Z-boson production at ILC.pdf
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Triple Z
0
-boson production in large extra dimensions model at
ILC
Jiang Ruo-Cheng, Li Xiao-Zhou, Ma Wen-Gan, Guo Lei, and Zhang Ren-You
Department of Modern Physics, University of Science and Technology
of China (USTC), Hefei, Anhui 230026, P.R.China
ABSTRACT
We investigate the eects induced by the interactions of the Kaluza-Klein (KK) graviton
with the standard model (SM) particles on the triple Z
0
-boson production process at the ILC
in the framework of the large extra dimension (LED) model. We present the dependence
of the integrated cross sections on the electron-positron colliding energy
p
s, and various
kinematic distributions of nal Z
0
bosons and their subsequential decay products in both
the SM and the LED model. We also provide the relationship between the integrated cross
section and the fundamental scale M
S
by taking the number of the extra dimensions (d) as
3, 4, 5, and 6, respectively. The numerical results show that the LED eect can induce a
observable relative discrepancy for the integrated cross section (
LED
), which can reach the
value of 13.11% (9.27%) when M
S
= 3.5 (3.8) TeV and the colliding energy
p
s = 1 TeV .
We nd the relative discrepancy of LED eect can even reach few dozen percent in the high
transverse momentum area or the central rapidity region of the nal Z
0
-bosons and muons.
PACS: 11.10.Kk, 13.66.Fg, 14.70.Hp
1
I. Introduction
The large extra dimensions (LED) model, proposed by Arkani-Hamed, Dimopoulos and Dvali
(ADD) [1], is an attractive extension of the standard model (SM) because of its possible testable
consequences. In the LED model only graviton can propagate in a D = 4 +d dimensional space
with d being the dimension number of extra space, while the SM particles exist in the usual
(3 + 1)-dimensional space. The picture of a massless graviton propagating in D-dimensions is
equal to the scene where numerous massive Kaluza-Klein (KK) gravitons propagate in (3 + 1)-
dimensions. So we can expect that even though the gravitational interactions in the 4 space-time
dimensions are suppressed by a factor of 1/M
P
, they can be compensated by these numerous
KK states. Therefore, in either the case of real graviton emission or the case of virtual graviton
exchange, it is shown that the Plank mass M
P
cancels out of the cross section after summing
over the KK states, and we can obtain an interaction strength comparable to the electroweak
strength [2, 3]. Up to now, many works on the LED phenomenology at colliders have been
done, including vector boson pair productions and association productions of vector boson with
graviton [4, 5, 6].
The precision measurements of the trilinear gauge boson couplings (TGCs) are helpful for
verication of non-Abelian gauge structure, and the investigation of the quartic gauge boson
couplings (QGCs) can either conrm the symmetry breaking mechanism or present a direct test
on the new physics beyond the SM. The vector boson pair production processes were extensively
studied in the SM for probing the TGCs [7, 8, 9, 10, 11]. The direct study of QGCs requires
the investigation of the processes involving at least three external gauge bosons. Recently, the
triple Z
0
-boson production in the LED model at the LHC was studied in Ref.[12].
p
The International Linear Collider (ILC) is proposed with the colliding energy
s = 200
p
500 GeV which would be upgraded to
s = 1 TeV , and the integrated luminosity is required
to be 1000 fb
−1
in the rst phase of operation [11]. The triple gauge boson productions at the
ILC are important processes in probing the TGCs and QGCs of electroweak theory, which are
related to electroweak symmetry breaking mechanism. If their observables coincide with the LED
predictions on the triple gauge boson production processes, it means the coupling of graviton to
2
gauge bosons in the LED model would be the causation. Therefore, the understanding of the
LED phenomenology in triple gauge boson production processes at the ILC is necessary.
In this paper we study the LED eects on the process e
+
e
−
! Z
0
Z
0
Z
0
at the ILC. The
paper is organized as follows: In section II we present the calculation descriptions for the process
with a brief review of the related LED theory. The numerical results and discussions are given
in section III. In the last section a short summary is given.
II. Analytic calculations
In the LED model, the extra dimensions on a torus are compactied to a radius R/2. The
relationship between the usual Planck scale M
P
to the fundamental scale M
S
is
M
P
=
2(4)
−d/2
(d/2)
M
2+
S
R
d
(2.1)
p
G
N
1.22 × 10
19
GeV and G
N
is Newton’s constant. In our work we use the
where M
P
= 1/
de Donder gauge for the pure KK-graviton part and the Feynman gauge ( = 1) for the SM part.
The Feynman rules for the relevant vertices including KK graviton used in our calculations are
given below [13], where we assume that all the momenta ow to the vertices, except that the
fermionic momenta are set to be along the fermion ow directions.
• G
µ
KK
(k
3
) −
¯
(k
1
) − (k
2
) vertex :
− i
8
[
µ
(k
1
+ k
2
)
+
(k
1
+ k
2
)
µ
− 2
µ
(k
1
+ k
2
− 2m
)],
(2.2)
• G
µ
KK
(k
3
) − Z
(k
1
) − Z
(k
2
) vertex :
B
µ
m
Z
+ (C
µ
− C
µ
)k
1
k
2
+
1
E
µ
(k
1
,k
2
)
i
,
(2.3)
• G
µ
KK
(k
4
) −
¯
(k
1
) − (k
2
) − Z
(k
3
) vertex :
ie
4
(
µ
+
µ
− 2
µ
) (v
f
− a
f
5
),
(2.4)
where G
µ
KK
, and Z
µ
represent the elds of graviton, lepton, and Z
0
-boson, respectively,
e = g sin
W
is the positron electric charge, is the SU(2) gauge xing parameter, v
f
, a
f
are
3
the vector and axial-vector couplings which are the same as those dened in the SM, and
=
p
p
p
16G
N
=
2/M
P
where the reduced Planck scale M
P
= M
P
/
8. The explicit expressions
for the tensor coe
cients are given as
1
B
µ
2
(
µ
−
µ
−
µ
),
=
1
2
[
µ
− (
µ
+
µ
+
µ
+
µ
)],
E
µ
(k
1
,k
2
) =
µ
(k
C
µ
=
1
k
1
+ k
2
k
2
+ k
1
k
2
) − [
k
µ
1
k
1
+
k
µ
2
k
2
+ (µ $ )].
(2.5)
After summation over KK states the spin-2 KK graviton propagator can be expressed as [?]
=
1
2
G
µ
µ
+
µ
−
d + 2
µ
2
D(s)
,
(2.6)
KK
where
s
d/2−1
M
S
d+2
p
D(s) =
16
2
+ 2iI(
/
s)
,
(2.7)
and
Z
p
/
s
y
d−1
1 − y
2
.
p
I(
/
s) = P
dy
(2.8)
0
p
The integral I(
/
s) should be understood that a point y = 1 has been removed from the
integration path, and we set the ultraviolet cuto
to be the fundamental scale M
S
routinely.
We denote the process of the triple Z
0
-boson production at the ILC as
e
+
(p
1
) + e
−
(p
2
) ! Z
0
(p
3
) + Z
0
(p
4
) + Z
0
(p
5
).
(2.9)
The SM-like diagrams for the above process are depicted in Fig.1(a). In the LED model, the
KK graviton can couple to Z
0
-pair and fermion-pair. We present the four additional pure LED
Feynman diagrams in Fig.1(b).
In our calculations the developed FeynArts3.4 package [14] is adopted to generate all the low-
est order Feynman diagrams and convert them to the corresponding amplitudes. Subsequently,
the amplitude calculations are mainly implemented by applying modied FormCalc5.4 programs
[15].
4
III. Numerical results and discussions
In our numerical calculations, we use the following set of input parameters [17]:
m
W
= 80.385 GeV, m
Z
= 91.1876 GeV,
ew
(0) = 1/137.036,
sin
2
W
= 1 − m
2
W
/m
Z
= 0.222897, m
e
= 0.510998928 MeV.
We know that the ATLAS and CMS experiments found several SM Higgs-like events at the
location of M
H
125 GeV [18]. Recently, ATLAS reported the searching for extra dimensions
p
by using diphoton events in
s = 7 TeV pp collisions [19]. The results provided 95% C.L. lower
limits on the fundamental Planck scale M
S
between 2.27 TeV and 3.53 TeV depending on the
number of extra dimensions d in the range of 7 to 3. The diphoton and dilepton results from
CMS set limits on M
S
in the range of 2.5 − 3.8 TeV as d varies from 7 to 2 at 95% C.L. [20].
In this work we take M
H
= 125 GeV , and set M
S
= 3.5 TeV (or M
S
= 3.8 TeV ) and d = 3 as
the representative ADD parameters in case otherwise stated.
In Refs.[10, 11] there exist the calculations for the SM one-loop electroweak corrections to
the e
+
e
−
! Z
0
Z
0
Z
0
process. We make comparison of our LO numerical results with theirs,
there we adopt the input parameters equal to those in Ref.[10] and use two dierent packages in
order to check the correctness of our LO calculations. In Table 1 we present the results of the
p
LO integral cross sections in the SM at the
s = 500 GeV ILC obtained by using CompHEP-
Z
Z
E
Z
H
E
E
Z
E
Z
E
Z
Z
E
(a)
Z
Z
Z
Z
E
G
KK
E
G
KK
G
KK
Z
Z
Z
E
E
E
G
KK
Z
E
Z
E
E
E
Z
Z
Z
E
Z
(b)
Figure 1: The Feynman diagrams for the process e
+
e
−
! Z
0
Z
0
Z
0
in the LED model. (a) The
SM-like diagrams. (b) The additional diagrams with KK graviton exchange.
5
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