Using methods of effective field theory, a systematic analysis of the fragmentation functions D_{a/H}(x,m_Q) of a hadron H containing a heavy quark Q is performed (with a=Q,Q_bar,q,q_bar,g). By integrating out pair production of virtual and real heavy quarks, the fragmentation functions are matched onto a single nonperturbative function describing the fragmentation of the heavy quark Q into the hadron H in "partially quenched" QCD. All calculable, short-distance dependence on x is extracted in this step. For x->1, the remaining fragmentation function can be matched further onto a universal function defined in heavy-quark effective theory in order to factor off its residual dependence on the heavy-quark mass. By solving the evolution equation in the effective theory analytically, large logarithms of the ratio mu/m_Q are resummed to all orders in perturbation theory. Connections with existing approaches to heavy-quark fragmentation are discussed. In particular, it is shown that previous attempts to extract log^n(1-x) terms from the fragmentation function D_{Q/H}(x,m_Q) are incompatible with a proper separation of short- and long-distance effects.
In the framework of the kT-factorization approach we have calculated in the fragmentation model the pT-spectra of B(c) mesons at the energies of the Tevatron and the LHC Colliders and at the large pT domain. We compare the obtained results with the existing experimental data and with the predictions obtained in the collinear parton model.
We study the fragmentation function of the gluon to color-octet $^3S_1$ heavy quark-antiquark pair using the soft gluon factorization (SGF) approach, which expresses the fragmentation function in a form of perturbative short-distance hard part convoluted with one-dimensional color-octet $^3S_1$ soft gluon distribution (SGD). The short distance hard part is calculated to next-to-leading order in $α_s$ and a renormalization group equation for the SGD is derived. By solving the renormalization group equation, threshold logarithms are resummed to all orders in perturbation theory. The comparison w…
The p_T-spectra for direct J/psi and psi' in hadroproduction at Tevatron energy have been calculated based on NRQCD formalism and fragmentation approximation in the collinear parton model and k_T-factorization approach. We have described the CDF data and obtained a good agreement between the predictions obtained in the parton model and k_T-factorization approach. We performed the calculations using the relevant leading order in alpha_s hard amplitudes and the equal values of the color-octet long-distance matrix elements for the both models.
Current all-orders proofs of factorization of hard processes are made by extracting the leading power behavior of Feynman graphs, i.e., by extracting asymptotics strictly order-by-order in perturbation theory. The resulting parton densities and fragmentation functions include non-perturbative effects. I show how there are missing elements in the proofs; these are related to and exemplified by string and cluster models of hadronization. The proofs rely on large rapidity differences between different parts of graphs for the process; but in reality large rapidity gaps are filled in
The universal fragmentation functions of gluon into the flavored quarkonia $B_c$ and (polarized) $B_c^*$ are computed within NRQCD factorization framework, at the lowest order in velocity expansion and strong coupling constant. It is mandatory to invoke the DGLAP renormalization program to render the NRQCD short-distance coefficients UV finite in a point-wise manner. The calculation is facilitated with the sector decomposition method, with the final results presented with high numerical accuracy. This knowledge is useful to enrich our understanding toward the large-$p_T$ behavior of $B_c^{(*)}…
We discuss baryon production in the nucleon fragmentation region in deep inelastic scattering. The dependence of the nucleon spectra on Bjorken-$x$ is evaluated within the parton model and in QCD, and ways to look for the break-down of the DGLAP approximation via baryon production are suggested. We argue also that the leading neutron production in these small $x$ processes is rather insensitive to the pion parton densities of the nucleon.
In the target fragmentation region of Semi-Inclusive Deep Inelastic Scattering, the diffractively produced hadron has small transverse momentum. If it is at order of $Λ_{QCD}$, it prevents to make predictions with the standard collinear factorization. However, in this case, differential cross-sections can be predicted by the factorization with fracture functions, diffractive parton distributions. If the transverse momentum is much larger than $Λ_{QCD}$ but much smaller than $Q$ which is the virtuality of the virtual photon, both factorizations apply. In this case, fracture functions can be fac…
We study the transverse single-spin asymmetry for single-hadron production in proton-proton collisions within the framework of collinear twist-3 factorization in Quantum Chromodynamics. By taking into account the contribution due to parton fragmentation we obtain a very good description of all high transverse-momentum data for neutral and charged pion production from the Relativistic Heavy Ion Collider. Our study may provide the crucial step towards a final solution to the longstanding problem of what causes transverse single-spin asymmetries in hadronic collisions within Quantum Chromodynamic…
The next-to-leading order (NLO) ($\mathcal{O}(α_s^3)$) corrections for gluon fragmentation functions to a heavy quark-antiquark pair in ${^{3}\hspace{-0.6mm}P_{J}^{[1,8]}}$ states are calculated within the NRQCD factorization. We use integration-by-parts reduction and differential equations to semi-analytically calculate fragmentation functions in full-QCD, and find that infrared divergences can be absorbed by the NRQCD long distance matrix elements. Thus, the NRQCD factorization conjecture is verified at two-loop level via a physical process, which is free of artificial ultraviolet divergence…
We present new sets of fragmentation functions for protons and inclusive charged hadrons obtained in combined NLO QCD analyses of single-inclusive hadron production in electron-positron annihilation, proton-proton collisions, and deep-inelastic lepton-proton scattering. These analyses complement previous results for pion and kaon fragmentation functions with charge and flavor discrimination. The Lagrange multiplier technique is used to assess the uncertainties in the extraction of the new sets of fragmentation functions.
We investigate the properties of interference fragmentation functions measurable from the distribution of two hadrons produced in the same jet in the current fragmentation region of a hard process. We discuss the azimuthal angular dependences in the leading order cross section of two-hadron inclusive lepton-nucleon scattering as an example how these interference fragmentation functions can be addressed separately.
Using the formalism of two-hadron fragmentation functions, we discuss single-spin asymmetries occurring in the production of two hadrons in the current region of deep inelastic scattering, with a particular emphasis on transversity measurements.
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…
We reconsider the option of extracting the transversity distribution by using interference fragmentation functions into two leading hadrons inside the same current jet. To this end, we perform a new study of two-hadron fragmentation functions. We derive new positivity bounds on them. We expand the hadron pair system in relative partial waves, so that we can naturally incorporate in a unified formalism specific cases already studied in the literature, such as the fragmentation functions arising from the interference between the s- and p-wave production of two mesons, as well as the production o…
In the hadronic decays of Z, gluon emission leads to the appearance of the longitudinal component of the fragmentation function, F_L. Measurement of F_L and the transverse component, F_T, could thus provide an insight into the gluon fragmentation function. However, hadronization corrections at low x can be significant. Here we present a method of accounting for such corrections, using the JETSET event generator as illustration.
A next-to-leading order (NLO) analysis of hadron-jet momentum correlations in p-p collisions at the LHC is carried out. We show that the inclusive charged hadron momentum distributions inside jets is a very sensitive observable which allows one to disentangle among various fragmentation function sets presently available. Correlations with identified hadrons (kaons, protons) are investigated as well.
We investigate the inclusive hadroproduction of a heavy quarkonium ($J/ψ$ or $Υ$), in association with a light-flavored jet, as a testing ground for the semi-hard regime of QCD. Our theoretical setup is the hybrid high-energy and collinear factorization, where the standard collinear approach is supplemented by the $t$-channel resummation of leading and next-to-leading energy logarithms à la BFKL. We present predictions for rapidity and azimuthal-angle differential distributions, hunting for stabilizing effects of the high-energy series under higher-order corrections. Our reaction represents an…
We study the inclusive production of two hadrons in deep inelastic processes, l N ---> l h_1 h_2 X, with h_1 in the current fragmentation region (CFR) and h_2 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.
We propose a modification of the mechanism of charmed quarks fragmentation into $D$ mesons in hadronic collisions. It is shown that the distinction in valence quark distributions in the initial $π^{\pm}$ and $K^{\pm}$ mesons leads to different inclusive spectra of $D$ and $D_s$ mesons produced in $π^{\pm}$ and $K^{\pm}$ beams.
We investigate the properties of interference fragmentation functions arising from the emission of two leading hadrons inside the same jet for semi-inclusive lepton-nucleon deep-inelastic scattering. Using an extended spectator model we give numerical estimates for the example of the fragmentation into a proton-pion pair with its invariant mass on the Roper resonance.
Using perturbative Quantum Chromodynamics, we compute dihadron fragmentation functions for a large invariant mass of the dihadron pair. The main focus is on the interference fragmentation function $H_1^{\open}$, which plays an important role in spin physics of the nucleon. Our calculation also reveals that $H_1^{\open}$ and the Collins fragmentation function have a closely related underlying dynamics. By considering semi-inclusive deep-inelastic scattering, we further show that collinear factorization in terms of dihadron fragmentation functions, and collinear factorization in terms of single …
We perform a re-analysis of Belle data for the transverse $Λ$ and $\barΛ$ polarization in $e^+ e^-$ annihilation processes within a transverse momentum dependent (TMD) factorization approach. We consider two data sets, one referring to the associated production of $Λ$'s with a light unpolarized hadron in an almost back-to-back configuration, and one for the inclusive $Λ$ production, with the reconstruction of the thrust axis. We adopt the Collins-Soper-Sterman framework and employ the recent formulation on the factorization of single-inclusive hadron production. This extends a previous phenome…
We demonstrate a factorization formula for semi-inclusive deep-inelastic scattering with hadrons in the current fragmentation region detected at low transverse momentum. To facilitate the factorization, we introduce the transverse-momentum dependent parton distributions and fragmentation functions with gauge links slightly off the light-cone, and with soft-gluon radiations subtracted. We verify the factorization to one-loop order in perturbative quantum chromodynamics and argue that it is valid to all orders in perturbation theory.
We apply collinear expansion to inclusive hadron production in $e^+e^-$ annihilation and derive a formalism suitable for systematic study of leading as well as higher twist contributions to fragmentation functions at the tree level. We make the calculations for hadrons with spin-0, spin-1/2 as well as spin-1 and obtain the results in terms of different components of fragmentation functions for the hadronic tensors, the differential cross section as well as hadron polarizations in different cases. The results show a number of interesting features such as the existence of transverse polarization…
We calculate cross section for production of $D$ mesons and $Λ_c$ baryons in proton-proton collisions at the LHC. The cross section for production of $c \bar c$ pairs is calculated within $k_T$-factorization approach with the Kimber-Martin-Ryskin unintegrated gluon distributions obtained on the basis of modern collinear gluon distribution functions. We show that our approach well describes the $D^0$, $D^+$ and $D_s$ experimental data. We try to understand recent ALICE and LHCb data for $Λ_c$ production with the $c \to Λ_c$ independent parton fragmentation approach. The Peterson fragmentation f…
Predictions for light charged hadron production data in the current fragmentation region of deeply inelastic scattering from the H1 and ZEUS experiments are calculated using perturbative Quantum Chromodynamics at next-to-leading order, and using fragmentation functions obtained by fitting to similar data from $e^+ e^-$ reactions. General good agreement is found when the magnitude $Q^2$ of the hard photon's virtuality is sufficiently large. The discrepancy at low $Q$ and small scaled momentum $x_p$ is reduced by incorporating mass effects of the detected hadron. By performing quark tagging, the…
I present a preliminary determination of the Fragmentation Functions (FFs) of unidentified charged hadrons at next-to-leading order in quantum chromodynamics. The analysis is based on hadron production cross section data in single-inclusive electron-positron annihilation. It extends a recent determination of the FFs of identified charged pions, charged kaons and protons/antiprotons performed by the NNPDF Collaboration. I illustrate the quality of the FFs determined in this analysis and show how they describe the charged hadron spectra measured in proton-(anti)proton collisions.
We present new sets of next-to-leading order fragmentation functions describing the production of charged pions, kaons and protons from the gluon and from each of the quarks, obtained by fitting to all relevant data sets from e+ e- annihilation. The individual light quark flavour fragmentation functions are obtained phenomenologically for the first time by including in the data the light quark tagging probabilities obtained by the OPAL Collaboration.
We investigate the role of heavy quarks in the production of light flavored hadrons and in the determination of the corresponding non perturbative hadronization probabilities. We define a general mass variable flavor number scheme for fragmentation functions that accounts for heavy quark mass effects, and perform a global QCD analysis to an up-to-date data set including very precise Belle and BaBar results. We show that the mass dependent picture provides a much more accurate and consistent description of data.
We present a novel study on the inclusive production of a heavy quarkonium ($J/ψ$ or $Υ$), in association with a light-flavored jet, as a test field of the high-energy QCD dynamics. The large transverse momenta at which the two final-state objects are detected permits us to perform an analysis in the spirit of the variable-flavor number scheme (VFNS), in which the cross section for the hadroproduction of a light parton is convoluted with a perturbative fragmentation function that describes the transition from a light quark to a heavy hadron. The quarkonium collinear fragmentation function is b…
We present a model calculation of leading order interference fragmentation functions arising from the distribution of two hadrons produced in the same jet of a fragmenting quark in a hard process. Using a simple spectator model for the mechanism of the hadronization process, for the first time we give a complete calculation and numerical estimates for the example of a proton-pion pair production with its invariant mass on the Roper resonance.
The light top-squark $\sto$ may be the lightest squark and its lifetime may be `long enough' in a kind of SUSY models which have not been ruled out yet experimentally, so colorless `supersymmetric hadrons (superhadrons)' $(\sto \bar{q})$ ($q$ is a quark except $t$-quark) may be formed as long as the light top-squark $\sto$ can be produced. Fragmentation function of $\sto$ to heavy `supersymmetric hadrons (superhadrons)' $(\sto \bar{Q})$ ($\bar{Q}=\bar{c}$ or $\bar{b}$) and the hadronic production of the superhadrons are investigated quantitatively. The fragmentation function is calculated prec…
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 …
At hadron colliders the dominant production mechanism of $(\bar bc)$ mesons with large transverse momentum is due to parton fragmentation. We compute the rates and transverse momentum spectra for production of S-wave and P-wave $(\bar b c)$ mesons at the Tevatron via the direct fragmentation of the bottom antiquark as well as the Altarelli-Parisi induced gluon fragmentation. Since all the radially and orbitally excited $(\bar b c)$ mesons below the $BD$ flavor threshold will cascade into the pseudoscalar ground state $B_c$ through electromagnetic and/or hadronic transitions, they all contribut…
We compute model-independently the production rates and transverse momentum spectra for the $B_c$ mesons in various spin-orbital states ($n\,^1S_0$, $n\,^3S_1$, $n\,^1P_1$, and $n\,^3P_J\,(J=0,1,2)$ ) at hadron colliders via the direct fragmentation of the bottom antiquark and via the Altarelli-Parisi-induced gluon fragmentation. Since all the radially and orbitally excited states below the $BD$ flavor threshold will decay, either electromagnetically, hadronically, or a combination of both, into the pseudoscalar ground state $1\,^1S_0$, they all contribute significantly to the inclusive $B_c$ …
We present the complete structure of the azimuthal dependences and polarization observables for two-hadron production in $e^+e^-$ annihilation processes within a transverse momentum dependent (TMD) approach adopting the helicity formalism. The leading-twist TMD fragmentation functions (TMD-FFs) for spin-1/2 hadrons are fully accounted for. The role of the polarizing FF, together with its extraction from Belle data for the transverse polarization of $Λ$'s, are discussed as well. Finally, predictions for SIDIS processes at EIC are presented.
We present a reliable estimate on the production rate of $B_c$ mesons in $1S$ and $2S$ states in the large transverse momentum region at hadronic colliders using heavy quark fragmentation functions derived within the framework of perturbative QCD. We also present the transverse momentum distribution for the $B_c$ mesons. The production rate is large enough for the $B_c$ mesons to be identified at the Tevatron. At the SSC/LHC the rate is so large that their properties can be studied in details.
We propose the characterization of fragmentation functions by the energy fraction x, a hadron takes away from the energy of the jet measured in the frame co-moving with the jet. Besides, we propose the usage of the jet mass as the fragmentation scale Q. We show that these two Lorentz-invariant variables emerge naturally in a microcanonical ensemble with conserved fourmomentum. Then, we construct a statistical hadronisation model, in which, two features of the hadronic final states in various high-energy reactions (power law spectra and negative-binomial multiplicity distributions) can be conne…
In the fragmentation of a color flux tube in high-energy $pp$ collisions or $e^+$-$e^-$ annihilations, the production of $q$-$\bar q$ pairs along a color flux tube precedes the fragmentation of the tube. The local conservation laws in the production of these $q$-$\bar q$ pairs will lead to the correlations of adjacently produced hadrons. As a consequence, the fragmentation of a flux tube will yield a many-hadron correlation in the form of a chain of hadrons ordered in rapidity, with adjacent hadrons correlated in charges, flavor contents, and azimuthal angles. It will also lead to a two-hadron…
In this paper we prove factorization of fragmentation function in non-equilibrium QCD by using Schwinger-Keldysh closed-time path integral formalism. We use the background field method of QCD in a pure gauge in path integral approach to prove factorization of fragmentation function in non-equilibrium QCD. Our proof is valid in any arbitrary gauge fixing parameter $α$. This may be relevant to study hadron production from quark-gluon plasma at high energy heavy-ion colliders at RHIC and LHC.
We calculate the spin density matrix of the hadron $h$ created via quark fragmentation in the process $e^-e^+ \to q\bar q \to h + X$. In the case of $h=Λ$ the experimental data could possibly elucidate the problem of $s$-quark contribution to the spin of the $Λ$-hyperon, to be compared with the case of the proton (``spin crisis''). Generally we find that for hadrons with spin 1/2 the parton description, using only probabilities, works much better than for vector particles, since in the former case the non diagonal matrix elements of the hadron spin density matrix are suppressed by a small fact…
We show that hard-scattering factorization is violated in the production of high-p_T hadrons in hadron-hadron collisions, in the case that the hadrons are back-to-back, so that k_T factorization is to be used. The explicit counterexample that we construct is for the single-spin asymmetry with one beam transversely polarized. The Sivers function needed here has particular sensitivity to the Wilson lines in the parton densities. We use a greatly simplified model theory to make the breakdown of factorization easy to check explicitly. But the counterexample implies that standard arguments for fact…
By expanding functions of parton fragmentation into a heavy hadron in the inverse of the heavy quark mass $m_Q$ we attempt to factorize them into perturbative- and nonperturbative parts. In our approach the nonperturbative parts can be defined as matrix elements in heavy quark effective theory, the shape of the functions is predicted by perturbative QCD. In this work we neglect effect at order of $m_Q^{-2}$ and calculate the perturbative parts at one-loop level for heavy quark- and gluon fragmentation. We compare our results from leading log approximation with experimental results from $e^+e^-…
We present an analytical next-to-leading order QCD calculation of the partonic cross sections for the process $pp\rightarrow ({\text{jet}} \,h)X$, for which a specific hadron is observed inside a fully reconstructed jet. In order to obtain the analytical results, we assume the jet to be relatively narrow. We show that the results can be cast into a simple and systematic form based on suitable universal jet functions for the process. We confirm the validity of our calculation by comparing to previous results in the literature for which the next-to-leading order cross section was treated entirel…
We present the systematic results for three dimensional fragmentation functions defined via the quark-quark correlator for hadrons with spin 0, 1/2 and 1 respectively. These results are presented in terms of a spin independent part, a vector polarization dependent part and a tensor polarization dependent part. For spin 0 hadrons, only the spin independent part is needed, for spin 1/2 hadron, the polarization independent and vector polarization dependent parts are present, while for spin 1 hadrons, all the three parts exist. We discuss the general properties of these fragmentation functions and…
Proof of transverse-momentum-dependent(TMD) factorization for hadron-hadron collision is given in this paper. We focus on processes without detected soft final hadrons or detected final hadrons that are collinear to initial hadrons. This contradicts the widely accepted viewpoint that TMD- factorization does not hold in such processes even in the generalized sense. The key point is that singular points of the type l+ = 0 can be absorbed into Wilson lines of soft gluons, where l is collinear to the plus direction. Thus one should subtract such singular points from the collinear region of l. Afte…
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.
We review the description of inclusive single unpolarized light hadron production using fragmentation functions in the framework of the factorization theorem. We summarize the factorization of quantities into perturbatively calculable quantities and these universal fragmentation functions, and then discuss some improvements beyond the standard fixed order approach. We discuss the extraction of fragmentation functions for light charged and neutral hadrons using these theoretical tools through global fits to experimental data from reactions at various colliders, in particular from accurate e+e- …
We study factorization of processes involving two fragmentation functions in the case of very small transverse momenta. We consider two-hadron production in inclusive $e^+ e^-$ annihilation and demonstrate a new simple and illustrative method of factorization for such processes including leading order $α_S$ corrections.
We investigate the properties of interference fragmentation functions arising from the emission of two leading hadrons inside the same jet for inclusive lepton-nucleon deep-inelastic scattering. Using an extended spectator model for the mechanism of the hadronization, we give a complete calculation and numerical estimates for the examples of a proton-pion pair produced with invariant mass on the Roper resonance, and of two pions produced with invariant mass close to the $ρ$ mass. We discuss azimuthal angular dependence of the leading order cross section to point up favourable conditions for ex…
We present nonperturbative fragmentation functions (FFs) for bottom-flavored ($B$) hadrons both at next-to-leading (NLO) and, for the first time, at next-to-next-to-leading order (NNLO) in the $\overline{\mathrm{MS}}$ factorization scheme with five massless quark flavors. They are determined by fitting all available experimental data of inclusive single $B$-hadron production in $e^+e^-$ annihilation, from the ALEPH, DELPHI, and OPAL Collaborations at CERN LEP1 and the SLD Collaboration at SLAC SLC. The uncertainties in these FFs as well as in the corresponding observables are estimated using t…
The fragmentation fractions of c and b quarks into the weakly decaying charmed hadrons D0, D+, D_s+ and Lambda_c+, and into the charmed vector meson D*+ have been derived from the LEP measurements and averaged. The c quark fragmentation fractions represent probabilities to hadronise as a given charmed hadron, while the b quark fragmentation fractions are defined as sums of probabilities to hadronise as a particular charmed hadron or its antiparticle.
Making use of the collinear expansion technique developed by Ellis, Furmanski and Petrozio, and the special propagator concept invented by Qiu, we present a factorization approach to the photon fragmentation tensor for the inclusive Lambda hyperon production. As a result, the structure function $\tilde F$, which is related to the transverse polarization of the inclusive Lambda hyperon in unpolarized electron-positron annihilation, is expressed as a combination of four parton fragmentation matrix elements. Since the inclusive $Λ$ production we considered is theoretically the simplest, our study…
The forward-backward asymmetry is an important measurable quantity which can enable independent determination of the neutral-current couplings of fermions. In this paper, we extend the definition of the asymmetry from the partonic level to the hadronic level by calculating this asymmetry of the hadron pair in the semi-inclusive electron position annihilation process. Semi-inclusive implies that a back-to-back jet is also measured in addition to the hadron pair. Due to the limitation of the factorization theorem, we calculate this process up to leading order twist-4 level by applying the collin…
