  The principle of multi-critical-points (PMCP) may be a convincing approach to
determine the emerging parameter values in different kinds of
beyond-standard-model (BSM) models. This could certainly be applied to solve
the problem of undetermined new parameters in the ALP-Higgs interaction models.
In this paper, we apply this principle to such model and investigate whether
there are suitable solutions. Then, using the 1-loop effective potential, we
study the phase transition property of this model under the PMCP requirement.
It is gratifying to find that under the requirement of PMCP, the phase
transition can be not only first-order, but also strong enough to serve as a
solution for electroweak baryongenesis (EWBG). Finally, we show the parameter
space of ALP and provide the parameter range that leads to the first-order
phase transition.
