Perturbative renormalization of $Delta F = 2$ four-fermion operators with the chirally rotated Schrodinger functional

The chirally rotated Schrodinger functional ($chi$SF) renders the mechanism of automatic $O(a)$ improvement compatible with Schrodinger functional (SF) renormalization schemes. Here we define a family of renormalization schemes based on the $chi$SF for a complete basis of $Delta F = 2$ parity-odd four-fermion operators. We compute the corresponding scale-dependent renormalization constants to one-loop order in perturbation theory 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 scaling functions (SSF) of these operators approach their continuum limit with $O(a^{2})$ corrections without the need of operator improvement.