Transverse Momentum Dependent Factorization Research Articles

Unraveling gluon TMDs in J/ψ and pion production at the EIC

We investigate the azimuthal asymmetries such as cos2ϕT and Sivers symmetry for J/ψ and π± production in electron-proton scattering, focusing on scenarios where the J/ψ and the pion are produced in an almost back-to-back configuration. The electron is unpolarized, while the proton can be unpolarized or transversely polarized. For the J/ψ formation, we use nonrelativistic QCD (NRQCD), while π± is formed due to parton fragmentation. In this kinematics, we utilize the transverse momentum-dependent factorization framework to calculate the cross sections and asymmetries. We consider both quark and gluon-initiated processes and show that the gluon contribution dominates. In this work, we used the generalized parton model (GPM) for the transverse momentum distribution (TMD) parametrizations and did not consider the effect of TMD evolution. We provide numerical estimates of the upper bounds on the azimuthal asymmetries, as well as employ a Gaussian parametrization for the gluon TMDs, within the kinematical region accessible by the upcoming Electron-Ion Collider (EIC). Published by the American Physical Society 2024

Spin asymmetries for C -even quarkonium production as a probe of gluon distributions

Within the framework of transverse momentum dependent factorization in combination with nonrelativistic quantum chromodynamics (QCD), we study charmonium and bottomonium production in hadronic collisions. We focus on quarkonium states with even charge conjugation, for which the color-singlet production mechanism is expected to be also dominant in the small transverse momentum region, qT2≪4Mc,b2. It is shown that the distributions of linearly polarized gluons inside unpolarized, longitudinally, and transversely polarized protons contribute to the cross sections for scalar and pseudoscalar quarkonia in a very distinctive, parity-dependent way, whereas their effects on higher angular momentum states are strongly suppressed. We derive analytical expressions for single and double spin asymmetries, which would allow for the direct extraction of the gluon transverse momentum dependent distributions, mirroring the phenomenological studies of the Drell-Yan processes aimed at the extraction of their quark counterparts. By adopting Gaussian models for the gluon transverse momentum dependent distributions, which fulfill without saturating everywhere their positivity bounds, we provide numerical predictions for the transverse single-spin asymmetries. These observables could be measured at LHCSpin, the fixed target experiment planned at the LHC. Published by the American Physical Society 2024

Extraction of transhelicity worm-gear distributions and opportunities at the Electron-Ion Collider in China

We present a global analysis of the transhelicity worm-gear distribution function, g1T⊥, by fitting the longitudinal-transverse double-spin asymmetry data of the semi-inclusive deep inelastic scattering. The analysis is performed within the framework of transverse momentum dependent factorization and evolution. It is found that the u-quark favors a positive distribution and the d-quark favors a negative distribution, which is consistent with previous model calculations and phenomenological extractions. Based on the fit to existing world data, we also study the impact of the proposed Electron-Ion Collider in China and conclude that it can significantly improve the precision of the worm-gear distribution function and hence enhance our understanding of nucleon spin structures. Published by the American Physical Society 2024

Transverse momentum dependent factorization for SIDIS at next-to-leading power

The semi-inclusive deep-inelastic scattering (SIDIS) is the golden process for investigating the nucleon’s transverse momentum-dependent (TMD) structure. We present the complete expression for SIDIS structure functions in the TMD factorization formalism at the next-to-leading power. All perturbative elements of the factorization theorem—coefficient functions and evolution kernels—are presented at one-loop accuracy. We found several differences with earlier derivations, which are due to accounting for nontrivial kinematics of the quark-gluon interference terms. As a side result, we present the definition and evolution of twist-three TMD fragmentation functions, including the leading-order evolution kernel. Published by the American Physical Society 2024

Single spin asymmetry AULsin(2ϕh) in dihadron production in SIDIS

The paper calculates the helicity-dependent dihadron fragmentation function by extending the dihadron spectator model and examine the single longitudinal spin asymmetry AULsin(2ϕh) from dihadron in semi-inclusive inelastic scattering. This function elucidates the relationship between the longitudinal polarization of the fragmented quark and the transverse momentum of the resulting hadron pairs. A study by the COMPASS collaboration detected a minimal signal in their experimental search for this azimuthal asymmetry in semi-inclusive inelastic scattering. Here, we use the spectator model to calculate the unknown T-odd dihadron fragmentation function H1⊥. Adopting collinear factorization to describe the data, avoiding the transverse momentum dependent factorization and the associated resummation effects, helping us understand the asymmetry and explaining why the signal is so weak. We involve the approach of transverse momentum dependence in the model calculations, in order to formulate the differential cross sections and the spin asymmetries in terms of the collinear parton distributions and the collinear dihadron fragmentation functions. A transverse momentum factor analysis method was used, in which the transverse momentum of the final hadron pairs was not integrated. The asymmetry of sin(2ϕh) in COMPASS kinematics was calculated and compared with experimental data. In addition, predictions for the same asymmetry are also presented for HERMES and the Electron Ion Collider. Published by the American Physical Society 2024

Renormalon cancellation and linear power correction to threshold-like asymptotics of space-like parton correlators

In this paper, we show that the common hard kernel of double-log-type or threshold-type factorization for certain space-like parton correlators that arise in the context of lattice parton distributions, the heavy-light Sudakov hard kernel, has linear infrared (IR) renormalon. We explicitly demonstrate how this IR renormalon correlates with ultraviolet (UV) renormalons of next-to-leading power operators in two explicit examples: threshold asymptotics of space-like quark-bilinear coefficient functions and transverse momentum dependent (TMD) factorization of quasi wave function amplitude. Theoretically, the pattern of renormalon cancellation complies with general expectations to marginal asymptotics in the UV limit. Practically, this linear renormalon explains the slow convergence of imaginary parts observed in lattice extraction of the Collins-Soper kernel and signals the relevance of next-to-leading power contributions. Fully factorized, fully controlled threshold asymptotic expansion for space-like quark-bilinear coefficient functions in coordinate and moment space has also been proposed.

Transverse-momentum-dependent (TMD) factorization in reactions with nuclei: From Drell-Yan to hadron production

We study cold nuclear matter effects on Drell-Yan production at small and moderate pT in proton/pion-nucleus collisions using a new transversemomentum dependent (TMD) factorization framework. Both collisional broadening and medium-induced radiative corrections in the initial state are considered in the soft-collinear effective theory with Glauber gluons (SCETG) approach. We demonstrate that in-medium bremsstrahlung exhibits rapidity divergences as x → 1 and collinear divergences at the endpoints x = 0, 1 of the medium-induced emission spectra. We further show that the rapidity divergences lead to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution of the collision kernel and can be resummed into the transverse momentum broadening of particle production. In turn, the endpoints divergences of in-medium radiation can be resummed through the collinear evolution of parton densities in nuclear matter. The TMD factorization framework is applied to understand the transverse-momentum spectra of Drell-Yan pair production in pA and πA collisions and provides calculations with improved accuracy for hadron production in cold QCD processes at RHIC and LHC.

Kinematic power corrections in TMD factorization theorem

This work is dedicated to the study of power expansion in the transverse momentum dependent (TMD) factorization theorem. Each genuine term in this expansion gives rise to a series of kinematic power corrections (KPCs). All terms of this series exhibit the same properties as the leading term and share the same nonperturbative content. Among various power corrections, KPCs are especially important since they restore charge conservation and frame invariance, which are violated at a fixed power order. I derive and sum a series of KPCs associated with the leading-power term of the TMD factorization theorem. The resulting expression resembles a hadronic tensor computed with free massless quarks while still satisfying a proven factorization statement. Additionally, I provide an explicit check of this novel form of factorization theorem at the next-to-leading order (NLO) and demonstrate the restoration of the frame-invariant argument of the leading-power coefficient function. Numerical estimations show that incorporating the summed KPCs into the cross-section leads to an almost constant shift, which may help to explain the observed challenges in the TMD phenomenology.

Transverse momentum dependent distribution functions in the threshold limit

We apply the joint threshold and transverse momentum dependent (TMD) factorization theorem to introduce new threshold-TMD distribution functions, including threshold-TMD parton distribution functions (PDFs) and fragmentation functions (FFs). We apply Soft-Collinear Effective Theory and renormalization group methods to carry out QCD evolution for both threshold-TMD PDFs and FFs. We show the universality of threshold-TMD functions among three standard processes, i.e. the Drell-Yan production in pp collisions, semi-inclusive deep-inelastic scattering and back-to-back two hadron production in e+e− collisions. In the end, we present the numerical predictions for different threshold-TMD functions and also transverse momentum distributions at pp, ep, and e+e− collisions.

Searching for saturation in forward dijet production at the LHC

We review recent results for forward jests at the LHC and EIC as obtained within small-x Improved Transverse Momentum Dependent factorization (ITMD). In addition to elementary overview of various approaches to perturbative QCD at high energy, including High Energy Factorization, Color Glass Condensate and ITMD, we describe the Monte Carlo implementation and discuss the existing and unpublished phenomenological results for forward dijets.

Forward gamma +jet production in proton-proton and proton-lead collisions at LHC within the FoCal calorimeter acceptance

Using the small-x improved transverse momentum dependent factorization (ITMD), which, for a general two-to-two massless scattering can be proved within the color glass condensate (CGC) theory for transverse momenta of particles greater than the saturation scale, we provide predictions for isolated forward photon and jet production in proton-proton and proton-nucleus collisions within the planned ALICE FoCal detector acceptance. We study azimuthal correlations, p_T spectra, as well as normalized ratios of proton-proton cross sections for different energies. The only TMD distribution needed for that process is the “dipole” TMD gluon distribution, which in our computations is based on HERA data and undergoes momentum space BK evolution equation with DGLAP corrections and Sudakov resummation. We conclude, that the process provides an excellent probe of the dipole TMD gluon distribution in saturation regime.

Full treatment of the thrust distribution in single inclusive e+e−→ h X processes

Extending the transverse momentum dependent factorization to thrust dependent observables entails a series of difficulties, ultimately associated to the behavior of soft radiation. As a consequence, the definition of the transverse momentum dependent functions has to be revised, while preserving (and possibly extending) their universality properties. Moreover, the regularization of the rapidity divergences generates non trivial correlations between rapidity and thrust. In this paper, we show how to deal with these correlations in a consistent treatment of the thrust dependence of e+e− → h X cross section, where the hadron transverse momentum is measured with respect to the thrust axis. In this framework all results obtained in the past few years properly fit together, leading to a remarkable phenomenological description of the experimental measurements.

Transverse momentum dependent shape function for J/ψ production in SIDIS

It has been shown previously that the transverse momentum dependent (TMD) factorization of heavy quarkonium production requires a TMD shape function. Its perturbative tail can be extracted by matching the cross sections valid at low and high transverse momenta. In this article we compare the order-αs TMD expressions with the order- {alpha}_s^2 collinear ones for J/ψ production in semi-inclusive deep inelastic scattering (SIDIS), employing nonrelativistic QCD in both cases. In contrast to previous studies, we find that the small transverse momentum limit of the collinear expressions contains discontinuities. We demonstrate how to properly deal with them and include their finite contributions to the TMD shape functions. Moreover, we show that soft gluon emission from the low transverse momentum Born diagrams provide the same leading order TMD shape functions as required for the matching. Their revised perturbative tails have a less divergent behavior as compared to the TMD fragmentation functions of light hadrons. Finally, we investigate the universality of TMD shape functions in heavy quarkonium production, identify the need for process dependent factorization and discuss the phenomenological implications.

Basics of factorization in a scalar Yukawa field theory

The factorization theorems of QCD apply equally well to most simple quantum field theories that require renormalization but where direct calculations are much more straightforward. Working with these simpler theories is convenient for stress testing the limits of the factorization program and for examining general properties of the parton density functions or other correlation functions that might be necessary for a factorized description of a process. With this view in mind, we review the steps of factorization in a real scalar Yukawa field theory for both deep inelastic scattering and semi-inclusive deep inelastic scattering cross sections. In the case of semi-inclusive deep inelastic scattering, we illustrate how to separate the small transverse momentum region, where transverse momentum dependent parton density functions are needed, from a purely collinear large transverse momentum region, and we examine the influence of subleading power corrections. We also review the steps for formulating transverse momentum dependent factorization in transverse coordinate space, and we study the effect of transforming to the well-known ${b}_{*}$ scheme. Within the Yukawa theory, we investigate the consequences of switching to a generalized parton model approach and compare it with a fully factorized approach. Our results highlight the need to address similar or analogous issues in QCD.

Transverse Λ polarization in e+e− processes within a TMD factorization approach and the polarizing fragmentation function

We perform a re-analysis of Belle data for the transverse Λ and overline{Lambda} polarization in e+e− annihilation processes within a transverse momentum dependent (TMD) factorization approach. We consider two data sets, one referring to the associated production of Λ’s with a light unpolarized hadron in an almost back-to-back configuration, and one for the inclusive Λ production, with the reconstruction of the thrust axis. We adopt the Collins-Soper-Sterman framework and employ the recent formulation on the factorization of single-inclusive hadron production. This extends a previous phenomenological study carried out in a more simplified TMD approach, leading to a new extraction of the polarizing fragmentation function (FF). While confirming several features of the previous analysis, here we include the proper QCD scale dependence of this TMD FF and carefully exploit the role of its nonperturbative component. The compatibility, within a unique TMD factorization scheme, of double and single-inclusive hadron production is discussed in detail. Moreover, we consider another data set for inclusive Λ production, at much larger energies, from the OPAL Collaboration, in order to test the consistency of the entire approach. Finally, we elaborate on some fundamental issues like the role of SU(2) isospin symmetry and the heavy quark contributions.

Determination of the Collins-Soper kernel from cross-sections ratios

The Collins-Soper kernel is a powerful tool for studying the properties of the QCD vacuum and an essential component of the transverse momentum dependent (TMD) factorization theorem. In this paper, we present a novel method for determining the Collins-Soper kernel directly from the comparison of differential cross sections measured at different energies. The method relies solely on the structure of the TMD factorization theorem and thus also provides a direct test of the theorem validity. With minor modifications, the procedure can be applied to the real measured data for Drell-Yan and SIDIS processes. As a demonstration, we analyze the pseudodata generated by the CASCADE event generator and determine the Collins-Soper kernel suggested by the parton branching model.

Dijet photoproduction at low x at next-to-leading order and its back-to-back limit

We compute the cross section for the inclusive photoproduction of a pair of jets at next-to-leading order accuracy in the Color Glass Condensate (CGC) effective theory. The aim is to study the back-to-back limit, to investigate whether transverse momentum dependent (TMD) factorization can be recovered at this perturbative order. In particular, we focus on large Sudakov double logarithms, which are dominant terms in the TMD evolution kernel. Interestingly, the kinematical improvement of the low-x resummation scheme turns out to play a crucial role in our analysis.

Transverse-momentum-dependent wave functions and soft functions at one-loop in large momentum effective theory

In large-momentum effective theory (LaMET), the transverse-momentum-dependent (TMD) light-front wave functions and soft functions can be extracted from the simulation of a four-quark form factor and equal-time correlation functions. In this work, using expansion by regions we provide a one-loop proof of TMD factorization of the form factor. For the one-loop validation, we also present a detailed calculation of mathcal{O} (αs) perturbative corrections to these quantities, in which we adopt a modern technique for the calculation of TMD form factor based the integration by part and differential equation. The one-loop hard functions are then extracted. Using lattice data from Lattice Parton Collaboration on quasi-TMDWFs, we estimate the effects from the one-loop matching kernel and find that the perturbative corrections depend on the operator to define the form factor, but are less sensitive to the transverse separation. These results will be helpful to precisely extract the soft functions and TMD wave functions from the first-principle in future.

Combining nonperturbative transverse momentum dependence with TMD evolution

Central to understanding the nonperturbative, intrinsic partonic nature of hadron structure are the concepts of transverse momentum dependent (TMD) parton distribution and fragmentation functions. A TMD factorization approach to the phenomenology of semi-inclusive processes that includes evolution, higher orders, and matching to larger transverse momentum is ultimately necessary for reliably connecting with phenomenologically extracted nonperturbative structures, especially when widely different scales are involved. In this paper, we will address some of the difficulties that arise when phenomenological techniques that were originally designed for very high energy applications are extended to studies of hadron structures, and we will solidify the connection between standard high energy TMD implementations and the more intuitive, parton model based approaches to phenomenology that emphasize nonperturbative hadron structure. In the process, we will elaborate on differences between forward and backward TMD evolution, which in the context of this paper, we call ``bottom-up'' and ``top-down'' approaches, and we will explain the advantages of a bottom-up strategy. We will also emphasize and clarify the role of the integral relations that connect TMD and collinear correlation functions. We will show explicitly how they constrain the nonperturbative ``$g$-functions'' of standard Collins-Soper-Sterman implementations of TMD factorization. This paper is especially targeted toward phenomenologists and model builders who are interested in merging specific nonperturbative models and calculations (including lattice QCD) with TMD factorization at large $Q$. Our main result is a recipe for incorporating nonperturbative models into TMD factorization and for constraining their parameters in a way that matches perturbative QCD and evolution.

The sin 2phi _h azimuthal asymmetry of pion production in SIDIS within TMD factorization

We study the sin 2phi _h azimuthal asymmetry of charged and neutral pion productions off the longitudinally polarized nucleon targets in semi-inclusive deeply inelastic scattering (SIDIS) process within the transverse momentum dependent (TMD) factorization. The asymmetry is contributed by the convolution of the TMD distribution h_{1L}^{perp } and the Collins fragmentation function H_{1}^perp . We adopt the Wandzura–Wilczek-type (WW-type) approximation for h_{1L}^{perp } and two different parametrizations on the nonperturbative Sudakov form factor for h_{1L}^{perp } and H_{1}^perp . We estimate the sin 2phi _h asymmetry at HERMES and CLAS and compare them with the corresponding experimental data. We also provide the prediction at kinematical configuration of CLAS12. It is shown that our theoretical calculation can describe the HERMES and CLAS data, except the asymmetry of the pi ^- production off proton target at HERMES and CLAS. We also find that different choices of the nonperturbative part of TMD evolution lead qualitatively similar results.