New gauge bosons from the littlest Higgs model and the process $e^{+}e^{-}\to t\bar{t}$

0.28 MB·English

by  Chong-Xing Yue

New gauge bosons from the littlest Higgs model
and the process e+e tt¯
−
→
Chong-Xing Yue, Lei Wang, Jian-Xing Chen
Department of Physics, Liaoning Normal University, Dalian 116029. P.R.China ∗
5
0
0 February 2, 2008
2

n
a
J

0
2 Abstract

1
v Inthecontext of thelittlest Higgs(LH) model, westudytheprocesse+e−
→
6
8 tt¯. WefindthatthenewgaugebosonsZH andBH canproducesignificantcorrection
1
1 effects on this process, which can be further enhanced by the suitably polarized
0
5 beams. In most of the parameter space preferred by the electroweak precision data,
0
/ the absolute value of the relative correction parameter R is larger than 5%. As
h BH
p
long as 1TeV M 1.5TeV and 0.3 c 0.5, the absolute value of the
- ≤ ZH ≤ ≤ ≤
p
e relative correction parameter RZH is larger than 5%. With reasonable values of
h
: the parameters of the LH model, the possible signals of the new gauge bosons B
v H
Xi and ZH can be detected via the process e+e− tt¯in the future LC experiments
→
r
a with the c.m. energy √S = 800GeV. BH exchange and ZH exchange can generate
significantly corrections to the forward-backward asymmetry A (tt¯) only in small
FB
part of the parameter space.
∗
E-mail:[email protected]
1
I. Introduction
Although the standard model(SM) that bases on thegaugegroup SU(2) U(1) has
L Y
×
been successful in describing the physics of electroweak interactions, the mechanism of the
electroweak symmetry breaking(EWSB) and the origins of the masses of the elementary
fermions are still unknown. Furthermore, its scalar sector suffers from the problems of
trivialityandunnaturalness, etc. Thus, itisquitepossiblethattheSM isonlyaneffective
theory validbelow some high energy scale. New physics(NP) should exist at energy scales
around TeV.
Recently, a kind of theory for EWSB was proposed to solve the hierarchy between
the TeV scale of possible NP and the electroweak scale v = 246GeV, which is known as
”little Higgs models”[1,2,3]. The key feature of these models is that the Higgs boson is
a pseudo-Goldstone boson of a global symmetry which is spontaneously broken at some
higher scale f and thus is naturally light. EWSB is induced by a Coleman-Weinberg
potential, which is generated by integrating out the heavy degrees of freedom. This type
of models can be regarded as one of the important candidates of the NP beyond the SM.
A high energy e+e− linear collider(LC) will offer an opportunity to make precision
measurement of the properties of the electroweak gauge bosons, top quarks, Higgs bosons
andalsotoconstrainNP [4]. IntheLC experiments, topquarkpairsaremainlyproduced
from the S-channel exchange of the SM gauge bosons γ and Z via the process e+e− tt¯
→
[5]. The total cross section is of the order of 1pb, so that top quark pairs will be produced
at large rates in a clean environment at LC. If we assume that the integrated luminosity
£ is about 100fb−1, there will be several times 104 top quark pairs to be generated in
int
the future LC experiments. Furthermore, the QCD and EW corrections to the process
e+e− tt¯are small and decrease as the centre-of-mass(c.m.) energy √S increasing. The
→
option of longitudinally polarized beams can help to improve the measurement precision
and reduce background in search for NP. Thus, theoretical calculations of new particles
contributions to the process e+e− tt¯are of much interest for testing of NP.
→
In general, the new gauge bosons are heavier than the current experimental limits on
direct searches. However, thesenew particlesmay producevirtualeffects onsome physical
2
observable, which may be detected in the present or future high energy experiments. In
Ref.[6], we discussed the possible of detecting the new gauge bosons Z and B predicted
H H
by the littlest Higgs(LH) model [1] in the future LC experiments with the c.m. energy
√S = 500GeV and the integrating luminosity £ = 340fb−1 and both beams polarized
int
via considering their contributions to the processes e+e− ff¯ with f = τ,µ,b and
→
c. Since the masses of these fermions are largely smaller than the c.m. energy √S, we
have neglected the masses of these fermions in our numerical estimations. Our results
show that the new gauge bosons Z and B can indeed produce significant contributions
H H
to these process in most of the parameter space preferred by the electroweak precision
data, which might be observable in the future LC experiments. The aim of this paper
is to consider the contributions of the Z and B to the process e+e− tt¯and discuss
H H
→
whether these new particles can be detected via this process in the future LC experiments
with the c.m. energy √S = 800GeV and the integrating luminosity £ = 580fb−1. We
int
find that the absolute value of the relative correction parameter R generated by B
BH H
exchange is larger than 8% in most of the parameter space of the LH model preferred by
the electroweak precision data. As long as 1TeV M 1.5TeV and 0.3 c 0.5, the
≤ ZH ≤ ≤ ≤
absolutevalueofR islargerthan5%. Ifweassumethattheinitialelectronandpositron
ZH
beams are suitably polarized, the absolute values of the relative correction parameters
R and R can be enhanced. Thus, with reasonable values of the parameters of the
BH ZH
LH model, the possible signals of the new gauge bosons B and Z can be detected in
H H
the future LC experiments with the c.m. energy √S = 800GeV, which is similar to the
conclusions given in Ref.[6]. We further calculate the contributions of these new gauge
bosons to the forward-backward asymmetry A (tt¯). We find that they can generate
FB
significantly corrections to the forward-backward asymmetry A (tt¯) only in small part
FB
of the parameter space.
In section II, we give the formula of the contributions of new gauge bosons B and Z
H H
to the process e+e− tt¯and estimate the values of the relative corrections parameters
→
R = σBH(tt¯)/σSM(tt¯) and R = σZH(tt¯)/σSM(tt¯). The dependence of the relative
BH ZH
correctionparametersR andR ontheinitialbeampolarizationisdiscussedinsection
BH ZH
3
III.InsectionIV,we calculate thecontributions ofthese newgaugebosonstotheforward-
backward asymmetry A (tt¯). Our conclusions and discussions are given in section V.
FB
II. Corrections of the new gauge bosons B and Z to the process e+e− tt¯
H H
→
The LH model [1] is one of the simplest and phenomenologically viable models, which
realizes the little Higgs idea. It consists of a non-linear σ model with a global SU(5)
symmetry, which is broken down to its subgroup SO(5) by a vacuum condensate f
∼
Λs/4π TeV. At the same time, the locally gauged group SU(2) U(1) SU(2)
1 1 2
∼ × × ×
U(1) is broken to its diagonal subgroup SU(2) U(1), identified as the SM electroweak
2
×
gauge group. This breaking scenario gives rise to four massive gauge bosons B , Z , and
H H
±
W , which might produce characteristic signatures at the present and future high energy
H
collider experiments [7,8,9].
Taking account of the gauge invariance of the Yukawa coupling and the U(1) anomaly
cancellation, the coupling expressions of the gauge bosons B and Z to ordinary parti-
H H
cles, which are related to our calculation, can be written as [7]:
3e 2 e 2
gBHee = (c′2 ), gBHee = (c′2 ); (1)
V 4C s′c′ − 5 A 4C s′c′ − 5
w w
e 5 2 1 e 1 2 1
gBHtt = [ ( c′2) x ], gBHtt = [ ( c′2) x ]; (2)
V 2C s′c′ 6 5 − − 5 L A 2C s′c′ 2 5 − − 5 L
w w
ec ec
gZHee = , gZHee = ; (3)
V −4S s A 4S s
w w
ec ec
gZHtt = , gZHtt = . (4)
V 4S s A −4S s
w w
WhereS = sinθ ,θ istheWeinbergangle. Usingthemixingparametersc(s = √1 c2)
w w w
−
and c′(s′ = √1 c′2), we can represent the SM gauge coupling constants as g = g s = g c
1 2
−
′ ′ ′ ′ ′
and g = g s = g c. The mixing angle parameter between the SM top quark t and the
1 2
vector-like quark T is defined as x = λ2/(λ2 +λ2), in which λ and λ are the Yukawa
L 1 1 2 1 2
coupling parameters.
Global fits to the eletroweak precision data produce rather severe constraints on the
parameter space of the LH model [10]. However, if the SM fermions are charged under
U(1) U(1) , the constraints become relaxed. The scale parameter f = 1 2TeV is
1 2
× ∼
′
allowed for the mixing parameters c, c, and x in the ranges of 0 0.5, 0.62 0.73, and
L
∼ ∼
4
0.3 0.6, respectively [11]. In this case, the masses of B and Z are allowed in the
H H
∼
ranges of 300GeV 900GeV and 1TeV 3TeV, respectively. Thus, we will take the Z
H
∼ ∼
′
mass M , B mass M and the mixing parameters c, c and x as free parameters in
ZH H BH L
our calculation.
200 250
180
160 200
140
120 xL=0.3 xL=0.4
150
R(%)BH 10800 ccc’’’===000...667581 R(%)BH 100
60
40
50
20
0
-20 0
-40
-60 -50
400 500 600 700 800 900 400 500 600 700 800 900
MBH(GeV) MBH (GeV)
(a)
(b)
300
250
250
200
200
xL=0.5 xL=0.6
%) %) 150
R(BH 150 R(BH
100
100
50
50
0
0
-50 -50
400 500 600 700 800 900 400 500 600 700 800 900
MBH(GeV) MBH(GeV)
(d)
(c)
Figure 1: The relative correction parameter R as a function of the B mass M for
BH H BH
′
different values of the mixing parameters c and x .
L
For the SM, top quark pair tt¯ can be produced in sufficient abundance in the LC
experiments. The main production mechanism proceed at the Born level by the S-channel
annihilation of an initial electron-position pair into virtual photon or neutral gauge boson
5
Z, and their subsequent splitting into top quark pairs, e+e− γ,Z tt¯. For the LH
→ →
model, the B exchange and Z exchange can also produce the top quark pairs. The
H H
production cross sections can be written as:
Nfβ β2 4 S(M2 S)
σBH(tt¯) = c (1 ) e2gBHeegBHtt BH −
8πS { − 3 3 V V (S M2 )2 +M2 Γ2
− BH BH BH
β2
+[(gBHee)2 +(gBHee)2][(1 )[(gBHtt)2 +(gBHtt)2] (1 β2)(gBHtt)2]
V A − 3 V A − − A
S2
+(gZeegBHee +gZeegBHee)
(S M2 )2 +M2 Γ2 V V A A
− BH BH BH
β2
[(1 )(gZttgBHtt +gZttgBHtt) (1 β2)(gBHtt)(gZtt)]
− 3 V V A A − − A A
2S2[(S M2)(S M2 )+M Γ M Γ ]
− Z − BH Z Z BH BH , (5)
[(S M2)2 +M2Γ2][(S M2 )2 +M2 Γ2 ]}
− Z Z Z − BH BH BH
Nfβ β2 4 S(M2 S)
σZH(tt¯) = c (1 ) e2gZHeegZHtt ZH −
8πS { − 3 3 V V (S M2 )2 +M2 Γ2
− ZH ZH ZH
β2
+[(gZHee)2 +(gZHee)2][(1 )[(gZHtt)2 +(gZHtt)2] (1 β2)(gZHtt)2]
V A − 3 V A − − A
S2
+(gZeegZHee +gZeegZHee)
(S M2 )2 +M2 Γ2 V V A A
− ZH ZH ZH
β2
[(1 )(gZttgZHtt +gZttgZHtt) (1 β2)(gZHtt)(gZtt)]
− 3 V V A A − − A A
2S2[(S M2)(S M2 )+M Γ M Γ ]
− Z − ZH Z Z ZH ZH (6)
[(S M2)2 +M2Γ2][(S M2 )2 +M2 Γ2 ]}
− Z Z Z − ZH ZH ZH
with
e e
gZee = ( 1+4S2), gZee = (7)
V 4S C − w A 4S C
w w w w
e 8 e
gZtt = (1 S2), gZtt = , (8)
V 4S C − 3 w A 4S C
w w w w
where β = 1 4m2t, m is the top quark mass. Γ represent the total decay widths of
q − S t i
the gauge bosons Z,Z , and B . Γ and Γ have been given in Ref.[6]. From above
H H ZH BH
equations, we cansee thatσBH(tt¯)mainlydependents thefreeparameters M , c′ andx ,
BH L
while σZH(tt¯) only dependents the free parameters c and M , which is differently from
ZH
those for the process e+e− ff¯with f = τ,µ,b and c. In that case, the contributions
→
of the gauge bosons B is independent of the mixing parameter x . Thus, in this paper,
H L
′
we will take the mixing parameters c,c and x as free parameters. Certainly, due to
L
6
the mixing between the gauge bosons Z and Z , the SM tree-level couplings Zee¯ and
H
Ztt¯ receive corrections at the order of v2/f2, which can also produce contributions to
the production cross section of the process e+e− tt¯. However, the contributions are
→
suppressed by the factor v4/f4, which are smaller than those of B or Z . Thus, we have
H H
neglected this kind of corrections in above equations.
40
c=0.10
30
c=0.30
%) c=0.50
(
H
Z
R 20
-
10
0
1000 1500 2000 2500 3000
M (GeV)
ZH
Figure 2: The relative correction parameter R as a function of the Z mass M for
ZH H ZH
three values of the mixing parameter c.
To see the correction effects of B exchange and Z exchange on the tt¯production
H H
cross section, we plot the relative correction parameters R = σBH(tt¯)/σSM(tt¯) and
BH
R = σZH(tt¯)/σSM(tt¯) as functions of M andM in Fig.1 and Fig.2, respectively.
ZH BH ZH
From these figures, we can see that the gauge boson Z decreases the SM tt¯production
H
crosssectionσSM(tt¯)inalloftheparameterspace,whichsatisfiestheelectroweakprecision
constraints. In most part of the parameter space, the absolute value of the relative
correction parameter R is smaller than 5%, which is very difficult to be detected in the
ZH
7
future LC experiments. This is consistent with the contributions of Z to the process
H
e+e− ff¯, which has been studied in Ref.[6]. However, for the gauge boson B , it
H
→
is not this case. For M 800GeV, B exchange produce positive corrections to the
BH ≤ H
tt¯ production cross section σSM(tt¯) and the value of R increase as M , x and c′
BH BH L
increasing. For 800GeV < M 900GeV, B exchange decrease the cross section
BH ≤ H
σSM(tt¯) and the absolute of R increase as M decreasing and x , c′ increasing. The
BH BH L
peak of the R resonance emerges when the B mass M is approximately equal to
BH H BH
the c.m. energy √S = 800GeV. In most part of the parameter space, the absolute value
of R is larger than 8%. Thus, the virtual effects of B on the process e+e− tt¯should
BH H →
be easy detected in the future LC experiment with √S = 800GeV and £ = 580fb−1.
int
III. The dependence of the relative correction parameters R and R on the
BH ZH
electron and positron beam polarization
An LC has a large potential of the discovery of new particles and is well suited for the
precise analysis of NP beyond the SM. At present, the existing proposals are designed
with high luminosity of about £ = 340fb−1 at √S = 500GeV and £ = 580fb−1 at
int int
√S = 800GeV [4]. An important tool of an LC is the use of polarized beams. Beam
polarization is not only useful for a possible reduction of the background, but might also
serve as a possible tool to disentangle different contributions to the signal and lead to
substantial enhancement of the produce cross sections of some processes [12]. To see
whether the contributions of the new gauge bosons B and Z to the process e+e− tt¯
H H
→
can indeed be detected, we discuss the dependence of the relative correction parameters
R and R on the initial electron and positron beam polarization in this section.
BH ZH
Considering the polarization of the initial electron and positron beams, the cross sec-
tion of the process e+e− tt¯can be generally written as:
→
σ(tt¯) = (1+P )(1 P )(σ (tt¯)+σ (tt¯))+(1 P )(1+P )(σ (tt¯)+σ (tt¯)), (9)
e e¯ RR RL e e¯ LL LR
− −
where P and P are the degrees of longitudinal electron and position polarization, respec-
e e¯
tively. σ are the chiral cross sections of this process. The relative correction parameters
ij
′
R andR areplottedasfunctionsofM andM forc = 0.65,x = 0.5,c = 0.3and
BH ZH BH ZH L
8
different beam polarizations in Fig.3 and Fig.4, respectively. In these two figures, we have
used the solid line, dashed line, and dotted line to represent (P ,P )=(0,0), (0.8, 0.6),
e e¯
−
and ( 0.8,0.6), respectively. Our calculation results show that the absolute values of
−
R [R ] for (P ,P ) = (0.8,0.6)[(-0.8,-0.6)] are smaller than those for (P ,P ) = (0,0).
BH ZH e e¯ e e¯
Thus, in Fig.3 and Fig.4 we do not plot these lines.
80
60
xL=0.5
40 ( 0 , 0 )
(0.8,-0.6)
) 20 (-0.8,0.6)
%
(
H
B 0
R
-20
-40
-60
-80
-100
400 500 600 700 800 900
M (GeV)
BH
Figure 3: The relative correction parameter R as a function of the B mass M for
BH H BH
′
c = 0.65, x = 0.5, and (P ,P ) = (0,0),(0.8, 0.6),( 0.8,0.6).
L e e¯
− −
From Fig.3 and Fig.4 we can see that the suitably polarized beams can indeed enhance
the virtual effects of the new gauge bosons B and Z on the process e+e− tt¯. In the
H H
→
whole parameter space preferred by the electroweak precision data, the value of R for
BH
(P ,P ) = (0.8, 0.6) is larger than that for (P ,P ) = (0,0), while the absolute values of
e e¯ e e¯
−
R for (P ,P ) = ( 0.8,0.6) is larger than that for (P ,P ) = (0,0). Varying the values
ZH e e¯ − e e¯
′
of the free parameters c, x , and c does not change this conclusion. So, in Fig.3 and
L
′
Fig.4 we have taken these parameters for fixed values x = 0.5, c = 0.65, and c = 0.3.
L
Certainly, the values of R and R change as the values of these parameters varying.
BH ZH
9
For example, for 0.3 c 0.5 and 1TeV M 2TeV, the absolute value of R for
≤ ≤ ≤ ZH ≤ ZH
(P ,P ) = ( 0.8,0.6) is larger than 6%. The absolute of R for (P ,P ) = (0.8, 0.6) is
e e¯ − BH e e¯ −
′
larger than 5% for x = 0.5, 0.68 c 0.73 and 500GeV < M 900GeV, but for
L ≤ ≤ BH ≤
′
x = 0.6 its value is larger than 5% for 0.65 c 0.73 and 450GeV M 900GeV.
L ≤ ≤ ≤ BH ≤
Thus, using the suitably polarization of the initial electron and positron beams, it is more
easy to detect the possible signals of the new gauge bosons B and Z in the future LC
H H
experiments.
50
40
30
)
% ( 0 , 0 )
(
H (0.8,-0.6)
Z
R
(-0.8,0.6)
-
20
10
0
1000 1500 2000 2500 3000
M (GeV)
ZH
Figure 4: The relative correction parameter R as a function of the Z mass M for
ZH H ZH
c = 0.3 and (P ,P ) = (0,0),(0.8, 0.6),( 0.8,0.6).
e e¯
− −
IV. Gauge bosons B , Z and the forward-backward asymmetry A (tt¯)
H H FB
The events generated by the process e+e− ff¯can be characterized by the momen-
→
tum direction of the emitted fermion. If we assume that the final state fermion travels
forward(F) or backward(B) with respect to the electron beam, thanthe forward-backward
asymmetry can be defined as:
σ σ
F B
A = − , (10)
FB
σ +σ
F B
10