aainstitutetext: Department of Physical Sciences, IISER Mohali, Knowledge
City, SAS Nagar, Mohali - 140306, Punjab, Indiabbinstitutetext: Center for
Theoretical Physics of the Universe, Institute for Basic Science (IBS),
Daejeon 34126, Koreaccinstitutetext: Institute of Physics, Sachivalaya Marg,
Bhubaneswar, Pin-751005, Odisha, Indiaddinstitutetext: Homi Bhabha National
Institute, BARC Training School Complex, Anushakti Nagar, Mumbai-400094, India

#  Exploring the new physics phases in 3+1 scenario in neutrino oscillation
experiments

Nishat Fiza b,c,2 Mehedi Masud c,d,3 Manimala Mitra<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The various global analyses of available neutrino oscillation data indicate
the presence of the standard $3+0$ neutrino oscillation picture. However,
there are a few short baseline anomalies that point to the possible existence
of a fourth neutrino (with mass in the eV-scale), essentially sterile in
nature. Should sterile neutrino exist in nature and its presence is not taken
into consideration properly in the analyses of neutrino data, the interference
terms arising due to the additional CP phases in presence of a sterile
neutrino can severely impact the physics searches in long baseline (LBL)
neutrino oscillation experiments. In the current work we consider one light
(eV-scale) sterile neutrino and probe all the three CP phases ($\delta_{13}$,
$\delta_{24}$, $\delta_{34}$) in the context of the upcoming Deep Underground
Neutrino Experiment (DUNE) and also estimate how the results improve when data
from NOvA, T2K and T2HK are added in the analysis. We illustrate the
$\Delta\chi^{2}$ correlations of the CP phases among each other, and also with
the three active-sterile mixing angles. Finally, we briefly illustrate how the
relevant parameter spaces in the context of neutrinoless double beta decay get
modified in light of the bounds in presence of a light sterile neutrino.

††preprint: CTPU-PTC-21-02

##  1 Introduction

The successful discovery of the phenomena of neutrino oscillation Fukuda et
al. (1998); Ahmad et al. (2002) has led to an impressive amount of research
with an aim to establish the standard 3-flavour (referred to as 3+0 hereafter)
oscillation over a wide range of energy $(E)$ and neutrino propagation length
$(L)$. In the commonly used Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
parametrization Zyla et al. (2020) of the leptonic mixing matrix, this
requires fitting of the six standard oscillation parameters, namely three
mixing angles ($\theta_{12},\theta_{13},\theta_{23}$), two mass squared
differences ($\Delta m_{21}^{2}=m_{2}^{2}-m_{1}^{2},\Delta
m_{31}^{2}=m_{3}^{2}-m_{1}^{2}$) and one Dirac CP phase ($\delta_{13}$). A
complete knowledge about the oscillation parameters will help to shed light on
the following still unresolved issues in the neutrino sector: whether there
exists CP violation (CPV) in the leptonic sector (i.e., whether
$\delta_{13}\neq 0,\pi$), whether the neutrino mass eigenstates are arranged
in normal ordering (NO, i.e., $\Delta m_{31}^{2}>0$) or inverted ordering (IO,
i.e., $\Delta m_{31}^{2}<0$), and, whether $\theta_{23}$ lies in the higher
octant (HO, i.e., $\theta_{23}>\pi/4$) or in the lower octant (LO, i.e.,
$\theta_{23}<\pi/4$). If leptonic CPV exists, it would provide a crucial
missing ingredient Sakharov (1991) in resolving another very fundamental
elusive puzzle that is baryon asymmetry in the observed universe via a
mechanism called Leptogenesis Fukugita and Yanagida (1986). Determination of
mass ordering and $\theta_{23}$ octant will help in understanding the origin
of neutrino mass Mohapatra and Senjanovic (1980); Schechter and Valle (1980);
Petcov (1982), its Dirac/Majorana nature via neutrinoless double beta decay
Haxton and Stephenson (1984) and in exploring a new symmetry called $\mu-\tau$
symmetry Lam (2001); Harrison and Scott (2002). Presently running long
baseline (LBL) experiments such as Tokai to Kamioka (T2K) Abe et al. (2014)
and NuMI Off-axis $\nu_{e}$ Appearance (NO${\nu}$A) Ayres et al. (2004) have
started uncovering a few of the open issues mentioned above. Latest T2K
results Abe et al. (2020) have been able to rule out a large range of values
of $\delta_{13}$ around $\pi/2$ at $3\sigma$ confidence level (C.L.)
irrespective of mass ordering. It also excludes CP conservation ($\delta=0$ or
$\pi$) at 95% C.L. NO$\nu$A, in its latest dataset Acero et al. (2019) using
both $\nu$ and $\bar{\nu}$ running mode hint towards NO at $1.9\sigma$ C.L.
and shows a weak preference for $\theta_{23}$ lying in HO at a C.L. of
$1.6\sigma$. Upcoming LBL experiments such as the Deep Underground Neutrino
Experiment (DUNE) Acciarri et al. (2015); Abi et al. (2020a, b), Tokai to
Hyper-Kamiokande (T2HK) Abe et al. (2015), Tokai to Hyper-Kamiokande with a
second detector in Korea (T2HKK) Abe et al. (2018a), European Spallation
Source $\nu$ Super Beam (ESS$\nu$SB) Baussan et al. (2014) as well as future
data from T2K and NO$\nu$A are expected to resolve the issues mentioned above
with an unprecedented level of precision.

Various global analyses de Salas et al. (2020); Valencia-Globalfit (2020);
Capozzi et al. (2018); Esteban et al. (2020), analyzing neutrino data from a
diverse array of sources such as atmosphere, particle accelerators, sun and
nuclear reactors have consolidated the effort to build up the 3+0 oscillation
picture. There are however, a few short-baseline (SBL) anomalies Aguilar-
Arevalo et al. (2001, 2009); Mueller et al. (2011); Mention et al. (2011);
Huber (2011) that hint towards the existence of oscillation governed by
$\mathcal{O}(\text{eV}^{2})$ mass squared difference ($\Delta
m_{41}^{2}=m_{4}^{2}-m_{1}^{2}\sim\text{eV}^{2}$) that cannot be accommodated
by the standard 3+0 scenario. The effort to explain the SBL anomalies (excess
of electron-like events at low energy) has led to the models with possible
presence of a sterile, fourth type of neutrino (this scenario is referred to
as 3+1 hereafter), which can have small mixing with the three active
neutrinos. This fourth generation of neutrino, if exists, has to be sterile
since the LEP experiment Decamp et al. (1989) restricts the number of active
neutrino flavours to three. In addition to the six standard oscillation
parameters mentioned above, 3+1 scenario is parametrized by the mass squared
difference $\Delta m_{41}^{2}$, three active-sterile mixing angles
($\theta_{14},\theta_{24},\theta_{34}$), and two additional CP phases which we
refer as $\delta_{24}$ and $\delta_{34}$. Recently the observation of a low
energy electron-like event excess at a statistically significant $4.8\sigma$
by the Mini Booster Neutrino Experiment (MiniBooNE) in its latest dataset
Aguilar-Arevalo et al. (2018, 2020) has further boosted the search for sterile
neutrinos. Various other existing and future facilities aim to search for
sterile neutrinos with high precision using different neutrino sources and
detection techniques. These facilities include IceCube Aartsen et al. (2020),
Karlsruhe Tritium Neutrino Experiment (KATRIN) Osipowicz et al. (2001),
FermiLab’s Short Baseline Neutrino (SBN) programme Antonello et al. (2015),
ANTARES Ageron et al. (2011), Neutrino Experiment for Oscillation at Short
baseline (NEOS) Ko et al. (2017), Short baseline neutrino Oscillations with a
novel Lithium-6 composite scintillator Detector(SoLid) Abreu et al. (2017),
Neutrino-4 Serebrov et al. (2019), Precision Reactor Oscillation and SPECTrum
Experiment (PROSPECT) Ashenfelter et al. (2018), Sterile Reactor Neutrino
Oscillations (STEREO) Allemandou et al. (2018); Almazán et al. (2020),
Detector of the reactor AntiNeutrino based on Solid Scintillator (DANSS)
Alekseev et al. (2018), J-PARC Sterile Neutrino Search at J-PARC Spallation
Neutron Source($\text{JSNS}^{2}$) Rott (2020); Maruyama (2020). Neutrino-4 has
recently claimed to observe active-sterile oscillation at $3.5\sigma$ around
the vicinity of $\Delta m_{41}^{2}\approx 7\text{ eV}^{2}$ and
$\theta_{14}\approx 18^{\circ}$ by analyzing the its reactor antineutrino data
accumulated since 2016 Serebrov and Samoilov (2020); Serebrov et al. (2020);
Serebrov (2020). Though it has later been argued in literature that
considering a more appropriate log-likelihood distribution Giunti (2020);
Coloma et al. (2020) or correct energy resolution Giunti et al. (2021), the
statistical significance of the active-sterile observation results in reactor
neutrino experiments actually come down to a much lower value. ANTARES with
its 10 years of data have found very mild ($\sim 1.6\sigma$) signature of
sterile neutrino in their analyses Albert et al. (2019). Experiments looking
for Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) Akimov et al.
(2017) can also act as very useful probes of sterile neutrinos, as shown by
the authors of Miranda et al. (2020).

The effect of sterile neutrino, should it exist with $\Delta m_{41}^{2}\sim
1\text{ eV}^{2}$, is most pronounced around $L/E\sim 1$ km/GeVGandhi et al.
(2015); Böser et al. (2020). In LBL experiments where $L/E\sim 500$ km/GeV at
the far detectors (FD), the high frequency induced by $\Delta m_{41}^{2}\sim
1\text{ eV}^{2}$ gets averaged out due to the finite energy resolution of the
detector. Nevertheless, it has been shown Klop and Palazzo (2015); Berryman et
al. (2015); Gandhi et al. (2015); Palazzo (2016); Agarwalla et al. (2016a, b);
Agarwalla et al. (2017); Dutta et al. (2016); Rout et al. (2017); Kelly
(2017); Ghosh et al. (2017); Choubey et al. (2017); Coloma et al. (2018); Tang
et al. (2017); Choubey et al. (2018); Agarwalla et al. (2018); Gupta et al.
(2018); Choubey et al. (2019); De Gouvêa et al. (2019); Ghoshal et al. (2019);
Kumar Agarwalla et al. (2019); Majhi et al. (2020); Ghosh et al. (2020);
Chatterjee and Palazzo (2020) that even at the FDs of these LBL experiments,
the interference effects provided by the additional CP phases play very
significant roles in spoiling the sensitivities to the crucial issues of CPV,
MH and $\theta_{23}$ octant111For a recent comprehensive status report of the
impact of a light sterile neutrino in probing these issues at LBL see, for
e.g., Giunti and Lasserre (2019); Diaz et al. (2020); Palazzo (2020) and the
references therein.. For instance, as shown in Dutta et al. (2016), the CPV
sensitivity becomes a wide band whose width depends on the unknown magnitudes
of the sterile phases $\delta_{24}$ and $\delta_{34}$, - leading to serious
confusion in interpreting the results as CP violation or CP conservation. The
constraints on the active-sterile mixing angles $\theta_{i4}$ ($i=1,2,3$) do
of course reduce such ambiguities in the interpretation of the results to some
extent. But, a clear idea about how the LBL experiments are able to measure
the sterile phases $\delta_{24}$ and $\delta_{34}$ given the constraints on
the active-sterile mixing angles, will certainly minimise the obfuscation when
their data are analyzed with a view to tackle the unresolved physics issues.
The issue of measurement of one sterile phase ($\delta_{24}$ or $\delta_{14}$,
depending on the parametrization used) has been addressed to some extent in
literature. Dutta et al. (2016) analyzes how the sterile phases impact the
standard CPV and mass ordering measurements and also probe the joint parameter
space $\delta_{13}-\delta_{24}$ for DUNE. The authors of Agarwalla et al.
(2016b) use a slightly different parameterization and discuss the CPV arising
from the individual sterile CP phases and probe the joint parameter space
$\delta_{13}-\delta_{14}$ at DUNE. They further extend their analysis by
combining simulated data from other LBL experiments such as T2HK Agarwalla et
al. (2018) and ESS$\nu$SB Kumar Agarwalla et al. (2019). Exploring the
parameter space of $\delta_{13}-\delta_{24}$ has also been addressed in
Choubey et al. (2018) for DUNE, T2HK, T2HKK and their combinations. The
authors of Choubey et al. (2018) further illustrate how the individual phase
$\delta_{24}$ can be measured by these experiments at various C.L. by assuming
four possible true values ($0,\pm\pi/2,\pi$). Chatterjee and Palazzo (2020)
shows how the recent data from T2K and NO$\nu$A can help to probe the
parameter space of $\delta_{13}-\delta_{14}$. Most recently, the authors of
Chatla and Bambah (2020) estimate how the difference
($\delta_{14}-\delta_{24}$) in the sterile phases can impact in constraining
the standard $\delta_{13}-\theta_{23}$ parameter space.

It is noteworthy that the sterile CP phase $\delta_{34}$ and its correlation
with the other phases has been little addressed in literature. In the present
manuscript we tackle the very relevant issue of estimating the capability to
reconstruct all three CP phases ($\delta_{13},\delta_{24},\delta_{34}$),
taking into consideration their $\Delta\chi^{2}$ correlations with each other
and also with the active-sterile mixing angles
($\theta_{14},\theta_{24},\theta_{34}$). We carry out this exercise in the
context of DUNE and illustrate the improvement when combined with T2K,
NO$\nu$A (both these currently running experiments are simulated upto their
present exposure) and T2HK. Apart from studying these CP phases in detail,
another crucial aspect in which our analysis differs from the existing studies
mentioned above is that we have taken into consideration the current $3\sigma$
hint of CP violation and the corresponding exclusion region of the CP phase
$\delta_{13}$ by T2K data Abe et al. (2020). Moreover, in addition to
illustrating how the 2-d parameter spaces for the CP phases can be probed, we
also analyze how the individual CP phases can be reconstructed (after
marginalizing all other relevant parameters) by the experiments, irrespective
of the actual value they might have in nature. Additionally we have also
considered the ${\nu_{\mu}}\rightarrow{\nu_{\tau}}$ channel (in addition to
$\nu_{\mu}\rightarrow\nu_{e}$ and ${\nu_{\mu}}\rightarrow\nu_{\mu}$) in our
study and estimated the capability of the projected data to measure all three
CP phases in detail. This enables us to probe the parameter spaces associated
to $\theta_{34}$ and $\delta_{34}$ with better sensitivity.

The present manuscript is organised as follows. In Sec. 2 we give a brief
account of the constraints on the sterile neutrino parameters and how the CP
phases affect the relevant probabilities. In Sec. 3 we describe the
methodology of our statistical analyses. In Sec. 4 we discuss the
$\Delta\chi^{2}$ correlations among various CP phases, taking a pair of phases
at a time. We also discuss the potential to reconstruct the three CP phases
for all possible true values by performing simulations of DUNE, T2K, NOvA and
T2HK. The role of individual oscillation channels in such reconstructions are
analyzed in Sec. 5. We estimate the $\Delta\chi^{2}$ correlations among all
the CP phases and the active-sterile mixing angles in Sec. 6. Finally in Sec.
7 we briefly discuss how the relevant parameter spaces associated to
Neutrinoless Double Beta Decay gets modified in light of the constraints on
one eV scale sterile neutrino, followed by conclusion.

##  2 Basics

We first discuss the oscillation probabilities for the three channels
($P({\nu_{\mu}\rightarrow\nu_{e}})$, $P(\nu_{\mu}\rightarrow\nu_{\mu})$ and
$P({\nu_{\mu}\rightarrow\nu_{\tau}})$) in 3+1 scenario. Since the expressions
become immensely complicated in matter, we show them in vacuum and these will
act as useful templates for gaining the physics insights in explaining our
subsequent sensitivity results. For the mixing matrix we follow the
parametrization scheme adopted in Gandhi et al. (2015):

$U^{3+1}=R(\theta_{34},\delta_{34})R(\theta_{24},\delta_{24})R(\theta_{14})R(\theta_{13})R(\theta_{23},\delta_{13})R(\theta_{12}),$
(1)

where $R(\theta_{ij},\delta_{ij})$ is a rotation in the $ij-$th plane with an
associated phase $\delta_{ij}$ such that, for e.g.,

$R(\theta_{34},\delta_{34})=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos\theta_{34}&e^{-i\delta_{34}}\sin\theta_{34}\\\
0&0&-e^{i\delta_{34}}\sin\theta_{34}&\cos\theta_{34}\end{array}\right).$ (2)

Now we discuss briefly on the allowed ranges of the sterile sector parameters,
as estimated in great detail in the global analysis of various neutrino data
Dentler et al. (2018). $|U_{e4}|^{2}$ is bounded by $\nu_{e}$ and
$\bar{\nu}_{e}$ disappearance searches and is equal to $\sin^{2}\theta_{14}$.
The combined atmospheric neutrino data from IceCube, DeepCore and SK, at
$99\%$ C.L. (2 DOF) put the bound $|U_{e4}|^{2}\lesssim 0.1$. This implies
$\theta_{14}\lesssim 18.4^{\circ}$. It can be seen from all $\nu_{e}$ and
$\bar{\nu}_{e}$ searches that the best fit value of $|U_{e4}|^{2}$ is
approximately equal to $0.01$, which gives $\theta_{14}\approx 5.7^{\circ}$.
The data from $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ disappearance searches put the
following $99\%$ C.L. (2 D.O.F.) constraints: $|U_{\mu 4}|^{2}\lesssim 0.01$
and $|U_{\tau 4}|^{2}\lesssim 0.17$. Since, in our parametrization $|U_{\mu
4}|=\cos\theta_{14}\sin\theta_{24}$ and $|U_{\tau
4}|=\cos\theta_{14}\cos\theta_{24}\sin\theta_{34}$, the corresponding bounds
on $\theta_{24}$ and $\theta_{34}$ can easily be translated to
$\theta_{24}\lesssim 6.05^{\circ}$ and $\theta_{34}\lesssim 25.8^{\circ}$. The
allowed values of $\Delta m_{41}^{2}$ roughly lie in the range of $1-10\text{
eV}^{2}$ and we consider it to be $1.3\text{ eV}^{2}$ in the present work as
per the global analysis222In our notation, $m_{1,2,3}$ are the masses of the
three active neutrinos, and $m_{4}$ denotes the mass of the sterile neutrino.
Also, $\Delta m^{2}_{ij}=m^{2}_{i}-m^{2}_{j}$..

Using the standard approach for deriving oscillation probability, we obtain
for the $\nu_{\alpha}\to\nu_{\beta}$ ($\alpha,\beta=e,\mu,\tau,s$ and
$\alpha\neq\beta$) transition probability,

$\displaystyle P_{\alpha\beta}^{3+1}$ $\displaystyle=4|U_{\alpha 4}U_{\beta
4}|^{2}\times 0.5$ $\displaystyle-4\text{Re}(U_{\alpha 1}U_{\beta
1}^{*}U_{\alpha 2}^{*}U_{\beta 2})\sin^{2}\Delta_{21}+2\text{Im}(U_{\alpha
1}U_{\beta 1}^{*}U_{\alpha 2}^{*}U_{\beta 2})\sin 2\Delta_{21}$
$\displaystyle-4\text{Re}(U_{\alpha 1}U_{\beta 1}^{*}U_{\alpha 3}^{*}U_{\beta
3})\sin^{2}\Delta_{31}+2\text{Im}(U_{\alpha 1}U_{\beta 1}^{*}U_{\alpha
3}^{*}U_{\beta 3})\sin 2\Delta_{31}$ $\displaystyle-4\text{Re}(U_{\alpha
2}U_{\beta 2}^{*}U_{\alpha 3}^{*}U_{\beta
3})\sin^{2}\Delta_{32}+2\text{Im}(U_{\alpha 2}U_{\beta 2}^{*}U_{\alpha
3}^{*}U_{\beta 3})\sin 2\Delta_{32},$ (3)

where $\Delta_{ij}=\frac{\Delta m^{2}_{ij}L}{4E}.$ In deriving Eq. 2, we have
used the unitarity of the $4\times 4$ mixing matrix $U$ and applied the usual
assumptions that the term containing mass square splitting between $m_{4}$ and
$m_{i}\ (i=1,2,3)$, i.e., $\sin^{2}\Delta_{4i}$ and $\sin 2\Delta_{4i}$
average out to 0.5 and 0 respectively at long baseline ($i=1,2,3$). From Eq. 2
one can use the long baseline approximation (i.e., neglecting the oscillation
effects due to $\Delta m_{21}^{2}$) and arrive at the following simplified
expression for the dominant channel $\nu_{\mu}\to\nu_{e}$.

$\displaystyle P_{\mu e}^{4\nu}\approx$
$\displaystyle\frac{1}{2}\sin^{2}{2\theta_{\mu e}^{4\nu}}$
$\displaystyle+(a^{2}\sin^{2}2\theta_{\mu
e}^{3\nu}-\frac{1}{4}\sin^{2}2\theta_{13}\sin^{2}{2\theta_{\mu
e}^{4\nu}})\sin^{2}\Delta_{31}$
$\displaystyle+\cos(\delta_{13}+\delta_{24})a\sin{2\theta_{\mu
e}^{3\nu}}\sin{2\theta_{\mu e}^{4\nu}}\cos 2\theta_{13}\sin^{2}\Delta_{31}$
$\displaystyle+\frac{1}{2}\sin(\delta_{13}+\delta_{24})a\sin{2\theta^{3\nu}_{\mu
e}}\sin{2\theta^{4\nu}_{\mu e}}\sin{2\Delta_{31}},$ (4)

where we have followed the convention of Gandhi et al. (2015) for the
following quantities.

$\sin{2\theta_{\mu e}^{3\nu}}=\sin{2\theta_{13}}\sin{\theta_{23}}$,

$b=\cos{\theta_{13}}\cos{\theta_{23}}\sin{\theta_{12}}$,

$\sin{2\theta_{\mu e}^{4\nu}}=\sin{2\theta_{14}}\sin{\theta_{24}}$,

$a=\cos{\theta_{14}}\cos{\theta_{24}}$.

Eq. 2 tells us that in vacuum $P({\nu_{\mu}\rightarrow\nu_{e}})$ is sensitive
to both $\delta_{13}$ and $\delta_{24}$, but not to $\delta_{34}$. As
explained in Gandhi et al. (2015), small dependence on $\delta_{34}$ creeps in
when matter effect is taken into account. The expressions for the less
dominant channels $P(\nu_{\mu}\rightarrow\nu_{\mu})$ and
$P({\nu_{\mu}\rightarrow\nu_{\tau}})$ can similarly be derived from Eq. 2, but
the expressions are quite lengthy. Interested readers can see De Gouvêa et al.
(2019); Yue et al. (2020) for those probability expressions.

Parameter | Best-fit-value | 3$\sigma$ interval |  $1\sigma$ uncertainty
---|---|---|---
$\theta_{12}$ [Deg.] | 34.3 | 31.4 - 37.4 | 2.9%
$\theta_{13}$ (NH) [Deg.] | 8.58 | 8.16 - 8.94 | 1.5%
$\theta_{13}$ (IH) [Deg.] | 8.63 | 8.21 - 8.99 | 1.5%
$\theta_{23}$ (NH) [Deg.] | 48.8 | 41.63 - 51.32 | 3.5%
$\theta_{23}$ (IH) [Deg.] | 48.8 | 41.88 - 51.30 | 3.5%
$\Delta m_{21}^{2}$ [$\text{eV}^{2}$] | $7.5\times 10^{-5}$ | [6.94 - 8.14]$\times 10^{-5}$ | 2.7%
$\Delta m_{31}^{2}$ (NH) [$\text{eV}^{2}$] | $+2.56\times 10^{-3}$ | [2.46 - 2.65] $\times 10^{-3}$ | 1.2%
$\Delta m_{31}^{2}$ (IH) [$\text{eV}^{2}$] | $-2.46\times 10^{-3}$ | -[2.37 - 2.55]$\times 10^{-3}$ | 1.2%
$\delta_{13}$ (NH) [Rad.] | $-0.8\pi$ | $[-\pi,0]\cup[0.8\pi,\pi]$ | $-$
$\delta_{13}$ (IH) [Rad.] | $-0.46\pi$ | $[-0.86\pi,-0.1\pi]$ | $-$
$\theta_{14}$ [Deg.] | 5.7, 10 | 0 - 18.4 | $\sigma(\sin^{2}\theta_{14})=5\%$
$\theta_{24}$ [Deg.] | 5, 6 | 0 - 6.05 | $\sigma(\sin^{2}\theta_{24})=5\%$
$\theta_{34}$ [Deg.] | 20, 25 | 0 - 25.8 | $\sigma(\sin^{2}\theta_{34})=5\%$
$\delta_{24}$ [Rad.] | $0,-0.5\pi$ | $[-\pi,\pi]$ | $-$
$\delta_{34}$ [Rad.] | $0,-0.5\pi$ | $[-\pi,\pi]$ | $-$

Table 1:  Standard oscillation parameters and their uncertainties used in our
study. The values of 3+0 parameters were taken from the global fit analysis in
de Salas et al. (2020) while the 3+1 parameter values were chosen from Dentler
et al. (2018) (see also Sec. 2). If the $3\sigma$ upper and lower limit of a
parameter is $x_{u}$ and $x_{l}$ respectively, the $1\sigma$ uncertainty is
$(x_{u}-x_{l})/3(x_{u}+x_{l})\%$ Abi et al. (2020a). For the active-sterile
mixing angles, a conservative $5\%$ uncertainty was used on
$\sin^{2}\theta_{i4}$ (i = 1, 2, 3).  Figure 1: We show the probability bands
due to individual variation of the CP phases $\delta_{13}$ (grey),
$\delta_{24}$ (blue) and $\delta_{34}$ (red) in the whole range of
$[-\pi,\pi]$ at a baseline of 1300 km. The three panels correspond to the
three channels
$P({\nu_{\mu}\rightarrow\nu_{e}}),P(\nu_{\mu}\rightarrow\nu_{\mu})$ and
$P({\nu_{\mu}\rightarrow\nu_{\tau}})$. The insets in the second and third
panels show magnified versions of the rectangular regions indicated. The three
active-sterile mixing angles were taken as
$\theta_{14}=10^{\circ},\theta_{24}=6^{\circ},\theta_{34}=25^{\circ}$. The
values of the rest of the oscillation parameters were taken from Tab. 1.
Normal hierarchy was assumed for generating this plot.

Using the widely used General Long Baseline Experiment Simulator (GLoBES)
Huber et al. (2005, 2007) and the relevant plugin snu.c Kopp (2008); Kopp et
al. (2008) for implementing sterile neutrinos, we now illustrate how the
probabilities for different oscillation channels depends on the three CP
phases ($\delta_{13},\delta_{24},\delta_{34}$) individually at the DUNE
baseline of 1300 km. In Fig. 1, we plot the bands for $P_{\mu
e},P_{\mu\mu},P_{\mu\tau}$ (in the three panels respectively) due to
individual variation of $\delta_{13},\delta_{24},\delta_{34}$, each being
varied in the range $[-\pi,\pi]$. The grey band shows the variation of the
standard Dirac CP phase $\delta_{13}$. The variation due to $\delta_{24}$
($\delta_{34}$) is shown with the blue (red) band. For the variation of each
CP phase, the other two phases are kept fixed: $\delta_{13}$ is kept fixed at
the bf value of $-0.8\pi$ while $\delta_{24},\delta_{34}$ were considered to
be zero. The sterile phases are associated with the active-sterile mixing
angles. We have a slightly high active-sterile mixing:
$\theta_{14}=10^{\circ},\theta_{24}=6^{\circ},\theta_{34}=25^{\circ}$. As
expected333In Eq. 2 for $\alpha=\beta$, the imaginary part in the RHS
vanishes, diminishing the effect of the CP phases., the CP phases have larger
impact on appearance channels rather than the disappearance channels.
$\nu_{\mu}\rightarrow\nu_{e}$ channel is most affected by the variation of
$\delta_{13}$. Among the sterile CP phases, $\delta_{24}$ has a visibly
greater impact on $P({\nu_{\mu}\rightarrow\nu_{e}})$ than that of
$\delta_{34}$. It is interesting to note this feature especially in light of
the fact that the value of the active-sterile mixing angle $\theta_{24}$
(taken as $6^{\circ}$ in Fig. 1) is almost 5 times smaller than $\theta_{34}$
($25^{\circ}$). We also observe that with increase in energy the effect of
$\delta_{34}$ on $P({\nu_{\mu}\rightarrow\nu_{e}})$ further reduces.
$P({\nu_{\mu}\rightarrow\nu_{\tau}})$ is less prone to variation of the CP
phases. Still it shows slight variation for all three CP phases with
$\delta_{24}$ and $\delta_{34}$ having almost similar effects but less than
that of $\delta_{13}$. $P(\nu_{\mu}\rightarrow\nu_{\mu})$ on the other hand,
shows almost negligible variation with the CP phases. After a brief
description of the simulation procedure we will estimate the capability of LBL
experiments to reconstruct these CP phases.

##  3 Simulation details

We simulate the long baseline neutrino experiments DUNE, NOvA, T2K and T2HK
using GLoBES Huber et al. (2005, 2007). DUNE is a 1300 km long baseline
experiment employing a liquid argon far detector (FD) of 40 kt fiducial mass
with a beam of power 1.07 MW and running 3.5 years each on $\nu$ and
$\bar{\nu}$ mode (resulting in a total exposure of roughly 300 kt.MW.yr
corresponding to $1.47\times 10^{21}$ protons on target or POT). We have used
the official configuration files Alion et al. (2016) provided by the DUNE
collaboration for its simulation. Following this, we have also taken into
account the presence of a near detector (ND) at 459 m from the source. The ND
helps in making a more precise measurement of the flux and cross-section,
thereby reducing the relevant systematic uncertainties at the FD. We should
mention here that we have done a $\Delta\chi^{2}$ analysis (discussed later)
with the simulated data at FD alone, rather than a joint $\Delta\chi^{2}$
analysis using simulated data both at ND and FD444It is worthwhile to note
here that an eV-scale sterile neutrino will have its signature at the ND due
to short baseline active-sterile oscillation and consequently a joint analysis
using both ND and FD data would probably constrain the active-sterile mixing
angles slightly more. But our main aim is the analyses of the CP phases, given
the already existing constraints on the mixing angles from the global analysis
Dentler et al. (2018), and this would have more observable signals at the FD,
especially in neutrino appearance measurements Gandhi et al. (2015). Electron
neutrino appearance signals (CC), muon neutrino disappearance signals (CC), as
well as neutral current (NC) backgrounds and tau neutrino appearance
backgrounds (along with the corresponding systematics/efficiencies etc.) are
already included in the configuration files.

In the present analysis, we have additionally incorporated tau neutrino
appearance as a separate signal following De Gouvêa et al. (2019); Ghoshal et
al. (2019). Charged current interaction of an incoming $\nu_{\tau}$ produces a
$\tau$ lepton (requires a threshold energy of $\gtrsim 3.4$ GeV for the
incoming $\nu_{\tau}$), which can decay hadronically (with a branching
fraction $\sim 65\%$) or leptonically (with a branching fraction of $\sim
35\%$). The analysis of the hadronic decay channel involves the capability of
the detector to study the resulting pions and kaons. More importantly, NC
neutrino scattering constitutes the biggest background for the hadronic decay
channel of $\tau$. Following De Gouvêa et al. (2019), we have used an
efficiency to separate $30\%$ hadronically decaying $\tau$ events (with about
$1\%$ NC events remaining). On the other hand, the leptonic decay channels of
$\tau$ ($\tau^{-}\to e^{-}\bar{\nu}_{e}\nu_{\tau}$;
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$) are more difficult to analyse,
due to the large background mainly consisting of $\nu_{e}$-CC and
$\nu_{\mu}$-CC respectively (along with backgrounds from NC and contaminations
due to wrong sign leptons.). Following Ghoshal et al. (2019), we have taken
the efficiency of the electron channel to be $15\%$. Due to the overwhelming
background, we have taken a nominal efficiency of $5\%$ in the muon channel.
Naturally, the contribution of the leptonic decay channel of $\tau$ is very
small. We should also mention here that the decay of $\tau$ at the detector
will involve missing energy in the form of an outgoing $\nu_{\tau}$, which in
turn makes the energy reconstruction of the incoming $\nu_{\tau}$ difficult.
From De Gouvêa et al. (2019), we use a Gaussian energy reconstruction with a
resolution of $20\%$ which is a conservative estimate. We acknowledge that our
implementation of the $\nu_{\tau}$ channel as a signal is conservative in
nature. Nevertheless, this provides a small but non-negligible statistics in
terms of events and $\Delta\chi^{2}$ sensitivity (see Sec. 5 and Appendix C).
Using a much more sophisticated analysis of $\nu_{\tau}$ appearance channel at
DUNE by implementing jet- clustering algorithms and machine learning
techniques, as has been pioneered in Machado et al. (2020), one certainly
expects to exploit the rich physics capabilities hidden within this channel.

We have simulated NOvA with a baseline of 800 km employing an FD of fiducial
mass of 14 kt and a beam of 742 kW. The simulation for NOvA was implemented
according to Acero et al. (2019) which generates $8.85\times 10^{20}$
$(12.33\times 10^{20})$ POT in $\nu$ ($\bar{\nu}$) mode. T2K is a 295 km
experiment with a 22.5 kt water cherencov FD. For T2K simulation we use the
inputs from Abe et al. (2011, 2020). We have used a beam of 515 kW and
simulating $1.97\times 10^{21}$ $(1.63\times 10^{21})$ POT in $\nu$
($\bar{\nu}$) mode. T2HK is an upgraded version of T2K with a higher beam of
1.3 MW and a much bigger fiducial mass of 187 kt of its water cherencov FD.
For T2HK we simulate a total of $2.7\times 10^{22}$ POT in 1:3 ratio of $\nu$
and $\bar{\nu}$ mode (with inputs taken from Abe et al. (2015, 2018b)). Note
that, for the future experiments DUNE and T2HK we have used the full expected
exposure, while for the currently running experiments T2K and NOvA we have
simulated upto their current exposure.

To estimate the capability of LBL experiments to reconstruct the CP phases, we
carry out a $\Delta\chi^{2}$ analysis using GLoBES and the relevant plugin
snu.c Kopp (2008); Kopp et al. (2008) for implementing sterile neutrinos. In
order to gain insight, let us examine the analytical form of the
$\Delta\chi^{2}$,

$\displaystyle\Delta\chi^{2}(p^{\text{true}})=\underset{p^{\text{test}},\eta}{\text{Min}}\Bigg{[}$
$\displaystyle
2\sum_{k}^{\text{mode}}\sum_{j}^{\text{channel}}\sum_{i}^{\text{bin}}\Bigg{\\{}N_{ijk}^{\text{test}}(p^{\text{test}};\eta)-N_{ijk}^{\text{true}}(p^{\text{true}})$
$\displaystyle+N_{ijk}^{\text{true}}(p^{\text{true}})\ln\frac{N_{ijk}^{\text{true}}(p^{\text{true}})}{N_{ijk}^{\text{test}}(p^{\text{test}};\eta)}\Bigg{\\}}+\sum_{l}\frac{(p^{\text{true}}_{l}-p^{\text{test}}_{l})^{2}}{\sigma_{p_{l}}^{2}}+\sum_{m}\frac{\eta_{m}^{2}}{\sigma_{\eta_{m}}^{2}}\Bigg{]},$
(5)

where $N^{\text{true}}$ ($N^{\text{test}}$) is the true (test) set of events,
while $p^{\text{true}}$ ($p^{\text{test}}$) is the set of true (test)
oscillation parameters. The index $i$ is summed over the energy bins of the
experiment concerned (as discussed above). The indices $j$ and $k$ are summed
over the oscillation channels
($\nu_{\mu}\rightarrow\nu_{e},{\nu_{\mu}}\rightarrow\nu_{\mu},{\nu_{\mu}}\rightarrow{\nu_{\tau}}$)
and the modes ($\nu$ and $\bar{\nu}$) respectively. The term
$(N^{\text{test}}-N^{\text{true}})$ takes into account the algebraic
difference while the log-term inside the curly braces considers the fractional
difference between the test and true sets of events. The term summed over
$i,j,k$ and written inside the curly braces is the statistical part of
$\Delta\chi^{2}$. $\sigma_{p_{l}}$ is the uncertainty in the prior measurement
of the $l$-th oscillation parameter $p_{l}$. The values of the true or best-
fit oscillation parameters and their uncertainties as used in the present
analysis are tabulated in Table 1. In the last term, $\sigma_{\eta_{m}}$ is
the uncertainty on the systematic/nuisance parameter $\eta_{m}$ and the sum
over $m$ takes care of the systematic part of $\Delta\chi^{2}$. This way of
treating the systematics in the $\Delta\chi^{2}$ calculation is known as the
method of pulls Huber et al. (2002); Fogli et al. (2002); Gonzalez-Garcia and
Maltoni (2004); Gandhi et al. (2007). For DUNE, the $\nu_{e}$ and
$\bar{\nu}_{e}$ signal modes have a normalization uncertainties of $2\%$ each,
whereas the $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ signals have a normalization
uncertainty of $5\%$ each. The $\nu_{\tau}$ and $\bar{\nu}_{\tau}$ signals
have a normalization uncertainties of $20\%$ each. The background
normalization uncertainties vary from $5\%-20\%$ and include correlations
among various sources of background (coming from beam $\nu_{e}/\bar{\nu}_{e}$
contamination, flavour misidentification, neutral current and $\nu_{\tau}$).
The final estimate of $\Delta\chi^{2}$ obtained after a marginalization over
the $3\sigma$ range of test parameters555The test parameters that are
marginalised include those listed in Tab. 1 along with the specified ranges
and priors. Note that, we have used a $5\%$ prior for the sine of each of the
active-sterile mixing angles while varying them over their entire range.
$p^{\text{test}}$ and the set of systematics ($\eta$) is a function of the
true values of the oscillation parameters. Technically this $\Delta\chi^{2}$
is the frequentist method of hypotheses testing Fogli et al. (2002); Qian et
al. (2012).

##  4 Reconstruction of the CP phase in correlation with other phases

Figure 2: Reconstruction of the CP phases, taken pairwise at a time, at a C.L.
of $1\sigma$ (2 D.O.F.) at DUNE (red), DUNE + T2K + NO$\nu$A (green), and DUNE
+ T2K + NO$\nu$A + T2HK (blue). The top (bottom) row corresponds to the choice
$\delta_{24}=\delta_{34}=0$ ($-\pi/2$). The true value of the standard Dirac
CP phase $\delta_{13}$ is fixed at $-0.8\pi$. The black dot indicates the true
values assumed. The true values of active-sterile mixing angles are taken as
$\theta_{14},\theta_{24},\theta_{34}=5.7^{\circ},5^{\circ},20^{\circ}$. Other
oscillation parameters were taken from Tab. 1.

In Fig. 2, we illustrate how the CP phases can be reconstructed at $1\sigma$
C.L. in the plane of (test $\delta_{13}$-test $\delta_{24}$), (test
$\delta_{13}$-test $\delta_{34}$) and (test $\delta_{24}$-test $\delta_{34}$)
in the three columns respectively. The red contour shows the reconstruction
capability of DUNE. The green and blue contours illustrate the reconstruction
when it is combined with (T2K + NO$\nu$A) and (T2K + NO$\nu$A + T2HK)
respectively. The top (bottom) row of Fig. 2 shows the choice of the CP
conserving (maximally CP violating) true values of $\delta_{24}$ and
$\delta_{34}$. Marginalisation has been carried over the test values of
$\theta_{13}$, $\theta_{23}$, $\Delta m_{31}^{2}$ (including both the mass
hierarchies) with prior uncertainties mentioned in Tab. 1. the active-sterile
mixing angles $\theta_{14},\theta_{24},\theta_{34}$ (with a prior uncertainty
of $5\%$ on each $\sin^{2}\theta_{i4}$) and the third CP phase not shown along
the axes of a particular panel has also been considered for marginalisation.
Clearly the reconstruction of $\delta_{13}$ is better (as evidenced by the
narrowness of the contours along the test $\delta_{13}$ axis) compared to the
other two phases. This happens since the associated mixing angle $\theta_{13}$
has been measured very precisely unlike the corresponding active-sterile
mixing angles $\theta_{24}$ and $\theta_{34}$ (see Table 1 and Sec. 2). For
the maximally CP violating choices of true $\delta_{24}$ and true
$\delta_{34}$, their reconstruction gets better at the cost of small
degeneracies appearing for DUNE around test $\delta_{24}\approx 135^{\circ}$,
$\delta_{34}\approx 90^{\circ}$. Adding data from other experiments lifts
these degeneracies. We would also like to refer the reader to Appendices A and
B, which show the effect of prior of $\theta_{23}$ and different true choices
of the standard oscillation parameters on the reconstruction of the CP phases.

Figure 3: Reconstruction of the CP phases $\delta_{13}$, $\delta_{24}$ and
$\delta_{34}$, for all the choices of their true values in $[-\pi,\pi]$ at a
C.L. of $1\sigma$ (1 D.O.F.) at DUNE (red), DUNE + T2K + NO$\nu$A (green), and
DUNE + T2K + NO$\nu$A + T2HK (blue). The top (bottom) row corresponds to the
choice of the active-sterile mixing angles as true
$\theta_{14},\theta_{24},\theta_{34}=5.7^{\circ},5^{\circ},20^{\circ}$
($10^{\circ},6^{\circ},25^{\circ}$). The true values of the phases not shown
in a panel are fixed at: $\delta_{13}$, $\delta_{24}$ and
$\delta_{34}=-0.8\pi,0$ and $0$ respectively. In the first column the grey
shaded regions depict the $3\sigma$ allowed values measured by T2K Abe et al.
(2020).

In Fig. 3, we illustrate how efficiently the combination of LBL experiments
can reconstruct the three CP phases $\delta_{13},\delta_{24}$ and
$\delta_{34}$ at $1\sigma$ C.L. in the three columns respectively given their
true value lying anywhere in the whole parameter space of $[-\pi,\pi]$. In
addition to the poorly measured 3+0 parameters ($\theta_{23},\Delta
m_{31}^{2}$) and the active-sterile mixing angles
($\theta_{14},\theta_{24},\theta_{34}$), in each panel we have also
marginalised over the two other CP phases ($\in[-\pi,\pi]$) not shown along
the axes. The top (bottom) row depicts small (large) active-sterile mixing
with true
$\theta_{14},\theta_{24},\theta_{34}=5.7^{\circ},5^{\circ},20^{\circ}$
($\theta_{14},\theta_{24},\theta_{34}=10^{\circ},6^{\circ},25^{\circ}$). Note
that, the latter choice of values represents the upper limits of the allowed
active-sterile mixing (See Sec. 2). For each true value of the CP phases
($\in[-\pi,\pi]$), the corresponding vertical width of the contours provide an
estimate of the precision of reconstructing that true value. It is evident
that in comparison to the reconstruction of the sterile phases, the standard
Dirac phase $\delta_{13}$ can be reconstructed much more efficiently by the
LBL experiments. We note that the reconstruction of $\delta_{13}$ does not
noticeably depend upon the size of the active-sterile mixing as assumed in the
two rows. As the T2K data Abe et al. (2020) suggests, if $\delta_{13}$ indeed
turns out to lie in the lower half plane of $[-\pi,0]$ with a best fit roughly
around the maximal CPV ($\approx-\pi/2$), the future analyses with a
combination of (DUNE + T2K + NO$\nu$A+ T2HK) with their projected runtime will
be able to measure this phase in a narrow approximate range of
$[-115^{\circ},-75^{\circ}]$ (at $1\sigma$). The precision is marginally
better if $\delta_{13}$ turns out to be close to the CP conserving value
($0$). As far as the reconstruction of the sterile phases are concerned,
$\delta_{24}$ can be reconstructed with a better precision than $\delta_{34}$.
For small active-sterile mixing (top row of Fig. 3) and without considering
the T2HK-projected data, $\delta_{24}$ can be better reconstructed only in the
lower half plane. But with T2HK-projected data, its reconstruction becomes
much better throughout the entire parameter space. T2HK, due to its shorter
baseline and much higher fiducial mass of its water cerenkov detector can
offer very high statistics which helps to alleviate the degeneracy. On the
other hand, the reconstruction of $\delta_{34}$ is bad almost for the entire
parameter space if T2HK is not considered. For e.g., if the true value of
$\delta_{34}$ in nature turns out to be around $[-135^{\circ},-90^{\circ}]$,
the range of values reconstructed at $1\sigma$ by the combination of DUNE +
T2K + NO$\nu$A can be anywhere between $-180^{\circ}$ to $180^{\circ}$. Large
active-sterile mixing (bottom row of Fig. 3) significantly improves the
sensitivities of the LBL experiments to the sterile CP phases, resulting in
significant improvement of the reconstruction of $\delta_{24}$ and
$\delta_{34}$. We have already seen in Fig. 1 that
$P({\nu_{\mu}\rightarrow\nu_{e}})$ is most sensitive to $\delta_{13}$ and
least sensitive to $\delta_{34}$. This leads to a better precision in
reconstructing $\delta_{13}$ in comparison to the others.

##  5 Role of individual channels in reconstruction

Fig. 4 illustrates the impact of different appearance and disappearance
channels on the reconstruction of the CP phases (taken pairwise, like in Fig.
2) at $1\sigma$ C.L. at DUNE. The innermost red contours consisting of all the
three channels for DUNE are the same as the red contours in the upper row of
Fig. 2. We can clearly see the decrease in uncertainty in the measurement of
the CP phases as we keep on adding $\nu_{\mu}\rightarrow\nu_{\tau}$ appearance
and $\nu_{\mu}\rightarrow\nu_{\mu}$ disappearance channel to the
$\nu_{\mu}\rightarrow\nu_{e}$ appearance channel.

Figure 4: Reconstruction of the CP phases, taken pairwise at a time for three
different channels at DUNE at a C.L. of $1\sigma$ (2 D.O.F.). for $\nu_{e}$
appearance channel (cyan), $\nu_{e}$ \+ $\nu_{\tau}$ appearance channel (grey)
and all channels (red). The three panels depict the test values of the three
CP phases. The true values for the CP phases were assumed as
$\delta_{13},\delta_{24},\delta_{34}=-0.8\pi,0,0$ and the true active-sterile
mixing angles were chosen as
$\theta_{14},\theta_{24},\theta_{34}=5.7^{\circ},5^{\circ},20^{\circ}$. The
black dot indicates the true values assumed.

The phase dependence of the ${\nu_{\mu}}\rightarrow{\nu_{\tau}}$ channel (see
also Fig. 1) helps somewhat in this case. Note that, the improvement in result
due to the addition of the $\nu_{\tau}$ appearance channel is not very
significant and this is due to the very low statistics provided by this
channel due to the difficulty in observing $\nu_{\tau}$. On the other hand,
though the phase dependence of ${\nu_{\mu}}\rightarrow\nu_{\mu}$ channel is
small, it offers a very large number of events compared to the other two
channels, thereby decreasing the uncertainty in reconstruction (see also,
Appendix C). The improvement in reconstruction along the $\delta_{13}$
direction is not significant with the addition of channels, underlining the
pivotal role here played by the $\nu_{e}$ appearance channel. In contrast, as
observed from the first two panels of Fig. 4, the reconstruction of
$\delta_{24}$ and $\delta_{34}$ is significantly better (more so in the
former) particularly by the addition of $\nu_{\mu}$ disappearance channel. In
the third panel, the role of $\nu_{\mu}$ disappearance is emphasized by the
shrinking of the reconstruction contour in both $\delta_{24}$ and
$\delta_{34}$ directions. Here we also note a slight improvement in
reconstruction along the $\delta_{34}$ axis (but not so much along
$\delta_{24}$) with the addition of $\nu_{\tau}$ appearance channel. In the
next section we are going to estimate the correlations of the phases with the
active-sterile mixing angles.

##  6 Reconstruction of the CP phase in correlation with active-sterile mixing
angles

In Fig. 5 we show how efficiently the LBL experiments can reconstruct the CP
phases in correlation with active-sterile mixing angles at $1\sigma$ C.L. (2
D.O.F.). The three phases (angles) are shown along the three rows (columns).
In the second and third column the ranges of $\theta_{24}$ and $\theta_{34}$
are not shown beyond their allowed values. This results in the contours not
being closed in these cases. As expected, the top row shows much less
uncertainty along the test $\delta_{13}$ direction for all the three angles
involved. In all the panels, we note that combining NO$\nu$A and T2K data to
DUNE gives mild improvement while a further addition of data projected by T2HK
significantly improves the reconstruction. We have varied the test values of
the angles upto their upper limit at $99\%$ C.L.: $\theta_{14}\lesssim
18.4^{\circ},\theta_{24}\lesssim 6.05^{\circ},\theta_{34}\lesssim
25.8^{\circ}$, and the true values to be reconstructed were
$\theta_{14}=5.7^{\circ},\theta_{24}=5^{\circ},\theta_{34}=20^{\circ}$ (as
discussed in Sec. 2).

Figure 5: Reconstruction of the CP phases in correlation with the mixing
angles $\theta_{14},\theta_{24},\theta_{34}$ at a C.L. of $1\sigma$ (2 D.O.F.)
at DUNE (red), DUNE + T2K + NO$\nu$A (green), and DUNE + T2K + NO$\nu$A + T2HK
(blue). The three columns (rows) depict the test values of the three active-
sterile mixing angles (three CP phases). The true values for the CP phases
were assumed as $\delta_{13},\delta_{24},\delta_{34}=-0.8\pi,0,0$ and the true
active-sterile mixing angles were chosen as
$\theta_{14},\theta_{24},\theta_{34}=5.7^{\circ},5^{\circ},20^{\circ}$. The
black dot indicates the true values assumed.

The horizontal (vertical) span of the resulting contours indicate the
uncertainties along the mixing angles (CP phases). We observe that the
contours involving $\theta_{24}$ and $\theta_{34}$ do not close as long as we
confine ourselves upto the allowed upper limit of $\theta_{24}$ and
$\theta_{34}$. It is interesting to note that even if $\theta_{34}$ is allowed
to vary in a larger range, the contours involving it show considerable
uncertainty. This signifies the difficulty in constraining $\theta_{34}$ in
the experiments. It is mainly this reason which also hinders a good
reconstruction of the associated phase $\delta_{34}$, as we have seen in Sec.
4. To have a quantitative idea about the potential to reconstruct the true
values of the active-sterile mixing angle $\theta_{i4}$ ($i=1,2,3$), we
calculate how much the total horizontal span of each blue contour is. From
this we estimate the maximum range of uncertainty (so as to obtain a
conservative range) in reconstructing $\theta_{i4}$ and tabulate these below
in Table 2.

Angle | Value to be reconstructed | Reconstructed range
---|---|---
| [Degree] | [Degree]
$\theta_{14}$ | 5.7 | $3.3\lesssim\theta_{14}\lesssim 9.1$
$\theta_{24}$ | 5 | $3.2\lesssim\theta_{24}\lesssim 6$
$\theta_{34}$ | 20 | $9.0\lesssim\theta_{34}\lesssim 25$
Table 2: The maximum reconstructed ranges for
$\theta_{14},\theta_{24},\theta_{34}$ as estimated from Fig. 5 for DUNE + T2K
+ NOvA + HK.

##  7 Impact of constraints on Neutrinoless Double Beta Decay

Before concluding, we include discussion of $3+1$ framework on other
observables, such as, neutrinoless double beta decay. The sterile neutrino, if
a Majorana particle, gives non-zero contribution in the lepton number
violating neutrinoless double beta decay (NDBD). In the presence of a non-zero
$\theta_{14}$, the effective mass of NDBD process becomesGiunti and Zavanin
(2015),

$\displaystyle
m_{\textrm{eff}}=|m_{1}|U_{e1}|^{2}+m_{2}|U_{e2}|^{2}e^{i\alpha_{2}}+m_{3}|U_{e3}|^{2}e^{i\alpha_{3}}+m_{4}|U_{e4}|^{2}e^{i\alpha_{4}}|,$
(6)

where $\alpha_{2}$, $\alpha_{3}$, $\alpha_{4}$ are the relevant CP phases.
Following the parametrisation given in Eq. 1, the relevant elements of the
mixing matrix are as follows.

$|U_{e1}|=c_{12}c_{13}c_{14},|U_{e2}|=s_{12}c_{13}c_{14},|U_{e3}|=s_{13}c_{14},|U_{e4}|=\sin\theta_{14}.$
(7)

The expression for half-life of $0\nu\beta\beta$ transition can be given as
Mitra et al. (2012),

$\frac{1}{T_{1/2}^{0\nu}}=G_{0\nu}\left|M_{\nu}\eta_{\nu}+M_{N}\eta_{N}\right|^{2},$
(8)

where,

$\eta_{\nu}=\frac{U_{e\;i}^{2}m_{i}}{m_{e}},~{}~{}~{}\eta_{N}=\frac{V_{e\;i}^{2}m_{p}}{M_{i}}.$
(9)

In the above, $m_{i}$ is the mass of active neutrino and $U_{ei}$ is the PMNS
mixing; whereas, $\theta_{ei}$ is the mixing among the active and sterile and
$M_{i}$ is the corresponding mass of heavy sterile. In the above, $M_{\nu}$
and $M_{N}$ are the nuclear matrix elements (NME) for exchange of light and
heavy neutrinos respectively.

Figure 6: Effective mass of NDBD versus the smallest neutrino mass. The left
(right) panel shows the case of NH (IH). The red (blue) region correspond to
3+0 (3+1) scenario, labelled as SM (Sterile). The region hatched with blue
cross lines represents $\Delta m_{41}^{2}=1.3\text{ eV}^{2}$. The vertical
light grey region is excluded by Planck data at $95\%$ C.L. Aghanim et al.
(2020), while the horizontal dark region shows the $90\%$ sensitivity from
GERDA Agostini et al. (2019).

The values of NME and phase space factor $G_{0\nu}$ can be find in Ref. Meroni
et al. (2013). The half-life of $0\nu 2\beta$ is can generally given as
Kovalenko et al. (2009)

$\frac{1}{T_{1/2}}=K_{0\nu}\left|\Theta_{ej}^{2}\frac{\mu_{j}}{\langle
p^{2}\rangle-\mu^{2}_{j}}\right|^{2},$ (10)

where $j$ represents the number of light neutrino states and the additional
heavy neutrino states. The parameters $\mu_{j}$ and $\Theta_{ej}$ represent
the masses of the neutrino states and the mixing with SM neutrinos
respectively. In the above, $K_{0\nu}=G_{0\nu}(\mathcal{M}_{N}m_{p})^{2}$ and
$\langle p^{2}\rangle\equiv-
m_{e}m_{p}\frac{\mathcal{M}_{N}}{\mathcal{M}_{\nu}}$. Over the decade, many
experiments on improving the lower limits on $T_{1/2}^{0\nu}$ for
$0\nu\beta\beta$ transition have been performed and best stringent bounds have
been acquired from germanium-76, Xenon-136 and tellurium-130 Alduino et al.
(2018) isotopes. The experiment GERDA-II, at $90\%$ C.L. has obtained the
corresponding lower limit as $T_{1/2}^{0\nu}>8.0\times 10^{25}$ year for
$Ge^{(76)}$ Agostini et al. (2018), whereas from KamLAND-Zen experiment
$Xe^{(136)}$, at $90\%$ C.L. the lower limit on the half-life is obtained as
$T_{1/2}^{0\nu}>1.07\times 10^{26}$ year Gando et al. (2016); Penedo and
Petcov (2018).

In presence of a sterile neutrino, the effective mass obtained from $3+0$
framework will be significantly changed. Using Eq. 6, in Fig. 6, we show the
effective mass for the standard $3+0$ scenario, and for $3+1$ scenario with
the variation of the lightest mass. The left and right panels represent NH and
IH, respectively. We consider a variation of $\Delta m^{2}_{41}$ (for NH), and
$\Delta m^{2}_{43}$ (for IH) in between (1-10) $\textrm{eV}^{2}$. The CP
phases $\alpha_{2}$, $\alpha_{3}$ and $\alpha_{4}$ have been varied in between
$-\pi$ to $\pi$. Other oscillation parameters have been varied in their
$3\sigma$ ranges as shown in Tab. 1 de Salas et al. (2020) as applicable for
NH and IH. Overall we have considered $10^{7}$ iterations at each value of the
lightest mass in generating Fig. 6. The red and blue regions represent the
variation of $|m_{eff}|$ for the standard $3+0$ scenario, and $3+1$ scenario
respectively. As can be seen from the figure the $|m_{eff}|$ can be
significantly large in the presence of a sterile neutrino with large mass
value, and hence constrained from the experimental constraint. To make a
connection with our oscillation analyses in the preceding sections, we shade
the regions corresponding to $\Delta m_{41}^{2}=1.3\text{ eV}^{2}$ with blue
cross lines. Consequently we note a slight shrinking of the blue regions and
it largely comes down to below the exclusion limit by Gerda. We note that for
NH, there is a complete cancellation of $m_{\textrm{eff}}$ in the 3+0 case
when the lightest mass approximately lies in the range of
$10^{-3}-10^{-2}\text{ eV}$. In 3+1 case, there is no complete cancellation in
this region (due to the dominance of the $m_{4}$-term in Eq. 6 in this range)
and it only happens when the lightest mass becomes $10^{-2}$ eV and beyond.
For IH, the standard 3+0 case shows no total cancellation (dominant
$m_{1}$-term in Eq. 6), while there is a total cancellation in the 3+1 case in
the range shown. Our results qualitatively agree with Giunti and Zavanin
(2015).

##  8 Summary and Conclusion

In this paper we have considered the presence of an eV-scale sterile neutrino
(the so called 3+1 scenario which might turn out to be a possible resolution
of the short baseline neutrino oscillation anomalies) and have analyzed how
the present and future long baseline experiments T2K, NOvA, DUNE and T2HK can
potentially probe the additional CP phases. We discuss how the three CP
phases, namely $\delta_{13},\delta_{24}$ and $\delta_{34}$ can individually
affect the oscillation channels under consideration and appear in the
probability expression. In light of the constraints on the active-sterile
mixing from the global analysis, we estimate how the LBL experiments can probe
the parameter spaces associated to the CP phases, by taking a pair of CP
phases at a time. Though $\nu_{\mu}\rightarrow\nu_{e}$ oscillation channel
contributes the most in probing these parameter spaces,
${\nu_{\mu}}\rightarrow\nu_{\mu}$ and to a lesser extent
${\nu_{\mu}}\rightarrow{\nu_{\tau}}$ channel also help in exploring the
$\delta_{24}-\delta_{34}$ parameter space in particularly. By marginalizing
over all other parameters we then show how the three individual CP phases can
be reconstructed for all possible true values in the whole range of
$[-\pi,\pi]$. We find that $\delta_{24}$ and $\delta_{34}$ cannot be
reconstructed very efficiently by DUNE and also even after adding data from
NOvA and T2K. But adding T2HK data removes much of the degeneracies and the
uncertainties in reconstruction become much less. We found that if the active-
sterile mixing angles turn out to be lying close to their current upper
limits, the enhanced sensitivities to the associated phases make the
reconstructions of $\delta_{24}$ and $\delta_{34}$ much better. In contrast
the reconstruction of the standard CP phase $\delta_{13}$ is much better even
in presence of a light sterile neutrino and this conclusion is almost
independent of the size of active-sterile mixing. We then analyze how
efficiently the experiments can probe all the parameter spaces associated to
one CP phase and one active-sterile mixing angle. It turns out that the
parameter regions connected to the angle $\theta_{14}$ can be probed
relatively better that those related to the other two mixing angles. Finally,
we briefly show how the relevant parameter spaces in $0\nu\beta\beta$ get
modified in light of the active-sterile constraints used in this analysis.

###### Acknowledgements.

N.F. is grateful for a visit to IOP, Bhubaneswar where this project was
initialized. N.F. acknowledges the support of Dr. Satyajit Jena of IISER
Mohali. M. Masud acknowledges Dr. S.K. Agarwalla of IOP, Bhubaneswar for
providing the financial support from the Indian National Science Academy
(INSA) Young Scientist Project [INSA/SP/YS/2019/269]. M. Masud is supported by
IBS under the project code IBS-R018-D1. M. Mitra acknowledges the financial
support from DST INSPIRE Faculty research grant (IFA-14-PH-99), and thanks
Indo-French Centre for the Promotion of Advanced Research for the support
(grant no: 6304-2). We are grateful to the anonymous referee for constructive
suggestions and inputs.

## Appendix A: Effect of $\theta_{23}$-uncertainty

Figure 7: The effect of taking a non-Gaussian prior uncertainty on
$\theta_{23}$ on the reconstruction of the CP phases, taken pairwise at a
time, at a C.L. of $1\sigma$ (2 D.O.F.) for DUNE (red), DUNE + T2K + NO$\nu$A
(green), and DUNE + T2K + NO$\nu$A + T2HK (blue). The black dots represent the
true values to be reconstructed. The top row shows the case of taking a prior
uncertainty of $3.5\%$ on $\sin^{2}2\theta_{23}$. The bottom row (same as that
of the bottom row of Fig. 2), depicts the case of a Gaussian prior uncertainty
of $3.5\%$ on $\theta_{23}$.

In all the results so far, we have used a Gaussian prior of $3.5\%$ on
$\theta_{23}$. Since a true value lying in the higher octant of $\theta_{23}$
has been used in the analyses, we also need to study the effect of using a
non-Gaussian prior in order to properly consider both the octants when we
marginalise over $\theta_{23}$. For this purpose, we take a prior uncertainty
of $3.5\%$ on $\sin^{2}2\theta_{23}$ in the reconstruction of the CP phases
and compare it with the case of considering a Gaussian prior in Fig. 7. We
note that, taking the prior on $\sin^{2}2\theta_{23}$ results in only a mild
spread in the contours, especially along the axes of the sterile CP phases
$\delta_{24}$ or $\delta_{34}$. But this does not change our overall results
qualitatively. For e.g., though the contours showing the degenerate solutions
at DUNE (red) gets bigger with a prior on $\sin^{2}2\theta_{23}$, addition of
data from other experiments still removes this degenerate fake solution as
before.

## Appendix B: Effect of different true choice of the standard parameters

Figure 8: Effect of considering different true $\theta_{23}$-octant while
probing the CP phases, taken pairwise at a time (same as the bottom row of
Fig. 7, but for lower octant and maximal mixing for $\theta_{23}$).

The best fit values of the standard oscillation parameters used in our
analyses (as mentioned in Tab. 1) were obtained under the 3+0 scenario. It has
been shown in literature that some of these standard parameters are prone to
significant changes in the presence of a light sterile neutrino. For e.g., the
values of $\theta_{23}$, $\Delta m_{31}^{2}$, $\delta_{13}$ can change so much
that the issues of correct octant, mass hierarchy or CP Violation become very
ambiguous (see for e.g., Agarwalla et al. (2017); Dutta et al. (2016)). The
reactor mixing angle $\theta_{13}$, on the other hand is quite robust even in
the 3+1 scenario Kopp et al. (2013). Though the marginalisation process during
$\Delta\chi^{2}$ calculation partially takes care of the uncertainties of
relevant parameters in the fit, a brief discussion regarding the effect of
different true values of the standard parameters are in order. In Fig. 3, we
had already illustrated our results for all possible true values of the CP
violating phases. Here we now show in Fig. 8, the impact of the true choice of
lower octant (top row) and maximal mixing (bottom row) of $\theta_{23}$ in
probing the CP phases. This is similar to the bottom row of Fig. 7, but for
two different possible true choices of $\theta_{23}$. We see that the contours
remain qualitatively similar when different true octants of $\theta_{23}$ are
considered. For the case of maximal mixing, a slight elongation of the
contours are observed, especially along the $\delta_{24}$ and $\delta_{34}$
directions. We have also checked that for inverted hierarchy (IH) of neutrino
masses our results remain similar. This is due to the fact that even in
presence of a sterile neutrino, the mass hierarchy sensitivity of DUNE, though
deteriorates, remains high enough ($\gtrsim 5\sigma$) Dutta et al. (2016).

## Appendix C: Understanding the role of channels to probe $\delta_{24}$ and
$\delta_{34}$

Figure 9: Top row shows the bands for event spectra at DUNE due to variations
of 3+1 CP phases for $\nu_{\mu}\rightarrow\nu_{e}$ ,
${\nu_{\mu}}\rightarrow\nu_{\mu}$ and ${\nu_{\mu}}\rightarrow{\nu_{\tau}}$
channels respectively. The band within blue (red) lines corresponds to the
variation of $\delta_{24}$ ($\delta_{34}$) in the range $[-\pi,\pi]$. The
bottom row shows the corresponding spread $\Delta$ (square of difference
between the two same coloured curves in the top row) over $N_{\text{true}}$
(number of events when $\delta_{24},\delta_{34}$ are fixed at zero). See text
for details.

Here we make an attempt to understand the relative role of the three
oscillation channels to probe the sterile CP phases $\delta_{24}$ and
$\delta_{34}$ as observed in Fig. 4. For a simpler and more intuitive
explanation we use the following Gaussian $\Delta\chi^{2}$ (the Poissonian
definition in Eq. 3 reduces to the Gaussian version for sufficiently large
number of events Zyla et al. (2020).):

$\Delta\chi^{2}\sim\underset{\text{test}}{\text{Min}}\sum_{\text{bin}}\Bigg{[}\frac{\Big{(}N_{\mu
e}^{\text{true}}-N_{\mu e}^{\text{test}}\Big{)}^{2}}{N_{\mu
e}^{\text{true}}}+\frac{\Big{(}N_{\mu\mu}^{\text{true}}-N_{\mu\mu}^{\text{test}}\Big{)}^{2}}{N_{\mu\mu}^{\text{true}}}+\frac{\Big{(}N_{\mu\tau}^{\text{true}}-N_{\mu\tau}^{\text{test}}\Big{)}^{2}}{N_{\mu\tau}^{\text{true}}}\Bigg{]}+\text{(prior
\& systematics)}.$ (C1)

Here $N_{\alpha\beta}^{\text{true}}$ ($N_{\alpha\beta}^{\text{test}}$) is the
true (test) set of events coming from the $\nu_{\alpha}\to\nu_{\beta}$
oscillation channel. Note that, we have sketched here only the relevant
statistical part of the $\Delta\chi^{2}$ and ignored the prior and
systematics. In order to understand Fig. 4, we first estimate the band of
$N_{\alpha\beta}^{\text{test}}$ corresponding to individual variations of
$\delta_{24}\in[-\pi,\pi]$ and $\delta_{34}\in[-\pi,\pi]$. These spectra are
illustrated for the three channels in Fig. 9 (top row). The $\Delta\chi^{2}$
for reconstructing $\delta_{24}$ and $\delta_{34}$ is governed by the
spread/width of such event bands and we calculate the square of the vertical
width at each energy bin and refer it by $\Delta$. In accordance with Fig. 4
we then generate $N_{\alpha\beta}^{\text{true}}$ corresponding to
$\delta_{24}=0$ and $\delta_{34}=0$. Now, following Eq. C1, we plot
$\Delta/N_{\alpha\beta}^{\text{true}}$ for the three channels in the bottom
row of Fig. 9, both for $\delta_{24}$ (blue) and $\delta_{34}$ (red). This
approximately gives us an idea of the relative contribution of the channels in
the reconstruction of the sterile CP phases. As expected, the
$\nu_{\mu}\rightarrow\nu_{e}$ channel gives the dominant contribution and
$\delta_{24}$-reconstruction is expected to be slightly better than
$\delta_{34}$. ${\nu_{\mu}}\rightarrow\nu_{\mu}$ channel also plays a role and
this mainly comes due to the large number of $\nu_{\mu}$ events. Finally
${\nu_{\mu}}\rightarrow{\nu_{\tau}}$ channel has a small but non-negligible
contribution. Both ${\nu_{\mu}}\rightarrow\nu_{\mu}$ and
${\nu_{\mu}}\rightarrow{\nu_{\tau}}$ channels provide similar capability of
reconstructing $\delta_{24}$ and $\delta_{34}$.

Figure 10: $\Delta\chi^{2}$ in probing $\delta_{24}$ (blue) or $\delta_{34}$
(red) for $\nu_{\mu}\rightarrow\nu_{e}$ (left) and
${\nu_{\mu}}\rightarrow\nu_{\mu}$ (right) channels. The true values of
$\delta_{24}$ or $\delta_{34}$ are considered to be zero while the test values
are shown along the horizontal axes.

However, since the ${\nu_{\mu}}\rightarrow\nu_{\mu}$ channel events are also
limited by systematic uncertainties, some remarks regarding the effect of
systematics in Eq. C1 are in order. Extending our analysis in this appendix
from the level of events to that of $\Delta\chi^{2}$, we show in Fig. 10, a
comparison between the $\nu_{\mu}\rightarrow\nu_{e}$ and
${\nu_{\mu}}\rightarrow\nu_{\mu}$ channels in probing $\delta_{24}$ or
$\delta_{34}$. As in Fig. 9, here we also consider the true values of
$\delta_{24}$ or $\delta_{34}$ to be 0 and then plot the $\Delta\chi^{2}$ as a
function of the test values of $\delta_{24}$ (blue) or $\delta_{34}$ (red),
after marginalising over all other relevant test parameters. The left (right)
panel shows the contribution of the $\nu_{\mu}\rightarrow\nu_{e}$
(${\nu_{\mu}}\rightarrow\nu_{\mu}$) channel alone. As mentioned in Sec. 3,
$2\%$ ($5\%$) systematics has been considered in the
$\nu_{\mu}\rightarrow\nu_{e}$ (${\nu_{\mu}}\rightarrow\nu_{\mu}$) signal,
following Alion et al. (2016) (along with various background systematics as
well). Due to this larger systematic uncertainty, we note from Fig. 10, that
the relative role of ${\nu_{\mu}}\rightarrow\nu_{\mu}$ channel in probing the
sterile phases with respect to $\nu_{\mu}\rightarrow\nu_{e}$ channel further
diminishes compared to that of Fig. 9. Nevertheless, even with such
systematics the contribution of the ${\nu_{\mu}}\rightarrow\nu_{\mu}$ channel
can still be roughly $\sim 15\%-20\%$ of that of $\nu_{\mu}\rightarrow\nu_{e}$
in the favourable region of the parameter space. Thus when all the oscillation
channels are included to do a combined $\Delta\chi^{2}$ analysis, the
interplay and complementarity among the channels improves the sensitivity to
probe the CP phases. These findings are consistent with the full statistical
analysis as discussed in Sec. 5.

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