The EOS of a degenerate hadronic matter at asymptotically high densities, where the hadronic energies » 1 GeV, is very simple. Under such conditions, quarks are no longer confined to hadrons. On the contrary, they constitute a weakly-interacting Fermi gas (Collins & Perry, 1975). Therefore, one can apply the QCD in the weak-coupling limit and calculate the reliable EOS in the one-gluon exchange approximation.
First let us consider this reliable asymptotic limit. Then we will discuss how to extrapolate this EOS to neutron star densities. This is the "top-down" approach (see, e.g., Rho, 2001), where one starts from the top, i.e., from a solid QCD result, and proceeds downwards in density, via an extrapolation combined with some phenomenology, to a phenomenological description of the deconfinement transition in neutron star cores.
Parameters needed for calculating the EOS in the weak-coupling limit are quark masses and the quark-gluon coupling constant. The quark masses cannot be measured directly but can be inferred indirectly from hadron properties. Therefore, they depend on a model used for inferring. We will restrict ourselves to three lightest quarks, because heavier quarks cannot appear in stable neutron star cores (§ 8.12). The masses of u, d, and s quarks given by the Particle Data Group are the estimates of the so called current quark masses (see Yao et al., 2006, and references therein) which will be marked by the upperscript "(c)". They are mUc)c2 = (1.5 — 3.0) MeV, m^c2 = (3 — 7) MeV, and mfCc2 = (70 — 120) MeV. As the chemical potentials of u and d quarks in the quark matter are much larger than mU c2 and m^c2, these quarks can be treated as ultra-relativistic and massless. However, one should account for the s quark mass, reflecting in this way the SU(3) (flavor) symmetry breaking.
The flavor symmetry breaking implies the presence of electrons in an electrically neutral quark matter T = 0.2 We are looking for a thermodynamic equilibrium of a four-component plasma, with four thermodynamic variables ni, where i = u, d, s, and e. We assume the equilibrium with respect to the weak-interaction processes d —> u + e + Ve , u + e —> d + ve , (7.21a)
Because the matter is thought to be transparent to neutrinos (vVe = VVe = 0), we come to the following relations between the chemical potentials:
It is advantageous to use the thermodynamic potential per unit volume n(Vu, Vd, Vs, Ve) = E — Vunu — Vdnd — Vsns — Ven . (7.23)
2 This statement is valid in the weak-coupling regime. Color-flavor locked quark superconductivity with a large gap (A > 100 MeV at msc2 < 200 MeV) will expel electrons (Rajagopal & Wilczek, 2000).
The electron gas can be treated as free and ultrarelativistic, so that
For massless noninteracting quarks u and d we have an additional factor of three due to the color degree of freedom,
In the weak-coupling limit, the quark contribution to Q, denoted as Qq, can be calculated using a perturbation expansion in the QCD (strong-interaction) coupling constant as = g2c/4n, where gc is the quark-gluon coupling constant.3 The expansion has to be performed using a renormalization scheme. In particular, we should renormalize as and the strange-quark mass ms. The renormalized quantities depend on the selected value of the renormalization point, denoted by pR, which has the dimension of energy.4 The renormalized constant as decreases with the growth of the mean quark energy. Let us restrict ourselves to the first-order approximation valid for a sufficiently small as. In this case, the contributions of all three flavors to Qq are additive,
The lowest-order formula for a plasma of ultrarelativistic u and d quarks
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