Quasi-Dirac neutrino oscillations at DUNE and JUNO

TL;DR

This paper investigates how quasi-Dirac neutrinos, which include small Majorana mass terms, affect neutrino oscillations and assesses the potential of DUNE and JUNO experiments to detect or constrain these effects.

Contribution

The study provides a sensitivity analysis of DUNE and JUNO to quasi-Dirac neutrino parameters, highlighting their potential to improve bounds and discover such neutrinos.

Findings

DUNE and JUNO can significantly tighten bounds on quasi-Dirac parameters.

The experiments have potential to discover quasi-Dirac neutrinos if they exist.

New mixing angles and phases influence oscillation probabilities.

Abstract

Quasi-Dirac neutrinos are obtained when the Lagrangian density of a neutrino mass model contains both Dirac and Majorana mass terms, and the Majorana terms are sufficiently small. This type of neutrinos introduces new mixing angles and mass splittings into the Hamiltonian, which will modify the standard neutrino oscillation probabilities. In this paper, we focus on the case where the new mass splittings are too small to be measured, but new angles and phases are present. We perform a sensitivity study for this scenario for the upcoming experiments DUNE and JUNO, finding that they will improve current bounds on the relevant parameters. Finally, we also explore the discovery potential of both experiments, assuming that neutrinos are indeed quasi-Dirac particles.

Tables

Table 1. Table 1: Standard neutrino oscillation parameters used in the analysis, taken from Ref.  de Salas et al. ( 2018 ) .

Parameter	Value
$\Delta m_{21}^2$	$7.5\times {10}^{-5}$ eV2
$\Delta m_{31}^2$	$2.50\times {10}^{-3}$ eV2
${\sin }^2{\theta }_{12}$	0.32
${\sin }^2{\theta }_{23}$	0.547
${\sin }^2{\theta }_{13}$	0.0216
$\delta$	1.5$\pi$

Equations

$$m_{\nu }=\left(\begin{array}{cc}{m}_L& m_D\\ m_D& m_R\end{array}\right).$$

$${\nu }_1=$$

$${\nu }_2=$$

$${\nu }_1$$

$${\nu }_2$$

$$m_{1,2}\simeq m_D\left(1\pm \epsilon \right).$$

$${\mathcal{L}}_{CC}=-\frac{g}{\sqrt{2}}{W}_{\mu }^{-}\sum _{l=1}^3\sum _{j=1}^6{\mathbf{V}}_{lj}{\bar{\ell }}_l{\gamma }^{\mu }{P}_L{\nu }_j+\text{h.c.},$$

$${\mathcal{L}}_{\text{mass}}=\frac{1}{2}{\bar{\nu }}_{\alpha }{M}_{\alpha \beta }{\nu }_{\beta }+\text{h.c.}$$

$$\tilde{\mathbf{U}}\left({\theta }_{ij},{\delta }_{ij}\right)={\hat{R}}_{56}{\hat{R}}_{46}{\hat{R}}_{36}{\hat{R}}_{26}{\hat{R}}_{16}{\hat{R}}_{45}{\hat{R}}_{35}{\hat{R}}_{25}{\hat{R}}_{15}{\hat{R}}_{34}{\hat{R}}_{24}{\hat{R}}_{14}{\hat{R}}_{23}{\hat{R}}_{13}{\hat{R}}_{12},$$

$${\hat{R}}_{14}$$

$$\tilde{\mathbf{U}}\left({\theta }_{ij},{\delta }_{ij}\right)=R_{36}{R}_{26}{\hat{R}}_{16}{R}_{35}{R}_{25}{R}_{15}{R}_{34}{R}_{24}{R}_{14}{R}_{23}{\hat{R}}_{13}{R}_{12},$$

$$\mathbf{U}\left({\theta }_{ij},{\delta }_{ij}\right)\equiv \tilde{\mathbf{U}}\left({\theta }_{ij},{\delta }_{ij}\right)W,\mathrm{with}W=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{I}_3& i{I}_3\\ I_3& -i{I}_3\end{array}\right),$$

$$P\left({\nu }_{\alpha }\to {\nu }_{\beta }\right)={\left|\sum _{j=1}^6{\mathbf{U}}_{\beta j}{\mathbf{U}}_{\alpha j}^{\ast }\exp \left(-\frac{i{m}_j^{2}L}{2E}\right)\right|}^2,$$

$${\mathcal{H}}_0=\frac{1}{2E}\mathbf{U}{\mathbb{M}}^2{\mathbf{U}}^{†},$$

$$\mathcal{H}=\frac{1}{2E}\left(\mathbf{U}{\mathbb{M}}^2{\mathbf{U}}^{†}+\mathbb{A}\right),$$

$$\mathbb{A}=\mathrm{diag}(V_{\text{CC}},0,0,V_{\text{CC}},0,0).$$

$$X_1$$

$$X_3$$

$$X_5$$

$$X_7$$

$$P\left({\nu }_e\to {\nu }_e\right)$$

$$P\left({\nu }_{\mu }\to {\nu }_{\mu }\right)$$

$$P\left({\nu }_e\to {\nu }_{\mu }\right)$$

$$X_5=X_1{X}_3,X_6=X_2{X}_4,$$

$$\text{Re}\left(X_7\right)=\frac{1}{2}\left(1-X_1-X_2-X_3-X_4+X_1{X}_4+X_2{X}_3\right).$$

$${\chi }_{\text{DUNE}}^2(p)=\underset{\vec{\alpha }}{\min }\sum _{\text{channels}}2\sum _n\left[N_n(p,\vec{\alpha })-N_n^{\text{dat}}+N_n^{\text{dat}}\log \left(\frac{N_n^{\text{dat}}}{N_n(p,\vec{\alpha })}\right)\right]+\sum _i{\left(\frac{{\alpha }_i}{{\sigma }_i}\right)}^2.$$

$${\chi }^2(p)={\chi }_{\text{DUNE}}^2(p)+{\chi }_{\text{JUNO}}^2(p)+{\text{f}}_{\text{DB}}(p).$$

$${\text{f}}_{\text{DB}}(p)={\left[\frac{\left((c_{14}{c}_{15}{c}_{16}{s}_{13}{)}^2+s_{16}^2-1\right)\left((c_{14}{c}_{15}{c}_{16}{s}_{13}{)}^2+s_{16}^2\right)-{\sin }^2{\theta }_{\text{DB}}}{{\sigma }_{\text{DB}}{\sin }^2{\theta }_{\text{DB}}}\right]}^2,$$

$$1-\frac{X_5}{X_1{X}_3}=0.$$

$$1-\frac{X_5}{X_1{X}_3}=0.5.$$

Full text

Neutrinos, quasi-Dirac, pseudo-Dirac, Majorana, DUNE, JUNO.

1. I Introduction
2. II Quasi-Dirac neutrino oscillations
3. III Simulation of experiments
4. IV Results and discussion
5. V Conclusions

I Introduction

Since the discovery of solar and atmospheric neutrino oscillations about two decades ago, neutrino oscillation experiments have become more and more sophisticated. Nowadays many of the parameters characterizing the conversion of neutrino flavors in the standard 3-neutrino picture are rather well measured de Salas et al. (2018). However, this framework might not be complete and might need to be extended. Several studies considering global oscillation data have been performed assuming the existence of new physics beyond the standard sector, see for example Refs. Gariazzo et al. (2017); Dentler et al. (2018); Esteban et al. (2018). One of these scenarios which will be considered here is the case of quasi-Dirac neutrinos Anamiati et al. (2018).

Since neutrino oscillations are blind to the Dirac or Majorana nature of neutrinos, one needs other types of experiments, for example, those searching for neutrinoless double beta decay to determine it Avignone et al. (2008); Deppisch et al. (2012). In general, one can say that the Dirac case, consisting of $n$ neutrinos, is a limiting case of the more general Majorana scenario, with $2n$ neutrinos. This limit is performed by putting the Majorana mass terms in the Lagrangian to zero. Quasi-Dirac neutrinos arise from the presence of both Majorana and Dirac mass terms in the Lagrangian simultaneously, where the Majorana terms are small, but not exactly zero. As we will show, the departure from Diracness — i.e. non-zero Majorana mass terms — leads to the presence of new mixing angles and new mass splittings, which will affect neutrino oscillation probabilities.

Along this work, we will use “quasi-Dirac neutrinos” to refer to active-sterile neutrino pairs Valle (1983). In order to distinguish this scenario from the one with active-active pairs, we denote the latter ones as pseudo-Dirac neutrinos Wolfenstein (1981). Many aspects of pseudo-Dirac neutrinos have been studied in the literature, see for example Refs. Petcov (1982); Doi et al. (1983); Bilenky and Pontecorvo (1983); Bilenky and Petcov (1987); Giunti et al. (1992); Dutta and Joshipura (1995); Joshipura and Rindani (2000); Nir (2000). Note, however, that models with pseudo-Dirac neutrinos do not fit oscillation data anymore Brahmachari and Choubey (2002); Frampton et al. (2002); He (2004). In the context of quasi-Dirac neutrinos, many papers appeared in the literature proposing explanations for the solar and atmospheric neutrino problems Geiser (1999); Krolikowski (2000); Balaji et al. (2002), as well as consistent descriptions of standard and short baseline neutrino oscillations Ma and Roy (1995); Goswami and Joshipura (2002). Several papers derived limits on quasi-Dirac neutrino properties from different data sets Cirelli et al. (2005); de Gouvea et al. (2009), while others discussed them in the context of neutrino telescopes Beacom et al. (2004); Esmaili (2010); Esmaili and Farzan (2012); Joshipura et al. (2014).

From a theoretical point of view, there are several options on how quasi-Dirac neutrinos can be created. They can be produced, for instance, in models with a singular seesaw Stephenson et al. (2005); McDonald and McKellar (2007), double seesaw Ahn et al. (2016) or Dirac-seesaw Chang and Kong (2000) mechanisms. Another possibility is to obtain them from extended gauge groups Sanchez et al. (2001); Fonseca and Hirsch (2016) or even in super-gravity theories Abel et al. (2005).

Because of the presence of new spinors in Dirac neutrino models, there is some overlap between the study of quasi-Dirac neutrinos and the scenario with sterile neutrinos. Several experimental hints point towards the existence of sterile neutrinos, which have been extensively investigated in many experiments. The possible observation of short baseline oscillations in some of these experiments Athanassopoulos et al. (1996); Aguilar-Arevalo et al. (2001); Abdurashitov et al. (2006); Laveder (2007); Giunti and Laveder (2007); Mention et al. (2011); Gariazzo et al. (2018); Dentler et al. (2017) together with the non-observation of neutrino oscillations in others Adamson et al. (2017); Ahmad et al. (2002); Aartsen et al. (2017); Adamson et al. (2016); Agafonova et al. (2015); Abe et al. (2015); Adamson et al. (2016, 2019); Albert et al. (2018); Abe et al. (2019) lead to large tensions in the global 3+1 picture Gariazzo et al. (2017); Dentler et al. (2018); Diaz et al. (2019); Gariazzo et al. (2019), which cannot be reconciled even adding more than one sterile neutrino Giunti and Zavanin (2015). For a recent review on this topic, we refer the reader to Refs. Giunti and Lasserre (2019); Böser et al. (2019). Even though there is some theoretical overlap, these results would point towards new mass splittings at the $\sim 1$ eV2 scale. Therefore, this type of oscillations cannot be explained with quasi-Dirac neutrinos, whose additional mass splittings are constrained to be much below the eV scale.

In this paper, we study the sensitivity of the upcoming Deep Underground Neutrino Experiment (DUNE) Abi et al. (2018a, b, c) and Jiangmen Underground Neutrino Observatory (JUNO) An et al. (2016) to quasi-Dirac neutrino oscillations. The DUNE experiment, hosted by Fermilab, will exploit the synergy of a very high intense neutrino beam and two massive argon detectors to carry on a broad research program in neutrino physics. DUNE will allow to perform tests of the three-neutrino paradigm with remarkable sensitivity, in particular concerning neutrino oscillation parameters Ghosh et al. (2016); De Romeri et al. (2016); Srivastava et al. (2018). The high intensity of the neutrino beam as well as the high resolution of the near and far detectors, which characterize DUNE, will make it a leading experiment also in the search for new physics. Hence, besides pursuing a comprehensive study of the neutrino mixing, DUNE will also allow to explore new physics scenarios, for instance, via the search for non-standard interactions Coloma (2016); de Gouvêa and Kelly (2016); Blennow et al. (2016) or sterile neutrinos Coloma et al. (2018); Agarwalla et al. (2016), among others Escrihuela et al. (2017); Masud et al. (2019); Barenboim et al. (2018, 2019). JUNO is a next generation reactor experiment and will be located at 53 km from the Yangjiang (six cores with 2.9 GW${}_{\text{th}}$ thermal power each) and Taishan (four cores with 4.6 GW${}_{\text{th}}$ thermal power each) nuclear power plants. The current Daya Bay complex will also contribute with roughly 3% to the total antineutrino flux. The JUNO detector will be made of 20 kton of liquid scintillator. With these powerful sources and an excellent energy resolution, JUNO will be expecting around $10^{5}$ inverse beta decay events in total. Huge statistics and the long baseline (for a reactor experiment) assure a measurement of $\sin^{2}\theta_{12}$ , $\Delta m_{21}^{2}$ and $\Delta m_{ee}^{2}$ Nunokawa et al. (2005); Parke (2016) at below 1% level, which makes it a very complementary experiment to DUNE.

Our paper is structured as follows. In Sec. II we present the theoretical framework for quasi-Dirac neutrinos. The simulation of the DUNE and JUNO experiments is described in Sec. III. Next, we discuss our results in Sec. IV and, finally, we draw our conclusions in Sec. V.

II Quasi-Dirac neutrino oscillations

A pair of quasi-Dirac neutrinos is a pair of Majorana neutrinos with a small mass splitting and a relative CP-sign between the two states. For the sake of illustration, let us start considering only one neutrino generation. In this case, in the basis $\left(\nu,N^{c}\right)$, where $\nu$ and $N^{c}$ are the active and the sterile neutrinos, respectively, the most general neutrino mass matrix is

[TABLE]

Here, $m_{L}$ and $m_{R}$ are the terms that violate lepton number, while $m_{D}$ is the standard Dirac neutrino mass term. In the limit in which $m_{L}$ and $m_{R}$ are equal to zero, lepton number is conserved and neutrinos are Dirac particles. This limiting case is characterized by two degenerate mass eigenstates

[TABLE]

where the factor $i$ is introduced such that both mass eigenvalues are positive. Note that, in this mass eigenstate basis, both $\nu_{1}$ and $\nu_{2}$ are equal mixtures of active and sterile neutrinos. Small deviations from the limit $m_{L}=m_{R}=0$ then lead to quasi-Dirac neutrinos. If we define the new variables $\varepsilon=(m_{L}+m_{R})/(2m_{D})$ and $\theta=(m_{L}-m_{R})/(4m_{D})$, in the limit $\varepsilon,\theta\ll 1$, one can rewrite Eq. (2) as

[TABLE]

where the quasi-degenerate pairs are nearly maximally mixed and $\theta$ is a small angle describing the departure from maximality. The masses are given by

[TABLE]

Quasi-Dirac neutrinos are therefore characterized by new mass splittings and new mixing angles.

Let us now consider the extension of the standard model (SM) with three sterile neutrinos $N^{c}$. In the physical mass eigenstate basis, the charged current SM Lagrangian is modified to

[TABLE]

where $P_{L,R}=(1\mp\gamma_{5})/2$ are the chirality projectors, $l=1,2,3$ denote the flavor of the charged leptons, and $j=1,\dots,6$ the physical neutrino states. The mixing is parameterized by a rectangular $3\times 6$ mixing matrix, ${\bf V}_{lj}$ Schechter and Valle (1980). Moreover, the addition of the three sterile neutrinos allows for the mass term

[TABLE]

Here, indices $\alpha,\beta=1,2,3$ ($4,5,6$) are for active (sterile) neutrinos and $M_{\alpha\beta}$ is the generalization of Eq. (1) for three generations. The full neutrino mass matrix is now diagonalized by a $6\times 6$ unitary matrix, ${\bf\tilde{U}}$. We parameterize the neutrino mixing matrix as

[TABLE]

where $\widehat{R}_{ij}$ are complex rotation matrices which depend on the mixing angles $\theta_{ij}$ and CP-violating phases $\delta_{ij}$. The rotation matrices $\widehat{R}_{ij}$ are parameterized in the usual way. For example, for $\widehat{R}_{14}$ we have

[TABLE]

Note that the matrix ${\bf\tilde{U}}$ in Eq. (6) contains the mixing among sterile neutrinos, not observable in neutrino oscillation experiments. Thus, we will neglect these rotations in the following. In the remaining rotations, we have in general 12 angles and 12 phases. However, in our numerical studies we will limit ourselves to two phases only, namely $\delta_{13}$ and $\delta_{16}$. This means that the mixing matrix above can be reduced to

[TABLE]

where $R_{ij}$ denote real rotations. It proves convenient to multiply ${\bf\tilde{U}}$ by the following $6\times 6$ rotation matrix (as in Eq. (16) of Ref. Anamiati et al. (2018)):

[TABLE]

with $I_{3}$ being the $3\times 3$ identity matrix. This redefinition allows to recover trivially the Dirac limit for the mixing matrix, by putting to zero all non-standard angles. The probability of a neutrino oscillating from a flavor $\alpha$ to a flavor $\beta$ can then be written as

[TABLE]

where $L$ is the length traveled by the neutrino and $E$ its energy. Therefore, neutrino oscillations are described by the Hamiltonian

[TABLE]

where $\mathbb{M}^{2}=\operatorname{diag}(0,\Delta m_{21}^{2},\Delta m_{31}^{2},\epsilon_{1}^{2},\Delta m_{21}^{2}+\epsilon_{2}^{2},\Delta m_{31}^{2}+\epsilon_{3}^{2})$***From this expression it is clear why the convention chosen in Eq. (9) is useful. Setting $\epsilon_{i}^{2}$ and the non-standard angles to zero, $P\left(\nu_{\alpha}\rightarrow\nu_{\beta}\right)$ reduces to the standard expression for three generations, despite the fact that we sum over six states. and the square of the lightest neutrino mass, $m_{1}^{2}$, has been subtracted from the diagonal elements in the $\mathbb{M}^{2}$ matrix, as usual. To include matter effects on the neutrino propagation, one should add the effective matter potential to the neutrino Hamiltonian above. Quasi-Dirac neutrinos feel the same potential in the 4-5-6 sector as in the 1-2-3 sector. Thus,

[TABLE]

where the potential is now given by $\mathbb{A}=\operatorname{diag}(V_{\text{CC}}+V_{\text{NC}},V_{\text{NC}},V_{\text{NC}},V_{\text{CC}}+V_{\text{NC}},V_{\text{NC}},V_{\text{NC}})$. The charged current potential is given by $V_{\text{CC}}=2E\sqrt{2}G_{F}n_{e}$, where $G_{F}$ is the Fermi constant and $n_{e}$ is the electron number density. The neutral current potential, $V_{\text{NC}}$, is a common term to all the diagonal entries and, therefore, it can be removed from the effective Hamiltonian, that will read as follows

[TABLE]

This Hamiltonian will lead to a different oscillation behaviour compared to the standard case, as soon as any $\epsilon_{i}$ or any non-standard mixing angle is different from zero. As an example, we show in Fig. 1 (top panels) the oscillation probabilities for the two channels relevant for DUNE, $\nu_{\mu}\rightarrow\nu_{e}$ and $\nu_{\mu}\rightarrow\nu_{\mu}$. The standard oscillation parameters in these plots are fixed to the ones in Tab. 1, taken from Ref. de Salas et al. (2018). In the left panel we show the disappearance probability $P_{\mu\mu}$ as a function of the neutrino energy, turning on one new mixing angle at a time – which is always set to $\sin^{2}\theta_{\text{new}}=0.2$. The new angle $\theta_{16}$ has no visible effect on the disappearance probability, while $\theta_{26}$ has a visible effect close to the oscillation minima. In the right panel of Fig. 1 we show the appearance probability $P_{\mu e}$. Here both angles have a visible impact in the oscillation probability. This is expected from the fact that the new angles $\theta_{16}$ and $\theta_{26}$ take the role of the standard angles $\theta_{13}$ and $\theta_{23}$, respectively. On the other hand, the lower panel of Fig. 1 shows the effect of the new mixing angles $\theta_{14}$ and $\theta_{15}$ on the survival probability of electron antineutrinos at JUNO. In this case, the two mixing angles have opposite effects. Note that the survival probability shown here does not include the experimental energy resolution, which is included in our simulation of JUNO in Sec. III.

Fig. 1 is meant for illustration purposes only: As we shall see later in Sec. IV, standard and non-standard angles are highly correlated in the quasi-Dirac neutrino scenario and one can obtain perfect degeneracies among certain parameters. That is, even very different combinations of angles can lead to similar oscillation probabilities, which makes the establishment of limits on quasi-Dirac angles particularly difficult experimentally. For this reason, Ref. Anamiati et al. (2018) introduced a particular set of variables, $X_{i}$, which are parameterization independent combinations of entries in the neutrino mixing matrix ${\bf U}$. Not considering transitions to $\nu_{\tau}$, due to the scarcity of $\nu_{\tau}$ appearance data, one can show that only seven independent combinations of neutrino mixing angles enter the oscillation probabilities. The corresponding $X_{i}$ are defined as

[TABLE]

where ${\bf U}$ is the full mixing matrix defined in Eq. (9). Note that $|X_{7}|^{2}=X_{5}X_{6}$, i.e. only the phase in $X_{7}$ is a free parameter. The oscillation probabilities in vacuum can be written in terms of the $X_{i}$ as Anamiati et al. (2018)

[TABLE]

where $\mathcal{A}_{ij}\equiv-4\sin^{2}\left[\left(m_{i}^{2}-m_{j}^{2}\right)L/\left(4E\right)\right]$ and $\mathcal{B}_{ij}\equiv 2\sin\left[\left(m_{i}^{2}-m_{j}^{2}\right)L/\left(2E\right)\right]$.

The usefulness of defining these $X_{i}$ lies in the fact that, for a three-generation Dirac scenario, there are only four independent parameters entering these seven quantities: the three standard mixing angles and the phase $\delta_{13}$. Thus, in the Dirac limit, one can find three relations among the seven $X_{i}$:

[TABLE]

Eq. (18) allows to formulate quantitative tests of “quasi-Diracness”. We will come back to this in Sec. IV. Here we note that, although seven $X_{i}$ are defined here, DUNE will not be sensitive to $X_{1}$ and $X_{2}$, since they depend on the solar parameters. However, JUNO (and Daya Bay, which we will include as a prior in our analysis) will provide stringent constraints on $X_{1}$ and $X_{2}$, see section IV. On the contrary, DUNE will be able to put severe restrictions on $X_{3}$ and $X_{5}$ and some improvements on the remaining parameters $X_{4}$, $X_{6}$ and $X_{7}$, as we will show below.

Beyond the new mixing angles, we show DUNE’s sensitivity to the new mass splittings $\epsilon_{i}$ in Fig. 2. These results have been obtained by varying only one of the new mass splittings at a time and fixing the new angles to zero. In comparison with previous results derived in Ref. Anamiati et al. (2018), one can see that DUNE will not be competitive with other current experiments, which give bounds on $\epsilon_{1}$ and $\epsilon_{2}$ several orders of magnitude stronger than the ones shown in Fig. 2. The only comparable bound is the one for $\epsilon_{3}$. Note, however, that in Ref. Anamiati et al. (2018) the Authors marginalized over some of the oscillation parameters, while we kept all of them fixed. Marginalizing over additional parameters would result in weaker bounds, also for $\epsilon_{3}$. Therefore, given the poor sensitivity of DUNE to the new splittings $\epsilon_{i}$, we will set them to very small values in our analysis. The sensitivity of JUNO to the mass splittings $\epsilon_{i}$ has been discussed in Ref. Anamiati et al. (2018). We can infer from Tab. 1 in Anamiati et al. (2018) that JUNO will neither be able to improve the current bounds on any of the new mass splittings.

III Simulation of experiments

The Deep Underground Neutrino Experiment Abi et al. (2018a, b, c) is one of the next generation long-baseline accelerator experiments. It will consist of two detectors exposed to a megawatt-scale neutrino beam produced at Fermilab. This beam will consist of (nearly) only muon neutrinos. The near detector will be placed approximately 600 meters away from the source of the beam. The second (far) detector, divided into four modules, each using 10 kton of argon as detection material, will be installed 1300 kilometres away deep underground at the Sanford Underground Research Facility in South Dakota.

To simulate the neutrino signal in DUNE we use the GLoBES package Huber et al. (2005, 2007) with the configuration file provided by the DUNE collaboration Alion et al. (2016). We assume DUNE to run 3.5 years in neutrino mode and other 3.5 years in antineutrino mode. Considering an 80 GeV beam with 1.07 MW beam power, this corresponds to an exposure of 300 kton-MW-years. In this configuration, DUNE will be using $1.47\times 10^{21}$ protons on target (POT) per year. Our analysis includes disappearance and appearance channels, simulating both signals and backgrounds. The simulated backgrounds include contamination of antineutrinos (neutrinos) in the neutrino (antineutrino) mode, and also misinterpretation of flavors.

To include quasi-Dirac neutrino oscillations in our simulation of DUNE, we use the GLoBES extension snu.c Kopp (2008); Kopp et al. (2008). This extension was originally made to include non-standard neutrino interactions and sterile neutrinos in GLoBES simulations. For this analysis, we have modified the definition of the neutrino oscillation probability function inside snu.c by adding the additional rotation matrix of Eq. (9) and the matter potential of Eq. (13).

For the statistical analysis, we create a fake DUNE data sample using the standard oscillation parameters from Tab. 1. Next, we try to reconstruct the simulated data varying the mixing angles $\theta_{13}$, $\theta_{23}$, $\theta_{16}$ and $\theta_{26}$ (most relevant for DUNE) and the two CP-violating phases $\delta_{13}$ and $\delta_{16}$. The remaining new mixing angles are fixed to zero and the new mass splittings $\epsilon_{i}^{2}$ are fixed to very small values. Note as well that, since DUNE has no sensitivity to the solar parameters, these are fixed at their best fit values, in Tab. 1. On the other hand, given that we are mostly interested in correlations between the standard and new mixing angles, we have also kept $\Delta m_{31}^{2}$ fixed to its best fit value. Currently, there is a preference for normal mass ordering slightly above $3\sigma$ de Salas et al. (2018); Gariazzo et al. (2018); de Salas et al. (2018), so we will not consider negative values of $\Delta m_{31}^{2}$ here. We use GLoBES to calculate the event numbers for a given set of oscillation parameters $p$ and then we calculate the $\chi^{2}$ value for this set using the following expression

[TABLE]

Here, $N_{n}^{\text{dat}}$ corresponds to the simulated event number in the $n$-th bin for the oscillation parameters in Tab. 1, $N_{n}(p,\vec{\alpha})$ is the event number predicted in the $n$-th bin associated to the oscillation parameters $p$ and to the nuisance parameters $\alpha_{i}$, with standard deviations given by $\sigma_{i}$. All the nuisance parameters are associated to normalization uncertainties of signal or background events and introduce modifications of the type $N_{n}\to N_{n}(1+\alpha_{i})$. The last term in Eq. (19) penalizes the deviation of the latter parameters from their expectation values, $\alpha_{i}=0$. Finally, the $\chi^{2}$ sums over disappearance and appearance channels in both neutrino and antineutrino modes. The simulation and sensitivity analysis of the JUNO reactor experiment are performed following the procedure described in Anamiati et al. (2018). The corresponding $\chi^{2}$ function, $\chi^{2}_{\text{JUNO}}$, is obtained by allowing the variation of only five parameters which are relevant for JUNO, namely $\theta_{12},\theta_{13},\theta_{14},\theta_{15}$ and $\theta_{16}$. The solar mass splitting $\Delta m_{21}^{2}$ is fixed to the best fit value. To get the global future sensitivity to the quasi-Dirac scenario, in our analysis we combine the individual sensitivities obtained for DUNE and JUNO. Besides the two $\chi^{2}$ functions discussed above, $\chi^{2}_{\text{DUNE}}$ and $\chi^{2}_{\text{JUNO}}$, we introduce a penalty function associated to some of the mixing angles under study. As it was shown in Anamiati et al. (2018), the current reactor experiments cannot univocally measure the reactor angle $\theta_{13}$ in presence of quasi-Dirac neutrinos. However, it is still possible to simultaneously constrain several of these angles. If not, Daya Bay would have observed a different signal. This penalty can be obtained from Eq. (15) by imposing $\left(1-X_{1}-X_{2}\right)X_{1}+X_{1}X_{2}=\sin^{2}\theta_{\text{DB}}$ where $\sin^{2}\theta_{\text{DB}}\approx 0.022$ is the value currently measured by the Daya Bay reactor experiment Adey et al. (2018). Hence, our global $\chi^{2}$ function can be written as

[TABLE]

The penalty function in terms of the relevant mixing angles is given by

[TABLE]

where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$ and $\sigma_{\text{DB}}$ is the expected uncertainty in the final measurement of the reactor mixing angle by Daya Bay, set to 3%. This is a generalization of the standard reactor prior used in several studies on neutrino oscillations.

IV Results and discussion

In this section, we present the results of the statistical analysis performed in this work. Before discussing the results of the combined analysis of DUNE and JUNO, we discuss the results of the two experiments separately. Note, however, that we always add the penalty term in Eq. (21) to the $\chi^{2}$ function obtained from the sensitivity analysis of each experiment. In Fig. 3 we show the two-dimensional allowed regions obtained by scanning over the parameters $\theta_{13}$, $\theta_{23}$, $\theta_{16}$, $\theta_{26}$, $\delta_{13}$ and $\delta_{16}$ in DUNE. The parameters not shown are marginalized over in each panel. The colored regions correspond to the 1 (cyan), 2 (blue), 3 (red) $\sigma$ confidence levels for 2 degrees of freedom.

In the upper panels, we see that the two reactor angles, $\theta_{13}$ and $\theta_{16}$, and their corresponding phases, $\delta_{13}$ and $\delta_{16}$, behave in a very similar way. In principle, small values of the phases are allowed, although these require very small values for the associated mixing angles. From the right panel of the second row, however, we see that both angles cannot be small at the same time: if $\theta_{13}$ is small, $\theta_{16}$ has to be large and vice versa. In the former case, $\delta_{13}$ can take any value in the interval $[0,2\pi]$, while $\delta_{16}$ is rather restricted around its maximal value $1.5\pi$. This is an interesting point, because the fake data were created with $\delta_{16}=0$. The reason behind this is that the new angles and phases are correlated to the standard angles, e.g. $\theta_{13}$ and $\theta_{16}$ (see the definitions in Eq. (14)), hence they are interchangeable. Note as well that all the sensitivity to the reactor angle is lost, since DUNE can only reproduce the prior Anamiati et al. (2018) that we introduced as an input for our analysis, as explained in Sec. III. The interchangeability of the mixing parameters can also be seen from the left panel of the second row in Fig. 3. There, we see that $\theta_{23}$ and $\theta_{26}$ are also fully correlated: having a large $\theta_{23}$ and a small $\theta_{26}$ is equivalent to having a small $\theta_{23}$ and a large $\theta_{26}$. The same applies to the CP-violating phases, as can be seen in the right panel of the last row. The left panel of the last row shows that there are correlations also between the atmospheric and reactor angles, which are not present in the standard case of three-neutrino oscillations anymore, given the very good level of precision achieved in the determination of the mixing angles. Since $\theta_{16}$ and $\theta_{13}$, as well as $\theta_{26}$ and $\theta_{23}$, are equivalent, a similar result is obtained in the two-dimensional plane ($\theta_{13}$, $\theta_{23}$).

In Fig. 4 we show the result of our simulation of JUNO. In this case, we find that the new angles $\theta_{14}$ and $\theta_{15}$ are highly correlated with the standard solar angle $\theta_{12}$. In particular, one sees that, for $\sin^{2}\theta_{14}=0$, all values of $\theta_{12}$ and $\theta_{15}$ lying along the correspondingly labeled line are possible, showing a similar correlation as in the case of $\theta_{13}$ and $\theta_{16}$ or $\theta_{23}$ and $\theta_{26}$. For different values of $\sin^{2}\theta_{14}$ the correlating line is shifted as indicated in the figure. If we now marginalize over all possible values for $\theta_{14}$, we find that a large region of parameter space is still allowed. It is however important to notice that a point in the ($\sin^{2}\theta_{12}$, $\sin^{2}\theta_{15}$) plane always corresponds to one specific value of $\sin^{2}\theta_{14}$.

Now let us discuss the results of the combined analysis of DUNE and JUNO in terms of the variables $X_{i}$ introduced in Eq. (14). Compared to Ref. Anamiati et al. (2018), where most of the parameter space was allowed by current neutrino oscillation data, here we find that DUNE and JUNO will be able to strongly constrain some of the $X_{i}$ parameters. This is illustrated in Fig. 5, where we show the $\Delta\chi^{2}$ profiles for the $X_{i}$ variables. In the left panel, we see how precise DUNE and JUNO could measure some of these quantities. Notably, $X_{1},X_{2}$ and $X_{3}$ can be measured with a precision below $\%$.The sensitivity to $X_{4}$, $X_{5},X_{6}$ and $X_{7}$ will also be improved with respect to the current results obtained in Anamiati et al. (2018), although not as dramatically as for the previous three parameters. Note, however, that DUNE will not be able to set a lower limit on $X_{4}$, $X_{6}$ and $X_{7}$, which are allowed to be zero in our combined fits. In the case of $X_{4}$, this can be traced to the fact that DUNE does not have the resolution to demonstrate that there are three independent oscillation frequencies contributing to $P\left(\nu_{\mu}\rightarrow\nu_{\mu}\right)$ (see Eq. (16)). An upper limit on $X_{4}$ can instead be obtained from the unitarity relation $X_{3}+X_{4}<1$. Similar comments apply to $X_{6}$ and $X_{7}$.

In the right panel of Fig. 5, we construct a quantity to test directly the Diracness of neutrino oscillations. This quantity is obtained by assuming that neutrinos are Dirac particles in Eq. (14), see Ref. Anamiati et al. (2018) for more details. In this case one can derive that

[TABLE]

Any deviation from zero in this expression would be an indication for quasi-Dirac neutrinos. Since we created our fake data assuming neutrinos to be Dirac particles, our best-fit point is automatically located at zero. However, DUNE could restrict the allowed deviation considerably, as shown in the plot.

Finally, to further investigate the discrimination power of the experiments to the quasi-Dirac scenario, we have created another fake data set using a quasi-Dirac point as an input. For the particular point we have chosen, we expect

[TABLE]

Our choice falls inside the 1$\sigma$ contours of Fig. 3 and it corresponds to $\sin^{2}\theta_{23}=0.30$, $\sin^{2}\theta_{26}=0.37$, $\sin^{2}\theta_{13}=0.0108$, $\sin^{2}\theta_{16}=0.0108$. The CP-violating phases are assumed as in the first analysis. The result for this simulation is shown in Fig. 6. In the left panel one can see how most of the $X_{i}$ are mostly unaffected by the selected input point, while there is a visible difference in the profiles corresponding to $X_{3}$ and $X_{5}$. Nevertheless, the most visible effect appears in the right panel of the figure. There, we see that the Dirac-point — with $1-\frac{X_{5}}{X_{1}X_{3}}=0$ — could be completely excluded in this scenario. This is an important result, because it means that DUNE and JUNO would be able to distinguish standard three-neutrino oscillations from quasi-Dirac oscillations. Note, however, that this statement is true for our benchmark point. If the true value lies very close to Diracness, it would be more difficult to discriminate between the two scenarios.

V Conclusions

We have studied the sensitivity of the DUNE and JUNO experiments to quasi-Dirac neutrino oscillations. We have found that, within this scenario, the determination of neutrino mixing angles becomes much more complicated, if not impossible, even for next-generation experiments as DUNE or JUNO. The main reason is that the new angles and phases are strongly correlated to the corresponding ones in the active sector, leading to very relevant degeneracies.

As a further comment, let us mention that many of the degeneracies observed here could be broken by including a $\nu_{\tau}$ appearance channel in the DUNE analysis. This possibility has been recently discussed in Ref. de Gouvêa et al. (2019); Ghoshal et al. (2019). If neutrinos are quasi-Dirac particles, $\sum_{\beta}P_{\alpha\beta}<1$, with $\beta=\{e,\mu,\tau\}$. Hence, a more precise observation of the unitarity of neutrino oscillations including the $\nu_{\tau}$ channel would be extremely helpful to test the quasi-Dirac neutrino hypothesis, as well as other non-unitary neutrino scenarios Escrihuela et al. (2015); Blennow et al. (2017).

Despite the degeneracies affecting the angles, we have seen that we can define new observables which clearly allow to distinguish the standard oscillation case from the quasi-Dirac neutrino scenario. While most of the parameter space for these observables is still allowed at present, we have shown that DUNE and JUNO can considerably improve the current bounds on these quantities. We have also seen that, if quasi-Dirac neutrino oscillations are real, the new generation of experiments will have the potential to discover quasi-Dirac neutrinos, which would be a big breakthrough in particle physics.

Acknowledgments

We would like to thank Stefano Gariazzo and Renato Fonseca for useful discussions. Work supported by the Spanish grants FPA2017-90566-REDC (Red Consolider MultiDark), FPA2017-85216-P and SEV-2014-0398 (MINECO/AEI/FEDER, UE), as well as PROMETEO/2018/165 (Generalitat Valenciana). VDR acknowledges financial support by the ”Juan de la Cierva Incorporacion” program (IJCI-2016-27736) funded by the Spanish MINECO. CAT is also supported by the FPI fellowship BES-2015-073593. MT acknowledges financial support from MINECO through the Ramón y Cajal contract RYC-2013-12438.