NARODOWE CENTRUM BADA! J" DROWYCH National Centre for Nuclear Research ! MONOGRAFIE NCBJ Understanding the Nucleon's Spin Structure The Direct Gluon Polarisation Measurement at the COMPASS Experiment Krzysztof Kurek National Centre for Nuclear Research Theoretical Physics Department D. Sc. Thesis Swierk,´ November 2011 Abstract A modern understanding of the nucleon's spin structure is presented. Polarisation depen- dent deep inelastic experiments have shown that the helicity contribution of quarks to the nucleon spin is much smaller than expected from the simple quark model. This observation and the role of the triangle anomaly lead to the conclusion that gluons - if strongly polarised, might solve the spin problem of the nucleon. Therefore a direct gluon polarisation measure- ment is the one of main goals of the COMPASS experiment spin physics programme. Gluon polarisation is determined from the longitudinal double spin asymmetry in the scattering of 160 GeV polarised muons off a polarised 6LiD and a polarised proton target. The gluons are accessible by the selection of photon-gluon fusion events. The well-known processes to tag the photon-gluon fusion events in the energy range covered by the COMPASS experiment are: the production of open-charm D0 mesons and of light hadron pairs with large transverse momenta. A special emphasis is put on a Neural Network approach that has been widely used in both gluon polarisation determination methods. Thanks to a weighting method and the Neural Network used in the signal selection, as well as in the parameterization of an analyzing powers, the statistical precision of the results is significantly increased. The appli- cation of Next-to-Leading Order Quantum Chromodynamics corrections to the polarisation dependent and polarisation averaged partonic cross sections for open-charm production is discussed. These higher order contributions are non-negligible in the COMPASS kinematical domain. New results for the gluon polarisation, including these contributions, are obtained. The results strongly support the hypothesis, that the gluons inside the nucleon are weakly or not polarised. Therefore, the possible importance of quark and gluon angular momenta are also discussed. Combining new results obtained in a Lattice Quantum Chromodynamics approach, the possible scenario of the nucleon spin decomposition as a sum of the valence quarks helicity and the orbital angular momentum of the gluons is discussed. The new con- cept that the presence of the angular momentum of quarks inside nucleon is related to the spatial deformation of the quark densities in the transverse plane is also reviewed. Table of contents 1 Spin degrees of freedom 1 2 Longitudinal spin structure of the nucleon 7 2.1 Measurement of the inclusive asymmetry A1 and structure function g1 . 7 2.1.1 Spin-dependent DIS cross section . 7 2.1.2 Longitudinal double-spin asymmetry . 15 2.1.3 The polarisation dependent structure function g1 . 17 2.2 Decomposition of the nucleon spin . 25 2.2.1 The Bjorken and Ellis-Jaffie sum rules . 25 2.2.2 The triangle anomaly and the role of gluons . 30 2.3 The Gerasimov-Drell-Hearn sum rule . 34 2.4 Semi-inclusive asymmetries and the flavour separation . 35 2.5 Summary of the nucleon's longitudinal spin structure . 46 3 Transverse spin structure of the nucleon 48 3.1 Transversity PDF, Collins and Sivers asymmetries . 48 3.2 Catalogue of twist-2 parton distribution functions . 59 3.3 Summary of the TMDs and transverse spin structure of the nucleon . 66 4 COMPASS experimental set-up 68 5 Open-charm analysis with an Artificial Neural Network 73 5.1 Spin cross section asymmetry for D meson production . 75 5.2 Data selection . 78 ii TABLE OF CONTENTS 5.3 The weighted analysis . 85 5.3.1 Asymmetry determination . 85 5.3.2 Parameterization of the analysing power . 89 5.3.3 The signal purity . 91 5.4 Results . 96 5.4.1 Leading Order results for the gluon polarisation . 97 5.4.2 Open-charm asymmetry in kinematic bins . 101 5.4.3 NLO QCD corrections for spin-dependent charm muoproduction . 107 5.5 The D0 and D¯0 meson production asymmetry . 115 5.6 Summary of the gluon polarisation measurement from open-charm D meson production . 118 6 Determination of the gluon polarization from DIS events with high-pT hadron pairs 120 6.1 Spin cross section asymmetry for high-pT hadron pairs . 122 6.2 Data sample and event selection . 126 6.3 The analysis method . 127 6.4 Monte-Carlo simulations and ANN training . 128 6.5 Systematic studies . 131 6.6 The gluon polarisation results . 134 6.7 Summary of the gluon polarisation measurement . 136 7 The spin structure of the nucleon; GPDs and orbital motion 138 7.1 Transverse distortion, QCD lensing and the Sivers effect . 141 7.2 Controversy on the definition of orbital and total angular momentum . 144 7.3 Lattice QCD results . 146 8 Summary and outlook 151 9 Appendices 155 9.1 The helicity dependent DIS cross sections . 155 9.2 A Neural Network approach . 156 TABLE OF CONTENTS iii 2 9.3 NLO QCD fits of g1(x; Q ) updated with the NLO open-charm result on ∆G=G ....................................... 158 9.4 Angular momentum operator in QED . 159 Bibliography 161 Acknowledgements 1 Chapter 1 Spin degrees of freedom Spin is a fundamental degree of freedom originating from a space-time symmetry. It plays a critical role in determining the basic structure of fundamental interactions. Spin is a relativistic, quantum object. Effects related to spin survive also in the high-energy limit. Spin also provides an opportunity to probe the inner structure of composite systems such as the nucleon. After more than 25 years of measurements of the spin-dependent structure functions of the nucleon the third generation of polarised experiments is now running and delivering more precise data. Although our knowledge about the spin decomposition in the frame of quark parton model (QPM) and Quantum Chromodynamics (QCD) is now more complete and the polarisation dependent parton distribution functions (polarised PDFs) are better constrained by data, the driving question for QCD spin physics still has no answer: where does the nucleon spin come from? Pioneering experiments on the spin structure were performed in the seventies at SLAC [1]. The famous EMC double spin asymmetry measurement [2] and the naive interpretation of the results based on the Ellis-Jaffe sum rule [3] have introduced the so-called "spin crisis" to Particle Physics : quark spins carry only a small fraction of the nucleon helicity. A lot of theoretical work has been done to understand the spin crisis in the framework of QCD, e.g. higher order corrections to the Ellis-Jaffe sum rule [4]. 2 The quark helicity distributions ∆qi(x; Q ) are related to the vector-axial quark current. This current is not conserved due to the Adler-Bell-Jackiw anomaly [5]. The anomaly can explain the spin crisis by changing the interpretation of the measurement: instead of the R 1 Pnf 2 quark spin content ∆Σ = 0 i=1 ∆qi(x; Q )dx the flavor-singlet axial current matrix el- 2 Spin degrees of freedom 3αS R 1 2 ement a0 = ∆Σ − 2π ∆G is measured, where ∆G = 0 ∆G(x; Q )dx is a gluon helicity inside the nucleon. The spin crisis can be then avoided if ∆G is large enough. This inter- pretation was the driving force in the preparation of a series of new polarisation dependent Deep Inelastic Scattering (DIS) experiments dedicated to the precise measurement of par- ton helicity distributions, transverse nucleon structure, and to the direct measurements of gluon polarization: HERMES [6] at DESY, SMC [7] and COMPASS [8] at CERN, E155 [9] at SLAC and CLAS [10] at JLAB. The collider experiments STAR [11] and PHENIX [12] at RHIC measure the gluon polarisation by observing helicity asymmetries for hadrons and jets in polarised proton-proton collisions. In the light of the results obtained by these new spin-dependent experiments the role of the axial anomaly seems to be marginal as data prefer the gluon helicity contribution to the nucleon's spin to be small. Beside the quark and gluon helicities, also Orbital Angular Momenta (OAM) can con- tribute the nucleon spin structure. The definition of the angular momentum of quarks and gluons, both orbital and total, is a very delicate and nontrivial topic. It should be gauge invariant and expressed in terms of well defined local QCD operators built from quark and gluon fields. A possible solution was recently been proposed [13] although also this approach has difficulty [14]. The presence of OAM inside nucleon requires the extension of the usual QPM beyond the longitudinal approximation. Complementary measurements to the longitudinal spin structure of the nucleon are per- formed on transversely polarized targets. New polarisation dependent parton distribution functions called "transversity" are associated with such a "transverse" spin nucleon struc- ture. They are defined by the difference between quark (antiquark) distributions for two different spin projection orientations relative to the transversely polarised target .Transver- sity probes the relativistic nature of quarks. For models with non-relativistic quarks there is no difference of the helicity and transversity distributions due to rotation symmetry. Rela- tivistic quarks make a difference (relativistic Lorent'z boosts and rotations don't commute) which can be easily calculated in relativistic models. A good "textbook" example is the MIT bag model (see e.g.[15]). This model explains why ∆Σ ∼ 0:6 is below the naive expectation ∆Σ = 1 and predicts the reduction factor of about 0:83 for transversity. There is no transver- sity analog of the gluon helicity distribution due to angular momentum conservation (the nucleon spin decomposition in the case of transversity does not contain gluons).