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The chirally rotated Schr\"odinger functional ($\chi$SF) renders themechanism of automatic $O(a)$ improvement compatible with Schr\"odingerfunctional (SF) renormalization schemes. Here we define a family ofrenormalization schemes based on the $\chi$SF for a complete basis of $\Delta F= 2$ parity-odd four-fermion operators. We compute the correspondingscale-dependent renormalization constants to one-loop order in perturbationtheory and obtain their NLO anomalous dimensions by matching to the$\overline{\textrm{MS}}$ scheme. Due to automatic $O(a)$ improvement, once the$\chi$SF is renormalized and improved at the boundaries, the step scalingfunctions (SSF) of these operators approach their continuum limit with$O(a^{2})$ corrections without the need of operator improvement.

Perturbative renormalization of $\Delta F = 2$ four-fermion operators with the chirally rotated Schr\"odinger functional