  The production of a mu+mu- pair from the scattering of a muon-neutrino off
the Coulomb field of a nucleus, known as neutrino trident production, is a
sub-weak process that has been observed in only a couple of experiments. As
such, we show that it constitutes an exquisitely sensitive probe in the search
for new neutral currents among leptons, putting the strongest constraints on
well-motivated and well-hidden extensions of the Standard Model gauge group,
including the one coupled to the difference of the lepton number between the
muon and tau flavor, L_mu-L_tau. The new gauge boson, Z', increases the rate of
neutrino trident production by inducing additional $(\bar\mu \gamma_\alpha
\mu)(\bar\nu \gamma^\alpha \nu)$ interactions, which interfere constructively
with the Standard Model contribution. Existing experimental results put
significant restrictions on the parameter space of any model coupled to muon
number L_mu, and disfavor a putative resolution to the muon g-2 discrepancy via
the loop of Z' for any mass m_Z' > 400 MeV. The reach to the models' parameter
space can be widened with future searches of the trident production at
high-intensity neutrino facilities such as the LBNE.
