  A classic result, originally due to Kluberg-Stern and Zuber, states that
operators that vanish by the classical equation of motion (eom) do not mix into
"physical" operators. Here we show that and explain why this result does not
hold in soft-collinear effective theory (SCET) for the renormalization of
power-suppressed operators. We calculate the non-vanishing mixing of eom
operators for the simplest case of $N$-jet operators with a single collinear
field in every direction. The result implies that---for the computation of the
anomalous dimension but not for on-shell matrix elements---there exists a
preferred set of fields that must be used to reproduce the infrared
singularities of QCD scattering amplitudes. We identify these fields and
explain their relation to the gauge-invariant SCET Lagrangian. Further checks
reveal another generic property of SCET beyond leading power, which will be
relevant to resummation at the next-to-leading logarithmic level, the
divergence of convolution integrals with the hard matching coefficients. We
propose an operator solution that allows to consistently renormalize such
divergences.
