Differences in $dE/dX$ for $\mu+$ and $\mu-$ and its Effect on the Underground Charge Ratio

Theoretical calculations predict a small fractional difference in energy loss for $\mu^+$ and $\mu^-$ of the order of $0.15$ at high energies. This is predominantly due to a $z^3$ 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 $\mu^+$ and $\mu^-$ at high energies results in an amplified charge asymmetry of about $0.6$ many thousand meters water equivalent deep underground.