#  Electron Scattering and Neutrino Physics
A NF06 Contributed White Paper
Submitted to the Proceedings of the US Community
Study on the Future of Particle Physics (Snowmass 2021)

A. M. Ankowski ContributorCurrently at Institute of Theoretical Physics,
University of Wrocław, Wrocław, Poland SLAC National Accelerator Laboratory,
Stanford University, Menlo Park, CA, USA  A. Ashkenazi∗ Tel Aviv University,
Tel Aviv, Israel  S. Bacca∗ Institut für Kernphysik and PRISMA+ Cluster of
Excellence, Johannes Gutenberg-Universität, 55128 Mainz, Germany  Helmholtz-
Institut Mainz, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany
J. L. Barrow∗ Tel Aviv University, Tel Aviv, Israel  Massachusetts Institute
of Technology, Cambridge, MA, USA  M. Betancourt∗ Fermi National Accelerator
Laboratory, Batavia, IL, USA  A. Bodek∗ University of Rochester, Rochester,
NY, USA  M. E. Christy∗ Hampton University, Hampton, VA, USA  Thomas Jefferson
National Accelerator Facility, Newport News, VA, USA  L. Doria∗ Institut für
Kernphysik and PRISMA+ Cluster of Excellence, Johannes Gutenberg-Universität,
55128 Mainz, Germany  S. Dytman∗ University of Pittsburgh, Pittsburgh, PA, USA
A. Friedland∗ SLAC National Accelerator Laboratory, Stanford University, Menlo
Park, CA, USA  O. Hen∗ Massachusetts Institute of Technology, Cambridge, MA,
USA  C. J. Horowitz∗ Center for Exploration of Energy and Matter and
Department of Physics, Indiana University, Bloomington, IN, USA  N. Jachowicz∗
Department of Physics and Astronomy, Ghent University, Proeftuinstraat 86,
B-9000 Gent, Belgium  W. Ketchum∗ Fermi National Accelerator Laboratory,
Batavia, IL, USA  T. Lux∗ Institut de Física d’Altes Energies (IFAE) - The
Barcelona Institute of Science and Technology (BIST), Campus UAB, 08193
Bellaterra (Barcelona), Spain  K. Mahn Editor<EMAIL_ADDRESS>Michigan State
University, East Lansing, MI, USA  C. Mariani∗ Center for Neutrino Physics,
Virginia Tech, Blacksburg, VA, USA  J. Newby∗ Oak Ridge National Laboratory,
Oak Ridge, TN, USA  V. Pandey Editor<EMAIL_ADDRESS>Fermi National
Accelerator Laboratory, Batavia, IL, USA  A. Papadopoulou∗ Argonne National
Laboratory, Lemont, IL, USA  E. Radicioni∗ INFN Sezione di Bari, via Amendola
173, Bari, Italy  F. Sánchez∗ University of Geneva, Section de Physique, DPNC,
1205 Genéve, Switzerland  C. Sfienti∗ Institut für Kernphysik and PRISMA+
Cluster of Excellence, Johannes Gutenberg-Universität, 55128 Mainz, Germany
J. M. Udías∗ Nuclear Physics Group and IPARCOS, Universidad Complutense de
Madrid, 28040 Madrid, Spain  L. Weinstein∗ Old Dominion University, Norfolk,
VA, USA
L. Alvarez-Ruso  Instituto de Física Corpuscular, Consejo Superior de
Investigaciones Científicas and Universidad de Valencia, E-46980 Paterna,
Valencia, Spain  J. E. Amaro  Department of Atomic, Molecular and Nuclear
Physics, University of Granada, E-18071 Granada, Spain  C. A. Argüelles
Department of Physics & Laboratory for Particle Physics and Cosmology, Harvard
University, Cambridge, MA, USA  A.B. Balantekin  University of Wisconsin,
Department of Physics, Madison, WI, USA  S. Bolognesi  IRFU, CEA, Université
Paris-Saclay, Gif-sur-Yvette, France  V. Brdar  Fermi National Accelerator
Laboratory, Batavia, IL, USA  Department of Physics & Astronomy, Northwestern
University, Evanston, IL, USA  P. Butti  SLAC National Accelerator Laboratory,
Stanford University, Menlo Park, CA, USA  S. Carey  Department of Physics and
Astronomy, Wayne State University, Detroit, MI, USA  Z. Djurcic  Argonne
National Laboratory, Lemont, IL, USA  O. Dvornikov  University of Hawaii,
Honolulu, HI, USA  S. Edayath  Department of Physics and Astronomy, Iowa State
University, Ames, IA, USA  S. Gardiner  Fermi National Accelerator Laboratory,
Batavia, IL, USA  J. Isaacson  Fermi National Accelerator Laboratory, Batavia,
IL, USA  W. Jay  Massachusetts Institute of Technology, Cambridge, MA, USA  A.
Klustová  The Blackett Laboratory, Imperial College London, London SW7 2BW, UK
K. S. McFarland  University of Rochester, Rochester, NY, USA  A.
Nikolakopoulos  Fermi National Accelerator Laboratory, Batavia, IL, USA  A.
Norrick  Fermi National Accelerator Laboratory, Batavia, IL, USA  S. Pastore
Department of Physics and the McDonnell Center for the Space Sciences at
Washington University in St. Louis, MO, USA  G. Paz  Department of Physics &
Astronomy, Northwestern University, Evanston, IL, USA  M. H. Reno  Department
of Physics & Astronomy, University of Iowa, Iowa City, IA, USA  I. Ruiz Simo
Instituto de Física Corpuscular, Consejo Superior de Investigaciones
Científicas and Universidad de Valencia, E-46980 Paterna, Valencia, Spain  J.
E. Sobczyk  Institut für Kernphysik and PRISMA+ Cluster of Excellence,
Johannes Gutenberg-Universität, 55128 Mainz, Germany  A. Sousa  Department of
Physics, University of Cincinnati, Cincinnati, OH, USA  N. Toro  SLAC National
Accelerator Laboratory, Stanford University, Menlo Park, CA, USA  Y.-D
University of California, Irvine, CA, USA  Tsai  University of California,
Irvine, CA, USA  M. Wagman  Fermi National Accelerator Laboratory, Batavia,
IL, USA  J. G. Walsh  Michigan State University, East Lansing, MI, USA  G.
Yang  University of California, Berkeley, CA, USA

###### Contents

  1. 1 Executive Summary
  2. 2 Introduction
  3. 3 Electron Scattering as a Vital Input to Neutrino Physics
    1. 3.1 Impact on Long-Baseline Oscillation Physics
      1. 3.1.1 Challenges of Interaction Modelling for LBL Programs
      2. 3.1.2 Unique Benefits of Electron Scattering for LBL
    2. 3.2 Applications to Searches for Exotica and Nuclear Physics
    3. 3.3 Impact on Low-Energy Neutrino Physics
      1. 3.3.1 Coherent Elastic Scattering
      2. 3.3.2 10s of MeV Inelastic Scattering
  4. 4 Connecting Electron- and Neutrino-Nucleus Scattering Physics
    1. 4.1 Vector and Axial Current
    2. 4.2 Nuclear Effects
    3. 4.3 CEvNS and PVES
    4. 4.4 Experimental Input
  5. 5 Experimental Landscape I: Input to Accelerator-Based Neutrino Program
    1. 5.1 Current and Planned Experiments
      1. 5.1.1 E12-14-012 at JLab
      2. 5.1.2 E4nu at JLab
      3. 5.1.3 LDMX at SLAC
      4. 5.1.4 A1 Collaboration at MAMI
      5. 5.1.5 A1 Collaboration at Spanish facilities
    2. 5.2 Summary of the ongoing and future efforts
  6. 6 Experimental Landscape II: Input to Low-Energy Neutrino Physics
    1. 6.1 Parity-Violating Electron Scattering Experiments
      1. 6.1.1 PREX and CREX at JLab
      2. 6.1.2 MREX at MESA
      3. 6.1.3 Identifying Connections and Gaps
    2. 6.2 Low Energy Electron Scattering
      1. 6.2.1 MAGIX Collaboration at MESA
  7. 7 Addressing NP and HEP Boundary Conditions
  8. 8 Summary

##  1 Executive Summary

A thorough understanding of neutrino-nucleus scattering physics is crucial for
the successful execution of the entire US neutrino physics program. Neutrino-
nucleus interaction constitutes one of the biggest systematic uncertainties in
neutrino experiments - both at intermediate energies affecting long-baseline
Deep Underground Neutrino Experiment (DUNE), as well as at low energies
affecting coherent scattering neutrino program - and could well be the
difference between achieving or missing discovery level precision. To this
end, electron-nucleus scattering experiments provide vital information to
test, assess and validate different nuclear models and event generators
intended to be used in neutrino experiments.

As mentioned, for instance, in the DUNE Conceptual Design Report,
“uncertainties exceeding 1% for signal and 5% for backgrounds may result in
substantial degradation of the sensitivity to CP violation and the mass
hierarchy” [1]. DUNE will use a highly capable set of “near” detectors to make
precision neutrino-nucleus scattering measurements. Complementary
probes111Pion scattering, for example, has been extensively used in current
oscillation programs to reduce systematic uncertainties in conjunction with
near detector data., such as precise electron-nucleus scattering data that can
be obtained using controlled kinematics, provide data to robustly test and
resolve discrepancies in neutrino interaction physics.

Similarly, for the low-energy neutrino program revolving around the coherent
elastic neutrino–nucleus scattering (CEvNS) physics at stopped pion sources,
such as at ORNL, the main source of uncertainty in the evaluation of the CEvNS
cross section is driven by the underlying nuclear structure, embedded in the
weak form factor, of the target nucleus. To this end, parity–violating
electron scattering (PVES) experiments, utilizing polarized electron beams,
provide vital model-independent information of determining weak form factors.
This information is vital in achieving a percent level precision needed to
disentangle new physics signals from the standard model expected CEvNS rate.

Finally, it is important to emphasize that a necessary condition in addressing
neutrino-nucleus interaction physics challenge is the close collaboration of
nuclear physics (NP) and high-energy physics (HEP) communities. Electron
scattering efforts that play vital role in the future of the neutrino program
are inherently considered NP program while the neutrino programs are supported
by HEP. Unfortunately, crossing the strictly defined boundaries between the NP
and HEP funding agencies has been a significant challenge to progress. It is
also worth noting that the electron-nucleus and neutrino-nucleus scattering
also provide complementary information on nuclear structure and dynamics that
is interesting in its own right. A close collaboration between the two
communities could help provide unique information on nuclear and hadronic
physics.

In this white paper, we highlight connections between electron- and neutrino-
nucleus scattering physics at energies ranging from 10s of MeV to a few GeV,
review the status of ongoing and planned electron scattering experiments,
identify gaps, and layout a path forward that benefits the neutrino community.
We also highlight the systemic challenges with respect to the divide between
the nuclear and high-energy physics communities and funding that presents
additional hurdle in mobilizing these connections to the benefit of neutrino
programs.

##  2 Introduction

The success of current and future neutrino experiments in achieving discovery
level precision will greatly depend on the precision with which the
fundamental underlying process – neutrino interaction with the target nucleus
in the detector – is known [2]. To this end, electron scattering experiments
have been playing a crucial role by providing high-precision data as the
testbed to assess and validate different nuclear models intended to be used in
neutrino experiments.

For the accelerator-based neutrino program, such as DUNE, the primary physics
goals are determining mass hierarchy and testing the three flavor paradigm
with precision measurements, including subtle effects of $\delta_{CP}$ [3].
DUNE is a powerful facility with enormous physics reach, the intense neutrino
source can also be used to probe for exotic physics, and the massive detectors
can be used to measure neutrinos from astrophysical phenomena like supernova.
Each of those physics thrusts relies on precision knowledge of neutrino-
nucleus scattering. The main challenges in constraining neutrino-nucleus
scattering physics stems from the fact that neutrino energies at these
experiments typically range from 100s of MeV to a few GeV where different
interaction mechanisms yield comparable contributions to the cross section.
One has to constrain an accurate picture of the initial state target nucleus,
its response to the electroweak probe that include several reaction mechanisms
resulting into various finals state particles, and final state interactions
that further modify the properties of the hadronic system created at the
primary interaction vertex. While electron and neutrino interactions are
different at the primary vertex, many underlying relevant physical processes
in the nucleus are the same in both cases, and electron scattering data
collected with precisely controlled kinematics (initial and final energies and
scattering angles), large statistics and high precision allows one to
constrain nuclear properties and specific interaction processes.

For the coherent elastic neutrino–nucleus scattering (CEvNS) program at
stopped pion sources, such as at ORNL, the main source of uncertainty in the
evaluation of the CEvNS cross section is driven by the underlying nuclear
structure, embedded in the weak form factor, of the target nucleus. The recent
detection of CEvNS process by the COHERENT collaboration [44] has opened up a
slew of opportunities to probe several Beyond the Standard Model (BSM)
scenarios in CEvNS experiments. In order to disentangle new physics signals
from the SM expected CEvNS rate, the weak form-factor which primarily depends
on the neutron density has to be known at percent level precision. Although,
the ground state proton density distributions are relatively well constrained
through elastic electron scattering datasets [46, 47], the neutron density
distributions are only poorly known due to historic experimental difficulties
in using electroweak probes such as polarized electron beams. Parity–violating
electron scattering (PVES) experiments, utilizing polarized electron beams,
provide relatively model-independent ways of determining weak form factors and
neutron distributions of a target nucleus, that can be used as direct input to
the CEvNS cross section.

For over five decades, the electron scattering experiments at different
facilities around the world have provided wealth of information on the
complexity of nuclear structure, dynamics and reaction mechanisms. Static form
factors and charge distributions have been extracted from elastic scattering
data, while measurements of inelastic cross sections have allowed for
systematic studies of the dynamic response functions, which shed light on the
role played by different reaction mechanisms. Decades of experimental work
have provided vital testbed to assess and validate theoretical approximations
and predictions that propelled the theoretical progress staged around [7, 8].
While previous and existing electron scattering experiments provide important
information, new measurements to support neutrino-nucleus modeling are
valuable. Dedicated electron scattering experiments with targets and
kinematics of interests to neutrino experiments (CEvNS, supernova, and
accelerator-based) will be vital in the development of neutrino-nucleus
scattering physics modeling that underpin neutrino programs.

This white paper is structured as follows. In Sec. 3, we briefly outline the
impact of electron scattering physics on neutrino programs. We then identify
connections between electron- and neutrino-nucleus scattering physics in Sec.
4. In Sec. 5, we survey the current and planned electron scattering
experiments that are input to accelerator-based neutrino program, and identify
connections and gaps between them. In Sec. 6, we survey the current and
planned electron scattering experiments that are input to CEvNS and supernova
program. Thereby, in Sec. 7, we highlight the challenges emerged from the
divide between the nuclear and high-energy physics communities and funding
that presents additional hurdle in mobilizing these connections and building a
community. We summarize in Sec. 8.

##  3 Electron Scattering as a Vital Input to Neutrino Physics

In this section, we briefly describe the neutrino interactions challenges
faced by neutrino experiments in their key physics goals of measuring
oscillations parameters, disentangling new physics signals and in detecting
supernova neutrino signals. We then outline the key input that electron
scattering experiments provide in tackling neutrino interactions modeling for
both the long-baseline experiments as well as for CEvNS experiments.

###  3.1 Impact on Long-Baseline Oscillation Physics

####  3.1.1 Challenges of Interaction Modelling for LBL Programs

We discuss the impact of electron scattering measurements to the future DUNE
program, acknowledging that a similar set of arguments will apply for the
Hyper-Kamiokande program. DUNE is a “long baseline” (LBL) neutrino experiment
with 1) an intense, accelerator driven (anti)neutrino beam, 2) a suite of
“near” detectors [9] close to the neutrino source production, and 3) a set of
“far” detectors located in the neutrino beam. DUNE has a broad physics
program, including precision measurements of neutrino oscillation physics
parameters, searches for exotic physics, neutrino interactions and
astrophysics. The DUNE beam covers a wide range of energies (approximately 500
MeV to 5 GeV). Also, DUNE will collect interactions in the far detector from
atmospheric (anti)neutrinos, which, when combined with beam data, provide
enhanced sensitivity to oscillation physics. DUNE’s beam intensity and
detector size will provide unprecedentedly large samples of
$\nu_{e}$/$\overline{\nu}_{e}$ appearance.

To correctly estimate neutrino oscillation parameters of interest requires
robust knowledge of neutrino interactions. Neutrino oscillation parameters are
inferred from event rates, which requires the correct mixture of processes.
Across the DUNE beam energies, processes like quasielastic scattering,
resonant production of pions, shallow inelastic scattering and deep inelastic
scattering are all relevant. A prediction of each process, including the
kinematics and multiplicity of outgoing particles is needed to estimate
detection efficiency and therefore expected event rates. Oscillation occurs as
a function of true neutrino energy, so the relationship between all final
state particles and neutrino energy for each process is required.

It is widely known that the predictions of interaction processes are not
necessarily well modelled. Many models are a combination of a theoretically
motivated model but often have empirical extrapolations in difficult to assess
kinematic regions. Simulations of interaction models may also have to make
approximations to make computations tractable. Interaction model relevant
degrees of freedom are treated as nuisance parameters in oscillation fit. If
this parameterization is incomplete or insufficient, this can also lead to
issues in oscillation results. Notably, comparisons of experimental
measurements of neutrino processes at different energies do not present a
consistent or clear picture as to the origin of inconsistencies. As is
summarized in Ref [2], experimental and theoretical groups have reported the
possibility of bias in oscillation results due to deficiencies in the
interaction model. One projection of this is shown in Fig. 1, from Ref. [10],
which shows the cross section as a function of the reconstructed energy (Ecal)
for 1.159, 2.257 and 4.453 GeV carbon events and 2.257 and 4.453 GeV iron
events. The reconstructed neutrino energy spectra show a sharp peak at the
real beam energy, followed by a large tail at lower energies. For carbon, only
30–40% of the events reconstruct to within 5% of the real beam energy. For
iron this fraction is only 20–25%, highlighting the crucial need to well model
the low-energy tail of these distributions. This disagreement in the tails
increases with energy and nuclei number.

Figure 1: The $1p0\pi$ cross section plotted as a function of the
reconstructed calorimetric energy Ecal for data (black points), SuSAv2 (black
solid curve) and G2018 (black dotted curve) models. Different panels show
results for different beam energy and target nucleus combinations: (top row)
Carbon target at (left to right) 1.159, 2.257 and 4.453 GeV, and (bottom) Iron
target at (left) 2.257 and (right) 4.453 GeV incident beam. The 1.159 GeV
yields have been scaled by 1/2 and the 4.453 GeV yields have been scaled by 5
to have the same vertical scale. Coloured lines show the contributions of
different processes to the GENIE SuSAv2: QE (blue), MEC (red), RES (green) and
DIS (orange). Error bars show the 68% confidence limits for the statistical
and systematic uncertainties (not shown when they are smaller than data
point). Figure from Ref. [10].

Oscillation experiments mitigate these issues partially with data from a “near
detector”, positioned close to the neutrino beam origin to constrain
interaction physics. DUNE’s design includes a movable set of precision
detectors with broad acceptance of particle composition and kinematics. The
detector motion (“PRISM”) feature will enable, within the same detector,
comparisons of event rates from neutrino energy spectra which are different.
This will provide information, akin to electron scattering, in order to
mitigate interaction model bias.

Electron scattering data is complementary to the information on the cross
sections from the near detectors. First, although the near detector measures
both the axial and the vector part of the cross-section, the vector part,
precisely provided by electron scattering must be correctly predicted for the
right inference in predicting asymmetries in the neutrino-antineutrino rate,
key for measurements like CPV. Second, the near detector does not have uniform
or complete acceptance. Therefore, predictions at the DUNE far detector still
rely on a prediction which depends on particle multiplicity and kinematics.
Third, detectors have finite thresholds222The use of the ND-GAr detector aims
to provide a lower threshold than is used at the far detector to test models.,
and some particles, like neutrons, may be challenging to associate to a given
neutrino interaction. Fourth, robust interaction models are necessary for a
variety of exotic physics searches which use the near detector to search for
signal, and therefore do not benefit from the constraint on the cross section
the near detector provides in oscillation analyses. Finally, information from
electron scattering is used to develop theoretical models in advance of early
data from DUNE near detectors.

####  3.1.2 Unique Benefits of Electron Scattering for LBL

Given the inherent complexity of the neutrino-nucleus scattering problem,
thorough validation of models against data is required. It has long been
recognized that a robust model must reproduce not only neutrino-scattering
data, but also measurements of electron-nucleus scattering [11]. While there
are several important differences between neutrino and electron scattering,
the two processes share many common elements. These range from modeling of the
nuclear ground state, to transport of final-state hadrons through the nucleus,
to the primary-vertex interaction physics in the deep-inelastic limit (see,
e.g., Ref. [24] for discussion). By exploiting electron-scattering data to fix
the common physics elements, one will then be able to isolate the ingredients
that are specific to neutrino interactions, such as axial couplings.

The idea of using electron data was further developed and strengthened since
the turn of the millennium [13, 14]. It was incorporated into the mission
statement of the GENIE generator, was mentioned prominently in the original
paper [15], and comparisons at low energies were presented in Ref. [16].
Likewise, the generator GiBUU was tested against electron-scattering data [17,
18, 19, 20, 21].

What makes comparisons with electron-scattering data such a powerful tool is
the precise control of the scattering kinematics it affords. Neutrino-
scattering measurements are by necessity reported integrated over a range of
energies and angles. The integrated kinematics may mask significant problems,
which may reappear on different targets or in different kinematic regimes in
future analyses. In contrast, by precisely controlling scattering parameters,
electron-scattering experiments are able to zero in on specific physical
processes.

The utility of validating a neutrino Monte Carlo generator against electron-
scattering data that it had not been tuned to was demonstrated recently in
Refs. [22, 83]. Figure 2 shows the kinematic coverage of available data for
inclusive electron scattering on carbon that was used in Ref. [22]. This
coverage is superimposed on the expected distribution of the charged-current
$\nu_{\mu}$Ar events in the DUNE near detector. The availability of this data
made it possible to undertake a systematic study benchmarking GENIE v2.12
predictions across the different scattering regimes relevant to NOvA and DUNE.
Identifying which regimes host largest discrepancies is the first necessary
step towards the program of improving generator physics.

One notable finding from the comparisons at multi-GeV energies is a pattern of
large discrepancies in the inelastic regime, including the region of higher
resonances and its transition to deep inelastic scattering (the RES-to-DIS
region). This pattern reflects an inherent difficulty of the physics problem,
where no first-principles treatment exists and prescriptions based on various
physics models must be combined in ways that are consistent with quark-hadron
duality and reproduce the available data. The large discrepancies exist not
only in inclusive cross sections, but also in exclusive variables [24].

Figure 2: Kinematic coverage of available data for inclusive electron
scattering on carbon [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], the
most extensively studied nucleus, overlaid on the charged-current
$\nu_{\mu}$Ar event distribution expected for the 2020 flux estimate [38] in
the DUNE near detector according to GENIE 3.0.6. The thick solid lines
representing the measured cross sections are shown in the (a)
$(|\mathbf{q}|,\omega)$ and (b) $(|\mathbf{q}|,W)$ planes, with momentum
transfer $\mathbf{q}$, energy transfer $\omega$, and hadronic mass $W$. The
dark and light shaded areas correspond to 68% and 95% of the expected events.
The thin solid, dashed, and dotted lines present the kinematics corresponding
to quasielastic scattering, $\Delta$ excitation, and the deep-inelastic
scattering at $W=1.7$ GeV on free nucleons. Only scarce information on the
cross sections is available in the whole deep-inelastic region (above the thin
dotted lines) and for higher-resonance production at $|\mathbf{q}|\gtrsim 1$
GeV (above the thin dashed lines) in the region of most importance for DUNE.
For argon, a single dataset is currently available [39].

The physics uncertainties in the RES-to-DIS regime may have a profound impact
on the performance of DUNE. Indeed, more than half of neutrino interactions
there are expected to occur in this regime and the resulting rich hadronic
final states will need to be reliably modeled to reconstruct the energy of the
incoming neutrino. This especially concerns the energy fraction carried out by
multiple neutrons created at various stages of the event development, as these
are very difficult to reconstruct [40, 41].

It is notable in this regard that recent measurements by MINERvA report
discrepancies between the Monte Carlo predictions and pion-production data,
and that no tune could be found that would reconcile all of the channels in
the data [42]. This puzzle may be difficult to disentangle with neutrino data
only, but corresponding electron scattering studies could help isolate the
source of the problem.

To develop and validate improved models of neutrino-nucleus scattering in the
RES-to-DIS will require new, high-precision electron-scattering data. This is
already evident from the right panel of Fig. 2, which reveals insufficient
coverage for large values of the hadronic mass $W$. There is therefore an
urgent need for new measurements that include both the outgoing electron and
the final-state hadronic system. In Sec. 5, we discuss a new program of this
type at Jefferson Laboratory ($e4\nu$), and outline the capabilities of LDMX,
an experiment proposed at SLAC.

Electron scattering measurements are also used as input to improved
theoretical modelling. In discussions at a Snowmass workshop, there was strong
demand for semi inclusive measurements of $(e,e^{\prime}p)$;
$(e,e^{\prime}\pi)$ scattering for neutrino modelling. There is also a lack of
data on deuterium or neutron targets. Finally, for both theorist and
experimentalists, new projections of kinematic variables of interest is
extremely important for model building and validation, as are updated and
detailed systematic uncertainty assessments.

###  3.2 Applications to Searches for Exotica and Nuclear Physics

DUNE also has significant physics reach in exotic physics. DUNE will search
for proton decay, light dark matter, heavy neutral leptons, and sterile
neutrinos. Each of these searches, like the oscillation program, relies on
(anti)neutrino rates. In some cases, near detector data will be able to
constrain the interaction model, but not always, especially for searches
performed using the near detector where any measurement of the signal process
may be complicated by new physics. Searches for signals similar to neutral
current interactions directly benefit from electron scattering measurements,
and which are sometimes notorious to measure in neutrino detectors.

DUNE will also be able to make new measurements of interest to nuclear
physics, which are enhanced by electron scattering. The DUNE near detector
complex, by virtue of sampling multiple energy spectra, can effectively make
pseudo-monochromatic fluxes akin to electron scattering [9]. Then, by
comparing neutrino and antineutrino rates, DUNE can then produce measurements
of the axial-vector interference terms. There are no studies of the axial
response at DUNE’s energies. These unique measurements will be improved by
correctly accounting for the vector piece of the cross section provided by
electron scattering.

###  3.3 Impact on Low-Energy Neutrino Physics

####  3.3.1 Coherent Elastic Scattering

In 1974 Fredman suggested that "if there is a weak neutral current, then the
elastic process $\nu+A\rightarrow\nu+A$ should have a sharp coherent forward
peak just as the $e+A\rightarrow e+A$ does" [43]. Due to the elusive nature of
neutrinos and the necessary technological advancement needed to detect such a
tiny signal, it took 43 years for the first coherent elastic neutrino
scattering (CEvNS) to be experimentally observed by the COHERENT collaboration
[44]. A stopped-pion neutrino beam from the Spallation Neutron Source at Oak
Ridge National Laboratory impinged on a sodium-doped CsI detector, allowing
for an experimental discovery of CEvNS at the $6.7\sigma$ confidence level. A
few years later, the COHERENT collaboration used a 24-kg active-mass liquid-
argon scintillator detector [45] and found that two independent analyses
preferred CEvNS over the background-only null hypothesis with a significance
larger than $3\sigma$.

Coherent elastic neutrino scattering off a nucleus occurs in the regime $qR\ll
1$, where $q=|{\bf q}|$ is the absolute value of the three-momentum transfer
from the neutrino to the nucleus, and $R$ is the nuclear radius. The CEvNS
cross section is

$\displaystyle\frac{d\sigma}{dT}(E_{\nu},T)$ $\displaystyle\simeq$
$\displaystyle\frac{G_{F}^{2}}{4\pi}M\left[1-\frac{MT}{2E_{\nu}^{2}}\right]Q^{2}_{W}F^{2}_{W}(q^{2})\,,$
(1)

where $G_{F}$ is the Fermi constant, $M$ the mass of the nucleus, $E_{\nu}$
and $T$ the energy of the neutrino and the nuclear recoil energy,
respectively. The weak form factor $F_{W}(q^{2})$ and weak charge $Q_{W}$ are
given by

$\displaystyle F_{W}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{Q_{W}}[NF_{n}(q^{2})-(1-4\sin^{2}\theta_{W})ZF_{p}(q^{2})]$
(2) $\displaystyle Q_{W}$ $\displaystyle=$ $\displaystyle
N-(1-4\sin^{2}\theta_{W})Z\,,$

where $\theta_{W}$ is the Weinberg mixing angle. In Eq. (2) the quantities
$F_{p}(q^{2})$ and $F_{n}(q^{2})$ are the proton ($p$) and neutron ($n$) form
factors, respectively. While the proton distributions are relatively well
known through elastic electron scattering experiments [46, 47, 48], neutron
distributions are much more difficult to constrain. Because
$1-4\sin^{2}\theta_{W}(0)\approx 0$, on the one hand the weak form factor
becomes $F_{W}(q^{2})\approx F_{n}(q^{2})$, and on the other hand the cross
section scales like ${d\sigma}/{dT}\approx N^{2}$ (in the limit of
$q\rightarrow 0$). So far, data on CsI and Ar established the $N^{2}$ scaling
of the cross section and, using some phenomenological parameterization of the
nuclear form factors, obtained a signal that is consistent with the standard
model [45]. Measuring CEvNS with increased precision opens up the possibility
to discover new physics beyond the standard model, such as non standard
neutrino interactions [49, 50, 51]. For this reason, several experiments are
being performed, or planned for, to measure CEvNS with increased precision
using a variety of detector technologies and neutrino beams that come either
from stopped pions or from reactors, see Table 1.

Table 1: Worldwide efforts in measuring CEvNS. List of planned experiments.
Experiment | Technology | Location | Source
---|---|---|---
COHERENT | CSI, Ar, Ge, NaI | USA |  $\pi$DAR
CCM | Ar | USA |  $\pi$DAR
CONNIE | Si CCDs | Brazil | Reactor
CONUS | HPGe | Germany | Reactor
MINER | Ge/Si cryogenic | USA | Reactor
NuCleus | Cryogenic CaWO4, Al2O3 calorimeter array | Europe | Reactor
$\nu$GEN | Ge PPC | Russia | Reactor
RED-100 | LXe dual phase | Russia | Reactor
Riochet | Ge,Zn | France | Reactor
TEXONO | p-PCGe | Taiwan | Reactor
NCC-1701 | p-PCGe | Germany | Reactor

In order to disentangle new physics signals from the SM expected CEvNS rate,
the weak form factor, which primarily depends on the neutron density, has to
be known at percent level precision. Nuclear theory calculations can be used
to obtain the weak form factors. Various theories based on the use of proton
and neutrons as degrees of freedom have recently computed weak form factors
for selected nuclei from 12C to 208Pb [52, 53, 54, 55, 56], using different
schemes that range from the mean field approximation to the shell model and
the ab initio methods. In particular for 40Ar, several many-body methods have
been used and even different nuclear Hamiltonians have been implemented within
the same many-body method (see e.g., Ref. [52]), leading to an estimate of the
size of nuclear structure effects of just a few percent in the low-$q$ regime
relevant for CEvNS. Knowing that nuclear structure uncertainties are small is
a very promising information for beyond standard model searches with CEvNS. In
the future, more efforts will be devoted towards theoretical calculations in
order to assess nuclear structure corrections in nuclei other than 40Ar. Every
theoretical calculation should be tested against elastic electron scattering
to prove that the proton distributions are well described. Finally, it will be
exciting to incorporate beyond standard model physics into modern nuclear
structure calculations for CEvNS.

####  3.3.2 10s of MeV Inelastic Scattering

Stars with a mass larger than 8 times that of the Sun undergo a core collapse
process, where 99$\%$ of the gravitational binding energy of the inner proto-
neutron star is shed within just 10 second of time into neutrinos and
antineutrinos of all flavors. The kinetic energy of the produced
neutrino/antineutrinos is maximally 50 MeV. Because core-collapse supernova
are still not well understood, the detection of supernova neutrino from an
event in our Galaxy could dramatically influence our understanding of such
astrophysical objects. Furthermore, even though they are rare events, in the
whole Universe thousands of core-collapse explosions occur every hour,
resulting in a diffuse supernova neutrino background. The latter has not yet
been seen, but an observation would greatly enhance our understanding of core-
collapse supernovae.

Solar neutrinos have already enabled very important discoveries, from the
neutrino oscillation [57] that solved the solar neutrino problem [58] to the
first detection of CNO neutrinos from the Sun [59]. Still, many of their
facets need to be unveiled. For example, we would like to know whether solar
and reactor neutrinos oscillate with the same parameters. Solar neutrinos,
mostly generated though the hydrogen burning reactions occurring in the center
of our Sun, are of electron kind and reach up to 15 MeV of energy in case of
the 8B neutrinos.

Experimentally, one can use the charge current inelastic process to detect
supernovae and solar neutrinos. For the supernovae neutrinos, multiple
detectors are set up worldwide and are all connected via an early warning
system [60]. For the solar neutrinos, DUNE [1] plans to measure them with
increased precision using the $\nu_{e}+^{40}{\rm Ar}\rightarrow
e^{-}+^{40}{\rm K}^{*}$ process.

Inelastic interactions of neutrinos with nuclei are still poorly understood:
theory is sparse and experiments have large error bars. Because inelastic
neutrino interactions have big uncertainties, in the future it will be crucial
to measure inelastic electron scattering cross sections at energies below the
50 MeV mark and use those data to calibrate theoretical models for the
neutrino scattering process. From the theory side, inelastic scattering has
been recently tackled with mean-field methods [56] and further efforts will
need to be devoted to compute these quantities from other theories, including
ab initio nuclear theory. Overall, we expect nuclear structure effects to be
definitely larger than in CEvNS and presumably at least at the 10$\%$ level.
To properly assess uncertainties in these observables, modern machine learning
techniques will be used [61, 55].

##  4 Connecting Electron- and Neutrino-Nucleus Scattering Physics

In this section, we briefly describe how the underlying nuclear physics,
probed by electrons and neutrinos, is intimately connected to each other. The
benefit of kinematically controlled beams and precise measurements makes the
electron scattering data a vital benchmark to constrain nuclear models
intended to be used in neutrino experiments.

###  4.1 Vector and Axial Current

The fact that electron and (anti)neutrino scattering processes both originate
from the same electroweak interaction opens opportunities to invest synergies
between both, and exploit similarities between these two types of processes to
gather information about the weak interaction that is much more difficult to
assess experimentally than its electromagnetic counterpart is. Whereas
electromagnectic interactions are in first-order mediated by a (virtual)
photon, weak processes are effectuated by the exchange of a $W^{\pm}$ or
$Z^{0}$ boson for charged and neutral weak processes, respectively. However,
the way electromagnetic and weak processes manifest themselves is often very
different. This is not in the least due to the fact that intrinisic strength
of the weak interaction is extremely small compared to that of the
electromagnetic one. On top of that, the massless photons lead to a $Q^{-4}$
dependence of the boson propagator and hence cross sections that are strongly
forward peaked in the direction of the outgoing lepton, whereas for neutrinos
transverse contributions and backward lepton scattering plays a more important
role over a broader kinematic range.

Electromagnetic interactions on the nucleon or nucleus, mediated by the
neutral photon, do not change charge, and consist of isoscalar and third
component isovector contributions. This combination is shared by neutral weak
interactions, but charge-changing weak processes on the other hand are of a
pure isovector nature. As a consequence of vector current conservation (CVC),
the vector form factors in weak scattering, notoriously difficult to
investigate experimentally, can be inferred by a mere rotation in isospin
space i.e. CVC provides an unambiguous relation between the vector form
factors for both types of interactions. In a similar way, the structure of the
hadronic response functions for the vector contributions is fully equivalent
for both the electromagetic and weak processes. For the axial current a softer
constraint is provided by the partial conservation of the axial current
(PCAC). As a consequence of the fact that the chiral symmetry of the
Langragian is here only approximate, the axial current conservation becomes
exact only in the limit of vanishing pion mass. The relative weight of vector
and axial contributions in neutrino-induced processes depends strongly on the
kinematics of the interaction.

In semi-leptonic processes, the leptonic probe, be it neutral or charged, is
of course always subject to weak interactions. Obviously, in the weak case the
interference between vector and axialvector contributions gives rise to parity
violating processes. In electron scattering processes the axial contribution
is however veiled by the much stronger electromagnetic vector contributions.
In parity violating electron scattering (PVES) the interference between
electromagnetic and weak processes i.e. the interference between processes in
which a photon or a Z boson are exchanged, dominates the process, thereby
providing a way to amplify the axial contribution by multiplying it with the
much larger vector electromagnetic term. PVES and experiments carried out with
polarized electrons hence provides a powerful tool to study axial
characteristics of the weak interaction in experiments with charged leptons as
probes, that allow for a better control of the kinematics.

###  4.2 Nuclear Effects

From the nuclear point of view, the influence of nuclear medium effects as
e.g. Pauli blocking, long- and short-range correlation, and medium
modifications of resonances at higher energies can be expected to be largely
the same for electron as for neutrino-induced processes. An important point of
attention is of course the charge-changing nature of charged current weak
processes. In CC quasi-elastic processes, the final nuclear states will be
members of the same isospin triplet as the initial one, but reside in a
different final nucleus. As a consequence, the effect of the changed nuclear
(and Coulomb) potential experienced by hadronic and leptonic ejectiles in
charged-current weak interactions has to be taken into account carefully.
Owing to the vector-axial and parity-violating character of the interaction,
weak processes are able to excite different nuclear states than
electromagnetic processes do, a feature that is particularly apparent for low-
energy processes. So can neutrinos e.g. induce $0^{-}$ transitions in nuclei,
which electromagnetic interactions cannot, thus opening a window to study
nuclear states that are difficult to reach otherwise. The effects of two-body
currents in nuclei, both in the vector and the axial sectors, are comparable
to other nuclear effects. Two-body currents often are inseparable to the one-
body current even in the one-particle emission channel and present an
additional challenge [62]. Likewise, at much higher energies, pion production
processes for neutrinos can lead to different final states and outgoing
hadrons than their electron-induced counterparts and will require hence more
elaborate modeling efforts with less experimental constraints available.

Whereas validating modeling efforts for neutrino processes by also applying
the models to electron scattering data is a general practice and essential for
a solid verification of the vector part of the model, some models go to
lengths to exploit the similarities between both interaction types to guide
the description of weak scattering processes. Standing out in this respect is
the SuSA model [63] that found its origin exactly in the analogies between
neutrino- and electron quasi-elastic processes, and relies on scaling
considerations as a strong tool to predict neutrino-induced cross sections.

###  4.3 CEvNS and PVES

CEvNS and PVES are intimately connected to each other. From the formal point
of view, both processes are described in first order perturbation theory via
the exchange of an electroweak gauge boson between a lepton and a nucleus.
While in CEvNS the lepton is a neutrino and a $Z^{0}$ boson is exchanged, in
PVES the lepton is an electron, but measuring the asymmetry allows one to
select the interference between the $\gamma$ and $Z^{0}$ exchange. As a
result, both the CEvNS cross section (see Section 3.3.1) and the PVES
asymmetry (see Section 6.1) depend on the weak form factor $F_{W}(Q^{2})$,
which is mostly determined by the neutron distribution within the nucleus. The
latter builds an even stronger anchor between CEvNS and PVES.

Parity violating electron scattering experiments offer the least model
dependent and most precise approach to experimentally probing the neutron
distribution. Any result that will come from the PVES program with the goal of
pinning down the neutron-skin thickness will help improving our understanding
of the weak form factor and hence influence CEvNS. However, CEvNS has also
been proposed as an alternative and attractive opportunity in the future to
constrain the neutron distribution and the neutron radius in nuclei [64, 65,
66], provided that enough statistics can be reached.

The main difference lies in the choice of the nuclear target, which is
determined by practical considerations. In the case of PVES, the targets need
to be stable (or almost stable) neutron-rich nuclei, such as 208Pb and 48Ca,
that do not present low-lying excited states that would contribute to the
background noise. In the case of CEvNS, isotopes of sodium, argon, germanium,
cesium and iodine will be used, as the low cost allows to build large
detectors with these materials.

Because various electroweak observable correlate [53] with each other,
theoretical calculations will help to further connecting the various nuclear
targets and the two endeavors of CEvNS and PVES. For example, we can expect
that constraints experimentally determined on the neutron-skin thickness of
one nuclear target will affect the prediction of the weak form factor of
another target (see also Section 3.3.1).

It is worth mentioning that another promising avenue in experimental
extractions of the neutron-skin thickness is measurements of coherent pion
photoproduction cross sections [67, 94]. Experiments with
${}^{58}\mathrm{Ni}$, ${}^{116,120,124}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$
were performed last decade [67, 69]. Both experimental and theoretical
developments are expected for this process. Similar to PVES, measurements on a
few isotopes can help to remove major theoretical uncertainties.

###  4.4 Experimental Input

Electron and neutrino scattering are strongly linked in theory because
electron interactions with hadrons are almost purely vector and neutrinos
interact through a mixture of vector and axial currents. However, there are
significant differences in the measurements that have been published so far
because electron beams are more intense and monoenergetic. Therefore, both
quality and depth of measurements are more complete and neutrino calculations
should take advantage of this as much as possible.

The vector electron interaction is closely linked to the vector neutrino
interaction through CVC (Sect. 4.1). There are various ways to accomplish this
union. Theorists tend to write separate codes for electron and neutrino
interactions consistent with the underlying symmetries. Event generators tend
to build the relationships into a single code. For example, GENIE [15, 70]
uses the relationship given in Quigg’s book [71] to transform neutrino
interactions into electron interactions.

The ideal implementation of electron interactions in event generators should
allow adjustment of the models to electron scattering data simultaneous with
the update of neutrino interactions. Vector form factors in neutrino models
should come from same code as electron models and the core interaction
strengths linked by theory. Neutral current (NC) neutrino interactions are
closely related to electron scattering, but charged current (CC) neutrino
interactions can be different. For example, the excitation of nucleon
resonances goes to different charge states for NC and CC neutrino interactions
but the decay of the resonances should be universal.

Although GENIE has a generally solid implementation of electron scattering, it
suffers from the fact that the code was originally designed for neutrino
interactions and electron interactions were added later. Recent changes [83]
have brought GENIE closer to the ideal but problems in implementation still
cause troubles with the excitation of nucleon resonances. Implementation of
electron scattering in NuWro and NEUT is still underway. Further discussion on
the importance of electron scattering data for improving the neutrino event
generators can be found in Ref. [72]. Significant effort will be required to
reach the proper level so that the knowledge gained is implemented in neutrino
event generators with the proper parameters and their associated
uncertainties. To accomplish this, a more complete set of electron semi-
inclusive (esp. $(e,e^{\prime}\pi)$) data are needed.

##  5 Experimental Landscape I: Input to Accelerator-Based Neutrino Program

For over five decades, the electron scattering experiments at different
facilities around the world have provided wealth of information on the
complexity of nuclear structure, dynamics and reaction mechanisms. Decades of
experimental work has provided vital testbed to assess and validate
theoretical approximations and predictions that propelled the theoretical
progress staged around [7, 8]. While previous and existing electron scattering
experiments provide important information, new measurements which expand
kinematic reach, final states and provide an updated estimate of uncertainties
are beneficial.

In this section, we briefly outline current and planned electron scattering
experiments, majority of them are motivated by the needs of the accelerator
neutrino experiments. For each experiment, we briefly lay out the experimental
setup, capabilities and describe their (performed or proposed) measurements.
We then attempt to identify connections and gaps between these experimental
programs in the context of their input to the accelerator neutrino program.

###  5.1 Current and Planned Experiments

####  5.1.1 E12-14-012 at JLab

#### Overview

The JLab E12-14-012 experiment, performed in Jefferson Lab Hall A, has
collected inclusive $(e,e^{\prime})$ and exclusive $(e,e^{\prime}p)$ electron-
scattering data in five different kinematical setups using a natural argon and
a natural titanium targets.

Under the basic assumption that scattering processes involve just individual
nucleons, and not considering FSI between the outgoing proton and the
spectator nucleons, the removal energy and momentum of the knocked out
particle, $E$ and ${\bf p}$, could be reconstructed from measured kinematical
variables, and the cross section can be written in simple factorized form in
terms of the spectral function of the target nucleus, $P({\bf p},E)$,
trivially related to the nucleon Green’s function, $G({\bf p},E)$, through
$P({\bf p},E)=\frac{1}{\pi}\ {\rm Im}\ G({\bf p},E)$ [73]. As a consequence,
the spectral function which represents the probability to remove a proton with
momentum ${\bf p}$ from the target nucleus leaving the residual system with
excitation energy $E-E_{\rm thr}$, with $E_{\rm thr}$ being the proton
emission threshold—is determined from the data. For nuclei as heavy as 40Ar,
FSI effects should be accurately taken into account using an optical potential
[74].

Over the past several years, a great deal of work has been devoted to applying
the spectral function formalism to the study of neutrino-nucleus interactions,
whose quantitative understanding is needed for the interpretation of
accelerator-based searches of neutrino oscillations, see, e.g., Refs. [73,
75].

#### Experimental Setup

The experiment E12-14-012 was performed at Jefferson lab in Spring 2017.
Inclusive $(e,e^{\prime})$ and exclusive $(e,e^{\prime}p)$ electron scattering
data were collected on targets of natural argon and natural titanium, the
average neutron numbers calculated according to the natural abundances of
isotopes are 21.98 for argon and 25.92 for titanium [76].

The E12-14-012 experiment used an electron beam of energy 2.222 GeV provided
by the Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Lab.
The average beam current was approximately 15 $\mu$A for the 40Ar target and
20 $\mu$A for the 48Ti target. The scattered electrons were momentum analyzed
and detected in the left high-resolution spectrometer (HRS) in Hall A and the
coincident protons were similarly analyzed in the right HRS.

The spectrometers are equipped with two vertical drift chambers (VDCs)
providing tracking information [77], two scintillator planes for timing
measurements and triggering, double-layered lead-glass calorimeter, a gas
Cherenkov counter used for particle identification [78], pre-shower and shower
detectors (proton arm only) [78] and pion rejectors (electron arm only) [78].
The HRSs were positioned with the electron arm at central scattering angle
$\theta_{e}=21.5^{\circ}$ and the proton arm at an angle
$\theta_{p^{\prime}}=-50{{}^{\circ}}$. The beam current and position, the
latter being critical for the electron-vertex reconstruction and momentum
calculation, were monitored by resonant radio-frequency cavities (beam current
monitors [78]) and beam position monitors [78]). The beam size was monitored
using harp scanners.

The experiment employed an aluminum target with a thickness of $0.889\pm
0.002$ g/cm2. One of the aluminum foils was positioned to match the entrance
and the other to match the exit windows of the argon gas target cell. The two
thick foils were separated by a distance of 25 cm, corresponding to the length
of the argon gas cell and the Al foil’s thickness.

Table 2 summarizes the kinematical conditions of the E12-14-012 experiment.

Table 2: Kinematics settings used to collect the data for the JLab E12-14-012 experiment. | $E_{e}^{\prime}$ | $\theta_{e}$ | $Q^{2}$ | $|P|$ | $T_{p^{\prime}}$ | $\theta_{p^{\prime}}$ | $|{\bf q}|$ | $p_{m}$ | $E_{m}$
---|---|---|---|---|---|---|---|---|---
| (GeV) | (deg) | (GeV${}^{2}/c^{2}$) | (MeV/$c$) | (MeV) | (deg) | (MeV/$c$) | (MeV/$c$) | (MeV)
kin1 | 1.777 | 21.5 | 0.549 | 915 | 372 | $-50.0$ | 865 | 50 | 73
kin2 | 1.716 | 20.0 | 0.460 | 1030 | 455 | $-44.0$ | 846 | 184 | 50
kin3 | 1.799 | 17.5 | 0.370 | 915 | 372 | $-47.0$ | 741 | 174 | 50
kin4 | 1.799 | 15.5 | 0.291 | 915 | 372 | $-44.5$ | 685 | 230 | 50
kin5 | 1.716 | 15.5 | 0.277 | 1030 | 455 | $-39.0$ | 730 | 300 | 50

The VDCs’ tracking information was used to determine the momentum and to
reconstruct the direction (in-plane and out-of-plane angles) of the scattered
electron and proton, and to reconstruct the interaction vertex at the target.
Both the electron and proton arm information was separately used to
reconstruct the interaction vertex and we found them in very good agreement.
The transformation between focal plane and target quantities was computed
using an optical matrix, the accuracy of which was verified using the carbon
multi-foil target data and sieve measurements as described in previous papers
[39, 37, 76]. Possible variations of the optics and magnetic field in both
HRSs are included in the analysis as systematic uncertainties related to the
optics.

Several different components were used to build the triggers: the scintillator
planes on both the electron and proton spectrometers, along with signals from
the gas Cherenkov (GC) detector, the pion rejector (PR), the pre-shower and
the shower detector (PS).

#### Measurements

The inclusive data analysis is summarized in Refs. [37, 39, 76] and presented
in Fig. 3 (left), while the exclusive data analysis for Ar is summarized in
Refs. [79] and represented in Fig. 3 (right).

Figure 3: (Left) Comparison of the cross sections per nucleon for aluminum,
argon [39], titanium [37], and carbon [37] measured at beam energy 2.222 GeV
and scattering angle $15.54^{\circ}$. The average nucleon number for every
target is calculated according to the natural abundances of isotopes, see
details in the text. The bands represent the total uncertainties. Figure from
Ref. [76]. (Right) Six-fold differential cross section for argon as a function
of missing momentum integrated over $E_{m}$ from 0 to 27 MeV. Figure from Ref.
[79].

The inclusive cross section has been measured on a variety of targets,
including aluminum, carbon, and titanium and a closed argon-gas cell. The data
extend over broad range of energy transfer, where quasielastic interaction,
$\Delta$-resonance excitation, and deep inelastic scattering yield
contributions to the cross section. The double differential cross sections are
reported with high precision ($\sim$3%) for all targets over the covered
kinematic range ([37, 39, 76]).

The exclusive cross section has been measured on Ar with $\approx$4%
uncertainty, and have been studied as a function of missing energy and missing
momentum, and compared to the results of Monte Carlo simulations, obtained
from a model based on the Distorted Wave Impulse Approximation [79].

The availability of the inclusive data will be essential for the development
of accurate models of the lepton scattering off argon and titanium [80]. The
ultimate goal of the exclusive analysis is the determination of the spectral
functions. The results of this effort will provide the input for sampling
momentum and energy of the target nucleons, thus allowing for a reliable
reconstruction of neutrino interactions in liquid argon detectors.

####  5.1.2 E4nu at JLab

#### Overview

The $e4\nu$ effort at JLab uses a large acceptance detector to measure wide
phase-space exclusive and semi-exclusive electron-nucleus scattering at known
beam energies to test energy reconstruction methods and interaction models.
This includes both existing data taken with the CLAS spectrometer in 1999 and
new data taken in winter 2021/22 with the CLAS12 spectrometer.

The incident electron energies of 1 to 6 GeV span the range of typical
accelerator-based neutrino beams, the nuclear targets are similar, and the
hadron detection thresholds are similar to those of tracking neutrino
detectors. The data analysis is as similar as possible to neutrino-based ones
[81]. For each exclusive channel, differential cross sections are extracted
using experimental neutrino methods. The electron–nucleon cross-section is
much more forward peaked than the neutrino cross-section due to the difference
in the photon and $W^{\pm}$ propagators. Therefore, each event is weighted by
$Q^{4}$, where $Q^{2}$ is the square of the four-momentum transfer. The cross
sections, typically as a function of the reconstructed lepton energy, are
compared to predictions from the electron mode of neutrino-nucleus interaction
event generators and to the true beam energy (see Fig. 1).

#### Experimental Setup

Data was taken in 1999 using the CEBAF large acceptance spectrometer (CLAS) at
the Thomas Jefferson National Accelerator Facility (JLab) to measure electron-
nucleus scattering with multi-hadron final states. Data was taken with 3He,
4He, C and Fe targets at 1.1, 2.2 and 4.4 GeV. CLAS used a toroidal magnetic
field with six sectors of drift chambers, scintillation counters, Cherenkov
counters and electromagnetic calorimeters to identify electrons, pions,
protons, and photons, and to reconstruct their trajectories[82]. CLAS covered
scattering angles from about $10^{\circ}$ to about $140^{\circ}$ with
azimuthal coverage of 50 to 80%. CLAS covered approximately 50% of $4\pi$ and
had reconstruction thresholds of approximately 150 MeV/c for charged pions and
300 MeV/c for protons. Photons of greater than 250 MeV were detected at
$8\leq\theta\leq 45^{\circ}$. The experiment used the standard inclusive
electron trigger, with Cherenkov Counters (CC) and Electromagnetic Calorimeter
(EC) at 1.1 and 2.2 GeV and EC only at 4.4 GeV. The experiment collected data
corresponding to $2.2\times 10^{9}$ triggers and 13 mC of beam charge.

More data was taken with CLAS12 in 11/2021–2/2022 as described below.

#### Modelling Development

In order to compare the $e4\nu$ measurement to models, the electron-scattering
mode of the GENIE event generator (e-GENIE) was revised [83]. Many of the
interaction models were debugged and updated, making e-GENIE self-consistent
with the neutrino codes for the first time [70]. Radiative effects necessary
for comparing QE-like electron scattering data to simulation was incorporated.
The simplistic implementation was vetted against electron-hydrogen scattering
data from JLab. In addition, models based on the SuSAv2 approach were
implemented for the QE and MEC processes.

Neutrino mode cross sections are very similar to electron-mode cross sections
scaled by $Q^{4}$, validating the use of electron scattering to improve
neutrino event generators. The e-GENIE predictions were compared with
inclusive electron scattering data, and agree moderately well at low energy
transfer. However there is significant disagreement at higher energy transfer,
which is dominated by resonance production [83].

#### Measurements

The first $e4\nu$ analysis used 1999 CLAS [82] data on 4He, C and Fe targets
with monochromatic electron beams of 1.159, 2.257 and 4.453 GeV. The analysis
focused on Quasi Elastic (QE) like events, by selecting events with one
electron, one proton of momentum larger than 300 MeV/c, and no charged pions
of momenta larger than 150 MeV/c. In addition, the minimum CLAS electron
scattering angle corresponded to minimum momentum transfers of
$Q^{2}>0.1,0.4,$ and $0.8$ GeV2 for beam energies of 1.159, 2.257, and 4.453
GeV, respectively. The QE-like event sample was corrected for events with
undetected pions, photons, and extra protons using a data-based method; more
details about the background subtraction can be found at[10].

This was the first test of incident lepton reconstruction. Two methods were
used to reconstruct the incident energy: (1) using only the scattered lepton
momentum and angle and assuming QE kinematics:

$E_{QE}=\frac{2M_{N}\epsilon+2M_{N}E_{l}-m_{l}^{2}}{2(M_{N}-E_{l}+k_{l}\cos\theta_{l})}.$

This method is used in experiments with Cherenkov-detectors, which are
sensitive only to the outgoing lepton [84]. (2) used the outgoing proton and
electron energies and is commonly used across tracking detectors:

$E_{cal}=E^{\prime}_{e}+T_{p}+E_{binding}.$

Cross sections as a function of reconstructed energies were extracted for C
and Fe at 1.159, 2.257, 4.453 GeV and compared to two different versions of
GENIE [10].

Fig. 1 from [10] shows the cross section as a function of the calorimetric
reconstructed energy. All spectra show a sharp peak at the real beam energy,
followed by a large tail at lower energies. For carbon, only 30–40% of the
events reconstruct to within 5% of the real beam energy. For iron this
fraction is only 20–25%, highlighting the crucial need to model precisely the
low-energy tail of these distributions. GENIE described the data
qualitatively, but there are significant quantitative disagreements. One
version of GENIE predicted a reconstructed energy that was about 20 MeV too
low. This disagreement in the tails increases with incident energy and nuclear
mass.

The data was then binned as a function of the transverse missing momentum,

$P_{T}=P^{e}_{T}+P^{p}_{T}$

(where $P_{T}^{e}$ and $P_{T}^{p}$ are the three-momenta of the detected
lepton and proton perpendicular to the direction of the incident lepton). At
$P_{T}\leq 200$ MeV/c almost all events reconstruct to the correct cross
section and GENIE described those events well. At intermediate $P_{T}$
($200\leq P_{T}\leq 400$ MeV/c) only a small fraction of events reconstruct to
the correct beam energy and GENIE estimates the cross section of events that
reconstruct to lower energies. At large $P_{T}$ ($P_{T}>400$ MeV/c), none of
the events reconstruct to the correct beam energy and GENIE significantly
overestimates the cross section of events that reconstruct to lower energies.
This shows that neutrino experiments can use $P_{T}$ to select events that
reconstruct to the correct energy and that GENIE requires further modeling
improvements for large $P_{T}$ [10].

A second analysis examining the $1p1\pi$ channel is in progress for this data
set. There is very little data for this channel in the resonance region for
nuclear targets and the largest disagreement between GENIE and inclusive
$A(e,e^{\prime})$ data [83] is seen there. These data will help improve all
generator models for nuclear targets.

#### New Data with CLAS12

A dedicated $e4\nu$ measurement with the upgraded CLAS12 detector was approved
by the JLab PAC with an A scientific rating for 45 beam days [85]. The
experiment will use targets from D to Sn, including neutrino-detector
materials (C, O, and Ar), at 1, 2, 4, and 6 GeV. During the fall/winter
2021/2022 data were collected with C and Ar at 2, 4 and 6 GeV, along with
measurements of H, D, 4He, 40Ca, 48Ca and Sn for calibration and for other
nuclear physics purposes.

The CLAS12 spectrometer operates at higher luminosity (ten times higher than
CLAS), and takes advantage of the lower scattering angle thresholds
($\theta_{e}\geq 8^{\circ}$), along with the neutron detection capabilities in
the central detector ($\approx 35^{\circ}\leq\theta\leq 135^{\circ}$) [86] .
This experiment will provide a much larger data set, with more event channels
($1p0\pi$, $1p1\pi$, etc), greater angular and kinematic coverage, and more
targets than the existing CLAS data.

####  5.1.3 LDMX at SLAC

LDMX (Light Dark Matter eXperiment) is a fixed-target electron-scattering
experiment planned to search for sub-GeV dark matter [87]. A signature
expected for dark-matter production is missing energy and missing momentum,
carried away from the event by undetected particle(s). To reach the parameter
space previously unexplored, LDMX needs to achieve an excellent background
rejection and analyze an extensive body of data.

The requirement of precise reconstruction of both charged and neutral hadrons,
in conjunction with the necessity to collect a vast number of electron-
scattering events makes the goals of LDMX highly synergistic with the physics
program of the neutrino community interested in improving current modeling of
electroweak interactions [24].

Dedicated studies of neutrino cross sections performed by the MINERvA
Collaboration at the kinematics similar to that of DUNE unambiguously show
that existing Monte Carlo generators are not able to explain available data
[88, 42, 89]. Particularly dire problems are observed for pion production,
where measurements for different channels within a single experiment cannot be
reconciled, and the origin of this issue is difficult to pinpoint [42]. This
is a consequence of the complexity of the problem: flux-averaging greatly
diminishes differences between different interaction channels at the inclusive
level, different interaction mechanisms yield the same final states, only
nuclear cross sections are available with the desired precision, and
scattering involves multicomponent vector and axial currents [22].

These difficulties can be addressed by making the full use of the similarities
of electron and neutrino interactions [24]. Electron-scattering measurements
feature monoenergetic and adjustable beams, cross sections higher by several
orders of magnitude, and feasible targets ranging from complex nuclei to
deuterium and hydrogen. As long as the cross sections for neutrinos and
electrons are calculated consistently, and they are (over)constrained by
precise electron data, the uncertainties of Monte Carlo simulations employed
in neutrino physics can be reduced to those stemming from the axial
contributions, to be constrained by the near-detector measurements. In the
context of an accurate calorimetric neutrino-energy reconstruction, the
exclusive cross sections—currently unavailable for electron scattering on the
targets of interest—play particularly fundamental role [40, 41].

Figure 4: (Left) Kinematics accessible in _inclusive and (semi)exclusive_
electron-scattering measurements at LDMX with a 4-GeV beam, overlaid on the
charged-current $\nu_{\mu}$Ar event distribution expected in the DUNE near
detector according to GiBUU [24]. Constant scattering angles $\theta_{e}$ and
transverse momenta $p^{e}_{T}$ of electrons correspond to the blue and green
lines, respectively. LDMX will probe the region with $\theta_{e}<40^{\circ}$
and $p^{e}_{T}>10$ MeV, extending beyond the plot. (Right) Breakup of hadronic
energy of events with $20.0^{\circ}\leq\theta_{e}\leq 22.5^{\circ}$,
$p^{e}_{T}>200$ MeV, and $\omega>1$ GeV, according to GiBUU and GENIE. The
contributions outside of the LDMX acceptance—outside the 40∘ cone or below
detection thresholds—are hatched [24].

Inclusive measurements. Currently, only one kinematics of relevance for DUNE
has been probed through inclusive electron scattering on argon [39] and
titanium [90], the proton structure of which mirrors the neutron structure of
argon. The data—collected for beam energy 2.22 GeV, scattering angle
$15.54^{\circ}$, and extending to energy transfers below 0.95 GeV—show that at
the inclusive level and within uncertainties, the cross sections per nucleon
do not differ between argon and titanium [76], see Fig. 3. Employing a
titanium target [24], LDMX can collect a wealth of inclusive data for beam
energy 4-GeV, scattering angles below $40^{\circ}$, and energy transfers above
1 GeV. To illustrate their importance for DUNE, in the left panel of Fig. 4,
the kinematics accessible in LDMX is overlaid on the distribution of charged-
current $\nu_{\mu}$ interactions with argon expected in the near detector of
DUNE, as calculated using GiBUU [24]. With its nominal selections, LDMX can
cover the deep-inelastic kinematics in great detail, and probe the tail of
resonance production. These cross sections, currently unavailable, would be
essential for validation of Monte Carlo simulations for DUNE [22].

(Semi)exclusive measurements. With full-event reconstruction being essential
for its core program and the envisioned large integrated luminosity, LDMX is
uniquely equipped to extract coincidence cross sections for a variety of final
states involving pions and nucleons. For example, of the $10^{14}$ electrons
on target envisioned for the first phase of low-luminosity running, $10^{8}$
will scatter at $20.0^{\circ}$–$22.5^{\circ}$ and pass the selection cuts. The
kinetic energy spectra are expected to be measured down to $\sim$60 MeV for
charged pions and protons, and to $\sim$500 MeV for neutrons. As summarized in
the right panel of Fig. 4, the collected data should allow discriminating
between the cross section models implemented in GiBUU and GENIE, performing
their extensive validations, and quantifying a number of uncertainties
originating from the rich physics of electroweak interactions.

Future potential. By employing titanium as the target, LDMX in the baseline
dark-matter configuration can perform measurements of both inclusive and
(semi)exclusive electron scattering, which are of great interest for DUNE. The
physics program can be further extended, as discussed in Ref. [24].

  * •

Varying the target material, LDMX would provide more data for nuclear
modeling, allowing for deeper understanding of the cross-sections’ dependence
on the atomic number. In particular, a high-pressure argon-gas cell would
directly address the needs of the neutrino community, complementing the
titanium results. Measurements for helium, deuterium, and hydrogen would
provide a handle on how nuclear transparency affects the exclusive cross
sections. They would not only allow a clean separation of hadronic and nuclear
effects, but also give an unprecedented insight into how the effects of final-
state interactions shape the observed hadronic spectra.

  * •

The nominal physics selections can be extended to energy transfers below 1
GeV, to fully cover the regions in which resonance production and meson-
exchange currents provide important contributions to the cross sections. Such
measurements could improve our understanding of resonant and nonresonant pion
production, and shed light on the composition of hadronic final states in the
dip region between the quasielastic and $\Delta$-production peaks.

####  5.1.4 A1 Collaboration at MAMI

The Mainz Microtron (MAMI) is an electron accelerator run by the Institute for
Nuclear Physics of the University of Mainz and it employed for nuclear hadron
physics experiments. MAMI is a multi-stage racetrack microtron with normal
conducting acceleration cavities. The accelerator is available for experiments
since 1991 and has continuously undergone further development. The last added
stage, MAMI-C, became operational in 2007 rising the beam energy up to 1.6 GeV
with a maximum current of 100 $\mu$A. The beam diameter is $\sim$0.1 mm with
an energy resolution of 13 keV. A beam stabilization system maintains the beam
position constant within 200$\mu m$. The beam has a continuous-wave structure
(100% duty cycle) and the possibility to be polarized at the $\sim 80\%$
level. The main parameters of the last two MAMI microtron stages are
summarized in Tab. 3.

Stage | MAMI B | MAMI C
---|---|---
Max. Energy | 855.1 MeV | 1508 MeV
Circulations | 90 | 43
Deflecting Magnets: |  |
Magnetic Field | 1.28 T | 0.95 - 1.53 T
Mass |  2$\times$450 t |  4$\times$250 t
Cavities: |  |
Frequency | 2.45 GHz | 2.45 / 4.90 GHz
Power | 102 kW | 117 / 128 kW
Linac Length | 8.9 m | 8.6 / 10.1 m
Table 3: Main properties of the MAMI cascaded microtron accelerator.

The MAMI beam can be delivered in different experimental halls. The A1
Collaboration hall in particular is devoted to experiments with electrons,
while the A2 collaboration hall converts the electrons into a tagged photon
beam. Another hall is devoted to experiments with X-rays and detector
development. The A1 collaboration operates a three magnetic spectrometer setup
for experiments in nuclar and hadron physics. The spectrometers are
conventionally called A, B, and C. Spectrometers A and C have the same
quadrupole-sextupole-dipole-dipole (QSDD) magnetic optics for maximizing
angular acceptance, while spectrometer B has only one dipole magnet: this
construction choice allows this spectrometer to reach smaller angles (i.e. to
move closer to the beamline) with the additional capability to move out-of-
plane.

| units | Spec. A | Spec. B | Spec. C
---|---|---|---|---
Magnetic Configuration | - | QSDD | D | QSDD
Maximum momentum | (MeV/c) | 735 | 870 | 551
Reference momentum | (MeV/c) | 630 | 810 | 459
Central Momentum | (MeV/c) | 665 | 810 | 490
Solid Angle Acceptance | (msr) | 28 | 5.6 | 28
minimum angle | (deg) | 18 | 7 | 8
maximum angle | (deg) | 160 | 62 | 160
Momentum acceptance | – | 20% | 15% | 25%
Angular acceptance: |  |  |  |
dispersive | (mrad) | 70 | 70 | 70
nondispersive | (mrad) | 100 | 20 | 100
momentum resolution | – |  10-4 |  10-4 |  10-4
angular resolution at target | (mrad) | $<3$ | $<3$ | $<3$
position resolution at target | (mm) | 3–5 | 1 | 3–5
Table 4: Main properties of the three magnetic spectrometers of the A1
Collaboration at MAMI.

The spectrometers have a momentum resolution of $10^{-4}$. The characteristics
of the spectrometers are summarized in Tab. 4. All spectrometers have the same
detector package composed of two vertical drift chambers (VDCs), two planes of
scintillator bars, and a Cherenkov detector. Each VDC is composed by 4 wire
planes, two perpendicular to the dispersive plane for momentum and out-of-
plane angle reconstruction, and the other 2 planes, with wires tilted by a few
degrees for completing the position determination in the focal plane. The
single plane efficiency is $\sim$99%, leading to an overall efficiency of
better than 99.9%. The scintillator planes serve as trigger detectors and for
particle identification, together with the Cherenkov detector. The
spectrometers can be employed in coincidence (double or triple) for exclusive
measurements. Different kinds of targets can be installed at the end of the
MAMI beamline at the pivot point around which the spectrometers can be
rotated. Solid-state targets (e.g. C, Ca, Si, Ta, Pb) are routinely used. A
cryogenic target is also available and has been in use with different elements
in liquid phase (H, 2H, 3He, 4He) and its use for noble gases (e.g. Ar, Xe) is
possible. A waterfall target is also available for measurements with oxygen.
Recently, a supersonic gas-jet target was successfully tested with hydrogen
and argon.

####  5.1.5 A1 Collaboration at Spanish facilities

#### Experimental setup and capabilities

The proposed instrument (eALBA) is a multipurpose electron beam facility,
where the electron beam from the ALBA synchrotron in Barcelona(Spain) is
extracted from the tunnel to the experimental hall, where a multipurpose
experimental area will be conditioned. This beam is expected to serve
different experimental applications: Detectors characterization for HEP, Gamma
detector characterization, Radiation hardness tests for space electronic,
Electron imaging, Microdosimetry and Radio-biology Experiments with
conventional and Ultra-high Dose electron, but also Education and Outreach.
This infrastructure will be unique in Europe covering wide beam energy range
(100 MeV - 3 GeV) with excellent repetition rate (3 Hz) and high intensity per
bunch (1 nC).

One of the proposed experimental facilities is devoted to Electron-Nucleus
scattering experiments for neutrino studies. One possible realization of this
experiment is an atmospheric pressure Time Projection Chamber (TPC)
surrounding a target located outside of the gas volume. The TPC operates in a
magnetic field pointing in the direction of the electron beam and the TPC
field. The magnetic field bends particles according to its transverse
momentum. The field strength is expected to be moderate, and it can be built
using conventional technology. A cylindrical TPC with a diameter of 1 m and a
length 2 m is technologically feasible and sufficient for the purpose of the
experiment. The inner wall of the TPC can be very thin when operating the TPC
at atmospheric pressure and allowing low momentum particles to enter in the
active TPC volume from the external beam target. There are available
technology to read TPC of this size using Micro Pattern Gas detector
technology as the one pioneered by the T2K near detector. By using existing
technology and a moderate magnetic field, the experiment cost can be reduced
to levels that be constructed by small collaborations.

The angular distribution of 4 GeV electrons scattered from the C nucleus as
predicted by the GIBUU event generator are shown in Figure 5. Requesting the
electrons to traverse at least 25 cm inside the TPC (a fourth of the total
length) with an inner TPC radius of 1cm, the minimum acceptance angle is
around $40$ mrads. Most of the interactions below this value will be
quasielastic and probably will not eject hadrons with a large transverse
momentum. Electrons ejected below $40$ mrads can be detected with a forward
calorimeter.

Figure 5: Angular distribution of 4 GeV electrons scattered from the C nucleus
as predicted by the GIBUU event generator

One of the drawbacks of TPC’s is the low drift velocity in the gas, of the
order of tenths of microseconds. According to the same GIBUU simulations, the
integrated cross-section above $4$ mrads is of the order of $2\times
10^{-33}\mbox{cm}^{2}/\mbox{nucleon}$. The interaction probability in a Carbon
target with a density of approximately $2g/$cm3 is of the order of $5\times
10^{-7}$ in a $1$ mm thick target. At a $1pA$ beam current, there will be
$\approx 3$ interactions per second with the scattered electron above $40$
mrads. This event rates are within the TPC timing capabilities. Most of the
interactions below $40$ mrads will leave no signature in the detector due to
the spiral trajectory caused by its low transverse momentum. Targets thinner
than 1mm will improve the performance, the lower interaction rate can be
compensated with the high beam intensities and the target will affect less the
hadrons produced in the interactions. A dedicated Monte Carlo study will be
carried out to understand more precisely the capabilities of this design.

By hosting an external target, the experiment can explore different nuclei
targets and energies. To reach levels interesting to neutrino experiments
based on the water Cherenkov technology, we will require electron beams with
energies from 500 MeV to few GeV.

#### Proposed measurements

The measurements will cover several target nuclei. We expect a minimum set of
targets: C, CH, Be, Ca. Gaseous and liquid targets (He, H20, Ar,…) are a
challenge due to the size of the containment vessel. The experiment can be
seen as a facility that could be run in the future serving the needs of
different experiments.

###  5.2 Summary of the ongoing and future efforts

Collaborations | Kinematics | Targets | Scattering |
---|---|---|---|---
E12-14-012 (JLab) |  $E_{e}$ = 2.222 GeV | Ar, Ti |  ($e,e^{\prime}$) |
(Data collected: 2017) | $15.5^{\circ}\leq{\theta_{e}}\leq 21.5^{\circ}$ | Al, C | $e,p$ |
| $-50.0^{\circ}\leq{\theta_{p}}\leq-39.0^{\circ}$ |  | in the final state |
e4nu/CLAS (JLab) |  $E_{e}$ = 1, 2, 4, 6 GeV | H, D, He, |  ($e,e^{\prime}$) |
(Data collected: 1999, 2022) | ${\theta_{e}}>5^{\circ}$ |  C, Ar, 40Ca, | $e,p,n,\pi,\gamma$ |
|  |  48Ca, Fe, Sn | in the final state |
LDMX (SLAC) |  $E_{e}$ = 4.0, 8.0 GeV |  |  ($e,e^{\prime}$) |
(Planned) | ${\theta_{e}}<40^{\circ}$ | W, Ti, Al | $e,p,n,\pi,\gamma$ |
|  |  | in the final state |
A1 (MAMI) |  50 MeV $\leq E_{e}\leq 1.5$ GeV | H, D, He |  ($e,e^{\prime}$) |
(Data collected: 2020) | $7^{\circ}\leq{\theta_{e}}\leq 160^{\circ}$ | C, O, Al | 2 additional |
(More data planned) |  | Ca, Ar, Xe | charged particles |
A1 (eALBA) |  $E_{e}$ = 500 MeV | C, CH |  ($e,e^{\prime}$) |
(Planned) |  \- few GeV | Be, Ca |  |
Table 5: Current and planned electron scattering experiments. Figure 6:
Kinematic coverage of the ongoing and planned experiments for inclusive and
exclusive electron scattering on targets including argon and titanium,
presented in the (a) $(|\mathbf{q}|,\omega)$ and (b) $|\mathbf{q}|,W)$ planes,
with momentum transfer $\mathbf{q}$, energy transfer $\omega$, and hadronic
mass $W$. The light and dark shaded areas cover 68% and 95% of charged-current
$\nu_{\mu}$Ar events expected in the DUNE near detector [38], according to
GENIE 3.0.6. The thin solid, dashed, and dotted lines correspond to the
kinematics of quasielastic scattering, $\Delta$ excitation, and the onset of
deep-inelastic scattering at $W=1.7$ GeV on free nucleons.

In Table 5, we present a summary of the current and planned electron-
scattering experiments, which cover a broad range of kinematics and carry
varied level of particle identification and other detection capabilities. This
kinematics is presented in Fig. 6, where it is overlaid on the regions
expected to contain 68% and 95% of charged current interactions of muon
neutrinos with argon in the DUNE near detector, as estimated using GENIE
3.0.6. Thanks to its broad coverage of scattering angles—from $7^{\circ}$ to
$160^{\circ}$—and beam energies—from $\sim$50 MeV to 1.5 GeV—A1 in MAMI is
uniquely equipped to perform extensive studies of the quasielastic and
$\Delta$ peaks. LDMX in SLAC is planned to feature a 4-GeV beam, closely
corresponding to the average energy in DUNE, and perform extensive studies of
pion production. The e4nu experiment in JLab has the potential to study a
large swath of the DUNE kinematics, thanks to the broad range of employed beam
energies. To probe the kinematic region of interest at higher momentum
transfers and perform studies in the deep-inelastic regime, an experiment at
even higher energies would be desirable, such as the 8-GeV configuration of
LDMX (not depicted). However, the issue deserving most attention should always
remain measuring exclusive cross sections, essential for accurate calorimetric
reconstruction of neutrino energy, especially for low-energy hadrons.

##  6 Experimental Landscape II: Input to Low-Energy Neutrino Physics

The tens-of-MeV neutrinos, from stopped pion sources or from core-collapse
supernova, primarily interact via two processes: coherent elastic neutrino-
nucleus scattering (CEvNS), and inelastic charged and neutral current
scattering. Precise determination of Standard Model CEvNS cross section will
enable new physics searches in CEvNS experiments, while precise inelastic
cross section determination will enable detection of supernova signals in DUNE
experiment.

The main source of uncertainty in the evaluation of the CEvNS cross section is
driven by the underlying nuclear structure, embedded in the weak form factor,
of the target nucleus. Weak form factors and neutron radii are historically
poorly known. Recent advancements in PVES experiments, utilizing polarized
electron beams, provide relatively model-independent ways of determining weak
form factors that can be used as direct input in determining CEvNS cross
section. The inelastic neutrino-nucleus cross sections in this tens-of-MeV
regime are quite poorly understood. There are very few existing measurements,
none at better than the 10% uncertainty level. As a result, the uncertainties
on the theoretical calculations of, e.g., neutrino-argon cross sections are
not well quantified at all at these energies. To this end, 10s of MeV electron
scattering data will be vital in constraining neutrino-nucleus interaction at
this energy scale.

Dedicated electron scattering experiments with targets and kinematics of
interests to low-energy neutrino experiments will be crucial in achieving
precision goals of low-energy neutrino programs.

###  6.1 Parity-Violating Electron Scattering Experiments

####  6.1.1 PREX and CREX at JLab

The parity violating asymmetry $A_{pv}$ for elastic electron scattering is the
fractional difference in cross section for positive helicity and negative
helicity electrons. In Born approximation $A_{pv}$ is proportional to the weak
form factor $F_{W}(q^{2})$ [92],

$A_{pv}=\frac{d\sigma/d\Omega_{+}-d\sigma/d\Omega_{-}}{d\sigma/d\Omega_{+}+d\sigma/d\Omega_{-}}=\frac{G_{F}q^{2}|Q_{W}|}{4\pi\alpha\sqrt{2}Z}\frac{F_{W}(q^{2})}{F_{ch}(q^{2})}\,.$
(3)

Here $F_{ch}(q^{2})$ is the (E+M) charge form factor that is typically known
from unpolarized electron scattering. Therefore, one can extract $F_{W}$ from
measurements of $A_{pv}$. Note that Eq. 3 must be corrected for Coulomb
distortions as discussed in ref. [93]. These effects are absent for neutrino
scattering.

The PREX experiment at JLAB measures $A_{pv}$ for $\approx 1$ GeV electrons
elasticly scattered from 208Pb [91]. This experiment was motivated both to aid
the interpretation of atomic parity experiments and to study neutron rich
matter in astrophysics.

##### Atomic parity nonconservation (PNC):

provides an important low energy test of the standard model. The weak
interaction involves the overlap of atomic electrons with the weak charge of
the nucleus (ie neutrons). The existing atomic Cs experiment measures the weak
charge to about 0.3% and is probably not limited by uncertainty in the neutron
radius. However, a future 0.1% atomic experiment may need $R_{n}$ to about 1%.
The best PNC test of the standard model may involve combining an atomic
experiment with a separate measurement of $F_{W}(q^{2})$ [92].

##### Neutron rich matter in astrophysics:

Compress almost anything enough and electrons capture on protons to make
neutron rich matter. This material is at the heart of many fundamental
questions in Physics and Astrophysics. What are the high density phases of
QCD? Where did chemical elements come from? What is the structure of many
compact and energetic objects in the heavens, and what determines their
electromagnetic, neutrino, and gravitational-wave radiations?

A heavy nucleus such as 208Pb has a skin of neutron rich matter [94]. Probing
this region with coherent neutrino scattering or parity violating electron
scattering allows the study of neutron rich matter in the laboratory . The
pressure of neutron rich matter pushes neutrons out against surface tension in
208Pb. This same pressure holds neutrons up against gravity in a neutron star.
A neutron star is 18 orders of magnitude larger than a nucleus but it is made
of the same neutrons, with the same strong interactions, and the same equation
of state or pressure as a function of density. Therefore measurement of the
neutron radius of 208Pb in the laboratory has important implications for the
structure of neutron stars [95, 96].

Table 6 lists three recent parity violating elastic electron scattering
experiments. PREX has determined the neutron radius $R_{n}$ of 208Pb to 1.3%
[PREX2]. The CREX experiment has measured $A_{pv}$ from 48Ca [CREX]. The
analysis of this measurement is ongoing but we expect a sensitivity to $R_{n}$
in 48Ca of $\pm$0.7%. Finally the Qweak collaboration has measured electron
scattering from 27Al [97]. Note because of the limited energy resolution of
the Qweak spectrometer $\pm 100$ MeV this experiment is limited by systematic
errors from inelastic contributions rather than from statistics.

We see that the best present parity violating experiments have determined
$R_{n}$ to around one %. This may be more accurate than present neutrino
experiments. Although, the PVES experiments are usually carried out at a
single value of the momentum transfer at a time. Nevertheless, one may be able
to use parity violating electron scattering (and nuclear theory) to constrain
nuclear structure and $F_{W}(q^{2})$. This could allow coherent neutrino
scattering experiments to better constrain nonstandard neutrino interactions.

Experiment | Target |  $q^{2}$ (GeV2) |  $A_{pv}$ (ppm) |  $\pm\delta R_{n}$ (%)
---|---|---|---|---
PREX |  208Pb | 0.00616 | $0.550\pm 0.018$ | 1.3
CREX |  48Ca | 0.0297 |  | 0.7
Qweak |  27Al | 0.0236 | $2.16\pm 0.19$ | 4
MREX |  208Pb | 0.0073 |  | 0.52
Table 6: Parity violating elastic electron scattering experiments.

####  6.1.2 MREX at MESA

A novel experimental program is foreseen at the future "Mainz Energy-
recovering Superconducting Accelerator" (MESA), which is being build at the
Institute of Nuclear Physics of the Johannes Gutenberg University (Mainz)[98].
MESA is a low-energy high-intensity electron accelerator for particle physics
at the precision and intensity frontier. The combination of high beam quality
and intensity allows precision measurements to explore the limits of the
standard model. The design of the MESA accelerator provides two modes of
operation, which serve as the basis for a totally different classes of
experiments. With a “Pseudo internal target” combined with two high resolution
spectrometers experiment using ERL-mode explorative searches for hitherto
unknown gauge bosons can be undertaken (see 6.2.1)

The second experiment (P2 [99]) is a parity violating electron scattering
experiment with spin-polarized beam using EB-mode. The aim of the experiment
P2 is the precise determination of a fundamental parameter of the Standard
Model: the effective electroweak mixing angle $\sin^{2}(\theta_{W})$. The
desired relative accuracy of 0.1% outperforms existing measurements at low-
energy scales by more than an order of magnitude and is thus comparable to the
measurements at LEP and SLAC from the 1990s at very high-energy scales. The
aimed sensitivity is similar to the best Z-pole measurement.

Preliminary feasibility studies have shown that the Mainz Radius EXperiment
(MREX) at MESA, by adopting the same detector setup intended for the P2
experiment will be able to extract the neutron skin thickness of 208Pb by a
determination of the neutron radius with a sensitivity of $\delta
R_{\text{n}}/R_{\text{n}}=$0.5\text{\,}\%$$ (or $\pm$ 0.03 fm). With the
necessary momentum resolution of 1%, MREX will thus outperform the PREX
experiment. Indeed although the maximum incident energy at MESA (thus the
cross section) will be lower compared to the JLab measurement, the experiment
will benefit on the one hand from the higher beam intensities and on the other
hand from the full azimuthal coverage of the detector. A peak sensitivity for
208Pb is expected for the parameter reported in table 7. Since the same
experimental setup will be used to determine the weak mixing angle
$\theta_{W}$ and the neutron skin of 208Pb the running time for the latter one
was restricted to about 1500 hours.

beam energy | $155\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$
---|---
beam current | $150\text{\,}\mu\mathrm{A}$
target density | $0.28\text{\,}\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$
polar angle step size | $\Delta\theta=$$$
polar angular range |  $None$ to $None$
degree of polarization | $85\text{\,}\%$
parity violating asymmetry | $0.66\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$
running time | 1440 hours
systematic uncertainty | $1\text{\,}\%$
$\delta A^{\text{PV}}/A^{\text{PV}}$ | $1.39\text{\,}\%$
$\delta R_{\text{n}}/R_{\text{n}}$ | $0.52\text{\,}\%$
Table 7: Kinematical values and general parameters used for the run time
estimate.

The ultimate precision on the neutron radius measurement at MESA will be
limited by the implicit model assumption in the method on the surface
thickness. The extraction of the neutron-skin thickness in parity violation
experiments is indeed based on the measurement of the weak form factor at a
single value of momentum transfer. Thus some assumptions concerning the
surface thickness have to be made in order to reliably extract the neutron-
skin thickness. In order to remove this model dependence a determination of
the surface thickness is necessary with a precision better than the current
model assumption ($25\text{\,}\%$)[100]. In particular since the parity-
violation asymmetry measurement at MESA will determine the neutron radius of
208Pb with a precision of $\pm$$0.03\text{\,}\mathrm{f}\mathrm{m}$ the surface
thickness must be known down to $10\text{\,}\%$. Preliminary sensitivity study
indicates that such a measurement can be performed at the A1 spectrometer at
MAMI (see 5.1.4). According to the construction of the spectrometers and their
arrangement on the pivot surrounding the scattering chamber, there are
limitations in the accessible angular range. Furthermore, only selected beam
energies at MAMI are equipped with a special stabilization system which is
essential for performing parity-violation experiments. Based on the experience
of the first successful experimental campaigns [101] and on the requirement
that the sensitivity of $A_{\mathrm{PV}}$ to the weak-charge radius must be
suppressed w.r.t. the sensitivity to the surface thickness for the given beam
energy and scattering angles the optimum kinematics is obtained for 570 MeV
electron energy and
$0.06\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}^{2}\mathrm{/}\mathrm{c}^{2}$ and
$0.02\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}^{2}\mathrm{/}\mathrm{c}^{2}$
momentum transfer for the two spectrometers respectively.

####  6.1.3 Identifying Connections and Gaps

The true breakthrough in our understanding of neutron-rich matter will be only
attained by combining results from both terrestrial laboratories and
astrophysical observations. The ultimate precision on the neutron-skin
thickness will be attained at the MESA accelerator. However already nowadays
from the comparison between the most recent PREX-II result and the
astrophysical observation from both gravitational wave and neutron star
observatories stronger constraints on the EOS of neutron-rich matter have been
determined [96].

On the other side a recent analysis of the combination of coherent elastic
neutrino-nucleus scattering data from the COHERENT experiment and atomic
parity violation data has determined the most precise measurement of the
neutron rms radius and neutron-skin values of 133Cs and 127I [102]. A
combination of complementary methods to extract the neutron radius will
naturally produce even stronger constraints.

As already pointed out in 4.3 CEvNS experiments are limited to detector
materials with low scintillation or ionisation thresholds in order to
efficiently measure low-energy nuclear recoils. Quite the contrary is needed
as target material in parity violation electron scattering experiments: in
this case the highest the excited state of the nucleus the lower the
contamination of the elastic asymmetries by inelastic contributions from the
excited state (as an example the first excited state in 208Pb is about 2.60
MeV). In addition due to the high intensity of the electron beam a high
melting temperature of the target material is as well desirable.

A direct comparison between the neutron radius extracted for the same nucleus
in both type of experiments is therefore technically not feasible in the near
future. Naturally a combination of these two different approaches can proceed
by theory. In particular parity violation electron scattering results can be
used as anchors for modern density functional theory calculations which can
then predict the coherent nuclear form factor as measured in CEvNS at low
momentum transfer. Ultimately CEvNS extraction of the neutron distribution for
heavy nuclei by benchmarking theoretical predictions can serve as additional
constraint to the EOS of neutron rich matter.

The measurement of the coherent nuclear form factor for medium-mass nuclei as
for example 40Ar could additionally provide a validation bridge for modern ab-
initio calculations based on effective field potential. Together with the
neutron density extraction for 48Ca and 27Al as determined by CREX and Qweak
these data will provide the most accurate benchmark of modern nuclear
structure calculations.

###  6.2 Low Energy Electron Scattering

####  6.2.1 MAGIX Collaboration at MESA

As discussed in Sec. 6.1.2, MESA, a new cw multi-turn energy recovery linac
for precision particle and nuclear physics experiments with a beam energy
range of 100-200 MeV is currently being built. MESA will be operated in two
modes: energy recovery mode (ERM) and external beam mode (XBM). In ERM, the
accelerator will provide a beam current of up to 1 mA at 105 MeV for the MAGIX
internal target experiment with multi-turn energy recovery capability. In XBM,
a polarized beam of 150 $\mu$A will be provided. The linac will have an energy
gain of 50 MeV/pass by using four ELBE-like 9-cell cavities installed in two
cryomodules. The MAGIX experiment will be installed on one of the
recirculating paths: in this way the beam will pass through a supersonic jet-
target and then re-injected in the accelerator for energy recovery. MAGIX will
employ two high-resolution magnetic spectrometers movable around the jet
target. The relatively low density of the target (areal thickness $\sim
10^{20}$ atoms/cm2/Z2 with Z the gas atomic number) will be compensated by the
large ($\sim$mA) beam current.

The two spectrometers will be used for the detection of scattered electrons
and other produced particles. They will be equipped with TPCs (time projection
chambers) with GEM readout for tracking, and with a trigger and veto system
consisting of a stack of plastic scintillation detectors planes and a variable
number of additional lead absorber layers. Silicon strip detectors will be
installed inside the scattering chamber for detecting protons and nuclear
fragments in coincidence with the spectrometers.

The MESA accelerator and the MAGIX setup are currently under construction and
the jet target was already built and tested at the University of Münster. In
2017 the target was successfully installed and operated with hydrogen at the
existing A1 three-spectrometer setup at the Mainz Microtron (MAMI) electron
accelerator in Mainz, Germany. In 2021, the target was successfully tested
with argon. The jet target will be operated with hydrogen, oxygen, and noble
gases (e.g. He, Ar, Xe). Currently, the opportunity to install a beam-dump is
investigated: this would allow also the use of solid-state targets (which are
too dense to allow the beam recirculation) like carbon, for example.

MAGIX was conceived with the main goals of measuring the nucleon form factor
at low $Q^{2}$ contributing to the "proton-radius puzzle", searching for light
dark sector particles, measure astrophysical S-factors, and precisely measure
exclusive and inclusive nuclear cross sections. The MESA relatively low beam
energy falls into the supernova neutrinos range and can thus be interesting
for the tuning and testing of neutrino generators in this energy range.

##  7 Addressing NP and HEP Boundary Conditions

A ongoing challenge of the work described in this white paper– adapting
electron scattering to neutrino physics– is the separation of HEP and NP
research communities and a lack of funding support to address these cross-cut
research topics. Theorists and experimentalists who work on electron
scattering are supported from NP, but oscillation experiments like DUNE are
supported by HEP. To avoid duplication of funding, funding agencies opt to
avoid cross cutting topics such as this work; this view was expressed by
multiple groups via a workshop survey. Further, as some of the experiments
(like DUNE) are DOE HEP projects, the work is viewed as therefore under only
HEP or DOE scope.

Unfortunately, this separation has historically harmed the science and
community members. For example, when MINERvA was founded, collaborators funded
from NP had trouble getting funding for the (HEP) MINERvA project, and had to
leave. This resulted in the loss of valuable cross field discussion. It is
difficult to train and retain early career scientists with expertise on
intersectional topics, as they do not ‘fit’ neatly into either category. This
separation does not seem to occur in the LHC, where there is a clear
recognition of the value of cross section calculation and generator effort.

We propose the following possible solutions to challenge raised:

  1. 1.

Recognition of electron scattering as a necessary piece of a robust neutrino
physics program First, we summarize the motivation for new electron scattering
measurements and theory as applied to neutrino physics (this white paper). The
next step would be to commission impact metrics with experimental programs to
refine needed precision and use of measurements.

  2. 2.

Expand, and use new funding opportunities The Neutrino Theory Network will
support proposals of theorists working on topics across NP and HEP. The
formation of a topical group within DOE would avoid duplication of effort and
review of related proposals.

##  8 Summary

Neutrino physics has entered a precision era and exciting neutrino
experimental programs at low and high energies can lead to discoveries. The
importance of constraining systematics resulted from neutrino-nucleus
interaction physics in key neutrino measurements, in particular at
accelerator-based experiments, cannot be overstated. The kinematically
controlled beams and precise measurements of electron scattering data makes it
a key benchmark to assess and validate different nuclear models intended to be
used in neutrino experiments.

As part of the Snowmass 2021 process, we organized two dedicated Electron
Scattering workshop: i) “NF06 Electron Scattering” [103] focused on long-
baseline neutrino experiments, and ii) “NF06 Low Energy Neutrino and Electron
Scattering” [104] focused on the low-energy (CEvNS and supernova) neutrinos
that formed the basis of this contribution. In this White Paper, we highlight
connections between electron- and neutrino-nucleus scattering physics at
energies ranging from 10s of MeV to a few GeV, review the status of ongoing
and planned electron scattering experiments, identified connections and gaps
in the context of their input to the accelerator neutrino program. We also
highlight the systemic challenges with respect to the divide between the
nuclear and high-energy physics communities and funding that presents
additional hurdle in mobilizing these connections to the benefit of neutrino
programs.

While previous and existing electron scattering experiments provide important
information, dedicated measurements and theoretical developments to support
neutrino-nucleus interactions physics is needed. The NP-HEP cross community
collective efforts will play a vital role in this endevour. The goal of these
collective efforts will be to validate and solidify our understanding of the
neutrino-nucleus interactions, enabling precision physics goals of neutrino
experimental programs.

## Acknowledgements

We thank the speakers and participants of the “NF06 Electron Scattering” and
“NF06 Low Energy Neutrino and Electron Scattering” Workshops [103, 104] who
have contributed both directly and indirectly to this White Paper.

The authors acknowledge the support of the U.S. Department of Energy (DOE)
Office of Science, U.S. National Science Foundation (NSF), the Deutsche
Forschungsgemeinschaft (DFG) through the Cluster of Excellence “Precision
Physics, Fundamental Interactions, and Structure of Matter" (PRISMA+ EXC
2118/1) funded by the DFG within the German Excellence Strategy, the Research
Foundation Flanders (FWO-Flanders), and the Spanish Ministerio de Economía y
Competitividad (SEIDI-MINECO) under Grant No. PID2019-107564GB-I00, IFAE is
partially funded by the CERCA program of the Generalitat de Catalunya.

## References

  * [1] R. Acciarri et al. [DUNE], [arXiv:1512.06148 [physics.ins-det]].
  * [2] L. Alvarez-Ruso et al. [NuSTEC], Prog. Part. Nucl. Phys. 100, 1-68 (2018) doi:10.1016/j.ppnp.2018.01.006 [arXiv:1706.03621 [hep-ph]].
  * [3] B. Abi et al. [DUNE], Eur. Phys. J. C 80, no.10, 978 (2020) doi:10.1140/epjc/s10052-020-08456-z [arXiv:2006.16043 [hep-ex]].
  * [4] D. Akimov et al. [COHERENT], Science 357, no.6356, 1123-1126 (2017) doi:10.1126/science.aao0990 [arXiv:1708.01294 [nucl-ex]].
  * [5] H. De Vries, C. W. De Jager and C. De Vries, Atom. Data Nucl. Data Tabl. 36, 495-536 (1987) doi:10.1016/0092-640X(87)90013-1
  * [6] G. Fricke, C. Bernhardt, K. Heilig, L. A. Schaller, L. Schellenberg, E. B. Shera and C. W. de Jager, Atom. Data Nucl. Data Tabl. 60, 177-285 (1995) doi:10.1006/adnd.1995.1007
  * [7] O. Benhar, D. day and I. Sick, Rev. Mod. Phys. 80, 189-224 (2008) doi:10.1103/RevModPhys.80.189 [arXiv:nucl-ex/0603029 [nucl-ex]].
  * [8] O. Benhar, D. Day and I. Sick, [arXiv:nucl-ex/0603032 [nucl-ex]].
  * [9] A. Abed Abud et al. [DUNE], Instruments 5, no.4, 31 (2021) doi:10.3390/instruments5040031 [arXiv:2103.13910 [physics.ins-det]].
  * [10] M. Khachatryan et al. [CLAS and e4v], Nature 599, no.7886, 565-570 (2021) doi:10.1038/s41586-021-04046-5
  * [11] R. A. Smith and E. J. Moniz, Nucl. Phys. B 43, 605 (1972) [erratum: Nucl. Phys. B 101, 547 (1975)] doi:10.1016/0550-3213(75)90612-4
  * [12] A. M. Ankowski, A. Friedland, S. W. Li, O. Moreno, P. Schuster, N. Toro and N. Tran, Phys. Rev. D 101, no.5, 053004 (2020) doi:10.1103/PhysRevD.101.053004 [arXiv:1912.06140 [hep-ph]].
  * [13] G. A. Miller, Nucl. Phys. B Proc. Suppl. 112, 223-225 (2002) doi:10.1016/S0920-5632(02)01786-3
  * [14] H. Gallagher, AIP Conf. Proc. 698, no.1, 153-157 (2004) doi:10.1063/1.1664215
  * [15] C. Andreopoulos, A. Bell, D. Bhattacharya, F. Cavanna, J. Dobson, S. Dytman, H. Gallagher, P. Guzowski, R. Hatcher and P. Kehayias, et al. Nucl. Instrum. Meth. A 614, 87-104 (2010) doi:10.1016/j.nima.2009.12.009 [arXiv:0905.2517 [hep-ph]].
  * [16] T. Katori, AIP Conf. Proc. 1663, no.1, 030001 (2015) doi:10.1063/1.4919465 [arXiv:1304.6014 [nucl-th]].
  * [17] O. Buss, T. Leitner, U. Mosel and L. Alvarez-Ruso, Phys. Rev. C 76, 035502 (2007) doi:10.1103/PhysRevC.76.035502 [arXiv:0707.0232 [nucl-th]].
  * [18] T. Leitner, O. Buss, L. Alvarez-Ruso and U. Mosel, Phys. Rev. C 79, 034601 (2009) doi:10.1103/PhysRevC.79.034601 [arXiv:0812.0587 [nucl-th]].
  * [19] K. Gallmeister, U. Mosel and J. Weil, Phys. Rev. C 94, no.3, 035502 (2016) doi:10.1103/PhysRevC.94.035502 [arXiv:1605.09391 [nucl-th]].
  * [20] U. Mosel and K. Gallmeister, Phys. Rev. C 97, no.4, 045501 (2018) doi:10.1103/PhysRevC.97.045501 [arXiv:1712.07134 [hep-ex]].
  * [21] U. Mosel and K. Gallmeister, Phys. Rev. C 99, no.6, 064605 (2019) doi:10.1103/PhysRevC.99.064605 [arXiv:1811.10637 [nucl-ex]].
  * [22] A. M. Ankowski and A. Friedland, Phys. Rev. D 102, no.5, 053001 (2020) doi:10.1103/PhysRevD.102.053001 [arXiv:2006.11944 [hep-ph]].
  * [23] A. Papadopoulou et al. [electrons for neutrinos], Phys. Rev. D 103, 113003 (2021) doi:10.1103/PhysRevD.103.113003 [arXiv:2009.07228 [nucl-th]].
  * [24] A. M. Ankowski, A. Friedland, S. W. Li, O. Moreno, P. Schuster, N. Toro and N. Tran, Phys. Rev. D 101, no.5, 053004 (2020) doi:10.1103/PhysRevD.101.053004 [arXiv:1912.06140 [hep-ph]].
  * [25] R. R. Whitney, I. Sick, J. R. Ficenec, R. D. Kephart and W. P. Trower, Phys. Rev. C 9, 2230 (1974)doi:10.1103/PhysRevC.9.2230
  * [26] P. Barreau, M. Bernheim, J. Duclos, J. M. Finn, Z. Meziani, J. Morgenstern, J. Mougey, D. Royer, B. Saghai and D. Tarnowski, et al. Nucl. Phys. A 402, 515-540 (1983)doi:10.1016/0375-9474(83)90217-8
  * [27] J. S. O’Connell, W. R. Dodge, J. W. Lightbody, X. K. Maruyama, J. O. Adler, K. H. Hansen, B. Schroder, A. M. Bernstein, K. I. Blomqvist and B. H. Cottman, et al. Phys. Rev. C 35, 1063 (1987)doi:10.1103/PhysRevC.35.1063
  * [28] D. S. Bagdasaryan, M. K. Boyadzhian, G. B. Kazarian, K. P. A. Kechian, E. R. Markarian, G. G. Mkrtchian, O. P. Petrosian, I. A. Troshenkova and V. O. Tatevosian, YERPHI-1077-40-88.
  * [29] D. T. Baran, B. F. Filippone, D. Geesaman, M. Green, R. J. Holt, H. E. Jackson, J. Jourdan, R. D. McKeown, R. G. Milner and J. Morgenstern, et al. Phys. Rev. Lett. 61, 400-403 (1988)doi:10.1103/PhysRevLett.61.400
  * [30] R. M. Sealock, K. L. Giovanetti, S. T. Thornton, Z. E. Meziani, O. A. Rondon-Aramayo, S. Auffret, J. P. Chen, D. G. Christian, D. B. Day and J. S. Mccarthy, et al. Phys. Rev. Lett. 62, 1350-1353 (1989)doi:10.1103/PhysRevLett.62.1350
  * [31] D. B. Day, J. S. McCarthy, Z. E. Meziani, R. Minehart, R. Sealock, S. T. Thornton, J. Jourdan, I. Sick, B. W. Filippone and R. D. McKeown, et al.Phys. Rev. C 48, 1849-1863 (1993)doi:10.1103/PhysRevC.48.1849
  * [32] J. Arrington, P. Anthony, R. G. Arnold, E. J. Beise, J. E. Belz, P. E. Bosted, H. J. Bulten, M. S. Chapman, K. P. Coulter and F. Dietrich, et al.Phys. Rev. C 53, 2248-2251 (1996)doi:10.1103/PhysRevC.53.2248[arXiv:nucl-ex/9504003 [nucl-ex]].
  * [33] J. R. Arrington,[arXiv:nucl-ex/0608013 [nucl-ex]].
  * [34] J. Arrington, C. S. Armstrong, T. Averett, O. K. Baker, L. de Bever, C. W. Bochna, W. Boeglin, B. Bray, R. D. Carlini and G. Collins, et al.Phys. Rev. Lett. 82, 2056-2059 (1999)doi:10.1103/PhysRevLett.82.2056[arXiv:nucl-ex/9811008 [nucl-ex]].
  * [35] N. Fomin,doi:10.18130/V39G19[arXiv:0812.2144 [nucl-ex]].
  * [36] N. Fomin, J. Arrington, D. B. Day, D. Gaskell, A. Daniel, J. Seely, R. Asaturyan, F. Benmokhtar, W. Boeglin and B. Boillat, et al.Phys. Rev. Lett. 105, 212502 (2010)doi:10.1103/PhysRevLett.105.212502[arXiv:1008.2713 [nucl-ex]].
  * [37] H. Dai et al. [Jefferson Lab Hall A],Phys. Rev. C 98, no.1, 014617 (2018)doi:10.1103/PhysRevC.98.014617[arXiv:1803.01910 [nucl-ex]].
  * [38] B. Abi et al. [DUNE],[arXiv:2103.04797 [hep-ex]].
  * [39] H. Dai, M. Murphy, V. Pandey, D. Abrams, D. Nguyen, B. Aljawrneh, S. Alsalmi, A. M. Ankowski, J. Bane and S. Barcus, et al.Phys. Rev. C 99, no.5, 054608 (2019)doi:10.1103/PhysRevC.99.054608[arXiv:1810.10575 [nucl-ex]].
  * [40] A. Friedland and S. W. Li, Phys. Rev. D 99, no.3, 036009 (2019) doi:10.1103/PhysRevD.99.036009 [arXiv:1811.06159 [hep-ph]].
  * [41] A. Friedland and S. W. Li, Phys. Rev. D 102, no.9, 096005 (2020) doi:10.1103/PhysRevD.102.096005 [arXiv:2007.13336 [hep-ph]].
  * [42] P. Stowell et al. [MINERvA], Phys. Rev. D 100, no.7, 072005 (2019) doi:10.1103/PhysRevD.100.072005 [arXiv:1903.01558 [hep-ex]].
  * [43] D. Z. Freedman, Phys. Rev. D 9, 1389-1392 (1974) doi:10.1103/PhysRevD.9.1389
  * [44] D. Akimov et al. [COHERENT], Science 357, no.6356, 1123-1126 (2017) doi:10.1126/science.aao0990 [arXiv:1708.01294 [nucl-ex]].
  * [45] D. Akimov et al. [COHERENT],Phys. Rev. Lett. 126, no.1, 012002 (2021)doi:10.1103/PhysRevLett.126.012002[arXiv:2003.10630 [nucl-ex]].
  * [46] H. De Vries, C. W. De Jager and C. De Vries,Atom. Data Nucl. Data Tabl. 36, 495-536 (1987)doi:10.1016/0092-640X(87)90013-1
  * [47] G. Fricke, C. Bernhardt, K. Heilig, L. A. Schaller, L. Schellenberg, E. B. Shera and C. W. de Jager,Atom. Data Nucl. Data Tabl. 60, 177-285 (1995)doi:10.1006/adnd.1995.1007
  * [48] C. R. Ottermann, C. Schmitt, G. G. Simon, F. Borkowski and V. H. Walther,Nucl. Phys. A 379, 396-406 (1982)doi:10.1016/0375-9474(82)90004-5
  * [49] S. Davidson, C. Pena-Garay, N. Rius and A. Santamaria,JHEP 03, 011 (2003)doi:10.1088/1126-6708/2003/03/011[arXiv:hep-ph/0302093 [hep-ph]].
  * [50] J. Barranco, O. G. Miranda and T. I. Rashba,JHEP 12, 021 (2005)doi:10.1088/1126-6708/2005/12/021[arXiv:hep-ph/0508299 [hep-ph]].
  * [51] J. Liao and D. Marfatia,Phys. Lett. B 775, 54-57 (2017)doi:10.1016/j.physletb.2017.10.046[arXiv:1708.04255 [hep-ph]].
  * [52] C. G. Payne, S. Bacca, G. Hagen, W. Jiang and T. Papenbrock, Phys. Rev. C 100, no.6, 061304 (2019) doi:10.1103/PhysRevC.100.061304 [arXiv:1908.09739 [nucl-th]].
  * [53] J. Yang, J. A. Hernandez and J. Piekarewicz, Phys. Rev. C 100, no.5, 054301 (2019) doi:10.1103/PhysRevC.100.054301 [arXiv:1908.10939 [nucl-th]].
  * [54] M. Hoferichter, J. Menéndez and A. Schwenk, Phys. Rev. D 102, no.7, 074018 (2020) doi:10.1103/PhysRevD.102.074018 [arXiv:2007.08529 [hep-ph]].
  * [55] G. Co’, M. Anguiano and A. M. Lallena, JCAP 04, 044 (2020) doi:10.1088/1475-7516/2020/04/044 [arXiv:2001.04684 [nucl-th]].
  * [56] N. Van Dessel, V. Pandey, H. Ray and N. Jachowicz, [arXiv:2007.03658 [nucl-th]].
  * [57] Q. R. Ahmad et al. [SNO], Phys. Rev. Lett. 87, 071301 (2001) doi:10.1103/PhysRevLett.87.071301 [arXiv:nucl-ex/0106015 [nucl-ex]].
  * [58] B. T. Cleveland, T. Daily, R. Davis, Jr., J. R. Distel, K. Lande, C. K. Lee, P. S. Wildenhain and J. Ullman, Astrophys. J. 496, 505-526 (1998) doi:10.1086/305343
  * [59] M. Agostini et al. [BOREXINO], Nature 587, 577-582 (2020) doi:10.1038/s41586-020-2934-0 [arXiv:2006.15115 [hep-ex]].
  * [60] SNEWS: https://snews.bnl.gov.
  * [61] R. J. Furnstahl, D. R. Phillips and S. Wesolowski, J. Phys. G 42, no.3, 034028 (2015) doi:10.1088/0954-3899/42/3/034028 [arXiv:1407.0657 [nucl-th]].
  * [62] V. L. Martinez-Consentino, J. E. Amaro and I. Ruiz Simo,Phys. Rev. D 104, no.11, 113006 (2021)doi:10.1103/PhysRevD.104.113006[arXiv:2109.00854 [nucl-th]].
  * [63] R. Gonzaléz-Jiménez, G. D. Megias, M. B. Barbaro, J. A. Caballero and T. W. Donnelly,Phys. Rev. C 90, no.3, 035501 (2014)doi:10.1103/PhysRevC.90.035501[arXiv:1407.8346 [nucl-th]].
  * [64] P. S. Amanik and G. C. McLaughlin, J. Phys. G 36, 015105 (2009) doi:10.1088/0954-3899/36/1/015105
  * [65] K. Patton, J. Engel, G. C. McLaughlin and N. Schunck, Phys. Rev. C 86, 024612 (2012) doi:10.1103/PhysRevC.86.024612 [arXiv:1207.0693 [nucl-th]].
  * [66] M. Cadeddu, C. Giunti, Y. F. Li and Y. Y. Zhang, Phys. Rev. Lett. 120, no.7, 072501 (2018) doi:10.1103/PhysRevLett.120.072501 [arXiv:1710.02730 [hep-ph]].
  * [67] C. M. Tarbert, D. P. Watts, D. I. Glazier, P. Aguar, J. Ahrens, J. R. M. Annand, H. J. Arends, R. Beck, V. Bekrenev and B. Boillat, et al. Phys. Rev. Lett. 112, no.24, 242502 (2014) doi:10.1103/PhysRevLett.112.242502 [arXiv:1311.0168 [nucl-ex]].
  * [68] M. Thiel, C. Sfienti, J. Piekarewicz, C. J. Horowitz and M. Vanderhaeghen, J. Phys. G 46, no.9, 093003 (2019) doi:10.1088/1361-6471/ab2c6d [arXiv:1904.12269 [nucl-ex]].
  * [69] M. I. F. Bondy, PoS Bormio2015, 008 (2015) doi:10.22323/1.238.0008
  * [70] L. Alvarez-Ruso et al. [GENIE], Eur. Phys. J. ST 230, no.24, 4449-4467 (2021) doi:10.1140/epjs/s11734-021-00295-7 [arXiv:2106.09381 [hep-ph]].
  * [71] C. Quigg, “Gauge Theories of the Strong, Weak, and Electromagnetic Interactions: Second Edition”.
  * [72] J. M. Campbell, M. Diefenthaler, T. J. Hobbs, S. Höche, J. Isaacson, F. Kling, S. Mrenna, J. Reuter, S. Alioli and J. R. Andersen, et al. [arXiv:2203.11110 [hep-ph]].
  * [73] O. Benhar, P. Huber, C. Mariani and D. Meloni, Phys. Rept. 700, 1-47 (2017) doi:10.1016/j.physrep.2017.07.004 [arXiv:1501.06448 [nucl-th]].
  * [74] C. Giusti and F. D. Pacati, Nucl. Phys. A 473, 717-735 (1987) doi:10.1016/0375-9474(87)90276-4
  * [75] A. M. Ankowski and C. Mariani, J. Phys. G 44, no.5, 054001 (2017) doi:10.1088/1361-6471/aa61b2 [arXiv:1609.00258 [hep-ph]].
  * [76] M. Murphy, H. Dai, L. Gu, D. Abrams, A. M. Ankowski, B. Aljawrneh, S. Alsalmi, J. Bane, S. Barcus and O. Benhar, et al. Phys. Rev. C 100, no.5, 054606 (2019) doi:10.1103/PhysRevC.100.054606 [arXiv:1908.01802 [hep-ex]].
  * [77] K. G. Fissum, W. Bertozzi, J. P. Chen, D. Dale, H. C. Fenker, J. Gao, A. Gavalya, S. Gilad, C. R. Leathers and N. Liyanage, et al. Nucl. Instrum. Meth. A 474, 108-131 (2001) doi:10.1016/S0168-9002(01)00875-0
  * [78] J. Alcorn, B. D. Anderson, K. A. Aniol, J. R. M. Annand, L. Auerbach, J. Arrington, T. Averett, F. T. Baker, M. Baylac and E. J. Beise, et al. Nucl. Instrum. Meth. A 522, 294-346 (2004) doi:10.1016/j.nima.2003.11.415
  * [79] L. Gu et al. [Jefferson Lab Hall A], Phys. Rev. C 103, no.3, 034604 (2021) doi:10.1103/PhysRevC.103.034604 [arXiv:2012.11466 [nucl-ex]].
  * [80] C. Barbieri, N. Rocco and V. Somà, Phys. Rev. C 100, no.6, 062501 (2019) doi:10.1103/PhysRevC.100.062501 [arXiv:1907.01122 [nucl-th]].
  * [81] P. Abratenko et al. [MicroBooNE], Phys. Rev. Lett. 125, no.20, 201803 (2020) doi:10.1103/PhysRevLett.125.201803 [arXiv:2006.00108 [hep-ex]].
  * [82] B. A. Mecking et al. [CLAS], Nucl. Instrum. Meth. A 503, 513-553 (2003) doi:10.1016/S0168-9002(03)01001-5
  * [83] A. Papadopoulou et al. [electrons for neutrinos], Phys. Rev. D 103, 113003 (2021) doi:10.1103/PhysRevD.103.113003 [arXiv:2009.07228 [nucl-th]].
  * [84] K. Abe et al. [T2K], Nature 580, no.7803, 339-344 (2020) [erratum: Nature 583, no.7814, E16 (2020)] doi:10.1038/s41586-020-2177-0 [arXiv:1910.03887 [hep-ex]].
  * [85] A. Ashkenaz et al. “Electrons For Neutrinos. Addressing Critical Neutrino-Nucleus Issues, A Run Group Proposal Resubmission to Jefferson Lab PAC 46”.
  * [86] V. D. Burkert, L. Elouadrhiri, K. P. Adhikari, S. Adhikari, M. J. Amaryan, D. Anderson, G. Angelini, M. Antonioli, H. Atac and S. Aune, et al. Nucl. Instrum. Meth. A 959, 163419 (2020) doi:10.1016/j.nima.2020.163419
  * [87] T. Åkesson et al. [LDMX], [arXiv:1808.05219 [hep-ex]].
  * [88] B. G. Tice et al. [MINERvA], Phys. Rev. Lett. 112, no.23, 231801 (2014) doi:10.1103/PhysRevLett.112.231801 [arXiv:1403.2103 [hep-ex]].
  * [89] M. V. Ascencio et al. [MINERvA], [arXiv:2110.13372 [hep-ex]].
  * [90] H. Dai et al. [Jefferson Lab Hall A], Phys. Rev. C 98, no.1, 014617 (2018) doi:10.1103/PhysRevC.98.014617 [arXiv:1803.01910 [nucl-ex]].
  * [91] D. Adhikari et al. [PREX], Phys. Rev. Lett. 126, no.17, 172502 (2021) doi:10.1103/PhysRevLett.126.172502 [arXiv:2102.10767 [nucl-ex]].
  * [92] C. J. Horowitz, S. J. Pollock, P. A. Souder and R. Michaels, Phys. Rev. C 63, 025501 (2001) doi:10.1103/PhysRevC.63.025501 [arXiv:nucl-th/9912038 [nucl-th]].
  * [93] C. J. Horowitz, Phys. Rev. C 57, 3430-3436 (1998) doi:10.1103/PhysRevC.57.3430 [arXiv:nucl-th/9801011 [nucl-th]].
  * [94] M. Thiel, C. Sfienti, J. Piekarewicz, C. J. Horowitz and M. Vanderhaeghen, J. Phys. G 46, no.9, 093003 (2019) doi:10.1088/1361-6471/ab2c6d [arXiv:1904.12269 [nucl-ex]].
  * [95] F. J. Fattoyev, J. Piekarewicz and C. J. Horowitz, Phys. Rev. Lett. 120, no.17, 172702 (2018) doi:10.1103/PhysRevLett.120.172702 [arXiv:1711.06615 [nucl-th]].
  * [96] B. T. Reed, F. J. Fattoyev, C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 126, no.17, 172503 (2021) doi:10.1103/PhysRevLett.126.172503 [arXiv:2101.03193 [nucl-th]].
  * [97] D. Androic et al. [QWeak], [arXiv:2112.15412 [nucl-ex]].
  * [98] Nucl. Phys. News 31, no.3, 5-10 (2021) doi:10.1080/10619127.2021.1954434
  * [99] D. Becker, R. Bucoveanu, C. Grzesik, K. Imai, R. Kempf, K. Imai, M. Molitor, A. Tyukin, M. Zimmermann and D. Armstrong, et al. Eur. Phys. J. A 54, no.11, 208 (2018) doi:10.1140/epja/i2018-12611-6 [arXiv:1802.04759 [nucl-ex]].
  * [100] B. Reed, Z. Jaffe, C. J. Horowitz and C. Sfienti, Phys. Rev. C 102, 064308 (2020) doi:10.1103/PhysRevC.102.064308 [arXiv:2009.06664 [nucl-th]].
  * [101] A. Esser, M. Thiel, P. Achenbach, K. Aulenbacher, S. Baunack, J. Beričič, D. Bosnar, L. Correa, M. Dehn and M. O. Distler, et al. Phys. Rev. Lett. 121, no.2, 022503 (2018) doi:10.1103/PhysRevLett.121.022503
  * [102] M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, E. Picciau, C. A. Ternes and Y. Y. Zhang, Phys. Rev. C 104, no.6, 065502 (2021) doi:10.1103/PhysRevC.104.065502 [arXiv:2102.06153 [hep-ph]].
  * [103] “NF06 Electron Scattering Workshop”:https://indico.fnal.gov/event/46620/.
  * [104] “NF06 Low Energy Neutrino and Electron Scattering Workshop”:https://indico.fnal.gov/event/51519/.