Determination of the $\theta_{23}$ octant in LBNO

According to the recent results of the neutrino oscillation experiment MINOS, the neutrino mixing angle $\theta_{23}$ may not be maximal ($45^{\circ}$). Two nearly degenerate solutions are possible, one in the lower octant (LO) where $\theta_{23}<45^{\circ}$, and one in the higher octant (HO) where $\theta_{23}>45^{\circ}$. Long baseline experiments measuring the $\nu_{\mu}\rightarrow\nu_{e}$ are capable of resolving this degeneracy. In this work we study the potential of the planned European LBNO experiment to distinguish between the LO and HO solutions.


Introduction
Neutrino oscillations are described in terms of six physical variables: the three mixing angles θ 12 , θ 23 and θ 13 , the Dirac CP phase δ CP , and the two squared-mass differences ∆m 2 21 = m 2 2 − m 2 1 and ∆m 2 31 = m 2 3 − m 2 1 . Most of these quantities are experimentally measured with good accuracy [1,2,3]. However, one still does not know the sign of ∆m 2 31 , that is, whether the masses obey the normal hierarchy (NH) m 1 , m 2 < m 3 or the inverted hierarchy (IH) m 1 , m 2 > m 3 , and the value of the CP phase δ CP also is undetermined, all values in the range −180 • to +180 • being still allowed. In addition of these two unknowns, which are expected to be resolved in the future long baseline oscillation experiments, there is a third intriguing question known as the octant or θ 23 -degeneracy problem [4,5,6,7]. In the leading order the muon neutrino disappearance in the transition ν µ → ν µ is not sensitive to the octancy of θ 23 , that is, whether this angle lies in the lower octant (LO) θ 23 < 45 • or in the higher octant (HO) θ 23 > 45 • , both alternatives giving the same disappearance probability. In contrast, the leading term of the probability of the electron neutrino appearance ν µ → ν e is octant sensitive [8]. Hence an accurate measurement of the transition probability P (ν µ → ν e ) in the future long baseline neutrino oscillation experiments might be capable of resolving the octant degeneracy.
Of course, the octant degeneracy problem would not exist if the angle θ 23 mixing were maximal, i.e. θ 23 = 45 • . The recent results of the MINOS oscillation experiment [9] seem to indicate, however, that this is not the case. Two degenerate solutions were found, one in the lower octant (LO) with sin 2 θ 23 0.4 and one in the higher octant (HO) with sin 2 θ 23 0.6. This corresponds to a deviation of about 5 • downwards or upwards, correspondingly, from the maximal-mixing value θ 23 = 45 • . On the other hand the T2K collaboration [10] has recently reported a θ 23 central value lying close to the borderline between both octants, being unable to exclude any of the two possibilities.
The octant affects the event rates both for the neutrino transition ν µ → ν e and the antineutrino transitionν µ →ν e , the HO corresponding to more events than the LO. This is true for both the NH and IH mass hierarchies. In addition to the dependence on the θ 23 octant, the event rates are quite strongly affected by the value of the CP phase δ CP . A balanced neutrino and anti-neutrino data is requisite for the separation of LO and HO (for a recent analysis, see [11]).
In this paper, we analyze the potential of the planned long baseline neutrino oscillation experiment LBNO [12] for resolving the θ 23 octant degeneracy. In LBNO the aim is to send neutrino and antineutrino beams, produced at the CERN SPS accelerator, towards the Pyhäsalmi mine, located in central Finland at the distance 2288 km from CERN, where they will be measured using a two-phase Liquid Argon Time Projection Chamber (LArTPC) [13,14] combined with a magnetized muon detector (MIND) [15,16]. In the first phase, the size of the LArTPC detector is planned to have 20 kton fiducial mass. In this phase a 0.75 MW conventional neutrino beam from the CERN SPS will be used. In the second phase the total detector mass will be extended to 70 kton, and a powerful 2 MW HPPS proton driver [17] is foreseen to be in use. We will determine for both phases the 1 σ, 2 σ and 3 σ sensitivity limit of the angle θ 23 that LBNO can achieve with 5+5 -years neutrino and antineutrino run, allowing the CP phase to vary in the range −180 • to +180 • .

Numerical analysis
The sensitivity for determining the θ 23 octant has been previously analysed for NOνA [18] and T2K [10], as well as for the proposed very long baseline experiment of the future, LBNO [19] and LBNE [8]. According to recent reviews (see e.g. [11]), the LBNO offers the best potential for determining the octancy of θ 23 . In this work we present a detailed numerical analysis for the LBNO. The numerical simulation method we use is in most parts adopted from [19], however, using for our calculations the GLoBES simulation program [20,21] instead of Monte Carlo simulations.
GLoBES is a simulator that predicts the propagation of neutrinos from the moment they are created in the source to the point they reach the detector and interact with its content. The software evaluates the effect of matter potentials induced by the traversed medium and calculates the resulting event rates that follow from the detection and reconstruction of neutrino events that take place in the detector. The estimated event rates are then used to evaluate the likelihood of different oscillation parameter values with χ 2 -distributions.
The muon neutrino beam is assumed to be produced in the CERN SPS accelerator with a power of 750 kW, shared between neutrino and antineutrino modes at a 50%/50% ratio. (This is the same set-up proposed in [19] for the determination of the mass hierarchy.) This corresponds to 1.125 × 10 20 POT per year for each beam, and it builds up a total yield of 1.125 × 10 21 over the course of the 5+5 -year running time. We also consider the HPPS setup by increasing the annual POT number to 3.0 × 10 21 . The key parameters concerning the LBNO are presented in Tab. 1.
The muon neutrino beam is assumed to be nearly pure, though it is contaminated by small numbers of electron neutrinos and antineutrinos. The contamination is an irreducible side product of the muon neutrino creation through pion decay, and it cannot be removed. We have obtained the respective neutrino fluxes from dedicated flux files based on a GEANT4 simulation [22]. In this work we assume the following detecting properties [23,24]. The LArTPC detector is capable of detecting electron and muon neutrinos by observing secondary electron and muon leptons at approximately constant 90% rate. The LBNO neutrinos are detected within [0.1 GeV, 10.0 GeV] energy range, which is divided into 80 energy bins, each bin 0.125 GeV wide. The detection and reconstruction process has the following parameters: Whenever a neutrino interacts with the detector substance, the counting system reconstructs the energy and flavor of the incident neutrino and identifies the event with the corresponding energy bin. The reconstructed energy is assumed to be normally distributed with a resolution of 0.15 × √ E, where E is the neutrino energy in GeV. The cross sections of the charged current (CC) and neutral current (NC) neutrino-nucleon interactions are given in cross section files simulated for LArTPC with a dedicated GENIE simulation [25]. The simulation is specifically dedicated to LArTPC systems, and it takes the oscillations to tau neutrinos into account better than any previous simulation.
The LBNO experiment is designed to study the electron appearance probabilities P (ν µ → ν e ) and P (ν µ →ν e ) by counting the corresponding CC events in the detector. These CC events constitute the signal, whereas background consists of any type of events that have similar final state properties. On one hand, the electron appearance channels gain background from CC and NC events with ν e andν e arising from the oscillations of the impurities in the muon neutrino beam. On the other hand, the detector is also assumed to have a 0.5% chance to accept ν µ andν µ from ν µ → ν µ andν µ →ν µ as ν e andν e from both CC and NC event categories. Lastly, ν τ andν τ neutrinos originated from ν µ → ν τ andν µ →ν τ oscillations also contribute to the background. The number of τ leptons produced in the detector from these neutrinos accounts for approximately 6% of the total number of leptons produced [12]. Inserting the branching ratio of the τ subsequent decay into electrons through τ → e ν e ν τ (∼17.8%) [26] together with the detector efficiency (90%), one sees that the corresponding ν e contamination is 1%, hence of the same order of magnitude as the intrinsic beam contamination.
The χ 2 values are calculated as follows (see e.g. [20,21]). The statistical part is computed with the Poissonian function where the number of observed events (O i ) in the ith bin is computed from the so called true values (ω 0 ) and the number of test events (T i ) from the test values (ω), respectively. The observed events is the category of events that would result from oscillation parameter values that one considers to be closest to the truth. They are based on the best-fit values obtained from the most recent experiments. We denote these values with ω 0 . Since all parameter values are not precisely known, such as the sign of ∆m 2 31 , the χ 2 values need to be computed for all possible scenarios. The number of observed events is taken to be the sum of events from signal and background components: where N sg i and N bg i stand for the numbers of signal and background events. The test values on the other hand stand for event numbers that are computed with whatever oscillation parameter values one wants to test. We denote these values with ω. We also apply systematic errors to both signal and background events by incorporating two nuisance parameters [20,21], ζ 1 and ζ 2 , with error weights π 1 and π 2 : The systematic errors are included by minimizing the χ 2 function over nuisance parameters ζ 1 and ζ 2 : where χ 2 (ω, ω 0 ) is the Poissonian function given by Eq. (1). We assume 5 % systematical error weights in both signal and background by setting π 1 , π 2 = 0.05. This corresponds to the normalization error in the LArTPC detectors [11]. We also assume that the values of θ 12 , θ 13 , ∆m 2 21 , ∆m 2 31 , δ CP and ρ are associated with standard deviations σ(θ 12 ), σ(θ 13 ), σ(∆m 2 21 ), σ(∆m 2 31 ) and σ(ρ). We include these parameter uncertainties via the so called priors [20,21]. The prior function is given by: The overall χ 2 value is calculated as the sum of the pull and prior parts from Eq. (4) and Eq. (5), which is then minimized over the test values: The matter density parameter ρ is taken into account as a variable in Eq. (6). The density distribution of the Earth's crust between CERN and Pyhäsalmi is known to a good accuracy [27], but for this study we consider it sufficient to evaluate the matter density function with a 20-step PREM distribution [28], and assume 2% error value (1 σ).
The final χ 2 value is calculated by minimizing χ 2 total over all oscillation parameters in the test values, that is, over ω. The prior function constrains the value ranges over which χ 2 total may converge, and the absence of δ CP in Eq. (5) indicates that no such constraints are assumed for δ CP . We also choose to keep θ 23 fixed in the minimization process.
We calculate the 1 σ, 2 σ and 3 σ confidence levels for the event that the LBNO experiment will be able to rule out one octant when the other octant is assumed to be correct. This is done by computing χ 2 first for θ 23 and then 90 • − θ 23 , and calculating the difference between the two χ 2 values, both calculated as given in Eq. (6): We take the true values from [29], which contains a recent compilation on experimentally determined parameter values. These values are also presented in Tab. 2. The Gaussian errors shown in Tab. 2 are distributed for parameters sin 2 θ 12 , sin 2 2θ 13 , ∆m 2 21 and ∆m 2 31 , respectively. The errors of δ CP and sin 2 θ 23 are not present in the prior function χ 2 prior and therefore they are both marked with zero. This follows from our choice that δ CP is not assigned with constraints and θ 23 is kept fixed in the minimization of χ 2 .
The minimization of the χ 2 function in Eq. (6) is carried out keeping θ 23 fixed and other parameters free. Since θ 23 and δ CP are not precisely known, we calculate the χ 2 values for different possible values of θ 23 and δ CP , and for both mass hierarchies as well.

Results and discussion
We have investigated the ability of the LBNO experiment to determine the octancy of the neutrino mixing angle θ 23 up to a 3 σ confidence limit (CL) for all values of the phase δ CP . This was done by computing the ∆χ 2 distribution for a range of θ 23 and δ CP values. The ∆χ 2 distribution was computed with a grid of 120×360 points, interpolating the intermediate values.
The contour plots were produced for four different setups: SPS beam with 20 kt LArTPC, SPS beam with 70 kt LArTPC, HPPS beam with 20 kt LArTPC and HPPS beam with 70 kt LArTPC. Figs. 1 and 2 present the resulting 1 σ, 2 σ and 3 σ CL contours for the normal and inverted hierarchy, respectively. In each figure the white regions in the It is seen that for all the different setups considered the right θ 23 octant can be asserted in NH with at least 3 σ CL. As for IH this limit is reached for the lower octant in all cases, whereas for the higher octant it fails to be reached in the sole case of the 20 kt setup with 0.75 MW. All other setup versions yield improved sensitivities so that the 3 σ limit can be reached for all of them regardless of the mass hierarchy and δ CP value. The graphs also show by themselves that increasing beam power (by a factor of 2.7 i.e. from 0.75 MW to 2 MW) with the same detector is a lot more effective than increasing detector size from 20kt to 70 kt with the same beam power.
In principle, any increase in the exposure moves the 3 σ CL contour closer to the θ 23 = 45 • value. If one is to expect that the real value of θ 23 is to be 5 • off from 45 • , then even the SPS setup with 20 kt detector may be sufficient to reach the 3 σ CL for both mass hierarchies. The sensitivity is worse near 45 • , however, and it would require an upgrade to reach the 3 σ CL benchmark. A future HPPS facility with a 70 kt LArTPC detector, for instance, could solve this problem as it would set the limit to less than ±0.6 • . Furthermore, LBNO will most likely be able to measure the mass hierarchy with a 0.75 MW SPS beam and a 20 kt LArTPC detector, in which case the acquired data could be used to narrow down the estimate on the θ 23 octant. The determination of the θ 23 octancy would hence be a logical follow-up of the mass hierarchy measurement.