Trivial Vacua, High Orders in Perturbation Theory and Nontrivial Condensates

Abstract

In the limit of an infinite number of colors, an analytic expression for the quark condensate in $QC{D}_{1+1}$ is derived as a function of the quark mass and the gauge coupling constant. For zero quark mass, a nonvanishing quark condensate is obtained. Nevertheless, it is shown 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. The sum-rule calculation is based on masses and coupling constants calculated from 't Hooft's solution to $QC{D}_{1+1}$ which employs LF quantization and is thus based on a trivial vacuum. Nevertheless the chiral condensate remains nonvanishing in the chiral limit which is yet another example that seemingly trivial LF vacua are {\it not} in conflict with QCD sum-rule results.

Comments: REVTEX, 6 pages, 1 postscript figure available from the author by email