Quark propagator in Landau and Laplacian gauges with overlap fermions

0.31 MB·English

by  J. B. Zhang

ADP-04-023/T605
Quark propagator in Landau and Laplacian gauges with overlap fermions
J.B.Zhanga,PatrickO.Bowmana,b,RyanJ.Coada,UrsM.Hellerc,DerekB.Leinwebera,andAnthonyG.Williamsa
aCSSM Lattice Collaboration,
Special Research Center for the Subatomic Structure of Matter (CSSM) and Department of Physics,
University of Adelaide 5005, Australia
bNuclear Theory Center, Indiana University,
Bloomington IN 47405, USA
cAmerican Physical Society, One Research Road,
Box 9000, Ridge, NY 11961-9000, USA
5
(Dated: February 1, 2008)
0
0 The propertiesof themomentumspace quarkpropagator in Landau gauge and Gribov copy free
2 Laplacian gauge are studied for the overlap quark action in quenched lattice QCD. Numerical cal-
culations are done on two lattices with different lattice spacing a and the same physical volume.
n
Wehavecalculated thenonperturbativewavefunction renormalization function Z(q) and thenon-
a
J perturbative mass function M(p) for a variety of bare quark masses and perform a simple linear
extrapolation tothe chiral limit. Wefocus on thecomparison of thebehavior of Z(q) and M(p) in
4
thechirallimitinthetwogaugefixingschemesaswellasthebehaviorontwolatticeswithdifferent
1
latticespacinga. WefindthatthemassfunctionsM(p)areverysimilarforthetwogaugeswhilethe

2 wave-function renormalization function Z(q) is more strongly infrared suppressed in the Laplacian
v gauge than in the Landau gauge on the finer lattice. For Laplacian gauge, it seems that the finite
5 a error is large on thecoarse lattice which has a lattice spacing a of about 0.124 fm.
4
0 PACSnumbers: 12.38.Gc,11.15.Ha,12.38.Aw,14.65.-q
0
1
4 I. INTRODUCTION Gribov problem in lattice gauge theory, see Ref. [15].
0 An alternative approach is to work with the so-called
t/ QuantumChromodynamics (QCD) is widely accepted Laplacian gauge [16]. This gauge is “Landau like” in
a as the correct theory of the strong interaction and the the sense that it has similar smoothness and Lorentz
l
- quarkpropagatorisoneofitsfundamentalquantities. By invariance properties [17], but it involves a non-local
p
studying the momentum-dependentquarkmassfunction gaugefixing procedure that avoidslattice Gribov copies.
e
h intheinfraredregionwecangainvaluableinsightintothe ThegluonpropagatorhasalreadybeenstudiedinLapla-
: mechanism of dynamical chiral symmetry breaking and cian gauge in Refs. [18, 19] and the improved staggered
v
the associated dynamical generation of mass. At high quark propagator in Laplacian gauge in Ref. [6]. It has
i
X momenta, one can use the quark propagator to extract been shown [20] that Landau and Laplacian gauges be-
r the running quark mass [1]. come equivalent in the perturbative (high-momentum)
a Lattice QCD provides a way to study the quark prop- regime and this has been confirmed by numerical stud-
agator nonperturbatively. There have been several lat- ies [6, 18, 19].
tice studies of the momentum space quark propaga- In this paper we study the overlap quark propagator
tor [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] using different fermion in the Laplaciangauge andcomparethe results with the
actions. The usual gauge for these studies has been Landau gauge to explore the effects of selecting a gauge
Landau gauge, because it is a (lattice) Lorentz covari- conditionfree of Gribov copies. Unlike Asqtad fermions,
ant gauge that is easy to implement on the lattice, and the overlap formalism provides a fermion action which
the results from the lattice Landau gauge can be easily is free of doublers and preserves an exact form of chiral
compared to studies that use different methods. Finite symmetry on the lattice. The latter feature makes over-
volume effects anddiscretizationerrorshavebeen exten- lap fermions the action of choice for studying dynamical
sivelyexploredinlatticeLandaugauge[10,11]. Unfortu- chiral symmetry breaking near the chiral limit.
nately, lattice Landaugaugesuffers fromthe well-known We also compare Laplacian gauge results on two lat-
problem of Gribov copies. Although the ambiguity orig- tices with the same physical volume and different lattice
inally noticed by Gribov [12] is not present on the lat- spacings a to explore the finite a error. In this work,
tice, since in practice one never samples from the same the (a2) mean-field improved gauge action is used to
O
gauge orbit twice, the maximization procedure used for generate the quenched gauge configurations.
gauge fixing does not uniquely fix the gauge. In gen-
eral, there are many local maxima for the algorithm to
choose from, each one corresponding to a Gribov copy, II. GAUGE FIXING
and no local algorithm can choose the global maximum
from among them. While various remedies have been We consider the quark propagator in Landau and
proposed [13, 14], they are either unsatisfactory or com- Laplacian gauges. Landau gauge fixing is performed by
putationallyveryintensive. Forarecentdiscussionofthe enforcing the Lorentz gauge condition, ∂ A (x) = 0
µ µ µ
P
2
onaconfigurationbyconfigurationbasis. Forthetadpole where ζ is the renormalization point. The renormaliza-
improved plaquette plus rectangle (Lu¨scher-Weisz [21]) tion point boundary conditions are chosen to be
gauge action which we use in the current work, we use
the (a2)improvedgaugefixingscheme,this isachieved Z(ζ2;ζ2) 1 M(ζ2) m(ζ2). (3)
O ≡ ≡
by maximizing the functional [22],
wherem(ζ2)istherenormalizedquarkmassattherenor-
4 1 malization point. The functions A(ζ2;p2) and B(ζ2;p2),
= , (1)
F 3F1− 12u F2 or alternatively Z(ζ2;p2) and M(p2), contain all of the
0
nonperturbative information of the quark propagator.
where 1 and 2 are Note that M(p2) is renormalization point independent,
F F
allofthe renormalization-pointdependence is carriedby
1
= Tr U (x)+U†(x) Z(ζ2;p2).
F1 2 µ µ
x,µ When all interactions for the quarks are turned off,
X (cid:8) (cid:9)
i.e., when the gluonfield vanishes (or the links are setto
and one), the quark propagator has its tree-level form
1
F2 = 2 Tr Uµ(x)Uµ(x+µ)+Uµ†(x+µ)Uµ†(x) S(0)(p)= 1 , (4)
x,µ i/p+m0
X (cid:8) (cid:9)
respectively,andu0 istheusualplaquettemeasureofthe wherem0 is the barequarkmass. Whentheinteractions
mean link. In this case, a Fourier accelerated, steepest- with the gluon field are turned on we have
descents algorithm [23] is used to find a local maximum.
There are, in general, many local maxima and these are S(0)(p) Sbare(a;p)=Z (ζ2;a)S(ζ2;p), (5)
2
→
called lattice Gribov copies. This ambiguity in principle
will remain a source of uncontrolled systematic error. where a is the regularization parameter - in this case,
Laplacian gauge fixing is a nonlinear gauge fixing the lattice spacing - and Z (ζ2;a) is the quark wave-
2
that respects rotational invariance, has been seen to be function renormalizationconstantchosenso as to ensure
smooth, yet is free of Gribov ambiguity. It is also com- Z(ζ2;p2)p2=ζ2 = 1. For simplicity of notation we sup-
|
putationallycheaperthenLandaugaugefixing. Thereis, press the a-dependence of the bare quantities.
however, more than one way of obtaining such a gauge On the lattice we expect the bare quark propagators,
fixinginSU(N)latticegaugetheory. Therearethreeim- in momentum space, to have a similar form as in the
plementations of Laplaciangauge fixing employedin the continuum, except that the O(4) invariance is replaced
literature: bya 4-dimensionalhypercubic symmetryonanisotropic
lattice. Hence, the inverse lattice bare quark propagator
1. ∂2(I) gauge (QR decomposition), used by Alexan-
takes the general form
drou et al. [18].
2. ∂2(II) gauge, where the Laplacian gauge transfor- (Sbare)−1(p) i C (p)γ +B(p). (6)
µ µ
mation is projected onto SU(3) by maximising its ≡ !
µ
trace [19]. X
With the periodic boundary conditions in the spatial di-
3. ∂2(III) gauge (Polar decomposition), the original
rections and anti-periodic in the time direction, the dis-
prescription described in Ref. [16] and tested in
crete lattice momenta will be
Ref. [17].
2π N 2π 1 N
i t
All three versions reduce to the same gauge in SU(2). pi = ni , and pt = Nt ,(7)
N a − 2 N a − 2 − 2
For a more detailed discussion, see Ref. [19]. For SU(3) i (cid:18) (cid:19) t (cid:18) (cid:19)
staggeredquarks,thestudyinRef.[6]indicatethat∂2(I)
where n =1,..,N and n =1,..,N , N and N are the
and ∂2(II) gauge give very similar results, and ∂2(III) i i t t i t
lattice extent in spatial and temporal direction respec-
gauge is very noisy. In this work we will only use the
tively.
∂2(II) gauge.
The overlap fermion formalism [24, 25] realizes an ex-
act chiral symmetry on the lattice and is automatically
(a) improved. The massive overlap operator can be
III. QUARK PROPAGATOR ON THE LATTICE O
written as [26]
In a covariant gauge in the continuum, the renormal- 1
D(η)= [1+η+(1 η)γ ǫ(H )] , (8)
ized Euclidean space quark propagator has the form 2 − 5 w
S(ζ2;p)= 1 = Z(ζ2;p2) , (2) where Hw(x,y) = γ5Dw(x,y) is the Hermitian Wilson-
i/pA(ζ2;p2)+B(ζ2;p2) i/p+M(p2) Dirac operator, ǫ(Hw) = Hw/ Hw2 is the matrix sign
p
3
function, and the dimensionless quark mass parameter η IV. NUMERICAL RESULTS
is
A. Simulation parameters
m0
η , (9)
≡ 2m
w
In this paper we work on two lattices with different
where m0 is the bare quark mass and m is the Wilson lattice spacing, a, and very similar physical volumes.
w
quark mass which, in the free case, must be in the range The gauge configurations are created using a tadpole
0 < m < 2. The bare quark propagator in coordinate improved plaquette plus rectangle (Lu¨scher-Weisz [21])
w
space is given by the equation gauge action through the pseudo-heat-bath algorithm.
For each lattice size, 50 configurations are used. Lat-
Sbare(m0) D˜−1(η), (10) tice parameters are summarized in Table I. The lattice
≡ c
spacing a is determined from the static quark potential
where with a string tension √σ =440 MeV [30].
Landau gauge fixing to the gauge configuration
1
D˜−1(η) D˜−1(η) and was done using a Conjugate Gradient Fourier Ac-
c ≡ 2m
w celeration [31] algorithm with an accuracy of θ
≡
∂ A (x)2 < 10−12. The improved gauge-fixing
µ µ
| |
1 scheme was used to minimize gauge-fixing discretization
D˜−1(η) D−1(η) 1 . (11) ePrrors [22]. For the Laplacian gauge fixing, we only use
≡ 1 η −
− the∂2(II)gauge[19]. Inthiscaseweconstructthegauge
(cid:2) (cid:3)
When all the interactions are turned off, the inverse transformationbyprojectingM(x)constructedfromthe
barelattice quarkpropagatorbecomesthe tree-levelver- three lowest lying eigenmodes, onto SU(3) by means of
sion of Eq. (6) trace maximisation. Effectively, we maximise the trace
ofG(x)M(x)† by iterationoverCabibbo-MarinariSU(2)
subgroups.
(S(0))−1(p)≡i Cµ(0)(p)γµ!+B(0)(p). (12) Ournumericalcalculationbeginswithanevaluationof
µ theinverseofD(η)withtheunfixedgaugeconfigurations,
X
where D(η) is defined in Eq. (8). We approximate the
WecalculateS(0)(p)directlybysettingthelinkstounity matrix sign function ǫ(H ) by the 14th order Zolotarev
w
in the coordinate space, doing the matrix inversion and approximation [32]. We then calculate Eq. (10) for each
then taking its Fourier transform. It is then possible to configurationandrotateittoLandauorLaplaciangauge
identify the appropriate kinematic lattice momentum q by using the corresponding gauge transformation matri-
directly from the definition ces G (x) . Afterward we take the ensemble averageto
i
{ }
obtain Sbare(x,y). The discrete Fourier transformation
qµ ≡Cµ(0)(p). (13) is then applied to Sbare(x,y) and the momentum-space
bare quark propagator,Sbare(p) is obtained finally.
The form of q (p ) is shown and its analytic form given
µ µ We use the mean-field improved Wilson action in the
inRef.[9]. Havingidentifiedtheappropriatekinematical
overlap fermion kernel. The value κ = 0.19163 is used
lattice momentum q, we can now define the bare lattice
in the Wilson action, which provides m a = 1.391 for
propagator as w
the regulator mass in the interacting case [9]. We calcu-
Z(p) late the overlap quark propagator for ten quark masses
Sbare(p) . (14) on each ensemble by using a shifted Conjugate Gradient
≡ i/q+M(p)
solver. For the two lattices considered here, the quark
This ensures that the free lattice propagator is identical mass parameter η was adjusted to make the tree level
to the free continuum propagator. Due to asymptotic bare quark mass in physical units, the same on the two
freedom the lattice propagator will also take the contin- lattices. For example,we chooseµ=0.018,0.021,0.024,
uum form at large momenta. In the gauge sector, this 0.030, 0.036, 0.045, 0.060, 0.075, 0.090, and 0.105 on
type of analysis dramatically improves the gluon propa- ensemble 1, i.e., the 163 32 lattice with a = 0.093
×
gator [27, 28, 29]. fm. This corresponds to bare masses in physical units
The two Lorentz invariants can then obtained by of m0 =2µmw = 106, 124, 142, 177, 212, 266, 354, 442,
531, and 620 MeV respectively.
Z−1(p)= 1 Tr /qS−1(p) (15) Theresultsoflattice2(123×24)inLandaugaugewere
12iq2 { } presentedin detailin Ref. [9], andthe results oflattice 1
Z(p) (163 32)inLandaugaugewerealsoreportedinRef.[11].
M(p)= Tr S−1(p) . (16) Here×wewillfocusonthecomparisonoftheresultsoftwo
12 { }
lattice gauge fixing schemes, i.e., the Landau gauge and
ThismeansthatZ(p)isdirectlydependentonourchoice the Gribov copy free Laplacian gauge, to probe the be-
of momentum, q, while M(p) is not. havior of the overlap fermion propagator with different
4
TABLE I:Lattice parameters.
Action Volume N N β a (fm) u Physical Volume (fm4)
Therm Samp 0
Improved 163×32 5000 500 4.80 0.093 0.89650 1.53×3.00
Improved 123×24 5000 500 4.60 0.124 0.88888 1.53×3.00
B. Laplacian gauge
In Fig. 1 we show the results for all ten masses for
both the mass and wave-function renormalization func-
tions, M(p) and Z(R)(q) Z(ζ;q) respectively. As was
≡
shown in Ref. [11], the continuum limit is more rapidly
approached when the mass function is plotted against
thediscretelatticemomentump,whilethewave-function
renormalization function Z(R)(q) is plotted against the
kinematic momentum q. The renormalization point in
Fig. 1 for Z(R)(q) has been chosen to be ζ = 5.31 GeV
in the q-scale.
In the plots of M(p), the data is orderedas one would
expect by the values for bare quark mass m0, i.e., the
larger the bare quark mass m0, the higher the M(p)
curve. For large momenta, the function Z(R)(q) demon-
strateslittlemassdependence. DeviationofZ(R)(q)from
itsasymptoticvalueof1isasignofdynamicalsymmetry
breaking, so we expect the infrared suppression to van-
ishinthelimitofaninfinitelyheavyquark. Inthefigure
for Z(R)(q), the smaller the bare mass, the more pro-
nounced is the dip at low momenta. Similarly, at small
bare masses M(q) falls off more rapidly with increasing
momenta,whichisunderstoodfromthefactthatalarger
proportionoftheinfraredmassisduetodynamicalchiral
symmetry breaking at small bare quark masses. These
results are much the same as in Ref. [11], which display
the data on the same lattices in Landau gauge. This
qualitative behavior is also consistent with what is seen
in Dyson-Schwinger based QCD models [33, 34].
In Fig. 2 we plot the data after a linear chiral extrap-
FIG. 1: (Color online). The functions M(p) and Z(R)(q) ≡ olation for both functions M(p) and Z(R)(q) Z(ζ;q)
Z(ζ2;q)renormalizedatζ =5.31GeV(intheqscale)forall ≡
in Laplacian gauge. The mass function M(p) is shown
ten quark masses in Laplacian gauge on the163×32 lattice.
against p and while the wave function renormalization
The mass function M(p) is plotted versus the discrete mo-
function Z(R)(q) is shown against q with the renormal-
mentum p defined in Eq. (7), p = p2, over the interval
µ ization point chosen as at 5.31 GeV in the q scale. We
[0,5] GeV, and Z(R)(q) is plotted against the kinematic mo-
pP seethatbothM(p)andZ(R)(q)deviatestronglyfromthe
mentum q defined in Eq. (13), q = q2, over the interval
µ tree-levelbehavior,whichareM(p)=m0 andZ(R)(q)=
[0,12] GeV.Thedata correspond tobarequarkmasses (from
bottom to top) µ = 0.018, 0.021, 0p.02P4, 0.030, 0.036, 0.045, 1. Inparticular,asinearlierstudiesoftheLandaugauge
0.060, 0.075, 0.090, and 0.105, which in physical units corre- quark propagator[4, 5, 6, 9, 11], we find a clear signal
spond to m0 = 2ηmw ≃ 106, 124, 142, 177, 212, 266, 354, of dynamical mass generation and a significant infrared
442, 531, and 620 MeV respectively suppression of the Z(ζ;q) function.
gauge fixings, and the effect of Gribov copies. Before we C. Gauge fixing comparison
make the comparison, we first briefly present some data
inLaplaciangaugeonthe163 32lattice,ourfinelattice. Next we present the results in the two gauge fixing
×
Alldatahasbeencylindercut[27]. Statisticaluncertain- schemes for comparison. All data have been cylinder
ties are estimated via a second-order, single-elimination cut [27]. First we give the results on the 163 32 lat-
×
jackknife. tice.
5
FIG.2: TheplotofthefunctionsM(p)andZ(R)(q)≡Z(ζ;q)
FIG. 3: (Color online). The comparison of two gauge fixing
afteralinearextrapolationtothechirallimit. Themassfunc- results at finite bare quark mass (m0 = 106 MeV) on the
tionsM(p)isplottedagainstthediscretemomentumpinthe
fine 163 ×32 lattice with a = 0.093 fm. Z(R)(q) is renor-
upper part of the figure and Z(R)(q) with the renormaliza-
malized to one at the renormalization point ζ =5.31 GeV
tionpointζ =5.31GeV(intheq-scale) isplottedagainst the
(in the q-scale). For the mass function M(p), Landau gauge
kinematic momentum q in the lower part of the figure.
and Laplacian gauge are very similar, while the wave func-
tion renormalization functions Z(R)(q) differ in the infrared
region.
Fig.3reportsresultsforthemassandrenormalization
functions at our lightest bare quark mass of m0 = 106
MeV. These results may be compared with the Asqtad Landau gauge or the Gribov-copy free Laplacian gauge.
results of Ref. [6], where Fig. 9 compares results of var- Wenowproceedtocomparethedatainthechirallimit.
ious gauge fixing schemes for the renormalization func- Fig.4 showsthe comparisonofthe massfunction, M(p),
tion. There, Landau gauge results are seen to lie sig- andthewavefunctionrenormalizationfunction,Z(R)(q),
nificantly higher than the Laplacian gauge results in the in Landaugaugeand Laplaciangauge. We see that they
infrared. However,withoverlapfermions,Fig.3indicates give similar performance in terms of rotational symme-
the Landau gauge results lie much closer to the Lapla- tryandstatisticalnoise. Lookingmoreclosely,wecansee
cian gauge results. Given the improved chiral properties that Landau gauge gives a slightly cleaner signal at this
of the overlap operator, the present results should pro- latticespacing. Wealsonotethatatverylargemomenta,
vide a better indicationofthe continuumlimit behavior. the two gauge fixing schemes give similar results as ex-
In either case, the same qualitative behavior of Landau pected. Although Laplacian gauge is a non-local gauge
Gauge sitting above Laplacian gauge in the infrared is fixing scheme and difficult to understand perturbatively,
observed. it is equivalent to Landau gauge in the asymptotic re-
ThemassfunctionofFig.3revealsanapproximatein- gion[20]. Intheinfraredregion,themassfunction,M(p),
varianceonthe selectionofLandauorLaplaciangauges. in the two gauges are very similar. For the mass func-
Figs. 12 and 13 of Ref. [6] indicate that the mass func- tionsthere isahintthatthe datainLaplaciangaugeare
tionofAsqtadfermionsisalsoinsensitivetothechoiceof a little higher than for the Landau gauge, although they
6
FIG. 4: (Color online). The comparison of two gauge fixing FIG. 5: (Color online). The comparison of two gauge fixing
results after a linear extrapolation to the chiral limit on the results in the chiral limit on the coarse lattice, i.e., 123×24
fine lattice, i.e., 163×32 at a = 0.093 fm. Z(R)(q) is renor- with a = 0.124 fm. The small gauge dependence of the in-
malized to one at the renormalization point ζ =5.31 GeV fraredbehavioroftheZ-functionissimilartothatonthefine
(in the q-scale). For the mass function M(p), Landau gauge lattice in Fig. 4. The infrared mass functions, M(p), appear
andLaplaciangaugeareverysimilar,whilethewavefunction different on this coarse lattice whereas they were similar on
renormalizationfunctionsZ(R)(q)aresimilarinthelargemo- thefinelattice. ThissuggestslargerO(a2)errorsinLaplacian
mentum region but differ in the infrared region. gauge.
agreewithinstatisticalerrors. With greaterstatistics we thetwogaugefixingschemesgivesimilarresults,whilein
may resolve a small difference. For the renormalization the infrared region, Z(R)(q) is more strongly suppressed
function Z(R)(q), there are systematic differences in the in Laplacian gauge than in Landau gauge. For the mass
infrared region. The Z(R)(q) is more strongly infrared function, M(p), the situation is different to that seen on
suppressed in the Laplacian gauge than in the Landau thefinelattice. Intheinfraredregion,thedatainLapla-
gauge. Thatis consistentwithwhatwasseeninthe case cian gauge sit higher than in Landau gauge. While this
oftheAsqtadquarkaction[6]whencomparingthesetwo is consistent with the infrared behavior of the renormal-
gauges. ization function, Z(R)(q), which is more suppressed in
Now we give the results fromthe coarselattice. Fig. 5 Laplacian gauge than in Landau gauge, there is no sim-
shows the comparison of the mass function M(p) and ilar signal for our fine lattice 163 32. In that case (see
the wave function renormalization function Z(R)(q) in Fig. 5), the data in Laplacian ga×uge agree with that in
the chirallimit inLandaugaugeandLaplaciangaugeon Landaugaugewithin errorbars,althoughthere is a hint
the 123 24 lattice with a = 0.124fm. As in the case of that the data in Laplaciangauge are a little higher than
the163 ×32lattice,atverylargemomenta,thetwogauge inLandaugaugeintheinfraredregion. Thisinfraredbe-
×
fixing schemes give similar results. For the renormaliza- havior of the mass function in Laplacian gauge is likely
tionfunction Z(R)(q), the situationis verysimilar to the caused by the finite lattice spacing errors, i.e., on fine
case of the 163 32 lattice, i.e., at very large momenta, enough lattices we expect Landau and Laplacian gauge
×
7
well at large momenta, but there is a substantial differ-
ence in the infrared region. A similar comparison was
made between these two lattices in Landau gauge [11].
Inthatcase,boththe renormalizationfunction, Z(R)(q),
and the mass function, M(p), agree well on the two lat-
tices. This indicates that in Landau gauge, the finite a
errors are small even for our coarse lattice with lattice
spacing a = 0.124fm. But in Laplaciangauge,the finite
a errorsarenot negligible for ourcoarselattice 123 24.
×
A likely explanation for this is the fact that our imple-
mentationoftheLaplaciangaugeisnot (a2)improved.
O
V. SUMMARY AND OUTLOOK
The momentum-space quark propagator has been
studied in Landau gauge as well as the Gribov copy free
Laplacian gauge on two lattices with the same physical
volume but with different lattice spacings a . We calcu-
lated the nonperturbative momentum-dependent wave-
function renormalization,Z(q), and the nonperturbative
mass function, M(p), for a variety of bare quark masses.
We also performed a simple linear extrapolation to the
chiral limit.
At very large momenta the two gauge-fixing schemes
givesimilarresultsasexpected. Laplaciangaugeisequiv-
alent to the Landau gauge in the asymptotic region. In
the infrared region, the mass function, M(p), in the two
gauges are very similar on the fine lattice, but differ on
the coarse lattice. The present Laplacian gauge fixing is
not (a2) improved. For our fine lattice in the infrared
FIG. 6: (Color online). The comparison of Laplacian gauge O
region,themassfunction,M(p),agreeswithinstatistical
resultsinthechirallimitontwolattices, i.e.,123×24with a
errors in the two gauge fixings. However, there is a hint
=0.124fmand163×32witha=0.093fm. Therenormaliza-
tion point for Z(R)(q)≡Z(ζ;q) is chosen to be ζ =5.31 GeV that the data in Laplacian gauge may be a little higher
than in Landau gauge, which would be consistent with
(in the q-scale). For the mass functions M(p), the value on
123×24 lattice is higher than that on the163×32 lattice in the behavior of the renormalization function, Z(R)(q).
the infrared region. while the wave function renormalization For the renormalizationfunction, Z(R)(q), there are sys-
functionZ(R)(q)ontwolatticesagreewithinerrorbars. This tematic differences inthe infraredregion. The renormal-
suggests that the finite a errors are not small on the coarse ization function is more strongly infrared suppressed in
lattice in Laplacian gauge. the Laplacian gauge than in the Landau gauge.
results for M(p) to be very similar.
VI. ACKNOWLEDGMENTS
Finally, we present the data for the two lattices in
Laplacian gauge to further explore possible finite a er-
rors. Fig. 6 shows the comparison of the mass function, We thank the Australian National Computing Facil-
M(p), and the wave function renormalization function ity for Lattice Gauge Theory and both the Australian
Z(R)(q) in the chiral limit in Laplacian gauge on the Partnership for Advanced Computing (APAC) and the
123 24 and the 163 32 lattices. For the renormal- South Australian Partnership for Advanced Computing
izati×on function, Z(R)(×q), results from the two lattices (SAPAC) for generous grants of supercomputer time
have small differences, but agree with each other within which have enabled this project. This work is supported
errors. For the mass function, M(p), the results agree by the Australian Research Council.
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