Atmospheric neutrino oscillation analysis with external constraints in Super-Kamiokande I-IV

An analysis of atmospheric neutrino data from all four run periods of Super-Kamiokande optimized for sensitivity to the neutrino mass hierarchy is presented. Confidence intervals for $\mathrm{\Delta }{m}_{32}^2$, ${\sin }^2{\theta }_{23}$, ${\sin }^2{\theta }_{13}$ and ${\delta }_{CP}$ are presented for normal neutrino mass hierarchy and inverted neutrino mass hierarchy hypotheses, based on atmospheric neutrino data alone. Additional constraints from reactor data on ${\theta }_{13}$ and from published binned T2K data on muon neutrino disappearance and electron neutrino appearance are added to the atmospheric neutrino fit to give enhanced constraints on the above parameters. Over the range of parameters allowed at 90% confidence level, the normal mass hierarchy is favored by between 91.9% and 94.5% based on the combined Super-Kamiokande plus T2K result.

Figure 1

The propagation of two neutrinos through the simplified model of the Earth used in the analysis below. Both ${\nu }_A$ and ${\nu }_B$ are produced in the atmosphere. ${\nu }_A$ then experiences 6 oscillation steps ($\mathrm{air}\to \mathrm{crust}\to \mathrm{mantle}\to \mathrm{outer}\mathrm{core}\to \mathrm{mantle}\to \mathrm{crust}$), while ${\nu }_B$ experiences 4 oscillation steps ($\mathrm{air}\to \mathrm{crust}\to \mathrm{mantle}\to \mathrm{crust}$).

Figure 2

Oscillation probabilities for neutrinos (upper panels) and antineutrinos (lower panels) as a function of energy and zenith angle assuming a normal mass hierarchy. Matter effects in the Earth produce the distortions in the neutrino figures between two and ten GeV, which are not present in the antineutrino figures. Distortions in the ${\nu }_{\mu }$ survival probability and enhancements in the ${\nu }_e$ appearance probability occur primarily in angular regions corresponding to neutrino propagation across both the outer core and mantle regions (cosine zenith $<-0.9$) and propagation through the mantle and crust ($-0.9<$ cosine zenith $<-0.45$). For an inverted hierarchy the matter effects appear in the antineutrino figures instead. Here the oscillation parameters are taken to be $\mathrm{\Delta }{m}_{32}^2=2.5\times {10}^{-3}{\mathrm{eV}}^2$, ${\sin }^2{\theta }_{23}=0.5$, ${\sin }^2{\theta }_{13}=0.0219$, and ${\delta }_{CP}=0$.

Figure 3

Absolute energy scale measurements for each SK period. Vertical error bars denote the statistical uncertainty and horizontal error bars the momentum range spanned by each analysis.

Figure 4

Energy scale stability measured as a function of date since the start of SK operations. The energy scale is taken as the average of the reconstructed momentum divided by range of stopping cosmic ray muon data in each bin. The vertical axis shows the deviation of this parameter from the mean value for each SK period separately. Error bars are statistical.

Figure 5

Data and MC comparisons for the entire Super-K data divided into 19 analysis samples. Samples with more than one zenith angle bin (cf. Table 2) are shown as zenith angle distributions (second through fifth column) and other samples are shown as reconstructed momentum distributions (first column). Lines denote the best fit MC assuming the normal hierarchy. Narrow panels below each distribution show the ratio of the data to this MC. In all panels the error bars represent the statistical uncertainty. In this projection each bin contains events of all energies, which obscures the difference between the hierarchies. If the inverted hierarchy MC were also drawn it would lie on top of the normal hierarchy line and for this reason it is not shown here. Figure 10 provides a better projection for comparing the hierarchies.

Figure 6

True Super-K atmospheric neutrino energy spectra from simulation without oscillations.

Figure 7

Final event rates as a function of time since the start of SK operations. The error bars are statistical. Circles denote the fully contained event rate and upward-facing (downward-facing) triangles show the partially contained (upward-going muon) event rates.

Figure 8

Likelihood distribution used to separate SK-IV Multi-ring $e$-like events into the neutrinolike and antineutrinolike samples. Error bars represent the statistical uncertainty of the data. Events with negative (positive) likelihood values are designated ${\overline{\nu }}_e$-like (${\nu }_e$-like).

Figure 9

Constraints on neutrino oscillation parameters from the Super-K atmospheric neutrino data fit with no external constraints. Orange lines denote the inverted hierarchy result, which has been offset from the normal hierarchy result, shown in blue, by the difference in their minimum ${\chi }^2$ values.

Figure 10

Upward- ($\cos \theta <-0.4$) to downward-going ($\cos \theta >0.4$) event ratio as a function of energy. The error bars are statistical. For the single-ring samples the energy is taken to be the visible energy assuming the light-producing particle was an electron. For the multi-ring samples the total energy is used after accounting for the particle type (electron or muon) of each reconstructed ring. The cyan line denotes the best fit from the normal hierarchy hypothesis, and the orange dashed line the best fit from the inverted hierarchy hypothesis. The error on the prediction is dominated by the uncertainty in the ${\nu }_{\tau }$ cross section and is not more than 3% (absolute) in any bin of the figure.

Figure 11

Constraints on the matter effect parameter $\alpha$ from the Super-K atmospheric neutrino data fit assuming ${\sin }^2{\theta }_{13}=0.0219\pm 0.0012$. Orange lines denote the inverted hierarchy result, which has been offset from the normal hierarchy result, shown in blue, by the difference in their minimum ${\chi }^2$ values. Vacuum corresponds to $\alpha =0$, while the standard matter profile used in the rest of the analyses presented here corresponds to $\alpha =1$.

Figure 12

Comparison of the T2K model (solid) with T2K results digitized from Fig. 33 of Ref. [27] (dashed). Cyan and orange lines correspond to the normal hierarchy and inverted hierarchy fits, respectively, with the offset in the two representing the difference in their minimum ${\chi }^2$ values.

Figure 13

Expected sensitivity to the normal mass hierarchy as a function of the true value of ${\sin }^2{\theta }_{23}$. Grey (lower) and cyan (upper) bands show the sensitivity of the atmospheric neutrino sample alone and when combined with the T2K model, respectively. The width of the bands corresponds to the uncertainty in ${\delta }_{CP}$. The inverted hierarchy shows a qualitatively similar improvement in sensitivity and is not shown.

Figure 14

Constraints on neutrino oscillation parameters from the Super-K atmospheric neutrino data fit assuming ${\sin }^2{\theta }_{13}=0.0219\pm 0.0012$. Orange lines denote the inverted hierarchy result, which has been offset from the normal hierarchy result, shown in cyan, by the difference in their minimum ${\chi }^2$ values.

Figure 15

Constraints on neutrino oscillation contours at the 90% C.L. from analyses assuming the normal mass hierarchy. The Super-K contour (cyan) is taken from the analysis with ${\sin }^2{\theta }_{13}$ assumed to be $0.0219\pm 0.0012$. Contours from the T2K (violet) [8], NOvA (dashed green) [7], $\mathrm{MINOS}+$ (dashed blue) [36], and IceCube (red) [39] experiments are also shown.

Figure 16

Constraints on neutrino oscillation contours from a combined fit of Super-K atmospheric neutrino data and a model of the T2K experiment assuming ${\sin }^2{\theta }_{13}=0.0219\pm 0.0012$. Orange lines denote the inverted hierarchy result, which has been offset from the normal hierarchy result, shown in cyan, by the difference in their minimum ${\chi }^2$ values.

Figure 17

Constraints on neutrino oscillation contours from a combined fit of Super-K atmospheric neutrino data and a model of the T2K experiment. The left figure shows 90% C.L. constraints from the atmospheric neutrino data (dotted), the T2K model (dashed), and their combination (solid) for the normal hierarchy. The right figure shows the same for the inverted hierarchy fit. In each contour ${\sin }^2{\theta }_{13}$ is constrained to be $0.0219\pm 0.0012$. Normal and inverted hierarchy contours for each analysis are drawn relative the best fit among the two.

Figure 18

Distributions of the difference in best fit ${\chi }^2$ values between normal- and inverted-hierarchy fits to pseudo data sets used in the generation of the ${\mathrm{CL}}_s$ value for the SK ${\theta }_{13}$ constrained analysis. In the cyan (orange) histogram the pseudo data have been generated assuming the normal (inverted) hierarchy at the analysis best fit shown in Table 5. Shaded portions of the histograms denote the fraction of pseudo data sets with more extreme values than that observed in the data, $\mathrm{\Delta }{\chi }_{\text{data}}^2=-4.33$.