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Some general aspects of Majorana fermions (inparticular, neutrinos)
Kazuo Fujikawa

Interdisciplinary Theoretical and Mathematical Sciences Program
RIKEN, Wako 351-0198, Japan
1. Majorana fermion and parity
2. Charge conjugation, ψM (x) = νL (x) + CνL T (x)Pseudo-C ⇒ No neutrinoless double-beta decay
3. CP to deﬁne Majorana
4. Seesaw and generalized Pauli-Gursey transformation( Naturalness of seesaw mechanism)
5. Parity of “emergent” Majorana neutrino in the extension of SM

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1. Majorana fermion and parity
Majorana fermion is deﬁned by the Dirac action but with purely imaginary Diracgamma matrices γ µ .
Then the Dirac equation[iγ µ ∂µ − m]ψ(x) = 0
is a real diﬀerential equation, and one can impose the reality condition on thesolution
ψ(x)⋆ = ψ(x)which implies the self-conjugate under the charge conjugation
The conventional parity transformation
ψ(x) → ψ p (t, −⃗x) = γ 0 ψ(t, −⃗x)with eigenvlues ±1 cannot maintain the reality condition for the purely imaginary
γ 0 . Thus the “iγ 0 -parity”
ψ(x) → ψ p (t, −⃗x) = iγ 0 ψ(t, −⃗x)is chosen as a natural parity transformation rule for the Majorana fermion.
E. Majorana(1937), B. Kayser(1982).

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In the generic representation of the Dirac matrices, the “iγ 0 -parity” satisﬁes thecondition with C = iγ 2 γ 0iγ 0 ψ(t, −⃗x) = Ciγ 0 ψ(t, −⃗x)

Tfor the ﬁeld which satisﬁes the classical Majorana condition
ψ(x) = Cψ(x)Tand thus iγ 0 -parity is a natural choice of the parity for the Majorana fermion inthis generic representation.
For consistency, we assign the iγ 0 -parity convention to charged leptons also when
we assign iγ 0 -parity to neutrinos, although this extra phase is cancelled in the lepton
number conserving terms in the charged lepton sector.

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2. Charge conjugation C
In the extension of the SM, we often encounter the Majorana neutrino of the form
ψ(x) = νL (x) + CνL T (x)This ﬁeld satisﬁes the classical Majorana conditionTψ(x) = Cψ (x)and in this operation, we seeνL (x) → CνL T (x),CνL T (x) → νL (x)It is thus tempting to identify the charge conjugation of νL (x), which is denotedas νLc (x), byνLc (x) = CνL T (x)We named this as “Pseudo-C”.
KF and A. Tureanu, Eur. Phys. J. C79 (2019) 752.

5We note that this pseudo-C diﬀers from the conventional charge conjugationνLc (x) = CνR T (x)which is related to the absence of the Majorana-Weyl fermion in d = 4.
In d = 4, charge conjugation changes the chiral operator γ5 → −γ5 .
This pseudo-C has several diﬃculties:
First of all, it is operatorially inconsistent. Suppose that we have an operator
which generates the pseudo-C
˜ L (x)C˜† = CνL T (x)Cν5then we have, by noting νL (x) = ( 1−γ2 )νL (x),˜ L (x)C˜† = ( 1 − γ5 )Cν˜ L (x)C˜† = ( 1 − γ5 )CνL T (x) = 0Cν
22since CνL T (x) is right-handed.

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Secondly, it would keep the Weyl fermion∫∫
4d xL = d4 xνL (x)iγ µ ∂µ νL (x)formally invariant under the charge conjugation. We know that C is not deﬁned forthe Weyl neutrino.
Thirdly, we have a diﬃculty to deﬁne a sensible CP for the pseudo-C.
We know that the parity (mirror reﬂection) should be of the form
νL (t, ⃗x) → iγ 0 νR (t, −⃗x)
We also know that CP should be of the form
νL (t, ⃗x) → iγ 0 CνL T (t, −⃗x)
to keep the Weyl neutrino invariant. But it is impossible to deﬁne this CP using
the pseudo-C.

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Furthermore, no neutrinoless double beta decay if the vacuum is invariant
under the pseudo-C.
KF. Eur. Phys. J. C80 (2020) 285.
∫∫
4d xLWeak ==∫√d x[(g/ 2)¯lL (x)γ µ Wµ (x)UP M N S νL (x) + h.c.]4√(1 − γ5 )ψM (x) + h.c.]d4 x[(g/ 2)¯lL (x)γ µ Wµ (x)UP M N S
2in the case of Weinberg’s model ψM (x) = νL (x) + CνL T (x).
A necessary condition of the neutrinoless double beta decay is that not all thetime-ordered correlations of the neutrino mass eigenstates(1 − γ5 )(1 − γ5 )ψM (x)ψM (y)|0⟩22vanish in the second order perturbation in LWeak .
If a unitary operator C˜ which generates the pseudo-C exists and if the (neutrino)
vacuum |0⟩ should be invariant⟨0|T ⋆ νL (x)νL (y)|0⟩ = ⟨0|T ⋆C˜† |0⟩ = |0⟩
and thus ⟨0|C˜ = ⟨0|, one can prove that all of the above correlations vanish
1 − γ5)νL ](x)νL (y)|0⟩2
˜ ⋆ [( 1 − γ5 )νL ](x)νL (y)C˜† |0⟩= ⟨0|CT2
1 − γ5= ⟨0|T ⋆ [()CνL T ](x)[CνL T ](y)|0⟩2= 0⟨0|T ⋆ νL (x)νL (y)|0⟩ = ⟨0|T ⋆ [(T5˜˜†
where we used νL (x) = ( 1−γ2 )νL (x) and CνL (x)C = CνL (x) and the fact thatCνL T (x) is right-handed.

8How to deﬁne Majorana ψM (x) = νL (x) + CνL T (x)?
KF. Eur. Phys. J. C80 (2020) 285.
1. The use of CP to deﬁne the Majorana
We know CP for the chiral fermionνL (x) → iγ 0 CνL T (t, −⃗x)ThenT

ψM (x) → iγ 0 CψM (t, −⃗x) = iγ 0 ψM (t, −⃗x)
Tfor the fermion which satisﬁes the classical Majorana condition CψM (x) = ψM (x)We thus deﬁne the Majorana by CP symmetrycp
ψM (x) → ψM(t, −⃗x) = iγ 0 ψM (t, −⃗x)

We do not assign C to the Majorana ψM (x), since ψM (x) → CνR T (x) + νR (x) underC.
If one wishes to have C and P for the Majorana neutrino ψM (x), one may deﬁne
modiﬁed symmetry
CM = 1,PM = CPThen†CM ψM (x)CM= ψM (x),†PM ψM (x)PM= iγ 0 ψM (t, −⃗x)

9For seesaw model, the use of generalized Pauli-Gursey
KF. Phys.Lett. B789 (2019) 76
We start with a model LagrangianL = ν L (x)iγ µ ∂µ νL (x) + ν R (x)iγ µ ∂µ νR (x)− ν L (x)mD νR (x) − (1/2)νLT (x)CmL νL (x)− (1/2)νRT (x)CmR νR (x) + h.c.
We write the mass term as((−2)Lmass =νR νRC)( mR mD
TmD mL)(νLCνL(1))+ h.c.,(2)

where we deﬁned
νLC ≡ CνR T ,νRC ≡ CνL T(3)We diagonalize the complex symmetric mass matrix using a 6 × 6 unitary matrix(Autonne-Takagi factorization)
()
()mm
M
0RD1UTU=,(4)TmD mL
0 −M2
where M1 and M2 are 3 × 3 real diagonal matrices.

10In passing, we comment on Naturalness of seesaw:
KF and A. Tureanu, Phys. Lett. B767 (2017) 199.
We denote that√
M1 , M2 = (mR /2)2 + m2D ± mR /2 ≃ mR or m2D /mR

for the special case of a single generation with real mD , mR , and mL = 0.
Hierarchy issue has been analyzed using the dimensional regularization which is
free of quadratic divergences (hierarchy and quadratic divergence are inependentnotions.)
In this interpretation, for the typical mR = 104 ∼ 1015 GeV, a hitherto unrecognized ﬁne tuning of the order mν /mR = 10−15 ∼ 10−26 .
If SUSY is discovered at some energy below GUT scale, this naive estimate is
modiﬁed.
“Naturalness” is a subjective concept.

11Coming back to seesaw, we thus have)( C ))(M(0ν̃L1(−2)Lmass = ν̃R ν̃RC+ h.c.,0 −M2ν̃L
where we deﬁned(νLCνL)(=Uν̃LCν̃L)(,νRνRC)(= U⋆ν̃Rν̃RC(5)).

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We then have
L = ν̃L (x)i ̸ ∂ ν̃L (x) + ν̃R (x)i ̸ ∂ ν̃R (x)− (1/2){ν̃RT CM1 ν̃R − ν̃LT CM2 ν̃L } + h.c..
Conventionally, one deﬁnesTψ+ (x) = ν̃R + ν̃LC = ν̃R + C ν̃R ,ψ− (x) = ν̃L − ν̃RC = ν̃L − C ν̃LTand one obtains
L = (1/2){ψ+ (x)i ̸ ∂ψ+ (x) + ψ− (x)i ̸ ∂ψ− (x)}− (1/2){ψ+ M1 ψ+ + ψ− M2 ψ− }One may use CP to deﬁne the Majorana ψ± (x), as we discussed.

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Alternatively, generalized Pauli-Gursey transformation: canonical transformation.
Arbitrary 6 × 6 unitary U and generic transformation))
(( C)
( C)
(ν̃R
νL
ν̃L
νR
⋆
=U,=U(8)
ν̃RC
νL
ν̃L
νRC
The kinetic part of the Lagrangian is form invariant under this transformation.
We thus consider a further 6 × 6 real generalized Pauli-Gursey transformation Oby( C)
( C)
()
()
ν̃LNLν̃RNR=O,=O,(9)ν̃LNLν̃RC
NRC
By choose a speciﬁc 6 × 6 orthogonal transformation()
11 1O=√
2 −1 1(10)where 1 stands for a 3 × 3 unit matrix, we have
L = (1/2){N (x)i ̸ ∂N (x) + N C (x)i ̸ ∂N C (x)}− (1/4){N (M1 + M2 )N + N C (M1 + M2 )N C }− (1/4)[N (M1 − M2 )N C + N C (M1 − M2 )N ](11)

which is invariant under C, P and CP
T

C : N (x) ↔ N C (x) = CN (x),P : N (x) → iγ 0 N (t, −⃗x), N C (x) → iγ 0 N C (t, −⃗x),
CP : N (x) → iγ 0 N C (t, −⃗x), N C (x) → iγ 0 N (t, −⃗x).

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Note that only the Dirac-type particles N (x) and N C (x) with well-deﬁned C,
P and CP properties appear after this Pauli-Gursey: γ5 disappears.

13We now make a renaming of variables11
ψ+ (x) = √ (N (x) + N C (x)), ψ− (x) = √ (N (x) − N C (x)),
22(13)

and we obtain
L = (1/2){ψ+ (x)i ̸ ∂ψ+ (x) + ψ− (x)i ̸ ∂ψ− (x)}− (1/2){ψ+ M1 ψ+ + ψ− M2 ψ− }.

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After this renaming of variables, we ﬁnd the transformation laws of ψ± (x) inducedby those of N and N C ,C : ψ+ (x) → ψ+ (x), ψ− (x) → −ψ− (x),P : ψ+ → iγ 0 ψ+ (t, −⃗x), ψ− (x) → iγ 0 ψ− (t, −⃗x),
CP : ψ+ (x) → iγ 0 ψ+ (t, −⃗x), ψ− (x) → −iγ 0 ψ− (t, −⃗x)(15)

which naturally keep the Lagrangian invariant.
One can deﬁne Majorana fermions in a natural manner by a suitable choice ofgeneralized Pauli-Gursey transformation in the seesaw model.

14Parity in the extension of SM and “emergent” Majorana neutrino
If iγ 0 parity is mandatory for the Majorana, how to formulate the Majorana emeging in an extension of SM, such as in the model of Weinberg?
One can in fact formulate the emergent Majorana neutrino consistently using either
γ parity or iγ 0 parity.
0The basic idea is thatψM (x) = eiα νL (x) + e−iα CνL T (x)with arbitrary real α satisﬁes the Majorana conditionTψM (x) = CψM (x) (x)A suitable choice of α can compensate the diﬀerent parity, γ 0 parity or iγ 0 parity,in the starting theory.
KF. Phys. Rev. D102 (2020) 105001.