Coupling unification and dark matter in a standard model extension with adjoint Majorana fermions

We revisit an extension of the Standard Model with Majorana fermions in adjoint representations, where $SU(2{)}_L$ triplet and $SU(3{)}_c$ octet fermions are introduced. The model provides not only a precise coupling unification but also a dark matter candidate [an $SU(2{)}_L$ triplet fermion]. In particular, we show that an $SU(3{)}_c$ octet fermion, which is required for successful unification, can be a good nonthermal source of the triplet fermion dark matter. We also show that the scenario predicts a rather short lifetime of the proton compared with the minimal supersymmetric grand unified theory, and the most parameter space can be explored by future experiments such as the Hyper-Kamiokande experiment.

Figure 1

(Left panel) An example of the one-loop renormalization group evolutions of the gauge coupling constants in the extended model (solid lines). For a comparison, we show the evolutions in the SM as the dashed lines. The masses of the wino-like and gluino-like fermions are taken to be $M_{\tilde{w}}=3\mathrm{TeV}$ and $M_{\tilde{g}}=10^{11}\mathrm{GeV}$, respectively. In the figures, we use ${\alpha }_3(m_Z{)}^{\overline{\text{MS}}}=0.1185(6)$, $N_H=1$. We have also taken $m_h=126\mathrm{GeV}$ and $m_{\text{top}}=173.2\mathrm{GeV}$, although the results do not depend on these parameters significantly. (Right panel) The adjoint fermion masses which are preferred for a precise unification. In the (light-)blue shaded region, the three gauge coupling constants unify rather precisely, i.e., $|N_{\text{th}}|<5(10)$, respectively. The horizontal lines show the contours of the $g_1–g_2$ unification scale.

Figure 2

(Left panel) The decay temperature of the gluino-like fermion for a given $M_{*}$ [Eq. (11)] (red lines) and the dominate temperature [Eq. (15)] (green line). For comparison, we also show a typical freeze-out temperature of the wino-like fermion, $T_F^{(\tilde{w})}\simeq M_{\tilde{w}}/25$, as a blue shaded region assuming $M_{\tilde{w}}\simeq 10^{-(6–8)}{M}_{\tilde{g}}$ [Eq. (4)]. (Right panel) The yield of the gluino after freeze-out (solid line). For comparison, we show the yield without the Sommerfeld enhancement factor as a dashed line.

Figure 3

The evolution of the energy densities of the radiation and the gluino-like fermion. Here, the energy density of the original radiation (red line) scales as $a^{-4}$, while that of the gluino-like fermion scales as $a^{-3}$ (blue line). The radiation energy released by the occasional decay of the gluino-like fermion scales as $a^{-3/2}$ (green line).

Figure 4

(Left panel) The ratio between $T_D$ and the effective temperature of the nonthermal production $T_{\text{eff}}$ for given $T_D$. The solid lines show the ratios for $P_{\tilde{w}}$ given in Eq. (21). For comparison, the ratios values of $P_{\tilde{w}}$ that are 10 times smaller (larger) than the one given in Eq. (21) are shown as the lower (upper) dotted lines. (Right panel) The evolution of the energy density for $M_{\tilde{w}}=1\mathrm{TeV}$, $M_{\tilde{g}}=10^{10}\mathrm{GeV}$, and $M_{*}=10^{15}\mathrm{GeV}$ obtained by solving the Boltzmann equations numerically.

Figure 5

(Left panel) Contour plots of ${\mathrm{\Omega }}_{\tilde{w}}{h}^2$ on the $T_{\text{eff}}\text{−}{M}_{\tilde{w}}$ plane. (Right panel) Contour plots of $M_{*}$, which provides an appropriate decay temperature of the gluino-like fermion for ${\mathrm{\Omega }}_{\tilde{w}}{h}^2\simeq 0.1146$ via the nonthermal production. Here, we use superscripts, ${\mathrm{\Omega }}_{\tilde{w}}^{\mathrm{TH}}{h}^2\simeq 0.1$ (thermal) and ${\mathrm{\Omega }}_{\tilde{w}}^{\mathrm{NT}}{h}^2\simeq 0.1$ (nonthermal), to indicate the main contributions to the dark matter abundance. The precise unification conditions in Fig. 1 are also shown.

Figure 6

(Left panel) The mass range of the GUT gauge boson as a function of the mass of the wino-like fermion. We have assumed $N_{\text{th}}^{(\max )}=5$ for the blue (dark) shaded region and $N_{\text{th}}^{(\max )}=10$ for the light-blue (light) shaded region in Eq. (40). (Right panel) The proton lifetime via $p\to {\pi }^0+e^{+}$ assuming the GUT gauge boson mass in Eq. (40). Here, we have fixed the form factor to be $W_0=-0.06{\mathrm{GeV}}^2$ so that the exclusion limit is conservative. The horizontally shaded region is excluded by the current lower limit ${\tau }_p\gtrsim 8.2\times 10^{33}\mathrm{yr}$. The vertically shaded region is excluded by the lower limit, $M_{\tilde{w}}\gtrsim 270\mathrm{GeV}$.