Properties of the general $N$-Higgs-doublet model. I. The orbit space

We study the scalar sector of the general $N$-Higgs-doublet model via geometric constructions in the space of gauge orbits. We give a detailed description of the shape of the orbit space both for general $N$ and, in more detail, for $N=3$. We also comment on remarkable analogies between the $N$-Higgs-doublet model and quantum information theory.

Figure 1

(Left) The $(n_3,n_8)$ plane; all other $n_i=0$. Shown are the unit circle (section of the light cone), the inner circle (dashed line), the three points $P$, $P^{\prime }$, $P^{\prime \prime }$, from the neutral manifold, and the three line segments from the charge-breaking manifold. (Right) The same plane, but with three sets of coordinates: $(n_3,n_8)$ shown in black solid lines, $(n_3^{\prime },n_8^{\prime })$ shown in dashed blue lines, and $(n_3^{\prime \prime },n_8^{\prime \prime })$ shown in dash-dotted red lines.

Figure 2

Generating a 4D cone from the triangle by mixing the first two doublets.