Exploring Partial $\mu$-$\tau$ Reflection Symmetry at DUNE and Hyper-Kamiokande

We study origin, consequences and testability of a hypothesis of `partial $\mu$-$\tau$' reflection symmetry. This symmetry predicts $ |U_{\mu i}|=|U_{\tau i}|~(i=1,2,3) $ for a single column of the leptonic mixing matrix $U$. Depending on whether this symmetry holds for the first or second column of $U$ different correlations between $\theta_{23}$ and $ \delta_{CP} $ can be obtained. This symmetry can be obtained using discrete flavour symmetries. In particular, all the subgroups of SU(3) with 3-dimensional irreducible representation which are classified as class C or D can lead to partial $\mu$-$\tau$ reflection symmetry. We show how the predictions of this symmetry compare with the allowed area in the $\sin^2\theta_{23} - \delta_{CP}$ plane as obtained from the global analysis of neutrino oscillation data. Furthermore, we study the possibility of testing these symmetries at the proposed DUNE and Hyper-Kamiokande (HK) experiments (T2HK, T2HKK), by incorporating the correlations between $\theta_{23}$ and $ \delta_{CP}$ predicted by the symmetries. We find that when simulated data of DUNE and HK is fitted with the symmetry predictions, the $\theta_{23}-\delta_{CP}$ parameter space gets largely restricted near the CP conserving values of $ \delta_{CP} $. Finally, we illustrate the capability of these experiments to distinguish between the two cases leading to partial $\mu-\tau$ symmetry namely $|U_{\mu1}| = |U_{\tau 1}|$ and $|U_{\mu 2}| = |U_{\tau 2}|$.


I. INTRODUCTION
Considerable theoretical and experimental efforts are being devoted towards predicting and determining the unknowns of the leptonic sectors namely CP violating phases, octant of the atmospheric mixing angle θ 23 (i.e. θ 23 < 45 • , named as lower octant (LO) or θ 23 > 45 • named as upper octant (HO)) and neutrino mass hierarchy (i.e. the sign of ∆m 2 31 , ∆m 2 31 > 0 known as normal hierarchy (NH) and ∆m 2 31 < 0 known as inverted hierarchy (IH)). Symmetry based approaches have been quite successful in predicting the interrelations among these quantities and the structure of the leptonic mixing matrix as discussed in Refs. [1][2][3][4][5] and the references therein. General approaches along this line assume some individual residual symmetries of the leptonic mass matrices which could arise from the breaking of some bigger symmetry of the leptonic interactions. One such symmetry, called µ-τ reflection symmetry,originally discussed by Harrison and Scott in Ref. [6] leads to very successful predictions of mixing angles which are close to the present experimental knowledge and . This symmetry may be stated as equality of moduli of the leptonic mixing matrix U : for all the columns i = 1, 2, 3. Both the origin and consequences of this relation have been discussed in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].
This relation thus allows for a nonzero θ 13 unlike the simple µ-τ symmetry which predicts vanishing θ 13 [24][25][26][27][28][29], see recent review [30] and references therein. For θ 13 = 0, one gets δ CP = ± π 2 . Both these predictions are in accord with observations. The maximal θ 23 is allowed within 1σ by the global fits to neutrino observables and δ CP = − π 2 is preferred by the T2K experiment [31] and global fit of all neutrino data [32][33][34]. However a sizable range is still allowed. Whereas, the best fit value of θ 23 in the global fit deviates from the maximal value for either mass hierarchy. Such deviations can be regarded as a signal for the departure from the µ-τ reflection symmetry. A theoretically well-motivated possibility is to assume a 'partial µ-τ ' reflection symmetry [35] and assume that eq.(1) holds only for a single column 1 of U . Assuming that it holds for the third column, one gets maximal θ 23 and δ CP remains unrestricted. Eq.(1) on the other hand predicts correlations among δ CP and mixing angles if it is true for any of the first two columns. These correlations are found from eq.(2) in respective cases i = 1 and i = 2 to be These equations correlate the sign of cos δ CP to the octant of θ 23 . θ 23 in the first (second) octant leads to a negative (positive) value of cos δ CP in case of eq. (4). It predicts exactly opposite behavior for eq. (5). The exact quadrant of δ is still not fixed by these equations but it can also be determined from symmetry considerations [20]. These correlations were also obtained in [36,37] in the context of Z 2 and Z 2 symmetries 2 . Henceforth we refer to these correlations as C 1 and C 2 respectively. The above equations also indirectly lead to information on the neutrino mass hierarchy since the best fit values of θ 23 lie in the first (second) octant in case of the normal (inverted) hierarchy according to the latest global fits reported in [32][33][34]. Thus precise verification of the above equations is of considerable importance and the long baseline experiments can provide a way for such study. Similar study has been performed in the context of the NOνA and T2K experiments in [39][40][41].
In this paper, we consider the testability of these relations at the forthcoming long baseline experiments Deep Under-ground Neutrino Experiment (DUNE) and Hyper-Kamiokande (HK). These potential high-statistics experiments will overcome the parameter degeneracies faced by the current experiments and lead us in to an era of precision measurements of the oscillation parameters [42][43][44][45][46][47][48][49][50][51]. Because of this, these experiments are ideal to test the parameter correlations like the ones given in eqs. (4,5). In the following, we obtain the allowed parameter range in the δ CP -sin 2 θ 23 plane by fitting the above symmetry relations to the simulated DUNE and HK data. This in turn implies using the correlations embodied in the eqs. (4,5) in the fit. We also discuss whether the correlations C 1 (eq. (4)) and C 2 (eq. (5)) can be distinguished at DUNE and HK. Recent studies on testing various models from future experiments can be found for instance in [52][53][54][55][56][57][58][59][60].
We begin by first discussing the origin of partial µ-τ reflection symmetry and discuss the robustness of the resulting predictions in a large class of models based on flavour symmetry in Section II. We give a brief overview of the experiments and simulation details in Section III. In Section IV, we perform a phenomenological analysis of the testability of the above symmetries in DUNE and HK. We use the extra correlations predicted by the symmetry in fitting the simulated data of these experiments and obtain the allowed areas in the δ CP − sin 2 θ 23 plane. In subsection IV B, we discuss the possibility of differentiating between the two correlations -C 1 and C 2 . We draw our conclusions in Section V.

II. PARTIAL µ-τ REFLECTION SYMMETRY AND DISCRETE FLAVOUR SYMMETRIES
We briefly review here the general approach based on flavour symmetry to emphasize that partial µ-τ reflection symmetry is a generic prediction of almost all such schemes barring few exceptions. Basic approaches assume groups G ν and G l as the residual symmetries of the neutrino mass matrix M ν and the charged lepton mass matrix M l M † l respectively. Both these groups are assumed to arise from the breaking of some unitary discrete group G f . The U PMNS matrix U gets fixed upto the neutrino Majorana phases if it is further assumed that G ν = Z 2 × Z 2 and G l = Z n , n ≥ 3. In addition, if we demand that all the predicted mixing angles are non-zero, then the following unique form is predicted for almost all the discrete groups G f [61,62] where θ n ≡ πa n is a discrete angle with a = 0, 1, 2.... n 2 . We have not shown here the unphysical phases which can be absorbed in defining charged lepton fields and unpredicted Majorana phases. All the discrete subgroups of SU (3) with three dimensional irreducible representation are classified as class C or D and five exceptional groups [63]. Eq.(6) follows in all the type D groups taken as G f . Type C groups lead instead to democratic mixing which shows full µ-τ reflection symmetry but predict large reactor angle. Eq.(6) arises even if G f is chosen as a discrete subgroup of U (3) having the same textures as class D groups [62].
Eq.(6) displays partial µ-τ reflection symmetry for the second column for all the values of θ n = 0, π 2 . In the latter case, one gets total µ-τ reflection symmetry but at the same time one of the mixing angles is predicted to be zero and one would need to break the assumed residual symmetries to get the correct mixing angles. More importantly, eq.(6) being essentially a real matrix also predicts trivial Dirac CP phase δ CP = 0 or π. Eq.(5) in this case implies a correlation among angles. Non-zero CP phase and partial µ-τ symmetry in other columns can arise in an alternative but less predictive approach in which the residual symmetry of the neutrino mass matrix is taken as Z 2 instead of Z 2 × Z 2 . In this case, one can obtain the following mixing matrix U with a proper choice of residual symmetries.
where U ij denotes a unitary rotation either in the ij th plane corresponding to partial symmetry in the k th i = j = k column. Examples of the required residual symmetries are discussed in [1][2][3][4][5] and minimal example of this occurs with G f = S 4 . The partial µ-τ symmetries obtained this way also lead to additional restrictions where the first (second) relation follows from the partial symmetries of the first(second) column. These predictions arise here from the requirement that G ν and G l are embedded in DSG of SU (3) and need not arise in a more general approach. It is then possible to obtain specific symmetries [36,37] in which solar angle is a function of a continuous parameter. Eq.(4) (Eq. (5)). The solid(dashed) red curves represent the 3σ allowed parameter space as obtained by the global analysis of data by the Nu-fit collaboration [34,64] considering hierarchy to be NH(IH) respectively. Fig. 1, shows the correlation plots between sin 2 θ 23 and δ CP as given by eqs. (4,5). These equations give two values of CP phase (namely, δ CP and 360 • − δ CP ) for each value of θ 23 except for δ CP ≡ 180 • . The width of the lines are due to the uncertainty of the angles θ 12 and θ 13 subject to the conditions s 2 12 c 2 13 = 1/3 and c 2 12 c 2 13 = 2/3 corresponding to eq. (4) and eq. (5) respectively. It is seen that the correlation between sin 2 θ 23 and δ CP is opposite in the class of symmetries that give eq. (4) vis-a-vis those that give eq. (5). The parameters, sin 2 θ 23 and δ CP are correlated between 0 • − 180 • and anti-correlated between 180 • − 360 • for eq. (4). The opposite is true for eq. (5). We also notice here that eq. (5) rules out regions around CP conserving (i.e. 0 • , 180 • , 360 • ) values. The red solid(dashed) contours represent the 3σ allowed region for NH(IH) as obtained from the global-fit data by the Nu-fit collaboration [64]. We observe that at 3σ some of the allowed regions of sin 2 θ 23 and δ CP as predicted by the symmetries are disfavoured by the current global-fit data. From the global-fit data we observe that the region 39 • < δ CP < 125 • is completely ruled out at 3σ for NH and the region δ CP < 195 • for IH. The symmetry predictions can further constrain the values of δ CP presently allowed by the global data.
In the next section, we study how far the allowed areas in δ CP − sin 2 θ 23 plane can be restricted if the simulated experimental data confronts the symmetry predictions.

III. SPECIFICATIONS OF THE EXPERIMENTS
In this paper, we have simulated all the experiments using the GLoBES package [65,66] along with the required auxiliary files [67,68]. We have considered the experimental set-up and the detector performance of DUNE and HK in accordance with ref. [69] and ref. [70] respectively.
• Deep Underground Neutrino Experiment (DUNE) : DUNE is a Fermilab based next generation long baseline superbeam experiment. This experiment will utilize the existing NuMI (Neutrinos at the Main Injector) beamline design. In this experiment, the muon-neutrinos from Fermilab will travel a baseline of 1300 km before it gets detected at the far detector situated at the "Sanford Underground Research Facility (SURF)" in Lead, South Dakota. The proposed far detector for DUNE is a LArTPC (liquid argon time-projection chamber) detector with the volume of 40 kT. The beam power will be initially 1.2 MW and later will be increased to 2.3 MW [71]. In our simulation we consider the neutrino flux [72] corresponding to 1.2 MW beam power which gives 1 × 10 21 protons on target (POT) per year. This corresponds to a proton energy of 120 GeV.

IV. PHENOMENOLOGICAL ANALYSIS
In this section, we perform a phenomenological analysis exploring the possibility of probing the correlations C 1 and C 2 at DUNE, T2HK and T2HKK. This is discussed in terms of correlation plots in sin 2 θ 23 −δ CP plane. We also discuss the possibility of distinguishing between the two models at these experiments. We perform a χ 2 test with χ 2 defined as, We assume Poisson distribution to calculate the statistical χ 2 stat , Here, 'i' refers to the number of bins and N test i , N true i are the total number of events due to test and true set of oscillation parameters respectively. In Table I, we list the values for the neutrino oscillation parameters that we have used in our numerical simulation. These values are consistent with the results obtained from global-fit of world neutrino data [32][33][34]. The systematic errors are taken into account using the method of pulls [73,74] as outlined in [75]. The systematic uncertainties (normalization errors) and efficiencies corresponding to signals and backgrounds of DUNE and HK are taken from [42,70]. For DUNE the signal normalization uncertainties on ν e /ν e and ν µ /ν µ are considered to be 2% and 5% respectively. While a range of 5% to 20% background uncertainty along with the correlations among their sources have also been included. On the other hand, for T2HK and T2HKK the signal normalization error on ν e (ν e ) and ν µ (ν µ ) are considered to be 3.2% (3.9%) and 3.6%(3.6%) respectively. The background normalization uncertainties range from 3.8% to 5%. Additionally, we have added 5% prior on sin 2 2θ 13 in our numerical simulation.
A. Testing the sin 2 θ23 − δCP correlation predicted by the symmetries at DUNE, T2HK and T2HKK For this analysis we simulate the experimental data by considering the true values of oscillation parameters given in Table I. For each true combination, in the theoretical fit we marginalize over sin 2 θ 13 , ∆m 2 31 and sin 2 θ 23 in the range given in Table I. However, the δ CP values in the fit are taken according to the predictions of symmetries. We consider both NH and IH separately in our analysis. Thus, the resultant plots in Fig. (2,3) show the extent to which these three experiments can test the correlations between the two yet undetermined variables sin 2 θ 23 and δ CP in conjunction with the symmetry predictions. The blue, grey and the yellow bands in the fig. (2,3) represent 1σ, 2σ, 3σ regions in the sin 2 θ 23 − δ CP plane respectively and the red contours show the 3σ allowed area obtained by the Nu-fit collaboration [34,64]. The topmost panel corresponds to DUNE -40 kT detector whereas the middle and the lowest panels correspond to T2HK and T2HKK experiments respectively. The left plots in all the rows are for testing C 1 whereas the right plots are for testing C 2 .
The figures show that because of the correlations predicted by symmetries, certain combinations of the true θ 23 and δ CP values get excluded by DUNE, T2HK and T2HKK. Owing to their high sensitivity to determine CP violation, T2HK and T2HKK constrain the range of δ CP better than that of DUNE. This can be seen from the figure (fig. 2,3) which shows that, as we go from top to bottom the contours gets thinner w.r.t δ CP . For instance for the C 1 correlation, the CP conserving values 0 and 360 • get excluded at 3σ for both the octants by all the three experiments as can be seen from the plots in the left panels. However, for the C 2 , these values are allowed at 3σ for all the three experiments. Whereas, δ CP = 0 • and 360 • are excluded by DUNE and HK experiments at 1σ and 2σ respectively. Again one can see from the right panels that for C 2 , δ CP = 180 • is allowed for sin 2 θ 23 > 0.55 (i.e. higher octant) by DUNE but gets barely excluded at 2σ by T2HK and T2HKK experiments. The correlations predicted by the symmetry considerations being independent of hierarchy, the allowed regions are not very different for NH and IH. But the region of parameter space allowed by current data for IH is more constrained and the symmetry predictions restrict it further. Some of the parameter space allowed by the current data can also be disfavoured by incorporating the correlations due to symmetry relations.

B. Differentiating between the C1 and C2 symmetries
In this subsection, we explore the possibility of differentiation between the C 1 and C 2 . This is presented in Fig. (4) where we plot ∆χ 2 vs true θ 23 . To find χ 2 stat (eq.(10)), true events are calculated by spanning the true values of θ 23 in the range (39 • −51 • ) with marginalization over sin 2 θ 12 and sin 2 θ 13 , and the true δ CP values are calculated using the C 1 , provided the conditions given in eqn. (8) are satisfied. The remaining oscillation parameters are kept fixed at their best-fit values as shown in Table (I). Thus two sets of true events are generated corresponding to δ CP and 360 • − δ CP . In the theoretical fit, to calculate test events, we marginalize over sin 2 θ 12 , sin 2 θ 13 , ∆m 2 31 , sin 2 θ 23 in the range given in Table (I) and test δ CP values are calculated using the C 2 . Here we are presenting the analysis considering the true hierarchy as NH, analysis was also done with true hierarchy as IH and we obtained similar results. The three panels from left to right represent DUNE, T2HK and T2HKK respectively. The solid blue curves in the plots are for predicted range δ CP ∈ (0 < δ CP < 180 • ) and the dashed blue curves in the plots are for complementary range 360 • − δ CP ∈ (180 • < δ CP < 360 • ) as predicted by the correlations. Both plots are with known normal hierarchy. The brown solid lines show the 3σ line for correlation differentiation. We observe from the figure that at maximal θ 23 both correlations are indistinguishable by all the three experiments as is expected from the equations (4,5). These equations show that at maximal θ 23 both the correlations predict maximal δ CP value. The capability of the experiments to differentiate between the two correlations increases as we move away from maximal value. The range of θ 23 for which the three experiments  [34,64] for Normal Hierarchy. The blue, gray and yellow shaded contours correspond to 1σ, 2σ, 3σ respectively.
can differentiate between the correlations at 3σ is given in table II. The lower limits signify the values of θ 23 below which the correlations can be differentiated at 3σ and the upper limits is for the values above which the same can be achieved. The left(right) panels represent the predictions from the symmetry relations C 1 (C 2 ) which corresponds to the equations 4 (5) respectively. The hierarchy is fixed as IH. The red contour in each panel represent the 3σ allowed area from the global analysis of neutrino oscillation data as obtained by the Nu-fit collaboration [34,64] for Inverted Hierarchy. The blue, gray and yellow shaded contours correspond to 1σ, 2σ, 3σ respectively.

V. CONCLUSION
We study here partial µ − τ reflection symmetry of the leptonic mixing matrix, U , which can arise from discrete flavour symmetry. Specific assumptions which lead to this symmetry were reviewed here. This symmetry implies |U µi | = |U τ i | (i = 1, 2, 3) for a single column of the leptonic mixing matrix U . If this is true for the third column of U then it leads to maximal value of the atmospheric mixing angle and CP phase δ CP . However, if this is true for the first or the second column then one obtains   definite correlations among θ 23 and δ CP . We call these scenarios C 1 (equality for the first column) and C 2 (equality of the 2nd column). We find that almost all the discrete subgroups of SU (3), except a few exceptional cases, having three dimensional irreducible representations display the form of partial µ − τ symmetry. We study the correlations among θ 23 and δ CP in the two scenarios. Each scenario gives two values of δ CP for a given θ 23 -one belonging to 0 < δ CP < 180 • and the other belonging to 180 • < δ CP < 360 • . The models also give specific correlations between θ 23 and δ CP and these are opposite for C 1 and C 2 .
We study how the allowed areas in the sin 2 θ 23 − δ CP plane obtained by the global analysis of neutrino oscillation data from the Nu-Fit collaboration compare with the predictions from the symmetries. We also expound the testability of these symmetries considering next generation accelerator based experiments, DUNE and Hyper-Kamiokande. This is illustrated in terms of plots in the sin 2 θ 23 (true) -δ CP (true) plane obtained by fitting the simulated experimental data with the symmetry predictions for δ CP . The values of θ 23 are found to be more constrained for the CP conserving values namely δ CP = 0, 180 • , 360 • . For the C 2 correlation, the θ 23 is found to be in the higher octant for δ CP = 180 • and in the lower octant for δ CP = 0 and 360 • . For the correlation C 1 , values of δ CP around all the three CP conserving values δ CP = 0, 180 • and 360 • are seen to be disfavoured. Finally, we illustrate the capability of DUNE and Hyper-Kamiokande to distinguish between the predictions of the two correlations. We observe that both the experiments can better differentiate between these two as one moves away from the maximal θ 23 value.
In conclusion, the future experiments provide testing grounds for various symmetry relations, specially those connecting θ 23 and δ CP . work.