Phase transition in matrix model with logarithmic action: toy-model for gluons in baryons

Abstract

We study the competing effects of gluon self-coupling and their interactions with quarks in a baryon, using the very simple setting of a hermitian 1-matrix model with action trA4−log det (ν+A2). The logarithmic term comes from integrating out N quarks. The model is a caricature of 2d QCD coupled to adjoint scalars, which are the transversely polarized gluons in a dimensional reduction. ν is a dimensionless ratio of quark mass to coupling constant. The model interpolates between gluons in the vacuum (ν = ∞), gluons weakly coupled to heavy quarks (large ν) and strongly coupled to light quarks in a baryon (ν→0). Its solution in the large-N limit exhibits a phase transition from a weakly coupled 1-cut phase to a strongly coupled 2-cut phase as ν is decreased below νc = 0.27. Free energy and correlation functions are discontinuous in their third and second derivatives at νc. The transition to a two-cut phase forces eigenvalues of A away from zero, making glue-ring correlations grow as ν is decreased. In particular, they are enhanced in a baryon compared to the vacuum. This investigation is motivated by a desire to understand why half the proton's momentum is contributed by gluons.