text:
\section{Motivation}

Among all the proposed mechanisms toward deeply understanding the true origin
of tiny masses of the three known neutrinos $\nu^{}_i$ (for $i = 1, 2, 3$),
whose flavor eigenstates are commonly denoted as $\nu^{}_\alpha$ (for
$\alpha = e, \mu, \tau$), the canonical seesaw mechanism~\cite{Minkowski:1977sc,
Yanagida:1979as,GellMann:1980vs,Glashow:1979nm,Mohapatra:1979ia} stands out as
being most economical and most natural. The simplicity of this mechanism lies
in two aspects: (a) it just takes into account the right-handed neutrino fields
$N^{}_{\alpha \rm R}$, the chiral counterparts of the left-handed neutrino fields
$\nu^{}_{\alpha \rm L}$ (for $\alpha = e, \mu, \tau$), which were originally
ignored from the particle content of the standard model (SM)~\cite{Weinberg:1967tq};
(b) it simply allows for lepton number violation or the Majorana nature of
massive neutrinos~\cite{Majorana:1937vz}, which is completely
harmless to the theoretical framework of the SM itself. The naturalness of this
mechanism is reflected in its attributing the small masses of $\nu^{}_i$
to the existence of three heavy Majorana neutrinos $N^{}_i$ (for $i = 1, 2, 3$),
whose masses are expected to be far above the fulcrum of the seesaw --- presumably
the electroweak symmetry breaking scale of the SM characterized by the vacuum
expectation value of the Higgs field. On the other hand,
the seesaw mechanism offers a big bonus to cosmology: the CP-violating and
out-of-equilibrium decays of heavy Majorana neutrinos may give rise to a net
lepton-antilepton asymmetry in the early Universe, and such a {\it leptogenesis}
mechanism~\cite{Fukugita:1986hr} can finally lead to {\it baryogenesis} as a
natural interpretation of the observed baryon-antibaryon asymmetry in today's
Universe~\cite{ParticleDataGroup:2022pth}. In this sense the seesaw
mechanism is the very {\it stone} that can kill two fundamental {\it birds}
in particle physics and cosmology.

Note that the seesaw mechanism is expected to take effect at a superhigh energy
scale $\Lambda$ which is essentially of the order of the heavy Majorana neutrino
masses. But the $\rm SU(2)^{}_{\rm L} \times U(1)^{}_{\rm Y}$ electroweak
gauge symmetry has been unbroken until the Higgs field develops a nonzero
vacuum expectation value $v$ of ${\cal O}\big(10^2\big)~{\rm GeV}$. In this
situation the three active neutrinos are actually impossible to acquire their
{\it true} masses of ${\cal O}(v^2/\Lambda)$ at the seesaw scale $\Lambda$ due to
the absence of a {\it real} fulcrum of the seesaw. On the other hand, thermal
leptogenesis can be realized via the lepton-number-violating decays of heavy
Majorana neutrinos into the leptonic and Higgs doublets at $\Lambda$. So we
are well motivated to ask a conceptually important question: how can the seesaw
mechanism {\it formally} survive with the unbroken electroweak gauge symmetry
and work together with the leptogenesis mechanism? If the answer to this question
is affirmative, we wonder whether the mass eigenstates of heavy Majorana neutrinos
associated with thermal leptogenesis are exactly the same as those associated
with the seesaw mechanism itself
\footnote{A mismatch of this kind has been observed and discussed in the seesaw
framework {\it after} spontaneous electroweak symmetry breaking and in an
{\it approximate} way (see, e.g., Refs.~\cite{Drewes:2013gca,Canetti:2012kh,
Drewes:2019byd,Chrzaszcz:2019inj,Klaric:2021cpi}). Here we shall take a new look
at it {\it before} electroweak symmetry breaking and in an {\it exact} way at
the tree level.}.
In case that there exists a mismatch between these two sets of mass bases, then
the question becomes how small this mismatch is likely to be.

To answer the above questions and clarify some conceptual ambiguities that have
never been taken seriously, we are going to study how to make the seesaw mechanism
formally work before spontaneous electroweak symmetry breaking. We show that an
exact seesaw relation between the light and heavy Majorana neutrinos can be
established far above the electroweak scale, and it becomes the realistic seesaw
relation after the Higgs field develops its vacuum expectation value.
In this way it is straightforward to elucidate the reason why there is a
mismatch between the mass eigenstates of heavy Majorana neutrinos associated
with thermal leptogenesis and those associated with the seesaw mechanism. With
the help of a full Euler-like parametrization of the flavor structure in the
seesaw framework, we illuminate such a mismatch in a more specific way.
The exact and explicit relations between the {\it original} and {\it derivational}
parameters of massive Majorana neutrinos are obtained as a by-product, and
they are expected to be useful in determining or constraining some of the
original seesaw parameters from the low-energy neutrino experiments.

\setcounter{equation}{0}
\section{A formal seesaw mechanism?}

\subsection{The leptonic Yukawa interactions}

Let us begin with the gauge-invariant leptonic Yukawa interactions and the
$\rm SU(2)^{}_{\rm L}$-singlet Majorana neutrino mass term of the canonical
seesaw mechanism at $\Lambda$
\footnote{Throughout this paper, our discussions are subject to the minimal
extension of the SM with three right-handed neutrino fields and lepton number
violation at {\it zero} temperature, so as to make our key point clear and
avoid possible complications (e.g., thermal corrections to the masses of
heavy Majorana neutrinos~\cite{Klaric:2021cpi}).}
\begin{eqnarray}
-{\cal L}^{}_{\Lambda} = \overline{\ell^{}_{\rm L}} \hspace{0.05cm} Y^{}_l
H \hspace{0.02cm} l^{}_{\rm R} + \overline{\ell^{}_{\rm L}} \hspace{0.05cm}
Y^{}_\nu \widetilde{H} N^{}_{\rm R}
+ \frac{1}{2} \hspace{0.05cm}
\overline{(N^{}_{\rm R})^c} \hspace{0.05cm} M^{}_{\rm R} N^{}_{\rm R}
+ {\rm h.c.} \; ,
\label{1}
\end{eqnarray}
where $\ell^{}_{\rm L} = \big(\begin{matrix} \nu^{}_{\rm L} & l^{}_{\rm L}
\end{matrix}\big)^T$ denotes the leptonic $\rm SU(2)^{}_{\rm L}$ doublet of
the SM with $\nu^{}_{\rm L} = \big(\begin{matrix} \nu^{}_{e \rm L} &
\nu^{}_{\mu \rm L} & \nu^{}_{\tau \rm L}\end{matrix}\big)^T$ and
$l^{}_{\rm L} = \big(\begin{matrix} l^{}_{e \rm L} & l^{}_{\mu \rm L} &
l^{}_{\tau \rm L}\end{matrix}\big)^T$ standing respectively for the column
vectors of the left-handed neutrino and charged lepton fields,
$\widetilde{H} \equiv {\rm i} \sigma^{}_2 H^*$ with
$H = \big(\begin{matrix} \phi^+ & \phi^0\end{matrix}\big)^T$ being the Higgs
doublet of the SM and $\sigma^{}_2$ being the second Pauli matrix,
$l^{}_{\rm R} = \big(\begin{matrix} l^{}_{e \rm R} & l^{}_{\mu \rm R} &
l^{}_{\tau \rm R}\end{matrix}\big)^T$ and $N^{}_{\rm R} = \big(\begin{matrix}
N^{}_{e \rm R} & N^{}_{\mu \rm R} & N^{}_{\tau \rm R}\end{matrix}\big)^T$
stand respectively for the column vectors of the right-handed charged lepton
and neutrino fields which are the $\rm SU(2)^{}_{\rm L}$
singlets, $(N^{}_{\rm R})^c \equiv {\cal C} \overline{N^{}_{\rm R}}^T$
with $\cal C$ being the charge-conjugation matrix and satisfying
${\cal C}^{-1} = {\cal C}^\dagger = {\cal C}^T = - {\cal C}$, $Y^{}_l$ and
$Y^{}_\nu$ represent the respective Yukawa coupling matrices of charged leptons
and neutrinos, and $M^{}_{\rm R}$ is the symmetric right-handed neutrino mass
matrix. In Eq.~(\ref{1}) the hypercharges of $\ell^{}_{\rm L}$, $l^{}_{\rm R}$,
$N^{}_{\rm R}$, $H$ and $\widetilde{H}$ are $-1/2$, $-1$, $0$, $+1/2$ and
$-1/2$, respectively.
Since $\overline{\nu^{}_{\rm L}} \hspace{0.05cm} Y^{}_\nu N^{}_{\rm R}$
is a Lorentz scalar and can be transformed into
\begin{eqnarray}
\overline{\nu^{}_{\rm L}} \hspace{0.05cm} Y^{}_\nu N^{}_{\rm R} = \left[
\overline{\nu^{}_{\rm L}} \hspace{0.05cm} Y^{}_\nu N^{}_{\rm R}\right]^T =
\overline{(N^{}_{\rm R})^c} \hspace{0.05cm} Y^T_\nu (\nu^{}_{\rm L})^c \; ,
\label{2}
\end{eqnarray}
where $(\nu^{}_{\rm L})^c \equiv {\cal C} \overline{\nu^{}_{\rm L}}^T$ is
the charge-conjugated counterpart of $\nu^{}_{\rm L}$, one may easily
rewrite Eq.~(\ref{1}) as
\begin{eqnarray}
-{\cal L}^{}_{\Lambda} \hspace{-0.2cm} & = & \hspace{-0.2cm}
\overline{l^{}_{\rm L}} \hspace{0.05cm} Y^{}_l \hspace{0.03cm}
l^{}_{\rm R} \phi^0 + \frac{1}{2} \hspace{0.05cm}
\overline{\big[\begin{matrix} \nu^{}_{\rm L} & (N^{}_{\rm R})^c\end{matrix}
\big]} \left(\begin{matrix} {\bf 0} & Y^{}_\nu \phi^{0*} \cr
Y^T_\nu \phi^{0*} & M^{}_{\rm R} \end{matrix}\right)
\left[\begin{matrix} (\nu^{}_{\rm L})^c \cr N^{}_{\rm R} \end{matrix}\right]
\hspace{0.5cm}
\nonumber \\
&& \hspace{-0.25cm} + \hspace{0.1cm}
\overline{\nu^{}_{\rm L}} \hspace{0.05cm} Y^{}_l \hspace{0.03cm} l^{}_{\rm R}
\phi^+ - \overline{l^{}_{\rm L}} \hspace{0.05cm} Y^{}_\nu N^{}_{\rm R} \phi^-
+ {\rm h.c.} \; .
\label{3}
\end{eqnarray}
This expression is highly nontrivial in the sense that it
clearly shows a direct correlation between the left- and right-handed neutrino
fields via their Yukawa couplings to the neutral component of the Higgs doublet
even though the $\rm SU(2)^{}_{\rm L} \times U(1)^{}_{\rm Y}$ gauge symmetry is
perfect at the seesaw scale $\Lambda$. In this situation the $3\times 3$
Yukawa coupling matrix $Y^{}_\nu$ can be regarded as a ``virtual" fulcrum of
the seesaw before spontaneous electroweak symmetry breaking.

Note that both the scalar field $\phi^0$ and its charge-conjugated counterpart
$\phi^{0 *}$ have the mass dimension and act like two complex numbers in
Eq.~(\ref{3}). But of course they possess the respective hypercharges $+1/2$
and $-1/2$ as $\phi^{\pm}$ do. After spontaneous symmetry breaking
$\phi^0$ and $\phi^{0 *}$ will acquire the same vacuum expectation value
$\langle \phi^0\rangle = \langle \phi^{0 *}\rangle = v/\sqrt{2}$ with
$v \simeq 246~{\rm GeV}$, together with $\langle \phi^-\rangle =
\langle \phi^+\rangle =0$, as in the SM. Then the formal seesaw will acquire
a real fulcrum which allows one to naturally attribute the smallness of
three active Majorana neutrino masses to the existence of three heavy
Majorana neutrinos, as can be seen later on.

\subsection{The leptogenesis-associated basis}

Now that all the SM particles are exactly massless in the early Universe when
the temperature is far above the electroweak scale, a realization of
thermal leptogenesis at $\Lambda \gg v$ only needs to calculate the
lepton-number-violating decays of heavy Majorana neutrinos into the
leptonic doublet and the Higgs doublet at the one-loop level by simply
starting from Eq.~(\ref{1}) instead of Eq.~(\ref{3}) (see, e.g.,
Refs.~\cite{Fukugita:1986hr,Luty:1992un,Covi:1996wh,Plumacher:1996kc,
Pilaftsis:1997jf}). In this case the column vector of the mass eigenstates of
three heavy Majorana neutrinos, denoted as
${\cal N}^\prime = \big(\begin{matrix} {\cal N}^{}_1 &
{\cal N}^{}_2 & {\cal N}^{}_3 \end{matrix}\big)^T$, can easily be obtained by
making the Autonne-Takagi transformation~\cite{Autonne1915,Takagi1924} as
follows:
\begin{eqnarray}
U^{\prime \dagger}_0 M^{}_{\rm R} U^{\prime *}_0 = {\cal D}^{}_{\cal N} \; , \quad
{\cal N}^\prime_{\rm R} = U^{\prime T}_0 N^{}_{\rm R} \; ,
\label{4}
\end{eqnarray}
where $U^\prime_0$ is a unitary matrix, and ${\cal D}^{}_{\cal N} \equiv
{\rm Diag}\big\{{\cal M}^{}_1 , {\cal M}^{}_2 , {\cal M}^{}_3 \big\}$ with
${\cal M}^{}_i$ being the masses of ${\cal N}^{}_i$ (for $i = 1, 2, 3$).
As a result, the Lagrangian ${\cal L}^{}_\Lambda$ in Eq.~(\ref{1}) becomes
\begin{eqnarray}
-{\cal L}^{}_{\Lambda} = \overline{\ell^{}_{\rm L}} \hspace{0.05cm} Y^{}_l
H \hspace{0.02cm} l^{}_{\rm R} + \overline{\ell^{}_{\rm L}} \hspace{0.05cm}
{\cal Y}^{}_\nu \widetilde{H} {\cal N}^{\prime}_{\rm R} + \frac{1}{2}
\hspace{0.05cm} \overline{({\cal N}^{\prime}_{\rm R})^c} \hspace{0.05cm}
{\cal D}^{}_{\cal N} {\cal N}^{\prime}_{\rm R}
+ {\rm h.c.} \; ,
\label{5}
\end{eqnarray}
where ${\cal Y}^{}_\nu \equiv Y^{}_\nu U^{\prime *}_0$ is defined for the sake
of simplicity. The rates of ${\cal N}^{}_i$ decaying into $\ell^{}_{\rm L}$ and
$H$ or their CP-conjugated states
are therefore determined by ${\cal M}^{}_i$ and ${\cal Y}^{}_\nu$, so are the
corresponding CP-violating asymmetries associated closely with thermal
leptogenesis~\cite{Luty:1992un,Covi:1996wh,Plumacher:1996kc,Pilaftsis:1997jf}
\footnote{Here we have used some {\it calligraphic} characters to denote the
relevant physical quantities in the basis where $M^{}_{\rm R}$ is diagonalized
by the unitary transformation made in Eq.~(\ref{4}). This basis is associated
with ${\cal N}^{}_i$ decays and thermal leptogenesis, and it is conceptually
different from the basis taken for the seesaw mechanism as can be seen below.}.
To be more specific, the flavor-dependent CP-violating asymmetries of
${\cal N}^{}_i$ decays are given by
\begin{eqnarray}
\varepsilon^{}_{i \alpha} \hspace{-0.2cm} & \equiv & \hspace{-0.2cm}
\frac{\Gamma\big({\cal N}^{}_i \to \ell^{}_\alpha + H\big)
- \Gamma\big({\cal N}^{}_i \to \overline{\ell^{}_\alpha} +
\overline{H}\big)}{\displaystyle \sum_\alpha \left[\Gamma\big({\cal N}^{}_i \to
\ell^{}_\alpha + H\big) + \Gamma\big({\cal N}^{}_i \to \overline{\ell^{}_\alpha}
+ \overline{H}\big)\right]}
\nonumber \\
\hspace{-0.2cm} & = & \hspace{-0.2cm}
\frac{1}{8\pi \big({\cal Y}^\dagger_\nu
{\cal Y}^{}_\nu\big)^{}_{ii}} \sum_{j \neq i} \left\{ {\rm Im}
\left[\big({\cal Y}^*_\nu\big)^{}_{\alpha i}
\big({\cal Y}^{}_\nu\big)^{}_{\alpha j}
\big({\cal Y}^\dagger_\nu {\cal Y}^{}_\nu\big)^{}_{ij} \xi(x^{}_{ji})
+ \big({\cal Y}^*_\nu\big)^{}_{\alpha i}
\big({\cal Y}^{}_\nu\big)^{}_{\alpha j} \big({\cal Y}^\dagger_\nu
{\cal Y}^{}_\nu\big)^*_{ij} \zeta(x^{}_{ji}) \right]\right\} \; , \hspace{0.4cm}
\label{6}
\end{eqnarray}
where the Latin and Greek subscripts run respectively over $(1, 2, 3)$
and $(e, \mu, \tau)$, $x^{}_{ji} \equiv {\cal M}^2_j/{\cal M}^2_i$ are defined,
$\xi(x^{}_{ji}) = \sqrt{x^{}_{ji}} \left\{1 + 1/\left(1 - x^{}_{ji}\right)
+ \left(1 + x^{}_{ji}\right) \ln \left[x^{}_{ji} / \left(1 + x^{}_{ji}\right)
\right] \right\}$ and $\zeta(x^{}_{ji}) = 1/\left(1 - x^{}_{ji}\right)$ are
the loop functions. A net lepton-antilepton asymmetry can therefore result
from $\varepsilon^{}_{i \alpha}$ in the early Universe, and later on it
can be partly converted into a net baryon-antibaryon asymmetry
via the sphaleron interactions (see Ref.~\cite{DiBari:2021fhs} for a recent
review).

At this point it is worth remarking that the right-handed neutrino fields
$N^{}_{\alpha \rm R}$ have zero weak isospin and hypercharge, and hence
they have no coupling with the charged and neutral gauge bosons of the SM.
As a consequence, the mass eigenstates of heavy Majorana neutrinos obtained from
Eq.~(\ref{4}) do not participate in the weak charged-current interactions
of the SM,
\begin{eqnarray}
-{\cal L}^{}_{\rm cc} = \frac{g}{2} \hspace{0.05cm}
\overline{\ell^{}_{\rm L}} \hspace{0.05cm} \gamma^\mu \left(\sigma^{}_1
W^1_\mu + \sigma^{}_2 W^2_\mu\right) \ell^{}_{\rm L}
= \frac{g}{\sqrt 2} \hspace{0.05cm} \overline{l^{}_{\rm L}} \hspace{0.05cm}
\gamma^\mu \hspace{0.03cm} W^-_\mu \nu^{}_{\rm L} + {\rm h.c.} \; ,
\label{7}
\end{eqnarray}
where $g$ denotes the weak gauge coupling constant, $\sigma^{}_{1,2}$ represent
the first and second Pauli matrices, $W^\mu_{1,2}$ are two of the original
$\rm SU(2)^{}_{\rm L}$ gauge fields, and $W^\pm_\mu \equiv
\left(W^1_\mu \mp {\rm i} \hspace{0.03cm} W^2_\mu\right)/\sqrt{2}$ stand for
the fields of the physical charged gauge bosons $W^\pm$. But in the seesaw
framework we shall see that the expression of ${\cal L}^{}_{\rm cc}$ in
Eq.~(\ref{7}) will get modified, and the corresponding mass eigenstates of
three heavy Majorana neutrinos can definitely take part in the weak
charged-current interactions.

\subsection{The seesaw-associated basis}

We proceed to show that the canonical seesaw mechanism can ``formally" work before
spontaneous electroweak symmetry breaking but the corresponding mass eigenstates
of three heavy Majorana neutrinos are not exactly the same as ${\cal N}^{}_i$
(for $i = 1, 2, 3$) obtained above for the neutrino decays and thermal leptogenesis.
To clarify this important point, let us diagonalize the symmetric $6\times 6$
matrix in Eq.~(\ref{3}) in the following Autonne-Takagi way:
\begin{eqnarray}
\mathbb{U}^\dagger \left ( \begin{matrix} {\bf 0} & Y^{}_\nu \phi^{0*}
\cr Y^T_\nu \phi^{0*} & M^{}_{\rm R} \cr \end{matrix} \right )
\mathbb{U}^* = \left( \begin{matrix} D^{}_\nu & {\bf 0} \cr {\bf 0} &
D^{}_N \cr \end{matrix} \right) \; ,
\label{8}
\end{eqnarray}
where $\mathbb{U}$ is a $6\times 6$ unitary matrix, and the
diagonal and real matrices $D^{}_\nu$ and $D^{}_N$ are defined as
$D^{}_\nu \equiv {\rm Diag}\big\{m^{}_1, m^{}_2, m^{}_3\big\}$ and
$D^{}_N \equiv {\rm Diag}\big\{M^{}_1, M^{}_2, M^{}_3 \big\}$. Meanwhile,
the column vectors of left- and right-handed neutrino fields
$\big[\begin{matrix} \nu^{}_{\rm L} & (N^{}_{\rm R})^c\end{matrix}
\big]^T$ and $\big[\begin{matrix} (\nu^{}_{\rm L})^c & N^{}_{\rm R}
\end{matrix}\big]^T$ undergo the transformations
\begin{eqnarray}
\left[\begin{matrix} \nu^{}_{\rm L} \cr (N^{}_{\rm R})^c\end{matrix}
\right] \longrightarrow \hspace{0.1cm} \mathbb{U}^\dagger
\left[\begin{matrix} \nu^{}_{\rm L} \cr (N^{}_{\rm R})^c\end{matrix}
\right] \; , \quad
\left[\begin{matrix} (\nu^{}_{\rm L})^c \cr N^{}_{\rm R} \end{matrix}\right]
\longrightarrow \hspace{0.1cm} \mathbb{U}^T
\left[\begin{matrix} (\nu^{}_{\rm L})^c \cr N^{}_{\rm R} \end{matrix}\right] \; ,
\label{9}
\end{eqnarray}
such that the Lagrangian ${\cal L}^{}_\Lambda$ in Eq.~(\ref{3}) keeps unchanged
and thus its {\it overall} gauge symmetry is unbroken. Now that $Y^{}_\nu$
is dimensionless and $\phi^0$ has the same mass dimension as $M^{}_{\rm R}$, one
may argue that $m^{}_i$ should be the ``working" or ``virtual" mass parameters of
three light Majorana neutrinos as the electroweak gauge symmetry is unbroken at the
seesaw scale $\Lambda$. In comparison, $M^{}_i$ are essentially the true masses of
three heavy Majorana neutrinos in the existence of the $\phi^{0 (*)}$-mediated
neutrino Yukawa interactions. Along this line of thought, we find that it is
useful to decompose $\mathbb{U}$ into the product of three matrices,
\begin{eqnarray}
\mathbb{U} = \left( \begin{matrix}
I & {\bf 0} \cr {\bf 0} & U^{\prime}_0 \cr \end{matrix} \right)
\left( \begin{matrix} A & R \cr S & B \cr \end{matrix} \right)
\left( \begin{matrix} U^{}_0 & {\bf 0} \cr {\bf 0} & I \cr
\end{matrix} \right) \; ,
\label{10}
\end{eqnarray}
where the $3\times 3$ unitary matrix $U^\prime_0$ has been defined in Eq.~(\ref{4})
to primarily describe flavor mixing in the sterile (heavy) neutrino sector,
$U^{}_0$ denotes the other $3\times 3$ unitary matrix that is mainly responsible
for flavor mixing in the active (light) neutrino sector, while the $3\times 3$
matrices $A$, $B$, $R$ and $S$ signify the interplay between these two
sectors~\cite{Xing:2007zj,Xing:2011ur,Xing:2020ijf}. The unitarity of $\mathbb{U}$
assures
\begin{eqnarray}
A A^\dagger + R R^\dagger \hspace{-0.2cm} & = & \hspace{-0.2cm}
B B^\dagger + S S^\dagger = I \; ,
\nonumber \\
A S^\dagger + R B^\dagger \hspace{-0.2cm} & = & \hspace{-0.2cm}
A^\dagger R + S^\dagger B = {\bf 0} \; ,
\nonumber \\
A^\dagger A + S^\dagger S \hspace{-0.2cm} & = & \hspace{-0.2cm}
B^\dagger B + R^\dagger R = I \; .
\hspace{1.cm}
\label{11}
\end{eqnarray}
On the other hand, the arbitrary charged-lepton Yukawa coupling matrix
$Y^{}_l$ in Eq.~(\ref{3}) can be diagonalized by a bi-unitary transformation:
\begin{eqnarray}
U^{\dagger}_l \left(Y^{}_l \hspace{0.03cm} \phi^0\right) V^{}_l = D^{}_l \; ,
\quad l^\prime_{\rm L} = U^{\dagger}_l \hspace{0.02cm} l^{}_{\rm L} \; ,
\quad l^\prime_{\rm R} = V^{}_l \hspace{0.03cm} l^{}_{\rm R} \; ,
\label{12}
\end{eqnarray}
where $U^{}_l$ and $V^{}_l$ are unitary, $D^{}_l \equiv {\rm Diag}\big\{
m^{}_e , m^{}_\mu , m^{}_\tau\big\}$ stands for the ``working" or ``virtual"
masses of three charged leptons {\it before} spontaneous electroweak symmetry
breaking
\footnote{Note that the scalar field $\phi^0$ in Eq.~(\ref{12}) carries a
hypercharge, and hence $D^{}_l$ cannot be simply understood as a diagonal
``mass" matrix. The physical meaning of $D^{}_l$ is actually vague in our
calculations which are mathematically exact and clear, so is the physical
meaning of $D^{}_\nu$ in Eq.~(\ref{8}). But this vagueness will automatically
disappear after spontaneous electroweak symmetry breaking, as can be
subsequently seen.},
and $l^\prime = \big(\begin{matrix}
e & \mu & \tau \end{matrix}\big)^T$ is defined as the column vector of the
mass eigenstates of three charged leptons versus the column vector of their
flavor eigenstates $l = \big(\begin{matrix} l^{}_e & l^{}_\mu & l^{}_\tau
\end{matrix}\big)^T$. Substituting Eqs.~(\ref{8})---(\ref{10}) and (\ref{12})
into Eq.~(\ref{3}), we immediately arrive at
\begin{eqnarray}
-{\cal L}^{}_{\Lambda} = \overline{l^{\prime}_{\rm L}} \hspace{0.05cm} D^{}_l
\hspace{0.03cm} l^{\prime}_{\rm R} +
\frac{1}{2} \hspace{0.05cm} \overline{\nu^{\prime}_{\rm L}}
\hspace{0.05cm} D^{}_\nu (\nu^\prime_{\rm L})^c +
\frac{1}{2} \hspace{0.05cm} \overline{(N^{\prime}_{\rm R})^c} \hspace{0.05cm}
D^{}_N N^{\prime}_{\rm R} +
\overline{\nu^{}_{\rm L}} \hspace{0.05cm} Y^{}_l \hspace{0.03cm} l^{}_{\rm R}
\phi^+ - \overline{l^{}_{\rm L}} \hspace{0.05cm}
Y^{}_\nu N^{}_{\rm R} \phi^- + {\rm h.c.} \; ,
\label{13}
\end{eqnarray}
where $\nu^\prime = \big(\begin{matrix} \nu^{}_{1} & \nu^{}_{2} & \nu^{}_{3}
\end{matrix}\big)^T$ denotes the column vector of the {\it working} mass
eigenstates of three light Majorana neutrinos far above the electroweak
scale, and $N^\prime = \big(\begin{matrix} N^{}_{1} & N^{}_{2}
& N^{}_{3}\end{matrix}\big)^T$ stands for the column vectors of the mass
eigenstates of three heavy Majorana neutrinos relevant to the seesaw mechanism
at $\Lambda \gg v$. In this case the flavor eigenstates $\nu^{}_{\rm L}$ and
$N^{}_{\rm R}$ can be expressed in terms of the mass eigenstates
$\nu^\prime_{\rm L}$ and $N^\prime_{\rm R}$ or their charge-conjugated states
as follows:
\begin{eqnarray}
\nu^{}_{\rm L} = U \nu^\prime_{\rm L} + R \hspace{0.03cm} (N^\prime_{\rm R})^c \; ,
\quad
N^{}_{\rm R} = S^{\prime *} (\nu^\prime_{\rm L})^c
+ U^{\prime *} N^\prime_{\rm R} \; ,
\label{14}
\end{eqnarray}
where $U \equiv A U^{}_0$, $U^\prime \equiv U^\prime_0 B$ and
$S^\prime \equiv U^\prime_0 S U^{}_0$ are defined. Taking account of the Majorana
property of $\nu^{}_i$ and $N^{}_i$ (i.e., $\nu^c_i = \nu^{}_i$ and
$N^c_i = N^{}_i$~\cite{Majorana:1937vz} for $i = 1, 2, 3$), one simply obtains
$(N^\prime_{\rm R})^c = (N^{\prime c})^{}_{\rm L} = N^\prime_{\rm L}$
and $(\nu^\prime_{\rm L})^c = (\nu^{\prime c})^{}_{\rm R} = \nu^\prime_{\rm R}$.
One may then substitute the expression of $l^{}_{\rm L}$ in Eq.~(\ref{12}) and
that of $\nu^{}_{\rm L}$ in Eq. (\ref{14}) into the standard form of
${\cal L}^{}_{\rm cc}$ in Eq.~(\ref{7}) and get at
\begin{eqnarray}
-{\cal L}^{}_{\rm cc} = \frac{g}{\sqrt{2}} \hspace{0.1cm}
\overline{\big(\begin{matrix} e & \mu & \tau\end{matrix}\big)^{}_{\rm L}}
\hspace{0.1cm} \gamma^\mu \left[ U^{}_{\rm PMNS} \left( \begin{matrix} \nu^{}_{1}
\cr \nu^{}_{2} \cr \nu^{}_{3} \cr\end{matrix} \right)^{}_{\hspace{-0.08cm} \rm L}
+ R^{}_{\rm PMNS} \left(\begin{matrix} N^{}_{1} \cr N^{}_{2} \cr N^{}_{3}
\cr\end{matrix}\right)^{}_{\hspace{-0.08cm} \rm L} \hspace{0.05cm} \right]
W^-_\mu + {\rm h.c.} \; ,
\label{15}
\end{eqnarray}
where $U^{}_{\rm PMNS} = U^\dagger_l U$ is just the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix~\cite{Pontecorvo:1957cp,Maki:1962mu,Pontecorvo:1967fh} used to
describe the flavor oscillations of three active neutrinos, and
$R^{}_{\rm PMNS} = U^\dagger_l R$ is an analogue of $U^{}_{\rm PMNS}$ in the
seesaw mechanism which characterizes the strengths of weak charged-current
interactions for three heavy Majorana neutrinos.

Without loss of generality, one may choose a convenient flavor basis in which
the mass eigenstates of three charged leptons are identified with their
corresponding flavor eigenstates (i.e., $l^{}_{\rm L} = l^\prime_{\rm L}$,
or equivalently $U^{}_l = I$). In this case we are simply left with
$U^{}_{\rm PMNS} = U$ and $R^{}_{\rm PMNS} = R$, namely the effects of
lepton flavor mixing originate purely from the active and sterile Majorana
neutrino sectors and from the interplay between these two sectors. We
shall take advantage of this flavor basis in the following discussions
unless otherwise specified.

\subsection{Mismatch between the two bases}

Before discussing a mismatch between the mass eigenstates of heavy
Majorana neutrinos associated with thermal leptogenesis and those associated
with the seesaw mechanism, let us take a look at the flavor structures of
active and sterile neutrinos in the case that the electroweak gauge symmetry
is unbroken at $\Lambda$. First of all, a combination of Eqs.~(\ref{8}) and
(\ref{10}) allows us to immediately derive the exact seesaw relation between
the working masses of three light Majorana neutrinos and the real masses
of three heavy Majorana neutrinos:
\begin{eqnarray}
U D^{}_\nu U^T + R D^{}_N R^T = {\bf 0} \; ,
\label{16}
\end{eqnarray}
in which $U$ and $R$ are also correlated with each other via the unitarity
condition $U U^\dagger + R R^\dagger = I$. Note that $U = A U^{}_0$ holds,
where the unitary matrix $U^{}_0$ is primarily responsible for flavor mixing
of the three active neutrinos. So we find it useful to rewrite
Eq.~(\ref{16}) as
\begin{eqnarray}
U^{}_0 D^{}_\nu U^T_0 =  \left({\rm i} \hspace{0.03cm} A^{-1} R\right) D^{}_N
\left({\rm i} \hspace{0.03cm} A^{-1} R\right)^T \; ,
\label{17}
\end{eqnarray}
whose left- and right-hand sides are composed of the {\it derivational} and
{\it original} seesaw parameters, respectively. This point will become more
obvious when a complete Euler-like parametrization of the $6\times 6$ unitary
matrix $\mathbb{U}$ in Eq.~(\ref{10}) is adopted, as can be seen in
section~\ref{section 3}. Needless to say, the active-sterile flavor mixing
matrix $R$ essentially plays the role of the neutrino Yukawa coupling matrix
$Y^{}_\nu$ in the canonical seesaw framework,
\begin{eqnarray}
Y^{}_\nu \phi^{0 *} = R D^{}_N \left[ I - \left(B^{-1} S A^{-1} R\right)^T
\right] U^{\prime T} \; ;
\label{18}
\end{eqnarray}
and the right-handed Majorana neutrino mass matrix $M^{}_{\rm R}$ can be
reconstructed into the form
\begin{eqnarray}
M^{}_{\rm R} = U^\prime \left[ D^{}_N - \left(B^{-1} S A^{-1} R\right)
D^{}_N \left(B^{-1} S A^{-1} R\right)^T\right] U^{\prime T} \; .
\label{19}
\end{eqnarray}
Note that all the quantities in Eqs.~(\ref{18}) and (\ref{19}) belong to the
{\it original} seesaw parameters in the sense that they have nothing to do
with $D^{}_\nu$ and $U^{}_0$ --- the working masses and the primary flavor
mixing matrix of three light Majorana neutrinos which are {\it derived} from
the seesaw mechanism.

Now we turn to an unavoidable mismatch between the mass eigenstates of three
heavy Majorana neutrinos associated with the seesaw and leptogenesis mechanisms.
Eq.~(\ref{14}) tells us that the mass eigenstates $N^{\prime}_{\rm R}$ in
the seesaw basis can be expressed as
\begin{eqnarray}
N^\prime_{\rm R} = (U^{\prime *})^{-1} \left[ N^{}_{\rm R} - S^{\prime *}
(\nu^\prime_{\rm L})^c\right] =
(B^*)^{-1} \left[ {\cal N}^\prime_{\rm R} - U^{\prime T}_0 S^{\prime *}
(\nu^\prime_{\rm L})^c\right] \; ,
\label{20}
\end{eqnarray}
where Eq.~(\ref{4}) has been used to link $N^\prime_{\rm R}$ to
${\cal N}^\prime_{\rm R}$. To be more explicit, Eq.~(\ref{20}) means
\begin{eqnarray}
\left(\begin{matrix} N^{}_1 \cr N^{}_2 \cr N^{}_3 \end{matrix}\right) =
(B^*)^{-1} \left[ \left(\begin{matrix} {\cal N}^{}_1 \cr {\cal N}^{}_2
\cr {\cal N}^{}_3 \end{matrix}\right) - U^{\prime T}_0 S^{\prime *}
\left( \begin{matrix} \nu^{}_{1} \cr \nu^{}_{2} \cr \nu^{}_{3} \cr\end{matrix}
\right)\right] \; ,
\label{21}
\end{eqnarray}
from which the differences between $N^{}_i$ in the seesaw basis and
${\cal N}^{}_i$ (for $i = 1, 2, 3$) in the thermal leptogenesis basis can be
clearly seen. Similarly, a combination of Eqs.~(\ref{4}) and (\ref{19})
leads us to
\begin{eqnarray}
{\cal D}^{}_{\cal N} = B \left[ D^{}_N - \left(B^{-1} S A^{-1} R\right)
D^{}_N \left(B^{-1} S A^{-1} R\right)^T\right] B^{T} \; ,
\label{22}
\end{eqnarray}
from which one may easily see the difference between ${\cal D}^{}_N$ and
$D^{}_N$. Although $N^\prime_{\rm R}$ (or $D^{}_N$) and ${\cal N}^\prime_{\rm R}$
(or ${\cal D}^{}_{\cal N}$) would exactly coincide with each other if the Yukawa
coupling matrix $Y^{}_\nu$ (or equivalently, $R$ or $S$) were switched off, such
a coincidence would make no sense because both the seesaw and leptogenesis
mechanisms would fail in this special case. In the presence of the neutrino Yukawa
interactions, thermal leptogenesis may take effect via the CP-violating and
out-of-equilibrium decays of heavy Majorana neutrinos into the leptonic and
Higgs doublets, while the seesaw mechanism can ``formally" work with the help
of an interplay between the active and sterile neutrino fields coupled only to
the neutral component of the Higgs doublet. That is the key reason why there
is an inevitable mismatch between the seesaw- and leptogenesis-associated bases
for heavy Majorana neutrinos before spontaneous electroweak symmetry breaking.

\subsection{After gauge symmetry breaking}

So far we have made some proper transformations of the charged lepton
and neutrino fields in the flavor space to obtain their respective working or
true mass eigenstates. All such unitary flavor basis transformations are
completely reversible, and hence they do not affect the gauge invariance of
${\cal L}^{}_\Lambda$ at the seesaw scale. As already shown in Eqs.~(\ref{20})
and (\ref{22}), a seeable mismatch between $N^\prime_{\rm R}$ and
${\cal N}^\prime_{\rm R}$ or between $D^{}_N$ and ${\cal D}^{}_N$ results
from the fact that the working seesaw mechanism itself is
only associated with the neutral component of the Higgs doublet while the
heavy Majorana neutrino decays and thermal leptogenesis at the seesaw scale
$\Lambda$ are associated with the whole Higgs doublet. This unavoidable
mismatch deserves to be conceptually clarified as we have done, because it
is an intrinsic issue of the seesaw and leptogenesis mechanisms.

It is now straightforward to prove that the {\it formal} seesaw mechanism
far above the electroweak scale will become {\it real} after the Higgs
potential of the SM is minimized at
$\langle H\rangle \equiv \langle 0|H|0\rangle = v/\sqrt{2}$ with a
special direction characterized by $\langle \phi^\pm\rangle = 0$ and
$\langle \phi^0\rangle = v/\sqrt{2}$, by which the electroweak gauge
symmetry is spontaneously broken and thus all the particles coupled to the
Higgs field acquire their nonzero masses. In this case the Lagrangian
in Eq.~(\ref{3}) can be simplified to a more popular form,
\begin{eqnarray}
-{\cal L}^{\prime}_{\Lambda} = \overline{l^{}_{\rm L}} \hspace{0.05cm}
M^{}_l \hspace{0.03cm} l^{}_{\rm R} + \frac{1}{2} \hspace{0.05cm}
\overline{\big[\begin{matrix} \nu^{}_{\rm L} & (N^{}_{\rm R})^c\end{matrix}
\big]} \left(\begin{matrix} {\bf 0} & M^{}_{\rm D} \cr
M^T_{\rm D} & M^{}_{\rm R} \end{matrix}\right)
\left[\begin{matrix} (\nu^{}_{\rm L})^c \cr N^{}_{\rm R} \end{matrix}\right]
+ {\rm h.c.} \; ,
\label{23}
\end{eqnarray}
where $M^{}_l \equiv Y^{}_l \langle \phi^0\rangle = Y^{}_l v/\sqrt{2}$ denotes
the charged lepton mass matrix, and $M^{}_{\rm D} \equiv Y^{}_\nu \langle
\phi^0\rangle = Y^{}_\nu v/\sqrt{2}$ is usually referred to as the Dirac
neutrino mass matrix. The expression of $M^{}_{\rm D}$ in terms of the seesaw
parameters can be directly read off from Eq.~(\ref{18}), namely
\begin{eqnarray}
M^{}_{\rm D} = R D^{}_N \left[ I - \left(B^{-1} S A^{-1} R\right)^T
\right] U^{\prime T} \; .
\label{24}
\end{eqnarray}
We find that the exact seesaw formula obtained in Eq.~(\ref{16}) and the
analytical results obtained in Eqs.~(\ref{19})---(\ref{22}) formally keep
unchanged after spontaneous gauge symmetry breaking, but they are now
subject to the electroweak scale. In other words, the electroweak symmetry
breaking itself does not really affect the flavor structures of the seesaw
mechanism. This observation implies that it is possible to determine or
constrain some of the original seesaw-associated flavor parameters in some
low-energy neutrino experiments, after the radiative corrections to such
parameters are properly taken into account with the help of the relevant
renormaliztion-group equations (RGEs) between a superhigh seesaw scale and the
electroweak scale~\cite{Ohlsson:2013xva}.

Note that the exact seesaw formula obtained in Eq.~(\ref{16})
can be simplified to the more popular form in the leading-order approximations
of Eqs.~(\ref{19}) and (\ref{24}).
That is, $M^{}_{\rm R} \simeq U^\prime D^{}_N U^{\prime T}$
and $M^{}_{\rm D} \simeq R D^{}_N U^{\prime T}$, so the {\it effective}
mass matrix for three active Majorana neutrinos is given by
\begin{eqnarray}
M^{}_\nu \equiv U^{}_0 D^{}_\nu U^T_0 \simeq -R D^{}_N R^T \simeq
-M^{}_{\rm D} M^{-1}_{\rm R} M^{T}_{\rm D} \; ,
\label{25}
\end{eqnarray}
where $A \simeq B \simeq I$ has been assumed (i.e., $U \simeq U^{}_0$ holds
in the neglect of the non-unitary effects characterized by $A \neq I$). In
this approximation the effective Majorana mass term for three
active neutrinos at low energies turns out to be
\begin{eqnarray}
-{\cal L}^{}_\nu = \frac{1}{2} \hspace{0.05cm} \overline{\nu^{}_{\rm L}}
\hspace{0.07cm} M^{}_\nu (\nu^{}_{\rm L})^c + {\rm h.c.} =
\frac{1}{2} \hspace{0.05cm} \overline{\nu^{\prime}_{\rm L}}
\hspace{0.05cm} D^{}_\nu (\nu^\prime_{\rm L})^c + {\rm h.c.} \; ,
\label{26}
\end{eqnarray}
where the column vector of the light neutrino mass eigenstates
$\nu^\prime_{\rm L}$ has already been defined below Eq.~(\ref{13}), and
the physical meaning of $D^{}_\nu$ as the diagonal Majorana neutrino mass
matrix becomes definite and obvious.

\section{How small is the mismatch?}
\label{section 3}

\subsection{An Euler-like parametrization}

To clearly see how small the difference between $N^\prime_{\rm R}$ (or $D^{}_N$)
and ${\cal N}^\prime_{\rm R}$ (or ${\cal D}^{}_{\cal N}$) is expected to be, let
us follow Refs.~\cite{Xing:2007zj,Xing:2011ur,Xing:2020ijf} to make an Euler-like
parametrization of the $6\times 6$ unitary matrix $\mathbb{U}$ in Eq.~(\ref{10}).
First of all we introduce fifteen $6\times 6$ Euler-like unitary
matrices of the form $O^{}_{ij}$ (for $1 \leq i < j \leq 6$): its $(i, i)$ and
$(j, j)$ entries are both identical to $c^{}_{ij} \equiv \cos\theta^{}_{ij}$
with $\theta^{}_{ij}$ being a flavor mixing angle and lying in the first quadrant,
its other four diagonal elements are all equal to one, its $(i, j)$ and $(j, i)$
entries are given respectively by $\hat{s}^{*}_{ij} \equiv e^{-{\rm i}\delta^{}_{ij}}
\sin\theta^{}_{ij}$ and $-\hat{s}^{}_{ij} \equiv -e^{{\rm i}\delta^{}_{ij}}
\sin\theta^{}_{ij}$ with $\delta^{}_{ij}$ being a CP-violating phase, and its
other off-diagonal elements are all equal to zero. These matrices are then grouped
in the following way to respectively describe the {\it active} flavor sector,
the {\it sterile} flavor sector and the {\it interplay} between these two sectors:
\begin{eqnarray}
\left( \begin{matrix} U^{}_0 & 0 \cr 0 & I \cr \end{matrix} \right)
\hspace{-0.2cm} & = & \hspace{-0.2cm} O^{}_{23} O^{}_{13} O^{}_{12} \; ,
\nonumber \\
\left( \begin{matrix} I & 0 \cr 0 & U^{\prime}_0 \cr \end{matrix} \right)
\hspace{-0.2cm} & = & \hspace{-0.2cm} O^{}_{56} O^{}_{46} O^{}_{45} \; ,
\nonumber \\
\left( \begin{matrix} A & R \cr S & B \cr \end{matrix} \right)
\hspace{-0.2cm} & = & \hspace{-0.2cm} O^{}_{36} O^{}_{26} O^{}_{16}
O^{}_{35} O^{}_{25} O^{}_{15} O^{}_{34} O^{}_{24} O^{}_{14} \; , \hspace{0.7cm}
\label{27}
\end{eqnarray}
where the pattern of $U^{}_0$ is quite similar to the standard parametrization
of a unitary PMNS matrix as advocated by the Particle Data
Group~\cite{ParticleDataGroup:2022pth}
\footnote{When $U^{}_0$ is applied to the phenomenology of neutrino physics
in the basis of $U^{}_l = I$, it is the phase parameter
$\delta \equiv \delta^{}_{13} - \delta^{}_{12} - \delta^{}_{23}$ that
characterizes the strength of CP violation in neutrino oscillations.},
\begin{eqnarray}
U^{}_0 \hspace{-0.2cm} & = & \hspace{-0.2cm}
\left( \begin{matrix} c^{}_{12} c^{}_{13} & \hat{s}^*_{12}
c^{}_{13} & \hat{s}^*_{13} \cr
-\hat{s}^{}_{12} c^{}_{23} -
c^{}_{12} \hat{s}^{}_{13} \hat{s}^*_{23} & c^{}_{12} c^{}_{23} -
\hat{s}^*_{12} \hat{s}^{}_{13} \hat{s}^*_{23} & c^{}_{13}
\hat{s}^*_{23} \cr
\hat{s}^{}_{12} \hat{s}^{}_{23} - c^{}_{12}
\hat{s}^{}_{13} c^{}_{23} & -c^{}_{12} \hat{s}^{}_{23} -
\hat{s}^*_{12} \hat{s}^{}_{13} c^{}_{23} & c^{}_{13} c^{}_{23}
\cr \end{matrix} \right) \; , \hspace{0.4cm}
\label{28}
\end{eqnarray}
and the expression of $U^\prime_0$ can be directly read off from that of $U^{}_0$
with the subscript replacements $12 \leftrightarrow 45$, $13 \leftrightarrow 46$
and $23 \leftrightarrow 56$ for the three rotation angles and three
CP-violating phases. The explicit expressions of $A$, $B$, $R$ and $S$ in
terms of $c^{}_{ij}$ and $\hat{s}^{}_{ij}$ (for $i = 1, 2, 3$ and
$j = 4, 5, 6$) are rather lengthy, and hence they are listed in
Eqs.~(\ref{A.1}) and (\ref{A.2}) in Appendix~\ref{A} for the same of simplicity.
Among the four active-sterile flavor mixing matrices, only $A$ and $R$ affect
the physical processes in which the light and heavy Majorana neutrinos take
part, as can be seen from Eq.~(\ref{15}). As both $U = A U^{}_0$ and $R$
appear in ${\cal L}^{}_{\rm cc}$ in the chosen flavor basis (i.e., $U^{}_l = I$),
three of the nice CP-violating phases (or their combinations) of $A$
and $R$ can always be rotated away by properly redefining the phases of
three charged lepton fields~\cite{Endoh:2000hc,Branco:2001pq}.

The PMNS matrix $U$ is obviously non-unitary because of $UU^\dagger =
A A^\dagger = I - RR^\dagger \neq I$, but its deviation from exact unitarity
(i.e., from $U^{}_0$) is found to be very small. A detailed and careful
analysis of currently available electroweak precision measurements and
neutrino oscillation data has put a stringent constraint on the non-unitarity
of $U$ --- the latter is below or far below
${\cal O}(10^{-2})$~\cite{Antusch:2006vwa,
Antusch:2009gn,Blennow:2016jkn,Hu:2020oba,Wang:2021rsi}.
This result implies that the deviation of $A A^\dagger$ from $I$ ought to
be smaller than ${\cal O}(10^{-2})$, and thus the nine active-sterile
flavor mixing angles in $R$ should be smaller than ${\cal O}(10^{-1})$. The
advantage of such a phenomenological observation is that $U \simeq U^{}_0$
can be a quite reliable approximation in most cases, but its disadvantage
is that an experimental exploration of the seesaw-induced non-unitary effects
of $U$ at low energies will be rather challenging.

\subsection{Smallness of the mismatch}

Eq.~(\ref{20}) tells us that a difference between the mass eigenstates of
three heavy Majorana neutrinos associated with the seesaw mechanism
(i.e., $N^\prime_{\rm R}$) and those associated with thermal leptogenesis
(i.e., ${\cal N}^\prime_{\rm R}$) is mainly characterized by the deviation
of $(B^*)^{-1}$ from the identity matrix $I$. With the help of Eq.~(\ref{A.2}),
we arrive at
\begin{eqnarray}
(B^*)^{-1} \hspace{-0.2cm} & = & \hspace{-0.2cm}
\left( \begin{matrix} c^{-1}_{14} c^{-1}_{24} c^{-1}_{34} & 0 & 0
\cr \vspace{-0.45cm} \cr
\begin{array}{l}
+ \hat{t}^{}_{14} c^{-1}_{24} c^{-1}_{34} \hat{t}^{*}_{15} +
\hat{t}^{}_{24} c^{-1}_{34} c^{-1}_{24} \hat{t}^{*}_{25} \\
+ \hat{t}^{}_{34} c^{-1}_{15} c^{-1}_{25} \hat{t}^{*}_{35} \end{array} &
c^{-1}_{15} c^{-1}_{25} c^{-1}_{35} & 0
\cr \vspace{-0.45cm} \cr
\begin{array}{l}
+ \hat{t}^{}_{14} c^{-1}_{24} c^{-1}_{34} c^{-1}_{24} \hat{t}^{*}_{16}
+ \hat{t}^{}_{24} c^{-1}_{34} \hat{t}^{}_{15} \hat{t}^{*}_{25} \hat{t}^{*}_{16} \\
+ \hat{t}^{}_{24} c^{-1}_{34} c^{-1}_{25} c^{-1}_{16} \hat{t}^{*}_{26}
+ \hat{t}^{}_{34} \hat{t}^{}_{15} c^{-1}_{25} \hat{t}^{*}_{35} \hat{t}^{*}_{16} \\
+ \hat{t}^{}_{34} c^{-1}_{35} c^{-1}_{16} c^{-1}_{26} \hat{t}^{*}_{36}
+ \hat{t}^{}_{34} \hat{t}^{}_{25} \hat{t}^{*}_{35} c^{-1}_{16} \hat{t}^{*}_{26}
\end{array} &
\begin{array}{l}
+ \hat{t}^{}_{15} c^{-1}_{25} c^{-1}_{35} \hat{t}^{*}_{16}
+ \hat{t}^{}_{25} c^{-1}_{35} c^{-1}_{16} \hat{t}^{*}_{26} \\
+ \hat{t}^{}_{35} c^{-1}_{16} c^{-1}_{26} \hat{t}^{*}_{36} \end{array} &
c^{-1}_{16} c^{-1}_{26} c^{-1}_{36}
\cr \end{matrix} \right) \; \hspace{0.2cm}
\nonumber \\
\hspace{-0.2cm} & \simeq & \hspace{-0.2cm}
I + \left(\begin{matrix}
\displaystyle\frac{1}{2} \left(s^2_{14} + s^2_{24} + s^2_{34}\right) & 0 & 0 \cr
\hat{s}^{}_{14} \hat{s}^*_{15} + \hat{s}^{}_{24} \hat{s}^*_{25} +
\hat{s}^{}_{34} \hat{s}^*_{35} & \displaystyle\frac{1}{2}
\left(s^2_{15} + s^2_{25} + s^2_{35} \right) & 0 \cr
\hat{s}^{}_{14} \hat{s}^*_{16} + \hat{s}^{}_{24} \hat{s}^*_{26} +
\hat{s}^{}_{34} \hat{s}^*_{36} &
~\hat{s}^{}_{15} \hat{s}^*_{16} + \hat{s}^{}_{25} \hat{s}^*_{26} +
\hat{s}^{}_{35} \hat{s}^*_{36}~ & \displaystyle\frac{1}{2}
\left(s^2_{16} + s^2_{26} + s^2_{36}\right) \cr \end{matrix}\right) \; ,
\label{29}
\end{eqnarray}
where $\hat{t}^{}_{ij} \equiv e^{{\rm i} \delta^{}_{ij}} \tan\theta^{}_{ij}$
is defined, and all the terms of ${\cal O}(s^4_{ij})$ or smaller have been
omitted from the second equation as an excellent approximation due to the
smallness of $\theta^{}_{ij}$ (for $i = 1, 2, 3$ and $j = 4, 5, 6$). We see that
$(B^*)^{-1}$ is also a lower triangular matrix like $B$ itself. On the other
hand, the factor $U^{\prime T}_0 S^{\prime *}$ appearing in Eq.~(\ref{20})
can be explicitly expressed as follows:
\begin{eqnarray}
U^{\prime T}_0 S^{\prime *} = B^T S^* U^*_0 \simeq
-\left(\begin{matrix} \hat{s}^{*}_{14} & \hat{s}^{*}_{24} & \hat{s}^{*}_{34} \cr
\hat{s}^{*}_{15} & \hat{s}^{*}_{25} & \hat{s}^{*}_{35} \cr
\hat{s}^{*}_{16} & \hat{s}^{*}_{26} & \hat{s}^{*}_{36} \cr \end{matrix}\right)
U^*_0 \; ,
\label{30}
\end{eqnarray}
where Eq.~(\ref{A.2}) has been used, and the terms of ${\cal O}(s^3_{ij})$ or
smaller have been omitted from the second equation as a very good approximation.
Now we conclude that the heavy Majorana neutrino mass basis
$N^\prime_{\rm R}$ is identical to ${\cal N}^\prime_{\rm R}$ up to the
accuracy of ${\cal O}(s^2_{ij})$, but it contains a small contribution of
${\cal O}(s^{}_{ij})$ from the light Majorana neutrino mass basis
$(\nu^\prime_{\rm L})^c$ in the seesaw framework. Since the magnitudes of
$\theta^{}_{ij}$ (for $i = 1, 2, 3$ and $j = 4, 5, 6$) are highly suppressed
in a realistic seesaw model with little fine-tuning, the mismatch between
$N^\prime_{\rm R}$ and ${\cal N}^\prime_{\rm R}$ is expected to be negligible
in most cases.

Let us proceed to examine how small the difference between $D^{}_N$ and
${\cal D}^{}_{\cal N}$ in Eq.~(\ref{21}) can be. First of all, Eq.~(\ref{A.2})
allows us to make the approximation
\begin{eqnarray}
B \simeq I - \left( \begin{matrix} \displaystyle\frac{1}{2} \left(s^2_{14} +
s^2_{24} + s^2_{34}\right) & 0 & 0 \cr \vspace{-0.4cm} \cr
\hat{s}^{*}_{14} \hat{s}^{}_{15} + \hat{s}^{*}_{24} \hat{s}^{}_{25}
+ \hat{s}^{*}_{34} \hat{s}^{}_{35} & \displaystyle\frac{1}{2}
\left(s^2_{15} + s^2_{25} + s^2_{35}\right)
& 0 \cr \vspace{-0.4cm} \cr
\hat{s}^{*}_{14} \hat{s}^{}_{16} + \hat{s}^{*}_{24}
\hat{s}^{}_{26} + \hat{s}^{*}_{34} \hat{s}^{}_{36} & ~\hat{s}^{*}_{15}
\hat{s}^{}_{16} + \hat{s}^{*}_{25} \hat{s}^{}_{26} + \hat{s}^{*}_{35}
\hat{s}^{}_{36}~ & \displaystyle\frac{1}{2} \left(s^2_{16} + s^2_{26} +
s^2_{36}\right) \cr \end{matrix} \right) \; ,
\label{31}
\end{eqnarray}
where the terms of ${\cal O}(s^4_{ij})$ or smaller have been omitted.
Secondly, we obtain
\begin{eqnarray}
A^{-1} R \hspace{-0.2cm} & = & \hspace{-0.2cm}
\left( \begin{matrix} \hat{t}^{*}_{14} & c^{-1}_{14} \hat{t}^{*}_{15} &
c^{-1}_{14} c^{-1}_{15} \hat{t}^{*}_{16}
\cr \vspace{-0.45cm} \cr
c^{-1}_{14} \hat{t}^{*}_{24} &
\hat{t}^{}_{14} \hat{t}^{*}_{15} \hat{t}^{*}_{24} + c^{-1}_{15} c^{-1}_{24}
\hat{t}^{*}_{25} &
\begin{array}{l}
+ \hat{t}^{}_{14} c^{-1}_{15} \hat{t}^{*}_{16} \hat{t}^{*}_{24}
+ \hat{t}^{}_{15} \hat{t}^{*}_{16} c^{-1}_{24} \hat{t}^{*}_{25} \\
+ c^{-1}_{16} c^{-1}_{24} c^{-1}_{25} \hat{t}^{*}_{26} \end{array}
\cr \vspace{-0.45cm} \cr
c^{-1}_{14} c^{-1}_{24} \hat{t}^{*}_{34} &
\begin{array}{l}
+ \hat{t}^{}_{14} \hat{t}^{*}_{15} c^{-1}_{24} \hat{t}^{*}_{34}
+ c^{-1}_{15} \hat{t}^{}_{24} \hat{t}^{*}_{25} \hat{t}^{*}_{34} \\
+ c^{-1}_{15} c^{-1}_{25} c^{-1}_{34} \hat{t}^{*}_{35}
\end{array} &
\begin{array}{l}
+ \hat{t}^{}_{14} c^{-1}_{15} \hat{t}^{*}_{16} c^{-1}_{24} \hat{t}^{*}_{34}
+ \hat{t}^{}_{15} \hat{t}^{*}_{16} \hat{t}^{}_{24} \hat{t}^{*}_{25}
\hat{t}^{*}_{34} \\
+ \hat{t}^{}_{15} \hat{t}^{*}_{16} c^{-1}_{25} c^{-1}_{34} \hat{t}^{*}_{35}
+ c^{-1}_{16} \hat{t}^{}_{24} c^{-1}_{25} \hat{t}^{*}_{26} \hat{t}^{*}_{34} \\
+ c^{-1}_{16} \hat{t}^{}_{25} \hat{t}^{*}_{26} c^{-1}_{34} \hat{t}^{*}_{35}
+ c^{-1}_{16} c^{-1}_{26} c^{-1}_{34} c^{-1}_{35} \hat{t}^{*}_{36}
\end{array}
\cr \end{matrix} \right) \; \hspace{0.2cm}
\nonumber \\
\hspace{-0.2cm} & \simeq & \hspace{-0.2cm}
\left(\begin{matrix} \hat{s}^*_{14} & \hat{s}^*_{15} & \hat{s}^*_{16} \cr
\hat{s}^*_{24} & \hat{s}^*_{25} & \hat{s}^*_{26} \cr
\hat{s}^*_{34} & \hat{s}^*_{35} & \hat{s}^*_{36} \cr \end{matrix}\right)
\label{32}
\end{eqnarray}
from Eq.~(\ref{A.1}), where the terms of ${\cal O}(s^3_{ij})$ or smaller
have been neglected in the second equation as a reasonably good
approximation. The exact expression of $B^{-1} S$ can be directly read off
from that of $-\left(A^{-1} R\right)^*$ with the help of Eq.~(\ref{32})
by making the subscript replacements $15 \leftrightarrow 24$,
$16 \leftrightarrow 34$ and $26 \leftrightarrow 35$, so can its approximate
expression. As a consequence,
\begin{eqnarray}
B^{-1} S A^{-1} R \simeq
-\left( \begin{matrix} s^2_{14} + s^2_{24} + s^2_{34} &
~\hat{s}^{}_{14} \hat{s}^{*}_{15} + \hat{s}^{}_{24} \hat{s}^{*}_{25}
+ \hat{s}^{}_{34} \hat{s}^{*}_{35}~ &
\hat{s}^{}_{14} \hat{s}^{*}_{16} + \hat{s}^{}_{24} \hat{s}^{*}_{26}
+ \hat{s}^{}_{34} \hat{s}^{*}_{36} \cr
\hat{s}^{*}_{14} \hat{s}^{}_{15} + \hat{s}^{*}_{24} \hat{s}^{}_{25}
+ \hat{s}^{*}_{34} \hat{s}^{}_{35} & s^2_{15} + s^2_{25} + s^2_{35} &
\hat{s}^{}_{15} \hat{s}^{*}_{16} + \hat{s}^{}_{25}
\hat{s}^{*}_{26} + \hat{s}^{}_{35} \hat{s}^{*}_{36} \cr
\hat{s}^{*}_{14} \hat{s}^{}_{16} + \hat{s}^{*}_{24} \hat{s}^{}_{26}
+ \hat{s}^{*}_{34} \hat{s}^{}_{36} &
~\hat{s}^{*}_{15} \hat{s}^{}_{16} + \hat{s}^{*}_{25} \hat{s}^{}_{26}
+ \hat{s}^{*}_{35} \hat{s}^{}_{36}~ &
s^2_{16} + s^2_{26} + s^2_{36} \cr \end{matrix} \right) \;
\label{33}
\end{eqnarray}
holds in the same approximation as made above. This result implies that
${\cal D}^{}_{\cal N}$ and $D^{}_N$ are identical to each other up to
the accuracy of ${\cal O}(s^2_{ij})$, simply because
on the right-hand side of Eq.~(\ref{22}) the second term is
suppressed in magnitude to ${\cal O}(s^4_{ij})$ as compared with
the first term.

It is worth remarking that our above analytical approximations are more
or less subject to the canonical
seesaw mechanism at an energy scale far above the electroweak scale, and
thus the mismatch between $N^\prime_{\rm R}$ (or $D^{}_N$)
and ${\cal N}^\prime_{\rm R}$ (or ${\cal D}^{}_{\cal N}$) is very small.
This situation will change when the low-scale seesaw and leptogenesis
scenarios, in which a mismatch between the two sets of mass bases for
heavy Majorana neutrinos is crucial, are taken into account (see, e.g.,
Refs.~\cite{Canetti:2012kh,Drewes:2019byd}).

\subsection{Determination of $D^{}_\nu$ and $U^{}_0$}

As already shown in Eq.~(\ref{17}), the nine effective flavor parameters
of three light Majorana neutrinos in $D^{}_\nu$ and $U^{}_0$ (i.e., three
effective masses, three flavor mixing angles and three CP-violating phases)
can be expressed in terms of the eighteen seesaw parameters hidden in $A$,
$R$ and $D^{}_N$ (i.e., three heavy Majorana neutrino masses, nine
active-sterile flavor mixing angles and six CP-violating phases). It is
obvious that all the derivational seesaw parameters on the left-hand side
of Eq.~(\ref{17}) would vanish if $R \propto Y^{}_\nu$ were switched off.
So this equation provides an unambiguous way to determine the light degrees
of freedom from the heavy degrees of freedom in the seesaw framework.

To be more specific, the six independent elements of the effective Majorana
neutrino mass matrix $M^{}_\nu \equiv U^{}_0 D^{}_\nu U^T_0$ are given as
follows:
\begin{eqnarray}
\big(M^{}_\nu\big)^{}_{11} \hspace{-0.2cm} & = & \hspace{-0.2cm}
m^{}_1 c^2_{12} c^2_{13} + m^{}_2 \hat{s}^{* 2}_{12} c^2_{13} +
m^{}_3 \hat{s}^{* 2}_{13} \; ,
\nonumber \\
\big(M^{}_\nu\big)^{}_{12} \hspace{-0.2cm} & = & \hspace{-0.2cm}
-m^{}_1 c^{}_{12} c^{}_{13} \big(\hat{s}^{}_{12} c^{}_{23} + c^{}_{12}
\hat{s}^{}_{13} \hat{s}^*_{23}\big) + m^{}_2 \hat{s}^*_{12} c^{}_{13}
\big(c^{}_{12} c^{}_{23} - \hat{s}^*_{12} \hat{s}^{}_{13}
\hat{s}^*_{23}\big) + m^{}_3 c^{}_{13} \hat{s}^{*}_{13}
\hat{s}^{*}_{23} \; , \hspace{0.3cm}
\nonumber \\
\big(M^{}_\nu\big)^{}_{13} \hspace{-0.2cm} & = & \hspace{-0.2cm}
m^{}_1 c^{}_{12} c^{}_{13} \big(\hat{s}^{}_{12} \hat{s}^{}_{23} - c^{}_{12}
\hat{s}^{}_{13} c^{}_{23}\big) - m^{}_2 \hat{s}^*_{12} c^{}_{13}
\big(c^{}_{12} \hat{s}^{}_{23} + \hat{s}^*_{12} \hat{s}^{}_{13}
c^{}_{23}\big) + m^{}_3 c^{}_{13} \hat{s}^{*}_{13} c^{}_{23} \; ,
\nonumber \\
\big(M^{}_\nu\big)^{}_{22} \hspace{-0.2cm} & = & \hspace{-0.2cm}
m^{}_1 \big(\hat{s}^{}_{12} c^{}_{23} + c^{}_{12} \hat{s}^{}_{13}
\hat{s}^*_{23}\big)^2 + m^{}_2 \big(c^{}_{12} c^{}_{23} -
\hat{s}^*_{12} \hat{s}^{}_{13} \hat{s}^*_{23}\big)^2 +
m^{}_3 c^2_{13} \hat{s}^{* 2}_{23} \; ,
\nonumber \\
\big(M^{}_\nu\big)^{}_{23} \hspace{-0.2cm} & = & \hspace{-0.2cm}
-m^{}_1 \big(\hat{s}^{}_{12} c^{}_{23} + c^{}_{12} \hat{s}^{}_{13}
\hat{s}^*_{23}\big) \big(\hat{s}^{}_{12} \hat{s}^{}_{23} - c^{}_{12}
\hat{s}^{}_{13} c^{}_{23}\big)
\nonumber \\
\hspace{-0.2cm} & & \hspace{-0.2cm}
- m^{}_2 \big(c^{}_{12} c^{}_{23} -
\hat{s}^*_{12} \hat{s}^{}_{13} \hat{s}^*_{23}\big)
\big(c^{}_{12} \hat{s}^{}_{23} + \hat{s}^*_{12} \hat{s}^{}_{13}
c^{}_{23}\big) + m^{}_3 c^{2}_{13} c^{}_{23} \hat{s}^{*}_{23} \; ,
\nonumber \\
\big(M^{}_\nu\big)^{}_{33} \hspace{-0.2cm} & = & \hspace{-0.2cm}
m^{}_1 \big(\hat{s}^{}_{12} \hat{s}^{}_{23} - c^{}_{12} \hat{s}^{}_{13}
c^{}_{23}\big)^2 + m^{}_2 \big(c^{}_{12} \hat{s}^{}_{23} +
\hat{s}^*_{12} \hat{s}^{}_{13} c^{}_{23}\big)^2 +
m^{}_3 c^2_{13} c^{2}_{23} \; .
\label{34}
\end{eqnarray}
On the other hand, Eq.~({\ref{17}) tells us that these six matrix elements
can originally be determined by $M^{}_\nu = -\big(A^{-1} R\big) D^{}_N
\big(A^{-1} R\big)^T$ thanks to the exact seesaw relation bridging the
big gap between the light and heavy Majorana neutrinos. With the help of the
explicit expression of $A^{-1} R$ given in Eq.~(\ref{32}), it is straightforward
to obtain the expressions for the elements of $M^{}_\nu$ in terms of $M^{}_i$,
$\theta^{}_{ij}$ and $\delta^{}_{ij}$ (for $i = 1, 2, 3$ and $j = 4, 5, 6$).
Instead of presenting the exact analytical results, which are rather lengthy
and hence less instructive, here we make the leading-order approximations for
the expressions of $A$ and $R$ given in Eq.~(\ref{A.1}) and then arrive at
\begin{eqnarray}
\big(M^{}_\nu\big)^{}_{11} \hspace{-0.2cm} & \simeq & \hspace{-0.2cm}
-\left[M^{}_1 \hat{s}^{* 2}_{14} + M^{}_2 \hat{s}^{* 2}_{15} +
M^{}_3 \hat{s}^{* 2}_{16}\right]  \; ,
\nonumber \\
\big(M^{}_\nu\big)^{}_{12} \hspace{-0.2cm} & \simeq & \hspace{-0.2cm}
-\left[M^{}_1 \hat{s}^{*}_{14} \hat{s}^{*}_{24} +
M^{}_2 \hat{s}^{*}_{15} \hat{s}^{*}_{25} +
M^{}_3 \hat{s}^{*}_{16} \hat{s}^{*}_{26}\right] \; ,
\nonumber \\
\big(M^{}_\nu\big)^{}_{13} \hspace{-0.2cm} & \simeq & \hspace{-0.2cm}
-\left[M^{}_1 \hat{s}^{*}_{14} \hat{s}^{*}_{34} +
M^{}_2 \hat{s}^{*}_{15} \hat{s}^{*}_{35} +
M^{}_3 \hat{s}^{*}_{16} \hat{s}^{*}_{36}\right] \; ,
\nonumber \\
\big(M^{}_\nu\big)^{}_{22} \hspace{-0.2cm} & \simeq & \hspace{-0.2cm}
-\left[M^{}_1 \hat{s}^{* 2}_{24} + M^{}_2 \hat{s}^{* 2}_{25} +
M^{}_3 \hat{s}^{* 2}_{26}\right]  \; ,
\nonumber \\
\big(M^{}_\nu\big)^{}_{23} \hspace{-0.2cm} & \simeq & \hspace{-0.2cm}
-\left[M^{}_1 \hat{s}^{*}_{24} \hat{s}^{*}_{34} +
M^{}_2 \hat{s}^{*}_{25} \hat{s}^{*}_{35} +
M^{}_3 \hat{s}^{*}_{26} \hat{s}^{*}_{36}\right] \; , \hspace{0.3cm}
\nonumber \\
\big(M^{}_\nu\big)^{}_{33} \hspace{-0.2cm} & \simeq & \hspace{-0.2cm}
-\left[M^{}_1 \hat{s}^{* 2}_{34} + M^{}_2 \hat{s}^{* 2}_{35} +
M^{}_3 \hat{s}^{* 2}_{36}\right]  \; .
\label{35}
\end{eqnarray}
Let us emphasize that there appear nine CP-violating phases in Eq.~(\ref{35}),
but three of them (or their combinations) are redundant and can always
be removed by rephasing the charged lepton fields in a proper way
\footnote{A straightforward way to remove the three redundant phase
parameters of $A$ and $R$ is just to switch off three of the nine phases
in the nine active-sterile flavor mixing matrices $O^{}_{ij}$ (for
$i = 1, 2, 3$ and $j = 4, 5, 6$) in Eq.~(\ref{27}) from the very beginning.
As there are many options in doing so, we do not go into details here.}.
A combination of Eqs.~(\ref{34}) and (\ref{35}) allows us to establish
the direct relations between the nine derivational and eighteen original
seesaw parameters. So the former can in principle be determined from the
latter for a given seesaw model (a top-down approach), and the latter
may be partly probed or constrained from the former with the help of some
low-energy neutrino experiments (a bottom-up approach). A careful and
detailed analysis of the parameter space along this line of thought will
be made elsewhere.

\section{Summary}

We have reformulated the canonical seesaw mechanism by considering the
fact that the electroweak gauge symmetry is unbroken at the seesaw
scale characterized by the masses of heavy Majorana neutrinos,
and shown that it can {\it formally} work and allow us to derive an
exact seesaw relation between the active (light) and sterile (heavy)
Majorana neutrinos. In this way we have elucidated the reason why
there is an unavoidable mismatch between the mass eigenstates of heavy
Majorana neutrinos associated with the seesaw and thermal leptogenesis
mechanisms. The smallness of this mismatch has been discussed with the
help of a complete Euler-like parametrization of the flavor structure in
the seesaw framework, and the exact and explicit relations between
the {\it original} and {\it derivational} seesaw parameters have been
established as a by-product.

We hope that this work may help clarify some conceptual ambiguities
associated with the validity of the seesaw mechanism before and after
spontaneous electroweak symmetry breaking, because such ambiguities have
never been taken serious in the literature. It should also be helpful
to clarify the ambiguities associated with the RGE evolution between
the ``virtual" flavor parameters of Majorana neutrinos at the seesaw
scale and those ``real" ones at the electroweak scale, which is crucial
to bridge the gap between a well-motivated UV-complete flavor theory
including the seesaw mechanism and all the possible low-energy flavor
experiments.

\section*{Acknowledgements}

I am greatly indebted to Di Zhang and Shun Zhou for numerous helpful
discussions and comments about this paper, a work dedicated to the 50th
birthday of my home institute, the Institute of High Energy Physics,
which was founded on 1 February 1973. I would also like to thank
Marco Drewes for very useful discussions and comments. My research is
supported in part by the National Natural Science Foundation of
China under grant No. 12075254 and grant No. 11835013.

\newpage

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\section*{Appendix}


text:
\section{LG beams and the orbital angular momentum of light}
Light has been shown by Woerdman et al. \cite{allen1992allen} to carry orbital angular momentum (OAM) as a consequence of its spatial structure. In particular, light beams possessing an azimuthal phase dependence of $\exp(i\ell\phi)$ where $\ell$ is an integer, carry OAM of $\ell\hbar$ per photon. The canonical example of an OAM carrying beam is the Laguerre-Gauss (LG) beam with an electric field given by \cite{saleh2007fundamentals},
\begin{equation}
u_{\ell}^{p}(\rho,\phi,z) \approx \left(\frac{\rho}{\omega}\right)^{|\ell|} L^{|\ell|}_p \left(\frac{2\rho^2}{\omega^2}\right) \exp\left(\frac{-\rho^2}{\omega^2}\right)\exp\left(i\ell\phi\right)
\label{eq:LG}
\end{equation}
where $\omega$ is the beam waist, $L^\ell_p \left(\frac{2\rho^2}{\omega^2}\right)$ is the associated Laguerre polynomial, $\ell$ is the azimuthal mode, and $p$ is the radial mode. The discontinuity of the azimuthal phase leads to a singularity or vortex at the center of the beam. The azimuthal mode $\ell$, also called the topological charge, defines size of the vortex while $p$ defines the number of radial nodes.

The OAM of the LG beam has led to its applications in quantum information \cite{molina2001management,torres2003preparation,straupe2010quantum}, micromachining \cite{ladavac2004microoptomechanical}, optical communication \cite{gibson2004free}, optical tweezing \cite{grier2003revolution}, and in astronomy \cite{foo2005optical}  and gravitational wave detection \cite{granata2010higher}. A common method to determine the OAM of an LG beam is through interferometry \cite{soskin1997topological,basistiy1993optics, basistiy1995optical, harris1994laser, padgett1996experiment}. Moreover, the diffraction of the beam through different apertures has been shown to also be practicable method of probing the OAM. For example, the diffraction of the LG beam through polygonal apertures \cite{silva2014unveiling, ambuj2014diffraction, hickmann2010unveiling, mourka2011visualization, de2011measuring} and multi-pinhole set-ups \cite{shi2012characterizing, berkhout2008method} are shown to be effective in probing their OAM . The classical single-slit \cite{ferreira2011fraunhofer, ghai2009single} and double-slit \cite{emile2014young, sztul2006double} diffraction also have been reexamined using LG beams instead of the ordinary Gaussian beam. However, in these works they used stationary slits and they were only able to probe the low ordered modes. In this paper, we study the diffraction of the LG beam through a dynamic single-slit using a digital micromirror device. We look at the fringe patterns formed and show that these are related to the topological charge of the LG beam. We explain the resulting structure of the diffraction using the method of stationary phase applied to the Huygens-Fresnel diffraction integral. Our results are valuable in the study of OAM specially for beams where the current technology for detection is not yet mature, for example, beams whose wavelengths are in the terahertz regime or beams of higher energy such as, electron vortex beams \cite{uchida2010generation, verbeeck2010production}.

\section{Slit diffraction of LG using stationary phase approximation}\label{sec:intro}
	The main assumption of the classical single slit experiment is that a plane wave impinges on the slit. That is, at the plane of the slit the waves are in phase. The resulting diffraction pattern can be derived by imagining secondary sources along the slit. The phase difference between these wavelets are caused by their path difference to the observation point. Mathematically, the field is given by the Huygens diffraction integral,
\begin{equation}
u = \int\int_S \frac{ik}{r} e^{ikr} dS
\end{equation}
	where $r$ is the distance from the source point to the observation point, $k$ is the wavevector, and the integral is over the entire surface $S$ containing the source points. Applying the conditions for the Fraunhofer approximations and taking the intensity gives \cite{born2013principles},
	\begin{equation}
I \propto sinc^2(\beta)
\label{eq:sinc}
	\end{equation}
    where $\beta = kbsin\theta /2$, $b$ being the thickness of the slit and $\theta$ the angle from the center of the slit to the observation point. The sinc function means that there are alternating bright and dark fringes perpendicular to the slit and that the maximum intensity is found along the axis which contains about 85\% of the power transmitted.

	If instead we use a higher order LG beam rather than a Gaussian beam, then the waves are no longer in phase in the plane of the slit. There is an additional phase difference due to the azimuthal phase dependence of the LG beam. This additional phase difference can be incorporated inside the exponential in equation 2. However, evaluating the integral can be a bit involved so instead we invoke the method of stationary phase. The method of stationary phase states that for an integral of the form,
    \begin{equation}
    \int g(z)e^{ikf(z)}dz,
    \end{equation}
    where function $f(z)$ changes rapidly compared to the wavelength in the domain of integration, the real and imaginary parts of the exponential oscillate rapidly and the contributions from each point in the domain of integration will cancel each other out except at the edge of the integration region where the cancellation is incomplete \cite{born2013principles,kathavate1945geometric}. For the case of a vertical single slit, this means that most of the contribution to the diffraction integral comes from the top and bottom endpoints. For the case of the plane wave, the sources from the top and bottom endpoints of a vertical single slit are in phase, and therefore will interfere constructively along the horizontal line along the center of the slit \cite{narag2018diffraction}. This is the ordinary pattern we see in the classical single slit diffraction. There is a bright band perpendicular to the slit. The horizontal fringe is not uniform but is a sinc function as in equation \ref{eq:sinc}. However for a very narrow slit we can approximate it to be a uniform band, because the width of the sinc function becomes wide.
    In the case of LG beams, the phase on the top and bottom edge of the slit will differ depending on the OAM and on the position $x$ of the slit with respect to the center of the beam. A simple calculation shows that the phase difference $\Delta \phi$ between the top and bottom edge of the slit is given by,
    \begin{equation}
\Delta \phi = 2 \ell \cos ^{-1} \left(\frac{x}{\sqrt{x^2+L^2}}\right)
\label{eq:delphi}
    \end{equation}
    where $\ell$ is the topological charge of the beam, $x$ is the displacement of the slit, and $2L$ is the length of the slit. We assume that the slit is very narrow. When the slit is centered, i.e. $x = 0$, equation \ref{eq:delphi} gives the phase difference to be $\Delta \phi = \ell \pi$. Thus, when $\ell$ is even the result is constructive interference and the central horizontal fringe is a bright band and when $\ell$ is odd, the interference is destructive and the central horizontal fringe is a dark band.  Moreover, $\Delta \phi$ also depends on the slit position $x$. Figure \ref{fig:cosineinverse} shows the trend of the $2\ell cos^{-1}$ as the slit is moved away from the center. When the slit is at the center of the beam, $\Delta \phi$ is an odd (even) multiple of $\pi$ for odd (even) $\ell$. As the slit moves away from the center, or equivalently as $cos^{-1}$ approaches 1, the phase difference decreases and goes to zero. The phase difference evolves from an odd to an even multiple of $\pi$ and this evolution depends on the topological charge $\ell$. For example, for $\ell = 1$ the phase difference will change from an odd to an even multiple of $\pi$ only once. For $\ell = 2$, $\Delta \phi$ starts as an even multiple of $\pi$ ($\Delta \phi = 2 \pi$) then changes to an odd multiple of $\pi$ ($\Delta \phi = \pi$) and finally back to an even multiple ($\Delta \phi = 0$) as the slit is moved. Thus, when the slit is moved the central horizontal intensity will transition from a dark band to a bright band only once for $\ell = 1$ while for $\ell = 2$ the central intensity will transition from a bright to a dark then again to a bright band. The number of transitions and how fast the transitions of the fringes from dark to bright (or bright to dark) depends on $\ell$. In general, the central fringe starts as a dark band for odd $\ell$ and a bright band for even $\ell$ and the evolutions of the fringes from dark to bright or vice versa is faster for larger $\ell$.

\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{inversecosine2.png}
\caption{$2 \ell cos^{-1}$ vs $\xi$. Here $\xi$ is a normalized distance, $\xi = \frac{x}{\sqrt{x^2+L^2}}$. The phase difference transitions from an odd to even (or even to odd) multiples of $\pi$ as the slit position $x$ becomes very large or as $\xi$ approaches 1. The number of transitions, from odd to even or even to odd multiples of $\pi$ depends on the OAM of the beam.}
\label{fig:cosineinverse}
\end{figure}

\section{Simulation}

We simulated the generation of $LG_\ell^{0}$ beams by approximating the beams using equation \ref{eq:LG}. The beams were then modulated in amplitude by a slit of thickness = 54 $\mu$m. This is done by multiplying the beam matrix element wise with the amplitude mask of the slit. The intensity of $|LG|^{2}$ beam (a) before the slit, (b) after the slit and (c) after the slit has been moved laterally are shown in Figure \ref{fig:beforeandafterslit}a to and \ref{fig:beforeandafterslit}c, respectively; while the corresponding phases are shown in  Figure \ref{fig:beforeandafterslit}d to Figure \ref{fig:beforeandafterslit}f.

\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{4_beforeandafterslit.png}
\caption{Slit Modulation. Figure a and b show the intensity the $LG_2$ beam respectively before and immediately after the slit while shows intensity when the slit is moved at a distance x. d), e) and f) are the corresponding phase distribution.}
\label{fig:beforeandafterslit}
\end{figure}

The field right after the slit was then Fourier transformed to simulate the effect of the slit being placed in the back focal plane of a lens and being imaged in the front focal plane. The images for LG beams with different $\ell$ values and $x = 0$ are shown Figure \ref{fig:DiffractionSimulation}. The diffraction patterns show that there is an obvious difference between the diffraction pattern between LG beams with odd and even $\ell$. There is a horizontal dark line that cuts through the center of diffraction pattern for odd $\ell$ while for even $\ell$ the central horizontal band is bright. The number of fringes is also proportional to $\ell$. In particular the number of fringes is $\ell + 1$, same as the result in \cite{ambuj2014diffraction}. For higher $\ell$'s, however, it becomes more difficult to resolve the fringes. Also, the diffraction pattern becomes overall darker for larger $\ell$s since the vortex is also larger. That is why in the experimental result for $\ell = 5$ in fig. \ref{fig:DiffractionSimulation}, we had to increase the camera sensitivity to be able to capture the diffraction pattern and the images are noiser with more grains.

\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{4_DiffractionSimulation1.png}
\caption{Diffraction through a single slit of $LG_0^{\ell}$. For odd $\ell$ there is a central horizontal dark band while for even $\ell$ the central horizontal band is bright.}
\label{fig:DiffractionSimulation}
\end{figure}

We also graphed in \ref{fig:LineScanSimu} the relative intensity of the central horizontal band as the slit is moved. Again, we see from the graphs that the central intensity when $x = 0$ for LG beams with odd (even) $\ell$s start at a minima (maxima). When the slit is moved the central horizontal band changes from an intensity maxima (minima) to a minima (maxima) for even (odd) $\ell$. This is explained by the variation of $\Delta \phi$ with respect to the slit position in equation \ref{eq:delphi}. The graph of equation \ref{eq:delphi} shows that for odd $\ell$, the phase difference starts as an odd multiple of $\pi$ and will alternately become an even then odd multiple of $\pi$ as the slit displacement becomes very large. For even $\ell$, the phase difference also alternates between odd and even multiples of $\pi$ but it starts, at $x = 0$, as an even multiple. This means that number of intensity change from maxima to minima (or minima to maxima) and how fast this change as a function of slit position are dependent of $\ell$. For example, for $\ell = 1$, the phase difference will change from an odd to an even multiple of $\pi$ only once. Thus the central intensity will only go once from a dark to a bright fringe. The rates are also faster for larger $\ell$. We see in figure \ref{fig:DiffractionSimulation} that for odd $\ell$ the central intensities are minima when $x=0$, but as we displace the slit the graph for $\ell = 5$ evolves faster than that of $l=3$ and that $\ell = 3$ is faster than that of $\ell =1$. The analogous behaviour is observed for even $\ell$'s: $\ell = 4$ evolves faster than $\ell = 2$ which evolves faster than $\ell = 0$.

\begin{figure}[h]
\centering
\includegraphics[width=0.9\linewidth]{linescansimulation.png}
\caption{Simulation of Central intensity vs slit position $x$. The central intensity for odd (even) $\ell$s start from a minima (maxima) and then changes to a maxima (minima) as the slit is moved from the center. Larger $\ell$s evolve faster than smaller $\ell$s.}
\label{fig:LineScanSimu}
\end{figure}

\section{Experiment}

We generated the LG beam using the setup shown in figure \ref{fig:setup}a. A 632.8 nm HeNe laser was collimated by lenses L1 and L2. The beam was then converted into LG beams of various topological charges by uploading their corresponding phase profiles into the spatial light modulator. These phases are shown in figure \ref{fig:setup}b. A digital micromirror device (DMD) was then used as a dynamic slit. The DMD is composed of the an array of 608 by 608 mirrors of dimension 10.8 $\mu m$ and is controllable through a computer. The slit was moved from -1 mm (left) to 1 mm (right) of the beam center. The slit thickness was 54 $\mu m$. Finally, the diffraction was focused using lens L3 with focal length 25 cm and was captured using a CCD camera in the back focal plane of L3. The experimental results are discussed in the next section.

\begin{figure}[h]
\centering
\includegraphics[width=0.9\linewidth]{setup3.png}
\caption{Experimental setup (a). The LG beams were produced by uploading their phase in the SLM shown in (b). We were able to vary the position of the single slit by using a digital micromirror device (DMD) as a dynamic slit. The diffraction was imaged in the far-field using a CCD camera.}
\label{fig:setup}
\end{figure}

\section{Results and Discussions}
The intensity of the central peak as a function of the transverse displacement of the slit is shown in figure \ref{fig:slitpositionexp} and figure for odd and even $\ell$. The measured intensity for each position is the average of when the slit is a displaced to the left and the same distance to the right. For odd $\ell$, the intensity is zero when the slit is centered and increases to 1 as the slit is moved; while for even $\ell$, the opposite is true and the intensity is maximum when the slit is at the center and goes to zero as the slit is moved. These are all consistent with our calculations and our simulation in the previous section. One difference the we notice is that the experimental intensities do not reach the minima at zero. This is because we set the camera setting to high gain to capture our images specially for high $\ell$. We had to set the camera to increasing gain for higher $
\ell$ since the vortex is larger and the diffraction patterns becomes overall dimmer. However, although the minimum intensities do not completely reach zero, our eyes can still recognize them as intensity minima because the fringes are still visible.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\linewidth]{linescanexperiment.png}
\caption{Experimental result of central intensity vs position. Our results match our calculations and simulations that 1) as slit position is varied the central minima becomes a maxima for odd $\ell$, while the central maxima becomes a minima for even $\ell$ and that 2) higher $\ell$'s evolve faster than lower $\ell$'s.}
\label{fig:slitpositionexp}
\end{figure}

In ref. \cite{ghai2009single}, the single slit diffraction of OAM beams was analyzed by taking the phase difference between the edge of the short axis of the slit. That is, the slit has some finite thickness. The analysis in ref. \cite{ghai2009single} essentially treats the single slit as a double slit. The result was that the fringes were not straight but are bent. In fact the same technique was used in \cite{sztul2006double} and in ref. \cite{emile2014young} for double slits and found the same bending of the fringes. Here, bending is not noticable since we used a very thin slit. For a very thin and long slit, the effect that there is a central minima and maxima for odd and even $\ell$ is more evident than the bending of the fringes.

In another paper, Ferreira et al \cite{ferreira2011fraunhofer} analyzed the diffraction of the LG beam through a single slit in terms of the phase difference between the end of the long axis of the slit. They explained the resulting pattern by approximating the LG beam by two separate Gaussian beams. Here we explain it by the method of stationary phase and considered only the phases at the endpoints of the slit. Recently, the method of stationary phase has been used to study sharp-edge diffraction of slits of various shapes\cite{narag2018diffraction,borghi2016catastrophe} . In this work, we also made use of a DMD as a dynamic slit in contrast with static slits in refs. \cite{ferreira2011fraunhofer,sztul2006double,emile2014young,ghai2009single}. We were also able to derive the relationship between the phase difference and the slit position in equation 5. In ref. \cite{ferreira2011fraunhofer} they also displaced the slit by some distance. However, they were only able to displace the slit to one certain position since they used a static slit.

We note that the OAM can still be probed using stationary slits. However, it becomes harder and harder to resolve the fringes just by looking at the diffraction pattern, as in figure \ref{fig:DiffractionSimulation}. The number of horizontal fringes = $\ell + 1$ only works for small $\ell$, $\ell < 3$ in our case. In ref. \cite{ambuj2011far}, the number of fringes = $\ell + 1$ trend was also observed but only for $\ell > 3$. The difference between our method and that of ref. \cite{ambuj2011far}, is that they displaced the slit along the beam axis to observe the effect of the radial variation of the phase of the beam. Here we have displace the slit transversely and we have shown that displacing the slit allows us to access higher $\ell$. Our method, however, has a limitation. We cannot distinguish between the handedness of the OAM. We can only determine the absolute value of the OAM but not whether it is positive or negative. We also only investigate the diffraction for beams with integer $\ell$s. A future study in this field can look at beams with non-integer $\ell$.
\section{Conclusions}
We demonstrated that the OAM of an LG beam can be determined by looking at its diffraction through a single slit. On the basis of the method of stationary phase, we derived the difference between the endpoints of the slit we showed that for odd $\ell$ there is a central minima and for an even $\ell$ there is a central maxima. Furthermore, we also investigated the evolution of this phase difference as the slit is moved transversely. We used a digital micromirror device as a dynamic slit whose position can be controlled through a computer. We showed that although OAM of the slit can be determined using stationary slits, but with a dynamic slit we can access higher OAM values by looking at how the central horizontal fringes evolves as we change the slit position. We found that the evolution from intensity minima to maxima or maxima to minima is faster for larger $\ell$.

\begin{acknowledgments}
The authors acknowledge the Office of the Chancellor of the University of the Philippines Diliman through the Office of the Vice Chancellor for Research and Development Outright Research Grant for funding support. N. Hermosa is a recipient of the Balik PhD Program of the Office of the Vice President for Academic Affairs of the University of the Philippines. J. Narag is a Science Research Specialist of the PCIEERD Project No. 04002 under the Department of Science and Technology of the Republic of the Philippines.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv