Violation of Sum Rules for Twist-3 Parton Distributions

Sum rules for twist-3 distributions are reexamined. Integral relations between twist-3 and twist-2 parton distributions suggest the possibility for a $\delta$-function at $x=0$. We confirm and clarify this result by constructing $h_L$ and $h_L^3$ (quark-gluon interaction dependent part of $h_L$) explicitly from their moments for a one-loop dressed massive quark. The physics of these results is illustrated by calculating $h_L(x,Q^2)$ using light-front time-ordered pQCD to ${O}(\alpha_S)$ on a quark target.