Vector and Axialvector Vacuum Polarization in Lattice QCD

Vector and Axial-vector Vacuum Polarization in Lattice QCD Eigo Shintani (KEK) (JLQCD Collaboration) KEKPH 0712, Dec. 12, 2007

Introduction 2

Target �We try to extract some physical information from vector (V) and axial-vector (A) vacuum polarization at different energy. �Low energy (q 2~mπ2) �Chiral perturbation theory (CHPT) [Peskin, Takeuchi. (1992)] Low energy constant, LEC (L 10) → S-parameter �Muon g-2 Leading hadronic contribution �High energy (q 2 >>mπ2) �Operator product expansion (OPE) chiral <qq>, gluon <GG>, 4 -quark <qΓqqΓq> condensate 3

<VV-AA> �Vacuum polarization of <VV-AA> is associated with spontaneous chiral symmetry breaking. �pion mass diffrence, and L 10 through CHPT and spectral sum rule �<O 1>, <O 8> which are corresponding to electroweak penguin operator �We require non-perturbative method in chiral symmetry. → Lattice QCD using overlap fermion is needed. 4

Vacuum polarization �Vacuum polarization of <JJ> �Current-current correlator: J=V/A Spin 0 Spin 1 vector (pseudo-)scalar in Lorentz inv. , Parity sym. , and �Contribution to ΠJ Low-energy (q 2 ~ mπ2) CHPT, resonance model, … Pion, rho, … meson High-energy (q 2 ≫ mπ2) OPE, perturbation Gluon, quark field 5

Pion mass difference [Das, et al. (1967)] �Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule �Spectral sum rule, providing pion mass difference where ρJ(s)=Im ΠJ(s) We need to know the -q 2= Q 2 dependence of ΠV-A from zero to infinity. �Pion mass difference �One loop photon correction to pion mass �using soft-pion theorem → DGMLY sum rule is correct in the chiral limit 6

Models and other lattice works �Low energy constant [DMO (1967)][Ecker (2007)] � Experiment (+ Das-Mathur-Okubo sum rule + CHPT(2 -loop)) � 4 -quark condensate � Fit ansatz using τ decay (ALEPH) , factorization method [Cirigliano, et al. (2003)] �Pion mass difference � Experiment � Resonance saturation model (DGMLY sum rule) [Das, et al. (1967)] � Lattice (2 flavor DW) [Blum, et al. (2007)] 7

Our works 8

Lattice parameters gauge action β a-1 fermion action Iwasaki 2. 3 1. 67 Ge. V 2 -flavor overlap m 0 quark mass Qtop ZA = Z V 1. 6 0. 015, 0. 025, 0. 035, 0. 050 0 1. 38 • Vector and axial vector current 9

Extraction of vacuum polarization �Current 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. 10

Momentum dependence �Example, mq=0. 015 Q 2ΠV and Q 2ΠA Q 2ΠV-A = Q 2ΠV - Q 2ΠA � Q 2ΠV and Q 2ΠA are very similar. � Signal of Q 2ΠV-A is order of magnitudes smaller, but under good control thanks to exact chiral symmetry. 11

How to extract LECs �One-loop in CHPT In CHPT(2 -flavor), 〈VV-AA〉correlator can be expressed as where LECs corresponds to L 10 in SU(2)×SU(2) CHPT. �DMO sum rule l 5 is a slope at Q 2 =0 in the chiral limit and it can be obtained by chiral extrapolation in the finite Q 2. 12

How to extract LECs (preliminary) CHPT formula at 1 -loop Fitting at smallest Q 2: cf. exp. -0. 00509(57) Except for the smallest Q 2, CHPT at one-loop will not be suitable because momentum is too large. 13

How to extract 4 -quark condensate �OPE for 〈VV-AA〉 At high momentum, one found at renormalization scale μ. a 6 and b 6 has 4 -quark condensate, related to K → ππ matrix element We notice 1. In the mass less limit, ΠV-A starts from O(Q-6) 2. b 6 is subleading order. b 6 / a 6 ~ 0. 03 Our ansatz: linear mass dependence for a 6, and constant for b 6 14

How to extract 4 -quark condensate (preliminary) • Fitting form: Free parameter, a 6, b 6, c 6. • range [0. 9, 1. 3] Result: cf. using ALEPH data (τ decay) a 6 ~ -4. 5× 10 -3 Ge. V 6 15

How to extract Δmπ2 �Two integration range � Q 2 ＞Λ 2 : � Q 2≦ Λ 2 : fit ansatz, x 1~6 are free parameters, using Weinberg’s spectral sum rule [Weinberg. (1967)] and , 16

How to extract Δmπ2 (preliminary) • Fit range: Q 2≦ 1=Λ 2 • good fitting in all quark masses • In the chiral limit: including OPE result. • smaller than exp. 1260 Me. V 2 about 30~40% Finite size and fixed topology effect ? 17

Summary �Vacuum polarization includes some non-perturbative physics. (e. g. Δmπ2 , 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. 18

Backup 19

Low energy scale �CHPT �describing the dynamics of pion at low energy scale in the expansion to O(p 2) �Low energy theory associating with spontaneous chiral symmetry breaking (SχV). �VV-AA vacuum polarization �<VV-AA>=<LR> → corresponding to SχV �important to non-pertubative effect �Low energy constant: NLO lagrangian π L 10 is also related to S-parameter. [Peskin, Takeuchi. (1992)] 20

High energy scale �OPE formula �expansion to some dimensional operators CO : analytic form from pertrubation (3 -loop) <O> : condensate, which is determined non-perturbatively �ΠV-A and one found (in the chiral limit) related to K → ππ matrix element 21

Resonance saturation �Spectral representation �Resonance saturation ΠV-A Non-perturbative effect CHPT OPE Resonance state 22