Consequences of $ \mu-\tau $ Reflection Symmetry at DUNE

We consider minimal type-I seesaw framework to realize $ \mu-\tau $ reflection symmetry in the low-energy neutrino mass matrix, $ M_{\nu} $. Considering DUNE experiment, we scrutinize its potential to measure the precision of 2-3 mixing angle, $ \theta_{23} $ and the Dirac CP-phase, $ \delta $ for the given symmetry. Later, we examine the precision of these two parameters considering NuFit-3.2 data as one of the important true points. To study the low-energy phenomenology, we further discuss various breaking patterns of such an exact symmetry. Moreover, for each breaking scenario we perform the capability test of DUNE for the determination of $ \theta_{23} $ and to establish the phenomenon of CP violation considering the true benchmark point arising from the breaking of $ \mu-\tau $ reflection symmetry. We also make remarks on the potential of DUNE to rule out maximal CP-violation or CP-conservation hypothesis at a certain confidence level for different scenarios.


I. INTRODUCTION
During last a few years, there have been remarkable progress in the field of neutrino physics which guided us to understand some intriguing aspects of neutrinos in a comprehensive manner. It is now well established phenomenon from different experimental results that neutrinos possess non-zero mass and their different flavors are mixed [1]. However, the dynamical origin associated with neutrinos mass generation as well as mixing patterns are still unknown. There have been numerous theoretical attempts to understand the nature of tiny neutrino masses, among which seesaw mechanism is considered as highly appreciated one [2][3][4][5][6]. The simplest way to generate neutrino masses are to add at least two SU (2) singlet right-handed neutrino fields (i.e. N µR , N τ R ) in the Standard Model (SM). The relevant SM gauge invariant Lagrangian containing the neutrino Yukawa matrix and the Majorana neutrino mass matrix can be written as where L αL = (ν α , α) T L is the left-handed lepton doublet, Y ν denotes the neutrino Yukawa matrix, H = iσ 2 H * with H being the Higgs doublet in the SM. Also, M R is the Majorana neutrino mass matrix and C denotes the charge-conjugation operator. After spontaneous symmetry breaking, one obtains the Dirac neutrino mass term as ν αL M D N R + h.c., where M D = vY ν is the Dirac neutrino mass matrix with vacuum expectation value (vev), v = H ≈ 174 GeV [1]. Employing seesaw mechanism, one gets light neutrino mass matrix in type -I seesaw formalism as, M ν ≈ −M D M −1 R M T D and diagonalization of such M ν leads to three active neutrino masses m i (for i = 1, 2, 3).
In this work, we embed µ-τ reflection symmetry in minimal seesaw formalism such that one can address both neutrino masses and mixing patterns (see Ref. [31] for recent review).
Later, we study its consequences considering next-generation super beam Deep Underground Neutrino Experiment (DUNE). This statistically high potential experiment will improve the precision of the atmospheric mixing angle, θ 23 and play a key role to probe the leptonic CP-violating phase, δ [32]. Because of this, DUNE can test various flavor symmetry models and helps us to understand some inherent physics associate with it.
At the given framework along with maximal δ and θ 23 , we also find remaining oscillation parameters both analytically as well as numerically. Considering this as true benchmark point, we depict the allowed area in (δ −sin 2 θ 23 ) plane for DUNE at various confidence levels which serves our intention to inspect precision of these two less known parameters. This also show the potential of DUNE to know how well it can measure δ and θ 23 . Moreover, latest results of global-fit of neutrino oscillation data from NuFit-3.2 collaboration [33, 34] favors higher octant of θ 23 along with non-maximal δ 1 . Also, results of on-going neutrino oscillation experiments (e.g., T2K [35] and NOνA [36]) are in well agreement with the predictions of the concerned symmetry but, still show large uncertainties in their measurement of δ and θ 23 . Therefore, it is tenacious to accept the exact nature of µ − τ reflection symmetry.
In that respect, it is worthwhile to study various broken scenarios of such a symmetry.
To proceed with phenomenological study, we first perform our analysis considering global best-fit values as our benchmark point [33,34]. Afterwards, we consider breaking of µ-τ reflection symmetry by introducing explicit breaking parameter in the high energy neutrino mass matrices M D , M R , respectively. For each scenario, we find the set of neutrino oscillation parameters and perform the capability test of DUNE in (δ − sin 2 θ 23 ) plane. Considering different cases, we analyze the potential of DUNE to rule out the possibility of maximal CPviolation (CPV) as well as CP-conservation hypothesis at a given confidence level. Some recent studies considering different flavor models in the context of long baseline experiments have been performed in [30,[37][38][39][40][41][42][43][44][45].
We organize rest of the paper as follows. In Section II, we present a general set-up of the µ − τ reflection symmetry and perform our analysis in the given scenario for DUNE. We also present our numerical details in this section. We proceed to discuss our results considering NuFit-3.2 data in Section III. Furthermore, in subsequent subsections of Section III, we discuss the breaking of µ−τ reflection symmetry by introducing explicit breaking parameter 1 Note that θ 23 < 45 • is called as lower octant (LO) whereas θ 23 > 45 • is called as higher octant (HO).
in M D and M R , respectively and their implications in the context of DUNE. Finally, we summarize our noteworthy results in Section IV.
The µ − τ reflection symmetry at the low energy neutrino mass matrix, M ν was first proposed in Ref. [12] which leads us to following four predictions : where M αβ , (with α, β = e, µ, τ ) are the elements of M ν . We consider minimal type -I seesaw mechanism to realize µ − τ reflection symmetry at M ν . To achieve such symmetry, we extend the SM fields content by adding two right-handed neutrino fields which are singlet under the SM gauge group. Without loss of generality, we consider the following texture of M D to realize µ − τ reflection symmetry, Also, we adopt diagonal M R of the form M R = diag(M 1 , M 1 ) with degenerate heavy Majorana neutrino masses 2 . Further, considering type-I seesaw mechanism, we obtain the effective neutrino mass matrix for the light neutrinos as We notice that the elements of M ν as given by Eq. (4) satisfy all the conditions of Eq. (2) and hence leads to µ − τ reflection symmetry 3 . In the standard PDG [1] parameterization, 2 It is possible to find the considered mass textures using a suitable flavor group along with preferred Z n cyclic group. As our intention is to study the impact of these textures rather their theoretical origin, hence we do not perform this study here. 3 Note that non-degenerate Majorana neutrino mass matrix does not satisfy all the conditions mentioned in Eq. (2) and thus does not lead to the concerned symmetry which we discuss in section III. the unitary mixing matrix which diagonalizes neutrino mass matrix, M ν , can be written as, where c ij = cos θ ij , s ij = sin θ ij (for i < j = 1, 2, 3). Here, P l contains three unphysical phases of the form, P l = diag(e iφe , e iφµ , e iφτ ) which can be absorbed by the rephasing of charged lepton fields whereas P ν = diag(e iρ , e iσ , 1) contains two Majorana phases.
With the above form of M ν as given by Eq. (4), one can find that there exist 6 predictions for the leptonic mixing angles and phases which are 4 , Note that under µ − τ reflection symmetry the value of θ 13 , θ 12 remain unspecified. We find their analytical form in terms of model parameters as 5 , ; for IMO (7) where Similarly, one can calculate masses of light neutrinos by diagonalizing M ν of Eq.(4) as where m i 's (i = 1, 2, 3) are the active neutrino masses. Further, the masses can be expressed for NMO as whereas, expressions for IMO can be written as Here, m 2 = m 2 e 2iσ and σ can take value either 0 • or 90 • . Also note that as the minimal seesaw formalism always predicts massless lightest neutrino, one has the freedom of eliminating one of the Majorana phases. Thus, in this study we do not consider phase, ρ.
To proceed further and to investigate low energy phenomenology, we first give here simulation and experimental details that are considered in this work. The principle strategy of our numerical analysis is to scan all the high energy variables of Y ν and M R as free variables and later constrain the allowed space of high energy variables to find neutrino oscillation parameters which are compatible with the latest NuFit-3.2 data [33, 34] at low energies.
We vary different parameters as, We use the nested sampling package Multinest [46][47][48] to guide the parameter scan with the built χ 2 function considering latest NuFit-3.2 data [33,34]. The analytical expression of the Gaussian-χ 2 min function that we use in our numerical simulation is defined as, We consider here DUNE, which is a proposed next generation superbeam experiment at Fermilab, USA [32,49] designing to detect neutrinos. This experiment will utilize existing NuMI (Neutrinos at the Main Injector) beamline design at Fermilab as a neutrino source.
The far detector of DUNE will be placed at Sanford Underground Research Facility (SURF) in Lead, South Dakota, at a distance of 1300 km (800 mile) from neutrino source. DUNE collaboration has planned to use LArTPC (liquid argon time-projection chamber) detector.
For the numerical simulation of the DUNE data, we use the GLoBES package [50,51] along with the required auxiliary files presented in Ref. [49]. We perform our simulation considering 40 kton fiducial mass far detector. We also consider the flux corresponding to 1.07 MW beam power which gives 1.47 × 10 21 protons on target (POT) per year due to 80 GeV proton beam energy. In addition, we adopt signal and background normalization uncertainties for appearance as well as disappearance channel as presented in DUNE CDR [49]. Further, we distribute the total exposure of DUNE (i.e., 300 kton-MW-years) in two scenarios ; (i) in first scenario, we perform our analysis only with neutrino mode considering 7 years of neutrino run, i.e., DUNE[7ν + 0ν], and (ii) in second scenario, we consider 3.5 years each of neutrino and antineutrino mode i.e., DUNE[3.5ν + 3.5ν]. We also add 5% prior on sin 2 2θ 13 in our analysis.
The main steps to carry out our numerical analysis are to calculate set of neutrino oscillation parameters corresponding to minimum χ 2 (= χ 2 min ), as defined by Eq.(12), using Multinest in this model. Later, considering this set of parameters as true benchmark value, we generate DUNE results using GLoBES and present the allowed parameter space in test (δ − sin 2 θ 23 )-plane. We utilize GLoBES inbuilt χ 2 -function for the data analysis. In this study, we marginalize all the oscillation parameters over their 3σ range as given by Table II. In addition, we marginalize δ in the range δ ∈ [0 • , 360 • ) for each scenario unless otherwise stated.
In Fig.1, we present our results in the framework of µ − τ reflection symmetry. We calculate the numerical values for the set of neutrino oscillation parameters in the given scenario corresponding to χ 2 min as given in Table I. Considering these true set of parameters, we find the allowed area in (δ − sin 2 θ 23 )-plane in case of DUNE which we have depicted in    Considering maximal value of (δ, sin 2 θ 23 ) as true benchmark point, we notice from the first row of Fig Here green, pink and blue color represent 1σ, 3σ and 5σ allowed contours and 'red- * ' signifies true value of (δ, sin 2 θ 23 ). Also left (right) column represents normal (inverted) mass ordering and top (bottom) row shows our results for DUNE[7ν + 0ν] ( DUNE[3.5ν + 3.5ν] ). mass orderings. In case of NMO (for true δ = 90 • ), we observe that DUNE can rule out Similarly, from bottom row we notice that the same conclusion remains true even at 5σ C.L. except a small regions for NMO.
Having discussed our results in the µ−τ reflection symmetry scenario considering DUNE, in the following section we proceed to perform our analysis by utilizing current oscillations data. Later, we also examine different symmetry breaking scenarios where we will discuss the impact of breaking parameter on the poorly measured parameters, δ and sin 2 θ 23 .

III. PHENOMENOLOGY BEYOND µ − τ REFLECTION SYMMETRY
In this section, we discuss our results beyond µ−τ reflection symmetry considering DUNE.
As the current best-fit value of neutrino oscillation data prefers non-maximal value of δ, sin 2 θ 23 , we start the discussion considering this as our true benchmark point. Furthermore, in subsequent subsections we perform our study considering different breaking scenarios of µ − τ reflection symmetry and its impact in the context of DUNE.

A. Analysis of global best-fit data
In Table II   We discuss here three different scenarios to break µ−τ reflection symmetry by introducing explicit breaking parameter in the Dirac neutrino mass matrix, M D . Further, for each case we perform precision study to determine δ, sin 2 θ 23 considering DUNE. We study them as follows.
• Broken Scenario-1 (BS1): After assigning breaking parameter in the (12) The above texture of M D leads to low energy neutrino mass matrix M ν of the form, Now to find masses and mixing angles in presence of breaking term , we diagonalize M ν with V 7 . Note that V has similar form as V in the absence of as described by Eq. (5). In Table III, we give the expressions of modified masses and mixing angles for both the mass orderings. Note that for simplicity, we only consider the leading order corrections in terms of , θ 13 and ξ 1 = m 2 /m 3 (ξ 2 = ∆m 2 21 /m 2 2 ) for NMO (IMO).
Afterwards, we proceed to find the set of neutrino oscillation parameters numerically in this scenario. We also emphasize here that the numerical analysis throughout this work are based on exact formula not on any leading order approximations. The numerical best-fit values at χ 2 min for both the mass orderings are tabulated in Table  IV. Considering these set of values as the true benchmark point, we present allowed area in test (δ − sin 2 θ 23 )-plane for DUNE in Fig. 3 8 . 7 We vary the breaking term in the range, [-1,1] along with other high energy parameters, as mentioned in Eq. (11). 8 Note that one can also perform various correlations study considering neutrino oscillation parameters in different broken scenarios. Recently, authors in Ref. [28] have performed different correlation study.    blue color represent 1σ, 3σ and 5σ allowed contours and 'red- * ' signifies true value of (δ, sin 2 θ 23 ).
C.L. but not at 5σ C.L. which is depicted in first plot of second column by pink contour.
From second plot of right column, we observe that DUNE can reject CP-conservation hypothesis even at 5σ C.L. Further, both the cases of IMO can not reject the value corresponding to maximal CPV even at 1σ C.L. We also notice that precision of δ improves significantly when one chooses IMO over NMO and it gets even better with the combined mode of DUNE run as shown in the last plot. Finally, here we point out that DUNE can exclude δ in the range, δ ∈ [180 • , 360 • ] at 3σ C.L. for IMO (see first plot of right column) whereas same conclusion remains permissible even at 5σ C.L.
with combined (ν + ν) analysis of DUNE (see second plot of right column).
• Broken Scenario-2 (BS2): In this scenario, we introduce breaking term, , in the (22) position of M D and this modifies M D (which we renamed M D ) as Using the form of M D as given by Eq. 15, we find modified M ν as, To find modified masses and mixing angles in the given scenario, we follow the similar steps as discussed in subsection III B. In the following Table V, we give their expressions for both the mass orderings. The subleading order term in shows the corrections in active neutrino masses and mixing angles for this broken pattern.
M D given by Eq. (17), leads us to the following M ν through type -I seesaw formalism, Now we diagonalize M ν as given by Eq. (18) to find corrections in masses and mixing angles. Here also we perform similar study as discussed in subsection III B. In Table   VII, we give analytical expressions for masses and mixing angles considering both the mass orderings where O( ) term shows the correction in active neutrino masses and mixing angles for the concerned scenario.

Parameters(S3)
NMO IMO  Having discussed analytical results, we proceed to find the set of neutrino oscillation parameters in the broken scenario BS3. We calculate the best-fit values corresponding to χ 2 min numerically and present them in Table VIII. Using this set of true benchmark point, we examine allowed parameter space of DUNE in (δ−sin 2 θ 23 )-plane for both the mass orderings as shown in Fig. 5 (see figure caption for the adopted color convention and other minutes details). line. We find that similar conclusion remains permissible for the combined effect of (ν + ν) run of DUNE as shown in first plot of second row. In case of IMO with 7-years neutrino run, we find that DUNE can reject both the concerned hypotheses at 1σ C.L. whereas at higher confidence levels it fails to rule out any of these hypothesis as depicted in the first plot of second panel. Investegeting right hand side plot of second row, we notice that at 1σ C.L. it shows similar behaviour as neutrino mode whereas at 3σ C.L. it is able to rule out CP-conservation hypothesis but not maximal CPV as shown by pink contour. Finally, we observe a noteworthy outcome in this scenario compare to former two breaking patterns that this scenario can exclude lower octant of θ 23 at 1σ C.L. for NMO even with 7-years of neutrino mode data of DUNE.
We discuss here the breaking of µ−τ reflection symmetry by introducing explicit breaking parameter in the Majorana neutrino mass matrix, M R . We discuss the scenario as below.
• Broken Scenario-4 (BS4): After assigning breaking parameter in the (22) Note here that in this scenario M R becomes non-degenerate. After integrating out heavy right-handed neutrino fields, the low energy neutrino mass matrix in the type -I seesaw formalism can be written as  In this framework, we notice from Eq. (20) that as all the entries of O( ) term are non-zero, it is highly non-trivial to perform analytical study and to find expressions for modified neutrino masses and mixing angles. Therefore, we proceed to employ only numerical study unlike previous subsections where both analytical as well as numerical study were performed. The set of neutrino oscillation parameters at χ 2 min for possible mass ordering are tabulated in Table IX. We notice from table that best-fit values corresponding to χ 2 min deviates from maximal (δ, θ 23 ) for NMO whereas for IMO the given mass textures still favor maximal θ 23 but not maximal δ. After finding set of best-fit values at χ 2 min , we proceed to analyze its impact on DUNE. Performing similar kinds of analysis as illustrated in the former broken scenarios, here also we show the allowed parameter space of DUNE considering two poorly determined parameters, viz, δ and sin 2 θ 23 . We show our results in Fig. 6 considering test (δ − sin 2 θ 23 )-plane. Now from both the plots of first column, we notice that as the given mass textures have chosen the value of Dirac CP-phase, δ slightly away from its maximal value at χ 2 min , DUNE fails to rule out the phenomenon of maximal CPV even at 1σ C.L. In fact, it can rule out CP-conservation hypothesis at 3σ C.L. even with only neutrino run as shown in first plot of top row by pink contour. We see similar conclusion from the second plot of first column. We also notice here that DUNE with 3.5 years of each neutrino and antineutrino mode data can approximately exclude δ in the range, δ ∈ [0 • , 180 • ] at 5σ C.L. for the normal mass ordering. In case of IMO as depicted in the right column, we find that DUNE can exclude both the concerned phenomena, viz, maximal CPV and CP-conservation at 1σ C.L. but not at higher confidence levels. Also, none of the cases are able to rule out lower octant of sin 2 θ 23 even at 1σ C.L. In addition, we find here that NMO shows better CP-precision over IMO.
We add a remark here that as the Majorana neutrino mass matrix is always symmetric, addition of non-zero off-diagonal entry still respect µ − τ flavor symmetry and predicts maximal δ, sin 2 θ 23 . Hence, here we do not include this as an additional scenario.  Finally, we calculate the precisions of the two poorly measured parameter δ and sin 2 θ 23 .
Here, max (min) refers to the maximum (minimum) value of the concerned parameter in a given contour. Also, we present the precision table considering 3σ confidence level for all the cases that we have considered here around their true values.

IV. CONCLUSION
In this paper we present an elaborate discussion on the capability of DUNE experiment to test the consequences of µ − τ reflection symmetry considering two different modes namely, (i) 7-years of neutrino run and (ii) 3.5-years each of neutrino and antineutrino run. In addition, to realize µ − τ reflection symmetry in the low energy neutrino mass matrix under minimal type -I seesaw formalism, we add two heavy right-handed neutrino fields in the SM. This symmetry predicts maximal atmospheric mixing angle (i.e., θ 23 = 45 • ) and Dirac CP phase (i.e., δ = ±90 • ) along with trivial Majorana phases in the leptonic sector. In this framework, we also find remaining oscillation parameters both analytically as well as numerically. Later, considering numerical best-fit values of neutrino oscillation parameters as our true benchmark point, we find the allowed area in (δ − sin 2 θ 23 ) plane for DUNE.
Further, as the latest global best-fit data prefer non-maximal δ as well as θ 23 , we perform our study considering global best-fit values as one of our true benchmark point in the context of DUNE. Subsequently, we extend our study to break µ − τ reflection symmetry by introducing explicit breaking term in the high energy Dirac and Majorana neutrino mass matrices, respectively. Given the breaking scenario, we calculate the set of neutrino oscillation parameters and considering this set as true benchmark point we find the allowed area in test (δ − sin 2 θ 23 ) plane for DUNE. It is noteworthy to make a note here that allowed parameter space in test (δ −sin 2 θ 23 ) plane also gives an idea about the precision of these two poorly determined parameters for DUNE. Later, we examine the potential of DUNE to rule out maximal CP-violation (CPV) or CP-conservation hypothesis in each broken scenario.
We summarize DUNE's capability to test concerned hypotheses for all considered cases in Afterwards, we also examine the precision of both the less known parameters, δ, θ 23 and as a case study we present our results at 3σ confidence level in Finally, we conclude this work with a remark that with the available data in the neutrino oscillation sector, µ − τ reflection symmetry possesses as one of the finest theoretically favored approach to study some intriguing aspects of neutrinos. On the other hand forthcoming experiment, like DUNE with its high statistics and ability to measure (δ, θ 23 ) with high precisions serves as an impeccable experiment to test numerous predictions of different models.