CP-odd tadpole renormalization of Higgs scalar-pseudoscalar mixing

We consider an Abelian model with a $\mathrm{CP}$-conserving Higgs potential spanned by two complex Higgs fields. The $\mathrm{CP}$ invariance of the Higgs potential is then broken explicitly beyond the Born approximation by introducing soft-$\mathrm{CP}$-violating Yukawa interactions. Based on the non-renormalization theorem, we derive the consistency conditions under which a $\mathrm{CP}$-odd counterterm exists and, at the same time, renders the one-loop-induced mixing of a $\mathrm{CP}$-even Higgs boson with a $\mathrm{CP}$-odd Higgs scalar ultra-violet finite. The novel $\mathrm{CP}$-odd tadpole renormalization is uniquely determined from the minimization constraints on the Higgs potential. The main phenomenological consequences of the so-generated $\mathrm{CP}$-violating scalar-pseudoscalar mixing are briefly discussed.