Quark number densities

Here n corresponds to the total quark number density, while ns and ns describe color asymmetries. Note that n/3 also describes the conserved baryon number. The charges are related to four chemical potentials, fi, /13, /tg, and rq, and the chemical potentials of all particles in the system can be expressed through these four chemical potentials. This implies /3-equilibrium,... 

If quark matter is in the ferromagnetic phase, it may produce the dipolar magnetic field by their magnetic moment. Since the total magnetic dipole moment Mq should be simply given as Mq = fjq (47t/3 rq)nq for the quark sphere with the quark core radius rq and the quark number density nq. Then the dipolar magnetic field at the star surface R takes the maximal strength at the poles,... 

For nonvanishing A in Eq. (1) the color symmetry is broken. Two of the three quark color degrees of freedom are coupled to bosonic Cooper pairs in the color antitriplet state which can form a Bose condensate.One can combine the chemical potentials Hu, Hd of u and d quarks by introducing Hq = (hu + Hd)/2 and hi = (hu — Hd)/% as the Lagrange multipliers related to, respectively, the quark number density nq and the isospin asymmetry n/. In thermal equilib-... 

Energy release due to (anti)neutrino untrapping. The configurations for the quark stars are obtained by solving the Tolman-Oppenheimer-Volkoff equations for a set of central quark number densities nq for which the stars are stable. In Fig. 13 the configurations for different antineutrino chemical potentials are shown. The equations of state with trapped antineutrinos are softer and therefore this allows more compact configurations. The presence of antineutrinos tends to increase the mass for a given central density. 

Table 2. Composition of electrically and color neutral mixed phases, corresponding quark number chemical potentials and average baryon number densities pB = n/3 in unities of nuclear matter saturation density po = 0.17/fm3. The various components are defined in Tab. 1.

To neutralize the electrical charge in the homogeneous dense u, d quark matter, roughly speaking, twice as many d quarks as u quarks are needed, i.e., rid — 2nu, where nv,d are the number densities for u and d quarks. This induces a mismatch between the Fermi surfaces of pairing quarks, i.e., pd — Hu = 10 25n, where pe is the electron chemical potential. 

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. 

Figure 1. Chemical potentials of the three phases of matter (H, Q, and Q ), as defined by Eq. (2) as a function of the total pressure (left panel) and energy density of the H- and Q-phase as a function of the baryon number density (right panel). The hadronic phase is described with the GM3 model whereas for the Q and Q phases is employed the MIT-like bag model with ms = 150 MeV, B = 152.45 MeV/fm3 and as = 0. The vertical lines arrows on the right panel indicate the beginning and the end of the mixed hadron-quark phase defined according to the Gibbs criterion for phase equilibrium. On the left panel P0 denotes the static transition point.

Figure 13. Quark star configurations for different antineutrino chemical potentials r = 0, 100, 150 MeV. The total mass M in solar masses (MsUn = M in the text) is shown as a function of the radius R (left panel) and of the central number density nq in units of the nuclear saturation density no (right panel). Asterisks denote two different sets of configurations (A,B,f) and (A ,B ,f ) with a fixed total baryon number of the set.

In the following chapter we shall see how the quark-parton model explains the occurrence of Bjorken scaling and we shall obtain expressions for all the scaling functions in terms of number densities for the quark-partons in a hadron. 

In the previous chapters we saw that in so far as deep inelastic lepton-nucleon scattering was concerned the nucleon could be visualized as a bound system of constituent quark-partons, with which the lepton interacted as if they were free particles. Our aim now is to try to give some sort of justification for such a picture, and to derive more reliable results for deep inelastic scattering in which allowance is made for the internal longitudinal and transverse (Fermi) motion of the quark-partons. The approach also allows us to evaluate the forward hadronic matrix elements of currents which appear in the sum rules discussed in Chapters 16 and 17 in terms of parton number densities. 

Because the proton is polarized we need to slightly generalize the number densities nj(/c) used previously. Thus we introduce Uf K, A S) d K as the number of flavour / quarks with momentum in the range k —> K -l-d/c and with helicity A inside a proton of momentmn P and covariant spin vector [see eqn (15.6.2)]. After some algebra one finds for the forward hadronic matrix elements. 

The quark-paxton model was developed in great detail in the previous chapter. Here we discuss the experimental situation and confront the theory with the vast amoimt of data on deep inelastic scattering and related reactions. It must be borne in mind that we have not yet discussed the QCD corrections to the model. These are not small, but their dominant effect can be taken into account by allowing the parton number densities to depend upon in a way calculable in QCD, so that, fortuitously, the entire formalism is basically unchanged, except that each qj x) —> qj x,Q ). As mentioned earlier this implies a dynamical breaking of perfect Bjorken scaling. It also implies that if one is seeking accurate information about the qj x) from experiment then care must be taken to specify the involved. 

In the previous chapter we expressed all the measurable scaling functions in terms of the quark distributions or number densities. Notice that there are many more experimental scaling functions than quark nmnber densities u, d, s, u,d,s = s, so that the predictive power is in principle very great. 

It is quite likely to find dense quark matter inside compact stars like neutron stars. However, when we study the quark matter in compact stars, we need to take into account not only the charge and color neutrality of compact stars and but also the mass of the strange quark, which is not negligible at the intermediate density. By the neutrality condition and the strange quark mass, the quarks with different quantum numbers in general have different chemical potentials and different Fermi momenta. When the difference in the chemical potential becomes too large the Cooper-pairs breaks or other exotic phases like kaon condensation or crystalline phase is more preferred to the BCS phase. 

The experimental signatures of a phase transition include (a) suppression of production of the heavy vector mesons J/XV and E and the upsilon states, (b) the creation of a large number of ss quark-antiquark pairs, and (c) the momentum spectra, abundance, and direction of emission of di-lepton pairs. The first phase experiments in this held have been carried out. Energy densities of 2 GeV/fm3 were created. Strong J/XV suppression has been observed relative to p-A collisions along with an increase in strangeness production. 

A one hundreds of a second later all the quarks were gone, and the Universe consisted of an approximately equal number of electrons, positrons, neutrinos and photons, and a small amount of protons and neutrons the ratio of protons to photons is assumed to have been about 10 . The temperature was about 10 and the density so high, about 4 X10 kg that even the unreactive neutrinos were hindered to escape. 

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