Anatomy of scalar mediated proton decays in $SO(10)$ models

Realistic models based on the renormalizable grand unified theories have varieties of scalars, many of which are capable of mediating baryon ($B$) and lepton ($L$) number non-conserving processes. We identify all such scalar fields residing in ${\bf 10}$, $\overline{\bf 126}$ and ${\bf 120}$ dimensional irreps of $SO(10)$ which can induce baryon and lepton number violating interactions through the leading order $d=6$ and $d=7$ operators. Explicitly computing their couplings with the standard model fermions, we derive the effective operators including the possibility of mixing between the scalars stemming from a given representation. We find that such interactions at $d=6$ are mediated by only three sets of scalars: $T(3,1,-1/3)$, ${\cal T} (3,1,-4/3)$ and $\mathbb{T}(3,3,-1/3)$ and their conjugates. In the models with ${\bf 10}$ and $\overline{\bf 126}$, only the first has appropriate couplings to mediate the proton decay. While ${\cal T}$ and $\mathbb{T}$ can induce baryon number violating interactions when ${\bf 120}$ is present, ${\cal T}$ does not contribute to the proton decay at tree level because of its flavour antisymmetric coupling. Three additional colour triplets and their conjugates can mediate nucleon decay via $d=7$ operators which violate also the $B-L$. We give general expressions for partial widths of proton in terms of the fundamental Yukawa couplings and use these results to explicitly compute the proton lifetime and branching ratios for the minimal non-supersymmetric $SO(10)$ model based on ${\bf 10}$ and $\overline{\bf 126}$ Higgs. We find that the proton preferably decays into $\overline{\nu}\, K^+$ or $\mu^+\, K^0$ and list several distinct features of scalar mediated proton decay. If the latter dominates over the gauge mediated contributions, the proton decay spectrum provides a direct probe to the flavour structure of the underlying grand unified theory.