Two-loop RGE of a general renormalizable Yang-Mills theory in a renormalization scheme with an explicit UV cutoff

Abstract

We perform a systematic one-loop renormalization of a general renormalizable Yang-Mills theory coupled to scalars and fermions using a regularization scheme with a smooth momentum cutoff $\Lambda$ (implemented through an exponential damping factor). We construct the necessary finite counterterms restoring the BRST invariance of the effective action by analyzing the relevant Slavnov-Taylor identities. We find the relation between the renormalized parameters in our scheme and in the conventional $\overline{\mathrm{MS}}$ scheme which allow us to obtain the explicit two-loop renormalization group equations in our scheme from the known two-loop ones in the $\overline{\mathrm{MS}}$ scheme. We calculate in our scheme the divergences of two-loop vacuum graphs in the presence of a constant scalar background field which allow us to rederive the two-loop beta functions for parameters of the scalar potential. We also prove that consistent application of the proposed regularization leads to counterterms which, together with the original action, combine to a bare action expressed in terms of bare parameters. This, together with treating $\Lambda$ as an intrinsic scale of a hypothetical underlying finite theory of all interactions, offers a possibility of an unconventional solution to the hierarchy problem if no intermediate scales between the electroweak scale and the Planck scale exist.

arXiv commentupdated references, 90 pages, many figures

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• Formula for two-loop divergent part of 4-D Yang-Mills effective action
• Explicit Cutoff Regularization in Coordinate Representation
• All-order existence of and recursion relations for the operator product expansion in Yang-Mills theory
• LSZ-reduction, resonances and non-diagonal propagators: gauge fields
• LSZ-reduction, resonances and non-diagonal propagators: fermions and scalars
• Divergences in anomalous dimension matrices of quarks at three loops: Explanation and simple solution