1. Introduction
The Standard model (SM) of particles can be used to describe the elementary particle
picture. However, some theoretical drawbacks in SM have been motivation for the
construction of new physical theories. Extra dimensional theory is one of many attempts to
extend the SM and solve the hierarchy problem. Although the first idea, the Kaluza Klein
(KK) theory, has difficulty in phenomenology, the KK idea is used as a base of the
following modern theories.
The Randall-Sundrum (RS) model is one of many attempts to extend the SM and solve
the hierarchy problem, one of theoretical drawbacks of SM [1]. The RS setup involves two
three-branes bounding a slice of 5D compact anti-de Sitter space taken to be on an
S Z / 2 orbifold. Gravity is localized UV 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 ( ), corresponding to the quantum
fluctuations of the distance between the two 3-branes [2]. The radion mass is considered in
the range of (10 GeV) ≤ m ≤ (TeV) [3]. The common origin of the radion and KaluzaKlein gravitons means a sensitivity to brane curvature terms. The radion couples with the
matter via the trace of the energy momentum tensor. Therefore, the structure of the coupling
of the radion with the SM fields is similar to that of the Higgs boson. The mixing of Higgs
with another particle will change the Higgs production or decay patterns. General
covariance allows a possibility of mixing between the radion and the Higgs boson.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0035
Mathematical and Physical Sci. 2017, Vol. 62, Iss. 8, pp. 89-95
This paper is available online at
PRODUCTION OF HIGGS IN TWO PHOTON COLLISION
Bui Thi Ha Giang and Dao Thi Le Thuy
Faculty of Physics, Hanoi National University of Education
Abstract. The Higgs production was studied using the Randall-Sundrum model from photon-
photon collision. The independent parameters in the Randall-Sundrum model: the vacuum
expectation value of the radionfield , the radion mass m , the Higgs mass hm and
must be specified to fix the state mixing parameters. The total cross-section has been evaluated
as the function of the parameters , , the collision energy s and the result shows the
advantageous direction to observe the Higgs boson.
Keywords: Higgs production, photon collision, Randall-Sundrum model.
1. Introduction
The Standard model (SM) of particles can be used to describe the elementary particle
picture. However, some theoretical drawbacks in SM have been motivation for the
construction of new physical theories. Extra dimensional theory is one of many attempts to
extend the SM and solve the hierarchy problem. Although the first idea, the Kaluza Klein
(KK) theory, has difficulty in phenomenology, the KK idea is used as a base of the
following modern theories.
The Randall-Sundrum (RS) model is one of many attempts to extend the SM and solve
the hierarchy problem, one of theoretical drawbacks of SM [1]. 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 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 ( ), corresponding to the quantum
fluctuations of the distance between the two 3-branes [2]. The radion mass is considered in
the range of (10 GeV) ≤ m ≤ (TeV) [3]. The common origin of the radion and Kaluza-
Klein gravitons means a sensitivity to brane curvature terms. The radion couples with the
matter via the trace of the energy momentum tensor. Therefore, the structure of the coupling
of the radion with the SM fields is similar to that of the Higgs boson. The mixing of Higgs
with another particle will change the Higgs production or decay patterns. General
covariance allows a possibility of mixing between the radion and the Higgs boson.
Received July 17, 2017. Accepted August 20, 2017.
Contact Dao Thi Le Thuy, e-mail: thuydtl@hnue.edu.vn
Bui Thi Ha Giang and Dao Thi Le Thuy
90
The mixing of radion with Higgs will modify the Higgs and radion phenomenology
significantly [4]. In Ref [5], the authors show that the Higgs-dominated state at 125 GeV, in
which the Higgs signal is discovered by the ATLAS and CMS collaborations [6, 7], is
preferred to the radion-dominated state at 125 GeV. Additionally, the associated production
of Higgs bosons with the radion was also studied in Ref [8–12].
2. Content
2.1. A review of Randall - Sundrum model
The RS model is based on a 5D spacetime with non - factorizable geometry [1]. The
single extra dimension is compactified on an 1 2/S Z orbifold of which two fixed points
accommodate two three-branes (4D hyper-surfaces): the UV brane and the IR brane. 1S is a
sphere in one dimension and 2Z is the multiplicative group {-1, 1}. The background metric
reads:
2 | |2 2 2 ,c
kr
cds e dx dx r d
(1)
with ( 0,1,2,3)x the coordinates on the 4D hyper-surfaces of constant c, r the
compactification radius and k the bulk curvature. The four dimensional effective action is
obtained by integrating out the extra dimension.
The Higgs action can be shown as
2
2
4 ,c
kr
HS d x D H D H H H e
(2)
where is a mass parameter, the Higgs field .c
kr
H e H
The gravity-scalar mixing is described by the following action
4 ˆ ˆ ( ) ,vis visS d x g R g H H
(3)
where is the mixing parameter [13–16], ( )visR g is the Ricci scalar for the metric
2 ( )( )vis bg x h
induced on the visible brane, 01ckrb x e
is the
warp factor,
0 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 0 , there is neither a pure Higgs boson nor pure radion mass eigenstate. This
term mixes the
0h and 0 into the mass eigenstates h and as given by [12]
0
0
1 6 / cos sin
,
0 1/ sin cos
h Z h d c h
Z b a
(4)
where 2 2 2 21 6 1 6 36 Z is the coefficient of the radion kinetic term after
undoing the kinetic mixing , /v , 246 GeV.
The mixing angle is
Production of Higgs in two photon collision
91
0
0 0
2
2 2 2 2 2
tan 2 12
36
h
h
m
Z
m Z m
,
(5)
where
0h
m and
0
m are the Higgs and radion masses before mixing.
The new physical fields h and in (4) are Higgs-dominated state and radion, respectively
[18]
0 0 0 0 0 02 2 2 2 2 2 2 2 2, 2
1
[ ] 4
2
h h h hm m m m m Z m m
Z
.
(6)
Therefore, there are four independent parameters , , , ,hm m that must be specified to fix
the state mixing parameters.
For the massless gauge bosons such as photon and gluon, there are no large couplings to
Higgs (or radion) because there are no brane-localized mass terms. However, these
couplings may come from the loop effects of the gauge bosons. We lay out the necessary
Higgs - photon coupling
1
2
h hc hF F
L ,
(7)
with
2 2( ) ,
2
i
h i c i i Y
i
c d b e N F b b b
(8)
where
2 19 / 6, 41/ 6Yb b [12].
The auxiliary functions of the h are given by
1/2 ( ) 2 1 1 ,F f (9)
1( ) 2 3 3 2 ,F f (10)
with
2
2
1
arcsin ( 1)
,
1 1 1
ln ( 1)
4 1 1
f
i
(11)
2
2
4 .ii
h
m
m
(12)
im is the mass of the internal loop particle (including quarks, leptons and W boson).
Here,
2 2
W
W2 2
4 , 4
f
f
h h
m m
m m
denote the squares of fermion and W gauge boson mass ratios,
respectively.
Bui Thi Ha Giang and Dao Thi Le Thuy
92
2.2. The creation of Higgs in the two photon collision
In this section, we consider the photon collision process
1 2 1 2( ) ( ) ( ) ( ),p p h k h k (13)
Here ,i ip k (i = 1,2) stand for the momentums. There are three Feynman diagrams
contributing to reaction (13), representing the s, u, t-channels exchange depicted in Fig.1.
Figure.1. Feynman diagrams for hh collision
We obtain the scattering amplitude in the s, u, t-channels, respectively
1 1 2 1 2 22 2 2 2
( ) ( ) ( ),
hh hhh
s h
s s h
g g
M C C p p p p p p
q m q m
(14)
2
2 2 2 1 1 12
1
( ) ( ) ( ) ( ),u h u u u u
u
M iC p q p q p p q p q p
q
(15)
2
1 1 1 2 2 22
1
( ) ( ) ( ) ( ),t h t t t t
t
M iC p q p q p p q p q p
q
(16)
where C , hC , hhg , hhhg are given by [12],
1 2 1 2 ,sq p p k k 1 2 1 2 ,uq p k k p 1 1 2 2tq p k k p .
The expressions of the differential cross-section [18]
21 | | | | ,
(cos ) 32 | |
fi
d k
M
d s p
(17)
where
1/2
2| | , | | ,
4 2
h
s s
k m p
2 2 2 2| | | | | | | | 2Re( ).fi s u t s u s t u tM M M M M M M M M M
We evaluate the
dependence of the differential cross-section on cos ( 1 1( , )p k the angle between
momenta of the initial photon and the final Higgs) and the dependence of the total cross-
section on the collision energy s , VEV of the radion and .
We give some estimates or the cross-sections in the process hh as follows
i) In Fig.2, we choose 125hm GeV (CMS), 5TeV , 1/ 6, 110m GeV
[2]. We plot the differential cross-section as the function of cos at the fixed
collision energy, 3s TeV (CLIC). The result shows that the differential cross-
section reaches maximum value when cos is about 0. Therefore, the advantageous
direction to collect Higgs is perpendicular direction to the initial photon beam. In
Ref [19], we worked the radion production in high energy photon collisions. The
Production of Higgs in two photon collision
93
cross-section in hh is about 6 times as large as that in in the same
condition.
Figure 2. The differential cross-section in hh as a function of cos with
125hm GeV , 5TeV , 1 / 6, 110m GeV , 3s TeV
ii) We plot the total cross-section as a function of the collision energy s in Fig.3. The
parameters are chosen as Fig.2. The total cross-section achieves the minimum value
0.0268 fbar at 1380s GeV and increases in the high energy region. In the region
3 5 ,TeV s TeV the cross-section is as a linear function of the collision energy.
Figure 3. The total cross-section in hh as a function of the collision
energy s with 125hm GeV , 5TeV , 1 / 6, 110m GeV
iii) In Table 1, we give the dependence of the total cross-section on . The range
is chosen as 1 5TeV TeV . We choose 125hm GeV (CMS), 110m GeV ,
1/ 6 , 3 .s TeV The total cross-section has a maximum value at 1TeV .
Table 1. The total cross-section in hh as a function of .
(TeV) 1 2 3 4 5
( )fbar 21.0198 1.1807 0.2649 0.0961 0.0475
Bui Thi Ha Giang and Dao Thi Le Thuy
94
iv) To have 2 0,Z must lie in the region
2 2
1 4 1 4
1 1 1 1
12 12
.
We choose 125hm GeV (CMS), 110 ,m GeV 3 ,s TeV 5TeV corresponding to
the range 3.31 3.47 . We plot the dependence of the total cross-section on in Fig.4.
The result shows that the total cross-section decreases in the region 3.31 0 and
increases in the region 0 3.47 . We also give some of the total cross-section value in
Table 2. The total cross-section achieves the minimum value ( 0.004608fbar ) in case
of 0.1146 .
Figure 4. The total cross-section in hh as a function of with 125hm GeV ,
5TeV , 110 ,m GeV 3s TeV .
Table 2. The total cross-section in hh as a function of .
-3 -2 -1 -1/6 0 1/6 1 2 3
( )fbar
4495
2
421.66
2
5.650
1
0.005
2
0.009
4
0.047
5
6.24
9
597.98
7
14882
6
3. Conclusion
In this paper, we have studied the Higgs production in two photon collision in high
energy. The ability of search the Higgs boson is the perpendicular direction to the initial
photon beam. We have evaluated the dependence of the total cross-section on the
parameters , ,s . Numerical evaluation gives observable cross-section in the future
colliders.
Acknowledgement: The work is supported in part by Hanoi National University of
Education project under Grant No. SPHN-16-05.
Production of Higgs in two photon collision
95
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