Four-loop QCD mixing of current-current operators

TL;DR

This paper computes the four-loop anomalous dimension of specific weak effective operators in QCD at NNNLO, advancing the precision of theoretical predictions for CP violation in kaon decays.

Contribution

It provides the first complete four-loop calculation of these operator anomalous dimensions, including analytic results and basis transformation expressions.

Findings

First four-loop anomalous dimension calculation in QCD for these operators.

Analytic expressions provided for basis transformations.

Results applicable to CP violation measurements in kaon and B physics.

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.