Constraints on gauged ${B-3L}_{\tau }$ and related theories

We consider extensions of the standard model with an extra U(1) gauge boson that couples to $B-(\alpha L_e+\beta L_{\mu }+\gamma L_{\tau })$ with $\alpha +\beta +\gamma =3.\mathrm{}$ We show that the extra gauge boson necessarily mixes with the Z, leading to potentially significant corrections to the $\mathrm{Zf}\overline{f}$ vertex. The constraints on the size of this correction imposed by the Z-pole data from CERN LEP and SLD are derived.