Transverse Momentum Dependent Gluon Distribution Research Articles

Probing gluon TMDs with reconstructed and tagged heavy flavor hadron pairs at the EIC

Study of the transverse structure of the proton is one of the major physics goals of the upcoming Electron Ion Collider (EIC). The gluon transverse momentum dependent distributions (TMD) form an essential focus of this effort and are important towards understanding the angular momentum contribution to proton spin as well as QCD factorization. However, very limited experimental constraints on the gluon TMD exist currently. As the heavy quark production in lepton-nucleon DIS gets a dominant contribution from the photon-gluon-fusion process, heavy quark production makes an attractive tool to probe gluon distributions in nucleons. In this paper we present a study of heavy flavor hadron pair reconstruction at a future EIC detector with MAPS based inner tracking and vertexing subsystems to constrain gluon TMD. We utilize the excellent track pointing resolution provided by the detector to exclusively reconstruct heavy flavor hadron pairs via their hadronic decay channels and also to develop a heavy flavor hadron tagging algorithm. Statistical uncertainty projections on azimuthal asymmetries corresponding to gluon TMD at the EIC is evaluated. The heavy flavor tagging is found to substantially enhance the purity of heavy flavor hadron pair selection, and the statistical precision of the measurement compared to that from exclusive reconstruction. The correlation between the azimuthal angle of the transverse momentum of the gluon initiating the process and that of the corresponding heavy flavor hadron pair was also studied and found to be well correlated. This study opens up heavy flavor hadron pair measurements as an attractive channel to access gluon TMD at the EIC.

J/ψ polarization in semi-inclusive DIS at low and high transverse momentum

We study the polar and azimuthal decay angular distributions of J/ψ mesons produced in semi-inclusive, deep-inelastic electron-proton scattering. For the description of the quarkonium formation mechanism, we adopt the framework of nonrelativistic QCD, with the inclusion of the intermediate color-octet channels that are suppressed at most by a factor v4 in the velocity parameter v relative to the leading color-singlet channel. We put forward factorized expressions for the helicity structure functions in terms of transverse momentum dependent gluon distributions and shape functions, which are valid when the J/ψ transverse momentum is small with respect to the hard scale of the process. By requiring that such expressions correctly match with the collinear factorization results at high transverse momentum, we determine the perturbative tails of the shape functions and find them to be independent of the J/ψ polarization. In particular, we focus on the cos 2ϕ azimuthal decay asymmetry, which originates from the distribution of linearly polarized gluons inside an unpolarized proton. We therefore suggest a novel experiment for the extraction of this so-far unknown parton density that could be performed, in principle, at the future Electron-Ion Collider.

Transverse momentum dependent gluon distribution within high energy factorization at next-to-leading order

We discuss transverse momentum dependent (TMD) gluon distributions within high energy factorization at next-to-leading order in the strong coupling within the framework of Lipatov's high energy effective action. We provide a detailed discussion of both rapidity divergences related to the TMD definition and its soft factor on the one hand, and rapidity divergences due to high energy factorization on the other hand, and discuss common features and differences between Collins-Soper (CS) and Balitsky-Fadin-Kuarev-Lipatov (BFKL) evolution. While we confirm earlier results which state that the unpolarized and linearly polarized gluon TMD agree in the BFKL limit at leading order, we find that both distributions differ, once next-to-leading order corrections are being included. Unlike previous results, our framework allows us to recover the complete anomalous dimension associated with the Collins-Soper-Sterman (CSS) evolution of the TMD distribution, including also single-logarithmic terms in the CSS evolution. As an additional result, we provide a definition of ${k}_{T}$ factorization, i.e., matching of off shell coefficients to collinear factorization at next-to-leading order within high energy factorization and the effective action framework. We furthermore establish a link between the QCD operator definition of the TMD gluon distribution and a previously derived off shell TMD gluon-to-gluon splitting function, which is within the present framework obtained as the real one-loop correction.

Parton Energy Loss in the Generalized High-Twist Approach

We calculate the radiative parton energy loss in the deeply inelastic scattering (DIS) off a large nucleus within a generalized high-twist approach. The final gluon radiation spectrum is a convolution of the hard partonic part and the transverse momentum dependent (TMD) quark-gluon correlation function, without the twist expansion in the transverse momentum of the initial gluons used in original high twist approach. The TMD quark-gluon correlation function can be factorized approximately as the product of initial quark distribution and TMD gluon distribution which can be used to define the generalized or TMD jet transport coefficient. The radiation spectrum will recover the Gylassy-Levai-Vitev (GLV) result in the first order of the opacity expansion, under the static scattering center and soft gluon radiation approximation. We also investigate numerically the difference as a result of the soft gluon radiation approximation, under the static scattering center approximation.

J/\u03c8 meson production in SIDIS: matching high and low transverse momentum

We consider the transverse momentum spectrum and the cos 2ϕ azimuthal distribution of J/ψ mesons produced in semi-inclusive, deep-inelastic electron-proton scattering, where the electron and the proton are unpolarized. At low transverse momentum, we propose factorized expressions in terms of transverse momentum dependent gluon distributions and shape functions. We show that our formulae, at the order αs, correctly match with the collinear factorization results at high transverse momentum. The latter are computed at the order {alpha}_s^2 in the framework of nonrelativistic QCD (NRQCD), with the inclusion of the intermediate {}^3{S}_1^{left[1right]} color-singlet Fock state, as well as the subleading color-octet ones that are relatively suppressed by a factor v4 in the NRQCD velocity parameter v. We show that the {}^1{S}_0^{left[8right]} and {}^3{P}_J^{left[8right]} (J = 0, 1, 2) contributions diverge in the small transverse momentum region and allow us to determine the perturbative tails of the shape functions, which carry the same quantum numbers. These turn out to be identical, except for the overall magnitude given by the appropriate NRQCD long distance matrix element.

Transverse-momentum-dependent gluon distribution functions in a spectator model

We present a model calculation of transverse-momentum-dependent distributions (TMDs) of gluons in the nucleon. The model is based on the assumption that a nucleon can emit a gluon, and what remains after the emission is treated as a single spectator particle. This spectator particle is considered to be on-shell, but its mass is allowed to take a continuous range of values, described by a spectral function. The nucleon-gluon-spectator coupling is described by an effective vertex containing two form factors. We fix the model parameters to obtain the best agreement with collinear gluon distributions extracted from global fits. We study the tomography in momentum space of gluons inside nucleons for various combinations of their polarizations. These can be used to make predictions of observables relevant for gluon TMD studies at current and future collider facilities.

Photoproduction of three jets in the CGC: gluon TMDs and dilute limit

We study the process γA → qoverline{q}g + X in the Color Glass Condensate (CGC) effective theory. After obtaining the cross section, we consider two kinematic limits which are encompassed in our result. In the so-called correlation limit, the vector sum of the transverse momenta of the three outgoing particles is small with respect to the individual transverse momenta; the cross section then simplifies considerably and can be written in a factorized form, sensitive to both the unpolarized and linearly-polarized Weizsäcker-Williams transverse momentum dependent gluon distribution function (gluon TMD). The second limit of the CGC cross section that we consider is the dilute limit, which we obtain after performing a weak-field expansion; we recover a typical linear-regime expression, involving a single unintegrated gluon distribution function. Using numerical simulations of the small-x QCD evolution of the TMDs, we investigate the rapidity dependence of the cross section in the correlation limit.

Azimuthal asymmetries in semi-inclusive J/ψ+jet production at an EIC

We consider transverse momentum dependent gluon distributions inside both unpolarized and transversely polarized protons and show how they can be probed by looking at azimuthal modulations in $e \, p \to e \, J/\psi \, \mathrm{jet} \, X$. We find that the contribution due to quark induced subprocesses is always suppressed in the considered kinematic regions, accessible in principle at a future Electron-Ion Collider. Our model-independent estimates of the maximal values of these asymmetries allowed by positivity bounds suggest the feasibility of their measurement. In addition, by adopting the McLerran-Venugopalan model for the unpolarized and linearly polarized gluon densities, we study the behavior of the $\cos2\phi$ asymmetries in the small-$x$ limit.

Parton energy loss and the generalized jet transport coefficient

We revisit radiative parton energy loss in deeply inelastic scattering (DIS) off a large nucleus within the perturbative QCD approach. We calculate the gluon radiation spectra induced by double parton scattering in DIS without collinear expansion in the transverse momentum of initial gluons as in the original high-twist approach. The final radiative gluon spectrum can be expressed in terms of the convolution of hard partonic parts and unintegrated or transverse momentum dependent (TMD) quark-gluon correlations. The TMD quark-gluon correlation can be factorized approximately as a product of initial quark distribution and TMD gluon distribution which can be used to define the generalized or TMD jet transport coefficient. Under the static scattering center and soft radiative gluon approximation, we recover the result by Gylassy-Levai-Vitev (GLV) in the first order of the opacity expansion. The difference as a result of the soft radiative gluon approximation is investigated numerically under the static scattering center approximation.

TMD gluon distributions for multiparton processes

We derive gauge invariant operators entering definitions of the Transverse Momentum Dependent (TMD) gluon distributions, for all five and six parton processes. Our calculations utilize color decomposition of amplitudes in the color flow basis. In addition, we find the general result for multi-gluon process (with arbitrary number of gluons) at large N_{c}. On phenomenological ground our results may be used for multi-jet production in the small-x regime, where the TMD gluon distributions can be derived from the Color Glass Condensate effective theory.

TMD gluon distributions at small x in the CGC theory

In this paper, we review recent progress in the description of unpolarized transverse-momentum-dependent (TMD) gluon distributions at small [Formula: see text] in the color glass condensate (CGC) effective theory. We discuss the origin of the nonuniversality of TMD gluon distributions in the TMD factorization framework and in the CGC theory and the equivalence of the two approaches in their overlapping domain of validity. We show some applications of this equivalence, including recent results on the behavior of TMD gluon distributions at small [Formula: see text], and on the study of gluon saturation. We discuss recent advances in the unification of the TMD evolution and the nonlinear small-[Formula: see text] evolution of gluon distributions.

Prompt photon-jet angular correlations at central rapidities in p+A collisions

Photon-jet azimuthal correlations in proton-nucleus collisions are a promising tool for gaining information on the gluon distribution of the nucleus in the regime of non-linear color fields. We compute such correlations from the process $g+A\to q \bar{q} \gamma$ in the rapidity regime where both the projectile and target light-cone momentum fractions are small. By integrating over the phase space of the quark which emits the photon, subject to the restriction that the photon picks up most of the transverse momentum (to pass an isolation cut), we effectively obtain a $g+A\to q \gamma$ process. For nearly back-to-back photon-jet configurations we find that it dominates over the leading order process $q+A\to q \gamma$ by two less powers of $Q_\perp / Q_S$, where $\boldsymbol{Q}_\perp$ and $Q_S$ denote the net photon-jet pair momentum and the saturation scale of the nucleus, respectively. We determine the transverse momentum dependent gluon distributions involved in $g+A\to q \gamma$ and the scale where they are evaluated. Finally, we provide analytic expressions for $\langle\cos n\phi\rangle$ moments, where $\phi$ is the angle between $\boldsymbol{Q}_\perp$ and the average photon-jet transverse momentum $\tilde{\boldsymbol{P}}_\perp$, and first qualitative estimates of their transverse momentum dependence.

The gluon structure of hadrons and nuclei from lattice QCD

I discuss recent lattice QCD studies of the gluon structure of hadrons and light nuclei. After very briefly highlighting new determinations of the gluon contributions to the nucleon’s momentum and spin, presented by several collaborations over the last year, I describe first calculations of gluon generalised form factors. The generalised transversity gluon distributions are of particular interest since they are purely gluonic; they do not mix with quark distributions at leading twist. In light nuclei they moreover provide a clean signature of non-nucleonic gluon degrees of freedom, and I present the first evidence for such effects, based on lattice QCD calculations. The planned Electron-Ion Collider, designed to access gluon structure quantities, will have the capability to test this prediction, and measure a range of gluon observables including generalised gluon distributions and transverse momentum dependent gluon distributions, within the next decade.

Positivity bounds on gluon TMDs for hadrons of spin \u2264 1

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 ≥ 1. 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 to estimate their maximal contribution to observables. Since the gluonic content of hadrons is particularly relevant in the small-x kinematic region, we also study these bounds in the small-x limit for the dipole-type gauge link structure using matrix elements of a single Wilson loop.

Back-to-back heavy quark pair production in semi-inclusive DIS

The one-loop correction to heavy quark pair back-to-back production in unpolarized semi-inclusive deep inelastic scattering is given in this work in the framework of transverse momentum dependent(TMD) factorization. Both unpolarized and linearly polarized TMD gluon distribution functions are taken into account. A subtraction method based on diagram expansion is used to get finite hard coefficients. It is found the soft and collinear divergences of one-loop amplitude is proportional to tree level ones and can be expressed through several basic scalar triangle and bubble integrals. The subtraction of these divergences is spin independent. Beyond tree level an additional soft factor related to final heavy quark pair must be added into the factorization formula. This soft factor affects the azimuthal angle distribution of virtual photon in a nonperturbative way. Integrating over virtual photon azimuthal angle we construct three weighted cross sections, which depend on only three additional integrated soft factors. These weighted cross sections can be used to extract linearly polarized gluon distribution function. In addition, lepton azimuthal angle is unintegrated in this work, which provides more observables. All hard coefficients relevant to lepton and virtual photon azimuthal angle distributions are given at one-loop level.

Transverse momentum dependent quark and gluon distributions of light nuclei

We investigate the unpolarized transverse momentum dependent (TMD) structure of light nuclei in the modified chiral quark exchange model (QEM), for the first time. To this end, we calculate the TMD quark and gluon distributions inside the bound state nucleons of the light nuclei based on the modified chiral quark model in which the TMD bare quark distributions of the bounded nucleons are needed. In order to compute these bare distributions, we first obtain the bare quark momentum densities using the QEM and then calculate the TMD bare distributions applying a theoretical method in which the light-cone variables are used. Finally, considering the nucleon structure of helium, tritium and deuteron nuclei, we obtain their TMD quark and gluon densities at low Q2 scale. It is shown that our results have appropriate properties that are expected for the TMD distribution functions.

Associated production of a dilepton and a ϒ(J/ψ) at the LHC as a probe of gluon transverse momentum dependent distributions

We discuss the impact on the study of gluon transverse momentum dependent distributions (TMDs) of the associated production of a lepton pair and a ϒ (or a J/ψ) in unpolarised proton–proton collisions, pp→Qℓℓ¯X, at LHC energies, where one can assume that such final states are dominantly induced by gluon fusion. If the transverse momentum of the quarkonium–dilepton system – namely, the transverse momentum imbalance of the quarkonium state and the lepton pair – is small, the corresponding cross sections can be calculated within the framework of TMD factorisation. Using the helicity formalism, we show in detail how these cross sections are connected to the moments of two independent TMDs: the distribution of unpolarised gluons, f1g, and the distribution of linearly polarised gluons, h1⊥g. We complete our exhaustive derivation of these general relations with a phenomenological analysis of the feasibility of the TMD extraction, as well as some outlooks.

Forward di-jet production in dilute-dense collisions

We propose a factorization formula for the cross section for forward di-jet production in dilute-dense collisions. The new formula is applicable for an arbitrary value of the momentum imbalance of the two jets, kt. It unifies the previously derived transverse momentum dependent (TMD) factorization for small kt (of the order of the saturation scale), and the High Energy Factorization (HEF) for large kt (of the order of the momentum of the jets). We extend the previous TMD formula, first to finite Nc, and then to all ranges of kt by including off-shell matrix elements. We present previously unpublished analytical expressions for the TMD gluon distributions in the Golec-Biernat-Wusthoff model, and their perturbative behaviour in the McLerran-Venugopalan model. In addition, we show directly the equivalence of the HEF and the color glass condensate formulae in the dilute target approximation.

Transverse-momentum-dependent gluon distributions from JIMWLK evolution

Transverse-momentum-dependent (TMD) gluon distributions have different operator definitions, depending on the process under consideration. We study that aspect of TMD factorization in the small-x limit, for the various unpolarized TMD gluon distributions encountered in the literature. To do this, we consider di-jet production in hadronic collisions, since this process allows to be exhaustive with respect to the possible operator definitions, and is suitable to be investigated at small x. Indeed, for forward and nearly back-to-back jets, one can apply both the TMD factorization and Color Glass Condensate (CGC) approaches to compute the di-jet cross-section, and compare the results. Doing so, we show that both descriptions coincide, and we show how to express the various TMD gluon distributions in terms of CGC correlators of Wilson lines, while keeping Nc finite. We then proceed to evaluate them by solving the JIMWLK equation numerically. We obtain that at large transverse momentum, the process dependence essentially disappears, while at small transverse momentum, non-linear saturation effects impact the various TMD gluon distributions in very different ways. We notice the presence of a geometric scaling regime for all the TMD gluon distributions studied: the "dipole" one, the Weizs\"acker-Williams one, and the six others involved in forward di-jet production.

Fourier-positivity constraints on QCD dipole models

Fourier-positivity, i.e. the mathematical property that a function has a positive Fourier transform, can be used as a constraint on the parametrization of QCD dipole-target cross-sections or Wilson line correlators in transverse position (r) space. They are Bessel transforms of positive transverse momentum dependent gluon distributions. Using mathematical Fourier-positivity constraints on the limit r -> 0 behavior of the dipole amplitudes, we identify the common origin of the violation of Fourier-positivity for various, however phenomenologically convenient, dipole models. It is due to the behavior r^{2+epsilon}, epsilon>0, softer, even slightly, than color transparency. Fourier-positivity seems thus to conflict with the present dipole formalism when it includes a QCD running coupling constant alpha(r).