Two Higgs doublet model with a complex singlet scalar and Multi-critical Point Principle

We study a two Higgs doublet model extended by a complex singlet scalar, in which the imaginary part of the singlet serves as a dark matter (DM) candidate. In this model, degenerate masses of the three neutral Higgs bosons are crucial for achieving consistency with current constraints from DM direct-detection experiments and Higgs searches. This is called the degenerate scalar scenario. To provide a theoretical motivation for such a degenerate Higgs spectrum, we impose the tree-level Multiple Point Principle (MPP), which requires the electroweak and singlet vacua to be degenerate, and analyze its implications for the scalar potential, DM phenomenology, and the electroweak phase transition. We show that the tree-level MPP favors large SU(2)$_L$ doublet-singlet mixing parameters, which compete with the degenerate scalar scenario. Nevertheless, we demonstrate that viable parameter regions still exist in which the observed DM constraints are satisfied. Furthermore, although the tree-level MPP forbids a tree-level-driven first-order electroweak phase transition, we show that thermal loop effects can induce a strong first-order transition compatible with electroweak baryogenesis.