Neutrino parameters from reactor and accelerator neutrino experiments

We revisit correlations of neutrino oscillation parameters in reactor and long-baseline neutrino oscillation experiments. A framework based on an effective value of ${\theta }_{13}$ is presented, which can be used to analytically study the correlations and explain some questions including why and when ${\delta }_{CP}$ has the best fit value of $-\pi /2$, why current and future long-baseline experiments will have less precision of ${\delta }_{CP}$ around $\pm \pi /2$ than that around zero, etc. Recent hints on the $CP$ phase are then considered from the point of view that different reactor and long-baseline neutrino experiments provide currently different best-fit values of ${\theta }_{13}$ and ${\theta }_{23}$. We point out that the significance of the hints changes for the different available best-fit values.

Figure 1

Contour plot of ${\sin }^{2}2{\theta }_{13}^{\mathrm{eff}}({\theta }_{13},{\delta }_{CP})$. The five contours from the left to the right correspond to ${\sin }^{2}2{\theta }_{13}^{\mathrm{eff}}=0.087$, 0.106, 0.139, 0.184 and 0.218, respectively. Normal mass ordering is assumed, with $({\theta }_{12},{\theta }_{23},\alpha )=(34°,45°,0.031)$.

Figure 2

The best-fit value of ${\delta }_{CP}$ from T2K ${\nu }_e$ (blue curve) and ${\overline{\nu }}_e$ (orange curve) data [4] as a function of ${\theta }_{13}$. Solid/dashed curves are for normal/inverted mass ordering. For $5.5°<{\theta }_{13}<9.2r°$, both ${\nu }_e$ and ${\overline{\nu }}_e$ data prefer a maximal $CP$ violation value ($-\pi /2$) of ${\delta }_{CP}$.

Figure 3

Left panel: Constraints on $({\theta }_{13},{\delta }_{CP})$ from T2K [4] and constraints on ${\theta }_{13}$ from reactor neutrino experiments. Solid/dashed curves are for normal/inverted mass ordering, blue/orange curves for ${\nu }_e/{\overline{\nu }}_e$ appearance data, respectively. Right panel: Fit results on $({\theta }_{13},{\delta }_{CP})$ when T2K neutrino and antineutrino data are combined with reactor data. The upper plots apply for the $\mathrm{NO}\nu {\mathrm{A}}^{-}$ solution of ${\theta }_{23}$, the lower plots for $\mathrm{NO}\nu {\mathrm{A}}^{+}$.

Figure 4

Constraints on $({\theta }_{23},{\delta }_{CP})$ from T2K appearance data combined with ${\nu }_{\mu }$ disappearance data from T2K and $\mathrm{NO}\nu \mathrm{A}$. Solid/dashed curves are for normal/inverted mass ordering, blue/orange curves for ${\nu }_e/{\overline{\nu }}_e$ appearance data, respectively. The upper plots are for the best-fit value of ${\theta }_{13}$ from Daya Bay/RENO, the lower ones for the best-fit value of ${\theta }_{13}$ from Double Chooz.

Figure 5

$CP$ violation (CPV) in T2K data. Upper left panel: current significance of CPV in T2K (data collected until May 2016 with $1.5\times {10}^{21}$ POT [4]). Other panels: future sensitivities of CPV with $20\times {10}^{21}$ POT [63] assuming ${\delta }_{CP}=-\pi /2$. The ratios [e.g. $(1:2)$, $(2:1)$] denote ratios of running times in neutrino and antineutrino modes.

Figure 6

Comparison of normal and inverted mass ordering by $\mathrm{\Delta }{\chi }_{\min }^2\equiv {\chi }_{\min \mathrm{N}}^2-{\chi }_{\min \mathrm{I}}^2$ from T2K appearance data. The black dotted curve corresponds to $\mathrm{\Delta }{\chi }_{\min }^2=0$. With the one-parameter fitting performed here, $\mathrm{\Delta }{\chi }_{\min }^2={\sigma }^2$.

Figure 7

The probability of ${\nu }_{\mu }\to {\nu }_e$ transitions and $F(E_{\nu })$ in T2K [cf. Eqs. (a2) and (a16)]. The definitions of $P_X$ and $p_X$ ($X=0,s,c,2$) are given in Eqs. (a3) to (a10), $P$ is the total oscillation probability for $({\theta }_{23},{\theta }_{13},{\theta }_{12})=(45°,8.8°,34°)$, $\mathrm{\Delta }{m}_{13}^2=2.44\times {10}^{-3}{\mathrm{eV}}^2$, $\alpha =0.31$, and ${\delta }_{CP}=-\pi /2$ (hence $P_c=0$). The function $F$ is presented in arbitrary units.

Figure 8

Comparison of ${\nu }_e$ event rates (in arbitrary units) computed (i) from the effective ${\theta }_{13}^{\mathrm{eff}}$ formula [cf. Eqs. (3) and (5)] and (ii) from the actual oscillation formula [cf. Eq. (1)] applied to T2K.

Figure 9

Reproduced constraints (red and black curves) on $({\theta }_{13},{\delta }_{CP})$ from our effective ${\theta }_{13}$ framework compared to the original T2K result [4] (red and black points) for the normal mass ordering. The agreement shows that the effective ${\theta }_{13}$ framework is very accurate to describe the sensitivity of T2K ${\nu }_{\mu }\to {\nu }_e$ data on $({\theta }_{13},{\delta }_{CP})$.