pion mass difference from vacuum polarization
DESCRIPTION
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] - PowerPoint PPT PresentationTRANSCRIPT
Pion mass difference from vacuum polarization
E. Shintani, H. Fukaya, S. Hashimoto, J. Noaki, T. Onogi, N. Yamada (for JLQCD Collaboration)
April 22, 20231 The XXV International Symposium on Lattice Field Theory
April 22, 2023The XXV International Symposium on Lattice Field Theory
2
Introduction
What’s it ?
The XXV International Symposium on Lattice Field Theory
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π+-π0 mass 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 vector and axial-vector correlator:
April 22, 2023
004
24
4222
|},{||},{|
)()2(
0
πJJTππJJTπexd
qDeπqdmmm
EMν
EMμ
EMν
EMμ
iqx
μνπππ
Δ
π π
Dμν
},{},{
)()2(
33334
4
4
22
νμνμiqx
μνπ
π
AATVVTexd
qDπqd
fαm
EMΔ[Das, et al. 1967]
Vacuum polarization (VP)
April 22, 2023The XXV International Symposium on Lattice Field Theory
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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
(0)(1)
01
JνμJνμμν
JνμJνμμννμ
sssssgεiqs
dsqqqqqgJJ
ΠΠ
ΠΠ
ImIm2
02
)()(2
0ImIm (2nd)
,ImIm (1st)
0
1)(1)(
2
0
1)(1)(
AV
AV
sds
fds
Spectral function (spin-1) of V-A. cf. ALEPH (1998) and OPAL (1999). [Zyablyuk 2004]
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
)()( 43 20)1(20)1(2
0
22
EM2 QQQdQf
m AV
Δmπ2, fπ
2, S-parameter from VP
April 22, 2023The XXV International Symposium on Lattice Field Theory
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)()(lim
,)()(lim
2(2(220
2(2(2
0
2
2
2
QQQQ
S
QQQf
AVQ
AVQπ
0)10)1
0)10)1
ΠΠ
ΠΠ
[Das, et al. 1967][Harada 2004]
[Peskin, et al. 1990]
About Δmπ2
April 22, 2023The XXV International Symposium on Lattice Field Theory
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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 Δmπ
2(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]
Lattice works
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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 Δmπ
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
mres (~a few MeV) contribution. (cf. [Sharpe 2007]) ⇒ overlap fermion is the best choice !
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Strategy
Overlap fermion
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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 mπ
2→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]
)(2
11)()(),(2
11)()(0
50
xqDm
tγγxqZxAxqDm
tγxqZxV ova
μAaμov
aμV
aμ
What can we do ?
April 22, 2023The XXV International Symposium on Lattice Field Theory
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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.
)()(43 22
0
22
2 ΛΔΠΔΛ
QQdQfπαm AV
ππ EM
Lattice artifacts
April 22, 2023The XXV International Symposium on Lattice Field Theory
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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 〉 .
))(())(()1()()(
42
2)(2)(2
aQOaQOOQQQQQQQδJJ JνμJνμμννμ
01 ΠΠ
)()())(()(,))((:))((,))((:))((
,)(:)1(
33211
422
4
221
2
2
νμνμμνμ
μνμ
μν
aQaQaQaQQCδaQQBaQOδaQQBaQO
δQAO
Lattice artifacts (con’t)
April 22, 2023The XXV International Symposium on Lattice Field Theory
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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
223
221
2112,1
424
223
221
2
)()(:,,)()()(:
QbQbbQgCBQaQaQaaQfA
O(1) O((aQ)4)O((aQ)2)
O((aQ)4)
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Results
Lattice parameters
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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
Q2ΠV-A in mq ≠ 0
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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-A = Q2ΠV - Q2ΠAQ2ΠV and Q2ΠA
Q2ΠV-A in mq = 0
April 22, 2023The XXV International Symposium on Lattice Field Theory
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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 〉 .
22
22
22
22
22
222
)()1(~
)(
PS
V
V
PS
πAV
mQmOcmFQ
mQfQ
mQfQmQ
Π
))/((/
)(
222
22
22
22
222
QmOQbmamQfQ
mQfQmQ
V
V
PS
πAV
Π
Δmπ2
= 956[stat.94][sys.(fit)44]+[ΔOPE(Λ)88] MeV2 = 1044(94)(44) MeV2
cf. experiment: 1242 MeV2
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.
Q2ΠV-A in mq = 0 (con’t)
April 22, 2023The XXV International Symposium on Lattice Field Theory
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ΔmΔmππ22
Λ
caQaQ
2
21
2
caQaQ
aQaQ
4
23
2
22
12
OPEOPE~ O(Q~ O(Q-4-4))
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
fπ2 and S-parameter
April 22, 2023The XXV International Symposium on Lattice Field Theory
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ffππ22
S-paramS-param
Summary
April 22, 2023The XXV International Symposium on Lattice Field Theory
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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 Δmπ
2 = 1044(94)(44) MeV2. Also we obtained fπ and S-parameter in the chiral limit from
the Weinberg sum rule.
Q2ΠV-A in mq ≠ 0
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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+…
0~022
22
22
222
2
Qπ
π
V
VAV mQ
fQmQfQQ Π
VP of vector and axial-vector
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After subtraction we obtain vacuum polarization: ΠJ = ΠJ0 + ΠJ
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
Comparison with OPE
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OPE at dimension 6
with MSbar scale μ, and strong coupling αs .
2
2
262
2
ln41
48891
964)(
μQ
πα
qqQα
πQ
s
sAV
pertΠ