Process $e^{+}{e}^{-}\to 3\pi (\gamma )$ with final state radiative corrections

The process of annihilation of $e^{+}{e}^{-}$ to the three pion final state in the case of moderately high energies is considered. The final state emission of virtual and real photons is considered explicitly. The calculations are performed in QED and the pion sector of the effective chiral Lagrangian. The initial state emission is considered in terms of the lepton structure function. Some numerical estimates are given.

Figure 1

Feynman diagram contributing to the process probability in the Born approximation.

Figure 2

Types of Feynman diagrams of emission of an additional virtual photon.

Figure 3

The value $\delta -{\overline{\delta }}_h$ [see (33)] in percents for typical experimental conditions as a function of $\beta$.