Low Q^2 measurements of the proton form factor ratio $mu_p G_E / G_M$
ABSTRACT We present an updated extraction of the proton electromagnetic form factor
ratio, mu_p G_E/G_M, at low Q^2. The form factors are sensitive to the spatial
distribution of the proton, and precise measurements can be used to constrain
models of the proton. An improved selection of the elastic events and reduced
background contributions yielded a small systematic reduction in the ratio mu_p
G_E/G_M compared to the original analysis.
- SourceAvailable from: John Robert Arrington[Show abstract] [Hide abstract]
ABSTRACT: A high precision measurement of the proton elastic form factor ratio muGE/GM in the range of Q^2=0.3 - 0.7 (GeV/c)^2 has been made using recoil polarimetry in Jefferson Lab Hall A. In this low Q^2 range, previous data (BLAST: C. B. Crawford et al. 2007, Phys. Rev. Lett. 98 052301, LEDEX: G. Ron et al. 2007, Phys. Rev. Lett. 99 202002) along with many fits and calculations indicate substantial deviations of the ratio from unity, and continue to suggest that structures might be present in the individual form factors, and in the ratio. In this new measurement, we used the high resolution sepctrometer to detect recoil protons incoincidence with the elastic scattered electrons tagged by BigBite calorimeter. With 80% polarized electron beam for 24 days, we are able to achieve ˜0.5% statistical uncertainty. This high precision result will confirm or refute all existing suggestions of few percent structures in the form factors ratio. Beyond the intrinsic interest in nucleon structure, the improved form factor measurements also have implications for DVCS, determinations of the proton Zemach radius and for parity violation experiments.01/2007;
arXiv:1103.5784v2 [nucl-ex] 30 Aug 2011
Low Q2measurements of the proton form factor ratio µpGE/GM
G. Ron,1,2,3X. Zhan,4J. Glister,5,6B. Lee,7K. Allada,8W. Armstrong,9J. Arrington,10A. Beck,4,11
F. Benmokhtar,12B.L. Berman,13W. Boeglin,14E. Brash,15A. Camsonne,11J. Calarco,16J. P. Chen,11
Seonho Choi,7E. Chudakov,11L. Coman,17B. Craver,17F. Cusanno,18J. Dumas,19C. Dutta,8R. Feuerbach,11
A. Freyberger,11S. Frullani,18F. Garibaldi,18R. Gilman,19, 11O. Hansen,11D. W. Higinbotham,11
T. Holmstrom,20C.E. Hyde,21H. Ibrahim,21Y. Ilieva,13C. W. de Jager,11X. Jiang,19M. Jones,11A. Kelleher,20
E. Khrosinkova,22E. Kuchina,19G. Kumbartzki,19J. J. LeRose,11R. Lindgren,17P. Markowitz,14
S. May-Tal Beck,4,11E. McCullough,5M. Meziane,20Z.-E. Meziani,9R. Michaels,11B. Moffit,20B.E. Norum,17
Y. Oh,7M. Olson,23M. Paolone,24K. Paschke,17C. F. Perdrisat,20E. Piasetzky,25M. Potokar,26
R. Pomatsalyuk,27,11I. Pomerantz,25A. J. R. Puckett,4V. Punjabi,28X. Qian,29Y. Qiang,4R. Ransome,19
M. Reyhan,19J. Roche,30Y. Rousseau,19A. Saha,11A.J. Sarty,5B. Sawatzky,17,9E. Schulte,19M. Shabestari,17
A. Shahinyan,31R. Shneor,25S.ˇSirca,32,26K. Slifer,17P. Solvignon,10J. Song,7R. Sparks,11R. Subedi,22
S. Strauch,24G. M. Urciuoli,33K. Wang,17B. Wojtsekhowski,11X. Yan,7H. Yao,9and X. Zhu34
(The Jefferson Lab Hall A Collaboration)
1The Weizmann Institute of Science, Rehovot 76100, Israel
2Lawrence Berkeley National Lab, Berkeley, CA 94720, USA
3Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, Israel 91904
4Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
5Saint Mary’s University, Halifax, Nova Scotia B3H 3C3, Canada
6Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada
7Seoul National University, Seoul 151-747, Korea
8University of Kentucky, Lexington, Kentucky 40506, USA
9Temple University, Philadelphia, Pennsylvania 19122, USA
10Argonne National Laboratory, Argonne, Illinois, 60439, USA
11Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
12University of Maryland, Baltimore, Maryland, USA
13George Washington University, Washington D.C. 20052, USA
14Florida International University, Miami, Florida 33199, USA
15Christopher Newport University, Newport News, Virginia, 2360X, USA
16University of New Hampshire, Durham, New Hampshire 03824, USA
17University of Virginia, Charlottesville, Virginia 22904, USA
18INFN, Sezione Sanit´ a and Istituto Superiore di Sanit´ a, Laboratorio di Fisica, I-00161 Rome, Italy
19Rutgers, The State University of New Jersey, Piscataway, New Jersey 08855, USA
20College of William and Mary, Williamsburg, Virginia 23187, USA
21Old Dominion University, Norfolk, Virginia 23508, USA
22Kent State University, Kent, Ohio 44242, USA
23Saint Norbert College, Greenbay, Wisconsin 54115, USA
24University of South Carolina, Columbia, South Carolina 29208, USA
25Tel Aviv University, Tel Aviv 69978, Israel
26Institute “Joˇ zef Stefan”, 1000 Ljubljana, Slovenia
27Kharkov Institue, Kharkov 310108, Ukraine
28Norfolk State University, Norfolk, Virginia 23504, USA
29Duke University, Durham, NC 27708, USA
30Ohio University, Athens, Ohio 45701, USA
31Yerevan Physics Institute, Yerevan 375036, Armenia
32Dept. of Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
33INFN, Sezione di Roma, I-00185 Rome, Italy
34Duke University, Durham, North Carolina 27708, USA
(Dated: September 1, 2011)
We present an updated extraction of the proton electromagnetic form factor ratio, µpGE/GM,
at low Q2. The form factors are sensitive to the spatial distribution of the proton, and precise
measurements can be used to constrain models of the proton. An improved selection of the elastic
events and reduced background contributions yielded a small systematic reduction in the ratio
µpGE/GM compared to the original analysis.
PACS numbers: 13.0.Gp, 13.60.Fz, 13.88.+e, 14.20.Dh
We present a detailed reanalysis of polarization trans-
fer measurements of the proton form factor ratio
µpGE/GM initially presented in Ref. , with improved
selection of elastic events and significantly reduced con-
tamination from quasielastic events in the target win-
dows. The new results are typically lower by ∼1%.
The electric and magnetic form factors, GE(Q2) and
GM(Q2), describe the distribution of charge and magne-
tization in the proton. The form factors are extracted in
elastic electron–proton scattering and mapped out as a
function of the four-momentum transfer squared, Q2, to
yield the momentum-space structure of the proton. Pre-
cision measurements of proton form factors over a large
kinematic range can provide important constraints on
models of the proton. However, when extracting the form
factors from unpolarized cross section measurements us-
ing the Rosenbluth separation technique, it is difficult to
precisely separate GE from GM in the proton for very
high or very low Q2values. The addition of polariza-
tion measurements [2–5] allows for a much better sepa-
ration of GEand GM. Initial measurements for the pro-
ton focused on the high-Q2region [6–10], which showed
a significant falloff in the ratio µpGE/GM with Q2, in
contrast to previous extractions from Rosenbluth sepa-
rations . This difference is now believed to be due
to the contribution of two-photon exchange effects which
have a large impact on the extractions from the unpo-
larized cross section measurements but have less impact
on the polarization measurements [12–17]. These signifi-
cantly improved measurements of GEled to a great deal
of theoretical work aimed at understanding this behav-
ior [18–21], which showed, among other things, the im-
portance of quark orbital angular momentum in under-
standing the proton structure at high momentum [22–24].
These results also had a significant impact on studies of
the correlations between the spatial distribution of the
quarks and the spin or momentum they carry, showing
that the spherically symmetric proton is formed from a
rich collection of complex overlapping structures .
While initial investigations focused on extending pro-
ton measurements to higher Q2, the polarization mea-
surements can also be used to improve extractions at low
Q2values, providing improved precision and less sensi-
tivity to two-photon exchange corrections. The low-Q2
form factors relate to large-scale structures in the pro-
ton’s charge and magnetization distributions. As such,
it has long been believed that the “pion cloud” contri-
butions, e.g. the fluctuation of a proton into a virtual
neutron–π+system, will be important at low Q2, as the
mass difference means that the pion will contribute to
the large distance distribution in the bound nucleon–pion
system. It was recently suggested that such structures
are present in all the nucleon form factors , centered
at Q2≈ 0.3 GeV2, and that these structures reflect con-
tributions from the pion cloud of the nucleon. However,
the significance of the proposed structures and their in-
terpretation as a pion cloud effect have been much dis-
puted. This low-Q2region is also important in parity vi-
olating electron scattering measurements [27–30] aimed
at investigating the strange-quark contributions to the
proton electromagnetic structure. Isolating the strange
quark contributions relies on precise determinations of
the proton form factors at low Q2, including the impact
of two-photon exchange corrections  (discussed fur-
While the form factors encode information on the spa-
tial structure of the proton, there are theoretical issues
in extracting the spatial charge and magnetization distri-
butions, discussed in detail elsewhere [32–36]. However,
the difficulty in extracting true rest-frame distributions
for the proton does not interfere with the comparison
of form factor measurements and proton size/structure
corrections to atomic levels in hydrogen. Extractions of
the proton charge radius [37–41] define the proton root-
mean-square (RMS) radius as the slope of the form fac-
tor at Q2= 0. This definition is consistent with the
RMS radius needed in Lamb shift measurements in hy-
drogen  and muonic hydrogen . Corrections to
the hyperfine splitting [44–46] are also extracted directly
from the form factors. The charge radius is of particular
interest at present, due to the conflicting results between
Lamb shift measurements on muonic hydrogen  and
the electron scattering results and measurements from
the Lamb shift in electronic hydrogen .
This experiment was motivated by the ideas discussed
above: mapping out the large scale proton structure, the
benefit of improved precision in proton form factors in or-
der to extract strange quark form factors (and ultimately
the proton weak radius) from parity violating measure-
ments, and the importance of reducing the uncertainty in
hyperfine splitting calculations arising from proton finite-
II. PREVIOUS MEASUREMENTS
Since the 1960s, measurements of the unpolarized
cross section for elastic e–p scattering have been used
to separate GE and GM.
portional to (τG2
ε = (1+ 2(1+(Q2/4m2
while varying ε allows for a “Rosenbluth separation” 
of the contributions from GE and GM.
the factor of τG2
Mdominates, as τ becomes large and
makes extraction of GE difficult, as it contributes only
a small, angle-dependent correction to the larger cross
section contribution from GM. Similarly, in the limit of
very small Q2, and thus very small τ, it is difficult to iso-
late GM except in the limit where ε → 0, i.e. scattering
angle → 180◦.
Polarization measurements are sensitive to the ratio
GE/GM and thus, when combined with cross section
measurements, can cleanly separate the electric and mag-
netic form factors, no matter how small their contribu-
tion to the cross section becomes. It has been known for
some time [2–5] that measurements of polarization ob-
servables would provide a powerful alternative to Rosen-
bluth separation measurements, but only in the last
The cross section is pro-
E), where τ = Q2/4m2
p))tan2θ/2)−1. Keeping Q2fixed
At high Q2,
E(with GM/GE = µp at Q2= 0).This
decade or so have the high polarization, high intensity
electron beams been available, combined with polarized
nucleon targets or high efficiency nucleon recoil polarime-
The first such measurements for the proton [6, 7]
showed a decrease in µpGE/GM with Q2, which dif-
fered from the existing Rosenbluth separation measure-
ments, which showed approximate form factor scaling,
i.e. µpGE/GM ≈ 1. This discrepancy appeared to be
larger than could be explained even accounting for the
scatter in the previous Rosenbluth measurements .
A measurement using a modified Rosenbluth extraction
technique  was able to extract the ratio µpGE/GM
with precision comparable to the polarization measure-
ments, and showed a clear discrepancy, well outside
of the experimental systematics for either technique.
Experiments extending polarization measurements to
higher Q2show a continued decrease of µpGE/GM with
Q2[7, 9, 10].
It was suggested that the two-photon exchange (TPE)
correction may be able to explain the discrepancy be-
tween the two techniques [12, 16]. While these correc-
tions are expected to be of order αEM ≈ 1%, they can
have a similar ε dependence to the contribution from GE.
Because the contribution to GE is small at large Q2, a
TPE correction of a few percent could still be significant
in the extraction of GE. It was estimated that a TPE
contribution of ∼5%, with a linear ε dependence, could
explain the difference [12, 49], and early calculations sug-
gested effects of a few percent, with just such a lin-
ear ε-dependence [16, 50]. These corrections should also
modify the polarized cross section measurements, but it
should be a percent-level correction in the extraction of
GE/GM, as there is is no equivalent amplification of the
effect. Including the best hadronic calculations available
yields consistency between the two techniques, and good
separation of GEand GM up to high Q2[13, 15]. Com-
parisons of electron–proton and positron–proton scatter-
ing can be used to isolate TPE contributions , and
a series of such measurements are currently planned or
At low Q2values, the TPE should be well described
by the hadronic calculations [13, 55], and in fact the con-
tributions are small for 0.3 < Q2< 0.7 GeV2. While
this is a region where high precision Rosenbluth separa-
tions are possible, measurements prior to 2010 had rel-
atively large uncertainties, typically 3–5% or more on
µpGE/GM. Measurements using polarization observ-
ables in this region can provide a significant improvement
in precision, even in this low Q2regime. The MIT-Bates
BLAST experiment made measurements of µpGE/GM
using a polarized target  for 0.15 < Q2< 0.6 GeV2,
with typical uncertainties around 2%, about a factor of
two improvement over most earlier data. The experi-
ment, which provided the best knowledge of the low Q2
proton form factor ratio when published, measured values
below unity for Q2> 0.2 GeV2, but concluded that the
overall results were consistent with unity over the range
of the experiment.
data, which showed a clear deviation of µpGE/GM for
Q2>∼0.8 GeV2, this suggested that the ratio was unity
at very low Q2and then began to fall somewhere in the
range of 0.2–0.7 GeV2. The fact that there was no clear
indication of where the ratio began to fall below unity
was one of the motivating factors for this measurement.
The updated results of this reanalysis of  provide an
independent extraction of µpGE/GM in this kinematic
region, with precision comparable to the BLAST results.
More recently, JLab experiment E08-007 , a high-
statistics follow up to the work we present here, used
the same polarization transfer techniques but with coin-
cident detection of the final-state electron and proton for
all kinematics, yielding measurements of µpGE/GMwith
average uncertainties below 1.2%.
Last year, new measurements in this Q2region were
also obtained by an experiment at Mainz .
experiment made high-precision measurements of un-
polarized cross sections at ∼1400 kinematic points for
Q2< 1 GeV2. While they do not provide direct Rosen-
bluth extractions of GE and GM, they show a global fit
to their cross section results. Their extraction of GM is
systematically 2–4% above previous world’s data, imply-
ing a difference of 4–8% in the extrapolation of the cross
section to ε = 0. It is difficult to determine how much
of their error band could be strongly correlated in Q2,
as there is no information given on the size or sources of
systematic uncertainty assumed in their analysis. While
they apply a very limited form of the two-photon ex-
change corrections , which is neglected in most pre-
vious extractions, this should only reduce their value of
GMrelative to the uncorrected results, implying that the
true discrepancy is even larger. At this point, it is not
clear why there is such a large discrepancy between their
fit and previous measurements.
Combined with the high-Q2JLab
III. EXPERIMENT DETAILS
This experiment was carried out in Hall A of the
Thomas Jefferson National Accelerator Facility (JLab),
in the summer and fall of 2006, as part of experiment
E05-103 . While the experiment was focused on po-
larization observables in low energy deuteron photodis-
integration , elastic electron–proton scattering mea-
surements used to calibrate the focal plane polarimeter
provided high statistics data that allowed for an improved
extraction of the proton form factor ratio µpGE/GM at
A polarized electron beam was incident on a cryogenic
liquid hydrogen target, nominally 10 cm in length for
the 362 MeV beam energy running and 15 cm for the
687 MeV settings (the target length was misstated as
15 cm for all runs in the previous publication ). The
target cells are Al, with beam entrance windows about
0.1 mm thick, and beam exit and sides ∼0.2mm thick
(with some variation between the different targets). Elas-
tic e–p scattering events were identified by detecting the
struck proton in one of the High Resolution Spectrom-
eters (HRS) . Data were taken with a longitudinal
polarization of approximately 40% and with the beam
helicity flipped pseudo-randomly at 30Hz. For some set-
tings, the scattered electron was detected in the other
The polarization of the struck protons is measured in
a focal plane polarimeter (FPP) in the proton spectrom-
eter. Operation and analysis of events in the FPP is
described in detail in Refs. [8, 60]. Analysis of the angu-
lar distribution of rescattering in the polarimeter allows
us to extract the transverse polarization at the detector,
which can be used to reconstruct the longitudinal and
transverse (in-plane) components of the polarization of
the elastically scattered protons. In the Born approx-
imation, the ratio of these polarization components is
directly related to the ratio GE/GM,
= −E0+ E′
where Cz,x are the longitudinal and transverse compo-
nents of the proton polarization, E0is the beam energy,
and θeand E′are the scattered electron’s angle and mo-
mentum (reconstructed from the measured proton kine-
matics). Because the extraction of µpGE/GM depends
on the ratio of two polarization components, knowledge
of the absolute beam polarization and FPP analyzing
power are not necessary, although high polarization and
analyzing power improve the figure of merit of the mea-
In the experiment, we measure the polarization not at
the target, but in the spectrometer focal plane, and the
asymmetry in the rescattering is sensitive only to polar-
ization components perpendicular to the proton direc-
tion. If we look at the central proton trajectory, where
the spectrometer is well represented by a simple dipole,
then the transverse component, Cx, will be unchanged,
while the longitudinal component, Cz, will be precessed
in the dipole field. If we chose a spin precession angle,
χ, near 90 degrees, the longitudinal and transverse polar-
ization components at the target will yield “vertical” and
“horizontal” components in the frame of the focal plane
polarimeter, allowing for both to be extracted by a mea-
surement of the azimuthal distribution of rescattering in
the carbon analyzer. In the analysis, we use a detailed
model of the spectrometer to perform the full spin pre-
cession, rather than taking a dipole approximation, as
described in detail in Ref. .
A follow-up experiment, JLab E08-007  was pro-
posed to make extremely high precision measurements in
this kinematic regime. The measurement was run in the
summer of 2008, and in the analysis of the E08-007 data,
it was observed that the result was somewhat sensitive to
the cuts applied to the proton kinematics when isolating
elastic e–p scattering.
In this experiment, only the proton was detected for
most kinematic settings, and the elastic scattering events
TABLE I: Kinematics and FPP parameters for the measure-
netic energy, respectively. Tanalyzer is the thickness of the
FPP carbon analyzer and χ is the spin precession angle for
the central trajectory. The final column shows which kine-
matics had single-arm (S), coincidence (C), or a combination
of both (C/S).
laband Tp are the proton lab angle and proton ki-
(GeV2) (GeV) (deg) (GeV) (inches) (deg)
were isolated using cuts on the over-determined elastic
kinematics. In the original analysis , relatively loose
cuts were applied because the measurement was statis-
tics limited and little cut dependence had been observed
in previous measurements [7, 8, 62]. Most of these mea-
surements had high resolution reconstruction of both the
proton and electron kinematics, and so loose cuts on the
combined proton and electron kinematics provided clean
isolation of the elastic peak. In addition, the previous
measurements were generally at higher Q2and so of sig-
nificantly lower statistical precision, typically 3–5%, so it
was difficult to make precise evaluation of the impact of
tight cuts on the proton kinematics. Because the elastic
events could be cleanly identified without tight cuts on
the proton kinematics, this was not considered to be a
In the follow-up experiment, E08-007 , the electron
was detected in a large acceptance spectrometer with lim-
ited momentum and angle resolution. The electron detec-
tion led to significant suppression of scattering from the
target windows, but the poor electron resolution required
that the elastic peak be defined using cuts on the proton
kinematics. Because of this, and the high statistics of
the data set, it was possible to make detailed studies of
the cut dependence of the result. It was found that there
were small but noticeable changes in the extracted form
factor ratio if the proton kinematic cuts were made too
loose, even in cases where the endcap contributions were
Motivated by these issues, we reanalyzed the data from
our experiment. We include a more careful examination
of cuts used to identify the proton events and an updated
evaluation of the contribution from the target endcaps.
With our new, more restrictive cuts, there were small but
systematic changes in the extracted form factors. These
were mainly due to the reduction in the contribution from
electron scattering in the Aluminum endcaps rather than
any changes in the events corresponding to scattering
One of the most important issues in the original analy-
sis was the correction for events that came from scatter-
ing in the Aluminum endcaps of the targets, and there
were several difficulties involved in making these correc-
tions. For systematic checks, we took data with the elas-
tic peak centered on the focal plane, but also with the
spectrometer momentum approximately 2% higher and
lower, to map out the response of the FPP across the
focal plane. Data on the Aluminum dummy targets were
typically taken for only one setting, and so there was
a systematic uncertainty associated with the stability of
the size of the background contamination and the possi-
ble variation of the polarization transfer coefficients mea-
sured from the dummy target. Data were taken using
both 15 cm and 10 cm cryogenic hydrogen targets, but
only dummy foils for the standard 4 cm and 15 cm cry-
otargets were available, yielding some additional system-
atic uncertainties for the 10 cm targets. Finally, for some
runs the beam position was not perfectly stabilized on the
dummy foils and the beam, rastered to a 4 by 4 mm2spot
at the target, either partially missed the dummy foils or
impinged on both the 4 cm and 15 cm dummy targets.
While the beam position is continuously monitored and
we correct for any deviation in the event reconstruction
based on the position, the luminosity is not well known
if the beam is partially missing the foils. Therefore, the
relative normalization of the contribution from the target
endcaps and the dummy foils had to be determined by
looking at quasielastic events that are above the thresh-
old for scattering from the proton, rather than being cal-
culated directly, yielding an additional systematic in the
relative normalization of the endcaps and dummy foils.
In the original analysis, endcap scattering typically
yielded 3–5% of the cross section after all cuts were ap-
plied (much less for the two coincidence settings), so there
was a small but significant correction. Because of the is-
sues mentioned above, there were very large systematic
uncertainties associated with these corrections. In ad-
dition, the original analysis applied the full set of cuts
for elastic scattering to data from the dummy target,
yielding measurements of the polarization coefficient for
endcap scattering with extremely poor statistics and thus
large fluctuations. We now use much tighter cuts on the
reconstructed target position to try to remove most of
the endcap contributions, resulting in contributions of
<∼0.5%. While the cuts reduce the statistics of the main
measurement somewhat, the final uncertainty is often
better, as the endcap subtraction, which had large statis-
tical and systematic uncertainties, is now much smaller.
We also use looser cuts when extracting Cxand Czfrom
scattering in the aluminum endcaps, with an extra sys-
tematic uncertainty applied to account for possible cut
For elastic scattering using an electron beam with a
known energy the complete scattering kinematics can be
determined from the measurement of a single kinematic
quantity, typically the angle or energy of the final state
electron or proton. If two quantities are measured, then
the consistency of the two kinematic variables can be
used to determine if the event was associated with elastic
scattering. For this analysis, we use the proton scattering
angle and momentum to reconstruct the kinematics and
to identify elastic events. For some kinematics, the elec-
tron was also detected, which allows for almost complete
suppression of events coming from quasielastic scattering
in the aluminum entrance and exit windows of the target.
To identify the elastic peak, we use the difference between
the measured proton momentum and the momentum cal-
culated based on the measured proton angle measured
proton angle. The specific variable we use is DpKin,
which is the momentum difference, pp− pelastic(θp), di-
vided by the central momentum setting of the proton
spectrometer. This yields a fractional momentum devia-
tion from the expectation for elastic scattering.
FIG. 1: Reconstructed target position Ytg vs.
(pp− pelastic(θp))/pHRS for the measurements on the 10 cm
liquid hydrogen target (top) and on the 4 cm and 15 cm Alu-
minum “dummy” foils (bottom). Note that Ytg is the position
transverse to the spectrometer optic axis, not the position
along the beamline; this difference leads to the target dimen-
sions being reduced by a factor of ≈2 here. The elastic peak
is clearly visible at DpKin ≈ 0 for the LH2 target, while the
broad quasielastic contributions from endcap scattering are
visible at the ends of the LH2 target.
Figure 1 shows the distribution of events versus DpKin
and the reconstructed target position, Ytg, as seen by the
spectrometer. For the hydrogen target (top panel), there
is a strong peak at DpKin ≈ 0, corresponding to elastic
events. At the extreme Ytg values, there is a faint but
broad distribution corresponding to quasielastic scatter-
ing in the endcaps. We apply a cut to Ytgto remove most
of the contribution from the endcap scattering, and use
the measurements from the 4 cm and 15 cm dummy tar-
get (bottom panel) to subtract the residual contribution.
Note that for the spectra shown in Fig. 1, the length of
the LH2 target does not match either the inner or the
outer pair of foils from the dummy target. This means
that the acceptance as a function of DpKin depends on
Ytg and so will not be identical for the endcaps and the
foils in the dummy target. This is clear for the outer foils
of the dummy target, where there is a significant loss of
events at extreme positive (negative) values of DpKin
for the upstream (downstream) dummy foils.
For each Q2setting, three measurements were taken;
one with the elastic peak positioned at the central mo-
mentum of the spectrometer, and two where the elastic
peak was shifted up (down) by 2% in momentum. This
allowed us to verify that the result was independent of
the position of the events on the focal plane. However,
dummy events were typically taken at only one of these
three settings, and the extracted endcap contribution and
quasielastic recoil polarizations taken from that measure-
ment were applied to all three settings, so the DpKin
distributions will not be exactly identical, especially far
away from the elastic peak. The dummy spectra are nor-
malized to match the observed “super-elastic” contribu-
tion (DpKin > 0.03 in Fig. 2) in the LH2 data, using
only the inner foils for the dummy target, as they have
a DpKin acceptance which better matches the endcaps.
Figure 2 shows the spectra for the LH2 target (thick black
histogram) and the dummy target (grey histogram), af-
ter the dummy target has been normalized in the region
indicated by the vertical dashed lines. After normalizing
the spectra in this region, we can determine the endcap
contribution under the elastic peak. The region used to
define elastic events in the analysis is indicated by the
vertical dotted lines. We take a conservative approach
and apply a 50% systematic uncertainty to the size of
the endcap contribution when making the correction for
these events to account for the impact of the different
DpKin spectra between the endcaps and the dummy foils
and possible variation for the settings which are shifted
by ±2% in momentum.
Having determined the contribution from endcap scat-
tering, we use the data from the dummy targets to de-
termine the contributions from quasielastic scattering to
the recoil polarization components Cxand Cz. If we ap-
ply the same cuts to the dummy target as we use in the
analysis of the hydrogen, there is very little data left, and
we can not make a reliable extraction of Cxand Cz. For
the quasielastic scattering, we use all four aluminum foils
and a broader cut on DpKin to determine the quasielas-
FIG. 2: (Color online) The DpKin distribution for the hy-
drogen target (thick black histogram) and the dummy targets
(thin red histogram). The dashed vertical lines indicate the
region used to normalize the dummy contribution to match
the contribution from the aluminum endcaps of the hydrogen
target and the vertical dotted lines indicate the part of the
elastic peak used in the analysis.
tic values for Cxand Cz, and then assume that the co-
efficients are identical when looking at the central part
of the quasielastic spectrum. Comparisons showed com-
plete consistency between the extracted values of Cxand
Czwhen comparing the inner and outer dummy foils for
all kinematics, or when comparing the central part of
the quasielastic peak to the off-peak contributions. Be-
cause we could not make a precise determination of Cx
and Czwithout averaging over a larger kinematic region,
we apply an uncertainty to Cxand Cz of 0.02 and 0.05
respectively, compared to typical values for these polar-
ization components in this experiment of 0.08–0.2 for Cx
and 0.15–0.3 for Cz.
In the original analysis , the Ytgcut was loose and so
there was a large (3–5%) contribution from endcap scat-
tering which had to be subtracted. Because the tight cuts
used on the elastic events were also applied to the dummy
spectra used to subtract endcap scattering contributions,
the statistical uncertainty on these subtractions could be
very large. Therefore, fluctuations in the low statistics
dummy measurements led to large uncertainties and sig-
nificant fluctuations in the dummy-subtracted measure-
ments. In the present analysis, the endcap contributions
are greatly reduced, with a maximum contribution well
below 1%, such that the conservative systematic uncer-
tainties assumed for the dummy normalization and po-
larization coefficients yield only small uncertainties in the
final result. While the tighter cuts yield slightly reduced
statistics in the elastic peak, the total statistical uncer-
tainty is sometimes smaller because the background con-
tribution was reduced. Note that for a few settings, ad-
ditional runs were included, improving the statistics by
5–15%, but this was a small effect compared to the mod-
There were also some small changes in the evaluation
of the systematic uncertainties. In the previous analysis,
the systematic uncertainty from the endcap contribution
was folded into the reported statistical uncertainties, and
these are now part of the quoted systematics. In addi-
tion, the estimated systematics are somewhat larger than
in the previous analysis, due to a more detailed analy-
sis of the uncertainty in the spin precession through the
The proton energy loss, which can be significant for
the low Q2kinematics, was also more carefully evalu-
ated, leading to a small change in the average Q2for
each bin. For the 362 MeV running, where the proton
was detected at small angles, the energy loss depends on
the position in the target where the scattering occurs.
Figure 3 shows DpKin vs. Ytg for one of the 362 MeV
runs with an average proton energy loss is applied to all
events. Positive Ytg values correspond to the upstream
portion of the target, where all events exit through the
side of the target and travel through a constant amount
of hydrogen and aluminum and can be well corrected
assuming a fixed energy loss. Events that exit through
the downstream end of the target lose less energy because
they pass through less material, yielding a Ytg-dependent
position for the elastic peak. This yields a reduced proton
energy loss, and thus a higher apparent proton momen-
tum, for events that occur near the exit window. For
the kinematics where this effect is important, we apply
a Ytg-dependent cut, cut corresponding to a two-sigma
region around the elastic peak for each region of Ytg, as
indicated by the graphical cut displayed in Fig. 3. The re-
constructed value of DpKin is only used to select elastic
events, so while a position-dependent energy loss could
have been applied, one would still end up with the same
set of good events passing the cuts. In our approach, we
are not sensitive to any imperfections in the energy loss
correction, since we use a two-sigma cut for all Ytgvalues.
TABLE II: Kinematic-dependent cuts applied to the data.
The Ytg cut is chosen to significantly suppress any contribu-
tions from the target endcap (as shown in Fig. 3).
28.3 −0.022 < Ytg < 0.018 |δp/p| < 0.045
23.9 −0.022 < Ytg < 0.018 |δp/p| < 0.045
18.8 −0.018 < Ytg < 0.012 |δp/p| < 0.045
14.1 −0.014 < Ytg < 0.010 |δp/p| < 0.045
47.0 −0.025 < Ytg < 0.020 |δp/p| < 0.040
44.2 −0.025 < Ytg < 0.020 |δp/p| < 0.040
40.0 −0.028 < Ytg < 0.022 |δp/p| < 0.040
34.4 −0.024 < Ytg < 0.020 |δp/p| < 0.040
The kinematic-dependent cuts are detailed in Table II.
In addition, several cuts were applied to all kinemat-
ics. A cut was applied on the out-of-plane angle, |θtg| <
0.06 rad, and the in-plane angle, |φtg| < 0.03 rad, to
ensure events were inside of the angular acceptance of
FIG. 3: (Color online) The reconstructed target position, Ytg
vs. DpKin, the deviation of the momentum from the ex-
pected elastic peak position.
energy loss is applied, but there is a significant difference for
events on the upstream side of the target, which exit through
the side wall of the target, and events which occur nearer
the downstream end of the target and have less energy loss.
The band indicates the graphical cut placed on these runs, to
approximate a two-sigma range for each Ytg value.
A correction for the average
the spectrometer. A two-sigma cut was applied on the
DpKin peak, with a Ytg-dependence cut for the low en-
ergy kinematics to account for the position-dependent
average energy loss as shown in Figure 3. The tracks
before and after the Carbon analyzer were used to deter-
mine the scattering location and scattering angle in the
analyzer. Events were required to have the secondary
scattering occur within the analyzer, and angle between
5 and 50◦were accepted. In addition, we apply a cone
test  to ensure that there is complete azimuthal ac-
ceptance in the FPP. We do this by requiring that the
FPP would have accepted events with any azimuthal an-
gle given the reconstructed vertex and scattering angle.
This ensures that any asymmetry in the acceptance or
distribution of events does not lead to a difference in the
scattering angle distribution for vertical and horizontal
rescattering. A significant difference between the rescat-
tering distribution for vertical and horizontal rescattering
events would yield a different average analyzing power,
and the analyzing power would not cancel out in the ratio
of polarization components.
The combination of the more restrictive cuts on the
elastic events and the associated reduction in contami-
nation due to scattering from the target windows leads
to a reduction in the extracted ratio that is typically
at the 1–2% level. The largest effect is due to the im-
proved correction for endcap scattering, mainly due to
cuts which significantly reduced the size of this contri-
bution. There is also a 1% reduction in the coincidence
settings, where there are negligible endcap contributions,
which is due to the tighter cuts on the proton kinemat-
ics. Tight elastic kinematics cuts using just the proton
will remove events where there is a larger than average
TABLE III: Systematic uncertainties on R = µpGE/GM. See
text for details.
(GeV2) (endcap) (optics) (cuts)
error in the reconstruction of the proton scattering an-
gle or momentum due to multiple scattering or imperfect
track reconstruction. While these errors are small, the
reconstructed kinematics are used to determine the spin
propagation through the spectrometer, and thus the im-
pact of the poor reconstruction may be amplified in eval-
uating the spin precession. Table III shows the various
contributions to the systematic uncertainty as a function
of Q2. At high Q2, the uncertainty in the spin precession
due to imperfect knowledge of the spectrometer optics
dominates. At low Q2, the uncertainty is dominated by
our ability to determine the cut dependence of the re-
sult. The cut-dependent uncertainties come mainly from
two sources; possible variation of the result due to the
cuts on ytgand DpKin. While no systematic cut depen-
dence with the ytg cut was observed, we apply a 0.4%
uncertainty as a conservative estimate based on examin-
ing the variation of µpGE/GM with the ytgcut, in par-
ticular for the coincidence data where the background
contributions are smaller. For DpKin, we estimate the
uncertainty based on varying the width of the cut around
the elastic peak. This was done for both these data and
the E08-007 results and no systematic cut dependence
was observed, so the scatter of the results was taken as
a conservative estimate of the systematic uncertainties.
Because the data taken at low beam energy do not have
sufficient statistics to set precise limits, we fit the un-
certainties from the higher energy measurements and the
E08-007 results and find a behavior consistent with 1/Q4,
which we use to obtain the quoted uncertainties for the
The results of the reanalysis are given in Table IV and
shown in Figure 4, which presents the updated results
along with previous measurements and a selection of fits.
The updated analysis yielded a systematic decrease of
∼1% in the extracted ratio, except for the highest Q2
point which decreased by 5%. The analyzing power has
been extracted from these data , but the quality of
this extraction does not impact these results, as the ana-
lyzing power cancels out in the ratio of Eq. 1. However,
because the FPP efficiency and analyzing power are sig-
nificantly lower for the data taken at a beam energy of
362 MeV (due to the lower proton momentum and thin-
ner analyzer), the statistical uncertainty in these points is
much larger. The two-photon exchange corrections from
the hadronic calculation of Blunden, et al.  are 0.35%
for the data below Q2= 0.3 GeV2and 0.2% for the
higher Q2points. This is well below the statistical and
systematic uncertainties for all points, and no correction
(or uncertainty) for the TPE effects is included in the
Punjabi et al.
Crawford et al.
Paolone et al.
X. Zhan et al.
Bernauer et al.
Friedrich & Walcher fit
Updated global fit
Arrington, Melnitchouk & Tjon fit
FIG. 4: (Color online) The proton form factor ratio as a func-
tion of Q2(with the four low-Q2measurements combined into
one data point) shown with previous extractions with total
uncertainties ≤3%. The curves are various fits [7, 13, 26, 41],
while the dot-dashed curve and associated error bands show
the result of the fit to the recent Mainz measurements .
The results in Fig. 4 show that the original conclu-
sions of  are largely unaffected. The new results sup-
port even more clearly the conclusion that the ratio
µpGE/GM is below one even for these low Q2values,
with the change from previous Rosenbluth separations
being driven mainly by a change in GE, with a smaller
change in GM. The previous hint of a local minimum
near Q2= 0.35 − 0.4 GeV2was a consequence of the
point near 0.5 GeV2, and there is no longer any indica-
tion for this in our measurement. These results further
support the observation that the decrease of the ratio
below unity occurs at low Q2, and thus we expect that
there will be a slightly larger impact on the extraction of
strange quark contributions, as discussed in the original
A comparison of the high-precision measurements at
low-Q2shows some small but systematic differences. The
TABLE IV: Experimental Results. R is given along with its
statistical and systematic uncertainties. The last column (f) is
the fractional contribution from scattering in the target end-
caps, along with the statistical uncertainty; a 50% systematic
uncertainty is also applied. The contribution is negligible for
the coincidence settings. For Q2= 0.474 GeV2, dummy mea-
surements were taken at all three sub-settings, and the range
of results is given.
low-statistics point below Q2= 0.3 GeV2.
†The final entry is the average of the four
R = µpGE/GM
0.8250 ± 0.0483 ± 0.0162
0.9433 ± 0.0414 ± 0.0144
0.9882 ± 0.0420 ± 0.0132
0.9833 ± 0.0349 ± 0.0124
0.9320 ± 0.0123 ± 0.0119
0.9318 ± 0.0098 ± 0.0108 –/0.40(2)
0.9172 ± 0.0109 ± 0.0105
0.9225 ± 0.0160 ± 0.0127
0.9465 ± 0.0204 ± 0.0137n/a
results from the Mainz cross section measurements 
are 1–2% above the recoil polarization measurements
from this work and the lower Q2results from the re-
cent JLab E08-007 measurement , although they are
in agreement with the E08-007 results at higher Q2val-
ues.One concern for the results extracted from the
Mainz cross section measurements is the sensitivity to
two-photon exchange (TPE) corrections .
kinematics of the Mainz experiment, these corrections
are fairly small,<∼2%, but this is very large compared to
the statistical (<∼0.2%) and systematic (<∼0.5%) uncer-
tainties applied in the global fit to GE and GM. Thus,
if ignored, this could yield significant corrections com-
pared to the quoted uncertainties. Coulomb corrections
were applied using the prescription of McKinley and Fes-
hbach , which corresponds to the Q2= 0 limit of
the Coulomb distortion correction (the soft-photon ap-
proximation of the full TPE corrections). However, over
much of the Q2range of the experiment, applying the
Q2= 0 correction is worse than neglecting the correction
altogether, as the Coulomb correction changes sign at
Q2≈ 0.15 GeV2. An estimate of the impact of these
corrections  suggests that an improved prescription
would decrease µpGE/GM extracted from the cross sec-
tions by 1–3% for Q2>∼0.1 GeV2in a direct Rosenbluth
separation, although a more complete analysis is required
to determine the impact on their global fit. Bernauer et
al., have examined the impact of these corrections 
in more detail, although the TPE prescription they ap-
ply  is only valid up to Q2= 0.1 GeV2. They find
that their extracted value for µpGE/GMchanges by more
than the total quoted uncertainty for all Q2values up to
0.1 GeV2, and while they find smaller changes at larger
Q2values, this is where the prescription is not expected
to be valid.
If the Coulomb corrections bring the Rosenbluth 
extractions into agreement with the recoil polarization
data, there is still a small systematic disagreement be-
tween these and the polarized target measurements from
BLAST . At this point, we are unaware of any the-
oretical argument that would explain a difference be-
tween the results of the two different polarization tech-
niques.This discrepancy can be further examined in
the second phase of the JLab E08-007 experiment 
which will make extremely high precision measurements
of µpGE/GM down to Q2≈ 0.015 GeV2, allowing for
a comparison with the BLAST measurements using the
same basic technique.
Punjabi et al.
Crawford et al.
Paolone et al.
X. Zhan et al.
Boffi et al. PFCCQM
Faessler et al. LCQM
Belushkin, Hammer & Meissner
de Melo et al.
µpGE/GM vs. Q2, compared to several low-Q2models. The
curves shown are Miller’s light-front cloudy-bag model calcu-
lation ; Boffi’s point-form chiral constituent quark model
calculation ; Faessler’s light-front quark model calcula-
tion ; Lomon’s vector-meson dominance model ; the
dispersion analysis of Belushkin, Hammer and Meissner ;
and the model of de Melo et al. .
(Color online) The proton form factor ratio
Figure 5 shows the measurements compared to a set of
theoretical curves. The first type of calculation is based
on the constituent quark models, which was quite suc-
cessful in describing the ground state baryon static prop-
erties. To calculate the form factors, relativistic effects
need to be considered. Miller  performed a calcula-
tion in the light-front dynamics including the effect from
the pion cloud. Boffi et al.  performed a point form
calculation in the Goldstone boson exchange model with
point-like constituent quarks. Faessler et al.  used a
chiral quark model where pions are included perturba-
tively and dress the bare constituent quarks by mesons
in a Lorentz covariant fashion. Another group of calcu-
lations is based on the Vector Meson Dominance (VMD)
picture, in which the scattering amplitude is written as
an intrinsic form factor of a bare nucleon multiplied by
an amplitude derived from the interaction between the
virtual photon and a vector meson. This type of mod-
els usually involve a number of free parameters for the
meson mass and coupling strength. Lomon [72, 75, 76]
performed the VMD fits by including additional vector
mesons and pQCD constraints at large Q2. Belushkin et
al.  performed a calculation using dispersion relation
analysis with additional contribution from ρπ and K¯K
continua. More recently, de Melo et al.  performed a
calculation in the light-front VMD model by considering
the non-valence contribution of the nucleon state. While
most of the theoretical curves are a few percent higher,
the calculations of Miller  and de Melo et al.  gen-
erally reproduce the large deviation from µpGE/GM= 1
in this low Q2region, emphasizing the pion cloud or non-
Our new results, as well as other high-precision mea-
surements at low Q2, show that µpGE/GM < 1 even
down to very low values of Q2. A global fit  to the
cross section and polarization measurements in this re-
gion, including the data presented in this work, indicates
that GEis ∼2% below previous fits that did not include
the low Q2polarization measurements, while GM is ap-
proximately 1% higher. These small changes in the low
Q2form factors impact other measurements as well. For
example, it was recently pointed out  that the re-
duction in the form factor yields an agreement between
studies of the asymmetry in the D(e,e′p)n reaction at low
missing momentum in polarized target measurements at
NIKHEF and MIT-Bates [78–80], and recent measure-
ments at Jefferson Lab . Similarly, this small shift in
GE and GM at low Q2modifies the expected asymme-
try in parity-violating elastic electron–proton scattering,
which serves as the baseline when extracting the strange-
quark contribution to the proton form factors [27–30].
The effect is relatively small for any given extraction,
especially at forward angles where there is a partial can-
cellation due to the changes in GE and GM. However,
because this is a systematic correction to all such mea-
surements, the updated form factors could have a small
net contribution on the extracted strange-quark contri-
The form factors at very low Q2are important in both
extracting the proton size and as input to other finite-size
corrections in atomic physics, e.g. the Zemach radius in
hyperfine splitting [44–46]. Measurements of the ratio
µpGE/GM are not, by themselves, sufficient to extract
the proton radius or calculate the finite-size corrections,
as these require the individual form factors GE(Q2) and
GM(Q2). However, the polarization measurements can
be used to improve global fits to cross section data, in
which determining the relative normalization of differ-
ent data sets is often the limiting factor in the system-
atic uncertainties . It is often possible to modify the
normalization of certain data sets and modify the ratio
µpGE/GM in such a way that the cross section mea-
surements are still relatively well fit. Thus, having direct
constraints on µpGE/GMwith high precision allows for a
better determination of these normalization factors, and
an improved extraction of the form factors. The fit of
Ref.  was updated in Ref.  to include the data
presented here as well as additional polarization mea-
surements [9, 41, 82], yielding improved extractions of
GE, GM, as well as the proton charge and magnetization
radii. While these polarization measurements do not go
as low in Q2as one would like for the extraction of the
radius, they nonetheless play an important role in the
extraction, mainly by providing improved determinations
of the relative normalization of the different experiments.
In addition, higher order terms in the Q2expansion need
to be constrained to obtain a measure of the radius .
This updated global analysis yielded an RMS charge
radius of 0.875(10) fm, consistent with the CODATA
value  and recent extraction from cross section mea-
surements from Mainz . Combined, these indepen-
dent extractions based on the electron–proton interaction
yield a radius of 0.8772(46)fm , more than seven stan-
dard deviations from the recently published PSI muonic
hydrogen Lamb shift measurement  of 0.8418(7) fm.
While higher order corrections, which depend on the val-
ues of the form factors at finite Q2, can modify the ex-
tracted radius, these corrections appear to be far too
small to explain the discrepancy, although this is still
under investigation [83–86]. A recent work  has pro-
posed a possible mechanism to explain a difference be-
tween electronic and muonic probes of the proton struc-
ture, due to off-shell effects in the hadronic intermediate
state in the two-photon exchange diagrams. However,
while this is an area that has received a great deal of at-
tention in the recent past [83–85, 88–92], the question is
The updated global fit of the low Q2data yields a mag-
netic RMS radius of 0.867(20) fm  . This is in reason-
able agreement with other extractions, 0.85(3) fm from
Kelly  and 0.857 from Hammer and Meissner .
However, neither of these included Coulomb distortion
corrections, and they are global fits that include a large
body of high-Q2data which do not contain useful in-
formation on the proton radius, and thus may not be
reliable when it comes to examining the radius. Sick 
extracted the Zemach radius, which depends on both the
charge and magnetic distributions, and converting this to
a magnetic radius yields a value of 0.855(35) fm , in
agreement with the updated fit. However, the data used
by Sick are also included in this global fit, so these can-
not be considered independent extractions. The Mainz
result  is 0.777(17) fm, well below the other electron
scattering analyses, but the strong Q2dependence of the
Coulomb distortion at very low Q2, neglected in their ex-
traction, may have a significant impact on the extraction
of the radius . An updated estimate yields a 1.5σ shift
in their magnetic radius (and almost no change in the
electric radius), yielding 0.803(17) fm , where the un-
certainty does not include any contribution related to the
two-photon exchange corrections. This increased radius
is now only 2.4σ from our result, but this updated value
still appears to be an underestimate, as their weighted
averaging of results from different fits  leads to a bias
towards fits with fewer parameters. They give systemat-
ically lower values for the magnetic radius than the fits
with more parameters, as one might expect given that
the precision of the extraction of GM tends to improve
as Q2increases, and so inclusion of the high-Q2data can
dominate the fit if there is not enough flexibility in the
fit. This does not appear to affect their extraction of the
charge radius, as the extracted radius does not show the
same systematic dependence on the number of fit param-
eters. This is not surprising, as precise extractions of GE
are not limited to higher Q2values.
Finally, one can use the discrepancy between calcula-
tions and measurements of the hyperfine splitting in the
hydrogen ground state to extract the magnetic radius, as-
suming that all of the other proton structure corrections,
including the charge radius, are well known. The analysis
by Volotka et al.  yields a radius of 0.778(29) fm, in
agreement with the Mainz result . However, an up-
dated calculation  which uses updated proton form
factors  and spin structure is also consistent with the
hyperfine splitting even though it takes a larger mag-
netic RMS radius of 0.858 fm, consistent with our re-
sults. Thus, it appears that the hyperfine splitting does
not have the sensitivity to precisely constrain the mag-
netic radius, given the current uncertainty in the other
portions of the proton structure corrections.
So while the charge radius has a clear discrepancy be-
tween muonic hydrogen lamb shift measurements and the
various measurements using the electron–proton interac-
tion, the situation for the magnetic radius is less clear.
The extractions of the magnetic radius are more sensitive
to small corrections, both for electron scattering and ex-
tractions from hyperfine splitting measurements, making
it difficult to tell at this point if there are, in fact, real
discrepancies between the techniques.
VI.SUMMARY AND CONCLUSIONS
In summary, we present an updated extraction of the
form factor ratio µpGE/GM from the data of Ref. .
We find a somewhat lower value for µPGE/GMthan the
initial extraction for the entire dataset, consistent with
two recent high precision measurements [40, 41]. The
new analysis does not change our previous conclusion,
i.e., that there is clear indication of a ratio smaller than
unity, even for low Q2, indicating the necessity of in-
cluding relativistic effects in any calculation of the form
factors in this region. Both the form factors and pro-
ton charge radius extraction from various e–p scattering
measurements are in agreement, and there is still a sig-
nificant disagreement with the charge radius as extracted
from muonic hydrogen .
This work was supported by the U.S. Department
of Energy, including contract DE-AC02-06CH11357, the
U.S. National Science Foundation, the Israel Science
Foundation, the Korea Research Foundation, the US-
Israeli Bi-National Scientific Foundation, and the Adams
Fellowship Program of the Israel Academy of Sciences
and Humanities. Jefferson Science Associates operates
the Thomas Jefferson National Accelerator Facility un-
der DOE contract DE-AC05-06OR23177.
 G. Ron et al., Phys. Rev. Lett. 99, 202002 (2007).
 A. I. Akhiezer, L. N. Rozentsweig, and I. M. Shmuske-
vich, Sov. Phys. JETP 6, 588 (1958).
 A. I. Akhiezer and M. P. Rekalo, Sov. Phys. Dokl. 13,
 N. Dombey, Rev. Mod. Phys. 41, 236 (1969).
 R. G. Arnold, C. E. Carlson, and F. Gross, Phys. Rev.
C 23, 363 (1981).
 M. K. Jones et al., Phys. Rev. Lett. 84, 1398 (2000).
 O. Gayou et al., Phys. Rev. Lett. 88, 092301 (2002).
 V. Punjabi et al., Phys. Rev. C 71, 055202 (2005),
Erratum-ibid. C 71, 069902 (2005).
 A. J. R. Puckett et al., Phys. Rev. Lett. 104, 242301
 A. J. R. Puckett et al. (2011), arXiv:1102.5737.
 J. Arrington, Phys. Rev. C 68, 034325 (2003).
 P. A. M. Guichon and M. Vanderhaeghen, Phys. Rev.
Lett. 91, 142303 (2003).
 J. Arrington, W. Melnitchouk, and J. A. Tjon, Phys.
Rev. C76, 035205 (2007).
 C. E. Carlson and M. Vanderhaeghen, Ann. Rev. Nucl.
Part. Sci. 57, 171 (2007).
 J. Arrington, P. G. Blunden, and W. Melnitchouk (2011),
 P. G. Blunden, W. Melnitchouk, and J. A. Tjon, Phys.
Rev. Lett. 91, 142304 (2003).
 J. Arrington, Phys. Rev. C 71, 015202 (2005).
 C. E. Hyde-Wright and K. de Jager, Ann. Rev. Nucl.
Part. Sci. 54, 217 (2004).
 J. Arrington, C. D. Roberts, and J. M. Zanotti, J. Phys.
G34, S23 (2007).
 C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen,
Prog. Part. Nucl. Phys. 59, 694 (2007).
 J. Arrington, K. de Jager, and C. F. Perdrisat, J. Phys.
Conf. Ser. 299, 012002 (2011).
 A. V. Belitsky, X.-d. Ji, and F. Yuan, Phys. Rev. Lett.
91, 092003 (2003).
 S. J. Brodsky, J. R. Hiller, D. S. Hwang, and V. A. Kar-
manov, Phys. Rev. D 69, 076001 (2004).
 J. P. Ralston and P. Jain, Phys. Rev. D 69, 053008
 G. A. Miller, Phys. Rev. C 68, 022201 (2003).
 J. Friedrich and T. Walcher, Eur. Phys. J. A17, 607
 D. S. Armstrong et al., Phys. Rev. Lett. 95, 092001
 K. A. Aniol et al., Phys. Lett. B 635, 275 (2006).
 K. A. Aniol et al., Phys. Rev. Lett. 96, 022003 (2006).
 A. Acha et al., Phys. Rev. Lett. 98, 032301 (2007).
 J. Arrington and I. Sick, Phys. Rev. C76, 035201 (2007).
 J. J. Kelly, Phys. Rev. C 66, 065203 (2002).
 G. A. Miller, Phys. Rev. Lett. 99, 112001 (2007).
 G. A. Miller and J. Arrington, Phys. Rev. C78, 032201
 G. A. Miller, E. Piasetzky, and G. Ron, Phys. Rev. Lett.
101, 082002 (2008).
 S. Venkat, J. Arrington, G. A. Miller, and X. Zhan,
Phys.Rev. C83, 015203 (2011).
 I. Sick, Phys. Lett. B 576, 62 (2003).
 P. G. Blunden and I. Sick, Phys. Rev. C 72, 057601
 R. J. Hill and G. Paz, Phys. Rev. D82, 113005 (2010).
 J. C. Bernauer et al., Phys. Rev. Lett. 105, 242001
 X. Zhan et al. (2011), arXiv:1102.0318.
 P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod.
Phys. 80, 633 (2008).
 R. Pohl et al., Nature 466, 213 (2010).
 S. J. Brodsky, C. E. Carlson, J. R. Hiller, and D. S.
Hwang, Phys. Rev. Lett. 94, 022001 (2005).
 C. E. Carlson, V. Nazaryan, and K. Griffioen, Phys. Rev.
A 78, 022517 (2008).
 C. E. Carlson, V. Nazaryan, and K. Griffioen, Phys. Rev.
A 83, 042509 (2011).
 M. N. Rosenbluth, Phys. Rev. 79, 615 (1950).
 I. A. Qattan et al., Phys. Rev. Lett. 94, 142301 (2005).
 J. Arrington, Phys. Rev. C 69, 022201(R) (2004).
 Y. C. Chen, A. Afanasev, S. J. Brodsky, C. E. Carl-
son, and M. Vanderhaeghen, Phys. Rev. Lett. 93, 122301
 J. Arrington, Phys. Rev. C 69, 032201(R) (2004).
 J. Arrington,D. M. Nikolenko,
for positron measurement at VEPP-3,
 W. Brooks, A. Afanasev, J. Arrington, K. Joo, B. Raue,
L. Weinstein, et al., Jefferson Lab experiment E04-116.
 M. Kohl, AIP Conf. Proc. 1160, 19 (2009).
 P. G. Blunden, W. Melnitchouk, and J. A. Tjon, Phys.
Rev. C 72, 034612 (2005).
 C. B. Crawford et al., Phys. Rev. Lett. 98, 052301 (2007).
 W. A. McKinley and H. Feshbach, Phys. Rev. 74, 1759
 R. Gilman, A. Sarty, S. Strauch, et al., Jefferson Lab
 J. Glister et al., Phys. Lett. B697, 194 (2011).
 J. Alcorn et al., Nucl. Inst. & Meth. A522, 294 (2004).
 G. Ron, J. Arrington, D. Day, R. Gilman, D. Higin-
botham, A. Sarty, et al., Measurement of the Proton
Elastic Form Factor Ratio at Low Q2, Jefferson Lab Ex-
periment Proposal E08-007 (2008).
 O. Gayou et al., Phys. Rev. C 64, 038202 (2001).
 X. Zhan, Ph.D. thesis, Massachusettes Institute of Tech-
nology (2010), arXiv:1108.4441.
 J. Glister et al., Nucl. Instrum. Meth. A606, 578 (2009).
 J. Arrington and I. Sick, Phys. Rev. C 70, 028203 (2004).
 J. Arrington (2011), arXiv:1108.3058.
 J. C. Bernauer et al. (2011), arXiv:1108.3533.
 D. Borisyuk and A. Kobushkin, Phys. Rev. C75, 038202
 G. A. Miller, Phys. Rev. C 66, 032201 (2002).
 S. Boffi et al., Eur. Phys. J. A14, 17 (2002).
 A. Faessler,T. Gutsche,
K. Pumsa-ard, Phys. Rev. D 73, 114021 (2006).
 E. L. Lomon (2006), arXiv:nucl-th/0609020.
 M. A. Belushkin, H. W. Hammer, and U. G. Meissner,
Phys. Rev. C 75, 035202 (2007).
 J. P. B. C. de Melo, T. Frederico, E. Pace, S. Pisano, and
G. Salme, Phys. Lett. B671, 153 (2009).
 E. L. Lomon, Phys. Rev. C 64, 035204 (2001).
 C. Crawford et al., Phys. Rev. C82, 045211 (2010).
 D. Higinbotham, AIP Conf.Proc. 1257, 637 (2010),
 I. Passchier, L. van Buuren, D. Szczerba, R. Alarcon,
T. Bauer, et al., Phys.Rev.Lett. 88, 102302 (2002).
 B. D. Milbrath et al. (Bates FPP), Phys. Rev. Lett. 80,
 B. D. Milbrath et al., Phys. Rev. Lett. 82, 2221(E)
 B. Hu, M. Jones, P. Ulmer, H. Arenhovel, O. Baker, et al.,
Phys.Rev. C73, 064004 (2006).
 M. Paolone et al., Phys. Rev. Lett. 105, 072001 (2010).
 A. De Rujula, Phys.Lett. B693, 555 (2010).
 I. C. Cloet and G. A. Miller, Phys.Rev. C83, 012201
 M. O. Distler, J. C. Bernauer, and T. Walcher, Phys.Lett.
B696, 343 (2011).
 B. Y. Wu and C. W. Kao (2011), arXiv:1108.2968.
 G. Miller, A. Thomas, J. Carroll, and J. Rafelski (2011),
 V. Barger, C.-W. Chiang, W.-Y. Keung, and D. Marfatia,
Phys. Rev. Lett. 106, 153001 (2011).
 U. Jentschura, Eur.Phys.J. D61, 7 (2011).
 A. De Rujula, Phys.Lett. B697, 26 (2011).
 R. J. Hill and G. Paz (2011), arXiv:1103.4617.
 J. D. Carroll,A. W. Thomas,
G. A. Miller,AIP Conf. Proc. 1354,
 H. W. Hammer and U.-G. Meissner, Eur. Phys. J. A20,
 J. C. Bernauer, Ph.D. thesis, Johannes Guttenbert-
Universitat Mainz (2010).
 A. V. Volotka, V. M. Shabaev, G. Plunien, and G. Soff,
Eur. Phys. J. D33, 23 (2005).
V. E. Lyubovitskij, and
J. Rafelski, and