Non-minimal approximation for the type-I seesaw mechanism

M. N. Dubinin1,2, E. Yu. Fedotova1

1 Skobeltsyn Institute of Nuclear Physics (SINP MSU), M.V. Lomonosov Moscow
State University, Leninskie gory, GSP-1, 119991 Moscow, Russia
2 MegaScience Center, National University of Science and Technology MISIS,
Leninskiy Prospekt, 4, 119049, Moscow, Russia
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

A non-minimal approximation for effective masses of light and heavy neutrinos
in the framework of a type-I seesaw mechanism with three generations of
sterile Majorana neutrinos is considered. The main results are: $(a)$ the
next-order corrections to the effective mass matrix of heavy neutrinos due to
terms ${\cal O}(\theta M_{D})$ are obtained, which modify the commonly used
representation for the effective mass ($M_{D}$ is a Dirac neutrino mass when
the electroweak symmetry is spontaneously broken); and $(b)$ the general form
of the mixing matrix is found in non-minimal approximation parametrized by a
complex $3\times 3$ matrix satisfying a nontrivial constraint. Numerical
analysis within the $\nu$MSM framework demonstrates the very small effect of
new contributions of direct collider observables as opposed to their possible
significance for cosmological models.

##  1 Introduction

Numerous consequences of the exceptionally successful standard model (SM) of
elementary particle physics based on the gauge symmetry $SU(3)\times
SU(2)_{L}\times U(1)$ have been confirmed by experiments on colliders and
extracted particle beams. No significant deviations from the SM predictions
were found, as a result of which it is considered as a mathematically
consistent effective field theory for a very large energy scale. At the same
time, it is obvious that SM is not a complete theory, since it cannot explain
a number of observed phenomena not only in particle physics, but also in
astrophysics and cosmology. One of the most well-known problems of SM is the
problem of neutrino masses and neutrino oscillations [1, 2, 3, 4, 5], which
will be considered in this paper. The essense of the problem is the absence of
clarity concerning what mechanism provides the observed neutrino mass spectrum
and how neutrinos of one flavor disappear and turn into neutrinos of another
flavor. Some still-unknown particles would be needed to analyse the neutrino
puzzle. Let us mention that the solution of other equally well-known problems
of SM, which are the problem of dark matter and dark energy in the Universe,
as well as the problem of its baryon asymmetry (BAU) generation, like the
neutrino problem, can also be associated with the SM extension by new
particles and/or interactions. The inability of modern collider experiments to
detect new particles may be due not only to their large masses, but also to
extremely elusive interactions with SM particles, making their production
extremely rare events. In other words, to detect feebly interacting particles,
the requirements for the integrated luminosity of the beam(s) in the
experiment must cross the ’intensity frontier’.

A well-known example of a field-theoretic model, within the framework of which
it is possible to solve a significant part of the above-mentioned problems by
extending the set of SM particles with relatively light and weakly interacting
new particles of the lepton sector, is the $\nu$MSM model (neutrino minimal
standard model) [6, 7]. It is an extension of the SM with the help of three
Majorana neutrinos whose flavor states do not participate in gauge
interactions (so-called sterile neutrinos, also called heavy neutral leptons,
HNL). The masses of sterile neutrinos do not exceed the electroweak scale
${\cal O}(10^{2})$ GeV and at the same time significantly exceed the value of
the mass parameter $Fv$ ($F$ is the neutrino Yukawa coupling constant and $v$
is the Higgs field vacuum expectation value equal to 174 GeV), as a result of
which the mass generation mechanism (known as the seesaw type-I mechanism [8,
9, 10, 11], see Section 2 below) works well for neutrino. Within the framework
of the $\nu$MSM, it is not necessary to introduce a new mass scale for heavy
fermions, the Majorana masses are comparable to the masses of light quarks and
charged leptons. The number of sterile neutrinos should be at least three. At
least two states of sterile neutrinos are necessary to explain the known
experimental parameters $\Delta m^{2}_{atm}$ and $\Delta m^{2}_{sol}$.
However, none of these states are suitable for the role of a dark matter
particle, since (1) their Yukawa constants are not sufficiently limited to
form a moderate amount of dark matter within the framework of the
Dodelson–Widrow mechanism [12], and (2) the one-loop radiative decays of such
states in the subdominant channel $N_{2,3}\to\nu\gamma$ with the energy
$E_{\gamma}=1/2M_{2,3}$ [13, 14] would produce too many gamma rays compared to
the available astrophysical experimental data111 Note that the unidentified
spectral line at the energy $E\sim 3.5$ keV in the stacked X-ray spectra of
dark-matter-dominated objects has been observed [15, 16]. The signal
distribution over the sky and the ratios of its strengths are consistent with
predictions for decaying dark matter with the mass $M_{1}\simeq 7.1\pm 0.1$
keV [17].. For these reasons, one more sterile neutrino (the HNL of the first
generation222 Throughout the paper we will assume that there is a normal
ordering in the HNL sector, i.e., $M_{1}<M_{2}\leq M_{3}$. added to the
$\nu$MSM provides a suitable candidate for the role of dark matter.

Adding three HNLs introduces 18 new parameters, 3 Majorana masses and 15
parameters of the Yukawa coupling matrix $F$. The mass of a DM sterile
neutrino can be varied in the range $\sim$1–50 keV in the case of $\nu$MSM
[17, 18]. The baryogenesis of the $\nu$MSM is provided by the oscillation
mechanism [17]. In the following, three generations of HNL are considered,
which makes it possible to explain simultaneously the origin of the masses of
standard (or active) neutrinos due to the seesaw mechanism, the existence of a
dark-matter particle and the generation of baryon asymmetry. Significant
limitations on the mixing parameter value for HNL in the $\nu$MSM model, as
well as on the second- and third-generation HNL masses, which should be split
very weakly [19], are reviewed in [17].

In extensions of the SM lepton sector, the active neutrino flavor states and
mass states can be related with the help of Pontecorvo–Maki–Nakagawa–Sakata
mixing matrix $U_{\rm PMNS}$, $\nu_{lL}(x)=\sum_{i}(U_{\rm
PMNS})_{li}\upnu_{iL}(x)$, where $\nu_{lL}$ is the left-handed (LH) component
of the flavour field, $\upnu_{iL}$ is the LH component of the state possessing
a mass $m_{i}$, $i=1,2,3$, and $l=e,\mu,\tau$ [1, 2, 3, 4, 5, 20]. The exact
values of neutrino masses are unknown, but it follows from the existing data
that there are at least three active neutrino mass eigenstates, neutrinos must
be light, $|(m_{\nu})_{ll^{\prime}}|\leq 1$ eV ($l,l^{\prime}=e,\mu,\tau$),
[21, 22] and the neutrino mass values are rather strongly split. The
differences between them measured experimentally give the central values of
mass parameters squared $\Delta m_{sol}^{2}\approx 7.54\times 10^{-5}$ eV2 and
$\Delta m_{atm}^{2}\approx 2.43\times 10^{-3}$ eV2 [23]. This leaves two
different possibilities of neutrino mass ordering: normal ordering (NO)
$m_{1}<m_{2}\ll m_{3}$ or inverse ordering (IO) $m_{3}\ll m_{1}<m_{2}$. The
mass of the lightest neutrino $m_{0}$ may vary from zero to 0.03 eV (NO) or
0.015 eV (IO) allowed by cosmological observations [24] (see also [25]).
Within the $\nu$MSM, the lightest active neutrino is lighter than ${\cal
O}(10^{-5})$ eV [6, 7, 26]. The HNL masses responsible for active neutrino
mass generation are rather strongly constrained both by direct searches [27,
28, 29, 30] and cosmological considerations [31, 32, 33]. The analysis of
lepton universality within the $\nu$MSM shows that $M_{2,3}>173$ MeV (264 MeV)
for NO (IO) [34], see also [35]. The recent investigation of BAU [36] implies
the masses of HNL to be $M_{2,3}\geq{\cal O}(3)$ GeV.

Historically, the motivation for considering right-handed (RH) neutrinos have
appeared from the left-right symmetric extension of the SM [37] where the
electroweak group of symmetry $SU(2)_{L}\times U(1)_{Y}$ was extended to the
$SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. The observed left chirality of
the low-energy weak interactions in the model with left–right symmetry,
usually attributed to the spontaneous breaking of this symmetry, implies the
existence of a right-handed neutrino.

It is commonly assumed that the effective mass $(m_{\nu})_{ll^{\prime}}$ can
be presented as

$(m_{\nu})_{ll^{\prime}}=-M_{D}M_{M}^{-1}M_{D}^{T}$ (1)

in the limit $M_{D}\ll M_{M}$, where $M_{D},M_{M}$ are Dirac and Majorana mass
matrices of the LH and RH neutrinos (see details below). However, Eq. (1) is a
standard approximation of a more complicated expression including an
additional $3\times 3$ matrix $\theta$ ($|\theta|\ll I$) describing mixing of
the active neutrinos with HNL. The contribution of the next-order terms in
effective neutrino mass representations requires a separate study. In the
following, Section 2, we investigate the standard approach for the type-I
seesaw mechanism and consider the non-minimal approximations for effective
masses of the active and sterile neutrinos. Nonstandard parametrization of the
mixing inspired by diagonalization of the active-sterile neutrino mass matrix
in the approach of [38] and a nontrivial relation for the parametrization
matrix $\Omega_{\rm nm}$ have been obtained. The numerical analysis within the
$\nu$MSM framework is performed in Section 3. Results are discussed in Section
4. A symmetric structure of effective and Majorana mass matrices as well as
relations for $M_{D}\theta^{T}$ and $\theta^{\dagger}M_{D}$ are analyzed in
Appendix A. Simple representations for the matrix $\Omega_{\rm nm}$ are
considered in Appendix B. A form of effective mass matrix of active neutrinos
$m_{\nu}$ as function of $U_{\rm PMNS}$ in standard and non-minimal
approximations is derived in Appendix C.

##  2 A model with three right-handed neutrinos

The lagrangian of the SM extension by three right-handed neutrinos $\nu_{R}$
has the form (see e.g., [39])

${\cal L}={\cal
L}_{SM}+i\overline{\nu}_{R}\partial_{\mu}\gamma^{\mu}\nu_{R}-\left(F\overline{l}_{L}\nu_{R}\tilde{H}+\frac{M_{M}}{2}\overline{\nu^{c}}_{R}\nu_{R}+\text{h.c.}\right),$
(2)

where $l_{L}=(\nu_{L}\,\,e_{L})^{T}$ is the left-handed lepton doublet,
$\nu_{R}$ is the right-handed neutrino [$(\nu_{R})^{c}\equiv
C\overline{\nu}_{R}^{T}$, where $C=i\gamma_{2}\gamma_{0}$ is the charge
conjugation matrix], $H$ is the Higgs doublet
($\tilde{H}=\epsilon_{ij}H^{\dagger}$, $\epsilon_{ij}$ antisymmetric tensor),
$F$ is the neutrino Yukawa coupling matrix, and $M_{M}$ is the $3\times 3$
Majorana mass matrix of the RH neutrinos, which does not break the SM gauge
symmetry.

After spontaneous electroweak symmetry breaking, the full neutrino mass term
in Equation (2) can be presented as

$\frac{1}{2}(\overline{\nu}_{L}\vspace{2mm}\overline{\nu^{c}}_{R}){\cal
M}\left(\begin{tabular}[]{c}$\nu_{L}^{c}$\\\
$\nu_{R}$\end{tabular}\right),\qquad\text{where}\qquad{\cal
M}=\left(\begin{tabular}[]{cc}0&$M_{D}$\\\
$M_{D}^{T}$&$M_{M}$\end{tabular}\right),$ (3)

$M_{D}=Fv$ is a $3\times 3$ neutrino Dirac mass matrix ($v$=174 GeV is a
vacuum expectation value of the Higgs doublet), and $M_{D},M_{M}$ are complex-
valued in the general case. In the neutrino mass basis with the help of a
unitary matrix ${\cal U}$

${\cal U}=W\cdot{\rm diag}(U_{\nu},U_{N}^{*}),$ (4)

($U_{\nu},U_{N}$ on the one hand side and $W$ on the other hand side are
$3\times 3$ and $6\times 6$ matrices, respectively), one can define the
diagonal mass matrix [39]

${\cal U}^{\dagger}{\cal M}{\cal
U}^{*}=\left(\begin{tabular}[]{cc}$\hat{m}$&0\\\
0&$\hat{M}$\end{tabular}\right),\qquad\begin{tabular}[]{c}$\hat{m}={\rm
diag}(m_{1},m_{2},m_{3})$,\\\ $\hat{M}={\rm
diag}(M_{1},M_{2},M_{3})$,\end{tabular}$ (5)

where $m_{i},M_{i}$ are masses of light and heavy neutrinos. The unitary
diagonalization matrix $W$ formally expressed as the exponential of an
antihermitian matrix, see [38],

$W=\exp\left(\begin{tabular}[]{cc}0&$\theta$\\\
$-\theta^{\dagger}$&0\end{tabular}\right)$ (6)

(where $\theta$ is a $3\times 3$ complex matrix) converts the matrix ${\cal
M}$ to a block diagonal form

$W^{\dagger}\left(\begin{tabular}[]{cc}0&$M_{D}$\\\
$M_{D}^{T}$&$M_{M}$\end{tabular}\right)W^{*}=\left(\begin{tabular}[]{cc}$m_{\nu}$&0\\\
0&$M_{N}$\end{tabular}\right)\equiv\tilde{{\cal M}}.$ (7)

Here, $m_{\nu},M_{N}$ are effective Majorana mass terms for the LH and RH
flavour neutrinos (see, e.g., [40, 41]) which are related with $\hat{m}$ and
$\hat{M}$ by

$\displaystyle m_{\nu}=U_{\nu}\hat{m}U_{\nu}^{T},\ \qquad
M_{N}=U_{N}^{*}\hat{M}U_{N}^{\dagger}.$ (8)

The flavour states $\nu_{L}$ and $\nu_{R}$ and the mass states $\upnu$ and $N$
then can be related as [39]

$\left(\begin{tabular}[]{c}$\nu_{L}$\\\
$\nu_{R}^{c}$\end{tabular}\right)=P_{L}{\cal
U}\left(\begin{tabular}[]{c}${\upnu}$\\\ $N$\end{tabular}\right),$ (9)

where $P_{L}$ is a left-helicity projector.

###  2.1 Standard approximation

Assuming that $\theta$ is “small,” it is sufficient to expand the rotation
matrix (6) up to ${\cal O}(\theta^{2})$ terms

$\displaystyle W=\exp\left(\begin{tabular}[]{cc}0&$\theta$\\\
$-\theta^{\dagger}$&0\end{tabular}\right)\simeq\left(\begin{tabular}[]{cc}$1-\frac{1}{2}\theta\theta^{\dagger}$&$\theta$\\\
$-\theta^{\dagger}$&$1-\frac{1}{2}\theta^{\dagger}\theta$\end{tabular}\right),$
(14)

thus, using (7) and the requirement $\tilde{\cal M}_{12}=0$, one can relate
Dirac, Majorana and $\theta$ matrices

$\theta\simeq M_{D}M_{M}^{-1},$ (15)

which yields the expressions

$\displaystyle m_{\nu}$ $\displaystyle\simeq$ $\displaystyle-
M_{D}\theta^{T}\simeq-M_{D}M^{-1}_{M}M_{D}^{T},$ (16) $\displaystyle M_{N}$
$\displaystyle\simeq$ $\displaystyle M_{M}+\frac{1}{2}(\theta^{\dagger}\theta
M_{M}+M_{M}^{T}\theta^{T}\theta^{*})\simeq M_{M},$ (17)

where, in some cases, the terms ${\cal O}(\theta^{2})$ in (17) are omitted, so
Majorana mass matrices before and after rotation coincide. One can represent
(17) in terms of $\theta$ and $M_{D}$ using the relations (76) and (78) (see
Appendix A)

$M_{N}\simeq(\theta^{-1}+\theta^{\dagger})M_{D}.$ (18)

In the representation (14), the flavour states of active neutrinos can be
expressed as [see (9)]

$\displaystyle\nu_{L}$ $\displaystyle\simeq$
$\displaystyle\left(1-\frac{1}{2}\theta\theta^{\dagger}\right)U_{\nu}\upnu_{L}+\theta
U_{N}^{*}N_{L},$ (19) $\displaystyle\nu_{R}^{c}$ $\displaystyle\simeq$
$\displaystyle-\theta^{\dagger}U_{\nu}\upnu_{L}+\left(1-\frac{1}{2}\theta\theta^{\dagger}\right)U_{N}^{*}N_{L},$
(20)

where terms in front of $\upnu_{L}$ and $N_{L}$ have to be identified with
PMNS and mixing matrices333 In general, the PMNS matrix can be presented as
$U_{\rm PMNS}=U_{l}^{\dagger}(1+\eta)U_{\nu}$ [41], where $U_{l}$ diagonalizes
the charged lepton mass matrix $m_{l}$,
$U_{l}m_{l}m_{l}^{\dagger}U_{l}^{\dagger}={\rm
diag}(m_{e}^{2},m_{\mu}^{2},m_{\tau}^{2})$, but throughout the paper, we
assume that $U_{l}=I$.

$U_{\rm PMNS}\simeq(1+\eta)U_{\nu},\qquad\Theta_{\rm m}\simeq\theta
U_{N}^{*}.$ (21)

Here, the parameter $\eta=-1/2(\theta\theta^{\dagger})$ is considered as a
deviation from the unitarity.

###  2.2 Non-minimal approximation

One can, however, consider the expansion of the matrix $W$, see Equation (6),
up to ${\cal O}(\theta^{3})$-terms

$\displaystyle
W\simeq\left(\begin{tabular}[]{cc}$1-\frac{1}{2}\theta\theta^{\dagger}$&$\theta-\frac{1}{6}\theta\theta^{\dagger}\theta$\\\
$-\theta^{\dagger}+\frac{1}{6}\theta^{\dagger}\theta\theta^{\dagger}$&$1-\frac{1}{2}\theta^{\dagger}\theta$\end{tabular}\right),$
(24)

where again, $|\theta|\ll I$. Then, the corresponding expressions (7) can be
presented as

$\displaystyle\tilde{\cal M}_{11}$ $\displaystyle=$ $\displaystyle-(\theta
M_{D}^{T}+M_{D}\theta^{T})+\theta
M_{M}\theta^{T}+\frac{1}{6}(\theta\theta^{\dagger}\theta
M_{D}^{T}+M_{D}\theta^{T}\theta^{*}\theta^{T})$ (25) $\displaystyle+$
$\displaystyle\frac{1}{2}(\theta\theta^{\dagger}M_{D}\theta^{T}+\theta
M_{D}^{T}\theta^{*}\theta^{T})\simeq m_{\nu},$ $\displaystyle\tilde{\cal
M}_{22}$ $\displaystyle=$ $\displaystyle
M_{M}+(\theta^{\dagger}M_{D}+M_{D}^{T}\theta^{*})-\frac{1}{2}(\theta^{\dagger}\theta
M_{M}+M_{M}\theta^{T}\theta^{*})$ (26) $\displaystyle-$
$\displaystyle\frac{1}{2}\theta^{\dagger}(\theta
M_{D}^{T}+M_{D}\theta^{T})\theta^{*}-\frac{1}{6}(\theta^{\dagger}\theta\theta^{\dagger}M_{D}+M_{D}^{T}\theta^{*}\theta^{T}\theta^{*})\simeq
M_{N},$ $\displaystyle\tilde{\cal M}_{12}$ $\displaystyle=$ $\displaystyle
M_{D}-\theta
M_{M}-\frac{1}{2}(\theta\theta^{\dagger}M_{D}+M_{D}\theta^{T}\theta^{*})$ (27)
$\displaystyle-$ $\displaystyle\theta M_{D}^{T}\theta^{*}+\frac{1}{2}\theta
M_{M}\theta^{T}\theta^{*}+\frac{1}{6}\theta\theta^{\dagger}\theta M_{M}\simeq
0.$

One can note that the relation between Dirac, Majorana, and $\theta$ matrices
can be defined in the form [see (27)]

$M_{M}-\frac{1}{2}M_{M}\theta^{T}\theta^{*}-\frac{1}{6}\theta^{\dagger}\theta
M_{M}\simeq\theta^{-1}M_{D}\left(1-\frac{1}{2}\theta^{T}\theta^{*}\right)-\frac{1}{2}\theta^{\dagger}M_{D}-M_{D}^{T}\theta^{*},$
(28)

which is more complicated than the expression (15) of the standard
approximation and can be converted to it if ${\cal O}(\theta^{2})$-terms are
neglected. As one can see, the relation between two mass scales $M_{D}$,
$M_{M}$ and $\theta$ is not so obvious, but analyzing the relation (28) one
can note that (see Appendix A)

${\cal O}(\theta^{2}M_{M})\sim{\cal O}(\theta M_{D}).$ (29)

This allows one to express Equations (25) and (26) in terms of ${\cal
O}(\theta^{n}M_{D})$, where $n=-1,1,2,3$. However, the ${\cal
O}(\theta^{2}M_{D})$ and ${\cal O}(\theta^{3}M_{D})$ terms are much smaller
than ${\cal O}(\theta M_{D})$, so in the following we will consider only the
last one. Thus, the expressions for $m_{\nu},M_{N}$ and $M_{M}$ take the forms

$\displaystyle m_{\nu}$ $\displaystyle\simeq$ $\displaystyle-\theta
M_{D}^{T}\simeq-M_{D}\theta^{T},$ (30) $\displaystyle M_{N}$
$\displaystyle\simeq$
$\displaystyle\left(\theta^{-1}-\frac{1}{3}\theta^{\dagger}\right)M_{D}\simeq
M_{M}+\theta^{\dagger}M_{D},$ (31)

where the right forms can be obtained if one takes into account the relations
(76) and (78), see Appendix A.

Analyzing the representation for the flavour states (9)

$\displaystyle\nu_{L}$ $\displaystyle\simeq$
$\displaystyle\left(1-\frac{1}{2}\theta\theta^{\dagger}\right)U_{\nu}\upnu_{L}+\left(\theta-\frac{1}{6}\theta\theta^{\dagger}\theta\right)U_{N}^{*}N_{L},$
(32) $\displaystyle\nu_{R}^{c}$ $\displaystyle\simeq$
$\displaystyle\left(-\theta^{\dagger}+\frac{1}{6}\theta^{\dagger}\theta\theta^{\dagger}\right)U_{\nu}\upnu_{L}+\left(1-\frac{1}{2}\theta^{\dagger}\theta\right)U_{N}^{*}N_{L},$
(33)

one can identify the PMNS and active-sterile mixing matrices with

$U_{\rm
PMNS}\simeq\left(1-\frac{1}{2}\theta\theta^{\dagger}\right)U_{\nu},\qquad\Theta_{\rm
nm}\simeq\left(\theta-\frac{1}{6}\theta\theta^{\dagger}\theta\right)U_{N}^{*}.$
(34)

As far as $|\theta|\ll 1$, one can omit the terms ${\cal O}(\theta^{n})$,
$n>1$ in the expressions (34), so these forms coincide with ones of the
standard approximation, see (21).

Using (18) and (31), one can obtain the difference between the effective mass
of heavy neutrinos in standard and non-minimal approximations

$M_{N{\rm m}}-M_{N{\rm nm}}=\frac{4}{3}\theta^{\dagger}Fv.$ (35)

###  2.3 Parametrization of the mixing matrix $\Theta$

####  2.3.1 Standard approximation

The expression $I=M_{N}M_{N}^{-1}$, with the help of (8), (16), (17) and the
assumption $U_{\rm PMNS}\simeq U_{\nu}$ can be presented as

$\displaystyle I=-M_{D}^{T}U_{\rm PMNS}^{*}\hat{m}^{-1}U_{\rm
PMNS}^{\dagger}M_{D}U_{N}\hat{M}^{-1}U_{N}^{T}.$ (36)

One can rewrite (36) in the form

$I=-(\sqrt{\hat{M}^{-1}}U_{N}^{T}M_{D}^{T}U_{\rm
PMNS}^{*}\sqrt{\hat{m}^{-1}})(\sqrt{\hat{m}^{-1}}U_{\rm
PMNS}^{\dagger}M_{D}U_{N}\sqrt{\hat{M}^{-1}})=\Omega_{\rm m}^{T}\Omega_{\rm
m},$ (37)

where

$\Omega_{\rm m}=i\sqrt{\hat{m}^{-1}}U_{\rm
PMNS}^{\dagger}M_{D}U_{N}\sqrt{\hat{M}^{-1}}$ (38)

is an arbitrary complex orthogonal matrix ($\Omega_{\rm m}^{T}\Omega_{\rm
m}=I$). Thus, the Dirac matrix can be parametrized as [38]

$M_{D}=-iU_{\rm PMNS}\sqrt{\hat{m}}\Omega_{\rm
m}\sqrt{\hat{M}}U_{N}^{\dagger},$ (39)

which together with (15) leads to the expression for the mixing matrix

$\Theta_{\rm m}\simeq\theta U_{N}^{*}\simeq-iU_{\rm
PMNS}\sqrt{\hat{m}}\Omega_{\rm m}\sqrt{\hat{M}^{-1}}.$ (40)

####  2.3.2 Non-minimal approximation

The same procedure $I=M_{N}M_{N}^{-1}$ in the non-minimal case $M_{M}\sim{\cal
O}(\theta M_{D})$ leads to

$\displaystyle-\left[\sqrt{\hat{M}^{-1}}U_{N}^{T}\left(\theta^{-1}-\frac{1}{3}\theta^{\dagger}\right)U_{\nu}\sqrt{\hat{m}}\right]\left[\sqrt{\hat{m}}U_{\nu}^{T}(\theta^{T})^{-1}U_{N}\sqrt{\hat{M}^{-1}}\right]\simeq
I,$ (41)

where $M_{N}^{-1}\equiv U_{N}\hat{M}^{-1}U_{N}^{T}$ and $M_{N}$ is defined by
Equations (31) and (30). If one denotes the expression in the right brackets
of (41) as $\Omega_{\rm nm}$

$\Omega_{\rm
nm}=i\sqrt{\hat{m}}U_{\nu}^{T}(\theta^{T})^{-1}U_{N}\sqrt{\hat{M}^{-1}}$ (42)

and takes into account that the left side of (41) concerns the unit matrix,
then one can conclude using the identity $\Omega_{\rm nm}^{-1}\Omega_{\rm
nm}=I$ that the expression in left brackets is $\Omega_{\rm nm}^{-1}$. Then,
from Equation (41) one can find the relation for $\Omega_{\rm nm}$

$\Omega_{\rm nm}^{-1}=\Omega_{\rm nm}^{T}+\frac{1}{3}\hat{M}^{-1}(\Omega_{\rm
nm}^{-1})^{*}\hat{m}.$ (43)

Defining $\theta$ with the help of (42)

$\theta\simeq-iU_{\nu}\sqrt{\hat{m}}(\Omega_{\rm
nm}^{-1})^{T}\sqrt{\hat{M}^{-1}}U_{N}^{T}$ (44)

and using the definition of the mixing matrix $\Theta$, Equation (21), one can
write

$\Theta_{\rm nm}\simeq\theta U_{N}^{*}\simeq-iU_{\rm
PMNS}\sqrt{\hat{m}}(\Omega_{\rm nm}^{-1})^{T}\sqrt{\hat{M}^{-1}},$ (45)

where again it was assumed that $U_{\nu}\simeq U_{\rm PMNS}$. Note that the
mixing matrix $\Theta$ in approximations where $M_{M}\simeq\theta^{-1}M_{D}$
and $M_{M}\simeq{\cal O}(\theta M_{D})$ is parametrized in the same way. The
difference is in the presence of requirements that must be met by an arbitrary
$3\times 3$ matrix $\Omega$. The corresponding Yukawa coupling matrix in terms
of $\Omega_{\rm nm}$ can be presented as

$F\simeq\frac{i}{v}U_{\rm PMNS}\sqrt{\hat{m}}\Omega_{\rm
nm}\sqrt{\hat{M}}U_{N}^{\dagger}.$ (46)

Note that the Dirac matrix in non-minimal approximation differs from (39) by
the parametrization matrix $\Omega_{\rm nm}$ and the sign because
sign($\Omega_{\rm nm})\neq$ sign($\Omega_{\rm nm}^{-1}$) whereas
sign($\Omega_{\rm m}$) = sign($\Omega_{\rm m}^{-1}$), see (38) and (42).

There is an infinite set of $\Omega$ matrices leading to the observed neutrino
parameters [41]. In various mixing scenario frameworks, the matrix
$\Omega_{\rm m}$ can be parametrized by means of three Euler angles (complex-
valued in general), see [42, 43]; one complex angle $\omega$ and the sign
factor $\xi=\pm 1$ [25, 36], or just by the unit matrix, see, e.g., [42, 44].
It is desirable to relate a minimal and non-minimal matrices $\Omega_{\rm m}$
and $\Omega_{\rm nm}$ in order to consider the limit of the discussed
approximations. However, one can find that the simple representation

$\Omega_{\rm nm}=\Omega_{\rm n}+\Omega_{\rm m},$ (47)

where $\Omega_{\rm m}\Omega_{\rm m}^{T}=I$, $\Omega_{\rm n}\Omega_{\rm
n}^{T}\neq I$ is not relevant, see Appendix B.

###  2.4 Interactions of the Majorana fields with SM particles

The charged-current (CC) and the neutral-current (NC) interactions of the
Majorana fields with SM particles in terms of mixing parameter and $U_{\rm
PMNS}$ can be obtained as follows (see also [44]). Using the SM normalization
conventions of [39]

$-\frac{g}{\sqrt{2}}\overline{\nu}_{L}\gamma^{\mu}l_{L}W_{\mu}^{+}-\frac{g}{\sqrt{2}}\overline{l}_{L}\gamma^{\mu}\nu_{L}W_{\mu}^{-}-\frac{g}{2\cos\theta_{W}}\overline{\nu}_{L}\gamma^{\mu}\nu_{L}Z_{\mu},$
(48)

where $\theta_{W}$ is the Weinberg angle, one can write the corresponding
neutral and charged currents in the mass basis

$\displaystyle{\cal L}_{NC}^{\upnu}$ $\displaystyle=$
$\displaystyle\frac{g}{2c_{W}}\overline{\upnu}_{L}\gamma^{\mu}U_{\rm
PMNS}^{\dagger}U_{\rm PMNS}\upnu_{L}Z_{\mu},$ (49) $\displaystyle{\cal
L}_{CC}^{\upnu}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\overline{l}_{L}\gamma^{\mu}U_{\rm
PMNS}\upnu_{L}W_{\mu}^{-}+\text{h.c.},$ (50) $\displaystyle{\cal L}_{NC}^{N}$
$\displaystyle=$
$\displaystyle-\frac{g}{2c_{W}}\overline{N}_{L}\gamma^{\mu}\Theta^{\dagger}\Theta
N_{L}Z_{\mu}-\left(\frac{g}{2c_{W}}\overline{\upnu}_{L}U_{\rm
PMNS}^{\dagger}\gamma^{\mu}\Theta N_{L}Z_{\mu}+h.c.\right),$ (51)
$\displaystyle{\cal L}_{CC}^{N}$ $\displaystyle=$
$\displaystyle-\frac{g}{\sqrt{2}}\overline{l}_{L}\gamma^{\mu}\Theta
N_{L}W_{\mu}^{-}+\text{h.c.},$ (52)

where $\Theta$ is a minimal $\Theta_{\rm m}$ or a non-minimal $\Theta_{\rm
nm}$ mixing matrix.

It is instructive to note that the active neutrino mass matrix $m_{\nu}$ has
the same representation for both $M_{M}\simeq\theta^{-1}M_{D}$ and
$M_{M}\sim{\cal O}(\theta M_{D})$ approximations (see Appendix C)

$m_{\nu}\simeq U_{\rm PMNS}\hat{m}U_{\rm PMNS}^{T}.$ (53)

##  3 Numerical examples

As mentioned above, signals of sufficiently light HNL’s are not observed in
collider or beam-dump experiments. Experimental information is reduced to
exclusion contours generated on the mass-mixing plane for different detectors.
It should be noted that these exclusion contours use a model-independent
phenomenological approach, in which only one HNL gives the observed signal,
and the remaining two are strongly decoupled and practically do not affect the
analysis. Within the framework of this approach, only two independent
parameters are considered, the HNL mass and its coupling constant with an
active neutrino, the generation of exclusion contours is carried out for the
square of the mixing parameter as a function of mass for one flavor only,
while the mixing for other flavors is assumed to be zero. Assumptions of this
kind seem to be very strong and need to be translated for the case of a
specific model. Suffice it to note that, even in the simple case of one
generation of Majorana neutrinos [45], the mixing parameter $U=M_{D}/M_{N}$
depends on the mass of HNL, as a result of which the mapping of the active
neutrino mass—HNL mass plane to the mixing squared—HNL mass plane is a
triangle, not a rectangle. A part of the triangle for a limited region of HNL
masses may look approximately like a rectangle due to the low mass of the
active neutrino.

In the following instructive example, we will vary the masses of the lightest
neutrino and heavy neutrinos in the ranges (see [6, 7, 26])

$\displaystyle m_{0}:$ $\displaystyle 0-10^{-5}\;\text{eV},$ (54)
$\displaystyle M_{i}:$ $\displaystyle 1-50\;\text{keV}\;\text{for
}M_{1},\qquad 1-100\;\text{GeV}\;\text{for }M_{2,3},$ (55)

parametrize the masses of active neutrinos as
$m_{1}^{2}=m_{0}^{2},m_{2}^{2}=m_{0}^{2}+\Delta
m_{sol}^{2},m_{3}^{2}=m_{0}^{2}+\Delta m_{sol}^{2}+\Delta m_{atm}^{2}$ for NO
or $m_{1}^{2}=m_{0}^{2}+\Delta m_{sol}^{2},m_{2}^{2}=m_{0}^{2}+\Delta
m_{sol}^{2}+\Delta m_{atm}^{2},m_{3}^{2}=m_{0}^{2}$ for IO and use the present
central values of the known active neutrino sector parameters [23]. In
addition, we will use phenomenologically convenient quantities (see, e.g.,
[25])

$U_{\alpha I}^{2}=|\Theta_{\alpha I}|^{2},\qquad
U_{I}^{2}=\sum_{\alpha}U_{\alpha I}^{2},\qquad U^{2}=\sum_{I}U_{I}^{2}.$ (56)

Among the infinite set of $\Omega_{\rm m}$ matrices satisfying the requirement
$\Omega_{\rm m}^{T}\Omega_{\rm m}=I$, we consider the following explicit forms

$\displaystyle\Omega^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
I,\qquad\Omega^{{}^{\prime\prime}}=-\left(\begin{tabular}[]{ccc}$\frac{1}{\sqrt{2}}$&$-\frac{1}{2}$&$-\frac{1}{2}$\\\
$\frac{1}{3\sqrt{2}}$&$\frac{5}{6}$&$-\frac{1}{2}$\\\
$\frac{2}{3}$&$\frac{1}{3\sqrt{2}}$&$\frac{1}{\sqrt{2}}$\end{tabular}\right),$
(60) $\displaystyle\Omega^{{}^{\prime\prime\prime}}_{\rm NO}$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{ccc}1&0&0\\\ 0&0&1\\\
0&$-1$&0\end{tabular}\right),\;\text{ }\Omega^{{}^{\prime\prime\prime}}_{\rm
IO}=\left(\begin{tabular}[]{ccc}0&0&1\\\ 0&$-1$&0\\\
1&0&0\end{tabular}\right),$ (67)

where $\Omega_{\rm m}=\Omega^{{}^{\prime}}$ corresponds to a naive explicit
form used, e.g., in [42, 44]; the form $\Omega_{\rm
m}=\Omega^{{}^{\prime\prime}}$ is not motivated by any special physical
scenarios and used for numerical estimations in this section only; and
$\Omega_{\rm m}=\Omega^{{}^{\prime\prime\prime}}$ is a form inspired by a
parametrization in [36] with fixed parameters $\omega=\pi/2$, $\xi=1$ which
(as was determined numerically) maximize the $CP$-asymmetry. The non-minimal
parametrization matrix satisfying (43) is chosen as (see, also, Appendix B)

$\Omega_{\rm nm}={\rm diag}(\omega_{1},\omega_{2},\omega_{3}),\;\text{where
}\omega_{i}=\sqrt{1-\frac{1}{3}\frac{m_{i}}{M_{i}}}.$ (68)

Results of evaluations for diagonal elements $(m_{\nu})_{ee}$,
$(m_{\nu})_{\mu\mu}$, $(m_{\nu})_{\tau\tau}$ with explicit forms $\Omega_{\rm
m}$ as
$\Omega^{{}^{\prime}},\Omega^{{}^{\prime\prime}},\Omega^{{}^{\prime\prime\prime}}$
are presented in Table 1, where first $M_{D}$ is found using (39) then
$(m_{\nu})_{ll}$ is evaluated with the help of (8) and (16). The results for
all cases coincide [in accordance with (53)] and are below the present
experimental bounds.

Table 1: Numerical estimations of diagonal elements $(m_{\nu})_{ee}$, $(m_{\nu})_{\mu\mu}$ and $(m_{\nu})_{\tau\tau}$ in units of $10^{-2}$ eV for explicit forms $\Omega_{\rm m}$: $\Omega^{{}^{\prime}},\Omega^{{}^{\prime\prime}},\Omega^{{}^{\prime\prime\prime}}$, see (67). Here, $m_{0}$ and $M_{1},M_{2},M_{3}$ are defined by (54) and (55).  Ordering | $l=e$ | $l=\mu$ | $l=\tau$
---|---|---|---
NO | $0.4$ | $2.7$ | $2.6$
IO | $4.9$ | $1.9$ | $2.6$

The dependence of $U^{2}$ as a function of masses of heavy neutrinos $M_{1}$
and $M_{2}=M_{3}=M$ at $m_{0}=10^{-5}$ eV for NO is presented in Figure 1a,
where one can see that its values are ${\cal O}(10^{-9})$ and practically
constant for all $M$. The qualitative behavior is similar for other cases. As
a rule, the contributions of $M_{1}$ to $U^{2}$ are neglected (as was assumed
e.g., in [36]); however, Figure 1a demonstrates that the dependence
$U^{2}(M_{1})$ can be sizable in some cases. One can note that the minimal
(maximal) value of $U^{2}$ is achieved at benchmark point BPmin (BPmax), where
BPmin: $m_{0}=0$, $M_{1}$=50 keV, $M$=100 GeV,
BPmax: $m_{0}=10^{-5}$ eV, $M_{1}$=1 keV, $M$=1 GeV.

(a)

(b)

Figure 1: Mixing parameter estimations for NO: (a) the dependence of $U^{2}$
as a function of masses of heavy neutrinos $M_{1}$ and $M_{2}=M_{3}=M$ at
$m_{0}=10^{-5}$ eV, $\Omega_{\rm m}=\Omega^{{}^{\prime}}$; (b) the relative
difference in $U^{2}$ obtained in minimal and non-minimal approximations as a
function of $M_{2}=M_{3}=M$ at BP1: $m_{0}=0$, $M_{1}$=1 keV in units of
$10^{-12}$ (black dotted line); BP2: $m_{0}=10^{-5}$ eV, $M_{1}$=1 keV in
units of $10^{-9}$ (red dashed line); BP3: $m_{0}=0$, $M_{1}$=50 keV in units
of $10^{-12}$ (green solid line); and BP4: $m_{0}=10^{-5}$ eV, $M_{1}$=50 keV
in units of $10^{-11}$ (blue dot-dashed line).

Numerical estimations of $U^{2}$ for different $\Omega_{\rm m}$
($\Omega^{{}^{\prime}},\Omega^{{}^{\prime\prime}},\Omega^{{}^{\prime\prime\prime}}$)
can be found in Table 2; based on these numbers, it is possible to come to a
conclusion about the great complexity of heavy Majorana neutrinos observation.

Table 2: Results for $U^{2}$ in the case of NO and IO for different $\Omega_{\rm m}$: $\Omega^{{}^{\prime}},\Omega^{{}^{\prime\prime}},\Omega^{{}^{\prime\prime\prime}}$, see (67). Here, $m_{0}$ and $M_{1},M_{2},M_{3}$ are defined by (54) and (55). Ordering | $\Omega^{{}^{\prime}}$ | $\Omega^{{}^{\prime\prime}}$ | $\Omega^{{}^{\prime\prime\prime}}$
---|---|---|---
NO | $\ 10^{-12}-10^{-8}$ | $10^{-7}-10^{-5}$ | $10^{-12}-10^{-8}$
IO | $10^{-6}-10^{-5}$ | $10^{-6}-10^{-5}$ | $10^{-12}-10^{-8}$

The relative difference $|U^{2}_{\rm nm}-U^{2}_{\rm m}|/U^{2}_{\rm nm}$ for
NO, IO and different $\Omega_{\rm m}$
($\Omega^{{}^{\prime}},\Omega^{{}^{\prime\prime}},\Omega^{{}^{\prime\prime\prime}}$)
at BPmin (BPmax) is presented in Table 3. One can see that it is quite small
for $\Omega^{{}^{\prime}},\Omega^{{}^{\prime\prime\prime}}_{\rm NO}$ and is
very large for $\Omega^{{}^{\prime\prime}}_{\rm NO}$, which is a consequence
of different representations for $\Omega_{\rm m}$ and $\Omega_{\rm nm}$. Thus,
the most relevant comparison can be performed for $\Omega_{\rm
m}=\Omega^{{}^{\prime}}$ and $\Omega_{\rm nm}$ defined by (67) and (68) since
for such representations of the parametrization matrix, it is possible to
trace the limit from the standard approximation to the non-minimal one.

Table 3:  The relative difference $|U^{2}_{\rm nm}-U^{2}_{\rm m}|/U^{2}_{\rm nm}$ for NO, IO and different $\Omega_{\rm m}$ ($\Omega^{{}^{\prime}},\Omega^{{}^{\prime\prime}},\Omega^{{}^{\prime\prime\prime}}$) (67) at BPmin (BPmax). Ordering | $\Omega^{{}^{\prime}}$ | $\Omega^{{}^{\prime\prime}}$ | $\Omega^{{}^{\prime\prime\prime}}$
---|---|---|---
NO | $\ 10^{-13}\;\text{ }(10^{-9})$ | $10^{6}\;\text{ }(10^{3})$ | $10^{-13}\;\text{ }(10^{-9})$
IO | $10^{-7}\;\text{ }(10^{-5})$ | $0.1\;\text{ }(0.1)$ | $1\;\text{ }(1)$

The relative difference of $U^{2}$ obtained in the minimal and non-minimal
approximations as a function of $M_{2}=M_{3}=M$ for
$\Omega^{{}^{\prime\prime\prime}}$ in the NO case is presented in Figure 1b.
As one can see, the results for BP1 and BP3 coincide which is explained by the
choice $m_{0}$=0 and the explicit form of $\Omega^{{}^{\prime\prime\prime}}$.
The behavior of the relative difference of $U^{2}$ at $m_{0}=0$ and $m_{0}\neq
0$ is different due to the contributions of $M_{1}$ to the mixing parameter.
Thus, the approximation with two RH neutrinos is quite reliable at $m_{0}$=0
and $\Omega^{{}^{\prime\prime\prime}}$, while one should use a model with
three HNL at $m_{0}\neq 0$. The relative difference for NO is rather small.
Thus, in the cases where the limit of discussed approximations can be traced,
the phenomenological consequences with a mixing parameter obtained in the non-
minimal approximation can be hardly distinguished from the results of a
standard approach (see examples in [42, 44, 46]) at present or future
facilities.

##  4 Conclusions

The paper is devoted to the investigation of the SM extension by the three
generations of sterile Majorana neutrinos which recover the left–right
symmetry between quarks and leptons. Experimental detection of particles of
this kind would greatly affect our understanding of particle physics and
cosmology. The analysis carried out is especially significant for models in
which there is no additional nonstandard scale of particle masses, so that a
well-known ’model-independent approach’ to calculating constraints for the
model parameter space may not be directly relevant to reality. A non-minimal
approximation for effective masses of light and heavy neutrinos in the
framework of a type-I seesaw mechanism with three generations of right-handed
Majorana neutrinos was considered. It was found that the obtained next-order
corrections to the effective mass matrix of HNL can influence the well-known
results of the current order modifying them with ${\cal O}(\theta M_{D})$
corrections. Additional contributions to the effective masses of neutrinos
were analysed and the regimes when they can be significant were defined.
Following the method proposed in [38], a parametrization of the mixing matrix
was performed with the help of a complex $3\times 3$ matrix $\Omega_{\rm nm}$
satisfying a nontrivial relation (43) in the non-minimal approximation. The
structure of effective and Majorana mass matrices was scrutinised and it was
demonstrated that they are symmetric matrices, a fact which has not been
discussed before. Useful relations were found for $M_{D}\theta^{T}$ and
$\theta^{\dagger}M_{D}$, which allow one to express effective masses in a
simple form. It is interesting to note that the modification of the mixing
matrix does not change results for the effective mass matrix of active
neutrinos $m_{\nu}$ in the non-minimal case.

The obtained results were analyzed in detail within the $\nu$MSM framework. It
was demonstrated that the approximation with two RH neutrinos is quite
reliable at $m_{0}$=0, while one should use a model with three HNL at
$m_{0}\neq 0$. New additional corrections to effective masses can be more
pronounced when heavy neutrino masses are small and the lightest mass of
active neutrino is of ${\cal O}(10^{-5})$ eV. Numerical examples demonstrate
that the relative difference in the mixing parameter obtained in standard and
non-minimal approximations is negligibly small for all benchmark scenarios of
the model where the limit of the discussed approximations can be traced. Thus,
phenomenological consequences for the mixing parameter obtained in the minimal
and non-minimal approximations can be hardly distinguished at current and
future facilities.

The question of the explicit form of the parametrization matrix $\Omega$ in
standard and non-minimal approximations is still ambiguous. Various forms
satisfying the requirement $|\theta|\ll I$ and $\Omega_{\rm m}\Omega_{\rm
m}^{T}=I$ were considered with significantly different results. Effective
masses of HNL in the standard and non-minimal approximations demonstrate
noticeable differences due to new sizeable terms of effective heavy neutrino
mass matrix. Note that ${\cal O}(\theta M_{D})$ corrections to the splitting
of eigenvalues of $M_{M}$ and $M_{N}$ matrices can impact both the lepton-
number-violating event rates at colliders [47] and leptogenesis in the early
Universe [48] (see also [25]).

The research was carried out within the framework of the scientific program of
the National Center for Physics and Mathematics, project ”Particle Physics and
Cosmology”.

##  Appendix A Symmetric matrices $m_{\nu},M_{N},M_{M}$ and relations for
$M_{D}\theta^{T}$ and $\theta^{\dagger}M_{D}$

The elements of the matrix $\tilde{{\cal M}}$, see (7), have the forms

$\displaystyle\tilde{{\cal M}}_{11}$ $\displaystyle=$ $\displaystyle
W_{11}^{\dagger}M_{D}W_{21}^{*}+W_{21}^{\dagger}(M_{D}^{T}W_{11}^{*}+M_{M}W_{21}^{*})\equiv
m_{\nu}$ (69) $\displaystyle\tilde{{\cal M}}_{22}$ $\displaystyle=$
$\displaystyle
W_{12}^{\dagger}M_{D}W_{22}^{*}+W_{22}^{\dagger}(M_{D}^{T}W_{12}^{*}+M_{M}W_{22}^{*})\equiv
M_{N},$ (70) $\displaystyle\tilde{{\cal M}}_{12}$ $\displaystyle=$
$\displaystyle
W_{11}^{\dagger}M_{D}W_{22}^{*}+W_{21}^{\dagger}(M_{D}^{T}W_{12}^{*}+M_{M}W_{22}^{*})\equiv
0,$ (71) $\displaystyle\tilde{{\cal M}}_{21}$ $\displaystyle=$ $\displaystyle
W_{12}^{\dagger}M_{D}W_{21}^{*}+W_{22}^{\dagger}(M_{D}^{T}W_{11}^{*}+M_{M}W_{21}^{*})\equiv
0.$ (72)

Analysing (69)–(72), one can note that $m_{\nu},M_{N}$ and $M_{M}$ are
symmetric matrices

$m_{\nu}=m_{\nu}^{T},\,\,M_{N}=M_{N}^{T},\,\,M_{M}=M_{M}^{T},$ (73)

where for the last expression the relaton $\tilde{\cal M}_{12}-\tilde{\cal
M}_{21}^{T}=0$ was used.

In the standard approximation, a requirement $\tilde{\cal M}_{12}=0$ [see (7)]
has a form

$\theta M_{M}\simeq
M_{D}-\frac{1}{2}(\theta\theta^{\dagger}M_{D}+M_{D}\theta^{T}\theta^{*})-\theta
M_{D}^{T}\theta^{*}.$ (74)

Multiplying (74) by $\theta^{T}$ from the right side

$\theta M_{M}\theta^{T}\simeq
M_{D}\theta^{T}-\frac{1}{2}(\theta\theta^{\dagger}M_{D}\theta^{T}+M_{D}\theta^{T}\theta^{*}\theta^{T})-\theta
M_{D}^{T}\theta^{*}\theta^{T}$ (75)

and omitting terms ${\cal O}(\theta^{3})$ (as far as $|\theta|\ll I$), one can
find

$M_{D}\theta^{T}\simeq\theta M_{D}^{T},$ (76)

where the relation $\theta M_{M}\theta^{T}=(\theta M_{M}\theta^{T})^{T}$ [see
(73)] was applied. The same relation (76) takes place for the non-minimal
approximation, where instead of (74) one should use Equation (28).

Multiplying (74) by $\theta^{-1}$ from the left side

$M_{M}\simeq\theta^{-1}M_{D}-\frac{1}{2}(\theta^{\dagger}M_{D}+\theta^{-1}M_{D}\theta^{T}\theta^{*})-M_{D}^{T}\theta^{*}$
(77)

and using the relation (76), one can rewrite the requirement $M_{M}=M_{M}^{T}$
in the form

$\theta^{\dagger}M_{D}\simeq M_{D}^{T}\theta^{*}.$ (78)

The relation (78) is also relevant in the non-minimal approximation. The
expression (28) can be written as

$\displaystyle\theta^{\dagger}\theta M_{M}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\theta^{\dagger}\theta
M_{M}\theta^{T}\theta^{*}-\frac{1}{6}\theta^{\dagger}\theta\theta^{\dagger}\theta
M_{M}\simeq$ (79)
$\displaystyle\theta^{\dagger}M_{D}-\frac{1}{2}\theta^{\dagger}\theta\theta^{\dagger}M_{D}-\frac{1}{2}\theta^{\dagger}M_{D}\theta^{T}\theta^{*}-\theta^{\dagger}\theta
M_{D}^{T}\theta^{*},$

so, in the limit ${\cal O}(\theta M_{D})$

$\theta^{\dagger}\theta M_{M}\simeq\theta^{\dagger}M_{D},\qquad
M_{M}\theta^{T}\theta^{*}\simeq M_{D}^{T}\theta^{*},$ (80)

which allows one to express (28) as

$M_{M}\simeq\theta^{-1}M_{D}-\frac{1}{3}\theta^{\dagger}M_{D}-\frac{1}{2}M_{D}^{T}\theta^{*}-\frac{1}{2}\theta^{-1}M_{D}\theta^{T}\theta^{*}.$
(81)

The requirement $M_{M}=M_{M}^{T}$, (73), with the help of (76), leads to

$\theta^{-1}M_{D}+\frac{2}{3}\theta^{\dagger}M_{D}\simeq
M_{D}^{T}(\theta^{-1})^{T}+\frac{2}{3}M_{D}^{T}R^{*}.$ (82)

Multiplying (82) by $\theta$ and $\theta^{T}$ from the left and right sides,
one can obtain

$\theta\theta^{\dagger}M_{D}\theta^{T}\simeq\theta
M_{D}^{T}\theta^{*}\theta^{T},$ (83)

which leads to the relation (78).

##  Appendix B A simple representation for the matrix $\Omega_{\rm nm}$

Suppose that

$\Omega_{\rm nm}=\Omega_{\rm n}+\Omega_{\rm m},$ (84)

where $\Omega_{\rm m}\Omega_{\rm m}^{T}=I$, $\Omega_{\rm n}\Omega_{\rm
n}^{T}\neq I$. Inserting (84) into (43), one can find

$\Omega_{\rm m}=3\hat{m}^{-1}[(\Omega_{\rm n}^{-1})^{\dagger}-\Omega_{\rm
n}^{*}]\hat{M}-(\Omega_{\rm n}^{-1})^{T},$ (85)

which must satisfy the requirement $\Omega_{\rm m}^{-1}=\Omega_{\rm m}^{T}$,
i.e.,

$3\hat{M}^{-1}[\Omega_{\rm
n}^{\dagger}-(\Omega_{n}^{*})^{-1}]\hat{m}-\Omega_{\rm
n}^{T}=3\hat{M}[(\Omega_{\rm n}^{-1})^{*}-(\Omega_{\rm
n}^{*})^{T}]\hat{m}^{-1}-\Omega_{\rm n}^{-1}.$ (86)

Assuming

$A=\Omega_{\rm n}^{T}-\Omega_{\rm n}^{-1},\qquad A\neq 0,$ (87)

Equation (86) takes the form

$3\hat{M}^{-1}A^{*}\hat{m}+3\hat{M}A^{*}\hat{m}^{-1}-A=0.$ (88)

One can note that there is no matrix $A$ satisfying the relation (88). Thus,
the matrix $\Omega_{\rm nm}$ cannot be expressed by the form (84).

However, if one assumes that $\Omega_{\rm nm}$ is a diagonal matrix, then its
form is defined explicitly as

$\Omega_{\rm nm}={\rm diag}(\omega_{1},\omega_{2},\omega_{3}),\;\text{where
}\omega_{i}=\sqrt{1-\frac{1}{3}\frac{m_{i}}{M_{i}}}.$ (89)

##  Appendix C The effective masses of active neutrinos in standard and non-
minimal approximations

In the standard approach using the relations (15), (17), (40) and $\Omega_{\rm
m}\Omega_{\rm m}^{T}=I$, one can find

$\displaystyle m_{\nu}\simeq-M_{D}\theta^{T}\simeq-\theta
M_{N}\theta^{T}\simeq-\theta
U_{N}^{*}\hat{M}U_{N}^{\dagger}\theta^{T}=-\Theta\hat{M}\Theta^{T}\simeq
U_{\rm PMNS}\hat{m}U_{\rm PMNS}^{T}.$ (90)

In the non-minimal approximation, the expression for $m_{\nu}$, (30), with the
help of (31), is

$M_{D}\simeq\left[\theta-\frac{1}{3}(\theta^{\dagger})^{-1}\right]M_{N},$ (91)

and (8) can be presented as

$m_{\nu}\simeq-\left[\theta-\frac{1}{3}(\theta^{\dagger})^{-1}\right]U_{N}^{*}\hat{M}\Theta^{T}.$
(92)

Using (44) and (45), one can find

$\displaystyle m_{\nu}$ $\displaystyle\simeq$
$\displaystyle-i^{2}U_{\nu}\sqrt{\hat{m}}\Omega_{\rm
nm}\sqrt{\hat{M}^{-1}}U_{N}^{T}U_{N}^{*}\hat{M}\sqrt{\hat{M}^{-1}}\Omega_{\rm
nm}^{-1}\sqrt{\hat{m}}U_{\nu}^{T}$ (93) $\displaystyle\simeq$ $\displaystyle
U_{\rm PMNS}\hat{m}U_{\rm PMNS}^{T},$

where, as far as $|\theta|\ll 1$, one can assume $U_{\nu}\simeq U_{\rm PMNS}$,
see (34).

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