Addressing neutrino mixing models with DUNE and T2HK
Abstract
We consider schemes of neutrino mixing arising within the discrete symmetry approach to the wellknown flavour problem. We concentrate on \(3\nu \) mixing schemes in which the cosine of the Dirac CP violation phase \(\delta _{\mathrm {CP}}\) satisfies a sum rule by which it is expressed in terms of three neutrino mixing angles \(\theta _{12}\), \(\theta _{23}\), and \(\theta _{13}\), and a fixed real angle \(\theta ^\nu _{12}\), whose value depends on the employed discrete symmetry and its breaking. We consider five underlying symmetry forms of the neutrino mixing matrix: bimaximal (BM), tribimaximal (TBM), golden ratio A (GRA) and B (GRB), and hexagonal (HG). For each symmetry form, the sum rule yields specific prediction for \(\cos \delta _{\mathrm {CP}}\) for fixed \(\theta _{12}\), \(\theta _{23}\), and \(\theta _{13}\). In the context of the proposed DUNE and T2HK facilities, we study (i) the compatibility of these predictions with present neutrino oscillation data, and (ii) the potential of these experiments to discriminate between various symmetry forms.
1 Introduction and motivation
The best fit values and \(1\sigma \), \(2\sigma \), 3\(\sigma \) ranges of the neutrino oscillation parameters obtained in the global analysis of the neutrino oscillation data performed in [3]. NO (IO) stands for normal (inverted) ordering of the neutrino mass spectrum
Parameter  Best fit  \(1\sigma \) range  \(2\sigma \) range  \(3\sigma \) range 

\(\dfrac{\sin ^2\theta _{12}}{10^{1}}\)  2.97  2.81–3.14  2.65–3.34  2.50–3.54 
\(\dfrac{\sin ^2\theta _{13}}{10^{2}}\) (NO)  2.15  2.08–2.22  1.99–2.31  1.90–2.40 
\(\dfrac{\sin ^2\theta _{13}}{10^{2}}\) (IO)  2.16  2.07–2.24  1.98–2.33  1.90–2.42 
\(\dfrac{\sin ^2\theta _{23}}{10^{1}}\) (NO)  4.25  4.10–4.46  3.95–4.70  3.81–6.15 
\(\dfrac{\sin ^2\theta _{23}}{10^{1}}\) (IO)  5.89  4.17–4.48 \(\oplus \) 5.67–6.05  3.99–4.83 \(\oplus \) 5.33–6.21  3.84–6.36 
\(\delta _{\mathrm {CP}}\) [\(^{\circ }\)] (NO)  248  212–290  180–342  0–31 \(\oplus \) 137–360 
\(\delta _{\mathrm {CP}}\) [\(^{\circ }\)] (IO)  236  202–292  166–338  0–27 \(\oplus \) 124–360 
\(\dfrac{\Delta m_{21}^{2}}{10^{5} \mathrm {eV}^2}\)  7.37  7.21–7.54  7.07–7.73  6.93–7.96 
\(\dfrac{\Delta m_{31}^{2}}{10^{3} \mathrm {eV}^2}\) (NO)  2.56  2.53–2.60  2.49–2.64  2.45–2.69 
\(\dfrac{\Delta m_{23}^{2}}{10^{3} \mathrm {eV}^2}\) (IO)  2.54  2.51–2.58  2.47–2.62  2.42–2.66 
Understanding the origin of the patterns of neutrino oscillation parameters revealed by the data is one of the most challenging problems in neutrino physics. It is a part of the more general fundamental problem in particle physics of understanding the origins of flavour, i.e., the patterns of quark, charged lepton, and neutrino masses, and quark and lepton mixing. There exists a possibility that the highprecision measurements of the oscillation parameters may shed light on the origin of the observed pattern of neutrino mixing and lepton flavour. This would be the case if the observed form of neutrino (and possibly quark) mixing were determined by an underlying discrete flavour symmetry. One of the most striking features of the discrete symmetry approach to neutrino mixing and lepton flavour (see, e.g., [5, 6, 7] for reviews), is that it leads to (i) fixed predictions of the values of some of the neutrino mixing angles and the Dirac CPV phase \(\delta _{\mathrm {CP}}\), and/or (ii) existence of correlations between some of the mixing angles and/or between the mixing angles and \(\delta _{\mathrm {CP}}\) (see, e.g., [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]). These correlations are often referred to as neutrino mixing sum rules.^{2} Most importantly, these sum rules can be tested using oscillation data [8, 9, 10, 11, 15, 19, 29, 30, 31].
Within the discrete flavour symmetry approach, the PMNS matrix is predicted to have an underlying symmetry form, where \(\theta _{12}\), \(\theta _{23}\), and \(\theta _{13}\) have values which differ by subleading perturbative corrections from their respective measured values. The approach seems very natural in view of the fact that \(U_{\mathrm{PMNS}} = U_e^{\dagger } U_{\nu }\), where \(U_{e}\) and \(U_{\nu }\) are \(3\times 3\) unitary matrices which diagonalise the charged lepton and neutrino mass matrices. Typically (but not universally) the matrix \(U_{\nu }\) has a certain symmetry form, while the matrix \(U_{e}\) provides the corrections necessary to bring the symmetry values of the angles in \(U_{\nu }\) to their experimentally measured values. A sum rule which relates \(\cos \delta _{\mathrm {CP}}\) with \(\theta _{12}\), \(\theta _{23}\), and \(\theta _{13}\), arising in this approach, depends on the underlying symmetry form of the PMNS matrix and on the form of the “correcting” matrix \(U_{e}\).
The best fit values of \(\cos \delta _{\mathrm {CP}}\) and \(\delta _{\mathrm {CP}}\) from the sum rule in Eq. (1.1) for the different symmetry forms. The mixing angles \(\theta _{12}\), \(\theta _{23}\), and \(\theta _{13}\) have been fixed to their NO best fit values from Table 1. The \(\phi \) stands for the golden ratio: \(\phi = (1+\sqrt{5})/2\). See text for further details
Symmetry form  \(\theta ^\nu _{12}\) [\(^{\circ }\)]  \(\cos \delta _{\mathrm {CP}}\)  \(\delta _{\mathrm {CP}}\) [\(^{\circ }\)] 

BM  45  Unphysical  Unphysical 
TBM  \(\arcsin (1/\sqrt{3})\approx 35\)  \(0.16\)  \(99 \vee 261\) 
GRA  \(\arctan (1/\phi )\approx 32\)  0.21  \(78 \vee 282\) 
GRB  \(\arccos (\phi /2)=36\)  \(0.24\)  \(104 \vee 256\) 
HG  30  0.39  \(67 \vee 293\) 
The intervals for \(\delta _{\mathrm {CP}}\) due to the present \(3\sigma \) uncertainties in the values of the neutrino mixing angles. The quoted intervals are obtained varying one mixing angle in its corresponding \(3\sigma \) range for the NO spectrum and fixing the other two angles to their NO best fit values
Symmetry form  Intervals for \(\delta _{\mathrm {CP}}\) [\(^{\circ }\)] obtained varying  

\(\theta _{12}\) in \(3\sigma \)  \(\theta _{23}\) in \(3\sigma \)  \(\theta _{13}\) in \(3\sigma \)  
BM  150–180 \(\vee \) 180–210  Unphysical  Unphysical 
TBM  79–119 \(\vee \) 241–281  98–107 \(\vee \) 253–262  98–101 \(\vee \) 259–262 
GRA  57–95 \(\vee \) 265–303  76–78 \(\vee \) 282–284  77.6–77.9 \(\vee \) 282.1–282.4 
GRB  84–125 \(\vee \) 235–276  102–114 \(\vee \) 246–258  103–106 \(\vee \) 254–257 
HG  45–84 \(\vee \) 276–315  60–68 \(\vee \) 292–300  66–68 \(\vee \) 292–294 
In the case of the BM symmetry form, the obtained best fit value of \(\cos \delta _{\mathrm {CP}}= 1.26\) is unphysical. This reflects the fact that the BM symmetry form does not provide a good description of the present best fit values of the neutrino mixing angles, as discussed in [13]. As can be seen from Table 3, current uncertainties on the mixing angles allow us to accommodate physical values of \(\cos \delta _{\mathrm {CP}}\) for the BM symmetry form. For instance, fixing \(\sin ^2\theta _{13}\) and \(\sin ^2\theta _{23}\) to their best fit values, \(\cos \delta _{\mathrm {CP}}= 1\) requires \(\sin ^2\theta _{12} = 0.3343\), which is the upper bound of the corresponding \(2\sigma \) allowed range of \(\sin ^2\theta _{12}\) (see Table 1).
A rather detailed analysis of the predictions for \(\cos \delta _{\mathrm {CP}}\) of the sum rule in Eq. (1.1) has been performed in Refs. [9, 10]. In particular, likelihood profiles for \(\cos \delta _{\mathrm {CP}}\) for each symmetry form have been presented using the current and prospective precision on the neutrino mixing parameters (see Figs. 12 and 13 in [9]). In the present work, using the potential of the future longbaseline (LBL) neutrino oscillation experiments,^{3} namely, Deep Underground Neutrino Experiment (DUNE) and Tokai to HyperKamiokande (T2HK), we study in detail (i) to what degree the sum rule predictions for \(\cos \delta _{\mathrm {CP}}\) are compatible with the present neutrino oscillation data, and (ii) how well the considered symmetry forms, BM, TBM, GRA, GRB, and HG, can be discriminated from each other.
The layout of the article is as follows. In Sect. 2, we take a first glance at the sum rule predictions. In Sect. 3, we give a short description of the planned DUNE and T2HK experiments and provide expected event rates for both the setups. Section 4 contains details of the statistical analysis. In Sect. 5, we present and discuss results of this analysis. More specifically, in Sect. 5.1, we test the compatibility between the considered symmetry forms and present oscillation data. Next, in Sect. 5.2, we explore the potential of DUNE, T2HK, and their combination to distinguish between the symmetry forms in question under the assumption that one of them is realised in Nature. In Sect. 5.3, we consider the BM symmetry form using the values of the mixing angles for which this form is viable, and study at which C.L. it can be distinguished from the other symmetry forms considered. We conclude in Sect. 6. Appendix A discusses the issue of external priors on \(\sin ^2\theta _{12}\) and \(\sin ^2\theta _{13}\). In Appendix B, we show the impact of marginalisation over \(\Delta m_{31}^{2}\). Finally, in Appendix C, we study the compatibility of the considered symmetry forms with any potentially true values of \(\sin ^2\theta _{23}\) and \(\delta _{\mathrm {CP}}\) in the context of DUNE and T2HK.
2 A first glance at the sum rule predictions

The parameter \(\theta ^\nu _{12}\) has a fixed value for each symmetry form as given in Table 2. For fixed choices of \(\theta _{12}\), \(\theta _{23}\), \(\theta _{13}\), and \(\theta ^\nu _{12}\), Eq. (1.1) predicts a certain value of \(\cos \delta _{\mathrm {CP}}\), which gives rise to two values of \(\delta _{\mathrm {CP}}\) (see fourth column of Table 2).

The complementarity [52] between the modern reactor (Daya Bay, RENO, and Double Chooz) and accelerator (T2K and NO\(\nu \)A) data has enabled us to probe the parameter space for \(\delta _{\mathrm {CP}}\). Already, the latest global data have disfavoured values of \(\delta _{\mathrm {CP}}\in (31^{\circ },137^{\circ })\) at \(3\sigma \) C.L. for NO (see Table 1). From Table 3, we see that out of the three mixing angles, the \(3\sigma \) allowed range of \(\theta _{12}\) causes the largest uncertainty in \(\delta _{\mathrm {CP}}\) predicted by Eq. (1.1). But, note that, all the symmetry forms except BM predict one of the ranges of \(\delta _{\mathrm {CP}}\) in the interval of \(31^{\circ }\)–\(137^{\circ }\), which has been already ruled out at \(3\sigma \). Therefore, we will not consider these ranges further in our study, and we will only consider the values of \(\delta _{\mathrm {CP}}\) in the interval of \(180^{\circ }\) to \(360^{\circ }\).

Further, we notice from Tables 1 and 3 that for TBM and GRB, the predicted intervals of \(\delta _{\mathrm {CP}}\) lie within the \(1\sigma \) experimentally allowed range. For BM, GRA, and HG, the intervals of interest fall within the \(2\sigma \) range. Now, it would be quite interesting to assess the sensitivity of the future LBL experiments in discriminating various symmetry forms, which is the main thrust of the present work.

T2HK and DUNE will not be able to constrain \(\theta _{12}\), which causes the largest uncertainty in predicting the range of \(\delta _{\mathrm {CP}}\) (see Table 3). However, the proposed JUNO experiment will provide a highprecision measurement of \(\sin ^2\theta _{12}\) with a relative \(1\sigma \) uncertainty of 0.7%. Therefore, we impose a prior on \(\sin ^2\theta _{12}\) expected from JUNO, which we will discuss later in detail in Sect. 4 and in Appendix A.

Table 3 shows that the sum rule predictions depend to some extent on \(\theta _{23}\). Therefore, we vary this angle both in data and in fit. The LBL experiments themselves are sensitive to \(\theta _{23}\). Thus, we do not impose any external prior on this angle (see details in Sect. 4).

The experiments under discussion are sensitive to \(\theta _{13}\) through the appearance channels \({\nu _{\mu }}\rightarrow {\nu _e}\) and \({\bar{\nu }_{\mu }}\rightarrow {\bar{\nu }_e}\). Therefore, the role of an external prior on \(\sin ^2\theta _{13}\) is negligible for the physics case under study, as we will see later in Fig. 4. However, we put a prior on \(\sin ^2\theta _{13}\) as expected from Daya Bay to speed up our simulations (see Sect. 4 for details).
3 Experimental features and event rates
Total signal and background event rates for DUNE and T2HK setups assuming NO, \(\delta _{\mathrm {CP}}= 248^{\circ }\), and \(\sin ^2\theta _{23}\) = 0.425. For all other oscillation parameters, we take the best fit values corresponding to NO (see Table 1). Here “Int” means intrinsic beam contamination, “Misid” represents misidentified muon events, and “NC” stands for neutral current. See text for other details
Mode (Channel)  DUNE (248 \({\mathrm {kt} \cdot \mathrm {MW} \cdot \mathrm {year}}\))  T2HK (4200 \({\mathrm {kt} \cdot \mathrm {MW} \cdot \mathrm {year}}\))  

Signal  Background  Signal  Background  
CC  Int+Misid+NC=Total  CC  Int+Misid+NC=Total  
\(\nu \) (appearance)  614  125+29+24=178  2852  530+13+173=716 
\(\nu \) (disappearance)  5040  0+0+24=24  20024  12+44+1003=1059 
\({\bar{\nu }}\) (appearance)  60  43+10+7=60  1383  627+11+265=903 
\({\bar{\nu }}\) (disappearance)  1807  0+0+7=7  27447  14+5+1287=1306 
3.1 The next generation experiments: DUNE and T2HK
The planned Deep Underground Neutrino Experiment (DUNE) aims to achieve new milestones in the intensity frontier with a new, highintensity, onaxis, wideband neutrino beam from Fermilab directed towards a massive liquid argon timeprojection chamber (LArTPC) far detector housed at the Homestake Mine in South Dakota over a baseline of 1300 km [53, 54, 55, 56, 57]. In our simulation, we consider a fiducial mass of 35 kt for the far detector and the detector characteristics have been taken from Table 1 of Ref. [58]. As far as beam specifications are concerned, we assume a modest proton beam power of 708 kW in its initial phase with 120 GeV proton energy, which can supply \(6 \times 10^{20}\) protons on target (p.o.t.) in 188 days per calendar year. In our calculation, we have used the fluxes which were generated assuming a decay pipe length of 200 m and 200 kA horn current [59]. We assume that DUNE will collect data for ten years (5 years in \(\nu \) mode and 5 years in \({\bar{\nu }}\) mode), which is equivalent to a total exposure of 248 \(\mathrm {kt} \cdot \mathrm {MW} \cdot \mathrm {year}\).^{4} In our simulation, we consider the reconstructed \(\nu \) energy range to be 0.5 GeV to 10 GeV for both appearance and disappearance channels. We take the same energy range for antineutrino as well.
The proposed HyperKamiokande (HK) water Cherenkov detector will serve as the far detector of a longbaseline neutrino experiment using an upgraded neutrino beam from the JPARC facility, commonly known as “T2HK” (Tokai to HyperKamiokande) experiment [60, 61, 62]. This setup is highly sensitive to the Dirac CPV phase \(\delta _{\mathrm {CP}}\) of the PMNS \(3\nu \) mixing matrix and holds promise to resolve the mystery of leptonic CP violation in neutrino oscillations at an unprecedented confidence level [61]. We perform the simulation for T2HK according to Refs. [61, 62]. To produce an intense \(\nu \)/\({\bar{\nu }}\) beam for HK, we consider an integrated proton beam power of 7.5 MW \(\times \) \(10^7\) seconds, which can deliver in total \(15.6 \times 10^{21}\) p.o.t. with a 30 GeV proton beam. We assume that these total p.o.t. will be shared among \(\nu \) and \({\bar{\nu }}\) modes with a runtime ratio of 1:3 to have almost equal statistics in both the modes. The huge 560 kt (fiducial) HK detector will be placed in the Tochibora mine, at a distance of 295 km from JPARC at an offaxis angle of \(\sim 2.5^{\circ }\), which will produce a narrow band beam with a sharp peak around the first oscillation maximum of 0.6 GeV. The total exposure that we consider for T2HK is 4200 \(\mathrm {kt} \cdot \mathrm {MW} \cdot \mathrm {year}\). In our simulation, we take the reconstructed \(\nu _e\) and \({\bar{\nu }_e}\) energy range of 0.1 GeV to 1.25 GeV for the appearance channel. As far as the disappearance channel is concerned, the assumed energy range is 0.1 GeV to 7 GeV for both \({\nu _{\mu }}\) and \({\bar{\nu }_{\mu }}\) candidate events.
Recently, the baseline design for T2HK has been revised [63]. According to this latest publication [63], the total beam exposure is \(27 \times 10^{21}\) p.o.t. and the HK design proposes the construction of two identical water Cherenkov detectors in stage with fiducial mass of 187 kt per detector. The possibility of placing the first detector near the SuperKamiokande site, 295 km away and \(2.5^{\circ }\) offaxis from the JPARC neutrino beam and the second detector in Korea having a baseline of 1100 km from JPARC at an offaxis angle of \(\sim 2.5^{\circ }\) has also been explored in Ref. [63], and this setup has been referred as “T2HKK”. We follow the details as given in Ref. [63] to simulate the T2HKK setup.
3.2 Event rates
Table 4 shows a comparison between the expected total signal and background event rates^{6} in the appearance and disappearance modes for DUNE and T2HK setups. We compute the same for both neutrino and antineutrino runs assuming a total exposure of 248 \(\mathrm {kt} \cdot \mathrm {MW} \cdot \mathrm {year}\) for DUNE and 4200 \(\mathrm {kt} \cdot \mathrm {MW} \cdot \mathrm {year}\) for T2HK. We consider a runtime ratio of 1:1 among neutrino and antineutrino modes in DUNE and the corresponding ratio is 1:3 in T2HK. The total event rates are calculated assuming NO, \(\delta _{\mathrm {CP}}= 248^{\circ }\), and \(\sin ^2\theta _{23}\) = 0.425. For all other oscillation parameters, we consider the best fit values which are applicable for NO (see Table 1). To compute the full threeflavour neutrino oscillation probabilities in matter, we take the lineaveraged constant Earth matter density of 2.80 (2.87) g/cm\(^{3}\) for the T2HK (DUNE) baseline following the Preliminary Reference Earth Model (PREM) [68].
The main sources of backgrounds while selecting the \({\nu _e}\) and \({\bar{\nu }_e}\) candidate events are the intrinsic \({\nu _e}\)/\({\bar{\nu }_e}\) component in the beam, the muon events which will be misidentified as electron events, and the neutral current (NC) events. Table 4 clearly depicts that in case of appearance searches, the dominant background component is the intrinsic \({\nu _e}\)/\({\bar{\nu }_e}\) in the beam. For the \({\nu _{\mu }}\)/\({\bar{\nu }_{\mu }}\) candidate events, the main backgrounds are the NC events. Though we present the total event rates in Table 4, but, in our simulation, we have performed a full spectral analysis using the binned events spectra for both the DUNE and T2HK setups.
4 Details of statistical analysis
This section is devoted to describe the strategy that we adopt for the statistical treatment to quantify the sensitivities of DUNE and T2HK in testing various lepton mixing schemes. To produce our results, we take the help of the widely used GLoBES software [69, 70] which calculates the median sensitivity of the experiment without considering the statistical fluctuations. Unless stated otherwise, we generate our simulated data considering the best fit values of the oscillation parameters obtained in the global analysis assuming NO for the neutrino mass spectrum (see second column of Table 1). We also keep the choice of the neutrino mass ordering to be fixed to NO in the fit.^{7} The solar and atmospheric mass squared differences are already very well measured [2, 3, 4] and moreover, they do not appear in the sum rule (see Eq. (1.1)) that relates \(\cos \delta _{\mathrm {CP}}\) with the mixing angles. Therefore, we also keep them fixed in the fit at their best fit values while showing our results. Only in Appendix B, we give a plot, where we marginalise over test \(\Delta m_{31}^{2}\) in the fit in its present 3\(\sigma \) allowed range of \((2.45  2.69) \times 10^{3}\) eV\(^2\). The mixing angles play an important role in the sum rule and we treat them very carefully in our analysis. In the fit, we marginalise over test \(\sin ^2\theta _{23}\) in its present 3\(\sigma \) allowed range of 0.381 to 0.615. We show few results where we also vary the true value of \(\sin ^2\theta _{23}\) or marginalise over it in the same 3\(\sigma \) range. We do not impose any external prior on \(\sin ^2\theta _{23}\) as it will be directly measured by the DUNE and T2HK experiments. The sum rule as given in Eq. (1.1) is very sensitive to the value of \(\sin ^2\theta _{12}\). We vary this parameter in the fit in its present 3\(\sigma \) allowed range of 0.25 to 0.354. Since both DUNE and T2HK cannot constrain the solar mixing angle (see the probability expressions in [74]), we impose an external Gaußian prior of 0.7% (at 1\(\sigma \)) on this parameter as the proposed mediumbaseline reactor oscillation experiment JUNO will be able to measure \(\sin ^2\theta _{12}\) with this precision [75]. We also marginalise over test \(\sin ^2\theta _{13}\) in its present 3\(\sigma \) allowed range of 0.019 to 0.024. While doing so, we apply an external Gaußian prior of 3% (at 1\(\sigma \)) on this parameter expecting that the Daya Bay experiment would be able to achieve this precision by the end of its run [76]. Both DUNE and T2HK can measure \(\theta _{13}\) with high precision using \({\nu _{\mu }}\rightarrow {\nu _e}\) and \({\bar{\nu }_{\mu }}\rightarrow {\bar{\nu }_e}\) oscillation channels and therefore, the prior on \(\theta _{13}\) is not very crucial in our study (see Fig. 4 and related discussion in Appendix A). Still, we use this prior in our analysis to speed up the marginalisation procedure. For a given test choice of \(\theta _{12}\), \(\theta _{13}\), and \(\theta _{23}\), the test value of \(\delta _{\mathrm {CP}}\) is calculated using the sum rule (see Eq. (1.1)) for a particular choice of the lepton mixing scheme which is characterised by a certain value of \(\theta ^{\nu }_{12}\). Since the best fit value of \(\delta _{\mathrm {CP}}\) may change in the future, we also show some results varying the true choice of \(\delta _{\mathrm {CP}}\) in the range 180\(^{\circ }\) to 360\(^{\circ }\).
5 Results and discussion
5.1 Compatibility between various symmetry forms and present neutrino oscillation data

the black dot corresponding to the current best fit value of \(\delta _{\mathrm {CP}}= 248^{\circ }\) which translates to \(\sin ^2\theta ^\nu _{12} = 0.364\) (\(\Delta \chi ^2 = 0\));

the coloured dots corresponding to the values of \(\sin ^2\theta ^\nu _{12}\) which characterise the GRB (violet), TBM (red), GRA (blue) and HG (green) symmetry forms.
From this figure we see that, if the present best fit values of \(\theta _{12}\), \(\theta _{13}\), \(\theta _{23}\), and \(\delta _{\mathrm {CP}}\) were the true values of these parameters, the GRB (TBM) symmetry form would be compatible with them at slightly less (more) than \(1\sigma \) C.L., while the GRA and HG schemes would be disfavoured at more than \(2.7\sigma \) and \(3.7\sigma \), respectively, for both the combined setups.
However, at present the CPV phase \(\delta _{\mathrm {CP}}\) is not severely constrained, and as can be seen from Table 1, any value between \(180^{\circ }\) and \(342^{\circ }\) is allowed at \(2\sigma \) C.L., and any value except for the ones between \(31^{\circ }\) and \(137^{\circ }\) is allowed at \(3\sigma \). Fixing the three mixing angles to their best fit values, we find from Eq. (1.1) that the full range of \(\cos \delta _{\mathrm {CP}}\in [1,1]\) (allowed at present at \(3\sigma \)) translates to the values of \(\sin ^2\theta ^\nu _{12} \in [0.157, 0.460]\). Thus, in principle, any value from this range may turn out to be favoured in the future. For instance, imagine that in the future the best fit value of \(\delta _{\mathrm {CP}}\) will shift from \(248^{\circ }\) to \(290^{\circ }\), while the best fit values of the mixing angles will remain the same. Then, the value of \(\sin ^2\theta ^\nu _{12} = 0.250\), and thus the HG symmetry form, will be favoured. With this said, one should keep in mind that the position of the black dot in Fig. 1 is likely to change in the future, but having more precise measurements of \(\delta _{\mathrm {CP}}\) and the mixing angles at our disposal, we will be able to repeat this analysis favouring some symmetry forms and disfavouring the others.

For each symmetry form a significant part of the parameter space gets disfavoured at more than \(3\sigma \). Should the true value of \(\delta _{\mathrm {CP}}\) lie in this part of the parameter space, the corresponding symmetry form will be disfavoured at \(3\sigma \) confidence level.

Now we can see at which C.L. any given symmetry form is compatible with any potentially true value of \(\delta _{\mathrm {CP}}\). We just need to draw a vertical line at \(\delta _{\mathrm {CP}}^{\mathrm{true}}\) of interest. The points where it crosses the \(\Delta \chi ^2\) curves will provide the confidence levels at which the TBM, GRA, GRB, and HG forms are compatible with this \(\delta _{\mathrm {CP}}^{\mathrm{true}}\). In particular, for \(\delta _{\mathrm {CP}}^{\mathrm{true}} = 248^{\circ }\), we find numbers which correspond to those extracted from Fig. 1.^{9}
5.2 How well can DUNE and T2HK separate between various symmetry forms?
In this subsection, we will answer the question of how well DUNE and T2HK can distinguish the discussed symmetry forms under the assumption that one of them is realised in Nature. Given the fact that the BM form is not compatible with the current best fit values of the neutrino mixing angles, which we are going to use first in our analysis, we end up with four best fit values of interest. Namely, from Table 2, we read \(\delta _{\mathrm {CP}}= 256^{\circ }\), \(261^{\circ }\), \(282^{\circ }\), and \(293^{\circ }\) for the GRB, TBM, GRA, and HG symmetry forms, respectively. Assuming one of them to be the true value of \(\delta _{\mathrm {CP}}\), we will test the remaining three values against the assumed true value using DUNE, T2HK, and their combination. Overall, we have 12 pairs of the values we want to compare.
We start with DUNE. After performing a statistical analysis of simulated data, as described in Sect. 4, we obtain that for all the 12 cases \(\Delta \chi ^2\) does not exceed approximately 3.5. This value of \(\Delta \chi ^2\) is found when the value of \(\delta _{\mathrm {CP}}\) predicted in the HG (GRB) case is tested against the value of \(\delta _{\mathrm {CP}}\) for the GRB (HG) form, which is assumed to be the true one. Therefore, the sensitivity of DUNE alone is not enough to make a \(3\sigma \) claim on discriminating between the symmetry mixing forms under investigation, and we will test next all the cases using simulated data from the T2HK experiment, whose overall sensitivity to CPV is better than that of DUNE.
Performing a statistical analysis for T2HK, we find that it can discriminate the GRB case from the HG case at approximately \(2.5\sigma \) confidence level. More specifically, if \(\delta _{\mathrm {CP}}= 256^{\circ }~(293^{\circ })\) turned out to be the true value of the CPV phase, then T2HK could disfavour the value of \(\delta _{\mathrm {CP}}= 293^{\circ }~(256^{\circ })\) with \(\Delta \chi ^2 \approx 7.5\). We also find that the TBM and HG symmetry forms, in turn, occur to be resolvable at slightly less C.L. with \(\Delta \chi ^2\) being around 5.5. Thus, the sensitivity of T2HK is not sufficient as well to discriminate between the cases of interest at \(3\sigma \) C.L. For that reason, we will test them further using the potential of combining DUNE and T2HK.
The combination of DUNE and T2HK provides better sensitivity to the CPV phase \(\delta _{\mathrm {CP}}\) than either of these two experiments in isolation (see, e.g., [72]). A combined analysis performed by us leads to the results described below. Firstly, the GRB and HG mixing forms can be now distinguished at more than \(3\sigma \) confidence level. If \(\delta _{\mathrm {CP}}= 256^{\circ }~(293^{\circ })\) is the true value, then \(\delta _{\mathrm {CP}}= 293^{\circ }~(256^{\circ })\) will be disfavoured with \(\Delta \chi ^2 \approx 11~(10.5)\). Secondly, the TBM and HG cases can be resolved at slightly less than \(3\sigma \), the corresponding values of \(\Delta \chi ^2\) being around 8. Thirdly, discriminating between the GRA and GRB forms can be claimed with \(\Delta \chi ^2 \approx 5.5\). Finally, the sensitivity of the combination of these two experiments is not enough to discern TBM from GRA, GRA from HG, and TBM from GRB at even \(2\sigma \). For these three pairs, we find \(\Delta \chi ^2 \approx 3.5\), 1.2, and 0.2, respectively, when the corresponding predictions for \(\delta _{\mathrm {CP}}\) are compared between themselves.
Confidence levels (in number of \(\sigma \)) at which the symmetry forms under consideration can be distinguished from each other assuming that one of them is realised in Nature. The result is obtained using the combination DUNE + T2HK. All the mixing angles have been fixed to their NO best fit values both in data and in test
True  Tested  

TBM  GRA  GRB  HG  
TBM  1.9  0.5  \(\mathbf 2 .\mathbf 9 \)  
GRA  1.9  2.3  1.1  
GRB  0.5  2.3  \(\mathbf 3 .\mathbf 3 \)  
HG  \(\mathbf 2 .\mathbf 9 \)  1.1  \(\mathbf 3 .\mathbf 3 \) 
Further, performing the more involved analysis described in Appendix A, we obtain the results summarised in Fig. 3. This figure allows us to see immediately at which C.L. a given pair of the symmetry forms can be distinguished, under the assumption that one form in the pair is realised in Nature. In particular, the numbers presented in Table 5 get clear graphic representation. Indeed, we see that using the combination DUNE + T2HK, GRB and HG can be resolved at more than \(3\sigma \) C.L., while TBM and HG can be distinguished at almost \(3\sigma \).
As we see from Appendix A, the external prior on \(\sin ^2\theta _{12}\) from JUNO is very important for the analyses performed in the present study. Usually, the present precision on \(\sin ^2\theta _{12}\) is sufficient for the LBL experiments to achieve their goals on determination of \(\delta _{\mathrm {CP}}\), neutrino mass ordering, and the octant of \(\theta _{23}\). However, in our case, the role of \(\theta _{12}\) is very important, since, as we have mentioned earlier, Eq. (1.1), and thus predictions for \(\delta _{\mathrm {CP}}\) provided by different symmetry forms, are very sensitive to the value of the solar angle. Thereby, there is a nice synergy between JUNO on the one hand and the LBL experiments on the other: DUNE and T2HK will be much more sensitive in addressing the questions posed in the present study, if they are provided with a precise measurement of \(\theta _{12}\) performed by JUNO.
Finally, we would like to notice that the \(\Delta \chi ^2\) values obtained in the case of DUNE + T2HK in Fig. 3 can also be inferred from Fig. 2. Namely, drawing a vertical line at the minimum of \(\Delta \chi ^2\) curve for a given symmetry form in Fig. 2, we can assess how much the other forms are disfavoured with respect to the chosen form. For example, let us assume that the HG form is realised in Nature. Then, we have \(\delta _{\mathrm {CP}}^{\mathrm{true}} = 293^{\circ }\) (see Table 2). Drawing a vertical line at this value of \(\delta _{\mathrm {CP}}^{\mathrm{true}}\), we read from the intersections with the GRA, TBM, and GRB curves: \(\Delta \chi ^2 \approx 1\), 7, and 10, respectively. These are to be compared with the bottom right panel of Fig. 3.
5.3 The BM symmetry form
The values of \(\cos \delta _{\mathrm {CP}}\) and \(\delta _{\mathrm {CP}}\) for different symmetry forms obtained from the sum rule in Eq. (1.1) fixing \(\sin ^2\theta _{12} = 0.3343\) (its upper \(2\sigma \) bound) and two other mixing angles to their NO best fit values
Symmetry form  \(\cos \delta _{\mathrm {CP}}\)  \(\delta _{\mathrm {CP}}\) 

BM  \(\) 1.00  \(180^{\circ }\) 
TBM  0.07  \(86^{\circ }\vee 274^{\circ }\) 
GRA  0.43  \(65^{\circ }\vee 295^{\circ }\) 
GRB  \(\) 0.01  \(91^{\circ }\vee 269^{\circ }\) 
HG  0.60  \(53^{\circ }\vee 307^{\circ }\) 
Confidence levels (in number of \(\sigma \)) at which the symmetry forms under consideration can be distinguished from each other by different experiments in the case of possibility to have viable BM mixing in the neutrino sector. “D” and “T” stand for DUNE and T2HK, respectively. When not explicitly specified, the results are for DUNE + T2HK. Both in data and in test, \(\sin ^2\theta _{12}\) has been set to 0.3343, while \(\sin ^2\theta _{23}\) and \(\sin ^2\theta _{13}\) have been fixed to their NO best fit values
True  Tested  

BM  TBM  GRA  GRB  HG  
BM  \(\mathbf{5 .\mathbf 1 }\) (D)  \(\mathbf{5 .\mathbf 3 }\) (D)  \(\mathbf{5 .\mathbf 0 }\) (D)  \(\mathbf{5 .\mathbf 4 }\) (D)  
\(\mathbf{9 .\mathbf 4 }\) (T)  \(\mathbf{9 .\mathbf 8 }\) (T)  \(\mathbf{9 .\mathbf 2 }\) (T)  \(\mathbf{9 .\mathbf 7 }\) (T)  
TBM  \(\mathbf{5 .\mathbf 2 }\) (D)  2.1  0.5  3.4  
8.9 (T)  
GRA  5.5 (D)  2.1  2.5  1.4  
\(\mathbf{9 .\mathbf 2 }\) (T)  
GRB  5.1 (D)  0.5  2.5  3.1 (T)  
8.7 (T)  
HG  5.6 (D)  3.4  1.4  3.1 (T)  
9.2 (T) 
6 Summary and conclusions
In the present study, we have explored in detail the sensitivity of the future LBL experiments DUNE and T2HK to test various lepton mixing schemes predicted by flavour models with nonAbelian discrete symmetries. These models provide a natural explanation of the observed pattern of neutrino mixing. We have concentrated on a particular sum rule for \(\cos \delta _{\mathrm {CP}}\) given in Eq. (1.1), which holds for a rather broad class of discrete flavour symmetry models. We have considered five different underlying symmetry forms of the neutrino mixing matrix, namely, bimaximal (BM), tribimaximal (TBM), golden ratio type A (GRA), golden ratio type B (GRB), and hexagonal (HG). Each of these mixing schemes is characterised by a specific value of the angle \(\theta ^\nu _{12}\) entering into the sum rule in Eq. (1.1). The values of \(\theta ^\nu _{12}\) for the BM, TBM, GRA, GRB, and HG forms are \(45^{\circ }\), \(35^{\circ }\), \(32^{\circ }\), \(36^{\circ }\), and \(30^{\circ }\), respectively. The BM symmetry form is disfavoured by the present best fit values of the mixing angles. Table 2 summarises the predictions for \(\delta _{\mathrm {CP}}\) for the other symmetry forms assuming the current best fit values of \(\theta _{12}\), \(\theta _{23}\), and \(\theta _{13}\). In our analysis, we have considered only the predicted values of \(\delta _{\mathrm {CP}}\) lying around \(270^{\circ }\), since they are preferred by the present oscillation data (see Table 1).
Based on the prospective DUNE + T2HK data, the GRB and TBM symmetry forms are compatible with the current best fit values of the mixing parameters at around \(1\sigma \) confidence level. Under the same condition, the GRA and HG forms are disfavoured at around \(3\sigma \) and \(4\sigma \), respectively (see Fig. 1). Next, in Fig. 2, we show up to what extent any given symmetry form is compatible with any true value of \(\delta _{\mathrm {CP}}\) lying in the range \(180^{\circ }\) to \(360^{\circ }\). In our analysis, we impose an external Gaußian prior of \(0.7\%\) (at \(1\sigma \)) on \(\sin ^2\theta _{12}\) as expected from the upcoming JUNO experiment, which improves our results significantly, as shown in Fig. 4 in Appendix A. This demonstrates a very important synergy between JUNO and LBL experiments like DUNE and T2HK, while testing various lepton mixing schemes in light of oscillation data.
The combined data from DUNE and T2HK can discriminate among GRB and HG at more than \(3\sigma \), if one of them is realised in Nature and the other form is tested against it (see Table 5). The same is true for TBM and HG at almost \(3\sigma \). Note, in these two cases, the differences between the predicted best fit values of \(\delta _{\mathrm {CP}}\) are \(37^{\circ }\) and \(32^{\circ }\), respectively (see Table 2). Similarly, the GRA symmetry form can be distinguished from GRB and TBM at around \(2\sigma \). The corresponding differences in these cases are \(26^{\circ }\) and \(21^{\circ }\), respectively. At the same time, there is a difference of \(11^{\circ }\) for GRA and HG, which can be discriminated only at \(1\sigma \). For TBM and GRB, the difference is only \(5^{\circ }\) and therefore, the significance of separation is very marginal (around \(0.5\sigma \)).
In conclusion, the detailed analyses performed in the present work can be applied to any flavour model leading to a sum rule which predicts \(\delta _{\mathrm {CP}}\). In this regard, our article can serve as a useful guidebook for further studies.
Footnotes
 1.
 2.
Combining the discrete symmetry approach with the idea of generalised CP invariance [20, 21, 22], which is a generalisation of the standard CP invariance requirement, allows one to obtain predictions also for the Majorana CPV phases [23] in the PMNS matrix in the case of massive Majorana neutrinos (see, e.g., [24, 25, 26, 27, 28, 29] and references quoted therein).
 3.
 4.
Note that, our assumptions on various components of the DUNE setup differ slightly in comparison to the reference design in the Conceptual Design Report (CDR) of DUNE [54]. However, it is expected that the reference design of the DUNE experiment is going to evolve with time as we will learn more about this setup with the help of ongoing R&D studies.
 5.
The number of positron events can be calculated with the help of Eq. (3.1), by considering relevant oscillation probability and cross section. The same is valid for \(\mu ^{\pm }\) events.
 6.
While estimating these event rates, we properly consider the “wrongsign” components which are present in the beam for both \({\nu _e}/{\bar{\nu }_e}\) and \({\nu _{\mu }}/{\bar{\nu }_{\mu }}\) candidate events.
 7.
DUNE will operate at multiGeV energies with 1300 km baseline and therefore, the matter effect is quite substantial for this setup. For this reason, DUNE can break the hierarchy\(\delta _{\mathrm {CP}}\) degeneracy completely [71] and can resolve the issue of neutrino mass ordering at more than 5\(\sigma \) C.L. [54, 72]. Due to the shorter baseline of 295 km, T2HK has lower sensitivity to the mass ordering. However, HK can settle this issue using the atmospheric neutrinos at more than 3\(\sigma \) C.L. for both NO and IO provided \(\sin ^2\theta _{23} > 0.45\) [73]. Combining beam and atmospheric neutrinos in HK, the mass ordering can be determined at more than 3\(\sigma \) (5\(\sigma \)) with five (ten) years of data [73].
 8.
Note that we consider both CC and NC background events in our analysis and the NC backgrounds are independent of oscillation parameters.
 9.
Notes
Acknowledgements
S.K.A. and S.S.C. are supported by the DST/INSPIRE Research Grant [IFAPH12], Department of Science & Technology, India. A part of S.K.A.’s work was carried out at the International Centre for Theoretical Physics (ICTP), Trieste, Italy. It is a pleasure for him to thank the ICTP for the hospitality and support during his visit via SIMONS Associateship. A.V.T. and S.T.P. acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreements No 674896 (ITN Elusives) and No 690575 (RISE InvisiblesPlus). This work was supported in part by the INFN program on Theoretical Astroparticle Physics (TASP) and by the World Premier International Research Center Initiative (WPI Initiative, MEXT), Japan (S.T.P.).
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