  Searches for neutrino-less double-beta decay ($0\nu2\beta$) place an
important constraint on models where light fields beyond the Standard Model
participate in the neutrino mass mechanism. While $0\nu2\beta$ experimental
collaborations often consider various massless majoron models, including
various forms of majoron couplings and multi-majoron final-state processes,
none of these searches considered the scenario where the "majoron" $\phi$ is
not massless, $m_\phi\sim$~MeV, of the same order as the $Q$-value of the
$0\nu2\beta$ reaction. We consider this parameter region and estimate
$0\nu2\beta\phi$ constraints for $m_\phi$ of order MeV. The constraints are
affected not only by kinematical phase space suppression but also by a change
in the signal to background ratio characterizing the search. As a result,
$0\nu2\beta\phi$ constraints for $m_\phi>0$ diminish significantly below the
reaction threshold. This has phenomenological implications, which we illustrate
focusing on high-energy neutrino telescopes. Our results motivate a dedicated
analysis by $0\nu2\beta$ collaborations, analogous to the dedicated analyses
targeting massless majoron models.

High Energy Physics - Phenomenology