Exact $m_{quark}\ne 0$ Condensates in $QC{D}_{1+1}(N_C\to \infty )$

Abstract

In the limit of an infinite number of colors, we derive an analytic expression for the quark condensate in $QC{D}_{1+1}$ as a function of the quark mass and the gauge coupling constant. For zero quark mass, a nonvanishing quark condensate is obtained. Nevertheless, we prove that there is no phase transition as a function of the quark mass. It is furthermore shown that the expansion of $⟨0|\overline{\psi }\psi |0⟩$ in the gauge coupling has zero radius of convergence but that the perturbation series is Borel summable with finite radius of convergence. The nonanalytic behavior $⟨0|\overline{\psi }\psi |0⟩\stackrel{m_q\to 0}{\sim }-N_C\sqrt{G^2}$ can only be obtained by summing the perturbation series to infinite order.

Comments: 5 pages, 1 postscript figure (available by email from the author), LATEX