Vanishing Higgs potential at the Planck scale in a singlet extension of the standard model

We discuss the realization of a vanishing effective Higgs potential at the Planck scale, which is required by the multiple-point criticality principle (MPCP), in the standard model with singlet scalar dark matter and a right-handed neutrino. We find the scalar dark matter and the right-handed neutrino play crucial roles for realization of the MPCP, where a neutrino Yukawa becomes effective above the Majorana mass of the right-handed neutrino. Once the top mass is fixed, the MPCP at the (reduced) Planck scale and the suitable dark matter relic abundance determine the dark matter mass, $m_S$, and the Majorana mass of the right-handed neutrino, $M_R$, as $8.5(8.0)\times {10}^2\mathrm{GeV}\le m_S\le 1.4(1.2)\times {10}^3\mathrm{GeV}$ and $6.3(5.5)\times {10}^{13}\mathrm{GeV}\le M_R\le 1.6(1.2)\times {10}^{14}\mathrm{GeV}$ within current experimental values of the Higgs and top masses. This scenario is consistent with current dark matter direct search experiments and will be checked by future experiments such as LUX with further exposure and/or the XENON1T.

Figure 1

Numerical results for the realization of the MPCP [$\lambda =0$ (blue/dark gray curve) and ${\beta }_{\lambda }=0$ (orange/light gray curve)]. The contours for the two conditions at the Planck $M_{\text{pl}}$ and the reduced Planck ${\tilde{M}}_{\text{pl}}$ scales are shown by the dashed and solid curves in all figures, respectively. Figures (a), (b), (c), and (d) correspond to the cases of $(M_t[\text{GeV}],{\lambda }_S(M_Z))=(172.6,{10}^{-3})$, $(174.1,{10}^{-3})$, (172.6, 0.5), and (174.1, 0.5), respectively.

Figure 2

The positions of the intersection points in the $[M_R,M_t(\text{or}{m}_S)]$ plane for the ${\lambda }_S(M_Z)={10}^{-3}$ case. The solid and dashed curves indicate the MPCP solutions at ${\tilde{M}}_{\text{pl}}$ and $M_{\text{pl}}$, respectively. The values in the parentheses on the $m_S$ axis correspond to values in the case of the MPCP at the Planck scale, while the $m_S$ values without parentheses are for the case of MPCP at the reduced Planck scale.

Figure 3

The current experimental bounds on $m_S$ and $k$ [XENON100, 225 live-days (blue/darkest gray solid line) and LUX, 85.3 live-days (orange/lightest gray solid line)], and the future detectability by the LUX 300-day projection (orange/lightest gray dashed line) and the XENON1T (blue/darkest gray dashed line) [36, 37]. The black solid curve indicates the contour of ${\mathrm{\Omega }}_S{h}^2={\mathrm{\Omega }}_{\text{DM}}{h}^2=0.12$. (The green/second-darkest gray and yellow/second-lightest gray dashed curves correspond to ${\mathrm{\Omega }}_S{h}^2=0.5{\mathrm{\Omega }}_{\text{DM}}{h}^2$ and $0.2{\mathrm{\Omega }}_{\text{DM}}{h}^2$, respectively.)

Figure 4

Numerical results for the realization of the MPCP [$\lambda =0$ (blue/dark gray curve) and ${\beta }_{\lambda }=0$ (orange/light gray curve)] in the cases of ${\mathrm{\Omega }}_S/{\mathrm{\Omega }}_{\text{DM}}=0.5$ [(a)–(d)] and ${\mathrm{\Omega }}_S/{\mathrm{\Omega }}_{\text{DM}}=0.2$ [(e)–(h)]. The meanings of the figures are the same as in Fig. 1.