Radius of the $\pi$-meson and analytic properties of its form factor

By solving the extremal problem for the functional
$$ \Phi\{F,f\}=\int_{4m_{\pi^2}}^{\infty}f(t)|F_\pi(t)|^2\,dt, $$
where $f(t)$ is a given position function and $F_\pi(t)$ is the form factor of the $\pi$-meson withknown analytic properties, upper bounds are established for the radius of the $\pi$-meson and the behavior of its form factor in the space-like region ($t\leqslant 0$). These are determined by the values of the form-factor modulus in the annihilation channel ($t\geqslant 4m_{\pi^2}$). It is assumed on the basis of experiments at Novosibirsk and Orsay with colliding beams in the interval $4m_{\pi^2}<t\lesssim1$ (BeV)$^2$ that the form factor can be represented by the Breit–Wigner formula, it is also assumed that the modulus of the form factor for $t\gtrsim1$ (BeV)$^2$ does not exceed a certain constant value. The following results are then obtained: $r_{\max}=0{,}69\pm0{,}14$ (Novosibirsk) and $r_{\max}=0{,}9\pm0{,}06$ (Orsay).