Can Coulomb-gauge boosted correlators compute nucleon parton distribution functions?

What the Coulomb-gauge method removes from the calculation

The paper tests a variant of lattice QCD calculations of parton distribution functions (PDFs), the one-dimensional momentum densities of quarks and gluons inside a hadron. In the standard Large-Momentum Effective Theory (LaMET) approach, a lattice correlator is made gauge invariant by connecting the quark fields with a Wilson line, a path-ordered gauge link. That link brings a linear ultraviolet divergence and an associated renormalon, which require dedicated nonperturbative renormalization such as hybrid renormalization and leading-renormalon resummation.

The Coulomb-gauge (CG) proposal instead fixes each gauge configuration to the Coulomb gauge and uses boosted correlators without any Wilson line. The authors state that this circumvents the need for Wilson lines and could improve efficiency and accuracy. The trade-off is that the operator is no longer manifestly gauge invariant, so the renormalization and matching structure must be rederived; the paper does this at one loop and supplies matching kernels for the unpolarized, helicity, and transversity channels.

One ensemble, three spin channels, momenta above 3 GeV

The calculation uses a 2+1 flavor Highly-Improved-Staggered-Quark ensemble with lattice spacing a = 0.0601(5) fm and volume 48^3 × 64. Valence quarks are tadpole-improved clover Wilson fermions on a HYP-smeared background, with a valence pion mass of 300 MeV. The gauge fixing minimizes a functional of the spatial link variables on each time slice.

Nucleon states are boosted with off-axis momenta of 2.43 GeV and 3.04 GeV, and the paper reports statistics for source-sink separations t_s/a from 8 to 12. The authors describe this as the first lattice QCD calculation of isovector nucleon PDFs using the CG method across all three polarization channels. Because the ensemble has a single lattice spacing and an unphysical pion mass, no continuum or chiral extrapolation is performed; the authors flag that comparisons with phenomenology should therefore be read with caution.

Real and imaginary parts probe different quark channels

A reader could easily conflate the two halves of the correlator. In this work the real part of the renormalized matrix elements yields the valence combination (q − q̄)/2, while the imaginary part yields the full combination (q + q̄)/2. The two are renormalized in different schemes: the real part uses a hybrid scheme, and the imaginary part uses an SRI′-MOM scheme, giving Z_MS(a, μ = 2 GeV) = 0.914(5).

This split matters for interpreting every result that follows. The valence channel and the full channel are not two estimates of the same quantity, and their agreement with phenomenology must be judged separately. The helicity case reverses which part supplies which combination, because the helicity quasi-distribution is even under charge conjugation; the paper notes that the full helicity distribution comes from the real part.

Valence results converge with momentum and match global fits

The central positive result concerns the valence-quark PDFs extracted from the real part. The authors report good convergence as the nucleon momentum increases, and compatibility with the most recent global analyses for all spin structures. At P_z = 3.04 GeV, the unpolarized lattice results agree well with the NNLO NNPDF4.0 determination for both the valence and full components.

For helicity, the P_z = 3.04 GeV results are consistent within one standard deviation with the phenomenological determinations shown, which include NNPDFpol1.1 at NLO, NNPDFpol2.0 at NNLO, BDSSV24, MAPPDFpol1.0, and JAMpol25. The lattice helicity results are normalized using the axial charge g_A from JAMpol25, and the phenomenological bands are normalized by their own axial charges. Transversity is compared with the LO JAM3D-22 extraction, normalized by its tensor charge g_T.

Two perturbative checks support the reading. Comparing NLO matching with NLL accuracy including DGLAP evolution indicates stable perturbative behavior. Separately, results obtained with different Dirac structures Γ̃ move closer together when the momentum rises from 2.43 to 3.04 GeV, which the authors interpret as suppression of power corrections, consistent with the large-momentum expansion.

The imaginary channel shows momentum-dependent discrepancies

The full-quark-channel PDFs from the imaginary part behave less cleanly. The paper reports discrepancies between the two large nucleon momenta considered, although the results at the higher momentum, 3.04 GeV, are consistent with phenomenology. The same pattern appears in the unpolarized and helicity channels: the difference between the two momenta is more pronounced in the contribution from the imaginary part than from the real part.

The authors attribute this to stronger excited-state contamination in the imaginary matrix elements and note that this is consistent with observations in the literature. This is the paper's interpretation rather than a directly isolated measurement; the analysis states that discretization effects and excited-state contamination are not disentangled, so a more detailed quantitative comparison between renormalization schemes is left for future studies. The practical consequence is that the imaginary sector, not the real sector, is where the CG method currently needs more careful treatment.

Where perturbative control runs out in x

The comparison with phenomenology is only meaningful where the matching expansion is under control. The paper shades kinematic domains where the strong coupling evaluated at NLO exceeds 0.5, namely x < 0.18 or x > 0.82, and cautions that lattice results there should be interpreted with care. Small-x logarithms become significant because the relevant scale is 2xP_z, and threshold logarithms matter as x approaches 1; the authors estimate the reachable x range by requiring α_s(2(1−x_max)P_z) ∼ 1 and postpone threshold resummation.

A second boundary sits in the renormalization window. The SRI′-MOM determination uses 2 GeV ≤ |k⃗| ≤ 6 GeV, with the central value from a correlated average over those points; below that window perturbative and nonperturbative errors grow, and above it lattice artifacts appear, since a·(6 GeV) ≈ 1.9. The flatness of the renormalization factor across this window is presented as evidence that these systematics are under control.

What this benchmark establishes and what it does not

The paper's own conclusion is that the CG method is a promising tool for precision-controlled nucleon parton distributions, and that this work serves as a benchmark for broader applications. The evidence supporting that claim is concentrated in the real-part, valence-channel results at the higher momentum, where convergence with momentum and agreement with global fits appear together. The imaginary sector supplies the caution: agreement at 3.04 GeV coexists with disagreement against 2.43 GeV, which points to residual excited-state contamination rather than a failure of the matching.

The authors also compare with gauge-invariant lattice calculations in an appendix and report broad consistency, with small deviations attributed mainly to power corrections. In the helicity case they observe that MS-scheme results agree better with existing determinations than hybrid-scheme results, while noting that discretization and excited-state effects are not separated. The forward path they describe is more lattice spacings for continuum extrapolation, higher statistics at large source-sink separations, refined excited-state treatment such as the Lanczos algorithm and current-inspired GEVP, and improved signal-to-noise techniques.

For a reader deciding how much weight to place on this method, the discriminating comparison is not CG versus phenomenology in general but real versus imaginary channel at fixed momentum. The real-part valence results at 3.04 GeV are the ones that currently carry the case for the Coulomb-gauge approach; the imaginary-part full distributions are the ones that must improve before that case extends to all quark channels.

Sources used in this answer