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...

We present the first direct calculation of the transversity parton distribution function within the nucleon from lattice QCD. The calculation is performed using simulations with the light quark mass fixed to its physical value and at one value of the lattice spacing. Novel elements of the calculations are non-perturbative renormalization and extraction of a formula for the matching to light-cone PDFs. Final results are presented in the MS scheme at a scale of √ 2 GeV.arXiv:1807.00232v1 [hep-lat]