  We describe on-shell methods for computing one- and two-loop anomalous
dimensions in the context of effective field theories containing
higher-dimension operators. We also summarize methods for computing one-loop
amplitudes, which are used as inputs to the computation of two-loop anomalous
dimensions, and we explain how the structure of rational terms and judicious
renormalization scheme choices can lead to additional vanishing terms in the
anomalous dimension matrix at two loops. We describe the two-loop implications
for the Standard Model Effective Field Theory (SMEFT). As a by-product of this
analysis we verify a variety of one-loop SMEFT anomalous dimensions computed by
Alonso, Jenkins, Manohar and Trott.
