Precision constraints for three-flavor neutrino oscillations from the full MINOS+ and MINOS data set

We report the final measurement of the neutrino oscillation parameters $\Delta m^2_{32}$ and $\sin^2\theta_{23}$ using all data from the MINOS and MINOS+ experiments. These data were collected using a total exposure of $23.76 \times 10^{20}$ protons on target producing $\nu_{mu}$ and $\overline{\nu_\mu}$ beams and 60.75 kt$\cdot$yr exposure to atmospheric neutrinos. The measurement of the disappearance of $\nu_{\mu}$ and the appearance of $\nu_e$ events between the Near and Far detectors yields $|\Delta m^2_{32}|=2.40^{+0.08}_{-0.09}~(2.45^{+0.07}_{-0.08}) \times 10^{-3}$ eV$^2$ and $\sin^2\theta_{23} = 0.43^{+0.20}_{-0.04} ~(0.42^{+0.07}_{-0.03})$ at 68% C.L. for Normal (Inverted) Hierarchy.

This Letter reports new measurements of ∆m 2 32 and sin 2 (θ 23 ) using the complete set of beam and atmospheric data taken with the MINOS detectors. Two distinct beam energy configurations of the NuMI neutrino beam at Fermi National Accelerator Laboratory (FNAL) corresponded to two phases of the MINOS (2005-2012) and MINOS+ (2013-2016) long-baseline, on-axis neutrino oscillation experiments. The MINOS+ dataset significantly increases the statistics of the MINOS measurements [4] in the energy region above the oscillation maximum in the standard model of oscillations. Over this region extending from 1.5 GeV to above 10 GeV in E ν , MINOS and MINOS+ monitor the increase of ν µ -flavor survival probability. This provides additional sensitivity, not available to narrow-band beam experiments, for measuring the extent to which θ 23 deviates from non-maximal mixing. Monitoring this revival rate supplements the measurement of the depth of the oscillation maximum which occurs within a small span of E ν . Effects from nonstandard neutrino interactions [13], neutrino decay [14,15], decoherence [16], or the existence of sterile neutrinos [17,18], could manifest themselves over the large energy range. Consequently, the measurements reported here simultaneously provide a stringent test for such phenomena that lie outside the purview of conventional three-flavor neutrino oscillations.
The MINOS and MINOS+ long-baseline, on-axis neutrino oscillation experiments recorded two distinct phases of exposure to the NuMI neutrino beam [19] at Fermilab utilizing the MINOS Near and Far detectors [20]. Both detectors were functionally equivalent magnetized steelscintillator, tracking, sampling calorimeters. The Near Detector (ND) was 1.04 km from the target, 103 m underground, and had a mass of 980 t. The Far Detector (FD) was 735 km from the target, 705 m underground, and had a mass of 5.4 kt. The detectors had average toroidal magnetic fields of 1.4 T, to enable the separation of ν µ from ν µ . The FD was also used to study atmospheric neutrinos [21] making use of the scintillator veto shield to improve cosmic muon background rejection.
During MINOS-phase data taking, the NuMI beam operated primarily in a low-energy beam configuration, producing muon neutrinos or antineutrinos, depending on the polarity of the pulsed magnetic horns, with a peak energy around 3 GeV. The MINOS low-energy beam exposure was 10.56 × 10 20 protons on target (POT) in ν µmode and 3.36 × 10 20 POT in ν µ -mode. The MINOS ν µmode sample included an additional 0.15 × 10 20 POT exposure in a high-energy ν µ -mode with a peak ν µ energy of 9 GeV. The analysis of this MINOS-phase data has been described previously, presenting a 37.88 kt · yr sample of atmospheric neutrinos, measurements of both ν µ and ν µ disappearance, and ν e and ν e appearance [4,12,21,22]. This analysis uses the complete MINOS data set described above and, in addition, includes 22.87 kt · yr of atmospheric-neutrino data from 2011-2016, along with the complete three years of MINOS+ ν µ -mode beam data corresponding to an exposure of 9.69 × 10 20 POT. In the MINOS+ phase, the NuMI beam operated in the medium-energy configuration, producing a ν µ beam peaking at near 7 GeV. The MINOS+ ν µ charged current (CC) interactions in the ND were composed of 96.9% ν µ , 1.9% ν µ , and 1.2% (ν e + ν e ). In comparison, the MINOS low-energy ν µ -mode ND data were composed of 92.9% ν µ , 5.8% ν µ , and 1.3% (ν e +ν e ) CC interactions [23]. There is no ν e appearance included in the MINOS+ phase analysis since the higher energy exposure increases the neutralcurrent (NC) backgrounds to the low energy ν e appearance signal.
The Monte Carlo (MC) modeling was unchanged compared to the most recent previous publications. The accelerator beam neutrino flux was simulated using the FLUGG package [24] and the atmospheric neutrino flux using the Bartol calculations [25]. Beam neutrino interactions are simulated using NEUGEN3 [26] and interactions of atmospheric neutrinos using NUANCE [27]. The detector response to final-state particles is simulated, for both beam and atmospheric neutrino interactions, using a combination of GEANT3 [28] and GCALOR [29]. For MINOS+, the beam-neutrino reconstruction algorithms were tuned to account for the higher occupancy in the ND arising from the increased beam-neutrino flux. The data observed in the ND are used to tune the MINOS and MINOS+ flux simulations. The MINOS flux tuning procedure, described previously [23], was improved upon to separate the effect of the rate of secondary hadron production in the target from that of the charged particle focussing by the horns. This procedure combined data from special ND data taken with horn currents between 0 and 200 kA.
During the MINOS+ running period, it was observed that the neutrino energy peak position was shifted in the ND from that predicted by the MC by about 400 MeV. Furthermore, during the final running period, the upstream support of the first magnetic horn moved downward 4 mm over a period of a few months. Checks on these effects, however, showed that the oscillation parameter measurement is robust against these MC/data differences once the ND data are used to correct the neutrino flux to less than 0.05σ in both oscillation parameters.
This analysis uses ν µ and ν µ events, which result in a µ − or µ + in the final state. Signal events have a characteristic muon track with a hadron shower near the interaction point. The major source of background is from NC events that produce hadron showers with short tracks. A multivariate k-Nearest Neighbor (kNN) algorithm [30] was used by MINOS to select CC ν µ and ν µ interactions based on event topology and the characteristic muon track energy deposition. For MINOS+, the algorithm was trained using representative MINOS+ CC and NC events from the MC [31]; the distribution of the kNN discriminant in MINOS+ ND MC is shown in Fig. 1. Events with a value of the kNN discriminant below 0.3 were removed. The selected CC ν µ and ν µ sample has a purity of 99.1% in the ND and 99.3% in the FD, with the impurities due to NC interactions.
The visible energy of the selected ν µ and ν µ events was reconstructed from the sum of the muon track and the hadronic shower energy. The muon energy was measured from the range in the detector for tracks that were fully contained in the detector, or from the curvature in the magnetic field for tracks that exited the detector. The shower energy was estimated using another kNN algorithm that compares the topology of the event to a library of MC events and uses the closest-matching MC events to estimate the true energy of the hadron shower [31,32].
The beam data consist of ν µ and ν µ events with reconstructed interaction vertices within the detector's fiducial volume. The MINOS ν µ dataset includes a sample of non-fiducial muons from neutrinos that interacted outside the detector's fiducial volume or in the rock surrounding the detector, identified by muons entering the front or sides of the detector in time with the beam. Since this sample has poorly reconstructed interaction energy, its impact on the oscillation measurement is limited and thus did not warrant selecting a similar sample from the MINOS+ data [33].
The MINOS+ reconstructed CC ν µ and ν µ energy spectra in the ND were used to predict the energy spectrum expected in the FD using a beam transfer matrix, as was done for MINOS [23]. This prediction for MINOS and MINOS+ combined without oscillations is shown in Fig. 2 (orange line) compared to the selected data (black points). Also shown is the ratio of observed FD events to the number of predicted events assuming no oscillations as a function of reconstructed neutrino energy. The energy-dependent deficit of ν µ and ν µ interactions is clearly observed, indicating the expected three-flavor oscillatory nature of the disappearance. The MINOS+ data provide significant additional statistical power integrated between 4 and 8 GeV energy range over the lower energy MINOS data. Atmospheric neutrinos were separated from the cosmic ray backgrounds into three separate samples [21,[34][35][36]. The first sample of ν µ and ν µ CC interactions required a reconstructed interaction vertex in the detector's fiducial volume. The second sample were also containedvertex, but shower-like events, primarily CC ν e , CC ν e , and NC interactions that were used to constrain the atmospheric neutrino flux. The third sample required a reconstructed upward-going muon track and contained nonfiducial events that were initiated by atmospheric ν µ and ν µ interactions that occurred in the rock around the detector. The number of observed and predicted neutrino events for MINOS and MINOS+ are given in Table I. The combined fit to the MINOS and MINOS+ ν µ disappearance data was carried out independently of the MINOS ν e appearance fit. To determine values of the oscillation parameters from the muon neutrino data, a maximum likelihood fit was performed by varying ∆m 2 32 , sin 2 θ 23 , sin 2 θ 13 , and δ CP and using the negative loglikelihood function: where µ j and n j are the numbers of expected and observed events in bin j of the reconstructed energy distribution, α k include the fitted systematic parameters and a constraint on sin 2 θ 13 with corresponding uncertainties of σ α k . The mixing angle θ 13 was constrained to sin 2 θ 13 = 0.0210 ± 0.0011 [37]. The solar parameters were fixed to ∆m 2 21 = 7.54 × 10 −5 eV 2 and sin 2 θ 12 = 0.307 [38] since they have no effect on the oscillation parameter measurement. The likelihood function contained 17 nuisance parameters that accounted for the largest systematic uncertainties as discussed in previous publications [15,21].
Uncertainties on the flux of beam neutrinos were obtained from the fits performed using the ND data to tune the flux simulation. Separate uncertainties were calculated for the MINOS and MINOS+ beam-neutrino data sets. All uncertainties related to the interactions of neutrinos in the detector and neutrino reconstruction were unchanged from MINOS to MINOS+. All uncertainties on the atmospheric-neutrino samples were unchanged between the previously analyzed data and the new data added for this publication.
The effects of the systematic uncertainties on the ν µ disappearance measurement were studied with MC samples modified by shifting the uncertainties by one standard deviation. Table II shows the largest of the systematic uncertainties on the ∆m 2 32 measurement. The dominant uncertainties associated with the beam data are the shower energy uncertainty and the relative normalization between the two detectors The shower energy uncertainty, which has the second largest impact on ∆m 2 32 , averages at about 8% below 3 GeV and approaches 6.6% at higher shower energies. The 1.6% relative normalization uncertainty accounts for differences in event selection and reconstruction between the ND and FD as well as uncertainties on each detector's fiducial mass and livetime. The uncertainty on the measurement of the muon energy is fully correlated between the beam and atmospheric samples and is 2% when calculated from range and 3% when calculated from curvature [15]. The 15% atmospheric normalization uncertainty for contained-vertex events comes from uncertainties on the flux and the neutrino cross section [21]. This normalization uncertainty has the largest effect on ∆m 2 32 . The atmospheric nonfiducial events have a normalization uncertainty of 25% due to larger flux uncertainties of the much higher energy cosmic muons associated with this sample. Atmospheric contained-vertex events have an additional 10% uncertainty on theν µ /ν µ ratio. These three uncertainties have the largest effects on measuring sin 2 θ 23 in the combined fit of beam and atmospheric data. The final result summed the likelihood contributions coming separately from the combined ν µ disappearance and the MINOS ν e appearance [12,22] data sets, treating the systematic uncertainties as uncorrelated. Fig.  2 shows the MC predictions for the best fit oscillation parameters for MINOS (hatched red) and MINOS+ (hatched blue). The combined MINOS and MINOS+ MC spectrum is also shown (cyan). All MC samples with expected neutrino oscillations include the small contri-bution of background events from ν τ and ν τ appearance. The oscillation parameters best-fit point obtained using only the MINOS+ neutrino beam data falls within the 1σ contour from the previous MINOS measurement [4].  The 68% and 90% confidence level intervals in sin 2 θ 23 and ∆m 2 32 parameter space for the normal hierarchy obtained for the beam and atmospheric samples separately are shown in Fig. 3. The confidence level intervals include the best fit points for the beam sample at sin 2 θ 23 = 0.38, ∆m 2 32 = 2.48 × 10 −3 eV 2 and for the atmospheric sample at sin 2 θ 23 = 0.52, ∆m 2 32 = 2.11 × 10 −3 eV 2 . Studies of the compatibility of the atmospheric and beam results show a probability of 22% that they come from the same oscillation parameters.

MINOS
The oscillation parameters at the best fit point and confidence limits from the overall combined fit for the normal and inverted hierarchy are shown in Table III. Fig. 4 shows the confidence limits on ∆m 2 32 and sin 2 θ 23 and the likelihood profiles as functions of ∆m 2 32 and sin 2 θ 23 . The best fit point at ∆m 2 32 = 2.40 +0.08 −0.09 × 10 −3 eV 2 and sin 2 θ 23 = 0.43 +0.20 −0.04 weakly favors nonmaximal mixing at 0.91 σ and the normal hierarchy at 0.45 σ.
In summary, analysis of the ν µ disappearance and ν e appearance samples from the complete beam and atmospheric data sets of the MINOS and MINOS+ run phases has been presented and provides new, stringent and competitive constraints on the oscillation parameters ∆m 2 32 and sin 2 θ 23 , weakly favors non-maximal mixing, and exhibits octant degeneracy.