Flavored leptogenesis in broken cyclic symmetric

model

Biswajit Adhikarya∗, Mainak Chakrabortyb†, Ambar Ghosalb‡

a)Department of Physics, Gurudas College, Narkeldanga, Kolkata-700054, India

b) Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India

July 28, 2014

Abstract

Cyclic symmetry in the neutrino sector with the type-I seesaw mechanism in the mass basis

of charged leptons and right chiral neutrinos (NiR, i = e, µ, τ) generates two fold degenerate light

neutrino and three fold degenerate heavy neutrino mass spectrum. Consequently, such scheme,

produces one zero light neutrino mass squared difference and vanishing lepton asymmetry. To

circumvent such unphysical outcome, we break cyclic symmetry in the diagonal right chiral

neutrino mass term by two small parameters ε1 and ε2. Such breaking generates both the mass

squared differences and all three mixing angles nonzero. Assuming complex Yukawa couplings,

we derive generalized CP asymmetry parameters εαi for flavored leptogenesis scenario. We

first restrict the parameter space utilizing the neutrino oscillation data and further constrain

the allowed parameter ranges from the value of observed baryon asymmetry. We have seen that

fully flavored leptogenesis fails to produce baryon asymmetry in the observed range for the cases

of single parameter cyclic symmetry breaking. Again we note that quasi degenerate nature of

NiR’s enhance the CP asymmetry parameters. It is also seen that, in such scenario, imposition

of resonant condition for lepton asymmetry generates excess amount of baryon asymmetry

which is beyond the experimental range. Finally, we predict the low energy CP phases(both

Majorana and Dirac), sum of the three light neutrino masses and the value of |mνee | relevant

for neutrinoless double beta decay utilizing the neutrino oscillation data and observed baryon

asymmetry.

∗biswajitadhikary@gmail.com†mainak.chakraborty@saha.ac.in‡ambar.ghosal@saha.ac.in

1

arX

iv:1

407.

6173

v1 [

hep-

ph]

23

Jul 2

014

1 Introduction

Many experimental observations suggest the excess of matter over antimatter in the universe. In

fact, no evidence of appreciable amount of antimatter has been found yet. Various considerations

indicate that the universe has started its evolution from a baryon symmetric state and the baryon

asymmetry observed in the present era is generated dynamically. The process responsible for

the generation of baryon asymmetry is known as Baryogenesis[1, 2, 3, 4, 5]. There are three

necessary conditions known as Sakharov conditions[6] which have to be satisfied in order to generate

baryon asymmetry dynamically. They are (i)Baryon number violation, (ii)C and CP violation,

(iii)departure from thermal equilibrium. The baryon asymmetry of the universe(ηB) is expressed

by the difference between the baryonic(nB) and the antibaryonic(nB) number densities normalized

to the photon number density(nγ) as

ηB =nB − nB

nγ. (1.1)

After the recent result of Planck satellite experiment, the value of ηB can vary mostly within the

range as (5.94 − 6.17) × 10−10[7]. The lower limit arises solely due to the analysis of the Planck

data at 68% limit whereas inclusion of gravitational lensing data with the above shifts the value of

ηB to the higher end.

Among the various existing mechanisms to generate baryon asymmetry at electroweak scale,

baryogenesis via leptogenesis[8, 9, 10, 11] is a simple and attractive mechanism. In this mechanism

lepton asymmetry generated at a high scale (> 109 Gev) gets converted into baryon asymmetry (ηB)

at electroweak scale due to B + L violating sphaleron interactions[12, 13]. Within the framework

of standard SU(2)L × U(1)Y model (SM) with atleast two right chiral neutrinos (NiR) there is a

Dirac type Yukawa interaction of NiR with standard model leptons and Higgs doublet. At a high

scale (> 109 Gev) where SU(2)L×U(1)Y is unbroken, NiR’s with definite mass can decay both into

charged lepton with charged scalar and light neutrino with neutral scalar. CP conjugate decays of

the above processes are also admitted due to Majorana property of NiR. If out of equilibrium decay

of NiR in conjugate process occur at different rate than the actual process, a net lepton number

asymmetry will be realized, which then gets converted into baryon asymmetry due to sphaleronic

interactions of SM.

In the present work, we investigate the interrelation between leptogenesis, heavy right chiral neu-

trinos and flavor mixing of light neutrinos. In fact, we first constrain the parameter space utilizing

extant neutrino oscillation data[14, 15, 16], and subsequently we further restrict the parameter space

incorporating the reported value of baryon asymmetry by Planck satellite experiment. Finally, we

predict for each category of leptogeneis, the quantities, (i) sum of the three light neutrino masses

2

(∑mi), which could be tested by Planck satellite experimental data, (ii) |mνee | in view of ββ0ν (neu-

trinoless double beta decay) experiment, (iii) |JCP | (Jarlskog measure) and δD (Dirac CP phase)

which will also be verified in the ongoing and forthcoming neutrino oscillation experiments[17].

In particular, we consider a well defined model based on SU(2)L × U(1)Y gauge symmetry with

three right chiral neutrinos NeR , NµR , NτR invoking type-I seesaw mechanism and discrete cyclic

symmetry. The model has already been investigated by the authors recently in the context of an

application of the general methodology developed to calculate three mass eigenvalues, three mixing

angles and Dirac and Majorana phases of a general complex 3× 3 Majorana neutrino mass matrix.

In this work, we study baryogenesis via leptogenesis in detail and further restrict the parameter

space obtained earlier. It is seen, that the sign of the phases obtained earlier are consistent with

the positive value of ηB (ηB > 0 ), however the parameter space gets shrunk due to this additional

constraint.

We briefly describe the model here. The cyclic symmetry considered as follows

νeL → νµL → ντL → νeL ,

NeR → NµR → NτR → NeR . (1.2)

The symmetry invariant neutrino mass matrix can generate nonzero θ13 and other two mixing angles

inside experimentally constrained range at the leading order. In spite of having those attractive

properties the effective neutrino mass matrix encounters a serious problem of degenerate eigenvalues

which is strictly forbidden by the neutrino oscillation experimental data. Due to such degeneracy

in eigenvalues the mixing angles can not be determined uniquely. To overcome those shortcomings

the cyclic symmetry is broken in the right chiral neutrino sector only and the effective neutrino

mass matrix is constructed again with this broken symmetric right handed neutrino mass matrix

and symmetry preserving Dirac neutrino mass matrix. The eigenvalues and mixing angles of this

broken symmetric effective neutrino mass matrix are calculated directly (without any perturbative

approach) using the generalized formulas[18] we have developed to diagonalize a most general

Majorana neutrino mass matrix having 12 independent real parameters.

We organize the present work as follows: In Section 2 we briefly discuss about the model under

consideration. Starting from a most general leptonic mass term we have generated the effective

neutrino mass matrix(mν) through type-I seesaw mechanism. Parameterizition and diagonalization

of the broken symmetric mass matrix is also described in brief in this section. Section 3 is devoted

to the discussion of baryogenesis via leptogenesis. In its different subsections we have shown

the expressions of CP asymmetry parameters for 3 flavor resonant leptogenesis formalism and

the procedure to get the baryon asymmetry(ηB) from the CP asymmetry parameters for different

3

energy regimes. Section 4 contains a detailed numerical analysis and the plots of allowed parameter

space for various cases. A summary of the present work is given in Section 5.

2 Cyclic symmetric model

The most general leptonic Yukawa terms of the Lagrangian in the present model is

− Lmass = (m`)ll′ lLl′R +mDll′νlLNl′R +MRll′N

clLNl′R (2.1)

where l, l′ = e, µ, τ . We demand that the neutrino part of the Lagrangian is invariant under the

cyclic permutation symmetry [18, 19, 20, 21] as given in eq.(1.2). The symmetry invariant Dirac

neutrino mass matrix mD takes the form

mD =

y1 y2 y3

y3 y1 y2

y2 y3 y1

(2.2)

where in general all the entries are complex. The matrix mD can be written in terms of Yukawa

couplings as (mD)ij = hνijv√2, where hνij are the Yukawa couplings and v is the VEV (v = 246GeV ).

Without loss of generality, we consider a basis in which the right handed neutrino mass matrix (MR)

and charged lepton mass matrix (m`) are mass diagonal. Further, imposition of cyclic symmetry

dictates the texture of MR as

MR =

m 0 0

0 m 0

0 0 m

. (2.3)

Invoking type-I seesaw mechanism the effective neutrino mass matrix mν ,

mν = −mDM−1R mT

D (2.4)

takes the following form as

mν = − 1

m

y2

1 + y22 + y2

3 y1y2 + y2y3 + y3y1 y1y2 + y2y3 + y3y1

y1y2 + y2y3 + y3y1 y21 + y2

2 + y23 y1y2 + y2y3 + y3y1

y1y2 + y2y3 + y3y1 y1y2 + y2y3 + y3y1 y21 + y2

2 + y23

. (2.5)

With a suitable choice of parameters mν can be rewritten as

mν = m0

1 + p2e2iα + q2e2iβ peiα + qeiβ + pqei(α+β) peiα + qeiβ + pqei(α+β)

peiα + qeiβ + pqei(α+β) 1 + p2e2iα + q2e2iβ peiα + qeiβ + pqei(α+β)

peiα + qeiβ + pqei(α+β) peiα + qeiβ + pqei(α+β) 1 + p2e2iα + q2e2iβ

(2.6)

4

where we have parametrized the different elements of mν in terms of two real parameters p, q and

two phase parameters α, β accordingly as

m0 = −y23

m, peiα =

y1

y3, qeiβ =

y2

y3. (2.7)

Upon diagonalization, mν yields degenerate mass eigenvalues[18]. The eigenvectors corresponding

to the degenerate eigenvalues can not be determined uniquely. Hence the Diagonalization matrix is

not unique and there by generates an ambiguity1 in the neutrino mixing angles. Thus, it is necessary

to break the discrete symmetry in order to accommodate neutrino oscillation data. Retaining the

flavor diagonal texture of MR, we introduce two symmetry breaking parameters ε1 and ε2 in any

two diagonal entries. This can be done in three ways as

(i)MR = diag(m, m(1 + ε1), m(1 + ε2)

),

(ii)MR = diag(m(1 + ε1), m(1 + ε2), m

),

(iii)MR = diag(m(1 + ε1), m, m(1 + ε2)

).

It is to be noted that instead of perturbative approach, we directly diagonalize the broken symmetric

mass matrix with the help of the results obtained in Ref[18]. Let us first consider case(i). The mν

obtained in this case as

mν = m0

p2e2iα + q2e2iβ

(1+ε1) + 1(1+ε2) peiα + qeiβ

(1+ε2) + pqei(α+β)

(1+ε1)peiα

(1+ε2) + qeiβ

(1+ε1) + pqei(α+β)

peiα + qeiβ

(1+ε2) + pqei(α+β)

(1+ε1) 1 + p2e2iα

(1+ε1) + q2e2iβ

(1+ε2)peiα

(1+ε1) + qeiβ + pqei(α+β)

(1+ε2)peiα

(1+ε2) + qeiβ

(1+ε1) + pqei(α+β) peiα

(1+ε1) + qeiβ + pqei(α+β)

(1+ε2)p2e2iα

(1+ε2) + q2e2iβ + 1(1+ε1)

.(2.8)

The other two cases, case (ii) and (iii) mimic the same mν given in eq.(2.8) with different set of

parametrizations given by

Case (ii)

m0 = −y21

m, peiα =

y2

y1, qeiβ =

y3

y1(2.9)

Case(iii)

m0 = −y22

m, peiα =

y3

y2, qeiβ =

y1

y2. (2.10)

All the experimentally measurable observables(mass squared differences and mixing angles) of this

broken symmetric neutrino mass matrix are obtained in terms of the Lagrangian parameters (m0,

p, q, α, β) and breaking parameters(ε1, ε2) using the methodology developed in Ref.[18] to calculate

the masses and mixing angles from the most general Majorana neutrino mass matrix.

1We put a brief explanation of this ambiguity in Ref.[18]

5

3 Baryogenesis through leptogenesis

Here we will discuss about the lepton asymmetry arising from a CP asymmetry[22] generated due to

the decay of heavy right handed Majorana neutrinos. At a high energy scale where SU(2)L×U(1)Y

symmetry is not broken, physical right handed neutrinos NiR with definite mass can decay into

(i) charged lepton with charged scalar and (ii) light neutrino with neutral scalar. The conjugate

decay process is also possible due to self conjugate nature of NiR. A net lepton asymmetry will

be generated if two decay processes occur at different rate. In the present case the right handed

Majorana neutrinos are not hierarchical. Before the explicit breaking of the cyclic symmetry the

mass spectrum of the right handed neutrinos is degenerate. After cyclic symmetry breaking the

masses of the heavy right handed neutrinos differ by the symmetry breaking parameters(ε1, ε2)

which have considered are within 10%. Hence in this case we have to consider contributions from

all three generation of right handed neutrinos [23] to calculate the CP asymmetry parameters.

3.1 Calculation of CP asymmetry parameters

Neglecting the self energy terms (that are ∼ O[(hν)4]) the resummed effective Yukawa couplings

(considering three generations of NiR) are given by[23, 24]

(hν+)αi = hναi + iBαi − i3∑

j,k=1

|εijk|hναj

×mNi(mNiAij +mNjAji) +Rik

[mNiAkj(mNiAik +mNkAki) +mNjAjk(mNiAki +mNkAik)

]m2Ni− m2

Nj+ 2im2

NiAjj + 2i ImRik

(m2Ni|Ajk|2 +mNjmNkReA2

jk

) ,

(3.1)

where

Rij =m2Ni

m2Ni−m2

Nj+ 2im2

NiAjj

, (3.2)

Aij =(hν†hν)ji

16π, (3.3)

Bαi = −∑

(j 6=i)

(hν†hν)ijhναj

16πf(m2Nj

p2) (3.4)

and |εijk| is the modulus of the usual Levi–Civita anti-symmetric tensor.

The resummed effective amplitudes for the decays NiR → lαΦ are denoted as T (NiR → lαΦ)

and are given by

T (NiR → lαΦ) = (hν+)αi uα PR uNi , (3.5)

6

where i (i = 1, 2, 3) and α (α = e, µ, τ) are the generation indices ofNiR and leptons respectively and

uα, uNi denote corresponding spinorial fields. The CP conjugate decay amplitudes T (NiR → lcαΦ†)

can be obtained easily from eq.(3.5) by replacing (hν+)αi with (hν−)αi which can be further recovered

from eq.(3.1) by taking complex conjugate of the Yukawa couplings. The CP asymmetry of the

decay is characterized by a parameter εαi defined by

εαi =Γ(NiR → lαΦ)− Γ(NiR → lcαΦ†)

Σα[Γ(NiR → lαΦ) + Γ(NiR → lcαΦ†)]

=(hν †+ )iα(hν+)αi − (hν †− )iα(hν−)αi

(hν †+ hν+)ii + (hν †− hν−)ii

. (3.6)

After a long algebraic manipulation the expression of εαi is presented in a simpler form by breaking

it into three parts as

εαi = (εαi )1 + (εαi )2 + (εαi )3 (3.7)

where

(εαi )1 =1

4πv2Hii

∑j

Im{Hij(m†D)iα(mD)αj}f(xij) (3.8)

(εαi )2 =1

4πv2Hii[∑j,k

|εijk|(1− xij)Im{Hji(m

†D)iα(mD)αj}

(1− xij)2 + δijk+

∑j,k

|εijk|√xij(1− xij)Im{Hij(m

†D)iα(mD)αj}

(1− xij)2 + δijk] (3.9)

(εαi )3 =v2

Hii

∑j,k

|εijk|(1− xij){(1− xik)Im(Zαijk)−

Re(Zαijk)Hkk

4πv2}

{(1− xij)2 + δijk}{(1− xik)2 +H2kk

16π2v4}

(3.10)

with

Zαijk =(m†D)jα(mD)αi

32π2v6{HjkHki +

√xikHjkHik +

√xijHkjHik +

√xijxikHkjHki}, (3.11)

δijk =1

16π2[H2jj

v4+

4HjjHkk

v4{(1− xik)2 +H2kk

16π2v4}{ |Hkj |2

64π2v4+√xik

Re(H2kj)

64π2v4}] (3.12)

where mD = vhν√2

, H = (m†DmD), xij =m2Nj

m2Ni

and f(xij) is the loop function given by

f(xij) =√xij{1− (1 + xij) ln(

1 + xijxij

)}. (3.13)

In flavored leptogeneis with non-degenerate right handed neutrinos, nonzero εαi depends on two

relevant factors Iαij = Im{Hij(m†D)iα(mD)αj} and J αij = Im{Hji(m

†D)iα(mD)αj} where the latter

one vanishes after summing over flavor index α. In our obtained expressions only (εαi )3 does not

7

depend on those two terms. Hence, nonzero leptogenesis can be obtained even for vanishing Iαijand J αij when Zαij 6= 0. Although it is suppressed by higher power of Yukawa couplings but may be

relevant for quasi degenerate mass spectrum where 1−xij '√δijk and/or 1−xik ' Hkk

4πv2conditions

are satisfied.

Again, derived expressions for εαi are quite general and can be used for hierarchical as well as

quasi-degenerate mass spectrum (without or with resonant conditions like 1−xij '√δijk ) of right

handed neutrinos. But, for hierarchical case one can simplify εαi to standard formula[25] neglecting

δijk compared to (1− xij) and also dropping the Yukawa suppressed (εαij)3 term.

3.2 Calculation of baryon asymmetry using CP asymmetry parameters

Lepton asymmetry is defined as

Y α =nαl − nαl

s(3.14)

where nαl (nαl ) is the lepton(antilepton) number density and s is the entropy density. The above

asymmetry is related to the CP asymmetry εαi through the relation[26]

Y α =∑i

εαi Kαig?i

(3.15)

where, g?i is the effective number of spin degrees of freedom of particles and antiparticles at a

temperature equal to mNi . In an energy regime, where all flavors are participating in the process

Kαi is expressed approximately as[27, 28, 29] (assuming contributions from diagonal entries of A

matrix)

(Kαi )−1 ' 8.25

|Aαα|Kαi

+

( |Aαα|Kαi

0.2

)1.16

. (3.16)

In the above equation, Kαi is the flavor washout factor and is defined by

Kαi =

Γ(NiR → lαΦ)

H(mNi)(3.17)

which explicitly leads to,

Kαi =

|mDαi|2

mNi

MPl

6.64π√g?iv2

(3.18)

following the Hubble expansion parameter H(mNi) at a temperature mNi

H(mNi) = 1.66√g?im

2NiM

−1Pl (3.19)

where MPl being the Planck mass and to the lowest order, Γ(NiR → lαΦ) = |mDαi|2mNi(4πv2)−1

.

In the above, the quantity Aαα is defined through the following relation

Y αL =

∑β

AαβY β∆ (3.20)

8

using Y αL = s−1(nαL − nαL), where nαL is the number density of the electroweak lepton doublets with

flavor index α, and Y α∆ = 1

3YB − Yα, with YB as the baryonic number density (normalized to the

entropy density s). Explicit expression of the matrix A depends upon the energy scale and we will

state precise form of A for different cases subsequently. Now to calculate ηB, it is necessary to get

relation between YB and Yl =∑αY

α. Utilizing the relation[30]

YB = − 8nF + 4nH22nF + 13nH

Yl (3.21)

where nF is the number of the electroweak matter fermion doublets and nH is the number of

doublet Higgs present in the model at electroweak temperature it is possible to calculate the value

of ηB in terms of Yl. In the present model, nF = 3 and nH = 1, and hence, we get from eq.(3.21)

YB = −28

79Yl (3.22)

which leads to

ηB =s

nγYB (3.23)

and using the value of sηγ' 7.04 at the present time, we get

ηB ' −2.495Yl. (3.24)

Leptogenesis takes place at temperature scale ∼ Mlowest, where Mlowest is the lowest among the

three mass eigenvalues of the three right handed neutrinos. The effective values of Aαα and Kαidepend on which flavors are active in the washout process. Now we discuss the leptogenesis scenario

in three different regimes separately.

3.2.1 Mlowest < 109 GeV.

In this regime all the three flavors are separately active and thus fully flavored leptogenesis takes

place. To calculate the lepton asymmetry for a specific flavor here we have to use the formulas

from eq.(3.15) to (3.18). Hence we need the 3× 3 A matrix, and for the SM which is given by[26]

A =

−151

17920179

20179

25358 −344

53714537

25358

14537 −344

537

. (3.25)

With g?i = 112, the final expression of baryon asymmetry comes out to be

ηB = −2.22× 10−2∑α

(εα1Kα1 + εα2Kα2 + εα3Kα3 ). (3.26)

9

3.2.2 109 GeV < Mlowest < 1012 GeV.

In this regime τ flavor gets decoupled but e and µ flavors act predominantly in an indistinguishable

manner. Thus to calculate the CP asymmetry parameters and the dilution factors we need to sum

over the e and µ flavors, i.e two new quantities

εe+µi = εei + εµi (3.27)

and

Ke+µi = Ke

i +Kµi (3.28)

have to be introduced and the quantity Ke+µi is related to the washout factor Ke+µi through

(Ke+µi )−1 =8.25

|A11|(Ke+µi )

+

(|A11|(Ke+µ

i )

0.2

)1.16

(3.29)

and for the τ flavor it is given by

(Kτi )−1 =8.25

|A22|Kτi

+

(|A22|(Kτ

i )

0.2

)1.16

. (3.30)

It is clear that in this case the A is 2× 2 (which we denote by A) and for the SM is given by[26]

A =

(−920

589120589

30589 −390

589

)(3.31)

following the final expression of ηB comes out as

ηB = −2.22× 10−2∑i

(εe+µi Ke+µi + ετiKτi ). (3.32)

3.2.3 Mlowest > 1012 GeV.

In this regime all the flavors are indistinguishable, and hence, unflavored leptogenesis takes place.

To find out the CP asymmetry we have to sum over the flavor index α on εαi , and the asymmetry

is given by

εi =∑α

εαi (3.33)

In this case we have

K−1i = 8.25K−1

i + (Ki/0.2)1.16, (3.34)

with

Ki =∑α

Kαi =

(m†DmD)iiMPl

(6.64π√g?imNiv

2). (3.35)

10

The baryon asymmetry ηB is given by

ηB = −2.495∑i

εiKig?i

= −2.22× 10−2∑i

εiKi. (3.36)

4 Numerical results and phenomenological discussion

For numerical analysis of baryon asymmetry we need to know CP asymmetry parameters εαi and

washout factors Kαi in terms of the parameters m, m0, p, q, α, β and ε1, ε2. However, dependencies

of those parameters on εαi and the washout factors Kαi arise through the expressions of mD, H(=

m†DmD) and xij . Obviously it is then necessary to express, mD, H and xij in terms of those

Lagrangian parameters. Utilizing eq.(2.7) we explicitly express the elements of mD in terms of the

aforesaid parameters through

y3 = i√mm0, y1 = i

√mm0pe

iα, and y2 = i√mm0qe

iβ (4.1)

which in effect gives

mD = i√mm0

1 peiα qeiβ

qeiβ 1 peiα

peiα qeiβ 1

,H = m†DmD

=

|y1|2 + |y2|2 + |y3|2 y∗1y2 + y1y

∗3 + y∗2y3 y1y

∗2 + y∗1y3 + y2y

∗3

y1y∗2 + y∗1y3 + y2y

∗3 |y1|2 + |y2|2 + |y3|2 y∗1y2 + y1y

∗3 + y∗2y3

y∗1y2 + y1y∗3 + y∗2y3 y1y

∗2 + y∗1y3 + y2y

∗3 |y1|2 + |y2|2 + |y3|2

= mm0

X Y Y ∗

Y ∗ X Y

Y Y ∗ X

(4.2)

with

X = 1 + p2 + q2

Y = peiα + qe−iβ + pqei(β−α). (4.3)

Again xij = m2Nj/m2

Niis estimated from MR = diag

(m, m(1 + ε1), m(1 + ε2)

)and spelt out

as

x12 =1

x21= (1 + ε1)2,

11

x13 =1

x31= (1 + ε3)2,

x23 =1

x32=

(1 + ε3)2

(1 + ε1)2. (4.4)

To find out the allowed parameter space we adopt the following methodology. In the present work

the parameter space is constrained due to the bound on the Baryon asymmetry parameter(ηB)

keeping in mind all the neutrino oscillation experimental data(Table 1). The parameter space is

constrained in two steps. At first all the neutrino physics observables (mass eigenvalues, mixing

angles) are expressed in terms of the Lagrangian parameters (m0, p, q, α, β) and the breaking

parameters (ε1, ε2). In the first step the parameters get restricted by the experimental ranges of

neutrino mass squared differences (solar and atmospheric) and mixing angles. These constrained set

of parameters are used thereafter to calculate the CP asymmetry parameters and hence the baryon

asymmetry parameter ηB for the different mass scale m for three different types of leptogenesis

namely, fully flavored, τ -flavored and unflavored leptogenesis. Hence, the parameters get second

round of restriction from the limits on ηB. We have observed that apart from the two parameter

Table 1: Input data from neutrino oscillation experiments [16]

Quantity 3σ ranges/other constraint

∆m221 7.12 < ∆m2

21(105 eV −2) < 8.20

|∆m231|(N) 2.31 < ∆m2

31(103 eV −2) < 2.74

|∆m231|(I) 2.21 < ∆m2

31(103 eV −2) < 2.64

θ12 31.30◦ < θ12 < 37.46◦

θ23 36.86◦ < θ23 < 55.55◦

θ13 7.49◦ < θ13 < 10.46◦

breaking of cyclic symmetry(ε1, ε2 6= 0), one parameter breaking (ε1 or ε2 6= 0) is also well fitted

by the extant data. In order to pin down the parameter space for each type of leptogeneis we have

considered three categories of cyclic symmetry breaking with i)ε1 = 0, ε2 6= 0, ii)ε1 6= 0, ε2 = 0,

and iii)ε1 6= 0, ε2 6= 0. For each category we have studied the leptogenesis phenomena in three

different energy regimes separately. Since the right handed neutrino masses are nearly degenerate,

we have used the leptogenesis formalism where three generations contribute for all three cases for

each category.

Before proceeding to carry out numerical analysis few remarks are in order.

1. The upper bound on the sum of the three neutrino masses can vary within the range as∑mi(= m1 +m2 +m3) < (0.23− 1.11)eV [32] as reported by a combined analysis of Planck

12

experimental results[7] with different other cosmological experiments. The upper value of the

above range arises from a set of experimental results comprises with Planck, WMAP low l

polarization[33], gravitational lensing and results prior on the Hubble constant H0 from the

Hubble space telescope data whereas the lower value obtained due to the inclusion of SDSS

DR8[34] result with the above combination. We predict the value of∑mi and individual

masses of three light neutrinos in each cases.

2. Neutrinoless double beta decay experiment[16, 35, 36] constrains the magnitude of neutrino

mass matrix element mνee . A lots of experiments are presently running/proposed. One of

them, EXO-200 experiment[37] has quoted a range on the upper limit of |mνee | as |mνee | <(0.14− 0.38)eV . In the present work, we also treat |mνee | as prediction to testify the present

model in foreseeable future.

3. Due to nonzero θ13, CP violation may occur in the neutrino sector which can be observed

through the difference in oscillation probabilities. We also predict in each case the low energy

CP violating Jarlskog measure (JCP ) and the Dirac CP violating phase(δD), where

JCP =Im(h12h23h31)

(m22 −m2

1)(m23 −m2

2)(m23 −m2

1)=

sin 2θ12 sin 2θ23 sin 2θ13 cos θ13 sin δD8

. (4.5)

4. We also estimate the Majorana phases (denoted by αM , βM ) using the parameters constrained

by the neutrino oscillation data as well as the bound on ηB. It is found that Majorana phases

remain unrestricted (both αM and βM can vary from −90◦ to 90◦).

5. Quasidegenaracy in the right handed neutrino mass spectrum opens up an option to study

resonant leptogenesis phenomena. Here, we vary one symmetry breaking parameter say ε2,

within a range −0.1 < ε2 < 0.1, to remove degeneracy between the light neutrino masses and

keep the other breaking parameter ε1 very small, to satisfy the resonant condition 1− x12 '√δ123 without affecting the neutrino oscillation data. But, it is seen that exact resonant

condition produces excess amount of lepton asymmetry for all three energy regimes of m (fully

flavored, partly flavored and unflavored). This in effect generates excess amount of baryon

asymmetry which goes beyond the present observed limit of ηB. So, in the present model,

it is hard to reconcile the observed value of baryon asymmetry with the neutrino oscillation

data. Hence, in the following numerical analysis, we consider non-resonant leptogenesis.

13

4.1 Single nonzero breaking parameter (with ε1 = 0, ε2 6= 0)

It is implemented through the choice of ε1 = 0 in eq.(2.8). The calculation of mixing angles and

mass eigenvalues using the resulting mass matrix is carried out thereafter. The parameter space,

constrained by the extant data, is used to find out the numerical value of the baryon asymmetry

parameter ηB. By varying the mass of the right chiral neutrinos, we have studied leptogeneis in all

three energy regimes as mentioned earlier. The symmetry breaking parameter is also varied from

a negative to a positive value and we have restricted the value of ε2 in the range (−0.1 to 0.1).

In the fully flavored case the mass of the lightest right handed neutrino is less than 109 Gev.

All three neutrino flavors (νe, νµ, ντ ) are separately active in this regime. The formula needed to

calculate the baryon asymmetry parameter (ηB) is given in subsection(3.2.1). After second round

of restriction (including bound on ηB) here we don’t get any allowed parameter space, i.e the

parameters allowed by the oscillation data fail to produce ηB within the prescribed range.

In the partly flavored or τ-flavored case the mass of the lightest right handed neutrino is

less than 1012 Gev but greater than 109 GeV. In this regime we can not distinguish between e and µ

flavors, whereas the τ flavor is decoupled. The full mathematical expressions of baryon asymmetry

parameter(ηB) for this regime is given in subsection(3.2.2). Allowed parameter space after accom-

modating ηB bound is shown in Fig.1. It is clear from the plot that the phase parameter space

consists of three disconnected patches in α vs β plane and thereby, their sign also get restricted.

The ranges of the phase angles of three different patches are clearly stated in the Table 2 below.

Similarly, the ranges of the other parameters p, q and ε2 are also presented in Table 2. We have also

calculated the Majorana phases (αM , βM ) using the constrained values of the Lagrangian parame-

ters (p, q, α, β). A plot of Majorana phases is presented in Fig.1. Finally for the unflavored case

Figure 1: (colour online) Plot of the allowed parameter space in p vs q (left), α vs β (middle) and

αM vs βM plane (right) for τ -flavored leptogenesis. The choice of ε1, ε2 parameters are given in

Table 2.

the mass of the lightest right handed neutrino is greater than 1012 GeV and we can not differentiate

14

between e, µ and τ flavors. The relevant formulas to get the baryon asymmetry (ηB) is given in

subsection(3.2.3). Again allowed parameter space in p,q plane and α,β plane are shown in Fig.2.

α vs β parameter space contains two disconnected patches one is mirror image to the other in two

opposite quadrants. Their ranges are given in Table 2. Majorana phases remain unrestricted in this

case too and their plot is given in Fig.2. The allowed ranges of fully constrained parameter space

Figure 2: (colour online) Plot of the allowed parameter space in p vs q (left), α vs β (middle) and

αM vs βM plane (right) for Unflavored leptogenesis. The choice of ε1, ε2 parameters are given in

Table 2.

for the above three subcases is given below in tabular form. Furthermore, predictions for∑mi,

each individual neutrino masses(m1, m2, m3), |mνee |, |JCP |, δD in this case are listed in Table 3.

Table 2: Constrained parameter space for single nonzero breaking parameter (with ε1 = 0, ε2 6= 0)

Cases constrained parameter space Allowed

p q α (deg) β (deg) ε1 ε2 M(GeV )

Fully flavored No allowed parameter space N.A

Leptogenesis

τ flavored 0.67-3.4 0.67-3.7 (−164.2◦)-(−140.4◦) 87.2◦-107◦ 0 (−0.10) 2× 1010

Leptogenesis 92.3◦-142.5◦ (−163.2◦)-(−115.2◦) -(−0.09) -1× 1012

90.4◦-121◦ (−115.2◦)-(−94.5◦)

Unflavored 0.75-3.44 0.76-3.59 (−158.1◦)-(−102.1◦) 96.3◦-150.8◦ 0 (−0.10) 6× 1012

Leptogenesis 100◦-151◦ (−160.1◦)-(−105.2◦) -(−0.09) -1× 1016

15

Table 3: Predictions of some important parameters and phases (with ε1 = 0, ε2 6= 0)

Cases Parameters/Phases

m1 (eV) m2 (eV) m3 (eV)∑mi (eV) |mνee | (eV) |JCP | |δD| (deg)

Fully flavored No allowed parameter space

Leptogenesis

τ flavored 0.031− 0.071 0.032− 0.072 0.057− 0.089 0.12− 0.23 0.014− 0.056 0− 0.04 (0− 90)◦

Leptogenesis

Unflavored 0.017− 0.070 0.02− 0.07 0.051− 0.087 0.09− 0.23 0.006− 0.051 0− 0.04 (0− 90)◦

Leptogenesis

4.2 Single nonzero breaking parameter (with ε2 = 0, ε1 6= 0)

In this case too we have broken the cyclic permutation symmetry by one nonzero breaking pa-

rameter. Instead of ε1, ε2 is set to zero where as ε1 is varied from a negative to a positive value

precisely, we consider the range of ε1 as previous, i.e −0.1 to 0.1. In this case also, we investigate all

three subcases of leptogeneis namely, fully flavored, partly flavored and unflavored. Interestingly,

similar to the previous case, admissible parameter space is forbidden in the case of fully flavored

leptogeneis. For partly flavored case the constrained parameter space and plot of Majorana phases

is given in Fig.3 and for unflavored case the same is shown in Fig.4. The allowed ranges of the pa-

Figure 3: (colour online) Plot of the allowed parameter space in p vs q (left), α vs β (middle) and

αM vs βM plane (right) for τ -flavored leptogenesis. The choice of ε1, ε2 parameters are given in

Table 4.

rameters in this case are tabulated (in Table 4) exactly in the same manner as done in earlier case.

We treat also in this case the parameters∑mi, m1, m2, m3, |mνee |, |JCP | and δD as predictions

and their individual values for each type of leptogenesis are given in Table 5.

16

Figure 4: (colour online) Plot of the allowed parameter space in p vs q (left), α vs β (middle) and

αM vs βM plane (right) for Unflavored leptogenesis. The choice of ε1, ε2 parameters are given in

Table 4.

Table 4: Constrained parameter space for single nonzero breaking parameter (with ε2 = 0, ε1 6= 0)

Cases constrained parameter space Allowed

p q α (deg) β (deg) ε1 ε2 M(GeV )

Fully flavored No allowed parameter space N.A

Leptogenesis

τ flavored 0.68-1.6 0.22-1.5 (−125.6◦)-(−102.8◦) 138.3◦-167.6◦ (−0.10) 0 2× 1010

Leptogenesis 100.1◦-114.1◦ (−143.6◦)-(−92.4◦) -(−0.03) -1× 1012

140.3◦-160.8◦ (−124.7◦)-(−91.2◦)

Unflavored 0.82-1.37 0.34-1.18 (−134.5◦)-(−101.6◦) 100.4◦-159.3◦ (−0.10) 0 2× 1013

Leptogenesis 100.5◦-134.6◦ (−160.3◦)-(−101.5◦) -(−0.01) -1× 1016

Table 5: Predictions of some important parameters and phases (with ε2 = 0, ε1 6= 0)

Cases Parameters/Phases

m1 (eV) m2 (eV) m3 (eV)∑mi (eV) |mνee | (eV) |JCP | |δD| (deg)

Fully flavored No allowed parameter space

Leptogenesis

τ flavored 0.032− 0.071 0.033− 0.072 0.058− 0.088 0.12− 0.23 0.011− 0.068 0− 0.04 (0− 90)◦

Leptogenesis

Unflavored 0.019− 0.068 0.021− 0.069 0.051− 0.085 0.088− 0.22 0.006− 0.053 0− 0.04 (0− 90)◦

Leptogenesis

4.3 Two nonzero breaking parameters ( ε1 6= 0, ε2 6= 0)

This is the most general case containing both the breaking parameters nonzero. As previous,

we vary both the parameters from a negative value to a positive value simultaneously. Unlike

the previous two categories, we get allowed parameter space in the fully flavored regime and the

17

phase parameter space is composed of two separate regions in α-β plane one mirror image to the

other. The allowed ranges of α, β, p, q and the breaking parameters(ε1, ε2) are stated clearly

in Table 6. Allowed space of parameters and Majorana phases for this regime is shown in Fig.5.

For partly flavored regime, plot of allowed parameter space is given in Fig.6 and for unflavored

Figure 5: (colour online) Plot of the allowed parameter space in p vs q (left), α vs β (middle) and

αM vs βM plane (right) for fully flavored leptogenesis. The choice of ε1, ε2 parameters are given in

Table 6.

Figure 6: (colour online) Plot of the allowed parameter space in p vs q (left), α vs β (middle) and

αM vs βM plane (right) for τ -flavored leptogenesis. The choice of ε1, ε2 parameters are given in

Table 6.

regime parameter space is depicted in Fig.7. Allowed Majorana phases in αM -βM plane for partly

flavored and unflavored regime are shown in Fig.6 and Fig.7 respectively. Allowed values of all the

parameters for the above three subcases are given in Table 6. The final predictions of our analysis

in this case are given in Table 7.

18

Figure 7: (colour online) Plot of the allowed parameter space in p vs q (left), α vs β (middle) and

αM vs βM plane (right) for Unflavored leptogenesis. The choice of ε1, ε2 parameters are given in

Table 6.

Table 6: Constrained parameter space for two nonzero breaking parameters ( ε1 6= 0, ε2 6= 0)

Cases constrained parameter space Allowed

p q α (deg) β (deg) ε1 ε2 M(GeV )

Fully flavored 0.66-2.2 0.33-2.4 (−162.1◦)-(−101.3◦) 92.3◦-160.5◦ (−0.10) (−0.01) 6× 108

Leptogenesis 96.5◦-132.3◦ (−156.3◦)-(−96.6◦) -(0.01) -(0.01) -7× 108

τ flavored 0.26-1.9 0.29-1.5 (−156.7◦)-(−90◦) 98.2◦-155.7◦ (−0.10) (−0.10) 1× 109

Leptogenesis 88.6◦-161.1◦ (−165.1◦)-(−98.8◦) -(0.09) -(0.10) -1× 1012

Unflavored 0.31-1.22 0.68-1.45 (−151.8◦)-(−93.6◦) 102.4◦-136.5◦ (−0.03) (−0.09) 2× 1013

Leptogenesis 94.3◦-151.8◦ (−136.4◦)-(−101.5◦) -(0.10) -(0.10) -1× 1016

Table 7: Predictions of some important parameters and phases ( ε1 6= 0, ε2 6= 0)

Cases Parameters/Phases

m1 (eV) m2 (eV) m3 (eV)∑mi (eV) |mνee | (eV) |JCP | |δD| (deg)

Fully flavored 0.015− 0.067 0.017− 0.069 0.050− 0.085 0.083− 0.22 0.003− 0.042 0− 0.04 (0− 90)◦

Leptogenesis

τ flavored 0.012− 0.070 0.015− 0.072 0.049− 0.088 0.077− 0.23 0.002− 0.064 0− 0.04 (0− 90)◦

Leptogenesis

Unflavored 0.018− 0.070 0.02− 0.072 0.051− 0.088 0.091− 0.23 0.004− 0.062 0− 0.04 (0− 90)◦

Leptogenesis

5 Summary

We consider an SU(2)L × U(1)Y model with three right chiral neutrinos invoking type-I seesaw

mechanism and cyclic symmetry in the neutrino sector. Since, the symmetry invariant model

generates two fold degeneracy in the light neutrino mass, the model forbids to determine three

19

mixing angles in an unique way as well as generates vanishing value of one mass squared difference.

A possible way to get rid of those shortcomings is due to the breaking of the cyclic symmetry

imposed. Symmetry breaking is incorporated in a minimal way through two small parameters

ε1 and ε2 in MR. Armed with such modifications, we apply the most general diagonalization

method to find out mass eigenvalues and mixing angles. First we restrict the parameter space by

fixing the neutrino oscillation experimental data. We further explore leptogenesis in detail, namely,

fully flavored, τ -flavored and unflavored, in different energy regime satisfying the constraint on

ηB. We further divide each subcases through the choice of ε1,2 parameters as (i)ε1 = 0, ε2 6= 0,

(ii)ε1 6= 0, ε2 = 0, (iii)ε1,2 6= 0. Interestingly, we find for both cases (i) and (ii), fully flavored

leptogeneis is completely forbidden allowing only unflavored and τ -flavored cases, whereas case (iii)

allows all three types of leptogenesis. Important consequence of our analysis is that, apart from

constrained magnitude of different Lagrangian parameters, the sign of the phase parameters α and

β are also restricted. Both of them cannot lie in the same quadrant simultaneously. We emphasize

that imposition of the resonant leptogeneis condition produces a large value ηB which is orders

of magnitude higher than the experimentally allowed one. The hierarchy obtained in each case

is normal and such result will be tested in future neutrino oscillation experiments. We further

predict the value of∑mi, |mνee | which are below the present experimental upper bounds. The

other predictions |JCP | and |δD| could also be verified through different proposed future neutrino

oscillation experiments.

20

References

[1] V. A. Rubakov and M. E. Shaposhnikov, Usp. Fiz. Nauk 166, 493 (1996) [Phys. Usp. 39, 461

(1996)] [hep-ph/9603208].

[2] M. Trodden, Rev. Mod. Phys. 71, 1463 (1999) [hep-ph/9803479].

[3] A. Riotto, hep-ph/9807454.

[4] J. M. Cline, hep-ph/0609145.

[5] M. Dine and A. Kusenko, Rev. Mod. Phys. 76, 1 (2003) [hep-ph/0303065].

[6] A.D. Sakharov, Zh. Eksp. Teor. Fiz. Pis’ma 5, 32 (1967); JETP Lett. 91B, 24 (1967).

[7] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5076 [astro-ph.CO].

[8] M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45.

[9] A. Riotto and M. Trodden, Ann. Rev. Nucl. Part. Sci. 49 (1999) 35 [ arXiv:hep-ph/9901362]

[SPIRES].

[10] M. Yu. Khlopov, Cosmoparticle Physics, World Scientific, Singapore (1999).

[11] S. Davidson, E. Nardi and Y. Nir, Phys. Rept. 466, 105 (2008) [arXiv:0802.2962 [hep-ph]].

[12] W. Buchmuller and M. Plumacher, Int. J. Mod. Phys. A 15, 5047 (2000) [hep-ph/0007176].

[13] W. Bernreuther, Lect. Notes Phys. 591, 237 (2002) [hep-ph/0205279].

[14] D. V. Forero, M. Tortola and J. W. F. Valle, arXiv:1405.7540 [hep-ph].

[15] M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz, JHEP 1212, 123 (2012)

[arXiv:1209.3023 [hep-ph]].

[16] D. V. Forero, M. Tortola and J. W. F. Valle, Phys. Rev. D 86 (2012) 073012 [arXiv:1205.4018

[hep-ph]].

[17] H. Minakata, Acta Phys. Polon. B 39, 283 (2008) [arXiv:0801.2427 [hep-ph]].

[18] B. Adhikary, M. chakraborty and A. Ghosal, arXiv:1307.0988 [hep-ph].

[19] Y. Koide, hep-ph/0005137.

[20] A. Damanik, M. Satriawan, P. Anggraita, A. Hermanto and Muslim, J. Theor. Comput. Stud.

8, (2008) 0102 [arXiv:0710.1742 [hep-ph]].

21

[21] A. Damanik, arXiv:1004.1457 [hep-ph].

[22] L. Covi, E. Roulet and F. Vissani, Phys. Lett. B 384, 169 (1996) [hep-ph/9605319].

[23] A. Pilaftsis and T. E. J. Underwood, Nucl. Phys. B 692, 303 (2004) [hep-ph/0309342].

[24] A. Pilaftsis, Phys. Rev. D 56, 5431 (1997) [hep-ph/9707235].

[25] B. Adhikary, A. Ghosal and P. Roy, JCAP 1101, 025 (2011) [arXiv:1009.2635 [hep-ph]].

[26] S. Antusch, S. F. King and A. Riotto, JCAP 0611, 011 (2006) [hep-ph/0609038].

[27] G. F. Giudice, A. Notari, M. Raidal, A. Riotto and A. Strumia, Nucl. Phys. B, 685 (2004) 89

[hep-ph/0310123]

[28] W. Buchmuller, P. Di Bari and M. Plumacher, Annals Phys. 315 (2005) 305 [hep-ph/0401240]

[29] A. Abada, S. Davidson, A. Ibarra, F. -X. Josse-Michaux, M. Losada and A. Riotto, JHEP

0609 (2006) 010 [hep-ph/0605281].

[30] S. Weinberg, Cosmology (Oxford university press, Great Clarendon Street, Oxford, OX2 6DP,

UK) (2008).

[31] E. W. Kolb and M. S. Turner, The Early Universe (Addison-Wesley, Redwood City, CA,

U.S.A) (1990).

[32] E. Giusarma, R. de Putter, S. Ho and O. Mena, arXiv:1306.5544 [astro-ph.CO].

[33] C. L. Bennett et al. [WMAP Collaboration], Astrophys. J. Suppl. 208, 20 (2013)

[arXiv:1212.5225 [astro-ph.CO]].

[34] H. Aihara et al. [SDSS Collaboration], Astrophys. J. Suppl. 193, 29 (2011) [Erratum-ibid.

195, 26 (2011)] [arXiv:1101.1559 [astro-ph.IM]].

[35] A. Giuliani, Acta Phys. Polon. B 41 (2010) 1447.

[36] W. Rodejohann, J. Phys. G 39 (2012) 124008 [arXiv:1206.2560 [hep-ph]].

[37] M. Auger et al. [EXO Collaboration], Phys. Rev. Lett. 109 (2012) 032505 [arXiv:1205.5608

[hep-ex]].

22