Single Flavor Color Superconductivity In A Magnetic Field And Mass Effect

Quantum Chromodynamics(QCD) is a basic dynamical theory to describe the strong interaction. There are rich phase structures in QCD phase diagram, such as hadronic mat-ter,quark gluon plasma(QGP) and superconductor etc. At asymptotically high density, one-gluon exchange can be decomposed into an color antisymmetic antitriplet channel and a symmetric color sextet channel, the former one provides the attractive interaction. At moderate density, the interaction between quarks induced by instanton is also attractive. Corresponding to the Cooper theory, quark matter at sufficiently high baryon density and low temperature becomes a color superconductor (CSC). In this dissertation, we investi-gated the single flavor color superconductivity in a magnetic and its correction to the condensation energy when an arbitrary quark mass is introduced.CSC is characterized by a diquark condensate, which is analogous to the Cooper pair in an ordinary superconductor, but the structure of the condensate is much richer because quarks have the nonabelian color and flavor charges. So in chapter two, as a first step to understand CSC, we reviewed the BCS theory. The structure of the CSC states depends sensitively on the number of quark flavors and their masses. For very high baryon density, where the masses of u, d and s quarks can be ignored, the ground state is in the color-flavor-locked(CFL) phase, where quarks of different flavors pair. The situation becomes more complicated in moderate density because of the strange quark mass, β equilibrium and the charge neutrality conditions. A substantial Fermi momentum mismatch among different quark flavors is introduced and thereby reduces the available phase space for the cross-flavor pairing. Different exotic scenarios for cross-flavor pairing proposed in the literature (gapless CSC, LOFF state etc.) either run into various instabilities or reduce significantly the condensation energy. This makes the single flavor pairing, which is free from the Fermi momentum mismatch and the instabilities, a competing alternative even though the pairing force here is expected to be weaker. There are a number of different paring states. The ones frequently discussed in the literature include the spherical color-spin-lock(CSL) and nonspherical planar, polar and A. In nature, the core of compact stars is very likely to be a color superconductor. Therefore, the mass-radius relation and the cooling rate of neutron stars have been discussed as applications of CSC at the end of this chapter. And for the preparation of the calculation in the next chapter, we give a short introduction to the NJL model.In chapter three, We investigate the single flavor color superconductivity in a magnetic field. The presence of a magnetic field in the interior of a compact star will offset the energy balance among the four canonical single flavor pairings. The spherical CSL phase has an electromagnetic Meissner effect, but nonspherical phases:polar, A and planar phases do not. So if a quark matter of single flavor parings cools down through the critical temperature in a magnetic filed, forming CSL state will cost extra work to exclude magnetic fluxes from the bulk. Therefore, the magnetic contribution to the free energy may favor the nonspherical states. we have explored the consequences of the absence of the electromagnetic Meissner effect in a nonspherical CSC phase of single flavor pairing and have obtained the phase diagram with respect to the magnetic field and the temperature. We found that under the plausible magnitude of the the magnetic field inside a compact star, the most favored state is not always CSL and these nonspherical phases do occupy a significant portion of the H-T phase diagram. And then we calculate the latent heat associated with first-order phase transition between different types of single flavor color superconductivity in a magnetic field. This order of the magnitude of the energy release may contribute to weaker energy bursts such as the X-ray radiation at the later stage of a compact star。Chapter four was devoted to the mass effect in single flavor CSC. For this purpose, we formulate the single flavor CSC for a nonzero quark mass in terms of the NJL-like ef-fective action and introduce the mean-field approximation for an arbitrary mass in section2. Unlike the ultra-relativistic limit, where the cross-helicity(transverse) pairing domi-nates, the nonzero quark mass couples the cross helicity pairing channel and the equal-helicity (longitudinal) pairing channel and thereby complicates the gap matrix underlying the excitation spectrum. Fortunately, as will be shown in section3, the gap matrix for an arbitrary mass can still be diagonalized analytically for all four canonical phases and our results interpolate both the ultra-relativistic limit and the non-relativistic limit in the literature. The ranking of the condensation energy in the massless limit remains intact when a nonzero quark mass is switched on. In section4we got the new H-T diagram in presence of a magnetic field of a three-flavor quark matter beyond the ultra-relativistic limit. Because the transition temperature of the nonzero ms strange quark paring is re-duced, phase diagram with respect to temperature and magnetic field contains a region where only u and d flavors condensate. The size of this region is tiny for ms-150MeV but cannot be ignored for ms～μ. Chapter five is our concluding remarks and outlooks. Some technical details in the calculations had been deferred to the appendix in order to avoiding the complexity of the main part of this thesis. However, it is convenient for the interesting readers to refer to.