Dark matter and gauge coupling unification in nonsupersymmetric SO(10) grand unified models

Unlike minimal SU(5), SO(10) provides a straightforward path towards gauge coupling unification by modifying the renormalization group evolution of the gauge couplings above some intermediate scale which may also be related to the seesaw mechanism for neutrino masses. Unification can be achieved for several different choices of the intermediate gauge group below the SO(10) breaking scale. In this work, we consider in detail the possibility that SO(10) unification may also provide a natural dark matter candidate, stability being guaranteed by a leftover ${\mathbb{Z}}_2$ symmetry. We systematically examine the possible intermediate gauge groups which allow a nondegenerate, fermionic, Standard Model singlet dark matter candidate while at the same time respecting gauge coupling unification. Our analysis is done at the two-loop level. Surprisingly, despite the richness of SO(10), we find that only two models survive the analysis of phenomenological constraints, which include suitable neutrino masses, proton decay, and reheating.

Figure 1

Diagram responsible for the decay of ${\psi }^{+}$ into the DM ${\psi }^0$. $f$ and $f^{\prime }$ denote the SM particles.

Figure 2

${\chi }^2$ as functions of ${\log }_{10}(M_{\mathrm{int}})$ (top), ${\log }_{10}(M_{\mathrm{GUT}})$ (middle), and $g_{\mathrm{GUT}}$ (bottom). Left and right panels are for $G_{\mathrm{int}}=\mathrm{SU}(4{)}_C\otimes \mathrm{SU}(2{)}_L\otimes \mathrm{SU}(2{)}_R$ and $G_{\mathrm{int}}=\mathrm{SU}(4{)}_C\otimes \mathrm{SU}(2{)}_L\otimes \mathrm{SU}(2{)}_R\otimes D$, respectively. $M_{\mathrm{int}}$ and $M_{\mathrm{GUT}}$ are given in GeV.

Figure 3

Contour plots for the allowed region in the $g_{\mathrm{GUT}}\text{−}{\log }_{10}(M_{\mathrm{int}})$, $g_{\mathrm{GUT}}\text{−}{\log }_{10}(M_{\mathrm{GUT}})$, and ${\log }_{10}(M_{\mathrm{GUT}})\text{−}{\log }_{10}(M_{\mathrm{int}})$ parameter planes in the top, middle, and bottom panels, respectively. Left panels are for $G_{\mathrm{int}}=\mathrm{SU}(4{)}_C\otimes \mathrm{SU}(2{)}_L\otimes \mathrm{SU}(2{)}_R$, while right ones are for $G_{\mathrm{int}}=\mathrm{SU}(4{)}_C\otimes \mathrm{SU}(2{)}_L\otimes \mathrm{SU}(2{)}_R\otimes D$. Stars represent the best-fit point. The colored regions correspond to 68%, 95%, and 99% C.L. limits determined from $\mathrm{\Delta }{\chi }^2\simeq 2.30,5.99,9.21$.

Figure 4

Running of gauge couplings. Solid (dashed) lines show the case with (without) DM and additional Higgs bosons. Blue, green, and red lines represent the running of the U(1), SU(2) and SU(3) gauge couplings, respectively.

Figure 5

Proton lifetimes as functions of $M_X/M_{\mathrm{GUT}}$. Blue solid and red solid lines represent model I and model II, respectively. Blue dashed and red dashed lines represent the cases for $G_{\mathrm{int}}=\mathrm{SU}(4{)}_C\otimes \mathrm{SU}(2{)}_L\otimes \mathrm{SU}(2{)}_R$ and $G_{\mathrm{int}}=\mathrm{SU}(4{)}_C\otimes \mathrm{SU}(2{)}_L\otimes \mathrm{SU}(2{)}_R\otimes D$ when the DM and extra Higgs multiplets are not included. The shaded area shows the region which is excluded by the current experimental bound, $\tau (p\to e^{+}{\pi }^0)>1.4\times 10^{34}\text{years}$ [40, 41].

Figure 6

Diagram responsible for the DM production in model II.

Figure 7

Examples of diagrams responsible for the DM production in model I.

Figure 8

Reheating temperature as a function of DM mass. Pink band shows the theoretical uncertainty (a)  Model I and (b) Model II.