Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 50 (2017)

Parity-doublet representation of Majorana fermions and neutron
oscillation†∗1∗2and A.
A. Tureanu
Tureanu∗2Fujikawa,∗1K. Fujikawa
andWe present a parity-doublet theorem in the representation of the intrinsic parity of Majorana fermions,which is expected to be useful in condensed matterphysics as well, and it is illustrated to provide a criterion of neutron-antineutron oscillation in a BCS-likeeﬀective theory with ΔB = 2 terms. The CP violation inﬂuences the neutron electric dipole moment, butno direct CP violating eﬀects appear in the oscillation
itself, as demonstrated by an exact solution. An analogue of Bogoliubov transformation, which preserves Pand CP, is crucial in the analysis.
The Majorana fermions are deﬁned by the condition ψ(x) = C ψ̄ T (x) = ψ c (x) where C = iγ 2 γ 0 denotes the charge conjugation matrix. We start witha neutral Dirac fermion n(x) and deﬁne the combinac
tions ψ± (x) = √12 [n(x)±nc (x)], which satisfy ψ±(x) =±ψ± (x), showing that ψ+ (x) and ψ− (x) are Majorana
cﬁelds. We treat the fermion with ψ−(x) = −ψ− (x)also as a Majorana fermion. The common parity operation, which is called ”γ 0 -parity” here, is deﬁned byn(x) → γ 0 n(t, −x) and nc (x) → −γ 0 nc (t, −x) andsatisﬁes P2 = 1; it leads to a doublet representation{ψ+ (x), ψ− (x)}, namely ψ± (x) → γ 0 ψ∓ (t, −x). Only
when the two fermions ψ± (x) are degenerate, this doublet representation is consistent with dynamics. Themass splitting in ψ± (x) inevitably breaks the γ 0 -parity
as a symmetry of the Lagrangian. The parity thus
becomes crucial in the analysis of the neutron oscillation1,2) , which is based on the neutron expressed asa superposition of two Majorana-type fermions with
diﬀerent masses.
In the analysis of possible baryon number violationand neutron oscillation, one can study the quadratic
eﬀective Hermitian-Lagrangian
L = n(x)iγ μ ∂μ n(x) − mn(x)n(x)i− 1 [eiα nT (x)Cn(x) − e−iα n(x)CnT (x)]2i− 5 [nT (x)Cγ5 n(x) + n(x)Cγ5 nT (x)],2∗1∗2ψ+ (x; 1 ) → γ 0 ψ− (t, −x; −1 ),ψ− (x; −1 ) → γ 0 ψ+ (t, −x; 1 ),(2)which satisfy P2 = 1. This representation is valid
irrespective of whether the γ 0 -parity is conserved ornot. The γ 0 -parity violation by 1 = 0 in (1) is a necessary condition of neutron oscillation which requires
M+ = M (1 ) = M (−1 ) = M− (parity-doublet theorem) .
The solution of this Lagrangian with 1 = 0 deﬁnes
the Bogoliubov transformation
n(x)
cos φN (x) − iγ5 sin φN c (x)=,(3)
nc (x)
cos φN c (x) − iγ5 sin φN (x)with a Dirac fermion N (x), the mass M = m2 + 25 ,and sin 2φ ≡ 5 / m2 + 25 . To solve (1), we ﬁrst use
the Bogoliubov transformation (3) as a change of variables, which preserves C and P. We then obtain theexact mass formula(M ± 1

1 − (˜1 /1 )2 ) + i˜
1 γ5 ≡ M± e2iθ± γ5 ,(4)1/2

1 /1 )2 ]2 + (˜1 )2forwith M± = [M ± 1 1 − (˜two majorana fermions, where ˜1 ≡ 1 sin α sin 2φ.
The oscillation probability is given by (θ = θ+ − θ− )P (n(p) → nc (p); t) = (1 − sin2 2φ cos2 α̃) cos2 θ
1× sin2 ( ΔEt),(5)2with sin α̃ = sin α cos 2φ/ 1 − (sin α sin 2φ)2 . Under
CP transformation, which is equivalent to α → −α,θ → −θ, this oscillation formula with energy diﬀerence(1)

where m, 1 , 5 and α are real parameters. The ﬁrst
ΔB = 2 term with a real 1 breaks the γ 0 -parity while
the second ΔB = 2 term with a real 5 preserves γ 0 parity. The term with 5 in (1) preserves C and P,
while α = 0 implies CP violation. An analogy of theLagrangian in (1) to BCS theory has been emphasized
at the early stage of the study of neutron oscillation.3)†The presence of the neutron oscillation P (n → n̄) ∝sin2 ((ΔM/2)t) implies the mass splitting of auxiliaryMajorana-type fermions. In connection with this, wehave established the γ 0 -parity doublet representation{ψ+ (x; 1 ), ψ− (x; −1 )} of the solutions of (1),Condensed from the article in K. Fujikawa and A. Tureanu,
Phys. Rev. D 94 (2016) 115009.
RIKEN Nishina Center
Department of Physics, University of Helsinki- 139 -2 −2 is invariant. We do
ΔE = p2 + M+p2 + M−
not have a manifestation of CP violation in oscillationin vacuum, which is typically expressed by P (n(p) →nc (p); t) = P (nc (p) → n(p); t).

References
1) R. N. Mohapatra, J. Phys. G: Nucl. Part. Phys. 36,104006 (2009).
2) D. G. Phillips II et al., arXiv:1410.1100.
3) L. N. Chang and N. P. Chang, Phys. Rev. Lett. 45,
1540 (1980).