Chiral symmetry restoration at finite density in largeNcQCD

Abstract

At large ${N}_{c}$, cold nuclear matter is expected to form a crystal and thus spontaneously break translational symmetry. The description of chiral symmetry breaking and translational symmetry breaking can become intertwined. Here, the focus is on aspects of chiral symmetry breaking and its possible restoration that are by construction independent of the nature of translational symmetry breaking---namely spatial averages of chiral order parameters. A system will be considered to be chirally restored provided all spatially averaged chiral order parameters are zero. A critical question is whether chiral restoration in this sense is possible for phases in which chiral order parameters are locally nonzero but whose spatial averages all vanish. We show that this is not possible unless all chirally invariant observables are spatially uniform. This result is first derived for Skyrme-type models, which are based on a nonlinear sigma model and by construction break chiral symmetry on a point-by-point basis. A no-go theorem for chiral restoration (in the average sense) for all models of this type is obtained by showing that in these models there exist chirally symmetric order parameters that cannot be spatially uniform. Next, we will show that the no-go theorem applies to large ${N}_{c}$ QCD in any phase that has a nonzero but spatially varying chiral condensate. The theorem is demonstrated by showing that in a putative chirally restored phase, the field configuration can be reduced to that of a nonlinear sigma model. It is also shown that this no-go theorem is fully consistent with the vanishing of the spatial average of the chiral condensate $\frac{1}{2}\mathrm{Tr}(U)$ (as happens in ``half-skyrmion'' configurations). This is because the chiral condensate is only one of an infinite set of chiral order parameters, some of which must be nonzero. It is also shown that while an approximation of a unit cell of a Skyrme crystal as a hypersphere does lead to a phase that is chirally restored (in the average sense), this is an artifact of the approximation.