More uses for Wilson loops: Perturbation theory without Feynman diagrams

Abstract

For many arbitrary lattices with arbitrary $\mathrm{SU}(N)$ actions, it is easy to estimate the perturbative value of $\frac{{\ensuremath{\Lambda}}_{\mathrm{latt}}}{{\ensuremath{\Lambda}}_{\mathrm{MOM}}}$ without calculating any Feynman diagrams. This observation, first made by Creutz, is based on the fact that perturbative expansions of Wilson loop ratios can be trivially extracted from Monte Carlo data at large $\ensuremath{\beta}$. Here, we extend Creutz's results to general loop ratios including those of polygons and parallelograms encountered on nonstandard lattices. In particular, we analytically compute the lowest-order quantum corrections to these loop ratios, discuss which ratios are free from divergences, and give specific Monte Carlo examples.