Flavor structure of supersymmetric $SO(10)$ GUTs with extended matter sector

We discuss in detail the flavor structure of the supersymmetric $SO(10)$ grand unified models with the three traditional 16-dimensional matter spinors mixed with a set of extra ten-dimensional vector multiplets which can provide the desired sensitivity of the standard model matter spectrum to the grand unified theory symmetry breakdown at the renormalizable level. We put the qualitative argument that a successful fit of the quark and lepton data requires an active participation of more than a single vector matter multiplet on a firm, quantitative ground. We find that the strict no-go obtained for the fits of the charged-sector observables in case of a single active matter 10 is relaxed if a second vector multiplet is added to the matter sector and excellent, though nontrivial, fits can be devised. Exploiting the unique calculable part of the neutrino mass matrix governed by the $SU(2{)}_L$ triplet in the 54-dimensional Higgs multiplet, a pair of genuine predictions of the current setting is identified: a nonzero value of the leptonic 1–3 mixing close to the current 90% C.L. limit and a small leptonic Dirac $CP$ phase are strongly preferred by all solutions with the global-fit ${\chi }^2$ values below 50.

Figure 1

An $n=2$ histogram of the relative frequency of fits obtained for the quark and charged-lepton masses and CKM mixing parameter with ${\chi }^2<1$ for different values of the decoupling parameter $P$. The sharp decline of the counts towards the high $P$ limit is a manifestation of the no-go discussed in Sec. 3b2 for the $n=1$ case. The shaded region on the left corresponds to the fine-tuned setting with $V^{54}$ dominating over the singlet mass parameter $M_{10}$ and its specific shape is an artefact of the numerical method we use.

Figure 2

A sample of the ${\chi }^2$ values for the $n=2$ fits with $\eta \to 0$ as a function of the decoupling parameter $P$. One can see clearly that the extra constraints from the leptonic sector make the $n=2$ fits nontrivial, see also Sec. 3c2 and Table .

Figure 3

A sample of the ${\chi }^2$ values for the full $n=2$ fits as a function of the decoupling parameter $P$. One can see clearly that the extra constraints from the leptonic sector make the $n=2$ fits troublesome, see also Sec. 3c2.

Figure 4

The predicted value of the leptonic 13 mixing as a function of the ${\chi }^2$ corresponding to the fits of all the other measured parameters. The current 90% C.L. upper limit $\sin {\theta }_{13}^l\le 0.18$ is indicated by the dashed line. The distribution of the calculated ${\theta }_{13}^l$ values for the lowest-${\chi }^2$ points clusters in the lower part of the available domain at around $\sin {\theta }_{13}^l\sim 0.2$.

Figure 5

The predicted value of the leptonic Dirac $CP$ phase ${\delta }_{CP}^l$ (in units of $\pi$) as a function of the ${\chi }^2$ corresponding to the fits of all the other measured parameters. The distribution of the calculated ${\delta }_{CP}^l$ values for the lowest-${\chi }^2$ points clusters at around zero, thus indicating a possible difficulty in revealing the leptonic $CP$ violation in the next generation of neutrino oscillation experiments.