Pion scalar form factor and another confirmation of the existence of thef0(500)meson

Abstract

An explicit form of the pion scalar form factor ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\pi}}(t)$ is constructed by using its phase representation and a correct description of the S-wave isoscalar $\ensuremath{\pi}\ensuremath{\pi}$ phase shift ${\ensuremath{\delta}}_{0}^{0}(t)$ data by the parametrization of $\mathrm{tan}{\ensuremath{\delta}}_{0}^{0}(t)$ in the absolute valued pion c.m. three-momentum $q=[(t\ensuremath{-}4{m}_{\ensuremath{\pi}}^{2})/4{]}^{1/2}$. This parametrization has been found starting from fully general considerations. Then a calculation of the corresponding integral in the framework of the theory of residua provides ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\pi}}(t)$ in the form of a rational function with one zero and four poles in the $q$ variable. Investigations of the latter poles demonstrate that two of them, to be conjugate according to the imaginary axis in the $q$ plane, clearly correspond to complex conjugate ${f}_{0}(500)$ meson poles on the second Riemann sheet in a momentum transfer squared $t$ variable. Another pair of poles, also conjugate according to the imaginary axis in the $q$ plane, can be identified with two complex conjugate poles on the second Riemann sheet in the $t$ variable, corresponding to the ${f}_{0}(980)$ meson.