Majorana propagator on de Sitter space

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.

mological spaces was discussed in Refs. [7,8], 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 onshell 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. [7,8].
Due to the Majorana nature of neutrinos, leptogenesis [9] (see Ref. [10] 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 [11][12][13][14]. This motivates a careful consideration of the Majorana propagator in outof-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 [15], also see [16], [17]. Next, the vector propagators on de Sitter space were initially constructed in covariant Fermi gauges in [18], for massive vectors of scalar electrodynamics in exact Lorenz gauge in [19], and some subtleties related gauge fermionic mass squared scales as the Ricci curvature scalar.
fixing were subsequently discussed in Ref. [20]. The graviton propagator was constructed and discussed at length in Refs. [21][22][23][24][25][26][27][28][29]. 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. [30] and the massive vector propagator for scalar electrodynamics was recently constructed in Ref. [31], 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. with m = m 1 + im 2 , with m 1 , m 2 ∈ 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 µν = diag −1, a 2 , a 2 ...., a 2 D , (II. 3) such that its determinant equals, g = −a 2(D−1) . The covariant derivative operator ∇ µ acts on a spinor as, and spin(or) connection Γ µ can be obtained from, where e ν c denote vielbein (used to project vectors (and tensors) on tangent space, V µ (x) = e µ a (x)V a , where we use Latin letters to denote indices on flat tangent space). By transforming to conformal time η as, dt = a(η)dη, (II. 6) in which the metric becomes, g µν = a 2 (η)η µν (η µν denotes Minkowski metric in D dimensions) and using vielbein formalism [6] one arrives at, Here a(η) is the scale factor for a spatially flat homogeneous expanding universe and η 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.

A. Dirac equation
The Dirac equation which follows from Eq. (II.1) is given by, Using Eq. (II.7) and upon performing a conformal rescaling of the fermion field, the Dirac equation (II.8) can be recast as, 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, The Lagrangian (II.1) then becomes (cf. Appendix A), Under a global U(1) transformation, χ L,R → e −iθ χ L,R , and therefore the fields Ψ andΨ transform infinitesimally as, Ψ → Ψ+θΦ θ andΨ →Ψ+θΦ θ , with Φ θ = −iΨ andΦ θ = iΨ, from which it follows that the Lagrangian (II.12) remains invariant. These transformations imply the following conserved (Noether) vector current for Dirac particles, Let us now consider the Majorana particles, defined by the Majorana condition: (II.14) Here we have defined iσ 2 = , where ab denotes the antisymmetric tensor in two dimensions, defined by 12 = 1 = − 21 and 11 = 22 = 1. Thus, the Majorana field Ψ M (x) is defined as The simplest way to arrive at the Lagrangian for Majorana fermions is to to replace Ψ by Ψ M into (II.15), which results in the following Lagrangian, 2 in the symmetry generators of conformal group in D = 4 [32]. 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. [5] and to the more general propagator discussed in Ref. [6]. 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. [33] would be fundamentally changed in the absence of a conserved vector current (cf. Eq. (5) of Ref. [33]).

Canonical quantization
The above considerations and Eq. (II.1) suggest the following action and Lagrangian for the Majorana fermions in its canonical form, The additional factor 1/2 can be justified by noting that -due to the Majorana condition -Ψ M andΨ M are not independent fields, which can be clearly seen from, It is now quite easy to show that, transposing (II. 19) and inserting γ 0 (iγ 2 )(−iγ 2 )γ 0 = 1 and which is therefore not independent from Eq. (II.19). 3 The canonically normalized Lagrangian for the Majorana field Ψ M (x) in (II.17) (when recast in terms of 2 (D−2)/2 -spinor fields) reads, 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 χ R ), and therefore the spinors χ L and χ * L in (II.21) are independent complex spinor fields. The corresponding canonical momenta are obtained by varying the action S M with respect to ∂ 0 χ L (x) and where b denotes a spinor index. When these are promoted to operators,χ L (x),χ * L (x),π χ L andπ χ * L , canonical quantization then implies the following anticommutators ( = 1), 3 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 Φ is, where Φ * and Φ are considered independent degrees of freedom. On the other hand, the canonically normalized Lagrangian for a real scalar field φ is, where the reality condition φ * (x) = φ(x) plays the role of the Majorana condition for fermions, which is just the charge transformation for scalar fields. or equivalently, 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 χ L and χ * 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.

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, the derivation of which is presented in Appendix B.Ψ M,α (x) is the rescaled Majorana field, The operatorsb h ( k) andb † h ( k) in the mode decomposition (II.24) are the Majorana fermion annihilation and creation operators, respectively.b h ( k) annihilates the vacuum state |Ω , b h ( k)|Ω = 0, andb † h ( k) creates a particle of momentum 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, (II.28) A h,α (η, k) andĀ h,α (η, k) in Eqs. (II.24) and (II.25) are defined as, where ρ h (η, k) is the rescaled helicity eigenspinor in momentum space, L h (η, k) is the Majorana particle mode function, and ξ h ( k) is the helicity eigenspinor defined by, whereĥ is the helicity operator, which in D = 4 has the simple form,ĥ = ( k/ k ) · σ 4 and 4 We 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. [6]. using Eq. (VII.21) we can write A h (η, k) andĀ h (η, k), In order to progress towards normalization of the Majorana mode functions, note that the following identity for the chiral eigenvectors holds, which is proved in Appendix C. Next, since the helicity eigenstates L ± (η, k) do not couple to the vacuum, their normalization should be independent on helicity, can see that the normalisation condition is given by, (II.34) Finally, from Eqs. (II.32) and (II.33) from the above equation it follows that, 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. [6], [34]. 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, Here it is important to notice that, due to the hemispherical construction of helicity eigenmodes (discussed in detail in section II B 3 below), for a given helicity and momentum (h = ±, k), only one of the mode function densities contributes (the other is zero). For example, for h = +1, |L + (η, k)| 2 contributes and |L − (η, 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.

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 ≥ 0: It is convenient to transform to the positive/negative frequency basis, in which the mode functions u h± obey a Bessel's differential equation, The Bunch-Davies vacuum solutions 5 can be written in terms of Hankel functions as follows (cf. Ref. [34]), is Hankel function of the first kind, which in the ultraviolet (equivalently in a distant past, kη → −∞), reduces to the positive frequency vacuum solution. The indices ν ± are given by, We can then write the set of equations in Eq. (II.38) for k z ≤ 0 as follows, for which we again transform into the positive/negative frequency basis, 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, (II.46) From Eq. (II.41)) and Eq. (II.45) one finds a complete set of solutions for the mode functions, ν (− k η) and thus, the solutions for e − iφ 2 L h (η, k) and e iφ 2 L * h (η, − k) are also defined on this hemisphere, as can be seen in Fig. 1 (right panel). However, when we Hence the solutions to the corresponding mode functions (right panel) are also defined on the upper hemisphere.
Right panel depicts the solutions to the mode functions, which are also defined on the lower hemisphere.
consider k → − k, then the solutions are defined on the lower hemisphere, shown in Fig. 2 (left panel), and described by u h± (η, − k), which in turn are given in terms of Hankel functions of the second kind H (2) ν (− k η), and therefore the solutions for e − iφ 2 L h (η, − k) and e iφ 2 L * h (η, k) are also defined on the lower hemisphere, which is illustrated in Figure 2 (right panel).

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, Next, the following invariant distance (biscalar) functions are useful, where iε denotes an imaginary infinitesimal time shift (  1), which are useful for definition of various two-point functions on de Sitter. Eqs. (III.1-III.4) imply the following identities, (III.6) These distance functions are related to the geodesic distance on de Sitter space (x; x ) as

A. Construction of the propagator
The Majorana propagator obeys the equation, and it ought to be symmetric under the exchange of x and x . 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, This results in, Further steps in the construction of the propagator are analogous to those in Ref. [6].
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, After several steps, 6 and in particular upon making use of Eq. (II.7), one arrives at the Majorana propagator in the form, and H is the Hubble rate, which is constant on de Sitter. The functions iS + (x; x ) and iS + (x; x ) are biscalars defined as follows, Bessel functions can be found in identity (6.578.10) of Ref. [35]. 6 For details we refer to Ref. [6].
The propagator (III.12) is still in the basis in which the phase φ = 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, (III.14) When commuted through the kinetic operator in (III.12), this then gives, for Majorana particles, cf. footnote 3.

B. 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 ) 7 Notice that these operators are still proper projectors, in the sense that they obey, [P 5 ± ] 2 = P 5 ± and P 5 + P 5 − = 0 = P 5 − P 5 + , where we made use of e iγ 5 φ = cos(φ) + iγ 5 sin(φ). 8 Recall that the creation and annihilation operators are identical for both frequency poles. and iS + (x; x ) in Eq. (III.15) using the identity (9.131.2) from Ref. [35], which transforms the propagator to the form that is convenient near the lightcone, where The Minkowski limit is defined by H → 0 and a(η), a(η ) → 1, which also means that, when expanding the hypergeometric functions we can use that (1 ± iζ) n (∓iζ) n (|m|/H) 2n and (y ++ (x; x )/4) n H 2n (∆x 2 ++ ) n , where ∆x 2 ++ ≡ ∆x 2 ++ (x; x ) is defined in Eq. (III.2). Inserting these into (III.16-III.17) yields (in the Minkowski limit), where I ±ν (z) denote modified Bessel functions of the first kind. To get the last piece ∝ I − D−2 2 in Eq. (III.19) we made use of the following asymptotic property of the gamma functions, Now using the following definition, where K ν (z) denotes modified Bessel's function of the second kind, one can recast Eq.
(III.19) as, From this it follows that the Minkowski limit the propagator (III.15) is, where i∆ |m| (x; x ) denotes the massive scalar propagator in Minkowski space (whose mass equals |m|), The propagator (III.23) obeys the Minkowski space Dirac equation with the correct source, which immediately follows from the Majorana conditions, other under charge conjugation, Applying the charge conjugation transformation to Eq. (III.28) one obtains, If we were to calculate the integrals in Eq. (III.34), we would obtain, where iS − (x; x ) and iS − (x; x ) in contrast to iS + (x; x ) and iS + (x; x ) are given by, (III. 36) Notice that, upon the charge conjugation transformation, we have the propagator in terms of (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,

(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 ) andS + (x; x ), 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 [38], [39].

IV. ONE-LOOP EFFECTIVE ACTION
The one-loop effective action for Majorana fermions is given by 10 where the second equality follows from Eq. (III.14) and the observation that Tr[γ 5 ] = 0.
Taking a derivative of (IV.1) with respect to |m| and integrating over |m| gives an equvalent 9 The 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. 10 When 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 thatψ M and ψ 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. wherem > 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. [6], whereζ =m/H. This action can be expanded around D = 4, and its convenient rewriting is, where we made use of H D−2 = µ D−4 H 2 1 + D−4 2 ln H 2 /µ 2 + O (D − 4) 2 . Now, keeping in mind that, where γ E = ψ(1) ≈ 0.57 · · · is the Euler-Mascheroni constant, one sees that the divergences in (IV.4) can be removed by adding the counterterm action of the form, resulting in the following renormalized one-loop effective action, where S HE is the Hilbert-Einstein action with a renormalized Newton constant and cosmological constant, whereas the finite parts of the counterterm couplings α f and β 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. [6]. 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. [6], a different (more complicated) counterterm action is needed. For details of the renormalization procedure in that case we refer to Ref. [6].

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.

A. Multiflavor Majorana Fermions
The Lagrangian for the multiflavor Majorana fermion, Λ I (x), (I = 1, 2, · · · , n), where n is the number of flavors, is the following natural generalization of Eq. (II.17), where M IJ is a complex symmetric Majorana mass matrix, where m is a symmetric complex scalar (non-spinorial) n × n mass matrix, and L int is an interaction Lagrangian, which in the simple case of Yukawa couplings can be written as, where Y IJ is the symmetric complex Yukawa matrix of couplings, where y is a symmetric scalar complex matrix of couplings, and φ(x) is a scalar field con- from which we can defineΛ I (x) as,Λ and the Dirac equation that results from Eq.(V.1) is given by, where we neglected the Yukawa term (V.3). Here the Majorana fieldΛ J (x) is in the flavor mixing basis and we can write it in the mass diagonal basis as follows, where unitary matrices V and U are defined as follows, Eq. (V.7) can be diagnonalized 11 by writing it as follows, then we can multiply this equation by Y from the left which gives us the following result, The term Y · γ b ∂ b X † can be reduced as follows, and thus Eq. (V.13) can be written as, The 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).
where |M M 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, are the annihilation and creation operators, which are vectors in the flavor space, A (d) h (η, k) is the mode function matrix, which is diagonal in the basis in which the mass is diagonal, and it is defined by, where ξ ξ ξ h ( k) builds an n−component flavor vector, and similarly L can be found in the same way as we have done for the single flavor case in section III, . . .

B. Multiflavor Majorana propagator
The multiflavor Majorana propagator obeys the equation of motion, where δ IJ denotes the Kronecker delta in flavor space and the propagator iS P J (x; x ) is defined as (see Eq. (III.9)), (V.28) Making use of Eqs. (V.8-V.9) and (V.11) we can write this as follows, where the propagator iS (d) is in the mass diagonal basis and satisfies the following equation, and |M (d) | (see Eqs. (VII.  in Appendix E) is the real mass matrix obtained after diagonalization and is defined as, Thus we can see that the propagator in Eq. (V.29) can be rotated to remove the phases arising from X and 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, where iS + (x; x ) and iS + (x; x ) denote a (diagonal) matrix generalization of solutions in Eq. (III.13), where ζ ζ ζ (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 iG F (x; x ). Firstly, one introduces unitary rotation matrices R and Q, which act on the propagator as, where iG (d) (x; x ) denotes the diagonal Dirac propagator and the rotation matrices are given by, are the diagonal chiral matrices with n arbitrary phases, which can be used to rotate away the phases from the diagonal elements of M (d) . The propagator can then be written as, where M M M (d) is a diagonal complex matrix, iS S S + (x; x ) and iS S S + (x; x ) are defined in Eq. (V. 33) and the unitary rotation matrices U and V are given by, One should keep in mind that U and V are not general unitary matrices, but from each of them n common phases have been removed by factoring out the diagonal matrices e i 2 θ θ θ (d) γ 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 = e iφ and U R = e iψ , such that a complex mass m = m 1 + im 2 = ρe iµ can be diagonalized by U L m(U R ) † = e iφ ρe iµ e −iψ = ρ. This fixes the phase difference, ψ − φ = µ, but leaves their sum φ + ψ unconstrained. In the general case of n Dirac flavors we have m ij = ρ ij e iµ ij .
Let us for simplicity consider how the unitary matrices of the form (U R ) ij = e iφ j δ ij and This then implies that, out of the 2n phases {φ i , ψ i } (i = 1, 2, · · · , n), one can only use n of the differences φ i −ψ j to remove some of the phases in µ ij . However, the n remaining linearly independent combinations φ i + ψ 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 φ i + ψ 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, U L (n) ⊗ U R (n), but instead it is the quotient group, [U L (n) ⊗ U R (n)]/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 princi- 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. 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, φ = 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 12 Our results differ from earlier works [7,8], where the Majorana propagator was constructed by imposing the left chiral projector P L = (1 − γ 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. 13 These 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. negative frequency projectors, P ± = (1 ± γ 0 )/2, in the propagator (III.15) to give them a chiral nature, P 5 ± = 1 ± e iγ 5 φ γ 0 /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 Γ Charge conjugation operator is given by the following, where is an antisymmetric matrix and the following property satisfied by the charge conjugation operator comes in handy In what follows we introduce some transformations of projection operators, These operators transform under the charge conjugation as follows,

Appendix B: Mode decomposition for Majorana fields
The rescaled classical Dirac fieldΨ α,D (x) can be represented in the spatial momentum space as folows,Ψ whereχ L (x) andχ R (x) are the rescaled chiral 2-spinors and C h (η, k) is the momentum and helicity eigenspinor defined by, where L h (η, k) and R h (η, k) are complex chiral mode functions ξ h ( 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 e i k· x are eigenstates of the spatial derivative operator ∂ i with the eigenvalue ik i . It is convenient (and customary) to recast the rescaled quantum Dirac fieldΨ α,D (x) in terms of the mode operators decomposed into distinct creation (d † h ( k)) and annihilation (b h ( k)) operators, where D h,α (η, − k) = −iγ 2 C * h,α (η, − k). In contrast, the rescaled Majorana fermions can be described by imposing the Majorana condition on Eq. (VII.7), where the momentum eigenspinor A h (η, k), which gives the positive energy solutions, is defined as, (VII.14) The quantized rescaled Majorana fermion field is then decomposed as, where we have defined, |k| = k2 x +k 2 y and θ(k x ,k y ) = tan −1 k ŷ kx . Now we can see that (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 iθ(kx,ky) ξ h ( k) under k → − k transforms as follows, then we can use Takagi diagonalization [36,37] which says that for every symmetric complex mass matrix m, there exists a unitary matrix U L such that, 11 , m (d) 22 , · · · , m (d) nn , (VII. 40) where |m (d) | is real and non-negative as given from the Takagi theorem (a special care must be taken when the mass matrix is degenerate [37]). And subsequently taking the hermitian