Preprint typeset in JHEP style - HYPER VERSIONSISSA 48/2009/EP

Majorana Phases and Leptogenesis in See-Saw
Models with A4 Symmetry

C. Hagedorn
SISSA and INFN-Sezione di Trieste, Trieste I-34014, Italy
E-mail:hagedorn@sissa.it

E. Molinaro
SISSA and INFN-Sezione di Trieste, Trieste I-34014, Italy
E-mail:molinaro@sissa.it

S. T. Petcov ∗SISSA and INFN-Sezione di Trieste, Trieste I-34014, Italy;
IPMU, University of Tokyo, Tokyo, Japan

Abstract: The related issues of Majorana CP violation in the lepton sector and leptogenesis are investigated in detail in two rather generic supersymmetric models with type I
see-saw mechanism of neutrino mass generation and A 4 flavour symmetry, which naturally
lead at leading order to tri-bimaximal neutrino mixing. The neutrino sector in this classof models is described at leading order by just two real parameters and one phase. This
leads, in particular, to significant low energy constraints on the Majorana phases α 21 andα31 in the PMNS matrix, which play the role of leptogenesis CP violating parameters in
the generation of the baryon asymmetry of the Universe. We find that it is possible togenerate the correct size and sign of the baryon asymmetry in both A 4 models. The signof the baryon asymmetry is directly related to the signs of sin α 21 and/or sin α31 .

∗Also at: INRNE, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.

Contents

1. Introduction
The presence of two large and one small mixing angles in the lepton sector [1, 2],
+0.17
2
2
sin2 θ12 = 0.304+0.066−0.054 , sin θ23 = 0.50−0.14 , sin θ13 < 0.056 (3σ) ,(1.1)

suggests a pattern of neutrino mixing which is remarkably similar to the so called “tribimaximal” (TB) one [3]. Indeed, in the case of TB mixing, the solar and atmospheric
neutrino mixing angles θ12 and θ23 have values very close to, or coinciding with, the best
fit ones in eq. (1.1), determined in global analyses of neutrino oscillation data,sin2 θ12 = 1/3 , sin2 θ23 = 1/2 .

(1.2)

The TB mixing scheme predicts also that θ 13 = 0. Correspondingly, the neutrino mixing
matrix is given by


(1.3)
Uν = UT B diag 1, eiα21 /2 , eiα31 /2
where

 p


√
2/3 1/ 3
0
√
√
√ 

UT B =  −1/ 6 1/ 3 1/ 2 √√√−1/ 6 1/ 3 −1/ 2(1.4)

and α21 and α31 are Majorana CP violating phases [4, 5].
TB mixing can appear naturally in models using the tetrahedral group A 4 as flavor
symmetry [6]. 1 The three generations of left-handed leptons and right-handed (RH)
neutrinos are unified into a triplet representation of the A 4 group, whereas the right-handedcharged leptons are A4 -singlets. To be concrete, in the following we focus on two models
which represent a class of A4 models, namely the models of Altarelli-Feruglio (AF) [8] andAltarelli-Meloni (AM) [9]. Both models are based on the Standard Model (SM) gauge
symmetry group and are supersymmetric. Additional degrees of freedom, flavons, areintroduced in order to appropriately break the A 4 flavor symmetry at high energies. Bothmodels have in common that they predict at leading order (LO) a diagonal mass matrix for
charged leptons and lead to exact TB mixing in the neutrino sector. The mass matrix of
the RH neutrinos contains only two complex parameters X, Z. Light neutrino masses are
generated through the type I see-saw mechanism. All low energy observables are expressed1

It can also be derived from models with an S4 flavor symmetry [7].

–1–through only three independent quantities: α = |3Z/X|, the relative phase φ between X
and Z, and the absolute scale of the light neutrino masses. The latter is a combination of
the unique neutrino Yukawa coupling and |X| which determines the scale of RH neutrino
masses. The main difference at LO between the AF and AM models is in the generation of
the charged lepton mass hierarchy: in [8] an additional Froggatt-Nielsen (FN) symmetry
U (1)F N is invoked, whereas in [9] the hierarchy between the masses of the charged leptons
arises through multi-flavon insertions 2 . As a result the next-to-leading (NLO) correctionsin these models are different. The expansion parameter in the A 4 models of interest is thevacuum expectation value (VEV) of a generic flavon field divided by the cut-off scale Λ ofthe theory. Its typical size is λ2c ≈ 0.04 with λc ≈ 0.22 being of the size of the Cabibbo
angle.
The fact that the properties of light as well as heavy neutrinos are essentially fixed by
the three parameters α, φ and |X| leads to strong constraints. We will be interested, inparticular, in the dependence of the Majorana CP violating phases in the Pontecorvo-MakiNakagawa-Sakata (PMNS) matrix [10], which are relevant in leptogenesis and neutrinolessdouble beta decay, on α and φ. Actually, α and φ, as was shown in [9, 11], are related
through the ratio r = ∆m2 /|∆m2A | = (m22 − m21 )/|m23 − m21 |, where ∆m2 and ∆m2A are
neutrino mass squared differences driving the solar and the dominant atmospheric neutrino
oscillations. As a consequence, the Majorana CP violating phases depend effectively only
on α. The related constraints on the neutrino mass spectrum have been studied in [9,11,12].
In the present article we investigate in detail the generation of baryon asymmetry of
the Universe YB within the AF and AM models. Although our analysis is done for these
two specific models, it has generic features which are common to models based on A 4
flavour symmetry. In the class of models under discussion the neutrino masses arise via thetype I see-saw mechanism. Correspondingly, one can implement the leptogenesis scenario
of matter-antimatter asymmetry generation. As is well known, in this scenario the baryon
asymmetry is produced through the out of equilibrium CP violating decays of the RH
neutrinos νic (and their SUSY partners - RH sneutrinos ν̃ ic ) in the early Universe [13, 14].
The observed value of YB to be reproduced, given as the ratio between the net baryon
number density and entropy density, reads [15]YB ≡nB − nB̄
= (8.77 ± 0.24) × 10−11
s
0

(1.5)

where the subscript “0” refers to the present epoch.
As has been discussed in [16], the CP asymmetries  i , originating in the decays of
the RH neutrinos and sneutrinos νic and ν̃ic and relevant for the generation of the baryon
asymmetry of the Universe, vanish at LO in the class of models under discussion. Thus,
i 6= 0 are generated due to the NLO corrections. These are, as already mentioned, different
in the AF and AM models, so that the results for  i differ in the two models.
2

This is due to a different choice in the AM model of the vacuum alignment of the flavon, ϕT , slightly
different transformation properties of the right-handed charged leptons under A4 and the presence of adifferent additional cyclic symmetry.

–2–

We calculate the CP asymmetries i in the AF and AM types of models and discuss thedependence of YB on the parameter α. This is done in versions of the two models in which
the RH neutrinos have masses in the range M i ∼ (1011 ÷ 1013 ) GeV. As discussed in [9],the natural range of RH neutrino masses in the AF and AM models is M i ∼ (1014 ÷ 1015 )GeV. Such large values of the RH neutrino masses can lead [17] in SUSY theories withsee-saw mechanism to conflict with the existing stringent experimental upper limits on therates of lepton flavour violating (LFV) decays and reactions 3 , like µ → e + γ, µ → 3e,µ− + (A, Z) → e− + (A, Z), etc. The investigation of the LFV processes in the class of
AF and AM models is beyond the scope of the present study. However, in order to avoid
potential problems related to the issue of LFV, we work in versions of the AF and AM
models in which the scale of the RH neutrino masses is lower than in the original AF and
AM models. This is achieved by minimally extending the AF and AM models through an
additional Z2 symmetry, which enables us to appropriately suppress the neutrino Yukawa
couplings. This in turn allows to lower the scale of RH neutrino masses down to (10 11 ÷1013 )GeV.
We perform the analysis of baryon asymmetry generation in the so-called “one flavour”
approximation. The latter is valid as long as the masses of the RH neutrinos satisfy [18,19]
11
2
Mi >
∼ 5 × 10 (1 + tan β) GeV, where tan β is the ratio of the vacuum expectation valuesof the two Higgs doublets present in the SUSY extensions of the Standard Model. The“one flavour” approximation condition is satisfied for, e.g. tan β ∼ 3 and M i ∼ 1013
GeV. Actually in the models we consider relatively small values of tan β are preferable [8].
Further, with masses of the RH neutrinos in the range of (10 11 ÷ 1013 ) GeV one can safely
neglect the effects of the ∆L = 2 wash-out processes in leptogenesis [20]. This allows us to
use simple analytic approximations in the calculation of the relevant efficiency factors η ii .
We perform the calculation of the baryon asymmetry for the two types of light neutrinomass spectrum - with normal and inverted ordering. Both types of spectrum are allowedin the class of models considered.
We find that it is possible to generate the correct size and sign of the baryon asymmetry YB in the versions of both the AF and AM models we discuss. The sign of Y B is
uniquely determined by the sign of sin φ, since all other factors in Y B have a definite sign.
Interestingly, in the low energy observables only cos φ is present, so that both sin φ ≶ 0 arecompatible with the low energy data. Conversely speaking, taking into account the signof the baryon asymmetry YB we are able to fix uniquely also the sign of sin φ, which isotherwise undetermined through the low energy data.
Leptogenesis is not studied for the first time in the class of models of interest. However,
our work overlaps little with the already existing publications on the subject. In [21] the CP
asymmetries and YB were also calculated. This is done, however, not within the context
of a self-consistent model since instead of computing the NLO corrections, the authors
introduce ad hoc random perturbations in the Dirac neutrino mass matrix. In [16] only
the CP asymmetries are calculated within the AF model without discussing the washout
effects which can change the results for Y B . In [9] results for the CP asymmetries are
3

The indicated problem typically arises if the SUSY particles have masses in the range of few to several
hundred GeV, accessible to the LHC experiments [17].

–3–also given, but the washout effects are not taken into account. Finally, in [22] a highly
degenerate spectrum of masses of RH neutrinos is considered and the baryon asymmetry
is produced via resonant leptogenesis.
The paper is organized as follows: in Section 2 we give a short introduction to the AF
and the AM models and discuss the changes due to adding the Z 2 symmetry. We also givethe expressions for the NLO corrections relevant for the calculation of the baryon asymmetry YB . In Section 3 we discuss the light and heavy Majorana neutrino mass spectra.
We study the Majorana phases and their dependence on the parameter α. Our analysis of
leptogenesis in both models for light neutrino mass spectrum with normal ordering (NO)and inverted ordering (IO) is given in Section 4. We summarize the results of the present
work in Section 5. The two appendices contain details on the flavon superpotential and thegeneration of an appropriate VEV for the additional flavon field ζ, present in the models
considered by us.

2. Variants of the Two A4 Models
In this section we recapitulate the main features of the AF [8] and the AM model [9].
We supplement them with an additional Z 2 symmetry to appropriately suppress the Dirac
Yukawa couplings of the neutrinos and to lower the mass scale of the RH neutrinos. We
explicitly check that changes in the models connected to the Z 2 extension do not affect the
LO results for the lepton masses and mixings and also only slightly affect the NLO results.
For an introduction to the group theory of A 4 we refer to [8, 9], whose choice ofgenerators for the A4 representations we follow.
2.1 Altarelli-Feruglio Type ModelIn this model the flavor symmetry A4 is accompanied by the cyclic group Z 3 and the
Froggatt-Nielsen symmetry U (1)F N . We add, as mentioned, a further Z2 symmetry to
suppress the Dirac couplings of the neutrinos. By assuming that the RH neutrinos acquirea sign under Z2 , the renormalizable coupling becomes forbidden 4 . To allow a Yukawa
coupling for neutrinos at all we introduce a new flavon ζ which only transforms under Z 2 .
We call the VEV of ζ hζi = z in the following and assume that z ≈ λ 2c Λ as all other flavon
VEVs. Clearly, the Majorana mass terms of the RH neutrinos remain untouched, at LO.
The symmetries and particle content of the AF variant are as given in table 1. At LO
neutrino masses are generated by the terms 5yν (ν c l)hu ζ/Λ + aξ(ν c ν c ) + b (ν c ν c ϕS )(2.1)

with (· · · ) denoting the contraction to an A 4 -invariant. Thus, the LO terms in the neutrino
sector are the same as in the original AF model, apart from the suppression of the Diraccoupling. Also the LO result that the charged lepton mass matrix is diagonal, is not
4

Alternatively, one could also let hu instead of ν c transform under the Z2 symmetry to forbid the Dirac
Yukawa coupling at the renormalizable level.
5
The field ξ̃ does not have a VEV at LO and thus is not relevant at this level.

–4–Field
A4
Z3
Z2

l
3
ω
+

ec
1
ω2
+

µc
100
ω2
+

τc
10
ω2
+

νc
3
ω2
−

hu,d
1
1
+

ϕT
3
1
+

ϕS
3
ω2
+

ξ, ξ˜ ζ
1
1
2
ω
1
+ −

Table 1: Particle content of the AF variant. Here we display the transformation properties of lepton superfields,
Minimal Supersymmetric SM (MSSM) Higgs and flavons under the flavor group A 4 × Z3 × Z2 . l denotes the three
lepton doublets, ec , µc and τ c are the three SU (2)L singlets and ν c are the three RH neutrinos forming an A4 -triplet.
Apart from ν c and the flavon ζ all fields are neutral under the additional Z2 symmetry. Note that ω is the third
2πiroot of unity, i.e. ω = e 3 . To accommodate the charged lepton mass hierarchy the existence a U (1) F N , underwhich only the RH charged leptons are charged, is assumed. The U (1) F N is broken by an FN field θ only charged
under U (1)F N with charge -1. Additionally, the model contains a U (1)R symmetry relevant for the alignment of
the vacuum.

changed compared to the original model. The mass matrices of the neutrinos are of theform (mD is given in the right-left basis)
au + 2bvS −bvS
−bvS
100
 z



(2.2)
mD = yν  0 0 1  vu and mM =  −bvS
2bvS au − bvS Λ010−bvS au − bvS 2bvS
with vu = hhu i, hξi = u and hϕSi i = vS according to the alignment in [8]. The lightneutrino mass matrix is given by the type I see-saw termmν = −mTD m−1
M mD

(2.3)

and has the generic size λ2c vu2 /Λ. At the same time, the effective dimension-5 operator
lhu lhu /Λ, 6 which can also contribute to the light neutrino masses, is only invariant under
the flavor group, if it involves two flavons of the type ϕ S and ξ (ξ̃). Thus, its contribution
to the light neutrinos masses scales as λ 4c vu2 /Λ, which is always subdominant compared tothe type I see-saw contribution. The size of the contribution from the effective dimension-5
operator is actually of the same size as possible NLO corrections to the type I see-saw
term.
Considering the NLO corrections note that these involve for the Dirac neutrino mass
matrix either the two flavon combination ϕ T ζ or the shift of the vacuum of ζ. The first
type gives rise to two different termsyA (ν c l)3S ϕT hu ζ/Λ2 + yB (ν c l)3A ϕT hu ζ/Λ2

(2.4)

with (· · · )3S (A) standing for the (anti-)symmetric triplet of the product ν c l. The correction
due to the shift in hζi can be simply absorbed into a redefinition of the coupling y ν . Thus,
6

We make the “conservative” assumption that all non-renormalizable operators are suppressed by thesame cutoff scale Λ.

–5–using the alignment of ϕT as given in [8], the structure of the NLO corrections to m D isthe same as in the original model


2yA
0
0
 vT zδmD =  0
(2.5)
0−yA − yB  2 vu .
Λ
0 −yA + yB
0
The NLO corrections to the Majorana mass matrix of the RH neutrinos are exactly thesame as in the original AF model, i.e.
˜ c ν c ) + b (ν c ν c δϕS )a δξ(ν c ν c ) + ã δ ξ(ν(2.6)+ xA (ν c ν c )(ϕS ϕT )/Λ + xB (ν c ν c )0 (ϕS ϕT )00 /Λ + xC (ν c ν c )00 (ϕS ϕT )0 /Λ
+ xD (ν c ν c )3S (ϕS ϕT )3S /Λ + xE (ν c ν c )3S (ϕS ϕT )3A /Λ + xF (ν c ν c )3S ϕT ξ/Λ + xG (ν c ν c )3S ϕT ξ̃/Λ˜ Taking
where δϕS , δξ and δ ξ˜ indicate the shifted vacua of the flavons ϕ S , ξ and ξ.
into account the possibility of absorbing these corrections partly into the LO result, theygive rise to four independent additional contributions to m M which can be effectively
parametrized as


2x̃D
x̃A
x̃B − x̃C
(2.7)
δmM =  x̃Ax̃B + 2x̃C −x̃D  λ4c Λ .
x̃B − x̃C
−x̃D
x̃A

Compared to these, NLO corrections involving the new flavon ζ are suppressed, sinceinvariance under the Z2 symmetry requires always an even number of ζ fields and invarianceunder the Z3 at least one field of the type ϕS , ξ or ξ̃. The NLO corrections to the charged
lepton masses are also the same as in the original model and effects involving ζ can only
arise at the level of three flavons.
As we show in appendix A, the VEV of the flavon ζ is naturally also of the order λ 2c Λas the VEVs of the other flavons and the shift of its VEV is of the size δVEV ∼ λ 2c VEV.
We also calculate its effect on the vacuum alignment of the other flavons and show thatthe results achieved in the original AF model, especially the alignment at LO, remainunchanged.
2.2 Altarelli-Meloni Type Model
The AM model, proposed in [9], possesses as flavor symmetry A 4 × Z4 . To this we add a Z2
symmetry under which only the Higgs field h u and the new flavon ζ transform. Comparedto the original model, we change the transformation properties of h u into 100 under A4 andit transforms now trivially under Z 4 . The flavon ζ is a 10 under A4 and acquires a phase iunder Z4 . The transformation properties of leptonic superfields, MSSM Higgs and flavons
can be found in table 2. The Dirac neutrino coupling is at LO given byyν (ν c l)hu ζ/Λ ,(2.8)

which leads to the same Dirac mass matrix m D as in the original model, suppressed by thefactor λ2c for z/Λ ≈ λ2c . The Majorana mass terms for the RH neutrinos remain unaffected–6–Field
A4
Z4
Z2

l
3
i
+

ec
1
1
+

µc
1
i
+

τc
1
−1
+

νc
3
−1
+

hd
1
1
+

hu
100
1
−

ϕT
3
i
+

ξ0
10
i
+

ϕS
3
1
+

ξ
1
1
+

ζ
10
i
−

Table 2: Particle content of the AM variant. Transformation properties of lepton superfields, MSSM Higgs andflavons under the flavor group A4 × Z4 × Z2 are shown. The nomenclature is as in table 1. Apart from hu and theflavon ζ all fields are neutral under the additional Z2 symmetry. Apart from A4 × Z4 × Z2 , the model also containsa U (1)R symmetry relevant for the alignment of the vacuum, similar to the AF variant.

by the changes of the model, at LO,
M (ν c ν c ) + aξ(ν c ν c ) + b (ν c ν c ϕS )(2.9)

so that the contribution from the type I see-saw to the light neutrino masses arises from
M + au + 2bvS
−bvS
−bvS
100



 z
mD = yν  0 0 1  vu and mM = 
−bvS
2bvS
M + au − bvS Λ−bvS
M + au − bvS
2bvS
010(2.10)
Thereby the flavon alignment given in [9] is used. Equation (2.10) leads to exact TB mixing
in the neutrino sector. For M ≈ λ2c Λ, as argued in [9], we find the generic size of the lightneutrino masses to be λ2c vu2 /Λ. The effective dimension-5 operator lh u lhu /Λ arises in our
variant only at the two flavon level(ϕT ϕT )00 (ll)h2u /Λ3 + (ϕT ϕT )0 (ll)0 h2u /Λ3 + (ϕT ϕT )(ll)00 h2u /Λ3 + ((ϕT ϕT )3S (ll)3S )00 h2u /Λ3+(ξ 0 )2 (ll)h2u /Λ3 + ξ 0 (ϕT ll)0 h2u /Λ3 + ζ 2 (ll)h2u /Λ3(2.11)

where we omit order one couplings. Thus, its contributions to the light neutrino masses
are of order λ4c vu2 /Λ, i.e. of the same size as the expected NLO corrections to the type I
see-saw contribution, and hence subdominant.
The effect of the introduction of the Z 2 symmetry and the new field ζ on the charged
lepton sector is the following: an insertion of three flavons, two of which are ζ, gives a new
LO contribution to the electron mass
ζ 2 (ec lϕT )0 hd /Λ3 .

(2.12)

Using the same alignment as in [9], its contribution resembles the one from the operator
with ξ 0 instead of ζ and thus gives also a non-vanishing term in the (11) entry of thecharged lepton mass matrix. The latter is of the same size as those already encounteredin the original version of the AM model. Therefore in the variant of the AM model weare considering the charged lepton mass matrix is also diagonal at LO and the correcthierarchy among the charged lepton masses is predicted.

–7–

At NLO, the Dirac couplings of the neutrinos areyν (ν c l)δζhu /Λ + yA (ν c l)ξζhu /Λ2 + yB (ν c l)3S ϕS ζhu /Λ2 + yC (ν c l)3A ϕS ζhu /Λ2 .

(2.13)

The first two contributions can be absorbed into the LO coupling y ν . Compared to theoriginal model, the other corrections are of the same type and generate the same structure
2yB−yB − yC −yB + yC
 vS z(2.14)
δmD =  −yB + yC
2yB
−yB − yC  2 vu .
Λ−yB − yC −yB + yC
2yB

Note that actually the contribution associated to the coupling y B is still compatible with
TB mixing so that only yC can lead to deviations from the TB mixing pattern. As we will
see in section 4.2, for this reason also the CP asymmetries only depend on the coupling y C .
All effects to the Majorana mass matrix of the RH neutrinos involving ζ are negligible,
since we always need at minimum two fields ζ and additionally have to balance the Z 4charge of the operator. Thus, the NLO corrections are only those already present in theoriginal modelxA (ν c ν c )ξ 2 /Λ + xB (ν c ν c )(ϕS ϕS )/Λ + xC (ν c ν c )3S (ϕS ϕS )3S /Λ + xD (ν c ν c )3S ϕS ξ/Λ
+ xE (ν c ν c )0 (ϕS ϕS )00 /Λ + xF (ν c ν c )00 (ϕS ϕS )0 /Λ .

(2.15)

The first four contributions can be absorbed into the LO result (or vanish). We do notmention effects from shifts in the vacua of ϕ S and ξ, since these effects can in this model
also be absorbed into the LO result. The new structures at NLO lead to δm M of the form

0 x E xF
 v2
δmM = 3  xE xF 0  S .
(2.16)
ΛxF 0 x E
For the charged leptons, additional NLO corrections to the muon and the electron massarise from three and four flavon insertions, respectively, involving the field ζ. The operator
ζ 2 (µc lϕS )0 hd /Λ3(2.17)

corrects the muon mass. This type of subleading contribution already exists in the originalmodel such that no new structures are introduced. The NLO corrections to the electronmass are induced through the operator ζ 2 (ec lϕT )0 hd /Λ3 , if the shifts in the vacua are taken
into account, as well as through the four flavon operators
ζ 2 (ec l(ϕT ϕS )3S )0 hd /Λ4 + ζ 2 (ec l(ϕT ϕS )3A )0 hd /Λ4 + ζ 2 ξ(ec lϕT )0 hd /Λ4 + ζ 2 ξ 0 (ec lϕS )hd /Λ4 .
(2.18)All structures arising from these corrections are already generated by the NLO corrections
present in the original model so that the analysis given in [9] for the NLO corrections isstill valid in the constructed variant.
In appendix B we discuss how to give a VEV of the desired size to the field ζ, the shiftof this VEV from NLO corrections as well as the effects of ζ on the flavon superpotentialof the original model at LO and NLO.

–8–

3. Neutrino Masses and CP Violating Phases in the A4 Models
The models discussed in the previous section have in common that the Majorana mass
matrix of RH neutrinos is of the form
X + 2Z −Z−ZmM =  −Z(3.1)
2Z X − Z −Z X − Z 2Z
and the neutrino Dirac mass matrix reads


100
z

mD = y ν  0 0 1  vu
Λ
010

(3.2)

M1 = |X + 3Z| ≡ |X| |1 + αeiφ |, ϕ1 = arg(X + 3Z)

(3.4)



The symmetry of this class of models implies that, at leading order, the neutrino part of
the Lagrangian depends only on few parameters: X, Z and y ν . These parameters are, in
general, complex numbers. One can set y ν real by performing a global phase transformationof the lepton doublet fields. As we will see, CP violating phases, which enter in the CP
asymmetries of the RH neutrino decays, are functions of the relative phase between X and
Z. The see-saw mechanism for the neutrino mass generation implies that the full parameter
space of the neutrino sector can be constrained significantly by the low energy data.
The RH neutrino mass matrix (3.1) is diagonalized by an orthogonal matrix U T B ,given in (1.4):
diag(M1 eiϕ1 , M2 eiϕ2 , M3 eiϕ3 ) = UTTB mM UT B ,
(3.3)
where

M2 = |X|,

ϕ2 = arg(X)iφM3 = |X − 3Z| ≡ |X| |1 − αe |,ϕ3 = arg(3Z − X).

(3.5)
(3.6)

Here α ≡ |3Z/X| and φ ≡ arg(Z) − arg(X).
A light neutrino Majorana mass term is generated after electroweak symmetry breaking
via the type I see-saw mechanism:

where

∗
†
mν = −mTD m−1
M mD = U diag (m1 , m2 , m3 ) U

(3.7)



U = i UT B diag eiϕ1 /2 , eiϕ2 /2 , eiϕ3 /2(3.8)

and m1,2,3 are the light neutrino masses,mi ≡(yν )2 vu2  z 2,MiΛi = 1, 2, 3

(3.9)

The i in eq. (3.8) correspond to a unphysical common phase and we will ignore it in what
follows. We observe also that one of the phases ϕ k , say ϕ1 , can be considered as a common

–9–

Α
2.0

1.5

1.0

0.5

-1.0

cos@ΦD

-0.5

0.5

1.0

Figure 1: The correlation between the real parameter α and the phase φ, which appear in the RH
neutrino Majorana mass matrix. The figure is obtained by using the 3σ range of the parameter r
given in eq. (3.10). See text for details.

phase of the neutrino mixing matrix, and thus has no physical relevance. In the following
we always set ϕ1 = 0.
The parameters |X|, α and φ defined in (3.4)-(3.6), which determine the RH neutrinomass matrix (3.1), can be constrained by the neutrino oscillation data. More specifically,
we have for the ratio [9]:r≡∆m2
(1 + α2 − 2 α cos φ)(α + 2 cos φ)=,
4 | cos φ||∆m2A |

(3.10)

where ∆m2 = ∆m221 ≡ m22 − m21 > 0 and |∆m2A | = |∆m231 | ∼= |∆m232 | are the ν−masssquared differences responsible respectively for solar and atmospheric neutrino oscillations.
Since the value of r is fixed by the data, this relation implies a strong correlation between
the values of the parameters α and cos φ. Let us note that the sign of sin φ cannot be
constrained by the low energy data. As we will see later, the sign of sin φ is fixed by thesign of the baryon asymmetry of the Universe, computed in the leptogenesis scenario.
At 3σ, the following experimental constraints must be satisfied [1]:∆m2 > 0|∆m2A | = (2.41 ± 0.34) × 10−3 eV2(3.11)

r = 0.032 ± 0.006 .

In Fig. 1 we show the correlation between α and cos φ, following from (3.10) taking
into account (3.11). Depending on the sign of cos φ, the parameter space is divided into
two physically distinctive parts: cos φ > 0 corresponds to light neutrino mass spectrumwith normal ordering (NO), whereas for cos φ < 0 one obtains neutrino mass spectrumwith inverted ordering (IO) .

– 10 –

The main difference between the models we are discussing and the original models
reported in [8] and [9] is in the mass scale of the RH neutrino fields. In the models
considered here the predicted RH neutrino masses are always rescaled by the additional
factor (λ2c )2 ∼ 10−3 . Depending on the value of the neutrino Yukawa coupling y ν , in ourmodel the lightest RH Majorana neutrino mass can be in the range from (10 11 ÷1012 ) GeV,and up to 1013 GeV for a neutrino Yukawa coupling y ν ∼ O(1).
For neutrino mass spectrum with NO, the RH neutrino masses show approximately
the following partial hierarchy [16]: M 1 ≈ 2M2 ≈ 10M3 . The lightest neutrino mass m1 ,compatible with the experimental constraints given in (3.11), takes values in the interval
3.8 × 10−3 eV . m1 . 6.9 × 10−3 eV. This implies that the light neutrino mass spectrum iswith partial hierarchy 7 . For the sum of the neutrino masses we have:
6.25 × 10−2 eV . m1 + m2 + m3 . 6.76 × 10−2 eV .

(3.12)In the case of IO spectrum, the overall range of variability of the lightest neutrino mass,
m3 , is the following: 0.02 eV . m3 ≤ 0.50 eV, where only the lower bound follows fromthe low energy constraints. The upper bound was chosen by us to be compatible with the“conservative” cosmological upper limit on the sum of the neutrino masses [24, 25]. Thus,
the light neutrino mass spectrum can be with partial hierarchy or quasidegenerate. If thespectrum is with partial hierarchy (i.e. 0.02 eV . m 3 < 0.10 eV), for the RH Majorana
neutrino masses, to a good approximation, we have: M 1 ∼= M2 ∼= M3 /3. Quasidegenerate
light neutrino mass spectrum implies that, up to corrections ∼ O(r), one has M 1 ∼= M2 ∼=
M3 . The sum of the light neutrino masses in the case of IO spectrum is predicted to satisfy:
m1 + m 2 + m 3 >∼ 0.125 eV .

(3.13)

We give below the expressions for the lightest neutrino mass in the NO and IO spectrum
as functions of α and r. Recall that for fixed value of r, all the parameter space and theassociated phenomenology is characterized by the parameter α. In the numerical examples
reported in the following, we always use the best fit value of the ratio r: r = 0.032. By
expanding with respect to r, we get for the square of the lightest neutrino mass:
2(1 + α2 )r1
22+, NO spectrum ;(3.14)
m1 = ∆mA r
1 + 2α2(1 + 2α2 )3m23= |∆m2A |

1(1 + α2 )r+
2α2 α2 (1 + 2α2 )

,IO spectrum .

(3.15)For α = 1 the expression for m21 reduces to the one obtained in [9].
In the class of models we are considering, the three light neutrino masses obey thegeneral sum rule (valid for both types of spectrum) [9, 11]:
1
2eiϕ2
eiϕ3
=
−
m3
m1
m2
7

This was noticed also in [23].

– 11 –(3.16)

This equation implies a strong correlation between the neutrino masses and the Majorana phases arising from the RH neutrino mass matrix. The Majorana phases are responsible for CP violation in leptogenesis and therefore we will discuss them in detail in thefollowing subsection.
3.1 The Majorana CP Violating Phases
In the following, we use the standard parametrization of the PMNS matrix (see, e.g. [26,27]):

c12 c13s12 c13iδ
iδ
UPMNS = 
 −s12 c23 − c12 s23 s13 e c12 c23 − s12 s23 s13 e

s13 e−iδ




α31
α
i 221
, ei 2 )s23 c13  diag(1, es12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 eiδ c23 c13(3.17)

where cij ≡ cos θij , sij ≡ sin θij , θij ∈ [0, π/2], δ ∈ [0, 2π] is the Dirac CP violating phaseand α21 and α31 are the two Majorana CP violating phases, α 21,31 ∈ [0, 2π]. From thesee-saw mass formula (3.7) one can read directly the form of the neutrino mixing matrix
that arises at leading order in perturbation theory. Taking into account the standard
parametrization (3.17) and (1.4), the PMNS matrix is indeed:
UPMNS = diag(1, 1, −1)UT B diag(1, eiϕ2 /2 , eiϕ3 /2 )(3.18)

From eqs. (3.17) and (3.18) we identify the ”low energy” Majorana phases as
α21 = ϕ2

(3.19)

α31 = ϕ3(3.20)We remark that, at this order of perturbation theory, the CHOOZ mixing angle, θ 13 , isalways zero as a consequence of the TB form of the neutrino mixing matrix, imposed by
the broken A4 discrete symmetry.
In the models under discussion the Majorana phases α 21 and α31 can also be constrained by using the neutrino oscillation data 8 . After some algebraic manipulation, we
arrive at the following relations between the CP violating phases α 21 and α31 and theparameters α and φ of the model:
α sin φ1 + α cos φα sin φtan α31 = 2 2,
α −1

tan α21 = −(3.21)(3.22)

where α and cos φ satisfy eq. (3.10).
8

Let us recall that the probabilities of oscillations involving the flavour neutrinos do not depend on the
Majorana phases of the PMNS matrix [4, 28]. Thus, the Majorana phases cannot be directly constrained by
the neutrino oscillation data.

– 12 –

Normal Ordering

3.0

6.0

2.5

5.5

2.0

5.0

Α31

Α21

Normal Ordering

1.5
1.0

4.5
4.0

0.5

3.5

0.0
0.8

0.9

1.0

1.1

0.8

1.2

0.9

1.0

1.1

1.2

Α

Α

Figure 2: The Majorana phases α21 and α31 in the case of a light neutrino mass spectrum withnormal ordering. The parameter r is set to its best fit value, r = 0.032. The solutions of equations(3.21) and (3.22) shown in the figure correspond to sin φ < 0. See text for details

Inverted Ordering

Inverted Ordering

3.0

6.0

2.5

5.5
5.0

Α31

Α21

2.0
1.5
1.0

4.5
4.0

0.5

3.5

0.0
0.5

1.0

1.5

2.0

Α

0.5

1.0

1.5

2.0

Α

Figure 3: The same as in Fig. 2, but for a light neutrino mass spectrum with inverted ordering.

In the case of NO spectrum, we have φ = 0 ± ε, 2π ± ε, with ε < 0.2 and 0.8 . α . 1.2(see Fig. 1). If ε ∼= 0, the two CP violating phases become unphysical. No CP violation ispossible in this case. As regards the IO light neutrino mass spectrum, it is easy to showthat [9] 2 cos φ ≈ −α. The correction, δ α (α), which appears in the right-hand side of thisequation, is given by
2αr2(1 + α2 )rδα (α) =1−(3.23)
1 + 2α2(1 + 2α2 )2For light neutrino mass spectrum with inverted ordering, the parameter α varies in theinterval 0.07 . α . 2, where the lower limit of α comes from the indicative upper boundon the absolute neutrino mass scale used by us, m 1,2,3 . 0.5 eV.
In Figs 2 and 3 we show the behavior of the Majorana phases α 21 and α31 as functionsof α, for the NO and IO mass spectrum, respectively. We choose sin φ < 0 in (3.21) and(3.22). As we will see, this is dictated by reproducing correctly the sign of the baryon– 13 –

asymmetry. On the other hand, the relative sign of sin α 21 and sin α31 is fixed by therequirement that the sum rule in eq. (3.16) is satisfied. In the case of NO spectrum, thephase α21 is close to zero. The maximum value of α 21 is obtained for α ∼= 1. At α = 1 wep∼have approximately α21 = r/3 = 0.1. The other Majorana phase α31 can assume large
CP violating values. The largest | sin α 31 | is reached for α = 1: at this value of α we havesin α31 = −1.
If the light neutrino mass spectrum is with IO, both phases can have large CP violating√
values. We get sin α21 = 1 and sin α31 = −1 for α ≈ 2 and α = 1, respectively.
The Majorana phase α21 can be probed, in principle, in the next generation of experiments searching for neutrinoless double beta decay [29]. Below we give the expression of
the effective Majorana mass mee predicted in the class of models we are considering in thecases of neutrino mass spectrum with normal and inverted ordering [26]:q2∼mee = m1 cos θ12 + m21 + ∆m2 sin2 θ12 eiα21 , NO ;q∼
mee = m23 + |∆m2A | cos2 θ12 + eiα21 sin2 θ12 , IO .

(3.24)(3.25)We recall that in the class of models under discussion, sin 2 θ12 = 1/3, cos2 θ12 = 2/3 anda non-zero value of θ13 arises only due to the NLO corrections. As a consequence, thepredicted value of θ13 is relatively small, θ13 ∼ O(λ2c ∼ 0.04). Thus, the terms ∼ sin2 θ13
in mee give a negligible contribution. Further, since the Majorana phase α 21 ∼= 0 (see Fig.
2), the two terms in the expression for m ee in the case of NO spectrum add up. As aconsequence, we have:
1
2
m1 +
mee ∼
=
3
3

q
m21 + ∆m2 ,NO ,(3.26)

where 3.8 × 10−3 eV . m1 . 6.9 × 10−3 eV.
We show in Fig. 4 the effective Majorana mass m ee and the lightest neutrino mass m1
versus α, for the NO spectrum. In this case, m ee takes values in the interval: 6.5×10 −3 eV .
mee . 7.5 × 10−3 eV. Similar conclusion has been reached in [23]. In what concerns the
IO spectrum, the full range of variability of the effective Majorana mass, compatible withneutrino oscillation data, is predicted to be
q
q
1
2
2
(3.27)
m3 + |∆mA | . mee . m23 + |∆m2A | , with m3 >∼ 0.02 eV , IO .
3>
For m3 >∼ 0.02 eV, this implies mee ∼ 0.018 eV (see also [9]).

4. Leptogenesis
In this section we compute the baryon asymmetry within the AF and AM type models
defined in Section 2. As we have already noticed earlier, leptogenesis cannot be realized if
we take into account only the leading order contribution to the neutrino superpotential. In– 14 –

Normal Ordering
0.0075mee , m1 HeVL

0.00700.00650.00600.00550.00500.0045
0.8

0.9

1.0

1.1

1.2

Α
Figure 4: The effective Majorana mass mee (blue continuous line) and lightest neutrino mass m1(red dashed line) in the case of a light neutrino mass spectrum with normal ordering. In both cases∆m2A and r are fixed to their best fit values.

order to generate a sufficiently large CP asymmetry, higher order corrections to the Diracmass matrix of neutrinos must be taken into account. The RH neutrino mass spectrum
in this class of models is not strongly hierarchical. Consequently, the standard thermal
leptogenesis scenario in which the relevant lepton CP violating asymmetry is generated in
the decays of the lightest RH (s)neutrino only is not applicable and one has to take into
account the contribution from the out of equilibrium decays of the heavier RH (s)neutrinos.
The lepton asymmetry thus produced in the decays of all heavy RH (s)neutrinos ν ic (ν̃ic), i = 1, 2, 3, is converted into a baryon number by sphaleron interactions. The neutrino
and sneutrino CP asymmetry i , which are equal for lepton and slepton final states, is thefollowing [30]: †mi
1 X Im[(m̂D m̂D )2ji ]
,
(4.1)
f
i =
†
8πvu2
m
j
(m̂D m̂D )ii
j6=i
where
m̂D = U † mD

(4.2)

is the neutrino Dirac mass matrix in the mass eigenstate basis of RH neutrinos, and m i ,i = 1, 2, 3 are the light neutrino masses. The matrix U and the masses m i coincide withthose given in eqs. (3.8) and (3.9), respectively. The loop function f (m i /mj ) is defined as
f (x) ≡ −x





2
1
+ log 1 + 2
x2 − 1x(4.3)

This function depends strongly on the hierarchy of light neutrino masses. It can lead toa strong enhancement of the CP asymmetries if the light neutrino masses m i and mj arenearly degenerate. As we have seen earlier, the neutrinos can be quasidegenerate in mass

– 15 –

in the case of IO spectrum. In this case we have to a good approximation f (m i /mj ) ∼=−f (mj /mi ).
We recall that in the case of IO spectrum, the lightest two heavy Majorana (s)neutrinos,cc ), have very close masses. However, the conditions for resonant leptogenesis [31]
ν1,2 (ν̃1,2
are not satisfied in the models under consideration. Indeed, in all the region of the relevant
parameter space we have 0.2 . α . 2. Correspondingly, the relative mass difference of thetwo heavy Majorana (s)neutrinos in question is†(m̂D m̂D )12
λ6c
m1 ∼
M2 − M 1≈≈ 10−5=1−= (2 ÷ 14) × 10−3  maxM1m216π 2 vu2
π2

(4.4)

Under the above condition, the CP asymmetries for each (s)neutrino decay can be computedin perturbation theory as the interference between the tree level and one loop diagrams(see, e.g. [32, 33]).
The general expression for the baryon asymmetry [34], in which each RH (s)neutrino
gives a non-negligible contribution, can be cast in the following form:
3XnB − n̄B= −1.48 × 10−3
i ηij
YB ≡
s

(4.5)

i,j=1

where ηij is an efficiency factor that accounts for the effects of washout due to the ∆L = 1
interactions of νjc and ν̃jc of the asymmetry Yli , generated in the decays νic → li hu , ˜li h̃uand ν̃ic → l̃i hu , li h̃u . They take into account also the decoherence effects on l i caused bythe νjc and ν̃jc (j 6= i) interactions. We refer in the following discussion only to the leptons
number densities. The same considerations apply for the interactions involving slepton
states.
The computation of the efficiency factors in the models under discussion is considerably
simplified [35, 36] (see also [37]). This is due to the fact that to leading order, the heavy
Majorana neutrinos ν1c , ν2c and ν3c , as can be shown, couple to orthogonal leptonic states.
As a consequence, the Boltzmann evolutions of the three lepton CP violating asymmetries,
associated with the indicated three orthogonal leptonic states, are practically independent.
Taking into account the above considerations, one can compute the total baryon asymmetry
as an incoherent sum of the contributions arising from decays of each of the three heavy
RH neutrinos:
3
X
YB ≈
YBi ,
(4.6)
i=1

where

YBi ≡ −1.48 × 10−3 i ηii .

(4.7)

In the class of models considered the RH neutrino mass scale is set below 10 14 GeV,
preventing possible washout effects from ∆L = 2 scattering processes. In this case, theefficiency factors ηii can be expressed only in terms of the washout mass parameters m
e i [20]:!
1.16 −1
3.3 × 10−3 eV
m
ei
ηii =
+
,
(4.8)
m
ei
0.55 × 10−3 eV– 16 –

where(m̂D m̂†D )ii
.
(4.9)
m
ei ≡
Mi
Here m̂D is the neutrino Dirac mass matrix in the basis in which the Majorana mass matrix
of RH neutrinos is diagonal with real eigenvalues (see eq. (4.2).
4.1 Leptogenesis in the Variant of AF ModelIn this Section we compute the baryon asymmetry for the AF type model, in the one-flavor
leptogenesis regime.
In the basis in which the RH Majorana neutrino mass term given in (2.2) is diagonal,
the relevant matrix that enters into the expression of the leptogenesis CP asymmetries(4.1) is given by z 2yν2 vu2m̂D m̂†D = 1
Λ(4.10)√ i α21α31√2 ei 2 Re(yB )2 Re(yA )2 2e 2 Re(yA )3q √  v   z 2
α
α −α


−i 2212 i 31 2 21
Re(yA )
0
−2 3 eRe(yB )  T+  2 2eyν vu2
q
Λ
Λ


α −α
α31
2 i 21 2 31
√2 e−i 2 Re(yB ) −2
Re(yB )
−2 Re(yA )3e3where yA and yB are the higher order (complex) Yukawa couplings defined in (2.4) and(2.5). We can take all flavon VEVs real without loss of generality.

The CP asymmetries k (k = 1, 2, 3) can be written in the following way:
1  z  2  vT 2
6f (m1 /m2 ) sin α21 Re(yA )2 + f (m1 /m3 ) sin α31 Re(yB )2 (4.11)
1 = −
6π Λ
Λ




1 z 2 vT 23f (m2 /m1 ) sin α21 Re(yA )2 + f (m2 /m3 ) sin(α21 − α31 ) Re(yB )2
2 =
3π Λ
Λ
(4.12)
1  z 2  vT  2
3 =(2f (m3 /m2 ) sin(α31 − α21 ) + f (m3 /m1 ) sin α31 ) Re(yB )2 (4.13)
6π ΛΛwhere mk are the LO neutrino masses and z/Λ ≈ v T /Λ ≈ λ2c . Thus, in the model underconsideration we have|k | ∝ λ8c ≈ 6 × 10−6 , k = 1, 2, 3 .
(4.14)
This is the order of magnitude we expect for the CP asymmetry if we require successful
leptogenesis. Depending on the loop factor f (m i /mj ) (eq. 4.3) and the values of the
Majorana phases, the CP asymmetry can be enhanced or suppressed.
The washout mass parameters, associated to each of the three lepton asymmetries, aregiven by:m
e 1 = m1 (1 + O(λ2c ))m
e 2 = m2 (1 + O(λ2c ))m
e 3 = m3 (1 + O(λ2c ))– 17 –(4.15)(4.16)(4.17)

We see that the washout mass parameters, to a good approximation, coincide with theneutrino masses.
Results for NO Spectrum
We study the baryon asymmetry in the region of the parameter space corresponding
to a neutrino mass spectrum with normal ordering: 0.8 . α . 1.2. The lightest RH Majorana neutrino in this scenario is ν3c . The Majorana phases, that provide the requisite CP
violation for a successful leptogenesis, are solutions of equations (3.21) and (3.22) corresponding to sin φ < 0. The dependence on α of each of the two CP violating phases isshown in Fig. 2. We recall that only the solutions corresponding to sin φ < 0 give thecorrect sign of the total baryon asymmetry.
We show in Fig. 5, left panel, the dependence of the baryon asymmetry Y B on theparameter α. The individual contributions to Y B from the decays of each of the three
RH Majorana neutrinos are also shown. The term Y B3 , originating from the lightest RH
neutrino decays, is suppressed by largest washout effects, with respect to Y B1 and YB2 (see(4.17)).
The contribution to the total baryon asymmetry given by Y B1 shows an interplay
between two independent terms proportional to y A and yB , respectively. These two terms
have always the same signs and are of the same order of magnitude. The suppression due
to the Majorana phase α21 . 0.1 of the term proportional to yA is compensated by theenhancement due to the loop factor: we find that 6f (m 1 /m2 )/f (m1 /m3 ) ∼= −(8 ÷ 20). The
same considerations apply to YB2 . Now sin α21 and sin(α21 −α31 ) have the same sign and theratio of the corresponding loop factors is approximately 3f (m 2 /m1 )/f (m2 /m3 ) ∼= (20÷30).
In conclusion, in the case of NO light neutrino mass spectrum, each of the two Majorana phases α21 and α31 , having values within the ranges allowed by neutrino oscillation
data (see Fig. 2), can provide the CP violation which is required in order to have successful leptogenesis. Even in the case in which the term proportional to sin α 31 in the CP
asymmetries is strongly suppressed (which corresponds to the case of strong fine-tuning ofyB  1), successful baryogenesis can be naturally realised for values of the Majorana phase
α21 ≈ (0.04 ÷ 0.10) and a moderately large neutrino Yukawa coupling y A ∼ (2.5 ÷ 3.0).
Results for IO Spectrum
We now study in detail the region of the parameter space for which the neutrino mass
spectrum is with inverted ordering and is hierarchical. This scenario is realized for 0.2 <α . 2. In the following, we report the behavior of the baryon asymmetry in all the intervalof variability of α, compatible with an IO neutrino spectrum and for which the computation of the CP asymmetry can be done in perturbation theory. Thus, the results we showfor 0.07 < α . 0.2 should be valid provided the renormalisation group (RG) effects [38]are sufficiently small in the indicated region.
In Fig. 5, right panel, we plot the different contributions to the baryon asymmetry, as
we have done previously for the normal hierarchical mass spectrum.

– 18 –

1.2 ´ 10-10

1.5 ´ 10-10

1. ´ 10-10

1. ´ 10-10

6. ´ 10-11

YB

YB

8. ´ 10-11

4. ´ 10-11
5. ´ 10-11

2. ´ 10-11
0

0
0.8

0.9

1.0

1.1

1.2

Α

0.5

1.0

1.5

2.0

Α

Figure 5: AF type model: baryon asymmetry versus α in the cases of neutrino mass spectrum withnormal (left panel) and inverted (right panel) ordering. In each plot we show: i) the total baryon
asymmetry YB (red continuous curve), ii) YB1 (green dashed curve), iii) YB2 (orange dotted curve)
and iv) YB3 (blue dot-dashed curve). On the right panel, the lines corresponding to YB1 and YB2
overlap. In both cases sin φ < 0 and ∆m2A and r are fixed at their best fit values. The results shownin the left (right) panel correspond to yA = 2.5 and yB = 3 (yA = 0.4 and yB = 2). The horizontal
dashed lines represent the allowed range of the observed value of YB , YB ∈ [8.5, 9] × 10−11 .

The Majorana CP violating phases which enter into the expressions for the CP asymmetries are reported in Fig. 3. The solutions of equations (3.21) and (3.22) corresponding
to sin φ < 0 must be used also in this case in order to obtain the correct sign of thebaryon asymmetry. Now ν3c is the heaviest RH Majorana neutrino and the washout effects for the CP asymmetry generated in the decays of this state are less strong since
they are controlled to LO by the lightest neutrino mass m 3 : me 3 = m3 . We note, however, that also in this scenario the contribution of the term Y B3 in YB is always muchsmaller than the contribution of the other two terms Y B1 and YB2 . This is a consequenceof the strong enhancement in the self energy part of the loop function that enters into
the expressions for YB1 and YB2 . Indeed, if the spectrum is inverted hierarchical, we havef (m1 /m2 ) ∼= −f (m2 /m1 ) ≈ 50f (m3 , m1,2 ). For this reason the CP violating phase α 31
gives, in general, a subdominant contribution in the CP asymmetries  1 and 2 when the
Yukawa couplings yA and yB are of the same order of magnitude. This conclusion is valid
even in the region of the parameter space where α 31 ≈ 3π/2.
The analysis of all the parameter space defined by α, compatible with low energy
neutrino oscillation data, in the AF type model, show that in both the normal and inverted
patterns of light neutrino masses, the Majorana phases can provide enough CP violationin order to have successful leptogenesis, even in the case in which only one of the phases
α21 and α31 , effectively, contributes in the generation of the CP asymmetry.
4.2 Leptogenesis in the Variant of AM ModelIn this section we study the generation of the baryon asymmetry of the Universe in thevariant of the AM model considered by us. We work in the one-flavour leptogenesis ap-– 19 –

2. ´ 10-10
1.5 ´ 10-10

YB

YB

1. ´ 10-10

5. ´ 10-11

1. ´ 10-10
5. ´ 10-11

0
0
0.8

0.9

1.0

1.1

1.2

0.5

1.0

1.5

2.0

Α

Α

Figure 6: The same as in Fig. 5 but for the AM type model. The Yukawa coupling yC is set toyC = 2 in both the figures.

proximation. The quantity relevant for the calculation of the CP asymmetries in this caseis:
 z 2m̂D m̂†D = 1yν2 vu2Λ(4.18)√ i α31
6 Re(yB )0 2 3e 2 Re(yC )    2
z

 vS
+ 
yν vu2000√ −i α31ΛΛ−6 Re(yB )2 3e 2 Re(yC ) 0
where the yB and yC are defined in eqs. (2.13) and (2.14). Again we can choose all flavon
VEVs to be real without loss of generality.
The CP asymmetries in this model are given by1 = −

3  z 2  vS 2
f (m1 /m3 ) sin(α31 ) Re(yC )22π Λ
Λ2 = 03  z 2  vS 2
f (m3 /m1 ) sin(α31 ) Re(yC )2
3 =
2π Λ
Λ

(4.19)(4.20)(4.21)

where m1,3 are again the LO neutrino masses (see eq. (3.9)). The leptogenesis CP violatingphase now coincides with the Majorana phase α 31 . Moreover, the CP asymmetries 1,3 6= 0
are controlled by only one parameter, y C , of the matrix of neutrino Yukawa couplings(2.14), the reason being that only this parameter breaks the TB form of the latter.
As we see, the heavy RH Majorana neutrino ν 2c “decouples”: the CP violating lepton
asymmetry is produced in the out of equilibrium decays of the heavy Majorana neutrinos
ν1c and ν3c alone. This constitutes a major difference with the variant of the AF model,
analyzed in the preceding subsection. After the lepton asymmetries are converted into abaryon asymmetry by sphaleron processes, the final matter-antimatter asymmetry of the– 20 –

Universe can be estimated as:
YB ≡ YB1 + YB3(4.22)

where YBi , for i = 1, 3, are given in eq. (4.7). The LO washout mass parameters m
e 1,3 arethe same as in the variant of the AF model:me 1 = m1 (1 + O(λ2c ))m
e 3 = m3 (1 + O(λ2c ))(4.23)(4.24)In Fig. 6 we show the dependence of the baryon asymmetry on the parameter α
in the cases of neutrino mass spectrum with normal and inverted ordering. Both typesof spectrum are allowed in the model considered. The ranges of possible values of the
Majorana phase α31 which provides the correct sign of the baryon asymmetry are shownfor the NO and IO spectra in Figs. 2 and 3 (they are the same as for the AF type model).
We observe that, as in the variant of the AF model, the suppression of the term Y B3
with respect to YB1 in the case of NO spectrum is due to the relatively larger washout effects
in the generation of the asymmetry  3 . The maximum of the total baryon asymmetry Y B
is reached for α ≈ 1 where the CP violating Majorana phase α 31 ∼= 3π/2. (see Fig. 2, rightpanel).
In what concerns the IO spectrum, the two terms Y B1 and YB3 enter with the same signin the total baryon asymmetry and are of the same order of magnitude. The enhancementof the asymmetry for α < 0.7 is explained by the increase of the loop function f (m 1 /m3 ) ∼=−f (m3 /m1 ) in the region of quasi-degenerate light neutrino mass spectrum.
In this class of models, successful leptogenesis can be naturally realized for both typesof spectrum - NO and IO, for an effective Yukawa coupling y C >∼ 1.5.

5. Summary
In the present work we studied the related issues of Majorana CP violating phases andleptogenesis in variants of two prominent (and rather generic) supersymmetric A 4 models[8,9] which naturally lead at leading order (LO) to tri-bimaximal (TB) mixing in the lepton
sector. The pattern of neutrino mixing suggested by the existing neutrino oscillation data
is remarkably similar to the TB one. Both models are supersymmetric and employ the typeI see-saw mechanism of neutrino mass generation. They predict at LO a diagonal mass
matrix for charged leptons and lead to exact TB mixing in the neutrino sector. The mass
matrix of the RH neutrinos contains only two complex parameters X, Z. All low energy
observables are expressed through only three independent quantities: the real parameter
α = |3Z/X|, the relative phase φ between X and Z, and the absolute scale of the lightneutrino masses. The latter is a combination of the neutrino Yukawa coupling and theparameter |X| which determines the scale of RH neutrino masses.
The main difference between the original models and those considered by us is in thescale of RH neutrino masses. In the original models this scale is around (10 14 ÷ 1015 ) GeV.
In order to avoid possible potential problems with LFV processes we consider versions ofboth models in which the scale of RH neutrino masses is lower, namely, is in the range– 21 –

of (1011 ÷ 1013 ) GeV. This is achieved by imposing an additional Z 2 symmetry capable of
suppressing sufficiently the neutrino Yukawa couplings. As a consequence, the mass scaleof the RH neutrinos is lowered as well. We discussed in detail the flavon superpotential in
the modified models of interest. The results obtained at leading order and next to leading
order (NLO) in the original models are still valid in the extensions we consider.
The two Majorana phases of the PMNS matrix, α 21 and α31 , effectively play the roleof leptogenesis CP violating parameters in the generation of the baryon asymmetry. In
the models considered both the phases α 21 and α31 and the ratio r ≡ ∆m2 /|∆m2A | arefunctions of the two parameters α and φ. We analyzed in detail the dependence of thetwo “low energy” Majorana phases α21 and α31 on α and φ. In contrast to the low energy
observables, like neutrino masses and the effective Majorana mass in neutrinoless double
beta decay, mee , we show that these phases depend both on sin φ and cos φ, and not only
on cos φ. We show also that the sign of the baryon asymmetry Y B uniquely determines thesign of sin φ, which has to be negative: sin φ < 0.
In the case of neutrino mass spectrum with normal ordering (NO), α 21 is shown tobe small, α21 . 0.1. In the types of models considered sin2 θ13 is also predicted to besmall, sin2 θ13 ∼ 10−3 . As a consequence, the contributions of the terms ∝ sin 2 θ13 inmee are strongly suppressed. The lightest neutrino mass is predicted to lie in the interval(3.8 ÷ 6.9) × 10−3 eV, thus the neutrino mass spectrum is with partial hierarchy. The
effective Majorana mass mee has a relatively large value, mee ∼ 7 × 10−3 eV. We note that
if α21 had a value close to π, one would have m ee  10−3 eV. Depending on α, the phase
α31 can take large CP violating values. For light neutrino mass spectrum with invertedordering (IO), the Majorana CP phases α 21 and α31 vary (for sin φ < 0) between 0 and π
and π and 2π, respectively.
Throughout this study we have neglected renormalization group effects on neutrino
masses and mixings which can be large for a quasi-degenerate (QD) light neutrino mass
spectrum. A QD spectrum can arise in the models considered if ∆m 2A < 0 (i.e. thespectrum is with inverted ordering) and α . 0.2. However, this corresponds only to asmall portion of the parameter space of the models.
As has been already discussed in the literature, in the models of interest leptogenesis
is not possible at LO: the corresponding CP asymmetries  i vanish. Thus, the inclusion of
NLO effects is crucial for the generation of the baryon asymmetry Y B . More precisely, the
NLO effects correcting the neutrino Dirac mass matrix m D give rise to non-vanishing iand therefore - to non-vanishing YB . Due to this fact the CP asymmetries are naturally ofthe order of 10−6 (independent of the precise value of the loop function). Further, although
the AF and AM type models lead to the same results at LO, they differ at NLO so that
the CP asymmetries generated in the two models are different.
We find that it is possible to generate the correct size and sign of the baryon asymmetry
YB in the versions of both the AF and AM models we discuss. The study of leptogenesis
was performed in the framework of the one flavor approximation and by using analytic
formulae for the relevant efficiency factors η ii . Since the mass spectrum of the RH neutrinos
is generically not strongly hierarchical, the decays of all three RH (s)neutrinos contributeto the generation of the baryon asymmetry. We find that the correct magnitude as well as– 22 –

the correct sign of the baryon asymmetry Y B can be easily obtained in the AF and AM
type models for most values of the parameter α and natural values of the NLO couplings.
As already mentioned, the sign of YB uniquely fixes the sign of sin φ. The latter cannot be
determined by low energy observables since they exhibit only a cos φ-dependence.
To conclude, the results of our detailed study show that SUSY models with A 4 flavour
symmetry and type I see-saw mechanism of neutrino mass generation, which gives rise totri-bimaximal mixing and Majorana CP violation in the lepton sector, can account also
successfully for the observed baryon asymmetry of the Universe.

Acknowledgements
We thank Werner Rodejohann and Yasutaka Takanishi for discussions. C.H. would like tothank the Max-Planck-Institut für Kernphysik for kind hospitality during the last stages of
the work. This work was supported in part by the Italian INFN under the program “Fisica
Astroparticellare” and by the European Network “UniverseNet” (MRTN-CT-2006-035863).

– 23 –

A. Flavon Superpotential in the AF Type ModelIn the construction of the flavon superpotential we closely follow [8] and introduce an additional U (1)R symmetry under which driving fields have charge +2, superfields containing
SM fermions +1 and flavons, hu,d and FN field(s) are uncharged. To give a VEV of orderλ2c Λ to ζ we introduce a new driving field ζ 0 which is a singlet under all symmetries ofthe model, apart from carrying a U (1) R charge +2. The terms contributing to the flavon
superpotential containing ζ0 at LO read 9
wdζ = Mζ2 ζ0 + ga ζ0 ζ 2 + gb ζ0 (ϕT ϕT ) .

(A.1)

Analogously to the original model, we demand a vanishing F − term of ζ 0
Mζ2 + ga ζ 2 + gb (ϕ2T 1 + 2ϕT 2 ϕT 3 ) = 0 .

(A.2)

At the same time, the field ζ does not couple to the other driving fields, ϕ T0 ∼ (3, 1),
ϕS0 ∼ (3, ω 2 ) and ξ0 ∼ (1, ω 2 ) under (A4 , Z3 ), in the model at LO. Thus, their F −terms
read as in [8]. We find as solutionz2 = −


1Mζ2 + gb vT2
ga

(A.3)

and the same results for the VEVs of ϕ T , ϕS , ξ and ξ̃ as in [8]. For the mass parameterMζ being of order λ2c Λ the VEV z is also of order λ2c Λ.
Concerning the NLO contributions stemming from ζ to the alignment of the flavons
ϕT , ϕS , ξ and ξ˜ we find just one term

which gives an additional contribution

tz 2 T ζ ϕ0 ϕT
Λ

3tz
2gga

gb +

Mζ2
vT2

!

(A.4)

vT2
Λ

(A.5)

to the shift δvT 1 of ϕT . Its size is λ4c Λ, as expected. Furthermore, the shifts δv T 2,3 remainunchanged and thus still equal. The shifts in the vacuum of ϕ S and ξ̃ are also unchanged
and the VEV of ξ is still a free parameter.
The NLO terms affecting wdζ read8

∆wdζ =

1X Z
zi Ii
Λ

(A.6)

i=1

with
I1Z = ζ0 (ϕT ϕT ϕT ) ,I5Z = ζ0 ξ 3 ,

I2Z = ζ0 (ϕS ϕS ϕS ) ,I6Z = ζ0 ξ 2 ξ̃ ,I3Z = ζ0 ξ(ϕS ϕS ) ,I7Z = ζ0 ξ ξ˜2 ,9I4Z = ζ0 ξ̃(ϕS ϕS ) ,I8Z = ζ0 ξ̃ 3 .
(A.7)

Terms such as ζ0 hu hd are not relevant, since we assume that the flavor symmetry is broken much above
the electroweak scale.

– 24 –

The result for the shift in the VEV of ζ, z + δz, in the usual linear approximation, is
!
 3

2
gb g̃43gb tzga t3 Mζ vT3
u
g̃42
δz =
−
gb −
+ 2
t11 + 2 (t6 + t7 + t8 )2gg̃3 ga
zΛ
2gga2
tz
zΛ
3g̃3
vT
  3

vT
g̃42
1
u3
z1
+
−
(A.8)
z
+
z
3
5
2
2ga
u3
zΛ
3g̃3
with g4 = −g̃42 and g3 = 3g̃32 as introduced in [8]. This shift δz in hζi is of order λ 4c Λ.
Additionally, we find that - unless some non-trivial relation among the couplings in theflavon superpotential is fulfilled - the VEVs of all driving fields vanish at the minimum.

B. Flavon Superpotential in the AM Type Model
In order to induce a VEV for the flavon ζ we add a driving field ζ 0 which transforms as 10
under A4 , with −1 under Z4 and is invariant under the Z2 symmetry. Since it is responsible
for the vacuum alignment, its charge under the U (1) R symmetry is +2. The LO potential
for ζ0 is of the form
wdζ = ga ζ0 ζ 2 + gb ζ0 (ϕT ϕT )00 + gc ζ0 (ξ 0 )2 .
(B.1)
From the F -term of ζ0 we can derivega ζ 2 + gb (ϕ2T 2 + 2ϕT 1 ϕT 3 ) + gc (ξ 0 )2 = 0 .
Thus, z takes the value
1
1
gb vT2 + gc (u0 )2 = −
z =−
ga
ga
2




gb h21+ gc (u0 )2
4h22

(B.2)

(B.3)

so that z ∝ u0 holds in case of no accidental cancellations. u 0 is a free parameter in [9]which is taken to be of order λ2c Λ.
As one can check, the field ζ does not have renormalizable interactions with the driving
fields, ϕT0 ∼ (3, −1), ϕS0 ∼ (3, 1) and ξ0 ∼ (1, 1) under (A4 , Z4 ), of the original model. Thus,
the results for the vacuum alignment found in [9] still hold.
At NLO the field ζ contributes to the flavon superpotential of the original modelthrough
1 2 T
ζ (ϕ0 ϕS )0 ,(B.4)
Λ
while it does not introduce any contribution at this level involving ϕ S0 or ξ0 .
The NLO effects on the vacuum alignment of the field ζ stem from (order one coefficients are omitted)
1
1
1
1
1
ζ0 ζ 2 ξ + ζ0 (ϕT ϕT ϕS )00 + ζ0 ξ(ϕT ϕT )00 + ζ0 ξ 0 (ϕT ϕS )0 + ζ0 ξ 0 ξ 0 ξ .
(B.5)
Λ
Λ
Λ
Λ
Λ
Computing the effect of all NLO terms on the vacuum alignment one finds that still all
shifts δvSi are equal, i.e. the shifts do not change the structure of the vacuum, that thegeneric size of all shifts - for mass parameters and VEVs of order λ 2c Λ - is λ4c Λ and the freeparameter u0 is still undetermined.
Eventually, we checked that all driving fields can have a vanishing VEV at the minimum.

– 25 –

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– 27 –