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PoS(LATTICE 2008)134
Strong coupling constant and four-quark
condensates from vacuum polarization functions
with dynamical overlap fermions
Eigo Shintani∗1†
, S. Aoki2,3, S. Hashimoto1,4, H. Matsufuru1, J. Noaki1, T. Onogi5, N.
Yamada1,4(for JLQCD Collaboration)
1High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
2Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571,
Japan,
3Riken BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA,
4School of High Energy Accelerator Science, The Graduate University for Advanced Studies
(Sokendai), Tsukuba 305-0801, Japan,
5Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We study the vacuum polarization functions on the lattice with exact chiral symmetry of over-
lap fermion by matching the lattice data at large momentum scales with the Operator Product
Expansion (OPE). We extract the strong coupling constant
α
s(
µ
)in two-flavor QCD as Λ(2)
MS =
0.234(9)(+16
−0)GeV. From the analysis of the difference between the vector and axial-vectorchan-
nels, we extract some of the four-quark (dimension-six) condensates.
The XXVI International Symposium on Lattice Field Theory
July 14 - 19, 2008
Williamsburg, Virginia, USA
∗Speaker.
†shintani@post.kek.jp
c
Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAli keLicence. http://pos.sissa.it/
PoS(LATTICE 2008)134
Strong coupling constant and four-quark condensates from vacuum polarization functions ... Eigo Shintani
1. Introduction
In Quantum Chromodynamics (QCD) the vacuum polarization, defined through the (axial-
)vector current correlator, contains rich information of its perturbative and non-perturbative dy-
namics. In the long distance regime it is sensitive to the low-lying particle spectrum. The short
distance regime, on the other hand, can be analyzed using perturbation theory supplemented by
the Operator Product Expansion(OPE). The current correlator can be expressed as an expansion in
terms of the strong coupling constant
α
stogether with power corrections of the form hO(n)i/Qn.
Here, the local operator O(n)has a mass dimension nand Qis the momentum scale flowing into
the correlator. If one can calculate the correlators non-perturbatively, theoretical determination of
those fundamental parameters is made possible.
Lattice QCD calculation offers such a non-perturbative technique. Two-point correlators can
be calculated for space-like separations. In this work we investigate the use of the perturbative
formulae of the correlators for the lattice data obtained in the high Q2regime. By inspecting the
numerical data, we find that this is indeed possible at a lattice spacing a≃0.12 fm if we subtract the
bulk of the discretization effects non-perturbatively. The remaining effect can be estimated using
the perturbation theory. To our knowledge quantitative analysis including the determination of
α
s
and hO(n)ihas been missing until recently.
2. Vacuum polarization function and OPE
While the vacuum polarizations ΠJ(Q2)(Jdenotes vector or axial-vector channel) are ultra-
violet divergent and their precise value depends on the renormalization scheme, their derivative
DJ(Q2) = −Q2dΠJ(Q2)/dQ2, called the Adler function [1], is finite and renormalization scheme
independent. Therefore, the continuum perturbative expansion of DJ(Q2)to order
α
3
s[2], can be
directly applied to the lattice data. They include the parameters describing the gluon condensate
h
α
sG2i/Q4, the quark condensate hm¯qqi/Q4, and four-quark condensates hO8i/Q6and hO1i/Q6.
(The explicit forms of O8and O1are given in [3].)
In the continuum theory, the vacuum polarization functions Π(ℓ)
J(Q2)are defined through
hJ
µ
J
ν
i(Q)≡Zd4xeiQ·xhT{Jij
µ
(x)Jji
ν
(0)}i
= (
δµν
Q2−Q
µ
Q
ν
)Π(1)
J(Q2)−Q
µ
Q
ν
Π(0)
J(Q2),(2.1)
where the current Jij
µ
may either be a vector current Vi j
µ
=¯qi
γµ
qjor an axial-vector current Aij
µ
=
¯qi
γµγ
5qjwith flavor indices i6=j.Π(1)
J(Q2)and Π(0)
J(Q2)denote the transverse and longitudinal
parts of the vacuum polarization, respectively. For the vector channel (J=V), Π(0)
V(Q2) = 0 is sat-
isfied due to current conservation. For the axial-vector channel (J=A), the longitudinal component
may appear when the quark mass is finite.
With the overlap fermion, we usethe currents Vij
µ
=ZV¯qi
γµ
(1−D/2m0)qjand Aij
µ
=ZA¯qi
γµγ
5(1−
D/2m0)qj, which are not conserving but form a multiplet of the axial transformation as they do in
the continuum theory. Ddenotes the massless overlap-Dirac operator, and the parameter m0is
fixed at 1.6. The renormalization factor Z≡ZV=ZAhas been calculated non-perturbatively, Z=
2
PoS(LATTICE 2008)134
Strong coupling constant and four-quark condensates from vacuum polarization functions ... Eigo Shintani
1.3842(3) [4]. Taking account of remaining symmetries on the lattice (parity and cubic symme-
tries), the correlators of these lattice currents can be expressed as
hJ
µ
J
ν
ilat(Q) = Π(1)
J(Q)Q2
δµν
−Π(0+1)
J(Q)Q
µ
Q
ν
−
∞
∑
n=0BJ
n(Q)Q2n
µδµν
−
∞
∑
m,n=1CJ
mn(Q)Q2m+1
µ
Q2n−1
ν
+Q2m+1
ν
Q2n−1
µ
,(2.2)
where Π(0+1)
J(Q)≡Π(0)
J(Q)+Π(1)
J(Q). The lattice momentum Q
µ
is defined as Q
µ
= (2/a)sin(
π
n
µ
/L
µ
)
with an integer four-vector n
µ
whose components take values in [0,L
µ
/2]on a lattice of size L
µ
in
the
µ
-th direction (Li=1,2,3=16 and Lt=32). On the lattice, Π(ℓ)
J(Q)is not just a function of Q2
but of Q
µ
in general due to the violation of the Lorentz symmetry. In Ref.[5] we describe the detail
of derivation of these contribution to vacuum polarization, ΠV+A(Q2)and ΠV−A(Q2).
We now discuss the fit of the lattice data to the OPE expression of the form [6]
Π(0+1)
J|OPE(Q2) = c+C0(Q2,
µ
2) + CJ
m(Q2)
Q2+CJ
¯qq(Q2)hm¯qqi
Q4
+CGG(Q2)h(
α
s/
π
)GGi
Q4.(2.3)
Instead of directly treating the Adler function, we analyze its indefinite integral Π(0+1)
J|OPE(Q2).
A constant cis scheme-dependent, while other terms are finite and well defined. The leading
term, C0(Q2,
µ
2), is known to O(
α
2
s)in the massless limit [2]. For a finite quark mass there is a
contribution of O(m2/Q2), which is represented by the term CJ
m(Q2)with running mass m=m(
µ
)
at scale
µ
, known to O(
α
2
s). We ignore terms of O(m4)and higher. The OPE corrections of
the form hO(n)i/Qnstart from the dimension-four operators m¯qq and (
α
s/
π
)GG. Their Wilson
coefficients CJ
¯qq(Q2)and CGG(Q2)are known to O(
α
2
s)[7]. The terms of order 1/Q6and higher
are not included. The perturbative expansions are consistently given in terms of the strong coupling
constant
α
s(
µ
)defined in the MS scheme.
Here we note that the “gluon condensate” h(
α
s/
π
)GGiis defined only through the perturbative
expression like (2.3). Due to an operator mixing with the identity operator, the operator (
α
s/
π
)GG
contains a quartic power divergence that cannot be unambiguously subtracted within perturbation
theory, which is known as the renormalon ambiguity [8]. Therefore, the term h(
α
s/
π
)GGiin
(2.3) only has a meaning of a parameter in OPE, that may depend on the order of the perturbative
expansion, for instance. The quark condensate h¯qqiis well-defined in the massless limit, since
it does not mix with lower dimensional operators, provided that the chiral symmetry is preserved
on the lattice. Power divergence may appear at finite quark mass as ma−2. In the OPE formula
(2.3), it thus leads to a functional dependence m2a−2/Q4. In our numerical analysis we neglect this
quadratic term in m, as it should be smaller than the already small leading mdependence from the
quark condensate.
In addition to the individual vector and axial-vector correlators, we consider the V−Avacuum
polarization function. For the difference Π(0+1)
V−A(Q)≡Π(0+1)
V(Q)−Π(0+1)
A(Q), the lattice data are
more precise than the individual Π(0+1)
J(Q), so that the 1/Q6and 1/Q8terms are also necessary:
Π(0+1)
V−A|OPE(Q2) = (CV
m−CA
m)(Q2)1
Q2+CV
¯qq −CA
¯qq(Q2)hm¯qqi
Q4
3
PoS(LATTICE 2008)134
Strong coupling constant and four-quark condensates from vacuum polarization functions ... Eigo Shintani
+a6(
µ
) + b6(
µ
)ln Q2
µ
2+c6mq1
Q6+a8
Q8.(2.4)
In the V−Acombinations the coefficients CV
m−CA
mandCV
¯qq −CA
¯qq start at O(
α
s). The dimension-six
operators a6(
µ
)and b6(
µ
)contain the expectation values of dimension-six operators O8and O1
[3]. The scale
µ
is set to 2 GeV. Unlike the dimension-four quark condensate hm¯qqi,hO8iand
hO1iremain finite in the massless limit, hence gives leading contribution. The term c6, which has
a mass-dimension five, describes their dependence on the quark mass. The term a8/Q8represents
the contributions from dimension eight operators.
3. Numerical results
We use the lattice data from two-flavor QCD simulation with dynamical overlap fermions [9].
The simulations are performed at lattice spacing a= 0.118(2) fm on a 163×32 lattice, correspond-
ing to the physical volume (1.9 fm)3. The quark masses mqin this analysis are 0.015, 0.025, 0.035
and 0.050 in the lattice unit, that cover the range [ms/6,ms/2]with msthe physical strange quark
mass. The main advantage of this data set is that both the sea and valence quarks preserve exact
chiral and flavor symmetries by the use of the overlap fermion formulation [10]. The perturbative
formulae for the vacuum polarizations can therefore be applied without any modification due to
explicit violation of the chiral symmetry.
In the fitting of the lattice data with the functions (2.3) and (2.4), we use the value of the quark
condensate obtained from a simulation in the
ε
-regime using the same lattice formulation at slightly
smaller lattice spacing, h¯qqi(2 GeV)=−[0.251(7)(11) GeV]3[11]. The renormalization scale
µ
is set to 2 GeV. The quark mass is renormalized in the MS scheme using the non-perturbative
matching factor Zm(2 GeV)= 0.838(17) [4] as m(
µ
) = Zm(
µ
)mq. The coupling constant
α
s(
µ
)is
transformed to the scale of two-flavor QCD, Λ(2)
MS, using the four-loop formula [12]. Then, the free
parameters are the scheme-dependent constant c,h(
α
s/
π
)GGi, and Λ(2)
MS for the fit of an average
Π(0+1)
V+A(Q)≡Π(0+1)
V(Q)+ Π(0+1)
A(Q). For the difference Π(0+1)
V−A(Q),Λ(2)
MS obtained above is used as
an input and the dimension-six condensates a6,b6and c6are free parameters.
The OPE analysis requires a window in Q2where the systematic errors are under control. The
upper limit (aQ)2
max ≃1.324 is set by taking the points where different definitions of the lattice
momentum, i.e. Q
µ
= (2/a)sin(
π
n
µ
/L
µ
)and Q
µ
= (2/a)
π
n
µ
/L
µ
, give consistent results within
one standard deviation. In the physical unit, this corresponds to 1.92 GeV. To determine (aQ)2
min,
we investigate the dependence of the fit parameters on (aQ)2
min in Figure 1. From the left three
panels, we observe that the results for Λ(2)
MS,h(
α
s/
π
)GGi, and care stable between (aQ)2
min ≃0.48
and 0.65, which correspond to the momentum scale 1.16–1.35 GeV. Above (aQ)2
min ≃0.65 the fit
becomes unstable; the results are still consistent within one standard deviation.
Similar plots (right panel) are shown for a6,b6and a8obtained from the fit of Π(0+1)
V−A(Q).
We attempt to fit with (filled symbols) and without (open symbols) the a8/Q8term in order to
investigate how stable the results are against the change of the order of the 1/Q2expansion. We
find that the fit with a8/Q8is stable down to (aQ)2
min ≃0.46, while the other could not be extended
below (aQ)2
min ≃0.58. The difference between filled and open symbols ismarginal for a6(circles),
4
PoS(LATTICE 2008)134
Strong coupling constant and four-quark condensates from vacuum polarization functions ... Eigo Shintani
0.1
0.2
0.3
ΛMS
(2) (GeV)
-0.02
-0.01
0
<αs/πGG> (lat. unit)
0.3 0.4 0.5 0.6 0.7 0.8
(aQ)2
min,(aQ)2
max=1.32381
-0.005
-0.004
-0.003
c
0.3 0.4 0.5 0.6 0.7 0.8 0.9
(aQ)2
min, (aQ)2
max=1.32381
-0.0002
-0.0001
0
0.0001
0.0002
lattice unit
a6 (w/o a8)
b6 (w/o a8)
a6 (w/ a8)
b6 (w/ a8)
a8
Figure 1: Fit range dependence of Λ(2)
MS,h(
α
s/
π
)GGiand the constant term c(left). Similar plots for a6(
µ
),
b6(
µ
)and a8(right). The horizontal axis denotes the minimum momentum squared (aQ)2
min.
0.4 0.6 0.8 1 1.2 1.4
(aQ)2
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
ΠV+A
(0+1)
mq=0.015
mq=0.025
mq=0.035
mq=0.050
c+C0(Q2,µ2)
c+C0(Q2,µ2)+<αs/πGG>/Q4
0.6 0.8 1 1.2
(aQ)2
-0.0005
-0.0004
-0.0003
-0.0002
-0.0001
0
mq=0.015
mq=0.025
mq=0.035
mq=0.050 Q6ΠV-A
(0+1)
Figure 2: Π(0+1)
V+A(Q)(left panel) and Q6Π(0+1)
V−A(Q)(right panel) as a function of (aQ)2. The lattice data at
different quark masses are shown by open symbols. Fit curves for each quark mass and in the chiral limit
are drawn. For Π(0+1)
V+A(Q)the full result in the chiral limit (dashed-dots curves are at the finite masses, and
solid curve is in the chiral limit), as well as that without h
α
sG2i/Q4term (dashed curve), are shown.
but too large to make a reliable prediction for b6(squares). To quote the results we set (aQ)2
min =
0.586 for both Π(0+1)
V+A(Q)and Π(0+1)
V−A(Q).
Figure 2 shows the lattice data for Π(0+1)
V+A(Q)(left) and Π(0+1)
V−A(Q)(right panel) at each quark
mass and corresponding fit curves. In Q6Π(0+1)
V−A(Q), the quark mass dependence is clearly observed.
The main contribution comes from a dimension-six term c6mq/Q6, while the dimension-four term
hm¯qqi/Q4is sub-dominant (∼20%), as its coefficient starts at O(
α
s). In the chiral limit, there is a
small but non-zero value remaining in Q6Π(0+1)
V−A|OPE(Q2)as shown by a dashed curve in the plot.
This is due to the four-quark condensates a6and b6.
The quark mass dependence of Π(0+1)
V+A(Q)is, on the other hand, not substantial as expected
from the fit function (2.3). Our fit with the known value of h¯qqireproduces the data well. In the chi-
ral limit, (2.3) is controlled by two parameters: Λ(2)
MS and h(
α
s/
π
)GGi(apart from the constant term
5
PoS(LATTICE 2008)134
Strong coupling constant and four-quark condensates from vacuum polarization functions ... Eigo Shintani
c). The fit result in the chiral limit is drawn by a solid curve. The dashed curve, on the other hand,
shows the result when the contribution from the h(
α
s/
π
)GGiterm is subtracted. It indicates that
the Q2dependence is mainly controlled by the perturbative piece while the dimension-four term
gives a sub-dominant contribution. Numerically, we obtain Λ(2)
MS = 0.234(9)GeV and h(
α
s/
π
)GGi
=−0.058(7)GeV4. In order to estimate the systematic error due to the discretization effect, we
calculate the vacuum polarization function at the one-loop level of lattice perturbation theory and
extract the term of O((aQ)2)out of the physical ln(Q2/
µ
2)dependence ofC0(Q2,
µ
2). We then add
this term to the fit function (2.3) and repeat the whole analysis, which yields Λ(2)
MS = 0.249(37)GeV
and h(
α
s/
π
)GGi=+0.11(15)GeV4. We find that Λ(2)
MS is not largely affected, while h(
α
s/
π
)GGi
is very sensitive to the lattice artifact and in fact changes its sign. Other (Lorentz-violating) dis-
cretization effects due to BJ
nand CJ
mn are subtracted non-perturbatively so that the associated error
should be negligible. (see ref.[5])
The truncation of the perturbative and operator product expansions is a possible source of the
systematic error. In order to estimate the size of the former, we repeat the analysis using the fit
formulae truncated at a lower order (two-loop level), and find that the change of Λ(2)
MS is much less
than one standard deviation. It indicates that the higher order effects are negligible. The error from
the truncation of OPE is estimated by dropping the terms of O(1/Q4)from (2.3). From fits with
higher (aQ)2
min (between 0.79 and 0.89) to avoid contaminations of the 1/Q4effects, we obtain
Λ(2)
MS = 0.247(3) GeV. The deviation of Λ(2)
MS is about the same size as that due to the discretization
effect. The errors due to finite physical volume and the fixed topological charge in our simulation
[13] are unimportant for the short-distance quantities considered in this work.
Our final result is
Λ(2)
MS =0.234(9)(+16
−0)GeV,(3.1)
where the first error is statistical and the second is systematic due to the discretization and trun-
cation errors. The result is compatible with previous calculations of
α
sin two-flavor QCD: Λ(2)
MS
= 0.250(16)(16) GeV [14] and 0.249(16)(25) GeV [15]. (The physical scale is normalized with
an input r0= 0.49 fm.) The four-quark condensate a6is obtained from Π(0+1)
V−A(Q)as a6(2 GeV)
=−0.0038(3)(+16
−0)GeV6, where the first error is statistical. The second error represents an un-
certainty due to the truncation of the 1/Q2expansion. The central value is taken from the fit with
a8/Q8in (2.4) and the error reflects the shift when this term is discarded. The result agrees with
the previous phenomenological estimates −(0.003 ∼0.009)GeV6[16]. The other condensate is
less stable; we obtain b6(2GeV)=+0.0017(7) GeV6or −0.0008(2) GeV6with or without the
O(1/Q8)term, respectively.
4. Summary
With the exact chiral symmetry realized by the overlap fermion formulation, the analysis of the
lattice data can be greatly simplified. For the case of the vacuum polarizations, the continuum form
of OPE may be applied without suffering from additional operator mixings, such as the additive
renormalization of the operator ¯qq. With an input for the chiral condensate from other sources,
we can fit the lattice data at short distances and extract the strong coupling constant. The analysis
does not require lattice perturbation theory, which istoo complicated to carry out to the loop orders
6
PoS(LATTICE 2008)134
Strong coupling constant and four-quark condensates from vacuum polarization functions ... Eigo Shintani
available in the continuum theory. Moreover we obtain the four-quark condensates a6and b6, which
are relevant to the analysis of kaon decays. An obvious extension of this work is the calculation in
2+1-flavor QCD, which is underway [17].
Acknowledgments
Numerical calculations are performed on IBM System Blue Gene Solution and Hitachi SR11000
at High Energy Accelerator Research Organization (KEK) under a support of its Large Scale
Simulation Program (No. 08-05). This work is supported by the Grant-in-Aid of the Japanese
Ministry of Education (No. 18340075, 18740167, 19540286, 19740121, 19740160, 20025010,
20340047, 20740156 ), and National Science Council of Taiwan (No. NSC96-2112-M-002-020-
MY3, NSC96-2112-M-001-017-MY3).
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