Non-linear non-renormalization theorems

We study the mixing of operators under renormalization group flow in quantum theories, and prove a non-renormalization theorem at non-linear order. It dictates zeros up to a certain number of loops in anomalous dimension tensors that control, for example, the mixing of operators at order dimension six squared into dimension eight. We obtain new results at up to three loops for the mass dimension eight anomalous dimension tensor of $\phi^4$ theory in $D=4-2\varepsilon$ dimensions and verify the zeros predicted by the theorem.