Phase structure of the generalized two dimensional Yang-Mills theories on sphere

Abstract

We find a general expression for the free energy of $G(ϕ)=ϕ^{2k}$ generalized 2D Yang-Mills theories in the strong ($A>A_c$) region at large $N$. We also show that in this region, the density function of Young tableau of these models is a three-cut problem. In the specific $ϕ^6$ model, we show that the theory has a third order phase transition, like $ϕ^2$ (YM_2) and $ϕ^4$ models. We have some comments for $k \geq 4$ cases. At the end, we study the phase structure of $ϕ^2 + g ϕ^4$ model for $g \leq A/4$ region.