<span class='text_page_counter'>(1)</span><div class='page_container' data-page=1>

1

The Bhabha Scattering in the Randall-sundrum Model

Le Nhu Thuc

*

<i>Hanoi National University of Education (HNUE), 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam </i>
Received 26 September 2018

<i>Revised 26 October 2018; Accepted 17 December 2018 </i>

<b>Abstract: The change in other two fermion processes is called Bhabha process. In this paper, we </b>

discuss the Bhabha process <i>e e</i>  <i>e e</i>  in the Randall –Sundrum (RS). We caculated the cross
section for photon  <i>, boson Z, radion </i>



<i> and Higgs h exchange and evaluated exchange </i>
contributions of the Bhabha process in detail.

<i>Keywords: DCS, cross-section, Bhabha, radion, Randall-Sundrum. </i>

<b>1. Introduction</b>

The RS model [1] can solve the hierarchy problem by localizing all the Standard model (SM)
particles on the IR brane. This model predicts two new particles beyond the SM. One is a spin-2 graviton
and another is a scalar-field radion which is a metric fluctuation along the extra dimension. The mass of
radion is expected to be of the order of GeV. Therefore, the radion is expected to be the first signature
of warped extra dimension models in direct search experiments such as the Large Hadron Collider
(LHC) [2 - 8].

The Bhabha scattering has been studing in models beyond the SM, and it is also compared ILC250
to LEP2 and LHC [9]. In this paper, we discuss the radion and Higgs exchange contributions in the
cross section of the Bhabha scattering in the RS model. We hope that, the Bhabha channel suggests the

best way to study radion and Higgs.

<b>2. The cross-section of the proces </b><i>e e</i>  <i>e e</i> 

The Feynman diagrams of the process <i>e e</i>  <i>e e</i>  are shown in Fig. 2.1.
________

_{Tel.: 84-982004689.}

Email:

</div>
<span class='text_page_counter'>(2)</span><div class='page_container' data-page=2>

Fig 2.1. The Feynman diagrams for the process <i>e e</i>  <i>e e</i>  .

Using the Feynman rules for Fig 2.1, the matrix element for the process <i>e e</i>  <i>e e</i>  is given by:
+ For s-channel (Fig 2.1a, b),













2

t 2 1 1 1 1 2 2 2 2

s

ie

M

g

u(k , r ) u p ,s

v p ,s

v k , r

q

 

 







 





_

_

_



_{ }





_





₍₁₎









2

5

tz 2 2 2 2 1 1 e e 1 1

w s z Z

q q

ig

M

g

u(k , r )[

v

a

]u p ,s

16c (q -m )

m

  





 

_{ }

_



_

_{ }



_{ }



 

_













_

v p ,s [



₂ ₂









v

_e

 

a

_e 5



] v k , r



₂ ₂





_

(2)

2

2
e

t 2 2 1 1 1 1 2 2 2 2

w
s

m

i

g

M

(c+ γa)

u(k , r ) u(p ,s )v(p ,s ) v(k , r )

2 m

(q - m )







_

_

_









_

_













(3)

2

2
e

th 2 2 1 1 1 1 2 2 2 2

w
s h

m

i

g

M

(d+ γb)

u(k , r ) u(p ,s )v(p ,s ) v(k , r )

2 m

(q - m )



_

_

_









_

_













(4)

</div>
<span class='text_page_counter'>(3)</span><div class='page_container' data-page=3>

















2

' '

t 2 ' ' 2 2 2 2 1 1 1 1

t

ie

M

g

v k , r

v p ,s

u p ,s

u k , r

q

  

  







_

_{ }

_



_

_

_



_{ }





_





(5)

2

' '

tz 2 2 2 ' ' 2

w t z Z

q q

ig

M

g

16c (q -m )

m

 


 



 





_

_{ }





_













_



v k , r [



₂ ₂





'



v

_e

 

a

_e 5



]v p ,s



₂ ₂



 

_{ }

u p ,s [



₁ ₁





'



v

_e

 

a

_e 5



]u k , r



₁ ₁





_

(6)
2

2
e

t 2 2 2 2 1 1

w
t

m

i

g

M

(c+ γa)

v(k )v(p )u(p ) u(k )

2 m

(q - m )







_

_









_

_













(7)
2
2
e

th 2 2 2 2 1 1

w

t h

m

g

i

M

(d+ γb)

v(k )v(p )u(p ) u(k )

2 m

(q - m )



_



_







_

















<i></i>

<i> (8) </i>

From (1-8), we have





 



 





4

1 2 2 1 1 1 2 2 1

2

s 4

2

2 1 2

16

M

[

2

<i>s</i>

<i>p k</i>

<i>p k</i>

<i>e</i>

<i>q</i>

<i>p k</i>

<i>p k</i>

<i>p p</i>

<i>k k</i>











 







2





 





4

2 1 1 2 2 2 1 1

]

2

<i>p p</i>

<i>k k</i>

<i>m</i>

<i>_e</i>

<i>p k</i>

<i>p k</i>

<i>m</i>

<i>_e</i>











(9)



2 1



1 2

 

1 2



2 1

 

1 1



2



4
2

sz 4 2 2 2

2

2

M

16

{

256c (q - m )

<i>_w</i> <i>_s</i> <i>_z</i>

<i>p k</i>

<i>p k</i>

2

<i>k k</i>

<i>p p</i>

3

<i>p k</i>

<i>p k</i>

<i>g</i>









_

_





























2 4

1 1 2 1 2 1 2 1 2 1 2

2

[2

<i>e</i> <i>s</i> <i>s</i> <i>s</i> <i>s</i>

<i>z</i>

<i>me</i>

<i>p k</i>

<i>m</i>

<i>p k</i>

<i>q k</i>

<i>q p</i>

<i>k k</i>

<i>q p</i>

<i>q p</i>

<i>m</i>



 











 













2

1 2 <i>s</i> 1 <i>s</i> 2

2

2 2 <i>s</i> 1 <i>s</i> 2

2

<i>s</i> 1 <i>s</i> 1

<i>p p</i>

<i>q k</i>

<i>q k</i>

<i>p k</i>

<i>q k</i>

<i>q k</i>

<i>me</i>

<i>q k</i>

<i>q p</i>













2





4











1 1 2 2 1 1

]

4 1 2 1 2

1

)

[

4

<i>_s</i> <i>_s</i> <i>_s</i> <i>_s</i>

<i>z</i>

<i>p k</i>

<i>p k</i>

<i>me</i>

<i>p k</i>

<i>me</i>

<i>q p</i>

<i>q p</i>

<i>q k</i>

<i>q k</i>

<i>m</i>









 









 





 







2 2 2

1 1 2 2 1 1 1 1 2 2

2

<i>q</i>

<i>_s</i>

<i>q p</i>

<i>_s</i>

<i>q k</i>

<i>_s</i>

<i>p k</i>

2

<i>m</i>

<i>_e</i>

<i>q</i>

<i>_s</i>

<i>q p</i>

<i>_s</i>

<i>q k</i>

<i>_s</i>

2

<i>p k</i>

<i>q p</i>

<i>_s</i>

<i>q k</i>

<i>_s</i>





























2 2 2 2 2 4 2 2

1 1 2 2 <i>s</i> <i>s</i> <i>e</i> 1 1 <i>s</i> <i>s</i> <i>e</i> <i>s</i> <i>s</i>

]}

<i>p k</i>

<i>p k</i>

<i>q q</i>

<i>m</i>

<i>p k</i>

<i>q q</i>

<i>m</i>

<i>q q</i>







(10)
4

2 ₄ ₂ ₂

s 2 2 2 2 1 1 2

1 g

M 16 (c+ γa) {p p }{ }

(q - m ) 2

<i>e</i>

<i>e</i> <i>e</i>

<i>s</i> <i>w</i>

<i>m</i>

<i>m</i> <i>k k</i> <i>m</i>

<i>m</i>


 _ _ _ 
 
 _ _  

   

</div>
<span class='text_page_counter'>(4)</span><div class='page_container' data-page=4>

4

2 ₄ ₂ ₂

sh 2 2 2 2 1 1 2

1

g

M

16

(d+ γb)

{p p

}{

}

2

(q - m )

<i>e</i>

<i>e</i> <i>e</i>

<i>w</i>

<i>s</i> <i>h</i>

<i>m</i>

<i>m</i>

<i>k k</i>

<i>m</i>

<i>m</i>



_

_

_









_

_

















(12)



1 2



2 1

 

1 1



2 2

 





4
1 2
4 1
2
2

2

3

16

M

<i>_t</i>

[

2

<i>s</i>

<i>p k</i>

<i>p k</i>

<i>p k</i>

<i>p k</i>

<i>p</i>

<i>k k</i>

<i>q</i>

<i>p</i>

<i>e</i>















2





 





4

2 2 1 1 <i>e</i> 2 1 1 2 <i>e</i>

]

<i>p k</i>

<i>p k</i>

<i>m</i>

<i>p p</i>

<i>k k</i>

<i>m</i>









(13)





 





4

2 1 1 2 1 2 2 1
2

tz 4 2 2 2

M

16

{

256c (q - m

<i>w</i> <i>s</i> <i>z</i>

)

2

3

<i>p k</i>

<i>p</i>

<i>k</i>

<i>p</i>

<i>g</i>

<i>k</i>

<i>k</i>

<i>p</i>









 













2





4









1 1 2 2 2 2 2

[

2

1 2 1 2

2

2

<i>_e</i> <i>_t</i> <i>_t</i>

<i>z</i>

<i>p k</i>

<i>p k</i>

<i>me</i>

<i>p k</i>

<i>m</i>

<i>p k</i>

<i>q k</i>

<i>q p</i>

<i>m</i>











1 2



1



2

 

1 2



1



2

 

2 2



1



2



2

<i>k k</i>

<i>q p</i>

<i>_t</i>

<i>q p</i>

<i>_t</i>

<i>p p</i>

<i>q k</i>

<i>_t</i>

<i>q k</i>

<i>_t</i>

<i>p k</i>

<i>q k</i>

<i>_t</i>

<i>q k</i>

<i>_t</i>















2





 





2





1 1 <i>t</i> 2 <i>t</i> 2

2

<i>e</i> <i>t</i> 2 <i>t</i> 1 1 2 1 2 <i>e</i> 1 2

<i>p k</i>

<i>q p</i>

<i>q k</i>

<i>m</i>

<i>q p</i>

<i>q p</i>

<i>p p</i>

<i>k k</i>

<i>m</i>

<i>p p</i>



















 











4 2

1 2 1 2 1 2 1 2

4

1

)

]

[

4

<i>_t</i> <i>_t</i> <i>_t</i> <i>_t</i>

2

<i>_t</i> <i>_t</i> <i>_t</i>

<i>z</i>

<i>me</i>

<i>q p</i>

<i>q p</i>

<i>q k</i>

<i>q k</i>

<i>q</i>

<i>q p</i>

<i>q p</i>

<i>k k</i>

<i>m</i>







 





 





 









2 2 2 2

1 2 1 2 1 2 1 2 1 2

2

<i>m</i>

<i>_e</i>

<i>q</i>

<i>_t</i>

<i>q p</i>

<i>_t</i>

<i>q p</i>

<i>_t</i>

2

<i>p p</i>

<i>q k</i>

<i>_t</i>

<i>q k</i>

<i>_t</i>

<i>p p</i>

<i>k k</i>

<i>q q</i>

<i>_t</i> <i>_t</i>



















2 2 2 4 2 2

1 2

]}

<i>e</i> <i>t</i> <i>t</i> <i>e</i> <i>t</i> <i>t</i>

<i>m</i>

<i>p p</i>

<i>q q</i>

<i>m</i>

<i>q q</i>





(14)

4

2 ₄ ₂ ₂

1 1 2 2

2 2 2

1

g

M

16

(c+ γa)

{ p

}{ p

}

2

(q - m )

<i>e</i>

<i>t</i> <i>e</i> <i>e</i>

<i>w</i>
<i>s</i>

<i>m</i>

<i>k</i>

<i>m</i>

<i>k</i>

<i>m</i>

<i>m</i>






_

_

_









_

_

















(15)
4

2 ₄ ₂ ₂

th 2 2 2 1 1 2 2

1

g

M

16

(d+ γb)

{ p

}{ p

}

2

(q - m )

<i>e</i>

<i>e</i> <i>e</i>

<i>w</i>

<i>t</i> <i>h</i>

<i>m</i>

<i>k</i>

<i>m</i>

<i>k</i>

<i>m</i>

<i>m</i>



_

_

_









_

_

















(16)

The cross section for process <i>e e</i>  <i>e e</i>  is given by
2
2

1

64

<i>fi</i>

<i>k</i>

<i>d</i>

<i>M</i>

<i>S</i>

<i>d</i>

<i>s</i>

<i>p</i>









. (19)

where <i>s</i> is the center-of-mass energy, <i>M_fi</i>2 is the

<b>square of matrix element</b>

,

(cos )

</div>
<span class='text_page_counter'>(5)</span><div class='page_container' data-page=5>

We choose

m

_e



0, 00051GeV,

m

_w



80GeV,

m

_Z



91, 2GeV,

m

_



10GeV,

0

v



246GeV,

5000,



 

0

v 123

,
2500


  

 w

S



0, 231,

2

w w

C



1 S ,



w
w
0

2m

g

C ,

v



e

1

a

,

2





2

e w

1

v

2s ,

2

  

a

cos

,

Z





c

sin

6

cos ,

Z





 



1

,

6

 







2



Z



1 6

 

1 6

 

.

The

cos



of the <i>e e</i>  <i>e e</i>  differential cross section is shown in
figure 2.2 and figure 2.3 at <i>s</i> 3000<i>GeV</i> .

Figure 2.a, b show the DCS via radion and Higgs exchange for s-channel, does not depend on

cos



. The DCS is

4, 2.10

29

pbarn

a for radion exchange contribution and 23

0,7.10 pbarn for Higgs

exchange contribution. The DCS via radion and Higgs exchange for t channel depend on

cos



.
However, the DSC decreases very small while

cos



increases from 1 to 1 (Fig 2.3 c, d).

For photon and Z boson exchange contributions, the DCS is very large, and it is much larger than
for radion and Higgs exchange contributions.

a) b)

</div>
<span class='text_page_counter'>(6)</span><div class='page_container' data-page=6>

Fig2.2. The <i>e e</i>  <i>e e</i>  differential cross section via photon (a), Z boson (b), radion (c),
Higgs (d) exchange for s-channel as a function of

cos



a) b)

c) d)

</div>
<span class='text_page_counter'>(7)</span><div class='page_container' data-page=7>

a) b)

c) d)

Fig 2.4. The <i>e e</i>  <i>e e</i>  total cross section via photon (a), Z boson (b), radion (c), Higgs (d) exchange for
s-channel as a function of <i>s</i>

</div>
<span class='text_page_counter'>(8)</span><div class='page_container' data-page=8>

c) d)

Fig 2.5. The <i>e e</i>  <i>e e</i>  total cross section via photon (a), Z boson (b),radion (c), Higgs (d) exchange for
s-channel as a function of <i>s</i>

a) b)

Fig 2.6. The <i>e e</i>  <i>e e</i>  total cross section as a function of <i>s</i> for s (a), t (b)-channel

Next, we plotted the <i>s</i> dependence of the <i>e e</i>  <i>e e</i>  cross section. The results are shown in
figure 2.4-2.6. Here, we see that, the total cross section for photon, radion and Higgs exchange
contributions decrease while <i>s</i> increases from

<i>1000GeV</i>

to

<i>5000GeV</i>

. For Z boson exchange
contribution, the total cross section increases while <i>s</i> increases from

<i>1000GeV</i>

to

<i>5000GeV</i>

.
However, in low energy region ( <i>s</i> 100<i>GeV</i>), the <i>e e</i>  <i>e e</i>  total cross section for s (a), t (b)
channel is very large.

<b>Conclusion </b>

</div>
<span class='text_page_counter'>(9)</span><div class='page_container' data-page=9>

suggests that if radion and Higgs were produced in the high energy region, so the ability to observe them
is possible by Bhabha scattering.

<b>References </b>

[1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370, 1999.

[2] H. Davoudiasl, T. McElmurry and A. Soni, Phys. Rev. D82, 115028, 2010.
[3] V. P. Goncalves and W. K. Sauter, Phys. Rev. D82, 056009, 2010.

<i>[4] W. -J. Zhang, W. -G. Ma, R. -Y. Zhang, X. -Z. Li, L. Guo, and C. Chen, Phys. Rev. D92, 116005, 2015, </i>
[arXiv:1512.01766].

[5] C. Cai, Z.-H. Yu and H.-H. Zhang, Phys. Rev. D93. 075033, 2016.

[6] F. Abu-Ajamieh, R. Houtz, and R. Zheng, [arXiv: 1607.01464v1 [hep-ph]], 2016.
[7] The CMS Collaboration, CERN, August 2016.

</div>

<!--links-->
Protocols in relation to the OSI model.doc

• 1
• 631
• 0