Sphalerons with $\mathrm{CP}$-violating Higgs potentials

We investigate the effect on the sphaleron in the two Higgs doublet electroweak theory of including $\mathrm{CP}$ violation in the Higgs potential. To have better control over the relation between the sphaleron energy and the physical quantities in the theory, we show how to parametrize the Higgs potential in terms of physical masses and mixing angles, one of which causes $\mathrm{CP}$ violation. By altering this $\mathrm{CP}$ violating angle (and keeping the other physical quantities fixed) the sphaleron energy increases by up to 10%. We also calculate the static minimum energy path between adjacent vacua as a function of Chern-Simons number, using the method of gradient flow. The only effect $\mathrm{CP}$ violation has on the barrier is the change in height. As a by-product of our work on parametrization of the potential, we demonstrate that $\mathrm{CP}$ violation in the Higgs sector favors nearly degenerate light Higgs boson masses.