###### Abstract

Our knowledge of neutrino cross sections at the GeV scale, instrumental to
test CP symmetry violation in the leptonic sector, has grown substantially in
the last two decades. Still, their precision and understanding are far from
the standard needed in contemporary neutrino physics. Nowadays, the knowledge
of the neutrino cross-section at $O(10\%)$ causes the main systematic
uncertainty in oscillation experiments and jeopardizes their physics reach. In
this paper, we envision the opportunities for a new generation of cross
section experiments to be run in parallel with DUNE and HyperKamiokande. We
identify the most prominent physics goals by looking at the theory and
experimental limitations of the previous generation of experiments. We
highlight the priorities in the theoretical understanding of GeV cross-
sections and the experimental challenges of this new generation of facilities.

###### keywords:

neutrino; neutrino cross sections; neutrino interactions, accelerator neutrino
experiments

1 1 0 31 July 2021 17 August 2021 https://doi.org/ A New Generation of
Neutrino Cross Section Experiments: Challenges and Opportunities A. Branca1,
G. Brunetti1, A. Longhin2,3, M. Martini4, F. Pupilli3, F. Terranova 1 Antonio
Branca, Giulia Brunetti, Andrea Longhin, Marco Martini, Fabio Pupilli,
Francesco Terranova Branca, A.; Brunetti, G.; Longhin, A., Martini, M.,
Pupilli, F., Terranova, F. Correspondence<EMAIL_ADDRESS>
##  1 Introduction

The physics of neutrino oscillations at accelerators Mezzetto and Terranova
(2020) has entered a new phase after the approval of DUNE Abi et al. (2020)
and HyperKamiokande (HK) Abe et al. (2018). These facilities have an
unprecedented level of complexity, cost, and timescale, which is comparable to
the most ambitious collider experiments performed in the last decades as the
B-factories or the LHC experiments. Similar to collider experiments, the
progress in the experimental sensitivity must be matched by the progress in
theory predictions to fully reap the physics potential of the new facilities.
This ideal scenario is yet to come in neutrino physics Katori and Martini
(2018).

The measurement of the size of the $\theta_{13}\,$ mixing angle An et al.
(2012); Ahn et al. (2012); Abe et al. (2012, 2014) not only opened up this era
but also pinned down the weakest point of the strategy to measure neutrino
mixing and test the minimally extended Standard Model ($\nu$SM), i.e. the SM
minimally modified to accommodate three generations of massive neutrinos. The
optimal measurement strategy is based on high-intensity neutrino beams, whose
energy is between a few hundreds of MeV up to some GeV. This is one of the
most critical regions for SM calculations of neutrino interactions with matter
and, in particular, neutrino cross sections on moderate-$Z$ nuclei. Even more,
the targets of DUNE and HK are, respectively, Argon ($Z=18$) and water
($Z=1,8$) and lay in the critical region because the mean energy of the DUNE
and HK beam is 3 and 0.8 GeV, respectively. As a consequence, the standard
interactions of neutrinos with matter in the most ambitious oscillation
experiments under construction nowadays are known with a precision (10-30%)
that is generally inappropriate for the aim of these facilities: a 1-4%
systematic uncertainty on the measurement of the
$\nu_{\mu}\rightarrow\nu_{e}\,$and
$\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e}\,$oscillation probability
Abi et al. (2020).

DUNE and HK mitigate this risk with a very careful design of the near Abud et
al. (2021); Abe et al. (2019) and far Abi et al. (2020); Abe et al. (2018)
detectors, comparing the observed neutrino interactions rates in the two
locations. Still, the time-honored near-far cancellation technique is reaching
its intrinsic limitations and the systematic uncertainty currently dominates
the physics reach.

This paper addresses this issue in the most direct manner and is mirrored by
the outcome of the European Strategy for Particle Physics Ellis et al. (2019),
which has pointed out such weakness in the global effort to measure the
neutrino properties. Instead of refining the near-far cancellation technique,
we discuss the opportunities and challenges of a new generation of short-
baseline cross section experiments, optimized to reach percent level precision
in the inclusive, differential, and doubly-differential cross-sections. These
experiments must address the region of interest for DUNE and HK but should not
be considered only ancillary facilities of long-baseline experiments. Neutrino
interactions at the GeV scale Atar and Singh (2020) provide a wealth of
information on the nuclear structure at moderate $Z$ and can be extended to
cover the simplest nuclear system (e.g. hydrogen or deuteron) and explore the
high-Z region of relevance for high-density detectors. They are the workhorse
to gain a deep understanding of weak interactions in nuclear physics and
provide important data for the study of nuclear structure, which can be tested
against nuclear effective theories and, in the long term, lattice QCD. We call
this type of facilities a new generation of cross section experiments because
major advances in this field has been achieved by the previous generation
based on dedicated experiments (SciBooNE Alcaraz-Aunion and Walding (2009),
MINER$\nu$A Rodrigues et al. (2016); Aliaga et al. (2015), WAGASCI Pintaudi
(2019), ArgoNeuT Acciarri et al. (2020), etc.), the near detectors of long-
baseline (K2K Gran et al. (2006), MINOS Evans (2013), T2K-ND280 Abe et al.
(2015), NO$\nu$A Acero et al. (2020)) and short-baseline experiments
(MiniBooNE Aguilar-Arevalo et al. (2013), MicroBooNE Abratenko et al. (2020)).
We expect the next generation of dedicated experiments – running on a
timescale comparable with the data-taking of DUNE and HK – to achieve major
improvements on $\nu_{e}\ $, $\nu_{\mu}\ $ and $\overline{\nu}_{\mu}$ cross-
sections but also contribute to the short-baseline program for the search of
physics beyond the $\nu$SM.

Section 2 reviews the state-of-the-art: the main achievements on cross-
sections in the critical few-GeV region, the precision reached so far and
prominent open issues. In Sec. 3, we anticipate the improvements that can be
gained by the Near Detectors of DUNE, HK and, possibly, ESS$\nu$SB Dracos et
al. (2020); Alekou et al. (2021) exploiting the high-intensity, moderate
precision, beams used in future long-baseline experiments: LBNF for DUNE,
JPARC for HK, and the European Spallation Source for ESS$\nu$SB. High-
precision beams of moderate-intensity are discussed in Sec. 4 with special
emphasis on monitored neutrino beams (NP06/ENUBET Acerbi et al. (2018)) and
muon beams (nuSTORM Adey et al. (2013)). A review of optimized detectors for
high-precision cross section measurements and their physics reach is presented
in Sec. 5. Finally, we summarize the perspectives for this new generation of
experiments to be run in the DUNE/HK era in Sec. 6.

##  2 The current generation of cross section experiments and their impact on
neutrino oscillations

In contemporary particle physics, neutrino cross sections play two major
roles. They provide ancillary information to extract the oscillation
probabilities and, at the same time, probe the nuclear media employing pure
weak processes: $\nu_{eL}$ , $\nu_{\mu L}$ , $\nu_{\tau L}$ and – if neutrinos
are Dirac particles – their right-handed (RH) antiparticles. Here, $L$
indicates the left-handed (LH) chirality component of the neutrino fields in
the $\nu$SM. For the class of experiments considered here, the cross sections
are the same even if neutrinos are Majorana particles, where the
$\bar{\nu}_{R}$ states can be replaced by the RH Majorana state $\nu_{R}$ for
each flavor ($\nu_{eR}$, $\nu_{\mu R}$, $\nu_{\tau R}$). For the sake of
definiteness, we will assume Dirac neutrinos in the rest of the paper.

The ancillary role of neutrino cross section is particularly evident in
oscillation experiments aimed at measuring the oscillation probability of
$P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(E,L)$ as a function of the neutrino
energy $E$ and source-to-detector distance, the baseline $L$. For the case of
DUNE ($L\simeq 1300$ km) and HK ($L\simeq 295$ km),
$\nu_{\mu}\rightarrow\nu_{e}\,$and
$\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e}\,$are of paramount
importance to establish CP violation in the neutrino sector and the number of
$\nu_{e}\ $charged-current (CC) interactions observed far from the source is

$N^{e}_{FD}=\tilde{M}^{\prime}\int dE\left[\phi_{\nu_{\mu}}^{FD}(E)\
\epsilon_{e}(E)\ \sigma_{\nu_{e}}(E)P(\nu_{\mu}\rightarrow\nu_{e})\
+\phi_{\nu_{\mu}}^{FD}(E)\epsilon_{\mu\rightarrow
e}\sigma_{\nu_{\mu}}(E)\right]$ (1)

where $\tilde{M}^{\prime}$ is the number of scattering centers (proportional
to the mass and material of the detector), $\phi_{\nu_{\mu}}^{FD}(E)$ is the
total flux integrated during the data taking at $L$ by the Far Detector (FD),
and $\epsilon_{e}(E)\sigma_{\nu_{e}}(E)$ is the visible cross section, i.e.,
the cross-section corrected by the efficiency of the detector.
$\epsilon_{\mu\rightarrow e}$ is the misidentification probability of tagging
a $\nu_{\mu}\ $as a $\nu_{e}$, e.g., due to neutral-currents (NC) events or
non-reconstructed muons. For low precision experiments ($>5$% systematic
uncertainty), the uncertainties on the flux and the cross-section times
detector efficiency can be removed by an identical detector located at short
$L$, where oscillation effects are negligible, the Near Detector (ND). For the
ND, the $\nu_{\mu}\ $CC events provide an estimate of the initial flux

$N_{ND}=\tilde{M}_{N}\int dE\
\phi^{ND}_{\nu_{\mu}}(E)\epsilon_{\mu}(E)\sigma_{\nu_{\mu}}(E)$ (2)

and the $\nu_{e}\ $CC events measure the $\nu_{e}\ $ contamination at source:

$N^{e}_{ND}=\tilde{M}_{N}\int dE\left[\phi_{\nu_{e}}^{ND}(E)\ \epsilon_{e}(E)\
\sigma_{\nu_{e}}(E)\ +\phi_{\nu_{\mu}}^{ND}(E)\epsilon_{\mu\rightarrow
e}(E)\sigma_{\nu_{\mu}}(E)\right].$ (3)

In this way, the main contribution to the systematic uncertainty should come
only from the

$\frac{\epsilon_{e}(E)\sigma_{e}(E)}{\epsilon_{\mu}(E)\sigma_{\mu}(E)}$ (4)

ratio Huber et al. (2008) and is strongly mitigated by the lepton
universality, i.e. the statement that $\sigma_{e}(E)=\sigma_{\mu}(E)$ except
for calculable kinematic corrections. Unfortunately, such an elegant solution
does not work at $<5$% level. The ND and FD are not perfectly identical. Their
efficiencies are different ($\epsilon^{ND}\neq\epsilon^{FD}$) both for
$\nu_{e}\ $CC and $\nu_{\mu}\ $CC. Even more, $\phi_{\nu_{\mu}}(E)$ and
$\phi_{\nu_{e}}(E)$ are different at the near and far location beyond the
expected geometrical reduction ($\phi\sim L^{-2}$) because the ND integrates
the flux in a much larger neutrino phase-space. The solid angle seen by the ND
is much bigger than the FD and this phase mismatch requires corrections both
on flux normalization and energy spectrum, which, in turn, increase the
systematic uncertainty.

It is worth noting that flux, efficiencies, oscillation probabilities, and
cross sections are all functions of the neutrino energy $E$. The neutrino
energy is not known a priori and must be reconstructed event-by-event by the
final-state particles of the neutrino interaction in the detector. This
reconstruction is not possible in NC events because of the outgoing neutrino
and troublesome in CC events every time a final-state particle is missed or
misreconstructed. Again, this procedure generates a bias affecting the
measured oscillation probability. As a consequence, assuming priors on the
neutrino beam (Monte Carlo simulation of the beamline), cross-sections
(neutrino-nucleus models), and detector response are mandatory even after the
near-far comparison. These priors, clearly, are not free of systematic
uncertainties.

Along the same line, there is little hope to predict the $\nu$SM interactions
with matter starting from theory, and all advances in this field require a
sophisticated interplay between model-building and high-precision cross
section measurements.

###  2.1 Standard candles

$\nu$SM provides firm predictions only for elementary fermions. For any
realistic experiment at accelerators, this limitation impedes any ab-initio
calculation of scatterings except for purely leptonic processes.

The $\nu_{\mu}\ e^{-}\rightarrow\mu^{-}\nu_{e}\ $ scattering is a purely
leptonic $\nu_{\mu}\ $CC process with an outgoing $\nu_{e}$. The process is
called inverse muon decay. The differential cross section in $\nu$SM is:

$\frac{d\sigma}{d\Omega}=\frac{G_{F}^{2}}{\pi^{2}m_{e}}\frac{E_{\mu}^{2}}{E_{\nu_{\mu}}}\left[E_{\nu_{\mu}}E_{\mu}-E_{\nu_{\mu}}|\mathbf{k}^{\prime}|\cos\theta+m_{e}E_{\mu}-m_{\mu}^{2}\right].$
(5)

In eq. 5, $G_{F}$ is the Fermi constant, $\mathbf{k}^{\prime}$ is the three-
momentum of the final-state muon measured in the laboratory frame (LAB) and
$\theta$ is the corresponding scattering angle with respect to the direction
of the incoming neutrino. Unfortunately, the kinematic threshold for the
inverse muon decay is very large:

$E^{thr}_{\nu_{\mu}}=\frac{m_{\mu}^{2}-m_{e}^{2}}{2m_{e}}\simeq 11\
\mathrm{GeV}$ (6)

and this process is not particularly interesting in the DUNE/HK region.
$\sigma_{\nu_{\mu}\ e^{-}\rightarrow\mu^{-}\nu_{e}\ }/E$ was measured at CERN
and Fermilab with a precision of about 6% up to 600 GeV Mishra et al. (1990).

$\nu_{\mu}\ e^{-}\rightarrow\nu_{\mu}\ e^{-}$ is the threshold-less neutrino-
electron scattering corresponding to an elastic scattering of neutrinos
mediated by a $Z^{0}$, i.e. a neutral current. The tree-level cross section
is:

$\displaystyle\frac{d\sigma}{d\Omega}=\frac{G_{F}^{2}}{4\pi^{2}m_{e}}\frac{(E^{\prime}_{e})^{2}}{E_{\nu}}\left[(g_{V}^{e}+g_{A}^{e})^{2}(E_{\nu}E^{\prime}_{e}-E_{\nu}|\mathbf{p}^{\prime}|\cos\theta+m_{e}E^{\prime}_{e}-m_{e}^{2})\right.$
$\displaystyle+(g_{V}^{e}-g_{A}^{e})^{2}(E_{\nu}+m_{e}-E^{\prime}_{e})(E^{\prime}_{e}-|\mathbf{p}^{\prime}|\cos\theta)$
$\displaystyle\left.-m_{e}\left[(g^{e}_{V})^{2}+(g_{A}^{e})^{2}\right](m_{e}-E^{\prime}_{e}+|\mathbf{p}^{\prime}|\cos\theta)\right].$
(7)

Again, $G_{F}$ is the Fermi constant, $\mathbf{p}^{\prime}$ is the three-
momentum of the final-state electron in LAB and $\theta$ is the corresponding
scattering angle with respect to the direction of the incoming neutrino.
$g_{V}^{e}=-\frac{1}{2}+2\cdot sin^{2}\theta_{w}$ and $g_{A}^{e}=-\frac{1}{2}$
are the $Z^{0}$ couplings to the electron, where $\theta_{w}$ is the Weinberg
angle. This process was tested by several experiments at CERN, BNL, and
Fermilab and the cross section turned out to be in good agreement with the
$\nu$SM expectations. In particular, the CHARM experiment provides a total
cross section of Dorenbosch et al. (1989):

$\displaystyle\sigma(\nu_{\mu}\ e^{-})=2.2\pm 0.4\pm 0.4\times 10^{-42}\
E_{\nu}\ \frac{\mathrm{cm}^{2}}{\mathrm{GeV}}$
$\displaystyle\sigma(\overline{\nu}_{\mu}e^{-})=1.6\pm 0.3\pm 0.3\times
10^{-42}\ E_{\nu}\ \frac{\mathrm{cm}^{2}}{\mathrm{GeV}}$

where $E_{\nu}$ is the neutrino energy in GeV. Note that, in the DUNE/HK
region, this cross-section is $\sim$1000 times smaller than the
$\nu_{\mu}$-nucleon cross section. Since this process is elastic, all
kinematic variables are accessible to the experimentalist if the detector can
identify electrons, measure their energy
$E^{\prime}_{e}=\sqrt{|\mathbf{p}^{\prime}|^{2}+m_{e}^{2}}$, and the $\theta$
angle. The incoming neutrino direction can be considered constant
($\theta\simeq 0$) if $L$ is sufficiently large. Applying four-momentum
conservation for a neutrino with four-momentum
$p_{\nu}=(E_{\nu},0,0,p)\simeq(p,0,0,p)$, an electron at rest in LAB
($p_{e}=(m_{e},0,0,0)$), and neglecting the electron mass, we get:

$1-\cos\theta=\frac{m_{e}(1-T_{e}/E_{\nu})}{E_{e}}$ (8)

where $T_{e}$ is the electron kinetic energy. As a consequence, the energy and
direction of the electron provide the value of $E_{\nu}$. The $\nu_{e}\ $may
contribute with an elastic $\nu_{e}\ e^{-}\rightarrow\nu_{e}\ e^{-}$
scattering that includes both the exchange of a $W^{+}$ and a $Z^{0}$, that is
both the CC and NC contributions mentioned above. However, $\nu_{e}\ $are
always much less than $\nu_{\mu}\ $(<5%) in conventional accelerator neutrino
beams and the CC contribution is negligible.

The $\nu_{\mu}\ e^{-}$ elastic scattering is the most prominent example of
standard candle, a process that is predicted with outstanding precision by the
$\nu SM$ and can be used to constrain $\phi_{\nu_{\mu}}(E)$. A precision test
of this channel to a level comparable with the theoretical predictions is out
of reach at the GeV scale and is generally used at other energies to test the
$\nu SM$ and perform independent measurements of the Weinberg angle. On the
other hand, it is an important tool to measure the flux in very high-intensity
beams Marshall et al. (2020). In the current generation of experiments, this
technique has been employed by MINER$\nu$A to constrain the neutrino flux from
2 to 20 GeV accumulating 810 events. The theory uncertainty on the expected
$\nu_{\mu}\ $flux normalization was reduced from 7.6 to 3.9% Valencia et al.
(2019).

###  2.2 Neutrino-nucleus scattering

Going from elementary particle scattering to neutrino-nucleon or neutrino-
nucleus scattering represents a major theory challenge. The scattering of the
elementary fermions (neutrino-quark) described by the $\nu$SM must be
corrected by the presence of the spectator quarks, sea quarks and gluons,
which impact both the formation of the final state hadrons and their
reinteraction in the nuclear medium. In this section, we summarize the main
achievements reached by current experiments: they allowed us to correct for
the sizable tensions between model predictions and experimental data and shed
new light on the complexity of the process. A complete ab-initio calculation
would require a lattice QCD treatment that is still far-fetched, even if
lattice QCD already provides interesting information on specific topics
Kronfeld et al. (2019). On the other hand, effective models have reached a
high level of sophistication boosted by a large amount of data currently
available, cross-fertilization between the experimental program and model
refinement, and the pressing needs of oscillation experiments at accelerators.

The double-differential cross section describing a neutrino-nucleus scattering
is Katori and Martini (2018):

$\frac{d^{2}\sigma}{d\Omega_{k^{\prime}}d\omega}=\frac{G_{F}^{2}\cos^{2}\theta_{C}}{32\pi^{2}}\frac{|\mathbold{k}^{\prime}|}{|\mathbold{k}|}L_{\mu\nu}W^{\mu\nu}(\mathbf{q},\omega)$
(9)

where $G_{F}$ is the Fermi constant. The variables describing the

$\nu_{l}+A\rightarrow l^{-}+X$ (10)

process are the solid angle of the final state lepton $\Omega_{k^{\prime}}$,
its three-momentum $\mathbf{k}^{\prime}$ and the energy $\omega$ transferred
by the neutrino to the nucleus. In Eq. 10, $l$ is a charged lepton
($e,\mu,\tau$) and $\nu_{l}$ the corresponding neutrino, $A$ is the target
nucleus and $X$ is a generic set of outgoing particles.
$\omega=E_{\nu}-E^{\prime}_{l}$ is the difference between the initial-state
neutrino and final-state lepton energies. We can then write the initial
neutrino four-momentum as $k=(E_{\nu},\mathbf{k})$, the final-state lepton
four-momentum as $k^{\prime}=(E^{\prime}_{l},\mathbf{k^{\prime}})$, and define
the four-momentum transfer as $q=k-k^{\prime}\equiv(\omega,\mathbf{q}$).

The $\nu$SM provides an exact expression for the leptonic tensor $L_{\mu\nu}$
because leptons are elementary fermions. For an incoming neutrino:

$L_{\mu\nu}=8(k_{\mu}k^{\prime}_{\nu}+k_{\nu}k^{\prime}_{\mu}-k_{\mu}k^{\mu}-i\epsilon_{\mu\nu\alpha\beta}k^{\alpha}k^{\prime\beta})$
(11)

while for an incoming antineutrino is

$L_{\mu\nu}=8(k_{\mu}k^{\prime}_{\nu}+k_{\nu}k^{\prime}_{\mu}-k_{\mu}k^{\mu}+i\epsilon_{\mu\nu\alpha\beta}k^{\alpha}k^{\prime\beta})$
(12)

i.e. the antineutrino leptonic tensor has a minus sign in front of the Levi-
Civita tensor. On the one hand that is great news because the main cause of
the neutrino-antineutrino asymmetries is located in the analytical part
$L_{\mu\nu}$ of the cross section. This sign difference implies that the
various components of the hadronic tensor weigh differently in the neutrino
and antineutrino cross sections. The name of the game is the evaluation of the
hadronic tensor, which mixes up the hard weak-scattering with non-perturbative
QCD and nuclear dynamics effects. The simplest process (one particle-one hole,
“1p-1h”) is the excitation that brings one nucleon of the nucleus to a higher
nuclear level leaving a hole in the nucleus ground state configuration. If the
energy and momentum transfer are smaller than the production threshold of the
lightest hadron (the pion), this process corresponds to a quasi-elastic (QE)
scattering where $X$ is the final state nucleus. Otherwise, $X$ is a generic
set of hadrons, which generally include the final state nucleus and a few
light hadrons or a real hadron cascade, the deep-inelastic scattering (DIS).
The hadronic tensor can hence be expressed as:

$\mathbf{W}^{\mu\nu}(\mathbf{q},\omega)=\mathbf{W}^{\mu\nu}_{1p1h}(\mathbf{q},\omega)+\mathbf{W}^{\mu\nu}(\mathbf{q},\omega)_{2p2h}+\mathbf{W}^{\mu\nu}_{1p1h1\pi}(\mathbf{q},\omega)+\ldots$
(13)

accounting for quasi-elastic (“1p-1h”) multiple nucleon excitations (two-
particles two-holes, “2p-2h”), pion production and higher inelastic channels.
According to Eq. 9 the above decomposition holds for the cross section.

The current generation of experiments has shown that the challenge of
describing $\mathbf{W}^{\mu\nu}$ and the cross sections has been overly
underestimated. A large part of the difficulties is related to the fact that
the neutrino beams are not monochromatic but wide-band. Hence, the full
reconstruction of kinematics of the neutrino-nucleus reaction is impossible
without assumptions. The cross sections in terms of traditional kinematic
variables, like the transferred energy $\omega$ or the square of the
4-momentum transfer $Q^{2}=-q^{2}$ in the case of electron scattering, are
replaced for neutrinos by the neutrino flux-integrated differential cross
sections on direct observables, such as:

$\frac{d^{2}\sigma}{d\Omega_{k^{\prime}}~{}dE^{\prime}_{l}}=\frac{1}{\int\Phi(E_{\nu})~{}dE_{\nu}}\int~{}dE_{\nu}\left[\frac{d^{2}\sigma}{d\Omega_{k^{\prime}}~{}d\omega}\right]_{\omega=E_{\nu}-E^{\prime}_{l}}\Phi(E_{\nu}).$
(14)

In this type of cross sections, for fixed values of the measured variables
$E^{\prime}_{l}$ and $\Omega_{k^{\prime}}$, one explores the whole
($\omega$,$|\mathbf{q}|$) plane where the different terms of the hadronic
tensor live, hence all values of the energy transfer $\omega$ contribute to
the cross sections. In other words, since $E_{\nu}=E^{\prime}_{l}+\omega$, for
a given set of values of $E^{\prime}_{l}$ and $\Omega_{k^{\prime}}$ one
explores the full energy spectrum of neutrinos above the charged lepton
energy. A crucial aspect related to this point is that all the reaction
channels (quasi-elastic, 2p-2h, pion production,…) are entangled and isolating
a primary vertex process from the measurement of neutrino flux-integrated
differential cross section is much more difficult than in the cases of
monochromatic (such as electron) beams, where the quasielastic and the
$\Delta$-resonance bumps as well as the so-called “dip”-region, populated by
2p-2h contributions, can be easily distinguished.

In the following, we describe three prominent cases, where the amount and
quality of new cross section data allowed us to grasp the complexity of
nuclear media and inspired new interpretations, although a complete
understanding is yet left to the next generation of experiments.

###  2.3 Quasi-elastic region and the axial mass

Elastic neutral currents ($\nu_{l}+A\rightarrow\nu_{l}+A$) and quasi-elastic
charged currents ($\nu_{l}+A\rightarrow l^{-}+A^{\prime}$), where the final
state is a lepton of flavor $l=e,\mu,\tau$, are virtually the simplest
neutrino-nucleus scattering. An even simpler interaction mode would be the
elastic and quasi-elastic neutrino-nucleon scattering. Even today, however,
our experimental knowledge of these basic nucleon processes is limited to a
set of experiments carried out with deuterium from 1981 to 1983 Miller et al.
(1982); Baker et al. (1981); Kitagaki et al. (1983) and early results on
liquid hydrogen, plus the ancillary data from electron-nucleon scattering. The
quasi-elastic cross section on nuclei plays a leading role in many neutrino
oscillation experiments, especially in T2K/HK that employ water both as a
target and a Cherenkov radiator and use quasi-elastic kinematics-based method
to reconstruct the neutrino energy. QE are the main process of interest in
ESS$\nu$SB Baussan et al. (2014); Dracos et al. (2020); Alekou et al. (2021)
where the mean energy is just 0.2 GeV.

The different components of the hadronic tensor
$\mathbf{W}^{\mu\nu}(\mathbf{q},\omega)$ contain the vector and axial form
factors (related to the nucleon properties) and the response functions, or
structure functions, (related to the nuclear dynamics). Concerning the form
factors, the conserved vector current hypothesis allows us to apply the vector
(electric and magnetic) form factors measured in electron scattering to
neutrino scattering. The axial form factor instead is usually described by a
dipole parameterization

$G_{A}(Q^{2})=G_{A}(0)\left(1-\frac{Q^{2}}{M_{A}^{2}}\right)^{-2}$ (15)

where
$G_{A}(0)\equiv G_{A}(Q^{2}\rightarrow 0)=g_{A}=1.2756\pm 0.0013$ is extracted
from neutron $\beta$-decay Zyla et al. (2020). The form factor thus runs
toward the energies of interest for neutrino oscillations at accelerators
(0.1-10 GeV) through an empirical constant called the axial mass $M_{A}$.
Before precision neutrino data on nuclei were available, this value was
considered quite stable around Bernard et al. (2002)

$M_{A}=1.026\pm 0.021\ \mathrm{GeV},$ (16)

a value extracted from charged current quasielastic experiments on deuterium
bubble chambers Mann et al. (1973); Barish et al. (1977); Baker et al. (1981);
Kitagaki et al. (1983) and confirmed by the few-GeV data from NOMAD Lyubushkin
et al. (2009) on carbon. However, modern neutrino scattering data on carbon at
lower energies coming from K2K Gran et al. (2006), SciBooNE Nakajima et al.
(2011), and MiniBooNE Aguilar-Arevalo et al. (2010) seemed to suggest larger
values of the axial mass in contradiction with the previous one. In
particular, the MiniBooNE data, the first-ever neutrino flux-integrated double
differential cross sections in terms of the measured muon variables, could be
reproduced by calculating the nuclear response functions using the
relativistic Fermi Gas model and increasing the axial mass to the value
$M_{A}=1.35(\pm 0.17)$ GeV, revealing a substantial discrepancy. The
introduction of more realistic theoretical models for the nuclear response
functions assuming the validity of the hypothesis that the neutrino interacts
with a single nucleon in the nucleus did not change this conclusion Alvarez-
Ruso (2012). This contradiction brought to a reconsideration of the physical
meaning of $M_{A}$, its interplay with the nuclear models employed to describe
the nucleus, and, even more, the experimental biases. In particular, it was
suggested Martini et al. (2009) that the glaring inconsistencies between
models and data recorded by MiniBooNE were due to a missing component in the
modeling of the cross sections: the reaction mechanism of multinucleon (2p-2h)
excitations due to short-range nuclear correlations, meson exchange currents
and their interference (also called one nucleon-two nucleon interference).
Thus, what MiniBooNE published was not genuine quasi-elastic data. To avoid
further ambiguities on signal definition, starting from the T2K measurement of
Ref. Abe et al. (2016) (see Fig. 1) data are no more classified in terms of
the initial vertex of the reaction, which would require largely model-
dependent background subtraction in the data analysis since the various
reaction mechanisms are entangled in the neutrino cross sections. Data are now
classified in terms of final state particles, such as “1 muon 0 pion and any
number of protons”, the so-called zero-pion CC ($0\pi$-CC) events which
include also the $1\pi$-CC vertex contribution if the final pion is not
detected. Examples of cross sections measurements with the same signal
definition are the ones of Ref. Ruterbories et al. (2019) for MINERvA and of
Ref. Abratenko et al. (2020) for MicroBooNE.

Figure 1: The T2K $0\pi$-CC $\nu_{\mu}\ $double-differential cross sections
measured in 2016. The flux uncertainty is shown in the gray area and the
distributions are compared with the models of Martini et al. Martini et al.
(2009, 2010) (red solid line) and Nieves et al. Nieves et al. (2012a, b) (red
dotted line). Reproduced under CC-BY-3.0 licence from Abe et al. (2016).
Copyright 2016 American Physical Society.

The MiniBooNE experience also boosted the development of neutrino scattering
generators that embed several (switchable) models plus the features needed by
the experiments to provide mock data samples. These generators and the
associated community effort Alvarez-Ruso et al. (2018) are one of the most
valuable outcomes of dedicated cross section experiments.

###  2.4 Pion production and the resonance region

The region of interest for long-baseline neutrino oscillation experiments is
one of the toughest from the point of view of cross section calculations. A
4-momentum transfer greater than the pion production threshold creates light
mesons that may re-interact in the nuclear medium being re-absorbed or
producing new particles. This region is called the resonance (RES) region
because pions are generally produced by the strong decay of nuclear resonances
toward the nucleus ground state Atar and Singh (2020). The same experimental
data that became available in the last two decades and challenged the
evaluation of $M_{A}$ have unveiled the obsolescence of the semi-empirical
models used by early generators in the RES region. This is particularly true
for the time-honored Rein-Seghal model used as the standard reference up to
$\sim$2010 Rein and Sehgal (1983). The new approaches and models that address
these items cannot be properly disentangled without high-precision flux-
integrated differential cross sections. Here, the large statistics needed for
single or doubly differential cross-section measurements must be deconvoluted
by detector biases (unobserved pions) and uncertainties in the flux shape.
These procedures can be carried on only to a limited extent. Ideally, they
would require dedicated facilities where the flux is known with unprecedented
precision and the detector migration matrix is extracted independently.
Decoupling $\sigma_{\nu}\epsilon_{\nu}$ from $\sigma_{\nu}$ can be achieved
e.g. using complementary detection techniques on the same beam and, possibly,
the same target like, for instance, liquid argon versus gas argon detectors
Abud et al. (2021). Again, this is a goal that has inspired a new generation
of experiments in the DUNE/HK era.

###  2.5 Biases in the determination of oscillation parameters

It is unfortunate that intense accelerator-based neutrino sources are indeed
sources of $\nu_{\mu}$ only, plus a per-cent contamination of $\nu_{e}$ and
that the “antineutrino” runs are heavily polluted by $\nu_{\mu}$. In turn,
neutrino oscillations measurements mostly rely on
$\nu_{\mu}\rightarrow\nu_{e}\,$and
$\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e}\,$transitions. The
$\nu_{e}\ $ total cross section is measured at $>20\%$ level Abe et al. (2015,
2014); Wolcott et al. (2016); Abe et al. (2020); Abratenko et al. (2021) and
no measurements are available on $\overline{\nu}_{e}$ . As a consequence, the
measurement of the oscillation probabilities only relies on lepton
universality and takes $\sigma_{\nu_{e}}(E)=\sigma_{\nu_{\mu}}(E)$ and
$\sigma_{\bar{\nu}_{e}}(E)=\sigma_{\bar{\nu}_{\mu}}(E)$ for granted except for
radiative corrections arising from the electron and muon mass difference.
Other phase-space effects propagate in the second-class vector current and are
non-negligible in the systematic uncertainty of long-baseline experiments Day
and McFarland (2012). This bias is amplified by detector effects, where the
reconstruction of the neutrino energy in $\nu_{e}\ $ CC events and the purity
of the selected sample differ from the ideal case of lepton universality.
These effects include differences in the neutrino spectrum, the detector
response between ND and FD, the reconstruction of the neutrino energy, and the
large energy spread of the neutrino beam. A quantitative study of the
corresponding systematic uncertainty based on data from current facilities is
detailed in Fernandez-Martinez and Meloni (2011); Katori and Martini (2018);
Alvarez-Ruso et al. (2018). Once more, designing high-precision facilities
Charitonidis et al. (2021) to decouple $\sigma_{\nu_{e}}(E)$ from
$\epsilon_{\nu_{e}}(E)\sigma_{\nu_{e}}(E)$ or, at least, perform a high
precision measurement of $\epsilon_{\nu_{e}}(E)\sigma_{\nu_{e}}(E)$ using the
same target as the long-baseline experiments (water or liquid argon) would be
a remarkable asset.

###  2.6 Open issues in the theoretical understanding of cross-sections

As already mentioned, the MiniBooNE experience led the community to focus on
neutrino flux-integrated differential cross sections in terms of the final
state topology of the reactions. Furthermore, after the suggestion Martini et
al. (2009) of the inclusion of 2p-2h excitations mechanism as the likely
explanation of the MiniBooNE anomaly, the interest of the neutrino scattering
and oscillation communities on the multinucleon emission channel rapidly
increased. This channel was not included in the generators used for the
analyses of the neutrino cross sections and oscillations experiments but
turned to be crucial in the reconstruction of neutrino energy via QE-based
method Martini et al. (2012, 2013); Nieves et al. (2012b); Lalakulich et al.
(2012); Ankowski and Mariani (2017). The effort of the theoretical community
particularly focused on this channels. Many theoretical calculations of
CCQE+2p-2h and CC$0\pi$ flux integrated differential cross sections have been
performed by different groups Martini et al. (2011); Martini and Ericson
(2013, 2014); Martini et al. (2016), Nieves et al. (2012a, 2013); Bourguille
et al. (2021), Megias et al. (2016, 2019a, 2019b); Ivanov et al. (2019);
Barbaro et al. (2021), Lalakulich et al. (2012); Gallmeister et al. (2016);
Mosel and Gallmeister (2018), Pandey et al. (2015); Van Cuyck et al. (2016,
2017), Lovato et al. (2020). Nowadays several calculations agree on the
crucial role of the multinucleon emission in order to explain the MiniBooNE
Aguilar-Arevalo et al. (2010, 2013), T2K Abe et al. (2016, 2018, 2020a, 2020b,
2020c), MINERvA Ruterbories et al. (2019); Carneiro et al. (2020) and
MicroBooNE Abratenko et al. (2020) cross sections. Nevertheless there are some
differences on the results obtained by the different theoretical approaches.
An illustration of the amount of the differences between the results obtained
by different theoretical approaches for the CCQE and the CCQE+2p-2h is given
in Fig. 2 where the T2K flux-integrated double-differential cross section on
carbon is shown as a function of the muon momentum for
$0.85<\cos\theta_{\mu}<0.90$ and it is compared to the experimental CC$0\pi$
T2K results, already introduced in Fig. 1.

Figure 2: Comparison between different theoretical predictions of the CCQE and
CCQE+2p-2h T2K flux-integrated muon neutrino double-differential cross section
on carbon as a function of the muon momentum for $0.85<\cos\theta_{\mu}<0.90$.
The experimental CC$0\pi$ results of T2K Abe et al. (2016) are also shown.The
results of Lovato et al. are taken from Ref. Lovato et al. (2020). The results
of of Martini et al. Martini et al. (2009) and Nieves et al. Nieves et al.
(2011) appear in Ref. Abe et al. (2016). The Megias et al. results are taken
from Ref. Megias et al. (2016). The Mosel et al. results are taken from Ref.
Mosel and Gallmeister (2018) where the pion absorption contribution (not shown
in this figure and in any case relatively small for the angle bin considered
here) is also separately plotted. The Van Cuyck et al. results are taken from
Ref. Van Cuyck et al. (2017).

At the present level of experimental accuracy quantifying the agreement
between the T2K data and the different models is not evident; the
uncertainties are too large for any conclusive statement. However, even if a
one-to-one correspondence between the different exclusive channel
contributions of different theoretical calculations is not always possible
(for example short-range correlations are considered as part of genuine QE in
the approaches of Refs. Megias et al. (2016),Lovato et al. (2020), while they
are separately calculated as part of the 2p-2h excitations in the case of
Refs. Martini et al. (2009),Nieves et al. (2011),Mosel and Gallmeister
(2018),Van Cuyck et al. (2017)), we can consider Fig. 2 as an illustration of
the fact that also T2K data seems to prefer CCQE+2p-2h with respect to pure
CCQE calculations.

A more complete comparison between different theoretical predictions, Monte
Carlo Andreopoulos et al. (2010); Hayato (2009); Golan et al. (2012), and the
data is published in Ref. Abe et al. (2020b). In this work, the T2K flux-
integrated double differential $\textrm{CC0}\pi$ cross sections, for muon
neutrino and antineutrino are given, as well as their combinations important
for CP-violation investigations. From Fig. 2 and Ref. Abe et al. (2020b) it
turns out that although the trends of the theoretical calculations are
similar, some significant differences remain, including even differences
between calculations of the same category, such as the ones of Martini et al.
Martini et al. (2009), Nieves et al. Nieves et al. (2011). These differences
are illustrated for instance in Ref.Katori and Martini (2018), where a
comparison is shown between data and predictions of models which calculate
several neutrino and antineutrino MiniBooNE, T2K, and MINERvA flux-integrated
differential cross sections. Moreover, the approximations made by the
different groups in the treatment of 2p-2h excitations are also discussed in
this work.

A particularly important region is the one of very forward muon scattering
angle. Here not only the different theoretical models and Monte Carlo show
differences among them but also they globally overestimate the data. This is
the case of the $\textrm{CC0}\pi$ T2K cross sections Abe et al. (2016, 2020b)
(as it appears for example in the $0.98<\cos\theta_{\mu}<1.00$ panel of Fig.
1), of the T2K CC inclusive cross sections of Ref. Abe et al. (2018),
characterized by an increased angular acceptance and higher statistics with
respect to the ones of Ref. Abe et al. (2013), and of the MicroBooNE
CC1p0$\pi$ Abratenko et al. (2020) and CC0$\pi$ Abratenko et al. (2020) cross
sections on argon. A deeper understanding of the very forward region is
important for CP-violation experiments and should be further investigated. For
example, by analyzing neutrino scattering in the giant-resonances and quasi-
elastic region for fixed values of neutrino energy, it has been raised Martini
et al. (2016) and investigated Ankowski (2017); Nikolakopoulos et al. (2019);
González-Jiménez et al. (2019) the surprisingly dominance of the $\nu_{\mu}$
cross sections over the $\nu_{e}$ ones for small scattering angles.

Several reasons contribute to the difficulty of a precise modeling of the flux
integrated differential cross sections in the very forward region:

  * •

The Pauli blocking effects play a significant role Megias et al. (2019a);
Ivanov et al. (2019); González-Jiménez et al. (2019)

  * •

The giant resonances, neglected by many theoretical models, contribute to the
cross section Pandey et al. (2016)

  * •

The neutrino cross sections, in general dominated by the spin-transverse
response, becomes sensitive to the spin-longitudinal one which is
characterized not only by low-energy giant resonances but also by high-energy
collective states such as coherent mixture of $\Delta$-hole states and pions
Delorme and Ericson (1985); Martini and Ericson (2014)

  * •

Even in neutrino beams, such as T2K/HK and MiniBooNE/MicroBooNE, which induce
nuclear excitations dominated by quasielatic, 2p-2h and $1\pi$ production,
higher energy excitations up to the DIS contribute, as illustrated for example
in Refs. Gallmeister et al. (2016); Bourguille et al. (2021).

The ultimate goal should be the development of a theoretical unified approach
for the treatment of all nuclear excitations, from low-energy giant resonances
to DIS. For the moment flux-integrated differential cross sections
calculations are available from QE to DIS in the case of the GiBUU Buss et al.
(2012) implementation of the transport theory Lalakulich et al. (2012);
Gallmeister et al. (2016); Mosel and Gallmeister (2018) and from and from QE
to 1 pion production induced by $\Delta$-resonance excitations in the
approaches of Martini et al. Martini et al. (2009, 2011); Martini and Ericson
(2014), Nieves et al. Nieves et al. (2011, 2012a); Bourguille et al. (2021)
and SuSA/SuSAv2 Ivanov et al. (2012, 2016); Megias et al. (2016); Amaro et al.
(2020); Barbaro et al. (2021). Promising approaches exploiting the
factorisation of the nuclear cross section, based on spectral functions Rocco
et al. (2016); Vagnoni et al. (2017); Rocco et al. (2019a, b) and superscaling
Amaro et al. (2020); Barbaro et al. (2021) have already been extended to
inelastic region beyond $\Delta$-resonance and employed to investigate
electron scattering and neutrino cross sections for fixed kinematics and as a
function of the neutrino energy. Neutrino flux-integrated differential cross
sections represent the next step.

The developed theoretical nuclear models should not only cover different
reaction channels and kinematics but also be applicable to various nuclear
targets. If many calculations exist for the doubly magic $N=Z$ nuclei 12C and
16O, the effort to investigate the 40Ar started only recently Meucci et al.
(2013); Gallmeister et al. (2016); Van Dessel et al. (2018); Akbar et al.
(2017); Barbaro et al. (2018); Barbieri et al. (2019); Van Dessel et al.
(2019, 2020); Butkevich (2021); Franco-Patino et al. (2021); Barbaro et al.
(2021) and often it requires formal generalization of the existing approaches,
being the 40Ar an open-shell nucleus with $N\neq Z$.

Precise predictions and measurements of hadronic final states represent
another very important next step. The community is moving toward to this path.
Beyond the inclusive cross section, largely discussed above, where only the
outgoing lepton is detected, there is a rapidly increasing interest on the
semi-inclusive cross section, which corresponds to detecting in coincidence
the outgoing lepton and an hadron (or a system of hadrons), and on the
exclusive cross section, corresponding to the simultaneous detection of all
the final scattering products. The last two processes are more challenging
from both experimental and theoretical points of view. In the recent years the
community has focused in particular on semi-inclusive reactions where a muon
and one proton are detected in the final state since they can be used to
identify relevant nuclear effects related to both the initial state dynamics
and to final state interactions, as well as to 2p-2h excitations. Special
attention has been paid to the study of kinematical variables obtained by
projecting the final lepton and the ejected nucleon momenta on the plane
transverse to the neutrino beam. The cross sections in terms of these
observables, called Single-Transverse (Kinematic) Variables (STVs) or
transverse kinematic imbalances (TKI) Lu et al. (2016), are useful to enhance
some nuclear effects, and therefore discriminate between different models. T2K
Abe et al. (2018) and MINERvA Lu et al. (2018); Cai et al. (2020) measurements
have been published in terms of these variables. Semi-inclusive CC1p0$\pi$
cross sections function of muon and proton kinematics have been published also
by MicroBooNE Abratenko et al. (2020, 2020). The theoretical investigation on
semi-inclusive cross sections Moreno et al. (2014); Van Cuyck et al. (2016,
2017); Van Orden and Donnelly (2019); Sobczyk et al. (2020); Franco-Patino et
al. (2020); González-Jiménez et al. (2021) and on transverse kinematic
imbalances Dolan et al. (2018); Bourguille et al. (2021); Franco-Patino et al.
(2021) are only at the beginning, as well as the studies related to the
emission of nucleon pairs induced by meson exchange currents and short range
correlations Van Cuyck et al. (2016, 2017); Ruiz Simo et al. (2016); Pastore
et al. (2020).

Semi-inclusive processes characterized by the detection of one muon and one
pion in the final state deserve also further deep investigations. In this case
data-theory agreement remains very unsatisfactory. Nowadays there is no model
which can describe MiniBooNE Aguilar-Arevalo et al. (2010, 2011), MINERvA
Eberly et al. (2015); McGivern et al. (2016) and T2K Abe et al. (2017) data
simultaneously. For example the model of Refs.Hernández et al. (2013);
Alvarez-Ruso et al. (2020) allows to get a reasonable description of MiniBooNE
data but over-predicts the MINERvA ones and lacks forward pions in the case of
T2K. On the contrary, the models of Refs.Lalakulich and Mosel (2013a, b);
Mosel (2015) and Refs.González-Jiménez et al. (2018); Nikolakopoulos et al.
(2018) reproduce well the shape and strength of the MINERvA and T2K data but
underpredict the MiniBooNE ones. This is the so called “pion puzzle”. The
complications of pion data analyses lay not only on their primary production
models, but also on the pion final state interactions and on the fact that all
hadronic processes related to shallow inelastic scattering (SIS) and DIS
regions have to be modeled correctly, another major challenge Katori and
Martini (2018); Alvarez-Ruso et al. (2018); Sajjad Athar and Morfín (2021);
Alvarez-Ruso et al. (2020).

##  3 Learning from the next generation of near detectors

The near detectors of long-baseline experiments like NO$\nu$A and T2K have
been instrumental to improve our knowledge of cross-sections and unveiled a
complexity that was not suspected at the time of the discovery of neutrino
oscillations. The new generation of near detectors for DUNE and HK will
exploit mega-watt class beams even if the level of control of these beams
(diagnostic, flux, flavor at source, $\nu_{e}\ $and wrong-sign contamination,
etc.) will be comparable to their predecessors. Since systematics is the main
limiting factor to CP sensitivity both in DUNE and HK, the corresponding
collaborations are devising a near detector complex much more complete than
what has been done in the past. It is worth stressing, however, that the main
aim of these detectors is to minimize the uncertainty on the measurement of
unoscillated $\nu_{\mu}\ $and $\nu_{e}\ $interactions, that is the difference
in the number of expected events if no oscillation occurs. If the near and far
unoscillated rates were identical, the near-far difference as a function of
the energy would provide a measurement of the oscillation probability free of
systematic uncertainties. This goal is not equivalent to cross-section
measurements even if cross sections play a major role. Cross section
measurements are thus a byproduct of the DUNE/HK systematic mitigation
strategy and reap the lessons learned by NO$\nu$A and T2K.

For the sake of illustration, we detail what the DUNE near detector (ND)
complex Abud et al. (2021) can offer to the physics of neutrino cross section,
identify its strength and point out to possible limitations. Similar
considerations hold in a narrower energy range for HyperKamiokande.

Figure 3: Left: Overview of the DUNE near detector complex with the gas argon
TPC replacing the TMS. Right: the HyperKamiokande Intermediate Water Cherenkov
Detector.

The DUNE ND complex is shown in Fig. 3 and it is made up of three subsystems:

  * •

SAND: an on-axis detector based on the KLOE magnet and calorimeter
complemented by a low-density tracker

  * •

ND-LAr: a non-magnetized liquid argon TPC capable to stand the high neutrino
rate of the DUNE beam

  * •

TMS: a muon catcher that closes the kinematics of $\nu_{\mu}\ $CC events in
the liquid argon. In a second stage, this muon catcher will be replaced by a
magnetized high-pressure argon TPC (ND-GAr) inside a superconducting solenoid

The expected rates at ND-LAr is $10^{8}$ $\nu_{\mu}\ $CC and about $2\times
10^{6}$ $\nu_{e}\ $CC per year. Similarly, the expected rate in SAND is
$10^{7}$ $\nu_{\mu}\ $CC and $\simeq 2\times 10^{5}$ $\nu_{e}\ $CC per year.
The liquid argon TPC and the muon catcher (or the gas argon TPC) are movable
detectors that run in the horizontal plane perpendicular to the beam. In this
way, DUNE can monitor its beam both on-axis and off-axis. Unlike current
systems like the on-axis INGRID detector and the off-axis ND280 detector at
T2K, the off-axis detectors can span from 0 m (on-axis) to 30 m far from the
beam axis. They can thus implement the PRISM technique Bhadra et al. (2014);
Abud et al. (2021) to sample the neutrino flux at several angles and, hence,
at several energies. The combination of these data provides a faithful
representation of the unoscillated
$\phi_{\nu_{\mu}}(E)\sigma_{\nu_{\mu}}(E)\epsilon_{\nu_{\mu}}(E)$ and
$\phi_{\nu_{e}}(E)\sigma_{\nu_{e}}(E)\epsilon_{\nu_{e}}(E)$ at the far
detector in South Dakota. This strategy is also envisaged in HK with a water
Cherenkov detector located at about 1 km from the source and moving in the
vertical plane perpendicular to the beam axis (1-4∘) Bhadra et al. (2014) as
shown in Fig. 3, right.

The triple product flux$\times$cross-section$\times$efficiency in multiple
energy bins is the main observable to cope with the DUNE and HK systematics
due to the near-versus-far difference. The triple product can be partially
deconvolved to attain information on the cross sections. The main tool for
deconvolution is an independent measurement of the neutrino flux. DUNE and HK
are planning to exploit the aforementioned standard candles: the neutrino-
electron scattering and the low-$\nu$ events Belusevic and Rein (1988). In the
most aggressive scenario, standard candles combined with the classical
techniques to reduce the $\nu_{\mu}\ $flux systematics (beam simulation and
hadroproduction) might achieve a precision of 2-3%, especially in the low-
density detectors: SAND and the gas argon TPC in DUNE, and the upgraded ND280
in HK. The $\nu_{e}\ $flux systematics cannot be estimated with standard
candles due to the low flux of $\nu_{e}\ $with respect to $\nu_{\mu}$. Its
estimated uncertainty ranges from 7% to 10%, depending on whether DUNE runs in
neutrino or antineutrino mode. Unfortunately, the PRISM technique is not
enough to cancel the near-versus-far systematics even if it is an important
mitigation tool. The main obstruction is the neutrino energy reconstruction
that is still obtained by the reconstructed final state particles in the
neutrino detector. Citing Ref. Abud et al. (2021), PRISM “will minimize
uncertainties in the extracted oscillation parameters arising from the
spectral differences as implemented in the imperfect interaction (cross
section) model. … DUNE will need a vibrant program of cross section
measurements as input to that work”. The near detectors themselves can perform
some of these measurements comparing the double-differential cross sections
measured in the liquid argon with theory expectations and tuning the
corresponding Monte Carlo codes. Fig. 4 shows the impact on the CP sensitivity
assuming modest (1 bias) and conservative (5 bias) biases in the interaction
models Abud et al. (2021).

Figure 4: Impact on the DUNE CP sensitivity from uncertainties in the neutrino
interaction models.

Decoupling the detector efficiency $\epsilon_{\nu_{\mu}}(E)$ and
$\epsilon_{\nu_{e}}(E)$ from the corresponding cross section in the triple
product is a major challenge. The most direct approach is measuring the same
cross section with a high-density and low-density detector simultaneously.
Ideally, the medium should be the same to avoid additional biases due to the
$Z$ dependence of nuclear reinteraction effects. In DUNE, this is the
rationale of using simultaneously a liquid argon and gas argon TPC, provided
that the flux decoupling is performed at the per-cent level. A compromise is
to use detectors with the same target nuclei interspersed by high-precision
trackers. This is done in HK combining the measurement on carbon and oxygen
(ND280 and WAGASCI) with the measurement in water (IWCD, NINJIA, ND280-FGD2
water layers). In DUNE, besides the gas/liquid argon TPC, the SAND and liquid
argon TPC data can be combined, as well. At present, however, none of these
studies have been performed at percent level precision.

Tuning the neutrino interaction models with data is not equivalent to building
reliable models, which should be based on low Z (hydrogen or deuteron) targets
and scaled up to argon and oxygen. No near detector is equipped with these
devices but SAND could perform some of these studies looking at the transverse
momentum imbalance and exclusive state reconstruction, already introduced in
Sec. 2.6, to isolate samples of events enriched in interactions on hydrogen
within a hydrocarbon target. This technique has been proposed by several
authors Lu et al. (2016); Duyang et al. (2019); Munteanu et al. (2020);
Hamacher-Baumann et al. (2020) and recently used with the T2K near detector
data in Abe et al. (2021). The systematic uncertainties associated with the
subtraction of the non-hydrogen components of the target (mostly carbon) are
under evaluation.

##  4 A new generation of neutrino beams

As discussed in Sec. 3, the disentangling of the flux, cross-section, and
efficiency triple product is not fully accomplished either by current
experiments or by the forthcoming near detectors and deserves a dedicated
facility. This facility will be the core of a new generation of cross section
experiments capable to provide firm guidance to model builders instead of
tuning approximate models with data.

The first hindrance toward high-precision cross section measurements is the
uncertainty on flux. This is a prominent motivation for a new generation of
short-baseline beams, which can provide controlled fluxes at a level
unattainable in the near detectors of Sec. 3. The MINER$\nu$A experiment
performed the best measurement of the flux in the 1-10 GeV range on the NuMI
beam Aliaga et al. (2016) and showed that classical tools like GEANT4/FLUKA
simulations combined with beam and muon monitors can achieve a precision of
10-25% depending on the energy. By including hadro-production data the
uncertainty can be reduced to 5.4%. By using an independent measurement as the
$\nu_{\mu}\ e^{-}$ scattering for the integrated flux the precision of cross-
section measurements in MINER$\nu$A can be further improved: the experiment
was able to reduce such uncertainty from 7.5% to 3.9% by using the
$\nu_{\mu}+e^{-}\rightarrow\nu_{\mu}+e^{-}$ scattering which is characterized
by a purely leptonic cross section known with great precision Valencia et al.
(2019). This technique can be used in DUNE and HK as well. However, it will
reach its intrinsic limitation (detector systematics on $\nu_{\mu}\ e^{-}$
scattering and hadroproduction data) most likely at 3% level and will not
provide information on the $\nu_{e}$ flux since their high-intensity neutrino
beams are almost completely made of $\nu_{\mu}$.

###  4.1 Monitored neutrino beams

A facility like the one envisaged by the NP06/ENUBET project addresses
directly this problem. It aims to monitor a neutrino beam by measuring
associated charged leptons produced in a narrow-band meson beam. Large-angle
leptons from kaon decays are recorded in an instrumented decay tunnel while
the neutrino travels toward the neutrino detector. Since we produce a neutrino
for each lepton in the tunnel, lepton counting allows for a direct flux
prediction that bypasses the uncertainties coming from hadro-production yields
in the targets, beamline simulation, and proton-on-target (POT) counting.
These facilities are called monitored neutrino beams and can reduce the
uncertainty on the flux of $\nu_{e}$ and $\nu_{\mu}$ below 1%.

ENUBET is the most advanced monitored neutrino beam and all technical
challenges have been considered in the design of the ENUBET beamline. It is a
conventional beam where protons from the accelerator are sent to a fixed
target producing secondary mesons that decay into neutrinos. Two different
focusing options are being studied: a purely static focusing system as well as
a horn-based transfer line. Several existing accelerator complexes can feed
the ENUBET beamline at different proton energies: the 400 GeV (CERN-SPS), the
120 GeV (Fermilab Main Ring), and the 30 GeV (JPARC) protons have all been
simulated to estimate the secondary yield with several targets.

To cover the region of interest of DUNE, the hadron transfer line design is
optimized for the transport of 8.5 GeV/$c$ kaons. The electrons produced in
the three-body $K_{e3}$ decays together with the $\nu_{e}$ ($K^{+}\rightarrow
e^{+}\pi^{0}\nu_{e}$) are measured in a 40 meters long instrumented decay
tunnel. Similarly, ENUBET measures the large-angle muons from the
$K^{+}\rightarrow\mu^{+}\nu_{\mu}$ decay with a signal to noise ratio larger
than 6 and can monitor at single-particle level the muons from pion decays by
replacing the hadron dump with a range-meter. A neutrino detector is placed 50
m from the end of the decay tunnel. The guiding principles for the beamline
design are the kaon flux maximization at the tunnel entrance, the reduction of
beam size and divergence to contain the hadron beam in the decay volume, and
the total length of the transfer line to minimize early kaon decays. Figure 5
shows the survival probability of pions and kaons produced in neutrino beams
below 10 GeV/$c$ as a function of the length of the beamline.

Figure 5: Survival probabilities for low momentum pions and kaons, as a
function of the beamline length.

The static transfer line design makes use of normal-conducting quadrupoles and
dipoles: as they run at constant current and do not need to be pulsed, they
can be used with second-long proton extractions (slow-extraction). For
example, the current slow extraction operation mode at the CERN SPS produces a
few second long proton spill for fixed target experiments like ProtoDUNE and
it is characterized by low particle rates. This offers several advantages for
monitored beams since pile-up at the instrumented decay volume poses stringent
limits on the instantaneous particle rate. Besides, standard focusing magnets
are simpler, more robust, and less expensive than horns. This comes at the
expense of the flux at the decay tunnel and the lower neutrino yield at the
neutrino detector because quadrupole multiplets have smaller acceptance at the
ENUBET hadron energies with respect to horns.

For this reason, the ENUBET collaboration is also pursuing a horn-based
beamline. To cope with pile-up requirements in the decay region, studies have
been carried out at the CERN SPS to combine the need of pulsing the horn with
a proton slow extraction scheme. A new pulsed slow extraction scheme (burst-
mode slow extraction) was successfully developed and tested: 2-to-10 ms proton
pulses were repeated at 10 Hz for the full duration of the extraction Acerbi
et al. (2020, 2021). An example of this burst-mode slow extraction is
presented in Fig. 6 and compared with a nominal slowly extracted spill. Note
that the same number of protons are extracted in both cases. The horn design
optimization to maximize the flux of focused secondaries in the sought-for
momentum range is carried out with a full GEANT4 simulation considering
various horn geometries (single and double parabolic, and MiniBooNE-like
geometries). It was implemented with a Genetic Algorithm (GA) in a standalone
way using a beamline-independent figure-of-merit. Further studies on the
dedicated beamline configuration for the horn-focused beam are underway to
determine the flux increase with respect to the static option.

Figure 6: Comparison between a nominal slow extracted spill and a “burst-mode
slow extraction” one. The spill profiles have been measured with a secondary
emission monitor at the SPS, during dedicated machine tests. The same
intensity is extracted in the two cases.

In Figure 7 we show the most recent static beamline configuration of the
ENUBET project: a quadrupole triplet (aperture radius of 15 cm) is placed
after the proton target and two identical normal-conducting bending dipoles
with a 1.8 T field provide a total bending of the beam with respect to the
primary proton line of 14.8°. Such a large bending angle suppresses background
neutrinos produced by the early decay of neutral kaons in the proximity of the
target. The hadron dump is placed at the tunnel exit. Optics was optimized
using the TRANSPORT code while particle transport and interactions are fully
simulated with G4Beamline considering 400 GeV/$c$ protons interactions on a
graphite target. Proton-graphite interactions were simulated with FLUKA. Doses
accumulated in the transfer line are estimated using FLUKA, too.

Figure 7: Schematic of the ENUBET static transfer line. Bending elements:
dipoles are shown in red while quadrupoles are shown in grey. The hadron dump
is placed downstream the tagger exit.

The beamline produces the $\nu_{e}$ CC spectrum shown in Fig. 8 when
considering a neutrino detector located 50 m downstream the tunnel exit with a
6 $\times$ 6 m2 size in the plane perpendicular to the beam axis. Classifying
the different components of the spectrum based on the neutrino creation point
along the transfer line, it is clear that a loose energy cut would already be
good enough to separate the monitored component coming from the decay tunnel
from other neutrinos. 73.5% of the total $\nu_{e}$ CC flux is generated inside
the tunnel (it exceeds 80% considering only energies above 1 GeV). By assuming
$4.5\times 10^{19}$ POT/year at the SPS, 104 $\nu_{e}$ CC are collected in a
500-ton neutrino detector (e.g. ProtoDUNE Abi et al. (2020) or ICARUS Acciarri
et al. (2015); Amerio et al. (2004)) in about 2 years. Below 1 GeV, the main
component is produced in the proton-dump region, which can be further
suppressed by optimizing the proton dump design and position. The remaining
12% of the flux is given by the straight section in front of the tagger plus
the decays or interactions occurring after the tunnel and in the hadron dump.
Its contribution to the flux is corrected relying on simulation and particle
monitoring along the beamline. A static line design for a monitored neutrino
beam would also pave the way to the so-called tagged neutrino beams Hand
(1969); Bernstein et al. (1988); Pontecorvo (1979) that will be briefly
discussed later.

Figure 8: Top: $\nu_{e}$-CC interactions considering a $6\times 6$ m2front
face of a neutrino detector located 50 m downstream the ENUBET decay tunnel
exit. The events are divided into categories corresponding to the position
along the transfer line where the neutrino was generated. The red spectrum
corresponds to neutrinos generated inside the tagger. Bottom: relative
fraction of each category to the total $\nu_{e}$-CC rate.

Detailed studies were performed by ENUBET also on the target, hadron, and
proton dumps. The most common materials chosen for targets at proton
accelerators are low-Z, radiation hard materials. ENUBET performed
optimization studies on graphite ($\rho$=2.2 g/cm3), beryllium ($\rho$=1.81
g/cm3) and Inconel ($\rho$=8.2 g/cm3). Inconel – an austenitic nickel-
chromium-based superalloy – is a novel target choice currently considered also
by the nuSTORM collaboration (see below). Each target was modeled as a
cylinder with a variable radius between 10 to 30 mm and with a different
length from 5 to 140 cm. Figure 9 shows the kaon yields obtained for graphite
targets. The figure-of-merit employed for the optimization is the number of
kaons with a $\Delta p/p=\pm 10\%$ that enter an angular acceptance of $\pm$20
mrad. The most promising materials turned out to be graphite and Inconel-718.
400 GeV/$c$ protons impinging on a graphite target 70 cm long and with a
radius of 3 cm were thus used for the static beamline simulation, while the
Inconel option is considered quite promising for the horn beamline: the small
loss of kaon flux is compensated by the larger acceptance of the horn.
Moreover, Inconel produces fewer positrons reducing the expected background in
the decay tunnel.

Figure 9: Kaon yields as a function of the graphite target length for a 400
GeV/c proton beam. Colors refer to different kaon momenta while the marker
style identifies the target radius.

The positron flux coming directly from the target region that hits the
instrumented decay tunnel walls represents the most critical background since
it cannot be rejected by particle identification. A tungsten thin foil placed
directly downstream the target reduces this background by orders of magnitude
and was optimized comparing the amount of background expected with respect to
the kaon flux for different thicknesses of the foil. The proton dump is made
by a 3 m long graphite core, surrounded by aluminum and covered by iron. This
design is inherited by several dumps of high-intensity proton beams operated
at CERN. Its final design and position will be further studied to reduce the
proton dump contribution to the neutrinos recorded in the neutrino detector.
The structure of the hadron dump is similar. Its optimization, however, is
aimed at reducing the back-scattering. Back-scattering results in particles
hitting the tunnel walls increasing the background and in the growth of
neutron fluence that contributes to the total dose received by the
instrumentation. The hadron dump cannot be placed too far away from the tunnel
exit to minimize kaon decays in a region where the leptons cannot be
monitored. A design that mitigates back-scattering events consists of a hadron
dump placed two meters after the decay tunnel with a graphite core (50 cm
diameter), inside a layer of iron (1 m diameter), covered by borated concrete
(4 m diameter). In addition, 1 m of borated concrete is placed in front of the
hadron dump leaving an opening for the beam. This has provided a significant
reduction of the flux coming from the dump all along the tagger, in particular
in the last few meters where the neutron fluence is particularly relevant.

The impact of the ENUBET data on the $\nu_{e}$ cross-section measurement
assuming 1% flux precision is shown in Figure 10 and compared with current
data.

Figure 10: Impact of the ENUBET data on existing $\nu_{e}$ cross-section
measurements assuming 1% flux precision.

The results obtained by ENUBET fostered the possibility to constrain the low-
energy $\nu_{\mu}$ flux from pion decays by monitoring associated muons
emitted at low angles by instrumenting the hadron dump. Muons that exit the
decay tunnel and go through the hadron dump can be measured by detectors
placed between absorber layers. With a static focusing beamline and fast muon
detectors, it is possible to reconstruct muons on an event-by-event basis and
measure their momentum from their range in the detector stations. The
instrumented hadron dump allows to separate the three different components of
the muon spectrum: as shown in Figure 11, the measured momentum can be
exploited to separate muons from pions, and beam-halo muons from muons of
$K_{\mu 2}$ ($K^{+}\rightarrow\mu^{+}\nu_{\mu}$) and $K_{\mu 3}$ decays that,
in turn, can be used to determine the $\nu_{\mu}$ flux. The energy of
$\nu_{\mu}$ can be studied in bins corresponding to muon momenta reconstructed
by range as they show a clear anti-correlation due to the 2-body decay
kinematics, allowing to constrain the $\nu_{\mu}$ shape and normalization. The
ENUBET collaboration is studying a system of 8 muon stations to be placed
right after the hadron dump with iron absorbers from 2 m (upstream) to 0.5 m
(downstream) depth (see Fig. 11). Muon and neutron fluences have already been
estimated: the most upstream detector needs to cope with a muon rate of 2
MHz/cm2 and the total neutron fluence integrated over the experiment lifetime
is 1012 1-MeV-eq/cm2. The radiation damage expected in a monitored beam is
much smaller compared to beams where the dump is used to stop non-interacting
primary protons. This is the reason while instrumented dumps are not an option
in conventional beams like the ones used for long-baseline experiments.

Figure 11: Left: Schematic of the muon stations and absorbers configuration to
be installed at the end of the ENUBET tagger calorimeter. The grey slabs
represent the absorbers (made out of iron or rock) while the white slices, 8
in total, are the muon detector planes. Center: spatial distribution along one
of the two dimensions of a detector layer. Right: energy distribution. Red and
blue are the signal contribution, from pion and kaon decays. The gray
distribution represents halo-muons.

It is worth mentioning that ENUBET is working also on a site-dependent multi-
momentum beamline to cover the lower energy region, which is of interest for
HK: the current design of the multi-momentum beamline uses existing CERN
magnets. The beamline optics is flexible enough to select hadron momenta down
to 4 GeV or less. The design consists of quadrupoles for focusing and two
dipoles for momentum selection and a large total bending to ensure the
separation of the different $\nu_{e}$ components at the neutrino detector.

The possibility of measuring the flux of $\nu_{\mu}$ with a precision
comparable with $\nu_{e}$ is a significant asset for ENUBET. Since ENUBET is a
narrow band beam and the pions produce muon-neutrinos in a two-body decay,
there is a strong correlation between the neutrino angle and its energy. As a
consequence, the measurement of the interaction vertex of the neutrino in the
detector (and, hence, its angle) provides a measurement of the energy with an
average precision of 10%. It is worth stressing that this indirect energy
measurement (narrow-band off-axis technique) can be performed at the level of
a single neutrino without resorting to the energy reconstruction at the
detector. This is the main tool to remove all reconstruction biases mentioned
in Sec. 3 and suppress the systematics on neutrino interaction modeling that
plague the ND measurements without any tuning with data.

###  4.2 Tagged neutrino beams

As anticipated, a monitored neutrino beam with a slow proton extraction scheme
would open the possibility for a time-tagged neutrino beam where the neutrino
recorded in the detector is associated in time with the observation of the
lepton from the parent hadron in the decay tunnel. Tagged beams require a
detector system with a time resolution of $\mathcal{O}$( 100 ps). Reaching
this level of accuracy in the decay volume would represent another prominent
asset. A precise time coincidence between the reconstructed lepton and the
neutrino interaction provides the neutrino flavor without looking at the
lepton in the neutrino detector produced by CC interactions. Besides, if the
other products of the decays (of $\pi$ or K) are measured, the energy of the
neutrino can be constrained with even higher precision than a monitored beam.
The neutrino sample selected with a time-tagged beam would have an
unprecedented purity. For every neutrino interaction in the detector, one has
to select the lepton candidates in the decay volume compatible in time with
the neutrino within the detector resolution. If the neutrino interaction time
is $t_{\nu}$ and the lepton is time-tagged at $t_{l}$, considering the
distances from the decay point of the two interactions ($d_{\nu}$ is the
decay-neutrino interaction distance and $d_{l}$ is the decay-lepton
interaction distance), we can write:

$t_{\nu}-t_{l}=d_{\nu}-d_{l}\frac{E_{l}}{p_{l}}$ (17)

The neutrino production point is unknown so $d_{\nu}-d_{l}\frac{E_{l}}{p_{l}}$
has to be approximated by $z_{\nu}-z_{l}$, the distance of the neutrino
interaction vertex from the lepton impact point in the decay volume projected
along the axis of the decay volume. This approximation introduces a time
spread due to the fact that the neutrino and the lepton are not exactly
collinear. This irreducible spread ($\delta_{irr}$) is about 100 ps and sets
the bar of the timing precision needed in a tagged beam. The time match
requires:

$\Delta t=\left|t_{\nu}-t_{l}-(z_{\nu}-z_{l}/c)\right|<\sqrt{\left(\delta_{\nu
tagger}\right)^{2}+\left(\delta_{\nu
Det}\right)^{2}+\left(\delta_{irr}\right)^{2}}$ (18)

where the time resolution of the lepton tagger in the decay tunnel and the
neutrino detector are $\delta_{tagger}$ and $\delta_{\nu Det}$, respectively.
The main background for a tagged beam is given by accidental coincidences.
They are proportional to the combined time resolution
$\delta=\sqrt{(\delta_{tagger})^{2}+(\delta_{\nu Det})^{2}}$, and the time
match $\delta$ must be smaller than $\delta_{irr}$ to achieve a high purity
sample. The number of true time-coincidences can also be improved by
increasing the geometrical acceptance of the detectors in the decay volume,
which implies that the technology used for these detectors must provide
excellent timing over large areas. A dedicated project, NUTECH NUT , is
exploring different technologies for fast timing in large areas like Cherenkov
Micromegas (PICOSEC Kordas et al. (2020)), Large Areas Picosecond
PhotoDetectors (LAPPD Lyashenko et al. (2020)) or LYSO(Ce) crystals. In fact,
even the current photon veto of ENUBET, which is based on standard plastic
scintillators, has achieved a 200 ps resolution and further improvements are
envisaged to reach the 100 ps level. The same precision can be achieved by the
neutrino detectors thanks to the Cherenkov signal of water-based detectors and
the 128 nm scintillation light of argon.

###  4.3 Muon beams

Moving away from conventional neutrino beams, a precise flux measurement can
be performed through the muon decays $\mu^{\pm}\rightarrow e^{\pm}+\nu_{e}\
(\overline{\nu}_{e}\ )+\overline{\nu}_{\mu}(\nu_{\mu}\ )$ by storing and
accelerating muons in a ring. This is the idea behind the Neutrino Factories
Geer (1998); De Rujula et al. (1999); Abrams et al. (2011) where the beam
composition is precisely known: 50% $\nu_{e}\ (\overline{\nu}_{e}\ )$ and 50%
$\overline{\nu}_{\mu}(\nu_{\mu}\ )$. There are three main advantages in this
approach: first, the number of muons in the storage ring can be easily
measured providing an excellent estimate of the flux. Then, as neutrino
flavors are determined purely by the muon decay, the beam is not contaminated
by neutrinos originating from other particles. Finally, a Neutrino Factory
offers a powerful source of $\nu_{e}\ (\overline{\nu}_{e}\ )$ unlike
conventional beam (including ENUBET) where they never exceed 10% of the total
flux. Building this kind of facility presents, however, major technical
challenges that have not been superseded despite decades of R&D. Muons must be
accelerated fast with a reduced transverse momentum $p_{T}$ before injecting
them in the storage ring. A muon-based neutrino beam (and, even more, a muon
collider) thus requires a strong muon cooling. Some encouraging results were
recently achieved by the analysis of the MICE data Bogomilov et al. (2020).
Still, the first full-fledged muon storage ring has not been built, yet. It
has been designed by the nuSTORM collaboration Adey et al. (2013) as a
powerful facility for neutrino cross sections.

Even if nuSTORM is significantly more complex than a monitored neutrino beam,
it offers much larger statistics of $\nu_{e}$ and can feed an ambitious cross
section measurement program. Figure 13 shows a schematics of the nuSTORM
facility. nuSTORM makes use of a horn to focus pions that are then transported
by a series of quadrupoles and dipoles to the injection point of the storage
ring. The ring is composed of several FODO cells: a large acceptance focusing
and defocusing quadrupole lattice. The circular ring has two long straight
sections where muons of $E_{\mu}\sim 5$ GeV from pion decays produce neutrinos
of both electron and muon flavors.

The three-body kinematics of muon decay does not allow for the measurement of
the neutrino energy at the single-particle level that can be employed by
ENUBET through the narrow-band off-axis technique. Still, nuSTORM can achieve
excellent precision on the fluxes operating an FBCT (Fast Beam Current
Transformers) in the decay ring to measure the muon intensity. These devices
are based on toroids and developed at CERN, and are needed for fast
extractions where muons are bunched in tens of $\mu$s up to a few ms.

A feasibility study for nuSTORM at CERN Ahdida et al. (2020) reports that the
experiment would use protons extracted from the SPS at 100 GeV and direct them
on a low-Z target placed inside a focusing horn. After the horn, particles are
collected by a pair of quadrupoles. Then the beam goes through a short
transfer line composed of dipoles, collimators, and quadrupoles. The detailed
design of the target and capture system has to cope with radiation safety and
full containment in the transport of a beam with a momentum spread of
$\pm$10%. To fulfill these requirements the experiment is considering the
scheme that has been successfully used in the CERN PS complex at the
Antiproton Decelerator as well as the work done for the CENF and LAGUNA-LBNO
projects for the target-horn system.

For the pion transfer, modular construction with simple quadrupole FODO cells
and achromatic dipoles was chosen to ensure more flexibility in the current
design phase. The design of the proton absorber will be based on already
existing SPS internal dumps. The first achromatic bending section bends
particles in the momentum bite of the beamline towards the ring and far from
the proton absorber. A quadrupole FODO lattice is used to transport the beam
to the second achromatic bending section which is followed by a beta-function
matching before injecting the beam into the ring. The length of this section
is short enough to transport also low momentum pions.

The nuSTORM decay ring is a storage ring with a circumference of $\sim$616 m
made by large apertures magnets (see Fig. 12). In addition, strong bending
magnets are used in the arcs to minimize the arc length and maximize the
number of useful muon decays. The nuSTORM beamline was recently redesigned to
serve the neutrino-scattering program and store muons in the 1-6 GeV momentum
range with an acceptance of $\pm$16% to increase the neutrino flux. A hybrid
concept was then adopted in order to simultaneously have large momentum
acceptance and high muon accumulation efficiency: the conventional FODO optics
of the production straight section are combined with Fixed-Field Alternating
gradient cells (FFA) with zero chromaticity in the arcs and the return
straight.

Figure 12: Schematic drawing of the hybrid concept adopted for the nuSTORM
muon storage ring. The beam circulates in an anticlockwise direction. The
production straight (at z$\sim$30 m) is composed of large aperture quadrupoles
that produce the large values of the betatron function required to minimize
the divergence of the neutrino beam produced in muon decay. The lattices of
the arcs and return straight are based on the FFA concept and allow a large
dynamic aperture to be maintained.

The magnets employed in different sections are: superconducting combined-
function magnets with fields up to $\sim$2.6 T in the arcs, combined-function
room temperature magnets in the return straight and large aperture room
temperature quadrupoles in the production straight. In both the production and
return straight the mean betatron function contribution to the angular spread
of the neutrino beam is minimized so that both can be used to serve a neutrino
physics program. Finally, the arcs are connected with the injection and return
straights using specific matching sections. At the end of the production
straight undecayed pions and muons outside the selected momentum will be
directed to a dump. The Near Detector is located 50 m downstream of the dump
in a shallow experimental hall. In the CERN implementation of nuSTORM, the
neutrino beam is directed to LHC Point 2 where it is possible to install an
additional neutrino detector at a larger distance (see Fig. 13).

Figure 13: Overview of the implementation of nuSTORM at CERN. The nuSTORM ring
and associated infrastructure is shown in the right panel. The possible
location for a far detector is shown in the left panel.

To evaluate the performance of the hybrid design for the storage ring the
nuSTORM collaboration performed tracking studies that showed that the
dynamical acceptance of the machine is about 1 $\pi$ mm rad in both transverse
planes matching the requirements of the experimental program. If implemented
at CERN, an upgrade of the present extraction kickers is needed to work with
two 10.5 $\mu$s proton pulses, and approximately 165 kW of beam power will be
delivered in a cycle of 3.6 s corresponding to a total intensity of $4\times
10^{13}$ per cycle.

The feasibility of executing nuSTORM at CERN is carried out in the framework
of Physics Beyond Colliders, which has highlighted interesting synergies with
the ENUBET project. In particular, three main common areas have been
identified: the target facility, the first stage of the meson focusing, and
the proton dump. The two collaborations are currently evaluating this possible
synergy at the facility level.

##  5 A new generation of neutrino-beam detectors

High-precision neutrino beams, as the NP06/ENUBET and nuSTORM facilities
described in the previous section, allow to disentangle the flux term and to
reduce the corresponding systematics in Eq. 2. In order to retrieve precise
information on the other two terms, the detector efficiency and the neutrino
cross section, a facility based on different neutrino detection techniques is
highly desirable. In particular a program able to boost the physics reach of
DUNE and HK by studying neutrino interactions in Argon and water, but at the
same time capable of providing high precision data on neutrino interactions
with low-Z targets should be envisaged.

Liquid argon TPC prototypes of moderate mass (protoDUNE-SP and protoDUNE-DP
Agostino et al. (2014)) have been installed and operated with charged particle
beams in EHN1 at CERN, with the main goal of demonstrating the detector
technology of the DUNE far detector. If protoDUNE-DP is based on a more
innovative concept, with a gaseous argon layer above the liquid phase used to
amplify drifting charges for a better signal-to-noise ratio, protoDUNE-SP
stands on the single-phase liquid argon technology, successfully operated by
ICARUS Amerio et al. (2004) and MicroBooNE Acciarri et al. (2017) and used by
the upcoming SBND detector McConkey (2017), in the Fermilab Short-Baseline
Neutrino program, and it has already achieved excellent performance Abi et al.
(2020).

The active volume of protoDUNE-SP extends for 7 m along the beam direction,
7.2 m in the drift direction and 6.1 m in height, for a total mass of about
400 tons (Fig. 14). The detector is divided into two drift regions by a
cathode plane in the middle and the readout is performed on each side by sense
wires placed on anode planes. The anode planes embed photon detectors to
collect scintillation light from ionized LAr used to provide the absolute
timing of the neutrino interaction and an independent measurement of the
deposited energy.

Figure 14: Left: Overview of the protoDUNE-SP detector. Right: a photo of one
of the two drift volumes with the anode plane embedding sense wires and
photodetection systems on the left and the cathode plane on the right.

Unlike the liquid argon detector of the DUNE ND complex (ND-LAr Abud et al.
(2021) \- see Sec. 3), that is composed by 35 small optically separated TPC
modules to withstand the large neutrino interaction rate at the near location,
the large size of the protoDUNE TPC allows for an almost full containment of
the neutrino interaction. Moreover, it is based on the same components and has
the same maximum drift length as the first DUNE far detector module. These
features make protoDUNE-SP an ideal detector for the precise determination of
the $\sigma(E)\times\epsilon(E)$ product for DUNE, provided that the neutrino
flux is known at the percent level, like in ENUBET and nuSTORM.

To decouple the neutrino cross section in argon from the detector efficiency,
the simultaneous use of a liquid and gas phase TPC would be an ideal solution.
Atmospheric pressure argon TPCs have been used in neutrino experiments, like
the ND280 near detector of T2K and its upcoming upgrade Abe et al. (2019). A
large volume TPC Dellacasa et al. (2000) is used in the ALICE experiment at
LHC. The TPC employed in the ND-GAr detector Abud et al. (2021) of the ND
complex of DUNE will be based on the same design of ALICE, but will be
operated at a ten times bigger pressure to enhance the amount of target nuclei
in the volume and, hence, the neutrino event rate. The active volume is
divided into two parts by a central cathode and has a radius of 2.6 m and a
length of 5 m, for a total mass of 1 ton. The endplates are instrumented with
chambers equipped with MWPCs and pad planes.

The main advantages of a high pressure gaseous TPC with respect to a liquid
argon TPC are the superior momentum resolution given by track bending in the
TPC magnetic field, a better particle ID, particularly for proton-pion
separation, and a significant lower threshold that allows the reconstruction
of low-kinetic-energy protons and pions. The full reconstruction of the
hadronic system is of paramount importance for the evaluation of final state
interactions and for neutrino-nucleus interaction modeling. As an example, ND-
GAr will be able to reconstruct protons with a kinetic energy above 20 MeV
with an efficiency exceeding 90% and preliminary studies of a machine learning
based reconstruction show that the threshold could be pushed further down to 5
MeV Abud et al. (2021).

Concerning the cross section measurement in water, an interesting opportunity
is represented by the proposed Water Cherenkov Test Beam Experiment (WCTE)
Barbi et al. (2020) that will be exposed to charged particle beams at CERN in
2022 or 2023. The setup is based on a Water Cherenkov tank of 4 m height and
4.1 m radius, for a total mass of $\sim$50 tons and the light will be
collected by 19 small (8 cm diamter) photomultipliers arranged in 128 multi-
PMT optical modules (Fig. 15). The photosensor size is driven by the much
smaller dimensions of the detector: Cherenkov rings will cover a reduced area
and the event reconstruction would be deteriorated by using the same 50 cm
diameter PMTs as SK and HK. The same arrangement will be used in the
Intermediate Water Cherenkov Detector (IWCD) of HK.

Figure 15: Left: Cross section drawing of WCTE water tank with the multi-PMT
layout inside the detector. Right: Top view of a multi-PMT prototype.

Like protoDUNE, a neutrino exposure of the WCTE tank would profit of the
characterization of the detector response by charged particles to be performed
at the CERN East Area in 2022-23. Moreover, the detector technology and the
event reconstruction are very similar to those of the HK far detector and
allow for a reduction in the systematic uncertainties budget in cross section
and oscillation analyses. Given the small fiducial mass of the detector we do
not expect to perform a statistically significant measurement with $\nu_{e}$
in a moderate intensity beam like ENUBET, but $\nu_{\mu}^{CC}$ interactions
will provide a large sample for the disentanglement of detector effects and
double-differential cross section measurements. The detector size also limits
the muon containment and a downstream spectrometer must be envisaged for
momentum measurements.

Disentangling the neutrino cross section in water from the detector efficiency
can be accomplished by using fine-grained detectors with water targets. The
WAGASCI experiment Pintaudi (2019) exploits a 0.6 ton water target encompassed
by a 3D grid-like structure of plastic scintillator strips enclosing cells of
$\mathcal{O}(cm)$ linear size. The setup is complemented by two side modules
with steel plates interleaved with scintillator slabs and a downstream
magnetised detector equipped with plastic scinitillators (BabyMIND) for muon
momentum measurement. An alternative option has been developed by the NINJA
collaboration Hiramoto et al. (2020): nuclear emulsion films and iron plates
(500 $\mu$m thick) are interleaved in a sandwich like structure with 2 mm
thick water layers and a downstream scintillating fiber tracker is used to
timestamp and match tracks in the emulsions. With this setup the momentum
threshold of hadrons generated in the neutrino interaction can be pushed down
to 200 MeV/c and 50 MeV/c for protons and pions respectively.

The precision measurement of neutrino scattering off hydrogen and deuterium
(the lowest Z isoscalar nucleus) is pivotal to build reliable cross section
models not affected by nuclear effects, that can be extrapolated to higher Z
materials, and to perform detailed studies on the nucleon structure exploiting
a bare weak probe. Indirect approaches to measure the $\nu$-H cross sections
in hydrocarbon targets exploiting the transverse momentum imbalance or
subtracting measurements on thin dedicated graphite targets have been already
discussed in Sec. 3. A straightforward way to perform the measurement is the
direct use of a liquid hydrogen/deuterium target. In the past, bubble chambers
were extensively used in this way: the Argonne 12’ bubble chamber with few GeV
neutrinos on H and D Barish et al. (1979), the series of experiments performed
at higher energies at the Big European Bubble Chamber (e.g. Allasia et al.
(1990)), and the Fermilab 15’ bubble chamber Kitagaki et al. (1983). The
proponents of Alvarez-Ruso et al. (2020) suggest to re-use the Fermilab
chamber for an upgraded experiment with liquid hydrogen that could profit of
modern digital camera technology and machine-assisted reconstruction
techniques to improve precision and data analysis speed. On the other hand,
the statistics is limited by the slow operation cycle of the chamber, and
auxiliary detectors are needed for event containment due to the long radiation
and interaction length of hydrogen.

##  6 A graded strategy toward high-precision neutrino physics

At the conclusion of this paper, we recap the main motivations and advances
toward a new generation of cross section experiments. The results shown in
Sec. 2 testify for the need of high-precision experiments:

  * •

even in the most precise measurements (see for instance Fig. 1) the flux
uncertainty contribute to a large fraction of the systematic uncertainty.
Double-differential cross section measurements are not yet systematically
limited but the DUNE and HK near detectors of Sec. 3 will soon reach this
limit. A per-cent knowledge of the flux is, then, mandatory to reduce the
systematic uncertainty once statistical errors will be lower than 10%. The use
of standard candles mitigates this issue but the most desirable solution is a
flux measurement independent of the neutrino detector.

  * •

the use of non-monochromatic beams is the root of the model dependence of
cross section results. Major advances have been achieved once experimenters
became aware of the theory priors implicit in the definition of QE, RES, and
DIS events (see for instance Sec. 2.3). Still, the use of a narrow band beam
is mandatory to decouple interaction modeling from experimental data and
reduce the systematic budget due to the energy-integrated flux and the bias
coming from energy reconstruction. The PRISM technique is an important
mitigation tool. Once more, the ideal solution is a narrow band beam where the
neutrino energy is measured with $\simeq 10$% precision independently from the
detector reconstruction.

  * •

the decoupling of cross-section and detector efficiency is a major
experimental challenge. It is not strictly necessary for long-baseline
experiments but it plagues the interpretation of experimental data for model
building. Decoupling can be achieved by combining high- and low-density
detectors using the same nuclei. On the other hand, the construction of
realistic models of neutrino interactions requires a deeper understanding of
low-Z nuclear effects. The use of hydrogen or deuterated materials would be a
major asset for electroweak nuclear physics and the study of nuclear media
and, in turn, would impact the precision of oscillation measurements.

  * •

the construction of an intense $\nu_{e}\ $beam is extremely valuable to test
lepton universality and ground the $\nu_{\mu}\rightarrow\nu_{e}\,$ and
$\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e}\,$oscillation measurements.
Beams like ENUBET can provide $\nu_{e}\ $measurements at the per-cent level,
which fulfill the needs of DUNE and HK, but double-differential $\nu_{e}\ $CC
measurements require novel muon beams like nuSTORM.

The technology for high-precision facilities is well-grounded for conventional
beams like ENUBET, while requires a significant R& D for muon beams.
Similarly, neutrino detectors in liquid and gas phase are available for argon
and can be envisaged for water. On the other hand, low-Z measurements are the
most direct path toward the understanding of nuclear effects, and low-Z
detectors should be reconsidered in a modern perspective.

While the R&D for high-precision facilities has achieved remarkable results, a
complete site-dependent study is still lacking. It is fortunate that the beam
power needed for these facilities is well within the reach of existing proton
synchrotrons at CERN, FNAL, and JPARC. A site-dependent study should include a
careful evaluation of the impact of a high-precision neutrino beam in the
laboratories hosting the proton driver, the corresponding infrastructures, and
the engineering assessment, which are mandatory for a reliable estimate of
costs. At the time of writing, the most advanced site-dependent study is the
one being pursued at CERN for ENUBET and nuSTORM. The CERN infrastructure,
accelerators, and experimental halls, like EHN1 serving ProtoDUNE-SP and
ProtoDUNE-DP, are ideal to host such a facility and, in a timescale comparable
with DUNE and HK, CERN has the opportunity to be the main hub for precision
neutrino physics. The laboratories hosting LBNF for DUNE (FNAL) and the HK
neutrino beam (JPARC) are natural candidates too. A site-dependent study for
FNAL and JPARC is highly desirable since precision experiments do not
interfere with the running of long baseline experiments given the limited
power needed for short baseline neutrino detectors. The ideas and proposals
discussed in this paper are currently under investigation at CERN in the
framework of Physics Beyond collider and in the US within the Snowmass
community planning exercise sno

Secs. 1 and 6 were drafted by A.B., A.L., and F.T. and reviewed by all
authors. G.B. and F.P. contributed to Secs. 4 and 5, respectively. Secs. 2 and
3 were drafted by A.B., M.M., and F.T. Furthermore, M.M. reviewed the status
of model building and theory challenges for the whole paper. All authors
contributed to the final revision of the manuscript.

This work has been supported by the European Union’s Horizon 2020 Research and
Innovation Programme under Grant Agreement no. 681647 and no. 777419, and by
the Italian Ministry for Education and Research through NUTECH (MIUR, bando
FARE, progetto NUTECH) and 2017-NAZ-0444 (MIUR bando PRIN2017, CUP
H45J17000460006). Tagged neutrino beam studies are also supported by the Dep.
of Physics of the Univ. of Milano-Bicocca under the 2018-CONT-0128 project.

###### Acknowledgements.

We are grateful to our colleagues of the DUNE, ENUBET, HK, and T2K
collaborations. Special thanks go to A. Bross, N. Charitonidis, L. Ludovici,
and M. Mezzetto for many enlightening discussions on the challenges of
neutrino cross section measurements. The authors declare no conflict of
interest. no References yes

## References

  * Mezzetto and Terranova (2020) Mezzetto, M.; Terranova, F.  Three-flavour oscillations with accelerator neutrino beam.  Universe 2020, 6.  doi:blackhttps://doi.org/10.3390/universe6020032.
  * Abi et al. (2020) Abi, B.; Acciarri, R.; Acero, M. et al.  Volume I. Introduction to DUNE.  J. of Instrumentation 2020, 15, T08008–T08008.  doi:black10.1088/1748-0221/15/08/t08008.
  * Abe et al. (2018) Abe, K.; Abe, K.; Aihara, H. et al.  Hyper-Kamiokande Design Report, 2018, [arXiv:physics.ins-det/1805.04163].
  * Katori and Martini (2018) Katori, T.; Martini, M.  Neutrino–nucleus cross sections for oscillation experiments.  J. Phys. G 2018, 45, 013001.  doi:black10.1088/1361-6471/aa8bf7.
  * An et al. (2012) An, F.P.; Bai, J.Z.; Balantekin, A.B. et al.  Observation of electron-antineutrino disappearance at Daya Bay.  Phys. Rev. Lett. 2012, 108, 171803.  doi:black10.1103/PhysRevLett.108.171803.
  * Ahn et al. (2012) Ahn, J.K.; Chebotaryov, S.; Choi, J.H. et al.  Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment.  Phys.Rev.Lett. 2012, 108, 191802.  doi:black10.1103/PhysRevLett.108.191802.
  * Abe et al. (2012) Abe, Y.; Aberle, C.; Akiri, T. et al.  Indication for the disappearance of reactor electron antineutrinos in the Double Chooz experiment.  Phys. Rev. Lett. 2012, 108, 131801.  doi:black10.1103/PhysRevLett.108.131801.
  * Abe et al. (2014) Abe, K.; Adam, J.; Aihara, H. et al.  Observation of Electron Neutrino Appearance in a Muon Neutrino Beam.  Phys.Rev.Lett. 2014, 112, 061802.  doi:black10.1103/PhysRevLett.112.061802.
  * Abi et al. (2020) Abi, B.; Acciarri, R.; Acero, M.A. et al.  Long-baseline neutrino oscillation physics potential of the DUNE experiment.  Eur. Phys. J. C 2020, 80, 978.  doi:black10.1140/epjc/s10052-020-08456-z.
  * Abud et al. (2021) Abud, A.A.; Abi, B.; Acciarri, R. et al.  Deep Underground Neutrino Experiment (DUNE) Near Detector Conceptual Design Report, 2021, [arXiv:physics.ins-det/2103.13910].
  * Abe et al. (2019) Abe, K.; Aihara, H.; Ajmi, A. et al.  T2K ND280 Upgrade - Technical Design Report, 2019, [arXiv:physics.ins-det/1901.03750].
  * Abi et al. (2020) Abi, B.; Acciarri, R.; Acero, M. et al.  Volume IV. The DUNE far detector single-phase technology.  JINST 2020, 15, T08010.  doi:black10.1088/1748-0221/15/08/T08010.
  * Ellis et al. (2019) Ellis, R.K.; Heinemann, B.; de Blas, J. et al.  Physics Briefing Book: Input for the European Strategy for Particle Physics Update 2020.  Technical Report CERN-ESU-004, CERN, Geneva, 2019.  254 p.
  * Atar and Singh (2020) Atar, M.S.; Singh, S.K.  The physics of neutrino interactions; Cambridge University Press, 2020.
  * Alcaraz-Aunion and Walding (2009) Alcaraz-Aunion, J.L.; Walding, J.  Measurement of the nu(mu)-CCQE cross-section in the SciBooNE experiment.  AIP Conf.Proc. 2009, 1189, 145–150.  doi:black10.1063/1.3274145.
  * Rodrigues et al. (2016) Rodrigues, P.A.; Demgen, J.; Miltenberger, E. et al.  Identification of nuclear effects in neutrino-carbon interactions at low three-momentum transfer.  Phys. Rev. Lett. 2016, 116, 071802.  doi:black10.1103/PhysRevLett.116.071802.
  * Aliaga et al. (2015) Aliaga, L.; Altinok, O.; Araujo Del Castillo, C. et al.  MINERvA neutrino detector response measured with test beam data.  Nucl. Instrum. Meth. 2015, A789, 28–42.  doi:black10.1016/j.nima.2015.04.003.
  * Pintaudi (2019) Pintaudi, G.  T2K-WAGASCI: First physics run of the WAGASCI-BabyMIND detector with full setup.  PoS 2019, Lepton-Photon 2019, 142.  doi:black10.22323/1.367.0142.
  * Acciarri et al. (2020) Acciarri, R.; Adams, C.; Asaadi, J. et al.  First measurement of electron neutrino scattering cross section on argon.  Phys. Rev. D 2020, 102, 011101.  doi:black10.1103/PhysRevD.102.011101.
  * Gran et al. (2006) Gran, R.; Jeon, E.J.; Aliu, E. et al.  Measurement of the quasi-elastic axial vector mass in neutrino-oxygen interactions.  Phys.Rev. 2006, D74, 052002.  doi:black10.1103/PhysRevD.74.052002.
  * Evans (2013) Evans, J.  The MINOS experiment: results and prospects.  Adv.High Energy Phys. 2013, 2013, 182537.  doi:black10.1155/2013/182537.
  * Abe et al. (2015) Abe, K.; Adam, J.; Aihara, H. et al.  Measurement of the $\nu_{\mu}$ charged-current quasielastic cross section on carbon with the ND280 detector at T2K.  Phys. Rev. 2015, D92, 112003.  doi:black10.1103/PhysRevD.92.112003.
  * Acero et al. (2020) Acero, M.A.; others.  Adjusting neutrino interaction models and evaluating uncertainties using NOvA near detector data.  Eur. Phys. J. C 2020, 80, 1119.  doi:black10.1140/epjc/s10052-020-08577-5.
  * Aguilar-Arevalo et al. (2013) Aguilar-Arevalo, A.A.; Brown, B.C.; Bugel, L. et al.  First measurement of the muon antineutrino double-differential charged-current quasielastic cross section.  Phys. Rev. D 2013, 88, 032001.  doi:black10.1103/PhysRevD.88.032001.
  * Abratenko et al. (2020) Abratenko, P.; Alrashed, M.; An, R. et al.  First Measurement of Differential Charged Current Quasielastic-like $\nu_{\mu}$-Argon Scattering Cross Sections with the MicroBooNE Detector.  Phys. Rev. Lett. 2020, 125, 201803.  doi:black10.1103/PhysRevLett.125.201803.
  * Dracos et al. (2020) Dracos, M.; Baussan, E.; Bouquerel, E. et al.  The ESS$\nu$SB Project.  Proceedings of The 21st international workshop on neutrinos from accelerators — PoS(NuFact2019), 2020, Vol. 369, p. 024.  doi:black10.22323/1.369.0024.
  * Alekou et al. (2021) Alekou, A.; Baussan, E.; Kraljevic, N.B. et al.  Updated physics performance of the ESSnuSB experiment, 2021, [arXiv:hep-ex/2107.07585].
  * Acerbi et al. (2018) Acerbi, F.; Ballerini, G.; Bonesini, M. et al.  The ENUBET project.  Technical Report CERN-SPSC-2018-034. SPSC-I-248, CERN, Geneva, 2018.
  * Adey et al. (2013) Adey, D.; Agarwalla, S.K.; Ankenbrandt, C.M. et al.  nuSTORM - Neutrinos from STORed Muons: Proposal to the Fermilab PAC, 2013, [arXiv:physics.acc-ph/1308.6822].
  * Huber et al. (2008) Huber, P.; Mezzetto, M.; Schwetz, T.  On the impact of systematical uncertainties for the CP violation measurement in superbeam experiments.  Journal of High Energy Physics 2008, 2008, 021–021.  doi:black10.1088/1126-6708/2008/03/021.
  * Mishra et al. (1990) Mishra, S.; Bachmann, K.; Blair, R. et al.  Inverse Muon Decay, $\nu_{\mu}e\to\mu^{-}\nu_{e}$, at the Fermilab Tevatron.  Phys. Lett. B 1990, 252, 170–176.  doi:black10.1016/0370-2693(90)91099-W.
  * Dorenbosch et al. (1989) Dorenbosch, J.; Udo, F.; Allaby, J.V. et al.  EXPERIMENTAL RESULTS ON NEUTRINO - ELECTRON SCATTERING.  Z. Phys. C 1989, 41, 567.  [Erratum: Z.Phys.C 51, 142 (1991)], doi:black10.1007/BF01564701.
  * Marshall et al. (2020) Marshall, C.M.; McFarland, K.S.; Wilkinson, C.  Neutrino-electron elastic scattering for flux determination at the DUNE oscillation experiment.  Phys. Rev. D 2020, 101, 032002.  doi:black10.1103/PhysRevD.101.032002.
  * Valencia et al. (2019) Valencia, E.; Jena, D.; Nuruzzaman. et al.  Constraint of the $\mathrm{MINER}\nu$A medium energy neutrino flux using neutrino-electron elastic scattering.  Phys. Rev. D 2019, 100, 092001.  doi:black10.1103/PhysRevD.100.092001.
  * Kronfeld et al. (2019) Kronfeld, A.S.; Richards, D.G.; Detmold, W. et al.  Lattice QCD and Neutrino-Nucleus Scattering.  Eur. Phys. J. A 2019, 55, 196.  doi:black10.1140/epja/i2019-12916-x.
  * Miller et al. (1982) Miller, L.; Barish, S.J.; Engler, A. et al.  STUDY OF THE REACTION muon-neutrino D —$>$ MU- P P(S).  Phys. Rev. D 1982, 26, 537–542.  doi:black10.1103/PhysRevD.26.537.
  * Baker et al. (1981) Baker, N.J.; Cnops, A.M.; Connolly, P.L. et al.  Quasielastic Neutrino Scattering: A Measurement of the Weak Nucleon Axial Vector Form-Factor.  Phys. Rev. 1981, D23, 2499–2505.  doi:black10.1103/PhysRevD.23.2499.
  * Kitagaki et al. (1983) Kitagaki, T.; Tanaka, S.; Yuta, H. et al.  High-Energy Quasielastic Muon-neutrino n —> mu- p Scattering in Deuterium.  Phys. Rev. 1983, D28, 436–442.  doi:black10.1103/PhysRevD.28.436.
  * Baussan et al. (2014) Baussan, E.; Blennow, M.; Bogomilov, M. et al.  A very intense neutrino super beam experiment for leptonic CP violation discovery based on the European spallation source linac.  Nuclear Physics B 2014, 885, 127 – 149.  doi:blackhttps://doi.org/10.1016/j.nuclphysb.2014.05.016.
  * Zyla et al. (2020) Zyla, P.A.; Barnett, R.M.; Beringer, J. et al.  Review of Particle Physics.  PTEP 2020, 2020, 083C01.  doi:black10.1093/ptep/ptaa104.
  * Bernard et al. (2002) Bernard, V.; Elouadrhiri, L.; Meissner, U.G.  Axial structure of the nucleon: Topical Review.  J. Phys. 2002, G28, R1–R35.  doi:black10.1088/0954-3899/28/1/201.
  * Mann et al. (1973) Mann, W.A.; Mehtani, U.; Musgrave, B. et al.  Study of the reaction nu n —> mu- p.  Phys. Rev. Lett. 1973, 31, 844–847.  doi:black10.1103/PhysRevLett.31.844.
  * Barish et al. (1977) Barish, S.J.; Campbell, J.; Charlton, G. et al.  Study of Neutrino Interactions in Hydrogen and Deuterium. 1. Description of the Experiment and Study of the Reaction Neutrino d –> mu- p p(s).  Phys. Rev. 1977, D16, 3103.  doi:black10.1103/PhysRevD.16.3103.
  * Lyubushkin et al. (2009) Lyubushkin, V.; Popov, B.; Kim, J.J. et al.  A Study of quasi-elastic muon neutrino and antineutrino scattering in the NOMAD experiment.  Eur. Phys. J. 2009, C63, 355–381.  doi:black10.1140/epjc/s10052-009-1113-0.
  * Nakajima et al. (2011) Nakajima, Y.; Alcaraz-Aunion, J.L.; Brice, S.J. et al.  Measurement of inclusive charged current interactions on carbon in a few-GeV neutrino beam.  Phys.Rev. 2011, D83, 012005.  doi:black10.1103/PhysRevD.83.012005.
  * Aguilar-Arevalo et al. (2010) Aguilar-Arevalo, A.A.; Anderson, C.E.; Bazarko, A.O. et al.  First Measurement of the Muon Neutrino Charged Current Quasielastic Double Differential Cross Section.  Phys.Rev. 2010, D81, 092005.  doi:black10.1103/PhysRevD.81.092005.
  * Alvarez-Ruso (2012) Alvarez-Ruso, L.  Neutrino interactions: challenges in the current theoretical picture.  Nucl. Phys. Proc. Suppl. 2012, 229-232, 167–173.  doi:black10.1016/j.nuclphysbps.2012.09.027.
  * Martini et al. (2009) Martini, M.; Ericson, M.; Chanfray, G. et al.  A Unified approach for nucleon knock-out, coherent and incoherent pion production in neutrino interactions with nuclei.  Phys.Rev. 2009, C80, 065501.  doi:black10.1103/PhysRevC.80.065501.
  * Abe et al. (2016) Abe, K.; Andreopoulos, C.; Antonova, M. et al.  Measurement of double-differential muon neutrino charged-current interactions on C8H8 without pions in the final state using the T2K off-axis beam.  Phys. Rev. 2016, D93, 112012.  doi:black10.1103/PhysRevD.93.112012.
  * Ruterbories et al. (2019) Ruterbories, D.; Hurtado, K.; Osta, J. et al.  Measurement of Quasielastic-Like Neutrino Scattering at $\langle{E_{\nu}}\rangle\sim 3.5$ GeV on a Hydrocarbon Target.  Phys. Rev. D 2019, 99, 012004.  doi:black10.1103/PhysRevD.99.012004.
  * Abratenko et al. (2020) Abratenko, P.; Alrashed, M.; An, R. et al.  Measurement of differential cross sections for $\nu_{\mu}$ -Ar charged-current interactions with protons and no pions in the final state with the MicroBooNE detector.  Phys. Rev. D 2020, 102, 112013.  doi:black10.1103/PhysRevD.102.112013.
  * Martini et al. (2010) Martini, M.; Ericson, M.; Chanfray, G. et al.  Neutrino and antineutrino quasielastic interactions with nuclei.  Phys.Rev. 2010, C81, 045502.  doi:black10.1103/PhysRevC.81.045502.
  * Nieves et al. (2012a) Nieves, J.; Ruiz Simo, I.; Vicente Vacas, M.  The nucleon axial mass and the MiniBooNE Quasielastic Neutrino-Nucleus Scattering problem.  Phys.Lett. 2012, B707, 72–75.  doi:black10.1016/j.physletb.2011.11.061.
  * Nieves et al. (2012b) Nieves, J.; Sánchez, F.; Simo, I.R. et al.  Neutrino energy reconstruction and the shape of the charged current quasielastic-like total cross section.  Phys. Rev. D 2012, 85, 113008.  doi:black10.1103/PhysRevD.85.113008.
  * Alvarez-Ruso et al. (2018) Alvarez-Ruso, L.; Athar, M.S.; Barbaro, M.B. et al.  NuSTEC White Paper: Status and challenges of neutrino–nucleus scattering.  Prog. Part. Nucl. Phys. 2018, 100, 1–68.  doi:black10.1016/j.ppnp.2018.01.006.
  * Rein and Sehgal (1983) Rein, D.; Sehgal, L.M.  Coherent pi0 Production in Neutrino Reactions.  Nucl. Phys. 1983, B223, 29.  doi:black10.1016/0550-3213(83)90090-1.
  * Abe et al. (2015) Abe, K.; Adam, J.; Aihara, H. et al.  Measurement of the electron neutrino charged-current interaction rate on water with the T2K ND280 $\pi^{0}$ detector.  Phys.Rev. 2015, D91, 112010.  doi:black10.1103/PhysRevD.91.112010.
  * Abe et al. (2014) Abe, K.; Adam, J.; Aihara, H. et al.  Measurement of the Inclusive Electron Neutrino Charged Current Cross Section on Carbon with the T2K Near Detector.  Phys. Rev. Lett. 2014, 113, 241803.  doi:black10.1103/PhysRevLett.113.241803.
  * Wolcott et al. (2016) Wolcott, J.; Aliaga, L.; Altinok, O. et al.  Measurement of electron neutrino quasielastic and quasielasticlike scattering on hydrocarbon at $\langle E_{\nu}\rangle$ = 3.6 GeV.  Phys. Rev. Lett. 2016, 116, 081802.  doi:black10.1103/PhysRevLett.116.081802.
  * Abe et al. (2020) Abe, K.; Akhlaq, N.; Akutsu, R. et al.  Measurement of the charged-current electron (anti-)neutrino inclusive cross-sections at the T2K off-axis near detector ND280.  JHEP 2020, 10, 114.  doi:black10.1007/JHEP10(2020)114.
  * Abratenko et al. (2021) Abratenko, P.; Alrashed, M.; An, R. et al.  Measurement of the Flux-Averaged Inclusive Charged-Current Electron Neutrino and Antineutrino Cross Section on Argon using the NuMI Beam and the MicroBooNE Detector, 2021, [arXiv:hep-ex/2101.04228].
  * Day and McFarland (2012) Day, M.; McFarland, K.S.  Differences in Quasi-Elastic Cross-Sections of Muon and Electron Neutrinos.  Phys. Rev. 2012, D86, 053003.  doi:black10.1103/PhysRevD.86.053003.
  * Fernandez-Martinez and Meloni (2011) Fernandez-Martinez, E.; Meloni, D.  Importance of nuclear effects in the measurement of neutrino oscillation parameters.  Phys. Lett. 2011, B697, 477–481.  doi:black10.1016/j.physletb.2011.02.043.
  * Charitonidis et al. (2021) Charitonidis, N.; Longhin, A.; Pari, M. et al.  Design and Diagnostics of High-Precision Accelerator Neutrino Beams.  Appl. Sciences 2021, 11, 1644.  doi:black10.3390/app11041644.
  * Martini et al. (2012) Martini, M.; Ericson, M.; Chanfray, G.  Neutrino energy reconstruction problems and neutrino oscillations.  Phys.Rev. 2012, D85, 093012.  doi:black10.1103/PhysRevD.85.093012.
  * Martini et al. (2013) Martini, M.; Ericson, M.; Chanfray, G.  Energy reconstruction effects in neutrino oscillation experiments and implications for the analysis.  Phys. Rev. 2013, D87, 013009.  doi:black10.1103/PhysRevD.87.013009.
  * Lalakulich et al. (2012) Lalakulich, O.; Mosel, U.; Gallmeister, K.  Energy reconstruction in quasielastic scattering in the MiniBooNE and T2K experiments.  Phys. Rev. 2012, C86, 054606.  doi:black10.1103/PhysRevC.86.054606.
  * Ankowski and Mariani (2017) Ankowski, A.M.; Mariani, C.  Systematic uncertainties in long-baseline neutrino-oscillation experiments.  J. Phys. 2017, G44, 054001.  doi:black10.1088/1361-6471/aa61b2.
  * Martini et al. (2011) Martini, M.; Ericson, M.; Chanfray, G.  Neutrino quasielastic interaction and nuclear dynamics.  Phys.Rev. 2011, C84, 055502.  doi:black10.1103/PhysRevC.84.055502.
  * Martini and Ericson (2013) Martini, M.; Ericson, M.  Quasielastic and multinucleon excitations in antineutrino-nucleus interactions.  Phys.Rev. 2013, C87, 065501.  doi:black10.1103/PhysRevC.87.065501.
  * Martini and Ericson (2014) Martini, M.; Ericson, M.  Inclusive and pion production neutrino-nucleus cross sections.  Phys. Rev. 2014, C90, 025501.  doi:black10.1103/PhysRevC.90.025501.
  * Martini et al. (2016) Martini, M.; Jachowicz, N.; Ericson, M. et al.  Electron-neutrino scattering off nuclei from two different theoretical perspectives.  Phys. Rev. 2016, C94, 015501.  doi:black10.1103/PhysRevC.94.015501.
  * Nieves et al. (2013) Nieves, J.; Ruiz Simo, I.; Vicente Vacas, M.  Two Particle-Hole Excitations in Charged Current Quasielastic Antineutrino–Nucleus Scattering.  Phys.Lett. 2013, B721, 90–93.  doi:black10.1016/j.physletb.2013.03.002.
  * Bourguille et al. (2021) Bourguille, B.; Nieves, J.; Sánchez, F.  Inclusive and exclusive neutrino-nucleus cross sections and the reconstruction of the interaction kinematics.  JHEP 2021, 04, 004.  doi:black10.1007/JHEP04(2021)004.
  * Megias et al. (2016) Megias, G.; Amaro, J.; Barbaro, M. et al.  Charged-current neutrino-nucleus reactions within the superscaling meson-exchange current approach.  Phys. Rev. 2016, D94, 093004.  doi:black10.1103/PhysRevD.94.093004.
  * Megias et al. (2019a) Megias, G.D.; Barbaro, M.B.; Caballero, J.A. et al.  Neutrino-Oxygen CC0$\pi$ scattering in the SuSAv2-MEC model.  J. Phys. G 2019, 46, 015104.  doi:black10.1088/1361-6471/aaf3ae.
  * Megias et al. (2019b) Megias, G.D.; Barbaro, M.B.; Caballero, J.A. et al.  Analysis of the MINERvA antineutrino double-differential cross sections within the SuSAv2 model including meson-exchange currents.  Phys. Rev. D 2019, 99, 113002.  doi:black10.1103/PhysRevD.99.113002.
  * Ivanov et al. (2019) Ivanov, M.V.; Antonov, A.N.; Megias, G.D. et al.  Realistic spectral function model for charged-current quasielastic-like neutrino and antineutrino scattering cross sections on 12C.  Phys. Rev. C 2019, 99, 014610.  doi:black10.1103/PhysRevC.99.014610.
  * Barbaro et al. (2021) Barbaro, M.B.; De Pace, A.; Fiume, L.  The SuSA Model for Neutrino Oscillation Experiments: From Quasielastic Scattering to the Resonance Region.  Universe 2021, 7, 140.  doi:black10.3390/universe7050140.
  * Lalakulich et al. (2012) Lalakulich, O.; Gallmeister, K.; Mosel, U.  Many-Body Interactions of Neutrinos with Nuclei - Observables.  Phys.Rev. 2012, C86, 014614.  doi:black10.1103/PhysRevC.86.014614.
  * Gallmeister et al. (2016) Gallmeister, K.; Mosel, U.; Weil, J.  Neutrino-Induced Reactions on Nuclei.  Phys. Rev. 2016, C94, 035502.  doi:black10.1103/PhysRevC.94.035502.
  * Mosel and Gallmeister (2018) Mosel, U.; Gallmeister, K.  Muon-neutrino-induced charged current cross section without pions: Theoretical analysis.  Phys. Rev. C 2018, 97, 045501.  doi:black10.1103/PhysRevC.97.045501.
  * Pandey et al. (2015) Pandey, V.; Jachowicz, N.; Van Cuyck, T. et al.  Low-energy excitations and quasielastic contribution to electron-nucleus and neutrino-nucleus scattering in the continuum random-phase approximation.  Phys. Rev. 2015, C92, 024606.  doi:black10.1103/PhysRevC.92.024606.
  * Van Cuyck et al. (2016) Van Cuyck, T.; Jachowicz, N.; González-Jiménez, R. et al.  Influence of short-range correlations in neutrino-nucleus scattering.  Phys. Rev. 2016, C94, 024611.  doi:black10.1103/PhysRevC.94.024611.
  * Van Cuyck et al. (2017) Van Cuyck, T.; Jachowicz, N.; González-Jiménez, R. et al.  Seagull and pion-in-flight currents in neutrino-induced $1N$ and $2N$ knockout.  Phys. Rev. C 2017, 95, 054611.  doi:black10.1103/PhysRevC.95.054611.
  * Lovato et al. (2020) Lovato, A.; Carlson, J.; Gandolfi, S. et al.  Ab initio study of $\boldsymbol{(\nu_{\ell},\ell^{-})}$ and $\boldsymbol{(\overline{\nu}_{\ell},\ell^{+})}$ inclusive scattering in 12C: confronting the MiniBooNE and T2K CCQE data.  Phys. Rev. X 2020, 10, 031068.  doi:black10.1103/PhysRevX.10.031068.
  * Aguilar-Arevalo et al. (2013) Aguilar-Arevalo, A.A.; Brown, B.C.; Bugel, L. et al.  First measurement of the muon antineutrino double-differential charged-current quasielastic cross section.  Phys.Rev. 2013, D88, 032001.  doi:black10.1103/PhysRevD.88.032001.
  * Abe et al. (2018) Abe, K.; Amey, J.; Andreopoulos, C. et al.  First measurement of the $\nu_{\mu}$ charged-current cross section on a water target without pions in the final state.  Phys. Rev. D 2018, 97, 012001.  doi:black10.1103/PhysRevD.97.012001.
  * Abe et al. (2020a) Abe, K.; Akutsu, R.; Ali, A. et al.  First measurement of the charged current $\overline{\nu}_{\mu}$ double differential cross section on a water target without pions in the final state.  Phys. Rev. D 2020, 102, 012007.  doi:black10.1103/PhysRevD.102.012007.
  * Abe et al. (2020b) Abe, K.; Akhlaq, N.; Akutsu, R. et al.  First combined measurement of the muon neutrino and antineutrino charged-current cross section without pions in the final state at T2K.  Phys. Rev. D 2020, 101, 112001.  doi:black10.1103/PhysRevD.101.112001.
  * Abe et al. (2020c) Abe, K.; Akhlaq, N.; Akutsu, R. et al.  Simultaneous measurement of the muon neutrino charged-current cross section on oxygen and carbon without pions in the final state at T2K.  Phys. Rev. D 2020, 101, 112004.  doi:black10.1103/PhysRevD.101.112004.
  * Carneiro et al. (2020) Carneiro, M.F.; Ruterbories, D.; Ahmad Dar, Z. et al.  High-Statistics Measurement of Neutrino Quasielasticlike Scattering at 6 GeV on a Hydrocarbon Target.  Phys. Rev. Lett. 2020, 124, 121801.  doi:black10.1103/PhysRevLett.124.121801.
  * Nieves et al. (2011) Nieves, J.; Ruiz Simo, I.; Vicente Vacas, M.  Inclusive Charged–Current Neutrino–Nucleus Reactions.  Phys.Rev. 2011, C83, 045501.  doi:black10.1103/PhysRevC.83.045501.
  * Andreopoulos et al. (2010) Andreopoulos, C.; Bell, A.; Bhattacharya, D. et al.  The GENIE Neutrino Monte Carlo Generator.  Nucl.Instrum.Meth. 2010, A614, 87–104.  doi:black10.1016/j.nima.2009.12.009.
  * Hayato (2009) Hayato, Y.  A neutrino interaction simulation program library NEUT.  Acta Phys.Polon. 2009, B40, 2477–2489.
  * Golan et al. (2012) Golan, T.; Juszczak, C.; Sobczyk, J.T.  Final State Interactions Effects in Neutrino-Nucleus Interactions.  Phys. Rev. 2012, C86, 015505.  doi:black10.1103/PhysRevC.86.015505.
  * Abe et al. (2018) Abe, K.; Amey, J.; Andreopoulos, C. et al.  Measurement of inclusive double-differential $\nu_{\mu}$ charged-current cross section with improved acceptance in the T2K off-axis near detector.  Phys. Rev. D 2018, 98, 012004.  doi:black10.1103/PhysRevD.98.012004.
  * Abe et al. (2013) Abe, K.; Abgrall, N.; Aihara, H. et al.  Measurement of the inclusive $\nu_{\mu}$ charged current cross section on carbon in the near detector of the T2K experiment.  Phys.Rev. 2013, D87, 092003.  doi:black10.1103/PhysRevD.87.092003.
  * Ankowski (2017) Ankowski, A.M.  Effect of the charged-lepton’s mass on the quasielastic neutrino cross sections.  Phys. Rev. C 2017, 96, 035501.  doi:black10.1103/PhysRevC.96.035501.
  * Nikolakopoulos et al. (2019) Nikolakopoulos, A.; Jachowicz, N.; Van Dessel, N. et al.  Electron versus Muon Neutrino Induced Cross Sections in Charged Current Quasielastic Processes.  Phys. Rev. Lett. 2019, 123, 052501.  doi:black10.1103/PhysRevLett.123.052501.
  * González-Jiménez et al. (2019) González-Jiménez, R.; Nikolakopoulos, A.; Jachowicz, N. et al.  Nuclear effects in electron-nucleus and neutrino-nucleus scattering within a relativistic quantum mechanical framework.  Phys. Rev. C 2019, 100, 045501.  doi:black10.1103/PhysRevC.100.045501.
  * Pandey et al. (2016) Pandey, V.; Jachowicz, N.; Martini, M. et al.  Impact of low-energy nuclear excitations on neutrino-nucleus scattering at MiniBooNE and T2K kinematics.  Phys. Rev. 2016, C94, 054609.  doi:black10.1103/PhysRevC.94.054609.
  * Delorme and Ericson (1985) Delorme, J.; Ericson, M.  Exploration of the Spin - Isospin Nuclear Response Function by Neutrinos.  Phys. Lett. 1985, B156, 263.  doi:black10.1016/0370-2693(85)91521-7.
  * Buss et al. (2012) Buss, O.; Gaitanos, T.; Gallmeister, K. et al.  Transport-theoretical Description of Nuclear Reactions.  Phys.Rept. 2012, 512, 1–124.  doi:black10.1016/j.physrep.2011.12.001.
  * Ivanov et al. (2012) Ivanov, M.V.; Udias, J.M.; Antonov, A.N. et al.  Superscaling predictions for neutrino-induced charged-current charged pion production at MiniBooNE.  Phys. Lett. 2012, B711, 178–183.  doi:black10.1016/j.physletb.2012.03.072.
  * Ivanov et al. (2016) Ivanov, M.V.; Megias, G.D.; González-Jiménez, R. et al.  Charged-current inclusive neutrino cross sections in the SuperScaling model including quasielastic, pion production and meson-exchange contributions.  J. Phys. 2016, G43, 045101.  doi:black10.1088/0954-3899/43/4/045101.
  * Amaro et al. (2020) Amaro, J.E.; Barbaro, M.B.; Caballero, J.A. et al.  Electron- versus neutrino-nucleus scattering.  J. Phys. G 2020, 47, 124001.  doi:black10.1088/1361-6471/abb128.
  * Rocco et al. (2016) Rocco, N.; Lovato, A.; Benhar, O.  Unified description of electron-nucleus scattering within the spectral function formalism.  Phys. Rev. Lett. 2016, 116, 192501.  doi:black10.1103/PhysRevLett.116.192501.
  * Vagnoni et al. (2017) Vagnoni, E.; Benhar, O.; Meloni, D.  Inelastic Neutrino-Nucleus Interactions within the Spectral Function Formalism.  Phys. Rev. Lett. 2017, 118, 142502.  doi:black10.1103/PhysRevLett.118.142502.
  * Rocco et al. (2019a) Rocco, N.; Barbieri, C.; Benhar, O. et al.  Neutrino-Nucleus Cross Section within the Extended Factorization Scheme.  Phys. Rev. C 2019, 99, 025502.  doi:black10.1103/PhysRevC.99.025502.
  * Rocco et al. (2019b) Rocco, N.; Nakamura, S.X.; Lee, T.S.H. et al.  Electroweak Pion-Production on Nuclei within the Extended Factorization Scheme.  Phys. Rev. C 2019, 100, 045503.  doi:black10.1103/PhysRevC.100.045503.
  * Meucci et al. (2013) Meucci, A.; Giusti, C.; Vorabbi, M.  Relativistic descriptions of final-state interactions in charged-current neutrino-nucleus scattering at ArgoNeuT kinematics.  Phys.Rev. 2013, D88, 013006.  doi:black10.1103/PhysRevD.88.013006.
  * Van Dessel et al. (2018) Van Dessel, N.; Jachowicz, N.; González-Jiménez, R. et al.  $A$-dependence of quasielastic charged-current neutrino-nucleus cross sections.  Phys. Rev. C 2018, 97, 044616.  doi:black10.1103/PhysRevC.97.044616.
  * Akbar et al. (2017) Akbar, F.; Sajjad Athar, M.; Singh, S.K.  Neutrino-nucleus cross sections in ${}^{12}C$ and ${}^{40}Ar$ with KDAR neutrinos.  J. Phys. G 2017, 44, 125108.  doi:black10.1088/1361-6471/aa9125.
  * Barbaro et al. (2018) Barbaro, M.B.; De Pace, A.; Donnelly, T.W. et al.  Asymmetric relativistic Fermi gas model for quasielastic lepton-nucleus scattering.  Phys. Rev. C 2018, 98, 035501.  doi:black10.1103/PhysRevC.98.035501.
  * Barbieri et al. (2019) Barbieri, C.; Rocco, N.; Somà, V.  Lepton Scattering from 40Ar and Ti in the Quasielastic Peak Region.  Phys. Rev. C 2019, 100, 062501.  doi:black10.1103/PhysRevC.100.062501.
  * Van Dessel et al. (2019) Van Dessel, N.; Jachowicz, N.; Nikolakopoulos, A.  Forbidden transitions in neutral and charged current interactions between low-energy neutrinos and Argon.  Phys. Rev. C 2019, 100, 055503.  doi:black10.1103/PhysRevC.100.055503.
  * Van Dessel et al. (2020) Van Dessel, N.; Nikolakopoulos, A.; Jachowicz, N.  Lepton kinematics in low energy neutrino-Argon interactions.  Phys. Rev. C 2020, 101, 045502.  doi:black10.1103/PhysRevC.101.045502.
  * Butkevich (2021) Butkevich, A.V.  Analysis of flux-integrated semi-exclusive cross sections for charged current quasi-elastic neutrino scattering off 40Ar at energies available at the MicroBooNE experiment, 2021, [arXiv:hep-ph/2107.01827].
  * Franco-Patino et al. (2021) Franco-Patino, J.M.; Barbaro, M.B.; Caballero, J.A. et al.  Theoretical description of semi-inclusive T2K, Miner$\nu$A and MicroBooNE neutrino-nucleus data in the relativistic plane wave impulse approximation, 2021, [arXiv:nucl-th/2106.02311].
  * Lu et al. (2016) Lu, X.G.; Pickering, L.; Dolan, S. et al.  Measurement of nuclear effects in neutrino interactions with minimal dependence on neutrino energy.  Phys. Rev. 2016, C94, 015503.  doi:black10.1103/PhysRevC.94.015503.
  * Abe et al. (2018) Abe, K.; Amey, J.; Andreopoulos, C. et al.  Characterization of nuclear effects in muon-neutrino scattering on hydrocarbon with a measurement of final-state kinematics and correlations in charged-current pionless interactions at T2K.  Phys. Rev. D 2018, 98, 032003.  doi:black10.1103/PhysRevD.98.032003.
  * Lu et al. (2018) Lu, X.G.; Betancourt, M.; Walton, T. et al.  Measurement of final-state correlations in neutrino muon-proton mesonless production on hydrocarbon at $\langle E_{\nu}\rangle=3$ GeV.  Phys. Rev. Lett. 2018, 121, 022504.  doi:black10.1103/PhysRevLett.121.022504.
  * Cai et al. (2020) Cai, T.; Lu, X.G.; Harewood, L.A. et al.  Nucleon binding energy and transverse momentum imbalance in neutrino-nucleus reactions.  Phys. Rev. D 2020, 101, 092001.  doi:black10.1103/PhysRevD.101.092001.
  * Abratenko et al. (2020) Abratenko, P.; Alrashed, M.; An, R. et al.  First Measurement of Differential Charged Current Quasielastic-like $\nu_{\mu}$-Argon Scattering Cross Sections with the MicroBooNE Detector.  Phys. Rev. Lett. 2020, 125, 201803.  doi:black10.1103/PhysRevLett.125.201803.
  * Moreno et al. (2014) Moreno, O.; Donnelly, T.W.; Van Orden, J.W. et al.  Semi-inclusive charged-current neutrino-nucleus reactions.  Phys. Rev. 2014, D90, 013014.  doi:black10.1103/PhysRevD.90.013014.
  * Van Orden and Donnelly (2019) Van Orden, J.W.; Donnelly, T.W.  Nuclear Theory and Event Generators for Charge-Changing Neutrino Reactions.  Phys. Rev. C 2019, 100, 044620.  doi:black10.1103/PhysRevC.100.044620.
  * Sobczyk et al. (2020) Sobczyk, J.E.; Nieves, J.; Sánchez, F.  Exclusive-final-state hadron observables from neutrino-nucleus multinucleon knockout.  Phys. Rev. C 2020, 102, 024601.  doi:black10.1103/PhysRevC.102.024601.
  * Franco-Patino et al. (2020) Franco-Patino, J.M.; Gonzalez-Rosa, J.; Caballero, J.A. et al.  Semi-inclusive charged-current neutrino-nucleus cross sections in the relativistic plane wave impulse approximation.  Phys. Rev. C 2020, 102, 064626.  doi:black10.1103/PhysRevC.102.064626.
  * González-Jiménez et al. (2021) González-Jiménez, R.; Barbaro, M.B.; Caballero, J.A. et al.  Neutrino energy reconstruction from semi-inclusive samples, 2021, [arXiv:nucl-th/2104.01701].
  * Dolan et al. (2018) Dolan, S.; Mosel, U.; Gallmeister, K. et al.  Sensitivity of Neutrino-Nucleus Interaction Measurements to 2p2h Excitations.  Phys. Rev. C 2018, 98, 045502.  doi:black10.1103/PhysRevC.98.045502.
  * Ruiz Simo et al. (2016) Ruiz Simo, I.; Amaro, J.E.; Barbaro, M.B. et al.  Emission of neutron–proton and proton–proton pairs in neutrino scattering.  Phys. Lett. 2016, B762, 124–130.  doi:black10.1016/j.physletb.2016.09.021.
  * Pastore et al. (2020) Pastore, S.; Carlson, J.; Gandolfi, S. et al.  Quasielastic lepton scattering and back-to-back nucleons in the short-time approximation.  Phys. Rev. C 2020, 101, 044612.  doi:black10.1103/PhysRevC.101.044612.
  * Aguilar-Arevalo et al. (2010) Aguilar-Arevalo, A.A.; Anderson, C.E.; Bazarko, A.O. et al.  Measurement of nu(mu) and anti-nu(mu) induced neutral current single pi0 production cross sections on mineral oil at E(nu) O(1- GeV).  Phys.Rev. 2010, D81, 013005.  doi:black10.1103/PhysRevD.81.013005.