Measurement of electron antineutrino oscillation based on 1230 days of operation of the Daya Bay experiment

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Measurement of electron antineutrino oscillation based on 1230 days of operation of the Daya Bay experiment

F. P. An et al. (Daya Bay Collaboration)

Phys. Rev. D 95, 072006 – Published 6 April 2017

Abstract

A measurement of electron antineutrino oscillation by the Daya Bay Reactor Neutrino Experiment is described in detail. Six 2.9-GWth nuclear power reactors of the Daya Bay and Ling Ao nuclear power facilities served as intense sources of ${\bar{ν}}_{e}$ ’s. Comparison of the ${\bar{ν}}_{e}$ rate and energy spectrum measured by antineutrino detectors far from the nuclear reactors ( $\sim 1500 - 1950 m$ ) relative to detectors near the reactors ( $\sim 350 - 600 m$ ) allowed a precise measurement of ${\bar{ν}}_{e}$ disappearance. More than 2.5 million ${\bar{ν}}_{e}$ inverse beta-decay interactions were observed, based on the combination of 217 days of operation of six antineutrino detectors (December, 2011–July, 2012) with a subsequent 1013 days using the complete configuration of eight detectors (October, 2012–July, 2015). The ${\bar{ν}}_{e}$ rate observed at the far detectors relative to the near detectors showed a significant deficit, $R = 0.949 \pm 0.002 (stat) \pm 0.002 (syst)$ . The energy dependence of ${\bar{ν}}_{e}$ disappearance showed the distinct variation predicted by neutrino oscillation. Analysis using an approximation for the three-flavor oscillation probability yielded the flavor-mixing angle $\sin^{2} 2 θ_{13} = 0.0841 \pm 0.0027 (stat) \pm 0.0019 (syst)$ and the effective neutrino mass-squared difference of $| Δ m_{ee}^{2} | = (2.50 \pm 0.06 (stat) \pm 0.06 (syst)) \times 10^{- 3} {eV}^{2}$ . Analysis using the exact three-flavor probability found $Δ m_{32}^{2} = (2.45 \pm 0.06 (stat) \pm 0.06 (syst)) \times 10^{- 3} {eV}^{2}$ assuming the normal neutrino mass hierarchy and $Δ m_{32}^{2} = (- 2.56 \pm 0.06 (stat) \pm 0.06 (syst)) \times 10^{- 3} {eV}^{2}$ for the inverted hierarchy.

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Received 31 October 2016

DOI:https://doi.org/10.1103/PhysRevD.95.072006

Physics Subject Headings (PhySH)

Neutrino oscillations Nuclear reactors

Particles & FieldsNuclear Physics

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Vol. 95, Iss. 7 — 1 April 2017

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Figure 1
Layout of the Daya Bay experiment. The Daya Bay and Ling Ao nuclear power plant (NPP) reactors (red circles) were situated on a narrow coastal shelf between the Daya Bay coastline and inland mountains. Two antineutrino detectors installed in each underground experimental hall near to the reactors (EH1 and EH2) measured the ${\bar{ν}}_{e}$ flux emitted by the reactors, while four detectors in the far experimental hall (EH3) measured a deficit in the ${\bar{ν}}_{e}$ flux due to oscillation. The detectors were built and initially tested in a surface assembly building (SAB), transported to a liquid scintillator hall (LS Hall) for filling, and then installed in an experimental hall.
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Figure 2
Cross-sectional diagram of an AD. Scintillation light was produced when a reactor ${\bar{ν}}_{e}$ interacted within the central 20-ton GdLS target, which was contained in a transparent acrylic vessel. The target was nested within an additional 20 tons of pure LS to increase efficiency for detection of gamma rays produced within the target. Scintillation light was detected by 192 photomultipliers mounted on the inner circumference of a 5 by 5 m stainless steel vessel, which was filled with MO.
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Figure 3
Diagram of a near site detector system. Two ADs were immersed in a water-Cherenkov muon detector which functioned as both a passive radiation shield and an active muon tag. Tyvek sheets divided the pool into two optically separate detectors, the inner and outer water shields. An RPC system covered the pool, providing additional muon identification.
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Figure 4
Example of the baseline-subtracted ADC charge distribution of uncorrelated PMT signals modeled using Eq. (6).
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Figure 5
Average number of ADC counts per single photoelectron, averaged over all PMT channels within each AD versus time. The shaded vertical band delineates the Summer 2012 shutdown period, during which the high-voltage mainframes were frequently power-cycled and data-taking was partially interrupted for installation activities.
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Figure 6
Top: Observed light yield versus time, in units of observed PEs per MeV, as obtained using weekly deployments of a ${}^{60}{Co}$ source at the center of each detector. Bottom: Observed light yield obtained using spallation neutron capture on Gd distributed throughout the target volume. A consistent $\sim 1 %$ per year decline in the light yield was observed for all detectors. The offset in light yield between these two calibration references was due to the nonlinear response of the detectors. The vertical shaded band indicates the summer of 2012 shutdown period, during which data-taking was partially interrupted for installation of the final two detectors.
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Figure 7
Variation in light yield versus vertical ( $z$ ) and radial ( $r$ ) position of an interaction within EH1-AD1 for reconstruction A (blue surface). The functional form of the variation $f_{b} (z, r)$ was motivated by simulation, and constrained using the relative light yield of ${}^{60}{Co}$ calibration data from ACU A ( $r = 0 mm$ ), ACU B ( $r = 1350 mm$ ) and ACU C ( $r = 1772.5 mm$ ) (red points). While the modeled correction was between-6% to 17% over the volume of a detector, only slight differences ( $< 3.2 %$ ) were observed between the eight detectors.
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Figure 8
Variation in light yield versus vertical ( $z$ ) and radial ( $r^{2}$ ) position of neutron capture interactions within EH1-AD1 for reconstruction B. Each pixel shows the ratio of the observed light yield over the average for the entire GdLS region. The dashed lines indicate the boundary between the GdLS and LS regions. Neutrons captured on Gd were used for the innermost GdLS region, while neutrons captured on hydrogen were used for the outermost LS region. Only slight differences ( $< 3.0 %$ ) in the nonuniformity were observed between the eight detectors.
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Figure 9
Variation in light yield versus radial (top) and vertical (bottom) position of neutron capture interactions for reconstruction B. Each point shows the ratio of the observed light yield over the average for the entire the GdLS region. The variation was primarily due to the optical acceptance versus position, and was reproduced by optical simulation. Only slight differences ( $< 3.0 %$ ) in the nonuniformity were observed between the eight detectors.
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Figure 10
Top: Stability of the mean reconstructed energy of signals from spallation neutron capture on H versus time, obtained from calibration using a ${}^{60}{Co}$ source located at the detector center (reconstruction A). Bottom: The same, but for the calibration method which relies on spallation neutrons (reconstruction B). Each point corresponds to one day. The shaded vertical band represents the 2012 shutdown period.
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Figure 11
Comparison of the mean reconstructed energy between antineutrino detectors for a variety of calibration references using (a) calibration with ${}^{60}{Co}$ sources (reconstruction A) and (b) calibration with spallation neutrons (reconstruction B). $E_{AD}$ is the reconstructed energy determined for each AD, while $⟨ E ⟩$ is the eight-detector average. The mean energy for each calibration reference was obtained from the corresponding peak in the energy spectrum of all regular data (natural alphas and gammas, neutrons from IBD and muon spallation) and all weekly calibration runs (gammas from ${}^{68}{Ge}$ and ${}^{60}{Co}$ sources, neutrons from Am-C sources) taken during the time period when all eight ADs were in operation. An effective fiducial volume selection has been applied on distributed sources to suppress interactions outside the antineutrino target where AD-to-AD differences are larger. Error bars are statistical only, and systematic variations between detectors for all calibration references were $< 0.2 %$ for both reconstruction methods.
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Figure 12
Energy resolutions measured for calibration sources, neutron captures following inverse beta decay, and natural $α$ radioactivity within the scintillator (open blue markers). The detector resolution was limited by the statistical uncertainty of photoelectron counting, as modeled using Eq. (14) (blue solid line). The resolutions were consistent with Monte Carlo simulation (solid orange markers). The simulation predicted that the intrinsic detector resolution (dashed orange line) was slightly better than that estimated for the calibration sources (solid orange line), since the latter suffered from optical shadowing by the source encapsulation. The energy resolutions for $α$ particles emitted by natural radioactivity (open blue triangles) confirmed the intrinsic energy resolution.
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Figure 13
Ratio of observed energy to true energy for $γ$ -ray reference data relative to the estimated model of the nonlinear scintillation and Cherenkov light emission. For sources which consist of multiple $γ$ rays, the mean energy is used as an effective energy for the purposes of modeling the scintillator nonlinearity. The best fit for a model of the scintillator nonlinearity is also shown (red solid line). For clarity, the estimated nonlinearity contributed by the electronics has been removed from both the data and the model.
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Figure 14
Comparison of the electron energy spectrum from ${}^{12}B$ decay with simulation, including the estimated nonlinearity of the scintillator and electronics.
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Figure 15
Estimated ratio of reconstructed over true energy, $E_{rec} / E_{true}$ , for positron interactions totally contained within the antineutrino detector target (red solid line). The two 0.511-MeV $γ$ rays from positron annihilation were included in the energy. The resulting 68% C.L. region (gray shaded band) constrains this ratio to better than 1% over most of the energy range of interest. An independent estimate, which relied on the $β + γ$ spectra from bismuth and thallium decay as well as the Michel electron spectrum endpoint, produced a consistent model (blue dotted line).
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Figure 16
Reconstructed energy spectra of all signals in EH1-AD1 as successive cuts from selection A were applied: (A) all signals, (B) rejection of signals from PMT light emission, (C) water shield muon veto, (D) IBD signal pair selection, prompt energy, delayed energy, prompt-delayed time separation, and multiplicity veto, and (E) AD muon and shower vetos. The spectra E are the final prompt and delayed energy spectra of the ${\bar{ν}}_{e}$ candidates for EH1-AD1.
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Figure 17
(Top) Distribution of the time separation between prompt and delayed ${\bar{ν}}_{e}$ inverse beta-decay interaction candidates for each detector. (Bottom) Ratio of the observed distribution to the normalized average for all eight detectors. The consistency of these distributions constrained potential systematic variation in the fraction of IBD neutrons which captured on Gd among the eight detectors.
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Figure 18
Comparison of the capture time of neutrons on Gd measured in each detector, for neutrons from IBD, ${}^{241}{Am} − {}^{13}C$ , and muon spallation. The IBD neutron data are combined for the four detectors in EH3 (AD4–AD7) in order to reduce the statistical uncertainty. The observed capture times vary by less than $\pm 0.2 μ s$ between detectors (blue band). The spatial distributions of neutrons emitted by these sources differed, introducing slight offsets in their absolute capture times.
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Figure 19
The distribution of prompt versus delayed energy for signal pairs which satisfied the ${\bar{ν}}_{e}$ inverse beta-decay selection criteria. Interactions with prompt energy of $0.7 MeV < E_{p} < 12 MeV$ and delayed energy of $6 MeV < E_{d} < 12 MeV$ were used for the neutrino oscillation analysis discussed here (red dashed box). A few-percent contamination from accidental backgrounds (symmetric under interchange of prompt and delayed energy) and ${}^{9}{Li}$ decay and fast neutron backgrounds (high prompt and $\sim 8 MeV$ delayed energy) are visible within the selected region. Inverse beta-decay interactions where the neutron was captured on ${}^{1}H$ provided an additional signal region with $E_{d} \sim 2.2 MeV$ , albeit with much higher background.
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Figure 20
Typical charge distribution from PMT light emission. The eight ring by 24 column cylindrical PMT array has been projected onto a plane. The logarithmic color scale, as well as numbers, provides the percentage of the total signal charge observed by each PMT. In this example, light was emitted by the PMT at column 19 in ring 5. The discharging PMT and those PMTs directly opposite in quadrant 3 observed a significant fraction of the total charge. The distinct charge pattern allowed efficient rejection of this instrumental background.
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Figure 21
Distributions of the discriminator $f_{ID}$ for all inverse beta-decay delayed signal candidates from the eight detectors. The distributions are normalized to the number of antineutrino candidates for each AD. This demonstrates the consistency of these distributions among the ADs for genuine antineutrino candidates ( $f_{ID} < 0$ ). This normalization artificially enhances the magnitude of the light-emission background distributions ( $f_{ID} \geq 0$ ) for the far (AD4–AD7) relative to the near ADs. These background distributions differed between detectors depending on the characteristics of the light-emitting PMTs in each AD.
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Figure 22
The reconstructed energy spectra of isolated uncorrelated signals for all eight detectors. The promptlike signal rate was dominated by natural radioactivity with energies below 3 MeV. The delayedlike signal rate was dominated by beta decay of muon-induced unstable isotopes, mainly ${}^{12}B$ , and $γ$ rays from capture of ${}^{241}{Am} − {}^{13}C$ calibration source neutrons on materials at the top of each detector. The energy spectrum of the accidental background prompt signals in each AD was equivalent to these distributions.
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Figure 23
The accidental background rate for selection A as a function of time for each antineutrino detector, as calculated from measurements of the rates of uncorrelated signals which satisfy the prompt ${\bar{ν}}_{e}$ signal selection, $R_{p}$ , and delayed ${\bar{ν}}_{e}$ signal selection, $R_{d}$ . The accidental rate primarily depended on experimental hall depth due to the relative rate of unstable isotope production by muons. The decline of the accidental background versus time was due to combination of a decrease in the natural radioactivity following detector installation and a reduction of the neutron emission by the ${}^{241}{Am} − {}^{13}C$ calibration sources. $R_{p}$ was sensitive to small changes in electrical noise, which resulted in the observed instability in the accidental background rate. Installation of the final two detectors in the summer of 2012 is evident as a gap during this time. Removal of two of the three ${}^{241}{Am} − {}^{13}C$ sources in each far detector during the installation period reduced the accidental background rate by $\sim 50 %$ .
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Figure 24
Distribution of distances between the reconstructed positions of the prompt and delayed signals of the antineutrino inverse beta-decay candidate signals from all detectors (black points). The positions were highly correlated for true ${\bar{ν}}_{e}$ interactions, as demonstrated by Monte Carlo simulation (blue line). Repeating the selection of ${\bar{ν}}_{e}$ interactions, but with the time window for selection of the delayed signals offset by 1 to 20 ms, enhanced the uncorrelated accidental background (red points) and suppressed correlated signals. The consistency of the distributions confirmed the estimate of contamination by accidental backgrounds, which varied from 1% to 2% depending on the detector.
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Figure 25
(Top) An expanded selection of ${\bar{ν}}_{e}$ candidate signals with prompt energies from 0.7 to 100 MeV revealed a broad continuum of background, attributed to energetic, or fast, neutrons. Actual ${\bar{ν}}_{e}$ signals were visible as the dominant signals below $\sim 12 MeV$ . The energy spectra of the fast neutron background above 12 MeV were similar in all three experimental halls (EH1, black; EH2, red; EH3, blue). (Bottom) The sum of the ${\bar{ν}}_{e}$ candidate spectra from all three halls (black line) was compared to a high-purity fast neutron sample obtained by selecting those ${\bar{ν}}_{e}$ candidate signals which had been vetoed in the $200 μ s$ after an OWS or an RPC, but not an IWS, muon signal (OWS, blue; RPC, red). The energy spectra of the vetoed samples were consistent with the background observed above 12 MeV in the ${\bar{ν}}_{e}$ candidate sample. The normalization of the vetoed samples were adjusted to match the ${\bar{ν}}_{e}$ candidates above 12 MeV. The fast neutron contamination in the ${\bar{ν}}_{e}$ candidate sample below 12 MeV was estimated using the vetoed sample.
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Figure 26
The distribution of ${\bar{ν}}_{e}$ inverse beta-decay candidates versus time since the most recent muon-induced particle shower ( $E_{rec} > 2.5 GeV$ ) in the same detector (blue points). A fit to this distribution statistically distinguishes the muon-uncorrelated ${\bar{ν}}_{e}$ and accidentals (green hatched region) from the muon-correlated background from $β − n$ decay (red lined region). A minor contribution from accidental background, where the delayed signal was from ${}^{12}B$ $β$ decay, is barely visible in the first bin (blue lined region). A more restrictive ${\bar{ν}}_{e}$ selection of $3.5 MeV < E_{p} < 12 MeV$ , and $1 μ s < Δ t < 100 μ s$ suppressed the accidental contribution, providing a more precise estimate of the $β − n$ background.
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Figure 27
A comparison of the observed (black points) and predicted (blue line) prompt reconstructed energy spectrum from ${}^{9}{Li}$ $β − n$ decay. The observed spectrum was obtained from those ${\bar{ν}}_{e}$ candidate interactions which followed muon-induced particle showers, with the expected background from ${\bar{ν}}_{e}$ and accidentals subtracted. The prediction was estimated considering the known $β − n$ decay branches and contribution from the nuclear break-up to a neutron and two $α$ ’s.
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Figure 28
Uncorrelated individual signals with $E_{rec} > 5 MeV$ generated by a $\sim 59 Hz$ ${}^{241}{Am} − {}^{13}C$ neutron source temporarily installed on the lid of EH3–AD5 (black points). Peaks in the spectra are attributable to neutron capture on Fe, Cr, or Ni nuclei present in the stainless steel of the lid, where an energetic capture $γ$ ray subsequently penetrates the scintillating region of the detector. The measurement was used to benchmark a Monte Carlo simulation of the uncorrelated and correlated background induced by this source (red solid line), and thereby estimate the correlated background induced by the regular weak $\sim 0.7 Hz$ ${}^{241}{Am} − {}^{13}C$ sources present in the ACUs, also on the detector lid.
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Figure 29
The prompt reconstructed energy spectrum from correlated ${\bar{ν}}_{e}$ -like signal pairs induced by a $\sim 59 Hz$ ${}^{241}{Am} − {}^{13}C$ neutron source temporarily installed on the lid of EH3–AD5 (black points). Inelastic collisions of a neutron with nuclei present in stainless steel produced $γ$ rays which occasionally penetrated to the scintillating region of the detector, generating signals with a steeply falling energy spectrum. A benchmarked Monte Carlo simulation predicted a consistent spectrum (solid blue region). The prompt energy spectrum for correlated background from the regular $\sim 0.7 Hz$ ${}^{241}{Am} − {}^{13}C$ sources present in the ACUs was estimated to be identical, within the statistical precision of the simulation (red line).
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Figure 30
The distributions of low-energy prompt versus delayed reconstructed energy for the time intervals (a) 1 to $3 μ s$ , (b) 10 to $400 μ s$ , and (c) 1 to 4 ms for all detectors combined. The distributions revealed time-correlated signal pairs from actinide contamination in the GdLS regions of each detector. While these interactions were not a background for ${\bar{ν}}_{e}$ detection, the resulting estimates of actinide $α$ -particle activity constrained potential background from subsequent ${}^{13}C (α, n) {}^{16}O$ interactions.
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Figure 31
Simulation of energy spectra of the prompt signals for the ${}^{13}C (α, n) {}^{16}O$ background, for each of the primary sources of natural $α$ activity in the detectors. Energetic neutrons produced a broad peak from proton recoils below 4 MeV, and a small peak near 5 MeV from inelastic scattering on ${}^{12}C$ . Higher-energy $α$ particles increasingly populated the excited states of ${}^{16}O$ , resulting in a broad peak from 5 to 8 MeV from ${{}^{16}O}^{*}$ deexcitation.
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Figure 32
(Top) The rates of ${\bar{ν}}_{e}$ inverse beta decay observed in each antineutrino detector, for both selection A and selection B. (Bottom) The rates obtained by selection B were $\sim 0.2 %$ lower than those of selection A, demonstrating a difference in absolute selection efficiency within expectations. The ratio of the far-versus-near detector rates is consistent for the two selections, considering the statistical uncertainty from candidate signals uncommon between the two samples.
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Figure 33
(Left) Ratios of the ${\bar{ν}}_{e}$ interaction rates observed by detectors within the same experimental hall, $r_{obs}$ (black points). The ratio of the rate in AD1 to that of AD2 is shown for EH1 for the entire data period. For the period following installation of AD8 in EH2, the ratio of AD3 to AD8 is provided. For EH3, the ratio of the rate in each detector relative to the site average is given separately for the period before and after installation of AD7. Slight differences in the distances of each detector relative to each reactor as well as small variations in detector target mass $Δ N_{p}$ predicted minor deviations in these ratios, $r_{\exp}$ (blue dashed line). The uncertainties are dominated by statistics and the estimated 0.13% variation in efficiency between detectors. The consistency of the side-by-side detector rates confirmed the stringent limits on this latter systematic uncertainty. (Right) The same figure, expressed as the double ratio $r_{obs} / r_{\exp}$ .
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Figure 34
(Top) Distributions of the reconstructed prompt energy for the selected ${\bar{ν}}_{e}$ candidates in each of the eight detectors. (Bottom) The ratio of the distributions for detectors within the same experimental hall showed no significant deviations between detectors.
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Figure 35
(Top) Distributions of the reconstructed delayed energy for the selected ${\bar{ν}}_{e}$ candidates in each of the eight detectors. (Bottom) The ratio of the distributions for detectors within the same experimental hall showed no significant deviations between detectors; the slight slopes in the ratios are consistent with the $≲ 0.2 %$ relative differences in the energy scale among the detectors.
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Figure 36
Results of a validation study of the independent statistical methods (A–E) used to compare observation with the three-flavor model of neutrino oscillation. An independent program was used to generate fake observations under the assumption of a range of true values for the oscillation parameters $θ_{13}$ and $| Δ m_{ee}^{2} |$ (gray dashed lines). All methods demonstrated a consistent unbiased estimation of the true input parameters (colored points) for $θ_{13}$ (left panel) and $| Δ m_{ee}^{2} |$ (right panel). The final estimated total uncertainties, given by the error bars on each point, were also consistent between methods.
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Figure 37
Ratio of the detected ${\bar{ν}}_{e}$ signal to that expected assuming no oscillation versus the effective baseline for all eight antineutrino detectors. Oscillation due to $θ_{13}$ introduced a deficit in the far detectors relative to the near detectors, and the best-fit three-flavor oscillation model from the rate-only analysis is shown (red line). Extrapolation of the model to a baseline of 0 determined the absolute normalization of the reactor ${\bar{ν}}_{e}$ flux, $R^{pred} (L = 0)$ . The points representing the near (far) detectors are displaced by $\pm 6 m$ ( $\pm 30 m$ ) for clarity.
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Figure 38
Reconstructed positron energy spectra for the ${\bar{ν}}_{e}$ candidate interactions (black points). The spectra of the detectors in each experimental hall are combined: EH1 (top), EH2 (middle), and EH3 (bottom). The best-fit three-flavor neutrino oscillation model (red line) is determined from the difference in rate and spectrum observed at the far hall relative to the near halls. A prediction with no oscillation (blue line) is obtained from the best-fit model, but with $θ_{13} = 0$ . The inset in semilogarithmic scale shows the backgrounds. The ratio of the background-subtracted spectra to the prediction with no oscillation is shown in the panel beneath each energy spectrum.
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Figure 39
A comparison of the estimated values of $\sin^{2} 2 θ_{13}$ (top) and $Δ m_{ee}^{2}$ (bottom) obtained using various combinations of the two selected ${\bar{ν}}_{e}$ samples, statistical methods, and reactor ${\bar{ν}}_{e}$ flux models. The horizontal dashed lines show the best estimate of each parameter, while the gray regions show the $\pm 1 σ$ confidence interval from the reference results (selection A, method D, and the Huber-Mueller reactor flux model). The $≲ 1 σ$ offsets for methods A, B, and E were due to their choice of binning of the prompt energy spectrum, particularly below 1.3 MeV. When all methods used the same binning as method C, consistent results were obtained (open circles). See the text for details.
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Figure 40
Measured reactor ${\bar{ν}}_{e}$ spectral distortion, displayed as the oscillation survival probability versus $L_{eff} / E_{ν}$ . The effective propagation distance $L_{eff}$ was estimated for each hall based on the distribution of reactors contributing to the signal [see Eq. (56)]. The average true ${\bar{ν}}_{e}$ energy $⟨ E_{ν} ⟩$ was determined for each bin in the observed prompt positron spectrum based on the model of the detector response. The ${\bar{ν}}_{e}$ survival probability was given by the observed signal in each bin divided by the prediction assuming no oscillation. The measurement sampled ${\bar{ν}}_{e}$ survival over almost one full cycle, demonstrating distinct evidence in support of neutrino flavor oscillation.
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Figure 41
Confidence regions of $\sin^{2} 2 θ_{13}$ and $| Δ m_{ee}^{2} |$ from a combined analysis of the prompt positron spectra and rates. The $1 σ$ , $2 σ$ , and $3 σ$ two-dimensional confidence regions are estimated using $Δ χ^{2}$ values of 2.30 (red), 6.18 (green), and 11.83 (blue) relative to the best fit. The upper panel provides the one-dimensional $Δ χ^{2}$ for $\sin^{2} 2 θ_{13}$ obtained by profiling $| Δ m_{ee}^{2} |$ (blue line), and the dashed lines mark the corresponding $1 σ$ , $2 σ$ , and $3 σ$ intervals. The right panel is the same, but for $| Δ m_{ee}^{2} |$ , with $\sin^{2} 2 θ_{13}$ profiled. The point marks the best estimates, and the error bars display their one-dimensional $1 σ$ confidence intervals.
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Figure 42
Comparison of measurements of $\sin^{2} 2 θ_{13}$ : this measurement (blue point), RENO [73] and Double Chooz [74] (red points), T2K [75] and MINOS [76] (green points). The T2K and MINOS values were deduced from $2 \sin^{2} θ_{23} \sin^{2} 2 θ_{13}$ , and are presented for the two cases of the normal (upper) and inverted (lower) mass hierarchy. The MINOS measurement assumed $\sin^{2} θ_{23} = 0.5$ , $δ_{CP} = 0$ , while the T2K measurement marginalized over these unknown parameters.
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Figure 43
Comparison of measurements of $Δ m_{32}^{2}$ : this measurement (blue point), RENO [73] (red point), T2K [75], MINOS [77], and $NO ν A$ [78] (green points), Super-Kamiokande [79] and IceCube [80] (cyan points). All values are given for the case of the normal mass hierarchy; the comparison for the inverted ordering was qualitatively similar.
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Figure 44
The Jacobian $d E_{ν} / d E_{true}$ as a function of positron energy $E_{e}$ for a range of values of the positron emission angle $\cos θ_{e}$ .
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