Beam-normal single spin asymmetry in elastic electron scattering off 28Si and 90Zr

Abstract

We report on a new measurement of the beam-normal single spin asymmetry An in the elastic scattering of 570 MeV transversely polarized electrons off ²⁸Si and ⁹⁰Zr at Q2=0.04 GeV2/c2. The studied kinematics allow for a comprehensive comparison with former results on ¹²C. No significant mass dependence of the beam-normal single spin asymmetry is observed in the mass regime from ¹²C to ⁹⁰Zr.

Full text

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Beam-normal single spin asymmetry in elastic electron scattering off
28Si and 90Zr
A. Esser a, M. Thiel a, P. Achenbach a, K. Aulenbacher a, S. Aulenbacher a, S. Baunack a,
D. Bosnar b, S. Caiazza a, M. Christmann a, M. Dehn a, M.O. Distler a, L. Doria a, P. Ecke r t a,
M. Gorchtein a, P. Gülker a, P. Herrmann a, M. Hoek a, S. Kegel a, P. K l a g a, H.-J. Kreidel a,
M. Littich a, S. Lunkenheimer a, F.E. Maas a, M. Makek b, H. Merkel a, M. Mihoviloviˇ
ca,c,
J. Müller a, U. Müller a, J. Pochodzalla a, B.S. Schlimme a, R. Spreckels a, V. Tioukine a,
C. Sfienti a
aInstitut für Kernphys ik, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany
bDepartment of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
cJožef Stefan Institute, SI-1000 Ljubljana, Slovenia
a r t i c l e i n f o a b s t r a c t
Article history:
Received 30 April 2020
Received in revised form 27 July 2020
Accepted 27 July 2020
Avail able online xxxx
Editor: D.F. Geesaman
Keywords:
Transverse asymmetry
Elastic scattering
Polarized beam
Multi-photon exchange
We report on a new measurement of the beam-normal single spin asymmetry Anin the elastic scattering
of 570 MeV transversely polarized electrons off 28Si and 90 Zr at Q2=0.04 GeV2/c2. The studied
kinematics allow for a comprehensive comparison with former results on 12C. No significant mass
dependence of the beam-normal single spin asymmetry is observed in the mass regime from 12 C to
90Zr.
©2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.
1. Beam-normal single spin asymmetry
Measurement of parity violation in weak interactions is a well
established experimental technique in atomic, particle and nuclear
physics. Over the past 30 years, precision experiments have probed
hadron [1–6] and nuclear structure [7] and new proposals have
recently been put forward that will considerably improve our un-
derstanding of the electroweak interaction and will allow us to
explore physics beyond the standard model [8–10].
The interpretation of these future measurements requires the-
oretical predictions with uncertainties below those of the ex-
periments. To that end it is mandatory to go beyond the one-
photon exchange approximation and include higher-order correc-
tions (such as γZ-[11], γW-, [12]or γγ-box graphs [13]) in the
calculations.
The measurement of observables sensitive to two-photon ex-
change processes is essential to benchmark such higher-order cal-
culations.
E-mail address: esser@uni-mainz.de (A. Esser).
For this purpose the beam-normal single spin asymmetry (the
so-called transverse asymmetry) Anin polarized electron-nucleus
scattering is an ideal candidate. Since Anis a parity conserv-
ing asymmetry, arising from the interference of one- and two (or
more)-photon exchange amplitudes, it gives direct access to the
imaginary part of the two-photon exchange process.
Ancan be observed when the polarization vector 
Peof the
electrons is aligned parallel or antiparallel to the normal vector
ˆ
n=(
k×
k)/|
k×
k|of the scattering plane, where 
k(
k) are the
three-momenta of the incident (scattered) electrons. The measured
beam-normal single spin asymmetry in the two-photon approxi-
mation can be expressed as
An=σ↑−σ↓
σ↑+σ↓=
2ImM∗
γ·Mγγ

Mγ

2,(1)
where σ↑(σ↓) denotes the cross section for electrons with spin
parallel (antiparallel) to the normal vector ˆ
n. In Eq. (1), Im(M∗
γ·
Mγγ)denotes the imaginary part of the one- and two-photon ex-
https://doi.org/10.1016/j.physletb.2020.135664
0370-2693/©2020 The Author(s). Publishe d by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP3.

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change amplitudes Mγand Mγγ [14], respectively. The measured
asymmetry is related to Anby
Aexp =An
Pe·ˆ
n.(2)
The transverse asymmetry roughly scales as me
Eαem, with methe
electron mass, Ethe beam energy, and αem the electromagnetic
coupling constant [15]. Asymmetries as small as 10−5to 10−6are
therefore expected for beam energies of several hundred MeV. This
makes the experiments particularly challenging, as statistical and
systematic errors in the measurement need to be kept well below
10−6.
The theoretical treatment of Anis nontrivial as well since the
absorptive part of the two-photon exchange amplitude has to be
related to the sum of all possible physical (on-mass-shell) interme-
diate states. While several approaches are available to calculate the
transverse asymmetry for the reaction p
e,ep[15–18], only two
different calculations, exploiting different ansatzes, allow for an ex-
tension to nuclei with Z≥2. Cooper and Horowitz [19]are numer-
ically solving the Dirac equation to calculate Coulomb distortion
effects. To do so, they assume that only the ground state con-
tributes, especially with increasing Z. In contrast, Gorchtein and
Horowitz [20], following the approach by Afanasev and Merenkov
[17,18], include a full range of intermediate states (elastic and in-
elastic) but limit their calculation to the very low four-momentum
transfer region (mecQE/c). In this model the asymmetry
can be written as:
An∼C0log Q2
m2
ec2FCompton(Q2)
Fch(Q2),(3)
with C0being the energy-weighted integral over the total pho-
toabsorption cross section. It can be model-independently obtained
from the optical theorem and it depends on mass number Aand
charge number Zof the target nucleus. The last term in Eq. (3),
the ratio of Compton to charge form factor, allows the model to be
generalized to nuclear targets. For this, the value of the Compton
slope parameter B, given in the exponential form of the Compton
form factor (exp(−BQ2)), needs to be determined. The available
high-energy Compton scattering data on 1H and 4He (see [20]
and references therein) suggest an approximate independence of
FCompton(Q2)/ Fch(Q2)from the target nucleus, using
B≈R2
Ch
6+4
GeV2/c2(4)
for the Compton slope parameter, with RCh being the charge radius
of the nucleus. This estimate has been adopted as a reference value
for the calculation in [20]. While at low momentum transfer the
Q2dependence of Anis dominated by the logarithmic term, at
larger Q2the knowledge of the exact value of the Compton slope
parameter Bbecomes more important.
2. Previous studies
So far, the transverse asymmetry at forward angles (θ<6◦) has
been measured at the Thomas Jefferson National Accelerator Fa-
cility (JLab) for 1H, 4He, 12C, and 208 Pb [21]. Although the data
span the entire nuclear chart, a systematic interpretation in terms
of Q2, Z, and Edependence is hindered by the different kine-
matics of each measurement. A comparison to available theoretical
calculations [17,18,20]shows a good agreement for light nuclei,
with the corresponding asymmetry being dominated by inelastic
contributions. At the same time, a striking disagreement in the
case of 208Pb was observed: this may indicate the inadequacy of
the two-photon exchange (TPE) approximation in [20]given that
Tabl e 1
Measured beam-normal single spin asymmetries for each spectrometer and kine-
matical setting. The difference in scattering angle for similar Q2values is due to
the much wider angular acceptance of spectrometer A. Statistica l and systematic
(individual and total) uncertainty contributions are given in units of parts per mil-
lion (ppm). The first 5 entries are derived from the asymmetry corrections. Cur-
rentDrop denotes the error from dismissing events with short drops in the beam
current. AIand AGL correspond to the errors originating in the asymmetry of
beam current and gate-length, respectively. Gain corresponds to variations of the
PMT gain, while Ta ils accounts for possible nonlinearities for large asymmetry cor-
rections. Inversion accounts for the different number of events in both states of
the half-wave plate. BeamProfile denotes the error associated with an asymmetric
beam profile. Pcorresponds to the error of the polarization measurement.
Targ et 28 Si 90Zr
Spectrometer A B A B
Scattering angle 23.51◦19.40◦23.51◦20.67◦
Q2(GeV2/c2) 0.038 0.036 0.042 0.042
An(ppm) −23.302 −21.807 −17.033 −16.787
(∂σ/∂ x)0.007 0.003 0.003 0.010
(∂σ/∂ y)0.013 0.007 0.136 0.131
(∂σ/∂ x)0.003 0.009 0.054 0.120
(∂σ/∂ y)0.013 0.014 0.343 0.189
(∂σ/∂ E)0.099 0.053 0.321 0.259
CurrentDrop 0.006 0.029 0.015 0.031
AI0.010 0.010 0.013 0.013
AGL 0.005 0.005 0.008 0.008
Gain 0.067 0.036 0.041 0.018
Tail s −0.070 −0.189 +0.108 +0.405
Inversion +0.092 −0.039 −0.406 −0.500
BeamProfile 0.028 0.023 - -
P0.210 0.197 0.348 0.343
Total systematic error +0.553 +0.386 +1.390 +1.527
−0.531 −0.614 −1.688 −1.622
Statistical error 1.366 1.389 3.524 5.466
the expansion parameter of the perturbation theory is not small
(Zα∼1). If this were the case, the breakdown of the TPE model
of Ref. [20]should already become noticeable with intermediate
heavy nuclei, thus calling for a systematic study in this mass range.
As a first step, the Q2dependence of Anfor carbon has been
measured in the range between 0.02 GeV2/c2and 0.05 GeV2/c2.
The obtained results show a reasonable agreement with the ex-
isting theoretical calculation [22]. The deviations from the the-
oretical description have been related to the assumption of the
dominance of the log(Q2/m2
ec2)term and the independence of
FCompton(Q2)/Fch(Q2)from the target nucleus. The result empha-
sizes that the Q2behavior of the asymmetry cannot be treated in-
dependently of the target nucleus. Even larger discrepancies could
be expected for heavier nuclei.
Therefore, a new experiment has been performed with the
same setup and within the same four-momentum transfer range
with the aim of investigating heavier target materials such as 28 Si
and 90Zr.
3. New measurements
These experiments were carried out at the Mainz Microtron
MAMI [23]using the spectrometer setup of the A1 Collaboration
[24], a well established facility for high resolution spectroscopy in
electron scattering experiments. To allow for a comparison with
previous results, the data were taken with the same kinematics as
reported in [22]. Minor adjustments due to the different target ma-
terials led to slightly different spectrometer angles and Q2values
as given in Table 1. In order to study the transverse asymme-
try An, the A1 setup was slightly modified by inserting additional
fused-silica Cherenkov detectors in the focal plane of the two high-
resolution spectrometers Aand B. These detectors are capable of
handling high data rates and they allow to separate elastic from
inelastic events. Corresponding to the different focal plane geome-

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tries of spectrometers Aand B, the size of the fused-silica bars
were chosen to be (300 ×70 ×10)mm3and (100 ×70 ×10)mm3,
respectively. The fused-silica detectors were oriented at 45◦with
respect to the direction of the elastically scattered electrons in the
spectrometer. The produced Cherenkov light was collected by pho-
tomultiplier tubes (PMTs) with fused-silica windows.
In the MAMI beam source, the primary electrons were pro-
duced by illuminating a strained GaAs/GaAsP super lattice photo-
cathode with circularly polarized laser light [25,26]. In order to
measure the transverse asymmetry with the described spectrom-
eter setup, the polarization vector of the emitted – longitudinally
polarized – electrons had to be aligned vertically in order to be
perpendicular to the scattering plane. In this two-step process, the
longitudinal spin is first rotated to transverse orientation in the
horizontal plane using a Wien filter [27]. Secondly, the polariza-
tion vector is rotated to the vertical orientation using a pair of
solenoids. The polarization was verified to be solely vertical to
within 1% using a Mott polarimeter [28] located downstream of
the 3.5 MeV injector linac and a Møller polarimeter [29]close to
the interaction point in the spectrometer hall. Details on this pro-
cedure can be found in [30]. Measurements with both polarimeters
determined the absolute degree of polarization. During each ex-
perimental campaign the degree of polarization was monitored by
frequent measurements with the Mott polarimeter. The full range
of variation of the absolute degree of polarization throughout the
different measurements was between 78.2% and 83.6%.
The polarized electron beam had an energy of 570 MeV and an
intensity of 20 μA. It was impinging on a 1.1 7 g/cm2(1.11 g/cm2)
28Si (90 Zr) target with an isotopic purity of 99.9% (97.7%). Both tar-
gets needed to be cooled during the measurement to avoid a vari-
ation of their densities due to melting. For this purpose a custom-
made cooling frame was constructed. The targets, with an active
area of 10 mm ×10 mm each, were attached to a copper support
structure, which was mounted on an outer aluminum frame. In
this outer frame a mixture of water and ethanol was circulated at
a stabilized temperature Tcirc =0.5◦C. To spread the heat load of
the point-like beam spot, the electron beam was rastered over an
area of 4mm×4mm. For this purpose wobbler magnets running
synchronized to the power grid with a harmonic frequency of 2000
(2050) Hz in horizontal (vertical) direction have been used.
The fused-silica detectors had to be positioned in such a way
that they completely covered the elastic line while minimizing the
contribution of the excited states. Given that the momentum ac-
ceptance of the fused-silica bar covered only part of the whole
spectrometer acceptance, the magnetic field of the spectrometer
was set such that the elastic peak appeared at the position of the
Cherenkov detector. This geometrical adjustment was complicated
by the small distance between the elastic peak and the first ex-
cited state of only E≈1.8MeV for both targets. To verify the
exact placement of the fused-silica detectors, a low beam current
of I≈20 nA was used. In this mode, the events were processed
individually by a conventional data acquisition system measuring
timing and charge of the PMT pulses in parallel with the other
detectors in the spectrometers. The accurate position information
obtained from a set of drift chambers allowed to match the posi-
tion of the elastic line of the scattered electrons to the Cherenkov
detectors. The resulting detector coverage is illustrated in Fig. 1.
In the integrating mode of data taking used for the asymmetry
measurements, the beam current was raised to 20 μA. The current
produced by each detector PMT was integrated over 20 ms long
periods (so-called polarization-state windows) synchronized to the
power-grid frequency. These windows were arranged in a random
sequence of quadruples with the orientation of the electron beam
polarization being either ↑↓↓↑ or ↓↑↑↓. The polarization state
was reversed by setting the high voltage of a fast Pockels cell in
the optical system of the polarized electron source. A 80 μs time
Fig. 1. The exci tation energy spectra of 28 Si (top panel) and 90Zr (bottom panel)
show the acceptance of the spectrometer without (black line) and with (filled areas)
a cut on the Cherenkov detector. By changing the magnetic field of the spectrometer
the elastic peak was aligned with the position of the Cherenkov detector.
window between the polarization-state windows allowed for the
high voltage of the Pockels cell to be switched. The integrated PMT
signal for each polarization-state window was then digitized and
recorded.
In order to identify and reduce polarity correlated instrumen-
tal asymmetries, several methods have been applied to reverse the
sign of the measured asymmetry. Besides reversing the polariza-
tion vector orientation between the measuring gates, the differen-
tial electrical signal switching the polarity at the beam source was
reversed every five minutes. Additionally, a half-wave plate in the
optical system at the beam source [31]was used to reverse the
beam polarity on a time scale of 24 hours.
Fluctuations of beam parameters, such as current (I), energy
(E), horizontal and vertical position (xand y), and horizontal and
vertical slope (xand y), are partly correlated to the reversal of
the polarization vector orientation. This can introduce instrumen-
tal asymmetries. Therefore, it is of utmost importance to constantly
control these beam parameters. They have been measured by a
set of monitors: PIMO (Phase and Intensity MOnitor), ENMO (EN-
ergy MOnitor), and XYMO (XY MOnitor). Their signals were used
in a dedicated stabilization system to minimize polarity correlated
beam fluctuations (see Fig. 2) [31,32].
In parallel, the output signals of the monitors were acquired in
the same way as the detector signals to correct for instrumental
asymmetries in the offline analysis.
4. Data analysis
As a first step, all acquired values were corrected for fluctua-
tions in the integration gate length. Secondly, after the detector
signals were offset-corrected, the raw asymmetry could be calcu-
lated:
Araw =N↑
e−N↓
e
N↑
e+N↓
e
,(5)

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Fig. 2. Comparison between the beam parameters observed in a run with beam
stabilization off (black) and with beam stabilization on (red).
where N↑(↓)
eis the corrected detector signal. Assuming a linear
behavior of detectors and data acquisition, N↑(↓)
eis proportional to
the number of elastically scattered electrons for each polarization
state. To determine the experimental asymmetry
Aexp =Araw −c1AI−c2x−c3y−
−c4x−c5y−c6E,(6)
the raw asymmetry needs to be corrected for instrumental asym-
metries. Here, ci(i =1, ..., 6) are the correction factors, AIis the
beam current asymmetry, xand yare the polarity correlated
differences of the horizontal and vertical beam position, xand
yare the differences in beam angle, and Eis the difference in
beam energy. The physical parameters of the beam have to be ex-
tracted from the beam monitor data and the correction factors ci
need to be determined. Due to the beam stabilization system, the
helicity-correlated changes of the beam parameters were small but
not negligible.
For the calibration of the beam monitors, dedicated runs were
performed. The PIMO signal together with the PMT gain was au-
tomatically calibrated in special runs, which have been performed
approximately every three hours. For these special runs the beam
current was ramped up in steps of 0.25 μA from 17.5 μA to 22.5 μA,
covering the nominal beam current setting. The integrated PMT
signal was calibrated against the beam current allowing for the ex-
traction of an individual offset for every PMT. This procedure also
allowed to constantly check the linearity of the PMT responses and
to monitor any gain variations. A precise calibration of the PIMO
was also essential for the calibration of XYMOs and ENMO, since
their signals scale with the beam current.
For the XYMO calibrations, the beam was slowly rastered over
wire targets with known wire positions. For the ENMO calibration
an electronic, polarity-correlated signal corresponding to a defined
energy variation was superimposed on the raw energy signal. XY-
MOs and ENMO were calibrated once per experimental campaign.
While the correction factor c1for the beam current asymmetry
was set to a fixed value of 1, the other factors ciwere obtained by
an iterative optimization procedure. Its purpose was to minimize
the dependence of the corrected asymmetry Aexp on the polar-
ity correlated differences of the beam parameters. Therefore, the
analysis was run repeatedly, and each time the resulting experi-
mental asymmetry (Aexp ) was linearly fitted against the polarity
correlated difference of each beam parameter. Then all correction
factors were modified simultaneously in small increments to coun-
teract this dependence. This procedure was repeated until a min-
imal dependence was achieved. 0.01% of all events were excluded
from the analysis due to either a wrong gate-length signal or the
beam current being outwith the calibration range.
5. Results
The results obtained for Antogether with their uncertainties
are shown in Table 1. Large beam fluctuations and short running
time affects the 90Zr result that exhibits larger statistical and sys-
tematic errors compared to both the 28 Si measurement and our
former 12 C result [22].
The systematic errors consist of a set of contributions arising
from different sources. The contributions introduced by fluctua-
tions of position, angle, and energy of the beam were determined
by varying the correction factors independently by ± 25% and cal-
culating the maximum change in the resulting asymmetry. The
threshold for excluding events with short drops in the beam cur-
rent was varied over a wide range around the lower boundary of
the beam current calibration range to evaluate the influence of
this threshold on the resulting transverse asymmetry. A negligi-
ble effect of no more than 0.03 ppm was found (CurrentDrop in
Table 1), which contributed to the systematic uncertainty.
The current and gate-length asymmetry was measured for ev-
ery event. In order to estimate a conservative systematic uncer-
tainty, the statistical error of their mean asymmetry was added to
the systematic uncertainty (AIand AGL in Table 1).
Fluctuations in the offset of the individual PMT signals affect
the measured transverse asymmetry. The mean value and the stan-
dard deviation of the PMT signal offsets for all calibration runs
(Ncalib) were calculated and the individual PMT signal offsets were
varied to both extremes of their standard deviation. The corre-
sponding changes of the asymmetry caused by the individual PMT
signal offset variations were then added. For the final determina-
tion of the systematic error (Gain in Table 1), the accumulated
effect of all offset variations was divided by √Ncalib to take into

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Fig. 3. Extracted transverse asymmetries Anfor 12C (from Ref. [22]), 28 Si and 90 Zr.
The error bars mark the statistical uncertainty and the boxes show the system-
atic error. The theoretical calculation for Eb=0.570 GeV and Q2=0.04 GeV/c2of
Ref. [20](black line) is shown for comparison. The given bands indicate the theo-
retical error for uncertainties of the Compton slope parameter of 10% (light grey)
and 20% (dark grey).
account that the variation represents a statistical fluctuation. The
contribution of possible nonlinearities in the asymmetry correction
was estimated by excluding 0.1% of the events with the largest ab-
solute correction for each term in Eq. 6, respectively. The absolute
values of the resulting changes in the asymmetry were then added
up (Tails in Table 1).
To identify and correct for possible instrumental asymmetries
due to the electronics changing the beam polarity, a half-wave
plate in the optical system at the beam source was used to re-
verse the beam polarity independently of the electronics. To this
purpose a similar number of events was acquired for both states
of the half-wave plate. Between both states a difference in the
measured asymmetry was noticeable. Even though, it was not sta-
tistically significant, since the presence of leakage currents in the
electronics could not be entirely excluded, the estimated change
in the resulting asymmetry for an equal number of events in both
states was added to the systematic uncertainty (Inversion in Ta-
ble 1).
In addition, during the XYMO calibration for the measurement
on 28Si, an asymmetric average beam profile in vertical direction
was identified. In order to estimate the maximum impact of this
condition on the result, the XYMO calibration factors were varied
from 0 to twice the value obtained in the calibration procedure
without changing the correction factors. This led to an additional
contribution to the systematic uncertainty (BeamProfile in Ta-
ble 1).
The resulting transverse asymmetries for 28Si and 90 Zr includ-
ing our recent result for 12C [22]are shown in Fig. 3together with
an extension of the theoretical calculation from Refs. [20,22]to 28 Si
and 90Zr. The assigned uncertainty of the theoretical prediction
comprises contributions related to the Compton slope parameter
and terms not enhanced by the large logarithm (see Ref. [20]for
details). Both are added in quadrature, since it is expected that
they are independent of each other. Since the latter one represents
the limitation of the approximation used, 100% uncertainty is as-
signed to it. The impact on the transverse asymmetry due to the
contamination caused by the small fraction of inelastic events in
the detector acceptance is considered to be covered by this conser-
vative uncertainty assessment. The error bands in Fig. 3are then
computed by varying the Compton slope parameter by 10% and
20%. For identical kinematics, the theoretical calculation depends
only on the mass to charge ratio of the nucleus. Thus, the same
asymmetry is expected for both 12C and 28 Si.
Within the estimated theoretical uncertainty, the measurements
are in agreement with the theoretical prediction. A dramatic dis-
agreement, as it was obtained for 208 Pb [21], has not been ob-
served for 90Zr. Though our result is affected by a large statisti-
cal uncertainty, its value is not compatible with zero, unlike for
the 208Pb measurement. While the mean value of the asymmetry
for the zirconium target slightly deviates from the values for the
lighter nuclei, the experimental statistical error and the theoreti-
cal uncertainty on the Compton slope parameter do not allow for
a quantitative statement concerning a clear dependence of Anon
the nuclear charge. The discrepancy to the theoretical prediction
seems to be approximately independent on the target nucleus.
The experimental results for the beam-normal single spin
asymmetry on 28Si and 90 Zr presented in this work contribute sig-
nificantly to the study of this observable across the nuclear chart
from hydrogen to lead. Our results are in agreement with all pre-
vious measurements on light and intermediate nuclei confirming
that the theoretical model of Ref. [20] correctly grasps the relevant
physics. Several explanations for the disagreement with the 208Pb
result [21]are conceivable.
The coefficient C0-Eq.(3)-in front of the logarithmically
enhanced term could be suppressed for 208 Pb. However, this coef-
ficient is fixed by the total photoabsorption cross section, which, to
a good approximation, is known to scale with the mass of the nu-
cleus [33]in the relevant energy range. Contributions from nuclear
excitations are suppressed as ENucl/Ebeam , with ENucl a character-
istic scale of nuclear excitations (of the order of several MeV).
A possible underestimation of the systematic uncertainty of the
theoretical calculation could also explain the observed disagree-
ment. This uncertainty arises from two sources: the term that is
not enhanced by the large logarithm was assigned a conservative
100% uncertainty; the Compton form factor has the exponential
form, and the respective slope parameter was allowed to vary by
10% -20%. Given the agreement of the model and the data for all
nuclei up to 90Zr, an abrupt change in at least one of these terms
is needed to reconcile the calculation with 208Pb.
Eventually, the two-photon approximation used in [20], while
appropriate for light and intermediate mass nuclei might be inad-
equate for heavy nuclei. However, the reasonable agreement of the
theory with the 90Zr data (see Fig. 3) as well as a preliminary re-
sult of new calculations accounting for Coulomb distortion effects
(thus summing corrections ∼Zαto all orders) [34] seem to dis-
prove this explanation.
A new experimental program on Compton scattering at MAMI
will permit to reduce the uncertainty of the Compton parame-
ter for intermediate mass nuclei. In addition, measurements of
Anwith a 12C target at different beam energies will allow to
benchmark the energy dependence of the beam-normal single spin
asymmetry in the theoretical treatment. Furthermore, a new mea-
surement of Anfor 208Pb by the PREX-II experiment [35]might
provide additional clues to the solution of the current tension.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgements
We acknowledge the MAMI accelerator group and all the work-
shop staff members for outstanding support. This work was sup-
ported by the PRISMA+ (Precision Physics, Fundamental Interac-
tions and Structure of Matter) Cluster of Excellence, the Deutsche

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Forschungsgemeinschaft through the Collaborative Research Center
1044, and the Federal State of Rhineland-Palatinate. The work of
M. Gorchtein was supported by the German-Mexican research col-
laboration grant No. 278017 (CONACYT) and No. SP 778/4- 1 (DFG),
and by the EU Horizon 2020 research and innovation programme,
project STRONG- 2020, grant agreement No. 824093.
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