We have presented the leading twist quark transverse momentum-dependent parton distribution functions (TMDs) for the spin-1 heavy vector mesons $J/ψ$-meson and $Υ$-meson using the overlap of the light-front wave functions. We have computed their TMDs in the light-front holographic model (LFHM) as well as the light-front quark model (LFQM) and further compared the results with the Bethe-Salpeter (BSE) model. We have discussed the behavior of the TMDs with respect to momentum fraction carried by active quark ($x$) and the transverse quark momenta ($k_\perp$) in both the models. We have also calculated the $k_\perp$ moments of the quark in both the models and have compared the results with the BSE model. The predictions of LFQM are found to be in accord with the BSE model. Further, we have analyzed the leading twist parton distribution functions (PDFs) for both the heavy mesons in both the models and the results are found to be in accord with the basic light-front quantization (BLFQ) and BSE model.
Transverse momentum dependent parton distribution functions, also abbreviated as TMDs, offer a three-dimensional picture of hadrons by taking the intrinsic transverse momentum of the parton into consideration. Hence, they are very important for us to understand the structure of hadrons. In this article, we calculate and summarize all TMDs of quark through the spectator model, from twist-2 to twist-4. Especially, we give complete results of TMDs at twist-4. We adopt a general analytical framework to calculate TMDs, with both scalar and axial-vector spectators being considered. All TMDs are calc…
The covariant parton model is generalized to describe quark correlators in a systematic way. Previous results are reproduced for the T-even leading-twist transverse momentum dependent parton distribution functions (TMDs), and for the first time all T-even twist-3 TMDs are evaluated in this model. We apply the approach to evaluate the fully unintegrated quark correlator which allows us to understand the model-specific relations between different TMDs. We verify the consistency of the approach, present numerical results and compare to available TMD parametrizations.
We present the leading-twist quark transverse momentum-dependent parton distribution functions (TMDs) for the spin-1 target, such as the $ρ$-meson, in the light-front framework. Specifically, we predict the TMDs in the light-front holographic model and compare with the light-front quark model predictions. We obtain the TMDs using the overlap of the light-front wave functions. We evaluate the ${\bf k}_\perp$ moments upto second order and compare with the available theoretical predictions. Further, we analyze the leading-twist parton distribution functions (PDFs) of the $ρ$-meson in the light-fr…
Transverse Momentum Dependent (TMD) parton distribution functions (PDFs), in short referred to as TMDs, also take into account the transverse momentum (pT) of the partons. Just as the pT-integrated analogues we want to link them to quark and gluon matrix elements using Operator Product Expansion methods in QCD, involving operators of definite twist. The TMDs also involve operators of higher twist, which are not suppressed by powers of the hard scale, however. Using the expression for TMDs involving nonlocal matrix elements of quark and gluon fields there is a gauge link dependence, which also …
Transverse-momentum-dependent parton distribution functions (TMDs) were investigated at the twists 3 and 4 for spin-1 hadrons in addition to the twist-2 ones. They were found by studying all the possible decomposition of a quark correlation function in a Lorentz-invariant way with the Hermiticity and parity invariance. The time-reversal invariance was not imposed for the TMDs due to an active role of gauge links; however, they were used for collinear parton distribution functions (PDFs) by integrating the TMDs over the transverse momentum. We found that 30 TMDs exist in the tensor-polarized sp…
The Drell-Yan hadronic tensor for electromagnetic (EM) current is calculated in the Sudakov region $s\gg Q^2\gg q_\perp^2$ with ${1\over Q^2}$ accuracy, first at the tree level and then with the double-log accuracy. It is demonstrated that in the leading order in $N_c$ the higher-twist quark-quark-gluon TMDs reduce to leading-twist TMDs due to QCD equation of motion. The resulting tensor for unpolarized hadrons is EM gauge-invariant and depends on two leading-twist TMDs: $f_1$ responsible for total DY cross section, and Boer-Mulders function $h_1^\perp$. The order-of-magnitude estimates of ang…
The Drell-Yan process is studied in the framework of TMD factorization in the Sudakov region $s\gg Q^2\gg q_\perp^2$ corresponding to recent LHC experiments with $Q^2$ of order of mass of Z-boson and transverse momentum of DY pair $\sim$ few tens GeV. The DY hadronic tensors are expressed in terms of quark and quark-gluon TMDs with ${1\over Q^2}$ and ${1\over N_c^2}$ accuracy. It is demonstrated that in the leading order in $N_c$ the higher-twist quark-quark-gluon TMDs reduce to leading-twist TMDs due to QCD equation of motion. The resulting hadronic tensors depend on two leading-twist TMDs: $…
We study the leading-twist time-reversal even transverse momentum dependent parton distribution functions (TMDs) of light and heavy vector mesons, i.e., the $ρ$, $J/ψ$ and $Υ$. We employ the leading Fock-state light front wave functions (LF-LFWFs) of $ρ$ and $J/ψ$ from our recent study, and supplement with $Υ$'s LF-LFWFs. These LF-LFWFs are extracted from dynamically solved Bethe-Salpeter wave functions. The vector meson TMDs are then studied with the light front overlap representation at leading Fock-state. All the obtained TMDs are non-vanishing and evolve with current quark mass, in particu…
The main properties of the leading-twist transverse momentum dependent parton distributions in a light-cone constituent quark model of the nucleon are reviewed, with focus on the role of the spin-spin and spin-orbit correlations of quarks. Results for azimuthal single spin asymmetries in semi-inclusive deep inelastic scattering are also discussed.
Role of higher twist distributions in the tomography of proton
Department of Physics, Dr. B.R. Ambedkar National Institute of Technology（物理系，B.R.安贝德卡尔国立理工学院）；Laboratory for Advanced Scientific Technologies of Mega-Science Facilities and Experiments, Moscow Institute of Physics and Technology (MIPT)（大型科学设施与实验先进技术实验室，莫斯科物理技术学院）
We explain a gluon transversity, transverse-momentum-dependent parton distribution functions (TMDs), and parton distribution functions (PDFs) for spin-1 hadrons. The gluon transversity exists in hadrons with spin more than or equal to one, and it does not exist in the spin-1/2 nucleons. Since there is no direct contribution from the nucleons, it is an appropriate quantity to probe an exotic component in the spin-1 deuteron beyond a simple bound system of the nucleons. We show how the gluon transversity can be measured at hadron accelerator facilities by the Drell-Yan process in addition to lep…
We revisit the model-independent decomposition of the gluon correlator, producing T-even and T-odd gluon transverse momentum distributions (TMDs), at leading twist. We propose an expansion of the gluon correlator, using a basis of four tensors (one antisymmetric and three symmetric), which are expressed through generators of the $U(2)$ group acting in the two-dimensional transverse plane. One can do clear interpretations of the two transversity T-odd TMDs with linear polarization of gluons: symmetric and asymmetric under permutation of the transverse spin of the nucleon and the transverse mome…
A generalized idea of gauge invariance, that embodies into the Wilson lines the spin-dependent Pauli term $\sim F^{μν}[γ_μ, γ_ν]$, is applied to set up a new framework for the operator definition of transverse-momentum-dependent parton densities (TMDs). We show that such a treatment of gauge invariance is justified, since it does not change the leading-twist behavior of the TMDs, albeit it contributes to their twist-three properties, in particular, to their anomalous dimensions. We discuss other consequences of this generalization and its possible applications to lattice simulations of the TMD…
Parton distributions in impact parameter space, which are obtained by Fourier transforming GPDs, exhibit a significant deviation from axial symmetry when target and/or quark are transversely polarized. In combination with the final state interactions, this transverse deformation provides a natural mechanism for naive-T odd transverse single-spin asymmetries in semi-inclusive DIS. The deformation of PDFs in impact parameter space can also be related to the transverse force acting on the active quark in polarized DIS at higher twist.
Generalized transverse-momentum dependent parton distributions (GTMDs) encode the most general parton structure of hadrons. Here we focus on two twist-2 GTMDs which are denoted by $F_{1,4}$ and $G_{1,1}$ in parts of the literature. As already shown previously, both GTMDs have a close relation to orbital angular momentum of partons inside a hadron. However, recently even the mere existence of $F_{1,4}$ and $G_{1,1}$ has been doubted. We explain why this claim does not hold. We support our model-independent considerations by calculating the two GTMDs in the scalar diquark model and in the quark-…
In high-energy scattering processes involving two or more hadrons one can measure observables that are sensitive to partonic transverse momentum, which is encoded in so-called transverse momentum dependent (TMD) parton distribution functions (PDFs), also called TMDs. These functions correspond to Fourier transforms of matrix elements that contain process-dependent gauge links. As the energy associated to the collision process increases, one becomes more sensitive to the small-$x$ region which is dominated by gluon rather than quark TMDs. In this paper we study the leading-twist gluon TMDs in t…
The understanding of the pion structure as described in terms of transverse-momentum dependent parton distribution functions (TMDs) is of importance for the interpretation of currently ongoing Drell-Yan experiments with pion beams. In this work we discuss the description of pion TMDs beyond leading twist in a pion model formulated in the light-front constituent framework. For comparison, we also review and derive new results for pion TMDs in the bag and spectator models.
We obtain the leading twist T-odd quark transverse momentum dependent parton distribution functions (TMDs) of the proton, namely the Sivers function, $f_{1T}^{\perp q}(x,p_\perp^2)$, and the Boer-Mulders function, $h_1^{\perp q}(x,p_\perp^2)$, in a light-front quark-diquark model constructed with the wave functions predicted by the soft-wall AdS/QCD. The gluon rescattering is crucial to predict a nonzero T-odd TMDs. We study the utility of a nonperturbative $SU(3)$ gluon rescattering kernel going beyond the usual approximation of perturbative $U(1)$ gluons. The spin asymmetries in semi-inclusi…
We consider the transverse momentum dependent gluon distribution functions (called gluon TMDs) by studying the light-front gluon-gluon correlator, extending the results for unpolarized and vector polarized targets to also include tensor polarized targets -- the latter type of polarization is relevant for targets of spin $\ge1$. The light-front correlator includes process-dependent gauge links to guarantee color gauge invariance. As from the experimental side the gluon TMDs are largely unknown, we present positivity bounds for combinations of leading-twist gluon distributions that may be used t…
The azimuthal distributions around the jet axis of leading pions produced in the jet fragmentation process in pp collisions are studied within the framework of the so-called generalized parton model. The observable leading-twist azimuthal asymmetries are estimated in kinematic configurations presently investigated at RHIC. It is shown how the main contributions coming from the Collins and Sivers effects can be disentangled. In addition, a test of the process dependence of the Sivers function is provided.
We predict the twist-2 Transverse Momentum Dependent parton distribution functions (TMDs) of the pion, namely the unpolarized quark TMD, $f_{1}(x, k_\perp)$, and the transversely polarized quark TMD, also known as the Boer-Mulders function, $h^\perp_{1}(x, k_\perp)$, using a holographic light-front pion wavefunction with dynamical spin effects. These spin effects, in conjunction with gluon rescattering, are crucial to predict a non-zero holographic Boer-Mulders function. We investigate the use of a non-perturbative SU(3) gluon rescattering kernel, thus going beyond the usual approximation of p…
Higher-twist transverse momentum dependent parton distribution functions (TMDs) are a valuable probe of the quark-gluon dynamics in the nucleon, and play a vital role for the explanation of sizable azimuthal asymmetries in hadron production from unpolarized and polarized deep-inelastic lepton-nucleon scattering observed in experiments at CERN, DESY and Jefferson Lab. The associated observables are challenging to interpret, and still await a complete theoretical explanation, which makes guidance from models valuable. In this work we establish the formalism to describe unpolarized higher-twist T…
We derive relations between transverse momentum dependent distribution functions (TMDs) and the usual parton distribution functions (PDFs) in the 3D covariant parton model, which follow from Lorentz invariance and the assumption of a rotationally symmetric distribution of parton momenta in the nucleon rest frame. Using the known PDFs f_1(x) and g_1(x) as input we predict the x- and pT-dependence of all twist-2 T-even TMDs.
We present a systematic study of twist-4 light-cone distribution amplitudes of vector mesons in QCD, which is based on conformal expansion. The structure of meson mass corrections is studied in detail. A complete set of distribution amplitudes is constructed, which satisfies all (exact) equations of motion and constraints from conformal expansion. Nonperturbative input parameters are estimated from QCD sum rules. Our study suggests that meson mass corrections may present a dominant source of higher twist effects in exclusive processes.
Leading and subleading twist transverse momentum dependent parton distribution functions (TMDs) are studied in a quark model framework provided by the bag model. A complete set of relations among different TMDs is derived, and the question is discussed how model-(in)dependent such relations are. A connection of the pretzelosity distribution and quark orbital angular momentum is derived. Numerical results are presented, and applications for phenomenology discussed. In particular, it is shown that in the valence-x region the bag model supports a Gaussian Ansatz for the transverse momentum depend…
We show possible transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons including twist-3 and 4 functions in addition to the leading twist-2 ones by investigating all the possible decomposition of a quark correlation function in the Lorentz-invariant way. The Hermiticity and parity invariance are imposed in the decomposition; however, the time-reversal invariance is not used due to an active role of gauge links in the TMDs. Therefore, there exist time-reversal-odd functions in addition to the time-reversal even ones in the TMDs. We list all the functions up to tw…
We investigate the leading-twist transverse momentum-dependent distribution functions (TMDs) for a physical electron, a spin-1/2 composite system consisting of a bare electron and a photon, using the Basis Light-front Quantization (BLFQ) framework. The light-front wave functions of the physical electron are obtained from the eigenvectors of the light-front QED Hamiltonian. We evaluate the TMDs using the overlaps of the light-front wave functions. The BLFQ results are found to be in excellent agreement with those TMDs calculated using lowest-order perturbation theory.
We study the small-$x$ asymptotics of the flavor non-singlet T-odd leading-twist quark transverse momentum dependent parton distributions (TMDs), the Sivers and Boer-Mulders functions. While the leading eikonal small-$x$ asymptotics of the quark Sivers function is given by the spin-dependent odderon, we are interested in revisiting the sub-eikonal correction considered by us earlier. We first simplify the expressions for both TMDs at small Bjorken $x$ and then construct small-$x$ evolution equations for the resulting operators in the large-$N_c$ limit, with $N_c$ the number of quark colors. Fo…
We present a covariant formulation and model calculations of transverse momentum-dependent quark distribution functions (TMDs) for spin-one hadrons. Emphasis is placed on a description of these 3-dimensional distribution functions which is independent of any constraints on the spin quantization axis. We apply our covariant spin description to all nine leading-twist time-reversal even $ρ$ meson TMDs in the framework provided by the Nambu--Jona-Lasinio model, incorporating important aspects of quark confinement via the infrared cut-off in the proper-time regularization scheme. In particular, the…
We derive relations between polarized transverse momentum dependent distribution functions (TMDs) and the usual parton distribution functions (PDFs) in the 3D covariant parton model, which follow from Lorentz invariance and the assumption of a rotationally symmetric distribution of parton momenta in the nucleon rest frame. Using the known PDF $g_{1}^{q}(x)$ as input we predict the $x$- and $\mathbf{p}_{T}$-dependence of all polarized twist-2 naively time-reversal even (T-even) TMDs.
We present a systematic study of twist three light-cone distribution amplitudes of vector mesons in QCD, which is based on conformal expansion and takes into account meson and quark mass corrections. A complete set of distribution amplitudes is constructed for rho, omega, K* and phi mesons, which satisfies all (exact) equations of motion and constraints from conformal expansion. Nonperturbative input parameters are calculated from QCD sum rules, including an update of SU(3) breaking corrections in the leading twist distributions.
We show that in the large N_c limit four-quark twist-4 distributions in the pion can be expressed in terms of twist-2 pion distribution amplitude. This allows us to compute the isospin-2 structure function of the pion F_2^{I=2}(x_B) in the large N_c limit. The method can be easily applied to other mesons as well.
Two- and three-particle distribution amplitudes of heavy pseudoscalar mesons of well-defined geometric twist are introduced. They are obtained from appropriately parametrized vacuum-to-meson matrix elements by applying those twist projectors which determine the enclosed light-cone operators of definite geometric twist and, in addition, observing the heavy quark constraint. Comparing these distribution amplitudes with the conventional ones of dynamical twist we derive relations between them, partially being of Wandzura-Wilczek type; also sum rules of Burkhardt-Cottingham type are derived.The de…
We present a systematic study of twist-3 light-cone distribution amplitudes of $K^*$ and $ϕ$ mesons in QCD. The structure of SU(3)-breaking corrections is studied in detail. Non-perturbative input parameters are estimated from QCD sum rules. As a by-product, we update the parameters describing the twist-3 distribution amplitudes of the $ρ$ meson. We also review and update predictions for the twist-2 distribution amplitudes of $ρ$, $K^*$ and $ϕ$.
We present all the leading twist T-even TMDs in the light-front quark diquark model(LFQDM) and study the relations among them. The model contains both the scalar and vector diquarks with the light front wave functions modeled from the soft-wall AdS/QCD prediction. The $x-p^2_\perp$ factorization used in phenomenological extraction for TMDs is observed in this model. We present the results for the quark densities and the transverse shape of proton. The shape of the transversely polarized proton is shown to be non-spherical for nonzero transverse momentum. The scale evolution of both integrated …
We discuss the two and three particle light-cone distribution amplitudes (DAs) of pseudoscalar nonsinglet mesons of twist 3 and 4. Using nonlocal operator identities and conformal expansion, we derive closed expressions for several DAs. We also include meson-mass corrections which prove to be dominant in the twist 4 DAs of K and eta mesons. Explicit parametrizations for the DAs of pi, K and eta mesons are given, with the numerical input parameters determined from QCD sum rules.
We present a systematic study of twist-3 and twist-4 light-cone distribution amplitudes of the K meson in QCD. The structure of SU(3)-breaking corrections is studied in detail. Non-perturbative input parameters are estimated from QCD sum rules and renormalons. As a by-product, we give a complete reanalysis of the twist-3 and -4 parameters of the pi-meson distribution amplitudes; some of the results differ from those usually quoted in the literature.
We present a systematic study of twist-4 light-cone distribution amplitudes of the $K^*$ and $ϕ$ meson in QCD. The structure of SU(3)-breaking corrections is studied in detail. Non-perturbative input parameters are estimated from QCD sum rules and a renormalon based model. As a by-product, we give a complete reanalysis of the parameters of the twist-4 $ρ$-meson distribution amplitudes.
We analyze light meson electroproduction within the handbag model. We study cross sections and spin asymmetries for various mesons. The essential role of the transversity $\tilde H_T$ and $\tilde E_T$ GPDs in electroproduction of pseudoscalar mesons is found. Our results are in good agrement with experiment.
