Condensates diquark

When diquarks condense for the three flavor case, we have the following symmetry breaking ... 

At very high quark density the ordinary Goldstone phase is no longer favored compared with a superconductive one associated to the following type of diquark condensates ... 

The most amazing are the results for weak coupling. It appears that the gap function could have sizable values at finite temperature even if it is exactly zero at zero temperature. This possibility comes about only because of the strong influence of the neutrality condition on the ground state preference in quark matter. Because of the thermal effects, the positive electrical charge of the diquark condensate is easier to accommodate at finite temperature. We should mention that somewhat similar results for the temperature dependence of the gap were also obtained in Ref. [21] in a study of the asymmetric nuclear matter, and in Ref. [22] when number density was fixed. 

Recently, the possible formation of diquark condensates in QCD at finite density has been re-investigated in a series of papers following Refs. [1, 2], It has been shown that in chiral quark models with nonperturbatrive 4-point interaction motivated from instantons [3] or nonperturbative gluon propagators [4, 5], the anomalous quark pair amplitudes can be very large - of the order of 100 MeV. The diquark pairs that are formed as a result of the attractive inter-... 

One expects the diquark condensate to dominate the physics at densities beyond the deconfinement/chiral restoration transition and below the critical temperature. Various phases are possible. E.g., the so called 2-color superconductivity (2SC) phase allows for unpaired quarks of one color. There may also exist a color-flavor locked (CFL) phase [7] for not too large value of the strange quark mass ms, for 2A > m2s/fiq, cf. [8], where the color superconductivity... 

CSC) is complete in the sense that the diquark condensation produces a gap for quarks of all three colors and flavors. The values of the gap are of the same order of magnitude for 2SC and CFL phases, whereas relations between critical temperature and the gap might be different, Tc 0.57A for 2SC and Tc OJA for CFL phase [9], There are also another possibilities, e.g., of pairing in the spin-one channel, for which the pairing gap proves to be small A < 1 MeV, see [9],... 

DIQUARK CONDENSATION EFFECTS ON HOT QUARK STAR CONFIGURATIONS... 

Finally we consider the question whether the effect of diquark condensation which occurs in the earlier stages of the compact star evolution (t 100 s) [8, 21, 22] at temperatures T Tc 20 — 50 MeV can be considered as an engine for explosive astrophysical phenomena like supernova explosions due... 

Diquark condensation effects on hot quark star configurations... 

The order parameters the diquark gap A, which can be seen as the gain in energy due to diquark condensation, and the mass gaps (pu,indicate chiral symmetry breaking. 

In [25, 26] it is shown that at given pq the diquark gap is independent of the isospin chemical potential for Pi ) < Pic(Pq), otherwise vanishes. Increase of isospin asymmetry forces the system to pass a first order phase transition by tunneling through a barrier in the thermodynamic potential (2). Using this property we choose the absolute minimum of the thermodynamic potential (2) between two /3-equilibrium states, one with and one without condensate for the given baryochemical potential Pb = Pu + 2pd-... 

Figure 1 displays the T = 0 solutions of the chiral gap 4> and diquark gap A for different form-factors. For the densities relevant for stable star configurations, nq < 450 MeV, the critical chemical potential ffq for the chiral transition and for the onset of diquark condensation does depend on the type of the form-factor. The maximal value of the diquark gap A 150 MeV, however, does not depend sensitively on it.. 

We compare results in the chiral limit (mo = 0) with those for finite current quark mass mo = 2.41 MeV and observe that the diquark gap is not sensitive to the presence of the current quark mass, which holds for all form-factors However, the choice of the form-factor influences the critical values of the phase transition as displayed in the quark matter phase diagram (/j,q — T plane) of Fig. 2, see also Fig. 1. A softer form-factor in momentum space gives lower critical values for Tc and at the borders of chiral symmetry restoration and diquark condensation. 

Here we compare configurations with and without CS, in order to investigate the effect of diquark condensation on the total energy of a quark star and to... 

Figure 5. Hot (T = 40 MeV) versus cold (T = 0) quark star configurations for the Gaussian model(Left graphic)and Lorentzian(Right graphic) case A with diquark condensation (dashed versus full lines) and case B without diquark condensation (dash-dash-dotted versus dash-dotted lines). When a quark star with initial mass Mi cools down from T = 40 MeV to T = 0 at fixed baryon number Nb the mass defect AM occurs.

We have investigated the influence of diquark condensation on the thermodynamics of quark matter under the conditions of /5-equilibrium and charge neutrality relevant for the discussion of compact stars. The EoS has been derived for a nonlocal chiral quark model in the mean field approximation, and the influence of different form-factors of the nonlocal, separable interaction (Gaussian, Lorentzian, NJL) has been studied. The model parameters are chosen such that the same set of hadronic vacuum observable is described. We have shown that the critical temperatures and chemical potentials for the onset of the chiral and the superconducting phase transition are the lower the smoother the momentum dependence of the interaction form-factor is. 

The phase transition to color superconducting quark matter from the lower density regions at small temperatures (T < 5 a 10 MeV) is of first order, while the melting of the diquark condensate and the corresponding transition to normal quark matter at high temperatures is of second order. The presence... 

Diquark condensation makes the EoS harder, which leads to an increase in the maximum mass of the quark star configuration when compared to the case without diquark condensation. For finite temperatures the masses are smaller than at T = 0. For asymptotically high temperatures and densities the EoS behaves like a relativistic ideal gas, where the relation pressure versus energy density is temperature independent. In contrast to the bag model where this behavior occurs immediately after the deconfinement transition, our model EoS has a temperature dependent P(e) relation also beyond this point. 

A mass defect of about 0.1 M occurs in the cooling of a hot configuration to a cold one already without diquark condensation. It is mainly determined by the change of the configuration due to changes in the EoS. Cooling the star with diquark condensation results even in a lowering of the mass defect contrary to naive expectations. 

However, the domain of the QCD phase diagram where neutron star conditions are met is not yet accessible to Lattice QCD studies and theoretical approaches have to rely on nonperturbative QCD modeling. The class of models closest to QCD are Dyson-Schwinger equation (DSE) approaches which have been extended recently to finite temperatures and densities [11-13], Within simple, infrared-dominant DSE models early studies of quark stars [14] and diquark condensation [15] have been performed. 

The diquark condensation in the 2SC phase induces a color asymmetry which is proportional to the chemical potential /./, . Therefore we can write... 

The volume fraction that is occupied by the subphase with diquark condensation is defined by the charges in the subphases... 

Figure 6. Solutions of the gap equations and the charge neutrality condition (solid black line) in the /// vs //, plane. Two branches are shown states with diquark condensation on the upper right (A > 0) and normal quark matter states (A = 0) on the lower left. The plateau in between corresponds to a mixed phase. The lines for the /3-equilibium condition are also shown (solid and dashed straight lines) for different values of the (anti-)neutrino chemical potential. Matter under stellar conditions should fulfill both conditions and therefore for //,( = 0 a 2SC-normal quark matter mixed phase is preferable.

Figure 7. Volume fraction °f the phase with nonvanishing diquark condensate obtained by a Glendenning construction of a charge-neutral mixed phase. Results are shown for three different formfactors introduced in the text.

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