Constraining scalars of $16_H$ through proton decays in non-renormalisable $SO(10)$ models

Non-renormalisable versions of $SO(10)$\, based on irreducible representations with lesser degrees of freedom, are free of running into the catastrophe of non-perturbativity of standard model gauge couplings in contrast to the renormalisable versions having tensors with many degrees of freedom. $16_H$ is the smallest representation, participates in Yukawa Lagrangian at the non-renormalisable level, contributing to the charged and neutral fermion masses, and has six distinct scalars with different $B-L$ charges. We computed the leptoquark and diquark couplings of different pairs of scalars stemming from all possible decomposition of the term resulting from the coupling of $16_{H}$ with the ${\mathbf{16}}$ dimensional fermion multiplet of $SO(10)$,\, i.e. $\frac{{\mathbf{16}}\,{\mathbf{16}}\,16_{H}\,16_{H}}{\Lambda}$. Computing the tree and loop level contribution of different pairs to the effective dimension six, $B-L$ conserving operators, it turns out only three pairs, viz $\sigma\big(1,1,0\big)- T\big(3,1,\frac{1}{3}\big)$, and $H\big(1,2,-\frac{1}{2}\big)-\Delta\big(3,2,\frac{1}{6}\big)$, and $H-T$ can induce proton decay at tree level. Assuming that the Yukawa couplings of the $16_{H}$ are comparable to those of the $\overline{126}_{H}$ of a realistic $SO(10)$ model and setting the cutoff scale to the Planck scale typically constrains the $B-L$ breaking scale to be $4\sim 5$ orders of magnitude less than the cutoff scale $(\Lambda)$. Moreover, analysing the branching pattern of the leading two-body decay modes of the proton, we observed a preference for the proton to decay into second-generation mesons due to the hierarchical nature of Yukawa couplings. In a realistic $SO(10)$\, scenario, we find that $M_T >10^{8}$ TeV, while $M_\Delta$ could be as light as a few TeV$s$.