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DOE OSTI2024

Transverse Momentum Distributions from Lattice QCD without Wilson Lines

The transverse-momentum-dependent distributions (TMDs), which are defined by gauge-invariant 3D parton correlators with staple-shaped lightlike Wilson lines, can be calculated from quark and gluon correlators fixed in the Coulomb gauge on a Euclidean lattice. These quantities can be expressed gauge invariantly as the correlators of Coulomb-gauge-dressed fields, which reduce to the standard TMD correlators under principal-value prescription in the infinite boost limit. In the framework of large-momentum effective theory, a quasi-TMD defined from such correlators in a large-momentum hadron state can be matched to the TMD via a factorization formula, whose exact form is derived using soft collinear effective theory and verified at one-loop order. Compared to the currently used gauge-invariant correlators, this new method can substantially improve statistical precision and simplify renormalization for the time-reversal-even TMDs, which will greatly enhance the predicative power of lattice QCD in the nonperturbative region.

Effective field theory↗
DOE OSTI2024

Flavor dependence of unpolarized quark transverse momentum distributions from a global fit

We present an extraction of the unpolarized transverse-momentum-dependent parton distribution and fragmentation functions that takes into account possible differences between quark flavors and final-state hadrons. The extraction is based on experimental measurements from Drell-Yan processes and semi-inclusive deep-inelastic scattering, whose combination is essential to distinguish flavor differences. The analysis is carried out at N 3 LL accuracy. The extracted flavor-dependent distributions give a very good description of the data (χ 2 /N dat = 1.08). The resulting error bands take fully into account also the uncertainties in the determination of the corresponding collinear distributions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Lattice QCD Benchmark of Proton Helicity and Flavor-Dependent Unpolarized Transverse Momentum-Dependent Parton Distribution Functions at Physical Quark Masses

We present the first lattice QCD calculations of the isovector helicity transverse momentum-dependent parton distribution function (TMDPDF) and the flavor-dependent unpolarized TMDPDFs for up and down quarks in the proton. Our computations utilize domain-wall fermion discretization with physical quark masses. Employing Coulomb-gauge-fixed quark correlation functions within the large-momentum effective theory framework, we access nonperturbative transverse quark separations 𝑏 𝑇 up to approximately 1 fm, corresponding to transverse momenta as low as 200 MeV. Based on the quasi-TMD factorization theorem, we construct renormalization-group–invariant ratios that are equal to the corresponding light-cone TMDPDF ratios. At moderate 𝑥, our results reveal that the isovector helicity and unpolarized TMDPDFs exhibit nearly identical transverse structure up to a normalization factor, and the unpolarized distributions display only mild flavor dependence. These findings not only support key trends observed in recent global analyses but also provide robust, nonperturbative constraints that can distinguish between different parameterizations. This Letter establishes a first-principles benchmark for TMDPDFs, offering valuable input for ongoing and future experimental efforts to map the proton’s three-dimensional structure.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Transverse momentum moments

We establish robust relations between Transverse Momentum Dependent distributions (TMDs) and collinear distributions. We define weighted integrals of TMDs that we call Transverse Momentum Moments (TMMs) and prove that TMMs are equal to collinear distributions evaluated in some minimal subtraction scheme. The conversion to the modified minimal subtraction ( MS ¯ ) scheme can be done by a calculable factor, which we derive up to three loops for some cases. We discuss in detail the zeroth, the first, and the second TMMs and provide phenomenological results for them based on the current extractions of TMDs. The results of this paper open new avenues for theoretical and phenomenological investigation of the three-dimensional and collinear hadron structures. Published by the American Physical Society 2024

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Jet Definition and Transverse-Momentum–Dependent Factorization in Semi-inclusive Deep-Inelastic Scattering

Using the color dipole picture of deep inelastic scattering (DIS) and the color glass condensate effective theory, we study semi-inclusive jet production in DIS at small x in the limit where the photon virtuality Q 2 is much larger than the transverse momentum squared P$^{2}_{⊥}$ of the produced jet. In this limit, the cross section is dominated by aligned jet configurations, that is, quark–antiquark pairs in which one of the fermions—the would-be struck quark in the Breit frame—carries most of the longitudinal momentum of the virtual photon. We show that physically meaningful jet definitions in DIS are such that the effective axis of the jet sourced by the struck quark is controlled by its virtuality rather than by its transverse momentum. For such jet definitions, we show that the next-to-leading order cross section admits factorization in terms of the (sea) quark transverse momentum dependent distribution, which in turn satisfies a universal Dokshitzer-Gribov-Lipatov-Altarelli-Parisi and Sudakov evolution.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Back-to-Back Inclusive Dijets in Deep Inelastic Scattering at Small x : Complete NLO Results and Predictions

We compute the back-to-back dijet cross section in deep inelastic scattering at small x to next-to-leading order (NLO) in the color glass condensate effective field theory. Our result can be factorized into a convolution of the Weizsäcker-Williams gluon transverse-momentum-dependent distribution function (WW gluon TMD) with a universal soft factor and an NLO coefficient function. The soft factor includes both double and single logarithms in the ratio of the relative transverse momentum P ⊥ of the dijet pair to the dijet momentum imbalance q ⊥ ; its renormalization group (RG) evolution is resummed into the Sudakov factor. Likewise, the WW TMD obeys a nonlinear RG equation in x that is kinematically constrained to satisfy both the lifetime and rapidity ordering of the projectile. Exact analytical expressions are obtained for the NLO coefficient function of transversely and longitudinally polarized photons. Our results allow for the first quantitative separation of the dynamics of Sudakov suppression from that of gluon saturation. They can be extended to other final states and provide a framework for precision tests of novel QCD many-body dynamics at the Electron-Ion Collider. Published by the American Physical Society 2024

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

A precise measurement of the Z -boson double-differential transverse momentum and rapidity distributions in the full phase space of the decay leptons with the ATLAS experiment at $\sqrt{s}=8$ TeV

This paper presents for the first time a precise measurement of the production properties of the Z boson in the full phase space of the decay leptons. This is in contrast to the many previous precise unfolded measurements per formed in the fiducial phase space of the decay leptons. The measurement is obtained from proton–proton collision data collected by the ATLAS experiment in 2012 at $\sqrt{s}$ = 8 TeV at the LHC and corresponding to an integrated luminosity of 20.2 fb -1 . The results, based on a total of 15.3 million Z boson decays to electron and muon pairs, extend and improve a previous measurement of the full set of angular coefficients describing Z-boson decay. The double-differential cross section distributions in Z-boson transverse momentum pT and rapidity y are measured in the pole region, defined as 80 < m ℓℓ < 100 GeV, over the range |y| < 3.6. The total uncertainty of the normalised cross-section measurements in the peak region of the pT distribution is dominated by statistical uncertainties over the full range and increases as a function of rapidity from 0.5–1.0% for |y| < 2.0 to 2 - 7% at higher rapidities. The results for the rapidity-dependent transverse momentum distributions are compared to state of-the-art QCD predictions, which combine in the best cases approximate N 4 LL resummation with N 3 LO fixed-order per turbative calculations. The differential rapidity distributions integrated over pT are even more precise, with accuracies from 0.2–0.3% for |y| < 2.0 to 0.4–0.9% at higher rapidities, and are compared to fixed-order QCD predictions using the most recent parton distribution functions. The agreement between data and predictions is quite good in most cases.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Parton distributions and lattice-QCD calculations: Toward 3D structure

The strong force which binds hadrons is described by the theory of quantum chromodynamics (QCD). Determining the character and manifestations of QCD is one of the most important and challenging outstanding issues necessary for a comprehensive understanding of the structure of hadrons. Within the context of the QCD parton picture, the parton distribution functions (PDFs) have been remarkably successful in describing a wide variety of processes. However, these PDFs have generally been confined to the description of collinear partons within the hadron. New experiments and facilities provide the opportunity to additionally explore the transverse structure of hadrons which is described by generalized parton distributions (GPDs) and transverse-momentum-dependent parton distribution functions (TMD PDFs). In our previous report Lin et al. (2018), we compared and contrasted the two main approaches used to determine the collinear PDFs: the first based on perturbative QCD factorization theorems, and the second based on lattice-QCD calculations. In the present report, we provide an update of recent progress on the collinear PDFs, and also expand the scope to encompass the generalized PDFs (GPDs and TMD PDFs). We review the current state of the various calculations, and consider what new data might be available in the near future. We also examine how a shared effort can foster dialog between the PDF and lattice-QCD communities, and yield improvements for these generalized PDFs.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Transverse momentum-dependent parton distributions for longitudinally polarized nucleons from domain wall fermion calculations at the physical pion mass

Previous Lattice QCD calculations of nucleon transverse momentum-dependent parton distribu?tions (TMDs) focused on the case of transversely polarized nucleons, and thus did not encompass two leading-twist TMDs associated with longitudinal polarization, namely, the helicity TMD and the worm-gear TMD corresponding to transversely polarized quarks in a longitudinally polarized nucleon. Based on a definition of TMDs via hadronic matrix elements of quark bilocal operators containing staple-shaped gauge connections, TMD observables characterizing the aforementioned two TMDs are evaluated, utilizing an RBC/UKQCD domain wall fermion ensemble at the physical pion mass.

Engelhardt, Michael↗
DOE OSTI2022

Factorization for azimuthal asymmetries in SIDIS at next-to-leading power

Differential measurements of the semi-inclusive deep inelastic scattering (SIDIS) process with polarized beams provide important information on the three-dimensional structure of hadrons. Among the various observables are azimuthal asymmetries that start at subleading power, and which give access to novel transverse momentum dependent distributions (TMDs). Theoretical predictions for these distributions are currently based on the parton model rather than a rigorous factorization based analysis. Working under the assumption that leading power Glauber interactions do not spoil factorization at this order, we use the Soft Collinear Effective Theory to derive a complete factorization formula for power suppressed hard scattering effects in SIDIS. This yields generalized definitions of the TMDs that depend on two longitudinal momentum fractions (one of them only relevant beyond tree level), and a complete proof that only the same leading power soft function appears and can be absorbed into the TMD distributions at this order. We also show that perturbative corrections can be accounted for with only one new hard coefficient. Factorization formulae are given for all spin dependent structure functions which start at next-to-leading power. Prospects for improved subleading power predictions that include resummation are discussed.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Counting linearly polarized gluons with lattice QCD

We outline an approach to calculate the transverse-momentum-dependent distribution of linearly polarized gluons inside an unpolarized hadron on the lattice with the help of large momentum effective theory. To achieve this purpose, we propose calculating a Euclidean version of the degree of polarization for a fast-moving hadron on the lattice, which is ultraviolet finite, and no soft function subtraction is needed. It indicates a practical way to explore the distribution of the linearly polarized gluons in a proton and the linearly polarized gluon effects in hadron collisions on the lattice.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Sudakov double logs in single-inclusive hadron production in DIS at small x from the color glass condensate formalism

We investigate the high Q 2 (photon virtuality) limit of single-inclusive hadron production in deep inelastic scattering at small x , using the color glass condensate formalism at next-to-leading order. We focus on the Λ QCD 2 ≪ p h 2 ≪ Q 2 kinematic regime, where p h is the produced hadron transverse momentum, and extract the Sudakov double logarithms. We further argue that compatibility between the color glass condensate calculation and transverse momentum dependent distribution factorization at one-loop order can only be achieved if the small-x evolution is kinematically constrained. Published by the American Physical Society 2024

Altinoluk, Tolga (ORCID:0000000182671331)↗
DOE OSTI2026

Extraction of the Collins-Soper Kernel from a Joint Analysis of Experimental and Lattice Data

We present a first joint extraction of the Collins-Soper kernel (CSK) combining experimental and lattice QCD data in the context of an analysis of transverse-momentum-dependent distributions (TMDs). Based on a neural-network parametrization, we perform a Bayesian reweighting of an existing fit of TMDs using lattice data, as well as a joint TMD fit to lattice and experimental data. We consistently find that the inclusion of lattice information shifts the central value of the CSK by approximately 10% and reduces its uncertainty by 40%–50%, highlighting the potential of lattice inputs to improve TMD extractions.

Avkhadiev, Artur [Massachusetts Inst. of Technolog↗
DOE OSTI2021

TMDlib2 and TMDplotter: a platform for 3D hadron structure studies

A common library, TMDlib2, for Transverse-Momentum-Dependent distributions (TMDs) and unintegrated parton distributions (uPDFs) is described, which allows for easy access of commonly used TMDs and uPDFs, providing a three-dimensional (3D) picture of the partonic structure of hadrons. The tool TMDplotter allows for web-based plotting of distributions implemented in TMDlib2, together with collinear pdfs as available in LHAPDF.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

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.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗