Gauge Dependences of Higher-Order Corrections to NMSSM Higgs Boson Masses and the Charged Higgs Decay $H^\pm \to W^\pm h_i$

In this paper we compute the electroweak corrections to the charged Higgs boson decay into a $W$ boson and a neutral Higgs boson in the CP-conserving NMSSM. We calculate the process in a general $R_\xi$ gauge and investigate the dependence of the loop-corrected decay width on the gauge parameter $\xi$. The gauge dependence arises from the mixing of different loop orders. Phenomenology requires the inclusion of mass and mixing corrections to the external Higgs bosons in order to match the experimentally measured mass values. As a result, we move away from a strict one-loop calculation and consequently mix orders in perturbation theory. Moreover, determination of the loop-corrected masses in an iterative procedure also results in the mixing of different loop orders. Gauge dependence then arises from the mismatch with tree-level Goldstone boson couplings that are applied in the loop calculation, and from the gauge dependence of the loop-corrected masses themselves. We find that the gauge dependence is significant.