Relic density of dark matter in the inert doublet model beyond leading order for the low mass region. I. Renormalization and constraints

Abstract

The present paper is the first in a series that addresses the calculation of the full one-loop corrections of dark matter (DM) annihilation cross sections in the low mass region of the inert doublet model (IDM). This series is a sequel to our recent publication concerning these corrections in the high mass region. We first review the renormalization of the model both in a fully on shell (OS) scheme as well as in a mixed scheme that combines on shell (for the masses) and a $\overline{\mathrm{MS}}$ approach when the partial invisible width is closed and does not allow the use of a full OS scheme. The scale dependence introduced by the mixed scheme is shown to be tracked through an analysis of a parametrization of the tree-level cross section and the $\ensuremath{\beta}$ constant of a specific coupling; this analysis could be followed in theories. We discuss how to minimize the scale dependence. The theoretical uncertainty brought by the scale dependence leads us to introduce a new criterion on the perturbativity of the IDM. This criterion further delimits the allowed parameter space, which we investigate carefully by including a host of constraints, both theoretical and experimental, including in particular, new data from the LHC. We come up with a set of benchmark points that cover three different mechanisms for a viable relic density of DM: (i) a dominance of coannihilation into a fermion pair, (ii) annihilation into two vector bosons of which one is off shell that requires the calculation of a $2\ensuremath{\rightarrow}3$ process, (iii) annihilation that proceeds through the very narrow standard model Higgs resonance. Since the $2\ensuremath{\rightarrow}3$ vector boson channel features in all three channels and is essentially a buildup on the simpler annihilation to OS vector bosons, we study the latter in detail in the present paper. We confirm again that the corrected cross sections involve a parameter that can be considered as rescattering in the dark sector, which a tree-level computation is not sensitive to. The setup of the renormalization detailed in the present paper will be the backbone of the accompanying papers where each mechanism requires, calculationally, a specific treatment. One-loop corrections to $2\ensuremath{\rightarrow}3$ processes for DM annihilation are technically challenging and have not been attempted before. We dedicate one of the accompanying papers to such a computation. The one-loop correction in the presence of a resonance will be presented separately since we need to supplement our general schemes with a complex scheme. For the coannihilation into fermions, our study will show that the annihilation cross sections can be excellently parametrized through simple effective couplings.