1. Eigo Shintani (KEK) (JLQCD Collaboration) KEKPH0712, Dec. 12, 2007

2. Introduction 2

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

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

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

6. Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule Spectral sum rule, providing pion mass difference where ρ J (s)=Im Π J (s) Pion mass difference Pion mass difference One 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 -q 2 = Q 2 dependence of Π V-A from zero to infinity. 6 [Das, et al.(1967)]

7. Low energy constant Low energy constant Experiment (+ Das-Mathur-Okubo sum rule + CHPT(2-loop)) 4-quark condensate 4-quark condensate Fit ansatz using τ decay (ALEPH), factorization method Pion mass difference Pion mass difference Experiment Resonance 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)] 7 [Cirigliano,et al.(2003)]

8. Our works 8

9. gauge actionIwasaki β2.3 a -1 1.67 GeV fermion action2-flavor overlap m0m0 1.6 quark mass0.015, 0.025, 0.035, 0.050 Q top 0 Z A = Z V 1.38 Vector and axial vector current Lattice parameters 9

10. Current correlator 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. Extraction of vacuum polarization 10

11. Example, m q =0.015 Example, m q =0.015 Q 2 Π V and Q 2 Π A are very similar. Signal of Q 2 Π V-A is order of magnitudes smaller, but under good control thanks to exact chiral symmetry. Q 2 Π V-A = Q 2 Π V - Q 2 Π A Q 2 Π V and Q 2 Π A Momentum dependence 11

12. One-loop in CHPT 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 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. How to extract LECs 12

13. 13 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.

14. OPE for 〈 VV-AA 〉 OPE for 〈 VV-AA 〉 At high momentum, one found at renormalization scale μ. a 6 and b 6 has 4-quark condensate, We notice 1. In the mass less limit, Π V-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 How to extract 4-quark condensate 14 related to K → ππ matrix element

15. 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 GeV 6 15

16. Two integration range 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 and, How to extract Δm π 2 16 [Weinberg.(1967)]

17. How to extract Δm π 2 (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 MeV 2 about 30~40% Finite size and fixed topology effect ? 17

18. Vacuum polarization includes some non-perturbative physics. (e.g. Δm π 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 Π V-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 18

19. Backup 19

20. CHPT 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 vacuum polarization = → corresponding to SχV important to non-pertubative effect Low energy constant: NLO lagrangian L 10 is also related to S-parameter. Low energy scale π 20 [Peskin, Takeuchi.(1992)]

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

22. Spectral representation Spectral representation Resonance saturation 22 OPE Resonance state Π V-A Non-perturbative effect CHPT Resonance saturation