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 spin-1 hadron at the twists 3 and 4, whereas there are 10 TMDs at the twist 2. We also showed that there are 3 collinear PDFs at the twists 3 and 4. The corresponding TMD fragmentation functions exist sat the twists 2, 3, and 4 simply by changing function names and variables. Since the time-reversal invariance is valid in the collinear PDFs, the integrals of time-reversal-odd TMDs over the transverse momentum should vanish. It leads to the sum rules $ \int d^2 k_T \, h_{1LT} (x, k_T^{\, 2}) = \int d^2 k_T g_{LT} (x, k_T^{\, 2}) = \int d^2 k_T h_{LL} (x, k_T^{\, 2}) = \int d^2 k_T h_{3LL}(x, k_T^{\, 2}) =0$ on the time-reversal odd TMDs at the twists 3 and 4.
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 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…
In this paper we consider the parametrizations of gluon transverse momentum dependent (TMD) correlators in terms of TMD parton distribution functions (PDFs). These functions, referred to as TMDs, are defined as the Fourier transforms of hadronic matrix elements of nonlocal combinations of gluon fields. The nonlocality is bridged by gauge links, which have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. For gluons, the specific correlator with one future and one past pointing gauge link is, in the limit of small $x$, related to a cor…
We show transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons including twist-3 and 4 functions by taking the decomposition of a quark correlation function in the Lorentz-invariant way with the conditions of Hermiticity and parity invariance. We found 30 new TMDs in the tensor-polarized spin-1 hadron at twists 3 and 4 in addition to 10 TMDs at twist 2. Since time-reversal-odd terms of the collinear correlation function should vanish after integrals over the partonic transverse momentum, we obtained new sum rules for the time-reversal-odd structure functions, $\b…
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…
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…
Generalised parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs) describe complementary aspects of the three-dimensional structure of hadrons. We discuss their relation to each other and recall important theory results concerning their properties and their connection with physical observables.
We present an overview on the current experimental and phenomenological status of transverse single spin asymmetries (tSSAs) in proton-proton collisions. In particular, we focus on large-$p_T$ inclusive pion, photon, jet, pion-jet production and Drell-Yan processes. For all of them theoretical estimates are given in terms of a generalised parton model (GPM) based on a transverse momentum dependent (TMD) factorisation scheme. Comparisons with the corresponding results in a collinear twist-3 formalism and in a modified GPM approach are also made. On the experimental side, a selection of the most…
Generalized transverse momentum dependent parton distributions (GTMDs) are the most general parton correlation functions of hadrons. By considering the exclusive double Drell-Yan process it is shown for the first time how quark GTMDs can be measured. Specific GTMDs can be addressed by means of polarization observables.
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 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…
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…
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…
Transverse momentum dependent parton distribution functions (TMDPDFs) provide a unique probe of the three-dimensional spin structure of hadrons. We construct spin-dependent quasi-TMDPDFs that are amenable to lattice QCD calculations and that can be used to determine spin-dependent TMDPDFs. We calculate the short-distance coefficients connecting spin-dependent TMDPDFs and quasi-TMDPDFs at one-loop order. We find that the helicity and transversity distributions have the same coefficient as the unpolarized TMDPDF. We also argue that the same is true for pretzelosity and that this spin universalit…
The focus of this talk is on the transverse components of parton momenta. Like for collinear parton distribution functions (PDFs), we are also in the case of transverse momentum dependent (TMD) PDFs, talking about forward matrix elements. While the collinear PDFs describe only spin-spin correlations, the TMD PDFs (or in short TMDs) include spin-momentum correlations, including also time-reversal-odd (T-odd) correlations. The latter are important in the description of single spin asymmetries. In this way TMDs open up new ways of studying the spin structure of hadrons or they can be used as tool…
The CASCADE3 Monte Carlo event generator based on Transverse Momentum Dependent (TMD) parton densities is described. Hard processes which are generated in collinear factorization with LO multileg or NLO parton level generators are extended by adding transverse momenta to the initial partons according to TMD densities and applying dedicated TMD parton showers and hadronization. Processes with off-shell kinematics within $k_t$-factorization, either internally implemented or from external packages via LHE files, can be processed for parton showering and hadronization. The initial state parton sho…
We study the Wigner functions of the nucleon which provide multidimensional images of the quark distributions in phase space and combine in a single picture all the information contained in the generalized parton distributions (GPDs) and the transverse-momentum dependent parton distributions (TMDs). In particular, we present results for the distribution of unpolarized quarks in a longitudinally polarized nucleon obtained in a light-cone constituent quark model. Treating the Wigner distribution as it was a classical distribution, we obtain the quark orbital angular momentum and compare it with …
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.
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-…
The covariant parton model (CPM) is a consequent application of the parton model concept to the nucleon structure. In this model, there is a choice to put quarks either in a pure-spin state or in a mixed-spin state. We show that the mixed-spin version of the CPM does not support the quark-model relations among transverse momentum dependent parton distributions (TMDs) which were shown to hold in a large class of quark models. One can enforce the quark-model relations to be valid in the CPM by imposing a condition which is equivalent to putting the quarks in a pure-spin state. This gives a compl…
Quarkonium production offers good possibilities to study gluon TMDs. In this proceedings contribution this topic is explored for the linearly polarized gluons inside unpolarized hadrons and unpolarized gluons inside transversely polarized hadrons. It is argued that $χ_{b0/2}$ and $η_b$ production at LHC are best to study the effects of linearly polarized gluons in hadronic collisions, by means of angular independent ratios of ratios of cross sections. This can be directly compared to $\cos 2ϕ$ asymmetries in heavy quark pair and dijet production in DIS at a future high-energy Electron-Ion Coll…
A common library, TMDlib2, for Transverse-Momentum-Dependent distributions (TMDs) and unintegrated parton distributions (uPDFs) is described, which allows for easy access of commonly used TMDs and uPDFs, providing a three-dimensional (3D) picture of the partonic structure of hadrons. The tool TMDplotter allows for web-based plotting of distributions implemented in TMDlib2, together with collinear pdfs as available in LHAPDF.
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: $…
All (un)polarized transverse momentum dependent functions (TMDs), both distribution and fragmentation functions, are defined with the same universal soft function, which cancels spurious rapidity divergences within an individual TMD and renders them well-defined hadronic quantities. Moreover, it is independent of the kinematics, whether it is Drell-Yan, deep inelastic scattering or $e^+e^-\rightarrow 2$ hadrons. In this paper we provide this soft function at next-to-next-to-leading order (NNLO), necessary for the calculation of all TMDs at the same order, and to perform the resummation of larg…
We present a summary of a recent workshop held at Duke University on Partonic Transverse Momentum in Hadrons: Quark Spin-Orbit Correlations and Quark-Gluon Interactions. The transverse momentum dependent parton distribution functions (TMDs), parton-to-hadron fragmentation functions, and multi-parton correlation functions, were discussed extensively at the Duke workshop. In this paper, we summarize first the theoretical issues concerning the study of partonic structure of hadrons at a future electron-ion collider (EIC) with emphasis on the TMDs. We then present simulation results on experimenta…
Transverse momentum dependent (TMD) distribution correlators can be parametrized in terms of TMD parton distribution functions (PDFs), or TMDs for short. We provide an overview of the leading-twist quark and gluon TMDs, both in momentum space and in coordinate space (also called $b_T$-space, where $b_T$ is Fourier conjugate to the partonic momentum $k_T$). We consider unpolarized, vector polarized, as well as tensor polarized hadrons, which is relevant for spin-$0$, spin-$1/2$, and spin-$1$ hadrons.
Previous Lattice QCD calculations of nucleon transverse momentum-dependent parton distributions (TMDs) focused on the case of transversely polarized nucleons, and thus did not encompass two leading-twist TMDs associated with longitudinal polarization, namely, the helicity TMD and the worm-gear TMD corresponding to transversely polarized quarks in a longitudinally polarized nucleon. Based on a definition of TMDs via hadronic matrix elements of quark bilocal operators containing staple-shaped gauge connections, TMD observables characterizing the aforementioned two TMDs are evaluated, utilizing a…
We discuss the possibility of non-minimal gauge invariance of transverse-momentum-dependent parton densities (TMDs) that allows direct access to the spin degrees of freedom of fermion fields entering the operator definition of (quark) TMDs. This is achieved via enhanced Wilson lines that are supplied with the spin-dependent Pauli term $\sim F^{μν}[γ_μ, γ_ν]$, thus providing an appropriate tool for the "microscopic" investigation of the spin and color structure of TMDs. We show that this generalization leaves the leading-twist TMD properties unchanged but modifies those of twist three by contri…
We present studies of double longitudinal spin asymmetries in semi-inclusive deep inelastic scattering using a new dedicated Monte Carlo generator, which includes quark intrinsic transverse momentum within the generalized parton model based on the fully differential cross section for the process. Additionally we employ Bessel-weighting to the MC events to extract transverse momentum dependent parton distribution functions and also discuss possible uncertainties due to kinematic correlation effects.
Three-dimensional tomography of hadrons can be investigated by generalized parton distributions (GPDs), transverse-momentum-dependent parton distributions (TMDs), and generalized distribution amplitudes (GDAs). The GDA studies had been only theoretical for a long time because there was no experimental measurement until recently, whereas the GPDs and TMDs have been investigated extensively by deeply virtual Compton scattering and semi-inclusive deep inelastic scattering. Here, we report our studies to determine pion GDAs from recent KEKB measurements on the differential cross section of $γ^* γ\…
Using our formalism of parton virtuality distribution functions (VDFs) we establish a connection between the transverse momentum dependent distributions (TMDs) ${\cal F} (x, k_\perp^2)$ and quasi-distributions $Q(y,P_z)$ introduced recently by X. Ji for lattice QCD extraction of parton distributions $f(x)$. We build models for PQDs from the VDF-based models for soft TMDs, and analyze the $P_z$ dependence of the resulting PQDs. We observe a strong nonperturbative evolution of PQDs for small and moderately large values of $P_z$ reflecting the transverse momentum dependence of TMDs. Thus, the stu…
In a high-energy scattering process with hadrons in the initial state, color is involved. Transverse momentum dependent distribution functions (TMDs) describe the quark and gluon distributions in these hadrons in momentum space with the inclusion of transverse directions. Apart from the (anti)-quarks and gluons that are involved in the hard scattering process, additional gluon emissions by the hadrons have to be taken into account as well, giving rise to Wilson lines or gauge links. The TMDs involved are sensitive to the process under consideration and hence potentially nonuniversal due to the…
The Drell-Yan process provides important information on the internal structure of hadrons including transverse momentum dependent parton distribution functions (TMDs). In this work we present calculations for all leading twist structure functions describing the pion induced Drell-Yan process. The non-perturbative input for the TMDs is taken from the light-front constituent quark model, the spectator model, and available parametrizations of TMDs extracted from the experimental data. TMD evolution is implemented at Next-to-Leading Logarithmic precision for the first time for all asymmetries. Our…
I report on our investigations into the impact of (un)polarized transverse momentum dependent parton distribution functions (TMD PDFs or TMDs) for gluons at hadron colliders, especially at A Fixed Target Experiment at the LHC (AFTER@LHC). In the context of high energy proton-proton collisions, we look at final states with low mass (e.g. $η_b$) in order to investigate the nonperturbative part of TMD PDFs. We study the factorization theorem for the $q_T$ spectrum of $η_b$ produced in proton-proton collisions relying on the effective field theory approach, defining the tools to perform phenomenol…
We present studies of double longitudinal spin asymmetries in semi-inclusive deep inelastic scattering using a new dedicated Monte Carlo generator, which includes quark intrinsic transverse momentum within the generalized parton model based on the fully differential cross section for the process. Additionally, we apply Bessel-weighting to the simulated events to extract transverse momentum dependent parton distribution functions and also discuss possible uncertainties due to kinematic correlation effects.
We discuss the dependence on the flavor of the intrinsic transverse momentum in unpolarized transverse-momentum-dependent distribution functions (TMD PDFs) and fragmentation functions (TMD FFs) analyzing data of semi-inclusive deep-inelastic scattering (SIDIS) released by the HERMES collaboration. Adopting a flavor-dependent Gaussian model in the transverse momentum and neglecting QCD evolution, we find interesting evidences concerning the flavor dependence in TMD FFs, whereas the indications are weaker in TMD PDFs and deserve further investigations. Inclusions of new data sets of SIDIS, elect…
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…
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.
We study the leading-power gluon transverse momentum dependent distributions (TMDs) of relevance to the study of asymmetries in the scattering off transversely polarized hadrons. Next-to-leading-order perturbative calculations of these TMDs show that at large transverse momentum they have common dynamical origins, but that in the limit of small longitudinal momentum fraction $x$ only one origin remains. We find that in this limit only the dipole-type gluon TMDs survive and become identical to each other. At small $x$ they are all given by the expectation value of a single Wilson loop inside th…
Transverse momentum dependent (TMD) parton correlators describing the partonic structure of hadrons contain gauge links, required by color gauge invariance. The required gauge links enter in the matrix elements that contain the parton fields and depend on the color flow in the hard process. The correlators are expanded in terms of transverse momentum dependent parton distribution functions, referred to as TMD PDFs, or in short TMDs. In this paper, we introduce gluon TMDs of definite rank, by making an expansion of the TMD gluon correlator with the help of irreducible tensors built from the tra…
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 investigate the transverse momentum dependent parton distributions (TMDs) in the quasi-parton-distribution framework. The long standing hurdle of the so-called pinch pole singularity from the space-like gauge links in the TMD definitions can be resolved by the finite length of the gauge link along the hadron moving direction. In addition, with the soft factor subtraction, the quasi-TMD is free of linear divergence. We further demonstrate that the energy evolution equation of the quasi-TMD a.k.a. the Collins-Soper evolution, only depends on the hadron momentum. This leads to a clear matching…
This is a brief overview of the spin physics opportunities at a high energy, high luminosity, polarized Electron-Ion Collider (EIC). It covers measurements of electroweak polarized structure functions, quark and gluon PDFs, TMDs, GPDs and GTMDs. Exploiting the many possible final states allows to probe various spin effects. Open and bound heavy quark production can be used to probe gluon TMDs, but also color-octet NRQCD long distance matrix elements. Spin-dependent fragmentation functions can be used too, but are also interesting in themselves. Especially for studies of the small-$x$ and the h…
We study the transverse momentum dependent (TMD) parton distributions in the newly proposed quasi-parton distribution function framework in Euclidean space. A soft factor subtraction is found to be essential to make the TMDs calculable on lattice. We show that the quasi-TMDs with the associated soft factor subtraction can be applied in hard QCD scattering processes such as Drell-Yan lepton pair production in hadronic collisions. This allows future lattice calculations to provide information on the non-perturbative inputs and energy evolutions for the TMDs. Extension to the generalized parton d…
Transverse-momentum dependent parton distributions (TMDs) are studied in the framework of quark models. In particular, quark-model relations among TMDs are reviewed, elucidating their physical origin in terms of the quark-spin structure in the nucleon. The formal aspects of the derivation of these relations are complemented with explicit examples, emphasizing how and to which extent the conditions which lead to relations among TMDs are implemented in different classes of quark models.
The three-dimensional momenta of quarks inside a hadron are encoded in transverse momentum-dependent parton distribution functions (TMDs). This work presents an exploratory lattice QCD study of a TMD observable in the pion describing the Boer-Mulders effect, which is related to polarized quark transverse momentum in an unpolarized hadron. Particular emphasis is placed on the behavior as a function of a Collins-Soper evolution parameter quantifying the relative rapidity of the struck quark and the initial hadron, e.g., in a semi-inclusive deep inelastic scattering (SIDIS) process. The lattice c…
