| The cross section for different processes induced by $e^+e^-$ annihilation, in the kinematical
limit $\\beta_{\\mu}\\approx\\beta_{\\pi}=\\left(1-m_{\\pi}^2/\\epsilon^2\\right )^{1/2}\\sim 1$,
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. |
