  We study the Multi-critical Point Principle (MPP) in a complex singlet scalar
extension of the Standard Model (CxSM). The MPP discussed in this study selects
model parameters so that two low-energy vacua realized by scalar fields are
degenerate. We further note that the MPP may inhibit the electroweak phase
transition (EWPT) in a certain class of models where the tree-level potential
plays an essential role in its realization. Despite that, we show that strong
first-order EWPT still occurs even after imposing the MPP to the scalar
potential of the CxSM due to the 1-loop corrections by the new scalar boson. We
study the allowed parameter space where a mass of the additional scalar is
degenerate with that of the Standard Model Higgs boson, which provides a
built-in mechanism to circumvent constraints from dark matter direct detection
experiments. The parameter space for the non-degenerate scalar scenario is also
studied for comparison.
