Perturbative unitarity for models with singlet and doublet scalars

TL;DR

The paper develops a general framework to impose perturbative unitarity bounds on the quartic scalar sector of models with arbitrary numbers of SU(2) doublets and singlets, motivated by beyond-Standard-Model scenarios including dark matter. It constructs a high-energy, 2→2 scattering analysis using partial-wave decomposition and the Equivalence Theorem, organizing two-particle states by $|Q,Y,T\rangle$ and deriving seven independent scattering matrices from the quartic potential. The authors provide analytic bounds when possible and perform extensive checks across representative models—from the SM to 2HDMs and multi-singlet setups—demonstrating how unitarity constrains quartic couplings such as $\lambda_i$, $\gamma_i$, $\beta_i$ and more. A central contribution is the Mathematica notebook BounDS, which automates potential construction, matrix generation, diagonalization, and extraction of eigenvalue bounds, enabling fast parameter-space scans for complex BSM scalar sectors. The results offer a practical, symmetry-adapted toolkit to ensure perturbative consistency of extended scalar sectors before performing mass/mixing analyses or collider/DM phenomenology.

Abstract

We provide a complete description of unitarity bounds on the gauge-scalar sectors of models with extra $SU(2)$ doublet, neutral singlet, and charged singlet scalars. Such additions are very frequent in models beyond the Standard Model, and, in particular, they are almost universal in models explaining the dark matter problem. We propose a specific classification and minimal set of scattering matrices containing all the relevant information. We also developed a Mathematica implementation of our results, BounDS, and we use it to fully study a number of simple cases, comparing with the literature, when available. The Mathematica notebook BounDS is provided via a public GitHub repository.