The Effect of a Light Sterile Neutrino at NO$\nu$A and DUNE

Now that the NOvA experiment is approaching two years run time and has released some preliminary data, some constraints for the oscillation parameters can be inferred. Currently the best fits for NOvA are three separate results, the reason they are indistinct is that they produce almost degenerate probability curves. It has been postulated that these degeneracies can be resolved by running antineutrinos at NOvA and/or combining its data with T2K. However, this degeneracy resolution power can be compromised if sterile neutrinos are present due to additional degrees of freedom that can significantly alter the oscillation probability for any of the current best fits. We aim to investigate this degradation in predictive power and the effect of the DUNE experiment on it. Now that the NOvA experiment has been running for a few years and has released some preliminary data, some constraints for the oscillation parameters can be inferred. The best fits for NOvA include three degenerate results, the reason they are indistinct is that they produce almost degenerate probability curves. It has been postulated that these degeneracies can be resolved by running antineutrinos at NOvA and/or combining its data with T2K. However, this degeneracy resolution power can be compromised if sterile neutrinos are present due to additional degrees of freedom that can significantly alter the oscillation probability for any of these best fits. We aim to investigate this degradation in predictive power and the effect of the DUNE experiment on it. In light of the 2018 NOvA data we also consider the same fits but with theta_{23}=45 to see if the sensitivity results are different.


I. INTRODUCTION
The existence of neutrino oscillation implies that the mass eigenstates (ν 1 , ν 2 , ν 3 ) and flavor eigenstates (ν e , ν µ , ν τ ) of neutrinos do not have one-to-one correspondence, instead each mass eigenstate has a different mix of each flavor eigenstate defined some mixing matrix (named the PMNS matrix after Pontecorvo Maki Nakagawa and Sakata). Solar, atmospheric and reactor experiments have put limits on oscillation parameters (primarily θ 12 , θ 13 , θ 23 and ∆m 2 21 ) but are unable to fully resolve the parameter space. Long baseline (LBL) experiments are required to determine some of the more elusive parameters including the mass hierarchy and the CP phase δ 13 but unfortunately this is where several degeneracies arise.
In addition to the aforementioned parameter uncertainties, several short baseline experiments have reported results inconsistent with the three flavor oscillation paradigm, for an overview of the anomalies we refer to [1]. A possible explanation is that oscillation is still the culprit and that this implies there is a third independent mass-squared difference which we label ∆m 2 41 . The caveat, though, is that this additional mass splitting must be much larger than the other two (roughly 1eV) to get such a significant effect over such short distances. Additionally this implies a fourth mass eigenstate (ν 4 ) and hence, due to unitarity, a new flavor eigenstate (ν s ) which we assume must be 'sterile' to not interfere with astrophysical and particle physics constraints on the sum of active neutrino masses. Once we have this new splitting we discover that in turn we must introduce new oscillation parameters: θ 14 , θ 24 , θ 34 , δ 14 , δ 34 and ∆m 2 41
For true values, we use the three best fits from The NOνA Collaboration [28] outlined in TABLE I with the rest of the oscillation parameters identical between each case. We aim to expand on the analyses of [29] and [30] to analyse all three true solutions in the case where a sterile neutrino is introduced.
We will refer to these three solutions using the shorthand from TABLE I. It is important to analyse these results because they are examples of solutions degenerate in probability and thus must be resolved by detector effects or combined analyses. Note: we do not include any true hypotheses with θ 23 = 45 • due to the best fits from [28] not including them, Despite this, 'maximal-mixing' solutions are allowed by T2K, MINOS and recently NOνA at 90% C.L. [31,32]. The main part of our analysis is introducing the sterile parameters then changing the new sterile phase δ 14 to be several values and investigating its effect on the octant and mass hierarchy sensitivity, specifically their degeneracies. The standard 3 neutrino (3ν) and the extended 3+1 parameters with the two representative values for θ 23 are in TABLE II.

II. OSCILLATION THEORY
Extending to 4ν requires modification to the standard neutrino oscillation equations, it is important to pay attention to the parametrization chosen, because comparing mixing angles and CP phases between different choices is non-trivial. We utilize the same parametrization as in [30], defined as: where U (θ ij , δ ij ) is a 2 × 2 mixing matrix:  in the i, j sub-block of an n × n identity array with trigonometric terms abbreviated with the notation: The four flavor parametrization is then: With new mixing angles: θ 14 , θ 24 , θ 34 and phases: δ 14 , δ 34 . The fourth independent is chosen to be ∆m 2 41 for consistency. The probability expression is simplified with approximations as detailed in [18,30]. Note that the ∆m 2 41 terms are averaged over to represent the limited detector resolution, removing explicit dependence, leaving: The ∆, δ 13 and δ 14 dependent terms can lead to the MH-CP degeneracies, due to the unconstrained 2 CP phases (δ 13 and δ 14 ) and sign of ∆. Note also that the antineutrino probability can be obtained by performing the replacements: δ 13 → −δ 13 and δ 14 → −δ 14 .

III. EXPERIMENT SPECIFICATION
We run our simulation for the currently running NOνA experiment [33,34] (with modified experimental set-up taken from Ref. [35]) as well as the future experiment DUNE [36,37]. To simulate these experiments we use the GLoBES package along with auxiliary files to facilitate sterile neutrino simulation [38][39][40][41].
NOνA is a USA based experiment with a long baseline of 812 km. It runs from Fermilab's NuMI complex in Illinois to a far detector in Ash River Minnesota. We assume that NOνA will run for a total of three years in neutrino mode and three years in antineutrino mode (3 +3).
If these degeneracies can be solved at all with the current experiments T2K [32] and NOνA [28] then they may give the first hints of the values of δ 13 , θ 23 and the sign of ∆m 2 31 at some significant confidence level.
The addition of sterile neutrinos to the oscillation model can greatly lower sensitivity to degeneracies for NOνA and T2K [22], DUNE is already predicted to have very good degeneracy resolution [42,43] for 3ν so it's therefore important to see if how much its affected by the sterile neutrino. In addition, to see the how the sensitivity scales for runtime, we simulate DUNE for 2 +2 and 5 +5.
It is predicted that DUNE, along with other proposed next generation long-baseline experiments such as T2HK (Tokai to Hyper-Kamiokande) [44] and/or T2HKK (Tokai to Hyper-Kamiokande and Korea) [45] will be very sensitive to sterile induced CP phases [46,47]. As such, they will contribute much further to oscillation physics once the current degeneracies and issues are resolved, especially if sterile neutrinos are present.

IV. IDENTIFYING DEGENERACIES IN THE 3+1 CASE
A. Degeneracies at the probability level After taking the standard best fits for oscillation parameters from sources such as global fits and oscillation experiments [48][49][50] and choose sterile parameters consistent with [11,14,51,52] we then set θ 34 and δ 34 to zero because they are not present in the vacuum equation for P µe , Eq. (10), and we are under the assumption that matter interactions will not add any significant dependence to these terms. Finally we smooth our curves with a moving box-windowed average to represent the small oscillations that will be present but cannot be seen in real data, as mentioned in section II.
When we plot the probability plots for our three true values into the 4ν sector and vary δ 14 from −90 • to +90 • , our lines will become bands. This may cause additional overlap where there was none before, thus introducing or re-introducing specific degenerate solutions. This is the primary feature we are interested in as it will determine the sensitivity degradation that would be present in the 3 + 1 case.
For the plots where they are not axis variables we marginalize |∆m 2 31 |, δ test 13 and δ test 14 to minimize χ 2 in the fit. All of the marginalization ranges are summarized in TABLE II.

NOνA
It can be seen that for 3ν, all three probability curves for NOνA running neutrinos are almost entirely degenerate, though in the antineutrino case only the B and C solutions are degenerate. Extending to 4ν shows bands that are also almost totally overlapping for neutrinos while for antineutrinos, 4ν the bands get closer together again but solution A is still mostly separate.

DUNE
In contrast with the NOνA plot, the 3ν DUNE plots show only the A and B neutrino curves overlapping and no overlap for the antineutrino case, as shown in [29]. This points to much better degeneracy resolution than NOνA, especially while running antineutrinos. The 4ν plots do show overlap, specifically A, B and some C for neutrinos; and B and C for antineutrinos. Thus it is possible that some degeneracies can be reintroduced by extending our parameter space, even with the DUNE detector. Comparing these plots with the NOνA ones shows that solution A is still the favored solution for degeneracy resolution. The probability plots do not tell the whole story however as they do not reflect the statistics of the detector, therefore we must do more analysis to get an idea of what significance degeneracies arise at.

B. Degeneracies at the detector level
We now analyse our test hypotheses using several χ 2 type analyses to see for which values we can resolve the MH degeneracy, see what regions are allowed at 90% C.L. and also to look at the CP sensitivity. This is necessary because we need to account for statistical effects, combined neutrino/antineutrino runs. Recall that θ 34 and δ 34 do not come into the vacuum expression for P µe so we set them to zero and do not marginalize. In matter these will have some contribution from extra terms but this is small at NOνA and DUNE. When performing the χ 2 analysis we take the true parameters to be A, B or C and the test parameters to be as specified in TABLE II including marginalization ranges for the free parameters.
Our test statistic comes from GLoBES and is defined as: where N true i is the distribution for whatever the current true value is and N test i is the distribution for the test values that are varied over. This is calculated automatically by functions in by the GLoBES program with marginalization performed manually.  the exclusion plots for NOνA (FIG. 5.) we can see that the excluded region for true NH (true IH) includes the δ 13 = +90 • (δ 13 = −90 • ) favored region, this should be expected because for the favored parameters it is predicted that in the 3ν case NOνA alone can resolve the mass hierarchy. It can be seen that extending into 4ν changes these regions somewhat. It can also be seen in FIG. 8. that solution A can be resolved more easily than the other cases, by relating the probability plots to the allowed regions, the particularly large separation of the curves for antineutrinos compared to neutrinos is what allows for this.

DUNE
Evaluating the exclusion plots for the reduced or partial run of DUNE 2 +2 (FIG. 6.) shows that the excluded region expands to include much of the unfavored half plane.
Extending DUNE's run to 5 +5 further increases the parameter space for which the wrong mass hierarchy can be excluded (FIG. 7.) and only small areas in the unfavored halfplanes remain for θ 23 < 40 • which is roughly 2σ to 3σ outside of NOνA's current fits depending on the value of |∆m 2 31 |. Evaluating the allowed regions for DUNE 2 +2 shows an almost complete disappearance of WH solutions. Many of the WO solutions are gone too, for example the 3ν IH scenario in FIG. 9. Though some cases are still particularly bad, e.g. also in FIG. 9; true  B with δ 14 = +90 • . This is due to the fact that hierarchy resolution ability is related to the baseline of the experiment and as seen in FIG. 4. (b)

V. CONCLUSION
We extend the analysis from [30] in light of the discussions from [29] regarding the results in [28]. We include a light sterile neutrino specified as such to rectify the short baseline oscillation anomalies. From our analysis we see that the degenerate solutions are predicted to be worse at probability level for the 4ν case due to the additional free parameter space. We find that for certain values of δ 14 the sensitivity of NOνA to the octant degeneracy and (to a much lesser extent) hierarchy degeneracy may be reduced. We also predict that DUNE 2 +2 can solve the MH degeneracy at 90% C.L. while some octant ambiguity still exists. However, extending to the full DUNE 5 +5 run removes almost all ambiguity at 90% C.L. in all cases regardless of δ 14 . So it can be seen that for any of these true values with the sterile hypothesis being correct or not, that DUNE can resolve these degeneracies at 90% C.L. whilst NOνA alone loses some potential for degeneracy resolution in the sterile case.
New preliminary results from NOvA have been presented recently [53,54] and indicate new 1σ parameter ranges: with best fits of: δ 13 = 1.21π ≈ −142.2 • , HO, NH. These align somewhat better with previous T2K and MINOS results and no-longer explicitly rule out θ 23 = 45 • at 90% C.L. We will still continue to analyse our three values despite the fact that neither A or B are fully favored and C is disfavored, because we are interested purely in degeneracy resolution. New analyses will, of course, account for this. With regards to these new preliminary best fits from NOνA, our sensitivity predictions do not really change, these results still fall into the favored area for mass hierarchy resolution and as such the 35  NOνA only loses MH sensitivity in the specific 4ν case with δ 14 = −90 • . The octant region does have more spread for this true value, but the allowed region doesn't include the wrong octant, instead including the maximal mixing (θ 23 = 45 • ) case. This case can therefore not be ruled out at 90% C.L. and may require a combined analysis to differentiate.

VI. ACKNOWLEDGEMENT
SG, ZMM, PS and AGW thank the support by the University of Adelaide and the Australian Research Council through the ARC Centre of Excellence for Particle Physics (CoEPP) at the Terascale (grant no. CE110001004).