The leading twist transverse momentum dependent parton distributions (TMDs) are studied in a light-cone description of the nucleon where the Fock expansion is truncated to consider only valence quarks. General analytic expressions are derived in terms of the six amplitudes needed to describe the three-quark sector of the nucleon light-cone wave function. Numerical calculations for the T-even TMDs are presented in a light-cone constituent quark model, and the role of the so-called pretzelosity is investigated to produce a nonspherical shape of the nucleon.
We discuss recent theoretical results on diffractive deeply inelastic scattering, focusing on the partonic picture of diffraction in configuration space and the predictions for the beta behavior and the scaling violation.
We show that the final-state interaction effects in the single target spin asymmetry discovered by Brodsky et al. can be reproduced by either a standard light-cone gauge definition of the parton distributions with a prescription of the light-cone singularities consistent with the light-cone gauge link, or a modified light-cone gauge definition with a gauge link involving the gauge potential at the spatial infinity.
We derive the space-time distribution of energy and momentum deposited by a fast parton traversing a weakly coupled quark-gluon plasma by treating the fast part on as the source of an external color field perturbing the medium. We then use our result as a source term for the linearized hydrodynamical equations of the medium. We show that the solution contains a sonic Mach cone and a dissipative wake if the parton moves at a supersonic speed.
I discuss the physical interpretation of GPDs in the limit $ξ=0$, where the $t$ dependence contains information about the distribution of partons in the transverse plane. GPDs thus allow a simultaneous determination of the longitudinal momentum and transverse position of partons in the infinite momentum frame, which also provides a physical interpretation for Ji's angular momentum sum rule.
We review the information on the spin and orbital angular momentum structure of the nucleon encoded in the T-even transverse momentum dependent parton distributions within light-cone quark models. Model results for azimuthal spin asymmetries in semi-inclusive lepton-nucleon deep-inelastic scattering are discussed, showing a good agreement with available experimental data and providing predictions to be further tested by future CLAS, COMPASS and HERMES data.
By working out the kinematics of double parton scattering at short relative transverse distances, we obtain an explicit link between the transverse centres of mass, of the two hard partonic interactions, and the contributions to the process, due to pairs of interacting partons generated by perturbative splitting. One my thus foresee the interesting possibility of discriminating experimentally between contributions to the double parton scattering cross section, due to interacting parton pairs originated by independent evolution, and contributions, due to interacting parton pairs generated by sp…
Generalized parton distributions (GPDs) with purely transverse momentum transfer can be interpreted as Fourier transforms of the distribution of partons in impact parameter space. The helicity-flip GPD $E(x,0,-\Dps)$ is related to the distortion of parton distribution functions in impact parameter space if the target is not a helicity eigenstate, but has some transverse polarization. This transverse distortion can be used to develop an intuitive explanation for various transverse single spin asymmetries.
We discuss the general formalism for the calculation in light-cone quark models of the fully unintegrated, off-diagonal quark-quark correlator of the nucleon. The corresponding distributions in impact parameter space are the Wigner or phase-space distributions. The results obtained in two different light-cone quark models in the case of unpolarized quarks in an unpolarized proton are very similar and present a non-trivial shape which can be understood as due to the orbital motion of the quarks.
We present a detailed study of the two-parton light-cone distribution amplitudes for $1\,^3P_2$ nonet tensor mesons. The asymptotic two-parton distribution amplitudes of twist-2 and twist-3 are given. The decay constants $f_T$ and $f_T^\bot$ defined by the matrices of non-local operators on the lightcone are estimated using the QCD sum rule techniques. We also study the decay constants for $f_2(1270)$ and $f_2^\prime(1525)$ based on the hypothesis of tensor meson dominance together with the data of $Γ(f_2\to ππ)$ and $Γ(f'_2\to K\bar K)$ and find that the results are in accordance with the sum…
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 discuss the parton branching (PB) evolution equation for transverse momentum dependent (TMD) parton distribution functions (PDFs), especially we concentrate on the angular ordering constrain and its effect on the TMD distributions. We discuss application of the PB TMDs to precise prediction of Drell-Yan transverse momentum spectrum at LHC, stressing the role of angular ordering in the low transverse momentum region. We compare the PB implementation of the angular ordering condition with the Kimber-Martin-Ryskin-Watt (KMRW), both at the analytical and numerical level.
The cross section for transverse momentum dependent double parton scattering involves transverse momentum dependent double parton distributions (DTMDs). In the region of perturbative transverse momentum the DTMDs can be matched onto collinear double parton distributions. We present the framework and results for this matching, as well as the evolution equations for DTMDs in the region of large distance between the two partons. We discuss explicit results for one-loop matching coefficients and evolution kernels.
We classify the hadron light-cone wave-function amplitudes in terms of parton helicity, orbital angular momentum, and quark flavor and color symmetries. We show in detail how this is done for the pion, $ρ$ meson, nucleon, and delta resonance up to and including three partons. For the pion and nucleon, we also consider four-parton amplitudes. Using the scaling law derived recently, we show how these amplitudes scale in the limit that all parton transverse momenta become large.
Quark and gluon distributions in the light-cone wavefunction of a high energy hadron or nucleus are calculated in the saturation regime. One loop calculations are performed explicitly using the equivalence between the parton distribution in the light-cone wavefunction and the production distribution of that parton in a current-nucleon (nucleus) scattering. We argue that, except for some overall numerical factors, the Weizsacker- Williams wavefunction correctly gives the physics of the gluon distribution in a light-cone wavefunction.
We derive a nonperturbative transverse momentum distribution for partons using a potential model to describe the quark-quark interaction inside the proton. We use this distribution to calculate the diferential cross-section of $π^0$-production for intermediate values of transverse momentum in $p-p$ collisions at high energies. Assuming a variable string tension constant for the quark-quark potential we obtain a very good description of the experimental data at different energies. The corresponding values of the mean transverse momentum of the partons are essentially lower than those obtained u…
We study the impact of transverse-momentum dependent parton distributions on detailed features of multi-jet final states, focusing on angular jet correlations in DIS data.
I report on precision comparisons of theoretical predictions for Drell-Yan (DY) transverse momentum spectra from two approaches: the analytical resummation approach based on Collins-Soper-Sterman (CSS) formalism and the parton branching (PB) approach to the evolution of transverse momentum dependent (TMD) parton distributions.
Off-shell, transverse-momentum dependent splitting functions can be defined from the high-energy limit of partonic decay amplitudes. Based on these splitting functions, we construct Sudakov form factors and formulate a new parton branching algorithm. We present a first Monte Carlo implementation of the algorithm. We use the numerical results to verify explicitly momentum sum rules for TMD parton distributions.
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.
Transverse-momentum-dependent parton distribution functions are analyzed in semi-inclusive deep inelastic scattering at low transverse momentum using soft-collinear effective theory. The transverse-momentum-dependent parton distribution functions are defined on the lightcone without distorting the lightcone path nor adding additional soft Wilson lines. In this approach, the comparison between the integrated and unintegrated parton distribution functions becomes transparent. The procedure of computing radiative corrections in dimensional regularization is explained in detail, and the divergence…
The renormalization-group properties of transverse-momentum dependent parton distribution functions in the light-cone gauge with the Mandelstam-Leibbrandt prescription for the gluon propagator are addressed. An expression for the transverse component of the gauge field at light-cone infinity, which plays a crucial role in the description of the final-/initial-state interactions in the light-cone axial gauge, is obtained. The leading-order anomalous dimension is calculated in this gauge and the relation to the results obtained in other gauges is worked out. It is shown that, using the Mandelsta…
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…
We explore the possibility to construct higher-twist parton distributions in a nucleon at some low reference scale from convolution integrals of the light-cone wave functions (WFs). To this end we introduce simple models for the four-particle nucleon WFs involving three valence quarks and a gluon with total orbital momentum zero, and estimate their normalization (WF at the origin) using QCD sum rules. We demonstrate that these WFs provide one with a reasonable description of both polarized and unpolarized gluon parton densities at large values of Bjorken variable x > 0.5. Twist-three parton di…
I give a brief historical account on the way I have been conducted to revive the concept of the transverse spin from previous studies on multiparton scatterings.
We propose the study of unpolarized transverse momentum dependent gluon parton distributions as well as the effect of linearly polarized gluons on transverse momentum and rapidity distributions of $J/ψ$ and $Υ$ production within the framework of transverse momentum dependent factorization employing color evaporation model (CEM) in unpolarized proton-proton collision. We estimate the transverse momentum and rapidity distributions of $J/ψ$ and $Υ$ at LHCb, RHIC and AFTER energies using TMD evolution formalism.
The Parton Branching (PB) approach describes the evolution of transverse momentum dependent (TMD) parton densities. We propose to extend the PB method by including TMD splitting functions, instead of the DGLAP splitting functions which assume strong ordering in transverse momentum. We present the evolution equations and their numerical solution, which is the first Monte Carlo implementation including TMD splitting functions.
Transverse-momentum spectra in hard processes are typically described either in terms of intrinsic transverse momentum of partons, or in terms of perturbative radiation. The relation between these descriptions is discussed for the example of semi-inclusive deep inelastic scattering, with special focus on the angular distribution of the observed hadron. This involves nontrivial theoretical issues, such as the proper definition of transverse-momentum dependent parton distributions, and has practical consequences for the description of pT spectra in phenomenology.
The conventional NLO parton model is enough for the description of total cross sections and one-particle distributions. In the case of two-particle correlations, the collinear approximation has failed, and it is necessary to account for the transverse momenta of initial partons. The different possibilities to do this are discussed.
The present knowledge on non-trivial relations between generalized parton distributions on the one hand and transverse momentum dependent distributions on the other is reviewed. While various relations can be found in the framework of spectator models, so far no model-independent non-trivial relations have been established.
Two factorization approaches have been proposed for single transverse spin asymmetries. One is the collinear factorization, another is the transverse-momentum-dependent factorization. They have been previously derived in a formal way by using diagram expansion at hadron level. If the two factorizations hold or can be proven, they should also hold when we replace hadrons with parton states. We examine these two factorizations at parton level with massless partons. It is nontrivial to generate these asymmetries at parton level with massless partons because the asymmetries require helicity-flip a…
We investigate the relations between transverse momentum dependent parton distributions (TMDs) and generalized parton distributions (GPDs) in a light-front quark-diquark model motivated by soft wall AdS/QCD. Many relations are found to have similar structure in different models. It is found that a relation between the Sivers function and the GPD $E_q$ can be obtained in this model in terms of a lensing function. The quark orbital angular momentum is calculated and the results are compared with the results in other similar models. Implications of the results are discussed. Relations among diffe…
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.
Fast thermalization and a strong build up of elliptic flow of QCD matter were investigated within the pQCD based 3+1 dimensional parton transport model BAMPS including bremsstrahlung $2 \leftrightarrow 3$ processes. Within the same framework quenching of gluonic jets in Au+Au collisions at RHIC can be understood. The development of conical structure by gluonic jets is investigated in a static box for the regimes of small and large dissipation. Furthermore we demonstrate two different approaches to extract the shear viscosity coefficient $η$ from a microscopical picture.
The transverse-momentum dependent quark distributions are studied in the limit of the large number of colors Nc. The leading orders of Nc are determined for all twist-2 distributions including the T-odd functions. The isospin structure of the dominant contributions is found.
We perform calculations of Z+jet cross-section taking into account the transverse momenta of the initial partons. Transverse Momentum Dependent (TMD) parton densities obtained with the Parton Branching method are used and higher order corrections are included via TMD parton showers in the initial state. The predictions are compared to measurements of forward Z+jet production of the LHCb collaboration at $\sqrt{s}=7$ TeV. We show that the results obtained in kT-factorization are in good agreement with results obtained from a NLO calculation matched with traditional parton showers. We also demon…
Transverse parton momentum dependent distribution functions (TMDs) of the nucleon are studied in a covariant model, which describes the intrinsic motion of partons in terms of a covariant momentum distribution. The consistency of the approach is demonstrated, and model relations among TMDs are studied. As a byproduct it is shown how the approach allows to formulate the non-relativistic limit.
We study the leading twist gluon generalized parton distributions (GPDs) and the gluon angular momentum inside the proton within a light-cone spectator model. Using the light-cone wave functions derived from the model, we provide the expressions of these distributions at the particular kinematical point $ξ=0$ in the overlap representation. The numerical results of the $H^g$, $E^g$, $\tilde{H}^g$, $H_T^g$ and $E_T^g$ as functions of $x$ at different $Δ_T$ are presented. Particularly, $H^g$, $\tilde{H}^g$ at non-zero $Δ_T$ are different from their forward counterparts, the unpolarized distributi…
We perform a calculation of inclusive $Z$ boson production in proton-lead collisions at the LHC taking into account the transverse momenta of the initial partons. We use the framework of $k_T$-factorization combining transverse momentum dependent parton distributions (TMDs) with off-shell matrix elements. In order to do it we need to construct appropriate TMDs for lead nuclei which is done using the parton branching method. Our computations are compared with data from CMS taken at $\sqrt{s}=5.02$ TeV. The results are in good agreement with the measurements especially the transverse momentum di…
Using partonic states we verify the transverse momentum dependent factorization for single transverse spin asymmetries in semi inclusive DIS at leading but nontrivial order of alpha_s. The factorization has been previously derived in a formal way by using diagram expansion at hadron level. We find with our partonic results that two relevant structure functions satisfy the factorization. Our results also satisfy the collinear factorization but with the perturbative coefficient different than that derived formally.
The proton's transverse polarization structure is examined in terms of the Lorentz-covariant Pauli-Lubanski vector in QCD. We find that there are contributions from leading, subleading, and next-to-subleading partonic contributions in the light-front system of coordinates. The subleading and next-to-subleading contributions are related to the leading one through Lorentz symmetry. And the leading contribution obeys a simple partonic angular momentum sum rule that gives a clear physical interpretation to a relation known previously.
An explicit evaluation of the two pion transverse momentum dependent parton distributions at leading twist is presented, in the framework of the Nambu-Jona Lasinio model with Pauli-Villars regularization. The transverse momentum dependence of the obtained distributions is generated solely by the dynamics of the model. Using these results, the so called generalized Boer-Mulders shift is studied and compared with recent lattice data. The obtained agreement is very encouraging, in particular because no additional parameter has been introduced. A more conclusive comparison would require a precise …
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.
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.
The hadron-energy evolution (Collins and Soper) equation for all the leading-twist transverse-momentum and spin dependent parton distributions is derived in the impact parameter space. Based on the result, we present resummation formulas for the spin structure functions in the semi-inclusive deep inelastic scattering.
Transverse-momentum-dependent parton distributions (TMDs) provide three-dimensional images of the partonic structure of the nucleon in momentum space. We made impressive progress in understanding TMDs, both from the theoretical and experimental point of view. This brief overview on TMDs is divided in two parts: in the first, an essential list of achievements is presented. In the second, a selection of open questions is discussed.
We introduce a systematic way to write down the Fock components of a hadronic light-cone wave function with $n$ partons and orbital angular momentum projection $l_z$. We show that the wave function amplitude $ψ_n(x_i,k_{i\perp},l_{zi})$ has a leading behavior $1/(k^2_\perp)^{[n+|l_z|+{\rm min}(n'+|l_z'|)]/2-1}$ when all parton transverse momenta are uniformly large, where $n'$ and $l_z'$ are the number of partons and orbital angular momentum projection, respectively, of an amplitude that mixes under renormalization. The result can be used as a constraint in modeling the hadronic light-cone wav…
We suggest a classification scheme for parton distribution models, clarify the geometrical origin of unintegrated parton distribution relations, which were observed in various models, present new model relations, and provide for a so-called "spherical" model the analogous constraints for generalized parton distributions. Our findings suggest that various classes of uPDF and GPD models can be obtained from effective two-body light-cone wave functions.
A general expression for the dead cone of gluons radiated by virtual partons has been derived. The conventional dead cone for massive on-shell quarks and the dead cone for the massless virtual partons have been obtained by using different limits of the general expression. Radiative suppression due to the virtuality of initial parton jets in Heavy-Ion Collisions (HIC) has been discussed. It is observed that the suppression caused by the high virtuality is overwhelmingly large as compared to that on account of conventional dead-cone of heavy quarks. The dead cone due to virtuality may play a cru…
Longitudinal and transverse parton distributions for pion and nucleon are calculated from hadron vertexes obtained by a study of form factors within relativistic quark models. The relevance of the one-gluon-exchange dominance at short range for the behavior of the form factors at large momentum transfer and of the parton distributions at the end points is stressed.
We show how some parton distribution functions related to the transverse spin of nucleons can be extracted point by point from combinations of proton and deuteron observables. In particular, we present a determination of the valence and sea Sivers functions from the single-spin asymmetries measured by COMPASS.
We employ models from effective two-body light-cone wave functions (LCWFs) to provide a link between generalized parton distributions (GPDs) and unintegrated parton distribution functions (uPDFs). Since we utilize the underlying Lorentz symmetry, GPDs can be entirely obtained from the parton number conserved LCWF overlaps. This also allows us to derive model constraints among GPDs. We illustrate that transversity distributions may be rather sizeable.
The extension of the statistical parton distributions to include their transverse momentum dependence (TMD) is revisited by considering that the proton target has a finite longitudinal momentum. The TMD will be generated by means of a transverse energy sum rule. The new results are mainly relevant for electron-proton inelastic collisions in the low $Q^2$ region. We take into account the effects of the Melosh-Wigner rotation for the helicity distributions.
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.
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.
We consider the prompt photon production in pp collisions using, within the framework of perturbative QCD, a non-Gaussian distribution for the transverse momentum distribution of the partons inside the proton. Our description adopts the widely used in the literature factorization of the partonic momentum distribution into longitudinal and transverse components. It is argued that the non-Gaussian distribution of the intrinsic transverse momenta of the partons is dictated by the asymptotic freedom as well as the 3D confinement of the partons in the proton. To make this association more transpare…
If large-angle multigluon radiation gives significant contributions to parton showers in the LHC high-energy region, appropriate generalizations of parton branching methods are required for Monte Carlo simulations of exclusive high-multiplicity final states. We discuss the use in this context of transverse-momentum dependent kernels which factorize in the region of high energies. We give examples based on ep and p pbar multi-jet data, and point to possible developments for distributions associated with massive final states at the LHC.
The partonic transverse momentum, k_t distribution plays a crucial role in driving high-energy hadron interactions. If k_t is limited we have old fashioned Regge Theory. If k_t increases with energy the interaction may be described by perturbative QCD. We use BFKL diffusion in ln k_t, supplemented by a stronger absorption of low k_t partons, to estimate the growth of the mean transverse momenta <k_t> with energy. This growth reveals itself in the distribution of secondaries produced at the collider energies. We present a simple, BFKL-based, model to demonstrate the possible size of the effect.…
