Chargino and neutralino production in the 3-3-1 supersymmetric model in e(-)e(-) scattering

Chargino and neutralino production in the 3-3-1 supersymmetric model in e

À

_e

À

_scattering

M. Capdequi-Peyrane`re and M. C. Rodriguez*

Physique Mathe´matique et The´orique, CNRS-UMR 5825 Universite´ Montpellier II, F-34095, Montpellier Cedex 5, France

~Received 21 June 2001; published 21 December 2001!

The goal of this article is to derive the Feynman rules involving single charginos, neutralinos, double charged gauge bosons, and sleptons in a 3-3-1 supersymmetric model. Using these Feynman rules we calculate the production of double charged charginos with neutralinos and also the production of a pair of single charged charginos, both in an electron-electron linear collider.

DOI: 10.1103/PhysRevD.65.035001 PACS number~s!: 12.60.Jv, 13.85.Lg

I. INTRODUCTION

The standard model is exceedingly successful in describ-ing leptons, quarks, and their interactions. Nevertheless, the standard model is not considered as the ultimate theory since neither the fundamental parameters, masses and couplings, nor the symmetry pattern are predicted. Even though many aspects of the standard model are experimentally supported to a high degree of accuracy, the embedding of the model into a more general framework is to be expected. The possi-bility of a gauge symmetry based on the symmetry SU(3)_C

^_SU(3)L^_{U(1)N~3-3-1! @1#is particularly interesting,}

be-cause it explains some fundamental questions that have not been eluded in the context of the standard model@2#.

Recently one of us proposed the supersymmetric exten-sion of the 3-3-1 model@2#; in this kind of model if we want to remain in a perturbative regime at all energies, then we need, in a natural way, a broken supersymmetry at a TeV scale, and the lightest scalar boson has an upper limit, which is 124.5 GeV for a given set of parameters at the tree level. We must remember that in the minimal supersymmetric stan-dar model~MSSM!the bound using radiative corrections to the mass of the lightest Higgs scalar is 130 GeV@3#. On the other hand, no direct observation of a Higgs boson has been made yet, and current direct searches made at the CERN

e1_e2 _{collider accelerator leads to the condition M}

h

.113 GeV@4#in the MSSM.

Linear colliders would be most versatile tools in experi-mental high-energy physics. A large international effort is currently under way to study the technical feasibility and physics possibilities of linear e1e2 colliders in the TeV range. A number of designs have already been proposed

@Next Linear Collider ~NLC!, Japan Linear Collider ~JLC!, DESY TeV Energy Superconducting Linear Accelerator TESLA, CERN Linear Collider CLIC, VLEPP, . . .#and sev-eral workshops have recently been devoted to this subject. They can provide not only e1_e2 _{collisions and high}

lumi-nosities, but also very energetic beams of real photons. One could thus exploit gg, e2_{g, and even e}2_e2 _{collisions for}

physics studies.

The last exciting prospects have prompted a growing number of theoretical studies devoted to the investigation of the physics potential of such e2_e2 _{accelerator experiments.}

Of course, in the realm of the standard model this option is not particularly interesting because Mo”ller scattering and bremsstrahlung events are mainly observed. However, it is just for that reason that e2_e2 _{collisions can provide crucial}

information on exotic processes, in particular on processes involving lepton and/or fermion number violation. Therefore, new perspectives emerge in detecting new physics beyond the standard model in processes having nonzero initial elec-tric charge ~and nonzero lepton number! like in electron-electron e2e2process.

In this paper we will explicitly work out two e2_e2

pro-cesses and we will show the difference between the MSSM and the 3-3-1 supersymmetric model in the chargino produc-tion. This paper is organized as follows. The model is out-lined in Sec. II, where leptons, sleptons, scalars, higgsinos, gauge bosons, and gauginos are defined. We derive the mass spectrum in Sec. III, while the interactions are in Sec. IV. In order to calculate cross sections, we derive the Feynman rules in the Sec. V, while the productions of charginos and neutralinos are worked out in Sec. VI. Our conclusions are given in Sec. VII.

The Lagrangian is given in Appendix A, while in Appen-dix B we show the mass matrices of the charginos and neu-tralinos. The procedure to write two-component spinors in terms of four-component spinors is given in Appendix C. The differential cross sections of the processes we have cal-culated are given in the Appendix D.

II. PARTICLE CONTENT

First of all, let us outline the model, following the nota-tion given in@2#. We are writing here the particle content that we are using in this study, so it means that we are not going to discuss quarks and squarks.

The leptons and sleptons are given by

Ll5~nlllc!L t _{, ˜L}

l5~˜nll˜ l˜c!L t _{, l}

5e,m,t. ~2.1!

The scalars of our theory are *Permanent address: Instituto de Fı´sica Teo´rica, Universidade

Es-tadual Paulista, Brazil.

h₅

S

h0

h12

h₂1

D

, r5

S

r1 r0 r11

D

, x5

S

x2 x22

x0

D

,

S₅

S

s1

0 h21

A

2

h12

A

2

h₂1

A

2

H₁11 s2 0

A

2

h₁2

A

2

s2 0

A

2

H₂22

D

, ~2.2!

the higgsinos, the superpartners of the scalars, are defined as

h ˜₅

S

h ˜0

h ˜ 1 2

h ˜ 2 1

D

, ˜r₅

S

r ˜1

r ˜0

r ˜11

D

, x˜₅

S

x ˜2

x ˜22

x ˜0

D

,

S ˜₅

S

s ˜ 1 0 ˜h2

1

A

2

h ˜ 1 2

A

2

h ˜ 2 1

A

2

H˜₁11 s ˜ 2 0

A

2

h ˜ 1 2

A

2

s ˜ 2 0

A

2

H˜₂22

D

. ~2.3!

To cancel chiral anomalies generated by these higgsinos we have to add the fieldsh˜

₈

_,r˜

₈

_,x˜

₈

_{, and S}˜

₈

_{and their respective}

scalars, see@2#.

When we break the 3-3-1 symmetry to the SU(3)C

^_{U(1)E M, the scalars get the following vacuum expectation}

values ~VEVs!:

^

h

&

5

S

v

0

0

D

,

^

r

&

5

S

0

u

0

D

,

^

x

&

5

S

0

0

w

D

,

^

S

&

5

S

0 0 0

0 0 z

A

2

0 z

A

2 0

D

, ~2.4!

and similar expressions to the primed fields, where v

5v_h/

A

2, u5v_r/

A

2, w5v_x/

A

2, z5v_s

2/

A

2, v

8

5v_h

8/

A

2, u

8

5v_r

8/

A

2, w

8

5v_x

8/

A

2, and z

8

5v_s 2

8/

A

2. In the 3-3-1 supersymmetric model the bosons of the symmetry SU(3)Lare Va and their superpartners, the

gaugi-nos, arel_Aa_{, with a}

51, . . . ,8. For the U(1)_Nsymmetry, we have the boson V and its superpartner written as l_B ~which we call gaugino too!.

In the usual 3-3-1 model@1#the gauge bosons are defined as

Wm6~x!52 1

A

2~Vm

1

~x!7iVm

2 ~x!!,

V_m6~x!₅₂ 1

A

2~Vm

4

~x!₆iV_m5~x!!,

U_m66~x!₅₂ 1

A

2~Vm

6

~x!₆iV_m7~x!!,

Am~x!5 1

A

114t2@~Vm

3

~x!₂

A

3V_m8~x!!t₁Vm#,

Z_m0~x!₅₂ 1

A

114t2

F

A

113t

2_V

m

3

~x!₁

A

3t 2

A

113t2Vm

8 ~x!

2

t

A

113t2Vm~x!

G

,

Z

₈

_m0~x!₅ 1

A

1₁3t2~Vm

8

~x!₁

A

3tVm~x!!, ~2.5!

where t[tanu₅g

₈

/g where g

₈

and g are the gauge coupling constants of U(1) and SU(3), respectively. We can define the charged gauginos, in analogy with the gauge bosons, in the following way

l_W6_~x!

52 1

A

2~ lA

1

~x!₇il_A2~x!!,

l_V6_~x!

52

_A

1

2~ lA

4

~x!₆il_A5~x!!,

l_U66_~x!

52

1

A

2~ lA

6

~x!₆il_A7~x!!. ~2.6!

III. MASS SPECTRUM

The higgsino mass term comes fromL_{H M T}, see Eq.~A7!, the gauginos mass term is given byL_{G M T}, see Eq.~A8!and the mixing term between gauginos and higgsinos is given by

L

HH˜ V˜

scalar

A. Double charged chargino

Doing the calculations we obtain the mass matrix of the double charged charginos in the following1way:

L_massdouble₅₂m_ll_U22_l

U

11

2m₂r˜r

₈

22˜_r11₂mx

2˜x

22˜_x

₈

11₂mS

2 ~H˜1

8

22_H˜ 1 11

1H˜222H˜

8

211!

2ig

S

u˜r11

2w

₈

x˜

₈

11

2

z

A

2H

˜ 1 11

1

z

₈

A

2H

˜

₈

2 11

D

_l

U

22

2ig

S

w˜x22

2u

₈

˜r

₈

22

2

z

A

2H

˜ 2 22

1

z

₈

A

2H

˜ 1

8

22

D

_l

U

11

1

f₁v

3 x˜

22_˜r11

2

f₃

3 ux˜

22_H˜ 1 11

2

A

2f3 3 zx˜

22_˜r11

2

f₃

3 wH˜2

22_˜r11

1

f₁

8

v

8

3 x˜

8

11_˜r

₈

22

2

f₃

8

3 u

8

x˜

8

11_H˜

₈

1 22

2

A

2

f₃

₈

3 z

8

x˜

8

11˜_r

₈

22₂f3

8

3 w

8

H˜

8

2

11_r˜

₈

22_{1H.c., ~3.1!}

which can be written in analogy with the MSSM,2 see Ap-pendix B 1, as follows

L mass double

52₂1~ C66_!t_Y66_C66_{1H.c. ~3.2!}

The double chargino mass matrix is diagonalized using two rotation matrices, A and B, defined by

x ˜

i

11

5A_{i j}C_j11_{, x˜}

i

22

5B_{i j}C22_{j , i, j}

51, . . . ,5,

~3.3!

where A and B are unitary matrices such that

MDCC5B*TA21, ~3.4!

the matrix T is defined in Eq.~B3!. To determine A and B, we note that

MDCC

2

5ATt•TA215B*T•Tt~B*!21_{, ~3.5!}

which means that A diagonalizes Tt•_{T while B diagonalizes}

T•Tt.

We define the following Dirac spinors to represent the mass eigenstates:

C~x˜

i

11_!

5~x˜

i

11_˜¯_x

i

22_!t_{, C}c_~x˜ i

22_!

5~˜x

i

22_x˜¯

i

11_!t_,

~3.6!

wherex˜

i

11 _{is the particle andx˜}

i

22_{is the antiparticle~we are}

using the same notation as in@5#!.

B. Single Charged Chargino

We can write the mass matrix of the charged chargino in the following way

L_masssingle₅₂ml~ lV2lV11lW2lW1!2

mh

2 ~h˜12h˜1

8

11h˜2

8

2h˜21!2 mr

2˜r

8

2˜_r1₂mx

2 ˜x

2˜_x

₈

1₂mS

2 ~˜h12˜h1

8

11˜h2

8

2˜h21!

1ig

S

vh˜₂12w

8

˜x

8

12

z

2h˜2

1

D

_l

V

2

1ig

S

wx˜2_2v

₈

_h˜

₈

2 2

1z₂

8

˜h₂

₈

2

D

l_V1

2ig

S

u˜r1_2v

₈

_h˜

₈

1 1

1z₂

8

˜h₁

₈

1

D

l_W2

2ig

S

vh˜₁22u

8

r˜

8

22

z

2˜h12

D

lW12

f₁u

3 x˜

2˜_h 2 1

1

f₁w

3 h˜12r˜12 f₁

8

u

₈

3 h˜2

8

2˜x

8

11 f₁

8

w

₈

3 ˜r

8

2_h˜ 1

8

1

2

f₃u

3

A

2x

˜2_h˜ 2 1

2

f₃w

3

A

2h

˜ 1 2_˜r1

2

f₃

₈

u

₈

3

A

2x

˜

₈

1˜_h 2

8

2₂f3

8

w

8

3

A

2 h

˜ 1

8

1˜_r

₈

2_{1H.c., ~3.7!}

but Eq. ~3.7!, see Appendix B 2, takes the form

L_masssingle₅₂1

2~ C

6_!t_Y6_C6_{1H.c. ~3.8!}

1

Here we are considering the conservation of the quantum numberF_{, defined as}F_{5L1B, where L is the lepton number and B is the}

baryon number, then f2, f2850 @2#.

The chargino mass matrix is diagonalized using two rota-tion matrices, D and E, defined by

x ˜

i

1

5Di jC1j , x˜i25Ei jC2j , i, j51, . . . ,8. ~3.9! Then we can write the diagonal mass matrix (D and E are unitary!as

MSCC5E*XD21, ~3.10! the matrix X is defined in Eq. ~B8!.

As in the previous subsection we will define the following Dirac spinors:

C~x˜

i

1_!

5~x˜

i

1_x˜¯

i

2_!t_{, C}c_~x˜ i

2_!

5~x˜

i

2_x˜¯

i

1_!t_{, ~3.11!}

where x˜

i

1 _{is the particle and ˜x}

i

2 _{is the antiparticle. This}

structure is the same as in the chargino sector of the MSSM

@5#.

C. Neutralino

For the neutralino case we have

L mass neutralino

52

m_l

2 ~ lA

3

l_A3₁l_A8l_A8!₂m

8

2 lBlB2

mh

2 h˜

0_h˜

8

0

2m₂r˜r0_r˜

8

0

2m₂x˜x0_x˜

8

0

2m₂S~s˜ 1

8

0 s ˜ 1 0

1s˜ 2

8

0 s ˜ 2 0

!₂ig

8

A

2~wx

˜0

2ur˜0

1u

₈

˜r

8

0

2w

₈

x˜

8

0_!l

B2

ig

A

2

S

ur

˜0

2vh˜0₁v

8

h˜

8

0₂u

8

˜r

8

0₁

z

2s˜2 0

2

z

₈

2s˜2

8

0

D

l_A3₂ ig

A

6

S

2wx

˜0

2ur˜0

2vh˜0₁u

8

˜r

8

0

1v

8

h˜

8

0₂2w

8

˜x

8

0₁

z

2s˜2 0

2

z

₈

2s˜2

8

0

D

l_A8₁f1

3 ~ux˜

0_h˜0

2vx˜0˜r0₂wh˜0˜r0!₁

f₁

8

3 ~u

8

x˜

8

0_h˜

8

0

2v

8

x˜

8

0˜r

8

0₂w

8

h˜

8

0˜r

8

0!

2 f3

3

A

2~ux

˜0_s˜ 2 0

1w˜r0_s˜ 2 0

12zx˜0_˜r0_!

2

f₃

₈

3

A

2~u

8

x

˜

₈

0_s˜ 2

8

0

1w

₈

˜r

₈

0_s˜ 2

8

0

12z

₈

x˜

₈

0_˜r

8

0_!

1H.c. ~3.12!

Equation~3.12!can be put in the following form, see Appen-dix B3:

L_massneutralino₅₂1

2~ C

0_!t_Y0_C0

1H.c. ~3.13!

The neutralino mass matrix is diagonalized by a 13₃13 rotation matrix N, a unitary matrix satisfying

MN5N*Y0N21, ~3.14!

and the mass eigenstates are

x ˜

i

0

5Ni jCj

0

, j51, . . . ,13. ~3.15!

We can define the following Majorana spinor to represent the mass eigenstates

C~˜x

i

0 !5~x˜

i

0_˜¯_x

i

0

!t_{. ~3.16!}

In supersymmetric left-right models, there are double charged higgsinos too@6#. There and in the present model the diagonalization of the chargino and neutralino sectors are numerically performed.

D. Sleptons

We can write the slepton mass term as

L_massslepton₅L_SMT1L_Fslepton₁L_Dslepton

52m˜˜_n

2 n ˜

l

*n˜

l

2

S

m˜_L2₁4ml 2

9

D

l˜*l˜2

S

m˜R

2

1

4m_l2

9

D

l˜ c_*l˜c

1m˜_LR2 ~l˜ l˜c₂l˜c*l˜*!, ~3.17!

where

m ˜

n ˜

2

5m_n˜

2

1z

2

12v22u22w2

6 ,

m˜_L2₅m2_l˜1l₂2

v21

z2₂2~w2₂v22u2!

12 ,

m˜_R2₅m_l˜c 2

1l₂2v2₁

z2

22~u2

22w2

2v2!

12 ,

m˜_LR2 59m₄ l

FS

mS

A

21

f₃uw

9z

D

2 4

Performing the diagonalization we find the following states:

S

l˜₁2 l˜₂2

D

5

S

cosuf sinuf

2sinuf cosuf

D

S

l˜

l˜c*

D

,

~l˜₁1 l˜₂1!5~l˜* l˜c_!

S

cosuf 2sinuf sinuf cosuf

D

,

~3.19!

where the mixing angle is given by

tan 2uf5 2m˜_LR2

m ˜

L

2

2m˜_R2, ~3.20!

and the masses of these states are

m_1,22 ₅4ml 2

9 1

1

2@m˜L

2

1m˜_R2₆

A

~m˜_L2₂m˜_R2!2

14m˜_LR4 #.

~3.21!

Substituting the diagonal slepton states, given in Eq.

~3.19!, in Eq. ~3.17!, we obtain the following diagonal La-grangian:

L_massslepton₅₂m˜˜n˜n_l*˜n_l2m₁2l˜₁1l˜₁2₂m₂2l˜₂1l˜₂2. ~3.22! We wish to stress that the slepton sector of this model is the same as in the MSSM @5#.

E. Neutral gauge bosons

The gauge mass term is given by L_HHVVscalar , see Eq. ~A6!, which we can divided in L_masschargedandL_massneutral.

The neutral gauge boson mass is given by

L_massneutral₅

~

V3m V8m Vm

!

M2

S

V₃m

V₈m

Vm

D

, ~3.23!

where

M2₅g 2

2

S

~v21u21z2!

1

A

3~

v22u21z2! 22tu2

1

A

3~

v22u21z2!

1 3~v

2

1u2₁4w2₁z2! 2t

A

3~u

2

12w2!

22tu2 2t

A

3~u

2

12w2! 4t2~u2₁w2!

D

, ~3.24!

with t₅g

₈

/g.

In the approximation that w2_@v2, u2, and z2, the masses

of the neutral gauge bosons are: 0, M_Z2, and M_Z 8

2

, and the masses are given by

MZ

2 '1

2

g2₁4g

₈

2

g2₁3g

₈

2~

v21u21z21v

8

21u

8

21z

8

2!,

M_Z

8

2 '1

3~g

2

13g

₈

2!~2w2₁2w

₈

2!. ~3.25!

F. Charged gauge bosons

The charged gauge boson mass term,L_massneutral, is

L mass charged

5M_W2W_m2W1m₁M_V2V_m2V1m₁M_U2U_m22U11m,

~3.26!

where

M_U2₅g 2

4 ~vr

2

1v

x

2

14v

s₂

2

1v

r₈

2

1v

x₈

2

14v

s

2

8

2 !,

M_W2₅g 2

4 ~vh

2

1v_r212v_s 2

2

1v

h₈

2

1v

r₈

2

12v

s

2

8

2 !,

M_V2₅g 2

4 ~vh

2

1v

x

2

12v

s₂

2

1v

h₈

2

1v

x₈

2

12v

s

2

8

2 !.

~3.27!

Using MWgiven in Eq. ~3.27!and MZ in Eq.~3.25!we get the following relation:

MZ

2

M_W2 5

1₁4t2

1₁3t25 1

1₂sin2uW

, ~3.28!

therefore we obtain

t2

5 sin

2_u

W

1₂4 sin2uW

. ~3.29!

IV. INTERACTIONS

In the previous section we have analyzed the physical spectrum of the model. Now we are going to get interactions between these particles. The procedure to write two-component spinors in terms of four-component spinors is given in Appendix C.

A. Lepton gauge boson interactions

We will define the following Dirac spinors for the leptons:

C~l!5~l l¯c!t_{, C}c_~l!

5~lcl¯!t_{, C~n}

l!5~nl0!t. ~4.1! The leptons and the gauge bosons are defined in the Eqs.~2.1!and~2.5!and using Eqs.~4.1!, we can rewrite Eq.~A3!as follows:

L_llVlep₅₂ g

A

2@C

¯c_~l!gm_LC~l!U m

11

1C¯c_~l!gm_LC~n

l!Vm11C¯~nl!gmLC~l!Wm11H.c.#

2 gt

A

1₁4t2C

¯_~l!gm_C~l!A m2

g

2

A

1₁4t2

1₁3t2C

¯_~nl!gm_LC~n l!

F

Zm2

1

A

3 1

A

1₁4t2Zm

8

G

2g₄

A

114t

2

113t2

FS

21

114t2C

¯_~l!gm_C~l!

1C¯_~l!gm_g5_C~l!

D

_Z

m

1

S

2

A

3

A

1₁4t2C

¯_~l!gm_C~l!

2 1

A

3 1

A

1₁4t2C

¯_~l!gm_g5_C~l!

D

_Z

m

8

G

5L_llVNC₁L_llVCC, ~4.2!

where t is defined in Eq.~3.29!, and L and R are the usual chiral projectors, see Eq.~C2!. Let us define also the following parameters:

e5 g sinu

A

1₁3 sin2u, h~t!5114t 2_{, v}

l52 1

h~t!,

al51, vl

8

52

A

3

A

h~t!, al

8

5

v_l

8

3 , g

2

5

8G_FM_W2

A

2 . ~4.3!

Then we can now rewrite the neutral part of Eq.~4.2!:

L_llVNC₅₂eC¯_~l!gm_C~l!A m2

g

2

MZ

MW

C¯_~nl!gm_LC~n l!

3

F

Zm2 1

A

3 1

A

h~t!Zm

8

G

2 g

4

MZ

M_W@C¯~l!g

m_~

v_l1a_lg5!

3C~l!Zm1C¯~l!gm~vl

8

1al

8

g

5_!C~l!Z

m

8

#. ~4.4!

The charged part of Eq. ~4.2!is

L_llVCC₅₂ g

A

2@C

¯c_~l!gm_LC~l!U m

11

1C¯c_~l!gm_LC~n

l!Vm11C¯~nl!gmLC~l!Wm11H.c.#, ~4.5! where g is defined in Eq.~4.3!. The Lagrangian in the Eqs.~4.4!and~4.5!is the same that was shown in Ref.@1#.

B. Chargino„neutralino…UÀ À_interactions

L_Ux˜11x˜052

g

2$@WD3Lg m_RU˜

2U˜¯cLgm_RW˜

31

A

3~UDcLgmRW˜82WD8LgmRU˜!

1

A

2~TD₅0Lgm_RT˜

2 11

1TD₄0Lgm_RT˜

1 11

1TD₂c11Lgm_RT˜

6 0

1TD₁c11Lgm_RT˜

3 0

!₁SD₂c11Lgm_RS˜

4 0

1SD₁c11Lgm_RS˜

3 0

1SD4 0

Lgm_RS˜

1 11_1SD

3 0

Lgm_RS˜

2 11_#U

m

22_1H.c.%

52g₂$$@~N_i1*₂

A

3N*_i2!B_j11

A

2~N_i8*B_j31N_i7*B_j2!1N_i12* B_j51N_i13* B_j4#C¯~˜x

i

0

!Lgm_RC~x˜ j

11_!

1@A_i1*~

A

3Nj22Nj1!1

A

2~Ai3*Nj91A*i2Nj6!1Ai5*Nj131A*i4Nj12#C¯c~x˜i22!Lg m_RC~x˜

j

0

!%U_m22₁H.c.%.

~4.6!

In a similar way we get for the charginos

L_Ux˜1x˜152 g

A

2

FS

W

Dc_Lgm_RV˜

2VDcLgm_RW˜

1TD₂c1Lgm_RT˜

1 1_1TD

1

c1_Lgm_RT˜

2 1₁1

2SD1

c1_Lgm_RS˜

2 1₁1

2SD2

c1_Lgm_RS˜

1 1

D

_U

m

22_1H.c.

G

5 g

A

2

FS

Di1*Ej22Di2*Ej11Di4*Ej31Di3*Ej41 1

2Di7*Ej81 1

2Di8*Ej7

D

C¯c~x˜i2!Lg m_RC~x˜

j

1_!U

m

22_1H.c.

G

_{. ~4.7!}

C. Lepton slepton chargino„neutralino…interactions

The interactions between lepton slepton chargino~neutralino!are given by

L_{l l˜x}˜₅$₂g cosu_fA_iC¯~x˜_i0!LC~l!₂g sinu_fA_i1C¯~l!RCc~x˜_i22!₁2l₃sinu_f@A_i5*C¯c~x˜_i22!LCc~l!₁N_i8*C¯~˜x_i0!LC~l!#%l˜₁1

1$g sinufAi*C¯~˜xi

0

!LC~l!2g cosufAi1C¯~l!RC c_~x˜

i

22_!

12l₃cosuf@Ai5*C¯ c_~x˜

i

22_!LCc

~l!1N_i8*C¯_~x˜

i

0

!LC~l!#%l˜₂1

1

H

2g2l3

A

2 @Ei8*C

¯_~x˜

i

1_!LC~l!

1D_i7*C¯c_~x˜ i

2_!LCc

~l!1

A

2N_i7*C¯_~x˜

i

0

!LC~nl!#

J

˜nl1H.c., ~4.8!

where A_i5(N_i1*/

A

2₁N_i2*/

A

6).

V. FEYNMAN RULES

In the previous section we got the interaction Lagrangians, mainly the following vertices: lepton-lepton-gauge boson; chargino-chargino ~neutralino!-gauge boson; slepton-lepton-chargino~neutralino!.

In the Table I we give the Feynman rules for the interac-tions mentioned above, and we have defined the following operators:

O_{i j}1₅A_i1*~

A

3N_j22N_j1!1

A

2~A_i3*N_j91A_i2*N_j6!1A_i5*N_j13

1A_i4*N_j12, ~5.1!

O_{i j}2₅₂

S

D_i1*Ej22Di2*Ej11Di4*Ej31Di3*Ej41 1 2Di7*Ej8

11₂D_i8*Ej7

D

. ~5.2!

To derive the differential and total cross sections, we will assume that the mass of the lightest particle of this model coincides with the mass parameter coming from the MSSM.

In Table II we shown two different solutions got in mSUGRA @7#, and they are the values that we used in our calculations.

VI. APPLICATIONS

In this section, we will perform the calculation of differ-ential and total-cross-sections of the lightest single charged chargino, double charged chargino, and neutralino, in e2e2

scattering. First we present the chargino production. In the two subsections below, we neglect the electron mass and we assume that electrons have energy E/2. We will consider that P1 and P2are the four momenta of the

incom-ing electrons while K₁ and K₂ are the four momenta of the outgoing particles.

A. Charginos production eÀ_eÀ

\x˜Àx˜À

We have calculated the differential cross section, see Eq.

~D1!, and the total cross section, and we have displayed sev-eral plots of the total cross section with MU>0.5 TeV and

A

s₅0.5,1.0, and 2.0 TeV.

The plots show that outside the U resonance, the total-cross-section is of the order of pb, like in the MSSM @8#. This result is displayed in Figs. 2, 3, and 4.

B. Double chargino and neutralino production eÀ_eÀ

\x˜À Àx˜0

Considering the Table I, we have at the tree level nine diagrams for this process, see Fig. 5. In the supersymmetric

left-right model @6# the production of a double charged higgsino occurs via a selectron exchange in the t-channel, like in the third diagram of Fig. 5.

As in the previous subsection, we have calculated the dif-ferential cross section, see Eq.~D2!, and the total cross sec-tion. We have done several plots using Mx0, M_l˜

1, and Ml˜2,

given in Table II, and MU50.5 TeV. Some of our results are shown in Figs. 6, 7, and 8.

VII. CONCLUSIONS

Because of the low level of standard model backgrounds, the total-cross-section s'1023 nb at

A

s₅500 GeV @9#,

e2_e2 _{collisions are an appropriate reaction for discovering} TABLE I. Feynman rules derived from Eqs.~4.5!,~4.6!,~4.7!,

and~4.8!.

Vertices Feynman rules

l2_l2_U22

2 ig

A

2Cg

m L

x

˜ j

22_x˜

i 0

U22 ig

2Oij 1

Cgm R

x

˜ i

2_x˜

j

2_U22 ig

2Oij 2

Cgm_R

l˜₁2_l2_˜x

i

22 _{22il3Ai5sinufR}

l˜22l2˜xi22 22il3Ai5cosufR l˜₁2_l2˜_x

i 0

i

F

g

S

Ni1

A

21

Ni2

A

6

D

cosufR2l3 2

A

2sinufNi8R

G

l˜₂2_l2_˜x

i 0

i

F

g

S

Ni1

A

21

Ni2

A

6

D

sinufR1l3 2

A

2cosufNi8R

G

l˜11l2˜xi 0

i

F

g

S

Ni1*

A

21

Ni2*

A

6

D

cosufL2l3 2

A

2sinufNi8*L

G

l˜21l2˜xi 0

i

F

g

S

Ni1*

A

21

N_i2*

A

6

D

sinufL1l3 2

A

2cosufNi8*L

G

l˜11l2˜xi22 2igAi1sinufRC l˜21l2˜xi22 2igAi1cosufRC n

˜

ll2x˜i2 _2il3 2

A2

Di7*L

n

˜ l *l2_x˜

i

2

2il3 2

A

2Di7R

TABLE II. Mass values of the lightest chargino and neutralino, sleptons, and of the sneutrino at electroweak scale corresponding to the mSUGRA solution.

m˜ @GeV# RR1: tanb53 RR2: tanb530

x

˜

1

6 _{128 132}

x

˜

1

0 _{70 72}

e ˜ L

2 _{176 217}

e ˜ R

2 _{132 183}

n

˜ 166 206

FIG. 1. Feynman diagrams for the process e2_e2→_x˜2_x˜2_.

FIG. 2. Total cross section e2_e2_→˜_x2˜_x2_at

_A

_{s50.5 TeV and}

and investigating new physics at linear colliders.

We have shown in this work that the production of single charged charginos have more contributions in this model than in the MSSM. Although in the MSSM the chargino pairs can be only produced through e2_e2 _{collisions by}

sneutrino exchanges in the u and t channels, in the 3-3-1 supersymmetric model we also have a s channel contribution due to the exchange of U22_{. This new contribution induces}

a peak at

A

s.MU, where MUis the mass of this boson, and gives a clear signal. Near the resonance the dominant term in the total-cross-section is given by uO2u2(m_x˜1

4

28sm˜_x1 2

14s2) coming from the U contribution. The total-cross-section outside the U resonance has the same order of mag-nitude than the cross section in the MSSM, as we should expected, because in this case we do not have an enhance-ment due to the s- channel contribution.

It was also shown that in this model we have double charged charginos, which are not present in the MSSM

framework. Therefore, it is a very useful way to distinguish the 3-3-1 supersymmetric model from the MSSM, and also from the usual 3-3-1 model because in this case we do not have double charged leptons. We have considered the double chargino mass in the range 700<M_x˜11<800 GeV, and we

could get cross section of the order of pb outside the U resonance, while in the resonance we have an enhancement in the cross section. We believe that these new states can be discovered, if they really exist, in linear colliders like NLC, JLC, TESLA, CLIC or VLEPP.

ACKNOWLEDGMENTS

This research was financed by Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo ~M.C.R.!. One of us

~M.C.R.! would like to thank the Laboratoire de Physique Mathe´matique et The´orique~Montpellier!for its kind hospi-tality and also G. Moultaka for useful discussions.

FIG. 3. Total cross section e2_e2_→˜x2_˜x2_at

_A

_{s51.0 TeV and}

Di75102

1

, O25102 1

.

FIG. 4. Total cross section e2_e2_→˜_x2˜_x2_at

_A

_{s52.0 TeV and}

Di751, O251021.

FIG. 5. Feynman diagrams for the process e2_e2_→x˜22˜_x0

and i51,2.

FIG. 6. Total cross section e2_e2_→x˜22_x˜0_at

_A

_s

APPENDIX A: LAGRANGIAN

With the fields introduced in Sec. II, we can built the following Lagrangian @2#

L_331S5L_SUSY1L_soft ~A1!

where

L_SUSY5L_lepton1L_gauge1L_scalar, ~A2!

is the supersymmetric part whileL_softbreaks supersymmetry. Now we are going to present all terms in the Lagrangian of the model, which we have used in this work.

1. Lepton Lagrangian

In the L_{Leptonterm, the interactions between leptons and}

gauge bosons are given in components by

L_llVlep₅g

2¯Ls¯ m_la_LV

m a

, ~A3!

where la _{are the usual Gell-Mann matrices.The next parts} we are interested in are lepton-slepton-gaugino interactions:

L

l l˜V˜

lep

52 ig

A

2~L

¯_la_˜Ll¯ A a

2LDla_Ll A a

!. ~A4!

2. Gauge Lagrangian

The part we are interested inL_gaugeis the interaction be-tween higgsino and gauge boson; that is

L_llgauge_{V 52}ig fabcl¯

A a_l

A b_sm_V

m

c _{, ~A5!} where fabc are the structure constants of the gauge group

SU(3).

3. Scalar Lagrangian

In the scalar sector, we are interested in the following three terms:

L

HH˜ V˜

scalar

52

_A

ig

2@hDl a_hl¯

A a

2h¯la_h˜l A a

1rDla_r¯l A a

2¯r_la_˜rl A a

1xDla_xl¯ A a

2x¯_la_˜xl A a

1SDla_Sl¯ A a

2¯Sla_˜Sl A a

2hD

8

l*a_h

8

¯l

A a

1h¯

8

l*a_h˜

8

l_Aa

2rD

8

l*a_r

8

¯l

A a

1¯r

₈

_l*a_˜r

8

l_Aa

2xD

8

l*a_x

8

l¯

A a

1¯x

₈

_l*a_x˜

8

l_Aa

2SD

₈

l*a_S

8

l¯

A a

1¯S

₈

l*a_˜S

8

l_Aa]₂ig

8

A

2@rDrl

¯_B2¯_r˜_rlB2xD_x¯_lB1x¯˜_xlB2rD

₈

_r

₈

¯_lB1¯_r

₈

˜_r

₈

_lB1xD

₈

_x

₈

¯_lB2x¯

₈

˜_x

₈

_lB#,

L

H ˜ H˜ V

scalar

5g₂@hDs¯m_la_h˜

1rDs¯m_la_˜r

1xDs¯m_la_x˜

1SDs¯m_la_˜S

2hD

₈

s¯m_l*a_h˜

8

2rD

₈

s¯m_l*a_r˜

8

2xD

₈

s¯m_l*a_˜x

8

2SD

₈

s¯m_l*a_˜S

8

#Vm

a

1g₂

8

@rDs¯m_˜r

2xD¯sm_x˜

2rD

8

s¯m_˜r

8

1xD

8

s¯m_x˜

8

#Vm,

FIG. 8. Total cross section e2_e2_→x˜22_x˜0_at

_A

_s

52.0 TeV and X1cosuf5X2cosuf5X3sinuf5X4sinuf51022, O151022.

FIG. 7. Total cross section e2_e2→_x˜22˜_x0

L_HHVVScalar₅1

4@g

2_V

m

a_Vbm_h¯la_lb_h

1g2V_maVbm¯r_la_lb_r

1g2V_maVbm¯x_la_lb_x

1g2V_maVbm¯h

₈

_l*a_l*b_h

8

1g2V_maVbm¯r

₈

_l*a_l*b_r

8

1g2V_maVbmx¯

₈

_l*a_l*b_x

8

1g2V_maVbm~ l_ika¯Sk j1ljk a

S ¯_{ki!~ l}

ik a

Sk j1ljk a

Ski!

1g2V_maVbm~ l_ik*a

S ¯

k j

8

1l*_jka

S ¯

ki

8

!~ l_ik*a

S_{k j}

8

1l*_jka

S_ki

8

!₁g

₈

2VmVm¯rr1g

8

2VmVm¯xx12gg

8

Vm a

Vm~¯r_la_r!

22gg

₈

V_maVm~x¯_la_x!

1g

₈

2VmVmr¯

8

r

8

1g

8

2VmVmx¯

8

x

8

12gg

8

Vm a

Vm~¯r

₈

_l*a_r

8

!₂2gg

₈

V_maVm~¯x

₈

_l*a_x

8

!].

~A6!

4. Superpotential

The terms of interest in this case are

L_{H M T52}mh

2 h˜ih˜i

8

2

mr 2 r˜i˜ri

8

2

mx 2 x˜i˜xi

8

2

mS

2˜Si j˜Sji

8

1H.c.,

L_F51

3@3l1eFL˜ LL˜1l2e~2FLh1FhL˜!L˜

1l₃~2F_LS₁F_SL˜!L˜₁f₁e~F_rxh1rF_xh1rxF_h!

1f2~2FhhS1hhFS!1f3~FrxS1rFxS1rxFS!

1f₁

₈

e~Fr₈x

8

h

8

1r

8

Fx₈h

8

1r

8

x

8

Fh₈!

1f₂

₈

~2F_h8h

₈

S

₈

₁h

₈

h

₈

FS₈!

1f₃

₈

~Fr₈x

8

S

8

1r

8

Fx₈S

8

1r

8

x

8

FS₈!#1H.c.,

L_{HH˜ H}˜₅₂ 1

3@f1e~˜rx˜h1rx˜h˜1˜rxh˜!

1f₂~˜h_{h˜ S1hh˜ S˜1h}˜_h˜_S!1f

3~r˜˜ Sx 1rx˜ S˜1˜rx˜S!

1f₁

₈

e~˜r

₈

_x˜

₈

_h

₈

_1r

₈

_x˜

₈

_h˜

₈

_1r˜

₈

_x

₈

_h˜

₈

_!

1f₂

8

~h˜

₈

_h˜

₈

_S

₈

_1h

₈

_h˜

₈

˜_S

₈

_1h˜

₈

_h

₈

_S˜

₈

_!

1f₃

₈

~˜r

₈

˜_x

₈

_S

₈

_1r

₈

˜_x

₈

˜_S

₈

₁˜_r

₈

_x

₈

˜_S

₈

_{!#1H.c. ~A7!}

5. Soft term

The soft term of this model is

L_{G M T52}1

2

F

mla

(

51 8

~ lA a

lA a

!1m

₈

lBlB1H.c.

G

,

L_scalar52m_h2h†_h

2m_r2r†_r

2m_x2x†_x

2m_S2Tr~S†_S!

2m_h

8

2 h

₈

†_h

8

2m_r

8

2 r

₈

†_r

8

2m_x

8

2 x

₈

†_x

8

2m_S

8

2

Tr~S

₈

†_S

8

!1~k₁ei jkrixjhk1k2hihjSi j

†

1k₃xirjSi j

†

1k₁

₈

ei jkri

8

x

8

jh

8

k1k2

8

hi

8

hj

8

S

8

i j

†

1k₃

₈

xi

8

rj

8

S

8

i j

†

1H.c.!,

L_{S M T52}m_L2˜L†L˜₁z0

(

i51 3

(

j51 3

~L˜iL˜jSi j1LDiLDjSi j*!.

~A8!

APPENDIX B: THE MASS MATRICES

In this appendix we display the mass matrices of the charginos and neutralinos.

1. Double charged chargino

Introducing the notation

c11_5~2il

U

11˜_r11˜_x

₈

11_H˜ 1 11_H˜

₈

2 11_!t_,

c22

5~₂il_U22˜_r

₈

22_x˜22_H˜

₈

1 22_H˜

2 22_!t_,

and

C66_5~c11_c22_!t_{, ~B1!}

we can write Eq. ~3.1!as follows:

L_massdouble₅₂1

2~ C

66_!t_Y66_C66_{1H.c., ~B2!}

where

Y66₅

S

0 T

t

T 0

D

, ~B3!

T₅

1

2m_{l 2}gu gw

₈

gz

A

2 2

gz

₈

A

2

gu

₈

mr

2 2

S

f₁

₈

v

8

3 2

A

2

f₃

₈

3 z

8

D

0

f₃

₈

3 w

8

2gw ₂

S

f1v

3 2

A

2

f3

3 z

D

mx 2

f3

3 u 0

2gz

8

A

2

0 f3

8

3 u

8

mS

2 0

gz

A

2

f3

3 w 0 0

mS

2

2

. ~B4!

The matrix Y66 in Eq. ~3.2!satisfies the following rela-tion:

det~Y66₂lI!₅det

FS

2

l Tt

T ₂l

DG

5det~ l 2

2Tt•T!,

~B5!

so we only have to calculate Tt•T to obtain the eigenvalues.

Since Tt•T is a symmetric matrix,l2_{must be real, and}

posi-tive because Y66 is also symmetric.

2. Single charged chargino

Introducing the notation

c1_5~2il

W

1

2il_V1_h˜ 1

8

1_h˜ 2

1_˜r1_x˜

₈

1_h˜ 1

8

1_˜h 2 1_!t_,

c2_5~2il

W

2

2il_V2_h˜ 1 2_h˜

2

8

2_˜r

₈

2_x˜2˜_h 1 2_˜h

2

8

2_!t_, and

C6_5~c1_c2_!t_{, ~B6!} Equation~3.7!takes the form

L_massunique₅₂1

2~ C

6_!t_Y6_C6_{1H.c., ~B7!}

where

Y6₅

S

0 Xt

X 0

D

, ~B8!

with

X₅

¨

2m_l 0 gv

8

0 2gu 0 ₂

gz

₈

2 0

0 ₂m_l 0 ₂gv 0 gw

8

0 ₂

gz

2

2gv 0

mh

2 0 2

f1w

3 0 0 0

0 gv

8

0

mh

2 0

f1

8

u

8

3 0 0

gu

₈

0 ₂f1

8

w

8

3 0

mr

2 0 0 0

0 ₂gw 0 f1u

3 0

mx

2 0 0

2gz₂ 0 0 0 f3w

3

A

2

0 mS

2 0

0 gz

8

2 0 0 0

f₃

8

u

₈

3

A

2

0 mS

2

The matrix X satisfies the same properties as the matrix T@Eq.~B5!#.

3. Neutralinos

Introducing the notation

C0

5~2il_A3

2il_A8

2il_Bh˜0_h˜

8

0_˜r0_˜r

8

0_˜x0_x˜

8

0_s˜ 1 0_s˜

1

8

0_s˜ 2 0_s˜

2

8

0 !t_, Equation~3.12!takes the following form:

L_massneutralino₅₂1

2~ C

0_!t_Y0_C0

1H.c., ~B10!

where

with

A₅f1

v

3 1

A

2

f3

3 z, B5

f₁

₈

v

8

3 1

A

2

f₃

₈

3 z

8

. ~B12!

APPENDIX C: CONNECTION BETWEEN THE TWO- AND FOUR- COMPONENT SPINORS

In this appendix we show the procedure to write two-component spinors in terms of four-two-component spinors.

1. Weak eigenstates

The weak interaction eigenstates are:

U˜₅

S

2 il_U11 i¯l

U

22

D

, U˜

c

5

S

2il_U22

i¯l

U

11

D

,

T ˜ 1 11₅

S

˜r

11

rD

₈

22

D

, ˜T1

c11₅

S

˜r

8

22

rD11

D

,

T ˜ 2 11

5

S

x

˜

₈

11

xD22

D

, ˜T2

c11

5

S

x

˜22

xD

8

11

D

,

S ˜ 1 11

5

S

H˜₁11

HD

₈

₁22

D

, S ˜ 1

c11

5

S

H˜

₈

₁22

HD₁11

D

,

S ˜ 2 11 5

S

H ˜

₈

2 11

HD₂22

D

, S ˜ 2

c11

5

S

H˜₂22

HD

₈

₂11

D

,

W˜₅

S

2 il_W1 i¯l

W

2

D

, W˜

c

5

S

2il_W2

i¯l

W

1

D

,

V

˜₅

S

2ilV 1

i¯lV2

D

, V˜c₅

S

2 il_V2 il¯V1

D

, T ˜ 1 1 5

S

h ˜

₈

1 1

hD₁2

D

, ˜T1

c1 5

S

h ˜ 1 2

hD

₈

₁1

D

,

T ˜ 2 1 5

S

h ˜ 2 1

hD

₈

22

D

, ˜T₂c1₅

S

h ˜

₈

2 2

hD21

D

,

T ˜ 3 1

5

S

r

˜1

rD

8

2

D

, T˜3

c1

5

S

r

˜

₈

2

rD1

D

,

T ˜ 4 1₅

S

x˜

8

1

xD2

D

, T˜4

c1₅

S

x˜ 2

xD

₈

1

D

,

S ˜ 1 1 5

S

h ˜ 1 2

hD

₈

₁1

D

, S ˜ 1 c1 5

S

h ˜

₈

1 1

hD₁2

D

,

S ˜ 2 1 5

S

h ˜ 2 1

hD

₈

₂2

D

, S ˜ 2 c1 5

S

h ˜

₈

2 2

hD₂1

D

,

W˜35

S

2il_A3 i¯l

A

3

D

, W˜85

S

2il_A8 i¯l

A

8

D

,

B

˜₅

S

2ilB il¯_B

D

,

T ˜ 1 0

5

S

h

˜0

hD0

D

, T˜2 0

5

S

h

˜

₈

0

hD

₈

0

D

, T˜3 0

5

S

r

˜0

rD0

D

,

T ˜ 4 0

5

S

r

˜

₈

0

rD

8

0

D

, T˜5 0

5

S

x

˜0

xD0

D

,

T ˜ 6 0

5

S

x

˜

₈

0

xD

₈

0

D

, ˜S1 0

5

S

s

˜ 1 0

sD₁0

D

, ˜S2 0

5

S

s

˜

₈

1 0

sD

₈

₁0

D

,

S ˜ 3 0 5

S

s ˜ 2 0

sD2 0

D

, ˜S4

0 5

S

s ˜

₈

2 0

sD

8

2

0

D

. ~C1!

With the states defined in Eqs.~C1!we get the following identities that allow us to write the two-component spinors in terms of the four-component weak eigenstates

l_U22_sm_¯l A

3

52UDLgm_RW˜

3,

l_U11_sm_¯l A

3

52UDcLgm_RW˜

3,

l_A8sm_¯l U

11

52WD8LgmRU˜c,

l_U11_sm_¯l A

8

52UDcLgm_RW˜

8,

x

˜

₈

11_sm_xD

8

0

52TD3

c11_Lgm_RT˜

6 0

,

r

˜11_sm_rD0

52TD₁c11Lgm_RT˜

3 0_,

x

˜0_sm_xD22_52TD 5 0

Lgm_RT˜

2 11_,

r ˜

₈

0_sm_rD

8

22_52TD 4 0

Lgm_RT˜

1 11_,

s ˜

₈

2 0_sm_HD

8

12252SD4 0

Lgm_RS˜

1 11_,

H˜₁11sm_sD

2 0

52SD₁c11Lgm_RS˜

3 0_,

H ˜

₈

2 11_sm_sD

8

2 0

52SD₂c11Lgm_RS˜

4 0 , s ˜ 2 0_sm_HD

2 22

52SD₃0Lgm_RS˜

2 11_,

l_V2_sm_l¯ W

1

52VDLgm_RW˜c_{, l} W

1_sm_l¯ V

2

52WDcLgm_{RV˜ ,}

h ˜ 2 1_sm_hD

1 2

52TD₂c1Lgm_RT˜