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)

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

3

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)


4

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


5

)()(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


6

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


7

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 !


8

Strategy

Overlap fermion


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 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 ?


10

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


11

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)


12

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)


13

Results

Lattice parameters


14

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


15

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


16

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

qq

V

V

PS

πAV

mQmOcmFQ

mQfQ

mQfQmQ

Π

))/((/

)(

222

22

22

22

222

QmOQbmamQfQ

mQfQmQ

qq

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)


17

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


18

ffππ22

S-paramS-param

Summary


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 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


20

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


21

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


22

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

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Pion mass difference from vacuum polarization