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 the 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 ${\nu }_{\mu \left(\tau \right)}{\nu }_{\mu \left(\tau \right)}$ and on off-diagonal ${\nu }_e{\nu }_{\mu \left(\tau \right)}$ couplings in various regions of the parameter space. We discuss the evaluation of the cross section for four-particle interactions $(\nu \overrightarrow{\nu }\mathrm{JJ}$ and $\nu \overrightarrow{J}\nu J).\mathrm{}$ We show that these are typically dominated by three-particle subprocesses and do not give new independent constraints.