Model-independent form-factor constraints for electromagnetic spin-1 currents

Using local gauge invariance in the form of the Ward-Takahashi identity and the fact that properly constructed current operators must be free of kinematic singularities, it is shown that the magnetic moment $\mu$ and the quadrupole moment $Q$ of an elementary spin-1 particle with mass $m$ and charge $e$ are related by $2m\mu +m^{2}Q=e$, thus constraining the normalizations of the Sachs form factors. This relation holds true as a matter of course at the tree level in the standard model, but we prove that it remains true in general for dressed spin-1 states derived from elementary fields. General expressions for spin-1 propagators and currents with arbitrary hadronic dressing are given, which show that the result is independent of any dressing effect or model approach.

Figure 1

Depiction of the electromagnetic current vertex for the $\rho$ meson, $\gamma (k)+\rho (q)\to \rho (q^{\prime })$, with associated four-momenta and Lorentz indices. (Time runs from right to left.)