Exceptional regions of the 2HDM parameter space

The exceptional region of the parameter space (ERPS) of the two Higgs doublet model (2HDM) is defined to be the parameter regime where the scalar potential takes on a very special form. In the standard parametrization of the 2HDM scalar potential with squared mass parameters $m_{11}^2$, $m_{22}^2$, $m_{12}^2$, and dimensionless couplings, ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_7$, the ERPS corresponds to ${\lambda }_1={\lambda }_2$, ${\lambda }_7=-{\lambda }_6$, $m_{11}^2=m_{22}^2$ and $m_{12}^2=0$, corresponding to a scalar potential with an enhanced generalized $CP$ symmetry called GCP2. Many special features persist if ${\lambda }_1={\lambda }_2$ and ${\lambda }_7=-{\lambda }_6$ are retained while allowing for $m_{11}^2\ne m_{22}^2$ and/or $m_{12}^2\ne 0$, corresponding to a scalar potential with a softly broken GCP2 symmetry, which we designate as the ERPS4. In this paper, we examine many of the special features of the ERPS4, as well as even more specialized cases within the ERPS4 framework in which additional constraints on the scalar potential parameters are imposed. By surveying the landscape of the ERPS4, we complete the classification of 2HDM scalar potentials that exhibit an exact Higgs alignment (where the tree-level couplings of one neutral scalar coincide with those of the Standard Model Higgs boson), due to a residual symmetry that is unbroken in the vacuum. One surprising aspect of the ERPS4 is the possibility that the scalar sector is $CP$-conserving despite the presence of a complex parameter of the scalar potential whose complex phase cannot be removed by separate rephasings of the two scalar doublet fields. The significance of the ERPS4 regime for custodial symmetry is also discussed, and the cases where a custodial symmetric 2HDM scalar potential preserves an exact Higgs alignment are elucidated.