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Spacelike pion form factor from analytic continuation and the onset of perturbative QCD

The factorization theorem for exclusive processes in perturbative QCD predicts the behavior of the pion electromagnetic form factor $F(t)$ at asymptotic spacelike momenta $t(=-Q^2)<0$. We address the question of the onset energy using a suitable mathematical framework of analytic continuation, which uses as input the phase of the form factor below the first inelastic threshold, known with great precision through the Fermi-Watson theorem from $ππ$ elastic scattering, and the modulus measured from threshold up to 3 GeV by the BaBar Collaboration. The method leads to almost model-independent upper and lower bounds on the spacelike form factor. Further inclusion of the value of the charge radius and the experimental value at $-2.45 \gev^2$ measured at JLab considerably increases the strength of the bounds in the region $ Q^2 \lesssim 10 \gev^2$, excluding the onset of the asymptotic perturbative QCD regime for $Q^2< 7\gev^2$. We also compare the bounds with available experimental data and with several theoretical models proposed for the low and intermediate spacelike region.