Witten’s mechanism in the flipped $SU(5)$ unification

We argue that Witten’s loop mechanism for the right-handed Majorana neutrino mass generation identified originally in the $SO(10)$ grand unification context can be successfully adopted to the class of the simplest flipped $SU(5)$ models. In such a framework, the main drawback of the $SO(10)$ prototype—in particular, the generic tension among the gauge unification constraints and the absolute neutrino mass scale—is alleviated, and a simple yet potentially realistic and testable scenario emerges.

Figure 1

The gauge structure of Witten’s loop in the flipped $SU(5)$ scenario under consideration. Note that we display just one representative out of several graphs that may be obtained from the one above by permutations.

Figure 2

The shape of the allowed parameter space (${\omega }_{23}$ and ${\omega }_{13}$ governing $U_{\nu }$ on the horizontal axes and the minus log of the lightest neutrino mass $m_1$ on the vertical; note that $m_1$ decreases from bottom to top) in the $CP$ conserving setting discussed in Sec. 3a for $K=10$ in the upper and $K=30$ in the lower panel, respectively. The allowed points are all those in the interior of the depicted structure. The straight cut in the lower part corresponds to the current cosmology limit on the lightest neutrino mass $m_1\lesssim 8\times {10}^{-2}\mathrm{eV}$ [53], see the discussion in the text.

Figure 3

The same as in Fig. 2 for the $CP$ violating setting with the “Dirac” phase in $U_{\nu }$ set to $\sigma =\pi /2$ and $K=20$. The net effect of a nonzero $\sigma$ is that ${\omega }_{23}$ remains unconstrained unless $m_1$ is really tiny [for which case the dominance of the 33 element in the RH neutrino mass formula (28) is assumed]. The effects of the “outer” phases of $U_{\nu }$ in the observables discussed in Sec. 3a are small so we conveniently fixed all of them to zero.

Figure 4

Contour plots of the ${\omega }_{12}$-extremes (cf. Sec. 3b) of the partial widths $\mathrm{\Gamma }(p\to {\pi }^0{\mu }^{+})$ (upper panels, decreasing with darkening color) and $\mathrm{\Gamma }(p\to {\pi }^0{e}^{+})+\mathrm{\Gamma }(p\to {\pi }^0{\mu }^{+})$ (lower panels) superimposed with the (dashed) boundaries of the regions allowed by Eq. (32) evaluated for $m_1=0.8\times {10}^{-2}\mathrm{eV}$ (left), and $m_1=0.8\times {10}^{-3}\mathrm{eV}$ (right), respectively. In all the plots the innermost and outermost dashed contours correspond to $K=7$ and $K=30$, respectively.

Figure 5

The global upper limits on $X_{\mu }$ (upper plot) and the global lower limits on $X_{e+\mu }$ (lower plot), cf. Eqs. (36) and (37), as functions of the lightest neutrino mass (in the normal hierarchy case). The lowermost line on the upper plot and the uppermost line on the lower plot correspond to $K=7$, with every consecutive contour for $K$ increased by 2. The dots represent an independent numerical calculation of the same decay rates for $K=7$ with randomly chosen real $U_{\nu }$’s; only those points satisfying (30) are permitted. The hatched area corresponds to $m_1>0.08\mathrm{eV}$ which is disfavored by cosmology [53].

Figure 6

The same as in Fig. 5 but for a complex $U_{\nu }$ and $K=8$. The “outer” phases ${\rho }_1$, ${\rho }_2$, and ${\rho }_3$ (cf. Eq. (34)) are varied randomly while the “Dirac” phase $\sigma$ of $U_{\nu }$ was fixed to zero. It is clear that the effect of ${\rho }_i$’s is very mild as the desired features in the partial widths remain essentially intact.

Figure 7

The same as in Fig. 6 but this time for entirely random phases in $U_{\nu }$ including $\sigma$. The effects in the partial widths are smeared until $m_1\lesssim {10}^{-6}\mathrm{eV}$ because, for larger $m_1$, the important constraints on ${\omega }_{23}$ from perturbativity and SM vacuum stability are lost, see Fig. 3.

Figure 8

The unification constraints on the mass of the $(X^{\prime },Y^{\prime })$ gauge bosons (the left ordinate) and ${\mathrm{\Gamma }}^{-1}(p^{+}\to {\pi }^{+}\overline{\nu })$ (the right ordinate) drawn for constant $x=\mu /\lambda V_G$ as functions of the masses of the scalar colored triplets ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, cf. (21) in the simplified case of ${\lambda }_2={\lambda }_5$. The upper part of the plot corresponds to the “fine-tuned” region with $x$ very close to 1 with the mass of ${\mathrm{\Delta }}_1$ significantly lower than $M_G$, cf. Eq. (26), while the lower part corresponds to $x<1$. The bands (one loop in dashed and two loops in solid) correspond to the 3-$\sigma$ uncertainty in ${\alpha }_s$ and their boundary on the right depicts the “perturbativity” limit $|{\lambda }_i|<4\pi$, cf. Sec. 2c4.