(B-L) Symmetry vs. Neutrino Seesaw

We compute the effective coupling of the Majoron to W bosons at $\cO(\hbar)$ by evaluating the matrix element of the (B-L) current between the vacuum and a $W^+W^-$ state. The (B-L) anomaly vanishes, but the amplitude does not vanish as a result of a UV finite and non-local contribution which is entirely due to the mixing between left-chiral and right-chiral neutrinos. The result shows how anomaly-like couplings may arise in spite of the fact that the (B-L) current remains exactly conserved to all orders in $\hbar$, lending additional support to our previous proposal to identify the Majoron with the axion.