Sự sinh radion từ va chạm vật lý

2
LỜI CẢM ƠN


  


 
. 
  viên    
mình.

 r


   
Cao Nguyên     


L              
công.  mình.
09 4
H


3

Contents
 6


MÔ HÌNH RANDALL  SUNDRUM 9
 9
 10
 17
 18
 20
 Wise 21
 24


γμ

KHI CHÙM
μ

 24
 24
 24
 25
 27
 Feynman theo kênh u 27
 27
 29
 29


4
 29
2.4.  

kênh u 31
 32


-
γμ
KHI CHÙM
-
μ
 33
 33

 33

 34
 35
 
 35

 36
ion theo kênh t 37

trái 37

 38
3.4.  39
3.4.1. 
 39

5

 40
 40



-
γμ
41
4.1.  41
 45
 47
 49













6
MỞ ĐẦU
1. Lý do chọn đề tài




trong              





 








 


  
               


7
Randall  

Mô hình Randall  


Sự sinh radion từ va chạm


” làm

2. Mục đích nghiên cứu
 






3. Phương pháp nghiên cứu
- 

- 
4. Đối tượng và phạm vi nghiên cứu
- 


.
-
                


sinh
radion.
5. Ý nghĩa khoa học và thực tiễn của luận văn



 Sundrum.


8
6. Bố cục khóa luận
 4 
 Sundrum.



khi
chùm
-
μ


Bình 


khi chùm
-
μ
phân 
IV: 



.

9

CHƯƠNG I
MÔ HÌNH RANDALL – SUNDRUM
1.1 Tác dụng và khoảng bất biến của mô hình
  

1
. Không  
th de Sitter space).

2

μ
(x , )

và
μ

(x ,- )

là

1
/ Z
2
chín
0



và
π


 
0


trong Brane
IR  Brane, SM  Brane,
hay TeV  
π



         

μ

(x , )


2 M N
MN
ds = G dx dx


μ ν μ 2
μν μ
= G dx dx + 2G dx dx + G d

 

. (1.1)

MN


μ
G

old, nên lúc này ta
có:
2 μ ν 2
μν
ds = G dx dx + G d


.

(1.2)
I
vis μ
μν MN
g = G (x , = π)

và
hid μ
μν MN
g = G (x , = 0)

.


10

gravity vis hid
S = S + S + S
. (1.3)
 
t
π
43
gravity
-π
S = d x d G( Λ 2M R)




(1.4a)
4
vis vis vis vis
S = d x g ( L V )

(1.4b)
4
hid hid hid hid
S = d x g ( L V )

(1.4c)

MN
,  

1.2 Lời giải phương trình Einstein và khoảng bất biến trong trường hợp cổ
điển
               

vis
= L
hid
= 0, còn V
vis
và V
hid





 


π
4 3 4
vis vis vid vid
-π
S = d x d G( Λ 2M R) d x( g V g V )

     
  


π
43
-π
= d x d G( Λ 2M R)








11

vis vis hid vid
gVδ( π) g V )δ( )



    

. (1.5)
  = 0
và  = 
MN
:
π
43
MN
-π
δ
S = δS = d x d G ( Λ 2M R)
δG







MN
vis vis hid vid
gVδ( π) g V )δ( ) δG .


    

(1.6)

Ta có:
MN
MN
GG
δG
δG 2

,
MN
MN
MN
GR
δ GR
(R ) G
δG 2

,
μν is
is
μν
MN
MN
Gg
δg
δδ
δG 2
v
v




,
μν hid
hid
μν
MN
MN
Gg
δg
δδ
δG 2



.

δS 0
có

MN
MN MN
3
GR
1
G(R ) GG
24M

   


vis μν
vis vis μν M N
+ V g g δ δ δ( π)




hid μν
hid hid μν M N
+ V g g δ δ δ( )


. (1.7)


12

μ ν μ ν
μν μν
G dx dx f( )η dx dx

,
f( )




2ζ( )
f( ) e





(

2ζ( )
μν μν
G = ηe


,

(1.8)




và


μν
η = diag( 1, 1, 1, 1)
và
2 2 2
C
G d = r d




hay
2
C
G = r



c
g
 
c


2ζ( )
2ζ( )
2ζ( )
MN
2ζ( )
2
C
e 0 0 0 0
0 e 0 0 0
G =
0 0 e 0 0
0 0 0 e 0
0 0 0 0 r



















. (1.9)
22ζ( ) μ ν 2 2
μν c
ds = e η dx dx r d




. (1.10)
Ta có:
vis μ
μν MN
g = G (x , = π)

μν
= G ( = π)





2ζ(π)
2ζ(π)
2ζ(π)
2ζ(π)
000
e
00
0e
=
0
0 0 e
0 0 0 e











2ζ(π)
μν
= ηe


. (1.11)

hid μ
μν MN
g = G (x , = 0)

μν
= G ( = 0)




13

2ζ(0)
2ζ(0)
2ζ(0)
2ζ(0)
e 0 0 0
0 e 0 0
=
0 0 e 0
0 0 0 e












2ζ(0)
μν
= ηe

. (1.12)

28ζ( ) 4
MN C C
G = detG = r e G r e




p p p p k p k
μν μpν p μν ν μp pk μν νk pμ
R = R = Γ Γ + Γ Γ Γ Γ   

p p p p k p k
p p p pk k p
R = R = Γ Γ + Γ Γ Γ Γ
        
   

p p p p k p k
μ μp p μ μp pk μ k pμ

R = R = Γ Γ + Γ Γ Γ Γ
     
   

p p p p k p k
μ pμ p μ ν p pk μ μk p
R = R = Γ Γ + Γ Γ Γ Γ
     
   

A AD
BC C DB B DC D BC
1
= G ( G G G )
2
     


μν μν
-2ζ( ) 2 -2ζ( )
μν
22
cc
2
μμ
ηη
R= ζ"( )e + 4ζ' ( ) e ,
rr
R4ζ"( ) ζ' ( ) ,
R =R =0.



















(1.13)

MN μν
MN μν
R = G R = G R + G R



μν μν
μν 2ζ( ) 2ζ( ) 2 2ζ( )
22
cc

ηη
R= η e ζ"( )e + 4ζ' ( ) e
rr
  










2
2
c
1
4 ζ"( ) ζ' ( )
r







14
2
2

2 2 2 2
c c c c
4 16ζ' ( ) 4 4
R= ζ"( ) + ζ"( ) ζ' ( )
r r r r

  

  



2
2
c
4
R = 2ζ"( ) 5ζ' ( )
r


  

( 1.14)

a có:
μν μν
24ζ( )
c
RG
00

1
r e R
RG
2
00

 


   


   

   


μν
4ζ( ) 4ζ(π) 2ζ(π) μ ν
c vis μν M N
3
G
0
1
r e V e η e δ δ δ( π)
G
4
0
M







   






4ζ(0) 2ζ(0) μ ν
hid μν M N
Ve η e δ δ δ( )


. (1.15)






24ζ( ) 4ζ( )
cc
3
11
r e (R G R) = r e G
2 4M


  

  

2
23
c
6ζ' '
=
r 4M



(1.16)


μν


24ζ( ) 4ζ( )
c μν μν c μν
3
11
r e (R G R) = ( r e G
2 4M


  

2ζ(π) 2ζ(0)
vis hid
+ V e ( ) + V e δ( )
   

. (1.17)

hid vis
2 3 3
c c c
VV
3 "( )
= δ( ) δ( π)
r 4M r 4M r



. (1.18)

(1.16) và (1.18).


15


2
23
c
6ζ' ( )

=
r 4M



 
ζ > 0
ζ( ) = ζ( 2π)
ζ( ) = ζ( )









(1.19)
,  

2
3
Λ
k = (k 0)
24M


(1.20)


c
ζ = kr + C

.

c
ζ(0) = 0 C 0
ζ(π) = kr π






c
ζ = kr

. (1.21)
 2), ( -, 
 



0
π
2π
-
π
-2π
σ(π )

σ


16

Hình 1.1
 Xét chu kì (-, ), t
c
c
ζ' = kr sign( )
ζ" = kr sign'( )







1 khi > 0
sign( ) =
-1 khi < 0







Và
sign'( ) = 2ζ( )




( π, π)


ta có:
c
ζ'' = 2kr δ(φ)
(1.22)
 Xét chu kì (0, 2), t
cc
ζ = kr ( π) + kr π



Nên
c
ζ" = 2kr ( π)


. (1.23)

c
ζ'' = 2kr [δ( ) δ( π)]


. (1.24)


17
2
c
c hid
33
c hid
23
c vis
c vis
3
c
r
2kr = V
12M r V = 24kM
r V = 24kM
2kr = V
12M r














(1.25)

23
= 24k M

c
-2kr
2 μ ν 2 2
μν c
ds = e η dx dx r d



. (1.26)
1.3 Khối lượng Planck trong 4D

c



2kT(x)
2 μ ν 2 2
μν
μν
ds = e η h ( ) dx dx T (x)dx







, (1.27)

μν
h


 
μν
G


μν
μν μν
g (x) = ηh
(1.28)
            
c
là VEV (vacuum

EV r
c

-4
eV

c

π

43
gravity
π
S = d x d G( 2M R)




(1.29)
Ta có:
2kr
2kr
μν μν
R e R
G = e g
c









(1.30)


18


R

μν
g (x)
và
4kr
c
MN
MN
G gr e
R = G R
c








(1.31)

π
4kr 2kr
43
gravity c
π
S = d x d gr e ( 2M e R)
cc







. (1.32)

42
gravity P1
S = d x g2M R

. (1.33)

c
π
2kr
23
P1 c
π
M = r M d e







c
2kr
3

c
c
π
1
= 2r M ( )e
0
2kr





c
3
2kr π
M
= (1 e )
k


. (1.34)
r
c

 

1.4 Khối lượng Higgs

trên 3  



c
2kr π
vis
μν μν
g = e g

(1.35)


19
Ta có:
2
4 μν + 2 2
Higgs vis μ ν 0
S = d x g g (D H) (D H) λ( v )H

  


(1.36)

0
, do
c
c
c
4kr π
vis
2kr π

vis
μν μν
2kr π
μν μν
vis
g e g
g e g
g e g



  








(1.37)
Nên ta có:
c
2
2kr π
4 μν + 2 2
Higgs vis μ ν 0
S = d x g e g (D H) (D H) λ( H v )

  



(1.38)


c
c
2kr π
phys
2kr π
22
0
H e H
v = v e







(1.39)
 sóng ta có:
4 μν + 2 2
Higgs μ phys ν phys phys phys 0
S = d x g g (D H ) (D H ) λ(H H v )


  



.
(1.40)


c
kr π
0
v e v


. (1.41)


c
kr π
0
m e m


(1.42)

0

0
= M
pl
= 10
19
GeV thì

m 1TeV
.




20
m
0
nào trên 3  

1.5 Tại sao phải cần có Orbifold
 


nhóm Lorentz. Trong không - 




 
M N MN
, = 2η
(1.43)



ây là:
5 5 0 1 2 3
iγ γ γ γ γ   

. (1.44)

5


0 1 2 3 4 5
γ γ γ γ γ γ 1
, nói cách khác




Hình 1.2: C
0
















0






0








21


2
[7]
các hàm Z
2

LR
Q(x, ) = Q (x, ) + Q (x, )
  


(n) (n)

LR
n=0
Q (x)cos(n ) Q (x)sin(n )






LL
RR
Q (x, ) = Q (x, )
Q (x, ) = Q (x, )








. (1.45)
  
L
   
L
trong MHC
.
Tuy nhiên mode
không 

R



cc
LL
cc
LR
cc
RR
q ( ) = q ( )
q = q + q
q ( ) = q ( )










. (1.46)



1.6 Cơ chế Goldberger – Wise

V

hid
và V
vis

bán kính compact r
c
có giá 
 

2kT(x)
2 μ ν 2 2
μν
ds = e η (x)dx dx T (x)d




(1.47)

ó  



22

b. Lý 
 

Trong  


               
Goldberger   

(x,y)



π
4 MN 2 2 2 2
M N h h
-π
1
S = d x d G g Φ Φ m λ ( v ) δ( )
2



      




2 2 2
vv
λ ( v ) δ( π)


   

(1.48)


MN


h hid
λV
,
ν vis
λV


4ζ
2 2 2 2 2
v v h h
2
e δ( π) δ( )
( ) m = 4λ ( v ) +4λ ( v )
T (x) T(x) T(x)


         

(1.49)
g
T(x)



2ζ vζ vζ
( ) = e Ae + Be






, (1.50)

2
2
m
v = 4 +
k
.




23
2 2vkT(x)π 2 2vkT(x)π
V [T(x)] = k(v + 2)A [e 1] + (k 2)B [1 e ]


  


2 2 2 4kT(x)π 2 2 2
h h v v
+ λ (Φ (0) v ) + λ e (Φ (π) v )



. (1.51)
  có
 và các brane.

22
hh
k[(v 2)A (2 v)B] 2λ Φ(0)(Φ (0) v )=0    
(1.52)

2kT(x)π vkT(x)π vkT(x)π 2 2
vv
ke [(2 ν)e A (2 v)e B] 2λ Φ(π)(Φ (π) v )=0

    

(1.53)
        () có trung bình chân không trên các
brane sao cho () = (0) = v
h
và () = v
v



h
, 
v


(2 v)kT(x)π 2vkT(x)π

vh
A = v e e v ,
  

(1.54)
(2 v)kT(x)π 2vkT(x)π
vh
B = v e (1 e )v ,
  
  
(1.55)
 
-kT(x)
  
2
2
m
4k
ò


4kT(x)π kT(x)π 2
vh
V [T(x)] = 4ke (v v e )



ò

2 4kT(x)π kT(x)π 2
h h V h
k[v v e (v v e )

  
ò
ò



(4 )kT(x)π kT(x)π 2
h v h
V e (2v v e )] ( )
  
  
ò
ò
. (1.56)


2
h
2
v
v
4k
<T(x)> = r = ln( ).
πm v
(1.57)

24

2
2
m
k

-1
( hàm 
kr 10


1.7. Kết luận
Mô hình Randall  Sundrum  

, 





CHƯƠNG II
BÌNH PHƯƠNG BIÊN ĐỘ TÁN XẠ CỦA SỰ SINH RADION TỪ VA CHẠM
γμ

KHI CHÙM
μ

CHƯA PHÂN CỰC

         radion   
γμ

khi các
chùm
μ

 


các kênh này.
2.1. Sự sinh radion theo kênh s
2.1.1. Giản đồ Feynman theo kênh s
muon và photon
radion và muo
       
1 2 1 2
μ p + γ p μ k k



,


25

1
, p
2


1
, k
2



μ









Hình 2.1. 
-
μ


-
γμ

2.1.2. Bình phương biên độ tán xạ
này là:
 
s μ
μ
s1 μ 2 1

22
s μ

q + m
1
M u(k ) 1 ε (p ) ieγ u(p )
λ' q - m




 
 
μ
μ 2 1 s μ 1
22
s μ
-ie

ε (p )u(k ) q + m γ u(p )
λ' q - m


, (2.1)

s 1 2
  
q = p + p
.


s
:

 
 
+* ν
s ν 2 1 s μ 1
22
s μ
ie

M ε (p ) u(k ) q + m γ u(p )
λ' q - m







 
 
* ν
ν 2 1 s μ 1
22
s μ
ie

ε (p )u(p )γ q + m .u(k )
λ' q - m



. (2.2)


2
+
s s s
spins,pol
M M M


 
1
μk


 
2
k


 
1
μp


 
2
γp

 
s
μq




26

 
2
μν
2
2 2 2
s μ
( g )
λ' q + m
e




   
μν
1s μ 1 1 s μ 1

u(k ) q + m γ u(p )u(p )γ q + m u(k )

 

   
 
2
μ
1 μ s μ 1 μ μ s μ
2
2 2 2
s μ

  
Sp (k + m ) q + m γ (p +m )γ q + m .
λ' q + m
e



(2.3)

   
 
μ
1 μ s μ 1 μ μ s μ

  
I Sp (k + m ) q + m γ (p + m )γ q + m
.
Ta có:

2
1s μ s μ 1 μ



I Sp (k q + m q + m k m )



μ μ μ 2 μ
1 μ s 1 μ μ μ μ s μ μ
   
(γ p γ q + γ p γ m + m γ γ q + m γ γ )


   
 
μ 2 μ 2 μ
1 s 1 μ s μ 1 s μ μ s 1 μ

     
Sp k q γ p γ q + m Sp k q γ γ + m Sp q γ p γ


 
   
2 μ 2 μ 2 μ
μ s μ s μ 1 1 μ μ 1 μ s

   
+ m Sp q γ γ q + m Sp k γ p γ + m Sp k γ γ q

   
2 μ 4 μ
μ 1 μ s μ μ

+ m Sp γ p γ q + m Sp γ γ


 
   
2 2 2
1 s 1 s s 1 1 μ 1 s μ s 1
8 2(k q )(p q ) q (k p ) +16 m k q 8m q p   


2 2 2 2 2 4
μ s μP 1 1 μ 1 s μ 1 s μ
+ 16m q 8m (k p )+ 16m (k q ) 8m (p q )+ 16m


 
2 2 2
1 s 1 s s 1 1 μ 1 s μ s 1
8 2(k q )(p q ) q (k p ) + 32 m (k q ) 16m (q p )   


2 2 2 4
μ s μ 1 1 μ
+ 16m q 8m (k p )+ 16m
.
3

 

2
2
22
s 1 s 1 s s 1 1 μ 1 s
2
2 2 2
s μ
M 8 2(k q )(p q ) q (k p ) + 32 m (k q )
λ' q + m
e


  



2 2 2 2 4
μ s 1 μ s μ 1 1 μ
16m (q p )+ 16m q 8m (k p )+ 16m
. (2.4)