Phenomenology of the inert ($2+1$) and ($4+2$) Higgs doublet models

We make a phenomenological study of a model with two inert doublets plus one Higgs doublet [$\mathrm{I}(2+1)\mathrm{HDM}$] which is symmetric under a $Z_2$ group, preserved after electroweak symmetry breaking by the vacuum alignment $(0,0,v)$. This model may be regarded as an extension to the model with one inert doublet plus one Higgs doublet [$\mathrm{I}(1+1)\mathrm{HDM}$], by the addition of an extra inert scalar doublet. The neutral fields from the two inert doublets provide a viable dark matter (DM) candidate which is stabilized by the conserved $Z_2$ symmetry. We study the new Higgs decay channels offered by the scalar fields from the extra doublets and their effect on the standard model Higgs couplings, including a new decay channel into (off-shell) photon(s) plus missing energy, which distinguishes the $\mathrm{I}(2+1)\mathrm{HDM}$ from the $\mathrm{I}(1+1)\mathrm{HDM}$. Motivated by supersymmetry, which requires an even number of doublets, we then extend this model into a model with four inert doublets plus two Higgs doublets [$\mathrm{I}(4+2)\mathrm{HDM}$] and study the phenomenology of the model with the vacuum alignment $(0,0,0,0,v,v)$. This scenario offers a wealth of Higgs signals, the most distinctive ones being cascade decays of heavy Higgs states into inert ones. Finally, we also remark that the smoking-gun signature of all the considered models is represented by invisible Higgs decays into the lightest inert Higgs bosons responsible for DM.

Figure 1

Schematic mass-squared spectrum of the $Z_2$ symmetric 3HDM, where $\mathrm{\Sigma }=4{\mu }_{12}^4+({\mu }_1^2-{\mathrm{\Lambda }}_{{\varphi }_1}-{\mu }_2^2+{\mathrm{\Lambda }}_{{\varphi }_2}{)}^2$, and ${\mathrm{\Sigma }}^{\prime }=4{\mu }_{12}^4+({\mu }_1^2-{\mathrm{\Lambda }}_{{\varphi }_1}^{\prime }-{\mu }_2^2+{\mathrm{\Lambda }}_{{\varphi }_2}^{\prime }{)}^2$ and ${\mathrm{\Sigma }}^{\prime \prime }=4{\mu }_{12}^4+({\mu }_1^2-{\mathrm{\Lambda }}_{{\varphi }_1}^{\prime \prime }-{\mu }_2^2+{\mathrm{\Lambda }}_{{\varphi }_2}^{\prime \prime }{)}^2$. Note that the third doublet plays the role of the SM Higgs doublet.

Figure 2

Parameter scan of the $\mathrm{I}(2+1)\mathrm{HDM}$ mapped in terms of $\mathrm{BR}(h\to H_1{H}_1)$ as a function of ${\mu }_1^2$ (top-left), ${\mu }_2^2$ (top-right) and ${\mu }_{12}^2$ (bottom).

Figure 3

Parameter scan of the $\mathrm{I}(2+1)\mathrm{HDM}$ mapped in terms of $\mathrm{BR}(h\to H_1{H}_1)$ as a function of ${\lambda }_2$ (top-left), ${\lambda }_3$ (top-right), ${\lambda }_{11}$ (bottom-left) and ${\lambda }_{22}$ (bottom-right).

Figure 4

Parameter scan of the $\mathrm{I}(2+1)\mathrm{HDM}$ mapped in terms of $\mathrm{BR}(h\to H_1{H}_1)$ as a function of ${\lambda }_{12}$ (top-left), ${{\lambda }^{\prime }}_{12}$ (top-right), ${\lambda }_{23}$ (middle-left), ${{\lambda }^{\prime }}_{23}$ (middle-right), ${\lambda }_{31}$ (bottom-left) and ${{\lambda }^{\prime }}_{31}$ (bottom-right).

Figure 5

The loop diagrams responsible for $H_2\to H_1{\gamma }^{*}$ decays, where $H_1$ is absolutely stable and hence invisible, while ${\gamma }^{*}$ is a virtual photon that couples to fermion-antifermion pairs.

Figure 6

Parameter scan of the $\mathrm{I}(2+1)\mathrm{HDM}$ mapped in terms of $\mathrm{BR}(h\to H_1{H}_2)+\mathrm{BR}(h\to H_2{H}_2)$ as a function of $m_{H_2}-m_{H_1}$. The plot indicates SM Higgs decay rates into photon(s) of up to about 30 GeV plus missing (transverse) energy.

Figure 7

Parameter scan of the $\mathrm{I}(4+2)\mathrm{HDM}$ mapped in terms of $\mathrm{BR}(h\to H_{1u}^0{H}_{1u}^0)+\mathrm{BR}(h\to A_{1u}^0{A}_{1u}^0)$ as a function of $m_{H_{1u}^0}$ (top-left) and $m_{A_{1u}^0}$ (top-right) as well as of the $m_{H_{1u}^0}$ vs $m_{A_{1u}^0}$ correlation (bottom) for $\tan \beta =1$ and ${\lambda }_{aa}=0(\ne 0)$ for the red-star(blue-cross) symbol. (Recall that the two channels $h\to H_{1u}^0{H}_{1u}^0$ and $h\to A_{1u}^0{A}_{1u}^0$ are mutually exclusive, as explained in the text.)

Figure 8

Parameter scan of the $\mathrm{I}(4+2)\mathrm{HDM}$ mapped in terms of $\mathrm{BR}(h\to H_{1u}^0{H}_{1u}^0)+\mathrm{BR}(h\to A_{1u}^0{A}_{1u}^0)$ as a function of ${\mu }_a^2$ (top-left), ${\mu }_a^{\prime 2}$ (top-right), ${\mu }_i^2$ (bottom-left) and ${\mu }_i^{\prime 2}$ (bottom-right) for $\tan \beta =1$ and ${\lambda }_{aa}=0(\ne 0)$ for the red-star(blue-cross) symbol. (Recall that the two channels $h\to H_{1u}^0{H}_{1u}^0$ and $h\to A_{1u}^0{A}_{1u}^0$ are mutually exclusive, as explained in the text.)

Figure 9

Parameter scan of the $\mathrm{I}(4+2)\mathrm{HDM}$ mapped in terms of $\mathrm{BR}(h\to H_{1u}^0{H}_{1u}^0)+\mathrm{BR}(h\to A_{1u}^0{A}_{1u}^0)$ as a function of ${\lambda }_{ai}$ (top-left), ${{\lambda }^{\prime }}_{ai}$ (top-right), ${\lambda }_{aa}$ (center-left), ${\lambda }_i$ (center-right), ${\lambda }_1$ (bottom-left) and ${{\lambda }^{\prime }}_1$ (bottom-right) for $\tan \beta =1$ and ${\lambda }_{aa}=0(\ne 0)$ for the red-star(blue-cross) symbol. (Recall that the two channels $h\to H_{1u}^0{H}_{1u}^0$ and $h\to A_{1u}^0{A}_{1u}^0$ are mutually exclusive, as explained in the text.)

Figure 10

Parameter scan of the $\mathrm{I}(4+2)\mathrm{HDM}$ mapped in terms of $\mathrm{BR}(h\to H_{1u}^0{H}_{1u}^0)+\mathrm{BR}(h\to A_{1u}^0{A}_{1u}^0)$ (top-left), $\mathrm{BR}(A\to A_x{H}_x)+\mathrm{BR}(A\to A_y{H}_y)+\mathrm{BR}(A\to A_x{H}_y)+\mathrm{BR}(A\to A_y{H}_x)+\mathrm{BR}(A\to A_z{H}_z)$ (top-right) and $\mathrm{BR}(H^{\pm }\to H_z^{\pm }{H}_z)+\mathrm{BR}(H^{\pm }\to H_z^{\pm }{A}_z)$ (bottom) as a function of the mass of the decaying Higgs state for $\tan \beta \ne 1$ and ${\lambda }_{aa}\ne 0$.