  We describe the on-shell method to deriving the Renormalization Group (RG)
evolution of Wilson coefficients of high dimensional operators at one loop,
which is a necessary part in the on-shell construction of the Standard Model
Effective Field Theory (SMEFT), and exceptionally efficient based on the
amplitude basis in hand. The UV divergence is obtained by firstly calculating
the coefficients of scalar bubble integrals by unitary cuts, then subtracting
the IR divergence in the massless bubbles, which can be easily read from the
collinear factors we obtained for the Standard Model fields. Examples of
deriving the anomalous dimensions at dimension six are presented in a
pedagogical manner. We also give the results of contributions from the
dimension-8 $H^4D^4$ operators to the running of $V^+V^-H^2$ operators, as well
as the running of $B^+B^-H^2D^{2n}$ from $H^4D^{2n+4}$ for general $n$.
