Pion mass difference from vacuum
polarization
E. Shintani, H. Fukaya, S. Hashimoto, J. Noaki, T. Onogi,
N. Yamada (for JLQCD Collaboration)
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The XXV International Symposium on Lattice Field Theory
July 31, 2017
Introduction
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The XXV International Symposium on Lattice Field Theory
July 31, 2017
What’s it ?
π+-π0 mass difference


One-loop electromagnetic contribution to self-energy of π+ and π0:
[Das, et al. 1967]
4
d
q 2
Δmπ2  mπ2   mπ2 0  
e Dμν (q)
4
(2 π )
  d 4 x e iqx  π  | T {J μEM , J νEM } | π     π 0 | T {J μEM , J νEM } | π 0 


Dμν
Using soft-pion technique (mπ→0) and equal-time commutation relation,
one can express it with vector and axial-vector correlator:
[Das, et al. 1967]
4
αE M d q
Dμν (q)
2 
4
f π (2 π )
  d 4 x e iqx T {V μ3 , Vν3}  T { Aμ3 , Aν3 }
Δmπ2 

π
π
3

The XXV International Symposium on Lattice Field Theory
July 31, 2017

Vacuum polarization (VP)

Spectral representation

Current correlator and spectral function


 J μ J ν   gμνq 2  qμ qν Π(J1)  qμ qν Π(J0)



ds
(0)
gμν s 2  s μ sν Im Π(1)
J  s μ s ν Im Π J
2
s  q  iε
0


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



(1st)  ds Im V(1)  Im  (A1)  f2 ,
0


Spectral function (spin-1)
of V-A. cf. ALEPH (1998)
and OPAL (1999).
[Zyablyuk 2004]

(2nd)  ds s Im V(1)  Im  (A1)  0
0
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July 31, 2017
Δmπ2, fπ2, S-parameter from VP

Das-Guralnik-Mathur-Low-Young (DGMLY) sum rule

3 EM
2
2
(1 0)
2
(1 0)
2
m  
dQ
Q

(
Q
)


(
Q
)
V
A
2 
4f 0

2

[Das, et al. 1967]
[Harada 2004]
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

[Peskin, et al. 1990]

2
(10 )
2
(10 )
2
f π2   lim
Q
Π
(
Q
)

Π
(
Q
),
V
A
2
Q 0
Q


2
(10 )
2
(10 )
2
Q
Π
(
Q
)

Π
(
Q
)
V
A
2
0 Q
S   lim
2

where S ~ -16πL10
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July 31, 2017
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



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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
July 31, 2017
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 !
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July 31, 2017
Strategy
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The XXV International Symposium on Lattice Field Theory
July 31, 2017
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






1
1
Vμa ( x )  Z V q ( x ) γ μt a 1 
Dov q( x ), Aμa ( x )  Z A q ( x ) γ μ γ5t a 1 
Dov q( x )
 2m0

 2m0

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]

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July 31, 2017
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
Λ
3α
Δm   EM2  dQ 2 Q 2ΠV  A (Q 2 )  Δ( Λ )
4 πf π 0
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.
2
π

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The XXV International Symposium on Lattice Field Theory
July 31, 2017
Lattice artifacts

Current correlator

Our currents are not conserved at finite lattice spacing, then current
correlator 〈JμJν〉 J=V,A can be expanded as


 J μ J ν   δμνQ 2  QμQν Π(J1) (Q 2 )  QμQν Π(J0) (Q 2 )
 O(1)  O((aQ) 2 )  O((aQ) 4 )  
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
O (1) :
A(Q 2 )δ μν ,
O ((aQ ) 2 ) : B1 (Q 2 )(aQ μ ) 2 δ μν ,

O ((aQ ) 4 ) : B2 (Q 2 )(aQ μ ) 4 δ μν , C11 (Q 2 ) (aQ μ )(aQν ) 3  (aQ μ ) 3 (aQν )
We fit with these terms at each q2 and then subtract from 〈JμJν〉.
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July 31, 2017

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:
O((aQ)4)
A
: f (Q 2 )  a1  a2Q 2  a3 (Q 2 ) 2  a4 (Q 2 ) 4 ,
B1, 2 , C11 : g(Q 2 )  b1  b2Q 2  b3 (Q 2 ) 2
 no difference between V and A
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The XXV International Symposium on Lattice Field Theory
July 31, 2017
Results
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The XXV International Symposium on Lattice Field Theory
July 31, 2017
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
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Q2ΠV-A in mq ≠ 0

VP for vector and axial vector current
Q2ΠV and Q2ΠA


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Q2ΠV-A = Q2ΠV - Q2ΠA
Q2ΠV and Q2ΠA are very similar.
Signal of Q2ΠV-A is order of magnitudes smaller, but under good control
thanks to exact chiral symmetry.
The XXV International Symposium on Lattice Field Theory
July 31, 2017
Q2ΠV-A in mq = 0

Chiral limit at each momentum

Linear function in mq/Q2 except for the
smallest momentum,
Q 2 f π2
Q 2 fV2
Q ΠV  A (m)  2
 2

2
2
Q  mPS Q  mV
 a  bmq / Q 2  O((mq / Q 2 ) 2 )
2

At the smallest momentum, we use
Q 2 f π2
Q 2 fV2
Q ΠV  A (m)  2
 2

2
2
Q  mPS Q  mV
Q 2 F (1  cmq )  O (mq2 )
~
2
Q 2  mPS
2
for fit function. mPS is measured
value with 〈PP〉.
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July 31, 2017
Q2ΠV-A in mq = 0 (con’t)
Λ
 Fit function
one-pole fit (3 params)
Q 2 a1
c
2
Q  a2
two-pole fit (5 params)
Q 2 a3
Q 2a1
 2
c
2
Q  a2 Q  a4
OPE
~ O(Q-4)
Δmπ2
= 956[stat.94][sys.(fit)44]+[ΔOPE(Λ)88] MeV2
= 1044(94)(44) MeV2 2
Δmπ
cf. experiment: 1242 MeV2
 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.
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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)
S-param
cf.
fπ (exp) = 130.7 MeV,
fπ (mq=0) ~ 110 MeV
[talk by Noaki]
S(exp.) ~ 0.684
f π2
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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 Δmπ2 = 1044(94)(44) MeV2.
Also we obtained fπ and S-parameter in the chiral limit from
the Weinberg sum rule.
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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
Q 2ΠV  A
Q 2 fV2
Q 2 f π2
~ 2
 2
 0
2
2 2
Q
Q  mV Q  mπ 0
 In high momentum, OPE: ~m2Q-2 + m〈qq〉Q-4+〈qq〉2Q-6+…
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VP of vector and axial-vector
 After subtraction we obtain vacuum polarization: ΠJ = ΠJ0 + ΠJ1 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
64 αs
 q q 2
2
6
9π Q
 αs  89 1 Q 2 
 1    ln 2 
 π  48 4 μ 
2
Πpert
V  A (Q )  
with MSbar scale μ, and
strong coupling αs .
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Field Theory
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