  We study the renormalization of four-quark operators in one-loop perturbation
theory. We employ a coordinate-space Gauge-Invariant Renormalization Scheme
(GIRS), which can be advantageous compared to other schemes, especially in
nonperturbative lattice investigations. From our perturbative calculations, we
extract the conversion factors between GIRS and the modified Minimal
Subtraction scheme ($\overline{\rm MS}$) at the next-to-leading order. A
formidable issue in the study of the four-quark operators is that operators
with different Dirac matrices mix among themselves upon renormalization. We
focus on both parity-conserving and parity-violating four-quark operators,
which change flavor numbers by two units ($\Delta F = 2$). The extraction of
the conversion factors entails the calculation of two-point Green's functions
involving products of two four-quark operators, as well as three-point Green's
functions with one four-quark and two bilinear operators. The significance of
our results lies in their potential to refine our understanding of QCD
phenomena, offering insights into the precision of Cabibbo-Kobayashi-Maskawa
(CKM) matrix elements and shedding light on the nonperturbative treatment of
complex mixing patterns associated with four-quark operators.
