Understanding the degeneracies in NO$\nu$A data

The combined analysis of $\nu_\mu$ disappearance and $\nu_e$ appearance data of NO$\nu$A experiment leads to three nearly degenerate solutions. This degeneracy can be understood in terms of deviations in $\nu_e$ appearance signal, caused by unknown effects, with respect to the signal expected for a reference set of oscillations parameters. We define the reference set to be vacuum oscillations in the limit of maximal $\theta_{23}$ and no CP-violation. We then calculate the deviations induced in the $\nu_e$ appearance signal event rate by three unknown effects: (a) matter effects, due to normal or inverted hierarchy (b) octant effects, due to $\theta_{23}$ being in higher or lower octant and (c) CP-violation, whether $\delta_{CP} \sim - \pi/2$ or $\delta_{CP} \sim \pi/2$. We find that the deviation caused by each of these effects is the same for NO$\nu$A. The observed number of $\nu_e$ events in NO$\nu$A is equivalent to the increase caused by one of the effects. Therefore, the observed number of $\nu_e$ appearance events of NO$\nu$A is the net result of the increase caused by two of the unknown effects and the decrease caused by the third. Thus we get the three degenerate solutions. We also find that further data by NO$\nu$A can not distinguish between these degenerate solutions but addition of one year of neutrino run of DUNE can make a distinction between all three solutions. The distinction between the two NH solutions and the IH solution becomes possible because of the larger matter effect in DUNE. The distinction between the two NH solutions with different octants is a result of the synergy between the anti-neutrino data of NO$\nu$A and the neutrino data of DUNE.


I. INTRODUCTION
The data from the solar [1,2] and the atmospheric [3,4] neutrino experiments led to the discovery of neutrino oscillations. Both the solar and the atmospheric neutrino anomalies can be explained in terms of the oscillations of the three neutrino flavours, ν e , ν µ and ν τ , into one another. The oscillation probabilities depend on two independent mass-squared differences, ∆ 21 and ∆ 31 , three mixing angles, θ 12 , θ 13 and θ 23 , and a CP-violating phase δ CP . Among these parameters, there are two small quantities: the angle θ 13 and the ratio During the past decade and a half, a number of experiments with man-made neutrino sources have made precision measurements of the mass-squared differences and the mixing angles. This was possible because the expressions for the three flavour survival probabilities reduce to those of effective two flavour survival probabilities, under appropriate approximations. For example, setting θ 13 = 0 in P (ν e →ν e ) expression for KamLAND [5,6] experiment reduces it to an effective two flavour survival probability in terms of ∆ 21 and θ 12 . A similar effective two flavour survival probability, in terms of ∆ 31 and θ 23 , for MINOS [7] experiment can be obtained by setting θ 13 = 0 = ∆ 21 in the expression for P (ν µ → ν µ ).
For the short baseline reactor neutrino experiments, Double-CHOOZ [8], Daya-Bay [9] and RENO [10], an effective two flavour expression in terms of ∆ 31 and θ 13 is obtained by setting ∆ 21 = 0 in the expression for P (ν e →ν e ). This reduction to effective two flavour expressions leads to accurate measurement of the modulus of the mass-squared differences and sin 2 2θ ij .
The solar neutrino data requires ∆ 21 to be positive but the sign of ∆ 31 is still unknown.
The value of sin 2 2θ 23 is measured quite accurately but, since it is close to 1, there is a large uncertainty in the value of sin 2 θ 23 . There is no measurement yet of the CP-violating phase δ CP . The best-fit values and the allowed 1 σ and 3 σ of the mass-squared differences and the mixing angles from the disappearance data of the above experiments plus the solar and the atmospheric data is given in table I.
The dominant oscillations for these experiments are driven by ∆ 31 . These experiments are also designed to be sensitive to CP-violation in neutrino oscillations. Hence they are also sensitive to ∆ 21 dependent sub-dominant term in the oscillation probability. Thus the data  atmospheric, reactor and accelerator data [11]. Note that NOνA data is not included in this analysis.
of these two experiments must necessarily be analysed using the full three flavour expressions for the neutrino survival (ν µ disappearance) and oscillation (ν e appearance) probabilities.
Since these probabilities depend on a number of parameters, degenerate solutions arise when they are fit to the data. In particular, the ν µ → ν e appearance probability depends on three unknowns: (a) neutrino mass hierarchy (∆ 31 > 0 or ∆ 31 < 0), (b) θ 23 octant (θ 23 > π/4 or θ 23 < π/4) and (c) value of δ CP . In this report, we study how the three degenerate solutions of NOνA arise due to the above three unknowns. We also investigate how the DUNE [16] experiment can fully resolve this three fold degeneracy.
In T2K and NOνA experiments, the neutrinos travel long distances through earth matter and undergo coherent forward scattering. The effect of this scattering is taken into account through the Wolfenstein matter term [17] A (in eV 2 ) = 0.76 × 10 −4 ρ (in gm/cc) E (in GeV), where E is the energy of the neutrino and ρ is the density of the matter. The interference between A and ∆ 31 leads to the modification of neutrino oscillation probability due to matter 4 effects. This modified expression for P (ν µ → ν e ) is given by [18,19] P (ν µ → ν e ) = P µe = sin 2 2θ 13 sin 2 θ 23 sin 2∆ (1 −Â) (1 −Â) 2 +α cos θ 13 sin 2θ 12 sin 2θ 13 sin 2θ 23 cos(∆ + δ CP ) sin∆Â appropriately. Thus there is an eight-fold degeneracy in interpreting the expression for P µe , if the value of sin 2 2θ 13 is not known precisely. We will show below that the present precision measurement of θ 13 breaks this eight-fold degeneracy into (1 + 3 + 3 + 1) pattern.

A. 2017 analysis
NOνA [15] is a long baseline neutrino oscillation experiment capable of measuring the survival probability P (ν µ → ν µ ) and the oscillation probability P (ν µ → ν e ). The NuMI beam at Fermilab, with a power of 700 kW which corresponds to 6 × 10 20 protons on target (POT) per year, produces the neutrinos. The far detector consists of 14 kton of totally active scintillator material and is located 810 km away at a 0.8 • off-axis location. Due to the off-axis location, the flux peaks sharply at 2 GeV, which is close to the energy of maximum oscillation of 1.4 GeV. It has started taking data in 2014 and is expected to run three years in neutrino mode and three years in anti-neutrino mode. The combined analysis of ν µ disappearance and ν e appearance data is given in ref. [21], which is based on a neutrino run with 6.05 × 10 20 POT. This analysis gives the following three (almost) degenerate solutions for the unknown quantities: • normal hierarchy (NH), which increases P µe or inverted hierarchy (IH) which decreases it, • higher octant (HO), which increases P µe or lower octant (LO) which decreases it and • δ CP = −90 • , which increases P µe or δ CP = +90 • , which decreases it.
The event numbers are calculated using GLoBES software [22,23]. The following inputs are used for the well-measured neutrino parameters: ∆ 21 = 7.5 × 10 −5 eV 2 , sin 2 θ 12 = 0.306,   increase in P µe . We label this case as (+ + +). In such a case, we get the maximum number of ν e appearance events. Another case is that two of the undetermined parameters shift so as to increase P µe whereas the third parameter shifts to lower it. This can occur in three         fig. 1. We have also checked the discrimination capabilities of (4ν + 2ν) and (2ν + 4ν) runs. They are not noticeably different from those of the (3ν + 3ν) run.  From fig. 2, we see that the addition of one year of neutrino data of DUNE to NOνA data of (3ν + 3ν) run leads to an essentially unique identification of the correct solution at 3σ level. The average neutrino energy for the DUNE experiment is larger than the energy of NOνA and hence its matter effect is larger. Therefore, the change in ν e appearance events induced by matter effects is larger compared to the changes induced by octant effects or by δ CP . This sets apart the IH solution from the two NH solutions. There is a modest difference in the prediction of ν e appearance events for the two NH solutions with different octants of θ 23 , as shown in table VI. This difference, combined with the discriminating power of NOνA anti-neutrino data, leads to a 3 σ distinction between the two NH solutions.
Thus the synergy between the anti-neutrino data of NOνA and the neutrino data of DUNE plays an important role in distinguishing between the two NH solutions. In ref. [26] the combination of NOνA (3ν + 3ν) run along with DUNE (1ν + 1ν) run was considered.
Their results are very similar to our results. We have not included T2K in these simulations because its best-fit value of sin 2 θ 23 [14] does not agree with any of the solutions given in ref. [21].

B. 2018 data
During the past year, the NOνA collaboration has re-calibrated their signal identification algorithms [27]. In addition they have accumulated more data. As a result of the analysis with the new procedure, NOνA finds a best-fit solution in the higher octant at (NH, sin 2 θ 23 = 0.56, δ CP = −144 • ). There is a nearly degenerate solution in the lower octant at (NH, the ability of long baseline neutrino experiments to distinguish between these two solutions.
We study this discrimination ability using the same procedure as before. The parameters of the HO solution are used as input to GLoBES and the neutrino and anti-neutrino event spectra are simulated. We also use GLoBES to simulate these spectra for various 'test' values of the neutrino oscillation parameters and compute the χ 2 between the spectrum of the HO solution and each of the test spectra. This computation is done for for three different situations: for NOνA simulations alone, for NOνA + T2K simulations and for NOνA + T2K + DUNE simulations. The same procedure is repeated for the LO solution. In dealing with these new solutions, we have included the simulation of T2K data also, because the newly allowed values of sin 2 θ 23 agree with T2K best-fit value [14].
The results are shown in fig. 3 and fig. 4. Turning our attention to the discrimination between the two different octant solutions, we find that the data from NOνA alone can not distinguish between them. Addition of T2K data helps in reducing the allowed regions a little but still does not provide a discrimination between the two octants. T2K data strongly discriminates against δ CP ≈ 90 • hence the test values around this region are ruled out at 3 σ, though they are allowed by NOνA data. One year neutrino data of DUNE, which has a modest octant discrimination power, is able to rule out the wrong octant at 1 σ but not at 3 σ. Neither NOνA nor NOνA + T2K can establish CP violation at 3 σ.
We have also done a simulation of DUNE (5ν + 5ν) run to check how well CP violation can be established. The results are shown in fig. 5 allowed contours. We note that, for LO solution, CP-violation can be established at 5 σ.
But, for the HO solution, δ CP = 180 • is not ruled out at 5 σ. The addition of (5ν + 5ν) run of DUNE also helps in distinguishing between the two solutions at 3 σ level.   number of ν e events are well above or well below those expected from the reference point of vacuum oscillations with maximal θ 23 and no CP-violation, then one can uniquely determine the hierarchy, octant and δ CP . If the difference between observed events and those expected from reference point is moderate, then, in general, there will be three degenerate solutions: One solution whose octant is distinct from that of the other two, a second solution whose half plane of δ CP is distinct from that of other two and a third solution whose hierarchy is distinct from that of the other two. The early neutrino data of NOνA [21], which showed In this report, we have shown that NOνA will not be able to make a distinction between any of these three solutions. The two higher octant solutions are completely degenerate with respect to both neutrino and anti-neutrino data of NOνA. The lower octant solution is distinct from the point of view of anti-neutrino data but the expectedν e events are quite small. The corresponding large statistical errors prevent a clean isolation of this solution.
However, the addition of one year of neutrino data from DUNE can effectively isolate each of these three solutions at 3 σ. The two NH solutions can be discriminated from the IH solution because of the large matter effects in DUNE. Between the two NH solutions of different octants, both the anti-neutrino data of NOνA and the neutrino data of DUNE have a moderate discriminating capability. The synergy between these two sets of data is capable of providing a 3 σ discrimination between these two NH solutions.
Later data of NOνA, based on a more refined signal identification algorithm [27], has only two degenerate solutions: both with NH but with different octants, where sin 2 θ 23 values in both cases are closer to maximal mixing. This is a consequence of the new procedure, which has identified a larger number of signal events leading to a fairly large excess of ν e events compared to the expectation from the reference point. A significant part of this excess occurs due to the matter effects of NH. There are two possibilities to explain the remainder of the excess: • part of it is due to higher octant value of θ 23 and part of it is due to the δ CP in lower half plane but well away from the maximal CP-violation of −90 • • a small suppression due to lower octant value of θ 23 and a moderately large increase due to δ CP being in the neighbourhood of the maximal CP-violation value −90 • .
We found that neither NOνA nor NOνA + T2K is capable of distinguishing between these two solutions at 3 σ level nor can they rule out the wrong hierarchy. But addition of one year of neutrino data of DUNE is capable of ruling out the wrong hierarchy at 3 σ level but is unable to provide a similar discrimination between the two solutions. Addition of a (5ν + 5ν) run of the DUNE experiment can distinguish between the solutions at 3 σ. It can also establish CP-violation at 5 σ level for the lower octant solution but not for the higher octant solution. This occurs because the δ CP value of the lower octant solution is closer to maximal CP-violation.