Model independent bounds on the modulus of the pion form factor on the unitarity cut below the $\omega\pi$ threshold

TL;DR

This paper derives model-independent bounds on the pion electromagnetic form factor below the $ ext{-}$ threshold, using phase information and experimental data, providing consistency checks and improved low-energy estimates.

Contribution

It introduces a method to bound the pion form factor without relying on specific models or assumptions above the inelastic threshold.

Findings

Bounds are consistent with experimental data below the $ ext{-}$ threshold.

At low energies, bounds are more precise than existing experimental measurements.

Method provides a nontrivial check on recent $e^+ e^-$ and $ au$ decay data.

Abstract

We calculate upper and lower bounds on the modulus of the pion electro magnetic form factor on the unitarity cut below the $\omega\pi$ inelastic threshold, using as input the phase in the elastic region known via the Fermi-Watson theorem from the $\pi\pi$ $P$-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 $\omega\pi$ threshold from $e^+ e^-$ annihilation and $\tau$-decay experiments. In particular, at low energies the calculated bounds offer a more precise description of the modulus than the experimental data.