Precision Measurements of the Proton Elastic Form Factor Ratio

Precision Measurements of the Proton Elastic Form Factor Ratio

D. W. Higinbotham

New high precision polarization measurements of the proton elastic form factor ratio in the Q range from 0.3 to 0.7 [GeV/c] have been made. These elastic H(e,e’p) measurements were done in Jefferson Lab’s Hall A using 80 longitudinally polarized electrons and recoil polarimetry. For Q greater than 1 [GeV/c], previous polarization data indicated a strong deviation of the form factor ratio from unity which sparked renewed theoretical and experimental interest in how two-photon diagrams have been taken into account. The new high precision data indicate that the deviation from unity, while small, persists even at Q less than 1 [GeV/c].

Proton Form Factor

1 Introduction

Understanding the four-momentum transfer squared, Q, dependence of the proton’s electro-magnetic form factors is fundamental to understanding the proton and as well as a necessary input parameter in many calculations. Cross section measurements generally show that the ratio of the proton’s electric to magnetic form factor is basically unity Qattan et al. (2005), while at Q[GeV/c] recoil polarization measurements Jones et al. (2000); Gayou et al. (2002, 2001); Punjabi et al. (2005) as well a beam-target polarization measurement Jones et al. (2006) have observed a deviation from unity. At this time, the most likely cause for the difference between the cross section results and the polarization results is the two-photon part of the radiative corrections Guichon and Vanderhaeghen (2003).

2 Radiative Corrections

It is important to understand that two-photon diagrams have been included in the standard cross section radiative corrections such as Mo and Tsai Mo and Tsai (1969), but they have been included with varying degrees of approximation. For example, the Feynman diagrams for Mo and Tsai’s approach are shown in Fig. 1 where the proton’s structure is neglected and the two-photons were only allowed to have either all the four-momentum transfer or none.

Figure 1: Shown are the standard Feynman diagrams included in the calculation of radiative corrections in a lepton-hadron scattering experiment. The thin line represents the lepton, the thick line the hadron and the wavy line(s) the virtual photon(s).

Recent calculations of radiative corrections not only integrate over all possible photon energies and the proton’s structure, but even allow the proton within the box-diagrams to be off-shell as shown in Fig. 2. These effects make the calculations particularly challenging as it is the proton’s structure that one is trying to determine Afanasev et al. (2005); Blunden et al. (2005); Borisyuk and Kobushkin (2009); Kivel and Vanderhaeghen (2009) In fact, the unexpected discrepancy in the cross section and asymmetry measurements should, in the long run, dramatically improve our understanding of not only radiative corrections, but also the proton’s structure.

Figure 2: Shown are the handbag and pQCD diagrams for handling the two-photon corrections in a more detailed picture. In the handbag diagram the virtual photons couple to a single parton, while in the pQCD diagram, the virtual photons couple to different partons with a gluon coupling the last two partons together.

3 Low Q Form Factor Measurements

The observed discrepancy between cross section and asymmetry measurements as well as the calculation of the two-photon corrections has focused on Q [GeV/c]. Previous low Q polarization measurements Crawford et al. (2007); Ron et al. (2007) perhaps saw hints of an effect for Q [GeV/c], but certainly nothing definitive. Experiment E08-007 at Jefferson Lab Arrington et al. (2008) made a high precision survey of the form factor ratio in the Q range from 0.3 to 0.7 [GeV/c] using the recoil polarization technique. The experiment was done in Hall A Alcorn et al. (2004) with a High Resolution Spectrometer (HRS) detecting the elastically scattered proton and the BigBite magnet de Lange et al. (1998a, b) along with a lead glass calorimeter used for tagging the elastic electrons. In addition, a 6 cm long liquid hydrogen target was used along with 80 longitudinally polarized electrons from the Continuous Electron Beam Accelerator Facility (CEBAF) Leeman et al. (2001).

As knowledge of the spin precession within the high resolution spectrometer is critical for the extraction of the form factors, three different HRS momentum settings were used for each of the kinematics shown in Table 1. This was possible since the nominal momentum bite of the HRS is 4.5%, so by choosing momenta of and around the nominal elastic kinematic settings, we changed the spin precession while still staying well within the nominal acceptance of the device. Each of these settings was also measured to 1-2% statistics so that systematic effects could be studied and not confused with statical fluctuations. It is worth noting that spin precession of spin-1/2 particles though a dipole magnetic field is extremely well understood; it is possible to calculate the thousands of degrees of spin precession the polarized electrons at CEBAF undergo as they travel five times around the accelerator Higinbotham (2009). In order to also take into account the effects of the super-conducting quadrupoles in the HRS, COSY, a spin transport modelling program, is used. For events with rays that pass near the central ray of the spectrometer, the COSY model and the simple dipole model agree. Without using COSY, as one goes away from the central ray in angle, a strong slope can be seen in the form factor ratio. This slope goes away once the COSY spin matrix is applied.

The preliminary results of this experiment indicate that the deviation of the ratio of the form factor smoothly continues into the low Q and that there is no sharp transition to unity around Q equal to one. It does appear that a rapid change in the ratio must occur for the Q less then 0.3 [GeV/c] either with a rise to zero such as in relativistic pQCD models de Melo et al. (2009) or with a rise above one near Q of 0.1 [GeV/c] and then a return to unity at a Q of zero.






0.30 60.0 0.565 30.0 1.03
0.35 57.5 0.616 30.0 1.01
0.41 55.0 0.668 35.0 0.978
0.45 53.0 0.710 35.0 0.954
0.50 51.0 0.752 40.0 0.928
0.55 49.0 0.794 40.0 0.901
0.60 47.0 0.836 45.0 0.874
0.70 43.5 0.913 50.0 0.823
Table 1: Shown are the central angles and momentum for the high resolution spectrometer used for proton detection and the kinematics for the BigBite spectrometer used for electron detection. For each , three different spectrometer momentum settings were used; all within the nominal momentum acceptance of the spectrometer.

4 Nucleons in Deuterium

The simplest system in which to study a bound nucleon is the deuteron. Polarized beam and vector polarized target experiments at NIKHEF showed that at low Q the D(e,e’p)n reaction asymmetry was the same as the H(e,e’p) elastic asymmetry and only at missing momentum greater than approximately  [MeV/c] was an appreciable deviation observed Passchier et al. (2002). The MIT-Bates recoil polarization experiment also didn’t see any large difference between hydrogen elastic and low missing momentum deuteron quasi-elastic scattering Milbrath et al. (1998, 1999). This seems to contradict the recoil polarization results of B. Hu et al. Hu et al. (2006) where a deviation was reported. Since the data of B. Hu was taken with the same equipment as the new high precision hydrogen data, it is straightforward to include the new results and look at the ratios again as shown in Table 2. By doing this, the of the hypothesis of a ratio of one goes from 1.7 to 0.7. Thus, within experimental uncertainties, the low missing momentum data are consistent with no effect. It is now clear that what drove the original deviation was a high free proton form factor ratio. The current result again validates the common technique of extracting neutron properties from reactions on neutrons in a deuteron target, with small theoretical corrections.




Old Ratios

New Proton

New Ratio

0.43 0.92 0.03 0.99 0.03 0.92 0.04 0.93 0.01 0.99 0.03
1.00 0.88 0.01 0.88 0.02 1.00 0.03
1.61 0.93 0.04 0.87 0.04 1.08 0.07
Table 2: Shown is the results from B. Hu Hu et al. (2006) along with the new proton form factor data from E08-007. By including the new data, the of a flat line fit of the ratio goes from 1.7 to 0.7; thus indicating the low (e,e’p) reaction from deuterium is consistent with the free proton case. Do note that for the high point of 1.61, there are large systematics errors in determining the individual ratios that cancel in the ratio.

5 Future Experiments

It would be very interesting to push high precision proton form factor measurements to lower Q, but the technique of recoil polarization requires protons with reasonable momenta, so it is not feasible to go to lower Q with the setup described herein. With an electron-proton collider, it would be possible to get to lower Q due to the boost of the proton’s momentum, or one can do the measurement with a polarized beam and a fixed target where only the scattered electron needs to be detected. Such an experiment is planned for 2012 at Jefferson Lab as the second half of experiment E08-007 Arrington et al. (2008) and will cover the Q range from 0.015 to 0.4 [GeV/c].

If two-photon diagrams are truly the solution to the discrepancy between the cross section and the asymmetry experiments, this should be clearly seen in the upcoming high precision electron-proton and positron-proton cross section ratio experiments such as the Jefferson Lab’s Hall B experiment Brooks et al. (2004) presented herein by B. Raue (FIU), the VEPP-III experiment Arrington et al. (2004) and the Olympus experiment at DESY Kohl (2009).

I thank Carl Carlson (W&M) for teaching me about the history of two-photon corrections and Ph.D. student Xiaohui Zhan (MIT) for her outstanding work analyzing the new low Q data. This work was supported by the U.S. Department of Energy, and Jefferson Science Associates which operates the Thomas Jefferson National Accelerator Facility under DOE contract DE-AC05-06OR23177.


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