Effects of Dense Quark Matter on Gluon Propagators in Lattice QC 2 D
V. G. Bornyakov, V. V. Braguta, A. A. Nikolaev, R. N. Rogalyov
aa r X i v : . [ h e p - l a t ] M a y Effects of dense quark matter on gluon propagators in lattice QC D V. G. Bornyakov
NRC “Kurchatov Institute” - IHEP, 142281 Protvino, Russia,School of Biomedicine, Far East Federal University, 690950 Vladivostok, Russia
V. V. Braguta
NRC “Kurchatov Institute” - ITEP, 117259 Moscow, RussiaSchool of Biomedicine, Far East Federal University, 690950 Vladivostok, RussiaBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980 RussiaMoscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow region, 141700 Russia
A. A. Nikolaev
Department of Physics, College of Science, Swansea University, Swansea SA2 8PP, United Kingdom
R. N. Rogalyov
NRC ”Kurchatov Institute” - IHEP, 142281 Protvino, Russia
The transverse and longitudinal gluon propagators in the Landau gauge are studied in the two-color lattice QCD at nonzero quark chemical potential µ q . Parameterization of the momentumdependence of the propagators is provided for all values of chemical potential under study. We findthat the longitudinal propagator is infrared suppressed at nonzero µ q with suppression increasingwith increasing µ q . The transverse propagator dependence on µ q was found to be opposite: it isenhanced at large µ q . It is found, respectively, that the electric screening mass is increasing whilethe magnetic screening mass is decreasing with increasing µ q . Nice agreement between the electricscreening mass computed from the longitudinal propagator and the Debye mass computed earlierfrom the singlet static quark-antiquark potential was found. We discuss how the dependence ofthe propagators on the chemical potential correlates with the respective dependence of the stringtension. Additionally, we consider the difference between two propagators as a function of themomentum and make interesting observations. PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.AwKeywords: gauge field theory, gluon propagator
I. INTRODUCTION
Understanding of the phase diagram of the strong interactions is of high importance for experimental studies ofhadronic matter created in relativistic heavy ion collisions. The most difficult for theoretical investigation part ofthis phase diagram is at low temperature and high density. Lattice QCD being the nonperturbative first principlesapproach is very successful at zero baryon density but is inapplicable at high baryon density due to the so calledsign problem [1]. This makes important to study the theories similar to QCD (QCD-like) but without sign problem.In particular, two popular QCD-like theories are QCD with SU (2) gauge group [2] (to be called below QC D) andQCD with nonzero isospin chemical potential [3]. QCD with the isospin chemical potential was intensively studiedboth within lattice and other approaches (see, for instance, [3–8] ). In this paper we are going to focus on QC D atnonzero quark chemical potential µ q . Although two-color QCD differs from three-color QCD, lattice study of QC Dat nonzero quark chemical potential can provide us with important information about the properties of QCD withnon-zero baryon density.QC D was studied using various approaches: chiral perturbation theory [2, 9, 10], Nambu-Jona-Lasinio model [11–13], quark-meson-diquark model [14, 15], random matrix theory [16, 17], Dyson-Schwinger equations [18], massiveperturbation theory [19, 20]. These studies suggested the following phase structure of low-temperature QC D. Thereis a hadronic phase at µ q < µ c = µ π /
2, Bose-Einstein condensation phase at µ c < µ q < µ d , and the phase withdiquark condensation due to the Bardeen-Cooper-Schrieffer mechanism at µ q > µ d .It is worth to note that these approaches are also applicable to QCD at high baryon density. It is thus importantto check them in the case of QC D confronting respective results with first principles lattice results.Lattice studies of QC D were undertaken with both staggered fermions [21–28] for N f = 4 or, more recently, N f = 2and Wilson fermions [29–35] for N f = 2 mostly. In general the lattice results supported the phase structure describedabove.The question of the confinement-deconfinement transition in QC D at low temperature is still under debate. In ourrecent paper [26] we studied N f = 2 lattice QC D with staggered fermionic action at high quark density and T = 0and demonstrated that the string tension σ decreases with increasing µ q and becomes compatible with zero for µ q above 850 MeV. The simulations were carried out at small lattice spacing a = 0 .
044 fm which was few times smallerthan in all other lattice studies. This allowed to reach the domain of large quark chemical potentials avoiding stronglattice artifacts. In a more recent paper [36], where N f = 2 lattice QC D with Wilson fermionic action was studied,the authors did not find the confinement-deconfinement transition at low temperature. It is worth to note that in[36] rather coarse lattices were used with lattice spacings three times or more larger than in our study [26]. Thus therange of large µ q where we found the transition to deconfinement was reached in [36] at parameter aµ q > . N f = 2 lattice QC Dat zero temperature and varying quark chemical potential. We use the same lattice action as in [26, 27] and infact the same set of lattice configurations. Our goal is to study how the gluon propagators change when QC Dgoes through its transitions mentioned above: from hadron phase to superfluid phase, confinement-deconfinementtransition, disappearance of the spatial string tension. Some results of our study of the gluon propagators werepresented in [37]. Here we extend the range of µ q values, make more detailed comparison of two definitions of thescreening masses and consider in more detail the momentum dependence of the gluon propagators. We also study anew observable, the difference between the (color-)electric and magnetic propagators and study its dependence on themomentum and quark chemical potential.The gluon propagators are among important quantities to study, e.g. they play crucial role in the Dyson-Schwingerequations approach. Landau gauge gluon propagators in non-Abelian gauge theories at zero and nonzero temperaturewere extensively studied in the infrared range of momenta by various methods. We shall note lattice gauge theory,Dyson-Schwinger equations, Gribov-Zwanziger approach. At the same time the studies in the particular case ofnonzero quark chemical potential are restricted to a few papers only. For the lattice QCD this is explained by thesign problem mentioned above.The gluon propagators in lattice QC D at zero and nonzero µ q were studied for the first time in [30]. This study wascontinued in [34, 38, 39]. The main conclusion of Ref. [34] was that the gluon propagators practically do not changefor the range µ q < . µ q ∼
300 MeV) and increasing with increasing µ q .Part of our results were presented in [37]. The gluon propagators in QC D at nonzero µ q were also studied in Ref. [18]with help of the Dyson-Schwinger equations approach and in Ref. [20] using the massive Yang-Mills theory approachat one-loop. The authors emphasize that after the agreement with the lattice results for the gluon propagators willbe reached their methods could be applied to real QCD at nonzero baryon density. Thus to provide unbiased latticeresults is very important.The paper is organized as follows. In Section II we specify details of the lattice setup to be used: lattice action,definition of the propagators and details of the simulation. In the next Section we present the numerical results forthe momentum dependence of the propagators and our fits to the data. Section IV is devoted to the screening massescomputation and study of their dependence on the chemical potential. In Section V results for the difference betweenthe longitudinal and the transverse propagators are presented. The last section is devoted to the discussion of theresults and to conclusions to be drawn. II. SIMULATION DETAILS
We carry out our study using 32 lattices for a set of the chemical potentials in the range aµ q ∈ (0 , . λ = 0 . am q = 0 . m π / √ σ = 1 . m /σ using the value √ σa = 0 . r = 0 . r /a = 10 . a into physical units.To reach high quark densities without lattice artifacts one needs sufficiently small lattice spacing to satisfy condition aµ q ≪
1. At the same time, to study the gluon propagators in the infrared region it is necessary to employ largephysical volume. As a result of a compromise between these two requirements our lattice size is rather moderate: L = 3 . / √ σ = 1 . D at zero temperature. Here we want totranscribe the boundaries of this phase diagram in units of √ σ using results obtained in our previous papers [25–27].For small values of the chemical potential µ q < µ c , where µ c = m π / ≈ . · √ σ , the system is in the hadronic phase.In this phase the system exhibits confinement and chiral symmetry is broken. At µ q = µ c there is a second order phasetransition to a phase where scalar diquarks form a Bose-Einstein condensate (BEC phase). Enhancing the baryondensity further, we proceed to dense matter. At sufficiently high baryon density some observables of the system understudy can be determined using Bardeen-Cooper-Schrieffer theory (BCS phase). In particular, the baryon density iswell described by the density of noninteracting fermions which occupy a Fermi sphere of radius r F = µ q . The diquarkcondensate, which plays the role of a condensate of Cooper pairs, is proportional to the Fermi surface.In addition to the transition to the BCS phase we found [26] the confinement-deconfinement transition at µ q / √ σ ∼ .
1. This transition manifests itself in a rise of the Polyakov loop and vanishing of the string tension. It is interestingthat the transition to the BEC phase and the confinement-deconfinement transition are located close to each other asshow our preliminary results. It was also observed in [26] that above the deconfinement transition the spatial stringtension σ s monotonously decreases and vanishes at µ q / √ σ ∼ . A x,µ [42]: A x,µ = 12 iag (cid:16) U xµ − U † xµ (cid:17) ≡ A ax,µ σ a . (1)The lattice Landau gauge fixing condition is( ∇ B A ) x ≡ a X µ =1 ( A x,µ − A x − a ˆ µ,µ ) = 0 , (2)which is equivalent to finding an extremum of the gauge-fixing functional F U ( ω ) = 14 V X xµ Tr U ωxµ , (3)with respect to gauge transformations ω x . To fix the Landau gauge we use the simulated annealing (SA) algorithmwith finalizing overrelaxation [43]. To estimate the Gribov copy effect, we employ five gauge copies of each configura-tion; however, the difference between the ”best-copy” and ”worst-copy” values of each quantity under considerationlies within statistical errors.The gluon propagator D abµν ( p ) is defined as follows: D abµν ( p ) = 1 V a h e A aµ ( q ) e A bν ( − q ) i , (4)where e A bµ ( q ) = a X x A bx,µ exp (cid:16) iq ( x + ˆ µa (cid:17) , (5) q i ∈ ( − N s / , N s / q ∈ ( − N t / , N t /
2] and the physical momenta p µ are defined by the relations ap i = 2 sin ( πq i /N s ), ap = 2 sin ( πq /N t ).At nonzero µ q the O (4) symmetry is broken and there are two tensor structures for the gluon propagator [44] : D abµν ( p ) = δ ab (cid:0) P Tµν ( p ) D T ( p ) + P Lµν ( p ) D L ( p ) (cid:1) . (6)We consider the soft modes p = 0 and use the notation D L,T ( p ) = D L,T (0 , | ~p | ).Next we come back to discussion of the finite volume effects. At sufficiently high density the chromoelectric screeninglength determined as the inverse of the chromoelectric mass is estimated in perturbation theory as follows: l E = 1 m E ∼ g ( µ q ) µ q Our results are in agreement with this prediction as will be demonstrated in Section IV. Thus we expect that forsufficiently large µ q there should be no large finite volume effects for the longitudinal propagator D L ( p ).The screening length associated with the transverse propagator D T ( p ) is defined as the inverse of the chromomag-netic screening mass m M . Perturbation theory predicts zero value of the magnetic screening mass at large chemicalpotentials [45]; for this reason, the nonperturbative estimates of m M are of particular interest.Perturbation theory gives some evidence that, at sufficiently large µ q , the chromomagnetic screening mass goesdown, the respective screening length becomes large, and to study the infrared behavior of D T ( p ) large lattices areneeded. It should be noticed that these arguments apply to QCD at high baryon density as well. III. MOMENTUM DEPENDENCE
In this section we consider the momentum dependence of the gluon propagators for various values of µ q . Thepropagators are renormalized according to the MOM scheme to satisfy the condition D L,T ( p = κ ) = 1 /κ (7)at κ = 12 . √ σ .In Fig.1(left) we present the momentum dependence for the longitudinal propagator D L ( p ) for seven selected valuesof µ q . One can see that the infrared suppression of the propagator is clearly increasing with increasing µ q . Thisinfrared suppression hints on the increasing of the electric screening mass. We will study the screening mass in thenext section. The increasing of the infrared suppression of D L ( p ) with increasing µ q is analogous to the well establishedbehavior of D L ( p ) with increasing temperature in the deconfinement phase of both gluodynamics and QCD.In Fig.1(right) the momentum dependence for the transverse propagator D T ( p ) for the same values of µ q is shown.It is clear that D T ( p ) is much less sensitive to changes of µ q . We found decreasing of the respective screening massat large µ q as will be discussed in the next section. It is known that at a finite temperature the propagator D T ( p )has a clear maximum at the value of momentum increasing with temperature. Our data give no evidence for suchmaximum at a small momentum, however, we cannot exclude its existence. D L ( p ) σ p/ √σµ q / √σ = 0.00.91.42.42.83.33.8 0.01 0.1 1 0 2 4 6 8 10 D T ( p ) σ p/ √σµ q / √σ = 0.00.91.42.42.83.33.8 FIG. 1: The propagators D L (left) and D T (right) as functions of p at different values of µ q . The curves show results of the fitto eq. (8) We would like to provide an interpolation function for our data. It was demonstrated many times [46–50] that theinfrared behavior of the gluon propagators at zero and finite temperature can be well described by the fit functionwhich is the tree level prediction of the Refined Gribov-Zwanziger approach, [51] D L,T ( p ) = Z L,T δ L,T p p + 2 R L,T p + M L,T . (8)Our data for nonzero momentum start at rather large value p min / √ σ = 1 . µ q , in particular for D L at large µ q , see relevant discussion in Section III.We found [37] that the fit of the data based on a one-loop perturbative expression works well for p > p cut , where p cut = 3 . √ σ + µ q . (9)for D L and p cut = 6 . √ σ for D T . We perform the fit (8) over the domain p < p cut ; extending the fitting range above p cut results in a substantial decrease of the fit quality in most cases.The results for the fit parameters for D L ( p ) are presented in Appendix A, Table I. The fits for large µ q were notsuccessful. Using the Table the practitioners of other approaches to QC D can compare their results with ours[63].In practice we fitted the ratio D L,T ( p ) /D L,T ( p ) with p / √ σ = 6 .
3. This allowed us to decrease uncertainties in thefit parameters M L,T , R
L,T , δ
L,T . Respectively, the parameters Z L,T were not determined from the fitting procedurebut recomputed (for renormalized propagator) via the relation Z L,T = D L,T ( p ) p + 2 R L,T p + M L,T δ L,T p . (10)Results of the fits for D L ( p ) are also shown in Fig.1(left) together with the lattice data. In the hadron phase, thepropagators change insignificantly with increasing µ q . For this reason, absence of a systematic dependence of theparameters on µ q at small µ q is not a surprise. Beyond the hadron phase, the parameters M L , R L , and 1 /δ L show asimilar behavior: they increase with µ q .In the case of the transverse propagator the fits were successful for µ q / √ σ < .
0, see Table II. The fit parameters M T , R T and 1 /δ T again show qualitatively similar dependence on µ q . Their values are lower at the intermediatevalues 1 . < µ q / √ σ < . µ q / √ σ & . M T and R T ) or to higher values (1 /δ T ) than in the hadron phase. J L p / √σ p, GeV µ q / √σ =0.001.412.83 J T p / √σ p, GeV µ q / √σ =0.001.412.83 FIG. 2: Dressing functions J L and J T as functions of p at different values of µ q . Empty symbols in the left panel are thosebeyond our fitting range (9). It is instructive to look also at the respective dressing functions J L,T ( p ) defined as J L,T ( p ) = p D L,T ( p ) (11)It is seen in Fig.2 (left) that with increasing µ q the maximum of the longitudinal dressing function goes down andshifts to the right, thus approaching dressing function of a massive scalar particle. We note once more that thisdependence on µ q is very similar to dependence on the temperature, see e.g. Ref. [52].As can be seen in Fig. 2(right) the transverse dressing function shows instead infrared enhancement with increasing µ q . This is in agreement with the disappearance of the magnetic field screening at extremely large quark chemicalpotential predicted in [45]. IV. SCREENING MASSES
The widely used definition of the screening mass, see the review [53] and references therein, is through the inverseof the propagator at zero momentum m E = 1 D L (0) , m M = 1 D T (0) . (12)It is clear, that the screening mass defined by eq. (12) depends on renormalization. Moreover, it is rather sensitive tothe finite volume effects. Loosely speaking, eq. (12) characterizes “the total amount” of the interaction since1 m E,M = Z dx d~xD L,T ( x , ~x ) , (13)where D L,T ( x , ~x ) are the propagators in the coordinate representation.We also consider another definition of the screening mass using fitting of D − L,T ( p ) at low momenta by Taylorexpansion in p : D − L,T ( p ) = Z − ( ˜ m E,M + p + c · ( p ) + ... ) . (14) This method was used in [54] in the studies of lattice QCD at finite temperatures and we applied it to QC D in [37].In fact it would be more consistent to use the Yukawa type fitting function D − L,T ( p ) = Z − ( ˜ m E,M + p ) (15)as was done in [55–57] in the studies of lattice gluodynamics at zero and finite temperatures. It was shown in [57] thatthe Yukawa type function (15) provides a constant value for ˜ m E over rather wide range of momenta in the infrared.The reason we are using function (14) rather than function (15) is that we have no enough data points in the infraredregion where the propagator can be described by the function (15). Thus, to obtain a reasonable fit results we haveto use terms up to ( p ) for D L ( p ) and terms up to ( p ) for D T ( p ). Still, we hope that making use of the fit function(14) provides reasonably good estimates of the parameters in eq. (15).Let us note that the definition of ˜ m E,M can be related to the definition of the correlation length:˜ m E,M = ξ − E,M , (16)where the correlation length ξ E,M is conventionally defined in terms of the correlation function (propagator in ourcase) by the expression [58] ξ = 12 R V dx d~x ˜ D ( x , ~x ) | ~x | R V dx d~xD ( x , ~x ) = − D (0 ,~ X i =1 (cid:18) ddp i (cid:19) (cid:12)(cid:12)(cid:12) ~p =0 D (0 , ~p ) . (17)Even after the propagators are renormalized the definitions of the screening mass (16-17) and (12) differ in generalby a factor which may depend on the chemical potential or temperature. Its temperature dependence was found in SU (3) gluodynamics [57].In Fig.3 we show the electric (left panel) and magnetic (right panel) masses defined according to these two definitions.Our value for ˜ m E / √ σ at µ q = 0 is 1.50(4). This value can be compared with the value 1.47(2) obtained in SU(3)gluodynamics at zero temperature [56] by fitting the inverse propagator to the form (15) at small momenta[64] Wealso quote a value 1.48(5) obtained for a mass dominating the small momentum behavior of a gluon propagator in SU (2) lattice gluodynamics in [59].One can see that m E and ˜ m E show qualitatively very similar dependence on µ q . They do not change much at small µ q corresponding to the hadron phase. Above µ q / √ σ ≈ . µ q values. This behavior is similar to increasing of the electric screening mass with increasing temperature in QCD at T > T c as was demonstrated by lattice simulations with definition (12) in [52, 57, 60] and with definition (16-17) in[55, 57]. No such increasing was reported in Ref. [34].In Ref. [37] we found that the ratio ˜ m E /m E can be well approximated by a constant 1.6 for the range 0 . <µ q / √ σ < .
0. Now we can confirm this conclusion for larger µ q included in this paper. The lower curve in this Figureshows fit of m E values by a polynomial of degree two. The upper curve is obtained by multiplication with factor 1.6.One can see that the upper curve agree well with ˜ m E . The visible deviation is observed for the hadron phase only aswe reported in Ref. [37].From Fig.3 (right) one can see that the magnetic screening masses m M and ˜ m M also have qualitatively similardependence on µ q , although with one exception: ˜ m M shows increasing in the range 1 . . µ q / √ σ . . . while m M isnot increasing. Further, Fig.3 shows that for µ q / √ σ & . m M and m M are smaller than their valuesat lower µ q . Thus, we find an indication that the magnetic screening length is increasing at large chemical potentialin opposite to the electric screening length and in agreement with perturbation theory. No similar decreasing of m M was observed in the high temperature QCD or high temperature gluodynamics. Note, that the range of µ q / √ σ & . σ s is zero, see Fig.5 in Ref. [26]. m E / √ σ µ q / √σµ q a m~ E m E m M / √ σ µ q / √σµ q a m~ M m M FIG. 3: Electric (left panel) and magnetic (right panel) screening masses defined by eq. (16-17) (squares) and by eq. (12)(circles) as functions of µ q . The curves in the left panel are described in the text.) m ~ E / µ q µ q / √σµ q a our data FIG. 4: Comparison of the electric screening masses ˜ m E and Debye mass m D computed in Ref. [27] Comparing with results of Ref. [34] we note that the fluctuation of m M around a constant value at smaller valuesof µ q was also observed in that paper. At large values of µ q no decreasing of m M was found in Ref. [34]. In opposite,the results of Ref. [34] hint to increasing of m M at large µ q .In Ref. [27] we computed the Debye screening mass m D from the singlet quark-antiquark potential at large distancesusing the Coulomb gauge. It is expected that m D should agree with the electric screening mass computed from thegluon propagator. In Fig.4 we compare ˜ m E and m D . One can see the agreement within a standard deviation atall values of µ q in the deconfinement phase, i.e. at µ q / √ σ > .
9. Thus, the values of the electric screening masscomputed using two different approaches in two different gauges coincide over a wide range of µ q . We consider thisas an important result because it gives some evidence for gauge invariance of the electric screening mass. Note alsothat the ratio ˜ m E /µ q is a slowly varying function of µ q in a qualitative agreement with perturbation theory.We end this section with a remark on the reason for the differences between our results for the screening massesand results of Ref. [34]. We use a very small value of the lattice spacing in our simulations. This allows us to reachlarge physical values of µ q keeping aµ q small. In opposite, the values of lattice spacing used in Ref. [34]) are at leastthree times greater and this might cause large lattice artifacts at large µ q . Another source of the difference in resultsis the difference in the fermion action discretization used in this paper and in Ref. [34]). Thus results with the Wilsonfermions and small lattice spacing are highly needed. V. D L − D T AS AN INDICATOR OF TRANSITIONS
In the previous two sections we demonstrated that the propagators D L ( p ) and D T ( p ) become more and moredifferent in the infrared region when the chemical potential is increasing. At the same time they approach each otherat high momenta for fixed µ q . In this section we study how fast they approach each other with increasing momentumand how the picture changes with increasing µ q . Similar comparison of these two propagators was made in Ref. [57]in finite-temperature SU (3) gluodynamics where their ratio computed. It was demonstrated that D L ( p ) dominatesover D T ( p ) in the confinement phase at all momenta, whereas D T ( p ) becomes dominating at high enough momentain the deconfinement phase.We show below that, in the theory under study, the difference between the transverse and longitudinal propagators,∆( p ) = D T ( p ) − D L ( p ) has interesting dependence both on momentum and on chemical potential. The importantfinding is that the soft mode ∆( p ) , p = 0 which is studied here shows clear exponential dependence on p , which wasobserved recently also in SU (2) gluodynamics at finite temperatures [62].Our numerical results for ∆( p ) are presented in Fig.5. We show data at µ q / √ σ = 1 . , .
2, and 4.2. The exponential σ ∆ p / √σ p, GeV µ q a=0.110.230.45 FIG. 5: Difference D T − D L as functions of p at few values of µ q . decreasing is well established starting from some momentum p depending on µ q . We found that p = p min for1 . ≤ µ q / √ σ ≤ . p / √ σ ≈ . µ q .Thus we arrive at a simple fit function to describe the momentum dependence of ∆( p ) at p > p .∆( p ) = c exp( − ν · p ) , (18)As a check we compared the fit by function (18) with the fit by function∆( p ) = d · p v (19)motivated by a power-like behavior of both gluon propagators when p → ∞ .We cannot perform fitting for µ q / √ σ < . p ) differs from zero at two values of the momentum only. For0 . ≤ µ q / √ σ ≤ . p ) does not vanish at a very few momenta. For this reason, both fit functions work well. At µ q / √ σ > . µ q in Fig.5.The dependence of the parameters c and ν on the quark chemical potential is shown in Fig. 6. The exponent ν islinearly decreasing over the range 1 . ≤ µ q / √ σ ≤ . ν ( µ q ) can be fitted by the linear function ν = ν − ν µ q , (20)where ν = 1 . / √ σ , ν = 0 . /σ , χ N d.o.f = 2 . p − value = 0 . ν √ σ µ q / √σµ q a c σ µ q / √σµ q a FIG. 6: Parameters of the fit (18) as functions of µ q . VI. CONCLUSIONS
We presented results of our study of the longitudinal and transverse propagators in the Landau gauge of the QC Dwith N f = 2 lattice staggered quark action at nonzero quark chemical potential. In contrast to earlier studies ofthe gluon propagators in this theory [30, 34, 38, 39], we employed lattices with a very small lattice spacing and thusreached large physical values of µ q keeping lattice values of aµ q small.We checked the effects of Gribov copies and found no such effects even in the infrared region. This is different fromthe results of lattice gluodynamics. There are two reasons for this difference. The Z center symmetry which is asource of the Gribov copies in the gluodynamics with periodic boundary conditions is broken in a theory with thematter field. Another reason is that the physical volume of our lattices is rather small.Our main observations are as follows. We found that the longitudinal propagator D L ( p ) is more and more suppressedin the infrared with increasing µ q . This is reflected in particular in the increasing of the electric screening mass. Suchdependence of D L ( p ) on µ q is analogous to its dependence on the temperature at T > T c . In opposite, we found muchweaker dependence on µ q for the transverse propagator D T ( p ) with indication of the infrared enhancement at large µ q .We considered two definitions of the screening mass. The definition eq. (12) is widely used though it has somedrawbacks, in particular it depends on renormalization. The other definition eq. (14) is renormgroup invariant. Wefound that both electric masses increase with µ q and their ratio is a constant factor. A similar relation between themagnetic masses ˜ m M and m M is not ruled out although our results for ˜ m M have rather large statistical errors.It is encouraging that our value ˜ m E / √ σ = 1 . µ q = 0 is in a good agreement with respective valuesfound in SU (2) [59] and SU (3) [56] lattice gluodynamics.Another important result concerning the electric screening mass is a very good agreement between ˜ m E and theDebye screening mass m D determined from the singlet quark-anti-quark potential at large distances, see Fig. 4. Thisresult indicates gauge invariance of the electric screening mass (14).For the magnetic screening masses we found that they show only a weak dependence on µ q at µ q . . √ σ withclearly lower values at µ q & . √ σ . As we know from our previous study [26], this is the range where the spatialstring tension becomes zero. This decreasing of the magnetic screening mass is also in agreement with disappearanceof the magnetic field screening at extremely large quark chemical potential predicted in [45].Both increasing of the electric screening mass and decreasing of the magnetic screening mass at high quark densitieswere not observed before in simulations with Wilson fermions on coarse lattices [30, 34, 38, 39].We also studied the difference ∆( p ) = D L ( p ) − D T ( p ) and found that it decreases exponentially with momentumat large p . The respective exponent is decreasing linearly with µ q thus indicating that asymmetry between thepropagators survives for higher momenta with increasing µ q . Acknowledgments
The work was completed due to support of the Russian Foundation for Basic Research via grant 18-02-40130 mega.V. V. B. acknowledges the support from the BASIS foundation. A. A. N. acknowledges the support from STFC via0grant ST/P00055X/1. The authors are thankful to Andrey Kotov for participation in the project at the earlier stageand to Jon-Ivar Skullerud and Etsuko Itou for useful discussions.The research is carried out using the Central Linux Cluster of the NRC ”Kurchatov Institute” - IHEP, the equipmentof the shared research facilities of HPC computing resources at Lomonosov Moscow State University, the Linux Clusterof the NRC ”Kurchatov Institute” - ITEP (Moscow). In addition, we used computer resources of the federal collectiveusage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC Kurchatov Institute,http://ckp.nrcki.ru/. [1] S. Muroya, A. Nakamura, C. Nonaka, and T. Takaishi, Prog. Theor. Phys. , 615 (2003), hep-lat/0306031.[2] J. B. Kogut, M. A. Stephanov, D. Toublan, J. J. M. Verbaarschot, and A. Zhitnitsky, Nucl. Phys.
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