We present the first open-source analysis of parton distribution functions (PDFs) of charged pions using xFitter, an open-source QCD fit framework to facilitate PDF extraction and analyses. Our calculations are implemented at next-to-leading order (NLO) using applgrids generated by the mcfm generator. Using currently available Drell-Yan and photon production data, we find the valence distribution is well constrained; however, the considered data are not sensitive enough to unambiguously determine sea and gluon distributions. Fractions of momentum carried by the valence, sea and gluon components are discussed, and we compare with the results of the JAM collaboration and the GRV group.

Parton Distribution Functions of the Charged Pion Within The xFitter Framework

We extract the pion valence quark distribution ${q}_{\mathrm{v}}^{\ensuremath{\pi}}(x)$ from lattice QCD (LQCD) calculated matrix elements of spacelike correlations of one vector and one axial vector current analyzed in terms of QCD collinear factorization, using a new short-distance matching coefficient calculated to one-loop accuracy. We derive the Ioffe time distribution of the two-current correlations in the physical limit by investigating the finite lattice spacing, volume, quark mass, and higher-twist dependencies in a simultaneous fit of matrix elements computed on four gauge ensembles. We find remarkable consistency between our extracted ${q}_{\mathrm{v}}^{\ensuremath{\pi}}(x)$ and that obtained from experimental data across the entire $x$ range. Further, we demonstrate that the one-loop matching coefficient relating the LQCD matrix computed in position space to the ${q}_{\mathrm{v}}^{\ensuremath{\pi}}(x)$ in momentum space has well-controlled behavior with Ioffe time. This justifies that LQCD-calculated current-current correlations are good observables for extracting partonic structures by using QCD factorization, which complements to the global effort to extract partonic structure from experimental data.

Pion valence quark distribution from current-current correlation in lattice QCD

We present a high-statistics lattice QCD determination of the valence parton distribution function (PDF) of the pion, with a mass of 300 MeV, using two very fine lattice spacings of $a=0.06$ fm and 0.04 fm. We reconstruct the $x$-dependent PDF, as well as infer the first few even moments of the PDF using leading-twist 1-loop perturbative matching framework. Our analyses use both RI-MOM and ratio-based schemes to renormalize the equal-time bi-local quark-bilinear matrix elements of pions boosted up to 2.4 GeV momenta. We use various model-independent and model-dependent analyses to infer the large-$x$ behavior of the valence PDF. We also present technical studies on lattice spacing and higher-twist corrections present in the boosted pion matrix elements.

Valence parton distribution of the pion from lattice QCD: Approaching the continuum limit

The leading-twist parton distribution functions of the pion and kaon are calculated for the first time using a rainbow-ladder truncation of QCD's Dyson-Schwinger equations (DSEs) that self-consistently sums all planar diagrams. The nonperturbative gluon dressing of the quarks is thereby correctly accounted for, which in practice means solving the inhomogeneous Bethe-Salpeter equation (BSE) for the quark operator that defines the spin-independent quark distribution functions. An immediate consequence of using this dressed vertex is that gluons carry 35% of the pion's and 30% of the kaon's light-cone momentum, with the remaining momentum carried by the quarks. The scale associated with these DSE results is μ_{0}=0.78  GeV. The gluon effects generated by the inhomogeneous BSE are inherently nonperturbative and cannot be mimicked by the perturbative QCD evolution equations. A key consequence of this gluon dressing is that the valence quarks have reduced support at low-to-intermediate x, where the gluons dominate, and increased support at large x. As a result, our DSE calculation of the pion's valence quark distribution is in excellent agreement with the Conway et al. pion-induced Drell-Yan data, but nevertheless exhibits the q_{π}(x)≃(1-x)^{2} behavior as x→1 predicted by perturbative QCD.

Distinguishing Quarks and Gluons in Pion and Kaon Parton Distribution Functions.

We present the first lattice-QCD calculation of the kaon valence-quark distribution functions using the large-momentum effective field theory (LaMET) approach, a method that has been applied to a wide variety of isovector nucleon distributions and valence pion distributions. This is the first such lattice calculation with multiple pion masses with the lightest one around 220~MeV, 2 lattice spacings $a=0.06$ and 0.12~fm, $(M_\pi)_\text{min} L \approx 5.5$, and high statistics ranging from 11,600 to 61,312 measurements. We also find the valence-quark distribution of pion to be consistent with the FNAL E615 experimental results. Our ratio of the $u$ quark PDF in the kaon to that in the pion agrees with the CERN NA3 experiment. We make predictions of the strange quark distribution of the kaon.

The Valence-Quark Distribution of the Kaon from Lattice QCD

Nucleon quark distribution functions from the Dyson–Schwinger equations

We determine the leading Fock state light front wave functions (LFWFs) of the pion and kaon via light front projections of the covariant Bethe-Salpeter wave function. Using these LFWFs we study the multidimensional images of the valence quarks in the pion and kaon that are provided by their generalized parton distribution functions (GPDs) and transverse momentum dependent parton distribution functions (TMDs). Moments of the GPDs are taken to obtain the electromagnetic and gravitational form factors of the pion and kaon, and comparisons to available experimental and lattice data are made. Highlights from this study include predictions that the mean-squared impact parameter for the quarks in the pion and kaon are: ⟨bT2⟩uπ=0.11  fm2, ⟨bT2⟩sK=0.08  fm2, and ⟨bT2⟩uK=0.13  fm2, and therefore the s quark in the kaon is much closer to the center of transverse momentum than the u quark. From the electromagnetic and gravitational form factors we find that the light-cone energy radii are about 60% smaller than the light-cone charge radii for each quark sector in the pion and kaon. A quantitative measure of the importance of the leading Fock state is obtained via comparison with a full Dyson-Schwinger equations calculation (containing an infinite tower of Fock states) for the pion form factor.

Spatial and Momentum Imaging of the Pion and Kaon

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Distinguishing Quarks and Gluons in Pion and Kaon Parton Distribution Functions.

Nucleon quark distribution functions from the Dyson–Schwinger equations

Spatial and Momentum Imaging of the Pion and Kaon