Production cross sections of gamma-rays, electrons, and positrons in p-p collisions

Because the production cross sections of gamma-rays, electrons, and positrons made in p-p collisions, $\sigma_{pp\rightarrow \gamma}$ and $\sigma_{pp\rightarrow {e}^\pm}$, respectively, are kinematically equivalent with respect to the parent pion-production cross section $\sigma_{pp\rightarrow \pi}$, we obtain $\sigma_{pp\rightarrow {e}^\pm}$ directly from the machine data on $\sigma_{pp\rightarrow \gamma}$. In Sato et al. (2012), we give explicitly $\sigma_{pp\rightarrow \gamma}$, reproducing quite well the accelerator data with LHC, namely $\sigma_{pp\rightarrow {e}^\pm}$ is applicable enough over the wide energy range from GeV to 20\,PeV for projectile proton energy. We dicuss in detail the relation between the cross sections, and present explicitly $\sigma_{pp\rightarrow {e}^\pm}$ that are valid into the PeV electron energy.