A perturbative study of two four-quark operators in finite volume renormalization schemes

TL;DR

This paper develops a family of renormalization schemes for two four-quark operators in finite volume QCD, relating them to continuum schemes and calculating anomalous dimensions and cutoff effects relevant for lattice QCD studies.

Contribution

It introduces new SF renormalization schemes for four-quark operators, computes their one-loop anomalous dimensions, and assesses cutoff effects for improved and unimproved Wilson quarks.

Findings

Nine SF schemes related to continuum schemes at one-loop order.

Two-loop anomalous dimensions inferred for the SF schemes.

One-loop cutoff effects characterized for Wilson quarks.

Abstract

Starting from the QCD Schroedinger functional (SF), we define a family of renormalization schemes for two four-quark operators, which are, in the chiral limit, protected against mixing with other operators. With the appropriate flavour assignments these operators can be interpreted as part of either the $\Delta F=1$ or $\Delta F=2$ effective weak Hamiltonians. In view of lattice QCD with Wilson-type quarks, we focus on the parity odd components of the operators, since these are multiplicatively renormalized both on the lattice and in continuum schemes. We consider 9 different SF schemes and relate them to commonly used continuum schemes at one-loop order of perturbation theory. In this way the two-loop anomalous dimensions in the SF schemes can be inferred. As a by-product of our calculation we also obtain the one-loop cutoff effects in the step-scaling functions of the respective renormalization constants, for both O(a) improved and unimproved Wilson quarks. Our results will be needed in a separate study of the non-perturbative scale evolution of these operators.