Decay dynamics of $N\to \ell \pi $ and $N\to \ell \gamma$

SuperKamiokande experiment tightly bounds the lifetimes of the baryon number violating proton decays. The decay widths for the nucleon to an antilepton and a photon are not so well bounded experimentally as $\ell \pi$ modes. Using an effective Lagrangian approach, we relate the decay widths of $N\to \ell \gamma $ to the decay widths of $N\to \ell \pi$. Our result points out factor $10^{3}$ suppression of the decay widths $\Gamma (p \to \ell^{+} \gamma)$ and $\Gamma (n \to \bar \nu \gamma)$ compared to the decay widths $\Gamma (p \to \ell^{+} \pi^{0})$ and $\Gamma (n \to \bar \nu \pi^{0})$, respectively. This result is independent of the model of new physics. Then, we investigate the dynamics of $N\to \ell \pi $ and $N\to \ell \gamma$ amplitudes at the tree and loop level in which scalar leptoquarks are mediators of a new interaction. At the tree level, leptoquarks probe the physics beyond the Standard Model at a scale of $10^{16}$ GeV, while at the loop level, such decay can occur at a scale of $10^{7}$ GeV.