vector and axial-vector vacuum polarization in lattice qcd
DESCRIPTION
Vector and Axial-vector Vacuum Polarization in Lattice QCD. Eigo Shintani (KEK) (JLQCD Collaboration) KEKPH0712, Dec. 12, 2007. Introduction. Target. We try to extract some physical information from vector (V) and axial-vector (A) vacuum polarization at different energy. - PowerPoint PPT PresentationTRANSCRIPT
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
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[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
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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
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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