Off-lightcone Wilson-line operators in gradient flow at next-to-next-to-leading order

We consider gauge-invariant off-lightcone Wilson-line operators constructed from local fields connected by spacelike or timelike Wilson lines. The lattice evaluation of their matrix elements requires a suitable renormalization prescription. Gradient flow offers such a prescription and, by smoothing the fields, suppresses ultraviolet fluctuations and can reduce statistical noise. We use the previously established matching framework based on the one-dimensional auxiliary-field formalism. In the small-flow-time limit, this framework reduces the matching of nonlocal off-lightcone Wilson-line operators to that of local current operators. We calculate the matching coefficients for these local current operators from the gradient-flow scheme to the modified minimal subtraction ($\MS$) scheme through next-to-next-to-leading order (NNLO) in QCD. Our results can be applied to lattice calculations of unpolarized and polarized quark and gluon quasi-parton distributions, as well as meson and baryon quasi-distribution amplitudes. They also apply to gluonic correlators entering quarkonium transport and heavy-quark momentum diffusion in the quark-gluon plasma (QGP). Further applications include quarkonium decay and production in potential nonrelativistic QCD (pNRQCD) and spin-dependent relativistic corrections to the static potential.