Two-Loop Renormalization of a Chiral $SU(2)$ Gauge Theory in Dimensional Regularization with Non-Anticommuting $\gamma_5$

Higher order calculations in chiral gauge theories such as the Electroweak Standard Model require a sound treatment of the notoriously problematic $\gamma_5$-matrix in Dimensional Regularization (DReg). In the all-order consistent BMHV scheme anticommutativity has to be sacrificed, resulting in spurious breakings of BRST invariance, the restoration of which necessitates finite, symmetry-restoring counterterms. Following recent advances in successfully applying this scheme to multi-loop calculations for Abelian models, we shall here present the first complete non-Abelian two-loop result for the case of $SU(2)$, which is of particular interest to the Standard Model. We provide the complete list of finite, two-loop symmetry restoring counterterms and discuss intricacies of the non-Abelian implementation. Except for one novel term, the finite counterterm action exhibits the same structure as at one-loop order.