Scalar Lie point symmetries of the Standard Model with one or two real gauge singlets

TL;DR

This paper classifies all scalar Lie point symmetries of the Standard Model extended with one or two real gauge singlet scalars, providing algorithms for symmetry determination and characterizing different types of symmetry generators.

Contribution

It introduces a comprehensive classification of scalar Lie point symmetries in extended Standard Model models and develops efficient algorithms for their identification.

Findings

Identified all realizable Lie point symmetry algebras for SM+S and SM+2S models.

Developed parameter-based algorithms to determine symmetry algebras without solving determining equations.

Proved general results characterizing types of symmetry generators for broad classes of Lagrangians.

Abstract

We present a classification of all scalar Lie point symmetries of the Standard Model with one or two real gauge-singlet scalars (SM+S and SM+2S). By analyzing the associated field equations, we identify all realizable and inequivalent Lie point symmetry algebras of these models, distinguishing strict variational, variational (including divergence symmetries), and Euler--Lagrange cases. In addition, we devise efficient algorithms that, for any given numerical instance of the models, determine the Lie point symmetry algebra in each of the three categories by a parameter-based decision procedure using affine reparametrizations and simple parameter tests, thereby avoiding explicit symmetry analysis and the need to derive and solve the determining equations. Finally, we prove several relevant general results, including a characterization of the three disjoint types of Lie point symmetry generators -- strict variational, divergence, and non-variational -- for a broad class of Lagrangians with potentials, including the SM+S and SM+2S.