Neutrino emissivities in 2SC color-superconducting quark matter
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eutrino Emissivities in 2SC Color-Superconducting Quark Matter¶ J. Berdermann Institut für Physik der Universität, D-18051 Rostock, Germany Abstract—The phase structure and equation of state for two-flavor quark matter under compact star constraints is studied within a nonlocal chiral quark model. Chiral symmetry breaking leads to rather large, density dependent quark masses at the phase transition to quark matter. The influence of diquark pairing gaps and quark masses on density dependent emissivities for the direct URCA is discussed. Since mu > md, the direct URCA process due to quark masses cannot occur. We present cooling curves for model quark stars and discuss their relation to observational data. PACS numbers: 21.65.Qr DOI: 10.1134/S1063779608070320
1. INTRODUCTION
Ω ( µ B, µ Q, µ 8, T )
Emissivities and mean free paths of photons and neutrinos are essential for most astrophysical phenomena (supernovae, neutron star cooling, gamma ray bursts(GRB), pulsar kicks [8], etc.). Their correct treatment is one of the challenging tasks in astrophysics. Theoretical predictions of quark matter properties inside compact stellar objects, including the possibility of different color-superconducting phases, have resulted in a number of emissivity calculations in quark matter (e.g., [1–4]). The first calculation of neutrino emissivities in quark matter was done by Iwamoto [1]. He found that the matrix element of the direct URCA process would vanish and this important cooling process in quark stars could not occur if one neglects quark–quark interactions and quark masses. Most calculations consider the effect of quark–quark interactions to obtain a finite matrix element since the current up and down quark masses are small and their influence on the emissivity is negligible. However, the up and down quark masses can be up to almost two orders of magnitude larger than the current quark mass in the vicinity of the phase transition to quark matter. This can be shown within NJL-type chiral quark models [5]. The influence of both quark masses and diquark pairing on the direct URCA emissivities is discussed in the following sections.
2 φu + φd φu φd ∆ - + α ---------= ( 1 – α ) ---------------+ ---------8G S 4G S 4G D
2. RELATIVISTIC CHIRAL QUARK MODEL The grand canonical potential for quark matter in a 2SC superconductor is
¶ The
text was submitted by the author in English.
2
2
(1)
12
3 – λ a /T λa d p – 2 -------------3 ) + Ωe – Ω0 , ----- + T ln ( 1 + e ( 2π ) a = 1 2
∫
∑ 4
2
2
with Ωe = – µ Q /12π2 – µ Q T /6 – 7π2T 4/180 being the thermodynamic potential of ultrarelativistic electrons, where µQ = –µe and Ω0 is the divergent vacuum contribution which has to be subtracted to assure vanishing energy and pressure of the vacuum. Out of the twelve eigenvalues λa, four belong to the ungapped blue quarks and can be determined easily by using textbook methods [6] as λ1 ... 4 = Ef (p) ± µfb, with the dispersion 2
2
relation Ef (p) = p + M f ( p ) containing the dynamical mass function Mf (p) = mf + g(p)φf for the two fl
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