JHEP12(2018)111
addition to the low-energy CP-phases, there are also phases associated with the R matrix.
As the R matrix is complex and orthogonal, it may be parametrised in terms of three
complex rotation matrices
R =
1 0 0
0 c
θ
1
s
θ
1
0 −s
θ
1
c
θ
1
c
θ
2
0 s
θ
2
0 1 0
−s
θ
2
0 c
θ
2
c
θ
3
s
θ
3
0
−s
θ
3
c
θ
3
0
0 0 1
, (2.3)
where c
θ
i
= cos θ
i
, s
θ
i
= sin θ
i
for i = 1, 2, 3; and θ
i
are complex angles. Therefore, in
general, the baryon asymmetry produced from thermal leptogenesis is a function of 18
parameters of the Casas-Ibarra parametrisation.
2.1 Leptogenesis with two heavy Majorana neutrinos
In order to simplify our discussion, we consider the decoupling limit of the heaviest Ma-
jorana neutrino, N
3
. From the seesaw relation, this causes the lightest neutrino to be
massless, which is consistent with data from oscillation experiments. In this limit, the R
matrix assumes the following simplified form [32–34]
R
NO
=
0 cos θ ±sin θ
0 ±sin θ cos θ
1 0 0
, R
IO
=
cos θ ±sin θ 0
±sin θ cos θ 0
0 0 1
, (2.4)
for normal (NO) and inverted (IO) ordering respectively. As a consequence, the Yukawa
matrix, for normal ordering, may be written as
Y
N
=
1
v
U
e1
U
e1
U
e3
U
µ1
U
µ2
U
µ3
U
τ1
U
τ2
U
τ3
0 0 0
0
√
m
2
0
0 0
√
m
3
0 c
θ
s
θ
0 −s
θ
c
θ
1 0 0
T
√
M
1
0 0
0
√
M
2
0
0 0 0
=
1
v
√
M
1
U
e2
√
m
2
c
θ
+ U
e3
√
m
3
s
θ
√
M
2
U
e3
√
m
3
c
θ
− U
e2
√
m
2
s
θ
0
√
M
1
U
µ2
√
m
2
c
θ
+ U
µ3
√
m
3
s
θ
√
M
2
U
µ3
√
m
3
c
θ
− U
µ2
√
m
2
s
θ
0
√
M
1
U
τ2
√
m
2
c
θ
+ U
τ3
√
m
3
s
θ
√
M
2
U
τ3
√
m
3
c
θ
− U
τ2
√
m
2
s
θ
0
,
(2.5)
where θ is a complex angle. The scenario of thermal leptogenesis with two heavy Majorana
neutrinos is particularly appealing as the dimensionality of the model parameter space
is greatly reduced (from 18 to 11) and for this reason has been investigated using the
unflavoured approximation [32, 35–37] and also including flavour effects [38–41]. As we have
assumed the best-fit values for the low-energy neutrino parameters, the lepton asymmetry
depends only on four parameters (M
1
, M
2
, Re[θ] and Im[θ]) which we assume take the
ranges shown in table 1. It is worth noting that as we assume a strictly hierarchical
mass spectrum for the heavy neutrinos (M
2
≥ 10M
1
) we find the lower range for viable
leptogenesis to be M
1
≈ 10
9
GeV. Although non-resonant thermal leptogenesis is possible
at lower scales, a milder mass hierarchy is required.
– 4 –
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