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), transversemomentum dependent parton 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 information on these 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 within 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 the 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-distributions, OPE without OPE and good lattice cross sections. In the last part of the paper, we provide a summary and prospects of the field, with emphasis on the necessary conditions to obtain results with controlled uncertainties.

Pion valence structure from Ioffe-time parton pseudodistribution functions

“…Although significant achievements in the lattice QCD implementations of these approaches have been made in recent years [37][38][39][40][41][42][43][44][45][46][47][48], a proper understanding and control-ling various sources of systematics in these calculations still require further exploration and theoretical development. The status of current lattice QCD calculations of the x-dependent hadronic structure can be found in the following review articles [49][50][51]. Recently, an attempt to incorporate lattice QCD determination of PDFs together with experimental data to obtain the non-singlet quark distribution of the nucleon has been discussed in [52].…”

Parton distribution functions from Ioffe time pseudo-distributions

“…Other Good Lattice Cross Sections have been calculated to extract the pion DA [36,37] and the pion PDF [38]. We refer the reader to [39][40][41][42] for detailed reviews of these topics. Possible difficulties with these approaches were raised in [43,44].…”

Parton distributions from lattice data: the nonsinglet case

“…Following these novel ideas, a number of publications have appeared (see Refs. [8,9] for recent reviews), from both lattice and high-energy communities, addressing the main theoretical problems of the new lattice approach: the definition and renormalization of the non-local operators involved in the lattice simulation [10][11][12][13][14][15][16][17][18][19][20], the proof of the factorization theorem between PDFs and quasi-PDFs [7,[21][22][23][24][25], the computation of the matching coefficients relating latticecomputable quantities to PDFs in different renormalization schemes [21,22,[24][25][26][27][28][29][30][31][32][33]. Also, data coming from first lattice QCD simulations have started appearing and have gotten into already a relatively advanced stage over the last few years [13,[30][31][32][33][34][35][36][37][38][39][40][41][42]…”