Minkowski space structure of the Higgs potential in the two-Higgs-doublet model. II. Minima, symmetries, and topology

We continue to explore the consequences of the recently discovered Minkowski space structure of the Higgs potential in the two-Higgs-doublet model. Here, we focus on the vacuum properties. The search for extrema of the Higgs potential is reformulated in terms of 3-quadrics in the $3+1$-dimensional Minkowski space. We prove that 2HDM cannot have more than two local minima in the orbit space and that a twice-degenerate minimum can arise only via spontaneous violation of a discrete symmetry of the Higgs potential. Investigating topology of the 3-quadrics, we give concise criteria for existence of noncontractible paths in the Higgs orbit space. We also study explicit symmetries of the Higgs potential/Lagrangian and their spontaneous violation from a wider perspective than usual.

Figure 1

Examples of the manifolds ${\mathcal{M}}^0$ in the $2+1$-dimensional picture. Left pane: all ${\Lambda }_i>0$, right pane: ${\Lambda }_1>0$, ${\Lambda }_2<0$. In each case, the light cone $L{C}^{+}$ is also shown for comparison.

Figure 2

A $2+1$-dimensional illustration of the contact of ${\mathcal{M}}^{C_{\min }}$ with $L{C}^{+}$. The thick point indicates the position of $m^{\mu }$.

Figure 3

The possible 2-sections of ${\mathcal{M}}^C$ and $L{C}^{+}$ for the case when all ${\Lambda }_i>0$. The left and middle plots correspond to the minimum and the saddle point, respectively, while the right plot cannot happen in 2HDM.

Figure 4

$2+1$-dimensional illustration of the nontrivial topology of the valley when $m^{\mu }$ lies inside $L{C}^{+}$. The shaded band is the bottom of the valley on the surface of $L{C}^{+}$.

Figure 5

The four possible cases of how the two-dimensional section (spanned by $e_1$ and $e_2$) of ${\mathcal{M}}^{C_{\min }}$ and $L{C}^{+}$ can look like; (a): ${\Lambda }_1$, ${\Lambda }_2>0$; (b) and (c): ${\Lambda }_1>0$, ${\Lambda }_2<0$; (d): ${\Lambda }_1>0$, ${\Lambda }_2=0$. In all cases ${\Lambda }_1>{\Lambda }_2$. The shaded region is the section of ${\mathcal{M}}^{<C_{\min }}$.