  We introduce an operator depending on the "transverse velocity" r that
describes the effect of hadron masses on the leading 1/Q power correction to
event-shape observables. Here, Q is the scale of the hard collision. This work
builds on earlier studies of mass effects by Salam and Wicke and of operators
by Lee and Sterman. Despite the fact that different event shapes have different
hadron mass dependence, we provide a simple method to identify universality
classes of event shapes whose power corrections depend on a common
nonperturbative parameter. We also develop an operator basis to show that at a
fixed value of Q, the power corrections for many classic observables can be
determined by two independent nonperturbative matrix elements at the 10% level.
We compute the anomalous dimension of the transverse velocity operator, which
is multiplicative in r and causes the power correction to exhibit non-trivial
dependence on Q. The existence of universality classes and the relevance of
anomalous dimensions are reproduced by the hadronization models in Pythia 8 and
Herwig++, though the two programs differ in the values of their low-energy
matrix elements.
