Three loop neutrino model with isolated $k^{\pm \pm }$

We propose a three loop radiative neutrino mass scenario with an isolated doubly charged singlet scalar $k^{\pm \pm }$ without couplings to the charged leptons, while two other singly charged scalars $h_1^{\pm }$ and $h_2^{\pm }$ attach to them. In this setup, the lepton flavor violation originating from $k^{\pm \pm }$ exchanges is suppressed and the model is less constrained, where some couplings can take sizable values. As reported in our previous work [1], the loop suppression factor at the three loop level would be too strong and realized neutrino masses in a three loop scenario could be smaller than the observed minuscule values. The sizable couplings can help us to enhance neutrino masses without drastically large scalar trilinear couplings appearing in a neutrino mass matrix, which tends to drive the vacuum stability to become jeopardized at the one loop level. Now the doubly charged scalar $k^{\pm \pm }$ has less constraint via lepton flavor violation and the vacuum can be quite stable, and thus a few hundred GeV mass in $k^{\pm \pm }$ is possible, which is within the LHC reach and this model can be tested in the near future. Note that the other $h_1^{\pm }$ and $h_2^{\pm }$ should be heavy at least around a few TeV. We suitably arrange the charges of an additional global $U(1)$ symmetry, where the decay constant of the associated Nambu-Goldstone boson can be around a TeV scale consistently. Also, this model is indirectly limited through a global analysis on results of the LHC Higgs search and issues on a dark matter candidate, the lightest Majorana neutrino. After $h_1^{\pm }$ and $h_2^{\pm }$ are decoupled, this particle couples to the standard model particles only through two charge parity even scalars in theory and thus information on this scalar sector is important. Consistent solutions are found, but a part of them is now on the edge.

Figure 1

A schematic description for the radiative generation of neutrino masses.

Figure 2

Histograms showing distributions of $\delta$, $\varphi$, $(y_L{)}_{23}$, $|(y_R{)}_{22}|$ and $\mu$ of the result of a scanning with the charged scalar masses $m_{k^{\pm \pm }}=500\mathrm{GeV}$, $m_{h_1^{\pm }}=m_{h_2^{\pm }}=4.8\mathrm{TeV}$. The total number of the consistent data points is 2804.

Figure 3

Correlations between $\varphi$ and $\delta$, $(y_L{)}_{23}$ and $|(y_R{)}_{22}|$, and $\mu$ and $|(y_R{)}_{22}|$ are shown. The data set is the same as in Fig. 2.

Figure 4

The $1\sigma$ (blue) and $2\sigma$ (red) allowed regions of the global analysis based on the LHC data summarized in Table 6. From top left to bottom, we choose the parameters $({\lambda }_{\mathrm{\Phi }k},{\lambda }_{\mathrm{\Sigma }k}{v}^{\prime })=(1.0,v),(\pi ,v),(0.1,v)$, respectively.

Figure 5

The $1\sigma$ (blue) and $2\sigma$ (red) allowed regions of the global analysis based on the LHC data summarized in Table 6. From left to right, we choose the parameters $({\lambda }_{\mathrm{\Phi }k},{\lambda }_{\mathrm{\Sigma }k}{v}^{\prime })=(1.0,5v),(1.0,10v)$, respectively.

Figure 6

Realized present-day relic densities of the dark matter candidate $\chi$ are shown as functions of $m_{\chi }$. The other effective parameters are fixed as $\sin \alpha =0.4$, $v^{\prime }=800\mathrm{GeV}$, $m_H=250\mathrm{GeV}$, $m_{k^{\pm \pm }}=500\mathrm{GeV}$ (left panel) and $\sin \alpha =0.5$, $v^{\prime }=5\mathrm{TeV}$, $m_H=250\mathrm{GeV}$, $m_{k^{\pm \pm }}=500\mathrm{GeV}$ (right panel) for demonstration purposes, respectively. The area inside the two horizontal dashed lines (where the splitting is almost invisible) suggests the $2\sigma$ consistent region with the Planck data [$0.1196\pm 0.0031$ (68% C.L.) 120]. In the current setup, the relic density sharply drops around the two resonant regions around $m_h/2\simeq 62.5\mathrm{GeV}$ and $m_H/2\simeq 125\mathrm{GeV}$, where in the latter case, shown in the right panel, dropping is not sufficient for obtaining a proper amount of dark matter even around $m_H/2$ at present. We neglect the three- and four-body final states via virtual $W$ and $Z$ boson decays, which gives sizable modifications near the thresholds for producing gauge boson pairs [130, 131], since our interest is only around $m_h/2$ and $m_H/2$ where this correction is expected to be insignificant.

Figure 7

The matrix plots indicate suitable choices of the two parameters $v^{\prime }$ and $\sin \alpha$ when we consider the situation of $m_H=250\mathrm{GeV}$, $m_{k^{\pm \pm }}=250\mathrm{GeV}$. We obtain the observed relic density in the red (blue) region in the left (right) panel when $m_{\chi }$ is around $m_h/2\simeq 62.5\mathrm{GeV}$ ($m_H/2\simeq 125\mathrm{GeV}$), respectively. The green region is excluded via excess of the invisible decay channel of the observed Higgs boson. No excluded region is found by the direct detection in the shown parameter range.

Figure 8

The matrix plots for showing the region where a proper amount of the relic density is realized when $m_H=250\mathrm{GeV}$, $m_{k^{\pm \pm }}=500\mathrm{GeV}$. Conventions are the same with ones in Fig. 7. No excluded region is found by the direct detection in the shown parameter range.

Figure 9

Similar plots as Figs. 7 and 8 when $m_H=500\mathrm{GeV}$, $m_{k^{\pm \pm }}=250\mathrm{GeV}$ (left) and $m_H=500\mathrm{GeV}$, $m_{k^{\pm \pm }}=500\mathrm{GeV}$ (right), respectively. Here, no solution is found around $m_{\chi }\simeq m_H/2\simeq 250\mathrm{GeV}$, while one around $m_{\chi }\simeq m_h/2\simeq 62.5\mathrm{GeV}$ is still there. The green region is excluded by the invisible decay of the 125 GeV Higgs. No excluded region is found by the direct detection in the shown parameter range.