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 2Ele 3 GeV

The cross section for different processes induced by $e^+e^-$ annihilation, in the kinematical limit $\beta_{\mu}\approx\beta_{\pi}=(1-m_{\pi}^2/\epsilon^2)^{1/2}\sim 1$, is calculated taking into account first order corrections to the amplitudes and the corrections due to soft emitted photons, with energy $\omega\le\Delta E\le \epsilon$ 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 $\pi^+\pi^-(\gamma)$ and $\mu^+\mu^-(\gamma)$. In case of pions, form factors are taken into account. The differential cross sections for the processes: $e^++e^-\to e^++e^-(+\gamma)$, $\to \pi^++\pi^-(\gamma)$, $\to \mu^++\mu^-(\gamma),\to \gamma\gamma(\gamma)$ have been calculated and the corresponding formula are given in the ultrarelativistic limit $\sqrt{s}/2= \epsilon \gg m_{\mu}\sim m_{\pi}$ . For a quantitative evaluation of the contribution of higher order of the perturbation theory, the production of $\pi^+\pi^-$, 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.