Semi-leptonic three-body proton decay modes from light-cone sum rules

Using light-cone sum rule techniques, we estimate the form factors which parametrise the hadronic matrix elements that are relevant for semi-leptonic three-body proton decays. The obtained form factors allow us to determine the differential rate for the decay of a proton ($p$) into a positron ($e^+$), a neutral pion ($\pi^0$) and a graviton ($G$), which is the leading proton decay channel in the effective theory of gravitons and Standard Model particles (GRSMEFT). The sensitivity of existing and next-generation neutrino experiments in detecting the $p \to e^+ \pi^0 G$ signature is studied and the phenomenological implications of our computations for constraints on the effective mass scale that suppresses the relevant baryon-number violating GRSMEFT operator are discussed.