1. Pion mass difference from vacuum polarization E. Shintani, H. Fukaya, S. Hashimoto, J. Noaki, T. Onogi, N. Yamada (for JLQCD Collaboration) December 5, 20151 The XXV International Symposium on Lattice Field Theory

2. December 5, 2015 The XXV International Symposium on Lattice Field Theory 2 Introduction

3. What’s it ? The XXV International Symposium on Lattice Field Theory 3  π + -π 0 mass difference  One-loop electromagnetic contribution to self-energy of π + and π 0 : [Das, et al. 1967]  Using soft-pion technique (m π →0) and equal-time commutation relation, one can express it with vector and axial-vector correlator: December 5, 2015 π π Dμν [Das, et al. 1967]

4. Vacuum polarization (VP) December 5, 2015 The XXV International Symposium on Lattice Field Theory 4  Spectral representation  Current correlator and spectral function with VP of spin-1 (rho, a 1,…) and spin-0 (pion).  Weinberg sum rules [Weinberg 1967]  Sum rules for spectral function in the chiral limit Spectral function (spin-1) of V-A. cf. ALEPH (1998) and OPAL (1999). [Zyablyuk 2004]

5.  Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule with q 2 = -Q 2. Δm π 2 is given by VP in the chiral limit.  Pion decay constant and S-parameter (LECs, L 10 )  Using Weinberg sum rule, one also gets where S ~ -16πL 10 Δm π 2, f π 2, S-parameter from VP December 5, 2015 The XXV International Symposium on Lattice Field Theory 5 [Das, et al. 1967] [Harada 2004] [Peskin, et al. 1990]

6. About Δm π 2 December 5, 2015 The XXV International Symposium on Lattice Field Theory 6  Dominated by the electromagnetic contribution. Contribution from (m d – m u ) is subleading (~10%).  Its sign in the chiral limit is an interesting issue, which is called the “vacuum alignment problem” in the new physics models (walking technicolor, little Higgs model, …). [Peskin 1980] [N. Arkani-Hamed et al. 2002]  In a simple saturation model with rho and a 1 poles, this value was reasonable agreement with experimental value (about 10% larger than Δm π 2 (exp.)=1242 MeV 2 ). [Das, et al. 1967]  Other model estimations  ChPT with extra resonance: 1.1×(Exp.) [Ecker, et al. 1989]  Bethe-Salpeter (BS) equation: 0.83×(Exp.) [Harada, et al., 2004]

7. Lattice works December 5, 2015The XXV International Symposium on Lattice Field Theory7  LQCD is able to determine Δm π 2 from the first principles.  Spectoscopy in background EM field  Quenched QCD (Wilson fermion) [Duncan, et al. 1996]: 1.07(7)×(Exp.),  2-flavor dynamical domain-wall fermions [Yamada 2005]: ~1.1×(Exp.)  Another method  DGMLY sum rule provides Δm π 2 in chiral limit.  Chiral symmetry is essential, since we must consider V-A, and sum rule is derived in the chiral limit. [Gupta, et al. 1984]  With domain-wall fermion 100 % systematic error is expected due to large m res (~a few MeV) contribution. (cf. [Sharpe 2007]) ⇒ overlap fermion is the best choice !

8. December 5, 2015 The XXV International Symposium on Lattice Field Theory 8 Strategy

9. Overlap fermion December 5, 2015 The XXV International Symposium on Lattice Field Theory 9  Overlap fermion has exact chiral symmetry in lattice QCD; arbitrarily small quark mass can be realized.  V and A currents have a definite chiral property (V ⇔ A, satisfied with WT identity) and m π 2 →0 in the chiral limit.  We employed V and A currents as where t a is flavor SU(2) group generator, Z V = Z A = 1.38 is calculated non-perturbatively and m 0 =1.6.  The generation of configurations with 2 flavor dynamical overlap fermions in a fixed topology has been completed by JLQCD collaboration. [Fukaya, et al. 2007][Matsufuru in a plenary talk]

10. What can we do ? December 5, 2015 The XXV International Symposium on Lattice Field Theory 10  V-A vacuum polarization  We extract Π V-A = Π V － Π A from the current correlator of V and A in momentum space.  After taking the chiral limit, one gets where Δ(Λ) ~ O(Λ － 1 ). (because in large Q 2, Q 2 Π V-A ~O(Q -4 ) in OPE.)  We may also compute pion decay constant and S-parameter (LECs, L 10 ) in chiral limit.

11. Lattice artifacts December 5, 2015 The XXV International Symposium on Lattice Field Theory 11  Current correlator  Our currents are not conserved at finite lattice spacing, then current correlator 〈 J μ J ν 〉 J=V,A can be expanded as O(1, (aQ) 2, (aQ) 4 ) terms appear due to non-conserved current and violation of Lorentz symmetry.  O(1, (aQ) 2, (aQ) 4 ) terms  Explicit form of these terms can be represented by the expression We fit with these terms at each q 2 and then subtract from 〈 J μ J ν 〉.

12. Lattice artifacts (con’t) December 5, 2015 The XXV International Symposium on Lattice Field Theory 12  We extract O(1, (aQ) 2, (aQ) 4 ) terms by solving the linear equation at same Q 2.  Blank Q 2 points (determinant is vanished) compensate with interpolation:  no difference between V and A O(1)O((aQ) 4 )O((aQ) 2 ) O((aQ) 4 )

13. December 5, 2015 The XXV International Symposium on Lattice Field Theory 13 Results

14. Lattice parameters December 5, 2015 The XXV International Symposium on Lattice Field Theory 14  N f =2 dynamical overlap fermion action in a fixed Q top = 0  Lattice size: 16 3 ×32, Iwasaki gauge action at β=2.3.  Lattice spacing: a -1 = 1.67 GeV  Quark mass  m q = m sea = m val = 0.015, 0.025, 0.035, 0.050, corresponding to m π 2 = 0.074, 0.124, 0.173, 0.250 GeV 2  #configs = 200, separated by 50 HMC trajectories.  Momentum: aQ μ = sin(2πn μ /L μ ), n μ = 1,2,…,L μ -1

15. Q 2 Π V-A in m q ≠ 0 December 5, 2015The XXV International Symposium on Lattice Field Theory15  VP for vector and axial vector current  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

16. Q 2 Π V-A in m q = 0 December 5, 2015 The XXV International Symposium on Lattice Field Theory 16  Chiral limit at each momentum  Linear function in m q /Q 2 except for the smallest momentum,  At the smallest momentum, we use for fit function. m PS is measured value with 〈 PP 〉.

17. Δm π 2 = 956[ stat. 94][ sys.(fit) 44]+[ Δ OPE (Λ) 88] MeV 2 = 1044(94)(44) MeV 2 cf. experiment: 1242 MeV 2  Fit function one-pole fit (3 params) two-pole fit (5 params)  Numerical integral: cutoff (aQ) 2 ~ 2 = Λ which is a point matched to OPE  Δ OPE (Λ) ~ α/Λ ; α is determined by OPE at one-loop level. Q 2 Π V-A in m q = 0 (con’t) December 5, 2015 The XXV International Symposium on Lattice Field Theory 17 Δm π 2 Λ OPE ~ O(Q -4 )

18.  f π 2 : Q 2 = 0 limit  S-param.: slope at Q 2 = 0 limit  results (2-pole fit) f π = 107.1(8.2) MeV S = 0.41(14) cf. f π (exp) = 130.7 MeV, f π (m q =0) ~ 110 MeV [talk by Noaki] S(exp.) ~ 0.684 f π 2 and S-parameter December 5, 2015 The XXV International Symposium on Lattice Field Theory 18 fπ2fπ2fπ2fπ2 S-param

19. Summary December 5, 2015 The XXV International Symposium on Lattice Field Theory 19  We calculate electromagnetic contribution to pion mass difference from the V-A vacuum polarization tensor using the DGMLY sum rule.  In this definition we require exact chiral symmetry and small quark mass is needed.  On the configuration of 2 flavor dynamical overlap fermions, we obtain Δm π 2 = 1044(94)(44) MeV 2.  Also we obtained f π and S-parameter in the chiral limit from the Weinberg sum rule.

20. Q 2 Π V-A in m q ≠ 0 December 5, 2015 The XXV International Symposium on Lattice Field Theory 20  In low momentum (non-perturbative) region, pion and rho meson pole contribution is dominant to Π V-A, then we consider  In high momentum, OPE: ~m 2 Q -2 + m 〈 qq 〉 Q -4 + 〈 qq 〉 2 Q -6 +…

21. VP of vector and axial-vector December 5, 2015 The XXV International Symposium on Lattice Field Theory 21  After subtraction we obtain vacuum polarization: Π J = Π J 0 + Π J 1 which contains pion pole and other resonance contribution.  Employed fit function is “pole + log” for V and “pole + pole” for A.  Note that VP for vector corresponds to hadronic contribution to muon g-2. ⇒ going under way

22. Comparison with OPE December 5, 2015The XXV International Symposium on Lattice Field Theory 22  OPE at dimension 6 with MSbar scale μ, and strong coupling α s.