Proton lifetime in the minimal $SO(10)$ GUT and its implications for the LHC

We review the current status of the minimal nonsupersymmetric $SO(10)$ grand unified theory and perform a detailed next-to-leading-order analysis of the gauge unification and proton lifetime constraints on the part of its parameter space supporting a ZeV-scale color sextet scalar. This, together with a TeV-scale color octet studied in detail in a preceding work, represents one of the two minimally fine-tuned settings compatible with all the relevant consistency and phenomenology limits. Both these scenarios can be extensively tested at the future megaton-scale proton-decay facilities. On top of that, the light octet solution can be accessible in the TeV-scale collider searches.

Figure 1

Masses of the $(8,2,+\frac{1}{2})$ scalar field allowed by the NLO unification and matter stability constraints [the points consistent with the limits (12), (13) and (14) are plotted in light gray, dark gray and black color, respectively]; cf. [30]. In the left panel $|\sigma |\ge {10}^{12}\mathrm{GeV}$ is assumed while, in the right panel, we admit for $|\sigma |\ge {10}^{13}\mathrm{GeV}$. Remarkably, focusing on the black area, either the octet mass is below 20 TeV and, thus, potentially, within the reach of the LHC or its near future successors or proton lifetime should be seen at the HK before 2045. For more discussion see [30].

Figure 2

The points consistent with two-loop unification constraints and the limits (12), (13) and (14) on the proton lifetime plotted in light gray, dark gray and black color, respectively. The light-gray band in the background encloses the settings identified in the previous one-loop analysis [31] that are consistent with the current SK proton lifetime limit (12).

Figure 3

The values of $|{\omega }_R|$ and $|\sigma |$ for the points fulfilling the unification and proton lifetime constraints at the NLO level (color code as in Fig. 2). The dashed line corresponds to $|{\omega }_R|=|\sigma |$; above and below this line different symmetry breaking chains have been implemented; cf. Eqs. (17), (20).

Figure 4

$|\sigma |$ as a function of $M(6,3,+\frac{1}{3})$ for the NLO solutions consistent with all the requirements of Sec. 3 (color code as in Fig. 2). The sharp boundary in the NW direction reflects the unification constraints in the case of $|\sigma |>|{\omega }_R|$; on the other hand, the settings with $|{\omega }_R|>|\sigma |$ stretch far from this edge since the value of $|\sigma |$ does not affect the unification pattern in such cases. The solid lines correspond to $|\sigma |={10}^{12}\mathrm{GeV}$ and $|\sigma |={10}^{13}\mathrm{GeV}$ levels, respectively. Recall that $\sigma$ governs the seesaw scale as well as the amount of fine-tuning necessary to obtain a realistic light neutrino mass spectrum and mixing.

Figure 5

The same as in Fig. 2 with an extra assumption of $\sigma >{10}^{12}\mathrm{GeV}$ (on the left) and $\sigma >{10}^{13}\mathrm{GeV}$ (on the right). Notice that in the latter case the entire allowed parameter space may be probed by Hyper-K by 2030; see Eq. (13).

Figure 6

The gauge unification pattern for the sample Point 1 in Table 1; the RG evolution passes through the $SO(10)$, $3_c{2}_L{2}_R{1}_X$ and SM stages, respectively.

Figure 7

The same as in Fig. 6 but for Point 2 in Table 1.