Flavour-Dependent Type II Leptogenesis
FTUAM 07-07
IFT-UAM/CSIC 07-18
Flavour-Dependent Type II Leptogenesis
S. Antuscha
Departamento de F́ısica Teórica C-XI and Instituto de F́ısica Teórica C-XVI,
Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain
Abstract
We reanalyse leptogenesis via the out-of-equilibrium decay of the lightest right-
handed neutrino in type II seesaw scenarios, taking into account flavour-dependent
effects. In the type II seesaw mechanism, in addition to the type I seesaw con-
tribution, an additional direct mass term for the light neutrinos is present. We
consider type II seesaw scenarios where this additional contribution arises from
the vacuum expectation value of a Higgs triplet, and furthermore an effective
model-independent approach. We investigate bounds on the flavour-specific de-
cay asymmetries, on the mass of the lightest right-handed neutrino and on the
reheat temperature of the early universe, and compare them to the corresponding
bounds in the type I seesaw framework. We show that while flavour-dependent
thermal type II leptogenesis becomes more efficient for larger mass scale of the
light neutrinos, and the bounds become relaxed, the type I seesaw scenario for
leptogenesis becomes more constrained. We also argue that in general, flavour-
dependent effects cannot be ignored when dealing with leptogenesis in type II
seesaw models.
aE-mail: antusch@delta.ft.uam.es
http://arxiv.org/abs/0704.1591v3
1 Introduction
Leptogenesis [1] is one of the most attractive and minimal mechanisms for explaining
the observed baryon asymmetry of the universe nB/nγ ≈ (6.0965 ± 0.2055) × 10−10 [2].
A lepton asymmetry is dynamically generated and then converted into a baryon asym-
metry due to (B + L)-violating sphaleron interactions [3] which exist in the Standard
Model (SM) and its minimal supersymmetric extension, the MSSM. Leptogenesis can be
implemented within the type I seesaw scenario [4], consisting of the SM (MSSM) plus
three right-handed Majorana neutrinos (and their superpartners) with a hierarchical
spectrum. In thermal leptogenesis [5], the lightest of the right-handed neutrinos is pro-
duced by thermal scattering after inflation, and subsequently decays out-of-equilibrium
in a lepton number and CP-violating way, thus satisfying Sakharov’s constraints [6].
In models with a left-right symmetric particle content like minimal left-right sym-
metric models, Pati-Salam models or Grand Unified Theories (GUTs) based on SO(10),
the type I seesaw mechanism is typically generalised to a type II seesaw [7], where
an additional direct mass term mIILL for the light neutrinos is present. From a model
independent perspective, the type II mass term can be considered as an additional con-
tribution to the lowest dimensional effective neutrino mass operator. In most explicit
models, the type II contribution stems from seesaw suppressed induced vevs of SU(2)L-
triplet Higgs fields. One motivation for considering the type II seesaw is that it allows
to construct unified flavour models for partially degenerate neutrinos in an elegant way,
e.g. via a type II upgrade [8], which is otherwise difficult to achieve in type I models.
For leptogenesis in type II seesaw scenarios with SU(2)L-triplet Higgs fields, there
are in general two possibilities to generate the baryon asymmetry: via decays of the
lightest right-handed neutrinos or via decays of the SU(2)L-triplets [9,10,11,12]. In the
first case, there are additional one-loop diagrams where virtual triplets are running in
the loop [9, 13, 14, 15, 16]. In the following, we focus on this possibility, and assume
hierarchical right-handed neutrino masses (and that the triplets are heavier than ν1R).
In this limit, to a good approximation the decay asymmetry depends mainly on the
low energy neutrino mass matrix mνLL = m
LL + m
LL and on the Yukawa couplings
to the lightest right-handed neutrino and its mass [16]. It has been shown that type
II leptogenesis imposes constraints on the seesaw parameters, which, in the flavour-
independent approximation, differ substantially from the constraints in the type I case.
For instance, the bound on the decay asymmetry increases with increasing neutrino mass
scale [16], in contrast to the type I case where it decreases. As a consequence, the lower
bound on the mass of the lightest right-handed neutrino from leptogenesis decreases for
increasing neutrino mass scale [16]. One interesting application of type II leptogenesis
is the possibility to improve consistency of classes of unified flavour models with respect
to thermal leptogenesis [17]. Finally, since the type II contribution typically does not
effect washout, there is no bound on the absolute neutrino mass scale from type II
leptogenesis, as has been pointed out in [15]. For further applications and realisations
of type II leptogenesis in specific models of fermion masses and mixings, see e.g. [18].
In recent years, the impact of flavour in thermal leptogenesis has merited increasing
attention [19] - [38]. In fact, the one-flavour approximation is only rigorously correct
when the interactions mediated by the charged lepton Yukawa couplings are out of
equilibrium. Below a given temperature (e.g. O(1012GeV) in the SM and (1+tan2 β)×
O(1012GeV) in the MSSM), the tau Yukawa coupling comes into equilibrium (later
followed by the couplings of the muon and electron). Flavour effects are then physical
and become manifest, not only at the level of the generated CP asymmetries, but also
regarding the washout processes that destroy the asymmetries created for each flavour.
In the full computation, the asymmetries in each distinguishable flavour are differently
washed out, and appear with distinct weights in the final baryon asymmetry.
Flavour-dependent leptogenesis in the type I seesaw scenario has recently been ad-
dressed in detail by several authors. In particular, flavour-dependent effects in lepto-
genesis have been studied, and shown to be relevant, in the two right-handed neutrino
models [24] as well as in classes of neutrino mass models with three right-handed neu-
trinos [26]. The quantum oscillations/correlations of the asymmetries in lepton flavour
space have been included in [22, 32, 33, 35] and the treatment has been generalised to
the MSSM [26,29]. Effects of reheating, and constraints on the seesaw parameters from
upper bounds on the reheat temperature, have been investigated in [29]. Leptogenesis
bounds on the reheat temperature [29] and on the mass of the lightest right-handed
neutrino [29, 36] have also been considered including flavour-dependent effects. Strong
connections between the low-energy CP phases of the UMNS matrix and CP violation for
flavour-dependent leptogenesis have been shown to emerge in certain classes of neutrino
mass models [26] or under the hypothesis of no CP violation sources associated with the
right-handed neutrino sector (real R) [25,27,28,31]. Possible effects regarding the decays
of the heavier right-handed neutrinos for leptogenesis have been discussed in this con-
text in [21, 34], and flavour-dependent effects for resonant leptogenesis were addressed
in [38]. Regarding the masses of the light neutrinos, assuming hierarchical right-handed
neutrinos and considering experimentally allowed light neutrino masses (below about 0.4
eV), there is no longer a bound on the neutrino mass scale from thermal leptogenesis if
flavour-dependent effects are included [24].
In view of the importance of flavour-dependent effects on leptogenesis in the type I
seesaw case, it is pertinent to investigate their effects on type II leptogenesis. In this
paper, we therefore reanalyse leptogenesis via the out-of-equilibrium decay of the lightest
right-handed neutrino in type II seesaw scenarios, taking into account flavour-dependent
effects. We investigate bounds on the decay asymmetries, on the mass of the lightest
right-handed neutrino and on the reheat temperature of the early universe, and discuss
how increasing the neutrino mass scale affects thermal leptogenesis in the type I and
type II seesaw frameworks.
2 Type I and type II seesaw mechanisms
Motivated by left-right symmetric unified theories, we consider two generic possibilities
for explaining the smallness of neutrino masses: via heavy SM (MSSM) singlet fermions
(i.e. right-handed neutrinos) [4] and via heavy SU(2)L-triplet Higgs fields [7]. In both
cases, the effective dimension five operator for Majorana neutrino masses in the SM or
p2≪M2
Figure 1: Generation of the dimension 5 neutrino mass operator in the type I seesaw mechanism.
the MSSM, respectively,
κgf (LC
· φ) (Lf · φ) + h.c. , (1a)
κ = −
κgf (L̂
g · Ĥu) (L̂f · Ĥu)
+ h.c. , (1b)
is generated from integrating out the heavy fields. This is illustrated in figures 1 and 2. In
equation (1), the dots indicate the SU(2)L-invariant product, (L̂
f · Ĥu) = L̂fa(iτ2)ab(Ĥu)b,
with τA (A ∈ {1, 2, 3}) being the Pauli matrices. Superfields are marked by hats. After
electroweak symmetry breaking, the operators of equation (1) lead to Majorana mass
terms for the light neutrinos,
Lν = −12m
LLνLν
L , with m
LL = −
(κ)∗ . (2)
In the type I seesaw mechanism, it is assumed that only the singlet (right-handed)
neutrinos νRi contribute to the neutrino masses. With Yν being the neutrino Yukawa
matrix in left-right convention,1 MRR the mass matrix of the right-handed neutrinos
and vu = 〈φ0〉 (= 〈H0u〉) the vacuum expectation value of the Higgs field which couples
to the right-handed neutrinos, the effective mass matrix of the light neutrinos is given
by the conventional type I seesaw formula
mILL = −v2u Yν M−1RR Y
ν . (3)
In the type II seesaw mechanism, the contributions to the neutrino mass matrix from
both, right-handed neutrinos νRi and Higgs triplet(s) ∆L, are considered. The additional
contribution to the neutrino masses from ∆L can be understood in two ways: as another
contribution to the effective neutrino mass operator in the low energy effective theory
or, equivalently, as a direct mass term after the Higgs triplet obtains an induced small
vev after electroweak symmetry breaking (c.f. figure 2). The neutrino mass matrix in
the type II seesaw mechanism has the form
mνLL = m
LL +m
LL = m
LL − v2uYνM−1RRY
ν , (4)
1The neutrino Yukawa matrix corresponds to −(Yν)fi(Lf · φ) νiR in the Lagrangian of the SM and,
analogously, to (Yν)fi(L̂
f · Ĥu) ν̂Ci in the superpotential of the MSSM (see [16] for further details).
p2≪M2
Figure 2: Extra diagram generating the dimension 5 neutrino mass operator in the type II seesaw
mechanism from a SU(2)L-triplet Higgs field.
where mIILL is the additional term from the Higgs triplet(s). In left-right symmetric
unified theories, the generic size of both seesaw contributionsmILL andm
LL isO(v2u/vB−L)
where vB−L is the B-L breaking scale (i.e. the mass scale of the right-handed neutrinos
and of the Higgs triplet(s)).
3 Baryogenesis via flavour-dependent leptogenesis
Flavour-dependent effects can have a strong impact in baryogenesis via thermal lep-
togenesis [19] - [38]. The effects are manifest not only in the flavour-dependent CP
asymmetries, but also in the flavour-dependence of scattering processes in the thermal
bath, which can destroy a previously produced asymmetry.
The relevance of the flavour-dependent effects depends on the temperatures at which
thermal leptogenesis takes place, and thus on which interactions mediated by the charged
lepton Yukawa couplings are in thermal equilibrium. For example, in the MSSM, for
temperatures between circa (1+tan2 β)×105GeV and (1+tan2 β)×109GeV, the µ and τ
Yukawa couplings are in thermal equilibrium and all flavours in the Boltzmann equations
are to be treated separately. For tan β = 30, this applies for temperatures below about
1012 GeV and above 108GeV, a temperature range which is of most interest for thermal
leptogenesis in the MSSM. In the SM, in the temperature range between circa 109 GeV
and 1012 GeV, only the τ Yukawa coupling is in equilibrium and is treated separately in
the Boltzmann equations, whereas µ and e flavours are indistinguishable. A discussion
of the temperature regimes in the SM and MSSM, where flavour is important, can be
found, e.g., in [26].
We now briefly review the estimation of the produced baryon asymmetry in flavour-
dependent leptogenesis.2 For definiteness, we focus on the temperature range where
all flavours are to be treated separately. In the following discussion of thermal type II
leptogenesis, we will assume that the mass M∆L of the triplet(s) is much larger than
MR1. In this limit, the flavour-dependent efficiencies calculated in the type I seesaw
scenario can also be used in the type II framework. The out-of-equilibrium decays of the
heavy right-handed (s)neutrinos ν1R and ν̃
R give rise to flavour-dependent asymmetries
in the (s)lepton sector, which are then partly transformed via sphaleron conversion into
2For a discussion of approximations which typically enter these estimates, and which also apply to
our discussion, see e.g. section 3.1.3 in [29].
a baryon asymmetry YB.
3 The final baryon asymmetry can be calculated as
Y SMB =
Y SM∆f , (5)
Y MSSMB =
Ŷ MSSM∆f , (6)
where Ŷ∆f ≡ YB/3 − YLf are the total (particle and sparticle) B/3 − Lf asymmetries,
with YLf the lepton number densities in the flavour f = e, µ, τ . The asymmetries Ŷ
and Y SM∆f , which are conserved by sphalerons and by the other SM (MSSM) interactions,
are then usually calculated by solving a set of coupled Boltzmann equations, describing
the evolution of the number densities as a function of temperature.
It is convenient to parameterise the produced asymmetries in terms of flavour-specific
efficiency factors ηf and decay asymmetries ε1,f as
Y SM∆f = η
f ε1,f Y
, (7)
Ŷ MSSM∆f = η
(ε1,f + ε1, ef) Y
(εe1,f + εe1, ef) Y
. (8)
and Y
are the number densities of the neutrino and sneutrino for T ≫ M1 if
they were in thermal equilibrium, normalised with respect to the entropy density. In
the Boltzmann approximation, they are given by Y
≈ Y eq
≈ 45/(π4g∗). g∗ is the
effective number of degrees of freedom, which amounts 106.75 in the SM and 228.75 in
the MSSM.
ε1,f , ε1, ef , εe1,f and εe1, ef are the decay asymmetries for the decay of neutrino into Higgs
and lepton, neutrino into Higgsino and slepton, sneutrino into Higgsino and lepton, and
sneutrino into Higgs and slepton, respectively, defined by
ε1,f =
− Γν1
f (Γν1RLf + Γν1RLf
1, ef
f (Γν1
εe1,f =
Γeν∗1
f (Γeν∗1R Lf + Γeν1RLf
, εe1, ef =
f(Γeν1
. (9)
The flavour-dependent efficiency factors ηf in the SM and in the MSSM are defined
by Eqs. (7) and (8), respectively. As stated above, we assume that the mass M∆L of the
triplet(s) is much larger than MR1. In this limit, the efficiencies for flavour-dependent
thermal leptogenesis in the type I and type II frameworks are mainly determined by
the properties of ν1R, which means in particular that the flavour-dependent efficiencies
3In the following, Y will always be used for quantities which are normalised to the entropy density
s. The quantities normalised with respect to the photon density can be obtained using the relation
s/nγ ≈ 7.04k.
-2 -1 0 1 2
ÈAff K f È
�������������������������������
È Aff  K f È
�������������������������������
È Aff  K f È
�������������������������������
È Aff  K f È
= 100
Figure 3: Flavour-dependent efficiency factor η(AffKf ,K) in the MSSM as a function of AffKf , for
fixed values of K/|AffKf | = 2, 5 and 100, obtained from solving the flavour-dependent Boltzmann
equations in the MSSM with zero initial abundance of right-handed (s)neutrinos (figure from [26]).
A is a matrix which appears in the Boltzmann equations (see [19, 24] for A in the SM and [26] for
the MSSM case), and which has diagonal elements |Aff | of O(1). The small off-diagonal entries of A
have been neglected, which is a good approximation in most cases. In general, however, they have to
be included. More relevant than the differences in the flavour-dependent efficiency factors for different
K/|AffKf | is that the total baryon asymmetry is the sum of each individual lepton asymmetries, which
is weighted by the corresponding efficiency factors.
calculated in the type I seesaw scenario can also be used in the type II framework. In
the definition of the efficiency factor, the equilibrium number densities serve as a nor-
malization: A thermal population νR1 (and ν̃R1) decaying completely out of equilibrium
(without washout effects) would lead to ηf = 1.
The efficiency factors can be computed by means of the flavour-dependent Boltzmann
equations, which can be found for the SM in [19,22,23,24] and for the MSSM in [26,29].
In general, the flavour-dependent efficiencies depend strongly on the washout parameters
m̃1,f for each flavour, and on the total washout parameter m̃1, which are defined as
m̃1,f =
v2u |(Yν)f1|2
, m̃1 =
m̃1,f . (10)
Alternatively, one may use the quantities Kf , K, which are related to m̃1,f , m̃1 by
m̃1,f
, K =
Kf , (11)
with m∗SM ≈ 1.08 × 10−3 eV and m∗MSSM ≈ sin2(β) × 1.58 × 10−3 eV. Figure 3 shows
the flavour-specific efficiency factor ηf in the MSSM. Maximal efficiency for a specific
flavour corresponds to Kf ≈ 1 (m̃1,f ≈ m∗).
The most relevant difference between the flavour-independent approximation and
the correct flavour-dependent treatment is the fact that in the latter, the total baryon
asymmetry is the sum of each individual lepton asymmetries, which is weighted by the
corresponding efficiency factor. Therefore, upon summing over the lepton asymmetries,
the total baryon number is generically not proportional to the sum over the CP asymme-
tries, ε1 =
f ε1,f , as in the flavour-independent approximation where the lepton flavour
is neglected in the Boltzmann equations. In other words, in the flavour-independent ap-
proximation the total baryon asymmetry is a function of
f ε1,f
× ηind (
g Kg). In
the correct flavour treatment the baryon asymmetry is (approximately) a function of∑
f ε1,fη (AffKf , K). From this, it is already clear that flavour-dependent effects can
have important consequences also in type II leptogenesis.
The most important quantities for computing the produced baryon asymmetry are
thus the decay asymmetries ε1,f and the efficiency factors ηf (which depend mainly on
m̃1,f and m̃1 (or Kf and K)). While the efficiency factors can be computed similarly
to the type I seesaw case, important differences between leptogenesis in type I and type
II seesaw scenarios arise concerning the decay asymmetries as well as concerning the
connection between leptogenesis and seesaw parameters.
4 Decay asymmetries
4.1 Right-handed neutrinos plus triplets
Regarding the decay asymmetry in the type II seesaw mechanism, where the direct
mass term for the neutrinos stems from the induced vev of a Higgs triplet, there are
new contributions from 1-loop diagrams where virtual SU(2)L-triplet scalar fields (or
their superpartners) are exchanged in the loop. The relevant diagrams for the decay
ν1R → LfaHub in the limit M1 ≪ MR2,MR2,M∆ are shown in figure 4. Compared to
the type I seesaw framework, the new contributions are the diagrams (c) and (f). The
calculation of the corresponding decay asymmetries for each lepton flavour yields
1,f =
j 6=1
Im [(Y †)1f(Y
ν Yν)1j(Y
T )jf ]
ν Yν)11
1− (1 + xj) ln
xj + 1
, (12a)
1,f =
j 6=1 Im [(Y
†)1f(Y
ν Yν)1j(Y
T )jf ]
ν Yν)11
1− xj
, (12b)
1,f = −
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
ν Yν)11
−1 + y ln
y + 1
, (12c)
1,f =
j 6=1 Im [(Y
†)1f(Y
ν Yν)1j(Y
T )jf ]
ν Yν)11
−1 + xj ln
xj + 1
, (12d)
1,f =
j 6=1 Im [(Y
†)1f(Y
ν Yν)1j(Y
T )jf ]
ν Yν)11
1− xj
, (12e)
1,f = −
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
ν Yν)11
1− (1 + y) ln
y + 1
, (12f)
∆̃1, ∆̃2
Figure 4: Loop diagrams in the MSSM which contribute to the decay ν1
→ LfaHub for the case of a
type II seesaw mechanism where the direct mass term for the neutrinos stems from the induced vev of
a Higgs triplet. In diagram (f), ∆̃1 and ∆̃2 are the mass eigenstates corresponding to the superpartners
of the SU(2)L-triplet scalar fields ∆ and ∆̄. The SM diagrams are the ones where no superpartners
(marked by a tilde) are involved and where Hu is renamed to the SM Higgs φ.
where y := M2∆/M
R1 and xj := M
R1 for j 6= 1 and where we assume hierarchical
right-handed neutrino masses and M∆ ≫ MR1.4
The MSSM results for the type II contributions have been derived in [16]. In the SM,
the results in [16] correct the previous result of [15] by a factor of −3/2. In equation
(12) they have been generalised to the flavour-dependent case. The results for the
contributions to the decay asymmetries from the triplet in the SM and from the triplet
superfield in the MSSM are
SM,II
1,f = ε
1,f , (13a)
MSSM,II
1,f = ε
1,f + ε
1,f . (13b)
In the MSSM, we furthermore obtain
MSSM,II
1,f = ε
MSSM,II
1, ef
MSSM,II
MSSM,II
e1, ef
. (14)
The results corresponding to the diagrams (a), (b), (d) and (e) which contribute to
εI1 in the type I seesaw in the SM and in the MSSM, have been presented first in [39].
The results for the type I contribution to the decay asymmetries in the SM and in the
4Integrating out the heavy particles ν2
,∆ (and their superpartners) in figure 4 leads to an
effective approach involving the dimension 5 neutrino mass operator (c.f. figures 1, 2 and 5), as will be
discussed in section 4.2. We note that there are additional diagrams not shown in figure 4 (since they
are generically suppressed for M1 ≪ MR2,MR2,M∆) which are related to the dimension 6 operator
containing two lepton doublets, two Higgs doublets and a derivative.
MSSM are
1,f = ε
1,f + ε
1,f , (15a)
MSSM,I
1,f = ε
1,f + ε
1,f + ε
1,f + ε
1,f . (15b)
Again, in the MSSM, the remaining decay asymmetries are equal to ε
MSSM,I
1,f :
MSSM,I
1,f = ε
MSSM,I
1, ef
MSSM,I
MSSM,I
e1, ef
. (16)
Finally, the total decay asymmetries from the decay of ν1R in the type II seesaw,
where the direct mass term for the neutrinos stems from the induced vev of a Higgs
triplet, are given by
εSM1,f = ε
1,f + ε
SM,II
1,f , (17)
εMSSM1,f = ε
MSSM,I
1,f + ε
MSSM,II
1,f . (18)
It is interesting to note that the type I results can be brought to a form which
contains the neutrino mass matrix using
j 6=1 Im [(Y
†)1f (Y
ν Yν)1j(Y
T )jf ]
8π (Y
ν Yν)11
= −MR1
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
8π (Y
ν Yν)11
.(19)
In the limit y ≫ 1 and xj ≫ 1 for all j 6= 1, which corresponds to a large gap between
the mass MR1 and the masses MR2, MR3 and M∆, we obtain the simple results for the
flavour-specific decay asymmetries εSM1,f and ε
1,f [16]
εSM1,f =
g Im [(Y
ν )f1(Y
ν )g1(m
LL +m
LL)fg]
ν Yν)11
, (20a)
εMSSM1,f =
g Im [(Y
ν )f1(Y
ν )g1(m
LL +m
LL)fg]
ν Yν)11
. (20b)
In the presence of such a mass gap, the calculation can also be performed in an effective
approach after integrating out the two heavy right-handed neutrinos and the heavy
triplet, as we now discuss.
4.2 Effective approach to leptogenesis
Let us now explicitly use the assumption that the lepton asymmetry is generated via
the decay of the lightest right-handed neutrino and that all other additional particles, in
particular the ones which generate the type II contribution, are much heavier than MR1.
Furthermore, we assume that we can neglect their population in the early universe, e.g.
that their masses are much larger than the reheat temperature TRH and that they are
Figure 5: Loop diagrams contributing to the decay asymmetry via the decay ν1
→ LfaHub in the
MSSM with a (lightest) right-handed neutrino ν1
and a neutrino mass matrix determined by κ′ [16].
Further contributions to the generated baryon asymmetry stem from the decay of ν1
into slepton and
Higgsino and from the decays of the sneutrino ν̃1
. With Hu renamed to the SM Higgs, the first diagram
contributes in the extended SM.
not produced non-thermally in a large amount. Under these assumptions we can apply
an effective approach to leptogenesis, which is independent of the mechanism which
generates the additional (type II) contribution to the neutrino mass matrix [16].
For this minimal effective approach, it is convenient to isolate the type I contribution
from the lightest right-handed neutrino as follows:
(mνLL)fg = −
2(Yν)f1M
R1 (Y
ν )1g + (κ
′∗)fg
. (21)
κ′ includes type I contributions from the heavier right-handed neutrinos, plus any ad-
ditional (type II) contributions from heavier particles. Examples for realisations of the
neutrino mass operator can be found, e.g., in [40].
At MR1, the minimal effective field theory extension of the SM (MSSM) for lepto-
genesis includes the effective neutrino mass operator κ′ plus one right-handed neutrino
ν1R with mass MR1 and Yukawa couplings (Yν)f1 to the lepton doublets L
f , defined as
−(Yν)f1(Lf · φ) ν1R in the Lagrangian of the SM and, analogously, as (Yν)f1(L̂f · Ĥu) ν̂C1
in the superpotential of the MSSM.
The contributions to the decay asymmetries in the effective approach stem from the
interference of the diagram(s) for the tree-level decay of νR1 (and ν̃R1) with the loop
diagrams containing the effective operator, shown in figure 5. In the SM, we obtain the
simple result [16] for the flavour-specific effective decay asymmetries (corresponding to
diagram (a) of figure 5)
εSM1,f =
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
ν Yν)11
. (22)
For the supersymmetric case, diagram (a) and diagram (b) contribute to εMSSM1,f and we
obtain [16]:
εMSSM1,f =
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
ν Yν)11
. (23)
Explicit calculation furthermore yields
εMSSM1,f = ε
1, ef
= εMSSM
= εMSSM
e1, ef
. (24)
The results are independent of the details of the realisation of the neutrino mass operator
κ′. Note that, since the diagrams where the lightest right-handed neutrino runs in the
loop do not contribute to leptogenesis, we have written mνLL = −v2u(κ)∗/2 instead of
m′νLL := −v2u(κ′)∗/2 in the formulae in equations (22) - (23). The decay asymmetries
are directly related to the neutrino mass matrix mνLL.
For neutrino masses via the type I seesaw mechanism, the results are in agreement
with the known results [39], in the limit MR2,MR3 ≫ MR1. The results obtained in the
effective approach are also in agreement with our full theory calculation in the type II
scenarios with SU(2)L-triplets in equation (12) [16], in the limit M∆ ≫ MR1.
5 Type II bounds on decay asymmetries and on MR1
In the limit MR2,MR3,M∆ ≫ MR1 (or alternatively in the effective approach), upper
bounds for the total decay asymmetries in type II leptgenesis, i.e. for the sums |εSM1 | =
1,f | and |εMSSM1 | = |
1,f |, have been derived in [16]. For the flavour-specific
decay asymmetries εSM1,f and ε
1,f , the bounds can readily be obtained as
|εSM1,f | ≤
mνmax , |εMSSM1,f | ≤
mνmax . (25)
They are thus identical to the bounds for the total asymmetries. In particular, they also
increase with increasing mass scale of the light neutrinos. Note that, compared to the
low energy value, the neutrino masses at the scale MR1 are enlarged by renormalization
group running by ≈ +20% in the MSSM and ≈ +30% in the SM, which raises the
bounds on the decay asymmetries by the same values (see e.g. figure 4 of [41]).
A situation where an almost maximal baryon asymmetry is generated by thermal
leptogenesis can be realised, for example, if the total decay asymmetry nearly saturates
its upper bound and if, in addition, the washout parameters m̃1,f for all three flavours
approximately take its optimal value. Classes of type II seesaw models, where this
can be accommodated, have been considered in [8, 42, 17]. In these so-called “type-
II-upgraded” seesaw models, the type II contribution to the neutrino mass matrix is
proportional to the unit matrix (enforced e.g. by an SO(3) flavour symmetry or by one
of its non-Abelian discrete subgroups). From equation (20), one can readily see that if
the type II contribution (∝ 1) dominates the neutrino mass matrix mνLL, and if (Yν)f1
are approximately equal for all flavours f = 1, 2, 3 and chosen such that the resulting
m̃1,f are approximately equal to m
∗, we have realised ηf ≈ ηmax for all flavours and
simultaneously nearly saturated the bound for the total decay asymmetry.5
5We further note that the bound for one of the flavour-specific decay asymmetries can be nearly
saturated in this scenario if, for instance, (Yν)21 ≈ (Yν)31 ≈ 0.
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
-0.75
-0.25
type I seesaw
type II seesaw
Figure 6: Bound on the decay asymmetry ε1,f in type II leptogenesis (solid blue line) and type I lepto-
genesis (dotted red line) as a function of the mass of the lightest neutrino mν
:= min (mν1 ,mν3 ,mν3)
in type I and type II seesaw scenarios (see also [29]). The washout parameter |Aff |m̃1,f is fixed to
m∗ (close to optimal), and the asymmetry is normalised to ε
max,0
= 3MR1 (∆m
)1/2/(16π v
), where
≈ 2.5× 10−3 eV2 is the atmospheric neutrino mass squared difference. We have considered the
MSSM with tanβ = 30 as an explicit example.
Assuming a maximal efficiency factor ηmax for all flavours in a given scenario, and
taking an upper bound for the masses of the light neutrinos mνmax as well as the observed
value nB/nγ ≈ (6.0965 ± 0.2055) × 10−10 [2] for the baryon asymmetry, equation (25)
can be transformed into lower type II bounds for the mass of the lightest right-handed
neutrino [16]:
MSMR1 ≥
mνmax
nB/nγ
0.99 · 10−2 ηmax
, MMSSMR1 ≥
mνmax
nB/nγ
0.92 · 10−2 ηmax
. (26)
The bound on MR1 is lower for a larger neutrino mass scale.
The situation in the type II framework differs from the type I seesaw case: In the
latter, the flavour-specific decay asymmetries are constrained by [24]
|εI,SM1,f | ≤
mνmax
m̃1,f
, |εI,MSSM1,f | ≤
mνmax
m̃1,f
. (27)
Note that compared to the type II bounds, there is an extra factor of (m̃1,f/m̃1)
which depends on the washout parameters. As we shall now discuss, this factor implies
that it is not possible to have a maximal decay asymmetry ε1,f and an optimal washout
parameter m̃1,f simultaneously. Let us recall first that in the type I seesaw, in contrast
to the type II case, the flavour-independent washout parameter has the lower bound [43]
m̃1 ≥ mνmin , (28)
with mνmin = min (mν1 , mν3 , mν3). On the contrary, in the type I and type II seesaw, the
flavour-dependent washout parameters m̃1,f are generically not constrained. Note that in
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
type I seesaw
type II seesaw
Figure 7: Lower bound on MR1 in type II leptogenesis (solid blue line) and type I leptogenesis (dotted
red line) as a function of the mass of the lightest neutrinomν
:= min (mν1 ,mν3 ,mν3). For definiteness,
the MSSM with tanβ = 30 has been considered as an example.
the flavour-independent approximation, Eq. (28) leads to a dramatically more restrictive
bound on ε1 =
f ε1,f [44] for quasi-degenerate light neutrino masses, and finally even
to a bound on the neutrino mass scale [43]. This can be understood from the fact that
for m̃1 ≫ m∗ in the flavour-independent approximation, washout effects strongly reduce
the efficiency of thermal leptogenesis. Similarly, in the flavour-dependent treatment,
m̃1,f ≫ m∗ would lead to a strongly reduced efficiency for this specific flavour. This
strong washout for quasi-degenerate light neutrinos can be avoided in flavour-dependent
type I leptogenesis, and m̃1,f ≈ m∗ can realise a nearly optimal scenario regarding
washout (c.f. figure 3). However, we see from equation (27) that the decay asymmetries
in this case are reduced by a factor of (m∗/mνmin)
1/2 when compared to the optimal value,
leading to a reduced baryon asymmetry. On the other hand, realizing nearly optimal
ε1,f requires m̃1,f ≈ m̃1 ≥ mνmin, leading to large washout effects for quasi-degenerate
light neutrinos and even to a more strongly suppressed generation of baryon asymmetry
(c.f. figure 3). As a consequence, increasing the neutrino mass scale increases the lower
bound on MR1 (also in the presence of flavour-dependent effects), in contrast to the type
II seesaw case.
Comparing the type II and type I seesaw cases, in the latter the baryon asymmetry
is suppressed for quasi-degenerate light neutrino masses either by a factor (m∗/mνmin)
in the decay asymmetries or by a non-optimal washout parameter much larger than m∗
(or Kf ≫ 1, c.f. figure 3). The bounds on the decay asymmetries in type I and type
II leptogenesis are compared in figure 6, where m̃1,f has been fixed to m
∗, close to its
optimal value. From figure 6 we see that in the type I case the maximal baryon asymme-
try is obtained for hierarchical neutrino masses, whereas in the type II case, increasing
the neutrino mass scale increases the produced baryon asymmetry and therefore allows
to relax the bound on MR1, as shown in figure 7. In addition, for the same reason,
increasing the neutrino mass scale also relaxes the lower bound on the reheat tempera-
ture TRH from the requirement of successful type II leptogenesis. Including reheating in
the flavour-dependent Boltzmann equations as in Ref. [29] (for the flavour-independent
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
2´109
4´109
6´109
8´109
type I seesaw
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
2´109
4´109
6´109
8´109
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
5´108
1´109
1.5´109
2´109
type II seesaw
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
5´108
1´109
1.5´109
2´109
Figure 8: Lower bound on the reheat temperature TRH in type I leptogenesis (left panel) and in type II
leptogenesis (right panel) as a function of the mass of the lightest neutrino mν
= min (mν1 ,mν3 ,mν3),
in the MSSM with tanβ = 30. In the grey regions, values of TRH are incompatible with thermal
leptogenesis for the corresponding mν
case, see [45]), we obtain the mνmin-dependent lower bounds on TRH in type I and type
II scenarios shown in figure 8. While the bound decreases in type II leptogenesis by
about an order of magnitude when the neutrino mass scale increases to 0.4 eV, it in-
creases in the type I seesaw case. In the presence of upper bounds on TRH, this can
lead to constraints on the neutrino mass scale , i.e. on mνmin = min (mν1 , mν3, mν3). For
instance, with an upper bound TRH ≤ 5× 109 GeV, values of mνmin in the approximate
range [0.01 eV, 0.32 eV] would be incompatible with leptogenesis in the type I seesaw
framework (c.f. figure 8).
6 Summary, discussion and conclusions
We have analysed flavour-dependent leptogenesis via the out-of-equilibrium decay of
the lightest right-handed neutrino in type II seesaw scenarios, where, in addition to
the type I seesaw, an additional direct mass term for the light neutrinos is present.
We have considered type II seesaw scenarios where this additional contribution stems
from the vacuum expectation value of a Higgs triplet, and furthermore an effective
approach, which is independent of the mechanism which generates the additional (type
II) contribution to neutrino masses. We have taken into account flavour-dependent
effects, which are relevant if thermal leptogenesis takes place at temperatures below
circa 1012 GeV in the SM and below circa (1 + tan2 β) × 1012 GeV in the MSSM. As
in type I leptogenesis, in the flavour-dependent regime the decays of the right-handed
(s)neutrinos generate asymmetries in in each distinguishable flavour (proportional to the
flavour-specific decay asymmetries ε1,f), which are differently washed out by scattering
processes in the thermal bath, and thus appear with distinct weights (efficiency factors
ηf ) in the final baryon asymmetry.
The most important quantities for computing the produced baryon asymmetry are
the decay asymmetries ε1,f and the efficiency factors ηf (which mainly depend on
washout parameters m̃1,f and m̃1 =
f m̃1,f). With respect to the flavour-specific
efficiency factors ηf , in the limit that the mass M∆L of the triplet is much larger than
MR1 (and MR1 ≪ MR2,MR3), they can be estimated from the same Boltzmann equa-
tions as in the type I seesaw framework. Regarding the decay asymmetries ε1,f , in the
type II seesaw case there are additional contributions where virtual Higgs triplets (and
their superpartners) run in the 1-loop diagrams. Here, we have generalised the results
of [16] to the flavour-dependent case. The most important effects of flavour in leptoge-
nesis are a consequence of the fact that in the flavour-independent approximation the
total baryon asymmetry is a function of
f ε1,f
× ηind (
g m̃1,g), whereas in the cor-
rect flavour-dependent treatment the baryon asymmetry is (approximately) a function
f ε1,fη (Affm̃1,f , m̃1).
We have then investigated the bounds on the flavour-specific decay asymmetries ε1,f .
In the type I seesaw case, it is known that the bound on the flavour-specific asymmetries
εI1,f is substantially relaxed [24] compared to the bound on ε
1,f [44] in the case
of a quasi-degenerate spectrum of light neutrinos. For experimentally allowed light
neutrino masses below about 0.4 eV, there is no longer a bound on the neutrino mass
scale from the requirement of successful thermal leptogenesis. In the type II seesaw case,
we have derived the bound on the flavour-specific decay asymmetries ε1,f = ε
1,f + ε
1,f ,
which turns out to be identical to the bound on the total decay asymmetry ε1 =
f ε1,f .
We have compared the bound on the flavour-specific decay asymmetries in type I and
type II scenarios, and found that while the type II bound increases with the neutrino
mass scale, the type I bound decreases (for experimentally allowed light neutrino masses
below about 0.4 eV). The relaxed bound on ε1,f (figure 6) leads to a lower bound on the
mass of the lightest right-handed neutrino MR1 in the type II seesaw scenario (figure
7), which decreases when the neutrino mass scale increases. Furthermore, it leads to
a relaxed lower bound on the reheat temperature TRH of the early universe (figure 8),
which helps to improve consistency of thermal leptogenesis with upper bounds on TRH
in some supergravity models. This is in contrast to the type I seesaw scenario, where
the lower bound on TRH from thermal leptogenesis increases with increasing neutrino
mass scale. Constraints on TRH can therefore imply constraints on the mass scale of the
light neutrinos also in flavour-dependent type I leptogenesis, although a general bound
is absent.
We have furthermore argued that these relaxed bounds on ε1,f MR1 and TRH in the
type II case can be nearly saturated in an elegant way in classes of so-called “type-
II-upgraded” seesaw models [8], where the type II contribution to the neutrino mass
matrix is proportional to the unit matrix (enforced e.g. by an SO(3) flavour symmetry
or by one of its non-Abelian subgroups). One interesting application of these type II
seesaw scenarios is that the consistency of thermal leptogenesis with unified theories
of flavour is improved compared to the type I seesaw case. This effect, investigated in
the flavour-independent approximation in [17], is also present analogously in the flavour-
dependent treatment of leptogenesis. The reason is that if the type II contribution (∝ 1)
dominates, the decay asymmetries ε1,f become approximately equal and the estimate for
the produced baryon asymmetry is similar to the flavour-independent case. Nevertheless,
an accurate analysis of leptogenesis in this scenario requires careful inclusion of the
flavour-dependent effects. In many applications and realisations of type II leptogenesis
in specific models of fermion masses and mixings (see e.g. [18]), flavour-dependent effects
may substantially change the results and they therefore have to be taken into account.
In summary, type II leptogenesis provides a well-motivated generalisation of the
conventional scenario of leptogenesis in the type I seesaw framework. We have argued
that flavour-dependent effects have to be included in type II leptogenesis, and can change
predictions of existing models as well as open up new possibilities for for successful
models of leptogenesis. Comparing bounds on ε1,f MR1 and TRH in flavour-dependent
thermal type I and type II leptogenesis scenarios, we have shown that while type II
leptogenesis becomes more efficient for larger mass scale of the light neutrinos, and the
bounds become relaxed, leptogenesis within the type I seesaw framework becomes more
constrained.
Acknowledgments
I would like to thank Steve F. King, Antonio Riotto and Ana M. Teixeira for useful
discussions and for their collaboration on leptogenesis issues. This work was supported
by the EU 6th Framework Program MRTN-CT-2004-503369 “The Quest for Unification:
Theory Confronts Experiment”.
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	Introduction
	Type I and type II seesaw mechanisms
	Baryogenesis via flavour-dependent leptogenesis
	Decay asymmetries
	Right-handed neutrinos plus triplets
	Effective approach to leptogenesis
	Type II bounds on decay asymmetries and on bold0mu mumu MR1MR13.91663pt plus 1.95831pt minus 1.30554ptMR1MR1MR1MR1
	Summary, discussion and conclusions
