Study of the lepton flavor-violating Z0 ! decay

J. I. Aranda,1 J. Montano,2 F. Ramırez-Zavaleta,1 J. J. Toscano,3 and E. S. Tututi1,*1Facultad de Ciencias Fısico Matematicas, Universidad Michoacana de San Nicolas de Hidalgo,

Avenida Francisco J. Mujica S/N, 58060, Morelia, Michoacan, Mexico2Departamento de Fısica, CINVESTAV, Apartado Postal 14-740, 07000, Mexico, D. F., Mexico

3Facultad de Ciencias Fısico Matematicas, Benemerita Universidad Autonoma de Puebla,Apartado Postal 1152, Puebla, Puebla, Mexico

(Received 27 February 2012; published 8 August 2012)

The lepton flavor-violating Z0 ! decay is studied in the context of several extended models that

predict the existence of a new gauge boson named Z0. A calculation of the strength of the lepton flavor-

violating Z0 coupling is presented by using the most general renormalizable Lagrangian that includes

lepton flavor violation. We used the experimental value of the muon magnetic dipole moment to bound

this coupling, from which the ReðLRÞ parameter is constrained, and it was found that

ReðLRÞ 102 for a Z0 boson mass of 2 TeV. We also employed the experimental restrictions

over the ! and ! þ processes in the context of several models that predict the existence

of the Z0 gauge boson to bound the mentioned coupling. The most restrictive bounds come from the

calculation of the three-body decay. For this case, it was found that the most restrictive result is provided

by a vectorlike coupling, denoted as jj2, for the Z case, finding around 102 for a Z0 boson mass of

2 TeV. We used this information to estimate the branching ratio for the Z0 ! decay. According to the

analyzed models the least optimistic result is provided by the sequential Z model, which is of the order of

102 for a Z0 boson mass around 2 TeV.

DOI: 10.1103/PhysRevD.86.035008 PACS numbers: 12.60.Cn, 11.30.Hv, 14.70.Pw

I. INTRODUCTION

Besides the search for the Higgs boson or the quark-gluon plasma, experiments at the LHC are also dedicatedto looking for clues on the origin of the masses and extradimensions [1,2]. Experiments are also focused on findingnew particles beyond the standard model (SM) such asmagnetic monopoles, strangelets, supersymmetric parti-cles [3–5], etc. In fact, recent results found by the CMSand ATLAS collaborations [6–9] do not exclude the exis-tence of primed massive gauge bosons (Z0 or W 0) for amass approximately above the energy of 2 TeV. In par-ticular, the CMS Collaboration [6] has excluded theexistence of a Z0 gauge boson for a mass below 1.94 TeVat the 95% confidence level. Moreover, the ATLASCollaboration has established a lower limit of 2.21 TeVon the Z0 gauge boson mass [9]. The existence of the Z0boson is predicted in several extensions of the SM [10–14].The simplest one is that it simply adds an extra gaugesymmetry group U0ð1Þ [13–17] to the SM.

For the SM, flavor changing neutral current transitions inthe quark sector are allowed, but they are very suppressedby the Glashow-Iliopoulos-Maiani mechanism and be-cause they emerge at the one-loop level. However, thesetransitions could be enhanced greatly by the new physicseffects produced by the Z0qiqj couplings where qi and qjare up- or down-quarks [17,18]. On the other hand, in thelepton sector, the SM Lagrangian has an exact lepton flavor

symmetry. Although experimental data show that this sym-metry is not satisfied, since it has already been evidenced in

different situations in neutrino oscillations, in which onlythe total leptonic number is conserved [19]. Calculation oftransitions between charged leptons is reasonably simplerthan calculations of transitions in the quark sector, sincethe former leads us without the complications of theCabibbo-Kobayashi-Maskawa elements nor the complica-tions of the QCD matrix elements.Transitions between charged leptons is an important

issue since if transitions between charged leptons occur,it is a clear signal of lepton flavor violation (LFV). In theminimally extended standard model, at which right-handedneutrinos are added, the branching ratio Brð ! Þ be-haves as the mass of the neutrino over the W gauge bosonmass to the fourth power; the branching ratio is less than1040 [20] which is very far from present experimental

capabilities. This value is such that, when compared withthe present experimental data bound given by the ParticleData Group [21], namely Brð ! Þ< 108, is not asrestrictive as that obtained from the minimally extendedSM. In order to make the theoretical value of the branchingratio consistent with the experimental results, it is neces-sary to go into a theory beyond the SM which accounts forthe LFV. There exist several theories beyond the SM thatpredict LFV [10,13,22]. However, the simplest one isprovided by making minimal extensions of the SM [13].In this work we calculate a bound for the coupling Z0

by using the most general renormalizable Lagrangian,which includes lepton flavor violation mediated by a new*tututi@umich.mx

PHYSICAL REVIEW D 86, 035008 (2012)

1550-7998=2012=86(3)=035008(10) 035008-1 2012 American Physical Society

neutral massive gauge boson (namely, the Z0 gauge boson).We bound the foregoing coupling in the spirit of a model-independent approach (since no assumptions or extraparameters are required) by means of the current experi-mental value of the magnetic dipole moment of the-lepton and the experimental restrictions for the leptonflavor-violating ! and ! þ decays. For the ! and ! þ decay transitions, we calcu-lated it by resorting to different grand unification theories[13,17]. Then we determine bounds for the branching ratioof the process Z0 ! in the same context of grandunification theories used.

II. BOUNDING THE Z0 COUPLING

In order to calculate the branching ratio of the Z0 ! decay process, we employ the most general renormalizableLagrangian, which includes the lepton flavor violationmediated by a new neutral massive gauge boson, comingfrom any extended or grand unification model [23–25],which can be cast in the following way:

L NC ¼ Xi;j

½ fiðLfifjPL þRfifjPRÞfj

þ fjð

LfjfiPL þ

RfjfiPRÞfiZ0

; (2.1)

where fi is any fermion of the SM, PL;R ¼ 12 ð1 5Þ are

the chiral projectors, and Z0 is a new neutral massive

gauge boson predicted by several extensions of the SM[23–26]. The Llilj , Rlilj parameters represent the

strength of the Z0lilj coupling, where li is any charged

lepton of the SM. In the rest of the paper we will assumethat Llilj ¼ Lljli and Rlilj ¼ Rljli . The Lagrangian

in Eq. (2.1) includes both flavor-conserving and flavor-violating couplings mediated by a Z0 gauge boson. Thiswork is oriented to study the impact of lepton flavor-violating couplings mediated by a Z0 boson in theZ0 ! decay. For this purpose we need to bound thelepton flavor-violating coupling Z0. This task will berealized by using the experimental result of the muonanomalous magnetic dipole moment and the experimentalrestrictions for the lepton flavor-violating ! and ! þ decays. Moreover, in order to calculate thebranching ratio of the Z0 ! process it is necessary toknow the Z0 total decay width, which mainly depends onthe Z0 ! fi fi decays. These flavor-conserving couplingsare model dependent.

Here, we only consider the following Z0 bosons: theZS of the sequential Z model, the ZLR of the left-rightsymmetric model, the Z arising from the breaking

of SOð10Þ ! SUð5Þ Uð1Þ, the Zc resulting in E6 !SOð10Þ Uð1Þ, and the Z appearing in many

superstring-inspired models [16]. The flavor-conserving

couplings, QfiL;R [15–17], whose values are shown in

Table I for different extended models, are related to the

couplings as Lfifi ¼ g2QfiL and Rfifi ¼ g2Q

fiR ,

where g2 is the gauge coupling of the Z0 boson. For thevarious extended models we are interested in the gaugecouplings of Z0’s are

g2 ¼ffiffiffi5

3

ssinWg1g; (2.2)

where g1 ¼ g= cosW . g depends of the symmetry

breaking pattern being of Oð1Þ [27] and g is the weakcoupling constant. In the sequential Z model, the gaugecoupling g2 ¼ g1.

A. Muon anomalous magnetic dipole moment

The contribution of the Z0 vertex to the muonanomalous magnetic dipole moment is given through theFeynman diagram shown in Fig. 1.The flavor changing amplitude for the on-shell lili

vertex can be written as follows:

M ¼ uðp2Þuðp1Þðq; Þ; (2.3)

where the vertex function is given by

¼ eZ dDk

ð2ÞDT

; (2.4)

with

T ¼ ðð6kþ 6p2Þð6kþ 6p1Þ þm2j

Þ ðGVij þGAij5Þ þ 4mjð2k þ p1 þ p2Þ ðCVij þ CAij5Þ; (2.5)

TABLE I. Chiral-diagonal couplings of the extended models.

ZS ZLR Z Zc Z

QuL 0.3456 0:08493 1

2ffiffiffiffi10

p 1ffiffiffiffi24

p 22ffiffiffiffi15

p

QuR 0:1544 0.5038 1

2ffiffiffiffi10

p 1ffiffiffiffi24

p 22ffiffiffiffi15

p

QdL 0:4228 0:08493 1

2ffiffiffiffi10

p 1ffiffiffiffi24

p 22ffiffiffiffi15

p

QdR 0.0772 0:6736 1

2ffiffiffiffi10

p 1ffiffiffiffi24

p 22ffiffiffiffi15

p

QeL 0:2684 0.2548 3

2ffiffiffiffi10

p 1ffiffiffiffi24

p 12ffiffiffiffi15

p

QeR 0.2316 0:3339 3

2ffiffiffiffi10

p 1ffiffiffiffi24

p 12ffiffiffiffi15

p

QL 0.5 0.2548 3

2ffiffiffiffi10

p 1ffiffiffiffi24

p 12ffiffiffiffi15

p

FIG. 1. Diagram contributing to the magnetic and electricdipole moments of the li fermion.

J. I. ARANDA, et al. PHYSICAL REVIEW D 86, 035008 (2012)

035008-2

¼ðk2m2Z0 Þððkþp1Þ2m2

j Þððkþp2Þ2m2j Þ; (2.6)

where

GVij ¼jLijj2 þ jRijj2

2; (2.7)

GAij ¼jLijj2 jRijj2

2; (2.8)

CVij ¼Lij

Rij þRij

Lij

2; (2.9)

CAij ¼Rij

Lij Lij

Rij

2: (2.10)

After using the well-known Gordon identities, itis easy to see that there are contributions to themonopole ½Gðq2Þ, the magnetic dipole moment

½iðai=2miÞq, and the electric dipole moment

ðdi5qÞ. The contribution to the monopole is diver-

gent, but we only are interested in the magnetic and electricdipole moments, for which the contribution is free ofultraviolet divergences. After some algebra, the form fac-tors associated with the electromagnetic dipoles of li canbe written as follows:

ai ¼ x2i42

ðjLijj2 þ jRijj2Þfðxi; xjÞ

2xjxi

ReðLijRijÞgðxi; xjÞ

; (2.11)

di ¼exj

42mZ0ImðLij

RijÞgðxi; xjÞ; (2.12)

where

fðxi;xjÞ

¼Z 1

0dx

Z 1x

0dy

ð1 x yÞ2ð1þ x2i x2j Þðxþ yÞ x2i ðxþ yÞ2 1

;

(2.13)

gðxi;xjÞ¼

Z 1

0dx

Z 1x

0dy

1 2x 2y

ð1þ x2i x2j Þðxþ yÞ x2i ðxþ yÞ2 1:

(2.14)

In the above expressions the dimensionless variablesxi ¼ mi=mZ0 and xj ¼ mj=mZ0 have been introduced.

If we take mi ¼ m, and mj ¼ me, m, m and after

numerical evaluation it can be appreciated that fðx; xjÞis suppressed with respect to gðx; xjÞ by about 3 orders

of magnitude. Consequently, the x2fðx; xjÞ term in

Eq. (2.11) can be neglected, as it is irrelevant comparedwith xxjgðx; xjÞ. In particular, the result xxgðx; xÞ10xxegðx; xeÞ holds in the interval 1 TeV mZ0 3 TeV, thus we can neglect the electron contribution.Therefore, we can write only the dominant contributiongiven by

a ¼ xx

22ReðL

RÞgðx; xÞ (2.15)

d ¼ ex42mZ0

ImðLRÞgðx; xÞ: (2.16)

The anomalous magnetic moment of the muon is oneof the physical observables best measured. We will usethe experimental uncertainty on this quantity to boundReðL

RÞ. We will assume that the one-loop contri-

bution of the Z0 vertex to a is less than the experimen-

tal uncertainty, which is [21]

jaExp j< 6 1010: (2.17)

As ImðLRÞ is concerned, we can use the existing

experimental limit on the muon electric dipole moment,which is [21]

jdExp j< 0:1 1019 e cm: (2.18)

From the expressions given by Eqs. (2.15) and (2.16),one can write

jReðLRÞj< 22

aExp

xxgðx; xÞ; (2.19)

jImðLRÞj< 42

dExp

xgðx; xÞ: (2.20)

B. The two-body ! decay

The contribution of the flavor-violating Z0 vertex tothe ! decay is depicted in the diagrams shown inFig. 2. Notice that this transition is of a dipolar type andis model dependent since the coupling Z0lilj depends onboth the g2 coupling constant and the chiral couplingsQe

R, QeL [16–18,23]. The corresponding amplitude can be

reduced to

Mð ! Þ ¼ ieg2642m

uðpÞ q ½F1ðQeL Qe

RÞðL RÞ þ F2ðQeLL þQe

RRÞ

þ ðF1ðQeL Qe

RÞ ðL þRÞ þ F2ðQeLL Qe

RRÞÞ5uðpÞ; (2.21)

STUDY OF THE LEPTON FLAVOR-VIOLATING . . . PHYSICAL REVIEW D 86, 035008 (2012)

035008-3

with

F1 ¼ x2þ6

13x2

x2ð1x2Þlnx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14x2

px2

arcsechð2xÞþ1

;

F2 ¼ 2þZ 1

0dx

Z x

0dy

4x2ð1x2ÞðxyÞþð1þxyy2Þx2

4

x2

13x2

x2ð1x2Þlnx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14x2

px2

arcsechð2xÞþ1

;

where x ¼ m=mZ0 , with m being the mass of the taulepton. Furthermore, it can be observed that only a mag-netic dipole contribution is present, which is finite. Also,the muon mass has been neglected. After squaring theamplitude, one obtains the associated branching ratio

Brð ! Þ ¼ g2240964

½jF1ðQeL Qe

RÞ þ F2QeLj2jLj2

þ jF1ðQeR Qe

LÞ þ F2QeRj2jRj2m

;

(2.22)

where is the total decay width of the tau lepton. Thisbranching ratio must be less than the experimental con-straint BrExpð ! Þ< 4:4 108 [21,28], so this re-striction allows us to bound the jLj2 and jRj2parameters.

C. The three-body ! þ decay

We also can compute the contribution of the flavor-violating Z0 vertex to the ! þ decay (seeFig. 3). Since the three-body decay of the tau lepton comesfrom a tree-level Feynman diagram we only present theircorresponding branching ratio

Brð ! þÞ

¼ g223843

ðh1ðmZ0 ÞðjQeLLj2 þ jQe

RRj2Þ

þ h2ðmZ0 ÞðjQeLRj2 þ jQe

RLj2ÞÞm

; (2.23)

where

h1ðmZ0 Þ ¼Z 1

0dx

2x 1

ðx 1þm2Z0=m2

Þð2ð7 4xÞx 5Þ;

h2ðmZ0 Þ ¼Z 1

0dx

2x 1

ðx 1þm2Z0=m2

Þð1 2ðx 1ÞxÞ:

The branching ratio computed in Eq. (2.23) must be lessthan the corresponding experimental restriction to theprocess !þ, BrExpð ! þÞ< 2:1 108

[21,29], which allow us to get a bound on the flavor-violating jLj2 and jRj2 parameters.

For a Z0 boson mass ranging in the interval 1 TeV<mZ0 < 2 TeV the h2ðmZ0 Þ function is suppressed withrespect to the h1ðmZ0 Þ function by about 6 orders of mag-nitude. Thus, the interference term, h2ðmZ0 ÞðjQe

LRj2 þjQe

RLj2Þ, can be neglected, which simplifies the calcu-

lation of jLj2 and the jRj2 parameters. Therefore,

Eq. (2.23) becomes

Brð ! þÞ ¼ g223843

h1ðmZ0 ÞðjQeLLj2

þ jQeRRj2Þm

: (2.24)

III. NUMERICAL RESULTS AND DISCUSSION

Let us now introduce the numerical computations fordifferent cases in which we can bound the ReðL

RÞ,

jLj2, jRj2, and jLj2 þ jRj2 parameters,

which represent the strength of the Z0 coupling. Weuse the most restrictive bounds to predict the branchingratio of the Z0 ! decay in the context of the variousextended models mentioned above.

A. Bound according to the muon anomalousmagnetic dipole moment result

From the experimental uncertainty of the anomalousmagnetic dipole moment measurement and usingEq. (2.19) we bound the flavor-violating ReðL

RÞ

parameter as a function of the Z0 boson mass. In Fig. 4we can appreciate the behavior of the maximum ofReðL

RÞ as a function of the Z0 boson mass.

Notice that the growth is of the order of 102 for a Z0mass interval 1:9 TeV<mZ0

< 3 TeV.

The flavor-violating ImðLRÞ parameter, which is

computed from the experimental limit on the muon electricdipole moment and by using Eq. (2.20), gives a rather badconstraint since its least value is of the order of unity for aZ0 mass interval 1 TeV<mZ0 < 2 TeV.

FIG. 2. Feynman diagrams contributing to the lepton flavor-violating ! decay.

FIG. 3. Feynman diagram corresponding to the ! þdecay.

J. I. ARANDA, et al. PHYSICAL REVIEW D 86, 035008 (2012)

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B. Bounds according to the two- andthree-body decays

1. Vectorlike coupling

As concerns the two-body ! decay, considering avectorlike coupling implies that L ¼ R, and for

simplicity, in this case we define L . Solving

for jj2 from Eq. (2.22) we can write the following

inequality:

jj2 < 40964

m

BrExpð ! ÞT1

; (3.1)

where

T1 ¼ g22½jF1ðQeL Qe

RÞ þ F2QeLj2

þ jF1ðQeR Qe

LÞ þ F2QeRj2:

The graph in Fig. 5 shows the behavior of the maximum ofjj2 parameter provided by the inequality in Eq. (3.1).

We observe that the Z case offers the most restrictive

constraint, which ranges from 1 to 10 in the Z0 boson massinterval 1:9 TeV<mZ0 < 3 TeV. In order to compare thearising bounds from the different models, it is sufficient touse only a value for a mass of the Z0 boson since thedifferent curves in Fig. 5 are all monotonically increasing.Let us consider the T1 factor in Eq. (3.1) for the differentmodels with mZ0 ¼ 2 TeV. For the Z case T1 ¼1:17453 1013, for the ZLR case T1 ¼ 6:133351013, for the ZS case T1 ¼ 1:14458 1012, and finallyfor the Z case T1 ¼ 1:58561 1012. From these num-

bers we may infer that the Z case offers the most restric-

tive bound.In relation to the three-body ! þ decay, taking

a vectorlike Z0 boson and after solving for the jj2parameter in Eq. (2.24) we reach the following inequality:

jj2 < 3843

m

BrExpð ! þÞT2h1ðmZ0 Þ ; (3.2)

where

T2 ¼ g22ðQeL2 þQe

R2Þ:

In Fig. 6, the maximum of the jj2 parameter derived

from Eq. (3.2) can be appreciated. We note that its behavioris monotonically increasing for all the models analyzed.Let us emphasize that the Z case gives the most restrictive

constraint, which is about 3 orders of magnitude less thanthe respective previous bound derived from the ! decay. The strength of the jj2 parameter varies from

103 to 102 for a Z0 boson mass within the range1:9 TeV<mZ0 < 3 TeV. The T2 factor in Eq. (3.2) deter-mines the difference between the bounds coming from theanalyzed models. For the Z case, T2 ¼ 6:62832 103,

for the ZLR case T2 ¼ 3:50795 102, for the ZS caseT2 ¼ 6:48551 102, and for the Z case T2 ¼8:94823 102. From this analysis we conclude that theZ case provides the strongest bound.

1.5x10−2

2.0x10−2

2.5x10−2

3.0x10−2

3.5x10−2

4.0x10−2

4.5x10−2

2000 2250 2500 2750 3000

Re(

ΩLµ

τΩ* R

µτ)

mZ’ [GeV]

FIG. 4 (color online). Behavior of ReðLRÞ as a func-

tion of the Z0 boson mass.

100

101

102

103

2000 2250 2500 2750 3000

|Ωµτ

|2

mZ’ [GeV]

ZSZLR

ZχZη

FIG. 5 (color online). Behavior of jj2 as a function of theZ0 boson mass. This graph corresponds to the bound obtainedfrom the ! decay.

10−3

10−2

10−1

100

2000 2250 2500 2750 3000

|Ωµτ

|2

mZ’ [GeV]

ZSZLR

ZχZη

FIG. 6 (color online). Behavior of jj2 as a function of theZ0 boson mass. This graph corresponds to the bound obtainedfrom the ! þ decay.

STUDY OF THE LEPTON FLAVOR-VIOLATING . . . PHYSICAL REVIEW D 86, 035008 (2012)

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2. Maximal parity violation coupling

Returning to the case of the ! decay, consideringa vector-axial coupling, for simplicity, we take L ¼ 0.

Solving for jRj2 from Eq. (2.22) we can write the

following inequality:

jRj2 < 40964

m

BrExpð ! ÞT3

; (3.3)

where

T3 ¼ g22jF1ðQeR Qe

LÞ þ F2QeRj2:

The graph in Fig. 7 shows the behavior of the maximumof the jRj2 parameter provided by the inequality in

Eq. (3.3). We observe that the Z case offers the most

restrictive constraint, which ranges from 2 to 20 in the Z0boson mass interval 1:9 TeV<mZ0 < 3 TeV. To comparethe arising bounds from the different models, it is sufficientto use only a value for a mass of the Z0 boson since thedifferent curves in Fig. 7 are all monotonically increasing.For mZ0 ¼ 2 TeV, the T3 factor in Eq. (3.3) determines thediscrepancies among the bounds of the different modelsstudied. For the Z case T3 ¼ 5:87263 1014, for the

ZLR case T3 ¼ 2:65648 1013, for the ZS case T3 ¼6:14351 1013, and for the Z case T3 ¼ 7:928051013. From last analysis it is evident that the Z case

gives the most restrictive bound.In relation to the three-body ! þ decay,

taking a vector-axial coupling and after solving for thejRj2 parameter in Eq. (2.24) we reach the following

inequality:

jRj2 < 3843

m

BrExpð ! þÞT4h1ðmZ0 Þ ; (3.4)

where

T4 ¼ g22QeR2:

In Fig. 8 we show the maximum of the jRj2 parameter

derived from the inequality in Eq. (3.4). We note that itsbehavior is monotonically increasing for the all modelsanalyzed. Again the Z case gives the most restrictive

constraint, which is about 2 orders of magnitude less thanthe respective previous bound derived from the ! decay. The strength of the jRj2 parameter

varies from 103 to 102 for a Z0 boson mass within therange 1:9 TeV<mZ0 < 3 TeV. The T4 factor in Eq. (3.4)determines the difference between the bounds comingfrom the analyzed models. For the Z case T4 ¼3:31416 103, for the ZLR case T4 ¼ 2:21696 102,for the ZS case T4 ¼ 2:76799 102, and for the Z case

T4 ¼ 4:47412 102. The analysis evidences that the Z

case again gives the strongest bound.

3. General coupling

Here we analyze the most general bound, jLj2 þjRj2, which can be constrained from the ! and

! þ decays. Notice that the jLj2 þ jRj2parameter can only be obtained in this particular combi-nation for the Zc gauge boson derived from the E6 model,

since the jQeL;Rj diagonal couplings satisfy the special

relation jQeLj ¼ jQe

Rj.Regarding the two-body ! decay, from Eq. (2.22)

we can write the following inequality:

jLj2 þ jRj2 < 40964

g22

m

BrExpð ! Þj2F1 þ F2j2jQe

Lj2:

(3.5)

The graph in Fig. 9 shows the behavior of the maximum ofthe jLj2 þ jRj2 parameter provided by the inequal-

ity in Eq. (3.5). The intensity of the jLj2 þ jRj2parameter ranges from 16 to 106 in the Z0 boson massinterval 1:9 TeV<mZ0 < 3 TeV.

100

101

102

103

2000 2250 2500 2750 3000

| ΩR

µτ|2

mZ’ [GeV]

ZSZLR

ZχZη

FIG. 7 (color online). Behavior of jRj2 as a function of theZ0 boson mass. This graph corresponds to the bound obtainedfrom the ! decay.

10−3

10−2

10−1

100

2000 2250 2500 2750 3000

|ΩR

µτ|2

mZ’ [GeV]

ZSZLR

ZχZη

FIG. 8 (color online). Behavior of jRj2 as a function of theZ0 boson mass. This graph corresponds to the bound obtainedfrom the ! þ decay.

J. I. ARANDA, et al. PHYSICAL REVIEW D 86, 035008 (2012)

035008-6

In relation to the three-body ! þ decay, aftersolving for the jLj2 þ jRj2 parameter in Eq. (2.24)

we can write the following inequality:

jLj2 þ jRj2 < 3843

g22Qe2L

m

BrExpð ! þÞh1ðmZ0 Þ :

(3.6)

In Fig. 10 it can be appreciated that the maximum of thejLj2 þ jRj2 parameter as a function of the Z0 bosonmass is provided by the inequality in Eq. (3.6). We note thatits behavior is monotonically increasing for the Zc case.

The strength of the jLj2 þ jRj2 parameter varies

from 102 to 101 for a Z0 boson mass within the range1:9 TeV<mZ0 < 3 TeV. Let us emphasize that this con-straint is about 3 orders of magnitude less than the respec-tive previous bound derived from the ! decay.

C. The branching ratio of the Z0 ! process

In order to make predictions, we resort to the boundsobtained from the previous analysis. As discussed above,we are interested in studying the Z0 ! decay, whosebranching ratio can be written as

BrðZ0 ! Þ ¼ 1

24

mZ0

Z0

1 ðm þmÞ2

m2Z0

1 ðm mÞ2

m2Z0

1=2

2m2 þm2

m2Z0

ðm2 m2

Þ2m4

Z0

ðjLj2 þ jRj2Þþ 12

mm

m2Z0

ReðLRÞ

: (3.7)

where Z0 is the Z0 total decay width. On account of Z0 weinclude the total possible flavor-conserving and flavor-

violating decay modes [17,18], namely e e, ,

, e e, , , u u, c c, tt, d d, ss, b b, ucþ u c, tcþt c, and þ . To compute the branching ratios, we willuse the most restrictive bounds for the jLj2, jRj2,and jLj2 þ jRj2 parameters.

In Fig. 11 we show the branching ratio for the Z0 ! decay in the context of the sequential Z model, the left-right symmetric model, the SOð10Þ ! SUð5Þ Uð1Þmodel, and the superstring-inspired models in which E6

breaks to a rank-5 group [13,16]. From this figure, it ismore feasible to observe lepton flavor violation for a Z

gauge boson, since the related branching ratio can be ashigher as 7:4 101, while the more restrictive branchingratio corresponds to the sequential model, in which itsstrength can be as higher as 6:97 102.The graph in the Fig. 12 shows the branching ratio for

the Z0 ! decay in the context of the sequential Zmodel, the left-right symmetric model, the SOð10Þ !SUð5Þ Uð1Þ model, and the superstring-inspired modelsin which E6 breaks to a rank-5 group [13,16]. From this

5.0x10−2

1.0x10−1

1.5x10−1

2.0x10−1

2.5x10−1

3.0x10−1

3.5x10−1

2000 2250 2500 2750 3000

| ΩLµ

τ|2 +

|ΩR

µτ|2

mZ’ [GeV]

FIG. 10 (color online). Behavior of jLj2 þ jRj2 as afunction of the Z0 boson mass. This graph corresponds to thebound obtained from the ! þ decay.

20

30

40

50

60

70

80

90

100

110

2000 2250 2500 2750 3000

| ΩLµ

τ|2 +

|ΩR

µτ|2

mZ’ [GeV]

FIG. 9 (color online). Behavior of jLj2 þ jRj2 as afunction of the Z0 boson mass. This graph corresponds to thebound obtained from the ! decay.

10−2

10−1

100

2000 2250 2500 2750 3000

Br(

Z’ →

τµ)

mZ’ [GeV]

ZSZLR

ZχZη

FIG. 11 (color online). The branching ratio for the Z0 ! asfunction of the Z0 boson mass. This graph corresponds to avectorlike Z0 coupling.

STUDY OF THE LEPTON FLAVOR-VIOLATING . . . PHYSICAL REVIEW D 86, 035008 (2012)

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graph, again it is more feasible to observe lepton flavorviolation for a Z gauge boson, since the related branching

ratio can be as high as 7:4 101. However, we canobserve that the branching ratio for the Z gauge boson

has a similar strength, which makes it difficult to say,from a experimental point of view, to which Z0 modelit corresponds. In contrast, the more restrictive boundfor BrðZ0 ! Þ corresponds to the sequential model,in which the respective numerical value can be as highas 8 102.

In Fig. 13 we show the branching ratio for the Z0 ! decay in the context of the E6 ! SOð10Þ Uð1Þ model[13,16]. From this figure, we observe that the branchingratio can be as high as 6 101. This is the more intensebranching ratio for all the models studied, which impliesthat lepton flavor violation for a Z0 gauge boson in thecontext of the E6 ! SOð10Þ Uð1Þ model has a highprobability of being observed.

Let us estimate the discovery potential of leptonflavor violation mediated by a Z0 boson at present and inthe future at the LHC. Here, we use a Z0 boson mass of

2 TeV according to the current lower limit imposed by theCMSCollaboration [6]. The expected number of events,N,is given by N ¼ Lðpp ! Z0ÞBrðZ0 ! Þ, where L isthe integrated luminosity of the LHC,ðpp ! Z0Þ is the Z0production cross section at the LHC, which was calculatedin a recent work [30], and the corresponding branchingratio is 1:47 102. Currently, the most conservative in-tegrated luminosity corresponds to the ATLAS detector[31], which is L ¼ 6:57 fb1 at 4 TeV at the LHC.Therefore, we may assume L 10 fb1 at 7 TeV energycollision, where ðpp ! Z0Þ 40 fb [30], which impliesthat N 6 events. In the future, for a long run it would befeasible to reach an integrated luminosityL 100 fb1 at14 TeVat the LHC. At this energy collisionðpp ! Z0Þ 700 fb [30] and we obtain N 1029 events.

IV. FINAL REMARKS

The current experimental data by the CMSCollaboration evidence the possibility of the existenceof aZ0 gauge boson which is predicted in several theories.Whether this boson exists, it could give a solution to thedisagreement between the experimental results and thecorresponding results predicted by the SM on leptonflavor-violating transitions. In this work we employedthe most general renormalizable Lagrangian which in-cludes lepton flavor violation mediated by a new neutralmassive gauge boson to bound the Z0 coupling in thespirit of a model-independent approach. We used theexperimental uncertainty of the muon magnetic dipolemoment, which is measured with great precision, to ob-tain the maximum of ReðL

RÞ as a function of the

Z0 boson mass, which is of the order of 102 for1:9 TeV<mZ0 < 3 TeV. Additionally, we performed acalculation of the ! and ! þ transitionsand used their respective experimental restrictions inorder to bound the Z0 coupling. Specifically, we com-pute the maximum of the jLj2, jRj2, and

jLj2 þ jRj2 parameters as a function of the Z0

boson mass. We study three different cases: vectorlikecoupling, vector-axial coupling, and general coupling. Wefound that the most restrictive constraint corresponds tothat coming from the ! þ decay. From thisdecay, the most restrictive bound corresponds to the vec-torlike coupling, which is determined in the context of thesequential Z model, where jj2 ranges from 103 to

102 for 1:9 TeV<mZ0 < 3 TeV.To calculate the Z0 ! branching ratio we employ

the most restrictive bounds for the jLj2, jRj2, andjLj2 þ jRj2 parameters. Among the analyzed

models (ZS, ZLR, Z, Zc , and Z) we found that the

most restrictive branching ratio corresponds to the se-quential model in which BrðZ0 ! Þ is of the orderof 102 for a Z0 boson mass in the interval 1:9 TeV<mZ0 < 3 TeV.

10−2

10−1

100

2000 2250 2500 2750 3000

Br(

Z’ →

τµ)

mZ’ [GeV]

ZSZLR

ZχZη

FIG. 12 (color online). The branching ratio for the Z0 ! asfunction of the Z0 boson mass. This graph corresponds to avector-axial Z0 coupling.

1

2

3

4

5

6

7

2000 2250 2500 2750 3000

Br(

Z’ →

τµ)

x 1

0−1

mZ’ [GeV]

Zψ

FIG. 13 (color online). The branching ratio for the Z0 ! asfunction of the Z0 boson mass. This graph corresponds to a Zc

gauge boson.

J. I. ARANDA, et al. PHYSICAL REVIEW D 86, 035008 (2012)

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Surprisingly the most restrictive values for BrðZ0 ! Þare of the order of 102, in the broad range of the Z0 gaugeboson mass studied. This relaxed bound enablesthe possibility of LFV being observed by Z0 mediatedprocesses. Our results show that current experimental con-straints on lepton flavor violation weakly restrict this phe-nomenon mediated by an extra Z gauge boson. In fact, atpresent for a Z0 boson mass of 2 TeV we estimated around

6 events associated with the Z0 ! process for 7 TeV atthe LHC. In the future, for the same Z0 boson mass weobtained around 1029 events for 14 TeV at the LHC.

ACKNOWLEDGMENTS

This work has been partially supported by CONACYTand CIC-UMSNH.

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