Prompt atmospheric neutrinos and muons: NLO versus LO QCD predictions

We compare the leading and next-to-leading order QCD predictions for the flux of atmospheric muons and neutrinos from decays of charmed particles. We find that the full NLO lepton fluxes can be approximated to within $\sim 10\%$ by the Born-level fluxes multiplied by an overall factor of $2.2-2.4,$ which depends slightly on the PDF. This supports the approach of Thunman, Ingelman and Gondolo. We also find that their very low lepton fluxes are due to the mild slope they used for the gluon distribution function at small momentum fractions, and that substantially larger lepton fluxes result when the slope of the gluon distribution function at small momentum fractions is larger.