Potential of octant degeneracy resolution in JUNO

This research continues to focus on the idea using cyclotronic antineutrino source for purposes of neutrino physics. Long baseline experiments suffer from degeneracies and correlations between $\Theta_{23}$, $\delta_{\rm CP}$ and the mass hierarchy (MH). However the combination of a superconductive cyclotron and a big liquid scintillator detector like JUNO in a medium baseline experiment, which does not depend on the MH, may allow us to determine whether the position of the mixing angle $\Theta_{23}$ is in the lower octant (LO) or upper octant (UO). Such an experiment would improve the precision of the $\Theta_{23}$ measurement to a degree which depends on the CP-phase.


The problem of octant degeneracy
In the framework of 3-flavor neutrino mixing through Pontecorvo-Maki-Nakagawa-Sakata [1] unitary mixing matrix: in the standard parametrization sin 2 (Θ 23 ) and cos 2 (Θ 23 ) can be expressed as: It is clear that if Θ 23 = 45 o , then mixing between ν µ and ν τ becomes maximal. This would indicate symmetry between the ν e → ν µ and ν e → ν τ oscillation processes. The octant problem refers to the degeneracy between Θ 23 and π/2 − Θ 23 , because the mixing angle enters in the oscillation probability as a term within sin(2Θ 23 ). However the degeneracy between the lower octant (LO) and the upper octant (UO) can be eliminated, if a measurement is sensitive to terms with sin(Θ 23 ) or cos(Θ 23 ). Until recently, there was a quite a large uncertainty in the measurements of sin 2 (Θ 23 ). sin 2 (Θ 23 ) = 0.35 − 0.65 (90%C.L.) for normal hierarchy (NH) and sin 2 (Θ 23 ) = 0.34 − 0.67 (90%C.L.) for inverted hierarchy (IH) from combined analysis of MINOS experiment [2]. After that T2K reported the best fit value of sin 2 (Θ 23 ) = 0.532(NH) and sin 2 (Θ 23 ) = 0.534(IH) with smaller uncertainty and consistent with hypothesis of maximal mixing [3]. But recent data from the NOνA experiment now favors Θ 23 in either LO or UO, and disfavors of maximal mixing at 0.8σ significance [4].
Since the leading approximation of oscillation probability for reactor experiments does not depend on the mixing angle Θ 23 , the current scientific program of JUNO [5] will not allow for a solution to the problem of octant degeneracy. However precise measurements of ν e appearance fromν µ disappearance would provide a good possibility for improvement of this issue.
2 Methodology of the numerical analysis

Proposal of the experimental setup
The full description of our proposal is presented in [6], which is based on the DAEδALUS experiment project [7]. It is worthwhile to summarize the main components of the previous research. We suggested using the appearance channel for electron antineutrinos from muon antineutrinos. In the framework of standard three neutrino mixing theory the oscillation probability can be expressed as [8]: P ν µ →ν e = sin 2 θ 23 sin 2 2θ 13 sin 2 ∆ 31 + cos 2 θ 23 sin 2 2θ 12 sin 2 ∆ 21 + + sin 2θ 13 sin 2θ 23 sin 2θ 12 sin ∆ 31 sin ∆ 21 · cos(∆ 31 − δ CP ), (2.1) where ∆ ij = ∆m 2 ij ·L/(4E ν ); ∆m 2 ij -the neutrino mass squared difference; L -the distance between source and detector; E ν -neutrino energy; δ CP -Dirac phase of CP violation. The source ofν µ is three-body decay of µ + from decay at rest of the stopped π + , which will be produced by a superconductive cyclotron [9]. The contribution to antineutrino spectrum is around 10 −4 from π − , which are created together with π + [7]. Two cyclotrons (near and far) will be located at distances 1.5 km and 20 km respectively. The power of the near cyclotron is 1 MW. It is needed as a flux monitor. There are two options for the power of the far cyclotron: 5 MW and 10 MW. We are planning to use JUNO as a liquid scintillator detector, which has a total mass of 20 kt. The expected exposure time of the experiment is 10 years. NH is assumed, because at the distance 20 km the experiment is insensitive to mass hierarchy. The estimated IBD-event spectrum as a function of energy is depicted in Figure 1. It is clear, that neutrino rate increases with the mixing angle Θ 23 .

Statistical evaluation of MC simulations
Event rate analysis is based on statistical treatment of the expected IBD signal rate inside the detector. Initial muon antineutrinos have a continuous spectrum with an endpoint of 52.8 MeV. In order to exclude a significant part of the atmospheric background, we chose the energy window between 20 and 52.8 MeV. However this is not sufficient to ignore the background completely.
The current statistical analysis is split on two parts. The first part is the sensitivity to octant degeneracy. The second part is about the precise measurement of Θ 23 .

Sensitivity to discovery of true octant
Here we follow the so-called classical method of calculating a confidence level. This method is based on the calculation of a ∆χ 2 function, which, as Wilks's theorem predicts [10], should follow a chi-square distribution. The number of degrees of freedom can be calculated as the difference between degrees of freedom of initial chi-square functions. Usually, this number is equal to the quantity of estimating parameters. In our case, there is only one parameter -Θ 23 . A χ 2 distribution with one degree of freedom has the same distribution as the square of a single normally distributed variable [11]. Therefore standard Gaussian confidence levels 1σ(68.3%), 2σ(95.4%), 3σ(99.7%) etc. correspond to values of χ 2 : 1, 4, 9 and etc.
In general the sensitivity to octant degeneracy can be calculated through the minimization of a ∆χ 2 function, which is given by: where "min" means, that both chi-square functions χ 2 (90 o − Θ 23 ) and χ 2 (Θ 23 ) have to be minimized through their parameter space; Θ 23 is a scanning parameter, which is fixed for each iteration of MC cycle. In our case, the chi-square function has only one minimum, which is close to the test-true value of Θ 23 . In the opposite octant this function always increases. Consequently we need to redefine the ∆χ 2 function as: where 45 o corresponds to a border between two octants. We use the chi-square function presented in [12,13].
where pull-term includes Poisson statistics while taking into account the background and flux normalization. Additional Gaussian penalties are also added.  [14]. Most of them were used in prior-term of the chi-square function for our calculations, except the parameter of interest -Θ 23 . The normal hierarchy is assumed.
Here N b -total number of bins in the histogram; µ i -predicted counts in the i-th bin; n i -observed counts in the i-th bin; s and b -so-called nuisance parameters for signal and background respectively; σ s and σ b -systematic errors for signal and background counts. µ i is represented by next equation: where N i s and N i bkg -number of counts in the i-th bin for signal and background respectively. The second prior-term of the function (2.4) corresponds to uncertainties of oscillation parameters and can be written as: where N p -quantity of oscillation parameters; η jj-th oscillation parameter; η o j -best fit value of η j ; δη j -one sigma error of η o j .

The accuracy of Θ 23 measurement
The estimation of the accuracy of measurement for the current best fit value of Θ 23 can be obtained by minimizing the chi-square function (2.4) through the whole parameter space. It should be emphasized, that from recent experimental data [4] the best fit value of Θ 23 is split between LO and UO. And we also use two values of Θ 23 in the calculation of precision. Further, we give a set of the oscillation parameters and their uncertainties taken from PDG in Table 1.

Monte-Carlo simulations
The expected electron antineutrino event spectra at a distance of 20 km were simulated using the Monte-Carlo method including oscillation. The energy resolution of the JUNO detector is 3% per 1 MeV. The beam power of the far cyclotron is 5 or 10 MW with systematic flux uncertainty σ s = 2%, which includes the uncertainties of shape and normalization. We treat neutral current events (NC) as a background. The initial estimation gives 439 NC events for an exposure time of 200 kt·year with a duty factor of 33%. Using a technique from [15] which is based on the signals coincidence and pulse shape discrimination, this background can be reduced significantly to 33 NC events. Adding also fast neutron and charge current atmospheric events, the total background will equal 45 events. The last number is used in simulations with systematic uncertainty σ b = 5%.
To investigate the sensitive region of octant degeneracy, 1K MC "fake" experiments were calculated for each sample with particular fixed values of δ CP . We did not apply any constraints to the parameter Θ 23 . Both parts of ∆χ 2 in equation (2.2) were minimized using the ROOT package Minuit [16,17]. Finally, the sensitivity region was calculated as defined in section 2.2.1.
In order to evaluate the potential of JUNO to accurately measure the mixing angle Θ 23 , 5K MC "fake" experiments were simulated for each sample with particular fixed value of δ CP . The chi-square function (2.4) was minimized through all parameter space. Then a histogram was filled by the extracted values of Θ 23 . The shape of the histogram is Gaussian, because we supposed all uncertainties of parameters have Gaussian distribution. The 1σ error of Θ 23 was obtained as a standard deviation of the aforementioned histogram. This procedure has been repeated for the whole range of CP-phase from −π to π.

Results
Experimental sensitivity to octant degeneracy is depicted in Figure 2. The yellow area shows the 68.3% confidence interval, within which the experiment is insensitive to octant degeneracy. The green area shows the insensitive region with confidence level 99.7%. In    by maximum of the probability function (2.1) for δ CP = π/2 and minimum for δ CP = −π/2. As can be seen in Figure 3, the main uncertainty comes from oscillation parameters. Our estimation shows that the dominant uncertainty comes from mixing angle Θ 13 . The influence of background is quite small, especially for higher statistics with a 10 MW source. Statistically the improvement in the result is possible only for the 10 MW source. However, in reality only values from the LO can improve the result in the case of negative CP-phase.

Conclusion
Current work has demonstrated another application of using a superconductive cyclotron for measurements in neutrino physics. The transition channelν µ →ν e allows us to explore not only the problem of CP violation, but at the same time to realize the precise measurement of Θ 23 and partially resolve octant degeneracy.
It was shown that the distinction between LO and UO is comparable with combined analysis of T2K and NOνA especially for negative values of δ CP . Regarding the precision of Θ 23 measuring, the current best fit value can be improved only for the 10 MW case, especially if the mixing angle lies in LO. There are two main difficulties for precision measurements: uncertainties in oscillation parameters and small statistics. The problem of statistics can be reduced through the use of a small water detector instead of the near cyclotron for monitoring neutrino flux. This allows us to use the far cyclotron continuously, as proposed for the TNT2K experiment [19].
The combination of JUNO and superconductive cyclotrons can be a good alternative to conventional beam experiments. It will allow for the measurement of Θ 23 and δ CP in the current scientific program without the spoiling of JUNO's main goals.