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]:

(1)

where

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

\(G_{\mathrm {F}}\) 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.

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).

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

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 fact that the low energy transfer (low-\(\nu \)) cross section is roughly constant with neutrino energy [65,66,67,68,69,70]; and using interactions on the \(\hbox {gaseous argon (GAr)}\) in the \(\hbox {MPD}\). There is also the potential to include events where the muon does not pass through the \(\hbox {MPD}\), e.g. using multiple Coulomb scattering to estimate the muon momentum [71].

The \(\hbox {LArTPC}\) is a modular detector based on the ArgonCube design [72], with fully-3D pixelated readout [73] and optical segmentation [74]. These features greatly reduce reconstruction ambiguities that hamper monolithic, projective-readout \(\hbox {time projection chambers (TPCs)}\), and enable the \(\hbox {ND}\) to function in the high-intensity environment of the \(\hbox {DUNE}\) \(\hbox {ND}\) site. Each module is itself a \(\hbox {LArTPC}\) with two anode planes and a shared central cathode. The active dimensions of each module are \(1 \times 3 \times 1\) m (\(x \times y \times z\)), where the z direction is along the neutrino beam axis, and the y direction points upward. Charge drifts in the \(\pm x\) direction, with a maximum drift distance of 50 cm for ionization electrons. The full \(\hbox {liquid argon (LAr)}\) detector consists of an array of modules in a single cryostat. The minimum active size required for full containment of hadronic showers in the \(\hbox {LBNF}\) beam is \(3 \times 4 \times 5\) m. High-angle muons can also be contained by extending the width to 7 m. For this analysis, 35 modules are arranged in an array 5 modules deep in the z direction and 7 modules across in x so that the total active dimensions are \(7 \times 3 \times 5\) m. The total active \(\hbox {LAr}\) volume is 105 m\(^{3}\), corresponding to a mass of 147 tons.

The \(\hbox {MPD}\) used in the analysis consists of a high-pressure \(\hbox {gaseous argon time-projection chamber (GArTPC)}\) in a cylindrical pressure vessel at 10 bar, surrounded by a granular, high-performance electromagnetic calorimeter, which sits immediately downstream of the \(\hbox {LAr}\) cryostat. The pressure vessel is 5 m in diameter and 5 m long. The \(\hbox {TPC}\) is divided into two drift regions by a central cathode, and filled with a 90%:10% Ar:\(\hbox {CH}_{{4}}\) gas mixture, such that 97% of neutrino interactions will occur on the Ar target. The \(\hbox {GArTPC}\) is described in detail in Ref. [5]. \(\hbox {The electromagnetic calorimeter (ECAL)}\) is composed of a series of absorber layers followed by arrays of scintillator and is described in Ref. [75]. The entire \(\hbox {MPD}\) sits inside a magnetic field, which allows the \(\hbox {MPD}\) to precisely measure the momentum and discriminate the sign of particles passing through it.

Neutrino interactions are simulated in the active volume of the \(\hbox {LArTPC}\). The propagation of neutrino interaction products through the \(\hbox {LArTPC}\) and \(\hbox {MPD}\) detector volumes is simulated using a \(\hbox {Geant4}\)-based model [23]. Pattern recognition and reconstruction software has not yet been developed for the \(\hbox {ND}\). Instead, we perform a parameterized reconstruction based on true energy deposits in active detector volumes as simulated by \(\hbox {Geant4}\).

The analysis described here uses events originating in the \(\hbox {LAr}\) component, within a a \(\hbox {fiducial volume (FV)}\) that excludes 50 cm from the sides and upstream edge, and 150 cm from the downstream edge of the active region, for a total of \(6 \times 2 \times 3\) m\(^{2}\). Muons with kinetic energy greater than \(\sim \)1 GeV typically exit the \(\hbox {LAr}\). An energetic forward-going muon will pass through the \(\hbox {ECAL}\) and into the gaseous \(\hbox {TPC}\), where its momentum and charge are reconstructed by curvature. For these events, it is possible to differentiate between \(\mu ^{+}\) and \(\mu ^{-}\) event by event. Muons that stop in the \(\hbox {LAr}\) or \(\hbox {ECAL}\) are reconstructed by range. Events with wide-angle muons that exit the \(\hbox {LAr}\) and do not match to the \(\hbox {GArTPC}\) are rejected, as the muon momentum is not reconstructed. The asymmetric transverse dimensions of the \(\hbox {LAr}\) volume make it possible to reconstruct wide-angle muons with some efficiency.

The charge of muons stopping in the \(\hbox {LAr}\) volume cannot be determined event by event. However, the wrong-sign flux is predominantly concentrated in the high-energy tail, where leptons are more likely to be forward and energetic. In \(\hbox {FHC}\) beam running, the wrong-sign background in the focusing peak is negligibly small, and \(\mu ^{-}\) is assumed for all stopping muon tracks. In \(\hbox {RHC}\) beam running, the wrong-sign background is larger in the peak region. Furthermore, high-angle leptons are generally at higher inelasticity, which enhances the wrong-sign contamination in the contained muon subsample. To mitigate this, a Michel electron is required at the end of the track. The wrong-sign \(\mu ^{-}\) captures on Ar with 75% probability, effectively suppressing the relative \(\mu ^{-}\) component by a factor of four.

Fig. 5
figure 5

Top: LAr+MPD acceptance for \(\nu _{\mu }\) \(\hbox {CC}\) events as a function of muon transverse and longitudinal momentum. Bottom: Acceptance as a function of hadronic energy; the black line is for the full Fiducial Volume (FV) while the red line is for a \(1 \times 1 \times 1\) m\(^{3}\) volume in the center, where the acceptance is higher due to the better hadron containment. The blue curve shows the expected distribution of true hadronic energy in the \(\hbox {DUNE}\) \(\hbox {ND}\) flux normalized to unity; 56% of events have hadronic energy below 1 GeV where the acceptance is high

True muons and charged pions are evaluated as potential muon candidates. The track length is determined by following the true particle trajectory until it undergoes a hard scatter or ranges out. The particle is classified as a muon if its track length is at least 1 m, and the mean energy deposit per centimeter of track length is less than 3 MeV. The mean energy cut rejects tracks with detectable hadronic interactions. The minimum length requirement imposes an effective threshold on true muons of about 200 MeV kinetic energy, but greatly suppresses potential NC backgrounds with low-energy, non-interacting charged pions. Charged-current events are required to have exactly one muon, and if the charge is reconstructed, it must be of the appropriate charge.

As in the \(\hbox {FD}\) reconstruction described in Sect. 5, hadronic energy in the \(\hbox {ND}\) is reconstructed by summing all charge deposits in the \(\hbox {LAr}\) active volume that are not associated with the muon. To reject events where the hadronic energy is poorly reconstructed due to particles exiting the detector, a veto region is defined as the outer 30 cm of the active volume on all sides. Events with more than 30 MeV total energy deposit in the veto region are excluded from the analysis. This leads to an acceptance that decreases as a function of hadronic energy, as shown in the bottom panel of Fig. 5. Neutron energy is typically not observed, resulting in poor reconstruction of events with energetic neutrons, as well as in events where neutrons are produced in secondary interactions inside the detector. The reconstructed neutrino energy is the sum of the reconstructed hadronic energy and the reconstructed muon energy.

The oscillation analysis presented here includes samples of \(\nu _{\mu }\) and \(\bar{\nu }_{\mu }\) charged-current interactions originating in the \(\hbox {LAr}\) portion of the ND, as shown in Fig. 6. These samples are binned in two dimensions as a function of reconstructed neutrino energy and inelasticity, \(y_{\mathrm {rec}} = 1 - E^{\mathrm {rec}}_{\mu }/E^{\mathrm {rec}}_{\nu }\), where \(E^{\mathrm {rec}}_{\mu }\) and \(E^{\mathrm {rec}}_{\nu }\) are the reconstructed muon and neutrino energies, respectively. Backgrounds to \(\hbox {CC}\) arise from \(\hbox {NC}\) \(\pi ^{\pm }\) production where the pion leaves a long track and does not shower. Muons below about 400 MeV kinetic energy have a significant background from charged pions, so these \(\hbox {CC}\) events are excluded from the selected sample. Wrong-sign contamination in the beam is an additional background, particularly at low reconstructed neutrino energies in RHC.

The far detector simulation and reconstruction

The 40-kt \(\hbox {DUNE}\) \(\hbox {FD}\) consists of four separate \(\hbox {LArTPC}\) \(\hbox {detector modules}\), each with a \(\hbox {FV}\) of at least 10 kt, installed \(\sim \)1.5 km underground at the \(\hbox {Sanford Underground Research}\) \(\hbox {Facility (SURF)}\) [76]. \(\hbox {DUNE}\) is committed to deploying both single-phase [77] and dual-phase [78] \(\hbox {LArTPC}\) technologies, and is investigating advanced detector designs for the fourth \(\hbox {detector module}\). As such, the exact order of construction and number of modules of each design is unknown. In this work, the \(\hbox {FD}\) reconstruction performance is assessed assuming a single-phase design for all four modules, which does not fully exploit the benefits of different technologies with independent systematics in the sensitivity studies. A full simulation chain has been developed, from the generation of neutrino events in a \(\hbox {Geant4}\) model of the \(\hbox {FD}\) geometry, to efficiencies and reconstructed neutrino energy estimators of all samples used in the sensitivity analysis.

Fig. 6
figure 6

\(\hbox {ND}\) samples in both \(\hbox {FHC}\) (blue) and \(\hbox {RHC}\) (red), shown in the reconstructed neutrino energy and reconstructed inelasticity binning used in the analysis, shown for a 7 year staged exposure, with an equal split between \(\hbox {FHC}\) and \(\hbox {RHC}\). Backgrounds are also shown (dashed lines), which are dominated by \(\hbox {NC}\) events, although there is some contribution from wrong-sign \(\nu _\mu \) background events in \(\hbox {RHC}\)

The total active \(\hbox {LAr}\) volume of each single-phase \(\hbox {DUNE}\) \(\hbox {FD}\) \(\hbox {detector module}\) is 13.9 m long, 12.0 m high and 13.3 m wide, with the 13.3 m width in the drift direction subdivided into four independent drift regions, with two shared cathodes. Full details of the single-phase \(\hbox {detector module}\) design can be found in Ref. [79]. The total active volume of each module is \(\sim \)13 kt, the \(\hbox {FV}\) of at least 10 kt is defined by studies of neutrino energy resolution, using the neutrino energy estimators described below. At the anode, there are two wrapped-wire readout induction planes, which are offset by ±35.7\(^{\circ }\) to the vertical, and a vertical collection plane.

Neutrino interactions of all flavors are simulated such that weights can be applied to produce samples for any set of oscillation parameters. The interaction model described in Sect. 3 was used to model the neutrino-argon interactions in the volume of the cryostat, and the final-state particles are propagated in the detector through \(\hbox {Geant4}\). The electronics response to the ionization electrons and scintillation light is simulated to produce digitized signals in the wire planes and \(\hbox {photon detectors (PDs)}\) respectively.

Raw detector signals are processed using algorithms to remove the impact of the \(\hbox {LArTPC}\) electric field and electronics response from the measured signal, to identify hits, and to cluster hits that may be grouped together due to proximity in time and space. Clusters from different wire planes are matched to form high-level objects such as tracks and showers. These high level objects are used as inputs to the neutrino energy reconstruction algorithm.

The energy of the incoming neutrino in \(\hbox {CC}\) events is estimated by adding the lepton and hadronic energies reconstructed using the Pandora toolkit [80, 81]. If the event is selected as \(\nu _{\mu }\) \(\hbox {CC}\), the neutrino energy is estimated as the sum of the energy of the longest reconstructed track and the hadronic energy. The energy of the longest reconstructed track is estimated from its range if the track is contained in the detector. If the longest track exits the detector, its energy is estimated from multiple Coulomb scattering. The hadronic energy is estimated from the charge of reconstructed hits that are not in the longest track, and corrections are applied to each hit charge for recombination and the electron lifetime. An additional correction is made to the hadronic energy to account for missing energy due to neutral particles and final-state interactions.

If the event is selected as \(\nu _{e}\) \(\hbox {CC}\), the energy of the neutrino is estimated as the sum of the energy of the reconstructed electromagnetic (EM) shower with the highest energy and the hadronic energy. The former is estimated from the charges of the reconstructed hits in the shower, and the latter from the hits not in the shower; the recombination and electron lifetime corrections are applied to the charge of each hit. The same hadronic shower energy calibration is used for both \(\nu \) and \(\bar{\nu }\) based on a sample of \(\nu \) and \(\bar{\nu }\) events.

In the energy range of 0.5–4 GeV that is relevant for oscillation measurements, the observed neutrino energy resolution is \(\sim \)15–20%, depending on lepton flavor and reconstruction method. The muon energy resolution is 4% for contained tracks and 18% for exiting tracks. The electron energy resolution is approximately \(4\% \oplus 9\%/\sqrt{E}\), with some shower leakage that gives rise to a non-Gaussian tail that is anticorrelated with the hadronic energy measurement. The hadronic energy resolution is 34%, which could be further improved by identifying individual hadrons, adding masses of charged pions, and applying particle-specific recombination corrections. It may also be possible to identify final-state neutrons by looking for neutron-nucleus scatters, and use event kinematics to further inform the energy estimate. These improvements are under investigation and are not included in this analysis.

Event classification is carried out through image recognition techniques using a convolutional neural network, named \(\hbox {convolutional visual network (CVN)}\). Detailed descriptions of the \(\hbox {CVN}\) architecture can be found in Ref. [82]. The primary goal of the \(\hbox {CVN}\) is to efficiently and accurately produce event selections of the following interactions: \(\nu _{\mu }\) \(\hbox {CC}\) and \(\nu _{e}\) \(\hbox {CC}\) in \(\hbox {FHC}\), and \(\bar{\nu }_\mu \) \(\hbox {CC}\) and \(\bar{\nu }_e\) \(\hbox {CC}\) in \(\hbox {RHC}\).

Fig. 7
figure 7

A simulated \({2.2}\,{\mathrm{GeV}}\) \(\nu _{e}\) \(\hbox {CC}\) interaction shown in the collection view of the \(\hbox {DUNE}\) \(\hbox {LArTPCs}\). The horizontal axis shows the wire number of the readout plane and the vertical axis shows time. The colorscale shows the charge of the energy deposits on the wires. The interaction looks similar in the other two views. Reproduced from Ref. [82]

In order to build the training input to the \(\hbox {DUNE}\) \(\hbox {CVN}\) three images of the neutrino interactions are produced, one for each of the three readout views of the \(\hbox {LArTPC}\), using the reconstructed hits on individual wire planes. Each pixel contains information about the integrated charge in that region. An example of a simulated 2.2 GeV \(\nu _{e}\) \(\hbox {CC}\) interaction is shown in a single view in Fig. 7 demonstrating the fine-grained detail available from the \(\hbox {LArTPC}\) technology.

The \(\hbox {CVN}\) is trained using approximately three million simulated neutrino interactions. A statistically independent sample is used to generate the physics measurement sensitivities. The training sample is chosen to ensure similar numbers of training examples from the different neutrino flavors. Validation is performed to ensure that similar classification performance is obtained for the training and test samples to ensure that the \(\hbox {CVN}\) is not overtrained.

Fig. 8
figure 8

The distribution of \(\hbox {CVN}\) \(\nu _e\) \(\hbox {CC}\) (top) and \(\nu _\mu \) \(\hbox {CC}\) scores (bottom) for \(\hbox {FHC}\) shown with a log scale. Reproduced from Ref. [82]

For the analysis presented here, we use the \(\hbox {CVN}\) score for each interaction to belong to one of the following classes: \(\nu _{\mu }\) \(\hbox {CC}\), \(\nu _{e}\) \(\hbox {CC}\), \(\nu _{\tau }\) \(\hbox {CC}\) and \(\hbox {NC}\). The \(\nu _{e}\) \(\hbox {CC}\) score distribution, \(P(\nu _e \)CC), and the \(\nu _\mu \) \(\hbox {CC}\) score distribution, \(P(\nu _\mu \)CC), are shown in Fig. 8. Excellent separation between the signal and background interactions is seen in both cases. The event selection requirement for an interaction to be included in the \(\nu _e\) \(\hbox {CC}\) (\(\nu _\mu \) \(\hbox {CC}\)) is \(P(\nu _e \)CC\() > 0.85\) (\(P(\nu _\mu \)CC\() > 0.5\)), optimized to produce the best sensitivity to \(\hbox {charge parity (CP)}\) violation. Since all of the flavor classification scores must sum to unity, the interactions selected in the two event selections are completely independent. The same selection criteria are used for both \(\hbox {FHC}\) and \(\hbox {RHC}\) beam running.

Figure 9 shows the efficiency as a function of reconstructed energy (under the electron neutrino hypothesis) for the \(\nu _e\) event selection, and the corresponding selection efficiency for the \(\nu _\mu \) event selection. The \(\nu _e\) and \(\nu _\mu \) efficiencies in both \(\hbox {FHC}\) and \(\hbox {RHC}\) beam modes all exceed 90% in the neutrino flux peak.

Fig. 9
figure 9

Top: the \(\nu _e\) \(\hbox {CC}\) selection efficiency for \(\hbox {FHC}\) (left) and \(\hbox {RHC}\) (right) simulation with the criterion \(P(\nu _e \)CC\() > 0.85\). Bottom: the \(\nu _\mu \) \(\hbox {CC}\) selection efficiency for \(\hbox {FHC}\) (left) and \(\hbox {RHC}\) (right) simulation with the criterion \(P(\nu _\mu \)CC\() > 0.5\). The results from \(\hbox {DUNE}\)’s Conceptual Design Report (CDR) are shown for comparison [7]. The solid (dashed) lines show results from the \(\hbox {CVN}\) (\(\hbox {CDR}\)) for signal \(\nu _e\) \(\hbox {CC}\) and \(\bar{\nu }_e\) \(\hbox {CC}\) events in black and \(\hbox {NC}\) background interaction in red. The blue region shows the oscillated flux (A.U.) to illustrate the most important regions of the energy distribution. Reproduced from Ref. [82]

The ability of the \(\hbox {CVN}\) to identify neutrino flavor is dependent on its ability to resolve and identify the charged lepton. Backgrounds originate from the mis-identification of charged pions for \(\nu _{\mu }\) disappearance, and photons for \(\nu _{e}\) appearance. The probability for these backgrounds to be introduced varies with the momentum and isolation of the energy depositions from the pions and photons. The efficiency was also observed to drop as a function of track/shower angle (with respect to the incoming neutrino beam direction) when energy depositions aligned with wire planes. The shapes of the efficiency functions in lepton momentum, lepton angle, and hadronic energy fraction (inelasticity) are all observed to be consistent with results from previous studies, including hand scans of \(\hbox {LArTPC}\) simulations. The \(\hbox {CVN}\) is susceptible to bias if there are features in the data that are not present in the simulation, so before its use on data, it will be important to comprehensively demonstrate that the selection is not sensitive to the choice of reference models. A discussion of the bias studies performed so far, and those planned in future, can be found in Ref. [82].

Expected far detector event rate and oscillation parameters

In this work, \(\hbox {FD}\) event rates are calculated assuming the following nominal deployment plan, which is based on a technically limited schedule:

  • Start of beam run: two \(\hbox {FD}\) module volumes for total fiducial mass of 20 kt, 1.2 MW beam

  • After one year: add one \(\hbox {FD}\) module volume for total fiducial mass of 30 kt

  • After three years: add one \(\hbox {FD}\) module volume for total fiducial mass of 40 kt

  • After 6 years: upgrade to 2.4 MW beam

Table 5 shows the conversion between number of years under the nominal staging plan, and kt-MW-years, which are used to indicate the exposure in this analysis. For all studies shown in this work, a 50%/50% ratio of FHC to RHC data was assumed, based on studies which showed a roughly equal mix of running produced a nearly optimal \(\delta _{\mathrm{CP}}\) and mass ordering sensitivity. The exact details of the run plan are not included in the staging plan.

Table 5 Conversion between number of years in the nominal staging plan, and kt-MW-years, the two quantities used to indicate exposure in this analysis
Fig. 10
figure 10

\(\nu _e\) and \({\bar{\nu }}_e\) appearance spectra: reconstructed energy distribution of selected \(\nu _e\) \(\hbox {CC}\)-like events assuming 3.5 years (staged) running in the neutrino-beam mode (top) and antineutrino-beam mode (bottom), for a total of seven years (staged) exposure. Statistical uncertainties are shown on the datapoints. The plots assume normal mass ordering and include curves for \(\delta _{\mathrm{CP}}= -\pi /2, 0\), and \(\pi /2\)

Fig. 11
figure 11

\(\nu _\mu \) and \({\bar{\nu }}_\mu \) disappearance spectra: reconstructed energy distribution of selected \(\nu _{\mu }\) \(\hbox {CC}\)-like events assuming 3.5 years (staged) running in the neutrino-beam mode (top) and antineutrino-beam mode (bottom), for a total of seven years (staged) exposure. Statistical uncertainties are shown on the datapoints. The plots assume normal mass ordering