Diquarks

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 ... 

Table 1. Color superconducting phases and corresponding non-vanishing diquark gaps. The normal phase ( N ) is characterized by the fact that there is no nonzero diquark gap.

Figure 2. Phase diagram in the /u — (iq plane for T = /U3 = /us = 0. The various phases separated by the solid lines are characterized by different non-vanishing diquark gaps Ai as indicated in the figure. In the non-superconducting phase quarks are present only above the dashed line. The + and signs indicate the sign of the electric charge density in the corresponding region. The dotted line corresponds to electrically neutral matter in the CFL phase.

Figure 1. The effective potential as a function of the diquark gap A calculated at a fixed value of the electrical chemical potential // , = 148 MeV (dashed line), and the effective potential defined along the neutrality line (solid line). The results are plotted for /./ = 400 MeV with rj = 0.75.

Figure 2. The graphical representation of the solution to the charge neutrality conditions (thick dash-dotted line) and the solution to the gap equation for three different values of the diquark coupling constant (thick solid and dashed lines). The intersection points represent the solutions to both. The thin solid line divides two qualitatively different regions, A < S/i and A > S/i. The results are plotted for fi = 400 MeV and three values of diquark coupling constant Go = r/Gs with i] = 0.5, i] = 0.75, and i] = 1.0.

Figure 4 The temperature dependence of the diquark gap (solid line) and the value of Sfj, = /iie/2 (dashed line) in neutral quark matter. For comparison, the diquark gap in the model with /Lie = 0 and /us = 0 is also shown (dash-dotted line). The results are plotted for /r = 400 MeV and rj = 0.75.

For the g2SC phase, the typical results for the default choice of parameters H = 400 MeV and r/ = 0.75 are shown in Figure 4. Both the values of the diquark gap (solid line) and the mismatch parameter 5/j, = /i,./2 (dashed line) are plotted. One very unusual property of the shown temperature dependence of the gap is a nonmonotonic behavior. Only at sufficiently high temperatures, the gap is a decreasing function. In the low temperature region, T < 10 MeV, however, it increases with temperature. For comparison, in the same figure, the diquark gap in the model with /je = 0 and /./, = 0 is also shown (dash-dotted line). This latter has the standard BCS shape. 

Figure 5. The temperature dependence of the diquark gap in neutral quark matter calculated...

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-... 

Here // is the mass of a baryon, 2///3 - the mass of a diquark pair. The free energy in the rotahng frame now can be written as... 

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],... 

The Fourier component of the density of the thermodynamic potential (thermodynamic potential per unit volume V) in the superconducting quark matter with the diquark pairing can be written in the following form [12, 16], cf. also [1,11],... 

The Greek indices a,j3= II, B,G) count colors, the Latin indices i = u,d,s count flavors. The expansion is presented up to the fourth order in the diquark field operators (related to the gap) assuming the second order phase transition, although at zero temperature the transition might be of the first order, cf. [17], iln is the density of the thermodynamic potential of the normal state. The order parameter squared is D = d s 2 = dn 2 + dG 2 + de 2, dR dc dB for the isoscalar phase (IS), and D = 3 g cfl 2,... 

Above Tc, Eqs. (22) and (24) yield rather smooth functions of T except the region t 1. The fluctuation region is rather wide since Cy Cy even at T essentially larger then Tc. The appearance of an extra channel of the diquark decay width, beyond the Hartree approximation does not... 

Thus we see that fluctuations of the diquark gap may essentially contribute to the thermodynamic quantities even well below Tc and above Tc. 

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