Seesaw scale in the minimal renormalizable $SO(10)$ grand unification

Simple $SO(10)$ Higgs models with the adjoint representation triggering the grand unified symmetry breaking, discarded long ago due to inherent tree-level tachyonic instabilities in the physically interesting scenarios, have been recently brought back to life by quantum effects. In this work we focus on the variant with ${45}_H\oplus {126}_H$ in the Higgs sector and show that there are several regions in the parameter space of this model that can support stable unifying configurations with the $B-L$-breaking scale as high as ${10}^{14}\mathrm{GeV}$, well above the previous generic estimates based on the minimal survival hypothesis. This admits for a renormalizable implementation of the canonical seesaw and makes the simplest potentially realistic scenario of this kind a good candidate for a minimal $SO(10)$ grand unification. Last, but not least, this setting is likely to be extensively testable at future large-volume facilities such as Hyper-Kamiokande.

Figure 1

$M(6,3,+\frac{1}{3})-{\omega }_{BL}$ correlation in the case of a light $(6,3,+\frac{1}{3})$ multiplet in the desert. Various levels of gray correspond to domains accessible for different GUT-scale limits, cf. Sec. 3a2. $M(6,3,+\frac{1}{3})$ can vary only over a couple of orders of magnitude (for the current SK limit) and the range is likely to shrink considerably in future.

Figure 2

$|{\omega }_R|-{\omega }_{BL}$ correlation in the case of a light $(6,3,+\frac{1}{3})$ multiplet in the desert. The color code is the same as before, cf. Sec. 3a2. In all of the allowed region $|{\omega }_R|\ll {\omega }_{BL}$ so this setting prefers an intermediate $3_c{2}_L{2}_R{1}_{BL}$ stage.

Figure 3

$|{\omega }_R|-|\sigma |$ correlation in the case of a light $(6,3,+\frac{1}{3})$ multiplet in the desert. The color code is the same as before, cf. Sec. 3a2. $B-L$ as high as almost ${10}^{15}\mathrm{GeV}$ can be reached for the current SK limit, with the best Hyper-Kamiokande sensitivity limit the maximum lowers to $\mathrm{\text{few}}\times {10}^{14}\mathrm{GeV}$.

Figure 4

Correlation between the mass of the $(6,3,+\frac{1}{3})$ threshold and the allowed $B-L$ scale $\sigma$. There are two basic stability issues that bias the estimate of the span of the allowed domains: first, there is the technical requirement we impose on the hierarchy between the lightest and next-to-lightest thresholds, i.e., $(6,3,+\frac{1}{3})$ and the gauge sector associated with the $3_c{2}_L{2}_R{1}_{BL}$ breaking, which cuts the parametric space from below right; for large $\sigma$’s this, however, becomes irrelevant because some of the couplings (namely, ${\beta }_4$ and ${\beta }_4^{\prime }$) become nonperturbative yet before such a numerical instability can affect the relevant upper bound.

Figure 5

Unification of the effective SM gauge couplings in a sample setting with a light $(6,3,+\frac{1}{3})$ multiplet (here at around $5.6\times {10}^{11}\mathrm{GeV}$, cf. Sec. 3c1c) with the shaded area magnified on the lower panel. The short $3_c{2}_L{2}_R{1}_{BL}$ stage is clearly visible here. The small circles indicate the positions of various thresholds (for details, see Table ) inflicting changes in the three curves’ slopes. The almost vertical solid and dashed lines correspond to the current and future proton-decay limits, cf. Sec. 3a2. The displayed setting is compatible (at one loop) with the current SK limit, but it can be refuted by the Hyper-Kamiokande. The dotted vertical line indicates the position of the $B-L$ scale.

Figure 6

$M(8,2,+\frac{1}{2})-|{\omega }_R|$ correlation in the case of a light $(8,2,+\frac{1}{2})$ multiplet in the desert. The color code is the same as before, cf. Sec. 3a2. $M(8,2,+\frac{1}{2})$ can vary over many orders of magnitude in the lower part of the desert, and it is pushed down for increasing proton lifetime.

Figure 7

$|{\omega }_{BL}|-|{\omega }_R|$ correlation in the case of a light $(8,2,+\frac{1}{2})$ multiplet in the desert. Various levels of gray correspond to domains accessible for different GUT-scale limits, cf. Sec. 3a2. In the whole allowed region $|{\omega }_{BL}|\ll {\omega }_R$ so this setting always exhibits an intermediate $4_C{2}_L{1}_R$ stage.

Figure 8

An interesting $|{\omega }_{BL}|-|\sigma |$ correlation in the case of a light $(8,2,+\frac{1}{2})$ multiplet in the desert. The color code is the same as before, cf. Sec. 3a2. $B-L$ as high as ${10}^{14}\mathrm{GeV}$ can be reached and, remarkably enough, unlike in the $(6,3,+\frac{1}{3})$ case the maximum reach is insensitive to the proton-lifetime limit.

Figure 9

The same as in Fig. 5, here with a light $(8,2,+\frac{1}{2})$ multiplet (at around $2.3\times {10}^4\mathrm{GeV}$, cf. Sec. 3c2c) instead of $(6,3,+\frac{1}{3})$; for details of the spectrum see Table . The short $4_C{2}_L{1}_R$ stage as well as the “fake unification” feature is clearly visible here. The displayed setting is compatible (at one loop) with the current SK as well as possible future Hyper-Kamiokande limits.