Lattice continuum-limit study of nucleon quasi-PDFs
Constantia Alexandrou, Krzysztof Cichy, Martha Constantinou, Jeremy R. Green, Kyriakos Hadjiyiannakou, Karl Jansen, Floriano Manigrasso, Aurora Scapellato, Fernanda Steffens
备注：
24 pages, 21 figures. v2: additional references, discussion of systematics, and position-space comparison with phenomenology. v3: applied workaround for problem affecting Greek symbols in figures with some PDF viewers
The quasi-PDF approach provides a path to computing parton distribution functions (PDFs) using lattice QCD. This approach requires matrix elements of a power-divergent operator in a nucleon at high momentum and one generically expects discretization effects starting at first order in the lattice spacing $a$. Therefore, it is important to demonstrate that the continuum limit can be reliably taken and to understand the size and shape of lattice artifacts. In this work, we report a calculation of isovector unpolarized and helicity PDFs using lattice ensembles with $N_f=2+1+1$ Wilson twisted mass fermions, a pion mass of approximately 370 MeV, and three different lattice spacings. Our results show a significant dependence on $a$, and the continuum extrapolation produces a better agreement with phenomenology. The latter is particularly true for the antiquark distribution at small momentum fraction $x$, where the extrapolation changes its sign.
We present the first lattice-QCD calculation of the nucleon isovector unpolarized parton distribution functions (PDFs) at the physical-continuum limit using Large-Momentum Effective Theory (LaMET). The lattice results are calculated using ensembles with multiple sea pion masses with the lightest one around 135~MeV, 3 lattice spacings $a\in[0.06,0.12]$~fm, and multiple volumes with $M_πL$ ranging 3.3 to 5.5. We perform a simultaneous chiral-continuum extrapolation to obtain RI/MOM renormalized nucleon matrix elements with various Wilson-link displacements in the continuum limit at physical pion…
The recently proposed large momentum effective theory (LaMET) of Ji has led to a burst of activity among lattice practitioners to perform and control the first pioneering calculations of the quasi-PDFs of the nucleon. These calculations represent approximations to the standard PDFs defined as correlation functions of fields with lightlike separation, being instead correlations along a longitudinal direction of the operator $γ^z$; as such, they differ from standard PDFs by power-suppressed $1 \big/ p^2_z$ corrections, becoming exact in the limit $p_z \to \infty$. Investigating the systematics o…
We compute the master integrals for two-loop QCD corrections to quasi parton distribution functions (PDFs) in large momentum effective theory. Analytical results of the master integrals are derived using the method of differential equations, along with a proper choice of canonical basis. The results of master integrals are expressed in terms of Goncharov polylogarithms. These integrals allow to extract the two-loop short-distance matching coefficients between quasi and light cone PDFs in large momentum effective theory, and are helpful to extract the nucleon PDFs from first principles.
We review the Euclidean path-integral formulation of the nucleon hadronic tensor and classify the gauge invariant and topologically distinct insertions in terms of connected and disconnected insertions and also in terms of leading and higher-twist contributions in the DIS region. Converting the Euclidean hadronic tensor back to the Minkowski space requires solving an inverse problem of the Laplace transform. We have investigated several inverse algorithms and studied the pros and cons of each. We show a result with a relatively large momentum transfer ($Q^2 \sim 4\, {\rm GeV^2}$) to suppress t…
I review lattice-QCD calculations of the electromagnetic and generalized form factors (GFFs), which determine the transverse structure of the nucleon, and briefly comment on recent calculations related to transverse-momentum dependent parton distribution functions (TMDPDFs).
The parton distribution functions (PDFs) provide process-independent information about the quarks and gluons inside hadrons. Although the gluon PDF can be obtained from a global fit to experimental data, it is not constrained well in the large-$x$ region. Theoretical gluon-PDF studies are much fewer than those of the quark PDFs. In this work, we present the first lattice-QCD results that access the $x$-dependence of the gluon unpolarized PDF of the nucleon. The lattice calculation is carried out with nucleon momenta up to 2.16 GeV, lattice spacing $a\approx0.12$ fm, with valence pion masses of…
Nucleon Parton Distribution Functions from Boosted Correlations in the Coulomb gauge
Brookhaven National Laboratory（布鲁克海文国家实验室）；University of Maryland（马里兰大学）；Argonne National Laboratory（阿贡国家实验室）；Massachusetts Institute of Technology（麻省理工学院）
I explain the current status of parton-distribution-function (PDF) studies and future experimental prospects on their determinations. First, unpolarized PDFs of the nucleon are introduced as a field of precision QCD physics including higher-order $α_s$ corrections. Second, nuclear PDFs and polarized nucleonic PDFs are discussed. Third, the determination of fragmentation functions is explained. Forth, the three-dimensional (3D) structure functions are discussed in connection with the origin of the nucleon spin and gravitational form factors of hadrons. By the 3D structure functions, gravitation…
We present results for the flavor non-singlet u-d parton distribution functions within the nucleon using the quasi-PDF approach. The lattice calculation is performed by employing the twisted mass formulation and two gauge ensembles, having $N_f = 2$ and $N_f = 2 + 1 + 1$ dynamical fermions with masses tuned to their physical value. For the $N_f = 2$ physical point ensemble, the unpolarized, helicity and transversity distributions are computed for three values of the nucleon momentum, namely [6, 8, 10]$π/L$ corresponding to [0.83,1.11,1.38] GeV. Upon renormalization, we find that, as the nucleo…
Large momentum effective field theory (LaMET) enables the extraction of parton distribution functions (PDFs) directly on a Euclidean lattice through a factorization theorem that relates the computed quasi-PDFs to PDFs. We apply chiral perturbation theory (ChPT) to LaMET to further separate soft scales, such as light quark masses and lattice size, to obtain leading model independent extrapolation formulas for extrapolations to physical quark masses and infinite volume. We find that the finite volume effect is reduced when the nucleon carries a finite momentum. For nucleon momentum greater than …
We present the first Monte Carlo based global QCD analysis of spin-averaged and spin-dependent parton distribution functions (PDFs) that includes nucleon isovector matrix elements in coordinate space from lattice QCD. We investigate the degree of universality of the extracted PDFs when the lattice and experimental data are treated under the same conditions within the Bayesian likelihood analysis. For the unpolarized sector, we find rather weak constraints from the current lattice data on the phenomenological PDFs, and difficulties in describing the lattice matrix elements at large spatial dist…
We present the next-to-next-to-leading order (NNLO) calculation of quark quasi parton 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 quasi distribution and the lightcone 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 …
Tensor-polarized parton distribution functions of the deuteron by a convolution model
Quark Matter Research Center, Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所夸克物质研究中心）；Southern Center for Nuclear Science Theory, 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）；KEK Theory Center, Institute of Particle and Nuclear Studies, KEK（高能加速器研究机构粒子与核研究所学部KEK理论中心）
The large-momentum effective theory (LaMET) framework has been widely used to determine the Bjorken-$x$ dependence of parton distribution functions (PDFs) in lattice-QCD hadron-structure calculations. In this proceeding, I highlighted selected recent lattice-QCD results on parton distributions from MSULat group. We use clover valence fermions on ensembles generated by MILC Collaboration with lattice spacing $a \approx 0.06$, 0.09, and 0.12 fm, with $M_πL \approx 4$, and with pion masses including 135, 220 and 310 MeV and $N_f=2+1+1$ flavors of highly improved staggered dynamical quarks. Result…
Studying the structure of nucleons is not only important to understanding the strong interactions of quarks and gluons, but also to improving the precision of new-physics searches. Since a broad class of experiments, including the LHC and dark-matter detection, require Standard-Model backgrounds with parton distribution functions (PDFs) as inputs for disentangling SM contributions from potential new physics. For a long time, lattice calculations of the PDFs (as well as many hadron structures) has been limited to the first few moments. In this talk, we present a first direct calculation of the …
We present the first lattice results on isovector unpolarized and longitudinally polarized parton distribution functions (PDFs) at physical pion mass. 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 nonperturbatively in RI/MOM-scheme. However, the recent renormalized quasi-PDFs suffer from unphysical oscillations that alter the shape of the true distribution as a function of Bjorken-$…
In this letter, we report the results for the isovector polarized quark quasi-distribution functions $Δu(x,v) - Δd(x,v)$ in the large $N_c$, calculated within the chiral quark-soliton model. It is shown that the polarized quark quasi-distributions present good convergence to the light-cone PDFs in the limit of the nucleon boost momentum $P_N \to \infty$ (or the velocity $v \to 1$), compared to the case of the unpolarized isosinglet distributions.
We propose a new approach to determining quasi parton distribution functions (PDFs) from lattice quantum chromodynamics. By incorporating the gradient flow, this method guarantees that the lattice quasi PDFs are finite in the continuum limit and evades the thorny, and as yet unresolved, issue of the renormalization of quasi PDFs on the lattice. In the limit that the flow time is much smaller than the length scale set by the nucleon momentum, the moments of the smeared quasi PDF are proportional to those of the light-front PDF. We use this relation to derive evolution equations for the matching…
We provide an analysis of the x-dependence of the bare unpolarized, helicity and transversity iso-vector parton distribution functions (PDFs) from lattice calculations employing (maximally) twisted mass fermions. The x-dependence of the calculated PDFs resembles the one of the phenomenological parameterizations, a feature that makes this approach very promising. Furthermore, we apply momentum smearing for the relevant matrix elements to compute the lattice PDFs and find a large improvement factor when compared to conventional Gaussian smearing. This allows us to extend the lattice computation …
Quasi-PDFs provide a path toward an ab initio calculation of parton distribution functions (PDFs) using lattice QCD. One of the problems faced in calculations of quasi-PDFs is the renormalization of a nonlocal operator. By introducing an auxiliary field, we can replace the nonlocal operator with a pair of local operators in an extended theory. On the lattice, this is closely related to the static quark theory. In this approach, we show how to understand the pattern of mixing that is allowed by chiral symmetry breaking, and obtain a master formula for renormalizing the nonlocal operator that de…
We present the first lattice-QCD calculation of the unpolarized strange and charm parton distribution functions using large-momentum effective theory (LaMET). We use a lattice ensemble with 2+1+1 flavors of highly improved staggered quarks (HISQ) generated by MILC collaboration, with lattice spacing $a\approx 0.12$ fm and $M_π\approx 310$ MeV, and clover valence fermions with two valence pion masses: 310 and 690 MeV. We use momentum-smeared sources to improve the signal up to nucleon boost momentum $P_z=2.18$ GeV, and determine nonperturbative renormalization factors in RI/MOM scheme. We compa…
Precise exploration of the partonic structure of the nucleon is one of the most important aims of high-energy physics. In recent years, it has become possible to address this topic with first-principle lattice QCD investigations. In this talk, we focus on the so-called pseudo-distribution approach to determine the isovector unpolarized PDFs. In particular, we employ three lattice spacings to study discretization effects and extract the distributions in the continuum limit, at a pion mass of around 370 MeV. Also, for the first time with pseudo-PDFs, we explore effects of the 2-loop matching fro…
In this document, we summarize the status and challenges of calculating parton physics in lattice QCD for the US Particle Physics Community Planning Exercise (a.k.a. "Snowmass"). While PDF-moments calculations have been very successful and been continuously improved, new methods have been developed to calculate distributions directly in $x$-space. Many recent lattice studies have been focused on calculating isovector PDFs of the pion and nucleon, learning to control systematics associated with excited-state contamination, renormalization and continuum extrapolations, pion-mass and finite-volum…
One of the great challenges of QCD is to determine the partonic structure of the nucleon from first principles. In this work, we provide such a determination of the flavor non-singlet ($u-d$) unpolarized parton distribution function (PDF), utilizing the non-perturbative formulation of QCD on the lattice. We apply Radyushkin's pseudo-distribution approach to lattice results obtained using simulations with the light quark mass fixed to its physical value; this is the first ever attempt for this approach directly at the physical point. The extracted coordinate-space matrix elements are used to fi…
In this work we present an efficient construction of baryon interpolating fields for lattice QCD computations of two and three point functions. These are essential building blocks of computations of nucleon parton distribution functions (PDFs), generalized parton distribution functions (GPDs) and transverse momentum dependent distributions functions (TMDs). Lattice QCD computations of these quantities can provide additional input to assist with the global fits on experimental data for determining TMDs, GPDs and PDFs.
We present a detailed study of the helicity-dependent and helicity-independent collinear parton distribution functions (PDFs) of the nucleon, using the quasi-PDF approach. The lattice QCD computation is performed employing twisted mass fermions with a physical value of the light quark mass. We give a systematic and in-depth account of the salient features entering in the evaluation of quasi-PDFs and their relation to the light-cone PDFs. In particular, we give details for the computation of the matrix elements, including the study of the various sources of systematic uncertainties, such as exc…
Building upon the PDFSense framework developed in Ref. [1], we perform a comprehensive analysis of the sensitivity of present and future high-energy data to a number of quantities commonly evaluated in lattice gauge theory, with a particular focus on the integrated Mellin moments of nucleon parton distribution functions (PDFs), such as $\langle x \rangle_{u^+ - d^+}$ and $\langle x \rangle_{g}$, as well as $x$-dependent quark quasi-distributions -- in particular, that of the isovector combination. Our results demonstrate the potential for lattice calculations and phenomenological quark distrib…
We present an improved determination of the $ππ$ continuum contribution to the isovector spectral functions of the nucleon electromagnetic form factors. Our analysis includes the most up-to-date results for the $ππ\to\bar N N$ partial waves extracted from Roy-Steiner equations, consistent input for the pion vector form factor, and a thorough discussion of isospin-violating effects and uncertainty estimates. As an application, we consider the $ππ$ contribution to the isovector electric and magnetic radii by means of sum rules, which, in combination with the accurately known neutron electric rad…
Precise quantification of the structure of nucleons is one of the crucial aims of hadronic physics for the coming years. The expected progress related to ongoing and planned experiments should be accompanied by calculations of partonic distributions from lattice QCD. While key insights from the lattice are expected to come for distributions that are difficult to access experimentally, it is important that lattice QCD can reproduce the well-known unpolarized parton distribution functions (PDFs) with full control over systematic uncertainties. One of the novel methods for accessing the partonic …
We compute the second moments of pion and rho parton distribution functions (PDFs) in lattice QCD with $N_f = 2+1$ flavors of improved Wilson fermions. We determine both singlet and non-singlet flavor combinations and, for the first time, take disconnected contributions fully into account. In the case of the rho, we also calculate the additional contribution arising from the $b_1$ structure function. The numerical analysis includes 26 ensembles, mainly generated by the CLS effort, with pion masses ranging from 420MeV down to 214MeV and with 5 different lattice spacings in the range of 0.1fm to…
We present a next-to-next-to-leading order (NNLO) calculation of the quasi parton distribution functions (Quasi-PDFs) in the large momentum effective theory (LaMET). We focus on the flavor non-diagonal quark-quark channel and demonstrate the LaMET factorization at the NNLO accuracy in the modified minimal subtraction scheme. The matching coefficient between the quasi-PDF and the light-cone PDF is derived. This provides a first step towards a complete NNLO analysis of quasi-PDFs and to better understand the nucleon structures from the first principle of QCD.
We present the nonperturbatively renormalized nucleon gluon momentum fraction using ensembles with $2+1+1$ flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration. The calculation is done using clover fermions for the valence action with three pion masses, $220$, $310$ and $690$ MeV and three lattice spacings, 0.09, 0.12, and 0.15 fm. The renormalization is done using RI/MOM nonperturbative renormalization and using cluster-decomposition error reduction (CDER) to enhance the signal-to-noise ratio of the renormalization constant. We find the CDER technique is particu…
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 non-perturbatively 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 in…
We present results for the axial charge and root-mean-square (RMS) radii of the nucleon obtained from 2+1 flavor lattice QCD at the physical point with a large spatial extent of about 10 fm. Our calculations are performed with the PACS10 gauge configurations generated by the PACS Collaboration with the six stout-smeared $O(a)$ improved Wilson-clover quark action and Iwasaki gauge action at $β$ = 1.82 and 2.00 corresponding to lattice spacings of 0.085 fm and 0.063 fm respectively. We first evaluate the value of $g_A/g_V$ , which is not renormalized in the continuum limit and thus ends up with …
Generalizing the Soffer Bound: Positivity Constraints on Parton Distributions of Spin-3/2 Particles
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）；Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Quark Matter Research Center, Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所重离子研究中心）；KEK Theory Center, Institute of Particle and Nuclear Studies, KEK（日本高能加速器研究机构粒子与核研究部KEK理论中心）；Heavy Ion Science and Technology Key Laboratory, Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所重离子科学与技术重点实验室）；School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences（中国科学院大学核科学与技术学院）
Unpolarized gluon PDF of the nucleon from lattice QCD at physical point in the continuum limit
Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；University of Chinese Academy of Sciences, School of Nuclear Science and Technology（中国科学院大学核科学与技术学院）；Nanjing Normal University（南京师范大学）；Central South University（中南大学）
