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arXiv:hep-ph/0106110v3 7 Dec 2001

RCNP–Th01015

The Influence of an External Chromomagnetic Field on Color

Superconductivity

D. Ebert

Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki,Osaka 567,Japan

and Institut f¨ ur Physik, Humboldt-Universit¨ at zu Berlin, 10115 Berlin, Germany

V.V. Khudyakov and V.Ch. Zhukovsky

Faculty of Physics, Department of Theoretical Physics, Moscow State University, 119899,

Moscow, Russia

K.G. Klimenko

Institute of High Energy Physics, 142284, Protvino, Moscow Region, Russia

Abstract

We study the competition of quark-antiquark and diquark condensates under

the influence of an external chromomagnetic field modelling the gluon conden-

sate and in dependence on the chemical potential and temperature. As our

results indicate, an external chromomagnetic field might produce remarkable

qualitative changes in the picture of the color superconducting (CSC) phase

formation. This concerns, in particular, the possibility of a transition to the

CSC phase and diquark condensation at finite temperature.

PACS: 12.38.-t, 11.15.Ex, 97.60.Jd

Typeset using REVTEX

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I. INTRODUCTION

Low energy (large distance) effects in QCD can only be studied by approxi-

mate (nonperturbative) methods in the framework of various effective models

or in terms of lattice calculations. At present time, one of the most popu-

lar QCD-like effective theories is the well-known Nambu–Jona-Lasinio (NJL)

model [1], which is a relativistic quantum field theory with four-fermion in-

teractions. The physics of light mesons (see e.g. [2] and references therein),

diquarks [3,4] and meson-baryon interactions [5]- [7] based on dynamical

chiral symmetry breaking can be effectively described by NJL chiral quark

models. Moreover, NJL models are widely used in nuclear physics and astro-

physics (neutron stars) for the investigation of quark matter [8], to construct

alternative models of electroweak interactions [9] and in cosmological applica-

tions [10]. Moreover, its (2+1)-dimensional analogue serves as a satisfactory

microscopic theory for several effects in the physics of high-temperature su-

perconductors [11].

The NJL model displays the same symmetries as QCD. So it can be suc-

cessfully used for simulating some of the QCD vacuum properties under the

influence of external conditions such as temperature T and chemical potential

µ [12]. The role of such considerations significantly increases especially in the

cases, where numerical lattice calculations are not admissible in QCD, i.e. at

nonzero density and in the presence of external electromagnetic fields [13,14].

Recently, it was shown in the framework of a (2+1)-dimensional NJL model

that an arbitrary small external magnetic field induces the spontaneous chi-

ral symmetry breaking (χSB) even under conditions, when the interaction

between fermions is arbitrary weak [15]. Later it was shown that this phe-

nomenon (called magnetic catalysis effect) has a rather universal character

and gets its explanation on the basis of the dimensional reduction mechanism

[16]. (The recent reviews [17] consider the modern status of the magnetic

catalysis effect and its applications in different branches of physics.)

As an effective theory for low energy QCD, the NJL model does not con-

tain any dynamical gluon fields. Such nonperturbative feature of the real

QCD vacuum, as the nonzero gluon condensate < Fa

however, be mimicked in the framework of NJL models with the help of exter-

nal chromomagnetic fields. In particular, for a QCD-motivated NJL model

with gluon condensate (i.e. in the presence of an external chromomagnetic

field) and finite temperature, it was shown that a weak gluon condensate

µνFaµν>≡< FF > can,

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plays a stabilizing role for the behavior of the constituent quark mass, the

quark condensate, meson masses and coupling constants for varying temper-

ature [18]. Then, in a series of papers, devoted to the NJL model with gluon

condensate, it was shown that an external chromomagnetic field, similar to

the ordinary magnetic field, serves as a catalyzing factor in the fermion mass

generation and dynamical breaking of chiral symmetry as well [19]. The basis

for this phenomenon is the effective reduction of the space dimensionality in

the presence of external chromomagnetic fields [20].

There exists the exciting idea proposed more than twenty years ago [21]-

[23] that at high baryon densities a colored diquark condensate < qq > might

appear. In analogy with ordinary superconductivity, this effect was called

color superconductivity (CSC). The CSC phenomenon was investigated in

the framework of the one-gluon exchange approximation in QCD [24], where

the colored Cooper pair formation is predicted selfconsistently at extremely

high values of the chemical potential µ>

baryon densities are not observable in nature and not accessible in experi-

ments (the typical densities inside the neutron stars or in the future heavy

ion experiments correspond to µ ∼ 500 MeV). The possibility for the exis-

tency of the CSC phase in the region of moderate densities was proved quite

recently (see e.g. the papers [26]- [29] as well as the review articles [30] and

references therein). In these papers it was shown on the basis of different

effective theories for low energy QCD (instanton model, NJL model, etc.)

that the diquark condensate < qq > can appear already at a rather moder-

ate baryon density (µ ∼ 400 MeV), which can possibly be detected in the

future experiments on ion-ion collisions. Since quark Cooper pairing occurs

in the color anti-triplet channel, the nonzero value of < qq > means that,

apart from the electromagnetic U(1) symmetry, the color SUc(3) should be

spontaneously broken down inside the CSC phase as well. In the framework

of NJL models the CSC phase formation has generally be considered as a

dynamical competition between diquark < qq > and usual quark-antiquark

condensation < ¯ qq >. However, the real QCD vacuum is characterized in

addition by the appearence of a gluon condensate < FF > as well, which

might change the generally accepted conditions for the CSC observation. In

particular, one would expect that, similarly to the case of quark-antiquark

condensation, the process of diquark condensation might be induced by ex-

ternal chromomagnetic fields. For a (2+1)-dimensional quark model, this was

∼108MeV [25]. Unfortunately, such

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recently demonstrated in [31]. There, a SU(2)L×SU(2)Rchirally symmetric

(2+1)-dimensional NJL model with three colored quarks of two flavors was

considered at zero T,µ. It was shown that in this case for arbitrary fixed

values of coupling constants there exists a critical value of the external chro-

momagnetic field at which a CSC second order phase transition is induced in

the system.

are not in the same universality class of theories (QCD3with Nf = 2 has

a higher flavor symmetry SU(4)), the obtained results are intrinsic to real

QCD4rather than to QCD3. Indeed, our recent investigations on the basis of

a (3+1)-dimensional NJL model [32,33] and µ = 0 show that some types of

sufficiently strong external chromomagnetic fields may catalyze the diquark

condensation.

As argued above, CSC might occur inside neutron stars and possibly be-

come observable in ion-ion collisions, i.e. at nonzero baryon densities. Taking

into account the fact that at finite chemical potential the magnetic gener-

ation of dynamical χSB qualitatively differs from the µ = 0 case [14], one

might expect analogous effects for CSC, too. By this reason, the investiga-

tion of the chromomagnetic generation of CSC under the influence of a finite

chemical potential (finite particle density) is a very interesting and actual

physical problem.

The aim of the present paper is to study the influence of external con-

ditions such as chemical potential, temperature and especially of the gluon

condensate (as modelled by external color gauge fields) on the phase struc-

ture of quark matter with particular emphasize of its CSC phase. To this

end, we shall extend our earlier analysis of the chromomagnetic generation

of CSC at µ = 0 [31]- [33] to the case of an (3+1)-dimensional NJL type

model with finite chromomagnetic field, temperature and chemical potential

presenting a generalization of the free field model of [29].

The paper is organized as follows. In Sections II and III the extended NJL

model under consideration is presented, and its effective potential (≡ ther-

modynamic potential) at nonzero external chromomagnetic field, chemical

potential and temperature is obtained in the one-loop approximation. This

quantity contains all the necessary informations about the quark and diquark

1Since the two-flavored QCD3and the considered NJL model

1Strictly speaking, the CSC is induced by those components of external chromomagnetic fields,

which can stay massless inside the CSC phase.

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condensates of the theory. In the following Section IV the phase structure of

the model is discussed on the basis of numerical investigations of the global

minimum point of the effective potential. As our main result, it is shown

that the external chromomagnetic field can induce the transition to the CSC

phase and diquark condensation even at finite temperature. Thereby, the

characteristics of the CSC phase can significantly change in dependence on

the strength of the chromomagnetic field. Finally, Section V contains a sum-

mary and discussion of the results. Some details of the effective potential

calculation are relegated to an Appendix.

II. THE MODEL

Let us first give several (very approximative) arguments motivating the

chosen structure of our QCD-motivated extended NJL model introduced be-

low. For this aim, consider two-flavor QCD with nonzero chemical potential

and color group SUc(Nc) and decompose the gluon field Aa

densate background (“external”) field Aa

aa

functional of QCD over the quantum field aa

ing” the nonperturbative gluon propagator by a δ−function, one arrives at

an effective local chiral four-quark interaction of the NJL type describing

low energy hadron physics in the presence of a gluon condensate. Finally, by

performing a Fierz transformation of the interaction term, one obtains a four-

fermionic model with (¯ qq)–and (qq)–interactions and an external condensate

field Aa

ν(x) into a con-

ν(x) and the quantum fluctuation

ν(x). By integrating in the generating

ν(x) and further “approximat-

ν(x) around it, i.e. Aa

ν(x) =Aa

ν(x)+aa

µ(x) of the color group SUc(Nc) given by the following Lagrangian2

L = ¯ q[γν(i∂ν+ gAa

ν(x)λa

2) + µγ0]q +G1

2Nc[(¯ qq)2+ (¯ qiγ5? τq)2] +

+G2

Nc[i¯ qcε(iλb

as)γ5q][i¯ qε(iλb

as)γ5qc]. (1)

It is necessary to note that in order to obtain realistic estimates for masses

of vector/axial-vector mesons and diquarks in extended NJL–type of models

2The most general four-fermion interaction would include additional vector and axial-vector (¯ qq)

as well as pseudo-scalar, vector and axial-vector-like (qq) -interactions. For our goal of studying

the effect of chromomagnetic catalysis for the competition of quark and diquark condensates, the

interaction structure of (1) is, however, sufficiently general.

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