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A strongly first-order electroweak phase transition is a necessaryrequirement for Electroweak Baryogenesis. We investigate the plausibility ofobtaining a strong phase transition in a Two-Higgs-Doublet Model of type IIwith a minimal amount of $CP$ violation. By performing a Bayesian fit where weconstrain the scalar sector with indirect and direct measurements, we find thatcurrent data disfavours a first-order phase transition in this model. Thisresult is mainly driven by the interplay of three effects: Constraints from theLHC Higgs data on the magnitude of the quartic couplings, the requirement of a$H^\pm$ heavier than around 490 GeV to avoid large contributions to $BR(b\rightarrow s\gamma)$ and the fact that a first-order phase transition requiresrelatively light scalar states in addition to the 125 GeV Higgs. For similarreasons we find that a "hidden-Higgs" scenario, in which the 125 GeV state isidentified with the next-to-lightest scalar, is disfavoured by current dataindependent of any requirement on the phase transition strength.

Disfavouring Electroweak Baryogenesis and a hidden Higgs in a CP-violating Two-Higgs-Doublet Model