aainstitutetext: Department of Physics, Indian Institute of Technology Bhilai,
Raipur-492015, India

# Impact of CP violation searches at MOMENT experiment with sterile neutrinos

Kiran Sharma a , Sudhanwa Patra<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We examine the scope of the MOMENT experiment in the context of CP violation
searches with the presence of extra eV scale sterile neutrino. MOMENT is a
proposed short baseline neutrino oscillation experiment using muon beams for
neutrinos production, making it advantageous over $\pi_{0}$ background and
other technical difficulties. We work over the first oscillation maxima which
matches the peak value of flux with a run time of 5 years for both neutrino
and anti-neutrino modes. We perform the bi-probability studies for both 3 and
3+1 flavor mixing schemes. The CP violation sensitivities arising from the
fundamental CP phase $\delta_{13}$ and unknown CP phase $\delta_{14}$ are
explored at the firm footing. The slight deteriorates are observed in CP
violations induced by $\delta_{13}$ as the presence of sterile neutrino is
considered. We also look at the reconstruction of CP violations phases
$\delta_{13}$ and $\delta_{14}$ and the MOMENT experiment shows significant
capabilities in the precise measurement of $\delta_{13}$ phase.

###### Keywords:

Neutrino Oscillation, Medium-baseline, Sterile Neutrino, MOMENT

##  1 Introduction and Motivation

Neutrino physics has made tremendous progress in the past few years. The
discovery of the phenomenon of neutrino oscillations Wolfenstein:1977ue ;
Bilenky:1998dt ; Giunti:2006fr ; Nunokawa:2007qh ; Schwetz:2008er ;
Pascoli:2013wca ; Blennow:2013rca ; Bellini:2013wra  has not only answered the
solar- neutrino problem Mikheyev:1985zog ; Bethe:1986ej ; Haxton:2012wfz ;
Maltoni:2015kca  and atmospheric-neutrino problem Fogli:2003th ; Gonzalez-
Garcia:2004pka ; Kajita:2016gnc ; Kajita:2016cak , but also it has open up the
new doors to look at the beyond standard model physics. At present, as per the
standard model, there are three active flavors of neutrinos associated with
the electron, muon and, tau lepton. The neutrino oscillations in the presence
of three active neutrinos are described by six oscillation parameters: two
mass square differences i.e. $\Delta m^{2}_{21}=m^{2}_{2}-m^{2}_{1}$, $\Delta
m^{2}_{31}=m^{2}_{3}-m^{2}_{1}$, three neutrino mixing angles $\theta_{12}$,
$\theta_{13}$, $\theta_{23}$ and one Dirac CP phase $\delta_{CP}$. The values
of the oscillations parameters have been constrained by various neutrino
experiments DayaBay:2015ayh ; RENO:2015ksa ; DoubleChooz:2014kuw ;
DayaBay:2013yxg ; Nunokawa:2005nx ; deGouvea:2005hk ; MINOS:2013utc ;
Fogli:1996pv , the only unknown oscillation parameters left to be known
precisely are the sign of atmosphere mass -squared difference i.e. $\Delta
m^{2}_{31}$ which will resolve the issue of neutrino mass ordering, the
precise measurement of mixing angle $\theta_{23}$ which will help in
establishing the correct octant and the value of fundamental CP phase. The
long baseline experiments T2K T2K:2019bcf  and NO$\nu$A NOvA:2021nfi  have
independently reported the value of $\delta_{CP}$ but there is a slight
tension between the two values. This discrepancy may be because of the
systematic uncertainties or it may be an indication of the new physics arising
from the presence of sterile neutrinos Abazajian:2012ys ; Palazzo:2013me ;
Gariazzo:2015rra ; Giunti:2015jba ; Giunti:2015wnd ; Giunti:2019aiy ;
Boser:2019rta ; Kopp:2013vaa ; Sharma:2022qeo  or non-standard interactions
Ohlsson:2012kf ; Farzan:2017xzy ; Falkowski:2019kfn ; Bischer:2019ttk ;
Ge:2018uhz .

Certain short baseline anomalies are coming from the reactor and Gallium
Mention:2011rk ; GALLEX:1997lja , accelerator experiments LSND LSND:2001aii
and MiniBooNE MiniBooNE:2018esg  which hints towards the existence of a
hypothetical fourth sterile mass eigenstate. The presence of such right-handed
sterile neutrinos will extend the structure of the standard 3 flavor
framework. As a result, the number of neutrino oscillation parameters gets
enhanced in the minimal extended $3+1$ framework. There are now three
additional mixing angles i.e. $\theta_{14}$, $\theta_{24}$, $\theta_{34}$ and
two more CP phases. We have the new mass square differences arising from the
mixing among fourth mass eigen state $\nu_{4}$ with mass eigenstates of three
active flavors ($\nu_{e}$, $\nu_{\mu}$ and, $\nu_{\tau}$). The presence of
sterile neutrino has been widely explored in the literature (see references
Agarwalla:2018nlx ; Choubey:2017cba ; Choubey:2017ppj ; Haba:2018klh ;
Coloma:2017ptb ; Agarwalla:2016xxa ; Berryman:2015nua ).

The effect of sterile neutrinos over the standard oscillation parameters have
been widely explored in the current and future proposed neutrino oscillation
experiments like $\rm{T2HK}$ Hyper-KamiokandeProto-:2015xww ,$\rm{T2HKK}$
Hyper-Kamiokande:2016srs , and $\rm{DUNE}$ DUNE:2015lol ; DUNE:2016hlj ;
DUNE:2016evb ; DUNE:2016rla . An important point to remark is that all the
above-stated experimental facilities are using the neutrino beams originating
from the pions decay, whereas in the proposed work, we look at the potential
of upcoming neutrino oscillation experiment $\rm{MOMENT}$(MuOn-decay MEdium
baseline NeuTrino beam facility) Cao:2014bea  which observe a neutrino beam
produced from decaying muons at relatively low energies. As an advantage one
can skip the background effects and other technical difficulties at the
experimental level.

MOMENT is a proposed short baseline neutrino oscillation experiment with a
baseline of 150 km. A detailed description of the experimental facility is
provided in section 5. The matter effects arising from the interaction of
neutrinos with the matter potential as they propagate through it are
negligible in context with the long baseline neutrino experiments. As a
result, the fake CP violation scenarios arising from matter terms can be
avoided in the MOMENT. The comparative analysis for the precise measurement of
delta CP phase has been made among MOMENT, T2K, NO$\nu$A, T2HK, T2HKK, and
DUNE where it has been discussed that MOMENT can significantly improve the
bounds over $\delta_{CP}$ Tang:2019wsv . Furthermore, the effect of non-
standard interactions and their effect over the CP phase has been looked at in
references( Blennow:2015cmn ; Bakhti:2016prn ; Tang:2017qen ; Tang:2019wsv ).
To the best of our knowledge, the effect of sterile neutrinos in the MOMENT
facility is not been studied to a much large extent. In this work, we look at
the influence of sterile neutrino over the precise measurement of the
fundamental CP phase. This work aims to study the capabilities of the MOMENT
experiment and put it into context in the global experimental effort in
neutrino physics.

This motivates us to study the physics potential of MOMENT experiment in the
presence of a hypothetical right-handed eV scale sterile neutrino. We present
a detailed discussion of the behavior of the 4-flavor transition probabilities
and understand the bird’s eye view of CP trajectory curves in the standard
3-flavor scheme and extend it to our framework of the 3+1 flavor scheme. We
perform a prospective study addressing the CP violation sensitivities and look
at the degeneracies among different CP phases.

The paper is organized as follows: In the next section, we develop the
theoretical understanding of the neutrino and anti-neutrino oscillation
probabilities in presence of a sterile neutrino. The numerical results are
performed using a bi-probability detailed study in section 2. The experimental
details and the corresponding event spectrum are explored in the next section.
We focus on the CP violation study and the reconstruction among different CP
phases in section 6.1. The paper is concluded with a summary of prospective
studies perform in our work.

##  2 Transition probability in the 4-flavor scheme

###  2.1 Theoretical framework

The formalism of $3+1$ neutrino oscillation can be understood in terms of time
dependent Schrödinger equation in the mass basis as,

$\displaystyle i\frac{\partial\nu_{j}}{\partial t}$ $\displaystyle=$
$\displaystyle H_{0}\nu_{j},$ (1)

with $j=1,2,3,4$ and $H_{0}$ is defined as the effective Hamiltonian in the
mass basis and $\nu_{j}$ being the neutrino mass eigenstate. The effective
flavor dependent Hamiltonian for $3+1$ neutrino oscillation including matter
effects is given by

$H_{4\nu}\;=\;\underbrace{U_{4\nu}\left[\begin{array}[]{cccc}0&0&0&0\\\
0&\Delta m^{2}_{21}/2E&0&0\\\ 0&0&\Delta m^{2}_{31}/2E&0\\\ 0&0&0&\Delta
m^{2}_{41}/2E\end{array}\right]U^{\dagger}_{4\nu}}_{\displaystyle=H_{\rm
vac}}+\underbrace{\left[\begin{array}[]{cccc}V_{CC}&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ 0&0&0&-V_{NC}\end{array}\right]}_{\displaystyle=H_{\rm mat}}\;,$
(2)

In case of $3+1$ scenario, i.e, for three active neutrinos and one sterile
neutrino, the mixing matrix $U_{4\nu}$ can be parameterised as

$\displaystyle U_{4\nu}$ $\displaystyle=$ $\displaystyle
R\big{(}\theta_{34},\delta_{34}\big{)}\,R\big{(}\theta_{24},0\big{)}\,R\big{(}\theta_{14},\delta_{14}\big{)}R\big{(}\theta_{23},0\big{)}R\big{(}\theta_{13},\delta_{13}\big{)}R\big{(}\theta_{12},0\big{)}$
$\displaystyle\equiv$ $\displaystyle
R\big{(}\theta_{34},\delta_{34}\big{)}\,R\big{(}\theta_{24},0\big{)}\,R\big{(}\theta_{14},\delta_{14}\big{)}U_{3\nu}$

where
$U_{3\nu}=R\big{(}\theta_{23},0\big{)}R\big{(}\theta_{13},\delta_{13}\big{)}R\big{(}\theta_{12},0\big{)}$
is the standard three flavor neutrino mixing matrix. The mass eigenstates are
related to flavor eigenstates as
$\nu_{j}=\big{[}U_{4\nu}\big{]}_{j\alpha}\nu_{\alpha}$.

The $4\times 4$ real rotation matrices $R_{ij}$ (here R($\theta_{24}$),
R($\theta_{23}$), R($\theta_{12}$)) in the $(i,j)$ plane with $2\times 2$ sub
matrices are defined as:

$R_{ij}=\begin{pmatrix}c_{ij}&s_{ij}\\\ -s_{ij}&c_{ij}\\\ \end{pmatrix}$ (4)

while the complex rotation matrices (i.e. R($\theta_{34},\delta_{34}$),
R($\theta_{14},\delta_{14}$, and R($\theta_{13},\delta_{13}$))in the $(i,j)$
plane are defined by:

$\tilde{R_{ij}}=\begin{pmatrix}c_{ij}&s_{ij}\\\
-s_{ij}^{*}e^{-i\delta_{ij}}&c_{ij}\\\ \end{pmatrix}$ (5)

with $c_{ij}=\cos{\theta_{ij}}$ and $s_{ij}=\sin{\theta_{ij}}$. The
parametrization of unitary mixing matrix allows the conversion probability to
independent of mixing angle $\theta_{34}$ and the corresponding delta phase
$\delta_{34}$.

In the S-matrix formalism, the neutrino flavor changes after propagating a
distance $L$ is defined in terms of an evolution matrix $S$ as

$\nu_{\alpha}\big{(}L\big{)}=S_{\alpha\beta}\nu_{\beta}\big{(}0\big{)}\,.$ (6)

The key point is that the evolution matrix $S$ satisfies the same Schrödinger
equation. After some simplification, the form of evolution matrix can be
expressed in term $H_{4\nu}$ as

$\displaystyle S_{\beta\alpha}$ $\displaystyle=$
$\displaystyle\left[\,\exp\\!\left(-iH_{4\nu}L\right)\right]_{\beta\alpha}\;,$
(7)

The final expression of neutrino oscillation probability from $\nu_{\alpha}$
to $\nu_{\beta}$ with neutrino energy $E$ and baseline $L$ is expressed in
terms of evolution matrix as,

$\displaystyle P(\nu_{\alpha}\rightarrow\nu_{\beta})$ $\displaystyle=$
$\displaystyle\bigl{|}\,S_{\beta\alpha}\,\bigr{|}^{2}$ (8)

###  2.2 Appearance probability $P^{4\nu}_{\mu e}$

The $3+1$ appearance probability for $\nu_{\mu}\to\nu_{e}$ transition has been
derived in ref.Klop:2014ima  and is related to three flavour neutrino
transition probability $P^{3\nu}_{\mu e}$ and other interference terms arising
from sterile neutrino mixing parameters as follows,

$\displaystyle P^{4\nu}_{\mu e}\approx$
$\displaystyle\quad\bigg{(}1-\mathrm{s}^{2}_{14}-\mathrm{s}^{2}_{24}\bigg{)}P^{3\nu}_{\mu
e}$
$\displaystyle+4\mathrm{s}_{14}\mathrm{s}_{24}\mathrm{s}_{13}\mathrm{s}_{23}\sin\Delta_{31}\sin(\Delta_{31}+\delta_{13}-\delta_{14}),$
$\displaystyle-4\mathrm{s}_{14}\mathrm{s}_{24}\mathrm{c}_{23}\mathrm{s}_{12}\mathrm{c}_{12}\sin(\Delta_{21})\sin\delta_{14},$
$\displaystyle+2\mathrm{s}^{2}_{14}\mathrm{s}^{2}_{24}\,.$ (9)

with $\Delta_{ij}=\frac{\Delta m^{2}_{ij}L}{4E}$. The expression for
$P^{3\nu}_{\mu e}$ is sum of three contributions i.e, atmospheric
$P^{\mathrm{ATM}}$, solar $P^{\mathrm{SOL}}$ and their interference term
$P^{\mathrm{INT}}_{\mathrm{I}}$. Looking at the neutrino oscillation
parameters from recent global fit values displayed in Table.1

$\begin{array}[]{ll}s_{23}\;=\;0.76\;,&c_{23}\;=\;0.65\;,\\\
s_{12}\;=\;0.56\;,&c_{12}\;=\;0.83\;,\\\
s_{13}\;=\;0.15\;,&c_{13}\;=\;0.99\;.\\\ \end{array}$ (10)

The sine of the reactor mixing angle $s_{13}$ is treated as a small parameter
in comparison to other mixing angles and is taken to be of the order of
$O(\varepsilon)\approx 0.15$ while all other sines and cosines are $O(1)$. The
other parameter which is the ratio between two mass-square differences can be
considered as $|\alpha|\;=\;\dfrac{\Delta m^{2}_{21}}{|\Delta
m^{2}_{31}|}\;\approx\;0.03\simeq\varepsilon^{2}$. Also, parameters having
sine of the sterile mixing angles $\sin\theta_{14}$ and $\sin\theta_{24}$ are
considered to be small and are of the order of $\varepsilon$. However, the
other sterile neutrino mixing angle $\sin\theta_{34}$ and the corresponding
phase $\delta_{34}$ are taken to be zero as the vacuum probability expression
is independent of these contributions. Retaining term up to third order in
$\varepsilon$ the appearance probability $P^{4\nu}_{\mu e}$ can be simplified
as sum of three contributions,

$\displaystyle P^{4\nu}_{\mu e}\simeq P^{\rm{ATM}}+P^{\rm{INT}}_{\rm
I}+P^{\rm{INT}}_{\rm II}\,.$ (11)

with individual contributions are given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!P^{\rm{ATM}}$
$\displaystyle\\!\\!\simeq\,4s_{23}^{2}s^{2}_{13}\sin^{2}{\Delta}\,,$ (12)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!P^{\rm{INT}}_{\rm
I}$
$\displaystyle\\!\\!\simeq\,8s_{13}s_{12}c_{12}s_{23}c_{23}(\alpha\Delta)\sin\Delta\cos({\Delta+\delta_{13}})\,,$
(13)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!P^{\rm{INT}}_{\rm
II}$
$\displaystyle\\!\\!\simeq\,4s_{14}s_{24}s_{13}s_{23}\sin\Delta\sin(\Delta+\delta_{13}-\delta_{14})\,,$
(14)

where $\Delta\equiv\Delta_{31}$.

The first term coming from the atmospheric oscillating parameter $\Delta_{31}$
is a positive quantity providing the leading order contribution to the
transition probability. The sub-leading contribution are arising from the
interferences of different frequencies. The term $P^{\rm{INT}}_{\rm I}$
corresponds to the interference of solar and atmospheric frequencies while the
term $P^{\rm{INT}}_{\rm II}$ is connected with the interference of atmospheric
and sterile frequencies. One can infer from the expression of
$P^{\rm{INT}}_{\rm I}$ and $P^{\rm{INT}}_{\rm II}$ that the interference
induced by the presence of sterile neutrino is not proportional to $\Delta$
while the solar-atmospheric interference term is directly related with it. As
a result, the numerical simulation carried out for the MOMENT experiment
working over the first oscillation maxima has a good performance.

Parameters | True Value | Marginalization Range
---|---|---
$\sin^{2}\theta_{12}$ | 0.318 | Not Marginalized
$\sin^{2}\theta_{13}$ | 0.022 | Not Marginalized
$\sin^{2}\theta_{23}$ | 0.574 | [0.38,0.64]
$\sin^{2}\theta_{14}$ | 0.02 | Not Marginalized
$\sin^{2}\theta_{24}$ | 0.02 | Not Marginalized
$\sin^{2}\theta_{34}$ | 0 | Not Marginalized
$\delta_{13}$ | [-180,180] | [-180,180]
$\delta_{14}$ | [-180,180] | Not Marginalized
$\delta_{34}$ | 0 | Not Marginalized
$\Delta m^{2}_{21}(eV^{2})$ | $7.50\times 10^{-5}$ | Not Marginalized
$\Delta m^{2}_{31}(eV^{2})$ | $2.55\times 10^{-3}$ | $[2.4,2.7]\times 10^{-3}$
$\Delta m^{2}_{41}(eV^{2})$ | $1$ | Not Marginalized
Table 1: The various parameters and their true value used to simulate the data
in GLoBES are mentioned in columns 1 and 2. The third column depicts the range
over which $sin^{2}\theta_{23}$ , $\delta_{13}$ , and $\Delta m^{2}_{31}$ are
varied while minimizing the $\chi^{2}$ to obtain the final results.

##  3 Discussion at probability level

The impact of matter effects on the appearance probabilities is marginal as
MOMENT is a short baseline experiment with a baseline of $150$km. Thus, the
vacuum probabilities expressions can be used effectively while neglecting the
suppressed contributions from the MSW effect. For illustration, let us
consider the transition probability in the presence of matter effects up to
leading order as Klop:2014ima ; Cervera:2000kp ; Asano:2011nj ;
Agarwalla:2013tza ,

$P^{\rm{ATM}}_{m}\simeq(1+2v)P^{\rm{ATM}}\,,$ (15)

with the correction factor,

$\,v=\frac{V}{k}\equiv\frac{2VE}{\Delta m^{2}_{31}}\,,\quad\quad\mbox{with,
}\quad V=\sqrt{2}G_{F}N_{e}\,$ (16)

For the MOMENT experiment with baseline $150$km and considering first maxima
peak energy $E\approx 0.3$GeV, the correction factor is estimated to be
$v=0.048$. The correction factor for NO$\nu$A experiment is $v\sim 0.17$ while
taking the benchmark value of peak energy $E=2$ GeV and that is the reason why
matter effects are important for long baseline studies.

Figure 1: Transition probabilities for neutrinos and anti-neutrinos after
averaging over the fast oscillations are plotted against energy(varying from
0.1 to 0.8 GeV) with matter density kept fixed at 2.7 g/cc. The oscillation
probability for 3 flavor is shown by black curve while the colored curve
represents the oscillation probability in 3+1 mixing scenario for four
different values of $\delta_{14}$ i.e. $-90^{\circ}$,$0^{\circ}$,$90^{\circ}$,
and $180^{\circ}$. The left column corresponds to the neutrino transition
probabilities for two different values of $\delta_{13}$ whereas the right one
is for anti-neutrino transition probabilities.

We look at the variation of appearance probability channel
($\nu_{\mu}\rightarrow\nu_{e}$) and
$(\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e})$ as a function of energy
for the fixed value of the fundamental CP phase. The value of $\delta_{13}$ is
kept fixed at $0^{\circ}$ and $-90^{\circ}$ while the Dirac phase
$\delta_{14}$ is allowed to vary over different values as mentioned in the
legends in fig 1. The value of the remaining oscillation parameters has been
kept fixed at the benchmark values mentioned in table 1. The simulations are
performed over the constant value of matter density, 2.7 g/cc considered using
the Preliminary Reference Earth Model(PREM) Dziewonski:1981xy . We have taken
the value of the sterile mass square difference to be $\Delta
m^{2}_{41}=1eV^{2}$, as a result, the oscillation induced by the sterile
presence is averaged out by the finite energy resolution of the detector. We
look at the averaged-out behavior of the appearance probabilities. One can
also recall that the anti-neutrino appearance probability can be obtained from
that of neutrinos by changing the sigh of the matter potential and the
fundamental and sterile CP phases i.e.

$P_{\overline{\nu}_{\alpha}\rightarrow\overline{\nu}_{\beta}}(V,\delta_{13},\delta_{14})=P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(-V,-\delta_{13},-\delta_{14})$
(17)

As evident from eqn 15, the leading order contribution to the appearance
probability will be more for neutrinos with $V>0$ than anti-neutrinos with
$V<0$. The solid black line refers to the 3 flavor appearance probability
while the colored lines correspond to the different $3+1$ flavor mixing
scenario. The blue line indicates the contribution arising from
$\delta_{14}=-90^{\circ}$, orange for $\delta_{14}=0^{\circ}$, green for
$\delta_{14}=90^{\circ}$ and magenta for $\delta_{14}=180^{\circ}$. The left
column of the figure represents the appearance probability for neutrinos while
the right column explains the behavior of anti-neutrino appearance
probability. The curves are peaked around the first oscillation maxima for
MOMENT experiment i.e $\approx 0.3~{}GeV$. One can notice that the amplitude
and shape of probability are strongly dependent on the value of sterile CP
phase $\delta_{14}$. There is a decrease in the amplitude for the anti-
neutrino oscillation probability as expected in accordance to eqn 15 and which
will be further understood for the event spectrum in section 5. Moreover, the
plot also shows the mutual swapping for curves as one makes a transition from
neutrino to anti-neutrino probability.

##  4 Biprobability analysis

The bi-probability curves as the name suggest are the parametric curves that
allow us to trace the bi-probability space spanned together by the neutrino
and anti-neutrino oscillation probabilities. The idea of bi-probability curves
in the 3 flavor scheme was first introduced in the work Minakata:2001qm . The
oscillation probability in the 3 flavor scheme is defined as:

$P=P_{0}+A(\cos\Delta\cos\delta_{13}-\sin\Delta\sin\delta_{13})$ (18)

$\overline{P}=\overline{P_{0}}+\overline{A}(\cos\Delta\cos\delta_{13}-\sin\Delta\sin\delta_{13})$
(19)

where $A=8s_{13}s_{12}c_{12}s_{23}c_{23}(\alpha\Delta_{31})\sin\Delta_{31}.$
and $P_{0}\approx(1+2v)P^{ATM}$
Thus, the bi-probability curves display the effects coming from the
fundamental CP phase($\sin\delta_{13}$ term which is CP violating,
$\cos\delta_{13}$ term which is CP conserving), and the matter effects arising
from the presence of term A, all in a single diagram. Thus, bi-probability
provides us with a bird’s eye view important for the mass hierarchy and CP
violation studies. Since the MOMENT is a short baseline facility, it is not
worth performing the analysis for mass hierarchy sensitivities. In this work,
we fix ourselves to a normal hierarchy and look at the CP trajectory diagrams
for neutrino and anti-neutrino oscillation probabilities. The fundamental CP
phase $\delta_{13}$ is varied in the range from $-\pi$ to $\pi$ and
biprobability space is spanned. As the probabilities involve the periodic
function sine and cosine, as a result, the space spanned must form a closed
trajectory as the $\delta_{13}$ is varied. We generalize the bi-probability
representation to the 3+1 flavor scheme where the value of $\delta_{14}$ is
kept fixed at different values while $\delta_{13}$ is varied over the entire
range.

Under the adiabatic approximation, the equation 18 and 19, can be re-expressed
as:

$P=l\cos\delta_{13}+m\sin\delta_{13}+n$ (20)

$\overline{P}=\overline{l}\cos\delta_{13}-\overline{m}\sin\delta_{13}+\overline{n}$
(21)

A deeper understanding of the CP-trajectory curves for 3 flavor analysis can
be obtained by eliminating the $\delta_{13}$ factor from above equations and
under the assumption $A=\overline{A}$ which holds true in vacuum, we obtain
the equation followed by neutrino and anti-neutrino probability as:

$\bigg{(}\frac{P+\overline{P}-2n}{2l}\bigg{)}^{2}+\bigg{(}\frac{P-\overline{P}}{2m}\bigg{)}^{2}=1$
(22)

This is the equation of an ellipse where the lengths of the major and the
minor axes are measures for the coefficients of $\sin\delta_{13}$ and
$\cos\delta_{13}$, respectively, in the oscillation probability. Further, the
minor or major axis is always inclined at an angle of $45^{\circ}$ as visible
in bi-probability plots.

The idea of bi-probability plots in 3 flavor scheme can be extended to $3+1$
flavor analysis where we have some additional sources for CP violation arising
by sterile neutrino presence as explored in Agarwalla:2016mrc  and the
references therein. The analytic expression for the neutrino transition
probabilities in case of $3+1$ neutrino oscillation is recast into a compact
form as,

$\displaystyle P(\nu)\equiv
P=P_{0}+A\cos\big{(}\Delta+\delta_{13}\big{)}+B\sin\big{(}\Delta-\delta_{14}+\delta_{13}\big{)}\,$
(23)

where the first term
$P_{0}=P^{\rm{ATM}}\\!\\!\simeq\,4s_{23}^{2}s^{2}_{13}\sin^{2}{\Delta}$ is
independent of phase factor while the factors which are independent of phase
contained in second and third term are,
$A=8s_{13}s_{12}c_{12}s_{23}c_{23}(\alpha\Delta)\sin\Delta$ and
$B=4s_{14}s_{24}s_{13}s_{23}\sin\Delta$. After a few simplifications using
trigonometric relations, the transition probability given in eq.(38) modifies
to

$\displaystyle P=P_{0}+A^{\prime}\cos\delta_{13}+B^{\prime}\sin\delta_{13}$
(24)

Similarly, the simplified antineutrino transition probability is given by

$\displaystyle\overline{P}=\overline{P}_{0}+\overline{A}^{\prime}\cos\delta_{13}-\overline{B}^{\prime}\sin\delta_{13}$
(25)

where the coefficients $A^{\prime}$, $B^{\prime}$,$\overline{A}^{\prime}$ and
$\overline{B}^{\prime}$ are defined as follows

$A^{\prime}=A\cos\Delta+B\sin\big{(}\Delta-\delta_{14}\big{)}$ (26)

$B^{\prime}=-A\sin\Delta+B\cos\big{(}\Delta-\delta_{14}\big{)}$ (27)

$\overline{A}^{\prime}=\overline{A}\cos\Delta+\overline{B}\sin\big{(}\Delta+\delta_{14}\big{)}$
(28)

$\overline{B}^{\prime}=-\overline{A}\sin\Delta+\overline{B}\cos\big{(}\Delta+\delta_{14}\big{)}$
(29)

The detailed derivation for obtaining the CP-trajectory is mentioned in the
appendix LABEL: obtained by eliminating the $\delta_{13}$. The angle of
inclination is obtained by comparing the trajectory equation with the general
equation of ellipse and is obtained as:

$\displaystyle\tan 2\theta=\frac{(B^{2}-A^{2})\cos 2\Delta-2AB\sin
2\Delta\cos\delta_{14}}{2AB\sin\delta_{14}}$ (30)

It is clear from the expression that the orientation of the ellipse is
strongly dependent on the value of sterile CP phase $\delta_{14}$ and the
parameter $\Delta$. The following conclusions can be marked depending on the
values of these parameters:

  1. 1.

Firstly, under the vanishing condition of factor $\sin\delta_{14}$ (i.e.
$\delta_{14}=n\pi$), the inclination becomes $\theta=\pi/4$. The orientation
of the elliptical trajectory can be determined by looking at the sign of
numerator term. If the numerator is a positive definite quantity the
orientation is counter-clockwise while it is clockwise for negative numerator.

  2. 2.

Also, if $\Delta\approx n\pi/2$, which is true for MOMENT experiment which
works in the first oscillation maxima, the inclination angle simplifies as

$\tan 2\theta\approx\frac{0.366}{\sin\delta_{14}}$ (31)

Now there are again two possible inclinations for
$\delta_{14}\rightarrow(2n-1)\pi/2$, depending upon the value of n (where
$n={1,2,3...}$). For the positive sign in denominator, the inclination of
minor axis is $\approx 10^{\circ}$ whereas for the negative sign the major
axis is inclined by $\approx-10^{\circ}$.

Figure 2: Bi-probability plots under the normal hierarchy are shown for both 3
flavor and 3+1 flavor mixing scheme. The CP trajectory diagram for the
3-flavor is drawn by black color while the orange, magenta, blue and green
closed curves corresponds to the fixed values of $\delta_{14}$ phase as
mentioned in each legend. The neutrino energy is kept fixed at its first
oscillation peak value of 0.3 GeV. The CP-phase $\delta_{13}$ is varied in the
range $[-\pi,\pi]$. Four different values of $\delta_{13}$ phase (i.e.
$0^{\circ}$, $180^{\circ}$, $-90^{\circ}$, and $90^{\circ}$) are marked by
different symbols for comparing the orientation of ellipses.

The discussion can be easily confirmed from the bi-probability plots in fig 2.
We have shown the dependency of ellipse inclination for four different values
of $\delta_{14}$. The parameter $\delta_{13}$ is varied over the complete
allowed range $[-\pi,\pi]$ while $\delta_{14}$ is kept fixed for each
particular plot as mentioned in the legend. The solid black ellipse is drawn
for the variation of neutrino and anti-neutrino probabilities in the
$3$-flavor scheme. The centers of the ellipses in the $3+1$ flavor scheme are
almost coinciding with the centers in the $3$ flavor scheme. The slight
variation is arising from the negligible matter effects. We have marked
special symbols for highlighting four different fundamental CP phase values
i.e $\delta_{13}={-180^{\circ},-90^{\circ},0^{\circ},90^{\circ}}$. For
$\delta_{14}=-180^{\circ}$ and $\delta_{14}=0^{\circ}$, the inclination is
$45^{\circ}$ as seen by the orange and magenta curves. There is a swapping
among the values of $\delta_{13}$ phase pointing towards the tracing of
elliptical trajectory as expected. While the blue and green curves plotted
under $\delta_{14}=-90^{\circ},90^{\circ}$ are inclined by $-10^{\circ}$ and
$10^{\circ}$ respectively.

##  5 Experimental Details and Event Spectra

In this section, we shed light on the specifications of the experimental setup
MOMENT. As the name suggests MOMENT is a muon-decay medium-baseline neutrino
beam facility that uses a continuous proton beam of energy, $15$ MW. Since
ideally, it is very difficult for any solid target to withstand such a high
power, so mercury jets are adopted as the target. Neutrinos are produced from
the muons in a pion decay channel. The $\mu^{+}$ decays via channel
$\mu^{+}\rightarrow\bar{\nu}_{\mu}\nu_{e}e^{+}$ while $\mu^{-}$ decays as
$\mu^{-}\rightarrow\nu_{\mu}\bar{\nu}_{e}e^{-}$. Thus, we have the four
neutrino flavors i.e. ${\nu}_{e}$, ${\nu}_{\mu}$, $\bar{\nu}_{e}$ and
$\bar{\nu}_{\mu}$. Hence, the MOMENT setup would allow the study of eight
oscillation channels (i.e. $\nu_{e}\rightarrow\nu_{e}$,
$\nu_{e}\rightarrow\nu_{\mu}$, $\nu_{\mu}\rightarrow\nu_{e}$,
$\nu_{e}\rightarrow\nu_{\mu}$ as well as their corresponding CP-conjugate
partners). To provide sensitivity towards CP-violating, the distinction
between the neutrinos and anti-neutrinos is achieved by $500$ kilo ton
fiducial Gd-doped Water Cherenkov detector as the baseline detector. The final
neutrons captured on Gd provides us a way to distinguish neutrinos from anti-
neutrinos with the known interactions of neutrinos with nucleons as
$\nu_{e}+n\rightarrow p+e^{-}$, $\nu_{\mu}+n\rightarrow p+\mu^{-}$,
$\bar{\nu}_{e}+p\rightarrow n+e^{+}$, $\bar{\nu}_{\mu}+p\rightarrow
n+\mu^{+}$. The number of the proton on target (POT) with a proton beam power
of $15$ MW and with a proton energy of $1.5$ GeV can be calculated as:

$x=\frac{W\times y\times t\times 10^{16}}{1.6\times E_{p}}$ (32)

where $W$ is beam power and $E_{p}$ is the proton energy. The operational time
in one year is represented by t and is roughly $\approx 10^{7}$ seconds, while
$y$ is number of years protons deposited on the target. For MOMENT experiment,
POT will be $6.2\times 10^{22}$. The fluxes for neutrinos and anti-neutrinos
arising from $1.5$ GeV peaks around $0.030$ GeV while the total energy is
varied in range from $0.010-0.80$ GeV. The simulations are performed by
assuming the equal run rate for neutrino and anti-neutrino mode. We used an
uncertainty $2.5\%$ on the signal and $5\%$ uncertainty over the background
for both neutrinos and anti-neutrino modes.

Characteristics | MOMENT
---|---
Beam power | 15MW
Fiducial Detector mass | 500 kton Gd doped Water Cherenkov
Baseline | 150 km
Flux peaks at | 0.3 GeV
First Oscillation maxima |  $\approx 0.3$ GeV
Table 2: The experimental specifications of MOMENT experiment.

Figure 3: The expected number of signal events are plotted against the
reconstructed neutrino energy. The black curve refers to 3 flavor case while
the colored histograms are for (3+1) scheme with different values of
$\delta_{14}$ as mentioned in the legend. The left panel corresponds to
$\nu_{e}$ appearance events while the right one is for $\overline{\nu}_{e}$
appearance events.

From the existing literature, we know that the number of events in the i-th
energy bin are given by

$N_{i}=\frac{Tn_{n}\epsilon}{4\pi
L^{2}}\int_{0}^{E_{max}}dE\int_{E^{min}_{A_{i}}}^{E^{max}_{A_{i}}}dE_{A}\phi(E)\sigma_{\nu_{e}}(E)R(E,E_{A})P_{\mu
e}(E)$ (33)

where T is the total running time, $n_{n}$ is the number of target nucleons in
the detector, $\epsilon$ is the detector efficiency, $\phi(E)$ is the neutrino
flux, $\sigma_{\nu_{e}}$ is the neutrino interaction cross-section and
R(E,$E_{A}$) is the $Gau\beta ian$ energy resolution function of detector. The
quantities E and $E_{A}$ are the true and reconstructed (anti-)neutrino
energies respectively and L is the baseline.

The numerical results have been performed using the GLoBES software
Huber:2004ka ; Huber:2007ji  and its additional toolkit required to
incorporate the new physics arising from the presence of sterile neutrino. We
plot number of events against the reconstructed energy. It is found that the
maximum flux peaks about 0.3 GeV, so we have maximum number of events in that
energy bin. In the plot, the the thick black lines depicts the 3 flavor
scenario. The other colored histograms are drawn in the 3+1 scenario for the
different values of $\delta_{14}$. The number of $\nu_{e}$ appearance events
are almost double of the number of events of $\overline{\nu}_{e}$ events since
the cross section of neutrino interaction is very much different from the
anti-neutrino interaction cross-section.

##  6 CP violation sensitivities in presence of sterile neutrino

###  6.1 Chi-square analysis

The sensitivity of an experiment is to determine the precise values of the
oscillation parameters by performing the $\chi^{2}$ square analysis. It is
performed by comparing the simulated true event rates from the present best
fit data with the events generated by the test hypothesis. Also, the
theoretical uncertainties and the systematic errors at the experimental level
are incorporated using the method of pulls in calculating $\chi^{2}$ as
Huber:2002mx ; Fogli:2002pt ,

$\displaystyle\chi^{2}(p_{\rm true},p_{\rm
test})=\min_{\xi}\big{[}\sum_{i\,\in\,\mbox{bins}}\frac{(N^{\rm
true}_{i}-N^{\rm test}_{i}(\xi))^{2}}{N^{\rm
true}_{i}}+\frac{\xi^{2}}{\sigma^{2}_{\xi}}\big{]}$ (34)

where the nuisance parameter is denoted by $\xi$ and the corresponding
systematic error is presented by $\sigma_{\xi}$. The terms involving the
nuisance parameters are called pull terms. In order to counter the effect of
systematic errors, the penalty term $\xi^{2}$ is added. The nuisance
parameters are dependent on the fiducial mass of the detector used in a
particular experiment as well as on the other experimental properties like
flux normalization and cross section. The minimisation of $\chi^{2}$ is
obtained by marginalizing the oscillation parameter space. Therefore, one add
another penalty terms called priors to $\chi^{2}$. The mathematical expression
for the prior is given by

$\displaystyle\chi^{2}_{\rm prior}=\bigg{(}\frac{N^{\rm true}-N^{\rm
test}}{\sigma^{\rm true}}\bigg{)}^{2}$ (35)

In our analysis, we looked at the sensitivity of MOMENT experiment to
determine the precise value of CP phase in the presence of sterile neutrino.
In the three flavor scenario as seen by eq. (18), the CP violations are
induced by the presence of $\sin\delta_{13}$ term. The $\chi^{2}$ analysis is
carried out with the statistical significance at which we can reject no CP
violation test hypothesis as,

$\displaystyle\chi^{2}=\frac{(N(\delta^{\rm tr}_{\rm CP})-N(\delta^{\rm
test}_{\rm CP}=0,180))^{2}}{N(\delta^{\rm tr}_{\rm CP})}$ (36)

We have fixed the mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{14}$
to their best fit values as mentioned in table 1 in both true and test
spectrum. We look at the marginalization over the parameters $\theta_{23}$ and
the mass-squared difference $\Delta m^{2}_{31}$ in two different ways as
follows

  1. 1.

First Case: $\Delta m^{2}_{31}$ is kept fixed for both 3 flavor and 3+1 flavor
scheme while $\theta_{23}$ is marginalized over the range mentioned in table
1.

  2. 2.

Second Case: Both $\Delta m^{2}_{31}$ and $\theta_{23}$ are marginalized in 3
and 3+1 flavor scheme. Also the projected information on $\Delta m^{2}_{31}$
is added in the form of priors as

$\displaystyle\chi^{2}_{\rm prior}=\bigg{(}\frac{\Delta m^{2}_{31}(\rm
true)-\Delta m^{2}_{31}(\rm test)}{\sigma(\Delta m^{2}_{31})}\bigg{)}^{2}$
(37)

where $\sigma(\Delta m^{2}_{31})$ is the $1\sigma$ error on $\Delta
m^{2}_{31}$.

The numerical results are displayed in figure 4. The left-panel of the figure
corresponds to the fixed value of $\Delta m^{2}_{31}$ and varying
$\theta_{23}$ while right-panel of the figure corresponds to the second case
of marginalizing over both the mass hierarchy $\Delta m^{2}_{31}$ and
$\theta_{23}$. The solid black solid line represents the estimated value of
$\Delta\chi^{2}$ for three flavor scenario while all other color lines depict
the analysis for 3+1 flavor scenario. While performing the $\chi^{2}$ analysis
for discovery potential of CP phases we have considered equal neutrino and
anti-neutrino mode for a run time of (5+5) years. In each case we have
considered four different values of true $\delta_{14}$ phase as considered in
probability analysis while its test value is marginalized. The figure clearly
depicts that CP sensitivities decreases in the presence of sterile neutrino,
primarily because of the degeneracies between the fundamental and active-
sterile CP phase.

Figure 4: The figure indicated the potential of MOMENT experiment for the
discovery of CP violation induced by the fundamental phase $\delta_{13}$. The
black curve shows the behavior under 3 flavor scheme while the colored curves
in each panel are of different values of $\delta_{14}$. The left figure
$\Delta m^{2}_{31}$ is kept fixed for both 3 and 3+1 flavor mixing while we
marginalize over $\theta_{23}$ value whereas in the right figure, we
marginalised over both hierarchy $\Delta m^{2}_{31}$ and $\theta_{23}$.

###  6.2 Reconstructing the CP phases

In the last subsection, we looked at the CP-violation discovery potential of
the MOMENT experiment by performing $\chi^{2}$ analysis. But there is an
another way to extract the complementary information by reconstructing the
values of two CP phases $\delta_{13}$ and $\delta{14}$, independent of the
amount of CP violation. We look at the contour plots by reconstructing the CP
phases in $\delta_{13}-\delta_{14}$ plane. The test values of both the CP
phases are varied from $(-\pi,\pi)$ and contours are shown at one, two, and
three sigma level, simultaneously. The four plots represents the regions
reconstructed for the four benchmark values considered in figure 5. The solid
black dot in the upper row represents CP-conserving scenarios with values
(0,0), $(\pi,\pi)$ while the bottom panel has CP-violating picture with
$(-\pi/2,-\pi/2)$, $(\pi/2,\pi/2)$. Our results are predicts remarkable
sensitivity for determining the precise value of the CP phase $\delta_{13}$ in
the presence of sterile neutrino.

Figure 5:  Contour plots in the plane of $\delta_{13}$ (test) and
$\delta_{14}$ (test) for four different values of CP phases $\delta_{13}$ and
$\delta_{14}$. The discovery potential of $\delta_{13}$ in the contour plot of
$\delta_{13}$ vs $\delta_{14}$ plane has been presented by blue, red, and
green band with $1\sigma$, $2\sigma$, and $3\sigma$ confidence level
respectively. The left(right) panel of first row corresponds to (0,0) and
($\pi,\pi$) respectively while the left(right) panel of second row corresponds
to ($-\pi/2,-\pi/2$) and ($\pi/2,\pi/2$)

—————————–

##  7 Conclusions and Outlook

In the work, we have addressed the potential of MOMENT experiment to emphasis
the role of sterile neutrino presence over the standard three flavor
oscillation parameters. We have shown the transition probabilities for both
neutrinos and anti-neutrinos in 3+1 flavor scheme and looked at the space
spanned by the CP trajectory curves. The performance of MOMENT in
understanding the CP violation sensitivities induced by the fundamental CP
phase $\delta_{13}$ and new CP phase arising from active-sterile mixing has
been explored. We found that loss of CP sensitivity is dependent on the
different values of $\delta_{14}$ phase. The discovery potential of CP
violation in 3 flavor scheme is quite significiant at $7\,\sigma$ level though
it gets reduced in the presence of unknown CP phase $\delta_{14}$. The
reduction might has arose from the degeneracies among the two CP phases. We
have also assessed the capability of MOMENT experiment in reconstructing the
true values of such CP-phases.

#### Acknowledgments

Kiran Sharma would like to thank Ministry of Education for the financial
support for carrying out this research work. KS is very thankful to Dr. Sabya
Sachi Chatterjee for the fruitful discussion carried time to time for the
betterment of this work.

##  8 APPENDIX

##  Appendix A Detailed Description of Biprobability analysis.

The analytic expression for the neutrino transition probabilities in case of
$3+1$ neutrino oscillation are recasted into a compact form as,

$\displaystyle P(\nu)\equiv
P=P_{0}+A\cos\big{(}\Delta+\delta_{13}\big{)}+B\sin\big{(}\Delta-\delta_{14}+\delta_{13}\big{)}\,$
(38)

where the first term
$P_{0}=P^{\rm{ATM}}\\!\\!\simeq\,4s_{23}^{2}s^{2}_{13}\sin^{2}{\Delta}$ is
independent of phase factor while the factors which are independent of phase
contained in second and third term are,
$A=8s_{13}s_{12}c_{12}s_{23}c_{23}(\alpha\Delta)\sin\Delta$ and
$B=4s_{14}s_{24}s_{13}s_{23}\sin\Delta$. After few simplification using
trigonometric relations, the transition probability given in eq.(38) modifies
to

$\displaystyle P=P_{0}+A^{\prime}\cos\delta_{13}+B^{\prime}\sin\delta_{13}$
(39)

Similarly, the simplified antineutrino transition probability is given by

$\displaystyle\overline{P}=\overline{P}_{0}+\overline{A}^{\prime}\cos\delta_{13}-\overline{B}^{\prime}\sin\delta_{13}$
(40)

where the coefficients $A^{\prime}$, $B^{\prime}$,$\overline{A}^{\prime}$ and
$\overline{B}^{\prime}$ are defined as follows

$A^{\prime}=A\cos\Delta+B\sin\big{(}\Delta-\delta_{14}\big{)}$ (41)

$B^{\prime}=-A\sin\Delta+B\cos\big{(}\Delta-\delta_{14}\big{)}$ (42)

$\overline{A}^{\prime}=\overline{A}\cos\Delta+\overline{B}\sin\big{(}\Delta+\delta_{14}\big{)}$
(43)

$\overline{B}^{\prime}=-\overline{A}\sin\Delta+\overline{B}\cos\big{(}\Delta+\delta_{14}\big{)}$
(44)

The factors $\overline{A}^{\prime}$ and $\overline{B}^{\prime}$ are obtained
from known relations of $A^{\prime}$ and $B^{\prime}$ by replacing
$\delta_{14}\to-\delta_{14}$. Eliminating $\delta_{13}$ from the modified
neutrino and antineutrino transition probabilities, we obtain,

$\displaystyle\frac{1}{\bigg{[}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{]}^{2}}\bigg{(}\frac{P-P_{0}}{B^{\prime}}+\frac{\overline{P}-\overline{P}_{0}}{\overline{B}^{\prime}}\bigg{)}^{2}+\frac{1}{\bigg{[}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{]}^{2}}\bigg{(}\frac{P-P_{0}}{A^{\prime}}-\frac{\overline{P}-\overline{P}_{0}}{\overline{A}^{\prime}}\bigg{)}^{2}=1$
(45)

$\displaystyle\bigg{[}\frac{1}{{B^{\prime}}^{2}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{1}{{A^{\prime}}^{2}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}\bigg{]}P^{2}+\bigg{[}\frac{1}{{\overline{B}^{\prime}}^{2}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{1}{{\overline{A}^{\prime}}^{2}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}\bigg{]}\overline{P}^{2}+\bigg{[}\frac{2}{{B^{\prime}}{\overline{B}^{\prime}}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}$

$\displaystyle-\frac{2}{{A^{\prime}}{\overline{A}^{\prime}}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}\bigg{]}P\overline{P}^{\prime}+\bigg{[}\frac{-2P_{0}}{{B^{\prime}}^{2}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{-2P_{0}}{{A^{\prime}}^{2}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}+\frac{-2\overline{P}_{0}}{{B^{\prime}}{\overline{B}}^{\prime}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{-2\overline{P}_{0}}{{A^{\prime}}{\overline{A}^{\prime}}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}\bigg{]}P$

$\displaystyle+\bigg{[}\frac{-2\overline{P}_{0}}{{\overline{B}^{\prime}}^{2}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{-2\overline{P}_{0}}{{\overline{A}^{\prime}}^{2}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}+\frac{-2P_{0}}{{B^{\prime}}{\overline{B}}^{\prime}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{-2P_{0}}{{A^{\prime}}{\overline{A}^{\prime}}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}\bigg{]}\overline{P}+\bigg{[}\bigg{(}\frac{1}{{B^{\prime}}^{2}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}$

$\displaystyle++\frac{1}{{A^{\prime}}^{2}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}\bigg{)}P_{0}^{2}+\bigg{(}\frac{1}{{\overline{B}^{\prime}}^{2}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{1}{{\overline{A}^{\prime}}^{2}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}\bigg{)}\overline{P_{0}}^{2}+\bigg{(}\frac{2}{{B^{\prime}}{\overline{B}^{\prime}}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}$

$\displaystyle-\frac{2}{{A^{\prime}}{\overline{A}^{\prime}}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}\bigg{)}P_{0}\overline{P_{0}}-1\bigg{]}$
$\displaystyle=$ $\displaystyle 0$ (46)

Comparing eqn 46 with the general quadratic curve representing the equation of
ellipse

$ax^{2}+bxy+cy^{2}+cx+ey+f=0$ (47)

where the counterclockwise angle of rotation from the x-axis to the major axis
of the ellipse is defined by $\tan 2\theta=\frac{b}{a-c}$, we can understand
the inclination of ellipse in the biprobability plots.

$a=\frac{1}{{B^{\prime}}^{2}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{1}{{A^{\prime}}^{2}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}$
(48)

$b=\frac{2}{{B^{\prime}}{\overline{B}^{\prime}}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}-\frac{2}{{A^{\prime}}{\overline{A}^{\prime}}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}$
(49)

$c=\frac{1}{{\overline{B}^{\prime}}^{2}\bigg{(}\frac{A^{\prime}}{B^{\prime}}+\frac{\overline{A}^{\prime}}{\overline{B}^{\prime}}\bigg{)}^{2}}+\frac{1}{{\overline{A}^{\prime}}^{2}\bigg{(}\frac{B^{\prime}}{A^{\prime}}+\frac{\overline{B}^{\prime}}{\overline{A}^{\prime}}\bigg{)}^{2}}$
(50)

Simplifying the results under the assumption that matter effects induce
negligible perturbations on the interference terms(i.e. $A=\overline{A}$,
$B=\overline{B}$) and using equations 41, 42, 43 and 44, the angle of
inclination becomes

$\displaystyle\tan 2\theta=\frac{(B^{2}-A^{2})\cos 2\Delta-2AB\sin
2\Delta\cos\delta_{14}}{2AB\sin\delta_{14}}$ (51)

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