Two-dimensional QCD, instanton contributions and the perturbative Wu-Mandelstam-Leibbrandt prescription

TL;DR

This work analyzes two-dimensional QCD on a sphere to resolve discrepancies between perturbative calculations using the Wu–Mandelstam–Leibbrandt (WML) prescription and the nonperturbative instanton structure. By performing a Poisson resummation, the authors derive an explicit instanton expansion for the partition function and Wilson loops, with instanton actions $S_{inst}(n_i)=\frac{2\pi^2}{g^2A}\sum_i n_i^2$ and fluctuations encoded in $Z(n_i)$ and $W_k(n_i)$. They show that the zero-instanton sector reproduces the perturbative WML result, while the full sum over instantons restores the expected pure area-law exponentiation in the decompactified limit, thereby reconciling perturbative and nonperturbative pictures and clarifying confinement behavior at large $N$. The analysis highlights how decompactification and nonperturbative sectors crucially alter the Wilson loop, with instantons driving confinement in the strong coupling phase and perturbative sums capturing only a part of the physics. For $N=1$ area exponentiation persists across sectors, while in the large-$N$ limit the full instanton sum is essential for confinement.

Abstract

The exact Wilson loop expression for the pure Yang-Mills U(N) theory on a sphere $S^2$ of radius $R$ exhibits, in the decompactification limit $R\to \infty$, the expected pure area exponentiation. This behaviour can be understood as due to the sum over all instanton sectors. If only the zero instanton sector is considered, in the decompactification limit one exactly recovers the sum of the perturbative series in which the light-cone gauge Yang-Mills propagator is prescribed according to Wu-Mandelstam-Leibbrandt. When instantons are disregarded, no pure area exponentiation occurs, the string tension is different and, in the large-N limit, confinement is lost.