25 March 1999

Ž .Physics Letters B 450 1999 325–331

High density quark matter and the renormalization group in QCDwith two and three flavors

Thomas Schafer, Frank Wilczek¨Institute for AdÕanced Study, School of Natural Sciences, Princeton, NJ 08540, USA

Received 10 November 1998Editor: W. Haxton

Abstract

We consider the most general four fermion operators in QCD for two and three massless flavors and study theirrenormalization in the vicinity of the Fermi surface. We show that, asymptotically, the largest coupling corresponds to scalar

Ž .diquark condensation. Asymptotically the direct and iterated molecular instanton interactions become equal. We providesimple arguments for the form of the operators that diagonalize the evolution equations. Some solutions of the flowequations exhibit instabilities arising out of purely repulsive interactions. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 11.30.Rd; 12.38.Aw; 12.38.Mh

1. Recently there has been renewed interest in theproblem of hadronic matter at high baryon density. Itwas realized early on that at very high density,asymptotic freedom and the presence of a Fermisurface implies that attractive interactions betweenquarks cause a BCS instability, and cold quark mat-

w xter is a color superconductor 1 . More recently, theproblem was studied again, and it was emphasizedthat non-perturbative effects, instantons, could lead

w xto gaps on the order of 100 MeV 2,3 . A particularlyinteresting case is QCD with three light flavors. Inthis case the preferred order parameter involves acoupling between color and flavor degrees of free-

w xdom 4 . The resulting coherence leads to large gapseven for weak interactions. In addition to that,color-flavor locking implies that chiral symmetry isbroken. The usual quark condensate is non-vanishing

even in the high density phase, as is a gauge-in-variant D Bs2 condensate, signalling true superflu-

w xidity 4,5 .Most of these calculations were performed in the

Žmean field approximation. In the instanton model, aw x .number of refinements have been included, see 6,5 .ŽOne assumes a specific form of the interaction in-

.stantons, one gluon exchange, . . . , makes an ansatzfor the form of the condensate, and solves the gapequation in some approximation. Clearly, it is desir-able to analyze the structure of the theory for themost general form of the interaction, and to have aguiding principle for constructing a systematic ex-pansion scheme.

In this context, the renormalization group ap-proach to cold Fermi systems appears particularly

w xpromising 7,8 . In a cold Fermi system, the only

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00162-8

( )T. Schafer, F. WilczekrPhysics Letters B 450 1999 325–331¨326

relevant interactions take place in vicinity of theFermi surface. The corresponding excitations arequasi-particles and quasi-holes, described by an ef-fective action of the form

S s dt d3pc † iE y e p ye c . 1Ž . Ž .Ž .Ž .Heff t F

We can analyze the general structure of the quasi-particle interactions by studying the evolution of thecorresponding operators as we successively integrateout modes closer and closer to the Fermi surface.The main result of this analysis is that, in general,four fermion, six fermion, and higher order interac-tions are suppressed as we approach the Fermi sur-face. This fixed point corresponds to Landau liquid

w xtheory 9 . The only exception is the four-fermionoperator that corresponds to two particles from oppo-

Ž .site corners of the Fermi surface p ,yp scatter-F FŽ X X .ing into p ,yp . This kind of scattering leads to aF F

logarithmic growth of the coupling constant as weapproach the Fermi surface, and the well knownBCS instability.

2. For QCD, this approach was first employed byw xEvans, Hsu, and Schwetz 10 . These authors con-

centrated on the one gluon exchange interaction, butas we will see in a moment, their results are suffi-cient to deal with the most general quark-quark

Ž .interaction for QCD with three or more flavors. Weshould emphasize here that we will restrict ourselvesto massless QCD, a spherical Fermi surface, andlocal operators invariant under the appropriate chiralsymmetry.

w xFollowing 10 , we take the basic four fermionoperators in N s3 QCD to bef

20 0O s c g c , O s c g c c g cŽ . Ž . Ž .L L L 0 L L R L 0 L R 0 R

2Ž .

2i iO s c g c , O s c gc c gc .Ž . Ž . Ž .L L L i L L R L L R R

Each of these operators comes in two color struc-tures, for example color symmetric and color anti-symmetric

a b c dc c c c d d "d d . 3Ž . Ž .Ž . Ž . ab cd ad bc

Nothing essentially new emerges from consideringsuperficially different isospin structures, or differentDirac matrices. All such structures can be reduced to

Ž .linear combinations of the basic ones 2 , or theirparity conjugates, by Fierz rearrangements. In total,we have to consider eight operators.

These operators are renormalized by quark-quarkscattering in the vicinity of the Fermi surface. Thismeans that both incoming and outgoing quarks have

< < < <momenta p , p ,"p and p , p ,"q with p , q1 2 3 4

,p . We can take the external frequency to be zero.Fw xA graph with vertices G and G then gives 101 2

X XG G I G G y g gŽ . Ž . Ž . Ž .i j1 2 1 1 0 0i i k k k l

1X Xq g g G G 4Ž . Ž . Ž . Ž . Ž .i j j jk l 2 23 l l

i 2 Ž . w xwith Is m log L rL . Here L , L isIR UV IR UV28p

the range of momenta that was integrated out. Wewill denote the density of states on the Fermi surface

2 Ž 2 .by Nsm r 2p and the logarithm of the scaleŽ . w xts log L rL as in Ref. 10 . The renormaliza-IR UV

tion group does not mix LL and LR operators, and italso does not mix different color structures. Thismeans that the evolution equations contain at most2=2 blocks. For completeness, we reproduce the

w xresults of 10

d G L L qG L L NŽ . 20 i L L L Lsy G qG 5Ž .Ž .0 idt 3

d G L L y3G L LŽ . 20 i L L L LsyN G y3G 6Ž .Ž .0 idt

d G L R q3G L RŽ .0 is0 7Ž .

dt

d G L R yG L R 2 NŽ . 20 i L R L Rsy G yG 8Ž .Ž .0 idt 3

In this basis the evolution equations are alreadydiagonal. The coupling G sG L L qG L L evolves as1 0 i

1G t s 9Ž . Ž .1 1q Nr3 G 0 tŽ . Ž .1

( )T. Schafer, F. WilczekrPhysics Letters B 450 1999 325–331¨ 327

with analogous results for the other operators. Notethat the evolution starts at ts0 and moves towardsthe Fermi surface t™y`. If the coupling is attrac-

Ž .tive at the matching scale, G 0 )0, it will grow1

during the evolution, and reach a Landau pole atŽ Ž ..t s3r NG 0 . The location of the pole is con-c 1

trolled by the initial value of the coupling and thecoefficient in the evolution equation. If the initialcoupling is negative, the coupling decreases during

Ž .the evolution. The second operator in 5 has thelargest coefficient and will reach the Landau polefirst, unless the initial value is very small or nega-tive. In that case, subdominant operators may deter-mine the pairing.

The particular form of the operators that diagonal-ize the evolution equations is easily understood. Letus first order the operators according to the size ofthe coefficient in the evolution equations

2 2O s c g c y c gc , 10Ž .Ž . Ž .dom L 0 L L L

O s c g c c g cŽ . Ž .sub ,1 L 0 L R 0 R

1y c gc c gc , 11Ž .Ž . Ž .L L R R3

2 21O s c g c q c gc , 12Ž .Ž . Ž .sub ,2 L 0 L L L3

O s c g c c g c q c gc c gc .Ž . Ž . Ž . Ž .mar L 0 L R 0 R L L R R

13Ž .This result can be made more transparent by Fierzrearranging the operators. We find

O s2 c Cc c Cc , 14Ž . Ž .Ž .dom L L L L

1O s c Cgc c Cgc q . . . , 15Ž . Ž .Ž .sub ,1 L R R L3

4O s c CSc c CSc , 16Ž . Ž .Ž .sub ,2 L L L L3

1O s c Cg c c Cg c q . . . . 17Ž . Ž .Ž .mar L 0 R R 0 L2

This demonstrates that the linear combinations inŽ . Ž .10 – 13 correspond to simple structures in thequark-quark channel. It also means that it might havebeen more natural to perform the whole calculationdirectly in a basis of diquark operators.

Ž .The full structure of the LR operators isŽ .Ž .O ,O s c Cg t c c Cg t c qs u b ,1 m a r S , A S , A

Ž .Ž .c Cgg t c c Cgg t c where t are sym-5 A,S 5 A,S S, A

metricranti-symmetric isospin generators. Note thatbecause the two structures have different flavor sym-metry, the flavor structure cannot be factored out.

The dominant operator corresponds to pairing inthe scalar diquark channel, while the subdominantoperators contain vector diquarks. Note that from theevolution equation alone we cannot decide what thepreferred color channel is. To decide this question,we must invoke the fact that ‘‘reasonable’’ interac-tions, like one gluon exchange, are attractive in thecolor anti-symmetric repulsive in the color symmet-ric channel. Indeed, it is the color anti-symmetricconfiguration that minimizes the total color fluxemanating from the quark pair. If the color wavefunction is anti-symmetric, the dominant operatorfixes the isospin wave function to be anti-symmetricas well.

The dominant operator does not distinguish be-tween scalar and pseudoscalar diquarks. Indeed, forN G3 the basic 4-quark operators consistent withf

chiral symmetry also exhibit an accidental axialbaryon symmetry, under which scalar and pseu-doscalar diquarks are equivalent. For N s3 thisf

Ž .degeneracy is lifted by formally irrelevant six-w xfermion operators 5 . The form of the dominant

operator indicates the existence of potential instabili-ties, but does not itself indicate how they are re-solved. For example, to see whether color-flavor

w xlocking 4 is important we should use the operatoras input to a variational calculation.

3. For two flavors, we have to take into accountadditional operators. At first hearing it might seemodd that with fewer basic entities we encounter morebasic operators. It occurs because for N s2, but notf

for larger values, two quarks of the same chiralityŽ . Ž .can form a chiral SU 2 =SU 2 singlet. Related to

Ž .this, for N s2 we have additional U 1 violatingf A

four fermion operators. These operators are inducedby instantons.

For N s3, instantons are six fermion operators,f

and they are irrelevant in the technical sense. Thisdoes not mean they are physically irrelevant, particu-larly since they break a residual symmetry. Theformation of the gap will cause the evolution of thecouplings to stop, and the instanton coupling remainsat a finite value. Instantons have important physical

w xeffects, even for N s3 5 . Most notably, instantonsfŽcause quark-antiquark pairs to condense even in the

.high density phase , and lift the degeneracy betweenthe scalar and pseudoscalar diquark condensates.

( )T. Schafer, F. WilczekrPhysics Letters B 450 1999 325–331¨328

The new operators are

O s det c c , O s det c Sc 18Ž .Ž . Ž .S R L T R Lf f

Both operators are determinants in flavor space. Forquark-quark scattering, this implies that the twoquarks have to have different flavors. The fact thatthe flavor structure is fixed implies that the color

Ž .structure is fixed, too. For a given qq spin, onlyone of the two color structures contributes. Finally,both quarks have to have the same chirality, and thechirality is flipped by the interaction.

These considerations determine the structure ofŽ .the evolution equations see Fig. 1 . Two left handed

quarks can interact via one of the instanton opera-tors, become right handed, and then rescatter through

Ž .an anti-instanton, or through one of the U 1 sym-A

metric RR operators. The result will be a renormal-ization of the LL vertex in the first case, and arenormalization of the instanton in the second. Theiterated instanton-anti-instanton interaction was dis-

w xcussed in great detail in Ref. 11 , and was argued tow xplay an important role in the high temperature 11

w xand high density phase of QCD 5 . We should notethat the flavor structure will always remain a deter-minant. Even though instantons generate all the Dirac

Ž .structures in 2 , the color-flavor structure is morerestricted.

Evidently, instantons do not affect the evolutionof the LR couplings at all. The evolution equations

Žof the LL couplings are modified to become hence-.forth we drop the subscript LL :

dG N0 2 2 2s yG q2G G y5G yK� 0 0 i i Sdt 2

q2 K K y5K 2 19Ž .4S T T

dG Ni 1 10 13 12 2 2s G y G G q G q K� 0 0 i i S3 3 3 3dt 210 13 2y K K q K 20Ž .4S T T3 3

dK NSs 2 yG qG K q2 G y5G K� 4Ž . Ž .0 i S 0 i Tdt 2

21Ž .dK NT 2s G y5G KŽ .� 0 i S3dt 2

2q y5G q13G K 22Ž . Ž .40 i T3

These equations can be decoupled as

dG N1 2 2sy G qK , 23Ž .Ž .1 1dt 3dK 2 N1

sy G K , 24Ž .1 1dt 3dG2 2 2syN G qK , 25Ž .Ž .2 2dtdK2

sy2 NG K , 26Ž .2 2dt

where G sG qG , K sK qK and G sG y1 0 i 1 S T 2 0

Fig. 1. Chiral structure of the evolution equations.

( )T. Schafer, F. WilczekrPhysics Letters B 450 1999 325–331¨ 329

3G , K sK y3K . The equations for G, K decou-i 2 S T

ple even further. We have

d G qKŽ .2 2 2syN G qK 27Ž . Ž .2 2dt

d G yKŽ .2 2 2syN G yK , 28Ž . Ž .2 2dt

as well as the analogous equation for G , K . These1 1

differential equations are now trivial to solve, lead-ing to

1 1 1G t s q , 29Ž . Ž .2 ž /2 aqNt bqNt

1 1 1K t s y , 30Ž . Ž .2 ž /2 aqNt bqNt

again with the analogous result holding for G , K .1 1Ž Ž . Ž ..y1Here, a,bs G 0 "K 0 . The result implies2 2

that G and K will grow and eventually reach a2 2

Landau pole if either a or b is positive. The locationof the pole is determined by the smaller of thevalues, t syarN or t sybrN. The same is truec c

for G and K , but the couplings evolve more1 1

slowly, and the Landau pole is reached later.At this level a number of qualitatively different

scenarios are possible, depending on the sign andŽ . Ž .relative magnitude of G 0 and K 0 , see Fig. 2

Ž . Ž .henceforth we drop all subscripts . If G 0 andŽ .K 0 are both positive then they will both grow, and

the location of the nearest Landau pole is determinedŽ . Ž .by G 0 qK 0 . The asymptotic ratio of the two

Ž . Ž .couplings is 1. If G 0 and K 0 are both negative,Ž .and the magnitude of G 0 is bigger than the magni-

Ž .tude of K 0 , then the evolution drives both cou-plings to zero. These are the standard cases. Attrac-tion leads to an instability, and repulsive forces aresuppressed.

More interesting cases arise when the sign of thetwo parameters a,b are different. The caseŽ . Ž .G 0 , K 0 -0 and K 0 ) G 0 is especiallyŽ . Ž .

Ž . Ž .weird. Both G 0 , K 0 are repulsive, but the evolu-Ž .tion drives G 0 to positive values. Both couplings

reach a Landau pole, and near the pole their asymp-totic ratio approaches minus one. Similarly, we can

Ž . Ž . Ž .have a negative G 0 and positive K 0 with K 0

Ž .) G 0 . Again, the evolution will drive G 0 toŽ .positive values.

4. The dominant and sub-dominant instanton op-erators are

2 2O s det c c y c Sc , 31Ž .Ž . Ž .dom R L R L

f

2 21O s det c c q c Sc . 32Ž .Ž . Ž .sub R L R L3f

Upon Fierz rearrangement, we find

O s2 c Ct c c Ct c , 33Ž . Ž .Ž .dom L 2 L R 2 R

2O s c Ct Sc c Ct Sc , 34Ž . Ž .Ž .sub L 2 L R 2 R3

corresponding to scalar and tensor diquarks. Bothoperators are flavor singlet. Overall symmetry thenfixes the color wave functions, anti-symmetric 3 forthe scalar, and symmetric 6 for the tensor. Thedominant pairing induced by instantons is in thescalar diquark channel, the only other attractivechannel is the tensor. All this neatly confirms the

w xscenario discussed in Refs. 2,3 . Note that condensa-tion in the tensor channel violates rotational symme-try. As a result, the gap equation has additional

w xsuppression factors and the gap is very small 2 .Just as we found for N G3, there is an appealingf

heuristic understanding for the amazingly simple be-havior of the evolution equations, obtained by fo-cussing on the diquark channels. Instantons distin-guish between scalar diquarks with positive and neg-ative parity. GqK corresponds to the positive parity

Ž .operator c Cg c and GyK to the negative parity5Ž .c Cc . The asymptotic approach of GrK™1, thencorresponds to the fact that scalar diquark condensa-tion is favored over pseudoscalar diquark condensa-

Ž .tion. This is always the case if K 0 )0. We alsoŽ . Ž .understand the strange case G 0 , K 0 -0 and

K 0 ) G 0 . In this case the interaction forŽ . Ž .scalar diquarks is repulsive, but the interaction in thepseudoscalar channel is attractive and leads to aninstability. Note that this can only happen if we havethe ‘‘wrong’’ sign of the instanton interaction, i.e..for usp . Similarly, we can understand why the

Žasymptotic ratio of the molecular instanton-anti-in-.stanton and direct instanton couplings approaches

( )T. Schafer, F. WilczekrPhysics Letters B 450 1999 325–331¨330

Ž . Ž .Fig. 2. Solutions to the evolution equations for cases with different relative strength of the U 1 violating interaction K dashed line andAŽ . Ž .U 1 conserving interaction G solid line .A

GrKs"1. Instantons induce a repulsive interac-tion for pseudoscalar diquarks. During the evolution,this coupling will be suppressed, whereas the attrac-tive scalar interaction grows. But this means that in

Ž .the pseudoscalar channel, the repulsive instantonŽ .and attractive molecular forces have to cancel in

the asymptotic limit, so the effective couplings be-come equal.

We have not made an attempt to match all thecoupling constants to a realistic model at the UVscale. To do so would require an understanding of

the instanton density, the relevant value of a , thes

screening mechanism, and many other things. Fromthe form of the instanton vertex we can fix the ratioof the two instanton-like couplings, K rK sT S

Ž . w x1r 2 N y1 12,13 . This again shows that the ten-c

sor channel is not expected to be important. Also,both one gluon exchange and higher order instantoneffects give G L L )0,yG L L )0. Thus the favored0 i

Ž .scenario is that both instanton and U 1 symmetricA

couplings flow at the same rate, and pairing isdominated by scalar diquarks.

( )T. Schafer, F. WilczekrPhysics Letters B 450 1999 325–331¨ 331

5. In summary, we find that the renormalizationw xgroup analysis broadly supports the findings of 2,3 .

The dominant coupling corresponds to scalar diquarkcondensation, the sub-dominant coupling to tensor

Ž . Ž .diquarks. Asymptotically, U 1 breaking and U 1A A

symmetric couplings flow at the same rate. Sinceinstantons have a definite isospin structure, there isone flavor symmetric coupling that evolves indepen-dently of instantons. Asymptotically, this couplingalso flows at the same rate. In principle there is thepossibility of flavor and color symmetric diquarkcondensates, but none of the model interactions sofar proposed is attractive in that channel.

The analysis presented here is incomplete in sev-Žeral ways. Realistic interactions one-gluon ex-

.change, instantons, . . . are momentum dependent,and that should be included in the evolution. Thiscomplication is particularly important for one-gluonexchange, because in perturbation theory the diagramis not completely screened, and has a divergence forsmall momentum transfers. For N s3 a self-con-f

sistent calculation ought to be possible, becausecolor-flavor locking completely screens the interac-tion.

In any case, the renormalization group only deter-mines the running of the couplings from the match-ing point down to some infrared scale. If a subdomi-nant coupling is unusually large at the matchingscale, it might still dominate the pairing. To decide

the form of the pairing and determine the gap, giventhe couplings, still requires a variational calculation

w xalong the lines of 4,5 .

After this work was begun, we learned that thew xauthors of 10 have also extended their study to

include instanton operators. This work was supportedin part by NSF-PHY-9513835.

References

w x Ž .1 D. Bailin, A. Love, Phys. Rep. 107 1984 325.w x2 M. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B 422

Ž .1998 247.w x3 R. Rapp, T. Schafer, E.V. Shuryak, M. Velkovsky, Phys.¨

Ž .Rev. Lett. 81 1998 53.w x4 M. Alford, K. Rajagopal, F. Wilczek, hep-phr9804403.w x5 R. Rapp, T. Schafer, E.V. Shuryak, M. Velkovsky, in prepa-¨

ration.w x6 G.W. Carter, D. Diakonov, hep-phr9807219.w x7 J. Polchinski, hep-thr9210046.w x Ž .8 R. Shankar, Rev. Mod. Phys. 66 1995 129.w x9 A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods

of Quantum Field Theory in Statistical Physics, PrenticeHall, Englewood Cliffs, New Jersey, 1963.

w x10 N. Evans, S. Hsu, M. Schwetz, hep-phr9808444.w x11 T. Schafer, E.V. Shuryak, J.J.M. Verbaarschot, Phys. Rev. D¨

Ž .51 1995 1267.w x12 M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B

Ž .163 1980 46.w x Ž .13 T. Schafer, E.V. Shuryak, Rev. Mod. Phys. 70 1998 323.¨