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

TL;DR

This paper develops a perturbative renormalization scheme for $ riangle F=2$ four-fermion operators using the chirally rotated Schr"odinger functional, enabling more accurate continuum limit extrapolations with automatic $O(a)$ improvement.

Contribution

It introduces a family of $ ext{ extchi}$SF-based renormalization schemes for $ riangle F=2$ operators and computes their one-loop renormalization constants and anomalous dimensions.

Findings

Renormalization constants computed to one-loop order.

NLO anomalous dimensions obtained via scheme matching.

Step scaling functions approach continuum limit with $O(a^{2})$ corrections.

Abstract

The chirally rotated Schr\"odinger functional ($\chi$SF) renders the mechanism of automatic $O(a)$ improvement compatible with Schr\"odinger 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.