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About the Author

Frank Ellinghaus is a Research Associate at the University of Colorado, Boulder, and has been a member of the PHENIX experiment since 2005, and of the HERMES experiment at DESY since 2000.

Revealing the Spin Structure of the Nucleon

By Frank Ellinghaus

The main focus of the physics program at PHENIX and STAR that makes use of RHIC's polarized proton beams is to figure out how and if at all the gluons (the particles which carry the strong nuclear force) inside protons are polarized, or to put it another way, do the spin 1 gluons prefer to have their spins aligned or anti-aligned with the spin of the proton, or do they just not care? This question is an important part of the more general question of how the constituents of protons, gluons and quarks, conspire to make up the overall spin ½ of the proton. I will try to give a brief overview over the last three decades as well as the latest results from lattice QCD, lepton-proton scattering and of course proton-proton scattering.

In the most naive model in which the protons consist of three static quarks (the constituent quark model) things are easy. Two of the three spin ½ quarks will align their spins with the spin of the proton, the third one is anti-aligned, and voila, the three quarks account for the spin ½ of the proton. However, after pioneering experiments at SLAC starting in the mid-1970's it was the European Muon Collaboration (EMC) at CERN in the late 1980's that found the contribution of the quark spins to the spin of the nucleon to be small and actually consistent with zero within the errors [1].

The basic principle of this and other similar measurements is to use deep inelastic scattering (DIS) of longitudinally polarized leptons (muons in this case) off longitudinally polarized protons, which are usually embedded in a gaseous or solid state target. The polarized lepton will interact with a quark inside the polarized nucleon via the exchange of a spin 1 virtual photon. This photon can only be absorbed by the quark when the spins of the quark and the photon are anti-aligned due to angular momentum conservation. The cross section for this process can be measured for the two cases where the spin of the leptons and the nucleons is aligned and anti-aligned, and depending on which cross section turns out to be bigger, one can learn if the quark spins tend to be aligned or anti-aligned with the spin of the nucleon. This cross section difference is usually measured as a function of Q2, the negative four-momentum squared of the virtual photon, and of x, the fraction of the momentum of the (fast) proton carried by the struck quark, and it is given in terms of the so-called polarized structure function g1(x,Q2). Using different polarized targets like hydrogen, deuterium, and 3He, one can even disentangle the different quark flavors, and it turns out that up quarks tend to have their spins aligned with the proton spin while down quarks tend to have their spins anti-aligned.

The clear disagreement between the measurement and the naive expectation from the constituent quark model was maybe not too surprising even at the time of the EMC experiment, given that it was well known that the quarks are actually not static but move around at relativistic speed in a confined space. Early calculations in, for example, the MIT bag model, showed that only about 60% of the protons spin is carried by the quark spins and the rest comes from the orbital motion of the quarks. The so-called "spin crisis" was born with the value from the EMC experiment being compatible not only with zero but also being about 2 sigma away from the value derived from the bag model. The most recent results by the HERMES Collaboration at DESY [2] and the COMPASS Collaboration at CERN [3] show that the quarks contribute about 30% to the spin of the nucleon, leading to our current understanding that a substantial fraction of the proton's spin has to arise from the quark orbital angular momentum and/or the gluons.

The possible contribution of the gluon spin to the nucleon spin has first been studied in DIS. The problem with studying gluons in DIS is that the electromagnetic probe, the photon, cannot directly couple to the gluon since gluons carry no electric charge. An indirect way of accessing the polarized gluon parton distribution function (PDF) is via next-to-leading order (NLO) fits to the above mentioned structure function g1. The information on the polarized gluon PDF Δg(x) is contained in the dependence of g1 on Q2 at a given x, which is nearly constant. To be precise, it is the difference to the constant behavior, the so-called scaling violation, which contains the information. However, since all measurements of g1 have been done at fixed-target experiments the range in Q2 covered is very limited, leading to rather large uncertainties for Δg(x). Having said this, one of the big advantages of having a polarized ep collider like eRHIC is due to the large range of Q2 it makes observable. A more direct way to measure Δg(x) at the DIS experiments is via a NLO process called photon-gluon fusion (PGF), in which the virtual photon and the gluon interact via the creation of a quark-antiquark pair. However, the available results so far also suffer from the fact that the energy scale at the fixed-target experiments is rather low.

Thus, the motivation to study the gluon in pp collisions at RHIC is quite natural. The involved energies are high and the gluon participates in leading order (LO) since the basic processes we are interested in are the scattering of quarks off gluons (qg) and of gluons off gluons (gg). As in the case of g1 one measures the cross section difference between the two cases where the two polarized protons colliding have their spins aligned and anti-aligned. At LO the interpretation of this cross section difference is more complicated than in the case of g1, but in the end it is proportional to the polarized PDFs for quarks (Δq(x)) and gluons (Δg(x)) that enter the cross section in the three involved subprocess (gg, qg, qq) properly weighted by their respective contribution to the total cross section.

Figure 1. Longitudinal double spin asymmetry for neutral pion production as a function of the transverse momentum pT. The data from 2005 (Run 5) and 2006 (Run 6) are compared to two NLO pQCD calculations. See text for details.

Figure 1 shows an asymmetry, which is the difference of the two cross sections divided by their sum, as a function of the transverse momentum of neutral pions detected in the PHENIX detector. The data taken in 2005 (Run 5) and 2006 (Run 6) are compared to two theory curves which show the result of a NLO calculation using either Δq(x) and Δg(x) from the DIS data as input (GRSV-std) or using the assumption that the gluon spins do not contribute to the nucleon spin (GRSV ΔG=0). This result and the result from STAR, using measurements of jets, are consistent with the latter assumption in the x-range accessible by the RHIC experiments, see Refs. [4,5] for the respective publications from Run 5. Note that the presently accessible range is in between about 0.02 and 0.3, while the actual contribution from the gluon spins to the proton spin ΔG requires the integration of Δg(x) over the whole x-range between zero and unity. Also note that the GRSV-std curve does not indicate any uncertainties, and they are rather large mainly due to the poorly constrained Δg(x) from the DIS data as discussed above, hence the RHIC results remain consistent with the results from DIS. This has also been shown in a recent NLO pQCD fit [6], including for the first time the pp data (in addition to DIS and semi-inclusive DIS) in the extraction of the polarized PDFs, a so-called global fit. The fit also confirms that ΔG is small in the x-range between 0.05 and 0.2. The RHIC results were already very important in order to constrain ΔG in this fit and with more data to come will be the dominant source of information concerning ΔG in the probed x-range. A lower (higher) range in x can be probed running RHIC at higher (lower) energies. Also, with more data available in the future it is possible to advance from inclusive measurements, in which only one particle in the final state is detected, to measurements in which two or more leading particles or jets are detected in the final state, allowing one to get a much better handle on the x-dependence. A precise measurement of the functional from of the polarized gluon PDF Δg(x) will be possible at a polarized ep collider like eRHIC, since in DIS the momentum fraction x can be measured directly.

Finally, a word on the orbital angular momenta of quarks and gluons that should account for the missing spin not accounted for by the spin contribution of the quarks and gluons described above. About ten years ago Xiangdong Ji provided the first and so far only theoretical idea on how to access the total angular momentum (spin plus orbital) carried by the quarks (Jq) in the nucleon [7]. It involves new objects parameterizing the nucleon structure, the so-called generalized parton distributions (GPDs), which contain the well known parameterizations like the PDFs and nucleon form factors as certain kinematic limits and moments, respectively. As a word of caution, it should be noted that one can in principle deduct the total angular momentum of gluons Jg once Jq is known since ½ = Jq + Jg, however, Jg cannot be separated into gluon spin and gluon orbital contributions in a gauge invariant way, so it cannot be directly related to the measurements of ΔG mentioned above. Experimental access to the GPDs is gained in the hard exclusive production of photons in electron proton scattering (ep → epγ), i.e., the proton is still intact after its interaction with the highly energetic electron. The reaction in which an additional high energetic photon is produced in the final state is called deeply-virtual Compton scattering (DVCS). It is the most widely studied one since it is the theoretically cleanest process so far accessible. Another process is the exclusive production of mesons, which however so far suffers from the low scale at the fixed target experiments and thus opens another exciting application for eRHIC. A first model dependent extraction (see Figure 2) of the total angular momentum of u (Ju) and d quarks (Jd) has been performed by the HERMES and the Hall-A Collaborations, using DVCS data on transversely polarized hydrogen and unpolarized hydrogen/deuterium, respectively. As can be seen from the HERMES result, the results are very different using two different models, so for now the most important message is that a way to access Ju and Jd has been identified in principle. In the future the models used nowadays should be replaced by a global fit to all available hard exclusive data, similar to the global fit used for the extraction of Δg(x) described above.

Figure 2. Results on the total angular momentum of u-quarks Ju and d-quarks Jd based on the formalism of generalized parton distributions (GPDs). The model dependent HERMES and JLab results are based on DVCS data using 2 different GPD models and only one GPD model, respectively. The QCDSF and LHPC results are from Lattice-QCD calculations, the DFJK result is based on fits to nucleon form factor data. Theoretical systematic uncertainties are unavailable at present. See text and Ref [8] for details.

Alternatively moments of GPDs can be calculated in QCD on the lattice. The results by the QCDSF and LHPC Collaborations are shown in Figure 2. The result labeled as DFJK is based on the analysis of form factor data within the GPD formalism and is valid for valence quarks only. For details on Figure 2 see [8]. It should be noted that these results have theoretical systematic uncertainties that are not shown and are difficult to quantify at present. However, taking the results at face value, the total angular momentum carried by the quarks is about 40-50% and so the other half should come from the gluons. One might find another hint for a contribution of the glue of 50% if one recalls the discussion from above that actually only 30% of the proton's spin is carried by the quarks' spins, while in the bag model calculations 60% of the proton spin comes from the quarks. While these conclusions are premature, it is interesting to note that we know that half the momentum of the proton is carried by the gluons, so maybe indeed the same holds for the spin?

References

[1] European Muon Collaboration, J. Ashman et al., Phys.Lett.B206:364,1988

[2] HERMES Collaboration, A. Airapetian et al., Phys.Rev.D75:012007,2007, hep-ex/0609039

[3] COMPASS Collaboration, V.Yu. Alexakhin et al., Phys.Lett.B647:8,2007, hep-ex/0609038

[4] PHENIX Collaboration, A. Adare et al., Phys.Rev.D76:051106,2007, arXiv:0704.3599

[5] STAR Collaboration, B.I. Abelev et al., arXiv:0710.2048

[6] D. de Florian, R. Sassot, M. Stratmann, W. Vogelsang, arXiv:0804.0422

[7] X. Ji, Phys.Rev.Lett.78:610,1997, hep-ph/9603249

[8] HERMES Collaboration, A. Airapetian et al., arXiv:0802.2499