A Lattice Calculation of Parton Distributions

TL;DR

The paper tackles obtaining the $x$-dependent parton distributions $q(x)$ directly from lattice QCD by employing Ji's quasidistribution framework. It implements maximally twisted mass fermions on a $N_f=2+1+1$ ensemble and analyzes boosted nucleon matrix elements to form $\tilde{q}(x,\Lambda,P_3)$, which are then related to the light-cone PDF $q(x,\mu_R)$ through perturbative matching at ${cal O}(\alpha_s)$ and target-mass corrections via the Nachtmann variable $\xi$. The study finds a sizable imaginary part of the matrix elements that generates a quark–antiquark asymmetry, shows that Wilson-line gauge-link smearing enhances this effect and affects renormalization considerations, and demonstrates qualitative agreement with phenomenology when extrapolating toward larger $P_3$ or via mixed-momentum analyses. The work represents an important step toward ab initio, $x$-dependent PDFs in lattice QCD, while highlighting the need for nonperturbative renormalization, higher momenta, and physical-pion-mass simulations.

Abstract

We report on our exploratory study for the direct evaluation of the parton distribution functions from lattice QCD, based on a recently proposed new approach. We present encouraging results using Nf = 2 + 1 + 1 twisted mass fermions with a pion mass of about 370 MeV. The focus of this work is a detailed description of the computation, including the lattice calculation, the matching to an infinite momentum and the nucleon mass correction. In addition, we test the effect of gauge link smearing in the operator to estimate the influence of the Wilson line renormalization, which is yet to be done.

Figures (6)

• Figure 1: Schematic picture of a possible Wick contraction of the quark fields in the three-point function.
• Figure 2: We show the results for the unrenormalized matrix elements for different source-sink separations $t_s$, left: $P_3 = 2\pi/L$, right: $P_3 = 4\pi/L$.
• Figure 3: Results for the unrenormalized matrix elements with different steps of HYP smearing, left: real part, right: imaginary part, from top to bottom: $P_3=2\pi/L,4\pi/L,6\pi/L$.
• Figure 4: Comparison of results for $\tilde{q}$ obtained with five, two and no steps of HYP smearing, $P_3=4\pi/L$.
• Figure 5: The resulting quasidistribution $\tilde{q}$, PDF without subtracting the mass correction $q$, and final PDF $q^{(0)}$, left: $P_3=4\pi/L$, right: $P_3=6\pi/L$, from top to bottom: 0 steps, 2 steps, 5 steps of HYP smearing, negative region: $\bar{q}(x)=-q(-x)$, comparison with phenomenological $u(x)-d(x)$ curves at $Q^2=6.25\,\text{MeV}^2$ (MSTW Martin:2009iq, CJ12 Owens:2012bv, ABM11 Alekhin:2012ig).