  Using on-shell methods, we present a new perturbative non-renormalization
theorem for operator mixing in massless four-dimensional quantum field
theories. By examining how unitarity cuts of form factors encode anomalous
dimensions we show that longer operators are often restricted from
renormalizing shorter operators at the first order where there exist Feynman
diagrams. The theorem applies quite generally and depends only on the field
content of the operators involved. We apply our theorem to operators of
dimension five through seven in the Standard Model Effective Field Theory,
including examples of nontrivial zeros in the anomalous-dimension matrix at one
through four loops. The zeros at two and higher loops go beyond those
previously explained using helicity selection rules. We also include explicit
sample calculations at two loops.
