Improved constraints on neutrino mixing from the T2K experiment with $3.13\times {10}^{21}$ protons on target

The T2K experiment reports updated measurements of neutrino and antineutrino oscillations using both appearance and disappearance channels. This result comes from an exposure of $14.9(16.4)\times 10^{20}$ protons on target in neutrino (antineutrino) mode. Significant improvements have been made to the neutrino interaction model and far detector reconstruction. An extensive set of simulated data studies have also been performed to quantify the effect interaction model uncertainties have on the T2K oscillation parameter sensitivity. T2K performs multiple oscillation analyses that present both frequentist and Bayesian intervals for the Pontecorvo-Maki-Nakagawa-Sakata parameters. For fits including a constraint on ${\sin }^2{\theta }_{13}$ from reactor data and assuming normal mass ordering T2K measures ${\sin }^2{\theta }_{23}={0.53}_{-0.04}^{+0.03}$ and $\mathrm{\Delta }{m}_{32}^2=(2.45\pm 0.07)\times {10}^{-3}{\mathrm{eV}}^2{\mathrm{c}}^{-4}$. The Bayesian analyses show a weak preference for normal mass ordering (89% posterior probability) and the upper ${\sin }^2{\theta }_{23}$ octant (80% posterior probability), with a uniform prior probability assumed in both cases. The T2K data exclude CP conservation in neutrino oscillations at the $2\sigma$ level.

Figure 1

The SK flux prediction for runs 1–9a with horns operating in FHC (250 kA) mode (upper) and RHC ($-250\mathrm{kA}$) mode (lower).

Figure 2

The fractional systematic uncertainty on the ${\nu }_{\mu }$ flux at SK in FHC mode (top), on the right-sign ${\overline{\nu }}_{\mu }$ flux at SK in RHC mode (middle), and on the wrong-sign ${\nu }_{\mu }$ flux at SK in RHC mode (bottom). The solid black line shows the current total fractional uncertainty (NA61/SHINE 2009 data), while the dashed black line in the top panel shows the fractional uncertainty from an earlier flux prediction (NA61/SHINE 2007 data).

Figure 3

The total charged-current cross section for muon neutrinos interacting with a carbon nucleus, as predicted by neut, overlaid on the ND280 muon neutrino flux, and an example oscillated muon neutrino flux at SK. The oscillation parameters used here are the best fit from the previous analysis [26]. The total (Inc) cross section is separated into 1p1h, 2p2h, single-pion production (SPP), and DIS channels.

Figure 4

Top: the energy and momentum transfer distribution for the Nieves et al. 2p2h model in neut. The two-peak structure is clear, with QE-like kinematics corresponding to the lower left peak and Delta-like kinematics to the stronger central peak. Bottom: the reconstructed energy bias at SK is shown for 1p1h and 2p2h events for an oscillated muon neutrino flux. The different reconstructed energy smearing for 2p2h events with QE-like and Delta-like interaction kinematics can be seen.

Figure 5

The central value BeRPA suppression factor (solid line) and the total prior uncertainty (dashed lines) determined from ${10}^5$ uncorrelated throws of the free parameters, overlaid on the corresponding envelopes generated by varying each parameter. The continuity conditions result in a perhaps unintuitive total uncertainty envelope given the individual parameter variations.

Figure 6

The differential cross section for the two extreme variations of the 2p2h “shape” parameter—QE-like (top) and Delta-like (bottom)—cf. Fig. 4 (top) for the central-value-predicted cross section.

Figure 7

The predicted flux-averaged cross section for three different nuclear response models. The different models result in different predictions of the observed final-state lepton kinematics for the same oscillated neutrino flux. If such variations are not accounted for in the uncertainty model, extracted oscillation parameters may be biased. The contours contain the region of phase space with a differential cross section $\mathrm{d}\sigma /\mathrm{d}{p}_{\ell }\cos ({\theta }_{\ell })>0.05\times {10}^{-38}{\mathrm{cm}}^{2}c/\mathrm{GeV}/\mathrm{N}$.

Figure 8

Top: the predicted muon momentum spectrum for a number of different values of NRE in an oscillated SK flux. The shift toward lower momentum may be confused for a shift in $|\mathrm{\Delta }{m}_{32}^2|$. Bottom: the effect of varying NRE on the reconstructed energy bias; higher values result in more energy “feed down.”

Figure 9

The T2K data-taking periods, showing the accumulated beam protons on target delivered as a function of time.

Figure 10

Data in the muon monitors (MUMON) and INGRID on-axis detector for runs 1–9. The top row shows the event rate in both detectors, which is reduced as expected for RHC mode. The middle and bottom rows show the horizontal and vertical beam direction in both detectors.

Figure 11

Final-state muon momentum distributions of the FHC ${\nu }_{\mu }$ CC $0\pi$ (top) and ${\nu }_{\mu }$ CC $1{\pi }^{+}$ (bottom) data and simulation samples in FGD1 (left) and FGD2 (right).

Figure 12

Distributions of the final-state muon angle of the FHC ${\nu }_{\mu }$ CC $0\pi$ (top) and ${\nu }_{\mu }$ CC $1{\pi }^{+}$ (bottom) data and simulation samples in FGD1 (left) and FGD2 (right).

Figure 13

Final-state muon momentum distributions for the RHC ${\overline{\nu }}_{\mu }$ (top) and ${\nu }_{\mu }$ (bottom) CC 1-track (left) and CC $N$-track (right) FGD1 simulation samples. These distributions are before the ND280 fit.

Figure 14

Distributions of the final-state muon angle for the RHC ${\overline{\nu }}_{\mu }$ (top) and ${\nu }_{\mu }$ (bottom) CC 1-track (left) and CC $N$-track (right) FGD1 simulation samples. These distributions are before the ND280 fit.

Figure 15

Reconstructed event time distributions in the 1 ms (top) and $[-1.2,5.6]\mu \mathrm{s}$ (bottom) windows around the beam spill leading edge.

Figure 16

Distribution of number of reconstructed particle tracks for events passing the $wall>80\mathrm{cm}$ and $towall>170\mathrm{cm}$ FV criteria in FHC (top) and RHC (bottom) data.

Figure 17

Distribution of the $e/\mu$ PID discriminator for single-track events passing the $wall>80\mathrm{cm}$ and $towall>170\mathrm{cm}$ FV criteria in FHC (top) and RHC (bottom) data.

Figure 18

Distribution of the number of identified decay electrons for single-track events passing the $wall>80\mathrm{cm}$ and $towall>170\mathrm{cm}$ FV criteria in FHC (top) and RHC (bottom) data.

Figure 19

Distribution of the $e/{\pi }^0$ PID discriminator for $1{\mathrm{R}}_e$ events in FHC (top) and RHC (bottom) data.

Figure 20

Distribution of the $e/{\pi }^0$ PID discriminator for $1{\mathrm{R}}_e+1\mathrm{d}.\mathrm{e}.$ in FHC data.

Figure 21

Distribution of the $\mu /{\pi }^{+}$ PID discriminator for $1{\mathrm{R}}_{\mu }$ events in FHC (top) and RHC (bottom) data.

Figure 22

Reconstructed vertices of selected FHC data $1{\mathrm{R}}_e$ events projected in $z$ vs $r^2$ (top) and $y$ vs $x$ (bottom). The neutrino beam direction is shown as a red arrow and events which do not pass the fiducial volume criteria are shown as hollow crosses.

Figure 23

Reconstructed vertices of selected RHC data $1{\mathrm{R}}_e$ events projected in $z$ vs $r^2$ (top) and $y$ vs $x$ (bottom). The neutrino beam direction is shown as a red arrow and events which do not pass the fiducial volume criteria are shown as hollow crosses.

Figure 24

Reconstructed vertices of selected FHC data $1{\mathrm{R}}_e+1\mathrm{d}.\mathrm{e}.$ events projected in $z$ vs $r^2$ (top) and $y$ vs $x$ (bottom). The neutrino beam direction is shown as a red arrow and events which do not pass the fiducial volume criteria are shown as hollow crosses.

Figure 25

Reconstructed energy distribution for $1{\mathrm{R}}_e$ events in FHC (top) and RHC (bottom) data.

Figure 26

Reconstructed energy distribution for $1{\mathrm{R}}_e+1\mathrm{d}.\mathrm{e}.$ events in FHC data. The $\mathrm{\Delta }(1232)$ mass is used for the final-state nucleon in the reconstructed neutrino energy calculation.

Figure 27

Reconstructed energy distribution for $1{\mathrm{R}}_{\mu }$ events in FHC (top) and RHC (bottom) data.

Figure 28

The width of ${\sin }^2{\theta }_{23}$ $1\sigma$ confidence interval as a function of the slope and intercept of the $\mathrm{NC}{\pi }^{+}$ rejection cut line is shown on the top and the distribution of signal (box histogram, axis on top of plot) and background events (colored histogram) is shown on the bottom, along with the line below which $\mu$-like events are selected.

Figure 29

The sensitivity to exclude ${\delta }_{\mathrm{CP}}=0$ as a function of the slope and intercept of the $\mathrm{NC}{\pi }^0$ rejection cut line is shown on the top and the distribution of signal (box histogram, axis on top of plot) and background events (colored histogram) is shown on the bottom, along with the line below which $e$-like events are selected.

Figure 30

Detector regions used in detector systematic uncertainty estimation.

Figure 31

Posterior predictive distributions of SK atmospheric MCMC fit to determine detector systematic uncertainties. The total nominal MC is shown in gray, with components targeted by each of the distributions shown in red and blue. The 68.27% intervals of the posterior predictive distribution is shown in green. Observed data are shown as black circles. All distributions shown are for the largest detector region, 5. Distributions of the $e/\mu$ PID discriminator for zero and one decay-$e$ events are shown in (a) and (b). In (c) and (d), the track-counting parameter is shown for events with one and two decay electrons, respectively. The $e/{\pi }^0$ discriminator distribution is shown in (e) for $e$-like events, and the $\mu /{\pi }^{+}$ distribution is shown in (f) for $\mu$-like events.

Figure 32

Scans of the figures of merit used to optimize the FV criteria in the $wall$ vs $towall$ space that defines the FV for the analysis described here. Figures of merit corresponding to the $1{\mathrm{R}}_e$ (top left), $1{\mathrm{R}}_e+1\mathrm{d}.\mathrm{e}.$ (top right) and $1{\mathrm{R}}_{\mu }$ (bottom) samples are shown here. The chosen cut point is indicated in the figures as is the region which is ruled out by reconstruction performance considerations.

Figure 33

Square root of the diagonal elements of the covariance matrix describing the fractional uncertainty on the number of events due to the modeling of the SK detector.

Figure 34

The SK flux parameters for the ${\nu }_{\mu }$ (top left) and ${\nu }_e$ (top right) neutrino species in FHC and for the ${\overline{\nu }}_{\mu }$ (bottom left) and ${\overline{\nu }}_e$ (bottom right) neutrino species in RHC, as a fraction of the nominal value. The bands indicate the $1\sigma$ uncertainty on the parameters before (solid, red) and after (hatched, blue) the near detector fit. The results shown here are from the minuit-based framework.

Figure 35

The cross-section parameters as a fraction of the nominal value, taken from the minuit-based analysis. The bands indicate the $1\sigma$ uncertainty on the parameters before (solid, red) and after (hatched, blue) the near detector fit.

Figure 36

Distribution of the minimum negative log-likelihood values from fits to the mock datasets (black line), with the value from the fit to the data superimposed in red. The $p$ value is 47.3%.

Figure 37

Post-ND280-fit muon momentum distributions of the FHC ${\nu }_{\mu }$ CC $0\pi$ (top) and FHC ${\nu }_{\mu }$ CC $1\pi$ (bottom) samples in FGD1 (left) and FGD2 (right).

Figure 38

Post-ND280-fit distributions of the final state muon angle of the FHC ${\nu }_{\mu }$ CC $0\pi$ (top) and FHC ${\nu }_{\mu }$ CC $1\pi$ (bottom) samples in FGD1 (left) and FGD2 (right).

Figure 39

Post-ND280-fit muon momentum distributions for the RHC ${\overline{\nu }}_{\mu }$ (top) and ${\nu }_{\mu }$ (bottom) CC 1-track (left) and CC $N$-track (right) FGD1 samples.

Figure 40

Post-ND280-fit distributions of the final state muon angle for the RHC ${\overline{\nu }}_{\mu }$ (top) and ${\nu }_{\mu }$ (bottom) CC 1-track (left) and CC $N$-track (right) FGD1 samples.

Figure 41

Post-ND280-fit predicted event spectrum for the FHC beam SK samples as a function of $E_{\mathrm{QE}}^{\mathrm{Rec}}$. The samples are single-ring muonlike (top), single-ring electronlike without decay electrons (middle) and single-ring electronlike with a single decay electron (bottom).

Figure 42

Post-ND280-fit predicted event spectrum for the RHC SK samples as a function of $E_{\mathrm{QE}}^{\mathrm{Rec}}$. The samples are single-ring muonlike (top) and single-ring electronlike without decay electrons (bottom).

Figure 43

Comparisons of the fitted SK FHC flux parameters (top) and cross-section parameters (bottom) between the ND280 fit (red, solid) and the MCMC analysis (blue, hashed).

Figure 44

Observed kinematic distributions compared to the expectations generated with oscillation parameters set to the values in Table 3 for the different samples. The uncertainty shown around the data points in (a) and (b) accounts for statistical uncertainty only. The uncertainty range is chosen to include all points for which the measured number of data events is inside the 68% confidence interval of a Poisson distribution centered at that point.

Figure 45

The predicted $\mathrm{\Delta }{\chi }^2=-2\ln [L/L_{\max }]$ function as a function of ${\delta }_{\mathrm{CP}}$ with and without reactor constraint, for both mass orderings.

Figure 46

Predicted 90% C.L. regions for $\mathrm{\Delta }{m}^2$ vs ${\sin }^2{\theta }_{23}$ with and without the reactor constraint, for both mass orderings. Normal and inverted mass ordering contours are independent.

Figure 47

Comparison between analyses A, B and C. The normal mass ordering is assumed and the reactor constraint is applied. Top: the predicted $\mathrm{\Delta }{\chi }^2$ functions for ${\delta }_{\mathrm{CP}}$. Bottom: the $1\sigma$ (dashed lines) and 90% (solid lines) C.L. regions for $\mathrm{\Delta }{m}^2$ vs ${\sin }^2{\theta }_{23}$.

Figure 48

The momentum shift caused by increasing the nucleon removal energy in neut 5.3.3 by 18 MeV.

Figure 49

The ratio of the NRE simulated data to the nominal prediction for the ND280 CC $0\pi$ FGD1 sample. The histogram axes have been truncated for clarity.

Figure 50

Results of the fit to the near detector simulated data with an increased nucleon removal energy. The best-fit values for the ${\nu }_{\mu }$ flux parameters at SK (top) and cross-section parameters (bottom) are shown for the NRE fit (blue) compared to the Asimov data set fit (red).

Figure 51

Comparison of the nominal MC prediction (dashed black line), simulated data (solid blue line) and prediction from the near detector fit to the simulated data (shaded red). This is shown for the far detector muonlike (top) and electronlike (bottom) neutrino beam event samples.

Figure 52

Comparison of the nominal MC oscillation parameter $\mathrm{\Delta }{\chi }^2$ function (black lines) to those from the simulated data fits (red lines).

Figure 53

Comparison of the ${\sin }^2{\theta }_{23}-\mathrm{\Delta }{m}_{32}^2$ parameter contours using the nominal cross-section model (top), after the addition of the effective NRE parameter (middle) and after the additional smearing is applied to $\mathrm{\Delta }{m}_{32}^2$ (bottom). In all cases the expected result, as described in the text, is shown in black and the simulated data result is in red. The solid lines represent the 90% confidence region and the dashed lines represent the $1\sigma$ confidence region.

Figure 54

The observed $\mathrm{\Delta }{\chi }^2$ function of ${\delta }_{\mathrm{CP}}$, with and without the reactor constraint. The $\mathrm{\Delta }{\chi }^2$ is computed with respect to the best fit over the two mass orderings, and separate best-fit points are used for the T2K-only and the $\mathrm{T}2\mathrm{K}+\mathrm{reactor}$ cases.

Figure 55

The observed constant $\mathrm{\Delta }{\chi }^2$ 90% confidence regions of ${\sin }^2{\theta }_{23}$ and ${\delta }_{\mathrm{CP}}$ with normal and inverted mass orderings and with and without the reactor constraint. Normal and inverted mass ordering contours are independent. $\mathrm{\Delta }{\chi }^2$ values are calculated independently for the functions with and without the reactor constraint.

Figure 56

The observed T2K constant $\mathrm{\Delta }{\chi }^2$ 90% confidence regions (from analysis C) of ${\sin }^2{\theta }_{23}$ and $\mathrm{\Delta }{m}_{32}^2$ with the reactor constraint and compared to the results from Super-K [4], $\mathrm{NO}v\mathrm{A}$ [13] and IceCube [14].

Figure 57

Comparison between the observed $1\sigma$ (dashed lines) and 90% (solid lines) C.L. regions for ${\delta }_{\mathrm{CP}}$ vs ${\sin }^2{\theta }_{13}$ produced using analyses A, B and C, assuming normal mass ordering (top) or inverted mass ordering (bottom). The reactor constraint is applied.

Figure 58

Comparison between the observed $1\sigma$ (dashed lines) and 90% (solid lines) C.L. regions for $\mathrm{\Delta }{m}^2$ vs ${\sin }^2{\theta }_{23}$ produced using analyses A, B and C, assuming normal mass ordering (top) or inverted mass ordering (bottom). The reactor constraint is applied.

Figure 59

Feldman-Cousins critical values for ${\delta }_{\mathrm{CP}}$. The shaded bands represent the $\pm 1\sigma$ Monte Carlo statistical uncertainty on the calculated critical values.

Figure 60

The observed $3\sigma$ Feldman-Cousins (FC) confidence intervals for ${\delta }_{\mathrm{CP}}$. The $\mathrm{\Delta }{\chi }^2$ is computed with respect to the best fit over the two mass orderings.

Figure 61

Posterior probability density for ${\delta }_{\mathrm{CP}}$ and credible intervals obtained using a prior uniform in ${\delta }_{\mathrm{CP}}$ compared against credible intervals obtained with a prior uniform in $\sin ({\delta }_{\mathrm{CP}})$. The CP-conserving values are outside of the $2\sigma$ credible intervals in both cases.

Figure 62

Posterior probability distribution for $\mathrm{\Delta }{m}_{32}^2$ covering both mass orderings with 1, 2 and $3\sigma$ Bayesian credible intervals marked. The normal ordering contains 89% of the posterior probability.

Figure 63

Bayesian credible regions for $\mathrm{\Delta }{m}_{32}^2$ against ${\sin }^2{\theta }_{23}$ covering both mass orderings.

Figure 64

Posteriors probabilities together with 1, 2 and $3\sigma$ credible intervals for all the oscillation parameters of interest and their combinations in the normal mass ordering. A logarithmic scale is used for the axis corresponding to the posterior probability density, and darker colors correspond to larger probabilities.

Figure 65

The observed $\mathrm{\Delta }{\chi }^2=-2\mathrm{\Delta }\ln L$ functions compared with one-sided distributions of $\mathrm{\Delta }{\chi }^2$ values corresponding to 68.27% and 95.45% of 10 000 pseudoexperiments generated for ${\delta }_{\mathrm{CP}}=-\pi /2$ and normal mass ordering. Results for the normal mass ordering (top) and inverted mass ordering (bottom) are shown.

Figure 66

Change of the results of the fit of the T2K run 1–9 data for ${\delta }_{\mathrm{CP}}$ when observations get replaced by predictions for the different samples. The gray dashed lines correspond to the critical values for $2\sigma$ and $3\sigma$ obtained using the Feldman-Cousins method.

Figure 67

Lepton momentum and angle with respect to the beam direction for the 15 events observed in the FHC ${\nu }_e$ CC $1{\pi }^{+}$-like sample, overlaid with the MC predictions.

Figure 68

Change in the $\mathrm{\Delta }{\chi }^2$ function for ${\delta }_{\mathrm{CP}}$ when fitting data using an alternative model for DIS hadron multiplicity (red curves) and pion momentum spectra (black curves). The $\mathrm{\Delta }{\chi }^2$ was reduced by the value shown on the plot in each case. The difference is shown for the model change that gave the largest difference in each case.

Figure 69

Mass ordering posterior probabilities as a function of the prior probability assumed for the normal ordering, obtained in the data fit using analysis C. The dashed black line corresponds to the equal prior probability case used for the main result.

Figure 70

Ability to distinguish between the two mass orderings as a function of the true value of ${\delta }_{\mathrm{CP}}$. The bar indicates the range of values of $\mathrm{\Delta }{\chi }^2={\chi }_{\mathrm{NO}}^2-{\chi }_{\mathrm{IO}}^2$ which contains 95% of the values obtained for the pseudoexperiments generated in each case. The black line corresponds to the value obtained in the data fit by analysis A.

Figure 71

Expected distributions of the Bayes factor between the mass ordering hypotheses, compared to the value obtained in the data fit.

Figure 72

Candidate RHC one-ring $e$-like event rate vs the candidate FHC one-ring $e$-like event rate (including CC $1\pi$ events) for a variety of different oscillation parameter values. Predictions are generated for the given values of ${\delta }_{\mathrm{CP}}$, ${\sin }^2{\theta }_{23}$ and mass ordering with remaining oscillation parameters fixed at the central values of the prior probability density functions defined in Sec. 14a1. The uncertainty regions are created assuming that ${\delta }_{\mathrm{CP}}=-\pi /2$, ${\sin }^2{\theta }_{23}=0.5$, $\mathrm{\Delta }{m}_{32}^2=2.45\times {10}^{-3}\mathrm{eV}/c^2$ and normal mass ordering.

Figure 73

Distributions of the expected (top) and observed (bottom) rate-only (left) and $\mathrm{rate}+\mathrm{shape}$ (right) test statistics compared to the expected and observed test statistics. Here ${\mathrm{N}}_{\text{events}}$ denotes the number of observed events in the RHC single-ring $e$-like sample.