  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.
