Implications of the recent high statistics determination of the pion electromagnetic form factor in the timelike region

Abstract

The recently evaluated two-pion contribution to the muon $g\ensuremath{-}2$ and the phase of the pion electromagnetic form factor in the elastic region, known from $\ensuremath{\pi}\ensuremath{\pi}$ scattering by Fermi-Watson theorem, are exploited by analytic techniques for finding correlations between the coefficients of the Taylor expansion at $t=0$ and the values of the form factor at several points in the spacelike region. We do not use specific parametrizations, and the results are fully independent of the unknown phase in the inelastic region. Using for instance, from recent determinations, $⟨{r}_{\ensuremath{\pi}}^{2}⟩=(0.435\ifmmode\pm\else\textpm\fi{}0.005)\text{ }\text{ }{\mathrm{fm}}^{2}$ and $F(\ensuremath{-}1.6\text{ }\text{ }{\mathrm{GeV}}^{2})={0.243}_{\ensuremath{-}0.014}^{+0.022}$, we obtain the allowed ranges $3.75\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}4}\ensuremath{\lesssim}c\ensuremath{\lesssim}3.98\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}4}$ and $9.91\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}6}\ensuremath{\lesssim}d\ensuremath{\lesssim}10.46\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}6}$ for the curvature and the next Taylor coefficient, with a strong correlation between them. We also predict a large region in the complex plane where the form factor cannot have zeros.