Absence of CP Violation in the Scalar Sector of a Higgs Triplet Model

Abstract

We investigate the potential for spontaneous CP violation in a Higgs Triplet Model extension, including a complex singlet scalar. We demonstrate that the scalar potential strictly forbids spontaneous CP violation across the entire parameter space. For a non-zero trilinear coupling $\kappa$, minimization conditions combined with global symmetries enforce a phase alignment that yields a real vacuum. In the singular limit $\kappa \to 0$, we show that the enhancement of global symmetries renders unphysical vacuum phases. Consequently, we conclude that the scalar sector of this model cannot source CP violation.