Study ofωandφeffects on the pion form factor

The resonant $G$-parity-violation effect due to $\omega$ and $\phi$ on the pion form factor is studied using the solution of the Muskhelishvili-Omnes integral equation with inelastic unitarity. The relative phases of the so-called $\omega -\rho$ and $\phi -\rho$ interference effects are accurately determined by analyticity and time-reversal invariance. For the case of the $\omega$, the results are shown to be identical with those of the two-coupled-channel problem and also with those obtained from the modified propagator (or mass-matrix) approach. Similar results but with somewhat less accuracy are also valid for the $\phi$ effect on the pion form factor.