Degeneracy resolution capabilities of NO$\nu$A and DUNE in the presence of light sterile neutrino

We investigate the implications of a sterile neutrino on the physics potential of the proposed experiment DUNE and future runs of NO$\nu$A using latest NO$\nu$A results. Using combined analysis of the disappearance and appearance data, NO$\nu$A reported preferred solutions at normal hierarchy (NH) with two degenerate best-fit points one in the lower octant (LO) and $\delta_{13}$ = 1.48$\pi$ and other in higher octant (HO) and $\delta_{13}$ = 0.74$\pi$. Another solution of inverted hierarchy (IH) which is 0.46$\sigma$ away from best fit was also reported. We discuss chances of resolving these degeneracies in the presence of sterile neutrino.


Introduction
Sterile neutrinos are hypothetical particles that do not interact via any of the fundamental interactions other than gravity. The term sterile is used to distinguish them from active neutrinos, which are charged under weak interaction. The theoretical motivation for sterile neutrino explains the active neutrino mass after spontaneous symmetry breaking, by adding a gauge singlet term (sterile neutrino) to the Lagrangian under SU(3) c ⊗ SU(2) L ⊗ U(1) r where the Dirac term appears through the Higgs mechanism, and Majorana mass term is a gauge singlet, and hence appears as a bare mass term [1]. The diagonalization of the mass matrix gives masses to all neutrinos due to the See-Saw mechanism. okSome experimental anomalies also point towards the existence of sterile neutrinos.
Liquid Scintillator Neutrino Detector(LSND) detected ν µ → ν e transitions indicating ∆m 2 ≈ 1eV 2 which is inconsistent with ∆m 2 32 , ∆m 2 21 (LSND anomaly) [2]. Measurement of the width of Z boson by LEP gave number of active neutrinos to be 2.984±0.008 [3]. Thus the new neutrino introduced to explain the anomaly has to be a sterile neutrino. MiniBooNE, designed to verify the LSND anomaly, observed an unexplained excess of events in low-energy region of ν e , ν e spectra, consistent with LSND [4]. SAGE and GALLEX observed lower event rate than expected, explained by the oscillations of ν e due to ∆m 2 ≥ 1eV 2 (Gallium anomaly) [5][6][7]. Recent precise predictions of reactor anti-neutrino flux has increased the expected flux by 3% over old predictions. With the new flux evaluation, the ratio of observed and predicted flux deviates at 98.6 % C.L(Confidence level) from unity, this is called "Reactor anti-neutrino Anomaly" [8]. This anomaly can also be explained using sterile neutrino model. Short-baseline(SBL) experiments are running to search for sterile neutrinos. SBL experiments are the best place to look for sterile neutrino, as they are sensitive to new expected mass-squared splitting ∆m 2 ≃ 1eV 2 . However, SBL experiments cannot study all the properties of sterile neutrinos, mainly new CP phases introduced by sterile neutrino models. These new CP phases need long distances to become measurable [9,10], thus can be measured using Long baseline(LBL) experiments. With the discovery of relatively large value for θ 13 by Daya Bay [11], the sensitivity of LBL experiments towards neutrino mass hierarchy and CP phases increased significantly. Using recent global fits of oscillation parameters in the 3+1 scenario [12], current LBL experiments can extract two out of three CP phases (one of them being standard δ 13 ) [10]. Now, the sensitivity of LBL experiments towards their original goals decreases due to sterile neutrinos. It is seen in case of the CPV measurement; new CP-phases will decrease the sensitivity towards standard CP phase (δ 13 ). This will reduce degeneracy resolution capacities of LBL experiments. In this paper, we study hierarchy-θ 23 -δ 13 degeneracies using contours in θ 23 -δ 13 plane and how they are affected by the introduction of sterile neutrinos. We attempt to find the extent to which these degeneracies can be resolved in future runs of NOνA and DUNE.
The outline of the paper is as follows. In section 2, we present the experimental specifications of NOνA and DUNE used in our simulation. We introduce the effect of sterile neutrino on parameter degeneracies resolution in section 3. Section 4 contains the discussion about the degeneracy resolving capacities of future runs of NOνA and DUNE assuming latest NOνA results -NH(Normal hierarchy)-LO(Lower octant), NH-HO(Higher octant), and IH(Inverted hierarchy)-HO as true solutions for both 3 and 3+1 models. Finally, Section 5 contains concluding comments on our results.
NOνA [16,17] is an LBL experiment which started its full operation from October 2014.
NOνA has two detectors, the near detector is located at Fermilab (300 ton, 1 km from NuMI beam target) while the far detector(14 Kt) is located at Northern Minnesota 14.6 mrad off the NuMI beam axis at 810 km from NuMI beam target, justifying "Off-Axis" in the name.
This off-axis orientation gives us a narrow beam of flux, peak at 2 GeV [18]. For simulations, we used NOνA setup from Ref. [19]. We used the full projected exposure of 3.6 x 10 21 p.o.t (protons on target) expected after six years of runtime at 700kW beam power. Assuming the same runtime for neutrino and anti-neutrino modes, we get 1.8 x 10 21 p.o.t for each mode. Following [20] we considered 5% normalization error for the signal, 10 % error for the background for appearance and disappearance channels. DUNE (Deep Underground Neutrino Experiment) [21,22] is the next generation LBL experiment. Long Base Neutrino Facility(LBNF) of Fermilab is the source for DUNE. Near detector of DUNE will be at Fermilab. Liquid Argon detector of 40 kt to be constructed at Sanford Underground Research Facility situated 1300 km from the beam target, will act as the far detector. DUNE uses the same source as of NOνA; we will observe beam flux peak at 2.5GeV. We used DUNE setup give in Ref. [23] for our simulations. Since DUNE is still in its early stages, we used simplified systematic treatment, i.e., 5% normalization error on signal, 10 % error on the background for both appearance and disappearance spectra.
Oscillation parameters are estimated from the data by comparing observed and predicted ν e and ν µ interaction rates and energy spectra. Oscillation parameters are estimated from the data by comparing observed and predicted ν e and ν µ interaction rates and energy spectra.
GLoBES calculates event rates of neutrinos for energy bins taking systematic errors, detector resolutions, MSW effect due to earth's crust etc into account. The event rates generated for true and test values are used to plot χ 2 contours. GLoBES uses its inbuilt algorithm to calculate χ 2 values numerically considering parameter correlations as well as systematic errors. In our calculations we used χ 2 as: where R ij andR ij represent real and complex 4×4 rotation in the plane containing the 2×2 sub-block in (i,j) sub-block Where, c ij = cos θ ij , s ij = sin θ ij ,s ij = s ij e −iδ ij and δ ij are the CP phases.
There are three mass squared difference terms in 3+1 model-∆m 2 21 (solar)≃ 7.5×10 −5 eV 2 , ∆m 2 31 (atmospheric)≃ 2.4 × 10 −3 eV 2 and ∆m 2 41 (sterile)≃ 1eV 2 . The mass-squared difference term towards which the experiment is sensitive depends on L/E of the experiment. Since SBL experiments have small a very small L/E, sin 2 (∆m 2 ij L/4E) ≃ 0 for ∆m 2 21 and ∆m 2 31 . ∆m 2 41 term survives. Hence, SBL experiments depend only on sterile mixing angles and are insensitive to the CP phases. The oscillation probability, P µe for LBL experiments in 3+1 model, after averaging ∆m 2 41 oscillations and neglecting MSW effects, [26] is expressed as sum of the four terms These terms can be approximately expressed as follows: P 4 (δ 13 − (δ 14 − δ 24 )) = a sin 2θ 3ν µe sin 2θ 4ν µe cos 2θ 13 cos(δ 13 − (δ 14 − δ 24 )) sin 2 ∆ 31 With the parameters defined as The CP phases introduced due to sterile neutrinos persist in the P µe even after averaging out ∆m 2 41 lead oscillations. Last two terms of equation 4, give the sterile CP phase dependence terms. P 3 (δ 14 − δ 24 ) depends on the sterile CP phases δ 14 and δ 24 , while P 4 depends on a combination of δ 13 and δ 14 − δ 24 . Thus, we expect LBL experiments to be sensitive to sterile phases. We note that the probability P µe is independent θ 34 . One can see that θ 34 will effect P µe if we consider earth mass effects. Since matter effects are relatively small for NOνA and DUNE, their sensitivity towards θ 34 is negligible. The amplitudes of atmospheric-sterile interference term (eq.8) and solar-atmospheric interference term(eq.6), are of the same order. This new interference term reduces the sensitivity of experiments to the standard CP phase(δ 13 ).
In figure 1, we plot the oscillation probability(P µe ) as a function of energy for the three best fit values of latest NOνA results [24] i.e; NH-LO-1. In the next section, we explore how parameter degeneracies are affected in the 3+1 model and the extent to which these degeneracies can be resolved in future runs of NOνA and DUNE.

Results for NOνA and DUNE
We explore allowed regions in sin 2 θ 23 -δ cp plane from NOνA and DUNE simulation data with different runtimes, considering latest NOνA results as true values. Using combined analysis of the disappearance and appearance data, NOνA reported preferred solutions [24] at normal hierarchy (NH) with two degenerate best-fit points, one in the lower octant (LO) and δ cp = 1.48π, the other in higher octant (HO) and δ cp = 0.74π. Another solution of inverted hierarchy (IH), 0.46σ away from best fit is also reported. By studying the allowed regions, we understand the extent to which future runs of NOνA and DUNE will resolve WH-RO solution occupies a small region for 3ν case, covering half of δ 13 region for 4ν case.   In the second row of the figure, we plot allowed regions for NOνA [3+1]. We take true values as best fit points obtained by NOνA. We observe an increase in precision of parameter measurement, due to an increase in statistics, from added 1 yr of anti-neutrino run. In the first plot of the second row, the RH-RO octant region covers entire δ 13 range for both 3ν and 4ν case. RH-WO region includes −180 • to 45 • of δ 13 for 3ν case, while whole range of δ 13 is covered in 4ν case. A slight increase in the area of WH-RO is observed form 3ν to 4ν case. 4ν introduces WH-WO region which was resolved for 3ν case. In the second plot, RH-RO region allows full range of δ 13 for 4ν case, while it was restricted to lower half of CP range in 3ν case. We see WH-RO solution which was resolved in 3ν case, is reintroduced in 4ν case.
We also see a slight increase in the size of WH-WO solution from 3ν to 4ν. In third plot, RH-RO region covers whole CP range for 4ν while 35 • to 125 • of δ 13 are excluded in 3ν case.
The almost resolved RH-WO solution for 3ν doubles for 4ν case. WH-RO, WH-WO cover entire region of δ 13 for 4ν case.
In the third row, we show allowed regions for NOνA[3+3]. In the first plot, it can be seen that small area of RH-WO in case of 3ν case now covers the whole of δ 13 region for 4ν case.
While the 3ν case has WH-Wδ 13 degeneracy, 4ν case introduces equal sized WH-WO-Wδ 13 degeneracy. In second plot, for 3ν case: most of δ 13 values above 0 • are excluded, but for 4ν case we see contour covers whole of δ 13 range. Already present small area of RH-WO of 3ν is also increased for 4ν case. 4ν case also introduces a small region of WH solutions which were not present in 3ν case. In the third plot, we see that 4ν introduces RH-WO region of the almost equal size of RH-RO region of 3ν case. We observed a slight increase in WH-RO region for 4ν over 3ν case, while the WH-WO region almost triples for 4ν case.
In the figure 4, we show allowed parameter regions for DUNE experiment for different run-times. DUNE being the next generation LBL experiment it is expected to have excellent statistics. Hence, We plot 99% C.L regions for DUNE. In the first row of figure 4

Conclusions
We have discussed how the presence of a sterile neutrino will affect, the physics potential of the proposed experiment DUNE and future runs of NOνA, in the light of latest NOνA results [24]. The best-fit parameters reported by NOνA still contain degenerate solutions.
We attempt to see the extent to which these degeneracies could be resolved in future runs for the 3+1 model.