  We consider a new massive vector-boson Z' that couples to leptons through the
L_mu - L_tau current, and to quarks through an arbitrary set of couplings. We
show that such a model can be obtained from a renormalizable field theory
involving new heavy fermions in an anomaly-free representation. The model is a
candidate explanation for the discrepancy observed recently by the LHCb
collaboration in angular distributions of the final state particles in the rare
decay B \to K* mu^+ mu^-. Interestingly, the new vector-boson contribution to
the decay tau \to mu nu_tau \bar nu_mu can also remove a small tension in the
measurement of the corresponding branching ratio. Constraints from light flavor
meson-mixing restrict the coupling to the up- and down-quarks to be very small
and thus direct production of the vector-boson at hadron colliders is strongly
suppressed. The most promising ways to test the model is through the
measurement of the Z decay to four leptons and through its effect on neutrino
trident production of muon pairs. This latter process is a powerful but
little-known constraint, which surprisingly rules out explanations of (g-2)_mu
based on Z' gauge bosons coupled to muon number, with mass of at least a few
GeV.
