Transverse polarization functions

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Author: Dr. Mauro Anselmino, Univ Turin, Dipartimento Fis Teor, Sez Torino, I-10125 Turin and Ist Nazl Fis Nucl, I-10125 Turin, Italy

Dr. Mauro Anselmino accepted the invitation on 31 October 2009 (self-imposed deadline: 30 April 2010).

The exploration of the internal composition of nucleons - protons and neutrons - which form the overwhelming majority of all observed matter in the Universe, has undergone enormous progresses in the last decades. Starting from the end of the sixties, the nucleons have been probed by scattering point-like particles like electrons, positrons or muons (leptons) off protons or nuclear targets. The internal structure of the nucleons, with a size of the order of 1 femtometer $(10^{-15}$ m.), has been deeply investigated by the high energy leptons.

This long activity has lead to the discovery of the internal constituents of protons and neutrons and to the description of nucleons in terms of these new basic particles, quarks and gluons (partons). In parallel with the experimental work, a fundamental, quantum-relativistic theory, describing the quark and gluon interactions has been developed, Quantum Chromo Dynamics (QCD), and successfully tested. The experimental discovery of point-like quark constituents inside the nucleon was recognized with the 1990 Nobel Prize for Jerome Friedman, Henry Kendall and Richard Taylor, while David Gross, David Politzer and Frank Wilczek received the Prize in 2004 for their theoretical discovery of the “asymptotic freedom” of QCD which made it possible to obtain quantitative and testable predictions from the theory.

In a high energy experiment a fast moving nucleon is described as a set of co-moving partons, which interact dynamically (QCD parton model). The interpretation of the lepton-nucleon high energy scattering data within this QCD parton model has contributed fundamentally to our present knowledge about the structure of protons and neutrons: the number of quarks inside the proton, the way they share its fast motion and the way they share and contribute to its helicity (i.e, the spin component along the direction of motion), are rather well known. In particular the predictions of how these properties change with energy and depend upon the distance at which we probe the nucleon, have been very successfully confirmed by data, resulting in fundamental tests of QCD.

This information is encoded in the so-called Parton Distribution Functions (PDFs), usually denoted as $f_{q/p}(x,Q^2)$ , which give the number density of partons of type $q$ inside a proton $p$ . Here $x$ is the fraction of the nucleon momentum carried by the parton and $Q^2$ is the square of the four-momentum transfer from the initial to the final lepton (the bigger $Q^2$ is, the smaller is the spatial region of size $1/Q$ that we are exploring). Experiments have also accessed, albeit in much less detail, the helicity distributions of the nucleon, which count the difference between the number densities of partons with the same helicity and with opposite helicity as the proton’s.

However, many other important and interesting aspects of the structure of the nucleon are not revealed by the standard parton distributions, as these are essentially averaged over all degrees of freedom except the longitudinal one. They do not address questions such as: Do quarks orbit? How are they spatially distributed inside the proton (which for them is a huge 3-dimensional object)? Is there a connection between the motion of quarks, their spin and the spin of the proton?

A serious and systematic attempt to answer the above questions has started about a decade ago, with both dedicated experiments and new theoretical ideas. We have entered a new phase in our investigation of the basic structure of matter. The crucial innovation is that of looking at, and studying, physical observables which are sensitive to the transverse structure of the nucleon. Transverse and longitudinal refer to the direction of motion; for fast moving protons, for which the QCD parton model holds, the transverse properties, both in spin and motion, give novel information. This, when combined with the available longitudinal information, allows a true 3-dimensional understanding of the proton structure.

Semi-inclusive Deep Inelastic Scattering (SIDIS) and TMDs

The usual guiding experiments involve inelastic lepton-nucleon scattering at high energy: the lepton interacts with the quarks inside the nucleon and by observing the scattering angle and the energy of the outgoing lepton one obtains information about the quark content of the nucleon. In this process, denoted as Deep Inelastic Scattering (DIS, $\ell \, N \to \ell \, X$ , one only observes the final lepton, while the scattered quark and the remnant of the struck nucleon fragment into some final hadronic states $X$ not detected. These measurements allow one to obtain information about the $x$ and $Q^2$ dependences of the parton distributions, but offer no information about the transverse motion or spatial distribution of partons in the nucleon, which are integrated over.

Many additional possibilities for learning about the nucleon partonic structure arise if one looks at the so-called Semi-Inclusive Deep Inelastic Scattering processes (SIDIS, $\ell \, N \to \ell \, h \, X$ , in which one observes in the final state, in addition to the lepton, also one hadron, e.g. a pion. In this case the hadron, which results from the fragmentation of a scattered quark, “remembers” the original motion of the quark, including the transverse motion, and offers new information. The parton fragmentation process is described by a fragmentation function $D_{h/q}$ , which, analogously to the parton distribution functions, gives the number density of hadrons $h$ resulting from the hadronization of a parton $q$ . The cross-section data are analyzed according to a factorized theoretical expression:

$d\sigma^{\ell p \to \ell h X} = \sum_q f_{q/p}(x, \mathbf{k}_\perp; Q^2) \otimes d\hat\sigma^{\ell q \to \ell q} \otimes D_{h/q}(x, \mathbf{p}_\perp; Q^2)$

in which the non-perturbative, long-distance physics (contained in $f_{q/p}$ and $D_{h/q}$ ) is convoluted with the elementary, short-distance, hard-scattering interaction ( $d\hat\sigma^{\ell q \to \ell q}$ ). The parton distributions and fragmentation functions depend not only on $Q^2$ and the longitudinal momentum fraction (respectively, $x$ and $z$ ) but also on the transverse motion of partons inside the nucleon ( $\mathbf{k}_\perp$ ) and of the final hadron with respect to the fragmenting parton ( $\mathbf{p}_\perp$ ). These Transverse Momentum Dependent parton distributions and fragmentation functions are usually abbreviated as TMDs.

Polarized SIDIS and azimuthal asymmetries

The TMDs, in particular the parton distributions, contain information on both the longitudinal and transverse motion of partons and gluons inside a fast moving nucleon. When adding the spin degree of freedom, they may link the parton spin ( $\mathbf{s}_q$ ) to the parent proton spin ( $\mathbf{S}$ ) and to the transverse motion ( $\mathbf{k}_\perp$ ). The spin dependent TMDs, $f_{q/p}(x, \mathbf{k}_\perp; \mathbf{s}_q, \mathbf{S}; Q^2)$ , may depend on all appropriate combinations of the pseudo-vectors $\mathbf{s}_q$ , $\mathbf{S}$ and the vectors $\mathbf{k}_\perp$ , $\mathbf{p}$ (the nucleon momentum) which are allowed by parity invariance. At leading order in $1/Q$ , there are eight such combinations, leading to eight independent TMDs.