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Two-loop RGE of a general renormalizable Yang-Mills theory in a renormalization scheme with an explicit UV cutoff. (English) Zbl 1390.81312
Summary: 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.
MSC:
81T13 Yang-Mills and other gauge theories in quantum field theory
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