#  Zoom in muon survival probability with sterile neutrino for CP and
T-violation

Kiran Sharma<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Technology Bhilai, India  Sudhanwa Patra<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Bhilai, India

###### Abstract

We present the approximated analytic expressions for the muon survival
probability in a $3+1$ mixing scenario in the presence of matter effect using
the S-matrix formalism. We find that all the individual terms contributing to
the muon survival probability can significantly reduce to just three
contributions. The leading order contribution comes from the three flavor muon
survival probability followed by the two sub-leading contributions arising
from active-sterile mixing. Furthermore, to more simplify the results we adopt
the well known series expansion relations about mass-hierarchy parameter
$\alpha=\Delta m^{2}_{21}/\Delta m^{2}_{31}$ and the mixing angle
$\sin\theta_{13}$ in the vanishing limit of $\alpha^{2}$. We discuss the
relevance of muon survival probability to probe the CP and T-violation studies
coming from the new physics. We also compare the analytic relation between
vacuum and matter contributions to the muon survival probability at the
leading order. Finally, we comment on the probability behavior at the various
long baselines relevant to understand the atmospheric-neutrino sector and to
resolve the existing mass-hierarchy problem.

##  I Introduction

The famous solar-neutrino puzzle Bethe (1986); Ahmad  _et al._ (2001) and the
atmosphere neutrino problem Fukuda  _et al._ (1998) leads to the establishment
of the phenomenon of neutrino oscillations Bilenky  _et al._ (1999); Giunti
(2004). These oscillations occur among the three active neutrino flavor states
$\nu_{e}$, $\nu_{\mu}$ and $\nu_{\tau}$. The oscillation parameters in the $3$
flavor mixing scenario include the solar mass-square difference $\Delta
m^{2}_{21}$, the atmospheric mass-square difference $\Delta m^{2}_{31}$, three
mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{23}$ and one leptonic
CP-phase $\delta_{13}$. The neutrino oscillations are feasible only if at
least one of three neutrino mass eigen state is non-degenerate and the mixing
angles are non-zero. Thus neutrinos have a non-zero mass contrary to the
standard model prediction. With the discovery of neutrino oscillations,
revolutionary research begins looking for the beyond standard model physics.
In the current precision era, the main goals of current and proposed
experiments are to solve the unknowns in neutrino oscillations which include
the mass hierarchy problem, the octant issue, and determination of the CP-
violation phase. Moreover, with time certain anomalies Aguilar-Arevalo  _et
al._ (2001, 2018); Mention  _et al._ (2011); Hampel  _et al._ (1998) have
emerged which hints toward the existence of a fourth sterile neutrino state.
Including the presence of such sterile neutrino along with the three active
neutrinos increases the number of oscillation parameters, we have the sterile-
active neutrino mixing angles ($\theta_{14}$ ,$\theta_{24}$, $\theta_{34}$),
the new mass-squared differences ($\Delta m^{2}_{41}$, $\Delta m^{2}_{42}$,
$\Delta m^{2}_{43}$) and the new CP-violating phases ($\delta_{14}$,
$\delta_{24}$ and $\delta_{34}$). The presence of sterile neutrino may answer
the smallness of neutrino mass.

The probability level analysis in the presence of sterile neutrinos has
already been carried out in the literature Giunti and Li (2009); Abazajian
_et al._ (2012); Palazzo (2013); Kopp  _et al._ (2013); Gariazzo  _et al._
(2016); Giunti (2016, 2015); Choubey  _et al._ (2017); Agarwalla  _et al._
(2018); Giunti and Lasserre (2019); Böser  _et al._ (2020); Abe  _et al._
(2019). In the reference Klop and Palazzo (2015), authors have given
transition probability $P^{4\nu}_{\mu e}$ in the $3+1$ scheme by carrying out
the S-matrix analysis. In our recent work Sharma and Patra (2022), we have
also given a slightly different formalism which simplifies the transition
probability $P^{4\nu}_{\mu e}$ and survival probability $P^{4\nu}_{ee}$ using
One-scale mass dominance(OMSD) Choubey and Roy (2004). In the present work, we
provide the analytic expression for muon survival probability in the presence
of matter potential Mikheyev and Smirnov (1985); Wolfenstein (1978); Smirnov
(2005). Using the S-matrix formalism, we decompose the $4\times 4$ Hamiltonian
to the effective $3\times 3$ Hamiltonian. We perform the series expansion
Akhmedov  _et al._ (2004) about small parameters
$s_{13}\approx\mathcal{O}(\epsilon)$ and
$\alpha\approx\mathcal{O}(\epsilon^{2})$ such that the final probability
expression for muon appearance probability is of order
$\mathcal{O}(\epsilon^{3})$. We discuss the term-by-term analysis of leading
and sub-leading contributions to the final muon survival probability. The
analytic calculations involved in $P^{4\nu}_{\mu\mu}$ greatly simplify to just
$3$ significant contributions. We emphasis on the advantage of muon survival
probability for the study of CP violation Arafune  _et al._ (1997); Bilenky
_et al._ (1998); Dighe and Ray (2008); Gandhi  _et al._ (2015); Petcov and
Zhou (2018) and T-violation Cabibbo (1978); Kuo and Pantaleone (1987); Parke
and Weiler (2001); Akhmedov  _et al._ (2001); Xing (2013); Rout  _et al._
(2017); Petcov and Zhou (2018); Schwetz and Segarra (2022a, b) in context to
$T2K$ experiment Itow  _et al._ (2001); Abe  _et al._ (2011, 2021) as a case
study. We also look at the probability level analysis for the long baseline,
which can shed light on the mass-hierarchy sensitivity.

The paper is organized as follows: In the next section we give the analytic
expression for muon survival probability in vacuum followed by its treatment
in the presence of matter in section $\rm III$. In section $\rm IV$, we
provide a formalism which gives the terms contributing to total muon survival
probability. In the next leading section, we carry out the well known
$\alpha-s13$ approximation and perform the numerical analysis for different
baselines. The conclusion is marked in section $\rm VI$, followed by
appendices.

##  II Muon survival probability expression in Vaccum

In $3+1$ framework, we have three flavor states for active neutrinos
$\nu_{e},\nu_{\mu},\nu_{\tau}$ and additional sterile neutrino flavor state
$\nu_{s}$. Also there exists three mass eigenstates $\nu_{1},\nu_{2},\nu_{3}$
corresponding to the light neutrinos with masses $m_{1},m_{2},m_{3}$ plus one
more extra mass eigenstate $\nu_{4}$ with mass $m_{4}$. The time evolution of
these states in $3+1$ framework is defined as,

$i\frac{\partial}{dt}\begin{pmatrix}|\nu_{e}\rangle\\\ |\nu_{\mu}\rangle\\\
|\nu_{\tau}\rangle\\\
|\nu_{s}\rangle\end{pmatrix}=U\left[\begin{pmatrix}0&0&0&0\\\ 0&\frac{\Delta
m^{2}_{21}}{2E}&0&0\\\ 0&0&\frac{\Delta m^{2}_{31}}{2E}&0\\\
0&0&0&\frac{\Delta
m^{2}_{41}}{2E}\end{pmatrix}\right]U^{\dagger}\,\begin{pmatrix}|\nu_{e}\rangle\\\
|\nu_{\mu}\rangle\\\ |\nu_{\tau}\rangle\\\ |\nu_{s}\rangle\end{pmatrix}$

where $\Delta m^{2}_{21}$ is the difference between square of masses between
different neutrino mass eigenstates and $E$ is the typical energy of the
neutrinos of interest. The mixing matrix for $3+1$ framework is parametrized
in terms mixing angles $\theta_{i4}$ (with i=1,2,3),
$\theta_{12},\theta_{13}$, $\theta_{23}$ and CP-phases. It is important to
know the correct parametrization of the neutrino mixing matrix for $3+1$
framework. Although the choice of parametrization can differ depending upon
the placement of the CP-phase but the physics will remain same independent of
choice of parametrization. The standard parametrization considered for the
present work is given by

$\displaystyle U$ $\displaystyle=$ $\displaystyle
R\big{(}\theta_{34},\delta_{34}\big{)}\,R\big{(}\theta_{24},0\big{)}\,R\big{(}\theta_{14},\delta_{14}\big{)}$
(1) $\displaystyle\hskip 28.45274pt\times
R\big{(}\theta_{23},0\big{)}R\big{(}\theta_{13},\delta_{13}\big{)}R\big{(}\theta_{12},0\big{)}$

Here, $\theta_{14}$, $\theta_{24}$ and $\theta_{34}$ are the mixing angles
between sterile and active neutrinos while $\theta_{23}$, $\theta_{13}$ and
$\theta_{12}$ are known mixing angles of three flavor neutrinos. On top of
standard CP-phase $\delta_{13}$ we have two more CP-phases $\delta_{14}$ and
$\delta_{34}$. For completeness,the element-wise description of the unitary
mixing matrix is mentioned in the appendix A

The muon survival probability in vaccum using the general expression for the
oscillation probability Giunti and Kim (2007) for $N$ flavor neutrinos is
found to be,

$\displaystyle P_{\mu\mu}$ $\displaystyle=1-4\sum_{i<j}\text{Re}\big{(}U_{\mu
i}U_{\mu j}U_{\mu j}^{*}U_{\mu
i}^{*}\big{)}\sin^{2}\bigg{(}\Delta_{ij}L/4E\bigg{)}$
$\displaystyle+2\sum_{i<j}\text{Im}\big{(}U_{\mu i}U_{\mu j}U_{\mu
j}^{*}U_{\mu i}^{*}\big{)}\sin\bigg{(}2\Delta_{ij}L/4E\bigg{)},$ (2)

where $\Delta_{ij}=m_{i}^{2}-m_{j}^{2}$ is the mass-square difference between
any of the two mass eigenstates of the neutrinos. For $3+1$ scenario and using
the unitarity condition, we get

$\text{Im}\big{(}U_{\mu i}U_{\mu j}U_{\mu j}^{*}U_{\mu
i}^{*}\big{)}=\text{Im}\big{(}\big{|}U_{\mu i}\big{|}^{2}\big{|}U_{\mu
j}\big{|}^{2}\big{)}=0\,.$

Also, with the presence of eV scale sterile neutrino, the mass-square
difference terms are related as

$\Delta_{41}\simeq\Delta_{42}\simeq\Delta_{43}\,,\,\mbox{and}\,\Delta_{32}\simeq\Delta_{31}\,.$

It is to be noted that the terms involving solar mass square difference
$\Delta m^{2}_{21}=\Delta_{21}$ contributes negligible to the frequency part
of the survival probability in vaccum and hence, can be dropped from the
probability expression. With these simplifications, the muon survival
probability in $3+1$ scenario in vaccum reduces to

$\displaystyle P^{(3+1)}_{\mu\mu}$ $\displaystyle=1-4\,|U_{\mu
3}|^{2}\,\big{(}1+|U_{\mu 3}|^{2}-|U_{\mu
4}|^{2}\big{)}\,\sin^{2}\bigg{(}\frac{\Delta_{31}\,L}{4\,E}\bigg{)}$
$\displaystyle-4\,|U_{\mu 4}|^{2}\,\big{(}1-|U_{\mu
4}|^{2}\big{)}\,\sin^{2}\bigg{(}\frac{\Delta_{41}\,L}{4\,E}\bigg{)}$ (3)

By looking at the muon survival probability expression, it is interesting to
note that the long baseline experiments like $T2K$, $NOvA$ etc, can shed light
on sterile neutrino parameters like $\theta_{24}$ and $\Delta m^{2}_{41}$.
Thus, the study of survival probability is equally as crucial as transition
probability. However, the derived results are valid for vacuum but can not
hold true if we include matter effects. So we focus, in the present work, on
the analytic derivation of muon survival probability and its phenomenological
aspect within the $3+1$ neutrino oscillation scenario in the presence of
matter effects.

##  III General Treatment of $3+1$ neutrino oscillation in Matter

The neutrino oscillation probabilities can be modified significantly in
presence of matter due to interactions between neutrinos and the medium
particles. For example, the electron neutrino $\nu_{e}$ and muon neutrino
$\nu_{\mu}$ can scatter with the medium particles due to presence of electrons
$e^{-}$, protons $p$ and neutrons $n$ via charge and neutral current effects.
Thus,we must include matter effects on the oscillation probabilities for
neutrinos propagating via dense medium such as Earth. The effect is known as
Mikheyev-Smirnov-Wolfenstein (MSW) effect Mikheyev and Smirnov (1985);
Wolfenstein (1978); Smirnov (2005) and resonance enhancement of oscillations
can be important for long baseline experiments like T2K, NOvA, and DUNE as
well as for atmospheric neutrino detectors like Super-K, INO, and IceCube. The
present discussion will carry forward basic MSW effect in presence of extra
sterile neutrino while adopting various simplifications considered in refs
Akhmedov  _et al._ (2004); Klop and Palazzo (2015); Asano and Minakata (2011);
Sharma and Patra (2022); Choubey  _et al._ (2017); Agarwalla  _et al._ (2018).
The neutrino oscillation in presence of matter due to the coherent forward
scattering of neutrinos with the medium particles is characterized by
potential term $V_{\rm CC}$ as

$V_{\rm CC}=\sqrt{2}\,G_{F}\,N_{e}\,.$

where $G_{F}$ denotes the Fermi coupling constant, and $N_{e}$ is the electron
number density inside the matter. The general Hamiltonian for the $3+1$
scenario in the presence of matter is of the following form,

$\displaystyle H_{4\nu}\;$ $\displaystyle=$
$\displaystyle\;\underbrace{U\left[\begin{array}[]{cccc}0&0&0&0\\\ 0&\Delta
m^{2}_{21}/2E&0&0\\\ 0&0&\Delta m^{2}_{31}/2E&0\\\ 0&0&0&\Delta
m^{2}_{41}/2E\end{array}\right]U^{\dagger}}_{\displaystyle=H_{\rm vac}}$ (8)
$\displaystyle+$
$\displaystyle\underbrace{\left[\begin{array}[]{cccc}V_{CC}&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&-V_{NC}\end{array}\right]}_{\displaystyle=H_{\rm
mat}}\;,$ (13) $\displaystyle=$ $\displaystyle UKU^{\dagger}+V\,.$ (14)

where U is the unitary mixing matrix involving three active neutrinos and a
sterile neutrino with the form

$\displaystyle U$ $\displaystyle=$ $\displaystyle
R\big{(}\theta_{34},\delta_{34}\big{)}\,R\big{(}\theta_{24},0\big{)}\,R\big{(}\theta_{14},\delta_{14}\big{)}$
(15) $\displaystyle\hskip
56.9055ptR\big{(}\theta_{23},0\big{)}R\big{(}\theta_{13},\delta_{13}\big{)}R\big{(}\theta_{12},0\big{)}$
$\displaystyle\equiv$ $\displaystyle
R\big{(}\theta_{34},\delta_{34}\big{)}\,R\big{(}\theta_{24},0\big{)}\,R\big{(}\theta_{14},\delta_{14}\big{)}U_{3\nu}$

It is to be noted that $W$ boson mediated CC interaction potential $V_{CC}$
has played a crucial role in oscillation probability. In contrary, $Z$ boson
mediated NC potential $V_{\rm NC}$ is same for all flavors and hence, is
phased out from the probabilities. Since the sterile neutrino has no
interaction with the medium particles, the $V_{\rm NC}$ can not be phased out
and will play an important role when we consider the $3+1$ scenario of
neutrino oscillation. The diagonal matrix
$K=\mbox{diag}\left(0,k_{21},k_{31},k_{41}\right)$ with $k_{i1}=\Delta
m^{2}_{i1}/2E$ is for vacuum part of the Hamiltonian in mass basis in a
reduced form.

Let us introduce a change of neutrino flavor basis in $3+1$ framework as

$\overline{\nu}=\overline{U}^{\dagger}\nu$

where the modified mixing matrix in the new flavor basis is read as

  * •

The form of $\overline{U}$ is given by

$\overline{U}=R\big{(}\theta_{34},\delta_{34}\big{)}\,R\big{(}\theta_{24},0\big{)}\,R\big{(}\theta_{14},\delta_{14}\big{)}$

  * •

The $4\times 4$ Hamiltonian can be decomposed as $3\times 3$ effective
Hamiltonian Klop and Palazzo (2015)

$\displaystyle\overline{H}_{4\nu}$ $\displaystyle=$
$\displaystyle\overline{H}^{\rm kin}+\overline{H}^{\rm
dyn}=U_{3\nu}KU^{\dagger}_{3\nu}+\overline{U}^{\dagger}V\overline{U}$

with
$U_{3\nu}=R\big{(}\theta_{23},0\big{)}R\big{(}\theta_{13},\delta_{13}\big{)}R\big{(}\theta_{12},0\big{)}$.

It is to be noted that the first term describes the kinetic contribution while
relevant for neutrino oscillation in vacuum while the second term is the
dynamic contribution accommodating matter effects. The fact that the $k_{41}$
is very large as compared to other mass square difference terms, the (4,4)
entries of $\overline{H}$ is much larger than all other elements and hence,
can be decoupled from the $3+1$ framework leading to an effective three flavor
analysis. So the fourth eigenstate evolves independently of the other flavor
states. Thus, the $4\times 4$ complete Hamiltonian in $3+1$ framework can be
reduced to an effective projected Hamiltonian in three flavor scenario.

  * •

The dynamical contributions after some simplifications is described as,

$\displaystyle\overline{H}^{\rm dyn}$ $\displaystyle=$
$\displaystyle\overline{U}^{\dagger}V\overline{U}$ (16) $\displaystyle\simeq$
$\displaystyle
V_{C}\begin{pmatrix}1-(1-r)s^{2}_{14}&r\tilde{s}_{14}s_{24}&r\tilde{s}_{14}\tilde{s}^{*}_{34}\\\
\dagger&rs^{2}_{24}&rs_{24}\tilde{s}^{*}_{34}\\\
\dagger&\dagger&rs^{2}_{34}\end{pmatrix}$

The factor $r$ is the ratio of neutral current and charged current matter
potentials with a negative sign. For the Earth matter, it has a typical value
of $0.5$

  * •

The S-matrix evolution operator in terms projected Hamiltonian in three flavor
is as follows

$\displaystyle\overline{S}\;$ $\displaystyle=$
$\displaystyle\;\,\boldsymbol{e}^{-i\,\overline{H}\,L}\approx\left[\begin{array}[]{cc}\overline{S}_{3\nu}&{\bf
0}_{3\times 1}\\\ {\bf 0}_{1\times
3}&\exp\\!\left(-i\,k_{41}\,L\right)\end{array}\right]$ (19)

  * •

One can revert back to the original flavor basis by unitary transformations
$S=\overline{U}\,\overline{S}\,\overline{U}^{\dagger}$.

  * •

The probability expression can be readily obtain from the evolution matrix as

$P^{4\nu}_{\alpha\beta}\equiv
P^{4\nu}_{\alpha\beta}\big{(}\nu_{\alpha}\to\nu_{\beta};L\big{)}=\big{|}S_{\beta\alpha}\big{|}^{2}$
(20)

##  IV Muon survival probability in $3+1$ scenario

Using the general treatment mentioned in section III, we evaluate the muon
survival probability, with
$\overline{U}_{e2}=\overline{U}_{e3}=\overline{U}_{\mu 3}=0$ (exactly as per
parametrized), the relevant component $S_{\mu\mu}$ is given by

$\displaystyle S_{\mu\mu}$ $\displaystyle=$
$\displaystyle\bigg{[}\big{|}\overline{U}_{\mu
1}\big{|}^{2}\overline{S}_{ee}+\overline{U}^{*}_{\mu 1}\overline{S}_{\mu
e}\overline{U}_{\mu 2}+\overline{U}_{\mu
1}\overline{S}_{e\mu}\overline{U}^{*}_{\mu 2}+\big{|}\overline{U}_{\mu
2}\big{|}^{2}\overline{S}_{\mu\mu}\bigg{]}$ (21)
$\displaystyle+\overline{U}_{\mu 4}\overline{U}^{*}_{\mu 4}\overline{S}_{ss}$
$\displaystyle=$ $\displaystyle\mathcal{A}+\mathcal{B}$

and the complex conjugate part $S^{*}_{\mu\mu}$ is described as

$\displaystyle S^{*}_{\mu\mu}$ $\displaystyle=$
$\displaystyle\bigg{[}\big{|}\overline{U}_{\mu
1}\big{|}^{2}\overline{S}_{ee}+\overline{U}^{*}_{\mu 1}\overline{S}_{\mu
e}\overline{U}_{\mu 2}+\overline{U}_{\mu
1}\overline{S}_{e\mu}\overline{U}^{*}_{\mu 2}+\big{|}\overline{U}_{\mu
2}\big{|}^{2}\overline{S}_{\mu\mu}\bigg{]}^{*}$ (22)
$\displaystyle+\overline{U}^{*}_{\mu 4}\overline{U}_{\mu
4}\overline{S}^{*}_{ss}$ $\displaystyle=$
$\displaystyle\mathcal{A}^{*}+\mathcal{B}^{*}$

The disappearance probability is expressed in terms of evolution matrix and
other mixing matrices as follows

$\displaystyle P^{4\nu}_{\mu\mu}$ $\displaystyle=$ $\displaystyle
S_{\mu\mu}\cdot S^{*}_{\mu\mu}$ (23) $\displaystyle=$
$\displaystyle\bigg{(}\mathcal{A}+\mathcal{B}\bigg{)}\bigg{(}\mathcal{A}^{*}+\mathcal{B}^{*}\bigg{)}$
$\displaystyle=$
$\displaystyle\mathcal{A}\mathcal{A}^{*}+\mathcal{A}\mathcal{B}^{*}+\mathcal{B}\mathcal{A}^{*}+\mathcal{B}\mathcal{B}^{*}$

Averaging out terms containing
$\big{|}\overline{S}_{ss}=e^{-i\,k_{41}L}\big{|}^{2}$ gives a factor of 1/2
while terms containing only $\overline{S}_{ss}$ can be averaged out completely
from the general expression giving vanishing effects. The terms containing
only $\mathcal{B}$ or $\mathcal{B}$ (carries the sterile factor
$\overline{S}_{ss}$) are vanishing. The other factor
$\mathcal{B}\mathcal{B}^{*}$ results

$\displaystyle\mathcal{B}\mathcal{B}^{*}=\big{|}\overline{U}_{\mu
4}\big{|}^{4}\cdot\frac{1}{2}$ (24) $\displaystyle\approx$
$\displaystyle\frac{1}{2}\cos^{4}\theta_{14}\sin^{4}\theta_{24}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{2}\sin^{4}\theta_{24}\quad\mbox{of
order}\quad\mathcal{O}(\epsilon^{4})$

This term has negligible contribution as we keep only terms upto order
$\mathcal{O}(\epsilon^{3})$ in the probability expression. The only relevant
terms after averaging out terms involving $\Delta m^{2}_{41}$ are presented
below

$\displaystyle P^{4\nu}_{\mu\mu}$ $\displaystyle=$
$\displaystyle\mathcal{A}\mathcal{A}^{*}=\sum^{16}_{k=1}T_{k}$ (25)

while the individual terms are expressed as,

$\displaystyle T_{1}=\big{|}\overline{U}_{\mu
1}\big{|}^{4}\,\big{|}\overline{S}_{ee}\big{|}^{2}\,,\quad
T_{2}=\overline{S}_{ee}\overline{S}^{*}_{\mu\mu}\big{|}\overline{U}_{\mu
1}\big{|}^{2}\big{|}\overline{U}_{\mu 2}\big{|}^{2}\,,\quad
T_{3}=\overline{S}_{ee}\overline{S}^{*}_{\mu e}\big{|}\overline{U}_{\mu
1}\big{|}^{2}\overline{U}_{\mu 1}\overline{U}^{*}_{\mu 2}\,,\quad
T_{4}=\overline{S}_{ee}\overline{S}^{*}_{e\mu}\big{|}\overline{U}_{\mu
1}\big{|}^{2}\overline{U}^{*}_{\mu 1}\overline{U}_{\mu 2}$ $\displaystyle
T_{5}=\overline{S}_{\mu\mu}\overline{S}^{*}_{ee}\big{|}\overline{U}_{\mu
1}\big{|}^{2}\big{|}\overline{U}_{\mu 2}\big{|}^{2}\,,\quad
T_{6}=\big{|}\overline{U}_{\mu
2}\big{|}^{4}\quad\big{|}\overline{S}_{\mu\mu}\big{|}^{2}\,,\quad
T_{7}=\overline{S}_{\mu\mu}\overline{S}^{*}_{\mu e}\big{|}\overline{U}_{\mu
2}\big{|}^{2}\overline{U}_{\mu 1}\overline{U}^{*}_{\mu 2}\,,\quad
T_{8}=\overline{S}_{\mu\mu}\overline{S}^{*}_{e\mu}\big{|}\overline{U}_{\mu
2}\big{|}^{2}\overline{U}^{*}_{\mu 1}\overline{U}_{\mu 2}$ $\displaystyle
T_{9}=\overline{S}_{\mu e}\overline{S}^{*}_{ee}\big{|}\overline{U}_{\mu
1}\big{|}^{2}\overline{U}^{*}_{\mu 1}\overline{U}_{\mu 2}\,,\quad
T_{10}=\overline{S}_{\mu e}\overline{S}^{*}_{\mu\mu}\big{|}\overline{U}_{\mu
2}\big{|}^{2}\overline{U}^{*}_{\mu 1}\overline{U}_{\mu 2}\,,\quad
T_{11}=\big{|}\overline{U}_{\mu 1}\big{|}^{2}\big{|}\overline{U}_{\mu
2}\big{|}^{2}\big{|}\overline{S}_{\mu e}\big{|}^{2}\,,\quad
T_{12}=\overline{S}_{\mu e}\overline{S}^{*}_{e\mu}\big{(}\overline{U}^{*}_{\mu
1}\big{)}^{2}\big{(}\overline{U}_{\mu 2}\big{)}^{2}$ $\displaystyle
T_{13}=\overline{S}_{e\mu}\overline{S}^{*}_{ee}\big{|}\overline{U}_{\mu
1}\big{|}^{2}\overline{U}^{*}_{\mu 2}\big{(}\overline{U}_{\mu
1}\big{)}^{2}\,,T_{14}=\overline{S}_{e\mu}\overline{S}^{*}_{\mu\mu}\big{|}\overline{U}_{\mu
2}\big{|}^{2}\overline{U}^{*}_{\mu 2}\overline{U}_{\mu
1}\,,T_{15}=\overline{S}_{e\mu}\overline{S}^{*}_{\mu
e}\big{(}\overline{U}_{\mu 1}\big{)}^{2}\big{(}\overline{U}^{*}_{\mu
2}\big{)}^{2}\,,T_{16}=\big{|}\overline{U}_{\mu
1}\big{|}^{2}\big{|}\overline{U}_{\mu
2}\big{|}^{2}\big{|}\overline{S}_{e\mu}\big{|}^{2}$

If we have a closer look at these $16$ terms, either most of them are
suppressed by at least by order of $\mathcal{O}(\epsilon^{4})$ or combine to
form very simple relations. The first term $T_{1}=\big{|}\overline{U}_{\mu
1}\big{|}^{4}\,\big{|}\overline{S}_{ee}\big{|}^{2}$ is proportional to
$\sin^{8}\theta$ (assuming $\sin\theta_{14}\simeq\sin\theta_{24}$ ) and is
suppressed contribution at least by $\mathcal{O}(\epsilon^{8})$. We have
neglected contributions to the total muon survival probability at the leading
order by assuming that either these terms are suppressed by
$\mathcal{O}(\epsilon^{4})$ or higher order. We can see that terms like
$T_{1}$, $T_{2}+T_{5}$, $T_{3}+T_{9}$, $T_{4}+T_{13}$, $T_{11}+T_{16}$ and
$T_{12}+T_{15}$ are neglected from the probability analysis. The only non-zero
terms contributing to total muon survival probability are given by

$\displaystyle P^{4\nu}_{\mu\mu}$ $\displaystyle\approx$
$\displaystyle\big{|}\overline{U}_{\mu
2}\big{|}^{4}\quad\big{|}\overline{S}_{\mu\mu}\big{|}^{2}$ (27)
$\displaystyle+\big{|}\overline{U}_{\mu
2}\big{|}^{2}\,\mbox{Re}\bigg{(}\overline{S}_{\mu\mu}\,\overline{S}^{*}_{\mu
e}\,\overline{U}_{\mu 1}\overline{U}^{*}_{\mu 2}\bigg{)}$
$\displaystyle+\big{|}\overline{U}_{\mu
2}\big{|}^{2}\,\mbox{Re}\bigg{(}\overline{S}^{*}_{e\mu}\overline{S}_{\mu\mu}\overline{U}^{*}_{\mu
1}\overline{U}_{\mu 2}\bigg{)}$

The $T_{6}$, $T_{7}+T_{10}$, and $T_{8}+T_{14}$ are the main contributions to
the total muon survival probability as first, second and third terms,
respectively. The formalism we adopted has effectively simplified the
expression for muon survival probability in the $3+1$ flavor mixing scenario
under the presence of matter effects.

Parameter | Best Fit values | $1\sigma$
---|---|---
$\Delta_{21}/10^{-5}~{}\mathrm{eV}^{2}$ (NH ) | 7.50 |
$\sin^{2}\theta_{12}/10^{-1}$ (NH ) | 3.18 |
$\Delta_{31}/10^{-3}~{}\mathrm{eV}^{2}$ (NH) | 2.55 |
$\sin^{2}\theta_{13}/10^{-2}$ (NH) | 2.20 |
$\sin^{2}\theta_{23}/10^{-1}$ (NH) | 5.74 |
$\sin^{2}\theta_{14}$ | 0.02 |
$\sin^{2}\theta_{24}$ | 0.02 |
$\sin^{2}\theta_{34}$ | – |
Table 1: The value of the standard oscillation parameters are taken from the
global best fit values quoted in de Salas  _et al._ (2021). The value for
sterile mixing angles used for carrying out numerical analysis are mentioned
alongside.

##  V Muon survival probability using $\alpha-s13$ approximation

We have the projected Hamiltonian $\overline{H}^{\rm dyn}_{3\nu}$ extracted
from complete $4\times 4$ Hamiltonian $\overline{H}^{\rm dyn}_{4\nu}$ by
considering sterile neutrinos effectively decoupled from the $3+1$ scenario.
Also the frequency part of the oscillation probabilities involving
$\overline{S}_{ss}$ is averaged out and hence, there is no mass-square
difference $\Delta m^{2}_{41}$. As we are left with an effective $3\times 3$
effective Hamiltonian, we can now easily perform the series expansion.
Inspecting the neutrino oscillation parameters taken from the global fit
datade Salas  _et al._ (2021), one can mark two such oscillations parameters
that can be used to carry out the series expansion. These small parameters are
as mentioned in the table 1:

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\frac{\Delta
m^{2}_{21}}{\Delta m^{2}_{31}}\approx 0.026$ (28) $\displaystyle
s13=\sin\theta_{13}\approx 0.15$

Using the $\alpha-s13$ approximation, the series expansion has already been
carried out in the literature Akhmedov  _et al._ (2004) for the three flavor
scheme by expanding them up to second order. Moreover, expansions only in
$\alpha$ and $\sin\theta_{13}$ up to the first order have also been performed.
But in the presence of sterile neutrinos, it is quite tedious to arrive at the
exact analytic formula for muon survival probability keeping terms up to
$\alpha^{2}\approx{\cal O}(\epsilon^{4})$. In this work, we aim to retain
terms up to ${\cal O}(\epsilon^{3})$ and neglect all the high order
corrections. As we know, $\sin\theta_{13}\approx{\cal O}(\epsilon)$ and
sterile mixing angles, $\sin\theta_{i4}\approx{\cal O}(\epsilon)$ (with
i=1,2), all these approximations are considered while deriving the final
expression for muon survival probability. The expression for the muon survival
probability given in equation 27 can actually be simplified further while
keeping terms upto order ${\cal O}(\epsilon^{3})$.The resulting terms
contributing to the total muon survival probability are expressed as

$\displaystyle P^{4\nu}_{\mu\mu}$ $\displaystyle\approx$ $\displaystyle
P^{3\nu}_{\mu\mu}+P^{\rm I}_{\rm INT}+P^{\rm II}_{\rm INT}+P^{\rm III}_{\rm
INT}$ (29)

Here the contribution coming from the term $\big{|}\overline{U}_{\mu
2}\big{|}^{4}\quad\big{|}\overline{S}_{\mu\mu}\big{|}^{2}$ is actually
expressed in terms of two factors:

  1. 1.

The first one coming from three flavor muon survival probability i.e.
$P^{3\nu}_{\mu\mu}$ retaining terms upto ${\cal O}(\epsilon^{3})$.

  2. 2.

And the other factor $P^{\rm I}_{\rm INT}$ coming from the sterile-active
interference term after simplifying the factor $\big{|}\overline{U}_{\mu
2}\big{|}^{4}$ such that terms higher than ${\cal O}(\epsilon^{3})$ are safely
neglected.

The dominant contribution to muon survival probability independent of any
sterile mixing angles and phases is denoted as $P^{3\nu}_{\mu\mu}$. It has
been obtained under the vanishing limit of $\alpha^{2}$ from the known
expression of muon survival probability given in Akhmedov  _et al._ (2004) and
also mentioned in the appendix B for completeness.

$\displaystyle P^{3\nu}_{\mu\mu}\approx
1-\sin^{2}2\theta_{23}\sin^{2}{\Delta}+\alpha\cos^{2}\theta_{12}\sin^{2}2\theta_{23}\Delta\sin
2\Delta$
$\displaystyle~{}-4\sin^{2}\theta_{13}\sin^{2}\theta_{23}\frac{\sin^{2}(\hat{A}-1)\Delta}{(\hat{A}-1)^{2}}-\frac{2}{\hat{A}-1}\sin^{2}\theta_{13}\sin^{2}2\theta_{23}$
$\displaystyle~{}\times\big{(}\sin\Delta\cos\hat{A}\Delta\frac{\sin(\hat{A}-1)\Delta}{\hat{A}-1}-\frac{\hat{A}}{2}\Delta\sin
2\Delta\big{)}$ $\displaystyle~{}-2\alpha\sin\theta_{13}\sin 2\theta_{12}\sin
2\theta_{23}\cos\delta_{CP}\cos\Delta\frac{\sin\hat{A}\Delta}{\hat{A}}\frac{\sin(\hat{A}-1)\Delta}{\hat{A}-1}$
$\displaystyle~{}+\frac{2}{\hat{A}-1}\alpha\sin\theta_{13}\sin
2\theta_{12}\sin 2\theta_{23}\cos 2\theta_{23}\cos\delta_{CP}\sin\Delta$
$\displaystyle~{}\times\big{(}\hat{A}\sin\Delta-\frac{\sin\hat{A}\Delta}{\hat{A}}\cos(\hat{A}-1)\Delta\big{)}$
(30)

with $\hat{A}=\frac{A}{\Delta_{31}}$ with $A=2\sqrt{2}G_{F}N_{e}E$ and
$\Delta=\Delta_{31}L/4E$, where $\Delta_{ij}=m_{i}^{2}-m_{j}^{2}$, $G_{F}$ is
the Fermi constant, $N_{e}$ is the electron number density of the medium and
$E$ is the energy of the neutrinos.
The first interference term retaining terms $\sin^{2}\theta_{i4}$ from sterile
part while three flavor part is without having any $\alpha$ or
$\sin\theta_{13}$ as,

$\displaystyle P^{\rm I}_{\rm
INT}\approx-2\sin^{2}\theta_{14}\bigg{(}1-\sin^{2}2\theta_{23}\sin^{2}{\Delta}\bigg{)}$
(31)

The other sub-dominant terms are the interference terms $P^{\rm II}_{\rm INT}$
and $P^{\rm III}_{\rm INT}$ corresponding to $\big{|}\overline{U}_{\mu
2}\big{|}^{2}\,\mbox{Re}\big{(}\overline{S}_{\mu\mu}\,\overline{S}^{*}_{\mu
e}\,\overline{U}_{\mu 1}\overline{U}^{*}_{\mu 2}\big{)}$ and
$\big{|}\overline{U}_{\mu
2}\big{|}^{2}\,\mbox{Re}\big{(}\overline{S}^{*}_{e\mu}\overline{S}_{\mu\mu}\overline{U}^{*}_{\mu
1}\overline{U}_{\mu 2}\big{)}$ respectively, expanded upto first order in
$\sin\theta_{13}$.

$\displaystyle P^{\rm II}_{\rm
INT}\approx\frac{1}{4(\hat{A}-1)}\sin\theta_{13}\cos\theta_{24}\sin\theta_{14}\sin\theta_{24}$
$\displaystyle\bigg{[}\bigg{(}-3\cos\big{(}\delta_{14}+(\hat{A}-1)\Delta\big{)}+\cos\big{(}\delta_{14}+2\hat{A}\Delta+\delta_{13}\big{)}\bigg{)}\sin\theta_{23}$
$\displaystyle-2\sin(\delta_{14}+\delta_{13})\sin\big{(}2\Delta-\delta_{13}\big{)}\sin
2\theta_{23}$
$\displaystyle+2\cos\big{(}2\Delta-\delta_{13}\big{)}\times\bigg{(}\cos\big{(}\delta_{14}+(2\hat{A}-1)\Delta-\frac{\delta_{13}}{2}\big{)}$
$\displaystyle\sin
3\theta_{23}+\cos\big{(}\Delta-\frac{\delta_{13}}{2}\big{)}\cos(\delta_{14}+\delta_{13})\sin
4\theta_{23}\bigg{)}\bigg{]}$ (32)

$\displaystyle P^{\rm III}_{\rm
INT}\approx\frac{1}{2(\hat{A}-1)}\sin\theta_{14}\sin\theta_{13}\sin\theta_{23}\sin\theta_{24}\cos\theta_{24}$
$\displaystyle\bigg{[}\cos\bigg{(}\delta_{14}-2\Delta\bigg{)}-\cos(\delta_{14}-\delta_{13})$
$\displaystyle+2\cos\big{(}\Delta-\frac{\delta_{13}}{2}\big{)}\cos\big{(}\delta_{14}-\Delta-\frac{\delta_{13}}{2}\big{)}\cos
2\theta_{23}\bigg{]}$ (33)

The important point to notice is that these contributions (i.e. $P^{\rm
II}_{\rm INT}$ and $P^{\rm III}_{\rm INT}$) actually contain a factor of
$\cos^{3}\theta_{24}\sin\theta_{14}\sin\theta_{24}$ but since we have consider
$\sin\theta_{13}$ and sterile mixing angles, $\sin\theta_{i4}$ (with i=1,2)
with order $\approx{\cal O}(\epsilon)$, therefore we are left with
$\cos\theta_{24}\sin\theta_{14}\sin\theta_{24}$.

Figure 1:  The survival probabilities $P_{\nu\mu}$ for neutrino and anti-
neutrino mode as a function of neutrino energy in GeV. The baseline is kept
fixed at 295 km and the matter density $\rho$ is taken as 2.7g/cc.The sterile
mixing angles are assigned value
$\sin^{2}\theta_{14}=\sin^{2}\theta_{24}=0.02$

To look at the significant contribution coming from the various interference
terms (retained upto $\approx{\cal O}(\epsilon^{3})$) involved in the muon
survival probability under the $3+1$ scenario, we plot muon survival
probability for neutrinos and anti-neutrinos in FIG 1. The probability
variation with the energy of neutrinos is considered under the normal
hierarchy for the 295km baseline. The average matter density is taken as
$2.7g/cm^{3}$ in accordance to PREM profile of Earth matter density Dziewonski
and Anderson (1981). The solid blue curve represents the total probability,
including all kinds of interferences. The terms marked from $T0-T5$ are the
quantified contributions coming from each individual term involved in the
equation 30 for three-flavor probability. The terms Int ${\rm I}$, Int ${\rm
II}$ and Int ${\rm III}$ are the subdominant contributions as mentioned in the
legend. A similar analysis has been carried out for the anti-neutrinos case by
reversing the sign of the fundamental and sterile CP-phases i.e. $\delta_{13}$
and $\delta_{14}$ respectively. Also, the sign of matter potential is changed
to look at anti-neutrino probabilities. The value of the mass-squared
differences and mixing angles for active neutrinos are kept fixed, as
mentioned in Table.1. The active-sterile mixing angles $\sin^{2}\theta_{14}$
and $\sin^{2}\theta_{24}$ are also kept fixed at $0.02$ throughout the
analysis. It appears that the $T0$ term is majorly affecting the total muon
probability, with the other terms contributing at sub-leading order. But it is
emphasized later how these small contributions arising from active-sterile
mixings are widely affecting the behavior under CP and T transformations.

Figure 2: The linkage between CP and T- transformations

At the very first look, it may seem that the probability curves are almost
identical for both neutrinos and anti- neutrinos. But, zoom in picture marks
that they deviate from each other. To pinpoint this difference, which is very
important for explaining the matter-antimatter asymmetry, we carry out the CP-
violation and T-violation study. The difference between the muon and anti-muon
survival probability is defined as $\Delta
P_{CP}=P_{\alpha\beta}-P_{\overline{\alpha}\overline{\beta}}$ and the
difference under T transformation is given by $\Delta
P_{T}=P_{\alpha\beta}-P_{{\beta}{\alpha}}$. In vacuum, the CPT invariance
holds (i.e. $\Delta P_{CP}=\Delta P_{T}$ ) but it is not invariant in presence
of matter. But the CPT violation takes place only under an asymmetric density
profile. The CP and T transformation linkage has been established in FIG 2. We
are considering constant matter density for our analysis, so any violation of
CPT invariance directly hints toward the new physics coming from the sterile
presence.

Figure 3:  The survival probabilities $P_{\nu\mu}$ for neutrino and anti-
neutrino mode as a function of neutrino energy in GeV. The baseline is kept
fixed at 295 km and the matter density $\rho$ is taken as 2.7g/cc.The sterile
mixing angles are assigned value
$\sin^{2}\theta_{14}=\sin^{2}\theta_{24}=0.02$

We look at the variation of CP and T violations in FIG 3. The solid blue curve
represents the behavior when all the leading and sub-leading contributions are
considered, and it is marked by $\rm{\Delta P_{CP}(W)}$ in legend. The dotted
red curve corresponds to the case where we have neglected the contributions
from active-sterile interference terms $P^{\rm II}_{\rm INT}$ and $P^{\rm
III}_{\rm INT}$ and is represented by $\rm{\Delta P_{CP}(WO)}$. It is
interesting to note that the sub-leading terms are playing a significant role
towards the CP and T violations,particularly in the case of probability
difference under T-transformations. This difference is zero when the presence
of active-sterile interference is ignored. Thus, the muon survival probability
is highly sensitive to the T-violating effects arising from sterile-neutrino
presence. Also, it is evident from the figure that in the presence of matter
the CP violation is no more equivalent to the T violation. The violation
arises from the two kinds of effects- $\rm i)$ The asymmetry between particle
and anti-particle numbers in matter enhances the neutrino oscillations between
neutrinos while suppressing between anti-neutrinos. $\rm ii)$ The other one
arising from the leptonic CP phase $\delta_{13}$ and $\delta_{14}$. Thus the
muon survival probability $P_{\mu\mu}$ plays an important role in
understanding the CP-violation and T-violation studies.

Figure 4:  The survival probabilities $P_{\nu\mu}$ for neutrino and anti-
neutrino mode as a function of neutrino energy in GeV. The baseline is kept
fixed at 295 km and the matter density $\rho$ is taken as 2.7g/cc.The sterile
mixing angles are assigned value
$\sin^{2}\theta_{14}=\sin^{2}\theta_{24}=0.02$

Furthermore, the muon survival probability acts as an important channel for
the long path length/baseline study Gandhi  _et al._ (2006, 2005, 2007);
Agarwalla (2014); Abe  _et al._ (2015); Agarwalla  _et al._ (2018). We use our
analytic relation for muon survival probability to explore the physics
essential at the INO. Since, in the ICAL detector, we have neutrinos with
energy in range of 2-20 GeV over the path lengths (L) varying from 2000 km to
9000 km. At such longer baselines, the effect of the solar mass-squared
difference can be safely neglected in comparison with the atmospheric mass-
squared difference. Thus, under this approximation the parameter $\alpha$
tends to zero, and we are only left with $T0$, $T2$ and, $T3$ terms in
equation 30. Also, neglecting the contribution coming from the active-sterile
neutrino interference terms $P^{\rm II}_{\rm INT}$ and $P^{\rm III}_{\rm
INT}$, we show the variation of remaining terms in the muon and anti-muon
survival probability for baseline $5000$km in FIG 4. The value of oscillation
parameters is as mentioned in table 1, while the averaged constant matter
density is taken as $3.9g/cc$ based on the PREM profile. The terms $T0$, $T2$,
$T3$, and $P^{\rm I}_{\rm INT}$ are represented by dotted blue, red, green,
and magenta lines, respectively. The total survival probability is given by
solid black lines. The following important conclusions can be drawn from the
figure:

  1. 1.

The term $T0=\sin^{2}2\theta_{23}\sin^{2}{\Delta}$ and $P^{\rm I}_{\rm
INT}\approx-2\sin^{2}\theta_{14}\bigg{(}1-\sin^{2}2\theta_{23}\sin^{2}{\Delta}\bigg{)}$
are independent of any matter contributions, fundamental and sterile CP-
phases. As a result, they make an equal contribution to both muon and anti-
muon survival probability.

  2. 2.

The terms
$T2=-4s_{13}^{2}s_{23}^{2}\frac{\sin^{2}(\hat{A}-1)\Delta}{(\hat{A}-1)^{2}}$,
$T3=-\frac{2}{\hat{A}-1}s_{13}^{2}\sin^{2}2\theta_{23}\big{(}\sin\Delta\cos\hat{A}\Delta\frac{\sin(\hat{A}-1)\Delta}{\hat{A}-1}-\frac{\hat{A}}{2}\Delta\sin
2\Delta\big{)}$ are dependent on matter term $A$. As the sign of matter
contributions are different for neutrino and anti-neutrino interactions, these
terms have different contributions in muon and anti-muon survival probability
as seen from FIG 4

  3. 3.

One also notes that at $E\approx 5GeV$, the amplitude for total survival
probability is more for anti-neutrino mode than neutrino mode. This happens
because at this energy value, the terms $T0$ and $T3$ are zero and the
survival probability is dependent on $T2$ and $P^{\rm I}_{\rm INT}$ by
relation $P_{\mu\mu}=1-T2+P^{\rm I}_{\rm INT}$. Hence, we get a smaller muon
survival probability. Similarly, the complete behavior can be explained over
the entire energy range by taking the quantitative contributions from each
term.

Figure 5:  The survival probabilities $P_{\nu\mu}$ for neutrino and anti-
neutrino mode as a function of neutrino energy in GeV. The baseline is kept
fixed at 295 km and the matter density $\rho$ is taken as 2.7g/cc.The sterile
mixing angles are assigned value
$\sin^{2}\theta_{14}=\sin^{2}\theta_{24}=0.02$

Let us consider vacuum and matter contributions to muon survival probability
relevant for atmospheric neutrino studies in longer baselines at leading
order. For such larger baselines, typically of the order of $2000-9000$ km,
the contributions involving solar mass-square difference term can be neglected
safely and we have ignored the sub-leading sterile neutrino contributions. In
these limits, the simplified analytic relations for muon survival probability
for vacuum and matter effects become,

$\displaystyle P^{\rm vac}_{\mu\mu}$ $\displaystyle=$ $\displaystyle
1-\sin^{2}2\theta_{23}\sin^{2}\Delta+4s_{13}^{2}s_{23}^{2}\cos
2\theta_{23}\sin^{2}\Delta$ $\displaystyle P^{\rm mat}_{\mu\mu}$
$\displaystyle=$ $\displaystyle
1-\sin^{2}2\theta_{23}\sin^{2}{\Delta}-4s_{13}^{2}s_{23}^{2}\frac{\sin^{2}(\hat{A}-1)\Delta}{(\hat{A}-1)^{2}}$
(34)
$\displaystyle-\frac{2}{\hat{A}-1}\sin_{\theta_{13}}^{2}\sin^{2}2\theta_{23}\bigg{(}\sin\Delta\cos\hat{A}\Delta\frac{\sin(\hat{A}-1)\Delta}{\hat{A}-1}$
$\displaystyle\hskip 113.81102pt-\frac{\hat{A}}{2}\Delta\sin 2\Delta\bigg{)}$

After a series expansion upto linear order in $\hat{A}$, the matter
contribution reduces to,

$\displaystyle P^{\rm mat}_{\mu\mu}$ $\displaystyle=$ $\displaystyle P^{\rm
vac}_{\mu\mu}+8\hat{A}\cos\big{(}2\theta_{23}\big{)}\sin^{2}\theta_{13}\sin^{2}\theta_{23}\sin\Delta$
(35) $\displaystyle\times\big{(}-\Delta\cos\Delta+\sin\Delta\big{)}$

The atmospheric neutrino probability studies are carried out over the longer
baselines where the effect of $\delta_{CP}$ phase is quite insignificant.
Taking advantage of this feature of muon survival probability, we look at the
variation of probability under normal and inverted mass hierarchy. In FIG 5,
we show the variation for the baseline of $6000$ km and $7000$ km with the
approximate average matter densities $4.1g/cc$ and $4.15g/cc$ respectively. A
similar analysis can be performed over much longer baselines to build sharp
discrimination between normal and inverted mass hierarchy. However, beyond
$10,000$ km, the interferences among the mantle and core arise, and hence such
longer baselines are generally avoided for study. The analysis is done for the
probability at leading order, independent of active-sterile interferences
$P^{\rm II}_{\rm INT}$ and $P^{\rm III}_{\rm INT}$. The solid blue curves
represent the behaviour under normal hierarchy, while the dotted red curve
represents the inverted hierarchy behaviour. The behaviour in vacuum is marked
by a solid black line where we also consider the contribution coming from the
$\rm IntI$ term. It is find that the curve for vacuum is overlapping with the
probability behaviour under inverted hierarchy, while there is a significant
difference between the curves corresponding to both hierarchies. The
difference increases with the baseline, therefore it provides a clean way to
distinguish between the two hierarchies.

##  VI Conclusion

In this paper, we looked at the detailed analytic expressions for the muon
survival probability in the $3+1$ scenario under the influence of matter
potential. We carry forward the idea of references Klop and Palazzo (2015);
Sharma and Patra (2022), reducing the effective Hamiltonian of the $3+1$
scenario to the projected three- flavor Hamiltonian using S-matrix formalism.
We deduced that all the individual contributions to the total muon survival
probability could be expressed in just three terms- one leading contribution
coming from $3$ flavor muon survival probability and the other two arising
from active-sterile mixings. We then performed the well-known $\alpha-s13$
approximation to further simplify our analytic results. The term by term
contributions to the final muon survival probability are quantified
numerically for the $T2K$ experiment baseline for both neutrino and anti-
neutrino modes. We pointed out that T-violation in muon survival probability
arises purely from the new physics of active-sterile mixings. Furthermore, the
presence of sterile neutrino highlighted the violation of CPT invariance. We
discussed how matter contribution to muon survival probability is related to
vaccum contribution at the leading order. We also looked at the behaviour of
muon probability at the various longer baselines having implications for
exploring the atmospheric neutrino sector. Moreover, we discuss the role of
muon survival probability in resolving the mass-hierarchy degeneracy.

##  VII Acknowledgment

Kiran Sharma would like to acknowledge the Ministry of Education, Govt of
India for financial support.

##  VIII APPENDIX

##  Appendix A Parametrization of mixing matrix in 3+1 framework.

The standard parametrization considered for the present work is given by

$\displaystyle U$ $\displaystyle=$ $\displaystyle
R\big{(}\theta_{34},\delta_{34}\big{)}\,R\big{(}\theta_{24},0\big{)}\,R\big{(}\theta_{14},\delta_{14}\big{)}$
(36) $\displaystyle\hskip 28.45274pt\times
R\big{(}\theta_{23},0\big{)}R\big{(}\theta_{13},\delta_{13}\big{)}R\big{(}\theta_{12},0\big{)}$

Here, $\theta_{14}$, $\theta_{24}$ and $\theta_{34}$ are the mixing angles
between sterile and active neutrinos while $\theta_{23}$, $\theta_{13}$ and
$\theta_{12}$ are known mixing angles of three flavor neutrinos. The
fundamental CP phase $\delta_{13}$ and other CP-phases $\delta_{14}$ and
$\delta_{34}$ coming from active-sterile mixings. One can also write down the
complete $4\times 4$ mixing matrix in terms of usual three flavor matrix as,

$\displaystyle
U=R\big{(}\theta_{34},\delta_{34}\big{)}\,R\big{(}\theta_{24},0\big{)}\,R\big{(}\theta_{14},\delta_{14}\big{)}U_{3\nu}$
(37)

with
$U_{3\nu}=R\big{(}\theta_{23},0\big{)}R\big{(}\theta_{13},\delta_{13}\big{)}R\big{(}\theta_{12},0\big{)}$.

The general matrix structure of $U$ and $U^{\dagger}$ can be read as,

$\displaystyle U$ $\displaystyle=$
$\displaystyle\begin{pmatrix}U_{e1}&U_{e2}&U_{e3}&U_{e4}\\\ U_{\mu 1}&U_{\mu
2}&U_{\mu 3}&U_{\mu 4}\\\ U_{\tau 1}&U_{\tau 2}&U_{\tau 3}&U_{\tau 4}\\\
U_{s1}&U_{s2}&U_{s3}&U_{s4}\end{pmatrix}\,,U^{\dagger}=\begin{pmatrix}U^{*}_{e1}&U^{*}_{\mu
1}&U^{*}_{\tau 1}&U^{*}_{s1}\\\ U^{*}_{e2}&U^{*}_{\mu 2}&U^{*}_{\tau
2}&U^{*}_{s2}\\\ U^{*}_{e3}&U^{*}_{\mu 3}&U^{*}_{\tau 3}&U^{*}_{s3}\\\
U^{*}_{e4}&U^{*}_{\mu 4}&U^{*}_{\tau 4}&U^{*}_{s4}\end{pmatrix}$

The relation $UU^{\dagger}$ gives,

$\displaystyle\left[\begin{array}[]{cccc}\sum_{j=1}^{4}U_{ei}U^{*}_{ei}&\sum_{j=1}^{4}U_{ei}U^{*}_{\mu
i}&\sum_{j=1}^{4}U_{ei}U^{*}_{\tau i}&\sum_{j=1}^{4}U_{ei}U^{*}_{si}\\\
\sum_{j=1}^{4}U_{\mu i}U^{*}_{ei}&\sum_{j=1}^{4}U_{\mu i}U^{*}_{\mu
i}&\sum_{j=1}^{4}U_{\mu i}U^{*}_{\tau i}&\sum_{j=1}^{4}U_{\mu i}U^{*}_{si}\\\
\sum_{j=1}^{4}U_{\tau i}U^{*}_{ei}&\sum_{j=1}^{4}U_{\tau i}U^{*}_{\mu
i}&\sum_{j=1}^{4}U_{\tau i}U^{*}_{\tau i}&\sum_{j=1}^{4}U_{\tau
i}U^{*}_{si}\\\ \sum_{j=1}^{4}U_{si}U^{*}_{ei}&\sum_{j=1}^{4}U_{si}U^{*}_{\mu
i}&\sum_{j=1}^{4}U_{si}U^{*}_{\tau i}&\sum_{j=1}^{4}U_{si}U^{*}_{si}\\\
\end{array}\right]$ (42) $\displaystyle\hskip
56.9055pt=\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right]$ (47)

The unitarity condition of the PMNS mixing matrix in $3+1$ framework leads to

$\sum^{4}_{j=1}U_{\alpha j}U^{*}_{\beta
j}=\delta_{\alpha\beta}=\bigg{\\{}\begin{array}[]{cc}1\quad\mbox{if
$\alpha=\beta$}\\\ 0\quad\mbox{if $\alpha\neq\beta$}\end{array}$

The rowwise elements of the complete $4\times 4$ mixing matrix are expressed
as follows:

  1. I.

First column elements of $U$:

     * $\displaystyle U_{e1}=c_{21}c_{13}c_{14}$ $\displaystyle U_{\mu 1}=c_{12}\left(-c_{13}s_{14}s_{24}e^{\text{i$\delta$}_{14}}-c_{24}s_{13}s_{23}e^{\text{i$\delta$}_{13}}\right)-c_{23}c_{24}s_{12}$ $\displaystyle U_{\tau 1}=c_{12}\bigg{(}-c_{13}c_{24}s_{14}s_{34}e^{\text{i$\delta$}_{14}-\text{i$\delta$}_{34}}$ $\displaystyle\hskip 42.67912pt-s_{13}e^{\text{i$\delta$}_{13}}\left(c_{23}c_{34}-s_{23}s_{24}s_{34}e^{-\text{i$\delta$}_{34}}\right)\bigg{)}$ $\displaystyle\hskip 42.67912pt-s_{12}\left(-c_{23}s_{24}s_{34}e^{-\text{i$\delta$}_{34}}-c_{34}s_{23}\right)$ $\displaystyle U_{s1}=c_{12}\bigg{(}-s_{13}e^{\text{i$\delta$}_{13}}\left(-c_{23}s_{34}e^{\text{i$\delta$}_{34}}-c_{34}s_{23}s_{24}\right)$ $\displaystyle-c_{13}c_{24}c_{34}s_{14}e^{\text{i$\delta$}_{14}}\bigg{)}-s_{12}\left(s_{23}s_{34}e^{\text{i$\delta$}_{34}}-c_{23}c_{34}s_{24}\right)$ (48)

  2. II.

Second column elements of $U$:

     * $\displaystyle U_{e2}=c_{13}c_{14}s_{12}$ $\displaystyle U_{\mu 2}=s_{12}\left(-c_{13}s_{14}s_{24}e^{\text{i$\delta$}_{14}}-c_{24}s_{13}s_{23}e^{\text{i$\delta$}_{13}}\right)+c_{12}c_{23}c_{24}$ $\displaystyle U_{\tau 2}=c_{12}\left(-c_{23}s_{24}s_{34}e^{-\text{i$\delta$}_{34}}-c_{34}s_{23}\right)$ $\displaystyle\hskip 42.67912pt+s_{12}\bigg{(}-c_{13}c_{24}s_{14}s_{34}e^{\text{i$\delta$}_{14}-\text{i$\delta$}_{34}}$ $\displaystyle\hskip 42.67912pt-s_{13}e^{\text{i$\delta$}_{13}}\left(c_{23}c_{34}-s_{23}s_{24}s_{34}e^{-\text{i$\delta$}_{34}}\right)\bigg{)}$ $\displaystyle U_{s2}=c_{12}\left(s_{23}s_{34}e^{\text{i$\delta$}_{34}}-c_{23}c_{34}s_{24}\right)$ $\displaystyle\hskip 42.67912pt+s_{12}\bigg{(}-s_{13}e^{\text{i$\delta$}_{13}}\left(-c_{23}s_{34}e^{\text{i$\delta$}_{34}}-c_{34}s_{23}s_{24}\right)$ $\displaystyle\hskip 42.67912pt-c_{13}c_{24}c_{34}s_{14}e^{\text{i$\delta$}_{14}}\bigg{)}$ (49)

  3. III.

Third column elements of $U$:

     * $\displaystyle U_{e3}=c_{14}s_{13}e^{-\text{i$\delta$}_{13}}$ $\displaystyle U_{\mu 3}=c_{13}c_{24}s_{23}-s_{13}s_{14}s_{24}e^{\text{i$\delta$}_{14}-\text{i$\delta$}_{13}}$ $\displaystyle U_{\tau 3}=c_{13}\left(c_{23}c_{34}-s_{23}s_{24}s_{34}e^{-\text{i$\delta$}_{34}}\right)$ $\displaystyle\hskip 42.67912pt-c_{24}s_{13}s_{14}s_{34}e^{-\text{i$\delta$}_{13}+\text{i$\delta$}_{14}-\text{i$\delta$}_{34}}$ $\displaystyle U_{s3}=c_{13}\left(-c_{23}s_{34}e^{\text{i$\delta$}_{34}}-c_{34}s_{23}s_{24}\right)$ $\displaystyle\hskip 42.67912pt-c_{24}c_{34}s_{13}s_{14}e^{\text{i$\delta$}_{14}-\text{i$\delta$}_{13}}$ (50)

  4. IV.

Fourth column elements of $U$:

     * $\displaystyle U_{e4}=s_{14}e^{-\text{i$\delta$}_{14}}$ $\displaystyle U_{\mu 4}=c_{14}s_{24}$ $\displaystyle U_{\tau 4}=c_{14}c_{24}s_{34}e^{-\text{i$\delta$}_{34}}$ $\displaystyle U_{s4}=c_{14}c_{24}c_{34}$ (51)

##  Appendix B Three flavor oscillation probabilities in presence of matter
using $\alpha-s13$ approximations:

We will adopt the formalism of $\alpha-\sin\theta_{13}$ approximation as
discussed in ref Akhmedov  _et al._ (2004) to derive the muon survival
probability for three flavor neutrino oscillation. Usually, the oscillation
probability is expressed as a series expansion up to $\alpha^{2}$ with
$\alpha=\Delta m^{2}_{21}/\Delta m^{2}_{31}$. Let us define three key
parameters which we will use in expressing muon survival probability in
presence of matter,

• $\Delta=\frac{\Delta m^{2}_{31}L}{4E}$. • $\alpha=\Delta m^{2}_{21}/\Delta
m^{2}_{31}$. • $\left(\alpha\Delta\right)=\Delta m^{2}_{21}$.

The effective Hamiltonian in flavor basis can be written as

$\displaystyle
H=\frac{\Delta_{31}}{2E}\big{[}U\text{diag}(0,\alpha,1)U^{\dagger}+\text{diag}(\hat{A},0,0)\big{]},$
(52)

where $\hat{A}=A/\Delta_{31}$. In order to derive the double expansion, we
write the above Hamiltonian as

$\displaystyle
H=\frac{\Delta_{31}}{2E}R_{23}U_{\delta}MU_{\delta}^{\dagger}R_{23}^{T},$ (53)

where $U_{\delta}=\text{diag}(1,1,e^{i\delta_{CP}})$. We define,

$\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{\Delta_{31}}{2E}M$ (54)
$\displaystyle=$
$\displaystyle\frac{\Delta_{31}}{2E}\left[R_{13}R_{12}\text{diag}(0,\alpha,1)R_{12}^{T}R_{13}^{T}+\text{diag}(\hat{A},0,0)\right]$
$\displaystyle=$
$\displaystyle\begin{pmatrix}s_{12}^{2}c_{13}^{2}\alpha+s_{13}^{2}+\hat{A}&\alpha
c_{12}c_{13}s_{12}&s_{13}c_{13}(1-\alpha s_{12}^{2})\\\
s_{12}c_{12}c_{13}\alpha&\alpha c_{12}^{2}&-\alpha c_{12}s_{12}s_{13}\\\
s_{13}c_{13}(1-\alpha s_{12}^{2})&-s_{12}c_{12}s_{13}\alpha&\alpha
s_{12}^{2}s_{13}^{2}+c_{13}^{2}\\\ \end{pmatrix}.$

To start with the diagonalization of the above mass matrix, let us make the
desired approximation that $\alpha\approx\sin\theta_{13}\approx\epsilon$ for a
small parameter $\epsilon$. Although this approximation is not entirely
correct as $\alpha\approx\pm 0.03$ while $\sin\theta_{13}\equiv s_{13}\approx
0.15$. Also there are other ways to find oscillation probabilities using
expansion that accounts for different order of $\alpha$ and $\sin\theta_{13}$
Asano and Minakata (2011); Agarwalla  _et al._ (2014). However, these
parameters are being small and similar in order of magnitude, the simple
second order expansion in terms of $\epsilon$ is still valid. Using
perturbation theory up to second order in the small parameters $\alpha$ and
$s_{13}$, the resulting energy eigenvalues (
$E_{i}=\frac{\Delta_{31}}{2E}\lambda_{i}$) are given by

$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{31}}{2E}\big{(}\hat{A}+\alpha
s_{12}^{2}+s_{13}^{2}\frac{\hat{A}}{\hat{A}-1}+\alpha^{2}\frac{\sin^{2}2\theta_{12}}{4\hat{A}}\big{)},$
$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{31}}{2E}\big{(}\alpha
c_{12}^{2}-\alpha^{2}\frac{\sin^{2}2\theta_{12}}{4\hat{A}}\big{)},$
$\displaystyle E_{3}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{31}}{2E}\big{(}1-s_{13}^{2}\frac{\hat{A}}{\hat{A}-1}\big{)},$
(55)

Similarly, the resulting eigenvectors derived using the same approximation are
as follows,

$\displaystyle v_{1}=\begin{pmatrix}1\\\ \frac{\alpha\sin
2\theta_{12}}{2\hat{A}}+\frac{\alpha^{2}\sin 4\theta_{12}}{4\hat{A}^{2}}\\\
\frac{s_{13}}{\hat{A}-1}-\frac{\hat{A}\alpha
s_{13}s_{12}^{2}}{(\hat{A}-1)^{2}}\\\ \end{pmatrix},\hskip 8.53581pt$
$\displaystyle v_{2}=\begin{pmatrix}-\frac{\alpha\sin
2\theta_{12}}{2\hat{A}}-\frac{\alpha^{2}\sin 4\theta_{12}}{4\hat{A}^{2}}\\\
1\\\ \frac{\alpha s_{13}\sin 2\theta_{12}(\hat{A}+1)}{2\hat{A}}\\\
\end{pmatrix},$ $\displaystyle
v_{3}=\begin{pmatrix}-\frac{s_{13}}{\hat{A}-1}+\frac{\hat{A}\alpha
s_{13}s_{12}^{2}}{(\hat{A}-1)^{2}}\\\ \frac{\hat{A}\alpha s_{13}\sin
2\theta_{12}}{2(\hat{A}-1)}\\\ 1\\\ \end{pmatrix}.$ (56)

By stacking these eigenvectors, one can obtain the relevant mixing matrix is
$W=(v_{1},v_{2},v_{3})$ and the modified mixing matrix is read as,

$\displaystyle U_{M}=R_{23}U_{\delta}W,$ (57)

Let us now derive oscillation probability using calculated energy eigenvalues
and modified mixing matrix. With some simplifications, the muon survival
probability is found to be,

$\displaystyle P_{\mu\mu}$ $\displaystyle=$ $\displaystyle
1-\sin^{2}2\theta_{23}\sin^{2}{\Delta}+\alpha
c_{12}^{2}\sin^{2}2\theta_{23}\Delta\sin 2\Delta$ (58)
$\displaystyle~{}-\alpha^{2}\sin^{2}2\theta_{12}c_{23}^{2}\frac{\sin^{2}\hat{A}\Delta}{\hat{A}^{2}}-\alpha^{2}c_{12}^{2}\sin^{2}2\theta_{23}\Delta^{2}\cos
2\Delta$
$\displaystyle~{}+\frac{1}{2\hat{A}}\alpha^{2}\sin^{2}2\theta_{12}\sin^{2}2\theta_{23}$
$\displaystyle~{}\times\big{(}\sin\Delta\frac{\sin\hat{A}\Delta}{\hat{A}}\cos(\hat{A}-1)\Delta-\frac{\Delta}{2}\sin
2\Delta\big{)}$
$\displaystyle~{}-4s_{13}^{2}s_{23}^{2}\frac{\sin^{2}(\hat{A}-1)\Delta}{(\hat{A}-1)^{2}}-\frac{2}{\hat{A}-1}s_{13}^{2}\sin^{2}2\theta_{23}$
$\displaystyle~{}\times\big{(}\sin\Delta\cos\hat{A}\Delta\frac{\sin(\hat{A}-1)\Delta}{\hat{A}-1}-\frac{\hat{A}}{2}\Delta\sin
2\Delta\big{)}$ $\displaystyle~{}-2\alpha s_{13}\sin 2\theta_{12}\sin
2\theta_{23}\cos\delta_{CP}\cos\Delta\frac{\sin\hat{A}\Delta}{\hat{A}}\frac{\sin(\hat{A}-1)\Delta}{\hat{A}-1}$
$\displaystyle~{}+\frac{2}{\hat{A}-1}\alpha s_{13}\sin 2\theta_{12}\sin
2\theta_{23}\cos 2\theta_{23}\cos\delta_{CP}\sin\Delta$
$\displaystyle~{}\times\big{(}\hat{A}\sin\Delta-\frac{\sin\hat{A}\Delta}{\hat{A}}\cos(\hat{A}-1)\Delta\big{)},$

with $\alpha=\Delta_{21}/\Delta_{31}$, $\hat{A}=A/\Delta_{31}$,
$A=2\sqrt{2}G_{F}N_{e}E$ and $\Delta=\Delta_{31}/4E$. Where
$\Delta_{ij}=m_{i}^{2}-m_{j}^{2}$, $G_{F}$ is the Fermi constant, $N_{e}$ is
the electron number density of the medium and $E$ is the energy of the
neutrinos.
The vacuum oscillation probabilities up to second order in $\alpha$ and
$s_{13}$ can be readily derived with the approximation $\hat{A}\rightarrow 0$
in the above set of equation.

$\displaystyle P_{\mu\mu}$ $\displaystyle=$ $\displaystyle
1-\sin^{2}2\theta_{23}\sin^{2}\Delta+\alpha
c_{12}^{2}\sin^{2}\theta_{23}\Delta\sin 2\Delta$ (59)
$\displaystyle~{}-\alpha^{2}\Delta^{2}\big{[}\sin^{2}2\theta_{12}c_{23}^{2}+c_{12}^{2}\sin^{2}2\theta_{23}\big{(}\cos
2\Delta-s_{12}^{2}\big{)}\big{]}$ $\displaystyle~{}+4s_{13}^{2}s_{23}^{2}\cos
2\theta_{23}\sin^{2}\Delta$ $\displaystyle~{}-2\alpha s_{13}\sin
2\theta_{12}s_{23}^{2}\sin 2\theta_{23}\cos\delta_{CP}\Delta\sin 2\Delta,$

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