1 Introduction

The Deep Underground Neutrino Experiment (DUNE) is a next-generation, long-baseline neutrino oscillation experiment which will carry out a detailed study of neutrino mixing utilizing high-intensity \(\nu _\mu \) and \({\bar{\nu }}_\mu \) beams measured over a long baseline. \(\hbox {DUNE}\) is designed to make significant contributions to the completion of the standard three-flavor picture by measuring all the parameters governing \(\nu _1\)\(\nu _3\) and \(\nu _2\)\(\nu _3\) mixing in a single experiment. Its main scientific goals are the definitive determination of the neutrino mass ordering, the definitive observation of \(\hbox {charge-parity symmetry violation (CPV)}\) for more than 50% of possible true values of the charge-parity violating phase, \(\delta _{\mathrm{CP}}\), and precise measurement of oscillation parameters, particularly \(\delta _{\mathrm{CP}}\), \(\sin ^22\theta _{13}\), and the octant of \(\theta _{23}\). These measurements will help guide theory in understanding if there are new symmetries in the neutrino sector and whether there is a relationship between the generational structure of quarks and leptons [1]. Observation of \(\hbox {CPV}\) in neutrinos would be an important step in understanding the origin of the baryon asymmetry of the universe [2, 3].

The \(\hbox {DUNE}\) experiment will observe neutrinos from a high-power neutrino beam peaked at \(\sim \)2.5 GeV but with a broad range of neutrino energies, a \(\hbox {near detector (ND)}\) located at Fermi National Accelerator Laboratory, in Batavia, Illinois, USA, and a large \(\hbox {liquid argon time-projection}\) \(\hbox {chamber (LArTPC) far detector (FD)}\) located at the 4850 ft level of Sanford Underground Research Facility (SURF), in Lead, South Dakota, USA, 1285 km from the neutrino production point. The neutrino beam provided by \(\hbox {Long-Baseline Neutrino Facility (LBNF)}\) [4] is produced using protons from Fermilab’s Main Injector, which are guided onto a graphite target, and a traditional horn-focusing system to select and focus particles produced in the target [5]. The polarity of the focusing magnets can be reversed to produce a beam dominated by either muon neutrinos or muon antineutrinos. A highly capable \(\hbox {ND}\) will constrain many systematic uncertainties for the oscillation analysis. The 40-kt (fiducial) \(\hbox {FD}\) is composed of four 10 kt (fiducial) LArTPC modules [6,7,8]. The deep underground location of the \(\hbox {FD}\) reduces cosmogenic and atmospheric sources of background, which also provides sensitivity to nucleon decay and low-energy neutrino detection, for example, the possible observation of neutrinos from a core-collapse supernova [5].

The entire complement of neutrino oscillation experiments to date has measured five of the neutrino mixing parameters [9,10,11]: the three mixing angles \(\theta _{12}\), \(\theta _{23}\), and \(\theta _{13}\), and the two squared-mass differences \(\varDelta m^{2}_{21}\) and \(|\varDelta m^{2}_{31}|\), where \(\varDelta m^2_{ij} = m^2_{i} - m^{2}_{j}\) is the difference between the squares of the neutrino mass states in eV\(^{2}\). The neutrino mass ordering (i.e., the sign of \(\varDelta m^{2}_{31}\)) is unknown, though recent results show a weak preference for the normal ordering [12,13,14]. The value of \(\delta _{\mathrm{CP}}\) is not well known, though neutrino oscillation data are beginning to provide some information on its value [12, 15].

The oscillation probability of \(\nu _\mu \) \(\rightarrow \) \(\nu _e\) through matter in the standard three-flavor model and a constant density approximation is, to first order [16]:



$$\begin{aligned} a = \pm \frac{G_{\mathrm {F}}N_e}{\sqrt{2}} \approx \pm \frac{1}{3500~\mathrm {km}}\left( \frac{\rho }{3.0~\mathrm {g/cm}^{3}}\right) , \end{aligned}$$

\(G_{\mathrm {F}}\) is the Fermi constant, \(N_e\) is the number density of electrons in the Earth’s crust, \(\varDelta _{ij} = 1.267 \varDelta m^2_{ij} L/E_\nu \), L is the baseline in km, and \(E_\nu \) is the neutrino energy in GeV. Both \(\delta _{\mathrm{CP}}\) and a terms are positive for \(\nu _\mu \rightarrow \nu _e\) and negative for \(\bar{\nu }_\mu \rightarrow \bar{\nu }_e\) oscillations; i.e., a neutrino-antineutrino asymmetry is introduced both by \(\hbox {CPV}\) (\(\delta _{\mathrm{CP}}\)) and the matter effect (a). The origin of the matter effect asymmetry is simply the presence of electrons and absence of positrons in the Earth [17, 18]. The (anti-)electron neutrino appearance probability is shown in Fig. 1 at the \(\hbox {DUNE}\) baseline of \(1285\hbox { km}\) as a function of neutrino energy for several values of \(\delta _{\mathrm{CP}}\).

Fig. 1
figure 1

The appearance probability at a baseline of \(1285\hbox { km}\), as a function of neutrino energy, for \(\delta _{\mathrm{CP}}\) = \(-\pi /2\) (blue), 0 (red), and \(\pi /2\) (green), for neutrinos (top) and antineutrinos (bottom), for normal ordering

\(\hbox {DUNE}\) has a number of features that give it unique physics reach, complementary to other existing and planned experiments [19,20,21]. Its broad-band beam makes it sensitive to the shape of the oscillation spectrum for a range of neutrino energies. \(\hbox {DUNE}\)’s relatively high energy neutrino beam enhances the size of the matter effect and will allow \(\hbox {DUNE}\) to measure \(\delta _{\mathrm{CP}}\) and the mass ordering simultaneously. The unique \(\hbox {LArTPC}\) detector technology will enhance the resolution on \(\hbox {DUNE}\)’s measurement of the value of \(\delta _{\mathrm{CP}}\), and along with the increased neutrino energy, gives \(\hbox {DUNE}\) a different set of systematic uncertainties to other experiments, making \(\hbox {DUNE}\) complementary with them.

This paper describes studies that quantify DUNE’s expected sensitivity to long-baseline neutrino oscillation, using the accelerator neutrino beam. Note that atmospheric neutrino samples would provide additional sensitivity to some of the same physics, but are not included in this work. The flux simulation and associated uncertainties are described in Sect. 2. Section 3 describes the neutrino interaction model and systematic variations. The near and far detector simulation, reconstruction, and event selections are described in Sects. 4 and 5, respectively, with a nominal set of event rate predictions given in Sect. 6. Detector uncertainties are described in Sect. 7. The methods used to extract oscillation sensitivities are described in Sect. 8. The primary sensitivity results are presented in Sect. 9. We present our conclusions in Sect. 10.

2 Neutrino beam flux and uncertainties

The expected neutrino flux is generated using G4LBNF [5, 22], a \(\hbox {Geant4}\)-based [23] simulation of the \(\hbox {LBNF}\) neutrino beam. The simulation uses a detailed description of the \(\hbox {LBNF}\) optimized beam design [5], which includes a target and horns designed to maximize sensitivity to \(\hbox {CPV}\) given the physical constraints on the beamline design.

Fig. 2
figure 2

Neutrino fluxes at the \(\hbox {FD}\) for neutrino-enhanced, FHC, beam running (top) and antineutrino, RHC, beam running (bottom)

Neutrino fluxes for neutrino-enhanced, forward horn current (FHC), and antineutrino-enhanced, reverse horn current (RHC), configurations of \(\hbox {LBNF}\) are shown in Fig. 2. Uncertainties on the neutrino fluxes arise primarily from uncertainties in hadrons produced off the target and uncertainties in the design parameters of the beamline, such as horn currents and horn and target positioning (commonly called “focusing uncertainties”) [5]. Given current measurements of hadron production and \(\hbox {LBNF}\) estimates of alignment tolerances, flux uncertainties are approximately 8% at the first oscillation maximum and 12% at the second. These uncertainties are highly correlated across energy bins and neutrino flavors. The unoscillated fluxes at the \(\hbox {ND}\) and \(\hbox {FD}\) are similar, but not identical. The relationship is well understood, and flux uncertainties mostly cancel for the ratio of fluxes between the two detectors. Uncertainties on the ratio are dominated by focusing uncertainties and are \(\sim \) 1% or smaller except at the falling edge of the focusing peak (\(\sim \)4 GeV), where they rise to 2%. The rise is due to the presence of many particles which are not strongly focused by the horns in this energy region, which are particularly sensitive to focusing and alignment uncertainties. The near-to-far flux ratio and uncertainties on this ratio are shown in Fig. 3.

Fig. 3
figure 3

Ratio of \(\hbox {ND}\) and \(\hbox {FD}\) fluxes show for the muon neutrino component of the \(\hbox {FHC}\) flux and the muon antineutrino component of the \(\hbox {RHC}\) flux (top) and uncertainties on the \(\hbox {FHC}\) muon neutrino ratio (bottom)

Beam-focusing and hadron-production uncertainties on the flux prediction are evaluated by reproducing the full beamline simulation many times with variations of the input model according to those uncertainties. The resultant uncertainty on the neutrino flux prediction is described through a covariance matrix, where each bin corresponds to an energy range of a particular beam mode and neutrino species, separated by flux at the \(\hbox {ND}\) and \(\hbox {FD}\). The output covariance matrix has \(208 \times 208\) bins, despite having only \(\sim \)30 input uncertainties. To reduce the number of parameters used in the fit, the covariance matrix is diagonalized, and each principal component is treated as an uncorrelated nuisance parameter. The 208 principal components are ordered by the magnitude of their corresponding eigenvalues, which is the variance along the principal component (eigenvector) direction, and only the first \(\sim \)30 are large enough that they need to be included. This was validated by including more flux parameters and checking that there was no significant change to the sensitivity for a small number of test cases. By the 10th principal component, the eigenvalue is 1% of the largest eigenvalue. As may be expected, the largest uncertainties correspond to the largest principal components as shown in Fig. 4. The largest principal component (component 0) matches the hadron production uncertainty on nucleon-nucleus interactions in a phase space region not covered by data. Components 3 and 7 correspond to the data-constrained uncertainty on proton interactions in the target producing pions and kaons, respectively. Components 5 and 11 correspond to two of the largest focusing uncertainties, the density of the target and the horn current, respectively. Other components not shown either do not fit a single uncertain parameter or may represent two or more degenerate systematics or ones that produce anti-correlations in neighboring energy bins.

Fig. 4
figure 4

Select flux principal components are compared to specific underlying uncertainties from the hadron production and beam focusing models. Note that while these are shown as positive shifts, the absolute sign is arbitrary

Future hadron production measurements are expected to improve the quality of, and the resulting constraints on, these flux uncertainty estimates. Approximately 40% of the interactions that produce neutrinos in the \(\hbox {LBNF}\) beam simulation have no direct data constraints. Large uncertainties are assumed for these interactions. The largest unconstrained sources of uncertainty are proton quasielastic interactions and pion and kaon rescattering in beamline materials. The proposed EMPHATIC experiment [24] at Fermilab will be able to constrain quasielastic and low-energy interactions that dominate the lowest neutrino energy bins. The NA61 experiment at CERN has taken data that will constrain many higher energy interactions, and also plans to measure hadrons produced on a replica \(\hbox {LBNF}\) target, which would provide tight constraints on all interactions occurring in the target. A similar program at NA61 has reduced flux uncertainties for the T2K experiment from \(\sim \)10 to \(\sim \)5% [25]. Another proposed experiment, the \(\hbox {LBNF}\) spectrometer [26], would measure hadrons after both production and focusing in the horns to further constrain the hadron production uncertainties, and could also be used to experimentally assess the impact of shifted alignment parameters on the focused hadrons (rather than relying solely on simulation).

3 Neutrino interaction model and uncertainties

A framework for considering the impact of neutrino interaction model uncertainties on the oscillation analysis has been developed. The default interaction model is implemented in v2.12.10 of the \(\hbox {GENIE}\) generator [27, 28]. Variations in the cross sections are implemented in various ways: using \(\hbox {GENIE}\) reweighting parameters (sometimes referred to as “\(\hbox {GENIE}\) knobs”); with ad hoc weights of events that are designed to parameterize uncertainties or cross-section corrections currently not implemented within \(\hbox {GENIE}\); or through discrete alternative model comparisons. The latter are achieved through alternative generators, alternative \(\hbox {GENIE}\) configurations, or custom weightings, which made extensive use of the NUISANCE package [29] in their development.

The interaction model components and uncertainties can be divided into seven groups: (1) initial state, (2) hard scattering and nuclear modifications to the quasielastic, or one-particle one-hole (1p1h) process, (3) multinucleon, or two-particle two-hole (2p2h), hard scattering processes, (4) hard scattering in pion production processes, (5) higher invariant mass (W) and \(\hbox {neutral current (NC)}\) processes, (6) \(\hbox {final-state interactions (FSI)}\), (7) neutrino flavor dependent differences. Uncertainties are intended to reflect current theoretical freedom, deficiencies in implementation, and/or current experimental knowledge.

The default nuclear model in \(\hbox {GENIE}\) describing the initial state of nucleons in the nucleus is the Bodek–Ritchie global Fermi gas model [30]. There are significant deficiencies that are known in global Fermi gas models: these include a lack of consistent incorporation of the high-momentum tails in the nucleon momentum distribution that result from correlations among nucleons; the lack of correlation between location within the nucleus and momentum of the nucleon; and an incorrect relationship between momentum and energy of the off-shell, bound nucleon within the nucleus. They have also been shown to agree poorly with neutrino-nucleus scattering data [31]. \(\hbox {GENIE}\) modifies the nucleon momentum distribution empirically to account for short-range correlation effects, which populates the high-momentum tail above the Fermi cutoff, but the other deficiencies persist. Alternative initial state models, such as spectral functions [32, 33], the mean field model of GiBUU [34], or continuum random phase approximation (CRPA) calculations [35] may provide better descriptions of the nuclear initial state [36], but are not considered further here.

The primary uncertainties considered in 1p1h interactions (\(\nu _{l}+n \rightarrow l^{-}+p\), \(\bar{\nu }_{l}+p \rightarrow l^{+}+n\)) are the axial form factor of the nucleon and the nuclear screening—from the so-called \(\hbox {random phase approximation (RPA)}\) calculations—of low momentum transfer reactions. The Valencia group’s [37, 38] description of \(\hbox {RPA}\) comes from summation of \(W^\pm \) self-energy terms. In practice, this modifies the 1p1h (quasielastic) cross section in a non-trivial way, with associated uncertainties presented in Ref. [39], which were evaluated as a function of \(Q^2\). Here we use T2K’s 2017/8 parameterization of the Valencia RPA effect [12]. The shape of the correction and error is parameterized with a third-order Bernstein polynomial up to \(Q^2=1.2\text { GeV}^2\) where the form transitions to a decaying exponential. The BeRPA (Bernstein RPA) function has three parameters controlling the behavior at increasing \(Q^2\) (A, B and D), a fourth parameter (E) that controls the high-\(Q^2\) tail, and a fifth (U), which changes the position at which the behaviour changes from polynomial to exponential. The BeRPA parameterization modifies the central value of the model prediction, as decribed in Table 3. BeRPA parameters E and U are not varied in the analysis described here, the parameters A and B have a prefit uncertainty of 20%, and D has a prefit unertainty of 15%. The axial form factor parameterization we use, a dipole, is known to be inadequate [40]. However, the convolution of BeRPA uncertainties with the limited axial form factor uncertainties do provide more freedom as a function of \(Q^2\), and the two effects combined likely provide adequate freedom for the \(Q^2\) shape in quasielastic events. BBBA05 vector form factors are used [41].

The 2p2h contribution to the cross section comes from the Valencia model [37, 38], the implementation in \(\hbox {GENIE}\) is described in Ref. [42]. However, MINERvA [43] and NOvA [44] have shown that this model underpredicts observed event rates on carbon. The extra strength from the “MINERvA tune” to 2p2h is applied as a two-dimensional Gaussian in \((q_0,q_3)\) space, where \(q_0\) is the energy transfer from the leptonic system, and \(q_3\) is the magnitude of the three momentum transfer) to fit reconstructed MINERvA CC-inclusive data [43]. Reasonable predictions of MINERvA ’s data are found by attributing the missing strength to any of 2p2h from np initial state pairs, 2p2h from nn initial state pairs, or 1p1h (quasielastic) processes. The default tune uses an enhancement of the np and nn initial strengths in the ratio predicted by the Valencia model, and alternative systematic variation tunes (“MnvTune” 1-3) attribute the missing strength to the individual interaction processes above. We add uncertainties for the energy dependence of this missing strength based on the MINERvA results [43], and assume a generic form for the energy dependence of the cross section using the “A” and “B” terms taken from Ref. [45]. These uncertainties are labeled \(E_{2p2h}\) and are separated for neutrinos and antineutrinos. We add uncertainties on scaling the 2p2h prediction from carbon to argon on electron-scattering measurements of short-range correlated (SRC) pairs taken on multiple targets [46], separately for neutrinos (ArC2p2h \(\nu \)) and antineutrinos (ArC2p2h \(\bar{\nu }\)).

\(\hbox {GENIE}\) uses a modified version of the Rein–Sehgal (R–S) model for pion production [47], including only the 16 resonances recommended by the Particle Data group [48], and excluding interferences between resonances. The cross section is cut off at invariant masses, \(W \ge 1.7\) GeV (2 GeV in the original R-S model). No in-medium modifications to the resonances are included, and by default they decay isotropically in their rest frame, although there is a parameter denoted here as “\(\theta _{\pi }\) from \(\varDelta \)-decay”, for changing the angular distribution of pions produced through \(\varDelta \) resonance decays to match the experimentally observed distributions used in the original R-S paper [47]. Resonance decays to \(\eta \) and \(\gamma \) (plus a nucleon) are included from Ref. [48]. We use a tuning of the \(\hbox {GENIE}\) model to reanalyzed neutrino–deuterium bubble chamber data [49, 50] as our base model, as noted in Table 3. We note that an improved Rein–Sehgal-like resonance model has been developed [51], and has been implemented in Monte Carlo generators, although is not used as the default model in the present work.

The \(\hbox {deep inelastic scattering (DIS)}\) model implemented in \(\hbox {GENIE}\) uses the Bodek–Yang parametrization [52], using GRV98 parton distribution functions [53]. Hadronization is described by the AKGY model [54], which uses the KNO scaling model [55] for invariant masses \(W \le 2.3\) GeV and PYTHIA6 [56] for invariant masses \(W \ge 3\) GeV, with a smooth transition between the two for intermediate invariant masses. A number of variable parameters affecting \(\hbox {DIS}\) processes are included in \(\hbox {GENIE}\), as listed in Table 3, and described in Ref. [52]. In \(\hbox {GENIE}\), the \(\hbox {DIS}\) model is extrapolated to all values of invariant mass, and replaces the non-resonant background to pion production in the R-S model.

The NOvA experiment [57] developed uncertainties beyond those provided by \(\hbox {GENIE}\) to describe their single pion to \(\hbox {DIS}\) transition region data. We follow their findings, and implement separate, uncorrelated uncertainties for all perturbations of 1, 2, and \(\ge 3\) pion final states, CC/NC, neutrinos/antineutrinos, and interactions on protons/neutrons, with the exception of CC neutrino 1-pion production, where interactions on protons and neutrons are merged, following [50], which modifies the central value of the model prediction, as listed in Table 3. This leads to 23 distinct uncertainty channels with a label to denote the process it affects: NR [\(\nu \),\(\bar{\nu }\)] [CC,NC] [n,p] [1\(\pi \),2\(\pi \),3\(\pi \)]. Each channel has an uncertainty of 50% for \(W \le 3\) GeV, and an uncertainty which drops linearly above \(W = 3\) GeV until it reaches a flat value of 5% at \(W = 5\) GeV, where external measurements better constrain this process.

\(\hbox {GENIE}\) includes a large number of final state uncertainties on its final state cascade model [58,59,60], which are summarized in Table 2. A recent comparison of the underlying interaction probabilities used by GENIE is compared with other available simulation packages in Ref. [61].

The cross sections include terms proportional to the lepton mass, which are significant contributors at low energies where quasielastic processes dominate. Some of the form factors in these terms have significant uncertainties in the nuclear environment. Ref. [62] ascribes the largest possible effect to the presence of poorly constrained second-class current vector form factors in the nuclear environment, and proposes a variation in the cross section ratio of \(\sigma _\mu /\sigma _e\) of \(\pm 0.01/\mathrm{Max}(0.2~\mathrm{GeV},E_\nu )\) for neutrinos and \({\mp } 0.018/\mathrm{Max}(0.2~ \mathrm{GeV},E_\nu )\) for antineutrinos. Note the anticorrelation of the effect in neutrinos and antineutrinos. This parameter is labeled \(\nu _{e}\)/\(\bar{\nu }_{e}\) norm.

An additional normalization uncertainty (\(\hbox {NC}\) norm.) of 20% is applied to all \(\hbox {NC}\) events at the \(\hbox {ND}\) in this analysis to investigate whether the small contamination of \(\hbox {NC}\) events which passed the simple selection cuts had an effect on the analysis. Although a similar systematic could have been included (uncorrelated) at the \(\hbox {FD}\), it was not in this analysis.

Finally, some electron-neutrino interactions occur at four-momentum transfers where a corresponding muon-neutrino interaction is kinematically forbidden, therefore the nuclear response has not been constrained by muon-neutrino cross-section measurements. This region at lower neutrino energies has a significant overlap with the Bodek–Ritchie tail of the nucleon momentum distribution in the Fermi gas model [30]. There are significant uncertainties in this region, both from the form of the tail itself and from the lack of knowledge about the effect of RPA and 2p2h in this region. Here, a 100% uncertainty is applied in the phase space present for \(\nu _e\) but absent for \(\nu _\mu \) (labeled \(\nu _{e}\) phase space (PS)).

The complete set of interaction model uncertainties includes \(\hbox {GENIE}\) implemented uncertainties (Tables 1 and 2), and new uncertainties developed for this effort (Table 4) which represent uncertainties beyond those implemented in the \(\hbox {GENIE}\) generator.

Table 1 Neutrino interaction cross-section systematic parameters considered in \(\hbox {GENIE}\). \(\hbox {GENIE}\) default central values and uncertainties are used for all parameters except the CC resonance axial mass. The central values are the \(\hbox {GENIE}\) nominals, and the 1\(\sigma \) uncertainty is as given. Missing \(\hbox {GENIE}\) parameters were omitted where uncertainties developed for this analysis significantly overlap with the supplied \(\hbox {GENIE}\) freedom, the response calculation was too slow, or the variations were deemed unphysical
Table 2 The intra-nuclear hadron transport systematic parameters implemented in \(\hbox {GENIE}\) with associated uncertainties considered in this work. Note that the ‘mean free path’ parameters are omitted for both N–N and \(\pi \)–N interactions as they produced unphysical variations in observable analysis variables. Table adapted from Ref [28]

Tunes which are applied to the default model, using the dials described, which represent known deficiencies in \(\hbox {GENIE}\)’s description of neutrino data, are listed in Table 3.

Table 3 Neutrino interaction cross-section systematic parameters that receive a central-value tune and modify the nominal event rate predictions

The way model parameters are treated in the analysis is described by three categories:

  • Category 1: expected to be constrained with on-axis data; uncertainties are implemented in the same way for \(\hbox {ND}\) and \(\hbox {FD}\).

  • Category 2: implemented in the same way for \(\hbox {ND}\) and \(\hbox {FD}\), but on-axis \(\hbox {ND}\) data alone is not sufficient to constrain these parameters. They may be constrained by additional \(\hbox {ND}\) samples in future analyses, such as off-axis measurements.

  • Category 3: implemented only in the \(\hbox {FD}\). Examples are parameters which only affect \(\nu _e\) and \({\overline{\nu }}_e\) rates which are small and difficult to precisely isolate from background at the \(\hbox {ND}\).

All \(\hbox {GENIE}\) uncertainties (original or modified), given in Tables 1 and 2, are all treated as Category 1. Table 4, which describes the uncertainties beyond those available within \(\hbox {GENIE}\), includes a column identifying which of these categories describes the treatment of each additional uncertainty.

Table 4 List of extra interaction model uncertainties in addition to those provided by GENIE, and the category to which they belong in the analysis. Note that in this analysis, the NC norm. systematic is not applied at the \(\hbox {FD}\), as described in the text

4 The near detector simulation and reconstruction

The \(\hbox {ND}\) hall will be located at \(\hbox {Fermi National Accelerator}\) \(\hbox {Laboratory (Fermilab)}\), 574 m from where the protons hit the beam target, and 60 m underground. The baseline design for the \(\hbox {DUNE}\) \(\hbox {ND}\) system consists of a \(\hbox {LArTPC}\) with a downstream magnetized \(\hbox {multi-purpose detector (MPD)}\), and an on-axis beam monitor. Additionally, it is planned for the \(\hbox {LArTPC}\) and \(\hbox {MPD}\) to be movable perpendicular to the beam axis, to take measurements at a number of off-axis angles. The use of off-axis angles is complementary to the on-axis analysis described in this work through the DUNE-PRISM concept, originally developed in the context of the J-PARC neutrino beamline in Ref. [63]. We note that there are many possible \(\hbox {ND}\) samples which are not included in the current analysis, but which may either help improve the sensitivity in future, or will help control uncertainties to the level assumed here. These include: neutrino–electron scattering studies, which can independently constrain the flux normalization to \(\sim \)2% [64]; additional flux constraints from the low-\(\nu \) method, which exploits the fac