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

Abstract

The chirally rotated Schrödinger functional ($\chi$SF) 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 $\chi$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 $\overline{\text{MS}}$ 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.

Comments: 7 pages, 2 figures