Stability and Symmetry Breaking in the General Two-Higgs-Doublet Model

TL;DR

The paper develops a gauge-invariant, orbit-based formalism for the THDM scalar potential using bilinears $K_0$ and $\boldsymbol{K}$, and derives complete stability and EWSB conditions via the quartic form $J_4(\boldsymbol{k})$ and auxiliary functions $f(u)$ and $g(u)$. Stationary points are located through a unified framework that handles interior and boundary cases, enabling identification of global minima with $K_0=|\boldsymbol{K}|$ (neutral vacuum) and positivity of the charged-Higgs mass through $u_0>0$. The method is applied to MSSM and Gunion et al. THDM, reproducing known results for MSSM and revealing richer stationary-point structures and possible CP-violating minima in the Gunion THDM; it also clarifies conditions under D-flat directions do not destabilize the potential. The approach is set up for extension to multi-Higgs-doublet models and highlights the need to incorporate quantum corrections for a complete quantum-theory treatment.

Abstract

A method is presented for the analysis of the scalar potential in the general Two-Higgs-Doublet Model. This allows us to give the conditions for the stability of the potential and for electroweak symmetry breaking in this model in a very concise way. These results are then applied to two different Higgs potentials in the literature, namely the MSSM and the Two-Higgs-Doublet potential proposed by Gunion at al. All known results for these models follow easily as special cases from the general results. In particular, in the potential of Gunion et al. we can clarify some interesting aspects of the model with the help of the proposed method.

Figures (3)

• Figure 1: The stability determining functions $f'(u)$ and $f(u)$ as given by (\ref{['eq-fprd']}) and (\ref{['eq-fdiag']}) with $\eta_{00} = 0.05$, $(\mu_1,\mu_2, \mu_3) = (0.01, 0.02, 0.03)$ and $(\eta_1, \eta_2, \eta_3) = (0.002, 0.002, 0.002)$.
• Figure 2: The global minimum determining functions $\tilde{f}'(u)$ and $K_0(u)$ for the MSSM, see (\ref{['eq-mssmftp']}) and (\ref{['eq-mssmk0']}), with $\left\lvert {\mu} \right\rvert^2+m_{Hd}^2 = 157486\ \text{GeV}^2$, $\left\lvert {\mu} \right\rvert^2+m_{Hu}^2 = -2541\ \text{GeV}^2$, $\left\lvert {m_3^2} \right\rvert = 15341\ \text{GeV}^2$. The small boxes show the functions with enhanced ordinate resolution in the region around the physically relevant zero of $\tilde{f}'(u)$.
• Figure 3: The potential $V$ of Gunion:1989we, shifted to $V(\boldsymbol{\tilde{K}}=0)=0$, at all stationary points with $K_0 = \left\lvert {\boldsymbol{K}} \right\rvert > 0$ in dependence of $\lambda_1$, where $\lambda_2 = \lambda_1$. The other parameters are $\lambda_3=0.1,\ \lambda_4=0.2$, $\lambda_5=\lambda_6=0.4$, $v_1=30\text{~GeV},\ v_2=171\text{~GeV},\ \lambda_7=0,\ \xi=0$. The lines represent regular stationary points, where the solid curve corresponds to the "obvious" solution with $\tan\beta=v_2/v_1$ and $v_0=\sqrt{2(v_1^2+v_2^2)}$, which is a local minimum for the chosen parameters. For $\lambda_1=\lambda_2=0$ there are two degenerate exceptional minima and regular solutions only for the saddle points. Depending on $\lambda_1=\lambda_2$, the global minimum is given by that local minimum out of the two which has the lower potential.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5