The recently evaluated two-pion contribution to the muon g-2 and the phase of the pion electromagnetic form factor in the elastic region, known from ππscattering by Fermi-Watson theorem, are exploited by analytic techniques for finding correlations between the coefficients of the Taylor expansion at t=0 and the values of the form factor at several points in the spacelike region. We do not use specific parametrizations and the results are fully independent of the unknown phase in the inelastic region. Using for instance, from recent determinations, <r^2_π> = (0.435 \pm 0.005) fm^2 and F(-1.6 GeV^2)=0.243^{+0.022}_{-0.014}, we obtain the allowed ranges 3.75 GeV^{-4}\lesssim c \lesssim 3.98 GeV^{-4} and 9.91 GeV^{-6}\lesssim d \lesssim 10.46 GeV^{-6} for the curvature and the next Taylor coefficient, with a strong correlation between them. We also predict a large region in the complex plane where the form factor can not have zeros.
We perform a nonperturbative lattice calculation of the complex phase and modulus of the pion form factor in the timelike momentum region using the finite-volume technique. We use two ensembles of 2+1-flavor overlap fermion at pion masses m_pi = 380 and 290 MeV. By calculating the I = 1 correlators in the center-of-mass and three moving frames, we obtain the form factor at ten different values of the timelike momentum transfer around the vector resonance. We compare the results with the phenomenological model of Gounaris-Sakurai and its variant.
The elastic $I=1$ $p$-wave $ππ$ scattering amplitude is calculated together with the isovector timelike pion form factor using lattice QCD with $N_{\rm f}=2+1$ dynamical quark flavors. Wilson clover ensembles generated by the Coordinated Lattice Simulations (CLS) initiative are employed at four lattice spacings down to $a = 0.05\,\mathrm{fm}$, several pion masses down to $m_π = 200\,\mathrm{MeV}$, and spatial volumes of extent $L = 3.1-5.5\,\mathrm{fm}$. The set of measurements on these ensembles, which is publicly available, enables an investigation of systematic errors due to the finite latt…
The transverse charge density in the pion can be represented as a dispersion integral of the imaginary part of the pion form factor in the timelike region. This formulation incorporates information from e+e- annihilation experiments and allows one to reconstruct the transverse density much more accurately than from the spacelike pion form factor data alone. We calculate the transverse density using an empirical parametrization of the timelike pion form factor and estimate that it is determined to an accuracy of ~10% at a distance b ~ 0.1 fm, and significantly better at larger distances. The de…
We present a formula that allows one to calculate the pion form factor in the timelike region 2mpi <= sqrt{s} <= 4mpi in lattice QCD. The form factor quantifies the contribution of two-pion states to the vacuum polarization. It must be known very accurately in order to reduce the theoretical uncertainty on the anomalous magnetic moment of the muon. At the same time, the formula constitutes a rare example where, in a restricted kinematic regime, the spectral function of a conserved current can be determined from Euclidean observables without an explicit analytic continuation.
The $Δ(1232) \to γ^\ast N$ magnetic dipole form factor ($G_M^\ast$) is described here within a new covariant model that combines the valence quark core together with the pion cloud contributions. The pion cloud term is parameterized by two terms: one connected to the pion electromagnetic form factor, the other to the photon interaction with intermediate baryon states. The model can be used in studies of pp and heavy ion collisions. In the timelike region this new model improves the results obtained with a constant form factor model fixed at its value at zero momentum transfer. At the same time…
We consider a model with $ρππ$, $ρKK$, and gauge-invariant $ργ$ couplings and obtain the pion form factor $F_π$ at timelike momentum transfers by resummation of pion and kaon loops. We use a dispersion representation for the loop diagrams and analyse ambiguities related to subtraction constants. The resulting representation for $F_π$ is shown to have the form of the conventional vector meson dominance formula with one important distinction - the effective $ρ$-meson decay constant $f^{\rm eff}_ρ$ turns out to depend on the momentum transfer. For the electromagnetic pion form factor we include i…
New, accurate measurements of the pion and kaon electromagnetic form factors are expected in the near future from experiments at electron-positron colliders,using the radiative return method. We construct a model for the timelike pion electromagnetic form factor, that is valid also at momentum transfers far above the $ρ$ resonance. The ansatz is based on vector dominance and includes a pattern of radial excitations expected from dual resonance models.The form factor is fitted to the existing data in the timelike region, continued to the spacelike region and compared with the measurements there…
We show that measurements of the proton's polarization in electron-positron annihilation to a proton-antiproton pair $e^+ e^- \to p \bar p$ strongly discriminate between the analytic forms of models which fit the proton form factors in the spacelike region. In particular, the single-spin asymmetry normal to the scattering plane measures the relative phase difference between the timelike $G_E$ and $G_M$ form factors. The expected proton polarization in the timelike region is large, of order of several tens of percent.
The electromagnetic form factor of the pion is calculated within the use of functional formalism. We develop integral representation for the minimal set of Standard Model Green's functions and derive the dispersion relation for the form factor in the two flavor QCD isospin limit $m_u=m_d$. We use the dressed quark propagator as obtained form the gap equation in Minkowski space and within the Dyson-Schwinger equations formalism to derive the approximate dispersion relation for the form factor for the first time. We evaluate the form factor for the spacelike as well as for the timelike momentum …
The renormalization group applied to perturbation theory is ordinarily used to define the running coupling constant in the spacelike region. However, to describe processes with timelike momenta transfers, it is important to have a self-consistent determination of the running coupling constant in the timelike region. The technique called analytic perturbation theory (APT) allows a consistent determination of this running coupling constant. The results are found to disagree significantly with those obtained in the standard perturbative approach. Comparison between the standard approach and APT i…
A new model for the QCD analytic running coupling, which incorporates the effects due to the $π$ meson mass, is proposed. The properties of this invariant charge in spacelike and timelike regions are examined. Its main distinctive features are a finite infrared limiting value, which depends on the pion mass, and the "plateau-like" behavior in the deep infrared domain of the timelike region.
The pion electromagnetic form factor and two-pion production in electron-positron collisions are simultaneously fitted by a vector dominance model evolving to perturbative QCD at large momentum transfer. This model was previously successful in simultaneously fitting the nucleon electromagnetic form factors (spacelike region) and the electromagnetic production of nucleon-antinucleon pairs (timelike region). For this pion case dispersion relations are used to produce the analytic connection of the spacelike and timelike regions. The fit to all the data is good, especially for the newer sets of t…
We analyze the basic hard exclusive processes: πγ*γ- transition, pion and nucleon electromagnetic form factors, and discuss the analytic continuation of QCD formulas from the spacelike q^2<0 to the timelike region q^2 >0 of the relevant momentum transfers. We describe the construction of the timelike version of the coupling constant α_s. We show that due to the analytic continuation of the collinear logarithms each eigenfunction of the evolution equation acqiures a phase factor and investigate the resulting interference effects which are shown to be very small. We found no sources for the K-fa…
A new formula for the pion form factor F_pi is proposed which takes into account the pseudoscalar meson loops and the mixing among rho(770), rho(1450), and rho(1700). The expression possesses correct analytical properties and can be used in both timelike and spacelike kinematical regions. The comparison is made with the existing experimental data on F_pi collected by the detectors SND, CMD-2, KLOE, and BaBaR restricted to energies below 1 GeV. A good description of all four data sets is obtained. In the spacelike region, upon substituting the resonance parameters found in the timelike one, the…
Mechanism for generation of vector meson resonances is studied in the framework of QCD Dyson-Schwinger equations, which are defined and actually solved in Minkowski space. It is suggested that the timelike pion form factor is generated by the interference of the background quark loops together with the resonant structure predominantly created in the photon-quark-antiquark vertex. It is suggested that QCD Green's functions involving quark fields are oscillating for timelike arguments, which makes the interference effect among various QCD Green's functions quite strong and important for the corr…
We determine the values of the one- and two-loop low energy constants appearing in the SU(2) Chiral Perturbation Theory calculation of pion-pion scattering. For this we use a recent and precise sum rule determination of some scattering lengths and slopes that appear in the effective range expansion. In addition we provide sum rules for these coefficients up to third order in the expansion. Our results when using only the scattering lengths and slopes of the S, P, D and F waves are consistent with previous determinations, but seem to require higher order contributions if they are to accommodate…
In this paper a new comprehensive analysis of parton-to-pion fragmentation functions (FFs) is performed for the first time by including all experimental data sets on single inclusive pion as well as unidentified light charged hadron production in electron-positron ($e^+e^-$) annihilation. We determine the pion FFs along with their uncertainties using the standard "Hessian" technique at next-to-leading order (NLO) and next-to-next-to leading order (NNLO) in perturabative QCD. It is shown that the determination of pion FFs using simultaneously the data sets from pion and unidentified light charg…
A unified description of spacelike and timelike hadron form factors within a light-front model was successfully applied to the pion. The model is extended to the nucleon to study the role of $q \bar q$ pair production and of nonvalence components in the nucleon form factors. Preliminary results in the spacelike range $0 \le Q^2 \le 10 ~ (GeV/c)^2$ are presented.
Determination of charged pion unpolarised TMDPDFs from Drell-Yan measurements
Institute of Physics, Academia Sinica（中央研究院物理研究所）；Department of Physics, Chung-Yuan Christian University（中原大学物理系）；Centre for High Energy Physics, Chung-Yuan Christian University（中原大学高能物理中心）；Institute of Physics, National Yang Ming Chiao Tung University（国立阳明交通大学物理研究所）；Physics Division, National Centre for Theoretical Sciences（国家理论科学中心物理组）
We explore the feasibility of determining Mellin moments of the pion's light cone distribution amplitude using the heavy quark operator product expansion (HOPE) method. As the first step of a proof of principle study we pursue a determination of the second Mellin moment. We discuss our choice of kinematics which allows us to successfully extract the moment at low pion momentum. We describe the numerical simulation, and describe the data analysis, which leads us to a preliminary determination of the second Mellin moment in the continuum limit in the quenched approximation as $\langleξ^2\rangle=…
An explicitly crossing-symmetric decomposition of the pion-pion scattering amplitudes into low- and high-energy components is established. The high-energy components are entirely determined by absorptive parts at high energies. With the exception of the two leading-order parameters, all the parameters appearing in the one- and two-loop chiral amplitudes are determined by the high-energy components of the exact amplitudes.
We present a new method for the determination of the pion-pion scattering lenght combination a0-a2, based on the study of the pi0-pi0 spectrum in K->pi+ pi0 pi0 in the vicinity of the pi+ pi- threshold. The method requires a minimum of theoretical input, and is potentially very accurate.
An NLO QCD analysis of the final HERMES data on pion multiplicities is presented and a new set of pion fragmentation functions is extracted from the best fit to the data. We have studied the so-called [x,z] and [Q^2,z] presentations of their data, as given by HERMES, which, in principle, should simply be two different ways of presenting the experimental data. We have based our extraction on an excellent fit to the [Q^2,z] presentation of the data. We also draw attention to what appears to be a problem with the [x,z] presentation of the HERMES data.
We present a new determination of unpolarised charged pion and kaon fragmentation functions from a set of single-inclusive electron-positron annihilation and lepton-nucleon semi-inclusive deep-inelastic scattering data. The determination includes next-to-next-to-leading order QCD corrections to both processes, and is carried out in a framework that combines a neural-network parametrisation of fragmentation functions with a Monte Carlo representation of their uncertainties. We discuss the quality of the determination, in particular its dependence on higher order corrections.
We present a determination of the pion charge radius from high precision data on the pion vector form factor from both timelike and spacelike regions, using a novel formalism based on analyticity and unitarity. At low energies, instead of the poorly known modulus of the form factor, we use its phase, known with high accuracy from Roy equations for $ππ$ elastic scattering via the Fermi-Watson theorem. We use also the values of the modulus at several higher timelike energies, where the data from $e^+e^-$-annihilation and $τ$-decay are mutually consistent, as well as the most recent measurements …
The pion light-cone distribution amplitude (LCDA) is a central non-perturbative object of interest for the calculation of high-energy exclusive processes in quantum chromodynamics. In this article, we discuss the calculation of the second and fourth Mellin moment of the pion LCDA using a heavy-quark operator product expansion. The resulting value for the second Mellin moment is $ \langle{ ξ^2 }\rangle(μ= 2~\text{GeV})= 0.210 \pm 0.013\text{ (stat.)} \pm 0.034\text{ (sys.)}$. This result is compatible with those from previous determinations of this quantity.
We suggest to probe the pion light-cone distribution amplitude, applying a dispersion relation for the pion electromagnetic form factor. Instead of the standard dispersion relation, we use the equation between the spacelike form factor $F_π(Q^2)$ and the integrated modulus of the timelike form factor. For $F_π(Q^2)$, the QCD light-cone sum rule with a dominant twist-2 term is used. Adopting for the pion twist-2 distribution amplitude a certain combination of the first few Gegenbauer polynomials, it is possible to fit their coefficients $a_{2,4,6,...}$ (Gegenbauer moments) from this equation, e…
We re-evaluate our model predictions in Phys. Rev. D 79, 013002 (2009) for different observables in neutrino induced coherent pion production. This comes as a result of the new improved fit to old bubble chamber data of the dominant axial C_5^A nucleon-to-Delta form factor. We find an increase of 20%-30% in the values for the total cross sections. Uncertainties induced by the errors in the determination of C_5^A are computed. Our new results turn out to be compatible within about $1σ$ with the former ones. We also stress the existing tension between the recent experimental determination of the…
A model independent expression for the electromagnetic corrections to a phenomenological hadronic pion-nucleon scattering length, extracted from pionic hydrogen, is obtained. In a non-relativistic approach and using an extended charge distribution, these corrections are derived up to terms of order (alpha)**2 log(alpha) in the limit of a short-range hadronic interaction. We infer a charged pion-proton scattering length of 0.0870(5) in units of inverse pion mass, which gives for the charged pion-proton-neutron coupling, through the GMO relation, a value of 14.04(17).
We complete the set of QCD light-cone sum rules for $B\to ππ$ transition form factors, deriving a new sum rule for the timelike-helicity form factor $F_t$ in terms of dipion distribution amplitudes. This sum rule, in the leading twist-2 approximation, is directly related to the pion vector form factor. Employing a relation between $F_t$ and other $B\to ππ$ form factors we obtain also the longitudinal-helicity form factor $F_0$. In this way, all four (axial-)vector $B\to ππ$ form factors are predicted from light-cone sum rules with dipion distribution amplitudes. These results are valid for sma…
In this comment, we propose possible errors in constructing the continuum contribution in the sum rule studied by M. C. Birse and B. Krippa, Phys. Rev. C. {\bf 54} (1996) 3240.
This talk is dedicated to the memory of Paco Yndurain, the original speaker in the conference. After a short account of his scientific career, we briefly review our ongoing collaboration to determine precisely the $ππ$ scattering amplitude including the most recent data by means of Forward Dispersion Relations and Roy Equations. A remarkable improvement in precision over the intermediate energy region is obtained by using once-subtracted Roy Equations in addition to the standard twice-subtracted ones.
We revisit the evaluations for the spacelike and timelike $ρπ$ transition form factors $F_{ρπ}(Q^2)$ and $G_{ρπ}(Q^2)$ with the inclusion of the the next-to-leading order (NLO) QCD contributions in the framework of the $k_T$ factorization theorem. The infrared divergence is regularized by the transversal momentum carried by external valence quarks, and ultimately absorbed into the meson wave functions. In the region of $ Q^2 \leqslant 2 \, \textrm{GeV}^2 $, where PQCD factorization apporach applicable, the NLO contribution can bring no larger than $ 35\%$ enhancement to the spacelike form fact…
We critically evaluate the isovector GMO sum rule for forward pion-nucleon scattering using the recent precision measurements of negatively charged pion-proton and pion-deuteron scattering lengths from pionic atoms. We deduce the charged-pion-nucleon coupling constant, with careful attention to systematic and statistical uncertainties. This determination gives, directly from data a pseudoscalar coupling constant of 14.11+-0.05(statistical)+-0.19(systematic) or a pseudovector one of 0.0783(11). This value is intermediate between that of indirect methods and the direct determination from backwar…
We perform a complete next to leading order evaluation of inclusive neutral pion production from hadronic colliders which is based on the knowledge of the pion fragmentation functions at next to leading order. This problem has been tackled for the first time using different approaches. Predictions at LHC are given with the estimate of the theoretical uncertainty.
The pion transition form factor $G(q^2)$ is computed on the entire domain of spacelike and timelike momenta using a quantum field theory continuum approach. In analytical continuation of the function $G(Q^2)$ we utilized the Gauge Technique with the quark propagator determined from Minkowski space solution of QCD Dyson-Schwinger equations. The scale is set up by the phenomena of dynamical chiral symmetry breaking, which is a striking feature of low energy QCD.
We give the conversion equations which lead from experimental values of the $3p \to 1s$ transition energy in pionic hydrogen and the total width of the $1s$ level to values of the s-wave threshold scattering parameters for the processes $π^-p \to π^-p$ and $π^-p \to π^0n$ respectively. Using a three-channel potential model, we then calculate the electromagnetic corrections to these quantities, which remove the effects of the Coulomb interaction, the external mass differences and the presence of the $γn$ channel. We give the s-wave scattering parameters obtained from the present experimental da…
I present a first determination of a set of collinear fragmentation functions of charged pions using the NNPDF methodology. The analysis is based on a wide set of single-inclusive electron-positron annihilation data, including recent measurements from $B$-factory experiments, and is performed up to next-to-next-to-leading order accuracy in perturbative quantum chromodynamics. I discuss the results of the fits, highlighting their quality in the description of the data, their stability upon the inclusion of higher-order corrections, and their comparison to other sets of fragmentation functions.
Recent data from high statistics experiments that have measured the modulus of the pion electromagnetic form factor from threshold to relatively high energies are used as input in a suitable mathematical framework of analytic continuation to find stringent constraints on the shape parameters of the form factor at $t=0$. The method uses also as input a precise description of the phase of the form factor in the elastic region based on Fermi-Watson theorem and the analysis of the $ππ$ scattering amplitude with dispersive Roy equations, and some information on the spacelike region coming from rece…
We discuss the implications of a new determination of the pion-nucleon sigma term for the evaluation of the neutralino-nucleon cross-section, and, in turn, for detection rates for relic neutralinos in WIMP direct experiments and in some of the indirect searches. It is shown that the new range for the pion-nucleon sigma term, taken at its face value, favours values of the neutralino-nucleon cross-sections which are sizeably larger than some of the current estimates. Implications for neutralino cosmological properties are derived.
First-principles determination of anomaly-induced pion decay beyond the chiral limit
Peking University（北京大学）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所南方核科学理论中心（SCNT））；University of Connecticut（康涅狄格大学）
We provide a new value for the ratio $R={Γ(π^0\to e^+e^-γ(γ))}/{Γ(π^0\toγγ)}=11.978(6)\times10^{-3}$, which is by two orders of magnitude more precise than the current Particle Data Group average. It is obtained using the complete set of the next-to-leading-order radiative corrections in the QED sector, and incorporates up-to-date values of the $π^0$-transition-form-factor slope. The ratio $R$ translates into the branching ratios of the two main $π^0$ decay modes: $\mathcal{B}(π^0\toγγ)=98.8131(6)\,\%$ and $\mathcal{B}(π^0\to e^+e^-γ(γ))=1.1836(6)\,\%$.
The pion beta decay pi^+ -> pi^0 e^+ nu proceeds through pure weak vector hadronic currents and, therefore, the theoretical prediction for it is more reliable than for the processes with axial-vector current contribution. For example, recently the pion beta decay has been used for V_ud determination. The main aim of this letter is to point that kaon beta decay K^0 -> K^+(pi^+ pi^0) e^- nu-bar analogously can be used for this purpose.
Generalized parton distributions (GPDs) are 3-dimensional (3D) structure functions for hadrons, and they are important for solving the proton spin puzzle including partonic orbital-angular-momentum contributions. The $s$-$t$ crossed quantities of the GPDs are generalized distribution amplitudes (GDAs). Here, $s$ and $t$ are Mandelstam variables. The GDAs can be studied in two-photon processes ($γ^* γ\to h \bar h$) at KEKB. A GDA describes the amplitude from quark and antiquark to the hadron pair $h \bar h$. In 2016, the Belle collaboration reported measurements for pion-pair production in elec…
We present a determination of the pion-nucleon ($πN$) $σ$-term $σ_{πN}$ based on the Cheng-Dashen low-energy theorem (LET), taking advantage of the recent high-precision data from pionic atoms to pin down the $πN$ scattering lengths as well as of constraints from analyticity, unitarity, and crossing symmetry in the form of Roy-Steiner equations to perform the extrapolation to the Cheng-Dashen point in a reliable manner. With isospin-violating corrections included both in the scattering lengths and the LET, we obtain $σ_{πN}=(59.1\pm 1.9\pm 3.0)$ MeV $=(59.1\pm 3.5)$ MeV, where the first error …
We examine the isospin violation in the timelike pion form factor near threshold at next-to-leading order in the chiral expansion using the techniques of Chiral Perturbation Theory. This next-to-leading order contribution contains the first nonvanishing isospin violation. This isospin violation is found to be very small near threshold. In particular, the isospin violation at threshold is found to be of order 10^-4, which should be compared with the few percent level seen in the vector meson resonance region.
By fitting pion masses and decay constants from 2+1 flavor staggered lattice simulations to the predictions of NLO and NNLO SU(2) chiral perturbation theory we determine the low-energy constants l_3 and l_4. The lattice ensembles were generated by the Wuppertal-Budapest collaboration and cover pion masses in the range of 135 to 435 MeV and lattice scales between 0.7 and 2.0 GeV. By choosing a suitable scaling trajectory, we were able to demonstrate that precise and stable results for the LECs can be obtained from continuum ChPT to NLO. The pion masses available in this work also allow us to st…
The damped oscillating structures recently revealed by a three parametric formula from the proton ``effective'' form factor data extracted of the measured total cross section $σ^{bare}_{tot}(e^+e^-\to p\bar p)$ still seem to have an unknown origin. The conjectures of their direct manifestation of the quark-gluon structure of the proton indicate that they are not specific only of the proton and neutron, but they have to be one's own, similar to other hadrons. Therefore, the oscillatory structures from the charged pion electromagnetic form factor timelike data, extracted of the process $e^+e^-\t…
The pion-nucleon coupling constants determine the strength of the long-range nuclear forces and play a fundamental part in our understanding of nuclear physics. While the charged- and neutral-pion couplings to protons and neutrons are expected to be very similar, owing to the approximate isospin symmetry of the strong interaction, the different masses of the up- and down-quarks and electromagnetic effects may result in their slightly different values. Despite previous attempts to extract these coupling constants from different systems, our knowledge of their values is still deficient. In this …
The covariant spectator quark model, tested before in a variety of electromagnetic baryon excitations, is applied here to the $γ^\ast N \to N^\ast(1520)$ reaction in the timelike regime. The transition form factors are first parametrized in the spacelike region in terms of a valence quark core model together with a parametrization of the meson cloud contribution. The form factor behavior in the timelike region is then predicted, as well as the $N^\ast(1520) \to γN$ decay width and the $N^\ast (1520)$ Dalitz decay, $N^\ast (1520) \to e^+ e^- N$. Our results may help in the interpretation of die…
The impact of potential future measurements of beam charge asymmetries on the current knowledge of Compton form factors is evaluated. Elaborating on the results of a global neural network fit to deeply virtual Compton scattering data, a Bayesian reweighting analysis quantifies the improved determination of the real part of the Compton form factor $\mathcal{H}$. Such an improvement is particularly relevant for the experimental determination of the proton mechanical properties or for universality studies of generalized parton distributions through analyses of virtual Compton scattering in both s…
We present our preliminary results concerning the charged/neutral pion mass difference $M_{π^{+}} - M_{π^{0}}$ at order $\mathcal{O}(α_{em})$ in the QED interactions, and for $M_{π^{+}} - M_{π^{0}}$ at order $\mathcal{O}\left( (m_{d}-m_{u})^{2}\right)$ in the strong isospin-breaking term. The latter contribution provides a determination of the $\rm{SU}(2)$ chiral perturbation theory low-energy constant $\ell_{7}$, whose present estimate is affected by a rather large uncertainty. The disconnected contributions appearing in the diagrammatic expansion of $M_{π^{+}} - M_{π^{0}}$, being very noisy,…
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…
