PHYSICAL REVIEW D, VOLUME 64, 096002Masses of the pseudo Nambu-Goldstone bosons in the two flavor color superconducting phase
V. A. Miransky*Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
I. A. Shovkovy*School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455
L. C. R. WijewardhanaPhysics Department, University of Cincinnati, Cincinnati, Ohio 45221-0011
~Received 19 April 2001; published 3 October 2001!
The masses of the pseudo Nambu-Goldstone bosons in the color superconducting phase of dense QCD withtwo light flavors are estimated by making use of the Cornwall-Jackiw-Tomboulis effective action. Parametri-cally, the masses of the doublet and antidoublet bosons are suppressed by a power of the coupling constant ascompared to the value of the superconducting gap. This is qualitatively different from the mass expression forthe singlet pseudo Nambu-Goldstone boson, resulting from nonperturbative effects. It is argued that the~anti!doublet pseudo Nambu-Goldstone bosons form colorless@with respect to the unbrokenSU(2)c# charmonium-like bound states. The corresponding binding energy is also estimated.
DOI: 10.1103/PhysRevD.64.096002 PACS number~s!: 11.15.Ex, 12.38.Aw, 26.60.1cetoo
Although there is no reliable observational signature yfrom the theoretical point of view, it is quite reasonableassume that the cores of compact stars are made of csuperconducting quark matter@1#. If we take this assumptionseriously, it becomes quite important to study the properof the possible color superconducting phases in full de~for an up to date review on color superconductivity, sRefs.@2,3#!. In this paper, we continue our study@4,5# of thepseudo Nambu-Goldstone~NG! bosons, related to the approximate axial color symmetry, in theS2C phase of colddense QCD.
Let us recall that the axial color transformation is nosymmetry of the QCD action. However, its explicit breakiis a weak effect at sufficiently high quark densities wherecoupling constantas(m) is small~m is a chemical potential!.This was our main argument in Refs.@4,5# suggesting theexistence of five~rather than one@6#! light pseudo NGbosons in theS2C phase of cold dense QCD. No mass esmates for these pseudo-NG bosons were given in Refs@4,5#. In this paper, we fill in the gap by developing the formaism and calculating the masses.
This paper is organized as follows. In the next section,briefly introduce our model and notation. In Sec. III, wdescribe our method, based on the Cornwall-JackTomboulis~CJT! effective action, for calculating mass esmates of the pseudo-NG bosons. Then, in Sec. IV, the leing order diagram is approximately calculated, usianalytical methods. The fate of the colored pseudo-bosons in the doublet and antidoublet channels is discuin Sec. V. In Sec. VI, we give our conclusions. In Appendic
*On leave of absence from Bogolyubov Institute for TheoretiPhysics, 252143, Kiev, Ukraine.0556-2821/2001/64~9!/096002~17!/$20.00 64 0960t,
A and B we present the general expression for the glupolarization tensor and the calculation of the integrals tappear in its definition. In Appendix C the problem of thgauge invariance in the loop expansion of the CJT effecaction is discussed.
II. THE MODEL AND NOTATION
Here we consider cold dense QCD with two light quaflavors ~u and d! in the fundamental representation of thSU(3)c color gauge group. In order to keep the analyticcalculation under control, we assume that the chemicaltentialm is much larger thanLQCD . Of course, when we talkabout the compact stars, this is a far stretched assumpTherefore, while extending our analytical results to the reistic densities existing, for example, at the cores of compstars, one should be very careful. While most of the qualtive results may survive without being affected, most of tquantitative estimates would probably be valid only up toorder of magnitude. From the viewpoint of a theorist, itstill most interesting to study the predictions of the micrscopic theory, i.e. QCD in the problem at hand. The pricesuch a luxury is the necessity to work at asymptotically ladensities.
Instead of working with the standard four componeDirac spinors, in our analysis below, it is convenient totroduce the following eight component Majorana spinors:
& S cDcDCD , cDC5CcDT , ~1!wherecD is a Dirac spinor andC is a charge conjugationmatrix, defined byC21gmC52gm
T and C52CT. In thisnotation, the inverse fermion propagator in the color supconducting phase reads@711#l
2001 The American Physical Society02-1
MIRANSKY, SHOVKOVY, AND WIJEWARDHANA PHYSICAL REVIEW D 64 096002@G~p!#2152 i S ~p01m!g01pW DD ~p02m!g
52 i S g0@~p02ep2!Lp11~p01ep1!Lp2# DD g0@~p02ep
D , ~2!beefi
hiswhere D5g0Dg0, Dabi j [g5 i j ab3@D
1# andthe on-shell projectorsLp
2 S 16aW pWupW u D , where aW 5g0gW , ~3!~note thatD6 are complex valued gap functions!. Color andflavor indices are denoted by small latin letters from theginning and the middle of the alphabet, respectively. By dnition, ep
65upW u6m andpW 52pW gW . The effects of the quarkwave function renormalization are neglected here.
Now, after inverting the expression in Eq.~2!, we arrive atthe following propagator:
G~p!5 i S R11 R12R21 R22D , ~4!where
R11~p!5g0I1Fp01ep2ED2 Lp21 p02ep
1G1g0I2F 1p02ep2 Lp21 1p01ep1 Lp1G , ~5a!
R22~p!5g0I1Fp01ep1ED1 Lp21 p02ep
1G1g0I2F 1p02ep1 Lp21 1p01ep2 Lp1G , ~5b!
R12~p!5g5FD1Lp2ED1 1 D
ED2 G , ~5c!
R21~p!52g5F ~D2!* Lp2ED2 1 ~D
ED1 G , ~5d!
22(ep6)22uD6u2, and the three color-flavor ma
trices are defined as follows:
~I1!abi j 5~dab2da3db3!d i j , ~6!
~I2!abi j 5da3db3d i j , ~7!
abi j 52iTab
2 i j . ~8!
Notice that, when using this quark propagator in loop callations, one should make the substitutions09600--
ED6ED61 i e, ~9!
p06ep2p06ep27 i e sgn~ep2!, ~10!
p06ep1p06ep17 i e, ~11!
in the denominators, and take the limit of vanishinge at theend. This is important for preserving the causality of ttheory.
III. DESCRIPTION OF THE FORMALISM
Let us start from a simple observation. If the model undconsideration had real NG bosons in the spectrum, its eftive potential as a function of the order parameterDab
i j wouldhave a degenerate manifold of minima. The dimensionsuch a manifold would be equal to the number of the Nbosons. We know, however, that no global symmetriesbroken in theS2C phase and, therefore, the potential shouhave a single nondegenerate global minimum. The existeof the pseudo-NG bosons means, however, that the poteis nearly degenerate along selected directions. The curvaalong these directions defines the masses of the pseudobosons. In the limit of the zero curvature, masses go to zas it should be for the NG bosons.
In order to select the directions in the color-flavor spathat correspond to the five pseudo-NG bosons introduceRefs. @4,5#, we should recall their definition. Thespseudo-NG bosons correspond to breaking of the appromate axial color symmetry, given by the following transfomations of the quark fields:
cDUP1cD1UP2cD , ~12a!
cDCU* P2cDC1UTP1cDC , ~12c!
cDCcDCP1UT1cDCP2U* . ~12d!
Of course, this isnot an exact symmetry of the model. Foexample, the kinetic term of the Lagrangian density traforms as follows:
cD~ i ]1mg01A !cDcD~ i ]1mg01P1UA U
1P2UA U !cD , ~13!and no transformation of the vector field could promote ttransformation to a symmetry.2-2
MASSES OF THE PSEUDO NAMBU-GOLDSTONE BOSONS . . . PHYSICAL REVIEW D 64 096002The axial color transformation, as defined above, allous to explicitly extract the phase factors of the gap that crespond to the nearly degenerate directions of interest,
DP1UDU* 1P2UDUT, U5exp~ ivATA!. ~14!One could considervA as the dynamical fields of thpseudo-NG bosons~which, up to a factor of the decay constant, are related to the canonical fields!. Such a substitutionleads to the following changes of the components ofquark propagator:
R22R22~v!5P2UTR22U* 1P1U* R22UT, ~15b!
R12R12~v!5P1UR12U* 1P2UR12UT, ~15c!
R21R21~v!5P2UTR21U1P1U* R21U, ~15d!assuming that the fieldsvA are constant in space-time.
In order to construct the effective potential, we useCJT formalism@12#. The corresponding general expressireads
V5 i E d4p~2p!4
Tr@ ln G~p!S21~p!2S21~p!G~p!11#
where V2@G# represents the two-particle irreducible~withrespect to quark lines! contributions~we will discuss thispoint below!. There are, in general, an infinite numberdiagrams inV2@G#. In our analysis, we leave only a fewleading order diagrams, graphically shown in Fig. 1.
Before proceeding to the actual calculation, let us tryunderstand which type of diagrams could produce a ntrivial dependence of the potential on the pseudo-NG fievA. To this end, we have to recall the origin of thpseudo-NG bosons under consideration. In particular, icrucial that their appearance is related to the breaking ofapproximate axial color symmetry. It is clear, then, thatdependence of the effective potential on the pseudo-NG
FIG. 1. Vacuum energy diagrams.09600sr-
son fields results from some mixing between the left-hanand the right-handed sectors of the theory. Since there isleft-right mixing in the diagrams containing a single qualoop @diagrams~a!, ~b!, and ~c! in Fig. 1#, the first threediagrams in Fig. 1 turn out to be irrelevant for our calcution. The corresponding contributions to the effective potetial are free of any dependence on the constantvA fields.
Now, we move over to more complicated diagramsFigs. 1~d! and 1~e!. Notice, that these last two diagrams anot two-particle irreducible with respect to gluon lines. This consistent with the fact that we consider the CJT actiona functional of only the quark propagatorfor a discussion ofthis point see Ref.@13#. Both diagrams could potentiallyproduce nontrivial mass corrections for pseudo-NG bosoIn the diagrams in Figs. 1~d! and 1~e!, a non-trivial mixing ofthe left- and right-handed quark sectors is possible becathere are two separate quark loops.
A simple calculation shows that the diagram in Fig. 1~d!gives no correction to the effective potential. Thus, the leing order corrections come from the diagram in Fig. 1~e!.The details of our calculation are presented in the next stion.
IV. LEADING ORDER CALCULATIONS
As we argued in the preceding section, the leading orcorrections to the masses of the pseudo-NG bosons cfrom the diagram in Fig. 1~e!. In this section, we give thedetails of the calculation and derive an approximate analcal result for the masses.
The analytical expression for the vacuum diagram in F1~e! reads
V@Fig. 1~e!#522ip2as2E d4pd4kd4q
where the vertex is
GAm5gmS TA 00 2~TA!TD . ~18!The expression in Eq.~17! contains two factors of the fol-lowing type:
2ipasE d4p~2p!4 Tr@GAmG~p!GBkG~p2q!#. ~19!In order to extract the dependence of this quantity onpsedo-NG boson fieldsvA, we perform the substitutions oall component functions, given in Eq.~15!. At the end, weexpand the result in powers ofvA, keeping the terms up tothe second order. Thus, we arrive at the following result:2-3
MIRANSKY, SHOVKOVY, AND WIJEWARDHANA PHYSICAL REVIEW D 64 0960022ipasE d4p~2p!4 tr@P1gmTvAR12~p!gk~ TvB!TR21~p2q!1P2gmTvAR12~p!gk~TvB!TR21~p2q!1P1gm~TvA!TR21~p!gkTvBR12~p2q!1P2gm~ TvA!TR21~p!gkTvBR12~p2q!2P1gmTvAR11~p!gkTvBR11~p2q!2P2gmTvAR11~p!gkTvBR11~p2q!2P1gm~TvA!TR22~p!gk~TvB!TR22~p2q!2P2gm~ TvA!TR22~p!gk~ TvB!TR22~p2q!#
5PAB,mk~q!1vXvYf XACf YBDPCD,mk~q!11
2vXvYf XADf YDCPCB,mk~q!1
2vXvYf XBDf YDCPAC,mk~q!. ~20!su
letheHere we used the shorthand notation,
2vBvCf BADf CDETE1...,
2vBvCf BADf CDETE1...,
where f BAD are the structure constants ofSU(3). Also, weintroduced the one loop polarization tensor in the colorperconducting phase@14#,
PAB,mk~q!52ipasE d4p~2p!4 tr@gmTAR12~p!gk~TB!T3R21~p2q!1g
By performing the traces over the color and flavor indicwe arrive at the following expression for the polarizatitensor~see Appendices A and B for details!:
PAB,mn~q!uA,B54,5,6,75dABP4mn~q!1 i ~dA4dB52dA5dB4
After substituting the expansion~20! in Eq. ~17!, we arrive at
2 E d4q
It is noticeable that the right-hand side of the last expressis independent of thev8 field, related to the singlepseudo-NG boson. This means that the diagram in Fig.~e!does not give any non-trivial contribution to the value of tcorresponding mass. Only the~anti-!doublet pseudo-NGbosons get a nonzero mass.
In order to understand this point, it is instructive to cosider the ideal case, assuming that the axial color symmtry generated byg5T8 is a true ~rather than approximate!symmetry of dense QCD. Then, the singlet pseudo-NGson would be related to the breaking of the restricted asymmetry, defined by the following transformation:
cexp~ ivg5!I1c1I2c, ~25a!
ccI1 exp~ ivg5!1cI2 , ~25b!
cCexp~ ivg5!I1cC1I2cC, ~25c!
cCcCI1 exp~ ivg5!1cCI2 , ~25d!
acting on the first two color quarks. The other symme
@U5(1)#, acting on the third color quarks would remain ubroken. This is a simple consequence of the fact that the tcolor quarks do not participate in the color condensatiNow, the axial color transformation generated byg5T8 couldbe thought of as the ordinary axial transformationU5(1)accompanied by the unbrokenU5(1). Therefore, it appearsto be equivalent to say that either the restricted or the onary axial symmetry is spontaneously broken in this idlimit. In reality, both the restricted and the ordinary axisymmetries are explicitly broken. However, while the formis broken perturbatively, the latter is broken by much smanonperturbative~instantonlike! effects. Because of that, thsinglet pseudo-NG boson is connected with the ordinaxial transformationU5(1) and its mass is zero in any ordeof the expansion in the coupling constant.
Nonperturbative analysis reveals that the singpseudo-NG bosonh has a nonzero mass. The value of tmass was estimated in Ref.@15#. In our notation, it reads2-4
MASSES OF THE PSEUDO NAMBU-GOLDSTONE BOSONS . . . PHYSICAL REVIEW...