Transverse Momentum Dependent Parton Distribution Research Articles

Determination of the Collins-Soper Kernel from Lattice QCD.

This Letter presents a determination of the quark Collins-Soper kernel, which relates transverse-momentum-dependent parton distributions (TMDs) at different rapidity scales, using lattice quantum chromodynamics (QCD). This is the first such determination with systematic control of quark mass, operator mixing, and discretization effects. Next-to-next-to-leading logarithmic matching is used to match lattice-calculable distributions to the corresponding TMDs. The continuum-extrapolated lattice QCD results are consistent with several recent phenomenological parametrizations of the Collins-Soper kernel and are precise enough to disfavor other parametrizations.

Measurements of the Z0/γ⁎ cross section and transverse single spin asymmetry in 510 GeV p + p collisions

The differential cross section for Z0 production, measured as a function of the boson's transverse momentum (pT), provides important constraints on the evolution of the transverse momentum dependent parton distribution functions (TMDs). The transverse single spin asymmetry (TSSA) of the Z0 is sensitive to one of the polarized TMDs, the Sivers function, which is predicted to have the opposite sign in p+p→W/Z+X from that which enters in semi-inclusive deep inelastic scattering. In this Letter, the STAR Collaboration reports the first measurement of the Z0/γ⁎ differential cross section as a function of its pT in p+p collisions at a center-of-mass energy of 510 GeV, together with the Z0/γ⁎ total cross section. We also report the measurement of Z0/γ⁎ TSSA in transversely polarized p+p collisions at 510 GeV.

Simultaneous reweighting of Transverse Momentum Dependent distributions

The Bayesian reweighting procedure is extended to the case of multiple independent extractions of transverse momentum dependent parton distributions (TMDs). By exploiting the data on transverse single spin asymmetries, AN, for inclusive pion production in polarized proton-proton collisions measured at RHIC, we perform a simultaneous reweighting of the quark Sivers, transversity and Collins TMD functions extracted from semi-inclusive deep inelastic scattering (SIDIS) and e+e− annihilation into hadron pairs. The impact of the implementation of the Soffer bound, as well as the differences between older and newer AN data, are investigated. The agreement with AN data at large-xF values, a kinematical region complementary to those explored in SIDIS measurements, is enhanced, improving the knowledge of the polarized quark TMDs in the large-x region.

Wigner distributions of sea quarks in the light-cone quark model

We investigate the Wigner distributions of u¯ and d¯ quarks in a proton using the overlap representation within the light cone formalism. Using the light-cone wave functions which are obtained from the baryon-meson fluctuation model in terms of the |qq¯B⟩ Fock states, we calculate the Wigner distributions for the unpolarized/longitudinally polarized sea quarks in an unpolarized/longitudinally polarized proton. The Wigner distributions can be obtained through a Fourier transform on the generalized transverse-momentum dependent parton distributions (GTMDs). We also calculate the GTMDs of u¯ and d¯ quarks in the intermediate step. Numerical results for the Wigner distributions of u¯ and d¯ quarks in transverse momentum space, impact parameter space and the mixed plane are presented. We also study the orbital angular momentum and the spin-orbit correlations of the sea quarks. Published by the American Physical Society 2024

Extraction of unpolarized transverse momentum distributions from the fit of Drell-Yan data at N4LL

We present an extraction of unpolarized transverse momentum dependent parton distributions functions (TMDPDFs) and Collins-Soper kernel from the fit of Drell-Yan and weak-vector boson production data. The TMDPDF are parameterized, as commonly done, using their (large transverse momentum) asymptotic matching to PDF. The analysis is done at the next-to-next-to-next-to-next-to leading logarithmic accuracy (N4LL) (performed only approximately because PDF evolution is known so far at next-to-next-to leading order (NNLO)). The non-perturbative model used for TMDPDF is flavor dependent to reduce the colllinear PDF bias. The estimation of uncertainties is done with the replica method and, for the first time, it includes the propagation of uncertainties due to the collinear distributions.

Transverse momentum weighted Sivers asymmetries in SIDIS and Drell-Yan processes at COMPASS

We investigate the transverse momentum weighted Sivers asymmetries in the processes with transversely polarized proton target, including the PhTzMpsin⁡(ϕh−ϕS) weighted asymmetry in charged hadron production in semi-inclusive deeply inelastic scattering (SIDIS) and the qTMpsin⁡(ϕS) weighted asymmetry in π−p Drell-Yan process. These asymmetries can be expressed as the product of the transverse-moments of the transverse momentum dependent (TMD) parton distribution functions (PDFs) and fragmentation functions (FFs). Using the parametrization for the Sivers function of the proton, the unpolarized PDFs of proton and pion, and the unpolarized FF of charged hadron, we present the numerical calculation for weighted Sivers asymmetries in SIDIS and Drell-Yan processes at the kinematics of COMPASS. We find that our prediction for the SIDIS process is in agreement with the recent COMPASS measurement. Recent new high-statistics data of Sivers asymmetry measured in SIDIS process and Drell-Yan process from COMPASS favored the sign change of Sivers function. High-precision experimental data of weighted asymmetries are still needed to test the sign change property of the Sivers function between SIDIS and Drell-Yan processes and to constrain the sea quark Sivers function.

Polarized fragmenting jet functions in inclusive and exclusive jet production

In this work, we present a complete theoretical framework for analyzing the distribution of polarized hadrons within jets, with and without measuring the transverse momentum relative to the standard jet axis. Using soft-collinear effective theory (SCET), we derive the factorization and provide the theoretical calculation of both semi-inclusive and exclusive fragmenting jet functions (FJFs) under longitudinal and transverse polarization. With the polarized FJFs, one gains access to a variety of new observables that can be used for extracting both collinear and transverse momentum dependent parton distribution functions (PDFs) and fragmentation functions (FFs). As examples, we provide numerical results for the spin asymmetry ATU,TcosϕS−ϕ̂Sh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {A}_{TU,T}^{\\cos \\left({\\phi}_S-{\\hat{\\phi}}_{S_h}\\right)} $$\\end{document} from polarized semi-inclusive hadron-in-jet production in polarized pp collisions at RHIC kinematics, where a transversely polarized quark would lead to the transverse spin of the final-state hadron inside the jet and is thus sensitive to the transversity fragmentation functions. Similarly, another spin asymmetry, ATU,Lcosϕq−ϕS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {A}_{TU,L}^{\\cos \\left({\\phi}_q-{\\phi}_S\\right)} $$\\end{document} from polarized exclusive hadron-in-jet production in polarized ep collisions at EIC kinematics would allow us to access the helicity fragmentation functions. These observables demonstrate promising potential in investigating transverse momentum dependent PDFs and FFs and are worthwhile for further measurements.

Exploring twist-4 chiral-even GPDs in the light-front quark-diquark model

In this work, we delve into the realm of quantum chromodynamics (QCD) by calculating twist-4 chiral-even generalized parton distributions (GPDs), within the context of the light-front quark-diquark model (LFQDM), focusing specifically on the intriguing case of zero skewness. To shed light on the behavior of twist-4 chiral-even GPDs, we present comprehensive two-dimensional (2-D) and 3-dimensional (3-D) plots, mapping their dependence on the longitudinal momentum fraction (x) and the momentum transfer (t). Furthermore, our research looks into the profound connection between twist-4 chiral-even GPDs and other distribution functions (DFs), including parton distribution functions (PDFs), transverse momentum-dependent parton distributions (TMDs) and generalized transverse-momentum dependent parton distributions (GTMDs). Additionally, our investigation extends to the exploration of related higher twist form factors (FFs), which play a pivotal role in elucidating the internal structure of hadrons. To provide significant insights into the partons spatial distribution, we have also yielded impact parameter GPD plots.

Quark and gluon distributions in ρ-meson from basis light-front quantization

We solve for the ρ-meson's wave functions from a light-front QCD Hamiltonian determined for its constituent quark-antiquark and quark-antiquark-gluon Fock components, with a three-dimensional confinement using basis light-front quantization. From this, we obtain the leading-twist valence quark's parton distribution functions and transverse momentum-dependent parton distributions inside the ρ-meson. These results are qualitatively consistent with those of other models. We also demonstrate the important effects of a dynamical gluon on the ρ-meson's gluon densities, helicity, transversity, and tensor polarized distributions.

The small kT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{T}}$$\\end{document}region in Drell–Yan production at next-to-leading order with the parton branching method

The Parton Branching (PB) method describes the evolution of transverse momentum dependent (TMD) parton distributions, covering all kinematic regions from small to large transverse momenta kT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{T}}$$\\end{document}. The small kT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{T}}$$\\end{document}-region is very sensitive both to the contribution of the intrinsic motion of partons (intrinsic kT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{T}}$$\\end{document}) and to the resummation of soft gluons taken into account by the PB TMD evolution equations. We study the role of soft-gluon emissions in TMD as well as integrated parton distributions. We perform a detailed investigation of the PB TMD methodology at next-to-leading order (NLO) in Drell–Yan (DY) production for low transverse momenta. We present the extraction of the nonperturbative “intrinsic-kT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{T}}$$\\end{document} ” distribution from recent measurements of DY transverse momentum distributions at the LHC across a wide range in DY masses, including a detailed treatment of statistical, correlated and uncorrelated uncertainties. We comment on the (in)dependence of intrinsic transverse momentum on DY mass and center-of-mass energy, and on the comparison with other approaches.

T-even TMDs for the spin-0 pseudo-scalar mesons upto twist-4 using light-front formalism

We have investigated the pseudo-scalar meson structure in the form of transverse momentum-dependent parton distribution functions (TMDs) in the light-front based holographic model and quark model. Starting from leading order, we have calculated all the time-reversal even TMDs for pion and kaon up to twist-4 in these models. We have shown the 3-dimensional structure as well as the 2-dimensional structure of these particles along with their average quark transverse momenta. The parton distribution functions (PDFs) of pseudo-scalar pion have been compared with E615 and modified E615 results. The sum rules, TMD transverse dependence, inverse moments and Gaussian transverse dependence ratio in these models have also been studied. Further, the transverse quark densities have also been analyzed in the momentum space plane for these particles. The higher twist kaon properties in light-front framework have been predicted for the first time in this work.

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.

The neutral current deeply inelastic scattering process and a revisit on the intrinsic asymmetry

The intermediate particle in the neutral current deeply inelastic scattering process can be either a virtual photon or a Z0 boson. In this paper we calculate contributions from the weak interaction in the semi-inclusive deeply inelastic scattering process. Semi-inclusive implies that a jet in the current region is also measured in addition to the scattered electron. Jet can be a good candidate to study the three-dimensional structure of the nucleon in the semi-inclusive deeply inelastic scattering process. In the vector boson-nucleon collinear frame, the transverse momentum of the jet is the same to that of the quark in the nucleon under the leading order approximation. As a result, one can define the intrinsic asymmetry which reveals the imbalance of the transverse momentum of the incident quark to explore the transverse momentum dependent parton distribution functions (TMD PDFs). With the Gaussian ansatz for the TMD PDFs, we restudy the transverse momentum dependent behavior of the intrinsic asymmetry. We find that the asymmetry tends to a linear function with respect to k⊥ when Δu,d2 approach Δ′u,d2.

QCD resummation of dijet azimuthal decorrelations in pp and pA collisions

We study the azimuthal angular decorrelations of dijet production in both proton-proton (pp) and proton-nucleus (pA) collisions. By utilizing soft-collinear effective theory, we establish the factorization and resummation formalism at the next-to-leading logarithmic accuracy for the azimuthal angular decorrelations in the back-to-back limit in pp collisions. We propose an approach where the nuclear modifications to dijet production in pA collisions are accounted for in the nuclear modified transverse momentum dependent parton distribution functions (nTMDPDFs), which contain both collinear and transverse dynamics. This approach naturally generalizes the well-established formalism related to the nuclear modified collinear parton distribution functions (nPDFs). We demonstrate strong consistency between our methodology and the CMS measurements in both pp and pA collisions, and make predictions for dijet production in the forward rapidity region in pA collisions at LHC kinematics and for mid-rapidity kinematics at sPHENIX. Throughout this paper, we focus on the application of this formalism to a simultaneous fit to both collinear and transverse momentum dependent contributions to the transverse momentum dependent distributions.

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.

Transverse momentum dependent parton distributions of pion at leading twist

Transverse momentum dependent parton distributions of pion at leading twist

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.

Azimuthal correlations in Z +jets events in proton–proton collisions at sqrt{s} = 13,text {Te}hspace{-.08em}text {V}

The production of Z bosons associated with jets is measured in text {p}text {p} collisions at sqrt{s}=13,text {Te}hspace{-.08em}text {V} with data recorded with the CMS experiment at the LHC corresponding to an integrated luminosity of 36.3,text {fb}^{-1}. The multiplicity of jets with transverse momentum p_{textrm{T}} > 30,text {Ge}hspace{-.08em}text {V} is measured for different regions of the Z boson’s p_{textrm{T}} (text {Z }), from lower than 10,text {Ge}hspace{-.08em}text {V} to higher than 100,text {Ge}hspace{-.08em}text {V}. The azimuthal correlation varDelta phi between the Z boson and the leading jet, as well as the correlations between the two leading jets are measured in three regions of p_{textrm{T}} (text {Z }). The measurements are compared with several predictions at leading and next-to-leading orders, interfaced with parton showers. Predictions based on transverse-momentum dependent parton distributions and corresponding parton showers give a good description of the measurement in the regions where multiple parton interactions and higher jet multiplicities are not important. The effects of multiple parton interactions are shown to be important to correctly describe the measured spectra in the low p_{textrm{T}} (text {Z }) regions.