# Majorana propagator on de Sitter space

Tomislav Prokopec 111email<EMAIL_ADDRESS>Institute for Theoretical Physics
and EMME$\Phi$, Faculty of Science, Utrecht University, Princetonplein 5, 3584
CC Utrecht, The Netherlands  Vishnu Hari Unnithan 222email:
<EMAIL_ADDRESS> Institute for Theoretical Physics and EMME$\Phi$,
Faculty of Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The
Netherlands

###### Abstract

We study the dynamics of Majorana fermions in an expanding de Sitter space and
to that aim we construct the vacuum Feynman propagator for Majorana fermions
in de Sitter space. Surprisingly, the Majorana propagator is identical to that
of the Dirac fermions. We then use this propagator and calculate the one-loop
effective action for Majorana fermions, and show that it differs from that of
Dirac fermions by a factor of $1/2$, which accounts for the reduced number of
degrees of freedom of the Majorana particle. Finally, we derive the Majorana
and Dirac propagators for mixing fermionic flavors, which are suitable for
studying CP-violating effects in quantum loops.

##  I Introduction

Understanding Majorana particles has attracted considerable interest in the
last few decades. Understanding its dynamics have consequences for important
questions such as the nature of dark matter Chikashige:1980ui ;
Lattanzi:2014mia ; Beltran:2008xg , the origin and nature of matter anti-
matter asymmetry, the origin of neutrino masses, inflationary models, etc. The
field of quantum computing, topological insulators and other fields in low
energy physics have also wrestled with Majorana particles Beenakker:2011np ,
as pairs of Majorana particles could provide topologically protected quantum
gates.

The Dirac fermion propagator on de Sitter space was originally constructed by
Candelas and Raine Candelas:1975du , and then subsequently generalized to
accelerating, homogeneous cosmological spaces in Ref. Koksma:2009tc . 333The
Dirac propagator was constructed in spacetimes of a constant acceleration
parameter, $\epsilon=-\dot{H}/H^{2}\;(\dot{H}={\rm d}H/{\rm d}t)$, where
$H(t)$ denotes the expansion rate of the Universe and assuming that the
fermionic mass squared scales as the Ricci curvature scalar. How to construct
Majorana propagators in de Sitter space and in cosmological spaces was
discussed in Refs. Cotaescu:2018afn ; Cotaescu:2018zrq , where the Majorana
propagator was obtained by acting the left-handed chiral projectors on the
Dirac propagator.

In this work we provide an alternative derivation of the Majorana propagator
by using the method of mode sums and discuss in detail how to impose the
Majorana condition. We point out in particular at the difference that arises
when Majorana condition is imposed on-shell and off-shell, and stress the
importance of imposing it off-shell, as it can be important for accurate
perturbative calculations. One of the advantages of using the technique of
mode sums is that it brings to fore how the reduced number of degree of
freedom of Majorana particles influences construction of the propagator, which
is not evident in the procedure in Refs. Cotaescu:2018afn ; Cotaescu:2018zrq .

Due to the Majorana nature of neutrinos, leptogenesis Fukugita:1986hr  (see
Ref. Bodeker:2020ghk  for a review) is a key motivator for understanding the
dynamics of Majorana fermions as we know that leptogenesis is concerned with
lepton number generation in extensions of the standard model that include
heavy Majorana neutrinos. Lepton asymmetry produced via the neutrinos is
important for our understanding of the baryon asymmetry of the Universe, as
the one-loop decay processes are studied by making use of the resummed thermal
propagator for Majorana neutrinos Pilaftsis:2003gt ; Pilaftsis:2009pk ;
Buchmuller:1997yu ; Rangarajan:1999kt . This motivates a careful consideration
of the Majorana propagator in out-of-equilibrium conditions in which CP-
violating processes play an important role, one notable example being time
dependent backgrounds of the expanding Universe. Since leptogenesis process
involves mixing of lepton flavors, here we also consider the multiflavor
Majorana propagator in de Sitter space and compare it with that of the Dirac
fermions.

Motivated by the advances in precision observations of cosmic microwave
background radiation and Universe’s large scale structure, in recent decades
we have witnessed progress in perturbative results for interacting matter and
gravitational fields both in de Sitter space (which serves as the model space
for cosmological inflation), as well as in more general cosmological
spacetimes. The most important ingredient of these studies are the
propagators, as they are the essential building block for perturbative
studies. Scalar propagators in de Sitter space were first derived in
Chernikov:1968zm , also see Allen:1987tz , Fukuma:2013mx . Next, the vector
propagators on de Sitter space were initially constructed in covariant Fermi
gauges in Allen:1985wd , for massive vectors of scalar electrodynamics in
exact Lorenz gauge in Tsamis:2006gj , and some subtleties related gauge fixing
were subsequently discussed in Ref. Frob:2013qsa . The graviton propagator was
constructed and discussed at length in Refs. Antoniadis:1986sb ; Tsamis:1992xa
; Allen:1986ta ; Allen:1986tt ; Higuchi:2001uv ; Miao:2011fc ; Mora:2012zi ;
Frob:2016hkx ; Glavan:2019msf . Our knowledge on the propagators in more
general cosmological spacetimes is rather limited. The massless scalar
propagator for inflarionary spaces with constant deceleration parameter was
obtained in Ref. Janssen:2008px  and the massive vector propagator for scalar
electrodynamics was recently constructed in Ref. Glavan:2020zne , where the
authors discussed the subtleties of unitary gauge and various simpler limits
of the propagator. These propagators have been used to study various
perturbative processes in inflation, but the list of references is large, and
so we refrain from presenting it here.

The paper is organised as follows. In Section II we define the model that we
intend to study, and define Majorana fields in the helicity basis and solve
for the mode functions. Section III is devoted to the construction of the
propagator, which we check by showing that it reduces to the correct Minkowski
space propagator. In the same section we also investigate how the propagator
transforms under charge conjugation, as that is of interest for understanding
CP violation. As a simple application of the propagator in Section IV we
compute the one-loop effective action and show that it equals 1/2 of that for
the Dirac fermions. In order to better understand fermionic CP violation
(which is important for leptogenesis), in Section V we construct the
propagator for multiflavor Majorana fermions and compare with the
corresponding propagator for Dirac fermions. Our main results are discussed in
Section VI, where we also present an outlook to possible applications of the
propagator. Finally, in Section VII we present various technical derivations.

##  II The Model

To describe the dynamics of Majorana fermions in de Sitter space, we begin by
recalling the Lagrangian of Dirac fermions in general gravitational
backgrounds in D space-time dimensions, as this allows one to perform
perturbative calculations with renormalization and regularisation that
respects all the symmetries of the problem. The action and Lagrangian of the
free massive Dirac fermion in D dimensions are,

$S[\Psi,\bar{\Psi}]=\int d^{D}x\sqrt{-g}{\cal
L}\;;\quad\mathcal{L}=\frac{1}{2}i\bar{\Psi}\gamma^{\mu}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{\mu}\\!\\!\Psi-\bar{\Psi}\mathcal{M}\Psi\,,$
(II.1)

where $\Psi(x)$ denotes the Dirac spinor,
$\bar{\Psi}=\Psi^{\dagger}\gamma^{0}$,
$\bar{\Psi}\gamma^{\mu}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{\mu}\\!\\!\Psi=\bar{\Psi}\gamma^{\mu}\big{[}\nabla_{\mu}\Psi\big{]}-\big{[}\nabla_{\mu}\bar{\Psi}\big{]}\gamma^{\mu}\Psi$
(we use chiral representation for the Dirac matrices, whose basic properies
are listed in Appendix A), and $\mathcal{M}$ is the (complex) mass matrix
defined as:

$\mathcal{M}=(m_{1}\mathbb{1}_{2^{D/2}\times
2^{D/2}}+im_{2}\gamma_{5})=\begin{pmatrix}m^{*}\mathbb{1}_{2^{(D-2)/2}\times
2^{(D-2)/2}}&0\\\ 0&m\mathbb{1}_{2^{(D-2)/2}\times
2^{(D-2)/2}}\end{pmatrix}\,,$ (II.2)

with $m=m_{1}+im_{2}$, with $m_{1},m_{2}\in\mathbb{R}$. The metric of a
spatially flat, homogeneous universe can be written in terms of the time-
dependent scale factor $a=a(t)$ as,

$g_{\mu\nu}=\text{diag}\left(-1,a^{2},a^{2}....,a^{2}\right)_{\text{D}}\,,$
(II.3)

such that its determinant equals, $g=-a^{2(D-1)}$. The covariant derivative
operator $\nabla_{\\!\mu}$ acts on a spinor as,

$\nabla_{\\!\mu}\Psi=\left(\partial_{\mu}-\Gamma_{\mu}\right)\Psi,$ (II.4)

and spin(or) connection $\Gamma_{\mu}$ can be obtained from,

$\Gamma_{\mu}=-\frac{1}{8}e^{\nu}_{c}\left(\partial_{\mu}e_{\nu
d}-\Gamma^{\alpha}_{\mu\nu}e_{\alpha
d}\right)\left[\gamma^{c},\gamma^{d}\right]\,.$ (II.5)

where $e^{\nu}_{c}$ denote vielbein (used to project vectors (and tensors) on
tangent space, $V^{\mu}(x)=e^{\mu}_{a}(x)V^{a}$, where we use Latin letters to
denote indices on flat tangent space). By transforming to conformal time
$\eta$ as,

${\rm d}t=a(\eta){\rm d}\eta,$ (II.6)

in which the metric becomes, $g_{\mu\nu}=a^{2}(\eta)\eta_{\mu\nu}$
($\eta_{\mu\nu}$ denotes Minkowski metric in $D$ dimensions) and using
vielbein formalism Koksma:2009tc  one arrives at,

$i\gamma^{\mu}\nabla_{\mu}\Psi(x)=a^{-\frac{D+1}{2}}(\eta)i\gamma^{b}\partial_{b}\left(a^{\frac{D-1}{2}}(\eta)\Psi(x)\right)\,.$
(II.7)

Here $a(\eta)$ is the scale factor for a spatially flat homogeneous expanding
universe and $\eta$ is conformal time. The relation (II.7) is very useful as
it relates the covariant derivative acting on a spinor to the partial
derivative in (flat) tangent space.

###  II.1 Dirac equation

The Dirac equation which follows from Eq. (II.1) is given by,

$\left(i\gamma^{\mu}\nabla_{\mu}-\mathcal{M}\right)\Psi(x)=0\,.$ (II.8)

Using Eq. (II.7) and upon performing a conformal rescaling of the fermion
field,

$\Psi(x)\rightarrow a^{\frac{D-1}{2}}(\eta)\Psi(x)=\tilde{\Psi}(x)\,,$ (II.9)

the Dirac equation (II.8) can be recast as,

$\left(i\gamma^{b}\partial_{b}-a{\mathcal{M}}\right)\tilde{\Psi}(x)=0\,.$
(II.10)

###  II.2 Majorana fermions

The motivation to carefully investigate the dynamics of Majorana particles in
de Sitter space also arises, as stated above, from the observation that there
is a lack of conserved vector currents for Majorana particles. To see this we
can define the Dirac field in terms of its two chiral spinors as follows,

$\Psi(x)=\begin{pmatrix}\chi_{{}_{L}}(x)\\\ \chi_{{}_{R}}(x)\end{pmatrix}\,.$
(II.11)

The Lagrangian (II.1) then becomes (cf. Appendix A),

$\mathcal{L}=i\chi^{\dagger}_{{}_{L}}\bar{\sigma}^{\mu}\nabla_{\mu}\chi_{{}_{L}}+i\chi^{\dagger}_{{}_{R}}\sigma^{\mu}\nabla_{\mu}\chi_{{}_{R}}-m^{*}\chi^{\dagger}_{{}_{R}}\chi_{{}_{L}}-m\chi^{\dagger}_{{}_{L}}\chi_{{}_{R}}\,,$
(II.12)

where ${\sigma}^{\mu}=(\mathbb{1},\sigma^{i})$ and
$\bar{\sigma}^{\mu}=(\mathbb{1},-\sigma^{i})$, where $\sigma^{i}$ denote Pauli
matrices,
$i\sigma^{\mu}\nabla_{\mu}\chi_{{}_{R}}=a^{-\frac{D+1}{2}}i{\sigma}^{\mu}\partial_{\mu}\big{[}a^{\frac{D-1}{2}}\chi_{{}_{R}}\big{]}$
and
$i\bar{\sigma}^{\mu}\nabla_{\mu}\chi_{{}_{L}}=a^{-\frac{D+1}{2}}i\bar{\sigma}^{\mu}\partial_{\mu}\big{[}a^{\frac{D-1}{2}}\chi_{{}_{L}}\big{]}$.
Under a global U(1) transformation, $\chi_{{}_{L,R}}\rightarrow
e^{-i\theta}\chi_{{}_{L,R}}$, and therefore the fields $\Psi$ and $\bar{\Psi}$
transform infinitesimally as, $\Psi\rightarrow\Psi+\theta\Phi_{\theta}$ and
$\bar{\Psi}\rightarrow\bar{\Psi}+\theta\bar{\Phi}_{\theta}$, with
$\Phi_{\theta}=-i\Psi$ and $\bar{\Phi}_{\theta}=i\bar{\Psi}$, from which it
follows that the Lagrangian (II.12) remains invariant. These transformations
imply the following conserved (Noether) vector current for Dirac particles,

$j^{\mu}(x)\equiv\frac{\partial{\cal
L}}{\partial\partial_{\mu}\Psi}\Phi_{\theta}+\frac{\partial{\cal
L}}{\partial\partial_{\mu}\bar{\Psi}}\bar{\Phi}_{\theta}=\bar{\Psi}(x)\gamma^{\mu}\Psi(x)\,.$
(II.13)

Let us now consider the Majorana particles, defined by the Majorana condition:

$\Psi(x)=-i\gamma^{2}\Psi^{*}(x)=\begin{pmatrix}0&-\epsilon\\\
\epsilon&0\end{pmatrix}\begin{pmatrix}\chi^{*}_{{}_{L}}(x)\\\
\chi^{*}_{{}_{R}}(x)\end{pmatrix}\,.$ (II.14)

Here we have defined $i\sigma^{2}=\epsilon$, where $\epsilon^{ab}$ denotes the
antisymmetric tensor in two dimensions, defined by
$\epsilon^{12}=1=-\epsilon^{21}$ and $\epsilon^{11}=\epsilon^{22}=1$. Thus,
the Majorana field $\Psi_{M}(x)$ is defined as

$\Psi_{M}(x)=\begin{pmatrix}\chi_{{}_{L}}(x)\\\
\epsilon\chi^{*}_{{}_{L}}(x)\end{pmatrix}.$ (II.15)

The simplest way to arrive at the Lagrangian for Majorana fermions is to to
replace $\Psi$ by $\Psi_{M}$ into (II.15), which results in the following
Lagrangian, 444As argued below, there is a better way to write the Majorana
lagrangian as the normalization of the kinetic terms in (II.16) is not
canonical.

$\mathcal{L}_{M}=-i\chi_{{}_{L}}^{T}\epsilon\sigma^{\mu}\epsilon\,\nabla_{\\!\mu}\chi^{*}_{{}_{L}}+i\chi^{\dagger}_{{}_{L}}\bar{\sigma}^{\mu}\nabla_{\\!\mu}\chi_{{}_{L}}+m^{*}\chi^{T}_{{}_{L}}\epsilon\chi_{{}_{L}}-m\chi^{\dagger}_{{}_{L}}\epsilon\chi^{*}_{{}_{L}},$
(II.16)

It is clear that this Lagrangian is not invariant under the global U(1)
transformation discussed above, as the Majorana condition (II.14) is
inconsistent with the symmetry transformation discussed above.

An important consequence of this observation is that there is no conserved
vector current for Majorana fermions. This can be understood in terms of
Clifford algebra and the accompanying symmetry generators for Dirac and
Majorana particles. It can be noted that (in $D=4$) the Dirac particles have
16 generators of Clifford algebra whereas the Majorana particles have 15
generators, precisely due to the fact that the Majorana condition (II.14) is
inconsistent with the symmetry transformation discussed above. This means that
the symmetry algebra of Majorana particles is the Clifford algebra minus the
unity matrix, resulting in the symmetry generators of conformal group in $D=4$
BarrosoMancha:2019 . As a step towards understading ramifications of the lack
of conserved vector current for the dynamics of Majorana fermions, in this
paper we construct the Majorana propagator in de Sitter space, thereby paying
special attention to the implementation of the Majorana condition (II.14).
This will allow us to compare the Majorana propagator with that of Dirac
fermions originally constructed by Candelas and Raine in Ref. Candelas:1975du
and to the more general propagator discussed in Ref. Koksma:2009tc . We
already know that the absence of vector current plays an important role for
the CP-violating dynamics of mixing fermions, as the baryogenesis mechanism
from Ref. Garbrecht:2005rr  would be fundamentally changed in the absence of a
conserved vector current (cf. Eq. (5) of Ref. Garbrecht:2005rr ).

####  II.2.1 Canonical quantization

The above considerations and Eq. (II.1) suggest the following action and
Lagrangian for the Majorana fermions in its canonical form,

$S_{M}[\Psi]=\int d^{D}x\sqrt{-g}{\cal
L}_{M}\;;\quad\mathcal{L}_{M}=\frac{1}{4}\bar{\Psi}_{M}i\gamma^{\mu}\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{\mu}\\!\\!\Psi_{M}-\frac{1}{2}\bar{\Psi}_{M}\mathcal{M}\Psi_{M}\,,$
(II.17)

The additional factor 1/2 can be justified by noting that – due to the
Majorana condition – $\Psi_{M}$ and $\bar{\Psi}_{M}$ are not independent
fields, which can be clearly seen from,

$\Psi_{M}=(-i\gamma^{2})\Psi_{M}^{*}=(-i\gamma^{2})\gamma^{0}\left[\bar{\Psi}_{M}\right]^{T}\,\;\Longrightarrow\;\bar{\Psi}_{M}=\Psi_{M}^{T}\gamma^{0}(i\gamma^{2})\,.$
(II.18)

Since the condition is imposed at the level of the action, we refer to it as
the off-shell Majorana condition. This then means that, varying the action
(II.17) with respect to $\Psi(x)$ results in the Dirac equation that has the
same form as in Eq. (II.8), but with the Majorana condition imposed on
$\Psi_{M}$,

$\left(i\gamma^{\mu}\nabla_{\mu}-\mathcal{M}\right)\Psi_{M}(x)=0\,,\qquad\bar{\Psi}_{M}=\Psi_{M}^{T}\gamma^{0}(i\gamma^{2})\,.$
(II.19)

It is now quite easy to show that, transposing (II.19) and inserting
$\gamma^{0}(i\gamma^{2})(-i\gamma^{2})\gamma^{0}=1$ and commuting
$(-i\gamma^{2})\gamma^{0}$ with
$\big{[}i(\gamma^{\mu})^{T}\nabla_{\mu}-\mathcal{M}\big{]}$ (${\cal
M}^{T}={\cal M}$) results in

$\bar{\Psi}_{M}(x)\left(\\!-i\gamma^{\mu}\\!\stackrel{{\scriptstyle\leftarrow}}{{\nabla}}_{\\!\mu}-\mathcal{M}\right)=0\,,$
(II.20)

which is therefore not independent from Eq. (II.19). 555 A simple and well
studied analogy to the situation at hand is the case of complex and real
scalar fields. The canonically normalized Lagrangian for a complex scalar
$\Phi$ is, ${\cal
L}_{\Phi}=-g^{\mu\nu}(\partial_{\mu}\Phi^{*})(\partial_{\nu}\Phi)-m^{2}\Phi^{*}\Phi$,
where $\Phi^{*}$ and $\Phi$ are considered independent degrees of freedom. On
the other hand, the canonically normalized Lagrangian for a real scalar field
$\phi$ is, ${\cal
L}_{\phi}=-\frac{1}{2}g^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi)-\frac{1}{2}m^{2}\phi^{2}$,
where the reality condition $\phi^{*}(x)=\phi(x)$ plays the role of the
Majorana condition for fermions, which is just the charge transformation for
scalar fields.

The canonically normalized Lagrangian for the Majorana field $\Psi_{M}(x)$ in
(II.17) (when recast in terms of $2^{(D-2)/2}$-spinor fields) reads,

$\mathcal{L}_{M}=-\frac{1}{4}\chi_{{}_{L}}^{T}\epsilon
i\sigma^{\mu}\epsilon\,\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{\\!\mu}\chi^{*}_{{}_{L}}+\frac{1}{4}\chi^{\dagger}_{{}_{L}}i\bar{\sigma}^{\mu}\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{\\!\mu}\chi_{{}_{L}}+\frac{1}{2}m^{*}\chi^{T}_{{}_{L}}\epsilon\chi_{{}_{L}}-\frac{1}{2}m\chi^{\dagger}_{{}_{L}}\epsilon\chi^{*}_{{}_{L}}\,.$
(II.21)

Notice the 1/2 difference when compared with the naîve Majorana Lagrangian
(II.16). The Majorana condition was already used when writing (II.20) (to
remove $\chi_{{}_{R}}$), and therefore the spinors $\chi_{{}_{L}}$ and
$\chi^{*}_{{}_{L}}$ in (II.21) are independent complex spinor fields. The
corresponding canonical momenta are obtained by varying the action $S_{M}$
with respect to $\partial_{0}\chi_{{}_{L}}(x)$ and
$\partial_{0}\chi_{{}_{L}}^{*}(x)$ resulting in
$\left[\pi_{\chi_{L}^{*}}\right]_{b}=\frac{1}{2}a^{D-1}i\left[\chi_{L}^{T}\right]_{b}$
and
$\left[\pi_{\chi_{L}}\right]_{b}=\frac{1}{2}a^{D-1}i\big{[}\chi_{L}^{\dagger}\big{]}_{b}$,
where $b$ denotes a spinor index. When these are promoted to operators,
$\hat{\chi}_{{}_{L}}(x)$, $\hat{\chi}^{*}_{{}_{L}}(x)$, $\hat{\pi}_{\chi_{L}}$
and $\hat{\pi}_{\chi_{L}^{*}}$, canonical quantization then implies the
following anticommutators ($\hbar=1$),

$\begin{split}\\{\hat{\chi}_{{}_{L}}(\eta,\vec{x}),\hat{\pi}_{{\chi}_{{}_{L}}}(\eta,\vec{x}^{\,\prime})\\}_{bc}&=i\delta^{D-1}(\vec{x}-\vec{x}^{\,\prime})\delta_{bc}\\\
\\{\hat{\chi}^{*}_{{}_{L}}(\eta,\vec{x}),\hat{\pi}_{{\chi}^{*}_{{}_{L}}}(\eta,\vec{x}^{\,\prime})\\}_{bc}&=i\delta^{D-1}(\vec{x}-\vec{x}^{\,\prime})\delta_{bc},\end{split}$
(II.22)

or equivalently,

$\begin{split}\\{\hat{\chi}_{{}_{L}}(\eta,\vec{x}),\hat{\chi}^{\dagger}_{{}_{L}}(\eta,\vec{x}^{\,\prime})\\}_{bc}&=\frac{2\delta^{D-1}(\vec{x}-\vec{x}^{\,\prime})}{a(\eta)^{D-1}}\delta_{bc}\\\
\\{\hat{\chi}^{*}_{{}_{L}}(\eta,\vec{x}),\hat{\chi}^{T}_{{}_{L}}(\eta,\vec{x}^{\,\prime})\\}_{bc}&=\frac{2\delta^{D-1}(\vec{x}-\vec{x}^{\,\prime})}{a(\eta)^{D-1}}\delta_{bc},\end{split}$
(II.23)

and all other anti-commutators vanish. Note the conspicuous factor 2 on the
right hand side of these relations, which is absent in canonical quantization
of Dirac fermions. This factor is a consequence of our requirement to
canonically normalize the kinetic term for Majorana fermions, and it will
affect the normalization of our mode functions, and thus the propagator. There
is no deep physical meaning in it as this factor can be absorbed by a suitable
rescaling of $\chi_{L}$ and $\chi_{L}^{*}$. Finally, note that even though the
Lagrangian (II.21) suggests that there are two independent canonical momenta,
the canonical quantization reveals that there is only one independent anti-
commutator (they are related by transposition). This is to be contrasted with
the Dirac fermions, for which there are two independent canonical momenta,
each of them associated with the left and right handed fermion fields,
respectively. The Majorana condition imposes a dependency between the right-
and left-handed fermions, thus reducing the number of independent canonical
momenta to just one.

####  II.2.2 Helicity decomposition of Majorana fields

In this section we represent Majorana spinors in terms of mode sums, where the
modes are helicity eignespinors. Some attention is given to how to construct
helicity eigenspinors in $D$ dimensions, in which they are
$2^{(D-2)/2}-$component vectors.

Using the Majorana condition in Eq. (II.14) we can define the (conformally
rescaled) Majorana fields as follows,

$\displaystyle\hat{\tilde{\Psi}}_{M,\alpha}(x)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot\vec{x}}\left[\hat{b}_{h}(\vec{k})+\hat{b}^{\dagger}_{h}(-\vec{k})\right]A_{h,\alpha}(\eta,\vec{k}),$
(II.24) $\displaystyle\hat{\tilde{\bar{\Psi}}}_{M,\alpha}(x)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!{\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{-i\vec{k}\cdot\vec{x}}\left[\hat{b}_{h}(-\vec{k})+\hat{b}^{\dagger}_{h}(\vec{k})\right]\bar{A}_{h,\alpha}(\eta,\vec{k})}\,,$
(II.25)

the derivation of which is presented in Appendix B.
$\hat{\tilde{\Psi}}_{M,\alpha}(x)$ is the rescaled Majorana field,

$\hat{\tilde{\Psi}}_{M,\alpha}(x)=\begin{pmatrix}a^{\frac{D-1}{2}}\hat{\chi}_{{}_{L}}(x)\\\
\epsilon
a^{\frac{D-1}{2}}\hat{\chi}^{*}_{{}_{L}}(x)\end{pmatrix}=\begin{pmatrix}\hat{\rho}(x)\\\
\epsilon\hat{\rho}^{*}(x)\end{pmatrix}\,,$ (II.26)

and
$\hat{\tilde{\bar{\Psi}}}_{M,\alpha}(x)=\hat{\tilde{\Psi}}^{\dagger}_{M,\beta}(x)\gamma^{0}_{\beta\alpha}$.
From (II.23) one infers the (only) nontrivial anti-commutation relation for
the rescaled spinors,

$\begin{split}\\{\hat{\rho}(\eta,\vec{x}),\hat{\rho}^{\dagger}(\eta,\vec{x}^{\,\prime})\\}_{ab}=\\{\hat{\chi}_{{}_{L}}(\eta,\vec{x}),a^{D-1}\hat{\chi}^{\dagger}_{{}_{L}}(\eta,\vec{x}^{\,\prime})\\}_{ab}&=2\delta^{D-1}(\vec{x}-\vec{x}^{\,\prime})\delta_{ab}\\\
\end{split}$ (II.27)

The operators $\hat{b}_{h}(\vec{k})$ and $\hat{b}^{\dagger}_{h}(\vec{k})$ in
the mode decomposition (II.24) are the Majorana fermion annihilation and
creation operators, respectively. $\hat{b}_{h}(\vec{k})$ annihilates the
vacuum state $\lvert\Omega\rangle$,
$\hat{b}_{h}(\vec{k})\lvert\Omega\rangle=0$, and
$\hat{b}^{\dagger}_{h}(\vec{k})$ creates a particle of momentum $\vec{k}$ and
helicity $h$, and can be used to construct states that span the Hilbert space
of the problem. Note that the creation and annihilation operators in Eq.
(II.24) are identical, which is due to the fact that the positive and negative
frequency states (related by charge conjugation) do not independently
fluctuate, as it is dictated by the Majorana condition. The operators obey the
following anti-commutation relation,

$\begin{split}\\{\hat{b}_{h}(\vec{k}),\hat{b}^{\dagger}_{h^{\prime}}(\vec{k}^{\prime})\\}&=(2\pi)^{D-1}\delta^{D-1}(\vec{k}-\vec{k}^{\prime})\delta_{hh^{\prime}}\\\
\\{\hat{b}_{h}(\vec{k}),\hat{b}_{h^{\prime}}(\vec{k}^{\prime})\\}&=0\,,\hskip
51.21504pt\\{\hat{b}^{\dagger}_{h}(\vec{k}),\hat{b}^{\dagger}_{h^{\prime}}(\vec{k}^{\prime})\\}=0.\hskip
108.12054pt\end{split}$ (II.28)

$A_{h,\alpha}(\eta,\vec{k})$ and $\bar{A}_{h,\alpha}(\eta,\vec{k})$ in Eqs.
(II.24) and (II.25) are defined as,

$\begin{split}A_{h,\alpha}(\eta,\vec{k})&=\begin{pmatrix}\rho_{h}(\eta,\vec{k})\\\
\epsilon\rho^{*}(\eta,-\vec{k})\end{pmatrix}=\begin{pmatrix}L_{h}(\eta,\vec{k})\xi_{h}(\vec{k})\\\
L^{*}_{h}(\eta,-\vec{k})\epsilon\xi^{*}_{h}(-\vec{k})\end{pmatrix},\\\
\bar{A}_{h,\alpha}(\eta,\vec{k})&=\begin{pmatrix}-\rho^{T}_{h}(\eta,-\vec{k})\epsilon&\rho^{\dagger}_{h}(\eta,\vec{k})\end{pmatrix}=\begin{pmatrix}-L_{h}(\eta,-\vec{k})\xi_{h}^{T}(-\vec{k}\,)\epsilon\;&\;L_{h}(\eta,-\vec{k})\xi_{h}^{\dagger}(\vec{k}\,)\end{pmatrix}\,,\end{split}$
(II.29)

where $\rho_{h}(\eta,\vec{k})$ is the rescaled helicity eigenspinor in
momentum space, $L_{h}(\eta,\vec{k})$ is the Majorana particle mode function,
and $\xi_{h}(\vec{k})$ is the helicity eigenspinor defined by,

$\hat{h}\xi_{h}(\vec{k})=h\xi_{h}(\vec{k})\hskip 28.45274pt;\hskip
56.9055pth=\pm 1,$ (II.30)

where $\hat{h}$ is the helicity operator, which in $D=4$ has the simple form,
$\hat{h}=(\vec{k}/\|\vec{k}\|)\cdot\vec{\sigma}$ 666We have defined the
Majorana fields in D dimensions and the chirality operators can be readily
generalized to higher dimensions as was carefully done in Ref. Koksma:2009tc .
The properties listed for the helicity $2$-spinor in Appendix (Appendix C:
Properties of helicity 2-eigenspinor) Eqs. (VII.21–VII.23) and Eq. (VII.25)
stay the same in general $D$ dimensions. Thus, the definitions in $D=4$ that
we use in this section for the construction of the propagator remain valid in
$D$ dimensions. and using Eq. (VII.21) we can write $A_{h}(\eta,\vec{k})$ and
$\bar{A}_{h}(\eta,\vec{k})$,

$\begin{split}A_{h}(\eta,\vec{k})&=\begin{pmatrix}L_{h}(\eta,\vec{k}\,)\\\
he^{i\theta(\hat{k}_{x},\hat{k}_{y})}L^{*}_{h}(\eta,-\vec{k}\,)\end{pmatrix}\otimes\xi_{h}(\vec{k}\,)\\\
\bar{A}_{h}(\eta,\vec{k})&=\begin{pmatrix}he^{-i\theta(\hat{k}_{x},\hat{k}_{y})}L_{h}(\eta,-\vec{k}\,)\;&\;L^{*}_{h}(\eta,\vec{k}\,)\end{pmatrix}\otimes\xi^{\dagger}_{h}(\vec{k}\,)\,,\end{split}$
(II.31)

where
$\theta(\hat{k}_{x},\hat{k}_{y})=\tan^{-1}\big{(}\hat{k}_{y}/\hat{k}_{x}\big{)}$
(see Eq. (VII.20)).

In order to progress towards normalization of the Majorana mode functions,
note that the following identity for the chiral eigenvectors holds,

$\sum_{h=\pm}\left(\xi^{\dagger}_{h,a}(\vec{k})\otimes\xi_{h,b}(\vec{k})\right)=\delta_{ab}\hskip
11.38092pt,\hskip 28.45274pt\big{(}-\infty<\ k_{i}<+\infty\hskip
5.69046pt;\hskip 5.69046pti\in\\{x,y,z\\}\big{)}\,,$ (II.32)

which is proved in Appendix C. Next, since the helicity eigenstates
$L_{\pm}(\eta,\vec{k})$ do not couple to the vacuum, their normalization
should be independent on helicity,

$|L_{+}(\eta,\vec{k})|^{2}=|L_{-}(\eta,\vec{k})|^{2}.$ (II.33)

Making use of Eqs. (II.27–II.28) along with the definitions in Eq. (II.24) and
(II.26), one can see that the normalisation condition is given by,

$\sum_{h=\pm}\left[|L_{h}(\eta,\vec{k})|^{2}\left(\xi^{\dagger}_{h,a}(\vec{k})\otimes\xi_{h,b}(\vec{k})\right)\right]=\delta_{ab}\hskip
11.38092pt,\hskip 28.45274pt\big{(}-\infty<\ k_{i}<+\infty\hskip
5.69046pt;\hskip 5.69046pti\in\\{x,y,z\\}\big{)}\,.$ (II.34)

Finally, from Eqs. (II.32) and (II.33) from the above equation it follows
that,

$|L_{\pm}(\eta,\vec{k})|^{2}=1\hskip
28.45274pt\left(-\infty<k_{i}<\infty\hskip 5.69046pt;\hskip
5.69046pti\in\\{x,y,z\\}\right)\,.$ (II.35)

This normalization differs from the usual one for the Dirac fermions, in which
the mode functions squared are normalized to $1/2$, see e.g. Ref.
Koksma:2009tc ,Garbrecht:2006jm . This difference can be traced back to the
factor $2$ in the anticommutation relation (II.27). Eq. (II.33) can then be
written as follows,

$\sum_{h=\pm}\left[|L_{h}(\eta,\vec{k})|^{2}+|L_{h}(\eta,-\vec{k})|^{2}\right]=2\hskip
22.76228pt\hskip 28.45274pt\left(0<k_{i}<\infty\hskip 5.69046pt;\hskip
5.69046pti\in\\{x,y,z\\}\right)\,.$ (II.36)

Here it is important to notice that, due to the hemispherical construction of
helicity eigenmodes (discussed in detail in section II.2.3 below), for a given
helicity and momentum $(h=\pm,\vec{k})$, only one of the mode function
densities contributes (the other is zero). For example, for $h=+1$,
$|L_{+}(\eta,\vec{k})|^{2}$ contributes and $|L_{-}(\eta,\vec{k})|^{2}$ is
zero. This observation is of an essential importance for the correct
normalization of the Majorana propagator, which is one property that can be
used to distinguish between the Dirac and Majorana particles.

####  II.2.3 Mode function solutions

We can now describe the equations of motion for Majorana particles. For this
we use the Dirac equation given by Eq. (II.10) which gives the following
equations of motion for the mode functions for $k_{z}\geq 0$:

$\begin{split}i\partial_{\eta}L^{*}_{h}(\eta,-\vec{k})-h\|\vec{k}\|L^{*}_{h}(\eta,-\vec{k})-ahm^{*}L_{h}(\eta,\vec{k})e^{-i\theta}&=0\\\
i\partial_{\eta}L_{h}(\eta,\vec{k})+h\|\vec{k}\|L_{h}(\eta,\vec{k})-ahmL^{*}_{h}(\eta,-\vec{k})e^{i\theta}&=0\,.\end{split}$
(II.37)

Upon writing
$h=e^{i\frac{\pi}{2}\left(1-h\right)}=e^{-i\frac{\pi}{2}\left(1-h\right)}$ and
$m=|m|e^{i\phi}$, Eq. (II.37) becomes,

$\begin{split}ie^{\frac{i\bar{\phi}}{2}}\partial_{\eta}L^{*}_{h}(\eta,-\vec{k})-e^{\frac{i\bar{\phi}}{2}}h\|\vec{k}\|L^{*}_{h}(\eta,-\vec{k})-e^{-\frac{i\bar{\phi}}{2}}a|m|L_{h}(\eta,\vec{k})&=0\\\
ie^{-\frac{i\bar{\phi}}{2}}\partial_{\eta}L_{h}(\eta,\vec{k})+e^{-\frac{i\bar{\phi}}{2}}h\|\vec{k}\|L_{h}(\eta,\vec{k})-e^{\frac{i\bar{\phi}}{2}}a|m|L^{*}_{h}(\eta,-\vec{k})&=0,\end{split}$
(II.38)

where we have defined
$\bar{\phi}=\theta(\hat{k}_{x},\hat{k}_{y})+\frac{\pi}{2}\left(h-1\right)+\phi$.
It is convenient to transform to the positive/negative frequency basis,

$u_{h\pm}(\eta,\vec{k}\,)=\frac{1}{\sqrt{2}}\left(e^{-\frac{i\bar{\phi}}{2}}L_{h}(\eta,\vec{k}\,)\pm
e^{\frac{i\bar{\phi}}{2}}L^{*}_{h}(\eta,-\vec{k}\,)\right)\,,$ (II.39)

in which the mode functions $u_{h\pm}$ obey a Bessel’s differential equation,

$\partial^{2}_{\eta}u_{h\pm}(\eta,\vec{k}\,)+\left(\|\vec{k}\|^{2}+\frac{\frac{1}{4}-\left(\frac{1}{2}\mp\frac{i|m|}{H}\right)^{2}}{\eta^{2}}\right)u_{h\pm}(\eta,\vec{k}\,)=0.$
(II.40)

The Bunch-Davies vacuum solutions 777By judiciously introducing Bogolyubov
coefficients, one can quite straightforwardly construct positive and negative
frequency mode functions for more general (pure) Gaussian states. Details of
such a construction for Dirac fermions can be found in Ref. Koksma:2009tc .
Here we are primarily interested in the vacuum propagator in de Sitter, for
which the Bunch-Davies choice (II.41) sufficies. can be written in terms of
Hankel functions as follows (cf. Ref. Garbrecht:2006jm ),

$\begin{split}u_{h+}(\eta,\vec{k})&=e^{\frac{i\pi\nu_{+}}{2}}\sqrt{-\frac{\pi\|\vec{k}\|\eta}{4}}H^{(1)}_{\nu_{+}}(-\|\vec{k}\|\eta)\\\
u_{h-}(\eta,\vec{k})&=he^{\frac{i\pi\nu_{-}}{2}}\sqrt{-\frac{\pi\|\vec{k}\|\eta}{4}}H^{(1)}_{\nu_{-}}(-\|\vec{k}\|\eta)\,.\end{split}$
(II.41)

Here $H^{(1)}_{\nu}(-\|\vec{k}\|\eta)$ is Hankel function of the first kind,
which in the ultraviolet (equivalently in a distant past,
$k\eta\rightarrow-\infty$), reduces to the positive frequency vacuum solution.
The indices $\nu_{\pm}$ are given by,

$\nu_{\pm}=\frac{1}{2}\mp i\zeta\,,\qquad\zeta=\frac{|m|}{H}\,.\qquad$ (II.42)

We can then write the set of equations in Eq. (II.38) for $k_{z}\leq 0$ as
follows,

$\begin{split}ie^{\frac{i\bar{\phi}}{2}}\partial_{\eta}L^{*}_{h}(\eta,\vec{k}\,)-e^{\frac{i\bar{\phi}}{2}}h\|\vec{k}\|L^{*}_{h}(\eta,\vec{k}\,)+e^{-\frac{i\bar{\phi}}{2}}a|m|L_{h}(\eta,-\vec{k}\,)&=0\\\
ie^{-\frac{i\bar{\phi}}{2}}\partial_{\eta}L_{h}(\eta,-\vec{k}\,)+e^{-\frac{i\bar{\phi}}{2}}h\|\vec{k}\|L_{h}(\eta,-\vec{k}\,)+e^{\frac{i\bar{\phi}}{2}}a|m|L^{*}_{h}(\eta,\vec{k}\,)&=0\,,\end{split}$
(II.43)

for which we again transform into the positive/negative frequency basis,

$u_{h\pm}(\eta,-\vec{k})=\frac{1}{\sqrt{2}}\left(L_{h}(\eta,-\vec{k})e^{-\frac{i\bar{\phi}}{2}}\pm
L^{*}_{h}(\eta,\vec{k})e^{\frac{i\bar{\phi}}{2}}\right),$ (II.44)

where

$\begin{split}u_{h+}(\eta,-\vec{k}\,)&=e^{-\frac{i\pi\nu_{-}}{2}}\sqrt{-\frac{\pi\|\vec{k}\|\eta}{4}}H^{(2)}_{\nu_{-}}(-\|\vec{k}\|\eta)\\\
u_{h-}(\eta,-\vec{k}\,)&=-he^{\frac{-i\pi\nu_{+}}{2}}\sqrt{-\frac{\pi\|\vec{k}\|\eta}{4}}H^{(2)}_{\nu_{+}}(-\|\vec{k}\|\eta)\,.\end{split}$
(II.45)

Here $H^{(2)}_{\nu}(-\|\vec{k}\eta\|)$ is Hankel function of the second kind,
which in the ultraviolet regime reduces to the negative frequency vacuum
solution. From Eq. (II.36) one can infer that this set of solutions satisfies
the following normalization condition,

$\begin{split}|u_{h+}(\eta,\vec{k})|^{2}+|u_{h-}(\eta,\vec{k})|^{2}+|u_{h+}(\eta,-\vec{k})|^{2}+|u_{h-}(\eta,-\vec{k})|^{2}=2\,.\end{split}$
(II.46)

From Eq. (II.41)) and Eq. (II.45) one finds a complete set of solutions for
the mode functions,

$\displaystyle e^{-\frac{i\bar{\phi}}{2}}L_{h}(\eta,\vec{k})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\sqrt{-\frac{\pi\|\textbf{k}\|\eta}{8}}\left[e^{\frac{i\pi\nu_{+}}{2}}H^{(1)}_{\nu_{+}}(-\|\vec{k}\|\eta)+he^{\frac{i\pi\nu_{-}}{2}}H^{(1)}_{\nu_{-}}(-\|\vec{k}\|\eta)\right],$
(II.47) $\displaystyle
e^{\frac{i\bar{\phi}}{2}}L^{*}_{h}(\eta,-\vec{k})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\sqrt{-\frac{\pi\|\vec{k}\|\eta}{8}}\left[e^{\frac{i\pi\nu_{+}}{2}}H^{(1)}_{\nu_{+}}(-\|\textbf{k}\|\eta)-he^{\frac{i\pi\nu_{-}}{2}}H^{(1)}_{\nu_{-}}(-\|\vec{k}\|\eta)\right],$
(II.48) $\displaystyle e^{-\frac{i\bar{\phi}}{2}}L_{h}(\eta,-\vec{k})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\sqrt{-\frac{\pi\|\vec{k}\|\eta}{8}}\left[e^{\frac{-i\pi\nu_{-}}{2}}H^{(2)}_{\nu_{-}}(-\|\vec{k}\|\eta)-he^{\frac{-i\pi\nu_{+}}{2}}H^{(2)}_{\nu_{+}}(-\|\vec{k}\|\eta)\right],$
(II.49) $\displaystyle e^{\frac{i\bar{\phi}}{2}}L^{*}_{h}(\eta,\vec{k})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\sqrt{-\frac{\pi\|\vec{k}\|\eta}{8}}\left[e^{\frac{-i\pi\nu_{-}}{2}}H^{(2)}_{\nu_{-}}(-\|\vec{k}\|\eta)+h{\rm
e}^{\frac{-i\pi\nu_{+}}{2}}H^{(2)}_{\nu_{+}}(-\|\vec{k}\|\eta)\right].$
(II.50)

Figure 1:  Left panel shows the positive and negative frequency solutions
$u_{h\pm}(\eta,\vec{k})$, defined on the upper hemisphere ($k_{z}>0$) and
given by
$u_{h+}(\eta,\vec{k})=e^{\frac{i\pi\nu_{+}}{2}}\sqrt{-\frac{\pi\|\vec{k}\|\eta}{4}}H^{(1)}_{\nu_{+}}(-\|\vec{k}\|\eta)$,
$u_{h-}(\eta,\vec{k})=he^{\frac{i\pi\nu_{-}}{2}}\sqrt{-\frac{\pi\|\vec{k}\|\eta}{4}}H^{(1)}_{\nu_{-}}(-\|\vec{k}\|\eta)$.
Hence the solutions to the corresponding mode functions (right panel) are also
defined on the upper hemisphere.

Solutions in Eqs. (II.47)–(II.48) are valid for $k_{z}\geq 0$ and in Eqs.
(II.49)–(II.50) are valid for $k_{z}\leq 0$. To see this, we can look at Fig.
1 (left panel) where we see that on the upper hemisphere, the solutions are
described by $u_{h\pm}(\eta,\vec{k})$, which are given in terms of Hankel
functions of the first kind $H^{(1)}_{\nu}(-\|\vec{k}\|\eta)$ and thus, the
solutions for $e^{-\frac{i\bar{\phi}}{2}}L_{h}(\eta,\vec{k})$ and
$e^{\frac{i\bar{\phi}}{2}}L^{*}_{h}(\eta,-\vec{k})$ are also defined on this
hemisphere, as can be seen in Fig. 1 (right panel). However, when we consider
$\vec{k}\rightarrow-\vec{k}$, then the solutions are defined on the lower
hemisphere, shown in Fig. 2 (left panel), and described by
$u_{h\pm}(\eta,-\vec{k})$, which in turn are given in terms of Hankel
functions of the second kind $H^{(2)}_{\nu}(-\|\textbf{k}\|\eta)$, and
therefore the solutions for $e^{-\frac{i\bar{\phi}}{2}}L_{h}(\eta,-\vec{k})$
and $e^{\frac{i\bar{\phi}}{2}}L^{*}_{h}(\eta,\vec{k})$ are also defined on the
lower hemisphere, which is illustrated in Figure 2 (right panel).

Figure 2:  Left panel shows the positive and negative frequency solutions
$u_{h\pm}(\eta,-\vec{k})$ defined on the lower hemisphere ($k_{z}<0$) and
given by
$u_{h+}(\eta,-\vec{k})=e^{\frac{i\pi\nu_{-}}{2}}\sqrt{-\frac{\pi\|\vec{k}\|\eta}{4}}H^{(2)}_{\nu_{-}}(-\|\vec{k}\|\eta)$,
$u_{h-}(\eta,-\vec{k})=-he^{\frac{i\pi\nu_{+}}{2}}\sqrt{-\frac{\pi\|\vec{k}\|\eta}{4}}H^{(2)}_{\nu_{+}}(-\|\vec{k}\|\eta)$.
Right panel depicts the solutions to the mode functions, which are also
defined on the lower hemisphere.

##  III Majorana Propagator

Before we begin with construction of the Majorana propagator, let us define
some important quantities for de Sitter space. Firstly, the scale factor on
the expanding (Poincaré) patch of de Sitter space reads,

$a(\eta)=-\frac{1}{H\eta}\hskip 28.45274pt\big{(}\eta<0,\
H=\text{const.}\big{)}\,.$ (III.1)

Next, the following invariant distance (biscalar) functions are useful,

$\displaystyle y_{++}(x;x^{\prime})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\Delta
x^{2}_{++}(x;x^{\prime})}{\eta\eta^{\prime}}=\frac{1}{\eta\eta^{\prime}}\left(-\left(\lvert\eta-\eta^{\prime}\rvert-i\varepsilon\right)^{2}+\|\Delta\vec{x}\|^{2}\right)$
(III.2) $\displaystyle y_{+-}(x;x^{\prime})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{\eta\eta^{\prime}}\left(-\left(\eta-\eta^{\prime}+i\varepsilon\right)^{2}+\|\Delta\vec{x}\|^{2}\right)$
(III.3) $\displaystyle y_{-+}(x;x^{\prime})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{\eta\eta^{\prime}}\left(-\left(\eta-\eta^{\prime}-i\varepsilon\right)^{2}+\|\Delta\vec{x}\|^{2}\right)$
(III.4) $\displaystyle y_{--}(x;x^{\prime})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{\eta\eta^{\prime}}\left(-\left(\lvert\eta-\eta^{\prime}\rvert+i\varepsilon\right)^{2}+\|\Delta\vec{x}\|^{2}\right),$
(III.5)

where $i\varepsilon$ denotes an imaginary infinitesimal time shift
($\epsilon\ll 1$), which are useful for definition of various two-point
functions on de Sitter. Eqs. (III.1–III.4) imply the following identities,

$\begin{split}y_{++}(x;x^{\prime})&=\Theta(\eta-\eta^{\prime})y_{-+}(x;x^{\prime})+\Theta(\eta^{\prime}-\eta)y_{+-}(x;x^{\prime})\\\
y_{--}(x;x^{\prime})&=\Theta(\eta-\eta^{\prime})y_{+-}(x;x^{\prime})+\Theta(\eta^{\prime}-\eta)y_{-+}(x;x^{\prime})\,.\end{split}$
(III.6)

These distance functions are related to the geodesic distance on de Sitter
space $\ell(x;x^{\prime})$ as follows,
$4\sin^{2}[H\ell(x;x^{\prime})/2]=y_{ab}(x;x^{\prime})|_{\epsilon\rightarrow
0}\;(a,b=\pm)$.

###  III.1 Construction of the propagator

The Majorana propagator obeys the equation,

$\sqrt{-g}\left[i\gamma^{\mu}\nabla_{\mu}-{\cal
M}\right]iS_{F}(x;x^{\prime})=i\delta^{D}(x-x^{\prime})\,,$ (III.7)

and it ought to be symmetric under the exchange of $x$ and $x^{\prime}$. To
solve for the propagator on de Sitter space, we shall construct the propagator
by inserting the mode function decomposition (II.24) into the definition of
the propagator,

$\displaystyle iS_{\alpha\beta}(x;x^{\prime})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!a^{-\frac{D-1}{2}}(\eta)a^{-\frac{D-1}{2}}(\eta^{\prime})i\tilde{S}_{\alpha\beta}(x;x^{\prime})\,,\quad$
(III.8) $\displaystyle i\tilde{S}_{\alpha\beta}(x;x^{\prime})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\Big{\langle}\Omega\Big{\lvert}T\Big{[}\hat{\tilde{\Psi}}_{M,\alpha}(x)\hat{\tilde{\bar{\Psi}}}_{M,\beta}(x^{\prime})\Big{]}\Big{\rvert}\Omega\Big{\rangle}\hskip
213.39566pt$ $\displaystyle=$
$\displaystyle\\!\\!\Theta(\eta\\!-\\!\eta^{\prime})\Big{\langle}\Omega\Big{\lvert}\hat{\tilde{\Psi}}_{M,\alpha}(x)\hat{\bar{\tilde{\Psi}}}_{M,\beta}(x^{\prime})\Big{\rvert}\Omega\Big{\rangle}-\Theta(\eta^{\prime}\\!-\\!\eta)\Big{\langle}\Omega\Big{\lvert}\hat{\bar{\tilde{\Psi}}}_{M,\beta}(x^{\prime})\hat{\tilde{\Psi}}_{M,\alpha}(x)\Big{\rvert}\Omega\Big{\rangle}.\quad$

This results in,

$\begin{split}i\tilde{S}_{\alpha\beta}(x;x^{\prime})=\Theta(\eta\\!-\\!\eta^{\prime})\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}A_{h,\alpha}(\eta,\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},\vec{k})\\\
-\Theta(\eta^{\prime}\\!-\\!\eta)\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{-i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}A_{h,\alpha}(\eta,-\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},-\vec{k}),\end{split}$
(III.10)

where $A_{h,\alpha}(\eta,\vec{k})$ and
$\bar{A}_{h,\beta}(\eta^{\prime},\vec{k})$ are defined in Eq. (II.31) and we
have used the anti-commutation relation in Eq. (II.28) (with the help of which
we have performed one of the momentum integrals).

Further steps in the construction of the propagator are analogous to those in
Ref. Koksma:2009tc . The propagator is computed, component-by-component, using
the solutions for the mode functions from Eqs. (II.47–II.50). The propagator
components are defined in terms of the projection operators using the
following identity,

$\left(i\gamma^{b}\partial_{b}+a|m|\right)\frac{1\pm\gamma^{0}}{2}=\frac{1}{2}\begin{pmatrix}\pm
i\partial_{\eta}\pm
i\sigma^{i}\partial_{i}+a|m|&i\partial_{\eta}+i\sigma^{i}\partial_{i}\pm|m|\\\
i\partial_{\eta}-i\sigma^{i}\partial_{i}\pm a|m|&\pm i\partial_{\eta}\mp
i\sigma^{i}\partial_{i}+a|m|\end{pmatrix}.$ (III.11)

After several steps, 888For details we refer to Ref. Koksma:2009tc . and in
particular upon making use of Eq. (II.7), one arrives at the Majorana
propagator in the form,

$\left[iS_{F}(x;x^{\prime})\right]^{\tt
rot}=a\left(i\gamma^{\\!\mu}\nabla_{\mu}\\!+\\!|m|\right)\frac{H^{D-2}}{\sqrt{aa^{\prime}}}\left[\frac{1+\gamma^{0}}{2}iS^{+}(x;x^{\prime})\\!+\\!\frac{1-\gamma^{0}}{2}i\tilde{S}^{+}(x;x^{\prime})\right],$
(III.12)

where $H^{D-2}=\left(aa^{\prime}\eta\eta^{\prime}\right)^{\frac{2-D}{2}}$ and
$H$ is the Hubble rate, which is constant on de Sitter. The functions
$iS^{+}(x;x^{\prime})$ and $i\tilde{S}^{+}(x;x^{\prime})$ are biscalars
defined as follows,

$\begin{split}iS^{+}(x;x^{\prime})&=\frac{\Gamma\left(\frac{D}{2}\\!+\\!i\zeta\right)\Gamma\left(\frac{D-2}{2}\\!-\\!i\zeta\right)}{(4\pi)^{D/2}\Gamma(\frac{D}{2})}\times{}_{2}F_{1}\left(\frac{D}{2}\\!+\\!i\zeta,\frac{D-2}{2}\\!-\\!i\zeta;\frac{D}{2};1-\frac{y_{++}(x;x^{\prime})}{4}\right)\\\
i\tilde{S}^{+}(x;x^{\prime})&=\frac{\Gamma\left(\frac{D}{2}\\!-\\!i\zeta\right)\Gamma\left(\frac{D-2}{2}\\!+\\!i\zeta\right)}{(4\pi)^{D/2}\Gamma(\frac{D}{2})}\times{}_{2}F_{1}\left(\frac{D}{2}\\!-\\!i\zeta,\frac{D-2}{2}\\!+\\!i\zeta;\frac{D}{2};1\\!-\\!\frac{y_{++}(x;x^{\prime})}{4}\right)\,,\end{split}$
(III.13)

where ${}_{2}F_{1}$ denote Gauss’ hypergeometric functions and we have used
the definitions listed in Eq. (III.6). To calculate the integral we have also
made use of the properties of Hankel functions in Appendix D (the details of
the calculation can be found in an Appendix B of Koksma:2009tc , Eq. (B.1) and
(B.4–B.6)) and the relevant integrals needed for integration of products of
Bessel functions can be found in identity (6.578.10) of Ref. Gradshteyn:2014 .

The propagator (III.12) is still in the basis in which the phase $\phi={\rm
Arctan}[m_{2}/m_{1}]\;$ ($m=m_{1}+im_{2}$) of the mass term is removed, and it
is the propagator for Majorana fermions on de Sitter whose mass is real. When
the mass is complex however, one ought to bring it back to the standard form
by performing an additional chiral rotation,

$iS_{F}(x;x^{\prime})={\rm
e}^{-\frac{i\phi}{2}\gamma^{5}}\left[iS_{F}(x;x^{\prime})\right]^{\tt rot}{\rm
e}^{-\frac{i\phi}{2}\gamma^{5}}\,.$ (III.14)

When commuted through the kinetic operator in (III.12), this then gives,

$iS_{F}(x;x^{\prime})=a\left(i\gamma^{\\!\mu}\nabla_{\mu}+{\cal
M}^{\dagger}\right)\frac{H^{D-2}}{\sqrt{aa^{\prime}}}\left[\frac{1\\!+\\!{\rm
e}^{i\phi\gamma^{5}}\gamma^{0}}{2}iS^{+}(x;x^{\prime})\\!+\\!\frac{1\\!-\\!{\rm
e}^{i\phi\gamma^{5}}\gamma^{0}}{2}i\tilde{S}^{+}(x;x^{\prime})\right]\,,\quad$
(III.15)

where we made use of, ${\rm
e}^{-\frac{i\phi}{2}\gamma^{5}}\left(i\gamma^{\\!\mu}\nabla_{\mu}+|m|\right)=\left(i\gamma^{\\!\mu}\nabla_{\mu}+{\cal
M}^{\dagger}\right){\rm e}^{\frac{i\phi}{2}\gamma^{5}}$. This means that the
split between the positive and negative frequency poles in (III.15) is not
facilitated by the usual positive an negative energy projectors,
$P_{\pm}=\frac{1}{2}(1\pm\gamma^{0})$, but instead by more complicated
projectors that involve chiral rotations, $P_{\pm}^{5}=\frac{1}{2}(1\pm{\rm
e}^{i\gamma^{5}\theta}\gamma^{0})$. 999Notice that these operators are still
proper projectors, in the sense that they obey,
$[P_{\pm}^{5}]^{2}=P_{\pm}^{5}$ and $P_{+}^{5}P_{-}^{5}=0=P_{-}^{5}P_{+}^{5}$,
where we made use of ${\rm
e}^{i\gamma^{5}\phi}=\cos(\phi)+i\gamma^{5}\sin(\phi)$. This is a new result
for the vacuum propagators of both Dirac fermions and Majorana fermions on de
Sitter space.

When compared with the corresponding Dirac propagator on de Sitter, we see
that – in the special case when the mass is real – our Majorana propagator
(III.13–III.15) becomes identical to the Dirac propagator of Refs.
Candelas:1975du ; Koksma:2009tc , with the remark that our propagator
represents generalization to the complex mass case. The fact that the two
propagators agree should not surprise us in retrospect, because the
propagators measure statistical properties of vacuum fluctuations, which – in
the absence of CP violation – are identical for positive and negative
frequency modes for Dirac particles, thus yielding statistically identical
contributions from both frequency poles in the de Sitter vacuum. The same is
true by construction 101010Recall that the creation and annihilation operators
are identical for both frequency poles. for Majorana particles, cf. footnote
5.

###  III.2 Minkowski limit of the propagator

Here we consider the Minkowski limit of the propagator constructed in the
previous section in Eq. (III.15). To do this, we will first expand the
hypergeometric functions $iS^{+}(x;x^{\prime})$ and
$i\tilde{S}^{+}(x;x^{\prime})$ in Eq. (III.15) using the identity (9.131.2)
from Ref. Gradshteyn:2014 ,

$\begin{split}iS^{+}(x;x^{\prime})&=\frac{\Gamma\left(\frac{D}{2}+i\zeta\right)\Gamma\left(\frac{D-2}{2}-i\zeta\right)}{\left(4\pi\right)^{D/2}\Gamma(\frac{D}{2})}\\\
\times&\bigg{\\{}\frac{\Gamma(\frac{D}{2})\Gamma\left(\frac{D}{2}-1\right)}{\Gamma\left(\frac{D-2}{2}-i\zeta\right)\Gamma\left(\frac{D}{2}+i\zeta\right)}\left(\frac{y_{++}(x;x^{\prime})}{4}\right)^{\frac{2-D}{2}}\times{}_{2}F_{1}\left(1+i\zeta,-i\zeta;\frac{4-D}{2},\frac{y_{++}(x,x^{\prime})}{4}\right)\hskip
85.35826pt\\\
&+\frac{\Gamma(\frac{D}{2})\Gamma\left(\frac{2-D}{2}\right)}{\Gamma\left(1+i\zeta\right)\Gamma\left(-i\zeta\right)}\times{}_{2}F_{1}\left(\frac{D-2}{2}-i\zeta,\frac{D}{2}+i\zeta;\frac{D}{2},\frac{y_{++}(x;x^{\prime})}{4}\right)\bigg{\\}}\end{split}$
(III.16) $\displaystyle i\tilde{S}^{+}(x;x^{\prime})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\Gamma\left(\frac{D}{2}-i\zeta\right)\Gamma\left(\frac{D-2}{2}+i\zeta\right)}{\left(4\pi\right)^{D/2}\Gamma(\frac{D}{2})}$
(III.18)
$\displaystyle\hskip-56.9055pt\times\,\bigg{\\{}\frac{\Gamma(\frac{D}{2})\Gamma\left(\frac{D}{2}-1\right)}{\Gamma\left(\frac{D-2}{2}+i\zeta\right)\Gamma\left(\frac{D}{2}-i\zeta\right)}\left(\frac{y_{++}(x,x^{\prime})}{4}\right)^{\frac{2-D}{2}}\hskip-8.5359pt\times{}_{2}F_{1}\left(1-i\zeta,+i\zeta;\frac{4-D}{2},\frac{y_{++}(x;x^{\prime})}{4}\right)$
$\displaystyle\hskip-28.45274pt+\,\frac{\Gamma(\frac{D}{2})\Gamma\left(\frac{2-D}{2}\right)}{\Gamma\left(1-i\zeta\right)\Gamma\left(i\zeta\right)}\times{}_{2}F_{1}\left(\frac{D-2}{2}+i\zeta,\frac{D}{2}-i\zeta;\frac{D}{2},\frac{y_{++}(x;x^{\prime})}{4}\right)\bigg{\\}}\,,\quad$

which transforms the propagator to the form that is convenient near the
lightcone, where $y_{++}(x;x^{\prime})\sim 0$.

The Minkowski limit is defined by $H\rightarrow 0$ and
$a(\eta),a(\eta^{\prime})\rightarrow 1$, which also means that, when expanding
the hypergeometric functions we can use that $(1\pm i\zeta)_{n}(\mp
i\zeta)_{n}\simeq(|m|/H)^{2n}$ and $(y_{++}(x;x^{\prime})/4)^{n}\simeq
H^{2n}(\Delta x_{++}^{2})^{n}$, where $\Delta x_{++}^{2}\equiv\Delta
x_{++}^{2}(x;x^{\prime})$ is defined in Eq. (III.2). Inserting these into
(III.16–III.18) yields (in the Minkowski limit),

$\displaystyle iS^{+}(x;x^{\prime})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!i\tilde{S}^{+}(x;x^{\prime})$ $\displaystyle\simeq$
$\displaystyle\\!\\!\frac{\Gamma\left(\frac{D-2}{2}\right)\Gamma\left(\frac{4-D}{2}\right)\left(\frac{\lvert
m\rvert}{H}\right)^{D-2}}{2\left(2\pi\right)^{D/2}}\;\frac{I_{\frac{D-2}{2}}\left(\lvert
m\rvert\sqrt{\Delta x_{++}^{2}}\right)+I_{-\frac{D-2}{2}}\left(\lvert
m\rvert\sqrt{\Delta x_{++}^{2}}\right)}{\left(\lvert m\rvert\sqrt{\Delta
x_{++}^{2}}\right)^{\frac{D-2}{2}}}\,,\quad$

where $I_{\pm\nu}(z)$ denote modified Bessel functions of the first kind. To
get the last piece $\propto I_{-\frac{D-2}{2}}$ in Eq. (III.2) we made use of
the following asymptotic property of the gamma functions,

$\frac{\Gamma\left(\frac{D}{2}-i\zeta\right)\Gamma\left(\frac{D-2}{2}+i\zeta\right)}{\Gamma\left(1-i\zeta\right)\Gamma\left(i\zeta\right)}\approx(i\zeta)^{D-2}\qquad({\rm
correct\;when}\;|i\zeta|\rightarrow\infty)\,.\quad$ (III.20)

Now using the following definition,

$I_{\nu}(z)+I_{-\nu}(z)=\frac{2}{\Gamma\left(\nu\right)\Gamma\left(1-\nu\right)}K_{\nu}(z)\,,$
(III.21)

where $K_{\nu}(z)$ denotes modified Bessel’s function of the second kind, one
can recast Eq. (III.2) as,

$iS^{+}(x;x^{\prime})=i\tilde{S}^{+}(x;x^{\prime})\simeq\frac{\left(\frac{\lvert
m\rvert}{H}\right)^{D-2}}{\left(2\pi\right)^{D/2}}\;\frac{K_{\frac{D-2}{2}}\left(\lvert
m\rvert\sqrt{\Delta x_{++}^{2}}\right)}{\left(\lvert m\rvert\sqrt{\Delta
x_{++}^{2}}\right)^{\frac{D-2}{2}}}\,.\quad$ (III.22)

From this it follows that the Minkowski limit the propagator (III.15) is,

$iS_{F}(x;x^{\prime})\;\longrightarrow\;\left[iS_{F}(x;x^{\prime})\right]_{\rm
Mink}=\left(i\gamma^{\mu}\partial_{\mu}+\mathcal{M}^{\dagger}\right)i\Delta_{\lvert
m\rvert}(x;x^{\prime})\,,\quad$ (III.23)

where $i\Delta_{\lvert m\rvert}(x;x^{\prime})$ denotes the massive scalar
propagator in Minkowski space (whose mass equals $|m|$),

$i\Delta_{\lvert m\rvert}(x;x^{\prime})=\frac{\lvert
m\rvert^{D-2}}{\left(2\pi\right)^{D/2}}\;\frac{K_{\frac{D-2}{2}}\left(\lvert
m\rvert\sqrt{\Delta x_{++}^{2}}\right)}{\left(\lvert m\rvert\sqrt{\Delta
x_{++}^{2}}\right)^{\frac{D-2}{2}}}\,.\quad$ (III.24)

The propagator (III.23) obeys the Minkowski space Dirac equation with the
correct source,

$\left(i\gamma^{\mu}\partial_{\mu}-{\cal
M}\right)\left[iS_{F}(x;x^{\prime})\right]_{\rm
Mink}=i\delta^{D}(x\\!-\\!x^{\prime})\,.\quad$ (III.25)

As it is evident, the chiral parts of the projectors from Eq. (III.15) do not
contribute in the Minkowski limit. It is important to ask whether the chiral
part of the projector in the Majorana propagator in (III.15) can contribute to
CP-violation. This question can be properly addressed only when the propagator
is used in loop calculations, in which vertices contain CP violation. Since
such calculations are quite involved, this question is beyond the scope of
this paper.

###  III.3 Charge conjugation

Charge conjugation (CP) of the Majorana propagator is important for
understanding leptogenesis which often involves CP-violating processes during
the decay of massive Majorana neutrinos. To understand this better let us look
at the CP transformation of the propagator given by,

$iS_{F}(x;x^{\prime})\;\stackrel{{\scriptstyle\rm
CP}}{{\longrightarrow}}\;\left(-i\gamma_{2}\right)\left(iS_{F}(x;x^{\prime})\right)^{*}\left(-i\gamma_{2}\right)=-iS_{F}(x;x^{\prime})\,,$
(III.26)

which immediately follows from the Majorana conditions,

$\Psi(x)=(-i\gamma^{2})\Psi^{*}(x)\,,\qquad\bar{\Psi}(x)=-\bar{\Psi}^{*}(x)(-i\gamma^{2})$
(III.27)

and the propagator definition (III.1).

To see how the property (III.26) pans out for our propagator (III.15), it is
instructive to go through the mode decomposition (II.24—II.25) to understand
how CP-violation realises (III.26). Let us begin our analysis with Eq.
(III.10),

$\begin{split}\left\langle
T[\hat{\tilde{\Psi}}_{M,\alpha}(x)\hat{\tilde{\bar{\Psi}}}_{M,\beta}(x^{\prime})]\right\rangle=\Theta(\eta-\eta^{\prime})\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}A_{h,\alpha}(\eta,\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},\vec{k})\\\
-\Theta(\eta^{\prime}-\eta)\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{-i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}A_{h,\alpha}(\eta,-\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},-\vec{k})\,,\end{split}$
(III.28)

where $A_{h,\alpha}(\eta,-\vec{k})$ and
$\bar{A}_{h,\beta}(\eta^{\prime},-\vec{k})$ are defined as,

$\displaystyle A_{h,\alpha}(\eta,-\vec{k})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\begin{pmatrix}L_{h}(\eta,-\vec{k})\xi_{h}(-\vec{k})\\\
-he^{i\theta(\hat{k}_{x},\hat{k}_{y})}L^{*}_{h}(\eta,\vec{k})\xi_{h}(-\vec{k})\end{pmatrix}$
(III.29) $\displaystyle\bar{A}_{h,\beta}(\eta^{\prime},-\vec{k})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\begin{pmatrix}-he^{-i\theta(\hat{k}_{x},\hat{k}_{y})}L_{h}(\eta^{\prime},\vec{k})\xi_{h}^{\dagger}(-\vec{k})\;\;&L^{*}_{h}(\eta^{\prime},-\vec{k})\xi^{\dagger}_{h}(-\vec{k})\end{pmatrix}\,.\quad$
(III.30)

We can also see how
$A_{h,\alpha}(\vec{k,\eta})\bar{A}_{h,\beta}(\eta^{\prime},\vec{k})$ and
$A_{h,\alpha}(-\vec{k,\eta})\bar{A}_{h,\beta}(\eta^{\prime},-\vec{k})$
transform to each other under charge conjugation,

$(-i\gamma_{2})_{\alpha\lambda}\left(A_{h,\lambda}(\eta,\vec{k})\bar{A}_{h,\delta}(\eta^{\prime},\vec{k})\right)^{*}(-i\gamma_{2})_{\delta\beta}=-A_{h,\alpha}(\eta,-\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},-\vec{k})$
(III.31)
$(-i\gamma_{2})_{\alpha\lambda}\left(A_{h,\lambda}(\eta,-\vec{k})\bar{A}_{h,\delta}(\eta^{\prime},-\vec{k})\right)^{*}(-i\gamma_{2})_{\delta\beta}=-A_{h,\alpha}(\eta,\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},\vec{k}).$
(III.32)

Applying the charge conjugation transformation to Eq. (III.28) one obtains,

$\displaystyle\hskip-42.67912pt(-i\gamma_{2})_{\alpha\lambda}\Big{(}\Big{\langle}T\Big{[}\hat{\tilde{\Psi}}_{M,\lambda}(x)\hat{\tilde{\bar{\Psi}}}_{M,\delta}(x^{\prime})\Big{]}\Big{\rangle}\Big{)}^{*}(-i\gamma_{2})_{\delta\beta}$
(III.34) $\displaystyle=$
$\displaystyle\\!\\!\Theta(\eta\\!-\\!\eta^{\prime})\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{-i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}(-i\gamma_{2})_{\alpha\lambda}\left(A_{h,\lambda}(\eta,\vec{k})\bar{A}_{h,\delta}(\eta^{\prime},\vec{k})\right)^{*}(-i\gamma_{2})_{\delta\beta}\hskip
5.69046pt$
$\displaystyle\\!\\!-\,\Theta(\eta^{\prime}\\!-\\!\eta)\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}(-i\gamma_{2})_{\alpha\lambda}\left(A_{h,\lambda}(\eta,-\vec{k})\bar{A}_{h,\delta}(\eta^{\prime},-\vec{k})\right)^{*}(-i\gamma_{2})_{\delta\beta}\hskip
14.22636pt\;$ $\displaystyle=$
$\displaystyle\\!\\!-\Bigg{[}\Theta(\eta\\!-\\!\eta^{\prime})\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{-i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}A_{h,\alpha}(\eta,-\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},-\vec{k})\hskip
8.5359pt$ $\displaystyle\\!\\!\hskip
28.45274pt-\,\Theta(\eta^{\prime}\\!-\\!\eta)\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}A_{h,\alpha}(\eta,\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},\vec{k})\Bigg{]}.\hskip
5.69046pt$

If we were to calculate the integrals in Eq. (III.34), we would obtain,

$\begin{split}&\left(aa^{\prime}\right)^{-\frac{D-1}{2}}(-i\gamma_{2})_{\alpha\lambda}\Big{(}\Big{\langle}T\Big{[}\hat{\tilde{\Psi}}_{M,\lambda}(x)\hat{\tilde{\bar{\Psi}}}_{M,\delta}(x^{\prime})\Big{]}\Big{\rangle}\Big{)}^{*}(-i\gamma_{2})_{\delta\beta}\\\
&\hskip 14.22636pt=\
\bigg{\\{}a\big{(}i\gamma^{\mu}\nabla_{\mu}+|m|\big{)}\frac{H^{D-2}}{\sqrt{aa^{\prime}}}\bigg{[}\bigg{(}\frac{1+\gamma^{0}}{2}\bigg{)}S^{-}(x;x^{\prime})+\bigg{(}\frac{1-\gamma^{0}}{2}\bigg{)}\tilde{S}^{-}(x;x^{\prime})\bigg{]}\bigg{\\}}_{\alpha\beta}\,,\hskip
2.84544pt\end{split}$ (III.35)

where $iS^{-}(x;x^{\prime})$ and $i\tilde{S}^{-}(x;x^{\prime})$ in contrast to
$iS^{+}(x;x^{\prime})$ and $i\tilde{S}^{+}(x;x^{\prime})$ are given by,

$\begin{split}iS^{-}(x;x^{\prime})&=\frac{\Gamma\left(\frac{D}{2}\\!+\\!i\zeta\right)\Gamma\left(\frac{D-2}{2}\\!-\\!i\zeta\right)}{(4\pi)^{D/2}\Gamma(\frac{D}{2})}{}_{2}F_{1}\left(\frac{D}{2}\\!+\\!i\zeta,\frac{D\\!-\\!2}{2}\\!-\\!i\zeta;\frac{D}{2};1\\!-\\!\frac{y_{--}(x;x^{\prime})}{4}\right)\\\
i\tilde{S}^{-}(x;x^{\prime})&=\frac{\Gamma\left(\frac{D}{2}\\!-\\!i\zeta\right)\Gamma\left(\frac{D-2}{2}\\!+\\!i\zeta\right)}{(4\pi)^{D/2}\Gamma(\frac{D}{2})}{}_{2}F_{1}\left(\frac{D}{2}\\!-\\!i\zeta,\frac{D\\!-\\!2}{2}\\!+\\!i\zeta;\frac{D}{2};1\\!-\\!\frac{y_{--}(x;x^{\prime})}{4}\right).\end{split}$
(III.36)

Notice that, upon the charge conjugation transformation, we have the
propagator in terms of $S^{-}(x;x^{\prime})$ and
$\tilde{S}^{-}(x;x^{\prime})$, which is obtained when the time ordering for
the Dyson propagator is imposed. In order to understand this result, recall
the Stückelberg rule, which states that propagation of particles in the
positive time direction (governed by the time ordering in the Feynman
propagator) is identical to propagation of antiparticles in the negative time
direction. Reversing the direction of time, the rule tells us that propagation
of particles in the negative time direction is equivalent to propagation of
antiparticles in the positive time direction, which explains the Dyson time
order in Eq. (III.36). Since we are dealing with Majorana fermions,
propagation of antiparticles in the positive time direction must be related in
a simple way to propagation of particles in the positive time direction. To
see that, let us have a look at Eq. (III.34), in which we see that the
momentum integrals contain opposite phases when compared with that in Eq.
(III.10). This is cured by flipping the sign of the integrated momenta
resulting in,

$\begin{split}&\hskip-28.45274pt(-i\gamma_{2})_{\alpha\lambda}\Big{(}\Big{\langle}T\Big{[}\hat{\tilde{\Psi}}_{M,\lambda}(x)\hat{\tilde{\bar{\Psi}}}_{M,\delta}(x^{\prime})\Big{]}\Big{\rangle}\Big{)}^{*}(-i\gamma_{2})_{\delta\beta}\\\
&=-\bigg{[}\Theta(\eta-\eta^{\prime})\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}A_{h,\alpha}(\eta,\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},\vec{k})\\\
&\hskip
28.45274pt-\,\Theta(\eta^{\prime}-\eta)\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{-i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}A_{h,\alpha}(\eta,-\vec{k})\bar{A}_{h,\beta}(\eta^{\prime},-\vec{k})\bigg{]}.\end{split}\quad$
(III.37)

Evaluating this integral we get the result that we calculated before in Eq.
(III.15), however with an overall negative sign, as expected,

$\begin{split}&\hskip-28.45274pt(-i\gamma_{2})_{\alpha\lambda}\Big{(}\Big{\langle}T\Big{[}\hat{\tilde{\Psi}}_{M,\lambda}(x)\hat{\tilde{\bar{\Psi}}}_{M,\delta}(x^{\prime})\Big{]}\Big{\rangle}\Big{)}^{*}(-i\gamma_{2})_{\delta\beta}\\\
&=-\bigg{\\{}a\big{(}i\gamma^{\mu}\nabla_{\mu}+|m|\big{)}\frac{H^{D-2}}{\sqrt{aa^{\prime}}}\left[\frac{1+\gamma^{0}}{2}S^{+}(x;x^{\prime})+\frac{1-\gamma^{0}}{2}\tilde{S}^{+}(x;x^{\prime})\right]\bigg{\\}}_{\alpha\beta}\\\
&=-\Big{\langle}T\Big{[}\hat{\tilde{\Psi}}_{M}(x)\hat{\tilde{\bar{\Psi}}}_{M}(x^{\prime})\Big{]}\Big{\rangle}\,.\hskip
14.22636pt\end{split}$ (III.38)

Now our result is in accordance with our expectation in Eq. (III.26), in which
it is expressed in terms of time ordered $S^{+}(x;x^{\prime})$ and
$\tilde{S}^{+}(x;x^{\prime})$, which compose the Feynman propagator
(III.14–III.15).

An important question is whether the minus sign (III.38) can have any physical
significance. Here we refrain from making any detailed analysis, but just
remark that fermionic signs can lead to observable effect in two dimensional
($D=1+2=3$) topological systems, a notable example being fermionic systems
considered for building quantum computers in which braiding of the fermionic
wave function plays an important role Alicea:2010 ,Ivanov:2000mjr . The charge
conjugation operation of the Majorana propagator can be seen as the half-
braiding 111111The operation performed here is an exchange and thus it
corresponds to a half-braid. A full braid involves the majorana fermion going
around the other fermion, back to its original site. operation between two
Majorana fermions in two spatial dimensions, such that the transformation
$\mathcal{C}\hat{\tilde{\Psi}}_{M}(x)$ and
$\hat{\tilde{\bar{\Psi}}}_{M}(x^{\prime})\mathcal{C}^{\dagger}$ is similar to
exchange of particles as shown in Fig. 3 (left panel). After this half-
braiding transformation, the particles exchange their position, however, one
of the Majorana fermion operator incurs a negative sign which can be seen in
Fig. 3 (right panel). Therefore the half-braiding operation follows the
transformation rule
$\hat{\tilde{\Psi}}_{M}(x)\rightarrow\hat{\tilde{\Psi}}_{M}(x^{\prime})$ and
$\hat{\tilde{\Psi}}_{M}(x^{\prime})\rightarrow-\hat{\tilde{\Psi}}_{M}(x)$. It
must be noted that the negative sign that we see in the half-braiding
operation in two dimensions is physically realized as different qubit states
that are topologically protected Beenakker:2020 . The question whether the
charge conjugation operation of Majorana fermions in general $D$ dimensions
has any physical relevance, and whether it can be related to the braiding
operation in $2+1$ dimensions requires further analysis.

Figure 3:  Left panel shows that the two Majorana fermions $\Psi(x)$ and
$\Psi(x^{\prime})$ confined in two spatial dimensions at positions $x$ and
$x^{\prime}$, denoted by blue and red dots, respectively. The arrows indicated
how they are exchanged (half-braided) with each other. Right panel shows that,
after the exchange: $\Psi(x)\rightarrow\Psi(x^{\prime})$ and
$\Psi(x^{\prime})\rightarrow-\Psi(x)$. Although the exchange can be visualised
by how the blue and red dots have changed their position, this is only for a
visual representation. However, in reality such a distinction is not possible.

##  IV One-Loop Effective Action

The one-loop effective action for Majorana fermions is given by 121212When
compared with the Dirac fermions, the one-loop effective action for the
Majorana fermions (IV.1) contains a factor $1/2$ in front of the trace, which
can be understood by noting that $\bar{\psi}_{M}$ and $\psi_{M}$ are dependent
variables. Recall that analogous difference occurs between effective actions
for real and complex scalar fields, as the effective action for a complex
scalar field is equivalent to that of two real scalars.

$\Gamma^{(1)}_{\text{Maj}}=-\frac{i}{2}\text{Tr}\left[\text{log}\left(\sqrt{-g}\left(i\gamma^{\mu}\nabla_{\mu}-\tilde{\mathcal{M}}\right)\right)\right]=-\frac{i}{2}\text{Tr}\left[\text{log}\left(\sqrt{-g}\left(i\gamma^{\mu}\nabla_{\mu}-|m|\right)\right)\right]\,,\quad$
(IV.1)

where the second equality follows from Eq. (III.14) and the observation that
${\rm Tr}[\gamma^{5}]=0$. Taking a derivative of (IV.1) with respect to $|m|$
and integrating over $|m|$ gives an equvalent expression (up to an irrelevant
integration constant),

$\Gamma^{(1)}_{\text{Maj}}=\frac{1}{2}\int^{|m|}{\rm
d}\tilde{m}\text{Tr}\left[\sqrt{-g}iS^{ab}_{F}(x;x^{\prime})\right].$ (IV.2)

where $\tilde{m}>0$ is a real mass parameter. Since the integral in (IV.2) is
idential to that which occurs for the Dirac fermions, we can use Eq. (93) from
Ref. Koksma:2009tc ,

$\Gamma^{(1)}_{\text{Maj}}=\frac{1}{2}\int{\rm
d}^{D}x\sqrt{-g}\frac{H^{D-2}}{(2\pi)^{D/2}}\left\\{\Gamma\left(1\\!-\\!\frac{D}{2}\right)\int^{\rvert
m\lvert}{\rm
d}\tilde{m}\frac{\tilde{m}\Gamma\left(\frac{D}{2}\\!+\\!i\tilde{\zeta}\right)\Gamma\left(\frac{D}{2}\\!-\\!i\tilde{\zeta}\right)}{\Gamma\left(1\\!+\\!i\tilde{\zeta}\right)\Gamma\left(1\\!-\\!i\tilde{\zeta}\right)}\right\\},\quad$
(IV.3)

where $\tilde{\zeta}=\tilde{m}/H$. This action can be expanded around $D=4$,
and its convenient rewriting is,

$\displaystyle\Gamma^{(1)}_{\text{Maj}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int{\rm
d}^{D}x\sqrt{-g}\bigg{\\{}\frac{\mu^{D-4}}{2(2\pi)^{D/2}}\Gamma\left(1\\!-\\!\frac{D}{2}\right)\int^{\rvert
m\lvert}{\rm d}\tilde{m}\tilde{m}\bigg{[}H^{2}\\!+\\!\tilde{m}^{2}$ (IV.4)
$\displaystyle\\!\\!\hskip
73.97733pt+\,\frac{D\\!-\\!4}{2}H^{2}\left(\psi\left(1\\!+\\!i\tilde{\zeta}\right)+\psi\left(1\\!-\\!i\tilde{\zeta}\right)+2+\ln\Big{(}\frac{H^{2}}{\mu^{2}}\Big{)}\right)\bigg{]}\bigg{\\}}\,,\qquad$

where we made use of
$H^{D-2}=\mu^{D-4}H^{2}\Big{[}1+\frac{D-4}{2}\ln\big{(}H^{2}/\mu^{2}\big{)}+{\cal
O}\big{(}(D-4)^{2}\big{)}\Big{]}$. Now, keeping in mind that,

$\Gamma\Big{(}1-\frac{D}{2}\Big{)}=\frac{2}{D-4}+\gamma_{E}-1\,,$ (IV.5)

where $\gamma_{E}=\psi(1)\approx 0.57\cdots$ is the Euler-Mascheroni constant,
one sees that the divergences in (IV.4) can be removed by adding the
counterterm action of the form,

$S_{\rm ct}=\int{\rm
d}^{D}x\sqrt{-g}\bigg{[}\alpha\frac{R}{D(D-1)}+\beta\bigg{]}\,,\qquad
R=D(D-1)H^{2}\,,\qquad$ (IV.6)

where

$\displaystyle\alpha\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{\mu^{D-4}}{4(2\pi)^{D/2}}\Gamma\left(1\\!-\\!\frac{D}{2}\right)m^{2}+\alpha_{\rm
f}\,,\qquad$ $\displaystyle\beta\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{\mu^{D-4}}{8(2\pi)^{D/2}}\Gamma\left(1\\!-\\!\frac{D}{2}\right)m^{4}+\beta_{\rm
f}\,,\qquad$ (IV.7)

resulting in the following renormalized one-loop effective action,

$\displaystyle\Gamma^{(1)\rm ren}_{\text{Maj}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int{\rm
d}^{4}x\sqrt{-g}\bigg{\\{}\frac{H^{2}}{8\pi^{2}}\bigg{[}\int^{\rvert
m\lvert}{\rm
d}\tilde{m}\tilde{m}\left(\psi\left(1\\!+\\!i\tilde{\zeta}\right)\\!+\\!\psi\left(1\\!-\\!i\tilde{\zeta}\right)\right)\\!+\\!m^{2}\\!+\\!\frac{m^{2}}{2}\ln\bigg{(}\frac{H^{2}}{\mu^{2}}\bigg{)}\bigg{]}\bigg{\\}}\qquad$
(IV.8) $\displaystyle+$ $\displaystyle\\!\\!\hskip 0.0ptS_{\rm HE}\,,\qquad$

where $S_{\rm HE}$ is the Hilbert-Einstein action with a renormalized Newton
constant and cosmological constant, whereas the finite parts of the
counterterm couplings $\alpha_{\rm f}$ and $\beta_{\rm f}$ from (IV.7) were
absorbed in the (renormalized) Newton constant and cosmological constant,
respectively. The one-loop effective action (IV.8) equals $1/2$ of the
corresponding Dirac field one-loop effective action, cf. Ref. Koksma:2009tc .
This factor $1/2$ can be attributed to the reduced number of degrees of
freedom carried by the Majorana particle. In the above renormalization we used
a non-minimal subtraction scheme and assumed that the Majorana mass is a
parameter. If the mass is generated by a scalar field condensate, which is
what was assumed in Ref. Koksma:2009tc , a different (more complicated)
counterterm action is needed. For details of the renormalization procedure in
that case we refer to Ref. Koksma:2009tc .

##  V Multiflavor fermions and CP violation

After analyzing the propagator for a single Majorana fermion on de Sitter
space, here we extend the analysis to many mixing Majorana particles, which
can serve as a starting point for a better understanding of CP violation.

###  V.1 Multiflavor Majorana Fermions

The Lagrangian for the multiflavor Majorana fermion, $\Lambda{{}_{I}}(x)$,
$(I=1,2,\cdots,n)$, where $n$ is the number of flavors, is the following
natural generalization of Eq. (II.17),

$\mathcal{L}=\frac{1}{4}i\bar{\Lambda}_{{}_{I}}(x)\gamma^{\mu}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\nabla}}_{\mu}\\!\Lambda_{{}_{I}}(x)-\frac{1}{2}\bar{\Lambda}_{{}_{I}}(x)\mathcal{M}_{{}_{IJ}}\Lambda_{{}_{J}}(x)+{\cal
L}_{\rm int}\,,$ (V.1)

where $\mathcal{M}_{{}_{IJ}}$ is a complex symmetric Majorana mass matrix,

$\boldsymbol{\mathcal{M}}=\begin{pmatrix}\mathbf{m}^{\dagger}&0\\\
0&\mathbf{m}\end{pmatrix}\,,\qquad$ (V.2)

where $\mathbf{m}$ is a symmetric complex scalar (non-spinorial) $n\times n$
mass matrix, and ${\cal L}_{\rm int}$ is an interaction Lagrangian, which in
the simple case of Yukawa couplings can be written as,

${\cal L}_{\rm
int}=\bar{\Lambda}_{{}_{I}}(x)\phi(x)\mathcal{Y}_{{}_{IJ}}\Lambda_{{}_{J}}(x)\,,$
(V.3)

where $\mathcal{Y}_{{}_{IJ}}$ is the symmetric complex Yukawa matrix of
couplings,

$\boldsymbol{\mathcal{Y}}=\begin{pmatrix}\mathbf{y}^{\dagger}&0\\\
0&\mathbf{y}\end{pmatrix}\,,\qquad$ (V.4)

where $\mathbf{y}$ is a symmetric scalar complex matrix of couplings, and
$\phi(x)$ is a scalar field condensate (which in general situations could be a
matrix). Next, after rescaling the Majorana fields as,
$\Lambda_{{}_{I}}(x)\rightarrow
a^{\frac{D-1}{2}}\Lambda_{{}_{I}}(x)=\tilde{\Lambda}_{{}_{I}}(x)$. The
Majorana field obeys the Majorana condition ,

$\tilde{\Lambda}_{{}_{I}}(x)=\tilde{\Lambda}^{c}_{{}_{I}}(x)=-i\gamma^{2}\tilde{\Lambda}^{*}_{{}_{I}}(x),$
(V.5)

from which we can define $\tilde{\Lambda}_{{}_{I}}(x)$ as,

$\tilde{\Lambda}_{{}_{I}}(x)=\begin{pmatrix}\lambda_{{}_{I}}(x)\\\
\epsilon\lambda^{*}_{{}_{I}}(x)\end{pmatrix},$ (V.6)

and the Dirac equation that results from Eq.(V.1) is given by,

$\left(i\gamma^{b}\partial_{b}-a\mathcal{M}\right)_{{}_{IJ}}\tilde{\Lambda}_{{}_{J}}(x)=0\,,$
(V.7)

where we neglected the Yukawa term (V.3). Here the Majorana field
$\tilde{\Lambda}_{{}_{J}}(x)$ is in the flavor mixing basis and we can write
it in the mass diagonal basis as follows,

$\tilde{\Lambda}_{{}_{J}}(x)=V_{{}_{JK}}^{\dagger}\Lambda_{{}_{K}}^{(d)}(x)=\left[\mathbf{V}^{\dagger}\cdot\boldsymbol{\Lambda}^{(d)}(x)\right]_{J},$
(V.8)

and

$\bar{\tilde{\Lambda}}_{{}_{J}}(x)=\bar{\tilde{\Lambda}}^{(d)}_{{}_{K}}(x)U_{{}_{KJ}}=\left[\bar{\boldsymbol{\Lambda}}^{(d)}(x)\cdot\mathbf{U}\right]_{J}\,,$
(V.9)

where unitary matrices $\mathbf{V}$ and $\mathbf{U}$ are defined as follows,

$\mathbf{V}=\begin{pmatrix}\mathbf{U}_{{}_{L}}&0\\\
0&\mathbf{U}_{{}_{R}}\end{pmatrix}\,,\qquad\mathbf{U}=\begin{pmatrix}\mathbf{U}_{{}_{R}}&0\\\
0&\mathbf{U}_{{}_{L}}\end{pmatrix}\,.\qquad$ (V.10)

These matrices are used for the bi-unitary diagonalization of the flavor mass
matrix in the case of Dirac fermions. However in the case of Majorana
fermions, due to the Majorana condition the mass matrix is symmetric and
therefore we do not need a general bi-unitary transformation. In order to
understand the implication of this fact, note first that both
$\boldsymbol{\Lambda}$ and $\boldsymbol{\Lambda}^{(d)}$ should obey the
Majorana condition,
$\mathbf{V}^{\dagger}\cdot\mathbf{\Lambda}=-i\gamma^{2}\mathbf{V}^{T}\cdot\mathbf{\Lambda}^{*}$,
which translates to $\mathbf{U}_{{}_{R}}=\mathbf{U}^{*}_{{}_{L}}$, such that
Eq. (V.10) simplifies to,

$\begin{split}\mathbf{V}\rightarrow\mathbf{X}=\begin{pmatrix}\mathbf{U}_{{}_{L}}&0\\\
0&\mathbf{U}^{*}_{{}_{L}}\end{pmatrix}\,,\qquad\mathbf{U}\rightarrow\mathbf{Y}=\begin{pmatrix}\mathbf{U}^{*}_{{}_{L}}&0\\\
0&\mathbf{U}_{{}_{L}}\end{pmatrix}\,.\end{split}\qquad$ (V.11)

Eq. (V.7) can be diagnonalized131313The diagonalization of the Majorana mass
matrix uses the Takagi diagonalization, for which the mass matrix is complex
and symmetric and the resulting diagonalized mass matrix is real and non-
negative. Some details of the diagonalization procedure can be found in
Appendix E, see in particular Eqs. (VII.40–VII.41). by writing it as follows,

$\left(i\gamma^{b}\partial_{b}\mathbf{1}-a\boldsymbol{\mathcal{M}}\right)\cdot\mathbf{X}^{\dagger}\cdot\boldsymbol{\Lambda}^{(d)}(x)=0\,,$
(V.12)

then we can multiply this equation by $\mathbf{Y}$ from the left which gives
us the following result,

$i\mathbf{Y}\cdot\gamma^{b}\partial_{b}\mathbf{X}^{\dagger}\cdot\boldsymbol{\Lambda}^{(d)}(x)-a\mathbf{Y}\cdot\boldsymbol{\mathcal{M}}\cdot\mathbf{X}^{\dagger}\cdot\boldsymbol{\Lambda}^{(d)}(x)=0\,.$
(V.13)

The term $\mathbf{Y}\cdot\gamma^{b}\partial_{b}\mathbf{X}^{\dagger}$ can be
reduced as follows,

$\begin{split}\mathbf{Y}\cdot\gamma^{b}\partial_{b}\mathbf{X}^{\dagger}&=\begin{pmatrix}\mathbf{U}^{*}_{{}_{L}}&0\\\
0&\mathbf{U}_{{}_{L}}\end{pmatrix}\cdot\begin{pmatrix}0&\sigma^{\mu}\partial_{\mu}\\\
\bar{\sigma}^{\mu}\partial_{\mu}&0\end{pmatrix}\cdot\begin{pmatrix}\mathbf{U}^{\dagger}_{{}_{L}}&0\\\
0&\mathbf{U}^{T}_{{}_{L}}\end{pmatrix}\\\
&=\begin{pmatrix}0&\sigma^{\mu}\partial_{\mu}\\\
\bar{\sigma}^{\mu}\partial_{\mu}&0\end{pmatrix}=\gamma^{b}\partial_{b}\mathbf{1}\,,\hskip
156.49014pt\end{split}$ (V.14)

and thus Eq. (V.13) can be written as,

$\left(i\gamma^{b}\partial_{b}\mathbf{1}-a|\boldsymbol{\mathcal{M}}^{(d)}|\right)\cdot\boldsymbol{\Lambda}^{(d)}(x)=0\,,$
(V.15)

where $|\boldsymbol{\mathcal{M}}^{(d)}|$ is the diagonalized mass matrix,
which is real and non-negative. The diagonalization of the mass matrix from
Eq. (V.13) is given in Appendix E, from which we have made use of the result
in Eq. (VII.39). We can then express the multiflavor Majorana fermion field
operator in the mass diagonal basis as follows,

$\hat{\boldsymbol{\Lambda}}^{(d)}(x)=\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot\vec{x}}\mathbf{A}^{(d)}_{h}(\eta,\vec{k})\cdot\left[\hat{\mathbf{b}}_{h}(\vec{k})+\hat{\mathbf{b}}^{\dagger}_{h}(-\vec{k})\right],$
(V.16)

where $\hat{\mathbf{b}}_{h}(\vec{k})$ and
$\hat{\mathbf{b}}^{\dagger}_{h}(\vec{k})$ are the annihilation and creation
operators, which are vectors in the flavor space,
$\mathbf{A}^{(d)}_{h}(\eta,\vec{k})$ is the mode function matrix, which is
diagonal in the basis in which the mass is diagonal, and it is defined by,

$\begin{split}\mathbf{A}^{(d)}_{h}(\eta,\vec{k})=\begin{pmatrix}\mathbf{L}^{(d)}_{h}(\eta,\vec{k})\cdot\boldsymbol{\xi}_{h}(\vec{k})\\\
h\mathbf{L}^{*(d)}_{h}(\eta,-\vec{k})\cdot\boldsymbol{\xi}^{*}_{h}(-\vec{k})\end{pmatrix}=\begin{pmatrix}\mathbf{L}^{(d)}_{h}(\eta,\vec{k})\cdot\boldsymbol{\xi}_{h}(\vec{k})\\\
e^{i\phi^{\prime}}\mathbf{L}^{*(d)}_{h}(\eta,-\vec{k})\cdot\boldsymbol{\xi}_{h}(\vec{k})\end{pmatrix}\,,\quad\end{split}$
(V.17)

where $\boldsymbol{\xi}_{h}(\vec{k})$ builds an $n-$component flavor vector,

$[\boldsymbol{\xi}_{h}(\vec{k})]^{T}=\big{[}\xi_{h}(\vec{k}),\xi_{h}(\vec{k}),\cdots\big{]}_{{}_{n}},$
(V.18)

and similarly $\mathbf{L}^{(d)}_{h}(\eta,\vec{k})$ is the mode function
$(n\times n)$ matrix in the diagonal basis,

$\mathbf{L}^{(d)}_{h}(\eta,\vec{k})=\text{diag}\left[L^{(d)}_{{}_{h,11}},L^{(d)}_{{}_{h,11}},....,L^{(d)}_{{}_{h,nn}}\right]\,.\quad$
(V.19)

In Eq. (V.17)
$\phi^{\prime}=\frac{\pi(h-1)}{2}+\theta(\hat{k}_{x},\hat{k}_{x})$, and it is
identical to the phase in Eq. (II.31), making use of the property in (VII.21)
and the fact that helicity $h$ can be written as $h=e^{i\frac{\pi(h-1)}{2}}$.
Furthermore, the solutions of the Dirac equation in the mass diagonal basis in
Eq. (V.15) can be found in the same way as we have done for the single flavor
case in section III,

$\displaystyle{\rm
e}^{-\frac{i\phi^{\prime}}{2}}\mathbf{L}^{(d)}_{h}(\eta,\vec{k})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\sqrt{-\frac{\pi\|\textbf{k}\|\eta}{4}}\left[{\rm
e}^{\frac{i\pi\boldsymbol{\nu}^{\dagger}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(1)}_{\boldsymbol{\nu}^{\dagger}}(-\|\textbf{k}\|\eta)+h{\rm
e}^{\frac{i\pi\boldsymbol{\nu}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(1)}_{\boldsymbol{\nu}}(-\|\textbf{k}\|\eta)\right]$
(V.20) $\displaystyle{\rm
e}^{\frac{i\phi^{\prime}}{2}}\mathbf{L}^{*(d)}_{h}(\eta,-\vec{k})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\sqrt{-\frac{\pi\|\textbf{k}\|\eta}{4}}\left[{\rm
e}^{\frac{i\pi\boldsymbol{\nu}^{\dagger}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(1)}_{\boldsymbol{\nu}^{\dagger}}(-\|\textbf{k}\|\eta)-h{\rm
e}^{\frac{i\pi\boldsymbol{\nu}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(1)}_{\boldsymbol{\nu}}(-\|\textbf{k}\|\eta)\right]$
(V.21) $\displaystyle{\rm
e}^{\frac{i\phi^{\prime}}{2}}\mathbf{L}^{*(d)}_{h}(\eta,\vec{k})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\sqrt{-\frac{\pi\|\textbf{k}\|\eta}{4}}\left[{\rm
e}^{\frac{-i\pi\boldsymbol{\nu}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(2)}_{\boldsymbol{\nu}}(-\|\textbf{k}\|\eta)+h{\rm
e}^{\frac{-i\pi\boldsymbol{\nu}^{\dagger}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(2)}_{\boldsymbol{\nu}^{\dagger}}(-\|\textbf{k}\|\eta)\right]$
(V.22) $\displaystyle{\rm
e}^{\frac{-i\phi^{\prime}}{2}}\mathbf{L}^{(d)}_{h}(\eta,-\vec{k})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\sqrt{-\frac{\pi\|\textbf{k}\|\eta}{4}}\left[{\rm
e}^{\frac{-i\pi\boldsymbol{\nu}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(2)}_{\boldsymbol{\nu}}(-\|\textbf{k}\|\eta)-h{\rm
e}^{\frac{-i\pi\boldsymbol{\nu}^{\dagger}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(2)}_{\boldsymbol{\nu}^{\dagger}}(-\|\textbf{k}\|\eta)\right],\qquad$
(V.23)

where ${\rm
e}^{\frac{i\pi\boldsymbol{\nu}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(1),(2)}_{\boldsymbol{\nu}}(-\|\textbf{k}\|\eta)$
and ${\rm
e}^{\frac{i\pi\boldsymbol{\nu}^{\dagger}}{2}}\cdot\boldsymbol{\mathcal{H}}^{(1),(2)}_{\boldsymbol{\nu}^{\dagger}}(-\|\textbf{k}\|\eta)$
denote diagonal Hankel function matrices,

$\displaystyle{\rm
e}^{\frac{i\pi\boldsymbol{\nu}}{2}}\\!\cdot\boldsymbol{\mathcal{H}}^{(1),(2)}_{\nu}(-\|\textbf{k}\|\eta)\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\begin{pmatrix}\\!{\rm
e}^{\frac{i\pi\nu_{11}}{2}}&&\\\ &\ddots&\\\ &&{\rm
e}^{\frac{i\pi\nu_{nn}}{2}}\end{pmatrix}\begin{pmatrix}\\!\mathcal{H}^{(1),(2)}_{\nu_{11}}(-\|\textbf{k}\|\eta)&&\\\
&\ddots&\\\
&&\mathcal{H}^{(1),(2)}_{\nu_{nn}}(-\|\textbf{k}\|\eta)\end{pmatrix}$ (V.24)
$\displaystyle{\rm
e}^{\frac{i\pi\boldsymbol{\nu}^{\dagger}}{2}}\\!\cdot\boldsymbol{\mathcal{H}}^{(1),(2)}_{\boldsymbol{\nu}^{\dagger}}(-\|\textbf{k}\|\eta)\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\begin{pmatrix}\\!{\rm
e}^{\frac{i\pi\nu^{\dagger}_{11}}{2}}&&\\\ &\ddots&\\\ &&{\rm
e}^{\frac{i\pi\nu^{\dagger}_{nn}}{2}}\end{pmatrix}\begin{pmatrix}\\!\mathcal{H}^{(1),(2)}_{\nu^{\dagger}_{11}}(-\|\textbf{k}\|\eta)&&\\\
&\ddots&\\\
&&\mathcal{H}^{(1),(2)}_{\nu^{\dagger}_{nn}}(-\|\textbf{k}\|\eta)\end{pmatrix}\\!.\qquad\;\;$
(V.25)

where (cf. Eq. (VII.40)),

$\boldsymbol{\nu}=\frac{1}{2}+i\boldsymbol{\zeta}^{(d)}\,,\qquad\boldsymbol{\nu}^{\dagger}=\frac{1}{2}-i\boldsymbol{\zeta}^{(d)}\,,\qquad\boldsymbol{\zeta}^{(d)}=\frac{\lvert\boldsymbol{m}^{(d)}\rvert}{H}\,$
(V.26)

are $n\times n$ diagonal matrices in flavor space.

###  V.2 Multiflavor Majorana propagator

The multiflavor Majorana propagator obeys the equation of motion,

$\left(i\gamma^{b}\partial_{b}-a\mathcal{M}\right)_{{}_{IP}}iS_{{}_{PJ}}(x;x^{\prime})=i\delta_{IJ}\delta^{D}(x\\!-\\!x^{\prime}),$
(V.27)

where $\delta_{IJ}$ denotes the Kronecker delta in flavor space and the
propagator $iS_{{}_{PJ}}(x;x^{\prime})$ is defined as (see Eq. (III.1)),

$\begin{split}iS_{{}_{PJ}}(x;x^{\prime})&=\left[i\mathbf{S}_{F}(x;x^{\prime})\right]_{PJ}\,,\quad\\\
i\mathbf{S}_{F}(x;x^{\prime})&=\Big{\langle}\text{T}\Big{[}\hat{\tilde{\boldsymbol{\Lambda}}}(x)\hat{\tilde{\bar{\boldsymbol{\Lambda}}}}(x^{\prime})\Big{]}\Big{\rangle}=\Theta(\eta\\!-\\!\eta^{\prime})\Big{\langle}\hat{\tilde{\boldsymbol{\Lambda}}}(x)\hat{\tilde{\bar{\boldsymbol{\Lambda}}}}(x^{\prime})\Big{\rangle}-\Theta(\eta^{\prime}\\!-\\!\eta)\Big{\langle}\hat{\tilde{\bar{\boldsymbol{\Lambda}}}}(x^{\prime})\hat{\tilde{\boldsymbol{\Lambda}}}(x)\Big{\rangle}\,.\end{split}\quad$
(V.28)

Making use of Eqs. (V.8–V.9) and (V.11) we can write this as follows,

$\begin{split}i\mathbf{S}_{F}(x;x^{\prime})=\mathbf{X}^{\dagger}\cdot\big{\langle}\text{T}\big{[}\hat{\boldsymbol{\Lambda}}^{(d)}(x)\hat{\bar{\boldsymbol{\Lambda}}}^{(d)}(x^{\prime})\big{]}\big{\rangle}\cdot\mathbf{Y}\\\
=\mathbf{X}^{\dagger}\cdot
i\mathbf{S}_{F}^{(d)}(x;x^{\prime})\cdot\mathbf{Y}\,,\end{split}\qquad$ (V.29)

where the propagator $i\mathbf{S}^{(d)}$ is in the mass diagonal basis and
satisfies the following equation,

$\left(i\gamma^{b}\partial_{b}-a\lvert\mathcal{M}^{(d)}\rvert\right)_{{}_{IP}}iS^{(d)}_{{}_{PJ}}(x;x^{\prime})=i\delta_{IJ}\delta^{D}(x\\!-\\!x^{\prime}),$
(V.30)

and $\lvert\mathcal{M}^{(d)}\rvert$ (see Eqs. (VII.40–VII.41) in Appendix E)
is the real mass matrix obtained after diagonalization and is defined as,

$\lvert\boldsymbol{\mathcal{M}}^{(d)}\rvert=\begin{pmatrix}\lvert\textbf{m}^{(d)}\rvert&0\\\
0&\lvert\textbf{m}^{(d)}\rvert\end{pmatrix}\,.\quad$ (V.31)

Thus we can see that the propagator in Eq. (V.29) can be rotated to remove the
phases arising from $\mathbf{X}$ and $\mathbf{Y}$, therefore these phases are
not observable (at tree level) and there is no CP-violation at tree level.
Finally from the mode function solutions in Eq. (V.20–V.23) we can find the
propagator solution by following the same method that we used in section (III)
to obtain,

$i\mathbf{S}_{F}(x;x^{\prime})=\mathbf{X}^{\dagger}\\!\cdot\\!\left\\{a\left(i\gamma^{\mu}\nabla_{\mu}\\!+\\!\lvert\boldsymbol{\mathcal{M}}^{(d)}\rvert\right)\frac{H^{D-2}}{\sqrt{aa^{\prime}}}\left[\frac{1\\!+\\!\gamma^{0}}{2}i\mathbf{S}^{+}(x;x^{\prime})\\!+\\!\frac{1\\!-\\!\gamma^{0}}{2}i\tilde{\mathbf{S}}^{+}(x;x^{\prime})\right]\right\\}\\!\cdot\\!\mathbf{Y},\qquad$
(V.32)

where $i\mathbf{S}^{+}(x;x^{\prime})$ and
$i\tilde{\mathbf{S}}^{+}(x;x^{\prime})$ denote a (diagonal) matrix
generalization of solutions in Eq. (III.13),

$\begin{split}i\mathbf{S}^{+}(x;x^{\prime})&=\frac{\Gamma\left(\frac{D}{2}\\!+\\!i\boldsymbol{\zeta}^{(d)}\right)\Gamma\left(\frac{D-2}{2}\\!-\\!i\boldsymbol{\zeta}^{(d)}\right)}{(4\pi)^{D/2}\Gamma\big{(}\frac{D}{2}\big{)}}\times{}_{2}\boldsymbol{F}_{1}\left(\frac{D}{2}\\!+\\!i\boldsymbol{\zeta}^{(d)},\frac{D\\!-\\!2}{2}\\!-\\!i\boldsymbol{\zeta}^{(d)};\frac{D}{2};1-\frac{y_{++}(x;x^{\prime})}{4}\right)\\\
i\tilde{\mathbf{S}}^{+}(x;x^{\prime})&=\frac{\Gamma\left(\frac{D}{2}\\!-\\!i\boldsymbol{\zeta}^{(d)}\right)\Gamma\left(\frac{D\\!-\\!2}{2}\\!+\\!i\boldsymbol{\zeta}^{(d)}\right)}{(4\pi)^{D/2}\Gamma\big{(}\frac{D}{2}\big{)}}\times{}_{2}\boldsymbol{F}_{1}\left(\frac{D}{2}\\!-\\!i\boldsymbol{\zeta}^{(d)},\frac{D-2}{2}\\!+\\!i\boldsymbol{\zeta}^{(d)};\frac{D}{2};1\\!-\\!\frac{y_{++}(x;x^{\prime})}{4}\right)\,,\end{split}$
(V.33)

where $\boldsymbol{\zeta}^{(d)}$ is the diagonal matrix defined in Eq. (V.26).

For completeness and for comparison, here we sketch how to construct the
multiflavor Dirac propagator $i\mathbf{G}_{F}(x;x^{\prime})$. Firstly, one
introduces unitary rotation matrices $\mathbf{R}$ and $\mathbf{Q}$, which act
on the propagator as,

$\mathbf{R}\\!\cdot\\!i\mathbf{G}_{F}(x;x^{\prime})\\!\cdot\\!\mathbf{Q}=i\mathbf{G}^{(d)}(x;x^{\prime})\,,\qquad$
(V.34)

where $i\mathbf{G}^{(d)}(x;x^{\prime})$ denotes the diagonal Dirac propagator
and the rotation matrices are given by,

$\mathbf{R}={\rm
e}^{\frac{i}{2}\boldsymbol{\theta}^{(d)}\gamma^{5}}\mathbf{V}\,,\qquad\mathbf{Q}=\mathbf{U}^{\dagger}{\rm
e}^{-\frac{i}{2}\boldsymbol{\theta}^{(d)}\gamma^{5}}\,,\qquad$ (V.35)

where

${\rm
e}^{\pm\frac{i}{2}\boldsymbol{\theta}^{(d)}\gamma^{5}}=\cos\left(\frac{\boldsymbol{\theta}^{(d)}}{2}\right)\pm
i\gamma^{5}\sin\left(\frac{\boldsymbol{\theta}^{(d)}}{2}\right)$ (V.36)

are the diagonal chiral matrices with $n$ arbitrary phases, which can be used
to rotate away the phases from the diagonal elements of ${\mathcal{M}}^{(d)}$.
The propagator can then be written as,

$\displaystyle i\mathbf{G}(x;x^{\prime})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\mathbf{V}^{\dagger}{\rm
e}^{-\frac{i}{2}\boldsymbol{\theta}^{(d)}\gamma^{5}}\left\\{a\left(i\gamma^{\mu}\nabla_{\mu}\\!+\\!\lvert\boldsymbol{\mathcal{M}}^{(d)}\rvert\right)\frac{H^{D-2}}{\sqrt{aa^{\prime}}}\left[\frac{1\\!+\\!\gamma^{0}}{2}i\boldsymbol{S}^{+}(x;x^{\prime})\\!+\\!\frac{1\\!-\\!\gamma^{0}}{2}i\tilde{\boldsymbol{S}}^{+}(x,x^{\prime})\right]\right\\}$
$\displaystyle\times$ $\displaystyle{\rm
e}^{\frac{i}{2}\boldsymbol{\theta}^{(d)}\gamma^{5}}\mathbf{U}\,,$
$\displaystyle\\!\\!\hskip-51.21504pt=\,\mathbf{V}^{\dagger}\left\\{a\left(i\gamma^{\mu}\nabla_{\mu}\\!+\\!\boldsymbol{\mathcal{M}}^{(d)}\right)\frac{H^{D-2}}{\sqrt{aa^{\prime}}}\left[\frac{1\\!+\\!{\rm
e}^{i\boldsymbol{\theta}^{(d)}\gamma^{5}}\gamma^{0}}{2}i\boldsymbol{S}^{+}(x;x^{\prime})\\!+\\!\frac{1\\!-\\!{\rm
e}^{i\boldsymbol{\theta}^{(d)}\gamma^{5}}\gamma^{0}}{2}i\tilde{\boldsymbol{S}}^{+}(x,x^{\prime})\right]\right\\}\mathbf{U},$

where $\boldsymbol{\mathcal{M}}^{(d)}$ is a diagonal complex matrix,
$i\boldsymbol{S}^{+}(x;x^{\prime})$ and
$i\tilde{\boldsymbol{S}}^{+}(x;x^{\prime})$ are defined in Eq. (V.33) and the
unitary rotation matrices $\mathbf{U}$ and $\mathbf{V}$ are given by,

$\mathbf{U}=\begin{pmatrix}\mathbf{U_{{}_{R}}}&0\\\
0&\mathbf{U_{{}_{L}}}\end{pmatrix}\,,\qquad\mathbf{V}^{\dagger}=\begin{pmatrix}\mathbf{U}^{\dagger}_{{}_{L}}&0\\\
0&\mathbf{U}^{\dagger}_{{}_{R}}\end{pmatrix}\,.\qquad$ (V.38)

One should keep in mind that $\mathbf{U}$ and $\mathbf{V}$ are not general
unitary matrices, but from each of them $n$ common phases have been removed by
factoring out the diagonal matrices ${\rm
e}^{\frac{i}{2}\boldsymbol{\theta}^{(d)}\gamma^{5}}$. This means that we have
used $2n^{2}$ parameters from the two unitary matrices to diagonalize the
Dirac mass matrix, for which $n^{2}+n(n-1)=2n^{2}-n$ parameters are needed
($n^{2}$ do diagonalize the symmetric part of the mass matrix and $n(n-1)$ to
diagonalize the antisymmetric part), such that $n$ parameters remain unused.
However, none of these $n$ parameters can be used to remove phases in the
Yukawa mass matrix. To see this let us first look at the simplest case when
$n=1$, the rotation ‘matrices’ have the form, $U_{L}={\rm e}^{i\phi}$ and
$U_{R}={\rm e}^{i\psi}$, such that a complex mass $m=m_{1}+im_{2}=\rho{\rm
e}^{i\mu}$ can be diagonalized by $U_{L}m(U_{R})^{\dagger}={\rm
e}^{i\phi}\rho{\rm e}^{i\mu}{\rm e}^{-i\psi}=\rho$. This fixes the phase
difference, $\psi-\phi=\mu$, but leaves their sum $\phi+\psi$ unconstrained.
In the general case of $n$ Dirac flavors we have $m_{ij}=\rho_{ij}{\rm
e}^{i\mu_{ij}}$. Let us for simplicity consider how the unitary matrices of
the form $(U_{R})_{ij}={\rm e}^{i\phi_{j}}\delta_{ij}$ and $(U_{L})_{ij}={\rm
e}^{i\psi_{j}}\delta_{ij}$ (such that $(U_{L}^{\dagger})_{ij}={\rm
e}^{-i\psi_{i}}\delta_{ij}$) act,

$\displaystyle[(U_{R})\cdot m\cdot U_{L}^{\dagger}]_{ij}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!(U_{R})_{ik}m_{kl}(U_{L}^{\dagger})_{lj}=\delta_{ik}{\rm
e}^{i\phi_{k}}\rho_{kl}{\rm e}^{i\mu_{kl}}{\rm e}^{-i\psi_{l}}\delta_{lj}={\rm
e}^{i\phi_{i}}\rho_{ij}{\rm e}^{i\mu_{ij}}{\rm e}^{-i\psi_{j}}\,.$ (V.39)

This then implies that, out of the $2n$ phases
$\\{\phi_{i},\psi_{i}\\}\,(i=1,2,\cdots,n)$, one can only use $n$ of the
differences $\phi_{i}-\psi_{j}$ to remove some of the phases in $\mu_{ij}$.
However, the $n$ remaining linearly independent combinations
$\phi_{i}+\psi_{j}$ cannot be used to remove any phases as they appear with a
‘wrong sign’ in (V.39). For the same reason, none of the remaining $n$ phases
$\phi_{i}+\psi_{j}$ can be used to remove any of the phases in the Yukawa
matrix $y_{ij}$ (cf. Eq. (V.4)). This means that the group that diagonalizes a
general Dirac mass matrix is not direct product of the two unitary groups,
${\cal U}_{L}(n)\otimes{\cal U}_{R}(n)$, but instead it is the quotient group,
$[{\cal U}_{L}(n)\otimes{\cal U}_{R}(n)]/{\cal U}(1)^{n}$. This consideration
also shows that, even though after the diagonalization of the mass matrix the
unitary matrices $U_{L}$ and $U_{R}$ possess $n$ redundant parameters, none of
them can be used to remove any of the phases from the Yukawa matrix. With this
remark we conclude our analysis of the Dirac fermion propagator for mixing
flavors, which shows that there are CP violating effects at the tree level, as
was expected.

To understand the question of CP violation a bit better, let us summarize the
principal differences between the Majorana and Dirac fermions. The mass matrix
and Yukawa coupling matrix for multiflavor Majorana fermions are complex
symmetric matrices which consist of $n(n-1)$ off-diagonal real parameters and
$n$ diagonal phases (totalling $n^{2}$ parameters), which can be all removed
by the Tagaki diagonalization procedure by making use of a unitary matrix
which contains $n^{2}$ real parameters. Since this completely fixes the
unitary matrix, diagonalization of the mass matrix leaves the Majorana Yukawa
matrix completely general. Such a symmetric complex matrix has $n(n+1)/2$
phases and $n(n+1)/2$ real parameters, meaning that Majorana fermions can
harbor $n(n+1)/2$ CP-violating phases in the Yukawa matrix. This is to be
contrasted with the Dirac fermions, for which both the mass matrix and Yukawa
matrix are general complex matrices with $2n^{2}$ real parameters in total.
The diagonalization requires two independent unitary matrices, each of them
having $n^{2}$ real parameters, such that diagonalization requires $2n^{2}-n$
real parameters (after diagonalization $n$ real eigenvalues remain), leaving
$n$ parameters unfixed. As we have shown above, all of these $n$ parameters
are redundant, meaning that they cannot be used to remove any of the CP phases
in the Yukawa matrix. The general Yukawa matrix for Dirac fermions contains
$2n^{2}$ parameters, which can be decomposed into a symmetric matrix (with
$n(n+1)$ real parameters, $n(n+1)/2$ are phases) and an antisymmetric matrix
(with $n(n-1)$ real parameters, $n(n-1)/2$ are phases), such that in general
Dirac fermions harbor at most $n(n+1)/2+n(n-1)/2=n^{2}$ CP violating phases.
When this is compared with Majorana fermions, Dirac fermions can harbor
$n^{2}-n(n+1)/2=n(n-1)/2$ more CP violating phases than Majorana fermions.

##  VI Conclusion and Discussion

In this paper we construct the Majorana propagator in de Sitter space for a
complex Majorana mass (III.13–III.15), as well as for more general multiflavor
Majorana fields (V.32–V.33). 141414Our results differ from earlier works
Cotaescu:2018zrq ; Cotaescu:2018afn , where the Majorana propagator was
constructed by imposing the left chiral projector $P_{L}=(1-\gamma^{5})/2$ on
the Dirac propagator in de Sitter. While this projection correctly reduces the
number of degrees of freedom, it is inconsistent with the Majorana condition,
and therefore, in our opinion, questionable. Our work is relevant for the
dynamics of neutrinos in the early Universe setting, and more generally for
the dynamics of Majorana particles that may be involved e.g. in leptogenesis
scenarions, which constitute a popular explanation for the observed matter-
antimatter asymmetry of the Universe. In spite of various subtleties involved
in construction of the Majorana propagator 151515These subtleties include an
off-shell imposition of the Majorana condition, a careful account of canonical
quantization, a splitting of the momentum space mode functions into two
hemispheres, etc., the final result (III.13–III.15) is identical to the Dirac
propagator. This can be understood as follows. Even though at the operator
level there is a clear distinction between the Dirac fermions (for which the
positive and negative frequency poles fluctuate independently) and the
Majorana fermions (for which the same operator is responsible for fluctuations
at the positive and negative frequency poles), their statistical properties
are the same because the (gravitational) coupling of fermions to de Sitter
space does not violate CP symmetry. In other words, differences between
Majorana and Dirac fermions would arise in CP-violating backgrounds. Such CP-
violating backgrounds can occur, for example, during first order phase
transitions in the early Universe, and which may be suitable for dynamical
baryogenesis. One example of such a CP-violating background was considered in
Ref. Prokopec:2013ax , in which the dynamics of Dirac fermions in presence of
such a CP-violating background was studied, and in which indeed Dirac fermions
exhibit CP-violating effects. Our generalization to the case of multiflavor
Majorana fields (V.32–V.33) was done in the prescience of quantum loop
studies, in which the Majorana phases are expected to show CP-violating
effects. For comparison, we also quote the mutiflavor Dirac propagator, and
emphasise that mixing Dirac fermions contain more CP violating phases than
Majorana fermions, which could be a way to distinguish whether the nature of
fermions is Dirac or Majorana. One of these phases is the chiral phase,
$\phi={\rm Arctan}[m_{2}/m_{1}]$, of the complex mass term $m=m_{1}+im_{2}$,
which is present already at the single fermion level. Curiously, this phase
modifies the positive and negative frequency projectors,
$P_{\pm}=(1\pm\gamma^{0})/2$, in the propagator (III.15) to give them a chiral
nature, $P^{5}_{\pm}=\big{(}1\pm{\rm e}^{i\gamma^{5}\phi}\gamma^{0}\big{)}/2$.
Understanding the full significance of this observation requires quantum loop
studies, and it is thus beyond the scope of this work.

Even though the Dirac and Majorana vacuum propagators in de Sitter are
identical, one ought to be careful when using them in loop studies. To
illustrate this, in section IV we compute the one-loop effective action
$\Gamma^{(1)}_{\text{Maj}}$ for Majorana particles (IV.8), and show that it
differs from that of Dirac fermions (calculated earlier in e.g. Refs.
Candelas:1975du ; Koksma:2009tc ) by a factor 1/2,

$\Gamma^{(1)}_{\text{Maj}}=\frac{1}{2}\Gamma^{(1)}_{\text{Dirac}}\,,$ (VI.1)

which was to be expected since the Majorana fermions carry 1/2 of the degrees
of freedom when compared with the Dirac fermions.

## Acknowledgments.

The authors are grateful to Tanja Hinderer and Elisa Chisari for useful
comments. This work was partially supported by the D-ITP consortium, a program
of the Netherlands Organization for Scientific Research (NWO) that is funded
by the Dutch Ministry of Education, Culture and Science (OCW).

##  VII Appendices

## Appendix A: Dirac’s matrices in chiral representation

In this paper we use chiral representation for Dirac’s gamma matrices, which
in flat (tangent) space and in $D=4$ are of the form,

$\gamma^{a}=\begin{pmatrix}0&\sigma^{a}\\\ \bar{\sigma}^{a}&0\\\
\end{pmatrix}\,,\qquad\sigma^{a}=\left(\mathbb{1},\sigma^{i}\right)\,,\qquad\bar{\sigma}^{a}=\left(\mathbb{1},-\sigma^{i}\right)\,,\qquad(a=0,1,2,3)\,,\qquad$
(VII.1)

where $\sigma^{i}$ are Pauli matrices. They obey the anti-commutation
relation, $\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab}$ and $\gamma^{5}={\rm
diag}(-\mathbb{1},\mathbb{1})$. The following commutation properly is useful
for Majorana fermions,

$\gamma^{0}(-i\gamma^{2})\big{[}\gamma^{a}\big{]}^{T}=-\gamma^{a}\gamma^{0}(-i\gamma^{2})\,.\qquad$
(VII.2)

Charge conjugation operator is given by the following,

$\begin{split}\mathcal{C}=-i\gamma^{2}=\begin{pmatrix}0&-\epsilon\\\
\epsilon&0\end{pmatrix}\,,\qquad\epsilon=i\sigma^{2}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\,,\qquad\epsilon^{2}=-\mathbf{1}_{2\times 2}\\\
\mathcal{C}^{\dagger}=\mathcal{C}\qquad\qquad\qquad\qquad\qquad\qquad\end{split}$
(VII.3)

where $\epsilon$ is an antisymmetric matrix and the following property
satisfied by the charge conjugation operator comes in handy

$(-i\gamma^{2})(-i\gamma^{2})^{\dagger}=(-i\gamma^{2})(-i\gamma^{2})=\mathbf{1}_{4\times
4}.$ (VII.4)

In what follows we introduce some transformations of projection operators,

$P_{\pm}=\frac{1\pm\gamma^{0}}{2}\,,\qquad P^{5}_{\pm}=\frac{1\pm
e^{i\theta\gamma^{5}}\gamma^{0}}{2}.$ (VII.5)

These operators transform under the charge conjugation as follows,

$\mathcal{C}P_{\pm}\mathcal{C}^{\dagger}=P_{\mp}\,,\qquad\mathcal{C}P^{5}_{\pm}\mathcal{C}^{\dagger}=\left(P^{5}_{\mp}\right)^{\dagger}\,.\quad$
(VII.6)

## Appendix B: Mode decomposition for Majorana fields

The rescaled classical Dirac field $\tilde{\Psi}_{\alpha,D}(x)$ can be
represented in the spatial momentum space as folows,

$\hat{\tilde{\Psi}}_{\alpha,D}(x)=\begin{pmatrix}\hat{\tilde{\chi}}_{{}_{L}}(x)\\\
\hat{\tilde{\chi}}_{{}_{R}}(x)\end{pmatrix}=\int\frac{d^{D-1}k}{(2\pi)^{D-1}}{\rm
e}^{i\vec{k}\cdot\vec{x}}C_{h}(\eta,\vec{k})\,,$ (VII.7)

where $\hat{\tilde{\chi}}_{{}_{L}}(x)$ and $\hat{\tilde{\chi}}_{{}_{R}}(x)$
are the rescaled chiral 2-spinors and $C_{h}(\eta,\vec{k})$ is the momentum
and helicity eigenspinor defined by,

$C_{h}(\eta,\vec{k})=\begin{pmatrix}L_{h}(\eta,\vec{k})\xi_{h}(\vec{k})\\\
R_{h}(\eta,\vec{k})\xi_{h}(\vec{k})\end{pmatrix},$ (VII.8)

where $L_{h}(\eta,\vec{k})$ and $R_{h}(\eta,\vec{k})$ are complex chiral mode
functions $\xi_{h}(\vec{k})$ are the helicity eigenspinors, whose detailed
properties are discussed in Appendix C. The decomposition (VII.7) is
particularly convenient in spatially homogeneous spaces such as de Sitter
space, as ${\rm e}^{i\vec{k}\cdot\vec{x}}$ are eigenstates of the spatial
derivative operator $\partial_{i}$ with the eigenvalue $ik_{i}$. It is
convenient (and customary) to recast the rescaled quantum Dirac field
$\hat{\tilde{\Psi}}_{\alpha,D}(x)$ in terms of the mode operators decomposed
into distinct creation ($\hat{d}^{\,\dagger}_{h}(\vec{k})$) and annihilation
($\hat{b}_{h}(\vec{k})$) operators,

$\hat{\tilde{\Psi}}_{\alpha,D}(x)=\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot\vec{x}}\left[C_{h,\alpha}(\eta,\vec{k})\hat{b}_{h}(\vec{k})+D_{h,\alpha}(\eta,-\vec{k})\hat{d}^{\,\dagger}_{h}(-\vec{k})\right],$
(VII.9)

where
$D_{h,\alpha}(\eta,-\vec{k})=-i\gamma^{2}C^{*}_{h,\alpha}(\eta,-\vec{k})$. In
contrast, the rescaled Majorana fermions can be described by imposing the
Majorana condition on Eq. (VII.7),

$\hat{\tilde{\Psi}}_{\alpha,D}(x)=-i\gamma^{2}\hat{\tilde{\Psi}}^{*}_{\alpha,D}(x)\,.$
(VII.10)

From this condition we find the following relation between the two spinors
that form the Dirac spinor in Eq. (VII.8)

$\hat{\tilde{\chi}}_{{}_{R}}(x)=\epsilon\hat{\tilde{\chi}}_{{}_{L}}^{*}(x).$
(VII.11)

After imposing the condition in Eq. (VII.10) and the subsequent relation in
Eq. (VII.11) in Eq. (VII.7) we have for the rescaled Majorana field
$\hat{\tilde{\Psi}}_{M,\alpha}(x)$,

$\hat{\tilde{\Psi}}_{D,\alpha}(x)\;\stackrel{{\scriptstyle\rm
Maj.\;condition}}{{\longrightarrow}}\;\hat{\tilde{\Psi}}_{M,\alpha}(x)=\begin{pmatrix}\hat{\rho}(x)\\\
\epsilon\hat{\rho}^{*}(x)\end{pmatrix}\,,$ (VII.12)

Thus the classical momentum decomposition for
$\hat{\tilde{\Psi}}_{M,\alpha}(x)$ is defined by,

$\hat{\tilde{\Psi}}_{M,\alpha}(x)=\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot\vec{x}}A_{h}(\eta,\vec{k})\,,$
(VII.13)

where the momentum eigenspinor $A_{h}(\eta,\vec{k})$, which gives the positive
energy solutions, is defined as,

$A_{h}(\eta,\vec{k})=\begin{pmatrix}L_{h}(\eta,\vec{k})\xi_{h}(\vec{k})\\\
L^{*}_{h}(\eta,-\vec{k})\epsilon\xi^{*}_{h}(-\vec{k})\end{pmatrix}\,.$
(VII.14)

The quantized rescaled Majorana fermion field is then decomposed as,

$\hat{\tilde{\Psi}}_{M,\alpha}(x)=\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}\left[{\rm
e}^{i\vec{k}\cdot\vec{x}}A_{h}(\eta,\vec{k})\hat{b}_{h}(\vec{k})+{\rm
e}^{-i\vec{k}\cdot\vec{x}}B_{h}(\eta,\vec{k})\hat{b}^{\dagger}_{h}(\vec{k})\right],$
(VII.15)

with $\hat{b}_{h}(\vec{k})$ and $\hat{b}^{\dagger}_{h}(\vec{k})$ are the
identical annihilation and creation operators. The momentum eigenspinor
$B_{h}(\eta,\vec{k})$ obeys the Majorana condition,

$B_{h}(\eta,\vec{k})=-i\gamma^{2}A^{*}_{h}(\eta,\vec{k})=A_{h}(\eta,-\vec{k})\,.$
(VII.16)

We can now use this result to write Eq. (VII.15) as,

$\begin{split}\hat{\tilde{\Psi}}_{M,\alpha}(x)&=\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}\left[e^{i\vec{k}\cdot\vec{x}}A_{h}(\eta,\vec{k})\hat{b}_{h}(\vec{k})+e^{-i\vec{k}\cdot\vec{x}}A_{h}(\eta,-\vec{k})\hat{b}^{\dagger}_{h}(\vec{k})\right]\\\
&=\sum_{h=\pm}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}e^{i\vec{k}\cdot\vec{x}}\left[\hat{b}_{h}(\vec{k})+\hat{b}^{\dagger}_{h}(-\vec{k})\right]A_{h}(\eta,\vec{k})\,,\end{split}$
(VII.17)

which is the form used in the main text in Eqs. (II.24–II.25).

## Appendix C: Properties of helicity 2-eigenspinor

Here we elucidate the properties of the helicity 2-eigenspinor in $D=4$
dimensions for the purposes of keeping the calculations simple. This can be
extended to D dimensions as was done in Ref. Koksma:2009tc . We have seen the
definition of helicity 2-eigenspinor in Eq. (II.30). However
$\epsilon\xi^{*}_{h}(-\vec{k})$ can be written as follows

$\begin{split}\epsilon\xi^{*}_{h}(-\vec{k})=\frac{1}{\sqrt{2\left(1+h\hat{k}_{z}\right)}}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}-h\left(\hat{k}_{x}+i\hat{k}_{y}\right)\\\
1+h\hat{k}_{z}\end{pmatrix}\\\
\implies\epsilon\xi^{*}_{h}(-\vec{k})=\frac{1}{\sqrt{2\left(1+h\hat{k}_{z}\right)}}\begin{pmatrix}1+h\hat{k}_{z}\\\
h\left(\hat{k}_{x}+i\hat{k}_{y}\right)\end{pmatrix}.\end{split}$ (VII.18)

After massaging the terms in the matrix in Eq.(VII.18), we can write it as
follows

$\epsilon\xi^{*}_{h}(-\vec{k})=h\sqrt{\frac{\hat{k}_{x}+i\hat{k}_{y}}{\hat{k}_{x}-i\hat{k}_{y}}}\xi_{h}(\vec{k})$
(VII.19)

Then we can write the factor
$h\sqrt{\frac{\hat{k}_{x}+i\hat{k}_{y}}{\hat{k}_{x}-i\hat{k}_{y}}}$ as a
phase,

$\sqrt{\frac{\hat{k}_{x}+i\hat{k}_{y}}{\hat{k}_{x}-i\hat{k}_{y}}}=\sqrt{\frac{\lvert\tilde{k}\rvert
e^{i\tilde{\theta}}}{\lvert\tilde{k}\rvert
e^{-i\tilde{\theta}}}}=e^{i{\theta}(\hat{k}_{x},\hat{k}_{y})},$ (VII.20)

where we have defined,
$\lvert\tilde{k}\rvert=\sqrt{\hat{k}^{2}_{x}+\hat{k}^{2}_{y}}$ and
$\theta(\hat{k}_{x},\hat{k}_{y})=\tan^{-1}\left(\frac{\hat{k}_{y}}{\hat{k}_{x}}\right)$.
Now we can see that

$\epsilon\xi^{*}_{h}(-\vec{k})=he^{i\theta(\hat{k}_{x},\hat{k}_{y})}\
\xi(\vec{k})$ (VII.21)
$(\epsilon\xi^{*}_{h}(-\vec{k}))^{\dagger}=he^{-i\theta(\hat{k}_{x},\hat{k}_{y})}\xi^{\dagger}(\vec{k}),$
(VII.22)

where we have made use of Eq.(VII.20). We also find the following property
useful

$\sum_{h=\pm}\xi_{h,a}^{\dagger}(\vec{k})\otimes\xi_{h,b}(\vec{k})=\delta_{ab}\,,$
(VII.23)

which can be proved for the helicity 2-eigenspinor ($a,b\in\\{1,2\\}$) as
follows,

$\begin{split}\sum_{h=\pm}\xi_{h,a}^{\dagger}(\vec{k})\otimes\xi_{h,b}(\vec{k})&=\sum_{h=\pm}\frac{1}{2\left(1-h\hat{k}_{z}\right)}\begin{pmatrix}h\left(\hat{k}_{x}+i\hat{k}_{y}\right)&1-h\hat{k}_{z}\end{pmatrix}\otimes\begin{pmatrix}h\left(\hat{k}_{x}-i\hat{k}_{y}\right)\\\
\left(1-h\hat{k}_{z}\right)\end{pmatrix}\\\
&=\sum_{h=\pm}\begin{pmatrix}\frac{1+h\hat{k}_{z}}{2}&\frac{h\left(\hat{k}_{x}-i\hat{k}_{y}\right)}{2}\\\
\frac{h\left(\hat{k}_{x}+i\hat{k}_{y}\right)}{2}&\frac{1-h\hat{k}_{z}}{2}\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}=\delta_{ab}\,.\hskip 241.84842pt\end{split}$ (VII.24)

We also make note of the following property which is going to be crucial when
we discuss the equations of motion in section II B 3.
$he^{i\theta(\hat{k}_{x},\hat{k}_{y})}\xi_{h}(\vec{k})$ under
$\vec{k}\rightarrow-\vec{k}$ transforms as follows,

$he^{i\theta\left(\hat{k}_{x},\hat{k}_{y}\right)}\xi_{h}(\vec{k})\;\xrightarrow{\vec{k}\rightarrow-\vec{k}}\;-he^{i\theta\left(\hat{k}_{x},\hat{k}_{y}\right)}\xi_{h}(-\vec{k})\,.$
(VII.25)

This can be seen either by calculating $\epsilon\xi^{*}_{h}(\vec{k})$, which
yields $-he^{i\theta\left(\hat{k}_{x},\hat{k}_{y}\right)}\xi_{h}(-\vec{k})$,
or we can understand this from a topological viewpoint where under
$\vec{k}\rightarrow-\vec{k}$ we have the following transformation of the phase
$\theta(\hat{k}_{x},\hat{k}_{y})$,

$\theta(\hat{k}_{x},\hat{k}_{y})\;\xrightarrow{\vec{k}\rightarrow-\vec{k}}\;\theta(\hat{k}_{x}\,,\hat{k}_{y})+\pi\,.$
(VII.26)

## Appendix D: Properties of Hankel Functions

For convenience here we list some of the properties of Hankel functions used
in this work. Firstly we have,

$H^{(1)}_{-\nu}(z)=e^{i\pi\nu}H^{(1)}_{\nu}(z)$ (VII.27)
$H^{(2)}_{-\nu}(z)=e^{-i\pi\nu}H^{(2)}_{\nu}(z)$ (VII.28)
$\\{H^{(1)}_{\nu}(z)\\}^{*}=H^{(2)}_{\nu^{*}}(z^{*})$ (VII.29)
$H_{\nu}^{(1)}(e^{i\pi}z)=-H^{(2)}_{-\nu}(z)=-e^{-i\pi\nu}H^{(2)}_{\nu}(z)\,,$
(VII.30)
$H_{\nu}^{(2)}(e^{-i\pi}z)=-H^{(1)}_{-\nu}(z)=-e^{i\pi\nu}H^{(1)}_{\nu}(z)\,.$
(VII.31)

The Wronskian and the recurrence relation are given by,

$W[H^{(1)}_{\nu},H^{(2)}_{\nu}]=-\frac{4i}{\pi z}$ (VII.32)
$H^{(i)}_{\nu-1}(z)=\frac{d}{dz}H^{(i)}_{\nu}(z)+\frac{\nu}{z}H^{(i)}_{\nu}(z).$
(VII.33)

The Hankel functions are related to MacDonald functions $K_{\nu}(z)$ through
the following identities,

$H^{(1)}_{\nu}(z)=-\frac{2i}{\pi}e^{-\frac{i\pi\nu}{2}}K_{\nu}(-iz)$ (VII.34)
$H^{(2)}_{\nu}(z)=\frac{2i}{\pi}e^{\frac{i\pi\nu}{2}}K_{\nu}(iz).$ (VII.35)

From Eqs. (VII.27–VII.33) we can also write the following properties,

$\begin{split}H^{(1)}_{\nu_{+}}(-\|\textbf{k}\|\eta)=-\frac{e^{i\pi\nu_{-}}}{\|\textbf{k}\|}\left[\partial_{\eta}+\frac{\nu_{-}}{\eta}\right]H^{(1)}_{\nu_{-}}(-\|\textbf{k}\|\eta)\\\
H^{(1)}_{\nu_{-}}(-\|\textbf{k}\|\eta)=-\frac{e^{i\pi\nu_{+}}}{\|\textbf{k}\|}\left[\partial_{\eta}+\frac{\nu_{+}}{\eta}\right]H^{(1)}_{\nu_{+}}(-\|\textbf{k}\|\eta)\end{split}$
(VII.36)
$\begin{split}H^{(2)}_{\nu_{+}}(-\|\textbf{k}\|\eta)=-\frac{e^{-i\pi\nu_{-}}}{\|\textbf{k}\|}\left[\partial_{\eta}+\frac{\nu_{-}}{\eta}\right]H^{(2)}_{\nu_{-}}(-\|\textbf{k}\|\eta)\\\
H^{(1)}_{\nu_{-}}(-\lvert\textbf{k}\rvert\eta)=-\frac{e^{-i\pi\nu_{+}}}{\|\textbf{k}\|}\left[\partial_{\eta}+\frac{\nu_{+}}{\eta}\right]H^{(2)}_{\nu_{+}}(-\|\textbf{k}\|\eta).\end{split}$
(VII.37)

## Appendix E: Diagonalization of the Majorana mass matrix

We consider the mass matrix for the multiflavor Majorana mass from Eq. (V.1)
to be Hermitian and the Dirac equation in Eq. (V.7) can be written as,

$\left(i\gamma^{b}\partial_{b}\boldsymbol{1}-\mathbf{Y}\cdot\boldsymbol{\mathcal{M}}\cdot\mathbf{X}^{\dagger}\right)\cdot\mathbf{\Lambda}^{(d)}(x)=0\,.$
(VII.38)

Consider the term
$\mathbf{Y}\cdot\boldsymbol{\mathcal{M}}\cdot\mathbf{X}^{\dagger}$ (see Eq.
(V.2)):

$\mathbf{Y}\\!\cdot\\!\boldsymbol{\mathcal{M}}\\!\cdot\\!\mathbf{X}^{\dagger}=\begin{pmatrix}\mathbf{U}^{*}_{{}_{L}}&0\\\
0&\mathbf{U}_{{}_{L}}\end{pmatrix}\\!\cdot\\!\begin{pmatrix}\mathbf{m}^{\dagger}&0\\\
0&\mathbf{m}\end{pmatrix}\\!\cdot\\!\begin{pmatrix}\mathbf{U}^{\dagger}_{{}_{L}}&0\\\
0&\mathbf{U}^{T}_{{}_{L}}\end{pmatrix}=\begin{pmatrix}\mathbf{U}^{*}_{{}_{L}}\\!\cdot\\!\mathbf{m}^{\dagger}\\!\cdot\\!\mathbf{U}^{\dagger}_{{}_{L}}&0\\\
0&\mathbf{U}_{{}_{L}}\\!\cdot\\!\mathbf{m}\\!\cdot\\!\mathbf{U}^{T}_{{}_{L}}\end{pmatrix},$
(VII.39)

then we can use Takagi diagonalization Hahn:2006hr ; ChoiHaber  which says
that for every symmetric complex mass matrix $\mathbf{m}$, there exists a
unitary matrix $\mathbf{U}_{{}_{L}}$ such that,

$\mathbf{U}_{{}_{L}}\\!\cdot\\!\mathbf{m}\\!\cdot\\!\mathbf{U}^{T}_{{}_{L}}=\lvert\textbf{m}^{(d)}\rvert=\text{diag}\left[m^{(d)}_{11},m^{(d)}_{22},\cdots,m^{(d)}_{nn}\right],$
(VII.40)

where $\lvert\textbf{m}^{(d)}\rvert$ is real and non-negative as given from
the Takagi theorem (a special care must be taken when the mass matrix is
degenerate ChoiHaber ). And subsequently taking the hermitian conjugate of Eq.
(VII.40) gives,

$\mathbf{U}^{*}_{{}_{L}}\\!\cdot\\!\mathbf{m}^{\dagger}\\!\cdot\\!\mathbf{U}^{\dagger}_{{}_{L}}=\lvert\textbf{m}^{(d)\dagger}\rvert=\lvert\textbf{m}^{(d)}\rvert,$
(VII.41)

therefore, we can write Eq. (VII.39) as follows

$\mathbf{Y}\\!\cdot\\!\mathcal{M}\\!\cdot\\!\mathbf{X}^{\dagger}=\lvert\boldsymbol{\mathcal{M}}^{(d)}\rvert=\begin{pmatrix}\lvert\mathbf{m}^{(d)}\rvert&0\\\
0&\lvert\mathbf{m}^{(d)}\rvert\end{pmatrix},$ (VII.42)

and subsequently Eq. (VII.38) can be written as,

$\left(\boldsymbol{1}i\gamma^{b}\partial_{b}-\lvert\boldsymbol{\mathcal{M}}^{(d)}\rvert\right)\\!\cdot\\!\mathbf{\Lambda}^{(d)}(x)=0\,.\qquad$
(VII.43)

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