In a set of two papers, we propose to study an old-standing problem, namely the electromagnetic interaction for particles of arbitrary spin. Based on the assumption that light-cone helicity at tree level and $Q^2=0$ should be conserved non-trivially by the electromagnetic interaction, we are able to derive \emph{all} the natural electromagnetic moments for a pointlike particle of \emph{any} spin. In this second paper, we give explicit expressions for the light-cone helicity amplitudes in terms of covariant vertex functions, leading to the natural electromagnetic moments at $Q^2=0$. As an application of our results, we generalize the discussion of quark transverse charge densities to particles with arbitrary spin.
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 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 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.
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.
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 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.
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 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.
Using only the current empirical information on the nucleon electromagnetic form factors we map out the transverse charge density in proton and neutron as viewed from a light front moving towards a transversely polarized nucleon. These charge densities are characterized by a dipole pattern, in addition to the monopole field corresponding with the unpolarized density. Furthermore, we use the latest empirical information on the $N \to Δ$ transition form factors to map out the transition charge density which induces the $N \to Δ$ excitation. This transition charge density in a transversely polari…
We examine the spatial density within extended objects of the momentum component $p^+$, and find relativistically exact connections to Fourier transforms of gravitational form factors. We apply these results to obtain semi-empirical momentum density distributions within nucleons, and similar distributions for spin-1 objects based on theoretical results from the AdS/QCD correspondence. We find that the momentum density in the transverse plane is more compact than the charge density.
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 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…
We use the Bessel-inspired behavior of parton densities at small Bjorken $x$ values, obtained in the case of the flat initial conditions for DGLAP evolution equations in the double scaling QCD approximation (DAS), to evaluate the transverse momentum dependent (TMD, or unintegrated) quark and gluon distribution functions in a proton. The calculations are performed analytically using the Kimber-Martin-Ryskin (KMR) prescription with different implementation of kinematical constraint, reflecting the angular and strong ordering conditions. The relations between the differential and integral formula…
Different approaches to define transverse-momentum dependent parton distribution functions are considered from the point of view of their renormalization-group properties. The associated one-loop anomalous dimensions of these quantities are presented and compared to each other. We give arguments in favor of the "pure light-like" definition, and the use of the light-cone gauge.
The scale factor $σ_{eff}$, which characterizes double parton collisions in high energy hadron interactions, is a direct manifestation of the distribution of the interacting partons in transverse space, in such a way that different distributions give rise to different values of $σ_{eff}$ in different double parton collision processes. We work out the value of the scale factor in a few reactions of interest, in a correlated model of the multi-parton density of the proton recently proposed.
We investigate the transverse momentum dependent (TMD) quark and gluon distribution functions in the modified chiral quark model ($χQM$). Calculations of the TMD quark and gluon densities, using the modified $χQM$ are done for the first time in this article. For this propose we first formulate the TMD interactions that occur in the $χQM$ at low $Q^2$ scale ($Q^2=0.35~ GeV^2$) and then obtain the TMD parton distributions inside the proton, considering the interactions. To this end, we need to compute the TMD bare quark distributions. These TMD bare densities are calculated, using the solution o…
We investigate the flavour decomposition of the electromagnetic form factors of the nucleon, based on the chiral quark-soliton model ($χ$QSM) with symmetry-conserving quantisation. We consider the rotational $1/N_c$ and linear strange-quark mass ($m_s$) corrections. We discuss the results of the flavour-decomposed electromagnetic form factors in comparison with the recent experimental data. In order to see the effects of the strange quark, we compare the SU(3) results with those of SU(2). We finally discuss the transverse charge densities for both unpolarised and polarised nucleons. The transv…
We discuss the state-of-the-art of the theory of transverse-momentum dependent parton densities (TMD)s, paying special attention to their renormalization properties, the structure of the gauge links in the operator definition, and the role of the soft factor in the factorization formula within the TMD approach to the semi-inclusive processes. We argue that the use of the lightcone axial gauge offers certain advantages for a consistent definition of TMDs as compared to the off-the-light-cone gauges, or covariant gauges with off-the-lightcone gauge links.
The effect of lepton transverse polarization in the B^0 --> D^- l^+ nu_l, B^+ --> bar(D^0) l^+ nu_l decays (l= tau, mu) is analyzed within the framework of Standard Model in the leading order of HQET. It is shown that the non-zero transverse polarization appears due to the electromagnetic final state interaction. The diagrams with intermediate D, D^* - mesons contributing to the non-vanishing P_T are considered. Regarding only the contribution of these mesons, values of the tau-lepton transverse polarization, averaged over the physical region, in the B^0 --> D^- tau^+ nu_l and B^+ --> bar(D^0)…
We present a solution of the DGLAP evolution equations, written in terms of Sudakov form factors to describe the branching and no-branching probabilities, using a parton branching Monte Carlo method. We demonstrate numerically that this method reproduces the semi-analytical solutions. We show how this method can be used to determine Transverse Momentum Dependent (TMD) parton distribution functions, in addition to the usual integrated parton distributions functions. We discuss numerical effects of the boundary of soft gluon resolution scale parameter on the resulting parton distribution functio…
Using the recent empirical information on the deuteron electromagnetic form factors we map out the transverse charge density in the deuteron as viewed from a light front moving towards the deuteron. The charge densities for a transversely polarized deuteron are characterized by monopole, dipole and quadrupole patterns.
A phenomenological Lagrangian approach is employed to study the electromagnetic properties of deuteron. The deuteron is regarded as a loosely bound state of a proton and a neutron. The deuteron electromagnetic form factors are expressed in light-front representation in the transverse plane. The transverse charge density of the deuteron is discussed.
To study the heavy quark production processes, we use the transverse momentum dependent (TMD, or unintegrated) gluon distribution function in a proton obtained recently using the Kimber-Martin-Ryskin prescription from the Bessel-inspired behavior of parton densities at small Bjorken $x$ values. We obtained a good agreement of our results with the latest HERA experimental data for reduced cross sections $σ^{c\overline{c}}_{\rm red}(x,Q^2)$ and $σ^{b\overline{b}}_{\rm red}(x,Q^2)$, and also for deep inelastic structure functions $F_2^c(x,Q^2)$ and $F_2^b(x,Q^2)$ in a wide range of $x$ and $Q^2$ …
The transverse charge density in a fast-moving nucleon is represented as a dispersion integral of the imaginary part of the Dirac form factor in the timelike region (spectral function). At a given transverse distance b the integration effectively extends over energies in a range sqrt{t} ~< 1/b, with exponential suppression of larger values. The transverse charge density at peripheral distances thus acts as a low-pass filter for the spectral function and allows one to select energy regions dominated by specific t-channel states, corresponding to definite exchange mechanisms in the spacelike for…
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 first moment of longitudinal and transverse spin densities of quarks in the nucleon are calculated in a light-front constituent quark model for the different cases of quark and nucleon polarization. Significant distortions are found for the transverse spin densities. In particular the Sivers function is predicted with opposite sign for up and down quarks and the Boer-Mulders function is predicted large and negative for both up and down quarks, in agreement with lattice calculations. Quite a different spin distribution is obtained for up and down quarks in the cases of quarks and proton tra…
Generalized transversity distributions encode essential information on the internal structure of hadrons related to transversely polarized quarks. Lattice QCD allows us to compute the lowest moments of these tensor generalized parton distributions. In this talk, we discuss a first lattice study of the Soffer bound and show preliminary results for transverse spin densities of quarks in the nucleon.
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…
We extend the formalism relating electromagnetic form factors to transverse quark charge densities in the light-front frame to the case of a spin-3/2 baryon and calculate these transverse densities for the $Δ(1232)$ isobar using lattice QCD. The transverse charge densities for a transversely polarized spin-3/2 particle are characterized by monopole, dipole, quadrupole, and octupole patterns representing the structure beyond that of a pure point-like spin-3/2 particle. We present lattice QCD results for the $Δ$-isobar electromagnetic form factors for pion masses down to approximatively 350 MeV …
We present the first determination of transverse momentum dependent (TMD) photon densities with the Parton Branching method. The photon distribution is generated perturbatively without intrinsic photon component. The input parameters for quarks and gluons are determined from fits to precision measurements of deep inelastic scattering cross sections at HERA. The TMD densities are used to predict the mass and transverse momentum spectra of very high mass lepton pairs from both Drell-Yan production and Photon-Initiated lepton processes at the LHC.
An established phenomenology and theoretical interpretation of $p$-$p$ collision data at lower collision energies should provide a reference for $p$-$p$ and other collision systems at higher energies, against which claims of novel physics may be tested. The description of $p$-$p$ collisions at the relativistic heavy ion collider (RHIC) has remained incomplete even as claims for collectivity and other novelties in data from smaller systems at the large hadron collider (LHC) have emerged recently. In this study we report the charge-multiplicity dependence of two-dimensional (2D) angular correlat…
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…
Recently, Brodsky, Hwang and Schmidt have proposed a new mechanism that gives a transverse spin symmetry at leading twist in semi-inclusive deep-inelastic scattering. I show that the new mechanism is compatible with factorization and is due to an transverse-spin asymmetry in the k_T distribution of quarks in a hadron (the "Sivers asymmetry"). An earlier proof that the Sivers asymmetry vanishes because of time-reversal invariance is invalidated by the path-ordered exponential of the gluon field in the operator definition of parton densities. Instead, the time-reversal argument shows that the Si…
Using recent experimental data, we analyze the electromagnetic transition from the nucleon to the P11(1440) resonance. From the resulting empirical transition form factors, we map out the quark transverse charge densities which induce the N to P11(1440) transition. It is found that the transition from the proton to its first radially excited state is dominated by up quarks in a central region of around 0.5 fm and by down quarks in an outer band which extends up to about 1 fm.
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…
An unambiguous definition of the electromagnetic spatial densities for a spin-1/2 system is proposed and worked out in the zero average momentum frame and in moving frames. The obtained results are compared with the traditional definition of the densities in terms of the three-dimensional Fourier transforms of the electromagnetic form factors in the Breit frame.
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…
Collinear and transverse momentum dependent (TMD) parton densities are obtained from fits to precision measurements of deep inelastic scattering (DIS) cross sections at HERA. The parton densities are evolved by DGLAP evolution with next-to-leading-order (NLO) splitting functions using the parton branching method, allowing one to determine simultaneously collinear and TMD densities for all flavors over a wide range in $x$, $μ^2$ and $k_t$, relevant for predictions at the LHC. The DIS cross section is computed from the parton densities using perturbative NLO coefficient functions. Parton densiti…
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…
We evaluate both chirally even and odd generalized parton distributions(GPDs) in the leading twist in a recently proposed quark-diquark model for the proton where the light front wavefunctions are constructed from the soft-wall AdS/QCD prediction. The GPDs in transverse impact parameter space give the spin densities for different quark and proton polarizations. For longitudinally polarized proton only chiral even GPDs contribute but for transversely polarized proton both chiral even and chiral odd GPDs contribute to the spin densities. We present a detail study of the spin densities in this mo…
The electromagnetic emission rates at SPS energies satisfy spectral constraints in leading order in the pion and nucleon densities. These constraints follow from the strictures of broken chiral symmetry. We saturate these constraints using available data, leading to model independent emission rates from a hadronic gas. With a simple fire-ball scenario, only large nucleon densities may account for the present CERES data.
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.
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…
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 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…
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.
We show how generalized quark distributions in the nucleon describe the density of polarized quarks in the impact parameter plane, both for longitudinal and transverse polarization of the quark and the nucleon. This density representation entails positivity bounds including chiral-odd distributions, which tighten the known bounds in the chiral-even sector. Using the quark equations of motion, we derive relations between the moments of chiral-odd generalized parton distributions of twist two and twist three. We exhibit the analogy between polarized quark distributions in impact parameter space …
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.
There is considerable controversy about the size and importance of nonperturbative contributions to the evolution of transverse-momentum-dependent (TMD) parton distribution functions. Standard fits to relatively high-energy Drell-Yan data give evolution that when taken to lower Q is too rapid to be consistent with recent data in semi-inclusive deeply inelastic scattering. Some authors provide very different forms for TMD evolution, even arguing that nonperturbative contributions at large transverse distance b_T are not needed or are irrelevant. Here, we systematically analyze the issues, both …
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…
A simple, but fully-covariant model for describing neutral Vector Mesons, in both light and heavy sectors, is briefly illustrated. The main ingredients of our relativistic constituent model are i) an Ansatz for the Bethe-Salpeter vertex for Vector Mesons, and ii) a Mandelstam-like formula for the electromagnetic decay widths. The free parameters of our approach are fixed through a comparison with the valence transverse-momentum distribution, $n(k_\perp)$, obtained within phenomenological Light-Front Hamiltonian Dynamics models reproducing the mass spectra. Preliminary results for the transvers…
Skew density matrices can be diagonalized to yield probability interpretation. The power-counting prediction of perturbative QCD is found consistent with recent CEBAF data on $F_2(Q^2)/F_1(Q^2)$
I give an account of a new definition of transverse-momentum-dependent parton densities. The new definition solves a number of difficulties and inconsistencies in earlier definitions.
