We study the unintegrated, or transverse momentum dependent (TMD) gluon distribution obtained from the best description of the LHC data on the inclusive spectra of hadrons produced in the mid-rapidity region and low transverse momenta at starting scale Q0^2 = 1 GeV^2. To extend this gluon density at higher Q^2 we apply the Catani-Ciafoloni-Fiorani-Marchesini (CCFM) evolution equation. The influence of the initial (starting) non-perturbative gluon distribution is studied. The application of the obtained gluon density to the analysis of the ep deep inelastic scattering allows us to get the results which describe reasonably well the H1 and ZEUS data on the longitudinal proton structure function FL(x,Q^2). So, the connection between the soft processes at LHC and small x physics at HERA has been confirmed and extended to a wide kinematical region.
We revisit the model-independent decomposition of the gluon correlator, producing T-even and T-odd gluon transverse momentum distributions (TMDs), at leading twist. We propose an expansion of the gluon correlator, using a basis of four tensors (one antisymmetric and three symmetric), which are expressed through generators of the $U(2)$ group acting in the two-dimensional transverse plane. One can do clear interpretations of the two transversity T-odd TMDs with linear polarization of gluons: symmetric and asymmetric under permutation of the transverse spin of the nucleon and the transverse mome…
We study the role of the non-perturbative input to the transverse momentum dependent (TMD) gluon density in hard processes at the LHC. We derive the input TMD gluon distribution at low scale mu0^2 ~ 1 GeV^2 from the fit of the inclusive hadron spectra measured at low transverse momenta in pp collisions at the LHC and demonstrate that the best description of these spectra for larger hadron transverse momenta can be achieved by matching the derived TMD gluon distribution with the exact solution of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation obtained at low x and small gluon transverse mome…
Transverse momentum dependent (TMD) distribution correlators can be parametrized in terms of TMD parton distribution functions (PDFs), or TMDs for short. We provide an overview of the leading-twist quark and gluon TMDs, both in momentum space and in coordinate space (also called $b_T$-space, where $b_T$ is Fourier conjugate to the partonic momentum $k_T$). We consider unpolarized, vector polarized, as well as tensor polarized hadrons, which is relevant for spin-$0$, spin-$1/2$, and spin-$1$ hadrons.
I discuss how the rapidity evolution of gluon transverse momentum dependent distribution changes from nonlinear evolution at small $x\ll 1$ to linear evolution at moderate $x\sim 1$.
We study how the rapidity evolution of gluon transverse momentum dependent distribution changes from nonlinear evolution at small $x \ll 1$ to linear evolution at moderate $x \sim 1$.
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.
Transverse momentum dependent (TMD) parton distributions in a proton are important in high energy physics from both theoretical and phenomenological points of view. Using the latest RHIC and LHC data on the inclusive soft hadron production in $pp$ and $AA$ collisions at small transverse momenta, we determine the parameters of the initial TMD gluon density, derived in the framework of quark-gluon string model at the low scale $μ_0 \sim 1 - 2$ GeV and refine its large-$x$ behaviour using the LHC data on the $t \bar t$ production at $\sqrt s = 13$ TeV. Then, we apply the Catani-Ciafaloni-Fiorani-…
We study the evolution of the small $x$ gluon transverse momentum dependent(TMD) distribution in the dilute limit. The calculation has been carried out in the Ji-Ma-Yuan scheme using a simple quark target model. As expected, we find that the resulting small $x$ gluon TMD simultaneously satisfies both the Collins-Soper(CS) evolution equation and the Balitsky-Fadin-Kuraev-Lipatov(BFKL) evolution equation. We thus confirmed the earlier finding that the high energy factorization(HEF) and the TMD factorization should be jointly employed to resum the different type large logarithms in a process wher…
We study the rapidity evolution of gluon transverse momentum dependent distributions appearing in processes of particle production and show how this evolution changes from small to moderate Bjorken x.
Transverse Momentum Dependent (TMD) parton distribution functions (PDFs), in short referred to as TMDs, also take into account the transverse momentum (pT) of the partons. Just as the pT-integrated analogues we want to link them to quark and gluon matrix elements using Operator Product Expansion methods in QCD, involving operators of definite twist. The TMDs also involve operators of higher twist, which are not suppressed by powers of the hard scale, however. Using the expression for TMDs involving nonlocal matrix elements of quark and gluon fields there is a gauge link dependence, which also …
We consider the transverse momentum dependent gluon distribution functions (called gluon TMDs) by studying the light-front gluon-gluon correlator, extending the results for unpolarized and vector polarized targets to also include tensor polarized targets -- the latter type of polarization is relevant for targets of spin $\ge1$. The light-front correlator includes process-dependent gauge links to guarantee color gauge invariance. As from the experimental side the gluon TMDs are largely unknown, we present positivity bounds for combinations of leading-twist gluon distributions that may be used t…
We study how the rapidity evolution of gluon transverse momentum dependent distribution changes from nonlinear evolution at small $x\ll 1$ to linear double-logarithmic evolution at moderate $x\sim 1$.
Gluon TMDs can be accessed through the analysis of azimuthal asymmetries for heavy quark pair and dijet production in electron-proton collisions, similarly to the way quark TMDs are commonly extracted from semi-inclusive deep-inelastic scattering data. We calculate the upper bounds for these observables, showing in which kinematic regions they are large enough to be measured in future experiments at an Electron-Ion Collider. Moreover, we study their behavior in the small-$x$ region, adopting a McLerran-Venugopalan model for unpolarized and linearly polarized gluon distributions. By comparison …
Recently we obtained an evolution equation of gluon TMDs, which addresses a problem of unification of different kinematic regimes. It describes evolution in the whole range of Bjorken $x_B$ and the whole range of transverse momentum $k_\perp$. In this notes I study different limits of this evolution equation and show how it yields several well-known and some previously unknown results.
We investigate the dependence of the deep inelastic proton structure function $F_L(x, Q^2)$ on different forms of the transverse momentum dependent (TMD, or unintegrated) gluon distribution. We present a comparison of theoretical results with the latest experimental data taken by the H1 and ZEUS Collaborations at HERA. We demonstrate that these data, despite of having quite large uncertainties, can test different kinds of the gluon TMD in a proton. Moreover, different phenomenological models at low $x$ could be tested by future experiments on deep inelastic scattering.
We calculate in this paper the perturbative gluon transverse momentum dependent parton distribution functions (TMDPDFs) and fragmentation functions (TMDFFs) using the exponential regulator for rapidity divergences. We obtain results for both unpolarized and linearly polarized distributions through next-to-next-to leading order in strong coupling constant, and through ${\cal O}(ε^2)$ in dimensional regulator (finding discrepancy for the linearly polarized gluon TMDPDFs with a previous result in the literature). We find a nontrivial momentum conservation sum rule for the linearly polarized compo…
We study the energy or TMD evolution of the three leading twist dipole type T-odd gluon TMDs inside a transversely polarized nucleon, all of which at small $x$ dynamically originate from the spin dependent odderon. Their energy dependence presents a unique opportunity to study the polarization dependent TMD evolution in the small-$x$ region, where the distributions are identical up to a normalization constant at tree level. We further propose to study their evolution via azimuthal asymmetries in virtual photon-jet production in polarized proton-proton collisions at RHIC. We present model predi…
We present the study of accessing unpolarized and linearly polarized gluon TMDs in J/ψ and Υ(1S) production in unpolarized proton-proton collision at LHC, RHIC and AFTER energies. Non- relativistic QCD based color octet model (COM) is used for estimating quarkonium production rates within transverse momentum dependent factorization formalism. A comparison is drawn between the experimental data and the transverse momentum distribution of quarkonium obtained in COM and color evaporation model.
I report on our investigations into the impact of (un)polarized transverse momentum dependent parton distribution functions (TMD PDFs or TMDs) for gluons at hadron colliders, especially at A Fixed Target Experiment at the LHC (AFTER@LHC). In the context of high energy proton-proton collisions, we look at final states with low mass (e.g. $η_b$) in order to investigate the nonperturbative part of TMD PDFs. We study the factorization theorem for the $q_T$ spectrum of $η_b$ produced in proton-proton collisions relying on the effective field theory approach, defining the tools to perform phenomenol…
We propose a new analytical expression for the Transverse Momentum Dependent (TMD, or unintegrated) gluon density in the proton. Essential phenomenological parameters are extracted from the LHC data on inclusive hadron production in $pp$ collisions at low transverse momenta, $p_T \sim 1$~GeV. The latter are described in the framework of modified soft quark-gluon string model, where gluonic state and non-zero transverse momentum of partons inside the proton are taken into account. To determine the parameters important at moderate and large $x$ we used measurements of inclusive $b$-jet and Higgs…
We present exploratory studies of the 3D proton tomography through polarized $T$-odd gluon TMDs at leading twist, obtained in a spectator-model framework. We embody in our approach a flexible parameterization for the spectator-mass spectral function, suited to catch both small- and moderate-$x$ effects. All these studies are relevant to unveil the gluon dynamics inside hadrons, which represents a core research line of studies at new-generation colliders, such as the Electron-Ion Collider, NICA-SPD, the High-Luminosity LHC, and the Forward Physics Facility.
We show how transverse momentum dependent gluon distributions could be probed at a future Electron-Ion Collider through the analysis of transverse momentum spectra and azimuthal asymmetries in the inclusive electroproduction of $J/ψ$ and $Υ$ mesons. The maximum values of these asymmetries, obtained in a model-independent way by imposing the positivity bounds of the polarized gluon distributions, suggest the feasibility of the proposed measurements.
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.
This paper is a part of a series of works where we in detail examine the concept of Transverse Momentum Dependent (TMD), or k_T, factorization, which is frequently encountered in the literature and is widely used in the phenomenological applications of QCD at very high energies. We address the question of what exactly factorization is, as it is meant in different contexts and formalisms, and we compare the formalisms to each other. We clarify some basic concepts regarding factorization and how it exactly is applied in high energy QCD, and we make important notes on some key and fundamental poi…
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 …
We use the Color Glass Condensate (CGC) framework to study the production of forward heavy quark-antiquark pairs in unpolarized proton-nucleus or proton-proton collisions in the small-x regime. In the limit of nearly back-to-back jets, the CGC result simplifies into the transverse-momentum dependent (TMD) factorization approach. For massless quarks, the TMD factorization formula involves three unpolarized gluon TMDs: the Weizsäcker-Williams gluon distribution, the adjoint-dipole gluon distribution, and an additional one. When quark masses are kept non-zero, three new gluon TMDs appear -- each …
We consider the associated production of $J/ψ$ mesons and muons originating from the $b$-flavored hadron decays and non-prompt double $J/ψ$ production at the LHC using the $k_T$-factorization approach. To describe the inclusive $b$-hadron decays into the different charmonium states we apply fragmentation approach and adopt fragmentation functions based on the non-relativistic QCD factorization. The transverse momentum dependent (TMD) gluon densities in a proton are determined using the Catani-Ciafaloni-Fiorani-Marchesini equation and Kimber-Martin-Ryskin prescription. We investigate the effect…
We consider the transverse momentum dependent (TMD) quark densities of the proton which are very important ingredients for unpolarized Drell-Yan (DY) lepton pair production. We calculate the TMD sea quark density as a convolution of the Catani-Ciafaloni-Fiorani-Marchesini (CCFM)-evolved gluon distribution and the TMD gluon-to-quark splitting function. Based on the ${\cal O}(α^2)$ production amplitude $q^* + \bar q^* \to Z/γ^* \to l^+ + l^-$, calculated by taking into account the effective $q^*\bar q^* Z/γ^*$ - vertex, we analyze the distributions on the dilepton invariant mass, transverse mome…
We use the TMD quark densities of the proton to investigate unpolarized Drell-Yan lepton pair production in proton-proton collisions at the LHC energies. We investigate the case where the gluon-to-quark splitting occurs at the last evolution step and calculate the TMD sea quark density as a convolution of the CCFM-evolved gluon distribution and the TMD gluon-to-quark splitting function which contains all single logarithmic small-x corrections to the sea quark evolution for any order of perturbation theory. Based on the O(alpha) production amplitude of quark-antiquark annihilation calculated ac…
We study parton-branching solutions of QCD evolution equations and present a method to construct both collinear and transverse momentum dependent (TMD) parton densities from this approach. We work with next-to-leading-order (NLO) accuracy in the strong coupling. Using the unitarity picture in terms of resolvable and non-resolvable branchings, we analyze the role of the soft-gluon resolution scale in the evolution equations. For longitudinal momentum distributions, we find agreement of our numerical calculations with existing evolution programs at the level of better than 1 percent over a range…
We review recent progress in the description of unpolarized transverse-momentum-dependent (TMD) gluon distributions at small $x$ in the color glass condensate (CGC) effective theory. We discuss the origin of the non-universality 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 $x$, and on the study of gluon saturation. We discuss recent advances in the uni…
The back-to-back $J/ψ$ and $π$ associated production at $e^+ e^-$ colliders is proposed to detect the gluon transverse momentum dependent(TMD) fragmentation functions. TMD factorization is assumed for this process. With spinless pion, unpolarized and linearly polarized gluon TMD fragmentation functions can be defined. It is found at parton level the hadronic tensor can be described by four structure functions. As a result, there are three independent angular distributions, of which a $\cos{2ϕ}$ azimuthal asymmetry is sensitive to the linearly polarized gluon fragmentation function.
We study the process $γA \to q \bar{q} g+X$ 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 (gluo…
We calculate the transverse momentum dependent gluon-to-gluon splitting function within $k_T$-factorization, generalizing the framework employed in the calculation of the quark splitting functions in [1-3] and demonstrate at the same time the consistency of the extended formalism with previous results. While existing versions of $k_T$ factorized evolution equations contain already a gluon-to-gluon splitting function i.e. the leading order Balitsky-Fadin-Kuraev-Lipatov (BFKL) kernel or the Ciafaloni-Catani-Fiore-Marchesini (CCFM) kernel, the obtained splitting function has the important propert…
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 unpol…
We calculate the transverse momentum dependent gluon-to-gluon splitting function within kT-factorization [EPJC 78 (2018) 174], generalizing the framework of [NPB 175 (1980) 27; NPB 427 (1994) 475] and extending our previous works [JHEP 01 (2016) 181; PRD 94 (2016) 114013] for the quark splittings. While existing versions of kT factorized evolution equations contain already a gluon-to-gluon splitting function i.e. the leading order BFKL kernel or the CCFM kernel, the obtained splitting function has the important property that it reduces to the leading order BFKL kernel in the low z limit, to th…
We study the associated production of real (isolated) or virtual photons (with their subsequent leptonic decay) and hadronic jets in proton-proton collisions at the LHC using the $k_T$-factorization approach of QCD. The consideration is based on the off-shell quark-gluon QCD Compton scattering subprocesses. In the case of virtual photon production, the contributions from Z boson exchange as well as gamma*-Z interference with the full spin correlations are included. The transverse momentum dependent (TMD) quark and gluon densities in a proton are determined from the Kimber-Martin-Ryskin prescri…
We combine soft and collinear matrix elements to define transverse momentum dependent parton distributions (TMDs) for gluons, free from rapidity divergences. We establish a factorization theorem at next-to-leading order for the Higgs boson transverse momentum ($q_T$) spectrum, and use it to derive evolution equations for gluon TMDs. The evolution for all gluon TMDs is driven by a universal kernel, i.e. the same for all polarizations. In the region of intermediate $q_T$ we match the unpolarized, helicity and linearly polarized gluon distributions onto PDFs. We calculate the resummed Higgs trans…
We report on how $J/ψ$- and $Υ$-pair production are promising processes to access the polarised and unpolarised gluon TMDs at the LHC. We present the formalism used, as well as resulting observables that could be extracted from data.
The inclusive large-$p_T$ production of a single pion, jet or direct photon, and Drell-Yan processes, are considered for proton-proton collisions in the kinematical range expected for the fixed-target experiment AFTER, proposed at LHC. For all these processes, predictions are given for the transverse single-spin asymmetry, $A_N$, computed according to a Generalised Parton Model previously discussed in the literature and based on TMD factorisation. Comparisons with the results of a collinear twist-3 approach, recently presented, are made and discussed.
We investigate the gluon transverse momentum dependent correlators as Fourier transform of matrix elements of nonlocal operator combinations. At the operator level these correlators include both field strength operators and gauge links bridging the nonlocality. In contrast to the collinear PDFs, the gauge links are no longer unique for transverse momentum dependent PDFs (TMDs) and also Wilson loops lead to nontrivial effects. We look at gluon TMDs for unpolarized, vector and tensor polarized targets. In particular a single Wilson loop operators become important when one considers the small-x l…
We derive the transverse momentum dependent (TMD) factorization for heavy quark pair production in deep inelastic scattering, where the total transverse momentum is much smaller than the invariant mass of the pair. The factorization is demonstrated at one-loop order, in both Ji-Ma-Yuan and Collins-11 schemes for the TMD definitions, and the hard factors are calculated accordingly. Our result provides a solid theoretical foundation for the phenomenological investigations of the gluon TMDs in this process, and can be extended to other similar hard processes, including dijet (di-hadron) productio…
In a high-energy scattering process with hadrons in the initial state, color is involved. Transverse momentum dependent distribution functions (TMDs) describe the quark and gluon distributions in these hadrons in momentum space with the inclusion of transverse directions. Apart from the (anti)-quarks and gluons that are involved in the hard scattering process, additional gluon emissions by the hadrons have to be taken into account as well, giving rise to Wilson lines or gauge links. The TMDs involved are sensitive to the process under consideration and hence potentially nonuniversal due to the…
