  The complex singlet extension CxSM of the Standard Model (SM) is a simple
extension of the SM with two visible Higgs bosons in the spectrum and a Dark
Matter (DM) candidate. In this paper we complete the computation of the
next-to-leading (NLO) electroweak (EW) corrections to on-shell and
non-loop-induced Higgs decays. Our calculations are implemented in the code
EWsHDECAY which also includes the relevant QCD corrections. Performing an
extensive parameter scan in the model and including all relevant theoretical
and experimental single- and di-Higgs as well as DM constraints, we obtain a
viable parameter sample. We find that current DM constraints are able to test
the model in DM mass regions where collider searches are not sensitive. The
relative EW corrections turn out to be large for scenarios with relatively
large couplings, threshold effects or small leading-order (LO) widths.
Otherwise, they are of typical EW size and can amount up to about 20-25%. The
theory uncertainty derived from the change of the renormalization scheme
dependence then is of a few per cent. While the NLO corrections applied in the
constraints due to single- and di-Higgs searches impact the validity of
specific parameter points, the overall shape of the allowed parameter region is
not yet sensitive to the EW corrections. This picture will change with further
increased experimental precision in the future and necessitates precise
predictions on the theory side as presented in this paper.
