Spontaneous leptonic CP violation and nonzero �13

G.C. Branco,1,* R. Gonzalez Felipe,1,2,† F. R. Joaquim,1,‡ and H. Serodio1,§

1Departamento de Fısica and Centro de Fısica Teorica de Partıculas, Instituto Superior Tecnico,Universidade Tecnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

2Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emıdio Navarro, 1959-007 Lisboa, Portugal(Received 11 May 2012; published 8 October 2012)

We consider a simple extension of the Standard Model by adding two Higgs triplets and a complex

scalar singlet to its particle content. In this framework, the CP symmetry is spontaneously broken at high

energies by the complex vacuum expectation value of the scalar singlet. Such a breaking leads to leptonic

CP violation at low energies. The model also exhibits an A4 � Z4 flavor symmetry which, after being

spontaneously broken at a high-energy scale, yields a tribimaximal pattern in the lepton sector. We

consider small perturbations around the tribimaximal vacuum alignment condition in order to generate

nonzero values of �13, as required by the latest neutrino oscillation data. It is shown that the value of �13recently measured by the Daya Bay Reactor Neutrino Experiment can be accommodated in our frame-

work together with large Dirac-type CP violation. We also address the viability of leptogenesis in our

model through the out-of-equilibrium decays of the Higgs triplets. In particular, the CP asymmetries

in the triplet decays into two leptons are computed and it is shown that the effective leptogenesis and

low-energy CP-violating phases are directly linked.

DOI: 10.1103/PhysRevD.86.076008 PACS numbers: 11.30.Hv, 14.60.Pq, 98.80.Bp

I. INTRODUCTION

The solid evidence for neutrino oscillations and, con-sequently, for nonzero neutrino masses and mixing, hasestablished leptogenesis [1,2] as one of the most appealingmechanisms to explain the matter-antimatter asymmetryobserved today in our Universe. The most attractive featureof this mechanism relies on the fact that the interactionsrelevant for leptogenesis can simultaneously be respon-sible for the nonvanishing and smallness of neutrinosmasses, once the well-known seesaw mechanism is in-voked. Among the canonical seesaw realizations, the so-called triplet (or type II) seesaw [3] is probably the mosteconomical in the sense that it counts with a single sourceof flavor structure, namely, the symmetric complexYukawa coupling matrixY� that couples the SUð2ÞL scalartriplet � to leptons. Furthermore, in its minimal realiza-tion, with only one scalar triplet, the flavor pattern of Y�

uniquely determines the flavor structure of the low-energyeffective neutrino mass matrix m�. In this regard, it isworth recalling that the current solar and atmosphericneutrino oscillation data are consistent with the so-calledtribimaximal (TBM) leptonic mixing pattern [4], whichfrom the theoretical point of view seems to call for adiscrete family symmetry. Along this line many modelshave been proposed, with a vast majority relying on the A4

symmetry (for recent reviews on flavor symmetries seee.g., Refs. [5,6]).

Unfortunately, leptogenesis cannot be successfully im-plemented in the minimal triplet seesaw scenario. The CPasymmetry induced by the triplet decays is generatedbeyond the one-loop level and, consequently, is highlysuppressed. Thus, new sources for neutrino masses arenecessary to induce the required lepton asymmetry viathe out-of-equilibrium triplet decays [7–11]. In the lattercase, estimates of the thermal leptogenesis efficiency [8,9]as well as a more precise calculation of it by solving thefull set of Boltzmann equations [12] indicate that success-ful leptogenesis typically requires the lightest triplet massto be large enough, M� * 109 GeV, if no extra sources ofCP violation are present and lepton flavor effects are nottaken into account.A crucial ingredient of any dynamical mechanism which

aims at explaining the baryon asymmetry is the violation ofthe CP symmetry. For scalar triplet leptogenesis, the com-plex Yukawa couplings of the Higgs triplets to leptons, aswell as their complex couplings to the standard Higgsdoublet, usually provide the necessary source of (explicit)CP violation. Alternatively, the required amount of CPviolation can be generated if CP is spontaneously brokenby the complex vacuum expectation value (VEV) of ascalar field. Besides being more attractive from a theoreti-cal viewpoint, the latter framework is also more economi-cal since both the CP violation necessary to generate thebaryon asymmetry and leptonic CP violation [13] poten-tially observable at low energies come from a commonsource, namely, from the phase of the scalar field respon-sible for the spontaneous CP breaking at a high energyscale [14].The purpose of this work is to present a simple scenario

where the above three aspects (leptogenesis, leptonic

*gbranco@ist.utl.pt†ricardo.felipe@ist.utl.pt‡filipe.joaquim@ist.utl.pt§hserodio@cftp.ist.utl.pt

PHYSICAL REVIEW D 86, 076008 (2012)

1550-7998=2012=86(7)=076008(10) 076008-1 � 2012 American Physical Society

mixing and spontaneous CP violation) are related. To thisaim, we shall add to the Standard Model (SM) a minimalparticle content: two Higgs triplets �aða ¼ 1; 2Þ with unithypercharge and a complex scalar singlet S with zerohypercharge. In our framework, neutrinos acquire massesvia the well-known type II seesaw mechanism, imple-mented by the two scalar triplets, and leptogenesis be-comes viable due to the out-of-equilibrium decays of thelatter in the early universe. Furthermore, we shall assumethat the CP symmetry is spontaneously broken at highenergies by the complex VEV of the singlet S, leading tothe CP violation required for leptogenesis as well asto low-energy leptonic CP violation. By imposing anA4 � Z4 flavor symmetry which is spontaneously brokenat a high scale with a specific vacuum configuration, theTBM lepton mixing is obtained. Nonzero values of �13,required by the neutrino oscillation data from T2K [15] andMINOS [16], arise once we consider small perturbationsaround the vacuum alignment configuration which leads toexact TBM mixing. We show that in our framework thevalue of �13 measured by the Daya Bay Reactor NeutrinoExperiment [17] can be obtained, leading to large Dirac-type CP violation. Finally, we discuss how leptogenesis isrealized in order to show that sizable CP asymmetries canbe obtained. Moreover, the effective leptogenesis phasecan be directly linked to the low-energy leptonicCP-violating phases.

II. THE MODEL: PARTICLECONTENTAND SYMMETRIES

Let us consider the SM extended with two Higgstriplets �aða ¼ 1; 2Þ of unit hypercharge and a complexscalar singlet S with zero hypercharge. In the SUð2Þrepresentation:

�a ¼�0

a ��þa =

ffiffiffi2

p

��þa =

ffiffiffi2

p�þþ

a

!: (1)

We impose CP invariance at the Lagrangian level andintroduce a Z4 symmetry under which the scalar and leptonfields transform as follows:

S ! �S; � ! i�; �1 ! �1;

�2 ! ��2; Lj ! iLj; eRj ! �ieRj; (2)

where � ¼ �þ �0

� �is the standard Higgs doublet of

SUð2ÞL; Lj and eRj (j ¼ 1), 2, 3 are the SM lepton doublets

and right-handed singlets, respectively. The remainingfields transform trivially under the Z4 symmetry. As itturns out, the above symmetry assignment yields a quitesimple and predictive scenario.The most general scalar potential invariant under the

above symmetries can be written as

VCP�Z4 ¼ VS þ V� þ V� þ VS� þ VS� þ V�� þ VS��; (3)

where

VS ¼ �2SðS2 þ S�2Þ þm2

SS�Sþ �0

SðS4 þ S�4Þ þ �00SS

�SðS2 þ S�2Þ þ �SðS�SÞ2;V� ¼ m2

��y�þ ��ð�y�Þ2;

V� ¼ Xa

M2aTrð�y

a�aÞ þXa;b

½ð��ÞabTrð�ya�aÞTrð�y

b�bÞ þ ð�0�ÞabTrð�y

a�a�yb�b�;

VS� ¼ �SðS�SÞð�y�Þ þ �0SðS2 þ S�2Þð�y�Þ;

VS� ¼ Xa

Trð�ya�aÞ½�aðS2 þ S�2Þ þ �aS

�S�;

V�� ¼ Xa

½�0að�y�ÞTrð�y

a�aÞ þ �00að�y�y

a�a�Þ� þ ð�2M2~�T�2

~�þ H:c:Þ;

VS�� ¼ ~�T�1~�ð�1Sþ �0

1S�Þ þ H:c:; (4)

and ~� ¼ i�2��. Since CP invariance has been imposed at

the Lagrangian level, all the parameters are assumed to bereal. Notice that the choice of the Z4 symmetry given inEq. (2) forbids the term �1M1

~�T�1~� which is crucial for

the mechanism of leptogenesis to be viable in the presentmodel. Yet, once the singlet S acquires a complex VEV,this term is generated from the scalar potential contributionVS�� in Eq. (4).

In order to generate a realistic lepton mixing pattern, weshall also impose an A4 discrete symmetry at high energies(see discussion in Section IV). We recall that, in a particu-lar basis, the Clebsch-Gordan decompositions of the A4

group are 10 � 100 ¼ 1 and 3 � 3 ¼ 1 � 10 � 100 � 3s � 3a.Moreover, the Clebsch-Gordan coefficients for the productof two triplet fields with components a1;2;3 and b1;2;3 are

given by

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ða � bÞ1 ¼ a1b1 þ a2b3 þ a3b2;

ða � bÞ10 ¼ a3b3 þ a1b2 þ a2b1;

ða � bÞ100 ¼ a2b2 þ a1b3 þ a3b1;

ða � bÞ3s ¼1

3ð2a1b1 � a2b3 � a3b2; 2a3b3 � a1b2

� a2b1; 2a2b2 � a1b3 � a3b1Þ;ða � bÞ3a ¼

1

2ða2b3 � a3b2; a1b2 � a2b1; a1b3 � a3b1Þ;

(5)

while the symmetric product of three triplets reads

ðða � bÞ3s � cÞ1 ¼ 1

3ð2a1b1c1 þ 2a2b2c2 þ 2a3b3c3

� a1b2c3 � a1b3c2 � a3b1c2 � a3b2c1

� a2b1c3 � a2b3c1Þ: (6)

The spontaneous breaking of the A4 symmetry is thenguaranteed by adding to the theory two extra heavy scalar

fields, � and �, with a suitable VEV alignment. Thecomplete symmetry assignments of the fields underA4 � Z4 and SUð2ÞL �Uð1ÞY are given in Table I.Below the cutoff scale�, the flavor dynamics is encoded

in the relevant effective Yukawa Lagrangian L, whichcontains the lowest-order terms1 in an expansion in powersof 1=�,

L ¼ y‘e�ð �L�Þ1�eR þ y‘�

�ð �L�Þ100��R þ y‘

�ð �L�Þ10�R

þ y2��2ðLTL�Þ1 þ 1

��1ðLTLÞ1ðy1Sþ y01S�Þ þ H:c::

(7)

As soon as the heavy scalar fields develop VEVs along therequired directions,2 namely,

h�i ¼ ðr; 0; 0Þ; h�i ¼ ðs; s; sÞ; (8)

and the scalar singlet S acquires a complex VEV, hSi ¼vSe

i, the leptonic mass Lagrangian becomes

�L ¼ y‘er

��Le�eR þ y‘�r

��L���R þ y‘r

��L�R þ y2s

3��2ð2LT

eLe þ 2LT�L� þ 2LT

L � LTeL�

� LTeL � LT

�Le � LT�L � LT

Le � LTL�Þ þ vS

�0 �1ðLTeLe þ LT

�L þ LTL�Þðy1ei þ y01e

�iÞ þ H:c:

� Ye�

�L�eR� þ Y�1

�LTC�1L� þ Y�2

�LTC�2L� þ H:c:; (9)

with

Ye ¼ye 0 0

0 y� 0

0 0 y

0BB@

1CCA; Y�1 ¼ y�1

1 0 0

0 0 1

0 1 0

0BB@

1CCA;

Y�2 ¼ y�2

3

2 �1 �1

�1 2 �1

�1 �1 2

0BB@

1CCA (10)

and

ye;�;¼ r

�y‘e;�;; y�1

¼vS

�0 ðy1eiþy01e�iÞ; y�2

¼y2�s:

(11)

Notice that the Yukawa matrices Y�1 and Y�2 exhibit theso-called �� and magic symmetries, respectively.

III. SPONTANEOUS CP VIOLATION

In the present model, CP is conserved at the Lagrangianlevel because all the parameters are set to be real. Yet, thissymmetry can be spontaneously broken by the complexVEV of the scalar singlet S. To show that this is indeedthe case, let us analyze the scalar potential for S. Assumingthat this field is very heavy and decouples from the theory atan energy scale much higher than the electroweak and

TABLE I. Representations of the fields under the A4 � Z4 and SUð2ÞL �Uð1ÞY symmetries.

Field L eR, �R, R �1 �2 � S � �

A4 3 1, 10, 100 1 1 1 1 3 3Z4 i �i 1 �1 i �1 i 1

SUð2ÞL �Uð1ÞY (2, �1=2) (1, �1) (3, 1) (3, 1) (2, 1=2) (1, 0) (1, 0) (1, 0)

1In principle, one could also include the renormalizable 4-dimension term �2L

TL. This term is, however, easily removedby imposing an additional shaping Z4 symmetry.

2Following the standard procedure, we will assume the typicalvacuum alignment for this class of models. The way how thisvacuum configuration is achieved from the minimization of thecomplete flavon potential is out of the scope of the present work.

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Higgs-triplet scales, the relevant terms in the scalar poten-tial are simply those given by the contribution VS in Eq. (4).The tree-level potential then reads

V0 ¼ m2Sv

2S þ �Sv

4S þ 2ð�2

S þ �00Sv

2SÞv2

S cosð2Þþ 2�0

Sv4S cosð4Þ: (12)

Its minimization with respect to vS and yields theequations

@V0

@vS

¼ 2vS½m2S þ 2�Sv

2S þ 2ð�2

S þ 2�00Sv

2SÞ cosð2Þ

þ 4�0Sv

2S cosð4Þ� ¼ 0; (13)

@V0

@¼ �4v2

s sinð2Þ½ð�2S þ �00

Sv2SÞ

þ 4�0Sv

2S cosð2Þ� ¼ 0: (14)

Besides the trivial solution vS ¼ 0, which leads toV0 ¼ 0, there are other three possible solutions to theabove system of equations:

ðiÞ v2S ¼� m2

Sþ2�2S

2ð�Sþ2�0Sþ2�00

SÞ; ¼ 0;��;

ðiiÞ v2S ¼

�m2Sþ2�2

S

2ð�Sþ2�0S�2�00

SÞ; ¼��

2;

ðiiiÞ v2S ¼

�2�0Sm

2Sþ�00

S�2S

4�S�0S�8�02

S ��002S

; cosð2Þ¼��2Sþ�00

Sv2S

4�0Sv

2S

:

(15)

Note that in spite of the phase �=2 in (ii), in this case thevacuum does not violate CP [18]. Therefore, only the lastsolution is of interest to us since it leads not only to thespontaneous breaking of the CP symmetry but also to anontrivial CP-violating phase in the one-loop diagramsrelevant for leptogenesis. One can show that this solutionindeed corresponds to the global minimum of the potentialin a wide region of the parameter space. To illustrate this,let us consider m2

S < 0, �00S ’ 0, �S ’ 0 and �S > 2�0

S > 0.From Eqs. (15) we then obtain

v2S ’ � m2

S

2ð�S þ 2�0SÞ; ¼ 0;��

2;��; (16)

for the cases (i) and (ii), leading to

V0 ’ � m4S

4ð�S þ 2�0SÞ: (17)

In turn, solution (iii) yields

v2S ’ � m2

S

2ð�S � 2�0SÞ; ’ ��=4; (18)

implying

V0 ’ � m4S

4ð�S � 2�0SÞ: (19)

Clearly, the latter value corresponds to the absolute mini-mum of the potential. One can also easily show that bothmass eigenvalues are positive within the assumed parame-ter region: M2

S;1 ¼ �4m2S > 0 and M2

S;2 ¼ �16m2S�

0S=

ð�S � 2�0SÞ> 0.

IV. SEESAW MECHANISM, NEUTRINOMASSES AND LEPTONIC MIXING

In the present framework, neutrinos acquire massesthrough the well-known type II seesaw mechanism due tothe tree-level exchange of the heavy scalar triplets �a.From Eq. (9) it is straightforward to see that the effectiveneutrino mass matrix m� is given by

m � ¼ mð1Þ� þmð2Þ

� ; mðaÞ� ¼ 2uaY

�a ; (20)

where ua ¼ ��av

2=Ma are the VEVs of the neutral compo-nents �0

a of the scalar triplets, h�0i ¼ v ¼ 174 GeV and,in accordance with Eq. (4), �1¼ð�1e

iþ�01e

�iÞvS=M1.The Yukawa matrices Y�a are those already presented inEq. (10). Diagonalizingm� we obtain

m� ¼ U�d�Uy;

d� ¼ diagðjz1ei� þ z2j; z1; jz1ei� � z2jÞ� diagðm1; m2; m3Þ; (21)

where mi are the neutrino masses and

za¼2juay�aj; �¼ argðu1y�1

Þ

¼ arctan

��01��1

�1þ�01

tan

�þarctan

�y1�y01y1þy01

tan

�: (22)

Hereafter, we consider the relevant CP-violating phase asbeing �. The unitary matrix U is given by

U ¼ e�i�1=2UTBMK; (23)

where UTBM is the TBM mixing matrix,

UTBM ¼

2ffiffi6

p 1ffiffi3

p 0

� 1ffiffi6

p 1ffiffi3

p � 1ffiffi2

p

� 1ffiffi6

p 1ffiffi3

p 1ffiffi2

p

0BBBB@

1CCCCA; (24)

and

K ¼ diagð1; ei 1 ; ei 2Þ; 1 ¼ ð�1 � �Þ=2; 2 ¼ ð�1 � �2Þ=2; �1;2 ¼ argðz2 � z1e

i�Þ: (25)

Since at this point there is no Dirac-type CP violation(U13 ¼ 0), the Majorana phases 1;2 are the only source

of CP violation in the lepton sector.Let us now discuss how the experimental knowledge

on the neutrino mass squared differences constrains the

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076008-4

parameters z1 and z2. At 1� confidence level, the neutrinomass squared differences are [19]

�m221 ¼ ð7:59þ0:20

�0:18Þ � 10�5 eV2;

�m231 ¼ ð2:50þ0:09

�0:16Þ½�2:40þ0:09�0:08� � 10�3 eV2; (26)

for the normal [inverted] neutrino mass hierarchy. The signof the neutrino mass difference (m3 �m2) is dictated bythe ordering of the neutrino masses: positive for normalordering (m3 >m2 >m1) and negative for inverted order-ing (m3 <m2 <m1). In the first case, Eq. (21) togetherwith the condition m3 >m2 >m1 implies the constraints

z2 þ 2z1 cos�< 0; z2 � 2z1 cos�> 0; (27)

which, clearly, are satisfied only if �=2<�< 3�=2. Onthe other hand, for m3 <m2 <m1 one has

z2 þ 2z1 cos�< 0; z2 � 2z1 cos�< 0; (28)

which cannot be simultaneously fulfilled. Thus, the presentmodel cannot accommodate an inverted hierarchy for theneutrino mass spectrum.

The parameters z1 and z2 can be written in terms of�m2

21, �m231 and the angle � as

z1 ¼ � 1

2 cos�

�m231ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ð�m231 � 2�m2

21Þq ’ � 1

2 cos�

ffiffiffiffiffiffiffiffiffiffiffiffi�m2

31

2

s;

z2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2

31 � 2�m221

2

s’

ffiffiffiffiffiffiffiffiffiffiffiffi�m2

31

2

s; (29)

where the last equalities in the right-hand sides wereobtained using the fact that �m2

31 �m221. We notice

that z2 is completely fixed by the atmospheric neutrinomass squared difference. Moreover, z1 ’ �z2=ð2 cos�Þ.Using the best fit values given in Eqs. (26), we obtain

z1’�0:0175=cos�0:0175 eV; z2’0:035 eV: (30)

In turn, the neutrino masses defined in Eq. (21) can berewritten as

m1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz21 ��m2

21

q; m2 ¼ z1;

m3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz21 þ �m2

31 � �m221

q: (31)

Thus, the model predicts a lower bound for the lightestneutrino mass: m1 * 1:5� 10�2 eV. The neutrino masshierarchy is maximal when � ¼ �, while an almost de-generate spectrum is obtained for � ’ �=2 or � ’ 3�=2.Finally, the Majorana phases are approximately given by

1 ’ ��; 2 ’ ��

2� 1

2arctan

�tan�

3

�: (32)

The dependence of neutrino masses on the high-energyphase � is presented in Fig. 1 for the exact TBM case. Thelight red shaded area is currently disfavored by the recentWMAP seven-year cosmological observational data [20].

Although WMAP alone constrains the sum of light neu-trino masses below 1.3 eV (95% CL), when combined withbaryonic acoustic oscillation and type-Ia supernova datathis bound is more restrictive:

Pimi < 0:58 eV at 95% CL.

A. Vacuum-alignment perturbations and nonzero �13

The T2K [15] and MINOS [16] neutrino oscillation dataimply for the �13 mixing angle

sin 2�13 ¼ 0:013þ0:007�0:005ðþ0:015

�0:009Þ½þ0:022�0:012�; (33)

at 1�ð2�Þ½3��. Recently, through the observation ofelectron-antineutrino disappearance, the Daya BayReactor Neutrino Experiment has also measured the non-zero value [17]:

sin 2ð2�13Þ ¼ 0:092� 0:016ðstatÞ � 0:005ðsystÞ; (34)

with a significance of 5:2�. In light of these results, modelsthat lead to TBM mixing appear to be disfavored. Ingeneral, deviations from �13 ¼ 0 in these models cannotbring this angle into agreement with data without spoilingthe predictions for the solar and atmospheric mixing an-gles. Several alternative solutions have been recently putforward to explain a relatively large value of �13 in theframework of discrete flavor symmetries (see, for instance,Refs. [21–29]).Here we shall follow an approach based on considering

small perturbations around the TBM vacuum-alignmentconditions (8). Such perturbations could come from thepresence of higher dimensional operators in the flavonpotential. We consider two distinct cases:

Case A: Small perturbations around the flavon VEVh�i ¼ ðr; 0; 0Þ of the form h�i ¼ rð1; "1; "2Þ;

Case B: Small perturbations around the flavon VEVh�i ¼ sð1; 1; 1Þ of the form h�i¼sð1;1þ"1;1þ"2Þ; with j"1;2j � 1.

Since h�i is not perturbed in case A, the contribution tothe lepton mixing coming from the neutrino sector remains

m1

m2

m3

Exc

lude

dby

cosm

olog

ical

boun

d

5 4 3 210 2

10 1

1

mi

eV

FIG. 1 (color online). Neutrino masses mi as a function of thehigh-energy phase � in the exact TBM case.

SPONTANEOUS LEPTONIC CP VIOLATION AND . . . PHYSICAL REVIEW D 86, 076008 (2012)

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of TBM type [cf., Eq. (7)]. On the other hand, due to thenew form of h�i, the charged lepton Yukawa matrix is

Y‘ ¼ye y"1 y�"2

y"2 y� ye"1

y�"1 ye"2 y

0BB@

1CCA; (35)

which implies U‘ � 1, where U‘ is the unitary matrixwhich rotates the left-handed charged-lepton fields to the

their physical basis. The new lepton mixing matrix U ¼Uy

‘UTBM yields the perturbed mixing angles

sin2�12 ’ 1

3½1� 2ð"1 þ "2Þ�; sin2�23 ’ 1

2ð1þ 2"1Þ;

sin2�13 ’ ð"1 � "2Þ22

; (36)

at lowest order in "1;2. This leads to the following approxi-mate relation among the three lepton mixing angles

sin 2�13 ’ ð4sin2�23 � 3cos2�12Þ28

; (37)

compatible with neutrino data at the 1� level. Obviously,the rotation of the charged lepton fields does not affect theneutrino spectrum nor generate a Dirac-type CP-violatingphase. Since the flavon fields are real, the Majorana phases 1;2 also remain unaltered.

In Fig. 2 (left panel) we present the allowed regions inthe ("1, "2) plane taking into account the present neutrinooscillation data, at 1� (black), 2� (red) and 3� (green). Asit is apparent from the figure, with ð"1; "2Þ ’ ð�0:05; 0:1Þor ð"1; "2Þ ’ ð0:05;�0:05Þ one obtains agreement with thedata at the 1� level.

Neutrinoless double beta decay (0���) is an importantlow-energy process [30] which, if observed, will establishthe Majorana nature of neutrinos. The rate of this process isproportional to the modulus of the (11) entry of the effec-tive neutrino mass matrix, in the weak basis where thecharged-lepton mass matrix is diagonal and real. In ourframework, its value is given by

jmeej ¼��������Xi

miU21i

��������¼ 1

3j2m1ð1þ "1 þ "2Þ þm2e

�2i 1 j;

(38)

in leading order of "1;2. Although with large uncertainties

from the poorly known nuclear matrix elements, dataavailable at present set an upper bound on jmeej in therange 0.2–1 eV at 90% C.L. [31–33]. The existing limitswill be considerably improved in the forthcoming experi-ments, with an expected sensitivity of about 10�2 eV [34].In Fig. 3 we present the dependence of jmeej as a functionof the phase � for case A (scatter points) takinginto account the present neutrino oscillation data, at 1�(black), 2� (red) and 3� (green). We obtain jmeej ’ð0:004–0:2Þ eV, where the upper limit comes from thecosmological bound and it corresponds to an almost de-generate neutrino spectrum. We notice that the predictionsfor the effective Majorana mass parameter jmeej are withinthe reach of future experiments [30] and the model can beruled out if 0��� searches give jmeej< 4 meV.

FIG. 2 (color online). Allowed regions in the ("1, "2) plane corresponding to the VEV perturbations of the flavon field h�i ¼rð1; "1; "2Þ in case A (left panel) and h�i ¼ sð1; 1þ "1; 1þ "2Þ in case B (right panel). The scatter points were obtained consideringthe 1� (black), 2� (red) and 3� (green) neutrino oscillation data.

Future sensitivity region

Exc

lude

dby

cosm

olog

ical

boun

d

321

Case A TBM Case B

10 3

10 2

10 1

1

mee

eV

FIG. 3 (color online). Neutrinoless double beta decay parame-ter jmeej as a function of the high-energy phase � for case A(scatter points) and TBM and case B (cyan solid line).

BRANCO et al. PHYSICAL REVIEW D 86, 076008 (2012)

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We now discuss case B, in which h�i ¼ ðr; 0; 0Þ andsmall perturbations around the flavon VEV h�i ¼sð1; 1; 1Þ of the type h�i ¼ sð1; 1þ "1; 1þ "2Þ are con-sidered. This will induce corrections to the mixing comingfrom the neutrino sector. Notice that one can alternativelyconsider perturbations of the type h�i ¼ s0ð1þ "01; 1; 1þ"02Þ or h�i ¼ s00ð1þ "001 ; 1þ "002 ; 1Þ. The results obtainedbelow are obviously invariant under the choice of theperturbation, provided one takes into account the mappingof fs0; s00; "0i; "00i g into fs; "ig. At leading order this mappingcorresponds to s0 ¼ s00 ¼ sð1þ "1Þ, "01 ¼ �"1, "02 ¼"2 � "1, "

001 ¼ "2 � "1 and "001 ¼ �"1.

The Yukawa couplings Y�2 contributing to the neutrinomass matrix are now given by

Y�2 ¼ y�2

3

2 �1� "2 �1� "1

�1� "2 2þ 2"1 �1

�1� "1 �1 2þ 2"2

0BB@

1CCA: (39)

Consequently, at first order in "1;2, the neutrino mass

spectrum gets corrected in the following way

m21 ’

1

3½3z21 þ z22ð3þ 2"1 þ 2"2Þþ z1z2ð3þ "1 þ "2Þ cos��;

m22 ’ z21;

m23 ’

1

3½3z21 þ z22ð3þ 2"1 þ 2"2Þ� z1z2ð3þ "1 þ "2Þ cos��: (40)

As in the unperturbed case, it can be shown from the aboveequations that an inverted neutrino hierarchy is notallowed.

In the present case, the approximate analytic expressionsfor the mixing angles are

sin2�12 ’ 1

3þ 2

9ð"1 þ "2Þ; sin2�13 ’ ð"1 � "2Þ2

72cos2�;

sin2�23 ’ 1

2þ 1

6ð"1 � "2Þ; (41)

while for the Dirac-type CP-violating invariant JCPwe have

JCP ¼ Im½U11U22U�12U

�21� ’

"2 � "136

tan�:

Using the standard form JCP ¼ sinð2�12Þ sinð2�13Þ�sinð2�23Þ sin�=8 together with the relations (41), we getsin� ’ sin�, which means that the Dirac CP-violatingphase � is directly related with the phase of the singletVEV hSi, as shown in Eq. (22). The numerical results forthe allowed regions in the ("1, "2) plane in case B areshown in Fig. 2 (right panel). By comparing both panels inthe figure, it is clear that case B is less restrictive than caseA. We also note that the predictions for 0��� are exactlythe same as for the TBM case (cyan solid line in Fig. 3) sothat mee is obtained from Eq. (38) in the limit "1;2 ¼ 0.We now comment on the possibility of reproducing the

recent Daya Bay �13 value (34) in our framework. In theabsence of a 3-neutrino global analysis of the oscillationdata including the Daya Bay results, we take the 1� valuesfor �12, �23 and �m2

21;31 obtained in Ref. [19]. From

Eq. (37) one can see that the new Daya Bay value for �13is not compatible with the remaining mixing angles forcase A. Instead, for case B we get a perfect agreement withall data. This is apparent from Fig. 4 (left panel) where theallowed regions in the ("1, "2) plane are shown. In the rightpanel of the same figure the predictions for jJCPj are shownas a function of the CP-violating phase�which, as alreadymentioned, is approximately equal to the Dirac phase in thelepton mixing matrix. From these results we concludethat our framework predicts 11�=8 & � & 3�=2 with0:03 & jJCPj & 0:04, which is large enough to be mea-sured in future oscillation experiments.

V. HIGGS TRIPLET DECAYS AND LEPTOGENESIS

The mechanism of leptogenesis can be naturally realizedin the present model due to the presence of the scalartriplets �1 and �2. At tree level, the latter can decay intotwo leptons or two Higgs fields (cf., Fig. 5). In the presenceof CP-violating interactions, the decay of �a into two

0.2 0.1 0.0 0.1 0.20.2

0.1

0.0

0.1

0.2

1

2

Exc

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dby

cosm

olog

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boun

d

2.8

3.0

3.2

3.4

3.6

3.8

4.0

J CP

102

FIG. 4 (color online). Left: allowed regions in the ("1, "2) plane corresponding to the VEV perturbations of the flavon field� in caseB, taking into account the Daya Bay result for �13. The corresponding regions in the ðjJCPj; �Þ plane are shown in the right panel.

SPONTANEOUS LEPTONIC CP VIOLATION AND . . . PHYSICAL REVIEW D 86, 076008 (2012)

076008-7

leptons generates a nonvanishing leptonic asymmetry foreach triplet component (�0

a, �þa , �

þþa ),

�a¼2X�

�ð��a!LþL�Þ��ð�a! �Lþ �L�Þ

��aþ���

a

; (42)

where ��adenotes the total triplet decay width and the

overall factor of 2 arises because the triplet decay producestwo leptons. It is useful to define

BLa��a

� X;�

�ð��a ! L þ L�Þ ¼ Ma

8�TrðY�ayY�aÞ;

B�a ��a

� �ð��a ! �þ�Þ ¼ Ma

8�j�aj2;

(43)

where BLa and B�

a are the tree-level branching ratios toleptons and Higgs doublets, respectively. The total tripletdecay width is given by

��a¼ Ma

8�½TrðY�ayY�aÞ þ j�aj2�: (44)

When the triplet decays into leptons with given flavors

L and L�, a nonvanishing asymmetry ��a is generated

by the interference of the tree-level decay process withthe one-loop self-energy diagram shown in Fig. 5. Onefinds [13]

��a ’�gðxbÞ2�

c�Im½��a�bY

�a

�Y�b�� �

TrðY�ayY�aÞþj�aj2; ðb�aÞ; (45)

where

c� ¼8<: 2� �� for �0

a;�þþa

1 for �þa

; (46)

xb ¼ M2b=M

2a, and the one-loop self-energy function gðxbÞ

reads

gðxbÞ ¼ffiffiffiffiffixb

p ð1� xbÞðxb � 1Þ2 þ ð��b

=MaÞ2: (47)

Recalling that the effective light neutrino mass matrix isgiven by Eq. (20) in the type-II seesaw framework underdiscussion, Eq. (45) can be rewritten as

��a ’�gðxbÞ4�

MbðBLaB

�a Þ1=2

v2

c�Im½mðaÞ�;�m

��;��

½TrðmðaÞy� mðaÞ

� Þ�1=2: (48)

In the hierarchical limit Ma � Mb, it reduces to

��a ’ MaðBLaB

�a Þ1=2

4�v2

c�Im½mðaÞ�;�m

��;��

½TrðmðaÞy� mðaÞ

� Þ�1=2; (49)

which, after summing over the final lepton flavors, yieldsthe following expression for the unflavored asymmetry[12,35]:

�a ¼ MaðBLaB

�a Þ1=2

4�v2

Im½TrðmðaÞ� my

�Þ�½TrðmðaÞy

� mðaÞ� Þ�1=2

: (50)

One can then show that the following upper boundholds [12]:

j�aj � MaðBLaB

�a Þ1=2

4�v2½Trðmy

�m�Þ�1=2

¼ MaðBLaB

�a Þ1=2

4�v2

�Xk

m2k

�1=2

: (51)

This upper bound increases as the light neutrino mass scaleincreases. For hierarchical light neutrinos one obtains:

j�aj & 10�6ðBLaB

�a Þ1=2

�Ma

1010 GeV

�� ffiffiffiffiffiffiffiffiffiffiffiffi�m2

31

q0:05 eV

�: (52)

Clearly, the absolute maximum of the right-hand sides of

Eqs. (51) and (52) is obtained when BLa ¼ B�

a ¼ 1=2.Nevertheless, since the efficiency of leptogenesis is dic-tated by the solution of the relevant Boltzmann equations,the final baryon asymmetry is not necessarily maximal insuch a case.Due to the specific flavor structure of the Yukawa cou-

pling matrices Y�a given in Eqs. (10), the quantity

TrðmðaÞ� mðbÞy

� Þ vanishes in the exact TBM case. This con-clusion also holds when the flavon VEV perturbationscorresponding to cases A and B are considered.Therefore, the leptogenesis asymmetry defined inEq. (50) is equal to zero and unflavored leptogenesis issuppressed.3 This means that for leptogenesis to be viable

FIG. 5. Tree-level diagrams for the scalar triplet decays and one-loop diagram contributing to the CP asymmetry ��a .

3Unlike the type-I seesaw framework [36–38], imposing onthe Lagrangian a discrete symmetry, such as the A4 symmetry,would not necessarily lead to the vanishing of the leptogenesisasymmetry in the type II seesaw case [39]. Indeed, even if both

matrices, mðaÞ� and mðbÞ

� , are diagonalized by the TBM mixing

matrix UTBM, i.e., mðaÞ� ¼ UTBMK

�ad

ðaÞ� K�

aUTTBM, the quantity

Im½TrðmðaÞ� mðbÞy

� Þ� ¼ ImfTr½ðK�aKbÞ2dðaÞ

� dðbÞ� �g in general does

not vanish.

BRANCO et al. PHYSICAL REVIEW D 86, 076008 (2012)

076008-8

in our framework it must take place in the flavored regime.In particular, for T & 1012 GeV (T & 109 GeV) interac-tions involving the (�) Yukawa coupling are in thermalequilibrium and the corresponding lepton doublet is adistinguishable mass eigenstate.

AssumingMa � Mb and using Eqs. (10), (20), (22), and(43), the CP asymmetry given in Eq. (49) for each tripletcomponent can be rewritten as

��a ¼ c�Pa��

0a; (53)

where c� is defined in Eq. (46), and

�0a ¼ 1

3�

zazbjuaj2M2a sin�

z2atav4 þ 4juaj4M2

a

; (54)

with t1 ¼ 3 and t2 ¼ 2. The matrix Pa is given by

Pa ¼ ð�1Þa2

�2ð1þ "1 þ "2Þ "1 � "2 "2 � "1

"1 � "2 4ð"1 þ "2y2�=y

2Þ 1þ "1 þ "2

"2 � "1 1þ "1 þ "2 �4ð"1 þ "2y2�=y

2Þ

0BB@

1CCA; (55)

for case A, while

Pa ¼ ð�1Þa�1

2þ �a2

v4z22ð"1 þ "2Þ18M2

2u42 þ 9v4z22

� �2 0 0

0 0 1

0 1 0

0BB@

1CCA; (56)

in case B. Obviously, in the TBM limit ("1;2 ¼ 0), there is aunique matrix Pa. In this case, the flavor structure of Pa

dictates that the only allowed decay channels of�a are intothe ee and � flavors. Once the VEV perturbations areintroduced, new decay channels are opened in case Awiththe corresponding CP asymmetries suppressed by Oð"Þfactors.

Maximizing �0a with respect to the VEVof the decayingscalar triplet ua, one obtains

�01;max’M1

ffiffiffiffiffiffiffiffiffiffiffiffi�m2

31

q12

ffiffiffi6

p�v2

sin�; �02;max’M2

ffiffiffiffiffiffiffiffiffiffiffiffi�m2

31

q48�v2

tan�: (57)

In Fig. 6 we present the contours of j�0a;maxj in the (�, M)-

plane. One can see that sufficiently large values of the CP

asymmetries can be obtained in the flavored regime.Clearly, a more rigorous study which accounts for washouteffects would be necessary in order to estimate the finalvalue of the baryon asymmetry. Such analysis is beyondthe scope of the present work and will be presentedelsewhere.

VI. CONCLUSIONS

In this work we have presented an appealing scenariowhere spontaneous CP violation, leptonic mixing andthermal leptogenesis are related in a simple way. Weadded a minimal particle content to the SM, namely,two Higgs triplets �1;2 and a complex scalar singlet S.In this framework, neutrinos acquire masses via thetype II seesaw mechanism, implemented by the twoscalar triplets. Furthermore, assuming that the CP sym-metry is spontaneously broken at high energies by thecomplex VEV of the singlet scalar, the CP violationnecessary for leptogenesis as well as nonvanishing lep-tonic CP violation at low energies are obtained. Themodel also exhibits a spontaneously broken A4 � Z4

flavor symmetry which leads to a TBM pattern in theleptonic mixing at low energies. Deviations to the TBMcase were introduced in order to account for a nonzerovalue of the �13 mixing angle, recently reported by theT2K, MINOS and Daya Bay experiments. Such devia-tions were achieved by perturbing the TBM vacuumalignments of the flavon fields. Two cases have beenconsidered depending on whether the perturbationcomes from the charged lepton (case A) or neutrino(case B) sector. We concluded that the latest oscillationdata favor case B, leading to close-to-maximal DiracCP-violation with jJCPj ’ 0:03–0:04, which is large

enough to be measured in the near future by neutrino

. 10 8

1. 10

1. 10

1. 10 7

10 6

Unflavoured leptogenesis

Exc

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dby

cosm

olog

ical

boun

d

10 7

10 8

108

109

1010

1011

1012

M1,

2G

eV

108

109

1010

1011

1012

FIG. 6 (color online). Contours of the maximum valuesj�01;maxj (solid) and j�02;maxj (dash), as given by Eq. (57), in the

(�, M) plane.

SPONTANEOUS LEPTONIC CP VIOLATION AND . . . PHYSICAL REVIEW D 86, 076008 (2012)

076008-9

oscillation experiments. The model also leads to large

enough flavored CP-asymmetries in the decays of the

triplets into two leptons. Consequently, leptogenesis be-

comes viable due to the out-of-equilibrium decays of the

triplets at temperatures below 1012 GeV. This could

account for the observed value of the baryon asymmetry

of the Universe.

ACKNOWLEDGMENTS

We thank Mariam Tortola for private communication.The work of H. S. was supported by Fundacao para aCiencia e a Tecnologia (FCT, Portugal) under the GrantNo. SFRH/BD/36994/2007. This work was supported bythe projects No. CERN/FP/116328/2010, No. CFTP-FCTUNIT 777 and No. PTDC/FIS/098188/2008, which arepartially funded through POCTI (FEDER).

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