Tri-Bimaximal Lepton Mixing and Leptogenesis

Abstract

In models with flavour symmetries added to the gauge group of the Standard Model the CP-violating asymmetry necessary for leptogenesis may be related with low-energy parameters. A particular case of interest is when the flavour symmetry produces exact Tri-Bimaximal lepton mixing leading to a vanishing CP-violating asymmetry. In this paper we present a model-independent discussion that confirms this always occurs for unflavoured leptogenesis in type I see-saw scenarios, noting however that Tri-Bimaximal mixing does not imply a vanishing asymmetry in general scenarios where there is interplay between type I and other see-saws. We also consider a specific model where the exact Tri-Bimaximal mixing is lifted by corrections that can be parametrised by a small number of degrees of freedom and analyse in detail the existing link between low and high-energy parameters - focusing on how the deviations from Tri-Bimaximal are connected to the parameters governing leptogenesis. Comment: 29 pages, 6 figures; version 2: references added, minor corrections

Full text

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CFTP/09-030 DFPD-09/TH/14 IFIC/09-40
Tri-Bimaximal Lepton Mixing and Leptogenesis
D. Aristizabal Sierraa)1, F. Bazzocchib)2,
I. de Medeiros Varzielasc)3, L. Merlod)4and S. Morisie)5
a)INFN, Laboratori Nazionali di Frascati,C.P. 13, I-00044 Frascati, Italy
b)Department of Physics and Astronomy, Vrije Universiteit Amsterdam,
1081 HV Amsterdam, The Netherlands
c)CFTP, Departamento de F´ ısica, Instituto Superior T´ ecnico,
Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal
d)Dipartimento di Fisica ‘G. Galilei’, Universit` a di Padova
INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy
e)AHEP Group, Institut de F´ ısica Corpuscular – C.S.I.C./Universitat de Val` encia
Edificio Institutos de Paterna, Apt 22085, E–46071 Valencia, Spain
Abstract
In models with flavour symmetries added to the gauge group of the Standard
Model the CP-violating asymmetry necessary for leptogenesis may be related with
low-energy parameters. A particular case of interest is when the flavour symmetry
produces exact Tri-Bimaximal lepton mixing leading to a vanishing CP-violating
asymmetry. In this paper we present a model-independent discussion that confirms
this always occurs for unflavoured leptogenesis in type I see-saw scenarios, noting
however that Tri-Bimaximal mixing does not imply a vanishing asymmetry in general
scenarios where there is interplay between type I and other see-saws. We also consider
a specific model where the exact Tri-Bimaximal mixing is lifted by corrections that
can be parametrised by a small number of degrees of freedom and analyse in detail the
existing link between low and high-energy parameters - focusing on how the deviations
from Tri-Bimaximal are connected to the parameters governing leptogenesis.
1e-mail address: daristi@lnf.infn.it
2e-mail address: fbazzoc@few.vu.nl
3e-mail address: ivo@cftp.ist.utl.pt
4e-mail address: merlo@pd.infn.it
5e-mail address: morisi@ific.uv.es
arXiv:0908.0907v2 [hep-ph] 17 Sep 2009

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1 Introduction
Results from neutrino oscillation experiments [1] have firmly established that neutrinos
have tiny but non-zero masses. From a theoretical perspective the smallness of neutrino
masses can be well understood within the see-saw mechanism [2], in which the Standard
Model (SM) is extended by adding new heavy states. Light neutrino masses are generated
through effective operators which are typically suppressed by the masses of the states giving
rise to the see-saw. In type I see-saw the extra states are right-handed (RH) neutrinos with
large Majorana masses. Apart from providing an explanation for the origin of neutrino
masses, the mechanism contains all the necessary ingredients for a dynamical generation
of a cosmic lepton asymmetry through the decays of the heavy singlet neutrinos (lepto-
genesis): (a) Lepton number violation arising from the Majorana mass terms of the new
fermionic states; (b) CP-violating sources from complex Yukawa couplings; (c) departure
from thermal equilibrium in the hot primeval plasma at the time the singlet neutrinos start
decaying. This lepton asymmetry is then reprocessed into a baryon asymmetry through
B + L violating anomalous electroweak processes [3] thus yielding an explanation to the
origin of the baryon asymmetry of the Universe [4] i.e. baryogenesis through leptogenesis
(for a recent review see [5]).
The structure of mixing in the leptonic sector suggested by experimental data is in
sharp contrast with the small mixing that characterises the quark sector. Observations
indicate that solar neutrino oscillation is described by a large but non-maximal mixing
angle, atmospheric neutrino oscillation is described by maximal or nearly-maximal angle,
and reactor data puts a small upper bound on the third angle [6–8]. This mixing pattern
is well described by the so-called Tri-Bimaximal (TB) scheme [9] which corresponds to a
unitary matrix of the form

−1/√6 1/√3 +1/√2
and to the following mixing angles:
UTB=

?2/31/√30
−1/√6 1/√3 −1/√2

, (1)
sin2θTB
13= 0 sin2θTB
23= 1/2sin2θTB
12= 1/3 . (2)
This particular mixing structure can be interpreted as a signal of an underlying sym-
metry1and has motivated a great deal of studies aiming to determine the possible flavour
symmetry responsible for such a pattern. A large amount of discrete and continuous sym-
metries have been considered [11–43] and among them discrete non-Abelian ones have been
found to be particularly interesting as they can more naturally lead to the required pat-
tern. In the realization of explicit models, a general feature is the breaking of the flavour
symmetry: this is a well known result of a no-go theorem [16,44] that applies in the vast
majority of relevant cases; it could be evaded, for example using light Higgs fields charged
under the flavour symmetry, but inconsistencies related to flavour-changing neutral current
or lepton-flavour violating processes could appear. On the contrary, allowing only heavy
1for a different approach see [10].
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Higgs fields charged under the flavour symmetry, it is possible to avoid these dangerous
effects [45].
Global fits [7] to the data provides a subtle hint of a deviation from the TB scheme
and therefore it is desirable if the flavour symmetry predicts TB at leading order (LO)
and allows perturbations at higher orders. It is possible to constrain the amount of these
corrections by comparing the TB value of the mixing angles to their experimental measure-
ments: the solar angle is known with the lowest relative error and as a result it fixes the
upper bound of the deviations at about 0.05. Avoiding any parameter tuning or particular
relations among the deviations, we expect that the other LO mixing angles are perturbed
by quantities of the same order of magnitude: in particular the corrected θ13is expected
to be non-vanishing, but very small.2
In order to explain the baryon asymmetry of the Universe by leptogenesis, CP viola-
tion in the leptonic sector is needed. In principle it can be argued that leptogenesis is
supported by any observation of CP violation in the leptonic sector, e.g. in neutrino oscil-
lation experiments. However, since generically the baryon asymmetry is insensitive to the
low energy CP-violating phases [46,47] a definitive conclusion can not be established from
such an observation. In contrast, in models based on flavour symmetries that predict the
TB mixing pattern, the parameter space is further constrained and as a result one could
expect, quite generically, some link between low-energy observables and leptogenesis. As
pointed out in [33], in the context of an A4flavour symmetry model with type I see-saw the
CP-violating asymmetry (?Nα) vanishes in the limit of exact TB mixing, with leptogenesis
becoming viable only when deviations from this pattern are taken into account. The ex-
plicit structure of the corrections responsible for these deviations are model-dependent and
therefore whether a connection between ?Nαand low-energy parameters can be established
will depend on the particular realization.
In this paper we extend upon the work in [33]. In particular, we study the viability
of leptogenesis in the context of models based on an arbitrary flavour symmetry leading
to the TB lepton mixing pattern through the see-saw mechanism. When there is only
type I see-saw and independently of the nature of the underlying symmetry, we conclude
that ?Nα= 0 in the limit of exact TB mixing or any other exact mixing schemes where
the mixing matrix consists purely of numbers - such as Bi-maximal mixing [48], golden-
ratio mixing [49] and some (but not all) cases of Tri-maximal mixing [50,51]. Under these
conditions, only deviations from the flavour symmetry imposed pattern yield ?Nα?= 0. It
is important to note that this result is not in general valid in the presence of other types
of see-saw (e.g. with the interplay of type I and type II).
Following from the model-independent proof we consider particular cases. We check
our result by considering several models discussed in the literature. Finally, we also take
a specific simple A4 flavour model [39], where low-energy observables arising from TB
deviations can be linked to the CP-violating asymmetry in a straightforward manner and
analyse it in more detail.
Our discussion will be entirely devoted to “unflavoured” leptogenesis scenarios: in the
framework of flavour symmetry models predicting TB mixing the heavy singlet neutrinos
typically have masses above 1013GeV and for T ? 1012GeV lepton flavours are indistin-
2for an alternative proposal see [43].
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guishable [52,53].
This paper is organised as follows: in section 2 we fix our notation and briefly comment
upon some generic aspects of leptogenesis. For completeness of our results, in section 3
we present a brief analysis of randomly generated TB mixing and its implications for the
CP-violating asymmetry. We turn to the main subject of this paper in section 4, showing
that an exact mixing scheme enforced by a flavour symmetry in scenarios with just type
I see-saw leads to a vanishing CP-violating asymmetry. Leptogenesis becomes potentially
viable only when higher-order flavour symmetry corrections lift the pattern - or otherwise
if other types of see-saw (e.g. type II) are also present. In section 5 we confirm our model-
independent results in particular realizations, and in section 6 we analyse in detail a specific
model in which low-energy parameters and the CP-violating asymmetry are directly related
in a simple way. Finally in section 7 we conclude by summarizing our results.
2 The basic framework
In this section we will establish both the notation and a choice of a convenient basis. Let
us consider the leptonic part of the SM Lagrangian extended with three fermionic heavy
singlets Nα
3
− L = (Y )ijLiH?c
j+ (λ)iαLi? HNα+1
iare the complex conjugate charged lepton SU(2)
2(MR)αβ(Nc
α)TNβ+ h.c.. (3)
Here Liare the lepton SU(2) doublets, ?c
singlets and H (? H = iσ2H∗) is the Higgs SU(2) doublet. Latin indices i,j ... label lepton
in flavour space.
At energy scales well below the RH neutrino masses, light neutrino masses are generated
via effective operators. The effective Majorana neutrino mass matrix is
flavour, whereas Greek indices α,β ... denote RH species. Y , λ and MRare 3×3 matrices
mν= −mDM−1
RmT
D, (4)
where mD= λv/√2 (v ? 246 GeV). We then consider the unitary matrices U?, U?c and
Uν, which diagonalise the charged lepton and neutrino mass matrices:
ˆ m?= U†
?Y U?c
v
√2
ˆ mν= UT
νmνUν, (5)
where the “ˆ” refers to a diagonal matrix. The lepton mixing matrix is defined by U?and
Uν:
U = (U?)†Uν. (6)
From now on we will assume that in the basis in which the charged lepton mass matrix is
diagonal, mνis exactly diagonalised by the TB mixing matrix UTBand therefore
ˆ mν= DUT
TBmνUTBD, (7)
3The subsequent analysis is done for three RH neutrinos, but it can be generalised to an arbitrary
number with the conclusions being independent of it.
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where D accounts for the low-energy Majorana phases
D = diag(eiϕ1,eiϕ2,1). (8)
In general mDas well as MR(MR= MT
as follows
R) are complex matrices which can be diagonalised
ˆ mD
ˆ MR = VT
= U†
LmDUR,
RMRVR,
(9)
with UL,UR,VR3 × 3 unitary matrices, characterised in general by 3 rotation angles and
6 phases.
According to eq. (9) the effective neutrino mass matrix in (4) can be written as
mν= −ULˆ mD(U†
RVR)ˆ M−1
R(VT
RU∗
R) ˆ mDUT
L.(10)
The requirement of having exact TB diagonalisation can be written either in terms of
constraints over the light neutrino mass matrix entries, namely
mν12
mν22
mν11
= mν13,
= mν33,
= mν22+ mν23− mν12,
(11)
or, according to eqs. (7) and (10), requiring that
ˆ mν= −D(UT
TBUL) ˆ mD(U†
RVR)ˆ M−1
R(VT
RU∗
R) ˆ mD(UT
LUTB)D (12)
is diagonal and real. It is useful to introduce the notation of the Dirac neutrino mass
matrix in the basis in which the RH neutrino mass matrixˆ MRis real and diagonal:
mR
D≡ mDVR.(13)
2.1General remarks on leptogenesis
As mentioned in the introduction, singlet neutrinos in flavour symmetry models typically
have masses above 1013GeV. Thus, within these frameworks leptogenesis proceeds at
temperatures at which lepton flavour effects can be completely neglected. In the standard
thermal leptogenesis scenario singlet neutrinos Nα are produced by scattering processes
after inflation. Subsequent out-of-equilibrium decays of these heavy states generate a CP-
violating asymmetry given by [5,54]
?Nα=
1
4v2π(mR†
DmR
D)αα
?
β?=α
Im
??
(mR†
DmR
D)βα
?2?
f(zβ), (14)
where zβ= M2
β/M2
αand the loop function can be expressed as
?2 − zβ
f(zβ) =√zβ
1 − zβ
− (1 + zβ) log
?1 + zβ
zβ
??
. (15)
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Depending on the singlet neutrino mass spectrum the loop function can be further simpli-
fied. In the hierarchical limit (Mα? Mβ) this function becomes
f(zβ) → −
3
2√zβ
, (16)
whereas in the case of an almost degenerate heavy neutrino spectrum (zβ= 1+δβ, δβ? 1)
it can be rewritten as
f(1 + δβ) ? −1
δβ
. (17)
In any case, as can be seen from eq. (14), whether the CP-violating asymmetry vanishes
will be determined by the Yukawa coupling combination mR†
DmR
D.
3 CP asymmetry and exact TB mixing without any
underlying flavour symmetry
While the TB mixing pattern can be well understood as a consequence of an underlying
flavour symmetry, in principle it might be that it arises from a random set of parameters
(though quite unlikely). For completeness, in this section we consider this possibility and
study the consequences on the CP-violating asymmetry. Neutrino mixing angles are fixed
to satisfy the TB mixing pattern and in addition to the measured mass squared differences
we have a set of eight constraints on the parameter space: the TB mixing condition enforces
the relations in eq. (11), yielding six constraints (from the real and imaginary parts of the
mass matrix entries); the atmospheric and solar mass scales provide the remaining two.
To determine the effect of such constraints on ?Nαit is practical to use a parametrisation
of mDthat ensures that the TB mixing and the correct neutrino masses are obtained. In the
basis in which the RH neutrino mass matrix is diagonal and real it is convenient to introduce
the orthogonal complex matrix R defined by the so-called Casas-Ibarra parametrisation
[55], namely
R∗= (ˆ mν)−1/2UTmR
D(ˆ MR)−1/2. (18)
All low-energy observables are contained in the leptonic mixing matrix U and in the diag-
onal and real light neutrino mass matrix ˆ mν. The matrix R turns out to be very useful
in expressing the CP-violating asymmetry parameter. Considering for simplicity the case
of hierarchical RH neutrinos (M1 ? M2 ? M3 - thus validating the approximation in
eq. (16)), eq. (14) can be rewritten as
?Nα= −3Mα
8πv2
Im
??
jm2
jR2
jα
?
?
jmj|Rjα|2
, (19)
where mj≡ (ˆ mν)jj. Once the RH neutrino mass spectrum and low-energy observables are
fixed, random values of mR
that leptogenesis is completely insensitive to low-energy lepton mixing and CP-violating
phases [46]
Dcorrespond to random values of R. It is shown by eq. (19)
4and therefore the viability of leptogenesis is not at all related with any
4This statement is in general also true in flavoured leptogenesis [47].
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Figure 1: CP-violating asymmetry as a function of the angle ω for different values of σ.
M1is fixed to 1013GeV and ∆m2
atmto 2.39×10−3eV2[8] (see the text for further details).
accidental mixing pattern considered. The CP-violating asymmetry is determined by the
values of the entries of R which are arbitrary in the absence of any flavour symmetry, and
consequently ?Na?= 0 in general and its absolute value depends upon the heavy fermionic
singlet masses, the light neutrino masses and R.
To illustrate this point we consider the case in which only N1decays are relevant for
the generation of a lepton asymmetry. We assume normal hierarchy for the light neutrino
spectrum and a simple R = R13(ρ13) with ρ13= ω + iσ (i.e. R is a ρ13rotation matrix).
Under these assumptions the CP-violating asymmetry in eq. (19) becomes
?∆m2
?N1= −3M1
atm
2πv2
cosω sinhσ
√cosh2σ − cos2ω.
(20)
From figure 1 it can be seen that barring the cases ω = π/2 and/or σ = 0 the CP-violating
asymmetry does not vanish and its values are well within the range required for successful
leptogenesis, regardless of the mixing pattern.
4 Implications of flavour symmetries on the CP asym-
metry
We consider now the case in which an underlying flavour symmetry enforces an exact
mixing pattern. It will be evident throughout the proof that it holds for any mixing pattern
where the mixing matrix consists purely of numbers, but we will assume TB mixing for
definiteness.
Within the case considered the transformation properties of Liand Nαunder the flavour
symmetry group (Gf) determine the structure of mDand MR(which are no longer arbi-
trary). Indeed, these matrices can be regarded as form-diagonalisable matrices [44], i.e.
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the parameters which determine their eigenvalues are completely independent from the
parameters that define their diagonalising matrices. Accordingly, vanishing off-diagonal
elements of ˆ mνin eq. (12) can arise only if
UT
TBUL= PLODi
andU†
RVR= O†
DiPRORm, (21)
where PL,R= diag(eiαR,L
orthogonal matrices that arbitrarily rotate the i and m degenerate eigenvalues of mDand
MRsuch that if mD(MR) has no degenerate eigenvalues ODi= 1 (ORm= 1). Note that
the requirement of having canonical kinetic terms in addition to preserving the m-fold
degeneracy of the RH neutrino mass matrix enforce ORmto be real. Although ODiand
ORmdo not have any effect in eq. (12) they do affect the structure of UL,Rand VRand
correspondingly of mD(see eq. (9)). VRcan be defined in such a way thatˆ MRis real, and
the phases contained in ˆ mDare now denoted by γiand must obey: ϕi+αR
and αR
of a reductio ad absurdum. Let us consider for simplicity the case without any degeneracy
in the eigenvalues of ˆ mDandˆ MR: ODi= 1 and ORm= 1. If the products UT
are not diagonal, but simply unitary matrices with non-vanishing off-diagonal entries, then
the right-hand side of eq. (12) is in general a matrix whose entries are linear combinations
of the mass eigenvalues of ˆ mDand ofˆ MR. In order to have ˆ mνdiagonal, the off-diagonal
entries must vanish and this is possible only if the respective linear combinations cancel
out. However, there are no apriori reasons to have such cancellations, since it corresponds
to have well-defined relationships between the eigenvalues of ˆ mD and of ˆ MR, which is,
in other words, a fine-tuning. Avoiding this possibility, the only solution is to consider
eq. (21).
It is useful to classify the number of degenerate eigenvalues of mDand MR. There are
nine cases in total: 3 for mD(i=1, 2 or 3-fold degeneracy) and 3 for MR(m= 1, 2 or 3-fold
degeneracy). In the following we will identify each case by (i,m). The cases (3,3), (2,3)
and (3,2) are not consistent with experimental data on neutrino mass splittings, so we are
left with six viable cases:
1 ,eiαR,L
2 ,eiαR,L
3 ) whereas ODiand ORmare respectively unitary and
i+αL
i+γi= 2kπ
3+αL
3+γ3= 2nπ. It is easy to understand the conditions given in eq. (21) by the use
TBULand U†
RVR
a) (1,1): mDand MRhave no degenerate eigenvalues;
b)(2,1): mDwith 2 degenerate eigenvalues;
c)(1,2): MRwith 2 degenerate eigenvalues;
d) (2,2): mDand MRwith 2 degenerate eigenvalues;
e)(3,1): mDwith 3 degenerate eigenvalues;
f)(1,3): MRwith 3 degenerate eigenvalues.
We proceed to show that all the viable cases obey a common expression. In the ba-
sis in which the RH neutrinos are diagonal we use mR
δidiag(v1,v2,v3), where we have schematically indicated with δithe fact that i values of
D(see eq. (13)) and write ˆ mD =
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diag(v1,v2,v3) are equal. In other words for δi= δ3we have diag(v1,v1,v1) and for δi= δ2
we have diag(v1,v2,v1) or one of its possible permutations. We thus have
mR
D= UTBPLODiδidiag(v1,v2,v3)O†
DiPRORm. (22)
It is clear that in the subspace of the i degenerate eigenvalues the rotation ODiacts as
ODiδidiag(v1,v2,v3)O†
Di→ δidiag(v1,v2,v3). Therefore we simplify the expression of mR
mR
D:
D= UTBPLδidiag(v1,v2,v3) PRORm.(23)
The next step consists in the redefinition of the viby absorbing PL,PR. In this way the
degeneracy of the i eigenvalues is broken and we finally get
mR
D
= UTBdiag(v1,v2,v3)ORm


=

?
−v1
−v1
2
3v1
√6
√6
v2
√3
v2
√3
v2
√3
0
−v3
v3
√2
√2

 ORm.

(24)
According to our formalism, the RH neutrino mass matrix is trivially given by
ˆ MR= δmdiag(M1,M2,M3), (25)
where δmindicates that m eigenvalues of diag(M1,M2,M3) are degenerate.
We now rewrite eq. (23) according to the following parametrisation
mR
D= UTBP ˆ v ORm,(26)
with ˆ v = diag(|v1|,|v2|,|v3|) and all the phases absorbed in the diagonal unitary matrix P.
In this basis and using the parametrisation given in eq. (26) for mR
formula of eq. (12) is written as
D, the type I see-saw
ˆ mν
= −DUT
= (DP eiπ/2) ˆ vˆ M−1
TB(UTBP ˆ v ORm)ˆ M−1
Rˆ v (eiπ/2P D) = (ˆ vˆ M−1/2
R(OT
Rmˆ v P UT
TB)UTBD
R†)(R∗ˆ M−1/2
RR
ˆ v),(27)
where D = P∗e−iπ/2is a consequence of our definition of ˆ mνin eq. (12), and where we have
introduced the arbitrary orthogonal complex matrix R in the last part of eq. (27). ORm
acts only in the subspace of the degenerate right handed neutrinos and in this subspace
we have by definition ORmOT
Rm= 1. From eq. (27) we have that
ˆ m−1/2
ν
ˆ vˆ M−1/2
R
R†= 1, (28)
and remembering that R†R∗= RTR = 1 we arrive at our parametrisation for R∗
R∗= ˆ m−1/2
ν
ˆ vˆ M−1/2
R
.(29)
By comparing eq. (29) with the Casas-Ibarra parametrisation given in eq. (18) we deduce
that in the case of exact TB mixing the matrix R is real and according to eq. (19) the
CP-violating asymmetry vanishes.
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Nα
H
Li
∆
Lj
H
Figure 2: Vertex correction involving a triplet scalar ∆.
Note that so far we did not refer to any specific model realisation and we have assumed
just exact TB diagonalisation of mν within the context of type I see-saw. We not only
confirm the result in [33] (in which a model with the A4flavour symmetry has exact TB
mixing leading to a vanishing CP-violating asymmetry), but also extend it to any possible
flavour symmetry responsible for the exact TB scheme5.
It is also straightforward to check by replacing UTBwith the appropriate mixing matrix
that the matrix R still turns out to be real for other exact mixing schemes as long as their
mixing matrix also consists purely of numbers (e.g. the corresponding matrix for the
Bi-maximal mixing scheme). Note also that although we have only considered three RH
neutrinos our result is absolutely generalisable to models with either two RH neutrinos or
more than three such as [56].
The proof does not hold however in the presence of additional degrees of freedom, e.g.
in models involving type I and type II see-saw. Other contributions to the CP-violating
asymmetry will in general not vanish in the limit of exact TB mixing, rendering our result
invalid for situations which do not have only type I see-saw. In scenarios with type II
see-saw the details concerning the generation of the lepton asymmetry will depend upon
the hierarchies between the triplet (∆) and the lightest RH neutrino masses [57,58]. Even
in the case M∆> MNα(Nαbeing the lightest RH neutrino) the CP asymmetry will receive
an extra contribution from the loop diagram shown in figure 2. This contribution will not
necessarily vanish, although it is constrained by the TB mixing pattern [60].
An important consequence of our proof is that if the TB mixing pattern is due to
any underlying flavour symmetry in a type I see-saw scenario, the viability of leptogenesis
depends upon possible departures from the exact pattern. In the context of models based
on discrete flavour symmetries that predict TB mixing at LO this is achieved through
next to LO (NLO) corrections. Since the size of the deviations from TB mixing are not
arbitrary, in principle one might expect the CP-violating asymmetry to be constrained by
low-energy observables such as θ13and/or the CP-violating phases.
In order to see if this is the case let us consider the most generic situation, in which NLO
corrections affect m?, mDand MR. We can perform a linear expansion in the corrections
that appear at NLO. First, we note that m?is no longer diagonal and thus we have to
move to the basis in which the charged lepton mass matrix is diagonal:
U†
?m?m†
?U?=
?
m?m†
?
?
diag, (30)
5This result is basis independent and thus remains true even assuming a non-diagonal charged lepton
mass matrix.
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where U?= 1+U(1)
?
?, with U(1)
?
?
the matrix of the NLO shifts. Eq. (9) is modified as follows
?
≡
?
=
ˆ M?
1 + U(1)†
L
?
U†
L
mD+ m(1)
D
?
UR
1 + U(1)
R
?
?
ˆ mD+ U†
Lm(1)
DUR+ U(1)†
L
ˆ mD+ ˆ mDU(1)
R
ˆ m?
D= ˆ mD+ ˆ m(1)
D,
(31)
?
1 + V(1)T
R
?
VT
R
?
MR+ M(1)
R
?
VR
1 + V(1)
R
?
?
ˆ MR+ VT
RM(1)
RVR+ V(1)T
R
ˆ MR+ˆ MRV(1)
R
R=ˆ MR+ˆ M(1)
R.
Here the unitary matrices are parametrised as the LO terms shifted by the NLO ones. The
superscript “(1)” refers to the NLO corrections and “?” to the complete mass matrices up
to NLO. The corresponding shifts on the light neutrino masses due to the NLO corrections
can be estimated according to
O(ˆ m?
ν− ˆ mν) ∼ O(ˆ mDˆ m(1)
D/ˆ MR) ∼ O(ˆ m2
Dˆ M(1)
R/ˆ M2
R) . (32)
Similarly, we can parametrise the shift from the exact TB pattern in the neutrino mixing
matrix:
Uν= UTB
1 + U(1)
?
TB
?
D ,(33)
where U(1)
entries of U(1)
V(1)
We write now eq. (24) in the new basis in which the RH neutrinos and the charged
leptons are diagonal:
?
= mR
?
mR
TBarises by the interplay between all the corrections. When we constrain the
TBby neutrino experimental data, we obtain constraints on U(1)
R. Experimental data on neutrino mass splittings constrains m(1)
?, U(1)
R.
L, U(1)
R,
Dand M(1)
mR?
D
=
1 + U(1)†
?
?
UL
?
1 + U(1)
L
??
ˆ mD+ ˆ m(1)
D
??
1 + U(1)†
R
?
U†
RVR
?
R U†
1 + V(1)
R
?
D+ U(1)†
D+ ULU(1)
Lˆ mDU†
RVR+ ULˆ m?
DU†
RVR+ ULˆ mDU(1)†
RVR+ mR
DV(1)
R.
(34)
Thus after including NLO corrections the quantity relevant for leptogenesis becomes
?
mR?†
DmR?
D
= mR†
DmR
D+mR†
D
?
U(1)†
?
mR
D+ ULU(1)
Lˆ mDU†
RVR+ ULˆ m?
DU†
RVR+
?
+ULˆ mDU(1)†
R U†
RVR+ mR
DV(1)
R
+ h.c.
?
.
(35)
Some comments are in order concerning this expression. The combination mR†
by NLO corrections, and in general it is no longer real - leading to ?Nα?= 0 and enabling vi-
able leptogenesis. The combination of NLO corrections that defines the shift is not directly
related with any low-energy observable. Consequently, while we conclude that general
model-independent NLO corrections guarantee a non-vanishing CP-violating asymmetry,
correlations among low-energy observables in the leptonic sector and ?Nαcan not be es-
tablished unless the nature of the corrections is well known i.e. once the flavour model
realisation has been specified.
DmR
Dis shifted
10

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5 Model building realisations of the different possi-
bilities
In the previous section we have presented a model-independent proof: exact TB mixing
produced by any flavour symmetry in a type I see-saw scenario corresponds to vanishing
CP-asymmetry. In this section we gather the different models studied in literature which
fall under the validity of the proof, and verify that they correspond to one of the six viable
cases of section 4. We have also present a toy model exemplifying the (2,2) case (i.e. both
matrices have two degenerate eigenvalues) which has not been studied yet. We show that
all models lead to a vanishing CP-asymmetry and thus this analysis serves as an ample set
of examples of the validity our model-independent proof.
Before describing the flavour models proposed in the literature, it is useful to explain the
generic approach considered in flavour symmetry model building. The main goal of these
models is to explain the fermion mass hierarchies and mixing angles. To do so, an horizontal
flavour group Gfis added to the gauge group of the SM and the SM fields transform in a
non-trivial way under Gf. Extra fields (flavons) are added to the particle spectrum: the
flavons are invariant under SU(3) × SU(2) × U(1), but not under Gf; they can acquire
a non-vanishing vacuum expectation value (VEV) which spontaneously breaks the flavour
symmetry in a well determined breaking chain. It is through the specific realisation of the
breaking chain that one can achieve the goal of explaining fermion data: for example, the
lepton mixing matrix becomes the TB structure when Gf is broken down to two distinct
and specific subgroups, G?in the charged lepton sector and Gνin the neutrino one, with
the type of these subgroups defining the flavour structure of the mass matrices for the
leptons (which is model-dependent).
In the following analysis we specify only which Gfwas used, and the resulting neutrino
mass matrices. We leave all other details to the original papers.
a) (i,m) = (1,1)
There are only a few examples of this case in literature. This case is particularly attractive
within the context of a Grand Unified Theory (GUT). In some cases the models do not
have exact TB only because they account simultaneously for the quark sector [12], with
the Cabibbo angle generating LO deviations from exact leptonic TB - therefore they are
not as interesting for our current purpose, and in [59] leptogenesis within the sequential
dominance framework was considered in detail (note that there is no inconsistency with
our model-independent proof). Here we consider instead two other cases explicitly.
1. In [28] the authors present a model in the context of the SO(10) GUT with the
addition of the flavour group Gf= SU(3) × U(1). The breaking of Gf down to the
discrete non-Abelian group A4provides the TB pattern for the lepton mixing matrix.
The neutrino mass matrices have the flavour structure:

00ω2A
mD∝

A
B ωA
B0
0


andMR∝


A?
B?
0
B?
ωA?
0
0
0
ω2A?


(36)
11

Page 13

where ω = e
recovered by the diagonalisation of the charged lepton mass matrix and we refer to
the original paper for the details. For leptogenesis what is relevant are the imaginary
parts of the off-diagonal entries of the product mR†
diagonal matrix.
2iπ
3 . It is straightforward to show how the correct mixing pattern is
DmR
D, and in this case it is a
2. Another pattern has been presented in [34] in the context of an SO(10) GUT model
with A4 as the additional flavour group.
structure

B0A
The mass matrices have the following
mD∝

A
0
0
C
B
0


andMR∝


A?
0
B?
0B?
0
A?
C?
0

. (37)
After considering the charged leptons the TB mixing scheme is obtained. Computing
mR†
DmR
D, we find that the off-diagonal entries are real.
b) (i,m) = (2,1)
There are several papers in which the Dirac neutrino mass matrix has only two independent
mass eigenvalues: we can divide the discussion in terms of the flavour patterns used for
the mass matrices.
1. The first pattern is present in [19,24,29,33,39,41,43]. In the basis of diagonal charged
leptons, the neutrino mass matrices have the structure:

0 1 0
mD∝

1 0 0
0 0 1


andMR∝


A?+ 2B?
−B?
−B?
−B?
2B?
A?− B?
The product mR†
−B?
A?− B?
2B?

,
DmR
(38)
where MR is exactly diagonalisable by the TB mixing.
proportional to the identity.
Two different discrete groups have been used: A4 in [19,29,33,39,41,43] and T?
in [24].
Dis
2. The other pattern has been presented in [37] where the authors have used the S4
discrete symmetry and it differs from the previous one in the explicit form of the
Majorana mass matrix:

0 1 0
mD∝

1 0 0
0 0 1


andMR∝


2A?
B?− A?
2A?+ B?
−A?
B?− A?
−A?
2A?+ B?
B?− A?
B?− A?

. (39)
This pattern corresponds to a completely different neutrino oscillation phenomenol-
ogy, but the contribution to leptogenesis is still vanishing in the limit of exact TB
mixing.
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Page 14

c) (i,m) = (1,2)
There is only one pattern within this case [38]. The discrete group A4is used to construct
a Majorana mass matrix with two degenerate eigenvalues and a Dirac mass matrix of the
TB-type.6The mass matrices are given by:

−A
The product m†
that also the light neutrino mass matrix has this property. mR†
imaginary off-diagonal factor.
mD∝

2A + B
−A
−A
2A
B − A
−A
B − A
2A


andMR∝


1 0 0
0 0 1
0 1 0

. (40)
DmD is diagonalised by the TB mixing matrix and it is easy to verify
DmR
Ddoes not present any
d) (i,m) = (2,2)
There are no models of this kind in the literature. The difficulty consists in the possibility
that the degenerate eigenvalues of the Dirac and Majorana matrices conspire to give a
degenerate light neutrino spectrum. A fully developed model is beyond the scope of this
paper, but we present here an example. Although it requires some ad hoc conditions it
is sufficient to illustrate a possible setting in which both non-degenerate light neutrino
spectrum and TB mixing are achieved.
The flavour group consists of SO(3) (or a subgroup with an irreducible triplet repre-
sentation). The additional scalar content is a set of four flavon triplets, φ123, φ23, φ2and
φ3 which get non-vanishing VEVs. At this level we fix only the VEVs of the first two
flavons in such a way that ?φ123? ∝ (1,1,1) and ?φ23? ∝ (0,1,−1) (these VEVs must be
orthogonal). The structure is reminiscent of the models in [12].
The left and RH neutrinos transform as triplets under SO(3). We assume that any
additional symmetry allows the Dirac terms
(φ123iνi)(φ2αNα) ,(φ23iνi)(φ3αNα) (41)
and the Majorana terms
NαNα,(φ3αNα)(φ3βNβ) .(42)
The term NαNαby itself would lead to degenerate masses in the Majorana matrix. The
degeneracy is lifted only for one of the states by the VEV ?φ3? ∝ (0,0,1) (two eigenvalues
remain degenerate). Thus the RH neutrino mass matrix has structure:

0 0 x
MR∝

1 0 0
0 1 0

, (43)
where x parametrises that the entry receives contribution due to ?φ3?. In the Dirac sector
one of the eigenvalues is zero. For a non-trivial choice of parameters we end up with exactly
6We underline the absence of a relevant contribution to the Dirac mass matrix, the antisymmetric
contraction of the two triplets in a singlet [30]. In order to recover the TB pattern it is possible to either
assume a fine-tuning on the parameters or alternatively to adapt the model to use another discrete group
such as S4, in which case this problem is naturally solved by its properties.
13

Page 15

two non-zero degenerate eigenstates. With ?φ2? ∝ (0,1,0) and through the type I see-saw,
the term (φ123iνi)(φ2αNα) will give rise to the solar eigenstate and the term (φ23iνi)(φ3αNα)
will give rise to the atmospheric eigenstate. In this case the Dirac mass matrix is:

0 t −b
where t and b parametrise the contributions of (φ123ν)(φ2N) and (φ23ν)(φ3N) respectively.
The effective neutrino mass matrix is diagonalised by TB mixing, as this model fits within
the framework described in [12]. There is sufficient freedom to fit the squared mass dif-
ferences (as required by phenomenology), although only strongly hierarchical cases are
possible due to the vanishing eigenvalue of mD. The Dirac matrix has two degenerate
masses by requiring 3t2= 2b2(completely ad hoc, as it requires the conspiracy of the
VEVs of the flavons - we can express it as a very specific requirement on the magnitude of
?φ2?). It is straightforward to see that mR†
leptogenesis.
mD∝

0 t
0 t
0
b

,(44)
DmR
Dis a diagonal matrix, leading to vanishing
e) (i,m) = (3,1)
This case is the most studied in literature and there are some interesting flavour patterns.
1. The first pattern has been presented in [16,23,26,32] and the flavour group which
has been used is A4. The mass matrices appear as
mD∝ 1
andMR∝


A
0
0
0
A B
B
0
A

. (45)
The charged leptons need to be rotated in diagonal form, and the main result is that
the lepton mixing matrix is exactly the TB scheme.
2. The second pattern [17,20,22] is similar to the previous one and it still originates in
an A4context. The mass matrices are the following:
mD∝ 1
andMR∝


A
0
B
0
A
0
B
0
A

. (46)
In the basis of diagonal charged leptons, we obtain the TB pattern for the lepton
mixing matrix.
3. The third pattern [18] is also similar to the first one. Once again it is based on the
A4discrete symmetry. The mass matrices are given by
mD∝ 1
andMR∝


C
0
0
0
A B
B
0
A

.(47)
Like in the previous cases, when going to the basis of diagonal charged leptons it is
easy to see that the lepton mixing matrix is the TB pattern.
14