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arXiv:hep-ph/0003085v1 9 Mar 2000

IFT-P.026/2000

March 2000

Left-right asymmetries and exotic vector–boson discovery in

lepton-lepton colliders

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

Instituto de F´ ısica Te´ orica

Universidade Estadual Paulista

Rua Pamplona, 145

01405-900– S˜ ao Paulo, SP

Brazil

Abstract

By considering left-right (L-R) asymmetries we study the capabilities of lep-

ton colliders in searching for new exotic vector bosons. Specifically we study

the effect of a doubly charged bilepton boson and an extra neutral vector

boson appearing in a 3-3-1 model on the L-R asymmetries for the processes

e−e−→ e−e−, µ−µ−→ µ−µ−and e−µ−→ e−µ−and show that these asym-

metries are very sensitive to these new contributions and that they are in fact

powerful tools for discovery this sort of vector bosons.

PACS numbers: 13.88.+e; 12.60.-i 12.60.Cn;

Typeset using REVTEX

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I. INTRODUCTION

Any extension of the electroweak standard model (ESM) [1] implies necessarily the exis-

tence of new particles. We can have a rich scalar-boson sector if there are several Higgs-boson

multiplets [2] or have more vector and scalar fields in models with a larger gauge symmetry

as in the left–right symmetric [3] and in 3-3-1 models [4], or we also can have at the same

time more scalar, fermion, and vector particles as in the supersymmetric extensions of the

ESM [5].

If in a given model all the new particles contribute to all observables, it will be very dif-

ficult to identify their contribution in the usual and exotic processes. In some models [4,6]

the contributions of the scalar-bosons can not be suppressed by the fermion mass and they

can have same strength of the fermion–vector-boson coupling. Hence, we can ask ourselves

if there exist observables and/or processes which allow us to distinguish between the con-

tributions of charged and neutral scalar-bosons from those of the vector-bosons. In Ref. [7]

it was noted that the left-right (L-R) asymmetries in the lepton–lepton diagonal scattering

are insensible to the contribution of doubly-charged scalar fields but are quite sensible to

doubly-charged vector field contributions. On the other hand, in non-diagonal scattering

(as µ−e−) those asymmetries are sensible to the existence of an extra neutral vector-boson

Z′[8,9].

Here we will extend our previous analysis by considering a detailed study of the L-R

asymmetries in order to analyse their capabilities in detecting new physics. The outline of

the paper is the following: In Sec. II we define the asymmetries; in Sec. III we show the

lagrangian interaction of the models we are considering here. The results and experimental

considerations are given in Sec. IV and our conclusions appear in the last section.

II. THE L-R ASYMMETRIES

The left-right asymmetry for the process l−l′−→ l−l′−with one of the particles being

unpolarized is defined as

ARL(ll′→ ll′) ≡ ARL(ll′) =dσR− dσL

dσR+ dσL,(1)

where dσR(L)is the differential cross section for one right (left)-handed lepton l scattering

on an unpolarized lepton l′and where l,l′= e,µ. That is

ARL(ll′) =(dσRR+ dσRL) − (dσLL+ dσLR)

(dσRR+ dσRL) + (dσLL+ dσLR),(2)

where dσijdenotes the cross section for incoming leptons with helicity i and j, respectively,

and they are given by

dσij∝

?

kl

|Mij;kl|2,i,j;k,l = L,R.(3)

Notice that when the scattering is diagonal, l = l′= e,µ, dσRL= dσLR, so the asymmetry

in Eq. (2) is equal to the asymmetry defined as (dσRR−dσLL)/(dσRR+dσLL). For practical

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purposes, for the non-diagonal (eµ → eµ) case of the ARLasymmetry, we will focus on the

scattering of polarized muons by unpolarized electrons.

Another interesting possibility is the case when both leptons are polarized. We can

define an asymmetry AR;RLin 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;RL=dσRR− dσRL

dσRR+ dσRL,AL;RL=dσLR− dσLL

dσLL+ dσLR.(4)

In this case, when the non-diagonal scattering is considered, we will assume that the muon

beam has always the same polarization and the electron one can have both, the left and the

right polarizations.

We can integrate over the scattering angle and define the asymmetry

?ARLas

?

ARL=(?dσRR+?dσRL) − (?dσLL+?dσLR)

(?dσRR+?dσRL) + (?dσLL+?dσLR),

?175o

(5)

where

?dσij≡

5o

dσij. A similar expression can be written for AR;RL.

III. THE MODELS

We are studying here the asymmetries defined above in the context of two models: the

electroweak standard model (ESM) and in a model having a doubly charged bilepton vector

field (U−−

charged scalar bileptons but since their contributions cancel out in the numerator of the

asymmetries we are not consider them on this study. We identify the case under study by

using the {ESM}, {ESM + U}, and {ESM + Z′} labels in cross sections and asymmetries.

In the context of the electroweak standard model, at the tree level, the relevant part of

the ESM lagrangian is

µ ) and an extra neutral vector boson Z′[4]. The latter model has also two doubly

LF= −

?

i

g mi

2MW

¯ψiψiH0− e

?

i

qi¯ψiγµψiAµ−

g

2cosθW

ψiγµ(gi

V− gi

Aγ5)ψiZµ,(6)

θW≡ tan−1(g′/g) is the weak mixing angle, e = gsinθWis the positron electric charge with

g such that

g2=8GFM2

W

√2

; org2/α = 4πsin2θW,(7)

with α ≈ 1/128; and the vector and axial neutral couplings are

gi

V≡ t3L(i) − 2qisin2θW,gi

A≡ t3L(i),(8)

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

The charged current interactions in a model having a doubly charged vector boson [4],

in terms of the physical basis, are given by

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−

g

√2

?

¯ νLEν†

LEl

LlLW+

µ+¯lc

LγµElT

REν

LνLV+

µ−¯lc

LElT

REl

LlLU++

µ

?

+ H.c.,(9)

with l′

try (mass) eigenstates. We see from Eq. (9) that for massless neutrinos we have no mixing in

the charged current coupled to W+

to V+

the charged currents coupled to V+

µ

are not diagonal in flavor space and the mixing

matrix K = ElT

has three angles and six phases. In the present case, however, the matrix K is determined

entirely by the charged lepton sector, so we can rotate only three phases [10]).

The total width of the U-boson (Γtotal

U

) is a calculable quantity in the model once we

know all the U-boson couplings which are derived from the 3-3-1 gauge-invariant lagrangian.

However, we find that a complete computation of ΓUis out of the scope of this paper because

in this case some realistic hypotheses concerning the masses of the exotic scalars and quarks

should be made. Thus, we will only consider the partial width due to the U−−→ l−l−

decay. In the limit where all the lepton masses are negligible we have:

L= El

LlL,l′

R= El

RlR,ν′

L= Eν

LνL, the primed (unprimed) fields denoting symme-

µbut we still have mixing in the charged currents coupled

µ . That is, if neutrinos are massless we can always chose Eν†

µand U++

REν

µand U++

LEl

L= 1. However,

Lhas three angles and three phases. (An arbitrary 3 × 3 unitary matrix

Γtotal

U

∼ Γ(U−−→ leptons) =

?

i,j

GF

6√2πM2

WMU|Kij|2

(10)

where i,j run over the e,µ, and τ leptons and Kijis a mixing matrix in the flavor space. For

the expression above we can write Γtotal

U

=?

K is almost diagonal we can neglect Γijfor i ?= j and consider for practical purposes that

Γtotal

U

mass. For instance, for MU= 300 GeV we have Γtotal

In the model there is also a Z′neutral vector boson which couples with the leptons as

follows

g

2cW

W)1/2/√3 and Rl= 2Ll. Notice the leptophobic character of Z′

[11]. In this case we have no concerns about the Z′–width because this neutral boson is only

exchanged in the t–channel.

We will consider the process

iΓii+1

2

?

i?=jΓijand assuming that the matrix

= 3 × Γii. In our numerical applications Γtotal

U

is a varying function of the U-boson

∼ 2.5 GeV.

U

LZ′

NC= −

?¯laLγµLllaL+¯laRγµRllaR+ ¯ νaLγµLννaL

?

Z′

µ,(11)

with Ll= Lν = −(1 − 4s2

l−(p1,λ) + l′−(q1,Λ) → l−(p2,λ′) + l′−(q2,Λ′),

where q = p2−p1= q2−q1is the transferred momentum. As we said before, we will neglect

the electron mass but not the muon mass i.e., E = |? pe| for the electron and K2−|? qµ|2= m2

for the muon. In the non-diagonal elastic scattering in the standard model we have only the

t-channel contribution. The relevant amplitudes for the ESM, {ESM + U} and {ESM + Z′}

models are in the appendices of Ref. [7] (Ref. [8]) for the diagonal (non-diagonal) case.

(12)

µ

IV. RESULTS

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A. The U boson

We start this section by considering the contributions of the doubly charged vector boson

U to the asymmetries which contributes uniquely via the s-channel for a doubly charged

initial state. In Fig. 1 we see that the angular dependence of the ARLasymmetry, taken

into account the U contribution, presents a relatively different behavior with respect to the

ESM for a wide range of U-masses. Notice that the lines are considerably separated even for

those values of the U-mass that are not close to√s. Notice also that for U-mass lower than

√s we basically reproduce the ESM result for θ ≈ 0 and π/2; the largest difference with the

ESM occurs for θ in the interval 0.5–1. The behavior of ARLfor ESM+U as a function of

MUis showed in Fig. 2 for a fixed scattering angle and for several values of√s. We see that

this asymmetry is essentially negative and that its maximum value is zero and occurs at the

resonance point MU=√s. This is due to the fact that at the resonance the numerator of

ARL, as defined in Eq. (1), cancels out no mater the value of MU. On the other hand the

value of MU governs the width of the curves around the resonance point. This particular

feature is better seen in Fig. 3. In this figure we show the ARLasymmetry as a function of

the center-of-mass energy ECM =√s for some values of the U-mass and it is clearly seen

that not only at the peak but also for a considerably large range of masses around the peak,

the curves representing the respective U contribution are significantly separated from the

ESM one. It means that this asymmetry is very sensitive to the U-boson even in the case

where the U-mass is larger than√s; when the U-mass is lower than√s we reproduce the

ESM results.

In Fig. 4 and Fig. 5 we show the effects of the U-boson on the AR;RLasymmetry, defined

in Eq. (4), and as it behaves qualitatively like ARL we come to the same conclusions we

did for ARL. We must note that near the U-resonance the AR;RLasymmetry is negative.

However, in this case, polarization for both beams must be available.

The integrated asymmetry

?ARLdefined in Eq. (5) is shown in Fig. 6. There we can see

that while the ESM curve keeps an almost constant value (0.025-0.031) for 0.5 <√s < 2

TeV, the ESM+U curves go from zero, for MU?=√s, to a very pronounced peak (∼ −0.25)

at the resonance points. In Fig. 7, for the sake of detectability, we show the quantity δ%

defined by:

δ % =

?AESM+U

RL

−?AESM

?AESM

RL

RL

× 100, (13)

which in this case stands for the percent deviation of

can see that there is a wide range of U-masses that can be probed at e−e−colliders.

Next we study the effect of non-negligible initial and final fermion masses by considering

the µ−µ−→ µ−µ−process in a muon collider. The results are given in Fig. 8. There

we can see that, for MU = 500 GeV, below 300 GeV the muon mass effect is in evidence

differing sensibly from the electron-electron case, independently of the U-contribution for

higher energies the lepton mass has no effect at all for both models. Between 300 and

400 GeV all the curves are coincident and we cannot distinguish among both models or

leptons. Above 400 GeV it is the U-effect which dominates and it is the same for electrons

and muons. The effect of the U-resonance is well evident and even above the resonance

there is an almost constant difference between the ESM+U and ESM asymmetries showing

?AESM+U

RL

from

?AESM

RL . There we

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that the ARLasymmetry is still a sensitive parameter for the U discovery for√s > MU in

lepton-lepton colliders.

B. The Z′boson

In order to search for new physics in the neutral-vector boson sector it is worth to consider

de non-diagonal process µ−e−→ µ−e−. In this case, assuming that the couplings of the

U-boson with leptons are almost diagonal (Kii∼ 1 as in the ΓU), the s-channel U-boson

exchange will be negligible and provided that the Z′couples with leptons diagonally, the only

contributions to this process for ESM+Z′will be the t-channel ones, i.e., the contributions of

γ,Z, and Z′. The angular dependence of ARLis showed in Fig. 9 where we can see that a Z′

contribution is clearly distinguished from the ESM one for a wide range of Z′-masses around

a given Eµ=√s/2. (In the figures concerning the Z′-boson, once there is no s-channel, we

specify the energy by the muon-beam energy Eµ.) As expected the ARLasymmetry is more

sensitive to relatively light Z′boson. We come to the same conclusion from Fig. 10 in which

we show the ARLasymmetry as a function of Eµ. The sensitivity of the this asymmetry

with the Z′-mass is showed in Fig. 11.

Contrarily to the case of the search for the U-boson, the asymmetry AR;RLis not sensitive

to the extra neutral vector boson. In this case, the potential capabilities of the asymmetries

in discovering new neutral vector bosons are better explored by considering the integrated

asymmetry

?ARL. The angular integration over the scattering angle of the t-channel Z′

exchange contribution produces curves that are clearly separated, depending on the Z′-

mass, which are also clearly distinguishable from the ESM curve for a wide range of masses.

See Fig. 12.

We have also computed the asymmetries taking into account a Z′which couples to leptons

with the same couplings of the standard Z boson but with a different mass. Although these

ESM couplings are stronger than those of the 3-3-1 model they have no substantial effect

on the asymmetries: The results are very similar to the ones showed in Fig. 11.

For the eµ scattering there are also contributions coming from the neutral scalar sector

of the model. However as in all the scalar contributions the pure scalar terms cancel out in

the numerator of the asymmetry and the interference terms are numerically negligible [7].

C. Observability

Based on the figures we have shown throughout the text we have claimed that the

values of the asymmetries, when there is an extra contribution of a new vector boson, are

different enough from those of ESM ones to allow for the discovery of the referred bosons.

However, we must be sure that there is enough statistics to measure these asymmetries. In

order to provide some statistical analysis we assume a conservative value for the luminosity:

L = 1fb−1yr−1= 1032cm−2s−1for the e−e−, µ−µ−and the µ−e−colliders, and compute the

number of the expected number of events (N) based on the unpolarized integrated cross

section for each process.

For the e−e−→ e−e−or µ−µ−→ µ−µ−processes, the ESM cross section is relatively

small, it goes from 0.05 nb at√s = 0.5 TeV to 1.5 × 10−3nb at√s = 2 TeV. These values

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correspond to 5×104and 1.5×103events/yr respectively. Then, computing√N/N we get

∼ 4 × 10−3, for the first case, and ∼ 2 × 10−2, for the second one. This is an indication

that the ESM asymmetries can be measured. Note that the asymmetries we have computed

here are relatively large: AESM

RL

∼ AESM

order O(10−2) for the integrated ones as showed in Figs. 1-5 and Fig. 6, respectively. On

the other hand, this reaction is so sensitive to the U-boson contribution that there is an

enormous enlargement in the cross section and consequently in the statistics. In Tab. I we

show the relevant parameters depending on the U-boson mass and for only two values of√s

for shortness. There we can see that there is enough precision to measure the asymmetries,

for the range of masses and energies we have considered. For the U-boson discovery we can

study the cross section directly provided that it is considerably different from that of the

ESM [12]. However, the study of the asymmetry gives us more qualitative information once,

contrarily to the cross section, it filters the vector-nature contribution of the U-boson: it

was previously shown [7] that the scalar-boson contributions, which are present in the 3-3-1

model, cancel out in the asymmetry numerator.

For the µ−e−→ µ−e−process, the integrated cross sections for the ESM are larger than

for the electron diagonal process. We find 3 nb for Eµ= 0.5 TeV and ∼ 0.2 nb for Eµ= 2.0

TeV. The corresponding number of events are 3 × 106and ∼ 2 × 105, respectively, for the

same luminosity used before. In this case the ratio

respectively, which provide enough precision to measure the AESM

O(10−2), as can be seen from Fig. 9-12. For the ESM+Z′case we have that although the Z′-

contribution can affect significantly the values of the asymmetry ARL, as it has been shown,

it only slightly modifies the cross section values. It means that the number of events in for

ESM+Z′are similar to those of the ESM and hence we have enough precision to measure

AESM+Z′

RL

too. Once again we note that the asymmetries are more sensitive than the cross

section itself in looking for the Z′-discovery.

R;RL∼ O(10−1) for a fixed scattering angle and of the

√N/N gives 5.7 × 10−4and 2.2 × 10−3,

RL

once it is of the order

V. CONCLUSIONS

Here we have generalized the analysis of Ref. [7,8] and have shown that the L-R asym-

metries in the diagonal (e−e−,µ−µ−) lepton scattering can be the appropriate observable to

discover doubly charged vector bosons U even for values of MU and√s far away from the

resonance condition. Although the cross sections may have important contributions from

the scalar fields (doubly charged Higgs bosons) these contributions cancel out in the numer-

ator of the L-R asymmetries. On the other hand, the contribution of the extra neutral Z′,

leptophobic or with the same couplings of the standard model, gives small contributions to

the diagonal e−e−,µ−µ−scattering but it gives an important contribution to non-diagonal

µ−e−case.

Hence, both U−−and Z′vector bosons can be potentially discovered in these sort of

processes by measuring the L-R asymmetries. Since the couplings of both particles with

matter are known in a given model, once their masses were known other processes like

exotic decays could be used to study the respective contributions of the scalar fields present

in the model.

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ACKNOWLEDGMENTS

This work was supported by Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao Paulo

(FAPESP), Conselho Nacional de Ciˆ encia e Tecnologia (CNPq) and by Programa de Apoio

a N´ ucleos de Excelˆ encia (PRONEX).

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REFERENCES

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[7] J. C. Montero, V. Pleitez and M. C. Rodriguez, Phys.Rev. D 58, 094026 (1998); hep-

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FIGURES

FIG. 1. The ARLasymmetry for an e−e−collider with√s = 2 TeV for the ESM (solid line)

and for the ESM+U for several U–masses as a function of the scattering angle.

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FIG. 2. The ARLasymmetry for a fixed scattering angle θ = 0.5 rad and several values of√s

of e−e−colliders for ESM+U as a function of MU.

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FIG. 3. The ARLasymmetry for a fixed scattering angle θ = 0.5 rad for the ESM (solid line)

and for the ESM+U for several U-masses as a function of ECM.

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FIG. 4. The same as in Fig. 1 for the AR;RLasymmetry.

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FIG. 5. The same as in Fig. 3 for the AR;RLasymmetry .

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FIG. 6. The integrated asymmetry?ARLfor e−e−collider for the ESM (solid line) and for the

ESM+U for several U-masses as function of ECM.

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