Pion transverse charge density from $e^{+}$$e^{-}$ annihilation data and logarithmic dispersion relations

The transverse charge density of the pion is extracted from a dispersive analysis of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ exclusive annihilation data. A logarithmic dispersion relation is used to compute the unknown phase of the timelike pion form factor from the modulus obtained from the annihilation cross section. The method is model-independent and permits quantitative uncertainty estimates. The density is obtained with few-percent accuracy down to $b\sim 0.1$ fm; at smaller distances it depends qualitatively on the assumed high-energy behavior of the timelike form factor. Implications for pion structure and the relevance of pQCD asymptotics are discussed.