In the framework of quantum chromodynamics (QCD), parton distribution functions (PDFs) quantify how the momentum and spin of a hadron are divided among its quark and gluon constituents. Two main approaches exist to determine PDFs. The first approach, based on QCD factorization theorems, realizes a QCD analysis of a suitable set of hard-scattering measurements, often using a variety of hadronic observables. The second approach, based on first-principle operator definitions of PDFs, uses lattice QCD to compute directly some PDF-related quantities, such as their moments. Motivated by recent progress in both approaches, in this document we present an overview of lattice-QCD and globalanalysis techniques used to determine unpolarized and polarized proton PDFs and their moments. We provide benchmark numbers to validate present and future lattice-QCD calculations and we illustrate how they could be used to reduce the PDF uncertainties in current unpolarized and polarized global analyses. This document represents a first step towards establishing a common language between the two communities, to foster dialogue and to further improve our knowledge of PDFs.The detailed understanding of the inner structure of nucleons is an active research field with phenomenological implications in high-energy, hadron, nuclear and astroparticle physics. Within quantum chromodynamics (QCD), information on this structure -specifically on how the nucleon's momentum and spin are divided among quarks and gluons -is encoded in parton distribution functions (PDFs).There exist two main methods to determine PDFs. 1 The first method is the global QCD analysis [3][4][5][6][7][8][9][10][11][12]. It is based on QCD factorization of physical observables, i.e. the fact that a class of hard-scattering cross-sections can be expressed as a convolution between short-distance, perturbative, matrix elements and long-distance, nonperturbative, PDFs. By combining a variety of available hard-scattering experimental data with state-of-the-art perturbative calculations, complete PDF sets, including the gluon and various combinations of quark flavors, are currently determined for protons, in both the unpolarized [13][14][15][16][17] and the polarized [18][19][20][21] case.Recent progress in global QCD analyses has been driven, on the one hand, by the increasing availability of a wealth of high-precision measurements from Jefferson Lab, HERA, RHIC, the Tevatron and the LHC and, on the other hand, by the advancement in perturbative calculations of QCD and electroweak (EW) higher-order corrections. Parton distributions are now determined with unprecedented precision, in many cases at the few-percent level. A paradigmatic illustration of this progress is provided by both the unpolarized and polarized gluon PDFs, which were affected by rather large uncertainties until recently, due to the limited experimental information available. In the unpolarized case, the gluon PDF is now constrained quite accurately from small to large x thanks to the inclusion of processes such a...

We extract parton distribution functions (PDFs) of the nucleon from lattice QCD using an ensemble of gauge field configurations simulated with light quark masses fixed to their physical values. Theoretical and algorithmic improvements that allow such a calculation include momentum smearing to reach large nucleon boosts with reduced statistical errors, nonperturbative renormalization, target mass corrections, and a novel modified matching of lattice QCD results to connect to what is extracted from experimental measurements. We give results on the unpolarized and helicity PDFs in the modified minimal subtraction scheme at a scale of 2 GeV and reproduce the main features of the experimentally determined quantities, showing an overlap for a range of Bjorken-x values. This first direct nonperturbative evaluation opens a most promising path to compute PDFs in an ab initio way on the lattice and provides a framework for investigating also a wider class of similar quantities, which require the evaluation of hadronic matrix elements of nonlocal operators.

We present a detailed study of the helicity-dependent and helicity-independent collinear parton distribution functions (PDFs) of the nucleon, using the quasi-PDF approach. The lattice QCD computation is performed employing twisted mass fermions with a physical value of the light quark mass. We give a systematic and in-depth account of the salient features entering in the evaluation of quasi-PDFs and their relation to the light-cone PDFs. In particular, we give details for the computation of the matrix elements, including the study of the various sources of systematic uncertainties, such as excited states contamination. In addition, we discuss the non-perturbative renormalization scheme used here and its systematics, effects of truncating the Fourier transform and different matching prescriptions. Finally, we show improved results for the PDFs and discuss future directions, challenges and prospects for evaluating precisely PDFs from lattice QCD with fully quantified uncertainties. arXiv:1902.00587v1 [hep-lat] 1 Feb 2019of the theoretical and numerical developments, see Ref.[33].On more general grounds, it is very important to realize that quasi-PDFs and light-cone PDFs have been shown to share the same infrared physics [34,35], which is the fundamental observation that allows one to relate both quantities using perturbation theory, provided that the hadron is moving with a large, although necessarily finite, momentum in a chosen spatial direction. It has also been proven that quasi-PDFs can be extracted from lattice QCD in Euclidean spacetime [35] and that they do not suffer from power-divergent mixings with lower-dimensional operators [36][37][38]. A factorization formula makes it possible to extract the PDFs from the quasi-PDFs, an operation called matching [19,20,29,31,34,[39][40][41][42][43]. In general, this procedure is based on a newly developed large-momentum effective theory (LaMET) [44], and it is renormalizable to all orders in perturbation theory [45][46][47][48]. Other approaches for a direct computation of the x-dependence of PDFs include the hadronic tensor [49][50][51], fictitious heavy quark [52], auxiliary light quark [53], good lattice cross sections [54,55] (closely related to the auxiliary light quark method), "OPE without OPE" [56] and pseudo-PDFs [57][58][59][60], where the latter can be seen as a generalization of PDFs off the light-cone. These alternative approaches have been explored in lattice QCD, and recent results can be found in Refs. [38,56,[61][62][63][64][65][66][67]. The new formulation and its successful implementation within lattice QCD has also led to a wider interest on phenomenological studies using models and toy theories of QCD [57,58,[68][69][70][71][72][73][74][75]. A detailed overview of the current status of lattice QCD calculation of PDFs and other partonic distributions can be found in the recent reviews of Refs. [33,76].The remainder of the paper is organized as follows: In Sec. II, we provide the general theoretical aspects, lattice QCD action and parameters. We di...