Vector and Axial-vector Vacuum Polarization in Lattice QCD

1. Vector and Axial-vector Vacuum Polarization in Lattice QCD Eigo Shintani (KEK) (JLQCD Collaboration) KEKPH0712, Dec. 12, 2007

2. Introduction

3. Target • We try to extract some physical information from vector (V) and axial-vector (A) vacuum polarization at different energy. • Low energy (q2~mπ2) • Chiral perturbation theory (CHPT) Low energy constant, LEC (L10) → S-parameter • Muon g-2 Leading hadronic contribution • High energy (q2 >>mπ2) • Operator product expansion (OPE) chiral <qq>, gluon <GG>, 4-quark <qΓqqΓq> condensate [Peskin, Takeuchi.(1992)]

4. <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 operator • We require non-perturbative method in chiral symmetry. → Lattice QCD using overlap fermion is needed.

5. Vacuum polarization Spin 0 (pseudo-)scalar Spin 1 vector • Vacuum polarization of <JJ> • Current-current correlator: J=V/A in Lorentz inv., Parity sym., and • Contribution to ΠJ Low-energy (q2 ~ mπ2) CHPT, resonance model, … Pion, rho,… meson High-energy (q2≫ mπ2) OPE, perturbation Gluon, quark field

6. 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) • Pion mass difference • One loop photon correction to pion mass • using soft-pion theorem → DGMLY sum rule is correct in the chiral limit We need to know the -q2= Q2 dependence of ΠV-A from zero to infinity.

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

8. Our works

9. Lattice parameters • Vector and axial vector current

10. 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.

11. Momentum dependence • Example, mq=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 - Q2ΠA Q2ΠV and Q2ΠA

12. How to extract LECs • One-loop in CHPT In CHPT(2-flavor), 〈VV-AA〉correlator can be expressed as where LECs corresponds to L10 in SU(2)×SU(2) CHPT. • DMO 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.

13. How to extract LECs (preliminary) CHPT formula at 1-loop Fitting 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.

14. How to extract 4-quark condensate • OPE for 〈VV-AA〉 At high momentum, one found at renormalization scale μ. a6 and b6 has 4-quark condensate, We notice • In the mass less limit, ΠV-A starts from O(Q-6) • b6 is subleading order. b6 / a6 ~ 0.03 Our ansatz: linear mass dependence for a6, and constant for b6 related to K → ππ matrix element

15. 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

16. How to extract Δmπ2 • Two integration range • Q2＞Λ2 : • Q2≦ Λ2 : fit ansatz, x1~6 are free parameters, using Weinberg’s spectral sum rule and , [Weinberg.(1967)]

17. 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 and fixed topology effect ?

18. 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.

19. Backup

20. Low energy scale • CHPT • describing 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 polarization • <VV-AA>=<LR> → corresponding to SχV • important to non-pertubative effect • Low energy constant: NLO lagrangian L10 is also related to S-parameter. π [Peskin, Takeuchi.(1992)]

21. 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

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