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Co-positivity of tensors and bounded from below conditions of CP conserving two-Higgs-doublet potential

Yisheng Song

Abstract

In this paper, the analytic sufficient and necessary conditions are obtained for the CP conserving two-Higgs-doublet potential to be bounded from below by using the co-positivity of tensors. This is achieved by treating the potential as a quartic homogeneous polynomial about the moduli of the two Higgs doublet fields, where the 'angles' is described as the misalignment of the two doublets, then solving three minimum problems with respect to the misalignment. Finally, the analytic conditions are established with the help of the corresponding theory and methods of higher order tensors.

Co-positivity of tensors and bounded from below conditions of CP conserving two-Higgs-doublet potential

Abstract

In this paper, the analytic sufficient and necessary conditions are obtained for the CP conserving two-Higgs-doublet potential to be bounded from below by using the co-positivity of tensors. This is achieved by treating the potential as a quartic homogeneous polynomial about the moduli of the two Higgs doublet fields, where the 'angles' is described as the misalignment of the two doublets, then solving three minimum problems with respect to the misalignment. Finally, the analytic conditions are established with the help of the corresponding theory and methods of higher order tensors.
Paper Structure (6 sections, 10 theorems, 55 equations)

This paper contains 6 sections, 10 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. 2HDM potential and Real tensors
  3. Co-positivity of Matrices and Tensors
  4. Bounded-from-below conditions
  5. Conclusions
  6. Some remarks

Key Result

Theorem 1

Let $\Lambda_1>0$, $\Lambda_2>0$. Then $V_4(\Phi_1,\Phi_2)\geq0$ if and only if where $\Delta=4(12\Lambda_1\Lambda_2-12\Lambda_6\Lambda_7+(\Lambda_3+\Lambda_4+\Lambda_5)^{2})^{3}-(72\Lambda_1\Lambda_2(\Lambda_3+\Lambda_4+\Lambda_5)+36\Lambda_6\Lambda_7(\Lambda_3+\Lambda_4+\Lambda_5)-2(\Lambda_3+\Lambda_4+\Lambda_5)^{3}-108\Lambda_1\Lambda_7^{2}-108\Lambda_6^{2}\Lambda_2)^{2}$.

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 1
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • Proposition 6
  • proof
  • ...and 6 more