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Transversity and Collins functions from SIDIS and e+e- data
M. Anselmino, M. Boglione, U. D'Alesio, A. Kotzinian, F. Murgia, A. Prokudin, C. Turk
TL;DR
The paper addresses the lack of direct transversity measurements by globally analyzing SIDIS and e+e− azimuthal-asymmetry data. It simultaneously extracts Collins fragmentation and u- and d-quark transversity functions, finding opposite signs and magnitudes below their bounds, then predicts SIDIS asymmetries for forthcoming experiments.
Problem
Transversity is important for polarized-nucleon structure but remains unmeasured in inclusive DIS because its chiral-odd nature requires a quark helicity flip.
Method
The analysis globally fits HERMES and COMPASS SIDIS data together with Belle e+e−→h1h2 X data using a simultaneous parameterization of transversity and Collins functions.
Results
The fit extracts Collins functions and, for the first time, u- and d-quark transversity distributions, which have opposite signs and are smaller than their Soffer bounds.
Takeaways & Limitations
The extracted functions allow predictions for the SIDIS asymmetry A_UT^{sin(φS+φh)} in forthcoming COMPASS and JLab measurements.
Abstract
from arXiv · showhide
A 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, is performed. It results in the extraction of the Collins fragmentation function and, for the first time, of the transversity distribution function for u and d quarks. These turn out to have opposite signs and to be sizably smaller than their positivity bounds. Predictions for the azimuthal asymmetry A_{UT}^{sin(phi_h + phi_S)}, as will soon be measured at JLab and COMPASS operating on a transversely polarized proton target, are then presented.
I. INTRODUCTION
Transversity is essential for understanding polarized-nucleon structure but remains unmeasured in inclusive DIS because its chiral-odd nature requires a quark helicity flip. Combining SIDIS and Belle measurements enables a simultaneous extraction of transversity and Collins functions.
- Transversity describes transversely polarized quarks inside a transversely polarized nucleon and is necessary for a complete collinear description of polarized-nucleon structure.
- Inclusive DIS cannot measure transversity because helicity conservation in perturbative QED and QCD prevents the required quark helicity flip.
- The accessible SIDIS channel measures an azimuthal asymmetry involving the convolution of transversity with the Collins fragmentation function.
- Belle independently measured the Collins-function convolution in unpolarized e+e−→h1h2 X processes, supplying crucial information for the extraction strategy.
- A global fit combines HERMES and COMPASS SIDIS data with Belle data to extract u- and d-quark transversity distributions and Collins fragmentation functions.
II. TRANSVERSITY AND COLLINS FUNCTIONS FROM SIDIS PROCESSES
The SIDIS analysis formulates the transverse single-spin asymmetry using intrinsic transverse momenta and a Collins-weighted angular modulation. Gaussian transverse-momentum models and a bounded parameterization enable analytical integration and a simultaneous fit to experimental distributions.
- Kinematics and asymmetry: SIDIS kinematics are treated in the γ*−p center-of-mass frame with intrinsic quark and hadron transverse momenta retained at O(k⊥/Q).The analysis assumes PT ≃ ΛQCD ≃ k⊥ and neglects second-order corrections in k⊥/Q.
- Kinematics and asymmetry: The observed hadron transverse momentum satisfies p⊥ = PT − zk⊥, while x and z coincide with measurable SIDIS variables xB and zh.The soft transverse momentum is treated as mainly originating from intrinsic motion in the stated kinematic region.
- Kinematics and asymmetry: The transverse single-spin asymmetry compares cross sections for opposite proton transverse-spin directions and can be integrated over variables according to experimental coverage.
- Kinematics and asymmetry: The sin(φS + φh) weighted asymmetry isolates the spin-dependent fragmentation contribution of a transversely polarized quark and is measured by HERMES and COMPASS.
- Parameterization: Unpolarized TMD distribution and fragmentation functions are modeled with factorized Gaussian transverse-momentum dependence for small PT and analytical integration.
- Parameterization: The unknown transversity and Collins functions are parameterized with flavor-dependent normalization coefficients, while selected exponents and the dimensional parameter M are flavor independent.
- Parameterization: The parameterization enforces the Soffer bound for transversity and the positivity bound for the Collins function through normalized factors.
- Fit observables: The resulting asymmetry is expressed through the fit parameters and integrated over selected variables to obtain PT, x, and z distributions.
III. COLLINS FUNCTIONS FROM e+e− PROCESSES
The e+e−→h1h2X process provides access to Collins fragmentation through two-hadron azimuthal correlations, using either a reconstructed thrust axis or the observed h2 direction as reference. The resulting formalism relates measured cross sections and asymmetries to transverse-momentum-dependent fragmentation functions under stated kinematic approximations.
- Kinematics: The two detected hadrons are fragmentation products of a quark and antiquark produced in e+e− annihilation.Their lightcone fractions are z1 and z2, with intrinsic transverse momenta p⊥1 and p⊥2 relative to the fragmenting-quark directions.
- Kinematics: The thrust-axis frame places the back-to-back quark and antiquark along the z-axis and the e+e−→q q̄ scattering plane in the xz plane.Experimentally, this requires reconstruction of the jet thrust axis.
- Cross section and Collins functions: The unpolarized cross section is built from charge-weighted products of transverse-momentum-dependent fragmentation functions for h1 and h2.The helicity formalism identifies diagonal fragmentation-matrix elements with unpolarized fragmentation functions and non-diagonal elements with Collins functions.
- Cross section and Collins functions: Correlated two-hadron production in unpolarized e+e− collisions directly accesses the Collins functions and their z and p⊥ dependences.Integrating over intrinsic transverse momenta recovers the usual unpolarized cross section.
- Belle observables: The Belle observable is obtained by changing variables to (φ1, φ1+φ2), integrating selected intrinsic-momentum variables and φ1, and normalizing to the azimuthally averaged cross section.Belle data are integrated over the detector’s covered θ range, producing averaged angular factors, and unlike-sign to like-sign pion ratios are used to eliminate false asymmetries.
- Alternative reference frame: A second frame uses h2 to define the z-axis and the lepton-h2 plane as the xz plane, avoiding reconstruction of the quark direction.This choice has more complicated kinematics and is treated through an expansion in p⊥/(z√s), with second-order corrections neglected at large energies and not-too-small z.
IV. TRANSVERSITY AND COLLINS FUNCTIONS FROM A GLOBAL FIT
A global best fit combines HERMES, COMPASS, and Belle asymmetry data to extract transversity distributions and Collins fragmentation functions. The extracted u and d transversity distributions have opposite signs and remain below their Soffer bounds, while the fits describe the measured asymmetries with quantified parameter uncertainties.
- Global fit: The global fit simultaneously uses HERMES and COMPASS SIDIS asymmetries and Belle A12 or A0 data to determine transversity and Collins-function parameters.The two Belle data sets are fitted separately because they are strongly correlated, producing parameter sets for the A12 and A0 asymmetries.
- Global fit: The fit contains 9 parameters, with flavor-independent α and β and common γ and δ assumptions used to limit parameter proliferation.The parameter values and MINUIT errors are reported in Tables I and II, although the errors are strongly correlated.
- Fit comparison: The fitted curves describe HERMES, COMPASS, and Belle measurements, while shaded regions represent theoretical uncertainty from parameter errors and correlations.For Belle, agreement between fits to the two azimuthal correlations supports consistency between the two data sets.
- Transversity extraction: The extracted u and d transversity distributions are opposite in sign, with |∆T d(x)| smaller than |∆T u(x)| and both below their Soffer bounds.The distributions are plotted against x and k⊥, with the Soffer bound shown for comparison.
- Collins functions: The extracted Collins functions agree with earlier extractions and are well constrained at large valence z, remaining much smaller than their positivity bounds.The favored and unfavored functions are shown as functions of z and p⊥ together with previous results and the positivity bound.
V. PREDICTIONS FOR ONGOING AND FUTURE EXPERIMENTS
The extracted transversity and Collins functions are used to predict azimuthal asymmetries for forthcoming COMPASS and JLab measurements. The predictions are generally stable in transverse-momentum and fragmentation-fraction dependences but can vary substantially at moderately large x.
- The fitted transversity distributions and Collins functions provide predictions for new COMPASS and JLab measurements.
- COMPASS predictions: The predicted COMPASS hydrogen-target asymmetry is sizeable, reaching up to 5%.
- JLab predictions: JLab measurements use polarized proton and neutron targets at beam energies of 6 or 12 GeV, probing transversity up to x approximately 0.6.
- Prediction sensitivity: Predictions for the x dependence at JLab may vary drastically for 0.4 ≤ xB ≤ 0.6 because existing HERMES and COMPASS data sparsely constrain moderately large x.
- Prediction sensitivity: The predicted PT and zh dependences are more stable because they depend on the transversity distribution integrated over x.
- Kaon predictions: For kaon production, the calculation agrees fairly with K+ data but shows discrepancies for K−, where strange-quark transversity may be relevant.
VI. COMMENTS AND CONCLUSIONS
The combined fit describes the available asymmetry data and extracts Collins functions together with previously unmeasured u- and d-quark transversity distributions. These results support predictions for future SIDIS tests and indicate that kaon data can probe sea-quark transversity.
- The analysis fits HERMES, COMPASS, and Belle asymmetry data using nine parameters for u- and d-quark transversity and favored and unfavored Collins functions.
- All data are accurately described, yielding favored and unfavored Collins functions consistent with earlier results.
- The extracted u- and d-quark transversity distributions have opposite signs, with |∆T d(x)| smaller than |∆T u(x)| and both below their Soffer bounds.
- Knowledge of transversity and Collins functions enables azimuthal-asymmetry calculations for any SIDIS process and predictions for incoming COMPASS and JLab measurements.
- Kaon-production data may help disentangle the role of sea-quark transversity, especially given discrepancies in the K− asymmetry.
- Future Belle data can further constrain the z and p⊥ dependences of the Collins functions.