Electric and Magnetic Screening Masses around the Deconfinement Transition

0.15 MB·English

by  Attilio Cucchieri

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Electric and Magnetic Screening Masses around the
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2 Deconfinement Transition

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Attilio Cucchieriab, TerezaMendes∗a
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p aInstitutodeFísicadeSãoCarlos,UniversidadedeSãoPaulo,
e CaixaPostal369,13560-970SãoCarlos,SP,Brazil
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bGhentUniversity,DepartmentofPhysicsandAstronomy,
[
Krijgslaan281-S9,9000Gent,Belgium
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E-mail: [email protected],[email protected]
v
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0 We reportonthe status of ourstudy of gluonpropagatorsand screeningmasses aroundthe de-
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confiningtransitionforpureSU(2)gaugetheoryinLandaugauge.
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TheXXIXInternationalSymposiumonLatticeFieldTheory-Lattice2011
July10-16,2011
SquawValley,LakeTahoe,California
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ScreeningMassesaroundDeconfinement TerezaMendes
1. Introduction
Debye screening of the color charge, expected at high temperature, is signaled by screening
masses/lengths that can inprinciple be obtained from the gluon propagator [1]. Morespecifically,
chromoelectric (resp. chromomagnetic) screening will be related to the longitudinal (resp. trans-
verse) gluon propagator computed at momenta with null temporal component (soft modes). In
particular, we expect the real-space longitudinal propagator to fall off exponentially at long dis-
tances, defining a(real) electric screening mass, which canbe calculated perturbatively to leading
order. Also, according to the 3d adjoint-Higgs picture for dimensional reduction, we expect the
transversepropagatortoshowaconfiningbehavioratfinitetemperature, inassociationwithanon-
trivialmagneticmass(seee.g.[2]). Wenotethatthesepropagatorsaregauge-dependent quantities,
and the (perturbative) prediction that the propagator poles should be gauge-independent must be
checked, byconsidering different gauges.
Even though the nonzero-T behavior just described has been verified for various gauges and
established at high temperatures down to around twice the critical temperature T [2, 3], it is not
c
clearhowascreening masswoulddevelop around T . Atthesametime,latticestudies ofLandau-
c
gauge gluon propagators at finite temperature in pure SU(2) and SU(3) theory have observed a
sharp peak in the infrared value of the electric propagator around the deconfinement temperature,
suggesting an alternative order parameter for the QCD phase transition [4, 5, 6, 7, 8]. (Ofcourse,
a relevant question is, then, whether this singularity survives the inclusion of dynamical quarks
in the theory [9].) In the following, we investigate the critical behavior of electric and magnetic
gluon propagators and try to characterize the screening masses around the transition temperature
T by performing large-lattice simulations in pure SU(2) gauge theory. In particular, we use the
c
knowledge gained inthestudy ofthezero-temperature gluon propagator (see[10]forareview)to
identifysystematiceffectsintheinfraredlimitandtodefinetemperature-dependent massesforthe
regionaroundandbelowT . Amoredetailedanalysisofourdatawillbepresentedelsewhere[11].
c
(Preliminary resultswerereported in[12,13].)
2. Results
We have considered the pure SU(2) case, with a standard Wilson action. For our runs we
employacoldstart,performingaprojectiononpositive-Polyakov-loopconfigurations. Also,gauge
fixing is done using stochastic overrelaxation and the gluon dressing functions are normalized to
1 at 2 GeV. We take b values in the scaling region and lattice sizes ranging from N = 48 to
s
192 and from N = 2 to 16 lattice points, respectively along the spatial and along the temporal
t
directions. Our (improved) procedure for determining the physical temperature T is described
in [13]. The momentum-space expressions for the transverse and longitudinal gluon propagators
D (p)andD (p)canbefounde.g.in[4]. Wefirstdescribeourinvestigationofthecriticalbehavior
T L
for transverse and longitudinal gluon propagators and then discuss our proposal for computing
screening massesaroundT .
c
Ourdataforthetransverse (magnetic)propagator D (p)atthecriticaltemperature areshown
T
in Fig. 1. We clearly see the strong infrared suppression of the propagator, as expected, with a
turnover at around 400 MeV. Regarding systematic errors, there are considerable finite-physical-
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ScreeningMassesaroundDeconfinement TerezaMendes
6
3
T 48 X 4 2.299 (0.17, 8.2) fm
c
3
T 96 X 4 2.299 (0.17, 16.3) fm
c
3
T 192 X 4 2.299 (0.17, 32.6) fm
c
5 3
T 72 X 6 2.427 (0.11, 7.9) fm
c
3
T 144 X 6 2.427 (0.11, 15.8) fm
c
3
T 96 X 8 2.511 (0.08, 7.7) fm
c
3
4 T 192 X 16 2.740 (0.04, 7.7) fm
c
2)
-V
Ge 3
(T
D
2
1
0
0 0.5 1 1.5 2 2.5
p (GeV)
Figure 1: Transverse gluon propagator at T , for various choices of lattice size and b . Values for the
c
temperature, N3×N, b ,latticespacingaandspatiallatticesizeL(bothinfm,inparentheses)aregivenin
s t
theplotlabels. Thesolidlinesarefits,describedattheendofthissection.
size effects, evidenced by the fact that the lattices with the smaller physical spatial size (the red,
yellow,magentaandblackcurves)showdifferentbehaviorfromtheremainingcurves(green,blue
andcyan). ThenatureoftheseeffectsissimilartowhatisobservedatT =0[10]. Wementionthat
essentially thesamefeatures areseenforD (p)atothertemperatures, aboveandbelowT [13].
T c
The longitudinal (electric) propagator D (p) at T is shown in Fig. 2. We immediately see
L c
severe systematic effects for the smaller values of N. Let us note that our runs were initially
t
planned under the assumption that a temporal extent N = 4 might be sufficient to observe the
t
infraredbehaviorofthepropagatorsandourgoalwas,then,toincreaseN significantly,tocheckfor
s
finite-sizeeffects. AsseeninFig.2,thisassumption isnotjustifiedforthelongitudinal propagator
around the critical temperature, especially in the case of larger N . Indeed, as N is doubled from
s s
48to96andthento192,weseethattheinfraredvalueofD (p)changesdrastically, resulting ina
L
qualitatively different curve at N =192, apparently with a turnover in momentum. (Note that, in
s
this case, thereal-space longitudinal propagator manifestly violates reflection positivity.) Wetook
thisasanindication thatourchoiceofN =4wasnotvalidandtherefore considered larger values
t
ofN. WeassumeherethatdatapointsatN =16areessentiallyfreefromsystematiceffects,since
t t
(asshownin[13])thecurvesfortemperatures aroundT stabilizeforN >8. Asseeninthefigure,
c t
weobtain inthiswayadifferent picture forthecritical behavior ofD (p). Itisinteresting tonote
L
(seeFig.2)thattheN effectsatT aresignificantforN =6(withoppositesignwithrespecttothe
s c t
N =4case)andarestillpresentforN =8(andmaybealsoforN =16). Thisisalsotrueslightly
t t t
3
ScreeningMassesaroundDeconfinement TerezaMendes
40
3
T 48 X 4 2.299 (0.17, 8.2) fm
c
3
T 96 X 4 2.299 (0.17, 16.3) fm
c
35 T 1923 X 4 2.299 (0.17, 32.6) fm
c
3
T 72 X 6 2.427 (0.11, 7.9) fm
c
3
T 144 X 6 2.427 (0.11, 15.8) fm
30 c
3
T 96 X 8 2.511 (0.08, 7.7) fm
c
3
T 192 X 16 2.740 (0.04, 7.7) fm
c
25
2)
-V
Ge 20
(L
D
15
10
5
0
0 0.5 1 1.5 2 2.5
p (GeV)
Figure2: LongitudinalgluonpropagatoratandaroundT,forvariouschoicesoflatticesizeandb . Values
c
forthetemperature, N3×N, b ,latticespacingaandspatiallatticesizeL(bothinfm,inparentheses)are
s t
givenintheplotlabels. Thesolidlinesarefits,describedattheendofthissection.
belowT ,butnotimmediatelyaboveT [13].
c c
In summary, the transverse propagator D (p) shows significant finite-physical-size effects at
T
T ,whilethelongitudinal propagatorD (p)issubjecttotwosourcesofsystematicerrorsforsmall
c L
N: “pure” small-N effects (associated with discretization errors) and strong dependence on the
t t
spatial lattice size N at fixed N, when this value of N is smaller than 16. The latter effect was
s t t
observed onlyatT ∼<Tc,whereastheformerispresentinawiderrangeoftemperatures around Tc
(seebelow). Forallinvestigatedvaluesofthetemperature, D (p)seemstoreachaplateauatsmall
L
momentum p, while D (p) is infrared-suppressed, with a turnover in momentum roughly around
T
350MeVforallT 6=0.
Consideringtheinfraredplateauin D (p)—whichweestimateherebyD (0)—asafunction
L L
of temperature, the value observed at T = 0 increases as the temperature is switched on, drops
significantlyforT ∼>Tc andthenshowsasteadydecrease. InFig.3,weshowdataforDL(0)forall
ourrunsontheleft-hand side,andfortheregionaround T ontheright. Wegrouptogether results
c
from runs using the same value of N, and indicate them by the label “DL0_N”. The data points
t t
indicated with“sym”correspond tosymmetriclattices, i.e.tothezero-temperature case. Notethat
results fordifferent N ’satfixedN maynotfallontopofeachother, whichgivesusanindication
s t
of the systematic errors discussed above. These are especially serious for N =4 around T (red
t c
points). We see that, surprisingly, the maximum value of D (0) is not attained for T =T — as
L c
mighthaveappearedtobethecasefromtheN =4latticesonly—anditdoesnotcorrespond toa
t
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ScreeningMassesaroundDeconfinement TerezaMendes
45 50
"DL0_sym" "DL0_4"
40 "DL0_2" 45 "DL0_6"
"DL0_4" "DL0_8"
"DL0_6" 40 "DL0_16"
35
"DL0_8"
"DL0_16" 35
30
-2V) 25 -2V) 30
e e
0) (G 20 0) (G 25
D(L D(L 20
15
15
10
10
5
5
0
0 0.5 1 1.5 2 2.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
T/Tc T/Tc
Figure3:Infrared-plateauvalueforthelongitudinalgluonpropagator[estimatedbyD (0)]asafunctionof
L
thetemperatureforthefullrangeofT/T values(left)andfortheregionaroundT (right).Datapointsfrom
c c
runsatthesamevalueofN aregroupedtogetherandindicatedbythelabel“DL0_N”,where“sym”isused
t t
toindicatesymmetriclattices(i.e.T =0).
flatcurvefrom0.5T toT ,ascouldbeexpectedbylookingonlyatthesetwotemperatures. Rather,
c c
themaximumseemstolieatabout0.9T . Moreover, itclearlycorresponds toafinitepeak, which
c
doesnotturnintoadivergence asN isincreased atfixedN.
s t
Wehave also looked at the real-space propagators. We findclear violation of reflection posi-
tivity for the transverse propagator at all temperatures. For the longitudinal propagator, positivity
violation is observed unequivocally only at zero temperature and for a few cases around the crit-
ical region, in association with the severe systematic errors discussed above. For all other cases,
there is no violation within errors. Also, we always observe an oscillatory behavior, indicative of
acomplex-mass pole. Typical curves forthe longitudinal and transverse propagators inreal space
areshown(forT =0.25T )inFig.4.
c
We now address the problem of characterizing the screening masses around T . For all fits
c
shown in Figs. 1 and 2 we used the same expression, namely a five-parameter fitting form of the
Gribov-Stingl [14,15]type1
1+dp2h
D (p) = C . (2.1)
L,T (p2+a)2 +b2
This form allows for two (complex-conjugate) poles, with masses m2 = a ± ib, where m =
m +im . Themass m thus depends only on a, b and not on the normalizationC. The parameter
R I
h should be 1 if the fitting form also describes the large-momenta region (from our infrared data
we get h 6=1). Recall that at high temperatures one usually defines the electric screening mass
as the scale determining the exponential decrease of the real-space propagator at large distances,
whichisequivalentto D (0)−1/2 inthecaseofarealpole. Wethereforeexpecttoobserve m →0
L I
1Notethat,forgivenvaluesofa,b,d,h , theglobalconstantC isfixedbytherenormalizationcondition, sothat
thereareonlyfourfreeparametersin(2.1).
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ScreeningMassesaroundDeconfinement TerezaMendes
100 100
963 X 16 2.310 (0.16, 15.4) fm 963 X 16 2.310 (0.16, 15.4) fm
80 80
60 60
2) 2)
-V -V
e e
G G
(L 40 (T 40
D D
20 20
0 0
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
z (fm) z (fm)
Figure4: Longitudinal(left)andtransverse(right)gluonpropagatorin realspaceforT =0.25T. Values
c
for N3×N, b ,latticespacingaandspatiallatticesizeL(bothinfm,inparentheses)aregivenintheplot
s t
labels. Notethatthesolidlinesarenotfits.
(i.e. b→0)forthelongitudinal gluonpropagator athightemperature. Notethat,ifthepropagator
has the above form, then the screening mass defined by D (0)−1/2 = p(a2+b2)/C mixes the
L
complexandimaginarymasses m andm anddependsonthe(aprioriarbitrary)normalizationC.
R I
We generally find good fits to the Gribov-Stingl form (including the full range of momenta),
withnonzerorealandimaginarypartsofthepolemassesinallcases. Forthetransversepropagator
D (p), the masses m and m are of comparable size (around 0.6 and 0.4 GeVrespectively). The
T R I
same holds for D (p), but in this case the relative size of the imaginary mass seems to decrease
L
withincreasingtemperature. Adetaileddiscussionoftheassociatedmassesm ,m ispostponedto
R I
aforthcomingstudy[11],aswearepresentlyconsideringvariantsoftheabovefittingforminspired
by zero-temperature studies. Indeed, the useofaGribov-Stingl form ismotivated by thebehavior
of the gluon propagator at T =0, where this type of expression has been shown to describe well
the data in three space-time dimensions [16]. Recently, in [17], various fitting forms of this type
were used to describe large-lattice data for the 3d and 4d gluon propagator at T =0. Noting that
the3dcasemaybeconsidered astheT →¥ limitofthe4dcase, wepropose tointerpolate the3d
and4dzero-temperature formstodescribe ourfinite-T datain4d.
3. Conclusions
Westudythelongitudinal(electric)andtransverse(magnetic)gluonpropagatorsinmomentum
space,proposing thecalculation ofscreeningmassesthroughanAnsatzfromthezero-temperature
case. Going from zero to nonzero temperature, we see that the electric propagator D (p) is en-
L
hanced, with an apparent plateau value in the infrared, while the magnetic propagator D (p) gets
T
progressively more infrared-suppressed, with a clear turnover in momentum at all nonzero tem-
peratures considered. Severe systematic effects are observed for the electric propagator around
T , suggesting that lattices of temporal extent N >8 are needed for this study. Once these errors
c t
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ScreeningMassesaroundDeconfinement TerezaMendes
are removed, the data support a finite maximum (located at about 0.9 T ) for the infrared value of
c
D (p).
L
Acknowledgements
The authors thank agencies FAPESP and CNPq for financial support. A. Cucchieri also ac-
knowledges financial support from the Special Research Fundof Ghent University (BOFUGent).
Our simulations were performed on the new CPU/GPUcluster at IFSC–USP(obtained through a
FAPESPgrant).
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