Page 1

arXiv:nucl-ex/0601025v2 19 May 2006

Polarization transfer in the d(? e,e′? p)n reaction up to Q2=1.61 (GeV/c)2

B. Hu,1M.K. Jones,2P.E. Ulmer,2H. Arenh¨ ovel,3O.K. Baker,1W. Bertozzi,4E.J. Brash,5J. Calarco,6

J.-P. Chen,7E. Chudakov,7A. Cochran,1S. Dumalski,5R. Ent,7,1J.M. Finn,8F. Garibaldi,9S. Gilad,4

R. Gilman,10,7C. Glashausser,10J. Gomez,7V. Gorbenko,11J.-O. Hansen,7J. Hovebo,5C.W. de Jager,7

S. Jeschonnek,12X. Jiang,10C. Keppel,1A. Klein,2A. Kozlov,5S. Kuhn,2G. Kumbartzki,10M. Kuss,7

J.J. LeRose,7M. Liang,7N. Liyanage,4G.J. Lolos,5P.E.C. Markowitz,13D. Meekins,14R. Michaels,7

J. Mitchell,7Z. Papandreou,15C.F. Perdrisat,8V. Punjabi,16R. Roche,14D. Rowntree,4A. Saha,7

S. Strauch,10L. Todor,2G. Urciuoli,9L.B. Weinstein,2K. Wijesooriya,8B.B. Wojtsekhowski,7R. Woo17

1Hampton University, Hampton, VA 23668

2Old Dominion University, Norfolk, VA 23529

3Johannes Gutenberg-Universit¨ at, D-55099 Mainz, Germany

4Massachusetts Institute of Technology, Cambridge, MA 02139

5University of Regina, Regina, SK, Canada S4S 0A2

6University of New Hampshire, Durham, NH 03824

7Thomas Jefferson National Accelerator Facility, Newport News, VA 23606

8College of William and Mary, Williamsburg, VA 23187

9INFN, Sezione Sanit´ a and Istituto Superiore di Sanit´ a, Laboratorio di Fisica, I-00161 Rome, Italy

10Rutgers, The State University of New Jersey, Piscataway, NJ 08855

11Kharkov Institute of Physics and Technology, Kharkov 310108, Ukraine

12The Ohio State University, Lima, OH 45804

13Florida International University, Miami, FL 33199

14Florida State University, Tallahassee, FL 32306

15The George Washington University, Washington, DC 20052

16Norfolk State University, Norfolk, VA 23504

17TRIUMF, Vancouver, B.C., Canada V6T 2A3

(Dated: February 8, 2008)

The recoil proton polarization was measured in the d(? e,e′? p)n reaction in Hall A of the Thomas Jef-

ferson National Accelerator Facility (JLab). The electron kinematics were centered on the quasielas-

tic peak (xBj ≈ 1) and included three values of the squared four-momentum transfer, Q2=0.43, 1.00

and 1.61 (GeV/c)2. For Q2=0.43 and 1.61 (GeV/c)2, the missing momentum, pm, was centered

at zero while for Q2=1.00 (GeV/c)2two values of pm were chosen: 0 and 174 MeV/c. At low pm,

the Q2dependence of the longitudinal polarization, P′

calculation. Further, at higher pm, a 3.5σ discrepancy was observed in the transverse polarization,

P′

neutron electric form factor from the analogous d(? e,e′? n)p experiment.

z, is not well described by a state-of-the-art

x. Understanding the origin of these discrepancies is important in order to confidently extract the

PACS numbers: 25.30.Fj, 13.40.Gp, 13.88.+e, 14.20.Dh

In the loosely bound deuteron, the proton and neu-

tron are expected to behave essentially as free particles

in intermediate energy nuclear reactions with appropri-

ate kinematics. This expectation and the absence of suit-

able pure neutron targets make the deuteron a natural

choice for extracting properties of the neutron. Though

the neutron elastic electric form factor has been espe-

cially difficult to extract, the use of polarized beams and

targets in?d(? e,e′n)p [1, 2] and polarized beams with neu-

tron recoil polarimetry in d(? e,e′? n)p [3, 4, 5, 6, 7] has

allowed statistically precise measurements.

For elastic electron scattering from a free nucleon, it

was shown in [8, 9] that the polarizations transferred from

a longitudinally polarized electron beam to the recoil nu-

cleon (i.e., via the (? e,e′? p) or (? e,e′? n) reaction) can be

expressed in terms of the nucleon electromagnetic form

factors. This technique has been exploited to measure

the proton electric to magnetic form factor ratio for large

values of the squared four-momentum transfer, Q2, us-

ing a hydrogen target [10, 11, 12]. In order to extract the

neutron electric form factor, the d(? e,e′? n)p reaction has

been exploited at the MIT-Bates Laboratory [3], Mainz

[4, 5, 7] and Jefferson Lab (JLab) [6].

clear effects can compromise the direct connection be-

tween the polarization transfer coefficients and the neu-

tron form factors. This is especially true of the neutron

electric form factor, given its small size relative to pos-

sible competing effects. It is therefore essential that re-

action models be tested experimentally. The present ex-

periment, employing the d(? e,e′? p)n reaction, provides the

means for evaluating the validity of extracting form fac-

tors from the polarization transfer coefficients, since the

polarization observables can be compared directly with

those obtained from a free proton target via the elastic

However, nu-

Page 2

2

p(? e,e′? p) reaction. (In addition, our data may provide

useful information for the related4He(? e,e′? p)3H experi-

ments [13, 14, 15], where the higher nuclear density likely

leads to more important nuclear effects.)

In the simplest picture of the d(? e,e′? p)n reaction, the

plane wave impulse approximation (PWIA), the proton is

knocked out by the virtual photon and is detected with-

out any further interaction with the unobserved neutron.

In this picture, the transferred polarizations (see Fig. 1

for an illustration of the coordinate system) along the

momentum transfer direction, P′

plane, perpendicular to the momentum transfer, P′

be expressed in terms of various kinematical factors and

the ratio of the proton electric and magnetic form fac-

tors (GE and GM, respectively) [16]. Various calcula-

tions [17, 18, 19, 20, 21, 22, 23] predict that polarizations

measured in the d(? e,e′? n)p and d(? e,e′? p)n reactions for

kinematics close to zero missing momentum (pm, where

? pm≡ ? q − ? p with ? q the three-momentum transfer and ? p

the momentum of the detected nucleon) are expected to

be nearly free from the effects of interaction currents [me-

son exchange currents (MEC) and isobar configurations

(IC)] as well as final-state interactions (FSI) between the

outgoing nucleons. It is precisely the predicted insensi-

tivity to such effects which made the d(? e,e′? n)p reaction

a natural choice for the extraction of the neutron electric

form factor. However, the moderate experimental accep-

tances employed in these experiments entail an average

over kinematics outside the ideal limit of pm= 0. Polar-

izations measured in the d(? e,e′? p)n reaction can test some

of the model assumptions over the kinematical range of

interest.

To date only two other experiments on the d(? e,e′? p)n

reaction exist, one performed at the Mainz Microtron

(MAMI) facility [24] and the other at the MIT-Bates

Laboratory [25]. They were restricted to squared four-

momentum transfers of Q2=0.3 (GeV/c)2(Mainz) and

Q2=0.38 and 0.50 (GeV/c)2(Bates) and also to low pm.

The data from both experiments were well described by

theoretical models. The current JLab experiment was

able to achieve higher Q2and pm values with smaller

statistical uncertainties.

Three of our kinematics settings were centered at

pm = 0, roughly covering the Q2range of the JLab

d(? e,e′? n)p experiment [6]. At each of these kinematics,

both d(? e,e′? p)n and p(? e,e′? p) data were acquired. This

allowed forming ratios of the polarizations for deuterium

and hydrogen targets, providing a measure of nuclear ef-

fects. A fourth kinematics was selected at non-zero pm,

at the intermediate Q2value, in order to test reaction

models in a region where interaction effects are expected

to be somewhat larger. Furthermore, this kinematics is

relevant for the d(? e,e′? n)p experiment given that its ac-

ceptance includes pmvalues of this magnitude.

The experiment was performed in Hall A of JLab using

the high resolution spectrometer pair. The relevant kine-

z, and in the scattering

x, can

m

p

e

p

Y

Z

X

e’

q

FIG. 1:

components. The Z axis is along the momentum transfer ? q,

the Y axis is in the direction of ? e ×? e′(where ? e and ? e′are the

momenta of the incident and scattered electron, respectively)

and the X axis is in the electron scattering plane completing

the right-handed system. Here, ? p is the momentum of the

recoiling proton, and ? pm is the missing momentum. The “out-

of-plane” angle is the angle between the two depicted planes,

the scattering plane and the hadronic plane.

Coordinate system used to define the polarization

matical parameters are given in Table I. Details of the

Hall A instrumentation are given elsewhere [26]. Elec-

trons were detected in the “Left” spectrometer while pro-

tons were detected in the “Right” spectrometer. The

targets consisted of 15 cm long liquid hydrogen and deu-

terium cells. The Left spectrometer included an atmo-

spheric pressure CO2ˇCerenkov detector used to reject π−

events. In order to reduce other backgrounds, nominal

cuts were placed on the vertex and angular variables re-

constructed at the target. Uncorrelated ep coincidences

were removed via cuts on the coincidence time-of-flight,

as well as cuts on the missing mass and missing momen-

tum. The experiment used beam currents of up to 50

µA combined with a beam polarization of 76%, mea-

sured using a Møller polarimeter.

was flipped pseudo-randomly to reduce systematic un-

certainties of the extracted polarization transfer observ-

ables.The proton spectrometer was equipped with a

focal plane polarimeter (FPP) [12]. Polarized protons

scatter azimuthally asymmetrically in the carbon ana-

lyzer of the FPP. The analyzer thicknesses employed are

given in Table II. In order to reduce Coulomb scattering

for which the analyzing power is identically zero, cuts

restricting the polar angle of the second-scattering dis-

tribution were enforced and are shown in Table II. The

resulting distributions, in combination with information

on the beam helicity, were analyzed by means of a maxi-

mum likelihood method to obtain the transferred polar-

ization components. More details on the analysis can be

found in Refs. [12, 27].

As a check, our p(? e,e′? p) data were compared with the

extracted GE/GMratio from previous experiments which

also used the recoil polarization technique. Our results,

listed in Table III and plotted as filled diamonds in Fig. 2,

The beam helicity

Page 3

3

TABLE I: Kinematics (central values) for the present exper-

iment. The beam energy was 1.669 GeV for all kinematics.

Q2

pm

Electron

Momentum

GeV/c

1.429

1.127

0.804

1.127

Electron

θLAB

degrees

24.45

42.65

66.23

42.65

Proton

Momentum θLAB

GeV/c

0.692

1.128

1.525

1.128

Proton

(GeV/c)2MeV/c

0.43

1.00

1.61

1.00

degrees

−58.97

−42.68

−28.91

−33.88

0

0

0

174

TABLE II: Thickness of the FPP graphite analyzer for each

of our kinematics. Also shown are the cuts we placed on the

polar angle of the second scattering in the FPP.

Q2

pm

Analyzer Thickness

inches

3.0

9.0

16.5

9.0

θFPP Cut

degrees

3–30

3–30

3–40

3–30

(GeV/c)2

0.43

1.00

1.61

1.00

MeV/c

0

0

0

174

are seen to agree well with previous measurements. Also

shown in Fig. 2 is µGE/GM for the Lomon GKex(02S)

form factors [28]. The Lomon form factors agree well

with the polarization transfer data in this Q2range and

were therefore incorporated in our d(? e,e′? p)n calculations

(see below).

Fig. 3 and Table IV show results for the three mea-

surements centered at pm = 0.

els show P′

x, P′

calculation. The bottom panel shows the double ra-

tio, (P′

x/P′

d(? e,e′? p)n divided by the same ratio for p(? e,e′? p). Only

statistical uncertainties are shown in the figure; the sys-

tematic uncertainties are given in the table and are dis-

cussed in detail later in the paper.

shown are from Arenh¨ ovel [23]. The plane wave Born

approximation (PWBA) calculation includes scattering

from the neutron with detection of the spectator proton.

(As our kinematics involve relatively high momentum

transfers and are centered on pm= 0, the PWBA calcu-

lation is nearly identical to the PWIA calculation which

only includes scattering from the proton.) The distorted

wave Born approximation (DWBA) includes pn final-

The top three pan-

zcompared to the PWIA

zand P′

x/P′

z)D/(P′

x/P′

z)H, defined as the ratio P′

x/P′

zfor

The calculations

TABLE III: The form factor ratio obtained from our p(? e,e′? p)

data, scaled by the proton magnetic moment, µ. The uncer-

tainties are statistical and systematic respectively.

Q2

µGE/GM

(GeV/c)2

0.43

1.00

1.61

0.994 ± 0.034 ± 0.005

0.879 ± 0.022 ± 0.013

0.865 ± 0.039 ± 0.036

0.20.4 0.6 0.8

1

1.2

1.4

1.6 1.82

Q2 (GeV/c)2

0.6

0.7

0.8

0.9

1.0

1.1

µGE/GM

MIT-Bates

Mainz

JLAB00

JLAB01

JLAB03

This experiment

Lomon

FIG. 2: (Color online) The filled diamonds are µGE/GM for

this experiment. Data from other Jefferson Lab experiments

are labeled as JLAB00 [12], JLAB01 [29] and JLAB03 [27].

Data from other laboratories are labeled as MIT-Bates [25]

and Mainz [13]. The curve shows µGE/GM for the Lomon

GKex(02S) form factors [28].

0.40.60.811.21.41.6

Q2 ( GeV/c)2

0.9

1

1.1

0.9

1

1.1

0.9

1

1.1

0.9

1

1.1

Px’/Px’(PWIA)

Pz’/Pz’(PWIA)

(Px’/Pz’)/(Px’/Pz’)PWIA

(Px’/Pz’)D/(Px’/Pz’)H

FIG. 3: The open circles are the MIT-Bates data [25] and the

filled squares represent the data from the present experiment.

The dot-dashed curves are for PWBA, the dotted curves are

for DWBA, the dashed curves include MEC and IC and the

solid curves are the full calculations which also include rela-

tivistic corrections (RC). The top two panels show P′

normalized to the PWIA calculation. The third panel shows

P′

bottom panel shows the double ratio, defined in the text.

xand P′

z,

x/P′

zcompared to the same ratio calculated in PWIA. The

state rescattering (FSI). The DWBA+MEC+IC calcu-

lation includes also non-nucleonic currents (MEC and

IC) and the full calculation (DWBA+MEC+IC+RC)

further includes relativistic contributions of leading or-

der in p/m to the kinematical wave function boost and

to the nucleon current. The Bonn two-body interaction

Page 4

4

TABLE IV:

P′

centered at pm = 0. Also shown are the double ratios, de-

fined in the text. The uncertainties are statistical and sys-

tematic respectively. For P′

tainty includes a contribution from the statistical uncertainty

in our extraction of the analyzing power, Ac, amounting to

∆Ac/Ac = 2.7%, 1.4% and 2.3% for Q2= 0.43, 1.00 and 1.61

respectively.

The polarizations, P′

z, as a function of Q2for the d(? e,e′? p)n measurements

xand P′

z, and the ratio,

x/P′

xand P′

z, the statistical uncer-

Q2

P′

x

P′

z

(GeV/c)2

0.43

1.00

1.61

−0.218 ± 0.008 ± 0.0006 0.236 ± 0.008 ± 0.0009

−0.299 ± 0.006 ± 0.003

−0.279 ± 0.011 ± 0.011

0.557 ± 0.009 ± 0.003

0.722 ± 0.024 ± 0.004

P′

x/P′

z

(P′

x/P′

z)D/(P′

x/P′

z)H

0.43

1.00

1.61

−0.924 ± 0.029 ± 0.005 0.926 ± 0.044 ± 0.0005

−0.537 ± 0.010 ± 0.008 1.001 ± 0.030 ± 0.0007

−0.387 ± 0.015 ± 0.016 1.077 ± 0.070 ± 0.0015

[30] and the Lomon GKex(02S) nucleon form factors [28]

were used. The models were acceptance averaged using

MCEEP [31] via interpolation over a kinematical grid.

The polarizations computed by Arenh¨ ovel were rotated

from the center-of-mass system into the coordinate sys-

tem of Fig. 1 within MCEEP. Radiative folding was car-

ried out within the framework of Borie and Drechsel [32].

It can be seen that the predicted nuclear effects are quite

small for these kinematics. However, the full calculation

does not give the correct Q2dependence for P′

per degree of freedom of the three P′

to the full calculation is 5.9/3, implying a 12% probabil-

ity that our data are consistent with the theory. Given

the somewhat poorer statistical uncertainties, the χ2per

degree of freedom for the double ratio deviates from the

full calculation by 3.9/3, implying a 27% probability of

consistency. As can be seen from Fig. 2, our highest Q2

p(? e,e′? p) datum lies above the world average. Coupled

with the relatively larger uncertainty of this datum, the

double ratio at this Q2agrees better with theory than

the single ratio, P′

x/P′

lowest Q2point is the only one within the proton ki-

netic energy range used to determine the Bonn potential.

Two-photon exchange processes, not included in our cal-

culations, are estimated to have only minor effects on the

transferred polarizations in the elastic p(? e,e′? p) reaction

[33]. The effects on P′

xand P′

than 0.5% for Q2= 1 over the entire ǫ (longitudinal pho-

ton polarization) range. Since our d(? e,e′? p)n kinematics

are on the quasifree peak, we expect the effects of two-

photon exchange to be of similar size.

z. The χ2

zdata points relative

z. It should be cautioned that the

zare estimated to be less

In Fig. 4 and Table V the pm dependence of the po-

larizations, P′

xand P′

P′

x/P′

tistical uncertainties are plotted in the figure; the rela-

z, as well as the polarization ratio,

z, is shown for Q2= 1.00 (GeV/c)2. Only the sta-

tively smaller systematic uncertainties are given in the

table caption. The group of points at low pm were ob-

tained by binning the data for the pm = 0 kinematics

while the pair of data points at higher pmwere obtained

by binning the data for the pm= 174 MeV/c kinematics.

The proton spectrometer angles differ between the two

kinematics which gives rise to the discontinuities in the

calculations between low and high pm. At low pm nu-

clear effects are predicted to have little influence which

is consistent with the results shown in Fig. 3. This is

expected since the latter represents an average over the

four low pmpoints in Fig. 4. At high pmnuclear effects

and especially relativistic effects are significantly larger.

For P′

zat high pm, the data and full calculation agree

while for P′

xthere is a 3.5σ discrepancy, after combining

the two highest pmdata points.

The discrepancy observed at our high pm kinematics

may have serious implications for the d(? e,e′? n)p experi-

ment. In fact, since nuclear effects are predicted to be

larger for the neutron experiment (comparison between

Arenh¨ ovel’s calculations for the present experiment and

for the d(? e,e′? n)p experiment [34] suggest that nuclear

effects are four to six times larger for the neutron case

at the lowest and highest Q2kinematics, respectively),

one might expect any deviation from the calculation to

be larger as well. Without knowledge of the dependence

of the discrepancy on pmand on the out-of-plane angle

(see Fig. 1) one cannot quantitatively assess the effect on

the neutron experiment. However, under certain assump-

tions, one can make an estimate. To this end, we assume

that the discrepancy is proportional to pm(and therefore

zero at pm = 0) and has no dependence on the out-of-

plane angle. In this case, our discrepancy would imply a

(6±2)% effect on the neutron form factor at the interme-

diate Q2, where we have weighted over the acceptance

of the neutron experiment. This assumes that there is

no magnification in the effect between the d(? e,e′? p)n and

d(? e,e′? n)p experiments. If, on the other hand, we use the

ratio of nuclear effects within the model of Arenh¨ ovel as

a guide, the effect on the neutron form factor increases

to (27±8)%. We caution that these estimates involve a

host of assumptions. Only additional data can answer

the question definitively.

The breakdown of the systematic uncertainties for P′

P′

z, P′

The uncertainties are dominated by uncertainty in the

precession of the proton’s spin in the spectrometer mag-

netic fields. The spin precession is characterized by a

rotation matrix which relates the polarizations measured

with the FPP to the polarizations at the experimental

target, P′

xand P′

COSY [35] transport program applied to the magnetic el-

ements of the Hall A Right spectrometer. While COSY

employs a differential algebraic method to calculate the

transfer matrix, the spin matrix can also be calculated

using a geometric model [12]. In the latter approach the

x,

x/P′

zand (P′

x/P′

z)D/(P′

x/P′

z)His given in Table VI.

z. The matrix was obtained using the

Page 5

5

-0.3

-0.25

Px′

0.55

0.6

0.65

Pz′

-500 50 100 150 200

pm (MeV/c)

-0.5

-0.4

Px′/Pz′

FIG. 4:

a function of pm at Q2= 1.00 (GeV/c)2.

ues shown correspond to cross section weighted averages.

The labeling of the theoretical curves is the same as for

the previous figure. At low pm all curves except for the

solid (DWBA+MEC+IC+RC) are essentially indistinguish-

able. For P′

(DWBA+MEC+IC) curves are indistinguishable.

The polarizations P′

x, P′

z and the ratio P′

x/P′

z as

The pm val-

z at high pm the dotted (DWBA) and dashed

TABLE V:

along with their statistical uncertainties as a function of pm

at Q2= 1.00 (GeV/c)2. The statistical uncertainty includes

a contribution from the statistical uncertainty in Ac, amount-

ing to 1.35%, except for the highest two pm points where it is

negligible. The systematic uncertainties are essentially inde-

pendent of pm and are estimated to be 0.004, 0.002 and 0.008

for P′

The polarizations P′

x, P′

z and the ratio P′

x/P′

z

x, P′

zand P′

x/P′

zrespectively.

pm

P′

x

P′

z

P′

x/P′

z

MeV/c

−57

−26

−0.299 ± 0.010

−0.281 ± 0.009

−0.311 ± 0.010

−0.318 ± 0.013

−0.281 ± 0.026

−0.262 ± 0.027

0.556 ± 0.013

0.541 ± 0.013

0.570 ± 0.013

0.574 ± 0.017

0.578 ± 0.029

0.623 ± 0.033

−0.539 ± 0.019

−0.520 ± 0.018

−0.545 ± 0.019

−0.553 ± 0.025

−0.485 ± 0.052

−0.420 ± 0.050

26

56

135

170

elements of the spin matrix are based on the proton’s

bend angles in the spectrometer. Since the uncertainties

in the bend angle can be measured, this approach facili-

tates estimation of the precession-related systematic un-

certainties. So, although COSY was used to extract the

target polarizations from those measured at the FPP, the

geometric model was employed to estimate our system-

atic uncertainties. In order to improve the knowledge of

systematics for the general program of Hall A recoil po-

larization experiments, two dedicated experiments were

conducted to determine the magnitude of the bend an-

gle in the non-dispersive plane along with its uncertainty.

The uncertainty of the bend angle in the dispersive plane

was measured independently during the experiment of

Ref. [12]. The geometric model was then used to estimate

the resulting systematic uncertainties on P′

systematic uncertainties on P′

uncertainties in the bend angle in the non-dispersive and

dispersive planes, respectively).

(P′

x/P′

completely cancels since the outgoing protons from both

reactions travel through essentially the same magnetic

fields. Finally, especially for the lowest Q2measurement,

uncertainty in knowledge of the azimuthal angle of the

proton in the FPP makes a significant contribution to

the overall systematic uncertainty.

For p(? e,e′? p), both P′

xand P′

uct hAc (where h is the beam polarization and Ac is

the analyzing power of the FPP) and the proton form

factor ratio, GE/GM. Therefore, measurement of both

polarization components in p(? e,e′? p) allows determina-

tion of GE/GM and the product hAc. The analyzing

power can then be determined since h is measured in-

dependently with the Møller polarimeter. Note that an

uncertainty in h induces an uncertainty in Ac. However,

assuming that h does not change between the consecu-

tive p(? e,e′? p) and d(? e,e′? p)n measurements, any uncer-

tainty in this quantity will completely cancel against the

induced uncertainty in Acin our extraction of P′

for the d(? e,e′? p)n measurement. Our extraction of Acis

mostly sensitive to the uncertainty in P′

to uncertainty in the dispersive bend angle. However,

an uncertainty in the dispersive bend angle will induce

uncertainties in both Acand P′

tially cancel one another, thus, effectively reducing the

contribution of the dispersive bend angle to the total

systematic uncertainty on P′

ing power is relatively insensitive to P′

to the uncertainty in the non-dispersive bend angle and

so no such compensation exists for P′

systematic uncertainty in P′

xreceives contributions from

both Ac and the non-dispersive bend angle. The ana-

lyzing power cancels in P′

x/P′

certainty on P′

x/P′

dispersive and non-dispersive bend angles.

In conclusion, we have measured the d(? e,e′? p)n and

p(? e,e′? p) reactions at Q2= 0.43, 1.00 and 1.61 (GeV/c)2

for pm= 0 and at Q2= 1.00 (GeV/c)2for pmup to 170

MeV/c in Hall A of JLab. At low pm, the longitudinal

polarization, P′

with the reaction model for the deuteron. At high pm,

the same model fails to describe the transverse polariza-

tion, P′

x. These discrepancies indicate that nuclear effects

in the d(? e,e′? p)n reaction are not thoroughly understood

and further study of this reaction is needed. The dis-

crepancies also suggest that nuclear corrections in the

related neutron electric form factor experiments need to

be studied further.

We acknowledge the outstanding support of the staff of

the Accelerator and Physics Divisions at Jefferson Lab-

oratory that made this experiment successful. We also

xand P′

z(the

xand P′

zare dominated by

For the double ratio,

z)D/(P′

x/P′

z)H, the systematic uncertainty almost

zdepend on the prod-

xand P′

z

zand therefore

zfor d(? e,e′? p)n which par-

z. In contrast, the analyz-

xand therefore

x. Therefore, the

zand so the systematic un-

zreceives contributions from both the

z, exhibits a Q2dependence at variance

Page 6

6

TABLE VI: The breakdown of systematic uncertainties for

each kinematics. The values shown represent absolute uncer-

tainties on the various quantities. Here θbend and φbend refer

to the uncertainties arising from imperfect knowledge of the

dispersive and non-dispersive bend angles in the spectrom-

eter, respectively, while φFPP denotes the uncertainty from

the azimuthal angle in the FPP. The θbend contribution to

the uncertainty in P′

xis dominated by the uncertainty in our

extraction of the analyzing power (see the text for details).

The “Total” uncertainty is the quadrature sum of the various

contributions. Note that, due to correlations, the uncertainty

in P′

zis not simply the quadrature sum of the uncertain-

ties in P′

x/P′

xand P′

z.

Q2= 0.43 (GeV/c)2

pm = 0 MeV/c

θbend

φbend

φFPP

Total

P′

x

P′

z

P′

x/P′

z

(P′

(P′

x/P′

x/P′

z)D

z)H

0.00000

0.00015

0.00050

0.00056

0.00070

0.00015

0.00050

0.00087

0.0029

0.0015

0.0037

0.0050

0.00005

0.00045

0.00000

0.00045

Q2= 1.00 (GeV/c)2

pm = 0 MeV/c

θbend

φbend

φFPP

Total

P′

x

P′

z

P′

x/P′

z

(P′

(P′

x/P′

x/P′

z)D

z)H

0.0029

0.0005

0.0009

0.0031

0.0027

0.0003

0.0007

0.0028

0.0074

0.0012

0.0024

0.0079

0.00006

0.00064

0.00018

0.00067

Q2= 1.61 (GeV/c)2

pm = 0 MeV/c

θbend

φbend

φFPP

Total

P′

x

P′

z

P′

x/P′

z

(P′

(P′

x/P′

x/P′

z)D

z)H

0.011

0.001

0.001

0.011

0.0040

0.0002

0.0011

0.0042

0.016

0.001

0.002

0.016

0.0014

0.0006

0.0000

0.0015

Q2= 1.00 (GeV/c)2

pm = 174 MeV/c

θbend

φbend

φFPP

Total

P′

x

P′

z

P′

x/P′

z

0.0038

0.0007

0.0011

0.0040

0.0019

0.0002

0.0007

0.0020

0.0071

0.0023

0.0023

0.0078

acknowledge useful suggestions of R. Schiavilla.

work was supported in part by the U.S. Department

of Energy Contract No. DE-AC05-84ER40150 Modifi-

cation No. M175 under which the Southeastern Univer-

sities Research Association (SURA) operates the Thomas

Jefferson National Accelerator Facility.

edge additional grants from the U.S. DOE and NSF, the

Italian INFN, the Canadian NSERC and the Deutsche

Forschungsgemeinschaft (SFB 443).

This

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