Page 1

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

Page 2

2

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)

and ∆P?/PBorn

?

≈ ±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 [1], 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 [12]. 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

data.

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

Eand G2

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

separate G2

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 [22]:

Efrom G2

E/τG2

Mbe-

Pt = −hPe

σr

?

2?(1 − ?)

τ

?

GEGM+ GM?

?

M+ 2GM?

?

δ˜GE+

ν

M2˜F3

?

+GE?(δ˜GM) + O(e4)

hPe

σr

+ O(e4)?

σr = G2

τG2

P? =

?

1 − ?2

?

G2

?

δ˜GM+

ν

M2

?

1 + ?

˜F3

?

M+?

E+2?

?

(1 + ?)τ

2?

?

τGE?

M2˜F3

Pt

P?

?

δ˜GE+

?

GM?

??

ν

M2˜F3

?

+ 2GM?

?

+ν˜F3

M2

δ˜GM+

?ν

+ O(e4)

?

R ≡ −µp

= µpGE

1 −δ˜GM

GM

+δ˜GE

GE

1

GE

−

2?

1 + ?

1

GM

+ 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?

2

, 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.

e

ebeing the energy of the inci-

A direct

Page 3

3

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) [26].

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 [9], resulting in a very

small inelastic contamination for all three kinematics; at

? = 0.15, where it is the highest, the background fraction

is 0.7%.

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-

tively.

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 [27] 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

The difference

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

calculation.

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) [28] for R be-

?

, where PBorn

?

is

Page 4

4

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

approximation PBorn

?

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 > (◦)

1.87 2.493

2.842.490

3.632.490

?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

tot.P?/PBorn

−

1.007

1.023

?

± stat. ± p.t.p.

−

± 0.005

± 0.006

tot.

−

0.010

0.011

104.0

44.6

31.7

0.152 ±0.025

0.635 ±0.013

0.785 ±0.008

0.030

−

0.017

± 0.005

± 0.005

0.010

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

for P?/PBorn

?

. 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

P?/PBorn

?

. 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?

?

???∆P?/PBorn

?

??≤ 0.004?

. 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

[7]. The theoretical predictions are from: [14] (hadronic), [13]

(GPD), [19] (COZ and BLW) and [17] (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 [12] 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

Page 5

5

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. [14], where all

the proton intermediate states are taken into account

via a complete calculation of the loop integral using 4-

point Passarino-Veltman functions [29], shows a signifi-

cant positive TPEX contribution at small ?. The inclu-

sion of higher resonances makes almost no difference [15].

On the contrary, the partonic model of Afanasev et al.

[13], 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

[19], which uses two different light front proton distri-

bution amplitude parametrizations, one from Chernyak

et al. (COZ) [30] and the other one from Braun et al.

(BLW) [31], 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. [17], 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

P?/PBorn

?

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 [32]. 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

scattering.

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: mezianem@jlab.org

†Deceased.

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