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

TL;DR

This paper calculates the cross sections for various $e^+e^-$ annihilation processes in the 200 MeV to 3 GeV energy range, including first-order radiative corrections and form factors, with high precision outside narrow resonance regions.

Contribution

It provides detailed formulas for cross sections with radiative corrections using the lepton structure function approach, improving accuracy to better than 0.5% outside resonance regions.

Findings

Calculated differential cross sections for multiple processes.

Included radiative corrections and form factors in the analysis.

Achieved precision better than 0.5% outside narrow resonances.

Abstract

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