Gluon propagators in QC$_2$D at high baryon density

We study the transverse and longitudinal gluon propagators in the Landau-gauge lattice QCD with gauge group $SU(2)$ at nonzero quark chemical potential and zero temperature. We show that both propagators demonstrate substantial dependence on the quark chemical potential. This observation contradicts to earlier findings by other groups.


Introduction
The properties of nuclear matter at low temperature and high density and the location of the phase transition to deconfined quark matter are subjects of both experimental and theoretical studies. It is known that the non-perturbative first principles approach as lattice QCD is inapplicable at large baryon densities and small temperatures due to the so-called sign problem. This makes important to study the QCD-like models [1], in particular lattice SU (2) QCD (also called QC 2 D). The properties of this theory were studied with the help of various approaches: chiral perturbation theory [1,2,3], Nambu-Jona-Lasinio model [4,5,6], quark-meson-diquark model [7,8], random matrix theory [9,10]. Supported by agreement with high precision lattice results obtained in SU(2) QCD these methods can be applied to real QCD with higher confidence. Lattice studies were made with staggered fermions [11,13,12,14,15,16,17,18] for N f = 4 or, more recently, N f = 2 and Wilson fermions [19,20,21,22,23,24] for N f = 2 mostly.
The phase structure of N f = 2 QC 2 D at large baryon density and T = 0 was studied recently in [16]. The simulations were carried out at small lattice spacing and the range of large quark chemical potential was reached without strong lattice artifacts. The main result of Ref. [16] is the observation that the string tension σ is compatible with zero for µ q above 850 MeV. It was also found that the so called spatial string tension σ s started to decrease at approximately same value of µ q and went to zero at µ q > 2000 MeV.
The gluon propagators are among important quantities, e.g. they play crucial role in the Dyson-Schwinger equations approach. In this paper we present results of our study of dependence of the gluon propagators and respective screening masses on µ q , including large µ q values range. We also look for signals of the confinement-deconfinement transition in the propagators behavior.
Landau gauge gluon propagators were 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 of nonzero quark chemical potential are restricted to a few papers only. For the lattice QCD this is explained by the sign problem mentioned above.
The gluon propagators in lattice QC 2 D were recently studied for the first time in Ref. [24]. The main conclusion of Ref. [24] was that the gluon propagators practically do not change for the range of µ q values studied: µ q < 1.1 GeV. Our main conclusion is opposite. We found substantial influence of the quark chemical potential on the gluon propagators starting from rather low values (µ q ∼ 300 MeV) and increasing with increasing µ q . Thus results presented in Ref. [24] differ from our results presented in this paper in many respects. The reason for these rather drastic differences might be that the lattice action and lattice spacing differ from those used in our study.

Lattice setup
For numerical simulations we used the tree level improved Symanzik gauge action [25] and the staggered fermion action of the form whereψ, ψ are staggered fermion fields, a is the lattice spacing, m is the bare quark mass, and η ν (x) are the standard staggered phase factors. The quark chemical potential µ is introduced into the Dirac operator 2 through the multiplication of the lattice gauge field components U (x, 4) and U † (x, 4) by factors e ±µa , respectively. We have to add to the standard staggered fermion action a diquark source term [11]. This term explicitly violates U V (1) symmetry and allows to observe diquark condensation on finite lattices, because this term effectively chooses one vacuum from the family of U V (1)-symmetric vacua. Typically one carries out numerical simulations at a few nonzero values of the parameter λ and then extrapolates to λ = 0. The lattice configurations we are using are generated at one small value λ = 0.00075 which is much smaller than the quark mass in lattice units.
Integrating out the fermion fields, the partition function for the theory with the action S = S G + S F can be written in the form which corresponds to N f = 4 dynamical fermions in the continuum limit. Note that the pfaffian P f is strictly positive, such that one can use Hybrid Monte-Carlo methods to compute the integral. First lattice studies of the theory with partition function (3) have been carried out in Refs. [12,13,14]. We study a theory with the partition function corresponding to N f = 2 dynamical fermions in the continuum limit. We are using lattice gauge field configurations generated in Ref. [16] on 32 4 lattices for a set of the chemical potentials in the range aµ q ∈ (0, 0.3). At zero density scale was set using the QCD Sommer scale value r 0 = 0.468(4) fm [26]. We found [16] r 0 /a = 10.6(2). Thus lattice spacing is a = 0.044(1) fm while the string tension at µ q = 0 is √ σ 0 = 476(5) MeV. The pion is rather heavy with its mass m π = 740(40) MeV.
We employ the standard definition of the lattice gauge vector potential A x,µ [27]: where Z is the renormalization factor. The lattice Landau gauge fixing condition is which is equivalent to finding an extremum of the gauge functional with respect to gauge transformations ω x . To fix the Landau gauge we use the simulated annealing (SA) algorithm with finalizing overrelaxation [28]. To estimate the Gribov copy effect, we employed five gauge copies of each configuration; however, the difference between the "best-copy" and "worst-copy" values of each quantity under consideration lies within statistical errors. The gluon propagator D ab µν (p) is defined as follows: ] and the physical momenta p µ are defined by the relations At nonzero µ q the O(4) symmetry is broken and there are two tensor structures for the gluon propagator [29] : In what follows we consider the softest mode p 4 = 0 and use the notation p = | p| and D L,T (p) = D L,T (0, | p|). In this case, (symmetric) orthogonal projectors P T ;L µν (p) are defined as follows: Therefore, two scalar propagators -longitudinal D L (p) and transverse D T (p) -are given by is associated with the magnetic sector, D L (p) -with the electric sector.

Gluon propagators and screening masses
We begin with the analysis of the propagators in the infrared domain where their behavior is conventionally described in terms of the so called screening masses.

Definition of the screening mass
In the studies of the gluon propagators at finite temperatures/densities two definitions of the gluon screening mass are widely used. The first definition is as follows: chromoelectric(magnetic) screening mass is the parameterm that appears in the Taylor expansion of the respective (longitudinal or transverse) propagator at zero momentum (see Refs. [30,31]) The second one was proposed by A.Linde [32] for high orders of finite-temperature perturbation theory to make sense, it has the form Analogous quantity in the chromoelectric sector is often referred to as the chromoelectric screening mass [33]. These masses can be related by the factor ζ, If ζ is independent of the thermodynamical parameters, two definitions can be considered as equivalent (they differ by a constant factor and thermodynamical information is contained only in the dependence on the parameters). However, this is not always the case. To discriminate between them, we will label the former massm E,M as the proper screening mass and the latter m E,M as the Linde screening mass. We consider both masses in our study. Similar approach was considered in [31]. to compare its dependence on µ q with that ofm E , we show 1.6m E for the chromoelectric mass and 2.1m M for the chromomagnetic mass.

Screening masses in QC
We make fits over the extended range of momenta p < p cut = 2.3 GeV, comparatively high momenta are allowed for because our minimal momentum is as big as p min = 0.88 GeV.
We use the fit function for the chromoelectric sector and for the cromomagneic sector. These fit functions and the cutoff momentum p cut = 2.3 GeV are chosen for the following reasons: (i) fit function of the type (15) does not work for the transverse propagator: goodness-of-fit is not acceptable (typical p-value is of order 10 −5 ); (ii) it is unreasonable to use fit function of the type (16) in the chromoelectric sector because the parameters in this function are poorly determined, whereas satisfactory goodness-of-fit can be achieved with the 3-parameter fit; (iii) higher values of p cut results in a decrease of goodnessof-fit, whereas lower values result in large errors in the parameters, however, at µ < 0.3 GeV in the chromoelectric sector this is not the case and we choose 1 p cut = 1.8 GeV.
We checked stability of the proper chromoelectric screening mass against an exclusion of zero momentum from our fit domain. At µ q < 0.3 GeV this procedure results in an increase of m E by more than two standard deviations, whereas at higher µ q the value ofm E changes within statistical errors.
As for the chromomagnetic screening mass, an exclusion of zero momentum results in a dramatic increase of its uncertainty. Thus the longitudinal propagator considered over the momentum range 0.8 < p < 2.3 GeV does involve an information on the respective screening mass, whereas the transverse propagator -does not.
The dependence of bothm E and m E on the quark chemical potential is depicted in Fig.1, left panel. It is seen that at µ q < 0.3 GeV the difference betweenm E and m E is greater than that at larger values of µ q . At µ q > 0.3 GeV the ratiom  Fig.1, right panel. Moreover, as was mentioned above, discarding zero momentum we lose most information on the infrared behavior of the transverse propagator. For this reason, the proper magnetic screening mass can hardly be reliably extracted from our data. The dependence of the chromomagnetic Linde screening mass on µ q is shown in greater detail in Fig.2. together with the chromoelectric Linde screening mass. Our results on the dependence of Linde screening masses on µ q are in sharp disagreement with the results of Ref. [24]. It was found in Ref. [24] that at a = 0.138 fm m M increases by some 20% when µ q increases from 0 to 1.2 GeV and much faster growth was found at a = 0.186 fm. In opposite, we observe a trend to decreasing of the magnetic Linde screening mass with increasing µ q . The chromoelectric screening mass in Ref. [24] increases with µ q at a = 0.186 fm and fluctuates about a constant on a finer lattice with a = 0.138 fm. We find that on our lattices with much smaller lattice spacing a = 0.044 fm m E increases fast and this growth can be described by µ 2 q behavior predicted by the perturbation theory. From the results in Ref. [24] it follows that the chromoelectric and chromomagnetic screening masses come close to each other at all values of µ q , whereas we find that they coincide only at µ q = 0 and come apart from each other as µ q increases. Thus lattices with spacing a > 0.13 fm used in Ref. [24] might be not sufficiently fine for the studies of screening masses. The reason may stem from the fact that the condition µ q < < 1 a does not hold at large values of µ q on such rough lattices.

Perturbative behavior at high momenta and chemical potentials
At sufficiently high momenta it is natural to expect the RG-modified perturbative behavior of the gluon propagator at all values of µ q . In the one loop approximation, the asymptotic behavior of the gluon dressing function J(p) = D(p)p 2 has the form lim p→∞;g=const c and b are the coefficients of the RG functions, and κ is the normalization point. In the Landau-gauge SU (N c ) theories with N F flavors [34] we arrive at Thus we fit our data to the function where κ 0 = 6 GeV, over the domain p > p cut . The results are shown in Fig. 3. Goodness-of-fit is decreased by the effects of O(3) symmetry breaking, however, we do not perform a systematic study of these effects assuming that making use of the asymptotic standard error in the fitting parameter Λ takes these effects into account.
Each value of the cutoff momentum p cut is chosen so that (i) smaller values of p cut result in a substantial decrease of the respective p-value and (ii) greater values of p cut give no significant increase of the respective p-value. Thus we conclude that a domain of high momenta, where the longitudinal and transverse propagators can be described by the perturbatively motivated fit formula (21), does exist for each value of µ q . In the transverse case, this domain is bounded from below by the cutoff momentum p cut = 2.9 GeV uniformly on µ q . In the longitudinal case, the cutoff momenta can be roughly approximated by the formula p cut = 1.8GeV + 1.0µ q .
The dependence of the resulting parameters on µ q is shown in Fig.4. Λ L and Λ T designate the parameter Λ determined from the fit to J L and J T , respectively. Λ L gradually decreases with increasing µ q , whereas Λ T =const at µ q < µ b q ∼ 700 ÷ 800 MeV and shows a linear dependence on µ q , at µ q > µ b q . Fit over the range µ q > 0.65 GeV gives α 0 = 0.831 (17) GeV and α 0 = 0.468(18) with χ 2 /N d.o.f. = 0.19. Let us note that this sharp change in the behavior of Λ T (µ q ) occurs at µ q = µ b q , which is only a little smaller than the value µ s q ∼ 850 MeV, where σ s starts to decrease (see Ref. [16]). This value is also close to the chemical potential at which the string tension σ vanishes. Therefore, the high-momentum behavior of D T changes in the deconfinement phase.
At µ q > µ b q the scale parameter Λ T depends on the chemical potential and, if formula (23) holds true in the limit µ q → ∞, then That is, at sufficiently high µ q the scale parameter in the expression for J T is proportional to the chemical potential, as it is expected, whereas the scale parameter in the expression for J L depends only weakly on µ q . This controversial situation is very interesting and suggests further investigations.

Conclusions
We studied the gluon propagators in N f = 2 SU (2) QCD at T = 0 in the domain 0 < µ q < 1.4 GeV, 0 < p < 6.5 GeV. It was found that both longitudinal and transverse propagators depend on the chemical potential both at low and high momenta. At low momenta, we describe this dependence in terms of the chromoelectric (m E ) and chromomagnetic (m M ) screening masses using two definitions: Linde screening masses m E,M and proper screening massesm E,M . We found a good agreement between the two definitions of the chromoelectric screening mass at least at µ q > 0.3 GeV. m E increases substantially with µ q and can be fitted by the function (17).
The case of the chromomagnetic screening mass is more complicated: we find only a rough agreement between the two definitions. The Linde mass m M can be evaluated more precisely; it depends only weakly on µ q and can be fitted well by a constant at µ q < 0.8 GeV. At higher µ q one can see decreasing of m M which agrees with decreasing of σ s . Results for higher values of µ q are needed to decide whether m M goes to zero at large µ q as was argued in Ref. [35]. In any case, the difference between m E and m M shows a substantial growth with µ q starting at µ q ≈ 0.3 GeV (see Fig.2).
It should be emphasized that our findings contradict to the results of Ref. [24], where it was concluded that (i) m M comes close to m E for all µ q and (ii) both screening masses depend only weakly on µ q .
At high momenta, we used the perturbatively motivated fit function (21) and described µ q -dependence of the propagators D T,L in terms of the scaling parameters Λ T,L that appear in formulas like (21) for D T and D L .
Λ L shows a slow decrease with increasing µ q , whereas Λ T =const at µ q < 750 MeV and shows a linear growth at higher values of µ q . A sharp change in the behavior of Λ T (µ q ) occurs at µ q where the spatial string tension σ s peaks (see Ref. [16]).