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We perform a dispersive analysis of the $\omega\pi$ electromagnetictransition form factor, using as input the discontinuity provided by unitaritybelow the $\omega\pi$ threshold and including for the first time experimentaldata on the modulus measured from $e^+e^-\to\omega\pi^0$ at higher energies.The input leads to stringent parameterization-free constraints on the modulusof the form factor below the $\omega\pi$ threshold, which are in disagreementwith some experimental values measured from $\omega\to \pi^0\gamma^*$ decay. Wediscuss the dependence on the input parameters in the unitarity relation, usingfor illustration an $N/D$ formalism for the P partial wave of the scatteringprocess $\omega\pi \to \pi\pi$, improved by a simple prescription whichsimulates the rescattering in the crossed channels. Our results confirm theexistence of a conflict between experimental data and theoretical calculationsof the $\omega\pi$ form factor in the region around 0.6 GeV and bring furtherarguments in support of renewed experimental efforts to measure more preciselythe $\omega\to\pi^0\gamma^*$ decay.

Testing the consistency of the $\omega\pi$ transition form factor with unitarity and analyticity