Dependence of holomorphicity of the gauge coupling constant on the mass matrix in supersymmetric theories

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Abstract
We investigate the dependence of the holomorphicity of the gauge coupling constant function on the mass matrix at one- and two-loop levels in supersymmetric theories. Gauge invariance puts constraints on the mass matrix. These constraints at the one-loop level lead us to three cases of mass matrix that require different ways of regulating the infrared contributions: massive, pseudomassive, and intrinsically massless. The first two give rise to a holomorphic gauge coupling constant function whereas the last one does not. Two-loop contributions to super QED and super Yang-Mills theory are calculated using the super background field method and their dependence on the mass matrix is found to fall under the same three cases as at the one-loop level. Remarks concerning the general nature of this result to all orders in perturbation theory are included. Making use of our two-loop results we also verify the holomorphicity of the Wilson coupling based on general arguments of Shifman and Vainshtein.