Trivial Vacua, High Orders in Perturbation Theory and Nontrivial Condensates

In the limit of an infinite number of colors, an analytic expression for the quark condensate in $QCD_{1+1}$ 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 $\langle 0 | \overline{\psi}\psi |0\rangle$ 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 $\langle 0 | \overline{\psi}\psi |0\rangle \stackrel{m_q\rightarrow0}{\sim} - N_C \sqrt{G^2}$ 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 $QCD_{1+1}$ 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.