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,
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 , Mainz
[4, 5, 7] and Jefferson Lab (JLab) .
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
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) . 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
To date only two other experiments on the d(? e,e′? p)n
reaction exist, one performed at the Mainz Microtron
(MAMI) facility  and the other at the MIT-Bates
Laboratory . 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
Three of our kinematics settings were centered at
pm = 0, roughly covering the Q2range of the JLab
d(? e,e′? n)p experiment . 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
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 . 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) . 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
TABLE I: Kinematics (central values) for the present exper-
iment. The beam energy was 1.669 GeV for all kinematics.
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.
are seen to agree well with previous measurements. Also
shown in Fig. 2 is µGE/GM for the Lomon GKex(02S)
form factors . 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
Fig. 3 and Table IV show results for the three mea-
surements centered at pm = 0.
els show P′
calculation. The bottom panel shows the double ra-
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 . 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
z)H, defined as the ratio P′
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.
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
FIG. 2: (Color online) The filled diamonds are µGE/GM for
this experiment. Data from other Jefferson Lab experiments
are labeled as JLAB00 , JLAB01  and JLAB03 .
Data from other laboratories are labeled as MIT-Bates 
and Mainz . The curve shows µGE/GM for the Lomon
GKex(02S) form factors .
Q2 ( GeV/c)2
FIG. 3: The open circles are the MIT-Bates data  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
bottom panel shows the double ratio, defined in the text.
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
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
The polarizations, P′
z, as a function of Q2for the d(? e,e′? p)n measurements
z, and the ratio,
z, the statistical uncer-
−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
−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
 and the Lomon GKex(02S) nucleon form factors 
were used. The models were acceptance averaged using
MCEEP  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 .
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′
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
. The effects on 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-
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.
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  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′
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
COSY  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 . In the latter approach the
z)His given in Table VI.
z. The matrix was obtained using the
-500 50 100 150 200
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′
z and the ratio P′
The pm val-
z at high pm the dotted (DWBA) and dashed
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
The polarizations P′
z and the ratio P′
−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
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. . 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).
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′
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′
certainty on 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
with the reaction model for the deuteron. At high pm,
the same model fails to describe the transverse polariza-
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
zare dominated by
For the double ratio,
z)H, the systematic uncertainty almost
zdepend on the prod-
zfor d(? e,e′? p)n which par-
z. In contrast, the analyz-
x. Therefore, the
zand so the systematic un-
zreceives contributions from both the
z, exhibits a Q2dependence at variance
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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
zis not simply the quadrature sum of the uncertain-
ties in P′
Q2= 0.43 (GeV/c)2
pm = 0 MeV/c
Q2= 1.00 (GeV/c)2
pm = 0 MeV/c
Q2= 1.61 (GeV/c)2
pm = 0 MeV/c
Q2= 1.00 (GeV/c)2
pm = 174 MeV/c
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).
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