Toward precise $\xi$ gauge fixing for the lattice QCD

Lattice QCD provides a first-principles framework for solving Quantum Chromodynamics (QCD). However, its application to off-shell partons has been largely restricted to the Landau gauge, as achieving high-precision $\xi$-gauge fixing on the lattice poses significant challenges. Motivated by a universal power-law dependence of off-shell parton matrix elements on gauge-fixing precision in the Landau gauge, we propose an empirical precision extrapolation method to approximate high-precision $\xi$-gauge fixing. By properly defining the bare gauge coupling and then the effective $\xi$, we validate our $\xi$-gauge fixing procedure by successfully reproducing the $\xi$-dependent RI/MOM renormalization constants for local quark bilinear operators at 0.3\% level, up to $\xi \sim 1$.