Where we are on θ₁₃: addendum to 'Global neutrino data and recent reactor fluxes: status of three-flavor oscillation parameters'

Prompted by the recently published indication for electron neutrino appearance by the T2K experiment [1], we have updated the global neutrino oscillation analysis presented in [2]. The T2K experiment uses a neutrino beam consisting mainly of muon neutrinos, produced at the J-PARC accelerator facility and observed at a distance of 295 km and an off-axis angle of 2.5° by the Super-Kamiokande detector. The present data release corresponds to 1.43 × 10²⁰ protons on target [2]. Six events pass all selection criteria for an electron neutrino event. In a three-flavor neutrino oscillation scenario with θ₁₃ = 0 the expected number of such events is 1.5 ± 0.3 (syst). Under this hypothesis, the probability of observing six or more candidate events is 7 × 10⁻³, equivalent to a significance of 2.5σ. We investigate the implications of this result for the mixing angle θ₁₃ and the leptonic Dirac CP phase δ, focusing on long-baseline ν_μ → ν_e appearance data from T2K and MINOS in section 2, whereas the results of a combined analysis of global neutrino oscillation data are presented in section 3.

For our re-analysis of T2K, we use the spectral data shown in figure 5 of [2] given as 5 bins in reconstructed neutrino energy from 0 to 1.25 GeV. Using the neutrino fluxes predicted at Super-Kamiokande in the absence of oscillations provided in figure 1 of [2], we calculate the ν_μ → ν_e appearance signal by tuning our prediction to the corresponding prediction in figure 5 of [2]. In the fit we include the background distribution shown in that figure with a systematic normalization uncertainty of 23% and adopt the χ² definition based on the Poisson distribution. The calculation is performed by using the GLoBES simulation software [3]. Latest MINOS data on the ν_μ → ν_e channel have been presented in [4], corresponding to 8.2 × 10²⁰ protons on target, compared to the 7 × 10²⁰ used in [1]. MINOS finds 62 events with an expectation in the absence of oscillations of 49.6 ± 7.0(stat) ± 2.7(syst), showing no significant indication for ν_μ → ν_e transitions.

In figure 1, we show the region in the sin² θ₁₃ − δ plane indicated by T2K data in comparison to MINOS results. Whereas for T2K we obtain a closed region for sin² θ₁₃ at 90% CL (Δχ² = 2.7), for MINOS we found only an upper bound. As is clear from the figure, T2K and MINOS results are compatible with each other and we show the combined analysis as shaded regions, where the upper bound is determined by the MINOS constraint while the lower bound is given by T2K. Best fit values are in the range sin² θ₁₃ ≈ 0.015–0.023, depending on the CP phase δ, where the variation is somewhat larger for the inverted mass hierarchy. The dotted lines in this figure indicate the 90% CL upper bound on sin² θ₁₃ coming from a combined analysis of the remaining oscillation data, including global reactor, solar, atmospheric and long-baseline disappearance data.

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We move now to the combined analysis of the T2K and MINOS ν_e appearance searches with global neutrino oscillation data as described and referenced in [1]. For the reactor analysis we use the 'recommended' analysis from [1], which adopts the new reactor neutrino fluxes from [5] while including short-baseline reactor neutrino experiments with baselines ${\lesssim } 100$ km in the fit. The results for θ₁₃ are summarized in figure 2. For both neutrino mass hierarchies we found that the 2.5σ indication for θ₁₃ > 0 from T2K gets pushed to the 3σ level (Δχ² = 9) when combined with the weak hint for a nonzero θ₁₃ obtained from the remaining data [1]; see also [6]. We found best fit points at

$$\begin{equation} \begin{array}{cc} \sin^2\theta_{13} = 0.013,\quad \delta = -0.61\pi \quad {\rm (normal \ hierarchy),} \\[6pt] \sin^2\theta_{13} = 0.016,\quad \delta = -0.41\pi \quad {\rm (inverted \ hierarchy).} \end{array} \end{equation} \tag{ 1 }$$

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Due to some complementarity between T2K and MINOS, one obtains, after combining with the θ₁₃ limit from the rest of the data, a 'preferred region' for the CP phase δ at Δχ² = 1, as seen in figure 2. Obviously this preference for the CP phase is not significant³. Marginalizing over the CP phase δ (and all other oscillation parameters) we obtain for the best fit, one-sigma errors, and the significance for θ₁₃ > 0:

$$\begin{equation} \begin{array}{cl} \sin^2\theta_{13} = 0.013^{+0.007}_{-0.005},\quad \Delta\chi^2 = 10.1 \,(3.2\sigma) \quad {\rm (normal),} \\[10pt] \sin^2\theta_{13} = 0.016^{+0.008}_{-0.006},\quad \Delta\chi^2 = 10.1 \,(3.2\sigma) \quad {\rm (inverted).} \end{array} \end{equation} \tag{ 2 }$$

As expected the upper bound on sin² θ₁₃ is dominated by global data without long-baseline appearance data, whereas the lower bound comes mainly from T2K.

Let us briefly consider the sensitivity of these results to the assumptions on the analysis of reactor neutrino data. As discussed in detail in [1], there is a slight tension between reactor neutrino fluxes obtained in [5] and the results of short-baseline reactor neutrino experiments with baselines ${\lesssim } 100$ km. The increase of reactor neutrino fluxes compared to previous calculations in [5] has been confirmed qualitatively by an independent recent calculation [8]. To illustrate the impact on θ₁₃, we show the results for two alternative assumptions for the reactor analysis. Adopting the fluxes from [5] but omitting reactor experiments with baselines ${\lesssim } 100$ km, we obtain for the best fit, one-sigma errors, and the significance for θ₁₃ > 0:

$$\begin{equation} \begin{array}{cl} \hspace{-1.5pc}\sin^2\theta_{13} = 0.022 \pm 0.008,\quad \Delta\chi^2 = 13.5\,(3.7\sigma) \quad {\rm (NH)} \\[4pt] \hspace{-1.5pc}\sin^2\theta_{13} = 0.026 \pm 0.009,\quad \Delta\chi^2 = 15.2 \,(3.9\sigma) \quad {\rm (IH)} \end{array} \quad{\rm (no \ SBL \ react).} \end{equation} \tag{ 3 }$$

If instead we do include the short-baseline reactor data but leave the overall normalization of the reactor neutrino flux free, we obtain

$$\begin{equation} \begin{array}{cl} \sin^2\theta_{13} = 0.011^{+0.007}_{-0.004},\quad \Delta\chi^2 = 7.7 \,(2.8\sigma) \quad {\rm (NH)} \\[10pt] \sin^2\theta_{13} = 0.014^{+0.007}_{-0.006},\quad \Delta\chi^2 = 8.4 \,(2.9\sigma) \quad {\rm (IH)} \end{array} \quad{\rm (flux \ free).} \end{equation} \tag{ 4 }$$

We see that the precise value of the sin² θ₁₃ best fit point, as well as the significance for θ₁₃ > 0, still depend on assumptions on the reactor analysis, as discussed in detail in [1].

To summarize, we display the status for all neutrino oscillation parameters from the global analysis using the default reactor treatment in table 1. If θ₁₃ is indeed within the currently indicated range, we may expect a confirmation by future T2K data soon. Depending on whether its true value is close to the upper of the lower edge of the currently favored range, an independent confirmation of a nonzero θ₁₃ may be expected from reactor experiments within a few months to a few years [9]. After establishing the large mixing angle-Mikheyev–Smirnov–Wolfenstein (LMA-MSW) solution to the solar neutrino problem, the present 3σ indication for a nonzero θ₁₃ may turn out to be the first sign of the second necessary ingredient for observable CP violation in neutrino oscillations; see, e.g., [10] for a review.

Table 1. Summary of neutrino oscillation parameters. For Δm²₃₁, sin² θ₂₃, sin² θ₁₃ and δ, the upper (lower) row corresponds to normal (inverted) neutrino mass hierarchy. See [1] for details and references therein.

Parameter	Best fit ±1σ	2σ	3σ
Δm221 (10−5 eV2)	7.59+0.20−0.18	7.24–7.99	7.09–8.19
Δm231 (10−3 eV2)	2.50+0.09−0.16	2.25–2.68	2.14–2.76
	−(2.40+0.08−0.09)	−(2.23–2.58)	−(2.13–2.67)
sin2 θ12	0.312+0.017−0.015	0.28–0.35	0.27–0.36
sin2 θ23	0.52+0.06−0.07	0.41–0.61	0.39–0.64
	0.52±0.06	0.42–0.61
sin2 θ13	0.013+0.007−0.005	0.004–0.028	0.001–0.035
	0.016+0.008−0.006	0.005–0.031	0.001–0.039
δ	$\left (-0.61^{+0.75}_{-0.65}\right )\pi$	0–2π	0–2π
	$\left (-0.41^{+0.65}_{-0.70}\right )\pi$

This work was supported by the Spanish grants FPA2008-00319/FPA, MULTIDARK Consolider CSD2009-00064 and PROMETEO/2009/091 and by the EU network UNILHC through PITN-GA-2009-237920. MT acknowledges financial support from CSIC under the JAE-Doc program. This work was partially supported by the Transregio Sonderforschungsbereich TR27 'Neutrinos and Beyond' der Deutschen Forschungsgemeinschaft.