LEFT–RIGHT ASYMMETRIES AND EXOTIC VECTOR–BOSON DISCOVERY IN LEPTON–LEPTON COLLIDERS
ABSTRACT By considering left–right (L–R) asymmetries we study the capabilities of lepton 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 331 model on the L–R asymmetries for the processes ee→ee, μμ→ μμ and eμ→e μ and show that these asymmetries are very sensitive to these new contributions and that they are in fact powerful tools for discovery of this sort of vector bosons.

Article: Neutrinoless double beta decay with and without Majoronlike boson emission in a 331 model
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ABSTRACT: We consider the contributions to the neutrinoless double beta decays in a $SU(3)_L\otimes U(1)_N$ electroweak model. We show that for a range of the parameters in the model there are diagrams involving vectorvectorscalar and trilinear scalar couplings which can be potentially as contributing as the light massive Majorana neutrino exchange one. We use these contributions to obtain constraints upon some mass scales of the model, like the masses of the new charged vector and scalar bosons. We also consider briefly the decay in which besides the two electrons a Majoronlike boson is emitted. Comment: Revtex, 10 pages and 8 eps figures. Extended version to be published in Physical Review DPhysical Review D 03/2000; · 4.69 Impact Factor  SourceAvailable from: M.C. Rodriguez[Show abstract] [Hide abstract]
ABSTRACT: We build an supersymmetric version of the minimal 331 model with just two Higgs triplets using the superfield formalism. We study the mass spectrum of all particles in concordance with the experimental bounds. At the tree level, the masses of charged gauge bosons are the same as those of charged Higgs bosons. We also show that the electron, muon and their neutrinos as well as down and strange quarks gain mass through the loop correction. The narrow constraint on the ratio t_w = \frac{w}{w^\prime} is given by studying the new invisible decay mode of the Z boson.Nuclear Physics B 10/2012; 870(2). · 4.33 Impact Factor  SourceAvailable from: ArXiv[Show abstract] [Hide abstract]
ABSTRACT: The goal of this article is to derive the Feynman rules involving single charginos, neutralinos, double charged gauge bosons, and sleptons in a 331 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 electronelectron linear collider.Physical review D: Particles and fields 01/2002; 65(3).
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arXiv:hepph/0003085v1 9 Mar 2000
IFTP.026/2000
March 2000
Leftright asymmetries and exotic vector–boson discovery in
leptonlepton colliders
J. C. Montero, V. Pleitez and M. C. Rodriguez
Instituto de F´ ısica Te´ orica
Universidade Estadual Paulista
Rua Pamplona, 145
01405900– S˜ ao Paulo, SP
Brazil
Abstract
By considering leftright (LR) 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 331 model on the LR 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 scalarboson sector if there are several Higgsboson
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 331 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 scalarbosons can not be suppressed by the fermion mass and they
can have same strength of the fermion–vectorboson 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 scalarbosons from those of the vectorbosons. In Ref. [7]
it was noted that the leftright (LR) asymmetries in the lepton–lepton diagonal scattering
are insensible to the contribution of doublycharged scalar fields but are quite sensible to
doublycharged vector field contributions. On the other hand, in nondiagonal scattering
(as µ−e−) those asymmetries are sensible to the existence of an extra neutral vectorboson
Z′[8,9].
Here we will extend our previous analysis by considering a detailed study of the LR
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 LR ASYMMETRIES
The leftright 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;kl2,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 nondiagonal (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
righthanded, and the other is either right or lefthanded 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 nondiagonal 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 Uboson (Γtotal
U
) is a calculable quantity in the model once we
know all the Uboson couplings which are derived from the 331 gaugeinvariant 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
WMUKij2
(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 Uboson
∼ 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−? qµ2= m2
for the muon. In the nondiagonal elastic scattering in the standard model we have only the
tchannel 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 (nondiagonal) 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 schannel 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 Umasses. Notice that the lines are considerably separated even for
those values of the Umass that are not close to√s. Notice also that for Umass 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 centerofmass energy ECM =√s for some values of the Umass 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 Uboson even in the case
where the Umass is larger than√s; when the Umass is lower than√s we reproduce the
ESM results.
In Fig. 4 and Fig. 5 we show the effects of the Uboson 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 Uresonance 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 (0.0250.031) 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 Umasses that can be probed at e−e−colliders.
Next we study the effect of nonnegligible 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 electronelectron case, independently of the Ucontribution 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 Ueffect which dominates and it is the same for electrons
and muons. The effect of the Uresonance 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
leptonlepton colliders.
B. The Z′boson
In order to search for new physics in the neutralvector boson sector it is worth to consider
de nondiagonal process µ−e−→ µ−e−. In this case, assuming that the couplings of the
Uboson with leptons are almost diagonal (Kii∼ 1 as in the ΓU), the schannel Uboson
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 tchannel 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 schannel, we
specify the energy by the muonbeam 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 Uboson, 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 tchannel 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 331 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. 15 and Fig. 6, respectively. On
the other hand, this reaction is so sensitive to the Uboson 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 Uboson 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 Uboson 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 vectornature contribution of the Uboson: it
was previously shown [7] that the scalarboson contributions, which are present in the 331
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. 912. 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 LR 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 LR 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 nondiagonal
µ−e−case.
Hence, both U−−and Z′vector bosons can be potentially discovered in these sort of
processes by measuring the LR 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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(1967); A. Salam, in Elementary Particle Theory, Ed. by N. Svartholm (Almqviat and
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[2] J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, The Higgs Hunter’s Guide,
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[3] J. Pati and A. SAlam, Phys. Rev. D 10, 275 (1974); R. Mohapatra and J. Pati, Phys.
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[5] H. E. Haber and G. L. Kane, Phys. Rep. 117, 75 (1985).
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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 Umasses 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 Umasses as function of ECM.
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