612
K. Earl et al.
Higgs doublets that transform under the mirror EW group SU (2)
L × U(1)
Y and
˜
Φ = iσ 2 Φ ∗ . The fields N Ri and N
Ri are heavy fermions which are singlets under
both the SM and the mirror EW group. The number of generations in the mirror
sector is not fixed, though a minimum of two generations of mirror fermions as
well as N Ri and N
Ri are required for leptogenesis and to explain the two observed
neutrino squared mass differences. In Eq. (1), the Dirac mass term is responsible
for connecting the SM and the mirror sector. In addition, we have chosen, without
loss of generality, the basis where the Dirac mass matrix M, the charged lepton and
mirror lepton Yukawa couplings (not shown above) are real and diagonal. Other
portal interactions might exist, such as a Higgs portal or photon-mirror photon
mixing, but are more model dependent.
The structure of Eq. (1) can be obtained by imposing a global U(1) symmetry that
can be identified with the total lepton number L tot = L − L which is an extended
lepton number defined in terms of both SM lepton number (L) and mirror lepton
number (L ). We have
L tot (l Lα ) = L tot (N Ri ) = −L tot (l
Lα ) = −L tot (N
Ri ),
L tot (Φ) = L tot (Φ
) = 0.
(2)
In this case, the SM and twin right-handed neutrinos combine to form heavy Dirac
states
N = N R + (N
R )
c .
(3)
With this definition, we can rewrite the model Lagrangian as follows:
L = i ¯
N i /
∂N i − M i ¯
N i N i − y αj ¯
l α ˜
ΦP R N j − y
αj
¯
l α ˜
Φ
P R N
c
j + h.c.
(4)
1.1 Heavy N Decay and CP Violation
In the model considered in this study we can have the following decay processes
involving the heavy neutrinos N i → l α Φ, N i → ¯
l
α
¯
Φ , and its antiparticle ¯
N i →
¯
l α ¯
Φ, ¯
N i → l
α Φ . For generic complex Yukawa couplings y and y in Eq. (4) we can
have CP violation in the decays of the heavy neutrinos and this will imply non-zero
ΔΓ (N i ) α ≡ Γ (N i → l α Φ) − Γ ( ¯
N i → ¯
l α ¯
Φ) in the visible sector and non-zero
ΔΓ ( ¯
N i ) α ≡ Γ ( ¯
N i → l
α Φ ) − Γ (N i → ¯
l
α
¯
Φ ) in the mirror sector. Let us define
Γ (N i ) ≡
α [Γ (N i → l α Φ) + Γ (N i → ¯
l
α
¯
Φ )] and Γ ( ¯
N i ) ≡
α [Γ ( ¯
N i →
¯
l α ¯
Φ) + Γ ( ¯
N i → l
α Φ )]. CPT conservation implies that
Γ (N i ) = Γ ( ¯
N i ) ≡ Γ N i ,
(5)
where
Γ N i =
M i
16π
[(y
† y) ii + (y
y
) ii ].
(6)
K. Earl et al.
Higgs doublets that transform under the mirror EW group SU (2)
L × U(1)
Y and
˜
Φ = iσ 2 Φ ∗ . The fields N Ri and N
Ri are heavy fermions which are singlets under
both the SM and the mirror EW group. The number of generations in the mirror
sector is not fixed, though a minimum of two generations of mirror fermions as
well as N Ri and N
Ri are required for leptogenesis and to explain the two observed
neutrino squared mass differences. In Eq. (1), the Dirac mass term is responsible
for connecting the SM and the mirror sector. In addition, we have chosen, without
loss of generality, the basis where the Dirac mass matrix M, the charged lepton and
mirror lepton Yukawa couplings (not shown above) are real and diagonal. Other
portal interactions might exist, such as a Higgs portal or photon-mirror photon
mixing, but are more model dependent.
The structure of Eq. (1) can be obtained by imposing a global U(1) symmetry that
can be identified with the total lepton number L tot = L − L which is an extended
lepton number defined in terms of both SM lepton number (L) and mirror lepton
number (L ). We have
L tot (l Lα ) = L tot (N Ri ) = −L tot (l
Lα ) = −L tot (N
Ri ),
L tot (Φ) = L tot (Φ
) = 0.
(2)
In this case, the SM and twin right-handed neutrinos combine to form heavy Dirac
states
N = N R + (N
R )
c .
(3)
With this definition, we can rewrite the model Lagrangian as follows:
L = i ¯
N i /
∂N i − M i ¯
N i N i − y αj ¯
l α ˜
ΦP R N j − y
αj
¯
l α ˜
Φ
P R N
c
j + h.c.
(4)
1.1 Heavy N Decay and CP Violation
In the model considered in this study we can have the following decay processes
involving the heavy neutrinos N i → l α Φ, N i → ¯
l
α
¯
Φ , and its antiparticle ¯
N i →
¯
l α ¯
Φ, ¯
N i → l
α Φ . For generic complex Yukawa couplings y and y in Eq. (4) we can
have CP violation in the decays of the heavy neutrinos and this will imply non-zero
ΔΓ (N i ) α ≡ Γ (N i → l α Φ) − Γ ( ¯
N i → ¯
l α ¯
Φ) in the visible sector and non-zero
ΔΓ ( ¯
N i ) α ≡ Γ ( ¯
N i → l
α Φ ) − Γ (N i → ¯
l
α
¯
Φ ) in the mirror sector. Let us define
Γ (N i ) ≡
α [Γ (N i → l α Φ) + Γ (N i → ¯
l
α
¯
Φ )] and Γ ( ¯
N i ) ≡
α [Γ ( ¯
N i →
¯
l α ¯
Φ) + Γ ( ¯
N i → l
α Φ )]. CPT conservation implies that
Γ (N i ) = Γ ( ¯
N i ) ≡ Γ N i ,
(5)
where
Γ N i =
M i
16π
[(y
† y) ii + (y
y
) ii ].
(6)