First measurement of neutrino oscillation parameters using neutrinos and antineutrinos by NOvA

TL;DR

The NOvA experiment has for the first time measured neutrino oscillation parameters using both neutrinos and antineutrinos, providing new insights into neutrino properties and mass hierarchy.

Contribution

This paper presents the first combined measurement of neutrino and antineutrino oscillation parameters by NOvA, improving understanding of neutrino mass hierarchy and mixing angles.

Findings

Measured $| riangle m^2_{32}|$ with high precision

Favored normal neutrino mass hierarchy by 1.9$\sigma$

Excluded most $\delta_{CP}$ values near $rac{\pi}{2}$ for inverted hierarchy

Abstract

The NOvA experiment has made a $4.4\sigma$-significant observation of $\bar\nu_{e}$ appearance in a 2 GeV $\bar\nu_{\mu}$ beam at a distance of 810 km. Using $12.33\times10^{20}$ protons on target delivered to the Fermilab NuMI neutrino beamline, the experiment recorded 27 $\bar\nu_{\mu} \rightarrow \bar\nu_{e}$ candidates with a background of 10.3 and 102 $\bar\nu_{\mu} \rightarrow \bar\nu_{\mu}$ candidates. This new antineutrino data is combined with neutrino data to measure the oscillation parameters $|\Delta m^2_{32}| = 2.48^{+0.11}_{-0.06}\times10^{-3}$ eV$^2/c^4$, $\sin^2 \theta_{23} = 0.56^{+0.04}_{-0.03}$ in the normal neutrino mass hierarchy and upper octant and excludes most values near $\delta_{\rm CP}=\pi/2$ for the inverted mass hierarchy by more than 3$\sigma$. The data favor the normal neutrino mass hierarchy by 1.9$\sigma$ and $\theta_{23}$ values in the upper octant by 1.6$\sigma$.

Figures

Tables

Table 1. Table 1: Systematic uncertainties on the total predicted numbers of signal and beam-related background events at the best fit point (see Table  4 ) in the ν e subscript 𝜈 𝑒 \nu_{e} selected samples in the neutrino and antineutrino datasets.

	${\nu }_e$ Signal	${\nu }_e$ Bkg.	${\overline{\nu }}_e$ Signal	${\overline{\nu }}_e$ Bkg.
Source	(%)	(%)	(%)	(%)
Cross-sections	+4.7/-5.8	+3.6/-3.4	+3.2/-4.2	+3.0/-2.9
Detector model	+3.7/-3.9	+1.3/-0.8	+0.6/-0.6	+3.7/-2.6
ND/FD diffs.	+3.4/-3.4	+2.6/-2.9	+4.3/-4.3	+2.8/-2.8
Calibration	+2.1/-3.2	+3.5/-3.9	+1.5/-1.7	+2.9/-0.5
Others	+1.6/-1.6	+1.5/-1.5	+1.4/-1.2	+1.0/-1.0
Total	+7.4/-8.5	+5.6/-6.2	+5.8/-6.4	+6.3/-4.9

Table 2. Table 2: Systematic and statistical uncertainties on the oscillation parameters sin 2 ⁡ θ 23 superscript 2 subscript 𝜃 23 \sin^{2}\theta_{23} , Δ ​ m 32 2 Δ subscript superscript 𝑚 2 32 \Delta m^{2}_{32} , and δ CP subscript 𝛿 CP \delta_{\rm CP} , evaluated at the best fit point (see Table  4 ).

	${\sin }^2{\theta }_{23}$	$\left|\Delta m_{32}^2\right|$	${\delta }_{\mathrm{CP}}$
Source	($\times {10}^{-3}$)	($\times {10}^{-5}{\mathrm{eV}}^2/c^4$)	($\pi$)
Calibration	+5.4 / -9.2	+2.2 / -2.6	+0.03 / -0.03
Neutron model	+6.0 / -13.0	+0.5 / -1.3	+0.01 / -0.00
Cross-sections	+4.1 / -7.7	+1.0 / -1.1	+0.06 / -0.07
$E_{\mu }$ scale	+2.3 / -3.0	+1.0 / -1.1	+0.00 / -0.00
Detector model	+1.9 / -3.2	+0.4 / -0.5	+0.05 / -0.05
Normalizations	+1.3 / -2.7	+0.1 / -0.2	+0.02 / -0.03
ND/FD diffs.	+1.0 / -4.0	+0.2 / -0.2	+0.06 / -0.07
Beam flux	+0.4 / -0.8	+0.1 / -0.1	+0.00 / -0.00
Total syst.	+9.7 / -20	+2.6 / -3.2	+0.11 / -0.12

Table 3. Table 3: Event counts at the FD, both observed and predicted at the best fit point (see Table  4 ).

	Neutrino beam		Antineutrino beam
	${\nu }_{\mu }$ CC	${\nu }_e$ CC	${\overline{\nu }}_{\mu }$ CC	${\overline{\nu }}_e$ CC
${\nu }_{\mu }\to {\nu }_{\mu }$	112.5	0.7	24.0	0.1
${\overline{\nu }}_{\mu }\to {\overline{\nu }}_{\mu }$	7.2	0.0	70.0	0.1
${\nu }_{\mu }\to {\nu }_e$	0.1	44.3	0.0	2.2
${\overline{\nu }}_{\mu }\to {\overline{\nu }}_e$	0.0	0.6	0.0	16.6
Beam ${\nu }_e+{\overline{\nu }}_e$	0.0	7.0	0.0	5.3
NC	1.3	3.1	0.8	1.2
Cosmic	2.1	3.3	0.8	1.1
Others	0.7	0.4	0.6	0.3
Signal	120${}_{-12}{}^{}{}_{}{}^{+10}$	44.3${}_{-4.0}{}^{}{}_{}{}^{+3.5}$	93.9${}_{-8.2}{}^{}{}_{}{}^{+8.1}$	16.6${}_{-1.0}{}^{}{}_{}{}^{+0.9}$
Background	4.2${}_{-0.6}{}^{}{}_{}{}^{+0.5}$	15.0${}_{-0.9}{}^{}{}_{}{}^{+0.8}$	${2.2}_{-0.4}^{+0.4}$	10.3${}_{-0.5}{}^{}{}_{}{}^{+0.6}$
Best fit	124	59.3	96.2	26.8
Observed	113	58	102	27

Table 4. Table 4: Summary of oscillation parameters. The top three are inputs to this analysis  Patrignani  et al. ( 2016 ) , while the rest are the best fits for different choices of the mass hierarchy (NH, IH) and θ 23 subscript 𝜃 23 \theta_{23} octant (UO, LO), along with the significance (in units of σ 𝜎 \sigma ) at which those combinations are disfavored. In addition to the region indicated, for NH, LO a small range of sin 2 ⁡ θ 23 superscript 2 subscript 𝜃 23 \sin^{2}\theta_{23} 0.45 − 0.48 0.45 0.48 0.45-0.48 is allowed at 1 ​ σ 1 𝜎 1\sigma   sup ( 2019b ) .

$\Delta m_{21}^2/\left({10}^{-5}{\mathrm{eV}}^2/c^4\right)$	$7.53\pm 0.18$
${\sin }^2{\theta }_{12}$	${0.307}_{-0.012}^{+0.013}$
${\sin }^2{\theta }_{13}$	$0.0210\pm 0.0011$
	NH, UO	NH, LO	IH, UO	IH, LO
$\Delta m_{32}^2/({10}^{-3}{\mathrm{eV}}^2/c^4$)	$+{2.48}_{-0.06}^{+0.11}$	$+2.47$	$-2.54$	$-2.53$
${\sin }^2{\theta }_{23}$	${0.56}_{-0.03}^{+0.04}$	$0.48$	$0.56$	$0.47$
${\delta }_{\mathrm{CP}}/\pi$	${0.0}_{-0.4}^{+1.3}$	$1.9$	$1.5$	$1.4$
	-	$+1.6\sigma$	$+1.8\sigma$	$+2.0\sigma$

Full text

††preprint: FERMILAB-PUB-19-272-ND

The observations of neutrino oscillations by many experiments Fukuda et al. (1998, 2002); Ahmad et al. (2002); Eguchi et al. (2003); Michael et al. (2006); Abe et al. (2011, 2012); An et al. (2012); Ahn et al. (2012) are well described by the mixing of three neutrino mass eigenstates $\nu_{1}$, $\nu_{2}$, and $\nu_{3}$ with the flavor eigenstates $\nu_{e}$, $\nu_{\mu}$, and $\nu_{\tau}$. The mixing is parameterized by a unitary matrix, $U_{\rm PMNS}$, which depends on three angles and a phase, $\delta_{\rm{CP}}$, that may break Charge-Parity (CP) symmetry. The oscillation frequencies are proportional to the neutrino mass splittings, $\Delta m^{2}_{21}\equiv m_{2}^{2}-m_{1}^{2}\simeq$7.5\text{\times}{10}^{-5}\text{,}\mathrm{e}\mathrm{V}^{2}\mathrm{/}{\it c}^{4} and $|\Delta m^{2}_{32}|\simeq$2.5\text{\times}{10}^{-3}\text{\,}\mathrm{e}\mathrm{V}^{2}\mathrm{/}{\it c}^{4}, and the angles are known to be large: $\theta_{12}\simeq 34^{\circ}$, $\theta_{13}\simeq 8^{\circ}$, $\theta_{23}\simeq 45^{\circ}$ Patrignani et al. (2016); $\delta_{\rm{CP}}$, however, is largely unknown.

Within this framework, several questions remain unanswered. The angle $\theta_{23}$ produces nearly maximal mixing but has large uncertainties. If maximal, it would introduce an unexplained $\mu-\tau$ symmetry; should it differ from $45^{\circ}$, its octant would determine whether $\nu_{\tau}$ or $\nu_{\mu}$ couples more strongly to $\nu_{3}$. Furthermore, while it is known that the two independent mass splittings differ by a factor of 30, the sign of the larger splitting is unknown. The $\nu_{1}$ and $\nu_{2}$ states that contribute most to the $\nu_{e}$ state could be lighter (“normal hierarchy”, NH) or heavier (“inverted hierarchy”, IH) than the $\nu_{3}$ state. This question has important implications for models of neutrino mass Mohapatra and Smirnov (2006); Nunokawa et al. (2008); Altarelli and Feruglio (2010); King (2015); Petcov (2018) and for the study of the Dirac vs. Majorana nature of the neutrino Pascoli and Petcov (2002); Bahcall et al. (2004). Additionally, neutrino mixing may be a source of CP violation if $\sin\delta_{\rm CP}$ is non-zero.

These questions can be addressed by the measurement of $\nu_{\mu}\rightarrow\nu_{\mu}$, $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$, $\nu_{\mu}\rightarrow\nu_{e}$, and $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ oscillations in matter over baselines $L$ of order $(100-1000)\,\textrm{km}$, with neutrino energies $E{\rm[GeV]}\simeq L{\rm[km]}\cdot|\Delta m_{32}^{2}{\rm[eV}^{2}/c^{4}{\rm]}|$. Several long-baseline experiments have reported observations of $\nu_{\mu}\rightarrow\nu_{\mu}$ Ahn et al. (2006); Adamson et al. (2014); Abe et al. (2018a); Acero et al. (2018), $\nu_{\mu}\rightarrow\nu_{e}$ Adamson et al. (2014); Abe et al. (2018a); Acero et al. (2018), and $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$ Adamson et al. (2014); Abe et al. (2018a), but a statistically significant observation of $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ has not previously been made. This report combines the first antineutrino measurements using the NOvA detectors with the neutrino data reported in Ref. Acero et al. (2018) in a reoptimized analysis yielding a new determination of the oscillation parameters $|\Delta m^{2}_{32}|$, $\sin^{2}\theta_{23}$, $\delta_{\rm CP}$, and the neutrino mass hierarchy.

The NOvA experiment measures oscillations by comparing the energy spectra of neutrino interactions in two detectors placed in the Fermilab NuMI beam Adamson et al. (2016) at distances of $1\text{\,}\mathrm{k}\mathrm{m}$ (Near Detector, ND) and $810\text{\,}\mathrm{k}\mathrm{m}$ (Far Detector, FD) from the production target. The $14\text{\,}\mathrm{k}\mathrm{t}\mathrm{o}\mathrm{n}$ FD measures 15\text{\,}\mathrm{m}$\times$15\text{\,}\mathrm{m}$\times$60\text{\,}\mathrm{m} while the $290\text{\,}\mathrm{t}\mathrm{o}\mathrm{n}$ ND consists of a 3.8\text{\,}\mathrm{m}$\times$3.8\text{\,}\mathrm{m}$\times$12.8\text{\,}\mathrm{m} main detector followed by a muon range stack. Both detectors use liquid scintillator Mufson et al. (2015) contained in PVC cells that are 6.6\text{\,}\mathrm{c}\mathrm{m}$\times$3.9\text{\,}\mathrm{c}\mathrm{m} (0.15 radiation lengths $\times$ 0.45 Molière radii) in cross section and span the height and width of the detectors in planes of alternating vertical and horizontal orientation. The ND is located $100\text{\,}\mathrm{m}$ underground. The FD operates on the surface with modest shielding resulting in $130\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ of cosmic-ray activity. The detectors are located $14.6\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ off the beam axis where the neutrino energy spectrum peaks at $2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. Magnetic focusing horns in the beamline charge-select neutrino parents giving 96% (83%) pure $\nu_{\mu}$ ($\bar{\nu}_{\mu}$) event samples between 1 and $5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. Most contamination is wrong-sign ($\bar{\nu}$ in the $\nu$ beam, or vice versa) with $<1\%$ $\nu_{e}+\bar{\nu}_{e}$ contamination.

This Letter reports data from an antineutrino beam run spanning from June 29, 2016 to February 26, 2019, with an exposure of $12.33\times 10^{20}$ protons-on-target (POT) delivered during $317.0\text{\,}\mathrm{s}$ of beam-on time, combined with the previously reported Acero et al. (2018) neutrino beam exposure of $8.85\times 10^{20}$ POT and $438.2\text{\,}\mathrm{s}$. During these periods, the proton source achieved a peak hourly-averaged power of $742\text{\,}\mathrm{k}\mathrm{W}$.

The flux of neutrinos delivered to the detectors is calculated using a simulation of the production and transport of particles through the beamline components Adamson et al. (2016); Agostinelli et al. (2003) and reweighted Aliaga et al. (2016) to incorporate external measurements of hadron production and interactions Paley et al. (2014); Alt et al. (2007); Abgrall et al. (2011); Barton et al. (1983); Seun (2007); Tinti (2010); Lebedev (2007); Baatar et al. (2013); Skubic et al. (1978); Denisov et al. (1973); Carroll et al. (1979); Abe et al. (2013); Gaisser et al. (1975); Cronin et al. (1957); Allaby et al. (1969); Longo and Moyer (1962); Bobchenko et al. (1979); Fedorov et al. (1978); Abrams et al. (1970). Neutrino interactions in the detector are simulated using the genie event generator Andreopoulos et al. (2010). The cross section model has been tuned to improve agreement with external measurements and ND data, reducing uncertainties in the extrapolation of measurements in the ND to the FD. As in Ref. Acero et al. (2018), we set $M_{A}$ in the quasielastic dipole form factor to $1.04\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}{\it c}^{2}$ Meyer et al. (2016) and use corrections to the charged-current (CC) quasielastic cross section derived from the random phase approximation Nieves et al. (2004); Gran (2017). In this analysis, we also apply this effect to baryon resonance production as a placeholder for the unknown nuclear effect that produces a suppression observed at low four-momentum transfer in our and other measurements Adamson et al. (2015); Aguilar-Arevalo et al. (2011); McGivern et al. (2016); Altinok et al. (2017). Additionally, we increase the rate of deep-inelastic scattering with hadronic mass $W>$1.7\text{,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}{\it c}^{2}$$ by 10% to match our observed rates of short track-length $\nu_{\mu}$ CC events. We model multi-nucleon ejection interactions following Ref. Katori (2015) and adjust the rates in bins of energy transfer, $q_{0}$, and 3-momentum transfer, $|\vec{q}|$, for $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ separately to maximize agreement in the ND. The calculation of the $\nu_{e}$ and $\bar{\nu}_{e}$ rates uses these same models.

The energy depositions of final-state particles are simulated with geant4 Agostinelli et al. (2003) and input to a custom simulation of the production of, and the detector response to, scintillation and Cherenkov light Aurisano et al. (2015). The absolute energy scale of the detectors is calibrated to within $\pm 5\%$ using the minimum ionizing portion of cosmic-ray muon tracks that stop in the detectors.

Cells with activity above threshold (hits) are grouped based on their proximity in space and time to produce candidate neutrino events. Events are assigned a vertex, and clusters are formed from hits likely to be associated with particles produced there Baird et al. (2015). These clusters are categorized as electromagnetic or hadronic in origin using a convolutional neural network (CNN) Psihas (2018). Hits forming tracks are identified as muons by combining information on the track length, $dE/dx$, vertex activity, and scattering into a single particle identification (PID) score Raddatz (2016). The same reconstruction algorithms are applied to events from data and simulation in both detectors.

The $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ candidates are required to have a vertex inside the fiducial volume and no evidence of particles exiting the detector. The $\nu_{e}$ and $\bar{\nu}_{e}$ candidates are divided into a “core” sample which satisfies these containment requirements, and a “peripheral” sample which loosens these requirements for the most signal-like event topologies. A second CNN Aurisano et al. (2016) serves as the primary PID, classifying event topologies as $\nu_{e}$ CC, $\nu_{\mu}$ CC, $\nu_{\tau}$ CC, neutral-current (NC), or cosmic ray. The network is trained on simulated neutrino events and cosmic-ray data, separately for neutrino and antineutrino beam conditions. It has an improved architecture and higher rate of cosmic ray rejection over the previous network Acero et al. (2018). Events identified as $\nu_{\mu}$ CC are further required to contain at least one track classified as a muon.

Several requirements further reduce cosmic-ray backgrounds. For the $\nu_{\mu}$ CC sample, a boosted decision tree (BDT) algorithm based on vertex position and muon-like track properties is used. Events in the core $\nu_{e}$ sample not aligned with the beam direction and that are near the top of the detector are rejected. Events characterized as detached bremsstrahlung showers from cosmic tracks are also removed, as are events whose topology is consistent with photons entering from the detector north side where there is less shielding. Events in the $\nu_{e}$ peripheral sample are tested against a BDT classifier using event position and direction information to separate them from cosmic-ray topologies.

The selection of $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ CC events is 31.2% (33.9%) efficient relative to true interactions in the fiducial volume, resulting in 98.6% (98.8%) pure samples at the FD during neutrino (antineutrino) beam operation. Both $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ are counted as signal for the disappearance measurements. Selections against exiting particle tracks are the largest source of inefficiency. The efficiency for selecting signal $\nu_{e}$ CC ($\bar{\nu}_{e}$ CC) events is 62% (67%). Purities for the signal $\nu_{e}$ ($\bar{\nu}_{e}$) samples fall in the range 57–78% (55–77%) depending on the impact of oscillations on the signal and wrong-sign background levels. These efficiencies and purities differ from those quoted in Ref. Acero et al. (2018) due to a reoptimization of the selection algorithms Blackburn (2019). The wrong-sign component of the selected $\nu_{\mu}$ sample in the ND is calculated to be $2.8\pm 0.3\%$ and $10.6\pm 1.1\%$ for the neutrino and antineutrino beams. These fractions were found to be consistent with a data-driven estimate based on the rate of $\nu_{\mu}$ CC and NC interactions with associated detector activity indicative of neutron capture.

The incident neutrino energy is reconstructed from the measured energies of the final-state lepton and recoil hadronic system. The lepton energy is estimated from track length for muon candidates and from calorimetric energy for electron candidates. The hadronic energy is estimated from the sum of the calibrated hits not associated with the primary lepton. The neutrino energy resolution at the FD is 9.1% (8.1%) for $\nu_{\mu}$ CC ($\bar{\nu}_{\mu}$ CC) events and 10.7% (8.8%) for $\nu_{e}$ CC ($\bar{\nu}_{e}$ CC) events. The $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ events with the lowest hadronic energy fraction give the best energy resolution and lowest backgrounds, yielding the most precise measurement of the oscillated spectral shape, so we analyzed the spectra separately in quartiles of this variable Acero et al. (2018).

The energy spectra of the selected $\nu_{\mu}$ CC and $\nu_{e}$ CC interactions in the ND during neutrino and antineutrino beam operations are shown in Fig. 1. The selected ND $\nu_{e}$ sample consists entirely of background sources for the $\nu_{e}$ appearance measurement, predominantly the intrinsic beam $\nu_{e}$ component, along with misidentified $\nu_{\mu}$ CC and NC interactions. We analyze the $\nu_{e}$ candidate energy spectra in two bins of $\nu_{e}$ PID (“low” and “high”) to isolate a highly pure sample of $\nu_{\mu}\rightarrow\nu_{e}$ and $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ at the FD. In the ND, the high-PID sample is dominated by intrinsic beam $\nu_{e}$. A third bin containing the “peripheral” events is added for the FD.

The $\nu_{\mu}$ and $\nu_{e}$ signal spectra at the FD are predicted for the neutrino and antineutrino beams separately and are based on the observed spectra of $\nu_{\mu}$ candidate events in the ND. The true neutrino energy spectrum at the ND is estimated using the measured event rates in bins of reconstructed energy and the energy distributions of simulated events found to populate those bins. This true spectrum is corrected for differences in flux and acceptance between the ND and FD, as well as differences in the $\nu_{\mu}$ and $\nu_{e}$ cross sections; oscillations are then applied to yield predictions for the true $\nu_{\mu}$ and $\nu_{e}$ spectra at the FD. These spectra are then transformed into reconstructed energy using the underlying energy distributions from simulated neutrino interactions in the FD.

The predicted background spectra at the FD are also primarily data-driven. Data collected out-of-time with the NuMI beam provide a measurement of the rate of cosmic-ray backgrounds in the $\nu_{\mu}$ and $\nu_{e}$ samples. Neutrino backgrounds calculated to populate the FD $\nu_{e}$ spectra are corrected based on the reconstructed $\nu_{e}$ candidates at the ND. The procedure from Ref. Acero et al. (2018) is followed to determine corrections for each background component in the neutrino-mode beam, while for the antineutrino-mode beam a single scale factor is used. The remaining backgrounds, which include any misidentified neutrino events in the $\nu_{\mu}$ samples and misidentified $\nu_{\tau}$ interactions in the $\nu_{e}$ samples, make up less than 2% of the FD candidates and are taken directly from simulation.

To evaluate the impact of systematic uncertainties we recompute the extrapolation from the ND to the FD varying the parameters used to model the neutrino fluxes, neutrino cross sections, and the detector response. The procedure accounts for changes in the composition of the $\nu_{e}$ background, and for impact on the transformation to and from true and reconstructed energies due to variations in the model parameters. We parameterize each systematic variation and compute its effect in each analysis bin. These parameters are included in the oscillation fit constrained within their estimated uncertainties by penalty terms in the likelihood function.

The oscillation parameters that best fit the FD data are determined through minimization of a Poisson negative log-likelihood, $-2\ln\cal{L}$, considering three unconstrained parameters, $\Delta m^{2}_{32}$, $\sin^{2}\theta_{23}$, and $\delta_{\rm{CP}}$, as well as 53 constrained parameters covering the other PMNS oscillation parameters and the sources of systematic uncertainty summarized in Tables 1 and 2. The two-detector design and extrapolation procedure significantly reduce the effect of the $\simeq$10–20% a priori uncertainties on the beam flux and cross sections. The principal remaining uncertainties are neutrino cross sections, the energy scale calibration, the detector response to neutrons, and differences between the ND and FD that cannot be corrected by extrapolation.

The selection criteria and techniques used in the analysis were developed on simulated data prior to inspection of the FD data distributions. Figure 1 shows the energy spectra of the $\nu_{\mu}$ CC, $\bar{\nu}_{\mu}$ CC, $\nu_{e}$ CC, and $\bar{\nu}_{e}$ CC candidates recorded at the FD overlaid on their oscillated best-fit expectations. Table 3 summarizes the total event counts and estimated compositions of the selected samples. We recorded 102 $\bar{\nu}_{\mu}$ candidate events at the FD, reflecting a significant suppression from the unoscillated expectation of 476. We find 27 $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ candidate events with an estimated background of $10.3^{+0.6}_{-0.5}$, a excess over the predicted background. This observation is the first evidence of $\bar{\nu}_{e}$ appearance in a $\bar{\nu}_{\mu}$ beam over a long baseline. These new antineutrino data are analyzed together with 113 $\nu_{\mu}$ and 58 $\nu_{\mu}\rightarrow\nu_{e}$ candidates from the previous data set.

Table 4 shows the overall best-fit parameters, as well as the best fits for each choice of $\theta_{23}$ octant and hierarchy. The best-fit point is found for the normal hierarchy with $\theta_{23}$ in the upper octant where $-2\ln\cal{L}$ = 157.1 for 175 degrees of freedom (goodness-of-fit $p=0.91$ from simulated experiments). The measured values of $\theta_{23}$ and $\Delta m^{2}_{32}$ are consistent with the previous NOvA measurement Acero et al. (2018) that used only neutrino data, and are consistent with maximal mixing within $1.2\sigma$.

Confidence intervals for the oscillation parameters are determined using the unified approach Feldman and Cousins (1998), as detailed in Ref. Sousa et al. (2019). Figure 2 compares the 90% confidence level contours in $\Delta m^{2}_{32}$ and $\sin^{2}\theta_{23}$ with those of other other experiments Adamson et al. (2014); Aartsen et al. (2018); Abe et al. (2018b, a). Figure 3 shows the allowed regions in $\sin^{2}\theta_{23}$ and $\delta_{\rm{CP}}$. These results exclude most values near $\delta_{\rm CP}=\pi/2$ in the inverted mass hierarchy by more than 3$\sigma$; specifically the intervals between -0.040.97$$\pi in the lower $\theta_{23}$ octant and 0.040.91$$\pi in the upper octant. The data prefer the normal hierarchy with a significance of $1.9\sigma$ ($p=0.057$, $CL_{s}=0.091$ Read (2002)) and the upper $\theta_{23}$ octant with a significance of $1.6\sigma$ ($p=0.11$), profiling over all other parameter choices.