The proton's electromagnetic form factors are one of the most basic observables which characterize the proton's finite spatial extent. One may have thought that with over half a century of measurements we would know pretty much all that there is to know about the proton's charge and magnetic form factors, at least in the kinematics accessible to experiment. It was a great surprise, therefore, when several years ago a new type of experiment revealed an unexpectedly different behavior of the form factors than had previously been accepted.

For many years, the standard way to extract the electric and magnetic form factors in elastic electron-proton scattering was via the "Rosenbluth", or longitudinal-transverse (LT) separation method, by analyzing the angular (θ) dependence of the cross section at fixed momentum transfer squared, Q². At the most backward scattering angle (θ=180 degrees) the cross section depends only on the magnetic form factor, G_M, while from the slope in θ (or rather the virtual photon polarization, ε, which depends on both θ and Q²) one extracts the ratio of the electric to magnetic form factors, G_E /G_M. These experiments found that the Q² dependence of G_E approximately follows that of G_M, although the experimental uncertainties in G_E increase significantly at large Q².

Recent measurements at Jefferson Lab using the alternative, polarization transfer (PT) technique have found a dramatically different behavior of the ratio G_E /G_M compared with the Rosenbluth results [1], leading to much discussion about the possible origin of the discrepancy. In particular, the PT results reveal a linear drop in the ratio with Q², up to the largest available value of Q² ≈ 6 GeV². With its considerably smaller systematic uncertainties, the PT method is believed to be more reliable for extracting the form factor ratio, however, experimental error in the LT measurements has also been ruled out as an explanation.

To break the impasse, attention has subsequently turned to the theoretical assumptions that go into the analysis of the data. A likely candidate that has been identified is the two-photon exchange process, shown in Fig. 1, which forms part of the electromagnetic radiative corrections that must be applied to the raw data. While the exchange of two photons gives corrections to both the LT and PT measurements at the level of a few percent, because the LT method is very sensitive to small variations in the θ dependence of the cross section, two-photon effects have a dramatically larger impact on the results from LT separation.

The first detailed calculations of the two-photon exchange amplitudes, or more precisely their interference with the one-photon exchange or Born amplitude, were made using a hadronic basis including nucleon and Δ intermediate states [2]. These were also found to have a strong angular dependence when the finite size of the nucleon was taken into account, and had the correct sign and magnitude to resolve most of the difference between the LT and PT measurements, as Fig. 2 illustrates.

In a complementary approach, two-photon contributions have also been calculated at the partonic level, in terms of generalized parton distributions [3]. These contributions, which parameterize the high-mass part of the intermediate state (in contrast to the nucleon elastic and Δ intermediate states) were also found to reduce the Rosenbluth cross sections, bringing them closer to the PT results. Between the various contributions, it is very likely that two-photon exchange provides most, if not all, of the resolution of the form factor discrepancy.

In the future it will be important to confirm the role of two-photon exchange by measuring quantities that are directly sensitive to it. One such observable is the ratio of positron-proton to electron-proton elastic cross sections. The Born amplitude changes sign under the interchange e ⁻ ↔ e⁺, while the two-photon exchange amplitude does not. The interference of the one- and two-photon exchange amplitudes therefore has the opposite sign for e ⁻ p and e⁺ p scattering, which can show up in the experimental ratio. Existing positron scattering data have large errors which preclude any definitive conclusions, however, an experiment at Jefferson Lab using a tertiary e⁺/e ⁻ beam (produced by passing bremsstrahlung photons radiated from the primary electron beam through an e⁺/e ⁻ converter) aims to place tight constraints on the two-photon exchange amplitude up to Q² ≈ 2 GeV² [4]. A related experiment is also being planned at the VEPP-3 storage ring in Novosibirsk [5].

The imaginary part of the two-photon exchange amplitude can also be accessed by measuring the electron beam asymmetry for electrons polarized normal to the scattering plane. Because this is forbidden in the Born approximation, the normal polarization provides a clean signature of two-photon exchange effects, and experiments at MIT-Bates and MAMI have indeed observed non-zero asymmetries [6]. Knowledge of the imaginary part of the two-photon exchange amplitude can be used to constrain models of Compton scattering, although relating this to the real part (which enters in the cross section measurements) requires more detailed dispersion relation analysis.

What are the wider consequences of all this? Firstly, the new PT measurements have prompted a serious rethinking of our understanding of the distribution of charge in the proton (or more illustratively, its "shape"). The new data have spawned numerous theoretical speculations about the microscopic origin of the particular Q² dependence of the electric and magnetic form factors, and progress is being made towards understanding this behavior from first principles through lattice QCD simulations. Applications of two-photon effects have also found their way recently into atomic physics, in the study of polarizability effects on hyperfine splitting in hydrogen [7]. More broadly, the critical role played by two-photon exchange in elastic scattering has exposed the inherent limitations of the Born approximation in nuclear physics, which has hitherto been a ubiquitous tool with which to probe the structure of hadrons and nuclei. The lesson learned is that even in QED - long thought to be well understood and hence uninteresting - one cannot ignore the fact that the proton has finite size.

References

1. For a recent summary see V. Punjabi et al., Phys. Rev. C 71, 055202 (2005).

2. P. Blunden, W. Melnitchouk and J.A. Tjon, Phys. Rev. Lett. 91, 142304 (2003); S. Kondratyuk et al., Phys. Rev. Lett. 95, 172503 (2005).

3. Y.C. Chen et al., Phys. Rev. Lett. 93, 122301 (2004); A.V. Afanasev et al., Phys. Rev. D 72, 013008 (2005).

4. Jefferson Lab experiment E04-116, "Beyond the Born approximation: a precise comparison of e⁺ p and e ⁻ p scattering in CLAS", W.K. Brooks, contact person.

5. J. Arrington et al., "Two-photon exchange and elastic scattering of electrons/positrons on the proton", proposal for an experiment at VEPP-3, arXiv:nucl-ex/0408020.

6. S.P. Wells et al., Phys. Rev. C 63, 064001 (2001); F.E. Maas et al., Phys. Rev. Lett. 94, 082001 (2005).

7. C.E. Carlson and M. Vanderhaeghen, arXiv:hep-ph/0701272.

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Figures

Fig. 1: Two-photon exchange diagram for elastic electron-proton scattering.

 

Fig. 2: Reduced Rosenbluth cross section σ_R (scaled by the dipole form factor G_D²) versus the virtual photon polarization ε for several values of Q²: Q² = 2.64 (filled squares), 3.2 (open squares) and 4.1 GeV² (filled circles) [from Qattan et al., Phys. Rev. Lett. 94, 142301 (2005)]. The dotted curves are Born cross sections evaluated using form factors fitted to the PT data (hence they disagree with the Rosenbluth cross sections), while the solid curves account for two-photon contributions.