Electromagnetic Charge Radius of the Pion at High Precision

We present a determination of the pion charge radius from high precision data on the pion vector form factor from both timelike and spacelike regions, using a novel formalism based on analyticity and unitarity. At low energies, instead of the poorly known modulus of the form factor, we use its phase, known with high accuracy from Roy equations for $\pi \pi$ elastic scattering via the Fermi-Watson theorem. We use also the values of the modulus at several higher timelike energies, where the data from $e^{+}{e}^{-}$ annihilation and $\tau$ decay are mutually consistent, as well as the most recent measurements at spacelike momenta. The experimental uncertainties are implemented by Monte Carlo simulations. The results, which do not rely on a specific parametrization, are optimal for the given input information and do not depend on the unknown phase of the form factor above the first inelastic threshold. Our prediction for the charge radius of the pion is $r_{\pi }=(0.657\pm 0.003)\mathrm{fm}$, which amounts to an increase in precision by a factor of about 2.7 compared to the Particle Data Group average.

Figure 1

Statistical distributions of $⟨r_{\pi }^2⟩$ obtained using the Bern phase and one modulus measured by the BABAR experiment [16]. The left panel is obtained without spacelike input, the right one with input from $t=-1.60{\mathrm{GeV}}^2$. The vertical lines correspond to 68.3% confidence limit (C.L.)

Figure 2

68.3% C.L. intervals of $⟨r_{\pi }^2⟩$ for each input modulus in the region (0.65–0.71) GeV measured in $e^{+}{e}^{-}$-annihilation and $\tau$-decay experiments, using the Bern phase and the spacelike datum at $t=-1.60{\mathrm{GeV}}^2$.