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The normalized covariance matrix between the parameter of the "default" model fit. The real component is on the left and the imaginary component on the right. The IDs of the parameters used on the labels are given in tables 5 and 6.
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A bstract
In this study, we present continuum limit results for the unpolarized parton distribution function of the nucleon computed in lattice QCD. This study is the first continuum limit using the pseudo-PDF approach with Short Distance Factorization for factorizing lattice QCD calculable matrix elements. Our findings are also compared with the p...
Citations
... The coefficients ( ) for parton distribution functions (PDFs) can be seen in [13]. Then, we can replace equation (6) in equation (5) and compute the integrals using properties of the Gegenbauer polynomials, such that ...
... We combine these steps in a single fit, where we minimize a 2 employing a custom implementation of the variable projection algorithm, and we fit our data to the model [6] ( , 2 ) =˜( , 2 ) + | | 1 ( ) + Λ 1 ( ) + 2 Λ 2 1 ( ) which gathers all the aforementioned requirements. See [13] for a study of PDFs using a similar approach. We use Λ ≡ Λ (2) QCD = 330 MeV [15] to render all terms dimensionless. ...
In this proceeding we determine the distribution amplitude of the $\eta_c$-meson from first principles. This quantity appears as a consequence of factorization theorems, and it is necessary to compute the amplitude of multiple exclusive processes. Since it is defined along a light-cone, its calculation via lattice QCD was impossible until recently, when a generalization to Euclidean metric was proposed, and a connection to the physical limit was established. We briefly explain the method of short distance factorization, which allows us to compute the distribution amplitude, and our lattice setup. After summarizing the steps for the continuum and chiral extrapolation, we present our results and compare them to two alternative determinations, one using non-relativistic QCD and another solving the Dyson-Schwinger equations; we find a large discrepancy with the former.
... However, it was not until 2013 that fundamental limitations were overcome in the seminal paper by Ji [23]. At the beginning, many efforts focused on the parton distribution function (PDF) of the nucleon [24,25] and the PDF and DA of the pion [26][27][28], while other works aim now to compute their GPDs [29,30] as well as the structure of heavy mesons [31]. See [32] for a review on the lattice progress. ...
... Equation (2.6) is normalized to one and we can recover equation (2.5) replacing λ = 3/2. A similar approach has been used in the context of PDFs [25]. ...
... Once we have obtainedφ(ν, z) on every ensemble of table 1 at various Ioffe times ν and Wilson lines z, we remove the cutoff and match our results to the LCDA in one single step. See [25] for a study of PDFs using this approach. Although separating in two distinct steps the extrapolation and the problem with the inverse Fourier transform could be simpler, we would need several lattices for every momentum and Wilson line in physical units. ...
Distribution amplitudes are functions of non-perturbative matrix elements describing the hadronization of quarks and gluons. Thanks to factorization theorems, they can be used to compute the scattering amplitude of high-energy processes. Recently, new ideas have allowed their computation using lattice QCD, which should provide us with a general, fully relativistic determination. We present the first lattice calculation of the $\eta_c$-meson distribution amplitude at leading twist. Starting from the relevant matrix element in discrete Euclidean space on a set of $N_f=2$ CLS ensembles, we explain the method to connect to continuum Minkowski spacetime. After addressing several sources of systematic uncertainty, we compare to Dyson-Schwinger and non-relativistic QCD determinations of this quantity. We find significant deviations between the latter and our result even at small Ioffe times.
... The leading order DGLAP evolution was first used in [5], where numerical evidence suggested that the quenched lattice QCD results for M(ν, z 2 ) indeed follow this evolution in z 2 . More generally, a wealth of numerical analyses have shown that for z < 0.2 ∼ 0.3 fm, various truncations of the perturbative matching give rather similar results, mostly compatible with the phenomenological knowledge of unpolarized PDFs [40,52,[58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75]. ...
Lattice QCD offers the possibility of computing parton distributions from first principles, although not in the usual \( \overline{MS} \) factorization scheme. Calculations are therefore matched to \( \overline{MS} \) using a perturbative procedure which is the source of significant uncertainty within the currently accessible kinematics. We present the possibility of computing the z2 evolution of non-singlet pseudo-parton distribution functions within the short factorization scheme in a numerically improvable way. The goal is to have tools to evolve a calculation to a scale where perturbative uncertainties are less pronounced. We compare a numerical extraction of the evolution operator from lattice data to the computation of z2 dependence in perturbation theory. Finally, we discuss how this numerical work may be extended to address the two-scale problem that arises when the Ioffe time range must be made large to extend the reach of the calculation of the pseudo-PDF to smaller values of the momentum fraction.
... In order to cancel the renormalization factors, the z ¼ 0 matrix elements are not necessary, but this choice is favorable in that it enforces a normalization and cancels correlations. In this work, we only consider the case with P 0 z ¼ 0, commonly referred to as the reduced pseudo-ITD [43,[89][90][91][92][93][94]. Additionally, since there are no gluons involved in the case of the transversity distributions, the leading-twist OPE expansion of the pseudo-ITD does not depend on the flavor combination f 0 , even if f ≠ f 0 , and we, therefore, opt to omit the f 0 from our notation in order to not be overly cumbersome. ...
We present a lattice QCD calculation of the transversity isovector- and isoscalar-quark parton distribution functions (PDFs) of the proton utilizing a perturbative matching at next-to-leading-order (NLO) accuracy. Additionally, we determine the isovector and isoscalar tensor charges for the proton. In both calculations, the disconnected contributions to the isoscalar matrix elements have been ignored. The calculations are performed using a single ensemble of Nf=2+1 highly improved staggered quarks simulated with physical-mass quarks and a lattice spacing of a=0.076 fm. The Wilson-clover action, with physical quark masses and smeared gauge links obtained from one iteration of hypercubic smearing, is used in the valence sector. Using the NLO operator product expansion, we extract the lowest four to six Mellin moments and the PDFs via a neural network from the matrix elements in the pseudo-PDF approach. In addition, we calculate the PDFs in the quasi-PDF approach with hybrid-scheme renormalization and the recently developed leading-renormalon resummation technique, at NLO with the resummation of leading small-x logarithms.
... Unfortunately, this is not practical or possible at the moment. Instead, we follow an approach already used in the study of parton distribution functions [9], and parametrize the light-cone DA in terms of a basis of Jacobi polynomials, ...
... The lattice artifacts may also depend on the Ioffe time, and to model such dependence we can exploit the same basis of Jacobi polynomials used for the ITD. Following the example of [9], we define nuisance functions ...
... for quasidistributions, Refs. [44][45][46][47][48][49][50][51][52][53][54][55][56] for pseudodistributions and Refs. [57][58][59][60][61] for recent reviews. ...
In this paper, we present lattice QCD results for the x dependence of the unpolarized gluon parton distribution functions (PDFs) for the proton. We use one ensemble of Nf=2+1+1 maximally twisted mass fermions with a clover improvement, and the Iwasaki improved gluon action. The quark masses are tuned to produce a pion with a mass of 260 MeV. The ensemble has a lattice spacing of a=0.093 fm and a spatial extent of 3 fm. We employ the pseudodistribution approach, which relies on matrix elements of nonlocal operators that couple to momentum-boosted hadrons. In this work, we use five values of the momentum boost between 0 and 1.67 GeV. The gluon field-strength tensors of the nonlocal operator are connected with straight Wilson lines of varying length z. The light-cone Ioffe time distribution (ITD) is extracted utilizing data with z up to 0.56 fm and a quadratic parametrization in terms of the Ioffe time at fixed values of z. We explore systematic effects, such as the effect of the stout smearing for the gluon operator, excited states effects, and the dependence on the maximum value of z entering the fits to obtain the gluon PDF. Also, for the first time, the mixing with the quark singlet PDFs is eliminated using matrix elements with nonlocal quark operators that were previously analyzed within the quasi-PDF framework on the same ensemble. Here, we expand the dataset for the quark singlet and reanalyze within the pseudo-PDFs method eliminating the corresponding mixing in the gluon PDF.
... The GPDs allow to understand the contributions of different parton flavors to various observables which characterize the hadronic target. At present it is not possible to evaluate the GPDs directly from first principles, and for this reason studies of these objects rely on phenomenological extractions from experimental data or results of lattice simulations [7][8][9][10][11]. However, the existing lattice studies, due to technical challenges, at present mostly focus on the special zero-skewedness (ξ ¼ 0) limit and studies of some moments of GPDs, whereas phenomenological extractions suffer from various uncertainties, even for the cleanest and best understood channels [12]. ...
In this paper we analyze the exclusive photoproduction of the D-meson pairs with large invariant mass. We perform evaluations in the collinear factorization framework and in the leading order of the strong coupling αs, expressing the cross section in terms of generalized parton distributions (GPDs) of different parton flavors in the proton. We focus on the photoproduction of the pseudoscalar-vector pairs, like, e.g., D±D*∓, D0D¯*0, Ds+Ds*−, which gets the dominant contribution from the chiral even GPDs of the target, and estimate the cross section in the kinematics of the future Electron Ion Collider. In all channels the amplitude of the process obtains comparable contributions from gluons and only one of the light quark flavors. This finding signals that the process potentially could be used to single out the contributions of the individual chiral even GPDs of light flavors. We found that the process is mostly sensitive to the behavior of GPDs in the so-called Efremov-Radyushkin-Brodsky-Lepage region. Numerically, the cross section of the process is sufficiently large for experimental studies and thus can be used as a complementary probe for studies of the GPDs.
... Recently, progress has been made in the most-calculated isovector quark distribution of nucleon by MSULat [49], ETMC [51], and HadStruc Collaborations [103], who studied lattice-spacing dependence. MSULat studied three lattice spacings (0.09, 0.12, and 0.15 fm) and pion masses (135,220,310 MeV) and performed a simultaneous continuum-physical extrapolation using a third-order z-expansion on renormalized LaMET matrix elements [49] with nucleon boost momenta around 2.2 and 2.6 GeV. ...
... ETMC also uses three lattice spacings, 0.06, 0.08, and 0.09 fm, but with heavier pion mass (370 MeV) and investigated the continuum extrapolation of the data on renormalized LaMET matrix elements with boost momentum around 1.8 GeV [51]. HadStruc Collaboration studied three lattice spacings, 0.048, 0.065, and 0.075 fm with twoflavor 440-MeV lattice ensembles using the continuum pseudo-Ioffe-time distribution (ITD) [103]. Most of the works above found mild nonzero dependence on lattice spacing (varying with the Wilson-link displacement) in the nucleon case for LaMET or pseudo-ITD matrix elements. ...
We present the first physical-continuum limit x-dependent nucleon gluon distribution from lattice QCD using the pseudo-PDF approach, on lattice ensembles with 2+1+1 flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration. We use clover fermions for the valence action on three lattice spacings a≈0.9, 0.12 and 0.15 fm and three pion masses Mπ≈220, 310 and 690 MeV, with nucleon two-point measurements numbering up to O(106) and nucleon boost momenta up to 3 GeV. We study the lattice-spacing and pion-mass dependence of the reduced pseudo–Ioffe-time distribution matrix elements obtained from the lattice calculation, then extrapolate them to the continuum-physical limit before extracting xg(x)/⟨x⟩g. We use the gluon momentum fraction ⟨x⟩g calculated from the same ensembles to determine the nucleon gluon unpolarized PDF xg(x) for the first time entirely through lattice-QCD simulation. We compare our results with previous single-ensemble lattice calculations, as well as selected global fits.
... We extract signals from an excited two-pion state with the energy near the kaon mass, E ππ ≈ m K using the generalized eigenvalue problem (GEVP) method [26][27][28], which provides a convenient method for isolating the contributions of individual low-lying states to Euclidean correlation functions. The GEVP method has been used for several calculations of matrix elements, for example for nucleon structure [29][30][31][32][33][34][35], B physics [36][37][38][39], light-meson radiative transitions [40,41] and form factors [42]. We can utilize this method not only for removing excited-state contamination from the ground-state signal but also for extracting signals from low-lying excited states. ...
We present a lattice calculation of the $K\to\pi\pi$ matrix elements and amplitudes with both the $\Delta I = 3/2$ and 1/2 channels and $\varepsilon'$, the measure of direct $CP$ violation. We use periodic boundary conditions (PBC), where the correct kinematics of $K\to\pi\pi$ can be achieved via an excited two-pion final state. To overcome the difficulty associated with the extraction of excited states, our previous work~\cite{Bai:2015nea,RBC:2020kdj} successfully employed G-parity boundary conditions, where pions are forced to have non-zero momentum enabling the $I=0$ two-pion ground state to express the on-shell kinematics of the $K\to\pi\pi$ decay. Here instead we overcome the problem using the variational method which allows us to resolve the two-pion spectrum and matrix elements up to the relevant energy where the decay amplitude is on-shell. In this paper we report an exploratory calculation of $K\to\pi\pi$ decay amplitudes and $\varepsilon'$ using PBC on a coarser lattice size of $24^3\times64$ with inverse lattice spacing $a^{-1}=1.023$~GeV and the physical pion and kaon masses. The results are promising enough to motivate us to continue our measurements on finer lattice ensembles in order to improve the precision in the near future.
... As only a restricted range of Ioffe times can be probed numerically with acceptable noise levels, the inversion of the Fourier transform to reconstruct an x-dependent distribution is an ill-posed inversion problem, also known as an imputation problem. Several attempts have aleady been performed to try to handle this specific ill-posed problem [39][40][41]. ...
We present a systematic study demonstrating the impact of lattice QCD data on the extraction of generalised parton distributions (GPDs). For this purpose, we use a previously developed modelling of GPDs based on machine learning techniques fulfilling the theoretical requirements of polynomiality, a form of positivity constraint and known reduction limits. A special care is given to estimate the uncertainty stemming from the ill-posed character of the connection between GPDs and the experimental processes usually considered to constrain them, like deeply virtual Compton scattering (DVCS). Mock lattice QCD data inputs are included in a Bayesian framework to the prior model which is fitted to reproduce the most experimentally accessible information of a phenomenological model by Goloskov and Kroll. We highlight the impact of the precision, correlation and kinematic coverage of lattice data on GPD extraction at moderate $\xi$ which has only been brushed in the literature so far, paving the way for a joint extraction of GPDs.
