Coloured Scalar Mediated Nucleon Decays to Invisible Fermion

We investigate nucleon decays to light invisible fermion mediated by the coloured scalar $\bar S_1= (\bar 3, 1, -2/3)$ and compare them with the results coming from the mediation of $S_1 = (\bar 3,1,1/3)$. In the case of $\bar S_1= (\bar 3, 1, -2/3)$ up-like quarks couple to the invisible fermion, while in the case of $S_1 = (\bar 3,1,1/3)$ the down-like quarks couple to the invisible fermion. For the mass of invisible fermion smaller than the mass $m_p - m_K$, proton (neutron) can decay to $K$ and invisible fermion and the masses of $\bar S_1$ and $S_1$ are in the region $\sim 10^{15}$ GeV. The decays of nucleons to pions and invisible fermion can occur at the tree-level, but in the case of $\bar S_1$ they come from dimension-9 operator and are therefore suppressed by several orders of magnitude compared to the decays into kaons. For the invisible fermion mass in the range $(937.8 \, {\rm MeV},\, 938.8 \, {\rm MeV})$, decay of neutron $n \to\chi \gamma$ induced by $\bar S_1$ is possible at the loop level, while the proton remains stable. The branching ratio of such decay is $\le 10^{-6}$, which does not explain neutron decay anomaly, but is in agreement with the Borexino experiment bound. We comment on low-energy processes with the nucleon-like mass of $\chi$ in the final state as $\Lambda \to \chi \gamma$ and heavy hadron decays to invisibles.