Quark Fragmentation Functions Research Articles

Polarized TMD fragmentation functions for J/ψ production

We calculate the matching, at leading order, of the transverse momentum-dependent fragmentation functions (TMDFFs) for light quarks and gluons fragmenting to a J/ψ onto polarized nonrelativistic QCD (NRQCD) TMDFFs. The NRQCD TMDFFs have an operator product expansion in terms of nonperturbative NRQCD production matrix elements. Using the results we obtain, we make predictions for the light quark fragmentation contribution to the production of polarized J/ψ in semi-inclusive deep inelastic scattering (SIDIS) both for unpolarized and longitudinally polarized beams. These results are an important contribution to polarized J/ψ production in SIDIS and thus are needed for comparison with experiments at the future Electron-Ion Collider. Published by the American Physical Society 2024

Correlations of dihadron polarization in central, peripheral, and ultraperipheral heavy-ion collisions

While jet quenching in relativistic heavy-ion collisions has been extensively studied over decades, the polarization of quenched hadrons has rarely been discussed. It has recently been proposed that the correlations of dihadron polarization in e+e− and pp collisions provide a novel probe of the longitudinal spin transfer from hard partons to hadrons without requiring the colliding beams to be polarized. To support realistic experimental measurement of dihadron polarization with sufficient luminosity, we extend the aforementioned study to relativistic heavy-ion collisions by convoluting the vacuum fragmentation of partons with their energy loss inside the quark-gluon plasma. We find that while the correlation functions of Λ−Λ (or Λ−Λ¯) polarization in peripheral collisions is consistent with those in pp collisions, clear enhancement can be seen in central collisions. These correlation functions appear sensitive to different assumptions in the de Florian-Stratmann-Vogelsang (DSV) of parton fragmentation functions and, therefore, could place additional constraints on the spin-dependent fragmentation functions of quarks and gluons. The correlation of dihadron polarization has also been explored in ultraperipheral heavy-ion collisions, which provides a cleaner probe of fragmentation functions of quarks produced by energetic photon-photon and photon-Pomeron interactions. Published by the American Physical Society 2024

Developing predictions for pion fragmentation functions

Exploiting crossing symmetry, the hadron scale pion valence quark distribution function is used to predict the kindred elementary valence quark fragmentation function (FF). This function defines the kernel of a quark jet fragmentation equation, which is solved to obtain the full pion FFs. After evolution to a scale typical of FF fits to data, the results for quark FFs are seen to compare favourably with such fits. However, the gluon FF is markedly different. Notably, although FF evolution equations do not themselves guarantee momentum conservation, inclusion of a gluon FF which, for four quark flavours, distributes roughly 11% of the total light-front momentum fraction, is sufficient to restore momentum conservation under evolution. Overall, significant uncertainty is attached to FFs determined via fits to data; hence, the features of the predictions described herein could potentially provide useful guidance for future such studies.

Heavy quark fragmentation function in two-dimensional QCD in Nc→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_c\\rightarrow \\infty $$\\end{document} limit

We carry out a comprehensive study of the quark-to-meson fragmentation function in the ’t Hooft model, i.e., the two-dimensional quantum chromodynamics (QCD) in Nc→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_c\\rightarrow \\infty $$\\end{document} limit, following the operator definition pioneered by Collins and Soper. We apply the Hamiltonian approach as well as the diagrammatic approach to construct the functional form of the quark-to-meson fragmentation function in terms of the meson’s light-cone wave function. For the sake of comparison, we also investigate the heavy quark fragmentation into quarkonium in two-dimensional QCD within the framework of the nonrelativistic QCD (NRQCD) factorization, at the lowest order in quark velocity. In the heavy quark limit, the quark fragmentation function obtained from the ab initio method agrees well, both analytically and numerically, with that obtained from the NRQCD approach. This agreement might be regarded as a nontrivial justification for the validity of both field-theoretical approaches to compute the heavy quark fragmentation function.

Dead cone effect in charm and bottom quark jets

The evolution of a heavy quark initiated jet is mainly ruled by gluon bremsstrahlung. As a consequence of the dead-cone effect, this radiation is suppressed in the forward direction at angles smaller than that proportional to the heavy quark mass MQ, i.e. Θ0=MQ/EQ at energy EQ of the primary quark. In this paper, we unveil this effect in charm and bottom quark jets using DELPHI and OPAL data from Z0 boson decays in e+e− annihilation at center of mass energy 91.2 GeV. The analysis of the reconstructed heavy quark fragmentation function in momentum space shows the strong suppression of hadrons at high momenta in such events compared to light quark fragmentation by a factor ≲1/10. The amount of this suppression is well reproduced by perturbative QCD (pQCD) within the Modified Leading Logarithmic Aproximation and the compact scheme of Local Parton Hadron Duality (MLLA-LPHD). As a new result, we obtain an almost perfect agreement between the light quark fragmentation functions expected at W0∝MQ from DELPHI and OPAL data with Pythia8 and shed light on the reasons for the existence of the ultra-soft gluon excess at small momentum fraction in comparison with pQCD predictions.

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.

Unconventional Mechanisms of Heavy Quark Fragmentation

Heavy and light quarks produced in high-pT partonic collisions radiate differently. Heavy quarks regenerate their color field, stripped-off in the hard reaction, much faster than the light ones and radiate a significantly smaller fraction of the initial quark energy. This peculiar feature of heavy-quark jets leads to a specific shape of the fragmentation functions observed in e+e− annihilation. Differently from light flavors, the heavy quark fragmentation function strongly peaks at large fractional momentum z, i.e., the produced heavy–light mesons, B or D, carry the main fraction of the jet momentum. This is a clear evidence of the dead-cone effect, and of a short production time of a heavy–light mesons. Contrary to propagation of a small qq¯ dipole, which survives in the medium due to color transparency, a heavy–light Qq¯ dipole promptly expands to a large size. Such a big dipole has no chance to remain intact in a dense medium produced in relativistic heavy ion collisions. On the other hand, a breakup of such a dipole does not affect much the production rate of Qq¯ mesons, differently from the case of light qq¯ meson production.

Observation of the dead cone effect in charm and bottom quark jets and its QCD explanation

The production of a heavy quark is accompanied by gluon bremsstrahlung which is suppressed at small angles $\Theta\lesssim M_Q/E$ for mass $M_Q$ and high energy $E$ according to perturbative Quantum Chromo Dynamics (QCD) (``dead cone effect''). As particles at small angles typically have large momenta, the heavy quark mass also causes a suppression of high momentum particles. In this paper, we studied this effect in c- and b-quark events using data from Z boson decays in $e^+e^-$ annihilation. The heavy quark fragmentation function for charged particles is reconstructed in the momentum fraction variable $x$ or $\xi=\ln(1/x)$ by removing the decays of the heavy quark hadrons. Indeed, we find an increasing suppression of particles with rising $x$ down to a fraction of $\lesssim 1/10$ for particles with $x\gtrsim0.2$ in b-quark and $x\gtrsim0.4$ in c-quark jets in comparison to light quark fragmentation. The sensitivity to the dead cone effect in the present momentum analysis is considerably increased in comparison to the recently presented angular analysis. This amount of suppression and the differences between c- and b-quark fragmentation are in good quantitative agreement with the expectations based on perturbative QCD within the Modified Leading Logarithmic Approximation (MLLA) in the central kinematic region. The data also support a two parameter description in the MLLA of these phenomena (``Limiting Spectrum''). The sensitivity of these measurements to the heavy quark mass is investigated.

Dihadron production in DIS at small x at next-to-leading order: Transverse photons

We calculate the next-to-leading order corrections to dihadron production in deep inelastic scattering (DIS) at small $x$ using the color glass condensate formalism for the case when the virtual photon is transverse polarized. Similar to the case of longitudinal photon exchange all UV and soft singularities cancel while the collinear divergences are absorbed into quark and antiquark-hadron fragmentation functions. Rapidity divergences lead to Jalilian-Marian, Iancu, McLerran, Wiegert, Leonidov, Kovner (JIMWLK) evolution of dipoles and quadrupoles which describe multiple scatterings of the quark antiquark dipole on the target proton/nucleus and contain all the QCD dynamics of the target leading to a finite final result for the dihadron production cross section.

One-loop corrections to dihadron production in DIS at small x

We calculate the one-loop corrections to dihadron production in Deep Inelastic Scattering (DIS) at small x using the Color Glass Condensate formalism. We show that all UV and soft singularities cancel while the collinear divergences are absorbed into quark and anti quark-hadron fragmentation functions. Rapidity divergences lead to JIMWLK evolution of dipoles and quadrupoles describing multiple-scatterings of the quark anti-quark dipole on the target proton/nucleus. The resulting cross section is finite and can be used for phenomenological studies of dihadron angular correlations at small x in a future Electron-Ion Collider (EIC).

System of evolution equations for quark and gluon jet quenching with broadening

We propose a system of evolution equations that describe in-medium time-evolution of transverse-momentum-dependent quark and gluon fragmentation functions. Furthermore, we solve this system of equations using Monte Carlo methods. We then quantify the obtained solutions in terms of a few characteristic features, namely the average transverse momentum langle |k|rangle and energy contained in a cone, which allow us to see different behaviour of quark and gluon initiated final-state radiation. In particular, the later allows us to conclude that in the gluon-initiated processes there is less energy in a cone, so that the quark jet is more collimated.

Transverse polarization of hyperons produced in semi-inclusive deep inelastic scattering

We study the transverse polarization of hyperons produced in semi-inclusive deep inelastic scattering, $ep\to e\Lambda^\uparrow X$, in the framework of the collinear twist-3 factorization. The cross section from the twist-3 distribution functions and the twist-3 quark fragmentation functions is computed in the leading order with respect to the QCD coupling constant. The constraint relations among the twist-3 FFs are taken into account to simplify the formula. The formula is relevant to large-$P_T$ hyperon production in the future Electron-Ion-Collider experiment.

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.

Heavy quark jet production near threshold

In this paper, we study the fragmentation of a heavy quark into a jet near threshold, meaning that final state jet carries most of the energy of the fragmenting heavy quark. Using the heavy quark fragmentation function, we simultaneously resum large logarithms of the jet radius R and 1 − z, where z is the ratio of the jet energy to the initiating heavy quark energy. There are numerically significant corrections to the leading order rate due to this resummation. We also investigate the heavy quark fragmentation to a groomed jet, using the soft drop grooming algorithm as an example. In order to do so, we introduce a collinear-ultrasoft mode sensitive to the grooming region determined by the algorithm’s zcut parameter. This allows us to resum large logarithms of zcut/(1 − z), again leading to large numerical corrections near the endpoint. A nice feature of the analysis of the heavy quark fragmenting to a groomed jet is the heavy quark mass m renders the algorithm infrared finite, allowing a perturbative calculation. We analyze this for EJR ∼ m and EJR » m, where EJ is the jet energy. To do the latter case, we introduce an ultracollinear-soft mode, allowing us to resum large logarithms of EJR/m. Finally, as an application we calculate the rate for e+e− collisions to produce a heavy quark jet in the endpoint region, where we show that grooming effects have a sizable contribution near the endpoint.

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.

Next-to-leading-order QCD corrections to heavy quark fragmentation into {}^1S^{(1,8)}_0 quarkonia

Within NRQCD factorization framework, in this work we compute, at the lowest order in velocity expansion, the next-to-leading-order (NLO) perturbative corrections to the short-distance coefficients associated with heavy quark fragmentation into the {}^1S_0^{(1,8)} components of a heavy quarkonium. Starting from the Collins and Soper’s operator definition of the quark fragmentation function, we apply the sector decomposition method to facilitate the numerical manipulation. It is found that the NLO QCD corrections have a significant impact.

Fragmentation functions for a quark into a spin-singlet quarkonium: Different flavor case

In the paper, we calculate the fragmentation functions for a quark to fragment into a spin-singlet quarkonium, where the flavor of the initial quark is different from that of the constituent quark in the quarkonium. The ultraviolet divergences in the phase space integral are removed through the operator renormalization under the modified minimal subtraction scheme. The fragmentation function $D_{q \to \eta_Q}(z,\mu_F)$ is expressed as a two-dimensional integral. Numerical results for the fragmentation functions of a light quark or a bottom quark to fragment into the $\eta_c$ are presented. As an application of those fragmentation functions, we study the processes $Z \to \eta_c+q\bar{q}g(q=u,d,s)$ and $Z \to \eta_c+b\bar{b}g$ under the fragmentation and the direct nonrelativistic QCD approaches.

Power expansion for heavy quarkonium production at next-to-leading order in e+e\u2212 annihilation

We study heavy quarkonium production associated with gluons in e+e− annihilation as an illustration of the perturbative QCD (pQCD) factorization approach, which incorporates the first nonleading power in the energy of the produced heavy quark pair. We show how the renormalization of the four-quark operators that define the heavy quark pair fragmentation functions using dimensional regularization induces “evanescent” operators that are absent in four dimensions. We derive closed forms for short-distance coefficients for quark pair production to next-to-leading order ( {alpha}_s^2 ) in the relevant color singlet and octet channels. Using non-relativistic QCD (NRQCD) to calculate the heavy quark pair fragmentation functions up to v4 in the velocity expansion, we derive analytical results for the differential energy fraction distribution of the heavy quarkonium. Calculations for {}^3{S}_1^{left[1right]} and {}^1{S}_0^{left[8right]} channels agree with analogous NRQCD analytical results available in the literature, while several color-octet calculations of energy fraction distributions are new. We show that the remaining corrections due to the heavy quark mass fall off rapidly in the energy of the produced state. To explore the importance of evolution at energies much larger than the mass of the heavy quark, we solve the renormalization group equation perturbatively to two-loop order for the {}^1{S}_0^{left[8right]} case.

First extraction of the Λ polarizing fragmentation function from Belle e+e− data

We present a thorough phenomenological analysis of the experimental data from Belle Collaboration for the transverse $\Lambda$ and $\bar\Lambda$ polarisation, measured in $e^+e^-$ annihilation processes, for the case of inclusive (plus a jet) and associated production with a light charged hadron. This allows for the first ever extraction of the quark polarising fragmentation function for a $\Lambda$ hyperon, a transverse momentum dependent distribution giving the probability that an unpolarised quark fragments into a transversely polarised spin-1/2 hadron.

Towards Simulation of Quark and Diquark Fragmentations in the Quark-Gluon String Model (QGSM)

The NA61/SHINE collaboration has presented experimental data on $${{\rho }^{0}}$$, $$\omega $$ and $${{K}^{{0 * }}}$$ meson productions in $${{\pi }^{ - }}{\text{C}}$$ interactions at $$Plab = 158$$ and 350 GeV/c. As shown by the collaboration, the existing Monte Carlo event generators badly describe the data. In this paper a possibility to describe the data is considered in the Quark-Gluon String (QGS) model of Geant4. QGS model was proposed by A.B. Kaidalov and K.A. Ter-Martirosyan. Within the model, A.B. Kaidalov found a behaviour of quark and diquark fragmentation functions for $$z \to 0$$ and $$z \to 1$$, and proposed interpolation formulae for the functions in the whole region of $$z$$. These functions must be a solution of the well-known system of the integral equations. A simplified Monte Carlo estimation of the functions, based on usage of the fragmentation functions at $$z \to 1$$ as the kernel functions of the system, does not reproduce Kaidalov’s results. An improvement of the Monte Carlo simulations is proposed in this paper. It can be implemented in Monte Carlo event generators such as Los Alamos QGSM, QGSJet-II and the Geant4 QGS model. It will improve a description of experimental data in the models, especially, the description of the latest NA61/SHINE Collaboration data. Description of the data is a problem in DPMJet, QGSJet, EPOS and Sibyll models.