$χ$SF near the electroweak scale

We employ the chirally rotated Schr"odinger functional ($chi$SF) to study two-point fermion bilinear correlation functions used in the determination of $Z_{A,V,S,P,T}$ on a series of well-tuned ensembles. The gauge configurations, which span renormalisation scales from 4 to 70~GeV, are generated with $N_{mf}=3$ massless flavors and Schr"odinger Functional (SF) boundary conditions. Valence quarks are computed with $chi$SF boundary conditions. We show preliminary results on the tuning of the $chi$SF Symanzik coefficient $z_f$ and the scaling of the axial current normalization $Z_{m A}$. Moreover we carry out a detailed comparison with the expectations from one-loop perturbation theory. Finally we outline how automatically $mathrm{O}(a)$-improved $B_{m K}$ matrix elements, including BSM contributions, can be computed in a $chi$SF renormalization scheme.