Heavy Quark 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}.

On heavy-flavour jets with Soft Drop

We study hadronic jets that are tagged as heavy-flavoured, i.e. they contain either beauty or charm. In particular, we consider heavy-flavour jets that have been groomed with the Soft Drop algorithm. In order to achieve a deeper understanding of these objects, we apply resummed perturbation theory to jets initiated by a massive quark and we perform analytic calculations for two variables that characterise Soft Drop jets, namely the opening angle and the momentum fraction of the splitting that passes Soft Drop. We compare our findings to Monte Carlo simulations. Furthermore, we investigate the correlation between the Soft Drop energy fraction and alternative observables that aim to probe heavy-quark fragmentation functions.

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.

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.

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.

Fragmentation production of ground state spin-1/2 heavy baryons including the Fermi motion of constituents

We analytically compute the process-independent fragmentation functions of heavy quarks to fragment into ground state triply heavy baryons considering the Fermi motion effect of their constituent quarks. The effect of Fermi motion is basically included in the baryon wave functions so in all previous studies, a delta function form was approximated for the baryon wave functions which corresponds to ignoring this effect. Here, we study for the first time the effect of nonperturbative dynamics of baryon bound state considering a typical wave function which is different from the delta function and is the nonrelativistic limit of the solution of Bethe-Salpeter equation with the QCD kernel. Our analytical results will be presented for the fragmentation processes $b\rightarrow\Omega_{bbc}$ and $c\rightarrow\Omega_{ccb}$ through a scalar bc diquark intermediated.

Novel mechanism for suppression of heavy flavored mesons in heavy ion collisions

Production of heavy flavored hadrons from fragmentation of heavy quarks represents an alternative probe for a medium created after heavy ion collisions. We demonstrate that observed strong suppression of heavy flavored D and B mesons, produced with high transverse momenta pT, is caused by final state interactions with such a medium. The space-time pattern of hadronization of a highly virtual heavy quark is controlled predominantly by intensive gluon radiation, which is ceased at a short time scale in accordance with perturbative QCD calculations and LEP measurements of the fragmentation functions. However, production of heavy flavored hadrons lasts a long time due to prompt multiple breakups of produced colorless (pre)hadrons in the medium. This fact together with the specific shape of heavy quark fragmentation function, peaked at large z, allows to explain the observed strong suppression of D and B mesons in a good accord with data.

Transverse momentum spectra at threshold for groomed heavy quark jets

We present the transverse momentum spectrum for a heavy hadron at threshold in a groomed jet initiated by a heavy quark. The cross section is doubly differential in the energy fraction of an identified heavy hadron in the jet and its transverse momentum measured with respect to the groomed (recoil free) jet axis. The grooming is implemented using a soft-drop grooming algorithm and helps us in mitigating the effects of Non-Global logarithms and pile up. For the particular case of a B meson, we identify two distinct regimes of the transverse momentum spectrum and develop an EFT within the formalisms of Soft Collinear Effective Theory (SCET) and Heavy Quark Effective Theory (HQET) for each of these regions. We show how each region can be matched smoothly into the other to provide a prediction for the perturbative transverse momentum spectrum. The EFT also predicts the scaling behavior of the leading non-perturbative power corrections and implements a simple shape function to account for hadronization. We work in the threshold region where the heavy hadron carries most of the energy of the jet since in this regime, we have a very good discriminating power between heavy quark and gluon initiated jets. We observe that the shape of the spectrum is independent of the energy of the jet over a large range of transverse momentum. We propose that this spectrum can be used as a probe of evolution for heavy quark TMD fragmentation function. At the same time, it can be treated as a jet substructure observable for probing Quark-Gluon Plasma (QGP).

Revisiting heavy quark radiative energy loss in nuclei within the high-twist approach

We revisit the calculation of multiple parton scattering of a heavy quark in nuclei within the framework of recently improved high-twist factorization formalism, in which gauge invariance is ensured by a delicate setup of the initial partons' transverse momenta. We derive a new result for medium modified heavy quark fragmentation functions in deeply inelastic scattering. It is consistent with the previous calculation of light quark energy loss in the massless limit, but leads to a new correction term in the heavy quark case, which vanishes in the soft gluon radiation limit. We show numerically the significance of the new correction term in the calculation of heavy quark energy loss as compared to previous studies and with soft gluon radiation approximation.

Novel scenario for production of heavy flavored mesons in heavy ion collisions

The observed strong suppression of heavy flavored hadrons produced with high $p_T$, is caused by final state interactions with the created dense medium. Vacuum radiation of high-pT heavy quarks ceases at a short time scale, as is confirmed by pQCD calculations and by LEP measurements of the fragmentation functions of heavy quarks. Production of a heavy flavored hadrons in a dense medium is considerably delayed due to prompt breakup of the hadrons by the medium. This causes a strong suppression of the heavy quark yield because of the specific shape of the fragmentation function. The parameter-free description is in a good accord with available data.

Heavy quark fragmenting jet functions

Heavy quark fragmenting jet functions describe the fragmentation of a parton into a jet containing a heavy quark, carrying a fraction of the jet momentum. They are two-scale objects, sensitive to the heavy quark mass, $m_Q$, and to a jet resolution variable, $\tau_N$. We discuss how cross sections for heavy flavor production at high transverse momentum can be expressed in terms of heavy quark fragmenting jet functions, and how the properties of these functions can be used to achieve a simultaneous resummation of logarithms of the jet resolution variable, and logarithms of the quark mass. We calculate the heavy quark fragmenting jet function $\mathcal G_Q^Q$ at $\mathcal O(\alpha_s)$, and the gluon and light quark fragmenting jet functions into a heavy quark, $\mathcal G_g^Q$ and $\mathcal G_l^Q$, at $\mathcal O(\alpha_s^2)$. We verify that, in the limit in which the jet invariant mass is much larger than $m_Q$, the logarithmic dependence of the fragmenting jet functions on the quark mass is reproduced by the heavy quark fragmentation functions. The fragmenting jet functions can thus be written as convolutions of the fragmentation functions with the matching coefficients $\mathcal J_{i j}$, which depend only on dynamics at the jet scale. We reproduce the known matching coefficients $\mathcal J_{i j}$ at $\mathcal O(\alpha_s)$, and we obtain the expressions of the coefficients $\mathcal J_{g Q}$ and $\mathcal J_{l Q}$ at $\mathcal O(\alpha_s^2)$. Our calculation provides all the perturbative ingredients for the simultaneous resummation of logarithms of $m_Q$ and $\tau_N$.

Heavy quark perturbative QCD fragmentation functions in the presence of hadron mass

The dominant mechanism to produce hadronic bound states with large transverse momentum is fragmentation, that is the splitting of a high energy parton into a hadronic state and other partons. We review the present schemes to calculate the heavy quark fragmentation functions (FFs) and drive an exact analytical expression of FF which includes most of the kinematical and dynamical properties of the process. Using the perturbative QCD, we calculate the FF for c-quark to split into S-wave D-meson to leading order in the QCD coupling constant. Our result is compared with the current well-known phenomenological models which are obtained through a global fit to e+e- data from SLAC SLC and CERN LEP1 and we also compare the FF with experimental data form BELLE and CLEO. Specically, we study the effect of outgoing meson mass on the pQCD FF. Meson masses are responsible for the low-z threshold, where z is the scaled energy variable.

Fragmentation Functions for Heavy Baryons in the Recombination Model

Using the shower parton distributions determined by the recombination model, we predict the fragmentation functions for heavy baryons. Then we obtain the completed fragmentation functions of heavy quarks (c and b) splitting into their hadrons (mesons and baryons containing one heavy valence quark). The calculated process shows that the fragmentation functions for mesons and baryons are not independent if the hadronization of the shower partons is taken into account.

Heavy-quark fragmentation

Basic types of parametrizations are considered for the heavy-quark fragmentation function. The kinematical boundaries of the fragmentation-model applicability region are found. Nonfragmentation contributions to the production of heavy mesons are estimated.

Multiple parton scattering in nuclei: heavy quark energy loss and modified fragmentation functions

Multiple scattering, induced radiative energy loss and modified fragmentation functions of a heavy quark in nuclear matter are studied within the framework of generalized factorization in perturbative QCD. Modified heavy quark fragmentation functions and energy loss are derived in detail with illustration of the mass dependencies of the Landau–Pomeranchuk–Migdal interference effects and heavy quark energy loss. Due to the quark mass dependence of the gluon formation time, the nuclear size dependencies of nuclear modification of the heavy quark fragmentation function and heavy quark energy loss are found to change from a linear to a quadratic form when the initial energy and momentum scale are increased relative to the quark mass. The radiative energy loss of the heavy quark is also significantly suppressed due to limited cone of gluon radiation imposed by the mass. Medium modification of the heavy quark fragmentation functions is found to be limited to the large z region due to the form of heavy quark fragmentation functions in vacuum.

On the quark distribution in an on-shell heavy quark and its all-order relations with the perturbative fragmentation function

I present new results on the quark distribution in an on-shell heavy quark in perturbative QCD and explore its all-order relations with heavy-quark fragmentation. I first compute the momentum distribution function to all orders in the large-beta_0 limit and show that it is identical to the perturbative heavy-quark fragmentation function in the same approximation. I then analyze the Sudakov limit of the distribution and the fragmentation functions using Wilson lines and prove that the corresponding Sudakov exponents in the non-Abelian theory are the same to any logarithmic accuracy. The anomalous dimension is then determined to two-loop order, corresponding to next-to-next-to-leading logarithmic accuracy in the exponent, in two ways: the first by extracting the singular terms from a recent calculation of the fragmentation function and the second by performing the two-loop Wilson-line calculation in configuration space. I find perfect agreement between the two.