Simple formulas for the energy loss of ultrarelativistic muons by direct pair production

Abstract

Simple formulas for direct pair production are derived from the general equation for deeply inelastic lepton scattering. Applications to energy loss by ultrarelativistic muons are discussed. For muons above the critical energy, E c μ ≳ 200 GeV, where the radiative effects of direct pair production and bremsstrahlung are dominant, the expressions for energy losses are considerably simplified when quoted in terms of the fractional energy loss per radiation length. The differential probability for direct pair production in a thickness x of material of radiation length X 0 for an incident lepton of energy E , mass M , can be expressed as where ν is the energy of the produced pair, υ is the fractional energy loss, υ = ν E , m e is the electron mass and the variable z is defined by z = υ M 4m e For indicent muons, this simple expression agrees very well with the exact calculation to within ∼ 30% over the entire range of υ, for E ≳ 1 TeV. At higher energies complete screening occurs, and the agreement is further improved, to better than ∼ 15% (except for the range 0.005 ≤ υ ≤ 0.01 where it is ⋍ 25%). The integral of this expression gives the energy loss due to direct pair production by muons (complete screening) which is accurate to ∼ 10%: − 1 E d E pair d x = 7.32×10 −5 X 0 .