Contribution of the scalar unparticle on process e⁺ e⁻ -> hh in the randall-Sundrum model

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CONTRIBUTION OF THE SCALAR UNPARTICLE 
ON PROCESS e e hh+ − → IN THE RANDALL-SUNDRUM MODEL 
Bui Thi Ha Giang1, Dao Thi Le Thuy1, Dang Van Soa2 
1Hanoi National University of Education 
2Hanoi Metropolitan University 
Abstract: The pair production of Higgs is studied with the contribution of the scalar 
unparticle in the e e+ − collision in the Randall-Sundrum model in detail. We evaluate the 
observable cross-section which depends on the collision energy s and the scaling 
dimension of the unparticle operator Ud . The total cross-section with the unparticle 
contribution is compared to that without the unparticle. 
Keywords: Higgs, scalar unparticle, cross section , Randall-Sundrum model. 
Email: giangbth@hnue.edu.vn 
Received 15 July 2017 
Accepted for publication 10 September 2017 
1. INTRODUCTION 
The Standard model (SM) of particle is successful in describing the elementary 
particle picture. In the Lagrangian of the SM, the scale invariance is broken at or above the 
electroweak scale [1, 2]. Because of no particle states with a definite nonzero mass, there 
are no particles with a nonzero mass in a scale invariant sector in four space-time 
dimensions [1, 3]. Georgi has suggested that if the scale invariance exists, it is made of 
unparticles. Based on the Banks-Zaks theory [4], unparticle stuff with nontrivial scaling 
dimension is considered to exist in our world. The invariant Banks-Zaks field can be 
connected to the SM particles. Recently, the evidence of the unparticle has been studied 
with CMS detector at the LHC [5, 6]. 
Although the SM describes successfully almost all existing experimental data, the 
model suffers from many theoretical drawbacks. One of many attempts to extend the SM 
and solve the hierarchy problem, one of theoretical drawbacks of SM [7], is the Randall-
Sundrum (RS) model. The RS setup involves two three-branes bounding a slice of 5D 
compact anti-de Sitter space taken to be on an 1 2/S Z orbifold. Gravity is localized UV 
26 TRNG I HC TH  H NI 
brane, while the Standard Model (SM) fields are supposed to be localized IR brane. The 
separation between the two 3-branes leads directly to the existence of an additional scalar 
called the radion (φ ), corresponding to the quantum fluctuations of the distance between 
the two 3-branes [8]. In 2012, Higgs signal at 125 GeV is discovered by the ATLAS and 
CMS collaborations [9, 10]. 
However, the unparticle effects on the collisions have not been concerned in the RS 
model. In this paper, we study the Higgs couple production, which has been proposed as an 
option of e e+ − collisions. The layout of this paper is as follows. The unparticle and 
effective interactions are reviewed in Section 2 mostly cited on [1, 2, 3]. Section 3 is 
devoted to the creation of Higgs couple in e e+ − collision. Finally, we summarize our results 
and make conclusions in Section 4. 
2. THE UNPARTICLE AND EFFECTIVE INTERACTIONS 
The derivation of the virtual unparticle propagator is based on the scale invariance [2]. 
The unparticle propagators for scalar, vector and tensor operators are given by [2], 
respectively 
 22( ) ,
2sin( )pi
−∆ = −U Ud dscalar
U
iA
q
d
 (1) 
 22( ) ,
2sin( ) µν
pi
pi
−∆ = −U Ud dvector
U
iA
q
d
 (2) 
 22 ,( ) T ,2sin( ) µν ρσpi
−∆ = −U Ud dtensor
U
iA
q
d
 (3) 
where Ud is the noninteger scaling dimension of the unparticle operator, 
2
2
1
16 2
,
( 1) (2 )(2 )
pi pi
pi
 Γ + 
 =
Γ − ΓU U
U
d d
U U
d
A
d d
 (4) 
22 2
22
22 2
| | fors-channelprocess, is positive,
( )
| | for u-, t-channelprocess, is negative,
pi− −
−
−

− = 

U U
U
U
d id
d
d
q e q
q
q q
 (5) 
2
( ) ,
µ ν
µν µνpi = − +
q q
q g
q
 (6) 
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, 1 2( ) ( ) ( ) ( ) ( ) ( ) .
2 3
µν ρσ µρ νσ µσ νρ µν ρσpi pi pi pi pi pi = + − 
 
T q q q q q q (7) 
 The effective interactions for the scalar, vector and tensor unparticle operators are 
given by, respectively 
( )50 0 0 01 1
1 1 1 1
, , , ,
U U U U
U U U Ud d d d
U U U U
f f f i f f f G Gµ αβµ αβλ λ γ λ γ λ− −Ο Ο ∂ Ο ΟΛ Λ Λ Λ
 (8) 
1 1 51 1
1 1
, ,
U UU Ud d
U U
f f f fµ µµ µλ γ λ γ γ− −Ο ΟΛ Λ
 (9) 
( )2 21 1 1, ,4 U UU Ud dU U
i D D G Gµν α µνµ ν ν µ µα νλ ψ γ γ ψ λ− + Ο Ο
Λ Λ
t t
 (10) 
where iλ (i = 0, 1, 2) stand for the scalar, vector and tensor unparticle operators, 
respectively. '
2 2
a
a YD ig W ig Bµ µ µ µ
τ
= ∂ + + is the covariant derivative, B, iW are gauge fields, 
,
2
a
Y
τ
 correspond to the standard generators of (1)YU and (2)LSU . The corresponding 
coupling constants are denoted by g, g’. f stands for a standard model fermion, ψ stands 
for a standard model fermion doublet or singlet. Gαβ denotes the gauge field strength. 
3. THE HIGGS PRODUCTION 
In this section, we consider the + − →e e hh collision process 
1 2 1 2e ( ) ( ) ( ) ( ),
− ++ → +p e p h k h k (11) 
Here ,i ip k (i = 1,2) stand for the momentums. There are three Feynman diagrams 
contributing to reaction (11), representing the s, u, t-channels exchange depicted in Fig.1. 
Figure 1. Feynman diagrams for e e hh+ − → collision. 
28 TRNG I HC TH  H NI 
We obtain the scattering amplitude in the s, u, t-channels, respectively 
220 0
2 11
( ) ( ) ( )
2sin( )
λ λ
pi
−
−
= −
Λ Λ
U U
U U
d d
s sd d
UU U
Ai i
M q v p u p
d
 (12) 
2
2 12 2
ˆ( )( ) ( )= − +
−
eeh
u u e
u e
g
M i v p q m u p
q m
 (13) 
2
2 12 2
ˆ( )( ) ( )= − +
−
eeh
t t e
u e
g
M i v p q m u p
q m
 (14) 
where eehg are given by [11], 1 2 1 2 ,sq p p k k= + = + 1 2 1 2 ,uq p k k p= − = − 1 1 2 2.= − = −tq p k k p 
The expressions of the differential cross-section [12] 
21 | | | | ,
(cos ) 64 | | fi
d k
M
d s p
σ
ψ pi
=
r
r (15) 
where 2 2 2 2| | | | | | | | 2 Re( ).fi s u t s u s t u tM M M M M M M M M M
+ + += + + + + +
We give some estimates for the cross-sections as follows 
i) In Fig.2, we evaluate the dependence of the total cross-section on the collision 
energy s with the various dU. We choose 0 1, 1000GeVUλ = Λ = [2]. In case of the scalar 
unparticle, 1 2Ud< < [13]. The total cross-sections decrease when the collision energy s 
increases. 
Figure 2. Total cross-sections for e e hh+ − → versus s with the various Ud 
ii) The proportion of the total cross-sections with the unparticle contribution Uσ to 
that without the unparticle 0σ in Ref.14 is calculated in Table 1. The results show that the 
unparticle contribution is significant. 
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(GeV)s 
0/Uσ σ 
1.1Ud = 1.2Ud = 1.3Ud = 1.5Ud = 1.7Ud = 1.9Ud = 
300 4.3743 1.2514 0.3655 0.0353 0.0048 0.0022 
500 6.9403 2.4403 0.8731 0.1265 0.0261 0.0178 
800 9.2157 3.8971 1.6887 0.3578 0.1066 0.1063 
1000 10.4242 4.8242 2.2788 0.5751 0.2055 0.2454 
1500 13.2143 7.1753 3.9935 1.3961 0.6883 1.1396 
2000 16.0221 9.8011 6.1105 2.6851 1.6685 3.4696 
2500 19.5210 13.0539 8.8922 4.6706 3.4730 8.6227 
3000 23.9716 17.2340 12.6950 7.7305 6.6383 19.078 
4. CONCLUSION 
In this paper, the total cross-sections for the process + − →e e hh with the unparticle 
contribution are evaluated. The results indicate that the cross-sections depend on the 
parameter Ud and the collision energy s . They are larger in case of the small Ud and 
decrease when s increases. When 3000s = GeV and 1.1Ud = , the cross-section with the 
unparticle contribution is about 24 times as large as that without the unparticle. 
Acknowledgement: The work is supported in part by Hanoi National university of 
Education project under Grant No. SPHN-16-05. 
REFERENCES 
1. H. Georgi (2007), Phys. Rev. Lett. 98 p.221601. 
2. K. Cheung, W-Y. Keung and T-C. Yuan, Phys. Rev. D76 p.055003. 
3. S-L. Chen, X-G. He (2007), Phys. Rev. D76 p.091702. 
4. T. Banks and A. Zaks (1982), Nucl. Phys. B196, p.189. 
5. CMS Collaboration (2015), Eur. Phys. J. C75 p.235. 
6. CMS Collaboration (2016), Phys. Rev. D93 p.052011. 
7. L. Randall and R. Sundrum (1999), Phys. Rev. Lett. 83 p.3370. 
8. L. Randall and R. Sundrum (1999), Phys. Rev. Lett. 83 p.4690. 
9. M. Frank, K. Huitu, U. Maitra, M. Patra (2016), Phys. Rev. D94 p.055016. 
10. G. Aad et al. (2012), ATLAS Collaboration,Phys. Lett. B716 pp.1-29. 
11. S. Chatrchyan et al. (2012), CMS Collaboration, Phys. Lett. B716 pp.30-61. 
12. D. Dominici, B. Grzadkowski, J. F. Gunion and M. Toharia (2003), Nucl.Phys. B671 p.243. 
30 TRNG I HC TH  H NI 
13. M. E. Peskin and D. V. Schroeder (1995), “An Introduction to Quantum Field Theory”, 
Addision-Wesley Publishing. 
14. A. Friedland, M. Giannotti, M. Graesser (2009), Phys. Lett. B678 pp.149-155. 
15. B. T. H. Giang and D. T. L. Thuy (2016), Journal of science of HNUE, Vol. 61, No. 7, pp.58-64. 
ĐÓNG GÓP CỦA PHI HẠT VÔ HƯỚNG VÀO QUÁ TRÌNH 
e e hh+ − → TRONG MÔ HÌNH RANDALL-SUNDRUM 
Tóm tắt: Sự tạo cặp của Higgs trong mô hình Randall-Sundrum từ va chạm e e+ − với sự 
đóng góp của U-hạt vô hướng được nghiên cứu chi tiết. Chúng tôi đánh giá tiết diện quan 
sát phụ thuộc vào năng lượng va chạm và số chiều toán tử U-hạt. Tiết diện toàn phần có 
đóng góp của U-hạt được so sánh với trường hợp không có đóng góp của U-hạt. 
Từ khóa: Tạo Higgs, U-hạt vô hướng, tiết diện, mẫu Randall-Sundrum.