Pion mass difference from vacuum polarization

1
Pion mass difference from vacuum polarization

• E. Shintani, H. Fukaya, S. Hashimoto, J. Noaki,
  T. Onogi, N. Yamada (for JLQCD Collaboration)

2
Introduction
3
Whats it ?

• p-p0 mass difference
• One-loop electromagnetic contribution to
  self-energy of p and p0 Das, et al. 1967
• Using soft-pion technique (mp?0) and equal-time
  commutation relation, one can express it with
  vector and axial-vector correlator

Das, et al. 1967
Dµ?
p
p
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
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?mp2, fp2, S-parameter from VP

• Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule
• with q2 -Q2. ?mp2 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 -16pL10

Das, et al. 1967 Harada 2004
Peskin, et al. 1990
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About ?mp2

• 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
  ?mp2(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

7
Lattice works

• LQCD is able to determine ?mp2 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 ?mp2 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 !

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Strategy
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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 mp2?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
  m01.6.
• The generation of configurations with 2 flavor
  dynamical overlap fermions in a fixed topology
  has been completed by JLQCD collaboration.
  Fukaya, et al. 2007Matsufuru in a plenary talk

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What can we do ?

• V-A vacuum polarization
• We extract ?V-A ?V- ?A from the current
  correlator of V and A in momentum space.
• After taking the chiral limit, one gets
• where ?(?) O(?-1). (because in large Q2 ,
  Q2?V-AO(Q-4) in OPE.)
• We may also compute pion decay constant and
  S-parameter (LECs, L10) in chiral limit.

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Lattice artifacts

• Current correlator
• Our currents are not conserved at finite lattice
  spacing, then current correlator ltJµJ?gt JV,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 ltJµJ?gt.

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Lattice artifacts (cont)
O(1)
O((aQ)4)
O((aQ)2)

• 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)
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Results
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Lattice parameters

• Nf2 dynamical overlap fermion action in a fixed
  Qtop 0
• Lattice size 16332, Iwasaki gauge action at
  ß2.3.
• Lattice spacing a-1 1.67 GeV
• Quark mass
• mq msea mval 0.015, 0.025, 0.035, 0.050,
  corresponding to
• mp2 0.074, 0.124, 0.173, 0.250 GeV2
• configs 200, separated by 50 HMC trajectories.
• Momentum aQµ sin(2pnµ/Lµ), nµ 1,2,,Lµ-1

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Q2?V-A in mq ? 0

• VP for vector and axial vector current
• Q2?V and Q2?A are very similar.
• Signal of Q2?V-A is order of magnitudes smaller,
  but under good control thanks to exact chiral
  symmetry.

Q2?V-A Q2?V - Q2?A
Q2?V and Q2?A
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Q2?V-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 ltPPgt.

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Q2?V-A in mq 0 (cont)
?

• Fit function
• one-pole fit (3 params)
• two-pole fit (5 params)
• Numerical integral
• cutoff (aQ)2 2 ?
• which is a point matched
• to OPE
• ?OPE(?) a/?
• a is determined by OPE at one-loop level.

OPE O(Q-4)
?mp2 956stat.94sys.(fit)44?OPE(?)88
MeV2 1044(94)(44) MeV2 cf. experiment
1242 MeV2
?mp2
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fp2 and S-parameter

• fp2
• Q2 0 limit
• S-param.
• slope at Q2 0 limit
• results (2-pole fit)
• fp 107.1(8.2) MeV
• S 0.41(14)
• cf.
• fp (exp) 130.7 MeV,
• fp (mq0) 110 MeV
• talk by Noaki
• S(exp.) 0.684

S-param
fp2
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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 ?mp2 1044(94)(44)
  MeV2.
• Also we obtained fp and S-parameter in the chiral
  limit from the Weinberg sum rule.

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Q2?V-A in mq ? 0

• In low momentum (non-perturbative) region, pion
  and rho meson pole
• contribution is dominant to ?V-A , then we
  consider
• In high momentum, OPE m2Q-2
  mltqqgtQ-4ltqqgt2Q-6

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VP of vector and axial-vector

• After subtraction we obtain vacuum
  polarization ?J ?J0 ?J1 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

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Comparison with OPE

• OPE at dimension 6
• with MSbar scale µ, and
• strong coupling as .