Bounds on the Coupling of the Majoron to Light Neutrinos from Supernova Cooling

We explore the role of Majoron ($J$) emission in the supernova cooling process, as a source of upper bound on neutrino-Majoron coupling. We show that the strongest upper bound on the coupling to $\nu_e$ comes from the $\nu_e\nu_e \to J $ process in the core of a supernova. We also find bounds on diagonal couplings of the Majoron to $\nu_{\mu(\tau)}\nu_{\mu(\tau)}$ and on off-diagonal $\nu_e \nu_{\mu (\tau)}$ couplings in various regions of the parameter space. We discuss the evaluation of the cross-section for four-particle interactions ($\nu \nu \to JJ$ and $\nu J \to \nu J$). We show that these are typically dominated by three-particle sub-processes and do not give new independent constraints.