Analytical results for low-angle Bhabha scattering with pair production

We calculate the effect of pair production on the luminosity process of low-angle Bhabha scattering at the SLAC Linear Collider (SLC) and/or CERN $e^{+}{e}^{-}$ collider LEP. We work in the leading-logarithm approximation to orders ${\alpha }^2$ and ${\alpha }^3$ for the nonsinglet and the singlet contributions, respectively. In particular we calculate the size of the singlet contribution to be below 0.007% for the energy cuts $x_c>0.1\mathrm{}$, as was estimated by Jadach and co-workers. In this way, we remove the contribution of such pairs from the theoretical uncertainty in the SLC/LEP luminosity calculation of Jadach and co-workers via the Monte Carlo Bhlumi 2.01 at the precision level of 0.01%. Results are illustrated for typical SLC/LEP acceptance cuts.