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The cross section for different processes induced by $e^+e^-$ annihilation,in the kinematical limit$\beta_{\mu}\approx\beta_{\pi}=(1-m_{\pi}^2/\epsilon^2)^{1/2}\sim 1$, iscalculated taking into account first order corrections to the amplitudes andthe 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 resultsare given separately for charge--odd and charge--even terms in the finalchannels $\pi^+\pi^-(\gamma)$ and $\mu^+\mu^-(\gamma)$. In case of pions, formfactors are taken into account. The differential cross sections for theprocesses: $e^++e^-\to e^++e^-(+\gamma)$, $\to \pi^++\pi^-(\gamma)$, $\to\mu^++\mu^-(\gamma),\to \gamma\gamma(\gamma)$ have been calculated and thecorresponding formula are given in the ultrarelativistic limit $\sqrt{s}/2=\epsilon \gg m_{\mu}\sim m_{\pi}$ . For a quantitative evaluation of thecontribution of higher order of the perturbation theory, the production of$\pi^+\pi^-$, including radiative corrections, is calculated in the approach ofthe lepton structure functions. This allows to estimate the precision of theobtained results as better than 0.5% outside the energy region corresponding tonarrow resonances. A method to integrate the cross section, avoiding thedifficulties which arise from singularities is also described.

nuclear physics a

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