ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ TOWARD THE UNDERSTANDING OF QUARK MATTER FORMATION
© 2008 B. O. Kerbikov, E. V. Luschevskaya
Institute of Theoretical and Experimental Physics, Moscow, Russia Received December 19, 2006; in final form, March 9, 2007
The arguments are presented showing that crossover, fluctuation phenomena, and possible Anderson transition are the precursors of quark matter formation.
PACS:12.38.Mh, 12.38.-t, 11.10.Wx
1. INTRODUCTION
During the last decade, the investigation of quark matter at finite temperature and density has become one of the QCD focal points. It is expected that at densities which are 3—5 times larger than the normal nuclear density baryons are crushed into quarks. It is also expected that if the temperature of such quark matter is low enough (below few tens of MeV) the system is unstable with respect to the formation of quark—quark Cooper-pair condensate [1—3]. This phenomenon is called color superconductivity since diquarks belong to the 3-color channel. At present there is a fair understanding of color superconductivity physics in the regime of ultrahigh density when as is small [4]. Most interesting is, however, the region of moderate densities (3—5 times larger than the normal nuclear density). This region is important for physics of neutron stars and may possibly be investigated in the laboratory, in particular, in future experiments at the GSI heavy-ion machine^. In the moderate-density/strong-coupling regime the theory faces the well-known difficulties of the nonperturbative QCD. Lattice QCD calculations encounter serious obstacle at nonzero density, since in this case the determinant of the Dirac operator is complex, resulting in nonpositive measure of the corresponding path integral. Still, several attempts to perform lattice calculations at nonzero density have been performed, see the review paper [5]. The region of moderate densities has been extensively studied within the framework of the models of Nambu—Jona-Lasinio (NJL) type (see [6, 7] and references therein), using the instanton gas model [8] or chiral perturbation theory [9].
The main conclusion reached within the NJL-type models is that transition from the nuclear matter
!)In heavy-ion collisions high-density state is naturally formed with high temperature, thus preventing diquark condensation.
(NM) phase to the quark matter (QM) phase occurs very early, namely, when the quark density reaches the value only 3 times larger than the density of quarks in normal nuclear matter. This corresponds to the value ^ ~ 400 MeV of the quark chemical potential. In the NJL-type models the NM ^ QM transition has two main signatures, namely:
(i) the gap equation for the diquark channel acquires a nontrivial solution;
(ii) the quark constituent mass tends to zero2) and the chiral symmetry is restored. At T = 0 the transition is believed to be of the first order.
Both conclusions should be taken with reservations due to oversimplifications inherent for the models and certain arbitrariness in the interpretation of the results.
The principal deficiency of the existing approaches to the NM ^ QM transition is the lack of understanding, what happens to the gluon sector when the density increases. From thermodynamic arguments it follows that possible color diquark condensate and gluon condensate have the same energy scale, which results in competition between them somewhat similar to Meissner effect [10]. It also has been demonstrated within the Ginzburg—Landau approach (see below) that fluctuations of the gluon field are important and may significantly influence the character of the phase transition and the critical temperature [1, 11, 12]. From the fact that the would-be diquark condensate belongs to color antitriplet it follows that five of eight gluons become massive [3]. However, first principle derivation of the QCD string and gluon condensate evolution with the increase of the density is not currently available. An investigation aimed at the resolution of these problems has been attempted very recently [13].
2)In the density region under consideration only u and d quarks
participate in possible pairing.
In the present paper we specify the set of the key parameters which characterize the NM ^ QM transition and estimate the values of these parameters. In this way we obtain a model-independent, though schematic, picture which exhibits several nontrivial phenomena. The onset of the quark phase and its further evolution to higher densities may be viewed as a crossover from the strong-coupling regime of composite nonoverlapping bosons (diquarks) to the weak coupling regime of macroscopic overlapping Cooper-pair condensate3). Another feature of the transition region is the drastic increase of fluctuations. The NM ^ QM evolution possesses also some features of the Anderson transition. All these three properties are interrelated.
2. BEC-BCS CROSSOVER AND THE GINZBURG-LEVANYUK NUMBER
As already mentioned in Introduction, the gap equation for the diquark channel derived within the framework of the NJL-type models acquires a nontrivial solution (a gap) at i ~ 0.4 GeV [6, 7]. To describe the crossover we will need the corresponding value of the quark-number density. Relation between i and n is given by the well-known equation
n = —
on di'
(1)
where Q(T, A) is the thermodynamic potential, A is the gap parameter.
For the NJL type of models Q is easily calculated [6, 7] (see below). According to [6], for Nf = = 2 and T = 0 transition to A = 0 phase occurs at i = 0.292 GeV4). Then Eq. (1) yields n1/3 = = 0.18 GeV [6]. This number was obtained in the chiral limit, i.e., under the condition that the quark constituent mass goes to zero when A = 0. In our view the conclusion that m = 0 when A = 0 may be a specific feature of the NJL model (or some versions of this model). The behavior of the quark constituent mass throughout the NM ^ QM transition is still a moot point.
The above result for n may be with a good accuracy reproduced using the equation for free degenerate quarks
2
n2
(2)
3)This is a particular case of the Bose—Einstein condensate to Bardeen—Cooper—Schrieffer(BEC-BCS) crossover, see below.
4) Strange quark starts to participate in pairing at much higher densities when p ^ ms ^ 150 MeV.
1 /2
where i = (kF + m2) / and i = kF in the chiral limit. For i = 0.292 GeV Eq. (2) yields n1/3 = = 0.17 GeV in a good agreement with Eq. (1). For i = 0.4 GeV Eq. (2) gives n1/3 = 0.23 GeV. Thus we conclude that n1/3 ~ 0.2 GeV ~ 1 fm-1 in the transition region.
We now turn our attention to the physics behind the fact that the gap equation derived within the framework of the NJL model and in the mean-field approximation acquires a nontrivial solution starting from i ~ 0.4 GeV. It took quite some time before it was realized [14, 15] that a nonzero value of the gap does not mean the onset of the color superconductivity (the BCS regime). It is only a signal of the presence of fermion pairs. Depending on the dynamics of the system, on the fermion density, and on the temperature, such pairs may be either stable, or fluctuating in time, may form a BCS condensate, or a dilute Bose gas, or undergo a Bose—Einstein condensation. The fact that there is a continuous transition (crossover) from the strong-coupling/low-density regime of independent bound-state formation to the weak-coupling/high-density cooperative Cooper pairing is well known [16]. In contrast to macroscopic Cooper pairs, the compact molecular-like states which are formed in strong-coupling/low-density regime are called Schafroth pairs [17, 18].
The dimensionless crossover parameter is n1/3£ , where £ is the characteristic length of pair correlation when the system is in the BCS regime and the root of the mean-square radius of the bound state when the system is in the strong-coupling regime. Some arbitrariness occurs in the definition of the crossover parameter. For example, in [19] it is defined as kF£ ~ 1.7n1/3 £ ( see (2)). Another definition of the crossover parameter is x0 = i/A [20].
In the BCS theory £ is given by £ = vF/nA, where vF is the velocity at the Fermi surface. For a typical metal superconductor vF ~ c/137, A ~ 5 K, so that £ ~ 10_4 cm. The density of electrons is n ~ 1022 cm-3. Therefore in the BCS regime n1/3£ > 103. In coordinate space the wave function of the Cooper pair is proportional to (sin(kFr)/kFr)exp(—r/£) and, hence, it has ~103 nodes.
The crossover from the BCS to the strong-coupling regime occurs at [16, 19—22]
n1/3£ ~ 1. (3)
The width of the crossover region with respect to the above parameter is several units and is model-dependent [19—22]. It was first pointed out in [14, 15] that at i ~ 0.3—0.5 GeV the quark system is in the crossover regime and not in the BCS regime as it was
inferred from the fact that at such values of i the gap equation acquires a nontrivial solution.
Let us estimate the value of the crossover parameter at the onset of the phase with A = 0, i.e., at i ~ 0.3—0.5 GeV. We have seen that n1/3 ~ ~ 1 fm-1 in this region5). The value of £ in the strong-coupling regime cannot be evaluated from the first principles. One may expect that it is of a typical hadronic scale £ ~ 1—2 fm and that it grows with density asymptotically approaching the BCS value. Model calculations confirm this expectations [23— 25]. At zero density the root-mean-square radius of the diquark is ~1 fm [23]. At low density the single-gluon-exchange model leads to the result £ ~ ~ n-1/3 [25], while at i ~ 1 GeV £n1/3 ~ 10 [25]. It should be noted that the calculation of the pair size in the crossover region is a complicated task even for the "simpler" system of ultracold fermionic atoms [26]. To sum up, we conclude that £n1/3 ~ 1 at the onset of the A = 0 phase.
The NM ^ QM transition brings the system to the quark matter state in the crossover regime, but not in color superconducting state.
Next we turn to the structure of the diquark wav
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