Timelike behavior of the pion electromagnetic form factor in the functional formalism

Abstract

The electromagnetic form factor of the pion is calculated within the use of functional formalism. We develop integral representation for the minimal set of Standard Model Green's functions and derive the dispersion relation for the form factor in the two-flavor QCD isospin limit ${m}_{u}={m}_{d}$. We use the dressed quark propagator as obtained from the gap equation in Minkowski space and within the Dyson-Schwinger equations formalism to derive the approximate dispersion relation for the form factor for the first time. We evaluate the form factor for the spacelike as well as for the timelike momentum in the presented formalism. A new Nakanishi-like form of integral representation is proved on the basis of the vector Bethe-Salpeter equation for the quark-photon vector with a ladder-rainbow kernel. The gauge technique turns out to be a part of the entire structure of the vertex. In the analytic approach presented here, it is shown that a large amount of the $\ensuremath{\rho}$-meson peak in the cross section ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is governed by the gauge invariance of QED/QCD---i.e., by a gauge-technique-constructed quark-photon vertex. This approximation naturally explains the broad shape of the $\ensuremath{\rho}$-meson peak.