axis, and such transverse motion is not affected by the Lorentz transformation along this axis.

Although our decomposition looks quite satisfactory in many respects, one subtle question remained. It is a role of quantum-loop effects. Is the longitudinal gluon polarization $\Delta G$ gauge-invariant even at quantum level ? This is a fairly delicate question. In fact, despite the existence of several formal proof showing the gauge-invariance of $\Delta G$ [@Manohar90] -[@AEL95], it was sometimes claimed that $\Delta G$ has its meaning only in the light-cone gauge and infinite-momentum frame [@JTH96],[@HJL99]. More specifically, in an influential paper, Hoodbhoy, Ji, and Lu claim that $\Delta G$ evolves differently in the Feynman gauge and the LC gauge [@HJL99]. However, the gluon spin operator used in their Feynman gauge calculation is given by $$\begin{aligned}
 M^{+12}_{g-spin} &=& 2 \,\mbox{\rm Tr} \,
 [\,F^{+1} \, A^2 
 \ - \ F^{+2} \,A^1 
 \,] ,\end{aligned}$$ which is not gauge-invariant, and is delicately different from our gauge-invariant gluon-spin operator given as $$\begin{aligned}
 M^{+12}_{g-spin} &=& 2 \,\mbox{\rm Tr} \,
 [\,F^{+1} \, A^2_{phys} 
 \ - \ F^{+2} \,A^1_{phys} 
 \,] ,\end{aligned}$$ The problem is how to incorporate this difference into the Feynman rule for evaluating 1-loop anomalous dimension of the quark and gluon 