Nucleon Parton Distribution Functions from Boosted Correlations in the Coulomb gauge

Abstract

Recently, a novel approach has been proposed to compute parton distributions through the use of boosted correlators fixed in the Coulomb gauge from lattice QCD, within the framework of Large-Momentum Effective Theory (LaMET). This approach circumvents the need for Wilson lines, potentially enhancing the efficiency and accuracy of lattice calculations. In this work, we present the first exploratory implementation of the Coulomb gauge method for calculating nucleon unpolarized, helicity, and transversity parton distribution functions (PDFs). The calculations are performed on a Highly-Improved-Staggered-Quark ensemble with lattice spacing $a = 0.06$ fm, volume $L_s^3 \times L_t=48^3\times 64$, and valence pion mass $m_π=300$ MeV, employing boosted nucleon states with momenta up to 3.04 GeV. Our lattice predictions for the valence-quark PDFs -- extracted from the real part of the correlators -- show good convergence with increasing nucleon momentum and are compatible with the most recent global analyses for all spin structures. On the other hand, the full-quark-channel PDFs obtained from the imaginary part of the correlators exhibit discrepancies between the two large nucleon momenta considered, although the results at the higher momentum are consistent with phenomenology. The discrepancies are likely driven by stronger excited-state contamination in the imaginary matrix elements, which is consistent with the observation in the literature. Overall, this work demonstrates the efficacy of the Coulomb gauge approach for nucleon PDFs and serves as a benchmark for its broader applications.

Figures (19)

• Figure 1: Effective mass and dispersion relation of the nucleon. The left panel shows the effective masses extracted from the two-point correlators as functions of the source-sink separation $t_{\mathrm{sep}}$ for different hadron momenta. The dashed horizontal lines correspond to the energies computed from the relativistic dispersion relation $m_{\rm eff} = \sqrt{m_0^2 + (P^z)^2}$ with the static mass fixed to $m_0 = 1.1$ GeV. The right panel displays the ground-state energies obtained from two-state fits as functions of the hadron momentum, along with the fit to the dispersion relation $E = \sqrt{m_0^2 + c_1 P^2 + c_2 a^2 P^4}$. The shaded band denotes the fit uncertainty, and the fitted values of the parameters are indicated in the figure.
• Figure 2: Statistical summary of the ground-state fits. The left panel shows the density distribution of $\chi^2/{\rm d.o.f.}$ for the unpolarized, helicity, and transversity quasi-PDFs (qPDF), as well as the combined distribution. The right panel displays the cumulative distribution function (CDF) of the corresponding p-values.
• Figure 3: Examples of ground-state fits to the matrix elements of the unpolarized, helicity, and transversity quasi-PDFs at $z = 6\sqrt{2}~a$ and $P^z = 2.43$ GeV. The three columns from left to right correspond to the unpolarized, helicity, and transversity channels, respectively. The upper (lower) row shows the real (imaginary) parts of the matrix elements. The error bars denote the ratio data R. The colored bands represent the results of the joint fit over multiple source-sink separations, while the hatched gray bands indicate the joint-fit results and the solid gray bands show the corresponding ratio-only fit for comparison.
• Figure 4: Bare matrix elements of the unpolarized, helicity, and transversity quasi-PDFs in coordinate space. The three columns from left to right correspond to the unpolarized, helicity, and transversity channels, respectively. The upper (lower) row shows the real (imaginary) parts of the matrix elements. Results at two hadron momenta, $P^z=2.43$ GeV and $3.04$ GeV, are shown for comparison.
• Figure 5: Extrapolation of the real parts of the renormalized quasi-PDF matrix elements in coordinate space as functions of $\lambda = z P^z$. The three columns from left to right correspond to the unpolarized, helicity, and transversity channels, while the upper (lower) row shows the results at $P^z = 2.43$ GeV ($P^z = 3.04$ GeV). The data points are shown with error bars, and the shaded bands represent the extrapolation using the asymptotic form. The two red dashed vertical lines indicate the sub-asymptotic region used in the fit.