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Lie symmetry analysis of the two-Higgs-doublet model field equations

M. Aa. Solberg

Abstract

We apply Lie symmetry analysis of partial differential equations (PDEs) to the Euler-Lagrange equations of the two-Higgs-doublet model (2HDM), to determine its scalar Lie point symmetries. A Lie point symmetry is a structure-preserving transformation of the spacetime variables and the fields of the model, which is also continuous and connected to the identity. Symmetries of PDEs may in general be divided into strict variational symmetries, divergence symmetries and non-variational symmetries, where the first two are collectively referred to as variational symmetries. Variational symmetries are usually preserved under quantization, and variational Lie symmetries yield conservation laws. We demonstrate that there are no scalar Lie point divergence symmetries or non-variational Lie point symmetries in the 2HDM, and re-derive its well-known strict variational Lie point symmetries, thus confirming the consistency of our implementation of Lie's method. Moreover, we prove three general results which may simplify Lie symmetry calculations for a wide class of particle physics models. Lie symmetry analysis of PDEs is a broadly applicable method for determining Lie symmetries. As demonstrated here by example, it can be applied to models with many variables, parameters and reparametrization freedom, while any missing discrete symmetries may be identified through the automorphism groups of the resulting Lie symmetry algebras.

Lie symmetry analysis of the two-Higgs-doublet model field equations

Abstract

We apply Lie symmetry analysis of partial differential equations (PDEs) to the Euler-Lagrange equations of the two-Higgs-doublet model (2HDM), to determine its scalar Lie point symmetries. A Lie point symmetry is a structure-preserving transformation of the spacetime variables and the fields of the model, which is also continuous and connected to the identity. Symmetries of PDEs may in general be divided into strict variational symmetries, divergence symmetries and non-variational symmetries, where the first two are collectively referred to as variational symmetries. Variational symmetries are usually preserved under quantization, and variational Lie symmetries yield conservation laws. We demonstrate that there are no scalar Lie point divergence symmetries or non-variational Lie point symmetries in the 2HDM, and re-derive its well-known strict variational Lie point symmetries, thus confirming the consistency of our implementation of Lie's method. Moreover, we prove three general results which may simplify Lie symmetry calculations for a wide class of particle physics models. Lie symmetry analysis of PDEs is a broadly applicable method for determining Lie symmetries. As demonstrated here by example, it can be applied to models with many variables, parameters and reparametrization freedom, while any missing discrete symmetries may be identified through the automorphism groups of the resulting Lie symmetry algebras.

Paper Structure

This paper contains 34 sections, 4 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Conventions and notation
  4. Lie symmetry analysis of PDEs
  5. Point symmetries of systems of PDEs
  6. Infinitesimal generators and their prolongations
  7. Symmetries of the action $\mathcal{S}$
  8. Theories with potentials
  9. Scalar, variational symmetries in any spacetime dimension
  10. An illustrative example: real, scalar $\phi^4$ theory
  11. $\phi^4$ theory in one dimension
  12. Massive $\phi^4$ theory
  13. Massless $\phi^4$ theory
  14. Free, massive scalar theory
  15. Free, massless scalar theory
  16. ...and 19 more sections

Key Result

Theorem 1

Let $\mathcal{L}=T-V$ be a polynomial potential theory and let the infinitesimal generator where each $\eta^i$ is a polynomial, be a symmetry of $E(\mathcal{L})=0$. Moreover, assume that either $V(\varphi_1,\ldots,\varphi_m)$ does not contain any linear terms $\alpha_i \varphi_i$, or that $\alpha_i\ne 0 \Rightarrow \eta^i(y^1,\ldots,y^{q})$ does not contain a constant term. Then,

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • proposition 1
  • proof
  • Corollary 1
  • proposition 1
  • proof