IP/BBSR/2017-13
SISSA 53/2017/FISI
IPPP/17/75

Addressing Neutrino Mixing Schemes with DUNE and T2HK
Sanjib Kumar Agarwalla a,b,c,1 , Sabya Sachi Chatterjee a,b,2 ,arXiv:1711.02107v1 [hep-ph] 6 Nov 2017

S. T. Petcov d,e,3 and A. V. Titov d,f,4
a Institute of Physics, Sachivalaya Marg, Sainik School Post, Bhubaneswar 751005, Indiab Homi Bhabha National Institute, Training School Complex,

Anushakti Nagar, Mumbai 400085, India
c International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
d SISSA/INFN, Via Bonomea 265, 34136 Trieste, Italye Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, 277-8583 Kashiwa, Japanf Institute for Particle Physics Phenomenology, Department of Physics, Durham University,

South Road, Durham DH1 3LE, United Kingdom

Abstract
We consider schemes of neutrino mixing arising within the discrete symmetry approach to
the well-known flavour problem. We concentrate on 3ν mixing schemes in which the cosineof the Dirac CP violation phase δCP satisfies a sum rule by which it is expressed in terms
νof three neutrino mixing angles θ12 , θ23 , and θ13 , and a fixed real angle θ12, whose value
depends on the employed discrete symmetry and its breaking. We consider five underlying
symmetry forms of the neutrino mixing matrix: bimaximal (BM), tri-bimaximal (TBM),
golden ratio A (GRA) and B (GRB), and hexagonal (HG). For each symmetry form,
the sum rule yields specific prediction for cos δCP for fixed θ12 , θ23 , and θ13 . In thecontext of the proposed DUNE and T2HK facilities, we study (i) the compatibility ofthese predictions with present neutrino oscillation data, and (ii) the potential of theseexperiments to discriminate between various symmetry forms.

1

E-mail: sanjib@iopb.res.in
E-mail: sabya@iopb.res.in
3
Also at Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia,
Bulgaria.
4E-mail: arsenii.titov@durham.ac.uk
2

1

Introduction and Motivation

All compelling neutrino oscillation data are compatible with 3ν mixing [1], i.e., with existenceof three light neutrino states ν1,2,3 with definite masses m1,2,3 , three orthogonal linear combinations of which form the three flavour neutrino states νe , νµ and ντ . The flavour neutrino(flavour antineutrino) oscillation probabilities in the case of 3ν mixing are characterised, as
is well known, by six fundamental parameters. In the standard parametrisation of the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) neutrino mixing matrix [1], these are the solar,
reactor, and atmospheric mixing angles, θ12 , θ13 , and θ23 , respectively, the Dirac CP violation(CPV) phase, δCP , and the two independent mass squared differences, ∆m221 ≡ m22 − m21 and∆m231 = m23 − m21 . The mixing angles θ12 and θ13 as well as the solar and the absolute valueof the atmospheric neutrino mass squared differences, ∆m221 and |∆m231 |, respectively, havebeen measured in neutrino oscillation experiments with a relatively high precision [2–4]. The
precision on θ23 is somewhat worse, the 1σ relative uncertainty on sin2 θ23 being approximately 10%. At the same time, the octant of θ23 and the value of δCP remain unknown. The
sign of ∆m231 , which is also undetermined, allows, as is well known, for two possible typesof the neutrino mass spectrum: (i) with normal ordering (NO) if ∆m231 > 0, and (ii) withinverted ordering (IO) if ∆m231 < 0. In Table 1, we summarise the best fit values and 1σ,
2σ, 3σ ranges of the oscillation parameters obtained in one of the latest global analysis of
the neutrino oscillation data [3]1 . In the case of IO spectrum, instead of ∆m231 , we use thelargest positive mass squared difference ∆m223 ≡ m22 − m23 , which corresponds to ∆m231 of the
NO spectrum. In particular, we see from Table 1 that for the NO mass spectrum, the valuesof δCP ∈ (31◦ , 137◦ ) are already disfavoured at more than 3σ C.L., while the values of δCP
between 180◦ and 342◦ are allowed at 2σ.
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 moregeneral 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 high-precision 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 featuresof the discrete symmetry approach to neutrino mixing and lepton flavour (see, e.g., [5–7] forreviews), is that it leads to (i) fixed predictions of the values of some of the neutrino mixing
angles and the Dirac CPV phase δCP , and/or (ii) existence of correlations between some of
the mixing angles and/or between the mixing angles and δCP (see, e.g., [8–19]). These correlations are often referred to as neutrino mixing sum rules2 . Most importantly, these sum
rules can be tested using oscillation data [8–11, 15, 19, 29, 30].
Within the discrete flavour symmetry approach, the PMNS matrix is predicted to havean underlying symmetry form, where θ12 , θ23 , and θ13 have values which differ by sub-leading1

Global analyses of the neutrino oscillation data were also performed recently in Refs. [2] and [4]. The best
fit values and 1σ, 2σ, 3σ ranges of the parameters obtained in these two articles almost agree with the findings
in Ref. [3].
2Combining the discrete symmetry approach with the idea of generalised CP invariance [20–22], 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., [24–29]
and references quoted therein).

1ParameterBest fit

1σ range

2σ range

3σ range

2.97

2.81–3.14

2.65–3.34

2.50–3.54

2.15

2.08–2.22

1.99–2.31

1.90–2.40

2.16

2.07–2.24

1.98–2.33

1.90–2.42

4.25

4.10–4.46

3.95–4.70

3.81–6.15

5.89

4.17–4.48 ⊕ 5.67–6.05

3.99–4.83 ⊕ 5.33–6.21

3.84–6.36

δCP [◦ ] (NO)248

212–290

180–342

0–31 ⊕ 137–360

δCP [◦ ] (IO)

236

202–292

166–338

0–27 ⊕ 124–360

7.37

7.21–7.54

7.07–7.73

6.93–7.96

2.56

2.53–2.60

2.49–2.64

2.45–2.69

2.54

2.51–2.58

2.47–2.62

2.42–2.66

sin2 θ12
10−1
sin2 θ13(NO)
10−2
sin2 θ13(IO)
10−2
sin2 θ23(NO)
10−1
sin2 θ23
(IO)
10−1

∆m221
10−5 eV2∆m231(NO)10−3 eV2∆m223(IO)
10−3 eV2

Table 1: The best fit values and 1σ, 2σ, 3σ 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.

perturbative corrections from their respective measured values. The approach seems very
natural in view of the fact that UPMNS = Ue† Uν , where Ue and Uν are 3 × 3 unitary matrices which diagonalise the charged lepton and neutrino mass matrices. Typically (but notuniversally) the matrix Uν has a certain symmetry form, while the matrix Ue provides thecorrections necessary to bring the symmetry values of the angles in Uν to their experimentally measured values. A sum rule which relates cos δCP with θ12 , θ23 , and θ13 , arising in thisapproach, depends on the underlying symmetry form of the PMNS matrix and on the formof the “correcting” matrix Ue .
In the present work, we will concentrate on a particular sum rule for cos δCP derived in [8],which holds for a rather broad class of discrete flavour symmetry models. According to thissum rule, cos δCP is expressed in terms of the three measured neutrino mixing angles and one
ν determined by the underlying discrete symmetry. This sum rule has the
fixed parameter θ12
following form:cos δCP =tan θ23νν
cos 2θ12+ sin2 θ12 − cos2 θ12
1 − cot2 θ23 sin2 θ13 .
sin 2θ12 sin θ13(1.1)

In this study, we consider five widely discussed underlying symmetry forms of the neutrino
mixing matrix, namely, bimaximal (BM) [31–34], tri-bimaximal (TBM) [35–38] (see also [39]),2

Symmetry form
BM
TBM
GRA
GRB
HG

ν [◦ ]
θ12cos δCPδCP [◦ ]45√
arcsin(1/ 3) ≈ 35
arctan(1/φ) ≈ 32
arccos(φ/2) = 36
30

unphysical
−0.16
0.21
−0.24
0.39

unphysical99 ∨ 261
78 ∨ 282
104 ∨ 25667 ∨ 293

Table 2: The best fit values of cos δCP and δCP from the sum rule in eq. (1.1) for the different
symmetry forms. The mixing angles θ12 , θ23 , and θ13 have been
√ fixed to their NO best fit values
from Table 1. The φ stands for the golden ratio: φ = (1 + 5)/2. See text for further details.

Symmetry form
BM
TBM
GRA
GRB
HG

Intervals for δCP [◦ ] obtained varyingθ12 in 3σ
θ23 in 3σ
θ13 in 3σ
150–180 ∨ 180–210
79–119 ∨ 241–281
57–95 ∨ 265–303
84–125 ∨ 235–276
45–84 ∨ 276–315

unphysical
98–107 ∨ 253–262
76–78 ∨ 282–284
102–114 ∨ 246–258
60–68 ∨ 292–300

unphysical
98–101 ∨ 259–262
77.6–77.9 ∨ 282.1–282.4
103–106 ∨ 254–257
66–68 ∨ 292–294

Table 3: The intervals for δCP due to the present 3σ uncertainties in the values of the neutrino
mixing angles. The quoted intervals are obtained varying one mixing angle in its corresponding
3σ range for the NO spectrum and fixing the other two angles to their NO best fit values.

golden ratio type A (GRA) [40–42], golden ratio type B (GRB) [43, 44], and hexagonal (HG)ν[45, 46]. Each of these symmetry forms is characterised by a specific value of the angle θ12ν = 45◦ (or sin2 θ ν = 0.5) for
entering into the sumin eq. (1.1). Namely, θ1212√ rule given2 νν◦ν = arctan(1/φ) ≈ 32◦ (orBM; θ12 = arcsin(1/ 3) ≈ 35 (or sin
θ
=
1/3)
for
TBM;
θ
12
√ 12
ν ≈ 0.276) for GRA, φ = (1 + 5)/2 being the golden ratio; θ ν = arccos(φ/2) = 36◦
sin2 θ1212
ν ≈ 0.345) for GRB; and θ ν = 30◦ (or sin2 θ ν = 0.25) for HG.
(or sin2 θ121212First, in eq. (1.1), we use the best fit values of the neutrino mixing angles assuming NO
case from Table 1 to calculate the best fit value of cos δCP for a given symmetry form which
ν . We present the obtained values in Table 2. For each symmetry
has a fixed value of θ12
form, the predicted value of cos δCP gives rise to two values of δCP located symmetrically withrespect to zero, which are also given in Table 2. Further, we calculate errors on these valuesby varying θ12 , θ23 , and θ13 (one at a time) in their 3σ experimentally allowed ranges for NO
as given in Table 1 and fixing the two remaining angles to their best fit values. We summarise
the obtained intervals of values of δCP in Table 3.
In the case of the BM symmetry form, the obtained best fit value of cos δCP = −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 δCP for the BM symmetry form. For instance, fixing sin2 θ13 and sin2 θ23
to their best fit values, cos δCP = −1 requires sin2 θ12 = 0.3343, which is the upper bound of
the corresponding 2σ allowed range of sin2 θ12 (see Table 1).
3

A rather detailed analysis of the predictions for cos δCP of the sum rule in eq. (1.1) has
been performed in Refs. [9,10]. In particular, likelihood profiles for cos δCP 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
long-baseline (LBL) neutrino oscillation experiments3 , namely, Deep Underground Neutrino
Experiment (DUNE) and Tokai to Hyper-Kamiokande (T2HK), we study in detail (i) towhat degree the sum rule predictions for cos δCP 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 Section 2, we take a first glance at the sum
rule predictions. In Section 3, we give a short description of the planned DUNE and T2HK
experiments and provide expected event rates for both the set-ups. Section 4 contains detailsof the statistical analysis. In Section 5, we present and discuss results of this analysis. Morespecifically, in subsection 5.1, we test the compatibility between the considered symmetry
forms and present oscillation data. Next, in subsection 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 subsection 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 Section 6. Appendix A discusses the issue of external priors on sin2 θ12 andsin2 θ13 . Finally, in Appendix B, we study the compatibility of the considered symmetry forms
with any potentially true values of sin2 θ23 and δCP in the context of DUNE and T2HK.

2

A First Glance at the Sum Rule Predictions

Let us first take a closer look at the sum rule predictions for δCP summarised in Tables 2 and3. Several important points that can be made from these two tables and the current global
data (see Table 1) are in order.
ν has a fixed value for each symmetry form as given in Table 2. For• The parameter θ12
ν , eq. (1.1) predicts a certain value of cos δ
fixed choices of θ12 , θ23 , θ13 , and θ12
CP , which
gives rise to two values of δCP (see fourth column of Table 2).

• The complementarity [51] between the modern reactor (Daya Bay, RENO, and Double Chooz) and accelerator (T2K and NOνA) data has enabled us to probe the parameter space for δCP . Already, the latest global data have disfavoured values ofδCP ∈ (31◦ , 137◦ ) at 3σ C.L. for NO (see Table 1). From Table 3, we see that outof the three mixing angles, the 3σ allowed range of θ12 causes the largest uncertainty inδCP predicted by eq. (1.1). But, note that, all the symmetry forms except BM predict
one of the ranges of δCP in the interval of 31◦ to 137◦ , which has been already ruled out
at 3σ. Therefore, we will not consider these ranges further in our study, and we will
only consider the values of δCP in the interval of 180◦ to 360◦ .
• Further, we notice from Tables 1 and 3 that for TBM and GRB, the predicted intervalsof δCP lie within the 1σ experimentally allowed range. For BM, GRA, and HG, the3

Recently, the authors of Refs. [47–50] investigated the capabilities of current and future LBL experiments
to probe few flavour models, which lead to alternative correlations between the neutrino mixing parameters,
which differ from those considered by us.

4intervals of interest fall within the 2σ 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 θ12 , which causes the largest uncertainty
in predicting the range of δCP (see Table 3). However, the proposed JUNO experimentwill provide a high-precision measurement of sin2 θ12 with a relative 1σ uncertainty of
0.7%. Therefore, we impose a prior on sin2 θ12 expected from JUNO, which we will
discuss later in detail in Section 4 and in Appendix A.
• Table 3 shows that the sum rule predictions depend to some extent on θ23 . Therefore, wevary this angle both in data and in fit. The LBL experiments themselves are sensitive toθ23 . Thus, we do not impose any external prior on this angle (see details in Section 4).
• The experiments under discussion are sensitive to θ13 through the appearance channels
νµ → νe and ν̄µ → ν̄e . Therefore, the role of an external prior on sin2 θ13 is negligible
for the physics case under study, as we will see later in Fig. 4. However, we put a prior
on sin2 θ13 as expected from Daya Bay to speed up our simulations (see Section 4 fordetails).

3

Experimental Features and Event Rates

In this section, we first briefly describe the key experimental features of the proposed highprecision DUNE and T2HK facilities that we use in our numerical simulation.

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, high-intensity, on-axis, wide-band neutrino beamfrom Fermilab directed towards a massive liquid argon time-projection chamber (LArTPC) far
detector housed at the Homestake Mine in South Dakota over a baseline of 1300 km [52–56].
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. [57]. 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 × 1020 protons on target (p.o.t.) in 188 days per
calendar year. In our calculation, we have used the fluxes which were generated assuming adecay pipe length of 200 m and 200 kA horn current [58]. We assume that DUNE will collectdata for ten years (5 years in ν mode and 5 years in ν̄ mode), which is equivalent to a total
exposure of 248 kt · MW · year4 . In our simulation, we consider the reconstructed ν energy
range to be 0.5 GeV to 10 GeV for both appearance and disappearance channels. We takethe same energy range for antineutrino as well.
The proposed Hyper-Kamiokande (HK) water Cherenkov detector will serve as the far
detector of a long-baseline neutrino experiment using an upgraded neutrino beam from the
J-PARC facility, commonly known as “T2HK” (Tokai to Hyper-Kamiokande) experiment [59–4Note that, our assumptions on various components of the DUNE set-up differ slightly in comparison to
the reference design in the Conceptual Design Report (CDR) of DUNE [53]. However, it is expected that thereference design of the DUNE experiment is going to evolve with time as we will learn more about this set-up
with the help of ongoing R&D studies.

561]. This set-up is highly sensitive to the Dirac CPV phase δCP of the PMNS 3ν mixing matrix
and holds promise to resolve the mystery of leptonic CP violation in neutrino oscillations at
an unprecedented confidence level [60]. We perform the simulation for T2HK according toRefs. [60,61]. To produce an intense ν/ν̄ beam for HK, we consider an integrated proton beam
power of 7.5 MW × 107 seconds, which can deliver in total 15.6 × 1021 p.o.t. with a 30 GeV
proton beam. We assume that these total p.o.t. will be shared among ν and ν̄ modes witha run-time 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
J-PARC at an off-axis angle of ∼ 2.5◦ , 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 kt · MW · year. Recently, the baseline design for T2HK has been revised [62].
According to this latest publication [62], the total beam exposure is 27 × 1021 p.o.t. andthe fiducial mass for the HK detector is 374 kt, which are slightly different as compared towhat we have considered in the present work. In our simulation, we take the reconstructedνe and ν̄e energy range of 0.1 GeV to 1.25 GeV for the appearance channel. As far as thedisappearance channel is concerned, the assumed energy range is 0.1 GeV to 7 GeV for bothνµ and ν̄µ candidate events.

3.2

Event Rates

In this subsection, we present the expected total event rates for both the set-ups underconsideration. We compute the number of expected electron events5 in the i-th energy bin in
the detector using the following well-known expression:
Z E max

Z EmaxNi = T nn 

dE
0

Ai

min
EA

dEA φ(E) σνe (E) R(E, EA ) Pµe (E) ,

(3.1)

i

where φ(E) is the neutrino flux spectrum (m−2 · year−1 · GeV−1 ), T is the total running
time (year), nn is the number of target nucleons in the detector,  is the detector efficiency,
and R(E, EA ) is the Gaußian energy resolution function (GeV−1 ) of the detector. σνe is theneutrino interaction cross section (m2 ) which has been taken from Refs. [63, 64], where theauthors calculated the cross section for water and isoscalar targets. In order to have LArcross sections for DUNE, we have scaled the inclusive charged current (CC) cross sections ofwater by a factor of 1.06 (0.94) for neutrino (antineutrino) [65, 66]. We denote the true andreconstructed neutrino energies by the quantities E and EA , respectively.
Table 4 shows a comparison between the expected total signal and background event rates6
in the appearance and disappearance modes for DUNE and T2HK set-ups. We compute thesame for both neutrino and antineutrino runs assuming a total exposure of 248 kt · MW · year
for DUNE and 4200 kt · MW · year for T2HK. We consider a run-time 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, δCP = 248◦ , and sin2 θ23 = 0.425. For allother oscillation parameters, we consider the best fit values which are applicable for NO (seeTable 1). To compute the full three-flavour neutrino oscillation probabilities in matter, we
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 µ± events.
6
While estimating these event rates, we properly consider the “wrong-sign” components which are present
in the beam for both νe /ν̄e and νµ /ν̄µ candidate events.

6Mode (Channel)
ν (appearance)ν (disappearance)ν̄ (appearance)ν̄ (disappearance)

DUNE (248 kt · MW · year)
SignalBackground
CC
Int+Mis-id+NC=Total
614
125+29+24=178
5040
0+0+24=24
60
43+10+7=60
1807
0+0+7=7

T2HK (4200 kt · MW · year)
SignalBackground
CC
Int+Mis-id+NC=Total
2852
530+13+173=716
20024
12+44+1003=1059
1383
627+11+265=903
27447
14+5+1287=1306

Table 4: Total signal and background event rates for DUNE and T2HK set-ups assumingNO, δCP = 248◦ , and sin2 θ23 = 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,“Mis-id” represents mis-identified muon events, and “NC” stands for neutral current. See text
for other details.

take the line-averaged constant Earth matter density of 2.80 (2.87) g/cm3 for the T2HK
(DUNE) baseline following the Preliminary Reference Earth Model (PREM) [67].
The main sources of backgrounds while selecting the νe and ν̄e candidate events are the intrinsic νe /ν̄e component in the beam, the muon events which will be mis-identified 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 νe /ν̄e in the beam. For
the νµ /ν̄µ 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 set-ups.

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 [68, 69] whichcalculates the median sensitivity of the experiment without considering the statistical fluctuations. Unless stated otherwise, we generate our simulated data considering the best fit valuesof the oscillation parameters obtained in the global analysis assuming NO for the neutrinomass spectrum (see second column of Table 1). We also keep the choice of the neutrino massordering to be fixed to NO in the fit7 . The solar and atmospheric mass squared differences
are already very well measured [2–4] and moreover, they do not appear in the sum rule (seeeq. (1.1)) that relates cos δCP with the mixing angles. Therefore, we also keep them fixed in
the fit at their best fit values. The mixing angles play an important role in the sum rule andwe treat them very carefully in our analysis. In the fit, we marginalise over test sin2 θ23 in its
present 3σ allowed range of 0.381 to 0.615. We show few results where we also vary the true
7

DUNE will operate at multi-GeV energies with 1300 km baseline and therefore, the matter effect is quite
substantial for this set-up. For this reason, DUNE can break the hierarchy-δCP degeneracy completely [70]and can resolve the issue of neutrino mass ordering at more than 5σ C.L. [53, 71]. Due to the shorter baselineof 295 km, T2HK has lower sensitivity to the mass ordering. However, HK can settle this issue using theatmospheric neutrinos at more than 3σ C.L. for both NO and IO provided sin2 θ23 > 0.45 [72]. Combining
beam and atmospheric neutrinos in HK, the mass ordering can be determined at more than 3σ (5σ) with five(ten) years of data [72].

7value of sin2 θ23 or marginalise over it in the same 3σ range. We do not impose any external
prior on sin2 θ23 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 sin2 θ12 . We vary this parameterin the fit in its present 3σ allowed range of 0.25 to 0.354. Since both DUNE and T2HK
cannot constrain the solar mixing angle (see the probability expressions in [73]), we impose
an external Gaußian prior of 0.7% (at 1σ) on this parameter as the proposed medium-baseline
reactor oscillation experiment JUNO will be able to measure sin2 θ12 with this precision [74].
We also marginalise over test sin2 θ13 in its present 3σ allowed range of 0.019 to 0.024. While
doing so, we apply an external Gaußian prior of 3% (at 1σ) on this parameter expecting thatthe Daya Bay experiment would be able to achieve this precision by the end of its run [75].
Both DUNE and T2HK can measure θ13 with high precision using νµ → νe and ν̄µ → ν̄eoscillation channels and therefore, the prior on θ13 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 θ12 , θ13 , and θ23 , the test value of δCP
is calculated using the sum rule (see eq. (1.1)) for a particular choice of the lepton mixing
ν . Since the best fit value of δ
scheme which is characterised by a certain value of θ12
CP maychange in the future, we also show some results varying the true choice of δCP in the range
180◦ to 360◦ .
To perform our statistical analysis, we follow the procedure outlined in Refs. [76, 77], anduse the following Poissonian χ2 function:
" n
#
X
ỹ
i
χ2 = minξs ,ξb 2(ỹi − xi − xi ln ) + ξs2 + ξb2 ,
(4.1)
xi
i=1

where n is the total number of reconstructed energy bins andihỹi ({ω}, {ξs , ξb }) = Nith ({ω}) [1 + π s ξs ] + Nib ({ω}) 1 + π b ξb .

(4.2)

Above, Nith ({ω}) denotes the predicted number of CC signal events (estimated using eq. (3.1))
in the i-th energy bin for a set of oscillation parameters ω. Nib ({ω}) stands for the total
number of background events8 . The quantities π s and π b in eq. (4.2) represent the systematic
errors on the signal and background events, respectively. For DUNE (T2HK), we consider π s= 5% (5%) and π b = 5% (10%) in the form of normalisation errors for both the appearance
and disappearance channels. We take the same uncorrelated systematic uncertainties forboth the neutrino and antineutrino modes. The quantities ξs and ξb denote the “pulls” dueto the systematic error on the signal and background, respectively. The data in eq. (4.1)
are included through the variable xi = Niex + Nib , where Niex is the number of observed CC
signal events and Nib is the background as discussed earlier. To obtain the total χ2 , we add
the χ2 contributions coming from all the relevant oscillation channels for both neutrino andantineutrino modes in a given experiment in the following fashion9 :χ2total = χ2νµ →νe + χ2ν̄µ →ν̄e + χ2νµ →νµ + χ2ν̄µ →ν̄µ + χ2prior .
8

(4.3)

Note that we consider both CC and NC background events in our analysis and the NC backgrounds are
independent of oscillation parameters.
9
In eq. (4.3), we assume that all the oscillation channels for both neutrino and antineutrino modes arecompletely uncorrelated, all the energy bins in a given channel are fully correlated, and the systematic errors
on signal and background are fully uncorrelated.

8In the above expression, the last term appears due to the external Gaußian priors that we
impose on sin2 θ12 and sin2 θ13 in the following way:
χ2prior =true
sin2 θ12 − sin2 θ12σ(sin2 θ12 )

2


+

true
sin2 θ13 − sin2 θ13
σ(sin2 θ13 )

2
,

(4.4)

true and σ(sin2 θ ) = 0.03 × sin2 θ true as mentioned
where we take σ(sin2 θ12 ) = 0.007 × sin2 θ12
13
13
earlier. In our analysis, we assume that the true values of sin2 θ12 and sin2 θ13 will remainunchanged in the future. While implementing the minimisation procedure, χ2total is firstminimised with respect to the “pull” variables ξs , ξb , and then marginalised over the various
oscillation parameters in their allowed ranges in the fit as discussed above to obtain ∆χ2min .
In Fig. 5 in Appendix
q B, we quote the statistical significance of our results for 1 d.o.f. in termsof nσ, where n =∆χ2min , which is valid in the frequentist method of hypothesis testing [78].

5Results and Discussion

5.1

Compatibility between Various Symmetry Forms and Present Neutrino
Oscillation Data

Table 1 shows the current best fit values of the neutrino mixing angles and the CPV phaseν which characterises the symmetry
δCP . Equation (1.1) relates δCP with the parameter θ12
forms under consideration. The first question we want to answer is how much compatibility
there is between the mixing symmetry forms and the present best fit values of the oscillation
parameters. To this aim, we assume the current best fit values of θ12 , θ13 , θ23 , and δCP tobe their true values and generate prospective data using the DUNE and T2HK experimental
set-ups according to the procedure explained in Section 4. Further, in the test, we fix the
mixing angles to their best fit values and let δCP to vary between 180◦ and 360◦ . For eachν according to eq. (1.1) and the corresponding value of ∆χ2value of δCP we calculate sin2 θ12
combining DUNE and T2HK results. This latter value is ∆χ2 between a given value of δCPν andand its best fit value, i.e., δCP = 248◦ . In Fig. 1, we plot ∆χ2 as a function of sin2 θ12
put on the resulting curve:• the black dot corresponding to the current best fit value of δCP = 248◦ which translatesν = 0.364 (∆χ2 = 0);
to sin2 θ12ν which characterise the GRB• the coloured dots corresponding to the values of sin2 θ12(violet), TBM (red), GRA (blue) and HG (green) symmetry forms.

From this figure we see that, if the present best fit values of θ12 , θ13 , θ23 , and δCP were the true
values of these parameters, the GRB (TBM) symmetry form would be compatible with themat slightly less (more) than 1σ C.L., while the GRA and HG schemes would be disfavoured
at 3σ and almost 4σ, respectively.
However, at present the CPV phase δCP is not severely constrained, and as can be seenfrom Table 1, any value between 180◦ and 342◦ is allowed at 2σ C.L., and any value except forthe ones between 31◦ and 137◦ is allowed at 3σ. Fixing the three mixing angles to their best
fit values, we find from eq. (1.1) that the full range of cos δCP ∈ [−1, 1] (allowed at presentν ∈ [0.157, 0.460]. Thus, in principle, any value fromat 3σ) translates to the values of sin2 θ12
this range may turn out to be favoured in the future. For instance, imagine that in the future
9

DUNE + T2HK
16
TBM
GRA
GRB

12

HG

DΧ 2

Best fit
3Σ

8

4

0
0.15

2Σ

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Ν
sin2 Θ12

Figure 1: Status of the lepton mixing symmetry forms in light of the current best fit valuesof the mixing angles and the CPV phase δCP and using the potential of DUNE + T2HK. Theνblack dot corresponds to sin2 θ12= 0.364 obtained from eq. (1.1) using the present best fit
νvalues of θ12 , θ13 , θ23 , and δCP . The coloured dots correspond to the values of sin2 θ12
for the
TBM, GRA, GRB, and HG symmetry forms.

the best fit value of δCP will shift from 248◦ to 290◦ , while the best fit values of the mixingν = 0.250, and thus the HG symmetry
angles will remain the same. Then, the value of sin2 θ12
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 δCP
and the mixing angles at our disposal, we will be able to repeat this analysis favouring somesymmetry forms and disfavouring the others.
Having obtained an idea of how much the mixing symmetry forms in question are compatible with the present best fit values of the oscillation parameters, we go next to a more
involved analysis which will allow us to see the compatibility of the studied symmetry forms
with any value of δCP between 180◦ and 360◦ , should it turn out to be the true value. To thistrue , to be between 180◦ and 360◦ , the true valueaim, we fix the true value of the CPV phase, δCP
true , to a value from its 3σ range, and the true values of theof the atmospheric mixing angle, θ23
true
true , to their corresponding best fit values. Then,solar and reactor mixing angles, θ12 and θ13
we generate data with this input using the DUNE and T2HK set-ups. In the test, we assume
test , θ test , and θ test to values
a given symmetry form to hold and fix the three test values θ12
13
23
from the corresponding 3σ ranges. Using these test values and known for a given symmetry
ν , we calculate δ test from eq. (1.1). Each couple of the true and test oscillation vectors,
form θ12CP
true
true , θ true , θ true , δ true ) and ytest = (θ test , θ test , θ test , δ test ), provides a certain value ofy= (θ12
13
23
12
13
23
CP
CP

10

DUNE + T2HK
16
TBM
GRA
GRB

12

HG

DΧ 2

Best fit

8

4

3Σ

2Σ

0
180

225

true
∆ CP
@°D

270

315

360

Figure 2: Compatibility of the TBM, GRA, GRB, and HG symmetry forms with any potentially true value of the Dirac CPV phase δCP . The figure is obtained employing combined
potential of DUNE and T2HK. The black vertical line corresponds to the current best fit valueof δCP for the NO neutrino mass spectrum.

∆χ2 . Note that in calculating this value, we use external priors on sin2 θ12 and sin2 θ13 fromJUNO and Daya Bay, as explained in Section 4. A detailed discussion on the impact of thesetest (overtwo priors is presented in Appendix A. Further, for each ytrue we marginalise over θij
true we marginalise also over θ true . We repeat this procedure forytest ). Finally, for each δCP23each of the four symmetry forms in study. The obtained results are shown in Fig. 2.
Two comments on Fig. 2 are in order.
• For each symmetry form a significant part of the parameter space gets disfavoured atmore than 3σ. Should the true value of δCP lie in this part of the parameter space, thecorresponding symmetry form will be disfavoured at 3σ confidence level.
• Now we can see at which C.L. any given symmetry form is compatible with any potrue of interest.
tentially true value of δCP . We just need to draw a vertical line at δCP
2
The points where it crosses the ∆χ curves will provide the confidence levels at whichtrue . In particular, for
the TBM, GRA, GRB, and HG forms are compatible with this δCP
true◦δCP = 248 , we find numbers which correspond to those extracted from Fig. 1 10 .
10Note that the numbers we read from Fig. 2 are slightly smaller than those extracted from Fig. 1 due tothe fact that now we marginalise over the values of the mixing angles.

11

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 distinguishthe 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 theneutrino 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 δCP = 256◦ , 261◦ , 282◦ , and 293◦
for the GRB, TBM, GRA, and HG symmetry forms, respectively. Assuming one of them tobe the true value of δCP , 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 wewant to compare.
We start with DUNE. After performing a statistical analysis of simulated data, as described in Section 4, we obtain that for all the 12 cases ∆χ2 does not exceed approximately
3.5. This value of ∆χ2 is found when the value of δCP predicted in the HG (GRB) case istested against the value of δCP 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 3σ 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 2.5σ confidence level. More specifically, if δCP =256◦ (293◦ ) turned out to be the true value of the CPV phase, then T2HK could disfavour thevalue of δCP = 293◦ (256◦ ) with ∆χ2 ≈ 7.5. We also find that the TBM and HG symmetry
forms, in turn, occur to be resolvable at slightly less C.L. with ∆χ2 being around 5.5. Thus,
the sensitivity of T2HK is not sufficient as well to discriminate between the cases of interest
at 3σ 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 δCPthan either of these two experiments in isolation (see, e.g., [71]). 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 3σ confidence level. If δCP = 256◦ (293◦ ) is the true
value, then δCP = 293◦ (256◦ ) will be disfavoured with ∆χ2 ≈ 11 (10.5). Secondly, the TBM
and HG cases can be resolved at slightly less than 3σ, the corresponding values of ∆χ2 beingaround 8. Thirdly, discriminating between the GRA and GRB forms can be claimed with∆χ2 ≈ 5.5. Finally, the sensitivity of the combination of these two experiments is not enoughto discern TBM from GRA, GRA from HG, and TBM from GRB at even 2σ. For these threepairs, we find ∆χ2 ≈ 3.5, 1.2, and 0.2, respectively, when the corresponding predictions forδCP are compared between themselves.
We have checked that adding NOνA and T2K data to sets of simulated data obtainedusing the DUNE and T2HK set-ups leads to increase in the values of ∆χ2 of only several
tenths. Thus, inclusion of these data does not help to improve differentiating between theconsidered mixing schemes. We summarise the obtained results in Table 5, in which we
present confidence levels (in number of σ) at which the symmetry forms under consideration
can be distinguished from each other assuming that one of them is realised in Nature andusing the potential of combining DUNE with T2HK.
Further, performing the more involved analysis described in Appendix A, we obtain the12Tested
True
TBM
GRA
GRB
HG

TBM

1.9
0.5
2.9

GRA

GRB

HG

1.9

0.5
2.3

2.9
1.1
3.3

2.3
1.1

3.3

Table 5: Confidence levels (in number of σ) 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.

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 thepair 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 canbe resolved at more than 3σ C.L., while TBM and HG can be distinguished at almost 3σ.
As we see from Appendix A, the external prior on sin2 θ12 from JUNO is very important
for the analyses performed in the present study. Usually, the present precision on sin2 θ12 issufficient for the LBL experiments to achieve their goals on determination of δCP , neutrinomass ordering, and the octant of θ23 . However, in our case, the role of θ12 is very important,since, as we have mentioned earlier, eq. (1.1), and thus predictions for δCP provided by different
symmetry forms, are very sensitive to the value of the solar angle. Thereby, there is a nicesynergy between JUNO on the one hand and the LBL experiments on the other: DUNE andT2HK will be much more sensitive in addressing the questions posed in the present study, if
they are provided with a precise measurement of θ12 performed by JUNO.
Finally, we would like to notice that the ∆χ2 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 minimumof ∆χ2 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
true = 293◦ (see Table 2). Drawing a vertical line
form is realised in Nature. Then, we have δCPtrue , we read from the intersections with the GRA, TBM, and GRB curves:at this value of δCP2∆χ ≈ 1, 7, and 10, respectively. These are to be compared with the bottom right panel ofFig. 3.

5.3

The BM Symmetry Form

As we have mentioned in the Introduction, even though the BM symmetry form is not compatible with the current best fit values of the neutrino mixing angles, it turns out to be
viable, if the current 2σ ranges of the mixing angles are taken into account. For example, if
we keep sin2 θ13 and sin2 θ23 fixed to their best fit values for NO, we find that the value ofsin2 θ12 = 0.3343, which is the upper bound of the corresponding 2σ range (see Table 1), is
required to obtain cos δCP = −1.00 and thus, recover viability of the BM mixing form. For
this choice of values of the mixing angles, the values of cos δCP (and δCP ), predicted in the
TBM, GRA, GRB, and HG cases, change. We summarise them in Table 6.
We perform the analysis in this case and find that the BM form can be distinguished from
13

12

DΧ2

GRA HtrueL

TBM HtrueL

16

DUNE + T2HK

DUNE + T2HK

T2HK

T2HK

DUNE

DUNE

GRA

TBM

GRB

GRB

HG

HG

8

4

0
0.15

0.20

0.25

16

DΧ2

12

0.30

0.35

GRB HtrueL

0.40

0.20

0.25

3Σ

3Σ

2Σ

2Σ

0.45

0.50

0.15

0.20

0.25

0.30

T2HK

T2HK

DUNE

DUNE

TBM

TBM

GRA

GRA

HG

GRB

0.30

0.35

0.40

0.35

HG HtrueL

DUNE + T2HK

4

0
0.15

12

DUNE + T2HK

8

16

0.40

0.45

2Σ

2Σ

0.15

Ν
sin2Θ12

0.20

0.25

0
0.50

12

3Σ

0.50

4

16

3Σ

0.45

8

0.30

0.35

0.40

0.45

8

4

0
0.50

Ν
sin2Θ12Figure 3: Sensitivities of DUNE, T2HK, and their combination to distinguish between the
TBM, GRA, GRB, and HG symmetry forms under the assumption that one of them is realised
in Nature. In the top left (right) panel the assumed true symmetry form is TBM (GRA),
while in the bottom left (right) panel this form is GRB (HG). The GRB (HG) form can bediscriminated from HG (GRB) at more than 3σ C.L. using the combined potential of DUNE
and T2HK.

14

Symmetry formcos δCP

δCP

BM
TBM
GRA
GRB
HG

−1.00
0.07
0.43
−0.01
0.60

180◦
86◦ ∨ 274◦65◦ ∨ 295◦91◦ ∨ 269◦53◦ ∨ 307◦

Table 6: The values of cos δCP and δCP for different symmetry forms obtained from the sum
rule in eq. (1.1) fixing sin2 θ12 = 0.3343 (its upper 2σ bound) and two other mixing angles totheir NO best fit values.

Tested
True

BM

BM

TBM

GRA

GRB

HG

5.1 (D)
9.4 (T)

5.3 (D)
9.8 (T)

5.0 (D)
9.2 (T)

5.4 (D)
9.7 (T)

2.1

0.5

3.4

2.5

1.4

TBM

5.2 (D)
8.9 (T)

GRA

5.5 (D)
9.2 (T)

2.1

GRB

5.1 (D)
8.7 (T)

0.5

2.5

HG

5.6 (D)
9.2 (T)

3.4

1.4

3.1 (T)

3.1 (T)

Table 7: Confidence levels (in number of σ) at which the symmetry forms under consideration
can be distinguished from each other by different experiments in the case of possibility tohave 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 dataand in test, sin2 θ12 has been set to 0.3343, while sin2 θ23 and sin2 θ13 have been fixed to their
NO best fit values.

all the other forms at more than 5σ by DUNE alone. The corresponding ∆χ2 are between 25
and 31, and they translate to the numbers of σ presented in Table 7. T2HK provides evenbetter results, which we also show in Table 7.

6

Summary and Conclusions

In the present study, we have explored for the first time the sensitivity of the future LBL
experiments DUNE and T2HK to test various lepton mixing schemes predicted by flavour
models with non-Abelian discrete symmetries. These models provide a natural explanation of15

the observed pattern of neutrino mixing. We have concentrated on a particular sum rule forcos δCP 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), tri-bimaximal (TBM), golden ratio type A (GRA), golden
ratio type B (GRB), and hexagonal (HG). Each of these mixing schemes is characterised byν entering into the sum rule in eq. (1.1). The values of θ ν fora specific value of the angle θ1212
the BM, TBM, GRA, GRB, and HG forms are 45◦ , 35◦ , 32◦ , 36◦ , and 30◦ , respectively. The
BM symmetry form is disfavoured by the present best fit values of the mixing angles. Table 2
summarises the predictions for δCP for the other symmetry forms assuming the current best
fit values of θ12 , θ23 , and θ13 . In our analysis, we have considered only the predicted values ofδCP lying around 270◦ , 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 arecompatible with the current best fit values of the mixing parameters at around 1σ confidence
level. Under the same condition, the GRA and HG forms are disfavoured at around 3σand 4σ, 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 δCP lying in the range 180◦ to 360◦ . In
our analysis, we impose an external Gaußian prior of 0.7% (at 1σ) on sin2 θ12 as expectedfrom 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 ofoscillation data.
The combined data from DUNE and T2HK can discriminate among GRB and HG atmore than 3σ, 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 3σ. Note, in these two cases,
the differences between the predicted best fit values of δCP are 37◦ and 32◦ , respectively (seeTable 2). Similarly, the GRA symmetry form can be distinguished from GRB and TBM at
around 2σ. The corresponding differences in these cases are 26◦ and 21◦ , respectively. At thesame time, there is a difference of 11◦ for GRA and HG, which can be discriminated only at
1σ. For TBM and GRB, the difference is only 5◦ and therefore, the significance of separationis very marginal (around 0.5σ).
In conclusion, the detailed analyses performed in the present work can be applied to any
flavour model leading to a sum rule which predicts δCP . In this regard, our article can serve
as a useful guidebook for further studies.

Acknowledgements
S.K.A. is supported by the DST/INSPIRE Research Grant [IFA-PH-12], 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 Sklodowska-Curie grant agreements No 674896 (ITN Elusives) andNo 690575 (RISE InvisiblesPlus). This work was supported in part by the INFN program on
Theoretical Astroparticle Physics (TASP) and by the World Premier International ResearchCenter Initiative (WPI Initiative, MEXT), Japan (S.T.P.).

16

A

Issue of Priors on sin2 θ12 and sin2 θ13

In this appendix, we discuss the role of external priors on various mixing angles that we considered in our analysis. First of all, we do not consider any prior on sin2 θ23 since both DUNE
and T2HK will be able to measure this parameter with sufficient precision. However, sincethese experiments are not sensitive to θ12 (see probability expressions in [73]), we consider
an external Gaußian prior of 0.7% (at 1σ) on sin2 θ12 as expected from the proposed JUNO
experiment [74]. Even though both DUNE and T2HK can provide high precision measurement of θ13 using their appearance channels, but to speed up our computation, we also apply
a Gaußian prior of 3% (at 1σ) on sin2 θ13 as expected by the end of Daya Bay’s run [75].
Fig. 4 shows the impact of these priors in our analysis. We obtain this figure in thefollowing way. First, we assume one of the symmetry forms, e.g., TBM, to be realised in
Nature. For this symmetry form, we estimate the true value of δCP from eq. (1.1) using thecurrent best fit values of the mixing angles for NO (see Table 1) as their true values. We
generate data with this input. Then, in the test, we vary sin2 θij and δCP in their corresponding
3σ allowed ranges. For each set of test values of sin2 θij and δCP , we estimate ∆χ2 and also
ν using eq. (1.1). For the same sin2 θ ν , there cancalculate the corresponding value of sin2 θ1212
ν , we choose the minimal value of ∆χ2 .
be several values of ∆χ2 . From there, for each sin2 θ12
ν in Fig. 4. We present the results
Finally, we plot this minimum ∆χ2 as a function of sin2 θ12
for T2HK considering the following cases: (i) without assuming any priors on sin2 θ12 andsin2 θ13 , (ii) assuming priors on both the parameters, (iii) only the prior on sin2 θ12 , and(iv) only the prior on sin2 θ13 . From this figure, we can make the following few important
observations.
• First, we see that the curves corresponding to cases (i) and (iv) almost overlap witheach other. It suggests that for the physics case under study, T2HK does not need an
external prior on sin2 θ13 since it can provide a necessary precision on this parameter.
• Secondly, we observe that the curves corresponding to cases (ii) and (iii) also overlap
with each other. It indicates that for our purpose, T2HK needs an external prior on
sin2 θ12 from JUNO since it has a very mild sensitivity on this mixing parameter.
We have checked that the above observations are also valid for GRA, GRB, and HG
symmetry forms and for DUNE as well. We have also seen that cases (ii) and (iii) are almost
equivalent to the fixed parameter scenario, where we keep all the mixing angles to be fixedat their best fit values in the fit. Unless mentioned otherwise, we always impose both thesepriors in our statistical analysis, as described in Section 4.

B

Agreement between Various Mixing Schemes and Oscillatrue
truetion Data in (sin2 θ23, δCP) PlaneHere we will see which regions of the parameter space in the plane of true values of sin2 θ23
and δCP will be compatible at less than 3σ C.L. for each symmetry form of interest, if thatν ), we calculate δform is realised in Nature. To this aim, for each symmetry form (fixed θ12CP
testusing eq. (1.1) with the test values of the mixing angles θij . Then, we marginalise ∆χ2 overtest for given true values θ true and δ true . The coloured bands in Fig. 5 represent potentially
θij
23
CP
true values of δCP as well as sin2 θ23 with which the form under consideration is compatible
17

TBM H trueL

16w o priorsw priorsDΧ 2 for T2HK

12

only sin 2 Θ12 prior
only sin 2 Θ13 prior
3Σ

8

4

0
0.15

2Σ

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Ν
sin2 Θ12

Figure 4: Impact of external Gaußian priors on sin2 θ12 and sin2 θ13 on the resulting ∆χ2function in the case of T2HK. We use a prior on sin2 θ12 of 0.7% (at 1σ) from JUNO and a prior
on sin2 θ13 of 3% (at 1σ) from Daya Bay. While doing so, we assume the current best fit valuesνof these two parameters to be their true values. The black dot corresponds to sin2 θ12= 1/3,
which characterises the TBM symmetry form.

at 1σ, 2σ, 3σ confidence levels in the context of DUNE, T2HK and their combination. If
the true value of δCP turned out to lie outside these bands, this would imply that the given
symmetry form is disfavoured at more than 3σ C.L. For all the symmetry forms, a significant
part of the parameter space gets disfavoured at more than 3σ.
For each symmetry form, the black dashed line has been obtained using eq. (1.1) with θ12and θ13 fixed to their NO best fit values. Note that a given symmetry form is well compatible
with any point close to this line. The star denotes the present best fit values of sin2 θ23 andδCP for the NO spectrum as given in Table 1, namely, (sin2 θ23 , δCP ) = (0.425, 248◦ ). As we
can see from the right panels of Fig. 5, i.e., in the context of DUNE + T2HK, the HG (GRA)
form is disfavoured at more than (precisely) 3σ C.L., while the GRB and TBM forms arecompatible with the star at 1σ and 2σ, respectively. If the star moves in the future to adifferent point, we will immediately conclude which symmetry forms are (dis)favoured. Let
us assume, e.g., that the future best fit values are (sin2 θ23 , δCP ) = (0.58, 300◦ ). Then, in thecontext of DUNE + T2HK, the GRB and TBM forms would be disfavoured at more than 3σ,
while the HG (GRA) symmetry form would be compatible with this hypothetical position of
the star at 1σ (2σ) confidence level.
Finally, we would like to notice the compatibility between this figure and the numbers
in Table 5. Let us consider an example, in which GRB is the true form and HG is tested
against it. In this case, from Table 5, we read the C.L. at which these two symmetry forms
18

Figure 5: Compatibility of various symmetry forms with any potentially true values of sin2 θ23
and δCP in the context of DUNE (left panels), T2HK (middle panels), and DUNE + T2HK
ν), the black dashed line has been obtained(right panels). For a given symmetry form (fixed θ12
using eq. (1.1) and fixing θ12 and θ13 to their NO best fit values. The star indicates the present
best fit values for NO as given in Table 1. For all the symmetry forms, a significant part ofthe parameter space gets disfavoured at more than 3σ C.L. for DUNE + T2HK.

19

can be distinguished by DUNE + T2HK, namely, 3.3σ. We recall that the results in this table
have been obtained assuming the current best fit values of the mixing angles to be their true
values. Thus, for GRB we have (sin2 θ23 , δCP ) = (0.425, 256◦ ), which are the true values ofthese parameters in the case under consideration. Now, we put this point on the right bottom
panel of Fig. 5, corresponding to the HG symmetry form for DUNE + T2HK, and find that
this point falls just outside the dark green band representing δCP values compatible with the
HG form at 3σ C.L. It means that if δCP predicted by GRB together with the present best
fit values of θ23 , θ12 , and θ13 are realised in Nature, then the HG symmetry form will be
disfavoured by DUNE + T2HK at > 3σ. The same message is conveyed from Table 5 as well.

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