  The renormalization group equations (RGEs) in Standard Model effective theory
are usually either solved analytically, neglecting the scale dependence of
gauge and Yukawa couplings, or numerically without such approximations. We
present analytic solutions of RGEs that take into account the dominant scale
dependence of the anomalous-dimension matrix due to the running of the QCD
coupling and of the top-Yukawa coupling. We consider first the case for which a
given operator is generated directly through mixing with the parent operator
whose Wilson coefficient is non-vanishing at the new physics scale.
Subsequently we consider the case of two-step running, in which two operators
do not mix directly, but only via a third mediator operator. We generalize
these solutions to an arbitrary number of operators and show how even in this
case analytic solutions can be obtained.
