We discuss the recent interpretation of quark-distribution functions in the plane transverse to the light-cone direction. Such a mapping is model independent and allows one to build up multidimensional pictures of the hadron and to develop a semi-classical interpretation of the quark dynamics. We comment briefly the results obtained from the form factors of the nucleon. We show that a generalization to a target with arbitrary spin leads to a set of preferred values for the electromagnetic coupling characterizing structureless particles. Finally, we present the Wigner distribution for an unpolarized quark in an unpolarized proton and we propose an interpretation of the observed distortion as due to the orbital angular momentum of the quark.
Through an impulse approximation analysis of single spin Sivers and Collins asymmetries in the Bjorken limit, the possibility to extract the quark transverse-momentum distributions in the neutron from semi-inclusive deep inelastic electron scattering off polarized 3He is illustrated. The analysis is generalized to finite momentum transfers in a light-front Poincare' covariant framework, defining the light-front spin-dependent spectral function of a J=1/2 system. The definition of the light-front spin-dependent spectral function for constituent quarks in the nucleon allows us to show that, with…
The use of light front coordinates allows a fully relativistic description of a hadron's spatial densities to be obtained. These densities must be two-dimensional and transverse to a chosen spatial direction. We explore their relationship to the three-dimensional, non-relativistic densities, with a focus on densities associated with the energy momentum tensor. The two-dimensional non-relativistic densities can be obtained from the light front densities through a non-relativistic limit, and can subsequently be transformed into three-dimensional non-relativistic densities through an inverse Abel…
We explore the structure of the spin-1/2 flavor-octet baryons (hyperons) through their electromagnetic transverse densities. The transverse densities describe the distribution of charge and magnetization at fixed light-front time and enable a spatial representation of the baryons as relativistic systems. At peripheral distances b~1/M_pi the transverse densities are computed using a new method that combines chiral effective field theory and dispersion analysis. The peripheral isovector densities arise from two-pion exchange, which includes the rho-meson resonance through elastic unitarity. The …
The nucleon's electromagnetic form factors are expressed in terms of the transverse densities of charge and magnetization at fixed light-front time. At peripheral transverse distances $b = O(M_π^{-1})$ the densities are governed by chiral dynamics and can be calculated model-independently using chiral effective field theory (EFT). We represent the leading-order chiral EFT results for the peripheral transverse densities as overlap integrals of chiral light-front wave functions, describing the transition of the initial nucleon to soft pion-nucleon intermediate states and back. The new representa…
In the light-front description of nucleon structure the electromagnetic form factors are expressed in terms of frame-independent transverse densities of charge and magnetization. Recent work has studied the transverse densities at peripheral distances b = O(M_pi^{-1}), where they are governed by universal chiral dynamics and can be computed in a model-independent manner. Of particular interest is the comparison of the peripheral charge and magnetization densities. We summarize (a) their interpretation as spin-independent and -dependent current matrix elements; (b) the leading-order chiral effe…
In the partonic (or light-front) description of relativistic systems the electromagnetic form factors are expressed in terms of frame-independent charge and magnetization densities in transverse space. This formulation allows one to identify the chiral components of nucleon structure as the peripheral densities at transverse distances b = O(M_pi^{-1}) and compute them in a parametrically controlled manner. A dispersion relation connects the large-distance behavior of the transverse charge and magnetization densities to the spectral functions of the Dirac and Pauli form factors near the two-pio…
Transverse densities describe the distribution of charge and current at fixed light-front time and provide a frame-independent spatial representation of hadrons as relativistic systems. We calculate the transverse densities of the octet baryons at peripheral distances b = O(M_pi^{-1}) in an approach combining chiral effective field theory (ChEFT) and dispersion analysis. The densities are represented as dispersive integrals of the imaginary parts of the baryon electromagnetic form factors in the timelike region (spectral functions). The spectral functions on the two-pion cut at t > 4 M_pi^2 ar…
Electromagnetic form factors obtained from the overlap of light front wave functions (LFWFs) have been used to study the transverse densities of charge and magnetization. The calculations have been carried out to develop a relation between the charge distribution of the quarks inside nucleon in the transverse coordinate space as well as in the impact parameter space. When a comparison is carried out, it is found that the transverse distribution in the impact parameter space, where the longitudinal momentum fraction $x$ can be fixed, falls off faster than the spatial distribution in the transve…
The light front densities of momentum, angular momentum, and intrinsic pressure are calculated for the photon, both in the free case and at leading order in quantum electrodynamics. In the latter case, we moreover decompose the form factors into photon and electron contributions. Circularly and linearly polarized photons are both considered, with the latter containing significant azimuthal modulations in both the momentum density and in intrinsic stresses. We find that the D-term of the photon is positive instead of negative, and accordingly the intrinsic radial pressure of the photon is negat…
The transverse densities in a fast-moving nucleon offer a model-independent framework for analyzing the spatial structure of the pion cloud and its role in current matrix elements. We calculate the chiral large-distance component of the charge density using a dispersion representation of the form factor and discuss its partonic interpretation. The non-chiral core is dominant up to surprisingly large distances ~2 fm. The chiral component can be probed in precision low-Q^2 elastic eN scattering or in peripheral deep-inelastic processes which resolve its quark/gluon content.
The deuteron transverse charge density $ρ_C(b)$ is the two-dimensional Fourier transform of its charge form factor in the impact space. We show that different parameterizations of the charge form factors provide different $ρ_C(b)$, in particular at the central value of impact parameter ($b=0$), although all the parameterizations can well reproduce the form factors in the region of small $Q^2$. In addition, we also check the explicit contributions from the different coordinate intervals of the deuteron wave function to its root-mean-square radius.
We discuss the recent interpretation of quark distribution functions in the plane transverse to the light-cone direction. Such a mapping is model independent and allows one to build multidimensional pictures of the hadron and to develop a semi-classical intuition of the quark dynamics. We comment briefly the results obtained from the Form Factors of the nucleon. A generalization to a target with arbitrary spin led to a set of preferred values for the electromagnetic coupling characterizing structureless particles. Generalized polarizabilities can also be interpreted in that frame as the distor…
Transverse densities describe the spatial distribution of electromagnetic current in the nucleon at fixed light-front time. At peripheral distances b = O(M_pi^{-1}) the densities are governed by chiral dynamics and can be calculated model-independently using chiral effective field theory (EFT). Recent work has shown that the EFT results can be represented in first-quantized form, as overlap integrals of chiral light-front wave functions describing the transition of the nucleon to soft-pion-nucleon intermediate states, resulting in a quantum-mechanical picture of the peripheral transverse densi…
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.
I review some open questions relating to the large transverse momentum divergences in transverse moments of transverse momentum dependent (TMD) parton correlation func- tions. I also explain, in an abbreviated and summarized form, recent work that shows that the resulting violations of a commonly used integral relation are not perturbatively suppressed. I argue that this implies a need for more precise definitions for the correlation functions used to describe transverse moments.
Electromagnetic form factors have long been used to probe the underlying charge and magnetization densities of hadrons and nuclei. Traditional three-dimensional Fourier transform methods are not rigorously applicable for systems with constituents that move relativistically. The use of the transverse charge density is a new, rigorously defined way to analyze electromagnetic form factors of hadrons.This review is concerned with the following issues: what is a transverse charge density; how is one extracted one from elastic scattering data; the existing results; what is the relationship with othe…
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 …
The semi-inclusive correlator for a ${\cal J}$=1/2 bound-system, composed by A spin-1/2 fermions, is linearly expressed in terms of the light-front Poincaré covariant spin-dependent spectral function, in valence approximation. The light-front spin-dependent spectral function is fully determined by six scalar functions that allow for a complete description of the six T-even transverse-momentum distributions, suitable for a detailed investigation of the dynamics inside the bound system. The application of the developed formalism to a case with a sophisticated dynamical content, like $^3$He, reac…
We present a study of longitudinal momentum densities ($p^+ \rm densities$) in the transverse impact parameter space for $u$ and $d$ quarks in both unpolarized and transversely polarized nucleons by taking a two dimensional Fourier transform of the gravitational form factors with respect to the momentum transfer in the transverse direction. The gravitational form factors are obtained by the second moments of GPDs. Here we consider the GPDs of two different soft-wall models in AdS/QCD correspondence.
We have investigated the transverse charge density for longitudinal and transversely polarized $ρ$ meson in light-front quark model (LFQM). Charge densities are obtained from the elastic form factors of the $ρ$ meson calculated in LFQM including the zero-mode contributions. We have computed the different helicity matrix elements of the $ρ$ meson. In addition to this, we have also presented the results for the generalized parton distributions (GPDs) and impact-parameter dependent parton distribution functions (ipdpdfs) of the $ρ$ meson.
The possibility to extract the quark transverse-momentum distributions in the neutron from semi-inclusive deep inelastic electron scattering off polarized 3He is illustrated through an impulse approximation analysis in the Bjorken limit. The generalization of the analysis at finite momentum transfers in a Poincare' covariant framework is outlined. The definition of the light-front spin-dependent spectral function of a J=1/2 system allows us to show that within the light-front dynamics only three of the six leading twist T-even transverse-momentum distributions are independent.
We investigate the transverse charge densities of the deuteron in a soft-wall AdS/QCD model by considering both the unpolarized and the transversely polarized cases. The deuteron form factors are derived from the soft-wall AdS/QCD model and it is shown that the calculated results agree with the phenomenological parametrization and experimental data. The transverse charge densities of the deuteron are also derived from the soft-wall AdS/QCD model, and we notice slight deviations from those of the phenomenological parametrization.
We review the current status of collinear and Transverse Momentum Dependent densities based on the Parton Branching method. We investigate the performance of the PB-TMD evolution as well as PB-TMD parton shower with different configurations of the hard process in the four- and five-flavour schemes.
We propose a consistent treatment of divergences emerging in the computation of transverse-momentum-dependent parton densities in leading $α_s$-order of QCD perturbation theory.
We present a study of flavor structures of transverse charge and anomalous magnetization densities for both unpolarized and transversely polarized nucleons. We consider two different models for the electromagnetic form factors in holographic QCD. The flavor form factors are obtained by decomposing the Dirac and Pauli form factors for nucleons using the charge and isospin symmetry. The results are compared with two standard phenomenological parametrizations.
We investigated the transverse charge densities in the nucleon in nuclear matter within the framework of the in-medium modified Skyrme model. The medium modification of the nucleon electromagnetic form factors are first discussed. The results show that the form factors in nuclear matter fall off faster than those in free space, as the momentum transfer increases. As a result, the charge radii of the nucleon become larger, as the nuclear matter density increases. The transverse charge densities in the nucleon indicate that the size of the nucleon tends to bulge out in nuclear matter. This salie…
We investigate the transverse charge density in the longitudinally as well as transversely polarized deuteron using the recent empirical description of the deuteron electromagnetic form factors in the framework of holographic QCD. The predictions of the holographic QCD are compared with the results of a standard phenomenological parameterization. In addition, we evaluate GPDs and the gravitational form factors for the deuteron. The longitudinal momentum densities are also investigated in the transverse plane.
A specific value of light front variable $ζ_c(p_T,m)$ of the inclusively produced hadrons in a relativistic nuclear collision is constructed as a polynomial in transverse momentum $p_T$ of the particle within the UrQMD transport model. It is shown that those particles with the absolute value of their light front variable $ζ$ greater than the corresponding $ζ_c(p_T,m)$ follow a Fermi-Dirac or Bose-Einstein distribution depending upon the spin of the particle.
This talk provides a short overview of the phenomenology of transverse momentum depedent distribution and fragmention functions, focussing on the most recent phenomenological developments in the study of their scale evolution and energy depedence.
Using the light-front holographic model, we study the transverse momentum dependent parton distributions (TMDs) for the case of pion. At leading twist, the unpolarized parton distribution function $ f_{1π}(x,\bfk^{2}) $ and the Boer-Mulders function $ h_{1π}^{\bot}(x,\bfk^{2}) $ are obtained for pion. We calculate both the functions using the light-front holographic model with spin improved wave function and compare the predicted results with available results of other models. In order to provide inputs in predicting future experimental data, a LO evolution is performed from model scale to exp…
The current status of the theoretical understanding of the transverse-momentum dependent parton densities (TMDs) is discussed. Special attention is payed to the difference between the operator definitions of TMDs proposed so far, the treatment of specific divergences, the geometry of the gauge links, and the role of the soft factors.
We investigate the gravitational form factors (GFFs) and the longitudinal momentum densities ($p^+$ densities) for proton in a light-front quark-diquark model. The light-front wave functions are constructed from the soft-wall AdS/QCD prediction. The contributions from both the scalar and the axial vector diquarks are considered here. The results are compared with the consequences of a parametrization of nucleon generalized parton distributions (GPDs) in the light of recent MRST measurements of parton distribution functions (PDFs) and a soft-wall AdS/QCD model. The spatial distribution of angul…
The gravitational form factors are related to the matrix elements of the energy-momentum tensor $T^{μν}$. Using the light front wave functions of the scalar quark-diquark model for nucleon predicted by the soft-wall AdS/QCD, we calculate the flavor dependent $A(Q^2)$, $B(Q^2)$ and $\bar{C}(Q^2)$ form factors. We also present all the matrix element of the energy-momentum tensor in a transversely polarized state. Further, we evaluate the matrix element of Pauli-Lubanski operator in this model and show that the intrinsic spin sum rule involves the form factor $\bar{C}$. The longitudinal momentum …
We investigate the medium modification of the generalized vector form factors of the nucleon, which include the electromagnetic and energy-momentum tensor form factors, based on an in-medium modified $π$-$ρ$-$ω$ soliton model. We find that the vector form factors of the nucleon in nuclear matter fall off faster than those in free space, which implies that the charge radii of the nucleon become larger in nuclear medium than in free space. We also compute the corresponding transverse charge densities of the nucleon in nuclear matter, which clearly reveal the increasing of the nucleon size in nuc…
We extend the recently developed formalism to extract light-front quark charge densities from nucleon form factor data to the deformations of these quark charge densities when applying an external electric field. We show that the resulting induced polarizations can be extracted from proton generalized polarizabilities. The available data for the generalized electric polarizabilitiy of the proton yield a pronounced structure in its induced polarization at large transverse distances, which will be pinned down by forthcoming high precision virtual Compton scattering experiments.
We investigate the transverse quark spin densities of the nucleon with the lowest moment within the framework of the SU(3) chiral quark-soliton model, emphasizing the strange quark spin density. Based on previous results of the vector and tensor form factors, we are able to determine the impact-parameter dependent probability densities of transversely polarized quarks in an unpolarized nucleon as well as those of unpolarized quarks in a transversely polarized nucleon. We find that the present numerical results for the transverse spin densities of the up and down quarks are in good agreement wi…
We discuss different operator definitions of the transverse-momentum dependent (TMD) parton densities from the point of view of their renormalization-group (RG) properties and UV evolution. We also consider the structure of the gauge links (Wilson lines) in these operator definitions and examine the role of the soft factor in the factorization formula within the TMD approach to semi-inclusive processes.
We present a study of the generalized parton distributions for the quarks in a proton in both momentum and position spaces using the light-front wave functions of a quark-diquark model for the nucleon predicted by the soft-wall model of AdS/QCD. The results are compared with the soft-wall AdS/QCD model of proton GPDs for zero skewness. We also calculate the GPDs for nonzero skewness. We observe that the GPDs have a diffraction pattern in longitudinal position space, as seen before in other models. Then we present a comparitive study of the nucleon charge and anomalous magnetization densities i…
Densities associated with the energy-momentum tensor are calculated for spin-one targets. These calculations are done in a light front formalism, which accounts for relativistic effects due to boosts and allows for arbitrary spatial localization of the target. These densities include the distribution of momentum, angular momentum, and pressures over a two-dimensional plane transverse to the light front. Results are obtained for both longitudinally and transversely polarized targets, and the formalism is tailored to allow the possibility of massless targets. The momentum density and pressure di…
We present a calculation of the Mellin moments of the nucleon transverse quark spin densities extracted from the unpolarized and transversity generalized form factors. We use three $N_F=2+1+1$ ensembles of twisted mass fermions with quark masses tuned to their physical values and lattice spacings $a\sim 0.08$~fm, $a\sim 0.07$~fm and $a\sim 0.06$~fm and extrapolate the form factors to the continuum limit. Besides isovector densities we also include results for the tensor charge for each quark flavor using the ensemble with $a\sim 0.08$~fm for which we include the disconnected contributions.
We present a study of the flavor form factors in the framework of a hard-wall AdS/QCD model and compare with the available experimental data. We obtain the flavor form factors by decomposing the Dirac and Pauli form factors for the nucleons using the charge and isospin symmetry. Further, we present a detailed study of the flavor structures of the charge and anomalous magnetization densities in the transverse plane. Both the unpolarized and the transversely polarized nucleons are considered here. We compare the AdS/QCD results with two standard phenomenological parametrizations.
We investigate the nucleon to $Δ$ transition form factors in a soft-wall AdS/QCD model and a light-front quark-diquark model inspired by AdS/QCD. From the transition form factors we evaluate the transition charge densities which influences the nucleon to $Δ$ excitation. Here we consider both the unpolarized and the transversely polarized cases. The AdS/QCD predictions are compared with available experimental data and with the results of the global parameterization, MAID2007.
We develop a technique, denoted as the finite radius approximation (FRA), that uses a two-dimensional version of the Shannon-Nyquist sampling theorem to determine transverse densities and their uncertainties from experimental quantities. Uncertainties arising from experimental uncertainties on the form factors and lack of measured data at high $Q^2$ are treated. A key feature of the FRA is that a form factor measured at a given value of $Q^2$ is related to a definite region in coordinate space. An exact relation between the FRA and the use of a Bessel series is derived. The proton Dirac form f…
Transverse force tomography inside a proton from Basis Light-front Quantization
Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；School of Nuclear Science and Technology, University of Chinese Academy of Sciences（中国科学技术大学核科学与技术学院）
