Nuclear matrix elements for Majoron emitting double-$\beta$ decay

A complete calculation of the nuclear matrix elements (NME) for Majoron emitting neutrinoless double beta decay within the framework of IBM-2 for spectral indices $n=1,3,7$ is presented. By combining the results of this calculation with previously calculated phase space factors (PSF) we give predictions for expected half-lives. By comparing with experimental limits on the half-lives we set limits on the coupling constants $\left\langle g_{ee}^{M}\right\rangle$ of all proposed Majoron-emitting models.