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

Non-renormalisable versions of $SO\left(10\right)$ 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 16 dimensional fermion multiplet of $SO\left(10\right)$, i.e. $\frac{1616{16}_H{16}_H}{\Lambda }$. 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 $\sigma (1,1,0)-T(3,1,\frac{1}{3})$, and $H(1,2,-\frac{1}{2})-\Delta (3,2,\frac{1}{6})$, 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 $SO\left(10\right)$ 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 (Λ). 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 $SO\left(10\right)$ scenario, we find that $M_T>{10}^8$ TeV, while $M_{\Delta }$ could be as light as a few TeVs.