  We consider an anomaly free extension of the standard model gauge group
$G_{\rm SM}$ by an abelian group to $G_{\rm SM}\otimes U(1)_Z$. The condition
of anomaly cancellation is known to fix the $Z$-charges of the particles, but
two. We fix one remaining charge by allowing for all possible Yukawa
interactions of the known left handed neutrinos and new right-handed ones that
obtain their masses through interaction with a new scalar field with
spontaneously broken vacuum. We discuss some of the possible consequences of
the model. Assuming that the new interaction is responsible for the observed
differences between the standard model prediction for the anomalous magnetic
moment of the muon and its measured value, we constrain the size of the new
gauge coupling, the mass of the new gauge boson and the vacuum expectation
value of the new scalar field.
