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 (${\mathrm{QCD}}_{1+1}$) 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 ${\mathit{m}}_{\mathit{q}}$ 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‖ψ¯ψ‖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‖ψ¯ψ‖0〉${\mathrm{\sim }}_{\mathit{q}}^{\mathit{m}}$→0-${\mathit{N}}_{\mathit{c}}$ √${\mathit{G}}^2$ 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 ${\mathrm{QCD}}_{1+1}$ 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 semmingly trivial LF vacua are not in conflict with QCD sum-rule results. © 1996 The American Physical Society.