We find the exact nonperturbative expression for a simple Wilson loop of arbitrary shape for $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ and $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ Euclidean or Minkowskian two-dimensional Yang-Mills $\left({\mathrm{YM}}_2\right)$ theory regulated by the Wu-Mandelstam-Leibbrandt gauge prescription. The result differs from the standard pure exponential area law of ${\mathrm{YM}}_2$ theory, but still exhibits confinement as well as invariance under area-preserving diffeomorphisms and generalized axial gauge transformations. We show that the large $N$ limit is not a good approximation to the model at finite $N$ and conclude that Wu’s $N=\infty$ Bethe-Salpeter equation for two-dimensional QCD should have no bound state solutions. The main significance of our results derives from the importance of the Wu-Mandelstam-Leibbrandt prescription in higher-dimensional perturbative gauge theory.

• Received 15 September 1997

DOI:https://doi.org/10.1103/PhysRevD.57.2456