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

M. Aa. Solberg

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.

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

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.
Paper Structure (68 sections, 5 theorems, 294 equations, 2 figures)

This paper contains 68 sections, 5 theorems, 294 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the article
  3. Conventions and notation
  4. Lie symmetry analysis of PDEs
  5. Point symmetries of systems of PDEs
  6. Prolongations of infinitesimal generators
  7. Variational symmetries
  8. Characterization and preservation of symmetry types
  9. Affine reparametrizations
  10. SM+S
  11. Lagrangian
  12. Linear terms
  13. Solving the determining equations
  14. Nature of the symmetries
  15. Algorithm for determining SM+S symmetry algebras
  16. ...and 53 more sections

Key Result

Theorem 1

Let $\mathcal{L}=T-V$ be a Lagrangian with $T\in \mathbb{R}\{y\}$ and $V(\varphi_1,\ldots,\varphi_m) \in \mathbb{R}[\varphi]$ with $\varphi \subset y$, and let the infinitesimal generator where $\eta^i \in \mathbb{R}[y]$ for all $i$, be a symmetry of $E(\mathcal{L})=0$, with constant terms $a_i=\eta^i(0)$. Moreover, let all terms in $T$ be either at least quadratic in elements of the set $\varphi

Figures (2)

  • Figure 1: Reduction tree, Branches I and II. Each path starting from the root at the top and terminating at a leaf (marked in red) at the bottom corresponds to a reduced potential. For each leaf, the determining equations are solved and the corresponding symmetries are derived. Leaf 1 is discussed in Section \ref{['sec:BranchOne']}, while Leaf 2 is treated in Section \ref{['sec:BranchTwo']}.
  • Figure 2: Reduction tree, Branch IV (continued). Each path starting from the root in Fig. \ref{['fig:reduction-tree1']} and terminating at a leaf (numbered and marked in red) corresponds to a reduced potential. For each leaf, the determining equations for the reduced potential are solved and the associated symmetries are derived. The empty set $\varnothing$ denotes that no further reductions apply.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2: Affine reparametrizations
  • proof
  • Proposition 3
  • proof