One Loop Calculations in Gauge Theories Regulated on an $x^+$-$p^+$ Lattice

TL;DR

This paper performs one-loop calculations of gauge theories on a light-cone lattice, demonstrating proper renormalization of the cubic vertex and maintaining gauge invariance with a novel discretization scheme.

Contribution

It introduces a light-cone lattice scheme with discrete $p^+$ and $ au$, enabling finite diagram calculations and correct one-loop renormalization of gauge theories.

Findings

Successfully calculated the one-loop three gauge boson triangle diagram.

Confirmed the cubic vertex is correctly renormalized with massless gauge bosons.

Validated the scheme's ability to preserve gauge invariance at one loop.

Abstract

In earlier work, the planar diagrams of $SU(N_c)$ gauge theory have been regulated on the light-cone by a scheme involving both discrete $p^+$ and $\tau=ix^+$. The transverse coordinates remain continuous, but even so all diagrams are rendered finite by this procedure. In this scheme quartic interactions are represented as two cubics mediated by short lived fictitious particles whose detailed behavior could be adjusted to retain properties of the continuum theory, at least at one loop. Here we use this setup to calculate the one loop three gauge boson triangle diagram, and so complete the calculation of diagrams renormalizing the coupling to one loop. In particular, we find that the cubic vertex is correctly renormalized once the couplings to the fictitious particles are chosen to keep the gauge bosons massless.