How to Identify a Majoron: Effective Field Theories of Spontaneous Lepton Number Breaking
We revisit the traditional Type I, II and III Seesaw mechanisms in the presence of a complex scalar field charged under a global $U(1)$ symmetry that can be identified with lepton number and the Peccei-Quinn symmetry. After symmetry breaking, the radial mode becomes heavy while the angular mode appears as an axion-like particle, traditionally dubbed the Majoron. We construct the effective field theory obtained after integrating out the heavy states and analyse two matching orders: first removing the radial mode and then the Seesaw fields, and vice versa. Both procedures yield the same low-energy Lagrangian containing only Standard Model fields and the Majoron. Because a single vacuum expectation value fixes the mediator masses, the radial mode and every Majoron coupling, these models predict relations among observables rather than their individual size, and it is these relations that are testable. Indeed, the invisible Higgs width is locked to the universal suppression of the Higgs couplings, while the Majoron-lepton coupling is fixed by the measured non-unitarity of the leptonic mixing matrix, and the two independently give comparable lower bounds on the same lepton-number breaking scale, of order $1$-$10$ TeV. Neutrinoless double beta decay with Majoron emission, by contrast, has no sensitivity in this class of models. The framework is thus falsifiable even when the new states lie far beyond experimental reach.