Nonrenormalization and Operator Mixing via On-Shell Methods

Using on-shell methods, we present a new perturbative nonrenormalization 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 Feynman diagrams exist. 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.