On-Shell neutral Higgs bosons in the NMSSM with complex parameters

The Next-to-Minimal Supersymmetric Standard model (NMSSM) appears as an interesting candidate for the interpretation of the Higgs-measurement at the LHC and as a rich framework embedding physics beyond the Standard Model. We consider the renormalization of the Higgs sector of this model in its $\mathcal{CP}$-violating version, and propose a renormalization scheme for the calculation of on-shell Higgs masses. Moreover, the connection between the physical states and the tree-level ones is no longer trivial at the radiative level: a proper description of the corresponding transition thus proves necessary in order to calculate Higgs production and decays at a consistent loop order. After discussing these formal aspects, we compare the results of our mass calculation to the output of existing tools. We also study the relevance of the on-shell transition-matrix in the example of the $h_i \to \tau^+ \tau^-$ width. We find deviations between our full prescription and popular approximations that can exceed $10\%$.