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

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.

22 pages, 4 figures

References in corpus (4)

• Computation of the strong coupling in QCD with two dynamical flavours
• Non-perturbative renormalization of static-light four-fermion operators in quenched lattice QCD
• Precision computation of B_K in quenched lattice QCD
• NLO anomalous dimension of multiplicatively renormalizable four-fermion operators in Schroedinger Functional schemes

Cited by in corpus (8)

• Non-perturbative renormalization of static-light four-fermion operators in quenched lattice QCD
• Non-perturbative quark mass renormalisation and running in $N_f=3$ QCD
• A precise determination of $B_K$ in quenched QCD
• Non-perturbative renormalization of left-left four-fermion operators in quenched lattice QCD
• Non-perturbative renormalisation of Delta F=2 four-fermion operators in two-flavour QCD
• A strategy for implementing non-perturbative renormalisation of heavy-light four-quark operators in the static approximation
• Non-perturbative renormalization of tensor currents: strategy and results for $N_f=0$ and $N_f=2$ QCD
• Twisted mass QCD for weak matrix elements