Collins Fragmentation Function Research Articles

Collins and sivers effects in p↑p → jet πX: Universality and process dependence

In this paper we briefly review the transverse momentum dependent generalized parton model and its application to the study of azimuthal asymmetries in the distribution of leading hadrons (mainly pions) inside large transverse momentum jets inclusively produced in polarized proton-proton collisions. We put particular emphasis on the phenomenological interest of these observables, in combination with similar asymmetries measured in semi-inclusive deeply inelastic scattering, Drell-Yan processes and e+e- collisions, for the study of the universality properties of the transverse momentum dependent parton distribution and fragmentation functions. We present results for RHIC kinematics at center-of-mass energies sqrt{s} = 200 and 500 GeV, for central and mainly forward jet rapidities, in particular for the Sivers distribution and the Collins fragmentation function, that are believed to be responsible of many of the largest asymmetries measured in the last years. We also briefly discuss the case of inclusive jet production and recent phenomenological applications of other theoretical approaches, like the colour gauge invariant generalized parton model and the collinear twist-three approach, aiming at clarifying the issues of the universality and process dependence of transverse momentum dependent functions.

Single spin asymmetries in ℓp↑→hX processes and transverse momentum dependent factorization

Some estimates for the transverse single spin asymmetry, ${A}_{N}$, in the inclusive processes $\ensuremath{\ell}{p}^{\ensuremath{\uparrow}}\ensuremath{\rightarrow}hX$, given in a previous paper, are expanded and compared with new experimental data. The predictions are based on the Sivers distributions and the Collins fragmentation functions which fit the azimuthal asymmetries measured in semi-inclusive deep inelastic scattering (SIDIS) processes ($\ensuremath{\ell}{p}^{\ensuremath{\uparrow}}\ensuremath{\rightarrow}{\ensuremath{\ell}}^{\ensuremath{'}}hX$). The factorization in terms of transverse momentum dependent distribution and fragmentation functions (TMD factorization)---i.e. the theoretical framework in which SIDIS azimuthal asymmetries are analyzed---is assumed to hold also for the inclusive process $\ensuremath{\ell}p\ensuremath{\rightarrow}hX$ at large ${P}_{T}$. The values of ${A}_{N}$ thus obtained agree in sign and shape with the data. Some predictions are given for future experiments.

Transverse spin and transverse momentum structure of the nucleon from the COMPASS experiment

COMPASS is a fixed target experiment at the CERN SPS M2 beam line, taking data since year 2002. Part of the physics programme is dedicated to the study of the transverse spin and transverse momentum structure of the nucleon, by means of semi-inclusive deeply inelastic lepton scattering (SIDIS). For these measurements, data have been collected using transversely polarised proton (NH 3 ) and deuteron ( 6 LiD) targets. A selection of the results on the azimuthal asymmetries measured is presented, with particular focus on the most recent measurements from the data collected in 2007 and 2010 with the proton target. These results exhibit clear signals for the Collins asymmetry, interpreted as a convolution of a non-zero transversity PDF and Collins fragmentation function, and for the Sivers asymmetry which is related to the Sivers function, the most famous and discussed of the TMD PDFs. The data collected with the 6 LiD target, have also been recently analysed to search for the azimuthal modulations in the production of hadrons which are expected to be present in the unpolarised part of the SIDIS cross-section. The azimuthal hadron asymmetries, which are related to the Boer-Mulders TMD PDF, show strong and somewhat puzzling kinematical dependencies.

FRAGMENTATION FUNCTIONS AT BABAR

Inclusive hadron production cross section in e+e- annihilation shed light on fundamental questions of hadronization and fragmentation processes. We present measurements of inclusive spectra of charged pions, kaons, and protons (antiprotons) produced in e+e- collisions at the center-of-mass energy of 10.54 GeV, and tests of QCD predictions and hadronization models. We also report the results on the measurement of the azimuthal modulation induced by the Collins effect in inclusive production of charged pion pairs e+e- → ππX, where the two pions are produced in opposite hemispheres. These data allows the determination of the polarized Collins fragmentation functions.

Effects of Quark Spin Flip on the Collins Fragmentation Function in a Toy Model

The recent extension of the NJL-jet model to hadronization of transversely polarized quarks allowed the study of the Collins fragmentation function. Both favored and unfavored Collins fragmentation functions were generated, the latter purely by multiple hadron emissions, with 1/2 moments of opposite sign in the region of the light-cone momentum fraction z accessible in current experiments. Hints of such behavior has been seen in the measurements in several experiments. Also, in the transverse momentum dependent (TMD) hadron emission probabilities, modulations of up to fourth order in sine of the polar angle were observed, while the Collins effect describes just the linear modulations. A crucial part of the extended model was the calculation of the quark spin flip probability after each hadron emission in the jet. Here we study the effects of this probability on the resulting unfavored and favored Collins functions by setting it as a constant and use a toy model for the elementary single hadron emission probabilities. The results of the Monte Carlo simulations showed that preferential quark spin flip in the elementary hadron emission is needed to generate the favored and unfavored Collins functions with opposite sign 1/2 moments. For the TMD hadron emission modulations, we showed that the model quark spin flip probabilities are a partial source of the higher rode modulations, while the other source is the Collins modulation of the remnant quark from the hadron emission recoil.

Collins fragmentation function within the NJL-jet model

The NJL-jet model is extended to accommodate hadronization of a transversely polarized quark in order to explore the Collins effect within a multihadron emission framework. This is accomplished by calculating the polarized quark spin flip probabilities after a pseudoscalar hadron emission and the elementary Collins functions. The model is used to calculate the number densities of the hadrons produced in the polarized quark's decay chain. The full Collins fragmentation function is extracted from the sine modulation of the polarized number densities with respect to the polar angle between the initial quark's spin and hadron's transverse momentum. Two cases are studied here. First, a toy model for elementary Collins function is used to study the features of the transversely polarized quark-jet model. Second, a full model calculation of transverse momentum dependent pion and kaon Collins functions is presented. The remarkable feature of our model is that the 1/2 moments of the favored Collins fragmentation functions are positive and peak at large values of z but decrease and oscillate at small values of z. The 1/2 moments of the unfavored Collins functions have comparable magnitude and opposite sign to the favored functions, vanish at large z and peak at small values of z. This feature is observed for both the toy and full models and can be attributed to the quark-jet picture of hadronization. Moreover, the transverse momentum dependencies of the model Collins functions differ significantly from the Gaussian form widely used in the empirical parametrizations. Finally, a naive interpretation of the Schafer-Teryaev sum rule is proven not to hold in our model, where the transverse momentum conservation is explicitly enforced. This is attributed to the sizable average transverse momentum of the remnant quark that needs to be accounted for to satisfy the transverse momentum sum-rule.

Transverse target spin asymmetries at COMPASS

COMPASS is a fixed target experiment at the CERN M2 beamline where the nucleon spin structure is investigated using a 160 GeV/c polarized μ+ beam and polarized solid state targets. After taking data in the years 2002–2004 using a transversely polarized 6LiD (deuteron) target, in 2007 and 2010 additional data were collected on a transversely polarized NH3 (proton) target. The measurements of single spin asymmetries in semi-inclusive deep inelastic scattering (SIDIS) on a transversely polarized target are an important part of the COMPASS physics program. They allow to investigate the transversity distribution functions, e.g. coupled to the Collins fragmentation function, as well as transverse momentum dependent distribution functions, like the Sivers distribution function, by measuring azimuthal asymmetries in hadron production.

Single spin asymmetry scaling in the forward rapidity region at RHIC

We investigate the scaling properties in inclusive hadron production and the associated single transverse spin asymmetry in the forward rapidity region at RHIC. We find that the spin-averaged experimental data in both $pp$ and $dAu$ collisions demonstrates a transverse-momentum-dependent geometric scaling. We introduce the transverse momentum dependent Collins fragmentation function to study the scaling of the single transverse spin asymmetries. The general feature of the scaling analysis is consistent with the experimental observations, in particular, for the transverse momentum dependence of the spin asymmetries at RHIC energy. We further propose to probe the saturation scale of nucleus by measuring the spin asymmetry normalized by that in $pp$ scattering at low transverse momentum.

Constraining Quark Transversity through Collins Asymmetry Measurements in Mid-Rapidity Jets in p↑p Collisions at STAR

Quantitative insights have been obtained from past measurements of the Collins asymmetry in e+ e− collisions and deep-inelastic lepton-nucleon scattering. Further constraints can be added from measurements of the azimuthal asymmetry of leading charged pions in jets produced by transversely polarized proton collisions. This asymmetry can be expressed as a convolution of the quark transversity (δq(xQ2)), the Collins fragmentation function (ΔN D(z)) and a hard-scattering spin-transfer coefficient. Previous measurements constrain ΔN D(z), potentially allowing an independent extraction of δq(x,Q2). We present progress toward asymmetry measurements from √s = 200 GeV transversely polarized (∼58%) proton collision data collected at the Solenoidal Tracker at RHIC (STAR), with full azimuthal coverage at mid-rapidity (|η| < 1). A vailable data give a statistical precision of σ ∼ 0.01, for average quark momentum fraction ⟨x⟩ ∼ 0.2, at each of four intervals of the hadron-jet momentum fraction z over 0.1 < z < 0.6.

Azimuthal asymmetries for hadron distributions inside a jet in hadronic collisions

Using a generalized parton model approach including spin and intrinsic parton motion effects, and assuming the validity of factorization for large pT jet production in hadronic collisions, we study the azimuthal distribution around the jet axis of leading pions, produced in the jet fragmentation process. We identify the observable leading-twist azimuthal asymmetries for the unpolarized and single-polarized case related to both quark and gluon-originated jets. We account for all physically allowed combinations of the transverse momentum dependent (TMD) parton distribution and fragmentation functions, with special attention to the Sivers, Boer-Mulders, and transversity quark distributions, and to the Collins fragmentation function for quarks (and to the analogous functions for gluon partons).

Dihadron Fragmentation Functions for Large Invariant Mass

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)(∢), which plays an important role in spin physics of the nucleon. Our calculation also reveals that H(1)(∢) and the Collins fragmentation function have 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-hadron fragmentation functions provide the same result in the region of intermediate invariant mass.

Azimuthal asymmetries for hadron distributions inside a jet in hadronic collisions

Using a generalized parton model approach including spin and intrinsic parton motion effects, and assuming the validity of factorization for large-pT jet production in hadronic collisions, we study the azimuthal distribution around the jet axis of leading unpolarized or (pseudo)scalar hadrons, namely pions, produced in the jet fragmentation process. We identify the observable leading-twist azimuthal asymmetries for the unpolarized and single-polarized case related to quark and gluon-originated jets. We account for all physically allowed combinations of the transverse momentum dependent (TMD) parton distribution and fragmentation functions, with special attention to the Sivers, Boer-Mulders, and transversity quark distributions, and to the Collins fragmentation function for quarks (and to the analogous functions for gluons). For each of these effects we evaluate, at central and forward rapidities and for kinematical configurations accessible at BNL-RHIC, the corresponding potentially maximized asymmetry (for pi+ production), obtained by saturating natural positivity bounds (and the Soffer bound for transversity) for the distribution and fragmentation functions involved and summing additively all partonic contributions. We then estimate, for both neutral and charged pions, the asymmetries involving TMD functions for which parameterizations are available. We also study the role of the different mechanisms, and the corresponding transverse single spin asymmetries, for large-pT inclusive jet production.

Transverse spin structure of the nucleon through target single-spin asymmetry in semi-inclusive deep-inelastic (e, e’ $ \pi^{\pm}_{}$ ) reaction at Jefferson Lab

Jefferson Lab (JLab) 12 GeV energy upgrade provides a golden opportunity to perform precision studies of the transverse spin and transverse-momentum-dependent structure in the valence quark region for both the proton and the neutron. In this paper, we focus our discussion on a recently approved experiment on the neutron as an example of the precision studies planned at JLab. The new experiment will perform precision measurements of target Single Spin Asymmetries (SSA) from semi-inclusive electro-production of charged pions from a 40-cm long transversely polarized $^3$He target in Deep-Inelastic-Scattering kinematics using 11 and 8.8 GeV electron beams. This new coincidence experiment in Hall A will employ a newly proposed solenoid spectrometer (SoLID). The large acceptance spectrometer and the high polarized luminosity will provide precise 4-D ($x$, $z$, $P_T$ and $Q^2$) data on the Collins, Sivers, and pretzelocity asymmetries for the neutron through the azimuthal angular dependence. The full 2$\pi$ azimuthal angular coverage in the lab is essential in controlling the systematic uncertainties. The results from this experiment, when combined with the proton Collins asymmetry measurement and the Collins fragmentation function determined from the e$^+$e$^-$ collision data, will allow for a quark flavor separation in order to achieve a determination of the tensor charge of the d quark to a 10% accuracy. The extracted Sivers and pretzelocity asymmetries will provide important information to understand the correlations between the quark orbital angular momentum and the nucleon spin and between the quark spin and nucleon spin.

Transverse Spin Physics at COMPASS

The investigation of transverse spin and transverse momentum effects in deep inelastic scattering is one of the key physics programs of the COMPASS collaboration. Three channels have been analyzed at COMPASS to access the transversity distribution function: The azimuthal distribution of single hadrons, involving the Collins fragmentation function, the azimuthal dependence of the plane containing hadron pairs, involving the two-hadron interference fragmentation function, and the measurement of the transverse polarization of Lambda hyperons in the final state. Azimuthal asymmetries in unpolarized semi-inclusive deep-inelastic scattering give important information on the inner structure of the nucleon as well, and can be used to estimate both the quark transverse momentum k_T in an unpolarized nucleon and to access the so-far unmeasured Boer-Mulders function. COMPASS has measured these asymmetries using spin-averaged 6LiD data.

Effects of transversity in deep-inelastic scattering by polarized protons

Single-spin asymmetries for pions and charged kaons are measured in semi-inclusive deep-inelastic scattering of positrons and electrons off a transversely nuclear-polarized hydrogen target. The dependence of the cross section on the azimuthal angles of the target polarization (ϕS) and the produced hadron (ϕ) is found to have a substantial sin(ϕ+ϕS) modulation for the production of π+, π− and K+. This Fourier component can be interpreted in terms of non-zero transversity distribution functions and non-zero favored and disfavored Collins fragmentation functions with opposite sign. For π0 and K− production the amplitude of this Fourier component is consistent with zero.

Dihadron azimuthal correlation from the Collins effect in unpolarized hadron collisions

We study the dihadron azimuthal correlation produced nearly back-to-back in unpolarized hadron collisions, arising from the product of two Collins fragmentation functions. Using the latest Collins fragmentation functions extracted from the global analysis of available experimental data, we make predictions for the azimuthal correlation of two-pion production in $pp$ collisions at Relativistic Heavy Ion Collider energies. We find that the correlation is sizable in the midrapidity region for moderate jet transverse momentum.

Single spin asymmetries inlp→hXprocesses: A test of factorization

Predictions for the transverse single spin asymmetry (SSA), A_N, are given for the inclusive processes l p(transv. pol.) --> h X and l p(transv. pol.) --> jet + X, which could be measured in operating or future experiments. These estimates are based on the Sivers distributions and the Collins fragmentation functions which fit the azimuthal asymmetries measured in semi-inclusive deep inelastic scattering (SIDIS) processes (l p(transv. pol.) --> l' h X). The factorization in terms of transverse momentum dependent distribution and fragmentation functions (TMD factorization) - which supplies the theoretical framework in which SIDIS azimuthal asymmetries are analyzed - is assumed to hold also for the l p --> h X inclusive process at large P_T. A measurement of A_N would then provide a direct test of the validity of the TMD factorization in this case and would have important consequences for the study and understanding of SSAs in p p(transv. pol.) --> h X processes.

Transverse spin effects at COMPASS

The investigation of transverse spin and transverse momentum effects in deep inelastic scattering is one of the key physics programs of the COMPASS collaboration. In the years 2002–2004 COMPASS took data scattering 160 GeV muons on a transversely polarized 6 LiD target. In 2007, a transversely polarized NH 3 target was used. Three different channels to access the transversity distribution function have been analyzed: The azimuthal distribution of single hadrons, involving the Collins fragmentation function, the azimuthal dependence of the plane containing hadron pairs, involving the two-hadron interference fragmentation function, and the measurement of the transverse polarization of λ hyperons in the final state. Transverse quark momentum effects in a transversely polarized nucleon have been investigated by measuring the Sivers distribution function. Azimuthal asymmetries in unpolarized semi-inclusive deep-inelastic scattering give important information on the inner structure of the nucleon as well, and can be used to estimate both the quark transverse momentum k T in an unpolarized nucleon and to access the so-far unmeasured Boer-Mulders function. COMPASS has measured these asymmetries using spin-averaged 6 LiD data.

Update on transversity and Collins functions from SIDIS and [formula omitted] data

We present an update of a previous global analysis of the experimental data on azimuthal asymmetries in semi-inclusive deep inelastic scattering (SIDIS), from the HERMES and COMPASS Collaborations, and in e+ e- -> h1 h2 X processes, from the Belle Collaboration. Compared to the first extraction, a more precise determination of the Collins fragmentation function and the transversity distribution function for u and d quarks is obtained.

Latest HERMES results on transverse spin effects in hadron structure and formation⋆

Transverse target single-spin asymmetries in semi-inclusive deep inelastic scattering allow to study the so-called Collins and Sivers mechanisms. The first one connects the poorly known fundamental transversity distribution function, describing the transverse spin-polarization of quarks in a transversely polarized proton, to the Collins fragmentation function, describing spin-orbit correlations in the hadron formation process. The second one is sensitive to the Sivers function, which correlates the intrinsic transverse momentum of quarks with the proton’s spin orientation and is related to the orbital angular momentum of quarks. Preliminary results on azimuthal single target-spin asymmetries in semi-inclusive electro-production of pions and kaons at the HERMES experiment are presented. The full data set collected with a transversely polarized hydrogen target was analyzed providing the HERMES most precise results on the Collins and Sivers azimuthal moments.