Updated Lattice Results for Parton Distributions

TL;DR

This work advances lattice QCD determinations of parton distributions by computing unpolarized, helicity, and transversity iso-vector PDFs via quasi-distributions with spatial Wilson lines. Momentum smearing dramatically enhances signal quality, enabling higher nucleon momenta (up to P3 = 10π/L) and more reliable matching to light-cone PDFs, while HYP smearing helps mitigate divergences and renormalization effects. The results show qualitative agreement with phenomenological PDFs for unpolarized and helicity channels and demonstrate a nontrivial quark–antiquark asymmetry intrinsic to QCD; transversity is explored with promising, though less constrained, outcomes. The study highlights ongoing needs for nonperturbative renormalization and simulations at the physical pion mass to achieve quantitative alignment with experimental data, outlining a clear path for future refinements.

Abstract

We provide an analysis of the x-dependence of the bare unpolarized, helicity and transversity iso-vector parton distribution functions (PDFs) from lattice calculations employing (maximally) twisted mass fermions. The x-dependence of the calculated PDFs resembles the one of the phenomenological parameterizations, a feature that makes this approach very promising. Furthermore, we apply momentum smearing for the relevant matrix elements to compute the lattice PDFs and find a large improvement factor when compared to conventional Gaussian smearing. This allows us to extend the lattice computation of the distributions to higher values of the nucleon momentum, which is essential for the prospects of a reliable extraction of the PDFs in the future.

Figures (13)

• Figure 1: Free quark distributions for a lattice of size $48^3\times96$. As expected, the distributions tend to a Dirac delta at $x=1/3$ as the nucleon momentum grows.
• Figure 2: Real (left) and imaginary (right) parts of the matrix elements for the case of the vector operator.
• Figure 3: Real (left) and imaginary (right) parts of the matrix elements for the case of the axial-vector operator.
• Figure 4: Comparison of two different methods for the smearing of the quark fields for the computation of the matrix element $h^{u - d}(P_3,z)$. One is the standard Gaussian smearing, where 30000 measurements were used. The other is the new momentum smearing, where similar results to those of Gaussian smearing are achieved using only 150 measurements, for the case of momentum $P_3 = 6\pi/L$. In this plot, the real (imaginary) parts of the matrix elements were symmetrized (antisymmetrized).
• Figure 5: Energy of the nucleon as a function of the momentum boost $p\equiv P_3$. We fit the relativistic relation between the energy and the momentum, $E^2=m^2c^4 + p^2c^2$ and find the speed of light $c^2=1.046(81)$ and the (squared) nucleon mass in lattice units $a^2m^2c^4=0.267(23)$.