Low exposure long-baseline neutrino oscillation sensitivity of the DUNE experiment

Low exposure long-baseline neutrino oscillation sensitivity of the DUNE experiment A. Abed Abud et al.* (DUNE Collaboration) (Received 8 September 2021; accepted 14 March 2022; published 25 April 2022) The Deep Underground Neutrino Experiment (DUNE) will produce world-leading neutrino oscillation measurements over the lifetime of the experiment. In this work, we explore DUNE’s sensitivity to observe charge-parity violation (CPV) in the neutrino sector, and to resolve the mass ordering, for exposures of up to 100 kiloton-megawatt-calendar years (kt-MW-CY), where calendar years include an assumption of 57% accelerator uptime based on past accelerator performance at Fermilab. The analysis includes detailed uncertainties on the flux prediction, the neutrino interaction model, and detector effects. We demonstrate that DUNE will be able to unambiguously resolve the neutrino mass ordering at a 4σ(5σ) level with a 66 (100) kt-MW-CY far detector exposure, and has the ability to make strong statements at significantly shorter exposures depending on the true value of other oscillation parameters, with a median sensitivity of 3σfor almost all true δCP values after only 24 kt-MW-CY. We also show that DUNE has the potential to make a robust measurement of CPV at a 3σlevel with a 100 kt-MW-CY exposure for the maximally CP-violating values δCP ¼π=2. Additionally, the dependence of DUNE’s sensitivity on the exposure taken in neutrino-enhanced and antineutrino-enhanced running is discussed. An equal fraction of exposure taken in each beam mode is found to be close to optimal when considered over the entire space of interest. DOI: 10.1103/PhysRevD.105.072006 I. INTRODUCTION TheDeepUndergroundNeutrinoExperiment(DUNE)[1] is a next-generation long-baseline neutrino oscillation experiment which will utilize high-intensity νμand ¯ νμbeams with peak neutrino energies of ≈2.5GeV over a 1285 km baseline to carry out a detailed study of neutrino mixing. Some of DUNE’s key scientific goals are the definitive determination of the neutrino mass ordering, the definitive observation of charge-parity symmetry violation (CPV) for more than 50% of possible true values of the charge-parity violating phase, δCP, and the precise measurement of other three-neutrino oscillation parameters. These measurements will help guide theory in understanding if there are new symmetries in the neutrino sector and whether there is a relationshipbetween the generational structure of quarks and leptons [2]. Observation of CPV in neutrinos would be an important step in understanding the origin of the baryon asymmetry of the Universe [3,4]. DUNE has a rich physics program beyond the three-neutrino oscillation accelerator neutrino program described here. These include beyond standard model searches [5], supernova neutrino detection [6], and solar neutrino detection [7]. Additional physics possibilities with DUNE are discussed in Refs. [8,9]. Neutrino oscillation experiments have so far measured five of the three-neutrino mixing parameters [10–12]: the three mixing angles θ12,θ23, and θ13; and the two squaredmass differences Δm2 21 and jΔm2 31j, where Δm2 ij ¼m2 i− m2 jis the difference between the squares of the neutrino masses. The neutrino mass ordering (the sign of Δm2 31)is not currently known, though recent results show a weak preference for the normal ordering (NO), where Δm2 31 >0, over the inverted ordering (IO) [13–15]. The value of δCP is not well known, though neutrino oscillation data are beginning to provide some information on its value [13,16]. The oscillation probability of ν ð  Þ μ→ν ð  Þ ethrough matter in the standard three-flavor model and a constant matter density approximation can be written as [17] Pðν ð  Þ μ→ν ð  Þ eÞ≃sin2θ23sin22θ13 sin2ðΔ31 −aLÞ ðΔ31 −aLÞ2Δ2 31 þsin2θ23 sin2θ13 sin2θ12 sinðΔ31 −aLÞ ðΔ31 −aLÞΔ31 ×sinðaLÞ ðaLÞΔ21 cosðΔ31 δCPÞ þcos2θ23sin22θ12 sin2ðaLÞ ðaLÞ2Δ2 21;ð1Þ *Corresponding author. [email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 105, 072006 (2022) 2470-0010=2022=105(7)=072006(32) 072006-1 Published by the American Physical Society where a¼GFNe ffiffiffi 2 p≈1 3500 km ρ 3.0g=cm3; aL ≈0.367,GFis the Fermi constant, Neis the number density of electrons in Earth’s crust, Δij ¼1.267Δm2 ijL= Eν,Δm2 ij is in eV2,Lis the baseline in kilometers, and Eνis the neutrino energy in GeV. Both δCP and aterms are positive (negative) for νμ→νe(¯ νμ→¯ νe) oscillations. The matter effect asymmetry encapsulated in the aterms arises from the presence of electrons and absence of positrons in Earth’s crust [18,19]. In the analysis described here, the oscillation probabilities are calculated exactly [20]. DUNE has published sensitivity estimates [21] to CPV and the neutrino mass ordering, as well as other oscillation parameters, for large exposures which show the ultimate sensitivity of the experiment. Sophisticated studies with a detailed treatment of systematic uncertainties were carried out only at large exposures. In this work, DUNE’s sensitivity at low exposures is explored further, with a detailed systematics treatment, including an investigation into how the run plan may be optimized to enhance sensitivity to CPV and/or mass ordering. It is shown that DUNE will produce world-leading results at relatively short exposures, which highlights the need for a highperformance near detector complex from the beginning of the experiment. The DUNE far detector (FD) will ultimately consist of four modules, each with a 17 kt total mass. The neutrino beamline has an initial design intensity of 1.2 MW, with a planned upgrade to 2.4 MW. We assume a combined yearly Fermilab accelerator and neutrino beam-line uptime of 57% [8] and define kiloton-megawatt-calendar years (kt-MWCY) as the exposure that would be collected by DUNE per kiloton of FD mass per megawatt of beam power per calendar year of nominal running (not per year of 100% uptime). This is for ease of interpretation, so the reader can easily translate the given exposures into expected years of DUNE operation.1As the FD deployment schedule and beam power scenarios are both subject to change, the results shown in this work are consistently given in terms of exposure in units of kt-MW-CY, which is agnostic to the exact staging scenario, but can easily be expressed in terms of experiment years for any desired scenario. For example, with two FD modules, assuming a fiducial mass of 10 kt and a beam intensity of 1.2 MW, exposure would accumulate at a rate of 24 kt-MW-CY per calendar year, although a ramp-up in beam power is expected before reaching the design intensity in early running. In this work, the single-phase horizontal drift technology is assumed for all FD modules (see Sec. II D), which is a necessary simplification, but alternative technologies which may have slightly different performance are under investigation for some FD modules. The analysis framework is described in Sec. II, includinga description of the flux, neutrino interaction and detector models and associated uncertainties. A study of the dependence of the sensitivity to CPV and mass ordering on the fraction of data collected in neutrino-enhanced or antineutrino-enhanced running is given in Sec. III. A detailed study of the CPVand mass ordering sensitivities at low exposures are described in Secs. IV and V, respectively. Finally, conclusions are presented in Sec. VI. II. ANALYSIS FRAMEWORK This work uses the flux, neutrino interaction and detector model described in detail in Ref. [21], implemented in the CAFAna framework [22]. This section provides an overview of the key analysis features. Further details on all aspects can be found in Ref. [21]. A. Neutrino flux DUNE will operate with two different beam modes, which depend on the polarity of the electromagnetic horns used to focus secondary particles produced after protons from the primary beamline interact in the target. Forward horn current (FHC) corresponds to neutrino-enhanced running, and reverse horn current (RHC) corresponds to antineutrino-enhanced running. In both FHC and RHC there are significant contributions from neutrinos with energies between 0.5 and 6 GeV, with a flux peak at ≈2.5GeV. The neutrino flux prediction is generated with G4LBNF [8,23], using the LBNF optimized beam design [8]. Flux uncertainties are due to the production rates and kinematic distributions of hadrons produced in the target and the parameters of the beamline, such as horn currents and horn and target positioning (“focusing uncertainties”) [8]. They are evaluated using current measurements of hadron production and estimates of alignment tolerances, giving flux uncertainties of approximately 8% at the first oscillation maximum, which are highly correlated across energy bins and neutrino flavors. A flux covariance as a function of neutrino energy, beam mode, detector, and neutrino species is generated with a “toy throw”approach, which is built using variations (“throws”) of the systematics propagated through the full beamline simulation. 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. Only the first ≈30 principal components (out of 108) were found to have a significant effect in the analysis and were included. The shapes of the unoscillated fluxes at the ND and FD are 1In previous publications, DUNE has referred to the same quantity as kiloton-megawatt-years (kt-MW-yr). Here we use “calendar years”to make clear that we are referring to elapsed time rather than live time, at the request of the referee. We regret any confusion this change may cause. A. ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-2 similar, and the differences between them are understood at the percent level. B. Neutrino interaction model The interaction model used is based on GENIE v2.12.10 [24,25], although the combination of models used is much closer to some of the physics tunes available with GENIE v3.00.06, including a number of uncertainties beyond those provided by either GENIE version. These are motivated by data, although the available (anti)neutrino data taken on argon targets are sparse, leading to an uncertainty model that relies in a number of places on light target (mostly hydrocarbon) data. Variations in the cross sections are implemented either using GENIE reweighting parameters or with ad hoc weights of events designed to parametrize uncertainties or cross-section corrections currently not implemented within GENIE . The latter were developed using alternative generators or GENIE configurations, or custom weightings using the NUISANCE package [26]. Further details about the uncertainties used can be found in Ref. [21] (Sec. III). The nuclear model which describes the initial state of nucleons in the nucleus is the Bodek-Ritchie global Fermi gas model [27], which includes empirical modifications to the nucleon momentum distribution to account for shortrange correlation effects. The quasielastic model uses the Llewellyn-Smith formalism [28] with a simple dipole axial form factor, and BBBA05 vector form factors [29]. Nuclear screening effects and uncertainties are included based on the T2K 2017=8parametrization [30] of the Valencia group’s[31,32] random phase approximation model. The Valencia model of the multinucleon, 2p2h, contribution to the cross section [31,32] is used, as described in Ref. [33]. Both MINERvA [34] and NOvA [35] have shown that this model underpredicts observed event rates on carbon at relevant neutrino energies for DUNE. Modifications to the model are constructed to produce agreement with MINERvA CC-inclusive data [34], which are used in the analysis to introduce additional uncertainties on the 2p2h contribution, with energy-dependent uncertainties, and extra freedom between neutrinos and antineutrinos. Uncertainties are added on scaling the 2p2h prediction from carbon to argon on electron-scattering measurements of short-range correlated pairs taken on multiple targets [36], separately for neutrinos and antineutrinos. GENIE uses a modified version of the Rein-Sehgal (R-S) model for pion production [37]. A data-driven modification to the GENIE model is included based on reanalyzed neutrino-deuterium bubble chamber data [38,39]. The deep inelastic scattering (DIS) model implemented in GENIE uses the Bodek-Yang parametrization [40] and GRV98 parton distribution functions [41]. Hadronization is described by the Andreopoulos-Kehayias-GallagherYang model [42], which uses the Koba-Nielsen-Olesen scaling model [43] for invariant masses W≤2.3GeV and PYTHIA 6[44] for invariant masses W≥3GeV, with a smooth transition between the two models for intermediate invariant masses. Additional uncertainties developed by the NOvA Collaboration [45] to describe their resonance to DIS transition region data are also included. The final state interaction model and uncertainties available in GENIE are retained [46–48]. The cross sections include terms proportional to the lepton mass, which are significant at low energies where quasielastic processes dominate. Some of the form factors in these terms have significant uncertainties in the nuclear environment. Separate (and anticorrelated) uncertainties on the cross-section ratio σμ=σefor neutrinos and antineutrinos are adopted from Ref. [49]. Additionally, some νechargedcurrent (CC) interactions occur at four-momentum transfers where νμCC interactions are kinematically forbidden and so cannot be constrained by νμcross-section measurements. To reflect this, a 100% uncertainty is applied in the phase space present for νebut absent for νμ. Systematic effects beyond what can be obtained by shifting parameters of the GENIE model can be studied by fitting simulated data produced with entirely distinct interaction generators. In general, the fit is not able to exactly reproduce the alternate simulated data, resulting in biases to extracted oscillation parameters. Demonstrating that these biases are small is an important test of the robustness of the systematic uncertainties. Such a study has been performed using simulated data based on the NuWro generator [50]. Controlling these effects is critical for the long-term physics goals of DUNE, and Ref. [9] describes how near detector measurements can be used to mitigate the discrepancy observed between the reference model and the alternate simulated data. For the early physics milestones presented in this paper, the sensitivity is limited by far detector statistics, and the impact of additional uncertainties to cover these out-of-model effects is expected to be small, although they are not explicitly included. A detailed treatment of these issues can be expected in future works. C. Near detector simulation and reconstruction The near detector (ND) hall will be located 574 m downstream of the proton target and ≈60 m underground. The reference design for the DUNE ND system is fully described in Ref. [9] and consists of a liquid argon (LAr) time projection chamber (TPC) referred to as ND-LAr, a magnetized high-pressure gaseous argon TPC (ND-GAr), and an on-axis beam monitor (SAND). Additionally, NDLAr and ND-GAr are designed to move perpendicular to the beam axis in order to take data at various off-axis angles (the DUNE-PRISM technique). ND-LAr is a modular detector based on the ArgonCube design [51–53], with a total active LAr volume of 105 m3(a LAr massof 147 tons). ND-GAr is implementedinthisanalysisasacylindricalTPCfilledwitha 90=10 mixture of argon and CH4at 10 bar, surrounded by a granular, high-performance electromagnetic calorimeter LOW EXPOSURE LONG-BASELINE NEUTRINO OSCILLATION …PHYS. REV. D 105, 072006 (2022) 072006-3 (ECal). ND-GAr sits immediately downstream of the LAr cryostat and serves as a muon spectrometer for ND-LAr [9]. For the early physics milestones discussed in this paper, the uncertainties are dominated by FD statistics. While precision neutrino interaction measurements in ND-GAr areimportant for the long-term physics program, their impact on the measurements presented here is minimal, and the conclusions in this paper remain valid in a scenario where a temporary spectrometer is used at the start of the run. Neutrino interactions are simulated in the active volume of ND-LAr. The propagation of neutrino interaction products through the ND-LAr and ND-GAr detector volumes is simulated using a GEANT 4-based program [54]. As pattern recognition and reconstruction software has not yet been fully developed for the ND, this analysis uses a parametrized reconstruction based on the GEANT 4 simulated energy deposits in active detector volumes. Only CC-inclusive interactions originating in the LAr are considered in this analysis, with a fiducial volume (FV) which excludes 50 cm from the sides and upstream edge and 150 cm from the downstream edge of the active region, containing a total fiducial mass of ≈50 t. Most muons with kinetic energies greater than 1 GeVexit ND-LAr. Energetic forward-going muons pass into ND-GAr, where their momentum and charge are reconstructed by curvature. Muon energy is reconstructed by range for tracks that stop in the LAr, and the charge cannot be determined event by event. Events with muons that exit the LAr active volume and do not match to a track in ND-GAr are rejected, as the muon momentum is not well reconstructed. For FHC beam running, the wrong-sign background is small and the charge is assumed to be negative for all LAr-contained muons. For RHC beam running, a Michel electron is required at the end of these stopped tracks to suppress the wrong-sign μ−by a factor of 4. All generated muons and charged pions are evaluated as potential muon candidates. Tracks are classified as muons if their length is at least 1 m, and their mean energy deposit is less than 3MeV=cm. The minimum length requirement imposes an effective threshold on the true muon kinetic energy of about 200 MeV but greatly suppresses potential neutral current (NC) backgrounds with low-energy, noninteracting charged pions. Charged-current events are required to have exactly one muon candidate, and if the charge is reconstructed by curvature, it must be of the appropriate sign. Hadronic energy in the ND is reconstructed by summing all charge deposits in the LAr active volume that are not associated with the muon. To remove events where the hadronic energy is badly reconstructed due to charged particles exiting the detector, a veto region is defined as the outer 30 cm of the active volume on all sides, and events with more than 30 MeV total energy deposited in the veto region are rejected. Only a fraction of neutron kinetic energy is typically observed (24% on average with large fluctuations), resulting in poor energy reconstruction of events with energetic neutrons. The reconstructed neutrino energy Erec ν¼Erec μþErec had is the sum of the reconstructed hadronic energy Erec had and the reconstructed muon energy Erec μ. The reconstructed inelasticity yrec ¼ 1−Erec μ=Erec νis the fraction of the neutrino energy that is carried by hadrons. D. Far detector simulation and reconstruction The DUNE FD design consists of four separate LArTPC detector modules, each with a total LAr mass of 17 kt, installed ≈1.5km underground at the Sanford Underground Research Facility (SURF) [55]. The technologies to be deployed for the four modules and their order of construction are still under investigation, so in this analysis, only the single-phase design with a horizontal drift [56] is used. In this design, signals from drift electrons in the 13.3×12.0×57.5m3active volume are read out by ≈5mm spaced wires in anode readout planes. Scintillation light produced at the time of the neutrino interaction is detected and used to reconstruct the start time of the electron drift. We have developed a full simulation chain, which generates neutrino events in a GEANT 4 model of the FD geometry and simulates the electronics readout. We have developed a reconstruction package to calculate efficiencies and reconstructed neutrino energy estimators for the four CC-inclusive FD samples used in the analysis (νμ-like FHC, νe-like FHC, ¯ νμ-like RHC and ¯ νe-like RHC). The electronics response to the ionization electrons and scintillation light is simulated in the wire planes and photon detectors, respectively. Algorithms are applied to remove the impact of the LArTPC electric field and electronics response from the raw detector 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 using the Pandora toolkit [57,58]. Event classification is carried out through image recognition techniques using a convolutional neural network [59] which classifies events as ν ð  Þ μ-CC, ν ð  Þ e-CC, ν ð  Þ τ-CC, and NC. The ν ð  Þ eand ν ð  Þ μefficiencies in both beam modes exceed 90% in the flux peak. The neutrino energy for ν ð  Þ μ-CC ( ν ð  Þ e-CC) events is estimated by the sum of the energy of the longest reconstructed track (highest energy reconstructed electromagnetic shower) and the hadronic energy. For both event types, the hadronic energy is estimated from the charge of reconstructed hits that are not in the primary track or shower, and corrections are applied to each hit charge for recombination and electron lifetime effects. For ν ð  Þ μ-CC events, the energy of the longest track is estimated by range if the track is contained or by multiple Coulomb scattering if it is exiting. For 0.5–4 GeV neutrino energies, the observed neutrino energy resolution is 15%–20%. The muon energy resolution is 4% for contained tracks and 18% A. ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-4 for exiting tracks. The electron energy resolution is approximately 4% ⊕9%=ffiffiffiffi E p. The hadronic energy resolution is 34%. E. Detector systematics Detector effects impact the event selection efficiency as well as the reconstruction of neutrino energy and inelasticity (the variables used in the oscillation fits). The main sources of detector systematic uncertainties are limitations of the expected calibration and modeling of particles in the detector. Important differences between the ND and FD LArTPC design, size, detector environment, and calibration strategy lead to uncertainties that do not fully correlate between the two detectors. The degree of correlation is under active study, but in this analysis they are treated as being completely uncorrelated. Detailed simulations of detector effects are under development. In this analysis, uncertainties on the energy scale, energy resolution, particle responses, and detector acceptance are included to encapsulate these effects. The absolute scale uncertainties shift the reconstructed energy distributions, while the resolution uncertainties narrow or broaden them. An uncertainty on the overall energy scale is included in the analysis presented here, as well as particle energy scale and resolution uncertainties that are separate and uncorrelated between four particle classes: muons, charged hadrons, neutrons, and electromagnetic showers. In the ND, muons reconstructed by range in LAr and by curvature in the ND-GAr are treated separately and assigned uncorrelated uncertainties. For each class of particle, uncertainties on the energy scale are introduced as a function of the reconstructed particle energy E, with a constant term, a term proportionalto ffiffiffiffi E p,andatermproportionalto1=ffiffiffiffi E p.A10% uncertainty on the energy resolution is also included and treated as uncorrelated between the four particle classes. The parameters produce a shift to the kinematic variables in an event, as opposed to simply assigning a weight to each simulated event. The scale of the uncertainties is motivated by what has been achieved in recent experiments, including calorimetric-based approaches (NOvA, MINERvA) and LArTPCs (LArIAT, MicroBooNE, ArgoNeut). In addition to impacting energy reconstruction, the E-field model also affects the definition of the FD FV, which is sensitive to electron drift. An additional 1% uncertainty is therefore included on the total fiducial mass, which is conservatively treated as uncorrelated between the ν ð  Þ μand ν ð  Þ esamples due to the potential distortion caused by large electromagnetic showers in the electron sample. The FD is sufficiently large that acceptance is not expected to vary significantly as a function of event kinematics. However, the ND acceptance does vary as a function of both muon and hadronic kinematics due to various containment criteria. Uncertainties are evaluated on the muon and hadron acceptance at the ND based on the change in the acceptance as a function of muon kinematics and true hadronic energy. F. Sensitivity methods Systematics are implemented in the analysis using onedimensional response functions for each analysis bin, and oscillation weights are calculated exactly, in fine (50 MeV) bins of true neutrino energy. For a given set of inputs—flux, oscillation parameters, cross sections, detector energy response matrices, and detector efficiency—an expected event rate can be produced. Minimization is performed using the MINUIT [60] package. Oscillation sensitivities are obtained by simultaneously fitting the νμ-like FHC, νe-like FHC, ¯ νμ-like RHC and ¯ νelike RHC FD spectra along with the νμFHC and ¯ νμRHC samples from the ND. In the studies, all oscillation parameters shown in Table Iare allowed to vary. Gaussian penalty terms (taken from Table I) are applied to θ12,Δm2 21, and the matter density ρof Earth along the DUNE baseline [61]. Some studies presented in this work include a Gaussian penalty term on θ13 (also taken from NuFIT 4.0, given in Table I), which is precisely measured by experiments sensitive to reactor antineutrino disappearance [62–64]. The remaining parameters, sin2θ23,Δm2 32, and δCP are allowed to vary freely, with no penalty terms. The penalty terms are treated as uncorrelated with each other and uncorrelated with other parameters. Flux, cross-section, and FD detector parameters are allowed to vary in the fit but are constrained by a penalty term corresponding to the prior uncertainty. ND detector uncertainties are included via a covariance matrix based on the shape difference between ND prediction and the “data” TABLE I. Central value and relative uncertainty of neutrino oscillation parameters from a global fit [65,66] to neutrino oscillation data. The matter density is taken from Ref. [61]. Because the probability distributions are somewhat non-Gaussian (particularly for θ23), the relative uncertainty is computed using 1=6of the 3σallowed range from the fit, rather than 1=2of the 1σ range. For θ23,θ13, and Δm2 31, the best-fit values and uncertainties depend on whether NO or IO is assumed. The best fit for δCP is used as a test point in the analysis, but no uncertainty is assigned. Parameter Central value Relative uncertainty θ12 0.5903 2.3% θ23 (NO) 0.866 4.1% θ23 (IO) 0.869 4.0% θ13 (NO) 0.150 1.5% θ13 (IO) 0.151 1.5% Δm2 21 7.39 ×10−5eV22.8% Δm2 32 (NO) 2.451 ×10−3eV21.3% Δm2 32 (IO) −2.512 ×10−3eV21.3% ρ2.848 gcm−32% δCP (NO) −2.53 (rad)  δCP (IO) −1.33 (rad)  LOW EXPOSURE LONG-BASELINE NEUTRINO OSCILLATION …PHYS. REV. D 105, 072006 (2022) 072006-5 (which come from the simulation in this sensitivity study). The covariance matrix is constructed with a throwing technique. For each “throw,”all ND energy scale, resolution, and acceptance parameters are simultaneously thrown according to their respective uncertainties, and the modified prediction is produced by varying the relevant quantities away from the nominal prediction according to the thrown parameter values. The bin-to-bin covariance is determined by comparing the resulting spectra with the nominal prediction, in the same binning as is used in the oscillation sensitivity analysis. The compatibility of a particular oscillation hypothesis with both ND and FD data is evaluated using the standard Poisson log-likelihood ratio [67]: χ2ð  ϑ; xÞ¼−2log Lð  ϑ; xÞ ¼2X Nbins iMið  ϑ; xÞ−DiþDilnDi Mið  ϑ; xÞ þX Nsysts jΔxj σj2 þX NND bins kX NND bins lðMkð xÞ−DkÞV−1 kl ðMlð xÞ−DlÞ;ð2Þ where  ϑand  xare the vector of oscillation parameter and nuisance parameter values, respectively; Nbins is the total number of ND and FD bins used in the analysis; NND bins is the number of ND bins; Mið  ϑ; xÞand Diare the MC expectation and fake data in the ith reconstructed bin (summed over all selected samples), respectively, with the oscillation parameters neglected for the ND; Δxjand σjare the difference between the nominal and current value, and the prior uncertainty on the jth nuisance parameter, respectively; and Vkl is the covariance matrix between ND bins described previously. To protect against false minima, all fits are repeated starting at four different δCP values (−π, −π=2, 0, and π=2), in both mass orderings, and in both sin2θ23 octants, and the lowest obtained χ2value is taken as the true minimum. Two approaches are used for the sensitivity studies presented in this work. Asimov studies [68] are carried out (in Sec. III) in which the fake (Asimov) dataset is the same as the nominal MC. In these, the true value of all systematic uncertainties and oscillation parameters are set to their nominal value (see Table I) except the parameters of interest, which are set to a test point. Then a fit is carried out in which all parameters can vary, constrained by their prior uncertainty where applicable. For the smallest exposures investigated with an Asimov study in this work, all samples have at least 100 events, satisfying the Gaussian approximation inherent in the Asimov method. Toy throw studies are performed (in Secs. IVand V) in which an ensemble of systematic, oscillation parameter and statistical throws are made. Systematic throws are made according to their prior Gaussian uncertainties, oscillation parameters are randomly chosen as described in Table II, and Poisson fluctuations are then applied to all analysis bins, based on the mean event count for each bin after the systematic adjustments have been applied. For each throw in the ensemble, the test statistic is minimized, and the best-fit value of all parameters is determined. The expected resolution for parameters of interest are then determined from the spread in the distribution of their postfit values. Asimov studies are computationally efficient and, for Gaussian parameters and uncertainties, give a good sense of the median sensitivity of an experiment. Toy throwing studies are computationally expensive, fully explore the parameter space, and make fewer assumptions about the behavior of parameters and uncertainties. G. Near and far detector samples and statistics In this work, the sensitivity as a function of FD exposure is explored and results are reported in terms of kt-MW-CY, which does not assume any specific FD or beam intensity staging scenario. However, the ND used in this analysis (ND-LAr with a downstream muon spectrometer) is assumed not to be staged, and as such the ND sample size corresponding to a particular FD exposure will vary based on the staging scenario. The nominal staging scenario from Ref. [21] is therefore retained for the purpose of normalizing the ND samples at each FD exposure. In that scenario, a 7 year exposure corresponds to 336 kt-MW-CY at the FD and 480 t-MW-CY at the ND, summed over both beam modes. The ND statistics used in this analysis are scaled assuming this ratio throughout, using the same fraction of exposure in each beam mode as used at the FD. The ND samples used in this analysis are relatively quickly systematics limited in both beam modes, and so these approximations are unlikely to have a significant impact on the results. The oscillation analysis presented here includes two CCinclusive samples originating in the ND-LAr FV, an FHC νμand an RHC ¯ νμsample. These samples are both binned in two dimensions, as a function of reconstructed neutrino TABLE II. Treatment of the oscillation parameters for the simulated dataset studies. The value and uncertainty for θ13 in both NO and IO used in the analysis come from NuFIT 4.0 [65,66]. Parameter Prior Range sin2θ23 Uniform [0.4; 0.6] jΔm2 32j(×10−3eV2)Uniform j½2.3; 2.7j δCP (π) Uniform ½−1; 1 θ13 (NO) Gaussian 0.1503 0.0023 (rad) θ13 (IO) Gaussian 0.1510 0.0023 (rad) A. ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-6 (a) (b) FIG. 1. ND samples in both FHC and RHC, shown in the reconstructed neutrino energy and reconstructed inelasticity binning (yrec) used in the analysis, for a 105 t-MW-CY exposure (equivalent to a 100 kt-MW-CY exposure at the FD) with all relevant exposure assumptions including 57% accelerator uptime as described in the text, with an equal split between FHC and RHC. The size of the systematic uncertainty bands from all of the flux, cross-section and ND detector systematics used in the analysis are shown, as well as the postfit uncertainty bands obtained by performing an Asimov fit to the ND data. NC backgrounds and wrong-sign contributions to the total event rate are also shown. Statistical uncertainties are too small to be visible on this plot scale. (a) FHC (b) RHC. LOW EXPOSURE LONG-BASELINE NEUTRINO OSCILLATION …PHYS. REV. D 105, 072006 (2022) 072006-7 energy and inelasticity, yrec ¼1−Erec μ=Erec ν. The sample distributions for both FHC and RHC are shown in Fig. 1for an exposure of 105 t-MW-CY, corresponding to 100 ktMW-CY at the far detector with the assumptions stated above. The size of the systematic uncertainty bands from all of the flux, cross-section and ND detector systematics used in the analysis and described above are shown, as well as the postfit uncertainty bands obtained by performing an Asimov fit to the ND data. It is clear that, even after a relatively small exposure of 105 t-MW-CY, the ND samples are very high statistics and are systematics limited in the binning used in the analysis. Backgrounds in the ν ð  Þ μ-CC samples are also shown in Fig. 1. NC backgrounds are predominantly from NC πproduction where the pion leaves a long track and does not shower. Wrong-sign contamination in the beam is a background where the charge of the muon is not reconstructed, which particularly affects low reconstructed neutrino energies in RHC. The wrong-sign background is also larger at high reconstructed inelasticity yrec, due to the kinematics of neutrino and antineutrino scattering. The expected FD FHC νeand RHC ¯ νesamples are shown in Fig. 2for a 100 kt-MW-CY total FD exposure, split equally between FHC and RHC beam modes. The systematic uncertainty bands with and without the ND constraint applied are shown, as well as the background contributions. There are contributions from both νeand ¯ νe in both beam modes. The NC, intrinsic beam ν ð  Þ eand wrong flavor contamination is also shown; the largest background comes from the intrinsic ν ð  Þ ebeam contribution in both modes. After a 50 kt-MW-CYexposure in FHC, the νesample statistical uncertainty is close to the systematic uncertainty before the ND constraint, although it is still clearly statistics limited when the ND constraint is applied. The ¯ νesample is still strongly statistics limited after 50 ktMW-CY exposure in RHC. The difference is largely due to the difference in the νeand ¯ νecross sections. The expected FD FHC νμand RHC ¯ νμsamples are shown in Fig. 3for a 100 kt-MW-CY total FD exposure, split equally between FHC and RHC beam modes. The systematic uncertainty bands with and without the ND constraint applied are shown, as well as the background contributions. Although the wrong-sign νμcontribution to the RHC ¯ νμsample is shown separately, it still provides useful information for constraining the oscillation parameters and is included in the analysis. The statistics are much higher than in Fig. 2; the statistical uncertainty on the νμ FHC sample is smaller than the systematic uncertainty band for some regions of phase space, even after the ND constraint is applied, although the statistical uncertainty is larger than the constrained systematic uncertainty in the “dip”region, around 2.5 GeV, which is likely to have the most impact on the analysis. The statistical uncertainty on the ¯ νμRHC sample is larger, again due to the smaller ¯ νμ (than νμ) cross section and lower fluxes in RHC running. The statistical uncertainty around the 2.5 GeV dip region is significantly larger than the systematic uncertainty band, although, as for the FHC νμsample, the statistical uncertainty is smaller than the systematics for some regions of the parameter space. (a) (b) FIG. 2. Reconstructed energy distribution of selected CC ν ð  Þ elike events in the FD, for a 50 kt-MW-CY exposure in both FHC and RHC beam modes, for a total 100 kt-MW-CYexposure, with all relevant exposure assumptions including 57% accelerator uptime as described in the text. The plots are shown for NO, all other oscillation parameters are set to their NuFIT 4.0 best-fit values (see Table I). The size of the systematic uncertainty bands from all of the flux, cross-section and FD detector systematics used in the analysis are shown, as well as the postfit uncertainty bands with parameters constrained by ND data. Backgrounds are also shown, the largest contribution comes from intrinsic ν ð  Þ e contamination in the beam, although NC and other flavors, ν ð  Þ μþν ð  Þ τ, also contribute. (a) FHC (b) RHC. A. ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-8 Events with a reconstructed neutrino energy of less than 0.5 GeV (which are shown in Figs. 1–3) or a reconstructed neutrino energy greater than 10 GeVare not included in the analysis for any of the FD or ND samples. III. RUN PLAN OPTIMIZATION In previous DUNE sensitivity studies [21], equal running times in FHC and RHC were assumed, based on early sensitivity estimates for different scenarios. In this section, the dependence of the median CPV and mass ordering significances are studied, for different fractions of time spent in each beam mode, using the full analysis framework described in Sec. II. Figure 4shows DUNE’s Asimov sensitivity to CPV for a total 100 kt-MW-CY far detector exposure, with different fractions of FHC and RHC running, at the NuFIT 4.0 bestfit value in both NO and IO (see Table I), shown with and without a penalty on θ13 applied. For each point tested, all oscillation and nuisance parameters are allowed to vary, and three fits are carried out, two where δCP is set to the CPconserving values δCP ¼0and δCP ¼π, the minimum of which is the CP-conserving best-fit value, and another where δCP is allowed to vary. The difference in the best-fit χ2values is calculated: Δχ2 CPV ¼min fχ2 δCP¼0;χ2 δCP¼πg−χ2 CPV;ð3Þ and the square root of the difference is used as the figure of merit on the yaxis in Fig. 4. There are some caveats associated with this figure of merit, which are discussed in Sec. IV. A 100 kt-MW-CY exposure is shown as it was identified in Ref. [21] as the exposure at which DUNE’s median CPV significance exceeds 3σat δCP ¼π=2,an important milestone in DUNE’s physics program (with equal beam mode running). Figure 4shows that, when the reactor constraint on θ13 is included, the sensitivity to CPV can be increased in some regions of δCP parameter space with more FHC than RHC running. However, this degrades the sensitivity in other regions, most notably for δCP >0regardless of the true mass ordering. This is due to a degeneracy between δCP and the octant of sin22θ23 because both parameters impact the rate of νeappearance. The degeneracy is resolved by including antineutrino data; the octant of sin22θ23 affects the rate of νeand ¯ νeappearance in the same way, but the effect of δCP is reversed for antineutrinos. For regions of phase space where the octant degeneracy does not affect the result (e.g., sin2θ23 ≈0.5), there is no degradation in the sensitivity, and enhanced FHC running increases the sensitivity for all values of δCP. Increasing the fraction of RHC decreases the sensitivity for the entire δCP range when the reactor θ13 constraint is included, relative to equal beam mode running. This is due to the lower statistics of the ¯ νesample (see Fig. 2) because of the reduced antineutrino flux and cross section. For short exposures, DUNE will not have a competitive independent measurement of θ13, so the main analysis will include the reactor θ13 constraint. Nonetheless, it is instructive to look at the results without the penalty applied. In this case, the sensitivity is severely degraded (as expected) for 100% running in either beam mode. Figure 5shows DUNE’s Asimov sensitivity to the mass ordering for a total 24 kt-MW-CY far detector exposure, (a) (b) FIG. 3. Reconstructed energy distribution of selected CC ν ð  Þ elike events in the FD, for 50 kt-MW-CYexposure in both FHC and RHC beam modes, for a total 100 kt-MW-CY exposure, with all relevant exposure assumptions including 57% accelerator uptime as described in the text. The plots are shown for NO, all other oscillation parameters are set to their NuFIT 4.0 best-fit values (see TableI).Thesizeofthe systematicuncertaintybandsfromallof the flux,cross-section andND detectorsystematicsusedintheanalysis are shown, as well as the postfit uncertainty bands with parameters constrained by ND data. NC backgrounds and wrong-sign contributions to the event rate are also shown. (a) FHC (b) RHC. LOW EXPOSURE LONG-BASELINE NEUTRINO OSCILLATION …PHYS. REV. D 105, 072006 (2022) 072006-9 (a) (b) (c) (d) (e) (f) FIG. 11. Fraction of throws for which significance of DUNE’s CP-violation test (δCP ≠f0;πg) exceeds 1–3σ, calculated using both the FC (shaded histograms) and constant-Δχ2(dashed lines) methods, as a function of the true value of δCP. Shown for NO, for a number of different exposures. The number of throws used to make each figure is also shown. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) 24 kt-MW-CY (b) 66 kt-MW-CY (c) 100 kt-MW-CY (d) 150 kt-MW-CY (e) 197 kt-MWCY (f) 336 kt-MW-CY. 2 10 3 10 Exposure (kt-MW-CY) 0 0.2 0.4 0.6 0.8 1 Fraction of throws 2 π = - CP δ σ1with FC σ2with FC σ3with FC DUNE Simulation 2 10 3 10 Exposure (kt-MW-CY) 0 0.2 0.4 0.6 0.8 1 Fraction of throws σ1with FC σ2with FC σ3with FC DUNE Simulation (a) (b) FIG. 12. Fraction of throws for which the significance of DUNE’s CP-violation test (δCP ≠f0;πg) exceeds 1–3σ, for δCP ¼−π=2 and for 50% of δCP values, calculated with the FC (solid lines) and constant-Δχ2(dashed lines) methods, as a function of exposure. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) δCP ¼−π=2(b) 50% of δCP values. A. ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-16 (a) (b) (c) (d) (e) (f) FIG. 13. Fraction of throws for which the significance of DUNE’s CP-violation test (δCP ≠f0;πg) exceeds 1–5σ, as a function of the true value of δCP. Shown for NO, for a number of different exposures. The number of throws used to make each figure is also shown. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) 24 kt-MW-CY (b) 66 kt-MW-CY (c) 100 ktMW-CY (d) 150 kt-MW-CY (e) 197 kt-MW-CY (f) 336 kt-MW-CY. 2 10 3 10 Exposure (kt-MW-CY) 0 0.2 0.4 0.6 0.8 1 Fraction of throws 2 π = - CP δ σ1 σ2 σ3 σ4 σ5 DUNE Simulation 2 10 3 10 Exposure (kt-MW-CY) 0 0.2 0.4 0.6 0.8 1 Fraction of throws σ1 σ2 σ3 σ4 σ5 DUNE Simulation (a) (b) FIG. 14. Fraction of throws for which the significance of DUNE’s CP-violation test (δCP ≠f0;πg) exceeds 1–5σ, both assuming δCP ¼−π=2, and for 50% of δCP values, shown as a function of exposure, for NO. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) δCP ¼−π=2(b) 50% of δCPvalues. LOW EXPOSURE LONG-BASELINE NEUTRINO OSCILLATION …PHYS. REV. D 105, 072006 (2022) 072006-17 in the 3σΔχ2 cvalue toward the lowest exposures observed in Fig. 9. The number of throws carried out at each exposure is indicated on each plot. The number of throws decreases as a function of exposure because fixed computing resources were used for each configuration, and the time for the ensemble of fits carried out for each throw to complete increases slightly with exposure. The final 336 kt-MW-CY exposure has more throws because it was generated for the analysis presented in Ref. [21], where more than one projection was considered—requiring more throws to sample the space. Figure 12 shows the fraction of throws which exceed different significance thresholds atthe maximal δCP violation value of δCP ¼−π=2,andfor50%ofδCP values as a function of exposure, with and without FC corrections, for 1–3σ significance values. Figure 12 was produced using the same throws used for Fig. 11, with additional points from higher exposures used in Ref. [21], but not shown in Fig. 11 (646 and 1104 kt-MW-CY). After ≈200 kt-MW-CY, the median significance (including FC correction) for 50% of the δCP range is greater than 3σ. It is clear from Fig. 12 that the effect of the FC correction is not large, and ≈10% longer exposures are required for the median expected significance to cross each threshold than without correction, at both δCP ¼−π=2 and for the 50% range of δCP values. Calculating Δχ2 cvalues above 3σusing the FC method is challenging due to the large number of throws to explore the tails of the Δχ2 FC distribution and prohibitive computational cost. In Fig. 13, the fraction of throws that exceed 1–5σsignificance calculated only with the constant-Δχ2 method is shown in order to explore DUNE’s sensitivity at higher significance levels. All the caveats described above relating to the constant-Δχ2method still apply. Figure 13 shows that, although the median significance to CPV does not exceed 5σfor δCP ¼−π=2until ≈336 kt-MW-CY, there are significant fractions of throws at lower exposures which reach 5σsignificance. Figure 14 shows the fraction of throws which exceed different significance thresholds at the maximal CP-violating value of δCP ¼−π=2, and for 50% of all δCP values, as a function of exposure. By ≈200 kt-MW-CY, where the median significance for 50% of the δCP range is greater than 3σ, the sensitivity at δCP ¼ −π=2exceeds 4σ. V. NEUTRINO MASS ORDERING SENSITIVITY In this section, the toy-throwing approach described in Sec. II is used to explore the neutrino mass ordering sensitivity as a function of exposure in detail. In all cases, a joint ND þFD fit is performed, and the reactor θ13 constraint is always applied, as described in Sec. II.An equal split between FHC and RHC running is assumed based on the results obtained in Sec. III. Figure 15 shows the significance with which the neutrino mass ordering can be determined for both true NO and IO, for exposures of 66 and 100 kt-MW-CY. The sensitivity metric used is the square root of the difference between the best-fit χ2value obtained using each ordering, as shown in Eq. (4), which is calculated for each throw of the FIG. 15. Significance of the DUNE determination of the neutrino mass ordering, as a function of the true value of δCP, for 66 kt-MW-CY (blue) and 100 kt-MW-CY (orange) exposures. The width of the transparent bands cover 68% of fits in which random throws are used to simulate systematic, oscillation parameter and statistical variations, with independent fits performed for each throw constrained by prior uncertainties. The solid lines show the median significance. All exposures include an assumption of 57% accelerator uptime as described in the text. A. ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-18 systematics, other oscillation parameters and statistics. The characteristic shape of the MH sensitivity in Fig. 15 results from near degeneracy between matter and CPV effects that occurs near δCP ¼π=2(δCP ¼−π=2) for true normal (inverted) ordering. Dedicated studies have shown that special attention must be paid to the statistical interpretation of neutrino mass ordering sensitivities [72–74] because the Δχ2 MO metric does not follow the expected chi-square distribution for one degree of freedom, so the interpretation of the ffiffiffiffiffiffiffiffiffiffiffiffi Δχ2 MO pas the sensitivity is complicated. Given the complications with the interpretation of significance for mass ordering determination, it is instructive to look at the distribution of the test statistic [Eq. (4)], which gives more information than the 68% central band and median throw shown in Fig. 15. Figure 16 shows the distribution of Δχ2 MO obtained for a large ensemble of throws, for both true normal and inverted orderings, for a number of different exposures. There is a uniform distribution of true δCP used in the throws at each exposure. The change in shape at higher exposures in Fig. 16 is due to the degeneracy between δCP and the effect of the mass ordering, and as might be expected from Fig. 15, the separation between hierarchies is greater for some true values of δCP than others. This additional structure starts to become obvious from a ≈66 kt-MW-CY exposure, at which point the CPV sensitivity is not very strong (see Sec. IV). For all exposures, the shape of the throw distribution is highly nonGaussian, which makes it difficult to apply simple corrections to the sensitivity of the sort described in Ref. [74].As a result alternatives to ffiffiffiffiffiffiffiffiffiffiffiffi Δχ2 MO pas a sensitivity metric are not explored, as the full information is given in Fig. 16. Figure 16 also indicates the probability for the test statistic Δχ2 MO to be less (more) than zero from the toy throws for true normal (inverted) orderings at each (a) (b) (c) (d) (e) (f) FIG. 16. The distribution of Δχ2 MO ¼χ2 IO −χ2 NO values shown for both true normal (red) and true inverted (blue) hierarchies built using random throws of the systematic parameters, the oscillation parameters and with statistical variations. In each case, the χ2values are separately minimized with respect to all variable parameters before calculating the test statistic. The fraction of throws for which the value of Δχ2 MO is greater than (less than) 0 is also given for inverted (normal) hierarchies. For each ordering and exposure, approximately 100 000 throws were used. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) 6 kt-MW-CY (b) 12 kt-MW-CY (c) 24 kt-MW-CY (d) 66 kt-MW-CY (e) 100 kt-MW-CY (f) 336 kt-MW-CY. LOW EXPOSURE LONG-BASELINE NEUTRINO OSCILLATION …PHYS. REV. D 105, 072006 (2022) 072006-19 exposure. This information is summarized in Fig. 17. This marks the proportion of toys which appear more like the incorrect ordering than the true ordering for the toy and gives a sense of the ambiguity between the hierarchies, although it is not easily converted to a single number sensitivity. It is clear from Figs. 16 and 17 that DUNE is sensitive to the mass ordering even from very low (≈12 kt-MW-CY) exposures, with a small probability for preferring the incorrect ordering. By exposures of 66 kt-MW-CY, the overlap between the orderings is very small with ≈1% of toy throws which appear more like the incorrect ordering than the true ordering. Figure 18 shows an alternative way to present the result of the throws as a function of δCP, which is complementary to Fig. 15. The fraction of throws for which the simple figure of merit [the square root of Eq. (4)] exceeds different confidence levels are shown, for 1–5σsignificances, and a variety of exposures, all for true NO. The same throws are used as in Fig. 16. Despite the caveats regarding the interpretation of ffiffiffiffiffiffiffiffiffiffiffiffi Δχ2 MO pas units of σ, the general trend is clear and provides more information about the expected 10 2 10 Exposure (kt-MW-CY) 5− 10 4− 10 3− 10 2− 10 1− 10 Probability < 0) MO 2 χΔNO p( > 0) MO 2 χΔ IO p( DUNE Simulation FIG. 17. The probability for preferring the wrong neutrino mass ordering as a function of exposure, shown for both true NO and IO. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) (b) (c) (d) (e) (f) FIG. 18. Fraction of throws for which the DUNE sensitivity to the mass ordering exceeds 1–5σsignificance, as a function of the true value of δCP. Shown for NO, for a number of different exposures. The number of throws used to make each figure is also shown. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) 6 kt-MW-CY (b) 12 kt-MW-CY (c) 24 kt-MWCY(d) 66 kt-MW-CY (e) 100 kt-MW-CY (f) 336 kt-MW-CY. A. ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-20 DUNE sensitivity at low exposures. As with Figs. 11 and 13, the point at which the median significance (50% of throws) passes different significance thresholds can be easily read from the figures and can be compared with those shown in Fig. 15. The same general shape as a function of δCP as was observed in Fig. 15 can be seen. The general trend would be very similar in IO, reflected in the line δCP ¼0, although a slightly longer exposure is required to reach the same sensitivity.ThemediansignificanceforδCP ¼−π=2exceeds 5σfor 24 kt-MW-CY, at which point the fraction of throws for which the significance is 3σor smaller is only ≈2%.By 66 kt-MW-CY, 100% of the throws exceed 5σat δCP ¼ −π=2. By 100 kt-MW-CY exposures, the median significance approaches 5σfor all true values of δCP.Atlong exposures of 336 kt-MW-CY, almost 100% of the throws exceed 5σfor all values of δCP. VI. CONCLUSION InthisworkadetailedexplorationofDUNE’ssensitivityto CPV and the mass ordering at low exposures has been presented. The analysis uses the same framework, flux, cross section and detector models and selections as were used in Ref. [21], which showed the ultimate DUNE sensitivity to CPV, the neutrino mass ordering and other oscillation parameters,withlargestatisticssamplesafterlongexposures. The effect of operating with different run plans, involving different ratios of FHC and RHC beam modes, on the mass ordering and CPVAsimov sensitivities was explored. It was found that, for low exposures, the sensitivity to both CPV and the mass ordering can be increased for certain regions of parameter space, but at a cost to the sensitivity in other regions. This sensitivity increase is in part produced by leveraging the strong θ13 constraint available from reactor experiments. If there is a strong reason to favor the exploration of a given region of parameter space when DUNE begins to take data, this issue should be revisited. However, with no strong motivation to focus on a given ordering or region of δCP parameter space, equal FHC and RHC beam running provides a close to optimal sensitivity across all of the parameter space, so was used for the subsequent detailed sensitivity studies. The increase in sensitivity for unequal beam running is also a feature of low exposure running and degrades the sensitivity almost uniformly across the parameter space investigated for large exposures, with and without a θ13 constraint applied. The studies presented here demonstrate that a full treatment of DUNE’s sensitivity at low exposures supports the conclusions made in Refs. [8,21] using Asimov studies. In particular, the median CPV sensitivity is ≈3σfor δCP ¼ π=2after approximately a 100 kt-MW-CY FD exposure. Variations in the expected sensitivity around the median value were also explored. Additionally, it was shown that the CPV sensitivity is not significantly degraded when Feldman-Cousins corrections are included, leading to ≈10% longer exposures to reach a given significance level. Crucially, it was found that after an initial low exposure rise, the Feldman-Cousins Δχ2 cdo not change as a function of exposure, unlike the rise with exposure which has been observed by the T2K experiment [13]. It has also been shown that strong statements on the mass ordering can be expected with very short exposures of ≈12 kt-MW-CY, which supports the results shown in Refs. [8,21] with a more complete treatment of the systematic uncertainty. Although the analysis used here makes no assumptions about the FD staging scenario, and results are given as a functionofexposureonly,theresultsaredependentonhaving a performantND complex fromthe start of the experiment. In particular, the low exposures necessary to make worldleading statements about the massordering can only begiven with confidence with ND samples included in the fit. ACKNOWLEDGMENTS This document was prepared by the DUNE Collaboration using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359. This work was supported by CNPq, FAPERJ, FAPEG and FAPESP, Brazil; CFI, Institute of Particle Physics (IPP) and NSERC, Canada; CERN; MŠMT, Czech Republic; ERDF, H2020-EU and MSCA, European Union; CNRS/IN2P3 and CEA, France; INFN, Italy; FCT, Portugal; NRF, South Korea; Comunidad de Madrid (CAM), Fundación “La Caixa,”Junta de Andalucía-FEDER, and MICINN, Spain; SERI and SNSF, Switzerland; TÜBİTAK, Turkey; The Royal Society and UKRI/STFC, United Kingdom; DOE and NSF, United States of America. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC0205CH11231. APPENDIX: FELDMAN-COUSINS THROW DISTRIBUTIONS The distribution of throws used to calculate the Δχ2 c values for Fig. 9for nine different exposures with δCP ¼0 are shown in Fig. 19; for Fig. 10(a) for nine different values of δCP with an exposure of 100 kt-MW-CY in Fig. 20; and for Fig. 10(b) for nine different values of δCP with an exposure of 336 kt-MW-CY in Fig. 21. For each distribution shown in Figs. 19–21, the calculated Δχ2 cvalues corresponding to for 68.27% (1σ), 90%, 95.45% (2σ) and 99.73% (3σ) of the throws are given and indicated with a vertical line. The number of throws used is also given. The Δχ2 cvalues were only calculated up to the 3σ level due to the very large number of throws required for higher confidence levels. LOW EXPOSURE LONG-BASELINE NEUTRINO OSCILLATION …PHYS. REV. D 105, 072006 (2022) 072006-21 (a) (b) (c) (d) (e) (f) (g) (h) (i) FIG. 19. Distribution of Δχ2values, calculated using Eq. (5), for a large number of throws with true δCP ¼0, for a variety of exposures. The Δχ2 cvalues (vertical lines) obtained using the Feldman-Cousins method show the Δχ2 FC value below which 68.27% (1σ), 90%, 95.45% (2σ) and 99.73% (3σ) of throws reside, with the calculated values given in the legend. The number of throws used is also given. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) 24 kt-MW-CY (b) 66 kt-MW-CY (c) 100 kt-MW-CY (d) 150 kt-MW-CY(e) 197 kt-MW-CY (f) 336 kt-MW-CY(g) 500 kt-MW-CY (h) 646 kt-MW-CY (i) 936 kt-MW-CY. A. ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-22 (a) (b) (c) (d) (e) (f) (g) (h) (i) FIG. 20. Distribution of Δχ2values, calculated using Eq. (5), for a large number of throws for nine different values of true δCP,fora 100 kt-MW-CY exposure. The Δχ2 cvalues (vertical lines) obtained using the Feldman-Cousins method show the Δχ2 FC value below which 68.27% (1σ), 90%, 95.45% (2σ) and 99.73% (3σ) of throws reside, with the calculated values given in the legend. The number of throws used is also given. All exposures include an assumption of 57% accelerator uptime as described in the text. (a) δCP=π¼–1 (b) δCP=π¼–0.75 (c) δCP=π¼–0.5(d) δCP=π¼–0.25 (e) δCP=π¼0(f) δCP=π¼0.25 (g) δCP=π¼0.5(h) δCP=π¼0.75 (i) δCP=π¼1. LOW EXPOSURE LONG-BASELINE NEUTRINO OSCILLATION …PHYS. REV. D 105, 072006 (2022) 072006-23 (a) (b) (c) (d) (e) (f) (g) (h) (i) FIG. 21. DistributionofΔχ2values,calculatedusingEq.(5),foralargenumberofthrowsforninedifferentvaluesoftrueδCP,fora336ktMW-CYexposure. The Δχ2 cvalues (vertical lines) obtained using the Feldman-Cousins method show the Δχ2 FC value below which 68.27% (1σ), 90%, 95.45% (2σ) and 99.73% (3σ) of throws reside, with the calculated values given in the legend. The number of throws used is also given. 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ABED ABUD et al. PHYS. REV. D 105, 072006 (2022) 072006-32