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Are the Quirks of Quarkyons Lurking in a Low Energy Run?

by Rob Pisarski

The phase diagram of Quantum Chromodynamics (QCD) has proven to be a subject full of surprises. The wealth of experimental results obtained from RHIC to date show that the region which has been probed, the "strongly-coupled Quark Gluon Plasma," (sQGP) is much richer than anyone ever expected. This is a region of high temperature, but where the net baryon density is (very) low.

The proposed low energy run is interesting because by going to lower energies, it is possible to probe a different part of the phase diagram, where the temperature T is still high, but where the net baryon density is significant. As I explain in this article, it is possible that something qualitatively new happens in this region: one might reach a new phase, which has features of both quark and baryonic matter. Larry McLerran and I term this "quarkyonic" matter [1].

Quarkyonic matter has some unusual features. At zero net baryon density, numerical simulations on the lattice appear to indicate that the two phase transitions which are possible in QCD - for deconfinement, and for chiral symmetry restoration - occur at essentially the same point [2]. It is usually assumed that this remains true as one moves out to nonzero net baryon density. We argue, however, that this need not be true: in a sense which I discuss, quarkyonic matter is always confined. Even so, there can be two types of quarkyonic matter, one where chiral symmetry is broken, and one where it is restored. Such a confined, but chirally symmetric phase, is special to nonzero net baryon density.

QCD is a theory with three colors of gluons, and three flavors of light quarks. A few years after QCD was first invented, G. 't Hooft suggested considering the limit in which the number of colors, Nc, is very large, but the number of quark flavors is kept fixed. Generally, it is not possible to compute analytically in this limit, but even the qualitative insight is often most useful: for instance, Zweig's rule follows immediately. (Indeed, there is one limit where one can compute many things exactly: because of the "magic" of J. Maldacena, for the maximally supersymmetric generalization of QCD, when both the number of colors, and the coupling constant, are taken to be infinitely large [3].)

In a SU(Nc) gauge theory, gluons are SU(Nc) matrices, and quarks SU(Nc) vectors. Thus at large Nc, there are Nc×Nc  ~ Nc2 types of gluons, versus only  ~ Nc types of quarks. Thus purely on the basis of the number of degrees of freedom, the large Nc limit is one where gluons dominate.

For example, consider very high temperature. Once asymptotic freedom kicks in, up to (small) perturbative corrections, we just compare the free energy of ideal gases. Because there are so many more gluons, their give a huge contribution to the pressure,  ~ Nc2, versus  ~ Nc for quarks. Even when the perturbation theory is no longer a good approximation, it is natural to assume that the pressure from gluons remains huge,  ~ Nc2 times some function of the temperature.

In contrast, consider what happens at zero temperature, when there is no net density of baryons. Confinement tells us that all of these gluons and quarks have to combine into states which are color singlets. Now however they do so, the number of these color singlet states just involves the usual quantum numbers: spin, isospin, and so on. But since they are color singlets, their degeneracy can't involve the number of colors. Thus the pressure of the confined phase is tiny, only of order  ~ 1. This means that we can define the temperature at which the deconfining phase transition occurs, Td, just by the point where the pressure goes from being tiny,  ~ 1, to huge,  ~ Nc2. Because the pressure changes so sharply, one naturally suspects that the deconfining transition is of first order. This is supported by numerical simulations on the lattice [4].

Actually, at nonzero temperature, and with no net density of baryons, the QCD phase transition does look like that of large Nc [2]. There is a very rapid rise in the pressure in a narrow region of temperature. Now admittedly, the transition in QCD is crossover, but one can easily interpret this as due to corrections in 1/Nc.

Now let us consider a theory with net baryon density. Thermodynamics tells us that it is more convenient to speak of the baryon chemical potential, μB, instead of net baryon density. At large Nc, baryons are composed of Nc quarks, so the baryon mass is  ~ Nc. We find it useful to introduce the quark chemical potential, μq, which is simply related to that of baryons by μq = μB/Nc. With a little thought, one can convince oneself that like Td, the natural scale of μq is of order one at large Nc.

Given this, we can immediately say something interesting about small values of the quark chemical potential: because the large Nc limit is dominated by gluons, the critical temperature, Td, must be completely independent of μq! That is, as illustrated in Fig. 1, there is just a straight line which separates the confined and deconfined phases. As before, the pressure is tiny in the confined phase, and huge in the deconfined phase.

What if one stays at zero temperature, and increases the chemical potential? It is only possible to form a Fermi sea when the chemical potential exceeds the mass of the lightest particle. Thus if M is the mass of the lightest baryon (i.e., the nucleon), then at zero temperature, there is only a Fermi sea when μB > M. In terms of the quark chemical potential, this is μq > M/Nc. This is just a kinematic constraint. Now baryons are very heavy at large Nc, so they are Boltzmann suppressed even at nonzero temperature. Thus in the confined phase, this kinematic constraint is independent of temperature. This implies that in the plane of temperature and chemical potential, at large Nc the confined phase is a "box," as illustrated in Fig. 1. The tiny pressure in the confined phase is due only to glueballs and mesons, not to baryons.

Fig. 1: Possible phase diagram for a large number of colors, in the plane of temperature and quark chemical potential. In the quarkyonic phase, chiral symmetry is restored for temperatures above the dashed blue line.

Now let us move to larger values of the quark chemical potential, when there is a Fermi sea of baryons and/or quarks. When μq > M/Nc, the phase transition to deconfinement remains a straight line, with a huge pressure in the deconfined phase. This leaves a "strip" in the plane of temperature and chemical potential, where there is a Fermi sea, μq > M/Nc, but where the theory is confined, T < Td. This strip is labeled quarkyonic in Fig. 1.

Why do we use this term? Suppose that we crank the quark chemical potential to very large values. In QCD, there is a renormalization mass scale, ΛQCD  ~ 300 MeV. For any process, if the relevant momenta are much larger than ΛQCD, we can expect that perturbation theory should be a good approximation. Presumably ΛQCD isn't so different at large Nc. Now consider computing the pressure at VERY large values of μq; say, a thousand times ΛQCD. Then surely we should be able to compute the pressure in this regime, and find that up to (small) perturbative corrections, that it is just an ideal gas term for the quarks, proportional to the number of degrees of freedom. In the strip, this is big,  ~ Nc, but not huge,  ~ Nc2, as in the deconfined phase. It is also reasonable to assume that the pressure remains big,  ~ Nc, in the entire quarkyonic strip.

Now we run into the central puzzle. At large Nc, the quarkyonic strip represents a phase with confinement, since T < Td. But how can a big pressure,  ~ Nc, represent anything like a confined phase? This is especially confusing for a big Fermi sea, where the pressure is nearly that of ideal quarks; it seems like we are having both ways!

Our essential observation is that one can have it both ways. Assume again that the Fermi sea is very big. Then if one hits a quark deep in the Fermi sea with an energetic probe, and knocks it out, one ends up with a quark above the Fermi sea, and the hole deep in it. Since the quark has a large momentum, it will act like a quark for a very long time (and similarly for the quark hole). This is closely analogous to deep inelastic scattering at zero temperature, where one takes a high energy probe, and knocks a quark anti-quark pair out of the vacuum. The quark and anti-quark start out very fast, but then they radiate, slow down, and eventually turns into hadrons. The same thing happens to the pair of a quark and a quark hole. They eventually radiate, and end up forming a large number of quarks, and quark holes, near the Fermi surface. (Remember that whether or not particles are massive, near the Fermi surface it costs little energy to pop a particle above the Fermi surface; at the edge of the Fermi surface, it costs nothing.)

So what? Well now we remember that at large Nc, this is still a confined phase. That means that once quarks, or quark holes, are within  ~ ΛQCD of the Fermi surface - as they will surely be, sooner or later - then confinement kicks in, and one must deal instead with colorless states. That is, instead of a quarks, and quark holes, we must speak instead of baryons, and baryon holes.

Our picture is then the following. For a big Fermi sea, most of it is well described as that of quarks. For a skin" which is  ~ ΛQCD in width, though, the Fermi sea is that of baryons. The two smoothly interpolate from one to another, as there is no order parameter to distinguish the two. Because there is a baryonic skin to the quark Fermi sea, we term the combination quarkyonic. As one goes from large values of the chemical potential, to small values, the baryonic skin becomes larger and larger; once the quark chemical potential is of the same order as ΛQCD, the Fermi sea is then essentially entirely baryonic, which is nuclear matter.

The baryonic skin is especially important for pairing near the Fermi surface. Such pairing produces either superconductivity and/or superfluidity. For a quarkyonic phase, any pairing is dominated not by quarks, but by baryons.

The above picture holds for (very) massive quarks. Then there is a single quarkyonic phase in the entire strip. How much the quarkyonic Fermi sea is dominated by baryons, and how much by quarks, depends upon how big the chemical potential is, but the two smoothly pass over from one to the other.

What if there are light or massless quarks, so that there is a chiral phase transition? As one goes from the confined to a deconfined phase, for μq < M/Nc, presumably the chiral and deconfining transitions coincide. This is really automatic, since everything is driven by the gluons. What about the quarkyonic strip? I won't go into our (handwaving) arguments, but we strongly believe that quarkyonic matter includes two regions: one where chiral symmetry is broken, and one where it is restored; i.e., before  the theory deconfines! In Fig. 1, in the quarkyonic phase the line for the chiral phase transition is denoted by a dotted blue line: it is broken below it, and restored above. In Fig. 1, we have drawn it so that the chiral transition starts to separate from the deconfining transition as soon as one enters the quarkyonic strip, but this is just a guess.

This is exactly opposite to what people have expected. A deconfined phase with chiral symmetry breaking is easy to understand: it is just quarks with a constituent quark mass. In contrast, at zero net baryon number, it is very hard to understand how one could possibly have a confined phase which is also chirally symmetric. Indeed, it probably can't happen, due to constraints from the U(1) axial anomaly and the like.

However, we suggest that there is a confined, but chirally symmetric phase, only for nonzero net baryon density. Chiral symmetry is ensured by parity doubling: instead of one Fermi sea, we need two, one for the nucleon, and one for its parity partner, which is vacuum is probably the N*(1535). As a chirally symmetric phase, these two states must have equal (effective) masses, but that's all that is required.

What happens in QCD? Here we can only guess; one possibility is shown in Fig. 2. It is like that of large Nc, but with some curvature for the various lines. We do not know what the order of the phase transitions might be in QCD, or if there is a critical endpoint. The basic point is that as at large Nc, that there are three qualitatively distinct phases: the usual confined and deconfined phases, and now a quarkyonic phase as well. The quarkyonic phase includes two regions, in which the chiral symmetry is either broken, or restored; the line for chiral symmetry restoration is denoted by a dotted blue line in Fig. 2. As the net baryon density increases from zero, the deconfining and chiral phase transitions coincide for some range in chemical potential. At some value, however, they split, and quarkyonic matter appears. In the quarkyonic phase, the transition for chiral symmetry breaking occurring at a lower - not higher! - temperature than that for deconfinement.

Fig. 2: Possible phase diagram for QCD, in the plane of temperature and baryon chemical potential. Again, in the quarkyonic phase, chiral symmetry is restored for temperatures above the dashed blue line.

In QCD, when the chemical potential becomes very large, eventually the deconfining and quarkyonic phases can smoothly merge into one another. (At large Nc, this only happens for values of the quark chemical potential which grows like some power of Nc.) We don't know how the deconfining and quarkyonic phases merge, which is why the deconfining transition in Fig. 2 ends in a question mark. What we do suggest is that this may happen for baryon densities which are larger than will ever be probed either in heavy ion collisions, or in neutron/quark(yonic!) stars.

Could one hope to probe a quarkyonic phase in a low energy run? Well at large Nc, in Fig. 1, the deconfining temperature is independent of the chemical potential; this means that as shown in Fig. 2, things happen even at rather high temperature, close to that at μq = 0. A high temperature is certainly good for heavy ion collisions.

What about the baryon density? At large Nc, the chiral transition can't split from the deconfining transition until baryons begin to condense, for values of the baryon chemical potential larger than the nucleon mass. This might mean densities higher than probed in heavy ion collisions. Unlike large Nc, though, in QCD baryons aren't infinitely heavy, so at nonzero temperature, they can condense for μB < M. Whether enough baryons condense at sufficiently low chemical potential is not clear from such an elementary analysis.

Lastly, we suspect that if a critical end point does occur, then on the basis of effective models and the like, that it is natural for the chiral transition to split from the deconfining transition at this endpoint.

What about signals for a quarkyonic phase? Any phase transition associated with baryon number should have large scale fluctuations in baryon number. A quarkyonic phase with chiral symmetry breaking will look like "ordinary" nuclear matter, with light pions. A quarkyonic phase with chiral symmetry restoration will have parity doubled Fermi seas, with heavy pions. Thus the dilepton spectra should be very different from that of ordinary hadrons: pion contributions are suppressed, and it must exhibit explicit chiral symmetry.

Admittedly, all of our arguments are merely qualitative. Even so, they suggest that detailed studies at low energies may find evidence for a completely novel phase of QCD.

[1] L. McLerran and R. D. Pisarski, Nucl. Phys. A 796, 83 (2007) [arXiv:0706.2191 [hep-ph]].

[2] M. Cheng et al., Phys. Rev. D 77, 014511 (2008) [arXiv:0710.0354 [hep-lat]].

[3] D. T. Son and A. O. Starinets, Ann. Rev. Nucl. and Part. Sci. 57, 95 (2007) [arXiv:0704.0240 [hep-th]].

[4] B. Bringoltz and M. Teper, Phys. Lett. B628, 113 (2005) [arXiv:hep-lat/0506034].