Wilson Loops in Two-Dimensional Yang-Mills Theories

TL;DR

This thesis investigates the perturbative behavior of Wilson loops in two-dimensional and higher-dimensional Yang-Mills theories, revealing non-commuting limits and gauge-dependent effects, with implications for understanding gauge theories across dimensions.

Contribution

It provides a detailed perturbative analysis of Wilson loops in 2D and higher dimensions, highlighting non-commuting limits and gauge effects, and compares results across gauges.

Findings

Exponentiation holds in dimensions > 2 but not in 2 when limits are interchanged.

Finite results are obtained in 2+ε dimensions in light-cone gauge, matching Feynman gauge calculations.

Limits T→∞ and D→2 do not commute, affecting the behavior of Wilson loops.

Abstract

This Ph.D. thesis reaches two main results. The first one is represented by a detailed study, in Feynman gauge, of the perturbative ${\cal O}(g^4)$ contribution to a space-time Wilson loop, with respect to its (expected) Abelian-like time exponentiation when the temporal side goes to infinity. As soon as we are in dimensions greater than two, the expected behavior is found. But if we proceed first to the dimensional limit $D \to 2$, the exponentiation is not recovered. The limits $T \to \infty$ and $D \to 2$ do not commute. The other result is the computation in dimensions $D=2+\epsilon$ and in light-cone gauge with Mandelstam-Leibbrandt prescription of the perturbative ${\cal O}(g^4)$ contribution to the same Wilson loop, coming from diagrams with a self-energy correction in the vector propagator. In the limit $\epsilon \to 0$ the result is finite, in spite of the vanishing of the triple vector vertex in light-cone gauge, and provides the expected agreement with the analogous calculation in Feynman gauge. Consequences of these results concerning two and higher-dimensional gauge theories are pointed out.