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Testing the consistency of the $ωπ$ transition form factor with unitarity and analyticity

We perform a dispersive analysis of the $ωπ$ electromagnetic transition form factor, using as input the discontinuity provided by unitarity below the $ωπ$ threshold and including for the first time experimental data on the modulus measured from $e^+e^-\toωπ^0$ at higher energies. The input leads to stringent parameterization-free constraints on the modulus of the form factor below the $ωπ$ threshold, which are in disagreement with some experimental values measured from $ω\to π^0γ^*$ decay. We discuss the dependence on the input parameters in the unitarity relation, using for illustration an $N/D$ formalism for the P partial wave of the scattering process $ωπ\to ππ$, improved by a simple prescription which simulates the rescattering in the crossed channels. Our results confirm the existence of a conflict between experimental data and theoretical calculations of the $ωπ$ form factor in the region around 0.6 GeV and bring further arguments in support of renewed experimental efforts to measure more precisely the $ω\toπ^0γ^*$ decay.