Abstract
We complete the two-loop calculation of beta-functions for vacuum expectation values (VEVs) in gauge theories by the missing O(g^4)-terms. The full two-loop results are presented for generic and supersymmetric theories up to two-loop level in arbitrary R_xi-gauge. The results are obtained by means of a scalar background field, identical to our previous analysis. As a by-product, the two-loop scalar anomalous dimension for generic supersymmetric theories is presented. As an application we compute the beta-functions for VEVs and tan(beta) in the MSSM, NMSSM, and E6SSM.
Highlights
General gauge theory and scalar background fieldsThe renormalization of vacuum expectations can be cast in an elegant scheme by employing a scalar background field
As an application we compute the β-functions for vacuum expectation values (VEVs) and tan β in the MSSM, NMSSM, and E6SSM
The concrete results for anomalous dimensions and β-functions of VEVs and tan β are provided in the well-known supersymmetric models MSSM, NMSSM, and E6SSM
Summary
The renormalization of vacuum expectations can be cast in an elegant scheme by employing a scalar background field. The VEVs va are replaced in this formalism by scalar background fields (φa + va) These auxiliary fields allow to formulate a rigid (global) gauge invariant gauge fixing; analogous to ref. Without background fields, the most generic renormalization transformation of the scalar fields with shifts reads φa + va → Zab (φb + vb + δvb) = Zab (φb + vb) + δva. Kφb are the sources of the BRS transformation of the scalar field, and qa is the BRS transformation of φa Both of these unphysical fields appear in a very simple and well prescribed way in the Lagrangian. Our β-functions describe the running of the full VEV, which is required, for example, in many supersymmetry applications such as spectrum generators [7, 8]. Ref. [9] contains a diagram exposing the difference in the running between the different definitions
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