Renormalization of Higher-Dimensional Operators from On-shell Amplitudes

On-shell amplitude methods allow to derive one-loop renormalization effects from just tree-level amplitudes, with no need of loop calculations. We derive a simple formula to obtain the anomalous dimensions of higher-dimensional operators from a product of tree-level amplitudes. We show how this works for dimension-6 operators of the Standard Model, providing explicit examples of the simplicity, elegance and efficiency of the method. Many anomalous dimensions can be calculated from the same Standard Model tree-level amplitude, displaying the attractive recycling aspect of the on-shell method. With this method, it is possible to relate anomalous dimensions that in the Feynman approach arise from very different diagrams, and obtain non-trivial checks of their relative coefficients. We compare our results to those in the literature, where ordinary methods have been applied.