theoretical physics - lund university - lavi nethome.thep.lu.se/~bijnens/talks/odense08.pdfsymmetry...

EFFECTIVE FIELD THEORY:

INTRODUCTION AND APPLICATIONS

Johan Bijnens

Lund University

[email protected]://www.thep.lu.se/∼bijnens

Various ChPT: http://www.thep.lu.se/∼bijnens/chpt.html

lavinet

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.1/103

Overview

What is effective field theory?

Introduction to ChPT including possible problems

Results for two-flavour ChPT

Results for three flavour ChPT

Partial quenching and ChPT

η → 3πSome comments about Higgs sector and ChPT


Wikipedia

http://en.wikipedia.org/wiki/Effective_field_theory

In physics, an effective field theory is an approximate theory(usually a quantum field theory) that contains theappropriate degrees of freedom to describe physicalphenomena occurring at a chosen length scale, but ignoresthe substructure and the degrees of freedom at shorterdistances (or, equivalently, higher energies).


Wikipedia

http://en.wikipedia.org/wiki/Chiral_perturbation_theory

Chiral perturbation theory (ChPT) is an effective field theoryconstructed with a Lagrangian consistent with the(approximate) chiral symmetry of quantumchromodynamics (QCD), as well as the other symmetries ofparity and charge conjugation. ChPT is a theory whichallows one to study the low-energy dynamics of QCD. AsQCD becomes non-perturbative at low energy, it isimpossible to use perturbative methods to extractinformation from the partition function of QCD. Lattice QCDis one alternative method that has proved successful inextracting non-perturbative information.


Effective Field Theory

Main Ideas:

Use right degrees of freedom : essence of (most)physics

If mass-gap in the excitation spectrum: neglect degreesof freedom above the gap.

Examples:

Solid state physics: conductors: neglectthe empty bands above the partially filledoneAtomic physics: Blue sky: neglect atomicstructure


Power Counting

➠ gap in the spectrum =⇒ separation of scales➠ with the lower degrees of freedom, build the mostgeneral effective Lagrangian


Power Counting


➠ ∞# parameters➠ Where did my predictivity go ?


Power Counting



=⇒ Need some ordering principle: power counting


Power Counting



=⇒ Need some ordering principle: power counting➠ Taylor series expansion does not work (convergenceradius is zero)➠ Continuum of excitation states need to be taken intoaccount


Why is the sky blue ?

System: Photons of visible light and neutral atomsLength scales: a few 1000 Å versus 1 ÅAtomic excitations suppressed by ≈ 10−3




LA = Φ†v∂tΦv + . . . LγA = GF 2µνΦ†vΦv + . . .Units with h/ = c = 1: G energy dimension −3:





σ ≈ G2E4γ





σ ≈ G2E4γ

blue light scatters a lot more than red{

=⇒ red sunsets=⇒ blue sky

Higher orders suppressed by 1 Å/λγ.


Why Field Theory ?

➠ Only known way to combine QM and special relativity➠ Off-shell effects: there as new free parameters


Why Field Theory ?


Drawbacks• Many parameters (but finite number at any order)any model has few parameters but model-space is large

• expansion: it might not converge or only badly


Why Field Theory ?


Drawbacks• Many parameters (but finite number at any order)any model has few parameters but model-space is large

• expansion: it might not converge or only badly

Advantages• Calculations are (relatively) simple• It is general: model-independent• Theory =⇒ errors can be estimated• Systematic: ALL effects at a given order can be included• Even if no convergence: classification of models oftenuseful


Examples of EFT

Fermi theory of the weak interaction

Chiral Perturbation Theory: hadronic physics

NRQCD

SCET

General relativity as an EFT

2,3,4 nucleon systems from EFT point of view


references

A. Manohar, Effective Field Theories (Schladminglectures), hep-ph/9606222

I. Rothstein, Lectures on Effective Field Theories (TASIlectures), hep-ph/0308266

G. Ecker, Effective field theories, Encyclopedia ofMathematical Physics, hep-ph/0507056

D.B. Kaplan, Five lectures on effective field theory,nucl-th/0510023

A. Pich, Les Houches Lectures, hep-ph/9806303

S. Scherer, Introduction to chiral perturbation theory,hep-ph/0210398

J. Donoghue, Introduction to the Effective Field TheoryDescription of Gravity, gr-qc/9512024


School

PSI Zuoz Summer School on Particle PhysicsEffective Theories in Particle Physics, July 16 - 22, 2006

http://ltpth.web.psi.ch/zuoz_school/previous_summerschools/zuoz2006/index.html

R. Barbieri: Effective Theories for Physics beyond the Standard Model

M. Beneke: Concept of Effective Theories, Heavy Quark Effective Theory andSoft-Collinear Effective Theory

G. Colangelo: Chiral Perturbation Theory

U. Langenegger: B Physics and Quarkonia

H. Leutwyler: Historical and Other Remarks on Effective Theories

A. Manohar: Nonrelativistic QCD

L. Simons: Pion-Nucleon Interaction

M. Sozzi: Kaon Physics


Chiral Perturbation Theory

Exploring the consequences of the chiral symmetry of QCDand its spontaneous breaking using effective field theorytechniques



Exploring the consequences of the chiral symmetry of QCDand its spontaneous breaking using effective field theorytechniques

Derivation from QCD:H. Leutwyler, On The Foundations Of Chiral Perturbation Theory,Ann. Phys. 235 (1994) 165 [hep-ph/9311274]


The mass gap: Goldstone Modes

UNBROKEN: V (φ)

Only massive modesaround lowest energystate (=vacuum)

BROKEN: V (φ)

Need to pick a vacuum〈φ〉 6= 0: Breaks symmetryNo parity doubletsMassless mode along bottom

For more complicated symmetries: need to describe thebottom mathematically: G → H =⇒ G/H (explain)


The two symmetry modes compared

Wigner-Eckart mode Nambu-Goldstone mode

Symmetry group G G spontaneously broken to subgroup H

Vacuum state unique Vacuum state degenerate

Massive Excitations Existence of a massless mode

States fall in multiplets of G States fall in multiplets of H

Wigner Eckart theorem for G Wigner Eckart theorem for H

Broken part leads to low-energy theorems

Symmetry linearly realized Full Symmetry, G, nonlinearly realized

unbroken part, H, linearly realized


Some clarifications

φ(x): orientation of vacuum in every space-time point

Examples: spin waves, phonons

Nonlinear: acting by a broken symmetry operatorchanges the vacuum, φ(x) → φ(x) + αThe precise form of φ is not important but it mustdescribe the space of vacua (field transformationspossible)

In gauge theories: the local symmetry allows the vacuato be different in every point, hence the GoldstoneBoson must not be observable as a massless degree offreedom.


The power counting

Very important:

Low energy theorems: Goldstone bosons do notinteract at zero momentum

Heuristic proof:

Which vacuum does not matter, choices related bysymmetry

φ(x) → φ(x) + α should not matterEach term in L must contain at least one ∂µφ



Degrees of freedom: Goldstone Bosons from ChiralSymmetry Spontaneous Breakdown

Power counting: Dimensional countingExpected breakdown scale: Resonances, so Mρ or higher

depending on the channel






Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: SU(3)V

But LQCD =∑

q=u,d,s

[iq̄LD/ qL + iq̄RD/ qR − mq (q̄RqL + q̄LqR)]

So if mq = 0 then SU(3)L × SU(3)R.






Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: SU(3)V

But LQCD =∑

q=u,d,s

[iq̄LD/ qL + iq̄RD/ qR − mq (q̄RqL + q̄LqR)]

So if mq = 0 then SU(3)L × SU(3)R.Can also see that via v < c, mq 6= 0 =⇒

v = c, mq = 0 =⇒/Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.17/103


〈q̄q〉 = 〈q̄LqR + q̄RqL〉 6= 0SU(3)L × SU(3)R broken spontaneously to SU(3)V8 generators broken =⇒ 8 massless degrees of freedomand interaction vanishes at zero momentum



〈q̄q〉 = 〈q̄LqR + q̄RqL〉 6= 0SU(3)L × SU(3)R broken spontaneously to SU(3)V8 generators broken =⇒ 8 massless degrees of freedomand interaction vanishes at zero momentum

Power counting in momenta: (explain)

p2

1/p2

∫

d4p p4

(p2)2 (1/p2)2 p4 = p4

(p2) (1/p2) p4 = p4



Large subject:

Steven Weinberg, Physica A96:327,1979: 1789citations

Juerg Gasser and Heiri Leutwyler,Nucl.Phys.B250:465,1985: 2290 citations

Juerg Gasser and Heiri Leutwyler, AnnalsPhys.158:142,1984: 2254 citations

Sum: 3777

Checked on 18/2/2007 in SPIRES

For lectures, review articles: seehttp://www.thep.lu.se/∼bijnens/chpt.html


Chiral Perturbation Theor ies

Baryons

Heavy Quarks

Vector Mesons (and other resonances)

Structure Functions and Related Quantities

Light Pseudoscalar MesonsTwo or Three (or even more) FlavoursStrong interaction and couplings to externalcurrents/densitiesIncluding electromagnetismIncluding weak nonleptonic interactionsTreating kaon as heavy

Many similarities with strongly interacting HiggsOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.20/103

Lagrangians

U(φ) = exp(i√

2Φ/F0) parametrizes Goldstone Bosons

Φ(x) =

0

B

B

B

B

B

B

@

π0√2

+η8√6

π+ K+

π− − π0

√2

+η8√6

K0

K− K̄0 −2 η8√6

1

C

C

C

C

C

C

A

.

LO Lagrangian: L2 = F20

4 {〈DµU †DµU〉 + 〈χ†U + χU †〉} ,

DµU = ∂µU − irµU + iUlµ ,left and right external currents: r(l)µ = vµ + (−)aµ

Scalar and pseudoscalar external densities: χ = 2B0(s + ip)quark masses via scalar density: s = M + · · ·〈A〉 = TrF (A)


External currents?

in QCD Green functions derived as functionalderivatives w.r.t. external fields

Green functions are the objects that satisfy Wardidentities

By introducing a local Chiral Symmetry, Ward identitiesare automatically satisfied (great improvement overcurrent algebra)

QCD Green functions form a connection QCD⇔ChPT∫

[dGdqdq]eiR

d4xLQCD(q,q,G,l,r,s,p) ≈∫

[dU ]eiR

d4xLChP T (U,l,r,s,p)

so also functional derivatives are equal


Lagrangians

L4 = L1〈DµU †DµU〉2 + L2〈DµU †DνU〉〈DµU †DνU〉+L3〈DµU †DµUDνU †DνU〉 + L4〈DµU †DµU〉〈χ†U + χU †〉+L5〈DµU †DµU(χ†U + U †χ)〉 + L6〈χ†U + χU †〉2

+L7〈χ†U − χU †〉2 + L8〈χ†Uχ†U + χU †χU †〉−iL9〈FRµνDµUDνU † + FLµνDµU †DνU〉

+L10〈U †FRµνUFLµν〉 + H1〈FRµνFRµν + FLµνFLµν〉 + H2〈χ†χ〉

Li: Low-energy-constants (LECs)Hi: Values depend on definition of currents/densities

These absorb the divergences of loop diagrams: Li → LriRenormalization: order by order in the powercounting


Lagrangians

Lagrangian Structure:2 flavour 3 flavour 3+3 PQChPT

p2 F,B 2 F0, B0 2 F0, B0 2p4 lri , h

ri 7+3 L

ri , H

ri 10+2 L̂

ri , Ĥ

ri 11+2

p6 cri 52+4 Cri 90+4 K

ri 112+3

p2: Weinberg 1966p4: Gasser, Leutwyler 84,85p6: JB, Colangelo, Ecker 99,00

➠ replica method =⇒ PQ obtained from NF flavour➠ All infinities known➠ 3 flavour special case of 3+3 PQ: L̂ri ,K

ri → Lri , Cri

➠ 53 → 52 arXiv:0705.0576 [hep-ph]Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.24/103

Chiral Logarithms

The main predictions of ChPT:

Relates processes with different numbers ofpseudoscalars

Chiral logarithms

m2π = 2Bm̂ +

(

2Bm̂

F

)2 [1

32π2log

(2Bm̂)

µ2+ 2lr3(µ)

]

+ · · ·

M2 = 2Bm̂B 6= B0, F 6= F0 (two versus three-flavour)


LECs and µ

lr3(µ)

l̄i =32π2

γilri (µ) − log

M2πµ2

.

Independent of the scale µ.

For 3 and more flavours, some of the γi = 0: Lri (µ)

µ :

mπ, mK : chiral logs vanish

pick larger scale

1 GeV then Lr5(µ) ≈ 0 large Nc arguments????compromise: µ = mρ = 0.77 GeV


Expand in what quantities?

Expansion is in momenta and masses

But is not unique: relations between masses(Gell-Mann–Okubo) exists

Express orders in terms of physical masses andquantities (Fπ, FK)?

Express orders in terms of lowest order masses?

E.g. s + t + u = 2m2π + 2m2K in πK scattering

See e.g. Descotes-Genon talk

I prefer physical masses

Thresholds correct

Chiral logs are from physical particles propagatingOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.27/103

An example

mπ =m0

1 + am0f0fπ =

f0

1 + bm0f0


An example

mπ =m0

1 + am0f0fπ =

f0

1 + bm0f0

mπ = m0 − am20f0

+ a2m30f20

+ · · ·

fπ = f0

(

1 − bm0f0

+ b2m20f20

+ · · ·)


An example

mπ =m0

1 + am0f0fπ =

f0

1 + bm0f0

mπ = m0 − am20f0

+ a2m30f20

+ · · ·

fπ = f0

(

1 − bm0f0

+ b2m20f20

+ · · ·)

mπ = m0 − am2πfπ

+ a(b − a)m3π

f2π+ · · ·

mπ = m0

(

1 − amπfπ

+ abm2πf2π

+ · · ·)

fπ = f0

(

1 − bmπfπ

+ b(2b − a)m2π

f2π+ · · ·

)


An example

mπ =m0

1 + am0f0fπ =

f0

1 + bm0f0

mπ = m0 − am20f0

+ a2m30f20

+ · · ·

fπ = f0

(

1 − bm0f0

+ b2m20f20

+ · · ·)

mπ = m0 − am2πfπ

+ a(b − a)m3π

f2π+ · · ·

mπ = m0

(

1 − amπfπ

+ abm2πf2π

+ · · ·)

fπ = f0

(

1 − bmπfπ

+ b(2b − a)m2π

f2π+ · · ·

)

a = 1 b = 0.5 f0 = 1Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103

An example: m0/f0

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

mπ

m0

mπLO

NLONNLO


An example: mπ/fπ

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

mπ

m0

mπLO

NLOpNNLOp


Two-loop Two-flavour

Review paper on Two-Loops: JB, hep-ph/0604043 Prog. Part.Nucl. Phys. 58 (2007) 521

Dispersive Calculation of the nonpolynomial part in q2, s, t, u

Gasser-Meißner: FV , FS: 1991 numerical

Knecht-Moussallam-Stern-Fuchs: ππ: 1995 analytical

Colangelo-Finkemeier-Urech: FV , FS: 1996 analytical


Two-Loop Two-flavour

Bellucci-Gasser-Sainio: γγ → π0π0: 1994Bürgi: γγ → π+π−, Fπ, mπ: 1996JB-Colangelo-Ecker-Gasser-Sainio: ππ, Fπ, mπ:1996-97

JB-Colangelo-Talavera: FV π(t), FSπ: 1998

JB-Talavera: π → ℓνγ: 1997Gasser-Ivanov-Sainio: γγ → π0π0, γγ → π+π−:2005-2006

mπ, Fπ, FV , FS, ππ: simple analytical forms

Colangelo-(Dürr-)Haefeli: Finite volume Fπ,mπ2005-2006


LECs

l̄1 to l̄4: ChPT at order p6 and the Roy equation analysis inππ and FS Colangelo, Gasser and Leutwyler, Nucl. Phys. B 603 (2001)125 [hep-ph/0103088]

l̄5 and l̄6 : from FV and π → ℓνγ JB,(Colangelo,)Talavera

l̄1 = −0.4 ± 0.6 ,l̄2 = 4.3 ± 0.1 ,l̄3 = 2.9 ± 2.4 ,l̄4 = 4.4 ± 0.2 ,

l̄6 − l̄5 = 3.0 ± 0.3 ,l̄6 = 16.0 ± 0.5 ± 0.7 .

l7 ∼ 5 · 10−3 from π0-η mixing Gasser, Leutwyler 1984Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.33/103

LECs

Some combinations of order p6 LECs are known as well:curvature of the scalar and vector formfactor, two morecombinations from ππ scattering (implicit in b5 and b6)

Note: cri for mπ, fπ, ππ: small effect

cri (770MeV ) = 0 for plots shown

expansion in m2π/F2π shown

General observation:

Obtainable from kinematical dependences: known

Only via quark-mass dependence: poorely knownOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.34/103

m2π

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25

mπ2

M2 [GeV2]

LONLO

NNLO


m2π (l̄3 = 0)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05 0.1 0.15 0.2 0.25

mπ2

M2 [GeV2]

LONLO

NNLO


Fπ

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.05 0.1 0.15 0.2 0.25

Fπ

[GeV

]

M2 [GeV2]

LONLO

NNLO


Two-loop Three-flavour,≤2001

ΠV V π, ΠV V η, ΠV V K Kambor, Golowich; Kambor, Dürr; Amorós, JB, Talavera

ΠV V ρω Maltman

ΠAAπ, ΠAAη, Fπ, Fη, mπ, mη Kambor, Golowich; Amorós, JB, Talavera

ΠSS Moussallam Lr4, L

r6

ΠV V K , ΠAAK , FK , mK Amorós, JB, Talavera

Kℓ4, 〈qq〉 Amorós, JB, Talavera Lr1, Lr2, Lr3

FM , mM , 〈qq〉 (mu 6= md) Amorós, JB, Talavera Lr5,7,8,mu/md


Two-loop Three-flavour,≥2001

FV π, FV K+, FV K0 Post, Schilcher; JB, Talavera Lr9

Kℓ3 Post, Schilcher; JB, Talavera Vus

FSπ, FSK (includes σ-terms) JB, Dhonte Lr4, Lr6

K,π → ℓνγ Geng, Ho, Wu Lr10ππ JB,Dhonte,Talavera

πK JB,Dhonte,Talavera

relation lri and Lri , C

ri Gasser,Haefeli,Ivanov,Schmid

Finite volume 〈qq〉 JB,Ghorbani


Two-loop Three-flavour

Known to be in progress

η → 3π: being written up JB,GhorbaniKℓ3 iso: preliminary results available JB,Ghorbani

Finite Volume: sunsetintegrals being written up JB,Lähde

relation cri and Lri , C

ri Gasser,Haefeli,Ivanov,Schmid


Cri

Most analysis use:Cri from (single) resonance approximation

π

πρ, S

→ q2π

π|q2|

Cri

LV = −1

4〈VµνV µν〉 +

1

2m2V 〈VµV µ〉 −

fV

2√

2〈Vµνfµν+ 〉

− igV2√

2〈Vµν [uµ, uν ]〉 + fχ〈Vµ[uµ, χ−]〉

LA = −1

4〈AµνAµν〉 +

1

2m2A〈AµAµ〉 −

fA

2√

2〈Aµνfµν− 〉

LS =1

2〈∇µS∇µS − M2SS2〉 + cd〈Suµuµ〉 + cm〈Sχ+〉

Lη′ =1

2∂µP1∂

µP1 −1

2M2η′P

21 + id̃mP1〈χ−〉 .

fV = 0.20, fχ = −0.025, gV = 0.09, cm = 42 MeV, cd = 32 MeV, d̃m = 20 MeV,

mV = mρ = 0.77 GeV, mA = ma1 = 1.23 GeV, mS = 0.98 GeV, mP1 = 0.958 GeV

fV , gV , fχ, fA: experiment

cm and cd from resonance saturation at O(p4)


Cri

Problems:

Weakest point in the numerics

However not all results presented depend on this

Unknown so far: Cri in the masses/decay constants andhow these effects correlate into the rest

No µ dependence: obviously only estimate


Cri

Problems:

Weakest point in the numerics

However not all results presented depend on this

Unknown so far: Cri in the masses/decay constants andhow these effects correlate into the rest

No µ dependence: obviously only estimate

What we did about it:

Vary resonance estimate by factor of two

Vary the scale µ at which it applies: 600-900 MeV

Check the estimates for the measured ones

Again: kinematic can be had, quark-mass dependencedifficult


Inputs

Kℓ4: F (0), G(0), λ E865 BNL

m2π0, m2η, m

2K+, m

2K0 em with Dashen violation

Fπ+

FK+/Fπ+

ms/m̂ 24 (26) m̂ = (mu + md)/2

Lr4, Lr6


Outputs: Ifit 10 same p4 fit B fit D

103Lr1 0.43 ± 0.12 0.38 0.44 0.44

103Lr2 0.73 ± 0.12 1.59 0.60 0.69

103Lr3 −2.35 ± 0.37 −2.91 −2.31 −2.33

103Lr4 ≡ 0 ≡ 0 ≡ 0.5 ≡ 0.2

103Lr5 0.97 ± 0.11 1.46 0.82 0.88

103Lr6 ≡ 0 ≡ 0 ≡ 0.1 ≡ 0

103Lr7 −0.31 ± 0.14 −0.49 −0.26 −0.28

103Lr8 0.60 ± 0.18 1.00 0.50 0.54

➠ errors are very correlated➠ µ = 770 MeV; 550 or 1000 within errors➠ varying Cri factor 2 about errors➠ Lr4, L

r6 ≈ −0.3, . . . , 0.6 10−3 OK

➠ fit B: small corrections to pion “sigma” term, fit scalar radius

➠ fit D: fit ππ and πK thresholdsOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.45/103

Correlations

-5-4

-3-2

-10

L3r

00.2

0.40.6

0.81

L1r

0

0.2

0.4

0.6

0.8

1

1.2

1.4

L2r

(older fit)103 Lr1 = 0.52 ± 0.23103 Lr2 = 0.72 ± 0.24103 Lr3 = −2.70 ± 0.99


Outputs: II

fit 10 same p4 fit B fit D

2B0m̂/m2π 0.736 0.991 1.129 0.958

m2π: p4, p6 0.006,0.258 0.009,≡ 0 −0.138,0.009 −0.091,0.133

m2K : p4, p6 0.007,0.306 0.075,≡ 0 −0.149,0.094 −0.096,0.201

m2η: p4, p6 −0.052,0.318 0.013,≡ 0 −0.197,0.073 −0.151,0.197

mu/md 0.45±0.05 0.52 0.52 0.50F0 [MeV] 87.7 81.1 70.4 80.4FKFπ

: p4, p6 0.169,0.051 0.22,≡ 0 0.153,0.067 0.159,0.061

➠ mu = 0 always very far from the fits

➠ F0: pion decay constant in the chiral limit


ππ

p4

p6

-0.4 -0.2 0 0.2 0.4 0.6 103 L4

r -0.3-0.2

-0.10

0.10.2

0.30.4

0.50.6

103 L6r

0.195

0.2

0.205

0.21

0.215

0.22

0.225

a00

p4

p6

-0.4 -0.2 0 0.2 0.4 0.6 103 L4

r -0.3-0.2

-0.10

0.10.2

0.30.4

0.50.6

103 L6r

-0.048

-0.047

-0.046

-0.045

-0.044

-0.043

-0.042

-0.041

-0.04

-0.039

a20

a00 = 0.220 ± 0.005, a20 = −0.0444 ± 0.0010Colangelo, Gasser, Leutwyler

a00 = 0.159 a20 = −0.0454 at order p2


ππ and πK

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-0.4 -0.2 0 0.2 0.4 0.6

103

L6r

103 L4r

ππ constraints

a20C1

a03/2

C+10

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-0.4 -0.2 0 0.2 0.4 0.6

103

L6r

103 L4r

πK constraints

C+10a0

3/2

C1a20

preferred region: fit D: 103Lr4 ≈ 0.2, 103Lr6 ≈ 0.0

General fitting needs more work and systematic studies


Quark mass dependences

Updates of plots inAmorós, JB and Talavera, hep-ph/0003258, Nucl. Phys. B585 (2000) 293

Some new onesProcedure: calculate a consistent set of mπ,mK ,mη, fπ withthe given input values (done iteratively)

vary ms/(ms)phys, keep ms/m̂ = 24m2π, m

2K , Fπ, FK

vary ms/(ms)phys keep m̂ fixedm2π, Fπvary mπ, keep mK fixedf+(0): the formfactor in Kℓ3 decays

f+(0),f+(0)/(

m2K − m2π)2


m2π fit 10

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 0.2 0.4 0.6 0.8 1

mπ2

[GeV

2 ]

ms/(ms)phys

LONLO

NNLO


m2π fit D

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 0.2 0.4 0.6 0.8 1

mπ2

[GeV

2 ]

ms/(ms)phys

LONLO

NNLO


m2K fit 10

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

mK

2 [G

eV2 ]

ms/(ms)phys

LONLO

NNLO


m2K fit D

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

mK

2 [G

eV2 ]

ms/(ms)phys

LONLO

NNLO


Fπ fit 10

0.086

0.088

0.09

0.092

0.094

0.096

0.098

0.1

0 0.2 0.4 0.6 0.8 1

Fπ

[GeV

]

ms/(ms)phys

LONLO

NNLO


FK fit 10

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0 0.2 0.4 0.6 0.8 1

FK [G

eV]

ms/(ms)phys

LONLO

NNLO


FK/Fπ fit 10

1

1.05

1.1

1.15

1.2

1.25

0 0.2 0.4 0.6 0.8 1

FK/F

π

ms/(ms)phys

NLONNLO


m2π fit 10, fixed m̂

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 0.2 0.4 0.6 0.8 1

mπ2

[GeV

2 ]

ms/(ms)phys

LONLO

NNLO


Fπ fit 10, fixed m̂

0.086

0.088

0.09

0.092

0.094

0.096

0.098

0.1

0 0.2 0.4 0.6 0.8 1

Fπ

[GeV

]

ms/(ms)phys

LO

NLO

NNLO


f+(0) fit 10, fixed mK

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

corr

ectio

n to

f +(0

)

mπ2/(mK

2)phys

sump4

p6 2-loopsp6 Li

r


(f+(0))/(

m2K − m2π)2

fit 10, fixed mK

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1

(cor

rect

ion

to f +

(0))

/(m

K2 −

mπ2 )

2 [G

eV−4

]

mπ2/(mK

2)phys

sump4

p6 2-loopsp6 Li

r


f0(t) in Kℓ3

Main Result: JB,Talavera

f0(t) = 1 −8

F 4π(Cr12 + C

r34)

(

m2K − m2π)2

+8t

F 4π(2Cr12 + C

r34)

(

m2K + m2π

)

+t

m2K − m2π(FK/Fπ − 1)

− 8F 4π

t2Cr12 + ∆(t) + ∆(0) .

∆(t) and ∆(0) contain NO Cri and only depend on the Lri at

order p6

=⇒All needed parameters can be determined experimentally

∆(0) = −0.0080 ± 0.0057[loops] ± 0.0028[Lri ] .Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.62/103

≥ 3-flavour: PQChPT

Essentially all manipulations from ChPT go through toPQChPT when changing trace to supertrace and addingfermionic variables

Exceptions: baryons and Cayley-Hamilton relations

So Luckily: can use the n flavour work in ChPT at two looporder to obtain for PQChPT: Lagrangians and infinities

Very important note: ChPT is a limit of PQChPT=⇒ LECs from ChPT are linear combinations of LECs

of PQChPT with the same number of sea quarks.

E.g. Lr1 = Lr(3pq)0 /2 + L

r(3pq)1


PQChPT

One-loop: Bernard, Golterman, Sharpe, Shoresh, Pallante,. . .with electromagnetism: JB,Danielsson, hep-lat/0610127

Two loops:

m2π+ simplest mass case: JB,Danielsson,Lähde, hep-lat/0406017Fπ+: JB,Lähde, hep-lat/0501014Fπ+, m2π+ two sea quarks: JB,Lähde, hep-lat/0506004

m2π+: JB,Danielsson,Lähde, hep-lat/0602003

Neutral masses: JB,Danielsson, hep-lat/0606017

Lattice data: a and L extrapolations needed

Programs available from me (Fortran)Formulas: http://www.thep.lu.se/∼bijnens/chpt.html


Partial Quenching and ChPT

Mesons

=

Quark FlowValence

+

Quark FlowSea

+ · · ·

Lattice QCD: Valence is easy to deal with, Sea very difficult

They can be treated separately: i.e. different quark massesPartially Quenched QCD and ChPT (PQChPT)

One Loop or p4: Bernard, Golterman, Pallante, Sharpe,Shoresh,. . .

Two Loops or p6: This talk JB, Niclas Danielsson, Timo Lähde


PQChPT at Two Loops: General

Add ghost quarks: remove the unwanted free valence loops

Mesons

=

Quark FlowValence

+

Quark FlowValence

+

Quark FlowSea

+

Quark FlowGhost

Possible problem: QCD =⇒ ChPT relies heavily on unitarity

Partially quenched: at least one dynamical sea quark=⇒ Φ0 is heavy: remove from PQChPT

Symmetry group becomes SU(nv + ns|nv) × SU(nv + ns|nv)(approximately)


PQChPT at Two Loops: General

Essentially all manipulations from ChPT go through toPQChPT when changing trace to supertrace and addingfermionic variables

Exceptions: baryons and Cayley-Hamilton relations

So Luckily: can use the n flavour work in ChPT at two looporder to obtain for PQChPT: Lagrangians and infinities

Very important note: ChPT is a limit of PQChPT=⇒ LECs from ChPT are linear combinations of LECs

of PQChPT with the same number of sea quarks.

E.g. Lr1 = Lr(3pq)0 /2 + L

r(3pq)1


PQChPT at Two Loop: Papers

valence equal mass, 3 sea equal mass:

m2π+: JB,Danielsson,Lähde, hep-lat/0406017

Other mass combinations:

F 2π+: JB,Lähde, hep-lat/0501014

F 2π+, m2π+ two sea quarks: JB,Lähde, hep-lat/0506004

In progress: the other charged masses

Actual Calculations:

➠ heavy use of FORM Vermaseren➠ Main problem: sheer size of theexpressions

No fits to lattice data (yet): a and L extrapolations needed


Long Expressions

=⇒


Long Expressions

=⇒

δ(6)22loops = π16 L

r0

[

4/9 χηχ4 − 1/2 χ1χ3 + χ213 − 13/3 χ̄1χ13 − 35/18 χ̄2]

− 2 π16 Lr1 χ213− π16 Lr2

[

11/3 χηχ4 + χ213 + 13/3 χ̄2

]

+ π16 Lr3

[

4/9 χηχ4 − 7/12 χ1χ3 + 11/6 χ213 − 17/6 χ̄1χ13 − 43/36 χ̄2]

+ π216[

−15/64 χηχ4 − 59/384 χ1χ3 + 65/384 χ213 − 1/2 χ̄1χ13 − 43/128 χ̄2]

− 48 Lr4Lr5 χ̄1χ13 − 72 Lr24 χ̄21− 8 Lr25 χ213 + Ā(χp)π16

[

−1/24 χp + 1/48 χ̄1 − 1/8 χ̄1 Rpqη + 1/16 χ̄1 Rcp − 1/48 Rpqη χp − 1/16 Rpqη χq+ 1/48 Rηpp χη + 1/16 R

cp χ13

]

+ Ā(χp)Lr0

[

8/3 Rpqη χp + 2/3 Rcp χp + 2/3 R

dp

]

+ Ā(χp)Lr3

[

2/3 Rpqη χp

+ 5/3 Rcp χp + 5/3 Rdp

]

+ Ā(χp)Lr4

[

−2 χ̄1χ̄ppηη0 − 2 χ̄1 Rpqη + 3 χ̄1 Rcp]

+ Ā(χp)Lr5

[

−2/3 χ̄ppηη1 − Rpqη χp+ 1/3 Rpqη χq + 1/2 R

cp χp − 1/6 Rcp χq

]

+ Ā(χp)2

[

1/16 + 1/72 (Rpqη)2 − 1/72 RpqηRcp + 1/288 (Rcp)2

]

+ Ā(χp)Ā(χps)[

−1/36 Rpqη − 5/72 Rpsη + 7/144 Rcp]

− Ā(χp)Ā(χqs)[

1/36 Rpqη + 1/24 Rpsη + 1/48 R

cp

]

+ Ā(χp)Ā(χη)[

−1/72 RpqηRvη13 + 1/144 RcpRvη13]

+ 1/8 Ā(χp)Ā(χ13) + 1/12 Ā(χp)Ā(χ46)Rηpp

+ Ā(χp)B̄(χp, χp; 0)[

1/4 χp − 1/18 RpqηRcp χp − 1/72 RpqηRdp + 1/18 (Rcp)2 χp + 1/144 RcpRdp]

+ Ā(χp)B̄(χp, χη; 0)[

1/18 RηppRcp χp − 1/18 Rη13Rcp χp

]

+ Ā(χp)B̄(χq, χq; 0)[

−1/72 RpqηRdq + 1/144 RcpRdq]

− 1/12 Ā(χp)B̄(χps, χps; 0)Rpsη χps − 1/18 Ā(χp)B̄(χ1, χ3; 0)RqpηRcp χp+ 1/18 Ā(χp)C̄(χp, χp, χp; 0)R

cpR

dp χp + Ā(χp; ε)π16

[

1/8 χ̄1Rpqη − 1/16χ̄1 Rcp − 1/16 Rcp χp − 1/16 Rdp

]

+ Ā(χps)π16 [1/16 χps − 3/16 χqs − 3/16 χ̄1] − 2 Ā(χps)Lr0 χps − 5 Ā(χps)Lr3 χps − 3 Ā(χps)Lr4 χ̄1+ Ā(χps)L

r5 χ13 + Ā(χps)Ā(χη)

[

7/144 Rηpp − 5/72 Rηps − 1/48 Rηqq + 5/72 Rηqs − 1/36 Rη13]

+ Ā(χps)B̄(χp, χp; 0)[

1/24 Rpsη χp − 5/24 Rpsη χps]

+ Ā(χps)B̄(χp, χη; 0)[

−1/18 RηpsRzqpη χp− 1/9 RηpsRzqpη χps

]

− 1/48 Ā(χps)B̄(χq, χq; 0)Rdq + 1/18 Ā(χps)B̄(χ1, χ3; 0)Rqsη χs+ 1/9 Ā(χps)B̄(χ1, χ3; 0, k)R

qsη + 3/16 Ā(χps; ε)π16 [χs + χ̄1] − 1/8 Ā(χp4)2 − 1/8 Ā(χp4)Ā(χp6)

+ 1/8 Ā(χp4)Ā(χq6) − 1/32 Ā(χp6)2 + Ā(χη)π16[

1/16 χ̄1 Rvη13 − 1/48 Rvη13 χη + 1/16 Rvη13 χ13

]

+ Ā(χη)Lr0

[

4Rη13 χη + 2/3 Rvη13 χη

]

− 8 Ā(χη)Lr1 χη − 2 Ā(χη)Lr2 χη + Ā(χη)Lr3[

4Rη13 χη + 5/3 Rvη13χη

]

+ Ā(χη)Lr4

[

4 χη + χ̄1 Rvη13

]

− Ā(χη)Lr5[

1/6 Rηpp χq + Rη13 χ13 + 1/6 R

vη13 χη

]

+ 1/288 Ā(χη)2 (Rvη13)

2

+ 1/12 Ā(χη)Ā(χ46)Rvη13 + Ā(χη)B̄(χp, χp; 0)

[

−1/36 χ̄ppηηη1 − 1/18 RpqηRηpp χp + 1/18 RηppRcp χp+ 1/144 RdpR

vη13

]

+ Ā(χη)B̄(χp, χη; 0)[

−1/18 χ̄ηpηpη1 + 1/18 χ̄ηpηqη1 + 1/18 (Rηpp)2Rzqpη χp]

− 1/12 Ā(χη)B̄(χps, χps; 0)Rηps χps − Ā(χη)B̄(χη, χη; 0)[

1/216 Rvη13 χ4 + 1/27 Rvη13 χ6

]

− 1/18 Ā(χη)B̄(χ1, χ3; 0)R1ηηR3ηη χη + 1/18 Ā(χη)C̄(χp, χp, χp; 0)RηppRdp χp + Ā(χη; ε)π16 [1/8 χη− 1/16 χ̄1 Rvη13 − 1/8 Rη13 χη − 1/16 Rvη13 χη

]

+ Ā(χ1)Ā(χ3)[

−1/72 RpqηRcq + 1/36 R13ηR31η + 1/144 Rc1Rc3]

− 4 Ā(χ13)Lr1 χ13 − 10 Ā(χ13)Lr2 χ13 + 1/8 Ā(χ13)2 − 1/2 Ā(χ13)B̄(χ1, χ3; 0, k)+ 1/4 Ā(χ13; ε)π16 χ13 + 1/4 Ā(χ14)Ā(χ34) + 1/16 Ā(χ16)Ā(χ36) − 24 Ā(χ4)Lr1 χ4 − 6 Ā(χ4)Lr2 χ4+ 12 Ā(χ4)L

r4 χ4 + 1/12 Ā(χ4)B̄(χp, χp; 0) (R

p4η)

2 χ4 + 1/6 Ā(χ4)B̄(χp, χη; 0)[

Rp4ηRηp4 χ4 − Rp4ηRηq4 χ4

]

− 1/24 Ā(χ4)B̄(χη, χη; 0)Rvη13 χ4 − 1/6 Ā(χ4)B̄(χ1, χ3; 0)R14ηR34η χ4 + 3/8 Ā(χ4; ε)π16 χ4− 32 Ā(χ46)Lr1 χ46 − 8 Ā(χ46)Lr2 χ46 + 16 Ā(χ46)Lr4 χ46 + Ā(χ46)B̄(χp, χp; 0)

[

1/9 χ46 + 1/12 Rηpp χp

+ 1/36 Rηpp χ4 + 1/9 Rηp4 χ6

]

+ Ā(χ46)B̄(χp, χη; 0)[

−1/18 Rηpp χ4 − 1/9 Rηp4 χ6 + 1/9 Rηq4 χ6 + 1/18 Rη13 χ4]

− 1/6 Ā(χ46)B̄(χp, χη; 0, k)[

Rηpp − Rη13]

+ 1/9 Ā(χ46)B̄(χη, χη; 0)Rvη13 χ46 − Ā(χ46)B̄(χ1, χ3; 0) [2/9 χ46

+ 1/9 Rηp4 χ6 + 1/18 Rη13 χ4] − 1/6 Ā(χ46)B̄(χ1, χ3; 0, k)Rη13 + 1/2 Ā(χ46; ε)π16 χ46

+ B̄(χp, χp; 0)π16[

1/16 χ̄1 Rdp + 1/96 R

dp χp + 1/32 R

dp χq

]

+ 2/3 B̄(χp, χp; 0)Lr0 R

dp χp

+ 5/3 B̄(χp, χp; 0)Lr3 R

dp χp + B̄(χp, χp; 0)L

r4

[

−2 χ̄1χ̄ppηη0χp − 4 χ̄1Rpqη χp + 4 χ̄1Rcp χp + 3 χ̄1Rdp]

+ B̄(χp, χp; 0)Lr5

[

−2/3χ̄ppηη1χp − 4/3 Rpqη χ2p + 4/3 Rcp χ2p + 1/2 Rdp χp − 1/6 Rdp χq]

+ B̄(χp, χp; 0)Lr6

[

4 χ̄1χ̄ppηη1 + 8 χ̄1R

pqη χp − 8 χ̄1Rcp χp

]

+ 4 B̄(χp, χp; 0)Lr7 (R

dp)

2

+ B̄(χp, χp; 0)Lr8

[

4/3 χ̄ppηη2 + 8/3 Rpqη χ

2p − 8/3 Rcp χ2p

]

+ B̄(χp, χp; 0)2

[

−1/18 RpqηRdp χp + 1/18 RcpRdp χp+ 1/288 (Rdp)

2]

+ 1/18 B̄(χp, χp; 0)B̄(χp, χη; 0)[

RηppRdp χp − Rη13Rdp χp

]

plus several more pages


Why so long expressions

Many different quark and meson masses (χij)

Charged propagators: −iGcij(k) =ǫj

k2−χij+iε (i 6= j)

Neutral propagators: Gnij(k) = Gcij(k) δij − 1nsea G

qij(k)







qij(k)

−iGqii(k) =Rdi

(k2−χi+iε)2 +Rci

k2−χi+iε +Rπηii

k2−χπ+iε +R

ηπii

k2−χη+iε

Rijkl = Rzi456jkl, R

di = R

zi456πη,

Rci = Ri4πη + R

i5πη + R

i6πη − Riπηη − Riππη

Rzab = χa − χb, Rzabc =χa−χbχa−χc , R

zabcd =

(χa−χb)(χa−χc)χa−χd

Rzabcdefg =(χa−χb)(χa−χc)(χa−χd)(χa−χe)(χa−χf )(χa−χg)







qij(k)

−iGqii(k) =Rdi

(k2−χi+iε)2 +Rci

k2−χi+iε +Rπηii

k2−χπ+iε +R

ηπii

k2−χη+iε

Rijkl = Rzi456jkl, R

di = R

zi456πη,

Rci = Ri4πη + R

i5πη + R

i6πη − Riπηη − Riππη

Rzab = χa − χb, Rzabc =χa−χbχa−χc , R

zabcd =

(χa−χb)(χa−χc)χa−χd

Rzabcdefg =(χa−χb)(χa−χc)(χa−χd)(χa−χe)(χa−χf )(χa−χg)

Relations =⇒ order of magnitude smallerOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103

PQChPT at Two Loop

Problem: Plotting with many input parameters

Plot masses as a function of lowest order mass squared

I.e. of quark mass: χi = 2B0mi = m2(0)M

Remember: χi ≈ 0.3 GeV 2 ≈ (550 MeV )2 ∼ border ChPT


PQChPT at Two Loop

Problem: Plotting with many input parameters

Plot masses as a function of lowest order mass squared

I.e. of quark mass: χi = 2B0mi = m2(0)M

Remember: χi ≈ 0.3 GeV 2 ≈ (550 MeV )2 ∼ border ChPT

1+1 case: Valence: χ1 = χ2 = χ3Sea: χ4 = χ5 = χ6

Plot along curves: χ4 = tan θ χ1 or χ4 constantOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.71/103

PQChPT: 1+1 case, 3 sea quarksS

ea

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

χ 4 [G

eV2 ]

χ1 [GeV2]

15o

30o

45o

60o

75o

A

Valence



ea

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

χ 4 [G

eV2 ]

χ1 [GeV2]

15o

30o

45o

60o

75o

A

-0.04

-0.02

0

0.02

0.04

0.06

0 0.1 0.2 0.3 0.4 0.5

δ(4)

/F02

χ1 [GeV2]

15o

30o

45o

60o

75o

A

Valence p4 mass



ea

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

χ 4 [G

eV2 ]

χ1 [GeV2]

15o

30o

45o

60o

75o

A

-0.04

-0.02

0

0.02

0.04

0.06

0 0.1 0.2 0.3 0.4 0.5

δ(4)

/F02

χ1 [GeV2]

15o

30o

45o

60o

75o

A

Valence p4 mass

Notice the Quenched Chiral Logs:m2πχ1

= 1+α

F 2χ4 log χ1+· · ·


PQChPT: 1+1 case, 3 sea quarks

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5

δ(4)

/F02 +

δ(6)

/F04

χ1 [GeV2]

15o

30o

45o

60o

75o

A

p4 + p6 relative correction mass



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5

δ(4)

/F02 +

δ(6)

/F04

χ1 [GeV2]

15o

30o

45o

60o

75o

A

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5δ(

4)/F

02 +δ(

6)/F

04

χ1 [GeV2]

15o

30o

45o

60o

75o A

p4 + p6 relative correction mass decay constant



-0.2

-0.1

0

0.1

0.2

0.3

0.4

χ1 [GeV2]

χ 3 [G

eV2 ]

∆M

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Relative Correction: Mass

χ1: valence mass, χ3: sea massOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.74/103


-0.2

-0.1

0

0.1

0.2

0.3

0.4

χ1 [GeV2]

χ 3 [G

eV2 ]

∆M

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Relative Correction: Mass

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

χ1 [GeV2]

χ 3 [G

eV2 ]

∆F

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Decay Constant

χ1: valence mass, χ3: sea massOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.74/103

PQChPT: 2 sea quarks

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15 0.2

∆ M

χ1 [GeV2]

z = 1.4z = 1.2z = 1.0z = 0.8

z = 0.6z = 0.4

nf = 2dval = 1dsea = 2

θ = 70o

χ3 = χ1 tan(θ)

χ4 = z χ3

Relative Correction: Mass1+2 caseValence: χ1 6= χ2Sea: χ3 = χ4


PQChPT: 2 sea quarks

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15 0.2

∆ M

χ1 [GeV2]

z = 1.4z = 1.2z = 1.0z = 0.8

z = 0.6z = 0.4


θ = 70o

χ3 = χ1 tan(θ)

χ4 = z χ3

Relative Correction: Mass1+2 caseValence: χ1 6= χ2Sea: χ3 = χ4

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15 0.2

∆ Mχ1 [GeV

2]

x = 0.8x = 1.0x = 1.2x = 1.4

x = 1.6x = 1.8


θ = 70o

χ3 = χ1 tan(θ)

χ2 = x χ1

Mass2+1 caseValence: χ1 = χ2Sea: χ3 6= χ4


PQChPT: Fitting Strategy

For masses and Decay constants

At order p4: Lr4, Lr5, L

r6, L

r8

At order p6: all allowed quadratic quark masscombinations show up

Problem: Need Lr0, Lr1, L

r2, L

r3 from ππ, πK scattering but

only to order p4

Nonanalytic structure at p6 given (only one included innumerics shown)


Conclusions PQChPT Part

All relevant mass combinations for masses and decayconstants for charged pseudoscalar mesons nowknown to two loops

Decay constants and masses for two sea quarksconverge nicely

Masses for three sea quarks convergence slower

Looking forward to getting lattice data to fit

Expressions available fromhttp://www.thep.lu.se/∼bijnens/chpt.htmlFor the numerical programs contact the authors


Wishing list

General:

Quark-mass dependences everywhere

Not only fits but also continuum infinite volume resultsat a given quark-mass (so we can also fit ourselves forstudying other inputs)

More use of the existing two-loop calculations

Analytical NNLO only: not really fewer parameters

Only more in LECs: p4 scattering LECs also inmasses/decay constants

I.e. lr1, lr2 (nF = 2), L

r1, L

r2, L

r3 (nF = 3),

L̂ri , i = 0, 1, 2, 3 (PQChPT)


Wishing list: Two-flavour

(with input from Gasser et al.)

l̄3 and errors

l̄4: from Fπ(mq) and scalar radius: can lattice check thisrelation

a20 accurately predicted in terms of scalar radius: canlattice check this

Isospin breaking in ππ scattering (important for CPviolation in K → ππ)l̄5 − l̄6 Needed for π → ℓνγ, can be had from ΠAA


Wishing list: ≥ 3-flavour

Ideas on how to make all those calculations usable foryou

Three and more flavour: typically slow numerically

large Nc suppressed couplings: i.e. ms dependence ofmπ, Fπ

Lr4, Lr6

sigma terms and scalar radii


Conclusions and final comments

Lots of analytical work done in ChPT

Use the correct ChPT2-flavour for varying m̂ and possible for Nf = 2 andNf = 2 + 1 at fixed ms (but have different LECs)

otherwise 3-flavourthe various partially quenched versions

Remember at which order in ChPT you compare things


Conclusions and final comments

Lots of analytical work done in ChPT

Use the correct ChPT2-flavour for varying m̂ and possible for Nf = 2 andNf = 2 + 1 at fixed ms (but have different LECs)

otherwise 3-flavourthe various partially quenched versions

Remember at which order in ChPT you compare things

Seen a lot of lattice work, looking forward to seeingmore

LECs in many talks: Boyle, Matsufuru, Urbach,Kuramashi and many parallel session talks


η → 3π

Reviews: JB, Gasser, Phys.Scripta T99(2002)34 [hep-ph/0202242]JB, Acta Phys. Slov. 56(2005)305 [hep-ph/0511076]

ηpη

π+pπ+

π−pπ−

π0 pπ0

s = (pπ+ + pπ−)2 = (pη − pπ0)2

t = (pπ− + pπ0)2 = (pη − pπ+)2

u = (pπ+ + pπ0)2 = (pη − pπ−)2

s + t + u = m2η + 2m2π+ + m

2π0 ≡ 3s0 .

〈π0π+π−out|η〉 = i (2π)4 δ4 (pη − pπ+ − pπ− − pπ0) A(s, t, u) .

〈π0π0π0out|η〉 = i (2π)4 δ4 (pη − p1 − p2 − p3) A(s1, s2, s3)

A(s1, s2, s3) = A(s1, s2, s3) + A(s2, s3, s1) + A(s3, s1, s2) ,Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.82/103

η → 3π: Lowest order (LO)

Pions are in I = 1 state =⇒ A ∼ (mu − md) or αemαem effect is small (but large via mπ+ − mπ0)η → π+π−π0γ needs to be included directly



Pions are in I = 1 state =⇒ A ∼ (mu − md) or αem

ChPT:Cronin 67: A(s, t, u) =B0(mu − md)

3√

3F 2π

{

1 +3(s − s0)m2η − m2π

}





3√

3F 2π

{

1 +3(s − s0)m2η − m2π

}

with Q2 ≡ m2s−m̂2

m2d−m2uor R ≡ ms−m̂md−mu m̂ =

12(mu + md)

A(s, t, u) =1

Q2m2Km2π

(m2π − m2K)M(s, t, u)3√

3F 2π,

A(s, t, u) =

√3

4RM(s, t, u)





3√

3F 2π

{

1 +3(s − s0)m2η − m2π

}

with Q2 ≡ m2s−m̂2

m2d−m2uor R ≡ ms−m̂md−mu m̂ =

12(mu + md)

A(s, t, u) =1

Q2m2Km2π

(m2π − m2K)M(s, t, u)3√

3F 2π,

A(s, t, u) =

√3

4RM(s, t, u)

LO: M(s, t, u) = 3s − 4m2π

m2η − m2πM(s, t, u) =

1

F 2π

(

4

3m2π − s

)


η → 3π beyondp4: p2 and p4

Γ (η → 3π) ∝ |A|2 ∝ Q−4 allows a PRECISE measurement

Q ≈ 24 gives lowest order Γ(η → π+π−π0) ≈ 66 eV .





Other Source from m2K+ − m2K0 ∼ Q−2 =⇒ Q = 20.0 ± 1.5Lowest order prediction Γ(η → π+π−π0) ≈ 140 eV .





Other Source from m2K+ − m2K0 ∼ Q−2 =⇒ Q = 20.0 ± 1.5Lowest order prediction Γ(η → π+π−π0) ≈ 140 eV .

At order p4 Gasser-Leutwyler 1985:

∫

dLIPS|A2 + A4|2∫

dLIPS|A2|2= 2.4 ,

(LIPS=Lorentz invariant phase-space)

Major source: large S-wave final state rescattering

Experiment: 295 ± 17 eV (PDG 2006)Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.84/103

η → 3π beyondp4: DispersiveTry to resum the S-wave rescattering:

Anisovich-Leutwyler (AL), Kambor,Wiesendanger,Wyler (KWW)

Different method but similar approximationsHere: simplified version of AL

Up to p8: No absorptive parts from ℓ ≥ 2=⇒ M(s, t, u) =

M0(s)+(s−u)M1(t)+(s−t)M1(t)+M2(t)+M2(u)−2

3M2(s)

MI : “roughly” contributions with isospin 0,1,2


η → 3π beyondp4: Dispersive

3 body dispersive: difficult: keep only 2 body cuts

start from πη → ππ (m2η < 3m2π) standard dispersive analysisanalytically continue to physical m2η.

MI(s) =1

π

∫ ∞

4m2π

ds′ImMI(s

′)

s′ − s − iεImMI(s

′) −→ discMI(s) =1

2i(MI(s + iε) − MI(s − iε))

M0(s) = a0 + b0s + c0s2 +

s3

π

Z

ds′

s′3discM0(s′)

s′ − s − iε ,

M1(s) = a1 + b1s +s2

π

Z

ds′

s′2discM1(s′)

s′ − s − iε ,

M2(s) = a2 + b2s + c2s2 +

s3

π

Z

ds′

s′3discM2(s′)

s′ − s − iε .


η → 3π beyondp4

• Technical complications in solving• Only 4 relevant constants:M(s, t, u) = a + bs + cs2 − d(s2 + tu)

M0(s) +4

3M2(s) sM1(s) + M2(s) + s

24L3 − 1/(64π2)F 2π (m

2η − m2π)

converge better

c = c0 +4

3c2 =

1

π

Z

ds′

s′3

discM0(s′) +

4

3discM2(s

′)

ff

,

d = −4L3 − 1/(64π2)

F 2π(m2η − m2π)

+1

π

Z

ds′

s′3

˘

s′discM1(s′) + discM2(s

′)¯

Fix a, b by matching to tree level or p4 amplitudeOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.87/103

η → 3π beyondp4

-0.5

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8

Re

M (

KW

W)

s/mπ2

tree

p4

dispersive

Along s = u KWW

-0.5

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8

Re

M (

AL)

s/mπ2

Adler zeroAdler zero

tree

p4

dispersive

Along s = u AL


η → 3π beyondp4

-0.5

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8

Re

M

s/mπ2

p2

dispersive (match p2)

dispersive (match (|M|)

dispersive (match Re M)

Along s = u BG

Very simplified analysisJB, Gasser 2002looks more like AL


Two Loop Calculation: why

In Kℓ4 dispersive gave about half of p6 in amplitude

Same order in ChPT as masses for consistency checkon mu/mdCheck size of 3 pion dispersive part

At order p4 unitarity about half of correction

Technology exists:Two-loops: Amorós,JB,Dhonte,Talavera,. . .Dealing with the mixing π0-η:Amorós,JB,Dhonte,Talavera 01


Two Loop Calculation: why

In Kℓ4 dispersive gave about half of p6 in amplitude

Same order in ChPT as masses for consistency checkon mu/mdCheck size of 3 pion dispersive part

At order p4 unitarity about half of correction

Technology exists:Two-loops: Amorós,JB,Dhonte,Talavera,. . .Dealing with the mixing π0-η:Amorós,JB,Dhonte,Talavera 01

Done: JB, Ghorbani, arXiv:0709.0230 [hep-ph]Dealing with the mixing π0-η: extended to η → 3π


Diagrams

(a) (b) (c) (d)

(a) (b)(c)

(d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Include mixing, renormalize, pull out factor√

34R , . . .

Two independent calculations (comparison majoramount of work)


Dalitzplot

0.08

0.1

0.12

0.14

0.16

0.08 0.1 0.12 0.14 0.16

t [G

eV

2]

s [GeV2]

mpiavmpidiff

u=ts=u

t-thresholdu thesholds threshold

x=y=0

x variation:vertical

y variation:parallel to t = u


η → 3π: M(s, t = u)

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s

,t,u

=t)

s [GeV2]

Li fit 10 and Ci

Re p2

Re p2+p

4

Re p2+p

4+p

6

Im p4

Im p4+p

6

Along t = u

-2

0

2

4

6

8

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s

,t,u

=t)

s [GeV2]

Li fit 10 and Ci

Re p6 pure loops

Re p6 Li

r

Re p6 Ci

r

sum p6

Along t = u parts


η → 3π: M(s, t = u)

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

M(s

,t,u

=t)

s [GeV2]

Li = Ci = 0

Re p2

Re p2+p

4

Re p2+p

4+p

6

Im p4

Im p6

Along t = uLri = C

ri = 0

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s

,t,u

=t)

s [GeV2]

Li fit 10 and Ci

Re p2+p

4

µ= 0.6 GeVµ= 0.9 GeV

Re p2+p

4+p

6

µ= 0.6 GeVµ= 0.9 GeV

Along t = u: µ dependenceI.e. where Cri (µ) estimated


η → 3π: M(s = u, t)

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s

,t,u

=s)

s [GeV2]

Re p2

Re p2+p

4

Re p2+p

4+p

6

Im p4

Im p4+p

6

Along s = u

Shape agrees with AL

Correction larger:20-30% in amplitude


Dalitz plot

x =√

3T+ − T−

Qη=

√3

2mηQη(u − t)

y =3T0Qη

− 1 = 3 ((mη − mπo)2 − s)

2mηQη− 1 iso= 3

2mηQη(s0 − s)

Qη = mη − 2mπ+ − mπ0

T i is the kinetic energy of pion πi

z =2

3

∑

i=1,3

(

3Ei − mηmη − 3m0π

)2

Ei is the energy of pion πi

|M |2 = A20(

1 + ay + by2 + dx2 + fy3 + gx2y + · · ·)

|M |2 = A20 (1 + 2αz + · · · )


Experiment: charged

Exp. a b d

KLOE −1.090 ± 0.005+0.008−0.019 0.124 ± 0.006 ± 0.010 0.057 ± 0.006

+0.007−0.016

Crystal Barrel −1.22 ± 0.07 0.22 ± 0.11 0.06 ± 0.04 (input)Layter et al. −1.08 ± 0.014 0.034 ± 0.027 0.046 ± 0.031

Gormley et al. −1.17 ± 0.02 0.21 ± 0.03 0.06 ± 0.04

KLOE has: f = 0.14 ± 0.01 ± 0.02.Crystal Barrel: d input, but a and b insensitive to d


Theory: charged

A20

a b d f

LO 120 −1.039 0.270 0.000 0.000NLO 314 −1.371 0.452 0.053 0.027

NLO (Lri = 0) 235 −1.263 0.407 0.050 0.015NNLO 538 −1.271 0.394 0.055 0.025

NNLOp (y from T 0) 574 −1.229 0.366 0.052 0.023NNLOq (incl (x, y)4) 535 −1.257 0.397 0.076 0.004NNLO (µ = 0.6 GeV) 543 −1.300 0.415 0.055 0.024NNLO (µ = 0.9 GeV) 548 −1.241 0.374 0.054 0.025

NNLO (Cri = 0) 465 −1.297 0.404 0.058 0.032NNLO (Lri = C

ri = 0) 251 −1.241 0.424 0.050 0.007

dispersive (KWW) — −1.33 0.26 0.10 —tree dispersive — −1.10 0.33 0.001 —

absolute dispersive — −1.21 0.33 0.04 —error 18 0.075 0.102 0.057 0.160

NLO toNNLO:Littlechange

Error on|M(s, t, u)|2:

|M (6) M(s, t, u)|


Experiment: neutral

Exp. α

KLOE 2007 −0.027 ± 0.004+0.004−0.006

KLOE (prel) −0.014 ± 0.005 ± 0.004Crystal Ball −0.031 ± 0.004

WASA/CELSIUS −0.026 ± 0.010 ± 0.010Crystal Barrel −0.052 ± 0.017 ± 0.010GAMS2000 −0.022 ± 0.023

SND −0.010 ± 0.021 ± 0.010

A2

0 α

LO 1090 0.000

NLO 2810 0.013

NLO (Lri = 0) 2100 0.016

NNLO 4790 0.013

NNLOq 4790 0.014

NNLO (Cri = 0) 4140 0.011

NNLO (Lri = Cri = 0) 2220 0.016

dispersive (KWW) — −(0.007—0.014)tree dispersive — −0.0065

absolute dispersive — −0.007Borasoy — −0.031

error 160 0.032

Note: NNLO ChPT gets a00 in ππ correct


α is difficult

Expand amplitudes and isospin:

M(s, t, u) = A(

1 + ã(s − s0) + b̃(s − s0)2 + d̃(u − t)2 + · · ·)

M(s, t, u) = A(

3 +(

b̃ + 3d̃) (

(s − s0)2 + (t − s0)2 + (u − s0)2)

+ · ·

Gives relations (Rη = (2mηQη)/3)

a = −2Rη Re(ã) , b = R2η(

|ã|2 + 2Re(b̃))

, d = 6R2η Re(d̃) .

α =1

2R2η Re

(

b̃ + 3d̃)

=1

4

(

d + b − R2η|ã|2)

≤ 14

(

d + b − 14a2

)

equality if Im(ã) = 0

Large cancellation in α, overestimate of b likely the problemOdense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.100/103

r and decay rates

sin ǫ =√

34R + O(ǫ2)

Γ(η → π+π−π0) = sin2 ǫ · 0.572 MeV LO ,sin2 ǫ · 1.59 MeV NLO ,sin2 ǫ · 2.68 MeV NNLO ,sin2 ǫ · 2.33 MeV NNLOCri = 0 ,

Γ(η → π0π0π0) = sin2 ǫ · 0.884 MeV LO ,sin2 ǫ · 2.31 MeV NLO ,sin2 ǫ · 3.94 MeV NNLO ,sin2 ǫ · 3.40 MeV NNLOCri = 0 .


r and decay rates

r ≡ Γ(η → π0π0π0)

Γ(η → π+π−π0)

rLO = 1.54

rNLO = 1.46

rNNLO = 1.47

rNNLO Cri =0 = 1.46

PDG 2006

r = 1.49 ± 0.06 our average .r = 1.43 ± 0.04 our fit ,Good agreement


R and Q

LO NLO NNLO NNLO (Cri = 0)

R (η) 19.1 31.8 42.2 38.7

R (Dashen) 44 44 37 —

R (Dashen-violation) 36 37 32 —

Q (η) 15.6 20.1 23.2 22.2

Q (Dashen) 24 24 22 —

Q (Dashen-violation) 22 22 20 —

LO from R =m2K0 + m

2K+ − 2m2π0

2(

m2K0

− m2K+

) (QCD part only)

NLO and NNLO from masses: Amorós, JB, Talavera 2001

Q2 =ms + m̂

2m̂R = 12.7R (ms/m̂ = 24.4)


OverviewWikipediaWikipediaEffective Field TheoryPower CountingWhy is the sky blue ?Why Field Theory ?Examples of EFTreferencesSchoolChiral Perturbation TheoryThe mass gap: Goldstone ModesThe two symmetry modes comparedSome clarificationsThe power countingChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation TheoryChiral Perturbation Theor{�lue ies}LagrangiansExternal currents?LagrangiansLagrangiansChiral LogarithmsLECs and $mu $Expand in what quantities?An exampleAn example: $m_0/f_0$An example: $m_pi /f_pi $Two-loop Two-flavourTwo-Loop Two-flavourLECsLECs$m_pi ^2$$m_pi ^2$ ($�ar l_3 = 0$)$F_pi $Two-loop Three-flavour, $le $2001Two-loop Three-flavour, $ge $2001Two-loop Three-flavour$C_i^r$$C_i^r$$C_i^r$InputsOutputs: ICorrelationsOutputs: II$pi pi $$pi pi $ and $pi K$Quark mass dependences$m_pi ^2$ fit 10$m_pi ^2$ fit D$m_K^2$ fit 10$m_K^2$ fit D$F_pi $ fit 10$F_K$ fit 10$F_K/F_{pi }$ fit 10$m_pi ^2$ fit 10, fixed $hat m$$F_pi $ fit 10, fixed $hat m$$f_+(0)$fit 10, fixed $m_K$$(f_+(0))/left(m_K^2-m_pi ^2ight )^2$ fit 10, fixed $m_K$$f_0(t)$in $K_{ell 3}$$ge 3$-flavour: PQChPTPQChPTPartial Quenching and ChPTPQChPT at Two Loops: GeneralPQChPT at Two Loops: GeneralPQChPT at Two Loop: PapersLong ExpressionsWhy so long expressionsPQChPT at Two LoopPQChPT: 1+1 case, 3 sea quarksPQChPT: 1+1 case, 3 sea quarksPQChPT: 1+1 case, 2 sea quarksPQChPT: 2 sea quarksPQChPT: Fitting StrategyConclusions PQChPT PartWishing listWishing list: Two-flavourWishing list: $ge 3$-flavourConclusions and final comments$eta o 3pi $ $eta o 3pi $: Lowest order (LO)$eta o 3pi $ beyond $p^4$: $p^2$ and $p^4$$eta o 3pi $ beyond $p^4$: Dispersive$eta o 3pi $ beyond $p^4$: Dispersive$eta o 3pi $ beyond $p^4$$eta o 3pi $ beyond $p^4$$eta o 3pi $ beyond $p^4$Two Loop Calculation: whyDiagramsDalitzplot$eta o 3pi $: $M(s,t=u)$$eta o 3pi $: $M(s,t=u)$$eta o 3pi $: $M(s=u,t)$Dalitz plotExperiment: chargedTheory: chargedExperiment: neutral$alpha $ is difficult$r$ and decay rates$r$ and decay rates$R$ and $Q$

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