We show that $L_1=L_e-L_{\mu }$, $L_2=L_e-L_{\tau }$, and $L_3=L_{\mu }-L_{\tau }$ (where $L_{e,\mu ,\tau }$ are the family lepton numbers) are anomaly free and can thus be gauged. Three simple theories featuring a flavor-conserving second $Z$ boson ($Z_{1,2,3}^{\prime }$) result. Bounds on the $Z_i^{\prime }$ coupling constant ($g_i^{\prime }$) and mass are derived. The mass of $Z_1^{\prime }$ has a lower bound of 60-150 GeV at 90% confidence level for reasonable values of $g_1^{\prime }$. The mass of $Z_2^{\prime }$ has a best-fit value of about 60 GeV. The physics of these bosons in the KEK TRISTAN and CERN LEP window of 63-87 GeV is discussed. Constraints on $Z_3^{\prime }$ interactions are derived from ${\mathrm{}(g-2\mathrm{})\mathrm{}}_{\mu }$.

• Received 29 May 1990

DOI:https://doi.org/10.1103/PhysRevD.43.R22