#  HUPD-2213 Determination of Majorana type-phases from the time evolution of
lepton numbers

Nicholas J. Benoit1111E-mail address<EMAIL_ADDRESS>Yuta
Kawamura 2222E-mail address<EMAIL_ADDRESS>Saki Kawano 3333E-mail
address<EMAIL_ADDRESS>

Takuya Morozumi 4,5444E-mail address<EMAIL_ADDRESS>, Yusuke
Shimizu 4,5555E-mail address<EMAIL_ADDRESS>, Kei Yamamoto
6666E-mail address<EMAIL_ADDRESS>

1Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526,
Japan 2Yokkaichi city, Mie, Japan 3Tokyo prefecture, Japan 4Physics Program,
Graduate School of Advanced Science and Engineering,
Hiroshima University, Higashi-Hiroshima 739-8526, Japan 5Core of Research for
the Energetic Universe, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan 6Department of Global Environment Studies,
Hiroshima Institute of Technology,
Saeki-ku, Hiroshima 731-5193, Japan

( Abstract
We have investigated an approach to determine the Majorana type-phases using
the time evolution of lepton family numbers. The Majorana type-phases are
related to the orientation of unitarity triangles for the Pontecorvo-Maki-
Nakagawa-Sakata (PMNS) matrix, and the Majorana phases $\alpha_{21}$ and
$\alpha_{31}$. After taking the second-order time derivative of the lepton
family number expectation values, the dependencies on the summation of
Majorana type-phases can be determined. Thus allowing for the extraction of
the orientation of the unitarity triangles and the Majorana phases. We study
how to extract the Majorana type-phases and the lightest neutrino mass for
three massive neutrinos, and when a neutrino is massless, i.e., $m_{1,3}=0$.
Our result can be complimentary to using neutrinoless double-beta decay for
determining the orientation of PMNS unitarity triangles and the Majorana
phases. )

##  1 Introduction

Despite phenomenal experimental programs in neutrino physics, open questions
remain. We know the three active neutrinos of the Standard Model come in three
flavors (families) electron neutrino, muon neutrino, and the tauon
neutrino[1]. Those flavors are assigned based on the neutrinos weak charged
current lepton partner. We also know that neutrinos can oscillate between
flavors over macroscopic distances[2, 3]. To describe the oscillations the
three neutrino flavors are treated as linear combinations of massive
eigenstates, then to explain experimental data at least two of the eigenstates
must have small, but non-zero, masses[4, 5]. This is in contrast to the
Standard Model, in which the three neutrinos are massless particles. Because
neutrino flavor oscillation experiments can only constrain the mass squared
differences of the neutrinos $\Delta m^{2}_{ij}=m^{2}_{i}-m^{2}_{j}$, details
of neutrino masses remain unknown. For example neutrinos are neutrally charged
particles with non-zero mass, which means they could be their own
antiparticles. Under that setup neutrinos would be called Majorana particles
and have a Majorana mass[6, 7, 8]. A Dirac particle type and mass are also
possible for neutrinos, this would follow the usual Standard Model lepton
content. Neither particle type has not been experimentally established for
neutrinos despite years of searching for Majorana neutrinos. Perhaps the most
famous experiments are searching for neutrinoless double-beta decay[9, 10, 11,
12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], but there are also neutrinoless
quadruple-beta decay[24], neutrinoless double-electron capture[25, 26, 27, 28,
29, 30], proposed processes mediated by the exchange of a pair of virtual
neutrinos akin to a Casimir force[31, 32, 33], coherent scattering of
neutrinos on nuclei with bremsstrahlung[34], and proposed searches of quantum
statics for final state decay processes involving neutrinos[35, 36, 37, 38].

We explore the situation that neutrinos are Majorana particles. In section 2,
we summarize how the three neutrino flavors are constructed as linear
combinations of the massive eigenstates via a mixing matrix; and we introduce
how the mixing matrix can be formed with Majorana type-phases and unitarity
triangles. Then, we connect our previous work on the time evolution of lepton
family numbers to the Majorana type-phases in section 3. We explain how our
previous work can be used to determine the Majorana type-phases and the
lightest neutrino mass in section 4. In the last section, section 5, we
consider the lightest neutrino to be massless and illustrate how this enhances
the predictive power of our work. Finally, we provide some concluding
thoughts.

##  2 Majorana type-phases and unitarity triangles

Neutrino masses are a pillar for physics beyond the Standard Model that
started with the discovery of neutrino flavor oscillations [39, 40]. The
results of neutrino oscillation experiments are mathematically described by a
mixing of basis states via the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix
[41, 42, 43]

$\nu_{L\alpha}=U_{\alpha i}\nu_{Li}=\begin{pmatrix}U_{e1}&U_{e2}&U_{e3}\\\
U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\\ U_{\tau 1}&U_{\tau 2}&U_{\tau
3}\end{pmatrix}\begin{pmatrix}\nu_{L1}\\\ \nu_{L2}\\\ \nu_{L3}\end{pmatrix},$
(1)

where $\alpha=e,\mu,\tau$ and $i=1,2,3$. The PMNS matrix $U_{\alpha i}$ can be
formed for neutrinos with a Majorana mass though diagonalization of the
Majorana mass matrix,

$(U)_{\alpha i}(m_{\nu})_{\alpha\beta}(U)_{\beta j}=(m_{\nu})_{i}\delta_{ij},$
(2)

using a Takagi Factorization [44]. Then the leptonic weak charged current
interaction is written as,

$\mathcal{L}^{(CC)}_{I}=\frac{g}{\sqrt{2}}\overline{l_{L\alpha}}\gamma_{\mu}U_{\alpha
i}\nu_{Li}W^{\mu-}+\text{h.c.}\,.$ (3)

The charged leptons are free to be re-phased, because their mass term is
invariant under the phase transformations of $l_{\alpha}\rightarrow
l^{{}^{\prime}}_{\alpha}=\exp(i\phi_{\alpha})l_{\alpha}$. But, the neutrino
Majorana mass term is not invariant under the phase transformations. This
means the standard parametrization of the PMNS matrix [45] must be extended to
include two independent CP violating phases. Thus, the unitary $3\times 3$
PMNS matrix depends on three mixing angles and three CP violating phases.

$U_{\alpha
i}=U^{D}\operatorname{diag}\left(1,\,e^{i\frac{\alpha_{21}}{2}},\,e^{i\frac{\alpha_{31}}{2}}\right),$
(4)

where the Dirac portion $U^{D}$ is of the form,

$U^{D}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{pmatrix}.$
(5)

We use the usual notation of $c_{ij}=\cos\theta_{ij}$ and
$s_{ij}=\sin\theta_{ij}$ from PDG[45]. The CP violating phases in the diagonal
matrix, $\alpha_{21}$ and $\alpha_{31}$, are the Majorana phases, whereas the
CP violating phase of $U^{D}$ is the Dirac phase $\delta$. The CP violating
phases have physical meaning only if all the mixing angles are not $0$ or
$\pi/2$. This can be seen by constructing the invariants [46, 47, 48, 49] of
the PMNS matrix.

We consider an interesting aspect of Majorana neutrinos and the rephasing
invariant bilinears [48] that Branco and Rebelo name “Majorana type-
phases”[50]. The Majorana type-phases are defined by Branco and Rebelo to be
the argument of the bilinears $U_{\alpha k}U^{\ast}_{\alpha j}$ with no
summation over the repeated index $\alpha$. Then, Branco and Rebelo choose six
Majorana type-phases and prove the six phases can be used to exactly reproduce
the $3\times 3$ PMNS matrix Eq.(4). We reproduce their [50] six Majorana type-
phases for self-containment of our work,

$\displaystyle\beta_{1}=\arg{U_{e1}U^{\ast}_{e2}}\,,$
$\displaystyle\beta_{2}=\arg{U_{\mu 1}U^{\ast}_{\mu 2}}\,,$
$\displaystyle\beta_{3}=\arg{U_{\tau 1}U^{\ast}_{\tau 2}}\,,$ (6)
$\displaystyle\gamma_{1}=\arg{U_{e1}U^{\ast}_{e3}}\,,$
$\displaystyle\gamma_{2}=\arg{U_{\mu 1}U^{\ast}_{\mu 3}}\,,$
$\displaystyle\gamma_{3}=\arg{U_{\tau 1}U^{\ast}_{\tau 3}}\,.$ (7)

To reproduce the $3\times 3$ PMNS matrix of Eq.(4), from the Majorana type-
phases Eq.(6) and Eq.(7), we must consider the construction of unitarity
triangles. Two types of triangles can be created from the PMNS matrix, Dirac
and Majorana triangles. We are interested in Majorana triangles and their
dependence on the Majorana type-phases. The three Majorana triangles are
derived by multiplying the columns of the PMNS matrix,

$\displaystyle U_{e1}U^{\ast}_{e2}+U_{\mu 1}U^{\ast}_{\mu 2}+U_{\tau
1}U^{\ast}_{\tau 2}$ $\displaystyle=0,$ triangle 1, (8) $\displaystyle
U_{e1}U^{\ast}_{e3}+U_{\mu 1}U^{\ast}_{\mu 3}+U_{\tau 1}U^{\ast}_{\tau 3}$
$\displaystyle=0,$ triangle 2, (9) $\displaystyle U_{e2}U^{\ast}_{e3}+U_{\mu
2}U^{\ast}_{\mu 3}+U_{\tau 2}U^{\ast}_{\tau 3}$ $\displaystyle=0,$ triangle 3.
(10)

The equation for triangle 1, Eq.(8) is connected with the $\beta_{k}$ Majorana
type-phases of Eq.(6). We illustrate that connection between the triangle
vectors of Eq.(8) and $\beta_{k}$ in with figure 1. From figure 1, we can see
all sides of the triangle are constructed from the rephasing invariants
$U_{\alpha k}U^{\ast}_{\alpha j}$. Thus, the Majorana triangles are not free
to rotate in the complex plain and their orientation is physically meaningful
[50, 51].

Figure 1:  The first Majorana triangle depends on the Majorana type-phases
$\beta_{k}$. The sides of the triangle are constructed from the first two rows
of the PMNS matrix. The orientation is physically meaningful and can only be
determined by knowledge of the Majorana type-phases.

In addition, the internal angles of the triangle $\zeta_{k}$ are,

$\displaystyle\zeta_{1}=\pi-\left(\beta_{3}-\beta_{2}\right),$
$\displaystyle\zeta_{2}=\left(\beta_{3}-\beta_{1}\right)-\pi,$
$\displaystyle\zeta_{3}=\pi-\left(\beta_{2}-\beta_{1}\right).$ (11)

Triangle 2 is connected to the $\gamma_{k}$ of Eq.(7) and behaves similarly to
triangle 1 where the orientation is physically meaningful. The internal angles
of triangle 2 are calculated by replacing $\beta_{k}$ with $\gamma_{k}$ in
Eq.(11). Lastly, the connection to triangle 3 comes from subtraction between
the $\beta_{k}$’s and $\gamma_{k}$’s and its behavior depends on the other two
triangles.

Branco and Rebelo relate the mixing angles of the parametrized PMNS matrix in
Eq.(5) to the Majorana type-phases using the law of sines on the Majorana
triangles. For example, to find $\theta_{12}$ they take,

$\begin{split}\tan^{2}\theta_{12}&{}=\frac{\absolutevalue{U_{e2}U^{\ast}_{e3}}^{2}}{\absolutevalue{U_{e1}U^{\ast}_{e3}}^{2}}\\\
&{}=\frac{\absolutevalue{\sin(\gamma_{1}-\gamma_{3})}\absolutevalue{\sin(\gamma_{2}-\gamma_{1})}\absolutevalue{\sin(\gamma_{3}-\gamma_{2}-(\beta_{3}-\beta_{2}))}}{\absolutevalue{\sin(\gamma_{1}-\gamma_{3}-(\beta_{1}-\beta_{3}))}\absolutevalue{\sin(\gamma_{2}-\gamma_{1}-(\beta_{2}-\beta_{1}))}\absolutevalue{\sin(\gamma_{3}-\gamma_{2})}}.\end{split}$
(12)

Then they prove how the Dirac phase $\delta$ relates to the Majorana type-
phases via the common area of the triangles,

$\begin{split}A&{}=\frac{1}{16}\absolutevalue{\sin 2\theta_{12}\sin
2\theta_{13}\sin 2\theta_{23}\cos\theta_{13}\sin\delta}\\\
&{}=\frac{1}{2}\absolutevalue{\cos\theta_{12}\cos\theta_{13}\sin\theta_{13}}^{2}\frac{\absolutevalue{\sin(\gamma_{1}-\gamma_{2})}\absolutevalue{\sin(\gamma_{1}-\gamma_{3})}}{\absolutevalue{\sin(\gamma_{2}-\gamma_{3})}}.\end{split}$
(13)

We use the relations of Branco and Rebelo like Eq.(12) and Eq.(13) to update
their analysis [50] of the Majorana triangles based on the recent neutrino
oscillation global fit. Specifically, we use the best fit values of NuFITv5.1
calculated by the Nu-Fit collaboration [52]. Compared to the analysis of
ref.[50], we include additional data from neutrino oscillation experiments.
For simplicity, we consider the Majorana phases to be absent
$\alpha_{21}=\alpha_{31}=0$, and later we will include them. The vectors for
Triangle 1 in fig.2 are,

$\begin{split}U_{e1}U^{\ast}_{e2}&=0.4496-0.0i,\\\ U_{\mu 1}U^{\ast}_{\mu
2}&=-0.2295+0.05711i,\\\ U_{\tau 1}U^{\ast}_{\tau
2}&=-0.2200-0.05711i,\end{split}$ (14)

where we only write the first four significant figures with no rounding.

Figure 2:  We build the three Majorana triangle using the best fit values of
NuFITv5.1[52] for normal hierarchy. The orientation of the triangles depends
on the Majorana type-phases; $\beta_{k}$ for triangle 1 that is related to the
Majorana phase $\alpha_{21}$, $\gamma_{k}$ for triangle 2, that is related to
the Majorana phase $\alpha_{31}$, and for triangle 3 the subtraction between
the Majorana type-phases, $\gamma_{k}-\beta_{k}$. For this instance, we set
the Majorana phase $\alpha_{21}=\alpha_{31}=0$, because it is experimentally
not determined. NuFITv5.1 assumes the PMNS matrix is unitary, so all triangles
are completely closed.

We treat the vectors for Triangles 2 and 3 of fig.2 similarly as Triangle 1,

$\displaystyle\left.\begin{aligned} U_{e1}U^{\ast}_{e3}&=-0.07945-0.09469i,\\\
U_{\mu 1}U^{\ast}_{\mu 3}&=-0.2354+0.04261i,\\\ U_{\tau 1}U^{\ast}_{\tau
3}&=0.3149+0.05208i;\end{aligned}\right\\}\text{Triangle 2}$ (15)
$\displaystyle\left.\begin{aligned} U_{e2}U^{\ast}_{e3}&=-0.05251-0.06258i,\\\
U_{\mu 2}U^{\ast}_{\mu 3}&=0.4339+0.02816i,\\\ U_{\tau 2}U^{\ast}_{\tau
3}&=-0.3814+0.03442i.\end{aligned}\right\\}\text{Triangle 3}$ (16)

In all three triangles for fig.2 the vectors completely close to form a
triangle. This is because the data we are using from NuFITv5.1 assume the PMNS
matrix Eq.(5) is unitary. If the unitary assumption was relaxed, the triangles
could be open. Furthermore, all three triangles are scalene, which is
different from the isosceles triangles of Branco and Rebelo because we do not
consider perturbations around tri-bimaximal mixing.

Lastly, current and near future neutrino oscillation experiments can not
determine the orientation of the Majorana triangles. This is because the
oscillation experiments measure the Dirac phase $\delta$, but can not measure
the Majorana phases $\alpha_{21}$ and $\alpha_{31}$ that determine the
triangle orientation. For example, we can change the values of
$\Delta\alpha_{21}$ to rotate triangle 1 in figure 3.

Figure 3:  We rotate the first Majorana triangle from the best fit values of
NuFITv5.1[52] for normal hierarchy. The orientation of the triangle depends on
the Majorana type-phases $\beta_{k}$, thus it is rotated by varying between
$0<\alpha_{21}<\pi$.

Branco and Rebelo discussed this problem [50], stating that measurements of
the Dirac phase can only provide insight to the differences of the Majorana
type-phases, i.e.$\gamma_{1}-\gamma_{3}$; so the triangle orientation can not
be determined. Whereas, the addition of Majorana type-phases
$\beta_{i}+\beta_{j}$ and $\gamma_{i}+\gamma_{j}$ are connected to the
Majorana phases $\alpha_{21}$ and $\alpha_{31}$, respectively. They then go on
to discuss the possibility of determining the triangle orientation with
neutrinoless double-beta decay and flavor sensitive leptogenesis[53, 54, 55].
In this paper, we will describe an additional method to determine the triangle
orientation based on our work of lepton number oscillations[56, 57].

##  3 Time evolution of lepton number and Majorana type-phases

To determine the triangle orientation, we study the time evolution of lepton
number focusing on the dependence of the Majorana type-phases. In the works
[56, 57], the lepton family numbers $L_{\alpha}^{M}$ for the Majorana
neutrinos are defined. Then, the time evolution of their expectation values
are obtained. We rewrite the Majorana $L_{\alpha}^{M}(t)$

$\bra*{\sigma}L_{\alpha}^{M}(t)\ket*{\sigma}\\\
=\sum_{i,j}^{3}\left[\real(U^{\ast}_{\alpha i}U_{\sigma i}U_{\alpha
j}U^{\ast}_{\sigma
j})\left(\cos\\{E_{i}(\mathbf{q})t\\}\cos\\{E_{j}(\mathbf{q})t\\}+\frac{\mathbf{q}^{2}}{E_{i}(\mathbf{q})E_{j}(\mathbf{q})}\sin\\{E_{i}(\mathbf{q})t\\}\sin\\{E_{j}(\mathbf{q})t\\}\right)\right.\\\
-\imaginary(U^{\ast}_{\alpha i}U_{\sigma i}U_{\alpha j}U^{\ast}_{\sigma
j})\left(\frac{\absolutevalue{\mathbf{q}}}{E_{i}(\mathbf{q})}\sin\\{E_{i}(\mathbf{q})t\\}\cos\\{E_{j}(\mathbf{q})t\\}-\frac{\absolutevalue{\mathbf{q}}}{E_{j}(\mathbf{q})}\cos\\{E_{i}(\mathbf{q})t\\}\sin\\{E_{j}(\mathbf{q})t\\}\right)\\\
\left.-\real(U^{\ast}_{\alpha i}U^{\ast}_{\sigma i}U_{\alpha j}U_{\sigma
j})\frac{m_{i}}{E_{i}(\mathbf{q})}\frac{m_{j}}{E_{j}(\mathbf{q})}\sin\\{E_{i}(\mathbf{q})t\\}\sin\\{E_{j}(\mathbf{q})t\\}\right].$
(17)

For our notation the initial state $\sigma$ denotes the neutrino with a
definite flavor i.e., $\sigma=e,\mu,\tau$ and the momentum $\mathbf{q}$. The
energy of the mass states we define as
$E_{i,j}(\mathbf{q})=\sqrt{\mathbf{q}^{2}+m_{i,j}^{2}}$. To extract the
Majorana type-phases we are interested in the third term of Eq.(17) that
depends on the PMNS combination of $\real(U^{\ast}_{\alpha i}U^{\ast}_{\sigma
i}U_{\alpha j}U_{\sigma j})$. To isolate the third term, consider that the
first two terms can be completely determined from neutrino oscillation and
neutrino mass experiments. This means we can subtract those terms from the
Majorana expectation value to isolate the third term. We define the quantity
$L^{Q}_{\alpha}(t)$ which denotes the sum of the first and the second term of
Eq.(17),

$\bra*{\sigma}{L^{Q}}_{\alpha}(t)\ket*{\sigma}\\\
=\sum_{i,j}^{3}\left[\real(U^{\ast}_{\alpha i}U_{\sigma i}U_{\alpha
j}U^{\ast}_{\sigma
j})\left(\cos\\{E_{i}(\mathbf{q})t\\}\cos\\{E_{j}(\mathbf{q})t\\}+\frac{\mathbf{q}^{2}}{E_{i}(\mathbf{q})E_{j}(\mathbf{q})}\sin\\{E_{i}(\mathbf{q})t\\}\sin\\{E_{j}(\mathbf{q})t\\}\right)\right.\\\
\left.-\imaginary(U^{\ast}_{\alpha i}U_{\sigma i}U_{\alpha j}U^{\ast}_{\sigma
j})\left(\frac{\absolutevalue{\mathbf{q}}}{E_{i}(\mathbf{q})}\sin\\{E_{i}(\mathbf{q})t\\}\cos\\{E_{j}(\mathbf{q})t\\}-\frac{\absolutevalue{\mathbf{q}}}{E_{j}(\mathbf{q})}\cos\\{E_{i}(\mathbf{q})t\\}\sin\\{E_{j}(\mathbf{q})t\\}\right)\right].$
(18)

We represent that subtraction process with the difference between the Majorana
expectation value Eq.(17) and the quantity Eq.(18),

$\bra*{\sigma}L_{\alpha}^{M-Q}(t)\ket*{\sigma}\equiv\bra*{\sigma}L_{\alpha}^{M}(t)\ket*{\sigma}-\bra*{\sigma}L_{\alpha}^{Q}(t)\ket*{\sigma}.$
(19)

This results in the isolated third term of,

$\begin{split}\bra*{\sigma}L_{\alpha}^{M-Q}(t)\ket*{\sigma}={}&-\sum_{i,j}^{3}\real(U^{\ast}_{\alpha
i}U^{\ast}_{\sigma i}U_{\alpha j}U_{\sigma
j})\frac{m_{i}}{E_{i}(\mathbf{q})}\frac{m_{j}}{E_{j}(\mathbf{q})}\sin\\{E_{i}(\mathbf{q})t\\}\sin\\{E_{j}(\mathbf{q})t\\},\\\
={}&-\sum_{i=1}^{3}\absolutevalue{U_{\alpha i}U_{\sigma
i}}^{2}\left(\frac{m_{i}\sin\\{E_{i}(\mathbf{q})t\\}}{E_{i}(\mathbf{q})}\right)^{2}\\\
&-2\sum_{i<j}\real(U^{\ast}_{\alpha i}U_{\alpha j}U^{\ast}_{\sigma i}U_{\sigma
j})\frac{m_{i}}{E_{i}(\mathbf{q})}\frac{m_{j}}{E_{j}(\mathbf{q})}\sin\\{E_{i}(\mathbf{q})t\\}\sin\\{E_{j}(\mathbf{q})t\\}.\end{split}$
(20)

Let us investigate the difference by focusing on the dependence of the
Majorana type-phases from Eq.(6). We take $\sigma=\alpha=e$ to obtain,

$\begin{split}\bra*{e}L_{e}^{M-Q}(t)\ket*{e}={}&-\sum_{i=1}^{3}\absolutevalue{U_{ei}}^{4}\left(\frac{m_{i}\sin\\{E_{i}(\mathbf{q})t\\}}{E_{i}(\mathbf{q})}\right)^{2}\\\
&-2\real\\{(U^{\ast}_{e1}U_{e2})^{2}\\}\frac{m_{1}\sin\\{E_{1}(\mathbf{q})t\\}}{E_{1}(\mathbf{q})}\frac{m_{2}\sin\\{E_{2}(\mathbf{q})t\\}}{E_{2}(\mathbf{q})}\\\
&-2\real\\{(U^{\ast}_{e2}U_{e3})^{2}\\}\frac{m_{2}\sin\\{E_{2}(\mathbf{q})t\\}}{E_{2}(\mathbf{q})}\frac{m_{3}\sin\\{E_{3}(\mathbf{q})t\\}}{E_{3}(\mathbf{q})}\\\
&-2\real\\{(U^{\ast}_{e1}U_{e3})^{2}\\}\frac{m_{1}\sin\\{E_{1}(\mathbf{q})t\\}}{E_{1}(\mathbf{q})}\frac{m_{3}\sin\\{E_{3}(\mathbf{q})t\\}}{E_{3}(\mathbf{q})}.\end{split}$
(21)

In the last three terms of Eq.(21), the PMNS combinations can be written with
the Majorana type-phases $2\beta_{1}$ and $2\gamma_{1}$ based on Eq.(6) and
Eq.(7). The explicit dependence of Eq.(21) on the Majorana type-phases is
then,

$\begin{split}\bra*{e}L_{e}^{M-Q}(t)\ket*{e}={}&-\sum_{i=1}^{3}\absolutevalue{U_{ei}}^{4}\left(\frac{m_{i}\sin\\{E_{i}(\mathbf{q})t\\}}{E_{i}(\mathbf{q})}\right)^{2}\\\
&-2\absolutevalue{U_{e1}U_{e2}}^{2}\cos(2\beta_{1})\frac{m_{1}\sin\\{E_{1}(\mathbf{q})t\\}}{E_{1}(\mathbf{q})}\frac{m_{2}\sin\\{E_{2}(\mathbf{q})t\\}}{E_{2}(\mathbf{q})}\\\
&-2\absolutevalue{U_{e2}U_{e3}}^{2}\cos(2(\gamma_{1}-\beta_{1}))\frac{m_{2}\sin\\{E_{2}(\mathbf{q})t\\}}{E_{2}(\mathbf{q})}\frac{m_{3}\sin\\{E_{3}(\mathbf{q})t\\}}{E_{3}(\mathbf{q})}\\\
&-2\absolutevalue{U_{e1}U_{e3}}^{2}\cos(2\gamma_{1})\frac{m_{1}\sin\\{E_{1}(\mathbf{q})t\\}}{E_{1}(\mathbf{q})}\frac{m_{3}\sin\\{E_{3}(\mathbf{q})t\\}}{E_{3}(\mathbf{q})}.\end{split}$
(22)

We can obtain similar results as Eq.(22) for the muon and tauon numbers of
$\alpha=\mu,\tau$,

$\begin{split}\bra*{e}L_{\mu}^{M-Q}(t)\ket*{e}=&{}-\sum_{i=1}^{3}\absolutevalue{U_{\mu
i}U_{ei}}^{2}\left(\frac{m_{i}\sin\\{E_{i}(\mathbf{q})t\\}}{E_{i}(\mathbf{q})}\right)^{2}\\\
&-2\absolutevalue{U^{\ast}_{e1}U_{e2}U^{\ast}_{\mu 1}U_{\mu
2}}\cos(\beta_{1}+\beta_{2})\frac{m_{1}\sin\\{E_{1}(\mathbf{q})t\\}}{E_{1}(\mathbf{q})}\frac{m_{2}\sin\\{E_{2}(\mathbf{q})t\\}}{E_{2}(\mathbf{q})}\\\
&-2\absolutevalue{U^{\ast}_{e2}U_{e3}U^{\ast}_{\mu 2}U_{\mu
3}}\cos(\gamma_{1}-\beta_{1}+\gamma_{2}-\beta_{2})\frac{m_{2}\sin\\{E_{2}(\mathbf{q})t\\}}{E_{2}(\mathbf{q})}\frac{m_{3}\sin\\{E_{3}(\mathbf{q})t\\}}{E_{3}(\mathbf{q})}\\\
&-2\absolutevalue{U^{\ast}_{e1}U_{e3}U^{\ast}_{\mu 1}U_{\mu
3}}\cos(\gamma_{1}+\gamma_{2})\frac{m_{1}\sin\\{E_{1}(\mathbf{q})t\\}}{E_{1}(\mathbf{q})}\frac{m_{3}\sin\\{E_{3}(\mathbf{q})t\\}}{E_{3}(\mathbf{q})}.\end{split}$
(23)

In contrast to Eq.(22), the muon number $L_{\mu}^{M-Q}(t)$ of Eq.(23) depends
on the Majorana type-phases $\beta_{1}+\beta_{2}$ and $\gamma_{1}+\gamma_{2}$.
Finally, the tauon number $L_{\tau}^{M-Q}(t)$ is written as,

$\begin{split}\bra*{e}L_{\tau}^{M-Q}(t)\ket*{e}={}&-\sum_{i=1}^{3}\absolutevalue{U_{\tau
i}U_{ei}}^{2}\left(\frac{m_{i}\sin\\{E_{i}(q)t\\}}{E_{i}(q)}\right)^{2}\\\
&-2\absolutevalue{U^{\ast}_{e1}U_{e2}U^{\ast}_{\tau 1}U_{\tau
2}}\cos(\beta_{1}+\beta_{3})\frac{m_{1}\sin\\{E_{1}(\mathbf{q})t\\}}{E_{1}(\mathbf{q})}\frac{m_{2}\sin\\{E_{2}(\mathbf{q})t\\}}{E_{2}(\mathbf{q})}\\\
&-2\absolutevalue{U^{\ast}_{e2}U_{e3}U^{\ast}_{\tau 2}U_{\tau
3}}\cos(\gamma_{1}-\beta_{1}+\gamma_{3}-\beta_{3})\frac{m_{2}\sin\\{E_{2}(\mathbf{q})t\\}}{E_{2}(\mathbf{q})}\frac{m_{3}\sin\\{E_{3}(\mathbf{q})t\\}}{E_{3}(\mathbf{q})}\\\
&-2\absolutevalue{U^{\ast}_{e1}U_{e3}U^{\ast}_{\tau 1}U_{\tau
3}}\cos(\gamma_{1}+\gamma_{3})\frac{m_{1}\sin\\{E_{1}(\mathbf{q})t\\}}{E_{1}(\mathbf{q})}\frac{m_{3}\sin\\{E_{3}(\mathbf{q})t\\}}{E_{3}(\mathbf{q})}.\end{split}$
(24)

The last three terms are written with the Majorana type-phases
$\beta_{1}+\beta_{3}$ and $\gamma_{1}+\gamma_{3}$. All the lepton family
numbers Eqs.(22-24) depend on the sum of Majorana type-phases
$\beta_{1}+\beta_{j}$ and $\gamma_{1}+\gamma_{j}$ where $j=1,2,3$. In contrast
to the difference of Majorana type-phases in an unitarity triangle such as
$\beta_{i}-\beta_{j}$ and $\gamma_{i}-\gamma_{j}$, the sum of the Majorana
type-phases depend on the Majorana phases $\alpha_{21}$ and $\alpha_{31}$ of
Eq.(4).

As we have seen in Eqs.(22-24), one can identify the part which depends on the
summation of the Majorana type-phases, i.e., $\beta_{i}+\beta_{j}$ and
$\gamma_{i}+\gamma_{j}$. One may determine the combination of Majorana type-
phases by fitting the curve for Eq.(20) as we vary those unknown parameters.
This method may require the measurement of the long time behaviour of the time
evolution. In the next section, we propose other quantities directly related
to them from the derivatives of the expectation values with respect to a slice
of time.

##  4 Determination of Majorana type-phases and the lightest neutrino masses
with the lepton numbers

In this section, we derive the formula to determine the lightest neutrino mass
and Majorana phases with time dependence of the lepton numbers. We consider
two cases,

  1. 1.

there are three massive neutrinos where two Majorana phases are allowed,

  2. 2.

there is a massless neutrino and only one Majorana phase is allowed.

We take the second-order time derivative of Eq.(17) at the initial time $t=0$,

$\begin{split}\frac{d^{2}}{dt^{2}}\bra*{\sigma}L_{\alpha}^{M}(t)\ket*{\sigma}|_{t=0}&=-\sum_{i,j}^{3}\real(U_{\alpha
i}^{\ast}U_{\sigma i}U_{\alpha j}U_{\sigma
j}^{\ast})\left(m_{i}^{2}+m_{j}^{2}\right)-2\sum_{i,j}^{3}{\rm Re}(U_{\alpha
i}^{\ast}U_{\sigma i}^{\ast}U_{\alpha j}U_{\sigma j})m_{i}m_{j}\\\
&=-2\sum_{i}^{3}\delta_{\alpha\sigma}\absolutevalue{U_{\sigma
i}}^{2}m_{i}^{2}-2\sum_{i,j}^{3}\real(U_{\alpha i}^{\ast}U_{\sigma
i}^{\ast}U_{\alpha j}U_{\sigma j})m_{i}m_{j}.\end{split}$ (25)

The first term is independent of the Majorana phases and the second term
depends on the Majorana phases though the PMNS combination
$\real(U^{\ast}_{\alpha i}U^{\ast}_{\sigma i}U_{\alpha j}U_{\sigma j})$. Then,
the second-order derivative of the total lepton number
$L^{M}(t)=\sum_{\alpha}L_{\alpha}^{M}(t)$ is given by,

$\frac{d^{2}}{dt^{2}}\bra*{\sigma}L^{M}(t)\ket*{\sigma}|_{t=0}=-4\sum_{i}^{3}m_{i}^{2}|U_{\sigma
i}|^{2}.$ (26)

We use the total lepton number of Eq.(26) to rewrite the first term of Eq.(25)
resulting in,

$\frac{d^{2}}{dt^{2}}\bra*{\sigma}L_{\alpha}^{M}(t)\ket*{\sigma}|_{t=0}=\delta_{\alpha\sigma}\frac{1}{2}\frac{d^{2}}{dt^{2}}\bra*{\sigma}L^{M}(t)\ket*{\sigma}|_{t=0}-2\sum_{i,j}^{3}{\rm
Re}(U_{\alpha i}^{*}U_{\sigma i}^{*}U_{\alpha j}U_{\sigma j})m_{i}m_{j}.$ (27)

Using Eq.(26), one can derive the lightest neutrino mass for the normal
hierarchy and the inverted hierarchy cases in terms of the second-order
derivative of the total lepton number.

$\displaystyle
m_{1}^{2}=-\frac{1}{4}\frac{d^{2}}{dt^{2}}\bra*{\sigma}L^{M}(t)\ket*{\sigma}|_{t=0}-\Delta
m^{2}_{21}|U_{\sigma 2}|^{2}-\Delta m^{2}_{31}|U_{\sigma 3}|^{2}$ Normal, (28)
$\displaystyle
m_{3}^{2}=-\frac{1}{4}\frac{d^{2}}{dt^{2}}\bra*{\sigma}L^{M}(t)\ket*{\sigma}|_{t=0}-\Delta
m^{2}_{13}|U_{\sigma 1}|^{2}-\Delta m^{2}_{23}|U_{\sigma 2}|^{2}$ Inverted.
(29)

To investigate the Majorana type-phases, we consider two cases for the lepton
families. First we set $\alpha=\sigma$ in Eq.(27) and obtain the following
formula,

$\frac{d^{2}}{dt^{2}}\bra*{\sigma}L_{\sigma}^{M}(t)\ket*{\sigma}|_{t=0}=\frac{1}{2}\frac{d^{2}}{dt^{2}}\bra*{\sigma}L^{M}(t)\ket*{\sigma}|_{t=0}-2\sum_{i,j}^{3}(U_{\sigma
i}^{*}U_{\sigma j})^{2}m_{i}m_{j}.$ (30)

Then we set $\alpha\neq\sigma$ in Eq.(27) to obtain,

$\frac{d^{2}}{dt^{2}}\bra*{\sigma}L_{\alpha}^{M}(t)\ket*{\sigma}|_{t=0}=-2\sum_{i,j}^{3}{\rm
Re}(U_{\alpha i}^{*}U_{\alpha j}U_{\sigma i}^{*}U_{\sigma j})m_{i}m_{j}.$ (31)

Next, by specifying the family indices $\sigma$ and $\alpha$ in Eq.(30) and
Eq.(31) we can clarify the dependencies on the Majorana type-phases from
Eqs.(6-7). We take $\sigma=e$ in Eq.(30) to show the dependencies on the
Majorana type-phases $\beta_{1}$ and $\gamma_{1}$,

$\frac{d^{2}}{dt^{2}}\bra*{e}L_{e}^{M}(t)\ket*{e}|_{t=0}=\frac{1}{2}\frac{d^{2}}{dt^{2}}\bra*{e}L^{M}(t)\ket*{e}|_{t=0}-2\sum_{i=1}^{3}m_{i}^{2}\absolutevalue{U_{ei}}^{4}-4m_{1}m_{2}\absolutevalue{U_{e1}U^{\ast}_{e2}}^{2}\cos(2\beta_{1})\\\
-4m_{2}m_{3}\absolutevalue{U_{e2}U^{\ast}_{e3}}^{2}\cos(2\beta_{1}-2\gamma_{1})-4m_{3}m_{1}\absolutevalue{U_{e1}U^{\ast}_{e3}}^{2}\cos(2\gamma_{1}).$
(32)

In contrast, for Eq.(31) we take $\sigma=e$ and $\alpha=\mu$,

$\frac{d^{2}}{dt^{2}}\bra*{e}L_{\mu}^{M}(t)\ket*{e}|_{t=0}=-2\sum_{i}^{3}\absolutevalue{U_{\mu
i}}^{2}\absolutevalue{U_{ei}}^{2}m_{i}^{2}-4\absolutevalue{U_{\mu
1}^{\ast}U_{\mu 2}U_{e1}^{\ast}U_{e2}}\cos(\beta_{1}+\beta_{2})m_{1}m_{2}\\\
\qquad-4\absolutevalue{U_{\mu 2}^{\ast}U_{\mu
3}U_{e2}^{\ast}U_{e3}}\cos(\beta_{1}-\gamma_{1}+\beta_{2}-\gamma_{2})m_{2}m_{3}\\\
-4\absolutevalue{U_{\mu 3}^{\ast}U_{\mu
1}U_{e3}^{\ast}U_{e1}}\cos(\gamma_{1}+\gamma_{2})m_{3}m_{1},$ (33)

where we have dependence on the additional Majorana type-phases $\beta_{2}$
and $\gamma_{2}$ compared to Eq.(32). Lastly we take $\alpha=\tau$,

$\frac{d^{2}}{dt^{2}}\bra*{e}L_{\tau}^{M}(t)\ket*{e}|_{t=0}=-2\sum_{i}^{3}\absolutevalue{U_{\tau
i}}^{2}\absolutevalue{U_{ei}}^{2}m_{i}^{2}-4\absolutevalue{U_{\tau
1}^{\ast}U_{\tau 2}U_{e1}^{\ast}U_{e2}}\cos(\beta_{1}+\beta_{3})m_{1}m_{2}\\\
\qquad-4\absolutevalue{U_{\tau 2}^{\ast}U_{\tau
3}U_{e2}^{\ast}U_{e3}}\cos(\beta_{1}-\gamma_{1}+\beta_{3}-\gamma_{3})m_{2}m_{3}\\\
-4\absolutevalue{U_{\tau 3}^{\ast}U_{\tau
1}U_{e3}^{\ast}U_{e1}}\cos(\gamma_{1}+\gamma_{3})m_{3}m_{1},$ (34)

which results in the dependence on the Majorana type-phases $\beta_{3}$ and
$\gamma_{3}$.

As discussed near the end of section 2, the Majorana type-phases $\beta_{i}$
and $\gamma_{i}$ change for variation of the Majorana phases
$\Delta\alpha_{21}$ and $\Delta\alpha_{31}$,

$\displaystyle\beta_{i}+\beta_{j}\to\beta_{i}+\beta_{j}-\Delta\alpha_{21},$
(35)
$\displaystyle\gamma_{i}+\gamma_{j}\to\gamma_{i}+\gamma_{j}-\Delta\alpha_{31}.$
(36)

The time derivatives of all the three lepton family numbers in Eq.(32-34) are
dependent on summations of the Majorana type-phases i.e.,
$\beta_{i}+\beta_{j}$ and $\gamma_{i}+\gamma_{j}$. From Eqs.(35-36) we know
$\beta_{i}+\beta_{j}$ and $\gamma_{i}+\gamma_{j}$ are sensitive to the
Majorana phases $\Delta\alpha_{21}$ and $\Delta\alpha_{31}$. Thus, the two
Majorana phases $\alpha_{21}$ and $\alpha_{31}$ can be determined by knowing
the time derivatives of the two or more lepton family numbers. In addition, we
can resolve the lightest neutrino mass from the time derivatives of the total
lepton number in Eqs.(28-29). This is a method to determine the triangle
orientations and the lightest neutrino mass based on lepton number
oscillations.

##  5 Considerations when the lightest neutrino is massless

In this section, we consider the lightest neutrino to be massless. As an
example, the $(3,2)$ Type-I seesaw model can predict such a massless neutrino
with three active neutrinos and two heavy right-handed Majorana neutrinos[58].
In this framework, which neutrino is massless depends on the hierarchy,

$\displaystyle m_{1}=0,$ $\displaystyle 0<m_{2}<m_{3},$ Normal hierarchy, (37)
$\displaystyle m_{3}=0,$ $\displaystyle 0<m_{1}<m_{2},$ Inverted hierarchy.
(38)

Because a massless neutrino is free to be re-phased, a few of the six Majorana
type-phases from the three massive Majorana neutrinos in the previous section,
4, are no longer invariants. In the normal hierarchy case, the invariant
combinations of the three $\beta_{i}$ and three $\gamma_{i}$ become,

$\displaystyle\arg(U_{e2}U_{e3}^{\ast})=\gamma_{1}-\beta_{1},$
$\displaystyle\arg(U_{\mu 2}U_{\mu 3}^{\ast})=\gamma_{2}-\beta_{2},$
$\displaystyle\arg(U_{\tau 2}U_{\tau 3}^{\ast})=\gamma_{3}-\beta_{3}.$ (39)

This is because, $\beta_{i}$ and $\gamma_{i}$ are no longer re-phasing
invariants by themselves and one can form four invariant quartets,

$\displaystyle\arg(U_{e1}^{\ast}U_{e2}U_{\mu 1}U_{\mu
2}^{\ast})=\beta_{2}-\beta_{1}$ (40)
$\displaystyle\arg(U_{e1}^{\ast}U_{e2}U_{\tau 1}U_{\tau
2}^{\ast})=\beta_{3}-\beta_{1}$ (41)
$\displaystyle\arg(U_{e1}^{\ast}U_{e3}U_{\mu 1}U_{\mu
3}^{\ast})=\gamma_{2}-\gamma_{1}$ (42)
$\displaystyle\arg(U_{e1}^{\ast}U_{e3}U_{\tau 1}U_{\tau
3}^{\ast})=\gamma_{3}-\gamma_{1}.$ (43)

The Majorana type-phases defined in Eq.(39) and the four arguments defined in
Eqs.(40-43) are not independent. In addition to a Majorana type-phase
$\gamma_{1}-\beta_{1}$ in Eq.(39), one can choose the four arguments of the
quartets as independent re-phasing invariant combinations. This means the
other two Majorana type-phases can be written using them,

$\begin{split}\gamma_{i}-\beta_{i}&=(\gamma_{1}-\beta_{1})+(\gamma_{i}-\beta_{i})-(\gamma_{1}-\beta_{1})\\\
&=(\gamma_{1}-\beta_{1})+(\gamma_{i}-\gamma_{1})-(\beta_{i}-\beta_{1})\qquad(i=2,3).\end{split}$
(44)

A similar situation occurs for the inverted hierarchy massless neutrino of
$m_{3}=0$. The re-phasing invariant Majorana type-phases become $\beta_{i}$,
for $i=1,2,3$. Then, one can choose a Majorana type-phase and the four
arguments in Eqs.(40-43) as independent re-phasing invariants.

###  5.1 Neutrinoless double beta decay and lepton number

In this subsection, we relate one independent Majorana type-phase to physical
observables, such as $\absolutevalue{m_{\nu ee}}$ related to the neutrinoless
double beta decay rate and time evolution of lepton numbers. For the normal
hierarchy case, the $\absolutevalue{m_{\nu ee}}$ is given by the following
formula,

$\absolutevalue{m_{\nu
ee}}^{2}_{\text{norm}}=m_{2}^{2}\absolutevalue{U_{e2}}^{4}+m_{3}^{2}\absolutevalue{U_{e3}}^{4}+2m_{2}m_{3}\absolutevalue{U_{e2}}^{2}\absolutevalue{U_{e3}}^{2}\cos{2(\gamma_{1}-\beta_{1})}.$
(45)

Whereas for the inverted hierarchy case it is,

$\absolutevalue{m_{\nu
ee}}^{2}_{\text{inv}}=m_{1}^{2}\absolutevalue{U_{e1}}^{4}+m_{2}^{2}\absolutevalue{U_{e2}}^{4}+2m_{1}m_{2}\absolutevalue{U_{e1}}^{2}\absolutevalue{U_{e2}}^{2}\cos{2\beta_{1}}.$
(46)

From the expressions in Eq.(45-46), the determination of
$\absolutevalue{m_{\nu ee}}_{\text{norm},\text{inv}}$ is sufficient to
identify a single Majorana type-phase; provided the other moduli of the PMNS
matrix elements in Eq.(5) and two non-zero masses of the neutrinos are
extracted from the neutrino oscillation experiments.

Next, we investigate the time evolution of lepton number when the lightest
neutrino is massless, as in Eqs.(37-38). From Eq.(28), for the normal
hierarchy case, the second-order time derivative of the total lepton number is
related to the combination of the two non-vanishing masses;

$\frac{1}{4}\frac{d^{2}}{dt^{2}}\bra*{\sigma}L^{M}(t)\ket*{\sigma}|_{t=0}=-m^{2}_{2}\absolutevalue{U_{\sigma
2}}^{2}-m^{2}_{3}\absolutevalue{U_{\sigma 3}}^{2}.$ (47)

Then we can rewrite the second-order time derivatives of the electron, muon,
and tauon lepton numbers of Eqs.(32-34) as,

$\displaystyle\frac{d^{2}}{dt^{2}}\bra*{e}L_{e}^{M}(t)\ket*{e}|_{t=0}=\frac{d^{2}}{dt^{2}}\bra*{e}L^{M}(t)\ket*{e}|_{t=0}-4m_{2}m_{3}\absolutevalue{U_{e2}U^{\ast}_{e3}}^{2}\cos(2\beta_{1}-2\gamma_{1}),$
(48)
$\displaystyle\frac{d^{2}}{dt^{2}}\bra*{e}L_{\mu}^{M}(t)\ket*{e}|_{t=0}=-2\sum_{i=2}^{3}\absolutevalue{U_{\mu
i}}^{2}\absolutevalue{U_{ei}}^{2}m_{i}^{2}-4\absolutevalue{U_{\mu
2}^{\ast}U_{\mu
3}U_{e2}^{\ast}U_{e3}}\cos(\beta_{1}-\gamma_{1}+\beta_{2}-\gamma_{2})m_{2}m_{3},$
(49)
$\displaystyle\frac{d^{2}}{dt^{2}}\bra*{e}L_{\tau}^{M}(t)\ket*{e}|_{t=0}=-2\sum_{i=2}^{3}\absolutevalue{U_{\tau
i}}^{2}\absolutevalue{U_{ei}}^{2}m_{i}^{2}-4\absolutevalue{U_{\tau
2}^{*}U_{\tau
3}U_{e2}^{*}U_{e3}}\cos(\beta_{1}-\gamma_{1}+\beta_{3}-\gamma_{3})m_{2}m_{3}.$
(50)

The second-order time derivative for the normal hierarchy in Eq.(48) can be
used to determine one of the Majorana like-phases; $\gamma_{1}-\beta_{1}$. The
other two Majorana type-phases, $\gamma_{i}-\beta_{i}(i=1,2)$,can be
determined through Eq.(44) by knowing the arguments of the quartet in Eq.(40).
This allows us to learn of the orientation of triangle 3 in Fig.2.
Alternatively, with the three second-order time derivatives of Eqs.(48)-(50)
one can determine three Majorana type-phases $\gamma_{i}-\beta_{i}(i=1-3)$.

In the inverted hierarchy case $m_{3}$ is the lightest neutrino mass and the
second-order time derivative of the total lepton number becomes,

$\frac{1}{4}\frac{d^{2}}{dt^{2}}\bra*{\sigma}L^{M}(t)\ket*{\sigma}|_{t=0}=-m^{2}_{2}\absolutevalue{U_{\sigma
2}}^{2}-m^{2}_{1}\absolutevalue{U_{\sigma 1}}^{2}.$ (51)

The major difference between the normal hierarchy of Eq.(47) and the inverted
hierarchy of Eq.(51) is the last mass multiplied by PMNS matrix term. This
modifies the second-order time derivatives to become,

$\displaystyle\frac{d^{2}}{dt^{2}}\bra*{e}L_{e}^{M}(t)\ket*{e}|_{t=0}=\frac{d^{2}}{dt^{2}}\bra*{e}L^{M}(t)\ket*{e}|_{t=0}-4m_{1}m_{2}\absolutevalue{U_{e1}U^{\ast}_{e2}}^{2}\cos(2\beta_{1}),$
(52)
$\displaystyle\frac{d^{2}}{dt^{2}}\bra*{e}L_{\mu}^{M}(t)\ket*{e}|_{t=0}=-2\sum_{i}^{2}\absolutevalue{U_{\mu
i}}^{2}\absolutevalue{U_{ei}}^{2}m_{i}^{2}-4\absolutevalue{U_{\mu
1}^{\ast}U_{\mu 2}U_{e1}^{\ast}U_{e2}}\cos(\beta_{1}+\beta_{2})m_{1}m_{2},$
(53)
$\displaystyle\frac{d^{2}}{dt^{2}}\bra*{e}L_{\tau}^{M}(t)\ket*{e}|_{t=0}=-2\sum_{i}^{2}\absolutevalue{U_{\tau
i}}^{2}\absolutevalue{U_{ei}}^{2}m_{i}^{2}-4\absolutevalue{U_{\tau
1}^{\ast}U_{\tau 2}U_{e1}^{\ast}U_{e2}}\cos(\beta_{1}+\beta_{3})m_{1}m_{2}.$
(54)

Similar to the normal hierarchy equations of Eqs.(48-50), the second-order
time derivatives of the inverted hierarchy in Eqs.(52-54) allow for a complete
determination of the Majorana type-phases $\beta_{i}$ and the orientation of
triangle 1 in Fig.2.

###  5.2 Numerical illustration for two neutrino generations

In this subsection, we illustrate the analytical results of the relations
among the lightest neutrino mass, a Majorana type-phase, and the second-order
time derivative of lepton numbers. We illustrate them using the two generation
model. In the model, there is only two Majorana type-phases, $\beta_{1}$ and
$\beta_{2}$ defined in Eq.(6). They are not independent because the unitary
relation

$U_{e1}U_{e2}^{\ast}=-U_{\mu 1}U_{\mu 2}^{\ast},$ (55)

holds. It leads to the relation between the two Majorana type-phases,

$\beta_{1}=\arg(U_{e1}U_{e2}^{\ast})=\arg(U_{\mu 1}U_{\mu
2}^{\ast})-\pi=\beta_{2}-\pi.$ (56)

Adopting the following parametrization for a 2 by 2 unitary PMNS matrix,

$\begin{pmatrix}\cos\theta_{12}&\sin\theta_{12}e^{i\frac{\alpha_{21}}{2}}\\\
-\sin\theta_{12}&\cos\theta_{12}e^{i\frac{\alpha_{21}}{2}}\\\ \end{pmatrix},$
(57)

the Majorana type-phase $\beta_{1}$ is related to a Majorana phase as,
$\beta_{1}=-\frac{\alpha_{21}}{2}$. Contrary to the three generation model,
one can not write $\tan^{2}\theta_{12}$ in terms of the Majorana type-phase.

In this model, the lightest neutrino mass $m_{1}$ and a Majorana type-phase
$\beta_{1}$ are unknown parameters to be determined by the time evolution of
lepton numbers. We assume that a mass squared difference $\Delta m^{2}_{21}$
and a mixing angle $\theta_{12}$ are measured by the oscillation experiment.
The time evolution of the electron and muon numbers for two generation model
are given by,

$\begin{split}\bra{e}L_{e}(t)\ket{e}&=c_{12}^{4}\left(1-\frac{2m_{1}^{2}\sin^{2}(E_{1}t)}{E_{1}^{2}}\right)+s_{12}^{4}\left(1-\frac{2m_{2}^{2}\sin^{2}(E_{2}t)}{E_{2}^{2}}\right)\\\
&\quad+s_{12}^{2}c_{12}^{2}\left\\{\left(1+\frac{q^{2}-m_{1}m_{2}\cos(2\beta_{1})}{E_{1}E_{2}}\right)\cos\\{(E_{1}-E_{2})t\\}\right.\\\
&\quad\left.+\left(1-\frac{q^{2}-m_{1}m_{2}\cos(2\beta_{1})}{E_{1}E_{2}}\right)\cos\\{(E_{1}+E_{2})t\\}\right\\},\end{split}$
(58)
$\displaystyle\begin{split}\bra{e}L_{\mu}(t)\ket{e}=&c_{12}^{2}s_{12}^{2}\left(\left(1-\frac{2m_{1}^{2}\sin^{2}(E_{1}t)}{E_{1}^{2}}\right)+\left(1-\frac{2m_{2}^{2}\sin^{2}(E_{2}t)}{E_{2}^{2}}\right)\right)\\\
&-s_{12}^{2}c_{12}^{2}\left\\{\left(1+\frac{q^{2}-m_{1}m_{2}\cos(2\beta_{1})}{E_{1}E_{2}}\right)\cos\\{(E_{1}-E_{2})t\\}\right.\\\
&\left.+\left(1-\frac{q^{2}-m_{1}m_{2}\cos(2\beta_{1})}{E_{1}E_{2}}\right)\cos\\{(E_{1}+E_{2})t\\}\right\\}.\end{split}$
(59)

One can also compute the expectation values of the total lepton number
$L(t)=L_{e}(t)+L_{\mu}(t)$ and the difference
$L_{e-\mu}(t)=L_{e}(t)-L_{\mu}(t)$. The straightforward calculation leads to
the second order time derivatives for the total lepton number at $t=0$ as
follows,

$\frac{d^{2}}{dt^{2}}\bra{e}L(t)\ket{e}\Bigr{|}_{t=0}=-4(m_{1}^{2}c_{12}^{2}+m_{2}^{2}s_{12}^{2}).$
(60)

Figure 4 shows that total lepton number sharply decreases at $t=0$ for the
larger the lightest neutrino mass $m_{1}$.

Figure 4:  The time dependence of the total lepton number for different
lightest neutrino masses. The momentum $q=0.002$(eV). The dashed line shows
the case for $m_{1}=0.01$(eV) and the solid line shows the case for
$m_{1}=0.02$(eV). We use $\Delta m^{2}_{21}=7.42\times 10^{-5}\text{eV}^{2}$
and $\sin(\theta_{12})=0.551$.

This can be also understood by solving the lightest neutrino mass with the
second derivatives of Eq.(60), a mixing angle and a mass squared difference.

$m_{1}^{2}=-\frac{1}{4}\frac{d^{2}}{dt^{2}}\bra{e}L(t)\ket{e}\Bigr{|}_{t=0}-\Delta
m^{2}_{21}s_{12}^{2}.$ (61)

The second order time derivative for the electron minus muon number
$L_{e-\mu}$ at $t=0$ is given as follows,

$\frac{d^{2}}{dt^{2}}\bra{e}L_{e-\mu}(t)\ket{e}\Bigr{|}_{t=0}=-4\absolutevalue{m_{\nu
ee}}^{2}_{\text{two gene.}},$ (62)

where $\absolutevalue{m_{\nu ee}}^{2}_{\text{two gene.}}$ is given by the same
formula in Eq.(46) as the inverted hierarchy case with $m_{3}=0$. By
substituting the mixing angle to $|U_{ei}|$ ($i=1,2$) in Eq.(57), it is given
by,

$\absolutevalue{m_{\nu ee}}^{2}_{\text{two
gene.}}=m_{1}^{2}{c_{12}}^{4}+m_{2}^{2}{s_{12}}^{4}+2m_{1}m_{2}{c_{12}}^{2}{s_{12}}^{2}\cos{2\beta_{1}}.$
(63)

Eq.(62) tells us that $L_{e-\mu}$ sharply decreases for the larger
$\absolutevalue{m_{\nu ee}}$. Fig.(5) shows the dependence of $L_{e-\mu}$ on
Majorana type-phase $\beta_{1}$. Because $\absolutevalue{m_{\nu ee}}$ is the
largest for $\beta_{1}=0$ and the smallest for $\beta_{1}=\pi$, as shown in
Eq.(63), Fig.(5) numerically confirms the dependence on the
$\absolutevalue{m_{\nu ee}}$ of Eq.(62).

Figure 5:  The time dependence of $L_{e-\mu}=L_{e}-L_{\mu}$for different
choice of the Majorana type-phase $\beta_{1}$. $q=0.002$(eV) $m_{1}=0.01$(eV)
and $\sin(\theta_{12})=0.551$. The dashed line shows the case for
$\beta_{1}=0$. The dot-dashed line shows the case for
$2\beta_{1}=\frac{\pi}{2}$ and solid line shows the case for $2\beta_{1}=\pi$.
$\Delta m^{2}_{21}$ is the same as in Fig.4.

One can also solve for a Majorana type-phase $\beta_{1}$ with the second order
derivative of Eqs.(62), a mixing angle and a mass squared difference.

$\cos(2\beta_{1})=\frac{-\frac{1}{4}\frac{d^{2}}{dt^{2}}\bra{e}L_{e-\mu}(t)\ket{e}\bigr{|}_{t=0}-(m_{1}^{2}c_{12}^{4}+m_{2}^{2}s_{12}^{4})}{2m_{1}m_{2}c_{12}^{2}s_{12}^{2}}.$
(64)

Eqs.(61-64) can be used to determine the lightest neutrino mass and the
Majorana type-phase once the time derivatives for $\ddot{L}|_{t=0}$ and
$\ddot{L}_{e-\mu}|_{t=0}$ are measured.

##  6 Concluding Remarks

We have investigated an approach to extract the Majorana type-phases of Branco
and Rebelo using the time evolution of lepton numbers. The specific
combinations of Majorana type-phases in the same triangle are related to the
orientation of unitarity triangles for the PMNS matrix, and the Majorana
phases $\alpha_{21}$ and $\alpha_{31}$. After taking the second-order time
derivative of the lepton number expectation values, the dependencies on the
summation of Majorana type-phases i.e., $\beta_{i}+\beta_{j}$ and
$\gamma_{i}+\gamma_{j}$, can be determined. Thus allowing the extraction of
the orientation of unitarity triangles for the PMNS matrix, and the Majorana
phases.

We view our result as complimentary to using neutrinoless double-beta decay
for determining the orientation of unitarity triangles for the PMNS matrix,
and the Majorana phases. We also show the time derivative of the total lepton
number is sensitive to the lightest neutrino mass. The above features are also
numerically demonstrated for two generation toy model.

For the future, we are interested in identifying a possible experimental setup
used to measure the quantities discussed.

### Acknowledgement

This work is supported by Japan Society for the Promotion of Science (JSPS)
KAKENHI Grant Number JP17K05418 (T.M) and JP21K13923(K.Y). One author, N.J.B,
would like to express thanks to the Japanese government Ministry of Education,
Culture, Sports, Science, and Technology (MEXT) for the financial support
during the writing of this work.

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