Chromomagnetic Catalysis of Color Superconductivity and Dimensional Reduction

835 Progress of Theoretical Physics, Vol. 106, No. 4, October 2001 Chromomagnetic Catalysis of Color Superconductivity and Dimensional Reduction D. Ebert,1,2 K. G. Klimenko,3 Hiroshi Toki1 and V. Ch. Zhukovsky4 1 Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan 2 Institut für Physik, Humboldt-Universität zu Berlin, 10115 Berlin, Germany 3 Institute of High Energy Physics, 142284, Protvino, Moscow Region, Russia 4 Faculty of Physics, Department of Theoretical Physics, Moscow State University 119899, Moscow, Russia (Received June 11, 2001) We consider diquark condensation in external chromomagnetic fields at non-zero temperature. The general features of this process are investigated for various field configurations in relation to their symmetry properties and the form of the quark spectrum. According to the fields, there arises dimensional reduction by one or two units. In all cases there exists diquark condensation, even arbitrarily weak quark attraction, confirming the idea with regard to the universality of this mechanism in a chromomagnetic field. The possible influence of a nonzero chemical potential on the results obtained is also discussed. §1. Introduction Nonperturbative effects in QCD at low energies (large distances) can only be studied using approximate methods in the framework of various effective models proposed. Among such nonperturbative effects are the existence of the QCD vacuum with gluon and quark condensates 1) and the hadronization process. One of the possibilities to approximately describe the gluon condensate is to introduce background color fields of certain configurations. One may, in particular, study the influence of external (background) color fields on quarks. 2) In this case it is possible to find expressions for the quark Green’s functions with exact treatment of the gauge field strength. This approach enables one to make analytical calculations in order to obtain estimates of various nonperturbative processes, such as fermion condensate formation in constant non-Abelian fields, 3) thermodynamic stabilization of the vacuum state in an SU(2) model of QCD with condensate fields 4) deep inelastic hadron scattering influenced by gluon vacuum fields, 5) etc. As is well known, the physics of light mesons can be described by effective fourfermion models, such as the Nambu-Jona-Lasinio (NJL) quark model, which was successfully used to implement the ideas of dynamical chiral symmetry breaking (DχSB) and bosonization (see e.g. Ref. 6) and references therein; for a review of (2+1)-dimensional four-quark effective models see Ref. 7)). In particular, for a QCDmotivated NJL-model with gluon condensate at finite temperature, it was shown that a weak gluon condensate plays a stabilizing role for the behavior of the constituent quark mass, the quark condensate, meson masses and coupling constants for varying 836 D. Ebert, K. G. Klimenko, H. Toki and V. Ch. Zhukovsky temperature. 8) The influence of the temperature, the chemical potential, 9) and an external magnetic field 10) on the phase structure of various modifications of the Nambu-Jona-Lasinio model has also been discussed. It is in the framework of four-fermion models that a constant magnetic field was shown 11) to induce DχSB, as well as fermion mass generation, even under conditions for which the interaction between fermions is weak. Later, this phenomenon, i.e., the effect of magnetic catalysis, was explained on the basis of the idea of effective reduction of spatial dimensionality in the presence of a strong external magnetic field 12) (see also Ref. 13) and references therein). It was also demonstrated that a strong chromomagnetic (i.e., nonabelian) field catalyzes DχSB. 14) As shown in Ref. 15), this effect can be understood in the framework of the dimensional reduction mechanism as well, and it does not depend on the particular form of the constant chromomagnetic field configuration. Recently, the effect of diquark condensation and possible color superconductivity (CSC) has attracted much attention and has been studied in various publications (see, e.g. Refs. 16) – 19), and also the review paper Ref. 20) and references therein). One may expect that, in analogy to the case of the quark condensate, the process of diquark condensation can be catalyzed by intensive external (vacuum) gauge fields. For a (2+1)-dimensional model, this was recently discussed in Ref. 21). The purpose of the present paper is to further investigate this possibility, now for a (3+1)-dimensional model including (q̄q) and (qq) interactions, for various external chromomagnetic fields, including non-abelian axially-symmetric and rotationallysymmetric ones, as well as for abelian fields. In particular, we show that in all cases, even for a weak coupling of the quarks, the diquark condensation effect induced by external chromomagnetic fields exists and is related to an effective dimensional reduction. Moreover, we find a simple relation between the symmetry properties of external fields, the degeneracy of the quark energy spectra, and the phenomenon of dimensional reduction. The latter effect leads to a nonanalytic logarithmic dependence of the diquark condensate on the field strength in the strong field limit. We also consider the effect of finite temperature and show that in the strong field limit, there exists a finite critical temperature, at which a phase transition takes place, and color symmetry is restored in both abelian and non-abelian models of the gluon condensate. In particular, there arises the BCS relation TC1 = C|δ0 (0)| between the critical temperature and the zero temperature diquark condensate δ0 (0), with a universal constant C for different fields. Finally, we briefly discuss the influence of a nonzero chemical potential on the results obtained. §2. Quark and diquark condensates in external fields 2.1. General definitions Let us consider an NJL model that describes the interaction of flavored and colored quarks qi,α (i = 1, . . . , Nf , α = 1, . . . , Nc ) with Nf = 2 and NC = 3 the numbers of flavors and colors, respectively (for convenience, corresponding indices are sometimes suppressed in what follows), moving in an external chromomagnetic field. Chromomagnetic Catalysis of Color Superconductivity 837 The underlying quark Lagrangian is chosen to contain four-quark interaction terms, which are shown later to be responsible for spontaneous breaking of both chiral and color symmetries. Hence, two types of condensates characterize the ground state of the model: the quark condensate q̄q (spontaneous breaking of chiral symmetry), and the diquark condensate qq (spontaneous breaking of color symmetry). Upon performing the usual bosonization procedure 22), 6) and introducing meson and diquark fields σ and π and ∆b and ∆∗b , the four-quark terms are replaced by Yukawa interactions of quarks with these fields, and the Lagrangian takes the following form (our notation is that for four-dimensional Eucl (...truncated)