Dressing L_mu - L_tau in Color

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.