1 / 22

Pion mass difference from vacuum polarization

Pion mass difference from vacuum polarization . E. Shintani , H. Fukaya, S. Hashimoto, J. Noaki, T. Onogi, N. Yamada (for JLQCD Collaboration) . Introduction. What’s it ?. π + -π 0 mass difference One-loop electromagnetic contribution to self-energy of π + and π 0 : [Das, et al. 1967]

teague
Download Presentation

Pion mass difference from vacuum polarization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Introduction The XXV International Symposium on Lattice Field Theory

  3. What’s it ? • π+-π0mass difference • One-loop electromagnetic contribution to self-energy of π+ and π0: [Das, et al. 1967] • Using soft-pion technique (mπ→0) and equal-time commutation relation, one can express it with vectorand axial-vector correlator: [Das, et al. 1967] Dμν π π The XXV International Symposium on Lattice Field Theory

  4. Vacuum polarization (VP) • Spectral representation • Current correlator and spectral function with VP of spin-1 (rho, a1,…) 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] The XXV International Symposium on Lattice Field Theory

  5. Δmπ2, fπ2, S-parameter from VP • Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule with q2 = -Q2. Δmπ2 is given by VP in the chiral limit. • Pion decay constant and S-parameter (LECs, L10) • Using Weinberg sum rule, one also gets where S ~ -16πL10 [Das, et al. 1967] [Harada 2004] [Peskin, et al. 1990] The XXV International Symposium on Lattice Field Theory

  6. About Δmπ2 • Dominated by the electromagnetic contribution. Contribution from (md – mu) 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 a1 poles, this value was reasonable agreement with experimental value (about 10% larger than Δmπ2(exp.)=1242 MeV2). [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] The XXV International Symposium on Lattice Field Theory

  7. Lattice works • LQCD is able to determine Δmπ2 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 Δmπ2 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 mres (~a few MeV) contribution. (cf. [Sharpe 2007]) ⇒ overlap fermion is the best choice ! The XXV International Symposium on Lattice Field Theory

  8. Strategy The XXV International Symposium on Lattice Field Theory

  9. Overlap fermion • 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 mπ2→0 in the chiral limit. • We employed V and A currents as where ta is flavor SU(2) group generator, ZV = ZA = 1.38 is calculated non-perturbatively and m0=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] The XXV International Symposium on Lattice Field Theory

  10. What can we do ? • V-A vacuum polarization • We extract ΠV-A= ΠV－ ΠA 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 Q2 , Q2ΠV-A~O(Q-4) in OPE.) • We may also compute pion decay constant and S-parameter (LECs, L10) in chiral limit. The XXV International Symposium on Lattice Field Theory

  11. Lattice artifacts • Current correlator • Our currents are not conserved at finite lattice spacing, then current correlator 〈JμJν〉 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 q2 and then subtract from 〈JμJν〉. The XXV International Symposium on Lattice Field Theory

  12. Lattice artifacts (con’t) O(1) O((aQ)2) O((aQ)4) • We extract O(1, (aQ)2,(aQ)4) terms by solving • the linear equation at same Q2. • Blank Q2 points (determinant is vanished) compensate • with interpolation: • no difference between V and A O((aQ)4) The XXV International Symposium on Lattice Field Theory

  13. Results The XXV International Symposium on Lattice Field Theory

  14. Lattice parameters • Nf=2 dynamical overlap fermion action in a fixed Qtop = 0 • Lattice size: 163×32, Iwasaki gauge action at β=2.3. • Lattice spacing: a-1 = 1.67 GeV • Quark mass • mq = msea = mval = 0.015, 0.025, 0.035, 0.050, corresponding to mπ2 = 0.074, 0.124, 0.173, 0.250 GeV2 • #configs = 200, separated by 50 HMC trajectories. • Momentum: aQμ = sin(2πnμ/Lμ), nμ = 1,2,…,Lμ-1 The XXV International Symposium on Lattice Field Theory

  15. Q2ΠV-A in mq≠ 0 • VP for vector and axial vector current • 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 and Q2ΠA Q2ΠV-A = Q2ΠV - Q2ΠA The XXV International Symposium on Lattice Field Theory

  16. Q2ΠV-A in mq =0 • Chiral limit at each momentum • Linear function in mq/Q2 except for the smallest momentum, • At the smallest momentum, we use for fit function. mPS is measured value with 〈PP〉. The XXV International Symposium on Lattice Field Theory

  17. Q2ΠV-A in mq =0 (con’t) Λ • 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 • ΔOPE(Λ) ~ α/Λ; • α is determined by OPE at one-loop level. OPE ~ O(Q-4) Δmπ2 = 956[stat.94][sys.(fit)44]+[ΔOPE(Λ)88] MeV2 = 1044(94)(44) MeV2 cf. experiment: 1242 MeV2 Δmπ2 The XXV International Symposium on Lattice Field Theory

  18. fπ2 and S-parameter • fπ2 : • Q2 = 0 limit • S-param.: • slope at Q2 = 0 limit • results (2-pole fit) • fπ = 107.1(8.2) MeV • S = 0.41(14) • cf. • fπ (exp) = 130.7 MeV, • fπ (mq=0) ~ 110 MeV • [talk by Noaki] • S(exp.) ~ 0.684 S-param fπ2 The XXV International Symposium on Lattice Field Theory

  19. Summary • 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 Δmπ2 = 1044(94)(44) MeV2. • Also we obtained fπ and S-parameter in the chiral limit from the Weinberg sum rule. The XXV International Symposium on Lattice Field Theory

  20. Q2ΠV-A in mq≠ 0 • In low momentum (non-perturbative) region, pion and rho meson pole • contribution is dominant to ΠV-A , then we consider • In high momentum, OPE: ~m2Q-2 + m〈qq〉Q-4+〈qq〉2Q-6+… The XXV International Symposium on Lattice Field Theory

  21. VP of vector and axial-vector • After subtraction we obtain vacuum polarization: ΠJ = ΠJ0 + ΠJ1which • 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 The XXV International Symposium on Lattice Field Theory

  22. Comparison with OPE • OPE at dimension 6 • with MSbar scale μ, and • strong coupling αs . The XXV International Symposium on Lattice Field Theory

More Related