Search for effects beyond the Born approximation in polarization transfer observables
in ? ep elastic scattering
M. Meziane,1, ∗E. J. Brash,2,3R. Gilman,4,3M. K. Jones,3W. Luo,5L. Pentchev,1C. F. Perdrisat,1A. J. R.
Puckett,6,7V. Punjabi,8F. R. Wesselmann,8A. Ahmidouch,9I. Albayrak,10K. A. Aniol,11J. Arrington,12A.
Asaturyan,13O. Ates,10H. Baghdasaryan,14F. Benmokhtar,15W. Bertozzi,6L. Bimbot,16P. Bosted,3W.
Boeglin,17C. Butuceanu,18P. Carter,2S. Chernenko,19E. Christy,10M. Commisso,14J. C. Cornejo,11S. Covrig,3
S. Danagoulian,9A. Daniel,20A. Davidenko,21D. Day,14S. Dhamija,17D. Dutta,22R. Ent,3S. Frullani,23H.
Fenker,3E. Frlez,14F. Garibaldi,23D. Gaskell,3S. Gilad,6Y. Goncharenko,21K. Hafidi,12D. Hamilton,24D.
W. Higinbotham,3W. Hinton,8T. Horn,3B. Hu,5J. Huang,6G. M. Huber,18E. Jensen,2H. Kang,25C.
Keppel,10M. Khandaker,8P. King,20D. Kirillov,19M. Kohl,10V. Kravtsov,21G. Kumbartzki,4Y. Li,10V.
Mamyan,14D. J. Margaziotis,11P. Markowitz,17A. Marsh,2Y. Matulenko,21, †J. Maxwell,14G. Mbianda,26D.
Meekins,3Y. Melnik,21J. Miller,27A. Mkrtchyan,13H. Mkrtchyan,13B. Moffit,6O. Moreno,11J. Mulholland,14
A. Narayan,22Nuruzzaman,22S. Nedev,28E. Piasetzky,29W. Pierce,2N. M. Piskunov,19Y. Prok,2R. D.
Ransome,4D. S. Razin,19P. E. Reimer,12J. Reinhold,17O. Rondon,14M. Shabestari,14A. Shahinyan,13
K. Shestermanov,21, †S.ˇSirca,30I. Sitnik,19L. Smykov,19, †G. Smith,3L. Solovyev,21P. Solvignon,12R.
Subedi,14R. Suleiman,3E. Tomasi-Gustafsson,31,16A. Vasiliev,21M. Vanderhaeghen,32M. Veilleux,2B.
B. Wojtsekhowski,3S. Wood,3Z. Ye,10Y. Zanevsky,19X. Zhang,5Y. Zhang,5X. Zheng,14and L. Zhu10
1The College of William and Mary, Williamsburg, Virginia 23187, USA
2Christopher Newport University, Newport News, Virginia 23606, USA
3Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
4Rutgers, The State University of New Jersey, Piscataway, New Jersey 08855, USA
5Lanzhou University,222 Tianshui Street S., Lanzhou 730000, Gansu, People’s Republic of China
6Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
7Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
8Norfolk State University, Norfolk, Virginia 23504, USA
9North Carolina A&T state University, Greensboro, North Carolina 27411, USA
10Hampton University, Hampton, Virginia 23668, USA
11California State University, Los Angeles, Los Angeles, California 90032, USA
12Argonne National Laboratory, Argonne, Illinois 60439, USA
13Yerevan Physics Institute, Yerevan 375036, Armenia
14University of Virginia, Charlottesville, Virginia 22904, USA
15Carnegie Mellon University, Pittsburgh, PA 15213, USA
16Institut de Physique Nucl´ eaire, CNRS,IN2P3 and Universit´ e Paris Sud, Orsay Cedex, France
17Florida International University, Miami, Florida 33199, USA
18University of Regina, Regina, SK S4S OA2, Canada
19JINR-LHE, Dubna, Moscow Region, Russia 141980
20Ohio University, Athens, Ohio 45701, USA
21IHEP, Protvino, Moscow Region, Russia 142284
22Mississippi State University, Starkeville, Mississippi 39762, USA
23INFN, Sezione Sanit` a and Istituto Superiore di Sanit` a, 00161 Rome, Italy
24University of Glasgow, Glasgow G12 8QQ, Scotland, United Kingdom
25Seoul National University, Seoul 151-742, South Korea
26University of Witwatersrand, Johannesburg, South Africa
27University of Maryland, College Park, Maryland 20742, USA
28University of Chemical Technology and Metallurgy, Sofia, Bulgaria
29Unviversity of Tel Aviv, Tel Aviv, Israel
30Jozef Stefan Institute, 3000 SI-1001 Ljubljana, Slovenia
31CEA Saclay, F-91191 Gif-sur-Yvette, France
32Institut f¨ ur Kernphysik, Johannes Gutenberg-Universit¨ at, D-55099 Mainz, Germany
(Dated: March 1, 2011)
Intensive theoretical and experimental efforts over the past decade have aimed at explaining the
discrepancy between data for the proton electric to magnetic form factor ratio, GE/GM, obtained
separately from cross section and polarization transfer measurements. One possible explanation for
this difference is a two-photon-exchange (TPEX) contribution. In an effort to search for effects
beyond the one-photon-exchange or Born approximation, we report measurements of polarization
transfer observables in the elastic H(? e,e?? p) reaction for three different beam energies at a fixed
squared momentum transfer Q2= 2.5 GeV2, spanning a wide range of the virtual photon polariza-
tion parameter, ?. From these measured polarization observables, we have obtained separately the
arXiv:1012.0339v2 [nucl-ex] 28 Feb 2011
ratio R, which equals µpGE/GM in the Born approximation, and the longitudinal polarization trans-
fer component P?, with statistical and systematic uncertainties of ∆R ≈ ±0.01(stat) ± 0.013(syst)
≈ ±0.006(stat) ± 0.01(syst). The ratio R is found to be independent of ? at the
1.5% level, while the ? dependence of P? shows an enhancement of (2.3±0.6)% relative to the Born
approximation at large ?.
After decades of experimental and theoretical ef-
forts, the internal structure of the nucleon remains one
of the defining problems of nuclear physics. Based on the
generally accepted notion that the electromagnetic inter-
action is well understood from a theoretical point of view,
elastic electron-nucleon scattering has served as a pow-
erful tool to measure fundamental observables: the elec-
tromagnetic form factors. There are two experimental
methods for extracting the ratio of the electric to mag-
netic form factors of the proton, GE/GM, from electron-
proton elastic scattering. In the Rosenbluth separation
technique , G2
Mare determined from the an-
gular dependence of the reduced cross section σrat con-
stant Q2. Polarization experiments determine GE/GM
by using a polarized electron beam with either a polar-
ized proton target, or a measurement of the transferred
polarization to the scattered proton. At values of the
squared-momentum-transfer, Q2≤ 1 GeV2, data from
the Rosenbluth and polarization techniques are in good
agreement. At large Q2, however, the cross section data
[2–5] disagree with the ratios GE/GMobtained using the
polarization transfer method [6–9]. This systematic dif-
ference is a source of intense debate in both the theoret-
ical and experimental nuclear physics communities.
Recently, theoretical attention has been paid to the set
of radiative corrections [10, 11] that must be made to the
cross section data in order to extract the form factor ra-
tio. These corrections change the slope of the reduced
cross section by as much as 30% for larger Q2. In con-
trast, radiative corrections to the polarization data are
essentially negligible . Until recently, only “standard”
radiative corrections were taken into account. Based on
the observed discrepancy in the data, new efforts have
been made to include higher-order radiative mechanisms,
such as two-photon exchange (TPEX) [13–19]. Several of
these calculations have indicated that TPEX partially re-
solves the disagreement between the two data sets, but
further investigation is needed. What is still lacking is
a complete set of elastic ep scattering observables sensi-
tive to the TPEX amplitudes, with sufficient precision to
guide the development of a consistent theoretical frame-
work for the interpretation of the discrepancy in terms of
TPEX. This experiment is an effort to provide additional
When considering the exchange of two or more pho-
tons, the hadronic vertex function can generally be ex-
pressed, in terms of three independent and complex am-
plitudes, ˜GE,M ≡ GE,M(Q2) + δ˜GE,M(Q2,?) and˜F3
which are functions of Q2and the kinematical parameter
? = (1 + 2(1 + τ)tan2θe/2)−1, where τ ≡ Q2/4M2, M
is the proton mass, and θeis the electron scattering an-
gle. In the Born approximation, the first two amplitudes
equal the real electric and magnetic Sachs form factors
which depend only on Q2, and˜F3vanishes. The exper-
imental observables used to extract GEpand GMpfrom
cross section and polarization transfer measurements, as-
suming the validity of the Born approximation, are af-
fected in different ways by TPEX, as shown in Eq.(1).
The Rosenbluth method relies on measuring the ? de-
pendence of the reduced cross section σrat fixed Q2to
M, and becomes highly sensitive to
additive TPEX effects at large Q2when ?G2
comes small. The transferred polarization to the recoil
proton in H(? e,e?? p) has transverse (Pt) and longitudinal
(P?) components with respect to the momentum trans-
fer in the scattering plane [20, 21]. The ratio R defined
in Eq.(1) equals µpGE/GM in the Born approximation,
and is much less vulnerable to TPEX corrections. The
TPEX corrections appear in σr, Pt, P?, and R as interfer-
ence terms between the Sachs form factors and the real
part of the TPEX amplitudes :
Pt = −hPe
2?(1 − ?)
+GE?(δ˜GM) + O(e4)
σr = G2
1 − ?2
1 + ?
(1 + ?)τ
R ≡ −µp
1 + ?
+ O(e4) (1)
where ? stands for the real part, h = ±1 and Pe are
the helicity and polarization of the electron beam, and
ν = MEe+E?
, with Ee, E?
dent and scattered electron, respectively. In the Born ap-
proximation, these corrections vanish and the well known
expressions for these observables [20, 21] are recovered.
Other observables, such as the induced normal polariza-
tion component, and the target-normal and beam-normal
single-spin asymmetries, depend only on the imaginary
(absorptive) part of the TPEX amplitude.
ebeing the energy of the inci-
test of the TPEX effect is the comparison between e+p
and e−p elastic scattering cross sections. Since the two-
photon contributions (relative to the Born amplitudes)
are of opposite sign, a few percent deviation from unity
as a function of ? is predicted for the ratio σe+/σe−. Re-
cent analyses of e±p cross sections are inconclusive due
to large uncertainties in the data [23–25].
In this experiment, carried out at Jefferson Lab in Hall
C, a longitudinally polarized electron beam (82-86% po-
larization) was scattered elastically off a 20 cm liquid hy-
drogen target at Q2= 2.5 GeV2. Electrons were detected
by a 1744 channel lead-glass electromagnetic calorimeter
(BigCal), which measured their coordinates and energy.
Overlapping analog sums of up to 64 channels were used
to form the BigCal trigger with a threshold of about half
the elastic electron energy. Coincident protons were de-
tected in the High Momentum Spectrometer (HMS) .
The HMS trigger was formed from a coincidence between
a scintillator plane located behind the drift chambers and
an additional paddle placed in front of the drift chambers.
The polarization of scattered protons, after undergoing
spin precession in the HMS magnets, was measured by
the Focal Plane Polarimeter (FPP), which consists of an
assembly of two 55 cm thick CH2analyzer blocks, each
followed by a pair of drift chambers to track re-scattered
protons with an angular resolution of approximately 1
mrad. Elastic event selection was performed offline in
the same way as explained in , resulting in a very
small inelastic contamination for all three kinematics; at
? = 0.15, where it is the highest, the background fraction
The scattered proton polarization was obtained from
the angular distribution of protons scattered in the ana-
lyzer blocks of the FPP. The polar and azimuthal scat-
tering angles (ϑ,ϕ) of single-track events in the FPP
chambers were calculated relative to the incident track
defined by the HMS drift chambers.
and the sum of the azimuthal angular distributions for
positive and negative beam helicities give the physical
(helicity-dependent) and instrumental or false (helicity-
independent) asymmetries at the focal plane, respec-
Since the proton polarization components are mea-
sured at the focal plane, knowledge of the spin transport
matrix of the HMS is needed to obtain Ptand P?at the
target. The differential-algebra based modeling program
COSY  was used to calculate the spin-transport ma-
trix elements for each event from a detailed layout of
the HMS magnetic elements. The quantities PeAyPtand
PeAyP? were extracted using the maximum-likelihood
method described in [8, 9], with Aythe analyzing power
of ? p+CH2→ one charged particle+X scattering. Their
ratio gives Pt/P?independent of Ayand Pe.
As an example of the quality of the data, Fig.1 shows
R and AyP?as a function of the vertical (dx/dz) and hor-
izontal (dy/dz) slopes of the scattered proton trajectory
relative to the HMS optical axis. Owing to the small
acceptance of the HMS in both ? and Q2for all three
kinematics, R and AyP? are constant across the accep-
tance to a very good approximation.
anomalous dependence of the extracted R and AyP?on
the reconstructed kinematics is thus an important test of
the accuracy of the field description in the COSY calcu-
lations. In each panel of Fig. 1, the data are integrated
over the full acceptance of all other variables. The hori-
zontal line shows the one-parameter fit to the extracted
data. In all panels, the χ2per degree of freedom is close
to one, indicating the excellent quality of the precession
The absence of
FIG. 1: R (left column) and AyP? (right column) versus the
dispersive dx/dz (vertical) and non-dispersive dy/dz (horizon-
tal) slopes for the low-energy setting using the COSY model.
The main results of this experiment are given in Ta-
ble I and shown in Fig. 2. Figure 2a displays R as a
function of ? with selected theoretical estimates. The
data do not show any evidence of an epsilon dependence
of R at Q2= 2.5 GeV2. Both statistical and point-to-
point systematic uncertainties (relative to the largest ?
kinematic) are shown in the figure. The total system-
atic uncertainties in R are shown in Table I. For a given
data point, the point-to-point systematics are obtained
as the quadrature sum over the differences between each
of the systematic contribution and the corresponding one
for a reference kinematic. Because the dominant sources
of systematic uncertainty described below affect R for
all three kinematics in a strongly correlated fashion, the
systematic uncertainty on the relative variation of R as
a function of ?, characterized by the point-to-point un-
certainties, is very small.
Another sensitive probe of two-photon effects is taking
the ratio of the measured P?to PBorn
P? calculated in the Born approximation. In the limit
? → 0, angular momentum conservation requires P?→ 1,
independent of R (see Eq. 1); for our measurement at
? = .15, P?varies by only 1.4% (relative)  for R be-
, where PBorn
TABLE I: Kinematic table with the average quantities: the beam energy Ee, the momentum transfer squared Q2, the electron
scattering angle θe, and the kinematical parameter ?. Both the ratio R and longitudinal polarization P? divided by Born
are given with statistical (stat.), total systematic (tot.) and point-to-point (p.t.p.) uncertainties relative
to the highest ? point for R and to the lowest ? kinematic for P?/PBorn
< Ee >(GeV) < Q2>(GeV2) < θe > (◦)
?R ± stat. ± p.t.p.
0.696 ± 0.009 ± 0.006 0.013
0.688 ± 0.011 ± 0.001 0.009
0.692 ± 0.011 ± 0.000 0.009
± stat. ± p.t.p.
tween 0 and 1. Therefore, the measured value of hAyP?
at ? = .15 determines¯Ay = 0.15079 ± 0.00038 (specific
to this polarimeter), corresponding to a relative uncer-
tainty of 0.25%, included in the statistical error budget
. Applying the same phase space cuts at
the focal plane results in Aybeing the same for all three
kinematics, at the 10−3level. PBorn
the beam energy, the proton momentum, and the fitted
value of R from this experiment, with the errors in each
quantity accounted for in the total systematic error in
. In Fig. 2b, the ratio P?/PBorn
sus ?. The results show an enhancement at large epsilon
of 2.3 ± 0.6% relative to the Born approximation.
The number of sources of systematic uncertainty is
drastically reduced by the fact that the beam helicity
and the polarimeter analyzing power cancel out exactly
in the ratio of polarization components. Consequently,
the spin precession uncertainty is the dominant contri-
bution. Since the central proton momentum was fixed
across the three kinematics, the spin transport matrix is
identical, resulting in small point-to-point systematic un-
certainties. The error ∆φbend= ±0.5 mrad in the non-
dispersive bend angle, due to uncertainty in the HMS
quadrupole positions, represents the largest contribution.
The uncertainty of ±0.1% in the absolute determination
of the proton momentum, which has a negligible effect
on the precession uncertainty in Pt/P?, leads to a rel-
ative uncertainty in the kinematic factor
that is roughly 1/3 of the relative uncertainty in Pt/P?
at the lowest ?, but negligible at higher ?. Errors in the
dispersive bend angle, the beam energy, and the scatter-
ing angle in the FPP give smaller contributions. The in-
clusion of instrumental asymmetry terms obtained from
Fourier analysis of the helicity-independent asymmetries
in the likelihood function induces a negative correction to
R (|∆R| ≤ 0.013) and P?/PBorn
for each ? value. A systematic uncertainty equal to half
the false asymmetry correction was included in the to-
tal systematic uncertainty on R and P?/PBorn
absolute systematic uncertainty (0.5% point to point)
from the M¨ oller measurements of the beam polarization
was added to the error budget of P?/PBorn
minimize systematic differences in the spin transport cal-
culation among the three kinematics, cuts were applied
was calculated from
is plotted ver-
?τ(1 + ?)/2?
. A 1%
. In order to
FIG. 2: a) R as a function of ? with statistical uncertain-
ties, filled circles from this experiment and open triangle from
. The theoretical predictions are from:  (hadronic), 
(GPD),  (COZ and BLW) and  (SF) offset for clarity
by -0.006 with respect to the fit. The one parameter fit result
is: R = 0.6923±0.0058. b) P?/PBorn
point-to-point systematic uncertainties, shown with a band in
both panels, are relative to the largest ? kinematic in a) and
relative to the smallest ? kinematic in b). The star indicates
the ? value at which the analyzing power is determined.
as a function of ?. The
to the focal plane trajectories of the data for the two
larger ? points to match the smaller acceptance of the
point at ? = 0.15. The program MASCARAD  was
used to compute “standard” radiative corrections to R.
Small, positive corrections ∆R of 1.2×10−3, 1.4×10−4
and 0.7×10−4were found for ? = 0.15, 0.63, and 0.78,
respectively. The corrections to P?/PBorn
were found to
5 Download full-text
be even smaller. The results shown in Table I do not
include these corrections.
The theoretical curves shown in Figure 2a make widely
varying predictions for the epsilon dependence of R.
The hadronic model of Blunden et al. , where all
the proton intermediate states are taken into account
via a complete calculation of the loop integral using 4-
point Passarino-Veltman functions , shows a signifi-
cant positive TPEX contribution at small ?. The inclu-
sion of higher resonances makes almost no difference .
On the contrary, the partonic model of Afanasev et al.
, where the TPEX takes place in a hard scattering of
the electron by quarks which are embedded in the nucleon
through the GPDs, predicts a negative TPEX contribu-
tion. A pQCD calculation of Kivel and Vanderhaeghen
, which uses two different light front proton distri-
bution amplitude parametrizations, one from Chernyak
et al. (COZ)  and the other one from Braun et al.
(BLW) , presents a behaviour similar to the partonic
model. The limit of applicability of the GPD and pQCD
models is shown by the vertical dashed line on the fig-
ure. The electron structure function (SF) based model
developed by Bystritskiy et al. , which takes into ac-
count all high order radiative corrections in the leading
logarithm approximation, does not predict any measur-
able ? dependence of R. The GPD, hadronic and pQCD
models, while in good agreement with the available cross
section data, predict a deviation of R at small ? due to
modification in Pt; not seen in the results presented here.
Refering to Eq.(1), R is directly proportional to the Born
value GE/GM, so all the theory predictions, which use
a GE/GM value from [6–8], can be renormalized by an
overall multiplicative factor. The enhancement seen in
is not predicted by any models. The behiavior
of R at large ? implies the same deviation of Ptfrom its
Born value as the one observed in P?/PBorn
The high precision data presented in this letter add
significant constraints on possible solutions of Eq.(1) for
the real part of the TPEX amplitudes . In this exper-
iment, no ? dependence was found in R, suggesting that
the 2γ amplitudes are small or compensate each other in
the ratio. The study of the non-linearity of the Rosen-
bluth plot, a precise measurement of the single spin asym-
metries and the determination of the σe+/σe− ratio are
essential to fully understand, quantify and characterize
the two-photon-exchange mechanism in electron proton
The GEp2γ collaboration thanks the Hall C technical
staff and the Jefferson Lab Accelerator Division for their
outstanding support during the experiment. This work
was supported in part by the U.S. Department of En-
ergy, the U.S. National Science Foundation, the Italian
Institute for Nuclear research, the French Commissariat
` a l’Energie Atomique (CEA), the Centre National de la
Recherche Scientifique (CNRS), and the Natural Sciences
and Engineering Research Council of Canada. This work
is supported by DOE contract DE-AC05-06OR23177, un-
der which Jefferson Science Associates, LLC, operates the
Thomas Jefferson National Accelerator Facility.
∗Corresponding author: email@example.com
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