Notes on phase structure and non-vanishing 𝛽 functions of one-unitary matrix model

Unitary matrix models, among other things, play an important role in representing the irregular conformal block and hence low-energy effective action (LEEA) of some asymptotically free susy gauge theories. Here, we develop a method of how to determine the qualitative structure of phase diagram by the deformation of classical potential, and illustrate this by the examples that contain term up to cos n𝛼, n ≤ 3. We point out that a set of 𝛽 functions on critical “lines” is nowhere a vanishing vector. This generalizes the n = 1 GWW case and tells us the third order (rather than second order) phase transition in conformity with the range of values for the susceptibility exponent.