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Quasi-parton distribution functions: a study in the diquark spectator model

Leonard Gamberg, Zhong-Bo Kang, Ivan Vitev, Hongxi Xing

Abstract

A set of quasi-parton distribution functions (quasi-PDFs) have been recently proposed by Ji. Defined as the matrix elements of equal-time spatial correlations, they can be computed on the lattice and should reduce to the standard PDFs when the proton momentum $P_z$ is very large. Since taking the $P_z\to \infty$ limit is not feasible in lattice simulations, it is essential to provide guidance for what values of $P_z$ the quasi-PDFs are good approximations of standard PDFs. Within the framework of the spectator diquark model, we evaluate both the up and down quarks' quasi-PDFs and standard PDFs for all leading-twist distributions (unpolarized distribution $f_1$, helicity distribution $g_1$, and transversity distribution $h_1$). We find that, for intermediate parton momentum fractions $x$, quasi-PDFs are good approximations to standard PDFs (within $20-30%$) when $P_z\gtrsim 1.5-2$ GeV. On the other hand, for large $x\sim 1$ much larger $P_z > 4$ GeV is necessary to obtain a satisfactory agreement between the two sets. We further test the Soffer positivity bound, and find that it does not hold in general for quasi-PDFs.

Quasi-parton distribution functions: a study in the diquark spectator model

Abstract

A set of quasi-parton distribution functions (quasi-PDFs) have been recently proposed by Ji. Defined as the matrix elements of equal-time spatial correlations, they can be computed on the lattice and should reduce to the standard PDFs when the proton momentum is very large. Since taking the limit is not feasible in lattice simulations, it is essential to provide guidance for what values of the quasi-PDFs are good approximations of standard PDFs. Within the framework of the spectator diquark model, we evaluate both the up and down quarks' quasi-PDFs and standard PDFs for all leading-twist distributions (unpolarized distribution , helicity distribution , and transversity distribution ). We find that, for intermediate parton momentum fractions , quasi-PDFs are good approximations to standard PDFs (within ) when GeV. On the other hand, for large much larger GeV is necessary to obtain a satisfactory agreement between the two sets. We further test the Soffer positivity bound, and find that it does not hold in general for quasi-PDFs.

Paper Structure

This paper contains 11 sections, 20 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Standard PDFs and Quasi-PDFs: overview and definitions
  3. Standard PDFs and Quasi-PDFs in spectator diquark model
  4. The spectator diquark model
  5. Standard PDFs and quasi-PDFs in the diquark model
  6. Unpolarized distributions: $f_1$ and $\tilde{f}_1$
  7. Helicity distributions: $g_1$ and $\tilde{g}_1$
  8. Transversity distributions: $h_1$ and $\tilde{h}_1$
  9. Positivity bound: Soffer inequality
  10. Numerical studies for standard PDFs and Quasi-PDFs
  11. Summary

Figures (9)

  • Figure 1: The generic Feynman diagram representation for the leading-twist PDFs and the corresponding quasi-PDFs.
  • Figure 2: Feynman rules in the spectator diquark model: (a) vertex representing the interaction between the quark, the nucleon, and the diquark, (b) the diquark propagator.
  • Figure 3: The lowest order Feynman diagram for the leading-twist standard PDFs (or quasi-PDFs) in the spectator diquark model.
  • Figure 4: The unpolarized quasi-PDFs $x \tilde{f}_1(x, P_z)$ are plotted as a function of $x$ for $u$ (left) and $d$ (right) quark, respectively. Different lines are shown for $P_z=1$ GeV (purple), 2 GeV (green), 3 GeV (blue), and 4 GeV (red), respectively. The standard PDF $f_1(x)$ (black dashed) is also shown for comparison.
  • Figure 5: The helicity quasi-PDFs $x \tilde{g}_1(x, P_z)$ are plotted as a function of $x$ for $u$ (left) and $d$ (right) quark, respectively. Different lines are shown for $P_z=1$ GeV (purple), 2 GeV (green), 3 GeV (blue), and 4 GeV (red), respectively. The standard helicity distribution $g_1(x)$ (black dashed) is also shown for comparison.
  • ...and 4 more figures