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

Abstract

The cross section for different processes induced by $e^{+}{e}^{-}$ annihilation, in the kinematical limit ${\beta }_{\mu }\approx {\beta }_{\pi }=(1-m_{\pi }^2/{ϵ}^2{)}^{1/2}\sim 1$, is calculated taking into account first order corrections to the amplitudes and the corrections due to soft emitted photons, with energy $\omega \le \Delta E\le ϵ$ 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 ${\pi }^{+}{\pi }^{-}(\gamma )$ 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 $\sqrt{s}/2=ϵ\gg m_{\mu }\sim m_{\pi }$ . 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.

Comments: 25 pages 3 firgure