Lightlike Wilson loops and gauge invariance of Yang-Mills theory in 1+1 dimensions

Abstract

A light-like Wilson loop is computed in perturbation theory up to O(g 4 ) for pure Yang-Mills theory in 1+1 dimensions, using Feynman and light-cone gauges to check its gauge invariance. After dimensional regularization in intermediate steps, a finite gauge invariant result is obtained, which however does not exhibit abelian exponentiation. Our result is at variance with the common belief that pure YangMills theory is free in 1+1 dimensions, apart perhaps from topological effects.

Discontinuous behavior of perturbative Yang-Mills theories in the limit of dimensionsD→2

“…Since results in 2ω dimensions were available, in view of the peculiar features of Yang-Mills theories in 2 dimensions mentioned above, the interest arose in knowing the outcome of the check in the limit ω → 1. The following unexpected results were obtained in [17].…”

“…[18], where an anologous Wilson loop was calculated in Feynman gauge, but also with ref. [17], where the loop was oriented in a different direction. Moreover, in LCG, different families of diagrams ("crossed" and "bubble" diagrams) give the same contribution (C F C A (LT ) 2 3 and C F C A LT π 2 respectively) no matter the orientation of the loop: remarkably, invariance under area-preserving diffeomorphisms is recovered in the limit D → 2, even when the Wilson loop is first evaluated in higher dimensions, and then the limit D → 2 is taken.…”

“…This is not surprising, as the same phenomenon occurred in ref. [17], although for a different contour (contour with light-like sides).…”

“…[18] when taking the limit D → 2. The source of such a discrepancy is rooted in the mentioned self-energy diagram contribution, which is obviously missing at D = 2, but provides a finite term in the limit D → 2, thereby producing a discontinuity in the theory [17].…”

Two-dimensional Yang-Mills theory: perturbative and instanton contributions, and its relation to QCD in higher dimensions

“…Still topological degrees of freedom occur if the theory is put on a (partially or totally) compact manifold, whereas the simpler behavior on the plane enforced by the LCG condition entails a severe worsening in its infrared structure. These features are related aspects of the same basic issue: even in two dimensions (D = 2) the the-ory contains non trivial dynamics, as immediately suggested by other gauge choices as well as by perturbative calculations of gauge invariant quantities, typically of Wilson loops [2]. We can say that, in LCG, dynamics gets hidden in the very singular nature of correlators at large distances (IR singularities).…”