Full Next-To Leading-Order Electroweak and QCD Corrections to the Relic Density in the CxSM

Abstract

In this work, we present the calculation of the next-to-leading order (NLO) QCD and electroweak corrections to the thermally averaged cross section of Dark Matter (DM) annihilation in the framework of the complex singlet extended Standard Model (CxSM) with one DM candidate. We derive the corresponding NLO QCD and electroweak corrected relic density and discuss the impact of the corrections as well as the remaining theoretical uncertainty due to missing higher-order corrections. This is estimated through a variation of the renormalization schemes for the singlet vacuum expectation value $v_S$ and the Higgs mixing angle $α$. For the bulk of the scanned parameter points, the QCD and electroweak corrections are found to be of typical size with the remaining theoretical uncertainty ranging at the percent level. The larger corrections that are found, can be attributed to a large counterterm for $v_S$ emerging in the case of large mass gaps in the decay channel used for the renormalization. The corrections are of phenomenological impact. There are parameter points that are allowed at leading order (LO), but are excluded after including the NLO corrections to the relic density, and vice versa. Our corrections have been implemented in the new code RelExt@NLO based on the LO code RelExt, which has been made publicly available. This work marks a first step towards the calculation of the QCD and electroweak corrected relic densities in models with DM candidates stabilized by a discrete $\mathbb{Z}_2$ symmetry.