We calculate the fragmentation function for charm quark into J/psi at the QCD next-to-leading-order (NLO) and find that the produced J/psi is of larger momentum fraction than it is at the leading-order. Based on the fragmentation function and partonic processes calculated at the NLO, the transverse momentum distribution on J/psi hadroproduction associated with a charm c (or \bar{c}) jet are predicted. We find that the distribution is enhanced by a factor of 2.0--3.3 at the NLO as p_t increased from 10 GeV to 100 GeV and it is measurable at the LHC with charm tagger. The measurement at the LHC will supply a first chance to directly test the QCD fragmentation mechanism on heavy quarkonium production where the fragmentation function is calculable in perturbative QCD. It is also applied to J/psi (Upsilon) production in the decay of Z^0 (top quark).
The leading color-singlet contribution to the fragmentation function for a gluon to split into spin-triplet S-wave quarkonium is presented. In the case of charmonium, we find that this color-singlet term is always negligible compared to the leading color-octet contribution.
We calculate the polarisation of $ψ$'s and $Υ$'s produced by the fragmentation of heavy quarks. We find that fragmentation to transversely aligned quarkonium is slightly enhanced relative to longitudinally polarised. This net alignment corresponds to a $\sim 5\%$ asymmetry in the angular distribution of leptons produced in the subsequent decay of the quarkonium. We point out that the leading gluon contribution to $ψ$ and $Υ$ production arises from the evolution of the charm quark fragmentation function, rather than from direct gluon fragmentation.
We review the physics of quarkonium production at high energy colliders, with a specific emphasis on production $via$ fragmentation. We discuss the fragmentation picture for the production of quarkonia at large-$p_T$ at the Tevatron and emphasise the importance of the colour-octet components of the quarkonium wave function which are included in the fragmentation functions. Applications of these theoretical ideas to other processes, both at the Tevatron and at other experiments like LEP, HERA and the LHC are important as they will provide stringent tests of these ideas.
We calculate the functions of parton fragmentation into $^3P_J$ quarkonium at order $α_s ^2$, where the parton can be a heavy or light quark. The obtained functions explicitly satisfy the Altarelli-Parisi equation and they are divergent, behaving as $z^{-1}$ near $z=0$. However, if one choses the renormalization scale as twice of the heavy quark mass, the fragmentation functions are regular over the whole range of $z$.
The dominant production mechanism for heavy quark-antiquark bound states in very high energy processes is fragmentation, the splitting of a high energy parton into a quarkonium state and other partons. We show that the fragmentation functions $D(z,μ)$ describing these processes can be calculated using perturbative QCD. We calculate the fragmentation functions for a gluon to split into S-wave quarkonium states to leading order in the QCD coupling constant. The leading logarithms of $μ/m_Q$, where $μ$ is the factorization scale and $m_Q$ is the heavy quark mass, are summed up using Altarelli-Par…
The short-distance coefficients for the color-octet 3S1 term in the fragmentation function for a gluon to split into heavy quarkonium states is calculated to order alpha_s^2. The gauge-invariant definition of the fragmentation function by Collins and Soper is employed. Ultraviolet divergences are removed using the MS-bar renormalization procedure. The longitudinal term in the fragmentation function agrees with a previous calculation by Beneke and Rothstein. The next-to-leading order correction to the transverse term disagrees with a previous calculation.
The fragmentation function of a scalar leptoquark into possible S-wave bound states with a heavy anti-quark is calculated to the leading order in perturbative QCD for the high energy processes at large transverse momenta. The one-loop equations for the $q^2$-evolution of moments of the fragmentation function due to the hard gluon emission by the leptoquark are derived. The integral probabilities of fragmentation are evaluated.
The functions of the gluon fragmentation into $^3P_J$ quarkonium are calculated to order $α_s^2$. With the recent progress in analysing quarkonium systems in QCD we show explicitly how the socalled divergence in the limit of the zero-binding energy, which is related to $P$-wave quarkonia, is treated correctly in the case of fragmentation functions. The obtained fragmentation functions satisfy explicitly at the order of $α_s^2$ the Altarelli-Parisi equation and when $z\rightarrow 0$ they behave as $z^{-1}$ as expected. Some comments on the previous results are made.
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 $D_{q \to η_Q}(z,μ_F)$ 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 $η_c$ are presented. As an appli…
We analyze the uncertainties induced by different definitions of the momentum fraction $z$ in the application of gluon to heavy quarkonium fragmentation function. We numerically calculate the initial $g \to J / ψ$ fragmentation functions by using the non-covariant definitions of $z$ with finite gluon momentum and find that these fragmentation functions have strong dependence on the gluon momentum $\vec{k}$. As $| \vec{k} | \to \infty$, these fragmentation functions approach to the fragmentation function in the light-cone definition. Our numerical results show that large uncertainties remains w…
We study the transverse-momentum spectrum of quarkonium production from single light-parton fragmentation mechanism. In the case of semi-inclusive deep inelastic scattering, we observe that there are two possible initiating processes, namely photon-gluon fusion and light-quark fragmentation. For the second case we derive the factorization theorem, which involves a new hadronic quantity: the quarkonium transverse-momentum-dependent fragmentation functions in NRQCD. We calculate their matching onto the non-perturbative long distance matrix elements at the lowest order in the strong-coupling cons…
Following the operator definition of the fragmentation function developed by Collins and Soper, we compute the gluon-to-$h_c$ fragmentation function at the lowest order in the velocity expansion in NRQCD factorization approach. Utilizing some modern technique developed in the area of multi-loop calculation, we are able to analytically deduce the infrared-finite color-singlet short-distance coefficient associated with the fragmentation function. The fragmentation probability for gluon into $h_c$ is estimated to be order $10^{-6}$.
We present the calculation at next-to-leading order (NLO) in alpha_s of the fragmentation function of a gluon into heavy quarkonium in the color-octet spin-singlet S-wave channel. To calculate the real NLO corrections, we adapt a subtraction scheme introduced by Frixione, Kunszt, and Signer. Ultraviolet and infrared divergences in the real NLO corrections are calculated analytically by evaluating the phase-space integrals of the subtraction terms using dimensional regularization. The subtracted phase-space integrals are then evaluated in 4 space-time dimensions. The divergences in the virtual …
Study of the polarized heavy quarkonium production in recently proposed QCD factorization formalism requires knowledge of a large number of input fragmentation functions (FFs) from a single parton or a heavy quark-antiquark pair to a polarized heavy quarkonium. In this paper, we calculate these FFs at the input scale in terms of nonrelativistic QCD (NRQCD) factorization. We derive all relevant polarized NRQCD long-distance matrix elements based symmetries and propose a self-consistent scheme to define them in arbitrary $d$ dimensions. We then calculate polarized input FFs contributed from all …
Inspired by the recent measurements of the $η_c$ meson production at LHC, we investigate the relativistic correction effect for the fragmentation function of the gluon into $η_c$, which constitutes the crucial nonperturbative elements to understand $η_c$ production at high $p_T$. Employing three distinct methods, we calculate the leading relativistic correction to the $g\toη_c$ fragmentation function in the NRQCD factorization framework, as well as verify the existing NLO result for the $c\to η_c$ fragmentation function. We also study the evolution behavior of these fragmentation functions wit…
We present the first calculation at next-to-leading order (NLO) in $α_s$ of a fragmentation function into quarkonium whose form at leading order is a nontrivial function of $z$, namely the fragmentation function for a gluon into a spin-singlet S-wave state at leading order in the relative velocity. To calculate the real NLO corrections, we introduce a new subtraction scheme that allows the phase-space integrals to be evaluated in 4 dimensions. We extract all ultraviolet and infrared divergences in the real NLO corrections analytically by calculating the phase-space integrals of the subtraction…
We use the NRQCD factorization formalism to calculate the relativistic corrections to the fragmentation function for a gluon fragmenting into a spin-triplet S-wave heavy quarkonium. We make use of the gauge-invariant formulation of the fragmentation function of Collins and Soper. The color-octet contribution receives a large, negative relativistic correction, while the color-singlet contribution receives a large, positive relativistic correction. The considerable decrease in the color-octet contribution requires a corresponding increase in the phenomenological value of the leading color-octet …
We compute the contributions due to gluon and heavy-quark fragmentation to quarkonium production at large transverse momentum in $\bar p p$ and $p p$ collisions. For inclusive $J/ψ$ production, there is a large contribution from the $g \rightarrow χ_c$ fragmentation. At large $p_T$, this is comparable to the conventional charmonium model prediction via gluon fusion at the ISR and dominates over the latter at the Tevatron energy. This may help to explain, at least partly, the large $J/ψ$ production cross-section recently observed by the CDF experiment. However, the fragmentation contribution to…
We compute fragmentation functions for a quark to fragment to a quarkonium through an $S$-wave spin-triplet heavy quark-antiquark pair. We consider both color-singlet and color-octet heavy quark-antiquark ($Q\bar Q$) pairs. We give results for the case in which the fragmenting quark and the quark that is a constituent of the quarkonium have different flavors and for the case in which these quarks have the same flavors. Our results for the sum over all spin polarizations of the $Q\bar Q$ pairs confirm previous results. Our results for longitudinally polarized $Q\bar Q$ pairs agree with previous…
In the paper, we calculate the fragmentation functions for $c \to η_c$ and $b \to η_b$ up to next-to-leading-order (NLO) QCD accuracy. The ultraviolet divergences in the real corrections are removed through operator renormalization under the modified minimal subtraction scheme. We then obtain the fragmentation functions $D_{c \to η_c}(z,μ_F)$ and $D_{b \to η_b}(z,μ_F)$ up to NLO QCD accuracy, which are presented as figures and fitting functions. The numerical results show that the NLO corrections are significant. The sensitives of the fragmentation functions to the renormalization scale and th…
Fragmentation is the dominant production mechanism for heavy hadronic bound states with large transverse momentum. We numerically calculate the initial $g\rightarrow H(Q\bar{Q})$ fragmentation functions (FFs) using the nonrelativistic QCD factorization approach. Our analytical expression of FFs depends on both the momentum fraction $z$ and the transverse momentum of the gluon, and contains most of the kinematical and dynamical properties of the process. Specifically, using the perturbative QCD we present the FF for a gluon to split into $S$-wave charmonium meson $H_c$ to leading order in the Q…
The gluon fragmentation function of $g \to h_c$ at leading order of strong coupling constant $α_s$ and typical velocity $v$ is calculated in the framework of NRQCD, of which the contributions from both color-singlet and -octet processes are taken into account and hence the result is infrared safe. It is found that the fragmentation probability of a high energy virtual gluon to split into $h_c$ is about $7.1\times 10^{-7}$. The analytic fragmentation function presented in this work can be employed to the study of $h_c$ production in large transverse momentum at hadron colliders, such as Large H…
At the large transverse momentum region, the production of heavy-heavy bound-states such as charmonium, bottomonium, and $\bar bc$ mesons in high energy $e^+e^-$ and hadronic collisions is dominated by parton fragmentation. We calculate the heavy quark fragmentation functions into the D-wave quarkonium and $\bar bc$ mesons to leading order in the strong coupling constant and in the non-relativistic expansion. In the $\bar b c$ meson case, one set of its D-wave states is expected to lie below the open flavor threshold. The total fragmentation probability for a $\bar b$ antiquark to split into t…
Fragmentation is the dominant mechanism for hadron production with high transverse momentum. For spin-triplet S-wave heavy quarkonium production, contribution of gluon fragmenting to color-singlet channel has been numerically calculated since 1993. However, there is still no analytic expression available up to now because of its complexity. In this paper, we calculate both polarization-summed and polarized fragmentation functions of gluon fragmenting to a heavy quark-antiquark pair with quantum number $^3S_1^{[1]}$. Our calculations are performed in two different frameworks. One is the widely …
The fragmentation functions for gluons to split into P-wave heavy quarkonium states are calculated to leading order in the QCD coupling constant. Long-distance effects are factored into two nonperturbative parameters: the derivative of the radial wavefunction at the origin and a second parameter related to the probability for a heavy-quark-antiquark pair that is produced in a color-octet S-wave state to form a color-singlet P-wave bound state. The fragmentation probabilities for a high transverse momentum gluon to split into the P-wave charmonium states $χ_{c0}$, $χ_{c1}$, and $χ_{c2}$ are est…
We report the recent progress in the studies of D-wave heavy quarkonia production from gluon fragmentation at the Tevatron and in Z^0 decays, and the D-wave charmonia production in B decays. We show that the color-octet contributions are of crucial importance to the production of D-wave quarkonia. Signals, as strong as that for J/ψand ψ^\prime, should be seen for the D-wave charmonium in the processes. These will provide a test of the color-octet production mechanism.
We present a calculation of the leading order QCD fragmentation functions for gluons to split into spin-triplet D-wave quarkonia. We apply them to evaluate the gluon fragmentation contributions to inclusive ^3D_J quarkonium production at large transverse momentum processes like the Tevatron and find that the D-wave quarkonia, especially the charmonium 2^{--} state, could be observed through color-octet mechanism with present luminosity. Since there are distinctively large gaps between the contributions of two different (i.e, color-singlet and color-octet) quarkonium production mechanisms, our …
Quarkonium production at a given large p_T is dominated by parton fragmentation: a parton which is produced with transverse momentum p_T/z fragments into a quarkonium state which carries a fraction z of the parton momentum. Since parton production cross sections fall steeply with p_T, high z fragmentation is favored. However, quantum number constraints may require the emission of gluons in the fragmentation process, and this softens the z dependence of the fragmentation function. We discuss the possibility that higher- order processes may enhance the large z part of fragmentation functions and…
Dimensional regularization is incompatible with the standard covariant projection methods that are used to calculate the short-distance coefficients in inclusive heavy quarkonium production and annihilation rates. A new method is developed that allows dimensional regularization to be used consistently to regularize the infrared and ultraviolet divergences that arise in these perturbative calculations. We illustrate the method by calculating the leading color-octet terms and the leading color-singlet terms in the gluon fragmentation functions for arbitrary quarkonium states. We resolve a discre…
We present hadron-level predictions from the Monte-Carlo generator CASCADE and numerical calculations of beauty quark and quarkonium production in the framework of the kt-factorization approach for LHC energies. Our predictions are compared with the CMS experimental data.
It has recently been noticed that heavy quark ($Q$) and gluon ($g$) fragmentation to heavy quarkonium states ($Q\bar{Q}$) can be calculated in perturbative QCD. This technique allows for the inclusion of a subset of higher order corrections to quarkonium production which dominates at large transverse momentum. A comparison will be presented between the new calculation of quarkonium production to the existing data from the Tevatron. In addition some new results on predictions for double heavy quark baryons at various colliders will be presented.
We discuss heavy quarkonium production through parton fragmentation, including a review of arguments for the factorization of high-p_T particles into fragmentation functions for hadronic initial states. We investigate the further factorization of fragmentation functions in the NRQCD formalism, and argue that this requires a modification of NRQCD octet production matrix elements to include nonabelian phases, which makes them gauge invariant. We describe the calculation of uncanceled infrared divergences in fragmentation functions that must be factorized at NNLO, and verify that they are absorbe…
The photoproduction of quarkonium in coherent hadron-hadron (pp/pA/AA$) interactions for LHC energies is an important tool to investigate the QCD dynamics at high energies. In this paper we estimate the integrated cross section and rapidity distribution for J/Psi and Upsilon production using the Color Glass Condensate (CGC) formalism. We predict large rates, implying that the experimental identification could be feasible at the LHC.
The production of heavy quarkonia is a powerful tool to test our understanding of strong interaction dynamics. It is well-known that the dominant production mechanism for heavy quarkonia with large transverse momentum is fragmentation. In this work we, analytically, calculate the QCD leading order contribution to the process-independent fragmentation functions (FFs) for a gluon to split into the vector ($J/ψ$) and pseudoscalar ($η_c$) $S$-wave charmonium states. The analyses of this paper differ in which we present, for the first time, an analytical form of the $g\rightarrow J/ψ$ FF using a di…
The perturbative QCD approach to quarkonium decay into a photon and hadrons is reconsidered. It is shown that a consistent treatment within perturbative QCD calls for the introduction of a fragmentation contribution which has been neglected so far. The ensuing phenomenological implications are discussed, and, in particular, the possibility of measuring the gluon fragmentation function of the photon is addressed. *To appear in the proceedings of the workshop QCD94, Montpellier, July '94.
Heavy quarkonium production in the framework of the nonrelativistic quantum chromodynamics and leading order of the parton Reggeization approach at the Tevatron and LHC Colliders is discussed. The new results, which are reviewed in this report, include the comparison with recent data on chi_{c1,c2} production from ATLAS Collaboration and special discussion of heavy quarkonium polarization issues in the considered framework.
In this paper we discuss the systematics of quarkonium production at the LHC. In particular, we focus on the necessity to sum logs of the form log(Q/p_perp) and log(p_perp/m_Q). We show that the former contributions are power suppressed, while the latter, whose contribution in fragmentation is well known, also arise in the short distance (i.e., non-fragmentation) production mechanisms. Though these contributions are suppressed by powers of m_Q/p_perp, they can be enhanced by inverse powers of v, the relative velocity between heavy quarks in the quarkonium. In the limit p_perp >> m_Q short dist…
The hadronic production of $D$-wave states of $\bar b c$ is studied. The relative yield of such states is estimated for kinematic conditions of LHC experiments.
The quarkonium data suggest a qualitatively new production mechanism, enhancing the rate of the quarkonia produced in the fragmentation region of an incoming coloured particle.
The quarkonium plus photon production in coherent hadron-hadron interactions at the LHC is studied using the non-relativistic QCD factorization formalism. We investigate a set of kinematic distributions and compute the total cross sections for $J/Ψ+ γ$ production. Our results demonstrate the feasibility of such process in the LHC kinematic regime and explore the possibilities for the Future Circular Collider, where higher event yields can be achieved.
In this paper we study the inclusive and diffractive double quarkonium production. Based on the nonrelativistic QCD (NRQCD) factorization formalism for the quarkonium production mechanism we estimate the rapidity and transverse momentum dependence of the cross section for the $J/ΨJ/Ψ$ and $ΥΥ$ production in diffractive processes at LHC energies. The contributions of the color-singlet and color-octet channels are estimated and predictions for the total cross sections in the kinematical regions of the LHC experiments are also presented. Our results demonstrate that the contribution of diffractiv…
In this contribution we briefly discuss an ongoing phenomenological programme on quarkonium production in unpolarized and polarized proton-proton collisions in a fixed target setup at LHCb, the LHCSpin project. Within a TMD approach, we aim at considering in particular: the relative role of the NRQCD color-singlet and color-octet production mechanisms, both for unpolarized and polarized quarkonium production; the study of azimuthal and transverse single-spin asymmetries as a phenomenological tool for learning about the almost unknown gluon Sivers function; the role of initial- and final-state …
The rates for diffractive quarkonium plus prompt-photon associated production at the LHC are estimated. The theoretical framework considered in the J/Psi (or Upsilon) production associated with a direct photon at the hadron collider is the non-relativistic QCD (NRQCD) factorization formalism. The corresponding single diffractive cross section is computed based on the hard diffractive scattering factorization supplemented by absorptive corrections. Such processes are sensitive to the gluon content of the Pomeron at small Bjorken-x and they may also be a good place to test the different availabl…
Based on the factorization formalism of nonrelativistic quantum chromodynamics (NRQCD), we calculate the production cross section for the charmonium and the bottomonium produced by the hard photoproduction processes and fragmentation processes in relativistic heavy ion collisions. It is shown that the existing experimental data on heavy quarkonium production at the Large Hadron Collider (LHC) can be described in the framework of the NRQCD formalism, and the phenomenological values of matrix elements for color-singlet and color-octet components give the main contribution. The numerical results …
Heavy quarkonium production in the framework of the non-relativistic quantum chromodynamics and leading order of the parton Reggeization approach at the Tevatron and LHC is discussed. In this note, we compare our predictions for the bottomonium production at the LHC due to the color-singlet approximation of the non-relativistic quantum chromodynamics with CMS and LHCb data. It is found, that in the production of Upsilon(1S) states, the color-singlet mechanism is dominating, whereas to describe the data for the inclusive Upsilon(2S) and Upsilon(3S) production, the color- octet contributions sho…
We compute the relativistic corrections to the color-singlet contribution to gluon fragmentation into a J/psi at relative order v^4, making use of the nonrelativistic QCD (NRQCD) factorization approach. The corresponding full-QCD process exhibits infrared divergences that manifest themselves as single and double poles in epsilon in 4-2epsilon dimensions. We isolate the infrared-divergent contributions and treat them analytically. In the matching of full QCD to NRQCD, the pole contributions are absorbed into long-distance NRQCD matrix elements. The renormalizations of the ultraviolet divergence…
We discuss factorization in heavy quarkonium production in high energy collisions using NRQCD. Infrared divergences at NNLO are not matched by conventional NRQCD matrix elements. However, we show that gauge invariance and factorization require that conventional NRQCD production matrix elements be modified to include Wilson lines or non-abelian gauge links. With this modification NRQCD factorization for heavy quarkonium production is restored at NNLO.
Sizable amount of heavy-quarkonium events can be produced through $W$-boson decays at the LHC. Such channels will provide a suitable platform to study the heavy-quarkonium properties. The "improved trace technology", which disposes the amplitude ${\cal M}$ at the amplitude-level, is helpful for deriving compact analytical results for complex processes. As an important new application, in addition to the production of the lower-level Fock states $|(Q\bar{Q'})[1S]>$ and $|(Q\bar{Q'})[1P]>$, we make a further study on the production of higher-excited $|(Q\bar{Q'})>$-quarkonium Fock states $|(Q\ba…
Next-to-next-to-leading-order QCD corrections to ${}^3S_1^{(8)}$ gluon fragmentation function for quarkonium
China University of Mining and Technology（中国矿业大学）；Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；South China Normal University（华南师范大学）；Southwest University（西南大学）
The photonuclear production of vector mesons in ultraperipheral heavy ion collisions is investigated within the collinear approach using different parameterizations for the nuclear gluon distribution. The integrated cross section and the rapidity distribution for the $AA \to V AA$ ($V = J/Ψ, Υ$) process are computed for energies of RHIC and LHC. A comparison with the recent PHENIX data on coherent production of $J/Ψ$ mesons is also presented. We demonstrate that the study of the exclusive quarkonium photoproduction can be used to constrain the nuclear effects in the gluon distribution.
The value of quarkonium wave function at the origin is an important quantity while studying many physical problems concerning a heavy quarkonium. This is because that it is widely used to evaluate the production and decay amplitudes of the heavy quarkonium within the effective filed theory framework, e.g., the non-relativistic QCD (NRQCD). In this paper, the value of the Schr${\rm \ddot{o}}$dinger radial wave function or its first nonvanishing derivative at zero quark-antiquark separation, i.e., $|(|c\bar{c})[n]\rangle$-, $|(|b\bar{c})[n]\rangle$-, and $|(b\bar{b})[n]\rangle$-quarkonium, have …
