Table Of ContentITEP-LAT/2010-10
SU(2) lattice gluon propagators at finite temperatures
in the deep infrared region and Gribov copy effects
V. G. Bornyakov
High Energy Physics Institute, Protvino, Russia
and Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia
V. K. Mitrjushkin
Joint Institute for Nuclear Research, 141980 Dubna, Russia
and Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia
Westudynumerically theSU(2) Landaugauge transverseandlongitudinalgluon propagatorsat
1
non-zero temperatures T both in confinement and deconfinement phases. The special attention is
1
0 paid to theGribov copy effects in theIR-region. Applyingpowerful gauge fixingalgorithm we find
2 that the Gribov copy effects for the transverse propagator DT(p) are very strong in the infrared,
while the longitudinal propagator DL(p) shows very weak (if any) Gribov copy dependence. The
n
value DT(0) tends to decrease with growing lattice size; however, DT(0) is non-zero in the infinite
a
volumelimit, in disagreement with thesuggestion madein [1]. Weshowthat intheinfrared region
J
DT(p) is not consistent with the pole-type formula not only in the deconfinement phase but also
5
for T < Tc. We introduce new definition of the magnetic infrared mass scale (’magnetic screening
2
mass’) mM. The electric mass mE has been determined from the momentum space longitudinal
gluonpropagator. Westudyalsothe(finite)volumeandtemperaturedependenceofthepropagators
]
t as well as discretization errors.
a
l
- PACSnumbers: 11.15.Ha,12.38.Gc,12.38.Aw
p Keywords: Latticegaugetheory,gluonpropagator,finitetemperature,Gribovproblem,simulatedannealing
e
h
[ I. INTRODUCTION through the plasma (see, e.g., [3–7]). Also, the non-
perturbatively calculated lattice propagators are to
2
v be used to check the correctness of various analyti-
Oneofthemostinterestingfeaturesofthequantum
0 calmethodsinQCD,e.g.,Dyson–Schwingerequations
chromodynamics at finite temperature is the transi-
9 (DSE)method. Forstudyofthegluonpropagatorus-
7 tionfromtheconfinementtothedeconfinementphase. ing DSE approach at finite temperature see e.g. [8–
4 This transition separates a low-temperature phase,
10].
. which is expected to be highly non-perturbative and
1 The lattice study of the finite temperature SU(2)
1 characterized by quark and gluon confinement, from
gluon propagators in Landau gauge has been per-
0 a high-temperature - quark-gluon plasma (QGP) -
formed in a number of papers (see, e.g., papers [10–
1 phase,wherecolorchargesshouldbe deconfined. The
15]). In Ref. [13] the electric and magnetic propaga-
: conjecture of the existence of the QGP has been sup-
v tors were studied in both coordinate and momentum
i ported by recent observations of the collective effects
X spaces and in 4 and 3 dimensions. The conclusion
in ultrarelativistic heavy-ion collisions at SPS and
was made that the magnetic propagator had a com-
r RHIC (see, e.g., [2] and references therein).
a plicated infrared behavior which was not compatible
The non-perturbative - first principle - calculation with simple pole mass behavior. It was also found
of gauge variant gluon (as well as ghost) propagators that this propagator had strong volume and gauge
is of interest for various reasons. These propagators dependence. In Refs. [10, 15] results for both gluon
are expected to show different behaviorin eachphase and ghost propagators in momentum space were pre-
and,therefore,toserveasauseful’orderparameters’, sented, but some important questions, e.g. Gribov
detecting the phase transition point T . One expects copies effects, infrared behavior and scaling were not
c
that their study can shed the light on the mechanism addressed.
oftheconfinement-deconfinementtransition. Another In paper [1] it has been suggestedthat the proxim-
reason is that for the reliable phenomenological anal- ity of the Gribov horizon at finite temperature forces
ysis of high-energy heavy-ion collision data, it is im- the transverse gluon propagator DT(p~,p4 = 0) to
portant to obtain information on the momentum de- vanish at zero three-momentum. If this is the case,
pendence of the longitudinal (electric) gluon propa- then the finite-temperature analog of Gribov formula
gator D (p) and transverse (magnetic) gluon propa- |p~|2/(|p~|4+M4 ) ≡ 1/(|p~|2+m2 (p~)) suggests that
L M eff
gator D (p), especially in the (deep) infrared region. the effective magnetic screening mass m (p~) be-
T eff
One example is the study of the radiative energy loss comes infinite in the infrared (interpreted as a mag-
in dense nuclear matter (jet quenching) which results netic gluons ’confinement’).
from the energy loss of high energy partons moving TheGribovcopyeffectsstillremainoneofthemost
2
seriousproblemin the lattice calculations,atleast,in II. GLUON PROPAGATORS: THE
thedeepIR-region. Inourstudyweemploythegauge DEFINITIONS
condition which requires the Landau gauge fixing
functional F (see the definition in Section II) to take We study the SU(2) lattice gauge theory with the
extrema as close as possible to the global extremum. standard Wilson action
This choice for the gauge condition is supported by
thefollowingfacts: a)aconsistentnon-perturbative S =β 1− 1Tr U U U† U† ,
gauge fixing procedure proposed by Parrinello-Jona- Xx µX>ν(cid:20) 2 (cid:16) xµ x+µ;ν x+ν;µ xν(cid:17)(cid:21)
Lasinio and Zwanziger (PJLZ-approach)[16, 17] pre-
sumesthatthechoiceofauniquerepresentativeofthe where β = 4/g2 and g is a bare coupling constant.
gauge orbit should be through the global extremum 0 0
The link variables U ∈ SU(2) transform under
xµ
of the chosen gauge fixing functional; b) in the
gauge transformations g as follows:
x
case of pure gauge U(1) theory in the weak coupling
(Coulomb) phase some of the gauge copies produce
a photon propagator with a decay behavior inconsis- U 7→g Ug =g†U g ; g ∈SU(2). (1)
xµ xµ x xµ x+µ x
tent with the expected zero mass behavior [18–20].
The choice of the global extremum permits to obtain Our calculations were performed on the asymmetric
the physical - massless - photon propagator. lattices with lattice volume V = L ·L3, where L is
4 s 4
the number of sites in the 4th direction. The temper-
ature T is given by
Inaseriesofpapers[21–25]weinvestigatedtheGri-
bov copy effects in the Landau gauge gluon (and/or 1
ghost) propagators in zero temperature SU(2) gluo- T = , (2)
aL
4
dynamics. It has been demonstrated unambiguously
that these effects are strong in the infrared 1. Thus where a is the lattice spacing. We employ the stan-
theGribovcopyeffectsreductionisveryimportantfor dard definition of the lattice gauge vector potential
the infrared behavior studies. Recently it has been A [27]:
x+µˆ/2,µ
pointed out in Ref. [26] that lattice results for the
infrared gluon and ghost propagators free of Gribov
1 σ
copieseffectswouldhelptodiscriminatebetweenscal- A = U −U† ≡Aa a . (3)
ing and decoupling solutions of the Dyson-Schwinger x+µˆ/2,µ 2i (cid:16) xµ xµ(cid:17) x+µˆ/2,µ 2
equations. Inthis paperwe undertakeacarefulstudy
The Landau gauge fixing condition is
of the Gribov copy effects in the Landau gauge gluon
propagator at finite temperature. We employ the
gauge fixing procedure which we used recently in our 4
study at zero temperature [25] with changes dictated (∂A) = A −A =0, (4)
x x+µˆ/2;µ x−µˆ/2;µ
by nonzero temperature (see Section III). µX=1(cid:0) (cid:1)
which is equivalent to finding an extremum of the
gauge functional
Also we attempt to make a careful analysis of (fi-
nite) volume and temperature dependence of D and
T
D as well as of scaling violations. 1 1
L F (g)= Tr Ug , (5)
U 4V 2 xµ
Xxµ
Section II contains maindefinitions aswellas some with respect to gauge transformations gx . After re-
details of simulations and gauge fixing procedure we placingU ⇒Ug attheextremumthegaugecondition
use. SectionIIIisdedicatedtothestudyoftheGribov (4) is satisfied.
copy effects. Volume and temperature dependence of The (unrenormalized) gluon propagator Dµabν(p) is
thepropagatorsaswellasdiscretizationerrorsaredis- defined as follows
cussed in Section IV. Section V is dedicated to the
discussion of the screening masses and Section VI is
a2
reserved for conclusions and discussion. Dab(p)= hAa(k)Ab(−k)i
µν g2 µ ν
0
e e
where A(k) represents the Fourier transform of
1 Unfortunately, authors of [15] cite our paper [22] in a com- the gauge potentials defined in Eq.(3) after hav-
e
pletelymisleadingcontext. ing fixed the gauge, ki ∈ (−Ls/2,Ls/2] and k4 ∈
3
(−L /2,L /2]. The physical momenta p are given Inordertokeepfinite-volumeeffects undercontrol,
4 4 µ
by ap =2sin(πk /L ), ap =2sin(πk /L ). we considered a few different lattice volumes for each
i i s 4 4 4
Inwhatfollowsweconsideronlysoftmodesp =0. temperature. The choice of the 6×483 lattice at β =
4
The hard modes (p 6= 0) have an effective thermal 2.635isimportantforthecheckofthescalingbehavior
4
mass 2πT|k | and behave like massive particles2. (see Section IV).
4
As is well known, on the asymmetric lattice there
are two tensor structures for the gluon propagator β a−1[Gev] a[fm] L4 Ls T/Tc Nmeas Ncopy
[28] :
2.260 1.073 0.184 4 40 0.9 800 40
2.260 1.073 0.184 4 48 0.9 800 40
Dab(p)=δ PT (p)D (p)+PL(p)D (p) , (6) 2.300 1.192 0.165 4 26 1.0 1200 24
µν ab µν T µν L
(cid:0) (cid:1) 2.300 1.192 0.165 4 40 1.0 300 40
where (symmetric) orthogonalprojectorsPT;L(p) are 2.300 1.192 0.165 4 48 1.0 400 40
µν
defined at p=(p~6=0; p4 =0) as follows 2.350 1.416 0.139 4 16 1.1 2000 24
2.350 1.416 0.139 4 20 1.1 2000 24
2.350 1.416 0.139 4 26 1.1 1200 24
p p
PiTj(p) = (cid:18)δij − p~i2j(cid:19), PµT4(p)=0 ; (7) 2.350 1.416 0.139 4 32 1.1 800 40
2.350 1.416 0.139 4 40 1.1 800 40
PL(p) = 1 ; PL(p)=0. (8)
44 µi 2.350 1.416 0.139 4 48 1.1 800 40
Therefore, two scalar propagators - longitudinal 2.512 2.397 0.082 4 20 2.0 2000 24
D (p) and transverse D (p) - are given by 2.512 2.397 0.082 4 32 2.0 800 40
L T
2.512 2.397 0.082 4 40 2.0 800 40
2.512 2.397 0.082 4 48 2.0 800 40
3 3
1
D (p) = Daa(p) ; 2.635 3.596 0.055 6 40 2.0 800 40
T 6Xa=1Xi=1 ii 2.635 3.596 0.055 6 48 2.0 800 40
3
1
DL(p) = 3 D4a4a(p), TABLE I: Values of β, lattice sizes, temperatures, num-
Xa=1 ber of measurements and number of gauge copies used
(9) throughoutthispaper. Tofixthescale wetake√σ=440
MeV.
Forp~=0propagatorsD (0)andD (0)aredefined
T L
as follows
For gauge fixing we employ the Z(2) flip operation
as has been proposed in [23]. It consists in flipping
3 3 alllinkvariablesU attachedandorthogonaltoa3d
1 xµ
DT(0) = 9 Diaia(0), plane by multiplying them with −1.
Xa=1Xi=1 Such global flips are equivalent to non-periodic
3 gaugetransformationsandrepresentanexactsymme-
1
D (0) = Daa(0). (10) try of the pure gauge action. The Polyakov loops in
L 3 00
Xa=1 thedirectionofthechosenlinksandaveragedoverthe
3dplaneobviouslychangetheirsign. Atfinitetemper-
The transverse propagator D (p) is associated to
T ature we apply flips only to directions µ = 1,2,3. In
magnetic sector, and the longitudinal one D (p) - to
L the deconfinement phase, where the Z(2) symmetry
electric sector.
is restored, the Z(2) sector of the Polyakov loop in
Wegeneratedensemblesofuptotwothousandinde- the µ = 4 direction has to be chosen since on large
pendentMonteCarlolatticefieldconfigurations. Con- enough volumes all lattice configurations belong to
secutive configurations (considered as independent) the same sector, i.e. there are no flips between sec-
were separated by 100 (for L < 32) or 200 (for tors. We choose sector with positive Polyakov loop.
s
L ≥ 32) sweeps, each sweep being of one local heat- In the confinement phase one may use a flip in the
s
bathupdatefollowedbyL /2microcanonicalupdates. µ=4direction. However,inatestrunwehavefound
s
In Table I we provide the full information about the that at β = 2.26 studied in this paper the maximal
field ensembles used throughout this paper. gaugefixing functional(5) hasbeen foundin the pos-
itive Polyakov loop sector in more than 90 % cases.
To save computer time we stick to this sector for all
configurations at this β. Therefore, in our study the
2 Letusnotethat the4theuclidiancomponent p46=0hasno flip operations combine for each lattice field configu-
physicalmeaning. ration the 23 distinct gauge orbits (or Polyakov loop
4
sectors)ofstrictlyperiodicgaugetransformationsinto
one larger gauge orbit.
Following Ref.[25] in what follows we call the com-
10 ββ==22..33
bined algorithm employing simulated annealing (SA)
algorithm (with finalizing overrelaxation) and Z(2) 9
flips the ‘FSA’ algorithm. For every configuration
8
the Landau gauge was fixed Ncopy = 24(40) times -2v
(3(5) gauge copies for every flip–sector) on lattices e 7
G
with Ls ≤ 26(Ls ≥ 32), each time starting from a p); 6
random gauge transformation of the mother config- D(
uration, obtaining in this way Ncopy Landau-gauge 5
fixed copies. We take the copy with maximal value of 804; OR; fc
the functional (5) as our best estimator of the global 4 804; SA; fc
maximum and denote it as best (“bc ”) copy. In or- 3 444; FSA; bc
dertodemonstratetheGribovcopyeffectwecompare
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
with the results obtained from the randomly chosen
|p|; Gev
first (“fc ”) copy.
We present results for the unrenormalized propa-
gators and make comments on renormalization when FIG.1: ComparisonofdataobtainedforbcFSAgaugefix-
ingwiththoseobtainedwiththestandardfcOR-method
propagatorscomputedatdifferentvaluesofβarecom-
and thefc SA algorithm (thisFigure from paper [25])
pared or comparison with the renormalized results of
other groups is necessary. Other details of our gauge
fixing procedure are described in our recent papers
bc transverse propagators calculated for every config-
[23–25]
uration, this average being normalized to the bc (av-
To suppress ’geometrical’ lattice artifacts, we have
eraged) transverse propagator. In a similar way one
applied the “α-cut” [29], i.e. πk /L < α , for every
i s candefinealsothenormalizeddifferenceofthe fc and
component, in order to keep close to a linear behav-
bc longitudinal propagators∆ (p).
ior of the lattice momenta p ≈ (2πk )/(aL ), k ∈ L
i i s i The longitudinal propagators D (p) demonstrate
(−L /2,L /2]. We have chosen α = 0.5. Obviously, L
s s very weak dependence on the choice of Gribov copy.
this cut influences large momenta only. We did not
In Figure 2 we show the momentum dependence of
employ the cylinder cut in this work.
∆ (p) on the 4×483 lattice at β = 2.35, 2.512 and
L
β = 2.26, i.e. at temperatures both above and below
transition. One can see that values of ∆ (p) are con-
III. GRIBOV COPY EFFECTS AND LARGE L
sistent with zero for all β-values shown in the figure.
Ls BEHAVIOR OF DT(0)
This was observed on all other lattices employed in
our study.
Ashasbeenalreadypointedabove,theGribovcopy
problem still remains acute, at least, in the deep in- In contrast, the Gribov copy dependence of the
frared region and the choice of the efficient gauge fix- transverse propagator is rather strong. In Figure 3
ingmethodisveryimportant. Theimportanceofthis weshowthemomentumdependence of∆T(p)onvar-
choiceisdemonstratedinFig.1takenfromourrecent ious lattices at β = 2.35 (T/Tc = 1.1). One can see
paper [25]. In this Figure we compare our bc FSA re- that for fixed physical momentum p the effect of Gri-
sultsforthebaregluonpropagatorD(p)calculatedon bov copies tends to decrease with increasing volume.
a444 latticewiththoseofthestandardfc ORmethod Suchbehaviorisinagreementwiththe absenceofthe
obtainedforan804 latticeandalsowiththe fc SAre- Gribovcopieseffects(withinGribovregion)inthein-
sults. In particular, we observe that the OR method finite volume limit, suggested by Zwanziger [30]. On
with one gauge copy produces completely unreliable the other hand, on given lattice there are always 3 or
results for the range of momenta |p|<∼0.7 GeV. (The 4minimalvaluesofmomentumforwhichtheseeffects
detailed discussion can be found in [25]) are substantial. In particular, ∆T(p) varies between
0.35 and 0.55 for p = 0, between 0.09 and 0.12 for
Letusdefinethenormalizeddifferenceofthefc and
|p| = p ≡ (2/a)sin(π/L ) and between 0.03 and
min s
bc transverse propagators∆ (p) :
T 0.04 for momentum next after p . Similar observa-
min
tions were made at zero temperature as well [25].
Dfc(p)−Dbc(p) In Figure 4 we show the parameter ∆T(p) for two
∆T(p)= T Dbc(p)T , (11) temperatures, T/Tc = 1.1 and T/Tc = 2, on lattices
T with approximately equal physical volumes. One can
where the numerator has been obtained by averaging see that there is rather weak (if any) dependence of
overallconfigurationsofthedifferencebetweenfcand the Gribov copy effects on the temperature T.
5
0.50
0.4
3 0.45 β=2.350; 4·263
0.3 4·48 0.40 β=2.512; 4·483
0.35
0.2
0.30
0.1 p) 0.25
p) ∆(T0.20
∆(L0.0 0.15
-0.1 0.10
0.05
-0.2 β=2.35
0.00
β=2.512
-0.3 β=2.26 -0.05 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
|p|; Gev
-0.4
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
|p|; Gev FIG.4: Themomentumdependenceof∆T(p)on twolat-
tices with approximately equal physicalvolume.
FIG. 2: The momentum dependence of ∆L(p) for three
temperatures.
of Dbc(0) and Dfc(0) demonstrate a tendency to de-
T T
crease; moreover, our data, especially for T = 2T ,
c
suggestthat Dbc(0) andDfc(0) seemto (slowly)con-
T T
vergein the limit L →∞. This convergenceis again
s
0.55 β=2.35 L=16 in accordance with a conjecture made by Zwanziger
0.50 s
0.45 Ls=20 in [30] and in accordance with the zero-temperature
L=26
0.40 s case studied numerically in [22, 25].
L=32
∆(p)T000...323055 LLsss==4408 isnOonnt-zheerootihnerthheanindfi,noiuterdvaotluamalesolimsuigtgfeosrtbthotahtDvaTlu(0es)
0.20 of T, in disagreement with the suggestion made in
0.15 [1]. There is no indication for a vanishing transverse
0.10 propagatorat zero momentum for increasing volume,
0.05
similarlytothesituationforthezero-temperaturecase
0.00
-0.05 [22,25]. ThisisinagreementwiththerefinedGribov-
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Zwanziger formalism [31, 32].
|p|; Gev
Let us note that one still cannotexclude that there
areevenmoreefficientgaugefixingmethods,superior
FIG.3: Themomentumdependenceof∆T(p)atβ =2.35 to the one we use, which could make this decreasing
on various lattices.
more drastic.
The comparison of the results for ∆ (p) obtained
T
on lattices 4·323 at β = 2.512 and 6×483 at β = IV. VOLUME AND TEMPERATURE
2.635 (both corresponding to the same temperature DEPENDENCE AND FINITE SPACING
T/T = 2 and the same physical volume) shows that EFFECTS.
c
the Gribov copy effects depend weakly on the lattice
spacing a. A. Volume dependence
Let us note that results for ∆ (p) discussed above
T
are obtained with the gauge fixing algorithmwe have
As is well-known, the finite volume dependence is
chosen, i.e., FSA. The value of ∆ (p) will be essen-
T very strong near the second order phase transition
tially higher ifone uses the ORalgorithmto compute
point T . For the transverse propagator D (p) this
the fc propagatorDTfc(p). dependecnce can be seen from Figure 6 andT for the
In Figure 5 we show the 1/aLs dependence and longitudinal propagator DL(p) from Figure 7, both
Gribov copy sensitivity of the zero-momentum trans- calculated at β = 2.3 (slightly above T ) on various
c
verse propagator D (0) in the deconfinement phase lattices.
T
(for T = 1.1T and for T = 2T ). The difference be- Deeper inside inthe deconfinementphase the finite
c c
tween bc values (filled symbols) and fc values (open volume effects are much less pronounced, at least at
symbols) is rather big, as has been already discussed non-zero values of momentum. In Figure 8 we show
above. However, with increasing size L the values the momentum dependence of the transverse propa-
s
6
20
3.5 β=2.35; bc L=4 18 β=2.3
β=2.35; fc 4 16 4·263
3.0 β=2.512; bc 14 4·403
2.5 β=2.512; fc -2ev 12 4·483
-20); Gev 2.0 D(p); GL108
D(T1.5 6
4
1.0
2
0.5 0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
|p|; Gev
0.0
0 0.02 0.04 0.06 0.08 0.1 0.12
1/aL; Gev FIG. 7: The momentum dependence of the longitudinal
s
propagators DL(p) on various lattices at β=2.3.
FIG.5: The1/aLs dependenceandGribov copysensitiv-
ity of the transverse propagator DT(0) at β = 2.35 (cir-
cles) and β = 2.512 (squares). Values of Ls are given in
Table I. Filled symbols correspond to the bc ensemble,
open symbols to the fc ensemble. The lines are to guide 2.2 bc 4·203; β=2.35
the eye. 2.0
4·323; β=2.35
1.8
4·403; β=2.35
1.6
-2v 1.4 4·483; β=2.35
Ge 4·203; β=2.512
2.2 β=2.3 p); 11..02 4·323; β=2.512
2.0 4·263 D(T0.8 4·403; β=2.512
-2ev 111...468 44··440833 00..46 4·483; β=2.512
G
(p); T11..02 00..02
D 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.6 |p|; Gev
0.4
0.2
FIG. 8: The momentum and volume dependence of the
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 transverse propagators DT(p) at β=2.35 and β=2.512.
|p|; Gev
FIG. 6: The momentum dependence of the transverse maximumofthetransversepropagatorD (p)atnon-
T
propagators DT(p) on various lattices at β=2.3. zero momentum p with |p | ∼ 0.4÷0.5 Gev. Deep
0 0
in the deconfinement phase the maximum has been
found before [10, 13, 15]. But we observed it for the
gatorsDT(p)onvariouslattices in the deconfinement firsttime atT =Tc andT <Tc asonecanseeinFig-
phase at T = 1.1Tc and T = 2Tc. Apart from the ure10. Inourrecentpapers[24,25]wereportedabout
strongvolume dependence atp=0,we see thatfinite the existence of the maximum of the scalar propaga-
volume effects are rather weak, the sizable effects are tor D(p) at non-zero momentum on the symmetric
seen only for the minimal non-zero momentum pmin lattices (zero-temperature case) when lattice size is
on a given lattice. big enough. Therefore, we conclude that the trans-
The volume dependence of the longitudinal propa- verse propagator D (p) has its maximum at p 6= 0
T 0
gators D (p) at T = 1.1T and at T = 2T is pre- for all temperatures, and the behavior of the trans-
L c c
sentedinFigure9. Evidently,thevolumedependence versepropagatorinthedeepinfraredisnotconsistent
ofD (p)isevenweakerthanthatofD (p),itisrather with the simple pole-type behavior both in confine-
L T
weak even at p = 0. Note that at T = 1.1T D (0) ment and deconfinement phases.
c L
slowly increases with increasing volume, contrary to ItisinstructivetocompareourresultsforD (p)at
T
decreasing of DT(0). T = 1.1Tc on Ls = 48 lattices with respective results
For every lattice we observed a well-pronounced of Ref. [15] obtained at this temperature on the lat-
7
2.2
3.5 bc 4·203; β=2.35 2.1 4·483
3.0 4·323; β=2.35 21..09 ββ==22..3350
4·403; β=2.35 1.8 β=2.26
-2v 2.5 4·483; β=2.35 -2Gev 1.7
D(p); GeL21..05 444···234020333;;; βββ===222...555111222 D(p); T1111....4635
1.0 4·483; β=2.512 1.2
1.1
0.5 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
|p|; Gev
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
|p|; Gev FIG. 10: The momentum dependence of the transverse
propagators DT(p) near Tc on the4 483 lattices.
×
FIG. 9: The momentum dependence of the longitudinal
propagators DL(p) on various lattices at β = 2.35 and
β =2.512.
ThusthetransversegluonsinLandaugaugearenot
tices with L =4 and L =46 and presented in their directly related to confinement [10, 33].
4 s
Figure 1. To make this comparison we made renor-
malizationatµ=2GeVasitwasmadein[15]. Inthe In contrast, the longitudinal propagator DL(p)
infraredregionwefoundbothqualitativeandsubstan- demonstrates a drastic jump in its values in the in-
tialquantitativedisagreementbetweenourresultsand frared when the critical temperature is crossed from
results of Ref. [15]. In particular, the clear maximum above(seeFigure11). Therespectiverenormalization
which we see in our Figure 8 at T = 1.1T can not constantscomputedatµ=2GeVdifferbyafewper-
c
be seen from Figure 1 of Ref. [15]. These differences cent in this case and this difference has a tendency
betweenour resultsandresults ofRef. [15],whichare to decrease with increasing µ. The difference in the
justGribovcopieseffects, aremoreessentialthandif- renormalizationconstantsonlyslightlyaltersthetem-
ferences between our bc and fc results discussed in perature dependence of DL(p) after renormalization.
Section III. Therefore,DL(p)atsmallmomentacanbeconsidered
IncontrastwithD (p),thelongitudinalpropagator as an order parameter signaling the phase transition.
T
D (p)doesnotshowanytraceofmaximumat|p|6=0
L
both above and below Tc, as one can see in Figure 9 Note that the study of the related quantity ∆A2 ≡
hg2A2 − g2A2 i suggests that in the vicinity of T
and Figure 11 in agreement with results of Refs. [10, E M c
thetemperaturedependenceofD (p)canhaverather
15]. This gives an idea that it can be fitted by the L
nontrivial (non-monotonous) character [34]. Further
pole-type behavior (see Section V). The pole-type
studies at T very close to T are necessary to clar-
behavior at high temperature is not surprising since c
ify this issue. Let us note also that for the first
athighenoughtemperaturetheeffectivetheoryisthe
time the fast change of the longitudinal propagator
Higgs 3d theory with A playing a role of the Higgs
4
nearthe transitionpointhasbeenobservedin[33](in
field.
the SU(3) case). This fast change has been recently
demonstrated in Ref. [15] both in SU(2) and SU(3)
B. Temperature dependence theories.
The temperature dependence of the transverse Deep into the deconfinement phase the decreasing
propagator D (p) near the critical point T is very of the transverse propagator D (p) at p ∼ 0 with in-
T c T
smooth. Figure 10 makes comparison of the momen- creasingtemperature(see Figure8)is inaqualitative
tum dependence of D (p) for three temperatures : agreementwithdimensionalreductionsinceaccording
T
T = 0.9T , T = T and T = 1.1T . Indeed, there to dimensional reduction at high temperature D (p)
c c c T
is no sign of sensitivity to the phase transition. It is istobeproportionalto(g2(T)T)−2. Thequantitative
worthwhile to note that the effect of renormalization agreementisnotyetexpectedattemperaturesconsid-
do not alter this conclusion since the renormalization eredhere. Similarly,theelectricpropagatorD (p)for
L
constants computed at µ= 2 GeV differ by less than smallmomentadecreasesfastwithincreasingtemper-
0.5%. ature, see Figure 9.
8
24 4·483 1.0
2202 β=2.35 0.9 6·483; β=2.635; bc
18 β=2.30 0.8 4·323; β=2.512; bc
β=2.26
-2v 16 0.7
p); Ge 1124 -2Gev 0.6 T/Tc=2
(DL108 (p); L00..45
6 D
4 0.3
2 0.2
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1
|p|; Gev
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
FIG. 11: The momentum dependence of the longitudinal |p|; Gev
gluon propagators DL(p) near Tc on the4 483 lattices.
×
FIG. 13: The momentum dependence of the longitudinal
propagator DL(p) on two different lattices corresponding
to thesame temperature and thesame physical volume.
6·483; β=2.635; bc smaller spacings) as compared to that employed in
1.00
4·323; β=2.512; bc Ref. [10]. This explains the fact that the discretiza-
tion effects we find are much smaller than in that pa-
0.80
-2v T/T=2 per.
e c
G Contrary to the transverse propagator, the data
p); 0.60 for the longitudinal propagator DL(p) show substan-
D(T tial scaling violations in the infrared. This can be
0.40
seen from Figure 13 where propagators D (p) com-
L
puted on the same lattices as used in Figure 12 are
0.20
depicted. The renormalization constants computed
at µ = 2 Gev differ only by 4% so the scaling vio-
0.00
0.0 0.5 1.0 1.5 2.0 2.5 lations do not disappear after renormalization. Evi-
|p|; Gev dently, the discretization errorsare large at momenta
smallerthan1GeV.Toreducethefinitecut-offeffects
FIG. 12: The momentum dependence of the transverse one should increase L4 or use improved lattice action
propagator DT(p) on two different lattices corresponding (see, e.g. [12]).
to thesame temperature and thesame physicalvolume. The results of our study of the scaling behavior at
zerotemperature[25]suggestthatforsmallervaluesof
β (i.e., β =2.35, β =2.3 and β =2.26)discretization
C. On the discretization errors errors are substantial for both propagators.
To estimate discretization errors at T = 2T we
c
calculatedtransverseandlongitudinalpropagatorson V. ON THE SCREENING MASSES
two differentlatticescorrespondingtothe samephys-
ical 3d volume (aL )3 but with different lattice spac- One of the interesting features of the high temper-
s
ing. ThesearelatticesL =4,L =32,β =2.512and ature phase is the appearance of the infrared mass
4 s
L =6,L =48,β =2.635. In Figure 12 we show the scale parameters : m (’electric’) and m (’mag-
4 s E M
momentum dependence of the transverse propagator netic’). These parameters (or ’screening masses’) de-
D (p) for these two lattices. One cansee goodagree- fine screening of electric and magnetic fields at large
T
mentbetweenresultsobtainedontheselatticesforall distances and, therefore, control the infrared behav-
included momenta. This implies that at T = 2T the ior of D (p) and D (p). The electric screening mass
c L T
discretization effects are small even on lattices with m has been computed in the leading order of per-
E
L = 4. We expect that this is true also at higher turbation theory long ago: m2 = 2g2T2 for SU(2)
4 E 3
temperatures. gluodynamics. But at the next order the problem of
Let us note that at this temperature we have theinfrareddivergencieshasbeenfound. Ontheother
employed larger L -values and larger β-values (i.e., hand,themagneticmassm isentirelynonperturba-
4 M
9
tiveinnature. Thusafirst-principlesnonperturbative
calculations in lattice QCD should play an important
role in the determination of these quantities.
As has been already mentioned above, the momen- 0.7 β=2.35; bc
β=2.26; bc
tumdependenceofthelongitudinalpropagatorD (p)
L 0.6
in the deep infrared is expected to fit the pole-type
behavior. Indeed, as an illustration, in Figure 14 we 2v 0.5
e
show the momentum dependence of the inverse prop- G
vaoglautmoreDeffL−e1c(tps)aarteTsm=al0l.e9nTocuagnhd,aTt=lea1s.t1,Tact. pSi6=nc0e,twhee D(p); L00..43
use data obtainedonalllattices listedin Table I with 1/
0.2
exception for p=0. For this momentum we included
data for the largest lattice only. 0.1
Onecanseethatatsmallmomentathedependence
on p2 is linear. Thus in the infrared region we have 0.0
0.0 0.1 0.2 0.3 0.4 0.5
used the fitting formula |p|2; Gev2
D−1(p)=A·(p2+m2). (12) FIG.14: Themomentumdependenceoftheinverselongi-
L E
tudinalpropagator 1/DL(p) at β =2.35 and β =2.26.
The results of the fits are presented in the Table II.
At T = 2T we can compare our results with results
c
of Ref.[12] shown in their Figure 3. We find that mE has been used recently in Ref. [25] to fit the T = 0
computed on lattices with L4 = 4 is in good agree- gluon propagator in the infrared region of momenta.
ment with respective result of Ref.[12]. For finer lat- In(13)m isamassiveparameter,|p |ismomentum
M 0
ticespacing(L4 =6)ourvalueisonlyslightlysmaller shiftandC a normalizationconstant. Another fitting
than the value for L4 = 4 (see Table II) indicating function is shifted pole propagator of the form
small scaling deviations, while in Ref.[12] the value
obtained on the lattice with L =8 was substantially
4
C
higher. We believe that due to simplicity of the fit- f (p)= (14)
ting function (12) extracting of mE from DL(p) al- P (m2M +(|p|−|p0|)2)
lows more precise measurement of this quantity than
(we keep same notations for fitting parameters). The
its determination from the correlator D (z) used in
E
zero momentum was excluded from the fitting range,
Ref.[12], .
the maximal momentum was determined by require-
In Table II we also show the maximal momenta
mentthatrespectiveχ2/N wassmallerthan1. We
p included into a fit. This value was defined by dof
max
found that both fits work well in the infrared with
conditionthattheχ2/dofvalueforthefitwassmaller
better performance (larger range) for the fit function
than 2. One can see that p increases with T.
max
(14).
β T/Tc mE[Gev] mE/T p2max[GeV2]
2.260 0.9 0.41(1) 1.53(4) 0.15 β T/Tc mM[Gev] mM/T pmax[GeV] |p0|[Gev]
2.350 1.1 0.56(1)(4) 1.59(3)(12) 1.3 0.40(1)
2.300 1.0 0.46(1) 1.54(4) 0.10
2.512 2.0 0.78(1)(7) 1.30(2)(11) 1.3 0.51(1)
2.350 1.1 0.73(2) 2.06(6) 0.30
2.512 2.0 1.21(2) 2.02(4) 1.0
2.635 2.0 1.15(3) 1.92(6) 1.0 TABLE III: Values of the mass parameter mM obtained
from fitsto eq.(14) and maximal momenta used in thefit
pmax.
TABLEII:ValuesofthescreeningmassmE obtainedfrom
fitstoeq.(12)andmaximalmomentausedinthefitpmax.
InTableIII we showfitting parametersforfitfunc-
tion eq. (14). The second error for m is the differ-
In contrast,the transverse propagatorD (p) has a M
T
ence from result for fit to eq. (13) in which case m
form which is not compatible with the simple pole- M
wasbigger. Thedifferenceinvaluesof|p |fortwofits
type behavior,so form another,differentfrompole 0
M
was less than 1%.
mass, definition is necessary. We applied two fitting
We can compare our value for m at T/T = 2
functions to our data for D (p). One of them, Gaus- M c
T
with result from Ref.[12] presented in Figure 4. In
sian function with shifted argument
that paper resultfor T/T =2 was obtainedon 322×
c
64×8 lattice. They found m /T =2.0(3), i.e. much
M
f (p)=Ce−(|p|−|p0|)2/m2M (13) higher than our value. Apart from difference in the
G
10
definitionofm thisdeviationmightbeexplainedby the deconfinement phase but also for T ≤ T . Thus
M c
theGribovcopyeffect,whichismuchstrongerform we confirmed that there is no possibility to explain
M
thanform . Weneedtomakecomputationsathigher the IR-behavior of the transverse D (p) gluon prop-
E T
temperatures to make more detailedcomparisonwith agator on the basis of a simple pole-type behavior
results of Ref.[12]. ∼ 1/(p2 + m2). Instead we fitted this propagator
to fitting functions eq.(13) and eq.(14) with massive
parameter m . m /T is slowly decreasing with in-
M M
VI. CONCLUSIONS creasing temperature. To check if this decreasing is
compatiblewithgT behavior,asexpectedforthemag-
Inthis workwestudiednumericallythe behaviorof neticscreeningmass,aswellastocompareourresults
the Landau gauge longitudinal and transverse gluon forthis parameterwithresultsformagneticscreening
propagatorsinpuregaugeSU(2)latticetheoryinthe mass, obtained by other authors, we need to repeat
infrared region of momentum values. The special ac- our computations at higher temperatures.
cent has been made on the study of the dependence
ForD (p)wefoundgoodagreementwith’pole-like’
of these ’observables’ on the choice of Gribov copies. L
behavior at small enough momenta p < p with
The simulations have been performed using the max
p increasing with T. Our value for m at T =
standard Wilson action at temperatures from 0.9T max E
c 2T agrees well with result from [12] obtained also on
up to 2T on lattices with L = 4 and spatial linear c
c 4 latticeswithL =4. Again,weneedresultsathigher
sizes up to L= 48. For T =2T simulations were re- 4
c temperatures to compare with other results and with
peatedonlatticeswithL =6. Forgaugefixinggauge
4 the perturbation theory predictions. We shall note
orbits enlarged by Z(2) flip operations were consid-
that our method of computing m in the momentum
ered with up to 5 gauge copies in every flip-sector (in E
spaceratherthaninthecoordinatespacegivesriseto
total, up to 40 gauge copies). The maximization of
higher precision.
the gauge functional was achieved by the simulated
annealing method always combined with consecutive Away from the transition temperature the longi-
overrelaxation. tudinal propagators D (p) demonstrate very weak
L
Our findings can be summarized as follows. volume dependence. The volume dependence of the
transverse propagators D (p) is strong at p = 0 and
Similarly to the gluon propagator at T = 0, the T
it is weak at p>0.
Gribov copy dependence of the transverse propaga-
tors D (p) is very strong in the infrared, more pre-
T WefoundverysmallscalingviolationsforD (p)at
T
cisely, at a few minimal (for given lattice) momenta.
T = 2T comparing results obtained on lattices with
c
At the same time for fixed physical momentum p the
L =4and6. Inopposite,forD (p)scalingviolations
4 L
effect of Gribov copies decreases with increasing vol-
in the infrared are substantial.
umeinagreementwith[30]. Wefoundnodependence
oftheGribovcopieseffectsonthetemperatureorlat- We confirmed the observation made in Ref. [15]
tice spacing. that the longitudinal propagator D (p) in the in-
L
The Gribov copy dependence of the longitudinal fraredincreasesfastwhentemperature crossestransi-
propagatorsD (p) is very weak, at least, at non-zero tion from above. From results presented in Ref. [15]
L
momenta, and is comparable with the statistical er- for T < T it is clear that D (p) has a maximum
c L
rors (so called ’Gribov noise’). nearto T . This isalsoinagreementwithfindings for
c
We havetoemphasize thatour conclusionsforGri- ∆A2 ≡hg2A2E −g2A2Mi [34].
bov copies effects are relevant for our gauge fixing al-
gorithm and they might change if another, less effi-
cient, algorithm is used.
Acknowledgments
WithincreasingsizeL the bc -valuesofD (0)and
s T
fc -values of D (0) demonstrate a tendency to de-
T
Thisinvestigationhasbeenpartlysupportedbythe
crease;moreover,Dbc(0) andDfc(0)seem to (slowly)
T T Heisenberg-Landauprogramofcollaborationbetween
converge in the limit L →∞ which is in accordance
s the Bogoliubov Laboratory of Theoretical Physics of
withaconjecturemadebyZwanzigerin[30]andinac-
the JointInstitute for NuclearResearchDubna (Rus-
cordance with the zero-temperature case studied nu-
sia) and German institutes, partly by the Federal
merically in [22, 25]. However, D (0) is non-zero in
T Special-Purpose Programme ’Cadres’ of the Russian
the infinite volume limit, in disagreement with the
Ministry of Science and Education and partly by the
suggestion made in [1].
grant for scientific schools NSh-6260.2010.2. VB is
We observed the existence of the maximum of the supported by grants RFBR 09-02-00338-aand RFBR
D (p) at momenta |p| ∼ 0.4÷0.5 Gev not only in 08-02-00661-a.
T
