Lessons from two-dimensional QCD (N→∞): Vacuum structure, asymptotic series, instantons, and all that

We discuss two-dimensional QCD ($N_c\to \infty$) with fermions in the fundamental as well as adjoint representation. We find factorial growth $\sim \frac{{\left(g^2{N}_c\pi \right)}^{2k}\mathrm{}\left(2k\right)!{\mathrm{}(-1\mathrm{})\mathrm{}}^{k-1\mathrm{}}}{{\left(2\mathrm{}\pi \right)}^{2k}}$ in the coefficients of the large order perturbative expansion. We argue that this behavior is related to classical solutions of the theory, instantons; thus it has nonperturbative origin. Phenomenologically such a growth is related to highly excited states in the spectrum. We also analyze the heavy-light quark system $Q\overline{q}$ within the operator product expansion (which turns out to be an asymptotic series). Some vacuum condensates $〈\overline{q}{\left(x_{\mu }{D}_{\mu }\right)}^{2n}q〉\sim {\left(x^2\right)}^{n}n!$ which are responsible for this factorial growth are also discussed. We formulate some general puzzles which are not specific for two-dimensional physics, but are inevitable features of any asymptotic expansion. We resolve these apparent puzzles within two-dimensional QCD and we speculate that analogous puzzles might occur in real four-dimensional QCD as well.