[Paper Review] Parton distributions from lattice QCD

William Detmold, Wally Melnitchouk+1 more|arXiv (Cornell University)|Aug 2, 2001

Quantum Chromodynamics and Particle Interactions1 references4 citations

TL;DR

This paper presents a method to reconstruct the $x$-dependence of the valence nonsinglet $u-d$ parton distribution function (PDF) in the nucleon from lattice QCD calculations of low moments, using a chiral extrapolation formula that ensures correct behavior in both the chiral and heavy quark limits. The key result is a reconstructed PDF that agrees well with phenomenological parameterizations and reveals a connection between the small-$x$ behavior of valence quarks and the $m_q$-dependence of $1^{--}$ Regge trajectory meson masses.

ABSTRACT

We extract the x dependence of the nonsinglet u-d distribution function in the nucleon from the lowest few moments calculated on the lattice, using an extrapolation formula which ensures the correct behavior in the chiral and heavy quark limits. We discuss the implications for the quark mass dependence of meson masses lying on the J(PC)=1(--) Regge trajectory.

Motivation & Objective

• To determine the $x$-dependence of the valence $u-d$ parton distribution function from lattice QCD data on low moments.
• To address the longstanding discrepancy between lattice QCD results and experimental PDFs by incorporating non-linear, non-analytic chiral extrapolation.
• To explore the implications of the reconstructed PDF for the quark mass dependence of meson masses on the $J^{PC}=1^{--}$ Regge trajectory.
• To validate the method by comparing predictions for excited meson masses with existing lattice data.

Proposed method

• Uses the lowest three to four moments of the $u-d$ distribution function calculated in lattice QCD simulations at unphysical quark masses.
• Applies a chiral extrapolation formula that includes non-analytic, non-linear dependence on the pion mass to ensure correct behavior in both the chiral and heavy quark limits.
• Performs an inverse Mellin transform to reconstruct the $x$-dependence of the PDF from the extrapolated moments.
• Compares the reconstructed PDF with phenomenological parameterizations to validate the method and assess the impact of different extrapolation forms.
• Uses the reconstructed PDF to predict the $m_q$-dependence of the Regge trajectory parameters ($eta$ and $eta_0$) for $1^{--}$ mesons.
• Compares predictions for the masses of excited $a_2(2^{++})$ and $ ho_3(3^{--})$ mesons with existing lattice data from the UKQCD Collaboration.

Experimental results

Research questions

• RQ1 Can the $x$-dependence of the valence $u-d$ PDF be reliably reconstructed from only the first few lattice QCD moments?
• RQ2 How does using a non-linear chiral extrapolation formula improve agreement with phenomenological PDFs compared to linear extrapolation?
• RQ3 What is the connection between the small-$x$ behavior of the valence PDF and the quark mass dependence of meson masses on the $1^{--}$ Regge trajectory?
• RQ4 Are the predicted masses of excited $1^{--}$ mesons consistent with existing lattice data on $a_2$ and $ ho_3$ states?

Key findings

• The reconstructed $x$-dependence of $x(u_v - d_v)$ using the improved chiral extrapolation shows good agreement with phenomenological PDF parameterizations, particularly in the small-$x$ region.
• Linear extrapolation of lattice moments leads to unphysical behavior at small $x$, producing a spurious peak at $x \sim 1/3$, indicative of a heavy quark-like distribution.
• The chiral extrapolation formula successfully reproduces the correct small-$x$ behavior and ensures consistency with the heavy quark limit, where the distribution approaches a $δ$-function.
• The analysis predicts that the slope of the $1^{--}$ Regge trajectory increases with increasing quark mass, consistent with expectations from the heavy quark limit.
• Predicted masses for the $ ho_3(3^{--})$ and $a_2(2^{++})$ mesons are in reasonable agreement with existing lattice data, though with large uncertainties.
• The results suggest a quantitative link between the small-$x$ structure of valence quarks and the $m_q$-dependence of meson masses on the $1^{--}$ trajectory, which can be tested with future lattice calculations of excited hadron spectra.