Perturbative renormalization of DF = 2 four-fermion operators with the chirally rotated Schrödinger functional

The chirally rotated Schrödinger functional (xSF) renders the mechanism of automatic O(a) improvement compatible with Schrödinger functional (SF) renormalization schemes. Here we define a family of renormalization schemes based on the xSF for a complete basis of ΔF = 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 MS scheme. Due to automatic O(a) improvement, once the xSF is renormalized and improved at the boundaries, the step scaling functions (SSF) of these operators approach their continuum limit with O(a2) corrections without the need of operator improvement.