Cross section of the processes $e^++e^-\to e^++e^-(γ)$, $\to π^++π^-(γ)$, $ μ^++μ^-(γ)$, $ γ+γ(γ)$ in the energy region 200 MeV $\le 2E\le$ 3 GeV

Abstract

The cross section for different processes induced by $e^+e^-$ annihilation, in the kinematical limit $β_μ\approxβ_π=(1-m_π^2/ε^2)^{1/2}\sim 1$, is calculated taking into account first order corrections to the amplitudes and the corrections due to soft emitted photons, with energy $ω\leΔE\le ε$ in the center of mass of the $e^+e^-$ colliding beams. The results are given separately for charge--odd and charge--even terms in the final channels $π^+π^-(γ)$ and $μ^+μ^-(γ)$. In case of pions, form factors are taken into account. The differential cross sections for the processes: $e^++e^-\to e^++e^-(+γ)$, $\to π^++π^-(γ)$, $\to μ^++μ^-(γ),\to γγ(γ)$ have been calculated and the corresponding formula are given in the ultrarelativistic limit $\sqrt{s}/2= ε\gg m_μ\sim m_π$ . For a quantitative evaluation of the contribution of higher order of the perturbation theory, the production of $π^+π^-$, including radiative corrections, is calculated in the approach of the lepton structure functions. This allows to estimate the precision of the obtained results as better than 0.5% outside the energy region corresponding to narrow resonances. A method to integrate the cross section, avoiding the difficulties which arise from singularities is also described.