Lepton flavor violating � decays in the type-III seesaw mechanism

Abdesslam Arhrib,1,* Rachid Benbrik,2,3,† and Chuan-Hung Chen2,3,‡

1Depatement de Mathematique, Faculte des Sciences and Techniques, Universite Abdelmalek Essaadi, B. 416, Tangier, Morocco2Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan

3National Center for Theoretical Sciences, Hsinchu 300, Taiwan(Received 10 February 2010; published 15 June 2010)

In this paper, the lepton flavor violating � ! ‘PðVÞ (P, V ¼ �0, �, �0, �0, !, �) and � ! 3‘ (‘ ¼ e,

�) decays are studied in the framework of the type-III seesaw model, in which new triplet fermions with a

zero hypercharge (Y ¼ 0) interact with ordinary lepton doublets via Yukawa couplings, and affect tree-

level leptonic Z-boson couplings. We investigate the experimental bound from the leptonic Z decay to get

constraints on the existing parameters space. We predict that the upper limits on the branching ratios of

� ! ‘PðVÞ and � ! 3‘ can reach the experimental current limits.

DOI: 10.1103/PhysRevD.81.113003 PACS numbers: 13.35.Dx

I. INTRODUCTION

In the standard model (SM) with massless neutrinos,lepton flavor is conserved. However, the current neutrinooscillation data experiments indicate with very convincingevidence that neutrinos are massive and lepton flavor aremixed [1]. This is a powerful incentive for considering newparticles and interactions those of the SM of quarks andleptons. If the neutrino oscillation phenomenon actuallytakes place, lepton flavor symmetry would be broken. Inthat case, however, lepton flavor violating (LFV) processesare still highly suppressed because of the smallness ofneutrinos masses. Hence, any experimental signal ofcharged LFV would be a clear indication of physics be-yond the SM. This fact has led to a great amount oftheoretical effects for revealing the underlying new physicsin the leptonic flavor sector.

LFV appears in various extensions of the SM. In par-ticular LFV decays � ! 3‘ (where ‘ ¼ e or �) are dis-cussed in various models [2–5]. Some of these models withcertain combinations of parameters predict that the branch-ing fractions for � ! 3‘ can be as high as 10�7, which isalready accessible in high-statistics B factory experiments.Searches for LFV in charged lepton sector such as � !�PðVÞ decays with a pseudoscalar or vector meson arealso discussed in models with Higgs mediated LFV pro-cesses [6,7], heavy singlet Dirac neutrinos [8], dimension-six effective fermionic operators that induce ���mixing[9], R-parity violation in SUSY [10], type-III two-Higgsdoublet models [10], and flavor changing Z0 bosons [10].At the LHC, the � leptons are produced predominantlyfrom decays ofB andDmesons andW and Z bosons. In thelow-luminosity phase, corresponding to an integrated lu-minosity of 10 fb�1 per year, one expects approximately1012 and 108 � leptons produced per year from heavy

meson and weak boson decays, respectively [11]. If werestrict the � from weak bosons decays only, and assumingthat a branching ratio close to the current upper limit, wecan expect approximately 10 events within the acceptancerange of a typical LHC general purpose detector after 1 yrof low-luminosity running. With 30 fb�1 of data, it shouldbe possible to probe branching ratios down to a level ofBrð� ! 3‘; ‘P; ‘VÞ � 10�8 at the LHC.In order to give mass to the neutrinos, several ways have

been studied in [12–15]. An alternative, equally valid andrather economical possibility is to extend the lepton sectorof the SM by a heavy triplet fermions and allow them tointeract with the ordinary lepton doublets via Yukawacouplings. In this scenario, the Higgs sector is unmodified,and a set of self-conjugate SUð2ÞL triplets of exotic leptonswith zero hypercharge are added to that model’s so-calledtype-III seesaw mechanism.The model has many interesting features, including the

possibility of having low seesaw scale of order a TeV torealize leptogenesis [16] and detectable effects at LHC[17,18] due to the fact that the heavy triplet leptons havegauge interactions being nontrivial under the SUð2ÞL gaugegroup. In particular, if kinematically accessible, thecharged component of the triplet will be produced inhigh energy collision, and its decay into Higgs and lightlepton [17] provides a rather spectacular signature.Fermionic triplet effects have been studied in the lepton

sector [19] such as � ! 3‘, ‘ ! ‘0�, Z ! ‘‘0, �� econversion and the anomalous magnetic moment of leptons(g� 2) [20,21]. Several other processes have not beenstudied in the context of type-III seesawmodel such as � !‘P and � ! ‘V.In this paper we try to demonstrate that we can have a

contribution to lepton flavour violating decays even withone triplet and singlet fermions. The paper is organized asfollows: In the next section, we will recall the basic fea-tures of type-III seesaw model and discuss the motivationswhich adds a triplet. In Sec. III we will discuss the con-straint on the Yukawa couplings coming from neutrino

*aarhrib@ictp.it†rbenbrik@mail.ncku.edu.tw‡physchen@mail.ncku.edu.tw

PHYSICAL REVIEW D 81, 113003 (2010)

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experiments. Section IV then discusses the analytical LFV� decay rates with estimates the corresponding LFV ob-servables and conclusions are drawn in Sec. V.

II. Z-MEDIATED LFV IN THE TYPE-III SEESAWMODEL

To study the lepton flavor violating effects in the so-called type-III seesaw models [13,14,22], we consider theSUð2ÞL fermionic triplet with the quantum number of (1, 3,0) under SUð3Þc � SUð2ÞL �Uð1ÞY gauge symmetry [14].For explaining the current data in neutrino physics, a modelwith only one triplet fermion is not sufficient; therefore,more fermionic triplets and/or singlets should be consid-ered. Since our purpose is to illustrate the � LFV in theframework of type-III seesaw models, for simplicity wefocus on the minimal extension of the SM (MSM), i.e. thecase with one triplet and one singlet fermions [21]. Thedetailed analysis in the model with three triplets could bereferred to Ref. [23]. Let us describe the model in moredetail to identify new tree-level FCNC in the lepton sector,the component fields of the triplet fermion is chosen to be

T ¼ T0=ffiffiffi2

pT�

Tþ T0=ffiffiffi2

p !

: (1)

In order to keep invariance under SUð2ÞL gauge trans-formation, we have required the transformation of T tobe T ! U�TUy. For studying flavor changing effects, weneed to understand the gauge couplings of SM leptons andtriplet fermion. The Yukawa sector with respect to thegauge symmetry SUð2ÞL �Uð1ÞY can be written as [21]

�LY ¼ Hy �eaYabE Lb þ yaT Trð �Tc

LLaHTÞ þ yaSH

Ti�2LaS

þ 12mT Trð �TTÞ þ 1

2mSSTCSþ H:c: (2)

whereHT ¼ ð�þ; �0Þ is the SMHiggs doublet, ea denotesthe right-handed lepton, and aðbÞ is the correspondinglepton flavor, LT ¼ ð�; ‘ÞL is the weak gauge doublet oflepton, Yab

E and yaT;S are Yukawa couplings, C is the charge

conjugation operator, andmTðSÞ is the mass of the new stuff

in triplet (singlet) fermion. Similarly, the relevant gaugekinetic terms are written as

L kin ¼ �Li 6D2Lþ �‘Ri 6D1‘R � Tr½ �T0Li 6D3T

0L�; (3)

where D2� ¼ @� þ ig=2 ~� � ~W� � ig0=2B�, D1� ¼ @� �ig0B� and D3� ¼ @� þ ig ~� � ~W� are the covariant deriva-

tives, T0 ¼ i�2T and the associated gauge transformationis T0 ! UT0Uy. Since the singlet fermion does not coupleto gauge boson, we do not show it in Eq. (3). Althoughcharged currents will induce flavor changing effectsthrough box and penguin diagrams, however, due to loopsuppression, the Z-mediated LFV induced at tree level willbe dominant. Consequently, we focus on the Z-bosonrelated interactions. By Eq. (3), the interactions in weakeigenstates of lepton are found by

L Z ¼ g

2 cosW�‘��ðcos2WXLPL � 2sin2WXRPRÞ‘Z�

(4)

with ‘L ¼ ðeL;�L; �L; TcRÞ, ‘R ¼ ðeR;�R; �R; T

cLÞ and

XL ¼ 13�3 03�1

01�3 �2cos2W= cos2W

� �;

XR ¼ 13�3 03�1

01�3 cos2W=sin2W

� �:

(5)

Here, we have taken Tc as the charge-conjugation state ofTþ. Since the couplings of the new charged leptons toZ-boson differ from those of ordinary leptons, we willsee that LFV at tree level will be induced by the misalign-ment between weak and physical states. To understand theeffects of LFV, we first introduce two unitary matrices VL;R

that transform the weak states to physical states by ‘LðRÞ !VLðRÞ‘LðRÞ. Then the matrices in Eq. (5) become

Z� � V�X�Vy� ¼ V�IV

y� þ V�ðX� � IÞVy

� (6)

with � ¼ L, R and I being the unit matrix. Immediately,we can see that the lepton flavor changing effects areassociated with

Z�ij ¼ �V�i4V��j4; L ¼ �2cos2W= cos2W � 1;

R ¼ cos2W=sin2W � 1: (7)

To get the information on V�i4 and V�4i, we need to study

the detailed mass matrix for charged leptons.After SSB, the mass matrix for charged lepton could be

written as

�L‘Y ¼ �‘RM‘‘L þ H:c: (8)

with

M‘ ¼ðYEÞ3�3v=

ffiffiffi2

p j 03�1

��� j ��ðyyTÞ1�3v=

ffiffiffi2

p j mT

0B@

1CA: (9)

Moreover, by choosing a suitable basis, indeed Eq. (9) canbe further simplified. For instance, using the transforma-tion ‘LðRÞ ! ULðRÞ‘LðRÞ with

ULðRÞ ¼�ULðRÞ3�3 j 03�1

��� j ��01�3 j 1

0@

1A;

the matrix YE in Eq. (9) can be diagonalized and Eq. (9)becomes

M‘ ¼ðmEÞ3�3 j 03�1

��� j ��ðhy

TÞ1�3 j mT

0@

1A; (10)

where diagðmEÞ ¼ diagð �URðYEv=ffiffiffi2

p Þ �UyLÞ ¼

ðme;m�;m�Þ and hT ¼ �ULyTv=ffiffiffi2

p. Since M‘ still has

off-diagonal elements, clearly me;�;� are not physical ei-

ABDESSLAM ARHRIB, RACHID BENBRIK, AND CHUAN-HUNG CHEN PHYSICAL REVIEW D 81, 113003 (2010)

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genstates. In addition, from Eq. (10) one can expect that thelepton flavor violating effects will be associated with hT .To get the physical states, we use VR;L introduced early to

diagonalize the mass matrix of lepton, i.e. Mdia‘ ¼

VRM‘VyL . The individual information on VL and VR can

be obtained by

Mdiay‘ Mdia

‘ ¼ VLMy‘M‘V

yL; Mdia

‘ Mdiay‘ ¼ VRM‘M

y‘ V

yR

(11)

with

My‘M‘ ¼

myEmE þ hTh

yT j hTmT

��� j ��mTh

yT j m2

T

0B@

1CA;

M‘My‘ ¼

mEmyE j mEhT

��� j ��hyTm

yE j m2

T þ hyThT

0B@

1CA:

Clearly, VL and VR are the unitary matrices to diagonalize

the matrixMy‘M‘ andM‘M

y‘ , respectively. Expectably, the

off-diagonal elements of flavor mixing matrices will beassociated with (mEiihTi) and mThTi which reflect themixture of ordinary quarks and triplet fermion. Althoughin general the 4� 4 matrices will be complicated andunknown, however, since the introduced triplet fermionsare much heavier than SM leptons, i.e. mT � mEii, hTi v, for a good approximation we can expand VLðRÞ to be

VLðRÞ � 14�4 þ �LðRÞ where �LðRÞ is regarded as

OðhTi=mTÞ½OðmEiihTi=m2TÞ�. Comparing with Eq. (7), we

see V�i4ð4iÞ � ��i4ð4iÞ. From Eq. (11), we can derive the

leading order for flavor mixing as

�Li4 � ��L4i � � mThTi

m2T �m2

Ei � jhTij2; (12)

�Ri4 � ���R4i � � mEihTi

m2T þ hy

ThT �m2Ei

: (13)

Sincem2T � mThTi � mEihTi, it is clear that the effects of

�Ri4ð4iÞ are negligible. Hence, the significant LFV in type-

III seesaw model is only associated with left-handed neu-tral currents.

III. CONSTRAINTS ON THE PHYSICALPARAMETERS

In this section, we discuss the constraints coming fromthe neutrino experiments on the relevant Yukawa couplingsyaT . After spontaneous symmetry breaking (SSB), where

the Higgs field is driven to obtain the VEV i.e. h�0i ¼v=

ffiffiffi2

p, the light neutrino mass matrix is given by

ðm�Þab ¼ �v2

2

�yaTy

bT

mT

þ yaSybS

mS

�(14)

The unitary PMNS matrix that diagonalizes the neutrinomass matrix Eq. (14) is given by

U ¼c12c13 s12c13 s13e

�i�

�s12c23 � c12s23s13ei� c12c23 � s12s23s13e

i� s23c13s12s23 � c12c23s13e

i� �c12s23 � s12c23s13ei� c23c13

0B@

1CA� diagð1; ei�; 1Þ: (15)

where, sij ¼ sinij, cij ¼ cosij (i, j ¼ 1, 2, 3), � is theCP-violating Dirac phase and � denotes the Majoranaphase. The experimental constraints on the neutrino massesand mixing parameters, at 2� level [24,25] are

7:3� 10�5 eV2 < �m2S < 8:1� 10�5 eV2 (16)

2:1� 10�3 eV2 < j�m2Aj< 2:7� 10�3 eV2 (17)

0:28< sin212 < 0:37; 0:38< sin223 < 0:68;

sin213 < 0:033:(18)

In this section, we focus mainly on the case of normalhierarchy (NH, m�

1 ¼ 0), and Inverted Hierarchy (IH,m�

3 ¼ 0) neglecting the Majorana phase. Using the aboveexperimental constraints, the neutrino masses are given by[24,25]

m�2 ¼

ffiffiffiffiffiffiffiffiffiffi�m2

S

q; m�

3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2

S þ �m2A

q(19)

in the case of NH, and

m�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2

A ��m2S

q; m�

2 ¼ffiffiffiffiffiffiffiffiffiffiffi�m2

A

q(20)

in the case of IH. The constraints on the neutrino massmatrix elements direclty translate into the physical Yukawacouplings yaT . Using Casas-Ibarra parametrization [26], onecan find a formal solution for the Yukawa couplings yaT andyaS can be expressed as

yaT ¼ �i

ffiffiffiffiffiffiffiffiffi2mT

pv

ð ffiffiffiffiffiffiffim�

2

pcoszU�

a2 þffiffiffiffiffiffiffim�

3

psinzU�

a3Þ (21)

yaS ¼ �i

ffiffiffiffiffiffiffiffiffi2mS

pv

ð� ffiffiffiffiffiffiffim�

2

psinzU�

a2 þffiffiffiffiffiffiffim�

3

pcoszU�

a3Þ (22)

for NH, and

yaT ¼ �i

ffiffiffiffiffiffiffiffiffi2mT

pv

ð ffiffiffiffiffiffiffim�

1

pcoszU�

a1 þffiffiffiffiffiffiffim�

2

psinzU�

a2Þ (23)

yaS ¼ �i

ffiffiffiffiffiffiffiffiffi2mS

pv

ð� ffiffiffiffiffiffiffim�

1

psinzU�

a1 þffiffiffiffiffiffiffim�

2

pcoszU�

a2Þ (24)

for IH, where a ¼ 1, 2, 3 and z is a complex parameter. In

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order to study the effect of the z parameter, we show inFig. 1 the relative size of the Yukawa couplings yaT as afunction of ImðzÞ for IH (left panel) and NH (right panel)with fixedmT ¼ 1 TeV, when ignoring the influence of theMajorana phase �. As we can see from both panels, forlarge ImðzÞ, the Yukawa couplings remain large and evenOð1Þ, which can be understood from Eqs. (21)–(24) whereyaT are proportional to eImðzÞ. This allows to account for theexperimental values of neutrino masses without fine-tuningthe Yukawa couplings due to a cancellations in combina-tion of them, the observable effects are then possible.

IV. DECAY RATES FOR � ! ‘M AND � ! 3‘

Based on previous analysis, now we can study the LFVof the type-III seesaw model on semileptonic � ! ‘M withM ¼ ðP; VÞ and leptonic � ! 3‘ decays. According to theinteractions in Eq. (4) and Z couplings in the SM, therelevant effective Hamiltonian for � flavor changing de-cays can be written as

H ¼ 4GFffiffiffi2

p cos2WZLi3ðgfL �f��PLf �‘i��PL�

þ gfR�f��PRf �‘i��PL�Þ (25)

with

gfL ¼ I3f �Qfsin2W; gfR ¼ �Qfsin

2W; (26)

where we have used the equalities cosW ¼ mW=mZ and

g2=8m2W ¼ GF=

ffiffiffi2

p, f could be leptons and quarks, and I3f

and Qf denote the third component of weak isospin and

electric charge of the particle f, respectively. Since semi-

leptonic � decays are associated with meson productionwhere the decay constants of nonperturbative hadroniceffects involve, for dealing with the hadronic effects, asusual the decay constants of pseudoscalar (P) and vector(V) mesons are defined as

h0j �q���5qjPðpÞi ¼ ifPp�;

h0j �q��qjVðpÞi ¼ imVfV"�V

(27)

with "�V being the polarization vector of vector meson.

Moreover, for the modes associated with � and �0 mesons,we employ the quark-flavor scheme in which � and �0physical states are described by [27,28]

��0

� �¼ cos� � sin�

sin� cos�

� ��q

�s

� �(28)

with � being the mixing angle, �q ¼ ðu �uþ d �dÞ= ffiffiffi2

pand

�s ¼ s�s. Accordingly, the decay constant of�ð0Þ associatedwith �q���5q (q ¼ u, d) current is given by f�ð0Þ ¼cos�ðsin�Þf�q

. Consequently, for � ! ‘iP process, the

decay amplitude can be summarized by

hP‘ijH j�i ¼ ffiffiffi2

pGFfPm� cos2WYPZLi3

�‘iPR�; (29)

while for � ! ‘iV decay, it is

hV‘ijH j�i ¼ ffiffiffi2

pGFfVmV cos2WYVZLi3

�‘i"6 VPL�; (30)

where f�0 ¼ sin�ðcos�Þf�s,

10-5

10-4

10-3

10-2

10-1

100

6 8 10 12 14

|yi T

|

Im(z)

Inverted Hierarchy

10-5

10-4

10-3

10-2

10-1

100

6 8 10 12 14

|yi T

|

Im(z)

Normal Hierarchy

FIG. 1. The absolute value of Yukawa couplings yaT as a function of ImðzÞ for IH (left) and NH (right), withmT ¼ 1 TeV and� ¼ 0,in which the solid, dashed, and dotted lines represent a ¼ 1, 2, and 3, respectively.

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Y�0 ¼ � 1ffiffiffi2

p ; Y� ¼ � 1

2; Y�0 ¼ 1

2;

Y�0 ¼ cos2Wffiffiffi2

p ; Y! ¼ � 2

3ffiffiffi2

p sin2W;

Y� ¼ � 1

2þ 2

3sin2W:

(31)

Because of me;� m�, we have neglected the masses of

light leptons. Hence, the branching ratios (BRs) for semi-leptonic � LFV are found to be

Bð� ! ‘iPÞ ¼ G2F

16���

f2Pm3�cos

22WY2PjZLi3j2

��1�m2

P

m2�

�2; (32)

Bð� ! ‘iVÞ ¼ G2F

16���

f2Vm3�cos

22WY2V jZLi3j2

��1�m2

V

m2�

�2�1þ 2

m2V

m2�

�: (33)

For leptonic � ! 3‘ decays, although there involve nohadronic effects, however, they are three body decays andhave more complicated phase space. To simplify the for-mulation, we neglect the effects of light lepton masses.Hence, using the interactions in Eq. (4), the BR for � ! 3‘is given by

Bð� ! ‘i‘ �‘Þ ¼ cos22W jZLi3j2ð jg‘Lj2 þ jg‘Rj2Þ�Bð� ! ‘�� ��‘Þ (34)

where ¼ 2 for ‘i ¼ ‘ and ¼ 1 for ‘i � ‘.After introducing the contributions of Z-mediated LFV

in the type-III seesaw model, in order to constrain the freeparameters of ZLi3, we have to find out the possible strictlimits. As known that the Z-mediated effects at tree level inthe SM are flavor conserved, intuitively the flavor violating

decays Z ! ‘i �‘j with i � j, � ! ‘ðP; VÞ and � ! 3‘ etc.

could give strong constraints on the unknown parameters.Therefore, by examining the relation of the BR and the

associated parameter, one can easily find jZLijj ¼j�Li4�

�Lj4j /

ffiffiffiffiffiffiffiBR

p. By taking BðZ ! ‘i �‘jÞ 10�6 and

Bð� ! ½‘ðP; VÞ; 3‘�Þ 10�7, we see j�Li4��Lj4j / 10�3

and j�Li4��Lj4j / 10�4, respectively. However, if we think

further, the current high precision measurements of Z !‘i �‘i processes in fact will provide more severe limits.Roughly speaking, the key reason can be understood bywhich the allowed range for new physics is directly gov-

erned by small errors of data for Z ! ‘i �‘i, i.e. j�Li4j2 /��=�Z � ½�expðZ ! ‘i �‘iÞ � �SMðZ ! ‘i �‘iÞ�=�Z 10�5

[29]. Although the constraint in the real situation willdepend on the detailed characters of the process, however,

we will adopt current data of Z ! ‘i �‘i with 1� errors asthe inputs and the BRs for � ! ‘ðP; VÞ and � ! 3‘ decaysare our predictions.

In terms of the Z couplings in Eq. (4), the BR for Z !‘i �‘i decay with new effects can be formulated as

B ðZ ! ‘i �‘iÞ ¼ BSMðZ ! ‘i �‘iÞ þ �Zj�Li4j2 (35)

with

BSMðZ ! ‘i �‘iÞ ¼ GFm3Z

3ffiffiffi2

p��Z

��cos2W

2

�2 þ sin4W

�;

�Z ¼ m3ZGF

6ffiffiffi2

p��Z

cos22WL: (36)

Because of m‘ mZ, here we have dropped the mass oflepton. In addition, we also neglected the terms that powerin free parameter is higher than j�Li4j2. As a result, theallowed range for unknown parameters can be bound by

j�Li4j2 ¼ �Bi

�Z

¼ 1

�Z

½BexpðZ ! ‘i �‘iÞ �BSMðZ ! ‘i �‘iÞ�:(37)

By using GF ¼ 1:16634� 10�5 GeV�2, sin2W ¼ 0:223,and�Be;�;� ¼ ð4; 7; 8Þ � 10�5, where the values are taken

from 1� errors of data for Z ! ‘i �‘i [29], the upper limit onj�Li4j is given in Table I.Thus, based on Eqs. (32) and (33), the upper limits on

the BRs for � ! ‘ð�0; �; �0Þ and � ! ‘ð�0; !;�Þ areshown in Tables II and III, respectively. Here, we haveused the hadronic values as

f� ¼ 0:13; f� ¼ 0:11; f0� ¼ 0:135;

f� ¼ 0:216; f! ¼ 0:187; f� ¼ 0:237(38)

in units of GeV. From the values in the tables, we seeclearly that the upper limits on the BRs for � !‘ð�0; �0; �Þ could be Oð10�8Þ and the order in size isBð� ! ‘�0Þ>Bð� ! ‘�0Þ>Bð� ! ‘�0Þ. Therefore,the semileptonic � ! ‘ð�0; �0Þ decays could be the goodcandidates to probe the Z-mediated � LFV.With the same constraints shown in the Table I, the

values of BRs for leptonic � decays formulated byEq. (34) are presented in Table IV. It is clear that theBRs for all � ! 3‘ decays are of order 10�8 and thepredictions are close to each other. Furthermore, from the

TABLE I. Upper limit on �Li4 with 1� errors of BðZ ! ‘i �‘iÞ.Mode e�eþ ���þ ���þ

j�Li4j 0.016 0.021 0.023

TABLE II. Upper limits on the BRs (in units of 10�8) for � !‘ð�0; �; �0Þ decays with 1� errors of BðZ ! ‘i �‘iÞ as the con-straints.

Mode � ! ðe; �Þ�0 � ! ðe;�Þ� � ! ðe;�Þ�0

Current limit (8.0, 11) (9.2, 6.5) (16, 13)

This work (2.8, 5.0) (0.6, 1.0) (0.4, 0.7)

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Table IV, one can find that the value of Bð� ! 3�Þ is alittle bit larger than the current experimental upper limit. Itseems that � ! 3� provides the strictest constraint on thefree parameters. However, by reexamining the constraints

of Z ! ‘i �‘i, we find that the reverse situation is arisenfrom the errors of Z ! ð���þ; ���þÞ being larger thanthat of Z ! e�eþ, i.e. �B� �B� >�Be. If we adopt

3� of the world average BðZ ! ‘i �‘iÞ ¼ ð3:3658�0:0023Þ% for ‘ ¼ e, � and �, the new upper limits onthe BRs for semileptonic and leptonic decays are found tobe

B½� ! ‘ð�0; �; �0Þ�< ð4:2; 0:8; 0:6Þ � 10�8;

B½� ! ‘ð�0; !;�Þ�< ð2:5; 0:1; 1:6Þ � 10�8;

B½� ! ð3‘;�e�eþ; e���þÞ�< ð3:1; 2:0; 2:0Þ � 10�8:

(39)

Clearly, precision measurements of Z ! ‘i �‘i play an es-sential role on the constraints.

V. CONCLUSIONS

We have investigated the lepton flavor violating effectsin the framework of type-III seesaw model by extendingthe SM with one SUð2ÞL triplet and singlet fermions.Because of the difference in weak charges between newand ordinary leptons, intriguingly Z-mediated LFV is gen-erated at tree level. Moreover, it is found that the signifi-cant effects only occur in the left-handed leptons. AlthoughLFV could be induced by charged currents through oneloop, comparing with tree contributions, they are sublead-ing effects and neglected in our analysis. To illustrate thenovel effects, we study the semileptonic � ! ‘M andleptonic � ! 3‘ decays. For numerical calculations, we

find that the precision measurements of Z ! ‘i �‘i play animportant role on the constraints of the free parameters.Furthermore, we find that the upper limits on the BRs for� ! ‘ð�0; �0; �Þ and � ! 3‘ could reach Oð10�8Þ in themodel under discussion.

ACKNOWLEDGMENTS

We would like to thank Carla Biggio for useful discus-sions and Goran Senjanovic for illuminating discussionsduring the ‘‘Signaling the arrival of the LHC era’’ work-shop in ICTP where this work was initiated. R. B. issupported by National Cheng Kung University GrantNo. HUA 97-03-02-063 and C. C.H. is supported by theNational Science Council of R. O. C. under Grant Nos.:NSC-97-2112-M-006-001-MY3.

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Mode � ! 3e � ! 3� � ! �e�eþ � ! e���þ

Current limit 3.6 3.2 2.7 3.7

This work 2.1 3.7 2.3 1.3

TABLE III. Upper limits on the BRs (in units of 10�8) for � !‘ð�0; !;�Þ decays with 1� errors of BðZ ! ‘i �‘iÞ as the con-straints.

Mode � ! ðe; �Þ�0 � ! ðe;�Þ! � ! ðe;�Þ�Current limit (6.3, 6.8) (11, 8.9) (7.3, 13)

This work (1.7, 3.0) (0.09, 0.2) (1.1, 1.7)

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