Novemb er 1998 IASSNS-HEP 98/100 hep-ph/9811473 Continuity of Quark and Hadron Matter 1 Thomas Schafer and 2 Frank Wilczek School of Natural Sciences Institute for Advanced Study Princeton, NJ 08540 Abstract We review, clarify, and extend the notion of color- avor lo cking. We present evidence that for three degenerate avors the qualitative features of the color- avor lo cked state, reliably predicted for high density,match the exp ected features of hadronic matter at low density. This provides, in particular, a controlled, weak-coupling realization of con nement and chiral symmetry breaking in this (slight) idealization of QCD. 1 Research supp orted in part by NSF PHY-9513835. e-mail: [email protected] 2 Research supp orted in part by DOE grant DE-FG02-90ER40542. e-mail: [email protected] 1 In a recent study [1] of QCD with three degenerate avors at high density, a new form of ordering was predicted, wherein the color and avor degrees of freedom b ecome rigidly correlated in the groundstate: color- avor lo cking. This prediction is based on a weak coupling analysis using a four-fermion interaction with quantum numb ers abstracted from one gluon exchange. One exp ects that sucha weak coupling analysis is appropriate at high density,for the following reason [2,3,4].Tentatively assuming that the quarks start out in a state close their free quark state, i.e. with large Fermi surfaces, one nds that the relevantinteractions, which are scatterings the states near the Fermi surface, for the most part involve large momentum transfers. Thus, by asymptotic freedom, the e ective coupling governing them is small, and the starting assumption is con rmed. Of course, as one learns from the theory of sup erconductivity [5], even weak couplings near the Fermi surface can have dramatic qualitative e ects, fundamentally b ecause there are manylow-energy states, and therefore one is inevitably doing highly degenerate p erturbation theory. Indeed, the au- thors of [1] already p ointed out that their color- avor lo cked state, which is constructed by adapting the metho ds of sup erconductivity theory to the problem of high-density quark matter, displays a gap in all channels ex- cept for those asso ciated with derivatively coupled spin zero excitations, i.e. Nambu-Goldstone mo des. This is con nement. For massless quarks, they also demonstrated sp ontaneous chiral symmetry breaking. In very recentwork we [6], and others [7], have reinforced this circle of ideas by analyzing renormalization of the e ectiveinteractions as one integrates out mo des far from the Fermi surface. A fully rigorous treatment will have to deal with the extremely near-forward scatterings, which are singular due to the absence of magnetic mass for the gluons, at least in straightforward p erturbation theory. This problem, which is presumably technical, is any case ameliorated self-consistently for states of the color- avor lo cking typ e, wherein all the gluons acquire mass through the Anderson-Higgs mechanism. In the earlier work [1], several striking analogies b etween the calculated prop erties of the color- avor lo cked state and the exp ected prop erties of hadronic matter at low or zero density, based on standard lore and observed phenomenology,were noted. In addition to con nementandchiral symmetry breaking, the authors observed that the dressed elementary excitations in the color- avor lo cked state have the spin quantum numb ers of low-lying hadron 2 states and for the most part carry the exp ected avor quantum numb ers, including integral electric charge (in units of the electron charge). Thus, as we shall sp ell out immediately b elow, the gluons match the o ctet of vector mesons, the quark o ctet matches the baryon o ctet, and an o ctet of collective mo des asso ciated with chiral symmetry breaking matches the pseudoscalar o ctet. However there are also a few apparent discrepancies: there is an extra massless singlet scalar, asso ciated with the sp ontaneous breaking of baryon numb er (sup er uidity);thereareeight rather than nine vector mesons (no singlet); and there are nine rather than eightbaryons (extra singlet). We will argue that these \discrepancies" are sup er cial { or rather that they are features, not bugs. Let us rst brie y recall the fundamental concepts of color- avor lo cking. The case of three massless avors is the richest due to its chiral symme- try (and adding a common mass do es not change anything essential) so we shall concentrate on it. The primary condensate, which one calculates using the metho ds of sup erconductivity theory near the Fermi surface, involves diquarks. It takes the form [1] _ _ j k l i (1) + i = q i = hq q hq 2 1 ij _ _ a b b a Lb La Rka Rlb Here L, R lab el the helicity, i; j; k ; l are two-comp onent spinor indices, a; b are avor indices, and ; are color indices. A common space-time argu- ment is suppressed. ; are parameters (dep ending on chemical p otential, 1 2 coupling, :::) whose non-zero values emerge from a dynamical calculation. This equation must b e interpreted carefully. The value of any lo cal quan- tity which is not gauge invariant, taken literally, is meaningless, since lo cal gauge invariance parametrizes the redundantvariables in the theory,and cannot b e broken [8]. But as weknow from the usual treatment of the elec- troweak sector in the Standard Mo del, it can b e very convenient to use such quantities. The p oint is that we are allowed to x a gauge during interme- diate stages in the calculation of meaningful, gauge invariantquantities { indeed, in the context of weak coupling p erturbation theory,wemust do so. For our present purp oses however it is imp ortant to extract non-p erturbative results, esp ecially symmetry breaking order parameters, that we can match to our exp ectations for the hadronic side. To do this, we can take suitable pro ducts of the memb ers of (1) and their complex conjugates, and contract the color indices. In this waywe can pro duce the square of the standard chiral 3 symmetry breaking order parameter of typ e hq q i and a baryon numb er vio- L R 2 lating order parameter of typ e h(qqq) i, b oth scalars and singlets under color and avor. Atthislevel only the square of the usual chiral order parameter app ears, fundamentally b ecause our condensates preserve left-handed quark number modulo two. This conservation law is violated by the six-quark vertex asso ciated with instantons, and byconvolving that vertex with our four-quark condensate we can obtain the usual two-quark chiral symmetry breaking order parameter [9]. By demanding invariance of the diquark condensate directly,we infer the c L R symmetry breaking pattern SU (3) SU (3) SU (3) U (1) ! SU (3) . c Here among the initial microscopic symmetries SU (3) is lo cal color symme- try, while the remaining factors are chiral family and baryon numb er symme- tries. The nal residual unbroken symmetry is a global diagonal symmetry. Indeed, the Kronecker deltas in the nal term of (1) are invariantonlyunder simultaneous color and avor rotations, so the color and avor are \lo cked". This lo cking o ccurs separately for the left and right handed quarks, but since color symmetry itself is vectorial, the e ect is also to lo ckleftand righthanded avor rotations, breaking chiral symmetry. The global baryon numb er symmetry is, of course, manifestly broken, but quark number is con- served mo dulo two. Pro jecting onto the gauge invariant, color singlet, sector this implies that baryon numb er is violated only mo dulo two. The same conclusions would emerge from analysis of the gauge invariant symmetry generators only, up on consideration of the gauge invariant order parameters we constructed ab ove. Ordinary electromagnetic gauge invariance, like color symmetry, is vio- lated by (1), but a linear combination of hyp ercharge (diagonal matrix -2/3, 1/3, 1/3) and electromagnetic charge (diagonal matrix 2/3, -1/3, -1/3) anni- hilates the combinations correlated by color- avor lo cking, and generates a true symmetry.Thephysical result is that there is a massless gauge degree of freedom, representing the photon as mo di ed byitsinteraction with the condensate. As seen by this mo di ed photon, all the elementary excitations have appropriate charges to match the corresp onding hadronic degrees of freedom. In particular, their charges are all integral multiples of the electron charge [1]. This is, of course, another classic asp ect of con nement. It was essential, in this construction, that the charges of the quarks add up to zero. If that were not so, wewould not have b een able to nd a color generator capable of comp ensating the violation of naive electromagnetic 4 gauge invariance. Yet it seems somewhat accidental that these charges do add up to zero, and one would b e quite worried if any qualitative asp ect of con nement dep ended on this accident. This worry touches the form rather than the substance of our argument. If the quark charges did not up to zero, it would not b e valid to ignore Coulomb repulsion. One would havetoadda comp ensating charge background as a mathematical device, or contemplate inhomogeneous states. Insofar as wewant to use external gauge elds as a prob e of pure QCD, wemust restrict ourselves to those which preserve the overall neutrality of the QCD groundstate.