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Starting from the QCD Schroedinger functional (SF), we define a family ofrenormalization schemes for two four-quark operators, which are, in the chirallimit, protected against mixing with other operators. With the appropriateflavour assignments these operators can be interpreted as part of either the$\Delta F=1$ or $\Delta F=2$ effective weak Hamiltonians. In view of latticeQCD with Wilson-type quarks, we focus on the parity odd components of theoperators, since these are multiplicatively renormalized both on the latticeand in continuum schemes. We consider 9 different SF schemes and relate them tocommonly used continuum schemes at one-loop order of perturbation theory. Inthis 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 effectsin the step-scaling functions of the respective renormalization constants, forboth O(a) improved and unimproved Wilson quarks. Our results will be needed ina separate study of the non-perturbative scale evolution of these operators.Comment: 22 pages, 4 figure

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