A full parametrization of the $9\times 9$ active-sterile flavor mixing matrix
in the inverse or linear seesaw scenario of massive neutrinos

He-chong Han1,2 ***E-mail<EMAIL_ADDRESS>Zhi-zhong Xing1,2,3
†††E-mail<EMAIL_ADDRESS>
1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, China
2School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
3Center of High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

The inverse and linear seesaw scenarios are two typical extensions of the
canonical seesaw mechanism, which contain much more sterile degrees of freedom
but can naturally explain the smallness of three active neutrino masses at a
sufficiently low energy scale (e.g., the TeV scale). To fully describe the
mixing among three active neutrinos, three sterile neutrinos and three extra
gauge-singlet neutral fermions in either of these two seesaw paradigms, we
present the first full parametrization of the $9\times 9$ flavor mixing matrix
in terms of 36 rotation angles and 36 CP-violating phases. The exact inverse
and linear seesaw formulas are derived, respectively; and possible deviations
of the $3\times 3$ active neutrino mixing matrix from its unitary limit are
discussed by calculating the effective Jarlskog invariants and unitarity
nonagons.

##  1 Introduction

The absence of the right-handed neutrino fields in the standard model (SM)
makes it impossible to accommodate a gauge-invariant neutrino mass term like
the Yukawa-interaction term for the charged leptons [1]. Therefore, the
simplest way to go beyond the SM and generate finite masses for three active
neutrinos (i.e., $\nu_{e}$, $\nu_{\mu}$ and $\nu_{\tau}$) is to introduce
three right-handed neutrino fields and allow for the gauge-invariant neutrino
Yukawa interactions. But such a Dirac neutrino mass term has no way to explain
why the active neutrinos are so light as compared with the respective charged
leptons, which participate in the standard weak interactions in the same way.
Moreover, the right-handed neutrino fields and their charge-conjugate
counterparts (which are left-handed) can form a gauge-invariant but lepton-
number-violating Majorana mass term, which should not be discarded in general.
Combining this Majorana neutrino mass term with the aforementioned Dirac mass
term leads us to the canonical (type-I) seesaw mechanism [2, 3, 4, 5, 6], in
which all the six neutrino mass eigenstates have the Majorana nature (i.e.,
their charge conjugates are equal to themselves; or equivalently, the
neutrinos are their own antiparticles [7]) and the smallness of three active
neutrino masses $m_{i}$ can naturally be attributed to the largeness of three
sterile neutrino masses $M_{i}$ (for $i=1,2,3$). Note that $M_{i}$ act as the
cut-off scales in the SM effective field theory with a unique dimension-five
Weinberg operator for generating tiny masses of the active neutrinos [8], and
the “fulcrum” of such a seesaw is just around the electroweak scale
$\Lambda_{\rm EW}\sim 10^{2}$ GeV. Unfortunately, the naturalness prerequisite
$M_{i}\gg\Lambda_{\rm EW}$ implies that this conventional seesaw picture has
essentially lost its testability in any feasible high-energy experiments [9].

In this regard a possible way out is to lower the seesaw scale (i.e., the
values of $M_{i}$) by introducing more extra degrees of freedom, and the
typical examples of this kind include the inverse (or double) seesaw scenario
[10, 11] and the linear seesaw scenario [11, 12, 13, 14] 111A natural
extension of the inverse seesaw scenario to the multiple seesaw scenario has
been proposed in Ref. [15].. Besides the left-handed neutrino fields
$\nu_{\alpha{\rm L}}$ and the right-handed neutrino fields $N_{\alpha{\rm R}}$
(for $\alpha=e,\mu,\tau$), the inverse or linear seesaw mechanism requires the
adding of three neutral SM gauge-singlet fermions $S_{\alpha{\rm R}}$ (for
$\alpha=e,\mu,\tau$) and one scalar singlet $\Phi$. With such new degrees of
freedom, a generic gauge-invariant neutrino mass term can be written as
follows:

$\displaystyle-{\cal L}_{\rm mass}=\overline{\ell_{\rm
L}}Y_{\nu}\widetilde{H}N_{\rm R}+\frac{1}{2}\overline{(N_{\rm R})^{c}}M_{\rm
R}N_{\rm R}+\overline{\nu_{\rm L}}Y^{\prime}_{\nu}\Phi S_{\rm
R}+\overline{(N_{\rm R})^{c}}Y_{S}\Phi S_{\rm R}+\frac{1}{2}\overline{(S_{\rm
R})^{c}}\mu S_{\rm R}+{\rm h.c.}\;,$ (1)

where $\nu_{\rm L}=(\nu_{e\rm L}\hskip 4.26773pt\nu_{\mu\rm L}\hskip
4.26773pt\nu_{\tau\rm L})^{T}$, $N_{\rm R}=(N_{e\rm R}\hskip
4.26773ptN_{\mu\rm R}\hskip 4.26773ptN_{\tau\rm R})^{T}$ and $S_{\rm
R}=(\nu_{e\rm R}\hskip 4.26773pt\nu_{\mu\rm R}\hskip 4.26773pt\nu_{\tau\rm
R})^{T}$ stand respectively for the vector columns of left-handed neutrino
fields, right-handed neutrino fields and gauge-singlet fermion fields,
$\ell_{\rm L}$ denotes the lepton doublet of the SM, $\widetilde{H}$ is
defined as $\widetilde{H}\equiv{\rm i}\sigma_{2}H^{*}$ with $H$ being the
Higgs doublet of the SM, and the mass matrices $M_{\rm R}$ and $\mu$ are both
symmetric. After spontaneous symmetry breaking, Eq. (1) turns out to be

$\displaystyle-{\cal L}^{\prime}_{\rm
mass}=\frac{1}{2}\overline{\left[\nu_{\rm L}\hskip 5.69046pt(N_{\rm
R})^{c}\hskip 5.69046pt(S_{\rm R})^{c}\right]}\left(\begin{matrix}0&M_{\rm
D}&\varepsilon\cr M^{T}_{\rm D}&M_{\rm
R}&M_{S}\cr\varepsilon^{T}&M^{T}_{S}&\mu\cr\end{matrix}\right)\left(\begin{matrix}(\nu_{\rm
L})^{c}\cr N_{\rm R}\cr S_{\rm R}\cr\end{matrix}\right)+{\rm h.c.}\;,$ (2)

where $M_{\rm D}=Y_{\nu}\langle H\rangle$,
$\varepsilon=Y^{\prime}_{\nu}\langle\Phi\rangle$ and
$M_{S}=Y_{S}\langle\Phi\rangle$. Switching off the extra neutral gauge-singlet
fermion fields, we are immediately left with the neutrino mass term of the
canonical seesaw mechanism from Eqs. (1) and (2). To partly reduce the number
of new free parameters associated with the overall $9\times 9$ neutrino mass
matrix in Eq. (2), one has considered the following two simplified but
phenomenologically interesting cases.

  * •

The inverse seesaw scenario with $M_{\rm R}=0$ and $\varepsilon=0$ [10, 11],
where the mass scale of $\mu$ is naturally small because it is the only
lepton-number-violating term, and the mass scale of $M_{S}$ can be
considerably larger than that of $M_{\rm D}$ if $\langle\Phi\rangle\gg\langle
H\rangle$ and $Y_{S}\sim Y_{\nu}$ are taken. The effective Majorana neutrino
mass matrix for three active neutrinos is therefore given by

$\displaystyle M_{\nu}\simeq M_{\rm D}(M^{T}_{S})^{-1}\mu\hskip
1.70709pt(M_{S})^{-1}M^{T}_{\rm D}\;$ (3)

in the leading-order approximation. So the tiny mass eigenvalues of $M_{\nu}$
are mainly attributed to the smallness of $\mu$, and they are further
suppressed by the largeness of $M_{S}$ as compared with $M_{\rm D}$ even if
$M_{S}\sim{\cal O}(1)$ TeV is assumed.

  * •

The linear seesaw scenario with $M_{\rm R}=0$ and $\mu=0$ [11, 12, 13, 14],
where $\varepsilon$ is the only lepton-number-violating term, and thus its
mass scale is naturally small. In this case the effective Majorana neutrino
mass matrix for three active neutrinos is found to be

$\displaystyle M_{\nu}\simeq-\varepsilon M^{-1}_{S}M^{T}_{\rm D}-(\varepsilon
M^{-1}_{S}M^{T}_{\rm D})^{T}\;$ (4)

in the leading-order approximation. As a result, the smallness of $M_{\nu}$ is
mainly ascribed to that of $\varepsilon$ and further suppressed by the ratio
of the mass scales of $M_{\rm D}$ to $M_{S}$.

In either case the seesaw scale can be successfully lowered to the TeV scale
which is experimentally accessible at the Large Hadron Collider (LHC).

Although a lot of work has been done to explore various phenomenological
consequences of the inverse or linear seesaw scenario (see, e.g., Refs. [16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]), a complete
description of the $9\times 9$ active-sterile flavor mixing matrix for such a
seesaw picture has been lacking. Following the full Euler-like parametrization
of active-sterile flavor mixing in the type-I or type-(I+II) seesaw mechanism
with three sterile neutrinos done previously by one of us [31, 32, 33], here
we are going to present the first full parametrization of the $9\times 9$
flavor mixing matrix in terms of 36 Euler rotation angles and 36 CP-violating
phases in the inverse or linear seesaw scenario. The exact inverse or linear
seesaw formula will also be derived; and possible deviations of the $3\times
3$ active neutrino mixing matrix (i.e., the so-called Pontecorvo-Maki-
Nakagawa-Sakata lepton flavor mixing matrix [34, 35, 36]) from its unitary
limit will be discussed by calculating the effective Jarlskog invariants and
unitarity nonagons. This study is expected to be useful for generally
describing possible interplays between three active neutrino species and up to
six species of extra (light or heavy) degrees of freedom no matter whether a
specific seesaw scenario is taken into account or not.

The remaining parts of this paper are organized as follows. In section 2 we
parameterize the $9\times 9$ flavor mixing matrix with 36 Euler rotation
angles and 36 CP-violating phases in such a way that the primary unitary
$3\times 3$ flavor mixing submatrices of three active neutrinos, three sterile
neutrinos and three extra neutral fermions are linked and modified by the
intermediate flavor mixing matrices. Section 3 is devoted to deriving the
exact inverse or linear seesaw formula and reproducing its approximate
expression shown in Eq. (3) or Eq. (4). In section 4 we examine possible
deviations of the $3\times 3$ PMNS matrix from its unitary limit by
calculating the effective Jarlskog invariants and discussing the unitarity
nonagons. A brief summary of our main results, together with some further
discussions, is finally made in section 5.

##  2 Parametrization of the $9\times 9$ flavor mixing matrix

To fully describe the parameter space of flavor mixing between three active
neutrinos and their six sterile counterparts, one may write out the $9\times
9$ active-sterile flavor mixing matrix $\cal U$ in terms of 36 two-dimensional
unitary matrices $O_{ij}$ ($1\leq i<j\leq 9$) as follows:

$\displaystyle{\cal U}$ $\displaystyle=$
$\displaystyle\big{(}O_{89}O_{79}O_{69}O_{59}O_{49}O_{39}O_{29}O_{19}\big{)}\big{(}O_{78}O_{68}O_{58}O_{48}O_{38}O_{28}O_{18}\big{)}\big{(}O_{67}O_{57}O_{47}O_{37}O_{27}O_{17}\big{)}$
(5)
$\displaystyle\big{(}O_{56}O_{46}O_{36}O_{26}O_{16}\big{)}\big{(}O_{45}O_{35}O_{25}O_{15}\big{)}\big{(}O_{34}O_{24}O_{14}\big{)}\big{(}O_{23}O_{13}O_{12}\big{)}\;,$

where only the combination $\big{(}O_{23}O_{13}O_{12}\big{)}$ is purely
associated with the flavor mixing sector of three active neutrinos. Following
the example of Refs. [31, 32, 33], let us adjust the ordering of $O_{ij}$ in
$\cal U$ in such a way that

$\displaystyle{\cal U}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}I&0&0\cr 0&I&0\cr
0&0&S_{0}\end{matrix}\right)L_{3}\left(\begin{matrix}I&0&0\cr
0&U^{\prime}_{0}&0\cr
0&0&I\end{matrix}\right)L_{2}L_{1}\left(\begin{matrix}U_{0}&0&0\cr 0&I&0\cr
0&0&I\end{matrix}\right)\;,$ (6)

where $I$ denotes the $3\times 3$ identity matrix; the $3\times 3$ unitary
matrices

$\displaystyle U_{0}=O_{23}O_{13}O_{12}\;,\quad
U^{\prime}_{0}=O_{56}O_{46}O_{45}\;,\quad S_{0}=O_{89}O_{79}O_{78}$ (7)

describe the primary flavor mixing sectors of three active neutrinos, three
sterile neutrinos and three extra neutral fermions, respectively; and the
$9\times 9$ matrices

$\displaystyle L_{1}$ $\displaystyle=$ $\displaystyle
O_{36}O_{26}O_{16}O_{35}O_{25}O_{15}O_{34}O_{24}O_{14}=\left(\begin{matrix}A_{1}&R_{1}&0\cr
S_{1}&B_{1}&0\cr 0&0&I\end{matrix}\right)\;,$ $\displaystyle L_{2}$
$\displaystyle=$ $\displaystyle
O_{39}O_{29}O_{19}O_{38}O_{28}O_{18}O_{37}O_{27}O_{17}=\left(\begin{matrix}A_{2}&0&R_{2}\cr
0&I&0\cr S_{2}&0&B_{2}\end{matrix}\right)\;,$ $\displaystyle L_{3}$
$\displaystyle=$ $\displaystyle
O_{69}O_{59}O_{49}O_{68}O_{58}O_{48}O_{67}O_{57}O_{47}=\left(\begin{matrix}I&0&0\cr
0&A_{3}&R_{3}\cr 0&S_{3}&B_{3}\end{matrix}\right)\;\hskip 19.91684pt$ (8)

describe the interplays between any two of the three sectors. There are two
obvious advantages associated with the parametrization of $\cal U$ in Eq. (6):
on the one hand, the flavor mixing angles of $L_{1}$ and $L_{2}$ are naturally
small because they measure the strength of interplay between the active sector
and two sterile sectors; on the other hand, the three flavor sectors will
automatically become decoupled if the off-diagonal elements of $L_{1}$,
$L_{2}$ and $L_{3}$ are all switched off.

To be more specific, the three primary $3\times 3$ unitary flavor mixing
matrices are given by

$\displaystyle U_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{12}c_{13}&\hat{s}^{*}_{12}c_{13}&\hat{s}^{*}_{13}\cr-\hat{s}_{12}c_{23}-c_{12}\hat{s}_{13}\hat{s}^{*}_{23}&c_{12}c_{23}-\hat{s}^{*}_{12}\hat{s}_{13}\hat{s}^{*}_{23}&c_{13}\hat{s}^{*}_{23}\cr\hat{s}_{12}\hat{s}_{23}-c_{12}\hat{s}_{13}c_{23}&~{}-c_{12}\hat{s}_{23}-\hat{s}^{*}_{12}\hat{s}_{13}c_{23}~{}&c_{13}c_{23}\end{matrix}\right)\;,$
$\displaystyle U^{\prime}_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{45}c_{46}&\hat{s}^{*}_{45}c_{46}&\hat{s}^{*}_{46}\cr-\hat{s}_{45}c_{56}-c_{45}\hat{s}_{46}\hat{s}^{*}_{56}&c_{45}c_{56}-\hat{s}^{*}_{45}\hat{s}_{46}\hat{s}^{*}_{56}&c_{46}\hat{s}^{*}_{56}\cr\hat{s}_{45}\hat{s}_{56}-c_{45}\hat{s}_{46}c_{56}&~{}-c_{45}\hat{s}_{56}-\hat{s}^{*}_{45}\hat{s}_{46}c_{56}~{}&c_{46}c_{56}\end{matrix}\right)\;,$
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{78}c_{79}&\hat{s}^{*}_{78}c_{79}&\hat{s}^{*}_{79}\cr-\hat{s}_{78}c_{89}-c_{78}\hat{s}_{79}\hat{s}^{*}_{89}&c_{78}c_{89}-\hat{s}^{*}_{78}\hat{s}_{79}\hat{s}^{*}_{89}&c_{79}\hat{s}^{*}_{89}\cr\hat{s}_{78}\hat{s}_{89}-c_{78}\hat{s}_{79}c_{89}&~{}-c_{78}\hat{s}_{89}-\hat{s}^{*}_{78}\hat{s}_{79}c_{89}~{}&c_{79}c_{89}\end{matrix}\right)\;,$
(9)

where $c_{ij}\equiv\cos\theta_{ij}$ and $\hat{s}_{ij}\equiv e^{{\rm
i}\delta_{ij}}\sin\theta_{ij}$ with $\theta_{ij}$ and $\delta_{ij}$ (for
$1\leq i<j\leq 9$) standing respectively for the rotation angles and phase
angles. Moreover, the four $3\times 3$ submatrices of $L_{1}$ read as

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{14}c_{15}c_{16}&0&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{14}c_{15}\hat{s}_{16}\hat{s}^{*}_{26}-c_{14}\hat{s}_{15}\hat{s}^{*}_{25}c_{26}\\\
-\hat{s}_{14}\hat{s}^{*}_{24}c_{25}c_{26}\end{array}&c_{24}c_{25}c_{26}&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{14}c_{15}\hat{s}_{16}c_{26}\hat{s}^{*}_{36}+c_{14}\hat{s}_{15}\hat{s}^{*}_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
-c_{14}\hat{s}_{15}c_{25}\hat{s}^{*}_{35}c_{36}+\hat{s}_{14}\hat{s}^{*}_{24}c_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
+\hat{s}_{14}\hat{s}^{*}_{24}\hat{s}_{25}\hat{s}^{*}_{35}c_{36}-\hat{s}_{14}c_{24}\hat{s}^{*}_{34}c_{35}c_{36}\end{array}&\begin{array}[]{l}-c_{24}c_{25}\hat{s}_{26}\hat{s}^{*}_{36}-c_{24}\hat{s}_{25}\hat{s}^{*}_{35}c_{36}\\\
-\hat{s}_{24}\hat{s}^{*}_{34}c_{35}c_{36}\end{array}&c_{34}c_{35}c_{36}\cr\end{matrix}\right)\;,$
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{14}c_{24}c_{34}&0&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{14}c_{24}\hat{s}^{*}_{34}\hat{s}_{35}-c_{14}\hat{s}^{*}_{24}\hat{s}_{25}c_{35}\\\
-\hat{s}^{*}_{14}\hat{s}_{15}c_{25}c_{35}\end{array}&c_{15}c_{25}c_{35}&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{14}c_{24}\hat{s}^{*}_{34}c_{35}\hat{s}_{36}+c_{14}\hat{s}^{*}_{24}\hat{s}_{25}\hat{s}^{*}_{35}\hat{s}_{36}\\\
-c_{14}\hat{s}^{*}_{24}c_{25}\hat{s}_{26}c_{36}+\hat{s}^{*}_{14}\hat{s}_{15}c_{25}\hat{s}^{*}_{35}\hat{s}_{36}\\\
+\hat{s}^{*}_{14}\hat{s}_{15}\hat{s}^{*}_{25}\hat{s}_{26}c_{36}-\hat{s}^{*}_{14}c_{15}\hat{s}_{16}c_{26}c_{36}\end{array}&\begin{array}[]{l}-c_{15}c_{25}\hat{s}^{*}_{35}\hat{s}_{36}-c_{15}\hat{s}^{*}_{25}\hat{s}_{26}c_{36}\\\
-\hat{s}^{*}_{15}\hat{s}_{16}c_{26}c_{36}\end{array}&c_{16}c_{26}c_{36}\cr\end{matrix}\right)\;;\hskip
25.6073pt$ (10)

and

$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\hat{s}^{*}_{14}c_{15}c_{16}&\hat{s}^{*}_{15}c_{16}&\hat{s}^{*}_{16}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-\hat{s}^{*}_{14}c_{15}\hat{s}_{16}\hat{s}^{*}_{26}-\hat{s}^{*}_{14}\hat{s}_{15}\hat{s}^{*}_{25}c_{26}\\\
+c_{14}\hat{s}^{*}_{24}c_{25}c_{26}\end{array}&-\hat{s}^{*}_{15}\hat{s}_{16}\hat{s}^{*}_{26}+c_{15}\hat{s}^{*}_{25}c_{26}&c_{16}\hat{s}^{*}_{26}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-\hat{s}^{*}_{14}c_{15}\hat{s}_{16}c_{26}\hat{s}^{*}_{36}+\hat{s}^{*}_{14}\hat{s}_{15}\hat{s}^{*}_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
-\hat{s}^{*}_{14}\hat{s}_{15}c_{25}\hat{s}^{*}_{35}c_{36}-c_{14}\hat{s}^{*}_{24}c_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
-c_{14}\hat{s}^{*}_{24}\hat{s}_{25}\hat{s}^{*}_{35}c_{36}+c_{14}c_{24}\hat{s}^{*}_{34}c_{35}c_{36}\end{array}&\begin{array}[]{l}-\hat{s}^{*}_{15}\hat{s}_{16}c_{26}\hat{s}^{*}_{36}-c_{15}\hat{s}^{*}_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
+c_{15}c_{25}\hat{s}^{*}_{35}c_{36}\end{array}&c_{16}c_{26}\hat{s}^{*}_{36}\cr\end{matrix}\right)\;,$
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}-\hat{s}_{14}c_{24}c_{34}&-\hat{s}_{24}c_{34}&-\hat{s}_{34}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}\hat{s}_{14}c_{24}\hat{s}^{*}_{34}\hat{s}_{35}+\hat{s}_{14}\hat{s}^{*}_{24}\hat{s}_{25}c_{35}\\\
-c_{14}\hat{s}_{15}c_{25}c_{35}\end{array}&\hat{s}_{24}\hat{s}^{*}_{34}\hat{s}_{35}-c_{24}\hat{s}_{25}c_{35}&-c_{34}\hat{s}_{35}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}\hat{s}_{14}c_{24}\hat{s}^{*}_{34}c_{35}\hat{s}_{36}-\hat{s}_{14}\hat{s}^{*}_{24}\hat{s}_{25}\hat{s}^{*}_{35}\hat{s}_{36}\\\
+\hat{s}_{14}\hat{s}^{*}_{24}c_{25}\hat{s}_{26}c_{36}+c_{14}\hat{s}_{15}c_{25}\hat{s}^{*}_{35}\hat{s}_{36}\\\
+c_{14}\hat{s}_{15}\hat{s}^{*}_{25}\hat{s}_{26}c_{36}-c_{14}c_{15}\hat{s}_{16}c_{26}c_{36}\end{array}&\begin{array}[]{l}\hat{s}_{24}\hat{s}^{*}_{34}c_{35}\hat{s}_{36}+c_{24}\hat{s}_{25}\hat{s}^{*}_{35}\hat{s}_{36}\\\
-c_{24}c_{25}\hat{s}_{26}c_{36}\end{array}&-c_{34}c_{35}\hat{s}_{36}\cr\end{matrix}\right)\;.\hskip
25.6073pt$ (11)

The exact results in Eqs. (10) and (11) have already been obtained in Refs.
[31, 32, 33] for the canonical (type-I) seesaw mechanism. All the nine flavor
mixing angles associated with $A_{1}$, $B_{1}$, $R_{1}$ and $S_{1}$ are very
small and can at most reach the level of ${\cal O}(0.1)$ [37, 38, 39, 40], as
they describe the strength of active-sterile (or light-heavy) neutrino mixing
and thus are well constrained by current neutrino oscillation data and
precision measurements of various electroweak processes. In fact, the
smallness of $\theta_{i4}$, $\theta_{i5}$ and $\theta_{i6}$ (for $i=1,2,3$)
implies that $A_{1}$ and $B_{1}$ are essentially equal to the identity matrix
$I$, and the magnitude of each element of $R_{1}$ and $S_{1}$ is at most of
${\cal O}(0.1)$. Similarly, the four $3\times 3$ submatrices of $L_{2}$ can be
expressed as

$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{17}c_{18}c_{19}&0&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{17}c_{18}\hat{s}_{19}\hat{s}^{*}_{29}-c_{17}\hat{s}_{18}\hat{s}^{*}_{28}c_{29}\\\
-\hat{s}_{17}\hat{s}^{*}_{27}c_{28}c_{29}\end{array}&c_{27}c_{28}c_{29}&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{17}c_{18}\hat{s}_{19}c_{29}\hat{s}^{*}_{39}+c_{17}\hat{s}_{18}\hat{s}^{*}_{28}\hat{s}_{29}\hat{s}^{*}_{39}\\\
-c_{17}\hat{s}_{18}c_{28}\hat{s}^{*}_{38}c_{39}+\hat{s}_{17}\hat{s}^{*}_{27}c_{28}\hat{s}_{29}\hat{s}^{*}_{39}\\\
+\hat{s}_{17}\hat{s}^{*}_{27}\hat{s}_{28}\hat{s}^{*}_{38}c_{39}-\hat{s}_{17}c_{27}\hat{s}^{*}_{37}c_{38}c_{39}\end{array}&\begin{array}[]{l}-c_{27}c_{28}\hat{s}_{29}\hat{s}^{*}_{39}-c_{27}\hat{s}_{28}\hat{s}^{*}_{38}c_{39}\\\
-\hat{s}_{27}\hat{s}^{*}_{37}c_{38}c_{39}\end{array}&c_{37}c_{38}c_{39}\cr\end{matrix}\right)\;,$
$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{17}c_{27}c_{37}&0&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{17}c_{27}\hat{s}^{*}_{37}\hat{s}_{38}-c_{17}\hat{s}^{*}_{27}\hat{s}_{28}c_{38}\\\
-\hat{s}^{*}_{17}\hat{s}_{18}c_{28}c_{38}\end{array}&c_{18}c_{28}c_{38}&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{17}c_{27}\hat{s}^{*}_{37}c_{38}\hat{s}_{39}+c_{17}\hat{s}^{*}_{27}\hat{s}_{28}\hat{s}^{*}_{38}\hat{s}_{39}\\\
-c_{17}\hat{s}^{*}_{27}c_{28}\hat{s}_{29}c_{39}+\hat{s}^{*}_{17}\hat{s}_{18}c_{28}\hat{s}^{*}_{38}\hat{s}_{39}\\\
+\hat{s}^{*}_{17}\hat{s}_{18}\hat{s}^{*}_{28}\hat{s}_{29}c_{39}-\hat{s}^{*}_{17}c_{18}\hat{s}_{19}c_{29}c_{39}\end{array}&\begin{array}[]{l}-c_{18}c_{28}\hat{s}^{*}_{38}\hat{s}_{39}-c_{18}\hat{s}^{*}_{28}\hat{s}_{29}c_{39}\\\
-\hat{s}^{*}_{18}\hat{s}_{19}c_{29}c_{39}\end{array}&c_{19}c_{29}c_{39}\cr\end{matrix}\right)\;;\hskip
25.6073pt$ (12)

and

$\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\hat{s}^{*}_{17}c_{18}c_{19}&\hat{s}^{*}_{18}c_{19}&\hat{s}^{*}_{19}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-\hat{s}^{*}_{17}c_{18}\hat{s}_{19}\hat{s}^{*}_{29}-\hat{s}^{*}_{17}\hat{s}_{18}\hat{s}^{*}_{28}c_{29}\\\
+c_{17}\hat{s}^{*}_{27}c_{28}c_{29}\end{array}&-\hat{s}^{*}_{18}\hat{s}_{19}\hat{s}^{*}_{29}+c_{18}\hat{s}^{*}_{28}c_{29}&c_{19}\hat{s}^{*}_{29}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-\hat{s}^{*}_{17}c_{18}\hat{s}_{19}c_{29}\hat{s}^{*}_{39}+\hat{s}^{*}_{17}\hat{s}_{18}\hat{s}^{*}_{28}\hat{s}_{29}\hat{s}^{*}_{39}\\\
-\hat{s}^{*}_{17}\hat{s}_{18}c_{28}\hat{s}^{*}_{38}c_{39}-c_{17}\hat{s}^{*}_{27}c_{28}\hat{s}_{29}\hat{s}^{*}_{39}\\\
-c_{17}\hat{s}^{*}_{27}\hat{s}_{28}\hat{s}^{*}_{38}c_{39}+c_{17}c_{27}\hat{s}^{*}_{37}c_{38}c_{39}\end{array}&\begin{array}[]{l}-\hat{s}^{*}_{18}\hat{s}_{19}c_{29}\hat{s}^{*}_{39}-c_{18}\hat{s}^{*}_{28}\hat{s}_{29}\hat{s}^{*}_{39}\\\
+c_{18}c_{28}\hat{s}^{*}_{38}c_{39}\end{array}&c_{19}c_{29}\hat{s}^{*}_{39}\cr\end{matrix}\right)\;,$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}-\hat{s}_{17}c_{27}c_{37}&-\hat{s}_{27}c_{37}&-\hat{s}_{37}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}\hat{s}_{17}c_{27}\hat{s}^{*}_{37}\hat{s}_{38}+\hat{s}_{17}\hat{s}^{*}_{27}\hat{s}_{28}c_{38}\\\
-c_{17}\hat{s}_{18}c_{28}c_{38}\end{array}&\hat{s}_{27}\hat{s}^{*}_{37}\hat{s}_{38}-c_{27}\hat{s}_{28}c_{38}&-c_{37}\hat{s}_{38}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}\hat{s}_{17}c_{27}\hat{s}^{*}_{37}c_{38}\hat{s}_{39}-\hat{s}_{17}\hat{s}^{*}_{27}\hat{s}_{28}\hat{s}^{*}_{38}\hat{s}_{39}\\\
+\hat{s}_{17}\hat{s}^{*}_{27}c_{28}\hat{s}_{29}c_{39}+c_{17}\hat{s}_{18}c_{28}\hat{s}^{*}_{38}\hat{s}_{39}\\\
+c_{17}\hat{s}_{18}\hat{s}^{*}_{28}\hat{s}_{29}c_{39}-c_{17}c_{18}\hat{s}_{19}c_{29}c_{39}\end{array}&\begin{array}[]{l}\hat{s}_{27}\hat{s}^{*}_{37}c_{38}\hat{s}_{39}+c_{27}\hat{s}_{28}\hat{s}^{*}_{38}\hat{s}_{39}\\\
-c_{27}c_{28}\hat{s}_{29}c_{39}\end{array}&-c_{37}c_{38}\hat{s}_{39}\cr\end{matrix}\right)\;.\hskip
25.6073pt$ (13)

One can see that the triangular matrices $A_{2}$ and $B_{2}$ have the same
forms as $A_{1}$ and $B_{1}$, and the former can be directly obtained from the
latter with the replacements $\theta_{i4}\to\theta_{i7}$,
$\theta_{i5}\to\theta_{i8}$, $\theta_{i6}\to\theta_{i9}$ and
$\delta_{i4}\to\delta_{i7}$, $\delta_{i5}\to\delta_{i8}$,
$\delta_{i6}\to\delta_{i9}$ (for $i=1,2,3$). The same is true for the forms of
$R_{2}$ and $S_{2}$ as compared respectively with those of $R_{1}$ and
$S_{1}$. It is obvious that the flavor mixing angles $\theta_{i7}$,
$\theta_{i8}$ and $\theta_{i9}$ are all small because they describe the
interplay between three active neutrinos and three extra neutral fermions.
Finally, the four $3\times 3$ submatrices of $L_{3}$ are

$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{47}c_{48}c_{49}&0&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{47}c_{48}\hat{s}_{49}\hat{s}^{*}_{59}-c_{47}\hat{s}_{48}\hat{s}^{*}_{58}c_{59}\\\
-\hat{s}_{47}\hat{s}^{*}_{57}c_{58}c_{59}\end{array}&c_{57}c_{58}c_{59}&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{47}c_{48}\hat{s}_{49}c_{59}\hat{s}^{*}_{69}+c_{47}\hat{s}_{48}\hat{s}^{*}_{58}\hat{s}_{59}\hat{s}^{*}_{69}\\\
-c_{47}\hat{s}_{48}c_{58}\hat{s}^{*}_{68}c_{69}+\hat{s}_{47}\hat{s}^{*}_{57}c_{58}\hat{s}_{59}\hat{s}^{*}_{69}\\\
+\hat{s}_{47}\hat{s}^{*}_{57}\hat{s}_{58}\hat{s}^{*}_{68}c_{69}-\hat{s}_{47}c_{57}\hat{s}^{*}_{67}c_{68}c_{69}\end{array}&\begin{array}[]{l}-c_{57}c_{58}\hat{s}_{59}\hat{s}^{*}_{69}-c_{57}\hat{s}_{58}\hat{s}^{*}_{68}c_{69}\\\
-\hat{s}_{57}\hat{s}^{*}_{67}c_{68}c_{69}\end{array}&c_{67}c_{68}c_{69}\cr\end{matrix}\right)\;,$
$\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{47}c_{57}c_{67}&0&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{47}c_{57}\hat{s}^{*}_{67}\hat{s}_{68}-c_{47}\hat{s}^{*}_{57}\hat{s}_{58}c_{68}\\\
-\hat{s}^{*}_{47}\hat{s}_{48}c_{58}c_{68}\end{array}&c_{48}c_{58}c_{68}&0\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-c_{47}c_{57}\hat{s}^{*}_{67}c_{68}\hat{s}_{69}+c_{47}\hat{s}^{*}_{57}\hat{s}_{58}\hat{s}^{*}_{68}\hat{s}_{69}\\\
-c_{47}\hat{s}^{*}_{57}c_{58}\hat{s}_{59}c_{69}+\hat{s}^{*}_{47}\hat{s}_{48}c_{58}\hat{s}^{*}_{68}\hat{s}_{69}\\\
+\hat{s}^{*}_{47}\hat{s}_{48}\hat{s}^{*}_{58}\hat{s}_{59}c_{69}-\hat{s}^{*}_{47}c_{48}\hat{s}_{49}c_{59}c_{69}\end{array}&\begin{array}[]{l}-c_{48}c_{58}\hat{s}^{*}_{68}\hat{s}_{69}-c_{48}\hat{s}^{*}_{58}\hat{s}_{59}c_{69}\\\
-\hat{s}^{*}_{48}\hat{s}_{49}c_{59}c_{69}\end{array}&c_{49}c_{59}c_{69}\cr\end{matrix}\right)\;;\hskip
25.6073pt$ (14)

and

$\displaystyle R_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\hat{s}^{*}_{47}c_{48}c_{49}&\hat{s}^{*}_{48}c_{49}&\hat{s}^{*}_{49}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-\hat{s}^{*}_{47}c_{48}\hat{s}_{49}\hat{s}^{*}_{59}-\hat{s}^{*}_{47}\hat{s}_{48}\hat{s}^{*}_{58}c_{59}\\\
+c_{47}\hat{s}^{*}_{57}c_{58}c_{59}\end{array}&-\hat{s}^{*}_{48}\hat{s}_{49}\hat{s}^{*}_{59}+c_{48}\hat{s}^{*}_{58}c_{59}&c_{49}\hat{s}^{*}_{59}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}-\hat{s}^{*}_{47}c_{48}\hat{s}_{49}c_{59}\hat{s}^{*}_{69}+\hat{s}^{*}_{47}\hat{s}_{48}\hat{s}^{*}_{58}\hat{s}_{59}\hat{s}^{*}_{69}\\\
-\hat{s}^{*}_{47}\hat{s}_{48}c_{58}\hat{s}^{*}_{68}c_{69}-c_{47}\hat{s}^{*}_{57}c_{58}\hat{s}_{59}\hat{s}^{*}_{69}\\\
-c_{47}\hat{s}^{*}_{57}\hat{s}_{58}\hat{s}^{*}_{68}c_{69}+c_{47}c_{57}\hat{s}^{*}_{67}c_{68}c_{69}\end{array}&\begin{array}[]{l}-\hat{s}^{*}_{48}\hat{s}_{49}c_{59}\hat{s}^{*}_{69}-c_{48}\hat{s}^{*}_{58}\hat{s}_{59}\hat{s}^{*}_{69}\\\
+c_{48}c_{58}\hat{s}^{*}_{68}c_{69}\end{array}&c_{49}c_{59}\hat{s}^{*}_{69}\cr\end{matrix}\right)\;,$
$\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}-\hat{s}_{47}c_{57}c_{67}&-\hat{s}_{57}c_{67}&-\hat{s}_{67}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}\hat{s}_{47}c_{57}\hat{s}^{*}_{67}\hat{s}_{68}+\hat{s}_{47}\hat{s}^{*}_{57}\hat{s}_{58}c_{68}\\\
-c_{47}\hat{s}_{48}c_{58}c_{68}\end{array}&\hat{s}_{57}\hat{s}^{*}_{67}\hat{s}_{68}-c_{57}\hat{s}_{58}c_{68}&-c_{67}\hat{s}_{68}\cr\vspace{-0.4cm}\cr\begin{array}[]{l}\hat{s}_{47}c_{57}\hat{s}^{*}_{67}c_{68}\hat{s}_{69}-\hat{s}_{47}\hat{s}^{*}_{57}\hat{s}_{58}\hat{s}^{*}_{68}\hat{s}_{69}\\\
+\hat{s}_{47}\hat{s}^{*}_{57}c_{58}\hat{s}_{59}c_{69}+c_{47}\hat{s}_{48}c_{58}\hat{s}^{*}_{68}\hat{s}_{69}\\\
+c_{47}\hat{s}_{48}\hat{s}^{*}_{58}\hat{s}_{59}c_{69}-c_{47}c_{48}\hat{s}_{49}c_{59}c_{69}\end{array}&\begin{array}[]{l}\hat{s}_{57}\hat{s}^{*}_{67}c_{68}\hat{s}_{69}+c_{57}\hat{s}_{58}\hat{s}^{*}_{68}\hat{s}_{69}\\\
-c_{57}c_{58}\hat{s}_{59}c_{69}\end{array}&-c_{67}c_{68}\hat{s}_{69}\cr\end{matrix}\right)\;.\hskip
25.6073pt$ (15)

Different from the eighteen active-sterile flavor mixing angles $\theta_{1j}$,
$\theta_{2j}$ and $\theta_{3j}$ (for $j=4,5,\cdots,9$), which are all expected
to be strongly suppressed in magnitude, the nine sterile-sterile flavor mixing
angles $\theta_{4j}$, $\theta_{5j}$ and $\theta_{6j}$ (for $j=7,8,9$) are
completely unconstrained. Given the very fact that the right-handed neutrino
fields and the extra neutral fermion fields are both hypothetical, one may
naively conjecture that these two sterile sectors might be essentially
disconnected and thus their interplay might be very weak. Here we only focus
our attention on the issue of active-sterile flavor mixing. Without loss of
any generality, we have chosen the basis in which the flavor eigenstates of
three charged leptons are identical with their mass eigenstates throughout
this work.

After Eqs. (6), (7) and (8) are taken into account, the $9\times 9$ active-
sterile flavor mixing matrix $\cal U$ can be explicitly written as

$\displaystyle{\cal
U}=\left(\begin{matrix}A_{2}A_{1}U_{0}&A_{2}R_{1}&R_{2}\cr\big{(}R_{3}S_{2}A_{1}+A_{3}U^{\prime}_{0}S_{1}\big{)}U_{0}&R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}B_{1}&R_{3}B_{2}\cr
S_{0}\big{(}B_{3}S_{2}A_{1}+S_{3}U^{\prime}_{0}S_{1}\big{)}U_{0}&~{}S_{0}\big{(}B_{3}S_{2}R_{1}+S_{3}U^{\prime}_{0}B_{1}\big{)}~{}&S_{0}B_{3}B_{2}\end{matrix}\right)\;.$
(16)

The unitarity of $\cal U$ (i.e., ${\cal U}{\cal U}^{\dagger}={\cal
U}^{\dagger}{\cal U}=I_{9\times 9}$ with $I_{9\times 9}$ being the $9\times 9$
identity matrix) allows us to obtain a number of constraint equations, as
listed in appendix A. In the chosen flavor basis the key role of $\cal U$ is
to transform the flavor eigenstates of three active neutrinos, three sterile
neutrinos and three extra neutral fermions shown in Eq. (2) into their mass
eigenstates; namely,

$\displaystyle{\cal U}^{\dagger}\left(\begin{matrix}0&M_{\rm D}&\varepsilon\cr
M^{T}_{\rm D}&M_{\rm
R}&M_{S}\cr\varepsilon^{T}&M^{T}_{S}&\mu\cr\end{matrix}\right){\cal
U}^{*}=\left(\begin{matrix}D_{\nu}&0&0\cr 0&D_{N}&0\cr
0&0&D_{S}\end{matrix}\right)\;,$ (17)

where $D_{\nu}=\\{m_{1},m_{2},m_{3}\\}$, $D_{N}=\\{M_{1},M_{2},M_{3}\\}$ and
$D_{S}=\\{M^{\prime}_{1},M^{\prime}_{2},M^{\prime}_{3}\\}$ with $m_{i}$,
$M_{i}$ and $M^{\prime}_{i}$ being the respective masses of the active
neutrinos $\nu_{i}$, sterile neutrinos $N_{i}$ and extra neutral fermions
$N^{\prime}_{i}$ (for $i=1,2,3$). We are therefore left with

$\displaystyle\left(\begin{matrix}\nu_{e}\cr\nu_{\mu}\cr\nu_{\tau}\end{matrix}\right)_{\rm
L}=A_{2}A_{1}U_{0}\left(\begin{matrix}\nu_{1}\cr\nu_{2}\cr\nu_{3}\end{matrix}\right)_{\rm
L}+A_{2}R_{1}\left(\begin{matrix}N_{1}\cr N_{2}\cr
N_{3}\end{matrix}\right)_{\rm L}+R_{2}\left(\begin{matrix}N^{\prime}_{1}\cr
N^{\prime}_{2}\cr N^{\prime}_{3}\end{matrix}\right)_{\rm L}\;.$ (18)

The weak charged-current interactions of three active neutrinos, three sterile
neutrinos and three extra neutral fermions turn out to be

$\displaystyle-{\cal L}_{\rm cc}=\frac{g}{\sqrt{2}}\
\overline{\left(e~{}~{}\mu~{}~{}\tau\right)_{\rm L}}\
\gamma^{\mu}\left[U\left(\begin{matrix}\nu_{1}\cr\nu_{2}\cr\nu_{3}\end{matrix}\right)_{\rm
L}+R\left(\begin{matrix}N_{1}\cr N_{2}\cr N_{3}\end{matrix}\right)_{\rm
L}+R^{\prime}\left(\begin{matrix}N^{\prime}_{1}\cr N^{\prime}_{2}\cr
N^{\prime}_{3}\end{matrix}\right)_{\rm L}\right]W^{-}_{\mu}+{\rm h.c.}\;,$
(19)

where $U\equiv A_{2}A_{1}U_{0}$ represents the effective PMNS matrix, $R\equiv
A_{2}R_{1}$ and $R^{\prime}\equiv R_{2}$ characterize the contributions of new
degrees of freedom to ${\cal L}_{\rm cc}$. Once the extra neutral fermions are
switched off (i.e., $A_{2}=I$ and $R_{2}=0$), Eq. (19) will be reduced to the
canonical seesaw case [31, 32, 33].

Note that the $3\times 3$ PMNS matrix $U$ is not exactly unitary, and its
slight deviation from the unitarity limit can be clearly seen from

$\displaystyle
UU^{\dagger}=A_{2}A_{1}A^{\dagger}_{1}A^{\dagger}_{2}=I-RR^{\dagger}-R^{\prime}R^{\prime\dagger}\;,$
(20)

where the relation in Eq. (A.1) has been used. This issue will be further
discussed in section 4.

Note also that the full parametrization of the $9\times 9$ unitary flavor
mixing matrix $\cal U$ obtained in Eq. (16) is a quite general result, and it
is actually not subject to the seesaw scenario and the corresponding $9\times
9$ neutrino mass matrix considered in Eqs. (1) and (2). For example, the
texture zeros in the original neutrino mass matrix will help establish some
correlations between the mass and flavor mixing parameters, including the
exact seesaw formulas. This point will be clearly seen later on in the inverse
and linear seesaw mechanisms.

##  3 Exact and approximate seesaw formulas

###  3.1 The inverse seesaw scenario

As for the inverse seesaw scenario with $M_{\rm R}=0$ and $\varepsilon=0$, the
corresponding neutrino mass matrix takes the well-known Fritzsch texture [41]

$\displaystyle{\cal F}=\left(\begin{matrix}0&M_{\rm D}&0\cr M^{T}_{\rm
D}&0&M_{S}\cr 0&M^{T}_{S}&\mu\end{matrix}\right)={\cal
U}\left(\begin{matrix}D_{\nu}&0&0\cr 0&D_{N}&0\cr
0&0&D_{S}\end{matrix}\right){\cal U}^{T}\;.$ (21)

The texture zero ${\cal F}_{11}=0$ allows us to obtain the exact inverse
seesaw formula

$\displaystyle UD_{\nu}U^{T}+RD_{N}R+R^{\prime}D_{S}R^{\prime T}=0\;,$ (22)

where $U\equiv A_{2}A_{1}U_{0}$, $R\equiv A_{2}R_{1}$ and $R^{\prime}\equiv
R_{2}$ have been defined below Eq. (19), and they are correlated with one
another through $UU^{\dagger}+RR^{\dagger}+R^{\prime}R^{\prime\dagger}=I$ as
can be seen from Eq. (20). If the extra neutral fermion sector is switched off
(i.e., $A_{2}=I$ and $R^{\prime}=0$), one will reproduce the exact seesaw
formula of the canonical seesaw mechanism [31, 32, 33]. Moreover, the texture
zeros ${\cal F}_{22}=0$ and ${\cal F}_{13}={\cal F}_{31}=0$ lead us to the
constraint equations

$\displaystyle{\rm(a)}:$
$\displaystyle\big{(}R_{3}S_{2}A_{1}U_{0}+A_{3}U^{\prime}_{0}S_{1}U_{0}\big{)}D_{\nu}\big{(}U^{T}_{0}A^{T}_{1}S^{T}_{2}R^{T}_{3}+U^{T}_{0}S^{T}_{1}U^{\prime
T}_{0}A^{T}_{3}\big{)}$
$\displaystyle+\big{(}R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}B_{1}\big{)}D_{N}\big{(}R^{T}_{1}S^{T}_{2}R^{T}_{3}+B^{T}_{1}U^{\prime
T}_{0}A^{T}_{3}\big{)}+R_{3}B_{2}D_{S}B^{T}_{2}R^{T}_{3}=0\;,\hskip
14.22636pt$ $\displaystyle{\rm(b)}:$
$\displaystyle\big{(}S_{0}B_{3}S_{2}A_{1}U_{0}+S_{0}S_{3}U^{\prime}_{0}S_{1}U_{0}\big{)}D_{\nu}U^{T}_{0}A^{T}_{1}A^{T}_{2}$
(23)
$\displaystyle+\big{(}S_{0}B_{3}S_{2}R_{1}+S_{0}S_{3}U^{\prime}_{0}B_{1}\big{)}D_{N}R^{T}_{1}A^{T}_{2}+S_{0}B_{3}B_{2}D_{S}R^{T}_{2}=0\;.$

These two equations, together with Eq. (22), characterize the correlations
between the mass and flavor mixing parameters in this seesaw scenario.

We proceed to derive the approximate but more instructive inverse seesaw
formula given in Eq. (3) from Eqs. (22) and (23). To this end, we take into
account the fact that the interplay between three active neutrinos and those
hypothetical degrees of freedom must be strongly suppressed in magnitude. In
other words, the active-sterile flavor mixing angles $\theta_{1j}$,
$\theta_{2j}$ and $\theta_{3j}$ (for $j=4,5,\cdots,9$) are all very small, and
thus the eight $3\times 3$ matrices $A_{1,2}$, $B_{1,2}$, $R_{1,2}$ and
$S_{1,2}$ in Eqs. (10)—(13) can be reliably simplified to the forms listed in
appendix B. As very good approximations, Eqs. (22) and (23) become

$\displaystyle
U_{0}D_{\nu}U^{T}_{0}+R_{1}D_{N}R^{T}_{1}+R_{2}D_{S}R^{T}_{2}\simeq 0\;,$ (24)

and

$\displaystyle A_{3}U^{\prime}_{0}D_{N}U^{\prime
T}_{0}A^{T}_{3}+R_{3}D_{S}R^{T}_{3}\simeq 0\;,\hskip 28.45274pt$
$\displaystyle S_{3}U^{\prime}_{0}D_{N}R^{T}_{1}+B_{3}D_{S}R^{T}_{2}\simeq
0\;.$ (25)

Since $D_{\nu}$ is strongly suppressed in magnitude as compared with $D_{N}$
and $D_{S}$, the sum of the second and third terms on the left-hand side of
Eq. (24) should be as small as the first term and possess the opposite sign.
Moreover, $D_{N}\simeq-D_{S}$ is expected to hold due to the smallness of
$\mu$, as one can see when solving the eigenvalue equation of $\cal F$. This
observation implies that $R_{1}\simeq R_{2}$ is a good approximation. Then
$A_{3}U^{\prime}_{0}\simeq-R_{3}$ and $S_{3}U^{\prime}_{0}\simeq B_{3}$ can be
derived from Eq. (25), and their opposite signs will be explained later. In
this case the effective $3\times 3$ Majorana neutrino mass matrix of three
active neutrinos can be defined as

$\displaystyle M_{\nu}\equiv U_{0}D_{\nu}U^{T}_{0}\simeq-
R_{1}D_{N}R^{T}_{1}-R_{2}D_{S}R^{T}_{2}\;,$ (26)

where Eq. (24) has been used. To see how $M_{\nu}$ is related to $M_{\rm D}$,
$M_{S}$ and $\mu$, we start from Eq. (21) and obtain the following results:

$\displaystyle M_{\rm D}$ $\displaystyle=$ $\displaystyle
R_{1}D_{N}\left(R^{T}_{1}S^{T}_{2}R^{T}_{3}+U^{\prime
T}_{0}A^{T}_{3}\right)+R_{2}D_{S}R^{T}_{3}$ $\displaystyle\simeq$
$\displaystyle R_{1}D_{N}U^{\prime T}_{0}A^{T}_{3}+R_{2}D_{S}R^{T}_{3}$
$\displaystyle\simeq$ $\displaystyle R_{1}D_{N}\big{(}U^{\prime
T}_{0}A^{T}_{3}-R^{T}_{3}\big{)}$ $\displaystyle\simeq$ $\displaystyle
2R_{1}D_{N}U^{\prime T}_{0}A^{T}_{3}\;,$ $\displaystyle M_{S}$
$\displaystyle=$
$\displaystyle\big{(}R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}\big{)}D_{N}\left(R^{T}_{1}S^{T}_{2}B^{T}_{3}S^{T}_{0}+U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}\right)+R_{3}D_{S}B^{T}_{3}S^{T}_{0}$
$\displaystyle\simeq$ $\displaystyle A_{3}U^{\prime}_{0}D_{N}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}+R_{3}D_{S}B^{T}_{3}S^{T}_{0}$ $\displaystyle\simeq$
$\displaystyle\big{(}A_{3}U^{\prime}_{0}-R_{3}\big{)}D_{N}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}$ $\displaystyle\simeq$ $\displaystyle
2A_{3}U^{\prime}_{0}D_{N}U^{\prime T}_{0}S^{T}_{3}S^{T}_{0}\;,$
$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\big{(}S_{0}B_{3}S_{2}R_{1}+S_{0}S_{3}U^{\prime}_{0}B_{1}\big{)}D_{N}\left(R^{T}_{2}S^{T}_{2}B^{T}_{3}S^{T}_{0}+B^{T}_{1}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}\right)+S_{0}B_{3}B_{2}D_{S}B^{T}_{2}B^{T}_{3}S^{T}_{0}\hskip
14.22636pt$ (27) $\displaystyle\simeq$ $\displaystyle
S_{0}S_{3}U^{\prime}_{0}D_{N}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}+S_{0}B_{3}D_{S}B^{T}_{3}S^{T}_{0}$
$\displaystyle\simeq$ $\displaystyle
S_{0}B_{3}\big{(}B^{-1}_{3}S_{3}U^{\prime}_{0}D_{N}D^{-1}_{N}D_{N}U^{\prime
T}_{0}S^{T}_{3}B^{T-1}_{3}+D_{S}\big{)}B^{T}_{3}S^{T}_{0}$
$\displaystyle\simeq$ $\displaystyle
S_{0}B_{3}\big{(}-R^{-1}_{1}R_{2}D_{S}R^{T}_{2}R^{T-1}_{1}-D_{N}\big{)}B^{T}_{3}S^{T}_{0}\;.$

It is known that a natural TeV-scale inverse seesaw scenario requires $M_{\rm
D}\sim{\cal O}(1)$ GeV, $M_{S}\sim{\cal O}(1)$ TeV and $\mu\sim{\cal O}(1)$
keV (see, e.g., Ref. [16]). Hence the two terms in $M_{\rm D}$ and $M_{S}$
should be of the same sign to avoid a significant cancellation. That is why we
have chosen $A_{3}U^{\prime}_{0}\simeq-R_{3}$ below Eq. (25). On the other
hand, the two terms in $\mu$ should nearly cancel each other, implying that
$S_{3}U^{\prime}_{0}\simeq B_{3}$ is a reasonable choice. Substituting the
first equation in Eq. (25) into Eq. (26) and then making use of the
approximate expressions obtained in Eq. (27), we simply arrive at

$\displaystyle M_{\nu}$ $\displaystyle\simeq$ $\displaystyle-
R_{1}D_{N}R^{T}_{1}-R_{2}D_{S}R^{T}_{2}$ (28) $\displaystyle\simeq$
$\displaystyle
R_{1}\left(-D_{N}-R^{-1}_{1}R_{2}D_{S}R^{T}_{2}R^{T-1}_{1}\right)R^{T}_{1}$
$\displaystyle\simeq$ $\displaystyle
R_{1}U^{\prime-1}_{0}S^{-1}_{3}B_{3}\left(-D_{N}-R^{-1}_{1}R_{2}D_{S}R^{T}_{2}R^{T-1}_{1}\right)B^{T}_{3}S^{T-1}_{3}\left(U^{\prime
T}_{0}\right)^{-1}R^{T}_{1}$ $\displaystyle\simeq$
$\displaystyle\Big{[}R_{1}U^{\prime-1}_{0}S^{-1}_{3}S^{-1}_{0}\Big{]}\mu\Big{[}R_{1}U^{\prime-1}_{0}S^{-1}_{3}S^{-1}_{0}\Big{]}^{T}$
$\displaystyle\simeq$ $\displaystyle\Big{[}2R_{1}D_{N}U^{\prime
T}_{0}A^{T}_{3}\big{(}2S_{0}S_{3}U^{\prime}_{0}D_{N}U^{\prime
T}_{0}A^{T}_{3}\big{)}^{-1}\Big{]}\mu\Big{[}\big{(}2A_{3}U^{\prime}_{0}D_{N}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}\big{)}^{-1}\big{(}2R_{1}D_{N}U^{\prime
T}_{0}A^{T}_{3}\big{)}^{T}\Big{]}\hskip 14.22636pt$ $\displaystyle\simeq$
$\displaystyle M_{\rm
D}\left(M^{T}_{S}\right)^{-1}\mu\left(M_{S}\right)^{-1}M^{T}_{\rm D}\;.$

This leading-order result is just the well-known inverse seesaw formula given
in Eq. (3). While such an approximate expression may be more instructive in
some explicit model-building exercises, our exact inverse seesaw formula and
full parametrization of the $9\times 9$ active-sterile flavor mixing matrix
will be more useful for a generic study of the inverse seesaw picture.

At this point it is worth remarking that we have only paid attention to the
leading-order effects when defining the effective Majorana neutrino mass
matrix of three active neutrinos in Eq. (26) and reproducing the inverse
seesaw formula for it in Eqs. (27) and (28). The next-to-leading-order
corrections to the leading-order inverse seesaw formula have been discussed at
the tree level [20]. We find that it will be a messy and tangled business to
include such next-to-leading-order effects into our approach, because both our
generic Euler-like parametrization of the $9\times 9$ active-sterile flavor
mixing matrix and our exact inverse seesaw formula are actually independent of
the sub-leading effects of this kind. This is also true of our discussions
about the non-unitary effects on flavor mixing and CP violation based on our
parametrization in section 4.

###  3.2 The linear seesaw scenario

Since $M_{\rm R}=0$ and $\mu=0$ are taken for the linear seesaw scenario, one
may reconstruct the corresponding traceless neutrino mass matrix in the
following way:

$\displaystyle{\cal F}^{\prime}=\left(\begin{matrix}0&M_{\rm D}&\varepsilon\cr
M^{T}_{\rm D}&0&M_{S}\cr\varepsilon^{T}&M^{T}_{S}&0\end{matrix}\right)={\cal
U}\left(\begin{matrix}D_{\nu}&0&0\cr 0&{D_{N}}&0\cr
0&0&{D_{S}}\end{matrix}\right){\cal U^{T}}\;.$ (29)

In this case the texture zero ${\cal F}^{\prime}_{11}=0$ leads us to the exact
linear seesaw formula

$\displaystyle UD_{\nu}U^{T}+RD_{N}R+R^{\prime}D_{S}R^{\prime T}=0\;,$ (30)

where $U\equiv A_{2}A_{1}U_{0}$, $R\equiv A_{2}R_{1}$ and $R^{\prime}\equiv
R_{2}$ are defined and
$UU^{\dagger}+RR^{\dagger}+R^{\prime}R^{\prime\dagger}=I$ holds. One can
immediately see that Eq. (30) is formally the same as Eq. (22) obtained in the
inverse seesaw case, although their physical contexts are somewhat different.
This interesting observation tells us that such an exact seesaw formula is
actually universal and valid for the more general seesaw scenario described by
the mass terms in Eqs. (1) and (2).

Taking account of the texture zeros ${\cal F}^{\prime}_{22}={\cal
F}^{\prime}_{33}=0$ in the linear seesaw scenario, we arrive at two further
constraint equations

$\displaystyle{\rm(a)}:$
$\displaystyle\big{(}R_{3}S_{2}A_{1}U_{0}+A_{3}U^{\prime}_{0}S_{1}U_{0}\big{)}D_{\nu}\big{(}U^{T}_{0}A^{T}_{1}S^{T}_{2}R^{T}_{3}+U^{T}_{0}S^{T}_{1}U^{\prime
T}_{0}A^{T}_{3}\big{)}$
$\displaystyle+\big{(}R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}B_{1}\big{)}D_{N}\big{(}R^{T}_{1}S^{T}_{2}R^{T}_{3}+B^{T}_{1}U^{\prime
T}_{0}A^{T}_{3}\big{)}+R_{3}B_{2}D_{S}B^{T}_{2}R^{T}_{3}=0\;,$
$\displaystyle{\rm(b)}:$
$\displaystyle\big{(}S_{0}B_{3}S_{2}A_{1}U_{0}+S_{0}S_{3}U^{\prime}_{0}S_{1}U_{0}\big{)}D_{\nu}\big{(}U^{T}_{0}A^{T}_{1}S^{T}_{2}B^{T}_{3}S^{T}_{0}+U^{T}_{0}S^{T}_{1}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}\big{)}$ (31)
$\displaystyle+\big{(}S_{0}B_{3}S_{2}R_{1}+S_{0}S_{3}U^{\prime}_{0}B_{1}\big{)}D_{N}\big{(}B^{T}_{1}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}+R^{T}_{1}S^{T}_{2}B^{T}_{3}S^{T}_{0}\big{)}+S_{0}B_{3}D_{S}B^{T}_{3}S^{T}_{0}=0\;.\hskip
14.22636pt$

Once the approximations made in Eqs. (B1)—(B4) are taken into consideration,
Eqs. (30) and (31) can be easily simplified to

$\displaystyle
U_{0}D_{\nu}U^{T}_{0}+R_{1}D_{N}R^{T}_{1}+R_{2}D_{S}R^{T}_{2}\simeq 0\;,$ (32)

and

$\displaystyle A_{3}U^{\prime}_{0}D_{N}U^{\prime
T}_{0}A^{T}_{3}+R_{3}D_{S}R^{T}_{3}\simeq 0\;,$ $\displaystyle
S_{3}U^{\prime}_{0}D_{N}U^{\prime T}_{0}S^{T}_{3}+B_{3}D_{S}B^{T}_{3}\simeq
0\;.\hskip 31.2982pt$ (33)

As in the inverse seesaw scenario, we similarly have $D_{N}\simeq-D_{S}$ and
$R_{1}\simeq R_{2}$ in the linear seesaw scenario. The relations
$A_{3}U^{\prime}_{0}\simeq-R_{3}$ and $S_{3}U^{\prime}_{0}\simeq B_{3}$ are
expected to remain valid. The reason why these two approximate relations have
the opposite signs is also that the terms in $M_{\rm D}$ and $M_{S}$ should
avoid significant cancellations while $\varepsilon$ should be strongly
suppressed in magnitude.

Now let us use Eq. (32) to define the effective $3\times 3$ Majorana neutrino
mass matrix of three active neutrinos:

$\displaystyle M_{\nu}\equiv U_{0}D_{\nu}U^{T}_{0}\simeq-
R_{1}D_{N}R^{T}_{1}-R_{2}D_{S}R^{T}_{2}\;.$ (34)

Starting from Eq. (29) and taking into account the approximate relations
obtained in and below Eq. (33), we have

$\displaystyle M_{\rm D}$ $\displaystyle=$ $\displaystyle
R_{1}D_{N}\big{(}R^{T}_{1}S^{T}_{2}R^{T}_{3}+U^{\prime
T}_{0}A^{T}_{3}\big{)}+R_{2}D_{S}R^{T}_{3}$ $\displaystyle\simeq$
$\displaystyle 2R_{1}D_{N}U^{\prime T}_{0}A^{T}_{3}\;,$ $\displaystyle M_{S}$
$\displaystyle=$
$\displaystyle\big{(}R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}\big{)}D_{N}\big{(}R^{T}_{1}S^{T}_{2}B^{T}_{3}S^{T}_{0}+U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}\big{)}+R_{3}D_{S}B^{T}_{3}S^{T}_{0}\hskip 14.22636pt$
$\displaystyle\simeq$ $\displaystyle 2A_{3}U^{\prime}_{0}D_{N}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}$ $\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle R_{1}D_{N}\big{(}R^{T}_{1}S^{T}_{2}B^{T}_{3}S^{T}_{0}+U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}\big{)}+R_{2}D_{S}B^{T}_{3}S^{T}_{0}$ (35)
$\displaystyle\simeq$ $\displaystyle R_{1}D_{N}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}+R_{2}D_{S}B^{T}_{3}S^{T}_{0}\;.$

It is then easy to arrive at

$\displaystyle R_{1}D_{N}R^{T}_{1}-R_{1}D_{N}R^{T}_{2}$ $\displaystyle\simeq$
$\displaystyle
R_{1}U^{\prime-1}_{0}S^{-1}_{3}\left(S_{3}U^{\prime}_{0}D_{N}R^{T}_{1}+B_{3}D_{S}R^{T}_{2}\right)$
$\displaystyle\simeq$ $\displaystyle\left(2R_{1}D_{N}U^{\prime
T}_{0}A^{T}_{3}\right)\left(2S_{0}S_{3}U^{\prime}_{0}D_{N}U^{\prime
T}_{0}A^{T}_{3}\right)^{-1}\left(R_{1}D_{N}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}+R_{2}D_{S}B^{T}_{3}S^{T}_{0}\right)^{T}$
$\displaystyle\simeq$ $\displaystyle M_{\rm
D}\left(M^{T}_{S}\right)^{-1}\varepsilon^{T}\;,$ $\displaystyle
R_{1}D_{N}R^{T}_{2}+R_{2}D_{S}R^{T}_{2}$ $\displaystyle\simeq$ $\displaystyle
R_{1}D_{N}\left(R_{2}B^{-1}_{3}S_{3}U^{\prime}_{0}\right)^{T}+R_{2}D_{S}R^{T}_{2}$
(36) $\displaystyle\simeq$ $\displaystyle\left(R_{1}D_{N}U^{\prime
T}_{0}S^{T}_{3}S^{T}_{0}+R_{2}D_{S}B^{T}_{3}S^{T}_{0}\right)\left(2R_{3}D_{S}B^{T}_{3}S^{T}_{0}\right)^{-1}\left(2R_{2}D_{S}R^{T}_{3}\right)^{T}$
$\displaystyle\simeq$
$\displaystyle\varepsilon\left(M_{S}\right)^{-1}M^{T}_{\rm D}\;.$

Combining Eq. (36) with Eq. (34) allows us to obtain the linear seesaw formula

$\displaystyle M_{\nu}$ $\displaystyle\simeq$ $\displaystyle-
R_{1}D_{N}R^{T}_{1}-R_{2}D_{S}R^{T}_{2}$ (37) $\displaystyle\simeq$
$\displaystyle-M_{\rm
D}\left(M^{T}_{S}\right)^{-1}\varepsilon^{T}-\varepsilon\left(M_{S}\right)^{-1}M^{T}_{\rm
D}\;.\hskip 22.76228pt$

This result is fully consistent with Eq. (4).

##  4 Jarlskog invariants and unitarity nonagons

###  4.1 The Jarlskog invariants

In the standard three-flavor scheme without any new degrees of freedom, the
strength of CP violation in neutrino oscillations is measured by the unique
Jarlskog invariant [42]

$\displaystyle{\cal J}_{0}={\cal J}^{ij}_{\alpha\beta}\equiv{\rm
Im}\left(U_{\alpha i}U_{\beta j}U^{*}_{\alpha j}U^{*}_{\beta i}\right)\;,$
(38)

where the Latin (or Greek) indices $i$ and $j$ (or $\alpha$ and $\beta$) run
cyclically over $1$, $2$ and $3$ (or $e$, $\mu$ and $\tau$). Since $U=U_{0}$
holds in this case, the parametrization of $U_{0}$ in Eq. (9) leads us to

$\displaystyle{\cal
J}_{0}=c_{12}s_{12}c^{2}_{13}s_{13}c_{23}s_{23}\sin\delta\;,$ (39)

where $\delta=\delta_{13}-\delta_{12}-\delta_{23}$ is sometimes referred to as
the “Dirac” CP-violating phase. Note that this phase parameter may also
manifest itself in the lepton-number-violating neutrino-antineutrino
oscillations [43, 44], simply because it is a nontrivial CP phase of Majorana
neutrinos.

In the inverse or linear seesaw scenario, however, the $3\times 3$ PMNS matrix
$U=A_{2}A_{1}U_{0}$ is not exactly unitary. As a consequence, some additional
effects of CP violation described by the deviations of ${\cal
J}^{ij}_{\alpha\beta}$ from ${\cal J}_{0}$ will manifest themselves in
neutrino oscillations. To explicitly calculate the nine Jarlskog invariants
${\cal J}^{ij}_{\alpha\beta}$, let us rewrite the approximate expressions of
$A_{1}$ in Eq. (B1) and $A_{2}$ in Eq. (B3) as follows:

$\displaystyle A_{1\left(2\right)}\simeq\left(\begin{matrix}1&0&0\cr-
X_{1\left(2\right)}&1&0\cr-
Y_{1\left(2\right)}&-Z_{1\left(2\right)}&1\end{matrix}\right)\;,$ (40)

where only the leading term of each matrix element is kept, and

$\displaystyle X_{1}$ $\displaystyle\equiv$
$\displaystyle\hat{s}_{14}\hat{s}^{*}_{24}+\hat{s}_{15}\hat{s}^{*}_{25}+\hat{s}_{16}\hat{s}^{*}_{26}\;,$
$\displaystyle Y_{1}$ $\displaystyle\equiv$
$\displaystyle\hat{s}_{14}\hat{s}^{*}_{34}+\hat{s}_{15}\hat{s}^{*}_{35}+\hat{s}_{16}\hat{s}^{*}_{36}\;,$
$\displaystyle Z_{1}$ $\displaystyle\equiv$
$\displaystyle\hat{s}_{24}\hat{s}^{*}_{34}+\hat{s}_{25}\hat{s}^{*}_{35}+\hat{s}_{26}\hat{s}^{*}_{36}\;,\hskip
22.76228pt$ (41)

together with

$\displaystyle X_{2}$ $\displaystyle\equiv$
$\displaystyle\hat{s}_{17}\hat{s}^{*}_{27}+\hat{s}_{18}\hat{s}^{*}_{28}+\hat{s}_{19}\hat{s}^{*}_{29}\;,$
$\displaystyle Y_{2}$ $\displaystyle\equiv$
$\displaystyle\hat{s}_{17}\hat{s}^{*}_{37}+\hat{s}_{18}\hat{s}^{*}_{38}+\hat{s}_{19}\hat{s}^{*}_{39}\;,$
$\displaystyle Z_{2}$ $\displaystyle\equiv$
$\displaystyle\hat{s}_{27}\hat{s}^{*}_{37}+\hat{s}_{28}\hat{s}^{*}_{38}+\hat{s}_{29}\hat{s}^{*}_{39}\;.\hskip
22.76228pt$ (42)

Now that $A_{1}$ and $A_{2}$ signify the departure of $U$ from $U_{0}$, the
magnitudes of $X_{1\left(2\right)}$, $Y_{1\left(2\right)}$ and
$Z_{1\left(2\right)}$ are at most of the percent level. On the other hand, the
smallest flavor mixing angle of $U_{0}$ is $\theta_{13}$, and its size is
about $0.16$. So it is reasonable to omit the terms of ${\cal
O}(X_{1(2)}s_{13})$, ${\cal O}(Y_{1(2)}s_{13})$, ${\cal O}(Z_{1(2)}s_{13})$,
${\cal O}(X^{2}_{1(2)})$, ${\cal O}(Y^{2}_{1(2)})$ and ${\cal
O}(Z^{2}_{1(2)})$ in calculating ${\cal J}^{ij}_{\alpha\beta}$. Our results
are

$\displaystyle{\cal J}^{12}_{e\mu}$ $\displaystyle\simeq$ $\displaystyle{\cal
J}_{0}+c_{12}s_{12}c_{23}{\rm Im}\Big{[}\big{(}X_{1}+X_{2}\big{)}e^{-{\rm
i}\delta_{12}}\Big{]}\;,$ $\displaystyle{\cal J}^{12}_{\tau e}$
$\displaystyle\simeq$ $\displaystyle{\cal J}_{0}+c_{12}s_{12}s_{23}{\rm
Im}\Big{[}\big{(}Y_{1}+Y_{2}\big{)}e^{-{\rm
i}\left(\delta_{12}+\delta_{23}\right)}\Big{]}\;,$ $\displaystyle{\cal
J}^{12}_{\mu\tau}$ $\displaystyle\simeq$ $\displaystyle{\cal
J}_{0}+c_{12}s_{12}c_{23}s_{23}\Big{\\{}s_{23}{\rm
Im}\Big{[}\big{(}X_{1}+X_{2}\big{)}e^{-{\rm i}\delta_{12}}\Big{]}+c_{23}{\rm
Im}\Big{[}\big{(}Y_{1}+Y_{2}\big{)}e^{-{\rm
i}\left(\delta_{12}+\delta_{23}\right)}\Big{]}\Big{\\}}\;,$
$\displaystyle{\cal J}^{23}_{\mu\tau}$ $\displaystyle\simeq$
$\displaystyle{\cal J}_{0}+c_{12}c_{23}s_{23}\Big{\\{}s_{12}s_{23}{\rm
Im}\Big{[}\big{(}X_{1}+X_{2}\big{)}e^{-{\rm
i}\delta_{12}}\Big{]}+s_{12}c_{23}{\rm
Im}\Big{[}\big{(}Y_{1}+Y_{2}\big{)}e^{-{\rm
i}\left(\delta_{12}+\delta_{23}\right)}\Big{]}$ $\displaystyle+c_{12}{\rm
Im}\Big{[}\big{(}Z_{1}+Z_{2}\big{)}e^{-{\rm i}\delta_{23}}\Big{]}\Big{\\}}\;,$
$\displaystyle{\cal J}^{31}_{\mu\tau}$ $\displaystyle\simeq$
$\displaystyle{\cal J}_{0}+s_{12}c_{23}s_{23}\Big{\\{}c_{12}s_{23}{\rm
Im}\Big{[}\big{(}X_{1}+X_{2}\big{)}e^{-{\rm
i}\delta_{12}}\Big{]}+c_{12}c_{23}{\rm
Im}\Big{[}\big{(}Y_{1}+Y_{2}\big{)}e^{-{\rm
i}\left(\delta_{12}+\delta_{23}\right)}\Big{]}\hskip 14.22636pt$ (43)
$\displaystyle-s_{12}{\rm Im}\Big{[}\big{(}Z_{1}+Z_{2}\big{)}e^{-{\rm
i}\delta_{23}}\Big{]}\Big{\\}}\;,$

and ${\cal J}^{23}_{e\mu}\simeq{\cal J}^{31}_{e\mu}\simeq{\cal J}^{23}_{\tau
e}\simeq{\cal J}^{31}_{\tau e}\simeq{\cal J}_{0}$. One can see that five of
the nine Jarlskog invariants are sensitive to the active-sterile flavor mixing
angles and the associated CP-violating phases.

As for the probabilities of neutrino oscillations in vacuum, let us assume
that all the sterile particles are heavy enough and hence kinematically
forbidden to participate in a realistic long-baseline oscillation process. In
this case one does not have to worry about the masses of those hypothetical
particles and their differences from the masses of three active neutrinos, and
the Jarlskog invariants calculated above determine the CP-violating asymmetry
${\cal A}_{\alpha\beta}\equiv
P(\nu_{\alpha}\to\nu_{\beta})-P(\overline{\nu}_{\alpha}\to\overline{\nu}_{\beta})$
(for $\alpha,\beta=e,\mu,\tau$) as follows [31]:

$\displaystyle{\cal A}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{4}{\left(UU^{\dagger}\right)_{\alpha\alpha}\left(UU^{\dagger}\right)_{\beta\beta}}\sum_{i<j}{\cal
J}^{ij}_{\alpha\beta}\sin\frac{\Delta m^{2}_{ij}L}{2E}$ (44)
$\displaystyle\simeq$ $\displaystyle 4\left[{\cal
J}^{12}_{\alpha\beta}\sin\frac{\Delta m^{2}_{21}L}{2E}-{\cal
J}^{31}_{\alpha\beta}\sin\frac{\Delta m^{2}_{31}L}{2E}+{\cal
J}^{23}_{\alpha\beta}\sin\frac{\Delta m^{2}_{32}L}{2E}\right]\;,\hskip
14.22636pt$

where $\Delta m^{2}_{ij}\equiv m^{2}_{i}-m^{2}_{j}$ (for $i,j=1,2,3$) are
defined, $E$ and $L$ stand respectively for the neutrino beam energy and
baseline length, and the approximations $A_{1}A^{\dagger}_{1}\simeq I$ and
$A_{2}A^{\dagger}_{2}\simeq I$ have been used. In practice, however,
terrestrial matter effects must be taken into account for a long-baseline
neutrino oscillation experiment which is sensitive to leptonic CP violation
(see, e.g., Refs. [45, 46, 47, 48, 49, 50, 51]). That is why a careful
analysis of the non-unitary flavor mixing effects and matter effects is needed
for a realistic long-baseline neutrino (or antineutrino) oscillation
experiment [54], no matter whether CP violation is concerned or not.

###  4.2 The unitarity nonagons

It is well known that six orthogonality relations of the $3\times 3$ unitary
PMNS matrix $U=U_{0}$ define six triangles in the complex plane, which are
referred to as the unitarity triangles [52, 53]. Their areas are all equal to
${\cal J}_{0}/2$, thanks to the unitarity of $U_{0}$. Three of the six
triangles, defined by

$\displaystyle\triangle_{\tau}$ $\displaystyle:$ $\displaystyle
U_{e1}U^{*}_{\mu 1}+U_{e2}U^{*}_{\mu 2}+U_{e3}U^{*}_{\mu 3}=0\;,$
$\displaystyle\triangle_{e}$ $\displaystyle:$ $\displaystyle U_{\mu
1}U^{*}_{\tau 1}+U_{\mu 2}U^{*}_{\tau 2}+U_{\mu 3}U^{*}_{\tau 3}=0\;,\hskip
14.22636pt$ $\displaystyle\triangle_{\mu}$ $\displaystyle:$ $\displaystyle
U_{\tau 1}U^{*}_{e1}+U_{\tau 2}U^{*}_{e2}+U_{\tau 3}U^{*}_{e3}=0\;,$ (45)

where $U=U_{0}$ is implied, are directly associated with leptonic CP violation
in neutrino oscillations.

Given three sterile neutrinos and three extra neutral fermions which slightly
mix with the active neutrinos, the corresponding $9\times 9$ unitary flavor
mixing matrix $\cal U$ allows us to totally define 36 nonagons in the complex
plane. Among them, only three unitarity nonagons defined by the orthogonality
conditions $\left({\cal UU}^{\dagger}\right)_{e\mu}=0$, $\left({\cal
UU}^{\dagger}\right)_{\mu\tau}=0$ and $\left({\cal UU}^{\dagger}\right)_{\tau
e}=0$ are relevant to the flavor oscillations of three active neutrinos. Since
the sides associated with sterile flavors must be very short as compared with
the sides $|U_{\alpha i}U^{*}_{\beta i}|$ for $(\alpha,\beta)=(e,\mu)$,
$(\mu,\tau)$, $(\tau,e)$ and $i=1,2,3$, these three unitarity nonagons can be
regarded as the deformed versions of unitarity triangles $\triangle_{\tau}$,
$\triangle_{e}$ and $\triangle_{\mu}$. Taking account of Eqs. (40)—(42) in
this case, the three unitarity nonagons under discussion can therefore be
expressed as

$\displaystyle\triangle^{\prime}_{\tau}$ $\displaystyle:$ $\displaystyle
U_{e1}U^{*}_{\mu 1}+U_{e2}U^{*}_{\mu 2}+U_{e3}U^{*}_{\mu
3}\simeq-X^{*}_{1}-X^{*}_{2}\;,\hskip 14.22636pt$
$\displaystyle\triangle^{\prime}_{e}$ $\displaystyle:$ $\displaystyle U_{\mu
1}U^{*}_{\tau 1}+U_{\mu 2}U^{*}_{\tau 2}+U_{\mu 3}U^{*}_{\tau
3}\simeq-Z^{*}_{1}-Z^{*}_{2}\;,$ $\displaystyle\triangle^{\prime}_{\mu}$
$\displaystyle:$ $\displaystyle U_{\tau 1}U^{*}_{e1}+U_{\tau
2}U^{*}_{e2}+U_{\tau 3}U^{*}_{e3}\simeq-Y_{1}-Y_{2}\;.$ (46)

Comparing Eq. (46) with Eq. (45), one can see that the new sides originating
from those sterile degrees of freedom are all of ${\cal O}(s^{2}_{ij})$ with
$i=1,2,3$ and $j=4,5,6,7,8,9$. That is why we use
$\triangle^{\prime}_{\alpha}$ to denote the unitarity nonagon defined by
$\left({\cal UU}^{\dagger}\right)_{\beta\gamma}=0$ (for $\alpha$, $\beta$ and
$\gamma$ running cyclically over $e$, $\mu$ and $\tau$), simply because it
deviates only slightly from $\triangle_{\alpha}$.

In the lack of definite information about the active-sterile flavor mixing
parameters (i.e., $\theta_{ij}$ and $\delta_{ij}$ for $i=1,2,3$ and
$j=4,5,6,7,8,9$), one may simply treat the new sides of
$\triangle^{\prime}_{\alpha}$ (for $\alpha=e$, $\mu$ or $\tau$) on the right-
hand side of Eq. (46) as an effective “single” side. In this case the
unitarity nonagon is reduced to an effective unitarity quadrangle whose
shortest side signifies the existence of sterile neutrinos and extra neutral
fermions, as schematically illustrated by Fig. 1 for
$\triangle^{\prime}_{\tau}$. For simplicity, we have made the choice that the
side $U_{e1}U^{*}_{\mu 1}$ lies in the horizontal direction and forms a sharp
angle to the side $U_{e2}U^{*}_{\mu 2}$ in Fig. 1. Then the side
$U_{e3}U^{*}_{\mu 3}$ may form either a sharp angle or an obtuse angle to the
horizontal side, and the shortest side defined by $-X^{*}_{1}-X^{*}_{2}$ is
likely to link these two longer sides in several different topologies.

Figure 1: A schematic illustration of possible shapes of the effective
unitarity quadrangle $\triangle^{\prime}_{\tau}$ as an example in the complex
plane, where the three long sides correspond to $U_{e1}U^{*}_{\mu 1}$,
$U_{e2}U^{*}_{\mu 2}$ and $U_{e3}U^{*}_{\mu 3}$, and the fourth (effective)
side is measured by $-X^{*}_{1}-X^{*}_{2}$.

To constrain the shortest side of each effective unitarity quadrangle, one may
study those lepton-flavor-violating processes such as neutrino oscillations
and radiative decays of charged leptons. In the latter case the new degrees of
freedom can mediate the one-loop $\alpha^{-}\to\beta^{-}+\gamma$ transitions
for $(\alpha,\beta)=(\mu,e)$, $(\tau,e)$ and $(\tau,\mu)$, and the relevant
loop functions depend on the masses of three sterile neutrinos and three extra
neutral fermions (i.e., $M_{i}$ and $M^{\prime}_{i}$ for $i=1,2,3$) [26, 55,
56]. A systematic analysis of such lepton-flavor-violating processes in the
inverse and linear seesaw scenarios will be done elsewhere.

##  5 Concluding remarks

The inverse and linear seesaw scenarios are two simple extensions of the
canonical (type-I) seesaw mechanism aiming to lower the mass scales of those
hypothetical particles and hence enhance the experimental testability. In this
regard a high price that one has to pay is the introduction of three species
of extra neutral fermions besides three species of sterile neutrinos. How to
determine or constrain the flavor structure of such a complicated seesaw
scenario and describe the corresponding flavor mixing pattern turns out to be
a highly nontrivial issue.

Instead of trying to reduce the number of free parameters by imposing some
kind of flavor symmetry or empirical assumptions on the texture of the
$9\times 9$ mass matrix (or its $3\times 3$ submatrices) in either the inverse
seesaw scenario or the linear seesaw scenario [33, 57, 58, 59], here we have
focused on a full description of the $9\times 9$ active-sterile flavor mixing
matrix in terms of 36 rotation angles and 36 CP-violating phases. Such a
generic and model-independent work is certainly new. The most salient feature
of our parametrization that the primary unitary $3\times 3$ flavor mixing
submatrices of three active neutrinos, three sterile neutrinos and three extra
neutral fermions, which would look like three isolated islands if the Yukawa-
like interactions among them were absent, are linked and modified by the
intermediate flavor mixing matrices. This approach proves to be useful for
describing possible deviations of the $3\times 3$ PMNS matrix of three active
neutrinos from its unitary limit, as it allows us to calculate the effective
Jarlskog invariants and the deformed unitarity triangles in a good
approximation and without loss of any generality.

Starting from our generic result, one may easily arrive at a full Euler-like
parametrization of the $7\times 7$ active-sterile flavor mixing matrix in the
so-called minimal inverse seesaw model which contains only two species of
sterile neutrinos and two species of extra neutral fermions (see, e.g., Refs.
[16, 26, 60, 61, 62, 63, 64]). A similar simplification can also be applied to
the so-called minimal linear seesaw model [65] and the study of its various
phenomenological consequences.

Finally, it is also worth pointing out that our exact Euler-like
parametrization of the $n\times n$ unitary matrix (for $3\leq n\leq 9$) is
likely to find applications in some other physical systems.

## Acknowledgements

We are indebted to Jian-Wei Mei and Shun Zhou for useful discussions. This
work is supported in part by the National Natural Science Foundation of China
under grant No. 12075254, grant No. 11775231 and grant No. 11835013.

##  Appendix A The unitary conditions of $\cal U$

Given the expression of the $9\times 9$ active-sterile flavor mixing matrix
$\cal U$ in Eq. (16), one may obtain either the six unitary conditions for its
$3\times 3$ submatrices from ${\cal U}{\cal U}^{\dagger}=I_{9\times 9}$:

$\displaystyle(1)$ $\displaystyle
A_{2}A_{1}A^{\dagger}_{1}A^{\dagger}_{2}+A_{2}R_{1}R^{\dagger}_{1}A^{\dagger}_{2}+R_{2}R^{\dagger}_{2}=I\;,$
$\displaystyle(2)$
$\displaystyle\big{(}R_{3}S_{2}A_{1}+A_{3}U^{\prime}_{0}S_{1}\big{)}\big{(}A^{\dagger}_{1}S^{\dagger}_{2}R^{\dagger}_{3}+S^{\dagger}_{1}U^{\prime\dagger}_{0}A^{\dagger}_{3}\big{)}+\big{(}R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}B_{1}\big{)}\big{(}R^{\dagger}_{1}S^{\dagger}_{2}R^{\dagger}_{3}+B^{\dagger}_{1}U^{\prime\dagger}_{0}A^{\dagger}_{3}\big{)}$
$\displaystyle+R_{3}B_{2}B^{\dagger}_{2}R^{\dagger}_{3}=I\;,$
$\displaystyle(3)$
$\displaystyle\big{(}B_{3}S_{2}A_{1}+S_{3}U^{\prime}_{0}S_{1}\big{)}\big{(}A^{\dagger}_{1}S^{\dagger}_{2}B^{\dagger}_{3}+S^{\dagger}_{1}U^{\prime\dagger}_{0}S^{\dagger}_{3}\big{)}+\big{(}B_{3}S_{2}R_{1}+S_{3}U^{\prime}_{0}B_{1}\big{)}\big{(}B^{\dagger}_{1}U^{\prime\dagger}_{0}S^{\dagger}_{3}+R^{\dagger}_{1}S^{\dagger}_{2}B^{\dagger}_{3}\big{)}$
$\displaystyle+B_{3}B_{2}B^{\dagger}_{2}B^{\dagger}_{3}=I\;,$
$\displaystyle(4)$ $\displaystyle
A_{2}A_{1}\big{(}A^{\dagger}_{1}S^{\dagger}_{2}R^{\dagger}_{3}+S^{\dagger}_{1}U^{\prime\dagger}_{0}A^{\dagger}_{3}\big{)}+A_{2}R_{1}\big{(}R^{\dagger}_{1}S^{\dagger}_{2}R^{\dagger}_{3}+B^{\dagger}_{1}U^{\prime\dagger}_{0}A^{\dagger}_{3}\big{)}+R_{2}B^{\dagger}_{2}R^{\dagger}_{3}=0\;,$
$\displaystyle(5)$
$\displaystyle\big{(}R_{3}S_{2}A_{1}+A_{3}U^{\prime}_{0}S_{1}\big{)}\big{(}A^{\dagger}_{1}S^{\dagger}_{2}B^{\dagger}_{3}+S^{\dagger}_{1}U^{\prime\dagger}_{0}S^{\dagger}_{3}\big{)}+\big{(}R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}B_{1}\big{)}\big{(}B^{\dagger}_{1}U^{\prime\dagger}_{0}S^{\dagger}_{3}+R^{\dagger}_{1}S^{\dagger}_{2}B^{\dagger}_{3}\big{)}$
$\displaystyle+R_{3}B_{2}B^{\dagger}_{2}B^{\dagger}_{3}=0\;,$
$\displaystyle(6)$
$\displaystyle\big{(}B_{3}S_{2}A_{1}+S_{3}U^{\prime}_{0}S_{1}\big{)}A^{\dagger}_{1}A^{\dagger}_{2}+\big{(}B_{3}S_{2}R_{1}+S_{3}U^{\prime}_{0}B_{1}\big{)}R^{\dagger}_{1}A^{\dagger}_{2}+B_{3}B_{2}R^{\dagger}_{2}=0\;;$
(A.1)

or the six unitary conditions for its $3\times 3$ submatrices from ${\cal
U}^{\dagger}{\cal U}=I_{9\times 9}$:

$\displaystyle(1)$ $\displaystyle
U^{\dagger}_{0}A^{\dagger}_{1}A^{\dagger}_{2}A_{2}A_{1}U_{0}+U^{\dagger}_{0}\big{(}A^{\dagger}_{1}S^{\dagger}_{2}R^{\dagger}_{3}+S^{\dagger}_{1}U^{\prime\dagger}_{0}A^{\dagger}_{3}\big{)}\big{(}R_{3}S_{2}A_{1}+A_{3}U^{\prime}_{0}S_{1}\big{)}U_{0}$
$\displaystyle+U^{\dagger}_{0}\big{(}A^{\dagger}_{1}S^{\dagger}_{2}B^{\dagger}_{3}+S^{\dagger}_{1}U^{\prime\dagger}_{0}S^{\dagger}_{3}\big{)}\big{(}B_{3}S_{2}A_{1}+S_{3}U^{\prime}_{0}S_{1}\big{)}U_{0}=I\;,$
$\displaystyle(2)$ $\displaystyle
R^{\dagger}_{1}A^{\dagger}_{2}A_{2}R_{1}+\big{(}R^{\dagger}_{1}S^{\dagger}_{2}R^{\dagger}_{3}+B^{\dagger}_{1}U^{\prime\dagger}_{0}A^{\dagger}_{3}\big{)}\big{(}R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}B_{1}\big{)}$
$\displaystyle+\big{(}B^{\dagger}_{1}U^{\prime\dagger}_{0}S^{\dagger}_{3}+R^{\dagger}_{1}S^{\dagger}_{2}B^{\dagger}_{3}\big{)}\big{(}B_{3}S_{2}R_{1}+S_{3}U^{\prime}_{0}B_{1}\big{)}=I\;,$
$\displaystyle(3)$ $\displaystyle
R^{\dagger}_{2}R_{2}+B^{\dagger}_{2}R^{\dagger}_{3}R_{3}B_{2}+B^{\dagger}_{2}B^{\dagger}_{3}B_{3}B_{2}=I\;,$
$\displaystyle(4)$ $\displaystyle
A^{\dagger}_{1}A^{\dagger}_{2}A_{2}R_{1}+\big{(}A^{\dagger}_{1}S^{\dagger}_{2}R^{\dagger}_{3}+S^{\dagger}_{1}U^{\prime\dagger}_{0}A^{\dagger}_{3}\big{)}\big{(}R_{3}S_{2}R_{1}+A_{3}U^{\prime}_{0}B_{1}\big{)}$
$\displaystyle+\big{(}A^{\dagger}_{1}S^{\dagger}_{2}B^{\dagger}_{3}+S^{\dagger}_{1}U^{\prime\dagger}_{0}S^{\dagger}_{3}\big{)}\big{(}B_{3}S_{2}R_{1}+S_{3}U^{\prime}_{0}B_{1}\big{)}=0\;,$
$\displaystyle(5)$ $\displaystyle
R^{\dagger}_{1}A^{\dagger}_{2}R_{2}+\big{(}R^{\dagger}_{1}S^{\dagger}_{2}R^{\dagger}_{3}+B^{\dagger}_{1}U^{\prime\dagger}_{0}A^{\dagger}_{3}\big{)}R_{3}B_{2}+\big{(}B^{\dagger}_{1}U^{\prime\dagger}_{0}S^{\dagger}_{3}+R^{\dagger}_{1}S^{\dagger}_{2}B^{\dagger}_{3}\big{)}B_{3}B_{2}=0\;,\hskip
65.44142pt$ $\displaystyle(6)$ $\displaystyle
R^{\dagger}_{2}A_{2}A_{1}+B^{\dagger}_{2}R^{\dagger}_{3}\big{(}R_{3}S_{2}A_{1}+A_{3}U^{\prime}_{0}S_{1}\big{)}+B^{\dagger}_{2}B^{\dagger}_{3}\big{(}B_{3}S_{2}A_{1}+S_{3}U^{\prime}_{0}S_{1}\big{)}=0\;.$
(A.2)

Switching off the extra neutral fermion sector, for example, we arrive at
$A_{2}=I$ and $R_{2}=0$. In this case the first relation in Eq. (A.1) can be
simplified to $A_{1}A^{\dagger}_{1}+R_{1}R^{\dagger}_{1}=I$, which is valid
for the canonical seesaw mechanism [31, 32, 33].

##  Appendix B The approximate forms of $A_{1,2}$, $B_{1,2}$, $R_{1,2}$ and
$S_{1,2}$

Given the very fact that the eighteen active-sterile flavor mixing angles
$\theta_{1j}$, $\theta_{2j}$ and $\theta_{3j}$ (for $j=4,5,\cdots,9$) must be
strongly suppressed in magnitude, one may simplify the four $3\times 3$
matrices $A_{1}$, $B_{1}$, $R_{1}$ and $S_{1}$ in Eqs. (10) and (11) to the
following forms:

$\displaystyle A_{1}$ $\displaystyle\simeq$ $\displaystyle
I-\left(\begin{matrix}\frac{1}{2}\left(s^{2}_{14}+s^{2}_{15}+s^{2}_{16}\right)&0&0\cr\hat{s}_{14}\hat{s}^{*}_{24}+\hat{s}_{15}\hat{s}^{*}_{25}+\hat{s}_{16}\hat{s}^{*}_{26}&\frac{1}{2}\left(s^{2}_{24}+s^{2}_{25}+s^{2}_{26}\right)&0\cr\hat{s}_{14}\hat{s}^{*}_{34}+\hat{s}_{15}\hat{s}^{*}_{35}+\hat{s}_{16}\hat{s}^{*}_{36}&\hat{s}_{24}\hat{s}^{*}_{34}+\hat{s}_{25}\hat{s}^{*}_{35}+\hat{s}_{26}\hat{s}^{*}_{36}&\frac{1}{2}\left(s^{2}_{34}+s^{2}_{35}+s^{2}_{36}\right)\cr\end{matrix}\right)\;,$
$\displaystyle B_{1}$ $\displaystyle\simeq$ $\displaystyle
I-\left(\begin{matrix}\frac{1}{2}\left(s^{2}_{14}+s^{2}_{24}+s^{2}_{34}\right)&0&0\cr\hat{s}^{*}_{14}\hat{s}_{15}+\hat{s}^{*}_{24}\hat{s}_{25}+\hat{s}^{*}_{34}\hat{s}_{35}&\frac{1}{2}\left(s^{2}_{15}+s^{2}_{25}+s^{2}_{35}\right)&0\cr\hat{s}^{*}_{14}\hat{s}_{16}+\hat{s}^{*}_{24}\hat{s}_{26}+\hat{s}^{*}_{34}\hat{s}_{36}&\hat{s}^{*}_{15}\hat{s}_{16}+\hat{s}^{*}_{25}\hat{s}_{26}+\hat{s}^{*}_{35}\hat{s}_{36}&\frac{1}{2}\left(s^{2}_{16}+s^{2}_{26}+s^{2}_{36}\right)\cr\end{matrix}\right)\;,\hskip
14.22636pt$ (B.1)

where the terms of ${\cal O}(s^{4}_{ij})$ (for $i=1,2,3$ and $j=4,5,6$) have
been omitted; and

$\displaystyle
S_{1}\simeq-R^{\dagger}_{1}\simeq-\left(\begin{matrix}\hat{s}_{14}&\hat{s}_{24}&\hat{s}_{34}\cr\hat{s}_{15}&\hat{s}_{25}&\hat{s}_{35}\cr\hat{s}_{16}&\hat{s}_{26}&\hat{s}_{36}\cr\end{matrix}\right)\;,$
(B.2)

where the terms of ${\cal O}(s^{3}_{ij})$ (for $i=1,2,3$ and $j=4,5,6$) have
been omitted. Similarly, the four $3\times 3$ matrices $A_{2}$, $B_{2}$,
$R_{2}$ and $S_{2}$ in Eqs. (12) and (13) can be simplified as follows:

$\displaystyle A_{2}$ $\displaystyle\simeq$ $\displaystyle
I-\left(\begin{matrix}\frac{1}{2}\left(s^{2}_{17}+s^{2}_{18}+s^{2}_{19}\right)&0&0\cr\hat{s}_{17}\hat{s}^{*}_{27}+\hat{s}_{18}\hat{s}^{*}_{28}+\hat{s}_{19}\hat{s}^{*}_{29}&\frac{1}{2}\left(s^{2}_{27}+s^{2}_{28}+s^{2}_{29}\right)&0\cr\hat{s}_{17}\hat{s}^{*}_{37}+\hat{s}_{18}\hat{s}^{*}_{38}+\hat{s}_{19}\hat{s}^{*}_{39}&\hat{s}_{27}\hat{s}^{*}_{37}+\hat{s}_{28}\hat{s}^{*}_{38}+\hat{s}_{29}\hat{s}^{*}_{39}&\frac{1}{2}\left(s^{2}_{37}+s^{2}_{38}+s^{2}_{39}\right)\cr\end{matrix}\right)\;,$
$\displaystyle B_{2}$ $\displaystyle\simeq$ $\displaystyle
I-\left(\begin{matrix}\frac{1}{2}\left(s^{2}_{17}+s^{2}_{27}+s^{2}_{37}\right)&0&0\cr\hat{s}^{*}_{17}\hat{s}_{18}+\hat{s}^{*}_{27}\hat{s}_{28}+\hat{s}^{*}_{37}\hat{s}_{38}&\frac{1}{2}\left(s^{2}_{18}+s^{2}_{28}+s^{2}_{38}\right)&0\cr\hat{s}^{*}_{17}\hat{s}_{19}+\hat{s}^{*}_{27}\hat{s}_{29}+\hat{s}^{*}_{37}\hat{s}_{39}&\hat{s}^{*}_{18}\hat{s}_{19}+\hat{s}^{*}_{28}\hat{s}_{29}+\hat{s}^{*}_{38}\hat{s}_{39}&\frac{1}{2}\left(s^{2}_{19}+s^{2}_{29}+s^{2}_{39}\right)\cr\end{matrix}\right)\;,\hskip
14.22636pt$ (B.3)

where the terms of ${\cal O}(s^{4}_{ij})$ (for $i=1,2,3$ and $j=7,8,9$) have
been omitted; and

$\displaystyle
S_{2}\simeq-R^{\dagger}_{2}\simeq-\left(\begin{matrix}\hat{s}_{17}&\hat{s}_{27}&\hat{s}_{37}\cr\hat{s}_{18}&\hat{s}_{28}&\hat{s}_{38}\cr\hat{s}_{19}&\hat{s}_{29}&\hat{s}_{39}\cr\end{matrix}\right)\;,$
(B.4)

where the terms of ${\cal O}(s^{3}_{ij})$ (for $i=1,2,3$ and $j=7,8,9$) have
been omitted.

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