vector and axial-vector vacuum polarization in lattice qcd

Eigo Shintani (KEK)(JLQCD Collaboration)

KEKPH0712, Dec. 12, 2007

Introduction

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We try to extract some physical information from vector (V) and axial-vector (A) vacuum polarization at different energy.

Low energy (qLow energy (q22~m~mππ22))

Chiral perturbation theory (CHPT) Low energy constant, LEC (L10) → S-parameterMuon g-2 Leading hadronic contribution

High energy (qHigh energy (q22 >>m >>mππ22))

Operator product expansion (OPE) chiral <qq>, gluon <GG>, 4-quark <qΓqqΓq>

condensate

Target

3

[Peskin, Takeuchi.(1992)]

<VV-AA>Vacuum polarization of <VV-AA> is associated

with spontaneous chiral symmetry breaking.pion mass diffrence, and L10

through CHPT and spectral sum rule<O1>, <O8> which are corresponding to

electroweak penguin operatorWe require non-perturbative method in chiral

symmetry. → Lattice QCD using overlap fermion is needed.

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Vacuum polarizationVacuum polarization of <JJ>Vacuum polarization of <JJ>

Current-current correlator: J=V/A

in Lorentz inv., Parity sym., and

Contribution to ΠJ Low-energy (q2 ~ mπ

2)CHPT, resonance model, …Pion, rho,… meson

High-energy (q2 ≫ mπ2)

OPE, perturbationGluon, quark field

Spin 1 vectorSpin 1 vectorSpin 0Spin 0(pseudo-)scalar(pseudo-)scalar

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Das-Guralnik-Mathur-Low-Young (DGMLY) sum Das-Guralnik-Mathur-Low-Young (DGMLY) sum ruleruleSpectral sum rule, providing pion mass difference

where ρJ(s)=Im ΠJ(s)

Pion mass differencePion mass differenceOne loop photon correction to pion mass

using soft-pion theorem

→ DGMLY sum rule is correct in the chiral limit

Pion mass difference

We need to know the -q2= Q2 dependence of ΠV-A from zero to infinity.

We need to know the -q2= Q2 dependence of ΠV-A from zero to infinity.

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[Das, et al.(1967)]

Low energy constantLow energy constantExperiment (+ Das-Mathur-Okubo sum rule + CHPT(2-

loop))

4-quark condensate4-quark condensateFit ansatz using τ decay (ALEPH) , factorization method

Pion mass differencePion mass differenceExperimentResonance saturation model (DGMLY sum rule)

Lattice (2flavor DW)

Models and other lattice works[DMO (1967)][Ecker (2007)]

[Das, et al.(1967)]

[Blum, et al.(2007)]

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[Cirigliano,et al.(2003)]

Our works

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gauge action Iwasaki

β 2.3

a-1 1.67 GeV

fermion action 2-flavor overlap

m0 1.6

quark mass 0.015, 0.025, 0.035, 0.050

Qtop 0

ZA = ZV 1.38• Vector and axial vector current

Lattice parameters

9

Current correlatorCurrent correlator

Additional term, which corresponds to the contact term due to using non-conserving current

However, VV-AA is mostly canceled, so that we ignore these terms including higher order.

Extraction of vacuum polarization

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Example, mExample, mqq=0.015=0.015

Q2ΠV and Q2ΠA are very similar.

Signal of Q2ΠV-A is order of magnitudes smaller, but under good control thanks to exact chiral symmetry.

Q2ΠV-A = Q2ΠV - Q2ΠAQ2ΠV and Q2ΠA

Momentum dependence

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One-loop in CHPTOne-loop in CHPTIn CHPT(2-flavor), 〈 VV-AA 〉 correlator can be expressed

as

where

LECs corresponds to L10 in SU(2)×SU(2) CHPT.

DMO sum ruleDMO sum rule

l5 is a slope at Q2 =0 in the chiral limit and it can be obtained by chiral extrapolation in the finite Q2.

3

1ln2

1

1ln

248

2

22

2

2

522

222

mQl

mQ

fQQ r

AV

3

5ln

248lim,

41

2

2

2

2

522)10(2

02

2

2

QQlQfQ

Q

m rAV

m

rl5

rAV

mQlQdQd 5

)10(22

008/limlim

22

How to extract LECs

12

13

How to extract LECs (preliminary)CHPT formula at 1-loopFitting at smallest Q2:

cf. exp. -0.00509(57)

Except for the smallest Q2, CHPT at one-loop will not be suitable because momentum is too large.

OPE for OPE for 〈〈 VV-AAVV-AA 〉〉At high momentum, one found

at renormalization scale μ.a6 and b6 has 4-quark condensate,

We notice 1. In the mass less limit, ΠV-A starts from O(Q-6)

2. b6 is subleading order. b6 / a6 ~ 0.03Our ansatz: linear mass dependence for a6, and constant for

b6

How to extract 4-quark condensate

)(1

ln)()()()()( 862

2

662

42

02

22)10(

QO

Q

QbaQC

Q

qqmQC

Q

mQ qq

qAV

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related to K → ππ matrix element

How to extract 4-quark condensate (preliminary)• Fitting form:

Free parameter, a6, b6,c6.• range [0.9,1.3]

Result:cf. using ALEPH data (τ decay) a6 ~ -4.5×10-3 GeV6

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Two integration rangeTwo integration range

Q2 ＞ Λ2 :Q2≦ Λ2 : fit ansatz, x1~6 are free parameters,

using Weinberg’s spectral sum rule

and ,

How to extract Δmπ2

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[Weinberg.(1967)]

How to extract Δmπ2 (preliminary)

• Fit range: Q2≦1=Λ2

• good fitting in all quark masses• In the chiral limit:

including OPE result.• smaller than exp. 1260 MeV2 about 30~40%

Finite size andfixed topology effect ?

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Vacuum polarization includes some non-perturbative physics. (e.g. Δmπ

2 , LECs, 4-quark condensate, …)

Their calculation requires the exact chiral symmetry, since the behavior near the chiral limit is important.

Overlap fermion is suitable for this study.Analysis of ΠV-A is one of the feasible studies with

dynamical overlap fermion.JLQCD collaboration is doing 2+1 full QCD

calculation, and it will be available to this study in the future.

Summary

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Backup

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CHPTCHPTdescribing the dynamics of pion at low energy

scale in the expansion to O(p2)Low energy theory associating with

spontaneous chiral symmetry breaking (SχV).VV-AA vacuum polarizationVV-AA vacuum polarization

<VV-AA>=<LR> → corresponding to SχVimportant to non-pertubative effectLow energy constant: NLO lagrangian

L10 is also related to S-parameter.

Low energy scale

π

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[Peskin, Takeuchi.(1992)]

OPE formulaOPE formulaexpansion to some dimensional operators

CO : analytic form from pertrubation (3-loop)

<O> : condensate, which is determined non-perturbatively

ΠΠV-AV-A

and one found (in the chiral limit)

High energy scale

related to K → ππ matrix element

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Spectral representationSpectral representation

Resonance saturation

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OPE

Resonance state

ΠV-ANon-perturbativeeffect

CHPT

Resonance saturation