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 (1+1)-dimensional QCD (QCD(1+1)) is derived as a function of the quark mass and the gauge coupling constant. The vacuum condensate is perfectly well defined and is not zero for arbitrary m(q) despite the fact that the perturbative contribution is divergent. This calculation explicitly demonstrates the definition of the nonperturbative vacuum condensates. 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 in the 't Hooft regime of the model. It is furthermore shown that the expansion of ([0\<(psi)over bar>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 [GRAPHICS] 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 not in conflict with QCD sum-rule results.