Higgs-gauge boson interactions in the economical 3-3-1 model

PHYSICAL REVIEW D 73, 075005 (2006)

Higgs-gauge boson interactions in the economical 3-3-1 model

Phung Van Dong* and Hoang Ngoc Long†

Institute of Physics, VAST, P.O. Box 429, Bo Ho, Hanoi 10000, Vietnam

Dang Van Soa‡

Department of Physics, Hanoi University of Education, Hanoi, Vietnam(Received 13 March 2006; published 7 April 2006)

*Electronic†Electronic‡Electronic

1550-7998=20

Interactions among the standard model gauge bosons and scalar fields in the framework of the SU�3�C �SU�3�L � U�1�X gauge model with minimal (economical) Higgs content are presented. From thesecouplings, all scalar fields including the neutral scalar h and the Goldstone bosons can be identifiedand their couplings with the usual gauge bosons such as the photon, the charged W�, and the neutral Z,without any additional conditions, are recovered. In the effective approximation, the full content of thescalar sector can be recognized. The CP-odd part of the Goldstone associated with the neutral non-Hermitian bilepton gauge boson GX0 is decoupled, while its CP-even counterpart has the mixing in thesame way in the gauge boson sector. Masses of the new neutral Higgs boson H0

1 and the neutral non-Hermitian bilepton X0 are dependent on a coefficient of Higgs self-coupling (�1). Similarly, masses of thesingly charged Higgs boson H�2 and of the charged bilepton Y� are proportional through a coefficient ofHiggs self-interaction (�4). The hadronic cross section for production of this Higgs boson at the CERNLHC in the effective vector boson approximation is calculated. Numerical evaluation shows that the crosssection can exceed 260 fb.

DOI: 10.1103/PhysRevD.73.075005 PACS numbers: 12.60.Fr, 14.80.Cp

I. INTRODUCTION

Recent neutrino experimental results [1] establish thefact that neutrinos have masses and the standard model(SM) must be extended. Among the beyond-SM exten-sions, the models based on the SU�3�C � SU�3�L � U�1�X(3-3-1) gauge group have some intriguing features: First,they can give partial explanation of the generation numberproblem. Second, the third quark generation has to bedifferent from the first two, so this leads to the possibleexplanation of why the top quark is uncharacteristicallyheavy.

There are two main versions of the 3-3-1 models. In oneof them [2] the three known left-handed lepton componentsfor each generation are associated to three SU�3�L tripletsas ��l; l; lc�L, where lcL is related to the right-handed isospinsinglet of the charged lepton l in the SM. The scalar sectorof this model is quite complicated (three triplets and onesextet). In the variant model [3] three SU�3�L lepton tripletsare of the form ��l; l; �cl �L, where �cl is related to the right-handed component of the neutrino field �l (a model withright-handed neutrinos). The scalar sector of this modelrequires three Higgs triplets; therefore, hereafter we callthis version the 3-3-1 model with three Higgs triplets(331RH3HT). It is interesting to note that, in the331RH3HT, two Higgs triplets have the same U�1�X chargewith two neutral components at their top and bottom.Allowing these neutral components vacuum expectation

address: [email protected]: [email protected]: [email protected]

06=73(7)=075005(15)$23.00 075005

values (VEVs), we can reduce the number of Higgs tripletsto be two. As a result, the dynamics symmetry breakingalso affects the lepton number. Hence it follows that thelepton number is also broken spontaneously at a high scaleof energy. This kind of model was proposed in Ref. [4]; itsgauge boson mixing and currents have been considered indetail in Ref. [5].

Note that the mentioned model contains a very impor-tant advantage; namely, there is no new parameter, but itcontains a very simple Higgs sector. Hence, the significantnumber of free parameters is reduced. To mark the minimalcontent of the Higgs sector, this version is going to becalled the economical 3-3-1 model.

It is well known that the electroweak symmetry breakingin the SM is achieved via the Higgs mechanism. In theGlashow-Weinberg-Salam model there is a single complexHiggs doublet, where the Higgs boson h is the physicalneutral Higgs scalar which is the only remaining part ofthis doublet after spontaneous symmetry breaking (SSB).In the extended models there are additional charged andneutral scalar Higgs particles. The prospects for Higgscoupling measurements at the CERN LHC have recentlybeen analyzed in detail in Ref. [6]. The experimentaldetection of h will be a great triumph of the SM ofelectroweak interactions and will mark a new stage inhigh energy physics.

In extended Higgs models, which would be deduced inthe low energy effective theory of new physics models,additional Higgs bosons like charged and CP-odd scalarbosons are predicted. Phenomenology of these extra scalarbosons strongly depends on the characteristics of each newphysics model. By measuring their properties like masses,

-1 © 2006 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.73.075005

PHUNG VAN DONG, HOANG NGOC LONG, AND DANG VAN SOA PHYSICAL REVIEW D 73, 075005 (2006)

widths, production rates, and decay branching ratios, theoutline of physics beyond the electroweak scale can beexperimentally determined.

The mass spectrum of the mentioned scalar sector hasbeen presented in [4], and some couplings of the twoneutral scalar fields with the charged W and the neutral Zgauge bosons in the SM were presented. From explicitexpression for the ZZh vertex, the authors concluded thattwo vacuum expectations responsible for the second step ofSSB have to be in the same range: u� v, or theory needsone more third scalar triplet. As we will show in thefollowing, this conclusion is incorrect. That is why thiswork is needed.

The interesting feature compared with other 3-3-1 mod-els is the Higgs physics. In the 3-3-1 models, the generalHiggs sector is very complicated [7,8] and this prevents themodels’ predictability. The scalar sector of this model is asubject of the present study. As shown, by coupling of thescalar fields with the ordinary gauge bosons such as thephoton, the W, and the neutral Z gauge bosons, we are ableto identify the full content of the Higgs sector in the SMincluding the neutral h and the Goldstone bosons eaten bytheir associated massive gauge ones. All interactionsamong Higgs-gauge bosons in the SM are recovered.

Production of the Higgs boson in the 331RH3HT at theCERN LHC has been considered in [9]. In the scalar sectorof the considered model, there exists the singly chargedboson H�2 , which is a subject of intensive current studies(see, for example, Refs. [10,11]). The trilinear couplingZW�H�, which differs at the tree level from zero only inthe models with Higgs triplets, plays a special role in thestudy of phenomenology of these exotic representations.We shall pay particular attention to this boson.

The paper is organized as follows. Section II is devotedto a brief review of the model. The scalar fields and massspectrum are presented in Sec. III and their couplings withthe ordinary gauge bosons are given in Sec. IV. Productionof the H�2 at the CERN LHC are calculated in Sec. V. Weoutline our main results in the last section—Sec. VI.

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II. A REVIEW OF THE MODEL

The particle content in this model, which is anomalyfree, is given as follows:

aL � ��aL; laL; NaL�T � �1; 3;�1=3�;

laR � �1; 1;�1�;(1)

where a � 1, 2, 3 is a family index. Here the right-handedneutrino is denoted by NL ��R�c.

Q1L��u1;d1;U�TL��3;13�;

Q�L��d�;�u�;D��TL��3

;0�; ��2;3; uaR��1;23�;

daR��1;�13�; UR��1;

23�; D�R��1;�

13�: (2)

Electric charges of the exotic quarks U and D� are thesame as of the usual quarks, i.e. qU �

23 and qD�

� � 13 .

The SU�3�L � U�1�X gauge group is broken spontane-ously via two steps. In the first step, it is embedded in thatof the SM via a Higgs scalar triplet

� � ��01; �

�2 ; �

03�T � �3;�1

3� (3)

acquired with a VEV given by

h�i �1��2p �u; 0; !�T: (4)

In the last step, to embed the gauge group of the SM inU�1�Q, another Higgs scalar triplet

� � ��1 ; �02; �

�3 �

T � �3; 23� (5)

is needed with the VEV as follows:

h�i �1��2p �0; v; 0�T: (6)

The covariant derivative of a triplet is

D� � @� � igTaWa� � igXT9XB� @� � iP�

@� � iPNC� � iP

CC� ; (7)

where the matrices in (7) are given by [5]

P NC� �

g2

W3� �1��3p W8� � t

��23

qXB� 0 y�

0 �W3� �1��3p W8� � t

��23

qXB� 0

y� 0 � 2��3p W8� � t

��23

qXB�

0BBBB@

1CCCCA (8)

and

P CC� �

g��2p

0 c�W�� s�Y

�� X0

�c�W

�� s�Y

�� 0 c�Y

�� s�W

��

X0� c�Y

�� s�W

�� 0

0B@1CA: (9)

Here t gX=g � 3��2psW=

��4c2

W � 1q

, tan� � u! and W�� , Y�� and X0

� are the physical fields. The existence of y� is aconsequence of mixing among the real part (X0

� � X0�) with W3�, W8�, and B�; its expression is determined from mixing

matrix U given in the Appendix of Ref. [5],

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HIGGS-GAUGE BOSON INTERACTIONS IN THE . . . PHYSICAL REVIEW D 73, 075005 (2006)

y� U42Z� �U43Z0� � �U44 � 1��X0

� � X0��

2p (10)

with

U42��t�0�c’

��1�4s2

�0c2W

q�s’

��4c2

W�1q �

; U43��t�0�s’

��1�4s2

�0c2W

q�c’

��4c2

W�1q �

; U44��1�4s2

�0c2W

q: (11)

We remind the reader that ’ is the Z–Z0 mixing angle and �0 is the similar angle of W4, Z, Z0 mixing defined by [5]

t2’ �

��3� 4s2

W��1� 4t22��q

f�c2W � �3� 4s2W�t

22� u

2 � v2 � �3� 4s2W�t

22�!

2g

�2s4W � 1� �8s4

W � 2s2W � 3�t22� u

2 � �c2W � 2�3� 4s2W�t

22� v

2 � �2c4W � �8s

4W � 9c2W�t

22� !

2 ; (12)

TABLE I. Nonzero lepton number L of the model particles.

Fields �L lL;R NL �01 ��2 ��3 UL;R D2L;R D3L;R

L 1 1 �1 2 2 �2 �2 2 2

TABLE II. B and L charges of the model multiplets.

Multiplet � � Q1L Q�L uaR daR UR D�R aL laR

B charge 0 0 13

13

13

13

13

13 0 0

L charge 43�

23 �

23

23 0 0 �2 2 1

3 1

s�0 t2�

cW��1� 4t22�

q : (13)

After SSB the non-Hermitian physical gauge bosons W,X0, Y� gain masses given by

M2W �

g2v2

4; (14)

M2Y �

g2

4�u2 � v2 �!2�; M2

X � M2Y �M

2W: (15)

The Yukawa interactions which induce masses for thefermions can be written in the most general form as

L Y � �L�Y �L�

Y� �LmixY ; (16)

where

�L�Y�L�

Y��h011

�Q1L�UR�h0�� Q�L�D�R

�heab � �L�ebR�habpmn� � c�L�p� bL�m��n

�hd1a �Q1L�daR�hd�a �Q�L�uaR�H:c:;

(17)

LmixY � hu1a �Q1L�uaR � hu�a �Q�L�daR � h001� �Q1L�D�R

� h00�1�Q�L�UR � H:c: (18)

The VEV! gives the mass for the exotic quarksU andD�,u gives the mass for u1, d� quarks, while v gives the massfor u�, d1, and all ordinary leptons. As mentioned above,the VEV ! is responsible for the first step of symmetrybreaking, while the second step is due to u and v.Therefore the VEVs in this model have to be satisfied bythe constraints

u; v < !: (19)

The Yukawa couplings of Eq. (17) possess an extraglobal symmetry which is not broken by VEVs v, ! butby u. From these Yukawa couplings, one can find thefollowing lepton symmetry L as in Table I (only the fieldswith nonzero L are listed; all other fields have vanishingL). Here, L is broken by u which is behind L��0

1� � 2 (see

075005

also [12]), i.e., u is a kind of the lepton-number violatingparameter. It is interesting that the exotic quarks also carrythe lepton number. Thus, this L obviously does not com-mute with gauge symmetry. One can construct a newconserved charge L through L by making the linear com-bination L � xT3 � yT8 � zX�LI where T3 and T8 areSU�3�L generators. One finds the following solution [8]:x � 0, y � 4��

3p , z � 0, and

L �4��3p T8 �LI: (20)

Another useful conserved charge B, which is not broken byu, v, and !, is the usual baryon number B � BI. The Land B charges for the fermion and Higgs multiplets arelisted in Table II. Moreover, the Yukawa couplings of (18)conserve B and violate L with�2 units which implies thatthese interactions are very small.

Taking into account the famous experimental data [13]

Rmuon �� ! e��e~��

�� ! e�~�e��< 1:2% 90% CL (21)

we get the constraint Rmuon ’ M4W=M

4Y < 0:012.

Therefore, it follows that MY > 230 GeV. However, thestronger bilepton mass bound has been derived from con-sideration of the experimental limit on lepton-numberviolating charged lepton decays [14] of 440 GeV.

In the case of u! 0, analyzing the Z decay width [15],the Z–Z0 mixing angle is constrained by �0:0015 � ’ �

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0:001. From atomic parity violation in cesium, bounds forthe mass of the new exotic Z0 and the Z–Z0 mixing angles,again in the limit u! 0, are given [15],

�0:001 56 � ’ � 0:001 05; MZ2� 2:1 TeV: (22)

These values coincide with the bounds in the usual331RH3HT [16].

From the W width, one gets an upper limit [5]:

sin� � 0:08: (23)

III. HIGGS POTENTIAL

In this model, the most general Higgs potential has verysimple form

V��;�� 21�y��2

2�y�� 1��

y��2 � �2��y��2

� �3��y��y�� 4��y��y��: (24)

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Note that there is no trilinear scalar coupling and thismakes the Higgs potential much simpler than those in the331RH3HT [8,17] and closer to that of the SM. Theanalysis in Ref. [4] shows that, after symmetry breaking,there are eight Goldstone bosons and four physical scalarfields. One of two physical neutral scalars is the SM Higgsboson.

Let us shift the Higgs fields into physical ones,

� ��P0

1 �u��2p

��2�P0

3 �!��2p

0B@1CA; � �

��1�P0

2 �v��2p

��3

0B@1CA: (25)

The subscript P denotes physical fields as in the usualtreatment. However, in the following, this subscript willbe dropped. By substitution of (25) into (24), the potentialbecomes

V��;�� 21

��0

1 �u��2p

��0

1 �u��2p

�� 2 �

�2 �

��0

3 �!��2p

��0

3 �!��2p

��2

2

��1 �

�1 �

��0

2 �v��2p

��0

2 �v��2p

�

��3 ��3

�� 1

��0

1 �u��2p

��0

1 �u��2p

�� 2 �

�2 �

��0

3 �!��2p

��0

3 �!��2p

��2� �2

��1 �

�1 �

��0

2 �v��2p

�

�

��0

2 �v��2p

��3 �

�3

�2� �3

��0

1 �u��2p

��0

1 �u��2p

�� 2 �

�2 �

��0

3 �!��2p

��0

3 �!��2p

��1 �

�1

�

��0

2 �v��2p

��0

2 �v��2p

��3 �

�3

�� 4

��0

1 �u��2p

��1 � �

�2

��0

2 �v��2p

��

��0

3 �!��2p

��3

�

�

��1

��0

1 �u��2p

��

��0

2 �v��2p

��2 ��

�3

��0

3 �!��2p

��: (26)

From the above expression, we get constraint equations atthe tree level,

�21 � �1�u2 �!2� � �3

v2

2� 0; (27)

�22 � �2v

2 � �3�u2 �!2�

2� 0; (28)

which imply that the Higgs vacuums are not SU�3�L �U�1�X singlets. As a result, the gauge symmetry is brokenspontaneously. The nonzero values of � and � at theminimum value of V��;�� can be easily obtained by

�� u2 �!2

2��3�2

2 � 2�2�21

4�1�2 � �23

;

�� v2

2��3�2

1 � 2�1�22

4�1�2 � �23

:

(29)

It is worth noting that any other choice of u, ! for thevacuum value of � satisfying (29) gives the same physicsbecause it is related to (4) by an SU�3�L � U�1�X trans-formation. Thus, in general, we assume that u � 0.

Since u is a parameter of lepton-number violation, theterms linear in u violate the latter. Applying the constraintequations (27) and (28) we get the minimum value, massterms, lepton-number conserving and violating interac-tions:

V��;�� Vmin � VNmass � VC

mass � VLNC � VLNV; (30)

where

Vmin � ��2

4v4 �

1

4�u2 �!2��1�u

2 �!2� � �3v2 ;

VNmass � �1�uS1 �!S3�

2 � �2v2S2

2

� �3v�uS1 �!S3�S2; (31)

VCmass �

�4

2�u��1 � v�

�2 �!�

�3 ��u�

�1 � v�

�2 �!�

�3 �;

(32)

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VLNC � �1��2 � �2��

��2 � �3��

� �4�� 2�1!S3��

� 2�2vS2�� 3vS2��

� �3!S3��

�4��2p �v��2 �!�

�3 ��

��

��4��

2p �v��2 �!�

�3 ��

��; (33)

VLNV � 2�1uS1�� 3uS1�� 4��

2p u��1 ��

��

��1 �� : (34)

In the above equations, we have dropped the subscript Pand used � � ��0

1; ��2 ; �

03�T , � � ��1 ; �

02; �

�3 �

T .Moreover, we have expanded the neutral Higgs fields as

�01 �

S1 � iA1��2p ; �0

3 �S3 � iA3��

2p ;

�02 �

S2 � iA2��2p :

(35)

In the literature, the real parts �Si; i � 1; 2; 3� are alsocalled CP-even scalar fields and the imaginary parts�Ai; i � 1; 2; 3�, CP-odd scalar fields. In this paper, forshort, we call them scalar and pseudoscalar fields, respec-tively. As expected, the lepton-number violating part VLNC

is linear in u and trilinear in scalar fields.In the pseudoscalar sector, all fields are Goldstone bo-

sons: G1 � A1, G2 � A2, and G3 � A3 [cf. Eq. (31)]. Thescalar fields S1, S2, and S3 gain masses via (31), thus we getone Goldstone boson G4 and two neutral physical fields—the SM H0 and the new H0

1 with masses

m2H0 � �2v2 � �1�u2 �!2�

��2v

2 � �1�u2 �!2� 2 � �2

3v2�u2 �!2�

q’

4�1�2 � �23

2�1v2; (36)

M2H0

1� �2v2 � �1�u2 �!2�

��2v2 � �1�u2 �!2� 2 � �2

3v2�u2 �!2�

q’ 2�1!

2: (37)

In terms of scalars, the Goldstone and Higgs fields aregiven by

G4 �1��

1� t2�q �S1 � t�S3�; (38)

H0 � cS2 �s��

1� t2�q �t�S1 � S3�; (39)

075005

H01 � sS2 �

c��1� t2�

q �t�S1 � S3�; (40)

where

t2 �3MWMX

�1M2X � �2M2

W

: (41)

From Eq. (37), it follows that the mass of the new Higgsboson MH0

1is related to the mass of the bilepton gauge X0

(or Y� via the law of Pythagoras) through

M2H0

1�

8�1

g2 M2X

�1�O

�M2W

M2X

��

�2�1s2

W

��M2X

�1�O

�M2W

M2X

�� 18:8�1M

2X: (42)

Here, we have used � � 1128 and s2

W � 0:231.In the charged Higgs sector, the mass terms for

��1; �2; �3� are given by (32), thus there are twoGoldstone bosons and one physical scalar field:

H�2 1��

u2 � v2 �!2p �u��1 � v�

�2 �!�

�3 � (43)

with mass

M2H�2��4

2�u2 � v2 �!2� � 2�4

M2Y

g2 �s2W�4

2��M2Y

’ 4:7�4M2Y: (44)

Two remaining Goldstone bosons are

G�5 �1��

1� t2�q ��1 � t��

�3 �; (45)

G�6 �1��

�1� t2��u2 � v2 �!2�

q �v�t��1 ��

�3 �

�!�1� t2��2 : (46)

Thus, all pseudoscalars are eigenstates and massless(Goldstone). Other physical fields are related to the scalarsin the weak basis by the linear transformations:

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H0

H01

G4

0B@

1CA � �ss� c �sc�

cs� s cc�c� 0 �s�

0@

1A S1

S2

S3

0@

1A; (47)

H�2G�5G�6

0BB@

1CCA� 1��

!2�c2�v

2q

�

!s� vc� !c�

c��!2�c2

�v2

q0 �s�

��!2�c2

�v2

qvs2�

2 �! vc2�

0BBB@1CCCA

�

��1��2��3

0BB@1CCA: (48)

From (36) and (37), we come to the previous result inRef. [4],

�1 > 0; �2 > 0; 4�1�2 > �23: (49)

Equation (44) shows that the mass of the massive chargedHiggs boson H�2 is proportional to those of the chargedbilepton Y through a coefficient of Higgs self-interaction�4 > 0. Analogously, this happens for the SM Higgs bosonH0 (MH0 �MW) and the newH0

1 (MH01�MX). Combining

(49) with the constraint equations (27) and (28) we get aconsequence: �3 is negative (�3 < 0).

To finish this section, let us comment on our physicalHiggs bosons. In the effective approximation w� v, u,from Eqs. (47) and (48) it follows that

H0 � S2; H01 � S3; G4 � S1;

H�2 ��3 ; G�5 ��

�1 ; G�6 � �

�2 :

(50)

This means that, in the effective approximation, thecharged bosonH�2 is a scalar bilepton (with lepton numberL � 2), while the neutral scalar bosons H0 and H0

1 do notcarry lepton number (with L � 0).

TABLE III. Trilinear coupling constants of W�W� with neu-tral Higgs bosons.

Vertex Coupling

W�W�H g2

2 vc

W�W�H01

g2

2 vs

IV. HIGGS—SM GAUGE BOSON COUPLINGS

There are a total of nine gauge bosons in the SU�3�L �U�1�X group and eight of them are massive. As shown inthe previous section, we have just eight massless Goldstonebosons—the justified number for the model. One of theneutral scalars is identified with the SM Higgs boson,therefore its couplings to ordinary gauge bosons such asthe photon, the Z, and the W� bosons have to have, in theeffective limit, usual known forms. To search Higgs bosonsat future high energy colliders, one needs their couplingswith ordinary particles, especially with the gauge bosons inthe SM.

The interactions among the gauge bosons and the Higgsbosons arise in part from

075005

XY��;�

�D�Y��D�Y�:

In the following, the summation over Y is default and onlythe terms giving interesting couplings are explicitlydisplayed.

First, we consider the relevant couplings of the SM Wboson with the Higgs and Goldstone bosons. The trilinearcouplings of the pair W�W� with the neutral scalars aregiven by

�P CC� h�i��P CC�� P CC

� h�i��PCC�� H:c:

�g2v

2W��W��S2: (51)

Because S2 is a combination of only H and H01 , there are

two couplings, which are given in Table III.Couplings of the single W with two Higgs bosons exist

in

i�Y�P CC� @�Y�@�Y�PCC

� Y�

�ig��

2p W�� Y2�c�@

�Y1�s�@�Y3�

�@�Y2�c�Y1�s�Y3� �H:c:

�ig��

2p W��

�2 �c�@

��01�s�@

��03��@

��2 �c��01�s��

03�

��02 �c�@

��1 �s�@��3 ��@

��02 �c��

�1 �s��

�3 �

�H:c: (52)

The resulting couplings of the single W boson with twoscalar fields are listed in Table IV, where we have used a

notation A@�$B � A�@�B� � �@�A�B. Vanishing couplings

are

V �W�H�2 H0� � V �W�H�2 H

01� � V �W�H0G�6 �

� V �W�H01G�6 � � V �W�H�2 G2�

� V �W�G�6 G2� � 0:

Quartic couplings of W�W� with two scalar fields arisein part from

-6

TABLE IV. Trilinear coupling constants of W� with two Higgs bosons.

Vertex Coupling Vertex Coupling

W��H�2 @�$G4 �igvc��=�2

��!2 � c2

�v2

q� W��G�6 @�

$G1 �gc�!�=�2

��!2 � c2

�v2

q�

W��G�5 @�$H ��igc=2� W��G�5 @�

$G2 � g

2

W��G�6 @�$G4 �ig!�=�2

��!2 � c2

�v2

q� W��G�5 @�

$H0

1 � ig2 s

W��H�2 @�$G1 ��gvc2

��=�2��!2 � c2

�v2

q� W��G�6 @�

$G0

3 ��gs�!�=�2��!2 � c2

�v2

q�

W��H�2 @�$G3 �gvs2��=�4

��!2 � c2

�v2

q�


�PCC� Y��PCC�Y� �

g2

2W��W

��2 ��2 � c

2��

01 �

01

� s2��

03 �

03 � c�s��

01 �

03 � �

01�

03 �

��02 �

02 � c

2��1 �

�1 � s

2��3 �

�3

� c�s��1 ��3 ��

�1 �

�3 � : (53)

With the help of (A1) and (A2), we get the interestedcouplings of W�W� with two scalars, which are listed inTable V. Our calculations give the following vanishing

TABLE V. Nonzero quartic coupling co

Vertex Coupling

W�W�H�2 H�2 �g2c2

�v2�=�2�!2 � v2c2

��

W�W�G�5 G�5 g2=2

W�W�G�6 G�6 �g2!2�=�2�!2 � c2

�v2�

W�W�H�2 G�6 ��g2c�v!�=�2�!

2 � c2�v

W�W�HH g2c2=2

W�W�H01H

01 g2s2

=2

075005

couplings:

V �W�W�H�2 G�5 � � V �W�W�G�5 G

�6 �

� V �W�W�H0G04�

� V �W�W�H01G

04� � 0: (54)

Now we turn to the couplings of neutral gauge bosonswith Higgs bosons. In this case, the interested couplingsexist in

i�Y�PNC� @�Y�@�Y�PNC

� Y��ig2

�W�

3 �@��01 �

01�@��

�2 ��2 �@��

�1 �

�1 �@��

02 �

02��

W�8��3p �@��

01 �

01

�@��2 ��2 �@��

�1 �

�1 �@��

02 �

02�2@��

03 �

03�2@��

�3 �

�3 �

� t

��2

3

sB�

��

1

3�@��

01 �

01�@��

�2 ��2 �@��

03 �

03��

2

3�@��1 �

�1 �@��

02 �

02�@��

�3 �

�3 �

�

�y��@��01 �

03�@��

03 �

01�@��

�1 �

�3 �@��

�3 �

�1 �

��H:c: (55)

It can be checked that, as expected, the photon A� does notinteract with neutral Higgs bosons. Other vanishing cou-plings are

V �AH�2 G�5 � � V �AH�2 G

�6 � � V �AG�6 G

�5 � � 0 (56)

and

V �AAH0� � V �AAH01� � V �AAG4� � 0;

V �AZH0� � V �AZH01� � V �AZG4� � 0;

V �AZ0H0� � V �AZ0H01� � V �AZ0G4� � 0:

The nonzero electromagnetic couplings are listed inTable VI. It should be noticed that the electromagneticinteraction is diagonal, i.e. the nonzero couplings, in thismodel, always have a form

ieqHA�H@�

$H: (57)

For the Z bosons, the following observation is useful:

W�3 � U12Z� � � � � ; W�

8 � U22Z� � � � � ;

B� � U32Z� � � � � ; y� � U42Z

� � � � � :(58)

nstants of W�W� with Higgs bosons.

Vertex Coupling

W�W�G01G

01 g2c2

�=2W�W�G0

3G03 g2s2

�=2W�W�G0

4G04 g2=2

2� W�W�HH01 g2s2=4

W�W�G01G

03 ��g2s2�=4�

W�W�G02G

02 g2=2

-7

TABLE VI. Trilinear electromagnetic coupling constants ofA� with two Higgs bosons.

Vertex A�H�2 @�$H�2 A�G�5 @�

$G�5 A�G�6 @�

$G�6

Coupling ie ie ie


Here

U12 � c’c�0cW;

U22 �c’�s

2W � 3c2

Ws2�0 � � s’

��1� 4s2

�0c2W��4c

2W � 1�

q��3pcWc�0

;

(59)

U32 � �tW�c’

��4c2

W � 1q

� s’��1� 4s2

�0c2W

q��

3pc�0

(60)

are elements in the mixing matrix of the neutral gaugebosons given in the Appendix of Ref. [5]. From (55) and(58), it follows that the trilinear couplings of the single Zwith charged Higgs bosons exist in part from theLagrangian terms

�ig2Z��U12 �

U22��3p �

t3

��2

3

sU32

�@��

�2 ��2 �

�U12 �

U22��3p

�2t3

��2

3

sU32

�@��1 �

�1 �

��

2��3p U22 �

2t3

��2

3

sU32

�

� @��3 �

�3 �U42�@��

�1 �

�3 � @��

�3 �

�1 �

�� H:c:

(61)

From (61) we get trilinear couplings of the Z with thecharged Higgs bosons which are listed in Table VII. Thelimit sign (! ) in the tables is the effective one.

In the effective limit, the ZG5G5 vertex gets an exactexpression as in the SM. Hence G5 can be identified withthe charged Goldstone boson in the SM (GW�).

Now we search couplings of the single Z� boson withneutral scalar fields. With the help of the following equa-tions,

TABLE VII. Trilinear coupling constants

Vertex

Z�H�2 @�$H�2 fig=�2�!2 � v2c2

�� gf�v2c2� �!

2s2��U12 � �!

Z�G�5 @�$G�5

ig2 �c

2�U12 � �1� 3s2

��U22��

3p

Z�G�6 @�$G�6 fig=�2�!2 � c2

�v2� gf�!2 � v2s2

�c2��U12 � �v

2

Z�H�2 @�$G�5 �ig!=�4

��!2 � c2

�v2

qZ�H�2 @�

$G�6 f�ig!vc��=�2�!

2 � c2�v

2� g��c

Z�G�5 @�$G�6 ��igvc��=�4

��!2 � c2

�vq

075005

�01@�$�0

1 � iG1@�$S1;

�03@�$�0

3 � iG3@�$S3;

�02@�$�0

2 � iG2@�$S2;

@��01 �

03 � @��

03 �

01 �

12�@�S1S3 � @�S3S1 � @�G1G3

� @�G3G1 � iG3@�$S1

� iG1@�$S3 ; (62)

the necessary parts of the Lagrangian are

g2

��U12 �

U22��3p �

t3

��2

3

sU32

�G1@�

$S1 �U42G1@�

$S3

�

��

2��3p U22 �

t3

��2

3

sU32

�G3@�

$S3 �U42G3@�

$S1

�

��U12 �

U22��3p �

2t3

��2

3

sU32

�G2@�

$S2

�: (63)

The resulting couplings are listed in Table VIII. FromTable VIII, we conclude that G2 should be identified toGZ in the SM. For the Z0 boson, the following remark isagain helpful:

W�3 � U13Z0� � � � � ; W�

8 � U23Z0� � � � � ;

B� � U33Z0� � � � � ; y� � U43Z0� � � � � ;(64)

where

U13 � s’c�0cW;

U23 �s’�s2

W � 3c2Ws

2�0 � � c’

��1� 4s2

�0c2W��4c

2W � 1�

q��3pcWc�0

;

(65)

U33 � �tW�s’

��4c2

W � 1q

� c’��1� 4s2

�0c2W

q��

3pc�0

: (66)

Thus, with the replacement Z! Z0 one just replaces col-umn 2 by column 3; for example, trilinear coupling con-

of Z� with two charged Higgs bosons.

Coupling2�1� 3c2

�� v2c2�

U22��3p � �v2c2

� � 2!2� t3 �23�

1=2U32 �!2s2�U42g

! �igsWtW� 2t

3 �23�

1=2U32 � s2�U42 !ig

2cW�1� 2s2

W�

c2��1� 3c2

�� !2 U22��

3p � t

3 �23�

1=2�!2 � 2v2c2��U32 � 2v2s�c

3�U42g

! ig2cW�1� 2s2

W�

� �s2�U12 ��3ps2�U22 � 2c2�U42� ! 0

2�U12 � �2� 3c2

��U22��

3p � t

3 �23�

1=2U32 � s2�U42 ! 0��2� �s2�U12 �

��3ps2�U22 � 2c2�U42� ! 0

-8

TABLE VIII. Trilinear coupling constants of Z� with two neutral Higgs bosons.

Vertex Coupling

Z�G1@�$H �

gs2 ��U12 �

U22��3p � t

3 �23�

1=2U32�s� �U42c� ! 0

Z�G2@�$H g

2 ��U12 �U22��

3p � 2t

3 �23�

1=2U32�c ! �g

2cW

Z�G3@�$H gs

2 ��2��3p U22 �

t3 �

23�

1=2U32�c� �U42s� ! 0

Z�G1@�$H0

1gc

2 ��U12 �U22��

3p � t

3 �23�

1=2U32�s� � U42c� ! 0

Z�G2@�$H0

1g2 ��U12 �

U22��3p � 2t

3 �23�

1=2U32�s ! 0

Z�G3@�$H0

1 �gc

2 ��2��3p U22 �

t3 �

23�

1=2U32�c� � U42s� ! 0

Z�G1@�$G4

g2 ��U12 �

U22��3p � t

3 �23�

1=2U32�c� � U42s� !g

2cW

Z�G2@�$G4 0

Z�G3@�$G4

g2 ��

2��3p U22 �

t3 �

23�

1=2U32�s� � U42c� ! 0


stants of the Z0� with two neutral Higgs bosons are given inTable IX.

Next, we search couplings of two neutral gauge bosonswith scalar fields which arise, in part, from

Y�PNC� PNC�Y �

g2

4�Y1�A

�11A11� � y�y

�� Y3�A11�y�

� A33�y�� Y1 � �A�22A22��Y2Y2

� �Y1�A11�y� � A33�y��

� Y3�A�33A33� � y�y�� Y3; (67)

�g2

4f��0

1 �A��11 A

�11��y�y

��03 �A

�11�y

��A�33�y�� 0

1

��A��22 A�22��

�2 ��2 ��

01 �A

�11�y

��A�33�y��

��03 �A

��33 A

�33��y�y

�� 03��

�1 �A

��11 A

�11�

�y�y��3 �A

�11�y

��A�33�y�� 1

��A��22 A�22��

02 �

02��

�1 �A

�11�y

��A�33�y��

��3 �A��33 A

�33��y�y

�� 3 g: (68)

TABLE IX. Trilinear coupling constants

Vertex

Z0�G1@�$H �

gs2

Z0�G2@�$H g

2 ��

Z0�G3@�$H gs

2

Z0�G1@�$H0

1gc

2 �

Z0�G2@�$H0

1

Z0�G3@�$H0

1 �gc

2 ��

Z0�G1@�$G4

g2 ��U13

Z0�G2@�$G4

Z0�G3@�$G4

g2

075005

Here A�ii �i � 1; 2; 3� is a diagonal element in the matrix2gP

NC� which is dependent on the U�1�X charge:

A��11 � W�3 �

W�8��3p �

t3

��2

3

sB�;

A��11 � W�3 �

W�8��3p �

2t3

��2

3

sB�;

A��22 � �W�3 �

W�8��3p �

t3

��2

3

sB�;

A��22 � �W�3 �

W�8��3p �

2t3

��2

3

sB�;

A��33 � �2W�

8��3p �

t3

��2

3

sB�;

A��33 � �2W�

8��3p �

2t3

��2

3

sB�:

(69)

of Z0� with two neutral Higgs bosons.

Coupling

��U13 �U23��

3p � t

3 �23�

1=2U33�s� �U43c� ! 0

U13 �U23��

3p � 2t

3 �23�

1=2U33�c !g

2cW��4c2

W�1p

�� 2��3p U23 �

t3 �

23�

1=2U33�c� �U43s� ! 0

�U13 �U23��

3p � t

3 �23�

1=2U33�s� �U43c� ! 0g2 ��U13 �

U23��3p � 2t

3 �23�

1=2U33�s ! 0

2��3p U23 �

t3 �

23�

1=2U33�c� �U43s� ! �gcW��4c2

W�1p

� U23��3p � t

3 �23�

1=2U33�c� �U43s� !gc2W

2cW��4c2

W�1p

0

�� 2��3p U23 �

t3 �

23�

1=2U33�s� �U43c� ! 0

-9

TABLE X. Quartic coupling constants of ZZ with two scalar bosons.

Vertex Coupling

ZZG1G1g2

2 ��U12 �U22��

3p � t

3 �23�

1=2U32�2 �U2

42 !g2

2c2W

ZZG2G2g2

2 ��U12 �U22��

3p � 2t

3 �23�

1=2U32�2 ! g2

2c2W

ZZG3G3g2

2 ��2��3p U22 �

t3 �

23�

1=2U32�2 �U2

42 ! 0

ZZG1G3g2

2 �U12 �U22��

3p � 2t

3 �23�

1=2U32�U42 ! 0

ZZHH g2

2 fs2 �s

2��U12 �

U22��3p � t

3 �23�

1=2U32�2 � c2

��2��3p U22 �

t3 �

23�

1=2U32�2 �U2

42 � s2�U42�U12 �U22��

3p � 2t

3 �23�

1=2U32�

�c2 �U12 �

U22��3p � 2t

3 �23�

1=2U32�2g ! g2

2c2W

ZZH01H

01

g2

2 fc2 �s

2��U12 �

U22��3p � t

3 �23�

1=2U32�2 � c2

��2��3p U22 �

t3 �

23�

1=2U32�2 �U2

42 � s2�U42�U12 �U22��

3p � 2t

3 �23�

1=2U32�

� s2 �U12 �

U22��3p � 2t

3 �23�

1=2U32�2g ! 0

ZZG4G4g2

2 �c2��U12 �

U22��3p � t

3 �23�

1=2U32�2 � s2

��2��3p U22 �

t3 �

23�

1=2U32�2 � s2��U12 �

U22��3p � 2t

3 �23�

1=2U32�U42 � U242 !

g2

2c2W

ZZHH1 �g2s2

4 �s2��U12 �

U22��3p � t

3 �23�

1=2U32�2 � c2

��2��3p U22 �

t3 �

23�

1=2U32�2 �U2

42 � �U12 �U22��

3p � 2t

3 �23�

1=2U32�2

� s2��U12 �U22��

3p � 2t

3 �23�

1=2U32�U42 ! 0

ZZHG4 �g2s

4 �U12 �U22��

3p � 2t

3

��23

qU32��2c2�U42 � s2��U12 �

��3pU22� ! 0

ZZH1G4g2c

4 �U12 �U22��

3p � 2t

3

��23

qU32��2c2�U42 � s2��U12 �

��3pU22� ! 0


Quartic couplings of two Z with neutral scalar fields aregiven by

g2

4f��0

1 �A��11 A

�11� � y�y

�� 03 �A

�11�y

� � A�33�y�� 0

1

� ��01 �A

�11�y

� � A�33�y�� 0

3 �A��33 A

�33�

� y�y�� 03 � �A

��22 A

�22��

02 �

02g

�g2

4f�A��11 A

�11� � y�y

��01 �

01 � �A

��33 A

�33�

� y�y��03 �

03 � �A

�11�y

� � A�33�y��0

1 �03

� �03 �

01� � �A

��22 A

�22��

02 �

02g: (70)

In this case, the couplings are listed in Table X.Trilinear couplings of the pair ZZ with one scalar field

are obtained via the following terms:

g2

4�vS2A

�22�A

��22 � uS1A

�11�A

��11 �!S3A

�33�A

��33

� �uS1 �!S3�y�y� � �!S1 � uS3�y

�A�22� : (71)

The obtained couplings are given in Table XI.

TABLE XI. Trilinear coupling const

Vertex

ZZH g2

2 �vc �U12 �U22��

3p � 2t

3 �23�

1=2U32�2 � uss��U12

� 2!ss��U

ZZH01

g2

2 �vs �U12 �U22��

3p � 2t

3 �23�

1=2U32�2 � ucs��U12

� 2!cs��U

ZZG4g2!

2 �s��U12 ��3pU22�

075005

Because of (64), for the ZZ0 couplings with scalar fields,the above manipulation is good enough. For example,Table X is replaced by Table XII.

Now we turn to the interested coupling ZW�H�2 whicharose, in part, from

Y�PNC� P CC�Y � H:c: �

g2

2��2p fW��A

�22Y

2�c�Y1 � s�Y3�

�W�� c�A�11 � s�y

��Y1

� �c�y� � s�A�33�Y

3 Y2g � H:c:

(72)

For our Higgs triplets, one gets

g2

2��2p fW�� A

��22 �

�2 �c��

01 � s��

03� � A

��22 �

02 �c��

�1

� s��3 � �W��2 ��c�A

��11 � s�y

��01

� �c�y� � s�A��33 ��

03 �W

��0

2��c�A��11

� s�y��1 � �c�y

� � s�A��33 ��

�3 g � H:c: (73)

From Eq. (73), the trilinear couplings of the W bosonwith one scalar and one neutral gauge boson are given by

ants of ZZ with one scalar bosons.

Coupling

� U22��3p � t

3 �23�

1=2U32�2 �!sc��

2��3p U22 �

t3 �

23�

1=2U32�2 �! s

c�U2

42

12 �U22��

3p � 2t

3 �23�

1=2U32�U42 !g2v2c2

W

� U22��3p � t

3 �23�

1=2U32�2 �!cc��

2��3p U22 �

t3 �

23�

1=2U32�2 �! c

c�U2

42

12 �U22��

3p � 2t

3 �23�

1=2U32�U42 ! 0

� c2�c�U42 �U12 �

U22��3p � 2t

3 �23�

1=2U32 ! 0

-10

TABLE XII. Trilinear coupling constants of ZZ0 with one scalar bosons.

Vertex Coupling

ZZ0H g2

2 �vc �U12 �U22��

3p � 2t

3 �23�

1=2U32��U13 �U23��

3p � 2t

3 �23�

1=2U33� � uss��U12 �U22��

3p � t

3 �23�

1=2U32��U13 �U23��

3p � t

3 �23�

1=2U33�

�!sc��2��3p U22 �

t3 �

23�

1=2U32��2��3p U23 �

t3 �

23�

1=2U33� �!sc�U42U43 �!ss��U12 �

U22��3p � 2t

3 �23�

1=2U32�U43

�!ss��U13 �U23��

3p � 2t

3 �23�

1=2U33�U42 !g2vc2W

2cW��4c2

W�1p

ZZ0H01

g2

2 �vs �U12 �U22��

3p � 2t

3 �23�

1=2U32��U13 �U23��

3p � 2t

3 �23�

1=2U33� � ucs��U12 �U22��

3p � t

3 �23�

1=2U32��U13 �U23��

3p � t

3 �23�

1=2U33�

�!cc��2��3p U22 �

t3 �

23�

1=2U32��2��3p U23 �

t3 �

23�

1=2U33� �!cc�U42U43 �!cs��U12 �

U22��3p � 2t

3 �23�

1=2U32�U43

�!cs��U13 �U23��

3p � 2t

3 �23�

1=2U33�U42 ! 0

ZZ0G4g2!s�

2 ��U12 �U22��

3p � t

3 �23�

1=2U32��U13 �U23��

3p � t

3 �23�

1=2U33� � �2��3p U22 �

t3 �

23�

1=2U32��2��3p U23 �

t3 �

23�

1=2U33�

�cot2�U42�U13 �U23��

3p � 2t

3 �23�

1=2U33� � cot2�U43�U12 �U22��

3p � 2t

3 �23�

1=2U32� ! 0


g2

4W��

�v��1

�c�

�2��3p W�

8 �4t3

��2

3

sB�

�� s�y�

�

� v��3

�c�y� � s�

��W�

3 �W�

8��3p �

4t3

��2

3

sB�

��

�!��2

�s��W

�3 �

��3pW�

8 � �c2�

c�y�� H:c: (74)

From the above equation, we get necessary nonzero cou-plings, which are listed in Table XIII. Vanishing couplingsare

V �AW�H�2 � � V �AW�G�6 � � 0: (75)

TABLE XIII. Trilinear coupling constants of neutral g

Vertex

AW�G�5ZW�H�2 �

Z0W�H�2 ��g

ZW�G�5g2v

4 ��

ZW�G�6 f�g2�v2c

TABLE XIV. The SM coupling c

Vertex Coupling

WWhh g2

2

WWh g2

2 v

WGWhig2

WGWGZg2

ZZhh g2

2c2W

AWGWg2

2 vsW

ZGZh � g2cW

WGWh � ig2

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Equation (75) is consistent with an evaluation in Ref. [11],where authors neglected the diagrams with the �W�H�

vertex.From (9), it follows that, to get couplings of the bilepton

gauge boson Y� with ZH�2 , one just makes the followingreplacement in (74): c� ! �s�, s� ! c�.

Finally, we can identify the scalar fields in the consid-ered model with those in the SM as follows:

H $ h; G�5 $ GW� ; G2 $ GZ: (76)

In the effective limit !� v, u, our Higgs can be repre-sented as

auge bosons with W� and the charged scalar boson.

Couplingg2

2 vsW�g2v!�=�2

��!2 � c2

�v2

q� �s�c��U12 �

��3pU22� � c2�U42

2v!�=�2��!2 � c2

�v2

q� �s�c��U13 �

��3pU23� � c2�U43 ! 0

s2�U12 � �2� 3s2

��U22��

3p � 4t

3 �23�

1=2U32 � s2�U42 ! �g2

2 vsWtW2� �!

2� =�8c��!2 � c2

�v2

q�g�s2��U12 �

��3pU22� � 2c2�U42 ! 0

onstants in the effective limit.

Vertex Coupling

GWGWA ie

WWGZGZg2

2

WWGWGWg2

2

ZZh g2

2c2Wv

ZZGZGZg2

2c2W

ZWGW � g2

2 vsWtW

ZGWGWig

2cW�1� 2s2

W�

-11


� �

1��2p u�GX0

GY�1��2p �!�H0

1 � iGZ0 �

0B@1CA;

� �GW�

1��2p �v� h� iGZ�

H�2

0B@1CA

(77)

where G3 �GZ0 , G�6 �GY� , and

G4 � iG1 ��2pGX0 : (78)

Note that identification in (78) is possible due to the factthat both the scalar and pseudoscalar parts of �0

1 are mass-less. In addition, the pseudoscalar part is decoupled fromothers, while its scalar part mixes with the same ones as inthe gauge boson sector (for details, see [5]).

We emphasize again, in the effective approximation, thatall Higgs-gauge boson couplings in the SM are recovered(see Table XIV). In contradiction with the previous analy-sis in Ref. [4], the condition u� v or the introduction ofthe third triplet is not necessary.

V. PRODUCTION OF H�2 VIA WZ FUSION AT LHC

The possibility of detecting the neutral Higgs boson inthe minimal version at e�e� colliders was considered in[18] and the production of the SM-like neutral Higgs bosonat the CERN LHC was considered in Ref. [9]. This sectionis devoted to the production of the charged H�2 at theCERN LHC.

Let us first discuss the mass of this Higgs boson.Equation (44) gives us a connection between its massand those of the singly charged bilepton Y through thecoefficient of Higgs self-coupling �4. Note that in theconsidered model, the nonzero Majoron couplings of GX0

with the leptons exist only in the loop levels. To keep thesmallness of these couplings, the massMH�2

can be taken inthe electroweak scale with �4 � 0:01 [19]. From (44),taking the lower limit for MY to be 1 TeV, the mass ofH�2 is in the range of 200 GeV.

Taking into account that, in the effective approximation,H�2 is the bilepton, we get the dominant decay channels asfollows:

H�2 ! l�l; ~Uda; D�~ua;

& ZW�; Z0W�; XW�; ZY�: (79)

Assuming that masses of the exotic quarks �U;D�� arelarger thanMH�2

, we come to the fact that the hadron modesare absent in the decay of the charged Higgs boson.Because the Yukawa couplings of H�2 l

�� are very small,the main decay modes of the H�2 are in the second line of(79). Note that the charged Higgs bosons in doublet mod-els, such as the two-Higgs doublet model or the minimalsupersymmetric standard model, have both hadronic and

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leptonic modes [10]. This is a specific feature of the modelunder consideration.

Because the exotic X, Y, Z0 gauge bosons are heavy, thecoupling of a singly charged Higgs boson (H�2 ) with theweak gauge bosons, H�2 W

�Z, may give the main decaymode of H�2 . Here, it is of particular importance for theelectroweak symmetry breaking. Its magnitude is directlyrelated to the structure of the extended Higgs sector underglobal symmetries [20]. This coupling can appear at thetree level in models with scalar triplets, while it is inducedat the loop level in multiscalar doublet models. The cou-pling, in our model, differs from zero at the tree level due tothe fact that the H�2 belongs to a triplet.

Thus, for the charged Higgs bosonH�2 , it is important tostudy the couplings given by the interaction Lagrangian

L int � fZWHH�2 W

��Z

�; (80)

where fZWH, at tree level, is given in Table XIII. The sameas in [11], the dominant rate is due to the diagram con-nected with the W and Z bosons. Putting necessary matrixelements in Table XIII, we get

fZWH��g2v!s2�

4��!2�c2

�v2

q c’�s’��4c2

W�1��1�4t22��q

��1�4t22��c

2W��4c

2W�1�t22�

q :

Thus, the form factor, at the tree level, is obtained by

F fZWHgMW

� �!s2��c’ � s’

��4c2

W � 1��1� 4t22��q

2��!2 � c2

�v2��1� 4t22��c

2W � �4c

2W � 1�t22�

q :

(81)

The decay width of H�2 ! W�i Zi, where i � L, T repre-sent, respectively, the longitudinal and transverse polar-izations, is given by [11]

��H�2 ! W�i Zi� � MH�2

�1=2�1; w; z�16�

jMiij2; (82)

where ��1; w; z� � �1� w� z�2 � 4wz, w � M2W=M

2H�2

,

and z � M2Z=M

2H�2

. The longitudinal and transverse contri-

butions are given in terms of F by

jMLLj2 �

g2

4z�1� w� z�2jFj2; (83)

jMTT j2 � 2g2wjFj2: (84)

For the case of MH�2� MZ, we have jMTTj

2=jMLLj2 �

8M2WM

2Z=M

4H�2

which implies that the decay into a longi-

tudinally polarized weak boson pair dominates that into atransversely polarized one. The form factor F and themixing angle t’ are presented in Table XV, where we

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TABLE XV. Values of F, t’, and MmaxH�2

for given s�.

s� 0.08 0.05 0.02 0.009 0.005

t’ �0:0329698 �0:0156778 �0:00598729 �0:00449063 �0:00422721F �0:087481 �0:0561693 �0:022803 �0:0102847 �0:00571598MmaxH�2

[GeV] 1700 1300 700 420 320


have used s2W � 0:2312, v � 246 GeV, ! � 3 TeV (or

MY � 1 TeV) as the typical values to get five cases corre-sponding to the s� values under the constraint (23) whichwas given in [5].

Next, let us study the impact of the H�2 W�Z vertex on

the production cross section of pp! W�ZX ! H�2 Xwhich is a pure electroweak process with high pT jetsgoing into the forward and backward directions from thedecay of the produced scalar boson without color flow inthe central region. The hadronic cross section for pp!H�2 X via W�Z fusion is expressed in the effective vectorboson approximation [21] by

eff�s;M2H�2� ’

16�2

��1; w; z�M3H�2

�X

��T;L

��H�2 ! W�� Z��dLd�

��pp=W�� Z�

;

(85)

where � � M2H�2=s, and

dLd�

��pp=W�� Z�

�Xij

Z 1

�

d�0

�0Z 1

�0

dxxfi�x�fj��0=x�

�dLd�

��qiqj=W�� Z�

; (86)

10-5

10-4

10-3

10-2

10-1

100

101

102

103

400 800 1200 1600 2000

σ eff

[fb]

MH2[GeV]

sθ=0.08sθ=0.05sθ=0.02

sθ=0.009sθ=0.005

FIG. 1 (color online). Hadronic cross section of the W�Zfusion process as a function of the charged Higgs boson massfor five cases of sin�. The horizontal line is the discovery limit(25 events).

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with �0 � s=s and � � �=�0. Here fi�x� is the partonstructure function for the ith quark, and

dLd�

��qiqj=W�T ZT

�c

64�4

1

�ln�s

M2W

�ln�s

M2Z

��2� ��2

� ln�1=�� 2�1� ��3� �� ; (87)

dLd�

��qiqj=W�L ZL

�c

16�4

1

��1� �� ln�1=�� 2�� 1� ;

(88)

where c � �g4c2�=16c2

W��g21V�qj� � g

21A�qj� with g1V�qj�,

g1A�qj� for quark qj are given in Table I of Ref. [5]. UsingCTEQ6L [22], in Fig. 1, we have plotted eff�s;M2

H�2� at��

sp� 14 TeV, as a function of the Higgs boson mass

corresponding to the five cases in Table XV.Assuming a discovery limit of 25 events corresponding

to the horizontal line, and taking the integrated luminosityof 300 fb�1 [23], from the figure, we come to the conclu-sion that, for s� � 0:08 (the line on top), the charged Higgsboson H�2 with mass larger than 1700 GeV cannot be seenat the LHC. These limiting masses are denoted by Mmax

H�2and listed in Table XV. If the mass of the above-mentionedHiggs boson is in the range of 200 GeV and s� � 0:08, thecross section can exceed 260 fb: i.e., 78 000 of H�2 can beproduced at the integrated LHC luminosity of 300 fb�1.This production rate is about 10 times larger than those inRef. [11]. The cross sections decrease rapidly as the massof the Higgs boson increases from 200 GeV to 400 GeV.

VI. CONCLUSIONS

In this paper we have considered the scalar sector in theeconomical 3-3-1 model. The model contains eightGoldstone bosons—the justified number of the masslessones eaten by the massive gauge bosons. Couplings of theSM-like gauge bosons such as of the photon, the Z, and thenew Z0 gauge bosons with physical Higgs ones are alsogiven. From these couplings, the SM-like Higgs boson aswell as Goldstone ones are identified. In the effectiveapproximation, the full content of the scalar sector can berecognized. The CP-odd part of the Goldstone associatedwith the neutral non-Hermitian bilepton gauge bosons GX0

is decoupled, while its CP-even counterpart has the mixingin the same way in the gauge boson sector. Despite themixing among the photon with the non-Hermitian neutral

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bilepton X0 as well as with the Z and the Z0 gauge bosons,the electromagnetic couplings remain unchanged.

It is worth mentioning that masses of all physical Higgsbosons are related to those of gauge bosons through thecoefficients of Higgs self-interactions. All gauge-scalarboson couplings in the standard model are recovered.The couplings of the photon with the Higgs bosons arediagonal.

It should be mentioned that in Ref. [4], to get the non-zero coupling ZZh at the tree level, the authors suggestedthe following solution: (i) u� v or (ii) introducing thethird Higgs scalar with VEV (� v). This problem doesnot happen in our consideration.

After all, we focused our attention on the singly chargedHiggs boson H�2 with mass proportional to the bileptonmass MY through the coefficient �4. Mass of the H�2 isestimated and is in the range of 200 GeV. This boson, incontrast to those that arose in the Higgs doublet models,does not have the hadronic and leptonic decay modes. Thetrilinear coupling ZW�H�2 which differs, at the tree level,while the similar coupling of the photon �W�H�2 as ex-pected, vanishes. In the model under consideration, thecharged Higgs boson H�2 with mass larger than1700 GeV cannot be seen at the LHC. If the mass of theabove-mentioned Higgs boson is in the range of 200 GeV,however, the cross section can exceed 260 fb: i.e., 78 000of H�2 can be produced at the LHC for the luminosity of

075005

300 fb�1. By measuring this process we can obtain usefulinformation to determine the structure of the Higgs sector.

Detailed analysis of the discovery potential of all theseHiggs bosons will be presented elsewhere.

APPENDIX: MIXING MATRICES OF THE SCALARSECTOR

For the sake of convenience in practical calculations, wegive here some mixing matrices:

(1) N

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eutral scalar bosons,

S1

S2

S3

0@ 1A � �ss� cs� c�c s 0�sc� cc� �s�

0B@1CA H

H01

G4

0@ 1A: (A1)

(2) S
ingly charged scalar bosons,
��1��2��3

0B@1CA� 1��

!2�c2�v

2q

�

!s� c��!2�c2

�v2

qvs2�

2vc� 0 �!

!c� �s��!2�c2

�v2

qvc2

�

0BB@1CCA H�2

G�5G�6

0B@

1CA:

(A2)

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Higgs-gauge boson interactions in the economical 3-3-1 model