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We show that the transverse momentum dependent transversity function is proportional to the longitudinal polarization of a quark in a transversely polarized proton. This result suggests an alternative, convenient method for determining transversity, without knowing unusual fragmentation functions. The method consists of measuring the double spin azimuthal asymmetry in semi-inclusive pion leptoproduction by a transversely polarized proton target. The asymmetry, which is twist 3, is estimated to be more than 10% under the most favorable conditions. The experiment we suggest is feasible in facilities like DESY and CERN.

We calculate QCD corrections to transversely polarized Drell-Yan process at a measured Q_T of the produced lepton pair in the dimensional regularization scheme. The Q_T distribution is discussed resumming soft gluon effects relevant for small Q_T.

The HERMES experiment at DESY has been designed to study the spin structure of the nucleon. Impressive results have been obtained not only in inclusive processes but also for semi-inclusive and exclusive processes in deep-inelastic lepton scattering, helping to clarify the spin puzzle of the nucleon.

The investigation of the partonic degrees of freedom beyond collinear approximation (3D description) has been gained increasing interest in the last decade. At the HERMES experiment, azimuthal asymmetries in hard exclusive reactions and in semi-inclusive deep-inelastic scattering of electrons and positrons off a (polarized) hydrogen and deuterium target have been measured. Such asymmetries provide new insights on crucial aspects of the parton dynamics. By measuring various hadron types in the initial and final states, flavor sensitivity is achieved. Non zero signals are reported for azimuthal asymmetries with respect the transverse target polarization in real-photon exclusive-electroproduction, which are related (still in a model dependent way) to the elusive quark orbital motion. Evidence is reported of the poorly known transversity function and of naive-T-odd transverse-momentum-dependent parton distribution functions related to spin-orbit effects. Evidence of spin-orbit effects in quark fragmentation is also observed, which are opposite in sign for favored and disfavored processes.

Our original suggestion to investigate exclusive π⁰ electroproduction as a method for extracting from data the tensor charge, transversity, and other quantities related to chiral odd generalized parton distributions is further examined. We now explain the details of the process: i) the connection between the helicity description and the cartesian basis; ii) the dependence on the momentum transfer squared, Q², and iii) the angular momentum, parity, and charge conjugation constraints (J^PC quantum numbers).

We argue that due to parity constraints, the helicity combination of the purely momentum space counterparts of the Wigner distributions — the generalized transverse momentum distributions — that describes the configuration of an unpolarized quark in a longitudinally polarized nucleon, can enter the deeply virtual Compton scattering amplitude only through matrix elements involving a final state interaction. The relevant matrix elements in turn involve light cone operators projections in the transverse direction, or they appear in the deeply virtual Compton scattering amplitude at twist three. Orbital angular momentum or the spin structure of the nucleon was a major reason for these various distributions and amplitudes to have been introduced. We show that twist three contributions to deeply virtual Compton scattering provide observables related to orbital angular momentum.

The exploration of the structure of the nucleon reaches a new level of sophistication when explicit dependences on transverse momenta of partons in both parton distributions and fragmentation functions are included. These dependences lead to a manifold of new correlations between the nucleon and quark spin as well as their momenta and thus a wealth of experimental observables. A particular example is the novel Sivers function that—according to firm QCD predictions—has to change sign when going from the Drell-Yan process to semi-inclusive deep-inelastic scattering. Here a review is given on which knowledge has been gathered about transverse-momentum-dependent (TMD) distribution and fragmentation functions from semi-inclusive deep-inelastic scattering experiments.

Our original suggestion to investigate exclusive π⁰ electroproduction as a method for extracting the tensor charge, transversity, and other quantities related to chiral odd generalized parton distributions from cross section and asymmetry data is reviewed. We explain some of the details of the process: i) the connection between the helicity description and the cartesian basis; ii) the dependence on the momentum transfer squared, Q², and iii) the angular momentum, parity, and charge conjugation constraints (J^PC quantum numbers).

The valence transversity distributions of the u- and the d-quarks have been extracted point-by-point from single-hadron production and dihadron production data measured in semi-inclusive deep inelastic scattering and in e⁺e annihilation. The extraction is based on some simple assumptions and does not require any parametrization. The transversity distributions are found to be compatible with each other and with previous analyses.

The COMPASS measurements of the azimuthal asymmetries in the hadron pairs production in DIS on transversely polarised targets are here presented. Results on oppositely charged pion and kaon pairs produced on both deuteron and proton targets are shown. They have been used to update the calculation of the Transversity function which was previously extracted from the asymmetries measured from un-identified hadron pairs. Also a new method for the direct measurement of the integral of Transversity using both COMPASS and BELLE results is presented.

We present the preliminary results of the measurement of the azimuthal asymmetries of two charged pions in the inclusive process $e^{+}{e}^{-}\to \pi \pi X$ in the BESIII experiment. These asymmetries are attributed to the so-called Collins fragmentation function, which depict the behavior of a hadron produced from a transversely polarized quark. This spin-dependent function is an important input for the global analysis of extracting the transversity inside the nucleon. In addition, by comparison with the measured asymmetries at Belle, this measurement provides the first data to explore the $Q^2$ evolution of the spin-dependent fragmentation function. This work is performed based on about 62${\mathrm{\text{pb}}}^{-1}$ data collected with the BESIII detector at BEPCII at $\sqrt{s}$=3.65GeV.

The valence transversity distributions of the u- and the d-quarks have been extracted point-by-point from single-hadron production and dihadron production data measured in semi-inclusive deep inelastic scattering and in $e^{+}{e}^{}$ annihilation. The transversity distributions are found to be compatible with each other and with previous analyses.

The transversity distribution, which describes transversely polarized quarks in a transversely polarized nucleon, is a fundamental component of the current picture of the spin structure of the nucleon, and is only loosely constrained by existing semi-inclusive deep inelastic scattering data. The di-hadron interference fragmentation function (IFF), which describes the fragmentation of transversely polarized quarks, is expected to give rise to spin-dependent di-hadron correlations in p${}^{\uparrow }$+p collisions. Significant di-hadron correlations have recently been measured at RHIC in p${}^{\uparrow }$+p collisions at $\sqrt{s}$ = 200 GeV at mid-rapidity. In 2011, STAR collected p${}^{\uparrow }$+p data at $\sqrt{s}$ = 500 GeV, allowing STAR to extend these di-hadron asymmetries into a previously unexplored kinematic region. The status of the charge-ordered pion pair asymmetry measurement from $\sqrt{s}$ = 500 GeV p${}^{\uparrow }$+p collisions will be presented.

In the fragmentation of a transversely polarized quark a left-right asymmetry, the Collins asymmetry, is expected for each hadron produced in the process $\mu N\to {\mu }^{\prime }{h}^{+}{h}^{-}X$. Similarly, an asymmetry is also expected for the hadron pair, the dihadron asymmetry. Both asymmetries have been measured to be different from zero on transversely polarised proton targets and have allowed for first extractions of the transversity distributions. From the high statistics COMPASS data we have further investigated these asymmetries getting strong indications that the two mechanisms are driven by a common physical process.

Single spin asymmetry measurements ($A_{UT}$) of the azimuthal distribution of charged pions inside jets produced in transversely polarized proton collisions are sensitive to the transversity distribution and the Collins fragmentation function. The STAR Detector at the Relativistic Heavy Ion Collider is well suited for these types of measurements as it is capable of full jet reconstruction and charged pion identification in the mid-rapidity region ($|$$\eta$$|$$<1$). We report here the first observation of Collins $A_{UT}$ asymmetries in $\sqrt{s}=200$ GeV $p^{\uparrow }p$ collisions.

The cross section of the exclusive ${\pi }^0$ and $\eta$ electroproduction reaction $ep\to e^{\prime }{p}^{\prime }{\pi }^0/\eta$ was measured at Jefferson Lab with a 5.75-GeV electron beam and the CLAS detector. Differential cross sections $d^4\sigma /dtd{Q}^{2}d{x}_{B}d{\varphi }_{\pi }$ and structure functions ${\sigma }_T+ϵ{\sigma }_L,{\sigma }_{TT}$ and ${\sigma }_{LT}$ as functions of $t$ were obtained over a wide range of $Q^2$ and $x_B$. The data are compared with the GPD based theoretical models. Analyses find that a large dominance of transverse processes is necessary to explain the experimental results. Generalized form factors of the transversity GPDs ${〈H_T〉}^{\pi ,\eta }$ and ${〈{Ē}_T〉}^{\pi ,\eta }$ were directly extracted from the experimental observables for the first time. It was found that GPD ${Ē}_T$ dominates in pseudoscalar meson production. The combined ${\pi }^0$ and $\eta$ data opens the way for the flavor decomposition of the transversity GPDs. The first ever evaluation of this decomposition was demonstrated.

Enlarging the set of hard exclusive reactions to be studied in the framework of QCD collinear factorization opens new possibilities to access generalized parton distributions (GPDs). We studied the photoproduction of a large invariant mass photon-photon or photon-meson pair in the generalized Bjorken regime which may be accessible both at JLab and at the EIC.

Within the framework of light-cone quantization we derive the overlap representation of generalized parton distributions for transversely polarized quarks using the Fock-state decomposition in the transverse-spin basis. We apply this formalism to the case of light-cone wave functions in a constituent quark model giving numerical results for the four chiral-odd generalized parton distributions in a region where they describe the emission and reabsorption of a quark by the nucleon. With the transversity distribution obtained in the forward limit of the generalized distribution, we provide some predictions for the double transverse-spin asymmetry in Drell-Yan dilepton production in the kinematics of the experiment.

Preliminary results on azimuthal single-spin asymmetries in semi-inclusive electro-production of pions and charged kaons at the HERMES experiment are presented. Significant amplitudes for both the Collins and the Sivers mechanisms are extracted from the full data set collected with a transversely polarized target. A new method for the evaluation of the acceptance effects, developed in view of a future extraction of the P_h⊥-weighted moments, is also discussed.

The measuring of transverse single spin asymmetries (SSA) is part of the physics program at COMPASS, a fixed target experiment at CERN SPS. In 2002-04 the COMPASS experiment has collected data with a transversely polarised ⁶LiD target using a 160 GeV/c polarised µ⁺ beam. By measuring transverse SSA one has access to the transversity distribution function Δ_Tq(x). This is one of the three quark distribution functions, which are needed to fully describe the spin structure of the nucleon at leading twist. At COMPASS three different quark polarimeters have been used to access transversity: the Collins effect, which produces an azimuthal asymmetry in the single hadron distribution, the azimuthal target spin asymmetries of charged hadron pairs and the transverse polarisation of Λ hyperons.

In addition the SSA arising from the correlation between the transverse nucleon spin and the quark intrinsic transverse momentum (Sivers effect), was measured, together with six more transverse target spin asymmetries.

All the asymmetries measured on the deuteron target are small and compatible with zero.