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Model independent bounds on the modulus of the pion form factor on the unitarity cut below the $ωπ$ threshold

We calculate upper and lower bounds on the modulus of the pion electro magnetic form factor on the unitarity cut below the $ωπ$ inelastic threshold, using as input the phase in the elastic region known via the Fermi-Watson theorem from the $ππ$ $P$-wave phase shift, and a suitably weighted integral of the modulus squared above the inelastic threshold. The normalization at $t=0$, the pion charge radius and experimental values at spacelike momenta are used as additional input information. The bounds are model independent, in the sense that they do not rely on specific parametrizations and do not require assumptions on the phase of the form factor above the inelastic threshold. The results provide nontrivial consistencychecks on the recent experimental data on the modulus available below the $ωπ$ threshold from $e^+ e^-$ annihilation and $τ$-decay experiments. In particular, at low energies the calculated bounds offer a more precise description of the modulus than the experimental data.