Phase Transition In Matrix Model Research Articles

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

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.

Remarks on phase transitions in Matrix models and [formula omitted] supersymmetric gauge theory

A Hermitian one-matrix model with an even quartic potential exhibits a third-order phase transition when the cuts of the matrix model curve coalesce. We use the known solutions of this matrix model to compute effective superpotentials of an N=1, SU(N) supersymmetric Yang–Mills theory coupled to an adjoint superfield, following the techniques developed by Dijkgraaf and Vafa. These solutions automatically satisfy the quantum tracelessness condition and describe a breaking to SU(N/2)×SU(N/2)×U(1). We show that the value of the effective superpotential is smooth at the transition point, and that the two-cut (broken) phase is more favored than the one-cut (unbroken) phase below the critical scale. The U(1) coupling constant diverges due to the massless monopole, thereby demonstrating Ferrari's general formula. We also briefly discuss the implication of the Painlevé II equation arising in the double scaling limit.