Instantons and the renormalisation group in supersymmetric Yang-Mills theories

Abstract

The renormalisation group requires that in supersymmetric Yang-Mills theories the one- and two-loop quantum corrections to the vacuum energy (and related quantities) in an instanton background field vanish, at least for the terms proportional to In μ, where μ is the renormalisation point. It has recently been suggested that this result can be generalised to all orders in perturbation theory. We verify the two-loop calculation result by explicit calculation, both in the Wess-Zumino gauge and using covariant supergraphs. This requires a careful treatment of both the bosonic and fermionic (including the supergauge) zero modes. The subtleties and compliccations of our calculation and the cancellation mechanism make it difficult to imagine howa general all-orders argument might be constructed.