Exceptional regions of the 2HDM parameter space

The exceptional region of the parameter space (ERPS) two Higgs doublet model (2HDM) is defined to be regime where scalar potential takes on a very special form. In standard parametrization 2HDM with squared mass parameters ${m}_{11}^{2}$, ${m}_{22}^{2}$, ${m}_{12}^{2}$, and dimensionless couplings, ${\ensuremath{\lambda}}_{1},{\ensuremath{\lambda}}_{2},\dots{},{\ensuremath{\lambda}}_{7}$, ERPS corresponds ${\ensuremath{\lambda}}_{1}={\ensuremath{\lambda}}_{2}$, ${\ensuremath{\lambda}}_{7}=\ensuremath{-}{\ensuremath{\lambda}}_{6}$, ${m}_{11}^{2}={m}_{22}^{2}$ ${m}_{12}^{2}=0$, corresponding an enhanced generalized $CP$ symmetry called GCP2. Many features persist if ${\ensuremath{\lambda}}_{1}={\ensuremath{\lambda}}_{2}$ ${\ensuremath{\lambda}}_{7}=\ensuremath{-}{\ensuremath{\lambda}}_{6}$ are retained while allowing for ${m}_{11}^{2}\ensuremath{\ne}{m}_{22}^{2}$ and/or ${m}_{12}^{2}\ensuremath{\ne}0$, softly broken GCP2 symmetry, which we designate as ERPS4. this paper, examine many ERPS4, well even more specialized cases within ERPS4 framework in additional constraints imposed. By surveying landscape complete classification potentials that exhibit exact alignment (where tree-level couplings one neutral coincide those Standard Model boson), due residual unbroken vacuum. One surprising aspect possibility sector $CP$-conserving despite presence complex whose phase cannot removed by separate rephasings fields. significance custodial also discussed, symmetric preserves elucidated.