Transverse Momentum Dependent Parton Research Articles

Polarized J/ψ production in semi-inclusive DIS at large Q2 : Comparing quark fragmentation and photon-gluon fusion

We compare the relative importance of different mechanisms for polarized J/ψ production in semi-inclusive deep inelastic scattering processes at large Q2. The transverse momentum dependent (TMD) factorization framework and nonrelativistic quantum chromodynamics are used to study the leading contributions from light quark fragmentation to polarized J/ψ, and compared to direct production via photon-gluon fusion, which can proceed through color-singlet as well as color-octet mechanisms. We identify kinematic regimes where light quark fragmentation dominates, allowing for the extraction of the S31[8] matrix element, as well as regimes where photon gluon fusion dominates, suggesting that the gluon TMD parton distribution function can be probed. Published by the American Physical Society 2024

The Sivers asymmetry in charged Kaon and Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} hyperon produced SIDIS process at electron ion colliders

We investigate the single transverse-spin asymmetry with a sin(ϕh-ϕS)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sin (\\phi _h-\\phi _S)$$\\end{document} modulation in the charged Kaon produced and in Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} hyperon produced SIDIS process within the theoretical framework of transverse momentum dependent (TMD) factorization at the next-to-leading-logarithmic order. The asymmetry is contributed by the convolution of Sivers function and the unpolarized fragmentation function D1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_1$$\\end{document} for the produced hadron. The parametrization for the proton Qiu–Sterman function, which is closely related to the Sivers function, is adopted to numerically estimate the Sivers asymmetry at the kinematical region of Electron Ion Collider (EIC) and Electron Ion Collider in China (EicC). The TMD evolution of the TMD parton distribution functions are considered by employing the non-perturbative Sudakov form factor. It is found that the predicted Sivers asymmetries AUTsin(ϕh-ϕS)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{UT}^{\\sin (\\phi _h-\\phi _S)}$$\\end{document} as functions of x, z and PhT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{hT}$$\\end{document} are sizable at the kinematical configurations of both EIC and EicC. The strange constituent of the produced charged Kaon and Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} hyperon in the final state can be a promising probe of the sea quark Sivers function as well as the flavor dependence in the proton target. Therefore, it is important to utilize the future EIC facilities to constrain the sea quark distribution functions as well as the validity of the generalized universality of the Sivers function.

Transverse momentum distributions of heavy hadrons and polarized heavy quarks

We initiate the study of transverse momentum-dependent (TMD) fragmentation functions for heavy quarks, demonstrate their factorization in terms of novel nonperturbative matrix elements in heavy-quark effective theory (HQET), and prove new TMD sum rules that arise from heavy-quark spin symmetry. We discuss the phenomenology of heavy-quark TMD FFs at B factories and find that the Collins effect, in contrast to claims in the literature, is not parametrically suppressed by the heavy-quark mass. We further calculate all TMD parton distribution functions for the production of heavy quarks from polarized gluons within the nucleon and use our results to demonstrate the potential of the future EIC to resolve TMD heavy-quark fragmentation in semi-inclusive DIS, complementing the planned EIC program to use heavy quarks as probes of gluon distributions.

Transformation of transverse momentum distributions from Parton Branching to Collins-Soper-Sterman framework

Two main frameworks for defining transverse momentum dependent (TMD) parton densities are the Collins-Soper-Sterman (CSS) formalism, and the Parton Branching (PB) approach. While PB-TMDs have an explicit dependence on a single scale which is used to evolve PB-TMDs in momentum space, TMDs defined in CSS formalism present a double-scale evolution in renormalization and rapidity scales, via a pair of coupled evolution equations. In this letter I leverage the Collins-Soper kernel determined from simulated Drell Yan transverse momentum spectra using PB-TMDs, and provide, for the first time, the transformation of TMD parton distributions from the PB framework to the CSS formalism. The evolved PB-TMDs in b-space are compared to the recently released, unpolarized TMD distribution ART23.

Collinear matching for next-to-leading power transverse-momentum distributions

Phenomenological studies of transverse momentum dependent (TMD) parton distributions rely on the expansion in small values of the transverse separation of fields, where TMD parton distributions match onto collinear parton distribution functions. In this work, we derive this expansion at tree-level for the genuine next-to-leading power quark-gluon-quark TMDs, taking into account all the target mass corrections. We find that only a limited number of TMD parton distributions exhibit matching to twist-three collinear distributions, leading to a significant simplification in the analysis of the structure functions of semi-inclusive deep inelastic scattering.

Lattice calculation of the intrinsic soft function and the Collins-Soper kernel

We calculate the soft function using lattice QCD in the framework of large momentum effective theory incorporating the one-loop perturbative contributions. The soft function is a crucial ingredient in the lattice determination of light cone objects using transverse-momentum-dependent (TMD) factorization. It consists of a rapidity-independent part called intrinsic soft function and a rapidity-dependent part called Collins-Soper kernel. We have adopted appropriate normalization when constructing the pseudoscalar meson form factor that is needed in the determination of the intrinsic part and applied Fierz rearrangement to suppress the higher-twist effects. In the calculation of CS kernel we consider a CLS ensemble other than the MILC ensemble used in a previous study. We have also compared the applicability of determining the CS kernel using quasi TMDWFs and quasi TMDPDFs. As an example, the determined soft function is used to obtain the physical TMD wave functions (WFs) of pion and unpolarized iso-vector TMD parton distribution functions (PDFs) of proton.

Multi-jet merging with TMD parton branching

One of the main theoretical systematic uncertainties in studies of final states with large jet multiplicities at high-energy hadron colliders is associated with the merging of QCD parton showers and hard-scattering matrix elements. We present a method to incorporate the physics of transverse momentum recoils due to initial-state shower evolution into multi-jet merging algorithms by using the concept of transverse momentum dependent (TMD) distributions and the associated parton branching. We investigate the dependence on the merging scale and illustrate the impact of the new method at the level of both exclusive and inclusive final-state observables by studying differential jet rates, transverse momentum spectra and multiplicity distributions, using vector boson + jets events at the LHC as a case study.

Drell-Yan transverse spectra at the LHC: a comparison of parton branching and analytical resummation approaches

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.

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.

Associated Higgs boson+ jets production at the LHC and Catani-Ciafaloni-Fiorani-Marchesini gluon dynamics in a proton

We consider the associated production of Higgs boson and hadronic jet(s) in $pp$ collisions at the LHC for the first time using the $k_T$-factorization approach. Our analysis is based on the off-shell gluon-gluon fusion subprocess, where non-zero transverse momenta of initial gluons are taken into account and covers $H \to \gamma \gamma$, $H \to ZZ^* \to 4l$ (with $l = e,\mu$) and $H \to W^+ W^- \to e^\pm \mu^\mp \nu \bar \nu$ decay channels. The transverse momentum dependent (TMD) gluon densities in a proton are taken from Catani-Ciafaloni-Fiorani-Marchesini evolution equation. To simulate the kinematics of the produced jets the TMD parton shower implemented into the Monte-Carlo event generator \textsc{cascade} is applied. The comparison of our results with the latest experimental data taken by the CMS and ATLAS Collaborations at $\sqrt s = 8$ and $13$ TeV and conventional higher-order perturbative QCD calculations is presented. We highlight observables, which are sensitive to the TMD gluon densities in a proton.

CASCADE3 A Monte Carlo event generator based on TMDs

The Cascade3 Monte Carlo event generator based on Transverse Momentum Dependent (TMD) parton densities is described. Hard processes which are generated in collinear factorization with LO multileg or NLO parton level generators are extended by adding transverse momenta to the initial partons according to TMD densities and applying dedicated TMD parton showers and hadronization. Processes with off-shell kinematics within k_{{t}}-factorization, either internally implemented or from external packages via LHE files, can be processed for parton showering and hadronization. The initial state parton shower is tied to the TMD parton distribution, with all parameters fixed by the TMD distribution.

Joint thrust and TMD resummation in electron-positron and electron-proton collisions

We present the framework for obtaining precise predictions for the transverse momentum of hadrons with respect to the thrust axis in e+e− collisions. This will enable a precise extraction of transverse momentum dependent (TMD) fragmentation functions from a recent measurement by the Belle Collaboration. Our analysis takes into account, for the first time, the nontrivial interplay between the hadron transverse momentum and the cut on the thrust event shape. To this end, we identify three different kinematic regions, derive the corresponding factorization theorems within Soft Collinear Effective Theory, and present all ingredients needed for the joint resummation of the transverse momentum and thrust spectrum at NNLL accuracy. One kinematic region can give rise to non-global logarithms (NGLs), and we describe how to include the leading NGLs. We also discuss alternative measurements in e+e− collisions that can be used to access the TMD fragmentation function. Finally, by using crossing symmetry, we obtain a new way to constrain TMD parton distributions, by measuring the displacement of the thrust axis in ep collisions.

Factorization of e+e\u2212 \u2192 H X cross section, differential in zh, PT and thrust, in the 2-jet limit

Factorizing the cross section for single hadron production in e+e− annihilations is a highly non trivial task when the transverse momentum of the outgoing hadron with respect to the thrust axis is taken into account. We work in a scheme that allows to factorize the e+e−→ H X cross section as a convolution of a calculable hard coefficient and a Transverse Momentum Dependent (TMD) fragmentation function. The result, differential in zh, PT and thrust, will be given to all orders in perturbation theory and explicitly computed to Next to Leading Order (NLO) and Next to Leading Log (NLL) accuracy. The predictions obtained from our computation, applying the simplest and most natural ansatz to model the non-perturbative part of the TMD, are in exceptional agreement with the experimental measurements of the BELLE Collaboration. The factorization scheme we propose relates the TMD parton densities defined in 1-hadron and 2-hadron processes, restoring the possi- bility to perform global phenomenological studies of TMD physics including experimental data from semi-inclusive deep inelastic scattering, Drell-Yan processes, e+e−→ H1H2X and e+e−→ H X annihilations.

Universality-breaking effects in e^+e^- hadronic production processes

Recent BELLE measurements provide the cross section for single hadron production in e^+e^- annihilations, differential in thrust and in the hadron transverse momentum with respect to the thrust axis. Universality breaking effects due to process-dependent soft factors make it very difficult to relate this cross sections to those corresponding to hadron-pair production in e^+e^- annihilations, where transverse momentum dependent (TMD) factorization can be applied. The correspondence between these two cross sections is examined in the framework of the Collins-Soper-Sterman factorization, in the collinear as well as in the TMD approach. We propose a scheme that allows to relate the TMD parton densities defined in 1-hadron and in 2-hadron processes, neatly separating, within the soft and collinear parts, the non-perturbative terms from the contributions that can be calculated perturbatively. The regularization of rapidity divergences introduces cut-offs, the arbitrariness of which will be properly reabsorbed by means of a mechanism closely reminiscent of a gauge transformation. In this way, we restore the possibility to perform global phenomenological studies of TMD physics, simultaneously analyzing data belonging to different hadron classes.

Transverse moments of TMD parton densities and ultraviolet divergences

I review some open questions relating to the large transverse momentum divergences in transverse moments of transverse momentum dependent (TMD) parton correlation functions. I also explain, in an abbreviated and summarized form, recent work that shows that the resulting violations of a commonly used integral relation are not perturbatively suppressed. I argue that this implies a need for more precise definitions for the correlation functions used to describe transverse moments.

Efficient Fourier transforms for transverse momentum dependent distributions

Hadron production at low transverse momenta in semi-inclusive deep inelastic scattering can be described by transverse momentum dependent (TMD) factorization. This formalism has also been widely used to study the Drell–Yan process and back-to-back hadron pair production in e+e− collisions. These processes are the main ones for extractions of TMD parton distribution functions and TMD fragmentation functions, which encode important information about nucleon structure and hadronization. One of the most widely used TMD factorization formalism in phenomenology formulates TMD observables in coordinate b⊥-space, the conjugate space of the transverse momentum. The Fourier transform from b⊥-space back into transverse momentum space is sufficiently complicated due to oscillatory integrands that it requires a careful and computationally intensive numerical treatment in order to avoid potentially large numerical errors. Within the TMD formalism, the azimuthal angular dependence is analytically integrated and the two-dimensional b⊥ integration reduces to a one-dimensional integration over the magnitude b⊥. In this paper we develop a fast numerical Hankel transform algorithm for such a b⊥-integration that improves the numerical accuracy of TMD calculations in all standard processes. Libraries for this algorithm are implemented in Python 2.7 and 3, C++, as well as FORTRAN77. All packages are made available open source. Program summaryProgram Title: Fast Bessel Transform (FBT)CPC Library link to program files:http://dx.doi.org/10.17632/65zkb53z8d.1Developer’s repository link:http://github.com/UCLA-TMD/OgataLicensing provisions: MITProgramming language: Python 2/3, C++, FORTRAN77Nature of problem: In order to perform extractions of transverse momentum distribution functions, numerical Hankel transforms must be performed from b⊥-space to momentum space. However, these numerical Hankel transforms are a huge bottleneck in these extractions, making these extractions extremely computationally intensive.Solution method: We develop a numerical Hankel transform algorithm by optimizing Ogata quadrature formula. This algorithm improves the performance of these numerical Hankel transforms by nearly an order of magnitude.

Relation between the parton branching approach and Catani-Ciafaloni-Fiorani-Marchesini evolution

We consider the associated production of electroweak gauge bosons and charm or beauty quark jets at the LHC using the $k_T$-factorization framework. We apply the transverse momentum dependent (TMD) parton distributions in a proton obtained from the Parton Branching (PB) method as well as from the Catani-Ciafaloni-Fiorani-Marchesini (CCFM) evolution equation. For the PB approach, our prescription merges the standard leading order ${\cal O}(\alpha \alpha_s)$ $k_T$-factorization calculations with several tree-level next-to-leading order ${\cal O}(\alpha \alpha_s^2)$ off-shell production amplitudes. For the CCFM scenario, our consideration is based on the ${\cal O}(\alpha \alpha_s^2)$ off-shell gluon-gluon fusion subprocess $g^* g^* \to Z/W + Q\bar Q$ and some subleading ${\cal O}(\alpha \alpha_s^2)$ subprocesses involving quark interactions, taken into account in conventional (collinear) QCD factorization. We find that the $W + c$ and $Z + b$ cross sections, calculated within the PB and CCFM-based schemes with the proper choice of leading and next-to-leading subprocesses, are in good agreement with each other, thus establishing a correspondence between these two scenarios. A comparison with the latest LHC experimental data is given and the necessity for the proper off-shell treatment of the production amplitudes in determination of the parameters of the TMD parton density is demonstrated.

Azimuthal asymmetries in unpolarized SIDIS and Drell-Yan processes: A case study towards TMD factorization at subleading twist

We consider the azimuthal distribution of the final observed hadron in semi-inclusive deep-inelastic scattering and the lepton pair in the Drell-Yan process. In particular, we focus on the cos⁡ϕ modulation of the unpolarized cross section and on its dependence upon transverse momentum. At low transverse momentum, for these observables we propose a factorized expression based on tree-level approach and conjecture that the same formula is valid in transverse-momentum dependent (TMD) factorization when written in terms of subtracted TMD parton distributions. Our formula correctly matches with the collinear factorization results at high transverse momentum, solves a long-standing problem and is a necessary step towards the extension of the TMD factorization theorems up to the subleading twist.

Probing transverse-momentum distributions with groomed jets

We present the transverse momentum spectrum of groomed jets in di-jet events for e+e− collisions and semi-inclusive deep inelastic scattering (SIDIS). The jets are groomed using a soft-drop grooming algorithm which helps in mitigating effects of non-global logarithms and underlying event. At the same time, by reducing the final state hadronization effects, it provides a clean access to the non-perturbative part of the evolution of transverse momentum dependent (TMD) distributions. In SIDIS experiments we look at the transverse momentum of the groomed jet measured w.r.t. the incoming hadron in the Breit frame. Because the final state hadronization effects are significantly reduced, the SIDIS case allows to probe the TMD parton distribution functions. We discuss the sources of non-perturbative effects in the low transverse momentum region including novel (but small) effects that arise due to grooming. We derive a factorization theorem within SCET and resum any large logarithm in the measured transverse momentum up to NNLL accuracy using the ζ-prescription as implemented in the artemide package and provide a comparison with simulations.

Transverse Momentum Dependent Observables from Low to High Energy: Factorization, Evolution, and Global Analyses

Transverse momentum dependent (TMD) observables in high-energy particle collisions allow one to study the motion of hadron constituents (partons) in three dimensions. The intense theoretical, phenomenological, and experimental work over the last several decades has seen great progress in modeling and extracting TMD parton distribution functions (PDFs) and fragmentation functions (FFs) [1, 2]. The wealth of experimental data, both current and future [3–6], across various reactions and energy ranges, along with substantial progress in other approaches, such as Lattice QCD [7], has increased the demand to rigorously study the intrinsic transverse motion of hadrons. Such issues currently being addressed include factorization, evolution, and operator definitions of TMD distributions, as well as how they influence phenomenological and Lattice QCD studies. Therefore, in this special issue we attempt to further the effort of understanding TMD structure through a series of review articles that address some of the significant areas of ongoing research.