Difference in dE/dX for μ+ and μ- and its effect on the underground charge ratio

Theoretical calculations predict a small fractional difference in energy loss for μ+ and μ- of the order of 0.15% at high energies. This is predominantly due to a z3 term in an extended ionization dE/dX relation, in analogy to the Barkas effect at low energies around the Bethe-Bloch maximum. The atmospheric muon energy spectrum is steeply falling off with approximately E-3.7 and thus the small difference in dE/dX between μ+ and μ- at high energies results in an amplified charge asymmetry of about 0.6% a few thousand meters water equivalent deep underground.