PHYSICAL REVIEW D 70, 014006 ~2004!Abnormal number of Nambu-Goldstone bosons in the color-asymmetric dense colorsuperconducting phase of a NambuJona-Lasiniotype model
D. Blaschke*Fachbereich Physik, Universitat Rostock, D-18051 Rostock, Germany
Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
Institut fur Physik, Humboldt-Universitat zu Berlin, D-12489 Berlin, Germany
K. G. Klimenko
Institute of High Energy Physics, 142281 Protvino, Moscow Region, Russia
M. K. Volkov and V. L. Yudichev
Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia~Received 16 March 2004; published 27 July 2004!
We consider an extended NambuJona-Lasinio model including both (qq) and ~qq! interactions with twolight-quark flavors in the presence of a single~quark density! chemical potential. In the color superconductingphase of the quark matter the colorSUc(3) symmetry is spontaneously broken down toSUc(2). If the usualcounting of Goldstone bosons would apply, five Nambu-Goldstone~NG! bosons corresponding to the fivebroken color generators should appear in the mass spectrum. Unlike that expectation, we find only threegapless diquark excitations of quark matter. One of them is anSUc(2) singlet; the remaining two form anSUc(2) ~anti!doublet and have a quadratic dispersion law in the small momentum limit. These results are inagreement with the Nielsen-Chadha theorem, according to which NG bosons in Lorentz-noninvariant systems,having a quadratic dispersion law, must be counted differently. The origin of the abnormal number of NGbosons is shown to be related to a nonvanishing expectation value of the color charge operatorQ8 reflecting thelack of color neutrality of the ground state. Finally, by requiring color neutrality, two massive diquarks areargued to become massless, resulting in a normal number of five NG bosons with the usual linear dispersionlaws.
DOI: 10.1103/PhysRevD.70.014006 PACS number~s!: 12.39.2x, 11.30.Qc, 21.65.1fne
, toI. INTRODUCTION
It is well known that, in accordance with the Goldstotheorem@1,2#, N Nambu-Goldstone~NG! bosons appear inLorentz-invariant systems if an internal continuous symmtry groupG is spontaneously broken down to a subgroupH~hereN is the number of generators in the coset spaceG/H);i.e., the number of NG modes is equal to the numberbroken generators. However, in Lorentz-noninvariant stems the number of NG bosons can be less thanN. In thiscase, the counting of NG bosons is regulated by the NielsChadha~NC! theorem@3#: Let n1 andn2 be the numbers ogapless excitations that in the limit of long wavelengths hathe dispersion lawsE;upW u and E;upW u2, respectively; then,N
BLASCHKE et al. PHYSICAL REVIEW D 70, 014006 ~2004!appear in the NJL model. We shall prove that in the 2phase only three gapless excitations that can be identwith NG bosons appear. Two of them have a quadraticpersion law, thus yielding no contradiction with the NC therem. For further applications, notice also the following important criterion which is sufficient for the equality betwethe number of NG bosons and the number of broken gentors @5#: If Qi , i 51, . . . ,N, is the full set of broken generators and if^@Qi ,Qj #&50 for any pair (i , j ), then the numberof NG bosons is equal to the numberN of broken generators
Recall, that there is an alternative approach to study csuperconductivity, which is based on weak coupling perbative QCD@10#. In this case, using the Schwinger-Dysoequation with one-gluon exchange, the color superconding phase was proved to exist at asymptotically high deties. In the two-flavored QCD investigations of color supconductivity five scalar NG bosons are shown to exist, whare mixing with gluons and required as longitudinal compnents for five massive gluons~color Meissner effect! @11#.Hence, the results concerning the number of NG bosonboth the QCD and the NJL model approaches seem to bcontradiction.1 The origin of the above discrepancy is relatto the fact that the ground state of the considered simNJL-type model is not color neutral. Indeed, since the exptation value of the color charge operatorQ8 is nonvanishing,^Q8&0, the criterion@5# is not applicable, and an abnormnumber of NG bosons must arise. On the other handquark models with a color-neutral ground state, accordingthe above criterion, there should arise a normal numbefive NG bosons all having normal~linear! dispersion laws.Most interestingly, we find from the analysis of the maspectrum in our model that the masses of two scalar diquare proportional to the ground-state expectation value ofcolor charge Q8&. Thus, when color neutralityi.e.,^Q8&50is imposed in NJL-type models as an additional codition ~which can be realized by introducing a colochemical potentialm8 by hand @12#!, then two additionalmassless particles should appear, and the total number obosons would be equal to 5. Recently, it was shown thatground state of the 2SC phase of QCD is automatically coneutral @13# due to a dynamical generation of the colochemical potentialm8 by gluon condensation. So we see ththe original discrepancy in the number of NG bosonstween the QCD approach and an extended NJL approwould disappear.
The paper is organized as follows. In Sec. II we invesgate the considered NJL-type model in the framework ofNambu-Gorkov formalism and derive an effective mesodiquark action for a finite chemical potential. Moreover, tgap equations for the quark and diquark condensates aremerically studied. Sections III and IV contain a detailanalysis of the massless and massive excitations in the
1Of course, it is necessary to point out that the QCD weak cpling considerations are valid at asymptotic values of the chempotentiali.e., atm.1 GeVwhereas NJL model results are corect for sufficiently lower valuesm,1 GeV. Thus, a direct comparison of results seems to be problematic.01400eds--
quark sectors and explicitly show the influence of the coproperties of the ground state on the diquark mass spectSection V contains a summary and discussions. Finally,persion laws for gapless diquarks are derived in the Appdix.
II. MODEL AND ITS EFFECTIVE ACTION
It is well known that perturbative methods are not appcable in the low-energy and -density QCD regions. Insteeffective field theories are usually considered. The mpopular effective theories are based on Lagrangians wfour-fermion interactions, like the NJL-type models. Letfirst give several~very approximate! arguments somehowjustifying the chosen structure of our QCD-motivated Nmodel introduced below. For this aim, consider two-flavQCD with a nonzero chemical potential and the color groSUc(3). By integrating in the generating functional of QCover gluons and further approximating the nonperturbtive gluon propagator by ad function, one arrives at an effective local chiral four-quark interaction of the NJL typdescribing low-energy hadron physics. Finally, by perforing a Fierz transformation of the interaction term and takinto account only scalar and pseudoscalar (qq)- as well asscalar~qq!-interaction channels, one obtains a four-fermionmodel given by the Lagrangian2 ~in Minkowski space-timenotation!
L5q@gni ]n1mg02m0#q1G1@~ qq!
1G2@ qCedg5t2q#@ qedg5t2qC#. ~1!
In Eq. ~1!, m>0 is the quark chemical potential, which iisotopically symmetric quark matter is the same for boquark flavors, qC5Cqt, qC5qtC are charge-conjugatespinors, andC5 ig2g0 is the charge conjugation matrix~thesymbol t denotes the transposition operation!. The quarkfield q[qia is a flavor doublet and color triplet as well asfour-component Dirac spinor, wherei 51,2; a51,2,3.~Latin and Greek indices refer to flavor and color indicerespectively; spinor indices are omitted.! Furthermore, weuse the notationtW[(t1,t2,t3) for Pauli matrices in flavorspace; (ed)ab[eabd is the totally antisymmetric tensor incolor space, respectively. Clearly, Lagrangian~1! is invariantunder the chiralSU(2)L3SU(2)R ~at m050) and colorSUc(3) symmetry groups. The physics of light mesons@1417#, diquarks@18,19#, and meson-baryon interactions@20,21#was successfully described in the framework of different Nmodels. These effective theories were involved for the invtigation of both ordinary@2224#, hot and dilute@16#, andcolor superconducting dense quark matter@2528#. Usually,
2The most general Fierz-transformed four-fermion interaction
cludes additional vector and axial-vector (qq) as well as pseudo-scalar, vector and axial-vector-like~qq! interactions. However, thesterms are omitted here for simplicity.6-2