Consequences of $\mu$-$\tau$ reflection symmetry for $3+1$ neutrino mixing

We investigate the consequences of $\mu-\tau$ reflection symmetry in presence of a light sterile neutrino for the $3+1$ neutrino mixing scheme. We discuss the implications of total $\mu-\tau$ reflection symmetry as well partial $\mu-\tau$ reflection symmetry. For the total $\mu-\tau$ reflection symmetry we find values of $\theta_{23}$ and $\delta$ remains confined near $\pi/4$ and $\pm \pi/2$ respectively. The current allowed region for $\theta_{23}$ and $\delta$ in case of inverted hierarchy lies outside the area preferred by the total $\mu-\tau$ reflection symmetry. However, interesting predictions on the neutrino mixing angles and Dirac CP violating phases are obtained considering partial $\mu-\tau$ reflection symmetry. We obtain predictive correlations between the neutrino mixing angle $\theta_{23}$ and Dirac CP phase $\delta$ and study the testability of these correlations at the future long baseline experiment DUNE. We find that while the imposition of $\mu-\tau$ reflection symmetry in the first column admit both normal and inverted neutrino mass hierarchy, demanding $\mu-\tau$ reflection symmetry for the second column excludes the inverted hierarchy. Interestingly, the sterile mixing angle $\theta_{34}$ gets tightly constrained considering the $\mu-\tau$ reflection symmetry in the fourth column. We also study consequences of $\mu-\tau$ reflection symmetry for the Majorana phases and neutrinoless double beta decay.


I. INTRODUCTION
Over the past years non-zero neutrino masses and mixings have been well established by several neutrino oscillation experiments and most of the parameters have been measured with considerable precision. The parameters governing the three generation neutrino oscillation phenomena are the three mixing angles (namely, solar mixing angle θ 12 , atmospheric mixing angle θ 23 and rector mixing angle θ 13 ), two mass-squared differences (namely, solar mass-squared difference ∆m 2 sol = m 2 2 − m 2 1 and atmospheric mass-squared difference ∆m 2 atm = m 2 3 − m 2 1 ) and Dirac CP phase δ.  [1], from cosmology. From the theoretical perspective, lots of effort have been exercised in last few decades to realize the observed neutrino mixing pattern. In this regard, many discrete flavor symmetry groups were exploited to understand the dynamics of this mixing pattern in the lepton sector by extending the Standard Model gauge group with some additional symmetry. A review on lepton masses and mixing based on such discrete groups can be found for instance in [2][3][4][5][6].
The observational data guided by θ 23 ≈ 45 • is indicative of a simple µ-τ flavor symmetry.
The simplest realization of such µ-τ flavor symmetry is known as µ-τ permutation symmetry.
For a review on µ-τ flavor symmetry and its phenomenological implications see [12].
In addition to three active neutrinos, there may exist a light sterile neutrino (Standard Model gauge singlets) at the eV scale (for a review see [39]) which can address anomalies inν µ → ν e oscillations observed in some short-baseline neutrino oscillation experiments. Initially the anomaly was found in the antineutrino flux measurement of LSND accelerator experiment [40,41] at Los Alamos which was subsequently confirmed by MiniBooNE [42] (a short baseline experiment at Fermilab). Very recently MiniBooNE experiment again refurbished their earlier results with ν e appearance data reinstating the presence of a light sterile neutrino [43]. Results from few experiments like gallium solar experiments [44][45][46] with artificial neutrino sources, reactor neutrino experiments [47,48] with recalculated fluxes also support the hypothesis of at-least one sterile neutrino. In this context the 3+1 scenario [49] consisting of three active neutrinos and mixing with one eV scale sterile neutrino is considered to be most viable [50,51]. Here, we have to keep in mind that inclusion of sterile neutrinos must face tight cosmological hurdles coming from the Cosmic Microwave Background observations, Big Bang Nucleosynthesis and Large Scale Structures. Al-though fully thermalized sterile neutrinos with mass ∼ 1 eV are not cosmologically safe, they can still be generated via 'secret interactions' [52][53][54]. For a brief review on eV scale sterile neutrinos see [55]. Despite many constrains as well as tension between disappearance and appearance data from oscillation experiments the sterile neutrino conjecture is still a topic of intense research.
In the context of 3 + 1 neutrino mixing exact µ-τ permutation symmetry would still give θ 13 zero. Studies have been accomplished in the literature examining the possible role of activesterile mixing in generating a breaking of this symmetry starting from a µ-τ symmetric 3 × 3 neutrino mass matrix [28,[56][57][58][59][60][61]. In this paper we concentrate on the ramifications of µ-τ reflection symmetry for the 4 × 4 neutrino mass matrix in presence of one sterile neutrino. We study the consequences of total as well as partial µ-τ reflection symmetry in the 3+1 framework and obtain predictions and correlations between different parameters. We also formulate the 4 × 4 neutrino mass matrix which can give rise to such a µ-τ reflection symmetry. Further we study the experimental consequences of µ-τ reflection symmetry at the future long baseline neutrino oscillations experiment DUNE. In addition we discuss the implications of µ-τ reflection symmetry for Majorana phases and neutrinoless double β decay.
Rest of this paper is organized as follows. In Section II we first construct the generic structure of the 4 × 4 mass matrix which can give rise to µ-τ reflection symmetry for sterile neutrinos. In the next section, we find the correlation among the active and sterile mixing angles and Dirac CP phases. In Section IV we study the experimental implications of such µ-τ reflection symmetry for DUNE experiment and also calculate the effective neutrino mass which can be probed through neutrinoless double β decay experiments. Then finally in Section V we summarize the findings.

II. µ-τ REFLECTION SYMMETRY FOR 3+1 NEUTRINO MIXING
Guided by the atmospheric neutrino data, the µ-τ reflection symmetry was first proposed for 3-generation neutrino mixing back in 2002 [13,16]. Under such symmetry the elements of lepton mixing matrix satisfy : This indicates that the moduli of µ and τ flavor elements of the 3 × 3 neutrino mixing matrix are equal. With these constraints, the neutrino mixing matrix can be parameterised as [13,16] where the entries in the first row, u i 's are real (and non-negative) 2 . v i satisfy the orthogonality [12]. In [16], it was argued that the mass matrix leading to the mixing matrix given in Eq. 2 can be written as where a, b are real and d, c are complex parameters. As a consequence of the symmetry given in Eq. 1-3, we obtain the predictions for maximal θ 23 = 45 • and δ = 90 • or 270 • in the basis where the charged leptons are considered to be diagonal. This scheme however still leaves room for nonzero θ 13 . Several attempts were made in this direction to explain correct mixing (for three active neutrinos) with µ-τ reflection symmetry and to study their origin and consequences in various scenarios [21, 24-26, 31, 33, 63-73].
Although, µ-τ reflection symmetry is well studied for three active neutrinos, it lacks a comprehensive study considering sterile neutrinos. Now such a mixing scheme can easily be extended for a 3 + 1 scenario incorporating sterile neutrinos. Under such circumstances, the 4 × 4 neutrino mixing matrix can be parameterised as where u i , w i are real but v i are complex. Within this extended scenario, the mass matrix can now be written as where a, b, e, g are real and d, c, f are complex parameters. Such a complex symmetric mass matrix can be obtained from the Lagrangian with where m j 's are the real positive mass eigenvalues.
Here the matrix M is characterized by the transformation and respects the mixing matrix given in Eq Now, using Eq. 7, we find Following the above equation, one can therefore find another diagonalizing matrix, U = SU * .
Now it can be shown that if both U and U satisfy the diagonalization relation U T M ν U = diag (m 1 , m 2 , m 3 , m 4 ) with non-degenerate mass eigenvalues, then there exists a diagonal unitary matrix X such that here X jj is an arbitrary phase factor for m j = 0 and X = ±1 for m j = 0. Therefore the constraint obtained in Eq. 10 leads to 3 The above equation can also be verified in an alternate way. Let us first define an Hermitian matrix as, considering the form of M ν given in Eq. 5 one can easily find Now, one can write the diagonalization relation in this case as : which follows only if masses are degenerate or |U µi | = |U τ i | [16]. Therefore, it is now clear to us that the mass matrix given in Eq. 5 actually leads to a mixing matrix of the form in Eq. 4. In the following section we discuss the consequences of this µ-τ reflection symmetry involving the active and sterile mixing angles and phases in details.
It is important to note that the mixing matrix given in Eq. 2 should correspond to the standard neutrino mixing matrix U P M N S for three generation case. Now, depending upon the choice of the arbitrary phase factor X given in Eq. 10 the Majorana phases can be fixed in the context of µ-τ reflection symmetry. With the choice of X ii = 1 or -1 the Majorana phases are fixed at 0 • or 90 • [12,25]. Such fixed values of phases can have implication for neutrinoless double beta decay which will be discussed later.

III. CONSTRAINING 3+1 NEUTRINO MIXING WITH µ-τ REFLECTION SYMMETRY
For 3+1 neutrino mixing scenario the neutrino mixing matrix U can be written in terms of a 4 × 4 unitary matrix. This unitary matrix can be parameterized by three active neutrino mixing angles θ 13 , θ 12 , θ 23 and three more angles originating from active-sterile mixing, namely, θ 14 , θ 24 and θ 34 . It will also contain three Dirac CP violating phases, such as, δ, δ 14 and δ 24 . Hence this 4 × 4 unitary PMNS matrix U can be given by where the rotation matrices R andR's can be written as Along with the parameterization defined in Eq. 14, there also exists a diagonal phase matrix, Note that, the correspondence of the mixing matrix in Eq. 4 along with the diagonal phase matrix P in Eq. 16 implies that the Majorana phases are zero or ± π 2 . However, in light of Eq. 1 this diagonal phase matrix do not play any role in the present analysis. But they can play role in neutrinoless double β decay which will be discussed in Section IV. Following these conditions, one can obtain four different equalities among six mixing angles and three Dirac CP violating phases. To keep the present analysis simple, first we have assumed the sterile Dirac CP violating phases (δ 14 and δ 24 ) to be zero. For this case, δ 14 = δ 24 = 0 • , from Eq. 1 these four correlations can be written as, Here, the first three equalities enable us to study the correlation among the mixing angles  In this subsection we present the results assuming µ − τ reflection symmetry to be valid for all the four columns simultaneously; we call this as the total µ − τ reflection symmetry. The magenta shaded region in Fig. 1 represents the allowed region for total µ − τ symmetry for 3+1 neutrino scenario in θ 23 −δ plane. The analysis is performed by varying the other mixing parameters in their 3σ range as in Tab. I and the sterile CP phases δ 14 and δ 24 between 0 • to 360 •4 . The grey solid and green dashed contours denote the currently allowed parameter space for NH and IH respectively in this and the subsequent figures. The application of the total µ − τ symmetry significantly restricts the parameters θ 23 and δ. θ 23 is primarily restricted around the maximal while δ falls in the close vicinity of 90 • and 270 • . Comparing the results with 3 neutrino scenario [32,33] where θ 23 is strictly restricted to be maximal and δ to 90 • and 270 • we conclude that the involvement of the sterile mixing angles and phases lead to slight deviations in θ 23 and δ from their 3 generation predictions. However, the current global fit results from [76] suggests that the best fit for θ 23 is 49.5 • for both normal and inverted hierarchies. Thus, even with inclusion of sterile neutrinos, total µ − τ reflection symmetry cannot explain the current best-fit. This motivates us to consider the partial µ − τ reflection symmetry for the 3+1 scenario.
B. Partial µ − τ reflection symmetry In this section we discuss the implications partial µ − τ reflection symmetry which implies that the condition |U µi | = |U τ i | is satisfied for individual columns. In Fig. 4 we show the consequence of |U µ3 | = |U τ 3 | , and it is seen that the allowed value of In this 3+1 neutrino framework, the fourth equality |U µ4 | = |U τ 4 | establishes a powerful correlation between the two sterile mixing angles θ 24 and θ 34 . Here we obtain a one-to-one correspondence between θ 24 and θ 34 as given in Eq. 20. Even with the involvement of sterile CP phase this relation remains same as evident from Eq. 11 and 14. This correlation yields a linear dependence between the two sterile mixing angles θ 24 and θ 34 as given in Fig. 5. Once we impose the constraints coming from the the current allowed value of θ 24 [78], the sterile mixing angle θ 34 becomes restricted from below significantly. Note that so far there only exists a upper limit on θ 34 [78]. In Fig. 5 we plot this correlation and find that θ 34 lies within the range 4.9 • − 9.8 • corresponding to the 3σ allowed range of θ 24 [78]. Therefore, the µ − τ reflection symmetry presented here restricts the sterile mixing angle θ 34 considerably. The allowed parameter space in the θ 24 − θ 34 plane also gets restricted. This is one of the most crucial finding in this µ−τ reflection symmetric framework for 3+1 neutrino scenario. It is to be noted that among the current constraints on sterile mixing angles, the bound on θ 34 is much weaker, there being only an upper limit on this. The reactor neutrino experiments are sensitive to the mixing matrix element U 2 e4 or s 2 14 in our parametrization. The short baseline oscillation experiments using appearance channel are sensitive to the product |U e4 | 2 |U 2 µ4 | which contains the product s 2 14 s 2 24 . Bounds on θ 34 have been obtained from atmospheric neutrinos at SuperKamiokande [82], DeepCore detector at Icecube [83], and from Neutral Current data at MINOS [84], NOνA [85] and T2K [86] experiments. The constraints on θ 34 from individual experiments are somewhat weaker (in the ballpark of 20 • − 30 • ) than what is obtained in the global analysis of [78]. In our analysis the later has been used. Neutral current events at the DUNE detector can also improve on the bound on θ 34 coming from a single experiment [87]. The ν τ appearance channel is also sensitive to θ 34 and the potential of DUNE experiment to constrain this mixing angle has been studied in [88][89][90]. Thus it is expected that future data can test this correlation and the allowed parameter space.
These discussions lead us to the inference that partial µ − τ reflection symmetry is more favorable scenario. However it is to be noted that in this scenario the case |U µ3 | = |U τ 3 | is disfavored because it fixes θ 23 around the maximal value. Again, the simultaneous application of equalities |U µ1 | = |U τ 1 | and |U µ2 | = |U τ 2 | restricts θ 23 ∼ 45 • hence both these equalities cannot be satisfied together. Such experimental constraints do not apply on |U µ4 | = |U τ 4 | therefore this equality may still hold. So, the favorable scenarios are:

A. Neutrino Oscillations Experiments
In this section we explore the consequences of partial µ − τ reflection symmetry in the 3+1 scenario for the Deep Underground Neutrino Experiment (DUNE).

Experimental and Simulation details
DUNE is a proposed future long baseline accelerator experiment which is expected to lead the endeavor in determination of the unknown neutrino oscillation parameters. The neutrino source for DUNE is proposed to be the Long Baseline Neutrino Facility (LBNF) at Fermilab, which will provide intense 1.2 MW neutrino beams. The detector is a 40 kt liquid Argon detector located at South Dakota with a baseline of 1300 km. The total POT is expected to be 10 × 10 21 over a period of 10 years with 5 years each of neutrino and anti-neutrino run. The simulation have been performed using the package General Long Baseline Experiment Simulator (GLoBES) [91,92], and the sterile neutrino effects have been applied using the sterile neutrino engine as described in [93].
To test the correlations at DUNE we define χ 2 as where, the statistical χ 2 is χ 2 stat while the systematic uncertainties are incorporated by χ 2 pull . The later is calculated by the method of pulls with pull variables given by ξ [94][95][96]. The oscillation parameters {θ 23 , θ 12 , θ 13 , δ CP , ∆m 2 21 , ∆m 2 31 , θ 14 , θ 24 , θ 34 , δ 14 δ 24 } are represented by ω. The statistical χ 2 stat is calculated assuming Poisson distribution, where k = s(b) represent the signal(background) events. The effect of the pull variable ξ norm (ξ tilt ) on the number of events are denoted by c norm i (c i tilt ). The bin by bin mean reconstructed energy is represented by E i where i represents the bin. E min , E max andĒ = (E max + E min )/2 are the minimum energy, maximum energy and the mean energy over this range. The signal normalization uncertainties used are as follows: for ν e /ν e -2% and ν µ /ν µ -5%. While the background uncertainties vary from 5% to 20%.
2. µ-τ reflection symmetry at DUNE for 3+1 neutrino mixing • The test events are generated by marginalization of the parameters θ 12 , θ 13 , |∆m 2 31 |, θ 23 , θ 14 , θ 24 , θ 34 over the range given in Tab. I subject to the condition embodied in Eq. 17 for the left panel and Eq. 18 for the right panel. The other parameters are held fixed at their true values for calculation of N test . In this study we have assumed normal hierarchy as the true hierarchy. We have checked that marginalizing over test hierarchy do not have significant effect because the correlations are independent of hierarchy.
• For each true value of θ 23 and δ the χ 2 is minimized and the allowed regions defined by min + ∆χ 2 are plotted corresponding to 1σ, 2σ and 3σ values of ∆χ 2 .
The experimental consequences at DUNE are presented in Fig. 6. Here the first(second) column represents the correlation |U µ1 | = |U τ 1 |(|U µ2 | = |U τ 2 |). The condition |U µ4 | = |U τ 4 | is also incorporated in both the plots. Each plot consists of 1σ, 2σ & 3σ confidence regions considering partial µτ reflection symmetry which are shaded as blue, grey and green respectively.
The current 3σ permitted region from NuFIT [76,77] data is drawn with red solid line. The plots show a similar nature as the correlation plots in Fig. 2  as observed in 2νββ decay. The half-life (T 1/2 ) for the 0νββ process is given as, where G contains the lepton phase space integral, m e is the mass of electron, M ν is the nuclear matrix element (NME) which takes into consideration all the nuclear structure effects, m ββ stands for the effective neutrino mass and can be expressed as Here m i are the real positive neutrino mass eigenvalues with i = 1, 2, 3 for three generation and for NH (IH) and ∆m 2 LSND = m 2 4 − m 2 1 . Predictions for m ββ with respect to the lightest neutrino mass for 3+1 scheme along with the three neutrino case is presented in Fig. 7. The plot in the left panel shows the effective neutrino mass for NH while the right panel is for IH. In generating these plots we have varied the oscillations parameters within their 3σ range as given in Tab. I with ∆m 2 LSND = 1.7 eV 2 [78]. In both panels It is to be noted that in the red shaded region in the right panel of Fig. 7 the value of m ββ is > 0.06eV, while, the current experimental bound is m ββ < 0.07eV. Therefore, a portion of It is well known that near future 0νββ experiments like SNO+ Phase I [102], KamLAND-Zen 800 [103], and LEGEND 200 [104] can test the IH region in the context of the three generation scenario. These experiments will be able to test the predictions for α = β = γ = 0 • further. However, in presence of an extra sterile neutrino, a null signal in these experiments cannot exclude IH because of the occurence of the cancellation regions.
* Normal Hierarchy: From the first panel of Fig. 7 we observe that for α = β = γ = 0 • , the red shaded region, m ββ stays is in the range ∼ 0.01 − 0.05 eV rising upto 0.
In this case for α = β = γ = 0 • , the value of m ββ for t  few benchmark points of the lightest neutrino mass (m 1 for NH and m 3 for IH) in Table   IV B.
In our analysis we have considered ∆m 2 LSN D = 1.7 eV 2 which gives the physical mass of sterile neutrino m 4 = m s ph ∼ 1.3 eV. The Cosmic Microwave Background analysis in Λ CDM + r 0.05 + N eff + m s eff model using the Planck 2015 data [1] gives the N eff < 3.78 and m s eff < 0.78 eV [105]. The bounds including other datasets are more stringent than this. As the effective mass in terms of the N eff and physical mass of sterile neutrino is given as We formulate the mass matrix compatible with the lepton mixing matrix which can give rise to µ − τ reflection symmetry, defined via |U µi | = |U τ i | where i = 1, 2, 3, 4. We obtain and plot the correlations connecting the mixing angle θ 23 and the CP phase δ for the case when sterile phases are assumed to be zero, as well as present the correlation plots with the sterile phases varied in their full range. We find that if we consider total µ − τ reflection symmetry i.e. |U µi | = |U τ i | is simultaneously satisfied for all the four columns then the mixing angle θ 23 is confined in a narrow region around θ 23 = 45 • and δ is restricted around the maximal CP violating values. However, the deviation of θ 23 from maximal value with the inclusion of the sterile mixing is not sufficient to account for the observed best fit value. This prompts us to consider partial µ − τ reflection symmetry and study the consequences for each column individually.
The equalities |U µ1 | = |U τ 1 | and |U µ2 | = |U τ 2 | yield important correlations among the neutrino mixing angle θ 23 and Dirac CP phase δ. Interestingly we find that the best-fit value for (θ 23 , δ) shows a good agreement with inverted neutrino mass hierarchy for |U µ1 | = |U τ 1 | and normal mass hierarchy for |U µ2 | = |U τ 2 |. With precise measurement of θ 23  We also explore the possibility of testing the µ-τ reflection symmetry for 3+1 neutrino mixing at the future LBL experiment DUNE. The application of the correlations constrains a significant area of the parameter space yet unconstrained by the present global fit data. In particular the constraint is more stringent for the relation |U µ1 | = |U τ 1 | and all the CP conserving values δ = 0 • , 180 • , 360 • are excluded at 3σ. However, for |U µ2 | = |U τ 2 | CP conserving values of δ remain allowed. In conclusion, µ − τ reflection symmetry for sterile neutrinos in a 3+1 picture gives some interesting predictions which can be tested in future neutrino oscillation and neutrinoless double beta experiments and the scenario can be confirmed or falsified.