Gauge and scalar boson mediated proton decay in a predictive SU(5) GUT model

We assess proton decay signatures in the simplest viable $SU\left(5\right)$ model with regard to constraints on parameters governing the Standard Model fermion mass spectrum. Experimental signals for all eight two-body proton decay processes result from exchange of two gauge bosons, a single scalar leptoquark, or their combination. Consequently, it enables us to delve into an in-depth anatomy of proton decay modes and anticipate future signatures. Our findings dictate that observing a proton decay into $p\to {\pi }^0{e}^{+}$ indicates gauge boson mediation, with the potential for observation of $p\to {\eta }^0{e}^{+}$ mode. Alternatively, if decay is through the $p\to K^{+}\overline{\nu }$ process, it is mediated by a scalar leptoquark, possibly allowing the observation of $p\to {\pi }^0{\mu }^{+}$. Detection of both $p\to {\pi }^0{e}^{+}$ and $p\to K^{+}\overline{\nu }$ could enhance $p\to {\pi }^0{\mu }^{+}$ through constructive interference. The model predicts inaccessibility of $p\to {\pi }^{+}\overline{\nu }$, $p\to {\eta }^0{\mu }^{+}$, $p\to K^0{e}^{+}$, and $p\to K^0{\mu }^{+}$, regardless of the dominant mediation type, in the coming decades. In summary, through a comprehensive analysis of proton decay signals, gauge coupling unification, and fermion masses and mixing, we both accurately and precisely constrain the parameter space of the $SU\left(5\right)$ model in question.