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arXiv:1003.1468v2 [hep-ph] 27 Oct 2010

Neutrino Masses and a TeV Scale Seesaw Mechanism

Wei Chao∗

Center for High Energy Physics, Peking university, Beijing 100871, China

Abstract

A simple extension of the Standard Model providing TeV scale seesaw mechanism is presented.

Beside the Standard Model particles and right-handed Majorana neutrinos, the model contains a

singly charged scalar, an extra Higgs doublet and three vector like singly charged fermions. In our

model, Dirac neutrino mass matrix raises only at the loop level. Small but non-zero Majorana

neutrino masses come from integrating out heavy Majorana neutrinos, which can be at the TeV

scale. The phenomenologies of the model are investigated, including scalar mass spectrum, neutrino

masses and mixings, lepton flavor violations, heavy neutrino magnetic moments as well as possible

collider signatures of the model at the LHC.

∗Electronic address: chaow@pku.edu.cn

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I.INTRODUCTION

The observation of neutrino oscillations [1–4] has revealed that neutrinos have small

but non-zero masses and lepton flavors are mixed, which can not be accommodated in the

Standard Model (SM) without introducing extra ingredients. As such, neutrino physics

offer an exciting window into new physics beyond the SM. Perhaps the most attractive

approach towards understanding the origin of small neutrino masses is using the dimension-

five weinberg operator [5]:

1

4κgfℓC

Lc

gεcdφdℓf

Lbεbaφa+ h.c. , (1)

which comes from integrating out new superheavy particles.

A simple way to obtain the operator in Eq. (1) is through the Type-I seesaw mechanism

[6], in which three right-handed neutrinos with large Majorana masses are introduced to

the SM. Then three active neutrinos may acquire tiny Majorana masses through the Type-

I seesaw formula, i.e., the mass matrix of light neutrinos is given by Mν= −MDM−1

where MDis the Dirac mass matrix linking left-handed light neutrinos to right-handed heavy

RMT

D,

neutrinos and MRis the mass matrix of heavy Majorana neutrinos. Actually, there are three

tree-level seesaw scenarios (namely type-I, Type-II [7] and Type-III [8] seesaw mechanisms)

and one loop-level seesaw scenario (namely Ma [9] model), which may lead to the effective

operator in Eq. (1).

Although seesaw mechanisms can work naturally to generate Majorana neutrino masses,

they lose direct testability on the experimental side. A direct test of seesaw mechanism would

involve the detection of these heavy seesaw particles at a collider and the measurement of

their Yukawa couplings with the electroweak doublets. In the canonical seesaw mechanism,

heavy seesaw particles turn out to be too heavy, i.e., 1014∼16GeV, to be experimentally

accessible. One straightforward way out is to lower the seesaw scale “by hand” down to the

TeV scale, an energy frontier to be explored by the Large Hadron Collider (LHC). However

this requires the structural cancellation between the Yukawa coupling texture and the heavy

Majorana mass matrix, i.e. MDM−1

RMT

D≈ 0 [10–15] at the tree level, and is thus unnatural!

To solve this unnaturalness problem, we propose a novel TeV-scale seesaw mechanism in

this paper. The model includes, in addition to the SM fields and right-handed Majorana

neutrinos, a charged scalar singlet, an extra Higgs doublet and three vector like singly

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charged fermions. Due to Z2discrete flavor symmetry, right-handed Majorana neutrinos

don’t couple to left-handed lepton doublets, such that Dirac mass matrix only raises at the

loop level and is comparable with the charged lepton mass matrix. This drives down the

seesaw scale to the TeV, and thus the model is detectable at the LHC.

The paper is organized as follows: In section II, we describe our model. Section III is

devoted to investigate the phenomenologies of the model, including neutrino masses and

mixings, lepton flavor violations, transition magnetic moments of heavy Majorana neutrinos

as well as possible collider signatures. We conclude in Section IV. An alternative settings to

the model is presented in appendix A.

II. THE MODEL

In our model, we extend the SM by introducing three right-handed Majorana neutrinos

NR, three singly charged vector-like fermion SL,SR, an extra Higgs doublet Hn, a singly

charged scalar Φ as well as discrete Z2flavor symmetry. The Z2charges for these fields is

given in table I. Due to Z2symmetry, right-handed neutrinos don’t couple to SM Higgs.

TABLE I: Z2charges of particles.

fields ℓLeRNRSLSRH HnΦ

Z2

+1 +1 -1 -1 +1 +1 +1 +1

As a result the new lagrangian can be written as

LN= V (H,Hn,Φ) − ℓLYSHSR− SLMSSR− YNSLΦNR−1

2NC

RMRNR+ h.c. ,(2)

where YSand YNare new Yukawa couplings, MSand MRare mass matrices of S and NR,

respectively. Z2symmetry is explicitly broken by SLMSSRterm. It can be recovered by

adding an extra scalar singlet η, with Z2charge −1 and Yukawa coupling SLηSR. We will

not consider Yukawa couplings ℓLHneRand ℓLHnSR, which can be forbidden by another Z

′

2

symmetry. The following is the full Higgs potential:

V = −m2

1H†H − m2

+λ3(H†H)(H†

2H†

nHn) +λ4

nHn− m2

4(H†Hn+ H†

3Φ†Φ + λ1(H†H)2+ λ2(H†

nH)2−λ5

nHn) +?λnΦ(HTiσ2Hn) + h.c.?

3

nHn)2

4(H†Hn− H†

nH)2

+λ6(Φ†Φ)(H†H) + λ7(Φ†Φ)(H†

. (3)

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We define ?H? = v1/√2 and ?Hn? = v2/√2.

minimum, one finds that

After imposing the conditions of global

v2

1=λ2m2

1− λom2

λ1λ2− λ2

2

o

;v2

2=λ1m2

2− λom2

λ1λ2− λ2

1

o

,(4)

where λo= 1/2(λ3+ λ4).

In the basis (h−,h−

n,S−), we can derive the mass matrix for charged scalars:

3+λ6v2

MC=

2. MCcan be diagonalized by the 3×3 unitary transformation

V : V†MCV∗= diag(MG+,MH+,MS+). The mass eigenvalues for these charged scalars are

−λ4v2

λ4v1v2

√2λnv2

2

λ4v1v2

√2λnv2

−√2λnv1

ρ

−λ4v2

−√2λnv1

1

,

(5)

where ρ ≡ −2m2

1+λ7v2

then

MG+ = 0 ;

MH+ =

1

2

1

2

????B −

?

?

B2+ 4(v2

1+ v2

2)(ρλ4+ 2λ2

n)

????;

MS+ =

????B +

B2+ 4(v2

1+ v2

2)(ρλ4+ 2λ2

n)

????,(6)

where B ≡ ρ − λ4(v2

matrix for CP-even scalars in the basis (h,hn)Tand CP-odd scalars in the basis (G,Gn)T:

?2λ1v2

λov1v2

2λ2v2

2

1+ v2

2). Here G+is the SM goldstone boson. We also derive the mass

MN=

1

λov1v2

?

;MG=

?−λsv2

λsv1v2

2

λsv1v2

−λsv2

1

?

,(7)

where λs= 1/2(λ4− λ5).

We also derive the masses for gauge bosons, which are M2

W = g2(v2

1+ v2

2)/4 and

M2

Z= g2(v2

1+ v2

2)/4cos2θw, separately. Such that electroweak precision observable ρ ≡

Zcos2θw= 1 in our theory. Our scalar field sector is similar to that in Zee modelM2

W/M2

[16]. We present in appendix A a different setting to the particle contents, by replacing

scalar singlet with triplet.

III.PHENOMENOLOGICAL ANALYSIS

In this section, we devote to investigate some phenomenological implications of our model.

We focus on (A) neutrino masses and mixings; (B)lepton flavor violations; (C) electromag-

netic properties and (D) collider signatures of heavy majorana neutrinos, which will be

deployed in the following:

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A.Neutrino masses and lepton mixing martrix

In our model, there is no Dirac neutrino mass term at the tree level. However we can

derive a small Dirac neutrino mass matrix at the loop level. The relevant feynman diagram

is shown in Fig. 1.

MS

ℓL

NR

SR

SL

H

Φ

Hn

FIG. 1: One-loop correction to the Dirac neutrino mass matrix.

A direct calculation results in

(MD)loop

ab

=λn?Hn?

32π2

(YS)ac(Y†

N)cbMScF?M2

H,M2

Φ,M2

Sc

?

,(8)

where the one loop function appearing in upper equation is given by

F(m2

1,m2

2,m2) =

1

m2

1− m2

2

?lnβ1

β1− 1−

lnβ2

β2− 1

?

,(9)

with βi= m2/m2

i. When m1= m2= m, F reduces to 1/2m2.

is the Dirac neutrino mass matrix linking the left and right hand neutrinos,Here Mloop

D

which only raises at the loop level in our model. If neutrinos are Dirac particles, then

Eq. (8) is just neutrino mass formula, whose predication must be consistent with present

neutrino oscillation data. In this paper, we assume that neutrinos are Majorana particles,

i.e., left-handed and right-handed neutrinos have different mass eigenvalues. Then three

active neutrino masses can be generated from seesaw mechanism. In this case, we can write

down the 6 × 6 neutrino mass matrix:

M =

?

0Mloop

D

MloopT

D

MR

?

,(10)

which can be diagonalized by the unitary transformation U†MU∗=?

R

STMloopT

D

M; or explicitly,

0

ˆ MN

?V

?†?

0Mloop

D

MR

??VR

ST

?∗

=

?ˆ Mν

0

?

.(11)

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Given Mloop

D

≪MR, the light Majorana neutrino mass formula is then Mν

T. Notice that Dirac neutrino mass matrix is suppressed by loop fac-

=

−Mloop

tor, we assume Mloop

D

M−1

RMloop

D

D

∼ O(MeV), which will not cause any fine-tune problem. Then, to

generate electron-volt scale active neutrino masses, heavy Majorana neutrinos would be of

the order of several hundred GeV.

We also obtain the charged lepton mass matrix in the basis (EL,SL)T,

?ME

0MS

where ME= v/√2YEand MC= v/√2YS.

Mℓ=

MC

?

=

?I

MCM−1

S

0

I

??ME

0

0MS

?

,(12)

According to Eqs. (11) and (12), we may derive the lepton mixing matrix (MNS), which

comes from the mismatch between the diagonalizations of the neutrino mass matrix and

charged lepton mass matrix, i.e., U = V†

eVν:

U ≈ (1 −1

2|MCM−1

S|2)V .(13)

As a result, the effective charged and neutral current interactions for charged leptons can

be written as

− LCC≈

g

√2eαγµPLUαiνiWµ+ h.c. ;

g

cosθw

(14)

−LNC≈eαγµ

?

(UU†)αβ(−1

2+ sin2θw)PL+ δαβsin2θwPR

?

eβZµ. (15)

The MNS matrix in Eq. (13) is non-unitary, which is mainly because the large mixing

between charged leptons and heavy vector like fermions. To a better degree of accuracy,

we have UU†≈ 1 − |MCM−1

precision electroweak data (e.g., on the invisible width of the Z0boson, universality tests

S|2. A global analysis of current neutrino oscillation data and

and rare decays) has yield quite strong constraints on the unitarity of U. Translating the

numerical results of Refs. [17–20] into the restriction on |MCM−1

at the 90% confidence level. In addition, interactions in Eqs. (14) and (15) will lead to tree

S|2, we have

< 1.6 · 10−2

< 1.0 · 10−2

< 1.0 · 10−2

|MCM−1

S|2=

< 1.1 · 10−2

< 7.0 · 10−5

< 1.6 · 10−2

< 7.0 · 10−5

< 1.0 · 10−2

< 1.0 · 10−2

,

(16)

level lepton flavor violations (as can be seen in Eq. (15)) and “ zero distance” effects [17] in

neutrino oscillations, which can be verified in the future long baseline neutrino oscillation

experiments.

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B. Lepton flavor violations (LFV)

Notice that the emergence of big unitary violation of MNS matrix can lead to observable

LFV effects. In this subsection, we investigate constraints on parameter space from LFV

processes.

In our model, ℓi→ 3ℓjmay occur at the tree level, just like the case in type III seesaw

mechanism. The branching ratios for the µ → 3e can be given by

?|UU†|ee(−1 + 2sin2θw)2+ 4sin4θw

Here Ω is the final states phase space integration

?zu

where zd= 4z0, zu= (1−√z0)2, z0= m2

Radiative decays, i.e., ℓi→ ℓj+γ occur at one-loop level. The branching ratios for these

processes can be written as

?????

with

√zαzβ(1 − zh) − 2zβ(1 − zh) + 2√zαzβ

(1 − zh)2

where zα= m2

BR(µ → e−e+e−) = |UU†|2

eµ

?Ω .(17)

Ω =

zd

?

1 −4z0

z

?

−2z2+ (1 + 3z0)z − 4z0(1 + z0) +2z0(1 − z0)2

z

?

λ

1

2(1,z0,z)dz , (18)

e/m2

µand λ(x,y,z) = x2+y2+z2−2xy−2xz−2yz.

BR(ℓβ→ ℓαγ) =

3πα

Fm4

16G2

β

?

i

YSαiY∗

SβiSi

?????

2

BR(ℓβ→ νβℓα¯ να) ,(19)

Sρ=

−zβ(1 − zh) −√zαzβ

(1 − zh)3

lnzh,(20)

α/M2

Si; zβ= m2

β/M2

Si; zh= M2

h/M2

Si.

In Fig. 2 (a), we plot BR(µ → 3e) as function of |UU†|2

current experimental constraints. Our result shows that, to meet the experimental data,

eµ. The horizon line stands for

|UU†|2

like fermion, we plot, in Fig. 2 (b), BR(µ → eγ)/|YSY†

Mh= 120 GeV. We find that, to get big Yukawa coupling YS, MSmust lie around 270 GeV

eµmust lie below 8.5 × 10−6. Assuming that there is only one generation vector

S|2

eµas function of MSby setting

or be heavier than several TeV.

C.Electromagnetic properties of heavy Majorana neutrinos

The electromagnetic properties of Majorana neutrinos show up, in a quantum field the-

ory, as its interaction with the photon, and is described by the following effective interaction

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1.?10?6

1.5?10?62.?10?6

3.?10?6

5.?10?6

7.?10?6

0.00001

5.?10?14

1.?10?13

5.?10?13

1.?10?12

5.?10?12

|UU†|eµ

BR(µ → 3e)

(a)

300500 7001000

1.?10?6

0.00001

0.0001

0.001

Mi

BR(µ → eγ)/|(YSY†

S)|2

eµ

(b)

FIG. 2: Branching ratios for (a) µ → 3e and (b) µ → eγ.

vertex: Leff=¯ψΓµψAµ. The most general matrix element of Leffbetween two one-particle

states, i.e., ?p′,s′|Jµ(0)|p,s? = ¯ us′(p′)Γµus(p), which is consistent with the Lorentz invari-

ance, can be written as

¯ u(p2,s2)Γµu(p1,s1) = ¯ u(p2,s2)?E(q2)γµ− M(q2)iσµνqν+ H(q2)qµ?u(p1,s1)

+¯ u(p2,s2)?G(q2)γµγ5− T (q2)iσµνqνγ5+ S(q2)qµγ5?u(p1,s1)

where q = p2− p1. 2MM(0) and 2MT (0) correspond to the magnetic moment and electro

dipole moment of heavy neutrinos, respectively.

, (21)

Due to the Majorana nature, the magnetic moment of heavy Majorana neutrinos is zero.

There is only transition magnetic moment for them. In the model considered, we have four

diagrams contributing to the transition magnetic moment, which are depicted in Fig. 3.

The Yukawa interactions of heavy Majorana neutrinos with Φ and S can be rewritten in the

following way

?

through which we can derive relevant feynman rules. Assuming that heavy Majorana neu-

1

2

NC

αΦ−(YT)αiPRSC

i+ NαΦ+(Y†)αiPLSi

?

+1

2

?

SiYiαΦ−PRNα+ SC

i(Y∗)iαΦ+PLNC

α

?

, (22)

trinos are nearly degenerate, i.e., Mα≈ Mβ≈ M, we derive the transition magnetic moment

for heavy Majorana neutrinos

M2

64π2

aN

αβ=

?

(Y†

N)βi(YN)iα− (YT

N)βi(YN)∗

iα

??I(M2

Φ,M2,M2

i) − I(M2

i,M2,M2

Φ)?

,(23)

with

I(A,B,C) =

?

dx

x(1 − x)2

(1 − x)A + x(x − 1)B + xC,

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B

Nα

Nβ

Si

Φ

(a)

B

Nα

Nβ

SC

i

Φ

(b)

B

Nα

Nβ

Φ

SC

i

(c)

B

Nα

Nβ

Φ

Si

(d)

FIG. 3: Feynman diagrams contributing to heavy Majorana neutrino transition magnetic moment.

where Miand MΦare the mass eigenvalues of heavy vector-like fermion S and scalar Φ,

respectively.

Now, we turn to some numerical analysis.As shown in Eq. (2), YNSLΦNRis to-

tally the interaction of new fields beyond the SM, so that there is no experimental con-

√4π (to satisfy the perturbation theory). We plot, in Fig.

straint on YNexcept O(YN) <

4, |aN

(YT

αβ/[(Y†

N)βi(YN)iα− (YT

N)βi(YN)∗

N)βi(YN)∗

iα]| as function of MN. Assuming O([(Y†

N)βi(YN)iα−

iα]) ∼ 1, We can find that the transition magnetic moment of heavy Majorana

neutrinos can be of O(10−2) for special parameter settings. Our result in Eq. (23) is different

from that in Ref. [22] for not considering the Yukawa coupling NC

RΦSR, which is forbidden

by the Z2symmetry in our model. Given the large electromagnetic form factors, heavy

Majorana neutrinos can be produced at the LHC through the electromagnetic interaction.

D.Collider signatures

We switch to comment on the collider signatures of our model. All of the new particles

introduced in the model lie around several hundred GeV. Singly charged scalar and vector

like fermions can be produced through the electromagnetic interaction at the LHC. The

most promising production channel may be pp → S+S−→ ℓ+ℓ−jjjj for heavy charged

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|aN

αβ/[(Y†

N)βi(YN)iα− (YT

N)βi(YN)∗

iα]|

MN (GeV)

FIG. 4: Numerical illustration for the transition magnetic moment of heavy Majorana neutrinos

as function of MN. We assume there is only one generation vector-like fermion S and choose

(MΦ, Mi) = (350, 150) (GeV) for solid line, (300, 200) (GeV) for dashed line and (300, 250) (GeV)

for dotted line.

fermions and pp → Φ+Φ−→ jjjj for heavy charged scalar. The production cross sections

for these charged particles at the LHC (with√s = 14TeV) are about several fb when heavy

particle masses lie around 300 GeV [21, 22]. The large transition magnetic moment can help

to produce the heavy Majorana neutrinos at the LHC. Its signatures are similar to that in

Type-III seesaw model [23–25]. The only distinguish is that, heavy neutrinos can not be

produced through weak interactions and must be produced in pair in our model.

IV. CONCLUDING REMARKS

In this paper, we have proposed a novel TeV-scale seesaw mechanism. One salient feature

of our model is that Dirac neutrino mass matrix raises only at the loop level. As a result,

the heavy Majorana neutrinos can be several hundred GeV. Another salient feature is that

heavy Majorana neutrinos can get large electromagnetic form factors, through which they

can be produced and detected at the LHC. We have derived light Majorana neutrino mass

formula and calculated constraints on parameter space from LFV processes. At last we

have discussed the signatures of heavy fermions (vector-like fermions and heavy Majorana

neutrinos) and scalar at the LHC.

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Acknowledgments

The author thanks to Tong Li, Yi Liao and Shu Luo for useful discussion. This work was

supported in part by the National Natural Science Foundation of China.

Appendix A: An alternative setting to the model

Beside the model presented in section II, we can extending the SM with different particle

contents, which may lead to the same TeV seesaw mechanism. For example, we can sub-

stitute SL,SR, Φ, with vector like fermion triplets ΣL,ΣRand scalar triplet ∆. In this case

the lagrangian can be written as

L = LSM− ℓLYΨ˜HΣR− ΣLMΣΣR− YNTr[ΨL∆]NR+ V (H,∆) .(A1)

Here the weak hypercharge of the Ψ and Σ are zero. ΨL, Φ and NRare odd , while the

other fields are even under Z2transformation.

The Higgs potential can be written as

V =

1

2m2

HH†H +1

2m∆Tr[∆†∆] +1

4λ(H†H)2+ λXH†Σiσ2H + ··· , (A2)

where dots denote Higgs potential terms we don’t concern.

[1] SNO Collaboration, Q. P. Ahamed et al., Phys. Rev. Lett. 89, 011301 (2002).

[2] For a review, see: C. K. Jung et al., Ann. Rev. Nucl. Part. Sci. 51, 451 (2001).

[3] KamLAND Collaboration, K. Eguchi et al., Phys. Rev. Lett. 90, 021802 (2003).

[4] K2K Collaboration, M. H. Ahn et al., Phys. Rev. Lett. 90, 041801 (2003).

[5] S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979).

[6] P. Minkowski, Phys. Lett. B 67, 421 (1977); T. Yanagida, in Workshop on Unified Theories,

KEK report 79-18 p.95 (1979); M. Gell-Mann, P. Ramond, R. Slansky, in Supergravity (North

Holland, Amsterdam, 1979) eds. P. van Nieuwenhuizen, D. Freedman, p.315; S. L. Glashow,

in 1979 Cargese Summer Institute on Quarks and Leptons (Plenum Press, New York, 1980)

eds. M. Levy, J.-L. Basdevant, D. Speiser, J. Weyers, R. Gastmans and M. Jacobs, p.687;

R. Barbieri, D. V. Nanopoulos, G. Morchio and F. Strocchi, Phys. Lett. B 90, 91 (1980);

11

Page 12

R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980); G. Lazarides, Q. Shafi

and C. Wetterich, Nucl. Phys. B 181, 287 (1981).

[7] W. Konetschny and W. Kummer, Phys. Lett. B 70, 433 (1977); T. P. Cheng and L. F. Li,

Phys. Rev. D 22, 2860 (1980); G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181,

287 (1981); J. Schechter and J. W. F. Valle, Phys. Rev. D 22, 2227 (1980); R. N. Mohapatra

and G. Senjanovic, Phys. Rev. D 23, 165 (1981).

[8] R. Foot, H. Lew, X. G. He and G. C. Joshi, Z. Phys. C 44, 441 (1989).

[9] E. Ma and U. Sarkar, Phys. Rev. Lett. 80, 5716 (1998).

[10] M. Y. Keung and G. Senjanovic, Phys. Rev. Lett. 50, 1427 (1983); A. Pilaftsis, Z. Phys. C

55, 275 (1992); B. Bajc, M. Nemevsek and G. Senjanovic, Phys. Rev. D 76, 055011 (2007).

[11] J. Bernabeu, A. Santamaria, J. Vidal, A. Mendez, and J.W.F. Valle, Phys. Lett. B 187, 303

(1987); W. Buchmuller and D. Wyler, Phys. Lett. B 249, 458 (1990); W. Buchmuller and

C. Greub, Nucl. Phys. B 363, 345 (1991); A. Datta and A. Pilaftsis, Phys. Lett. B 278,

162 (1992); G. Ingelman and J. Rathsman, Z. Phys. C 60, 243 (1993); C.A. Heusch and P.

Minkowski, Nucl. Phys. B 416, 3 (1994); D. Tommasini, G. Barenboim, J. Bernabeu, and C.

Jarlskog, Nucl. Phys. B 444, 451 (1995).

[12] J. Gluza, Acta Phys. Polon. B 33, 1735 (2002); J. Kersten and A.Yu Smirnov, Phys. Rev. D

76, 073005 (2007); X. G. He, S. Oh, J. Tandean and C. C. Wen, Phys. Rev. D 80, 073012

(2009).

[13] T. Han and B. Zhang, Phys. Rev. Lett. 97, 171804 (2006).

[14] F.del Aguila, J.A. Aguilar-Saavedra, A.M. de la Ossa, and M. Meloni, Phys. Lett. B 613, 170

(2005); F.del Aguila and J.A. Aguilar-Saavedra, JHEP 0505, 026 (2005); F.del Aguila, J. A.

Aguilar-Saavedra, and R. Pittau, JHEP 0710, 047 (2007); N. Haba, S. Matsumoto and K.

Yoshioka, Phys. Lett. B 677, 291 (2009).

[15] W. Chao, S. Luo, Z.Z. Xing, and S. Zhou, Phys. Rev. D 77, 016001 (2008); W. Chao, Z. Si,

Z.Z. Xing, and S. Zhou, Phys. Lett. B 666, 451 (2008); Z. Z. Xing, Phys. Lett. B 679, 255

(2009); W. Chao, Z. Si, Y. J. Zheng and S. Zhou, Phys. Lett. B 683, 26 (2010).

[16] A. Zee, Phys. Lett. B 93, 389 (1980), Erratum-ibid.B 95, 461 (1980).

[17] S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela and J. Lopez-Pavon, JHEP 0610,

084 (2006).

[18] E. Fernandez-Martinez, M. B. Gavela, J. Lepez-Pavon and O. Yasuda, Phys. Lett. B 649, 427

12

Page 13

(2007).

[19] Z. Z. Xing, Phys. Lett. B 660, 515 (2008).

[20] A. Abada, C. Biggio, F. Bonnet, B. Gavela and T. Hambye, JHEP, 0712, 061 (2007).

[21] A. Aparici, K. Kim, A. Santamaria and J. Wudka, Phys. Rev. D 80, 013010 (2010).

[22] A. Aparici, K. Kim, A. Santamaria and J. Wudka, arXiv: 0911.4103 [hep-ph].

[23] R. Franceschini, T. Hambye and A. Strumia, Phys. Rev. D 78, 033002 (2008).

[24] F. del Agulia and J. A. Aguilar-Saavedra, Nucl. Phys. B 813, 22 (2009).

[25] Tong Li and X. G. He, Phys. Rev. D 80. 093003 (2009).

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