Radion effects on Bhabha scattering

TRƯỜNG ĐẠI HỌC THỦ ĐÔ H

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NỘI

RADION EFFECTS ON BHABHA SCATTERING
Ha Huy Bang, Pham Que Duong,
Nguyen Thi Thu Huyen, Nguyen Thi Thuy Linh
Ha Noi University of Sciences
Abstract: In this article, we have considered the possible signatures of radion through
Bhabha scattering. The numerical results show that the total cross section with radion
effects are abont 0.62-0.65 pb.
This could have inportant implications for radion searches and for the measurement of
the cross – section of the Bhabha scattering.
PACS number: 12.60.Cn, 12.20.-m, 42.55.-f.
Keywords:
Keywords radion, Bhabha scattering.
Email:
Received 24 March 2019
Accepted for publication 25 May 2019

1. INTRODUCTION
As well known, there are many convincing evidences that 80% of the matters in the
universe is composed of dark matters (DM).
In several extensions of the Standard Model, radion or u-boson is postulated [1-5].
On the other hand, the Randall – Sundrum (RS) Model is one of the attractive
condidates to solve the gauge hierarchy problem in the standard Model Many works have
been done on the phenomenological aspects of radion in various colliders [6-9].
As we well known, Bhabha scattering is among the key processes in particle physics.
Recently, the authors have presented the results of the SANC group on the complete oneloop calculation of the electroweak rediative corrections to Bhabha scattering with

polarized beams [10].
Very recently, we have investigated unparticle effects on Bhabha scattering [11] and
on axion-like particles production in e+e- collisions [12]. In this paper, we investigate
virtual radion effects via Bhabha scattering.


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69

2. RADION EXCHANGE AND CROSS SECTION
In this cestion, we will derive a formula for the cross-section of the process presented
in Figure.2, which shows on of the possible processes, where a radion may intermediate a
creation of e+e- in the e+e- scattering.

Í´

Í

Fig 1. Feynman diagram for Bhabha scattering via radion and photon

The amplitude for this process is given by

M = v ( k 2 )ieγ µ u ( k1 )
2

= ie ε

2

− ie 2 ( g µν − q µ qν / mu2 )
2

q −m
2
u

( g µν − q µ qν / m )
q 2 − mu2

2
u

u ( p1 )i eγ ν v ( p 2 )
(1)

v ( k 2 )γ u ( k1 )u ( p1 )γ v ( p 2 ),
µ

µ

From this, we get
2

M =

16e4ε 4
{ 2( p1k1 )( p2 k2 ) + 2( p1k2 )( p2 k1 )
(q 2 − mu2 ) 2

−

qk2
[( p1k1 )(qp2 ) + (qp1 )( p2 k1 ) − 2( p1 p2 )(qk1 )]
mu2

−

qk1
[ ( p1k2 )(qp2 ) + (qp1 )( p2 k2 ) − 2( p1 p2 )(qk2 )]
mu2

+2

−

k1k2
(qk )(qk )
 2(qp1 )(qp2 ) − ( p1 p2 )q 2  + 2 1 4 2  2(qp1 )(qp2 ) − ( p1 p2 )q 2 
2 
mu
mu

q 2 (k1k2 )
 2(qp1 )(qp2 ) − ( p1 p2 )q 2  }
mu4 

(2)

In center of mass frame, four-moments of particles are defined

k1 = ( E , k ), k2 = ( E , −k ), p1 = ( E , p ), p2 = ( E , − p )
and

S = ( k1 + k 2 ) 2 = ( p1 + p2 ) 2 = q 2 = 4 E 2 .

Where S is the center of mass energy.The differential cross-section can be obtained as
follows.Neglecting the mass of electron, we have


TRƯỜNG ĐẠI HỌC THỦ ĐÔ H

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2

M =

4e4ε 4 s 2 
s 
1 + cos 2 θ + 2  ,
2
2 2 
(q − mu ) 
mu 

NỘI
(3)

So, the differential cross section can be obtained as follows

dσ

α 2ε 4 s 
s 
=
1 + cos 2 θ + 2  ’
2
2 2 
d Ω 4(q − mu ) 
mu 

(4)

where
Therefore, the total cross section is

σ=

α 2πε 4 s  4

s 
 + 2 .
(q − m )  3 mu 
2

2 2
u

(5)

Finally, from (4) and (5) we get


s 
2
 1 + cos θ + 2 
mu 
dσ
=
σ dΩ
4 s 
4π  + 2 
 3 mu 

(6)

Numerical results and discussions
Let us now turn to the numerical analysis. We take ε = 10 −2 , mn = 10 GeV
As input parameters.
In Fig.2 we plot the

dσ
with respect to cosθ . As we can observer from Fig.2 the
σ dΩ

dσ
has a minimum for cosθ = 0 .
σ dΩ

Fig 2. The

dσ

with respect to cosθ .
σ dΩ


TẠP CHÍ KHOA HỌC − SỐ 31/2019
Table 1. The

cosθ

-1.0

dσ
(×10 2 ) 7.959
σ dΩ

71

dσ
at different cosθ
σ dΩ

-0.8

-0.6

-0.4

-0.2

0.0

7.958

7.957

7.957

7.956

7.956

0.2

0.4

0.6

0.8

1.0

7.956 7.957 7.957 7.958 7.959

In the figure 3, we plot the differential cross sections and the toatal cross sections as a
function of

s for cos θ = 1

Fig 3. The variation of

dσ
as a function of
dΩ

s for cos θ = 1

As we see from the Figure 3 that the radion effects quickly go down as s becomes
larger.
In the following, we give the numerical values of the differential cross section with
radion effects in Table 2.
Table 2. The differential cross sections with radion effects for cos θ = 1 at different energies

s GeV

500

650

800

950

1100

1250

1400

1550

1700

1850

2000

dσ
(×10 2 ) 5.195 5.191 5.190 5.189 5.188 5.188 5.187 5.187 5.187 5.187 5.187
σ dΩ

For the next step, we give the numerical values of the total cross-sections with
radion effects at different energies in Figure 4 and Table 3.


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Fig 4. The variation of

NỘI

σ as a function of s .

So, direct computations have showed that the total cross-sections should be about
0.006-0.007 fem to barn.
Therefore, in the range

s =500GeV to 2000GeV the total cross-sections are slightly

different.
Table 3. The total cross sections with radion effects at different energies

s GeV

σ ×103 ( fb)

500

650

800

950

1100

1250

1400

1550

1700

1850

2000

6.5266 6.5230 6.5213 6.5203 6.5197 6.5192 6.5190 6.5188 6.5186 6.5185 6.5184

3. CONCLUSION
Our results are attractive because of possible connection to radion. We hope that future
experiments will confirm the existence of radion.

REFERENCES
1.

R.Foot, X. G. He (1991), Phys.Lett. B 267, 509.

2.

J. D. Bjorken, R. Essig, P. Schuster, and N. Toro (2009), Phys. Rev. D 80, 075018.

3.

Davoudiasl, H. S. Lee and W. J. Marciano (2012), Phys. Rev. Lett. 109, 031802.

4.

M.Anelli et al. [SHiP Collaboration], arXiv:1504.04956[physics.ins-det].

5.

P. Ilten, Y. Soreq, J. Thaler, M. Williams and W. Xue, Phys. Rev. Lett. 116, no. 25, (2016)
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D.V. Soa, D.T.L. Thuy, N.H. Thao and T.D. Tham, Mod. Phys. Lett. A27 (2012) 1250126.

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10. Andrey Arubuzov, wt al. (LL2018), proceedings of Science, POS 010.
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ẢNH HƯỞNG CỦA RADION LÊN TÁN XẠ BHABHA
Tóm tắ
tắt: Trong bài báo này chúng tôi xem xét các tín hiệu của hạt radion qua tán xạ
Bhabha. Các kết quả đánh giá số chỉ ra tiết diện tán dạ toàn phần với ảnh hưởng của
radion là khoảng 0.62-0.65 pb. Điều này là quan trọng cho việc tìm kiếm radion và cho
việc đo tiết diện tán xạ của tán xạ Bhabha.
Từ khóa:
khóa radion, tán xạ Bhabha