Vacuum Structure in the Standard Model with a Complex Scalar Singlet

Abstract

We analyze the vacuum structure in an extension of the Standard Model with a complex singlet scalar field. Beyond the Standard Model, this model shows a much richer vacuum structure and a variety of interesting mass spectra. In this paper, we first show that the Higgs mass matrix can be simplified using the proper basis, and it can comprehensively reflect the change in the tadpole conditions due to the soft symmetry-breaking terms. Thus, this framework makes it easy to obtain a mass matrix through the corresponding tadpole conditions when considering new symmetry such as $\mathbb{Z}^\prime_2$. Finally, we introduce the Higgs potential for the trilinear and quartic Higgs self couplings, and summarize the vacuum structure taking into account all possible cases with a corresponding set of inputs and miscellaneous relations.