The Gerasimov-Drell-Hearn Sum Rule and the Spin Structure of the Proton

The Gerasimov-Drell-Hearn Sum Rule and the Spin Structure of the Proton

G. Krein

Instituto de Física Teórica, Universidade Estadual Paulista,

Rua Pamplona 145, 01405-900, São Paulo, SP

Received on 22 February, 2001 2

We present an overview of the present theoretical status of the Gerasimov-Drell-Hearn (GDH) sum rule. It is emphasized the importance of near threshold s-wave photoproduction contributions to the sum rule. We also discuss the experimental verification of the sum rule, with particular emphasis on the recent experiment at the Mainz Microtron (MAMI). The spin polarizabilities and the generalized GDH sum rule, which are of interest to issues related to the spin of the proton as measured at high energies, are discussed.

I Introduction

The response of the internal degrees of freedom of the nucleon to an external electromagnetic field can be described in terms of polarizabilities. For real photons, polarizabilities and other ground-state properties can be related to integrals over photoabsorption cross sections by sum rules. One of the most prominent examples is the Gerasimov-Drell-Hearn (GDH) sum rule [1], which provides a relationship between the anomalous magnetic moment k_N of the nucleon and the photoabsorption cross sections for parallel and antiparallel alignments of the nucleon and photon helicities, s_3/2 and s_1/2

where I_N is the GDH integral, given by

The importance of the sum rule stems from the fact that it is based on general principles of physics, such as Lorentz and gauge invariance, crossing symmetry, causality and unitarity.

Closely related to the GDH sum rule is the forward spin polarizability [2]

This quantity is related to the Ragusa polarizabilities [3] g_i, i=1¼4, as

The Ragusa polarizabilities appear as low-energy contants of the terms (w³) in the expansion of the Compton amplitude in powers of the photon energy w.

The use of virtual photons from electron scattering processes provides us with even more detailed information on the internal degrees of freedom of the nucleon. In particular, as we increase the four-momentum Q² of the virtual photon from the real-photon point (Q² = 0) to large values of Q², we can investigate the transition from the nonperturbative to the perturbative regime of quantum chromodynamics (QCD). The generalizations of real-photon sum rules to virtual photons provide an interesting possibility to study this transition and the varying role of the relevant degrees of freedom.

In this contribution we present an overview of the present theoretical and experimental status of the GDH sum rule. We also discuss generalisations of the spin polarizability and GDH sum rule. These generalizations are of interest to issues related to the spin of the proton as measured at high energies. At the end of this contribution, we also discuss a generalization of the Baldin sum rule as a function of Q² and compare predictions based on chiral perturbation theory and a phenomenological resonance model.

II The GDH sum rule

Until recently, there was no direct measurement of the cross sections s_3/2 and s_1/2 in the integrands of the integrals above. The situation changed with the recent experiment at MAMI [4], which measured the cross-section difference Ds = - for photon lab. energies in the range 200 - 800 MeV. Before this experiment, one had to rely on estimates using pion photoproduction amplitudes for Ds, as compiled, for example, by the SAID group [5].

The decomposition of Ds in Eqs. (2) and (3) in terms of photoproduction multipoles is given as [6]

where q is the c.m. momentum of the pion and k in the c.m. momentum of the photon. Because of the weighting factor 1/w in I_GDH, it is to be expected that a large fraction of the the sum rule is saturated by s-wave near-threshold pion photoproduction and by D(1232) resonance production. This seems particularly true for g, because of the 1/w³ in the integrand of Eq. (3).

The status of the GDH sum rule until recently was, as discussed in Refs. [7] and [8], that there seemed to be a severe discrepancy between the prediction of the sum rule and the saturation of the integral with photoproduction data. For the proton, for example, the sum rule predicts

while the SAID multipoles, together with an old estimate [9] of the pNN contribution, predicted [7] I_p = -289 mb. A more recent SAID analysis, SAID-SP97K, gave [8]

The discrepancy between the prediction of the sum rule and the SAID multipoles was on the order of 30%.

Very recently, the present author together with D. Drechsel [10] showed that the discrepancy is drastically reduced when using the HDT [11] multipoles, instead of the SAID multipoles. The HDT multipoles are based on fixed-t dispersion relations and unitarity. In the HDT dispersive approach the threshold region is not included in the fitting of free parameters; the threshold values for the s-wave amplitudes are genuine predictions. The HDT analysis is limited to photon energies up to 400 MeV, s, p, and d waves for isospin 1/2, and s and p waves for isospin 3/2. Although the differences between the SAID-SP97K and HDT multipoles are very small for energies above 400 MeV, large differences occur for the E₀₊ multipole for p⁺ production close to threshold. In Fig. 1 we show the results for the integrand of the sum rule using the HDT (solid) and SAID (dashed) multipoles.

Figure 1.
Multipole contributions to the integrand of I_p: (a) E_0⁺, (b) M_1⁺, and (c) all (up to L = 5) partial waves. The solid lines correspond to the HDT multipoles and the dotted to the SP97K-SAID multipoles.

The HDT multipoles lead to I_p (1p) = - 196 mb. This implies in a change of 20 mb, compared to SAID. If we use the (very) old estimate estimate of Karliner [9] for the two-pion contribution, I_p(2p) = - 65 mb, the earlier discrepancy [7] reduces from 38 % to 28 %. Moreover, if one uses the estimate of Ref. [12] of the contributions beyond one-pion production, -32 mb, together with the single-pion HDT prediction, the discrepancy falls to only 12 % .

This value is not too far the result of the recent GDH experiment at Mainz [4], for energies up to 800 MeV. Concerning the remaining discrepancy, it will be interesting to see the outcome of the GDH experiment at ELSA at higher energies.

Regarding the spin polarizability, Eq. (3), because of the 1/w³ in the denominator, one expects that near-threshold s-wave pion photoproduction and D(1232) resonance production saturate the above integral even more rapidly than the GDH integral. This was actually verified in Ref. [13]: the integrand is already negligible at w 500 MeV.

The HDT multipoles lead for the proton

while the SP97K-SAID give

The recent Mainz experiment [4] of Ds = s_1/2- s_3/2 for energies larger than 200 MeV gives a value of g_p~ -1.7 × 10^-4fm⁴. The HDT multipoles predict that the region very close to threshold, w_th < w < 200 MeV, contributes to g_p with a positive value equal to +0.9. Therefore, to the extent that the HDT multipoles provide an accurante description of the low-energy region, one can predict that g_p = -0.8 × 10^-4fm⁴.

It is instructive to compare [13] these numbers with predictions of chiral perturbation theory. Two recent calculations, Refs. [14] and [15], in (p⁴) of HBChPT find g_p = -3.9 × 10^-4fm⁴. Another recent (p⁴) calculation [16], which differs from the previous two in the way the single-particle reducible contributions are treated, leads to g_p = -1.0 × 10^-4fm⁴. A calculation [17] using dispersion relations together with large N_c arguments leads to g_p = -0.1 × 10^-4fm⁴ .

III Generalized GDH integral and spin polarizability

One of the many possible generalizations of the GDH sum rule [18] to virtual photons is the expression

where n₀ = m_p + (m+ Q²)/2m is the threshold lab energy of one-pion production. Here, x₀ is the threshold value of x = Q²/2Mn, x₀ = Q²/(Mm_p + m + Q²), and g = Q/n. Also, s_3/2 and s_1/2 are the cross sections for the scattering of polarized electrons on polarized nucleons with parallel and antiparallel alignments of the electron and nucleon helicities, and s¢_LT is the longitudinal-transverse cross section.

Contact with deep inelastic scattering experiments can be made by expressing I₁(Q²) in terms of an integral over the spin structure function g₁(x, Q²)

A generalized polarizability can also be defined in terms of electroproduction cross sections s_3/2 and s_1/2 as [19]

There is very little experimental information on the electroproduction cross-sections in the resonance region. The situation will change in the near future with the new data from the Jefferson Lab. For the time being, one has to rely on phenomenological models. Here, we review our recent work [20] [21] on generalized GDH sum rule and spin polarizabilities. In these works we used the recently developed isobar model for electroproduction, known as MAID. The model is well documented and accessible via the Internet [22]. It is based on an effective phenomenological Lagrangian for Born terms ("background") and resonances up to the third resonance region. For each partial wave the multipoles satisfy the constraints of gauge invariance and unitarity, and in the real photon case the results agree well with the predictions of dispersion theory [11]. The model is able to describe the correct energy dependence of the multipoles for photon energies up to w 1 GeV, and it provides a good description of all experimentally measured differential cross sections and polarization observables.

The prediction of MAID is showm in Fig.??. We compare results with a recent HBChPT calculation [23] at (p⁴). For the proton, the result obtained in Ref. [23] for G₁ is given by

Being this a result of a HBChPT calculation, it is expected to be valid only for small values of Q². In the Figure we compare the predictions of MAID (solid line) and Eq. (13) (dashed line). Is is woth mentioning the the MAID result is very close to recent SLAC data [20] at Q² = 0.5 GeV². From the figure, it is seen that for Q² > 0.05 GeV², the HBChPT result becomes considerably different from the MAID prediction.

For the generalized spin polizability, a recent calculation [19] has been made in the framework of the one-loop approximation to (relativistic) chiral perturbation theory (ChPT), supplemented by tree graphs for the excitation of the D(1232) resonance in the relativistic Rarita-Schwinger formalism.

In Fig. 3 we compare results again with MAID [21]. The most striking difference refers to the slope of g_p(Q²) close to the real photon point. The pronounced slope in g_p(Q²) observed in the MAID result is due to the interference between background and D(1232) terms, which has its the physical origin in the dynamical dressing of the gN D vertex [24].

Figure 3.
Comparison of results for g_p from MAID (solid) and relativistic ChPT (dashed).

In summary, one of the striking features of the generalized GDH integral is its rapid fluctuation with Q² and in particular a change of sign at Q² 0.5 (GeV)², which imposes severe constraints on any model for the nucleon structure. This zero-crossing separates the region dominated by resonance-driven coherent processes from a region of essentially incoherent scattering off the nucleon's constituents. A similar zero-crossing is predicted by ChPT for the generalized spin polarizability, g(Q²) [19], while we found that MAID excludes such a cross-over for Q²£ (1 GeV)².

Acknowledgments Work partially supported FAPESP and CNPq.

References

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Publication Dates

• Publication in this collection
  05 Mar 2002
• Date of issue
  June 2001

History

• Received
  22 Feb 2001