Three-Loop Mixing of Dipole Operators

We calculate the complete three-loop $\mathcal{O}({\alpha }_s^3)$ anomalous dimension matrix for the dimension-five dipole operators that arise in the standard model after integrating out the top quark and the heavy electroweak bosons. Our computation completes the three-loop anomalous dimension matrix of operators that govern low-energy $|\Delta F|=1$ flavor-changing processes, and represents an important ingredient of the next-to-next-to-leading order QCD analysis of the $\overline{B}\to X_s\gamma$ decay.