Seesaw mechanism and pseudo C-symmetry

It is shown that the specific"charge conjugation"symmetry used to define the Majorana fermions in the conventional seesaw mechanism, namely $(\nu_{R})^{C}=C\overline{\nu_{R}}^{T}$ for a chiral fermion $\nu_{R}$, is a hidden symmetry associated with CP symmetry, and thus it formally holds independently of the P and C violating terms in the CP invariant Lagrangian and it is applicable to charged leptons and quarks as well. This hidden symmetry, however, is not supported by a consistent unitary operator and thus it leads to mathematical (operatorial) ambiguities. To distinguish it from the standard charge conjugation symmetry, we suggest for it the name of pseudo C-symmetry. Using the pseudo-C transformation, it is impossible to define the antiunitary CPT operation and justify the CPT invariance of the Standard Model. A way to avoid the operatorially undefined pseudo C-operation is to reformulate the seesaw scheme by invoking a relativistic analogue of the Bogoliubov transformation.


Introduction
Recent impressive developments in neutrino physics are well summarized in [1][2][3][4][5][6][7]. The main remaining issue is a better understanding of the extremely small neutrino masses, and the seesaw mechanism [8][9][10] provides a convenient framework to analyze this fundamental problem. It is, however, puzzling that the Majorana neutrino, which is the exact eigenstate of C-symmetry, appears in the seesaw Lagrangian with left-right asymmetric mass terms which spoil P and C symmetries. Moreover, the customary charge conjugation used to define the Majorana neutrino in the conventional seesaw scheme, when carefully examined, leads to mathematical (operatorial) inconsistencies [11,12]. The purpose of the present paper is to clarify these puzzling aspects.
In the following, we shall use the term charge conjugation in seesaw (and, later on, pseudo C-transformation) for the operation used in defining Majorana neutrinos in the seesaw scheme, and denote it byC. This operation is explained in more detail below. In contrast, we shall name standard charge conjugation and denote it by C, the usual operation of charge conjugation as is stated in standard textbooks on field theory [13,14].
We start by recalling the essential facts about the charge conjugation analysis in the conventional seesaw scheme [2,[4][5][6]. In this framework, one constructs a Majorana fermion ν M from a chiral fermion ν R , for example, which diagonalizes the mass term of the Lagrangian, in the manner where with a charge conjugation matrix C = iγ 2 γ 0 in the convention of Ref. [13]. This charge conjugation in seesaw satisfies the relation and the Majorana-type condition The above charge conjugation in the seesaw thus satisfies the properties analogous to the standard charge conjugation. While the standard charge conjugation preserves the chirality of a Dirac spinor as in (53), the charge conjugation in seesaw is defined for ν L (x) and ν R (x) separately, as in (2), and changes the chirality of the field. The charge conjugation in seesaw is thus insensitive to the left-right mass asymmetry in the seesaw Lagrangian (see eq. (8)) but is operatorially undefined [11,12].
One may recall that, in Lagrangian field theory, we first define the classical symmetry operation and then look for the quantum operator to realize it by Noether theorem in the case of continuous symmetries or other methods. In any quantum field theory one should be able to define an operatorial realization of the charge conjugation transformation.
However, if one assumes the existence of a unitary operator C which satisfies and similarly for ν L (x). Here we used the fact that ν R (x) = ( 1+γ 5 2 )ν R (x) and the left-handedness of Cν R T (x) 1 . This sequence of equalities shows that there is a discrepancy between the classical definition of charge conjugation in seesaw and a possible quantum realization of it.
One finds further puzzling aspects arising from the ansatz (2). One can confirm that, using (1), If one assumes a transformation rule of charge conjugation in seesaw as suggested by (2), it turns out that the first and second expressions in (6) are invariant under the transformation, while the last expression leads to a vanishing Lagrangian. We emphasize that the puzzling aspect in (6) arises from the assumed classical transformation rule (7), irrespective of the existence or non-existence of the quantum operator C. Consequently, the example (6) shows that even as a classical operation, the charge conjugation in seesaw (2) is ambiguous. Remark that the three expressions in (6) are identical as long as one assumes the relation (1). The reason for the vanishing of the last expression in (6) is that we use the symmetry (2) which is not consistently defined for chiral fields; the symmetry is not compatible with the explicit presence or absence of chiral projection operators 1 Incidentally, for this definition of the seesaw charge conjugation operator, we formally have (1 ± γ 5 )/2 in front of chiral fields ν R,L (x), namely, [(1 ± γ 5 )/2]ν R,L (x) = ν R,L (x) in the Lagrangian. Any sensible definition of parity reverses the chirality, and thus the CP transformation defined as combination of the above charge conjugation in seesaw (2) and a suitable parity acts as and thus cannot be a symmetry of weak interactions, for example. The purpose of the present paper is to show that the charge conjugation in seesaw (2) is in fact a hidden symmetry associated with CP invariance. We suggest that the "charge conjugation" (2), which is used in the conventional formulation of seesaw mechanism but exhibits operatorial inconsistencies and no unitary operator, may be called pseudo C-symmetry. It is shown that a way to avoid the operatorially undefined pseudo C-symmetry but retain the physics contents of Majorana neutrinos unchanged is to use a relativistic analogue of the Bogoliubov transformation which has been formulated recently [11,12,15].
2 Derivation of pseudo C-symmetry

Seesaw Lagrangian
We study a generic Lagrangian for the three generations of neutrinos, where m D is a diagonal real 3 × 3 Dirac mass matrix (after an application of biunitary transformation), and m L and m R are 3 × 3 real symmetric matrices by assuming CP symmetry, for simplicity. The anti-symmetry of the matrix C and Fermi statistics imply that m L and m R are symmetric, and CP symmetry implies m L = m † L and m R = m † R . Thus m L and m R are real symmetric. This is the Lagrangian of neutrinos with Dirac and Majorana mass terms. For m L = 0, it represents the classical seesaw Lagrangian of type I. In the following, we shall call the expression (8) as the seesaw Lagrangian for the sake of generality, but all the physical analyses are performed for the case m L = 0.
To diagonalize the mass terms in (8), the two chiral fields ν L (x) and ν R (x) are usually taken as the basic variables in the seesaw scheme. One may equally adopt a Dirac-type variable ν(x) as a basic variable and define ν L (x) = [(1 − γ 5 )/2]ν(x) and ν R (x) = [(1 + γ 5 )/2]ν(x), respectively. These two choices are mutually invertible ν(x) = ν L (x) + ν R (x) and thus equivalent. This latter choice is used in the next section.
In the common seesaw scheme with chiral variables, which we follow in this section, the Lagrangian (8) is split into the left-handed part and the right-handed part with well-defined mass eigenvalues after the diagonalization of the mass terms. It is thus natural to define C, P and CP for those chiral variables. Historically, C, P and CP are defined for a Dirac fermion such as the electron in QED; these definitions together with their generalization to chirally projected variables are briefly summarized in Appendix A. Among those symmetry transformations, the CP transformation is defined for theories only with ν L or ν R . CP symmetry thus potentially provides a good symmetry for chiral theories. It is in fact known that CP symmetry is valid for a general class of chiral or mixed theories. In the present problem, we can show that the Lagrangian (8) is invariant under CP if certain constraints on the mass parameters are satisfied.
In the above definition of CP we adopted the transformation rule of "iγ 0 parity" which is defined, for a generic Dirac field, by such that ψ P L,R (t, x) = iγ 0 ψ R,L (t, − x) which were used to infer the classical transformation laws of chiral fermions above. The non-trivial phase freedom of the parity transformation in fermion number non-conserving theory has been analyzed by Weinberg [14]. This definition of parity operation is the natural choice in a theory with Majorana fermions. The reason is that a Majorana fermion ψ M (x), which is defined by ψ M (x)(x) = Cψ M T (x), stays Majorana after parity transformation, i.e. the parity transformation preserves the Majorana condition: [11,12]. The "iγ 0 parity" is crucial to assign a consistent intrinsic parity to an isolated Majorana fermion 2 . It is shown later in (39) that we can exactly diagonalize the Lagrangian (8) in the form (by suppressing the tilde symbol for the mass eigenstates) where These variables satisfy the classical Majorana conditions It is thus legitimate to look for some operatorC which satisfies From the comparison of (13) and (14), it may appear natural to guess that the operatorC acts as follows: This is precisely the pseudo C-symmetry transformation we discussed in (2). By the token of eq. (5), an operatorC with the above properties cannot be defined. Moreover, the inexistance of a unitary realization of the pseudo-C transformation makes it impossible to define the CPT symmetry in a way compatible with pseudo-C. Consequently, one arrives at the contradiction that in a relativistic quantum field theory like the Standard Model and its extensions, the CPT theorem apparently does not hold. (This does not preclude the definition of a well-behaved unitary charge conjugation operator corresponding to the classical transformations (13), as we show in Sect. 3. Having this, the antiunitary CPT transformation can be defined and the above-mentioned contradiction disappears.) It is remarkable that the pseudo-C symmetry is formally an exact symmetry of the Lagrangian (11) but operatorially undefined. We clarify the precise nature of the pseudo C-symmetry in the following.

Pseudo C-symmetry
We shall examine now explicitly the CP-symmetry of the Lagrangian (8). The analysis of CP invariance of the fermion number preserving terms in the Lagrangian is the usual one. We thus analyze the CP invariance of the fermion number violating terms: where we used {iγ 0 , C} = 0 and C T C = 1 [13]. This shows the CP invariance if m † L = m L and m † R = m R , and we can promote the above CP transformation rule (9) to a unitary operator in the context of the Lagrangian (8): Now we come to the crucial observation of the present paper. We examine the CP transformation of the entire Lagrangian, but stop at the level after cancelling the factor iγ 0 and changing the integration variables − x → x. We then have This relation shows a remarkable property, namely, the CP invariance of the seesaw Lagrangian implies that the action is formally invariant under the replacements independently of the values of mass parameters. Note that this symmetry is independent of space-time inversion, in spite of the fact that it changes the chirality of the field. This is precisely what we suggest to be called the pseudo C-symmetry (2) of the seesaw Lagrangian. A characteristics of the pseudo C-symmetry as a hidden symmetry associated with CP invariance is that it is defined for any CP invariant theory even if the separate well-defined P or C symmetries do not exist. Thus it is not influenced by the P and C violating left-right mass asymmetry of the seesaw Lagrangian. Moreover, this pseudo C-symmetry is also defined for "mass eigenstates" which diagonalize (8) as long as CP is preserved, for example, for the fields used in (1). One can confirm that the relation (1) when ν M (x) is formally treated as an independent field is "covariant" under CP symmetry up to the common factor iγ 0 on both sides together with spatial inversion, while the relation (1) is invariant under the pseudo C-symmetry without any spatial inversion. The pseudo C-symmetry is very general but unfortunately formal as is seen in the identity, for example, Both expressions in (20) are invariant under the CP symmetry (17), while the first expression is invariant under the pseudo C-symmetry (19) but the second expression vanishes under the same symmetry. Also, the operatorial inconsistency of the pseudo C-transformation in (5) does not occur for the well-defined CP transformation in (17). We thus understand the origin of the operatorial indefiniteness of the pseudo C-symmetry as arising from the arbitrary elimination of the factor iγ 0 of the CP transformation (17) and thus resulting in the absence of a consistent unitary operator (5) and the inconsistency in (6). The pseudo C-symmetry is defined for any CP invariant theory and thus for the Standard Model (except for the terms containing Kobayashi-Maskawa-type angles). One can introduce the pseudo C-transformation for charged leptons and quarks also; for example, in the case of the electron it will read: In contrast, the standard C transformation is defined by e L (x) → Ce R (x) T and

The CP invariant weak interaction Lagrangian (for a single generation model) is written as
One can confirm that the first expression in L W is invariant under the pseudo Csymmetry (19) and (21) together with W − µ (x) → W + µ (x), while the second identical expression of L W vanishes under the same pseudo C-symmetry [11,12]. The pseudo C-symmetry is thus operatorially ill-defined.
The chirality of the pseudo C-transformation is reversed relative to the ordinary C transformation and, to our knowledge, a "consistent CP symmetry" defined as the combination of the pseudo C-symmetry and a sensible parity operation, which may be used for weak interactions, has not been given.

Seesaw formulation with Bogoliubov transformation 3.1 Single generation model
A way to avoid the use of the pseudo C-symmetry in the analysis of the seesaw model is to use the idea of a relativistic analogue of the Bogoliubov transformation [11,12]. We first illustrate the basic procedure by analyzing the single generation model for which we can work out everything explicitly. We define a new Dirac-type variable in terms of which the above Lagrangian is re-written as where ǫ 1 = m R + m L and ǫ 5 = m R − m L . The C and P transformation rules of ν(x) are defined by and thus ν(x) ↔ ν C (x) under C and ν C (x) → iγ 0 ν C (t, − x) under P; CP is given by The above Lagrangian (24) is CP conserving, although C and P (iγ 0 -parity) are separately broken by the last term for real m D , m L and m R . Note that here we are using the standard charge conjugation and parity transformation for Dirac fields (see Appendix A).
After the Bogoliubov transformation, which diagonalizes the Lagrangian with ǫ 1 = 0, L in (24) becomes with the mass parameter The Lagrangian (27) in the form with masses M ± = M ± ǫ 1 /2. The charge conjugation and iγ 0 -parity properties, which are induced by the transformation properties of N(x), are and thus define massive Majorana fermions. It is straightforward to define the unitary charge conjugation operator C M for the free fermions ψ ± (x), which satisfies with C M |0 M = |0 M = |0 N , following the procedure in the textbook [13]; in fact, the operator charge conjugation has the form C M = exp[iπn ψ − ], with the number , and thus acts on ψ + (x) in a trivial manner. The original neutrino is expressed in terms of the Majorana fermions ψ ± if one uses (26) as ± (x)|0 M = 0 is thus sufficient for all practical applications. But we encountered an analogue of Bogoliubov transformations, and thus it is interesting to examine the possible multiple-vacua structure. If one should define the vacuum by C ν (0)|0 ν = |0 ν , then |0 M = |0 ν , since one notes that C ν = C M if one defines C ν ν(x)C † ν = ν c (x). In any case, C ν (t) is time-dependent since the C-transformation thus defined is not a symmetry of the Lagrangian (8). This implies that the vacuum of the Majorana fermions is different from the vacuum of the starting chiral fermions, when the latter are regarded as the chiral components of the Dirac neutrino field [11,12] 3 . A relation between the Majorana vacuum and the Dirac vacuum has been analyzed in detail in the infinitesimal neighborhood of the Dirac vacuum by taking ǫ 1 = 0 and ǫ 5 infinitesimal in [15].

Three generations model
For the sake of completeness, we briefly illustrate the basic procedure for the realistic three generations of leptons. We start with the Lagrangian (8) and write the mass term as where Since the mass matrix appearing is real and symmetric, we can diagonalize it by a 6 × 6 orthogonal transformation as where M 1 and M 2 are 3 × 3 real diagonal matrices. We denote one of the eigenvalues as −M 2 instead of M 2 to define the natural Majorana mass later (note that M 1 , M 2 = (m R /2) 2 + m 2 D ± m R /2, respectively, for the single flavor case with m L = 0). We thus have with Hence we can write of neutrino with CP invariance, one uses a 2 × 2 orthogonal transformation O of original chiral variables ν L,R to mass eigenstatesν L,R , This transformation mixes the fermion and anti-fermion and in this sense changes the definition of the vacuum.
In the present orthogonal transformation (38), one can confirm thatν C L = Cν R T , for example, holds after the transformation. The exact solution (39) is re-written as (by suppressing the tilde symbol for the mass eigenstates) if one defines the Dirac-type variable Introducing two 3 × 3 diagonal real matrices by Note that the terms with E 1 are C-invariant while the terms with E 5 are C-violating. One recognizes that the exact solution is three copies of the single flavor model with the vanishing Dirac mass.
A relativistic analogue of the Bogoliubov transformation (for a single flavor case) is given by (26) with the definition of the parameter sin 2θ = E 5 /2/ (E 5 /2) 2 + m 2 D with m D = 0 there. One thus obtains θ = π/4 [11], and θ = π/4 is used for all the flavors uniformly in the present example (42).
The Bogoliubov transformation is thus defined by The Lagrangian for the Bogoliubov quasiparticle N(x) is then given by [11]  When one defines the Majorana fields which satisfy the charge conjugation properties which agrees with the conventional seesaw formula [2][3][4][5][6][7], since M 1 = 1 2 (E 5 + E 1 ) and M 2 = 1 2 (E 5 − E 1 ). We bypassed the use of the pseudo C-symmetry using the Bogoliubov transformation (43). It is straightforward to define the unitary charge conjugation operator C M = C N for the free fermions ψ ± (x), which satisfies We have from the Bogoliubov transformation (43) and (45) (by restoring the tilde symbol for the mass eigenstates) and the flavor fields from (38) which shows that where we defined 3 × 3 real matrices Again we conclude C ν = C M , if one defines C ν ν(x)C † ν = ν C (x). One confirms that the CP operation of the Majorana fields The CP symmetry is preserved in the transition from original chiral fermions to Majorana fermions, although (formally defined) C ν and P ν are separately broken in (8) while both C M and P M are good symmetries in (46).

Conclusion
The conventional formulation of the seesaw mechanism customarily involves the definition of a "pseudo C-symmetry". In this paper we have clarified the origin of this pseudo C-symmetry in the CP invariance of the seesaw Lagrangian. Therefore it is defined for any fermions such as charged leptons and quarks in the SM also. This pseudo C-symmetry is thus very general, but it is operatorially undefined when carefully examined.
The operatorial indefiniteness of the pseudo C-symmetry motivated us to reformulate the seesaw Lagrangian by rewriting it in a form analogous to the BCS theory. Then a relativistic analogue of Bogoliubov transformation leads to Majorana fermions in an algebraically well-defined manner. The discrepancy between the C conjugation expected from the original Lagrangian in the Dirac neutrino limit and the C conjugation in the picture of Majorana neutrinos is taken care of by an analogue of Bogoliubov transformation [11,12]. We emphasize that in this treatment, connecting ν L and ν R by charge conjugation and parity into a Dirac field as in (23) is a justified option and it is in the spirit of Bogoliubov's approach to the BCS theory. In (26), the fields in the left-and right-hand sides are quantum fields, for which reason we could speak about the canonicity condition for them. The idea of Bogoliubov is that one can quantize the theory in terms of fields that do not diagonalize the Lagrangian and, by a canonical (Bogoliubov) transformation, arrive at the physical quantum fields in terms of which the Lagrangian is diagonal. The scheme with Bogoliubov transformation has the unique virtue of connecting two interesting quantum theories of massive neutrinos, namely the Dirac and Majorana types.
The present work was initiated when one of us (KF) was visiting Institute of High Energy Physics (IHEP), Chinese Academy of Sciences, Beijing. KF thanks Zhizhong Xing and Shun Zhou for critical comments and hospitality at IHEP. We thank Masud Chaichian for helpful comments. KF is supported in part by JSPS KAKENHI (Grant No.18K03633).

A C, P and CP transformation laws of a fermion Dirac field
We here summarize the basic definitions of C, P and CP trasnformations of a Dirac fermion together with their generalization to chirally projected components. The standard definitions of C, P and CP for a Dirac field ψ D (x) = ν(x) are given by [13,14] C : ν(x) → ν C (x) = Cν T (x), P : ν(x) → iγ 0 ν(t, − x), where we used the specific "iγ 0 -parity" instead of the more common γ 0 -parity for the reasons stated in section 2 of the present paper. The transformation laws for the chirally projected fields are defined by C : ν L,R (x) → ν C L,R (x) = Cν R,L T (x), P : ν L,R (x) → iγ 0 ν R,L (t, − x), ν C L,R (x) → iγ 0 ν C R,L (t, − x), CP : ν L,R (x) → iγ 0 Cν L,R (t, − x) T .
A salient feature of these transformation laws is that we have the doublet representations for C and P. On the other hand, we have a self-consistent transformation law for each chiral component in the case of CP symmetry. These facts play important roles in the discussions in the body of the paper.

Majorana field
Standard transformation laws of Majorana field can be inferred from the Majorana fermion defined in terms of a Dirac field ψ ± (x) ≡ 1