwyler, Baryon chiral perturbation theory inmanifestly Lorentz invariant form, Eur. Phys. J. C9 (1999), 643–671.

[BM06] V. Bernard and U.-G. Meißner, Chiral perturbation theory,arXiv:hep-ph/0611231.

[BP07] S. Boffi and B. Pasquini, Generalized parton distributions and thestructure of the nucleon, arXiv:0711.2625.

[BPT03] S. Boffi, B. Pasquini, and M. Traini, Linking generalized parton dis-tributions to constituent quark models, Nucl. Phys. B649 (2003),243–262.

[BR05] A. V. Belitsky and A. V. Radyushkin, Unraveling hadron structurewith generalized parton distributions, Phys. Rept. 418 (2005), 1–387.

[Bur01] V. D. Burkert, Comment on the generalized Gerasimov-Drell-Hearn sum rule in chiral perturbation theory, Phys. Rev. D63(2001), 097904.

[CD04] G. Colangelo and S. Durr, The pion mass in finite volume, Eur.Phys. J. C33 (2004), 543–553.

[CDM05] J.-P. Chen, A. Deur, and Z.-E. Meziani, Sum rules and momentsof the nucleon spin structure functions, Mod. Phys. Lett. A20(2005), 2745–2766.

155

BIBLIOGRAPHY

[Che08] J-P. Chen, Highlights and Perspectives of the JLab Spin PhysicsProgram, arXiv:0804.4486.

[CJ01] J.-W. Chen and X.-D. Ji, Is the Sullivan process compatible withQCD chiral dynamics?, Phys. Lett. B523 (2001), 107–110.

[CJ02] J.-W. Chen and X.-D. Ji, Leading chiral contributions to the spinstructure of the proton, Phys. Rev. Lett. 88 (2002), 052003.

[D+06] K. V. Dharmawardane et al., Measurement of the x- and Q2-Dependence of the Asymmetry A1 on the Nucleon, Phys. Lett.B641 (2006), 11–17.

[Dav91] Andrei I. Davydychev, A simple formula for reducing Feynmandiagrams to scalar integrals, Phys. Lett. B263 (1991), 107–111.

[DGH07] M. Dorati, T. A. Gail, and T. R. Hemmert, Chiral Analysis of theGeneralized Form Factors of the Nucleon, PoS LAT2007 (2007),071.

[DGH08] M. Dorati, T. A. Gail, and T. R. Hemmert, Chiral Perturba-tion Theory and the first moments of the Generalized Parton Dis-triputions in a Nucleon, Nucl. Phys. A798 (2008), 96–131.

[DH] M. Dorati and T.R. Hemmert, Generalized Axial Form Factors ofthe Nucleon, in preparation.

[DH66] S. D. Drell and A. C. Hearn, Phys. Rev. Lett. 16 (1966), 908.

[DHH] M. Dorati, Ph. Hagler, and T.R. Hemmert, Moments of nucleonGPDs at next-to-leading one-loop order, in preparation.

[Die03] M. Diehl, Generalized parton distributions, Phys. Rept. 388(2003), 41–277.

[dJ06] K. de Jager, Nucleon form factor experiments and the pion cloud,arXiv:nucl-ex/0612026.

[DJM94] R. F. Dashen, E. E. Jenkins, and A. V. Manohar, The 1/N(c)expansion for baryons, Phys. Rev. D49 (1994), 4713–4738.

[DKT01] D. Drechsel, S. S. Kamalov, and L. Tiator, The GDH sum ruleand related integrals, Phys. Rev. D63 (2001), 114010.

[DL91] J. F. Donoghue and H. Leutwyler, Energy and momentum in chiraltheories, Z. Phys. C52 (1991), 343–351.

[DMN+01] W. Detmold, W. Melnitchouk, J. W. Negele, D. B. Renner, andA. W. Thomas, Chiral extrapolation of lattice moments of protonquark distributions, Phys. Rev. Lett. 87 (2001), 172001.

156

BIBLIOGRAPHY

[DMS05] M. Diehl, A. Manashov, and A. Schafer, Generalized parton dis-tributions for the pion in chiral perturbation theory, Phys. Lett.B622 (2005), 69–82.

[DMS06] M. Diehl, A. Manashov, and A. Schafer, Chiral perturbation theoryfor nucleon generalized parton distributions, Eur. Phys. J. A29(2006), 315–326.

[DMS07] M. Diehl, A. Manashov, and A. Schafer, Generalized parton dis-tributions for the nucleon in chiral perturbation theory, Eur. Phys.J. A31 (2007), 335–355.

[DMW91] R. M. Davidson, N. C. Mukhopadhyay, and R. S. Wittman, Effec-tive Lagrangian approach to the theory of pion photoproduction inthe Delta (1232) region, Phys. Rev. D43 (1991), 71–94.

[Dor05] M. Dorati, Laurea Thesis, University of Pavia, Pavia, Italy (2005).

[DPV03] D. Drechsel, B. Pasquini, and M. Vanderhaeghen, Dispersion re-lations in real and virtual Compton scattering, Phys. Rept. 378(2003), 99–205.

[DT04] Dieter Drechsel and Lothar Tiator, The Gerasimov-Drell-Hearnsum rule and the spin structure of the nucleon, Ann. Rev. Nucl.Part. Sci. 54 (2004), 69–114.

[E+06a] R. G. Edwards et al., Nucleon structure in the chiral regimewith domain wall fermions on an improved staggered sea, PoSLAT2006 (2006), 121.

[E+06b] R. G. Edwards et al., The nucleon axial charge in full lattice QCD,Phys. Rev. Lett. 96 (2006), 052001.

[Eck94] G. Ecker, Chiral invariant renormalization of the pion - nucleoninteraction, Phys. Lett. B336 (1994), 508–517.

[F+03] R. Fatemi et al., Measurement of the proton spin structure functiong1(x,Q2) for Q2 from 0.15−GeV 2 to 1.6−GeV 2 with CLAS, Phys.Rev. Lett. 91 (2003), 222002.

[FGJS03] T. Fuchs, J. Gegelia, G. Japaridze, and S. Scherer, Renormal-ization of relativistic baryon chiral perturbation theory and powercounting, Phys. Rev. D68 (2003), 056005.

[FJ01] B. W. Filippone and X.-D. Ji, The spin structure of the nucleon,Adv. Nucl. Phys. 26 (2001), 1.

[FLMS97] H. W. Fearing, R. Lewis, N. Mobed, and S. Scherer, Muon captureby a proton in heavy baryon chiral perturbation theory, Phys. Rev.D56 (1997), 1783–1791.

157

BIBLIOGRAPHY

[FMMS00] N. Fettes, U.-G. Meißner, M. Mojzis, and S. Steininger, The chiraleffective pion nucleon Lagrangian of order p4, Ann. Phys. 283(2000), 273–302.

[FMS98] N. Fettes, U.-G. Meißner, and S. Steininger, Pion nucleon scatter-ing in chiral perturbation theory. I: Isospin-symmetric case, Nucl.Phys. A640 (1998), 199–234.

[G+04] M. Gockeler et al., Generalized parton distributions from latticeQCD, Phys. Rev. Lett. 92 (2004), 042002.

[G+05] M. Gockeler et al., Nucleon electromagnetic form factors on thelattice and in chiral effective field theory, Phys. Rev. D71 (2005),034508.

[G+07] K. Goeke et al., The pion mass dependence of the nucleon form-factors of the energy momentum tensor in the chiral quark-solitonmodel, arXiv:hep-ph/0702031.

[Gai07] T. A. Gail, PhD Thesis, Technical University of Munich, Munich,Germany (2007).

[Ger66] S. B. Gerasimov, Sov. J. Nucl. Phys. 2 (1966), 430.

[GF04] T. Gorringe and H. W. Fearing, Induced pseudoscalar coupling ofthe proton weak interaction, Rev. Mod. Phys. 76 (2004), 31–91.

[GH06] T. A. Gail and T. R. Hemmert, in Proceeding of ECT* WorkshopLATTICE QCD, CHIRAL PERTURBATION THEORY ANDHADRON PHENOMENOLOGY, Trento, Italy (2-6 Oct 2006), 30.

[GH07] T. A. Gail and T. R. Hemmert, The electromagnetic nucleon toDelta transition in chiral effective field theory, AIP Conf. Proc.904 (2007), 151–157.

[GL84] J. Gasser and H. Leutwyler, Chiral Perturbation Theory to OneLoop, Ann. Phys. 158 (1984), 142.

[GMG54] M. Gell-Mann and M. L. Golberger, Scattering of Low-Energy Pho-tons by Particles of Spin 1/2, Phys. Rev. 96 (1954), 1433–1438.

[Gol61] J. Goldstone, Nuovo Cimento 19 (1961), 154.

[GRS99] M. Gluck, E. Reya, and I. Schienbein, Pionic parton distributionsrevisited, Eur. Phys. J. C10 (1999), 313–317.

[GSS88] J. Gasser, M. E. Sainio, and A. Svarc, Nucleons with chiral loops,Nucl. Phys. B307 (1988), 779.

[H+] T.R. Hemmert et al., The energy momentum tensor of a nucleonfrom the viewpoint of ChPT, in preparation.

158

BIBLIOGRAPHY

[H+03] P. Hagler et al., Moments of nucleon generalized parton distribu-tions in lattice QCD, Phys. Rev. D68 (2003), 034505.

[H+08] Ph. Hagler et al., Nucleon Generalized Parton Distributions fromFull Lattice QCD, Phys. Rev. D77 (2008), 094502.

[HGH04] R. P. Hildebrandt, H. W. Griesshammer, and T. R. Hemmert, Spinpolarizabilities of the nucleon from polarized low energy Comptonscattering, Eur. Phys. J. A20 (2004), 329–344.

[HGHP04] R. P. Hildebrandt, H. W. Griesshammer, T. R. Hemmert, andB. Pasquini, Signatures of chiral dynamics in low energy Comptonscattering off the nucleon, Eur. Phys. J. A20 (2004), 293–315.

[HGHP05] R. P. Hildebrandt, H. W. Griesshammer, T. R. Hemmert, andD. R. Phillips, Explicit Delta(1232) degrees of freedom in Comptonscattering off the deuteron, Nucl. Phys. A748 (2005), 573–595.

[HHK97] Thomas R. Hemmert, Barry R. Holstein, and Joachim Kambor,Delta(1232) and the polarizabilities of the nucleon, Phys. Rev. D55(1997), 5598–5612.

[HHK98] T. R. Hemmert, B. R. Holstein, and J. Kambor, Chiral La-grangians and Delta(1232) interactions: Formalism, J. Phys. G24(1998), 1831–1859.

[HPW03a] T. R. Hemmert, M. Procura, and W. Weise, Quark mass depen-dence of the nucleon axial-vector coupling constant, Phys. Rev.D68 (2003), 075009.

[HPW03b] T. R. Hemmert, M. Procura, and W. Weise, Quark mass depen-dence of the nucleon axial-vector coupling constant, Phys. Rev.D68 (2003), 075009.

[HW02] T. R. Hemmert and W. Weise, Chiral magnetism of the nucleon,Eur. Phys. J. A15 (2002), 487–504.

[IKL] B. L. Ioffe, V. A. Khoze, and L. N. Lipatov, HARD PROCESSES.VOL. 1: PHENOMENOLOGY, QUARK PARTON MODEL, El-sevier Science Publishers B.V.

[Ji97] X.-D. Ji, Gauge invariant decomposition of nucleon spin, Phys.Rev. Lett. 78 (1997), 610–613.

[Ji98] X.-D. Ji, Off-forward parton distributions, J. Phys. G24 (1998),1181–1205.

[JKO00] X.-D. Ji, C.-W. Kao, and J. Osborne, Generalized Drell-Hearn-Gerasimov sum rule at order O(p**4) in chiral perturbation theory,Phys. Lett. B472 (2000), 1–4.

159

BIBLIOGRAPHY

[JM91a] E. E. Jenkins and A. V. Manohar, Baryon chiral perturbation the-ory using a heavy fermion Lagrangian, Phys. Lett. B255 (1991),558–562.

[JM91b] E. E. Jenkins and A. V. Manohar, Chiral corrections to the baryonaxial currents, Phys. Lett. B259 (1991), 353–358.

[JO01] X. Ji and J. Osborne, Generalized sum rules for spin-dependentstructure functions of the nucleon, J. Phys. G27 (2001), 127.

[K+06] A. A. Khan et al., Axial coupling constant of the nucleon for twoflavours of dynamical quarks in finite and infinite volume, Phys.Rev. D74 (2006), 094508.

[KM01] B. Kubis and Ulf-G. Meißner, Low energy analysis of the nucleonelectromagnetic form factors, Nucl. Phys. A679 (2001), 698–734.

[KP02] N. Kivel and M. V. Polyakov, One loop chiral corrections to hardexclusive processes. I: Pion case, arXiv:hep-ph/0203264.

[KSV03] C. W. Kao, T. Spitzenberg, and M. Vanderhaeghen, Burkhardt-Cottingham sum rule and forward spin polarizabilities in heavybaryon chiral perturbation theory, Phys. Rev. D67 (2003), 016001.

[L+04] LHPC et al., Transverse structure of nucleon parton distributionsfrom lattice QCD, Phys. Rev. Lett. 93 (2004), 112001.

[Low54] F. E. Low, Scattering of Light of Very Low Frequency by Systemsof Spin 1/2, Phys. Rev. 96 (1954), 1428–1432.

[LQ] LHPC and QCDSF, - FORTHCOMING -.

[M+05] Z. E. Meziani et al., Higher twists and color polarizabilities in theneutron, Phys. Lett. B613 (2005), 148–153.

[MMD96] P. Mergell, U.-G. Meißner, and D. Drechsel, Dispersion theoreticalanalysis of the nucleon electromagnetic form-factors, Nucl. Phys.A596 (1996), 367–396.

[MMS00] U.-G. Meissner, G. Muller, and S. Steininger, Renormalization ofthe chiral pion nucleon Lagrangian beyond next-to-leading order,Ann. Phys. 279 (2000), 1–64.

[P+08] Y. Prok et al., Moments of the Spin Structure Functions gp1 and

gd1 for 0.05 < Q2 < 3.0GeV 2, arXiv:0802.2232.

[PHW04] M. Procura, T. R. Hemmert, and W. Weise, Nucleon mass, sigmaterm and lattice QCD, Phys. Rev. D69 (2004), 034505.

[PMHW07] M. Procura, B. U. Musch, T. R. Hemmert, and W. Weise, Chiralextrapolation of g(A) with explicit Delta(1232) degrees of freedom,Phys. Rev. D75 (2007), 014503.

160

BIBLIOGRAPHY

[PMW+06] M. Procura, B. U. Musch, T. Wollenweber, T. R. Hemmert, andW. Weise, Nucleon mass: From lattice QCD to the chiral limit,Phys. Rev. D73 (2006), 114510.

[PP03] V. Pascalutsa and D. R. Phillips, Effective theory of theDelta(1232) in Compton scattering off the nucleon, Phys. Rev.C67 (2003), 055202.

[PPV06] C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen, Nucleon elec-tromagnetic form factors, arXiv:hep-ph/0612014.

[RS41] W. Rarita and J. S. Schwinger, On a theory of particles with halfintegral spin, Phys. Rev. 60 (1941), 61.

[Sch03] S. Scherer, Introduction to chiral perturbation theory, Adv. Nucl.Phys. 27 (2003), 277.

[Sch05] M. Schumacher, Polarizability of the nucleon and Compton scat-tering, Prog. Part. Nucl. Phys. 55 (2005), 567–646.

[SFGS07] M. R. Schindler, T. Fuchs, J. Gegelia, and S. Scherer, Axial,induced pseudoscalar, and pion nucleon form factors in mani-festly Lorentz-invariant chiral perturbation theory, Phys. Rev. C75(2007), 025202.

[SGS05] Matthias R. Schindler, Jambul Gegelia, and Stefan Scherer, Elec-tromagnetic form factors of the nucleon in chiral perturbation the-ory including vector mesons, Eur. Phys. J. A26 (2005), 1–5.

[TE96] H.-B. Tang and P. J. Ellis, Redundance of Delta isobar parametersin effective field theories, Phys. Lett. B387 (1996), 9–13.

[Ter99] O. V. Teryaev, Spin structure of nucleon and equivalence principle,arXiv:hep-ph/9904376.

[Wei58] S. Weinberg, Charge Symmetry of Weak Interactions, Phys. Rev.112 (1958), 1375 – 1379.

[Wei76] S. Weinberg, Implications of Dynamical Symmetry Breaking, Phys.Rev. D13 (1976), 974–996.

[Wei79] S. Weinberg, Implications of Dynamical Symmetry Breaking: AnAddendum, Phys. Rev. D19 (1979), 1277–1280.

[Wil00] D. H. Wilkinson, Limits to second-class nucleonic and mesoniccurrents, Eur. Phys. J. A7 (2000), 307–315.

[xpr] http://www-spires.dur.ac.uk/hepdata/pdf3.html.

[Y+03] J. Yun et al., Measurement of inclusive spin structure functions ofthe deuteron with CLAS, Phys. Rev. C67 (2003), 055204.

161

BIBLIOGRAPHY

162

Acknowledgements

I would like to thank my supervisor in Pavia Prof. Sigfrido Boffi for his supportand help during these three years of my PhD. I am grateful to dr. BarbaraPasquini for her kind ability to give me always precious advices. I would alsolike to thank the Department of Theoretical and Nuclear Physics of the Uni-versity of Pavia and all the PhD students who shared with me this experience.

I would like to thank Prof. Wolfram Weise for giving me the opportunity towork within the T39 Research Group of the Technische Universitat Munchen,where most of the work summarized in the second chapter of this thesis wasdone. I would also like to thank the whole working group for all their supportand hospitality, in particular dr. Massimiliano Procura for his useful discus-sions and the warm working atmosphere and Bernard Musch for his precioushelp and his patience in answering all my questions. A very special thank to dr.Thomas Hemmert for his guidance and encouragement and to my collaboratordr. Tobias Gail for his enthusiastic contribution to our work. Their supportwas essential for the realization of this thesis.

I would like to thank Prof. Ulf-G. Meißner for giving me the opportunity towork at the theoretical division of the Helmholtz Institut fur Strahlen undKernphysik in Bonn, where the work discussed in the third chapter of thisthesis was done. I am also very grateful to dr. Hermann Krebs for his keycontribution to the project. A very special thank to Prof. Veronique Bernardfor her kindness and very appreciated help, it was really a great pleasure tocollaborate with her.

I am very grateful to Prof. Jiunn-Wei Chen for acting as referee for this thesis.

Un grazie infinito va ai miei genitori e a mia sorella, che da sempre appoggianoogni mia scelta e la cui presenza ho sempre sentito vicina nonostante la dis-tanza che spesso ci ha separato in questi ultimi anni.

This research activity was in part supported by DAAD, German Academic Ex-change Service.

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