Exploring Partonic Structure of Hadrons Using ab initio Lattice QCD Calculations

Nucleon Transversity Distribution in the Continuum and Physical Mass Limit from Lattice QCD

We report a state-of-the-art lattice QCD calculation of the isovector quark transversity distribution of the proton in the continuum and physical mass limit using large-momentum effective theory. The calculation is done at four lattice spacings a={0.098,0.085,0.064,0.049}  fm and various pion masses ranging between 220 and 350 MeV, with proton momenta up to 2.8 GeV. The result is nonperturbatively renormalized in the hybrid scheme with self-renormalization, which treats the infrared physics at large correlation distance properly, and extrapolated to the continuum, physical mass, and infinite momentum limit. We also compare with recent global analyses for the nucleon isovector quark transversity distribution.

Lattice QCD Determination of the Bjorken-x Dependence of Parton Distribution Functions at Next-to-Next-to-Leading Order

We report the first lattice QCD calculation of pion valence quark distribution with next-to-next-to-leading order perturbative matching correction, which is done using two fine lattices with spacings a=0.04 and 0.06 fm and valence pion mass m_{π}=300  MeV, at boost momentum as large as 2.42 GeV. As a crucial step to control the systematics, we renormalize the pion valence quasidistribution in the recently proposed hybrid scheme, which features a Wilson-line mass subtraction at large distances in coordinate space, and develop a procedure to match it to the MS[over ¯] scheme. We demonstrate that the renormalization and the perturbative matching in Bjorken-x space yield a reliable determination of the valence quark distribution for 0.03≲x≲0.80 with 5%-20% uncertainties.

Overview of lattice calculations of the x-dependence of PDFs, GPDs and TMDs

For a long time, lattice QCD was unable to address the x-dependence of partonic distributions, direct access to which is impossible in Euclidean spacetime. Recent years have brought a breakthrough for such calculations when it was realized that partonic light-cone correlations can be accessed through spatial correlations computable on the lattice. Appropriately devised observables can be factorized into physical PDFs via a perturbative procedure called matching, analogous to the standard factorization of experimental cross sections. In this short review, aimed at a broader high-energy and nuclear physics community, we discuss the recent highlights of this research program. Key concepts are outlined, followed by a case study illustrating the typical stage of current lattice extractions and by a brief review of the most recent explorations. We finalize with a number of messages for the prospects of lattice determinations of partonic structure.

Quark and Gluon Momentum Fractions in the Pion from N_{f}=2+1+1 Lattice QCD

We perform the first full decomposition of the pion momentum into its gluon and quark contributions. We employ an ensemble generated by the Extended Twisted Mass Collaboration with N_{f}=2+1+1 Wilson twisted mass clover fermions at maximal twist tuned to reproduce the physical pion mass. We present our results in the MS[over ¯] scheme at 2 GeV. We find ⟨x⟩_{u+d}=0.601(28), ⟨x⟩_{s}=0.059(13), ⟨x⟩_{c}=0.019(05), and ⟨x⟩_{g}=0.52(11) for the separate contributions, respectively, whose sum saturates the momentum sum rule.

Distribution Amplitudes of K^{*} and ϕ at the Physical Pion Mass from Lattice QCD

We present the first lattice QCD calculation of the distribution amplitudes of longitudinally and transversely polarized vector mesons K^{*} and ϕ using large momentum effective theory. We use the clover fermion action on three ensembles with 2+1+1 flavors of highly improved staggered quarks action, generated by the MIMD Lattice Computation Collaboration, at physical pion mass and {0.06,0.09,0.12}  fm lattice spacings and choose three different hadron momenta P_{z}={1.29,1.72,2.15}  GeV. The resulting lattice matrix elements are nonperturbatively renormalized in a recently proposed hybrid scheme. An extrapolation to the continuum and infinite momentum limit is carried out. We find that, while the longitudinal distribution amplitudes tend to be close to the asymptotic form, the transverse ones deviate rather significantly from the asymptotic form. Our final results provide crucial ab initio theory inputs for analyzing pertinent exclusive processes.

Flavor Decomposition for the Proton Helicity Parton Distribution Functions

We present, for the first time, an ab initio calculation of the individual up, down, and strange quark helicity parton distribution functions for the proton. The calculation is performed within the twisted mass clover-improved fermion formulation of lattice QCD. The analysis is performed using one ensemble of dynamical up, down, strange, and charm quarks with a pion mass of 260 MeV. The lattice matrix elements are nonperturbatively renormalized and the final results are presented in the MS[over ¯] scheme at a scale of 2 GeV. We give results for Δu^{+}(x), Δd^{+}(x) Δu^{-}(x), Δd^{-}(x), including disconnected quark loop contributions, as well as for Δs^{+}(x) and Δs^{-}(x). For the latter we achieve unprecedented precision compared to the phenomenological estimates.

The x-dependence of hadronic parton distributions: A review on the progress of lattice QCD

In this article, we review recent lattice calculations on the x-dependence of parton distributions, with the latter providing information on hadron structure. These calculations are based on matrix elements of boosted hadrons coupled to non-local operators and can be related to the standard, light-cone distribution functions via an appropriate factorization formalism. There is significant progress in several directions, including calculations of flavor-singlet parton distribution functions (PDFs), first calculations of generalized parton distributions (GPDs), as well as the implementation of some of the approaches for the transverse-momentum-dependent PDFs (TMD PDFs). This new direction of lattice calculations is particularly interesting for phenomenological fits on experimental data sets, as the lattice results can help improve the constraints on the distribution functions.

Extraction of Next-to-Next-to-Leading-Order Parton Distribution Functions from Lattice QCD Calculations

We present for the first time complete next-to-next-to-leading-order coefficient functions to match flavor nonsinglet quark correlation functions in position space, which are calculable in lattice QCD, to parton distribution functions (PDFs). Using PDFs extracted from experimental data and our calculated matching coefficients, we predict valence-quark correlation functions that can be confronted by lattice QCD calculations. The uncertainty of our predictions is greatly reduced with higher order matching coefficients. By performing Fourier transformation, we also obtain matching coefficients for corresponding quasi-PDFs and pseudo-PDFs. Our method of calculations can be readily generalized to evaluate the matching coefficients for sea-quark and gluon correlation functions, making the program to extract partonic structure of hadrons from lattice QCD calculations comparable with and complementary to that from experimental measurements.

Next-to-Next-to-Leading Order Calculation of Quasiparton Distribution Functions

We present the next-to-next-to-leading order (NNLO) calculation of quark quasiparton distribution functions (PDFs) in the large momentum effective theory. The nontrivial factorization at this order is established explicitly and the full analytic matching coefficients between the quasidistribution and the light-cone distribution are derived. We demonstrate that the NNLO numerical contributions can improve the behavior of the extracted PDFs sizably. With the unprecedented precision study of nucleon tomography at the planned electron-ion collider, high precision lattice QCD simulations with our NNLO results implemented will enable to test the QCD theory and more precise results on the PDFs of nucleons will be obtained.

Unpolarized and Helicity Generalized Parton Distributions of the Proton within Lattice QCD

We present the first calculation of the x dependence of the proton generalized parton distributions (GPDs) within lattice QCD. Results are obtained for the isovector unpolarized and helicity GPDs. We compute the appropriate matrix elements of fast-moving protons coupled to nonlocal operators containing a Wilson line. We present results for proton momenta 0.83,1.25,1.67 GeV, and momentum transfer squared 0.69,1.38  GeV^{2}. These combinations include cases with zero and nonzero skewness. The calculation is performed using one ensemble of two degenerate mass light quarks, a strange quark, and a charm quark of maximally twisted mass fermions with a clover term. The lattice results are matched to the light-cone GPDs using one-loop perturbation theory within the framework of large momentum effective theory. The final GPDs are given in the MS[over ¯] scheme at a scale of 2 GeV.

Parton Distribution Functions from Ioffe Time Pseudodistributions from Lattice Calculations: Approaching the Physical Point

We present results for the unpolarized parton distribution function of the nucleon computed in lattice QCD at the physical pion mass. This is the first study of its kind employing the method of Ioffe time pseudodistributions. Beyond the reconstruction of the Bjorken-x dependence, we also extract the lowest moments of the distribution function using the small Ioffe time expansion of the Ioffe time pseudodistribution. We compare our findings with the pertinent phenomenological determinations.

Accessing Gluon Parton Distributions in Large Momentum Effective Theory

Gluon parton distribution functions (PDFs) in the proton can be calculated directly on Euclidean lattices using large momentum effective theory (LaMET). To realize this goal, one has to find renormalized gluon quasi-PDFs in which power divergences and operator mixing are thoroughly understood. For the unpolarized distribution, we identify four independent quasi-PDF correlators that can be multiplicatively renormalized on the lattice. Similarly, the helicity distribution can be derived from three independent multiplicatively renormalizable quasi-PDFs. We provide a LaMET factorization formula for these renormalized quasi-PDFs from which one can extract the gluon PDFs.

Multiplicative Renormalizability of Operators defining Quasiparton Distributions

Extracting parton distribution functions (PDFs) from lattice QCD calculation of quasi-PDFs has been actively pursued in recent years. We extend our proof of the multiplicative renormalizability of the quasiquark operators of Ishikawa et al. [Phys Rev. D 96, 094019 (2017)] to quasigluon operators, and demonstrated that quasigluon operators could be multiplicatively renormalized to all orders in perturbation theory, without mixing with other operators. We find that using a gauge-invariant UV regulator is essential for achieving this proof. With the multiplicative renormalizability of both quasiquark and quasigluon operators, and QCD collinear factorization of hadronic matrix elements of there operators into PDFs, extracting PDFs from lattice QCD calculated hadronic matrix elements of quasiparton operators could have a solid theoretical foundation.

Parton Distribution Functions and Lattice QCD

Recently, there have been rapid developments in lattice-QCD calculations of proton structure, especially in the parton distribution functions (PDFs). We overcame a longstanding obstacle and for the first time in lattice-QCD are able to directly calculate the Bjorken-x dependence of the quark, helicity and transversity distributions. The PDFs are obtained using the large-momentum effective field theory (LaMET) framework where the full Bjorken-x dependence of finite-momentum PDFs, called “quasi-PDFs”, can be calculated on the lattice. The quasi-PDF nucleon matrix elements are renormalized non-perturbatively in RI/MOM-scheme. Following a nonperturbative renormalization of the parton quasi-distribution in a regularization-independent momentum-subtraction scheme, we establish its matching to the $ \overline {{\rm{MS}}} $ PDF and calculate the non-singlet matching coefficient at next-to-leading order in perturbation theory. In this proceeding, I will show the progress that has been made in recent years, highlighting the latest state-of-the art PDF calculations at the physical pion mass. Future impacts on the large-x global PDF fits are also discussed.

Light-Cone Parton Distribution Functions from Lattice QCD

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.

Universality of Generalized Parton Distributions in Light-Front Holographic QCD

The structure of generalized parton distributions is determined from light-front holographic QCD up to a universal reparametrization function w(x) which incorporates Regge behavior at small x and inclusive counting rules at x→1. A simple ansatz for w(x) that fulfills these physics constraints with a single-parameter results in precise descriptions of both the nucleon and the pion quark distribution functions in comparison with global fits. The analytic structure of the amplitudes leads to a connection with the Veneziano model and hence to a nontrivial connection with Regge theory and the hadron spectrum.