Four-loop QCD mixing of current-current operators

Abstract

We calculate the anomalous dimension of the $|\Delta S| = 1$ current-current operators of the weak effective Lagrangian at next-to-next-to-next-to-leading order (NNNLO) in QCD. This constitutes the first step towards a full four-loop calculation of the QCD correction to $\epsilon_K$, the measure for indirect CP violation in the neutral kaon system. We present fully analytic results, together with the expressions necessary to transform our results to a basis with an arbitrary different definition of evanescent operators. As an application, we calculate the corresponding results in the ``standard'' operator basis used in $B$ physics.