The Complex Gap Function And Angular Momentum Mixing In Color Superconductivity

Cooper's theorem implies that if there is an attractive interaction in a cold Fermi sea,the system is unstable with respect to the formation of a particle-particle condensate.In QCD case at asymptotically high density,single-gluon exchange can be decomposed into a antisymmetic color antitriplet and a symmetric color sextet,the former one provides the attractive interaction.As for the moderate density,the interaction between quarks induced by instanton is also attractive. Thus,it is unavoidable that cold dense quark matter is a color superconductor,which in turn will generate a lot of non-Fermi liquid behaviors.In this dissertation,we employed the perturbative properties of QCD in weak coupling to investigate the complex gap function and the angular momentum mixing in color superconductivity.In the first chapter,we briefly introduced the basics of QCD lagrangian and its main symmetries as well as the QCD phase diagram of our present knowledges.In chapter two,as a first step of understanding color superconductivity(CSC),we reviewed the BCS theory in ordinary superconductivity,many definitions of which will also be used in CSC.Since quarks are not only electrically charged but also carry non-abelian color and flavor, CSC has a more rich phase structure than that of the ordinary superconductivity.Within this chapter,2SC,CFL phase and the single flavor CSC have been discussed in detail.We also briefly reviewed some exotic states regarding the presence of the Fermi momentum mismatch,such as gapless CSC,LOFF and BP state.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.In chapter three,we calculated the complex gap function of CSC.The HDL gluon propagator is effectively screened by a Debye mass m_D in its electric part,while the magnetic part is poorly screened via Landau damping and thus it is still a long range interaction.Taking into account the long-range magnetic interaction,the leading order of gap is proportional to exp(-c/g).This behavior is different from the naive expectation from BSC theory,which predicts exp(-c/g~2). In addition,the general pairing potential containing landau damping has a branch cut along the real axis of the complex energy plane.Consequently,the gap function acquires a nontrivial imaginary part along the axis of real energy,which means the gap is complex.Before embarking on the calculation of the complex gap,we clarified some general properties of the gap regarding its functional dependence on the energy and momentum dictated by the invariance under a space inversion or a time reversal.With these properties,we solved some confusion regarding the analytic continuation of the quasi-particle pole from Matsubara energy to real energy.Eliashberg theory formulated for an electronic superconductor of strong pairing force regards both the complex gap and the quai-particle weight.They are determined at equal footing from a pair of self consistent equations of the electron self energy.For QCD at asymptotic density,however, the full complexity of the Eliashberg equations is unnecessary.In the leading approximation,we may ignore the imaginary part in the gap equation of the real part.In section two,we solved the nonlinear gap equation by iterations starting with a constant gap.In each step of iterations,the integral equation defines an eigenvalue problem,which can be analyzed with the perturbation method developed in literatures.We found only the first iteration is required for our purpose. To determine the imaginary part in leading order,we also need only one Eliashberg equation, which is the analytic continuation from the gap equation of Euclidean energy.We found that the imaginary part of the gap is down by g relative to the real part and is therefore corresponds to the sub-sub-leading contribution to the complex gap function.At the end of this chapter,we discussed some questions regarding the imaginary part,especially the pole of the quasi-particle.Chapter four was devoted to the angular momentum mixing in single flavor CSC.Cooper pair in single flavor pairing should be implemented at a higher total angular momentum as required by Pauli principle.The obvious choice is the p-wave pairing analogous to ~3He superfluidity,which had been called Spin-1 CSC in literatures.There are four main phases in single flavor pairing correspond to different pairing patterns,except for the color-spin-locked(CSL) phase,the other gaps including polar,A and planar are non-spherical.In QCD,the pairing potential mediated by one-gluon exchange contains all partial waves and they are equal to the leading order because of the forward singularity.Therefore,in principle,the non-spherical gaps can not be restricted in a single non-s-wave channel and the mixing among different channels will occur because of the nonlinearity of the gap equation.This is what we called angular momentum mixing.In the first section,we clarified the concept of the angular momentum mixing with a toy model of non-relativistic and spinless fermion.In the next section,we started from the CJT effective action to examine the angular momentum mixing in non-spherical polar,A and planar phases in single flavor CSC.We found that the mixture of angular momenta indeed occured and all non-spherical phases would be modified by including all partial waves,although the contribution from higher angular momentum was small because the pairing strength of each partial wave fell off with increasing J in sub-leading order.The free energy would be brought down by angular momentum mixing compared with that contained p-wave only.However,the drop amount of the free energy was too small to favor the non-spherical pairing.Even the transverse planar phase,which exhibited high potential to be able to compete CSL phase since the gain of the condensation energy from the former to the latter was only two percent falling within the range of the percentage increment by angular momentum mixing,can not become the favored state by including the mixing.Therefore,we conjecture that angular momentum mixing of various nonspherical CSC is not sufficient to compete with the CSL state energetically in ultra relativistic limit.A rigorous proof was presented at the end of this section.In the last section,we proposed two possible mechanisms,i.e.s quark mass and the strong magnetic field,which may offset the energy balance between non-spherical states and CSL.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.