The transverse structure of protons and neutrons: TMDs

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Mauro Anselmino (2011), Scholarpedia, 6(2):10209. doi:10.4249/scholarpedia.10209 revision #91873 [link to/cite this article]
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The Nucleon Momentum Structure

The exploration of the internal composition of nucleons - protons and neutrons - which form the overwhelming majority of the mass of all the observed matter in the Universe, has undergone enormous progress in the last decades. In the sixties only the global structure of protons and neutrons was known, their charges and magnetic moments as well as their size, of the order of 1 femtometer \((10^{-15}\) m.). Starting from the end of the sixties, the nucleons have been probed by scattering high energy point-like particles like electrons, positrons, muons or neutrinos (leptons) off protons or nuclear targets, leading to a thorough investigation of the deep inner structure of the nucleons.

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 description works so well, the transverse properties, both in spin and motion, give novel information. Combined with the available longitudinal information, this 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 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 (\(\mathbf{k}_\perp\)), and also the parton transverse momentum to the nucleon spin. 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. Three of them - the unpolarized, the helicity and the transversity distributions - survive in the collinear limit (but only the first two can be accessed in the usual DIS experiments); the other five - related to different couplings between spin and transverse motion - describe new, so far unexplored, properties of the motion of quarks and gluons inside protons and neutrons.

For example, there exists a \(\mathbf{S} \cdot (\mathbf{P} \times \mathbf{k}_\perp)\) term involving the so-called Sivers function (Sivers, 1990), related to the number density of unpolarized partons inside a transversely polarized proton. Notice that such a term vanishes in the longitudinal spin (\(\mathbf{S} \propto \mathbf{P}\)) and in the collinear (\( \mathbf{k}_\perp = 0\)) cases. This term is special because it is not a spin-spin transfer, but rather couples spin \( \mathbf{S} \) and momenta. It generates azimuthal Single-Spin Asymmetries (SSAs), that is differences of the cross-section upon changing the nucleon spin orientation (\( \mathbf{S} \to -\mathbf{S} \)), discussed further in the next paragraph. The appearance of multiple momenta (see Fig. 1) in the \(\mathbf{S} \cdot (\mathbf{P} \times \mathbf{k}_\perp)\) term induces observable azimuthal dependences. In particular, the Sivers function describes the correlation between the momentum direction of the struck parton and the spin of its parent nucleon and is hence related to the orbital motion of partons inside the proton.

A similar correlation between spin and transverse motion can occur in the fragmentation process of a transversely polarized quark, with spin vector \( \mathbf{s}_q \) and three-momentum \( \mathbf{p}_q \ ,\) into a hadron with longitudinal momentum fraction \(z\) and transverse momentum \( \mathbf{p}_\perp \) (with respect to the quark direction); such a mechanism is called the Collins effect (Collins, 1993) and appears in the fragmentation function via a \(\mathbf{s}_q \cdot (\mathbf{p}_q \times \mathbf{p}_\perp)\) term. The Collins function couples to the distribution of transversely polarized quarks inside a transversely polarized nucleon – the transversity PDF – and also generates SSAs and azimuthal dependences in SIDIS processes.

Pioneering experiments measuring azimuthal distributions in SIDIS processes related to the Sivers and Collins effects have been performed by the HERMES (DESY) and COMPASS (CERN) Collaborations. Different combinations of various distribution and fragmentation functions are extracted by exploring the cross-section dependence on the two azimuthal angles \(\phi\) and \(\phi_S\) defined in Fig. 1 (Bacchetta et al., 2004). The Sivers and Collins effects, for example, manifest themselves, respectively, as \(\sin(\phi-\phi_S)\) and \(\sin(\phi+\phi_S)\) modulations in the azimuthal distribution of the produced hadrons.

Figure 1: SIDIS kinematical variables and azimuthal angles in the \(\gamma^*-p\) c.m. frame.

The first measurements with transversely polarized protons, reported a few years ago by HERMES (HERMES, 2005), provided evidence for both Sivers and Collins effect. These non-zero asymmetries are now firmly established by recent data of much improved statistical precision (HERMES, 2009, COMPASS, 2007, 2010), in particular for the production of \(\pi^+\) and \(K^+\ .\) As scattering off \(u\)-quarks dominates these data taken with a proton target, the positive Sivers asymmetries for \(\pi^+\) and \(K^+\) leads to a large Sivers function for \(u\)-quarks, and hence suggests a sizeable orbital angular momentum for \(u\)-quarks. A global analyses of the SIDIS data, aiming at the extraction of the TMDs is in progress and first results are already available (Anselmino et al., 2007, 2009). The final outcome, similarly to what has been achieved, during the last 40 years, for the longitudinal degrees of freedom, will be a 3-dimensional imaging of the motion of polarized partons inside polarized nucleons.

Single Spin Asymmetries

As indicated, an important issue is the appearance of Single-Spin Asymmetries. The existence of transverse Single-Spin Asymmetries has been known in high energy physics for many years and has always posed a challenging problem to QCD. The first large SSAs where observed more than three decades ago in inclusive hadronic processes, like \(p\,p \to \Lambda^\uparrow X\) and \(p^\uparrow p \to \pi X \ .\) In the first case it was discovered that the Lambda (and other hyperon) particles, inclusively produced in the high energy scattering of unpolarized nucleons, were strongly polarized; the second group of processes showed large SSAs in the production of pions in the scattering of transversely polarized protons off unpolarized nucleons. There is no way of relating such large SSAs to the elementary QCD dynamics with co-moving partons, which predicts negligible single spin asymmetries at the partonic level. The common expectation that such SSAs should vanish at higher energies has been disproved by recent RHIC data, which clearly show a persistence of these subtle spin effects up to the highest available energies.

The TMDs, introducing nonperturbative spin-orbit correlations are able to explain the observed single spin effects (D'Alesio et al., 2004). The key property here is time reversal (T) invariance, a symmetry that is respected by QCD. T-invariance allows making a distinction between observables that are even (T-even) or odd (T-odd) under time reversal and basically implies that T-even distributions, among them all collinear distribution functions, only show up in T-even measurements such as double spin asymmetries. It basically implies that T-odd distributions, among them the Sivers function, show up in T-odd observables such as SSAs. But all T-odd distribution functions necessarily involve transverse momenta of partons being correlated either with the spin of the quarks or with the spin of the nucleon (Boer, Mulders, 1998).

The transverse structure of the nucleon, thus, plays a crucial role in physical observables. Another class of processes which, although difficult to measure, might yield plenty of new information on TMDs is the inclusive production of lepton pairs in hadronic collisions, like \(p\,p \to \ell^+ \ell^- X\ ,\) the so-called Drell-Yan process. Spin asymmetries and even azimuthal dependences in the unpolarized case can be generated by TMDs, so that their measurements would give information on the transverse nucleon structure (Arnold et al., 2009).

Generalized Parton Distributions

Complementary information towards a genuine 3-dimensional space and momentum resolution of the nucleon structure is offered by the so-called Generalized Parton Distributions (GPDs), which depend on more variables than the TMDs (which are particular functional limits of the GPDs). These functions offer opportunities to study a uniquely new aspect of the nucleon substructure: the localization of partons in the plane transverse to the motion of the nucleon. The GPDs are partonic amplitudes which contain the non-perturbative, long distance physics of factorized hard exclusive scattering processes. In such hard exclusive processes, the nucleon stays intact and the final state is fully observed. Since the factorization takes place on the level of amplitudes, phase information is uniquely accessible in exclusive processes. Measurements of hard exclusive processes that provide information about GPDs are much more challenging than traditional inclusive and semi-inclusive scattering experiments. These exclusive processes require a difficult full reconstruction of final state particles and their cross-sections are small, demanding high luminosity machines. Nevertheless, after the first signal of GPDs, exclusive processes are now being systematically investigated by experiments at CERN and JLab.

Future experiments

The study of the three-dimensional structure of the nucleon is being actively pursued at several experimental facilities; ongoing experiments are collecting data at CERN (COMPASS), Jefferson Laboratory (JLab, USA) and Brookhaven National Laboratory (RHIC, USA), while HERMES at DESY (Germany), although not running any more, is still analyzing previously collected data. A rich program of future dedicated experiments has either been approved like the CLAS12 (at JLab, with energy upgraded up to 12 GeV) or is being proposed, like PANDA and PAX at the large facility for hadron physics FAIR (GSI, Germany). Spin oriented measurements, aiming at TMD dependents observables, are also being considered at existing facilities, like Drell-Yan processes with pion beams at COMPASS and jet studies at RHIC and LHC. Drell-Yan processes are proposed also at the new JPARC (Japan) facility and at Fermilab. A real breakthrough in our understanding of the nucleon structure in terms of spin degrees of freedom and the role of orbital motion of partons, can only be obtained with dedicated future electron-ion (EIC) or electron-nucleon (ENC) colliders. These new machines, with polarized leptons and nucleons, high luminosity and energy, are under active consideration within the hadron physics scientific community (EIC, 2009).


  • D.W. Sivers, Phys. Rev. D41, 83 (1990); Phys. Rev. D43, 261 (1991).
  • J.C. Collins, Nucl. Phys. B396, 191 (1993).
  • A. Bacchetta et al., Phys. Rev. D70, 117504 (2004).
  • A. Airapetian et al. (HERMES), Phys. Rev. Lett. 94, 012002 (2005).
  • A. Airapetian et al. (HERMES), Phys. Rev. Lett. 103, 152002 (2009).
  • E.S. Ageev et al. (COMPASS), Nucl. Phys. B 765, 31 (2007).
  • M.G. Alekseev et al. (COMPASS), Phys. Lett. B 692, 240 (2010).
  • M. Anselmino et al., Phys. Rev. D75, 054032 (2007); Eur. Phys. J. A39, 89 (2009).
  • U. D'Alesio et al., Phys.Rev. D70, 074009 (2004).
  • D. Boer and P.J. Mulders, Phys. Rev. D57, 5780 (1998).
  • S. Arnold et al., Phys. Rev. D79, 034005 (2009).

Further reading

  • A. Airapetian et al. (HERMES), Phys. Rev. Lett. 103, 152002 (2009).

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