Collins-Soper-Sterman Research Articles

Unified description of DGLAP, CSS, and BFKL evolution: TMD factorization bridging large and small x

This paper introduces a transverse-momentum dependent (TMD) factorization scheme designed to unify both large and small Bjorken-x regimes. We compute the next-to-leading order quantum chromodynamics (QCD) corrections to the gluon TMD operator for an unpolarized hadron within this proposed scheme. This leads to the emergence of a new TMD evolution, incorporating those in transverse momentum, rapidity, and Bjorken-x. When matched to the collinear factorization scheme, our factorization scheme faithfully reproduces the well-established Dokshitzer-Gribov-Lipatov-Altarelli-Parisi and Collins-Soper-Sterman (CSS) evolutions. Conversely, matching with high-energy factorization not only yields the Balitsky-Fadin-Kuraev-Lipatov evolution but also reveals distinctive signatures of CSS logarithms. The development of this novel TMD factorization scheme, capable of seamlessly reconciling disparate Bjorken-x regimes and faithfully reproducing established QCD evolution equations, has the potential to significantly advance our comprehension of high-energy processes and three-dimensional parton structures of hadrons. Published by the American Physical Society 2024

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.

Measurement of the angle between jet axes in pp collisions at sqrt{s} = 5.02 TeV

This article reports measurements of the angle between differently defined jet axes in pp collisions at sqrt{s} = 5.02 TeV carried out by the ALICE Collaboration. Charged particles at midrapidity are clustered into jets with resolution parameters R = 0.2 and 0.4. The jet axis, before and after Soft Drop grooming, is compared to the jet axis from the Winner-Takes-All (WTA) recombination scheme. The angle between these axes, ∆Raxis, probes a wide phase space of the jet formation and evolution, ranging from the initial high-momentum-transfer scattering to the hadronization process. The ∆Raxis observable is presented for 20 < {p}_{textrm{T}}^{textrm{ch} textrm{jet}} < 100 GeV/c, and compared to predictions from the PYTHIA 8 and Herwig 7 event generators. The distributions can also be calculated analytically with a leading hadronization correction related to the non-perturbative component of the Collins-Soper-Sterman (CSS) evolution kernel. Comparisons to analytical predictions at next-to-leading-logarithmic accuracy with leading hadronization correction implemented from experimental extractions of the CSS kernel in Drell-Yan measurements are presented. The analytical predictions describe the measured data within 20% in the perturbative regime, with surprising agreement in the non-perturbative regime as well. These results are compatible with the universality of the CSS kernel in the context of jet substructure.

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.

Nonperturbative functions for SIDIS and Drell–Yan processes

We update the well-known BLNY fit to the low transverse momentum Drell–Yan lepton pair productions in hadronic collisions, by considering the constraints from the semi-inclusive hadron production in deep inelastic scattering (SIDIS) from HERMES and COMPASS experiments. We follow the Collins–Soper–Sterman (CSS) formalism with the [Formula: see text]-prescription. A nonperturbative form factor associated with the transverse momentum dependent quark distributions is found in the analysis with a new functional form different from that of BLNY. This releases the tension between the BLNY fit to the Drell–Yan data with the SIDIS data from HERMES/COMPASS in the CSS resummation formalism.

Connecting different TMD factorization formalisms in QCD

In the original Collins-Soper-Sterman (CSS) presentation of the results of transverse-momentum-dependent (TMD) factorization for the Drell-Yan process, results for perturbative coefficients can be obtained from calculations for collinear factorization. Here we show how to use these results, plus known results for the quark form factor, to obtain coefficients for TMD factorization in more recent formulations, e.g., that due to Collins, and apply them to known results at order $\alpha_s^2$ and $\alpha_s^3$. We also show that the "non-perturbative" functions as obtained from fits to data are equal in the two schemes. We compile the higher-order perturbative inputs needed for the updated CSS scheme by appealing to results obtained in a variety of different formalisms. In addition, we derive the connection between both versions of the CSS formalism and several formalisms based in soft-collinear effective theory (SCET). Our work uses some important new results for factorization for the quark form factor, which we derive.

Extracting the QCD ΛMS¯ parameter in Drell–Yan process using Collins–Soper–Sterman approach

In this work, we directly fit the QCD dimensional transmutation parameter, [Formula: see text], to experimental data of Drell–Yan (DY) observables. For this purpose, we first obtain the evolution of transverse momentum dependent parton distribution functions (TMDPDFs) up to the next-to-next-to-leading logarithm (NNLL) approximation based on Collins–Soper–Sterman (CSS) formalism. As is expecting the TMDPDFs are appearing at larger values of transverse momentum by increasing the energy scales and also the order of approximation. Then we calculate the cross-section related to the TMDPDFs in the DY process. As a consequence of global fitting to the five sets of experimental data at different low center-of-mass energies and one set at high center-of-mass energy, using CETQ06 parametrizations as our boundary condition, we obtain [Formula: see text] MeV corresponding to the renormalized coupling constant [Formula: see text] which is within the acceptable range for this quantity. The goodness of [Formula: see text] shows the results for DY cross-section are in good agreement with different experimental sets, containing E288, E605 and R209 at low center-of-mass energies and [Formula: see text], CDF data at high center-of-mass energy. The repeated calculations, using HERAPDFs parametrizations is yielding us numerical values for fitted parameters very close to what we obtain using CETQ06 PDFs set. This indicates that the obtained results have enough stability by variations in the boundary conditions.

Relating transverse-momentum-dependent and collinear factorization theorems in a generalized formalism

We construct an improved implementation for combining transverse-momentum-dependent (TMD) factorization and collinear factorization. TMD factorization is suitable for low transverse momentum physics, while collinear factorization is suitable for high transverse momenta and for a cross section integrated over transverse momentum. The result is a modified version of the standard $W+Y$ prescription traditionally used in the Collins-Soper-Sterman (CSS) formalism and related approaches. We further argue that questions regarding the shape and $Q$-dependence of the cross sections at lower $Q$ are largely governed by the matching to the $Y$-term.

An overview of transverse-momentum-dependent factorization and evolution

I review TMD factorization and evolution theorems, with an emphasis on the treatment by Collins and originating in the Collins-Soper-Sterman (CSS) formalism. I summarize basic results while attempting to trace their development over that past several decades.

Extraction of quark transversity distribution and Collins fragmentation functions with QCD evolution

We study the transverse momentum dependent (TMD) evolution of the Collins azimuthal asymmetries in $e^+e^-$ annihilations and semi-inclusive hadron production in deep inelastic scattering (SIDIS) processes. All the relevant coefficients are calculated up to the next-to-leading logarithmic (NLL) order accuracy. By applying the TMD evolution at the approximate NLL order in the Collins-Soper-Sterman (CSS) formalism, we extract transversity distributions for $u$ and $d$ quarks and Collins fragmentation functions from current experimental data by a global analysis of the Collins asymmetries in back-to-back di-hadron productions in $e^+e^-$ annihilations measured by BELLE and BABAR Collaborations and SIDIS data from HERMES, COMPASS, and JLab HALL A experiments. The impact of the evolution effects and the relevant theoretical uncertainties are discussed. We further discuss the TMD interpretation for our results, and illustrate the unpolarized quark distribution, transversity distribution, unpolarized quark fragmentation and Collins fragmentation functions depending on the transverse momentum and the hard momentum scale. We make detailed predictions for future experiments and discuss their impact.

Scheme dependence and transverse momentum distribution interpretation of Collins–Soper–Sterman resummation

Following an earlier derivation by Catani, de Florian and Grazzini (2000) on the scheme dependence in the Collins–Soper–Sterman (CSS) resummation formalism in hard scattering processes, we investigate the scheme dependence of the Transverse Momentum Distributions (TMDs) and their applications. By adopting a universal C-coefficient function associated with the integrated parton distributions, the difference between various TMD schemes can be attributed to a perturbative calculable function depending on the hard momentum scale. We further apply several TMD schemes to the Drell–Yan process of lepton pair production in hadronic collisions, and find that the constrained non-perturbative form factors in different schemes are consistent with each other and with that of the standard CSS formalism.

A study on the interplay between perturbative QCD and CSS/TMD formalism in SIDIS processes

We study the Semi-Inclusive Deep Inelastic Scattering (SIDIS) cross section as a function of the transverse momentum, q T . In order to describe it over a wide region of q T , soft gluon resummation has to be performed. Here we will use the original Collins-Soper-Sterman (CSS) formalism; however, the same procedure would hold within the improved Transverse Momentum Dependent (TMD) framework. We study the matching between the region where fixed order perturbative QCD can successfully be applied and the region where soft gluon resummation is necessary. We find that the commonly used prescription of matching through the so-called Y-factor cannot be applied in the SIDIS kinematical configurations we examine. In particular, the non-perturbative component of the resummed cross section turns out to play a crucial role and should not be overlooked even at relatively high energies. Moreover, the perturbative expansion of the resummed cross section in the matching region is not as reliable as it is usually believed and its treatment requires special attention.

Phenomenology of TMDs

We present a review of current Transverse Momentum Dependent (TMD) phenomenology focusing our attention on the unpolarized TMD parton distribution function and the Sivers function. The paper introduces and comments about the new Collins-Soper-Sterman (CSS) TMD evolution formalism. We make use of a selection of results obtained by several groups to illustrate the achievements and the failures of the simple Gaussian approach and the TMD CSS evolution formalism.

Nonperturbative contributions to a resummed leptonic angular distribution in inclusive neutral vector boson production

We present an analysis of nonperturbative contributions to the transverse momentum distribution of $Z/\gamma^*$ bosons produced at hadron colliders. The new data on the angular distribution $\phi^*_\eta$ of Drell-Yan pairs measured at the Tevatron is shown to be in excellent agreement with a perturbative QCD prediction based on the Collins-Soper-Sterman (CSS) resummation formalism at NNLL accuracy. Using these data, we determine the nonperturbative component of the CSS resummed cross section and estimate its dependence on arbitrary resummation scales and other factors. With the scale dependence included at the NNLL level, a significant nonperturbative component is needed to describe the angular data.

Energy evolution for the Sivers asymmetries in hard processes

We investigate the energy evolution of the azimuthal spin asymmetries in semi-inclusive hadron production in deep inelastic scattering (SIDIS) and Drell-Yan lepton pair production in pp collisions. The scale dependence is evaluated by applying an approximate solution to the Collins-Soper-Sterman (CSS) evolution equation at one-loop order which is adequate for moderate Q^2 variations. This describes well the unpolarized cross sections for SIDIS and Drell-Yan process in the $Q^2$ range of 2.4-100GeV^2. A combined analysis of the Sivers asymmetries in SIDIS from HERMES and COMPASS experiments, and the predictions for the Drell-Yan process at RHIC at \sqrt{S}=200GeV are presented. We further extend to the Collins asymmetries and find, for the first time, a consistent description for HERMES/COMPASS and BELLE experiments with the evolution effects. We emphasize an important test of the evolution effects by studying di-hadron azimuthal asymmetry in e^+e^- annihilation at moderate energy range, such as at BEPC at \sqrt{S}=4.6GeV.

Calculation of Transverse-Momentum-Dependent Evolution for Sivers Transverse Single Spin Asymmetry Measurements

The Sivers transverse single spin asymmetry (TSSA) is calculated and compared at different scales using the transverse-momentum-dependent (TMD) evolution equations applied to previously existing extractions. We apply the Collins-Soper-Sterman (CSS) formalism, using the version recently developed by Collins. Our calculations rely on the universality properties of TMD functions that follow from the TMD-factorization theorem. Accordingly, the nonperturbative input is fixed by earlier experimental measurements, including both polarized semi-inclusive deep inelastic scattering (SIDIS) and unpolarized Drell-Yan (DY) scattering. It is shown that recent preliminary COMPASS measurements are consistent with the suppression prescribed by TMD evolution.

QCD evolution of the Sivers function

We extend the Collins-Soper-Sterman (CSS) formalism to apply it to the spin dependence governed by the Sivers function. We use it to give a correct numerical QCD evolution of existing fixed-scale fits of the Sivers function. With the aid of approximations useful for the nonperturbative region, we present the results as parametrizations of a Gaussian form in transverse-momentum space, rather than in the Fourier conjugate transverse coordinate space normally used in the CSS formalism. They are specifically valid at small transverse momentum. Since evolution has been applied, our results can be used to make predictions for Drell-Yan and semi-inclusive deep inelastic scattering at energies different from those where the original fits were made. Our evolved functions are of a form that they can be used in the same parton-model factorization formulas as used in the original fits, but now with a predicted scale dependence in the fit parameters. We also present a method by which our evolved functions can be corrected to allow for twist-3 contributions at large parton transverse momentum.

NONPERTURBATIVE CONTRIBUTIONS TO A RESUMMED LEPTONIC ANGULAR DISTRIBUTION IN INCLUSIVE Z/γ* BOSON PRODUCTION

We summarize a new analysis of the distribution [Formula: see text] of charged leptons produced in decays of Z and γ* bosons in the Collins-Soper-Sterman (CSS) formalism for transverse momentum resummation. By comparing the [Formula: see text] distribution measured at the Tevatron with the resummed CSS cross section with approximate [Formula: see text] Wilson coefficients, we constrain the magnitude of the nonperturbative Gaussian smearing factor and analyze its uncertainty caused by variations in scale parameters. We find excellent agreement between the [Formula: see text] data and our theoretical prediction, provided by the RESBOS resummation program. The nonperturbative factor that we obtained can be used to update resummed QCD predictions for precision measurements in inclusive W and Z production and for comparisons to various models of nonperturbative dynamics.

Transverse momentum dependent parton distribution and fragmentation functions with QCD evolution

We assess the current phenomenological status of transverse momentum dependent (TMD) parton distribution functions (PDFs) and fragmentation functions (FFs) and study the effect of consistently including perturbative QCD (pQCD) evolution. Our goal is to initiate the process of establishing reliable, QCD-evolved parametrizations for the TMD PDFs and TMD FFs that can be used both to test TMD-factorization and to search for evidence of the breakdown of TMD-factorization that is expected for certain processes. In this article, we focus on spin-independent processes because they provide the simplest illustration of the basic steps and can already be used in direct tests of TMD-factorization. Our calculations are based on the Collins-Soper-Sterman (CSS) formalism, supplemented by recent theoretical developments which have clarified the precise definitions of the TMD PDFs and TMD FFs needed for a valid TMD-factorization theorem. Starting with these definitions, we numerically generate evolved TMD PDFs and TMD FFs using as input existing parametrizations for the collinear PDFs, collinear FFs, non-perturbative factors in the CSS factorization formalism, and recent fixed-scale fits. We confirm that evolution has important consequences, both qualitatively and quantitatively, and argue that it should be included in future phenomenological studies of TMD functions. Our analysis is also suggestive of extensions to processes that involve spin-dependent functions such as the Boer-Mulders, Sivers, or Collins functions, which we intend to pursue in future publications. At our website we have made available the tables and calculations needed to obtain the TMD parametrizations presented herein.

TMD FACTORIZATION AND EVOLUTION FOR TMD CORRELATION FUNCTIONS

We discuss the application of transverse momentum dependent (TMD) factorization theorems to phenomenology. Our treatment relies on recent extensions of the Collins-Soper-Sterman (CSS) formalism. Emphasis is placed on the importance of using well-defined TMD parton distribution functions (PDFs) and fragmentation functions (FFs) in calculating the evolution of these objects. We explain how parametrizations of unpolarized TMDs can be obtained from currently existing fixed-scale Gaussian fits and previous implementations of the CSS formalism in the Drell-Yan process, and provide some examples. We also emphasize the importance of agreed-upon definitions for having an unambiguous prescription for calculating higher orders in the hard part, and provide examples of higher order calculations. We end with a discussion of strategies for extending the phenomenological applications of TMD factorization to situations beyond the unpolarized case.