The A^2 Asymmetry and Gluon Propagators in Lattice SU(3) Gluodynamics at T\simeq T_c
V. G. Bornyakov, V. A. Goy, V. K. Mitrjushkin, R. N. Rogalyov
TThe (cid:10) A (cid:11) Asymmetry and Gluon Propagatorsin Lattice SU (3) Gluodynamics at T (cid:39) T c . V. G. Bornyakov
Institute for High Energy Physics NRC “Kurchatov Institute”, 142281 Protvino, RussiaInstitute of Theoretical and Experimental Physics NRC “Kurchatov Institute”, 117259 Moscow, Russia
V. A. Goy
Institut Denis Poisson CNRS/UMR 7013, Universit´e de Tours, 37200 FrancePacific Quantum Center, Far Eastern Federal University, Sukhanova 8, 690950 Vladivostok, Russia
V. K. Mitrjushkin
Joint Institute for Nuclear Research, 141980 Dubna, Russia
R. N. Rogalyov
Institute for High Energy Physics NRC “Kurchatov Institute”,142281 Protvino, Russia
We study numerically the chromoelectric-chromomagnetic asymmetry of the dimension two A gluon condensate as well as the infrared behavior of the gluon propagators at T (cid:39) T c in the Landau-gauge SU (3) lattice gauge theory. We find that a very significant correlation of the real partof the Polyakov loop with the asymmetry as well as with the longitudinal propagator makes itpossible to determine the critical behavior of these quantities. We obtain the screening masses indifferent Polyakov-loop sectors and discuss the dependence of chromoelectric and chromomagneticinteractions of static color charges and currents on the choice of the Polyakov-loop sector in thedeconfinement phase. PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.AwKeywords: Lattice gauge theory, critical behavior, gluon propagator, dimension 2 gluon condensate
I. INTRODUCTION
It is widely hoped that the behavior of the Green’sfunctions of gauge fields encodes the confinementmechanism [1–3]. Thus their dependence on the vol-ume, temperature and momentum at temperaturesclose to the confinement-deconfinement transition at-tracts particular interest.In the one-gluon exchange approximation, theFourier transform of the gluon propagator measuresinteraction potential between static color charges. Wealso should mention the relation of the low-momentumlongitudinal and transverse propagators to the chro-moelectric and chromomagnetic screening masses and,therefore, to the properties of strongly interactingquark-gluon matter. Motivation for the studies of theasymmetry and gluon propagators is also discussed in[4–7] and references therein.Recently, significant correlations between thechromoelectric-chromomagnetic asymmetry and thePolyakov loop as well as between the zero-momentumlongitudinal propagator and the Polyakov loop werefound in SU(2) gluodynamics [8, 9]. This made itpossible to describe critical behavior of the asymme-try and the propagator and to reliably evaluate finite-volume effects.Our attention here is concentrated on the behavior of these quantities in the Landau-gauge SU (3) latticegauge theory. It is well known that the first-orderphase transition occurs in this model and the Polyakovloop jumps from zero to a nonzero value [10, 11] whichis associated with the spontaneous breaking of the Z center symmetry.Though the behavior of the asymmetry and thegluon propagators at T ∼ T c have received much at-tention in the literature, the situation with their tem-perature and volume dependence in a close vicinityof T c is far from being clear. The behavior of thegauge-field vector potentials under the Z symmetrytransformation is also poorly understood.We suggested a new approach to the studies of thelongitudinal propagator at zero momentum D L (0),which makes it possible to clarify its critical behav-ior in the infinite-volume limit of the SU (2) gluody-namics and evaluate the respective critical exponentto 6-digit precision [9]. It was shown that the criti-cal exponent γ is unrelated to the critical behavior of D L (0).Our approach is based on correlations between thePolyakov loop P and D L (0) and between P and theasymmetry A . In the studies of these correlationswe employ well-established properties of the Polyakovloop.The paper is organized as follows. In the next sec- a r X i v : . [ h e p - l a t ] J a n tion we introduce the definition and describe the de-tails of our numerical simulations. The correlationbetween the asymmetry A and the Polyakov loop P forms the subject of Section 3. Our analysis beginswith the observation that, in a finite volume, the val-ues of P are distributed in a finite range making itpossible to collect a sufficient number of configura-tions generated at different temperatures, but givingthe same value of the Polyakov loop. This allows usto study the dependence of conditional distributionsof A on the temperature and conclude that such dis-tributions are governed by the value of the real part ofthe Polyakov loop rather than by the temperature it-self. This finding and the knowledge of the critical be-havior of the Polyakov loop enables one to determinethe critical behavior of the asymmetry. The propa-gators are studied in a similar way in Section 4. Weobtain the critical behavior of only the longitudinalpropagator because it correlates with the real part ofthe Polyakov loop much more significantly than thetransverse propagator. Therewith, we evaluate bothchromoelectric and chromomagnetic screening massesin all Polyakov-loop sectors and obtain their depen-dence on the temperature. It turns out that, in the de-confinement phase, the chromoelectric screening massdepends crucially on the choice of the Polyakov-loopsector. We discuss consequences of this in the con-text of the center-cluster scenario of the deconfine-ment transition [12]. In Conclusions we summarizeour findings. II. DEFINITIONS AND SIMULATIONDETAILS
We study SU(3) lattice gauge theory with the stan-dard Wilson action in the Landau gauge. Definitionsof the chromo-electric-magnetic asymmetry and thepropagators can be found e.g. in [5, 6, 8, 13].We use the standard definition of gauge vector po-tential A xµ lattice [14] : A xµ = 12 i (cid:16) U xµ − U † xµ (cid:17) traceless ≡ A ax,µ T a , (1)Transformation of the link variables U xµ ∈ SU (3)under gauge transformations g x ∈ SU (3) has the form U xµ g (cid:55)→ U gxµ = g † x U xµ g x + µ . The lattice Landau gauge condition is given by( ∂ A ) x = (cid:88) µ =1 ( A xµ − A x − ˆ µ ; µ ) = 0 . (2)It represents a stationarity condition for the gauge-fixing functional F U ( g ) = 14 V (cid:88) xµ Re Tr U gxµ , (3) with respect to gauge transformations g x .The bare gluon propagator D abµν ( p ) is defined as D abµν ( p ) = a g (cid:68) (cid:101) A aµ ( k ) (cid:101) A bν ( − k ) (cid:69) , (4)where (cid:101) A ( k ) is the Fourier transform of the gauge po-tentials (1). The physical momenta p are given by p i = (cid:0) /a (cid:1) sin ( πk i /N s ) , p = (2 /a ) sin ( πk /N t ) , k i ∈ ( − N s / , N s / , k ∈ ( − N t / , N t / p = 0.The gluon propagator on an asymmetric lattice in-volves two tensor structures [15]: D abµν ( p ) = δ ab (cid:16) P Tµν ( p ) D T ( p ) + P Lµν ( p ) D L ( p ) (cid:17) , (5)where the longitudinal P Lµν ( p ) and the transverse P Tµν ( p ) projectors are defined at p = 0 as follows: P L ( p ) = 1 , P Lµi ( p ) = P Liµ ( p ) = 0 ; (6) P Tij ( p ) = (cid:18) δ ij − p i p j (cid:126)p (cid:19) , P Tµ ( p ) = P T µ ( p ) = 0 . Therefore, the longitudinal D L ( p ) and the transverse D T ( p ) form factors (also referred to as the longitudi-nal and transverse propagators) are given by D L ( p ) = 18 (cid:88) a =1 D aa ( p ); D T ( p ) = 116 (cid:88) a =1 3 (cid:88) i =1 D aaii ( p ) . (7)At (cid:126)p = 0 the zero-momentum propagators D T (0) and D L (0) have the form D T (0) = 124 (cid:88) a =1 3 (cid:88) i =1 D aaii (0) ; D L (0) = 18 (cid:88) a =1 D aa (0) . (8)The longitudinal propagator D T ( p ) is associated withthe electric sector and the transverse propagator D L ( p ) is associated with the magnetic sector.Our calculations are performed on asymmetric lat-tices N t × N s , where N t is the number of sites inthe temporal direction (in our study, N t = 8 and N s = 24). The temperature T is given by T = 1 /aN t where a is the lattice spacing.We use the parameter τ = T − T c T c (9)useful at temperatures close to T c . We rely on thescale fixing procedure proposed in [16] and use thevalue of the Sommer parameter r = 0 . β c = 6 .
06 and T c √ σ = 0 .
63 [17] gives T c = 294 MeV and √ σ = 0 .
47 GeV.In Table I we provide information on lattice spac-ings, temperatures and other parameters used in thiswork. β a fm a − , GeV p min , MeV τ TABLE I: Parameters associated with lattices understudyIn order to consider all three Polyakov-loop sectorsin detail, we generate ensembles of 200 independentMonte Carlo gauge-field configurations for each of thesectors: ( I ) − π < arg P < π II ) π < arg P < π ( III ) − π < arg P < − π . Consecutive configurations (considered as indepen-dent) were separated by 200 ÷
400 sweeps, each sweepconsisting of one local heatbath update followed by N s / Z (3) flips for space directionswith the simulated annealing (SA) algorithm followedby overrelaxation.Here we do not consider details of the approach tothe continuum limit and renormalization consideringthat the lattices with N t = 8 (corresponding to spac-ing a (cid:39) .
08 fm at T ∼ T c ) are sufficiently fine.In terms of lattice variables, the asymmetry has theform A = 6 a N t β (cid:88) b =1 (cid:32)(cid:68) A bx, A bx, (cid:69) − (cid:88) i =1 (cid:68) A bx,i A bx,i (cid:69)(cid:33) , (11)It can also be expressed in terms of the gluon propa-gators: A = 16 N t βa N s (cid:104) D L (0) − D T (0)) (12)+ (cid:88) p (cid:54) =0 (cid:18) | (cid:126)p | − p p D L ( p ) − D T ( p ) (cid:19) (cid:105) where D L ( D T ) is the longitudinal (transversal) gluonpropagators. Thus the asymmetry A , which is nothingbut the vacuum expectation value of the respectivecomposite operator, is multiplicatively renormalizableand its renormalization factor coincides with that ofthe propagator . Assuming that both D L ( p ) and D T ( p ) are renormalized by III. A ASYMMETRY NEAR T c Critical behavior of the asymmetry in SU (2) gluo-dynamics was studied in [9], where the distribution ofthe configurations in the asymmetry was consideredand the correlation between the asymmetry A andthe Polyakov loop P was found. Then the regressionanalysis based on the conditional cumulative distri-bution function F ( A|P ) was employed to determinethe dependence of the conditional expectation of theasymmetry (cid:104)A(cid:105) P ≡ E ( A|P ) = (cid:90) dF ( A|P ) d A A d A (13)on the Polyakov loop, lattice volume, and the temper-ature. It was found that, in the leading order in τ , thevolume and temperature dependence of the asymme-try is accounted for by its dependence on the Polyakovloop.In the SU (3) case, the correlation between theasymmetry and the real part of the Polyakov loop isclearly seen on the scatter plot in the left panel ofFig.1. In view of this observation, we employ regres-sion analysis to estimate a relationship between A and P using the linear regression model based on the fitfunction E ( A| Re P ) (cid:39) A + A Re P + A (cid:0) Re P (cid:1) , (14)where A is a predicted variable (regressand) and Re P is an explanatory variable (regressor). The parame-ters A , A , and A extracted from our data are pre-sented in Table II. The residuals e n = A n − A − A Re P n − A (Re P n ) , (15)where subscript n numbers gauge-field configurations,show correlation with neither Re P nor Im P . Henceour data give no evidence for a correlation between A and Im P or for a correction to the relation (14)between A and Re P .We have more to say on the temperature de-pendence of the asymmetry. In the infinite-volumelimit, the width of the distribution of field configu-rations in the Polyakov loop tends to zero and, con-sequently, P n = (cid:104)P(cid:105) . Thus the expectation value E (cid:0) A (cid:12)(cid:12) P = P ( τ ) (cid:1) determines the asymmetry in theinfinite-volume limit: (cid:104)A(cid:105) = A ( τ ) if τ < , (16) (cid:104)A(cid:105) = A ( τ ) + A ( τ ) Re P ( τ ) ++ A ( τ )(Re P ( τ )) if τ > . The coefficients A i evaluated on the lattices underconsideration show rather smooth dependence on τ in the same factor. A sy mm e t r y Re P
T/T c =0.974T/T c =1.104T/T c =0.974T/T c =1.104 A sy mm e t r y Im P
T/T c =0.974T/T c =1.104 FIG. 1: Correlation between the asymmetry and the real part of the Polyakov loop (left); scatter plot“imaginary part of the Polyakov loop—asymmetry” is consistent with the absence of correlation between them(right).a neighborhood of the point τ = 0 associated with thedeconfinement transition: say, A changes by some5% as τ changes from − . . SU (2) case,they not only posses this property but also dependvery weakly on the lattice volume [9]. For this reason,it is natural to assume that the lattice size ∼ SU (2) case, now we employ our knowl-edge of the critical behavior of the Polyakov loop forthe investigation of the critical behavior of the asym-metry. At τ > P is somefunction of τ such thatlim τ → + |P ( τ ) | = P c > . (17)The discontinuity P c > E (cid:0) A (cid:12)(cid:12) P = P c (cid:1) − E (cid:0) A (cid:12)(cid:12) P = 0 (cid:1) = G + A < , (18)when we choose the Polyakov-loop sector with arg P =0 and E (cid:0) A (cid:12)(cid:12) P = e ıπ P c (cid:1) − E (cid:0) A (cid:12)(cid:12) P = 0 (cid:1) = (19)= E (cid:0) A (cid:12)(cid:12) P = e − ıπ P c (cid:1) − E (cid:0) A (cid:12)(cid:12) P = 0 (cid:1) = G −A > T = T c gives rise to the discontinuity of the asym-metry.Our regression analysis indicates that the depen-dence of A on Re P is much stronger than on Im P and τ ; that is, temperature dependence of A at τ > P . Scatter plot in theright panel of Fig.1 demonstrates that the values of A plotted against Im P look like the values of Re P plot-ted against Im P . Such pattern agrees well with theconclusion that A is independent of Im P . τ A A A -0.096 33.98(14) -962.8(19.4) - 603(2182)-0.026 35.54(14) -1060.5(13.5) 7645(644)0.025 37.26(24) -1104.6(8.7) 7773(393)0.104 39.78(35) -1040.3(8.5) 5969(342) TABLE II: Results of the fit (14).
IV. GLUON PROPAGATORS NEARCRITICALITY
We begin with the observation that the zero-momentum longitudinal propagator is strongly corre-lated with the real part of the Polyakov loop, see thescatter plot in Fig.2. Some correlation between D T (0)and Re P also takes place, whereas neither D L (0) nor D T (0) has a correlation with Im P .We prove this relying on a procedure analogous tothat used in the case of asymmetry. Namely, we alsobegin with the conditional distribution of the propa-gator values and find the average value of the propa-gator as a function of the Polyakov loop using a lin-ear regression model. An important difference fromthe case of asymmetry is that homoscedasticity ofconditional distributions in D L (0) at various valuesof P is severely broken. The heteroscedasticity is sogreat that it can hardly be evaluated on the basis of -1 0 1 2 3 4-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Log [ D L ( ) σ ] Re P
T/T c =0.9041.0250.9041.025 -1 0 1 2 3 4-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Log [ D T ( ) σ ] Re P
T/T c =0.9041.025 FIG. 2: Correlation between the longitudinal (left) and transverse (right) gluon propagator at zero momentumand the real part of the Polyakov loop.our limited data set. This stems mainly from non-Gaussian character of the distribution of configura-tions in D L (0), which also holds for the conditionaldistributions at fixed P . To obviate this problem, weconsider the quantity D = ln (cid:0) D L (0) σ (cid:1) (20)such that the conditional distributions of configura-tions in it are normal (at least approximately) andthe heteroscedasticity can be evaluated. Having suchevaluation, we find the conditional average (cid:104)D(cid:105) P ≡ E ( D|P ) = (cid:90) dF ( D|P ) d D D d D , (21)where F ( D|P ) is the cumulative distribution functionof D at a given value of the Polyakov loop P . For thispurpose, we employ the linear regression model basedon the fit function D (cid:39) D ( τ )+ D ( τ ) Re P ( τ )+ D ( τ ) (cid:0) Re P ( τ ) (cid:1) . (22)The results of our analysis are presented in Table III.Over the range − . < τ < . D ( τ )caused by the change of the coefficients D and D ismuch smaller than the variation caused by the changeof P according to formula (22), whereas the coefficient D is poorly determined on our statistics. Residu-als show correlation with neither Re P nor Im P in-dicating that D L (0) does not depend on Im P and,within precision available on our data, its dependenceon Re P is accounted for by formula (22) with thecoefficients from Table III.In [18] it was concluded on the basis of simulationson the 34 × T < T c (seenext subsection for more detail); therewith, in a finitevolume the center symmetry is not broken and no dis-continuities in the propagator should emerge. Nev-ertheless, one can see a discontinuity in the tempera-ture dependence of the propagator even on a small-sizelattice at any given temperature T fake provided thatone takes into account all three Polyakov-loop sectorsat T < T fake and only one Polyakov-loop sector at
T > T fake . Obviously, such a jump is unrelated to thephase transition. The proper jump of the longitudinalpropagator should appear only in the infinite-volumelimit and we argue that such a jump does emerge.Our reasoning is based on • smooth dependence of the quantity (cid:104)D(cid:105) P (and,therefore, D L (0)) on the Polyakov loop • the fact that, in the infinite-volume limit, thedistribution in P becomes infinitely narrow (thePolyakov-loop susceptibility tends to infinity as V → ∞ ); • the assumption that the coefficients D i in for-mula (22) depend on the lattice size only weakly(this does occur in the SU (2) gluodynamics,however, should be verified in the SU (3) case).Thus regression analysis gives some evidence thatthe dependence of D (and, therefore, of D L (0)) on P near the criticality is rather smooth and it is rea-sonable to draw some consequences of this. Having τ D ± δ STAT ± δ SY ST D ± δ STAT ± δ SY ST D ± δ STAT ± δ SY ST -0.096 1 . ± . ± . − . ± . ± . − ± ± . ± . ± . − . ± . ± . ± ± . ± . ± . − . ± . ± . ± ± . ± . ± . − . ± . ± . − ± ± TABLE III: Results of the fit (22). For each coefficient we present both the statistical error δ ST AT and thesystematic error δ SY ST associated with different methods of evaluation of the heteroscedasticity.regard to the fact that, in the infinite-volume limit,the Polyakov loop is a discontinuous function of thetemperature, we conclude that the zero-momentumlongitudinal gluon propagator is also discontinuous.It should be noted that the quantity exp (cid:0) (cid:104)D(cid:105) (cid:1) σ givesa biased estimate of (cid:104) D L (0) (cid:105) . However, here we focusonly on qualitative reasoning and it is sufficient forour purposes that this bias can in principle be eval-uated and smooth dependence of (cid:104)D(cid:105) P on P impliessmoothness of (cid:104) D L (0) (cid:105) P .Here one should make a comment on the tempera-ture dependence of the zero-momentum longitudinalpropagator shown in the left panel of Fig.2 in Ref.[5],where it is seen that D L (0) increases with tempera-ture at 0 < T < T x and approaches its peak at some T x such that T x < T c . When | τ | ∼
1, this growth isunrelated to the correlation with the Polyakov loopbecause the values of P are distributed close to zeroand we ignore a reason of such growth. However, atsubcritical temperatures ( τ < , | τ | <<
1) the widthof the |P| distribution begins to rise. It should beemphasized that, at a nonvanishing width of the |P| distribution, D L (0) depends substantially on whichPolyakov-loop sectors are taken into consideration.Since only sector ( I ) was taken into accountin Ref.[5], the correlation shown in Fig.2 indicates thata broadening of the |P| distribution results in a de-crease of D L (0). However, in another Polyakov-loopsector, D L (0) increases with broadening of the |P| dis-tribution. When all three sectors are taken into con-sideration, D L (0) continues to increase up to T = T c and decreasing of D L (0) with temperature at T > T c occurs only due to the restriction to sector ( I ).In a finite volume both the Polyakov loop and thepropagator are smooth functions of T . However, whenone takes an average (cid:104) D L (0) (cid:105) over configurations in allthree sectors at T < T c and over configurations only insector ( I ) at T > T c , the above reasoning implies thatsuch an average jumps down at T = T c and this is afake discontinuity associated with an abrupt artificialrestriction to only one sector: it is precisely what wasshown in [18]. Such restriction is justified only in theinfinite-volume limit.In this limit, D L (0) jumps precisely at T = T c ex-actly opposite to the Polyakov loop: it jumps down provided that sector I is chosen by the system at T > T c and jumps up when the system at T > T c chooses sector II or III .In any case, for a comprehensive investigation of thecritical behavior of Green functions all Polyakov-loopsectors should be taken into account in some neigh-borhood of the critical temperature.
A. Propagators in different Polyakov-loopsectors
Our statistics is not sufficient for a detailed study ofthe gluon propagators at a given value of the Polyakovloop or, more precisely, at a given value of Re P sincethey are independent of Im P . However, certain con-clusions on the behavior of the propagators below andabove T c can be drawn simply from considering themin different Polyakov-loop sectors. Since the propa-gators computed in sector ( II ) coincide with those insector ( III ), we compare the propagators evaluated insector ( I ) referred to as “Re P >
0” with those eval-uated on configurations from sectors ( II ) and ( III )referred to as “Re P < P > P <
0. For thelattice size under consideration ( ∼ j = D Re P < L (0) D Re P > L (0)runs up to 3 well below the critical temperature ( τ = − . j shows arapid growth and reaches 30 at τ ≈ . P > P < D L ( p ) , G e V - p, GeV τ =-0.096, Re P>0 τ =-0.096, Re P<0 τ =0.104, Re P>0 τ =0.104, Re P<0 D T ( p ) , G e V - p, GeV τ =-0.096, Re P>0 τ =-0.096, Re P<0 τ =0.104, Re P>0 τ =0.104, Re P<0 FIG. 3: Longitudinal (left) and transverse (right) gluon propagator as functions of the momentum in differentPolyakov-loop sectors below and above critical temperature. Notice logarithmic scale on the ordinate axis onthe left panel. τ D L (0) D L (0) D T (0) D T (0)Re P > P < P > P < TABLE IV: Average values of the zero-momentumpropagators in different Polyakov-loop sectors. Nodifference between sectors ( II ) and ( III ) has beenfound, they are referred to as “Re P < p < . D ( p ) (cid:39) c p + d ( p + M ) + b (23)over the momentum range p < . d T ( p ) = D Re P < T ( p ) − D Re P > T ( p )which rapidly decreases with increasing momentumand moderately increases with increasing tempera-ture. This being so, the transverse propagator is inde-pendent of temperature in the sector Re P > P < d L ( p ) = D Re P < L ( p ) − D Re P > L ( p )in a deep infrared, which is, however, characterizedby a very sharp decrease with the momentum, es-pecially at τ >
0. Though d L ( p ) increases with in-creasing temperature, the behavior of the longitudi-nal propagator by itself is more complicated. In thesector Re P >
0, the longitudinal propagator sub-stantially decreases with increasing temperature overthe range − . < τ < .
1, whereas in the sectorRe P < p (cid:38) . − . < τ < . B. Screening masses
We also evaluate screening for different sectors ofthe Polyakov loop. The chromoelectric and chromo-magnetic screening masses are obtained from the fitof the formula1 D L,T ( p ) (cid:39) Z (cid:0) m E,M + p + rp (cid:1) (24)to the data on inverse longitudinal and transversegluon propagators at low momenta, respectively; formore detail about this definition of screening massessee Refs.[19, 20]. We performed fit over the range0 ≤ p < . m E and D L (0) relative to their chromo-magnetic counterparts. τ m E m E m M m M Re P > P < P > P < TABLE V: Values of the chromoelectric andchromomagnetic screening masses (in GeV )obtained by the fit formula (24) in differentPolyakov-loop sectors. No difference between sectors( II ) and ( III ) has been found, they are referred toas “Re P < T /T c increases from 0.9 to 1.1, the chromo-electric screening mass in the sector Re P < P < T < T c can be attributed tofinite-volume effects related to a finite width of the dis-tribution in Re P . In the deconfinement phase screen-ing of color charges is different in different sectors. At T = 1 . T c , as an example, screening radii differ sub-stantially: 1.2 fm in the sector Re P > P < V E = m E D L (0) or V M = m M D T (0) (25)rather than by D L (0) or D T (0), respectively. Thescreening masses in the case under consideration arerather small, nevertheless, in Table VI we present thevalues of V E,M which have the meaning of the depthof the static-quark potential well in the limit of largescreening masses. Though these quantities are nor-malization dependent, their ratios give information,in particular, on the strength of chromoelectric inter-actions relative to the strength of the chromomagneticinteractions in different Polyakov-loop sectors. As the temperature increases from 0 . T c to 1 . T c ,the ratio V E V M (cid:12)(cid:12)(cid:12)(cid:12) Re P > decreases by some 30% and theratio V E V M (cid:12)(cid:12)(cid:12)(cid:12) Re P < decreases more than by a factor offour. τ V E , GeV V M , GeVRe P > P < P > P < TABLE VI: Strength of the chromoelectric ( V E ) andchromomagnetic ( V M ) interactions determined byformula (25) in different Polyakov-loop sectors.Errors are not shown because the presented valuescan be used only for rough qualitative estimates.Thus in the deconfinement phase the relativestrength of chromoelectric interactions decreasesslightly in the sector Re P > P <
0. This being so, the chromoelec-tric screening radius decreases in the sector Re P > P < P > P <
0, we arrive at a medium with strong and short-range chromomagnetic interactions and weak andlong-range chromoelectric interactions.
C. Speculations on the deconfinement phasetransition
Thus the longitudinal propagator in the sectorRe P > P <
0. In this connection,it is reasonable to recollect the confinement scenarioproposed, in particular, in Refs.[12, 22] and investi-gated in [23]. In these works, the properties of thegluon medium responsible for confinement of heavystatic quarks were discussed. Such medium can becharacterized in terms of center clusters (the domainswhere the Polyakov loop takes the values mainly fromone sector).In the deconfinement phase, there exists a percolat-ing cluster associated with some center element of thegauge group, and the remaining space is either occu-pied by finite-size clusters associated with some centerelement or characterized by the values of the Polyakovloop that does not clearly favor a definite center ele-ment. As the temperature decreases, the part of spaceoccupied by the percolating cluster decreases until itdisappears at the critical temperature.Let us proceed to some qualitative speculationsto outline directions of further investigations. Ourfinding that the Polyakov-loop sectors differ in theinfrared behavior of the longitudinal gluon propa-gator gives some evidence that static color chargesinteract differently in different clusters. Thus thePolyakov-loop sectors are not equivalent for gauge-dependent quantities even in a pure gauge theory. Inthe Landau gauge, we obtain that the “trivial” sec-tor Re P > P < d L ( p ) and d T ( p ) which rapidly decrease with increasing momen-tum. It is interesting to find out whether there re-ally is a momentum p J , common to all temperatures,such that both d L ( p ) and d T ( p ) become negligible at p > p J . Such a momentum would indicate the bound-ary between nonperturbative infrared and perturba-tive ultraviolet domains in gluodynamics. V. CONCLUSIONS
We have studied the asymmetry A and the longi-tudinal gluon propagator in the Landau-gauge SU (3)gluodynamics on lattices 24 × . T c < T < . T c . Our findings can besummarized as follows: • Both the asymmetry A and the zero-momentumlongitudinal propagator D L (0) have a significant correlation with the real part of the Polyakovloop P . The correlation between D T (0) andRe P is non-negligible. Neither A nor D L (0) nor D T (0) has a correlation with Im P . • We suggest a method to substantially re-duce finite-volume effects. In the deconfine-ment phase, the conditional averages (cid:104)A(cid:105) P = z or (cid:104) D L (0) (cid:105) P = z give a close approximation tothe infinite-volume limit of A or D L (0) at thetemperature τ determined from the equation P ∞ ( τ ) = z provided that z is an allowed infinite-volume value of the Polyakov loop in a chosensector and P ∞ is the infinite-volume expectationvalue of P . • We determined critical behavior of A and D L (0)in the infinite-volume limit. Regression analysisreveals that the conditional averages (cid:104)A(cid:105) P and (cid:104) D L (0) (cid:105) P are smooth functions of the Polyakovloop. Discontinuity in the Polyakov loop at T = T c in the infinite-volume limit implies dis-continuity of the asymmetry and the longitudi-nal gluon propagator. The discontinuities of A and D L (0) at T = T c are readily determinedfrom the dependencies of (cid:104)A(cid:105) P and (cid:104) D L (0) (cid:105) P on Re P . • The infrared behavior of the longitudinal propa-gator depends significantly on the Polyakov-loopsector; a moderate dependence of the transversepropagator in the infrared on the Polyakov-loopsector is also observed. • In the deconfinement phase, distinctions be-tween gauge-dependent quantities in differentPolyakov-loop sectors are significant. We haveconsidered as an example chromoelectric in-teractions relative to chromomagnetic interac-tions, whose dependence on the temperatureand the Polyakov-loop sector is not very signif-icant. They are weakly suppressed and short-range in the sector Re P > P < Acknowledgments
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