Transverse Momentum Dependent Distribution Research Articles

The Spin Structure of the Nucleon; A COMPASS Perspective

The COMPASS experiment at CERN is carrying on an experimental investigation of the spin structure of the nucleon, covering both longitudinal and transverse spin phenomena. In the first case, the central topic is the direct measurement of the gluon polarisation with the hope to solve the spin crisis first observed by EMC. The result shows that Δg/g is small around xg≃0.1, and its first moment should not be larger than 0.2—0.3 in absolute value. About transverse spin effects, evidence is given for new phenomena, associated with transverse momentum dependent distribution and fragmentation functions.

NONTRIVIAL RELATIONS BETWEEN GPDs AND TMDs>

The present knowledge about nontrivial relations between generalized parton distributions for a spin-1/2 hadron on the one hand and transverse momentum dependent distributions on the other is reviewed. While various relations can be found in the framework of simple spectator models, so far no model-independent nontrivial relations have been established. In fact, by relating the two types of parton distributions to the fully unintegrated, off-diagonal quark-quark correlator for a spin-1/2 hadron, we argue that none of the nontrivial relations can be promoted to a model-independent status.

Measurement of the [formula omitted] and [formula omitted] Azimuthal Moments of the Deep Inelastic Scattering Cross-section

The cross section for hadron production in deep-inelastic lepton scattering contains azimuthal modulations which can be related to transverse momentum dependent (TMD) distribution and fragmentation functions. The former provide a picture of how the quarks are moving within nucleons. Specifically, the cos ϕ h and cos 2 ϕ h modulations of the unpolarized cross section relate quark spin and quark transverse momentum. These moments have been carefully measured at the HERMES experiment in a fully differential way, as a function of x, y, z, and P h ⊥ for positive and negative hadrons produced from hydrogen and deuterium targets. These measurements give new access to the flavor dependent TMDs via their charge and target dependence.

Transverse momentum dependent distribution functions in a covariant parton model approach with quark orbital motion

Transverse parton momentum dependent distribution functions (TMDs) of the nucleon are studied in a covariant model, which describes the intrinsic motion of partons in terms of a covariant momentum distribution. The consistency of the approach is demonstrated, and model relations among TMDs are studied. As a by-product it is shown how the approach allows to formulate the nonrelativistic limit.

A method of experimentally probing transverse momentum dependent distributions

We calculate the double spin asymmetry ALL for the π0 production in semi-inclusive deep inelastic lepton–proton scattering with a spectator model of power-law and a model based on the factorization ansatz. We also calculate the double spin asymmetry for the integration over parts of the kinematic range for the setups of the experiments of COMPASS, HERMES, and JLab. We find that the results are characteristically dependent on the model used. Therefore, we suggest that the measurements of the double spin asymmetry provides a method of experimentally probing the transverse momentum dependent distributions.

Are there approximate relations among transverse momentum dependent distribution functions?

Certain exact relations among transverse momentum dependent parton distribution functions due to QCD equations of motion turn into approximate ones upon the neglect of pure twist-3 terms. On the basis of available data from HERMES, we test the practical usefulness of one such 'Wandzura-Wilczek-type approximation', namely, of that connecting h{sub 1L}{sup perpendicular}({sup 1a})(x) to h{sub L}{sup a}(x), and discuss how it can be further tested by future CLAS and COMPASS data.

Predictions of the QCD Parton Model

The QCD parton model allows to establish approximate relations among transverse momentum dependent quark distribution functions and, simultaneously, to fix an energy scale, involved in the parametrization of the correlator, which determines the Q 2 dependence of the azimuthal asymmetries in inclusive reactions, like SIDIS and Drell-Yan. Some of the present data support such model predictions. We suggest further measurements of asymmetries and an alternative way of determining (approximately) the proton transversity.

THE QCD PARTON MODEL: A USEFUL APPROXIMATION

Approximate relations among transverse momentum dependent quark distribution functions are established in the framework of the QCD parton model. The validity of such results survives QCD evolution effects, owing to the Politzer theorem on equations of motion. Furthermore the model fixes an energy scale, involved in the parametrization of the correlator, which determines the Q2 dependence of the azimuthal asymmetries in inclusive reactions. Some of the present data — in particular the sin 2ϕ single spin asymmetry in semi-inclusive deep inelastic scattering (SIDIS) and the cos ϕ asymmetry in unpolarized SIDIS — support model predictions. Further measurements of SIDIS and Drell–Yan asymmetries are suggested, in particular the SIDIS double spin asymmetry, which allows to determine approximately the proton transversity.

Relations between generalized and transverse momentum dependent parton distributions

Recent work suggests nontrivial relations between generalized parton distributions on the one hand and (naive time-reversal odd) transverse momentum dependent distributions on the other. Here we review the present knowledge on such type of relations. Moreover, as far as spectator model calculations are concerned, the existing results are considerably extended. While various relations between the two types of parton distributions can be found in the framework of spectator models, so far no nontrivial model-independent relations have been established.

Gluonic pole cross sections and single spin asymmetries in hadron-hadron scattering

The gauge-links connecting the parton field operators in the hadronic matrix elements appearing in the transverse momentum dependent distribution functions give rise to T-odd effects. Due to the process-dependence of the gauge-links the T-odd distribution functions appear with different pre-factors. A consequence is that in the description of single spin asymmetries the parton distribution and fragmentation functions are convoluted with gluonic pole cross sections rather than the basic partonic cross sections. In this paper we calculate the gluonic pole cross sections encountered in single spin asymmetries in hadron-hadron scattering. The case of back-to-back pion production in polarized proton-proton scattering is worked out explicitly. It is shown how T-odd gluon distribution functions originating from gluonic pole matrix elements appear in twofold.

Beam single-spin asymmetry in semiinclusive deep inelastic scattering

We calculate, in a model, the beam spin asymmetry in semi-inclusive jet production in deep inelastic scattering. This twist-3, $T$-odd observable is nonzero due to final state strong interactions. With reasonable choices for the parameters, one finds an asymmetry of several percent, about the size seen experimentally. We present the result both as an explicit asymmetry calculation and as a model calculation of the new transverse-momentum dependent distribution function ${g}^{\ensuremath{\perp}}$.

Single transverse-spin asymmetries at largex

The large-x behavior of the transverse-momentum dependent quark distributions is analyzed in the factorization-inspired perturbative QCD framework, particularly for the naive time-reversal-odd quark Sivers function which is responsible for the single transverse-spin asymmetries in various semi-inclusive hard processes. By examining the dominant hard gluon exchange Feynman diagrams, and using the resulting power counting rule, we find that the Sivers function has power behavior (1-x){sup 4} at x {yields} 1, which is one power of (1-x) suppressed relative to the unpolarized quark distribution. These power-counting results provide important guidelines for the parameterization of quark distributions and quark-gluon correlations.

Single-spin observables and orbital structures in hadronic distributions

Within the light-quark sector of the standard model, P-odd observables are generated from point-like electroweak processes while A_t- odd observables (neglecting quark mass parameters) come from dynamic spin-orbit correlations within hadrons or within larger composite systems, such as nuclei. The effects of A_t-odd dynamics can be inserted into transverse-momentum dependent constituent distribution functions and, in this paper, we construct the contribution from an orbital quark to the A_t odd quark parton distribution. Using this distribution, we examine the crucial role of initial- and final-state interactions in the observation of the scattering asymmetries in different hard-scattering processes. This construction provides a geometrical and dynamical interpretation of the Collins conjugation relation between single-spin asymmetries in semi-inclusive deep inelastic scattering and the asymmetries in Drell-Yan production. Finally, our construction allows us to display a significant difference between the calculation of a spin asymmetry generated by a hard scattering mechanism involving color-singlet exchange (such as a photon) and a calculation of an asymmetry with a hard-scattering exchange involving gluons. This leads to an appreciation of the process dependence inherent in measurements of single-spin observables.

Transverse momentum in semi-inclusive deep inelastic scattering

Within the framework of perturbative quantum chromodynamics we derive the evolution equations for transverse momentum dependent distributions and apply them to the case of semi-inclusive deep inelastic scattering. The evolution equations encode the perturbative component of transverse momentum generated by collinear parton branchings. The current fragmentation is described via transverse momentum dependent parton densities and fragmentation functions. Target fragmentation instead is described via fracture functions. We present, to leading logarithmic accuracy, the corresponding semi-inclusive deep inelastic scattering cross-section, which applies to the entire phase space of the detected hadron. Some phenomenological implications and further developments are briefly outlined.

Color Gauge Invariance in Hard Processes

Within the theoretical framework we apply, a suggested origin for single spin asymmetries is the presence of gauge links in transverse momentum dependent distribution functions. Recently we found new gauge link structures in a number of hard processes. These structures need to be considered in the evolution of parton distribution functions and for establishing factorization.

Spin densities in the transverse plane and generalized transversity distributions

We show how generalized quark distributions in the nucleon describe the density of polarized quarks in the impact parameter plane, both for longitudinal and transverse polarization of the quark and the nucleon. This density representation entails positivity bounds including chiral-odd distributions, which tighten the known bounds in the chiral-even sector. Using the quark equations of motion, we derive relations between the moments of chiral-odd generalized parton distributions of twist two and twist three. We exhibit the analogy between polarized quark distributions in impact parameter space and transverse momentum dependent distribution functions.

Azimuthal asymmetry in unpolarized πN Drell–Yan process

Taking into account the effect of final-state interaction, we calculate the non-zero (naïve) T-odd transverse momentum dependent distribution h1⊥(x,k⊥2) of the pion in a quark-spectator-antiquark model with effective pion-quark-antiquark coupling as a dipole form factor. Using the model result we estimate the cos2ϕ asymmetries in the unpolarized π−N Drell–Yan process which can be expressed as h1⊥×h¯1⊥. We find that the resulting h1π⊥(x,k⊥2) has the advantage to reproduce the asymmetry that agrees with the experimental data measured by NA10 Collaboration. We estimate the cos2ϕ asymmetries averaged over the kinematics of NA10 experiments for 140, 194 and 286 GeV π− beam and compare them with relevant experimental data.

Nonzero transversity distribution of the pion in a quark-spectator-antiquark model

We calculate the non-zero (na\"{i}ve) T-odd transverse momentum dependent transversity distribution $h_1^{\perp}(x,\kp^2)$ of the pion in a quark-spectator-antiquark model. The final-state interaction is modelled by the approximation of one gluon exchange between the quark and the antiquark spectator. Using our model result we estimate the unsuppressed cos2$\phi$ azimuthal asymmetry in unpolarized $\pi^-p$ Drell-Yan process. We find that the transverse momentum dependence of $h_1^{\perp}(x,\kp^2)$ of the pion is the same as that of $h_1^{\perp}(x,\kp^2)$ of the proton calculated from the quark-scalar-diquark model, although the $x$ dependencies of them are different from each other. This suggests a connection between cos2$\phi$ asymmetries in Drell-Yan processes with different initial hadrons.

Gauge link structure in quark–quark correlators in hard processes

Distribution functions in hard processes can be described by quark–quark correlators, nonlocal matrix elements of quark fields. Color gauge invariance requires inclusion of appropriate gauge links in these correlators. For transverse momentum dependent distribution functions, in particular important for describing T-odd effects in hard processes, we find that new link structures containing loops can appear in Abelian and non-Abelian theories. In transverse moments, e.g., measured in azimuthal asymmetries, these loops may enhance the contribution of gluonic poles. Some explicit results for the link structure are given in high-energy leptoproduction and hadron–hadron scattering.

Sivers function in the MIT bag model

The Sivers function, an asymmetric transverse-momentum distribution of the quarks in a transversely polarized nucleon, is calculated in the MIT bag model. The bag quark wave functions contain both S-wave and P-wave components, and their interference leads to nonvanishing Sivers function in the presence of the final state interactions. We approximate these interactions through one-gluon exchange. An estimate of another transverse momentum dependent distribution h1⊥ is also performed in the same model.