Two-loop beta function for complex scalar electroweak multiplets
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Springer
Received: August 7, 2020 Accepted: August 24, 2020 Published: September 24, 2020
Joachim Brod and Zachary Polonsky Department of Physics, University of Cincinnati, Cincinnati, OH 45221, U.S.A.
E-mail: [email protected], [email protected] Abstract: We present the general form of the renormalizable four-point interactions of a complex scalar field furnishing an irreducible representation of SU(2), and derive a set of algebraic identities that facilitates the calculation of higher-order radiative corrections. As an application, we calculate the two-loop beta function for the SM extended by a scalar multiplet, and provide the result explicitly in terms of the group invariants. Our results include the evolution of the Higgs-portal couplings, as well as scalar “minimal dark matter”. We present numerical results for the two-loop evolution of the various couplings. Keywords: Renormalization Group, Gauge Symmetry ArXiv ePrint: 2007.13755
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)158
JHEP09(2020)158
Two-loop beta function for complex scalar electroweak multiplets
Contents 1 Introduction
1
2 Construction of the operator basis
2
3 Beta function for a scalar multiplet
5 16
5 Numerics
26
6 Conclusions
29
A Analytic checks of our calculation
30
B Renormalization constants
31
1
Introduction
Complex scalar fields furnishing a general representation of the electroweak gauge group SU(2) × U(1) of the standard model (SM) received increased interest in recent years. For instance, they can provide a viable dark matter candidate in so-called minimal dark-matter models [1]. The renormalization group (RG) evolution of coupling constants is an invaluable tool in phenomenological analyses [2]. It plays a particularly important role when interpreting and comparing the results of experiments performed at widely different energy scales, such as dark matter direct detection and production of dark matter at particle colliders. A framework for consistent RG analysis for fermionic dark matter in the context of effective field theories has been presented in ref. [3]. The first consistent and complete basis of effective operators for scalar dark matter up to mass dimension six has been written down in ref. [4]; however, the RG evolution has not yet been calculated. For scalar dark matter it is possible to write down self interactions, as well as interactions with the SM, at the renormalizable level — the so-called Higgs-portal dark matter [5–7]. To our knowledge, the first classification of the self interactions of scalar fields with electroweak charges has been given in ref. [8]. In this work, we rederive the scalar potential in a slightly different form that is well suited for the calculation of radiative corrections. We then calculate the beta functions for all scalar couplings, as well as the new scalar contributions to all SM couplings, at the two-loop level. To this end, we prove a set of algebraic relations that allows to express all two-loop matrix element
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