Two-loop beta function for complex scalar electroweak multiplets

Joachim Brod,Zachary Polonsky

doi:10.1007/jhep09(2020)158

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.

Highlights

JHEP09(2020)158 relations rely on the algebra of Clebsch-Gordan coefficients as well as SU(2) gauge symmetry, many of them turn out to be quite non-trivial, and have not been derived before, to the best of our knowledge
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
While these results are known in principle [9, 10], we present them in closed form and explicitly in terms of group invariants for the first time

Summary

Construction of the operator basis

The form of the real coefficients vijrφks must be determined such that the operator Oφ is invariant under the SU(2) gauge group (the U(1) invariance is immediately apparent). The symmetry properties of the Clebsch-Gordan coefficients imply the corresponding properties of the Sigma matrices, Σ(mJm),a (−1)J. This restricts the number of independent operators in the basis. The only non-zero operators in our basis are those involving Sigma matrices that are symmetric in their lower indices, Σ(mJm)a. The sum over J in eq (2.14) effectively runs only over even values for integer jφ, while for half-integer jφ only terms with odd J contribute.

Beta function for a scalar multiplet

Group theory relations

Numerics

Conclusions

A Analytic checks of our calculation

B Renormalization constants