aainstitutetext: Physical Research Laboratory, Navrangpura,Ahmedabad-380009,
Indiabbinstitutetext: Assam Don Bosco University, Tapesia Campus, Sonapur,
Assam, 782402, Indiaccinstitutetext: Instituto de Física Gleb Wataghin -
UNICAMP, 13083-859, Campinas, São Paulo, Brazil

# Invisible neutrino decay : First vs second oscillation maximum

Kaustav Chakraborty b Debajyoti Dutta a Srubabati Goswami a,c Dipyaman
Pramanik<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the physics potential of the long-baseline experiments T2HK, T2HKK
and ESS$\nu$SB in the context of invisible neutrino decay. We consider normal
mass ordering and assume that the state $\nu_{3}$ as unstable, decaying into
sterile states during the flight and obtain constraints on the neutrino decay
lifetime ($\tau_{3}$). We find that T2HK, T2HKK and ESS$\nu$SB are sensitive
to the decay-rate of $\nu_{3}$ for $\tau_{3}/m_{3}\leq 2.72\times
10^{-11}$s/eV, $\tau_{3}/m_{3}\leq 4.36\times 10^{-11}$s/eV and
$\tau_{3}/m_{3}\leq 2.43\times 10^{-11}$s/eV respectively at 3$\sigma$ C.L. We
compare and contrast the sensitivities of the three experiments and specially
investigate the role played by the mixing angle $\theta_{23}$. It is seen that
for experiments with flux peak near the second oscillation maxima, the poorer
sensitivity to $\theta_{23}$ results in weaker constraints on the decay
lifetime. Although, T2HKK has one detector close to the second oscillation
maxima, having another detector at the first oscillation maxima results in
superior sensitivity to decay. In addition, we find a synergy between the two
baselines of the T2HKK experiment which helps in giving a better sensitivity
for $\theta_{23}$ in the higher octant. We discuss the octant sensitivity in
presence of decay and show that there is an enhancement in sensitivity which
occurs due to the contribution from the survival probability $P_{\mu\mu}$
which is more pronounced for the experiments at the second oscillation maxima.
We also obtain the combined sensitivity of T2HK+ESS$\nu$SB and
T2HKK+ESS$\nu$SB as $\tau_{3}/m_{3}\leq 4.36\times 10^{-11}$s/eV and
$\tau_{3}/m_{3}\leq 5.53\times 10^{-11}$s/eV respectively at 3$\sigma$ C.L.

##  1 Introduction

Neutrinos are one of the most fascinating particles in nature. Observation of
oscillation of neutrinos signifies evidence of neutrino mass. The very
existence of neutrino mass is the reason of neutrinos being so special since
in the Standard Model (SM) of particle physics there is no explanation of
neutrino mass. The standard formalism of neutrino oscillation requires three
mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{23}$, two mass-squared
differences $\Delta m^{2}_{21}$ and $\Delta m^{2}_{31}$ and one CP phase
$\delta_{CP}$. The parameters $\theta_{12}$, $\theta_{13}$, $\Delta
m^{2}_{21}$ and the absolute value of $\Delta m^{2}_{31}$ have been measured
very precisely by oscillation experiments. However, the octant of
$\theta_{23}$, the sign of $\Delta m^{2}_{31}$ (mass-ordering) and
$\delta_{CP}$ are not yet determined with certainty. Global analysis results
Esteban:2020cvm ; deSalas:2020pgw ; Capozzi:2017ipn  indicate towards normal
hierarchy of $\Delta m^{2}_{31}$ and $\theta_{23}$ in the higher octant. The
global analysis also disfavours $\delta_{CP}=\frac{\pi}{2}$ with more than
3$\sigma$ CL.

Many experiments are planned for the near future with the goal of determining
these quantities. Some examples of future experiments are DUNE
Acciarri:2015uup ; Acciarri:2016crz ; Acciarri:2016ooe ; Abi:2018dnh ;
Strait:2016mof , T2HK/T2HKK Abe:2015zbg ; Abe:2016ero , ESS$\nu$SB
Baussan:2013zcy , JUNO An:2015jdp , INO Kumar:2017sdq , PINGUAartsen:2014oha ,
KM3Net-ORCAMargiotta:2014gza  etc. It is a well known fact that if there is a
new physics beyond the SM, then that can result in modification of the
oscillation probabilities in these experiments. So, these experiments can be
sensitive to the new physics. Also, the new physics can affect the sensitivity
of these experiments to the standard parameters. Invisible neutrino decay
during the flight is one such new physics idea.

The idea of neutrino decay was first proposed to explain the solar neutrino
problem in the very early days Bahcall:1972my . Later, neutrino oscillation
with decay solutions were studied as an explanation of the depletion of solar
neutrinos Acker:1993sz ; Berezhiani:1991vk ; Berezhiani:1992xg ;
Choubey:2000an ; Bandyopadhyay:2001ct ; Joshipura:2002fb ;
Bandyopadhyay:2002qg ; Picoreti:2015ika . These assumed $\nu_{2}$ to be the
unstable state and were able to put bound on the lifetime of $\nu_{2}$. The
bound from the solar neutrino data is $\tau_{2}/m_{2}>8.5\times 10^{-7}$ s/eV
Bandyopadhyay:2002qg . A recent study analysed low energy solar neutrino data
and put bounds on both $\tau_{2}$ and $\tau_{1}$ Berryman:2014qha . Supernova
neutrinos also give bounds on $\tau_{1}$ and $\tau_{2}$. SN1987A data gives
the bound of $\tau/m>10^{5}$ s/eV Frieman:1987as . $\tau_{1}$ and $\tau_{2}$
can also be constrained from high resolution multi-ton Xenon detector.
Recently, in ref. Huang:2018nxj , the authors show that such a detector can
give very strong bounds
$\tau_{1}/m_{1}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>3\times
10^{-2}$ s/eV and
$\tau_{2}/m_{2}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>8\times
10^{-3}$ s/eV at 2$\sigma$ level using solar neutrinos.

Atmospheric and long-baseline experiments give bounds on the $\nu_{3}$
lifetime. A neutrino decay solution (without any oscillation) was proposed in
ref. LoSecco:1998cd  to explain the atmospheric neutrino problem but this
solution fitted the data poorly. Ref. Barger:1998xk ; Lipari:1999vh
considered neutrino decay and neutrino mixing together. This was successful to
reproduce the $L/E$ distribution of the Super-Kamiokande (SK) data. However,
when zenith angle dependence was used instead of L/E distribution, this model
was found to give poorer fit to the SK data Fogli:1999qt . Ref. Barger:1998xk
; Fogli:1999qt  assumed $\Delta m^{2}>0.1$ eV2 to comply with the K-decay
bounds Barger:1998xk . Therefore $\Delta m^{2}$ dependent terms were averaged
out. These constraints can be relaxed if the unstable state decays to some
invisible state with which it has no mixing. There are two scenarios in the
literature. The first kept $\Delta m^{2}$ unconstrained and it explicitly
appeared in the probabilities Choubey:1999ir . This fits the SK data with a
best-fit value of non-zero decay parameter and $\Delta m^{2}\sim 0.003$ eV2.
Ref. Barger:1999bg  considered $\Delta m^{2}<<10^{-4}$ eV2. Here, the
probability does not contain $\Delta m^{2}$ explicitly. This was able to fit
the SK data. However, independent analysis by SK collaboration showed that
this scenario gives a poorer fit to the data than only oscillation
Ashie:2004mr . Global analysis of atmospheric and long-baseline experiments
were performed in ref. GonzalezGarcia:2008ru . Only oscillation gave best-fit
to the SK data and the fit for the oscillation plus decay was not bad. But
addition of LBL data from MINOS reduced the fit quality. This analysis put
bound on $\tau_{3}/m_{3}\geq 2.9\times 10^{-10}$ s/eV at the 90 % C.L. Ref.
Gomes:2014yua  studied the oscillation plus decay scenario with unconstrained
$\Delta m^{2}$ for MINOS and T2K data and it found $\tau_{3}/m_{3}>2.8\times
10^{-12}$ s/eV at 90 % C.L. Most of these analyses are done using two
generation approximation without matter effect. Authors of ref.
Choubey:2018cfz  performed a complete three generation study of the
oscillation plus decay scenario assuming matter effect for the NOvA and T2K
preliminary data. This study put a bound on the lifetime on the
$\tau_{3}/m_{3}$ as $\tau_{3}/m_{3}\geq 1.50\times 10^{-12}$ s/eV at
3$\sigma$. Recently, ref. Denton:2018aml  addressed the IceCUBE track and
cascade tension using invisible neutrino decay. They showed that an unstable
neutrino with $\tau/m=100$ s/eV solves the track vs cascade tension.

There have been many studies in literature which discuss the potential of
future experiments to the invisible decay. Medium baseline reactor neutrino
experiment like JUNO can give a bound on $\tau_{3}/m_{3}>7.5\times 10^{-11}$
s/eV (95 % C.L.) Abrahao:2015rba . Decay of ultrahigh energy astrophysical
neutrinos can give constraints on decay Beacom:2002vi ; Maltoni:2008jr ;
Pakvasa:2012db ; Pagliaroli:2015rca . Recently, ref Bustamante:2016ciw  showed
that IceCUBE can probe $\tau/m$ upto 10 s/eV for both mass-orderings for 100
TeV neutrinos coming from a source at a distance of 1 Gpc. The ref.
Choubey:2017dyu  studied the invisible neutrino decay in the context of DUNE.
It showed that using charged-current electron type and muon type events, DUNE
after 5+5 years of running can put a bound $\tau_{3}/m_{3}>4.50\times
10^{-11}$ s/eV at 90 % confidence level. More recently ref. Ghoshal:2020hyo
performed a multi-channel analysis using charged-current, neutral current and
tau-channel events and showed that DUNE’s sensitivity would be
$\tau_{3}/m_{3}>5.2\times 10^{-11}$ s/eV at 90 % confidence level. For the
potential of MOMENT experiment to probe invisible neutrino decay see ref
Tang:2018rer . There are also studies on the expected sensitivities from the
future atmospheric experiments. See ref. Choubey:2017eyg ; Mohan:2020tbi  for
INO and ref. deSalas:2018kri  for KM3Net-ORCA.

Invisible neutrino decay can happen for both Dirac and Majorana neutrinos. If
neutrinos are Dirac, there can be a coupling between the neutrinos and a light
scalar boson Acker:1991ej ; Acker:1993sz . This gives the decay channel
$\nu_{j}\rightarrow\bar{\nu}_{iR}+\chi$, where $\bar{\nu}_{iR}$ is a right-
handed singlet and $\chi$ is an iso-singlet scalar. If neutrinos are Majorana
particles, neutrino can couple with a Majoron and a sterile neutrino via a
pseudo-scalar coupling Gelmini:1980re ; Chikashige:1980ui . This gives
$\nu_{j}\rightarrow\nu_{s}+J$. LEP data on the Z-decay to invisible particles
constraints the Majoron to be dominantly singlet Pakvasa:1999ta . Neutrinos
can also decay to another active state Kim:1990km ; Acker:1992eh ;
Lindner:2001fx .This is called the visible neutrino decay scenario. This type
of decay can happen in the following way. $\nu_{j}\rightarrow\bar{\nu}_{i}+J$
or $\nu_{j}\rightarrow\nu_{i}+J$. If neutrinos are Majorana, the decay product
can be observed in the detector. Ref. Coloma:2017zpg ; Gago:2017zzy ;
Ascencio-Sosa:2018lbk  discusses the visible decay in the context of long-
baseline experiments. In ref. Porto-Silva:2020gma , the authors discuss the
visible neutrino decay for reactor experiments KAMLAND and JUNO. For visible
decay of the astrophysical neutrinos at IceCUBE, see ref Abdullahi:2020rge .

Invisible neutrino decay can be constrained using Cosmological observations.
In ref. Escudero:2019gfk , the authors put a bound on neutrino lifetime
$\tau_{\nu}>(1.3-0.3)\times 10^{9}$ s ($m_{\nu}/0.05$ eV)3 at 95% C.L. using
Planck2018 data.

In this paper we study the constraints on invisible neutrino decay which can
come from future planned/proposed long baseline experiments – T2HK/T2HKK
Abe:2015zbg ; Abe:2016ero  and ESS$\nu$SB Baussan:2013zcy . We perform a full
three flavour study using matter effect and obtain the sensitivity to
$\tau_{3}/m_{3}$ for these experiments. The salient feature of the T2HKK and
ESS$\nu$SB experiment is that they are both designed to have energy peak near
the second-oscillation maximum. Since the second oscillation maximum occurs at
a lower energy for a particular baseline the effect of decay is expected to be
more at the second oscillation maximum. We examine this aspect and delve into
the detail of whether the experiments at the second oscillation maximum stand
to gain in sensitivity in presence of decay. We also check if the
determination of $\theta_{23}$ can get affected if we assume decay in the
data, while the fit does not not assume any decay to be present. In
particular, we investigate how the experiments at first and second oscillation
maximum fare in this respect and what are the important factors on which the
measurement of $\theta_{23}$ can depend in presence of decay. We also explore
how the octant sensitivity of these experiments change in presence of neutrino
decay.

The paper is organized in the following way. The next section discusses the
neutrino oscillation in presence of invisible neutrino decay. In Section 3 we
give our experimental and numerical details, in section 4 we present our
results and in section 5, we finally draw our conclusions.

##  2 Neutrino oscillation in presence of invisible decay of neutrino

In this section we discuss the propagation of neutrinos in presence of
invisible neutrino decay. We assume that the $\nu_{3}$ is unstable and it
decays into a sterile neutrino and a singlet scalar
($\nu_{3}\rightarrow\bar{\nu}_{4}+J$) with lifetime $\tau_{3}$. In this case
we can extend the mass and flavour bases by
$\big{(}\nu_{i}\quad\nu_{4}\big{)}^{T}$ and
$\big{(}\nu_{\alpha}\quad\nu_{s}\big{)}^{T}$, where $i=1,2,3$ and
$\alpha={\nu_{e}},{\nu_{\mu}},{\nu_{\tau}}$. They are related by the following
unitary relation.

$\begin{pmatrix}\nu_{\alpha}\\\ \nu_{s}\end{pmatrix}=\begin{pmatrix}U&0\\\
0&1\end{pmatrix}\begin{pmatrix}\nu_{i}\\\ \nu_{4}\end{pmatrix}.$ (1)

$U$ is the standard PMNS matrix describing the standard 3 neutrino
oscillation. We assume normal hierarchy and $m_{4}$ to be the least massive
state. We assume that the decay eigenstates and the mass-eigenstates are same.
Under these assumptions, we can write the neutrino evolution in presence of
matter in the following way.

$i\frac{d}{dx}\nu_{f}=\frac{1}{2E}\big{[}U\tilde{H}U^{\dagger}+A\big{]}\nu_{f}.$
(2)

Where,

$\tilde{H}=\begin{pmatrix}0&0&0\\\ 0&\Delta m^{2}_{21}&0\\\ 0&0&\Delta
m^{2}_{31}-i\frac{m_{3}}{\tau_{3}}\end{pmatrix},$ (3)

and

$A=2\sqrt{2}G_{F}n_{e}E.$ (4)

Here, A is the matter potential, $G_{F}$ is the Fermi constant, $E$ is the
energy and $n_{e}$ density of electrons in the earth. We define
$\alpha_{3}=m_{3}/\tau_{3}$ as the decay rate of the $\nu_{3}$ state. Since we
are considering the decay of $\nu_{3}$ only, from here onward, we use $\alpha$
instead of $\alpha_{3}$ for the $\nu_{3}$ state. The probability of getting a
neutrino in the flavour state $\nu_{b}$ for the initial flavour state of
$\nu_{a}$ is given by

$P_{ab}=\mid\langle\nu_{b}\mid\nu_{a}\rangle\mid^{2}.$ (5)

Here, a,b denote the flavour states e,$\mu,\tau$.

The effect of decay comes as the $\exp(-\alpha L/E)$ factor in the
probability. So an experiment is sensitive to the values of $\alpha$ where,
$\alpha\sim E/L$. It is clear that, smaller values of $\alpha$ and hence a
higher sensitivity can be achieved for longer baseline and for a particular
baseline the sensitivity is more for a lower energy. Thus sensitivity is
expected to be more at the second oscillation maxima as compared to the first
oscillation maxima.

##  3 Experimental and simulation details

In this section we describe various experiments and the specifications used in
our analysis with the summary of the experiments presented in Tab. 1. First we
give the brief descriptions of the experiments followed by details of our
numerical simulations.

###  3.1 Experimental details

####  3.1.1 T2HK

T2HK Abe:2015zbg  is a proposed upgradation plan of the currently running T2K
experiment in Japan. The neutrinos will be generated at Tokai by an upgraded
version of the J-PARC beam. Currently J-PARC gives a beam power of 470 KW, but
before T2HK becomes operational, the beam power will be increased beyond 1.3
MW. Although near detector of T2HK is yet to be finalized, there are several
ideas. Some of these are like upgradation of the current near detector ND280,
building a water Čerenkov detector similar to the far detector but in a
smaller scale. The far detector will be the upgradation of the currently
running Super-Kamiokande (SK) Fukuda:2002uc  detector. SK is situated at 295
km away from Tokai at the Kamioka village and it is slightly (2.5∘) off-axis
to the beam-axis. Thus this will give a narrow beam centered around 0.56 GeV
which is at the first oscillation maximum of the neutrino oscillation. The
upgraded SK, called Hyper-Kamiokande (HK), will consist of two identical
cylindrical tanks placed upright. Each tank will contain pure water which will
have a fiducial mass of 187 kt each thus giving a total fiducial mass of 374
kt. The new detector will also have better resolution and efficiency compared
to SK.

####  3.1.2 T2HKK

One of the main aim of the future long-baseline experiments is to measure the
CP-violation, and T2HK is no exception. T2HK has a very good sensitivity to
the CP-violation if the mass-hierarchy is known. However, the CP-violation
sensitivity of T2HK drops if mass-hierarchy is not known beforehand. This is
due to a degeneracy between mass-hierarchy and $\delta_{CP}$ Kajita:2006bt .
In order to solve this problem, there is a proposal to shift one of the water
tanks of HK to Korea at about a distance of 1100 km away from the source at
Tokai. This proposal is called the T2HKK Abe:2016ero  (the second K is for
Korea.). This will have more matter effect. Thus the mass-hierarchy
sensitivity will increase significantly breaking the CP-hierarchy degeneracy.
However, the second detector will be at the second oscillation maximum and as
the baseline is larger and the flux will also decrease.

The Korean detector site is not yet final and there are many possibilities
Abe:2016ero . However, all these sites are in the southern part of the Korean
peninsula and lie within a range of 1-3∘ off-axis angle with the J-PARC beam
line.

####  3.1.3 ESS$\nu$B

ESS$\nu$B Baussan:2013zcy  is another future long-baseline super-beam
experiment. The neutrino beam will be generated from the ESS facility at Lund,
Sweden. This will have a 2 GeV proton beam with 5 MW beam power. The neutrino
beam created by this will have a peak around 0.25 GeV. The far detector will
be situated at a distance of about 540 km away from Lund at a mine in
Garpenberg. The far detector is proposed to be 500 kt water Čerenkov detector.
At 540 km, the second oscillation maximum occurs at 0.35 GeV. Thus the peak of
the neutrino beam will be close to the second oscillation maximum for
ESS$\nu$B.

Experiment | Baseline (L in km) | L/E (in km/GeV) | Fiducial Volume (in kton)
---|---|---|---
T2HK | 295 km | 527 | 187 $\times$ 2
T2HKK | 295 km; 1100 km | 527(295 km); 1964(1100 km) | 187(295 km) + 187(1100 km)
ESS$\nu$SB | 540 km | 1543 | 500
Table 1: The baselines, L/E and fiducial volumes of each detector for
T2HK(L1), T2HK(L2) and ESS$\nu$SB. The energies for T2HK and ESS$\nu$SB are
0.56 GeV and 0.35 GeV respectively.

###  3.2 Simulation details

We have used Global Long-Baseline Experiments Simulator (GLoBES)globes1 ;
globes2  for simulating the long-baseline experiments. We present our results
in terms of statistical $\chi^{2}$, where,

$\displaystyle\chi^{2}_{{\rm stat}}=2\sum_{i}\\{N_{i}^{{\rm test}}-N_{i}^{{\rm
true}}+N_{i}^{{\rm true}}\ln\frac{N_{i}^{{\rm true}}}{N_{i}^{{\rm test}}}\\},$
(6)

where $(N)_{i}^{{\rm true}}$ corresponds to the simulated data and
$(N)_{i}^{{\rm test}}$ is the number of events predicted by the theoretical
model. The effect of systematic errors are included by the “pull” method
through the “pull” variables $\xi$. We have incorporated signal normalization
error, background normalization error, signal “tilt” error & background “tilt”
error in our analysis. Incorporating the errors, the signal and background
events can be written as,

$N^{\rm
test}_{i}=\sum_{s(b)}N^{s(b)}_{i}\bigg{(}1+{c_{i}^{s(b)}}^{norm}{\xi^{s(b)}}^{norm}+{c_{i}^{s(b)}}^{tilt}{\xi^{s(b)}}^{tilt}\frac{E_{i}-\bar{E}}{E_{max}-E_{min}}\bigg{)}$
(7)

where, $s(b)$ denotes signal(background). $c_{i}^{norm}$(${c_{i}}^{tilt}$)
denotes the change in number of events by the variation of the “pull” variable
$\xi^{norm}$(${\xi}^{tilt}$). The signal and background normalization errors
are presented in Tab. 2. In the above equation $E_{i}$ represents the mean
reconstructed energy of the $i^{th}$ bin. The maximum and minimum energy of
the energy range are $E_{min}$ and $E_{max}$ respectively. The mean energy is
given by, $\bar{E}=({E_{max}+E_{min}})/{2}$.

Channel | T2HK(295 km) | T2HK(1100 km) | ESS$\nu$SB
---|---|---|---
$\nu_{e}$ appearance | 3.2%(5%) | 3.8%(5%) | 3.2%(5%)
$\nu_{\bar{e}}$ appearance | 3.9%(5%) | 4.1%(5%) | 3.9%(5%)
$\nu_{\mu}$ disappearance | 3.6%(5%) | 3.8%(5%) | 3.6%(5%)
$\nu_{\bar{\mu}}$ disappearance | 3.6%(5%) | 3.8%(5%) | 3.6%(5%)
Table 2: The signal(background) normalization uncertainties of the experiments
for the various channels for T2HK, T2HKK and ESS$\nu$SB.

The detector type proposed for T2HK, T2HKK and ESS$\nu$SB are water Čerenkov
detectors. T2HK & T2HKK have been simulated based on Abe:2016ero  and the
ESS$\nu$SB experiment has been simulated according to Baussan:2013zcy . The
detector of ESS$\nu$SB is based on the MEMPHYS detector Agostino:2012fd ,
which is a water Čerenkov detector. The neutrino (antineutrino) appearance
channels for signal events are $\nu_{e}$ ($\bar{\nu_{e}}$) and the background
events in this channel are neutral current background, intrinsic beam
background, mis-identified muons and wrong-sign signal errors. The neutrino
(antineutrino) disappearance channels are $\nu_{\mu}$ ($\bar{\nu_{\mu}}$), the
background channels are neutral current background and wrong-sign muons. The
signal and background normalization uncertainties have been considered as
presented in Tab. 2. Additionally, a 5% (10%) signal (background) “tilt” error
have been considered for T2HK & T2HKK and 0.1% (0.1%) in case of ESS$\nu$SB.

Oscillation parameters | True Values | Marginalization Range
---|---|---
$\theta_{13}$ | $8.6^{\circ}$ | Fixed
$\theta_{12}$ | $33.82^{\circ}$ | Fixed
$\sin^{2}\theta_{23}$ |  $0.447$(LO), $0.56$(HO) |  $0.413$ — $0.671$
$\Delta m^{2}_{21}$ (eV2) | $7.39\times 10^{-5}$ | Fixed
$|\Delta m^{2}_{31}|$ (eV2) | $2.52\times 10^{-3}$ |  $2.3\times 10^{-3}$ — $2.6\times 10^{-3}$
$\delta$ | $-90^{\circ}$ |  $-180^{\circ}$ — $180^{\circ}$
Table 3: The values of the 3 neutrino oscillation parameters nufit ;
Esteban:2020cvm  used in the present analysis unless stated otherwise..

For the calculation of probability in presence of invisible neutrino decay we
modified the probability code of the GLoBES. We used
$\theta_{12}=33.82^{\circ}$, $\theta_{13}=8.6^{\circ}$,
$\sin^{2}\theta_{23}=0.447(0.56)$ for lower (higher) octant,
$\delta_{CP}=-90^{\circ}$,$\Delta m^{2}_{21}=7.39\times 10^{-5}$ eV2 and
$\Delta m^{2}_{31}=2.52\times 10^{-3}$ eV2 for generating the simulated data
(true values) as shown in the Tab. 3. These are consistent with the current
ranges allowed by the latest global fits Esteban:2020cvm . For the statistical
studies we have marginalized over $\delta_{CP}$ and $\theta_{23}$ in their
allowed ranges. We kept other standard parameters fixed as we found they have
little effects on the marginalization. We also assumed the mass-hierarchy is
known and it is normal hierarchy.

##  4 Results

In this section, we present the results of our study. First, we give the plots
at the probability level in presence of decay for T2HK, T2HKK and ESS$\nu$B.
These figures can be helpful in understanding many of the results presented in
the next sections. Then we give the sensitivity to the decay rate $\alpha$ for
each experiment. We also discuss a particular synergy which we observed
between the two detectors in T2HKK experiment – one at the first oscillation
maximum and one at the second oscillation maximum. We also present the results
on the effect of decay on the measurement of $\theta_{23}$ and octant
sensitivity for these experiments

###  4.1 Probability at T2HK/T2HKK and ESS$\nu$SB baselines

---
Figure 1:  The probabilities as a function of energy at L = 295 km, 1100 km
and 540 km from top to bottom. The left (right) panel shows
(${\nu_{\mu}}\rightarrow{\nu_{\mu}}$)${\nu_{\mu}}\rightarrow{\nu_{e}}$
probabilities. The black (red dashed) curves are for $\alpha=0$ and
$\theta_{23}$ in the lower (higher) octant. The blue solid (green dashed)
curves are for $\alpha=3\times 10^{-5}$ eV2 and $\theta_{23}$ in the lower
(higher) octant. Where, $\theta_{23}$ = $40^{\circ}$ for LO and $\theta_{23}$
= $51^{\circ}$ for HO.

In fig.1, we present the appearance and disappearance probabilities as a
function of energy for the three different baselines under investigation. We
give the plots for the no-decay case ($\alpha=0$) and also for a
representative value of $\alpha=3\times 10^{-5}$ eV2. The choice of this
$\alpha$ is motivated from the current bounds as obtained from T2K and
NO$\nu$A data analysis Choubey:2018cfz . In both cases we give the plots for
two values of $\theta_{23}$ – one in lower octant $(40^{\circ})$ and one in
the higher octant ($51^{\circ}$). The probabilities for other values of
$\theta_{23}$ within this range will lie between these two curves. The figures
show the interplay between $\theta_{23}$ and $\alpha$ for the various
baselines. The left (right) panel shows ${\nu_{\mu}}\rightarrow{\nu_{e}}$
(${\nu_{\mu}}\rightarrow{\nu_{\mu}}$) probabilities. The black (blue) solid
and red (green) dashed curves are for $\alpha=0$ ($\alpha=3\times 10^{-5}$
eV2) and $\theta_{23}$ in the lower octant and higher octant respectively. The
top, middle and the bottom panel are for the baselines 295 km, 1100 km and 540
km respectively.

In the left panels of fig. 1, we see that for all the baselines, at the
oscillation maxima, $P_{\mu e}$ for the the no decay case is higher for a
fixed value of $\theta_{23}$. As decay is introduced, the probabilities
reduce. On the other hand, for a fixed value of $\alpha$, lower octant gives
smaller probabilities compared to the higher octant. Thus the peak appearance
probability can get reduced due to increase in the value of $\alpha$ and/or
decrease in the value of the mixing angle $\theta_{23}$.

The top panel is for T2HK and here the first oscillation maximum is at
$\sim$0.6 GeV which is also where the flux peaks. For the baseline and
energies involved the effect of decay for T2HK is small for the sample value
of $\alpha$ considered in the plot. However, we see that the higher and lower
octant bands are well separated for T2HK.

Next we focus on the middle and the bottom panels which are relevant baselines
for T2HKK (second detector) and ESS$\nu$SB. For these experiments the flux
peaks near the second oscillation maxima – $\sim$0.7 GeV for the baselines
1100 km and $\sim$0.35 GeV 540 km. Since these are higher baselines and/or
lower energies the effect of decay is more pronounced. The HO and LO bands are
much closer in these cases and a reduction in probability due to non-zero
values of $\alpha$ can be compensated by increasing $\theta_{23}$.

In the right panels of fig. 1 we show the ${\nu_{\mu}}\rightarrow{\nu_{\mu}}$
disappearance probability $P_{\mu\mu}$ which plays a crucial role in
determining the value of $\theta_{23}$ in long baseline experiments. In this
case, the curves with $\alpha=0$ are overlapping irrespective of the octant of
$\theta_{23}$ which shows that the $P_{\mu\mu}$ channel do not have any octant
sensitivity in absence of neutrino decay, being dependent on
$\sin^{2}2\theta_{23}$. So, the red-dashed and black-solid curves (no-decay)
are well separated from the blue-solid and green-dashed curves which reveals
the effect of decay. This feature is true for all the three baselines
considered in our study. In case of the 295 km baseline the blue curve is
closer to the black and red curves compared to the 1100 km and 540 km cases
since the effect of decay is less for the 295 km baseline for the $\alpha$
value chosen. Also, it has to be noted that, the otherwise octant degenerate
$P_{\mu\mu}$ channel begins to have octant sensitivity once $\alpha\neq 0$,
resulting in a gap between the blue and green curves.

This can be simply understood if we consider the expression for two-generation
survival probability in vacuum

$P_{\mu\mu}=\left[1-\sin^{2}\theta_{23}(1-e^{-\frac{\alpha
L}{E}}\right]^{2}-\sin^{2}2\theta_{23}e^{-\frac{\alpha
L}{2E}}\,\sin^{2}\left(\frac{\Delta m_{31}^{2}L}{4E}\right)\ $ (8)

In absence of decay the $e^{-\frac{\alpha L}{E}}$ term is 1 and the
probability depends on $\sin^{2}2\theta_{23}$. But in presence of decay octant
sensitivity ensues due to the $e^{-\frac{\alpha L}{E}}$ factor.

---
Figure 2:  The probabilities as a function of $\sin^{2}\theta_{23}$ at L = 540
km for ESS$\nu$SB. The left (right) panel shows
${\nu_{\mu}}\rightarrow{\nu_{e}}$(${\nu_{\mu}}\rightarrow{\nu_{\mu}}$)
probabilities. The black dashed (blue) curves are for $\alpha=0$ and
$\alpha=3\times 10^{-5}$ eV2 respectively.

For our later analysis, it is important to understand the dependence of the
probability on $\theta_{23}$ at the survival probability maxima (SPMAX) and
minima (SPMIN). At SPMIN, though the probability value is small, the flux
peaks near this energy since it corresponds to the oscillation maxima. In
figure 2 we show the behaviour of the probability for the 540 km baseline as a
function of $\theta_{23}$ for energies corresponding to SPMAX and SPMIN.
Similar behaviour is also true for the two other baselines – 295 km and 1100
km. It is seen that at SPMAX, the no-decay probability is always higher than
the decay probability. The no-decay probability has no dependence on
$\theta_{23}$ while the decay probability reduces with increasing
$\theta_{23}$ monotonically. This can be easily understood from the expression
8. At SPMAX $\sin^{2}\left(\frac{\Delta m_{31}^{2}L}{4E}\right)\ =0$ and the
remaining term decreases monotonically with increasing $\theta_{23}$ for non-
zero $\alpha$. At SPMIN, the $\theta_{23}$ dependence is more complicated. For
lower octant the decay probability is larger as compared to the no-decay
probability and the probability reduces with increasing $\theta_{23}$ till a
certain value (depending on $\alpha$), after which in higher octant it
increases with $\theta_{23}$. For higher octant, no-decay probability is
higher than the decay probability. As we will see, these features of the
$P_{\mu\mu}$ probability will play a role in the $\chi^{2}$ fit.

###  4.2 Sensitivity to the decay

In this section we study the capability of the T2HK, T2HKK and ESS$\nu$B
experiments in constraining invisible neutrino decay. We assume the neutrinos
to be stable while generating the simulated data. In the fit we have
marginalized over $|\Delta m^{2}_{31}$, $\theta_{23}$ and $\delta_{CP}$ in the
range given in Tab. 3. The fig. 3 shows the value of $\chi^{2}$ for different
values of $\alpha$ taken in the fit (test). A higher value of $\chi^{2}$ is
more disfavoured compared to a lower values of $\chi^{2}$.

---
Figure 3:  The $\chi^{2}$ as a function of $\alpha$, assuming no decay in the
simulated data. The left (right) panel is for $\theta_{23}$ (true) in the
lower (higher) octant. The red solid, blue dashed dotted and the black dashed
curves are for T2HK, T2HKK and ESS$\nu$B respectively.

In the left panel we assumed $\sin^{2}\theta_{23}\rm{(true)}=0.447$ (lower
octant) while generating the data. In the right panel we assumed
$\sin^{2}\theta_{23}\rm{(true)}=0.56$ (higher octant). The red solid, blue
dashed dotted and black dashed curves are for T2HK, T2HKK and ESS$\nu$B
respectively. Naively, the sensitivity to decay depends on the $L/E$. From
Tab. 1, we see that the second detector of T2HKK has highest $L/E$ and the
detector at 295 km has lowest $L/E$ (See Tab. 1). So, we expect T2HKK to have
the best sensitivity and T2HK to have the worst, but we observe that
ESS$\nu$SB has a lower sensitivity than T2HK. Here, marginalization over
$\theta_{23}$ plays an important role. Determination of $\theta_{23}$ is
governed mainly by the $P_{\mu\mu}$ channel. In absence of decay there is no
sensitivity to $\theta_{23}$ at the maxima of survival probability which
corresponds to ${\sin^{2}{\Delta m^{2}_{31}L}/{4E}}=0$. The maximum precision
of $\theta_{23}$ comes at the oscillation maximum or the survival probability
minimum (SPMIN) where, ${\sin^{2}{\Delta m^{2}_{31}L}/{4E}}=1$ and can be
expressed as,

$\Delta({\sin^{2}\theta_{23}})=-\frac{\Delta(P_{\mu\mu})}{4\cos^{2}2\theta_{23}}$
(9)

In presence of decay, the survival probability maxima also acquires some
sensitivity to $\theta_{23}$ which increases with increasing value of the
decay constant $\alpha$ for a fixed baseline.

As seen in the earlier section, for T2HK the octant bands are well separated
while the effect of decay is not very significant. Hence marginalization over
$\theta_{23}$ does not play much role. For ESS$\nu$SB on the other hand, the
octant bands are not so well separated and the effect of decay is more. As a
result, marginalization over $\theta_{23}$ tends to make the probabilities
closer by shifting $\theta_{23}$ to a different value and the sensitivity
reduces. The direction of shift depends on the initial value of $\theta_{23}$.
For SPMIN, decay in the fit, can give same probability as no-decay in data for
an increased $\theta_{23}$ for both lower and higher octant. This can be seen
by drawing a horizontal line in fig. 2. For SPMAX, on the other hand, if we
fit decay with no-decay in data, then shift of $\theta_{23}$ towards lower
values make the probabilities closer, giving a lower $\chi^{2}$. Thus,
depending on which energy bins contribute most, the $\theta_{23}$ in the fit
comes at a higher or lower value, reducing the $\chi^{2}$. We have checked
that if we keep $\theta_{23}$ as fixed then ESS$\nu$SB gives a better
sensitivity.

In case of T2HKK, other than a detector close to the second oscillation
maximum, there is also a detector at the first oscillation maximum which can
measure $\theta_{23}$ with a good precision. So, the combination of the two
detectors, one at the first oscillation maximum and other at the second
oscillation maximum gives the best sensitivity. This will be discussed further
in the next section. There is also another interesting feature for ESS$\nu$SB,
which is that for true octant as higher octant we observe a sudden fall of
sensitivity for $\alpha$ above $\alpha\sim 3\times 10^{-5}$ eV2. We will see
in the next section that this is a unique feature of second oscillation maxima
and we will discuss this in detail in the coming section.

###  4.3 Synergy between 295 km and 1100 km baselines

---
Figure 4:  The $\chi^{2}$ as a function of $\alpha$, assuming no decay in the
simulated data. The left (right) panel is for $\theta_{23}$ (true) in the
lower (higher) octant. The dark red solid, cyan dashed, blue dashed-dotted
curves are for a 187 kt detector at 295 Km, 187 kt detector at 1100 km and
their combination respectively. The grey dotted curve shows the $\chi^{2}$ if
we just add the $\chi^{2}$s of the dark red and cyan curves.

In this section we discuss a synergy between the two baselines $L1=295$ km and
$L2=1100$ km of T2HKK in constraining neutrino decay. The first one
corresponds to the Japanese detector (JD) and the second one corresponds to
the Korean detector (KD). It can happen that the $\chi^{2}$ obtained by
combining two or more experiments can be different from the naive sum of their
individual $\chi^{2}$s. We call such enhancement of $\chi^{2}$ as synergy
between the experiments. In fig. 4, we present the sensitivity to the $\alpha$
for one of the HK detectors situated at L1 (dark red solid) and L2 (cyan
dashed). We also show their combined sensitivity ( blue dashed-dotted) which
is nothing but the sensitivity of T2HKK. The grey dotted curve shows the naive
summation of the $\chi^{2}$s i.e $\chi^{2}_{L1}+\chi^{2}_{L2}$. The left panel
is for $\sin^{2}\theta_{23}=0.447$ and the right panel is for
$\sin^{2}\theta_{23}=0.56$. We see that although there is no significant
synergy for the left panel, there is some significant synergy for the right
panel for $\alpha$ greater than $2.268\times 10^{-5}$ eV2, i.e., the
$\chi^{2}_{L1+L2}$ of T2HKK is larger than the $\chi^{2}_{L1}+\chi^{2}_{L2}$.

This synergy can be understood from the fig. 5 and fig. 6. These figures give
the $\chi^{2}$ for a given value of $\theta_{23}$ in the fit with a fixed
value of $\alpha$ in the fit. The decay constant $\alpha=0$ while simulating
the data and $|\Delta m^{2}_{31}|$ and $\delta_{CP}$ are marginalized for all
the cases. Therefore global minima of the curves in these figures give the
$\chi^{2}$ values for the corresponding $\alpha$ (test) in fig. 4. Fig. 5 is
for the case when $\sin^{2}\theta_{23}=0.447$ and fig.6 is for
$\sin^{2}\theta_{23}=0.56$ in the simulated data. The dark red solid curves
assume $\alpha=0$ in the fit and green dashed curves assume $\alpha=3\times
10^{-5}$ eV2 in the fit.

---
Figure 5:  The $\chi^{2}$ as a function of $\theta_{23}$ assuming the true
$\sin^{2}\theta_{23}=0.447$. The left (right) panels are for baselines 295
(1100) km respectively. The dark red solid curves are for the standard cases
and the green dashed curves are for the case when $\alpha=0$ in the simulated
data but assumed to be $3\times 10^{-5}$ eV2 in the fit.
---
Figure 6:  The $\chi^{2}$ as a function of $\theta_{23}$ assuming true
$\sin^{2}\theta_{23}=0.56$. The left (right) panels are for baselines 295
(1100) km respectively. The dark red solid curves are for the standard cases
and the green dashed curves are for the case when $\alpha=0$ in the simulated
data but assumed to be $3\times 10^{-5}$ eV2 in the fit.

First let us consider the fig. 5. Here, $\sin^{2}\theta_{23}(true)=0.447$. We
see that for L1, the global minimum, for a test value of $\alpha=3\times
10^{-5}$ eV2 in the fit, shifts slightly towards a higher
$\sin^{2}\theta_{23}$ but stays in the same octant as the $\alpha=0$. The
shift towards a higher value can be understood by looking at the curve for
$P_{\mu\mu}$. It is seen that at the oscillation maxima (i.e SPMIN), the no-
decay probability (black) curve is lower than that with decay (blue curve) for
lower octant. Hence when we try to fit data with no-decay with decay in
theory, the probabilities can come closer by increasing $\theta_{23}$
slightly. Similar feature is also observed at L2 in the right panel though.
the shift is almost un-noticeable. Sensitivity to the parameter $\alpha$ for
the combination L1 and L2 will be given by the global minimum of the sum of
$\chi^{2}$s of two green dashed curves of fig. 5. As the global minima of
these two curves approximately coincide (42∘ for L2 and 42.5∘ for L1), the
global minimum of the sum of their $\chi^{2}$s will be approximately equal to
the sum of $\chi^{2}$s at the global minima of L1 and L2. Therefore, the
sensitivity to the combination of L1 and L2 will be approximately equal to the
sum of the sensitivities for L1 and L2. Thus there is not significant synergy
for this case.

Next we consider the fig. 6. Here, $\sin^{2}\theta_{23}=0.56$. We see that for
L1, like previous case, the global minimum for $\alpha=3\times 10^{-5}$ eV2
(test) shifts a little towards higher value since at SPMIN, no-decay
probability is larger, it can be matched by increased value of $\theta_{23}$
(cf. 2).

However, for L2, we see that the position of the global minimum drastically
changes and now it is in the opposite octant, i.e., lower octant. Again, like
the previous case, the sensitivity to the combination of L1 and L2 will be
determined by the sum of $\chi^{2}$s of two green dashed curves. But here, the
positions of the global minima for L1 and L2 are completely different.
Therefore, the position of the global minimum of the sum of $\chi^{2}$s of L1
and L2 curves is non-trivial and is not equal to the sum of the $\chi^{2}$s at
the global minima of L1 and L2 curves. Hence, we see the synergy in this case.
The shift of the global minimum can be attributed to the interplay between the
appearance and disappearance channels.

---
Figure 7:  The $\chi^{2}$ as a function of $\theta_{23}$. The left (right)
panels are for true $\sin^{2}\theta_{23}=0.447$ ($\sin^{2}\theta_{23}=0.56$)
respectively. The red and blue curve are for appearance and disappearance
channels respectively. While the total contribution from both the channels is
represented by green curves. The solid curves are for $\alpha\rm(test)=3\times
10^{-5}eV^{2}$ and the dashed curves for $\alpha\rm(test)=10^{-5}eV^{2}$ in
the fit. The data have been generated for $\alpha=0$.

In order to understand this interplay in fig. 7 we have plotted the appearance
and disappearance $\chi^{2}$ s for two different values of the decay constant
$\alpha$ for L2. From the left panel of the figure it is seen that for true
$\sin^{2}\theta_{23}$ in the lower octant, the minima for both appearance and
disappearance channels stay in the correct octant. The octant sensitivity is
seen to come from both disappearance and appearance channel. However, for
appearance channels, the octant bands are not very widely separated for
experiments close to second oscillation maxima, and with increase in $\alpha$
the effect of disappearance channel becomes more pronounced and there is a
sharp rise in the octant sensitivity coming from the disappearance channel.

The right panel depicts the situation for the higher octant. In this case when
the no-decay in data is fitted with decay, the minima for the appearance
channel stays in the same octant, shifting slightly towards higher
$\theta_{23}$. This behaviour can be understood from the appearance
probability since for this HO and no-decay gives a higher probability and
hence if we fit with decay, the $\theta_{23}$ tends to shift to a larger value
to get closer to the data. As the decay constant increases, the shift in the
value of $\theta_{23}$ to get closer to the no-decay curve will be higher.

On the other hand the disappearance minima is seen to shift to the lower
octant. For a lower value of the decay constant, the difference between the
disappearance $\chi^{2}$ minimum in the two octants is not very high. On the
other hand, the appearance $\chi^{2}$ rises steeply in the opposite octant and
when the appearance $\chi^{2}$ is added to the disappearance $\chi^{2}$, the
overall minima comes in the correct octant. But for a higher value of the
decay constant, the minima in the wrong octant is much lower than the minima
in the correct octant and thus the overall minima comes in the wrong octant,
being driven by disappearance channel. The reason for disappearance channel
preferring lower octant can be understood from the probability figure 1, by
noticing that for peak flux energies of around 0.6 GeV, the red-curves (no-
decay, HO) are closer to the blue curves (decay, lower octant). Hence, when
data is generated in the higher octant with no decay, disappearance channel
prefers the lower octant if $\alpha\neq 0$ in the fit, the shift being higher
as the decay constant increases. This effect is not present at the detector
tuned for the first oscillation maximum at 295 km. From the top-left panel of
Fig. 1, we see that the octant bands are wide enough such that the appearance
channel gives high octant sensitivity and therefore the appearance channel
always dominates and we do not get any false octant solution.

The above discussion can be now used to explain the kink in the sensitivity of
ESS$\nu$SB (fig. 3 right panel). For ESS$\nu$SB there is only a single
detector. For lower values of $\alpha$, the disappearance channel is still
weak and the appearance channel dominates and we get the octant in the higher
octant. However, as the $\alpha$ becomes more than some critical value, the
disappearance channel begins to dominate and the octant flips to the wrong
side. Thus the $\chi^{2}$ is abruptly increased, giving a lower sensitivity.

###  4.4 Role of decay in the $\theta_{23}$ determination

---
Figure 8:  The $\Delta\chi^{2}$ as a function of $\theta_{23}$ (test) assuming
$\sin^{2}\theta_{23}$(true) $=0.447$. The left, middle and the right panels
are for T2HK, T2HKK and ESS$\nu$B respectively. The dark red solid curves are
for standard cases and the green dashed curves are for the cases where
$\alpha$ is assumed to be $3\times 10^{-5}$ eV2 in the simulated data but
decay is not considered in the fit.

In this section, we will see how presence of invisible neutrino decay can
affect the measurement of $\theta_{23}$ in T2HK, T2HKK and ESS$\nu$B
experiments.

Fig. 8 gives $\Delta\chi^{2}$ as a function of $\theta_{23}$ (test). Here, we
assumed$\sin^{2}\theta_{23}=0.447$ in the data. The left, middle and right
panels are for T2HK, T2HKK and ESS$\nu$B respectively. In the fit, we
marginalized over, $|\Delta m^{2}_{31}|$, $\delta_{CP}$ and kept $\alpha$
fixed at zero. For dark-red solid curves, we assumed stable neutrino in the
data and for green dashed curves, we assumed $\alpha=3\times 10^{-5}$ eV2 in
the data.

We see that for the three cases, the best-fit values for data generated for
$\alpha=3\times 10^{-5}$ eV2 and $\alpha=0$ are different. We notice that for
true $\alpha=3\times 10^{-5}$ eV2 the best-fit of $\theta_{23}$ is shifted
towards lower values for T2HK and T2HKK. However, for ESS$\nu$SB, the shift is
in the opposite direction. The shift towards lower values for T2HK and T2HKK
is governed by the behaviour of $P_{\mu\mu}$ at the oscillation maxima i.e at
SPMIN, where the flux peaks. The data is generated for decay which gives a
higher probability than no-decay in the lower octant. Therefore when data is
fitted with $\alpha=0$, a reduced value of $\theta_{23}$ gives a better fit
since the probability increases with decreasing $\theta_{23}$.

The reason for different behaviour for ESS$\nu$SB is more complicated. Here,
resolution plays a big role. The resolution of the detector smears the imprint
of probability over all energy bins. As a result, the feature near the
oscillation minima is lost due to the larger bin width of 0.1 GeV. A bin from
0.3 GeV to 0.4 GeV contains events from the survival probability minima as
well as the features of the probability around 0.3 GeV where the probabilities
corresponding to decay and no decay curves show opposite behaviour than at
$\sim$ 0.35 GeV. As an upshot if one plots the events then the no-decay events
are higher than the events with decay in most bins. When the simulated event
spectrum is fitted without decay, $\theta_{23}$ is shifted towards higher
value to bring the probability down. The effect of smearing affects the
experiments close to second oscillation maxima more since the variation with
probability is sharper compared to the first oscillation maxima.

---
Figure 9:  The $\Delta\chi^{2}$ as a function of $\theta_{23}$ (test) assuming
$\sin^{2}\theta_{23}$(true) $=0.56$. The left, middle and the right panels are
for T2HK, T2HKK and ESS$\nu$B respectively. The dark red solid curves are for
standard cases and the green dashed curves are for the cases where $\alpha$ is
assumed to be $3\times 10^{-5}$ eV2 in the simulated data but decay is not
considered in the fit.

Fig. 9 is similar to fig. 8, but here, we assume $\sin^{2}\theta_{23}=0.56$ in
the true data. We see like the previous figure, the best-fit values of
$\theta_{23}$ is changed but in this case, for all three experiments, the
shifts of $\theta_{23}$ are towards a lower value. This can be explained
since, for $\theta_{23}$ in the higher octant the probability for non-zero
$\alpha$ is smaller than the probability with $\alpha=0$ for both SPMIN and
SPMAX and hence no-decay gives larger number of events. For $\theta_{23}$ in
higher-octant, lowering $\theta_{23}$ reduces the probability and hence gives
a better fit when we fit decay in data with no-decay.

---
Figure 10:  The octant sensitivity $\chi^{2}$ as a function of
$\delta_{CP}$(true) for ESS$\nu$SB(top row) and T2HKK-L2 (bottom row). The
left panel represents the octant sensitivity for the case without decay i.e.
$\alpha=0$ and the right panel for the case with decay $\alpha=3\times
10^{-5}$. The $\alpha$ values are same for both data and fit in each plot. The
red and blue plots are for appearance-CC, disappearance-CC respectively, while
the green plots shows the contribution from both appearance-CC and
disappearance-CC. The true $\sin^{2}\theta_{23}=0.56$ and the test
$\theta_{23}$ is varied in the lower octant in the range
$40^{\circ}<\theta_{23}<45^{\circ}$.

In Fig. 10, we show the octant sensitivities as a function of the CP phase
$\delta_{CP}$ for experiments which are tuned to the second oscillation maxima
in presence and absence of decay. The top panels are for ESS$\nu$SB and the
bottom panels are for a hypothetical T2HKK experiment whose both detectors are
at 1100 km. The left panels are for no decay and the right panels are for the
case when $\alpha\neq 0$. The red lines are for appearance channels, the blue
lines are for disappearance channels and the green lines are for both
appearance and disappearance channels together. In absence of decay
disappearance channel does not have any octant sensitivity. However when
appearance $\chi^{2}$ is added to disappearance $\chi^{2}$, then the
appearance probability being dependent on $\sin^{2}\theta_{23}$ rises
monotonically in the opposite octant and gives a large octant-sensitive
contribution. In presence of decay, the disappearance channel itself has
octant sensitivity, which enhances the overall octant sensitivity.

###  4.5 Combined Analysis

Here in Fig. 11, we have shown sensitivities of different experiments to
constrain the decay parameter $\alpha$(test) with (left) and without (right)
decay in ‘data’ assuming the higher octant as the true octant. The plots in
the left panel show how precisely these experiments can measure $\alpha$ and
hence for a given true value how they can exclude the no decay scenarios. On
the other hand, plots in the right panel show the constraints these
experiments can put on invisible neutrino decay. The results are shown for all
the three experiments ESS$\nu$SB, T2HK, T2HKK and the combination of
ESS$\nu$SB with both T2HK and T2HKK. In the Fig. 11, the black dashed line
shows the sensitivity of ESS$\nu$SB, the blue dashed-dotted line is for T2HKK,
while the red solid line represent the same for T2HK. The pink and cyan dashed
line stands for the combination of ESS$\nu$SB with T2HK and T2HKK
respectively.

Figure 11:  The $\chi^{2}$ as a function of test $\alpha$ for various
combination of experiments. The left panel is for the case where
$\alpha=3\times 10^{-5}$ (eV2) is assumed for the data and the right panel is
for the case where $\alpha=0$ in the decay. The black dashed curves are for
ESS$\nu$SB, red solid curves are for T2HK, blue dashed-dotted curves for
T2HKK, the magenta dashed for combination of ESS$\nu$SB and T2HK and the cyan
curves are for combination of ESS$\nu$SB and T2HKK.

It is observed from the left panel where we have assumed decay in the
simulated data, that T2HKK has the highest precision out of the three and at
3$\sigma$ it can precisely measure $\alpha$ in the range $1.278\times
10^{-5}<\alpha<4.614\times 10^{-5}$ eV2 for the given true value of
$\alpha=3\times 10^{-5}eV^{2}$. For the same true value of $\alpha$ T2HK gives
better precision than ESS$\nu$SB and at 3$\sigma$ the range is $3.408\times
10^{-6}<\alpha<5.524\times 10^{-5}$ eV2 and $4.945\times
10^{-6}<\alpha<7.303\times 10^{-5}$ eV2 respectively. Combining T2HKK with
ESS$\nu$SB improves the bounds further and at 3$\sigma$ the range is
$1.597\times 10^{-5}<\alpha<4.231\times 10^{-5}$ eV2. Although the combination
of T2HK and ESS$\nu$SB improves the bound, still it comes to be similar to
that of T2HKK. In the right panel of Fig. 11, we observe that by combining two
experiments it is possible to improve the constraint on $\alpha$. Specially,
the combination of T2HKK and ESS$\nu$SB gives the highest limit on $\alpha$
and at 3$\sigma$ the constraint is $\alpha\leq 1.193\times 10^{-5}$ eV2.
Similarly, T2HK+ ESS$\nu$SB also improves the constrain and the derived limit
in this case is very much equal to that of T2HKK.

Figure 12:  The 95 % confidence contours in the $\theta_{23}$-$\alpha$ plane
for various combination of experiments. The left panel is for the case where
$\alpha=3\times 10^{-5}$ (eV2) is assumed for the data and the right panel is
for the case where $\alpha=0$ in the decay. The black dashed curves are for
ESS$\nu$SB, red solid curves are for T2HK, blue dashed-dotted curves for
T2HKK, the magenta dashed for combination of ESS$\nu$SB and T2HK and the cyan
curves are for combination of ESS$\nu$SB and T2HKK.

In Fig. 12 we present two complementary plots to Fig. 11. If neutrino decays
in nature in addition to oscillation, then how it affects the measurements of
$\theta_{23}$ is shown in the left panel of Fig. 12. We have shown the results
in the $\theta_{23}$(test)-$\alpha$(test) parameter space at 95$\%$ C.L. We
observe that the parameter space shrinks noticeably when we combine two
experiments. Again in the right panel, we show the allowed region in
$\theta_{23}$(test)-$\alpha$(test) parameter space assuming no decay scenario
in the simulated data. Here also, the combination of experiments gives better
results that the individual experiments.

##  5 Summary & Conclusions

We have examined the physics potential of future long-baseline experiments
T2HK/T2HKK and ESS$\nu$SB in the context of invisible neutrino decay. We
performed our study for the case of normal hierarchy where $\nu_{3}$ is the
highest mass state and this is assumed to be unstable, decaying into lighter
sterile states. We compared and contrasted the sensitivities to decay of these
experiments with special emphasis on the location of the experiments at first
and second oscillation maximum and the various factors that can affect the
sensitivities.

The effect of decay appears as $\exp{(-\alpha L/E)}$ where, $\alpha$ is the
decay constant, $L$ is the distance traveled and $E$ is the energy. Hence, it
is expected that second oscillation maxima experiments having higher baselines
and/or lower energy than the first effect of decay depends on the $L/E$ factor
and second oscillation maximum occurs for lower energy it can be expected that
effect of decay can be more pronounced for second oscillation maxima. However,
the experimental sensitivities for the decay rate also depend on other vital
factors like neutrino flux, resolution of the detector and the sensitivity of
the experiment to resolve $\theta_{23}$. We expounded in detail which factors
play important roles in determining the sensitivity to decay at the first and
second oscillation maximum. Among the three experiments T2HK is designed to
have its flux peak at the first oscillation maxima, T2HKK has one detector at
the first oscillation maximum and one near the second oscillation maximum,
while ESS$\nu$SB is an experiment at the second oscillation maximum. T2HK has
a lower L/E as compared to the other experiments and has less sensitivity to
decay. On the other hand being at the first oscillation maxima it has a better
sensitivity to $\theta_{23}$ and its octant. In comparison, ESS$\nu$SB has a
higher L/E, but because of lower energy resolution as well as poor sensitivity
to $\theta_{23}$, it has a reduced sensitivity to the decay constant $\alpha$.
The two detector set up of T2HKK helps since the detector at the first
oscillation maximum helps in determining $\theta_{23}$ while the second helps
in better constraining $\alpha$. We have studied the sensitivity of these
experiments to the decay rate $\alpha$ for true $\theta_{23}$ in both lower
and higher octant.

We have also presented the octant sensitivity as a function of $\delta_{CP}$
for the three experiments. We have obtained an interesting result that in
presence of decay, the overall octant sensitivity is enhanced. This can be
attributed to the octant sensitive contribution coming from the disappearance
channel in presence of decay. Since T2HK is at the first oscillation maxima,
the appearance channel already gives a good sensitivity and the effect of
disappearance channel is not very significant. But since ESS$\nu$SB is at the
second oscillation maximum the octant sensitivity coming from appearance
channel is not very high and the disappearance channel plays a consequential
role in enhancing the octant sensitivity in presence of decay. Similar feature
is also observed for the detector at 1100 km baseline with flux peak near the
second oscillation maxima.

The octant sensitivity coming from the disappearance channel increases with
the decay constant $\alpha$ and also leads to an interesting interplay with
the appearance channel in determining the overall minimum for experiments at
second oscillation maxima. At the first oscillation maxima on the other hand,
the appearance channel dominates. As a result of this different behaviour, for
T2HKK, which has one detector at first and one at second oscillation maxima, a
synergy is observed for true $\theta_{23}$ in the higher octant leading to a
$\chi^{2}$ greater than the naive sum of the $\chi^{2}$s at two different
baselines. Since at second oscillation maximum the appearance channel does not
have very good octant sensitivity the $\chi^{2}$ minimum for the second
detector comes in the wrong octant driven by the disappearance channel. The
$\chi^{2}$ minima for the detector at the first oscillation maximum on the
other hand stays in the correct octant (the wrong octant solution being
disfavoured by the appearance channel). As an upshot, the global minima while
combining the two baselines come in a position different from the individual
minima resulting in the synergy. On the other hand, for ESS$\nu$SB, the first
detector is not present and the leaning of the disappearance channel towards
the wrong octant, weaken the sensitivity after a certain value of the decay
constant $\alpha$.

Given the correlation between $\alpha$ and $\theta_{23}$, the determination of
$\theta_{23}$ can also get affected, if there is decay in nature but it is
ignored in the fit resulting in a wrong determination of $\theta_{23}$. We
have examined this issue in detail and found that the shift of $\theta_{23}$
from its true value, can be in either direction, depending on the octant of
true $\theta_{23}$ and whether the contribution is coming from bins centered
at the maxima or minima of the survival probability, which in turn depends on
the smearing and bin-width. This effect is more for the second oscillation
maxima experiments since the probability band is narrower.

Apart from the sensitivity study for individual experiments we have also
performed sensitivity studies for the two combinations of T2HK+ESS$\nu$SB and
T2HKK+ESS$\nu$SB. We found that the sensitivity increases significantly if we
combine the experiments. The sensitivity to the decay-rate attained by the
experiments T2HK, T2HKK, ESS$\nu$SB and their combinations of T2HK+ESS$\nu$SB,
T2HKK+ESS$\nu$SB have been summarized in the Tab.4 .

Experiment(s) | $\alpha~{}\rm{(eV^{2})}$ | $\tau_{3}/m_{3}~{}\rm{(s/eV)}$
---|---|---
T2HK | $\geq~{}2.42\times 10^{-5}$ | $\leq~{}2.72\times 10^{-11}$
T2HKK | $\geq~{}1.51\times 10^{-5}$ | $\leq~{}4.36\times 10^{-11}$
ESS$\nu$SB | $\geq~{}2.71\times 10^{-5}$ | $\leq~{}2.43\times 10^{-11}$
T2HK+ESS$\nu$SB | $\geq~{}1.51\times 10^{-5}$ | $\leq~{}4.36\times 10^{-11}$
T2HKK+ESS$\nu$SB | $\geq~{}1.19\times 10^{-5}$ | $\leq~{}5.53\times 10^{-11}$
Table 4: The sensitivity to the decay rate $\alpha$ and the neutrino decay
lifetime at 3$\sigma$ CL for the standalone experiments T2HK, T2HKK,
ESS$\nu$SB and also for the combinations of T2HK+ESS$\nu$SB, T2HKK+ESS$\nu$SB.

We have also done a precision study assuming $\alpha=3\times 10^{-5}$ eV2 in
the data and found that the precision is maximum for the combination of
T2HKK+ESS$\nu$SB while standalone T2HKK has the best precision. This also
reiterates the fact that combination of first and second oscillation maximum
can give the best sensitivity to decay. Although combination of the
experiments improve the sensitivity to $\alpha$, no significant improvement is
observed for $\theta_{23}$.

In conclusion, we studied the sensitivity to decay at experiments at the first
and second oscillation maxima and discussed the salient features. Overall the
combination of first and second oscillation maxima is best for decay since at
the first oscillation maximum the precision of $\theta_{23}$ is more and at
the second oscillation maximum the sensitivity to decay could be more because
of enhanced baseline (for same energy) and/or lower energy (for same
baseline). Our study underscores the importance of the disappearance channel
in giving enhanced octant sensitivity for second oscillation maxima
experiments in presence of decay and the importance of better energy
resolution in obtaining a higher sensitivity to decay.

Note Added: While this work was being written a paper Choubey:2020dhw  came on
arXiv, which also considered neutrino decay in the context of ESS$\nu$SB
experiment. Our study contains a comparative analysis of T2HK, T2HKK and
ESS$\nu$SB experiment.

## Acknowledgment

The authors would like to thank Enrique Fernandez-Martinez for providing them
with the GLoBES files used for the simulation of the ESS$\nu$SB experiment. DD
and DP would like to acknowledge the organisers of WHEPP-XVI for their
hospitality where some initial discussions about the work took place.

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