Left-right asymmetries in polarized lepton-lepton scattering

Left-right asymmetries in polarized lepton-lepton scattering

J. C. Montero, V. Pleitez, and M. C. Rodriguez

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 Sa˜o Paulo, SP, Brazil

~Received 12 February 1998; published 2 October 1998!

Several parity-violating left-right asymmetries in Mo_{”ller electron-electron and muon-muon scattering are} considered in the context of the electroweak standard model at the tree level in fixed target and collider experiments. We show that in colliders the asymmetry with only one of the beams polarized is large enough to compensate the smaller cross section at high energies. We also show that these asymmetries are very sensitive to a doubly charged vector bilepton resonance but they are insensitive to scalar ones.@S0556-2821~98!09219-4# PACS number~s!: 13.88.1e, 12.60.Cn

I. INTRODUCTION

It is well known that the so-called Next Linear Collider ~NLC! will provide opportunities for both discovery and pre-cision measurements @1#. With the construction of the next generation of e1e2 linear colliders, with a center of mass energy up to 1500 GeV and which will be able to operate also ingg,ge2, and e2e2modes, new perspectives arise in detecting new physics beyond the electroweak standard model in processes having nonzero initial electric charge ~and nonzero lepton number!. For instance, new resonances of doubly charged scalar and vector bosons will enhance the cross section and in this case it is possible that the measure-ment of the left-right asymmetry (ARL) in electron-electron scattering can be a better parameter to signal out those reso-nances~if any! than the total cross section itself. On the other hand, although the First Muon Collider~FMC! will be one of the type m2m1 ~it is a circular machine! @2#, if this sort of technology were mastered, perhaps a Linear Muon Collider ~LMC! will be feasible. In this work we will consider mainly e2e2processes but we will also comment on the muon case.

A. Lepton Mo_{”ller scattering at high energies}

The appealing features for studying parity-violating asym-metries between the scattering of left- and right-handed po-larized electrons on a variety of targets were pointed out some years ago by Derman and Marciano@3#. In particular, in the context of the electroweak standard model~ESM! @4# the measurement of the A_RL asymmetry determined the rela-tive sign between the weak and electromagnetic interactions. It is well known that the left-right asymmetry in electron-electron Møller scattering ~fixed target! and an unpolarized target is rather small,;1027~see below! @3#. Notwithstand-ing, since the scattering has a large cross section, we can expect that fixed target experiments like those at SLAC @5# will have a number of events enough to determine such a small number. The typical beam energy of such experiments is 50 GeV and both the incident electron beam and the target electron can be polarized. On the other hand, collider experi-ments were stopped years ago at very low energies, around 1.2 GeV @6#. Although the ARL asymmetry is larger in col-liders than in fixed target experiments, the former have a smaller cross section. Hence, it seems at first sight that a

measurement of that asymmetry with large statistics in col-liders will be difficult.

In general we have processes such as

l1~p1,l1!1l2~p2,l2!→l

8

1~q1,L1!1l

8

2~p2

8

,L2!, ~1!

when l1,25e,m and l1,2

8

5e,m,t,L, with L an exotic lepton.

In Eq.~1!, pi,qi (li,Li) denote the initial and final lepton momenta~helicities!. In this work we will consider only pro-cesses such as

l~p1,l1!1l~p2,l2!→l~q1,L1!1l~q2,L2!. ~2!

Note that if we were considering quarks in the final state, there must be a mixing of ordinary quarks with exotic ones since with the known quarks we cannot build a doubly charged state.

Our main goal is to compare left-right asymmetry in the context of the electroweak standard model in fixed target experiments with the respective one in colliders experiments. Since we are not neglecting the masses of the fermions, we can use the same amplitudes for the case of e2e2 and

m2_m2 _{scattering. We also study the behavior of that} quan-tity when a vector or scalar bilepton is considered in collider experiments.

B. Four-spinor helicity eigenstates and left-right asymmetry

In this work we will not neglect the ESM Higgs-lepton interactions because in collider experiments with muons the Higgs contributions are not, in principle, negligible. Besides, in the future we will consider processes in which the final lepton can bet or even an exotic lepton L2 ~or even exotic quarks J). Hence, we will work with massive four-spinor eigenvectors of the helicity operator, defined as

S5

S

sW•pW/upWu 0

0 sW_•pW/upWu

D

, ~3! with the gamma matrices in the chiral representation, which are given by

ulR5Nl

S

~11hl!cos~ul/2!e2ifl/2 ~11hl!sin~ul/2!eifl/2 ~211hl!cos~ul/2!e2ifl/2 ~211hl!sin~ul/2!eifl/2

D

, u_lL5N_l

S

~211hl!sin~ul/2!e2ifl/2 2~211hl!cos~ul/2!eifl/2 ~11hl!sin~ul/2!e2ifl/2 2~11hl!cos~ul/2!eifl/2

D

, ~4!

where we have defined

Nl5

A

El ~11hl 2_!, hl 2₅El1ml El2ml , ~5!

Eland mlbeing the total energy and rest mass for the lepton l. (\51 and c51 throughout this work.! In Eq. ~4!, uR and uL denote positive energy spinors with positive ~right! and negative ~left! helicity, respectively. Similarly expressions can be obtained for the negative energy solutions but we will not write them explicitly.

The left-right asymmetry is defined as

ARL~ll→ll!5

dsR2dsL dsR1dsL

, ~6!

where dsR(L) is the differential cross section for one right-~left-! handed lepton l scattering on an unpolarized lepton l. That is,

A_RL~ll→ll!5~dsRR1dsRL!2~dsLL1dsLR! ~dsRR1dsRL!1~dsLL1dsLR!

, ~7! where dsi j denotes the cross section for incoming leptons with helicity i and j , respectively, and they are given by

dsi j}

(

kl uMi j;klu

2_{, i, j ;k,l5L,R. ~8!}

When both leptons are identical we can use the property MRL;RL5MLR;LR ~which implies dsRL5dsLR) assured by rotational invariance in the former case. Hence, for calculat-ing the asymmetry A_RLin Eq.~7! we use

dsRR}uMRR;RRu21uMRR;LRu21uMRR;RLu21uMRR;LLu2, ~9a! dsRL}uMRL;RRu21uMRL;RLu21uMRL;LRu21uMRL;LLu2,

~9b! ds_LR}uM_LR;RRu21uM_LR;RLu21uM_LR;LRu21uM_LR;LLu2,

~9c! and

dsLL}uMLL;RRu21uMLL;RLu21uMLL;LRu21uMLL;LLu2. ~9d!

Another interesting possibility is the case when both lep-tons are polarized. We can define an asymmetry AR;RL in which one beam is always in the same polarization state, say, right handed, and the other is either right- or left-handed polarized~similarly we can define AL;LR):

AR;RL5 ds_RR2ds_RL dsRR1dsRL , AL;RL5 ds_LR2ds_LL dsLL1dsLR . ~10!

We can define also an asymmetry when one incident par-ticle is right handed and the other is left handed and the final states are right and left or left and right handed:

ARL;RL,LR5

dsRL;RL2dsRL;LR

dsRL;RL1dsRL;LR ~11! or, similarly, ALR;RL,LR. These asymmetries, in Eqs. ~10! and ~11!, are also dominated by QED contributions. How-ever, this will not be the case if a bilepton resonance exists at typical energies of the NLC. To show this fact is the goal of the next section. These asymmetries can be calculated for both fixed target~FT! and colliders ~CO! experiments.

This work is organized as follows. In Sec. II we study the left-right asymmetries for both fixed target and collider ex-periments in the context of the ESM and for e2e2 →e2_e2_{. We also briefly consider the case ofm}2_m2 collid-ers. The same asymmetries for colliders are shown in Sec. III but considering the contributions of vector and scalar bilep-tons. Our conclusions appear in the last section while we left Appendixes A–E for showing the several amplitudes and cross sections we have used in this work.

II. LEFT-RIGHT ASYMMETRIES IN THE ELECTROWEAK STANDARD MODEL

Let us consider the lepton-lepton~diagonal! process in the context of the electroweak standard model at the tree level. The relevant part of the Lagrangian of this model is

LF52

(

i gmi 2 MWc ¯_iciH0_2e

(

i qic¯igmciAm 2 g 2 cosuWcig m_~g V i_2g A i_g5_!c iZm. ~12!

uW[tan21(g

8

/g) is the weak mixing angle, and e5g sinuWis the positron electric charge with g such that

g258GFMW

2

A

2 or g

2_/a54psin2_u

W, ~13!

witha'1/128. The vector and axial neutral couplings are g_Vi[t3L~i!22qisin2uW, gA

i_[t

3L~i!, ~14!

where t3L(i) is the weak isospin of the fermion i and qiis the charge of ci in units of e.

A. e2e2fixed target experiments

With the spinors in Eqs. ~4! we have obtained all the helicity amplitudes of Ref. @3# when me→0; even with the actual value of m_e the numerical value for the A_RL(ee) asymmetry coincides with that obtained in Ref. @3# ~i.e., at the tree level! as it should be. In fact, neglecting the lepton masses we obtain the asymmetry for a FT experiment in the ESM context: A_RLFT,ESM~ll→ll!'2GFQ 2

A

2pa 12y 11y41~12y!4~124 sin 2_u W!, ~15! where Q252(p₁

8

2p1)2, y5sin2(u/2), and the other

constants are the well knowing a, GF, and sin2uW. For FT experiments we have Q252q25y(p11p1

8

)

5y(2ml₁

2_12m

l₁Ebeam). In Eq. ~15! the approximation me

2

!Q2_!M

Z

2 _{was used. Also terms of order m}

l/Ebeam were

neglected. This approximation is valid even for E_beam'1 TeV ~typical energies of the NLC! since in this case Q2 '0.5 (GeV)2_.

We see that for a FT experiment, the ARL asymmetry is small; in addition to the factor 124 sin2uWwe have a small Q2 @7#. Using the helicity amplitudes given in Appendixes A and B, with Ebeam550 GeV, u'90 ° (y51/2), and for

100% beam polarization, we obtain

A_RLFT,ESM~ee→ee!'2331027. ~16! Hence, for FT experiments we have confirmed the value obtained in Ref.@3# for the case of e2e2 at the tree level:

A_RLFT,ESM~ee→ee!'2631029~Ebeam/1 GeV!. ~17!

Radiative corrections reduce the tree level value for the A_RLFT,ESM asymmetry in e2e2 by 4063% @7#. Although the asymmetry is small in FT experiments, the cross section of the Mo_{”ller scattering is high:}s_T'103 _{nb. In Eq. ~C1! we}

show the differential cross section for the fixed target case. The A_R;RLFT,ESM asymmetry defined in Eq. ~10! is shown in Fig. 1 as a function of sin2(u/2). This asymmetry is domi-nated in the context of the ESM by purely QED effects@8,9#. However, we will see that it sensible for purely electroweak effects. In fact, for massless electrons we have~see Table I! A_R;RLFT,ESM~ee→ee! 5 12y42~12y!4 y2~12y!2 1sbW~gV 2_1g A 2_14g VgA!1O~bW 2_! 11y41~12y!4 y2~12y!2 1sbW~3gV 2_1g A 2_14g VgA!1O~bW 2_! , ~18! where bW5g2/4paMZ 2_'5.231024

. Notice, however, that there are electroweak contributions which do not have the suppression factor 124sW

2

. Hence, the last effects are greater than in the case when the target is unpolarized, i.e.,

for the case of the asymmetry defined in Eq.~7!. This is due to the fact that the contributions proportional to g_A2 do not cancel out. When we calculate the asymmetry with the target unpolarized the cancellation occurs and we obtain the ex-pression given in Eq. ~15!. In Fig. 1 the AR;RL asymmetry appears when all amplitudes in Appendix A 1 are consid-ered. In Fig. 1 we also show the same asymmetry but only considering the photon amplitudes which are given in Ap-pendix A 1. We note that, in fact, electroweak contributions are important as can be seen by comparing Figs. 1 and 2.

For theoretical purely QED calculations in the Born ap-proximation see Refs.@8,10# and for radiative correction see Ref.@10#. For experimental data see Ref. @9#. The difference with the theoretical calculations @8,10# is due to the elec-troweak effects that we have considered. This indicates that this asymmetry is sensitive to massive vector boson ex-changes. Measurements of the QED contribution to the AR;RL asymmetry~i.e., with both the incident beam and tar-get longitudinally polarized! in a fixed target experiment with a beam energy up to near 20 GeV are given in Ref.@9#.

B. e2e2collider experiments

Next, let us consider a collider experiment with the inter-actions given in Eq.~12!. In this case the transferred momen-tum is Q252~p1

8

2p1!2 52m_l 1 8 2 2ml₁ 2₁s 2 22 cosu

FS

₄s2ml₁ 2_2m l 1 8 2

DS

s 422ml1 2

DG

1/2 →s sin2u 2, when mli→0. ~19!

Hence, now Q2 is not a small factor.

FIG. 1. The asymmetry with both beams polarized defined in Eq. ~10! in a fixed target experiment for the process e2e2

→e2_e2 _{as a function of y5sin}2

(u/2), where u is the center of mass scattering angle ~solid line! and the same quantity but only considering the photon contributions given in Appendix A 1

As we said before, in the ESM the total cross section is '1000 nb in FT experiments and '1023 _{nb in collider} experiments at

A

s5300 GeV. @See Eqs. ~C1! and ~C2! for the differential cross section in the context of the elec-troweak standard model for collider and fixed target experi-ments, respectively.# So the required luminosity, or the pa-rameter to be measured, in colliders must be larger than in fixed target experiments. According to Czarnecki and Mar-ciano@7# the number of scattering events required for a 1028 statistical accuracy in FT experiments is of the order of 1016. However, since the cross section for Mo_{”ller scattering in FT} experiments is large at low Q2, the above requirement does not intimidate us. On the other hand, in collider experiments the cross section is rather smaller than that of FT experi-ments at high energies. Notwithstanding, as in this case the asymmetry is ;105 _{times the corresponding value for FT}

experiments @11# @see Eq. ~20! below#, the 104 events/yr, which is the number of events expected for a luminosity of L'1033 _cm22_s21_{, will provide statistics just enough}

(

A

N/N;1022) to measure, although poorly measure, the A_RL asymmetry in collider experiments. In addition, if new physics does really exist at the hundreds of GeV level, the

cross section in colliders can serve to detect new resonances. If this is indeed the case, the cross section will have typical resonance enhancement~s!. Recall also that the collider mea-surements of ARL in Mo”ller scattering unlike the case of FT experiments will not have a background from eN scattering. Thus, it is worth considering the contribution to the ARL asymmetry at the energies of next linear colliders ~electrons up to 1.5 TeV! and also in muon colliders ~muons up to 4 TeV!. Although a large polarization implies sacrifice in lu-minosity at a muon collider@2#, new physics ~if any! with a clear signature could be distinguished even with low lumi-nosity. Anyway, the asymmetry depends more on the polar-ization than on the luminosity.

At energies

A

s5300 GeV andu'90 ° ~in the center of mass frame! the asymmetry is

ARL

CO, ESM_{~ee→ee!'20.05. ~20!}

Compare with the value for the FT experiment given in Eq. ~16!. In Fig. 2 we show the ARL

CO,ESM

asymmetry as a function of

A

s andu. Notice that this asymmetry is a smooth function of both variables.

TABLE I. Nonvanishing contributions in the ESM when the lepton mass is zero. Here bW 5g2 /4paMZ 2 . Initial\Final RR RL LL LR RR 1 y (y21) 12sbW(gV1gA)2 0 0 0 RL 0 y 12y 1s ybW(gV 2 2gA 2 ) 0 12y y 1sbW(12y)(gV 2 2gA 2 ) LR 0 12y y 1sbW(12y)(gV 2_2g A 2 ) 0 y 12y 1s ybW(gV 2_2g A 2 ) LL y 12y 12sbW(gV1gA) 2 0 0 0

FIG. 2. The asymmetry ARL

CO,ESM

We can integrate in the scattering angle and define the asymmetry A¯RL as A ¯_RL~ll→ll! 5

S

E

dsRR1

E

dsRL

D

2

S

E

dsLL1

E

dsLR

D

S

E

ds_RR1

E

ds_RL

D

1

S

E

ds_LL1

E

ds_LR

D

, ~21! where *dsi j[*5 ° 175 °

dsi j. This integrated asymmetry for ee→ee, in the electroweak standard model @A¯RL

CO,ESM

(ll→ll)#, appears in Fig. 3 as a function of

A

s. We see that this integrated asymmetry varies slowly with

A

s and it is almost constant at the value 20.02 above

A

s 5400 GeV.

C. µ2µ2experiments

A hypothetical fixed targetm2m2experiment at the same Ebeamenergy used in Sec. II A gives ARL(mm)'5.431025 for u'90 °. In this case Q_muon2 5(me/mm)Qelectron

2

'4.2 GeV2_{. A more realistic case is the m2m2 collider}

where, for the same experimental conditions as in the previ-ous subsection, we obtain

ALR

CO,ESM_{~mm→mm!'20.1436. ~22!}

Notice the large value for the asymmetry for the muons with respect to the electron case given in Eq. ~20!.

In the electron case the Higgs contributions are negligible. For the muon case, if we do not take into account the Higgs amplitudes given in Appendix A 1, the asymmetry in Eq. ~22! has the value of 20.1437, showing a slight dependence on the Higgs contributions. There are also effects of the lep-ton mass in theg,Z contributions to the asymmetry that are

responsible for the different values in Eqs. ~20! and ~22!. Independently of the numerical value of the Higgs contribu-tions, the important issue is that all its main contributions ~see the amplitudes given in Sec. A 3! cancel out in the ARL asymmetry defined in Eq. ~7!. In fact, as we will see in the next section, even in models with larger scalar-lepton cou-plings the extra scalar contributions cancel out.

III. LEFT-RIGHT ASYMMETRIES IN BILEPTON MODELS

An example of possible new physics are models with a doubly charged vector boson@12#. Then, the charged current interaction in terms of the physical basis is given by

2 g

A

2@n ¯_LE L n†_E L l lLW_m11 l¯L c_gm E_RlTE_LnnLV_m1 2 l¯L c_E R lT_E L l_l LUm11#1H.c., ~23! with l_L

8

5E_LllL, lR

8

5ER l_l R, nL

8

5ELnnL, the primed ~unprimed! fields denoting symmetry ~mass! eigenstates. We see from Eq. ~23! that for massless neutrinos we have no mixing in the charged current coupled to W_m1 but we still have mixing in the charged currents coupled to V_m1 and U_m11. That is, if neutrinos are massless, we can always choose E_Ln†E_Ll51. However, the charged currents coupled to V_m1and U_m11are not diagonal in flavor space and the mixing matrix K5E_RlTE_Ln has three angles and three phases. ~An arbitrary 333 unitary matrix has three angles and six phases. In the present case, however, the matrix K is deter-mined entirely by the charged lepton sector; so we can rotate only three phases @13#.!

If the bilepton U_m11 is not too heavy, say, with a mass of the order of a few hundred GeV, it is possible that it will appear as a resonance in the next linear colliders. Assuming that the matrixK is almost diagonal we can neglect the mix-ing effects in processes involvmix-ing the same initial and final FIG. 3. The angular-integrated asymmetry A¯RL

CO,ESM

leptons as l2l2→l

8

2l

8

2. The helicity amplitudes for U ex-change in the s channel are given, at the tree level, in Ap-pendix D. We do not know yet the total width GU of the U22bilepton; however, we know that at least it has to decay into l2l

8

2: If these were the only channels, we have, for the width, G~U22_→l i 2_l j 2_!5GFMU 3 6

A

2puKi ju 2_{, ~24!}

and if K_{i j} is almost diagonal, the main decays will be U →e2_e2_,m2_m2_,t2_t2_{; with an illustrative value of M}

U 5300 GeV, we have G(U→leptons)'36 GeV. Recent analyses based on the cross section for the reaction e2e2 →m2_m2 _{using effective interactions consider doubly} charged vector bileptons of mass 200 GeV for an effective constant coupling of the order of 0.1@14# which corresponds to our Ki j'1 case. In models with such a vector bilepton there are several Higgs fields and also exotic quarks. Hence, we should have to take into account U22→scalars and also U22→d₃¯J25/3,u¯₁(u¯₂)J24/3, with d₃and u_1,2denoting sym-metry eigenstates of the known quarks and JQ(J¯Q) denotes an exotic quark ~antiquark! of electric charge Q. The left-handed mixing matrices, defined by the relation among the symmetry eigenstates ~primed fields! with mass eigenstates ~unprimed fields! i.e., UL

8

5VL

u_U

L and DL

8

5VL d_D

L, survive in the Lagrangian density. Thus, the partial width is of the form~neglecting the masses of the quarks J)

G~U22_→d 3¯J25/3!5 CGFMU 3 6p

A

2 u~VL d_! 3iu2, ~25!

and similarly for the other decays into u quarks. Notice that the mixing matrix in Eq.~25! is not necessarily equal to the Cabibbo-Kobayashi-Maskawa mixing matrix which is de-fined as V_CKM5V_Lu†V_Ld. The parameters in both Eqs. ~24! and~25! are constrained for other processes but here we will not consider these constraints. A realistic calculation will have to take into account the masses of the exotic quarks J. At present, however, there are no experimental data bounds on the values of these masses. Of course, there are also tri-linear vector interactions U22→W2V2that it will occur if MV1MW,MU (V2 is a singly charged bilepton present in the model!. As a detailed calculation of GU is out of the scope of this work, we will considerGUas a free parameter. In this model the Yukawa couplings between leptons and a sextet of scalars transforming as (6,0) under SU(3)L

^U(1)N are of the form gab¯(caiL)ccb jLSi j where a,b 5e,m,t; i, j are SU~3! indices and the matrix gab denotes a symmetrical matrix. If this is the only lepton-Yukawa in-teraction in the model, by using a discrete symmetry we can avoid the coupling to the tripleth;(3,0); there is no flavor-changing neutral current ~FCNC! in this sector and the lepton-Higgs sextet couplings are proportional to ml/vS where ml is the lepton mass andvS is the vacuum expecta-tion value ~VEV! of the neutral component of the sextet. Since there are already two other scalar triplets which give

the appropriate mass to the W6 and Z0 vector bosons, it is not necessary thatvSbe of the same order of magnitude than the other vacuum expectation values that are present in the model. Hence,vSmay be of the order of a few GeV since the heavier lepton is the twith a mass of 1.777 GeV @15#. No-tice that since the neutral part of the sextet is a doublet of SU~2!, these fields have r[M_W2/ M_Z2cos2uW51 at the tree level. The doubly charged member H111 is part of a triplet

with its neutral partner having vanishing vacuum expectation value ~if neutrinos do not get Majorana masses!. There is also a doubly charged H₂11 which is a singlet of SU~2!. Strictly speaking the VEV which is in control of r ~and flavor-changing neutral currents coupled to Z) is that of the triplet which breaks the SU(3)L^U(1)N symmetry. For an arbitrary high value for this VEV we can obtain the ESM predictions. At the one-loop level it is possible that a fine-tuning is required but this deserve a more detailed study.

From the experimental point of view a lower bound on the mass of doubly charged scalars is mH11.45 GeV @16,17# but we recall that the phenomenology of the H11 derives from its couplings. The processes which have to be consid-ered are g/Z→H11H22 and H22→l2l

8

2 @18#. In the model of Ref.@12# the coupling W2W2with doubly charged scalars does not exist ~the vertex always includes a single charged vector bilepton V2; i.e., the coupling which really exists is with V2W2).

Although the bilepton model we consider here is a gauge-invariant theory which has its own scalar spectrum, we will consider in this work the bilepton contributions as extra con-tributions to the electroweak standard model ones due to the fact that in any case the neutral Higgs contributions are smaller than that of the doubly charged vector bilepton. We denote this by using the $ESM% and $ESM1U% labels in cross sections and asymmetries. However, as we will show, the doubly charged scalar bilepton contributions, denoted by

$ESM1H%, cancel out in the numerator of the ARL asymme-try. Hence, in practice our calculations can be considered a true effect of the bilepton model of Ref. @12#.

A. e2e2high energy colliders

By using the expression in Eqs. ~C1! and ~D6! we can compute the effect of such bilepton fields. In Fig. 4 we show the ratiosCO,ESM1U/sCO,ESMfor the vector bilepton case as a function of

A

s andGU. We observe that this ratio becomes appreciable for

A

s near the U resonance as expected. In this case we have chosen an arbitrary ~but reasonable! value of MU5300 GeV to illustrate the cross section behavior. We also note that this quantity is smoothly dependent on theGU values. Since the cross section could be large at the U peak, if the mass of the U is lower than the energies of future collider experiments, it is interesting to know the value of the ARL asymmetry at these energies.

Here, we will not show the analytic expression of A_RLCO, but at the same conditions as Eq. ~20! it gives

ARL

and we see that it is larger than the respective value obtained in the context of the ESM@see Eq. ~20!#. A more illustrative quantity is the angular-integrated asymmetry defined in Eq. ~21! and denoted here by ARLCO,ESM1U(ee→ee). We show it in Fig. 5 as a function of

A

s andGU. We can observe that the dependence of GU cancels out in ARLCO,ESM1U(ee →ee), so that even if we do not use a realistic calculation forGUbased in all possible decay channels predicted by the

model, the result will be, in fact, independent of it. As ex-pected the asymmetry is larger in the U pole. In Fig. 6 we quote the quantity

dA¯RL~ee→ee![~A¯RL CO,ESM1U_2A¯ RL CO,ESM_!/A¯ RL CO,ESM , ~27! with the A¯ ’s defined in Eq.~21!, as a function of

A

s andGU. FIG. 4. The ratiosCO,ESM1U/sCO,ESMfor the total cross section obtained from Eq.~D6! which includes the s-channel U22contribution and Eq.~C2! for the ESM for the reaction e2e2→e2e2as a function of

A

s andGU.

As observed above this quantity is independent ofGU.dARL is large at the U resonance; however, we would like to stress that it remains appreciably large even far from the U peak. That particular behavior suggests that this quantity could be the one to be considered in the search for new physics, like the bilepton U22, in future colliders.

Next let us consider the scalar contribution in the model. The s-channel amplitudes appear in Appendix E. In this model the amplitudes in Appendix A 3 are used by making

v→vS, and, as we said before, in models with several scalar multiplets it is possible that one of the neutral scalars does not contribute significantly to the W2and Z masses. It means that the VEV of such a scalar can be of the order of a few GeV. In this case there are some scalar contributions which are proportional to ml/vS with ml denoting the lepton mass andvS is a VEV which gives mass for the leptons.

Here we will use M_H

i

11_{550,300 GeV ~for both i51,2} although the masses of the triplet and singlet might be dif-ferent! and vS510 GeV only as an illustration. We also take for colliders

A

s5300 GeV. Taking into account the ESM plus the scalar contribution ~but not the vector U22 contri-butions! we obtain

A_RLCO,ESM1H~ee→ee!520.05, ~28! for y51/2. We see that this value is the same as the pure ESM contribution given in Eq. ~20!. This is not a surprise since the purely scalar contributions cancel out in both the neutral Higgs ~as in Appendix A 3! and also in the doubly charged Higgs contributions as can be seen from the ampli-tudes in Appendix E and the definition of the asymmetry in Eq. ~7!. However, in the muon-muon case the interference terms among the neutral Higgs and g,Z contributions are relatively significant as we will see in the next subsection.

B. µ2µ2high energy colliders

We can calculate all the above asymmetries in the case of a possible m2m2 collider. For instance taking into account the contributions of the U22 vector bilepton we obtain

A_RLCO,ESM1U~mm→mm!520.1801, ~29! while the contribution of the H22 scalar bilepton gives

A_RLCO,ESM1H~mm→mm!520.1436, ~30! for MH11550 and 300 GeV.

As discussed above, in the model with bileptons the VEV of the neutral scalar coupled mainly with leptons does not need to be equal to the VEV of the ESM (v;246 GeV) but

it can have a lower value, say, vS;10 GeV. For the muon case we calculate the asymmetry in Eq. ~29! with and with-out the ESM Higgs scalar contributions given in Appendix A 3. The respective values are20.1436 and 20.1442. If the mass of the U22 were 500 GeV, the asymmetry in Eq.~29! has the value20.0133.

IV. CONCLUSIONS

We have shown in this work that collider experiments can be suitable for studying the left-right asymmetries in lepton-lepton scattering. In fact, we have shown that in this case the asymmetries are larger than those which appear in fixed tar-get experiments. In particular we have shown that there are significant differences between electron and muon asymme-tries, the muon ones being larger than the electron ones due to mass effects. However, more details were given here for the electron case. For instance, the ARLasymmetry, when the contributions of the U22 vector bilepton are considered, is FIG. 6. dARLdefined in Eq.~27! as a function of GUand

A

s for e2e2→e2e2.

larger than the ESM value, as can be seen from Fig. 6 for the electron-electron case. On the other hand, the purely scalar bilepton contributions cancel out in the asymmetries as can be seen from the expression given in Appendixes A 3 and E. Only the interference terms between the scalars andg,Z am-plitudes survive and these are not negligible in the muon-muon case.

Concerning the asymmetry A_R;RL defined in Eq.~10! i.e., when both beams are polarized, we see from Table II that it is given by

AR;RL

CO,ESM1U_{~ee→ee!'211O~b}

W!, ~31! and comparing with Eq.~18! we see that the maximal value of A_R;RLCO,ESM1U(ee→ee) and its dependence on y are different in models with vector bileptons and the electroweak standard model ~almost purely QED! case. At the U resonance dsRL @dsRR, and hence AR;RL

CO,ES M1U

(ee→ee)'21. In the ESM on the contrary we have dsRR@dsRL, and hence A_R;RLCO,ESM(ee→ee)'11. Similarly, A_L;RLCO,ESM1U(ee→ee)' 21 and AL;RL

CO,ESM

(ee→ee)'11. Although this asymmetry is large in both the ESM~see Fig. 1! and the bilepton model the sign of it is opposite and could be useful for discovering such a vector field.

Summarizing, we have calculated in this work several left-right asymmetries in the context of the ESM and in mod-els with both vector and scalar bileptons.

Finally, a remark: in some models with vector bileptons such as U22 the decay m2→e2e2e1 ~and similar ones! does occur. The branching ratio for this decay is ;10212 @15#. This bound strongly limits the Kem couplings. The branching fraction form→3e decay is

B~m→3e!}

S

KemKee M_U2

D

2

1

G_F2. ~32! For MU5300 GeV and Kee'1 the experimental value for the above branching ratio implies Kem;1026. The doubly charged scalar will also contribute to these rare decays. For a comprehensive analysis of the constraints on bilepton masses and their coupling constants with leptons and for several types of bileptons see Ref. @19#.

ACKNOWLEDGMENTS

This work was supported by Fundac¸a˜o de Amparo a` Pes-quisa do Estado de Sa˜o Paulo~FAPESP!, Conselho Nacional de Cieˆncia e Tecnologia~CNPq! and by Programa de Apoio a Nu´cleos de Exceleˆncia ~PRONEX!.

APPENDIX A: INVARIANT AMPLITUDES IN THE ESM

In this appendix we give the invariant amplitudes for the Mo_{”ller scattering in the context of the SU(2)}L^U(1)Y ~ESM! model. Although the Higgs boson in context of the ESM gives contributions that are negligible for practical pro-poses, we include its contribution. In models with a rich scalar spectrum and/or with exotic heavy fermions the scalar contribution may become important. We omit a common fac-tor of (2El)2 in all amplitudes since it cancels out in the asymmetries for diagonal processes such as e2e2→e2e2.

1. Photon amplitudes

Up to a factor ie2/t the t-channel amplitudes are@recall that y5sin2(u/2), 122y5cosu, and 2 y1/2(122y)5sinu]

M_RR;RRg ~t!54El 2 1@~122y!23#ml 2 2E_l2 , MRR;LR g _~t!5M RR;RL g _~t!5 ml 2El 2 y1/2~122y!, ~A1a! M_RR;LLg ~t!5ml 2 E_l2y , ~A1b! M_LR;RRg ~t!5M_LR;LLg ~t!5M_RR;LRg ~t!, M_LR;LRg ~t!5 2E_l22m_l 8 2 E_l2 ~12y!, MLR;LR g _~t!5M RR;LL g _{~t!, ~A1c!} M_RL;RRg ~t!5M_RL;LLg ~t!52M_RR;LRg ~t!, Mg_RL;LR~t!5M_LR;LRg ~t!, M_RL;LLg ~t!5M_RR;LLg ~t!, ~A1d! M_LL;RRg ~t!5M_RR;LLg ~t!, 2M_LL;LRg ~t!5M_LL;RLg ~t!5M_RR;LRg ; M_LL;LLg ~t!5M_RR;RRg ~t!, ~A1e!

and the photon u-channel amplitudes, up to a factor ie2/u, are, TABLE II. Nonvanishing contributions of the U-vector bosons when the lepton mass is zero.

Initial\Final RR RL LL LR

RR 0 0 0 0

RL 0 22(12y) 0 2y

LR 0 22y 0 2(12y)

M_RR;RRg ~u!54El 2_2@31~122y!#m l 2 2E_l2 , MRR;LR g _~u!52M RR;RL g _~u!5M RR;LR g _{~t!, ~A2a!} M_RR;LLg 5ml 2 El 2~12y!, ~A2b!

2MLR;RRg ~u!5MLR;LLg ~u!5MRR;LRg ~t!, MLR;LRg ~u!5 m_l2 E_l2y , MLR;RL g _~u!52El 2_2m l 2 E_l2 y , ~A2c! M_RL;RRg ~u!52M_RL;LLg 5M_RR;LRg ~t!, Mg_RL;LR~u!5M_LR;RLg ~u!, M_RL;RLg ~u!5M_RR;LLg ~u!, ~A2d! MLL;RRg ~u!5MRR;LLg ~u!, MLL;LRg 5MgLL;RL~u!5MRR;LRg ~t!, MLL;LLg ~u!5MRR;RRg ~u!. ~A2e!

2. Z amplitudes

Up to a factor ig2/(t2M_Z2), the t-channel Z-exchange amplitudes are M_RR;RRZ ~t!5

F

4El 2_{1@231~122y!#m} l 2 2E_l2

G

~gV 2_1g A 2_!24

S

El 2_2m l 2 E_l2

D

1/2 g_Vg_A, ~A3a! M_RR;LRZ ~t!5 ml 2El~gV 2_1g A 2_!2y1/2_{~122y!, M} RR;RL Z ~t!5_2Eml l~gV 2_2g A

2_!2y1/2_{~122y!, ~A3b!}

M_RR;LLZ ~t!5 ml 2 2E_l2$gV 2 y1g_A2@31~122y!#%, ~A3c! MLR;RR Z _~t!5M LR;LL Z _~t!5M RR;RL Z _{~t!, ~A4a!} M_LR;LRZ ~t!5

S

2El 2_2m l 2 E2

D

~gV 2_2g A 2_!c u/2 2 _{, M} LR;RL Z _~t!5ml 2 E_l2~gV 2_2g A 2_{!y, ~A4b!} 2MRL;RR Z _~t!5M RL;LL Z _~t!5M RR;RL Z _{~t!, M} RL;LR Z _~t!5M LR;RL Z _{~t!, M} RL;RL Z _5M LR;LR Z _{~t!, ~A5!} MLL;RR Z _~t!5M RR;LL Z _{~t!, M} LL;LR Z _~t!5M LL,RL Z _~t!5M RR;LR Z _{~t!, M} LL;LL Z _5M RR;RR Z _{~t!. ~A6!} Up to a factor ig2_/(u2M Z

2_{), the u-channel Z-exchange amplitudes are}

M_RR;RRZ ~u!

F

4El 2_2@31~122y!#m l 2 E_l2

G

~gV 2_1g A 2_!24

S

El 2_2m l 2 E_l2

D

1/2 gVgA, ~A7a!

M_RR;LRZ ~u!5M_RR;LRZ ~t!, M_RR;RLZ ~u!52M_RR;RLZ ~t!, ~A7b! M_RR;LLZ ~u!5 ml 2 2E_l2$gV 2 ~12y!1gA 2 @~122y!23#%, ~A7c!

M_LR;RRZ ~u!5M_LR;LLZ ~u!5M_RR;RLZ ~t!, M_LR;LRZ ~u!5ml

2 E_l2~gV 2_2g A 2_{!~12y!, ~A8a!} M_LR;RLZ ~u!5

S

2El 2_2m l 2 El 2

D

~gV 2_2g A 2_{!y, ~A8b!}

MLL;RR Z _~u!5M RR;LL Z _{~u!, M} LL;LR Z _~u!5M LL;RL Z _~u!5M RR;RL Z _{~t!, ~A10a!} M_LL;LLZ ~u!5

F

4El 2 2@31~122y!#ml 2 E_l2

G

~gV 2_1g A 2_!14

S

El 2_2m l 2 E_l2

D

1/2 gVgA. ~A10b! 3. H0amplitudes

Here H0denotes a neutral Higgs boson with couplings like the ESM Higgs boson. Up to a common im_l2/v2(t2M_H2) factor, the t-channel H-exchange amplitudes are

MRR;RR H _~t!52ml 2 E_l2~12y!, MRR;LR H _~t!5M RR;RL H ₅₂ ml 2El 2 y1/2~122y!, MRR;LL H _{~t!5y, ~A11a!} 2MLR;RR H _~t!5M LR;LL H _~t!5M RR;LR H _{~t!; M} LR;LR H _~t!5M RR;RR H _{~t!, M} LR;RL H _~t!5M RR;LL H _{~t!, ~A11b!} 2MRL;RR H _~t!5M RL;LL H _~t!5M RR;LR H _{~t!, M} RL;LR H _~t!5M RR;LL H _{~t!, M} LR;RL H _~t!5M RR;RR H _{~t!, ~A11c!} 2MLL;RR H _~t!5M RR;LL H _{~t!; 2M} LL;LR H _~t!52M LL;RL H _~t!5M RR;LR H _{~t!, M} LL;LL H _~t!5M RR;RR H _{~t!. ~A11d!} Up to a common im_l2/v2_(u2M H 2

) factor the u-channel H-exchange amplitudes are

M_RR;RRH ~u!52ml 2 E_l2y , 2MRR;LR H _~u!5M RR;RL H _5M RR;LR H _{~t!, M} RR;LL H _{~u!512y, ~A12a!}

M_LR;RRH ~u!5M_LR;LLH ~u!5M_RR;LRH ~t!; MH_LR;LR~u!5M_RR;LLH ~u!, M_LR;RLH ~u!5M_RR;RRH ~u!, ~A12b! M_RL;RRH ~u!52M_RL;LLH ~u!5M_RR;LRH ~t!, 2MH_RL;LR~u!5M_RR;RRH ~u!, M_RL;RLH ~u!5M_RR;LLH ~u!, ~A12c! M_LL;RRH ~u!5M_RR;LLH ~u!, M_LL;LRH ~u!5MH_LL;RL~u!5M_RR;LRH ~t!, M_LL;LLH ~u!52M_RR;RRH ~u!. ~A12d!

APPENDIX B: TOTAL AMPLITUDES IN THE ESM

We define Mi j;kl5

(

X Mi j;kl X _~u!1

(

X Mi j;kl X _~t!1

(

X Mi j;kl X _{~s! ~B1!}

as the total amplitudes for fixed i j ;kl polarization and exchanged particles X5g,Z0,H0, and others. In the context of the standard model only t and u channels contribute. Notice also that these contributions arise only when the lepton-lepton scattering conserves flavors, l₁l₂→l₁l₂ with l₁ being equal or not to l₂:

MRR;RR5ie2

H

S

4E223m_l2 2E2

D

S

u1t tu

D

1 m_l2 2E2~122y!

S

u2t tu

D

1 g2 e2

S

4E223m_l2 2E2

D

~gV 2_1g A 2_!

F

u1t22MZ 2 ~t2MZ 2_!~u2M Z 2_!

G

24g 2 e2gVgA

A

E22m_l2 E2

F

u1t22M_Z2 ~t2MZ 2_!~u2M Z 2_!

G

1 g2 e2 m_l2 2E2~122y!~gV 2_1g A 2_!

F

u2t ~t2MZ 2_!~u2M Z 2_!

G

2 ml 4 E2e2v2

S

12y t2MH 21 y u2MH 2

D

J

, ~B2a! MRR;LR5ie2

F

u1t tu 1 g2 e2 u2t ~t2MZ 2_!~u2M Z 2_!~gV 2_1g A 2_!1 ml 2 e2_v2 t2u ~t2MH 2_!~u2M H 2_!

G

ml 2E2 y 1/2_{~122y!, ~B2b!} M_RR;RL5ie2

F

u2t tu 1 g2 e2 u2t ~t2MZ 2_!~u2M Z 2_!~gV 2_2g A 2_!2 ml 2 e2v2 t2u ~t2MH 2_!~u2M H 2_!

G

ml 2E2y 1/2_{~122y!, ~B2c!}

MRR;LL5ie2

F

S

y t 1 12y u

D

m_l2 E21 g2 e2

S

g_V2y1g_A2~422y! t2M_Z2 1 g_V2~12y!2g_A2~212y! u2M_Z2

D

m_l2 2E21 m_l2 e2v2

S

y t2M_H2 1 12y u2M_H2

D

G

, ~B2d! MLR;RR5ie2

F

u2t tu 1 g2 e2 u1t22M_Z2 ~t2MZ 2_!~u2M Z 2_!~gV 2_2g A 2_!1 ml 2 e2v2 u2t ~t2MH 2_!~u2M H 2_!

G

ml 2E2 y 1/2_{~122y!, ~B2e!} MLR;LR5ie2

F

2 t 2 m_l2 E2

S

u2t tu

D

1 g2 e2

F

2 t2M_Z22 m_l2 E2

S

u2t ~t2MZ 2_!~u2M Z 2_!

D

G

~gV 2_2g A 2_!2 ml 2 e2v2

S

m_l2 E2~t2M_H2!2 1 u2M_H2

DG

~12y!, ~B2f! MLR;RL5ie2

F

2 u2 m_l2 E2

S

u2t tu

D

1 g2 e2

F

2 u2M_Z22 m_l2 E2

S

t2u ~t2MZ 2_!~u2M Z 2_!

D

G

~gV 2_2g A 2_!2 ml 2 e2_v2

S

m_l2 E2_~u2M H 2_!2 1 t2M_H2

DG

y , ~B2g! MLR;LL5ie2

F

u1t tu 1 g2 e2 u1t22M_Z2 ~t2MZ 2_!~u2M Z 2_!~gV 2_2g A 2_!2 ml 2 e2v2 t1u22M_H2 ~t2MH 2_!~u2M H 2_!

G

ml 2E2y 1/2_{~122y!, ~B2h!} M_RL;RR5ie2

F

u1t tu 1 g2 e2 u2t ~t2MZ 2_!~u2M Z 2_!~gV 2_2g A 2_!2 ml 2 e2v2 t2u ~t2MH 2_!~u2M H 2_!

G

ml 2E2 y 1/2_{~122y!, ~B2i!} MRL;LR52ie2

F

22 u 1 m_l2 E2

S

t2u tu

D

2 g2 e2

F

2 u2M_Z22 m_l2 E2

S

t2u ~t2MZ 2_!~u2M Z 2_!

D

G

~gV 2_2g A 2_!1 ml 2 e2v2

S

m_l2 E2~u2M_H2!1 1 t2M_H2

DG

y , ~B2j! MRL;RL5ie2

F

2 t 2 m_l2 E2

S

u2t tu

D

1 g2 e2

F

2 t2M_Z22 m_l2 E2

S

u2t ~t2MZ 2_!~u2M Z 2_!

D

G

~gV 2_2g A 2_!2 ml 2 e2v2

S

m_l2 E2~t2M_H2!2 1 u2M_H2

DG

~12y!, ~B2k! MRL;LL5ie2

F

u2t ut 1 g2 e2 u1t22MZ 2 ~t2MZ 2_!~u2M Z 2_!~gV 2_2g A 2_!1 ml 2 e2_v2 t2u ~t2MH 2_!~u2M H 2_!

G

ml 2E2y 1/2_{~122y!, ~B2l!} MLL;RR5ie2

F

S

y t1 12y u

D

m_l2 E21 g2 e2

S

g_V2y1g_A2~422y! t2M_Z2 1 g_V2~12y!1g_A2~212y! u2M_Z2

D

m_l2 2E22 m_l2 e2v2

S

y t2M_H2 2 12y u2M_H2

D

G

, ~B2m! MLL;LR5ie2

F

t2u tu 1 g2 e2 u1t22M_Z2 ~t2MZ 2_!~u2M Z 2_!~gV 2_2g A 2_!2 ml 2 e2v2 t2u ~t2MH 2_!~u2M H 2_!

G

m_l2 2E2y 1/2_{~122y!, ~B2n!} M_LL;RL5ie2

F

u1t tu 1 g2 e2 u1t22M_Z2 ~t2MZ 2_!~u2M Z 2_!~gV 2_2g A 2_!2 ml 2 e2v2 t2u ~t2MH 2_!~u2M H 2_!

G

ml 2E2y 1/2_{~122y!, ~B2o!} M_LL;LL5ie2

F

S

4E 2_23m l 2 2E2

D

S

u1t tu

D

1 m_l2 2E2~122y!

S

u2t tu

D

1 g2 e2

S

4E223m_l2 2E2

D

~gV 2_1g A 2_!

S

u1t22MZ 2 ~t2MZ 2_!~u2M Z 2_!

D

14g 2 e2gVgA

A

E22m_l2 E2

S

u1t22M_Z2 ~t2MZ 2_!~u2M Z 2_!

D

1 g2 e2 m_l2 2E2~122y!~gV 2_1g A 2_!

S

u2t ~t2MZ 2_!~u2M Z 2_!

D

1 ml 4 E2e2v2

S

12y t2M_H21 y u2M_H2

D

G

. ~B2p!

APPENDIX C: DIFFERENTIAL CROSS SECTION FOR FIXED TARGETS AND COLLIDERS IN THE ESM

In the context of the standard model in a collider experiment we have dsESM dV

U

_CO5 e4 128p2s

H

11y41~12y!4 y2~12y!2 1 2g2 e2

F

~gV 2_1g A 2_! 112a y~12y!~y1a!@12y1a#1~gV 2_2g A 2_! 3

S

~12y! 2 y~y1a!1 y2 ~12y!@12y1a#

DG

1 g4 e4~gV 2_2g A!2

S

~12y!2 ~y1a!21 y2 ~12y1a!2

D

1g 4 e4~gV 4_16g V 2_g A 2_1g A 4_! ~112a! 2 ~y1a!2_@12y1a#

J

, ~C1! where a[M_Z2/s.

For the fixed target experiment we have dsESM dV

U

_FT' e4 128p2mepe p_e 8 2 ~Ee1me!pe82peEe8cosuee8

F

11y1~12y!4 y2~12y!2

G

, ~C2!

where we have written only the main contribution. Here pe (Ee) and pe8 (Ee8) denote the momentum~energy! of the initial and final electron, respectively; cosuee8is the cosine of the angle between the incident electron and one of the final electrons.

APPENDIX D: U AMPLITUDES

In this appendix we give the invariant amplitudes for ll→l

8

l

8

scattering in the context of a model with a doubly charged vector bilepton U22. In principle, the Higgs contributions are not negligible. However, for the sake of simplicity we will not take the extra scalar contributions into account.

Up to a common ig2/2(s2M_U2) factor, the s-channel U22-exchange amplitudes are MRR;RR U _{~s!50, M} RR;LR U _{~s!50, M} RR;RL U _{~s!50, M} RR;LL U _{~s!50, ~D1!} M_LR;RRU ~s!521 2

S

El2ml El E_l

8

2m_l

8

E_l

8

D

1/2

F

12

S

El1ml El2ml E_l

8

1m_l

8

E_l

8

2m_l

8

D

1/2

G

2 y1/2~122y!, ~D2a! M_LR;LRU ~s!5

S

El2ml El E_l

8

2m_l

8

E_l

8

D

1/2

F

11

S

El1ml El2ml E_l

8

1m_l

8

E_l

8

2m_l

8

D

1/2

G

~12y!, ~D2b! M_LR;RLU ~s!52

S

El2ml El E_l

8

2m_l

8

El

8

D

1/2

F

12

S

El1ml El2ml E_l

8

1m_l

8

El

8

2ml

8

D

1/2

G

y , ~D2c! M_LR;LLU ~s!521 2

S

El2ml El E_l

8

2m_l

8

E_l

8

D

1/2

F

12

S

El1ml El2ml E_l

8

1m_l

8

E_l

8

2m_l

8

D

1/2

G

2 y1/2~122y!. ~D2d! MRL;RR U _~s!521 2

S

El2ml El E_l

8

2m_l

8

E_l

8

D

1/2

F

12

S

El1ml El2ml E_l

8

1m_l

8

E_l

8

2m_l

8

D

1/2

G

2 y1/2~122y!, ~D3a! M_RL;LRU ~s!5

S

El2ml E_l E_l

8

2m_l

8

E_l

8

D

1/2

F

11

S

El1ml E_l2m_l E_l

8

1m_l

8

E_l

8

2m_l

8

D

1/2

G

y , ~D3b! M_RL;RLU ~s!52

S

El2ml E_l E_l

8

2m_l

8

E_l

8

D

1/2

F

11

S

El1ml E_l2m_l E_l

8

1m_l

8

E_l

8

2m_l

8

D

1/2

G

y , ~D3c!

M_RL;LLU ~s!51 2

S

El2ml El E_l

8

2m_l

8

E_l

8

D

1/2

F

12

S

El1ml El2ml E_l

8

1m_l

8

E_l

8

2m_l

8

D

1/2

G

2 y1/2~122y!, ~D3d! M_LL;RRU ~s!50, M_LL;LRU ~s!50, M_LL;RLU ~s!50, M_LL;LLU ~s!50, ~D4! M_RRU ~s!5M_RR;RRU ~s!1MU_RR;LR~s!1M_RR;RLU ~s!1M_RR;LLU ~s!50, ~D5a! M_RLU ~s!5M_RL;RRU ~s!1M_RL;LRU ~s!1M_RL;RLU ~s!1M_RL;LLU ~s! 5 ig 2 2~s2M_U2!

S

El2ml El E_l

8

2m_l

8

E_l

8

D

1/2

F

11

S

El1ml El2ml E_l

8

1m_l

8

E_l

8

2m_l

8

D

1/2

G

~122y!, ~D5b! M_LRU ~s!5M_LR;RRU ~s!1M_LR;LRU ~s!1M_LR;RLU ~s!1M_LR;LLU ~s! 5 ig 2 2~s2M_U2!

S

El2ml El El

8

2ml

8

E_l

8

D

1/2

F

11

S

El1ml El2ml El

8

1ml

8

E_l

8

2m_l

8

D

1/2

G

~122y!, ~D5c! M_LLU~s!5M_LL;RRU ~s!1MU_LL;LR~s!1M_LL;RLU ~s!1M_LL;LLU ~s!50. ~D5d! For the collider case, if there exists a doubly charged vector boson U22 which is not too heavy, it will appear at the energies of the next linear colliders ~for electron beams up to 1.5 TeV!. In this case the cross section at the U peak will be larger than in the case of the standard model. Explicitly, the collider cross section in the case of a U exchange besides the ESM contributions is dsESM1U dV

U

CO 5ds ESM dV

U

CO 1 e 4 128p2s

F

g2 4e4~12b!2@~12y! 2_1y2_#1 g 2 e2~12b! 3

S

y 3_2~12y!3 y~12y! 1 g2 e2~gV 2_2g A 2_!y 2_~y1a!2~12y!2_~12y1a! ~y1a!~12y1a!

D

G

, ~D6! where b5M_U2/s. APPENDIX E: H11 AMPLITUDES

Here we will consider the amplitudes of the two doubly charged scalars H₁11and H₂11which are present in the model of Ref.@12#. The former is part of a triplet of SU~2! and the latter one is a singlet of SU~2!. Hence, the experimental lower bound on their masses will be different. Up to a common 2m_l2(ml2E)2/vS

2

(s2M_H

i

2

), i51,2, factor the amplitudes are

M_RR;RRHi 52M RR;LL H_i 5MLL;LL H_i 52MLL;RR H_i 5~ml2E!~E1ml! E2_1m l 2 , ~E1!

with the other amplitudes vanishing.

@1# The NLC Accelerator Design Group and The NLC Physics

Working Group, S. Kuhman et al., ‘‘Physics and Technology of the Next Linear Collider,’’ Report No. FERMILAB-PUB96/112. See also http://nlc.physics.upenn.edu/nlc/nlc.html and http://pss058.psi.ch/cuypers/e-e-.htm.

@2# J. F. Gunion, ‘‘Muon Colliders: The Machine and the

Phys-ics,’’ hep-ph/9707379. See also http//www.cap.bnl.gov/mumu and http://www.fnal.gov/projects/muon_collider.

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@5# See htpp://www.salc.stanford.edu/FIND/explist.html.

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School on Actual Problems of Particle Physics, Gmel, Belarus, 1997, hep-ph/9712407.

@11# Recently the fact that the left-right asymmetry is large in

col-liders has been pointed out by A. Czarnecki and W. Marciano, Int. J. Mod. Phys. A 13, 2235~1998!.

@12# F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 ~1992!. @13# J. T. Liu and D. Ng, Phys. Rev. D 50, 548 ~1994!. @14# F. Cuypers and M. Raidal, Nucl. Phys. B501, 3 ~1997!. @15# Particle Data Group, R. M. Barnett et al., Phys. Rev. D 54,

1~1996!.

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347~1992!.

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@18# J. F. Gunion, C. Loomis, and K. T. Pitts, in Proceedings of

1996 DPF/DPB Summer Study on New Directions for High-Energy Physics (Snowmass 96), Snowmass, Colorado, 1996, edited by D. G. Cassel, L. Trindle Gennari, and R. H. Siemann

~SLAC, Stanford, 1997!, hep-ph/9610237.

@19# F. Cuypers and S. Davidson, Eur. Phys. J. C2, 503 ~1998!, and