# Neutrino mass matrix in neutrino-related processes

M. I. Krivoruchenko<EMAIL_ADDRESS>National Research Centre
”Kurchatov Institute”, Moscow, Russia  F. Šimkovic
<EMAIL_ADDRESS>Faculty of Mathematics, Physics and Informatics,
Comenius University in Bratislava, Bratislava, Slovakia
Institute of Experimental and Applied Physics, Czech Technical University in
Prague, Prague, Czech Republic

###### Abstract

Techniques are developed for constructing amplitudes of neutrino-related
processes in terms of the neutrino mass matrix, with no reference to the
neutrino mixing matrix. The amplitudes of neutrino oscillations in vacuum and
medium, quasi-elastic neutrino scattering, $\beta$ decays and double-$\beta$
decays are considered. The proposed approach makes extensive use of Frobenius
covariants within the framework of Sylvester’s theorem on matrix functions.
The in-medium dispersion laws are found in quadratures for three flavors of
Majorana neutrinos as an application of the developed formalism. The in-medium
dispersion laws for Dirac neutrinos can be determined in the general case by
searching for the roots of a polynomial of degree 6. In the rest frame of
baryonic matter, the minimum energy of both Majorana and Dirac neutrinos is
achieved at a neutrino momentum equal to half the mean-field potential. In
such cases, Dirac neutrinos occupy a hollow Fermi sphere at zero temperature
and low densities. Fitting experimental data in terms of the neutrino mass
matrix can provide better statistical accuracy in determining the neutrino
mass matrix compared to methods using the neutrino mixing matrix at
intermediate stages.

##  I Introduction

The predictions of the Standard Model (SM) are effectively confirmed by a vast
number of experiments. The first discrepancy with SM was discovered in 1998 as
a result of observations of atmospheric neutrino oscillations Fukuda:1988 ,
proving the existence of finite neutrino masses. By adding the right neutrinos
to SM while maintaining the renormalizability of the model, it is possible to
generate Dirac neutrino masses. When the non-renormalizable $d=5$ interaction
of left leptons with the Higgs boson is turned on, masses of Majorana
neutrinos appear Weinberg:1979 . A plausible hierarchy of masses is suggested
by the seesaw mechanism involving both Dirac and Majorana neutrinos
Bilenky:2018 . The mechanism of neutrino mass generation by the spontaneous
chiral symmetry breaking is also discussed Babic:2021 . Only in isolated cases
the theory encounters challenges in adjacent SM sectors, the interpretation of
measurements of the muon’s anomalous magnetic moment Labe:2022  or the decays
of beauty mesons LHCb:2022 .

In the searches for physics beyond SM, neutrinos occupy a unique place. The
mass matrices of Majorana and Dirac neutrinos have 9 and 7 free parameters,
respectively, already in the simplest case of three generations Kobzarev:1982
; Ismael:2021 . This is roughly half of the total number of SM free
parameters. The elements of neutrino mass matrix may contain a great deal of
information on potential SM extensions.

Cross-sections of processes involving neutrinos, probabilities of oscillations
and decay rates are usually expressed in terms of the
Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix and the diagonal neutrino
masses. Significant progress in the experimental determination of the elements
of the mixing matrix and obtaining restrictions on neutrino masses has been
achieved over the past decade due to the analysis of neutrino oscillations,
including data from atmospheric experiments, accelerator data on the modes of
appearance and disappearance of neutrinos, and data from reactor experiments
deSalas:2017kay ; Capozzi:2017ipn . The elements of the mass matrix, in
principle, are accessible to experimental determination in terms of the mixing
matrix and the diagonal neutrino masses, but contain uncertainties associated
with arbitrariness of the absolute mass scale. An indirect approach to the
determination of the mass matrix can lead to loss of information at
intermediate stages of statistical analysis. In this paper, we show that the
processes associated with neutrinos are explicitly described in terms of the
neutrino mass matrix, without using a mixing matrix. The existence of such
representations makes it possible to fit the elements of neutrino mass matrix
directly from experimental data, which should provide an advantage when
conducting statistical analysis and, potentially, distinguish the Majorana
mass matrix from the Dirac one.

Another purpose of this paper is to point out effective techniques of matrix
algebra that significantly simplify the work with neutrinos of three or more
flavors. The paper uses extensively Sylvester’s theorem on the representation
of matrix functions Roger:1991 , Frobenius covariants, which determine
projection operators in terms of the matrix under consideration, and trace
identities Kondratyuk:1992 ; Brown:1994 , which provide explicit solutions to
the Faddeev–LeVerrier algorithm Frame:1949 ; Faddeev:1972 . The described
techniques allow to effectively construct the neutrino dispersion laws. The
cases of $n\leq 4$ flavors of Majorana neutrinos in a matter of arbitrary
composition and $n\leq 4$ flavors of Dirac neutrinos in a baryonic matter are
solved in quadratures.

The outline of the paper is as follows. Neutrino oscillations in a vacuum are
discussed in the next section. Neutrino oscillations in the medium are covered
in Section III. Majorana neutrinos and Dirac neutrinos are considered
separately. The neutrino dispersion laws are derived, and it is demonstrated
that the mass matrix explicitly describes the physical amplitudes of neutrino
propagation in all cases considered. The neutrino dispersion law in the medium
takes an unusual form: the minimum neutrino energy is achieved at a finite
momentum. Other processes covered briefly in Section IV include tritium
$\beta$ decay, electron capture, double-$\beta$ decays, and quasi-elastic
neutrino scattering. In all circumstances, probabilities are explicitly
expressed in terms of the neutrino mass matrix rather than the mixing matrix.

##  II Neutrino oscillations in vacuum

###  II.1 Majorana neutrino

Let us first consider the problem of oscillations of Majorana neutrinos. The
Lagrangian of free Majorana neutrinos is written as

$\mathcal{L}_{M}=\frac{1}{2}\bar{\nu}_{\alpha}i\hat{\nabla}\nu_{\alpha}-\frac{1}{2}\bar{\nu}_{\alpha}\left(M_{\alpha\beta}+i\gamma_{5}N_{\alpha\beta}\right)\nu_{\beta}.$
(II.1)

Here $\nu_{\alpha}$ are the real four-component spinors of Majorana neutrinos,
the gamma matrices are defined in the Majorana representation where
$\gamma^{*}_{\mu}=-\gamma_{\mu}$, $\gamma^{T}_{\mu}=-\gamma^{\mu}$,
$\gamma^{*}_{5}=\gamma^{T}_{5}=-\gamma_{5}$, $\gamma^{\dagger}_{5}=\gamma_{5}$
and $\nu_{\alpha}^{c}=\nu_{\alpha}^{*}=\nu_{\alpha}$,
$\bar{\nu}_{\alpha}^{c}={\nu}_{\alpha}^{T}\gamma_{0}$. The left and right
neutrinos are defined by $\nu_{\alpha L}=\Pi_{-}\nu_{\alpha}$ and $\nu_{\alpha
R}=\Pi_{+}\nu_{\alpha}$, respectively, where $\Pi_{\pm}=(1\pm\gamma_{5})/{2}$
are the projection operators onto states with the definite chirality. The
transformation properties of the spinor bilinear forms determine the mass term
symmetry. We denote by $\Gamma_{A}$ the matrices
$1,i\gamma_{5},\gamma^{\mu},\gamma_{5}\gamma^{\mu},\sigma^{\mu\nu}$, defined
in the same way as in Bjorken:1964 . Taking into account
$\gamma_{0}\Gamma_{A}^{\dagger}\gamma_{0}=\Gamma_{A}$ and the fermionic
character of the fields, for both Dirac and Majorana neutrinos

$\bar{\nu}_{\alpha}\Gamma\nu_{\beta}=-\bar{\nu}_{\beta}^{c}\gamma^{0}(\gamma^{0}\Gamma_{A})^{T}\nu_{\alpha}^{c}=\eta\bar{\nu}_{\beta}^{c}\Gamma_{A}\nu_{\alpha}^{c}.$
(II.2)

The sign $\eta$ is read from the equation (cf. Appendix B in
Krivoruchenko:2011 )

$-\gamma^{0}(\gamma^{0}\left(\begin{array}[]{l}1\\\ i\gamma_{5}\\\
\gamma^{\mu}\\\ \gamma_{5}\gamma^{\mu}\\\
\sigma^{\mu\nu}\end{array}\right))^{T}=\left(\begin{array}[]{l}1\\\
i\gamma_{5}\\\ -\gamma^{\mu}\\\ \gamma_{5}\gamma^{\mu}\\\
-\sigma^{\mu\nu}\end{array}\right).$ (II.3)

The conditions $\bar{\nu}_{\alpha}\nu_{\beta}=\bar{\nu}_{\beta}\nu_{\alpha}$
and
$\bar{\nu}_{\alpha}i\gamma_{5}\nu_{\beta}=\bar{\nu}_{\beta}i\gamma_{5}\nu_{\alpha}$
valid for Majorana fields imply that the mass matrices $M$ and $N$ are
symmetric. The hermiticity of the Lagrangian ensures the hermiticity of
$\mathbb{M}$ and $\mathbb{N}$. Since they are symmetric, they are real. We
define for $n$ flavors a $4n\times 4n$ real matrix

$\hat{\mathbb{M}}=\mathbb{M}+i\gamma_{5}\mathbb{N}$ (II.4)

and $n\times n$ complex symmetric matrices

$\mathbb{M}_{\pm}=\mathbb{M}\pm i\mathbb{N}.$ (II.5)

One can verify that
$\hat{\mathbb{M}}^{\dagger}=\hat{\mathbb{M}}^{T}=\gamma_{0}\hat{\mathbb{M}}\gamma_{0}=\mathbb{M}-i\gamma_{5}\mathbb{N}$
and $\mathbb{M}_{+}^{\dagger}=\mathbb{M}^{*}_{+}=\mathbb{M}_{-}$. We also use
notations $\mathbb{M}_{R}=\mathbb{M}_{+}$, $\mathbb{M}_{L}=\mathbb{M}_{-}$ and
write $\mathbb{M}_{\pm 1}$ for $\mathbb{M}_{\pm}$ and $\Pi_{\pm 1}$ for
$\Pi_{\pm}$. The matrices $\mathbb{M}$ and $\mathbb{N}$ are specified by
$2\times(n(n+1)/2)$ real parameters. Phase rotations of Majorana spinors
$\nu_{\alpha}\to\exp(i\gamma_{5}\phi_{\alpha})\nu_{\alpha}$ keep the kinetic
term in the Lagrangian and the weak neutral current $j_{\mu}^{\mathrm{NC}}$
unchanged but modifies the mass matrix and introduces the phase multipliers
into the weak charged current $j_{\mu}^{\mathrm{CC}}$. The phase multipliers
in $j_{\mu}^{\mathrm{CC}}$ are absorbed further by the charged leptons. The
number of remaining free parameters in the mass matrix equals $n^{2}$.

In the representation of two-component Weyl spinors $\nu_{\alpha L}$, the
Lagrangian (II.1) has the form

$\mathcal{L}_{M}=\frac{1}{2}\bar{\nu}_{\alpha L}i\hat{\nabla}\nu_{\alpha
L}-\frac{1}{2}\bar{\nu}_{\alpha L}M_{\alpha\beta}^{M}\nu_{\beta
L}^{c}+\mathrm{h.c.},$ (II.6)

where $\nu_{L\alpha}^{c}=(\nu_{L\alpha})^{c}$ and
$M^{M}_{\alpha\beta}=M_{\alpha\beta}+iN_{\alpha\beta}$ is a symmetric complex
$n\times n$ matrix $\mathbb{M}_{R}$ defined above. It is useful to identify
phases of the mass matrix elements:
$M^{M}_{\alpha\beta}=|M^{M}_{\alpha\beta}|e^{i\phi^{\prime}_{\alpha\beta}}.$
The chiral rotations
$\nu_{\alpha}\to\exp(-i\gamma_{5}\phi^{\prime}_{\alpha\alpha}/2)\nu_{\alpha}$
enable the diagonal elements of $\mathbb{N}$ to be set to zero and the
diagonal elements of $\mathbb{M}_{R}$ to be considered as real and positive.
For three generations

$\mathbb{M}_{R}=\left\|\begin{array}[]{lll}M^{M}_{ee}&|M^{M}_{e\mu}|e^{i\phi_{e\mu}}&|M^{M}_{e\tau}|e^{i\phi_{e\tau}}\\\
|M^{M}_{e\mu}|e^{i\phi_{e\mu}}&M^{M}_{\mu\mu}&|M^{M}_{\mu\tau}|e^{i\phi_{\mu\tau}}\\\
|M^{M}_{e\tau}|e^{i\phi_{e\tau}}&|M^{M}_{\mu\tau}|e^{i\phi_{\mu\tau}}&M^{M}_{\tau\tau}\end{array}\right\|,$
(II.7)

where
$\phi_{\alpha\beta}=\phi^{\prime}_{\alpha\beta}-(\phi^{\prime}_{\alpha\alpha}+\phi^{\prime}_{\beta\beta})/2$.

The free Lagrangian (II.1) is invariant with respect to the charge
conjugation: $\nu_{\alpha}\rightarrow\nu_{\alpha}^{c}=\nu_{\alpha}^{\dagger}$,
however, due to the special type of the mass matrix, it is non-invariant with
respect to $P$-parity transformations
$\nu_{\alpha}\rightarrow\mathcal{P}\nu_{\alpha}\mathcal{P}^{-1}=i\gamma_{0}\nu_{\alpha}$,
and time reversal
$\nu_{\alpha}\rightarrow\mathcal{T}\nu_{\alpha}\mathcal{T}^{-1}=\gamma_{0}\gamma_{5}\nu_{\alpha}$,
When weak interactions are turned on, only the combined CPT symmetry survives.
$CP$ invariance would call for $\mathbb{N}=0$ and forbid phase rotations with
the $\gamma_{5}$ matrix, which would leave $n(n+1)/2$ free parameters in the
mass matrix.

The Feynman propagator has the form

$\hat{\mathbb{S}}_{F}(x-y)=-i\langle
0|T{\nu}(x)\otimes\bar{\nu}(y)|0\rangle=\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\hat{\mathbb{S}}_{F}(p),$
(II.8)

where

$\displaystyle\hat{\mathbb{S}}_{F}(p)=\frac{1}{\hat{p}-\hat{\mathbb{M}}}$
(II.9)

is determined by the quadratic form of the Lagrangian. The matrix
$\hat{p}-\hat{\mathbb{M}}$ has the dimension $4n$. The energy eigenvalues are
grouped by pairs $\pm\sqrt{m_{i}^{2}+\mathbf{p}^{2}}$ for different masses
$m_{i}$ ($i=1,..,n$). The additional doubling is due to the spin degeneracy.
Using the identity $A^{-1}=B(AB)^{-1}=(BA)^{-1}B$ for
$A=\hat{p}-\hat{\mathbb{M}}$ and $B=\hat{p}+\hat{\mathbb{M}}^{\dagger}$ and
$\hat{p}\hat{\mathbb{M}}=\hat{\mathbb{M}}^{\dagger}\hat{p}$, we obtain

$\displaystyle\hat{\mathbb{S}}_{F}(p)=(\hat{p}+\hat{\mathbb{M}}^{\dagger})\frac{1}{p^{2}-\hat{\mathbb{M}}\hat{\mathbb{M}}^{\dagger}}=\frac{1}{p^{2}-\hat{\mathbb{M}}^{\dagger}\hat{\mathbb{M}}}(\hat{p}+\hat{\mathbb{M}}^{\dagger}).$
(II.10)

The chirality-state projections can be simplified with the use of the
relationships
$f(\hat{\mathbb{M}})\Pi_{\varsigma}=f(\mathbb{M}_{\varsigma})\Pi_{\varsigma}$
and
$f(\hat{\mathbb{M}}^{\dagger})\Pi_{\varsigma}=f(\mathbb{M}_{-\varsigma})\Pi_{\varsigma}$,
which hold true for any function $f(x)$ ($\varsigma=\pm 1$). The projection of
$\hat{\mathbb{S}}_{F}(p)$ involved in the creation of a neutrino with
chirality $-\varsigma$ has the form:

$\displaystyle\hat{\mathbb{S}}_{F}(p)\Pi_{\varsigma}=(\hat{p}+\mathbb{M}_{-\varsigma})\frac{1}{p^{2}-\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}}\Pi_{\varsigma}.$
(II.11)

The advantage of this representation is that in the denominator there is a
$n\times n$ matrix multiplied by a unit matrix in the space of bispinors,
while the original expression contains a $4n\times 4n$ matrix with the mixed
bispinor and flavor indices. The mass spectrum is determined by poles of the
propagator, in this instance by zeros of the characteristic polynomial

$\displaystyle
p(\lambda)=\det||\lambda-\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}||=0.$
(II.12)

For three flavors, $p(\lambda)$ is a polynomial of the third degree that has,
respectively, either three real roots, or one real and two complex roots. The
mass spectrum does not depend on $\varsigma$, because
$\det||\lambda-\mathbb{M}_{L}\mathbb{M}_{R}||=\det||\lambda-\mathbb{M}_{R}\mathbb{M}_{L}||$.
Given that $(\mathbb{M}_{L}\mathbb{M}_{R})^{*}=\mathbb{M}_{R}\mathbb{M}_{L}$,
the complex conjugation of $p(\lambda)$ gives the characteristic equation
$p(\lambda^{*})=0$. Thus, either all three roots are real, or one root is
real, and two roots have imaginary parts of opposite sign. The product
$\mathbb{M}_{L}\mathbb{M}_{R}$, however, is a Hermitian matrix. Hence all
eigenvalues of $\mathbb{M}_{L}\mathbb{M}_{R}$ are real. Let
$\lambda_{i}=m_{i}^{2}$ be one of them. There exists a normalized vector
$|i\varsigma\rangle$ such that

$\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}|i\varsigma\rangle=\lambda_{i}|i\varsigma\rangle.$
(II.13)

Multiplying on the left by $\langle i\varsigma|$ yields $\lambda_{i}=\langle
i\varsigma|(\mathbb{M}_{-\varsigma})^{\dagger}\mathbb{M}_{-\varsigma}|i\varsigma\rangle>0$.
Furthermore, if $|i\varsigma\rangle$ is a normalized eigenvector of
$\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}$, then

$|i-\varsigma\rangle=\frac{1}{m_{i}}\mathbb{M}_{-\varsigma}|i\varsigma\rangle$
(II.14)

is the normalized eigenvector of
$\mathbb{M}_{-\varsigma}\mathbb{M}_{\varsigma}$ with the same eigenvalue
$\lambda_{i}$. In the basis of flavor states $|\alpha\rangle$, the coordinates
of these eigenvectors appear to be complex conjugate:
$\langle\alpha|i\varsigma\rangle^{*}=\langle\alpha|i-\varsigma\rangle$. One
can check the validity of this relation by writing Eq. (II.13) in the flavor
basis. After performing a complex conjugation, an equation follows for the
eigenvectors of $\mathbb{M}_{-\varsigma}\mathbb{M}_{\varsigma}$, which means
the eigenvectors of $\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}$ and
$\mathbb{M}_{-\varsigma}\mathbb{M}_{\varsigma}$ differ by phase factors only.
The phase factors of $|i\varsigma\rangle$, however, can always be chosen so
that the relationship

$\langle\alpha|i\varsigma\rangle^{*}=\langle\alpha|i-\varsigma\rangle$ (II.15)

is fulfilled. The mixing of Majorana neutrinos is described thereby in terms
of a single mixing matrix. Equation (II.14) is equivalent to the condition
$\langle i-|\mathbb{M}_{L}|j+\rangle=m_{i}\delta_{ij}$, which shows that for
$n$ flavors the mass matrix $\mathbb{M}_{L}$ is diagonalized by two unitary
matrices:

$\mathbb{V}^{\dagger}\mathbb{M}_{L}\mathbb{U}=\mathrm{diag}(m_{1},\ldots,m_{n}),$
(II.16)

with $V_{\alpha i}=\langle\alpha|i-\rangle$ and $U_{\alpha
i}=\langle\alpha|i+\rangle$. Moreover, by virtue of Eq. (II.15), $\mathbb{V}$
is expressed in terms of $\mathbb{U}$: $\mathbb{V}=\mathbb{U}^{*}$, so that

$\mathbb{U}^{T}\mathbb{M}_{L}\mathbb{U}=\mathbb{V}^{T}\mathbb{M}_{R}\mathbb{V}=\mathrm{diag}(m_{1},\ldots,m_{n}).$
(II.17)

By means of the $4n\times 4n$ matrix
$\hat{\mathbb{U}}=\mathbb{U}\Pi_{-}+\mathbb{V}\Pi_{+}$, the mass term of the
Lagrangian transforms into a diagonal matrix,

$\hat{\mathbb{M}}\rightarrow\gamma^{0}\hat{\mathbb{U}}^{\dagger}\gamma^{0}\hat{\mathbb{M}}\hat{\mathbb{U}}=\mathbb{V}^{\dagger}\mathbb{M}_{L}\mathbb{U}\Pi_{-}+\mathbb{U}^{\dagger}\mathbb{M}_{R}\mathbb{V}\Pi_{+}=\hat{1}\otimes\mathrm{diag}\left(m_{1},...,m_{n}\right),$
(II.18)

where $\hat{1}$ is a unit matrix in the space of bispinors. Since
$\hat{\mathbb{U}}$ is real, the neutrality of bispinors in the flavor basis is
a necessary and sufficient condition for the neutrality of bispinors
diagonalizing the mass matrix. When the PMNS matrix $\mathbb{U}$ is employed,
Eq (II.17) yields a mass matrix $\mathbb{M}_{R}$ with non-zero phases on the
diagonal. These phases are further eliminated by chiral rotation, as indicated
below Eq. (II.6), after which the mass matrix takes on the form (II.7).

Using trace identities Kondratyuk:1992 , the characteristic polynomial (II.12)
for three neutrino flavors can be written as follows

$\displaystyle
p(\lambda)=(\lambda-\lambda_{0})^{3}-\frac{1}{2}(\lambda-\lambda_{0})\mathrm{Tr}[\mathbb{C}^{2}]-\frac{1}{3}\mathrm{Tr}[\mathbb{C}^{3}],$
(II.19)

where

$\displaystyle\mathbb{C}=\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}-\frac{1}{3}\mathrm{Tr}[\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}]$
(II.20)

is a traceless Hermitian matrix, and

$\displaystyle\lambda_{0}=\frac{1}{3}\mathrm{Tr}[\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}]$
(II.21)

is a real positive number that represents the average of neutrino masses
squared. The coefficients of the characteristic polynomial are traces of the
powers of $\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}$.

The matrix $\mathbb{C}$ can be decomposed over the Gell–Mann–Okubo matrices:
$\mathbb{C}=\omega_{a}\lambda^{a}$, where $\omega_{a}$ are real numbers
($a=1,\ldots,8$). Suppose we reduce $\mathbb{C}$ to a diagonal form:
$\mathbb{C}=O\Lambda O^{\dagger}$, where $O$ is unitary. The diagonal
traceless matrix $\Lambda$ has the form: $\Lambda=a\lambda^{3}+b\lambda^{8}$.
The eigenvalues of $\mathbb{C}$ are $a+b/\sqrt{3}$, $-a+b/\sqrt{3}$, and
$-2b/\sqrt{3}$. For quantities $a$ and $b$, solving the cubic equation by the
Viet method, we find

$\displaystyle a$ $\displaystyle=$
$\displaystyle~{}~{}\sqrt{\frac{\mathrm{Tr}[\mathbb{C}^{2}]}{2}}\sin\Theta,$
(II.22) $\displaystyle b$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{\mathrm{Tr}[\mathbb{C}^{2}]}{2}}\cos\Theta,$
(II.23)

where

$\displaystyle\Theta=\frac{1}{3}\arccos\left(\frac{\sqrt{6}\mathrm{Tr}[\mathbb{C}^{3}]}{(\mathrm{Tr}[\mathbb{C}^{2}])^{3/2}}\right).$
(II.24)

The masses of diagonal neutrinos are finally expressed in terms of
$\mathbb{M}^{\dagger}\mathbb{M}$:

$\displaystyle m_{1}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{0}+a+\frac{b}{\sqrt{3}},$ $\displaystyle m_{2}^{2}$
$\displaystyle=$ $\displaystyle\lambda_{0}-a+\frac{b}{\sqrt{3}},$
$\displaystyle m_{3}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{0}-\frac{2b}{\sqrt{3}}.$ (II.25)

The condition of $m_{i}^{2}$ to be real also follows from the inequality
$(\mathrm{Tr}[\mathbb{C}^{2}])^{3}\geq 6(\mathrm{Tr}[\mathbb{C}^{3}])^{2}$,
which holds for all $3\times 3$ traceless Hermitian matrices. The eigenvalues
of $\mathbb{M}_{R}\mathbb{M}_{L}$ coincide with the eigenvalues of
$\mathbb{M}_{L}\mathbb{M}_{R}$. In general, there are three mass eigenstates
for three Majorana neutrinos. The ordering of $m_{i}^{2}$ depends on the angle
(II.24).

Using Sylvester’s matrix theorem Roger:1991 , the propagator (II.10) can be
written in the form:

$\hat{\mathbb{S}}_{F}(p)=\sum_{i\varsigma}\frac{1}{p^{2}-m_{i}^{2}}\left(\hat{p}+\mathbb{M}_{-\varsigma}\right)|i\varsigma\rangle\langle
i\varsigma|\Pi_{\varsigma}.$ (II.26)

The neutrino states are projected onto the eigenstates of
${\mathbb{M}}_{\varsigma}{\mathbb{M}}_{-\varsigma}$ with the use of the
Frobenius covariants Roger:1991 :

$\displaystyle|i\varsigma\rangle\langle
i\varsigma|\equiv\mathbb{F}_{i\varsigma}=\prod_{j\neq
i}\frac{\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}-\lambda_{j}}{\lambda_{i}-\lambda_{j}}.$
(II.27)

The mass eigenstates with $\varsigma=\pm 1$ form complete bases in flavor
space:

$\sum_{i}|i\varsigma\rangle\langle i\varsigma|=1.$ (II.28)

To illustrate the statements made, consider

Example 1. Let the mass matrix be given by the equation

$\hat{\mathbb{M}}=\lambda^{3}+i\gamma_{5}\lambda^{6},$ (II.29)

where $\lambda^{a}$ are the Gell-Mann matrices. The components of
$\hat{\mathbb{M}}$ are measured in mass units. The standard representation for
$\lambda^{a}$ is assumed to be identical with the flavor basis, so that

$\mathbb{M}_{L}=\mathbb{M}_{R}^{\ast}=\lambda^{3}-i\lambda^{6}=\left\|\begin{array}[c]{lll}1&0&0\\\
0&-1&-i\\\ 0&-i&0\end{array}\right\|.$ (II.30)

The eigenvalues and eigenvectors can readily be found using the MAPLE symbolic
computation software package MAPLE :

$(m_{Li})^{T}=(m_{Ri}^{*})^{T}=\left\|\begin{array}[]{c}1\\\ -1/\eta\\\
-\eta\end{array}\right\|,\ \ \ \ \tilde{W}_{\alpha i}=\
\left\|\begin{array}[]{ccc}1&0&0\\\ 0&-i/\eta&-i\eta\\\
0&1&1\end{array}\right\|,$ (II.31)

where $\eta=$ $e^{i\pi/3}$, the index $\alpha$ numbers the rows, the index $i$
numbers the columns. After normalizing the eigenvectors of $\mathbb{M}_{L}$:

$\tilde{W}_{\alpha i}\rightarrow W_{\alpha i}=(\sum_{\beta}\tilde{W}_{\beta
i}^{2})^{-1/2}\tilde{W}_{\alpha i},$ (II.32)

the matrix $\mathbb{W}$ becomes a complex orthogonal matrix obeying

$\sum_{\alpha}{W}_{\alpha i}{W}_{\alpha j}=\delta_{ij}$

or, in matrix notation, $\mathbb{W}^{T}\mathbb{W}=1$. The mass matrix is
converted to the diagonal form in two steps. On the vector space of dimension
$4n$, we define first

$\hat{\mathbb{W}}=\mathbb{W}\Pi_{-}+\mathbb{W}^{\ast}\Pi_{+}.$ (II.33)

The action of $\hat{\mathbb{W}}$ on the mass matrix gives

$\hat{\mathbb{M}}\rightarrow\gamma^{0}\hat{\mathbb{W}}^{T}\gamma^{0}\hat{\mathbb{M}}\hat{\mathbb{W}}={\mathbb{W}}^{T}\mathbb{M}_{L}{\mathbb{W}}\Pi_{-}+{\mathbb{W}}^{{\dagger}}\mathbb{M}_{R}{\mathbb{W}^{*}}\Pi_{+}=\mathrm{diag}\left(e^{i\gamma_{5}\vartheta_{1}}|m_{L1}|,...,e^{i\gamma_{5}\vartheta_{n}}|m_{Ln}|\right),$
(II.34)

where $e^{-2i\vartheta_{i}}=m_{Li}/m_{Ri}$. At the second stage, the chiral
rotations $\nu_{i}\rightarrow\exp(-i\gamma_{5}\vartheta_{i}/2)\nu_{i}$
eliminate the phase factors. $\hat{{\mathbb{W}}}$ is a real matrix, therefore,
it converts Majorana fields to Majorana fields. The drawback of
${\hat{\mathbb{W}}}$ is its non-unitarity.

The matrices included in the denominator of the neutrino propagator can be
found to be

$\mathbb{M}_{R}\mathbb{M}_{L}=\left(\mathbb{M}_{L}\mathbb{M}_{R}\right)^{\ast}=\left\|\begin{array}[c]{ccc}1&0&0\\\
0&2&i\\\ 0&-i&1\end{array}\right\|.$ (II.35)

The corresponding eigenvalues and eigenvectors are equal

$(m_{i}^{2})^{T}=\left\|\begin{array}[]{c}1\\\ 2-g\\\ 1+g\end{array}\right\|,\
\ \ \ \tilde{U}_{\alpha i}=\tilde{V}_{\alpha
i}^{*}=\left\|\begin{array}[]{ccc}1&0&0\\\ 0&-i/g&ig\\\
0&1&1\end{array}\right\|,$ (II.36)

where $g=(1+\sqrt{5})/2$. The normalization gives

$\langle\alpha|i+\rangle=\langle\alpha|i-\rangle^{\ast}=(\sum_{\beta}|\tilde{U}_{\beta
i}|^{2})^{-1/2}\tilde{U}_{\alpha i}.$

It can be verified that Eq. (II.14) applied to $\langle\alpha|i+\rangle$
yields vectors proportional to $\langle\alpha|i-\rangle$. Calculation of the
parameters entering Eqs. (II.21)–(II.24) gives $\lambda_{0}=4/3$,
$a=\frac{2}{\sqrt{3}}\cos(\frac{1}{3}\arccos(\frac{11}{16}))$,
$b=-\frac{2}{\sqrt{3}}\sin(\frac{1}{3}\arccos(\frac{11}{16}))$. Using
relations (II.25) leads to values $m_{i}^{2}$ that coincide with those given
above.

Table 1: Four pairs of the lepton currents at points $x$ and $y$, leading to neutrino oscillations of various types, are presented in the first two columns; the next column shows the time ordering, the last column shows the processes described by the lepton currents  $j_{\mu}^{\textrm{CC}}(x)$ | $j_{\mu}^{\textrm{CC}}(y)$ | Time ordering | Process
---|---|---|---
$\bar{\ell}_{\beta}^{-}\gamma_{\mu}(1-\gamma_{5})\nu_{\beta}$ | $\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\ell_{\alpha}^{-}$ |  | $x^{0}-y^{0}>0$
---
$x^{0}-y^{0}<0$
| $\nu_{\beta L}\leftarrow\nu_{\alpha L}$
---
$\nu_{\beta R}\rightarrow\nu_{\alpha R}$
$\bar{\ell}_{\beta}^{-c}\gamma_{\mu}(1+\gamma_{5}){\nu}_{\beta}$ | $\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\ell_{\alpha}^{-}$ |  | $x^{0}-y^{0}>0$
---
$x^{0}-y^{0}<0$
| $\nu_{\beta R}\leftarrow\nu_{\alpha L}$
---
$\nu_{\beta L}\rightarrow\nu_{\alpha R}$
$\bar{\ell}_{\beta}^{-}\gamma_{\mu}(1-\gamma_{5})\nu_{\beta}$ | $\bar{\nu}_{\alpha}\gamma_{\mu}(1+\gamma_{5})\ell_{\alpha}^{-c}$ |  | $x^{0}-y^{0}>0$
---
$x^{0}-y^{0}<0$
| $\nu_{\beta L}\leftarrow\nu_{\alpha R}$
---
$\nu_{\beta R}\rightarrow\nu_{\alpha L}$
$\bar{\ell}_{\beta}^{-c}\gamma_{\mu}(1+\gamma_{5}){\nu}_{\beta}$ | $\bar{\nu}_{\alpha}\gamma_{\mu}(1+\gamma_{5})\ell_{\alpha}^{-c}$ |  | $x^{0}-y^{0}>0$
---
$x^{0}-y^{0}<0$
| $\nu_{\beta R}\leftarrow\nu_{\alpha R}$
---
$\nu_{\beta L}\rightarrow\nu_{\alpha L}$

Integrating the time-like component of the four-momentum in Eq. (II.8), we
find neutrino propagator in the coordinate representation

$\displaystyle i\hat{\mathbb{S}}_{F}(x-y)$ $\displaystyle=$
$\displaystyle\theta(x^{0}-y^{0})\sum_{i\varsigma}\int\frac{d\mathbf{p}}{(2\pi)^{3}2E_{i}(\mathbf{p})}e^{-iE_{i}(\mathbf{p})(x^{0}-y^{0})+i\mathbf{p}(\mathbf{x}-\mathbf{y})}\left(\hat{p}_{i}+\mathbb{M}_{-\varsigma}\right)\mathbb{F}_{i\varsigma}\Pi_{\varsigma}$
(II.37) $\displaystyle-$
$\displaystyle\theta(y^{0}-x^{0})\sum_{i\varsigma}\int\frac{d\mathbf{p}}{(2\pi)^{3}2E_{i}(\mathbf{p})}e^{+iE_{i}(\mathbf{p})(x^{0}-y^{0})-i\mathbf{p}(\mathbf{x}-\mathbf{y})}\left(\hat{p}_{i}-\mathbb{M}_{-\varsigma}\right)\mathbb{F}_{i\varsigma}\Pi_{\varsigma},$

where
$\hat{p}_{i}=E_{i}(\mathbf{p})\gamma_{0}-\mbox{\boldmath$\gamma$}\mathbf{p}$
and $E_{i}(\mathbf{p})=\sqrt{m_{i}^{2}+\mathbf{p}^{2}}$.

Massive neutrinos change chirality during oscillations. The right
antineutrinos take part in weak interaction. Vertices with right neutrinos
should be taken into account because Majorana neutrinos are neutral. The
matrix elements shown in the first row of Table 1 control the emission and
absorption of neutrinos. The charged leptons are represented by
${\ell}_{\alpha}^{-}=(e^{-},\mu^{-},\tau^{-})$. The next rows of Table 1
contain three more pairs of matrix elements produced by permuting fermionic
fields. The oscillations of the left neutrino created at point $y$ into the
left neutrino absorbed at point $x$ are described by the matrix elements in
the first row of Table 1. The oscillations of the right neutrino generated at
point $x$ into the right neutrino absorbed at point $y$ occur with the reverse
ordering of time $x^{0}-y^{0}<0$.

It is not difficult to identify the component of the neutrino propagator that
triggers each of the processes shown in the table because projection operators
$\Pi_{\pm}$ enter the vertices. The component of
$\hat{\mathbb{S}}_{F}(x-y)\Pi_{+}$ proportionate to $\hat{p}_{i}\Pi_{+}$ is
suitable in the case of $\nu_{\beta L}\leftarrow\nu_{\alpha L}$. When there
are oscillations $\nu_{\beta R}\rightarrow\nu_{\alpha L}$, the term is
proportional to $\mathbb{M}_{L}\Pi_{+}$ is relevant, and so on. The vertices
listed in the table only partially act independently. For instance, the
transformation of the time inversion relates the process $\nu_{\beta
L}\leftarrow\nu_{\alpha L}$ to the process $\nu_{\beta
R}\rightarrow\nu_{\alpha R}$ in the second row: $\mathcal{T}\hat{A}(\nu_{\beta
L}\leftarrow\nu_{\alpha L})\mathcal{T}^{-1}=\hat{A}(\nu_{\beta
R}\rightarrow\nu_{\mathrm{R}\alpha})$. In relation to processes $B^{c}\to
A^{c}$, all processes $A\to B$ share this feature.

The component of the amplitudes,
$\Pi_{\upsilon}i\hat{\mathbb{S}}_{F}(x-y)\Pi_{\varsigma}$
($\upsilon,\varsigma=\pm 1$), that describes the neutrino propagation, is
represented by a $4\times 4$ matrix in the space of bispinors of the charged
leptons. Neglecting the spread in the neutrino momenta, the neutrino component
can be represented as follows

$\displaystyle\hat{A}(\nu_{\beta L}\leftarrow\nu_{\alpha
L})=-\hat{A}(\nu_{\beta R}\rightarrow\nu_{\alpha R})$ $\displaystyle\propto$
$\displaystyle-\sum_{i}e^{-iE_{i}(\mathbf{p})\tau}\langle\beta|i+\rangle\langle
i+|\alpha\rangle\frac{\hat{p}_{i}}{E_{i}(\mathbf{p})}\Pi_{+}$ $\displaystyle=$
$\displaystyle-\sum_{i}e^{-iE_{i}(\mathbf{p})\tau}\langle\beta|\mathbb{F}_{i+}|\alpha\rangle\frac{\hat{p}_{i}}{E_{i}(\mathbf{p})}\Pi_{+},$

where $\tau=|x^{0}-y^{0}|$ and $|\alpha\rangle$ are the neutrino flavor
states. The matrix element

$\langle\alpha|i+\rangle=U_{\alpha i}$ (II.39)

can be recognized as the PMNS matrix element. The ratio
${\hat{p}_{i}}/{E_{i}(\mathbf{p})}$ is sensitive to the neutrino masses for
$|\mathbf{p}|\lesssim m_{i}$, the exponential factor
$\exp(-iE_{i}(\mathbf{p})\tau)$ is sensitive to the neutrino masses in a broad
range of $\mathbf{p}$ due to the macroscopic time factor $\tau$, the Frobenius
covariant (II.27) entering the second line depends on the neutrino masses from
the start, while the other components of the amplitude have purely kinematic
origin and are the same for all types of neutrinos.

In recent decades, the theory of neutrino oscillations in a vacuum and in a
medium has been studied in many details Wolfenstein:1978 ; Mikheev:1985 ;
Kobzarev:1982 ; deSalas:2017kay ; Bilenky:2018 ; Kovalenko:2023 ; Gribov:1969
; Naumov:2020 . In the limit of large values of neutrino momenta, the
kinematic factors of reactions involving neutrinos are factorized, and an
expression for the probability of oscillations can be found:

$P(\nu_{\beta L}\leftarrow\nu_{\alpha
L})=\sum_{ij}e^{-i(E_{i}(\mathbf{p})-E_{j}(\mathbf{p}))t}U_{i\beta}^{\ast}U_{i\alpha}U_{j\beta}U_{j\alpha}^{\ast},$
(II.40)

where $t$ is the oscillation time. It is assumed that the momentum
$\mathbf{p}$ of neutrinos is fixed. This equation becomes

$P(\nu_{\beta L}\leftarrow\nu_{\alpha
L})=\sum_{ij}e^{-i(E_{i}(\mathbf{p})-E_{j}(\mathbf{p}))t}\langle\beta|\mathbb{F}_{i+}|\alpha\rangle\langle\alpha|\mathbb{F}_{j+}|\beta\rangle$
(II.41)

once formulae (II.27) and (II.39) are taken into account. This equation is
also a consequence of Eq. (II.1).

The probability has a similar form in the spatial picture with fixed energy
$E$. The exponential factor changes to

$\exp(-i(E_{i}(\mathbf{p})-E_{j}(\mathbf{p}))t)\approx\exp(-i\frac{m_{i}^{2}-m_{j}^{2}}{2|\mathbf{p}|}t)\rightarrow\exp(-i\frac{m_{i}^{2}-m_{j}^{2}}{2E}L),$
(II.42)

where $L$ is the distance between the source and the detector. A description
of the complete kinematic structure of the oscillation amplitudes in the
spatial representation can be found in Kovalenko:2023 .

In this paper, the analysis of oscillations is based on the plane-wave
approximation Gribov:1969 . Despite the successful application of the plane-
wave approximation, modeling processes using wave packets of the involved
particles encompasses some challenges. Wave packets are employed in quantum
field theory to accurately define asymptotic states, while scattering theory
includes wave packets from the outset. When the problem is reformulated using
the wave packet techniques, the standard formulas for the probability of
neutrino oscillations are reproduced; however, neutrino oscillations are
expected to be suppressed at distances greater than the coherence and
dispersion lengths typical for neutrinos arriving from galactic sources
Naumov:2020 .

Chirality-changing transitions of neutrinos are of interest in the context of
double-$\beta$ decay. The process $\nu_{\beta R}\leftarrow\nu_{\alpha L}$ is
described by the currents shown in the second raw of Table 1. The same matrix
elements are responsible for the process $\nu_{\beta R}\rightarrow\nu_{\alpha
L}$. The component of the amplitude associated with the neutrino propagation
equals $\Pi_{+}i\hat{\mathbb{S}}_{F}(x-y)\Pi_{+}$, so we obtain

$\displaystyle\hat{A}(\nu_{\beta R}\leftarrow\nu_{\alpha
L})=-\hat{A}(\nu_{\beta R}\rightarrow\nu_{\alpha L})$ $\displaystyle\propto$
$\displaystyle-\sum_{i}e^{-iE_{i}(\mathbf{p})\tau}\ m_{i}\langle
i+|\beta\rangle\langle i+|\alpha\rangle\frac{1}{E_{i}(\mathbf{p})}\Pi_{+}$
$\displaystyle=$
$\displaystyle-\sum_{i}e^{-iE_{i}(\mathbf{p})\tau}\langle\beta|\mathbb{M}_{L}\mathbb{F}_{i+}|\alpha\rangle\frac{1}{E_{i}(\mathbf{p})}\Pi_{+}.$

Initially, under the sum sign, the expression
$\langle\beta|\mathbb{M}_{L}|i+\rangle\langle i+|\alpha\rangle$ occurs. In
order to obtain the first line, we use the identities
$\langle\beta|\mathbb{M}_{L}|i+\rangle$ = $m_{i}\langle\beta|i-\rangle$ =
$m_{i}\langle\beta|i+\rangle^{*}$ = $m_{i}\langle i+|\beta\rangle$, which also
allow calculating the mass matrix $\langle\beta|\mathbb{M}_{L}|\alpha\rangle$
in terms of the neutrino masses and the mixing matrix:

$\langle\beta|\mathbb{M}_{L}|\alpha\rangle=\sum_{i}m_{i}\langle
i+|\beta\rangle\langle i+|\alpha\rangle.$ (II.44)

This equation is equivalent to Eq. (II.17). The mass matrix of the right
chirality is determined by
$\langle\beta|\mathbb{M}_{R}|\alpha\rangle=\langle\alpha|\mathbb{M}_{L}|\beta\rangle^{\ast}$.
We remark out that, as can be seen from Example 1, the eigenvalues of the
matrix $\mathbb{M}_{L}$ do not match the masses $m_{i}$ obtained by
diagonalizing the matrix $\mathbb{M}_{R}\mathbb{M}_{L}$. Section IV gives
further details of the double-$\beta$ decay process.

Thus, the amplitude of processes associated with the emission and absorption
of Majorana neutrinos is expressed explicitly in terms of the mass matrix
using Eqs. (II.20)–(II.25) and (II.27).

###  II.2 Dirac neutrino

The results can be generalized to the case of Dirac neutrinos. The Lagrangian
for Dirac neutrinos has the form

$\mathcal{L}_{D}=\bar{\nu}_{\alpha}i\hat{\nabla}\nu_{\alpha}-\bar{\nu}_{\alpha}\left(M_{\alpha\beta}+i\gamma_{5}N_{\alpha\beta}\right)\nu_{\beta},$
(II.45)

where $\nu_{\alpha}$ are the four-component complex spinors of Dirac
neutrinos. Hermiticity of the Lagrangian implies that the matrices
$\mathbb{M}$ and $\mathbb{N}$ are Hermitian, whereas
$\mathbb{M}_{\pm}=\mathbb{M}\pm i\mathbb{N}$ are arbitrary complex matrices
satisfying $\mathbb{M}_{+}^{\dagger}=\mathbb{M}_{-}$. The mass matrix is
defined by $\hat{\mathbb{M}}=\mathbb{M}+i\gamma_{5}\mathbb{N}$. The neutrino
propagator has the form of Eq. (II.10). Because $\mathbb{M}_{R}\mathbb{M}_{L}$
is Hermitian, Eqs. (II.8)–(II.28) apply to Dirac neutrinos, except for (II.15)
and (II.17).

In the Weyl representation of two-component spinors $\nu_{\alpha L}$ and
$\nu_{\alpha R}$, the Lagrangian (II.45) takes the form

$\mathcal{L}_{M}=\bar{\nu}_{\alpha L}i\hat{\nabla}\bar{\nu}_{\alpha
L}-\bar{\nu}_{\alpha L}M_{\alpha\beta}^{{D}}\nu_{\beta R}+\mathrm{h.c.},$
(II.46)

where $M^{{D}}_{\alpha\beta}=M_{\alpha\beta}+iN_{\alpha\beta}$ is a complex
$n\times n$ matrix $\mathbb{M}_{R}$.

The matrix $\mathbb{M}_{R}\mathbb{M}_{L}$ for Majorana and Dirac neutrinos is
controlled by $n^{2}$ independent real parameters. Majorana neutrinos have
$n(n+1)$ and Dirac neutrinos have $2n^{2}$ parameters in the matrix
$\mathbb{M}_{L}$ (or $\mathbb{M}_{R}$). The diagonalization of
$\mathbb{M}_{R}\mathbb{M}_{L}$ does not automatically imply that in the
derived basis $\mathbb{M}_{L}$ (or $\mathbb{M}_{R}$) is diagonal. The only
result that follows from arguments used in the previous subsection is Eq.
(II.18), where in the case of Dirac neutrinos $\mathbb{U}$ and $\mathbb{V}$
are two distinct unitary matrices.  The unitary matrices $\mathbb{V}$ and
$\mathbb{U}$ in Eq. (II.16) can be expressed as
$\mathbb{V}=\mathbb{P}_{1}\mathbb{V}^{\prime}\mathbb{P}_{2}$ and
$\mathbb{U}=\mathbb{P}_{3}\mathbb{U}^{\prime}\mathbb{P}_{4}$, respectively,
where $\mathbb{P}_{i}=\mathrm{diag}(e^{i\chi_{i1}},\ldots,e^{i\chi_{in}})$.
Without reducing generality, $\chi_{21}=\chi_{41}=0$ can be used, resulting in
the unitary matrices $\mathbb{V}^{\prime}$ and $\mathbb{U}^{\prime}$ being
defined by $n^{2}-2n+1$ independent parameters. Since the right neutrinos do
not participate in interactions, it is possible to proceed to their linear
combination by a unitary transformation. Rotating the right neutrinos with
$\mathbb{P}_{3}^{\dagger}\mathbb{V}$ removes $n^{2}$ parameters. By phase
rotations $\nu_{L}\to\mathbb{P}_{4}^{\dagger}\nu_{L}$, the $n-1$ additional
parameters can be removed. The number of independent parameters of the mass
matrix becomes $n^{2}-n+1$. This result is the number of diagonal masses
$m_{i}$ plus the number of parameters of $\mathbb{U}^{\prime}$. For $n=3$ the
mass matrix is determined by $9-3+1=7$ parameters.

To illustrate the similarities and differences between the Dirac and Majorana
mass matrices, consider

Example 2. Let the mass matrix be given by the equation

$\hat{\mathbb{M}}=\lambda^{3}+i\gamma_{5}\lambda^{7}.$ (II.47)

The components of $\hat{\mathbb{M}}$ are measured in mass units. The matrix is
not symmetric, but it satisfies the condition of Dirac self-conjugacy
$\gamma^{0}\hat{\mathbb{M}}^{{\dagger}}\gamma^{0}=\hat{\mathbb{M}}$. The left
and righ chiral projections of $\hat{\mathbb{M}}$ read

$\mathbb{M}_{L}=\lambda^{3}-i\lambda^{7}=\left\|\begin{array}[]{rrr}1&0&0\\\
0&-1&1\\\ 0&-1&0\end{array}\right\|,\ \ \ \
\mathbb{M}_{R}=\lambda^{3}+i\lambda^{7}=\left\|\begin{array}[]{rrr}1&0&0\\\
0&-1&-1\\\ 0&1&0\end{array}\right\|.$ (II.48)

Unlike Majorana neutrinos, $\mathbb{M}_{L}$ and $\mathbb{M}_{R}$ are not
complex conjugate. The eigenvalues and unnormalized eigenvectors can be found
to be:

$(m_{Li})^{T}=(m_{Ri})^{T}=\left\|\begin{array}[]{c}1\\\ -1/\eta\\\
-\eta\end{array}\right\|,\ \ \ \ \tilde{W}_{\alpha
i}^{L}=\left\|\begin{array}[]{ccc}1&0&0\\\ 0&1/\eta&\eta\\\
0&1&1\end{array}\right\|,\ \ \ \ \tilde{W}_{\alpha
i}^{R}=\left\|\begin{array}[]{ccc}1&0&0\\\ 0&-1/\eta&-\eta\\\
0&1&1\end{array}\right\|,$

where $\eta=$ $e^{i\pi/3}$. The matrices entering the denominators of the
neutrino propagator are equal to

$\mathbb{M}_{R}\mathbb{M}_{L}=\left\|\begin{array}[]{rrr}1&0&0\\\ 0&2&-1\\\
0&-1&1\end{array}\right\|,\ \ \ \
\mathbb{M}_{L}\mathbb{M}_{R}=\left\|\begin{array}[]{rrr}1&0&0\\\ 0&2&1\\\
0&1&1\end{array}\right\|.$ (II.49)

The corresponding eigenvalues and eigenvectors become

$(m_{i}^{2})^{T}=\left\|\begin{array}[]{c}1\\\ 2-g\\\ 1+g\end{array}\right\|,\
\ \ \ \tilde{U}_{\alpha i}=\left\|\begin{array}[]{ccc}1&0&0\\\ 0&1/g&-g\\\
0&1&1\end{array}\right\|,\ \ \ \ \tilde{V}_{\alpha
i}=\left\|\begin{array}[]{ccc}1&0&0\\\ 0&-1/g&g\\\ 0&1&1\end{array}\right\|,$
(II.50)

where $g=(1+\sqrt{5})/2$. Calculation of the parameters entering Eqs.
(II.21)–(II.24) gives $\lambda_{0}=4/3$,
$a=\frac{2}{\sqrt{3}}\sin(\frac{1}{3}\arccos(\frac{11}{16}))$,
$b=-\frac{2}{\sqrt{3}}\cos(\frac{1}{3}\arccos(\frac{11}{16}))$. Using the
relations (II.25) leads to values $m_{i}^{2}$ that coincide with those given
above. A significant difference from the Majorana mass case is that two chiral
projections require two mixing matrices,

$\displaystyle\langle\alpha|i+\rangle$ $\displaystyle=$
$\displaystyle(\sum_{\beta}|\tilde{U}_{\beta i}|^{2})^{-1/2}\tilde{U}_{\alpha
i},$ (II.51) $\displaystyle\langle\alpha|i-\rangle$ $\displaystyle=$
$\displaystyle(\sum_{\beta}|\tilde{V}_{\beta i}|^{2})^{-1/2}\tilde{V}_{\alpha
i},$ (II.52)

to diagonalize the propagator. The relationship (II.15) for Dirac neutrinos is
no longer valid, as can be seen from the example considered.

Table 2: First two columns show the lepton currents at points $x$ and $y$, leading to neutrino oscillations; the next column shows the time ordering, the last column shows the process corresponding to the lepton currents; the second and third rows show the lepton currents that produce oscillations of Majorana neutrinos, but not Dirac neutrinos  $j_{\mu}^{\textrm{CC}}(x)$ | $j_{\mu}^{\textrm{CC}}(y)$ | Time ordering | Process
---|---|---|---
$\bar{\ell}_{\beta}^{-}\gamma_{\mu}(1-\gamma_{5})\nu_{\beta}$ | $\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\ell_{\alpha}^{-}$ |  | $x^{0}-y^{0}>0$
---
$x^{0}-y^{0}<0$
| $\nu_{\beta L}\leftarrow\nu_{\alpha L}$
---
$\left(\nu^{c}\right)_{\beta R}\rightarrow\left(\nu^{c}\right)_{\alpha R}$
$\bar{\nu}_{\beta}\gamma_{\mu}(1-\gamma_{5})\ell_{\beta}^{-}$ | $\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\ell_{\alpha}^{-}$ | $x^{0}-y^{0}\gtrless 0$ | -
$\bar{\ell}_{\beta}^{-}\gamma_{\mu}(1-\gamma_{5})\nu_{\beta}$ | $\bar{\ell}_{\alpha}^{-}\gamma_{\mu}(1-\gamma_{5})\nu_{\alpha}$ | $x^{0}-y^{0}\gtrless 0$ | -
$\bar{\nu}_{\beta}\gamma_{\mu}(1-\gamma_{5})\ell_{\beta}^{-}$ | $\bar{\ell}_{\alpha}^{-}\gamma_{\mu}(1-\gamma_{5})\nu_{\alpha}$ |  | $x^{0}-y^{0}>0$
---
$x^{0}-y^{0}<0$
| $\left(\nu^{c}\right)_{\beta R}\leftarrow\left(\nu^{c}\right)_{\alpha R}$
---
$\nu_{\beta L}\rightarrow\nu_{\alpha L}$

Table 2 shows the possible types of neutrino oscillations caused by the weak
lepton currents at the space–time points $x$ and $y$. The combinations are
also shown that could lead to the oscillations of Dirac neutrinos in the
presence of right currents. One part of the amplitudes is expressed through
the other by means of time inversion. The transition amplitudes represented in
the last row are obtained by permuting the arguments $x\leftrightarrow y$ of
the amplitudes in the first row. The oscillation amplitudes corresponding to
the first row are determined by Eqs. (II.1) with the right Majorana neutrinos
$\nu_{\alpha R}$ identified as the left Dirac antineutrinos $(\nu_{\alpha
L})^{c}$.

The conclusion that is crucial to this discussion is that the neutrino mass
matrix explicitly enters the amplitude of the oscillations of Dirac neutrinos
in general.

Dirac neutrino is a superposition of two Majorana neutrinos with masses that
are equal in absolute values and opposite in signs (see, e.g.,
Krivoruchenko:1996 ). Although the case described here is essentially a
modified version of the one previously discussed, it is highlighted because of
its significance in application. The mixed case of Majorana and Dirac
neutrinos is reduced to Majorana neutrinos and is not discussed here.

##  III Neutrino oscillations in medium

The electroweak mean-field potential created by the $Z^{0}$ boson and the
exchange potential of the $W^{\pm}$ boson generated by charged leptons in the
medium have both an effect on neutrinos. The mean field in the rest frame of
matter has a time-like element:
$(V_{\mu})_{\alpha\beta}=(V_{\alpha\beta},\mathbf{0)}$. The potential in the
flavor basis is diagonal: $V_{\alpha\beta}=V_{(\alpha)}\delta_{\alpha\beta}$,
as illustrated in Fig. 1. Hereafter, $\mathbb{V}$ stands for a matrix
containing the components $V_{\alpha\beta}$.

Figure 1:  Diagrams contributing to the mean-field potential of neutrino in
matter. $f$ is a Standard Model fermion, $\ell^{-}_{\alpha}$ is a charged
lepton, and $\nu_{\alpha}$ are neutrinos with the $\alpha$ flavor. The $Z$
boson exchange in the $t$-channel $a$) produces a diagonal potential
independent of neutrino flavor. Charged leptons and neutrinos interact
additionally through the $s$-channel exchange of the $W^{-}$ boson, as shown
in the diagram $b$). The individual components of the mean-field potential
$b$) are determined by the partial lepton densities. The total potential is
diagonal in the neutrino flavor. The initial and final states of the medium
particles are identical, since the mean-field potential is determined by the
forward scattering amplitude at zero transmitted momentum.

###  III.1 Majorana neutrino

The effective Lagrangian of Majorana neutrinos in the medium has the form

$\mathcal{L}_{M}^{\prime}=\frac{1}{2}\bar{\nu}_{\alpha}\left(i\hat{\nabla}\delta_{\alpha\beta}+\hat{V}_{\alpha\beta}\gamma_{5}\right)\nu_{\beta}-\frac{1}{2}\bar{\nu}_{\alpha}\left(M_{\alpha\beta}+i\gamma_{5}N_{\alpha\beta}\right)\nu_{\beta}.$
(III.1)

The mean-field potential
$\hat{V}_{\alpha\beta}=(\gamma_{0}V_{\alpha\beta},\mathbf{0})$ is determined
by the forward scattering amplitude of neutrinos and depends on the
composition of the substance. The initial and final states of the scattering
particles are identical. Neutrino exchange with the medium particles via the
$Z$ boson is independent of the neutrino flavor. W± bosons contributes to
$\mathbb{V}$ through the $s$-channel exchange involving lepton component of
the substance. Figure 1 shows both types of diagrams, each generating a
flavor-diagonal mean-field potential. Consequently, the mean-field potential
is generally represented as
$V_{\alpha\beta}=V_{(\alpha)}\delta_{\alpha\beta}$. For example, there are no
$\tau^{-}$ mesons in interiors of neutron stars, therefore, the component of
$V_{(\alpha)}$ associated with the $\tau^{-}$ meson is zero. The mean-field
potential influences the left neutrino component. The mean field is a
symmetric matrix in flavor indices, so its vector component disappears (cf.
Eq. (II.3)) and only its axial component remains in Eq. (III.1). In the Weyl
basis

$\mathcal{L}_{M}^{\prime}=\bar{\nu}_{L\alpha}\left(i\hat{\nabla}\delta_{\alpha\beta}-\hat{V}_{\alpha\beta}\right)\nu_{L\beta}-\frac{1}{2}\bar{\nu}_{L\alpha}^{c}\left(M_{\alpha\beta}-iN_{\alpha\beta}\right)\nu_{L\beta}-\frac{1}{2}\bar{\nu}_{L\alpha}\left(M_{\alpha\beta}+iN_{\alpha\beta}\right)\nu_{L\beta}^{c},$

where $\nu_{L\alpha}^{c}=(\nu_{L\alpha})^{c}$ and
$\bar{\nu}_{L\alpha}^{c}=\overline{\nu_{L\alpha}^{c}}$.

In the momentum representation, the Feynman propagator takes the form

$\displaystyle\hat{\mathbb{S}}_{F}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{p}-\hat{\mathbb{M}}+\hat{\mathbb{V}}\gamma_{5}}$
(III.2) $\displaystyle=$
$\displaystyle\left(\hat{p}+\hat{\mathbb{M}}^{\dagger}+\hat{\mathbb{V}}\gamma_{5}\right)\frac{1}{p^{2}-\hat{\mathbb{M}}\hat{\mathbb{M}}^{\dagger}-\hat{\mathbb{V}}^{2}+\left(\hat{p}\hat{\mathbb{V}}-\hat{\mathbb{V}}\hat{p}\right)\gamma_{5}}.$

Unlike the case discussed in Section II, the chiral projections do not
eliminate $\gamma$-matrices completely from the denominator. Given
$\hat{p}\hat{\mathbb{V}}-\hat{\mathbb{V}}\hat{p}=2\mathbb{V}\mbox{\boldmath$\alpha$}\mathbf{p}$
in the rest frame of matter, it is helpful to decompose
$\hat{\mathbb{S}}_{F}(p)$ further using projections into states where the
scalar product of the matrix $\alpha$ and the momentum $\mathbf{p}$ equals
$\pm|\mathbf{p}|$. The corresponding projection operators are defined by
$P_{\pm}=(1\pm\mbox{\boldmath$\alpha$}\mathbf{n})/{2}$, where
$\mathbf{n}=\mathbf{p}/|\mathbf{p}|$, $P_{\sigma}$ commute with
$\Pi_{\varsigma}$. In the standard representation and in the Weyl
representation of the $\gamma$-matrices, the relationship
$\mbox{\boldmath$\alpha$}=\gamma_{5}\otimes\mbox{\boldmath$\sigma$}$ holds
true, so that $\mbox{\boldmath$\alpha$}\mathbf{n}$ can be interpreted as
chirality times helicity ($\sigma=\varsigma\eta$, $\eta$ is helicity).

The neutrino propagator is represented by sum of the four projections:

$\displaystyle\hat{\mathbb{S}}_{F}(p)$ $\displaystyle=$
$\displaystyle\sum_{\varsigma\sigma}\left(\hat{p}+{\mathbb{M}}_{-{\varsigma}}+{\varsigma}\hat{\mathbb{V}}\right)\frac{1}{p^{2}-{\mathbb{M}}_{{\varsigma}}{\mathbb{M}}_{-{\varsigma}}-{\mathbb{V}}^{2}+2{\varsigma}\sigma\mathbb{V}|\mathbf{p}|}\Pi_{\varsigma}P_{\sigma}.$
(III.3)

The denominator is a Hermitian matrix, so the eigenvalues are real. They are
determined from the characteristic equation

$\displaystyle\det\left\|\lambda-{\mathbb{M}}_{{\varsigma}}{\mathbb{M}}_{-{\varsigma}}-{\mathbb{V}}^{2}+2\eta\mathbb{V}|\mathbf{p}|\right\|=0.$
(III.4)

The value of $\lambda$ depends on the momentum. The complex conjugation of the
characteristic equation permutes the matrices ${\mathbb{M}}_{R}$ and
${\mathbb{M}}_{L}$, so the poles of the expression (III.3) depend on the
helicity $\eta$ only. The dispersion law takes the form

$\displaystyle E_{i\eta}(\mathbf{p})=\sqrt{\lambda_{i\eta}+\mathbf{p}^{2}},$
(III.5)

where $\lambda_{i\eta}$ is one of the solutions. Entering the determinant
(III.4) is a $3\times 3$ matrix. This suggests that we are dealing with a
cubic equation. To find the roots, one can use the Vieta method, which has
already been employed to analyze neutrino oscillations in vacuum.

The procedure is equivalent to that used in Section II. We redefine the matrix
$\mathbb{C}$ and the parameter $\lambda_{0}$:

$\displaystyle\mathbb{C}$ $\displaystyle=$
$\displaystyle\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+{\mathbb{V}}^{2}-2\eta\mathbb{V}|\mathbf{p}|-\frac{1}{3}\mathrm{Tr}\left[\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+{\mathbb{V}}^{2}-2\eta\mathbb{V}|\mathbf{p}|\right],$
(III.6) $\displaystyle\lambda_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\mathrm{Tr}\left[\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+{\mathbb{V}}^{2}-2\eta\mathbb{V}|\mathbf{p}|\right].$
(III.7)

The formulas (II.20)–(II.25) and (II.27) allow to find the eigenvalues
$\lambda_{i\eta}$ and the in-medium dispersion law.

The eigenstates determined by the equation

$\displaystyle\left(\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+{\mathbb{V}}^{2}-2\eta\mathbb{V}|\mathbf{p}|\right)|i\varsigma\eta\rangle=\lambda_{i\eta}|i\varsigma\eta\rangle$
(III.8)

depend on the momentum. The values of $E_{i\eta}(\mathbf{p})$ are real, since

$E_{i\eta}^{2}(\mathbf{p})=\langle{i\varsigma\eta}|\left((\mathbb{M}_{-\varsigma})^{\dagger}\mathbb{M}_{-\varsigma}+\left(|\mathbf{p}|-\eta\mathbb{V}\right)^{2}\right)|i\varsigma\eta\rangle\geq
0.$

The eigenvalues of Eq. (III.8) are real and coincide for $\varsigma=\pm 1$,
while the coordinates of $|i\pm\eta\rangle$ in the flavor basis are related by
the equation

$\langle\alpha|i\varsigma\eta\rangle^{*}=\langle\alpha|i-\varsigma\eta\rangle.$
(III.9)

In the simplest case of the mean-field potential proportional to the unit
matrix, which occurs when the diagram in Fig. 1 b) is turned off, the momentum
dependence of the dispersion law is quite transparent. Assuming
$V_{(\alpha)}=V$ for $\alpha=1,2,3$, we find

$\displaystyle E_{i\eta}(\mathbf{p})=\sqrt{m_{i}^{2}+\left(|\mathbf{p}|-\eta
V\right)^{2}},$ (III.10)

where $m_{i}^{2}$ are the eigenvalues of
$\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}$. Since $\mathbb{V}$ is
commutative with $\mathbb{M}_{\varsigma}$, the dependence on $\eta$ disappears
from $|i\varsigma\eta\rangle$.

The equation for eigenvalues in an arbitrary potential admits a simple
solution for neutrinos with a large momentum. In the left side of Eq. (III.8),
the leading term is $2\eta\mathbb{V}|\mathbf{p}|$ with $\eta=\pm 1$. Since the
potential is diagonal in flavor, the eigenstates are also diagonal in flavor.
The first two terms on the left side of the equations give a correction to
$\lambda_{i\eta}$. In the leading order, the eigenvalues, the dispersion law,
and the eigenstates take the form

$\displaystyle\lambda_{\alpha\eta}$ $\displaystyle=$ $\displaystyle-2\eta
V_{\alpha}|\mathbf{p}|+m_{\alpha}^{2}+{V}_{\alpha}^{2}+\ldots,$ (III.11)
$\displaystyle E_{\alpha\eta}$ $\displaystyle=$
$\displaystyle|\mathbf{p}|-\eta
V_{\alpha}+\frac{m_{\alpha}^{2}}{2|\mathbf{p}|}+\ldots,$ (III.12)
$\displaystyle|\alpha\varsigma\eta\rangle$ $\displaystyle=$
$\displaystyle|\alpha\rangle+\frac{\eta}{2|\mathbf{p}|}\sum_{\beta\neq\alpha}|\beta\rangle\frac{1}{V_{(\beta)}-V_{(\alpha)}}\langle\beta|\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}|\alpha\rangle+\ldots,$
(III.13)

where
$m_{\alpha}^{2}=\langle\alpha|\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}|\alpha\rangle$.
In the limit of $|\mathbf{p}|\to\infty$, the oscillations disappear for
$V_{(\beta)}\neq V_{(\alpha)}$. The degeneracy of the potential
$V_{(\beta)}\approx V_{(\alpha)}$ for any pair of $\alpha\neq\beta$ restores
the mixing Wolfenstein:1978 ; Mikheev:1985 . The mixing matrix (II.39) is
split in the medium into two components depending on the parameter $\eta$:

$U^{\dagger}_{i\alpha}\to\langle
i+|\alpha\varsigma\eta\rangle=U^{\dagger}_{i\alpha}+\frac{\eta}{2|\mathbf{p}|}\sum_{\beta\neq\alpha}U^{\dagger}_{i\beta}\frac{1}{V_{(\beta)}-V_{(\alpha)}}\langle\beta|\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}|\alpha\rangle+\ldots$
(III.14)

The correction described by the second term can be significant for low values
of $|\mathbf{p}|$ or in the presence of degeneracy $V_{(\beta)}\approx
V_{(\alpha)}$.

Integrating the time-like component of four-momentum in the coordinate
representation, we obtain

$\displaystyle i\hat{\mathbb{S}}_{F}(x-y)$ $\displaystyle=$
$\displaystyle\theta(x^{0}-y^{0})\sum_{i\varsigma\eta}\int\frac{d\mathbf{p}}{(2\pi)^{3}2E_{i\eta}(\mathbf{p})}e^{-iE_{i\eta}(\mathbf{p})(x^{0}-y^{0})+i\mathbf{p}(\mathbf{x}-\mathbf{y})}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(\hat{p}_{i\eta}+{\mathbb{M}}_{-\varsigma}+\varsigma\hat{\mathbb{V}}\right)|i\varsigma\eta\rangle\langle
i\varsigma\eta|\Pi_{\varsigma}P_{\sigma}$ $\displaystyle-$
$\displaystyle\theta(y^{0}-x^{0})\sum_{i\varsigma\eta}\int\frac{d\mathbf{p}}{(2\pi)^{3}2E_{i\eta}(\mathbf{p})}e^{+iE_{i\eta}(\mathbf{p})(x^{0}-y^{0})-i\mathbf{p}(\mathbf{x}-\mathbf{y})}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(\hat{p}_{i\eta}-{\mathbb{M}}_{-\varsigma}-\varsigma\hat{\mathbb{V}}\right)|i\varsigma\eta\rangle\langle
i\varsigma\eta|\Pi_{\varsigma}P_{\sigma},$

where $\sigma=\varsigma\eta$.

The Frobenius covariants are given by

$\displaystyle|i\varsigma\eta\rangle\langle
i\varsigma\eta|\equiv\mathbb{F}_{i\varsigma\eta}=\prod_{j\neq
i}\frac{\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+{\mathbb{V}}^{2}-2\eta\mathbb{V}|\mathbf{p}|-\lambda_{j\eta}}{\lambda_{i\eta}-\lambda_{j\eta}}.$
(III.16)

Like in the previous section we consider first neutrino oscillations
$\nu_{\beta L}\leftarrow\nu_{\alpha L}$, described by the current
$j_{\mu}^{\textrm{CC}}=\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\ell_{\alpha}^{-}$
at the neutrino creation point $y$ and by the current
$j_{\mu}^{\textrm{CC}}=\bar{\ell}_{\beta}^{-}\gamma_{\mu}(1-\gamma_{5})\nu_{\beta}$
at the neutrino absorption point $x$. A process is also possible in which the
right neutrino $\nu_{\alpha R}$ is created at the point $x$ and propagates to
the point $y$ without chirality flip, where it is absorbed ($x^{0}-y^{0}<0$).
Neglecting the neutrino momentum spread, the neutrino component of the
amplitude, $\Pi_{-}i\hat{\mathbb{S}}_{F}(x-y)\Pi_{+}$, becomes

$\displaystyle\hat{A}(\nu_{\beta L}\leftarrow\nu_{\alpha L})$
$\displaystyle\propto$
$\displaystyle-\sum_{i\eta}e^{-iE_{i\eta}(\mathbf{p})\tau}\langle\beta|\left(\hat{p}_{i\eta}+\hat{\mathbb{V}}\right)\mathbb{F}_{i\varsigma\eta}|\alpha\rangle\frac{1}{E_{i\eta}(\mathbf{p})}\Pi_{\varsigma}P_{\eta},$
(III.17) $\displaystyle\hat{A}(\nu_{\beta R}\rightarrow\nu_{\alpha R})$
$\displaystyle\propto$
$\displaystyle~{}~{}~{}\sum_{i\eta}e^{-iE_{i\eta}(\mathbf{p})\tau}\langle\beta|\left(\hat{p}_{i\eta}-\hat{\mathbb{V}}\right)\mathbb{F}_{i\varsigma\eta}|\alpha\rangle\frac{1}{E_{i\eta}(\mathbf{p})}\Pi_{\varsigma}P_{\eta},$
(III.18)

where $\varsigma=+1$.

The mixing matrices with chirality $\varsigma=\pm 1$ are connected by Eq.
(III.9), however to characterize the mixing of neutrinos with helicity
$\eta=\pm 1$, two distinct matrices $\langle\alpha|i++\rangle$ and
$\langle\alpha|i+-\rangle$ are required. In the limit of vanishing matter
density, the matrices $\langle\alpha|i+\pm\rangle$ become the PMNS matrix.

The process $\nu_{\beta R}\leftarrow\nu_{\alpha L}$ with chirality flip is
associated with the lepton current
$j_{\mu}^{\textrm{CC}}=\bar{\nu}_{\alpha}\gamma_{\mu}(1-\gamma_{5})\ell_{\alpha}^{-}$
at the creation point $y$ and by the lepton current
$j_{\mu}^{\textrm{CC}}=\bar{\nu}_{\beta}\gamma_{\mu}(1-\gamma_{5})\ell_{\beta}^{-}$
at the absorption point $x$. The process $\nu_{\beta R}\rightarrow\nu_{\alpha
L}$ is described for $x^{0}-y^{0}>0$ by the same currents. The component of
the amplitudes responsible for chirality-flip propagation of neutrinos equals
$\Pi_{+}i\hat{\mathbb{S}}_{F}(x-y)\Pi_{+}$, so we obtain

$\hat{A}(\nu_{\beta R}\leftarrow\nu_{\alpha L})=-\hat{A}(\nu_{\beta
R}\rightarrow\nu_{\alpha
L})\propto-\sum_{i\eta}e^{-iE_{i\eta}(\mathbf{p})\tau}\langle\beta|\mathbb{M}_{L}\mathbb{F}_{i\varsigma\eta}|\alpha\rangle
e^{-i\phi_{\alpha}-i\phi_{\beta}}\frac{1}{E_{i\eta}(\mathbf{p})}\Pi_{\varsigma}P_{\eta},$
(III.19)

where $\varsigma=+1$. The explicit dependence on the neutrino mass matrix is
obvious. The arguments given after Eq. (II.1) prove the equation
$\langle\beta|\mathbb{M}_{L}|i\varsigma\eta\rangle=m_{i}\langle
i\varsigma\eta|\beta\rangle$ for $\mathbb{V}$ commutative with
$\mathbb{M}_{\pm}$.

###  III.2 Dirac neutrino

The in-medium Lagrangian for Dirac neutrinos is written as

$\mathcal{L}_{D}^{\prime}=\bar{\nu}_{\alpha}\left(i\hat{\nabla}\delta_{\alpha\beta}-\hat{V}_{\alpha\beta}\Pi_{-}\right)\nu_{\beta}-\bar{\nu}_{\alpha}\left(M_{\alpha\beta}+i\gamma_{5}N_{\alpha\beta}\right)\nu_{\beta}.$
(III.20)

The mean-field term $\hat{V}_{\alpha\beta}\Pi_{-}$ distinguishes it from the
vacuum Lagrangian, $\mathcal{L}_{D}$. The mass matrices $\mathbb{M}$ and
$\mathbb{N}$ are Hermitian. The Feynman propagator has the following form in
the momentum representation:

$\displaystyle\hat{\mathbb{S}}_{F}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{\hat{p}-\hat{\mathbb{M}}-\hat{\mathbb{V}}\Pi_{-}}.$
(III.21)

Figure 2:  Two branches of dispersion law for Majorana neutrinos in a medium
with a mean-field potential of $V=-10$ eV, typical for nuclear matter at
saturation, and a neutrino mass of $m_{i}=0.1$ eV. The upper and lower lines
correspond to neutrinos with left and right helicity $\eta=\mp 1$,
respectively. The neutrino energy takes a minimum value
$E_{i+}(\mathbf{p})=m_{i}$ at $|\mathbf{p}|=10$ eV. The dispersion law of
positive-energy Dirac neutrinos takes the same form for $V=-20$ eV and is
shifted along the vertical axis accordingly.

Example 3. Given that $\hat{\mathbb{V}}$ is determined by time-like component
of the four-vector, the substitutions $E\to E-\mathbb{V}$ and
$\mathbb{V}\to\mathbb{V}/2$ reduce the problem to the case of Majorana
neutrinos provided $\mathbb{V}$ is proportional the unit matrix. In such a
case, the dispersion law reads

${E}_{i\eta}^{D\pm}(\mathbf{p})=\frac{1}{2}V\pm E_{i\eta}(\mathbf{p}),$
(III.22)

where $E_{i\eta}(\mathbf{p})$ simplifies to (III.10) with $V$ replaced by
$V/2$. In a neutron star, the mean-field potential generated by neutrons is
$V\sim-20$ eV (see, e.g., Krivoruchenko:2022 ). At zero temperature and for
$V/2+m_{i}<0$ the potential well is filled first by Dirac neutrinos with a
negative energy at $|\mathbf{p}|\sim|V|/2$, then states with higher energy are
filled. At a low density of neutrinos, the design of the Fermi sphere turns
out to be quite unusual: the neutrino states form in the momentum space a
hollow sphere with a radius close to $|V|/2$. Figure 2 depicts the dispersion
law for the Majorana neutrino for $V=-10$ eV. The positive-energy Dirac
neutrino dispersion law is similar, but it is displaced vertically and has a
modified scale of the horizontal axis.

The BCS mechanism causes a gap to appear in the neutrino excitation spectrum
at ultra-low temperatures Kapusta:2004 . The pairing of Dirac neutrinos is
caused by the exchange of the Higgs boson, which generates an attraction
between fermions. In the case under consideration, the BCS mechanism is
applicable to the right–helicity neutrinos adjacent to both the outer surface
of the hollow Fermi sphere and the inner. At the temperature of old neutron
stars of the order of 100 eV, condensation has not yet formed, so the
appearance of superfluidity can be expected only in the very distant future as
a result of further cooling of neutron stars. The right–helicity neutrinos
contain an admixture of the left–chiral neutrinos, which interact in a
repulsive manner due to the $Z^{0}$–boson exchange. The admixture of the left-
chiral states is suppressed as $m_{i}/E_{i+}(\mathbf{p})$, so a sufficiently
small neutrino mass is required to ensure the dominance of attraction due to
the Higgs boson exchange.

The bottom curve of the dispersion law reaches the border of the continuum of
states with negative energy at a depth of the potential well of $-4m_{i}$.
This leads to the creation of neutrino-antineutrino pairs from vacuum and
filling of levels with energy $E_{i}^{D+}(\mathbf{p})<-m_{i}$.

The technique for determining the dispersion law in a general scenario for in-
medium Dirac neutrinos is more involved. We decompose first the neutrino
propagator (III.21) in a series by powers of the potential. The chirality-
state projections of each of the sum terms can be simplified by using Eq.
(II.10). The summation of the outcome results in

$\displaystyle\hat{\mathbb{S}}_{F}(p)$ $\displaystyle=$
$\displaystyle\left(\hat{p}+\mathbb{M}_{L}\right)\frac{1}{p^{2}-\mathbb{M}_{R}\mathbb{M}_{L}-\hat{\mathbb{V}}\hat{p}}\Pi_{+}$
(III.23) $\displaystyle+$
$\displaystyle\left(\hat{p}+\mathbb{M}_{R}\right)\frac{1}{p^{2}-\mathbb{M}_{L}\mathbb{M}_{R}}\Pi_{-}+\left(\hat{p}+\mathbb{M}_{L}\right)\frac{1}{p^{2}-\mathbb{M}_{R}\mathbb{M}_{L}-\hat{\mathbb{V}}\hat{p}}\hat{\mathbb{V}}\mathbb{M}_{R}\frac{1}{p^{2}-\mathbb{M}_{L}\mathbb{M}_{R}}\Pi_{-}.$

The second line, which refers to $\hat{\mathbb{S}}_{F}(p)\Pi_{-}$, has two
types of poles. The first corresponds to a neutrino in a medium, while the
second to a neutrino in a vacuum. The second type pole is inactive. To
demonstrate this, we employ the chirality-state projectors to act on
$\hat{\mathbb{S}}_{F}(p)\Pi_{-}$ from the left. Basic algebra gives

$\displaystyle\hat{\mathbb{S}}_{F}(p)\Pi_{-}$ $\displaystyle=$
$\displaystyle\mathbb{M}_{L}\frac{1}{p^{2}-\mathbb{M}_{R}\mathbb{M}_{L}-\hat{\mathbb{V}}\hat{p}}\left(\hat{p}-\hat{\mathbb{V}}\right)\mathbb{M}_{L}^{-1}\Pi_{-}$
(III.24) $\displaystyle+$
$\displaystyle\frac{1}{p^{2}-\mathbb{M}_{R}\mathbb{M}_{L}-\hat{p}\hat{\mathbb{V}}}\mathbb{M}_{R}\Pi_{-}.$

The transformations are carried out under the assumptions that the matrix
$\mathbb{V}$ only has a time-like component, all matrices are non-singular,
and the matrix $\mathbb{V}$ generally does not commute with the mass matrix.
Since the poles of the free neutrino propagator vanish, the vacuum poles of
the expression (III.23) are fictitious. With the substitutions $E\rightarrow
E-\mathbb{V}$ and $\mathbb{V}\to\mathbb{V}/2$, the propagator poles for
Majorana neutrino and Dirac neutrino formally coincide.

The denominators of propagators turn into $n\times n$ matrices in the flavor
space using projectors $P_{\sigma}$. Using the commutativity of $P_{\sigma}$
and $\Pi_{\varsigma}$, the relationships
$\hat{\mathbb{V}}\hat{p}P_{\sigma}=\mathbb{V}(E-\sigma|\mathbf{p}|)P_{\sigma}$,
$P_{\sigma}\gamma_{0}=\gamma_{0}P_{-\sigma}$ and
$(\hat{p}-\hat{\mathbb{V}})P_{\sigma}=P_{-\sigma}(\hat{p}-\hat{\mathbb{V}})$,
the denominators of $\hat{\mathbb{S}}_{F}(p)\Pi_{\varsigma}P_{\sigma}$ can be
transformed into
$p^{2}-\mathbb{M}_{R}\mathbb{M}_{L}-\mathbb{V}(E-\eta|\mathbf{p}|)$. The
excitation spectrum is determined thereby from the equation:

$\left(\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+\mathbb{V}(E-\eta|\mathbf{p}|)\right)|i\sigma\varsigma\rangle=\lambda_{i\eta}(E)|i\varsigma\eta\rangle.$
(III.25)

It is useful to compare this equation to Eq. (III.8).

The characteristic equation of Dirac neutrinos differs from that of Majorana
neutrinos in terms depending on the potential:

$\displaystyle\det\left\|\lambda-{\mathbb{M}}_{{\varsigma}}{\mathbb{M}}_{-\varsigma}-\mathbb{V}(E-\eta|\mathbf{p}|)\right\|=0.$
(III.26)

For completeness, we also give the expressions of the matrix $\mathbb{C}$ and
the parameter $\lambda_{0}$:

$\displaystyle\mathbb{C}$ $\displaystyle=$
$\displaystyle\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+\mathbb{V}(E-\eta|\mathbf{p}|)-\frac{1}{3}\mathrm{Tr}\left[\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+\mathbb{V}(E-\eta|\mathbf{p}|)\right],$
(III.27) $\displaystyle\lambda_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\mathrm{Tr}\left[\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+\mathbb{V}(E-\eta|\mathbf{p}|)\right].$
(III.28)

The relations (II.22 )–(II.24) with $n=3$ are fully applicable for Dirac
neutrinos in matter, in the relations (II.25) it is necessary to replace
$m_{i}^{2}\to\lambda_{i\eta}(E)$. Unlike the Majorana neutrino,
$\lambda_{i\eta}(E)$ depends on the energy. The characteristic equation,
therefore, is a polynomial of degree $6$ with respect to the energy, and for a
matrix $\mathbb{V}$ of general form it is not solved in quadratures.

The Frobenius covariants read:

$\displaystyle|i\varsigma\eta\rangle\langle
i\varsigma\eta|\equiv\mathbb{F}_{i\varsigma\eta}(E,\mathbf{p})=\prod_{j\neq
i}\frac{\mathbb{M}_{\varsigma}\mathbb{M}_{-\varsigma}+\mathbb{V}(E-\eta|\mathbf{p}|)-\lambda_{j\eta}(E)}{\lambda_{i\eta}(E)-\lambda_{j\eta}(E)}.$
(III.29)

Let us consider the case of large momenta. The dominant term
$\mathbb{V}(-\eta|\mathbf{p}|)$ in Eq. (III.25) if not entirely compensated by
the term $\mathbb{V}E$ implies that the zeroth approximation is formed by the
flavor states. In the leading order, we obtain

$\displaystyle\lambda_{\alpha\eta}(E_{\alpha\eta}^{D+})$
$\displaystyle=(1-\eta)V_{\alpha}|\mathbf{p}|+m_{\alpha}^{2}+\frac{1}{2}(1-\eta)V_{\alpha}^{2}+\ldots,$
(III.30) $\displaystyle E_{\alpha\eta}^{D+}$
$\displaystyle=|\mathbf{p}|+\frac{1}{2}(1-\eta)V_{\alpha}+\frac{m_{\alpha}^{2}}{2|\mathbf{p}|}+\ldots,$
(III.31) $\displaystyle|\alpha\varsigma\eta\rangle$
$\displaystyle=|\alpha\rangle-\frac{1}{2|\mathbf{p}|}\sum_{\beta\neq\alpha}|\beta\rangle\frac{1}{V_{(\beta)}-V_{(\alpha)}}\langle\beta|\mathbb{M}_{R}\mathbb{M}_{L}|\alpha\rangle+\ldots~{}~{}~{}~{}(\eta=-1),$
(III.32)

where
$m_{\alpha}^{2}=\langle\alpha|\mathbb{M}_{R}\mathbb{M}_{L}|\alpha\rangle.$ As
noted earlier, the poles of the Majorana neutrino and Dirac neutrino
propagators formally coincide after the energy shift $E\rightarrow
E-\mathbb{V}$ and redefinition $\mathbb{V}\to\mathbb{V}/2$; this is reflected
in the dispersion laws (III.12) and (III.31). According to Eq. (III.31), a
Dirac neutrino with right helicity does not feel the external potential, which
physically distinguishes a Dirac neutrino from a Majorana neutrino. Moreover,
for $\eta=+1$, the term $\mathbb{V}(E-\eta|\mathbf{p}|)$ in Eq. (III.25) is
not dominant, so the validity of the expressions (III.30)–(III.32) is
restricted to the case $\eta=-1$.

We omit the intermediate steps and give the final expression for neutrino
component of the oscillation amplitudes of Dirac neutrinos

$\displaystyle\hat{A}(\nu_{\beta L}\leftarrow\nu_{\alpha L})$ $\displaystyle=$
$\displaystyle-\hat{A}((\nu^{c})_{\beta R}\rightarrow(\nu^{c})_{\alpha R})$
(III.33) $\displaystyle\propto$
$\displaystyle-\sum_{i\eta}e^{-iE_{i\eta}^{D+}(\mathbf{p})\tau}\langle\beta|\mathbb{F}_{i\varsigma\eta}(E_{i\eta}^{D+},\mathbf{p})|\alpha\rangle\frac{2\hat{p}_{i\eta}^{D+}}{2{E}_{i\eta}^{D+}(\mathbf{p})-\lambda_{i\eta}^{\prime}({E}_{i\eta}^{D+}(\mathbf{p}))}\Pi_{\varsigma}P_{\eta},$

where $\varsigma=+1$,
$\lambda_{i\eta}^{\prime}(E)={\partial}\lambda_{i\eta}(E)/{\partial E}$, and
$E$ is replaced by $E_{i\eta}^{D+}$ in
$\mathbb{F}_{i\varsigma\eta}(E,\mathbf{p})$. By virtue of Eq. (III.25)
$\lambda_{i\eta}^{\prime}(E)=\langle
i\varsigma\eta|\mathbb{V}|i\varsigma\eta\rangle$. Expressions in quadratures
complicate the analysis of the analytical properties of the propagator as a
function of energy. Returning to expressions (III.23) and (III.24), we note
that the inversion of matrices in the denominators results in the denominators
becoming a polynomial of degree $2n=6$, which, in the absence of degeneracy,
has simple zeros on the real axis of energy. The expression shown in (III.33)
is the result of integration with respect to the time-like variable $p^{0}$,
while the integration contour is closed in the lower half-plane, where the
propagator has no singularities.

##  IV Other neutrino-related processes

It was demonstrated in the preceding sections that experiments involving
neutrino oscillations in a medium and in a vacuum can both be explained solely
in terms of the neutrino mass matrix without the need for the mixing matrix.
This is demonstrated via Sylvester’s theorem, which makes use of Frobenius
covariants. Additional neutrino-related processes are covered below. It turns
out that the probabilities are also directly expressed using the neutrino mass
matrix in all of the scenarios that are considered to be the most typical.

###  IV.1 $\beta$-decay of tritium

Tritium is used as a $\beta$ emitter in advanced experiments on the direct
measurement of the mass of an electron antineutrino Kraus:2005 ; Aseev:2011 ;
KATRIN:2022 . For the last decade, precise measurements of the electron
antineutrino mass have been carried out by the KATRIN collaboration. As a
result of the analysis of the edge of the experimental spectrum, a restriction
on the neutrino mass $<0.8$ eV at CL = 90% was obtained KATRIN:2022 .

The edge of the $\beta$ electron energy spectrum is not sensitive to decay
channels associated with the electron shell excitation. However, when the
$\beta$ electron energy exceeds the binding energy of an electron in a tritium
atom, it is necessary to take into account new channels available in energy.
The loss of energy for ionization leads to a distortion of the shape of the
$\beta$ spectrum. In the $\beta$ decay of tritium, the Coulomb interaction of
the knocked-out electron with the nucleus and the channels associated with the
population of higher discrete levels by an atomic electron are also important
to construct a realistic $\beta$ decay spectrum over the entire range of the
energy spectrum of $\beta$ electrons. Theoretical corrections arising on the
atomic and nuclear level are also discussed in Mertens:2015 ; Tyirn:2019 .

Neglecting electron shell structure effects the $\beta$-decay probability of
tritium equals (see, e.g., Okun:1981 )

$d\Gamma_{\beta}=\frac{G_{\mathrm{F}}^{2}\cos^{2}\Theta_{\mathrm{C}}}{2\pi^{3}}|\mathcal{M}_{\mathrm{nucl}}|^{2}F(E,Z=2)\sum_{i}\left|\langle
i+|\alpha\rangle\right|^{2}p_{\nu_{i}}E_{\nu_{i}}p_{e}E_{e}dE_{e},$ (IV.1)

where $G_{\mathrm{F}}$ denotes the Fermi constant, $\Theta_{\mathrm{C}}$ is
the Cabibbo angle, $\mathcal{M}_{\mathrm{nucl}}$ is the nuclear matrix
element, $F(E,Z=2)$ is the Fermi function, $p_{\nu_{i}}$ and $E_{\nu_{i}}$ are
the momentum and energy of the neutrino with a mass of $m_{i}$, $p_{e}$ and
$E_{e}$ are the momentum and energy of the $\beta$ electron,
$Q=E_{\nu_{i}}+E_{e}$ is the decay energy, $|\alpha\rangle$ is the electron-
flavor neutrino state, and $\langle i+|\alpha\rangle$ is the mixing matrix.
The kinematic factor is the phase space volume. Using the formula (II.27), we
obtain

$d\Gamma_{\beta}=\frac{G_{\mathrm{F}}^{2}\cos^{2}\Theta_{\mathrm{C}}}{2\pi^{3}}|\mathcal{M}_{\mathrm{nucl}}|^{2}F(E,Z=2)\sum_{i}\langle\alpha|\mathbb{F}_{i+}|\alpha\rangle
p_{\nu_{i}}E_{\nu_{i}}p_{e}E_{e}dE_{e}.$ (IV.2)

The decay probability is expressed in terms of the neutrino mass matrix in the
flavor basis.

###  IV.2 Electron capture

The inverse to the $\beta$ decay of an atom is the capture of an orbital
electron. The following formula for the total probability is used in the limit
of zero width of excited electron shells Behrens:1982 :

$\lambda_{x}=\frac{G_{\mathrm{F}}^{2}\cos^{2}\Theta_{\mathrm{C}}}{(2\pi)^{2}}n_{x}\mathcal{B}_{x}\beta_{x}^{2}C_{x}\sum_{i}\left|\langle
i+|\alpha\rangle\right|^{2}p_{\nu_{i}}E_{\nu_{i}},$ (IV.3)

where $n_{x}$ is the relative occupation number of the $x$ shell, $\beta_{x}$
is proportional to the wave function of the captured electron inside the
parent nucleus. The overlap-exchange correction $\mathcal{B}_{x}$ is a square
of the atomic transition matrix element which takes into account collective
electron effects. The phase-space volume is proportional to
$p_{\nu_{i}}E_{\nu_{i}}$. The shape function of the transition, $C_{x}$,
contains information about the nuclear structure and transformation properties
of the decay amplitude. Other notations as in the previous subsection.

In terms of the Frobenius covariant (II.27), the decay probability becomes

$\lambda_{x}=\frac{G_{\mathrm{F}}^{2}\cos^{2}\Theta_{\mathrm{C}}}{(2\pi)^{2}}n_{x}\mathcal{B}_{x}\beta_{x}^{2}C_{x}\sum_{i}\langle\alpha|\mathbb{F}_{i+}|\alpha\rangle
p_{\nu_{i}}E_{\nu_{i}}.$ (IV.4)

Electron capture is used or is planned to be used to measure the electron
neutrino mass Gastaldo:2014 ; Faverzani:2016 ; Gastaldo:2017 ; Ge:2022a ;
Ge:2022b .

###  IV.3 Neutrinoless double-$\beta$ decay

Experimental observation of neutrinoless double-$\beta$ decays
($0\nu\beta^{-}\beta^{-}$, $0\nu\beta^{+}\beta^{+}$), neutrinoless double
electron capture ($0\nu$2EC) or mixed decays $0\nu\beta^{-}$EC could prove the
non-conservation of the total lepton number and, accordingly, Majorana nature
of neutrino mass. These processes are also of interest for determining the
absolute scale and the hierarchy type of neutrino masses. In view of the value
of processes sensitive to the non-conservation of the total lepton number,
extensive literature is devoted to the physics of $0\nu\beta\beta$ decay,
$0\nu$2EC, and the corresponding models of the nuclear structure Suhonen:2007
; Simkovic:2011 ; Vergados:2012 ; Raduta:2015 ; Engel:2017 ; Ejiri:2019 ;
Blaum:2020 .

The $0\nu\beta^{-}\beta^{-}$, $0\nu\beta^{+}\beta^{+}$, $0\nu\beta^{+}$EC
decays and $0\nu$2EC are caused by the chirality flip of a massive Majorana
neutrino. The amplitudes of the $0\nu\beta^{+}\beta^{+}$, $0\nu\beta^{+}$EC
and $0\nu$2EC processes are proportional to the propagator projected onto the
left state of the emitted and the left state of the absorbed neutrino. The
projection of the propagator can be written as follows

$\langle\alpha|\Pi_{+}\hat{\mathbb{S}}_{F}(p)\Pi_{+}|\alpha\rangle=\sum_{i}\frac{1}{p^{2}-m_{i}^{2}}\langle\alpha|\mathbb{M}_{L}\mathbb{F}_{i+}|\alpha\rangle\Pi_{+},$
(IV.5)

where $|\alpha\rangle$ is the electron-flavor state of the neutrino in the
initial and final states.111In terms of the PMNS matrix (II.39), Eq. (IV.5)
reads
$\langle\alpha|\Pi_{+}\hat{\mathbb{S}}_{F}(p)\Pi_{+}|\alpha\rangle=\sum_{ij\beta}\frac{m_{j}}{p^{2}-m_{i}^{2}}U_{\alpha
j}^{\dagger}U_{\beta j}^{\dagger}U_{\beta i}U_{\alpha i}^{\dagger}\Pi_{+}.$ We
use the condition of completeness of mass eigenstates $|i\varsigma\rangle$ and
take into account that the transmitted momentum of neutrino has the order of
the Fermi momentum at saturation $p_{\mathrm{F}}=270$ MeV. Since this momentum
is much higher than the neutrino masses $m_{i}\lesssim 1$ eV, one can replace
$m_{i}^{2}$ by the average value of the squares of the neutrino masses,
$\langle m_{i}^{2}\rangle$. By performing the summation, one has

$\langle\alpha|\Pi_{+}\hat{\mathbb{S}}_{F}(p)\Pi_{+}|\alpha\rangle\approx\frac{1}{p^{2}-\langle
m_{i}^{2}\rangle}\langle\alpha|\mathbb{M}_{L}|\alpha\rangle\Pi_{+}.$ (IV.6)

The decay rate is determined by a single element of the mass matrix:

$d\Gamma\varpropto\left|\langle\alpha|\mathbb{M}_{L}|\alpha\rangle\right|^{2}=\left|\sum_{i}m_{i}\langle
i+|\alpha\rangle^{2}\right|^{2}.$ (IV.7)

Contributions to the double-$\beta$ processes of Majorana neutrinos with a
mass greater than $p_{\mathrm{F}}$ are also discussed Babic:2017 . For heavy
neutrinos, it is necessary to perform summation in Eq. (IV.5) with the exact
Frobenius covariants (II.27).

Equation (IV.5) shows that the $0\nu\beta\beta$, $0\nu\beta^{-}$EC and
$0\nu$2EC probabilities are expressed in terms of the mass matrix alone.

###  IV.4 Quasi-elastic neutrino scattering

As the next example, consider a neutrino scattered on a lepton:
$\nu_{i}+\ell_{i}^{-}\to\nu_{j}+\ell_{f}^{-}$. According to the scattering
theory, there is an asymptotic state $|i+\rangle\otimes|\ell_{i}^{-}\rangle$
of a neutrino and a lepton with masses $m_{i}$ and $\mu_{i}$ at infinity
$t\to-\infty$, and an asymptotic state $|j+\rangle\otimes|\ell_{f}^{-}\rangle$
of a neutrino and a lepton with masses $m_{j}$ and $\mu_{f}$ at infinity
$t\to+\infty$. Since the interaction is formulated in electroweak basis, the
amplitude $\mathcal{A}_{\alpha\beta}$ is calculated first for flavor
neutrinos, then converted to the basis of the diagonal in mass asymptotic
states:

$\mathcal{A}_{ji}=\langle
j+|\alpha\rangle\mathcal{A}_{\alpha\beta}\langle\beta|i+\rangle.$ (IV.8)

The summation is implied over the repeated indices. The cross section is
represented as

$d\sigma=\frac{1}{4\sqrt{(p_{i}p_{\ell^{-}_{i}})^{2}-m_{i}^{2}\mu_{i}^{2}}}\left|\mathcal{A}_{ji}\right|^{2}\frac{1}{(2\pi)^{2}}\Phi_{2}(\sqrt{s},m_{j},\mu_{f}),$
(IV.9)

where $p_{\ell^{-}_{i}}$ is the initial four-momentum of lepton,
$s=(p_{i}+p_{\ell^{-}_{i}})^{2}$, and $\Phi_{2}(\sqrt{s},m_{j},\mu_{f})$ is
the relativistic phase space of the final particles. The identity
$|\mathcal{A}_{ji}|^{2}=\mathrm{Tr}\left[\mathcal{A}|i+\rangle\langle
i+|\mathcal{A}^{{\dagger}}|j+\rangle\langle j+|\right]$ allows the cross
section for the non-polarized particles to be written in terms of the
Frobenius covariants (II.27):

$\displaystyle
d\sigma=\frac{1}{4\sqrt{(p_{i}p_{\ell^{-}_{i}})^{2}-m_{i}^{2}\mu_{i}^{2}}}\mathrm{Tr}\left[\mathcal{A}\mathbb{F}_{i+}\mathcal{A}^{{\dagger}}\mathbb{F}_{j+}\right]\frac{1}{(2\pi)^{2}}\Phi_{2}(\sqrt{s},m_{j},\mu_{f}).$
(IV.10)

The trace can be calculated in flavor basis. Explicit use of a basis diagonal
in neutrino masses is not mandatory. The corresponding expressions under the
trace sign contain only the neutrino mass matrix. These conclusions are valid
for both Majorana neutrinos and Dirac neutrinos.

After completing this paper, our attention was drawn to the papers
Bellandi:1997 ; Aquino:1999 ; Fogli:1999 ; Denton:2020 ; Denton:2022 , where
the problem of constructing the mixing matrix in terms of the mass matrix is
discussed, including the use of analogues of Frobenius covariants in a less
general framework. Equation (28) of the paper Fogli:1999  is reproduced from
Eqs. (III.8) and (III.25) with the help of perturbation theory in the limit of
small mean-field potential $|\mathbf{p}||V_{\alpha}|\ll m_{i}^{2}$ and
$|\mathbf{p}|\gg m_{i}$. In Eqs. (III.13) and (III.32) we give similar
expressions in the limit of $|\mathbf{p}||V_{\alpha}|\gg m_{i}^{2}$ and
$|\mathbf{p}|\gg m_{i}$, which is interesting for physics of newly born
neutron stars. In Ref. Denton:2022 , an interesting theorem is discussed on
the relashionship of the module of the mixing matrix elements with the
eigenvalues of the mass matrix and the eigenvalues of the diagonal minors of
the mass matrix. The Frobenius covariants, apparently, successfully solve the
problem of constructing the mixing matrix, including determining the phases of
matrix elements. In the simplest case of neutrinos in vacuum, Eqs. (II.27) and
(II.39) give, in particular,

$U_{\alpha i}U_{\beta i}^{*}=\langle\alpha|\mathbb{F}_{i+}|\beta\rangle.$
(IV.11)

The main message of our paper, on the other hand, is that in neutrino physics,
all physical characteristics of the processes are expressed through the
neutrino mass matrix without reference to the mixing matrix or assumptions
about the hierarchy of neutrino masses. The proposed procedure works for the
Majorana mass term, a complex symmetric matrix (II.7) specified by nine
parameters for three neutrino flavors. Of course, an observation of the
$0\nu\beta\beta$-decay or an assumption about the value of $M^{M}_{ee}$ is
necessary.

##  V Conclusion

In light of the lack of an explanation for the observed violation of $CP$
symmetry, the abundance of baryons in the universe, the presence of dark
matter, etc., the limitations of SM are obvious. The characteristics of
neutrinos, such as their nature (Majorana or Dirac), and the mass matrix with
a large number of parameters, can point the right path when looking for SM
generalizations. This paper demonstrates that the amplitudes of neutrino-
related processes can be constructed by methods, which do not require a mixing
matrix and instead allow explicit representations to be built purely in terms
of the neutrino mass matrix.

Sections II and III analyze in detail the involved processes of neutrino
oscillations in a vacuum and in a medium. Section IV demonstrates that the
probabilities of the most typical processes described by diagrams with
external neutrino lines are also expressed in terms of only the neutrino mass
matrix without reference to the mixing matrix. We discuss quasi-elastic
neutrino scattering, and single- and double-beta decays. Although the
scattering theory is formulated in terms of asymptotic states, the
transformation of scattering amplitudes into the basis of massive diagonal
states is not mandatory.

Compared to methods using a mixing matrix, the neutrino mass matrix can be
sensitive to alternative experimental data sets. Indirect determination of the
mass matrix through the mixing matrix involves loss of information, which
leads typically to an increase in statistical errors. The components of the
neutrino mass matrix can apparently be determined with reduced errors if the
measurement data are directly compared with the mass matrix. Using such a
fitting, it is potentially possible to gain access to the structure of the
mass matrix - Majorana or Dirac.

###### Acknowledgements.

The authors are grateful to Prof. E. Lisi for reading the manuscript and
valuable comments.

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