Since the parton model was introduced by Feynman more than fifty years ago, we have learned much about the partonic structure of the proton through a large body of high-energy experimental data and dedicated global fits. However, calculating the partonic observables such as parton distribution function (PDFs) from the fundamental theory of strong interactions, QCD, has made limited progress. Recently, the authors have advocated a formalism, large-momentum effective theory (LaMET), through which one can extract parton physics from the properties of the proton travelling at a moderate boost-factor, e.g., $γ\sim (2-5)$. The key observation behind this approach is that Lorentz symmetry allows the standard formalism of partons in terms of light-front operators to be replaced by an equivalent one with large-momentum states and time-independent operators of a universality class. With LaMET, the PDFs, generalized PDFs or GPDs, transverse-momentum-dependent PDFs, and light-front wave functions can all be extracted in principle from lattice simulations of QCD (or other non-perturbative methods) through standard effective field theory matching and running. Future lattice QCD calculations with exa-scale computational facilities can help to understand the experimental data related to the hadronic structure, including those from the upcoming Electron-Ion Colliders dedicated to exploring the partonic landscape of the proton. Here we review the progress made in the past few years in development of the LaMET formalism and its applications, particularly on the demonstration of its effectiveness from initial lattice QCD simulations.
Partons are effective degrees of freedom describing the structure of hadrons involved in high-energy collisions. Familiar theories of partons are QCD light-front quantization and soft-collinear effective theory, both of which are intrinsically Minkowskian and appear unsuitable for classical Monte Carlo simulations. A ``new'' form of the parton theory has been formulated in term of the old-fashioned, Feynman's infinite momentum frame, in which the parton degrees of freedom are filtered through infinite-momentum external states. The partonic structure of hadrons is then related to the matrix ele…
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…
In a large-momentum nucleon state, the matrix element of a gauge-invariant Euclidean Wilson line operator accessible from lattice QCD can be related to the standard light-cone parton distribution function through the large-momentum effective theory (LaMET) expansion. This relation is given by a factorization theorem with a non-trivial matching coefficient. Using the operator product expansion we prove the large-momentum factorization of the quasi-parton distribution function in LaMET, and show that the more recently discussed Ioffe-time distribution approach also obeys an equivalent factorizat…
LaMET-Agent: An Agent Framework for Large-Momentum Effective Theory Analysis
Fermi National Accelerator Laboratory（费米国家加速器实验室）；University of Maryland（马里兰大学）；Argonne National Laboratory（阿贡国家实验室）；Indiana University（印第安纳大学）；Brookhaven National Laboratory（布鲁克海文国家实验室）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
Although equivalent in the infinite-momentum limit, large-momentum effective theory (LaMET) and short-distance operator product expansion (SD-OPE) are two different approaches to extract parton distribution functions (PDFs) from coordinate-space correlation functions in large-momentum hadrons. LaMET implements a momentum-space expansion in $Λ_{\rm QCD}/[x(1-x)P^z]$ to directly calculate PDFs $f(x)$ in a middle region of Bjorken $x\in [x_{\rm min}\sim Λ_{\rm QCD}/P^z, x_{\rm max}\sim 1-x_{\min}]$. SD-OPE applies perturbative QCD at small Euclidean distances $z$ to extract a range $[0,λ_{\rm max…
The evaluation of partonic distributions presents a challenge for QCD, and in particular for its Euclidean lattice realization. Recently, objects called quasi-distributions (which become standard distributions in a limit of the longitudinal momentum of the target hadron going to infinity) have been proposed. We present a non-perturbative, dynamical evaluation of the quark quasi-distribution amplitude (QDA) of the pion in the framework of chiral quark models (the Nambu--Jona-Lasinio model and the spectral quark model). We arrive at simple but nontrivial analytic expressions, where the dependenc…
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 …
Towards power corrections in the factorization of baryon quasi-distribution amplitudes in LaMET
School of Physics, East China University of Science and Technology（华东理工大学物理学院）；College of Physics and Electronic Engineering, Hainan Normal University（海南师范大学物理与电子工程学院）；State Key Laboratory of Dark Matter Physics, Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院暗物质物理全国重点实验室粒子天体物理与宇宙学教育部重点实验室上海粒子物理与宇宙学重点实验室）
We show that transverse-momentum-dependent parton distribution functions (TMDPDFs), important non-perturbative quantities for describing the properties of hadrons in high-energy scattering processes such as Drell-Yan and semi-inclusive deep-inelastic scattering with observed small transverse momentum, can be obtained from Euclidean QCD calculations in the framework of large-momentum effective theory (LaMET). We present a LaMET factorization of the Euclidean quasi-TMDPDFs in terms of the physical TMDPDFs and off-light-cone soft function at leading order in $1/P^z$ expansion, with the perturbati…
Gluon parton distribution functions (PDFs) in the proton can be calculated directly on Euclidean lattices using large momentum effective theory (LaMET). To realize this goal, one has to find renormalized gluon quasi-PDFs in which power divergences and operator mixing are thoroughly understood. For the unpolarized distribution, we identify four independent quasi-PDF correlators that can be multiplicatively renormalized on the lattice. Similarly, the helicity distribution can be derived from three independent multiplicatively renormalizable quasi-PDFs. We provide a LaMET factorization formula fo…
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…
Within the framework of large momentum effective theory (LaMET), genenaralized parton distributions (GPDs) can be extracted from lattice calculations of quasi-GPDs through a perturbative matching relation, up to power corrections that are suppressed by the hadron momentum. In this paper, we focus on isovector quark GPDs, including the unpolarized, longitudinally and transversely polarized cases, and present the one-loop matching that connects the quasi-GPDs renormalized in a regularization-independent momentum subtraction (RI/MOM) scheme to the GPDs in MS scheme. We find that the matching coef…
As a sequel of our preceding work [1] we carry out a comprehensive comparative study between the quasi parton distribution functions (PDFs), distribution amplitudes (DAs) and their light-cone counterparts for various flavor-neutral mesons, in the context of the 't Hooft model, that is, the two-dimensional QCD in the large $N$ limit. In contrast to the original derivation via diagrammatic techniques exemplified by Dyson-Schwinger and Bethe-Salpeter equations, here we employ the Hamiltonian operator approach to reconstruct the celebrated 't Hooft equation in light-front quantization, and Bars-Gr…
For a proton in the infinite momentum frame, its wave function contains a zero-momentum part (light-front zero-modes) originated from the modification of the QCD vacuum in the presence of the valence quarks, exhibiting a light-front long-range order in the quantum state. This ``non-travelling'' component of the proton contributes to its fundamental properties, including the mass and spin, as well as the naive-time-reversal-odd correlations between transverse momentum and spin. The large momentum effective theory (LaMET) provides a theoretical approach to study the physical properties of the pr…
Large-Momentum Effective Theory (or LaMET) advocated by the present authors provides a direct approach to simulate parton physics in Eulidean lattice QCD theory. Recently, there has been much interest in this theory in the literature, with some questioning its validity and effectiveness. Here we provide some discussions aiming at a further exposition of this approach. In particular, we explain why it does not have the usual power divergence problem in lattice QCD calculations for the moments of parton distributions. We show that although the Ioffe-time distribution provides an alternative way …
We study the nucleon's partonic angular momentum (AM) content at peripheral transverse distances $b = \mathcal{O}(M_π^{-1})$, where the structure is governed by chiral dynamics. We compute the nucleon form factors of the energy-momentum tensor in chiral effective field theory (ChEFT) and construct the transverse densities of AM at fixed light-front time. In the periphery the spin density is suppressed, and the AM is predominantly orbital. In the first-quantized representation of ChEFT in light-front form, the field-theoretical AM density coincides with the quantum-mechanical orbital AM density…
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…
We complete the procedure of extracting parton distribution functions (PDFs) using large momentum effective theory (LaMET) at leading power accuracy in the hadron momentum. We derive a general factorization formula for the quasi PDFs in the presence of mixing, and give the corresponding hard matching kernel at $\mathcal O(α_s)$, both for the unpolarized and for the polarized quark and gluon quasi-PDFs. Our calculation is performed in a regularization-independent momentum subtraction scheme. The results allow us to match the nonperturbatively renormalized quasi-PDFs to normal PDFs in the presen…
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.
Our knowledge on the three-dimensional momentum structure of hadrons is encoded in the Transverse Momentum Dependent partonic distribution and fragmentation functions (TMDs). A brief and updated review of the TMDs and of the processes in which they might play a role is presented.
Parton physics, when formulated as light-front correlations, are difficult to study non-perturbatively, despite the promise of light-front quantization. Recently an alternative approach to partons have been proposed by re-visiting original Feynman picture of a hadron moving at asymptotically large momentum. Here I formulate the approach in the language of an effective field theory for a large hadron momentum $P$ in lattice QCD, LaMET for short. I show that using this new effective theory, parton properties, including light-front parton wave functions, can be extracted from lattice observables …
Recent experimental data on semi-inclusive deep-inelastic scattering from the HERMES collaboration allow us to discuss for the first time the flavor dependence of unpolarized transverse-momentum dependent distribution and fragmentation functions. We find convincing indications that favored fragmentation functions into pions have smaller average transverse momentum than unfavored functions and fragmentation functions into kaons. We find weaker indications of flavor dependence in the distribution functions.
We present lattice results for the isovector unpolarized parton distribution with nonperturbative RI/MOM-scheme renormalization on the lattice. In the framework of large-momentum effective field theory (LaMET), the full Bjorken-$x$ dependence of a momentum-dependent quasi-distribution is calculated on the lattice and matched to the ordinary lightcone parton distribution at one-loop order, with power corrections included. The important step of RI/MOM renormalization that connects the lattice and continuum matrix elements is detailed in this paper. A few consequences of the results are also addr…
We present first results on selected twist-3 quark GPDs using the quasi-distribution method. This approach relates lattice QCD data and light-cone distribution functions using Large Momentum Effective Theory (LaMET). We calculate quark-antiquark correlators of boosted nucleons coupled to non-local operators with vector and axial Dirac structure, which is transverse to the momentum boost. We use three values of the momentum boost, namely 0.83, 1.25, 1.67 GeV. The GPDs are defined in the symmetric (Breit) frame, which we implement here with 4-vector momentum transfer squared of 0, 0.69 and 1.39 …
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 present an extraction of unpolarized partonic transverse momentum distributions (TMDs) from a simultaneous fit of available data measured in semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production. To connect data at different scales, we use TMD evolution at next-to-leading logarithmic accuracy. The analysis is restricted to the low-transverse-momentum region, with no matching to fixed-order calculations at high transverse momentum. We introduce specific choices to deal with TMD evolution at low scales, of the order of 1 GeV$^2$. This could be considered as a first attemp…
Partonic intrinsic transverse momentum can be essential for the explanation of large single spin asymmetries in hadronic reactions in the framework of perturbative QCD. The status of an ongoing program investigating in a consistent way the role of intrinsic transverse momentum both in unpolarized and polarized processes is discussed. We compute inclusive cross sections for hadron and photon production in hadronic collisions and for Drell-Yan processes; the results are compared with available experimental data in several different kinematical situations.
We discuss the general formalism for the calculation in light-front quark models of the fully unintegrated, off-diagonal quark-quark correlator of the nucleon, parametrized in terms of generalized transverse momentum dependent parton distributions (GTMDs). By taking specific limits or projections, these GTMDs yield various transverse-momentum dependent and generalized parton distributions, thus providing a unified framework to simultaneously model different observables. The corresponding distributions in impact-parameter space are the Wigner functions which provide multidimensional images of t…
The parton quasi-distribution functions (QDFs) of Ji have been found by Radyushkin to be directly related to the transverse momentum distributions (TMDs), to the pseudo-distributions, and to the Ioffe-time distributions (ITDs). This makes the QDF results at finite longitudinal momentum of the hadron interesting in their own right. Moreover, the QDF-TMD relation provides a gateway to the pertinent QCD evolution, with respect to the resolution scale Q, for the QDFs. Using the Kwieciński evolution equations and well established parameterizations at a low initial scale, we analyze the QCD evolutio…
We consider the prompt photon production in pp collisions using, within the framework of perturbative QCD, a non-Gaussian distribution for the transverse momentum distribution of the partons inside the proton. Our description adopts the widely used in the literature factorization of the partonic momentum distribution into longitudinal and transverse components. It is argued that the non-Gaussian distribution of the intrinsic transverse momenta of the partons is dictated by the asymptotic freedom as well as the 3D confinement of the partons in the proton. To make this association more transpare…
We provide a theoretical basis for understanding the spin structure of the proton in terms of the spin and orbital angular momenta of free quarks and gluons in Feynman's parton picture. We show that each term in the Jaffe-Manohar spin sum rule can be related to the matrix element of a gauge-invariant, but frame-dependent operator through a matching formula in large-momentum effective field theory. We present all the matching conditions for the spin content at one-loop order in perturbation theory, which provide a basis to calculate parton orbital angular momentum in lattice QCD at leading loga…
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…
We estimate the two-photon exchange contribution to elastic electron-proton scattering at large momentum transfer through the scattering off a parton in the proton. We relate the process on the nucleon to the generalized parton distributions which also enter in other wide angle scattering processes. We find that when taking the polarization transfer determinations of the form factors as input, adding in the 2 photon correction, does reproduce the Rosenbluth data.
We present a first attempt at a global fit of unpolarized quark transverse momentum dependent distribution and fragmentation functions from available data on semi-inclusive deep-inelastic scattering, Drell-Yan and $Z$ boson production processes. This analysis is performed in the low transverse momentum region, at leading order in perturbative QCD and with the inclusion of energy scale evolution effects at the next-to-leading logarithmic accuracy.
We evaluate nonperturbatively the quark quasidistribution amplitude and the valence quark quasidistribution function of the pion in the framework of chiral quark models, namely the Nambu--Jona-Lasinio model and the Spectral Quark Model. We arrive at simple analytic expressions, where the nonperturbative dependence on the longitudinal momentum of the pion can be explicitly assessed. The model results for the quark quasidistribution amplitude of the pion compare favorably to the data obtained from the Euclidean lattice simulations. The quark distribution amplitude, arising in the limit of infini…
We calculate $γZ$ exchange corrections to the parity-violating asymmetry of the elastic electron-proton scattering in a parton model using the formalism of generalized parton distributions(GPDs). We also examine the validity of the zero-momentum-transfer approximation adopted in the literatures and find that it overestimates the $γZ$ exchange effect significantly at the forward scattering angles.
We study momentum imbalance as a function of jet asymmetry in high-energy heavy-ion collisions. To implement parton production during the collision, we include all Leading Order (LO) $2\to 2$ and $2\to 3$ parton processes in pQCD. The produced partons lose energy within the quark gluon plasma and hadronize collinearly when they leave it. The energy and momentum deposited into the plasma is described using linear viscous hydrodynamics with a constant energy loss per unit length and a total energy loss given by a Gaussian probability centered around a mean value $\bar{\mathcal{E}}$ and a half-wi…
We argue that due to parity constraints, the helicity combination of the purely momentum space counterparts of the Wigner distributions -- the generalized transverse momentum distributions -- that describes the configuration of an unpolarized quark in a longitudinally polarized nucleon, can enter the deeply virtual Compton scattering amplitude only through matrix elements involving a final state interaction. The relevant matrix elements in turn involve light cone operators projections in the transverse direction, or they appear in the deeply virtual Compton scattering amplitude at twist three.…
Within the framework of transverse-momentum-dependent factorization, we investigate for the first time the impact of a flavor-dependent intrinsic transverse momentum of quarks on the production of $W^{\pm}$ bosons in proton-proton collisions at $\sqrt{s}$ = 7 TeV. We estimate the shift in the extracted value of the $W$ boson mass $M_W$ induced by different choices of flavor-dependent parameters for the intrinsic quark transverse momentum by means of a template fit to the transverse-mass and the lepton transverse-momentum distributions of the $W$-decay products. We obtain $-6\leq ΔM_{W^+} \leq …
We study kinematic effects due to the approximation of on-shell, collinear partons in shower Monte Carlo event generators. We observe that the collinearity approximation, combined with the requirements of energy-momentum conservation, gives rise to a kinematic shift, event by event, in longitudinal momentum distributions. We present numerical results in the case of jet and heavy flavor production processes measured at the LHC.
We present the first attempt to access the $x$-dependence of the gluon unpolarized parton distribution function (PDF), based on lattice simulations using the large-momentum effective theory (LaMET) approach. The lattice calculation is carried out with pion masses of 340 and 678 MeV on a 2+1-flavor DWF configuration with lattice spacing $a=0.111$~fm, for the gluon quasi-PDF matrix element with the nucleon momentum up to 0.93 GeV. Taking the normalization from similar matrix elements in the rest frame of the nucleon and pion, our results for these matrix elements are consistent with the Fourier …
We investigate the impact of three-body forces on the transverse-momentum distribution of partons inside the proton. This is achieved by considering the three-body problem in a class of hyper-central quark potential models. Solving the corresponding Schrödinger equation, we determine the quark wave function in the proton and with appropriate transformations and projections we find the transverse-momentum distribution of a single quark. In each case the parameters of the quark potentials are adjusted in order to sufficiently describe observable properties of the proton. Using a factorization an…
We present the first lattice-QCD calculation of the isovector polarized parton distribution functions (both helicity and transversity) using the large-momentum effective field theory (LaMET) approach for direct Bjorken-$x$ dependence. We first review the detailed steps of the procedure in the unpolarized case, then generalize to the helicity and transversity cases. We also derive a new mass-correction formulation for all three cases. We then compare the effects of each finite-momentum correction using lattice data calculated at $M_π\approx 310$ MeV. Finally, we discuss the implications of thes…
The dynamics of partons, hadrons and strings in relativistic nucleus-nucleus collisions is analyzed within the novel Parton-Hadron-String Dynamics (PHSD) transport approach, which is based on a dynamical quasiparticle model for partons (DQPM) matched to reproduce recent lattice-QCD results -- including the partonic equation of state -- in thermodynamic equilibrium. The transition from partonic to hadronic degrees of freedom is described by covariant transition rates for the fusion of quark-antiquark pairs or three quarks (antiquarks), respectively, obeying flavor current-conservation, color ne…
In large-momentum effective theory (LaMET), calculating parton physics starts from calculating coordinate-space-$z$ correlation functions $\tilde h(z, a,P^z)$ in a hadron of momentum $P^z$ in lattice QCD. Such correlation functions involve both linear and logarithmic divergences in lattice spacing $a$, and thus need to be properly renormalized. We introduce a hybrid renormalization procedure to match these lattice correlations to those in the continuum $\overline{\rm MS}$ scheme, without introducing extra non-perturbative effects at large $z$. We analyze the effect of ${\cal O}(Λ_{\rm QCD})$ a…
We present the first lattice-QCD calculation of the kaon distribution amplitude using the large-momentum effective theory (LaMET) approach. The momentum-smearing technique has been implemented to improve signals at large meson momenta. We subtract the power divergence due to Wilson line to high precision using multiple lattice spacings. The kaon structure clearly shows an asymmetry of the distribution amplitude around $x=1/2$, a clear sign of its skewness. We also study the leading SU(3) flavor symmetry breaking relations for the pion, kaon and eta meson distribution amplitudes, and the result…
We propose a revised definition of quasi-distributions within the framework of large-momentum effective theory (LaMET) that improves convergence towards the large-momentum limit. Since the definition of quasi-distributions is not unique, each choice goes along with a specific matching function, we can use this freedom to optimize convergence towards the large-momentum limit. As an illustration, we study quasi-distributions with a Gaussian weighting factor that naturally suppresses long-range correlations, which are plagued by artifacts. This choice has the advantage that the matching functions…
We review the latest progress in lattice QCD calculations of the partonic structure of hadrons. This structure is, in particular, described in terms of $x$-dependent distributions, the simplest of which are the standard parton distribution functions (PDFs). The lattice calculations rely on matrix elements probing spatial correlations between partons in a boosted hadron, that can be matched to light-cone correlations defining the relevant distributions. We discuss the recent theoretical and practical refinements of this strategy, as well as new exploratory directions. The latter include general…
We calculate the ${\cal O}(α_s^3)$ short-distance, QCD collinear-factorized coefficient functions for all partonic channels that include the production of a heavy quark pair at short distances. This provides the first power correction to the collinear-factorized inclusive hadronic production of heavy quarkonia at large transverse momentum, $p_T$, including the full leading-order perturbative contributions to the production of heavy quark pairs in all color and spin states employed in NRQCD treatments of this process. We discuss the role of the first power correction in the production rates and…
Transverse-momentum-dependent parton distribution and fragmentation functions describe the partonic structure of the nucleon in a three-dimensional momentum space. They are subjects of flourishing theoretical and experimental activity. They provide novel and intriguing information on hadronic structure, including evidence of the presence of partonic orbital angular momentum.
