Quasi-PDFs, momentum distributions and pseudo-PDFs

Abstract: We show that quasi-PDFs may be treated as hybrids of PDFs and primordial rest-frame momentum distributions of partons. This results in a complicated convolution nature of quasi-PDFs that necessitates using large $p_3 \sim 3$ GeV momenta to get reasonably close to the PDF limit. As an alternative approach, we propose to use pseudo-PDFs $P(x, z_3^2)$ that generalize the light-front PDFs onto spacelike intervals and are related to Ioffe-time distributions $M (\nu, z_3^2)$, the functions of the Ioffe time $\nu = p_3 z_3$ and the distance parameter $z_3^2$ with respect to which it displays perturbative evolution for small $z_3$. In this form, one may divide out the $z_3^2$ dependence coming from the primordial rest-frame distribution and from the problematic factor due to lattice renormalization of the gauge link. The $\nu$-dependence remains intact and determines the shape of PDFs.

Insights into Quasi-PDFs and Pseudo-PDFs

The paper by A. V. Radyushkin titled "Quasi-PDFs, momentum distributions and pseudo-PDFs" offers an advanced discourse on the intricacies of parton distribution functions (PDFs), quasi-PDFs, and pseudo-PDFs, primarily in the context of quantum chromodynamics (QCD). This work builds upon X. Ji's method, transforming how PDFs are extracted from lattice QCD calculations, enabling a more direct computation as functions of the momentum fraction variable, $x$.

Quasi-PDFs: Dynamics and Limitations

Quasi-PDFs are introduced as hybrids of standard PDFs and primordial momentum distributions of partons in a hadron's rest frame. The necessity to employ large momenta, specifically $p_3 \gtrsim 3$ GeV, is emphasized to achieve an approximation near the PDF limit. This requirement arises due to quasi-PDFs' convolution nature involving both the partonic momentum and the intrinsic primordial rest-frame distribution. The dependence on transverse momentum distributions (TMDs) and their implications is methodically articulated. The quasi-PDF formalism accounts for motion-dependent sources of partonic momentum ($k_3$), dissecting it into contributions from the hadron's movement as a whole ($xp_3$) and its intrinsic rest-frame distribution ($k_3-xp_3$).

Pseudo-PDFs: An Alternative Approach

Radyushkin proposes pseudo-PDFs as a more refined alternative. They extend light-front PDFs to spacelike intervals and maintain the canonical $x$-support $[-1, 1]$, independent of $z_3^2$. This feature grants pseudo-PDFs certain advantages, notably in simplifying the extraction of PDFs from lattice QCD, avoiding some of the complications intrinsic to quasi-PDFs. The pseudo-PDFs relate Fourier-transformed matrix elements depending on both Ioffe time $\nu$ and spatial component $z_3^2$, exhibiting a simpler perturbative evolution. The extraction of PDFs through pseudo-PDFs involves forming reduced Ioffe-time distributions, where the problematic lattice gauge link renormalization effects and inherent rest-frame artifacts are systematically divided out.

Theoretical Implications and Future Directions

The research underscores important distinctions between the light-cone and quasi-evolution of PDFs in QCD. The pseudo-PDF approach potentially streamlines the lattice QCD procedures by minimizing lattice artifacts. This concept aligns with the need to resolve limitations imposed by spacetime separations constrained within Euclidean lattices. Furthermore, this framework opens potential avenues for accurately mapping the nuances of parton dynamics, suggesting that future lattice computations could effectively incorporate pseudo-PDFs to enhance the fidelity of parton distribution approximations.

Conclusion

Radyushkin's work contributes a significant reinterpretation of parton momentum distributions in terms of quasi- and pseudo-PDFs, underlining the nuanced convolution mechanisms inherent in these constructs. By doing so, it not only challenges prevailing methods but also offers a pragmatic solution set to improve computational strategies in lattice QCD. Advancements in numerical techniques and lattice simulations will undoubtedly benefit from the theoretical underpinnings articulated here, promising a more precise determination of parton distributions.

This paper does not claim to revolutionize the field but rather provides subtle yet impactful refinements that hold promise for more accurate lattice QCD methodologies. Researchers in high-energy physics and lattice QCD should find these explorations to be of practical importance, contributing to a deeper understanding of intrinsic hadronic structures and their distributions.