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
ABNORMAL NUMBER OF NAMBU-GOLDSTONE BOSONS . . . PHYSICALREVIEW D 70, 014006 ~2004!on the basis of light-meson and baryon phenomenology,following restrictions on the coupling constants are consered:
G1.p2/~6L2!, G2,G1 , ~2!
where L is the cutoff parameter in the three-dimensionmomentum space, necessary to eliminate the ultravioletvergences appearing when quantum effects~loops! are takeninto account~usually L,1 GeV). Since Eq.~1! is a low-energy effective theory for QCD, the external parameterm isrestricted from the above:m,L. In the region of couplingconstants~2! and at small values ofm ~the case of lowbaryon densities!, only the operatorqq has a nonzerovacuum expectation value. Thus, the chiral symmetry ofmodel is spontaneously broken in this case, whileSUc(3)remains intact. The behavior of quark and meson masseestablished rather well in the quark matter of low dens@22,23#. In particular, the number ofp mesons, which are thethree NG bosons in this phase, is equal to the numbebroken symmetry generators. Hence, this example is a gdemonstration of the fact that Lorentz noninvariance isnecessary but not sufficient condition for the abnormal nuber of NG bosons to appear in the system. At sufficienhigh values of the chemical potentialm;300350 MeV, thetwo-flavor color superconductivity phase occurs in model~1!@27#. So a nonzero diquark condensate (^qq&0) is formedin this case, and as a consequence, color symmetry is staneously broken down to theSUc(2) subgroup. A naivecounting gives us five NG bosons in this case@it is just thenumber of broken generators of theSUc(3) group#. How-ever, as will be shown further, there are only three gapbosonic excitations of the 2SC quark-matter ground stTwo of them satisfy the quadratic dispersion law, in agrment with the NC theorem.
The linearized version of Lagrangian~1! that containsauxiliary bosonic fields has the form
L5q@gni ]n1mg02s2m02 ig
5tWpW #q2s21pW 2
2@ qedg5t2qC#. ~3!
Lagrangians~1! and ~3! are equivalent to the equationsmotion for bosonic fields, from which it follows that
Dd;qCedg5q, s;qq, pW ; i qg5tWq. ~4!
However, it is more convenient to start our consideratfrom Lagrangian~3!. Clearly, thes and pW fields are colorsinglets. Besides, the~bosonic! diquark fieldDd is a colorantitriplet and a~isoscalar! singlet under the chiralSU(2)L3SU(2)R group. Note further thats and D
d are ~Lorentz!scalars, butpW are pseudoscalar fields. Hence, if^s&0, thenthe chiral symmetry of the model is spontaneously broken~atm050), whereas D
d&0 indicates the dynamical breakinof color symmetry. In the framework of the Nambu-Gorko01400e-
formalism ~for a recent relativistic treatment see, e.g., R@29#! quark fields are represented by a bispinor
C5S qqCD .Integrating in the generating functional based on thegrangian~3! over the quark fields, one obtains an effectimeson-diquark action of the original model~1!:
Seff~s,pW ,Dd,D* d!52E d4xFs21pW 24G1 1DdD* d
2Trs f cxlnS D1 K2K1 D2D . ~5!
Besides of an evident trace over the two-dimensioNambu-Gorkov matrix, the Tr operation in Eq.~5! standsfor calculating the trace in spinor (s), flavor (f ), and color~c! as well as four-dimensional coordinate~x! spaces, corre-spondingly. We have used also the notation
K15 iD* dedg5, K252 iDdedg5,
6, S65s6 ig5pW tW . ~6!
Let us introduce notation for the ground-state expectatvalues of the meson and diquark fields:^s&5s0 , ^pW &5pW 0 , ^D*
d&5D0*d , ^Dd&5D0
6 ,S06!5~K6,D6,S6!us5s0 ,pW 5pW 0 ,Dd5D0
d , . . .
The quantitiess0 ,p0k ,D0
d ,D0*d (k,d51,2,3) correspond to
the global minimum of the effective potentialVeff and can befound as a solution of the system of equations
]D* d50, ~7!
usually called gap equations. In Eqs.~7! we used the follow-ing definition of the effective potential:
Seffu s,pW ,Dd,D* d5const52Veff~s,pW ,Dd,D* d!E d4x, ~8!where, in the spirit of the mean-field approximation, all fielare considered to be independent ofx. Without losing anygenerality of the consideration, one can search for a soluof Eqs.~7! in the form (D0
1 ,D02 ,D0
3)[(0,0,D). Moreover, we
suppose also thatpW 050. ~The last assumption is maintaineby the observation that in the theory of strong interactionPparity is a conserved quantity.! If D50, the color symmetryof the model remains intact. IfD0, thenSUc(3) symmetryis spontaneously broken down toSUc(2), and the 2SCphaseis realized in the model. In this case, the system of gap eqtions ~7! is reduced to6-3
BLASCHKE et al. PHYSICAL REVIEW D 70, 014006 ~2004!s02G1
54iM E d4q~2p!4E
D2~q0!J , ~9!
~2p!4H 1D1~q0! 1 1D2~q0!J , ~10!
where D6(q0)5q022(E6)22uDu2, E65E6m, E
5AqW 21M2, andM5m01s0 is the constituent quark masIn these and other similar expressions,q0 is a shorthandnotation forq01 i sgn(q0), where the limit01 must betaken at the end of all calculations. This prescription crectly implements the role ofm as chemical potential anpreserves the causality of the theory~see, e.g.,@30#!.
Let us now make a field shifts(x)s(x)1s0 ,D* d(x)D* d(x)1D0* d , Dd(x)Dd(x)1D0d and then ex-pand the expression~5! into a power series of the meson andiquark fields. The second-order termSeff(2) of this expansionis responsible for the mass spectrum of mesons and diquIt looks like
S eff(2)~s,pW ,Dd,D* d!52E d4xFs21pW 24G1 1DdD* d
4Trs f cxH S0S S1, 2K22K1, S2 D
3S0S S1, 2K22K1, S2 D J , ~11!whereS0 is the quark propagator, which is an evident 232matrix in the Nambu-Gorkov space
S05S D01 K02K01 D02D21
[S a bc dD , ~12!whose matrix elements can be found by means of the protion operator technique@31#:
e2 iq(x2y)H q02E1q0
g0L2J , ~13!b52 iDe3E d4q
g5L1J , ~14!
c5 iD* e3E d4q~2p!4
g5L2J , ~15!d5E d4q
~2p!4e2 iq(x2y)H q01E1
g0L1J , ~16!whereL65
0(gW pW 2M )/E# are projectors on the solutions of the Dirac equation with positive-negative energande3e35diag(21,21,0) is the projector~up to a sign! inthe color space. Quantities~13!~16! are nontrivial operatorsin the coordinate, spinor, and color spaces, but they areoperators in flavor space.
A tedious but straightforward analysis ofSeff(2) shows thatthere arises a mixing between thes andD3 fields. We have
S eff(2)~s,pW ,Dd,D* d!5S pp(2)~pW !1S s3(2)~s,D3,D* 3!1S 1(2)~D1,D* 1!1S 2(2)~D2,D* 2!,
S r(2)~Dr,D* r!52E d4xDrD* r4G21
2Trs f cx$aD
rerg5dD* rerg5%. ~18!
In the following we are going to search for gapless bosoexcitations of the 2SC quark-matter ground state ofmodel only. Since we are mainly interested in the diquasector, the expression for thep-meson effective actionS pp(2)(pW ) is omitted here. Moreover,S s3(2)(s,D3,D* 3) has arather cumbersome form, so we also do not show it.@Ifneeded, both these expressions can be obtained immedifrom Eq. ~11!.#
Before proceeding further, we should fix the parametof our model. Here, we choose the parametrization procedgiven in @15#. We simplify that model by ignoring the corrections to the pion-quark coupling constant that come frthe mixing between the pion and axial vector. To fix tmodel parameters, we require the model to reproduceGoldberger-Treiman relation Mgp5Fp , where Fp'93 MeV is the pion weak-decay constant, andgp is thepion-quark coupling constant determined in the local Nmodel as follows:6-4
ABNORMAL NUMBER OF NAMBU-GOLDSTONE BOSONS . . . PHYSICALREVIEW D 70, 014006 ~2004!gp225
~This integral is divergent, and a regularization is suppoto be implemented here. In our work we use a thrdimensional cutoff to make such integrals meaningful.!
Then, we require the observed pion mass (;140 MeV) tobe reproduced by the local NJL model with our parameteAfter this, one more condition is needed to completeparameter fixing procedure. We consider two possibilitto finalize it: ~i! fix the model parameters so that the QCsum rules estimate for the chiral condensate^qq&'(2245 MeV)3 is reproduced in our model,~ii ! choose themodel parameters in a way that allows one to obtaincorrect description of therpp decay, as was done in@15#.In the first case~let us refer to this set of parameters asA!, the cutoffL is about 618 MeV, and the four-quark inteaction constantG1 is equal to 5.86 GeV
22. The remainingconstantG2 is chosen according to the Fierz transformatiasG253G1/454.40 GeV
22. The current quark massm0 isfixed via the gap equations to the value: 5.7 MeV. The cstituent quark mass in the vacuum~at zeroth temperature anchemical potential! amounts to 350 MeV.
In the second case, we obtain a different set of moparameters, which will be referred to, throughout the papas set B. Imposing the condition that an extension of tmodel to the vector channel would give the experimenvalue for ther meson width~ther meson decays mostly inta couple of charged pions, and the decay is described byconstantgr'6.1.! and keeping in mind that in the local NJmodel one gets a natural connection betweengr and gp~discarding pionaxial-vector transitions!, A6gr5gp , weobtain L5856 MeV, G152.49 GeV
22, G253G1/451.87 GeV22, andm053.6 MeV. In the vacuum the constituent quark mass is 233 MeV, which is small, comparedthe case, where the chiral condensate is used to fix the mparameters.3 A similar procedure has been implemented@28# in the chiral limit. Here, we keep the current quark manonzero in order to reproduce the pion mass in the vacu
Solving the gap equations at a fixed chemical potentwe obtain the constituent quark massM and the color gapDthat satisfy the global minimum of the effective potential.Figs. 1 and 2, one can see the quark massM and the colorgapD versus the chemical potential for parameter sets AB. For parameter set A, one observes a first-order phasesition from hadron matter to the 2SC phase. The gaps~con-stituent quark mass and color gap! change abruptly when thchemical potential reaches the vacuum value ofM ~see Fig.1!. On the contrary, if parameter set B is chosen, the greveal a rather smooth character near the phase transwhich is typical for a phase transition of second order~seeFig. 2!.
3A larger quark mass can be obtained if one takes into accounmixing between the pion and axial vector. However, we omit itour paper for simplicity and for the reason that this mixing is neligible in the 2SC phase.01400d-
III. GAPLESS EXCITATIONS IN THE DIQUARK SECTORS
SinceD1 in Eq. ~17! is not mixed with other fields, let usfirst of all, focus on the diquarkD1 sector. If D1(x)5@w1(x)1 iw2(x)#/A2, D* 1(x)5@w1(x)2 iw2(x)#/A2,then the effective actionS 1(2)(D1,D* 1) from Eq.~17! can bepresented in the form
S 1(2)~D1,D* 1![S 1(2)~w1 ,w2!
2E d4xd4ywk~x!Gkl~x2y!w l~y!,~20!
where the 232 matrix G(x2y) is the inverse propagator othe w1 ,w2 fields @summation overk,l 51,2 is implied in Eq.~20!#. Its matrix elements can be found via a second variatof S 1(2) :
FIG. 1. The constituent quark massM ~solid line! and the colorgapD ~dashed line! as functions of the chemical potential~param-eter set A!.
FIG. 2. The constituent quark massM ~solid line! and the colorgapD ~dashed line! as functions of the chemical potential~param-eter set B!.6-5
BLASCHKE et al. PHYSICAL REVIEW D 70, 014006 ~2004!Gkl~x2y!52d2S 1(2)
dw l~y!dwk~x!. ~21!
In momentum space, the Fourier-transformed componenthe G matrix have the structure
2@GD* D~p!1GD* D~2p!#,
2@GD* D~p!2GD* D~2p!#. ~22!
The derivation of these relations is given in the Appendwhere the quantityGD* D(p) is also presented@see Eq.~A5!#.The particle dispersion laws are defined by zeros of detG(p)in the p0 planei.e., by the equation
5GD* D~p!GD* D~2p!50. ~23!
Clearly, the solutions of Eq.~23! with positive and negativep0 correspond to particles and antiparticles, respectively.us first putp5(p0,0,0,0). In this case, detG(p0) is an evenfunction ofp0, so solutions of Eq.~23! in thep0
2 plane mightbe searched for. They are the squared masses of exstates in theD1 sector of the model. Moreover, all thesformulas are then greatly simplified. Indeed, it follows froEq. ~A5! at pW 250 that
GD* D~p0!54ip0E d4q~2p!4
where it is possible to integrate overq0, using the followingprescription: (p01q0)(p01q0)1 i sgn(p01q0) and q0q01 i sgn(q0) with 01 @see also comments after fomula ~10!#, and
In Eq. ~25! ED65A(E6)21uDu2. Since the integral on the
right-hand side~RHS! of this equation is ultraviolet divergent, we regularize it, as the other divergent integrals,using a three-dimensional cutoffLi.e.,qW 2
ABNORMAL NUMBER OF NAMBU-GOLDSTONE BOSONS . . . PHYSICALREVIEW D 70, 014006 ~2004!Fig. 4 the axism begins with 250 MeV. At smallerm ~butgreater than 233 MeV!, the value ofm1 is almost zero. Com-mon to both parameter sets A and B is that the values ofm1are small and thus justify the series expansion of the funcH(p0) aroundp050.
4 Analogous results are obtained for thD2 sector of the model. There, as in theD1 sector, only onemassless boson with the quadratic dispersion law as wellmassive one with the massm2[m1 is found. Evidently, thetwo massless and two massive bosons in theD1, D2 sectorsform antidoublets with respect to the unbrokenSUc(2) sym-metry.
Now let us consider the sector with mixing of the complex diquarkD3 field and the scalars meson. Introducingnew real fields f(x)5@D3(x)1D* 3(x)#/A2, c(x)5 i @D* 3(x)2D3(x)#/A2, we rewrite the effective actionS s3(2) @here we omit the cumbersome expression for this qutity, however it can be easily reproduced from Eq.~11!# inthe form
S s3(2)~s,D3,D* 3!521
2E d4xd4ys~x!,f~x!,c~x! P~x2y! s~y!,f~y!,c~y!t, ~28!
whereP(x2y) is the inverse propagator of thes meson andthe diquark fieldsf,c. Clearly, it is a 333 matrix, whoseelements can be obtained from Eq.~28! by taking all secondvariational derivatives ofS s3(2) over the fieldss, f, andc.We have calculated the momentum-space matrixP(p) ~cor-responding explicit formulas and relations are omitted!, andthen, puttingp5(p0,0,0,0), found that det@P(p0)# has onlyone zero atp0
250. Therefore, only one NG boson, whichan SUc(2) singlet, exists in this sectoras in theD
4The numerical investigation shows that the mass valuem1 foundin the low-p0 approximation~see Figs. 3 and 4! differs from theexact solution of the equationH(m1)50 by no more than 10%.
FIG. 4. The diquark massm15H(0)/H8(0) as a function of thechemical potential~parameter set B!.01400n
sectors, each zero of det@P(p0)# in the p02 plane means the
mass squared of the ground-state excitation, this time inmixed s-D3 sector.5
As was mentioned above, in the 2SC phase of the mowhereD0, the colorSUc(3) symmetry of the ground statis spontaneously broken down to theSUc(2) group. Hence,in accordance with the Goldstone theorem, it is generaexpected that five NG bosons would appear in the theHowever, we have proved that in theD1,D2,D3 sectors ofthe model only three massless bosons exist, which canidentified with NG bosons. Therefore, there seems to bdeficiency of two NG bosons in the framework of the modunder consideration. In spite of this fact, there are, howeno contradictions with the NC theorem, since two of tthree found NG bosons have quadratic dispersion lawsupW u0. In addition, the model contains also light massidiquarks.
Usually, in the framework of the NJL model, the 2Sphase is studied in the region of coupling constants~2!. Inthis case, atm50 we have a chirally noninvariant phaseven ifm00, and the transition to the 2SC phase occurssome finite value of the chemical potential. Formally, hoever, one can consider the region
v5$~G1 ,G2!:G2.p2/~4L2!, p2~G22G1!.2G1G2L
where~even atm50) the 2SC phase is realized@32#. As caneasily be seen from Eq.~27!, in the case ofm50 we haveH(0)50 andm15m250, and the total number of gaplesexcitations~NG bosons! in the D1 and D2 sectors of themodel equals 4. They form two masslessSUc(2) antidou-blets. Apart from this, there is one NG boson, which isSUc(2) singlet, in the mixeds-D
3 sector. Naturally, all theseexcitations have a linear dispersion law, as should be inrelativistically invariant case~see the Appendix!. However,at an arbitrary smallm, particles from one of these antidoublets acquire nonzero masses, and the dispersion laws fotwo massless particles from the remaining antidoubletchanged essentially from the linear to quadratic laws.
Note finally, that in the above consideration we usedreal-imaginary part parametrization for diquark fields:
D2~x!5~ w1~x!1 i w2~x!!/A2,
D3~x!2D5~f~x!1 ic~x!!/A2, ~30!
whereDr(x) are the unshifted diquark fields from Lagranian ~3! andD5^D3&. However, one can use also an alterntive polar coordinate parametrization
5A detailed investigation of the meson-diquark spectrum includthe s-D3 mixing is in preparation.6-7
BLASCHKE et al. PHYSICAL REVIEW D 70, 014006 ~2004!S D1~x!D2~x!D3~x!
D 5expH 2 i(a lat ja~x!A2D J 1A2 S 00A2D1h~x!D5
A2 S j52 i j4j72 i j6A2D1h1 i 2A3
j8D 1o~j i ,h!, ~31!
wherela are Gell-Mann matrices and the summation ova54, . . . ,8 isimplied @the form of an exponential multipliein Eq. ~31! corresponds to the fact thatDr(x) is anSUc(3)antitriplet#. Comparing Eqs.~30! and~31! at smallj i ,h, wesee that
w15j5 , w252j4 , w15j7 ,
w252j6 , f5h, c52
A3j8 . ~32!
Further, one should insert Eq.~31! into Eq. ~5! and expandthe resulting expression into a series of meson andja ,hfields. It is easily seen that in the second order of the nvariables the effective action is a sum similar to Eq.~17!. Inparticular, it means that the diquark fieldsj4 ,j5 are decou-pled from other fields, and their effective action looks liEq. ~20! with the replacement~32! used. So some elemenof the inverse propagator matrix forj4 ,j5 fields differ by asign from Gkl(x2y), Eq. ~21!, but the determinant of thewhole momentum space matrix is not changed. Hence,mass spectrum, the dispersion laws, etc., in thej4 ,j5 fieldsare the same, as in the case with old fieldsw1 ,w2. Similarconclusions are valid for the rest of new variables; i.e., sphysical characteristics of the model, such as particle masdispersion laws, etc., do not depend on the choice of fiparametrizations, Eq.~30! or ~31!. In particular, the numbeof abnormal NG bosons, which equals 3 in the model unconsideration, is a parametrization-invariant quantity.6
IV. COLOR ASYMMETRY AND MASSESOF SCALAR DIQUARKS
As was noted in@13#, the color superconducting phaseautomatically color neutral in QCD, but this does not hofor the NJL model under consideration. Massive scalarquarks and, as a consequence, the abnormal number obosons in the model discussed in previous sections testifythis. As mentioned in the Introduction, for the number of Nbosons to equal the number of broken generators of the
6In Refs. @4,5# an abnormal number of NG bosons in the saSU(2)3U(1) toy model was discovered. But in@4# the parametri-zation~30! was used for the scalar doublet, whereas in@5# a polarcoordinate parametrization, similar to Eq.~31!, was used. Evi-dently, their results coincide; i.e., the abnormal number of Nbosons, particle masses, formulas for dispersion laws are the s01400r
derlying symmetry group@in our caseSUc(3)], theground-state expectation value for the commutator of any two bken symmetry generatorsQa and Qb must be zero:^@Qa ,Qb#&50 @5#. This criterion is not fulfilled for the NJLmodel considered herenamely, the ground-state expetion value of the color charge operatorQ85qg
0l8q does notvanishand, therefore, the usual counting rule for Nbosons is not applicable.
Let us look at the functionH(p0) at p050. One canrewrite the expression~27! in a different form:
D2S 22u~m2E!1 E1
This formula can be expressed in terms of quark densitFor this purpose, we need the expression of the thermonamical potential of the system, which in the mean-field aproximation is equal to the value ofVeff in its global mini-mum point, supplied by the gap equationsi.e.,
V~m!5Veff~s0 ,pW 0 ,D0d ,D0*
whereVeff was given in Eq.~8!. Further, one can divide thethermodynamical potential into several terms
whereVc(m) is the contribution from the color-c quark, de-fined as follows:
~ uE1u1uE2u!. ~36!
The contributions from color-1 and color-2 quarkV1(2)(m), are equal due to the remainingSUc(2) color sym-metry after the globalSUc(3) symmetry group is spontaneously broken in the color superconducting phase. The thcolor quark, which does not condensate, givesV3(m). Then,one can calculate the densitiesnc of color-1, -2, and -3quarks separately without introducing additional chemipotentials for each color simply by differentiatingVc(m)over mi.e.,
]m, c51,2,3, ~37!
ABNORMAL NUMBER OF NAMBU-GOLDSTONE BOSONS . . . PHYSICALREVIEW D 70, 014006 ~2004!Equation~33! can thus be transformed to
From Eq. ~39! and the fact that the diquark massm15H(0)/H8(0) is found to be nonzero~see Figs. 3 and 4!,one concludes that the 2SC ground state in the NJL mounder consideration has finite color charge^Q8&. Evidently,this fact is related to the inequality of color quark densitn1,2.n3 for paired ~1,2! and unpaired~3! quarksi.e., tocolor asymmetry. Moreover, as discussed in the Appenthe nonzero value ofH(0) is responsible for the abnormadispersion lawp0;pW
2/H(0) for two NG bosons.To restore the local color neutrality in the NJL model, o
can, e.g., include an additional termm8Q8 and impose thecolor-neutrality condition
or consider the superconducting matter as composed ofored domains, with the global color charge being equal zThe latter, however, goes beyond the mean-field approxition exploited here and should be considered as an appmation for the case of large colored domains. Anyway,true ground state should be checked to give the absominimum of the thermodynamical potential.
The inclusion of an additional chemical potentialm8 byhand, however, does not contradict QCD. As discussed@13#, a nonvanishing expectation value of the gluon fieldA0
8,corresponding to the eighth generator in theSUc(3) group,gives rise to a nonvanishing value ofm8, which then cancelscertain tadpole contributions and restores the color neutraThis cannot occur in the NJL model automatically becathere are no gluons in it.
V. SUMMARY AND DISCUSSIONS
Recently, it has been shown that in QCD with a nonzstrangeness chemical potential@4,5#, as well as in some models for relativistic massive vector field condensation@6#, thenumber of NG bosons can be less than the number of brosymmetry generators, so that the usual counting rule forbosons evidently does not hold. In the present paper, we hpresented another relativistic model providing us with annormal number of NG bosons. This is an NJL-type mowith two light-quark flavors, where a single quark-chemicpotential is taken into account. In the color superconductphase of this model, the colorSUc(3) group is spontaneously broken down toSUc(2); i.e., the number of brokensymmetry generators equals 5. Despite this fact, therepeared only three NG bosons in the 2SC phase; two of thform an antidoublet with respect to the unbrokenSUc(2)subgroup and have quadratic dispersion laws. The remaione is anSUc(2) singlet, so there are no contradictions wthe NC theorem@3#. Moreover, there exists anSUc(2) anti-doublet of light diquarksD1,D2 with massesm1, the behav-01400el
ior of which at final chemical potentialm is shown in Figs. 3and 4 for the parameter sets A and B, respectively.
Most interestingly, we found that the diquark massesm15H(0)/H8(0) are proportional to the ground-state averaof the color charge operatorQ8&51/A3(n11n222n3).Here, the quark densitiesn1 , n2 of paired quarks with color1,2 are equal due to the remainingSUc(2) symmetry andlarger than the densityn3 of the unpaired color-3 quarkThus, the ground state of the considered NJL-type modeevidently color asymmetric. It is just the appearance of tnonvanishing expectation value^Q8& which simultaneouslyleads to a violation of the criterion for the equality of thnumber of NG bosons and broken symmetry generators@5#and to the appearance of the quadratic dispersion law forNG diquarks@compare Eqs.~38!, ~A6!#.
In the investigations of color superconductivity in thtwo-flavored QCD, five scalar NG bosons are argued toist. They are mixing with gluons and required as longitudincomponents for five massive gluons~color Meissner effect!@11#. The fact that we find an abnormal number of three Nbosons does not directly contradict QCD. Namely, in tcolor superconducting phase of QCD a nonvanishing cdensate of the eighth gluon field componentA0
8 appears,which induces a color chemical potentialm8 and cancelscertain tadpole contributions, responsible for the nonvaning color charge of the ground state@13#. This mechanismjust leads to color neutrality. Obviously, there arisesquestion, how can one reconcile the considered NJLproach with QCD? Insofar as NJL-type models do not cotain gluon fields, the required contribution from the codensed eighth gluon field does not follow automaticaTherefore, the tadpole contribution becomes unbalancedgluon contributions. It can be cancelled by hande.gwith the help of the additional color chemical potentialm8simulating the omitted gluon contribution. In such enlargNJL models the condition of color neutrality of the grounstatei.e., Q8&50can be imposed as an additional phycal requirement@12#. According to our results, one gets ithis case two additional NG bosons as well as a changquadratic dispersion laws into normal linear ones. Thus,abnormal number of three NG bosons might, in principbecome converted into the normal number five foundQCD.
Nevertheless, what to do now with these five NG bosoarising in the color neutral 2SC phase of an enlarged Ntype model? Clearly, in the framework of the standard Nmodel one cannot apply the color Meissner effect, in ordeabsorb these NG bosons into longitudinal degrees of freedof massive gluons. A possible way out of this problem coube an extension of the NJL mode via the inclusion of pertbative gluons, fluctuating around the original low-energluonic fields~realized, e.g., by instantons or other nonpturbative background fields, which were already integraout to yield the effective four-quark interactions! @33#. Obvi-ously, the color Meissner effect could then be realizedsuch perturbative gluons.
In addition, let us remark that the above discussions reto the case of locally neutral quark matter, where color ntrality should indeed be required. On the other hand, glo6-9
BLASCHKE et al. PHYSICAL REVIEW D 70, 014006 ~2004!color neutrality does not demand local color neutrality:example, one can assume the ground state as beingposed of colored domains. Anyway, to decide which ofconsidered ground states is in favor, the absolute minimof the thermodynamic potential should be calculated in ecase.~Note that the presence of domains means a violatiotranslational invariance and is not consistent with the simmean-field approach.!
Finally, notice that in the framework of the considerNJL-type of model there arises a mixing between thes me-son andD3 diquark in the 2SC phase. The behavior ofs-,D3- as well asp-meson masses versusm in the 2SC phase isnow under consideration.
This work has been supported in part by DFG proj436 RUS 113/477/0-2, RFBR grant 02-02-16194,Heisenberg-Landau Program, and the Dynasty Foundtion.
APPENDIX: THE G-MATRIX ELEMENTSAND DISPERSION LAWS
Using the effective action~18!, let us introduce auxiliaryquantities
GD* D~z!52d2S 1(2)
4G22 i Trsc$a~z!e
GDD* ~z!52d2S 1(2)
4G22 i Trsc$d~z!e
whered(z) is the Dirac delta function,z5x2y, and a(z)and d(z) are operators~13! and ~16!, respectively. Sinced2S 1(2)/(dD1dD1)5d2S 1(2)/(dD* 1dD* 1)50 and
A2 S ddD 1 ddD* D , ddw2 5 iA2 S ddD 2 ddD* D ,the matrix elements of the inverse propagator matrix~21!have the form
2@GD* D~z!1GDD* ~z!#,
2@GD* D~z!2GDD* ~z!#. ~A2!
For an arbitrary functionF(z), it is possible to define theFourier-transformed one,F(p), by the relation01400rm-
F~p!5E d4zF~z!eipz, i.e. F~z!5E d4p~2p!4
Now, using Eq.~A3!, it is possible to get, from Eq.~A1!,
4G22 iTrscE d4q
4G22 iTrscE d4q
where the Fourier-transformed expressionsa(q), d(q) canbe easily derived from Eqs.~13!, ~16!. It follows from Eqs.~A4! that GD* D(2p)5GDD* (p). So, taking into accounEqs.~A2!, one can easily obtain the relations~22!.
After tedious calculations we arrive at the expression
3Fp01q01m1EqD2~q0! 1 p01q01m2EqD1~q0! G2
3FEqpW 21~q01m1Eq!~pW qW !EqD2~q0!
22~q01m2Eq!~pW qW !
EqD1~q0!G . ~A5!
Here, the notation that was introduced after Eq.~10! is used.
Moreover,Eq[E5AqW 21M2. Note that expression~A5! isobtained under the assumption thatD0; i.e., it is valid onlyin the 2SC phase. In this case, the gap equation~10! wasused to eliminate the coupling constantG2 from relation~A4! in favor of other model parameters.
Quasiparticle dispersion laws are defined by Eq.~23! or,equivalently, by the two equationsGD* D(6p)50. One caneasily see that atpW 250 the first integral in RHS of Eq.~A5!coincides with Eq.~24! having an evident zero atp050.This is just the dispersion law of a gapless particle~NGboson! at pW 250. Let us find the dispersion law for the NGboson at small nonzero values ofpW 2i.e., try to solve theequationGD* D(p)50 at p0. Expanding expression~A5!into a Taylor series of (p0 ,pW ) at the point (0,0W ) and takinginto account only leading terms, we have, atm0, the fol-lowing equation connecting the energy and momentum ofmassless particle~in Sec. III, it was proved that such a paticle, NG boson, exists in theD1 sector of the model!:6-10
ABNORMAL NUMBER OF NAMBU-GOLDSTONE BOSONS . . . PHYSICALREVIEW D 70, 014006 ~2004!GD* D~p![4ip0H~0!24ipW 2
3H 12 2qW 23Eq~q01m2Eq!J2
3H 11 2qW 23Eq~q01m1Eq!J 1. D
where the quantityH(0) ~which is nonzero atm0) is givenin Eq. ~27!. Obviously, the quadratic dispersion law,p0;pW 2, for a massless particle follows from Eq.~A6!. How-ever, in case of relativistic invariance of the firstname.lastname@example.org., atm50 and for coupling constants from the regionv, Eq.~29!# or for a color-neutral ground state, whenH(0)50, thedispersion law for a NG boson changes. Indeed, in this cthe term 4ip0H(0) from Eq. ~A6! should be replaced by4ip0
2H8(0) @H8(0) is presented in Eq.~27!#, and one arrivesat a linear dispersion law for NG bosons..
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