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Bounds on the spacelike pion electromagnetic form factor from analyticity and unitarity

We use the recently measured accurate BaBaR data on the modulus of the pion electromagnetic form factor, $F_π(t)$, up to an energy of 3 GeV, the I=1 $P$-wave phase of the $ππ$ scattering amplitude up to the $ω-π$ threshold, the pion charge radius known from Chiral Perturbation Theory, and the recently measured JLAB value of $F_π$ in the spacelike region at $t=-2.45 {\rm GeV}^2$ as inputs in a formalism that leads to bounds on $F_π$ in the intermediate spacelike region. We compare our constraints with experimental data and with perturbative QCD along with the results of several theoretical models for the non-perturbative contributions proposed in the literature.