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

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.