Transverse Momentum Dependent Gluon Research Articles

Two-loop bottom mass effects on the Higgs transverse momentum spectrum in top-induced gluon fusion

We compute bottom mass (mb) corrections to the transverse momentum (qT) spectrum of Higgs bosons produced by gluon fusion in the regime qT ∼ mb ≪ mH at leading power in mb/mH and qT/mH, where the gluons couple to the Higgs via a top loop. To this end we calculate the quark mass dependence of the transverse momentum dependent gluon beam functions (aka gluon TMDPDFs) at two loops in the framework of SCET. These functions represent the collinear matrix elements in the factorized gluon-fusion cross section for small qT. We discuss in detail technical subtleties regarding rapidity regulators and zero-bin subtractions in the calculation of the virtual corrections present for massive quarks. Combined with the known soft function for mb ≠ 0 our results allow to determine the resummed Higgs qT distribution in the top-induced gluon fusion channel at NNLL′ (and eventually N3LL) with full dependence on mb/qT. We perform a first phenomenological analysis at fixed order, where the new corrections to the massless approximation lead to percent-level effects in the peak region of the Higgs qT spectrum. Upon resummation they may thus be relevant for state-of-the-art precision predictions for the LHC.

Cos2ϕt azimuthal asymmetry in back-to-back J/ψ -jet production in ep→eJ/ψ jet X at the EIC

In this article, we investigate the $\cos2\phi_t$ azimuthal asymmetry in $e ~p\rightarrow e ~J/\psi ~Jet~ X$, where the $J/\psi$-jet pair is almost back-to-back in the transverse plane, within the framework of the generalized parton model(GPM). We use non-relativistic QCD(NRQCD) to calculate the $J/\psi$ production amplitude and incorporate both color singlet(CS) and color octet(CO) contributions to the asymmetry. We estimate the asymmetry using different parameterizations of the gluon TMDs in the kinematics that can be accessed at the future electron-ion collider (EIC) and also investigate the impact of transverse momentum dependent (TMD) evolution on the asymmetry. We present the contributions coming from different states to the asymmetry in NRQCD.

Rapidity evolution of TMDs with running coupling

The scale of a coupling constant for rapidity-only evolution of transverse-momentum dependent (TMD) operators in the Sudakov kinematic region is calculated using the Brodsky-Lepage-Mackenzie (BLM) optimal scale setting. The effective argument of a coupling constant is halfway in the logarithmical scale between the transverse momentum and energy of TMD distribution. The resulting rapidity-only evolution equation is solved for quark and gluon TMDs.

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.

Parton rescattering and gluon saturation in dijet production at EIC

Large angle gluon radiations induced by multiple parton scatterings contribute to dijet production in deeply inelastic scattering off a large nucleus at the Electron-Ion Collider. Within the generalized high-twist approach to multiple parton scattering, such contributions at the leading order in perturbative QCD and large Bjorken momentum fraction $x_B$ can be expressed as a convolution of the multiple parton scattering amplitudes and the transverse momentum dependent (TMD) two-parton correlation matrix elements. We study this medium-induced dijet spectrum and its azimuthal angle correlation under the approximation of small longitudinal momentum transfer in the secondary scattering and the factorization of two-parton correlation matrix elements as a product of quark and gluon TMD parton distribution function (PDF). Using a simple model for gluon saturation based on the parametrized gluon TMD PDF, we can calculate the $x_B$ and $Q^2$ dependence of the saturation scale and parton transport coefficient $\hat q$. Contributions to dijet cross section from double scattering are power-suppressed and only become sizable for mini-jets at small transverse momentum. We find that the total dijet correlation for these mini-jets, which also includes the contribution from single scattering, is sensitive to the transverse momentum broadening in the quark TMD PDF at large $x$ and saturation in the gluon TMD PDF at small $x$ inside the nucleus. The correlation from double scattering is also found to increase with the dijet rapidity gap and have a quadratic nuclear-size dependence because of the Landau-Pomeranchuk-Migdal (LPM) interference in gluon emission induced by multiple scattering. Experimental measurements of such unique features in the dijet correlation can shed light on the LPM interference in strong interaction and gluon saturation in large nuclei.

General helicity formalism for two-hadron production in e+e\u2212 annihilation within a TMD approach

We present the complete leading-order results for the azimuthal dependences and polarization observables in e+e−→ h1h2 + X processes, where the two hadrons are produced almost back-to-back, within a transverse momentum dependent (TMD) factorization scheme. We consider spinless (or unpolarized) and spin-1/2 hadron production and give the full set of the corresponding quark and gluon TMD fragmentation functions (TMD-FFs). By adopting the helicity formalism, which allows for a more direct probabilistic interpretation, single- and double-polarization cases are discussed in detail. Simplified expressions, useful for phenomenological analyses, are obtained by assuming a factorized Gaussian-like dependence on intrinsic transverse momenta for the TMD-FFs.

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.

The importance of kinematic twists and genuine saturation effects in dijet production at the Electron-Ion Collider

We compute the differential yield for quark anti-quark dijet production in high-energy electron-proton and electron-nucleus collisions at small x as a function of the relative momentum P⊥ and momentum imbalance k⊥ of the dijet system for different photon virtualities Q2, and study the elliptic and quadrangular anisotropies in the relative angle between P⊥ and k⊥. We review and extend the analysis in [1], which compared the results of the Color Glass Condensate (CGC) with those obtained using the transverse momentum dependent (TMD) framework. In particular, we include in our comparison the improved TMD (ITMD) framework, which resums kinematic power corrections of the ratio k⊥ over the hard scale Q⊥. By comparing ITMD and CGC results we are able to isolate genuine higher saturation contributions in the ratio Qs/Q⊥ which are resummed only in the CGC. These saturation contributions are in addition to those in the Weizsäcker-Williams gluon TMD that appear in powers of Qs/k⊥. We provide numerical estimates of these contributions for inclusive dijet production at the future Electron-Ion Collider, and identify kinematic windows where they can become relevant in the measurement of dijet and dihadron azimuthal correlations. We argue that such measurements will allow the detailed experimental study of both kinematic power corrections and genuine gluon saturation effects.

Unpolarized quark and gluon TMD PDFs and FFs at N3LO

In this paper we calculate analytically the perturbative matching coefficients for unpolarized quark and gluon Transverse-Momentum-Dependent (TMD) Parton Distribution Functions (PDFs) and Fragmentation Functions (FFs) through Next-to-Next-to-Next-to-Leading Order (N3LO) in QCD. The N3LO TMD PDFs are calculated by solving a system of differential equation of Feynman and phase space integrals. The TMD FFs are obtained by analytic continuation from space-like quantities to time-like quantities, taking into account the probability interpretation of TMD PDFs and FFs properly. The coefficient functions for TMD FFs exhibit double logarithmic enhancement at small momentum fraction z. We resum such logarithmic terms to the third order in the expansion of αs. Our results constitute important ingredients for precision determination of TMD PDFs and FFs in current and future experiments.

Investigating the Transverse Momentum Dependent Gluon Sivers Function in Quarkonium Production at pp Colliders

In this contribution, we will present a short overview of the transverse momentum dependent (TMD) approach as a tool for studying the 3-dimensional structure of hadrons in high-energy (un)polarized hadron collisions. We will then summarize the present status of a running research programme that aims at constraining the poorly known transverse momentum dependent gluon Sivers function, through the study of single spin asymmetries in quarkonium (mainly $J/\psi$), pion, and $D$-meson production in polarized proton-proton collisions at RHIC. Finally, we will shortly discuss perspectives for this field of research, emphasizing in particular its role in the physics programme of LHC in the fixed-target setup and NICA.

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.

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.

Bottomonia production and polarization in the NRQCD with k_T-factorization. II: Upsilon (2S) and chi _b(2P) mesons

The Upsilon (2S) production and polarization at high energies is studied in the framework of k_T-factorization approach. Our consideration is based on the non-relativistic QCD formalism for bound states formation and off-shell production amplitudes for hard partonic subprocesses. The direct production mechanism, feed-down contributions from radiative chi _b(3P) and chi _b(2P) decays and contributions from Upsilon (3S) decays are taken into account. The transverse momentum dependent gluon densities in a proton were derived from the Ciafaloni–Catani–Fiorani–Marchesini evolution equation, Kimber-Martin-Ryskin prescription and Parton Branching method. Treating the non-perturbative color octet transitions in terms of the mulitpole radiation theory, we extract the corresponding non-perturbative matrix elements for Upsilon (2S) and chi _b(2P) mesons from a combined fit to Upsilon (2S) transverse momenta distributions measured by the CMS and ATLAS Collaborations at the LHC energies sqrt{s} = 7 and 13 TeV and from the relative production rate R^{chi _b(2P)}_{Upsilon (2S)} measured by the LHCb Collaboration at sqrt{s} = 7 and 8 TeV. Then we apply the extracted values to investigate the polarization parameters lambda _theta , lambda _phi and lambda _{theta phi }, which determine the Upsilon (2S) spin density matrix. Our predictions have a good agreement with the currently available data within the theoretical and experimental uncertainties.

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.

Scale dependence of the small x transverse momentum dependent gluon distribution

We investigate the scale dependence of transverse momentum dependent(TMD) gluon distribution in saturation regime. We found that in the Collins-2011 scheme, the scale dependence of small x gluon TMD is governed by the same renormalization group(RG) equation that holds at moderate or large $x$. Following the standard procedure, one then can resum both double leading logarithm and single leading logarithm in saturation regime by jointly solving the Collins-Soper equation and the RG equation.

Accessing the gluon Sivers function at a future electron-ion collider

In this work, we present a systematic study on the feasibility of probing the largely unexplored transverse momentum dependent gluon Sivers function (GSF) in open charm production, high p$_T$ charged di-hadron and di-jet production at a future high energy, high luminosity Electron-Ion Collider (EIC). The Sivers function is a measure for the anisotropy of the parton distributions in momentum space inside a transversely polarized nucleon. It is proposed that it can be studied through single spin asymmetries in the photon-gluon fusion subprocess in electron proton collisions at the EIC. Using a well tuned Monte Carlo model for deep inelastic scattering, we estimate the possible constraints of the GSF from the future EIC data. A comparison of all the accessible measurements illustrates that the di-jet channel is the most promising way to constrain the magnitude of the GSF over a wide kinematic range.

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.

Linearly polarized small- x gluons in forward heavy-quark pair production

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 Weizs\"{a}cker-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 partnered to one of the aforementioned distributions -- which describe the distribution of linearly-polarized gluons in the unpolarized small-x target. We show how these six gluon TMDs emerge from the CGC formulation and we determine their expressions in terms of Wilson line correlators. We calculate them analytically in the McLerran-Venugopalan model, and further evolve them towards smaller values of x using a numerical implementation of JIMWLK evolution.

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.