Precision calculations for $h\to WW/ZZ\to 4$ fermions in a singlet extension of the Standard Model with PROPHECY4F

We consider an extension of the Standard Model by a real singlet scalar field with a $Z_2$-symmetric Lagrangian and spontaneous symmetry breaking with vacuum expectation value for the singlet. Considering the lighter of the two scalars of the theory to be the $125GeV$ Higgs particle, we parametrize the scalar sector by the mass of the heavy Higgs boson, a mixing angle $\alpha$, and a scalar Higgs self-coupling ${\lambda }_{12}$. Taking into account theoretical constraints from perturbativity and vacuum stability, we compute next-to-leading-order electroweak and QCD corrections to the decays $h\to WW/ZZ\to 4$ fermions of the light Higgs boson for some scenarios proposed in the literature. We formulate two renormalization schemes and investigate the conversion of the input parameters between the schemes, finding sizeable effects. Solving the renormalization-group equations for the $\overline{\text{MS}}$ parameters $\alpha$ and ${\lambda }_{12}$, we observe a significantly reduced scale and scheme dependence in the next-to-leading-order results. For some scenarios suggested in the literature, the total decay width for the process $h\to 4f$ is computed as a function of the mixing angle and compared to the width of a corresponding Standard Model Higgs boson, revealing deviations below $10\%$. Differential distributions do not show significant distortions by effects beyond the Standard Model. The calculations are implemented in the Monte Carlo generator Prophecy4f, which is ready for applications in data analyses in the framework of the singlet extension.