We investigate how charmonium hadroproduction at fixed-target energies can be used to constrain the gluon distribution in pions. Using nonrelativistic QCD (NRQCD) formulation, the $J/ψ$ and $ψ(2S)$ cross sections as a function of longitudinal momentum fraction $x_F$ from pions and protons colliding with light targets, as well as the $ψ(2S)$ to $J/ψ$ cross section ratios, are included in the analysis. The color-octet long-distance matrix elements are found to have a pronounced dependence on the pion parton distribution functions (PDFs). This study shows that the $x_F$ differential cross sections of pion-induced charmonium production impose strong constraints on the pion's quark and gluon PDFs. In particular, the pion PDFs with larger gluon densities provide a significantly better description of the data. It is also found that the production of the $ψ(2S)$ state is associated with a larger quark-antiquark contribution, compared with $J/ψ$.
We present an analysis of hadroproduction of $J/ψ$ and $ψ(2S)$ at fixed-target energies in the framework of non-relativistic QCD (NRQCD). Using both pion- and proton-induced data, a new determination of the color-octet long-distance matrix elements (LDMEs) is obtained. Compared with previous results, the contributions from the $q \bar{q}$ and color-octet processes are significantly enhanced, especially at lower energies. A good agreement between the pion-induced $J/ψ$ production data and NRQCD calculations using the newly obtained LDMEs is achieved. We find that the pion-induced charmonium pro…
The QCD collinear factorization framework allows to describe exclusive backward production of a $J/ψ$ meson in pion-nucleon collisions in terms of pion-to-nucleon transition distribution amplitudes. We calculate the scattering amplitude at the leading order in the strong coupling constant and estimate the cross section of this reaction in the backward kinematical region for a medium energy pion beam available at the J-Parc experimental facility.
We perform a non-perturbative calculation of the $J/Ψ-π$ cross section using a $SU(2)\times SU(2)$ effective Lagrangian. Our results differ from those of previous calculations, specially in the description of vertices involving pions.
In this article, we study the two-pion decays in the $P$-wave to $S$-wave charmonium transitions with the heavy meson effective theory, and make qualitative predictions for the ratios among the two-pion decay widths.
We propose the study of unpolarized transverse momentum dependent gluon parton distributions as well as the effect of linearly polarized gluons on transverse momentum and rapidity distributions of $J/ψ$ and $Υ$ production within the framework of transverse momentum dependent factorization employing color evaporation model (CEM) in unpolarized proton-proton collision. We estimate the transverse momentum and rapidity distributions of $J/ψ$ and $Υ$ at LHCb, RHIC and AFTER energies using TMD evolution formalism.
The analysis of the $πγ$ transition form factor provides severe constraints on the pion's wave function. This information is used to examine charmonium decays into two pions critically. It will be argued that the standard perturbative QCD analysis of these reactions fails, i.e. the need for additional contributions can convincingly be demonstrated. Colour-octet admixtures to the charmonium states are proposed as a possible dynamical mechanisms to solve the puzzle. Consequences of the $πγ$ analysis for the electromagnetic form factor of the pion are also discussed.
Meson-charmonium dissociation reactions governed by the quark interchange are studied with temperature-dependent quark potentials. Quark-antiquark relative-motion wave functions and masses of charmonia and charmed mesons are determined by the central spin-independent part of the potentials or by the central spin-independent part and a smeared spin-spin interaction. The prominent temperature dependence of the masses is found. Based on the potentials, the wave functions, and the meson masses, we obtain temperature-dependent cross sections for fifteen pion-charmonium and rho-charmonium dissociati…
Charmonium production in association with a forward or backward pion in nucleon-antinucleon annihilation is one of the most promising reaction to access nucleon-to-pion transition distribution amplitudes (TDAs) at PANDA. We briefly review the description of this reaction in terms of πN TDAs within the collinear factorization approach and present the first results of dedicated feasibility studies for the PANDA experimental setup.
In this work, we investigate the decay widths and the line shapes of the open-charm radiative and pionic decays of Y(4274) with the $D_s\bar{D}_{s0}(2317)$ molecular charmonium assignment. Our calculation indicates that the decay widths of $Y(4274)\to D^{+}_{s}D^{*-}_{s}γ$ and $Y(4274)\to D^+_{s}D^-_{s}π^0$ can reach up to 0.05 keV and 0.75 keV, respectively. In addition, the result of the line shape of the photon spectrum of $Y(4274)\to D_s^+ {D}_s^{*-} γ$ shows that there exists a very sharp peak near the large end point of photon energy. The line shape of the pion spectrum of $Y(4274)\to D_…
We compute the pion quark Generalized Parton Distribution H and quark Double Distributions F and G in a coupled Bethe-Salpeter and Dyson-Schwinger approach in terms of quark flavors or isospin states. We use simple analytic expressions inspired by the numerical resolution of Dyson-Schwinger and Bethe-Salpeter equations. We explicitly check the support and polynomiality properties, and the behavior under charge conjugation or time invariance of our model. We obtain analytic expressions for the pion Double Distributions and Generalized Parton Distribution at vanishing pion momentum transfer at a…
The $J/ψπ\to \bar{D} D^*, \bar{D} D$, ${\bar D}^* D^*$ and ${\bar D} D^*$ cross sections as a function of $\sqrt{s}$ are evaluated in a QCD sum rule calculation. We study the Borel sum rule for the four point function involving pseudoscalar and vector meson currents, up to dimension four in the operator product expansion. We find that our results are smaller than the $J/ψπ\to {charmed mesons}$ cross sections obtained with models based on meson exchange, but are close to those obtained with quark exchange models.
Generalized parton distributions (GPDs) encoding multidimensional information of hadron partonic structure appear as the building blocks in a factorized description of hard exclusive reactions. The nucleon GPDs have been accessed by deeply virtual Compton scattering and deeply virtual meson production with lepton beam. A complementary probe with hadron beam is the exclusive pion-induced Drell-Yan process. In this paper, we discuss recent theoretical advances on describing this process in terms of nucleon GPDs and pion distribution amplitudes. Furthermore, we address the feasibility of measurin…
The production of the resonance $X(3823)$, identified as the charmonium $^3D_2$ state, in the process $e^+e^- \to ππX(3823)$ has been recently reported by BESIII. It is pointed out that this process is fully described, up to one overall coupling constant, in the soft pion limit. An interpretation of the available and possible future data within the discussed theoretical framework may reveal new features of the charmoniumlike states. In particular, the observed relative yield for this process at different energies strongly suggests a very significant enhancement of the amplitude at the charmoni…
Pion exchange in S-wave between hadrons that are themselves in a relative S-wave is shown to shift energies by hundreds of MeV, leading to deeply bound quasi-molecular states. In the case of charmed mesons $D^*,D_1$ a spectroscopy arises consistent with enigmatic charmonium states observed above 4 GeV in $e^+e^-$ annihilation. A possible explanation of $Y(4260)\to ψππ$ and $Y(4360) \to ψ'ππ$ is found. We give results for all isospin and charge-conjugation combinations, and comment on flavor exotic doubly charmed states and bottomonium analogs. A search in $D\bar{D}3π$ is recommended to test th…
K*-charmonium dissociation reactions in hadronic matter are studied in the Born approximation, in the quark-interchange mechanism, and with a temperature-dependent quark potential. We obtain the temperature dependence of unpolarized cross sections for K*-charmonium dissociation reactions which produce charmed mesons and charmed strange mesons. We use the cross sections for charmonium dissociation in collisions with pion, rho meson, kaon, vector kaon, and eta meson to calculate dissociation rates of charmonium with the five types of mesons. Because of the temperature dependence of the meson mas…
Despite the success of modern nuclear parton distribution functions (nPDFs) in describing nuclear hard-process data, they still suffer from large uncertainties. One of the poorly constrained features is the possible asymmetry in nuclear modifications of valence $u$ and $d$ quarks. We study the possibility of using pion-nucleus Drell-Yan dilepton data as a new constraint in the global analysis of nPDFs. We find that the nuclear cross-section ratios from the NA3, NA10 and E615 experiments can be used without imposing significant new theoretical uncertainties and, in particular, that these datase…
Inclusive production of $ΥJ/ψ$ pair in proton-proton interation at LHCb is considered. This process is forbidden at leading order of perturbation theory, so such channels as double parton scattering, $χ_bχ_c$ pair production with subsequent radiative decays of $P$-wave quarkonia, contributions of color-octet states, and NLO corrections are studied in details. For all these channels we present theoretical predictions of total cross sections at LHCb and distributions over different kinematical variables. According to presented in the paper results, double parton interaction gives main contributi…
Quarkonium production at a given large p_T is dominated by parton fragmentation: a parton which is produced with transverse momentum p_T/z fragments into a quarkonium state which carries a fraction z of the parton momentum. Since parton production cross sections fall steeply with p_T, high z fragmentation is favored. However, quantum number constraints may require the emission of gluons in the fragmentation process, and this softens the z dependence of the fragmentation function. We discuss the possibility that higher- order processes may enhance the large z part of fragmentation functions and…
The X(3872) with quantum numbers J(PC) = 1(++) is considered as a composite hadronic state comprised of the dominant molecular D(0) D*(0) component and other hadronic pairs - D(+-) D*(-+), Jpsi omega and Jpsi rho. Applying the compositeness condition we constrain the couplings of the X(3872) to its constituents. We calculate two- and three-body hadronic decays of the X(3872) to charmonium states chi(cJ) and pions using a phenomenological Lagrangian approach. Next using the estimated X-Jpsi-omega and and X-Jpsi-rho couplings we calculate the widths of X(3872) to Jpsi + h, where h = pi(+) pi(-),…
A rainbow-ladder truncation of QCD's Dyson-Schwinger equations, constrained by existing applications to hadron physics, is employed to compute the valence-quark parton distribution functions of the pion and kaon. Comparison is made to pi-N Drell-Yan data for the pion's u-quark distribution and to Drell-Yan data for the ratio u_K(x)/u_π(x): the environmental influence of this quantity is a parameter-free prediction, which agrees well with existing data. Our analysis unifies the computation of distribution functions with that of numerous other properties of pseudoscalar mesons.
In this paper, we calculate the next-to-leading order (NLO) quantum chromodynamics (QCD) corrections to the exclusive processes $B_c^+\to χ_{cJ}(h_c)π^+$ in the framework of the nonrelativistic QCD (NRQCD) factorization formalism. The results show that NLO QCD corrections markedly enhance the branching ratios with $K$ factors of about 2.5. In combination with the study of $B_c^+\to J/ψπ^+$, we find that the NLO NRQCD prediction for the ratio of branching fractions $\frac{\mathcal{B}(B_c^+\to χ_{c0}π^+)}{\mathcal{B}(B_c^+\to J/ψπ^+)}$ is then compatible with the experimental measurement.
Supposing only that there is an effective charge which defines an evolution scheme for parton distribution functions (DFs) that is all-orders exact, strict lower and upper bounds on all Mellin moments of the valence-quark DFs of pion-like systems are derived. Exploiting contemporary results from numerical simulations of lattice-regularised quantum chromodynamics (QCD) that are consistent with these bounds, parameter-free predictions for pion valence, glue, and sea DFs are obtained. The form of the valence-quark DF at large values of the light-front momentum fraction is consistent with predicti…
We derive expression for the large b_perp asymptotic of the 3D parton distributions q(x,b_perp) in the pion. The asymptotic depends exclusively on the mass scales F_pi and m_pi. Therefore it provides us with a nice example of a strict non-perturbative result for the partonic structure of Nambu-Goldstone bosons in QCD. Analyzing the x-dependent pion transverse radius we reveal a new phenomenon of "chiral inflation"-- in the parametrically wide region of Bjorken x (m_pi^2/(4 pi F_pi)^2 << x << 1) the pion radius grows exponentially fast with the rapidity eta=ln(1/x). We show that the partons in …
A Continuum Schwinger Method to Study the Pion's Generalized Parton Distribution
Universidad de Valencia（瓦伦西亚大学）；IFIC-CSIC（伊菲克-西班牙高等科学研究委员会）；Universidad Pablo de Olavide（巴勃罗·德·奥拉维德大学）；Universidad de Huelva（韦尔瓦大学）；Irfu, CEA, Université Paris-Saclay（IRFU，法国原子能和替代能源委员会，巴黎萨克雷大学）
We perform the first global QCD analysis of parton distribution functions (PDFs) in the pion, combining $πA$ Drell-Yan data with leading neutron electroproduction from HERA within a Monte Carlo approach based on nested sampling. Inclusion of the HERA data allows the pion PDFs to be determined down to much lower values of $x$, with relatively weak model dependence from uncertainties in the chiral splitting function. The combined analysis reveals that gluons carry a significantly higher pion momentum fraction, $\sim 30\%$, than that inferred from Drell-Yan data alone, with sea quarks carrying a …
In the first part of the talk, we briefly review the problem of parton-energy loss and thermalization at the Relativistic Heavy-Ion Collider and discuss how heavy quarks (charm and bottom) can help to resolve the existing experimental and theoretical puzzles. The second part of the talk is devoted to the properties of heavy quarkonia in the (strongly interacting) Quark-Gluon Plasma (sQGP) and their consequences for observables in heavy-ion collisions.
Pion and kaon structural properties provide insights into the emergence of mass within the Standard Model and attendant modulations by the Higgs boson. Novel expressions of these effects, in impact parameter space and in mass and pressure profiles, are exposed via $π$ and $K$ generalised parton distributions, built using the overlap representation from light-front wave functions constrained by one-dimensional valence distribution functions that describe available data. Notably, e.g. $K$ pressure profiles are spatially more compact than $π$ profiles and both achieve near-core pressures of simil…
We perform a new Monte Carlo QCD analysis of pion parton distribution functions, including, for the first time, transverse momentum dependent pion-nucleus Drell-Yan cross sections together with $p_{\rm T}$-integrated Drell-Yan and leading neutron electroproduction data from HERA. We assess the sensitivity of the Monte Carlo fits to kinematic cuts, factorization scale, and parametrization choice, and we discuss the impact of the various datasets on the pion's quark and gluon distributions. This study provides the necessary step towards the simultaneous analysis of collinear and transverse momen…
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 la…
The quantum statistical approach proposed more than one decade ago was used to determine the parton distributions for the proton by considering a large set of accurate Deep Inelastic Scattering experimental results. We propose to extend this work to extract the parton distributions for the pion by using data on lepton pair production from various experiments. This global next-to-leading order QCD analysis leads to a good description of several Drell-Yan $π^- W$ data. The resulting parton distributions are compared with earlier determinations. We will also discuss the difference between nucleon…
We compute the pion quark Generalised Parton Distribution H and quark Double Distributions in a coupled Bethe-Salpeter and Dyson-Schwinger approach in terms of quark flavors or isospin states. We use analytic expressions inspired by the numerical resolution of Dyson-Schwinger and Bethe-Salpeter equations. We obtain an analytic expression for the pion Generalised Parton Distribution at a low scale. Our model fulfils most of the required symmetry properties and compares very well to experimental pion form factor or valence parton distribution function experimental data. In addition, we have high…
We compare the jet-path length and beam-energy dependence of the pion nuclear modification factor and a parton-jet nuclear modification factor at RHIC and LHC. We contrast predictions based on a linear pQCD and a highly non-linear hybrid-AdS holographic model of jet-energy loss. We find that both models require a reduction of the jet-medium coupling from RHIC to LHC to account for the measured pion nuclear modification factor. In case of the parton-jet nuclear modification factor, however, which serves as a lower bound for the LO jet nuclear modification factor of reconstructed jets, the extra…
We present a lattice-QCD calculation of the unpolarized isovector parton distribution function (PDF) using ensembles at the physical pion mass with large proton boost momenta $P_z \in \{2.2,2.6,3.0\}$~GeV within the framework of large-momentum effective theory (LaMET). In contrast to our previous physical-pion PDF result, we increase the statistics significantly, double the boost momentum, increase the investment in excited-state contamination systematics, and switch to $γ_t$ operator to avoid mixing with scalar matrix elements. We use four source-sink separations in our analysis to control th…
We have investigated the charmonium survival probability, in a high baryon density parton plasma, expected to be produced in nuclear collisions at FAIR. Charmonia are assumed to undergo complete dissociation by color screening, if the in-medium Debye radius becomes comparable to the spatial size of the corresponding bound state. Results indicate a non-trivial dependence of the suppression pattern on the plasma evolution dynamics. A much larger magnitude of suppression is foreseen induced by cold nuclear matter compared to that due to plasma screening.
We analyze the moments of parton distribution functions in the pion calculated in lattice QCD, paying particular attention to their chiral extrapolation. Using the lowest three non-trivial moments calculated on the lattice, we assess the accuracy with which the x dependence of both the valence and sea quark distributions in the pion can be extracted. The resulting valence quark distributions at the physical pion mass are in fair agreement with existing Drell-Yan data, but the statistical errors are such that one cannot yet confirm (or rule out) the large-x behavior expected from hadron helicit…
We estimate the contributions by double-parton interactions to the cross sections for pp->pi^0 pi^0 X and dA->pi^0 pi^0 X at RHIC. We find that such contributions become important at large forward rapidities of the produced pions. This is in particular the case for dA scattering, where they strongly enhance the azimuthal-angular independent "pedestal" component of the cross section, providing a natural explanation of this feature of the RHIC dA data. We argue that the discussed processes open a window to studies of double quark distributions in nucleons. We also briefly address the roles of sh…
Within the Dyson-Schwinger equation formulation of QCD, a rainbow ladder truncation is used to calculate the pion valence-quark distribution function(PDF). The gap equation is renormalized at a typical hadronic scale, of order 0.5GeV, which is also set as the default initial scale for the pion PDF. We implement a corrected leading-order expression for the PDF which ensures that the valence-quarks carry all of the pion's light-front momentum at the initial scale. The scaling behavior of the pion PDF at a typical partonic scale of order 5.2GeV is found to be $(1-x)^ν$, with $ν\simeq 1.6$, as $x$…
Charmonia produced in initial hard parton scatterings during heavy ion collisions move with respect to the medium rather than flow with the medium. Lattice studies suggest that charmonium bound states at the rest are dissociated at $T\gtrsim 1.5 T_c$. We present results on momentum dependences of charmonium properties in a hot medium from lattice QCD Monte Carlo simulations. The dispersion relation of the screening mass and the change of correlation and spectral functions at various temperatures and momenta are discussed.
We present a lattice QCD study of valence parton distribution inside the pion within the framework of Large Momentum Effective Theory. We use a mixed action approach with 1-HYP smeared valence Wilson clover quarks on 2+1 flavor HISQ sea with the valence quark mass tuned to 300 MeV pion mass. We use $48^3 \times 64$ lattice at a fine lattice spacing $a=0.06$ fm for this computation. We renormalize the quasi-PDF matrix element in the non-perturbative RI-MOM scheme. As a byproduct, we test the validity of 1-loop matching procedure by comparing the RI-MOM renormalized quasi-PDF matrix element with…
We investigate the inclusive production of prompt J/psi mesons in polarized hadron-hadron, photon-hadron, and photon-photon collisions in the factorization formalism of nonrelativistic quantum chromodynamics (NRQCD) providing all contributing partonic cross sections in analytic form. In the case of photoproduction, we also include the resolved-photon contributions. We present numerical results appropriate for BNL RHIC-Spin, the approved SLAC fixed-target experiment E161, and the e^+e^- and gamma gamma modes of TESLA. Specifically, we assess the feasibility to access the spin-dependent parton d…
In statistical QCD, color deconfinement and the properties of the quark-gluon plasma determine the in-medium behavior of heavy quark bound states. In high energy nuclear collisions, charmonia probe the partonic medium produced in the early stages of the interaction. We survey the present theoretical status and provide a critical evaluation of the charmonium production measurements in experiments at the CERN-SPS and the BNL-RHIC.
Two-parton correlations in the pion are investigated in terms of double parton distribution functions. A Poincaré covariant Light-Front framework has been adopted. As non perturbative input, the pion wave function obtained within the so-called soft-wall AdS/QCD model has been used. Results show how novel dynamical information on the structure of the pion, not accessible through one-body parton distribution, are encoded in double parton distribution functions.
Inclusive cross sections for pion production in $pp$ collisions are calculated for the first time fully based on unintegrated gluon, quark and antiquark distributions (uPDF). We use recently developed Kwieciński uPDF's and different phenomenological fragmentation functions (FF) from the literature. In addition to the $gg \to g$ diagram we include also $g q \to q$ and $q g \to q$ diagrams for inclusive parton production. The cross section for pions is obtained then by convoluting the inclusive parton distributions and FFs. Applications for SPS and RHIC are shown.
We compute the pion Generalized Parton Distribution (GPD) in a valence dressed quarks approach. We model the Mellin moments of the GPD using Ansätze for Green functions inspired by the numerical solutions of the Dyson-Schwinger Equations (DSE) and the Bethe-Salpeter Equation (BSE). Then, the GPD is reconstructed from its Mellin moment using the Double Distribution (DD) formalism. The agreement with available experimental data is very good.
We discuss a general diagrammatic description of n-point functions in the QCD instanton vacuum that resums planar diagrams, enforces gauge invariance and spontaneously broken chiral symmetry. We use these diagrammatic rules to derive the pion and kaon quasi-parton amplitude and distribution functions at leading order in the instanton packing fraction for large but finite momentum. The instanton and anti-instanton zero modes and non-zero modes are found to contribute to the quasi-distributions, but the latter are shown to drop out in the large momentum limit. The pertinent pion and kaon parton …
We re-examine one-pion exchange as a possible binding mechanism in the X(3872) charmonium-like state and argue it to be not sufficiently binding for this purpose. We conclude therefore that other short-range dynamics are responsible for the X formation.
We present the first direct lattice-QCD calculation of the Bjorken-$x$ dependence of the valence quark distribution of the pion. Using large-momentum effective theory (LaMET), we calculate the boosted pion state with long Wilson link operators. After implementing the one-loop matching and meson mass corrections, our result at $m_π\approx 310$ MeV is in agreement with those extracted from experimental data as well as from Dyson-Schwinger equation in small $x$ region, but a sizeable discrepancy in the large $x$ region. This discrepancy provides a nice opportunity to systematically study and dise…
We study pion production in proton-proton collisions within a pQCD-improved parton model in next-to-leading order augmented by intrinsic transverse momentum ($k_\perp$) of the partons. We find the introduction of $k_\perp$ necessary to reproduce the experimental data in the CERN SPS to RHIC energy range. We study the influence of intrinsic $k_\perp$ on the so-called $K$ factor, the ratio of the NLO cross section to the Born term. A strong $p_T$ dependence is seen, especially in the 3-6 GeV transverse momentum region of the outgoing pion, where nuclear effects (e.g. the Cronin effect) play an i…
We consider two simple covariant models for pions (one with scalar and the other with spin-1/2 constituents). Pion generalized parton distributions are derived by integration over the light-cone energy. The model distributions are consistent with all known properties of generalized parton distributions, including positivity. We also construct the corresponding double distributions by appealing to Lorentz invariance. These ostensibly constructed double distributions lead to different generalized parton distributions that need not respect the positivity constraints. This inconsistency arises fro…
We use detailed balance for a hadron composed of quark and gluon Fock states to obtain parton distributions in the proton and pion on the basis of a simple statistical model.
The initial single pion emission (ISPE) mechanism is applied to the processes $Υ(5S)\to πB^{(*)}\bar{B}^{(*)}$ whose details have been recently reported at ICHEP2012 and we obtain reasonable agreement with Bell's measurements, i.e., we succeed in reproducing the enhancement structures of $Z_b(10610)$ and $Z_b(10650)$. Inspired by this success, we predict the corresponding enhancement structures in higher charmonia open charm pion decay near the thresholds of $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$.
We apply hard pion Chiral Perturbation Theory to charmonium decays to $ππ$, $KK$ and $ηη$. We first discuss why we expect to be able to provide results for the chiral logarithms in $χ_{c0}$ and $χ_{c2}$ decays to two pseudoscalars while for the decays from $J/ψ$, $ψ(nS)$ and $χ_{c1}$ no simple prediction is possible. The leading chiral logarithm turns out to be absent for $χ_{c0},χ_{c2}\to PP$. This result is true for all fully chiral singlet states of spin zero and two.
We present the first determination of the $x$-dependent pion gluon distribution from lattice QCD using the pseudo-PDF approach. We use lattice ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration, at two lattice spacings $a\approx 0.12$ and 0.15~fm and three pion masses $M_π\approx 220$, 310 and 690 MeV. We use clover fermions for the valence action and momentum smearing to achieve pion boost momentum up to 2.29 GeV. We find that the dependence of the pion gluon parton distribution on lattice spacing and pion mass is mild. We compare our resu…
