ЯДЕРНАЯ ФИЗИКА, 2012, том 75, № 5, с. 670-674
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
CONFINING BUT CHIRALLY SYMMETRIC DENSE AND COLD MATTER
©2012 L. Ya. Glozman*
Institute for Physics, Theoretical Physics Branch, University of Graz, Austria
Received March 31, 2011
The possibility for existence of cold, dense chirally symmetric matter with confinement is reviewed. The answer to this question crucially depends on the mechanism of mass generation in QCD and interconnection of confinement and chiral symmetry breaking. This question can be clarified from spectroscopy of hadrons and their axial properties. Almost systematical parity doubling of highly excited hadrons suggests that their mass is not related to chiral symmetry breaking in the vacuum and is approximately chirally symmetric. Then there is a possibility for existence of confining but chirally symmetric matter. We clarify a possible mechanism underlying such a phase at low temperatures and large density. Namely, at large density the Pauli blocking prevents the gap equation to generate a solution with broken chiral symmetry. However, the chirally symmetric part of the quark Green function as well as all color non-singlet quantities are still infrared divergent, meaning that the system is with confinement. A possible phase transition to such a matter is most probably of the first order. This is because there are no chiral partners to the lowest lying hadrons.
1. INTRODUCTION
A key question to QCD at high temperatures and densities is whether and how deconfinement and chiral restoration transitions (crossovers) are connected to each other. In order to answer this question we need understanding of hadron mass generation in QCD, how both confinement and chiral symmetry breaking influence the origin of mass. We know from the trace anomaly in QCD that the hadron mass (we discuss here only the light quark sector) almost entirely consists of the energy of quantized gluonic field. However, this tells us nothing about the effect of chiral symmetry breaking on hadron mass. The chiral symmetry is dynamically broken in the QCD vacuum and this phenomenon is crucially important for the mass origin of the lowest-lying hadrons, such as pion, nucleon or rho-meson. Phenomenologically it follows from the well established (pseudo) Nambu— Goldstone nature of pion as well as from the absence of chiral partners to the lowest-lying hadrons. Their mass is determined to large degree by the quark condensate of the vacuum, which can be seen from the SVZ sum rules [1, 2], as well as from many different microscopical models.
At the same time almost systematical parity doubling in both highly excited baryons [3] and mesons [4] suggests that chiral symmetry breaking in the vacuum is almost irrelevant to the mass generation of these hadrons, i.e. the chiral (and U(1)^) symmetry gets effectively restored, for a review see [5].
E-mail: leonid.glozman@uni-graz.at
This conjecture is strongly supported by the pattern of strong decays of excited hadrons [6]. Experimental discovery of the still missing chiral partners to some high-lying states is required, however, for the unambiguous conclusion [7, 8].
If effective chiral restoration in excited hadrons is correct, then there is a possibility in QCD for a phase with confinement (i.e. elementary excitations are of the color-singlet hadronic type) and where at the same time chiral symmetry is restored.
2. QUARKYONIC MATTER
In the large Nc world with quarks in the fundamental representation there are neither dynamical quark—antiquark nor quark—quark hole loops. Consequently, at low temperatures there is no Debye screening of the confining gluon propagator and gluodynamics is the same as in vacuum. Confinement persists up to arbitrary large density. In such case it is possible to define quarkyonic matter [9]. In short, it is a strongly interacting matter with confinement and with a well-defined Fermi sea of baryons or quarks. At smaller densities it should be a Fermi sea of nucleons (so it matches with standard nuclear matter), while at higher densities, when nucleons are in a strong overlap, a quark Fermi surface should be formed. While a quark Fermi sea is formed, the system is still with confinement and excitation modes are of the color-singlet hadronic type. Obviously, the question of existence or nonexistence of the quarkyonic phase in the real world Nc = 3 cannot be formulated and
studied with models that lack explicit confinement of quarks.
Note that nothing can be concluded from this simple argument about existence or nonexistence of the chiral restoration phase transition, i.e., whether there is or not a "subphase" with restored chiral symmetry within the quarkyonic matter (quite often, unfortunately, a quarkyonic matter is confused with confining but chirally symmetric phase).
In the large-Nc 't Hooft limit such a matter persists at low temperatures up to arbitrary large densities. At which densities in the real Nc = 3 world will we have a deconfining transition (which could be a very smooth crossover) to a quark matter with uncorrelated single quark excitations? Lattice results for Nc = 2 suggest that such a transition could occur at densities of the order 100 x nuclear matter density [10]. If correct, then it would imply that at all densities relevant to future experiments and astrophysics we will have a dense quarkyonic (baryonic) matter with confinement.
The most interesting question concerns the fate of chiral symmetry breaking in this dense, cold quarkyonic matter with confinement. Indeed, in a dense matter one expects that all lowest-lying quark levels (that are required for the very existence of the quark condensate) are occupied by valence quarks and Pauli blocking prevents dynamical breaking of chiral symmetry. Consequently, one would obtain a confining but chirally symmetric phase within the quarkyonic matter [11]. To this end one needs a quark that is confined but at the same time its Green function is chirally symmetric. Is it possible?
There is no way to answer today this question within QCD itself. What can be done is to clarify the issue whether it is possible or not in principle. If possible, a key question is about the physical mechanism that could be behind such a phase. Then to address this question one needs a model that is manifestly confining, chirally symmetric and provides dynamical breaking of chiral symmetry. Such a model does exist. The answer to the question above is "yes", at least within the model.
3. CONFINING AND CHIRALLY SYMMETRIC MODEL
We will use the simplest possible model that is manifestly confining, chirally symmetric and guarantees dynamical breaking of chiral symmetry in a vacuum [12, 13]. It is assumed within the model that the only gluonic interaction between quarks is a linear instantaneous potential of the Coulomb type. (This model can be considered as a (3 + 1)-dim generalization of the't Hooft model [14]. In the
't Hooft model, that is the large-Nc QCD in (1 + 1)-dimensions, the only interaction between quarks is a linear confining potential of the Coulomb type.) Such potential in (3 + 1)-dimensions is a main ingredient of the Gribov—Zwanziger scenario in Coulomb gauge [15] and is indeed observed in Coulomb gauge variational calculations [16] as well as in Coulomb gauge lattice simulations [17].
A key point is that the quark Green function (that is a solution of the gap equation in a vacuum) contains not only the chiral symmetry breaking part Ap, but also the manifestly chirally symmetric part Bp:
£(p) = + (7 • p)[Bp - p].
(1)
The linear potential requires the infrared regu-larization. Otherwise all loop integrals are infrared divergent. All observable color-singlet quantities are finite and well defined in the infrared limit (i.e., when the infrared cutoff approaches zero). These are hadron masses, the quark condensate, etc. In contrast, all color-nonsinglet quantities are divergent. E.g., single quarks have infinite energy and consequently are removed from the spectrum. This is a manifestation of confinement within this simple model.
Given a quark Green function obtained from the gap equation, one is able to solve the Bethe—Salpeter equation for mesons. A very important aspect of this model is that it exhibits the effective chiral restoration in hadrons with large J [5, 18, 19]. This is because chiral symmetry breaking is important only at small momenta of quarks. But at large J the centrifugal repulsion cuts off the low-momenta components in hadrons and consequently the hadron wave function and its mass are insensitive to the chiral symmetry breaking in the vacuum. The chiral symmetry breaking in the vacuum represents only a tiny perturbation effect: Practically the whole hadron mass comes from the chiral invariant dynamics. This explicitly demonstrates that it is possible to construct hadrons in such a way that their mass origin is not the quark condensate. If so, it is clear a priori that there are good chances to obtain a confining but chirally symmetric matter within this model.
Now we want to see what will happen with confinement and chiral symmetry at zero temperature and large density. There are no quark loops within this model and consequently Debye screening of the confining potential is absent. Confinement persists up to arbitrary large density. Will it be possible to restore chiral symmetry at some density?
This (3 + 1)-dim model is complicated enough and it is not possible to solve it exactly for a dense baryonic matter. What can be done is a kind of a mean-field solution. To obtain such a solution we need an additional assumption. Namely, we assume
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0.0010
0.0005
0.05 0.10 0.15 0.20
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Fig. 1. Chiral restoration phase transition in a dense quarkyonic matter with unbroken translational and rotational symmetries.
that both rotational and translational invariances are not broken in a medium. This impl
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