On the renormalized scalar density in quenched QCD

Abstract

We present a non-perturbative determination of the renormalization factor Z_S of the scalar density in quenched QCD with overlap fermions. Results are obtained at four values of the lattice spacing. By combining Z_S with results for the low-energy constant Sigma we are able to compute the renormalization group invariant scalar condensate in the continuum limit with a total accuracy of 7%, excluding dynamical quark effects. Our result translates to Sigma_msbar(2 GeV)=(285+/-9 MeV)^3 if the scale is set by the kaon decay constant. We have also performed scaling studies of the pseudoscalar decay constant and the vector mass. Our results indicate that quantities computed using overlap quarks exhibit excellent scaling behaviour, with small residual lattice artifacts. Comment: 15 pages, 7 figures

Full text

Page 1

arXiv:hep-lat/0507026v1 20 Jul 2005
DESY 05-129
On the renormalized scalar density in quenched QCD
J. Wennekers and H. Wittig
DESY, Theory Group, Notkestrasse 85,
D-22603 Hamburg, Germany
Abstract
We present a non-perturbative determination of the renormalization factor ZSof the
scalar density in quenched QCD with overlap fermions. Results are obtained at four
values of the lattice spacing. By combining ZSwith results for the low-energy constant
Σ we are able to compute the renormalization group invariant scalar condensate?Σ in the
result translates to ΣMS(2GeV) = (285 ± 9MeV)3if the scale is set by the kaon decay
constant. We have also performed scaling studies of the pseudoscalar decay constant
and the vector mass. Our results indicate that quantities computed using overlap quarks
exhibit excellent scaling behaviour, with small residual lattice artifacts.
continuum limit with a total accuracy of 7%, excluding dynamical quark effects. Our
July 2005

Page 2

1Introduction
The expectation value of the scalar density at vanishing quark mass, commonly named
the quark or chiral condensate, plays a central rˆ ole in QCD at low energies. Spontaneous
chiral symmetry breaking is signalled by the formation of a non-vanishing condensate,
and an accurate determination of its value is of great practical interest. Lattice sim-
ulations of QCD appear well suited for this task, but in order to guarantee a reliable
error estimation, it is crucial to have control over systematic effects. In particular, to
ensure that the quark condensate approaches the continuum limit as a power series in
the lattice spacing a, the renormalization of the bare scalar density must be known with
good accuracy. It is well known that renormalization factors computed in perturbation
theory at one loop are not reliable. Further complications arise if the lattice formula-
tion breaks chiral symmetry explicitly. For instance, in the case of Wilson fermions, a
cubically divergent term must be subtracted before multiplicative renormalization can
be applied [1].
In this paper we report on a non-perturbative calculation of the renormalization
factor ZS of the scalar density, using the overlap operator [2] as our fermionic dis-
cretization in the quenched approximation. We employ the method proposed in [3] and
compute ZSat four different values of the lattice spacing, ranging from a ≈ 0.12fm to
0.075fm. By identifying the bare condensate with the low-energy constant Σ, which
appears in effective low-energy descriptions of QCD, we can compute the renormalized
quantity, given results for Σ at the corresponding values of the bare coupling in the
quenched theory.
Our analysis of the scaling properties of the renormalized condensate indicates the
presence of only very small cutoff effects of order a2, provided that the non-perturbative
estimates for the renormalization factor ZSare used throughout. Thus, an extrapolation
to the continuum limit can be performed in a controlled way.
Moreover, we have extended the scaling analysis to other quantities, such as the
pseudoscalar meson decay constant and the mass in the vector channel. In all cases we
observe an excellent scaling behaviour, with leading cutoff effects of order a2, and thus
consistent with expectation. To our knowledge, these results represent the first detailed
scaling study for overlap fermions.
Results for the quark condensate have already been published by a number of au-
thors [4–13]. The novelty in this paper is the extension of previous simulations with
overlap fermions [5,7,8] to considerably finer lattice spacings, as well as the strict ap-
plication of non-perturbative renormalization, enabling us to take the continuum limit.
Overlap fermions, despite their larger numerical cost, have clear conceptual advantages
when it comes to studying the problem of chiral symmetry breaking, which is encoded
in the value for the quark condensate. We stress, though, that our results are valid
for quenched QCD, and thus great care must be taken if they are to be interpreted in
the context of the full theory. In particular, the chiral condensate is ill-defined in the
quenched approximation [14].
1

Page 3

2 Renormalization conditions
Here we briefly recall the conditions that fix the renormalization of the scalar and
pseudoscalar densities in simulations using fermionic discretizations that satisfy the
Ginsparg-Wilson relation [15,16]. Full details can be found in refs. [3,9].
If the regularization preserves chiral symmetry, then the chiral Ward identities
imply that
ZS= ZP= 1/Zm. (2.1)
The renormalization factor?ZS, which relates the bare scalar density to the renormal-
?ZS(g0) =(r0m)(g0)
In this expression UMdenotes the RGI quark mass in the continuum limit, in units of
the hadronic radius r0[17], while m is the bare quark mass that appears in the lattice
Dirac operator satisfying the Ginsparg-Wilson relation. The expression on the right is
evaluated at a given reference value, xref, of the square of the pseudoscalar meson mass
in units of r0. A convenient choice, which we also adopt here, is xref = 1.5736. For
r0= 0.5fm this corresponds to mP= mK= 495MeV. The original data required for
the determination of UMwere published in [18], and in eq. (3.1) of [3] UMis listed for
several choices of xref.
Since ZS= ZPan alternative renormalization condition can be formulated in terms
of the matrix element of the pseudoscalar density. If we introduce the shorthand nota-
tion
Gbare
P
= ?0|Pa(0)|PS?,
where λais some flavour matrix, then?ZPcan be defined via
?ZP=
The universal factor UPdenotes the RGI matrix element of the pseudoscalar density in
the continuum limit. Its value can be determined, for instance, using O(a) improved
Wilson fermions, and the results presented in refs. [18,19] then yield
ization group invariant (RGI) density, can then be defined by [3]
UM
????(r0mP)2=xref
.(2.2)
Pa(x) = (ψλaγ5ψ)(x), (2.3)
UP
(r2
0Gbare
P
)(g0)
?????(r0mP)2=xref
.(2.4)
UP= 1.802(42)at(r0mP)2= 1.5736.(2.5)
In order to compute?ZSor?ZP, it is clear from eqs.(2.2) and(2.4) that the main task is
at the point where (r0mP)2= xref, for a fermionic discretization based on the overlap
operator.
the determination of the value of the bare quark mass, m, and the matrix element Gbare
P
2

Page 4

3 Numerical simulations
In our simulations we have computed mesonic two-point correlation functions in the
pseudoscalar and vector channels. We have used the massive overlap operator Dm,
defined by [2]
Dm=
?
1 −1
2¯ am
?
D + m,D =1
¯ a
?
1 −
A
A†A
√
?
,(3.1)
where
A = 1 + s − aDw,¯ a =
a
1 + s,
|s| < 1,(3.2)
and Dwis the Wilson-Dirac operator.
The calculation of the quark propagator proceeds as usual by solving
Dmψ = η (3.3)
for a source field η. As pointed out in [20], the determination of both chiralities of the
solution ψ requires some care in the presence of zero modes of the massless operator D,
especially as the quark mass becomes small. To separate off the zero mode contribution
we have implemented the strategy outlined in section7 of ref. [20], which we briefly
review here. To this end we shall consider a gauge configuration which has a number of
zero modes with positive chirality.
The solution to eq. (3.3) with negative chirality is given by
P−ψ = (D†
mDm)−1P−D†
mη, (3.4)
and thus the inversion of D†
zero modes. The components with positive chirality are obtained from
mDmtakes place in the chirality sector that does not contain
P+ψ =1
mP0P+η + (P+DmP+)−1?
(1 − P0)P+η − P−DmP−ψ
?
,(3.5)
where P0is a projector onto the subspace spanned by the zero modes, and whose calcu-
lation is described in [20]. When implemented in a computer program, eq. (3.5) offers
complete control over the zero mode contribution. It is also clear that the necessary
inversion of (P+DmP+) is performed on a source where all zero mode contributions have
been projected out. The rˆ oles of the positive and negative chirality sectors are obviously
reversed in the above expressions if the zero modes have negative chirality.
In our programs we compute P−ψ and P+ψ using the Generalized Minimum Resid-
ual (GMRES) algorithm [21], which also allows for an inversion of Dmitself. To speed up
the inversion we have incorporated “low-mode preconditioning”, a technique designed to
protect against numerical instabilities caused by very small eigenvalues of D†
As we shall see later, the quark masses considered in this work are relatively large and
hence provide an infrared cutoff, but we found that the inversion can nevertheless be
accelerated in this way. The presence of zero modes in conjunction with the fact that the
mDm[20].
3

Page 5

low (non-zero) modes are only known with a certain numerical accuracy requires some
care in the implementation of low-mode preconditioning for the solution in eq. (3.5).
Details will be described elsewhere [22].
Since the goal of our study is the computation of the renormalized condensate in
the continuum limit, we have chosen our simulation parameters to coincide with those of
previous determinations of the bare condensate. To this end we have identified the latter
with the parameter Σ computed by matching the spectrum of low-lying eigenvalues of D
to the predictions of Random Matrix Theory [23]. More precisely, we have concentrated
on the dataset labeled “B” in that reference, which comprises three different lattice
spacings at a fixed box size of L = 1.49fm. We note that a spatial volume of this size
is sufficiently large to avoid large finite volume effects for masses and decay constants
at mP ≈ mK. In order to improve the accuracy of the continuum extrapolation we
added a fourth β-value, β = 5.9256, tuned to reproduce the same physical box size for
L/a = 14. Following the same procedure as in [23], we have determined the low-lying
spectrum of the Dirac operator and extracted the parameter Σ.
The computation of fermionic two-point functions proceeded by setting T = 2L, to
control the exponential decay of the correlation function in a more reliable way. Our
simulation parameters are listed in Table 1. As in ref. [23], the parameter s in the
definition of the overlap operator (c.f. eq. (3.2)) was set to s = 0.4.
β L/ar0/aa [fm]#cfgs
5.8458
5.9256
6.0000
6.1366
12
14
16
20
4.026
4.697
5.368
6.710
0.124
0.106
0.093
0.075
200
174
200
100
Table 1: Simulation parameters for the determination of mesonic two-point functions
At each value of the bare coupling we computed quark propagators for three bare
masses straddling the reference point corresponding to xref= (r0mP)2= 1.5736. We
added a fourth, heavier value at all but the finest lattice spacing we considered, to study
the quark mass dependence of mesonic quantities in more detail. Since the quark masses
here are relatively large, the low-lying spectrum of D cannot induce large fluctuations in
correlation functions like those observed in the so-called ǫ-regime [24,25]. Therefore, we
did not apply the method known as low-mode averaging [25,26] to enhance the signal.
In the pseudoscalar channel we used both the left-handed axial current Jµand the
pseudoscalar density P as interpolating operators, i.e.
Jµ(x) = (ψrγµP−ψs)(x),P(x) = (ψrγ5ψs)(x), (3.6)
where P±=1
2(1 ± γ5), and r, s denote flavour labels. Choosing r ?= s, both of these
4