The Amplitude of the Сompton
Scattering of the Low-Energy
Photons at Quark-Antiquark
System on the Basis of the
Bethe-Salpeter Equation with
Instantaneous Interaction.
Maksimenko N.V., Vakulina E.V., Deryuzkova О.М. Kuchin S.М.
GSU, GOMEL
In the basis of the low-energy theorems for interaction of the photons with
hadrons lay common relativistic principles of quantum field theory.
Therefore these theorems concern to the fundamental theorems.
At the same time hadrons are composite quark systems. Therefore the
research of the process of the photons interaction with quark systems
allows to obtain new information on structure of hadrons in the region of
energies, where methods of the theories of perturbation quantum
chromodynamics can not be sequentially used.
For the description of properties of bound systems in quantum field theory
at present time the covariant relativistic Bethe-salpeter equation, is mainly
applied. However its practical use is associated with a lot of difficulties.
Therefore, solution of the concrete tasks and physical intrpretation of the
obtained solutions are limited to reduction of BSE three-dimensional form,
in which the vertex function of the system of two particles is projected on
the equal times surface. Such approach has shown to be effective and for
the description of bound states of particles in an external electromagnetic
field.
In the given work the Сompton
scattering amplitude on the two quark
system at low energies is obtained on the
basis of the diagram approach in the
simultaneous approximations of the
meson-quark transition vertex functions,
that allows to express factors of lowenergy decomposition of the amplitude in
frequence of radiation trough wave
functions of compound system.
In the given work, the tensor of the Сompton scattering
at quark-antiquark system in the relativistic impulse
approximation can be presented as follows

^
d4p
'
''
Tv  
Sp
[

S
(
p
)(
iQ

)
S
(
p
1
1 
1 )(iQ1 v ) S ( p1 ) Г S ( p2 )] (1)
4
(2 )
In this expression the following denotations are entered:
S ( pi ) is the quark propagator,
Q1 is the quark charge with which a photon interacts,

is the Dirac matrix,
Гˆ и Гˆ   0 Гˆ  0
p1, p2 , k1
и
k2
are vertex functions of meson-quark transition at the initial and
terminal states,
are impulses of quark, antiquark, falling and dispersed
accordingly.
On the basis of conservation laws of four impulses also the
following denotations are entered
p  p1  k1 and p  p1  k1  k2
''
1
.
Let us introduce variables
from which it follows, that
'
1
p1  p 2
p
,   p1  p 2
2
Р
Р
p1 
 p, p 2 
p
2
2

Let .us take into consideration that the vertex function
doesn’t . depend on p0
Such approximation is valid in the case when
the potential of iteraction between quarks
doesn’t depend on p0

Let us select an integral in (1)
dp0 ' ' 


'
~


ˆ
ˆ
 adlm   S ac ( p1 )( a )cd S de ( p1 )( a )ef S fg ( p1 ) Slm ( p2 ) (2)
2i
where the
 indexes a, c, d, e, f, g and 1 take values 1, 2, 3, and 4 and the
matrix a   iQ1 
In fact, the function ~, defined in the expression (2),
is a Green two particles function in the two-times
approximation of the photon scattering on one of the quarks.
Let us put down the equation (2) in a more compact form.
.


~

dp0  '   ''  

S ( p1 )aˆ S ( p1 )aˆ S ( p1 )  S ( p2 )
2i
(3)

~

in the low energy approximation,
Now let us calculate the tensor
that is we take into consideration that the impulse of the photon k is a little
value in comparison with rest energy of a quark. Then with the help of the
relation



 
S ( p1  k )  S ( p1 )  S ( p1  k ) k S ( p1 )

(4)

'
1
we can transform the propagators S ( p ) and S ( p1'' )
in the expression (3) on the four-measured impulse k.
First let us inspect a more simple expression of the form
dp0 
'
S
(
p
1 )  S ( p 2 )
 2i
(5)

'
Being limited by the second order of transformation S ( p1 )
on k with the help of the rellation (4) we obtain
~




 


 
S ( p1  k )  S ( p1 )  S ( p1  k ) k S ( p1 )  S ( p1 ) k S ( p1 ) k S ( p1 )  ...
(6)
In this case the equation (5) has the form (7)





dp0
~
 (k )   [S ( p1 )  S ( p1 ) k S ( p1 )  S ( p1 ) k S ( p1 ) k S ( p1 )  ...]  S ( p2 )  ~ (0)  ~ (1)  ~ (2)  ..
2i
To calculate
~
(n)

let us present the propagators

S ( p1 ) and S ( p2 )
can be presented as follows:

 
() 
( )

m  p1
 ( p1 )
 ( p1 )
m
S ( p1 )  2
 

 (8)
2
p1  m  i E1  p10  E1  i p10  E1  i 



() 
()

m  p2
 ( p2 )
 ( p 2 ) 
m
S ( p2 )  2




2
p 2  m  i E 2  p 20  E1  i p 20  E 2  i  (9)

The greatest contributions to the integrals
~ ( n )
in (7) are given by the products
of the terms of propagators of the forms (8) and (9):
() 

( p1 )
m
()
ˆ
S ( p1 ) 
E1 p10  E1  i
(10)
( ) 

( p2 )
m
()
ˆ
S ( p 2 ) 
E2 p20  E2  i
,
(11)
In the right part of the expression (7) we use the Gordon identity
  () 
1 () 


() 
 ( p1 )  (q ) 
 ( p1 )[( p1  q)  i ( p1  q) ] (q )
. 2m
()
where
  
i  
(       )
2
Using (12), (10) and (11) the expression
can obtain the form (13)
~ ( k )
(7)
(12)
2
2

 () 
m
(
p
k
)
(
p
k
)
( ) 
~
1
1
 (k )  
 2
 ( p1 )   ( p2 )
1 
2
E1E2 ( P0  E1  E2 )  E1 ( P0  E1  E2 ) E1 ( P0  E1  E2 ) 
where
  2
2
2
p1k  E1  p1k , p1  E1  p1  m 2
.
In the transformation (13) in brackets there is the sum of the geometric
progression


1
n n
  (1) x   ( x) n
(14)
1 x
n 0
Using (14), the Green function (13) can be presented in the form of


m   ( p1 )    ( p 2 )
~
 (k )  (1)


( p1k )
E1 E 2 ( P0  E1  E 2 ) 1 
 (15)
 E1 ( P0  E1  E 2 ) 
2
With the help of the Green function (15)let us set the relation of the vertex
function ( q q ) - with the simultaneous wave function of the system ( q q )

 
dp0
'
' dp0
'
( P , P, p)   ˆ ( p1 , p2 ,P )
  S ( p1 )  S ( p2 )
2i
2i

where
'
0
(16)
p1'  p1  k , P '  P  k.

Let us take into consideration that the vertex function  doesn’t depend on
. In this case according to (15) we obtain




(1)  m    ( p1 )    ( p 2 )

Ф( P0' , P , p ) 


( p1 k )
E1 E 2 ( P0  E1  E 2 ) 1 

E
(
P

E

E
)
1
0
1
2 

2

(17)

Let the vertex
p0

have the form


'
ˆ
Г  g ( P0 , P, p) 5
(18)

Then The function of the  expression (17) can be presented as follows
 
 


'
( P , P , p )   ( P0 , P , p )  ( p1 ) 5   ( p 2 )

'
0
(19)
In the equation (19) the function is defined
 
'
( P0 , P, p ) 
 
(1)  m g ( P , P, p )


( p1k )
E1 E2 ( P0  E1  E2 ) 1 

E
(
P

E

E
)
1
0
1
2 

2
'
0
With the help of the bond of the scalar part of the vertex function
 
and the wave function ( P , P, p )
'
0
(20)
 
g ( P , P, p)
'
0
we can calculate a form-factor
of the compound system in the relativistic impulse approximation
and define normalization of the function 
Indeed, if the

 e
is the amplitude of interaction of a photon
( )
with the compound system ( e is a vector of polarization of a photon)
.

can be presented as follows

dp0
dp
 ˆ


(TP)  ( P 2  PP)QF (k 2 )  Q(P)  
g
g
Sp
(

S
(
p
5
1 ) PS ( p1 ) 5 S ( p 2 )) (21)
3

2i
(2 )
2
In the equation (21) F ( k )
functions g and
g'
is a formfactor of the system, and the vertex
have the following form
(1) E1 E2 ( P0  E1  E2 )

g
 ( M , p)
2
m
(22)


( p1k )

(1) E1 E 2 ( P0  E1  E 2 )1 
 
E1 ( P0  E1  E 2 ) 

'
'
g 
  ( P0 , P , p )
2
m
If in the equation (21) we take into consideration the evident form of the
functions g and
g ',(22) and (23) accordingly, in such case
(23)

 
 ( p, P) E2
dp
*
'
() 
() 


(P)  Q 

(
P
,
P
,
p
)

(
M
,
p
)

Sp
(

(
p
)

( p2 )
0
1
3
2
(2 )
m
(24)
In the system of rest target the following cinematic relations are valid


'  '  k 

P  0, P  k , p  p  , p1   p 2 , E1  E 2  E
2
m2 M '
Let us redefine in (24) the wave functions 


E1 E 2
(25)
Then the equation (24) can be presented as follows

2
 
dp
2
'*
' 


M (2M   ) F (k )  2M 
 ( P0 , k , p ) ( p)
3
(2 )
From the relation (26) is seen that under
k 0
the condition of the normalization of function

dp
'

 (2 ) 3
(26)
2
'
1
(27)
Now with an analogous method we can calculate green function of tensor
of the Compton scattering (1)
The two-times Green function for tensor of the  expression has the form
 () 
(1)  m ( p1 a 2 )( p1 a1 )(  ( p1 ) ( p 2 )
~(a 2 , a1 ) 


p1Q
( p1 k )
3
3
1 

E1 E 2 ( P0  E1  E 2 ) 1 
 E1 ( P0  E1  E 2 )  E1 ( P0  E1  E 2 ) 
2
()
In the expression (28) the following indications are introduced:

2

1 2

1

1 1
a  iQ e , a  iQ e , Q  k1  k 2
e2 , e1, k2 , k1 are the vectors of polarisation and the impulses of the dispersed
and falling photons accordingly.
(28)
In the system of rest target the amplitude of the Compton scattering
in the impulse approximation has the form

 

d
p
2M ( p1e2 )( p1e1 )
2
2
'


e2e1  i(Q1  Q2 ) 
 ( P0 , P , p ) 
 ( M , p) (29)
3
p
k
(2 )
1 1
E 2 ( M  2 E )(1 
)
E (M  2E )
In formula (29)
Q1 and Q2 are the quark charges,

'  '  Q
P  Q, p  p 
2
.
Thus, knowing the wave functions

we can define the amplitude of the
Compton scattering in the field of low energies (   M ), and therefore
also the coefficients of the low-energy transformation,which are bound
with the definite electromagnetic characteristics of the bound system.