Transverse Momentum Dependent Fragmentation Functions Research Articles

Transverse momentum-dependent heavy-quark fragmentation at next-to-leading order

The transverse momentum-dependent fragmentation functions (TMD FFs) of heavy (bottom and charm) quarks, which we recently introduced, are universal building blocks that enter predictions for a large number of observables involving final-state heavy quarks or hadrons. They enable the extension of fixed-order subtraction schemes to quasi-collinear limits, and are of particular interest in their own right as probes of the nonperturbative dynamics of hadronization. In this paper we calculate all TMD FFs involving heavy quarks and the associated TMD matrix element in heavy-quark effective theory (HQET) to next-to-leading order in the strong interaction. Our results confirm the renormalization properties, large-mass, and small-mass consistency relations predicted in our earlier work. We also derive and confirm a prediction for the large-z behavior of the heavy-quark TMD FF by extending, for the first time, the formalism of joint resummation to capture quark mass effects in heavy-quark fragmentation. Our final results in position space agree with those of a recent calculation by another group that used a highly orthogonal organization of singularities in the intermediate momentum-space steps, providing a strong independent cross check. As an immediate application, we present the complete quark mass dependence of the energy-energy correlator (EEC) in the back-to-back limit at Oαs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\alpha}_s\\right) $$\\end{document}.

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.

T-even transverse momentum dependent gluon fragmentation functions in a spectator model

We present a model calculation of transverse momentum dependent (TMD) gluon fragmentation functions in both spin-1/2 and spin-0 hadrons. Our approach is based on the postulation that a time-like off-shell gluon can fragment into a hadron and a single spectator particle. So far, such spectator models have been restricted to TMD distribution functions of quarks and gluon as well TMD fragmentation function of quarks. The effective coupling between the gluon-hadron-spectator system is described by a vertex composed of two form factors. We obtain analytical expressions for the four T-even TMD fragmentation functions of the gluon. We also provide numerical results on the z-dependence and kT-dependence of the fragmentation functions. Our analysis reveals the non-negligible impact of these fragmentation functions, which could potentially be explored in forthcoming experimental measurements.

Nuclear Modification of Transverse Momentum Dependent Parton Distribution Functions by a Global QCD Analysis.

We perform the first simultaneous global QCD extraction of the transverse momentum dependent (TMD) parton distribution functions and the TMD fragmentation functions in nuclei. We have considered the world set of data from semi-inclusive electron-nucleus deep inelastic scattering and Drell-Yan dilepton production. In total, this data set consists of 90 data points from HERMES, Fermilab, RHIC, and LHC. Working at next-to-leading order and next-to-next-to-leading logarithmic accuracy, we achieve a χ^{2}/d.o.f.=1.196. In this analysis, we perform the first extraction of nuclear modified TMDs and compare these to those in free nucleons. We also make predictions for the ongoing JLab 12GeV program and future electron-ion collider measurements.

General helicity formalism for two-hadron production in e+e\u2212 annihilation within a TMD approach

We present the complete leading-order results for the azimuthal dependences and polarization observables in e+e−→ h1h2 + X processes, where the two hadrons are produced almost back-to-back, within a transverse momentum dependent (TMD) factorization scheme. We consider spinless (or unpolarized) and spin-1/2 hadron production and give the full set of the corresponding quark and gluon TMD fragmentation functions (TMD-FFs). By adopting the helicity formalism, which allows for a more direct probabilistic interpretation, single- and double-polarization cases are discussed in detail. Simplified expressions, useful for phenomenological analyses, are obtained by assuming a factorized Gaussian-like dependence on intrinsic transverse momenta for the TMD-FFs.

TMD fragmentation functions at N3LO

We compute the unpolarized quark and gluon transverse-momentum dependent fragmentation functions (TMDFFs) at next-to-next-to-next-to-leading order (N3LO) in perturbative QCD. The calculation is based on a relation between the TMDFF and the limit of the semi-inclusive deep inelastic scattering cross section where all final-state radiation becomes collinear to the detected hadron. The required cross section is obtained by analytically continuing our recent computation of the Drell-Yan and Higgs boson production cross section at N3LO expanded around the limit of all final-state radiation becoming collinear to one of the initial states. Our results agree with a recent independent calculation by Luo et al.

Unpolarized quark and gluon TMD PDFs and FFs at N3LO

In this paper we calculate analytically the perturbative matching coefficients for unpolarized quark and gluon Transverse-Momentum-Dependent (TMD) Parton Distribution Functions (PDFs) and Fragmentation Functions (FFs) through Next-to-Next-to-Next-to-Leading Order (N3LO) in QCD. The N3LO TMD PDFs are calculated by solving a system of differential equation of Feynman and phase space integrals. The TMD FFs are obtained by analytic continuation from space-like quantities to time-like quantities, taking into account the probability interpretation of TMD PDFs and FFs properly. The coefficient functions for TMD FFs exhibit double logarithmic enhancement at small momentum fraction z. We resum such logarithmic terms to the third order in the expansion of αs. Our results constitute important ingredients for precision determination of TMD PDFs and FFs in current and future experiments.

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.

Collinear factorization in wide-angle hadron pair production in e+e− annihilation

We compute the inclusive unpolarized dihadron production cross section in the far from back-to-back region of $e^+ e^-$ annihilation in leading order pQCD using existing fragmentation function fits and standard collinear factorization, focusing on the large transverse momentum region where transverse momentum is comparable to the hard scale (the center-of-mass energy). We compare with standard transverse-momentum-dependent (TMD) fragmentation function-based predictions intended for the small transverse momentum region with the aim of testing the expectation that the two types of calculation roughly coincide at intermediate transverse momentum. We find significant tension, within the intermediate transverse momentum region, between calculations done with existing non-perturbative TMD fragmentation functions and collinear factorization calculations if the center-of-mass energy is not extremely large. We argue that $e^+ e^-$ measurements are ideal for resolving this tension and exploring the large-to-small transverse momentum transition, given the typically larger hard scales ($\gtrsim 10$ GeV) of the process as compared with similar scenarios that arise in semi-inclusive deep inelastic scattering and fixed-target Drell-Yan measurements.

Jet fragmentation functions for Z-tagged jets

Recently the LHCb collaboration has measured both longitudinal and transverse momentum distribution of hadrons produced inside Z-tagged jets in proton-proton collisions at the Large Hadron Collider. These distributions are commonly referred to as jet fragmentation functions and are characterized by the longitudinal momentum fraction zh of the jet carried by the hadron and the transverse momentum j⊥ with respect to the jet direction. We derive a QCD formalism within Soft-Collinear Effective Theory to describe these distributions and find that the zh-dependence provides information on standard collinear fragmentation functions, while j⊥-dependence probes transverse momentum dependent (TMD) fragmentation functions. We perform theoretical calculations and compare our results with the LHCb data. We find good agreement for the intermediate zh region. For j⊥-dependence, we suggest binning in both zh and j⊥, which would lead to a more direct probing of TMD fragmentation functions.

The transverse momentum distribution of hadrons within jets

We study the transverse momentum distribution of hadrons within jets, where the transverse momentum is defined with respect to the standard jet axis. We consider the case where the jet substructure measurement is performed for an inclusive jet sample pp → jet + X. We demonstrate that this observable provides new opportunities to study transverse momentum dependent fragmentation functions (TMDFFs) which are currently poorly constrained from data, especially for gluons. The factorization of the cross section is obtained within Soft Collinear Effective Theory (SCET), and we show that the relevant TMDFFs are the same as for the more traditional processes semi-inclusive deep inelastic scattering (SIDIS) and electron-positron annihilation. Different than in SIDIS, the observable for the in-jet fragmentation does not depend on TMD parton distribution functions which allows for a cleaner and more direct probe of TMDFFs. We present numerical results and compare to available data from the LHC.

Jet axes and universal transverse-momentum-dependent fragmentation

We study the transverse momentum spectrum of hadrons in jets. By measuring the transverse momentum with respect to a judiciously chosen axis, we find that this observable is insensitive to (the recoil of) soft radiation. Furthermore, for small transverse momenta we show that the effects of the jet boundary factorize, leading to a new transverse-momentum-dependent (TMD) fragmentation function. In contrast to the usual TMD fragmentation functions, it does not involve rapidity divergences and is universal in the sense that it is independent of the type of process and number of jets. These results directly apply to sub-jets instead of hadrons. We discuss potential applications, which include studying nuclear modification effects in heavy-ion collisions and identifying boosted heavy resonances.

Unpolarized transverse momentum dependent parton distribution and fragmentation functions at next-to-next-to-leading order

The transverse momentum dependent parton distribution/fragmentation functions (TMDs) are essential in the factorization of a number of processes like Drell-Yan scattering, vector boson production, semi-inclusive deep inelastic scattering, etc. We provide a comprehensive study of unpolarized TMDs at next-to-next-to-leading order, which includes an explicit calculation of these TMDs and an extraction of their matching coefficients onto their integrated analogues, for all flavor combinations. The obtained matching coefficients are important for any kind of phenomenology involving TMDs. In the present study each individual TMD is calculated without any reference to a specific process. We recover the known results for parton distribution functions and provide new results for the fragmentation functions. The results for the gluon transverse momentum dependent fragmentation functions are presented for the first time at one and two loops. We also discuss the structure of singularities of TMD operators and TMD matrix elements, crossing relations between TMD parton distribution functions and TMD fragmentation functions, and renormalization group equations. In addition, we consider the behavior of the matching coefficients at threshold and make a conjecture on their structure to all orders in perturbation theory.

Quark-jet model for transverse momentum dependent fragmentation functions

In order to describe the hadronization of polarized quarks, we discuss an extension of the quark-jet model to transverse momentum dependent fragmentation functions. The description is based on a product ansatz, where each factor in the product represents one of the transverse momentum dependent splitting functions, which can be calculated by using effective quark theories. The resulting integral equations and sum rules are discussed in detail for the case of inclusive pion production. In particular, we demonstrate that the 3-dimensional momentum sum rules are satisfied naturally in this transverse momentum dependent quark-jet model. Our results are well suited for numerical calculations in effective quark theories, and can be implemented in Monte-Carlo simulations of polarized quark hadronization processes.

Transverse momentum dependent gluon fragmentation functions from $$J/\psi \ \pi $$ J / ψ π production at $$e^+ e^-$$ e + e - colliders

The back-to-back $$J/\psi $$ and $$\pi $$ associated production at $$e^+ e^-$$ colliders is proposed to detect the gluon transverse momentum dependent (TMD) fragmentation functions. TMD factorization is assumed for this process. With a spinless pion, unpolarized and linearly polarized gluon TMD fragmentation functions can be defined. It is found that at parton level the hadronic tensor can be described by three independent structure functions. As a result, there are three independent angular distributions, of which the $$\cos {2\phi }$$ azimuthal asymmetry is sensitive to the linearly polarized gluon fragmentation function.

Phenomenology from SIDIS ande+e−multiplicities: multiplicities and phenomenology - part I

This study is part of a project to investigate the transverse momentum dependence in parton dis- tribution and fragmentation functions, analyzing (semi-)inclusive high-energy processes within a proper QCD framework. We calculate the transverse-momentum-dependent (TMD) multiplicities for e + e − annihilation into two hadrons (considering different combinations of pions and kaons) aiming to investigate the impact of in- trinsic and radiative partonic transverse momentum and their mixing with flavor. Different descriptions of the non-perturbative evolution kernel (see, e.g., Refs. (1-5)) are available on the market and there are 200 sets of flavor configurations for the unpolarized TMD fragmentation functions (FFs) resulting from a Monte Carlo fit of Semi-Inclusive Deep-Inelastic Scattering (SIDIS) data at Hermes (see Ref. (6)). We build our predictions of e + e − multiplicities relying on this rich phenomenology. The comparison of these calculations with future experimental data (from Belle and BaBar collaborations) will shed light on non-perturbative aspects of hadron structure, opening important insights into the physics of spin, flavor and momentum structure of hadrons.

Double hadron lepto-production in the current and target fragmentation regions

We study the inclusive production of two hadrons in deep inelastic processes, ℓN→ℓh1h2X, with h1 in the current fragmentation region (CFR) and h2 in the target fragmentation region (TFR). Assuming a factorized scheme, the recently introduced polarized and transverse momentum dependent fracture functions couple to the transverse momentum dependent fragmentation functions. This allows the full exploration of the fracture functions for transversely polarized quarks. Some particular cases are considered.

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.

Test of the Universality of Naive-Time-Reversal-Odd Fragmentation Functions

We investigate the "spontaneous" hyperon transverse polarization in e+ e- annihilation and semi-inclusive deep inelastic scattering processes as a test of the universality of the naive-time-reversal-odd transverse momentum dependent fragmentation functions. We find that universality implies definite sign relations among various observables. This provides a unique opportunity to study initial or final state interaction effects in the fragmentation process and test the associated factorization.

Spin asymmetries in jet-hyperon production at LHC

We consider polarized Λ hyperon production in proton–proton scattering, pp→(Λ↑jet)jetX, in the kinematical region of the LHC experiments, in particular the ALICE experiment. We present a new Λ polarization observable that arises from the Sivers effect in the fragmentation process. It can be large even at midrapidity and therefore, is of interest for high energy hadron collider experiments. Apart from its potential to shed light on the mechanisms behind the phenomenon of Λ polarization arising in unpolarized hadronic collisions, the proposed observable in principle also allows to test the possible color flow dependence of single spin asymmetries and the (non)universality of transverse momentum dependent fragmentation functions.