# Geometrical Interpretation of Neutrino Oscillation with decay

Rajrupa Banerjee<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology (IIT) Bhilai, Bhilai 491001, India  Kiran Sharma
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
(IIT) Bhilai, Bhilai 491001, India  Institut für Astroteilchenphysik,
Karlsruher Institut für Technologie(KIT), Hermann-von-Helmholtz-Platz 1,
76344, Eggenstein-Leopoldshafen, Germany.  Sudhanwa Patra
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
(IIT) Bhilai, Bhilai 491001, India  Prasanta K. Panigrahi
<EMAIL_ADDRESS>Indian Institute of Science Education and Research
Kolkata, 741246, WB, India

###### Abstract

The geometrical representation of two-flavor neutrino oscillation represents
the neutrino’s flavor eigenstate as a magnetic moment-like vector that evolves
around a magnetic field-like vector that depicts the Hamiltonian of the
system. In the present work, we demonstrate the geometrical interpretation of
neutrino in a vacuum in the presence of decay, which transforms this circular
trajectory of neutrino into a helical track that effectively makes the
neutrino system mimic a classical damped driven oscillator. We show that in
the absence of the phase factor $\xi$ in the decay Hamiltonian, the neutrino
exactly behaves like the system of nuclear magnetic resonance(NMR); however,
the inclusion of the phase part introduces a $CP$ violation, which makes the
system deviate from NMR. Finally, we make a qualitative discussion on under-
damped, critically-damped, and over-damped scenarios geometrically by three
different diagrams. In the end, we make a comparative study of geometrical
picturization in vacuum, matter, and decay, which extrapolates the
understanding of the geometrical representation of neutrino oscillation in a
more straightforward way.

###### pacs:

98.80.Cq,14.60.Pq

##  I Introduction

The experimental shreds of evidence of neutrino oscillation have proven the
existence of tiny but nonzero masses of neutrino, which leads to a new era of
exploration of fundamental physics SNO:2002tuh ; Super-Kamiokande:2002ujc ;
IceCube:2017lak ; DayaBay:2012fng ; MINOS:2011amj . But for a better
understanding of neutrino oscillation visually, the geometrical interpretation
of neutrino oscillation had been proposed where neutrino oscillation was
described as the rotation of a unit vector depicting the flavor state of the
neutrinos around a magnetic field-like vector denoted by the Hamiltonian of
the neutrino system Giunti:2007ry ; Kim:1987bv ; Smirnov_1986jht .
Analogously, this picturization has been compared with the precession of the
magnetic moment vector in the presence of an external magnetic field. The
projection of the flavor state $\ket{\nu_{\alpha}(t)}$ at time t, over the
mass basis $\ket{\nu_{i}}$ is connected to the amplitude of oscillation
Giunti:2007ry .

Although the geometrical interpretation makes it easier to understand the
neutrino oscillation and realize it pictorially, such representation is not
unique. Depending on the different choices of basis, some mathematical
approaches have come up with a heuristic overview of the flavor state
oscillation. For example, in Giunti. et al. Giunti:2007ry  an orthogonal basis
$e_{F}^{i}$, $e_{M}^{i}$ describing the flavor and the mass basis of the
neutrino has been investigated, where i stands for 1,2,3. The flavor state
$\nu_{e}$ revolves around the Hamiltonian vector $\vec{B}$ with an angle
$2\theta$. A similar kind of picturization has been done in Smirnov. et al
Smirnov_1986jht  with the choices of basis with axes $\nu_{1}$, $Re\nu_{1}$,
$Im\nu_{2}$ to describe the two generation neutrino oscillation. On the other
hand, Kim. et al. Kim:1987bv ; Sen:2018put  discussed the geometrical
interpretation of flavor oscillation of neutrino in three-dimensional
Euclidean space deduced from the two-valued representation of the flavor
space. They implemented their interpretation to study the MSW effect in
adiabatic and non-adiabatic scenarios.

Oscillation experiments JUNO:2021ydg ; Ghoshal:2020hyo ; Gonzalez-
Garcia:2008mgl  and extensive literature Bahcall:1972my ; Lindner:2001fx ;
Blasone:2020qbo  ambiguously prove that neutrinos possibly decay to lighter
invisible states. The literature proclaims an extensive study of the
analytical approach to neutrino oscillation in the presence of decay
Chattopadhyay:2021eba ; Dixit:2022izn . However, due to the non-Hermiticity of
decay-Hamiltonian, it is challenging to analyze the probability of oscillation
analytically, making it harder to visualize the picture of the damped neutrino
oscillation Chattopadhyay:2021eba ; Goswami:2023hux . Due to this, a
geometrical approach to understanding the oscillation in the presence of decay
Hamiltonian has not been proposed yet.

In the present work, we address the above adversity of the non-hermitian
nature of decay Hamiltonian by a simple mathematical approach analogous to the
NMR system and illustrate the neutrino oscillation geometrically in the
presence of decay Hamiltonian. The motivation for adopting such a procedure is
to consider neutrino as an open quantum system with a similar viewpoint to an
NMR system where the external magnetic field acts as a beam splitter; in the
case of neutrino, the unitary transformation matrix performs the same
KumarJha:2020pke ; Blasone:2010ta ; Blasone:2019chv ; Blasone:2020qbo . The
result of the damped oscillation of neutrino also goes hand in hand with the
NMR. However, introducing the phase factor $\xi$ with the non-Harmitian decay
Hamiltonian the introduces $CP$ violation in two flavor neutrino oscillation
even in absence of the Majorana phase which modifies the dynamic of the flavor
state that introduces a significant difference from the NMR. For a simplistic
overview, we restrict ourselves to only two-flavor neutrino oscillation.

The construction of the paper is as follows. We start our work with the
general theoretical framework(Section II) for the neutrino oscillation, where
we define the two mass eigen states $\ket{\nu_{1}}$ and $\ket{\nu_{2}}$ as a
two-level system. We also introduce the decay Hamiltonian along with it. Next,
we follow the density matrix formalism to define the neutrino as an open
quantum system in the presence of decay. To picture the oscillation
vectorially, we vectorized the operator using the bra-flipper operator. We use
the Pauli Matrix basis to represent the vectorized operator and construct the
dynamical evolution equation. In the result and discussion session(section
III), we evaluate the probabilities and equation of motion. We explain our
result with two different schemes. First, we approximated the decay parameters
to be zero for the ideal case. In the next, we derive the equation of motion,
including the decay parameters. In this section, we also explain the result
graphically and present the same using geometrical schematics. In the
successive section(section IV), we compare geometrical interpretation in
different bases following Kim:1987bv ; Sen:2018put  with the present work. We
also briefly discuss the matter effect and illuminate the geometrical
interpretation of neutrino oscillation in matter. The paper concludes with an
overall summary of the total work and sheds light on some of the other
perspectives of the work for future reference.

##  II Theoretical Framework

Consider an ultra-relativistic neutrino (left-handed) with flavor $\alpha$
with $\alpha=e,\mu,\tau$ and momentum $\vec{p}$. The present theoretical
framework extends to considering the following two eigenstates of neutrinos,
i.e., mass eigenstate, which is the propagating eigenstate, and flavor
eigenstate, which is the interacting eigenstate. These two eigenstates are
expressed by the two state vectors $\ket{\nu_{\alpha}}$ and $\ket{\nu_{i}}$
which are related by the unitary transformation transformation.

$\ket{\nu_{\alpha}}=\sum_{k}U^{*}_{\alpha k}\ket{\nu_{k}}$ (1)

Where $\ket{\nu_{\alpha}}$ is the flavor state of the neutrino, and
$\ket{\nu_{k}}$ is the mass eigenstate of the neutrino. As we deal only with
the two flavors, no additional $CP$ phase part is needed. Hence, the unitary
transformation $U^{*}$ will be replaced by the orthogonal transformation
$\mathcal{O}^{T}$. where,

$\mathcal{O}=\begin{pmatrix}\cos{\theta}&\sin{\theta}\\\
-\sin{\theta}&\cos{\theta}\end{pmatrix}$ (2)

In this work, we consider only Dirac-type neutrino. It is to be mentioned here
that for Majorana type of neutrino $CP$ violation phase needed to be
considered even for the two-flavor neutrino oscillation as wellDixit:2022izn ,
i.e., $\ket{\nu_{\alpha}}=\mathcal{O}U_{ph}\ket{\nu_{i}}$, where
$U_{ph}=\begin{pmatrix}1&0\\\ 0&e^{i\phi}\\\ \end{pmatrix}$. Associated with
any physical system, there is a complex Hilbert space known as the system’s
state space. In Hilbert space, the state vector entirely describes the
system’s state. These state vectors of a finite Hilbert space correspond to
the possible physical condition of the system, and they are the pure state.

Considering the fact that a column matrix gives any state vector in Hilbert
space, we consider the mass eigenstates of the neutrino as a two-level system
with state $\ket{\nu_{1}}=\begin{pmatrix}1\\\ 0\\\ \end{pmatrix}$ and state
$\ket{\nu_{2}}=\begin{pmatrix}0\\\ 1\\\ \end{pmatrix}$. In the present work,
we restrict ourselves to the two-flavor oscillation only. In Hilbert space any
operator is given by a square matrix,
$\hat{A}=\begin{pmatrix}\bra{1}\hat{A}\ket{1}&\bra{1}\hat{A}\ket{2}\\\
\bra{2}\hat{A}\ket{1}&\bra{2}\hat{A}\ket{2}\\\ \end{pmatrix}$ With diagonal
elements as the eigenvalues and the off-diagonal elements as the transition
elements between the two levels. In resemblance with the neutrino system
states $\ket{1}$ and $\ket{2}$ replace by the mass eigenstates $\ket{\nu_{1}}$
and $\ket{\nu_{2}}$. To start with, consider two flavor neutrino states as
$\psi=\big{(}\nu_{e}\,\quad\nu_{\mu}\big{)}^{T}$ and the corresponding
effective Hamiltonian in vacuum is expressed in the limit of ultra-
relativistic energy approximation $E_{i}\simeq p+m_{i}^{2}/2p\approx
E+m_{i}^{2}/2E$ as,

$\mathcal{H}_{0}=\left[E\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)+\frac{1}{2E}\left(\begin{array}[]{cc}m^{2}_{1}&0\\\
0&m^{2}_{2}\end{array}\right)\right]$ (3)

It is well known that the term proportional to identity has no effect on
neutrino oscillation and hence can be dropped from now on. The neutrino mass
eigenstates are related to the corresponding flavor eigenstates by a unitary
mixing matrix. Thus, the vacuum part of effective Hamiltonian in flavor basis
is read as,

$\displaystyle\mathcal{H}_{\rm F}$ $\displaystyle=$
$\displaystyle\frac{1}{2E}\left[\begin{array}[]{cc}m^{2}_{1}\cos^{2}\theta+m^{2}_{2}\sin^{2}\theta&-\cos\theta\sin\theta\Delta
m^{2}\\\ -\cos\theta\sin\theta\Delta
m^{2}&m^{2}_{1}\sin^{2}\theta_{+}m^{2}_{2}\cos^{2}\theta\end{array}\right]$
(6) $\displaystyle=$ $\displaystyle\frac{\Delta m^{2}}{4E}\begin{pmatrix}-\cos
2\theta&\sin 2\theta\\\ \sin 2\theta&\cos 2\theta\\\ \end{pmatrix}\big{]}$ (7)

with $\Delta m^{2}=m^{2}_{2}-m^{2}_{1}$ is the mass square difference between
the two mass eigenvalues of the neutrinos. From the eigenvalue equation for
mass eigenstate have the eigenvalues

$\vspace{-0.3cm}\mathcal{H}_{0}\ket{\nu_{k}}=E_{k}\ket{\nu_{k}}$ (8)

$E_{1}=\frac{m^{2}_{1}}{2E}$ and $E_{2}=\frac{m^{2}_{2}}{2E}$. Here we
consider the same energy approach for the ultra-relativistic process
$p_{i}\sim E+\frac{m^{2}_{i}}{2E}$ so that the oscillation phase reduces to
$\Delta\phi=-\Delta p\cdot L\sim\frac{\Delta m^{2}}{2E}L$ Akhmedov:2019iyt .
It is sufficient to notice no transition between the two energy levels in
vacuum oscillation. In Schrodinger’s picture, any neutrino state follows the
evolution equation of the initial flavor $\alpha$ as

$i\frac{d}{dt}\ket{\nu_{\alpha}}=\mathcal{H}_{F}\ket{\nu_{\alpha}}$ (9)

Where $\mathcal{H}_{F}$ is the Hamiltonian of the flavor state. For simplified
of the mathematical description we take $\frac{m^{2}_{1}}{2E}=a_{1}$ and
$\frac{m^{2}_{2}}{2E}=a_{2}$.

From the experimental evidence of neutrino oscillations, it has already been
proven that neutrino has a finite mass. All the particles having finite rest
mass have some finite decay width. The decay probability increases with the
mass of the particles Halzen:1984mc ; Baerwald:2012kc . The decay probability
becomes very small for a very low-mass particle, and we get quasi-stationary
states $i.e.$, the particles are localized for a long time. Without decay,
each flavor eigenstate is the coherent superposition of mass eigenstates. The
Hermitian nature of the Hamiltonian gives rise to real eigenvalues.
Nevertheless, the non-Hermitian attribute of the decay Hamiltonian generates
discrete complex eigenvalues. The Decay Hamiltonian for neutrino is given
byDixit:2022izn ; Borsten:2008wd ; Berryman:2014yoa ; Chattopadhyay:2021eba

$\Gamma/2=\begin{pmatrix}b_{1}&\frac{1}{2}\eta e^{i\zeta}\\\ \frac{1}{2}\eta
e^{-i\zeta}&b_{2}\end{pmatrix}$ (10)

Incorporating the decay Hamiltonian with the standard Hamiltonian term, we get
the effective Hamiltonian $\mathcal{H}_{m}$ in mass basis,

$\begin{split}\mathcal{H}_{m}&=\mathcal{H}_{0}-i\Gamma/2\\\
&={\begin{pmatrix}a_{1}&0\\\
0&a_{2}\end{pmatrix}}-{\begin{pmatrix}ib_{1}&\frac{i}{2}\eta e^{i\zeta}\\\
\frac{i}{2}\eta e^{-i\zeta}&ib_{2}\end{pmatrix}}\end{split}$ (11)

$\mathcal{H}_{0}$ is the vacuum Hamiltonian, associated with the mass square
terms and energy. Considering a finite decay width of the single neutrino
system, a decay Hamiltonian is also associated with the vacuum Hamiltonian
given by the $\Gamma/2$ matrix. As mentioned in Dixit:2022izn  a bound of
$\tau_{\nu}\geq 5.7\times 10^{5}s(m_{\nu}/eV)$ is derived from the neutrino
data of Supernova 1987A. which leads to $\Gamma_{\nu}\equiv b\approx
10^{-21}eV$ for a neutrino of mass 1 eV. Also, in Soni:2023njf  suggested an
approximate value of $b$. Although the literature suggests a rough estimation
of the value of the decay parameter, it converges towards the ultrahigh-energy
neutrino from the astrophysical sources Dixit:2022izn . Here, $b_{1},b_{2}$
and $\eta$ are real parameters depicting the cause of the decay of mass
eigenstate. Since there is no established numerical value of decay parameters,
the motivation of the work is to visualize the decay geometrically and
following the defined value of the parameters and probability inDixit:2022izn
; Chattopadhyay:2021eba ; Soni:2023njf ; Shafaq:2021lju  plot the appearance
and disappearance probabilities in the presence of decay. For the presence
work, we restrict ourselves only to the invisible neutrino decay for which we
approximate $\xi=0$Shafaq:2021lju . Also, for analytical simplicity, we
consider $b_{1}=b_{2}=b$ Dixit:2022izn .
The claim is that the Hamiltonian of the system doesn’t change with time. The
decaying system makes the standard Hamiltonian of the system,
$\mathcal{H}_{0}$, non-commutative with the decay Hamiltonian, $\Gamma/2$, as
a non-hermitian off-diagonal element $\eta$ is associated with the decay
Hamiltonian. However, in spite of the anti-Hermitian nature of the diagonal
element $b$ of $\Gamma/2$, it commutes with $\mathcal{H}_{0}$. This is so
because, in the absence of the off-diagonal element, the decay basis matches
with the mass basis Hence, the information about the eigenstates of such a
decaying is not explicitly deterministic. Hence, the intuition is to find a
basis for this modified Hamiltonian for a system of decaying neutrino
oscillation and to interpret the system’s dynamics geometrically. We represent
the Hamiltonian for the decaying system of the neutrino in the mass basis as
$\mathcal{H}_{m}$, which comprise both $\mathcal{H}_{0}$ and $\Gamma/2$,
Figuratively,

$\begin{split}\mathcal{H}_{m}&=\begin{pmatrix}a_{1}&0\\\ 0&a_{2}\\\
\end{pmatrix}-\frac{i}{2}\begin{pmatrix}b&0\\\ 0&b\\\
\end{pmatrix}-\frac{i}{2}\begin{pmatrix}0&\eta/2\\\ \eta/2&0\\\
\end{pmatrix}\end{split}$ (12)

We redefine eq.(13) as
$\frac{(a_{1}+a_{2})}{2}\sigma_{0}+\frac{(a_{1}-a_{2})}{2}\sigma_{3}-ib\sigma_{0}-i\frac{\eta}{4}\sigma_{1}$,
Dixit:2022izn . Explitice mathematical form of this description is,

$\begin{split}\mathcal{H}_{m}&=\omega\sigma_{3}+(-ib)\sigma_{0}+(-i\eta)/2\sigma_{1}\end{split}$
(13)

The term, $\omega=\frac{\Delta m^{2}}{4E}$, incorporates the mass square
differences of the neutrinos, which is the most essential component for the
phenomenon of oscillation. We neglect here the term $\frac{(a_{1}+a_{2})}{2}$
in $\sigma_{0}$ as it has no explicit physical significance in flavor-changing
neutrino oscillation. This decomposition makes it possible to represent the
Hamiltonian structurally as the inner product of a vector with the Pauli
matrices. To address this affirmation, we introduce a vector $\mathcal{B}$,
which denotes the vector representing $\mathcal{H}_{m}.$ This Hamiltonian is
now analogous to the inner product of two vectors. Now define a vector
$\mathcal{B}$, depicting the Hamiltonian of the system. The above expression
is analogous to the inner product of the vectorized Hamiltonian, $\mathcal{B}$
and the Pauli matrices, which gives as follows,

$\mathcal{H}_{m}=\mathcal{B}\cdot\mathbf{\sigma}$ (14)

However, the ambiguity arises in defining the basis. Instead of relating
Cartesian coordinates to define the Pauli matrix, we use the mass eigenstates,
$\ket{\nu_{1}}$ and $\ket{\nu_{2}}$ to expound the Pauli matrices. This
characterization is feasible for the present framework as we consider mass and
flavor basis. In this context, we refurbish the Pauli matrices,

$\begin{split}\sigma_{x}&=\ket{\nu_{1}}\bra{\nu_{2}}+\ket{\nu_{2}}\bra{\nu_{1}}\\\
\sigma_{y}&=i\ket{\nu_{1}}\bra{\nu_{2}}-i\ket{\nu_{2}}\bra{\nu_{1}}\\\
\sigma_{z}&=\ket{\nu_{2}}\bra{\nu_{2}}-\ket{\nu_{1}}\bra{\nu_{1}}\\\
I_{2}&=\ket{\nu_{1}}\bra{\nu_{1}}+\ket{\nu_{2}}\bra{\nu_{2}}\\\ \end{split}$
(15)

Eq.(15) suggests the operator representation of the Pauli matrices.
Consequently, this definition motivates us to represent a new basis for the
Pauli matrix. However, two points need to be highlighted. Firstly, the
dimension of this space, i.e., $\ket{\nu_{i}}\bra{\nu_{j}}$ will differ from
the previously defined state space $\ket{\nu_{i}}$. Secondly, a vector is
represented by a column matrix, whereas a square matrix represents an
operator. When we modify the Hamiltonian operator formalism to vector
representation, the square matrix must also be converted to the column matrix.
This is possible using a bra-flipper operator $\mho$, Gyamfi_2020 ,

$\mho[\ket{\nu_{i}}\bra{\nu_{j}}]=\ket{\nu_{i}}\otimes\ket{\nu_{j}}^{*}\equiv|\nu_{i},\nu_{j}\rangle\rangle$

This allows us to define a linear space of matrices, converting the matrices
effectively into vectors Gyamfi_2020 ; Manzano:2020abc . But the
representation also has an extra complex factor $i$, making it difficult to
interpret the $\mathcal{B}$ vectorially. Here, we initiate the concept of
quaternion Wharton:2014fdg . The quaternion is the three-dimensional
projection of a four-dimensional complex hyperspace, similar to stereographic
projection in complex analysis Rotelli:1988fc . It is noteworthy that Pauli
matrices form the quaternion groups Shirokov_2018oiu . Hence, it is physically
more relevant to use the Pauli matrices as the basis for this vectorization of
the operator, which inherently carries the complex factor within. Observing
such fact in Pauli basis , the complex $i$ factor is nothing but a phase
factor $e^{-i\pi/2}$ which denotes the rotation of the axis Bohm_1986uoi .
This implies that the inner product with Pauli matrices leads to the
vectorization of the operator. Following this, the Hamiltonian modifies to,

$\mathbf{B}=\omega\hat{\sigma}_{3}+(-ib)\sigma_{0}+e^{-i\pi/2}\eta\hat{\sigma}_{1}$
(16)

$\hat{\sigma}$ is the correspond $\mathbf{B}$ denotes the vectorized
Hamiltonian in mass basis. Therefore, moving it in the flavor basis changes
the basis vectors, i.e., the Pauli operator. The rotational transformation of
the Pauli basis which is
$\sigma_{0}^{M}=\sigma_{0}^{F}$,$\sigma_{1}^{M}=\sin{2\theta}\sigma_{3}^{F}+\cos{2\theta}\sigma_{1}^{F}$,
$\sigma_{2}^{M}=\sigma_{2}^{F}$,
$\sigma_{3}^{M}=\cos{2\theta}\sigma_{3}^{F}-\sin{2\theta}\sigma_{1}^{F}$.
Defining this, we next move to analyze the flavor state and its evolution on
this new basis.
Now consider the neutrino is created in the flavor eigen state as electron
neutrino in Hilbert space $\mathcal{H}_{1}$. We have already mentioned that
each flavor eigenstate of neutrino does not have any definitive eigenstate.
Instead, it is the linear superposition of the two different mass eigenstates.
The dynamical propagation of the mass basis acquires an amount of flavor basis
during propagation, due to which a certain amount of mass also gets mixed
associated with each flavor state. More details can be followed from
smirnov2019mikheyevsmirnovwolfenstein ; Smirnov:2003da . As we have already
mentioned, each flavor state of neutrino can be thought of as a two-level
system consisting of two orthonormal mass eigenstates, $\ket{\nu_{1}}=\ket{0}$
and $\ket{\nu_{2}}=\ket{1}$, defined over the Pauli bases. For the present
discussion on the two-flavor oscillation, we restrict the superposition to the
two mass eigenstates only, i.e.,
$\ket{\nu_{\alpha}}=\cos{\theta}\ket{0}+\sin{\theta}\ket{1}$. Hence, each
flavor state of neutrino is an entangled state of the mass eigenstate
KumarJha:2020pke ; Blasone:2019chv . From the basic postulates of the quantum
mechanics corresponding to every system’s physical state, a state vector is
associated with it. The state vector represents the pure states, which specify
all the known physical information about the system. This is an ideal case. In
most cases, the system is not in a pure state. Most generally, we may know
that a quantum system can be in one state of a set $\\{\ket{\nu_{\alpha}}\\}$
with probabilities $w_{i}$. In this case, the mathematical tool that describes
our knowledge of the system is the density operator (or density matrix). Due
to the oscillatory property of the neutrinos, instead of using a state vector
for the flavor state, we use the density operator $\hat{\rho}_{\alpha}$ where
$\hat{\rho}_{\alpha}\in\mathcal{H}_{1}$ Shafaq:2021lju . Consequently, instead
of investigating the dynamics of single state vector we study the evolution
equation the density operator $\rho_{\alpha\alpha}=\ket{\nu_{\alpha}}$, we use
$\ket{\nu_{\alpha}}\bra{\nu_{\alpha}}$ for pure states and
$\rho_{\alpha\beta}=\ket{\nu_{\alpha}}\bra{\nu_{\beta}}$ for the mixed state,
that depicts an ensemble of bi-flavored neutrino system. The state
$\ket{\nu_{\alpha}}$ is from Hilbert space $\mathcal{H}_{1}$ and the state
$\ket{\nu_{\beta}}$ is from Hilbert space $\mathcal{H}_{2}$. The dynamics of
the combined system $\rho=\ket{\nu_{\alpha}}\bra{\nu_{\beta}}$ is defined over
the Hilbert space $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$Blasone:2019chv
which has the dimension of $\mathcal{H}_{2}\otimes\mathcal{H}_{2}$. The
manifestation of this idea makes our choice basis more tangible. The density
operator $\ket{\nu_{e}}\bra{\nu_{e}}$ for pure electron neutrino state is,

$\begin{split}\ket{\nu_{e}}\bra{\nu_{e}}=\rho_{ee}^{m}&=\cos^{2}{\theta}\ket{\nu_{1}}\bra{\nu_{1}}+\sin{\theta}\cos{\theta}\ket{\nu_{1}}\bra{\nu_{2}}\\\
&+\sin{\theta}\cos{\theta}\ket{\nu_{2}}\bra{\nu_{1}}+\sin^{2}{\theta}\ket{\nu_{2}}\bra{\nu_{2}}\\\
&=\cos^{2}{\theta}\ket{00}+\cos{\theta}\sin{\theta}\ket{01}\\\
&+\cos{\theta}\sin{\theta}\ket{10}+\sin^{2}{\theta}\ket{11}\end{split}$ (17)

While propagation, the flavor state is mixed with the mass eigenstate, and
since each flavor state consists of two mass eigenstates, the purity of the
system is not conserved, and the neutrino oscillates from one flavor state to
the other. Redefined mass eigen state consists of
$\ket{00}=\ket{0}_{s}\otimes\ket{0}_{e}$ ,
$\ket{11}=\ket{1}_{s}\otimes\ket{1}_{e}$
,$\ket{11}=\ket{0}_{s}\otimes\ket{1}_{e}$ and
$\ket{10}=\ket{1}_{s}\otimes\ket{0}_{e}$. From eq.(17), the density operator
denoting the flavor state,
$\rho_{ee}=\begin{pmatrix}\cos^{2}{\theta}&\cos{\theta}\sin{\theta}\\\
\cos{\theta}\sin{\theta}&\sin^{2}{\theta}\\\ \end{pmatrix}$. We exploit the
fact that $\mathcal{R}^{2}\otimes\mathcal{R}^{2}$ is isomorphic to
$\mathcal{R}^{4}$ to vectorise the density operator. The objective is to
vectorize the Hamiltonian and density operator over a common basis to study
the dynamics pictorially. This will give us a comparative understanding of the
neutrino evolution and the precision of an NMR in an external magnetic field.
To express the vectorized form of the density operator, we introduce a
momentum-like vector $\vec{F}$ such that
$F_{i}=\mbox{Tr}\big{[}\frac{1}{2}\sigma_{i}\rho\big{]}$, where $i$ stands for
the 1, 2, and 3 components.

$\begin{split}F_{1}&=\mbox{Tr}\bigg{[}\frac{1}{2}\sigma_{1}\rho\bigg{]}\\\
&=\mbox{Tr}\bigg{[}\frac{1}{2}\begin{pmatrix}0&1\\\ 1&0\\\
\end{pmatrix}\begin{pmatrix}\rho_{11}&\rho_{12}\\\ \rho_{21}&\rho_{22}\\\
\end{pmatrix}\bigg{]}\\\ &=\frac{1}{2}(\rho_{12}+\rho_{21})\end{split}$ (18)

Similarly $F_{2}=\frac{i}{2}(\rho_{12}-\rho_{21})$ and
$F_{3}=\frac{1}{2}(\rho_{11}-\rho_{22})$. Hence, the density matrix is
constructed as,

$\rho_{ij}=\begin{pmatrix}F_{3}&F_{1}-iF_{2}\\\ F_{1}+iF_{2}&-F_{3}\\\
\end{pmatrix}$ (19)

$\rho_{ij}$ stands for the components of the density operator in Pauli bases.
Hence, $\mathbf{F}$ signifies a vector that stands for the flavor state of the
system that aligns toward the flavor basis. We consider the neutrinos created
a pure electron flavor state. The two flavor states corresponding to the bi-
flavored neutrino oscillation are orthogonal to each other. Accordingly, we
take the $\nu_{e}$ state along $\sigma_{3}^{F}$ and $\nu_{\mu}$ along
$-\sigma_{3}^{F}$. In flavor basis at $t=0$, $F_{1}^{F}=0$, $F_{2}^{F}=0$ and
$F_{3}^{F}=1$. We move $\mathbf{F}$ from flavor to mass basis to study the
evolution equation to avoid mathematical ambiguity. However, time evolved
$F^{F}(t)$ will be applicable for understanding the scenario at the
probability level. Constructing so, we move to evaluate the evolution of the
density operator.
Now we have two vectors $\mathbf{F}$ and $\mathbf{B}$ representing the two
operators, density operator $\rho$ and the total Hamiltonian including decay
$\mathcal{H}_{m}$ respectively, defined over the Pauli basis. The Hamiltonian
vector $\mathbf{B}$ has a Hermitian component, $\mathbf{B}_{h}$, which is
associated with mass square differences and a non-hermitian component,
$\mathbf{B}_{nh}$, which incorporates the decay component of $\mathbf{B}$.
Although we initiated the study of the Schrodinger equation, which states the
dynamics of the flavor state, as soon as we introduced the concept of density
operator, we moved from the Schrodinger picture to Heisenberg’s picture, where
the operator has the dynamical evolution instead of a state vector.

$i\frac{d}{dt}\rho\big{(}t\big{)}=\bigg{(}\mathbf{B}_{h}\rho(t)-\rho(t)\mathbf{B}_{h}\bigg{)}+\bigg{(}\mathbf{B}_{nh}\rho(t)-\rho(t)\mathbf{B}_{nh}^{\dagger}\bigg{)}$
(20)

For the non-hermitian nature of the decay Hamiltonian
$\mathbf{B}_{nh}\neq\mathbf{B}_{nh}^{\dagger}$. However, discrepancies arise
as the Hamiltonian is vectored but not the density operator. To rectify this
inconsistency, we convert the density operator $\rho$ to the vectorized form
mentioned previously by the vector $\vec{F}$. Using the form of $\rho$ from
eq.(19) the form of LHS of eq.(20) modified to,

$\begin{split}\frac{d}{dt}\rho&=\begin{pmatrix}0&-2i\omega(F_{1}-iF_{2})\\\
2i\omega(F_{1}+iF_{2})&0\\\ \end{pmatrix}\\\
&-2b\begin{pmatrix}F_{3}&F_{1}-iF_{2}\\\ F_{1}+iF_{2}&-F_{3}\\\
\end{pmatrix}\\\ &+\begin{pmatrix}-\eta F_{1}&0\\\ 0&-\eta F_{1}\\\
\end{pmatrix}\\\ \end{split}$ (21)

The RHS of the above equation changes its form following the form
$\frac{d}{dt}F_{i}=Tr\frac{1}{2}\sigma_{i}\frac{d}{dt}\rho$. The component
wise evolution equation of $\mathbf{F}$ is thus,

$\begin{split}\dot{F}_{1}&=-\omega F_{2}-bF_{1}\\\ \dot{F}_{y}&=\omega
F_{1}-bF_{2}\\\ \dot{F}_{2}&=-2bF_{3}\end{split}$ (22)

In matrix form,

$\begin{pmatrix}\dot{F}_{1}\\\ \dot{F}_{2}\\\ \dot{F}_{3}\\\
\end{pmatrix}=\begin{pmatrix}-b&-\omega&0\\\ \omega&-b&0\\\ 0&0&-2b\\\
\end{pmatrix}\begin{pmatrix}F_{1}\\\ F_{2}\\\ F_{3}\\\ \end{pmatrix}$ (23)

it is noteworthy that although we have vectorized the density operator as
$\mathbf{F}$, we do not mention explicitly on which basis it is defined. In
general, $\mathbf{F}$ illustrates the density operator of the flavor state of
neutrino. We can denote flavor part of the $\mathbf{F}$ as
$\mathbf{F}_{i}^{F}$ while in mass part $\mathbf{F}_{i}^{M}$. As we consider
neutrino is created as the pure electron flavor, hence at $t=0$,
$\mathbf{F}_{1}^{F}=0$,$\mathbf{F}_{2}^{F}=0$, $\mathbf{F}_{3}^{F}=1$. In mass
basis at $t=0$, $\mathbf{F}_{1}^{M}=\sin{2\theta}$,$\mathbf{F}_{2}^{M}=0$,
$\mathbf{F}_{3}^{M}=\cos{2\theta}$. On that account eq.(23) donates the
evolution equation of $\mathbf{F}^{M}$ in order to find time evolved
$\mathbf{F}^{M}$ at $t=0$ Giunti:2007ry . Rewriting eq.(23),

$\begin{pmatrix}\dot{F}_{1}^{M}\\\ \dot{F}_{2}^{M}\\\ \dot{F}_{3}^{M}\\\
\end{pmatrix}=\begin{pmatrix}-b&-\omega&0\\\ \omega&-b&0\\\ 0&0&-2b\\\
\end{pmatrix}\begin{pmatrix}F_{1}^{M}\\\ F_{2}^{M}\\\ F_{3}^{M}\\\
\end{pmatrix}$ (24)

Recall we have mentioned earlier that $\omega=\frac{\Delta m^{2}}{4E}$.
Accordingly, $2\omega=\frac{\Delta m^{2}}{2E}$, which defines the standard
oscillation frequency of the flavor oscillation of neutrinos. Thus, we redraft
$2\omega$ as $\omega_{osc}$ for mathematical simplification. It is to be noted
that the time evolution of the density operator is not independent of $\eta$,
i.e., the off-diagonal part of the decay Hamiltonian, which is also referred
to in eq.(21). However, the time evolution of $\mathbf{F}$, i.e., vector
depicting the flavor state, is independent of the off-diagonal element $\eta$.
Choosing the Pauli basis makes the evolution equation independent of the off-
diagonal element.

##  III Results and Discussions

In this section, we discuss the evolution of $\mathbf{F}^{M}$ analytically and
illustrate the result geometrically. We interpret the dynamical equation as a
second-order differential equation, an easier way to understand the
oscillation pictorially. We scrutinize two different scenarios. Firstly, we
consider the damping parameters zero and give the geometrical representation
figuratively. Next, we include the decay parameters to reduce the differential
equation of neutrino oscillation to the modified form and represent it
geometrically.

  * •

Scenario 1: We first consider the standard neutrino oscillation without
including decay parameters. For the standard neutrino oscillation, we set
$\eta=0,b=0$. Hence, from the eq.(24), we can observe that the evolution
equation of the vector describing the density matrix $\mathbf{F}$ becomes
$\dot{F}_{1}^{M}=-\omega_{osc}F^{M}_{2}$,
$\dot{F}_{2}^{M}=\omega_{osc}F^{M}_{1}$ and $\dot{F}_{3}^{M}=0$. Individually
$F_{1}^{M}$ and $F_{2}^{M}$ satisfy the equation of simple harmonic
oscillator. Hence, in compact form

$\ddot{F}_{1,2}^{M}=-\omega_{osc}^{2}F_{1,2}^{M}$ (25)

However, to have a more rigorous mathematical picture, it will be convenient
to express them in terms of $\mathbf{B}$ and $\mathbf{F}$, although the in
eq.(25) $\omega_{osc}$ is nothing but the coefficient of $\mathbf{B}$ itself.
Eq.(24) can be expressed as,
$\dot{\mathbf{F}}_{1}^{M}=-\omega_{osc}F_{2}^{M}\hat{\sigma}_{1}$ and
$\dot{\mathbf{F}}_{2}^{M}=\omega_{osc}F_{1}^{M}\hat{\sigma_{2}}$, which is
contracted to the form,

$\begin{split}\dot{\mathbf{F}}&=-\omega_{osc}F_{2}^{M}\hat{\sigma_{1}}+\omega_{osc}F_{1}^{M}\hat{\sigma}_{2}\\\
&=\bigg{(}\omega_{osc}\hat{\sigma}_{3}\times
F^{M}_{2}\hat{\sigma}_{2}\bigg{)}+\bigg{(}\omega_{osc}\hat{\sigma}_{3}\times
F_{1}\hat{\sigma}_{1}\bigg{)}\\\ &=\mathbf{B}\times\mathbf{F}\\\ \end{split}$
(26)

Figure 1: Geometrical representation of basis in flavor basis and mass basis
in the presence of decay(left-panel) and geometrical representation of
standard neutrino oscillation without decay(right-panel). Here we follow the
Pauli basis, i.e., $\hat{\sigma}_{1}$, $\hat{\sigma}_{2}$, $\hat{\sigma}_{3}$
as the basis vector by which we define the coordinate axes. Plot legends in
the box define the color representation of different components of the
illustration. As the diagonal terms of the decay Hamiltonian match with the
mass bases, it acts as a displacement element to the $\hat{\sigma}_{3}$
component of the mass basis. The Hamiltonian depicting vector
$\vec{\mathbf{B}}$ is along $\hat{\sigma}_{3}$. The $\sigma_{3}$ direction
denotes the $\nu_{1}$ mass eigenstate and $-\sigma_{3}$ directs toward the
$\nu_{2}$. On the other hand, in the flavor basis, the two orthogonal flavor
states $\nu_{e}$ and $\nu_{\mu}$ align along positive and negative
$\sigma_{3}$ directions, respectively, as illustrated in the left figure. The
figure in the right panel shows that in the absence of decay, the flavor state
representing vector $\vec{\mathbf{F}}$ rotates around the Hamiltonian vector
$\vec{\mathbf{B}}$ following in the equation
$\dot{\mathbf{F}}=\mathbf{B}\times\mathbf{F}$. The initial flavor state is of
electron neutrino; hence, $\mathbf{F}$ is along $+\hat{\sigma}_{3}$. After a
finite time $t$, $\mathbf{F}$ is along $\mathbf{F}(t)$, which implies the
mixed state. After one complete revolution, it comes back to the initial pure
state. In the relativistic limit $t\equiv L$, the angular velocity of the
rotation is equivalent to the oscillation length $L_{osc}$, shown in the
adjacent figure on the right.

This leads to the standard form of the Bloch equation with oscillation
frequency $\omega_{osc}=\Delta m^{2}/2E$. Also, eq.(26) implies the
gyromagnetic ratio is 1 for this precessional motion of neutrinos. The diagram
in the right panel of fig.1 figuratively refers to this precessional motion.
At time $t=0$, the initial electron flavor state representing vector
$\mathbf{F}$(green line) is aligned along the $\sigma_{3}^{F}$ and is denoted
by $F(0)$. After a certain time $t$, the $\mathbf{F}$ will rotate to a
position and be aligned to the position $F(t)$. Considering the ultra-
relativistic limit for the neutrinos $t$ can be substituted with the
oscillation length. After the completion of one Oscillation length,
$\mathbf{F}$ will acquire the state $F(0)$. In the absence of decay
Hamiltonian, this circular motion of $\mathbf{F}$ around $\mathbf{B}$(yellow
line) will remain unaffected. Flavor and the mass basis are mentioned in the
plot legend. Bloch equation represents the evolution equation of the spin-
angular moment vector or magnetic moment vector along a direction of the
externally applied magnetic field. Analogically, it can be inferred that the
Hamiltonian in the mass basis acts as a magnetic field, and the flavor state
revolves around it with the precessional frequency $\omega$.

The applied magnetic field acts as a perturbation to the system, which exerts
a torque in the equilibrium condition of the system, driving the system to
rotate around. A similar phenomenon occurs for the system of neutrino as well,
where the Hamiltonian, which contains the squared difference term, illustrates
the linear combination of the two different masses; one is the system mass,
and the other is the mass acquired from the vacuum while propagation.

Figure 2: Plot of appearance probability $P_{ee}$(red line), disappearance
probability $P_{e\mu}$(black line) and the total probability $P$(Blue-dashed
line). The plot in the right panel shows the probabilities with the variation
of $L(km)/E(GeV)$ in vacuum. In the absence of decay, the two-flavor state
acts as a two-level system with a constant oscillation amplitude, which keeps
oscillating without deviation. Meanwhile, the figure in the right panel shows
the probabilities in the presence of decay. Graphically it describes that the
decay Hamiltonian which coincides with the mass basis is responsible for the
amplitude damping of probabilities according to the expression
$P_{ee}^{d}=e^{-2bL}P_{ee}$, $P_{e\mu}^{d}=e^{-2bL}P_{e\mu}$,
$P^{d}=e^{-2bL}$. The plot legends mention the figure indicates the
appearance, disappearance, and total probability variation. The figure
manifests the fact that with the increasing length $L(km)$, the amplitude of a
$P_{ee}^{d}$, $P_{e\mu}^{d}$ decreases in an oscillatory manner while the
total probability $P^{d}$ decreases exponentially, with $E=1GeV$. This figure
also conveys the graphical layout of under-damped oscillation when the decay
parameter magnitude of the decay parameter $(b)$ is much less than angular
velocity $\omega L$. The value of the parameter $b$ is $10^{-21}eV$ for
considering $1eV=\Delta m^{2}$Dixit:2022izn

Usually, this phenomenon cannot be seen when a particle has a definite mass. A
single flavor state consisting of a single mass eigenstate may not possibly
execute the oscillation. This adds the unique dynamical feature of eq.(26),
making it distinct from any other fermions.
$\ket{\nu_{\alpha}}=c_{1}\ket{\nu_{1}}+c_{2}\ket{\nu_{2}}$. This torque
appears due to the superposition of two mass eigenstates. In addition to this,
the existence of different flavor states makes the phenomenon more prominent.
Illustratively, it is described by Fig.1. We have considered that neutrino is
created as a pure electron flavor state. In flavor basis, the pure electron
flavor state is lined up along $\sigma^{f}_{3}$, and the muon flavor state is
along $-\sigma^{f}_{3}$. Considering the transformation of the basis vectors,
the mass and the flavor basis are inclined by an angle $2\theta$. This is
known as the mixing angle of neutrino. eq.(16) suggests that in mass basis
$\mathbf{B}$ is aligned along $\sigma^{M}_{3}$ with the states $\nu_{1}$ in
$\sigma^{M}_{3}$ and $\nu_{1}$ in $-\sigma^{M}_{3}$. Imposing the initial
condition at $t=0$, i.e., in flavor basis
$\mathbf{F}_{1}^{F}=0$,$\mathbf{F}_{2}^{F}=0$, $\mathbf{F}_{3}^{F}=1$ and in
mass basis, $\mathbf{F}_{1}^{M}=\sin{2\theta}$,$\mathbf{F}_{2}^{M}=0$,
$\mathbf{F}_{3}^{M}=\cos{2\theta}$, the solution of eq.(26) is reduced to,

$\begin{split}F_{1}^{M}(t)&=F_{1}^{M}(0)\cos{\omega_{osc}t}+F_{2}^{M}(0)\sin{\omega_{osc}t}\\\
F_{2}^{M}(t)&=F_{1}^{M}(0)\cos{\omega_{osc}t}-F_{2}^{M}(0)\sin{\omega_{osc}t}\\\
F_{3}^{M}(t)&=F_{3}^{M}(0)\\\ \end{split}$ (27)

Which follows, $F^{M}_{1}=-\sin{2\theta}\cos{\omega_{osc}t}$,
$F^{M}_{2}=-\sin{2\theta}\sin{\omega_{osc}t}$, $F^{M}_{3}=\cos{2\theta}$. So
far, we are in the mass basis where the vector defining the density matrix is
also defined over the mass basis. This refers to the solution of the evolution
equation of $\mathbf{F}$ around $\mathbf{B}$ in mass basis. To understand how
the dynamics affect the probability level $\mathbf{F}^{M}(t)$ has to be
transformed into the flavor basis $\mathbf{F}^{F}(t)$ and
$\mathbf{F}^{F}_{3}(t)$ will give the probability of obtaining the initial
flavor $\nu_{e}$ after a time t.

Figure 3: Figures illustrate the geometrical representation of neutrino
oscillation in the presence of decay for critically damped case(left panel)
and under damped case(right panel). For the under-damped case, the oscillation
amplitude damped down in an oscillatory way. Consequently, there is
oscillation with decreasing amplitude, which directs towards the fact that
after one full revolution, it is impossible to get back to the initial flavor
state as the presence of decay parameter $b$, which causes the flavor state
depicting vector $\mathbf{F}$ execute a helical motion. Moreover, this helical
motion doesn’t confine to a plane because this decay parameter affects the
$\hat{\sigma}_{3}$ components as well, which makes $\mathbf{F}$ drag toward
$\hat{\sigma}_{3}^{md}$ which the mass basis in the presence of decay. In
addition, with the gradual decrement of the pure state, the flavor state dies
down, demonstrating that it ultimately dissipates to the mass state and
$\mathbf{F}$ aligns itself towards $\mathbf{B}$. In a critically damped case,
the magnitude of the damping parameter is equal to the natural frequency of
the oscillation. Due to this equivalence, the $\mathbf{F}$ vector is set off
for rotational motion, although it can not complete one full rotation and
decays to the mass eigenstate. Figure 4: Figures illustrate the geometrical
representation of neutrino oscillation in the presence of decay for the over-
damped scenario. The oscillation amplitude damped down in an oscillatory way
in the presence of decay. Consequently, there is oscillation with decreasing
amplitude, which directs towards the fact that after one full revolution, it
is impossible to get back to the initial flavor state as the presence of decay
parameter $b$, which causes the flavor state depicting vector $\mathbf{F}$
execute a helical motion. Moreover, this helical motion doesn’t confine to a
plane because this decay parameter affects the $\hat{\sigma}_{3}$ components
as well, which makes $\mathbf{F}$ drag toward $\hat{\sigma}_{3}^{md}$ which
the mass basis in the presence of decay. In addition, with the gradual
decrement of the pure state, the flavor state dies down, demonstrating that it
ultimately dissipates to the mass state and $\mathbf{F}$ aligns itself towards
$\mathbf{B}$. In a critically damped case, the magnitude of the damping
parameter is equal to the natural frequency of the oscillation. Due to this
equivalence, the $\mathbf{F}$ vector is set off for rotational motion,
although it can not complete one full rotation and decays to the mass
eigenstate. The scenario of decay of flavor state to the mass eigenstate is
much more dominant for the over-damped case where $\mathbf{F}$ directly
dissipates to $\mathbf{B}$ without executing any rotational motion as shown by
the figure in the left panel.

Following the transformation,
$\hat{\sigma}^{F}_{3}=\cos{2\theta}\hat{\sigma}_{3}^{M}-\sin{2\theta}\hat{\sigma}_{1}^{M}$,
of the basis vectors, the $\vec{F}$ can be transformed into the flavor basis,
i.e., $F_{i}^{F}(t)$. We are only interested in the $\sigma_{3}$ component of
the $\vec{F}$ because the initial flavor state is along the $\hat{\sigma_{3}}$
component. The time-evolved component of $F_{z}$ provides the flavor state,
which eventually gives the appearance probability of the particular flavor
state, i.e., $P_{\alpha}=F_{3}^{M}(t)=1-\sin^{2}{2\theta}\sin^{2}{\frac{\Delta
m^{2}L}{4E}}$. The plot at the left of Fig.2 depicts the same behavior of
standard oscillation. The appearance (black line) and disappearance (red line)
probabilities are plotted with respect to the $L(km)/E(GeV)$ ratio. This
figure points to the un-deviated oscillatory nature of the precessional motion
of neutrinos, mentioned in Fig.1 which also implies the same picturization of
any standard two-level system Manzano:2020abc . It also predicts after which
length we should get back the same flavor. Intuitively, this refers to the
fact that the flavor state of the neutrino acts as a magnetic moment vector
rotating along the magnetic field where the Hamiltonian depicts the magnetic
field.

  * •

Scenario 2: $\eta\neq 0,b\neq 0$. In the next scenario, we analyze considering
both the decay parameters to be non-zero of the decay Hamiltonian, which
effectively modifies the evolution equation eq.(22) to,

$\begin{split}\ddot{F}_{1}^{M}&=\omega_{osc}^{2}F_{1}^{M}-2b\dot{F_{1}}^{M}\\\
\ddot{F}_{2}^{M}&=\omega_{osc}^{2}F_{2}^{M}-2b\dot{F_{2}}^{M}\\\
\dot{F}_{3}^{3}&=-2bF_{3}^{M}\end{split}$ (28)

Analogically, the above equation is the equation of motion of a damped
harmonic oscillator. Physically, one such instance can be noticed in NMR. In
the case of NMR, the $2b$ factor acts as the relaxation time $1/T$. It is
noted from eq.(28) that the $\eta$ factor doesn’t play a role here. If the
off-diagonal term is hermitian, it contributes to the evolution equation,
which can be noted following Am-Shallem_2015 . For the invisible neutrino
decay, the Bloch equation is independent of the diagonal terms of the
perturbation Hamiltonian. Moreover, the contribution from the $\sigma_{3}$
component makes the equation different from the standard oscillation, where no
contribution comes from the $\sigma_{3}$ component. To visualize the dynamics
geometrically, let us look through the differential Bloch equation form of
eq.(27).

$\frac{d}{dt}\mathbf{F}=\mathbf{B}\times\mathbf{F}-2b\mathbf{F}$ (29)

Eq.(28) implies that due to the presence of the decay factor, the circular
motion is converted to a spiral motion coiling into the mass basis of the
Hamiltonian. Moreover, due to the occurrence of damping along the
$\sigma_{3}^{M}$-axis $\mathbf{B}$ drag $\mathbf{F}$, which enforced
$\mathbf{F}$ to execute a helical motion in the $\sigma_{3}$ direction with
gradually decreasing screw pitch. We must quantify parameter b with an
approximated value mentioned in Dixit:2022izn ; Soni:2023njf  for a complete
pictorial overview. For consistency, $b$ has to be the dimension of
energy(eV). Next, we derive the oscillation probability using eq.(28), the
solution of which,

$\begin{split}F_{1}^{M}(t)&=e^{-2bt}F_{1}^{M}(0)\cos{\omega_{osc}t}\\\
&+e^{-2bt}F_{2}^{M}(0)\sin{\omega_{osc}t}\\\
F_{2}^{M}(t)&=e^{-2bt}F_{1}^{M}(0)\cos{\omega_{osc}t}\\\
&-e^{-2bt}F_{2}^{M}(0)\sin{\omega_{osc}t}\\\
F_{3}^{M}(t)&=e^{-2bt}F_{3}^{M}(0)\\\ \end{split}$ (30)

Here, we impose the same boundary condition as before,i.e.,
$F_{1}^{M}(0)=-\sin{2\theta}$, $F_{2}^{M}(0)=0$, $F_{2}^{M}(0)=\cos{2\theta}$.
Transforming the basis vector mass to flavor basis following
$\hat{\sigma}^{F}_{3}=\cos{2\theta}\hat{\sigma}_{3}^{M}-\sin{2\theta}\hat{\sigma}_{1}^{M}$,
furnish the expression of $F_{3}^{F}(t)$ which is the probability of detecting
flavour state $\nu_{e}$. Explicit mathematics follows,

$F_{3}^{F}(t)=\cos{2\theta}F^{M}_{3}-\sin{2\theta}F^{M}_{1}$ (31)

This produces the oscillation probability in the presence of decay Hamiltonian

$F_{3}^{F}(t)=e^{-2bt}\bigg{[}1-\sin^{2}{2\theta}\sin^{2}\bigg{(}{\frac{\omega_{osc}t}{2}}\bigg{)}\bigg{]}$
(32)

We have approximated $\omega_{osc}>>b$, so the oscillation frequency remains
unchanged. Nevertheless, the inclusion of $b$ modifies the oscillation to
$\sqrt{\omega_{osc}^{2}+b^{2}}$. The expression refers to the phenomenon of
amplitude damping of neutrino oscillation apart from imposing a perturbation
to the mass basis. The plot at the right panel of Fig.2 implies the same fact,
while the figure in the right panel of Fig.3 implements the same factuality
diagrammatically.
A parallel comparison can be drawn with the NMR system. The nuclear spin
system needs $T_{1}$ to reach the thermodynamic equilibrium in the presence of
the applied transverse external magnetic field. This results in exponential
damping. Moreover, the relaxation time depends on the material of the medium
and the direction of the applied field Kim2008PulsedN . The Bloch matrix for
the NMR system can be followed from Am-Shallem_2015 , which shows.

$\begin{pmatrix}\dot{S}_{x}\\\ \dot{S}_{y}\\\ \dot{S}_{z}\\\
\end{pmatrix}=\begin{pmatrix}-\frac{1}{T_{1}}&\Delta&0\\\
-\Delta&-\frac{1}{T_{1}}&\epsilon\\\ 0&-\epsilon&-\frac{1}{T_{2}}\\\
\end{pmatrix}\begin{pmatrix}S_{x}\\\ S_{y}\\\ S_{z}\\\
\end{pmatrix}+\begin{pmatrix}0\\\ 0\\\ S_{z}^{0}\\\ \end{pmatrix}$ (33)

Dependence of the relaxation time over the direction of the applied magnetic
field introduces the two relaxation times $T_{1}$ and $T_{2}$. Equivalently,
the system of the decay parameter defines the lifetime of the flavor state
Dixit:2022izn ; Chattopadhyay:2021eba  that follows a straightforward analytic
definition of the decay parameter, which is subjected to a dependence on the
mass of the neutrino $m_{\nu}$ and lifetime of neutrino $\tau_{\nu}$. Since no
definite magnetic field applies to the neutrino system, only one damping
factor $b$ exists.

  * •

Scenario 3: $\eta\neq 0,b\neq 0,\xi\neq 0$. This is the most general scenario
where we do not approximate the $\xi$ to be zero for the invisible neutrino
decay. Inclusion of $\xi$ term in the off-diagonal part of the decay
Hamiltonian further bifurcates $\mathbf{B}_{nh}$ in the following way,

$\mathbf{B}_{nh}=\frac{-i}{2}(\sigma_{0}b)-\frac{-i}{4}\cos{\xi}\hat{\sigma}_{1}+\frac{i}{4}\sin{\xi}\hat{\sigma}_{2}$
(34)

In our present choice of basis, again, the $\hat{\sigma}_{1}$ component will
be decoupled from the dynamics of $\dot{F}^{M}$. However, a contribution
remains from the $\hat{\sigma}_{2}$ component. From the commutation relation
$\frac{d}{dt}\rho=-\frac{\sin{\xi}}{2i}\begin{pmatrix}-F_{1}&F_{3}\\\
F_{3}&F_{1}\\\ \end{pmatrix}$ which modifies the the evolution equation of
$\mathbf{F}^{M}$ to,

$\begin{split}\dot{F}_{1}&=-\omega
F_{2}-bF_{1}+\eta\frac{\sin{\xi}}{2i}F_{3}\\\ \ \dot{F}_{2}&=\omega
F_{1}-bF_{2}\\\
\dot{F}_{3}&=-2bF_{3}-\eta\frac{\sin{\xi}}{2i}F_{1}\end{split}$ (35)

In more compact form, the matrix representation of eq.(35) is written as,

$\begin{pmatrix}\dot{F}_{1}^{M}\\\ \dot{F}_{2}^{M}\\\ \dot{F}_{3}^{M}\\\
\end{pmatrix}=\begin{pmatrix}-b&-\omega&\eta\frac{\sin{\xi}}{2i}\\\
\omega&-b&0\\\ -\eta\frac{\sin{\xi}}{2i}&0&-2b\\\
\end{pmatrix}\begin{pmatrix}F_{1}^{M}\\\ F_{2}^{M}\\\ F_{3}^{M}\\\
\end{pmatrix}$ (36)

The above equation indicates that the inclusion of the $\xi$ term can not
decouple the $\eta$ term from the derivation, which is a clear-cut indication
of the presence of $CP$ phase even in the absence of the Majorana phase
$\phi$. The effect of the factor $\xi$ in a more explicit way can be shown in
the following way.

$\begin{split}\dot{\mathbf{F}}&=\bigg{(}\omega_{osc}\hat{\sigma}_{3}\times
F_{2}^{M}\hat{\sigma}_{2}\bigg{)}+\bigg{(}\omega_{osc}\hat{\sigma}_{3}\times
F_{1}^{M}\hat{\sigma}_{1}\bigg{)}\\\
&-2b\mathbf{F}+\bigg{(}e^{-i\pi/2}\frac{\eta}{2}\sin{\xi}\hat{\sigma}_{2}\times
F_{3}^{M}\hat{\sigma}_{3}\\\
&+e^{-i\pi/2}\frac{\eta}{2}\sin{\xi}\hat{\sigma}_{2}\times
F_{1}^{M}\hat{\sigma}_{1}\bigg{)}\\\
&=\mathbf{B}\times\mathbf{F}-2b\mathbf{F}+e^{-i\pi/2}(\mathbf{E}\times\mathbf{F})\end{split}$
(37)

Where we have denoted
$\mathbf{E}=\frac{\eta}{2}\sin{\xi}\hat{\sigma}_{2}+\frac{\eta}{2}\sin{\xi}\hat{\sigma}_{2}$.
This above equation depicts that in the presence of $\xi$, $\mathbf{F}$ not
only tends to execute circular motion around $\mathbf{B}$, but also it does
evolve around $\mathbf{E}$. This modifies the evolution equation

$\begin{split}\ddot{F}_{1}^{M}+2b\dot{F}_{1}^{M}+\omega_{1}^{2}F_{1}^{M}&=0\\\
\ddot{F}_{2}^{M}+2b\dot{F}_{2}^{M}+\omega_{osc}^{2}F_{2}^{M}&=0\\\
\ddot{F}_{3}^{M}+2b\dot{F}_{3}+\eta\frac{\sin{\xi}}{2i}\dot{F}_{1}^{M}&=0\end{split}$
(38)

Where,
$\omega_{1}=\sqrt{\omega_{osc}^{2}+b^{2}-\bigg{(}\frac{\eta^{2}\sin{\xi}^{2}}{4}\bigg{)}}$.
This clearly shows that the presence of $\xi$ profoundly modifies the
oscillation frequency, affecting the system as a perturbation introducing
quantum fluctuation. The relative dependence of the values of the parameters
$\omega$ and $b$ guide us to probe the result in the following three
categories.

1\. Under damped oscillation: The most general case is when the oscillation
parameter $\omega_{osc}$ takes over the damping parameter $b$, i.e.,
$\omega_{osc}>>\omega_{1}$. The larger value of the oscillation parameter than
the damping parameter signifies the existence of both appearance and
disappearance probability with an exponential decrease in amplitude. The
amplitude of the flavor state gradually dies down. Consequently, $\mathbf{F}$
tries to align itself along the $\mathbf{B}$, which signifies that the flavor
state exponentially damps down to the mass state. Geometrically, the
phenomenon is depicted in the right panel of Fig.3. In the presence of the
decay, the $\sigma_{3}^{M}$ axis is displaced by an amount b(red dashed line).
The yellow dashed line denotes the system’s total Hamiltonian, $\mathbf{B}$.
The initial flavor state is aligned along $\sigma_{3}^{F}$, i.e., $F(0)$. The
existence of decay enforces $\mathbf{F}$ to execute a spiral motion. It is to
be mentioned here that for the critically damped case, the angular velocity
modifies to $\sqrt{\omega^{2}_{osc}+b^{2}}$, although $\omega_{osc}>>b$.
Moreover, this motion is not confined to the $\sigma_{2}\sigma_{1}$ plane. The
appearance of $e^{-2bt}$ along the $\sigma_{3}$ drags $F(t)$ towards
$\sigma_{3}$ compel the spiral rotation to follow a helical trajectory as
shown in the figure.

2.Critically damped oscillation: In this scenario, the oscillation parameter
and damping parameter, i.e., $\omega$ and $b$ merge together,i.e.,
$\omega_{osc}=b$. The appearance probability exponentially damped down in this
case. There is no oscillation of the flavor state associated with it. This
implies that the flavor state does not execute the rotational motion around
$\mathbf{B}$. With the decrement of the appearance probability, the
disappearance probability increases and reaches a maximum, after which it
reduces to zero. This implies that neutrinos tend to oscillate, but due to
high perturbation, they damped down quickly to the mass basis. The diagram in
the left panel of Fig.3 interprets this phenomenon geometrically. Despite the
fact that $\sqrt{\omega^{2}_{osc}+b^{2}}$, as the damping factor $2b$ equals
the standard oscillation frequency $\omega$, the $\mathbf{F}$ damps down
quickly to the mass basis. The figure suggests that $F(0)$ is set to rotate
around $\mathbf{B}$. But due to the high damping coefficient, which merges
with the standard oscillation frequency, the $\mathbf{F}$ can not complete one
complete revolution and dies down quickly to the mass basis.

3\. Overdamped oscillation: the appearance probability decreases rapidly in
this scheme. Initially, the neutrino tends to oscillate. Due to the negligible
value of the disappearance probability, it can be neglected. Without spiraling
in, the flavor state decays directly to the mass basis and loses its identity,
as explained by Fig.4. It refers that due to the overtaking of the value of
$b$ than $\omega$, $\mathbf{F}$ directly falls onto the mass basis. Without
spiraling in.

From the above analysis, it is clear that the neutrino system behaves like a
mechanical oscillator. Although the neutrino has been treated as an ultra-
relativistic particle with quantum mechanical properties throughout the
derivation, it starts to behave as a classical oscillator in the presence of
an external perturbing Hamiltonian. For a mechanical system, the equation of
motion in the presence of the viscous drag is
$\ddot{x}+2\gamma\dot{x}+\omega_{0}x=0$, where $\gamma$ factor is associated
with the damping of the system. In the case of the amplitude of displacement
getting damped down, as in the case of the neutrino, the amplitude of the
oscillation probability dies down. This implies that the presence of decay
neutrino starts to mimic the mechanical oscillator.

The above discussion encompasses the fact that the flavor state neutrino
decays to its mass state, which causes the flavor of the neutrino to disappear
after a certain distance completely, and the neutrino propagates as a
flavorless massive particle. In the critically damped oscillation, the flavor
state of the neutrino set off its journey towards accomplishing the circular
trajectories. However, the marginal value of the decay parameter, which
coincides with the decay parameter, restricts its motion, and they could not
complete one complete revolution. Consequently, there is some finite
probability of disappearance occurring, although the oscillation length can
not be achieved, and the trajectory spirals towards the mass basis similar to
the Fibonacci sequence Sudipta_2019yyu . Fig.3 depicts the same phenomenology
as described. For the over-damped oscillation, due to the substantial value of
the decay parameter $b$, i.e., greater than the standard oscillation, the
flavor state directly dies down to the mass basis without performing
oscillation. It is also to be mentioned that the system of the decaying flavor
state of neutrino is precisely similar to that of a nuclear spin state in an
external Magnetic Am-Shallem_2015 . This also indicates the non-adiabaticity
of the flavor state oscillation in the appearance of a perturbing Hamiltonian
in the neutrino system. However, in the probability regime, we have considered
the adiabatic transition. But figuratively, there is a clear-cut indication
that the mixing angle should also vary in the presence of decay. The flavor
vacuum oscillation of the neutrino indicates that the flavor state picked up
from the vacuum during propagation, and the initial flavor state of the
neutrino acts as a coupled harmonic oscillator.
We find that the appearance probability and disappearance probabilities in the
presence of decay are dependent on phase $\zeta$. The Majorana phase($\phi$)
also appears in the probability analysis if the off-diagonal decay terms are
present. Thus, we have two phases $\zeta$ and $\phi$ which can induce CP
violation in a vacuum. The anti-neutrino oscillation probabilities can be
easily obtained by replacing $\zeta\rightarrow-\zeta$ and
$\phi\rightarrow-\phi$. To quantify the amount of CP violation associated with
the presence of decay, we define the quantity
$A^{CP}_{\alpha\beta}=P^{decay}_{\alpha\beta}-\overline{P}^{decay}_{\alpha\beta}$.
The asymmetry associated with muon appearance probability, $A^{CP}_{e\mu}$ is
proportional to factor $e^{-2bt}\sin(\zeta-\phi)B$, where
$B=\frac{\sin{2\theta}\sin^{2}({\Delta m^{2}L/4E})}{\Delta
m^{2}/2E}$Dixit:2022izn .

##  IV Comparison with neutrino oscillation in vacuum and matter

We provide an overview of the geometric representation of neutrino
oscillations in both vacuum and in the presence of matter, as previously
explored in references Kim:1987bv ; Kim:1993byl ; Giunti:2007ry .
Subsequently, we compare these findings with our current analysis, which
incorporates the geometric representation of neutrino oscillations along with
decay effects.

###  IV.1 Description of neutrino oscillations in vacuum

Before going to our results for geometrical interpretation in the presence of
decay, we provide an overview of the geometrical interpretation in vacuum and
matter explored in the literature.

Let’s briefly outline the geometric representation of two-flavor neutrino
oscillations, as initially discussed by Kim et al. (1988) Kim:1987bv ;
Sen:2018put . This illustration utilizes the evolution equation for two
generations of neutrinos. Using orthonormal unit vectors, the authors
explained the geometrical picture of neutrino oscillations in vacuum and
matter. The propagation and time evolution of neutrino flavor states are
described as follows:

$\displaystyle i\frac{\partial}{\partial t}\left[\begin{array}[]{c}\nu_{e}\\\
\nu_{\mu}\end{array}\right]$ $\displaystyle=$ $\displaystyle H_{\rm
F}\left[\begin{array}[]{c}\nu_{e}\\\
\nu_{\mu}\end{array}\right]=-\frac{1}{2}\overrightarrow{\sigma}\cdot\overrightarrow{B}\left[\begin{array}[]{c}\nu_{e}\\\
\nu_{\mu}\end{array}\right]$ (45)

where

$\overrightarrow{B}=\frac{1}{2E}\bigg{[}-\Delta m^{2}\sin
2\theta\,\hat{x}+(\Delta m^{2}\cos 2\theta-A)\,\hat{z}\bigg{]}$

here $A=2\sqrt{2}G_{F}N_{e}E$ with $G_{F}$ Fermi constant, $N_{e}$ the
electron number density in matter, and $E$ neutrino energy. The vector B in
the presence of matter is linked to its value $B_{0}$ in vacuum ($A=0$) as

$B=B_{0}-(A/2E)\hat{z}$

Defining
$\psi\psi^{\dagger}=\frac{1}{2}\big{[}\mathbb{1}+\overrightarrow{\sigma}\cdot\overrightarrow{P}\big{]}$,
where $\overrightarrow{P}$ represents the equivalent magnetic moment vector
(or polarization vector), the pure electron neutrino state $\nu_{e}$ can be
visually depicted with $\overrightarrow{P}=\hat{z}$, while the pure muon
neutrino state $\nu_{\mu}$ is oriented along the negative z-axis
($\overrightarrow{P}=-\hat{z}$). The concept of neutrino oscillation can be
interpreted as the precession of the magnetic moment vector
($\overrightarrow{P}$) about an external magnetic field $\overrightarrow{B}$.

$\frac{d\overrightarrow{P}}{dt}=\overrightarrow{P}\times\overrightarrow{B}\,.$
(46)

B is a constant vector ($B_{0}$) in vacuum, and P describes the cone’s surface
with axis B and an opening angle 2$\theta$ as shown in Fig.5. Within the core
of the Sun, we have mainly neutrino associated with electrons such that
$\overrightarrow{P}$ is directed along the z-axis. Also, due to high electron
density for the propagating neutrinos, $\overrightarrow{B}$ is almost parallel
to the negative z-axis, so the precession cone is close to $180^{\circ}$. One
can visualize it as if P is rotating about $-B$ in an anticlockwise direction
with an angle close to zero. When the neutrinos emerge from the Sun, the
effective matter potential decreases, and $\overrightarrow{B}$ shifts to its
vacuum value $\overrightarrow{B_{0}}$. P will precise around
$\overrightarrow{B_{0}}$ with opening angle $180^{\circ}$ for adiabatic
migration. Thus, in the final state, we have a neutrino state dominated by
$\nu_{\mu}$ explaining the geometrical picture of the MSW effect inside the
Sun. The work carried out by Kim et al. was mainly focused on one initial
neutrino flavor. However, it is useful to understand the geometrical
representation for an ensemble of neutrinos with different flavors. We can
define such incoherent neutrino beam by using density matrix formalism used by
Giunti:2007ry . A Hermitian density operator can define the neutrino beam.

$\rho(x)=\sum_{\alpha}|\nu_{\alpha}(x)\rangle\,W_{\alpha}\,\langle\nu_{\alpha}(x)|$
(47)

The parameter $W_{\alpha}$ is defined as the initial statistical weight
factor, which is a measure of the probability of particular neutrino flavor
$\alpha$ at $x=t=0$.

Figure 5: Illustration of the geometrical interpretation of two flavor
neutrino oscillations in vacuum and matter effect. The choice of orthonormal
basis is in the Cartesian coordinate system where the vectors $\vec{\sigma}$,
$\vec{B}$, $\vec{P}\equiv\vec{F}$ etc. are derived. This is the picturization
of the precession of the polarisation vector or neutrino flavor state around
the effective Hamiltonian in vacuum as well as in matter. Initially, the pure
electron-neutrino state is defined as $\vec{P}(t=0)$ along the direction of
the z-axis. The evolution of this flavor state can be understood by the
quantity $\vec{H}\times\vec{P}$, where $\vec{H}$ is defined for effective
Hamiltonian for vacuum or matter. The other muon-neutrino flavor state is
depicted in the -z axis. Kim.et.al (1988) Kim:1987bv ; Sen:2018put

The key property of density matrix operator is
$\mbox{Tr}\big{[}\rho(x)\big{]}=\sum_{a}\langle\nu_{a}(x)|\,\rho(x)\,|\nu_{a}(x)|=1$
where $\big{\\{}|\nu_{a}\rangle\big{\\}}$ is a complete set of states. The
evolution equation of the density matrix in the flavor basis is derived in
Giunti:2007ry  and is given by

$\displaystyle i\frac{d\rho_{\rm F}}{dx}=\big{[}H_{\rm F},\rho_{\rm F}\big{]}$
(48)

Figure 6: Geometrical representation of neutrino oscillations in vacuum (in
left-panel) and in presence of matter (right-panel). The choice of basis
adopted here is the flavor basis of neutrinos with the three orthonormal
vectors ${\boldsymbol{e}}_{1F}$, ${\boldsymbol{e}}_{2F}$ and
${\boldsymbol{e}}_{3F}$ represented by black solid lines.The mass eigenstates
of neutrinos are derived from equivalent orthonormal vectors represented by
dashed magenta color lines and are related to flavor ones by mixing angle
$2\theta$. The effective Hamiltonian in two neutrino systems in a vacuum is
described in flavor basis by $H_{F}=-1/2\,{\boldsymbol{\sigma}}_{F}$ while the
vector ${\boldsymbol{B}}=-\frac{\Delta m^{2}\,\sin
2\theta}{2E}\,{\boldsymbol{e}}_{1F}+\frac{\Delta m^{2}\,\cos
2\theta}{2E}{\boldsymbol{e}}_{3F}$ is depicted in the second quadrant. The
initial neutrino flavor state, let us say $\nu_{e}$, is oriented along
${\boldsymbol{e}}_{3F}$ direction and is represented by a flavor state vector
${\boldsymbol{S}}(t=0)$. The evolution of the neutrino state vector
${\boldsymbol{S}}$ from distance $x=0$ to $x=L_{osc}$ is described by the
quantity ${\boldsymbol{S}}\times{\boldsymbol{B}}$ and thereby, the precision
of ${\boldsymbol{S}}$ about ${\boldsymbol{B}}$ describes neutrino oscillation.

with initial boundary condition $[\rho_{\rm F}]_{rs}(0)=W_{r}\delta_{rs}$.
With two flavor scenario with $\nu_{e}-\nu_{\mu}$, the effective Hamiltonian
$H_{\rm F}$ in flavor basis can be written down as

$H_{\rm F}=-\frac{1}{2}\overrightarrow{\sigma_{\rm
F}}\cdot\overrightarrow{\text{B}}\quad\,\rho_{\rm
F}=\frac{1}{2}\big{[}\sigma_{0}+\overrightarrow{\sigma_{\rm
F}}\cdot\overrightarrow{P}\big{]}\,.$ (49)

Here, the choice of basis is essential and as adopted in Giunti:2007ry ,
$\overrightarrow{e}_{1F}$, $\overrightarrow{e}_{2F}$,
$\overrightarrow{e}_{3F}$ are three orthonormal basis vectors with property
$\overrightarrow{e}_{aF}\cdot\overrightarrow{e}_{bF}=\delta_{ab}$ in which
flavor states of neutrinos can be constructed.

The components of vectors $\overrightarrow{B}$ (can be interpreted as a
magnetic field) defined in the flavor basis are

$\displaystyle B_{1F}=-\frac{\Delta m^{2}\,\sin 2\theta}{2E}\,$ $\displaystyle
B_{2F}=0\,$ $\displaystyle B_{3F}=\frac{\Delta m^{2}\,\cos 2\theta}{2E}\,.$
(50)

For the evolution of the system in the presence of matter, the mass square
difference($\Delta m^{2}$) and the mixing angle($\theta$) should be replaced
by the effective mass square difference( $\Delta\tilde{m}^{2}$) and modified
mixing angle ($\theta_{M}$) in the presence of matter potential. The
components for equivalently interpreted Polarisation vector
($\overrightarrow{P}$) arising from the density matrix operator as,

$\displaystyle P_{1F}=2\mbox{Re}[\rho_{\rm F}]_{e\mu}\,$ $\displaystyle
P_{2F}=-2\mbox{Im}[\rho_{\rm F}]_{e\mu}\,$ $\displaystyle P_{3F}=[\rho_{\rm
F}]_{ee}-[\rho_{\rm F}]_{\mu\mu}\,.$ (51)

With the orthonormal set of basis vectors $\overrightarrow{e}_{a}$, the
various vectors defined as $\overrightarrow{\sigma}_{\rm
F}=\sum^{3}_{a=1}\sigma^{a}\overrightarrow{e}^{a}_{\rm F}$,
$\overrightarrow{B}=\sum^{3}_{a=1}B^{a}_{\rm F}\overrightarrow{e}^{a}_{\rm
F}$, and $\overrightarrow{P}=\sum^{3}_{a=1}P^{a}_{\rm
F}\overrightarrow{e}^{a}_{\rm F}$. They derived from evolution equations of
the density matrix operator, and the corresponding translated initial boundary
conditions are $\overrightarrow{P}_{1F}(0)=0$, $\overrightarrow{P}_{2F}(0)=0$
and $\overrightarrow{P}_{3F}(0)=W_{e}-W_{\mu}$.

Let us understand the physical meaning of neutrino oscillations in vacuum in
terms of $\overrightarrow{\sigma}$, $\overrightarrow{B}$ and
$\overrightarrow{P}$.

Assume that the initial neutrino state is a pure electron neutrino produced in
Sun, i.e., $W_{e}=1$ and $W_{\mu}=0$. The Polarisation vector
$\overrightarrow{P}(0)$ is aligned along $\overrightarrow{e}_{3F}$ direction.
For pure muon type neutrino state ($W_{e}=0$ and $W_{\mu}=1$) the Polarisation
vector $\overrightarrow{P}(0)$ will be aligned along negative
$\overrightarrow{e}_{3F}$ direction. For an incoherent mixture of electron and
muon neutrino states, the orientation of the Polarisation vector
$\overrightarrow{P}(0)$ will be along $\pm\overrightarrow{e}_{3F}$ depending
upon the relative strength of $W_{r}$ with $r=e,\mu$. The norm of the
Polarization vector will decide whether the neutrino beam is a pure state or
incoherent mixture. The precision of $\overrightarrow{P}$ around the
$\overrightarrow{B}$ can have different geometrical viewpoints depending upon
the value of matter density compared to the resonance density. We present one
such scenario in Fig.6, where the transition is adiabatic for solar neutrinos.
The right panel of Fig.6 suggests the geometrical interpretation of neutrino
oscillation in matter in over $e^{F}_{i}$ and $e^{M}_{i}$ as described in
Giunti:2007ry . The presence of matter Hamiltonian $A=2\sqrt{2}G_{F}N_{e}E$,
the standard Hamiltonian of the neutrino is dragged down as the figure
indicates. However, the $\vec{P}$ tends to follow the precessional motion
around $\vec{B}$. In order to retain its circular motion $\vec{P}$ starts to
revolves round $-\vec{B}$ as a results of which instead of $\nu_{1}$ it is now
coincide with $\nu_{2}$ and $\vec{P}$ describes a narrow cone $(\sim
180^{\circ})$ round $\vec{B}$. The resonance crossing of $\vec{P}$ in the
nonadiabatic case represents $\nu_{2}\rightarrow\nu_{1}$Giunti:2007ry  We find
that both the works by Kim and Guinti describe neutrino oscillations in vacuum
and matter but with different basis choices. The choice of orthogonal basis is
not unique. For example, Mikheyev and Smirnov Mikheev:1987jp  have described
the two-flavor neutrino oscillation using an orthogonal mass eigenstates as
the choice of basis as $\nu_{1}$, $Re~{}\nu_{2}$, and $Im~{}\nu_{2}$.
Nevertheless, in the following subsection, we present the geometrical
interpretation of neutrino oscillations in the presence of decay, which is not
yet clearly explored in the literature.

###  IV.2 Description of neutrino oscillation with decay

The standard Hamiltonian of the neutrino oscillation is defined as
$\mathcal{H}_{0}=\begin{pmatrix}\frac{m^{2}_{1}}{2E}&0\\\
0&\frac{m^{2}_{2}}{2E}\\\ \end{pmatrix}$. FollowingDixit:2022izn  this
Hamiltonian can be decomposed as
$\mathcal{H}_{0}=\frac{1}{2}(\omega_{0}\mathbf{I}+\omega\cdot\sigma_{z})$ with
$\omega_{0}=\frac{(m^{2}_{1}+m^{2}_{2})}{2E}$ and
$\omega=\frac{(m^{2}_{1}-m^{2}_{2})}{2E}$. Since the $\omega_{0}$ term doesn’t
contribute to the flavor-changing neutrino oscillation, the first term of the
$\mathcal{H}_{M}$ is ruled out. As we have already discussed, we can unify the
effect of invisible neutrino decay by the Hamiltonian,

$\begin{split}\mathcal{H}_{d}&=\frac{-i}{2}\begin{pmatrix}b&\eta/2\\\
\eta/2&b\\\
\end{pmatrix}=\frac{-i}{2}(b\mathbf{I}+\eta\cdot\sigma)\end{split}$ (52)

Which modifies the description of the Hamiltonian of the neutrino system to
$\mathcal{H}_{m}=\frac{1}{2}(-ib\mathbf{I}+\omega\cdot\sigma-i\eta/2\cdot\sigma)$.
This results in a modification of the neutrino system’s Hamiltonian to
$\mathcal{H}_{m}=\frac{1}{2}(-ib\mathbf{I}+\omega\cdot\sigma-i\eta/2\cdot\sigma)$.
In summary, when considering the presence of decay, the Hamiltonian for the
neutrino system can be vectorized by introducing a vector that represents
decay in addition to the standard Hamiltonian for neutrino oscillation,
denoted as $\mathcal{H}_{m}=\mathcal{H}_{0}+\mathcal{H}_{d}$. This is
mathematically elucidated in the following equation.

$\begin{split}\mathcal{H}_{0}&=\frac{1}{2}(\omega_{0}\mathbf{I}+\omega\cdot\sigma_{3})\\\
\mathcal{H}_{d}&=\frac{-i}{2}(b\mathbf{I}+\eta\cdot\sigma_{1})\\\ \end{split}$
(53)

For a geometric representation, we have opted for the Pauli matrix basis. This
choice is motivated by the ability to expand the Pauli matrices over the mass
basis, which can be conceptualized as a two-level system featuring two mass
eigenstates, namely $\ket{\nu_{1}}$ and $\ket{\nu_{2}}$. The explicit form of
$\sigma_{i}$ (where $i=1,2,3$) over the mass basis is provided in eq.(15). The
unique aspect of the superposition of two non-degenerate mass eigenstates over
the flavor eigenstate of the neutrino system motivates the adoption of this
formalism. This contrasts with other particles, where a unique mass eigenstate
exclusively describes each flavor eigenstate. The selection of this basis
allows for the vectorization of the Hamiltonian in the mass basis, represented
by the basis vector $\hat{\sigma}{i}$. In the mass basis, we denote the vector
representation of the Hamiltonian for standard oscillation as $\mathbf{B}_{h}$
and the decay Hamiltonian as $\mathbf{B}_{nh}$.

$\begin{split}\mathbf{B}_{h}&=\frac{1}{2}\omega\hat{\sigma}_{3}\\\
\mathbf{B}_{nh}&=\frac{-i}{2}(b\mathbf{I}+\eta/2\hat{\sigma}_{1})\\\
\mathbf{B}&=\frac{1}{2}(\omega\hat{\sigma}_{3}+e^{\frac{-i\pi}{2}}(b\mathbf{I}+\eta\hat{\sigma}_{1}))\end{split}$
(54)

The imaginary $-i$ factor physically represents the rotation of the coordinate
axes by the angle of $\frac{-i\pi}{2}$. After elucidating the vectorized
Hamiltonian of the neutrino system, we have introduced the density matrix
formalism to express the flavor states. Similarly, as we have denoted the
Hamiltonian in vector representation, we have demonstrated the density
operator of the neutrinos depicting the flavor state as,

$\displaystyle\rho=\mathbf{F}\cdot\sigma=F_{1}\hat{\sigma}_{1}+F_{2}\hat{\sigma}_{2}+F_{3}\hat{\sigma}_{3}$
(55)

The Magnitude of the $\mathbf{F}$ is
$F_{i}=\mbox{Tr}\frac{1}{2}\sigma_{i}\rho$. For neutrino system,
$\rho(t=0)=\begin{pmatrix}\cos^{2}{\theta}&\cos{\theta}\sin{\theta}\\\
\cos{\theta}\sin{\theta}&\sin^{2}{\theta}\\\
\end{pmatrix}=\begin{pmatrix}F_{3}&F_{1}-iF_{2}\\\ F_{1}+iF_{2}&-F_{3}\\\
\end{pmatrix}$ The interpretation directly shows $\mathbf{F}$ in terms of the
component of $\rho$ as,

$\displaystyle\begin{split}F_{1}&=(\rho_{12}+\rho_{21})\\\
F_{2}&=i(\rho_{12}-\rho_{21})\\\ F_{3}&=(\rho_{11}-\rho_{22})\end{split}$ (56)

Thus, two operators, Hamiltonian, and density operator, are represented by two
vectors $\mathbf{B}$ and $\mathbf{F}$ respectively. The dynamical evolution
equation of $\mathbf{F}$ describes how these two vectors are interrelated. In
this demonstration, instead of denoting the Hamiltonian as an inner product
with the Pauli matrices on some arbitrary basis, we have expressed the
vectorized Hamiltonian in Pauli bases. In this context, it is to be mentioned
that the main reason for introducing the Pauli basis is that during the change
of basis from mass to flavor or vice-versa only these basis vector
$\hat{\sigma}_{i}$ will change which makes the visualization of the phenomenon
of oscillation and mathematics much simpler. This flavor-mass bases conversion
is given as,

$\begin{split}\sigma_{0}^{M}&=\sigma_{0}^{F}\\\
\sigma_{1}^{M}&=\sin{2\theta}\sigma_{3}^{F}+\cos{2\theta}\sigma_{1}^{F}\\\
\sigma_{2}^{M}&=\sigma_{2}^{F}\\\
\sigma_{3}^{M}&=\cos{2\theta}\sigma_{3}^{F}-\sin{2\theta}\sigma_{1}^{F}\\\
\end{split}$ (57)

The scenario also addresses that the Hamiltonian is responsible for the
evolution of flavor state with either time. Following the Von-Neumann
equation, the evolution equation of the density operator is explicitly of the
form,

$i\frac{d}{dt}\rho(t)=\frac{-i}{2}[\mathcal{H}_{0},\rho]-\frac{1}{2}\\{\mathcal{H}_{d},\rho\\}$
(58)

This follows from the Schrodinger equation, which gives the dynamical
evolution of the operator. In the present representation, this equation is
written as $\frac{d}{dt}F_{i}=\mbox{Tr}\frac{1}{2}\sigma_{i}\frac{d}{dt}\rho$
which gives the dynamical flavor evolution over Pauli basis in terms of state
vector as,

$\frac{d}{dt}\mathbf{F}=\mathbf{F}\times\mathbf{B}-2b\mathbf{F}$ (59)

In this framework, the dependency of the evolution of the state for the two-
flavor neutrino oscillation over the off-diagonal terms of the decay
Hamiltonian is automatically canceled out.

Figure 7: Figures illustrate the geometrical representation of neutrino
oscillation in decay under damped case in Pauli basis. For the under-damped
case, the oscillation amplitude damped down in an oscillatory way.
Consequently, there is oscillation with decreasing amplitude, which directs
towards the fact that after one full revolution, it is impossible to get back
to the initial flavor state as the presence of decay parameter $b$, which
causes the flavor state depicting vector $\mathbf{F}$ execute a helical
motion. Moreover, this helical motion doesn’t confine to a plane because this
decay parameter affects the $\hat{\sigma}_{3}$ components as well, which makes
$\mathbf{F}$ drag toward $\hat{\sigma}_{3}^{Md}$ which the mass basis in the
presence of decay. In addition, with the gradual decrement of the pure state,
the flavor state dies down, demonstrating that it ultimately dissipates to the
mass state and $\mathbf{F}$ aligns itself towards $\mathbf{B}$.

It is also to be mentioned here that the flavor state representing vector
$\mathbf{F}$ is defined over the mass basis. At an initial time $t=0$, we have
considered the neutrinos are produced in the pure electron flavor state.
Therefore, it is clear from eq.(22) that solving the equation and transforming
it to the flavor basis following the change of bases from eq.(57) give the
appearance and the disappearance probabilities.

$\begin{split}P_{e\mu}&=e^{-2bt}P_{e\mu}^{vac}\\\
P_{ee}&=e^{-2bt}P_{ee}^{vac}\\\ \end{split}$ (60)

Where $P_{e\mu}^{vac}=\sin^{2}{2\theta}\sin^{2}{\omega L}$. The geometrical
description of the eq.(59) is given in Fig.7, which implements the damped
oscillatory nature of the neutrino flavor state. Figuratively, it suggests
that the flavor eigenstate of neutrino decays to the mass eigenstates
following helical trajectory due to the decay factor $e^{-2bt}$. The work’s
uniqueness lies in the choice, simple mathematical description, and effective
geometrical visualization.

This illustration also shows that the neutrino system mimics the NMR system in
the context of the similar format of the Bloch equation. From Am-Shallem_2015
, the Bloch equation for an NMR system in an external magnetic field is

$\begin{pmatrix}\dot{S}_{x}\\\ \dot{S}_{y}\\\ \dot{S}_{z}\\\
\end{pmatrix}=\begin{pmatrix}-\frac{1}{T_{2}}&\Delta&0\\\
-\Delta&-\frac{1}{T_{2}}&\epsilon\\\ 0&-\epsilon&-\frac{1}{T_{2}}\\\
\end{pmatrix}\begin{pmatrix}S_{x}\\\ S_{y}\\\ S_{z}\\\
\end{pmatrix}+\begin{pmatrix}0\\\ 0\\\ S_{z}^{0}\\\ \end{pmatrix}$ (61)

Where the Hamiltonian of the system, including the perturbation, is given by
$\hat{\mathbf{H}}=\Delta\hat{\mathbf{S}}_{z}+\epsilon\mathbf{S}_{x}$. $S_{x}$,
$S_{y}$, $S_{z}$ denotes the nuclear spin under the influence of an external
magnetic field with $T_{1}$ and $T_{2}$ as the dissipation and dephasing
relaxation parameters. Similarly, for the Bloch equation for the neutrino
system as derived in eq.(23),

$\begin{pmatrix}\dot{F}_{x}\\\ \dot{F}_{y}\\\ \dot{F}_{z}\\\
\end{pmatrix}=\begin{pmatrix}-b&-\omega&0\\\ \omega&-b&0\\\ 0&0&-2b\\\
\end{pmatrix}\begin{pmatrix}F_{x}\\\ F_{y}\\\ F_{z}\\\ \end{pmatrix}$ (62)

Where the role of the external magnetic field is played by the mass square
differences $\omega$ and the dissipation parameter is $b$. Although the $x$
component of the Hamiltonian is present in both cases, for NMR, it is
Hermitian; hence, it is present, $\epsilon$, and for neutrino it is non-
Hermitian hence is doesn’t show up in the Bloch matrix. However, in the
Liouville super-operator formalism it will be present in the matrix defining
the Liouville super-operator Shafaq:2021lju .

##  V Conclusion

We discussed the geometrical interpretation of two-flavor neutrino oscillation
in the presence of neutrino decay with a detailed analysis of the theoretical
framework. A comparative study of the geometrical description of neutrino
oscillation performed in the present work along with earlier works Kim:1987bv
; Kim:1993byl ; Giunti:2007ry  has been presented for better understanding of
the reader. We followed the density matrix formalism to define the neutrino as
an open quantum system in the presence of decay. We vectorized the operators,
the Hamiltonian, and the density operator of the system to analyze the
neutrino oscillation, including decay geometrically. Implementing such
formalism, we devised a simple mathematical analysis following Am-Shallem_2015
to produce the expression of probabilities. Using the Pauli basis, we
interpret the results schematically, which validates the decay of the flavor
state to the mass state. we also showed the modified evolution of the flavor
state depicting vector in presence of phase factor $\xi$ in the non-hermitian
Hamiltonian. This introduced a $CP$ violation phase in two flavor neutrino
oscillation. We also qualitatively discussed the three scenarios of under-
damped, over-damped, and critically-damped motion. To give the work an overall
completeness, we did a comparative study of the geometrical interpretation of
neutrino oscillation in vacuum, matter, and decay that includes the MSW
effect. In a nutshell, we have found that in the presence of decay, the flavor
state depicting vector $\mathbf{F}$ executes a spiral motion around the
Hamiltonian depicting vector $\mathbf{B}$, through which the flavor state of
neutrino decays to the mass eigenstate. The choice of Pauli basis has made the
interpretation of the phenomenon of decay in a more convenient way. However,
such a choice of basis has made the oscillation probability independent of the
off-diagonal factor, which stipulates the event of amplitude damping of the
oscillation in addition to introducing a perturbation to the mass eigenstate.

The geometrical interpretation of two-flavor neutrino oscillation provides an
easier way to understand why such oscillations get enhanced in matter- an
effect known as MSW (Mikheyev-Smirnov-Wolfenstein) effect that could explain
the well-known solar neutrino puzzle. This description can also apply to
understanding the governing dynamics of Supernovae and the fast flavor
conversion of Supernovae neutrinos Dasgupta:2018ulw ; Glas:2019ijo ;
Sen:2017ogt . The three-flavor neutrino oscillations can be approximated with
two-flavor neutrino oscillations ($\nu_{e}$ and $\nu_{x}$ with $x=\mu,\tau$)
within Supernovae as the muon and tau neutrinos have identical interactions
and thereby, behaves similarly Bhattacharyya:2020dhu ; Bhattacharyya:2020jpj .
Additionally, one may consider neutrino as an open quantum system with a
similar viewpoint to the nuclear magnetic resonance (NMR) system Am-
Shallem_2015 , where the external magnetic field acts as a beam splitter. In
contrast, in the case of neutrino, the unitary transformation matrix performs
the same. In addition to this, the proper characterization of the decay
parameter with its value and its consequence on neutrino oscillation is still
a broad area to investigate. Furthermore, the investigation of the proper
reason behind the neutrino decay will make the understanding of neutrino
oscillation more effective.

###### Acknowledgements.

Rajrupa Banerjee would like to thank the Ministry of Electronics and IT for
the financial support through the Visvesvaraya fellowship scheme for carrying
out this research work. RB is very thankful to Prof. Prasant. K. Panigrahi for
the fruitful discussion carried from time to time for the betterment of this
work. KS acknowledges the financial support received via the ”Bi-nationally
Supervised Doctoral Degree Program” from the German Academic Exchange Service
(DAAD). Additionally, KS extends thanks to the ”Karlsruhe School of Elementary
and Astroparticle Physics: Science and Technology (KSETA),” where the
concluding phase of this work took place.

## References

  * (1) SNO, Q. R. Ahmad et al., “Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory,” Phys. Rev. Lett. 89 (2002) 011301, arXiv:nucl-ex/0204008.
  * (2) Super-Kamiokande, S. Fukuda et al., “Determination of solar neutrino oscillation parameters using 1496 days of Super-Kamiokande I data,” Phys. Lett. B 539 (2002) 179–187, arXiv:hep-ex/0205075.
  * (3) IceCube, M. G. Aartsen et al., “Measurement of Atmospheric Neutrino Oscillations at 6–56 GeV with IceCube DeepCore,” Phys. Rev. Lett. 120 (2018) no. 7, 071801, arXiv:1707.07081.
  * (4) Daya Bay, F. P. An et al., “Observation of electron-antineutrino disappearance at Daya Bay,” Phys. Rev. Lett. 108 (2012) 171803, arXiv:1203.1669.
  * (5) MINOS, P. Adamson et al., “Improved search for muon-neutrino to electron-neutrino oscillations in MINOS,” Phys. Rev. Lett. 107 (2011) 181802, arXiv:1108.0015.
  * (6) C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics.  2007\.
  * (7) C. W. Kim, J. Kim, and W. K. Sze, “On the Geometrical Representation of Neutrino Oscillations in Vacuum and Matter,” Phys. Rev. D 37 (1988) 1072.
  * (8) S. P. Mikheyev and A. Y. Smirnov, Massive Neutrinos in Astrophysics and Particle Physics.  Rencontre de Moriond (VI Moriond Workshop), Tignes, France, 1986, edited by O. Fackler and J. Tran Than Van, 1986.
  * (9) M. Sen, New Aspects of Supernova Neutrino Flavor Conversions: In the Standard Model and Beyond.  PhD thesis, Tata Inst., 8, 2018.
  * (10) JUNO, J. Wang et al., “Damping signatures at JUNO, a medium-baseline reactor neutrino oscillation experiment,” JHEP 06 (2022) 062, arXiv:2112.14450.
  * (11) A. Ghoshal, A. Giarnetti, and D. Meloni, “Neutrino Invisible Decay at DUNE: a multi-channel analysis,” J. Phys. G 48 (2021) no. 5, 055004, arXiv:2003.09012.
  * (12) M. C. Gonzalez-Garcia and M. Maltoni, “Status of Oscillation plus Decay of Atmospheric and Long-Baseline Neutrinos,” Phys. Lett. B 663 (2008) 405–409, arXiv:0802.3699.
  * (13) J. N. Bahcall, N. Cabibbo, and A. Yahil, “Are neutrinos stable particles?,” Phys. Rev. Lett. 28 (1972) 316–318.
  * (14) M. Lindner, T. Ohlsson, and W. Winter, “A Combined treatment of neutrino decay and neutrino oscillations,” Nucl. Phys. B 607 (2001) 326–354, arXiv:hep-ph/0103170.
  * (15) M. Blasone, P. Jizba, and L. Smaldone, “Flavor neutrinos as unstable particles,” J. Phys. Conf. Ser. 1612 (2020) no. 1, 012004.
  * (16) D. S. Chattopadhyay, K. Chakraborty, A. Dighe, S. Goswami, and S. M. Lakshmi, “Neutrino Propagation When Mass Eigenstates and Decay Eigenstates Mismatch,” Phys. Rev. Lett. 129 (2022) no. 1, 011802, arXiv:2111.13128.
  * (17) K. Dixit, A. K. Pradhan, and S. U. Sankar, “CP violation due to a Majorana phase in two flavor neutrino oscillations with decays,” Phys. Rev. D 107 (2023) no. 1, 013002, arXiv:2207.09480.
  * (18) S. Goswami, D. S. Chattopadhyay, K. Chakraborty, and A. Dighe, “Analytical treatment of neutrino oscillation and decay in matter,” PoS NOW2022 (2023) 059.
  * (19) A. Kumar Jha, S. Mukherjee, and B. A. Bambah, “Tri-Partite entanglement in Neutrino Oscillations,” Mod. Phys. Lett. A 36 (2021) no. 09, 2150056, arXiv:2004.14853.
  * (20) M. Blasone, F. Dell’Anno, S. De Siena, and F. Illuminati, “On entanglement in neutrino mixing and oscillations,” J. Phys. Conf. Ser. 237 (2010) 012007, arXiv:1003.5486.
  * (21) M. Blasone, P. Jizba, and L. Smaldone, “Dynamical Generation of Flavour Vacuum,” Springer Proc. Math. Stat. 335 (2019) 497–502.
  * (22) E. Akhmedov, “Quantum mechanics aspects and subtleties of neutrino oscillations,” in International Conference on History of the Neutrino: 1930-2018.  1, 2019.  arXiv:1901.05232.
  * (23) F. Halzen and A. D. Martin, QUARKS AND LEPTONS: AN INTRODUCTORY COURSE IN MODERN PARTICLE PHYSICS.  1984\.
  * (24) P. Baerwald, M. Bustamante, and W. Winter, “Neutrino Decays over Cosmological Distances and the Implications for Neutrino Telescopes,” JCAP 10 (2012) 020, arXiv:1208.4600.
  * (25) L. Borsten, D. Dahanayake, M. J. Duff, H. Ebrahim, and W. Rubens, “Black Holes, Qubits and Octonions,” Phys. Rept. 471 (2009) 113–219, arXiv:0809.4685.
  * (26) J. M. Berryman, A. de Gouvêa, D. Hernández, and R. L. N. Oliveira, “Non-Unitary Neutrino Propagation From Neutrino Decay,” Phys. Lett. B 742 (2015) 74–79, arXiv:1407.6631.
  * (27) B. Soni, S. Shafaq, and P. Mehta, “Distinguishing between Dirac and Majorana neutrinos using temporal correlations,” arXiv:2307.04496.
  * (28) S. Shafaq, T. Kushwaha, and P. Mehta, “Investigating Leggett-Garg inequality in neutrino oscillations – role of decoherence and decay,” arXiv:2112.12726.
  * (29) J. A. Gyamfi, “Fundamentals of quantum mechanics in Liouville space,”European Journal of Physics 41 (oct, 2020) 063002, arXiv:2003.11472.
  * (30) D. Manzano, “A short introduction to the Lindblad master equation,”AIP Advances 10 (feb, 2020) , arXiv:1906.04478.
  * (31) K. B. Wharton and D. Koch, “Unit quaternions and the Bloch sphere,” J. Phys. A 48 (2015) no. 23, 235302, arXiv:1411.4999.
  * (32) P. Rotelli, “The Dirac Equation on the Quaternion Field,” Mod. Phys. Lett. A 4 (1989) 933.
  * (33) D. Shirokov, “Clifford Algebras and Their Applications to Lie Groups and Spinors,” Geometry, Integrability and Quantization 19 (2018) 11–53.
  * (34) A. Bohm and M. Loewe, Quantum mechanics : foundations and applications / Arno Bohm.  Springer-Verlag New York, 2nd ed., rev. and enlarged / prepared with m. loewe. ed., 1986.
  * (35) A. Y. Smirnov, “The Mikheyev-Smirnov-Wolfenstein (MSW) Effect,” 2019.
  * (36) A. Y. Smirnov, “The MSW effect and solar neutrinos,” in 10th International Workshop on Neutrino Telescopes, pp. 23–43.  5, 2003.  arXiv:hep-ph/0305106.
  * (37) M. Am-Shallem, R. Kosloff, and N. Moiseyev, “Exceptional points for parameter estimation in open quantum systems: analysis of the Bloch equations,”New Journal of Physics 17 (nov, 2015) 113036.
  * (38) T. H. Kim, “Pulsed NMR : Relaxation times as function of viscocity and impurities,”  2008\.  https://api.semanticscholar.org/CorpusID:54017928.
  * (39) S. Sinha, “The Fibonacci Numbers and Its Amazing Applications,”.
  * (40) C. W. Kim and A. Pevsner, Neutrinos in physics and astrophysics, vol. 8.  1993\.
  * (41) S. P. Mikheev and A. Y. Smirnov, “Neutrino Oscillations in an Inhomogeneous Medium: Adiabatic Regime,” Sov. Phys. JETP 65 (1987) 230–236.
  * (42) B. Dasgupta, A. Mirizzi, and M. Sen, “Simple method of diagnosing fast flavor conversions of supernova neutrinos,” Phys. Rev. D 98 (2018) no. 10, 103001, arXiv:1807.03322.
  * (43) R. Glas, H. T. Janka, F. Capozzi, M. Sen, B. Dasgupta, A. Mirizzi, and G. Sigl, “Fast Neutrino Flavor Instability in the Neutron-star Convection Layer of Three-dimensional Supernova Models,” Phys. Rev. D 101 (2020) no. 6, 063001, arXiv:1912.00274.
  * (44) M. Sen, “Supernova neutrinos: fast flavor conversions near the core,” Springer Proc. Phys. 203 (2018) 533–537, arXiv:1702.06836.
  * (45) S. Bhattacharyya and B. Dasgupta, “Late-time behavior of fast neutrino oscillations,” Phys. Rev. D 102 (2020) no. 6, 063018, arXiv:2005.00459.
  * (46) S. Bhattacharyya and B. Dasgupta, “Fast Flavor Depolarization of Supernova Neutrinos,” Phys. Rev. Lett. 126 (2021) no. 6, 061302, arXiv:2009.03337.