QCD Evolutions of Twist-3 Chirality-Odd Operators

We study the scale dependence of twist-3 distributions defined with chirality-odd quark-gluon operators. To derive the scale dependence we explicitly calculate these distributions of multi-parton states instead of a hadron. Taking one-loop corrections into account we obtain the leading evolution kernel in the most general case. In some special cases the evolutions are simplified. We observe that the obtained kernel in general does not get simplified in the large-$N_c$ limit in contrast to the case of those twist-3 distributions defined only with chirality-odd quark operators. In the later, the simplification is significant.