1. Introduction
The gauge hierarchy between the eletroweak scale and the Plank scale was solved naturally
by the Randall-Sundrum (RS) model in 1999 [1]. The RS model involves two three-branes
bounding a slice of 5D compact anti-de Sitter space taken to be on an S1/Z2 orbifold extra
dimension. Gravity is localized at the UV brane, while the Standard model fields are supposed to
be localized at the IR brane. The additional scalar called radion (φ) corresponds to the fluctuations
of the size of the extra dimension [2-5]. Radion can be the lightest particle in the RS model. In the
effective four dimensional theory, the Higgs boson and the radion can be mixed since radion and
Higgs fields have the same quantum numbers [6, 7].
In 2012, the 125 GeV Higgs signal is discovered by the ATLAS and CMS collaborations
[8, 9], which has completed the particle spectrum of the Standard model. The international linear
collider (ILC) [10], which is an e−e+ collider with high energy and luminosity, will supply the
chance to detect new physics. The potential of the ILC can be enhanced by considering γγ and
γe− collisions with the photon beam generated by the backward Compton scattering of electron
and laser beams [11-14]. In this work, we investigate the e+e− production via γγ collision taking
into account the Higgs and radion propagators. The layout of this paper is as follows. In Section
II, we review the Higgs-radion mixing in the RS model. The e−e+ production of γγ collision is
calculated in Section III. Finally, we summarize our results and make conclusions in Section IV.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0066
Natural Sciences, 2018, Volume 63, Issue 11, pp. 28-33
This paper is available online at
THE e−e+ PRODUCTION VIA γγ COLLISION
IN THE RANDALL-SUNDRUM MODEL AT THE ILC
Bui Thi Ha Giang
Faculty of Physics, Hanoi National University of Education
Abstract. Taking into account the 125 GeV Higgs boson and radion propagators,
we investigate the e−e+ production via γγ collision in the Randall-Sundrum model at
International Linear Collider (ILC). The observable cross-sections are calculated for the
diphoton collision. The cross-sections are evaluated with dependence on the collision
energy
√
s, the polarization of electron and positron beams.
Keywords: ILC, diphoton collision, cross-section.
1. Introduction
The gauge hierarchy between the eletroweak scale and the Plank scale was solved naturally
by the Randall-Sundrum (RS) model in 1999 [1]. The RS model involves two three-branes
bounding a slice of 5D compact anti-de Sitter space taken to be on an S1/Z2 orbifold extra
dimension. Gravity is localized at the UV brane, while the Standard model fields are supposed to
be localized at the IR brane. The additional scalar called radion (φ) corresponds to the fluctuations
of the size of the extra dimension [2-5]. Radion can be the lightest particle in the RS model. In the
effective four dimensional theory, the Higgs boson and the radion can be mixed since radion and
Higgs fields have the same quantum numbers [6, 7].
In 2012, the 125 GeV Higgs signal is discovered by the ATLAS and CMS collaborations
[8, 9], which has completed the particle spectrum of the Standard model. The international linear
collider (ILC) [10], which is an e−e+ collider with high energy and luminosity, will supply the
chance to detect new physics. The potential of the ILC can be enhanced by considering γγ and
γe− collisions with the photon beam generated by the backward Compton scattering of electron
and laser beams [11-14]. In this work, we investigate the e+e− production via γγ collision taking
into account the Higgs and radion propagators. The layout of this paper is as follows. In Section
II, we review the Higgs-radion mixing in the RS model. The e−e+ production of γγ collision is
calculated in Section III. Finally, we summarize our results and make conclusions in Section IV.
2. Content
2.1. The Higgs-radion mixing in the Randall-Sundrum model
In the RS model, the values of the bare parameters are determined by the Planck scale
and the applicable value for size of the extra dimension is assessed by krcpi ≃ 35 (rc - the
Received September 4, 2018. Revised: November 9, 2018. Accepted November 16, 2018
Contact Bui Thi Ha Giang, e-mail address: giangbth@hnue.edu.vn
28
The e−e+ production via γγ collision in the randall-sundrum model at the ILC
compactification radius and k - the bulk curvature). Thus the weak and the gravity scales can
be naturally generated. The Higgs boson and radion can be mixed with the gravity-scalar mixing
described by the following action[6]
Sξ = ξ
∫
d4x
√
gvisR(gvis)Hˆ
+Hˆ, (2.1)
where ξ is the mixing parameter, R(gvis) is the Ricci scalar for the metric gµνvis = Ω2b(x)(ηµν +
εhµν) induced on the visible brane, Ωb(x) = e−krcpi(1 + φ0Λφ ) is the warp factor, φ0 is the
canonically normalized massless radion field, Hˆ is the Higgs field in the 5D context before
rescaling to canonical normalization on the brane. With ξ 6= 0, there is neither a pure Higgs
boson nor pure radion mass eigenstate. This ξ term mixes the h0 and φ0 into the mass eigenstates
h and φ as given by
h0 = dh+ cφ,
φ0 = bh+ aφ,
(2.2)
where a = −cosθ
Z
, b =
sinθ
Z
, c = sinθ +
6ξγ
Z
cosθ, d = cosθ − 6ξγ
Z
sinθ, Z2 = 1 +
6γ2ξ (1− 6ξ) is the coefficient of the radion kinetic term after undoing the kinetic mixing,
γ = υ/Λφ, υ = 246 GeV. The mixing angle θ is
tan 2θ = 12γξZ
m2h0
m2φ0 −m2h0 (Z2 − 36ξ2γ2)
, (2.3)
where mh0 and mφ0 are the Higgs and radion masses before mixing.
The new physical fields h and φ in (2.2) are Higgs-dominated state and radion, respectively
m2h,φ =
1
2Z2
[
m2φ0 + βm
2
h0 ±
√
(m2φ0 + βm
2
h0
)2 − 4Z2m2φ0m2h0
]
. (2.4)
There are four independent parameters Λφ, mh, mφ, ξ that must be specified to fix the state
mixing parameters. We consider the case of Λφ = 5 TeV and m0MP = 0.1, which makes the radion
stabilization model most natural [5].
Feynman rules for the couplings of Higgs and radion are showed as follows
V (h, f, f) = igffh = −i
gmf
2mW
(d+ γb) , (2.5)
V (φ, f, f) = igffφ = −i
gmf
2mW
(c+ γa) , (2.6)
V (h, γµ(k1), γµ(k2)) =iCγh [(k1k2)η
µν − kν1kµ2 ]
=− i α
2piυ0
(
(d+ γb)
∑
i
e2iN
i
cFi(τi)− (b2 + bY )γb
)
[(k1k2)η
µν − kν1kµ2 ] ,
(2.7)
V (φ, γµ(k1), γµ(k2)) =iCγφ [(k1k2)η
µν − kν1kµ2 ]
=− i α
2piυ0
(
(c+ γa)
∑
i
e2iN
i
cFi(τi)− (b2 + bY )γa
)
[(k1k2)η
µν − kν1kµ2 ] ,
(2.8)
29
Bui Thi Ha Giang
where b3 = 7, b2 = 19/6, bY = −41/6 are the SU(2)L ⊗ U(1)Y β-function coefficients in the
SM.
The auxiliary functions of the h and φ are given by
F1/2(τ) = −2τ [1 + (1− τ)f(τ)], (2.9)
F1(τ) = 2 + 3τ + 3τ(2 − τ)f(τ), (2.10)
with
f(τ) =
(
sin−1
1√
τ
)2
(for τ > 1), (2.11)
f(τ) = −1
4
(
ln
η+
η−
− ipi
)2
(for τ < 1), (2.12)
η± = 1±
√
1− τ , τi =
(
2mi
ms
)2
. (2.13)
mi is the mass of the internal loop particle (including quarks, leptons and W boson), ms is the
mass of the scalar state (h or φ). Here, τf =
(
2mf
ms
)2
, τW =
(
2mW
ms
)2
denote the squares of
fermion and W gauge boson mass ratios, respectively.
2.2. The e−e+ production of γγ collision
We consider the collision process in which the initial state contains diphoton, the
final state contains electron and positron in the RS model,
γ(p1) + γ(p2) → e−(k1) + e+(k2). (2.14)
Here, pi, ki (i = 1,2) stand for the momentums. There are three Feynman diagrams contributing to
reaction (2.14), representing the s, u, t-channels exchange depicted in Figure1.
Figure 1. The Feynman diagrams for γγ → e−e+
The transition amplitude can be given as follows:
30
The e−e+ production via γγ collision in the randall-sundrum model at the ILC
Ms =
[
−Cγφgeeφ
q2s −m2φ
− Cγhgeeh
q2s −m2h
]
εµ(p1) [(p1p2) η
µν − pν1pµ2 ] εν(p2)u(k1)v(k2), (2.15)
Mu = − ie
2
q2u −m2e
u(k1)γ
νεν(p2)(/qu +me)v(k2)γ
µεµ(p1), (2.16)
Mt = − ie
2
q2t −m2e
u(k1)γ
νεν(p1)(/qt +me)v(k2)γ
µεµ(p2). (2.17)
Here, qs = p1 + p2 = k1 + k2, qu = p1 − k2 = k1 − p2, qt = p1 − k1 = k2 − p2.
The differential cross-section for the subprocess γγ → e−e+ is written as [16]
dσ̂(ŝ)
dcosψ
=
1
32piŝ
|−→k 1|
|−→p 1| |Mfi|
2, (2.18)
where |Mfi|2 = |Ms|2+|Mu|2+|Mt|2+Re (M∗sMu +M∗uMs +M∗sMt +M∗tMs +M∗uMt +M∗tMu),
ψ = (−→p 1,−→k 1) is the scattering angle. The effective cross-section σ(s) for the subprocess
γγ → e−e+ at the ILC can be calculated as follows
σ(s) =
∫ 0.83
4m2e/s
dxfγ/e(x)
∫ (cosψ)max
(cosψ)min
dcosψ
dσ̂(ŝ)
dcosψ
, (2.19)
where x = ŝ/s in which
√
ŝ is center of mass energy of the subprocess γγ → e−e+,√s is center
of mass energy of the ILC. The photon distribution function fγ/e is given by [12]
fγ/e =
1
D(ζ)
[
(1− x) + 1
1− x −
4x
ζ(1− x) +
4x2
ζ2(1− x)2
]
(2.20)
where
D(ζ) =
(
1− 4
ζ
− 8
ζ2
)
ln(1 + ζ) +
1
2
+
8
ζ
− 1
2(1 + ζ)2
. (2.21)
xmax =
ζ
1 + ζ
, for ζ = 4.8, xmax = 0.83. With the model parameters chosen as mh = 125 GeV,
mφ = 10 GeV, we give some estimates for the cross-sections as follows:
(i) In Figure 2, we evaluate the dependence of the differential cross-section on the cosψ
in case of Pe− , Pe+ , which are the polarization coefficients of e−, e+ beams, respectively. The
collision energy
√
s is chosen as 500 GeV. The polarization coefficients are chosen as Pe− =
Pe+ = 1; Pe− = 1, Pe+ = −1; Pe− = Pe+ = 0. To make the scattered particles be detected,
the scattering angle is chosen as 10o ≤ ψ ≤ 170o [11]. The figure shows that the differential
cross-sections dσ/dcosψ decrease first and then increase.
(ii) We evaluate the dependence of the total cross-sections on the collision energy √s in
Figure 3. The polarization coefficients are chosen as Fig.2. The collision energy is chosen in
the range of 500 GeV≤ √s ≤ 1000 GeV (ILC). The figure shows that the total cross-sections
decrease rapidly when the collision energy
√
s increases.
(iii) In Figure 4, the total cross-section is plotted as the function of Pe− , Pe+ . The figure
indicates that the total cross-section achieves the minimum value when Pe− = Pe+ = ±1 and the
31
Bui Thi Ha Giang
maximum value when Pe− = 1, Pe+ = −1 or Pe− = −1, Pe+ = 1. Numerical values for the
production cross-section and the number of events in a year with the contribution of radion and
Higgs propagators in the γγ → e−e+ collision at the ILC in case of Pe− = 1, Pe+ = - 1 are given
in detail in Table 1. The high integrated luminosity is chosen as L = 10−34cm−1s−1.
Figure 2. The differential cross-section as a function of cosψ in case of Pe− , Pe+ ,
which are the polarization coefficients of e−, e+ beams, respectively
Figure 3. The cross-section as a function of the center of mass energy √s of the ILC
experiments in case of Pe− , Pe+ , which are the polarization coefficients
of e−, e+ beams, respectively
Figure 4. The cross-section as a function of the Pe− , Pe+ polarization coefficients
of the e−, e+ beams at ILC
32
The e−e+ production via γγ collision in the randall-sundrum model at the ILC
Table 1. Numerical values for the production cross-section
and the number of events in a year in case of Pe− = 1, Pe+ = −1 at the ILC
√
s (GeV) 500 600 700 800 900 1000
σ (fbar) 15667.1 10995.6 8150.21 6287.65 5001.23 4075.05
N ( 106 events) 4.941 3.468 2.570 1.983 1.577 1.285
3. Conclusions
In this paper, we have evaluated the production of e−e+ in the RS model via diphoton
collision. The result shows that with the high integrated luminosity and the high polarization of
electron and positron beams, the signal of Higgs-radion mixing might be detected in future ILC
experiments. The differential cross-section is peaked in the backward and forward direction. We
show that, in the region 500 GeV≤ √s ≤ 1000 GeV, the total cross-section is in the range of
4075.05 fbar ∼ 15667.1 fbar in case of Pe− = 1, Pe+ = −1. Finally, it is worth noting that the
number of events in a year can reach at about 106 events with the high integrated luminosity and
at the high degree of polarization.
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