  Recently, the validity of the guiding-center approach to model relativistic
runaway electrons in tokamaks has been challenged by full-orbit simulations
that demonstrate the breakdown of the standard magnetic moment conservation. In
this paper, we derive a new expression for the magnetic moment of relativistic
runaway electrons, which is conserved significantly better than the standard
one. The new result includes one of the second-order corrections in the
standard guiding-center theory which, in case of runaway electrons with
$p_{\parallel}\gg p_{\perp}$, can peculiarly be of the same order as the
lowest-order term. The better conservation of the new magnetic moment also
explains the collisionless pitch-angle-scattering effect observed in full-orbit
simulations since it allows momentum transfer between the perpendicular and
parallel directions when the runaway electron is accelerated by an electric
field. While the derivation of the second-order correction to the magnetic
moment in general case would require the full extent of the relativistic
second-order guiding-center theory, we exploit the Lie-perturbation method at
the limit $p_{\parallel}\gg p_{\perp}$ which simplifies the computations
significantly. Consequently, we present the corresponding guiding-center
equations applicable to the highly relativistic runaway electrons.
