Neutrino Mixing and the Axion-Gluon Vertex

We present detailed arguments and calculations in support of our recent proposal to identify the axion arising in the solution of the strong CP problem with the Majoron, the (pseudo-)Goldstone boson of spontaneously broken lepton number symmetry. At low energies, the associated $U(1)_L$ becomes, via electroweak parity violation and neutrino mediation, indistinguishable from an axial Peccei-Quinn symmetry in relation to the strong interactions. The axionic couplings are then fully computable in terms of known SM parameters and the Majorana mass scales. The determination of these couplings involves certain three-loop diagrams, with a UV finite neutrino triangle taking over the role of the usual triangle anomaly. A main novelty of our proposal is thus to solve the strong CP problem by a non-local term that produces an anomaly-like term in the IR limit.