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This paper shows that one can compute parton distribution functions (PDFs) directly in momentum space on a finite lattice. PDFs describe how quarks and gluons share a hadron’s momentum and are central inputs for many particle experiments. The authors use a version of the correlator defined in Coulomb gauge, which avoids a technical obstruction that normally forces lattice calculations to work in coordinate space and then invert a Fourier transform.
The key idea is that the Coulomb-gauge correlator does not include the long Wilson lines that produce a linear ultraviolet divergence. Because the renormalization (removing short-distance infinities) does not depend on the spatial separation, the discrete Fourier transform and renormalization commute. That makes a momentum-space operator on the finite lattice well defined, and removes the formal inverse problem that appears when one must reconstruct a smooth momentum distribution from a limited set of coordinate-space points.
To demonstrate the method the authors computed the pion valence quasi-PDF on a MILC gauge ensemble with 2+1+1 flavors of highly improved staggered quarks. The lattice size was 48^3×144 with spacing a≈0.06 fm and a valence pion mass of about 310 MeV. They used a Wilson-clover action for valence quarks, fixed the gauge to Coulomb gauge, applied HYP smearing, and measured boosted pion states with longitudinal momenta Pz≈2.2 and 2.6 GeV. The analysis used 100 gauge configurations with 16 sources each and techniques to control excited-state contamination.
Their momentum-space quasi-PDFs agree with results obtained by the traditional coordinate-space LaMET (Large Momentum Effective Theory) approach after that approach’s asymptotic extrapolation of long-distance data. The authors renormalized their quasi-PDFs in the MS (modified minimal subtraction) scheme and matched them to light-cone PDFs at next-to-leading order. They also extended the momentum-space construction to three dimensions and report the first direct 3D image of the pion obtained from lattice QCD.
Important caveats remain. The calculation uses a heavier-than-physical pion (mπ≈310 MeV), a single lattice spacing and volume, and a finite statistical sample, so further study is needed to quantify discretization, volume and mass dependence. Although significant progress has been made toward proving renormalizability of Coulomb-gauge observables, the formal proof is still under active work in the literature. For transverse-momentum-dependent distributions a soft-function subtraction is still required to connect to continuum factorization formulas. The authors estimate finite-volume corrections are small here (their lattice has mπL>4), but systematic uncertainties must be reduced before precision phenomenology can be claimed.