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Nuclear Physics B 761 (2007) 1–62 Two-loop electroweak next-to-leading logarithmic corrections to massless fermionic processes A. Denner a,, B. Jantzen a , S. Pozzorini b a Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland b Max-Planck-Institut für Physik, Föhringer Ring 6, D-80805 München, Germany Received 1 September 2006; accepted 13 October 2006 Available online 7 November 2006 Abstract We consider two-loop leading and next-to-leading logarithmic virtual corrections to arbitrary processes with external massless fermions in the electroweak Standard Model at energies well above the electroweak scale. Using the sector-decomposition method and alternatively the strategy of regions we calculate the mass singularities that arise as logarithms of Q 2 /M 2 W , where Q is the energy scale of the considered process, and 1poles in D = 4 2ε dimensions, to one- and two-loop next-to-leading logarithmic accuracy. The derivations are performed within the complete electroweak theory with spontaneous symmetry breaking. Our results indicate a close analogy between the form of two-loop electroweak logarithmic corrections and the singular structure of scattering amplitudes in massless QCD. We find agreement with the resum- mation prescriptions that have been proposed in the literature based on a symmetric SU(2) × U(1) theory matched with QED at the electroweak scale and provide new next-to-leading contributions proportional to ln(M 2 Z /M 2 W ). © 2006 Elsevier B.V. All rights reserved. 1. Introduction The electroweak radiative corrections to high-energy processes are characterized by the pres- ence of logarithms of the type ln(Q 2 /M 2 ), which involve the ratio of the typical scattering energy Q over the gauge-boson mass scale M = M W M Z [1]. These logarithmic corrections affect every reaction that involves electroweakly interacting particles and has a characteristic scale * Corresponding author. E-mail addresses: [email protected] (A. Denner), [email protected] (B. Jantzen), [email protected] (S. Pozzorini). 0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.10.014

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Page 1: Nucl.Phys.B v.761

Nuclear Physics B 761 (2007) 1–62

Two-loop electroweak next-to-leading logarithmiccorrections to massless fermionic processes

A. Denner a,∗, B. Jantzen a, S. Pozzorini b

a Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerlandb Max-Planck-Institut für Physik, Föhringer Ring 6, D-80805 München, Germany

Received 1 September 2006; accepted 13 October 2006

Available online 7 November 2006

Abstract

We consider two-loop leading and next-to-leading logarithmic virtual corrections to arbitrary processeswith external massless fermions in the electroweak Standard Model at energies well above the electroweakscale. Using the sector-decomposition method and alternatively the strategy of regions we calculate the masssingularities that arise as logarithms of Q2/M2

W, where Q is the energy scale of the considered process,and 1/ε poles in D = 4 − 2ε dimensions, to one- and two-loop next-to-leading logarithmic accuracy. Thederivations are performed within the complete electroweak theory with spontaneous symmetry breaking.Our results indicate a close analogy between the form of two-loop electroweak logarithmic correctionsand the singular structure of scattering amplitudes in massless QCD. We find agreement with the resum-mation prescriptions that have been proposed in the literature based on a symmetric SU(2) × U(1) theorymatched with QED at the electroweak scale and provide new next-to-leading contributions proportional toln(M2

Z/M2W).

© 2006 Elsevier B.V. All rights reserved.

1. Introduction

The electroweak radiative corrections to high-energy processes are characterized by the pres-ence of logarithms of the type ln(Q2/M2), which involve the ratio of the typical scattering energyQ over the gauge-boson mass scale M = MW ∼ MZ [1]. These logarithmic corrections affectevery reaction that involves electroweakly interacting particles and has a characteristic scale

* Corresponding author.E-mail addresses: [email protected] (A. Denner), [email protected] (B. Jantzen),

[email protected] (S. Pozzorini).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2006.10.014

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2 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

Q � M . They start to be sizable at energies of the order of a few hundred GeV and their impactincreases with energy. At the LHC [2] and future lepton colliders [3–5], where scattering ener-gies of the order of 1 TeV will be reached, logarithmic electroweak effects can amount to tens ofper cent at one loop and several per cent at two loops. Thus, this class of corrections will be veryimportant for the interpretation of precision measurements at future high-energy colliders.

For sufficiently high Q, terms of O(M2/Q2) become negligible and, at l loops, the elec-troweak corrections assume the form of a tower of logarithms,

(1.1)αl lnj

(Q2

M2

), with 0 � j � 2l,

where the leading logarithms (LLs), also known as Sudakov logarithms [6], have power j = 2l,and the subleading terms with j = 2l − 1,2l − 2, . . . are denoted as next-to-leading logarithms(NLLs), next-to-next-to-leading logarithms (NNLLs), and so on. The complete asymptotic limitincludes all logarithmic contributions (j > 0) as well as terms that are not logarithmically en-hanced at high energy (j = 0).1

Electroweak logarithmic corrections have a twofold origin. On the one hand they can appearas terms of the form ln(Q2/μ2

R) resulting from the renormalization of ultraviolet (UV) singu-larities at the renormalization scale μR ∼ M . These logarithms can easily be controlled throughthe running of the coupling constants. The other source of electroweak logarithms are mass sin-gularities, i.e. logarithms of the form ln(Q2/M2) that are formally singular in the limit wherethe gauge-boson masses tend to zero. As is well known, in gauge theories, mass singularitiesresult from the interactions of the initial- and final-state particles with soft and/or collinear gaugebosons. This permits, in principle, to treat mass singularities in a process-independent way andto derive universal properties of mass-singular logarithmic corrections.

At one loop, it was proven that the electroweak LLs and NLLs are universal, and a generalformula was derived that expresses the logarithmic corrections to arbitrary processes in terms ofthe electroweak quantum numbers of the initial- and final-state particles [7,8]. In recent years, theimpact of one-loop electroweak corrections was studied in detail for various specific processes athigh-energy colliders [9–15]. At the LHC, in general, large negative corrections are observed thatappear at transverse momenta pT around 100 GeV and grow with pT. Depending on the process,the size of these corrections can reach up to 10–40% at pT ∼ 500–1000 GeV. In Refs. [12,14],the predictions based on NLL and NNLL high-energy approximations were compared with exactone-loop calculations. In both cases it turned out that in the high-pT region, where the correc-tions are large, the NNLL predictions deviate from the exact calculation by much less than 1%,i.e. the numerical effect of O(M2/Q2) contributions is negligible. Also the NLL approxima-tion provides a correct description of the bulk of the corrections and their energy dependence.However, it was found that the actual precision of the NLL approximation, better than 1% forpp → jet+Z/γ [14] and about 5% for pp → Wγ [12], depends relatively strongly on the process.As it was shown for pp → jet + Z/γ , also the two-loop electroweak logarithms can have an im-pact at the several per-cent level on high-pT measurements at the LHC [14].

In the recent years, the properties of electroweak logarithmic corrections beyond one loophave been studied with two complementary approaches.2 On the one hand, evolution equations,

1 In the literature, such terms are sometimes denoted as “constants” since they do not involve logarithms that grow withenergy. However, in general they are functions of the ratios of kinematical invariants, which depend on the scatteringangles. At one loop such terms are formally classified as NNLLs.

2 For recent developments in the exact numerical calculation of complete two-loop integrals see, for instance, Ref. [16].

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 3

which are well known in QED and QCD, have been applied to the electroweak theory in orderto obtain the higher-order terms through a resummation of the one-loop logarithms3 [17–23]. Onthe other hand, explicit diagrammatic calculations based on the electroweak Feynman rules havebeen performed [24–31].

Fadin et al. [17] have resummed the electroweak LL corrections to arbitrary matrix elementsby means of the infrared evolution equation (IREE), and this approach has been extended byMelles to the NLL terms for arbitrary processes [18–21]. In Refs. [22,23] Kühn et al. have re-summed the logarithmic corrections to neutral-current massless 4-fermion processes up to theNNLL level. These resummations [17–23] are based on the IREE [17], which describes the all-order LL dependence of matrix elements with respect to the cut-off parameter μ⊥. This cut-offfixes the minimal transverse momentum for the gauge bosons that couple to the initial- andfinal-state particles and acts as a regulator of mass singularities. The IREE, which was originallyderived within symmetric gauge theories (QED and QCD), has been applied to the spontaneouslybroken electroweak theory under the assumption that this latter can be split into two regimes withexact gauge symmetry. In practice, in the regime Q � μ⊥ � M the masses of the electroweakgauge bosons, which result from the breaking of gauge symmetry, are supposed to be negligibleand exact SU(2) × U(1) symmetry is assumed. Instead, in the regime M � μ⊥ the weak gaugebosons are supposed to be frozen owing to their masses such that only photons contribute to theevolution, and this is characterized by exact U(1)em symmetry.

As a result, the resummed electroweak corrections factorize into two parts corresponding tothese two regimes: (i) a symmetric-electroweak part that can be computed within a symmetricSU(2) × U(1) gauge theory using the same cut-off parameter M to regularize the mass singu-larities that result from all gauge bosons, i.e. assuming that the photon is as heavy as the weakgauge bosons, and (ii) an electromagnetic part that originates from the mass gap between photonsand massive gauge bosons and can be computed within QED. The electromagnetic part containsdivergences that are due to massless photons and depend on the scheme adopted for their reg-ularization. These divergences cancel when the contributions of virtual and real photons arecombined, and then the electromagnetic contribution depends on the cuts imposed on real pho-tons. Instead the symmetric-electroweak part involves logarithms of the form ln(Q2/M2) whichare formally mass singular but numerically finite since M does not vanish. These logarithms arepresent in all physical observables that are exclusive with respect to real radiation of Z and Wbosons. Moreover, the electroweak logarithms remain present even in inclusive observables dueto a lack of cancellation between virtual and real contributions from electroweak gauge bosons.This is due to the fact that the conditions of the Bloch–Nordsieck theorem are not fulfilled sincefermions and gauge bosons carry non-Abelian weak-isospin charges [32].

The resummations [17–23] rely on the assumption that all relevant implications of electroweaksymmetry breaking are correctly taken into account by simply splitting the evolution equationinto two regimes with exact gauge symmetry. In particular, the following assumptions are explic-itly or implicitly made. (i) In the massless limit M/Q → 0, all couplings with mass dimension,which originate from symmetry breaking, are neglected. (ii) The weak-boson masses are intro-duced in the corresponding propagators as regulators of soft and collinear singularities from Wand Z bosons without spontaneous symmetry breaking. Since these masses are of the same or-der, one uses MW = MZ as an approximation. (iii) The regimes above and below the electroweak

3 In order to resum the LLs and NLLs it is sufficient to determine the kernel of the evolution equations to one-loopaccuracy. However, starting from the NNLLs also the two-loop contributions to the β-function and to the anomalousdimensions are needed.

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scale are treated as an unmixed SU(2) × U(1) theory and QED, respectively, and mixing effectsin the gauge sector, i.e. the interaction of photons with W bosons, are neglected.

It is important to understand to which extent the above assumptions are legitimate andwhether the resulting resummation prescriptions are correct. This can be achieved by explicitdiagrammatic two-loop calculations based on the electroweak Lagrangian, where all effectsrelated to spontaneous symmetry breaking are consistently taken into account. At the LLlevel, these checks have already been completed. A calculation of the massless fermionic sin-glet form factor [24,25] and then a Coulomb-gauge calculation for arbitrary processes [26]have supported the exponentiation of the electroweak LLs predicted by the IREE. Also theangular-dependent subset of the NLL corrections for arbitrary processes [27] has been shownto be consistent with the exponentiation anticipated in Refs. [21–23]. At present the com-plete set of electroweak NLLs is known only for the so-called fermionic form factor, whichcorresponds to the gluon–fermion–antifermion vertex [28]. This calculation has confirmedthat the IREE approach provides a correct description of the terms that do not depend onthe difference between the masses of the Z and the W bosons and has provided first in-sights into the behavior of the NLL terms of the type α2 ln(M2

Z/M2W) ln3(Q2/M2). The

complete tower of two-loop logarithms for the fermionic form factor has been computedin Refs. [29,30] adopting an unmixed SU(2) × U(1) theory with MZ = MW as approxi-mation. Within this framework it was found that the soft–collinear singularities resultingfrom massless photons factorize as suggested by the IREE and that, in absence of mix-ing, symmetry breaking is negligible up to the NNLL level and its first non-trivial effectsappear through a Higgs-mass dependence of the NNNLL terms. Using QCD resummationtechniques, the NNNLL two-loop results for the fermionic form-factor have been extendedto the amplitude for neutral-current four-fermion scattering [30,31]. Here terms of the typeln(M2

Z/M2W) were included through an expansion in s2

w = 1 − M2W/M2

Z up to the first orderin s2

w.In this paper we develop a formalism to derive virtual electroweak two-loop LL and NLL

corrections for arbitrary processes and apply it to the case of massless n-fermion reactions. Thecalculation is performed diagrammatically using the electroweak Feynman rules in the ’t Hooft–Feynman gauge. The mass-singular logarithms are extracted from the relevant Feynman diagramsby means of a soft–collinear approximation for the interaction of initial- and final-state fermionswith gauge bosons. Using Ward identities we prove that the LL and NLL mass singularities forgeneric n-fermion amplitudes factorize from the corresponding Born amplitudes. All relevantloop integrals are evaluated in the high-energy region Q � M to NLL accuracy using an autom-atized algorithm based on the sector-decomposition technique [33] and alternatively the methodof expansion by regions combined with Mellin–Barnes representations [34]. We do not assumeMZ = MW and we include all relevant contributions depending on the difference MZ − MW. Inaddition to the logarithms of the type (1.1), which arise from massive virtual particles, we includealso mass singularities from massless virtual photons. The latter are regularized dimensionallyand arise as 1/ε poles in D = 4 − 2ε dimensions. In our result the photonic singularities are fac-torized in a gauge-invariant electromagnetic term. The remaining part of the corrections, which isalso gauge invariant and does not depend on the scheme adopted to regularize photonic singular-ities, contains only finite log(Q2/M2) terms. The divergences contained in the electromagneticterm cancel if real-photon emission is included.

The paper is organized as follows. Section 2 contains definitions and conventions used in thecalculation. In Section 3 we identify the diagrams that produce ultraviolet and mass singulari-ties, split them into factorizable and non-factorizable parts and discuss our method to treat these

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 5

contributions. Using collinear Ward identities, we prove in Section 4 that all non-factorizablecontributions cancel. Explicit expressions for the factorizable contributions and for the counter-terms needed for the renormalization are provided in Sections 5 and 6, respectively. The completeone- and two-loop results are presented in Section 7 and discussed in Section 8. Our conclusionsare contained in Section 9, and various appendices are devoted to technical aspects of the calcu-lation.

2. Definitions and conventions

We consider a generic n → 0 process involving an even number n of polarized fermionicparticles,

(2.1)ϕ1(p1) · · ·ϕn(pn) → 0.

The symbols ϕi represent n/2 antifermions and n/2 fermions: ϕi = fκiσi

for i = 1, . . . , n/2 andϕi = f

κiσi

for i = n/2+1, . . . , n. These particles are massless chiral eigenstates with p2k = m2

k = 0and chirality κi = R or L. In practice every fermion or antifermion can be a lepton or a lightquark. The indices σi characterize the lepton/quark nature, the isospin and the generation of thefermions, fσi

= νe, e, u, d, . . . .The matrix element for the process (2.1) reads

(2.2)Mϕ1···ϕn =[

n/2∏i=1

v(pi, κi)

]G

ϕ1···ϕ

n(p1, . . . , pn)

[n∏

j=n/2+1

u(pj , κj )

],

where Gϕ

1···ϕ

n is the corresponding truncated Green function. The spinors for chiral fermionsand antifermions fulfill

(2.3)ωρu(p,κ) = δκρu(p, κ), v(p, κ)ωρ = δκρv(p, κ)

for ρ,κ = R,L, and

(2.4)ωR = ωL = 1

2

(1 + γ 5), ωL = ωR = 1

2

(1 − γ 5).

The matrix elements (2.2) are often abbreviated as M≡Mϕ1···ϕn .The amplitudes for physical scattering processes, i.e. 2 → n− 2 reactions, are easily obtained

from our results for n → 0 reactions using crossing symmetry.

2.1. Perturbative and asymptotic expansions

For the perturbative expansion of the matrix elements in α = e2/(4π), where e is the electro-magnetic coupling constant, we write

(2.5)M=∞∑l=0

Ml ,

with

(2.6)Ml =(

αε

)l

Ml ,

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6 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

and

(2.7)αε =(

4πμ2D

eγEQ2

α.

Note that, for convenience, in D = 4 − 2ε dimensions we include in the definition of αε a nor-malization factor depending on ε, the scale μD of dimensional regularization, the characteristicenergy Q of the scattering process and Euler’s constant γE. For μ2

D = Q2eγE/(4π) we haveαε = α.

The electroweak corrections are evaluated in the region where all kinematical invariants, rjk =(pj + pk)

2, are much larger than the squared masses of the heavy particles that enter the loops,

(2.8)|rjk| ∼ Q2 � M2W ∼ M2

Z ∼ m2t ∼ M2

H.

This means that we consider a situation with essentially two different scales, one fixed by Q,which can be identified with (the square root of the absolute value of) one of the kinematicalinvariants, Q =√|rjk|, and one fixed by the heavy masses of the Standard Model. In this region,the electroweak corrections are dominated by mass-singular logarithms,

(2.9)L = ln

(Q2

M2W

),

and logarithms of UV origin. Mass singularities that originate from soft and collinear masslessphotons and UV singularities are regularized dimensionally and give rise to 1/ε poles in D =4 − 2ε dimensions. Therefore, the l-loop contributions to the polarized matrix elements can bewritten as an expansion in L and ε,

(2.10)Ml =2l∑

m=0

∞∑n=−m

Ml,m,nεnLm+n.

The logarithmic terms in (2.10) are classified according to their degree of singularity, defined asthe total power m of logarithms L and 1/ε poles. The maximal degree of singularity at l-looplevel is m = 2l and the corresponding terms are denoted as leading logarithms (LLs). The termswith m = 2l − 1,2l − 2, . . . represent the next-to-leading logarithms (NLLs), next-to-next-to-leading logarithms (NNLLs), and so on.

In this paper we systematically neglect mass-suppressed corrections of order M2W/Q2 and we

calculate the one- and two-loop corrections to NLL accuracy, i.e. including LLs and NLLs. For

this approximation we use the symbolNLL= . The two-loop corrections are expanded in ε up to

the finite terms, i.e. contributions of order ε0L4 and ε0L3. Instead, the one-loop corrections areexpanded up to order ε2, i.e. including terms of order ε2L4 and ε2L3. These higher-order termsin the ε-expansion must be taken into account when expressing two-loop mass singularities interms of one-loop ones.

Since the loop corrections depend on various masses, MW ∼ MZ ∼ mt ∼ MH, and differentinvariants rjk , the coefficients Ml,m,n in (2.10) depend on mass ratios and ratios of invariants.4

Actually, as we will see in the final result, they depend only on logarithms of these ratios,

(2.11)li = ln

(M2

i

M2W

), ljk = ln

(−rjk

Q2

).

4 This dependence only appears at the next-to-leading level, i.e. for m = 2l − 1.

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 7

The logarithms of −rjk are extracted in the region rjk < 0, where the corrections are real. Theimaginary parts that arise in the physical region can be obtained via analytic continuation replac-ing rjk by rjk + i0.

2.2. Symmetry breaking and mixing

In the electroweak Standard Model, the physical gauge bosons result from the gauge bosonsW 1,W 2,W 3 and B associated with the SU(2) and U(1) gauge groups via the unitary transfor-mation

(2.12)W±μ = 1√

2

(W 1

μ ∓ iW 2μ

), Zμ = cwW 3

μ + swBμ, Aμ = −swW 3μ + cwBμ,

where cw = cos θw, sw = sin θw, and θw is the weak mixing angle. The generators associated withthe physical gauge bosons are related to the weak isospin generators T i and the weak hyperchargeY through

eIW± = g2√2

(T 1 ± iT 2), eIZ = g2cwT 3 − g1sw

Y

2,

(2.13)eIA = −g2swT 3 − g1cwY

2,

where g1 and g2 are the coupling constants associated with the U(1) and SU(2) groups, respec-tively. The electromagnetic charge operator is defined as

(2.14)Q = −IA,

and the generator associated with the Z boson can be expressed as

(2.15)eIZ = g2

cwT 3 − e

sw

cwQ.

The SU(2) × U(1) Casimir operator is given by

(2.16)∑

V =A,Z,W±IV IV = g2

1

e2

(Y

2

)2

+ g22

e2C,

where V denotes the complex conjugate of V , and C = ∑3i=1(T

i)2 is the Casimir operator ofthe SU(2) group with eigenvalues 3/4 and 0 for left- and right-handed fermions, respectively.The generators (2.13) obey the commutation relations

(2.17)e[IV1, IV2

]= ig2

∑V3=A,Z,W±

εV1V2V3IV 3 ,

with

(2.18)εV1V2V3 = i ×

⎧⎪⎨⎪⎩

(−1)p+1cw if V1V2V3 = π(ZW+W−),

(−1)psw if V1V2V3 = π(AW+W−),

0 otherwise,

and (−1)p represents the sign of the permutation π .The SU(2) × U(1) symmetry is spontaneously broken by a scalar Higgs doublet which ac-

quires a vacuum expectation value v. We parametrize the Higgs doublet in terms of the four

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8 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

degrees of freedom Φi = H,χ,φ+, φ−, where φ− = (φ+)+. In this representation the gauge-group generators are 4 × 4 matrices with components IV

ΦiΦj. The gauge-boson masses and the

mixing parameters are fixed by the condition that the gauge-boson mass matrix is diagonal,

(2.19)M2V V ′ = 1

2e2v2{IV , IV ′}

HH= δV V ′M2

V ,

for V,V ′ = A,Z,W±. The curly brackets in (2.19) denote an anticommutator. The mass eigen-values are given by

(2.20)MW± = 1

2g2v, MZ = 1

2cwg2v, MA = 0.

The weak mixing angle θw is related to the gauge-boson masses via

(2.21)cw = MW

MZ.

The vanishing mass of the photon is connected to the fact that the electric charge of the vacuumexpectation value is zero. This provides the relation

(2.22)cwg1YΦ = swg2,

between the weak mixing angle, the coupling constants g1, g2 and the hypercharge YΦ of theHiggs doublet. Identifying e = cwg1, the Gell-Mann–Nishijima relation for general YΦ readsQ = Y/2 + YΦT 3. In the calculation we keep YΦ as a free parameter, which determines thedegree of mixing in the gauge sector. In this way our analysis applies to the Standard Modelcase, YΦ = 1, as well as to the case YΦ = 0 corresponding to an unmixed theory with

(2.23)sw = 0, cw = 1, Zμ = W 3μ, Aμ = Bμ, MW = MZ.

2.3. Gauge interactions of massless fermions

For the Feynman rules and various group-theoretical quantities we adopt the formalismof Ref. [8] (see Appendices A and B therein). The Feynman rules for the vector-boson–fermion–antifermion vertices read

(2.24)= ieγ μ∑

κ=R,L

ωκIVf κ

σ ′f κσ,

where V = A,Z,W±, and IVf κ

σ ′f κσ

are the generators that describe SU(2) × U(1) transformations

of fermions in the fundamental (κ = L) or trivial (κ = R) representation. For antifermions wehave

(2.25)IV

f κσ ′ f κ

σ= −IV

f κσ f κ

σ ′ .

The chiral projectors ωκ that are associated with the gauge couplings (2.24) can easily be shiftedalong the fermionic lines using anticommutation relations5 until they can be eliminated using

5 In our calculation we use {γ μ,γ 5} = 0 in D = 4 − 2ε dimensions.

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 9

(2.3). As an example, for the coupling of a vector boson V to an incoming fermion ϕi = fκiσi

oneobtains

(2.26)i(/pi + /q)

(pi + q)2ieγ μIV

ϕ′iϕi

u(pi, κi) with IVϕ′

iϕi= IV

fκi

σ ′i

fκiσi

,

where the gauge coupling IVϕ′

iϕidepends on the chirality κi of the fermion. Similarly, for the

coupling of a vector boson to an incoming antifermion ϕi = fκiσi

one obtains

(2.27)v(pi, κi)ieγμIV

ϕ′iϕi

i(/pi + /q)

(pi + q)2with IV

ϕ′iϕi

= IV

fκi

σ ′i

fκiσi

= −IV

fκiσi

fκi

σ ′i

,

where the minus sign of the propagator is absorbed in the coupling of the antifermion.In our results, the matrix elements (2.2) are often multiplied by various matrices resulting

from the couplings of gauge bosons to the external fermions. For such expressions we introducethe shorthands

MIV1k =

∑ϕ′

k

Mϕ1···ϕ′k ···ϕnI

V1ϕ′

kϕk,

(2.28)MIV1k I

V2k =

∑ϕ′

k,ϕ′′k

Mϕ1···ϕ′′k ···ϕnI

V1ϕ′′

k ϕ′k

IV2ϕ′

kϕk,

etc. The gauge couplings in (2.28) satisfy the commutation relations

(2.29)e[I

V1k , I

V2k′]= ig2δkk′

∑V3=A,Z,W±

εV1V2V3IV 3k ,

and the ε-tensor is defined in (2.18).Global gauge invariance implies the charge-conservation relation

(2.30)Mn∑

k=1

IVk = 0,

which is fulfilled up to mass-suppressed terms in the high-energy limit.

3. Treatment of ultraviolet and mass singularities

Large logarithms and 1/ε poles originate from UV and from mass singularities. In this sec-tion we present the technique that we use to extract these singularities from one- and two-loopFeynman diagrams. In Section 3.1 we identify the diagrams that are responsible for mass sin-gularities within the ’t Hooft–Feynman gauge and classify their contributions into factorizableand non-factorizable ones. In Section 3.2 we introduce an approximation that describes the ex-change of virtual gauge bosons in the soft and collinear limit. Our treatment of UV singularitiesis discussed in Section 3.3.

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10 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

3.1. Mass singularities

In gauge theories, as is well known, mass singularities appear in loop diagrams involvingvirtual gauge bosons that couple to the on-shell external legs.6 These singularities originate fromthe integration over the loop momenta in the regions where the momenta of the virtual gaugebosons become soft and/or collinear to the external momenta.

3.1.1. Mass-singular diagrams at one loopAt one loop, the mass singularities of the n-fermion amplitude originate from diagrams of the

type

(3.1),

where an electroweak gauge boson, V = A,Z,W±, couples to one of the external fermions,i = 1, . . . , n, and to any other line of the diagram. These diagrams can be classified into threetypes:

(i) The diagrams where the virtual gauge boson V couples to the external leg i and to anotherexternal leg j with j �= i,

(3.2).

These diagrams produce single- and double-logarithmic mass singularities that originatefrom the regions where the gauge-boson momentum becomes soft and/or collinear to themomentum of the external leg i or j .

(ii) The external-leg self-energy insertions

(3.3).

These diagrams constitute a subset of the general mass-singular diagrams depicted in (3.1).However, they have to be omitted in our calculation since we express S-matrix elements interms of truncated Green functions and on-shell renormalized fields.

(iii) The diagrams where the virtual gauge boson V couples to the external line i and to aninternal line, i.e. an internal propagator of the tree subdiagram that is represented as thegrey blob in (3.1). These diagrams produce only single-logarithmic mass singularities thatoriginate from the region where the momenta of the gauge boson and the external fermioni become collinear.

6 In principle it is possible to adopt particular gauge fixings in order to isolate mass singularities in a smaller subset ofdiagrams. In the Coulomb gauge, for instance, collinear singularities appear only in external-leg self-energy corrections[26]. However, we prefer to use the ’t Hooft–Feynman gauge for our analysis of mass singularities. Thus, we mustconsider the most general set of Feynman diagrams which gives rise to mass singularities.

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 11

3.1.2. Factorizable and non-factorizable contributions at one loopThe diagrams of type (i) involve an n-fermion tree subdiagram. It is thus possible to extract

from the diagrams (3.2) mass-singular contributions that factorize from the n-fermion Born ma-trix element. Such contributions are called factorizable and are defined as7

= −ie2μ4−DD

∑ϕ′

i ,ϕ′j

IVϕ′

iϕiIVϕ′

j ϕj

∫dDq

(2π)D

1

(q2 − M2V )(pi + q)2(pj − q)2

× v(pj , κj )γμ(/pj − /q)G

ϕ1···ϕ′

i···ϕ′

j···ϕ

n(p1, . . . , pi, . . . , pj , . . . , pn)

(3.4)× (/pi + /q)γμu(pi, κi).

Here Gϕ

1···ϕ′

i···ϕ′

j···ϕ

n denotes the truncated Green function corresponding to the n-fermion treesubdiagram. By definition, in the factorizable contributions (F) we include only those parts of theabove diagrams that are obtained by performing the loop integration with the momentum q ofthe gauge boson V set to zero in the tree subdiagram. This prescription is indicated by the labelF in the tree subdiagram. In (3.4) the spinors of the external fermions k = 1, . . . , n with k �= i, j

are implicitly understood.By construction, the factorizable terms (3.4) contain all one-loop soft singularities, since the

soft singularities originate only from the diagrams of type (3.2) in the region where the gauge-boson momentum tends to zero. Actually, as we will show, the factorizable contributions (3.4)contain all one-loop mass singularities, i.e. not only all soft singularities but also all collinearsingularities.

The combination of all factorizable one-loop contributions is obtained by summing over allgauge bosons V = A,Z,W± and external legs i, j in (3.4),

(3.5)MF1 = 1

2

n∑i=1

n∑j=1j �=i

∑V =A,Z,W±

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

qμ→xpμi ,xp

μj

.

Note that the symmetry factor 1/2 avoids double counting of the pairs of external legs i, j .The limit qμ → xp

μi , xp

μj in (3.5) indicates that the above diagrams are evaluated in the ap-

proximation where the four-momentum qμ of the gauge boson V is soft and/or collinear toone of the momenta of the external legs i and j . This approximation, which is defined inSection 3.2, permits to extract all mass singularities, which result from soft and/or collinearregions.8 As we will see, these terms can be expressed as products of n-fermion Born matrixelements and one-loop integrals. The factorizable one-loop terms (3.5) are computed in Sec-tion 5.1.

7 Here we consider the case where the particles i and j are a fermion and an antifermion, respectively. The generaliza-tion to other combinations of particles or antiparticles is obvious.

8 Here we adopt a different approach as compared to Ref. [7]. There the diagrams of type (3.5) were evaluated usingthe eikonal approximation, which extracts only soft singularities, and the collinear singularities where treated sepa-rately.

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12 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

The remaining one-loop mass singularities are obtained by subtracting the factorizable con-tributions (3.4) and the external self-energy contributions (3.3) from the diagrams (3.1),

MNF1 =

n∑i=1

∑V =A,Z,W±

⎡⎢⎢⎢⎢⎢⎣ −

(3.6)−n∑

j=1j �=i

⎤⎥⎥⎥⎥⎥⎦

qμ→xpμi

.

These contributions are called non-factorizable (NF). We note that these NF terms are free ofsoft singularities since all soft singularities are contained in the factorizable parts and are sub-tracted in (3.6). We also observe that, in contrast to (3.5), the sum over pairs of external legs i, j

appearing in front of the factorizable contributions that are subtracted in (3.6) is not multipliedby a symmetry factor 1/2. This is due to the fact that in the subtraction terms in (3.6) we includeonly the contribution of the collinear region qμ → xp

μi , whereas in (3.5) every diagram contains

the mass singularities resulting from the two regions qμ → xpμi and qμ → xp

μj .

In Section 4 we will prove that the non-factorizable one-loop terms (3.6) vanish.

3.1.3. Mass-singular diagrams at two loopsThe diagrams that give rise to NLL mass singularities at two loops, i.e. terms with triple and

quartic logarithmic singularities, can be obtained from the one-loop diagrams of type (3.2), whichproduce double logarithms in the soft–collinear region, by inserting

• a second soft and/or collinear gauge boson that couples to an external line or to the virtualgauge boson in (3.2) providing an additional single or double logarithm,

• or a self-energy subdiagram in the propagator of the virtual gauge boson in (3.2), whichprovides an additional single logarithm.

There are five types of such diagrams:

, , ,

(3.7), .

Here the NLL mass singularities originate from the regions where the gauge boson V1, whichcouples to two external legs or to an external leg and another virtual gauge boson, is soft and

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 13

collinear and the gauge bosons V2 and V3, in the first four diagrams in (3.7), are soft and/orcollinear.

3.1.4. Factorizable and non-factorizable contributions at two loopsThe two-loop mass singularities resulting from the diagrams (3.7) are split into factorizable

and non-factorizable contributions. The factorizable contributions result from the diagrams thatcontain n-fermion tree subdiagrams,

MF2 =

n∑i=1

n∑j=1j �=i

∑Vm=A,Z,W±

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1

2

⎡⎢⎢⎢⎣ +

⎤⎥⎥⎥⎦

+ + + + 1

2

+n∑

k=1k �=i,j

⎡⎢⎢⎣ + 1

6

⎤⎥⎥⎦

(3.8)+ 1

8

n∑k=1

k �=i,j

n∑l=1

l �=i,j,k

⎫⎪⎪⎬⎪⎪⎭

qμ→xpμ

.

As in the one-loop case, these factorizable contributions are defined as those parts of the diagrams(3.8) that are obtained by performing the loop integrations with the momenta of the gauge bosonsV1,V2,V3 set to zero in the tree subdiagrams. This prescription is indicated by the label F in thetree subdiagrams in (3.8).

By construction, the factorizable terms (3.8) contain all two-loop soft–soft singularities, i.e.the singularities that originate from the region where all gauge bosons V1,V2,V3 are soft. Ac-tually, as we will show, the factorizable contributions (3.8) contain all two-loop NLL masssingularities, i.e. not only all soft singularities but also all collinear singularities.

The symmetry factors 1/2, 1/6 and 1/8 in (3.8) avoid double counting in the sums overcombinations of external legs i, j , k, l. The limit qμ → xpμ in (3.8) indicates that the abovediagrams are evaluated in the approximation where each of the four-momenta qμ of the variousgauge bosons is collinear to one of the momenta pμ of the external legs and/or soft. Whererelevant, also the contributions of hard regions are taken into account (see Section 3.2). Thefactorizable two-loop terms (3.8) are computed in Section 5.2.

The remaining two-loop NLL mass singularities, which we call non-factorizable (NF), areobtained by subtracting from the diagrams of type (3.7) the factorizable terms (3.8) and externalself-energy diagrams. There are four sets of non-factorizable contributions that are associatedwith the first four diagrams in (3.7), whereas the last diagram in (3.7) does not produce non-factorizable NLL terms. This is due to the fact that all NLL mass singularities resulting from thelast diagram in (3.7) originate from the region where the momenta of the gauge bosons V1,V2are soft and are thus included in the factorizable part of this diagram.

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14 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

The non-factorizable terms associated with the first diagram in (3.7) read

MNF,A2 =

n∑i=1

n∑j=1j �=i

∑Vm=A,Z,W±

⎡⎢⎢⎢⎢⎣ −

(3.9)− −n∑

k=1k �=i,j

⎤⎥⎥⎥⎥⎦

qμ2 →xp

μi

qμ1 →0

.

The non-factorizable contributions associated with the second diagram in (3.7) are

MNF,B2 =

n∑i=1

n∑j=1j �=i

∑Vm=A,Z,W±

⎡⎢⎢⎢⎢⎢⎣ −

(3.10)− −n∑

k=1k �=i,j

⎤⎥⎥⎥⎥⎦

qμ2 →xp

μi

qμ1 →0

.

The non-factorizable contributions associated with the third diagram in (3.7) are

MNF,C2 = 1

2

n∑i=1

n∑j=1j �=i

n∑k=1

k �=i,j

∑Vm=A,Z,W±

⎡⎢⎢⎣ −

(3.11)

− − −n∑

l=1l �=i,j,k

⎤⎥⎥⎦

qμ2 →xp

μi

qμ1 →0

.

Here the first subtraction term represents an external-leg self-energy insertion and the symmetryfactor 1/2 is introduced in order to avoid double counting of the terms resulting from the permu-tation of the external legs j and k. The non-factorizable contributions associated with the fourth

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 15

diagram in (3.7) read

MNF,D2 =

n∑i=1

n∑j=1j �=i

∑Vm=A,Z,W±

⎡⎢⎢⎢⎢⎣ −

(3.12)− −n∑

k=1k �=i,j

⎤⎥⎥⎥⎥⎦

qμ2 →xp

μi

qμ1 →0

.

As indicated by the limits qμ2 → xp

μi , q

μ1 → 0 in (3.9)–(3.12), we extract the NLL mass singu-

larities that appear in the above combinations of diagrams in the regions where the momentumq

μ1 of the gauge boson V1 is soft and the momentum q

μ2 of the gauge boson V2 is collinear to the

external momentum pμi . The terms (3.9)–(3.12) are free of singularities in the soft limit x → 0

since all soft–soft singularities are included in the factorizable parts that are subtracted.The subtraction of the factorizable terms must be performed without double-counting of dia-

grams and regions, i.e. the terms that are subtracted in (3.9)–(3.12) must correspond exactly tothe ones that are included in (3.8). This correspondence is not obvious at first sight, since cer-tain diagrams appear a different number of times or with different symmetry factors in (3.8) and(3.9)–(3.12). This is due to the fact that in (3.9)–(3.12) the contributions of certain diagrams aresplit into various terms that result from different singular regions, whereas in (3.8) every diagramincludes the contributions of all singular regions. For instance, let us consider the last diagram in(3.9). This diagram appears only once in (3.8) but is subtracted four times in (3.9)–(3.12). Thesefour subtraction terms can be rewritten as

n∑i=1

n∑j=1j �=i

n∑k=1

k �=i,j

∑Vm=A,Z,W±

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

⎡⎢⎢⎣

⎤⎥⎥⎦

qμa →0

qμb →xp

μi

+

⎡⎢⎢⎣

⎤⎥⎥⎦

qμa →xp

μi

qμb →0

(3.13)+

⎡⎢⎢⎣

⎤⎥⎥⎦

qμa →0

qμb →xp

μk

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

,

where the last term in (3.13) corresponds to the two terms that are multiplied with a symmetryfactor 1/2 in (3.11). As can easily be seen in (3.13), these three contributions are associated withthe three non-overlapping singular regions of the diagram that give rise to NLL terms,9 and theirsum corresponds to the complete contribution included in (3.8). Similarly, one can verify thatall other subtraction terms in (3.9)–(3.12) correspond exactly to the factorizable contributionsincluded in (3.8).

9 The region qμa → xp

μj

, qμb

→ 0 does not give rise to NLL mass singularities.

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16 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

In Section 4 we will prove that the non-factorizable contributions (3.9)–(3.12) cancel.

3.2. Soft–collinear approximation for virtual gauge-boson exchange

As discussed in the previous section, mass singularities appear when the external fermionsemit virtual gauge bosons in the soft and collinear regions. Here we introduce a soft–collinearapproximation that describes the gauge-boson–fermion–antifermion vertices in these regions.This approximation permits to derive the soft and collinear singularities of generic n-fermionamplitudes in a process-independent way.

Let us first consider an n-fermion diagram involving the exchange of a virtual gauge bosonV = A,Z,W± between the external leg i and some other (external or internal) line,10

(3.14)=∑ϕ′

i

GV ϕ′

iμ (−q,pi + q)

i(/pi + /q)

(pi + q)2ieγ μIV

ϕ′iϕi

u(pi, κi).

Here GV ϕ′

iμ (−q,pi + q) represents the truncated Green function corresponding to the internal

part of the diagram, which is depicted as a grey blob. The contraction with the spinors of theexternal fermions j = 1, . . . , n with j �= i is implicitly understood. The loop momentum of thegauge boson is denoted as q and the propagator that connects the gauge boson lines V μ and V μ

has been omitted. Commuting the Dirac matrices associated with the fermion propagator and thegauge-boson–fermion vertex and using the massless Dirac equation we have

(3.15)i(/pi + /q)ieγ μu(pi, κi) = −e[2(pi + q)μ − γ μ/q

]u(pi, κi).

In the collinear limit qμ → xpμi the /qu(pi, κi) term vanishes owing to the Dirac equation. Thus,

we obtain

(3.16)limqμ→xp

μi

= GV iμ (−q,pi + q)u(pi, κi)

−2eIVi (pi + q)μ

(pi + q)2,

where we have introduced the shorthand GV iμ (−q,pi + q)IV

i = ∑ϕ′

iG

V ϕ′i

μ (−q,pi + q)IVϕ′

iϕi.

The approximation (3.16) is applicable also to the case where the gauge boson V becomescollinear to another leg j �= i. In this case, the term γ μ/q on the right-hand side of (3.15) iscontracted with another soft–collinear factor (pj − q)μ resulting from the coupling of the gaugeboson V to the leg j and

(3.17)limqμ→xp

μj

(pj − q)μγ μ/q = (1 − x)x/p2j = 0.

Similarly, (3.16) applies also to the case where the gauge boson V splits into two gauge bosons,V ′ and V ′′, with V ′ being collinear to an external leg j �= i and V ′′ soft. Here, the combinationof the soft–collinear factor associated with the external leg j , the triple gauge-boson vertex and

10 For the moment, we consider the case where the external particle i is a fermion. The generalization to antifermionsis discussed below.

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 17

the term γ μ/q on the right-hand side of (3.15) yields

(3.18)limqμ→xp

μj

(pj − q)μ′ [−2gμμ′qμ′′ + gμ′μ′′qμ + gμμ′′qμ′ ]γ μ/q = 0.

For the multiple emission of collinear gauge bosons V1, . . . , Vn by an external fermion i weobtain

limq

μk →xkp

μi

= GV 1···V niμ1···μn (−q1, . . . ,−qn,pi + qn)u(pi, κi)

(3.19)× −2eIVn

i (pi + qn)μn

(pi + qn)2· · · −2eI

V1i (pi + q1)

μ1

(pi + q1)2,

where qj = q1 + · · · + qj . This approximation is also applicable to all cases where one ofthe gauge bosons V1, . . . , Vn becomes collinear to one of the other external legs j �= i and allremaining gauge bosons are soft. Therefore (3.19) provides a correct description of the gauge-boson–fermion couplings in all regions that are relevant for our NLL analysis.

Also for the case where the external line i represents an antifermion, the emission of collineargauge bosons produces factors −2eI

Vk

i (pi + qk)μk and, apart from the obvious replacement of

the spinor u(pi, κi) by v(pi, κi), we obtain exactly the same formula as in (3.19).The soft–collinear approximation (3.19) permits to replace the Dirac matrices associated

with each gauge-boson emission by simple four-vector factors −2(pi + qk)μk . In the soft limit,

qμj → 0, these factors correspond to the well-known factors −2p

μk

i that are used to derive softsingularities in the eikonal approximation. The soft–collinear approximation (3.19) can be re-garded as an extension of the eikonal approximation that permits to describe the emission ofgauge bosons in the soft and the collinear regions. When applied to the one- and two-loop factor-izable contributions (3.5) and (3.8), this approximation permits to factorize the mass singularitiesfrom the n-fermion Born amplitude explicitly.

We note that the soft–collinear approximation is not applicable in the case where NLLtwo-loop contributions arise as a combination of logarithmic singularities originating from soft–collinear and UV regions. In particular, for the two-loop factorizable terms of the type

(3.20), , ,

where a soft–collinear singularity resulting from the exchange of the gauge bosons V (and V ′)appears in combination with an UV singularity resulting from hard particles in one-loop sub-diagrams, the approximation (3.19) can be applied only to the vertices that occur outside theUV-divergent one-loop subdiagrams whereas for the vertices and propagators inside the one-loop subdiagrams we have to apply the usual Feynman rules.

In the case of the last two diagrams in (3.20) this approximation is not sufficient to eliminatethe chain of Dirac matrices along the external fermionic leg i. However, this chain of Diracmatrices can be contracted with the spinor of the external fermion by means of a simple projector.

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18 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

Let us illustrate how this is achieved for the second diagram in (3.20),

(3.21)= −2eIVϕ′

j ϕjG

ϕ′i (pi)X

V ϕi ϕ′i (pi,pj )u(pi, κi)

with

(3.22)XV ϕiϕ′i (pi,pj ) = μ4−D

D

∫dDq

(2π)D

(pj − q)μ(/pi + /q)iΓV ϕiϕ

′i

μ (q,pi)

(q2 − M2V )(pi + q)2(pj − q)2

.

Here the factor (pj − q)μ comes from the soft–collinear vertex along the leg j , and iΓV ϕiϕ

′i

μ

represents the one-loop vertex. The truncated Green function Gϕ′

i (pi) in (3.21) represents theinternal part of the diagram, where the momentum q of the gauge boson V is set to zero accordingto the definition of the factorizable part. The contraction with the spinors of the external fermionsj = 1, . . . , n with j �= i is implicitly understood. If one contracts this Green function with thespinor of the fermion i one simply obtains the on-shell Born matrix element,

(3.23)Gϕ′

i (pi)u(pi, κi) =Mϕ1···ϕ′

i ···ϕ′j ···ϕn

0 .

The chain of Dirac matrices occurring in (3.22) can be projected on the Dirac spinor using11

(3.24)XV ϕiϕ′i (pi,pj )u(pi, κi) = 1

2pipj

Tr[XV ϕiϕ

′i (pi,pj )ωκi

/pi/pj

]u(pi, κi),

so that one obtains

(3.25)

Gϕ′

i (pi)XV ϕiϕ

′i (pi,pj )u(pi, κi) = 1

2pipj

Tr[XV ϕiϕ

′i (pi,pj )ωκi

/pi/pj

]M

ϕ1···ϕ′i ···ϕ′

j ···ϕn

0 .

The same trace projector can be applied to the last diagram in (3.20),

(3.26)= 2e2IVϕ′

iϕiI Vϕ′

j ϕjG

ϕ′i (pi)X

ϕ′iϕ

′i (pi,pj )u(pi, κi),

where

(3.27)Xϕ′iϕ

′i (pi,pj ) = μ4−D

D

∫dDq

(2π)D

(pj − q)μ(/pi + /q)iΣϕ′iϕ

′i (pi + q)(/pi + /q)γμ

(q2 − M2V )[(pi + q)2]2(pj − q)2

,

11 This identity can easily be verified by means of the general decomposition

XV ϕiϕ

′i (pi ,pj )ωκ =

N∑k=0

∑l1=i,j

· · ·∑

l2k=i,j

Al1···l2k(pi ,pj )/pl1 · · ·/pl2k

ωκ ,

the Dirac equation, /piu(pi , κi ) = 0, the anticommutation relation {/pi, /pj } = 2pipj , the identities /p2i

= /p2j

= 0, and the

fact that, for massless fermions, the subamplitude XV ϕiϕ

′i (pi ,pj ) contains only Dirac chains with even numbers 2k of

Dirac matrices.

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 19

and iΣϕ′iϕ

′i represents the fermionic self-energy. Again we can project (3.27) on the Dirac spinor

using

(3.28)

Gϕ′

i (pi)Xϕ′

iϕ′i (pi,pj )u(pi, κi) = 1

2pipj

Tr[Xϕ′

iϕ′i (pi,pj )ωκi

/pi/pj

]M

ϕ1···ϕ′i ···ϕ′

j ···ϕn

0 .

3.3. Ultraviolet singularities

Let us first discuss our treatment of UV singularities at one loop. The UV singularities ap-pearing in the bare loop amplitudes are canceled by corresponding singularities provided by thecounterterms. These cancellations give rise to logarithmic contributions of the form

(3.29)

(μ2

D

Q2

)ε[1

ε

(Q2

μ2loop

− 1

ε

(Q2

μ2R

)ε]= ln

(μ2

R

μ2loop

)+O(ε),

where the first and the second term between the brackets result from bare loop diagrams andcounterterms, respectively. Here μD is the scale of dimensional regularization and we have fac-torized the term (μ2

D/Q2)ε that we always absorb in αε [see (2.7)]. The renormalization scaleis denoted as μR, and μloop represents the characteristic scale of the UV-singular loop diagram.This latter is related to the momenta of the lines that enter the loop.

At one loop, since in the high-energy limit all combinations of external momenta are hard, thecharacteristic scale of UV-divergent diagrams that contribute to truncated n-fermion Green func-tions is always of the order μ2

loop ∼ Q2. This permits us to isolate all logarithms that result fromUV singularities, i.e. terms of the type (3.29), in the counterterms. To this end, in our calculationwe perform a minimal subtraction of all UV poles that appear in the bare loop diagrams and inthe counterterms. The combination of these subtracted terms corresponds to

(3.30)

(μ2

D

Q2

)ε{1

ε

[(Q2

μ2loop

− 1

]− 1

ε

[(Q2

μ2R

− 1

]}

and is obviously equivalent to (3.29). As a result of the minimal subtraction the logarithmiccontributions ε−1[(Q2/μ2

loop)ε − 1] originating from bare loop diagrams that are characterized

by a hard scale μ2loop ∼ Q2 vanish.

Thus, at one loop we can restrict ourselves to the calculation of the mass-singular bare dia-grams and the counterterms. The minimal subtraction of the UV singularities appearing in thesetwo types of contributions permits us to ignore any other bare diagram that produces UV singu-larities.

At two loops, pure UV singularities produce only NNLL contributions since every UV-singular loop produces only a single-logarithmic term. The only NLL two-loop terms result-ing from UV singularities are combinations of one-loop UV logarithms with one-loop doublelogarithms resulting from soft–collinear gauge bosons. These terms originate from one-loop in-sertions in the one-loop diagrams (3.2). Here, as in the one-loop case, we perform a minimalsubtraction of the UV singularity, such that the logarithms associated with one-loop subdiver-gences from hard subdiagrams with μ2

loop ∼ Q2 are completely isolated in the counterterms. Asa consequence, the UV contributions associated with one-loop insertions in the internal part ofthe one-loop diagram (3.2) become irrelevant. In particular the UV logarithmic contributions

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20 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

resulting from the two-loop ladder diagrams of type

(3.31), , ,

in the region where V1 is soft and V2 is hard vanish. Minimal subtraction of the UV singularitiespermits us to use the soft–collinear approximation for all gauge bosons V1,V2 in the abovediagrams despite of the fact that this approximation is not appropriate to describe the hard regionand can in principle produce fake logarithms of UV origin. Instead, the diagrams of the type(3.20), which result from the insertion of one-loop subdiagrams in the lines that are not hard(μ2

loop Q2) in (3.2), give rise to non-negligible NLL contributions of UV type. These UVcontributions are correctly taken into account in our calculation as explained in Section 3.2.

4. Collinear Ward identities and cancellation of non-factorizable terms

In this section we prove that the non-factorizable subset of mass singularities, i.e. the one-loopterms (3.6) and the two-loop terms (3.9)–(3.12), vanish. The proof is based on the collinear Wardidentities that have been derived in Ref. [7].

Let us start with the non-factorizable one-loop terms (3.6). This contribution can be writtenas

MNF1 =

n∑i=1

∑V =A,Z,W±

∑ϕ′

i

μ4−DD

∫dDq

(2π)D

1

(q2 − M2V )(pi − q)2

limqμ→xp

μi

2(pi − q)μ

×{G

[V ϕ′i]

μ (q,pi − q)u(pi, κi)

(4.1)+n∑

j=1j �=i

∑ϕ′

j

2(pj + q)μ

(pj + q)2M

ϕ1···ϕ′i ···ϕ′

j ···ϕn

0 eIVϕ′

j ϕj

}ieIV

ϕ′iϕi

,

where we have factorized the two propagators that appear in all three diagrams in (3.6). Hereu(pi, κi) is the spinor of the external fermion i and we have introduced the abbreviation

(4.2)G[V ϕ

i]

μ (q,pi − q) = −for the tree subdiagrams that are associated with the first two terms on the right-hand side of(3.6). In (4.2) the contraction with the spinors of the external fermions j = 1, . . . , n with j �= i isimplicitly understood, and the incoming momenta associated with the lines Vμ and i are q andpi − q , respectively.

The combination of tree subdiagrams (4.2) fulfills the collinear Ward identities [7]

(4.3)limqμ→xp

μi

qμG[V ϕ

i]

μ (q,pi − q)u(pi, κi) =∑ϕ′

i

Mϕ1···ϕ′i ···ϕn

0 eIVϕ′

iϕi.

Using the charge-conservation relation (2.30) and

(4.4)limqμ→xp

μ

2qμ(pj + q)μ

(pj + q)2= 1 for j �= i,

i

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 21

we can rewrite the identities (4.3) as

(4.5)

limqμ→xp

μi

{G

[V ϕi]

μ (q,pi − q)u(pi, κi) +n∑

j=1j �=i

∑ϕ′

j

2(pj + q)μ

(pj + q)2M

ϕ1···ϕ′j ···ϕn

0 eIVϕ′

j ϕj

}= 0,

where the expression between the curly brackets is identical to the one that appears on the right-hand side of (4.1). These collinear Ward identities can be represented diagrammatically as

(4.6)limqμ→xp

μi

qμ ×

⎡⎢⎢⎢⎢⎢⎢⎣

− −n∑

j=1j �=i

⎤⎥⎥⎥⎥⎥⎥⎦

= 0,

where the contraction with all fermionic spinors, including u(pi, κi), is implicitly understood.The cancellation of the non-factorizable terms (4.1) is simply due to the fact that in the

collinear limit qμ → xpμi the four-vector (pi − q)μ on the right-hand side of (4.1) becomes

proportional to the gauge-boson momentum qμ and its contraction with the expression betweenthe curly brackets vanishes as a result of the collinear Ward identities (4.5).

Let us now consider the two-loop non-factorizable terms (3.9)–(3.12). The contribution (3.9)yields

MNF,A2 =

n∑i=1

n∑j=1j �=i

∑Vm=A,Z,W±

∑ϕ′

i ,ϕ′′i ,ϕ′

j

μ2(4−D)D

∫dDq1

(2π)D

×∫

dDq2

(2π)D

8e3(pi − q1)(pj + q1)

(q21 − M2

V1)(q2

2 − M2V2

)

× 1

(pi − q1)2(pj + q1)2(pi − q1 − q2)2lim

qμ1 →0

limq

μ2 →xp

μi

(pi − q1 − q2)μ

×{

G[V 2 ϕ′′

i]

μ (q2,pi − q1 − q2)u(pi, κi)

+ 2(pj + q1 + q2)μ

(pj + q1 + q2)2

∑ϕ′′

j

Mϕ1···ϕ′′

i ···ϕ′′j ···ϕn

0 eIV 2ϕ′′

j ϕ′j

(4.7)+n∑

k=1k �=i,j

2(pk + q2)μ

(pk + q2)2

∑ϕ′

k

Mϕ1···ϕ′′

i ···ϕ′j ···ϕ′

k ···ϕn

0 eIV 2ϕ′

kϕk

}I

V2ϕ′′

i ϕ′i

IV 1ϕ′

j ϕjI

V1ϕ′

iϕi= 0.

This cancellation can easily be verified by means of the collinear Ward identities (4.5) by ob-serving that in the soft–collinear limit q

μ1 → 0, q

μ2 → xp

μi the four-vector (pi − q1 − q2)

μ tendsto (1/x − 1)q

μ2 and the expression within the curly brackets in (4.7) becomes equivalent to the

one in (4.5). Similarly, for the contributions (3.10) and (3.11) we obtain

MNF,B2 =

n∑i=1

n∑j=1

∑Vm=A,Z,W±

∑ϕ′

i ,ϕ′′i ,ϕ′

j

μ2(4−D)D

∫dDq1

(2π)D

j �=i

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22 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

×∫

dDq2

(2π)D

8e3(pi − q1 − q2)(pj + q1)

(q21 − M2

V1)(q2

2 − M2V2

)

× 1

(pi − q2)2(pj + q1)2(pi − q1 − q2)2lim

qμ1 →0

limq

μ2 →xp

μi

(pi − q2)μ

×{

G[V 2 ϕ′′

i]

μ (q2,pi − q1 − q2)u(pi, κi)

+ 2(pj + q1 + q2)μ

(pj + q1 + q2)2

∑ϕ′′

j

Mϕ1···ϕ′′

i ···ϕ′′j ···ϕn

0 eIV 2ϕ′′

j ϕ′j

(4.8)+n∑

k=1k �=i,j

2(pk + q2)μ

(pk + q2)2

∑ϕ′

k

Mϕ1···ϕ′′

i ···ϕ′j ···ϕ′

k ···ϕn

0 eIV 2ϕ′

kϕk

}I

V 1ϕ′

j ϕjI

V1ϕ′′

i ϕ′i

IV2ϕ′

iϕi= 0,

and

MNF,C2 = 1

2

n∑i=1

n∑j=1j �=i

n∑k=1

k �=i,j

∑Vm=A,Z,W±

∑ϕ′

i ,ϕ′j ,ϕ′

k

μ2(4−D)D

∫dDq1

(2π)D

∫dDq2

(2π)D

1

(q21 − M2

V1)

× 8e3(pk − q1)(pj + q1)

(q22 − M2

V2)(pi − q2)2(pj + q1)2(pk − q1)2

limq

μ1 →0

limq

μ2 →xp

μi

(pi − q2)μ

×{

G[V 2 ϕ′

i]

μ (q2,pi − q2)u(pi, κi)

+ 2(pj + q1 + q2)μ

(pj + q1 + q2)2

∑ϕ′′

j

Mϕ1···ϕ′

i ···ϕ′′j ···ϕ′

k ···ϕn

0 eIV 2ϕ′′

j ϕ′j

+ 2(pk − q1 + q2)μ

(pk − q1 + q2)2

∑ϕ′′

k

Mϕ1···ϕ′

i ···ϕ′j ···ϕ′′

k ···ϕn

0 eIV 2ϕ′′

k ϕ′k

(4.9)

+n∑

l=1l �=i,j,k

2(pl + q2)μ

(pl + q2)2

∑ϕ′

l

Mϕ1···ϕ′

i ···ϕ′j ···ϕ′

k ···ϕ′l ···ϕn

0 eIV 2ϕ′

lϕl

}I

V 1ϕ′

j ϕjI

V1ϕ′

kϕkI

V2ϕ′

iϕi= 0.

Finally, for the contribution (3.12) we have

MNF,D2 =

n∑i=1

n∑j=1j �=i

∑Vm=A,Z,W±

∑ϕ′

i ,ϕ′j

μ2(4−D)D

∫dDq1

(2π)D

∫dDq2

(2π)D

4ie2g2εV1V2V3

(q21 − M2

V1)(q2

2 − M2V2

)

× 1

(q23 − M2

V3)(pi − q2)2(pj − q1)2

limq

μ1 →0

limq

μ2 →xp

μi

(pi − q2)μ2(pj − q1)

μ1

× [gμ1μ2(q1 − q2)

μ3 + gμ3μ2

(q2 + q3)μ1 − gμ3μ1

(q3 + q1)μ2

]×{

G[V 3 ϕ′

i]

μ3 (q3,pi − q2)u(pi, κi) + 2(pj + q2)μ3

(pj + q2)2

∑ϕ′′

Mϕ1···ϕ′

i ···ϕ′′j ···ϕn

0 eIV 3ϕ′′

j ϕ′j

j

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 23

(4.10)+n∑

k=1k �=i,j

2(pk + q3)μ3

(pk + q3)2

∑ϕ′

k

Mϕ1···ϕ′

i ···ϕ′j ···ϕ′

k ···ϕn

0 eIV 3ϕ′

kϕk

}I

V 1ϕ′

j ϕjI

V 2ϕ′

iϕi,

where q3 = q1 + q2. In the soft–collinear limit we observe that

limq

μ1 →0

limq

μ2 →xp

μi

(pi − q2)μ2(pj − q1)

μ1[gμ1μ2(q1 − q2)

μ3 + gμ3μ2

(q2 + q3)μ1

(4.11)− gμ3μ1

(q3 + q1)μ2

]= (1 − x)(pipj )qμ33 ,

and again the contraction of this four-vector with the expression between the curly brackets in(4.10) cancels as a result of the collinear Ward identities (4.5).

5. Factorizable contributions

In this section, we present explicit results for the one- and two-loop factorizable contribu-tions defined in Sections 3.1.2 and 3.1.4. These are evaluated within the ’t Hooft–Feynmangauge, where the masses of the Faddeev–Popov ghosts uA, uZ , uW±

and would-be Goldstonebosons χ,φ± read MuA = MA = 0, Mχ = MuZ = MZ, and Mφ± = MuW = MW. Using thesoft–collinear approximation introduced in Section 3.2, we express the factorizable contributionsresulting from individual diagrams as products of the n-fermion Born amplitude with matrix-valued gauge couplings and loop integrals. The definitions of these loop integrals are providedin Appendix A. The integrals are computed in NLL accuracy, and the result is expanded in ε

up to O(ε2) at one loop and O(ε0) at two loops. The UV poles are eliminated by means of aminimal subtraction as explained in Section 3.3 such that the presented results are UV finite. Allloop integrals have been solved and cross-checked using two independent methods: an automa-tized algorithm based on the sector-decomposition technique [33] and the method of expansionby regions combined with Mellin–Barnes representations [34].

5.1. One-loop diagrams

The one-loop factorizable contributions originate only from one type of diagram,12

(5.1)Mij

1 = = −M0

∑V1=A,Z,W±

IV 1i I

V1j D0(MV1; rij ).

The corresponding loop integral D0 is defined in (A.6) and to NLL accuracy yields

D0(MW; rij ) NLL= −L2 − 2

3L3ε − 1

4L4ε2 + 2(2 − lij )

(L + 1

2L2ε + 1

6L3ε2

),

D0(MZ; rij ) NLL= D0(MW; rij ) + lZ(2L + 2L2ε + L3ε2),

(5.2)D0(0; rij ) NLL= −2ε−2 − 2(2 − lij )ε−1,

12 The l-loop diagrams depicted in this section are understood without factors (αε/4π)l .

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24 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

where the UV singularities

(5.3)DUV0 (MW; rij ) NLL= DUV

0 (MZ; rij ) NLL= DUV0 (0; rij ) NLL= 4ε−1

have been subtracted.

5.2. Two-loop diagrams

The two-loop NLL factorizable terms (3.8) involve fourteen different types of diagrams. Thediagrams 1–3, 12, and 14 in this section give rise to LLs and NLLs, whereas all other diagramsyield only NLLs. The loop integrals associated with the various diagrams are denoted with sym-bols of the type Dh(m1, . . . ,mj ; rkl, . . .) and depend on various kinematical invariants rkl andmasses mi . The symbols mi are always used to denote generic mass parameters, which can as-sume the values mi = MW,MZ,mt,MH or mi = 0. Instead we use the symbols Mi to denotenon-zero masses, i.e. Mi = MW,MZ,mt,MH. The integrals are often singular when certain massparameters tend to zero, and the cases where such parameters are zero or non-zero need to betreated separately. For every integral we first evaluate Dh(MW, . . . ,MW; rkl, . . .), i.e. the casewhere all mass parameters are equal to MW. The dependence of the integral on the variousmasses is then described by subtracted functions of the type

(5.4)�Dh(m1, . . . ,mj ; rkl, . . .) = Dh(m1, . . . ,mj ; rkl, . . .) − Dh(MW, . . . ,MW; rkl, . . .).

Diagram 1

(5.5)M1,ij

2 = =M0

∑V1,V2=A,Z,W±

IV 2i I

V 1i I

V2j I

V1j D1(MV1,MV2; rij ),

where the loop integral D1 is defined in (A.6) and yields

D1(MW,MW; rij ) NLL= 1

6L4 − 2

3(2 − lij )L

3,

�D1(M1,M2; rij ) NLL= −2

3l1L

3,

�D1(0,M2; rij ) NLL= 2L2ε−2 + 8

3L3ε−1 + 11

6L4 − (2 − lij )

(4Lε−2 + 4L2ε−1 + 2L3)

− l2(4Lε−2 + 8L2ε−1 + 8L3),

�D1(M1,0; rij ) NLL= −2

3l1L

3,

(5.6)�D1(0,0; rij ) NLL= ε−4 − 1

6L4 + (2 − lij )

(2ε−3 + 2

3L3)

.

Here the UV singularities

DUV1 (M1,m2; rij ) NLL= −4L2ε−1 − 8

3L3,

(5.7)DUV1 (0,m2; rij ) NLL= −8ε−3

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 25

have been subtracted.

Diagram 2

(5.8)M2,ij

2 = =M0

∑V1,V2=A,Z,W±

IV 2i I

V 1i I

V1j I

V2j D2(MV1 ,MV2; rij ),

where the loop integral D2 is defined in (A.6). This integral is free of UV singularities and yields

D2(MW,MW; rij ) NLL= 1

3L4 − 4

3(2 − lij )L

3,

�D2(M1,M2; rij ) NLL= −2

3(l1 + l2)L

3,

�D2(0,M2; rij ) NLL= −2

3L3ε−1 − 7

6L4 + (2 − lij + l2)

(2L2ε−1 + 10

3L3)

,

�D2(M1,0; rij ) NLL= −2

3L3ε−1 − 7

6L4 + (2 − lij + l1)

(2L2ε−1 + 10

3L3)

,

(5.9)�D2(0,0; rij ) NLL= ε−4−1

3L4 + (2 − lij )

(2ε−3 + 4

3L3)

.

Diagram 3

M3,ij

2 =

(5.10)= −ig2

eM0

∑V1,V2,V3=A,Z,W±

εV1V2V3IV 2i I

V 1i I

V 3j D3(MV1 ,MV2,MV3; rij ),

where the ε-tensor is defined in (2.18). The loop integral D3 is defined in (A.6) and yields

D3(MW,MW,MW; rij ) NLL= 1

6L4 −

(3 − 2lij

3

)L3,

�D3(M1,M2,M3; rij ) NLL= −1

3(l1 + l3)L

3,

�D3(0,M2,M3; rij ) NLL= −1

3L3ε−1 − 7

12L4 + (2 − lij + l3)

(L2ε−1 + 5

3L3)

,

�D3(M1,0,M3; rij ) NLL= −1

3(l1 + l3)L

3,

�D3(M1,M2,0; rij ) NLL= −1

3L3ε−1 − 7

12L4 − 6Lε−2 − (2 + lij − l1)L

2ε−1

(5.11)+(

11

3− 5lij

3+ 5l1

3

)L3,

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26 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

where the UV singularities

DUV3 (m1,m2,M3; rij ) NLL= −3L2ε−1 − 2L3,

(5.12)DUV3 (m1,m2,0; rij ) NLL= −6ε−3

have been subtracted.

Diagram 4

(5.13)

M4,ij

2 = = −M0

∑V1,V2=A,Z,W±

IV2i I

V 2i I

V1i I

V 1j D4(MV1 ,MV2; rij ),

where the loop integral D4 is defined in (A.6) and yields

D4(MW,MW; rij ) NLL= 1

3L3,

�D4(M1,M2; rij ) NLL= 0,

�D4(0,M2; rij ) NLL= 2Lε−2 + 2L2ε−1 + L3,

�D4(M1,0; rij ) NLL= 0,

(5.14)�D4(0,0; rij ) NLL= −ε−3 − 1

3L3.

Here the UV singularities

DUV4 (M1,m2; rij ) NLL= L2ε−1 + 2

3L3,

(5.15)DUV4 (0,m2; rij ) NLL= 2ε−3

have been subtracted.

Diagram 5

(5.16)

M5,ij

2 = = −M0

∑V1,V2=A,Z,W±

IV2i I

V1i I

V 2i I

V 1j D5(MV1 ,MV2; rij ),

where the loop integral D5 is defined in (A.6) and yields

D5(MW,MW; rij ) NLL= −1

3L3,

�D5(M1,M2; rij ) NLL= 0,

�D5(0,M2; rij ) NLL= −2Lε−2 − 2L2ε−1 − L3,

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 27

�D5(M1,0; rij ) NLL= 0,

(5.17)�D5(0,0; rij ) NLL= ε−3 + 1

3L3.

Here the UV singularities

DUV5 (M1,m2; rij ) NLL= −L2ε−1 − 2

3L3,

(5.18)DUV5 (0,m2; rij ) NLL= −2ε−3

have been subtracted.

Diagram 6

M6,ij

2 = +

= 1

2

g22

e2M0

∑V1,V2,V3,V4=A,Z,W±

IV 1i I

V 4j εV1V 2V 3εV4V2V3

(5.19)× D6(MV1 ,MV2,MV3 ,MV4; rij ),where the loop integral D6 is defined in (A.6) and yields

D6(MW,MW,MW,MW; rij ) NLL= 20

9L3,

�D6(M1,M2,M3,M4; rij ) NLL= 0,

�D6(0,M2,M3,M4; rij ) NLL= M22 + M2

3

2M24

[−16Lε−2 − 8L2ε−1 + 16

3L3],

�D6(M1,0,M3,M4; rij ) NLL= 0,

�D6(M1,M2,0,M4; rij ) NLL= 0,

�D6(M1,M2,M3,0; rij ) NLL= M22 + M2

3

2M21

[−16Lε−2 − 8L2ε−1 + 16

3L3],

(5.20)�D6(0,M2,M3,0; rij ) NLL= 20

3Lε−2 + 10

3L2ε−1 − 20

9L3.

Here the UV singularities

DUV6 (M1,m2,m3,M4; rij ) NLL= 10

3L2ε−1 + 20

9L3,

DUV6 (0,m2,m3,M4; rij ) NLL= 10

3L2ε−1 + 20

9L3

+ m22 + m2

3

2M24

[−16ε−3 + 8L2ε−1 + 16

3L3],

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28 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

DUV6 (M1,m2,m3,0; rij ) NLL= 10

3L2ε−1 + 20

9L3

+ m22 + m2

3

2M21

[−16ε−3 + 8L2ε−1 + 16

3L3],

(5.21)DUV6 (0,m2,m3,0; rij ) NLL= 20

3ε−3

have been subtracted. We observe that the loop integrals associated with A–Z mixing-energysubdiagrams give rise to the contributions

(5.22)�D6(0,MW,MW,MZ; rij ) = M2W

M2Z

[−16Lε−2 − 8L2ε−1 + 16

3L3],

which depend linearly on the ratio M2W/M2

Z.

Diagram 7

M7,ij

2 =

= −g22

e2M0

∑V1,V2,V3=A,Z,W±

IV 1i I

V 3j

∑V =A,Z,W±

εV1V 2V εV3V2V

(5.23)× (D − 1)D7(MV1 ,MV2,MV3; rij ),where D = 4 − 2ε. The loop integral D7 is defined in (A.6) and yields

D7(MW,MW,MW; rij ) NLL= 0,

�D7(M1,M2,M3; rij ) NLL= 0,

�D7(0,M2,M3; rij ) NLL= M22

M23

[−2Lε−2 − L2ε−1 + 2

3L3],

�D7(M1,0,M3; rij ) NLL= 0,

�D7(M1,M2,0; rij ) NLL= M22

M21

[−2Lε−2 − L2ε−1 + 2

3L3],

(5.24)�D7(0,M2,0; rij ) NLL= 0,

where the UV singularities

DUV7 (0,M2,M3; rij ) NLL= M2

2

M23

[−2ε−3 + L2ε−1 + 2

3L3],

(5.25)DUV7 (M1,M2,0; rij ) NLL= M2

2

M21

[−2ε−3 + L2ε−1 + 2

3L3]

have been subtracted.

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 29

Diagram 8

M8,ij

2 =

= −e2v2M0

∑V1,V3,V4=A,Z,W±

IV 1i I

V 4j

∑Φi2 =H,χ,φ±

{IV1 , IV 3

}HΦi2

(5.26)× {IV3, IV4

}Φi2 H

D8(MV1 ,MΦi2,MV3,MV4; rij ),

where the curly brackets denote anticommutators and v is the vacuum expectation value. Theloop integral D8 is defined in (A.6) and yields

M2WD8(MW,MW,MW,MW; rij ) NLL= 0,

M2W�D8(M1,M2,M3,M4; rij ) NLL= 0,

�D8(0,M2,M3,M4; rij ) NLL= 1

M24

[−2Lε−2 − L2ε−1 + 2

3L3],

M2W�D8(M1,M2,0,M4; rij ) NLL= 0,

�D8(M1,M2,M3,0; rij ) NLL= 1

M21

[−2Lε−2 − L2ε−1 + 2

3L3],

(5.27)M2W�D8(0,M2,M3,0; rij ) NLL= 0.

Here the UV singularities

DUV8 (0,M2,M3,M4; rij ) NLL= 1

M24

[−2ε−3 + L2ε−1 + 2

3L3],

(5.28)DUV8 (M1,M2,M3,0; rij ) NLL= 1

M21

[−2ε−3 + L2ε−1 + 2

3L3]

have been subtracted. The above diagram represents the only contribution involving couplingsproportional to v, which originate from spontaneous symmetry breaking.

Diagram 9

M9,ij

2 =

= −1

2M0

∑V1,V4=A,Z,W±

IV 1i I

V 4j

∑Φi2 ,Φi3 =H,χ,φ±

IV1Φi3 Φi2

IV4Φi2Φi3

(5.29)× D9(MV1 ,MΦi2,MΦi3

,MV4; rij ),

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30 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

where the loop integral D9 is defined in (A.6) and yields

D9(MW,MW,MW,MW; rij ) NLL= 2

9L3,

�D9(M1,M2,M3,M4; rij ) NLL= 0,

�D9(0,M2,M3,M4; rij ) NLL= M22 + M2

3

2M24

[4Lε−2 + 2L2ε−1 − 4

3L3],

�D9(M1,M2,M3,0; rij ) NLL= M22 + M2

3

2M21

[4Lε−2 + 2L2ε−1 − 4

3L3],

(5.30)�D9(0,M2,M3,0; rij ) NLL= 2

3Lε−2 + 1

3L2ε−1 − 2

9L3.

Here the UV singularities

DUV9 (M1,M2,M3,M4; rij ) NLL= 1

3L2ε−1 + 2

9L3,

DUV9 (0,M2,M3,M4; rij ) NLL= 1

3L2ε−1 + 2

9L3 + M2

2 + M23

2M24

[4ε−3 − 2L2ε−1 − 4

3L3],

DUV9 (M1,M2,M3,0; rij ) NLL= 1

3L2ε−1 + 2

9L3 + M2

2 + M23

2M21

[4ε−3 − 2L2ε−1 − 4

3L3],

(5.31)DUV9 (0,M2,M3,0; rij ) NLL= 2

3ε−3

have been subtracted.

Diagram 10

M10,ij

2 =

= −1

2M0

∑V1,V3=A,Z,W±

IV 1i I

V 3j

∑Φi2 =H,χ,φ±

{IV1, IV3

}Φi2 Φi2

(5.32)× D10(MV1 ,MΦi2,MV3; rij ),

where

(5.33)D10 ≡ D7.

Also this diagram, which yields NLL contributions only through A–Z mixing-energy subdia-grams, gives rise to a correction proportional to M2

W/M2Z originating from

�D7(0,Mφ± ,MZ; rij ). This correction cancels the contribution proportional to M2W/M2

Z thatoriginates from diagram 9.

Diagram 11For the diagrams involving fermionic self-energy subdiagrams we consider the contributions

of a generic fermionic doublet Ψ with components Ψi = u,d . The sum over the three generations

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 31

of leptons and quarks is denoted by∑

Ψ , and color factors are implicitly understood. Assumingthat all down-type fermions are massless, md = 0, and that the masses of up-type fermions aremu = 0 or mt, we have

M11,ij

2 =

= −1

2M0

∑V1,V4=A,Z,W±

IV 1i I

V 4j

×∑Ψ

{ ∑Ψi2 ,Ψi3 =u,d

∑κ=R,L

IV1Ψ κ

i3Ψ κ

i2I

V4Ψ κ

i2Ψ κ

i3D11,0(MV1 ,mi2,mi3,MV4; rij )

(5.34)− (I

V1uRuRI

V4uLuL + I

V1uLuLI

V4uRuR

)m2

uD11,m(MV1 ,mu,mu,MV4; rij )},

where D11,m ≡ −4D8 represents the contribution associated to the mu-terms in the numerator ofthe up-type fermion propagators, whereas the integral D11,0, which is defined in (A.6), accountsfor the remaining contributions. This latter integral yields

D11,0(MW,MW,MW,MW; rij ) NLL= 8

9L3,

�D11,0(M1,m2,m3,M4; rij ) NLL= 0,

�D11,0(0,m2,m3,M4; rij ) NLL= m22 + m2

3

2M24

[8Lε−2 + 4L2ε−1 − 8

3L3],

�D11,0(M1,m2,m3,0; rij ) NLL= m22 + m2

3

2M21

[8Lε−2 + 4L2ε−1 − 8

3L3],

�D11,0(0,M2,M3,0; rij ) NLL= 8

3Lε−2 + 4

3L2ε−1 − 8

9L3,

(5.35)�D11,0(0,0,0,0; rij ) NLL= −2ε−3 − 8

9L3,

where the UV singularities

DUV11,0(M1,m2,m3,M4; rij ) NLL= 4

3L2ε−1 + 8

9L3,

DUV11,0(0,m2,m3,M4; rij ) NLL= 4

3L2ε−1 + 8

9L3 + m2

2 + m23

2M24

[8ε−3 − 4L2ε−1 − 8

3L3],

DUV11,0(M1,m2,m3,0; rij ) NLL= 4

3L2ε−1 + 8

9L3 + m2

2 + m23

2M21

[8ε−3 − 4L2ε−1 − 8

3L3],

(5.36)DUV11,0(0,m2,m3,0; rij ) NLL= 8

3ε−3

have been subtracted. As a consequence of

�D11,0(0,m2,m3,M4; rij ) = m22 + m2

3 �D11,m(0,m2,m3,M4; rij ),

2
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32 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

(5.37)�D11,0(M1,m2,m3,0; rij ) = m22 + m2

3

2�D11,m(M1,m2,m3,0; rij ),

all terms proportional to the fermion masses in M11,ij

2 cancel.

Diagram 12

(5.38)

M12,ijk

2 = =M0

∑V1,V2=A,Z,W±

IV 2i I

V 1i I

V1j I

V2k D12(MV1 ,MV2; rik),

where the loop integral D12 is defined in (A.6) and yields

D12(MW,MW; rik) NLL= 1

2L4 − 2(2 − lik)L

3,

�D12(M1,M2; rik) NLL= −2

3(2l1 + l2)L

3,

�D12(0,M2; rik) NLL= 2L2ε−2 + 2L3ε−1 + 2

3L4 − (2 − lik)

(4Lε−2 + 2L2ε−1 − 4

3L3)

− l2

(4Lε−2 + 6L2ε−1 + 14

3L3)

,

�D12(M1,0; rik) NLL= −2

3L3ε−1 − 7

6L4 + (2 − lik)

(2L2ε−1 + 10

3L3)

+ l1

(2L2ε−1 + 8

3L3)

,

(5.39)�D12(0,0; rik) NLL= 2ε−4 − 1

2L4 + (2 − lik)

(4ε−3 + 2L3).

Here the UV singularities

DUV12 (M1,m2; rik) NLL= −4L2ε−1 − 8

3L3,

(5.40)DUV12 (0,m2; rik) NLL= −8ε−3

have been subtracted. Note that, to NLL accuracy, the above diagram does not depend on rij andrjk .

Diagram 13

M13,ijk

2 =

= −ig2

eM0

∑V1,V2,V3=A,Z,W±

εV1V2V3IV 1i I

V 2j I

V 3k

(5.41)× D13(MV1 ,MV2,MV3; rij , rik, rjk),

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 33

where the loop integral D13 is defined in (A.6). This integral is free of UV singularities and yields

D13(MW,MW,MW; rij , rik, rjk)NLL= 0,

�D13(M1,M2,M3; rij , rik, rjk)NLL= 0,

�D13(0,M2,M3; rij , rik, rjk)NLL= (lij − lik)

(L2ε−1 + 5

3L3)

,

�D13(M1,0,M3; rij , rik, rjk)NLL= (ljk − lij )

(L2ε−1 + 5

3L3)

,

(5.42)�D13(M1,M2,0; rij , rik, rjk)NLL= (lik − ljk)

(L2ε−1 + 5

3L3)

.

Diagram 14

M14,ijkl

2 =

(5.43)=M0

∑V1,V2=A,Z,W±

IV 1i I

V1j I

V 2k I

V2l D14(MV1 ,MV2; rij , rkl),

where the loop integral D14 is simply given by the product of one-loop integrals (5.2),

(5.44)D14(MV1 ,MV2; rij , rkl) = D0(MV1; rij )D0(MV2; rkl).

6. Renormalization

In this section we discuss the NLL counterterm contributions that result from the renormal-ization of the gauge-boson masses,13

(6.1)MV,0 = MV +∞∑l=1

(αε

)l

δM(l)V ,

and the electroweak couplings,

(6.2)g0,i = gi +∞∑l=1

(αε

)l

δg(l)i , e0 = e +

∞∑l=1

(αε

)l

δe(l),

as well as from the renormalization constants associated with the wave functions of the externalfermions k = 1, . . . , n,

(6.3)Zk = 1 +∞∑l=1

(αε

)l

δZ(l)k .

The renormalized one- and two-loop amplitudes are presented in Section 7.

13 Note that in (6.1)–(6.3) we use the expansion parameter αε defined in (2.7).

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34 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

6.1. One-loop contributions

At one loop, the mass counterterms δM(1)V can be neglected, since the gauge-boson mass terms

in the Born amplitude give only contributions of order M2V /Q2, which are suppressed in the high-

energy limit, and the same holds for the contributions resulting from their renormalization.The electroweak couplings are renormalized in the MS scheme with an additional subtraction

of the UV singularities as explained in Section 3.3. Assuming that the renormalization scale14

μR is of the order of or larger than MW, this yields the counterterms

(6.4)δg(1)i

NLL= −gi

2

1

εb

(1)i

[(Q2

μ2R

− 1

], δe(1) NLL= − e

2

1

εb(1)e

[(Q2

μ2R

− 1

],

and the one-loop β-function coefficients b(1)1 , b

(1)2 , and b

(1)e are defined in Appendix C. The de-

pendence of the counterterms (6.4) on the factor (Q2/μR)ε is due to the normalization of theexpansion parameter αε in (6.2). As explained in Section 3.3, the expressions (6.4) are obtainedby subtracting the UV poles from the usual MS counterterms. Since the same subtraction is per-formed in the bare loop diagrams, the resulting renormalized amplitudes correspond to the usualMS renormalized amplitudes. The renormalization of the mixing parameters cw and sw can bedetermined from the renormalization of the coupling constants via (2.22). In NLL approxima-tion, this prescription is equivalent to using the on-shell renormalization condition, i.e. relation(2.21), for the weak mixing angle.

Here and in the following we assume that the Born amplitude M0 is expressed in terms ofcoupling constants renormalized at the scale μR = Q. As a consequence, the contribution of thecounterterms (6.4) to the one-loop amplitude M1 vanishes.

The only one-loop counterterm contribution arises from the on-shell wave-function renormal-ization constants δZ

(1)k for the massless fermionic external legs. These receive contributions only

from massive weak bosons, whereas the photonic contribution vanishes owing to a cancellationbetween UV and mass singularities within dimensional regularization. After subtraction of theUV poles we find

(6.5)δZ(1)k

NLL= −1

ε

{ ∑V =Z,W±

IVk IV

k

[(Q2

M2V

− 1

]− IA

k IAk

}.

Finally, the one-loop counterterm for a process with n external massless fermions in NLL ap-proximation is obtained as

(6.6)MWF1 =M0

n∑k=1

1

2δZ

(1)k .

6.2. Two-loop contributions

At two loops, the mass renormalization leads to non-suppressed logarithmic terms onlythrough the insertion of the one-loop counterterms δM

(1)V in the one-loop logarithmic corrections.

However, these contributions are of NNLL order and can thus be neglected in NLL approxima-tion [28]. In this approximation also the purely two-loop counterterms that are associated with

14 We do not identify the renormalization scale μR and the scale of dimensional regularization μD.

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 35

the renormalization of the external-fermion wave functions and the couplings, i.e. δZ(2)k , δg

(2)i

and δe(2), do not contribute.The only NLL two-loop counterterm contributions are those that result from the combination

of the one-loop amplitude with the one-loop counterterms δZ(1)k , δg

(1)i and δe(1). The wave-

function counterterms yield

(6.7)MWF2 = MF

1

n∑k=1

1

2δZ

(1)k .

The wave-function renormalization constants δZ(1)k and the unrenormalized one-loop amplitude

MF1 are given in (6.5) and in (D.1), respectively. In NLL approximation the one-loop counter-

terms MWF1 do not contribute, and only the LL part of the one-loop amplitude MF

1 is relevantfor (6.7). This is easily obtained from (D.1) by neglecting the NLL terms depending on rij andlZ, and using global gauge invariance (2.30) as

(6.8)e2MF1

LL= 1

2M0

n∑i=1

{ ∑V =A,Z,W±

e2IVi IV

i D0(MW;−Q2)+ e2IA

i IAi �D0

(0;−Q2)}

(6.9)LL= 1

2M0

n∑i=1

{[g2

1

(Yi

2

)2

+ g22Ci

]D0(MW;−Q2)+ e2Q2

i �D0(0;−Q2)}.

In the second line we used the identity (2.16). Note that only the LL contributions of D0 and�D0 contribute in (6.8) and (6.9).

The remaining NLL two-loop counterterms result from the insertion of the one-loop coupling-constant counterterms (6.4) in the LL one-loop amplitude (6.9) and read

e2MPR2

NLL= − 1

[(Q2

μ2R

− 1

]M0

n∑i=1

{[g2

1b(1)1

(Yi

2

)2

+ g22b

(1)2 Ci

]

(6.10)× D0(MW;−Q2)+ e2b(1)

e Q2i �D0

(0;−Q2)}.

The various one-loop β-function coefficients in (6.10), b(1)1 , b

(1)2 , and b

(1)e , are defined in Appen-

dix C.

7. Complete one- and two-loop results

In this section we combine the unrenormalized and the counterterm contributions presentedin Sections 5 and 6 and provide explicit NLL results for the renormalized one- and two-loopn-fermion amplitudes. As we have seen in Sections 5 and 6, the NLL corrections factorize, i.e.they can be expressed through correction factors that multiply the Born amplitude. Moreover,we find that the two-loop correction factors can be entirely expressed in terms of one-loop quan-tities. In this section we concentrate on the results. Details on the combination of the variouscontributions can be found in Appendices D and E.

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36 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

7.1. Renormalized one-loop amplitude

The renormalized one-loop matrix element for a process with n external massless fermions isgiven by

(7.1)M1 = MF1 + MWF

1 ,

where

(7.2)MF1 = 1

2

n∑i=1

n∑j=1j �=i

Mij

1

represents the bare one-loop contribution, which is given by the factorizable part (3.5), and MWF1

is the wave-function-renormalization counterterm (6.6). Using the explicit results presented inSection 5.1 we can write (more details can be found in Appendix D)

(7.3)M1NLL= M0

[F sew

1 + �F em1 + �F Z

1

].

Here the corrections are split into a symmetric-electroweak (sew) part,

(7.4)F sew1 = −1

2

n∑i=1

n∑j=1j �=i

∑V =A,Z,W±

IVi IV

j I (ε,MW;−rij ),

which is obtained by setting the masses of all gauge bosons, A,Z and W±, equal to MW in theloop diagrams, an electromagnetic (em) part

(7.5)�F em1 = −1

2

n∑i=1

n∑j=1j �=i

IAi IA

j �I (ε,0;−rij ),

resulting from the mass gap between the W boson and the massless photon, and an MZ-dependentpart

(7.6)�F Z1 = −1

2

n∑i=1

n∑j=1j �=i

IZi IZ

j �I (ε,MZ;−rij ),

describing the effect that results from the difference between MW and MZ. For the functions I ,including contributions up to the order ε2, we obtain

I (ε,MW;−rij )NLL= −L2 − 2

3L3ε − 1

4L4ε2

+ (3 − 2lij )

(L + 1

2L2ε + 1

6L3ε2

)+O

(ε3),

I (ε,MZ;−rij )NLL= I (ε,MW;−rij ) + lZ

(2L + 2L2ε + L3ε2)+O

(ε3),

(7.7)I (ε,0;−rij )NLL= −2ε−2 − (3 − 2lij )ε

−1,

and the subtracted functions �I are defined as

(7.8)�I (ε,m;−rij ) = I (ε,m;−rij ) − I (ε,MW;−rij ).

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 37

In LL approximation we have

(7.9)I (ε,m;−rij )LL= D0(m; rij ),

such that we can replace D0 and �D0 by I and �I , respectively, in (6.8), (6.9), and (6.10).

7.2. Renormalized two-loop amplitude

The renormalized two-loop matrix element reads

(7.10)M2 = MF2 + MWF

2 + MPR2 ,

where

MF2 =

n∑i=1

n∑j=1j �=i

[1

2

(M1,ij

2 + M2,ij

2

)+ M3,ij

2 + M4,ij

2 + M5,ij

2

(7.11)+ 1

2

11∑m=6

Mm,ij

2 +n∑

k=1k �=i,j

(M12,ijk

2 + 1

6M13,ijk

2 + 1

8

n∑l=1

l �=i,j,k

M14,ijkl

2

)]

represents the bare two-loop contribution, which is given by the factorizable terms (3.8), andMWF

2 and MPR2 are the wave-function- and parameter-renormalization counterterms given in

(6.7) and (6.10), respectively. As shown in Appendix E, the two-loop amplitude (7.10) can beexpressed in terms of the Born matrix element M0 and the one-loop correction factors (7.4)–(7.6) as

M2NLL= M0

{1

2

[F sew

1

]2 + F sew1 �F em

1 + 1

2

[�F em

1

]2 + F sew1 �F Z

1 + �F Z1 �F em

1

(7.12)+ Gsew2 + �Gem

2

},

where the additional terms

e2Gsew2 = 1

2

n∑i=1

[b

(1)1 g2

1

(Yi

2

)2

+ b(1)2 g2

2Ci

]J(ε,MW,μ2

R

),

(7.13)�Gem2 = 1

2

n∑i=1

Q2i

{b(1)e

[�J

(ε,0,μ2

R

)− �J(ε,0,M2

W

)]+ b(1)QED�J

(ε,0,M2

W

)}contain one-loop β-function coefficients, defined in Appendix C, and the combinations

J(ε,m,μ2

R

)= 1

ε

[I(2ε,m,Q2)−

(Q2

μ2R

I(ε,m,Q2)],

(7.14)�J(ε,m,μ2

R

)= J(ε,m,μ2

R

)− J(ε,MW,μ2

R

)of one-loop I -functions (7.7), (7.8) for m = MW,MZ,0. The relevant J -functions read explicitly

J(ε,MW,μ2

R

) NLL= 1

3L3 − lμRL2 +O(ε),

�J(ε,0,M2

W

) NLL= 3ε−3 + 2Lε−2 + L2ε−1 +O(ε),

2

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38 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

�J(ε,0,μ2

R

)− �J(ε,0,M2

W

) NLL= lμR

(−2ε−2 + ε−1(lμR − 2L)

(7.15)+ lμRL − 1

3l2μR

)+O(ε),

where

(7.16)lμR = ln

(μ2

R

M2W

).

In order to be able to express (7.12) in terms of the one-loop operators (7.4)–(7.6) it is crucialthat terms up to order ε2 are included in the latter.

The coefficients b(1)e and b

(1)QED describe the running of the electromagnetic coupling above

and below the electroweak scale, respectively. The former receives contributions from all chargedfermions and bosons, whereas the latter receives contributions only from light fermions, i.e. allcharged leptons and quarks apart from the top quark.

The couplings that enter the one- and two-loop correction factors15 are renormalized at thescale μR. Instead, as discussed in Section 6.1, the coupling constants in the Born matrix elementM0 in (7.3) and (7.12) are renormalized at the scale Q, i.e.

(7.17)M0 ≡M0| gi=gi (Q2)

e=e(Q2)

with

g2i

(Q2) NLL= g2

i

(μ2

R

)[1 − αε

4πb

(1)i

1

ε

[(Q2

μ2R

− 1

]],

(7.18)e2(Q2) NLL= e2(μ2R

)[1 − αε

4πb(1)e

1

ε

[(Q2

μ2R

− 1

]].

Thus, by definition, the Born amplitude M0 is independent of the renormalization scale μR, andthe dependence of the one- and two-loop amplitudes on μR is described by the terms (7.13).

The contributions (7.13) originate from combinations of UV and mass singularities. We ob-serve that the term proportional to b

(1)e vanishes for μR = MW. Instead, the terms proportional to

b(1)1 , b

(1)2 , and b

(1)QED cannot be eliminated through an appropriate choice of the renormalization

scale. This indicates that such two-loop terms do not originate exclusively from the running ofthe couplings in the one-loop amplitude.

Combining the Born amplitude with the one- and two-loop NLL corrections we can write

(7.19)M NLL= M0FsewF ZF em,

where we observe a factorization of the symmetric-electroweak contributions,

(7.20)F sew NLL= 1 + αε

4πF sew

1 +(

αε

)2[1

2

(F sew

1

)2 + Gsew2

],

15 These are the coupling α in the perturbative expansion (2.5)–(2.7) and the couplings g1, g2 and e that appear in(7.12)–(7.14) and enter also (7.4)–(7.6) through the dependence of the generators (2.13) on the couplings and the mixingparameters cw and sw.

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 39

the terms resulting from the difference between MW and MZ,

(7.21)F Z NLL= 1 + αε

4π�F Z

1 ,

and the electromagnetic terms resulting from the mass gap between the photon and the W boson,

(7.22)F em NLL= 1 + αε

4π�F em

1 +(

αε

)2[1

2

(�F em

1

)2 + �Gem2

].

We also observe that the symmetric-electroweak and electromagnetic terms are consistent withthe exponentiated expressions

F sew NLL= exp

[αε

4πF sew

1 +(

αε

)2

Gsew2

],

(7.23)F em NLL= exp

[αε

4π�F em

1 +(

αε

)2

�Gem2

].

In particular, these two contributions exponentiate separately. This double-exponentiating struc-ture is indicated by the ordering of the one-loop operators F sew

1 and �F em1 in the interference

term F sew1 �F em

1 in our result (7.12). It is important to realize that the commutator of thesetwo operators yields a non-vanishing NLL two-loop contribution. This means that the double-exponentiated structure of the result is not equivalent to a simple exponentiation, i.e.

(7.24)F sewF em �= exp

[αε

(F sew

1 + �F em1

)+(

αε

)2(Gsew

2 + �Gem2

)].

Instead we observe that in NLL approximation the commutator of �F Z1 with all operators in

(7.19)–(7.22) vanishes, and also [�F Z1 ]2 does not contribute. Thus, we have

(7.25)F sewF ZF em NLL= F ZF sewF em NLL= F sewF emF Z,

and, in principle, F Z can be written in exponentiated form,

(7.26)F Z NLL= exp

(αε

4π�F Z

1

),

or absorbed in one of the other exponentials,

F sewF Z NLL= exp

[αε

(F sew

1 + �F Z1

)+(

αε

)2

Gsew2

],

(7.27)F ZF em NLL= exp

[αε

(�F em

1 + �F Z1

)+(

αε

)2

�Gem2

].

The one- and two-loop corrections (7.3)–(7.6) and (7.12)–(7.13) contain various combinationsof matrices IV

i , which are in general non-commuting and non-diagonal. These matrices have tobe applied to the Born amplitude M0 according to the definition (2.28). In order to express theresults in a form which is more easily applicable to a specific process, it is useful to split theintegrals I (ε,MV ;−rij ) and �I (ε,MV ;−rij ) in (7.4)–(7.6) into an angular-independent partI (ε,MV ;Q2) and �I (ε,MV ;Q2), which only involves ε and L, and an angular-dependent part,which additionally depends on logarithms of rij . This permits to eliminate the sum over j for the

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40 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

angular-independent parts of (7.4)–(7.6) using the charge-conservation relation (2.30),

(7.28)M0

n∑i=1

n∑j=1j �=i

I Vi IV

j I(ε,MV ;Q2)= −M0

n∑i=1

IVi IV

i I(ε,MV ;Q2).

Moreover, one can easily see that the angular-independent part of (7.4) leads to the Casimiroperator (2.16). After these simplifications, all operators that are associated with the angular-independent parts can be replaced by the corresponding eigenvalues, and the one- and two-loopresults can be written as

(7.29)M1NLL= M0

[f sew

1 + �f em1 + �f Z

1

],

and

M2NLL= M0

{1

2

[f sew

1

]2 + f sew1 �f em

1 + 1

2

[�f em

1

]2 + f sew1 �f Z

1

(7.30)+ �f Z1 �f em

1 + gsew2 + �gem

2

},

with

f sew1

NLL= −1

2

(L2 + 2

3L3ε + 1

4L4ε2 − 3L − 3

2L2ε − 1

2L3ε2

) n∑i=1

[g2

1

e2

(yi

2

)2

+ g22

e2ci

]

+(

L + 1

2L2ε + 1

6L3ε2

)Kad

1 +O(ε3),

�f em1

NLL= −1

2

(2ε−2 + 3ε−1 − L2 − 2

3L3ε − 1

4L4ε2 + 3L + 3

2L2ε + 1

2L3ε2

) n∑i=1

q2i

−(

ε−1 + L + 1

2L2ε + 1

6L3ε2

) n∑i=1

n∑j=1j �=i

lij qiqj +O(ε3),

�f Z1

NLL=(

L + L2ε + 1

2L3ε2

)lZ

n∑i=1

(g2

ecwt3

i − g1

esw

yi

2

)2

+O(ε3),

gsew2

NLL=(

1

6L3 − 1

2lμRL2

) n∑i=1

[g2

1

e2b

(1)1

(yi

2

)2

+ g22

e2b

(1)2 ci

]+O(ε),

�gem2

NLL={−lμR

[ε−2 +

(L − 1

2lμR

)ε−1 − lμR

(1

2L − 1

6lμR

)]b(1)e

(7.31)+(

3

4ε−3 + Lε−2 + 1

2L2ε−1

)b

(1)QED

} n∑i=1

q2i +O(ε),

where lμR = ln (μ2R/M2

W), and ci , t3i , yi , qi , represent the eigenvalues of the operators Ci , T 3

i ,Yi , and Qi , respectively. The only matrix-valued expression in (7.31) is the angular-dependent

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 41

part of the symmetric-electroweak contribution f sew1 ,

(7.32)Kad1 =

n∑i=1

n∑j=1j �=i

lij∑

V =A,Z,W±IVi IV

j .

The two-loop corrections (7.30) involve terms proportional to Kad1 and [Kad

1 ]2. However, thelatter are of NNLL order and thus negligible in NLL approximation. The combination of thematrix (7.32) with the Born amplitude,

M0Kad1 =

n∑i=1

n∑j=1j �=i

lij∑

V =A,Z,W±M

ϕ1···ϕ′i ···ϕ′

j ···ϕn

0 IVϕ′

iϕiI Vϕ′

j ϕj

=n∑

i=1

n∑j=1j �=i

lij

{Mϕ1···ϕi ···ϕj ···ϕn

0

[g2

1

e2

yiyj

4+ g2

2

e2t3i t3

j

]

(7.33)+∑

V =W±M

ϕ1···ϕ′i ···ϕ′

j ···ϕn

0 IVϕ′

iϕiI Vϕ′

j ϕj

},

requires the evaluation of matrix elements involving SU(2)-transformed external fermions ϕ′i ,

ϕ′j , i.e. isospin partners of the fermions ϕi , ϕj .

8. Discussion

In this section we compare our results presented in Section 7 to existing results from theliterature and apply them to specific processes.

8.1. Extension of previous results

In Ref. [27] the one- and two-loop LL and angular-dependent NLL contributions have beencalculated for arbitrary non-mass-suppressed Standard Model processes, using a photon mass andfermion masses for the regularization of soft and collinear singularities, respectively. The purelysymmetric-electroweak parts of our results, made up of F sew

1 or f sew1 and its square, confirm the

corresponding terms involving δsew in Ref. [27], which can be seen e.g. by comparing (7.29),(7.30) and (7.31) from our paper with (4.9), (4.10) and (4.24) from Ref. [27]. In addition to theexisting results we have added all the remaining (non-angular-dependent) NLL contributions,including those proportional to ln(M2

Z/M2W) and the terms involving β-function coefficients, as

well as higher orders in ε in the one-loop results. We cannot compare the electromagnetic partscontained in �F em

1 or �f em1 and δsem because of the different regularization schemes for the

photonic singularities, i.e. for the soft and collinear divergences resulting from massless photons.In Ref. [28] the complete one- and two-loop LL and NLL contributions for the electroweak

singlet form factors have been derived. We can easily reproduce these form factor contributions asa special case of our results, as for only two external fermions no angular-dependent logarithmsappear and the summation over external legs, using (7.28), is trivial. The functions I , �I , J ,and �J appearing in (4.69), (4.70), and (4.73) of Ref. [28] correspond to the equally namedfunctions from this paper for rij = −Q2 = s. In the form factor case, due to the absence of theangular-dependent term Kad in (7.31), the one-loop operators F sew and �F em commute, so that

1 1 1
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42 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

the two-loop result can be written as a single exponential and we have M2NLL= M0{ 1

2 [F sew1 +

�F em1 + �F Z

1 ]2 + Gsew2 + �Gem

2 }, corresponding to the form in Ref. [28]. Note that we haverenormalized all couplings of the loop corrections at the unique scale μw = μe = μR, where thelast two lines of (4.73) in Ref. [28] vanish.

8.2. Comparison to Catani’s formula in QCD

The structure of our results for electroweak logarithmic corrections is similar to the singularstructure of scattering amplitudes in QCD. In Ref. [35] the singular part of a QCD one-loopamplitude, i.e. the pole part with terms ε−2 and ε−1, is obtained from the Born amplitude byapplying an operator I (1)(ε,μ2; {p}) which, for only fermions and antifermions as external par-ticles, can be expressed through Eqs. (12)–(15) of Ref. [35] as

(8.1)αS(μ2

R)

2πI (1)

(ε,μ2

R; {p}) NLL= αS,ε

(−1

2

)∑i

∑j �=i

T i · T j I (ε,0;−rij ),

where I (ε,0;−rij ) is the function defined in (7.7), and αS,ε represents the strong couplingrenormalized at μ2

R and rescaled as in (2.7). The product T i · T j of the color charge opera-

tors corresponds to∑

V =A,Z,W± IVi IV

j for electroweak interactions, so that (8.1) has exactly theform of F sew

1 (7.4) for MW = 0, including the angular-dependent logarithms. Note that the factor(Q2/μ2

R)ε from the difference in the definitions of αS [35] and αε (2.7),

(8.2)αS(μ2

R

)= αS,ε

(Q2

μ2R

,

is multiplied by a factor (−μ2R/rij )

ε from I (1)(ε,μ2R; {p}), producing the correct angular-

dependent logarithms lij = ln(−rij /Q2) contained in I (ε,0;−rij ). The ε-dependent pre-factor

in the definition of I (1)(ε,μ2R; {p}), e−εψ(1)/Γ (1 − ε) = 1 + O(ε2), is irrelevant in NLL accu-

racy, and the same holds for the corresponding factors in the two-loop result.In two loops, the operator acting on the Born amplitude is given by Catani’s formula, Eqs.

(18)–(21) of Ref. [35], in NLL accuracy as

(8.3)

(αS(μ2

R)

)2{1

2

[I (1)

(ε,μ2

R; {p})]2 + 2πβ0

ε

[I (1)

(2ε,μ2

R; {p})− I (1)(ε,μ2

R; {p})]}.

The first term in the curly brackets originates from the combination of I (1)(ε,μ2R; {p}) applied

to the one-loop amplitude and − 12 [I (1)(ε,μ2

R; {p})]2 acting on the Born amplitude in Ref. [35].It can be identified with the term 1

2 [F sew1 ]2 from our two-loop result (7.12). Using color conser-

vation, the second term gives(αS(μ2

R)

)2 2πβ0

ε

[I (1)

(2ε,μ2

R; {p})− I (1)(ε,μ2

R; {p})]

(8.4)NLL=

(αS,ε

)2

4πβ0 · 1

2

∑i

T 2i J(ε,0,μ2

R

),

where J (ε,0,μ2R) is the function defined in (7.14). In the electroweak model, the expression

4πβ0T2i in (8.4) corresponds directly to the term [b(1)

1 g21(Yi/2)2 +b

(1)2 g2

2Ci]/e2 in (7.13). There-fore (8.4) can be identified with the contribution of Gsew for MW = 0.

2
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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 43

The symmetric-electroweak part of our results can thus be obtained from Catani’s formula byan obvious replacement of gauge-group quantities together with a simple substitution of masslessone-loop integrals by massive ones. The remaining parts of our results, which are due to thedifferences between the masses of the photon, Z boson, and W boson and which cannot beinferred from Catani’s formula, can be expressed as simple combinations of the same one-loopintegrals.

8.3. Comparison to electroweak resummation results

In Refs. [18–21] a resummation of electroweak one-loop results has been proposed, includingall LL and NLL corrections apart from the ln(M2

Z/M2W) terms.

The non-angular-dependent O(ε0)-terms of f sew1 are found in Ref. [18] in the sum

∑nf

k=1 ofEq. (48) [Eq. (49) in the hep-ph version] for the contribution of external fermionic lines abovethe weak scale. These expressions are extended in Eq. (48) of Ref. [20] in order to include theβ-function terms of gsew

2 . Here care must be taken to use the resummed one-loop running of thecouplings, i.e. α(s) = α(M2)/(1 + c ln s/M2) with c = α(M2)β0/π , in order to correctly arriveat

(8.5)1

cln

s

M2

(ln

α(M2)

α(s)− 1

)+ 1

c2ln

α(M2)

α(s)= 1

2ln2 s

M2− c

6ln3 s

M2+O

(c2),

and equivalently for α → α′, c → c′, reproducing the coefficient of the β-function terms in gsew2

(7.31) for μR = MW.The angular-dependent NLL corrections are treated in Ref. [21]. The last line of Eq. (13)

there [Eq. (11) in the hep-ph version] matches the Kad1 -term of (7.31) and (7.32) in O(ε0) if one

takes all gauge-boson masses to be mVa = MW. The symmetric-electroweak parts of the one-and two-loop amplitudes in Ref. [21] are thus in agreement with our results.

To summarize, our explicit one- and two-loop results (7.29), (7.30) and (7.31) confirm thesymmetric-electroweak parts of the resummed ansatz. The electromagnetic contributions, i.e. thecontributions from below the weak scale, cannot be compared due to the different regulariza-tion schemes for the photonic singularities. However, the factorization of the electromagneticcontributions from the symmetric-electroweak ones and the fact that the former can be ex-pressed in terms of one-loop QED corrections are in agreement with the approach proposedin Refs. [18–21].

8.4. Four-fermion scattering processes

We now apply our results to massless four-fermion processes

(8.6)ϕ1(p1)ϕ2(p2) → ϕ3(−p3)ϕ4(−p4),

where each of the ϕi may be a massless fermion, ϕi = fκiσi

, or antifermion, ϕi = fκiσi

, with thenotations from Section 2, provided that the number of fermions and antifermions in the initialand final state is equal. The scattering amplitudes for the processes (8.6) follow directly from ourresults for the generic n → 0 process (2.1) by crossing symmetry. The Mandelstam invariants aregiven by s = r12 = r34, t = r13 = r24, and u = r14 = r23 with rij = (pi + pj )

2.The following discussion focuses on s-channel processes of the form

(8.7)f κσ f κ

ρ → f κ ′′ f κ ′

σ ρ
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44 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

with one fermion and one antifermion in the initial as well as in the final state, where f κσ and

f κ ′σ ′ are neither identical nor isospin partners of each other, and the same holds for f κ

ρ and f κ ′ρ′ .

Therefore the external fermion lines are always connected between f κσ and f κ

ρ in the initial state

and between f κ ′σ ′ and f κ ′

ρ′ in the final state. The number of independent chiralities κi is thusrestricted to two, κ and κ ′, and the particle pairs in the initial and final state must either be an-tiparticles of each other or antiparticles of the mutual isospin partners. The first case correspondsto neutral-current four-fermion scattering, which is treated in Section 8.4.1. The second caserefers to charged-current four-fermion scattering and is treated in Section 8.4.2.

The scattering amplitudes of all other processes (8.6) can be obtained from (8.7) by crossingsymmetry and additive combinations of amplitudes. For instance, the full four-quark amplitudefor uu → dd is given by the sum of the s-channel neutral-current amplitude and the t -channelcharged-current amplitude, where the latter follows from the result of the s-channel charged-current amplitude for ud → ud by exchanging p2 and p3, and thus s and t . The neutral- andcharged-current results in Sections 8.4.1 and 8.4.2 together are thus sufficient to construct thescattering amplitudes for all massless four-fermion processes (8.6).

8.4.1. Neutral-current four-fermion scatteringThis section deals with the s-channel neutral-current four-fermion processes

(8.8)f κσ f κ

σ → f κ ′σ ′ f κ ′

σ ′ ,

where a fermion–antifermion pair annihilates and produces another fermion–antifermion pair.The electromagnetic charge quantum numbers of the external particles are given by qf = qf κ

σ=

−qf κσ

and qf ′ = qf κ′

σ ′= −q

f κ′σ ′

, the hypercharges by yf = yf κσ

= −yf κσ

and yf ′ = yf κ′

σ ′= −y

f κ′σ ′

,

the isospin components by t3f = t3

f κσ

= −t3f κ

σ

and t3f ′ = t3

f κ′σ ′

= −t3f κ′

σ ′, and the isospin by tf = |t3

f |,tf ′ = |t3

f ′ |. These electroweak quantum numbers depend on the flavor and the chirality of thefermions.

The Born amplitude reads

(8.9)M0,NC = 1

sANCC0,NC

in the high-energy limit, where

(8.10)ANC = v(p2, κ)γ μu(p1, κ)u(−p3, κ′)γμv(−p4, κ

′)

represents the spinor structure of the incoming and outgoing massless fermions with chiralitiesκ and κ ′, and

(8.11)C0,NC = g21

(Q2)yf yf ′

4+ g2

2

(Q2)t3

f t3f ′ .

As indicated, the couplings gi(Q2) in the Born amplitude are renormalized at the scale Q,

whereas the additional couplings and mixing angles in the loop corrections below are renor-malized at the scale μR.

We write the neutral-current amplitude in the form

(8.12)MNCNLL= M0,NCF sew

NC F ZNCF em

NC

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 45

according to (7.19). Applying the non-diagonal operator F sew (7.20) to the Born amplitude, wefind

M0,NCF sewNC

NLL= 1

sANC

{C0,NC + αε

[−(

L2 + 2

3L3ε + 1

4L4ε2 − 3L

− 3

2L2ε − 1

2L3ε2

)C0,NCCsew

1,NC

+(

L + 1

2L2ε + 1

6L3ε2

)Cad

1,NC +O(ε3)]

+(

αε

)2[(1

2L4 − 3L3

)C0,NC

(Csew

1,NC

)2 − L3Cad1,NCCsew

1,NC

(8.13)+ C0,NCgsew2,NC +O(ε)

]},

where

(8.14)Csew1,NC = g2

1

e2

y2f

4+ g2

2

e2tf (tf + 1) + (f ↔ f ′)

results from the non-angular-dependent contributions to f sew1 in (7.31), whereas the application

of the angular-dependent operator Kad1 on the Born amplitude yields

Cad1,NC = C0,NC

[4 ln

(u

t

)(g2

1

e2

yf yf ′

4+ g2

2

e2t3f t3

f ′

)− 2 ln

(−s

Q2

)Csew

1,NC

]

(8.15)+ 2g22

(Q2)g2

2

e2

[ln

(u

t

)tf tf ′ −

(ln

(t

s

)+ ln

(u

s

))t3f t3

f ′

].

The last missing part in (8.13),

(8.16)gsew2,NC

NLL=(

1

3L3 − lμRL2

)[g2

1

e2b

(1)1

y2f

4+ g2

2

e2b

(1)2 tf (tf + 1) + (f ↔ f ′)

]+O(ε),

results directly from (7.31), where the values for the β-function coefficients b(1)1 and b

(1)2 in the

electroweak Standard Model are given in (C.7).The symmetric-electroweak result (8.13) is multiplied with the diagonal factors

(8.17)F ZNC

NLL= 1 + αε

4π�f Z

1,NC

and

(8.18)F emNC

NLL= 1 + αε

4π�f em

1,NC +(

αε

)2[1

2

(�f em

1,NC

)2 + �gem2,NC

].

The factors F ZNC and F em

NC follow from (7.31),

(8.19)

�f Z1,NC

NLL= 2

(L + L2ε + 1

2L3ε2

)lZ

[(g2

ecwt3

f − g1

esw

yf

2

)2

+ (f ↔ f ′)]

+O(ε3),

�f em1,NC

NLL= −(

2ε−2 + 3ε−1 − L2 − 2L3ε − 1

L4ε2 + 3L

3 4
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46 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

+ 3

2L2ε + 1

2L3ε2

)(q2f + q2

f ′)

−(

ε−1 + L + 1

2L2ε + 1

6L3ε2

)

(8.20)×[

4 ln

(u

t

)qf qf ′ − 2 ln

(−s

Q2

)(q2f + q2

f ′)]+O

(ε3),

�gem2,NC

NLL={−lμR

[2ε−2 + (2L − lμR)ε−1 − lμR

(L − 1

3lμR

)]b(1)e

(8.21)+(

3

2ε−3 + 2Lε−2 + L2ε−1

)b

(1)QED

}(q2f + q2

f ′)+O(ε),

where the values for the β-function coefficients b(1)e and b

(1)QED in the electroweak Standard Model

are given in (C.7) and (C.8).Our result can be compared to Refs. [22,23,31], where the logarithmic two-loop contributions

to neutral-current four-fermion scattering have been determined by resummation techniques.These papers use the renormalization scale μR = MW for the couplings in the loop corrections,so that we have to set lμR = ln(μ2

R/M2W) = 0 in (8.16) and (8.21). The large electroweak log-

arithms in Refs. [22,23,31] are defined with the choice Q2 = −s, such that ln(−s/Q2) = 0 in(8.15) and (8.20).

In Refs. [22,23,31] the photonic singularities are regularized with a finite photon mass.Therefore we cannot directly compare their electromagnetic corrections with our dimension-ally regularized result. But Refs. [22,23,31] define a finite scattering amplitude by factorizing thecomplete QED corrections,

(8.22)MfinNC = MNC

UQEDNC

.

The factor UQEDNC represents the full QED corrections from photons coupling exclusively to fermi-

ons, with the photonic singularities regularized in the same way as in the amplitude MNC. Thesoft and collinear divergences resulting from massless photons cancel in the ratio (8.22), andMfin

NC is independent of the regularization scheme for the photonic singularities. Using our di-

mensional regularization scheme and μR = MW, Q2 = −s as above, UQEDNC is given by

(8.23)UQEDNC

NLL= 1 + αε

4πf

QED1,NC +

(αε

)2[1

2

(f

QED1,NC

)2 + gQED2,NC

],

with

(8.24)fQED1,NC

NLL= −(2ε−2 + 3ε−1)(q2f + q2

f ′)− 4ε−1 ln

(u

t

)qf qf ′ ,

(8.25)

gQED2,NC

NLL=[(

3

2ε−3 + 2Lε−2 + L2ε−1 + 1

3L3)

b(1)QED + 1

3L3b

(1)top

](q2f + q2

f ′)+O(ε).

In contrast to F emNC (8.18), U

QEDNC , according to its definition in Refs. [22,23,31], contains the

complete photon contribution without the subtractions at a photon mass equal to MW. This iswhy (8.24) and (8.25) differ from (8.20) and (8.21) by finite logarithmic terms. The expressionsf

QED and gQED can be obtained from �F em (7.5) and �Gem (7.13) by replacing �I → I and

1,NC 2,NC 1 2
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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 47

�J → J and adding the term proportional to b(1)top in (8.25), where b

(1)top is defined in (C.9) and

corresponds to the top-quark contribution to the electromagnetic β-function. In our result, thislatter term is implicitly contained in gsew

2,NC (8.16) and, by construction, cancels in the subtracted

expression �gem2,NC (8.21). The factor U

QEDNC in (8.23) is valid for mt ∼ MW, whereas the cor-

responding factor in Refs. [23,31] is only valid for a massless top quark. However, the finiteamplitude Mfin

NC is independent of the top-quark mass in NLL approximation.The one- and two-loop contributions to Mfin

NC (8.22) can be expressed as follows:

Mfin1,NC

NLL= 1

sANC

{−C0,NC[(

L2 − 3L)Csew

1,NC − �f em1,NC + f

QED1,NC

]+ LCad

1,NC + C0,NC�f Z1,NC

}+O(ε),

Mfin2,NC

NLL= 1

sANC

{1

2C0,NC

[(L2 − 3L

)Csew

1,NC − �f em1,NC + f

QED1,NC

]2

− (LCad

1,NC + C0,NC�f Z1,NC

)(L2Csew

1,NC − �f em1,NC + f

QED1,NC

)(8.26)+ C0,NC

(gsew

2,NC + �gem2,NC − g

QED2,NC

)}+O(ε).

The one-loop result reproduces Eq. (50) from Ref. [22], and the two-loop result agrees withEqs. (51), (52) of Ref. [22] and Eqs. (51), (52), (54) of Ref. [23] for Ng = 3 families of leptonsand quarks. Note that in these papers (α/4π)(g2

2/e2) is used as the parameter of the perturbativeexpansion. The corrections proportional to �f Z

1,NC, which account for the mass difference ofthe heavy electroweak gauge bosons, MZ �= MW, have to be compared to Eq. (62) of Ref. [31],where the first order of an expansion in the parameter δM = s2

w = 1 − M2W/M2

Z is presented.Using lZ = ln(M2

Z/M2W) = s2

w +O(s4w), Ref. [31] gives indeed the first order in s2

w of our result,provided that we set sw = 0 and cw = 1 in �f Z

1,NC (8.19), thus neglecting higher orders in s2w.

8.4.2. Charged-current four-fermion scatteringIn order to complete our predictions for massless four-fermion scattering, we apply our results

to the s-channel charged-current processes

(8.27)f Lσ f L

ρ → f Lσ ′ f L

ρ′ ,

where the fermions f Lσ and f L

σ ′ are the isospin partners of f Lρ and f L

ρ′ , respectively. The hy-percharge quantum numbers of the external particles are given by yf = yf L

σ= −yf L

ρand

yf ′ = yf Lσ ′ = −yf L

ρ′ and the isospin components by t3 = t3f L

σ= t3

f Lρ

= t3f L

σ ′= t3

f Lρ′

. All external

fermions have to be left-handed, so t = |t3| = 1/2.In the high-energy approximation, the Born amplitude reads

(8.28)M0,CC = 1

sACC

g22(Q2)

2with the spinor structure

(8.29)ACC = v(p2,L)γ μu(p1,L)u(−p3,L)γμv(−p4,L).

As in the previous section, the amplitude is written in the form

(8.30)MCCNLL= M0,CCF sew

CC F ZCCF em

CC .

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48 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

Applying the non-diagonal operator F sew (7.20) to the Born amplitude, we find

M0,CCF sewCC

NLL= 1

sACC

{g2

2(Q2)

2+ αε

[−(

L2 + 2

3L3ε + 1

4L4ε2

− 3L − 3

2L2ε − 1

2L3ε2

)g2

2(Q2)

2Csew

1,CC

+(

L + 1

2L2ε + 1

6L3ε2

)Cad

1,CC +O(ε3)]

+(

αε

)2[(1

2L4 − 3L3

)g2

2(Q2)

2

(Csew

1,CC

)2

(8.31)− L3Cad1,CCCsew

1,CC + g22(Q2)

2gsew

2,CC +O(ε)

]},

with

Csew1,CC = g2

1

e2

y2f + y2

f ′

4+ 3

2

g22

e2,

Cad1,CC = g2

2(Q2)

2

[4 ln

(u

t

)g2

1

e2

yf yf ′

4− 2

(ln

(t

s

)+ ln

(u

s

))g2

2

e2

− 2 ln

(−s

Q2

)Csew

1,CC

]+ 2 ln

(u

t

)g2

1

(Q2)yf yf ′

4

g22

e2,

(8.32)gsew2,NC

NLL=(

1

3L3 − lμRL2

)(g2

1

e2b

(1)1

y2f + y2

f ′

4+ 3

2

g22

e2b

(1)2

).

The diagonal factors

(8.33)F ZCC

NLL= 1 + αε

4π�f Z

1,CC

and

(8.34)F emCC

NLL= 1 + αε

4π�f em

1,CC +(

αε

)2[1

2

(�f em

1,CC

)2 + �gem2,CC

]

are expressed through

�f Z1,CC

NLL=(

L + L2ε + 1

2L3ε2

)lZ

(g2

2

e2c2

w + 2g2

1

e2s2

w

y2f + y2

f ′

4

)+O

(ε3),

�f em1,CC

NLL= −(

2ε−2 + 3ε−1 − L2 − 2

3L3ε − 1

4L4ε2 + 3L + 3

2L2ε + 1

2L3ε2

)Cem

1,CC

−(

ε−1 + L + 1

2L2ε + 1

6L3ε2

)C

ad,em1,CC +O

(ε3),

�gem2,CC

NLL={−lμR

[2ε−2 + (2L − lμR)ε−1 − lμR

(L − 1

3lμR

)]b(1)e

(8.35)+(

3

2ε−3 + 2Lε−2 + L2ε−1

)b

(1)QED

}Cem

1,CC +O(ε),

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 49

with

Cem1,CC = 1

2

(q2f L

σ+ q2

f Lρ

+ q2f L

σ ′+ q2

f Lρ′

)= g21

e2c2

w

y2f + y2

f ′

4+ 1

2

g22

e2s2

w,

Cad,em1,CC = 2

[ln

(−s

Q2

)(qf L

σqf L

ρ+ qf L

σ ′ qf Lρ′ ) − ln

(−t

Q2

)(qf L

σqf L

σ ′ + qf Lρqf L

ρ′ )

− ln

(−u

Q2

)(qf L

σqf L

ρ′ + qf Lσ ′ qf L

ρ)

]

(8.36)= 4 ln

(u

t

)g2

1

e2c2

wyf yf ′

4−(

ln

(t

s

)+ ln

(u

s

))g2

2

e2s2

w − 2 ln

(−s

Q2

)Cem

1,CC.

9. Conclusion

We have studied the one- and two-loop virtual electroweak corrections to arbitrary processeswith external massless fermions in the Standard Model. In the high-energy region, where all kine-matical invariants are at an energy scale Q that is large compared to the electroweak gauge-bosonmasses, we have calculated mass singularities in D = 4 − 2ε dimensions taking into account allleading logarithmic (LL) and next-to-leading logarithmic (NLL) contributions. This approxima-tion includes all combinations αlε−k lnj−k(Q2/M2

W) of mass-singular logarithms and 1/ε poleswith j = 2l,2l − 1 and 2l − 4 � k � j . All masses of the heavy particles have been assumedto be of the same order MW ∼ MZ ∼ MH ∼ mt but not equal, and all light fermions have beenassumed to be massless.

The calculation has been performed in the complete spontaneously-broken electroweak Stan-dard Model using the ’t Hooft–Feynman gauge. All contributions have been split into those thatfactorize the lowest-order matrix element and non-factorizable parts. The non-factorizable partshave been shown to vanish in NLL approximation owing to collinear Ward identities. All factoriz-able contributions have been evaluated using a suitable soft–collinear approximation and minimalsubtraction of the ultraviolet singularities. Explicit results have been given for all contributingfactorizable Feynman diagrams. The two-loop integrals have been solved by two independentmethods in NLL approximation. One makes use of sector decomposition to isolate the masssingularities, the other uses the strategy of regions. The fermionic wave functions are renormal-ized on shell, and coupling-constant renormalization is performed in the MS scheme, but can begeneralized easily.

In order to isolate the effects resulting from the mass gaps between the photon, the W boson,and the Z boson, all contributions have been split into parts corresponding to MA = MZ = MW

and remaining subtracted parts associated with the massless photon and the Z boson. By com-bining the results of all diagrams we found that the electroweak mass singularities assume aform that is analogous to the singular structure of scattering amplitudes in massless QCD. Thesum of the two-loop leading and next-to-leading logarithms is composed of terms that can bewritten as the second-order terms of exponentials of the one-loop contribution plus additionalNLL contributions that are proportional to the one-loop β-function coefficients. All terms canbe cast into a product of three exponentials. The first inner exponential contains the part of thecorrections corresponding to MA = MZ = MW, i.e. the SU(2) × U(1) symmetric part. The sec-ond exponential contains the part originating from the mass gap between the Z boson and the Wboson and contains only terms involving ln(M2

Z/M2W). The third outer exponential summarizes

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50 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

the contributions that originate from the mass gap between the photon and the W boson and cor-responds to the QED corrections subtracted by the corresponding corrections with MA = MW.While the second exponential commutes with the other two and does in fact get no second-ordercontribution in NLL approximation, the first and the last exponential do not commute.

If one neglects the NLL contributions proportional to ln(M2Z/M2

W), our result confirms the re-summation prescriptions that have been proposed in the literature. These prescriptions are basedon the assumption that, in the high-energy limit, the electroweak theory can be described by asymmetric, unmixed SU(2) × U(1) theory, where all electroweak gauge bosons have mass MW,matched with QED at the electroweak scale. Indeed, apart from the terms involving ln(M2

Z/M2W),

in the final result we observe a cancellation of all effects associated with symmetry breaking, i.e.gauge-boson mixing, the gap between MZ and MW, and couplings proportional to the vacuumexpectation value. This simple behavior of the two-loop NLL corrections is ensured by subtlecancellations of mass singularities from different diagrams, where the details of spontaneoussymmetry breaking cannot be neglected but have to be taken into account properly.

In massless fermionic processes, the symmetry-breaking effects are restricted to a small subsetof diagrams, since the Higgs sector is coupled to massless fermions only via one-loop insertionsin the gauge-boson self-energies. Using the techniques developed in this paper, which are to alarge extent process independent, we plan to extend our study of two-loop NLL mass singularitiesto processes involving heavy external particles that are directly coupled to the Higgs sector.

As an application of our results for general n-fermion processes we have presented explicitexpressions for the case of neutral-current and charged-current 4-fermion reactions. In the for-mer case we found agreement with existing predictions obtained with the help of resummationsprescriptions.

Our results are also applicable to reactions that involve massless fermions and (hard) gluons,such as 2-jet production at hadron colliders, since gluons do not couple to electroweak gaugebosons.

Appendix A. Loop integrals

In this appendix, we list the explicit expressions for the Feynman integrals that contribute tothe one- and two-loop diagrams discussed in Sections 5.1 and 5.2. In order to keep our expres-sions as compact as possible we define the momenta

k1 = pi + l1, k2 = pi + l2, k3 = pi + l1 + l2,

q1 = pj − l1, q2 = pj − l2, q3 = pj − l1 − l2, l3 = −l1 − l2,

(A.1)r1 = pk − l1, r2 = pk − l2, r3 = pk − l1 + l2, l4 = l1 − l2.

For massive and massless propagators we use the notation

(A.2)P(q,m) = q2 − m2 + i0, P (q) = q2 + i0,

and for triple gauge-boson couplings we write

(A.3)Γ μ1μ2μ3(l1, l2, l3) = gμ1μ2(l1 − l2)μ3 + gμ2μ3(l2 − l3)

μ1 + gμ3μ1(l3 − l1)μ2 .

The normalization factors occurring in (2.7) are absorbed into the integration measure

(A.4)dli = (4π)2(

4πμ2D

eγEQ2

)D/2−2

μ4−DD

dDli

(2π)D= 1

π2

(eγEQ2π

)2−D/2 dDli,

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 51

and for the projection introduced in (3.24) we use the shorthand

(A.5)Πij (Γ ) = 1

rijTr(Γ ωκi

/pi/pj ) = 1

2rijTr(Γ /pi/pj ),

where the second equality holds if Γ does not involve γ5 or ωR,L, as it is the case in the followingequations. With this notation we have

D0(m1; rij ) =∫

dl14ik1q1

P(l1,m1)P (k1)P (q1),

D1(m1,m2; rij ) =∫

dl1 dl2−16(k1q1)(k3q3)

P (l1,m1)P (l2,m2)P (k1)P (k3)P (q1)P (q3),

D2(m1,m2; rij ) =∫

dl1 dl2−16(k1q3)(k3q2)

P (l1,m1)P (l2,m2)P (k1)P (k3)P (q2)P (q3),

D3(m1,m2,m3; rij ) =∫

dl1 dl2−2Πij (/k3γ

μ2/k1γμ1)q

μ33 Γμ1μ2μ3(l1, l2, l3)

P (l1,m1)P (l2,m2)P (l3,m3)P (k1)P (k3)P (q3),

D4(m1,m2; rij ) =∫

dl1 dl22Πij (/k1γ

μ2/k3γμ2/k1/q1)

P (l1,m1)P (l2,m2)[P(k1)]2P(k3)P (q1),

D5(m1,m2; rij ) =∫

dl1 dl22Πij (/k1γ

μ2/k3/q1/k2γμ2)

P (l1,m1)P (l2,m2)P (k1)P (k3)P (k2)P (q1),

D6(m1,m2,m3,m4; rij ) =∫

dl1 dl2−4k

μ11 q1μ4

P(l1,m1)P (l2,m2)P (l3,m3)P (l1,m4)P (k1)P (q1)

× [Γμ1μ2μ3(l1, l2, l3)Γ

μ4μ2μ3(l1, l2, l3) + 2l2μ1 lμ43

],

D7(m1,m2,m3; rij ) =∫

dl1 dl2−4k1q1

P(l1,m1)P (l2,m2)P (l1,m3)P (k1)P (q1),

D8(m1,m2,m3,m4; rij ) =∫

dl1dl2−4k1q1

P(l1,m1)P (l2,m2)P (l3,m3)P (l1,m4)P (k1)P (q1),

D9(m1,m2,m3,m4; rij ) =∫

dl1 dl24k

μ11 q

μ41 (l2 − l3)μ1(l2 − l3)μ4

P(l1,m1)P (l2,m2)P (l3,m3)P (l1,m4)P (k1)P (q1),

D10(m1,m2,m3; rij ) = D7(m1,m2,m3; rij ),D11,0(m1,m2,m3,m4; rij )

=∫

dl1 dl24k

μ11 q

μ41 Tr(γμ1/l2γμ4/l3)

P (l1,m1)P (l2,m2)P (l3,m3)P (l1,m4)P (k1)P (q1),

D11,m(m1,m2,m3,m4; rij ) = −4D8(m1,m2,m3,m4; rij ),D12(m1,m2; rik) =

∫dl1 dl2

−16(k1q1)(k3r2)

P (l1,m1)P (l2,m2)P (k1)P (k3)P (q1)P (r2),

D13(m1,m2,m3; rij , rik, rjk)

=∫

dl1 dl28k

μ11 q

μ22 r

μ33 Γμ1μ2μ3(−l1, l2, l4)

P (l1,m1)P (l2,m2)P (l4,m3)P (k1)P (q2)P (r3),

(A.6)D14(m1,m2; rij , rkl) = D0(m1; rij )D0(m2; rkl).

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52 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

Appendix B. Relations between loop integrals in NLL approximation

In the following we list relations between one- and two-loop integrals defined in Appendix A.These relations are valid after subtraction of the UV singularities and in NLL approximation.They have been obtained from the explicit results listed in Sections 5.1 and 5.2 and are employedin Appendix E in order to simplify the sum over all NLL two-loop contributions.

As in Section 5.2, also in the following the symbols mi are used to denote generic massparameters, which can assume the values mi = MW,MZ,mt,MH or mi = 0, and the symbols Mi

are used to denote non-zero masses, i.e. Mi = MW,MZ,mt,MH.Combinations of the 2-leg ladder integrals in (5.5) and (5.8) can be expressed as products of

one-loop integrals using

D1(m1,m2; rij ) + D2(m1,m2; rij ) + D1(m2,m1; rij ) + D2(m2,m1; rij )NLL= D0(m1; rij )D0(m2; rij )NLL= D0(MW; rij )D0(MW; rij ) + �D0(m1; rij )D0(MW; rij )

(B.1)+ D0(MW; rij )�D0(m2; rij ) + �D0(m1; rij )�D0(m2; rij ),where the subtracted functions �Dh have been defined in (5.4). For the relations in (B.1) it iscrucial that the results for D0(mk; rij ) including terms up to order ε2 are used. Moreover, D1and D2 fulfill the relations

D1(M1,m2; rij ) NLL= D1(M1,MW; rij ),(B.2)D2(m1,m2; rij ) = D2(m2,m1; rij ).

The loop integrals corresponding to the non-Abelian diagrams involving two external legs(5.10) can be replaced as

D3(M1,m2,m3; rij ) NLL= 1

2D2(m3,M1; rij ) − D4(m3,M1; rij )

− 6D9(m3,M1,M1,m3; rij ),�D3(MW,MW,m1; rij ) NLL= 1

2�D12(MW,m1; rij ) − �D4(m1,MW; rij )

− 6�D9(m1,MW,MW,m1; rij ),�D3(m1,MW,MW; rij ) NLL= �D3(MW,m1,MW; rij ) + �D1(MW,m1; rij )

+ �D2(MW,m1; rij ) + �D4(MW,m1; rij )− 1

2�D12(MW,m1; rij ),

(B.3)�D3(MW,m1,MW; rij ) NLL= −1

2�D1(MW,m1; rij ) − �D4(MW,m1; rij ).

The first of these relations has been verified and is needed only if at most one of the masses m2and m3 is zero.

The loop functions for the diagrams (5.13) and (5.16) are equal up to a minus sign

(B.4)D5(m1,m2; rij ) NLL= −D4(m1,m2; rij ).

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 53

Next we give relations for the loop integrals appearing in the diagrams with self-energy inser-tions in the gauge-boson line, (5.19), (5.23), (5.26), (5.29), (5.32), and (5.34). The loop integralsD6, D7, D8, and D11 can be related to D9 as

D6(M1,M2,M3,M4; rij ) NLL= 10D9(M1,M2,M3,M4; rij ),D11,0(M1,M2,M3,M4; rij ) NLL= 4D9(M1,M2,M3,M4; rij ),�D6(0,MW,MW,0; rij ) NLL= 10�D9(0,MW,MW,0; rij ),�D11,0(0,MW,MW,0; rij ) NLL= 4�D9(0,MW,MW,0; rij ),�D6(0,M2,M3,M4; rij ) NLL= �D6(M4,M2,M3,0; rij )

NLL= −12M2

2 + M23

M24

�D9(0,MW,MW,0; rij ),

�D7(0,M2,M3; rij ) NLL= �D7(M3,M2,0; rij ) NLL= −3M2

2

M23

�D9(0,MW,MW,0; rij ),

�D8(0,M2,M3,M4; rij ) NLL= �D8(M4,M2,M3,0; rij )NLL= − 3

M24

�D9(0,MW,MW,0; rij ),

�D9(0,M2,M3,M4; rij ) NLL= �D9(M4,M2,M3,0; rij )NLL= 3

M22 + M2

3

M24

�D9(0,MW,MW,0; rij ),

�D11,0(0,M2,M3,M4; rij ) NLL= �D11,0(M4,M2,M3,0; rij )

(B.5)NLL= 6

M22 + M2

3

M24

�D9(0,MW,MW,0; rij ).

The loop integral D9 can be expressed as

3D9(MW,MW,MW,MW; rij ) NLL= −J(ε,MW,Q2),

(B.6)3�D9(0,MW,MW,0; rij ) NLL= −[�J(ε,0,Q2)− �J

(ε,0,M2

W

)],

and the loop integral D11 as

(B.7)3[�D11,0(0,0,0,0; rij ) − �D11,0(0,MW,MW,0; rij )

] NLL= −4�J(ε,0,M2

W

)in terms of the functions J and �J defined in (7.14).

Combinations of 3-leg ladder integrals in (5.38) can be expressed as products of one-loopintegrals

D12(m1,m2; rij ) + D12(m2,m1; rik)NLL= D0(m2; rij )D0(m1; rik)NLL= D0(MW; rij )D0(MW; rik) + �D0(m2; rij )D0(MW; rik)

(B.8)+ D0(MW; rij )�D0(m1; rik) + �D0(m2; rij )�D0(m1; rik).

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54 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

For these relations again the terms of O(ε2) in D0(mk; rij ) are crucial. Furthermore

(B.9)∑

π(i,j,k)

sgn(π(i, j, k)

)D12(MW,MW; rik) NLL= 0,

where the sum runs over all permutations π(i, j, k) of i, j, k, with sign sgn(π(i, j, k)). Thisidentity simply follows from the fact that D12 is symmetric under the interchange of i and k anddoes not depend on rij and rjk .

The loop functions appearing in the non-Abelian diagrams involving three external legs (5.41)can be replaced as

D13(M1,M2,M3; rij ) NLL= 0,

2�D13(M1,M2,m3; rij , rik, rjk)NLL= 2�D13(m1,M2,M3; rik, rjk, rij )

NLL= 2�D13(M1,m2,M3; rjk, rij , rik)

(B.10)

NLL= �D12(MW,m3; rjk) − �D12(MW,m3; rik).

Appendix C. β-function coefficients

In this appendix we give relations and explicit expressions for the one-loop β-function coef-ficients b

(1)V1V2

, b(1)1 , b

(1)2 , b

(1)e , b

(1)QED, and b

(1)top that have been used in the calculation. For more

details we refer to Refs. [8,28].The matrix of β-function coefficients is defined as

(C.1)b(1)V1V2

= 11

3TrV

(IV 1IV2

)− 1

6TrΦ

(IV 1IV2

)− 2

3

∑Ψ

∑κ=R,L

TrΨ κ

(IV 1IV2

),

where IVi are the generators defined in (2.13) and TrV , TrΦ , and TrΨ denote the traces in therepresentations for the gauge bosons, scalars, and fermionic doublets, respectively. The sum

∑Ψ

runs over all doublets of leptons and quarks including different colors. The traces read moreexplicitly16

(C.2)e2 TrV(IV 1IV2

)= g22

∑V3,V4=A,Z,W±

εV 1V 3V 4εV2V3V4 ,

(C.3)e2 TrΦ(IV 1IV2

)= e2∑

Φi,Φj =H,χ,φ±I

V 1ΦiΦj

IV2Φj Φi

,

(C.4)e2 TrΨ κ

(IV 1IV2

)= e2∑

Ψi,Ψj =u,d

IV 1Ψ κ

i Ψ κjI

V2Ψ κ

j Ψ κi.

Multiplying b(1)V1V2

with generators and summing over all gauge bosons yields

(C.5)e2∑

V1,V2=A,Z,W±b

(1)V1V2

IV1i I

V 2i = g2

1b(1)1

(Yi

2

)2

+ g22b

(1)2 Ci,

16 Note that our normalization for the trace of the scalar fields differs from the one used in Refs. [8,28].

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 55

from which the coefficients corresponding to the weak couplings g1 and g2 can be read off. Thecoefficient corresponding to the electric-charge renormalization is given by

(C.6)b(1)e ≡ b

(1)AA = c2

wb(1)1 + s2

wb(1)2 .

The explicit values in the electroweak Standard Model are (YΦ = 1)

(C.7)b(1)1 = − 41

6c2w

, b(1)2 = 19

6s2w

, b(1)e = −11

3.

The QED β-function coefficient is determined by the light-fermion contributions only, i.e.

(C.8)b(1)QED = −4

3

∑f �=t

Nfc Q2

f = −80

9,

where Nfc represents the color factor, i.e. N

fc = 1 for leptons and N

fc = 3 for quarks.

The top-quark contribution to the electromagnetic β-function coefficient reads

(C.9)b(1)top = −16

9.

Appendix D. Summing up the one-loop contributions

In NLL approximation, the contribution of all bare one-loop diagrams to the matrix elementfor a process with n external massless fermions is given by (7.2), which results from the factor-izable diagrams (3.5). Using the explicit results presented in Section 5.1 we can write

(D.1)MF1

NLL= M0[F

F,sew1 + �F

F,em1 + �F

F,Z1

]with

FF,sew1 = −1

2

n∑i=1

n∑j=1j �=i

∑V =A,Z,W±

IVi IV

j D0(MW; rij ),

�FF,em1 = −1

2

n∑i=1

n∑j=1j �=i

IAi IA

j �D0(0; rij ),

(D.2)�FF,Z1 = −1

2

n∑i=1

n∑j=1j �=i

IZi IZ

j �D0(MZ; rij ).

Using the charge-conservation identity (2.30), the counterterms (6.6) can be cast into the form

(D.3)MWF1

NLL= 1

2M0

n∑i=1

n∑j=1j �=i

{ ∑V =A,Z,W±

IVi IV

j C(MW;Q2)+ IA

i IAj �C

(0;Q2)}

with

C(MW;Q2) NLL= L + 1

L2ε + 1L3ε2 +O

(ε3),

2 6

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56 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

(D.4)�C(0;Q2) NLL= −1

ε− L − 1

2L2ε − 1

6L3ε2 +O

(ε3).

When adding (D.1) and (D.3) we find (7.3)–(7.7) with

I (ε,MW;−rij )NLL= D0(MW; rij ) − C

(MW;Q2),

�I (ε,MZ;−rij )NLL= �D0(MZ; rij ),

(D.5)�I (ε,0;−rij )NLL= �D0(0; rij ) − �C

(0;Q2).

Appendix E. Summing up the two-loop contributions

In this appendix, the two-loop results listed in Section 5.2 are summed and decomposed intoreducible contributions, which involve products of the one-loop integrals D0 (5.2), plus remain-ing irreducible parts. To this end, we split the integrals according to (5.4) and use the relationsgiven in Appendix B as well as the commutation relations (2.29) and, in particular, the fact thatIAi , IZ

j , and∑

V =A,Z,W± IVk IV

k commute with each other. As we show, all irreducible contribu-tions cancel apart from those that can be expressed in terms of the one-loop functions J (7.14)and β-function coefficients. To start with, we consider four separate subsets and combine thesein a later stage.

E.1. Terms related to two external lines not involving gauge-boson self-energies

We begin by considering the contributions that result from the diagrams where the soft–collinear gauge bosons couple to two on-shell external lines and that do not involve self-energycontributions to the soft–collinear gauge bosons, i.e. from the diagrams 1, 2, 3, 4, and 5 of Sec-tion 5.2,

(E.1)Mij

2,no-se = M1,ij

2 + M2,ij

2 + [M3,ij

2 + M4,ij

2 + M5,ij

2 + (i ↔ j)].

These can be summarized as

Mij

2,no-se = M0

∑V1,V2=A,Z,W±

{I

V 2i I

V 1i I

V2j I

V1j D1(MV1 ,MV2; rij )

+ IV 2i I

V 1i I

V1j I

V2j D2(MV1 ,MV2; rij )

−[

ig2

e

∑V3=A,Z,W±

εV1V2V3IV 2i I

V 1i I

V 3j D3(MV1,MV2 ,MV3; rij )

+ IV2i I

V 2i I

V1i I

V 1j D4(MV1 ,MV2; rij )

+ IV2i I

V1i I

V 2i I

V 1j D5(MV1 ,MV2; rij ) + (i ↔ j)

]}NLL= M0

{1

2

∑V1,V2=A,Z,W±

IV 2i I

V 1i I

V2j I

V1j D0(MW; rij )D0(MW; rij )

+∑

±IVi IA

i IVj IA

j D0(MW; rij )�D0(0; rij )

V =A,Z,W
Page 57: Nucl.Phys.B v.761

A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 57

+∑

V =A,Z,W±IVi IZ

i IVj IZ

j D0(MW; rij )�D0(MZ; rij )

+ 1

2IAi IA

i IAj IA

j �D0(0; rij )�D0(0; rij )+ IZ

i IAi IZ

j IAj �D0(MZ; rij )�D0(0; rij )

+[ ∑

V1,V2=A,Z,W±ig2

e

1

2εAV1V2

(I

V 1i IA

i IV 2j + IA

i IV 1j I

V 2j

)�D12(MW,0; rij )

+∑

V1,V2=A,Z,W±ig2

e

1

2εZV1V2

(I

V 1i IZ

i IV 2j + IZ

i IV 1j I

V 2j

)�D12(MW,MZ; rij )

+ (i ↔ j)

]

+ 6∑

V1,V2=A,Z,W±I

V 1i I

V 2j TrV

(IV1IV2

)D9(MW,MW,MW,MW; rij )

(E.2)

+ 3∑

V1=A,Z,W±

(IAi I

V 1j + I

V 1i IA

j

)TrV

(IAIV1

)�D9(0,MW,MW,0; rij )

},

where we have made use of the identities (B.1)–(B.4), and (C.2).

E.2. Terms involving gauge-boson self-energy contributions

The contributions where a soft–collinear gauge boson connects two external lines and involvesself-energy corrections result from diagrams 6–11 in Section 5.2 and read

(E.3)Mij

2,se =11∑

m=6

Mm,ij

2 .

They can be summarized as

Mij

2,se = M0

{1

2

g22

e2

∑V1,V2,V3,V4=A,Z,W±

IV 1i I

V 4j εV1V 2V 3εV4V2V3

× D6(MV1 ,MV2,MV3 ,MV4; rij )

− g22

e2

∑V1,V2,V3,V4=A,Z,W±

IV 1i I

V 4j εV1V 2V 3εV4V2V3(D − 1)

× D7(MV1 ,MV2,MV4; rij )− e2v2

∑V1,V3,V4=A,Z,W±

IV 1i I

V 4j

∑Φi2 =H,χ,φ±

{IV1 , IV 3

}HΦi2

{IV3, IV4

}Φi2 H

× D8(MV1 ,MΦ2 ,MV3,MV4; rij )− 1

2

∑V1,V4=A,Z,W±

IV 1i I

V 4j

∑Φi2 ,Φi3 =H,χ,φ±

IV1Φi3 Φi2

IV4Φi2 Φi3

× D9(MV1 ,MΦi,MΦi

,MV4; rij )

2 3
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58 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

− 1

2

∑V1,V4=A,Z,W±

IV 1i I

V 4j

∑Φi2 =H,χ,φ±

{IV1 , IV4

}Φi2 Φi2

× D10(MV1 ,MΦi2,MV4; rij )

− 1

2

∑V1,V4=A,Z,W±

IV 1i I

V 4j

×∑Ψ

{ ∑Ψi2 ,Ψi3 =u,d

∑κ=R,L

IV1Ψ κ

i3Ψ κ

i2I

V4Ψ κ

i2Ψ κ

i3D11,0(MV1 ,mi2,mi3 ,MV4; rij )

− (I

V1uRuRI

V4uLuL + I

V1uLuLI

V4uRuR

)m2

uD11,m(MV1,mu,mu,MV4; rij )}}

NLL= −M0

{ ∑V1,V2=A,Z,W±

IV1i I

V 2j b

(1)V1V2

J(ε,MW,Q2)

+ IAi IA

j b(1)AA

[�J

(ε,0,Q2)− �J

(ε,0,M2

W

)]+ IA

i IAj b

(1)QED�J

(ε,0,M2

W

)+

∑V1,V2=A,Z,W±

6IV 1i I

V 2j TrV

(IV1IV2

)D9(MW,MW,MW,MW; rij )

+ 6IAi IA

j TrV(IAIA

)�D9(0,MW,MW,0; rij )

(E.4)+ 3(IAi IZ

j + IZi IA

j

)TrV

(IAIZ

)�D9(0,MW,MW,0; rij )

},

where we used the relations (B.5)–(B.7), (C.1)–(C.4), and (C.8). For the simplification of thediagram (5.26) in addition (2.19) was employed. We note that in this calculation terms of theform

∑V2,V3=A,Z,W± IA

i IZj εAV 2V 3εZV2V3M2

V2cancel, where M2

V2results either from the loop

integrals or from (2.19).

E.3. Terms related to three external lines

The terms where the soft–collinear gauge bosons couple to three of the n on-shell externallines result from diagrams 12 and 13 in Section 5.2 and can be written as

(E.5)Mijk

2 =( ∑

π(i,j,k)

M12,ijk

2

)+ M13,ijk

2 ,

where the sum runs over all six permutations π(i, j, k) of external lines i, j, k. These contribu-tions yield

Mijk

2 = M0

{ ∑π(i,j,k)

∑V1,V2=A,Z,W±

IV 2i I

V 1i I

V1j I

V2k D12(MV1 ,MV2; rik)

− ig2

e

∑V1,V2,V3=A,Z,W±

εV1V2V3IV 1i I

V 2j I

V 3k D13(MV1 ,MV2,MV3; rij , rik, rjk)

}

NLL= M0

∑π(i,j,k)

{1

2

∑±I

V 2i I

V 1i I

V2j I

V1k D0(MW; rij )D0(MW; rik)

V1,V2=A,Z,W

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 59

+∑

V =A,Z,W±IVi IA

i IVj IA

k D0(MW; rij )�D0(0; rik)

+∑

V =A,Z,W±IVi IZ

i IVj IZ

k D0(MW; rij )�D0(MZ; rik)

+ 1

2IAi IA

i IAj IA

k �D0(0; rij )�D0(0; rik)+ IZ

i IAi IZ

j IAk �D0(MZ; rij )�D0(0; rik)

+∑

V1,V2=A,Z,W±ig2

e

1

2εAV1V2IA

i IV 2j I

V 1k �D12(MW,0; rij )

(E.6)+∑

V1,V2=A,Z,W±ig2

e

1

2εZV1V2IZ

i IV 2j I

V 1k �D12(MW,MZ; rij )

},

where we used (B.8)–(B.10).

E.4. Terms from four external lines

Finally, we have the contributions where the soft–collinear gauge bosons couple to four of then on-shell external lines, i.e. diagram 14 in Section 5.2:

(E.7)Mijkl

2 = M14,ijkl

2 .

These reduce according to (5.44) directly to products of one-loop integrals

Mijkl

2 = M0

∑V1,V2=A,Z,W±

IV 1i I

V1j I

V 2k I

V2l D14(MV1 ,MV2; rij , rkl)

(E.8)NLL= M0

∑V1,V2=A,Z,W±

IV 1i I

V1j I

V 2k I

V2l D0(MV1; rij )D0(MV2; rkl).

E.5. Complete two-loop correction

The contributions from the above subsets of diagrams can be combined for an arbitraryprocess involving n on-shell external massless fermions according to (3.8) or (7.11) as

(E.9)MF2 =

n∑i=1

n∑j=1j �=i

[1

2

(Mij

2,no-se + Mij

2,se

)+n∑

k=1k �=i,j

(1

6Mijk

2 +n∑

l=1l �=i,j,k

1

8Mijkl

2

)].

As a first step, we show that all terms that cannot be expressed by the one-loop functions D0and J cancel. The terms involving explicit factors TrV (IV1IV2) or TrV (IV1IA), i.e. the last twolines in (E.2) and the last three lines in (E.4) cancel directly if we use the fact that the latter traceis only non-vanishing for V1 = A,Z. The irreducible terms involving explicit εV1V2V3 tensorsappearing in (E.2) and (E.6) yield

M0

n∑i=1

n∑j=1

{ ∑V1,V2=A,Z,W±

ig2

e

1

2εAV1V2

(I

V 1i IA

i IV 2j + IA

i IV 1j I

V 2j

)�D12(MW,0; rij )

j �=i

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60 A. Denner et al. / Nuclear Physics B 761 (2007) 1–62

+∑

V1,V2=A,Z,W±ig2

e

1

2εZV1V2

(I

V 1i IZ

i IV 2j + IZ

i IV 1j I

V 2j

)�D12(MW,MZ; rij )

+n∑

k=1k �=i,j

[ ∑V1,V2=A,Z,W±

ig2

e

1

2εAV1V2IA

i IV 2j I

V 1k �D12(MW,0; rij )

(E.10)+∑

V1,V2=A,Z,W±ig2

e

1

2εZV1V2IZ

i IV 2j I

V 1k �D12(MW,MZ; rij )

]}.

These terms vanish upon using global gauge invariance (2.30).The complete two-loop correction is thus given by the contributions to (E.2), (E.6), and

(E.8) involving products of D0-functions and the terms involving β-function coefficients andJ -functions in (E.4). These can be summarized straightforwardly as

MF2

NLL= 1

4M0

n∑i=1

n∑j=1j �=i

n∑k=1

n∑l=1l �=k

{1

2

∑V1,V2=A,Z,W±

IV 2i I

V2j I

V 1k I

V1l

× D0(MW; rij )D0(MW; rkl)

+∑

V =A,Z,W±IVi IV

j IAk IA

l D0(MW; rij )�D0(0; rkl)

+∑

V =A,Z,W±IVi IV

j IZk IZ

l D0(MW; rij )�D0(MZ; rkl)

+ 1

2IAi IA

j IAk IA

l �D0(0; rij )�D0(0; rkl)

+ IZi IZ

j IAk IA

l �D0(MZ; rij )�D0(0; rkl)

}

− 1

2M0

n∑i=1

n∑j=1j �=i

{ ∑V1,V2=A,Z,W±

IV1i I

V 2j b

(1)V1V2

J(ε,MW,Q2)

(E.11)

+ IAi IA

j b(1)AA

[�J

(ε,0,Q2)− �J

(ε,0,M2

W

)]+ IAi IA

j b(1)QED�J

(ε,0,M2

W

)}.

In NLL accuracy, this result can be expressed in terms of the lowest-order matrix element M0

and the one-loop correction factors (D.2) as

MF2

NLL= M0

{1

2

[F

F,sew1

]2 + FF,sew1 �F

F,em1 + F

F,sew1 �F

F,Z1

(E.12)+ 1

2

[�F

F,em1

]2 + �FF,Z1 �F

F,em1 + G

F,sew2 + �G

F,em2

}.

Note that in NLL approximation FF,sew1 and �F

F,em1 do not commute, while �F

F,Z1 commutes

with the other terms. Using (2.30), (C.5), and (C.6), the terms resulting from the last two lines of

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A. Denner et al. / Nuclear Physics B 761 (2007) 1–62 61

(E.11) can be written as

e2GF,sew2 = 1

2

n∑i=1

[b

(1)1 g2

1

(Yi

2

)2

+ b(1)2 g2

2Ci

]J(ε,MW,Q2),

(E.13)

�GF,em2 = 1

2

n∑i=1

Q2i

{b(1)e

[�J

(ε,0,Q2)− �J

(ε,0,M2

W

)]+ b(1)QED�J

(ε,0,M2

W

)}.

Adding the term (6.10) resulting from parameter renormalization with D0 replaced by I andusing the definition of J , (7.14), the functions G

F,sew2 and �G

F,em2 in (E.12) get replaced by the

functions Gsew2 and �Gem

2 as defined in (7.13).Finally, when combining the wave-function counterterm (6.7) with the parts of (E.12) involv-

ing products of the functions F F1 these terms can be written in the form given in (7.12). In order

to arrive at this result we write MF1 in the form (6.8) and δZ

(1)k in the form (6.5). After arrang-

ing the Casimir operators in an appropriate order, we can use global gauge invariance (2.30) totransform the wave-function counterterm contributions to the form needed.

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M. Ciafaloni, P. Ciafaloni, D. Comelli, Nucl. Phys. B 589 (2000) 359, hep-ph/0004071;M. Ciafaloni, P. Ciafaloni, D. Comelli, Phys. Lett. B 501 (2001) 216, hep-ph/0007096;M. Ciafaloni, P. Ciafaloni, D. Comelli, Phys. Rev. Lett. 87 (2001) 211802, hep-ph/0103315;M. Ciafaloni, P. Ciafaloni, D. Comelli, Nucl. Phys. B 613 (2001) 382, hep-ph/0103316;M. Ciafaloni, P. Ciafaloni, D. Comelli, Phys. Rev. Lett. 88 (2002) 102001, hep-ph/0111109;P. Ciafaloni, D. Comelli, JHEP 0511 (2005) 022, hep-ph/0505047.

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Nuclear Physics B 761 (2007) 63–91

First lattice QCD study of the Σ− → n axial and vectorform factors with SU(3) breaking corrections

D. Guadagnoli a, V. Lubicz b,c, M. Papinutto b,c, S. Simula c,∗

a Physik Department, Technische Universität München, D-85748 Garching, Germanyb Dipartimento di Fisica, Università di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy

c INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy

Received 16 June 2006; received in revised form 28 September 2006; accepted 20 October 2006

Available online 9 November 2006

Abstract

We present the first quenched lattice QCD study of the form factors relevant for the hyperon semileptonicdecay Σ− → n�ν. The momentum dependence of both axial and vector form factors is investigated andthe values of all the form factors at zero-momentum transfer are presented. Following the same strategyalready applied to the decay K0 → π−�ν, the SU(3)-breaking corrections to the vector form factor at zero-momentum transfer, f1(0), are determined with great statistical accuracy in the regime of the simulatedquark masses, which correspond to pion masses above ≈ 0.7 GeV. Besides f1(0) also the axial to vectorratio g1(0)/f1(0), which is relevant for the extraction of the CKM matrix element Vus , is determined withsignificant accuracy. Due to the heavy masses involved, a polynomial extrapolation, which does not includethe effects of meson loops, is performed down to the physical quark masses, obtaining f1(0) = −0.948 ±0.029 and g1(0)/f1(0) = −0.287 ± 0.052, where the uncertainties do not include the quenching effect.Adding a recent next-to-leading order determination of chiral loops, calculated within the Heavy BaryonChiral Perturbation Theory in the approximation of neglecting the decuplet contribution, we obtain f1(0) =−0.988 ± 0.029lattice ± 0.040HBChPT. Our findings indicate that SU(3)-breaking corrections are moderateon both f1(0) and g1(0). They also favor the experimental scenario in which the weak electricity formfactor, g2(0), is large and positive, and correspondingly the value of |g1(0)/f1(0)| is reduced with respectto the one obtained with the conventional assumption g2(q2) = 0 based on exact SU(3) symmetry.© 2006 Elsevier B.V. All rights reserved.

* Corresponding author.E-mail address: [email protected] (S. Simula).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2006.10.022

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64 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

1. Introduction

Recently it has been shown that SU(3)-breaking corrections to the K → π vector form factorcan be determined from lattice simulations with great precision [1]. The approach has allowedto reach the percent level of accuracy in the extraction of Vus from K�3 decays, thus stimulatingnew unquenched lattice studies to reduce the systematic uncertainty [2]. An independent wayto extract Vus is provided by hyperon semileptonic decays, and Ref. [3] has shown that it ispossible to extract the product |Vus ·f1(0)| at the percent level from hyperon experiments, wheref1(0) is the vector form factor (f.f.) at zero-momentum transfer. The Ademollo–Gatto (AG)theorem [4] protects f1(0) from first-order SU(3)-breaking corrections that are thus suppressed.Experiments seem to be consistent with negligible channel-dependent SU(3) corrections in thisf.f. [3], but they cannot exclude sizable (i.e. larger than percent), channel-independent effectsin the extraction of Vus . Model dependent estimates based on quark models, 1/Nc and chiralexpansions give different results (see e.g. Ref. [5]), so that a lattice QCD determination of f1(0),as well as of all the other vector and axial f.f.’s which are not AG protected, is of great interest.

The aim of the present work is to investigate SU(3)-breaking corrections to both vector and ax-ial form factors relevant for the Σ− → n transition, completing and generalizing in this way thepreliminary lattice study of Ref. [6]. Though our simulation is carried out in the quenched approx-imation, our results represent the first attempt to evaluate hyperon f.f.’s using a non-perturbativemethod based only on QCD.

We first show that it is possible to determine the SU(3)-breaking corrections to f1(0) onthe lattice following the method of Ref. [1], which is based on the following three main steps:(i) evaluation of the scalar form factor f0(q

2) at q2 = q2max = (MΣ − Mn)

2; (ii) study of themomentum dependence of f0(q

2) to extrapolate f0(q2max) down to f0(0) = f1(0); and (iii) ex-

trapolation in the quark masses down to the physical point. The latter step is one of the mainsources of uncertainty in the quenched lattice calculations of the vector f.f., since the AG theo-rem makes this quantity dominated by meson loops. We make use of the recent, next-to-leadingorder (NLO) calculation of the chiral corrections to f1(0) performed in Ref. [7] within the HeavyBaryon Chiral Perturbation Theory (HBChPT). As in the meson sector, the AG theorem preventsthe contribution from local counter-terms up to O(p4) and makes the NLO corrections finite andfree from unknown low-energy parameters. However, the convergence of the chiral expansionturns out to be rather poor and the inclusion of the decuplet contribution appears to spoil theexpansion itself [7]. Thus, the third step of the procedure of Ref. [1], i.e. the correction of theleading quenched chiral logs with the full QCD ones, is not possible in the hyperon case. In thisrespect we note that the use of HBChPT in conjunction with lattice QCD is likely to representa serious limitation for achieving a precise determination of f1(0) also in future (unquenched)lattice calculations, until it will be possible to simulate on the lattice very light quark masses,close to the physical ones, and sufficiently large volumes [8].

Due to the rather heavy masses involved in our simulation, it is unlikely that meson loops cancontribute significantly at the simulated quark masses. Thus a polynomial extrapolation, whichdoes not include the effects of meson loops, is performed down to the physical quark masses,leading to f1(0) = −0.948 ± 0.029. This result for the SU(3)-breaking corrections, correspond-ing to f1(0) + 1 = (5.2 ± 2.9)%, represents mainly an estimate of the local terms of the chiralexpansion and appears to be opposite in both sign and size to the NLO estimate of chiral loops,(−4 ± 4)%, calculated in Ref. [7] in the approximation of neglecting the decuplet contributionand assuming a 100% overall uncertainty. The two contributions should be added, leading to

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 65

our final estimate: f1(0) = −0.988 ± 0.029lattice ± 0.040HBChPT, where the uncertainties do notinclude the quenching effect.

Second, we investigate the momentum dependence of the other two vector f.f.’s, the weakmagnetism f2(q

2) and the induced scalar f3(q2), as well as of the three axial f.f.’s, the axial-

vector g1(q2), the weak electricity g2(q

2) and the induced pseudoscalar g3(q2).1

The ratio g1(q2)/f1(q

2) can be determined with quite good statistical accuracy directly fromthe ratio of appropriate axial and vector three-point correlation functions. We show that such a ra-tio can be evaluated on the lattice at q2 = q2

max (i.e. with hyperons at rest) within ≈ 10% accuracyand that the extrapolation down to q2 = 0 does not modify significantly that level of precision.Adopting a polynomial extrapolation in the quark masses we get g1(0)/f1(0) = −0.287±0.052,which is consistent with the value g1(0)/f1(0) = −0.340 ± 0.017 adopted in the recent analy-sis of Ref. [3]. The study of the degenerate transitions also allows to determine the value ofg1(0)/f1(0) directly in the SU(3) limit; we get [g1(0)/f1(0)]SU(3) = −0.269±0.047. Comparedwith the above estimate, it implies that the SU(3)-breaking corrections on g1(0)/f1(0) are mod-erate, though this ratio is not protected by the AG theorem. Our finding is in qualitative agreementwith the exact SU(3)-symmetry assumption of the Cabibbo model [9].

Contrary to f1(q2) and g1(q

2), the other f.f.’s cannot be evaluated at q2 = q2max and conse-

quently the extrapolation of the lattice data to q2 = 0 is affected by larger uncertainties. In thecases of the weak magnetism and of the induced pseudoscalar f.f.’s, whose matrix elements do notvanish in the SU(3) limit, we obtain f2(0)/f1(0) = −1.52 ± 0.81 and g3(0)/f1(0) = 6.1 ± 3.3 atthe physical point. The central values agree well with the experimental result f2(0)/f1(0) =−1.71 ± 0.12stat. ± 0.23syst. from Ref. [10], as well as with the SU(3)-breaking analysis ofRef. [11], and with the value g3(0)/f1(0) = 5.5 ± 0.9, obtained using the axial Ward identityand the generalized Goldberger–Treiman relation [12], which relates g3(0) with g1(0) in thechiral limit.

For the weak electricity g2(q2) and the induced scalar f3(q

2) f.f.’s a non-vanishing result,which is entirely due to SU(3)-breaking corrections, is found, namely g2(0)/f1(0) = 0.63 ±0.26 and f3(0)/f1(0) = −0.42 ± 0.22. Note that experiments carried out with polarized Σ−hyperons [10] have determined the f.f. combination |g1(0)/f1(0)−0.133g2(0)/f1(0)| to be equalto 0.327 ± 0.007stat. ± 0.019syst.. Our result is 0.37 ± 0.08, which means that the lattice resultsfor both g1(0)/f1(0) and g2(0)/f1(0) are nicely consistent with the experimental data on theΣ− → n transition. Our findings favor the scenario in which g2(0)/f1(0) is large and positive,and correspondingly the value of |g1(0)/f1(0)| is reduced with respect to the one obtained withthe conventional assumption g2(q

2) = 0 (done in Ref. [3]) based on exact SU(3) symmetry. Sucha scenario is slightly preferred also by experimental data (see Ref. [10]).

The plan of the paper is as follows. In Section 2 we introduce the notation and give somedetails about the lattice simulation. Section 3 is devoted to the extraction of f0(q

2) at q2 = q2max,

to its extrapolation down to q2 = 0 and to the polynomial extrapolation in the quark masses downto their physical values, without taking into account the effects of meson loops. The estimate ofthese effects, based on HBChPT at the NLO, is described in Section 4, where the problematicrelated to the convergence of the chiral expansion and to the decuplet contribution is brieflyillustrated. In Section 5 we present our results for the ratio g1(0)/f1(0), while those for the othervector and axial f.f.’s are collected in Section 6. Finally our conclusions are given in Section 7.

1 Our definitions of the f.f.’s f2, f3, g2 and g3 [see Eq. (1)] differ by the ones adopted in Ref. [3] by a factor (MΣ +Mn)/MΣ , equal to � 1.785 at the physical hyperon masses.

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66 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

2. Notation and lattice details

In this study we restrict our attention to the Σ− → nlν decay. We are interested in the hadronicmatrix element of the weak (V − A) current, M ≡ 〈n|uγμ(1 − γ5)s|Σ−〉, which can be conve-niently expressed in Minkowski space in terms of f.f.’s and external spinors as

M= un(p′){γ μf1

(q2) − i

σμνqν

Mn + MΣ

f2(q2) + qμ

Mn + MΣ

f3(q2)

+[γ μg1

(q2) − i

σμνqν

Mn + MΣ

g2(q2) + qμ

Mn + MΣ

g3(q2)]γ5

}uΣ(p)

(1)= un(p′){O

μV + O

μA

}uΣ(p)

with q = p − p′. A detailed discussion of the properties of the six f.f.’s in Eq. (1) can be found,e.g., in Ref. [3]. We also introduce the scalar f.f. f0(q

2) from the divergence of the vector weakcurrent 〈n|∂μV μ|Σ−〉 ≡ (MΣ + Mn)f0(q

2). It is related to f1(q2) and f3(q

2) by

(2)f0(q2) = f1

(q2) + q2

M2Σ − M2

n

f3(q2),

and at q2 = 0 it coincides with the quantity of interest f1(0). Note that for the Σ− → n transitionf1(0) is normalized as f1(0) = −1 in the SU(3) limit.

In order to access the matrix element of Eq. (1) on the lattice, one can consider the followingtwo- and three-point correlation functions in Euclidean space

(3)[GΣ(n)(t, p )

]γ ′γ =

∑x

⟨J

Σ(n)

γ ′ (t, x )JΣ(n)γ (0, 0 )

⟩e−i p·x,

(4)[V Σn

μ (tx, ty, p, p′)]γ ′γ = ⟨

J nγ ′(ty, y )Vμ(tx, x )JΣ

γ (0, 0 )⟩e−i( p− p′)·xe−i p′·y,

(5)[AΣn

μ (tx, ty, p, p′)]γ ′γ = ⟨

J nγ ′(ty, y )Aμ(tx, x )JΣ

γ (0, 0 )⟩e−i( p− p′)·xe−i p′·y,

where Jn,Σγ are local interpolating operators for the neutron and the Σ− hyperon, which we

choose to be

(6)Jnα = εijk

[dTi Cγ5uj

]dkα, JΣ

α = εijk

[dTi Cγ5sj

]dkα,

with Latin (Greek) symbols referring to color (Dirac) indices. The operators in Eq. (6) are relatedto the external spinors uΣ,n (cf. Eq. (1)) via the relation

(7)〈0|JΣ(n)γ (0,0)|Σ(n)( p,σ)〉 = √

ZΣ(n)

[uΣ(n)(p,σ )

]γ,

where σ refers to the polarization of the Σ(n) baryon.In Eqs. (4)–(5) Vμ and Aμ are the renormalized, O(a)-improved lattice weak vector and

axial-vector current respectively:

(8)V μ = ZV

(1 + bV

ams + a m�

2

)(uγ μs + a cV ∂νuσμνs

),

(9)Aμ = ZA

(1 + bA

ams + am�

2

)(uγ μγ5s + acA∂μuγ5s

),

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 67

where ZV and ZA are the vector and axial-vector renormalization constants, bV , bA, cV and cA

are O(a)-improvement coefficients [13] and the subscript � refers to the light u or d quarks,which we consider to be degenerate in mass.

Taking the large-time limits tx , (ty − tx) → ∞ and using (7) one can rewrite Eqs. (3)–(4) asfollows

(10)[GΣ(n)

(t, p( p′)

)]γ ′γ −−−−→

t→∞ ZΣ(n)e−EΣ(n)t

(i/p(/p′) + MΣ(n)

2EΣ(n)

)γ ′γ

,

[V Σn

μ (tx, ty, p, p′)]γ ′γ −−−−−−−→

tx ,(ty−tx )→∞ e−Entx e−EΣ(ty−tx )√

ZnZΣ

(11)×(

i/p′ + Mn

2En

)γ ′ρ

(OV

μ

)ρσ

(i/p + MΣ

2EΣ

)σγ

,

[AΣn

μ (tx, ty, p, p′)]γ ′γ −−−−−−−→

tx ,(ty−tx )→∞ e−Entx e−EΣ(ty−tx )√

ZnZΣ

(12)×(

i/p′ + Mn

2En

)γ ′ρ

(OA

μ

)ρσ

(i/p + MΣ

2EΣ

)σγ

,

where En = √M2

n + | p′|2, EΣ =√

M2Σ + | p|2 and OV

μ (OAμ ) is the euclidean version of the

vector (axial-vector) contribution to Eq. (1). From Eqs. (10) and (11) it follows

(13)

V Σnμ (tx, ty, p, p′)γ ′γ

GΣ(tx, p)γ ′γ Gn(ty − tx, p′)γ ′γ

−−−−−−−→tx ,(ty−tx )→∞

1√ZnZΣ

[(i/p′ + Mn)OVμ (i/p + MΣ)]γ ′γ

(i/p′ + Mn)γ ′γ (i/p + MΣ)γ ′γ,

(14)

AΣnμ (tx, ty, p, p′)γ ′γ

GΣ(tx, p)γ ′γ Gn(ty − tx, p′)γ ′γ

−−−−−−−→tx ,(ty−tx )→∞

1√ZnZΣ

[(i/p′ + Mn)OAμ (i/p + MΣ)]γ ′γ

(i/p′ + Mn)γ ′γ (i/p + MΣ)γ ′γ.

Let us consider the following kinematics

(15)p =(√

M2Σ + |q |2, q

)≡ (Eq, q), p′ = (Mn, 0)

and study the matrix elements of the weak vector current. In order to minimize the number ofcalculated three-point correlation functions we consider only one pair of values of the Diracindices, namely γ = γ ′ = 0. Then we define the following quantities

W1(q2; tx, ty

) ≡ 2Eq

√ZnZΣ

Eq + MΣ

Re(V Σn0 (tx, ty, q, 0 )00)

GΣ(tx, q )00Gn(ty − tx, 0 )00,

W2(q2; tx, ty

) ≡ 2Eq

√ZnZΣ

|q|2Im(qkV

Σnk (tx, ty, q, 0 )00)

GΣ(tx, q )00Gn(ty − tx, 0 )00,

(16)W3(q2; tx, ty

) ≡ 2Eq

√ZnZΣ

q1

Re(V Σn2 (tx, ty, q, 0 )00)

GΣ(tx, q )00Gn(ty − tx, 0 )00,

which in terms of the three vector form factors f1,2,3 read as

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68 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

W1(q2) = f1

(q2) − Eq − MΣ

MΣ + Mn

f2(q2) + Eq − Mn

MΣ + Mn

f3(q2),

W2(q2) = f1

(q2) − Eq − Mn

MΣ + Mn

f2(q2) + Eq + MΣ

MΣ + Mn

f3(q2),

(17)W3(q2) = f1

(q2) + f2

(q2),

where Wi(q2) ≡ limtx ,(ty−tx )→∞ Wi(q

2; tx, ty) (i = 1,2,3). Inverting the above equations onegets

(18)f1(q2) =N

{W1

(q2) − Eq − Mn

Eq + MΣ

W2(q2) − q2

(MΣ + Mn)(Eq + MΣ)W3

(q2)},

(19)f2(q2) =N

{−W1

(q2) + Eq − Mn

Eq + MΣ

W2(q2) + MΣ + Mn

Eq + MΣ

W3(q2)},

(20)f3(q2) =N

{−W1

(q2) + Eq + Mn

Eq + MΣ

W2(q2) + q2

(MΣ − Mn)(Eq + MΣ)W3

(q2)}

with N ≡ (MΣ + Mn)/2Mn.Eqs. (16)–(20) provide the standard procedure for measuring f.f.’s on the lattice. Typically the

accuracy that can be reached in this way is not better than 10–20%, which is clearly not sufficientfor measuring SU(3)-breaking effects in f1(0) at the percent level. In the next section we describethe procedure to get both the scalar form factor (2) at q2 = q2

max and the ratios f2(q2)/f1(q

2)

and f3(q2)/f1(q

2) with quite small statistical fluctuations.In the case of the axial-vector weak current, choosing always γ = γ ′ = 0, we define the

following quantities

W(A)1

(q2; tx, ty

) ≡ 2Eq

√ZnZΣ

Eq + MΣ

Im(AΣn(tx, ty, q, 0 )00)

GΣ(tx, q )00Gn(ty − tx, 0 )00,

W(A)2

(q2; tx, ty

) ≡ 2Eq

√ZnZΣ

q3

Re(AΣn0 (tx, ty, q, 0 )00)

GΣ(tx, q )00Gn(ty − tx, 0 )00,

(21)W(A)3

(q2; tx, ty

) ≡ −2Eq(MΣ + Mn)√

ZnZΣ

q1q3

Im(AΣn1 (tx, ty, q, 0 )00)

GΣ(tx, q )00Gn(ty − tx, 0 )00,

where AΣn ≡ AΣn3 + (AΣn

1 /q1 + AΣn2 /q2) · (|q |2 − q2

3 )/2q3. In terms of the three axial-vectorform factors g1,2,3 one has

W(A)1

(q2) = g1

(q2) − Eq − Mn

MΣ + Mn

g2(q2) + Eq − MΣ

MΣ + Mn

g3(q2),

W(A)2

(q2) = g1

(q2) − Eq + MΣ

MΣ + Mn

g2(q2) + Eq − Mn

MΣ + Mn

g3(q2),

(22)W(A)3

(q2) = g2

(q2) − g3

(q2),

where W(A)i (q2) ≡ limtx ,(ty−tx )→∞ W

(A)i (q2; tx, ty) (i = 1,2,3). Inverting the above equations

one gets

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 69

g1(q2) =N

{W

(A)1

(q2) + Mn − MΣ

Mn + MΣ

W(A)2

(q2) + 1

(Mn + MΣ)2W

(A)3

(q2)

(23)× [(Mn − MΣ)(Eq − Mn) − (Mn + MΣ)(Eq − MΣ)

]},

(24)g2(q2) =N

{W

(A)1

(q2) − W

(A)2

(q2) + Mn − MΣ

Mn + MΣ

W(A)3

(q2)},

(25)g3(q2) =N

{W

(A)1

(q2) − W

(A)2

(q2) − W

(A)3

(q2)}.

We have generated 240 quenched gauge field configurations on a 243 × 56 lattice at β = 6.20(corresponding to an inverse lattice spacing equal to a−1 � 2.6 GeV), with the plaquette gaugeaction. We have used the non-perturbatively O(a)-improved Wilson fermions with cSW = 1.614[14] and chosen quark masses corresponding to four values of the hopping parameters, namelyk ∈ {0.1336,0.1340,0.1343,0.1345}. Using Σ and n baryons with quark content (ksk�k�) and(k�k�k�), respectively, twelve different Σ → n vector (axial) correlation functions V Σn

μ (AΣnμ )

have been computed, using both ks < k� and ks > k�, corresponding to the cases in which the Σ

(neutron) is heavier than the neutron (Σ ). In addition, using the same combinations of quarkmasses, also the three-point n → Σ correlation functions V nΣ

μ (AnΣμ ) have been calculated.

Finally, twelve elastic, non-degenerate V ΣΣμ (AΣΣ

μ ) and four elastic, fully degenerate V nnμ (Ann

μ )three-point functions have been evaluated.

The simulated quark masses are approximately in the range ≈ 1–1.5 × ms , where ms is thestrange quark mass, and correspond to pseudoscalar meson masses in the interval ≈ 0.70–1 GeVand to baryon masses in the range ≈ 1.5–1.8 GeV. Though the simulated meson masses are largerthan the physical ones, the corresponding values of q2

max = (MΣ − Mn)2 are taken as close as

possible to q2 = 0 (see Table 1 in the next section).To improve the statistics, two- and three-point correlation functions have been averaged with

respect to parity and charge conjugation transformations. We have chosen ty/a = 24 in thethree-point correlation functions, which have been computed for 5 different values of the initialmomentum p ≡ (2π/aL) · κ , namely κ = (0,0,0), (1,0,0), (1,1,0), (1,1,1), (2,0,0), puttingalways the final hadron at rest [ p′ = (0,0,0)]. The squared four-momentum transfer q2 is thus

given by q2 = (EΣ(n) − Mn(Σ))2 − |q |2, where EΣ(n) =

√M2

Σ(n) + |q |2 and |q |2 = 0, q2min,

2q2min, 3q2

min, 4q2min with qmin = 2π/aL � 0.7 GeV.

The statistical errors are evaluated using the jackknife procedure, which is adopted throughoutthis paper.

3. Results for f1(0)

3.1. Determination of f0(q2max)

The main observation is that f0(q2) can be extracted with a statistical accuracy better than

O(1%) at the kinematical point q2 = q2max = (MΣ − Mn)

2 through the following double ratio ofthree-point functions with both external baryons at rest

(26)R0(tx, ty) ≡ V Σn0 (tx, ty, 0, 0 )00V

nΣ0 (tx, ty, 0, 0 )00

V nn0 (tx, ty, 0, 0 )00V

ΣΣ0 (tx, ty, 0, 0 )00

.

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70 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

For large source and sink times, one has

(27)R0(tx, ty)−−−−−−−→tx ,(ty−tx )→∞

( 〈n|uγ0s|Σ−〉〈Σ−|sγ0u|n〉〈n|uγ0u|n〉〈Σ−|sγ0s|Σ−〉

)p=p′=0

= [f0

(q2

max

)]2.

The double ratio (26), originally introduced in Ref. [15] for the study of heavy-heavy semilep-tonic transitions, has a number of nice features, described in detail in Ref. [1] and here brieflycollected:

• Normalization to unity in the SU(3) limit for every value of the lattice spacing, so that thedeviation from one of the ratio (26) gives a direct measure of SU(3)-breaking effects onf0(q

2max);

• Large reduction of statistical uncertainties due to noise cancellation between the numeratorand the denominator;

• Cancellation of the dependence on the matrix elements√

ZΣ and√

Zn (see Eq. (11)) be-tween the numerator and the denominator;

• No need to improve and to renormalize the local vector current. This implies that discretiza-tion errors on the ratio (26) start at O(a2) and are of the form a2(ms − m�)

2 because theratio is symmetric under the exchange ms ↔ m�;

• Quenching error is also quadratic in the SU(3)-breaking quantity (ms − m�).

The quality of the plateaux for the double ratio (26) can be appreciated by looking at Fig. 1.In Fig. 2 and Table 1 we have collected our results for the quantities f0(q

2max), determined

through the ratio (26), the two mass combinations a2(M2Σ + M2

n) and a2(M2Σ − M2

n), the latterbeing proportional to the SU(3)-breaking quantity (ms − m�), and a2q2

max ≡ a2(MΣ − Mn)2.

One can appreciate the remarkable statistical precision obtained for f0(q2max), which is always

below 0.2%: the SU(3)-breaking corrections are clearly resolved with respect to the statisticalerrors.

Fig. 1. Time dependence of the quantity −√R0(t, ty = 24a), defined in Eq. (26) for the two sets of the hopping parame-

ters, given in the legend. Horizontal bars represent the time interval chosen for the fit.

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 71

Fig. 2. Results for [−f0(q2max)] versus the difference of the squared masses of the external baryons, in lattice units.

Table 1Values of the hopping parameters ks and k� , a2(M2

Σ + M2n), a2(M2

Σ − M2n), a2q2

max ≡ a2(MΣ − Mn)2 and f0(q2max),

obtained from the double ratio (26)

ks − k� a2(M2Σ + M2

n) a2(M2Σ − M2

n) a2q2max f0(q2

max)

0.1336−0.1340 0.956 (9) −0.0261 (6) 0.000354 (17) −0.99995 (19)

0.1336−0.1343 0.937 (10) −0.0452 (12) 0.001086 (65) −0.99941 (44)

0.1336−0.1345 0.924 (10) −0.0582 (17) 0.00183 (12) −0.99858 (71)

0.1340−0.1336 0.867 (12) +0.0250 (8) 0.000360 (24) −0.99972 (15)

0.1340−0.1343 0.824 (12) −0.0187 (8) 0.000211 (20) −0.99985 (12)

0.1340−0.1345 0.811 (13) −0.0313 (14) 0.000603 (58) −0.99952 (36)

0.1343−0.1336 0.783 (15) +0.0431 (16) 0.001180 (99) −0.99933 (68)

0.1343−0.1340 0.759 (15) +0.0187 (8) 0.000229 (23) −0.99987 (14)

0.1343−0.1345 0.728 (16) −0.0121 (8) 0.000101 (14) −0.99992 (9)

0.1345−0.1336 0.728 (18) +0.0548 (27) 0.00206 (22) −0.9991 (16)

0.1345−0.1340 0.704 (18) +0.0307 (17) 0.000666 (84) −0.99977 (57)

0.1345−0.1343 0.686 (19) +0.0125 (8) 0.000113 (16) −1.00000 (11)

3.2. Extrapolation to q2 = 0

The next step is to determine both f0(q2) and f1(q

2) for various values of q2 using thestandard f.f. analysis given by Eqs. (18)–(20). The quality of the plateaux for the quantityW1(q

2; tx, ty) (see Eq. (16)), which provides the dominant contribution to f1(q2), is shown in

Fig. 3. The f.f. f1(q2) turns out to be determined with quite good precision as it can be clearly

seen in Fig. 4. Note that in Fig. 4 the data points appear paired since both Σ → n and n → Σ

transitions are considered in the analysis. On the contrary, due to the large statistical noise in thequantities W2(q

2; tx, ty) and W3(q2; tx, ty), the results for f0(q

2) do not share the same level ofprecision as the one obtained for the f.f. f1(q

2), a finding similar to what was already observedin Ref. [1] for the K → π transition.

To improve the determination of f0(q2) we introduce the following two double ratios of three-

point correlation functions, corresponding to matrix elements of the spatial and time components

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72 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

Fig. 3. Time dependence of the quantity W1(q2; t, ty = 24a) (see Eq. (16)) for the two combinations of the hoppingparameters (ks , k�) = (0.1336,0.1340) (a) and (ks , k�) = (0.1345,0.1343) (b). The horizontal bars represent the timeintervals chosen for the fit, which provide the value of W1(q2) (see Eq. (17)). Dots, squares and triangles correspond to|q |2 = q2

min, 2q2min and 4q2

min, with qmin = 2π/aL, respectively.

Fig. 4. Values of [−f1(q2)] versus a2q2, as determined by Eq. (18), for the two combinations of the hopping parameters(ks , k�) = (0.1336,0.1340) (a) and (ks , k�) = (0.1345,0.1343) (b). The solid and dashed curves represent the “dipole”and “monopole” fits of Eq. (30), respectively, whereas the dotted curve is a dipole fit with the slope parameter fixed bythe K∗-meson mass.

of the weak vector current:

R1(q2; tx, ty

) ≡ Im(qkVΣnk (tx, ty, q, 0 )00)

Re(V Σn0 (tx, ty, q, 0 )00)

Re(V ΣΣ0 (tx, ty, q, 0 )00)

Im(qkVΣΣk (tx, ty, q, 0 )00)

,

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 73

R2(q2; tx, ty

) ≡ (Eq + Mn) Im(qkVΣnk ) + (MΣ − Mn)|q |2 Re(V Σn

2 )/q1

(Eq + MΣ)Re(V Σn0 (tx, ty, q, 0 )00)

(28)× Re(V ΣΣ0 (tx, ty, q, 0 )00)

Im(qkVΣΣk (tx, ty, q, 0 )00)

.

The advantages of the ratios (28) are similar to those already pointed out for the double ratio (26),namely: (i) a large reduction of statistical fluctuations; (ii) the independence of the renormaliza-tion constant ZV and the improvement coefficient bV ; (iii) the cancellation of the dependenceon the matrix elements

√ZΣ and

√Zn (see Eq. (16)) between the numerator and the denomina-

tor, and (iv) Ri(q2; tx, ty) → 1 in the SU(3) limit. We stress that the introduction of the matrix

elements of degenerate transitions in Eq. (28) is crucial to largely reduce statistical fluctuations,because it compensates the different fluctuations of the matrix elements of the spatial and timecomponents of the weak vector current.

In terms of the large-time limits Ri(q2) ≡ limtx ,(ty−tx )→∞ Ri(q

2; tx, ty) one has

R1(q2) = (MΣ + Mn)f1(q

2) − (Eq − Mn)f2(q2) + (Eq + MΣ)f3(q

2)

(MΣ + Mn)f1(q2) − (Eq − MΣ)f2(q2) + (Eq − Mn)f3(q2),

(29)R2(q2) = (MΣ + Mn)f1(q

2) − (Eq − MΣ)f2(q2) + (Eq + Mn)f3(q

2)

(MΣ + Mn)f1(q2) − (Eq − MΣ)f2(q2) + (Eq − Mn)f3(q2),

which can be solved in terms of f2(q2)/f1(q

2) and f3(q2)/f1(q

2). The latter quantities can bemultiplied in turn by the corresponding values of f1(q

2) evaluated with the standard procedure,obtaining in this way an improved determination of both f2(q

2) and f3(q2), and consequently

of f0(q2). Our results for f0(q

2) are shown in Fig. 5, where the very precise lattice point atq2 = q2

max (the rightmost one), calculated via the double ratio (26), is also reported. It can clearly

Fig. 5. Values of [−f0(q2)] versus a2q2, calculated using the double ratios (28), for the two combinations of the hoppingparameters (ks , k�) = (0.1336,0.1340) (a) and (ks , k�) = (0.1345,0.1343) (b). The full squares represent the very pre-cise lattice point at q2 = q2

max, calculated via the double ratio (26). The dashed and solid curves represent the monopoleand dipole fits of Eq. (30), respectively.

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74 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

be seen that the precision achieved for the scalar f.f. f0(q2) at q2 �= q2

max is comparable to theone obtained for the vector f.f. f1(q

2) for all the simulated quark masses.The momentum dependence of f0(q

2) and f1(q2) can be analyzed in terms of monopole and

dipole behaviors, namely

fmon.

(q2) = A

1 − q2/B2(monopole fit),

(30)fdip.

(q2) = C

(1 − q2/D2)2(dipole fit).

As it is clear from Figs. 4 and 5, both functional forms describe the lattice points quite well, witha slightly better quality in the case of the dipole form (χ2/d.o.f. � 0.8–1.2 for monopole fits andχ2/d.o.f. � 0.6–0.8 for dipole fits). In the case of f1(q

2), the monopole-fit parameter B turnsout to be unrelated to the value predicted by pole dominance (the K∗-meson mass), while thedipole-fit parameter D agrees with the K∗-meson mass within 15% accuracy (see dotted linesin Fig. 4). The latter finding is consistent with the experimental determinations of nucleon formfactors. Existing data on the proton and neutron form factors are indeed reproduced well eitherby using a single dipole form with a radius parameter governed by the ρ-meson mass, or byseveral monopole terms corresponding to the dominance of the coupling of the nucleon with ρ,ω and φ mesons and their resonances [16].

We have applied Eq. (30) performing a fit to all the lattice points for both f0(q2) and f1(q

2),and imposing the equality f0(0) = f1(0) at zero-momentum transfer. Here below we limit our-selves to present the results obtained in two representative cases, where both f0(q

2) and f1(q2)

are assumed to be described either by monopole forms or by dipole ones. Other combinationsprovide similar results. After fitting the parameters appearing in Eq. (30) we get the values off1(q

2 = 0) for the various combinations of quark masses used in the simulation. Our results arecollected in Table 2 and plotted in Fig. 6, where they are also compared with those obtainedin Ref. [1] in the case of the vector form factor at zero-momentum transfer, f+(0), for the K�3decay.

Table 2Values of the hopping parameters ks and k� , a2(M2

K+ M2

π ), a2(M2K

− M2π ) and f1(q2 = 0), obtained assuming either

a monopole or a dipole momentum dependence of both f0(q2) and f1(q2) to perform the extrapolation to q2 = 0 (seeEq. (30))

ks − k� a2(M2K

+ M2π ) a2(M2

K− M2

π ) f1(0)

(monopole fit)f1(0)

(dipole fit)

0.1336−0.1340 0.2819 (16) −0.01483 (12) −0.9970 (4) −0.9976 (3)

0.1336−0.1343 0.2712 (17) −0.02557 (23) −0.9905 (14) −0.9923 (11)

0.1336−0.1345 0.2640 (17) −0.03271 (28) −0.9835 (24) −0.9866 (18)

0.1340−0.1336 0.2529 (17) +0.01439 (17) −0.9966 (5) −0.9972 (4)

0.1340−0.1343 0.2279 (17) −0.01061 (15) −0.9979 (4) −0.9983 (3)

0.1340−0.1345 0.2209 (18) −0.01758 (18) −0.9939 (11) −0.9951 (9)

0.1343−0.1336 0.2209 (18) +0.02464 (14) −0.9875 (24) −0.9902 (18)

0.1343−0.1340 0.2068 (18) +0.01050 (10) −0.9975 (5) −0.9981 (4)

0.1343−0.1345 0.1895 (18) −0.00676 (10) −0.9989 (3) −0.9991 (2)

0.1345−0.1336 0.2002 (19) +0.03111 (17) −0.9766 (60) −0.9818 (45)

0.1345−0.1340 0.1862 (19) +0.01714 (13) −0.9925 (22) −0.9942 (17)

0.1345−0.1343 0.1759 (19) +0.00685 (9) −0.9988 (4) −0.9991 (3)

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 75

Fig. 6. Results for [−f1(0)] vs. the mass difference a4(M2K

−M2π )2, obtained after extrapolation to q2 = 0 through either

monopole (open squares) or dipole (full circles) fits to the lattice points for f0(q2) and f1(q2) (see Eq. (30)). The opendiamonds are the results obtained in Ref. [1] for the vector form factor at zero-momentum transfer, f+(0), correspondingto the K�3 decay. The dotted, dashed and solid lines are linear fits to the data, according to the AG theorem.

In agreement with the Ademollo–Gatto theorem our results for hyperon semileptonic de-cays exhibit an approximate linear behavior in the quadratic SU(3)-breaking parameter (M2

K −M2

π )2 ∝ (ms −m�)2, as shown in Fig. 6. Note that: (i) the amount of SU(3) breaking in the vector

form factor at zero-momentum transfer appears to be larger in the hyperon case with respect tothe K�3 decay; (ii) the statistical uncertainty on f1(0) is in the range 0.02–0.6% at the simulatedquark masses, and (iii) lattice artifacts on f1(0) due to the finiteness of the lattice spacing start atO(a2) and are proportional to (ms −m�)

2, like the physical SU(3)-breaking effects; we thereforeexpect that, by having in our lattice simulation a−1 � 2.6 GeV, discretization errors are sensiblysmaller than the physical SU(3)-breaking effects. Further investigations at different values of thelattice spacing could better clarify this point.

3.3. Chiral extrapolation

Following Ref. [1] we construct the AG ratio defined as

(31)R(MK,Mπ) ≡ 1 + f1(0)

a4(M2K − M2

π )2,

in which the leading meson mass dependence predicted by the AG theorem is factorized out.This quantity depends on both kaon and pion masses, or equivalently on the two independentmass combinations a2(M2

K + M2π ) and a2(M2

K − M2π ). As shown in Fig. 7, the dependence of

the AG ratio (31) on the meson masses is well described by a simple linear fit:

(32)R(MK,Mπ) = R0 + R1 · a2(M2K + M2

π

),

whereas the dependence upon the variable a2(M2K − M2

π ) is found to be negligible. In order toinvestigate the stability of extrapolation to the physical point we consider also a quadratic fit inthe meson masses:

(33)R(MK,Mπ) = R0 + R1 · a2(M2K + M2

π

) + R2 · a4(M2K + M2

π

)2.

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76 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

Fig. 7. Values of the AG ratio R(MK,Mπ) versus the mass combination a2(M2K

+ M2π ), for the cases of the monopole

and dipole fits of both f0(q2) and f1(q2). The dashed lines represent the results of the linear fit (32), while the open dotsindicate the value of the AG ratio extrapolated to the physical point. In (a) the full diamonds and the open squares are thecorresponding results obtained for the AG ratio in the case of the K�3 decay in Ref. [1] with and without the subtractionof the leading chiral correction, respectively.

The consequent increase of the number of parameters leads to larger uncertainties, while the shiftin the central values remains at the level of the statistical errors.

The extrapolation of R(MK,Mπ) to the physical point is performed using the physical valuesof meson masses in lattice units calculated by fixing the ratios MK/MK∗ and Mπ/Mρ to theirexperimental values, obtaining

(34)[aMK ]phys = 0.189(2), [aMπ ]phys = 0.0536(7).

From Fig. 7 it can clearly be seen that:

• As already observed, the AG ratio for hyperon decay is � 2–3 times larger than the corre-sponding ratio obtained for the K�3 decay in Ref. [1].

• The minimum value of a2(M2K + M2

π ) reached in our calculations is two times larger thanthe corresponding minimum value achieved in the case of the K�3 decay. The reason is thatthe quality of the signal coming from the ground state depends on the energy gap betweenthe ground and the excited states (sharing the same quantum numbers). As the quark massdecreases, the energy gap slightly decreases in the case of hyperons [17], while it increasesin the case of pseudoscalar mesons. The net result is an increase of the statistical errors atlarge-time distances, which is more pronounced in the case of hyperons with respect to thecase of pseudoscalar mesons. Note also that we have adopted single, local interpolating fieldsfor hyperons (see Eq. (6)). The use of smeared source and sink, as well as the use of several,independent interpolating fields may help in achieving a better isolation of the ground-statesignal.

Since in our simulation quark masses are rather large, we expect that the effects ofpseudoscalar meson loops may be suppressed, which means that chiral logarithms are unlikely to

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 77

affect significantly the lattice results. The analog case of the K → π transition, in which the lead-ing chiral corrections in the quenched approximation have been explicitly evaluated [1], showsindeed that their contributions are almost negligible at the quark masses used in the simulation, asit is illustrated in Fig. 7 (compare full diamonds with open squares). Therefore our results for theAG ratio of the vector f.f. should be considered mainly as an estimate of the contributions fromlocal terms in the chiral expansion. We then extrapolate the lattice results to the physical valuesof the meson masses by assuming a simple polynomial dependence, as in Eqs. (32) and (33).Effects of meson loops, evaluated in full QCD at the physical quark masses, should be eventuallyadded. Their impact and the actual limitations of HBChPT will be discussed in the next section.We also note that, in the case of K�3 decays, preliminary results of unquenched simulations [2]seem to indicate the smallness of quenching effects.

In Table 3 we collect the extrapolated values R(Mphys.K ,M

phys.π ) obtained for the various func-

tional forms assumed in fitting both the q2-dependence of the form factors (Eq. (30)) and themeson mass dependence given by Eqs. (32)–(33). Two sets of results are presented in Table 3.The first set, given in the upper table, is obtained by determining the matrix elements

√ZΣ and√

Zn (see Eq. (16)) from a fit of the two-point correlation functions in the same time intervalchosen for the three-point correlators, namely t/a ∈ [10,16]. For the second set, given in thelower table, the same time interval chosen to extract hadron masses, i.e. t/a ∈ [19,25], has beenconsidered. Note that the uncertainties in the extrapolated value R(M

phys.K ,M

phys.π ) turn out to be

quite large mostly because of the long chiral extrapolation. Simulations at lower quark masseswould be very beneficial in reducing such an uncertainty.

From the spread of the results given in Table 3 we quote R(Mphys.K ,M

phys.π ) = 48 ± 14stat. ±

23syst., which implies

1 + f1(0) = R(M

phys.K ,M

phys.π

) · [(aMphys.K

)2 − (aM

phys.π

)2]2

(35)= 0.052 ± 0.015stat. ± 0.025syst. = (5.2 ± 2.9)%,

where the systematic error includes the uncertainties due to the extrapolation in q2 and in themeson masses, while it does not include the uncertainty due to the quenching effects. In addition,the result (35) does not include the effects of chiral logarithms induced by meson loops, whichwill be discussed in the next section.

Table 3Results for the AG ratio at the physical point, R(M

phys.K

,Mphys.π ), obtained from linear (32) or quadratic (33) fits in the

meson masses, assuming either monopole or dipole functional forms for the extrapolation of the scalar f0(q2) and vectorf1(q2) form factors to q2 = 0. Upper and lower tables correspond to different choices of the time interval chosen for thefits of the two-point correlation functions (see text)

R(Mphys.K

,Mphys.π ) Linear fit Quadratic fit

Monopole fits in q2 41 ± 13 52 ± 29Dipole fits in q2 32 ± 10 35 ± 30

R(Mphys.K

,Mphys.π ) Linear fit Quadratic fit

Monopole fits in q2 55 ± 19 78 ± 39Dipole fits in q2 40 ± 13 49 ± 35

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78 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

4. Chiral corrections in HBChPT

The chiral behavior of the vector form factor at zero-momentum transfer f1(0) has been re-cently investigated in Ref. [7] using HBChPT, where baryons are treated as heavy degrees offreedom and a 1/MB expansion around the non-relativistic limit is performed. The chiral correc-tions to f1(0) can be schematically expressed as

f1(0) = fSU(3)1 (0)

{1 +O

(M2

K

(4πfπ)2

)+O

(M2

K

(4πfπ)2

πδMB

MK

)

(36)+O(

M2K

(4πfπ)2

πMK

MB

)+ O

(p4)},

where fSU(3)1 (0) is the value of f1(0) in the SU(3) limit which is fixed by the vector current

conservation. In the r.h.s. of Eq. (36) the second term in the brackets represents the one-loopO(p2) correction, while the subsequent two terms are the (parametrically) sub-leading O(p3)

and O(1/MB) corrections, respectively. The various chiral corrections in Eq. (36) have beencomputed in Ref. [7], which completes and corrects previous determinations presented in theliterature [18,19]. The corresponding numerical estimates are collected in Table 4 for varioushyperon decays. An important feature of the chiral expansion (36) is that the O(p2), O(p3) andO(1/MB) corrections are independent from unknown LECs thanks to the AG theorem.

The sum of all the contributions for f1(0)/fSU(3)1 (0), presented in Table 4, turns out to be

positive and of the order of few percent for the various hyperon decays considered. However thefinal results come from partial cancellation of larger terms. Thus higher order corrections areexpected to give non-negligible contributions, and the convergence of the chiral expansion forhyperon transitions turns out to be questionable [7].

In addition an important source of uncertainty is represented by the mixing with the JP =3/2+ decuplet in the effective field theory calculations. If the mass-shift � between the decupletand the octet hyperons were much larger than the interaction scale ΛQCD, the decuplet contribu-tion could be integrated out and reabsorbed into the LECs, so that no correction would appearup to O(p3) in the chiral expansion. However � � 230 MeV is of order ΛQCD and therefore thedecuplet may give non-negligible, non-analytic contributions to the chiral expansion.

The HBChPT with explicit decuplet degrees of freedom was firstly proposed in Ref. [20]and formalized as an expansion in Ref. [21]. Its impact at O(p2), O(p3) and O(1/MB) hasbeen investigated in Ref. [7]. As for the octet contributions, the AG theorem protects the de-cuplet corrections from unknown LECs and the only new parameter, besides �, is the knowndecuplet–octet–meson coupling C � 1.6 [22]. At O(p2) the dynamical decuplet gives an impor-tant contribution to the Σ− → n transition (−3.1%). At O(p3) there are two contributions. Thefirst one is due to the insertion of decuplet mass-shifts and it is of order −1.8%. The second

Table 4Chiral corrections at the physical point estimated in Ref. [7] for various hyperon decays

f1(0)/fSU(3)1 (0) f

SU(3)1 (0) O(p2) O(p3) O(1/MB) All

Σ− → n −1 +0.7% +6.5% −3.2% +4.1%Λ → p −√

3/2 −9.5% +4.3% +8.0% +2.7%Ξ− → Λ

√3/2 −6.2% +6.2% +4.3% +4.3%

Ξ− → Σ0 1/√

2 −9.2% +2.4% +7.7% +0.9%

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 79

contribution is due to baryon mass-shift insertions and gives a huge contribution of about −38%.This effect completely breaks the chiral expansion raising serious doubts on the consistency ofthe HBChPT with dynamical decuplet. The reason why this effect was not noticed before isbecause other quantities, at this order, contain a large number of LECs which are difficult to es-timate from the data. In the case of f1(0) there are no LECs and a true test of the convergence ofthe chiral expansion is possible.

It is therefore clear that a model-independent estimate of chiral corrections for hyperon decayscannot be given at present. We can limit ourselves to trust HBChPT without dynamical decuplet,making the Ansatz that the decuplet contributions can be reabsorbed into local terms. Underthis assumption the chiral correction to f1(0) for the Σ− → n transition can be estimated fromTable 4 to be (−4 ± 4)%, assuming a 100% overall uncertainty. Thus (at least) in the case of theΣ− → n transition there seems to be a partial cancellation between chiral loop corrections fromthe HBChPT and local contributions from the lattice calculation, Eq. (35), though within quitelarge uncertainties. Adding the two results we get

(37)f1(0) = −0.988 ± 0.029lattice ± 0.040HBChPT.

Unquenched simulations with light quark masses are required to provide a reliable estimate ofSU(3)-breaking corrections to hyperon form factors without relying on the chiral expansion.

5. Results for g1(0)/f1(0)

The ratio of the axial to the vector f.f. at zero-momentum transfer, g1(0)/f1(0), is an importantingredient in the analysis of experimental data on hyperon semileptonic decays (see Refs. [3]and [23]). In this section we first present the lattice results for the momentum dependence of theratio g1(q

2)/f1(q2), then we extrapolate it in q2 down to q2 = 0 and finally we analyze its mass

dependence to obtain the ratio g1(0)/f1(0) at the physical quark masses.Using the standard procedure, the two f.f.’s g1(q

2) and f1(q2) can be separately calculated

for q �= 0 through Eqs. (23) and (18) in terms of the quantities W(A)i (q2) and Wj (q

2) (i, j =1,2,3), defined in Eqs. (22) and (17), respectively. In this way one obtains values of g1(q

2) witha statistical precision similar to the one achieved for the f.f. f1(q

2).A reduction of the statistical noise can be obtained by considering directly the quantity

g1(q2)/f1(q

2) in terms of the ratios W(A)i (q2)/Wj (q

2). In this way the statical noise introducedby the two-point correlation functions in the denominators of Eqs. (21) and (16) is cancelledout. Moreover, also the systematic uncertainty due to different ways of determining the matrixelements

√ZΣ and

√Zn (typically a � 10% effect) is removed. The values of g1(q

2)/f1(q2)

obtained at q �= 0 (i.e., q2 �= q2max) are shown in Fig. 8 for two combinations of quark masses.

At q = 0, corresponding to q2 = q2max, neither g1 nor f1 can be calculated directly. For both

initial and final hyperons at rest there are only two non-vanishing matrix elements with γ = γ ′ =0, Im(AΣn

3 (tx, ty, 0, 0 )00) and Re(V Σn0 (tx, ty, 0, 0 )00). Their ratio provides a new combination

of f.f.’s, namely

(38)Im(AΣn

3 (tx, ty, 0, 0 )00)

Re(V Σn0 (tx, ty, 0, 0 )00)

= g1(q2max) + MΣ−Mn

MΣ+Mng2(q

2max)

f1(q2max) + MΣ−Mn

MΣ+Mnf3(q2

max)≡ g1(q

2max)

f0(q2max)

.

To get g1(q2max)/f1(q

2max) we apply two corrections to Eq. (38). The first one corresponds to mul-

tiply the values of g1(q2max)/f0(q

2max) by f0(q

2max)/f1(q

2max), where the values of f0(q

2max) are ob-

tained with the double ratio (26), while those for f1(q2max) can be evaluated using the dipole fit of

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80 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

Fig. 8. Values of g1(q2)/f1(q2) versus a2q2, obtained for the two combinations of the hopping parameters(ks , k�) = (0.1336,0.1340) (a) and (ks , k�) = (0.1345,0.1343) (b). The dashed and dotted lines represent linear andquadratic fits in q2, respectively.

Eq. (30). The second correction is the subtraction of the contribution due to g2(q2max)/f1(q

2max),

whose values can be obtained by fitting the momentum dependence of the ratio g2(q2)/f1(q

2)

(see next section). Thanks to the smallness of the difference MΣ − Mn in our simulation, bothcorrections turn out to be much smaller than the statistical errors, so that the ratio (38) can besafely corrected providing a determination of g1(q

2max)/f1(q

2max) with quite good statistical pre-

cision (� 10%), as it is shown by the rightmost point in Fig. 8.The addition of a precise determination at q2 = q2

max is crucial to improve the extrapolationto q2 = 0. Indeed, thanks to the closeness of the q2

max values to q2 = 0, the extrapolated valuesg1(0)/f1(0) are only slightly affected by the specific functional form adopted for describing themomentum dependence of g1(q

2)/f1(q2), as it is can be clearly seen from Fig. 8. In Table 5 and

in Fig. 9 we show the results obtained for g1(0)/f1(0) using a linear fit in q2.Since the axial-vector f.f. at zero-momentum transfer, g1(0), is not protected by the AG theo-

rem against first-order corrections in the SU(3)-breaking parameter (ms −m�) ∝ a2(M2K −M2

π),we start the analysis of the mass dependence of g1(0)/f1(0) by considering a linear fit in the twomass combinations a2(M2

K + M2π ) and a2(M2

K − M2π ):

(39)g1(0)

f1(0)= A0 + A1 · a2(M2

K + M2π

) + A2 · a2(M2K − M2

π

).

The resulting values of the parameters are: A0 = −0.281 (59), A1 = 0.05 (20) and A2 =−0.25 (12), which provide at the physical meson masses the value

(40)

[g1(0)

f1(0)

]phys.

= −0.287 ± 0.052,

where the error does not include the quenching effect. Assuming A2 = 0 one gets A0 =−0.300 (61), A1 = 0.14 (21) with [g1(0)/f1(0)]phys. = −0.295 ± 0.053, whereas adoptingA1 = 0 one gets A0 = −0.270 (21), A2 = −0.30 (26) with [g1(0)/f1(0)]phys. = −0.280±0.026.In both cases the values extrapolated to the physical point agree with the result given by Eq. (40)

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 81

Table 5Values of the ratio g1(0)/f1(0), obtained from a linear fit in q2 of the lattice results for g1(q2)/f1(q2)

ks − k� g1(0)/f1(0)

0.1336−0.1340 −0.266 (18)

0.1336−0.1343 −0.261 (19)

0.1336−0.1345 −0.257 (19)

0.1340−0.1336 −0.273 (20)

0.1340−0.1343 −0.267 (22)

0.1340−0.1345 −0.264 (22)

0.1343−0.1336 −0.273 (22)

0.1343−0.1340 −0.275 (23)

0.1343−0.1345 −0.270 (26)

0.1345−0.1336 −0.274 (27)

0.1345−0.1340 −0.278 (28)

0.1345−0.1343 −0.279 (31)

Fig. 9. Results for the ratio g1(0)/f1(0) vs. the meson mass combinations a2(M2K

+ M2π ) (a) and a2(M2

K− M2

π ) (b),

obtained after extrapolation to q2 = 0 through linear fits in q2. Full dots and squares correspond to lattice results calcu-lated with non-degenerate and degenerate quark masses, respectively (i.e. with and without SU(3)-breaking corrections).The dashed and dotted lines are linear fits in the meson masses putting in Eq. (39) A2 = 0 (a) and A1 = 0 (b), while theopen dot and square are the corresponding values extrapolated to the physical point. The diamond represents the valueof the ratio g1(0)/f1(0) adopted in Ref. [3].

within the statistical errors. The two separate linear fits in the mass combinations a2(M2K + M2

π )

and a2(M2K −M2

π ) are shown in Fig. 9 by the dashed lines. We have checked that the results pre-sented for [g1(0)/f1(0)]phys. remain unchanged if instead of the meson masses the hyperon onesare adopted in the fitting procedure based on the Ansatz (39). We also tried quadratic fits in themass combinations a2(M2

K + M2π ) and a2(M2

K − M2π ). The consequent increase of the number

of parameters leads to larger uncertainties, but the shift in the central values remains smaller thanthe statistical errors.

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82 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

The case of degenerate quark masses ms = m� (i.e., q2max = 0) allows us to access directly the

values of g1(0)/f1(0) in the case of exact SU(3) symmetry. Without any extrapolation in q2 weobtain the four values of g1(0)/f1(0) shown in Fig. 9(a) by the full dots. A linear extrapolationin the squared pion mass gives

(41)

[g1(0)

f1(0)

]phys.

SU(3)

= −0.269 ± 0.047,

which nicely agrees with the prediction of the linear fit shown in Fig. 9(b) at the SU(3)-symmetricpoint a2(M2

K −M2π) = 0. The comparison with result (40) indicates that at the physical point the

ratio g1(0)/f1(0) is affected by moderate SU(3)-breaking corrections, though it is not protectedby the AG theorem. Our finding is in qualitative agreement with the exact SU(3)-symmetryassumption of the Cabibbo model [9].

Our result (40) is also consistent within the errors with the value g1(0)/f1(0) = −0.340 ±0.017 adopted in the recent analysis of hyperon decays of Ref. [3], where the exact SU(3)-symmetry assumption for the ratio g1(0)/f1(0) is avoided.

6. Results for the other form factors

In this section we analyze the momentum dependence of the ratios of the remaining f.f.’sf2(q

2), g2(q2), f3(q

2) and g3(q2) to the vector f.f. f1(q

2). The contributions of both f3(q2) and

g3(q2) to the decay distributions are suppressed by the ratio of the lepton to the hyperon mass,

and therefore they are negligible in the case of hyperon decays with electrons, while they can berelevant for muonic decays. The latter however suffer from quite small branching ratios [23].

As described in Section 3 the ratios f2(q2)/f1(q

2) and f3(q2)/f1(q

2) can be calculated usingthe double ratios (28)–(29). The latter can be easily generalized to the axial sector to determinethe ratios g2(q

2)/f1(q2) and g3(q

2)/f1(q2). We introduce the following two double ratios of

three-point correlation functions, corresponding to matrix elements of spatial and time compo-nents of the weak axial current:

R(A)1

(q2; tx, ty

) ≡ Re(AΣn0 (tx, ty, q, 0 )00)

Im(AΣn(tx, ty, q, 0 )00)

Im(AΣΣ(tx, ty, q, 0 )00)

Re(AΣΣ0 (tx, ty, q, 0 )00)

,

(42)R(A)2

(q2; tx, ty

) ≡ Re(AΣn0 ) + (Mn − MΣ) Im(AΣn

1 )/q1

Re(AΣn0 (tx, ty, q, 0 )00)

Im(AΣΣ(tx, ty, q, 0 )00)

Re(AΣΣ0 (tx, ty, q, 0 )00)

.

We emphasize that the ratios (42) are exactly equal to unity in the SU(3) limit, and therefore thedeviation of these ratios from one are a measure of SU(3)-breaking corrections. This is crucialfor obtaining a determination of f.f.’s like the weak electricity g2(q

2) (and the induced scalarf3(q

2) in the vector case), which vanish identically in the SU(3) limit.In terms of the large-time limits R

(A)i (q2) ≡ limtx ,(ty−tx )→∞ R

(A)i (q2; tx, ty) one has

R(A)1

(q2) = (MΣ + Mn)g1(q

2) − (Eq + MΣ)g2(q2) + (Eq − Mn)g3(q

2)

(MΣ + Mn)g1(q2) − (Eq − Mn)g2(q2) + (Eq − MΣ)g3(q2),

(43)R(A)2

(q2) = (MΣ + Mn)g1(q

2) − (Eq + Mn)g2(q2) + (Eq − MΣ)g3(q

2)

(MΣ + Mn)g1(q2) − (Eq − Mn)g2(q2) + (Eq − MΣ)g3(q2),

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 83

which can be easily solved in terms of g2(q2)/g1(q

2) and g3(q2)/g1(q

2). The latter canbe multiplied in turn by the values of g1(q

2)/f1(q2) to get the ratios g2(q

2)/f1(q2) and

g3(q2)/f1(q

2).The study of the momentum dependence of the four ratios f2(q

2)/f1(q2), f3(q

2)/f1(q2),

g2(q2)/f1(q

2) and g3(q2)/f1(q

2) suffers however of a crucial difference with respect to the caseof the ratio g1(q

2)/f1(q2), namely the absence of a determination at q2 = q2

max and, in general,of a lattice point sufficiently close to q2 = 0. This limitation makes the extrapolation to q2 = 0plagued by larger uncertainties. A way to cure this problem may be the use of twisted boundaryconditions for the quark fields [24]. Indeed, as shown in Ref. [25], non-periodic boundary con-ditions may greatly help in determining form factors at zero-momentum transfer, when the lattercannot be determined sufficiently close to q2 = 0 using periodic boundary conditions.

We discuss first the ratios f2(q2)/f1(q

2) and g3(q2)/f1(q

2), and then the ratios g2(q2)/f1(q

2)

and f3(q2)/f1(q

2), because the former have a non-vanishing SU(3) limit, while the latter, beingdue to second-class currents [26], are totally generated by SU(3)-breaking corrections.

6.1. Results for f2(0)/f1(0) and g3(0)/f1(0)

The momentum dependence of the ratios f2(q2)/f1(q

2) and g3(q2)/f1(q

2) is shown inFigs. 10 and 11, respectively. In the case of g3(q

2)/f1(q2) the statistical error is very large for

the kinematical point corresponding to initial momentum p = (2π/aL) · (2,0,0) and thereforethis point is not reported in Fig. 11.

The absence of a determination sufficiently close to q2 = 0 as well as the limited number ofpoints introduces some sensitivity of the values extrapolated at zero-momentum transfer to thespecific functional form assumed for the q2-dependence of the f.f.’s. The values of the ratiosextrapolated to zero-momentum transfer adopting a linear fit in q2 are collected in Table 6 andshown in Fig. 12. Note that the uncertainties are always larger than 50%.

Fig. 10. Values of f2(q2)/f1(q2) versus a2q2 for the two combinations of the hopping parameters(ks , k�) = (0.1336,0.1340) (a) and (ks , k�) = (0.1345,0.1343) (b). The dashed and dotted lines represent linear andquadratic fits in q2, respectively.

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84 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

Fig. 11. The same as in Fig. 10 but for the ratio g3(q2)/f1(q2).

Table 6Values of the ratios f2(0)/f1(0) and g3(0)/f1(0), obtained by performing a linear fit in q2

ks − k� f2(0)/f1(0) g3(0)/f1(0)

0.1336−0.1340 −0.18 (20) 4.9 (2.7)

0.1336−0.1343 −0.20 (21) 4.5 (2.7)

0.1336−0.1345 −0.21 (22) 3.9 (2.6)

0.1340−0.1336 −0.31 (26) 5.6 (2.9)

0.1340−0.1343 −0.34 (30) 4.8 (3.1)

0.1340−0.1345 −0.36 (31) 4.3 (2.9)

0.1343−0.1336 −0.44 (31) 5.9 (3.2)

0.1343−0.1340 −0.46 (34) 5.7 (3.4)

0.1343−0.1345 −0.50 (37) 5.0 (3.5)

0.1345−0.1336 −0.56 (34) 6.0 (3.4)

0.1345−0.1340 −0.58 (37) 5.8 (3.6)

0.1345−0.1343 −0.60 (39) 5.5 (3.8)

The mass dependence of f2(0)/f1(0) is well described by a simple linear fit in a2(M2K +

M2π ), whereas the dependence upon the variable a2(M2

K − M2π ) is found to be negligible. In the

case of g3(0)/f1(0) the findings are opposite and the dependence upon the variable a2(M2K +

M2π ) is negligible. Using linear fits in a2(M2

K + M2π ) for f2(0)/f1(0) and in a2(M2

K − M2π ) for

g3(0)/f1(0), we obtain at the physical point: f2(0)/f1(0) = −1.14 ± 0.66 and g3(0)/f1(0) =6.3 ± 3.5. In order to investigate the stability of the extrapolation we consider also a quadratic fitin a2(M2

K +M2π ) for f2(0)/f1(0) and in a2(M2

K −M2π ) for g3(0)/f1(0), obtaining the following

results that we quote as our final estimates of these quantities

(44)

[f2(0)

f1(0)

]phys.

= −1.52 ± 0.81,

(45)

[g3(0)

f1(0)

]phys.

= +6.1 ± 3.3.

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 85

Fig. 12. Results for the ratios f2(0)/f1(0) (a) and g3(0)/f1(0) (b) vs. the meson mass combinations a2(M2K

+M2π ) in (a)

and a2(M2K

− M2π ) in (b), obtained after extrapolation to q2 = 0 through linear fits in q2. The dashed (dotted) lines are

linear (quadratic) fits and the open dots (squares) are the corresponding values extrapolated to the physical point. Thediamond represents: in (a) the experimental value of f2(0)/f1(0) from Ref. [10], and in (b) the value of g3(0)/f1(0),obtained using the generalized Goldberger–Treiman relation [12] and the axial Ward identity (see text).

In the case of the weak magnetism f.f. the experimental result for f2(0)/f1(0) is known fromRef. [10], namely f2(0)/f1(0) = −1.71 ± 0.12stat. ± 0.23syst., which is shown in Fig. 12(a). Inthe case of g3(0)/f1(0) instead there is no direct experimental information. By combining thegeneralized Goldberger–Treiman relation [12] and the axial Ward identity one can argue that thef.f. g3(q

2) should have a K-meson pole at low values of q2, which implies that close to the chirallimit one has

(46)g3(0)

f1(0)=

(MΣ + Mn

MK

)2g1(0)

f1(0).

Using the result (40) we get g3(0)/f1(0) = 5.5 ± 0.9, which is consistent with Eq. (44). Thisestimate is also presented in Fig. 12(b). Note that both the latter and the experimental value forf2(0)/f1(0) are quite closer to the results (45) and (44), respectively.

A calculation of the weak magnetism f2 and the induced pseudoscalar g3 f.f.’s for values ofq2 closer to q2 = 0 and at lower values of the quark masses is mandatory to improve the precisionof the lattice determination of these f.f.’s.

6.2. Results for g2(0)/f1(0) and f3(0)/f1(0)

The momentum dependence of the ratios g2(q2)/f1(q

2) and f3(q2)/f1(q

2) is presented inFigs. 13 and 14, respectively. In the SU(3) limit both the weak electricity g2(q

2) and the inducedscalar f3(q

2) f.f.’s vanish identically, so that the results shown in Figs. 13 and 14 are totallygenerated by the breaking of the SU(3) symmetry. As in the case of g3(q

2)/f1(q2) the statistical

error for g2(q2)/f1(q

2) becomes very large for the largest absolute q2 value, which therefore isnot presented in Fig. 13. Note that in Figs. 13 and 14 we have explicitly reversed the signs of thevalues of g2(q

2) and f3(q2) corresponding to the n → Σ transitions, because both g2(q

2) andf3(q

2) belong to second-class currents.

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86 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

Fig. 13. Values of g2(q2)/f1(q2) versus a2q2 for the two combinations of the hopping parameters(ks , k�) = (0.1336,0.1340) (a) and (ks , k�) = (0.1345,0.1343) (b). The dashed and dotted lines represent linear andquadratic fits in q2, respectively.

Fig. 14. The same as in Fig. 13 but for the ratio f3(q2)/f1(q2).

The sensitivity of the extrapolated values at zero-momentum transfer to the specific functionalform assumed for the q2-dependence of the two ratios g2(q

2)/f1(q2) and f3(q

2)/f1(q2) is much

more limited with respect to what has been observed in the case of the ratios f2(q2)/f1(q

2) andg3(q

2)/f1(q2). The values of g2(0)/f1(0) and f3(0)/f1(0), obtained adopting a linear fit in q2,

are presented in Table 7 and in Figs. 15(a) and 16(a). Note again that, as expected for second-classcurrents, the signs of both g2(0)/f1(0) and f3(0)/f1(0) are linked to the sign of a2(M2

K − M2π ).

Since the values of g2(0)/f1(0) and f3(0)/f1(0) are exactly known in the SU(3) limit (i.e.,g2(0)/f1(0) = f3(0)/f1(0) = 0), the analysis of the mass dependence of these ratios can proceedin a way similar to the one adopted for f1(0). In this case, however, taking into account that both

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 87

Table 7Values of the ratios g2(0)/f1(0) and f3(0)/f1(0), obtained by performing a linear fit in q2

ks − k� g2(0)/f1(0) f3(0)/f1(0)

0.1336−0.1340 −0.083 (38) +0.065 (14)

0.1336−0.1343 −0.139 (70) +0.123 (28)

0.1336−0.1345 −0.185 (92) +0.166 (40)

0.1340−0.1336 +0.100 (43) −0.071 (19)

0.1340−0.1343 −0.085 (37) +0.064 (19)

0.1340−0.1345 −0.132 (66) +0.112 (35)

0.1343−0.1336 +0.220 (88) −0.146 (47)

0.1343−0.1340 +0.094 (42) −0.070 (25)

0.1343−0.1345 −0.071 (32) +0.053 (22)

0.1345−0.1336 +0.326 (123) −0.213 (80)

0.1345−0.1340 +0.180 (77) −0.132 (56)

0.1345−0.1343 +0.066 (33) −0.057 (27)

Fig. 15. Results for the ratios g2(0)/f1(0) (a) and Rg2 (b) vs. the meson mass combinations a2(M2K

− M2π ) in (a) and

a2(M2K

+ M2π ) in (b), obtained after extrapolation to q2 = 0 through linear fits in q2. The dashed line is the result of

a linear fit and the open dot is the corresponding value extrapolated to the physical point.

g2 and f3 are not protected by the AG theorem, we introduce the following ratios:

(47)Rg2(MK,Mπ) = g2(0)

f1(0)

1

a2(M2K − M2

π ),

(48)Rf3(MK,Mπ) = f3(0)

f1(0)

1

a2(M2K − M2

π ).

The mass dependence of these quantities is well described by simple linear fits in the masscombination a2(M2

K + M2π ), whereas the dependence upon the other variable a2(M2

K − M2π )

turns out to be negligible. At the physical point we get Rg2(Mphys.K ,M

phys.π ) = 18.0 ± 7.5 and

Rf3(Mphys.K ,M

phys.π ) = −11.9±6.2. A quadratic fit in the mass combination a2(M2

K +M2π ) does

not modify the central value of Rg2 , but it increases the uncertainty, i.e. Rg2(Mphys.

,Mphys.π ) =

K
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88 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

Fig. 16. The same as in Fig. 15 but for the ratios f3(0)/f1(0) (a) and Rf3 (b).

18.3 ± 14.6. A slight shift in the central value within a much larger uncertainty is found in thecase of Rf3 , namely Rf3(M

phys.K ,M

phys.π ) = −18.4 ± 15.5. Thus, using the values of Rg2 and Rf3

from the linear fits, we derive our final estimates of g2(0)/f1(0) and f3(0)/f1(0) extrapolated tothe physical point

(49)

[g2(0)

f1(0)

]phys.

= +0.63 ± 0.26,

(50)

[f3(0)

f1(0)

]phys.

= −0.42 ± 0.22.

Our results indicate a positive, non-vanishing value of g2(0)/f1(0) due to SU(3)-breakingcorrections. In the experiments neither g1(0)/f1(0) nor g2(0)/f1(0) are separately determined,but only a specific combination of these ratios can be extracted. The following combination hasbeen determined for the Σ− → n transition [10]

(51)

∣∣∣∣g1(0) − 0.133 · g2(0)

f1(0)

∣∣∣∣exp.

= 0.327 ± 0.007stat. ± 0.019syst..

Using our results (40) and (49) we get

(52)

∣∣∣∣g1(0) − 0.133 · g2(0)

f1(0)

∣∣∣∣phys.

= 0.37 ± 0.08

in good agreement with the experimental value. Even though the experimental data are compati-ble with general fitting procedures which make the conventional assumption g2(q

2) = 0 (as donefor instance in Ref. [3]), it has been found that positive values for g2(0)/f1(0) combined witha corresponding reduced value for |g1(0)/f1(0)| are slightly preferred [10]. The lattice results(40) and (49) appear to favor the second scenario.

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D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91 89

7. Conclusions

We have presented a lattice QCD study of SU(3)-breaking corrections in the vector and axialform factors relevant for the hyperon semileptonic decay Σ− → n�ν. Though our simulation hasbeen carried out in the quenched approximation, our results represent the first attempt to evaluatehyperon form factors using a non-perturbative method based only on QCD.

For each form factor we have studied its momentum and mass dependencies, obtaining itsvalue extrapolated at zero-momentum transfer and at the physical point. Our final results arecollected in Table 8, where the errors do not include the quenching effect.

We conclude with few additional comments:

• The SU(3)-breaking corrections to the vector form factor at zero-momentum transfer, f1(0),have been determined with great statistical accuracy in the regime of the simulated quarkmasses, which correspond to pion masses above 0.7 GeV. The magnitude of the errors re-ported in Table 8 is mainly due to the chiral extrapolation and to the poor convergence of theHeavy Baryon Chiral Perturbation Theory [7]. Though within large errors the central valueof f1(0) arises from a partial cancellation between the contributions of local terms, evalu-ated on the lattice, and chiral loops. This may indicate that SU(3)-breaking corrections onf1(0) are moderate (at least for the transition considered), giving support to the analysis ofRef. [3].

• The ratio g1(0)/f1(0) is found to be negative and consistent with the value adopted inthe analysis of Ref. [3]. The study of the degenerate transitions also allowed us to deter-mine the value of g1(0)/f1(0) directly in the limit of exact SU(3) symmetry, obtaining[g1(0)/f1(0)]SU(3) = −0.269±0.047. This means that SU(3)-breaking corrections are mod-erate also on this ratio, though the latter is not protected by the Ademollo–Gatto theoremagainst fist-order corrections.

• The weak electricity form factor at zero-momentum transfer g2(0) is found to be non-vanishing because of SU(3)-breaking corrections. Our result for g1(0)/f1(0) combined withthat of g2(0)/f1(0) are nicely consistent with the experimental result from [10]. Our findingsfavor the scenario in which g2(0)/f1(0) is large and positive with a corresponding reducedvalue for |g1(0)/f1(0)| with respect to the conventional assumption g2(q

2) = 0 (done forinstance in Ref. [3]) based on exact SU(3) symmetry.

Finally, we discuss few possible improvements for future lattice QCD studies of the hyperonsemileptonic transitions:

• The quenched approximation should be removed and the simulated quark masses should belowered as much as possible in order to reduce the impact of the chiral extrapolation.

Table 8Results of our lattice calculations of the vector and axial form factors for the Σ− → n transition

f1(0) −0.988 ± 0.029lattice ± 0.040HBChPT

g1(0)/f1(0) −0.287 ± 0.052f2(0)/f1(0) −1.52 ± 0.81f3(0)/f1(0) −0.42 ± 0.22g2(0)/f1(0) +0.63 ± 0.26g3(0)/f1(0) +6.1 ± 3.3

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90 D. Guadagnoli et al. / Nuclear Physics B 761 (2007) 63–91

• The accuracy of the ratios f2(0)/f1(0), f3(0)/f1(0), g2(0)/f1(0) and g3(0)/f1(0) can beimproved by implementing twisted boundary conditions for the quark fields. In this wayvalues of the momentum transfer closer to q2 = 0 can be accessed in the simulations.

• The use of smeared source and sink for the interpolating fields as well as the use of several,independent interpolating fields may help in increasing the overlap with the ground-statesignal, particularly at low values of the quark masses.

Acknowledgements

The authors warmly thank G. Martinelli and G. Villadoro for many useful discussions, andare grateful to E.C. Swallow for useful comments. D.G. acknowledges the financial support of“Fondazione Della Riccia”, Florence (Italy).

References

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[9] N. Cabibbo, Unitary symmetry and leptonic decays, Phys. Rev. Lett. 10 (1963) 531.[10] S.Y. Hsueh, et al., A high precision measurement of polarized sigma–beta decay, Phys. Rev. D 38 (1988) 2056.[11] A. Sirlin, Nucl. Phys. B 161 (1979) 301.[12] M. Goldberger, S.B. Treiman, Phys. Rev. 110 (1958) 1178;

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Nuclear Physics B 761 [FS] (2007) 93–108

Stochastic ϕ4-theory in the strong coupling limit

N. Abedpour a, M.D. Niry a, A. Bahraminasab b,c,∗, A.A. Masoudi d,J. Davoudi b, Muhammad Sahimi e, M. Reza Rahimi Tabar a,f

a Department of Physics, Sharif University of Technology, PO Box 11365-9161, Tehran, Iranb International Center for Theoretical Physics, Strada Costiera 11, I-34100 Trieste, Italy

c Department of Physics, Lancaster University, Lancaster, LA1 4YB, UKd Department of Physics, Alzahra University, Tehran 19834, Iran

e Mork Family Department of Chemical Engineering & Materials Science,University of Southern California Los Angeles, CA 90089-1211, USA

f CNRS UMR 6529, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France

Received 19 July 2006; received in revised form 6 September 2006; accepted 27 September 2006

Available online 23 October 2006

Abstract

The stochastic ϕ4-theory in d dimensions dynamically develops domain wall structures within which theorder parameter is not continuous. We develop a statistical theory for the ϕ4-theory driven with a randomforcing which is white in time and Gaussian-correlated in space. A master equation is derived for theprobability density function (PDF) of the order parameter, when the forcing correlation length is muchsmaller than the system size, but much larger than the typical width of the domain walls. Moreover, exactexpressions for the one-point PDF and all the moments 〈ϕn〉 are given. We then investigate the intermittencyissue in the strong coupling limit, and derive the tail of the PDF of the increments ϕ(x2)−ϕ(x1). The scalinglaws for the structure functions of the increments are obtained through numerical simulations. It is shownthat the moments of field increments defined by, Cb = 〈|ϕ(x2) − ϕ(x1)|b〉, behave as |x1 − x2|ξb , whereξb = b for b � 1, and ξb = 1 for b � 1.© 2006 Elsevier B.V. All rights reserved.

PACS: 05.10.Gg; 11.10.Lm

* Corresponding author.E-mail address: [email protected] (A. Bahraminasab).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2006.09.026

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1. Introduction

Thirty five years ago Wilson and Fisher [1] emphasized the relevance of the ϕ4-theory to un-derstanding the critical phenomena. Since then, the theory has become one of the most appealingtheoretical tools for studying the critical phenomena in a wide variety of systems in statisticalphysics. In the strong coupling limit, the ϕ4-theory develops domain walls, a phenomenon whichis of great interest in the classical and quantum field theories [2–10]. The dynamical ϕ4-theory—what is usually referred to as the time-dependent Ginzburg–Landau (GL) theory—provides aphenomenological approach to, and plays an important role in, understanding dynamical phasetransitions and calculating the associated dynamical exponent [11–16]. The time-dependent GLtheory for superconductors was presented phenomenologically only in 1968 by Schmid [17] (andderived from microscopic theory shortly thereafter [18]), when the first modulational theory wasderived in the context of Rayleigh–Benard convection [19,20]. Moreover, the GL equation withan additional noise term has been studied intensively as a model of phase transitions in equilib-rium systems; see, for example, [21].

In the present paper we consider the stochastic ϕ4-theory in the strong coupling limit. Thislimit is singular in the sense that, the equation that describes the dynamics of the system developssingularities. Therefore, starting with a smooth initial condition, the domain-wall singularities aredynamically developed after a finite time. At the singular points the field ϕ(x, t) is not contin-uous. We derive master equations for the joint probability density functions (PDF) of ϕ and itsincrements in d dimensions. It is shown that in the stationary state, where the singularities arefully developed, the relaxation term in the strong coupling limit leads to an unclosed term in thePDF equations.

Using the boundary layer method [22–24], we show that the unclosed term makes no finitecontribution (anomaly) in the strong coupling limit, and derive the PDF of ϕ and its moments,〈ϕn〉, in the same limit. We also investigate the scaling behavior of the moments of the field’sincrements defined by, δϕ = ϕ(x2) − ϕ(x1), and show that when |x2 − x1| is small, fluctuationsof the ϕ field have a bi-fractal structure and are intermittent. The intermittency implies thatthe structure function defined by, Cb = 〈|ϕ(x2) − ϕ(x1)|b〉, scales as |x2 − x1|ζb , where ζb is anonlinear function of b. It is also shown numerically that the moments of the field’s increments,〈|ϕ(x2) − ϕ(x1)|b〉, behave as, |x2 − x1|ξb , where xb = b for b � 1, and ξb = 1 for b � 1.

The rest of this paper is organized as follows. In the next section we present the model that wewish to study, and describe some of its properties by solving it numerically. In Sections 3 and 4we derive master equations for the order parameter of the model, and for the field’s incrementsand its PDF tail. The numerical simulations for extracting the scaling exponents are describedin Section 5. The paper is summarized in Section 6, while Appendices A and B provide sometechnical details of the work that we present in the main part of the paper.

2. The model and the coupling constant

The standard GL ϕ4-theory describes a second-order phase transition in any system with aone-component order parameter ϕ(x) and the ϕ → −ϕ symmetry in a zero external field. Thetheory is described by the following action,

(1)S =∫ [

−k

2(∇ϕ)2 + τ

2!ϕ2 − g

4!ϕ4]

ddx,

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N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108 95

where τ = T − Tc , with Tc being the critical temperature, and k is the diffusion coefficient. Weconsider the case in which τ > 0. For d � 2, the critical temperature Tc is finite, while in onedimension (1D), Tc = 0. The parameter g characterizes the strength of the fluctuation interaction,or the coupling constant. The equation of motion is given by,

(2)k∇2ϕ + τϕ − g

6ϕ3 = 0.

The critical dynamics of the system is described by a stochastic equation of a particularform—the Langevin equation (for a comprehensive review on Langevin type equations see[25])—given by

(3)∂ϕ

∂t= k∇2ϕ + τϕ − g

6ϕ3 + η(x, t),

where η(x, t) is a Gaussian-distributed noise with zero average and the correlation function,

(4)⟨η(x, t)η(x′, t ′)

⟩ = D0D(x − x′)δ(t − t ′),with D(x−x′) being an arbitrary smooth function. Typically, the spatial correlation of the forcingterm is considered to be a delta function in order to mimic short-range correlations. Here, though,the spatial correlation is defined by

(5)D(x − x′) = 1

(πσ 2)d/2exp

[− (x − x′)2

σ 2

],

where σ � L endows a short-range character to the random forcing. It is useful to rescaleEq. (3) by writing, ϕ′ = ϕ/ϕ0, x′ = x/x0, and t ′ = t/t0. If we let t0 = 1/τ , ϕ0 = (6τ/g)1/2,and, x0 = [(d0g)/(6τ 2)]1/d , all the parameters are eliminated in Eq. (3) except for, k′ =[6/(D0g)]2/dτ 4/d−1k, and one finds that

(6)∂ϕ

∂t= k′∇2ϕ + ϕ − ϕ3 + η(x, t),

where k′ is now the effective coupling constant of the theory. The weak and strong coupling limitsof the theory are then defined, respectively, by, k′ → ∞ and k′ → 0. In the weak coupling limitone can use numerical simulations and the Feynman diagrams to calculate the critical exponents.On the other hand, to solve the problem in the strong coupling limit we need other techniquesto derive the stochastic properties of the fluctuation field [16]. The nonlinearity of Eq. (6) in thestrong coupling limit gives rise to the possibility of formation of singularity in a finite time. Thismeans that there is a competition between the smoothing effect of diffusion (the Laplacian term)and the ϕ3 term. Let us now describe the main properties of the GL theory in the limit, k′ → 0.

(i) The unforced GL model [η(x, t) = 0], with given initial conditions, develops singularitiesin any spatial dimension. In one spatial dimension (1D) the singularities are developed in a finitetime tc as k′ → 0. At such singular points the field ϕ, representing an order parameter, is notcontinuous. In 2D the unforced GL model develops domain walls, characterized by singularlines with finite lengths (that depend on the initial condition). Under these conditions, the fieldϕ is discontinuous when crossing the singular lines. In three and higher dimensions the structureof the singularities can be more complex. For example, in 3D the singularities are domain wallswhere the field ϕ is discontinuous.

In Fig. 1 we show the time evolution of the order parameter ϕ of the unforced GL model in 2D,in the limit k′ → 0 [Eq. (6) with η = 0]. We have used the finite-element method to numericallysolve the Langevin equation with k′ → 0 and the initial condition, ϕ(x, y,0) = sinx siny. Such

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initial conditions are typical, and were used only for simplicity. The time scale for reaching thesingularity is of the order of k′−1/2. As Fig. 1(a) and (b) indicate, it is evident that at timest < tc (in the limit, k′ → 0) the ϕ field is continuous. At t = tc the ϕ field becomes singular; seeFig. 1(c).

(ii) Similarly, for a forcing term which is white noise in time and smooth in space, singularitiesare developed in any spatial dimension in the strong coupling limit and in a finite time, tη = tc,η,as k′ → 0. For example, in 2D the boundaries of the domain walls are smooth curves. In Fig. 2we demonstrate the time evolution of the order parameter ϕ of the forced GL model in 2D inthe limit k′ → 0. Starting from a smooth initial condition, as shown in Fig. 2(a) and (b), it isevident that for times t < tc,η the ϕ field is continuous. At t = tc,η the field becomes singular; seeFig. 2(c).

3. Master equation of the order parameter

In this section we derive a master equation to describe the time evolution of the PDF P(ϕ, t)

of the order parameter ϕ. Defining a one-point generating function by, Z(λ) = 〈Θ〉, where Θ isdefined by, Θ = exp[−iλϕ(x, t)]. Using Eq. (6), the time evolution of Z is governed by

(7)Zt = −iλk′⟨∇2ϕ exp[−iλϕ(x, t)

]⟩ − iλ〈ϕΘ〉 + iλ⟨ϕ3Θ

⟩ − iλ〈ηΘ〉 − λ2k(0)Z,

where, k(x) = D0D(x), and we have invoked Novikov’s theorem (see Appendix A), which isexpressed via the relation,

(8)⟨η exp

[−iλϕ(x, t)]⟩ = −iλk(0)Z.

Now, using the identities, −iλ〈ϕ exp[−iλϕ(x, t)]〉 = λZλ, and −iλ〈ϕ3 exp[−iλϕ(x, t)]〉 =λZλλλ, the generating function Z satisfies the following unclosed master equation

(9)Zt = −iλk′⟨∇2ϕ exp[−iλϕ(x, t)

]⟩ + λZλ + λZλλλ − λ2k(0)Z.

The −iλk′〈∇2ϕ exp[−iλϕ(x, t)]〉 term of Eq. (9) is the only one which is not closed with respectto Z. The PDF of order parameter P(ϕ) is constructed by Fourier transforming the generatingfunction Z:

(10)P(ϕ, t) =∫

2πexp(iλϕ)Z(λ, t).

Thus,

(11)Pt = −[(ϕ − ϕ3)P ]

ϕ+ k(0)Pϕϕ − ik′

∫dλ

2πλ exp(iλϕ)

⟨∇2ϕ exp[−iλϕ(x, t)

]⟩.

It is evident that the governing equation for P(ϕ, t) is also not closed.Let us now use the boundary layer technique to prove that the unclosed term [the last term of

Eq. (11)] makes, in the strong coupling limit, no contribution to the governing equation for thePDF [22–24]. We consider two different time scales in the limit, k′ → 0. (i) Early stages beforedeveloping the singularities (t < tc,η), and (ii) in the regime of established stationary state withfully-developed sharp singularities (t � tc,η).

In regime (i), ignoring the relaxation term in the governing equation for the PDF, one finds,in the limit k′ → 0, the exact equation for the time evolution of the PDF for the order parameter(see below for more details). In contrast, the limit k′ → 0 is singular in regime (ii), leading toan unclosed term (the relaxation term) in the equation for the PDF. However, we show that the

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Fig. 1. Snapshots of the order parameter ϕ at varioustimes in the 2D unforced ϕ4-theory in the strong cou-pling limit, k′ → 0. Top two figures show the ϕ fieldbefore the singularity develops. In the bottom figure,which is for time scales greater than the time at whichsingularity develops, the ϕ(x, y) field is not continuous.The initial condition is, ϕ(x, y,0) = sinx siny.

Fig. 2. Snapshots of the order parameter ϕ of a ran-domly-driven 2D ϕ4-theory in the strong coupling limit.In the simulations the relation between the forcing lengthscale σ and the sample size L is, σ L/3. The forcingstrength D0 is 0.1.

unclosed term scales as k′1/2, implying that this term, in the strong coupling limit, makes no finitecontribution or anomaly to the solution of Eq. (11). It is known for such time scales (the stationarystate) that the ϕ-field, which satisfies the Langevin equation, gives rise to discontinuous solutionsin the limit, k′ → 0. Consequently, for finite σ the singular solutions form a set of points where

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98 N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108

the domain walls are located, and are continuously connected. We should note that k′ϕxx , in thelimit k′ → 0, is zero at those points at which there is no singularity. Therefore, in the limit k′ → 0only small intervals around the walls contribute to the integral in Eq. (11). Within these intervals,a boundary layer analysis may be used for obtaining accurate approximation of ϕ(x, t).

Generally speaking, the boundary layer analysis deals with problems in which the pertur-bations are operative over very narrow regions, across which the dependent variables undergovery rapid changes. The narrow regions, usually referred to as the domain walls, frequently ad-join the boundaries of the domain of interest, due to the fact that a small parameter (k′ in thepresent problem) multiplies the highest derivative. A powerful method for treating the boundarylayer problems is the method of matched asymptotic expansions. The basic idea underlying thismethod is that, an approximate solution to a given problem is sought, not as a single expansionin terms of a single scale, but as two or more separate expansions in terms of two or more scales,each of which is valid in some part of the domain. The scales are selected such that the expansionas a whole covers the entire domain of interest, and the domains of validity of neighboring ex-pansions overlap. In order to handle the rapid variations in the domain walls’ layers, we define asuitable magnified or stretched scale and expand the functions in terms of it in the domain walls’regions.

For this purpose, we split ϕ into a sum of inner solution near the domain walls and an outersolution away from the singularity lines, and use systematic matched asymptotics to construct auniform approximation of ϕ. For the outer solution, we look for an approximation in the form ofa series in k′,

(12)ϕ = ϕout = ϕ0 + k′ϕ1 + O(k′2),

where ϕ0 satisfies the following equation

(13)ϕ0t = ϕ0 − ϕ30 + η(x, t).

Indeed, ϕ0 satisfies Eq. (6) with k′ = 0. Far from the singular points or lines, the PDF of ϕ0 satis-fies the Fokker–Planck equation, with the drift and diffusion coefficients being, D(1)(ϕ0, t) =ϕ0 − ϕ3

0 , and D(2) = k(0), respectively. Ref. [16] gives the solution of the time-dependentFokker–Planck equation with such drift and diffusion coefficients. At long times and in the areafar from the singular points or lines, the PDF of ϕ0 will have two maxima at ±1. This means thatwe are dealing with the smooth areas in Fig. 2(c) in the stationary state.

In order to deal with the inner solution around the domain walls, we consider the x componentnormal to the domain wall or singularity line, and decompose the operator ∇2 as ∂xx + ∇2

d−1. Inthe strong coupling limit, k′ → 0, the term ∇2

d−1ϕ makes no contribution to the PDF equation,whereas the term ∂xxϕ is singular. To derive the long-time solution of Eq. (6), we rescale x toz ≡ x√

2k′ and suppose that complete solution of Eq. (6) has the form, ϕ(z, t) = f (z, t)+ tanh(z).

All the effects of the initial condition and time-dependence of ϕ(x, t) will then be contained inf (z, t). We now rewrite Eq. (6) with the new variables to obtain

∂tf (z, t) = 1

2∂zzf (z, t) + f (z, t) − f 3(z, t) − 3f (z, t) tanh(z)

[f (z, t) + tanh(z)

](14)+ 4

√2k′η(z, t).

The last term of Eq. (14) is zero in the limit k′ → 0. Multiplying Eq. (14) by f (z, t) and inte-grating over z, one finds that,

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N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108 99

Fig. 3. Time variations of∫

dzf 2(z, t) vs. t with different types of initial conditions. The results show that∫dzf 2(z, t) → 0 at long times.

∂t

∫dzf 2(z, t) = −

∫dz

(∂zf (z, t)

)2 + 2∫

dzf 2(z, t) − 2∫

dzf 4(z, t)

(15)− 6∫

dzf 3(z, t)a(z) − 6∫

dza2(z)f (z, t)2,

where a(z) = tanh(z). We show in Fig. 3 the time variations of∫

dzf 2(z, t) verses t with dif-ferent types of initial conditions. The results show that

∫dzf 2(z, t) vanishes at long times.

Therefore, ϕ(z, t) → tanh(z), in the limit of a stationary state. Let us now compute the contribu-tion of the unclosed term in Eq. (11) in the stationary state, i.e.,

−k′∫

2πiλeiλϕ

⟨∇2ϕe−iλϕ(x,t)⟩

= −k′(∫

2πeiλϕ

⟨∇2ϕe−iλϕ(x,t)⟩)

ϕ

(16)= −k′⟨∇2ϕδ[ϕ − ϕ(x, t)

]⟩ϕ.

In the second line of Eq. (16), we have replaced iλ with differentiation with respect to ϕ and inthe third line the integration of λ has been carried through. Now, assuming ergodicity, the termk′〈∇2ϕδ(ϕ − ϕ(x, t))〉 is converted to,

(17)= −k′ limV →∞

1

V

∫V

dx dvd−1 ∇2ϕδ[ϕ − ϕ(x, t)

].

In the limit k′ → 0, only at points where we have singularity the above term is not zero. There-fore, we approach the domain walls’ regions as

(18)= −k′ limV →∞

1

V

∑j

∫Ωj

dx dvd−1(ϕxx + ∇2

d−1ϕ)δ[ϕ − ϕ(x, t)

],

where Ωj is the space close to the domain walls. Therefore, Eq. (16) is written as

(19)= −k′ limV →∞

1

V

∑j

∫Ω

dx dvd−1 ϕxxδ[ϕ − ϕ(x, t)

].

j

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100 N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108

Changing the variables from x to z and integrating over dvd−1, we have

= −k′ limV →∞

Vd−1

V

∑j

+∞∫−∞

ε dz1

ε2ϕzzδ

[ϕ − ϕ(z, t)

]

= −k′

εlim

V →∞Vd−1

V

∑j

+∞∫−∞

dzϕzzδ[ϕ − ϕ(z, t)

],

where ε = (2k′)1/2. Assuming statistical homogeneity, one finds,

(20)= −k′

εlim

V →∞N × Vd−1

V

+∞∫−∞

dzϕzzδ[ϕ − ϕ(z, t)

],

where N is number of singular lines. The quantity NVd−1/V is the density of the singular lines,and k′/ε = (k′/2)1/2. In the limit, V → ∞, we denote the density of the singularities by ρ.Therefore,

(21)= −(k′/2)1/2ρ

+∞∫−∞

dzϕzzδ[ϕ − ϕ(z, t)

].

Now, by changing the integration variable form z to ϕ, we can determine the integral exactly. Wefind that,

(22)

+∞∫−∞

dzϕzzδ[ϕ − ϕ(z, t)

] =+1∫

−1

dϕϕzz

ϕz

δ[ϕ − ϕ(z, t)

].

Using Eq. (6) in the limit, t → ∞, we determine ϕzz and ϕz in terms of ϕ. Multiplying Eq. (6)by ϕz and integrating over z, we obtain,

(23)ϕ2z = 1

2ϕ4 − ϕ2 + C,

where C is an integration constant. In the limit, z → ±∞, ϕz = ϕ = ±1. Therefore, C = 1/2,and ϕzz/ϕz is written as,

ϕzz

ϕz

= ϕ3 − ϕ√| 1

2ϕ4 − ϕ2 + 12 |

=√

2ϕ(ϕ2 − 1)

|ϕ2 − 1| = √2ϕ × sign

(ϕ2 − 1

).

The integral in Eq. (22) is now given by,

+1∫−1

dϕϕzz

ϕz

δ[ϕ − ϕ(z, t)

] =+1∫

−1

dϕ(z)√

2ϕ(z) × sign(ϕ(z)2 − 1

)δ(ϕ − ϕ(z, t)

)

= √2ϕ sign

(ϕ2 − 1

)θ(1 − ϕ2).

Now, the unclosed term in Eq. (16) is written as

−k′⟨∇2ϕδ[ϕ − ϕ(x, t)

]⟩ = −(k′/2)1/2ρ{A + 2

√2ϕ2δ

(ϕ2 − 1

)},

ϕ

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N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108 101

where, A = 2√

2θ(1 − ϕ2)θ(ϕ2 − 1) − √2θ(1 − ϕ2). Therefore, in the limit k′ → 0, the master

equation takes on the following form

(24)0 = −[(ϕ − ϕ3)P ]

ϕ+ k(0)Pϕϕ − (k′/2)1/2ρ

[A + 2

√2ϕ2δ

(ϕ2 − 1

)],

where we set Pt = 0 in the stationary state. The PDF P is continuous at ϕ = ±1, but its derivativeis not. By integrating Eq. (24) in the interval [1 − ε,1 + ε] (or [−1 − ε,−1 + ε]), one finds

(25)�Pϕ |ϕ=±1 = (k′)1/2ρ

k(0).

In the limit, k′ → 0 the derivative of the PDF will also be continuous. Considering the factor(k′)1/2 in Eq. (24), we conclude that, in the strong coupling limit, the unclosed term is identicallyzero. This means that there is no anomaly or finite term in the strong coupling limit in the masterequation for the PDF of the order parameter ϕ. The stationary solution of Eq. (24), in the limit,k′ → 0, takes on the following expression,

(26)Pst = N exp

[−ϕ4 + 2ϕ2

4k(0)

],

where the normalization constant is given by

(27)1

N= 1√

2exp

[1

8k(0)

]K 1

4

(1

8k(0)

),

where the Ka(b) is the modified Bessel functions. To derive the moments of 〈ϕn〉 in the stationarystate, we multiply Eq. (25) by ϕn and integrate the result over ϕ to obtain

(28)n⟨ϕn

⟩ − n⟨ϕn+2⟩ + n(n − 1)k(0)n

⟨ϕn−2⟩ = 0.

Eq. (28) is a recursive equation for computing all the moments in terms of the second-order one.Direct calculation then shows that,

(29)⟨ϕ2⟩ = −

K 14( 1

8k(0)) − K 3

4( 1

8k(0))

2K 14( 1

8k(0))

,

and all the odd moments vanish, 〈ϕ2k+1〉 = 0. Therefore, using Eqs. (28) and (29), we are ableto derive all the moments of the order parameter in the d-dimensional Ginzburg–Landau theoryin the strong coupling limit.

4. Master equation for the increments and their PDF tail

In this section we derive the PDF and the scaling properties of the moments of the incre-ments, P(ϕ(x2) − ϕ(x1)), for the ϕ4-theory in the strong-coupling limit. Defining the two-pointgenerating function by, Z(λ) = 〈Θ〉, where Θ is defined as

(30)Θ = e−iλ1ϕ(x1,t)−iλ2ϕ(x2,t)

the time evolution of Z is related to that of ϕ by

(31)Zt = −iλ1⟨ϕt (x1, t)e

−iλ1ϕ(x1,t)−iλ2ϕ(x2,t)⟩ − iλ2

⟨ϕt (x2, t)e

−iλ2ϕ(x1,t)−iλ2ϕ(x2,t)⟩.

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102 N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108

Substituting ϕt (x1, t) and ϕt (x2, t) from Eq. (6), the governing equation for the generating func-tion satisfies,

(32)Zt = λ1Zλ1 + λ1Zλ1λ1λ1 + λ2zλ2 + λ2Zλ2λ2λ2 − (λ2

1 + λ22

)k(0)Z − 2λ1λ2k(x)Z,

where x = x1 −x2, and we have used the fact that in the strong-coupling limit, the Laplacian termmakes no contribution to the PDF equation (see Appendix B for more details). Moreover, we haveinvoked the generalized Novikov’s theorem for the two-point generating function, according towhich [16] (see also Appendix A),

−iλ1⟨η(x1)e

−iλ1ϕ(x1,t)−iλ2ϕ(x2,t)⟩ − iλ2

⟨η(x2)e

−iλ1ϕ(x1,t)−iλ2ϕ(x2,t)⟩

(33)= −(λ2

1 + λ22

)k(0)Z − 2λ1λ2k(x)Z.

Fourier transforming Eq. (32), the governing equation of the joint PDF will be given by

Pt (ϕ1, ϕ2) = −[(ϕ1 − ϕ3

1

)P

]ϕ1

− [(ϕ2 − ϕ3

2

)P

]ϕ2

+ k(0)(Pϕ1,ϕ1 + Pϕ2,ϕ2)

(34)+ 2k(x)Pϕ1,ϕ2 .

It is useful to change the variables as, ϕ1 = (w − u)/2, and, ϕ2 = (w + u)/2, and, therefore,d/dϕ1 = d/dw − d/du, and, d/dϕ2 = d/dw + d/du. Now, Eq. (34) can be written as

Pt (w,u) = −(wP )w − (uP )u +[

1

4

(w3 + 3wu2)P

]w

+[

1

4

(u3 + 3uw2)P

]u

(35)+ 2k(0)(Pww + Puu) + 2k(x)(Pww − Puu).

To derive the governing equation for the PDF of the increments, u = ϕ2 − ϕ1, we integrateover w to find that,

(36)Pt (u) = −(uP )u + 1

4

(u3P

)u+ 3

4

(u⟨w2

∣∣u⟩P

)u+ 2

[k(0) − k(x)

]Puu,

where we have used the fact that the joint PDF P(w,u) can be written as, P(w|u)P (u). It isevident that we cannot derive a closed equation for the PDF of u. Indeed, to determine P(u) weneed to know the conditional averaging 〈w2|u〉. However, one can derive the tail (both the leftand right ones) of the PDF in the limit, u → ∞. To determine the tail we note that only near thesingularities, in small separation in space, one finds a large difference in the field ϕ and, hence,large u. On the other hand, near such points or lines, the field w will be very small. Therefore, inthe limit u → ∞, we can ignore the conditional averaging to find that,

(37)limu→∞

⟨w2

∣∣u⟩ 0.

Therefore, in the limit, u → ∞, we obtain the following behavior for the tails of the P(u) in thestationary state,

(38)Pst(φ2 − φ1 → ∞) ∼ exp

{− (φ2 − φ1)

4

32[k(0) − k(x)]}.

To derive the scaling behavior of the moments 〈un〉 one needs to know the entire range of thebehavior of the increments’ PDF. Here, we are able to only derive the equation for the shape ofthe PDF tails. In the next section, we investigate by numerical simulation the scaling behavior ofthe moments 〈un〉 vs. the separation x.

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N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108 103

5. Scaling exponents of the moments: Numerical simulation

To calculate numerically the scaling behavior of the moments with x, when x � 1, we shalluse here the initial-value problem for the two-dimensional Langevin equation, Eq. (6), in thelimit, k′ → 0, when the force is concentrated at discrete times [17–20]:

(39)f (x, y, t) =∑j

fj (x, y)δ(t − tj ),

where both the “impulses” fj (x, y) and the “kicking times” tj are prescribed (deterministic orrandom). The kicking times are ordered and form a finite or an infinite sequence. The impulsesare always taken to be smooth and acting only at large scales. The precise meaning that weascribe to the dynamical Langevin equation with such a forcing is that, at time tj , the solutionϕ(x, y, t) changes discontinuously by the amount fj (x, y),

(40)ϕ(x, y, tj+) = ϕ(x, y, tj−) + fj (x, y),

whereas between tj+ and t(j+1)− the solution evolves according to the unforced ϕ4 equation,

(41)∂tϕ = ϕ − ϕ3 + k′∇2ϕ.

Without loss of generality, we may assume that the earliest kicking time is, tj0 = t0, providedthat we set, fj0 = f0, and, ϕ(x, y, tj0−) = ϕ(x, y, t0−) for t < t0. Therefore, starting from t0,according to Eq. (40) we obtain

(42)ϕ(x, y, t0+) = ϕ(x, y, t0−) + f0(x, y),

and beyond that up to t1−, according to Eq. (41),

ϕ(x, y, t1−) = (1 + h)ϕ(x, y, t0+) − ϕ3(x, y, t0+),

(43)h = t1 − t0,

where, h = t1 − t0.It is clear that any force f (x, y, t) which is continuously acting in time can be approximated in

this way by selecting the kicking times sufficiently close. Hereafter, we shall consider exclusivelythe case where the kicking is periodic in both space and time. Specifically, we assume that theforce in the ϕ4 equation is given by

(44)f (x, y, t) = g(x, y)

+∞∑j=−∞

δ(t − jT ),

(45)g(x, y) ≡ −∇G(x,y),

where G(x,y), the kicking potential, is a deterministic function of (x, y) which is periodic andsufficiently smooth (e.g., analytic), and T is the kicking period.

The numerical experiments reported hereafter were made with the kicking potential G(x,y) =G1(x)G1(y), where G1(q) is given by

(46)G1(q) = 1

3sin 3q + cosq,

and the kicking period, T = 10−6. The number of collocation points chosen for our simulationsis generally Nx = 103. In Fig. 4 we plot the PDF of ϕ according to Eq. (26), and compare it

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104 N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108

Fig. 4. Exact solution of the PDF of ϕ4-theory and its comparison with numerical calculation.

Fig. 5. The moments, 〈|ϕ(x2) − ϕ(x1)|b〉, as a function of |x2 − x1|, obtained via numerical simulation.

with the numerical results. In Fig. 5, the moments of ϕ increments, 〈|ϕ(x2 + δx) − ϕ(x1)|b〉, arecalculated numerically as a function of x = |x2 − x1| for several values of b (with 0 < b < 1, and1 � b) and its scaling exponents ξb for x � 1 are checked. The results indicate that with goodprecision 〈|δϕ|b〉 scales with x with an exponent 1 for b > 1; otherwise, it scales with x withexponents ξb = b. Values of ξb are given in Fig. 6.

The bi-fractal behavior of the exponents is a consequence of the presence of the domain walls.Indeed, the structure function,

(47)Cb = ⟨∣∣ϕ(x2) − ϕ(x1)∣∣b⟩,

for b > 0 behaves, for small �x = |x2 − x1| as,

(48)Cb ∼ Ab|�x|b + A′b|�x|,

where the first term is due to the regular (smooth) parts of the order parameter ϕ, while thesecond one is contributed by the O(|�x|) probability to have a domain wall somewhere in aninterval of length |�x|. For 0 < b < 1 the first term dominates as |�x| → 0, while, for b > 1 itis the second term that does so.

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N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108 105

Fig. 6. Scaling exponents ξb , checked numerically. The results indicate that, with a good precision, 〈|δϕ|b〉 scales with x

with an exponent 1 for b > 1 larger than one, and elsewhere scales with x with an exponent ξb = b.

6. Summary

We studied the domain wall-type solutions in the ϕ4-theory in the strong-coupling limit,k′ → 0, in which the equation develops singularities. The scaling behavior of the moments ofdifferences of ϕ, δϕ = ϕ(x2) − ϕ(x1), and the PDF of ϕ, i.e., P(ϕ), were all determined. It wasshown that in the stationary state, where the singularities are fully developed, the relaxation termin the strong-coupling limit leads to an unclosed term in the equation for the PDF. However, weshowed that the unclosed term can be omitted in the strong-coupling limit. We proved that toleading order, when |x2 − x1| is small, fluctuation of the ϕ field is intermittent for b � 1. Theintermittency implies that, Cb = 〈|ϕ(x1) − ϕ(x2)|b〉 scales as |x1 − x2|ξb , where ξb is a constant.It was shown, numerically, that for the space scale |x2 −x1| and b � 1, the exponents ξb are equalto 1.

Appendix A

In this appendix we provide a proof of Novikov’s theorem. Consider the general stochasticdifferential equation with the following form,

(A.1)∂

∂tϕ = −1

2L

[ϕ(x, t)

] + η(x, t),

where L is an operator acting on ϕ, and η is a Gaussian noise with the correlation,

(A.2)⟨η(x, t)η(x′, t ′)

⟩ = k(x − x′)δ(t − t ′).

The PDF of the random noise has the following form,

[dρ(η)

] = [dη] exp

[−1

2

∫ddx ddx′ dt dt ′η(x, t)B(x − x′)δ(t − t ′)η(x′, t ′)

],

where B(x − x′) is the inverse of k(x − x′), so that,

(A.3)∫

k(x − x′)B(x′ − x′′) ddx = δ(x − x′′).

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106 N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108

We write the average of η(x, t)F (η) over the noise realization as:

(A.4)⟨η(x, t)F (η)

⟩ =∫

η(x, t)F (η)[dρ(η)

].

By integrating by parts and using Eq. (A.3), one finds that,

(A.5)⟨η(x, t)F (η)

⟩ =∫

ddx′′ dt ′′⟨η(x, t)η(x ′′, t ′′)

⟩⟨ ∂F

∂η(x′′, t ′′)

⟩.

Now, let us assume the function F to have the following form,

(A.6)F [η] = exp(−iλϕ(x′, t)

).

So that one finds,

(A.7)∂F

∂η(x′′, t ′′)= −iλ

∂ϕ(x′, t)∂η(x′′, t ′′)

F [η].Integrating Eq. (A.1) with respect to t , we find that,

(A.8)ϕ(x ′, t) = ϕ(x′, t0) − 1

2

t∫t0

dt ′′L[ϕ(x′, t ′′)

] +t∫

t0

dt ′′ η(x′, t ′′).

This allows us to show that,

(A.9)∂ϕ(x′, t)∂η(x′′, t ′′)

= −1

2

t∫t0

dt ′′′ ∂L[ϕ(x, t ′′′)]∂η(x′′, t ′′)

+ δ(x′ − x′′)θ(t − t ′′),

where, in the limit t ′′ → t , the first term of the right-hand side of Eq. (A.9) will vanish, and wecan write

(A.10)⟨η(x, t) exp

[−iλϕ(x′, t)]⟩ = (−iλ)k(x − x′)

⟨exp

(−iλϕ(x′, t))⟩,

where we used, θ(0) = 1.

Appendix B

In this appendix we prove that, for example in Eq. (24), the relaxation term k′∇2ϕ makes nocontribution or anomaly to the PDF of the increments, in the limit k′ → 0.

The joint probability distribution P(ϕ1, ϕ2) satisfies the following equation,

Pt (ϕ1, ϕ2)

= −[(ϕ1 − ϕ3

1

)P

]ϕ1

− [(ϕ2 − ϕ3

2

)P

]ϕ2

+ k(0)(Pϕ1,ϕ1 + Pϕ2,ϕ2) + 2k(x)Pϕ1,ϕ2

− ik′∫

dλ1

dλ2

2πλ1 exp(iλ1ϕ1 + iλ2ϕ2)

⟨∇2ϕ(x1) exp[−iλ1ϕ(x1, t) − iλ2ϕ(x2, t)

]⟩

(B.1)

− ik′∫

dλ1

dλ2

2πλ2 exp(iλ1ϕ1 + iλ2ϕ2)

⟨∇2ϕ(x2) exp[−iλ1ϕ(x1, t) − iλ2ϕ(x2, t)

]⟩.

The last two terms in Eq. (47) are not closed with respect to the PDF. Let us then compute thecontribution of the unclosed terms. They can be written as,

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N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108 107

−ik′∫

dλ1

dλ2

2πλi exp(iλ1ϕ1 + iλ2ϕ2)

⟨∇2ϕ(xi)e−iλ1ϕ(x1,t)−iλ2ϕ(x2,t)

= −ik′(∫

dλ1

dλ2

2πexp(iλ1ϕ1 + iλ2ϕ2)

⟨∇2ϕ(xi)e−iλ1ϕ(x1,t)−iλ2ϕ(x2,t)

⟩)ϕi

(B.2)= −k′⟨∇2ϕ(xi)δ(ϕ1 − ϕ(x1, t)

)δ(ϕ2 − ϕ(x2, t)

)⟩ϕi

.

Consider one of the terms in the above equation, for example, −k′〈∇2ϕ(xi)δ[ϕ1 −ϕ(x1, t)]×δ[ϕ2 − ϕ(x2, t)]〉ϕi

. Assuming ergodicity, it is written as,

(B.3)= k′ limV →∞

1

V

∫V

dxi dvid−1 ∇2ϕ(xi)δ

[ϕi − ϕ(xi, t)

]

in the limit, k′ → 0 limit, only at the points where we have singularity this term is not zero.Therefore, we restrict ourselves to the space near the domain walls,

(B.4)= −k′ limV →∞

1

V

∑j

∫Ωj

dxi dvid−1

(ϕxixi

+ ∇2d−1ϕ

)δ[ϕi − ϕ(xi, t)

]

where Ωj is the space close to the domain walls. Therefore, Eq. (A.4), in the limit, k′ → 0, iswritten as,

(B.5)= −k′ limV →∞

1

V

∑j

∫Ωj

dxi dvid−1 ϕxixi

δ[ϕi − ϕ(xi, t)

].

Changing the variables from xi to zi and integrating over dvid−1, one finds,

= −k′ limV →∞

Vd−1

V

∑j

+∞∫−∞

ε dzi

1

ε2ϕzizi

δ[ϕi − ϕ(zi, t)

]

= −k′

εlim

V →∞Vd−1

V

∑j

+∞∫−∞

dzi ϕziziδ[ϕi − ϕ(zi, t)

],

where ε = (2k′)1/2. Assuming statistical homogeneity, we have

(B.6)= −k′

εlim

V →∞NVd−1

V

+∞∫−∞

dzi ϕziziδ[ϕi − ϕ(zi, t)

],

where N is number of singular lines. Moreover, k′ε

= (k′)1/2, and, NVd−1V

is the density of thesingular lines which, in the limit, V → ∞, is simply the singularity density ρ. Therefore,

(B.7)= −(k′)1/2ρ

+∞∫−∞

dzi ϕziziδ[ϕi − ϕ(zi, t)

].

In the same way in, for example Eq. (21), by changing the integration variable from zi to ϕi , wecalculate the integral exactly,

(B.8)= √k′/2ϕ

{A + 2

√2φ2

i δ(φi − 1)},

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108 N. Abedpour et al. / Nuclear Physics B 761 [FS] (2007) 93–108

where A = 2√

2θ(1 − φ2i )θ(φ2

i − 1) − √2θ(1 − φ2

i ). Therefore, in the limit, k′ → 0 the masterequation will give Eq. (35).

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Pergamon, London, 1965, p. 546.[17] A. Schmid, Z. Phys. 215 (1968) 210.[18] L.P. Gorkov, G.M. Eliashberg, Sov. Phys. JETP 27 (1968) 338.[19] A.C. Newell, J.A. Whitehead, J. Fluid Mech. 38 (1969) 279.[20] L.A. Segel, J. Fluid Mech. 38 (1969) 203.[21] P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435.[22] C.M. Bender, S.A. Orzag, Advanced Mathematical Methods for Scientists and Engineers, third ed., McGraw–Hill,

New York, 1987.[23] A.A. Masoudi, F. Shahbazi, J. Davoudi, M.R. Rahimi Tabar, Phys. Rev. E 65 (2002) 026132.[24] S.M.A. Tabei, A. Bahraminasab, A.A. Masoudi, S.S. Mousavi, M.R. Rahimi Tabar, Phys. Rev. E 70 (2004) 031101.[25] H. Risken, The Fokker–Planck Equation, Springer-Verlag, Berlin, 1984.

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Nuclear Physics B 761 [FS] (2007) 109–140

Topological excitations and their contribution toquantum criticality in 2 + 1 D antiferromagnets

Zaira Nazario a,∗,1, David I. Santiago a,b,2

a Department of Physics, Stanford University, Stanford, CA 94305, USAb Gravity Probe B Relativity Mission, Stanford, CA 94305, USA

Received 20 July 2006; received in revised form 20 October 2006; accepted 30 October 2006

Available online 9 November 2006

Abstract

It has been proposed that there are new degrees of freedom intrinsic to quantum critical points thatcontribute to quantum critical physics. We study 2 + 1 D antiferromagnets in order to explore possible newquantum critical physics arising from nontrivial topological effects. We show that skyrmion excitations arestable at criticality and have nonzero probability at arbitrarily low temperatures. To include quantum criticalskyrmion effects, we find a class of exact solutions composed of skyrmion and antiskyrmion superpositions,which we call topolons. We include the topolons in the partition function and renormalize by integratingout small size topolons and short wavelength spin waves. We obtain a correlation length critical exponentν = 0.9297 and anomalous dimension η = 0.3381.© 2006 Elsevier B.V. All rights reserved.

PACS: 75.10.-b; 75.40.Cx; 75.40.Gb; 75.40.-s

1. Introduction

There have been recently interesting suggestions that there are intrinsic degrees of freedom atquantum critical points and that these intrinsic critical excitations are different from the elemen-tary excitations the quantum critical point separates [1,2]. In order to explore a path toward thispossible new physics at quantum critical points, we study the approach to the quantum critical

* Corresponding author.E-mail address: [email protected] (Z. Nazario).

1 Present address: Max Planck Institute for the Physics of Complex Systems in Dresden, Germany.2 Present address: Instituut-Lorentz at Leiden University in Leiden, The Netherlands.

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2006.10.024

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110 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

point from the Néel ordered phase of 2 + 1 D antiferromagnets. A physics that is usually notincluded in the studies of criticality is the effects of the nontrivial topology of the O(3) group onthe quantum critical physics. In the present article we study this topology and include its effectson the critical physics. We also calculate its contribution to critical exponents. In the end weassess how close or how far we are from finding the effective field theory of the critical pointderiving from its intrinsic critical excitations.

2. 2 + 1 D O(3) nonlinear sigma model in the stereographic projection language

We now turn to 2+1 D antiferromagnets whose effective field theory is given by the nonlinearσ model augmented by Berry phase terms [3,4]:

Z =∫

D�nδ(�n2 − 1

)e−S,

(1)

S = SB + ρs

2

β∫0

∫d2 �x

[(∂�x �n)2 + 1

c2(∂τ �n)2

]

= SB + Λ

2g

β∫0

d(cτ)

∫d2 �x

[(∂�x �n)2 + 1

c2(∂τ �n)2

],

where a is the lattice constant, Λ = 1/a, S is the microscopic spin with h included (not to beconfused with the Euclidean action), and J is the microscopic spin exchange. ρs ≡ JS2/h isthe spin stiffness, c = 2

√2JSa is the spin-wave velocity, and the dimensionless microscopic

coupling constant is g = 2√

2/S. These values are obtained for the case of nearest neighborHeisenberg interactions only. For this case, in order to move the system from a Néel orderedphase (g � 1) to a disordered or quantum paramagnetic phase (g � 1), we tune the microscopicspin from large to small values. In real life one does not have this option as the microscopic spinis fixed. Moreover, the smallest available spin S = 1/2 is not enough to place the system in thequantum paramagnetic phase [5]. In order to quantum disorder the system in real life, one wouldneed to add frustrating next nearest neighbor interactions J ′, frustrating ring exchange interac-tions K , or other longer range but still short range interactions that compete with the nearestneighbor Néel order interaction. In this case, the dimensionless coupling g becomes a functionof the ratios of the different interactions g = g(J/J ′, J/K, . . .) and by tunning the competinginteractions, one can take the system from the Néel ordered to the quantum paramagnetic phase.

The Berry phase term is the sum of the areas swept by the vectors �ni(τ ) on the surface of aunit sphere at each lattice site as they evolve in Euclidean time [3]. The Berry phase terms arezero when the Néel magnetization is continuous, i.e. Néel ordered phase and critical point, as thecontributions from neighboring lattice sites cancel [3,6,7]. Since we will concentrate on criticalproperties as approached from the Néel ordered side, the Berry phase vanishes.

A particular parametrization of the nonlinear sigma model that will prove specially convenientwhen we shortly move to study the topological structure of the model is obtained by mappingthe staggered magnetization �n to the complex variable w via the stereographic mapping [8]

(2)n1 + in2 = 2w

|w|2 + 1, n3 = 1 − |w|2

1 + |w|2 , w = n1 + in2

1 + n3.

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 111

More details of the stereographic projection are given in Appendix A. In terms of w the nonlinearsigma model action is

(3)S = 2Λ

∫d3x

∂μw∂μw∗

(1 + |w|2)2= 2Λ

∫d3x

∂0w∂0w∗ + 2∂zw∂z∗w∗ + 2∂z∗w∂zw

(1 + |w|2)2,

where ∂0 = ∂/∂(cτ). We have renamed g as gΛ because this is the microscopic coupling con-stant. In order to study the phases of the model and the quantum critical point, we will performrenormalization group studies [5,9–13]. When we integrate degrees of freedom, the couplingconstants will become renormalized, but retain the same form since the nonlinear sigma modelis a renormalizable field theory [11,12,14]. Thus the renormalized theory takes the form

(4)S = 2μ

∫d3x

∂αw∂αw∗

(1 + |w|2)2

with μ is the sliding renormalization scale [15] and gμ the renormalized coupling constant. Wehave suppressed field renormalization factors.

2.1. Traditional Goldstone renormalization in the stereographic projection Language

In order to gain experience with stereographic projections, to show that the results are equiv-alent to more used approaches [5,10–13,16] and for later use we now perform a one looprenormalization expansion of the nonlinear sigma model.

In terms of w, the partition function is

(5)Z =∫ ∏

τ,�x

Dw(τ, �x)Dw(τ, �x)∗

(1 + |w(τ, �x)|2)2e−S,

where the Euclidean action S is given by (3). A tricky point in the partition function is thenontrivial measure arising from the nonlinearity of the sigma model and from the Jacobian of thetransformation between the �n variables and the stereographic projection w:

(6)∏τ,�x

1

(1 + |w|2)2= exp

[−2

∑τ,�x

ln(1 + |w|2)].

These products, or the sum in the exponentials, are not well defined in the continuum limit.How this term is treated depends on your regularization method. For example, in dimensionalregularization this term is just ignored with the price of some infrared divergences that must betreated carefully. On the other hand, direct passage to the continuum limit yields

(7)exp

[−2

∑τ,�x

ln(1 + |w|2)] = exp

[−2δ3(0)

∫d3x ln

(1 + |w|2)].

The delta functions are ill defined as they are infinite. This can be dealt with in at least twoequivalent ways, either by introducing a lattice cutoff in real space or a momentum cutoff inmomentum space. We thus have

(8)exp

[−2

∑τ,�x

ln(1 + |w|2)] = exp

[− 2

a3

∫d3x ln

(1 + |w|2)],

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112 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

where space integrals are cutoff at the short distance a. Then the partition function can be writtenas

(9)Z =∫ ∏

τ,�xDw(τ, �x)Dw∗(τ, �x) e−Seff ,

where the effective action is the nonlinear sigma model action augmented by the terms obtainedfrom the nontrivial measure:

(10)Seff = 2

a3

∫d3x ln

(1 + |w|2) + S.

We are interested in the Néel ordered phase and in the critical point, but this last will bestudied as it is approached from the Néel ordered phase. So gΛ will not be too large and, at leastin the Néel ordered phase, a perturbative expansion in gΛ is possible. Renormalization groupresummations of this expansion yield approximations to the critical theory. The lowest orderterms in gΛ will be given by the approximation

(11)Seff 2

a3

∫d3x |w|2 + 2Λ

∫d3x ∂μw ∂μw∗ − 4Λ

∫d3x |w|2∂μw∂μw∗.

We now work in momentum space. The effective action is then

Seff = 2V 2

a3

∫d3k

(2π)3

∣∣w(k)∣∣2 + 2V 2Λ

∫d3k

(2π)3k2

∣∣w(k)∣∣2

− 4V 4Λ

∫d3k1 d3k2 d3k3

(2π)9w(k1)w

∗(k2)

(12)× k3 · (k1 − k2 + k3)w(k3)w∗(k1 − k2 + k3),

where k = (ω, �k) and V is the volume where the system lies. We will slightly change our defini-tion of the cutoff in order to absorb some angular factors. Hence we renormalize the momentumsphere to

(13)V Λ3 = 4π

3Λ3 = 1

a3.

We will renormalize the theory via momentum shell integration. In order for this action tobe well defined, i.e. finite, it is cutoff at large momentum Λ and will integrate the degrees offreedom from Λ to a smaller cutoff μ. The bare action is

(14)S0 = 2V 2 Λ

∫d3k

(2π)3k2

∣∣w(k)∣∣2

,

which leads to the bare momentum space Green’s function or propagator

(15)G0 = (2π)3

2V 2

Λ

1

k2.

The interaction term is given by

SI = 2V 2V Λ3

∫d3k

(2π)3

∣∣w(k)∣∣2 − 4V 4 Λ

∫ {d3k1d

3k2d3k3

(2π)9w(k1)w

∗(k2)

(16)× k3 · (k1 − k2 + k3)w(k3)w∗(k1 − k2 + k3)

}.

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 113

We know integrate the high momentum degrees of freedom to obtain the effective theory at scaleμ. If the momentum shell is thin enough, a perturbative expansion is justified and to lowest orderin the interaction we have

(17)Z ∫Λ

DwDw∗ (1 − SI )e−S0 .

After integrating the high energy degrees of freedom (μ < k < Λ) we obtain

(18)Z ∫μ

DwDw∗ e−Sμeff ,

where the effective action at scale μ is given by

(19)Sμeff = S0 + 〈SI 〉

and the average of SI is taken over the large momentum degrees of freedom. The noninteractingaction S0 is the one obtained from S0 at scale Λ by integrating out the degrees of freedom withmomenta between μ and Λ and the irrelevant constant term are thrown out as they only modifythe overall normalization. Now we evaluate the expectation value of SI term by term. We firstobtain

(20)⟨S

(1)I

⟩ = 2V 2V Λ3

∫d3k

(2π)3

⟨∣∣w(k)∣∣2⟩ = 2V 2V Λ

3

∫|k|<μ

d3k

(2π)3

∣∣w(k)∣∣2 + C,

C is a constant; and

⟨S

(2)I

⟩ = −4V 4 Λ

∫ {d3k1 d3k2 d3k3

(2π)2

⟨w(k1)w

∗(k2)

× k3 · (k1 − k2 + k3)w(k3)w∗(k1 − k2 + k3)

⟩}

− 2V 2(V Λ3 − V

μ3

) ∫|k|<μ

d3k

(2π)3

∣∣w(k)∣∣2

(21)− V 2

π2[Λ − μ]

∫|k|<μ

d3k

(2π)3k2

∣∣w(k)∣∣2 + C,

where the constant terms will be neglected as they only provide a change of overall normalizationof the partition function. Putting everything together we obtain

Sμeff = 2V 2V

μ3

∫|k|<μ

d3k

(2π)3

∣∣w(k)∣∣2 + 2V 2 Λ

{1 − gΛ

2π2

[1 − μ

Λ

]} ∫|k|<μ

d3k

(2π)3k2

∣∣w(k)∣∣2

− 4V 4 Λ

∫|ki |<μ

d3k1 d3k2 d3k3

(2π)9

× {w(k1)w

∗(k2)k3 · (k1 − k2 + k3)w(k3)w∗(k1 − k2 + k3)

}

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114 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

2V 2Vμ3

∫|k|<μ

d3k

(2π)3

∣∣w(k)∣∣2 + 2V 2 μ

∫|k|<μ

d3k

(2π)3k2

∣∣w(k)∣∣2

− 4V 4 μ

∫|ki |<μ

d3k1 d3k2 d3k3

(2π)9

(22)× {w(k1)w

∗(k2)k3 · (k1 − k2 + k3)w(k3)w∗(k1 − k2 + k3)

}.

We thus see that, to lowest order, the action renormalizes [5,10–12], i.e. goes into itself when fastdegrees of freedom are integrated out. This result can be proved to be true to all orders by meansof the O(3) Ward identity of the O(3) nonlinear sigma model [14].

The renormalized coupling constant at scale μ is seen to be

(23)gμ =(

μ

Λ

)gΛ

1 − gΛ

2π2

[1 − μ

Λ

] .

This expression exemplifies quite a bit of the physics of the renormalization group and thenonlinear sigma model. We first see that the coupling constant gets renormalized to differentvalues at different scales. From this expression we see at least two fixed point values of thecoupling constant in the sense that for these two values the coupling constant is the same at allmomentum scales. One is gΛ = gN

IR = 0 and the other is at gΛ = gcUV = 2π2 as found long ago

[10]. gNIR corresponds to the Néel ordered or Goldstone phase and it is an infrared fixed point

which corresponds to a stable phase of matter. It is an infrared fixed point because if 0 < gΛ <

gcUV, gμ → gN

IR = 0 as μ → 0. That is, as the energy scale is lowered, the theory approaches theNéel ordered behavior.

The critical point where Néel order is lost corresponds to gcUV. That this is a critical point

follows since it is an infrared unstable fixed point. If gΛ is arbitrarily close but not exactly gcUV,

it deviates from gcUV as μ → 0. This is easily seen from our formula if gΛ is chosen to be a

value infinitesimally smaller than gcUV. More importantly, the critical point is an ultraviolet fixed

point because if 0 < gΛ < gcUV, gμ → gc

UV as μ → ∞. All critical points are ultraviolet fixedpoints. Therefore, the critical properties can be studied from the Néel ordered phase by studyingtheir high momentum or high energy behavior. Finally, the nonlinear sigma model has anotherinfrared fixed point at gΛ = ∞ = gP

IR, which corresponds to the paramagnetic phase. This fixedpoint cannot be accessed by expanding about the Néel ordered phase as it is not adiabaticallycontinuable to the Néel ordered phase. In order to access this paramagnetic fixed point, oneneeds to perform a strong coupling expansion.

The spin stiffness of the nonlinear sigma model is proportional to the inverse coupling constant

(24)ρs ∝ μ

.

Classically, gμ/μ = gΛ/Λ is a constant and does not get renormalized. The spin stiffness onlyvanishes when the bare coupling constant gΛ becomes infinite. When fluctuation effects areincluded, gμ becomes renormalized according to

(25)ρs ∝ Λ

[1 − gΛ

2π2

(1 − μ

Λ

)].

If we tune gΛ to the critical value where Néel order is lost, gcUV, the spin stiffness is

(26)ρs ∝ μ

gc .

UV
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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 115

We thus see that at the quantum critical point, gΛ = gcUV = 2π2, the spin stiffness vanishes at

arbitrarily low energy scales μ → 0.The beta function is calculated from our expression for the renormalized coupling constant g

as a function of μ, (23), to be

(27)β(g) = μ∂g

∂μ

∣∣∣∣Λ=μ

= g − g2

2π2.

The fixed points are given by the zeros of the β function, which are easily seen to occur at g = 0and g = 2π2 as found before. The first term, which is positive, arises because in more than twospace–time dimensions the theory is not truly scale invariant but becomes so at fixed points.In more than two space–time dimensions, the presence of the negative term makes possible theexistence of a critical point because at certain value of the coupling constant, the negative orasymptotic freedom term cancels the positive scaling term of the β function.

Since our renormalization of the nonlinear sigma model was carried out to one loop, we didnot obtain the renormalization of the fields. The situation is not as bad as it seems as it is possibleto obtain a field renormalization by a different one loop calculation. More important than therenormalization of the w fields, we are interested in the renormalization Z of the staggeredmagnetization. Because of the O(3) invariance, the Goldstone fields Π1 = n1 and Π2 = n2,and the ordering direction or σ = n3 component are renormalized by the same factor

√Z =√

1 + δ 1 + δ/2. We will fix the renormalized magnetization σ to one in the Néel orderedphase. On the other hand, the bare magnetization is related to the renormalized one via σ0 =√

Zσ (1 + δ/2)σ . In order to obtain the renormalized magnetization we calculate

σ0 = 〈n3〉 =⟨

1 − |w(τ, �x)|21 + |w(τ, �k)|2

⟩ 1 − 2

⟨∣∣w(τ, �x)∣∣2⟩

,

(28)σ0 = √Z 1 + δ

2,

where the average value is obtained by integrating the fast degrees of freedom between scales μ

and Λ in order to obtain the magnetization at scale μ. Therefore we obtain the counterterm

(29)δ = −4⟨∣∣w(τ, �x)

∣∣2⟩ = −4V 2∫

μ<|k|<Λ

d3k

(2π)3G0(k) = −gΛ

π2

[1 − μ

Λ

].

N -point Green’s functions of the nonlinear sigma model satisfy the Callan–Symanzik (CS)equation [11,12,15,17,18]

(30)

∂μ+ β(g)

∂g+ N

2γ (g)

]G(N)(p,g,μ) = 0,

where the anomalous dimension γ (g) is given by

(31)γ (g) = μ∂ lnZ

∂μ

∣∣∣∣μ=Λ

μ∂δ

∂μ

∣∣∣∣μ=Λ

= g

π2.

We are particularly interested in the critical properties of antiferromagnets and hence the non-linear sigma model. When tuned to criticality, we obtain that exactly at the UV fixed point orcritical point, gΛ = gc

UV = 2π2, the Green’s function has the form

(32)G(2)(p,gc

UV,μ) = 1

2h

(p2

2

),

p μ

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116 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

which means that if we exchange the μ derivative for a p derivative the CS equation becomes

(33)

[p

∂p− γ

(gc

UV

)]h

(p2

μ2

)= 0.

This integrates immediately to

(34)h

(p2

h2

)= A′

(μ2

p2

)−γ /2

,

where A′ is an arbitrary integration constant. The two point critical function is then

(35)G(2)(p,gc

UV,μ) = A′

μγ(gcUV)

(1

p2

)1−γ (gcUV)/2

≡ A

p2−η.

From Eq. (31), we immediately find γ (gcUV) = 2 in agreement with previous one loop results

[10,11]. This is obviously wrong as the propagator loses all momentum dependence. The reasonfor this nonsense is that higher order corrections are quite large and must be included. More care-ful approximations give less large and more sensical values [5,14,16]. We will evaluate below,for the first time, the corrections coming from topological effects in order to get a more accurateevaluation of the anomalous exponents.

Better than tuning the system exactly to criticality it is more realistic to study the Green’sfunction in the Néel ordered phase. The two point Green’s function in the Néel ordered phasetakes the form

(36)G(2)(p, g,μ) = 1

p2h

(μ2

Λ2

)= 1

p2h(gμ) = 1

p2h

(g,

μ2

p2

).

Because h is a function of μ2/Λ2 only through the coupling constant. Hence the CS equationcan be integrated immediately to give

(37)G(2)(p, g,μ) = A

p2exp

[−

gμ∫gp

dg′ γ (g′)β(g′)

]= A

p2

(1 − gμ

gcUV

[1 − p

μ

])2

.

We now see that the Green’s function interpolates between the Goldstones behavior at long wave-lengths, p → 0

(38)G(2)(p, g,μ) = A

p2

(1 − gμ

gcUV

)2

and critical behavior with anomalous exponent η = 2 at short distances, p → ∞.

(39)G(2)(p, g,μ) = A

p2

[gμ

gcUV

]2p2

μ2= A

μ2

[gμ

gcUV

]2

.

Despite the large incorrect value of γ , the qualitative conclusions obtained are true. There isa nonzero anomalous exponent at the critical point and the Green’s function satisfy the Callan–Symanzik equation. Moreover, the conclusions drawn from the Callan–Symanzik equation, i.e.the scaling behavior, becomes quantitatively correct at small energy scales [5]. In fact, the quanti-tative agreement between RG studies of the nonlinear sigma model and neutron scattering resultsin the cuprate high Tc superconductors demonstrated that these are Néel ordered at low temper-atures when sufficiently underdoped [5].

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 117

For completeness and perspective, we finish this section by extracting the information thatfollows from the CS equation (30). In order to integrate this equation, we introduce the magneti-zation σ = σ(g) obtained by integrating the equation

0 =(

β(g)∂

∂g+ 1

2γ (g)

)σ(g),

(40)σ(gμ) = B exp

{−

gμ∫γ (g)

2β(g)dg

} M

[1 − gμ

2π2

]= M

(1 − gμ/gc

UV

),

where M is an arbitrary integration constant. Before continuing with the analysis of the CSequation, we briefly digress to discuss the properties of the coupling-dependent renormalizedmagnetization σ(gμ). For gμ < 2π2 = gc

UV, i.e. in the Néel ordered phase, the renormalizedmagnetization σ(gμ) is a nonzero constant. We see that as the coupling constant is tuned to thecritical value gc

UV, the magnetization goes to zero with the critical exponent β = 1.3 Since this isa one loop result, the exponent is not accurate but the fact that the magnetization goes to zero atcriticality is true.

Let us go back to the analysis of the CS equation. In order to simplify it, we define thecorrelation length via the equation

0 =(

μ∂

∂μ+ β(g)

∂g

)ξJ (μ,g),

(41)ξJ (gμ,μ) = 1

μexp

{ gμ∫dg′

β(g′)

}= 1

μ

[gμ/gc

UV

1 − gμ/gcUV

].

This correlation length is the Josephson correlation length ξJ [5,16] which determines thecrossover from short distance critical behavior to long distance Goldstone behavior. Just as withthe magnetization, this determination of the correlation length is not accurate enough because itis a one loop result. We see that the correlation length diverges with the exponent ν = 1.

3. Topological excitations of Néel ordered 2 + 1 D antiferromagnets

2 + 1 D antiferromagnets and their effective description via the O(3) nonlinear sigma modelhave a classical “ground state” or lowest energy state with Néel order corresponding to a constantmagnetization. The equations of motion that follow from the action have approximate time de-pendent solutions, corresponding to Goldstone spin wave excitations. The equations of motion,in 2 + 1 D only, also have exact static solitonic solutions of finite energy [20]. Digressing fora moment from antiferromagnets, since the 1970’s, it has been known that when systems haveexact classical, time independent solutions which are stable against quantum fluctuations, thesesolutions are quantum particle excitations of the system [19]. The nonlinear sigma model, in2 + 1 D only, possesses time independent solutions which are of a topological nature [8,20] andhave finite energy. These excitations are disordered at finite length scales but relax into the Néelstate far away

(42)lim|�x|→∞

�n = (0,0,1), lim|�x|→∞

w = 0.

3 This is not the β function, but this exponent is called β for convention. We hope this does not cause confusion.

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118 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

They consist in the order parameter rotating a number of times as one moves from infinity towarda fixed but arbitrary position in the plane. Since two-dimensional space can be thought of as aninfinite 2-dimensional sphere, the excitations fall in homotopy classes of a 2D sphere into a 2Dsphere: S2 → S2. The topological excitations are thus defined by the number of times they mapthe 2D sphere into itself. They are characterized by the Jacobian

(43)q = 1

∫d2x εij �n · ∂i �n × ∂j �n

or in terms of the stereographic variable

(44)q = i

2πεij

∫d2x

∂iw∂jw∗

(1 + |w|2)2= 1

π

∫d2x

∂zw∂z∗w∗ − ∂z∗w∂zw∗

(1 + |w|2)2.

The number q will be an integer measuring how many times the n-sphere gets mapped into theinfinite 2D sphere corresponding to the plane where the spins live. If we define the space–timecurrent

(45)Jμ = 1

8πεμνσ �n · ∂ν �n × ∂σ �n = i

2πεμνσ ∂νw∂σ w∗

(1 + |w|2)2,

it is easily seen that it is conserved ∂μJμ = 0 and that the charge associated with it is our topo-logical charge

(46)q =∫

d2x J 0.

Thus q is a conserved quantum number. These topological field configurations were originallydiscovered by Skyrme [21] and are called skyrmions. The conserved charge is the skyrmionnumber.

Since the skyrmions are time independent, their energy is given by

(47)Es = 4Λ

∫d2x

∂zw∂z∗w∗ + ∂z∗w∂zw∗

(1 + |w|2)2.

From this expression and the one for the charge q , it is easily seen [8,20] that E � 4π |q|Λ/gΛ.We see that we can construct skyrmions with q > 0 by imposing the condition

(48)∂z∗w = 0

that is w is a function of z only. The magnetization, �n or w, is an analytic function of z almosteverywhere. The worst singularities it can have are poles. The skyrmions will have a locationgiven by the positions of the poles or of the zeros of w. Far away from its position, the field con-figuration will relax back to the original Néel order. Therefore we have the boundary conditionw(∞) = 0, which implies

(49)w = λq

q∏i=1

1

z − ai

,

which can easily be checked to have charge q and energy

(50)Es(q) = 4πΛ

q = 4πρsq.

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 119

λq is the arbitrary size and phase of the configuration and ai are the positions of the skyrmionsthat constitute the multiskyrmion configuration. The energy is independent of the size and phasedue to the conformal invariance of the configuration. We remark that since the multiskyrmionsenergy is the sum of individual skyrmion energies, the skyrmions do not interact among them-selves [8]. Similarly, the multiantiskyrmion configuration can be shown to be

(51)w = (λ∗)qq∏

i=1

1

z∗ − a∗i

with charge −q and energy 4πΛq/gΛ. Skyrmions and antiskyrmions do interact as shown inAppendix A. In that appendix we collect these results on skyrmions and provide further devel-opments.

We now investigate whether skyrmions and antiskyrmion configurations are relevant at thequantum critical point. As mentioned above, their classical energy is 4πΛ/gΛ, which is inde-pendent of the size of the skyrmion λ. On the other hand, in real physical systems there arequantum and thermal fluctuations. These renormalize the effective coupling constant of the non-linear sigma model and make it scale dependent. To one loop order the renormalized couplingconstant is

(52)gμ = μ

Λ

1 − (gΛ/2π2)(1 − μ/Λ).

Since the skyrmion has an effective size λ, spin waves of wavelength smaller than λ renormal-ize the energy of the skyrmion via the coupling constant renormalization. The energy E of theskyrmion at the scale set by its size μ = 1/λ is now

(53)Es = 4πμ

∣∣∣∣μ=1/λ

= 4πΛ

[1 − gΛ

2π2

(1 − 1

λΛ

)].

We see that the skyrmion energy now depends on its size through the renormalization effects andthus the conformal invariance of the configuration is broken. This is the well-known phenomenonof broken scale invariance in renormalizable theories due to the scale dependence of the couplingconstant [18,22]. We see that if we tune the system to criticality, gΛ = 2π2, the energy of askyrmion of size λ is Es = 2/(πλ) and the energy of excitation for skyrmions of arbitrarily largesize is zero and hence degenerate with the ground state. The quantum critical point thus seemto be associated with skyrmion gap collapse. In order to see if skyrmion fluctuations are indeedrelevant to the critical point, let us be a little bit more careful and explicit.

If the system is at temperature T = 1/β , this temperature sets the size of the skyrmion to bethe thermal wavelength λ = β . The skyrmion Euclidean action is then

(54)Ss = 4πβ

βg1/β

= 4πβΛ

[1 − gΛ

2π2

(1 − 1

βΛ

)].

The probability for skyrmion creation is given by P ∝ e−Ss . Let us see how this probabil-ity behaves at low temperatures. When in the Néel ordered phase, gΛ < gc = 2π2, the skyrmionEuclidean action diverges as T → 0. Therefore, the probability for skyrmion contributions is sup-pressed exponentially at low temperatures, vanishing at zero temperature. Skyrmions are gappedand hence irrelevant to low temperature physics in the Néel ordered phase.

At the quantum critical point gΛ = gc = 2π2, the skyrmion Euclidean action is

(55)Ss = 2.

π

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120 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

This action is finite and constant at all temperatures and in particular, it will have a nonzero limitas the temperature goes to 0: The skyrmion probability is nonzero and constant at arbitrarilylow temperatures and zero temperature. Hence there are skyrmion excitations at criticality atarbitrarily low energies and temperatures, including at zero temperature. Therefore skyrmionexcitations contribute to quantum critical physics.

4. Topologically nontrivial configurations with zero skyrmion number

We have seen that skyrmions are relevant at criticality as the critical point is associated withskyrmion gap collapse and they have a nonzero probability to be excited at arbitrarily low tem-perature at criticality. On the other hand, skyrmions have nonzero conserved topological numberwhile the ground state has zero skyrmion number. Absent any external sources that can coupledirectly to skyrmion number, they will always be created in equal numbers of skyrmions andantiskyrmions. Therefore, in order to study the effect of skyrmions and antiskyrmions we needto include configurations with equal number of skyrmions and antiskyrmions in the partitionfunction.

We need to find nontrivial solutions of the nonlinear sigma model equations of motion withzero skyrmion number that corresponds to superpositions of equal number of skyrmions andantiskyrmions. The classical equations of motion which follow by stationarity of the classicalaction are

�w = 2w∗

1 + |w|2 ∂μw∂μw,

(56)∂20 w − 4∂z∂z∗w = 2w∗

1 + |w|2{(∂0w)2 − 4∂zw∂z∗w

}.

We are interested in time independent and finite energy (or nonzero probability) solutions. Fortime independent solutions, the structure of the equations suggests a solution of the form

(57)w = eiϕ tan

[f (z) + (

f (z))∗ + θ

2

],

where ϕ and θ are arbitrary, constant angles and f (z) is an arbitrary function of z only and not ofz∗, which is analytic in z almost everywhere (just as with skyrmion and antiskyrmion solutions,the function can have poles but no worse singularities). These solutions are topologically trivialsince they have zero skyrmion number. On the other hand, as we shall see explicitly below, thefinite energy solutions will correspond to arbitrary superpositions of equal number of skyrmionsand antiskyrmions with q = ±n. Since, despite being topologically trivial, these solutions willbe composed of topologically nontrivial configurations (skyrmions and antiskyrmions), we dubthem topolons. We hope this name does not create confusion.

The first steps we need to check that the topolon solution written above is a solution is to com-pute the derivatives. We start with the simplest ones. Since our solutions are time independent,we immediately obtain

(58)∂0w = 0, ∂20 w = 0.

The next derivatives to calculate are

∂zw = eiϕf ′(z) sec2[f (z) + (

f (z))∗ + θ

],

2

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 121

∂z∗w = eiϕ(f ′(z)

)∗ sec2[f (z) + (

f (z))∗ + θ

2

],

(59)∂z∂z∗w = 2eiϕ∣∣f ′(z)

∣∣2 sec2[f (z) + (

f (z))∗ + θ

2

]tan

[f (z) + (

f (z))∗ + θ

2

].

Using these derivatives, we see that the left-hand side of the equation of motion (56) is

(60)LHS = −8eiϕ∣∣f ′(z)

∣∣2 sec2[f (z) + (

f (z))∗ + θ

2

]tan

[f (z) + (

f (z))∗ + θ

2

]

and the right-hand side is

RHS = 2e−iϕ tan[f (z) + (f (z))∗ + θ

2

]1 + tan2

[f (z) + (f (z))∗ + θ

2

] {−4e2iϕ

∣∣f ′(z)∣∣2 sec4

[f (z) + (

f (z))∗ + θ

2

]}

(61)= −8eiϕ∣∣f ′(z)

∣∣2 sec2[f (z) + (

f (z))∗ + θ

2

]tan

[f (z) + (

f (z))∗ + θ

2

].

The left-hand side and the right-hand side are identical and therefore the topolon solution writtenabove is indeed an exact solution.

The energy of this solution is easily calculated to be

(62)E = 8Λ

∫d2x

∣∣∣∣df (z)

dz

∣∣∣∣2

.

In order for the solution to have finite energy, f (z) must decay faster than 1/zξ as z → ∞, withξ > 0. But since f (z) is analytic except for poles or multipoles, f (z) decays at least as fast as1/z as z → ∞. Therefore, a general form that has finite energy is given by

(63)fn(z) = λnn∏

i=1

1

z − ai

with wn given as above with f (z) = fn(z). This is the general n-topolon. Two parameters then-topolon depends on are the two orientation angles. In general, skyrmions depend on twoorientation angles which are given by the orientation of the Néel order the skyrmion relaxeson far away. These are usually not counted when counting the parameters of a skyrmion asthey are usually fixed to constant values by the boundary conditions. Besides these two angles,a skyrmion depends on an arbitrary complex parameter λ and n complex positions when wehave an n-skyrmion. The n-topolon depends also on an arbitrary complex parameter λ and n

complex positions. Besides the arbitrariness of these parameters (the same arbitrariness as for ann-skyrmion), there is no other arbitrariness to the topolon as its form is dictated by solving theequations of motion.

Given that the argument of the topolon is precisely the sum of an n-skyrmion with ann-antiskyrmion, it is clear that a topolon is in general a superposition of skyrmions and anti-skyrmions. The antiskyrmions are at the same positions as the skyrmions, i.e. the argument ofthe topolon has paired skyrmions and antiskyrmions at the same position. If this was not the case,we would not have a solution of the equations of motion. This is perhaps not surprising becauseif we take a look at the skyrmion–antiskyrmion interaction we obtained in Appendix A, it hasa minimum when the skyrmion and antiskyrmion are at the same position, that is, the relativedistance is zero.

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122 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

5. Renormalization of the nonlinear sigma model including goldstone and topolons

We have just found time independent solutions to the equations of motion given by

(64)w(n)t = eiϕ tan

[n∏

i=1

λ

z − ai

+n∏

i=1

λ∗

z∗ − a∗i

+ θ

2

],

which corresponds to superpositions of equal number of skyrmions and antiskyrmions. Thesesolutions contribute to the partition function for the antiferromagnet, which is then given by

(65)Z =∞∑

n=0

Zn,

where

(66)Z0 =∫ DνDν∗

(1 + |ν|2)2e−S[ν]

is the usual partition function for the nonlinear sigma model with no topolons and only the spinwave like fields ν, and S is the Euclidean action for the nonlinear sigma model in terms of thestereographic projection variable w = ν. We also have that

(67)Zn�=0 =∫ DνDν∗

(1 + |w(n)t + ν|2)2

1/Λ

n∏i=1

d2ai

Ae−S[w(n)

t +ν].

The Zn�=0 is the path integral with the n topolons with spin waves ν. Besides integrating overthe spin wave configurations, we must integrate over the topolon parameters: Its size λ nor-malized to the lattice spacing 1/Λ, the positions ai of the topolon constituents skyrmions andantiskyrmions, normalized to the area A of the system, and over the solid angle of the topolonorientation normalized to 4π .

The next step to evaluate the partition function, or to evaluate expectation values from it,would be to expand the topolon action in a semiclassical expansion about the spin waves.This process can be simplified by realizing that the same physics follows if all the constituentsskyrmions are placed in the same position. This result is not obvious so we will review the mainpoints of how it comes about. When the semiclassical expansion is performed, as will be de-scribed below, we obtain a sum of terms in the partition function, each of which is weighted bye−βEn where En is the n-topolon energy. This energy is given by

(68)En = 8πΛn

R(a1, a2, . . . , an)

]2n

,

where R is a distance that depends on the particular positions of the skyrmions and antiskyrmionsmaking up the arguments of the n-topolon. All terms in the semiclassical expansion, after beingmultiplied by the weight factor, need to be integrated over the solid angle of the orientation ofthe topolons (dΩ/4π ) and over the positions of the n skyrmions and antiskyrmions that con-stitute the argument of the n-topolon (d2a1, d

2a2, . . . , d2an/(area)n). The only dependence on

the ai ’s in the weighting factor appears through R(a1, . . . , an). We also need to integrate over λ.Integration over solid angles of each of the terms of the semiclassical expansion yields a con-stant independent of ai ’s. That is, the angular average over powers of w

(n)t is independent of the

positions ai ’s.

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 123

We make a change of variables and interchange the variable a1 with the variable R. When wedo this, we are left with integrals with volume element

(69)∂a1

∂RdλdR

n∏i=2

dai.

We then perform a change of variables in the λ–R plane:

(70)λ′ = λ

R, R′ = R.

The volume element of our integral becomes

(71)R′ ∂a1

∂R′ dλ′ dR′n∏

i=2

dai.

Now the weighting factor e−βEn depends on λ′ only and is independent of R′ and the ai ’s. Hencethe integrations over R′ and the ai ’s can be performed yielding an irrelevant constant factor whichcan be absorbed in the normalization or dropped. We are left with an integration over λ′ and aweighting factor which is equal to the one obtained with the n skyrmions and antiskyrmions thatconstitute the n-topolon placed in the same position

(72)w(n)t = eiϕ tan

[(λ

z − a

)n

+(

λ∗

z∗ − a∗

)n

+ θ

2

]

since the Euclidean action of such a configuration is given by

St = βEn = 4βμ

∫d2x

∂zw(n)t ∂z∗w(n)∗

t + ∂z∗w(n)t ∂zw

(n)∗t

(1 + |w(n)t |2)2

(73)= 8βμ

n2λ2n

∫d2x

1

|z|2(n+1)= 8πβμ

n(λΛ)2n.

For the topolon the scale is set by the size of the topolon, so we choose μ = 1/λ. In the casewhere λ is set by the temperature (β = λ), we have

(74)St = βEn =λ∫

0

dτ8π

λg1/λ

n(λΛ)2n 8πλΛ

n(λΛ)2n.

Without loss of generality, from now on we only consider the topolon with all skyrmionsconstituting f (z) placed at the same position a. For this case we have

(75)Zn�=0 =∫ DνDν∗

(1 + |w(n)t + ν|2)2

d2a

A

1/Λe−S[w(n)

t +ν].

For convenience computing averages, we will normalize the partition somewhat differently, sothat we have

(76)Z =∞∑

n=0

∫ DνDν∗ dλd2a dϕ dθ sin θ

(1 + |w(n)t + ν|2)2

e−S(w(n)t +ν)

Z0.

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124 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

The weight in the partition function is the appropriate nonlinear sigma model Euclidean action

(77)S(w

(n)t + ν

) = 2Λ

∫d3x

∂μ(w(n)∗t + ν∗)∂μ(w

(n)t + ν)

(1 + |w(n)t + ν|2)2

.

Z0 is an arbitrary normalization factor since multiplying the partition function by a constantdoes not change the physics. We will choose Z0 conveniently to make some of our intermediatecalculations look simpler.

Of course, the partition function (76) cannot be calculated exactly and we will approximate itby doing perturbation theory about the noninteracting action

(78)S0 = 2Λ

∫dτ d2x

[∂μw

(n)∗t ∂μw

(n)t

(1 + |w(n)t |2)2

+ ∂μν∗∂μν

]= St + Sν,

where the first action is the unperturbed topolon action and the second is a free spin wave actionneglecting the nonlinearity which accounts for the interactions. Then the partition function is

(79)Z = 1

Z0

∞∑n=0

∫ DνDν∗ dλd2a dϕ dθ sin θ

(1 + |w(n)t + ν|2)2

e−S0−SI , SI = S(w

(n)t + ν

) − S0.

We choose

(80)Z0 =∞∑

n=0

∫ DνDν∗ dλd2a dϕ dθ sin θ

(1 + |w(n)t |2)2

e−S0 .

We perform perturbation theory in the interaction term by integrating out short distance de-grees of freedom. This is done by integrating large momentum and frequency spin waves, thosewith values between the microscopic cutoff Λ and a lower renormalization scale μ. In order forperturbation theory to be controlled we keep μ close to Λ and improve this perturbation theoryby using the renormalization group [11,12,15,23]. To integrate the short distance topolon config-urations we integrate those with sizes between the lattice constant 1/Λ and the renormalizationdistance 1/μ. Since μ is close to Λ we expand the action to lowest order in the interaction termand a simple manipulation leads to a partition function with a lower cutoff μ and with effectiveaction correct to first order given by

(81)Sμeff = S0 + 〈SI 〉.

The average 〈SI 〉 can be calculated by expanding the interacting action to fourth order in ν andwt as higher order terms in the expansion contribute higher orders in μ − Λ and are suppressedfor μ very close to Λ. This expansion yields the 13 terms

I = 2Λ

∫d3x

⟨∂μwn

t ∂μwn∗t

⟩,

II = −4Λ

∫d3x

⟨∂μwn

t ∂μwn∗t

∣∣wnt

∣∣2⟩,

III = −4Λ

∫d3x

⟨∂μwn

t ∂μwn∗t |ν|2⟩,

IV = −4Λ∫

d3x⟨∂μwn

t ∂μwn∗t wn

t ν∗⟩ + C.C.,

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 125

V = 2Λ

∫d3x

⟨∂μν∂μwn∗

t

⟩ + C.C.,

VI = −4Λ

∫d3x

⟨∂μν∂μwn∗

t

∣∣wnt

∣∣2⟩ + C.C.,

VII = −4Λ

∫d3x

⟨∂μν∂μwn∗

t |ν|2⟩ + C.C.,

VIII = −4Λ

∫d3x

⟨∂μν∂μwn∗

t wnt ν∗⟩ + C.C.,

IX = −4Λ

∫d3x

⟨∂μν∂μwn∗

t wn∗t ν

⟩ + C.C.,

X = 2Λ

∫d3x

⟨∂μν∂μν∗⟩,

XI = −4Λ

∫d3x

⟨∂μν∂μν∗∣∣wn∗

t

∣∣2⟩,

XII = −4Λ

∫d3x

⟨∂μν∂μν∗|ν|2⟩,

(82)XIII = −4Λ

∫d3x

⟨∂μν∂μν∗wn

t ν∗⟩ + C.C.

When we perform the averages, all terms with space derivatives of wt are irrelevant as thederivatives can be interchanged by a derivative over the position coordinates of the topolons andwhen averaged, will be smaller than the other terms by a factor of the lattice constant divided bythe linear dimensions of the system, a ratio that goes to zero in the thermodynamic limit. Hencewe are left only with the terms X to XIII. Term XIII being odd in the spin waves averages tozero. The average of term X over short distance gives a constant and is thus irrelevant. We areleft with the averaging terms XI and XII. If we go to higher powers in the expansion, there willbe similar terms to XI with higher powers of |w(n)

t |2. These can all be summed. We then obtainthat the effective action after averaging over short distance fluctuations is

(83)−4Λ

∫d3x ∂μν∂μν∗ ⟨|ν|2⟩ + 2Λ

∫d3x ∂μν∂μν∗

⟨1

(1 + |w(n)t |2)2

⟩.

The first term, when averaged over short distance fluctuations gives the already calculated Gold-stone renormalization

(84)−4Λ

4π2

[1 − μ

Λ

]∫d3x ∂μν∂μν∗.

To average over short distance topolons, we first perform the angular average and the averageover positions, which gives

(85)

⟨1

(1 + |w(n)t |2)2

⟩angles+positions

= 1

3.

For this average we do not include the weighting factor, as this factor is independent of the anglesand positions. We now integrate over topolon sizes λ from 1/Λ to 1/μ, including the weightingfactor to finally obtain

(86)2Λ

[1 − μ

]exp

[−n

8π]∫

d3x ∂μν∂μν∗.

3gΛ Λ gΛ
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126 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

Since there are topolon contributions from n = 1 to n = ∞, we sum these contributions from 1to ∞ to obtain

3gΛ

[1 − μ

Λ

] ∞∑n=1

exp

[−n

]∫d3x ∂μν∂μν∗

(87)= 2Λ

3gΛ

[1 − μ

Λ

]1

e8π/gΛ − 1

∫d3x ∂μν∂μν∗.

Both Goldstone and topolon renormalizations are proportional to 1 − μ/Λ, which can be madeas small as possible to make sure the perturbation expansion is controlled. The Goldstone renor-malization is linear in the coupling constant, but the topolon contribution is not analytic in thecoupling constant around gΛ = 0. It is thus nonperturbative. The renormalized action, renormal-ized spin stiffness and beta function are

Sren = 2μ

∫d3x

∂μν∂μν∗

(1 + |ν|2)2

∫d3x ∂μν∂μν∗

{1 −

(1 − μ

Λ

)gΛ

2π2+

(1 − μ

Λ

)1

3(e8π/gΛ − 1)

},

ρs(μ) = μ

= Λ

{1 −

(1 − μ

Λ

)gΛ

2π2+

(1 − μ

Λ

)1

3(e8π/gΛ − 1)

},

(88)β(g) = μ∂g

∂μ

∣∣∣∣Λ=μ

= g − g2

2π2+ g

3(e8π/g − 1).

The last term in the spin stiffness and in the beta function is the contribution from the topolonsand the rest is the contribution of the spin waves to fourth order in the coupling constant. Thecoupling constant at the quantum critical point gc 23.0764 is obtained from β(gc) = 0.

The important point is that the topolons add a new term to the beta function calculated above.This will lead to modification of the critical properties beyond those calculated from the spinwave expansion only. Below we estimate exponents from our approximations. These exponentswill not be suffiently accurate for two reasons. First, we calculated our spin wave expansion tofirst order in the coupling constant g. This will lead to large inaccuracies as g is not small. Onthe other hand, this would be easy to improve by including higher order corrections obtainablefrom the d = 4 − ε expansion, which is Borel summable and thus very accurate as far as thespin wave effects concern [23]. Inclusion of these terms will not change the new term we foundfrom the topolons and hence will not affect the fact that there is new critical physics coming fromsuperpositions of skyrmions and antiskyrmions. On the other hand such corrections are necessaryfor accurate enough exponents.

Another issue is that there might be more exact solutions that correspond to superpositions ofequal numbers of skyrmions and antikyrmions. We found no other solutions, but it is expectedthat there should be solutions that are time dependent in which the skyrmions and antiskyrmionsare in motion. These need to be included in order to have the most accurate possible exponents,but they do not invalidate the physics we have found: That skyrmions and antiskyrmions modifycritical properties. They further add to the physics we have found in the present work. Finally, forcompleteness we compare below with other approaches that people use to calcualte exponents.Such approaches are the 1/N expansion and the d = 2 + ε expansion. These are not as accurateas the d = 4 − ε expansion, which seems to give the accepted Heisenberg exponents [23], but

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 127

this last expansion cannot capture topological contributions as these exist only in two spatialdimensions.

The correlation length at scale μ is given by [10]

(89)ξ ∼ 1

μexp

[ gμ∫gc

dg

β(g)

]∼ (gc − gμ)1/β ′(gc).

The correlation length exponent is ν = −1/β ′(gc) ≡ −(dβ/dg|g=gc )−1. Including topolon con-

tributions, it evaluates to ν = 0.9297. The d = 2 + ε expansion of the O(N) vector model,which agrees with the 1/N expansion for large N , gives ν = 0.5 [23]. We note that our valueis larger than the accepted numerical evaluations of critical exponents in the Heisenberg model,ν = 0.71125 [24], but about as close to this accepted Heisenberg value than the 2 + ε expansionor the 1/N expansion. We conjecture that the difference between our value and the Heisenbergvalue is real and attributable to quantum critical degrees of freedom.

Goldstone renormalizations of the ordering direction σ = n3, and hence of the anomalousdimension η, are notoriously inaccurate. The one loop approximation leads to a value of η = 2,thousands of percent different from the accepted numerical Heisenberg value of η 0.0375 [24].The large N approximation, which sums bubble diagrams, is a lot more accurate. To order 1/N

one obtains η = 8/(3π2N) 0.09 for N = 3. We now calculate the value of η from topologicalnontrivial configurations

⟨n2

3

⟩ = Z = 1 − ⟨n2

1 + n22

⟩ = 1 −⟨

4|w|2(1 + |w|2)2

⟩ 1 − 2

3

1 − μ/Λ

e8π/gΛ − 1

(90)⇒ η(g) = μ

Z

∂Z

∂μ

∣∣∣∣Λ=μ

2

3(e8π/g − 1).

For the anomalous dimension at the quantum critical point we obtain η(gc) = 0.3381. On theother hand, we have seen that spin wave contributions tend to give quite large and nonsensicalvalues of η. In fact so large as to wash out the momentum dependence of the propagator. Hence,to calculate η, spin wave contributions prove to be tough to control. Our calculation gives a valuequite larger than the accepted numerical value. We have recently calculated [30] the unique valueof η that follows from quantum critical fractionalization into spinons and find η = 1. Whileour value obtained from topolons is far from 1, it is a lot closer than the accepted numericalHeisenberg value and the 1/N value.

The main purpose of this work is to look for possible new physics due to intrinsic critical exci-tations of quantum critical points. In this article we studied the approach to the quantum criticalpoint from the Néel ordered phase. We have uncovered some things that were not known be-fore. One thing which has been hinted at before was that skyrmion excitations are relevant to thequantum critical point. We have provided strong evidence that skyrmion fluctuations contributeto the quantum critical point. For the first time, we found new exact solutions that correspondsto superpositions of equal number of skyrmions and antiskyrmions. Also for the first time, wedeveloped and calculated the inclusion of these topological effects in the renormalization groupto yield critical exponents.

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Appendix A. Stereography and topology

A.1. 2 + 1 D antiferromagnets

In the present work we study 2-dimensional O(3) quantum antiferromagnets on bipartite lat-tices as described by the Heisenberg Hamiltonian

(A.1)H = J∑〈ij〉

�Si · �Sj

with J > 0 and 〈ij 〉 meaning that ij are next neighbors. Haldane [25] showed that in the largeS limit, the low energy universal physics of the Heisenberg antiferromagnet is equivalent to thatgiven by the O(3) nonlinear sigma model described by the Lagrangian and action

(A.2)L = Λ

2g

∫d2x ημν∂μ�n · ∂ν �n = Λ

2g

∫d2x ∂μ�n · ∂μ�n,

(A.3)S =∫

dtL,

where ημν is the 2 + 1 Lorentz metric and �n is a 3-dimensional unit vector, �n · �n = 1, that rep-resents the sublattice magnetization. Physically the inverse coupling constant is proportional tothe “spin” or magnetization stiffness ρs . The large S identification of the Heisenberg and nonlin-ear sigma model holds because the amplitude fluctuations of the spins are irrelevant for large S

[25]. Hence as long as the amplitude fluctuations are irrelevant to the long distance physics, theO(3) nonlinear sigma model will be an apt description of antiferromagnets regardless of S. Oneexpects amplitude fluctuations to be less relevant for lower dimensionality.

A very useful way of describing the O(3) nonlinear sigma model is through the stereographicprojection [8]:

(A.4)n1 + in2 = 2w

|w|2 + 1, n3 = 1 − |w|2

1 + |w|2 , w = n1 + in2

1 + n3.

The stereographic projection maps the sphere subtended by the Néel field or staggered magne-tization �n to the complex w plane by placing the Néel sphere below the w plane, with the northpole touching the center of the plane. If one draws a straight line joining the south pole and thepoint on the sphere corresponding to the Néel field, then the mapping to the w plane of this Néeldirection is given by extending the mentioned line until it intersects the w plane.

In terms of w the Lagrangian is

(A.5)L = 2Λ

g

∫d2x

∂μw∂μw∗

(1 + |w|2)2= 2Λ

g

∫d2x

∂0w∂0w∗ − 2∂zw∂z∗w∗ − 2∂z∗w∂zw

(1 + |w|2)2,

where z = x + iy and z∗ = x − iy is its conjugate. The classical equations of motion which followby stationarity of the classical action are ��n = 0, which in terms of the stereographic variable w

are

(A.6)�w = 2w∗

1 + |w|2 ∂μw∂μw or ∂20w − 4∂z∂z∗w = 2w∗

1 + |w|2[(∂0w)2 − 4∂zw∂z∗w

].

The quantum mechanics of the O(3) nonlinear sigma model is achieved either via path inte-gral or canonical quantization. The last is performed by defining the momentum conjugate to �n,

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or to w and w∗, by

(A.7)�Π(t, �x) ≡ δL

δ∂0�n(t, �x), Π∗(t, �x) ≡ δL

δ∂0w(t, �x), Π(t, �x) ≡ δL

δ∂0w∗(t, �x)

and then imposing canonical commutation relations among the momenta and coordinates. Dueto the nonlinear constraint �n · �n = 1, the momentum �Π is an angular momentum satisfying theSU(2) algebra:

(A.8)�Π · �n = 0, �Π × �Π = i �Π.

The Hamiltonian is then given by

H =∫

d2x ( �Π · ∂0�n − L) =∫

d2x

[g

2Λ�Π2 + Λ

2g∂i �n · ∂i �n

]

=∫

d2x(Π∗ · ∂0w + Π · ∂0w

∗ − L)

=∫

d2x

[g

(1 + |w|2)2

Π∗Π + 2Λ

g

∂iw∂iw∗

(1 + |w|2)2

]

(A.9)=∫

d2x

[g

(1 + |w|2)2

Π∗Π + 4Λ

g

(∂zw∂z∗w∗∂z∗w∂zw∗)

(1 + |w|2)2

].

The Heisenberg equations of motion that follow from this Hamiltonian, when properly or-dered, are identical to the classical equations. There are ordering ambiguities in this Hamiltonian.The usual prescription to deal with the ambiguities is by symmetrization, but the correct ordercan only be determined by comparison with experiment if there is a measurement that is sensi-tive to the operator order. Most results are insensitive to these ordering ambiguities as they onlyintroduce short distance modifications to the physics.

A.2. Excitations of the Néel ordered phase of O(3) nonlinear sigma model

We remind the reader that classically the lowest energy state is Néel ordered for all g < ∞,i.e. the spin stiffness, ρs , is never zero. Quantum mechanically the situation is different. In 2 + 1and higher dimensions, quantum mechanical fluctuations cannot destroy the Néel order for thebare coupling constant less than some critical value gc [10–12]. At gc the renormalized long-distance, low-energy coupling constant diverges [10], i.e. the system loses all spin stiffness. Atsuch a point quantum fluctuations destroy the Néel order in the ground state as the renormalizedstiffness vanishes. In the present section we concentrate in the excitations of the Néel orderedphase.

A.3. Magnons

Linearization of the equations of motion leads to the low energy excitations of the sigmamodel (magnons in the Néel phase and triplons in the disordered phase) when quantized. We nowturn our attention to the Néel ordered phase. When the system Néel orders, �n, or equivalently w,will acquire an expectation value:

(A.10)⟨na

⟩ = −δ3a,

⟨1

w

⟩= 0,

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where we have chosen the order parameter in the −3-direction as it will always point in anarbitrary, but fixed direction. Small fluctuations about the order parameter

(A.11)1

w= ν

are the magnons or Goldstone excitations of the Néel phase. To leading order the magnon La-grangian is

L = 2Λ

g

∫d2x

∂μν∂μν∗

(1 + |ν|2)2 2Λ

g

∫d2x ∂μν∂μν∗

(A.12)= 2Λ

g

∫d2x

(∂0ν∂0ν

∗ − 2∂zν∂z∗ν∗ − 2∂z∗ν∂zν∗)

leading to the equations of motion

(A.13)�ν = 0, ∂20 ν − 4∂z∂z∗ν = 0.

The linearized excitations of the Néel phase have relativistic dispersion that vanishes at longwavelengths as dictated by Goldstone’s theorem [26]. The magnons are of course spin 1 particles.They have only 2 polarizations as they are transverse to the Néel order.

A.4. Skyrmions

The Goldstones are not the only excitations of the ordered phase in the nonlinear sigma model.Since the 1970’s, it has been known that exact or approximate time independent solutions of theclassical equations of motion, when stable against quantum fluctuations, are quantum particleexcitations of the system [19]. The nonlinear sigma model possesses time independent solutionsof a topological nature [8,20]. These excitations are disordered at finite length scales but relaxinto the Néel state far away:

(A.14)lim|�x|→∞

�n = (0,0,−1), lim|�x|→∞

w = ∞.

They consist in the order parameter rotating a number of times as one moves from infinity towarda fixed but arbitrary position in the plane. Since two-dimensional space can be thought of as aninfinite 2-dimensional sphere, the excitations fall in homotopy classes of a 2D sphere into a 2Dsphere: S2 → S2. The topological excitations are thus defined by the number of times they mapthe 2D sphere into itself. They are thus characterized by the Jacobian

(A.15)q = 1

∫d2x εij �n · ∂i �n × ∂j �n

or

(A.16)q = i

∫d2x

εij ∂iw∂jw∗

(1 + |w|2)2= 1

π

∫d2x

∂zw∂z∗w∗ − ∂z∗w∂zw∗

(1 + |w|2)2.

The number q will be an integer measuring how many times the n-sphere gets mapped into theinfinite 2D sphere corresponding to the plane where the spins live. If we define the space–timecurrent

(A.17)Jμ = 1

8πεμνσ �n · ∂ν �n × ∂σ �n = i

2πεμνσ ∂νw∂σ w∗

(1 + |w|2)2,

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 131

it is easily seen that it is conserved ∂μJμ = 0 and that the charge associated with it is our topo-logical charge:

(A.18)q =∫

d2x J 0.

Thus q is a conserved quantum number. These topological field configurations were originallydiscovered by Skyrme [21] and are called skyrmions. The conserved charge is thus the skyrmionnumber.

From the expressions for the charge q and for the Hamiltonian, it is easily seen [8,20] thatE � 4πΛ|q|/g. We see that we can construct skyrmions with q > 0 by imposing the condition

(A.19)∂z∗w = 0,

that is w is a function of z only. Since the magnetization, �n or w, is a continuous function ofz, the worst singularities it can have are poles. The skyrmions will have a location given by thepositions of the poles or of the zeros of w. Far away from its position, the field configuration willrelax back to the original Néel order. Therefore we have the boundary condition w(∞) = ∞,which implies

(A.20)w = 1

λq

q∏i=1

(z − ai).

This can easily be check to have charge q and energy 4πΛq/g. λq is the arbitrary size and phaseof the configuration and ai are the positions of the skyrmions that constitute the multiskyrmionconfiguration. The energy is independent of the size and phase due to the conformal invarianceof the configuration. We remark that since the multiskyrmions energy is the sum of individualskyrmion energies, the skyrmions do not interact among themselves [8]. An example of the ex-plicit calculation of the charge and energy for a diskyrmion is shown in Appendix A.6. Similarly,the multiantiskyrmion configuration can be shown to be

(A.21)w = 1

(λ∗)qq∏

i=1

(z∗ − a∗i )

with charge −q and energy 4πΛq/g.We have just studied the skyrmion and antiskyrmion configurations which relax to a Néel

ordered configuration in the −3 direction far away from their positions. We shall call them −3-skyrmions. The skyrmion direction is given by the boundary conditions as z → ∞. For example,(z − a)/λ gives na(∞) = −δ3a , so it is a −3-skyrmion. The +3-skyrmion is λ/(z − a). The+1-skyrmion is (z − a)/(z − b). The −1-skyrmion is −(z − a)/(z − b). The +2-skyrmion isi(z − a)/(z − b). The −2-skyrmion is −i(z − a)/(z − b). Because of the rotational invariance ofthe underlying theory, they are all kinematically equivalent. They are not dynamically equivalentsince a Néel ordered ground state has skyrmions and antiskyrmions corresponding to its orderingdirection as excitations.

That the skyrmion configurations behave like particles follows easily by making them timedependent and examining their dynamics. We do so for a single skyrmion here:

(A.22)w = z − a

λ.

We make the skyrmion time dependent by allowing it to move (making its position, a(t), timedependent), and become fatter or slimmer with time (making its size, λ(t), time dependent). We

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132 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

substitute this time dependent configuration in the Lagrangian and obtain in Appendix A.7:

(A.23)L = 2πΛ

g|a|2 − 4πΛ

g.

Since the skyrmion Lagrangian acquired a term proportional to the skyrmion velocity squared, akinetic energy term, we see that the skyrmion behaves like a free particle of mass 4πΛ/g with anexcitation gap of 4πΛ/g. The skyrmion position is a dynamical variable. On the other hand, theconformal parameter λ does not have dynamics as it has infinite mass in the thermodynamic limit,see Appendix A.7. The conformal parameter is thus an arbitrary constant making the skyrmionconfiguration conformally invariant even when we allow time dependence of the configuration.Even though the sigma model does not have a microscopic length, real antiferromagnets willhave a microscopic length as a consequence of amplitude fluctuations. We thus physically expect|λ| to be cutoff at small values by a coherence length ξ . The long distance physics is, of course,insensitive to this cutoff.

A.5. Skyrmion–antiskyrmion states

Since the charge or skyrmion number is conserved, a configuration with nonzero skyrmionnumber cannot be excited out of the ground state in the absence of an external probe that couplesto skyrmion number. Therefore, skyrmions and antiskyrmions will be created in equal numbers.We thus have to study the interaction between skyrmions and antiskyrmions. A skyrmion–antiskyrmion configuration, which is not a solution to the equations of motion, is given by

(A.24)w = 1

λ2(z − a)(z∗ − b∗).

The energy of this static configuration is given exactly by

(A.25)Es = 4πΛ

g+ Λ

g

|a − b|4|λ|4

∞∫0

rK( 4r

(r+1)2

)(1 + r2|a−b|4

16|λ|4)2

(r + 1)dr,

where K(x) is the Jacobian elliptic function [27]. We reproduce the details of the calculationin Appendix A.8 because it has been calculated or approximated incorrectly in previous works[8,28,29]. The skyrmion–antiskyrmion interaction, or potential energy, is given by the differencebetween the static energy Es and the sum of the energies of the isolated skyrmion and skyrmion,V = Es − 8πΛ/g.

The skyrmion–antiskyrmion interaction has many interesting features. As the distance be-tween the skyrmion and antiskyrmion becomes small compared to the size of the configuration,|a − b|/|λ| � 1, their interaction is very soft; the energy goes like

(A.26)V −4πΛ

g+ π2Λ

2g

|a − b|2|λ|2 .

At short distances the skyrmions and antiskyrmions are bound by a harmonic potential. The min-imum of this classical energy occurs when the skyrmion–antiskyrmion form a bound state withzero “separation” between the skyrmion and antiskyrmion, or equivalently infinite conformalsize, i.e. |a − b|/|λ| = 0. This bound state resonance has energy −4πΛ/g, or a binding energyof 4πΛ/g. Therefore the skyrmion and antiskyrmion gaps get halved. When this skyrmion-antiskyrmion configuration has a large but finite size, i.e. |λ|/|a − b| � 1, the potential between

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 133

the skyrmion and antiskyrmion is very soft and vanishes when the configuration has arbitrarilylarge size. In this limit the skyrmion and antiskyrmion do not interact despite being “bound”.

At large distances or small size, |a − b|/|λ| � 1, the interaction is approximately

(A.27)V 64πΛ

g

|λ|4|a − b|4 ln

( |a − b|2|λ|

).

At large enough distances the skyrmion and antiskyrmion are almost free and repel each otherwith an interaction that vanishes at infinitely large separations. We see that the skyrmion-antiskyrmion potential is attractive at short distances or large sizes, while at larger distancesor small size it goes to a maximum energy which is higher than 0 and then vanishes at infinity.In order to unbind them one has to at least supply an energy 4πΛ/g. Classically one would haveto supply enough energy to get over the potential energy hump, but quantum mechanically onecan, of course, tunnel through the barrier.

Contrary to pure skyrmion or pure antiskyrmion configurations, the skyrmion–antiskyrmionconfigurations are not stationary solutions of the equations of motion. Therefore the dynamicswill not be restricted to center of mass motion alone. In order to study the dynamics of theskyrmion and antiskyrmion configurations we allow motion of the positions of the skyrmion,a(t), and the antiskyrmion, b(t), and permit time dependence of the conformal parameter, λ(t).We substitute this time dependent configuration in the sigma model Lagrangian in Appendix A.9and obtain the kinetic energy part of the Lagrangian to be

(A.28)T = mab

2

(|a|2 + |b|2) + mλ

2|λ|2

with

mλ = Λ

2g

|a − b|6|λ|6

∞∫0

R3K( 4R

(R+1)2

)dR

(1 + |a − b|4R2/16|λ|4)2(R + 1),

(A.29)mab = 4Λ

g

∞∫0

2π∫0

r3 dr dθ1

[1 + r2(r2 + |a − b|2/|λ|2 − 2r|a − b| cos θ)/|λ|]2

with K(x) the Jacobian elliptic function [27]. At short distances, or large sizes, |a − b| � |λ|,the a, b and λ masses have the asymptotic behavior

(A.30)mλ 4π2Λ

g, mab 2πΛ

g.

In this limit the mass of the skyrmion and antiskyrmion is equal to 1/2 the mass of an isolatedskyrmion or antiskyrmion. At large distances, or small sizes, |a − b| � |λ|, the masses go like

(A.31)mλ 64πΛ

g

|λ|2|a − b|2 ln

( |a − b|24|λ|2

), mab 29πΛ

g

|λ|8|a − b|8 .

The Lagrangian that describes the dynamics of the skyrmion–antiskyrmion configuration is thus

(A.32)L = mab

4

∣∣∣∣ d

dt(a + b)

∣∣∣∣2

+ mab

4

∣∣∣∣ d

dt(a − b)

∣∣∣∣2

+ mλ

2|λ|2 − V

( |a − b||λ|

).

We see that the center of mass coordinate decouples as required by the translational invarianceof the system. Contrary to the pure skyrmion configurations, here the conformal parameter, λ,

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has dynamics and is not an arbitrary parameter. That is, the skyrmion–antiskyrmion is not con-formally invariant.

A.6. The diskyrmion

In the present subsection we explicitly check that a diskyrmion configuration has charge 2and energy equal to the sum of the energies of the two single skyrmions that constitute thediskyrmion. This is important because there are false claims [28] in the literature that multi-skyrmion configurations interact through logarithmic potentials. Above we concluded that sincea multiskyrmion configuration has an energy that is the sum of the skyrmions that constitute it,skyrmions do not interact. This was originally concluded by Gross [8]. In the present section weshow this by explicit calculation for the diskyrmion energy.

The diskyrmion configuration is

(A.33)w = 1

λ2(z − a)(z − b).

The charge is given by

(A.34)q = 4

π

∫d2x

|z − (a + b)/2|2(1 + |z − a|2|z − b|2)2

,

where we have rescaled z and defined a = a/λ and b = b/λ in order to absorb the arbitrary sizeλ. The energy is given by

(A.35)E = 16Λ

g

∫d2x

|z − (a + b)/2|2(1 + |z − a|2|z − b|2)2

.

In order to calculate the energy and charge we define

(A.36)A ≡ a + b

2, B ≡ a − b

2,

and make the change of origin z → z + A, to obtain that the energy and charge are E = 16ΛI/g

and q = 4I/π , where I is the integral

I = 1

2

∫ |z|2 dzdz∗

[1 + |z + B|2|z − B|2]2= 1

2

∫z dz z∗ dz∗

{1 + [z2 − B2][(z∗)2 − (B∗)2]}2

(A.37)= 1

8

∫d[z2 − B2]d[(z∗)2 − (B∗)2]

{1 + [z2 − B2][(z∗)2 − (B∗)2]}2= 1

4

∫dudu∗

[1 + |u|2]2.

To obtain the last equality we made the variable change u = z2 − B2. The factor of 2 arisesbecause u is linearly related to z2, so one must cover the u complex plane twice in order to coverthe z complex plane once. Going over to polar coordinates of the u plane we get

(A.38)I = 1

2

∫r dr dθ

(1 + r2)2= π

2.

We finally obtain

(A.39)q = 4

πI = 2, E = 16Λ

gI = 8πΛ

g= 4πΛ

gq

as expected.

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 135

A.7. Skyrmion kinetic energy

When we substitute the time dependent skyrmion

(A.40)w = z − a(t)

λ(t)

into the sigma model Lagrangian

(A.41)L = 2Λ

g

∫d2x

1

(1 + |w|2)2(∂0w∂0w

∗ − 2∂zw∂z∗w∗ − 2∂z∗w∂zw∗)

we obtain

(A.42)L = −4πΛ

g+ 2Λ

g

∫d2x

1

(1 + |w|2)2

( |λ|2|λ|2 |w|2 + |a|2

|λ|2 + λa∗

|λ|2 w + λ∗a|λ|2 w∗

).

First we evaluate the integral

(A.43)∫

d2xw

(1 + |w|2)2= |λ|2

∫r2eiθ dr dθ

(1 + r2)2= 0,

where we made the variable change z → z + a, the conformal transformation z → λz, and wentto polar coordinates of the complex z plane. Similarly we have

(A.44)∫

d2xw∗

(1 + |w|2)2= |λ|2

∫r2e−iθ dr dθ

(1 + r2)2= 0.

The skyrmion Lagrangian is then

L = −4πΛ

g+ 2Λ

g

∫d2x

{ |a|2|λ|2

1

(1 + |w|2)2+ |λ|2

|λ|2|w|2

(1 + |w|2)2

}

(A.45)= ma

2|a|2 + mλ

2|λ|2 − 4πΛ

g.

We see that there is a kinetic energy term for a with mass

(A.46)ma = 4Λ

g|λ|2∫

d2x1

(1 + |w|2)2= 4Λ

g

∫r dr dθ

(1 + r2)2= 4πΛ

g

and a kinetic energy term for λ with mass

mλ = 4Λ

g|λ|2∫

d2x|w|2

(1 + |w|2)2= 4Λ

g

∫r3 dr dθ

(1 + r2)2

(A.47)= 4Λ

g

[π ln

(1 + r2)∣∣∞

0 − π] = ∞.

Since the mass for λ is infinite, λ = 0 exactly in order not to pay an infinite kinetic energy cost.Thus λ is not a dynamical variable, but a constant arbitrary parameter. This is true classically andquantum mechanically. Quantum mechanically, the term

(A.48)|pλ|22mλ

→ 0 as mλ → ∞

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with pλ the momentum conjugate to λ. Since the pλ is the only quantity that does not commutewith λ in the Hamiltonian H , it immediately follows that

(A.49)λ = − i

h[λ,H ] = 0

in the infinite mass limit.

A.8. Skyrmion–antiskyrmion static energy

We now calculate the skyrmion number q and energy E of the skyrmion–antiskyrmion staticconfiguration

(A.50)w = 1

λ2(z − a)(z∗ − b∗).

The charge is given by

(A.51)q = 1

π

∫d2x

|z − b|2 − |z − a|2(1 + |z − a|2|z − b|2)2

,

where we have made the conformal transformation z → λz and defined a = a/λ and b = b/λ inorder to absorb the arbitrary conformal parameter λ. The energy is given by

(A.52)E = 4Λ

g

∫d2x

|z − b|2 + |z − a|2(1 + |z − a|2|z − b|2)2

.

As in Appendix A.6, in order to calculate the energy and charge we define

(A.53)A ≡ a + b

2, B ≡ a − b

2and make the change of origin z → z + A, to obtain that the energy and charge are

E = 8Λ

g

∫1

2

|z|2 dzdz∗

(1 + |z + B|2|z − B|2)2+ 4Λ

g|B|2

∫dzdz∗

(1 + |z + B|2|z − B|2)2,

(A.54)q = 1

∫dzdz∗ (|z + B|2 − |z − B|2)

(1 + |z + B|2|z − B|2)2.

The charge is easily seen to be zero. It is the subtraction of two integrals. If on the second integralwe take z → −z, it becomes equal to the first and thus cancels it upon subtraction. Thereforeq = 0 for the skyrmion-antiskyrmion configuration as expected. The first term in the energy wasevaluated in Appendix A.6. We thus have

(A.55)E = 4πΛ

g+ 4Λ

g|B|2

∫dzdz∗

[1 + |z + B|2|z − B|2]2.

Making the variable change u = z2 − B2, we get

E = 4πΛ

g+ 2Λ

g|B|2

∫dudu∗

|u + B2|[1 + |u|2]2

(A.56)= 4πΛ

g+ 4Λ

g|B|2

∫r dr dθ

1

(1 + r2)2√

r2 + |B|4 + 2r|B|2 cos θ,

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 137

where for the last equality we went to polar coordinates of the complex u plane. Now

2π∫0

dθ√r2 + |B|4 + 2r|B|2 cos θ

= 2

π∫0

dθ√r2 + |B|4 + 2r|B|2 cos θ

= 2

π∫0

dθ√r2 + |B|4 + 2r|B|2 − 4r|B|2 sin2(θ/2)

(A.57)= 4

π/2∫0

dφ√r2 + |B|4 + 2r|B|2 − 4r|B|2 sin2 φ

= 4

r + |B|2 K

(4r|B|2

(r + |B|2)2

),

where K(x) is the Jacobian elliptic function [27]. After making the variable change R = |B|2r ,we then have

E = 4πΛ

g+ 16Λ

g|B|4

∞∫0

K( 4R

(R+1)2

)R dR

(R + 1)(1 + |B|4R2)2

(A.58)

= 4πΛ

g+ Λ

g

|a − b|4|λ|4

∞∫0

K

(4R

(R + 1)2

)R dR

1

[R + 1][1 + |a − b|4R2/(16|λ|4)]2.

Asymptotic approximations yield to leading order

E 4πΛ

g+ π2Λ

2g

|a − b|2|λ|2 for

|a − b||λ| � 1,

(A.59)E 8πΛ

g+ 64πΛ

g

|λ|4|a − b|4 ln

( |a − b|2|λ|

)for

|a − b||λ| � 1.

A.9. Skyrmion–antiskyrmion kinetic energy

We now move to determine the kinetic energy of the time dependent skyrmion–antiskyrmionconfiguration:

(A.60)w = [z − a(t)][z∗ − b∗(t)]λ2(t)

we substitute

∂0w∂0w∗ = 4

|λ|2|λ|2 |w|2 + |a|2

|λ|4 |z − b|2 + |b|2|λ|4 |z − a|2 + 2

λa∗

|λ|2λ∗ (z − b)w

+ 2λ∗a|λ|2λ(z∗ − b∗)w∗ + 2

λb

|λ|2λ∗ (z∗ − a∗)w + 2λ∗b∗

|λ|2λ(z − a)w∗

(A.61)+ ab

|λ|4 (z∗ − a∗)(z∗ − b∗) + a∗b∗

|λ|4 (z − a)(z − b)

into the kinetic term of the sigma model Lagrangian

(A.62)L = 2Λ∫

d2x∂0w∂0w

∗ − 2∂zw∂z∗w∗ − 2∂z∗w∂zw∗

2 2

g (1 + |w| )
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138 Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140

to obtain

L =∫

d2x

(8Λ

g

|λ|2|λ|2

|w|2(1 + |w|2)2

+ 2Λ

g

|a|2|λ|4

|z − b|2(1 + |w|2)2

+ 2Λ

g

|b|2|λ|4

|z − a|2(1 + |w|2)2

+ 4Λ

g

λa∗

|λ|2λ∗(z − b)w

(1 + |w|2)2+ 4Λ

g

λ∗a|λ|2λ

(z∗ − b∗)w∗

(1 + |w|2)2+ 4Λ

g

λb

|λ|2λ∗(z∗ − a∗)w(1 + |w|2)2

+ 4Λ

g

λ∗b∗

|λ|2λ(z − a)w∗

(1 + |w|2)2+ 2Λ

g

ab

|λ|4(z∗ − a∗)(z∗ − b∗)

(1 + |w|2)2

(A.63)+ 2Λ

g

a∗b∗

|λ|4(z − a)(z − b)

(1 + |w|2)2

).

We now evaluate term by term of the Lagrangian. The first is

(A.64)L1 = 8|λ|2g|λ|2

∫d2x

|w|2(1 + |w|2)2

= 4|λ|2g

∫dzdz∗ |z − a|2|z − b|2

(1 + |z − a|2|z − b|2)2,

where we made the conformal transformation z → λz, and defined a = a/λ and b = b/λ. As inAppendix A.6, we define

(A.65)A ≡ a + b

2, B ≡ a − b

2

and make the change of origin z → z + A, to obtain

L1 = 4Λ

g|λ|2

∫dzdz∗ |z − B|2|z + B|2

(1 + |z − B|2|z + B|2)2

= 2Λ

g|λ|2

∫dudu∗ |u|2

(1 + |u|2)2|u + B2|= 4Λ

g|λ|2

∫r3 dr dθ

(1 + r2)2√

r2 + |B|4 + 2r|B|2 cos θ

= 16Λ

g|λ|2

∫ r3K( 4r|B|2

(r+|B|2)2

)dr

(1 + r2)2(r + |B|2)

= Λ

4g

|λ|2|λ|6 |a − b|6

∞∫0

R3K( 4R

(R+1)2

)dR

(1 + |a − b|4R2/16|λ|4)2(R + 1)

(A.66)≡ mλ

|λ|22

,

where the second line follows from the variable change u = z2 − B2, and the third by going topolar coordinates in the complex u plane. K(x) is the Jacobian elliptic function [27]. We have

L1 2π2Λ

g|λ|2 for

|a − b||λ| � 1,

(A.67)L1 32πΛ

g

|λ|2|λ|2|a − b|2 ln

( |a − b|24|λ|2

)for

|a − b||λ| � 1.

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Z. Nazario, D.I. Santiago / Nuclear Physics B 761 [FS] (2007) 109–140 139

The second term of the Lagrangian (A.63) is

L2 = 2|a|2g|λ|4

∫d2x

|z − b|2(1 + |w|2)2

= |a|2g

∫dzdz∗ |z − b|2

(1 + |z − a|2|z − b|2)2

= |a|2g

∫dzdz∗ |z|2

(1 + |z − 2B|2|z|2)2

= 2|a|2g

∞∫0

2π∫0

r3 dr dθ

[1 + r2(r2 + 4|B|2 − 4r|B| cos θ)]2

(A.68)≡ |a|22

ma

( |a − b||λ|

),

where we made the shift z → z + A − B from the second to the third line. The mass for the a

coordinate is defined through the integral

(A.69)ma

( |a − b||λ|

)= 4

g

∞∫0

2π∫0

r3 dr dθ

[1 + r2(r2 + 4|B|2 − 4r|B| cos θ)]2

with B = (a − b)/(2λ). For |a − b| � |λ|, we have

(A.70)ma = 2π

g.

For |a − b| � |λ| with r = |B|R, we have

(A.71)ma

( |a − b||λ|

)= 4

g|B|4∞∫

0

2π∫0

R3 dR dθ

[(1/|B|4) + R4 + 4R2 − 4R3 cos θ)]2 29π |λ|8

g|a − b|8 .

Similarly the third term of the Lagrangian (A.63) is

(A.72)L3 = |b|22

mb

( |a − b||λ|

)with

(A.73)mb

( |a − b||λ|

)= ma

( |a − b||λ|

).

The rest of the terms of the Lagrangian (A.63) come out to be zero by rotational invariance inthe complex z plane.

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