The ${\cal O}(\alpha_t+\alpha_\lambda+\alpha_\kappa)^2$ Correction to the $\rho$ Parameter and its Effect on the W Boson Mass Calculation in the Complex NMSSM

We present the prediction of the electroweak $\rho$ parameter and the $W$ boson mass in the CP-violating Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM) at the two-loop order. The $\rho$ parameter is calculated at the full one-loop and leading and sub-leading two-loop order $\mathcal{O}(\alpha + \alpha_t\alpha_s + \left(\alpha_t+\alpha_\lambda+\alpha_\kappa\right)^2)$. The new $\Delta \rho$ prediction is incorporated into a prediction of $M_W$ via a full supersymmetric (SUSY) one-loop calculation of $\Delta r$. Furthermore, we include all known state-of-the-art SM higher-order corrections to $\Delta r$. By comparing results for $\Delta \rho$ obtained using on-shell (OS) and $\overline{\mathrm{DR}}$ renormalization conditions in the top/stop sector, we find that the scheme uncertainty is reduced at one-loop order by 55%, at two-loop $\mathcal{O}(\alpha_s\alpha_t)$ by 22%, and at two-loop $\mathcal{O}(\alpha_t+\alpha_\kappa+\alpha_\lambda)^2$ by 16%, respectively. The influence of the two-loop results on the $M_W$ mass prediction is found to be sub-leading. The new calculation is made public in the computer program $\mathrm{\tt NMSSMCALC}$. We perform an extensive comparison in the $W$-mass, Higgs boson mass and the muon anomalous magnetic moment prediction between our calculation and three other publicly available tools and find very good agreement provided that the input parameters and renormalization scales are treated in the same way. Finally, we study the impact of the CP-violating phases on the $W$-mass prediction which is found to be smaller than the overall size of the SUSY corrections.