#### Measuring the sterile neutrino CP phase at DUNE and T2HK

Eur. Phys. J. C
Measuring the sterile neutrino CP phase at DUNE and T2HK
Sandhya Choubey 0 1
Debajyoti Dutta 1
Dipyaman Pramanik 1
0 Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, AlbaNova University Center , 106 91 Stockholm , Sweden
1 Harish-Chandra Research Institute , HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019 , India
The CP phases associated with the sterile neutrino cannot be measured in the dedicated short-baseline experiments being built to test the sterile neutrino hypothesis. On the other hand, these phases can be measured in longbaseline experiments, even though the main goal of these experiments is not to test or measure sterile neutrino parameters. In particular, the sterile neutrino phase δ24 affects the charged-current electron appearance data in long-baseline experiment. In this paper we show how well the sterile neutrino phase δ24 can be measured by the next-generation longbaseline experiments DUNE, T2HK (and T2HKK). We also show the expected precision with which this sterile phase can be measured by combining the DUNE data with data from T2HK or T2HKK. The T2HK experiment is seen to be able to measure the sterile phase δ24 to a reasonable precision. We also present the sensitivity of these experiments to the sterile mixing angles, both by themselves, as well as when DUNE is combined with T2HK or T2HKK.
1 Introduction
Neutrino oscillation physics has reached precision era.
Observation of neutrino oscillations by the solar [
1
] and
atmospheric [
2
] neutrinos have been confirmed
independently at accelerator [
3,4
] and reactor [5] experiments. The
two mass squared differences and all three mixing angles of
the three-generation neutrino oscillation theory have been
well determined and the three-generation paradigm well
established [
6
]. The only remaining questions which are still
to be answered are CP violation in the leptonic sector,
neutrino mass hierarchy (that is, whether ν3 is the lightest or the
heaviest) and whether θ23 lies in the lower or in the higher
octant. Future experiments like DUNE [
7–10
] and T2HK
[
11,12
] are going to explore these questions and are expected
to come with definite answers.
Even though neutrino oscillation with three-generations is
well established, there are some hints of neutrino oscillations
at a higher frequency corresponding to a mass-squared
difference m2 ∼ 1 eV2 [
13
]. LSND experiment [
14,15
] in Los
Alamos, USA, first showed evidence for such oscillations,
where a ν¯μ beam was sent to a detector and the
observations showed a 3.8σ excess in the positrons, which could
be explained in terms of ν¯μ → ν¯e oscillations. For the L
and E applicable for the LSND experiment, this oscillation
corresponds to a mass-squared difference of m2 ∼ 1 eV2.
Experiments like KARMEN [16] and MiniBooNE [
17–19
]
tested the claim. While KARMEN data did not show any
evidence for ν¯μ → ν¯e oscillations, it could not rule out the
entire allowed region from LSND. More recently, the
MiniBooNE experiment ran in both neutrino as well as
antineutrino mode. MiniBooNE did not find any significant excess
in their neutrino mode, however they reported some excess
in the antineutrino mode consistent with the LSND result.
Apart from these, MiniBooNE also reported some excess
in the low energy bins for both neutrino and antineutrino
appearance channels, but these cannot be explained in terms
of neutrino flavour oscillations. The one additional sterile
neutrino can be fitted along with the three active neutrinos
in the so-called 3 + 1 [
20
] type neutrino mass spectrum. The
global fit of all the relevant short-baseline data shows severe
tension between the appearance and disappearance data sets
and the overall goodness of fit is only 31 %. However, if one
consider appearance and disappearance channels separately,
the goodness of fit improves slightly and it becomes 50 %
and 35 %, respectively [
21
].
In the 3 + 1 scenario we have 3 mass squared differences1
m241, m231 and m221, 6 mixing angles and 3 phases. ln
addition to the mixing angles θ12, θ23 and θ13 which appear
1 We define mi2j = mi2 − m2j .
also in the three-generation sector we have 3 additional angles
θ14, θ24 and θ34 involving the fourth generation. Also, there
are two new CP phases δ24 and δ34 in addition to the standard
CP phase δ13. At short-baseline experiments, the oscillation
probabilities Pee, Pμμ and Pμe in the 3 + 1 scenario can
be written in an effective two-generation framework which
depends only on m241 and an effective mixing term given as
a combination of the sterile mixing angles θ14, θ24 and θ34.
As a result, the short-baseline experiments are completely
insensitive to the sterile CP phases δ24 and δ34. On the other
hand, it is now well known that even though the m241-driven
oscillations get averaged out, these phases show up in the
oscillation probabilities at the long-baseline experiments. A
lot of effort in the last couple of years has gone into estimating
the impact of the sterile neutrino mixing angles and phases on
the measurement of standard oscillation parameters at
longbaseline experiments [
22–31
].2 These papers showed that in
presence of sterile neutrino mixing the sensitivity to the
measurement of CP violation, mass hierarchy, as well as octant of
θ23 becomes a band, where the width of the band comes from
the uncertainty on both the values of the sterile mixing angles
as well as the sterile phases. While the sterile neutrino
mixing angles are constrained by the global short-baseline data,
there are no constraints on the sterile phases. In the future,
bounds on the sterile neutrino mixing angles are expected to
improve by the data from forthcoming short-baseline
experiments [54–56]. Studies have shown that the long-baseline
experiments could also give constraints on the sterile neutrino
mixing at their near [57] and far [24,58] detectors. The sterile
phases on the other hand, can be constrained only in the
longbaseline experiments. A short discussion on the study of
sterile phases were done at T2K+reactor [
29
] and T2K+NOvA
[
28
] and the sensitivity was shown to be poor. In this paper,
we study how well the next-generation experiments DUNE
and T2HK will be able to measure the sterile phases. We give
the expected sensitivity of DUNE and T2HK alone as well
as by combining data from the two experiments. To the best
of our knowledge, this is the first time such a complete study
is being performed. While the authors in [
27
] did attempt to
present the expected precision on the sterile phase (which in
their parametrisation was δ14) in the DUNE experiment, the
analysis they performed has several short-comings. In their
analysis, the authors of [
27
] keep the sterile mixing angles
θ14, θ24 and θ34 fixed in the fit at their assumed true values.
We allow these angles to vary freely in the fit we perform in
this paper. This allows the uncertainty due to both the mixing
angles as well as the phases to impact out final results. We
also keep our sterile neutrino mixing angles within the
currently allowed limits, which have been updated following the
results from the NEOS, MINOS and MINOS+ experiments
2 Some recent studies on other new physics scenarios in the context of
DUNE and T2HK can be found here [32–53].
[
21
]. Expected precision on the sterile phase from T2HK has
never been studied before and we present them for the first
time. We will also show the combined expected sensitivity of
DUNE and T2HK to constrain the sterile mixing angles θ24
and θ14, both when the 3 + 1 scenario is true as well as when
there is no positive evidence for sterile neutrino oscillations.
The paper is organised as follows: In Sect. 2 we briefly
describe the sterile neutrino hypothesis and the simulation
procedure. In the same section we provide the details of the
DUNE, T2HK and T2HKK experiments. In Sect. 3 we give
our main results on how well the future long-baseline
experiments will constrain the sterile phase δ24 if the 3 + 1
scenario is indeed true. In Sect. 4 we present our results on the
expected constraints from future long-baseline experiments
if the experiments do not see any positive signal for sterile
neutrino oscillations. Finally, we conclude in Sect. 5.
2 Sterile neutrino mixing and simulation
As discussed above, extra light neutrino state(s) with mass ∼
O(1eV2) have been proposed to explain the LSND results. In
this article we have considered one additional sterile neutrino
within the so-called 3 + 1 scenario, where the three active
neutrinos are separated from the sterile neutrinos by a mass
gap of ∼ 1 eV. In this scenario the mixing matrix is 4 ×
4 and hence is defined in terms of six mixing angles and
three phases. The mixing matrix can be parametrised in the
following way:
U 3+1
P M N S = O(θ34, δ34)O(θ24, δ24)R(θ14)R(θ23)O(θ13,
δ13)R(θ12).
(1)
Here O(θi j , δi j ) are 4×4 orthogonal matrices with
associated phase δi j in the i j sector, and R(θi j ) are the rotation
matrices in the i j sector. There are three mass-squared
differences in the 3 + 1 scenario – the solar mass-squared
difference m221 7.5×10−5 eV2, the atmospheric mass-squared
difference m231 2.5 × 10−3 eV2, and the LSND
masssquared difference m241 1 eV2. One can of course write
the most general oscillation probabilities in terms of these
three mass-squared differences, six mixing angles and three
phases. However, since oscillation driven by a given
masssquared difference depends on the L/E of the experiment
concerned, the expression for the oscillation probabilities
simplify accordingly. In particular, the short-baseline
experiments have a very small L/E such that sin2( mi2j L/4E ) ∼
0 for m221 and m231 and only the terms for m241 survive.
This is the one-mass-scale-dominance case, where only one
oscillation frequency due to one mass scale survives. As a
result the short-baseline experiments depend on only
“effective” sterile mixing angles, which are combinations of the
mixing angles θi j in Eq. (1). More importantly, since they
have only one oscillation frequency, they do not depend on
any CP violation phase. Hence, short-baseline experiments
are completely insensitive to the sterile phases for the 3 + 1
scenario.3
In long-baseline experiments such as T2HK and DUNE,
the oscillations driven by m231 dominate while those driven
by m221 are sub-dominant, while the very fast oscillations
driven by m241 ∼ O(1eV2) get averaged out. The transition
probability Pμe in the limit sin2( m241 L/4E ) ∼ 1/2 and
neglecting earth matter effect is [
22
]:
Pμ4eν = P1 + P2(δ13) + P3(δ24) + P4(δ13 + δ24).
(2)
Here P1 is the term independent of any phase, P2(δ13)
depends only on δ13, P3(δ24) depends only on δ24 and
P4(δ13 + δ24) depends on the combination (δ13 + δ24). The
full expression of the different terms in Eq. (2) are as follows:
1
P1 = 2 sin2 2θμ4νe
P2(δ13) = ba2 sin 2θμ3νe cos(δ13) cos 2θ12 sin2
P3(δ24) = ba sin 2θμ4νe cos(δ24)
+
1
a2 sin2 2θμ3νe − 4 sin2 2θ13 sin2 2θμ4νe
×(cos2 θ12 sin2
31 + sin2 θ12 sin2
+ b2a2 − 41 a2 sin2 2θ12 sin2 2θμ3νe
1
− 4 cos4 θ13 sin2 2θ12 sin2 2θμ4νe sin2
+ sin2 31 − sin2 32
1
− 2 sin(δ13) sin 2 21 − sin 2 31
+ sin 2 32 ,
× cos 2θ12 cos2 θ13 sin2
− sin2 θ13(sin2 31 − sin2 32
1
+ 2 sin(δ24) cos2 θ13 sin 2 21
21
+ sin2 θ13(sin 2 31 − sin 2 32) ,
×
1
− 2 sin2 2θ12 cos2 θ13 sin2
+ cos 2θ13(cos2 θ12 sin2
31
+ sin2 θ12 sin2
32)
21
32)
21,
(3)
21
(4)
(5)
P4(δ13 + δ24) = a sin 2θμ3νe sin 2θμ4νe cos(δ13 + δ24)
3 In the 3+2 mass spectrum case, there are two sterile neutrinos and two
mass squared difference that affect the oscillations at very short
baselines. In this case therefore, the short baseline experiments are sensitive
to the sterile CP phases.
1
+ 2 sin(δ13 + δ24) cos2 θ12 sin 2 31
We can see from Eq. (2) that even though the m241-driven
oscillations are averaged out, the CP phases associated with
the sterile sector still appear in the neutrino oscillation
probability Pμe. This dependence comes in term P3(δ24) that
depend only on the sterile phase δ24 as well as in term
P4(δ13 + δ24) which depends on combination of δ13 and
δ24. Hence, we can expect the next-generation long-baseline
experiments to be sensitive to the sterile phases. We will
see the anti-correlation between δ13 and δ24 manifest in our
results on measurement of these phases in the long-baseline
experiments. In fact, as has been pointed above, the sterile CP
phases cannot be measured in the short-baseline experiments
which are dedicated to measuring the sterile neutrino
mixing. Hence, experiments like DUNE and T2HK are the only
place where δ24 can be measured in the 3 + 1 scenario. Note
that in Eq. (2) the probability Pμe does not depend on the
mixing angle θ34, hence the corresponding phase associated
with this angle δ34 also does not appear. Once earth matter
effects are taken into account the probability Pμe picks up a
θ34 dependence and hence depends on δ34 as well. However,
for DUNE and T2HK experiments earth matter effects are
rather weak and hence their corresponding sensitivity to δ34
cannot be expected to be strong. Therefore, as we will see in
the Results section, these experiments are mainly able to put
constraints on δ24.
Prior to proceeding, we briefly discuss our simulation
procedure as well as the present statues of the neutrino
oscillation parameters. For our analysis we have used GLoBES
(Global Long Baseline Experiment Simulator) [59,60] along
with the additional codes [61,62] for calculating
probabilities in the 3 + 1 scenario. We have used constant matter
density for all the cases. Throughout the analysis we choose
the true values4 of the standard oscillation parameters as:
θ12 = 33.56◦, θ13 = 8.46◦, θ23 = 45◦, m221 = 7.5 × 10−5
eV2, m231 = 2.5 × 10−5 eV2 and δ13 = − 90◦ unless
specified otherwise. This choice of parameters are consistent with
the current limits [63]. Although one should marginalize over
all the free parameters whenever one introduces some new
physics, but new physics scenarios often give large number
4 Throughout this paper we refer to the oscillation parameter values at
which the “data” is generated as the “true value” and values in the fit as
“test values”.
of parameters. Marginalisation over these large number of
parameters is computationally challenging, so one has to do
some approximation. For our analysis we have checked that
the effect of marginalisation over the standard three neutrino
parameters other than δ13 have no significant effect. So to
save computational time we did not marginalise over these
not so relevant set of parameters.
For the sterile neutrino mixing, we have considered two
scenarios. We first start by assuming that active-sterile
neutrino oscillations indeed exist and find the expected
constraints on the sterile neutrino mixing angles and phase
δ24 assuming non-zero sterile neutrino mixing angles in the
“data”. For this case we generate the “data” at true values
m241 ≈ 1.7 eV2, θ14 ≈ 8.13◦, θ24 ≈ 7.14◦, θ34 = 0◦,
which are the current best-fit values taken from [
21
]. The true
values of the sterile phases will be specified. We marginalise
our χ 2 over all sterile mixing parameters except m241 in the
fit. The χ 2 is marginalised by varying θ14, θ24 and θ34 in the
range [5◦, 10.5◦], [4◦, 9.5◦], and [0◦, 12◦], respectively [
21
]
without any Gaussian prior, while the phases δ24 and δ34 are
varied in their full range ∈ [−180◦, 180◦].
We next assume a scenario where the sterile neutrino
mixing does not exist in nature and we show how well would then
the long-baseline experiments DUNE and T2HK constrain
the sterile neutrino mixing angles. For this case the data of
course corresponds to true sterile mixing angles zero. The
marginalisation of the χ 2 is done over all the three sterile
mixing angles and the three phases. Mixing angles θ14, θ24
and θ34 are marginalised in the range [0◦, 10◦], [0◦, 10◦], and
[0◦, 12◦], respectively, while the phases are allowed to vary
in their full range ∈ [−180◦, 180◦].
2.1 DUNE
DUNE (Deep Underground Neutrino Experiment) [
7–10
] is
a future long baseline experiment proposed in US. Purpose
of DUNE is to address all the three unknowns of the neutrino
oscillation sector – the leptonic CP violation, mass-hierarchy
and the octant of the θ23. DUNE will consist of a source
facility at Fermilab and a far detector at Sanford Underground
Research facility in South Dakota at a distance of 1300 km
from the source. Hence, the baseline of the experiment is
1300 km. The accelerator facility at Fermilab will give a
proton beam of energy 80–120 GeV at 1.2–2.4 MW which
will eventually give a wide-band neutrino beam of energy
range 0.5–8.0 GeV. The far site will consist of 4 identical
detector of 10 kt each which will give fiducial mass of 34
kt. All the detectors will be LArTPC (Liquid Argon Time
Projection Chamber).
In this work we have considered a Liquid Argon detector
of fiducial mass 34 kt at a baseline of 1300 km. The neutrino
flux is given by the 120 GeV, 1.2 MW proton beam. Here we
have considered 5 years of neutrino and 5 years of
antineutrino. Appearance and disappearance channels are combined
for the analysis. The energy resolutions for the μ and e are
taken to be 20%/√E and 15%/√E , respectively. The
signal efficiency is taken to be 85%. The backgrounds are taken
from [
8
]. In the neutrino (antineutrino) mode, the signal
normalization error is 2% (5%), the background normalization
error is 10% (10%) and the energy calibration error is 5%
(5%). This choice of systematics is conservative compared
to the projected systematics in [
8
].5
2.2 T2HK
The Hyper-Kamiokande (HK) [
11,12
] is the upgradation of
the Super-kamiokande (SK) [65] program in Japan, where the
detector mass is projected to be increased by about twenty
times the fiducial mass of SK. HK will consist of two 187 kt
water Cherenkov detector modules, to be place near the
current SK site about 295 km away from source. The detector
will be 2.5◦ off-axis from the J-PARC beam which is
currently being used by the T2K experiment [66]. T2HK has
similar physics goals as DUNE, but since it will employ a
narrow-band beam, it can be complimentary to the DUNE
experiment.
For our analysis we take a beam power of 1.3 MW and
the 2.5◦ off-axis flux. We consider a baseline of 295 km and
the total fiducial mass of 374 kt (two tank each of which
is 187 kt). We consider 2.5 years of neutrino and 7.5 years
of anti-neutrino runs in both appearance and disappearance
channels. The energy resolution is taken to be 15%/√E .
The number of events are matched with the Tables 2 and 3
of [
12
]. The signal normalization error in νe(ν¯e) appearance
and νμ(ν¯μ) disappearance channel are 3.2% (3.6%) and 3.9%
(3.6%), respectively. The background and energy calibration
errors in all channels are 10 and 5%, respectively.
2.3 T2HKK
In [
12
], the collaboration has also discussed the
possibility of shifting one of the water tanks to a different location
in Korea at a distance of about 1100 km from the source.
This proposed configuration will consist of the same
neutrino source but with one detector of fiducial mass 187 kt at
a baseline of 295 km and another similar detector of fiducial
mass 187 kt at a baseline of 1100 km. The second
oscillation maximum takes place near Eν = 0.6 GeV at the second
detector. Both detectors are taken 2.5◦ off-axis in our study.
5 The experimental specifications for DUNE has been updated in [64].
However, we have explicitly checked that the physics results do not
differ much for the two specifications. The older version was for 5 + 5
years of run while the newer one is for 3.5 + 3.5 years of run and yet
inspite of lower statistics the newer version is able to achieve similar
physics goal because of the optimised fluxes, detector response and
systematics.
Since the flux peaks at the same energy for both detector
locations, the Japan detector sees the flux at the first oscillation
maximum while the Korea detector sees it at the second
oscillation maximum. This whole setup is called T2HKK. In our
analysis, we have considered signal normalization error of
3.2% (3.6%) in νe (ν¯e) appearance channel and 3.9% (3.6%)
in νμ (ν¯μ) disappearance channel, respectively. The
background and energy calibration errors are 10% and 5% in all
the channels, respectively.
3 Measurement of the sterile phases
In this section we discuss the ability of the long baseline
experiments to constrain the sterile phases. The “data” is
generated for the 3 + 1 scenario for the values of mixing
parameters discussed in Sect. 2. In particular, for the sterile neutrino
parameters we take the following values: m241(true) = 1.7
eV2, θ14(true) = 8.13◦, θ24(true) = 7.14◦, θ34(true) = 0◦.
The true values of the phases δ24 will be taken at some
benchmark values and will be mentioned whenever needed. The
true values of standard oscillation parameters are taken at
their current best-fit values, mentioned in Sect. 2. The χ 2 is
marginalised over the relevant oscillation parameters in the
3 + 1 scenario, as discussed in Sect. 2, where the parameters
are allowed to vary within their current 3σ ranges. Although
there are three phases in the 3 + 1 scenario, the role of the
phase δ34 is weak. As was discussed in the previous section,
the mixing angle θ34 affects the oscillation probability Pμe
only when matter effects become important. For DUNE and
T2HKK earth matter effects are not very strong while for
T2HK the effect of earth matter is even weaker. Since the
impact of the phase δ34 on Pμe is proportional to the mixing
angle θ34, the phase δ34 is also less important for Pμe for
the same reason. Moreover, the current global best-fit for the
angle θ34 turns out to be zero [
21
]. Therefore, in this work
we set θ34(true) = 0◦ in the data. As a result the phase δ34 is
not expected to be very crucial in our analysis and hence we
take and δ34(true) = 0◦ in the data and show our results only
in the δ13–δ24 plane. We reiterate that the χ 2 is marginalised
over the mixing angle θ34 and phase δ34 in the fit, where the
mixing angle is allowed to vary between [0◦, 12◦] [
21
].
The Fig. 1 shows the capability of DUNE, T2HK and
T2HKK to measure the phase δ24. We show the plots of
χ 2 as a function of δ24(test) for T2HK (dotted black lines),
T2HKK (dash-dotted red lines) and DUNE (dashed blue
lines) for δ24(true) of 0◦ (top-left panel), 90◦ (top-right
panel), − 90◦ (bottom-left panel) and 180◦ (bottom-right
panel). The true values of all other parameters are taken as
detailed in Sect. 2 and the previous paragraph. The χ 2 plot has
been marginalised over all relevant parameters as discussed
before. The green solid horizontal lines show the χ 2
corresponding to 2σ CL. Table 1 shows that T2HK can better
constrain the phase δ24 as compared to DUNE, while T2HKK is
expected to perform better than DUNE but worse than T2HK.
Note that the sensitivity of DUNE and T2HKK is marginally
better for δ24(true) = 90◦ than for δ24(true) = − 90◦ while
the reverse is true in case of T2HK (see Table 1).
In order to understand why the measurement of δ24 is
expected to be better at T2HK than DUNE, we show in Fig. 2
the expected electron events at DUNE (top left panel) and
T2HK (top right panel). The four lines in each panel show
the expected events for four values of δ24 = 0◦ (solid red
lines), 90◦ (dashed blue lines), − 90◦ (dotted green lines)
and 180◦ (dash-dotted dark red lines). The upper panels of
the figure reveal that the two experiments behave in almost
the same way as far as the dependence of the probability
Pμe to δ24 is concerned. However, there is a clear difference
between the two when it comes to the overall statistics. T2HK
expects to see nearly 14 times more events than DUNE due
to its bigger detector size. Hence the corresponding χ 2 for
T2HK is also expected to be higher. Of course the systematic
uncertainty for DUNE is considerably less than for T2HK and
that compensates the effect of the lower statistics, however,
the effect of statistic shows up in a non-trivial way for δ24
measurement at the long baseline experiments and T2HK
with its bigger detector emerges as a better option in this
regard.
The lower panels of Fig. 2 show the event rate at the
oscillation maximum for DUNE (lower left panel) and T2HK
(lower right panel) as a function of δ24. The black solid
curves show the expected number of events whereas the
green and yellow bands show the 1σ and 3σ statistical
deviation. The black short-dashed straight lines show the event
rate at oscillation maxima for the four benchmark values of
δ24 = 0◦, 180◦, 90◦ and − 90◦. The Fig. 1 had revealed that
the χ 2 corresponding to δ24(true) = 0◦ and δ24(true) = 180◦
are much lower compared to that for δ24(true) = ± 90◦. This
can be understood from the lower panels of Fig. 2 as
follows. Figure 2 shows that the predicted number of events at
oscillation maximum for δ24(true) = 0◦ and 180◦ lie between
the predicted events for δ24(true) = ± 90◦. Therefore, for the
cases where data is generated for δ24(true) = 0◦ and 180◦, it
is easier for other δ24 values to fit the data and give a smaller
χ 2. However, data corresponding to δ24(true) = ± 90◦ takes
a more extreme value and the difference between the data
and fit for other values of δ24 for these cases becomes larger,
giving larger χ 2.
Another interesting feature visible in the event plots in
Fig. 2 is that the maxima and minima of the events are not at
δ24 = 90◦ or −90◦. Rather they are slightly shifted towards
the right. For the same reason the χ 2 plots in Fig. 1 are also
asymmetric about the true value of δ24. One can explain this
using Eq. (2). By inspecting the probability one can see that
the correlation between m221 and δ24 is negligible. Also, we
have taken δ13 = − 90◦ everywhere. Hence, for m221 = 0
and δ13 = − 90◦, the Eq. (2) can be rearranged as,
1
Pμe = A + B cos 2θ13 sin δ24 − 2 B cos δ24 ,
(8)
where A and B are independent of δ24. In the absence of the
last term, the probability would be a sine function shifted by
the constant A. However, the presence of the cosine term
shifts the curve and the shift is towards right because of
the minus sign in front of the cosine term. In particular, the
extrema of the probability in Eq. (8) is given by the condition,
339
appearance event rates at the oscillation maximum as a function of δ24
for DUNE (left) and T2HK (right). While the black curves give the
expected number of events, the green and yellow bands show the 1σ
and 3σ statistical uncertainties
1
cos δ24 = − 2 sin δ24 ,
which corresponds to minimum at δ24 = −63.4◦ and
maximum at δ24 = 116.6◦. This agrees very well with the event
plots in the lower panels of Fig. 2 which is obtained using
the exact numerical probability.
We next show in Fig. 3 the expected 95% CL allowed areas
in the δ13(test)−δ24(test) plane, expected to be measured
by the next generation long-baseline experiments T2HK (or
T2HKK) and DUNE, and by combining them. The four
panels of Fig. 3 have been generated for four different choices
of δ24(true). The value of δ13(true)= −90◦ in all the
panels. In each panel the benchmark point where the data is
(9)
generated is shown by the black star. The four panels
correspond to δ24(true)= 0◦ (top left), 90◦ (top right), −90◦
(bottom left) and 180◦ (bottom right). In all the four cases
we have considered 3+1 scenario both in the ‘data’ and in the
‘fit’ or ‘theory’. The χ 2 thus generated is then marginalised
over the sterile mixing angles θ14, θ24, θ34 and δ34, as
discussed before. The black dotted, red dash-dotted and blue
dashed contours are for T2HK, T2HKK and DUNE,
respectively, while the grey and magenta solid contours are for
DUNE+T2HK and DUNE+T2HKK. As in Fig. 1 we note that
T2HK can constrain the phase δ24 much better than DUNE,
while T2HKK performs better than DUNE but worse than
T2HK. We also see, as before, that for DUNE the precision
δ24 = − 90◦ (δ24 = 180◦). The black dotted curve is for T2HK, the red
dash-dotted curve is for T2HKK, the blue dashed curve is for DUNE,
the grey solid curve is for DUNE+T2HK and the magenta solid curve
is for DUNE+T2HKK
on δ24 is expected to be better for δ24(true)= ±90◦
compared to when δ24(true)= 0◦ or 180◦. For T2HK this
dependence of precision on δ24 measurement on δ24(true) is less
pronounced. The effect of θ34 on the measurement of δ24 is
also minimal. Finally, note that there is an anti-correlation
between δ13 and δ24. This comes from the term P4(δ13 + δ24)
of Eq. (2).
The Fig. 3 also shows how the measurements of δ24
and δ13 improve as we combine DUNE with either T2HK
or T2HKK. We see that combining DUNE with T2HKK
improves the precision considerably, with the combined
precision of DUNE and T2HKK becoming slightly better than
the precision expected from T2HK alone. Combining DUNE
with T2HK improves the precision even further, albeit only
marginally, since T2HK alone can measure the phases rather
precisely.
The question on how the measurement of the standard CP
phase δ13 gets affected by the sterile mixing angle phases
in the 3 + 1 scenario is another pertinent question that one
can ask. The Fig. 4 shows how the expected precision on
δ13 changes in presence of sterile neutrinos. The left panel is
for T2HK, middle panel is for T2HKK and right panel is for
DUNE. The blue dashed curves are for the standard 3+0 case
with no sterile neutrinos while the red dash-dotted curves
are for the 3 + 1 case with δ24 = − 90◦ in data. The other
standard and sterile neutrino oscillation parameters are taken
in data as described above and the fit performed as before.
The Fig. 4 shows that the expected precision on δ13 worsens
when the sterile neutrino is present. From Eq. (6) one can see
that there is an anti-correlation between δ13 and δ24 which
makes the δ13 precision worse. For DUNE the effect is more
compared to T2HK and T2HKK. For T2HK the δ13
measurement is seen to be nearly unaffected. DUNE measures
δ24 worse than T2HK and T2HKK and hence the
corresponding measurement of δ13 worsens due to the anti-correlation
mentioned above. Table 2 summarises the expected precision
on δ13 for the 3 + 0 and 3 + 1 scenario for four benchmark
values of δ24(true). We see that DUNE’s measurement of δ13
gets affected for all δ24 while effect on T2HK’s measurement
of δ13 is negligible.
4 Measurement of the mixing angles
Prospects of measuring the sterile neutrino mixing angles
at long-baseline experiments DUNE [
24
] and T2HK [58]
has been studied before. Here we study how well the
sterile mixing can be constrained by combining data from these
experiments. We also present the sensitivity of the individual
experiment DUNE, T2HK and T2HKK. Here we consider
two complementary approaches. We first assume that sterile
neutrino mixing does exist (as in the last section) and see how
precisely the data from long-baseline experiments can
measure and constrain the angles θ14 and θ24.6 We next consider
6 We do not study the mixing angle θ34 in this work. As discussed
before, this affects Pμe and Pμμ only mildly through matter effects. To
constrain this angle, we need to consider the neutral current data, which
has been done in [67–69].
and the red dashed curves are for 3 + 1 case in both theory and data.
The curves are δ24(true) = −90◦
the alternate situation where the active-sterile oscillations do
not really exist and then we see how well DUNE, T2HK
and T2HKK, as well as their combination, could put upper
bounds on the sterile neutrino mixing angles θ14 and θ24.
4.1 Measuring the sterile mixing angles when 3 + 1 is true
In this section we assume that the 3 + 1 scenario is indeed
true in nature and the mixing angles θ14 and θ24 are indeed
non-zero. We perform a χ 2 analysis with prospective data
generated in the 3 + 1 scenario and fitted within the 3 + 1
scenario and give expected allowed CL regions in the
sterile neutrino parameter spaces. As before, we take the true
sterile oscillation parameters at the following benchmark
values: m241(true) = 1.7 eV2, θ14(true) = 8.13◦,
θ24(true) = 7.14◦, θ34(true)= 0◦ which are consistent
with [
21
]. The standard oscillation parameters are taken and
treated as discussed before. The χ 2 is marginalised over all
relevant parameters in the fit and no Gaussian priors are
included.
In Fig. 5, we show the contours in sin2 θ14(test)– m241(test)
plane (left panel) and sin2 θ24(test)– m241(test) plane (right
panel). The colour code is same as Fig. 3. The point where
the data is generated is shown by the black star in the two
panels. The results show that in both panels, T2HK gives
better results than both DUNE and T2HKK. Again, T2HKK
is better than DUNE. Combining T2HK/T2HKK and DUNE
experiments improves the results and the expected allowed
ranges for the sterile neutrino mixing parameters shrink. The
precision expected from the combined data from T2HK and
DUNE is nearly the same as that from T2HKK and DUNE,
which for the former is only marginally better.
Figure 6 shows the contours in the sin2 θ14(test)–sin2 θ24(test).
The colour code for the different data sets considered is the
same as Fig. 3. We note that the expected upper limit on
sin2 θ24 is the same for all the three individual experiments
for 0.001 < sin2 θ14 < 0.03. Also, the expected upper bound
on sin2 θ14 is seen to be better for T2HK than DUNE.
Combining the experiments can improve the measurement of the
sterile neutrino mixing angles as seen from the solid contours
in Fig. 6. In particular, we now see a lower bound on sin2 θ24.
4.2 Excluding the sterile hypothesis when 3 + 1 is not true
If the sterile neutrino hypothesis was wrong and there was
no mixing between the active and sterile neutrinos the
nextgeneration experiments would falsify it. There are a series
of new short-baseline experiments planned which will be
testing this hypothesis [54, 55, 57]. Even the near detector of
planned long-baseline experiments are well-suited to check
the sterile neutrino mixing as their baseline and energy match
well to correspond to the maximum of m241-driven
oscillations [57]. In the same vein it is pertinent to ask how well
the next-generation long-baseline experiments could
constrain this hypothesis, since the oscillation probabilities for
long-baseline experiments also depend on the sterile
neucorrespond to standard three-generation oscillation scenario with no
sterile mixing while the fit is done in the 3 + 1 framework to obtain the
exclusion contours. The colour code is same as Fig. 3
trino mixing and phases even though the m241-driven
oscillations themselves average out. While some work in this
direction has already been done in the literature [
24, 58
],
we will present here, for the first time, the sensitivity of
T2HKK set-up to the sterile neutrino mixing angles θ24 and
θ14. We will also present the expected sensitivity from the
combined prospective data-sets of T2HK (or T2HKK) and
DUNE, which has not been studied before.
In Fig. 7 we show the exclusion curves for the 3 + 1
hypothesis expected from the next-generation long-baseline
experiments. The left panel of Fig. 7 shows the 95% CL
exclusion plots in the m241(test)-sin2 θ14(test) plane while
the right panel shows the results in m241(test)-sin2 θ24(test)
plane. Here we generate the data assuming the standard
three-generation neutrino scenario and then fit it with the
3 + 1 scenario. The blue dashed, black dotted and red
dashdotted curves show the exclusion plots for DUNE, T2HK
and T2HKK, while the magenta and grey solid curves show
the expected exclusion sensitivity for DUNE+T2HKK and
DUNE+T2HK, respectively. If only three neutrinos exist
in the nature, then the parameter region in the top-right of
the plots are excluded at 95% CL. Again, as in the
previous results, T2HK constrains θ14 better than both T2HKK
and DUNE for all values of m241 in the range 10−3 eV2
to 10 eV2. Combining DUNE with T2HK and T2HKK can
improve the constraint on sin2 θ14 compared to the
individual experiments. Since for higher values of m241 the
oscillations average out, the experiments become almost
insensitive to the value of m241. For 0.1 eV2 < m241 < 10.0
eV2, DUNE, T2HKK and T2HK can exclude sin2 θ14 ∼> 0.4,
sin2 θ14 ∼> 0.27 and sin2 θ14 ∼> 0.21, respectively, at 95%
CL. For the same range of values of m241, DUNE+T2HK
and DUNE+T2HKK could put slightly tighter constrain
on sin2 θ14 and the excluded regions are expected to be
sin2 θ14 ∼> 0.165 and sin2 θ14 ∼> 0.18 at 95% CL,
respectively.
The results presented in the right panel show the
capability of these experiments to constrain sin2 θ24. If m241
is small and lies in the range 10−3 eV2 < m241 < 0.01
eV2, T2HK gives better constraint on sin2 θ24 than both
DUNE and T2HKK. But for higher values of m241, in the
range of 0.014 eV2 < m241 < 0.1 eV2, the performance
of DUNE is better than both T2HK and T2HKK. For 0.1
eV2 < m241 < 10.0 eV2, performance of T2HKK is almost
similar to that of DUNE. Similar behaviour can be seen in
the combined case. In the lower m241 region, DUNE+T2HK
could constrain sin2 θ24 slightly better than DUNE+T2HKK.
But in the higher m241 region, DUNE+T2HKK is expected
to perform better than DUNE+T2HK. The expected
exclusion sensitivity for DUNE, T2HKK and T2HK in the range
0.1 eV2 < m241 < 10.0 eV2 are given as sin2 θ24 ∼> 0.026,
sin2 θ24 ∼> 0.026 and sin2 θ24 ∼> 0.03, at 95% CL
Similarly, the expected exclusion limit for DUNE+T2HKK
and DUNE+T2HK at 95% CL are sin2 θ24 ∼> 0.017 and
sin2 θ24 ∼> 0.019, respectively, for 0.1 eV2 < m241 < 10.0
eV2.
In Fig. 8 we show the expected exclusion contour in
sin2 θ14(test)-sin2 θ24(test) plane. Here, the region outside
the contour is excluded at 95% CL. The figure represents
the slice at m241 = 1.7 eV2 of the contour in the sin214,
sin2 θ24, m241 space. The colour code is the same as in Fig. 7.
Here also, we observe better capability of T2HK to constrain
sin2 θ14-sin2 θ24 parameter space than DUNE and T2HKK
in most regions of the parameter space. The plot also shows
that constraint on sin2 θ24 is complicated. We see that T2HK
is better than DUNE and T2HKK in constraining sin2 θ24 for
sin2 θ14 ∼> 10−2. However, for sin2 θ14 ∼< 10−2 DUNE and
T2HKK perform better than T2HK in constraining sin2 θ24.
Combining the data-sets improves the expected sensitivity
on both the sterile mixing angles.
5 Conclusions
There are a number of observational hints that support the
existence of neutrino oscillations at short baselines. Since
the m2 needed for these frequencies is inconsistent with
the m2 needed to explain the solar and atmospheric
neutrino anomalies – both of which have been confirmed by
earth-based experiments – it has been postulated that there are
additional light neutrino states which are mixed with the three
standard neutrino states. Since the Z -decay width restricts the
number of light neutrino states coupled to the Z boson to 3,
the additional neutrino states should be “sterile”. In this paper
we considered one extra such sterile neutrino in the so-called
3 + 1 mass spectrum. In the 3 + 1 scenario the neutrino
oscillation parameter space is extended by one new mass squared
difference m241, three new active-sterile mixing angles θ14,
θ24 and θ34 and two new CP phases δ24 and δ34. We work
within a parametrisation of the mixing matrix such that the
phase δ24 is associated with the mixing angle θ24 and δ34 is
associated with θ34. It is now well known that even though
the m241-driven oscillations are averaged out in the
longbaseline experiments, the active-sterile mixing angles and
the additional phases appear in the oscillation probabilities
and modify it. The sensitivity of the long-baseline
experiments to the active-sterile mixing angles has been studied
before. The impact of the sterile neutrino parameters on the
physics reach of these experiments for standard parameter
measurement such as CP violation, mass hierarchy
measurement and octant of θ23 measurement has been investigated
in details before. In this work, for the first time, we looked
at the prospects of measuring the sterile CP phase δ24 in the
long-baseline experiment T2HK (and T2HKK) and DUNE
as well as when data from them is combined.
Dedicated short-baseline experiments are being built
to test the active-sterile neutrino oscillation hypothesis.
However, these experiments are sensitive to oscillation
probabilities that have one-mass-scale-dominance. In other
words, these experiments mainly work within effective
twogeneration scenarios and are sensitive to m241 and effective
two-generation mixing angles, that can be written as
combination of the mixing angles θ14 and θ24. Therefore, they
are completely insensitive to the CP phases δ24 and δ34. On
the other hand, these phases do affect the oscillation
probabilities of the long-baseline experiments even though they
are not sensitive to m241 since oscillations corresponding
to this frequency averages out. In particular, the probability
Pμe that affects the electron appearance data in long-baseline
experiments depends on δ24. We exploited this dependence to
show that the phase δ24 can be measured at DUNE, T2HK and
T2HKK and estimated the expected precision on this
parameter. Since Pμe depends on θ34 only through earth matter
effects, the effect of δ34 on the long-baseline data is expected
to be very small and we cannot constrain it easily. Hence, in
this paper we concentrated only on δ24. We performed a χ 2
analysis of the prospective data at these experiments and
presented the expected precision on δ24 expected at these
experiments. We showed that with (7.5 + 2.5) years of running
in (neutrino, antineutrino) mode, T2HK could constrain δ24
to within [−63.0◦,47.88◦], [30.10◦, 147.9◦] and [−145.95◦,
−31.93◦] if the true value of δ24(true) is 0◦, 90◦ and − 90◦.
For δ24(true) = 180◦, all test δ24 such that δ24 ≤ −128.43◦
and δ24 ≥ 136.22◦ are allowed in T2HK. The corresponding
constraints from T2HKK and DUNE were seen to be weaker,
with DUNE measurement on δ24 expected to be the weakest.
We explained why T2HK is expected to perform better than
DUNE, with the main reason being the higher statistics in
T2HK. We also showed the expected allowed areas in the
δ13(test)-δ24(test) plane from prospective data from DUNE,
T2HK and T2HKK. We again reiterated that the expected
constraints on δ24 were seen to be strongest from the T2HK
experiment, while DUNE was seen to be weakest. We also
presented the allowed areas in this plane expected from
combined prospective data of DUNE and T2HK (or T2HKK). We
showed that the combined data set could constrain the CP
parameters better. Expected constraints from DUNE+T2HK
was seen to be better than constraints from DUNE+T2HKK.
We also discussed the impact of the sterile neutrino mixing
angles and phases on the measurement of the standard CP
phase δ13. We showed that for DUNE the expected δ13
precision worsens more than for T2HK.
We also presented constraints on the mixing angles θ14
and θ24. Again, we did not consider θ34 since the appearance
and disappearance data in long-baseline experiments depend
only mildly on this angle, with the dependence coming solely
from matter effects. We took two complementary approaches
in this study. First we assumed the 3 + 1 scenario to be
correct and generated data assuming non-zero values of the
sterile neutrino oscillation parameters. This was used to present
allowed areas in the sterile neutrino parameter space expected
from full run of DUNE, T2HK, T2HKK and combinations of
DUNE+T2HK and DUNE+T2HKK. We made a comparison
between the expected precision reach of the different
experiments. We next took the complementary approach where
we assumed that the 3 + 1 scenario was not true. The data
in this case was generated for no sterile mixing and fitted
with the 3 + 1 hypothesis to yield expected exclusion
limits on the sterile neutrino mixing parameters. Again, we did
this analysis for DUNE, T2HK, T2HKK and combinations of
DUNE+T2HK and DUNE+T2HKK and made a comparative
analysis of the different data-sets.
In conclusion, the sterile neutrino phases can be measured
only in experiments that are sensitive to more than one
oscillation frequency other than m241. The long-baseline
experiments therefore are the best place to measure δ24. We showed
that the sterile phase can be measured to reasonable precision
in the next-generation long-baseline experiments. The T2HK
set-up is better suited to measure δ24 compared to DUNE due
to larger statistics.
Acknowledgements We acknowledge the HRI cluster computing
facility (http://cluster.hri.res.in). The authors would like to thank the
Department of Atomic Energy (DAE) Neutrino Project of
HarishChandra Research Institute. This project has received funding from the
European Union’s Horizon 2020 research and innovation programme
InvisiblesPlus RISE under the Marie Sklodowska-Curie grant
agreement No 690575. This project has received funding from the European
Union’s Horizon 2020 research and innovation programme Elusives
ITN under the Marie Sklodowska-Curie grant agreement No 674896.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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