A guide to light-cone PDFs from Lattice QCD: an overview of approaches, techniques and results

Abstract: Within the theory of Quantum Chromodynamics (QCD), the rich structure of hadrons can be quantitatively characterized, among others, using a basis of universal non-perturbative functions: parton distribution functions (PDFs), generalized parton distributions (GPDs), transverse-momentum dependent distributions (TMDs) and distribution amplitudes (DAs). For more than half a century, there has been a joint experimental and theoretical effort to obtain these partonic functions. However, the complexity of the strong interactions has placed severe limitations, and first-principle results on the distributions was extracted mostly from their moments computed in Lattice QCD. Recently, breakthrough ideas changed the landscape and several approaches were proposed to access the distributions themselves on the lattice. In this paper, we review in considerable detail approaches directly related to partonic distributions. We highlight a recent idea proposed by X. Ji on extracting quasi-distributions that spawned renewed interest in the whole field and sparked the largest amount of numerical studies of Lattice QCD. We discuss theoretical and practical developments, including challenges that had to be overcome, with some yet to be handled. We also review numerical results, including a discussion based on evolving understanding of the underlying concepts and theoretical and practical progress. Particular attention is given to important aspects that validated the quasi-distribution approach, such as renormalization, matching to light-cone distributions and lattice techniques. In addition to a thorough discussion of quasi-distributions, we consider other approaches: hadronic tensor, auxiliary quark methods, pseudo-PDFs, OPE without OPE and good lattice cross sections. In closing, we provide prospects of the field, with emphasis on the necessary conditions to obtain results with controlled uncertainties.

Overview of Light-Cone PDFs from Lattice QCD

The paper explores the methodologies for extracting parton distribution functions (PDFs) from Lattice Quantum Chromodynamics (LQCD), an approach pivotal for understanding hadron structure at a fundamental level. Quantum Chromodynamics (QCD) has allowed us to frame hadrons—particles composed of quarks and gluons—via non-perturbative functions like PDFs, generalized parton distributions (GPDs), transverse-momentum-dependent parton distributions (TMDs), and distribution amplitudes (DAs). Over the decades, both experimental and theoretical frameworks have evolved to characterize these distributions, but the complexity of strong interactions has historically limited direct characterization on the lattice, confining efforts mostly to calculations of moments. The advent of innovative techniques such as the quasi-distribution approach presents an opportunity to circumvent these limitations, aiming to offer comprehensive insights into the intrinsic dynamics within hadrons.

Methodological Advances and Results

The paper provides a detailed review of key theoretical advances and methodological innovations pertinent to distribution functions. Noteworthy is the introduction of quasi-distributions by X. Ji, which has revitalized the field and spawned an array of numerical studies in Lattice QCD. This concept involves boosting hadrons to high momentum, and aligning correlators along spatial directions—approaching light-like correlations—which become feasible on a Euclidean lattice setup. Specifically, the quasi-distribution functions (quasi-PDFs) have proved integral, offering potential to match lattice calculations to their light-cone counterparts through a perturbatively computable convolution framework.

The authors discuss the pioneering work from various groups that undertook exploratory lattice studies, showcasing significant numerical investigations. These include adopting various tactics to offset systematic errors like those arising from finite lattice spacing and volumes or heavy pion masses compared to the physical scenario. Experiments engaged Gaussian and momentum smearing techniques, and ses systematic lattice studies clarified viable paths forward, delineating necessary conditions for further explorations.

Key Insights and Challenges

Critical insights offered include:
- Renormalization challenges: Non-local operators in quasi-PDFs encapsulate a spectrum of divergences, notably linear UV divergences attributed to the Wilson line artifacts. The paper delineates strategies validating renormalizability of the operators, affording manifold methodologies both perturbative and non-perturbative.
- Numerical techniques: The report elaborates on numerical strategies to mitigate excited states contamination and enhance lattice outputs through refined smearing methods, arriving at matrices amenable to high momentum transfers.
- Matching frameworks: Pivotal to the findings are developments within Large Momentum Effective Theory (LaMET), ensuring that lattice quasi-distributions may be factorizably matched and transition to light-cone PDFs through rigorous perturbative methods documented for both non-singlet and singlet QCD cases.

Future Research Directions

Looking ahead, the authors anticipate extensive computational efforts to address remaining challenges in lattice studies. These include achieving higher precision calculations, reducing model dependence, accommodating realistic pion masses, and solving distribution extrapolations from intermediate and large momentum scales. The explorations extend beyond quasi-distributions to incorporate burgeoning methodologies that either build upon or differ from traditional lattice representations, such as pseudo-distributions, offering comparably promising yet distinct frameworks.

Indeed, the prospects for advancing hadron structure knowledge are entwined with lattice computational increases and theoretical expansions. The paper propels inquisitive archivists of QCD to harness the advancing techniques, anchoring future investigations towards increasingly precise depictions of fundamental particle interactions.

Overall, the paper represents a milestone synthesis of the recent developments. It scrutinizes the nexus of theoretical foundations, computational rigor, and collaborative explorations in Lattice QCD, advancing towards the meticulous characterization of hadron structures—integral to comprehending the very fabric of quantum chromodynamics and nuclear interactions.