V.V. ANDREEV, N.V. MAKSIMENKO, O.M. DERYUZHKOVA

argumentwildlifeUrban and Civil

Nov 16, 2013 (3 years and 9 months ago)

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V.V. ANDREEV, N.V. MAKSIMENKO, O.M. DERYUZHKOVA


The F.
Skorina

Gomel State University


At

present

there

are

many

electrodynamic

processes

on

the

basis

of

which

experimental

data

on

hadrons’

polarizabilities

can

be

obtained
.

In

this

context,

there

is

a

task

of

covariant

determination

of

the

polarizabilities

contribution

to

the

amplitudes

and

cross
-
sections

of

electrodynamic

hadron

processes
.




[
Carlson,

C
.
E
.

Constraining

off
-
shell

effects

using

low
-
energy

Compton

scattering

/

C
.
E
.

Carlson,

M
.

Vanderhaeghen

//



2011
.


http
:
//physics
.
atom
-
ph/
1109
.
3779
.
;


Birse
,

M
.
C
.

Proton

polarizability

contribution

to

the

Lamb

shift

in

muonic

hydrogen

at

fourth

order

in

chiral

perturbation

theory

/

M
.
C
.

Birse
,

J
.
A
.

McGovern

//


2012
.


:

http
:
//

hep
-
ph
/
1206
.
3030
.

]
.

2

3

This

problem

can

be

solved

in

the

framework

of

theoretical
-
field

covariant

formalism

of

the

interaction

of

electromagnetic

fields

with

hadrons

with

account

for

their

polarizabilities
.

In

the

papers

[
Moroz
,

L
.
G
.
,

Fedorov

F
.
I
.

Scattering

matrix

taking

into

account

the

interaction

Pauli

/

//

ZETF
.

-

1960
.

-

39
.

-

Vol
.

2
.

-

P
.

293
-
303
;

Krylov
,

V
.
,

Radyuk

B
.
V
.

,
Fedorov

F
.
I
.

Spin

particles

in

the

field

of

a

plane

electromagnetic

wave

/

//

Preprint

of

the

Academy

of

Sciences

of

BSSR
.

In
-
t

Physics
.

-

1976
.

-



113
.



P
.
60

;

Maksimenko,

N
.
V
.

and

Moroz
,

L
.
G

Polarizability

and

гирация

elementary

particles

///

Problems

of

atomic

science

and

technology
.

Series
:

General

and

nuclear

physics
.

-

1979
.

-



4
(
10
)
.

-

P
.

26
-
27
;

Levchuk
,

M
.
I
.

and

L
.
G
.

Moroz

Gyration

nucleon

as

one

of

the

characteristics

of

its

electromagnetic

structure

///

Proc

Academy

of

Sciences

of

BSSR
.

Ser
.

p
hys
.
-
mat
.

sc
.

-

1985
.

-



1
.

-

P
.

45
-
54
]

one

can

find

covariant

methods

of

obtaining

the

Lagrangians

and

equations

describing

interaction

of

the

electromagnetic

field

with

hadrons,

in

which

electromagnetic

characteristics

of

these

particles

are

fundamental
.

4

Effective

field

Lagrangians

describing

the

interaction

of

low
-
energy

electromagnetic

field

with

nucleons

based

on

expansion

in

powers

of

inverse

mass

of

the

nucleon

have

been

widely

used

recently

[
R
.
J
.
Hill
,

G
.
Lee
,

G
.
Paz
,

M
.
P
.
Solon

The

NRQED

lagrangian

at

order
-

2012
.

-

:

http
:
//

hep
-
ph
/
1212
.
4508
]
.


In

Ref
.

[
Maksimenko,

N
.
V
.

Moroz

L
.
G
.

Phenomenological

description

polarizabilities

of

elementary

particles

in

a

field
-
theory

//

Proceedings

of

the

XI

International

young

scientists

school

on

high

energy

physics

and

relativistic

nuclear

physics
.

D
2
-
11707
,

JINR,

Dubna
.

-

1979
.

-

P
.

533
-
543
]

on

the

basis

of

correspondence

principle

between

classical

and

quantum

theories

an

effective

covariant

Lagrangian

describing

the

interaction

of

electromagnetic

field

with

particles

of

spin

½

is

presented

in

the

framework

of

field

approach

with

account

for

particles’

polarizabilities
.

4
1
M
5

In

this

paper,

in

the

framework

of

the

covariant

theoretical
-
field

approach

based

on

the

effective

Lagrangian

presented

in

[
Maksimenko,

N
.
V
.

and

Moroz

L
.
G
.

Phenomenological

description

polarizabilities

of

elementary

particles

in

a

field
-
theory

//

Proceedings

of

the

XI

International

young

scientists

school

on

high

energy

physics

and

relativistic

nuclear

physics
.

D
2
-
11707
,

JINR,

Dubna
.

-

1979
.

-

P
.

533
-
543
]

a

set

of

equations

describing

the

interaction

of

electromagnetic

field

with

hadrons

of

spin

½

is

obtained

taking

into

account

their

polarizabilities

and

anomalous

magnetic

moments
.

Using

the

Green’s

function

method

for

solving

electrodynamic

equations

[
Baryshevsky
,

V
.
G
.

Nuclear

optics

of

polarized

media



M
.:

Energoatomizdat
,

1995
.

-

315

p
.;

Bogush
,

A
.
A
.

and

Moroz

L
.
G
.

Introduction

to

the

theory

of

classic

fields

/

Minsk
:

Science

and

technology,

1968
.

-

387

p
.;

Bogush
,

A
.
A
.

Introduction

in

the

gauge

field

theory

of

electroweak

interactions

/

Minsk
:

Science

and

technology,

1987
.

-

359

p
.;

J
.
D
.

Bjorken
,

E
.
D
.

Drell
.

Relativistic

quantum

field

theory

/

-

1978
.



Vol
.

1
.

-

295

p
.
],

amplitude

of

Compton

scattering

on

the

particles

of

spin

½

is

obtained

with

account

for

their

polarizabilities
.

Structures

of

the

amplitude

that

are

similar

to

polarizabilities
,

but

are

caused

by

electromagnetic

interactions,

are

obtained
.

The

analysis

of

these

structures’

contributions

to

hadrons

polarizability

is

performed
.


To

determine

the

covariant

equations

describing

the

electromagnetic

field

interaction

with

nucleon

taking

into

account

anomalous

magnetic

moments

and

polarizabilities

we

use

the

following

effective

Lagrangian
:




(1)



The

following

notations

were

introduced
:



(2)



(3)




(4)

6

.
2
1
2
1
4
1




























m
D
i
m
D
i
F
F
L




,
4






A
ie
F
m
ie
D











,
4






A
ie
F
m
ie
D













..
~
~
2












F
F
F
F
m
g



7

,
4
2
4
1









K
F
m
e
A
e
m
i
F
F
L





















,
~
~
2










F
F
F
F
m
K


,
2










i
,











If we substitute expressions (2)
-
(4) into (1), the effective
Lagrangian

will have the
form:



(5)





We

separate

the

part

related

to

nucleon

polarizabilities

in

the


Lagrangian

(
5
)




(6)









(7)





















,
~
~
2
,













F
F
F
F
m
L



,
2
1
~
~









F
F
F
F
F
F






.
2
2
,


























F
F
F
F
m
L


,
2












i
8


Expression

for

the

Lagrangian

(
7
)

is

consistent

with

the

effective

Lagrangian

presented

in

[
L’vov
,

A
.
I
.

Theoretical

aspects

of

the

polarizability

of

the

nucleon

/

A
.
I
.

L’vov

//

Jnter
.

Journ
.

Mod
.

Phys
.

A
.



1993
.



Vol
.

8
.





30
.



P
.

5267
-
5303
]
.

Formula

(
7
)

is

a

relativistic

field
-
theoretic

generalization

of

the

non
-
relativistic

relation




which

corresponds

to

the

polarizabilities

of

induced

dipole

moments

in

a

constant

electromagnetic

field

[
Schumacher,

M
.

Dispersion

theory

of

nucleon

Compton

scattering

and

polarizabilities

/

M
.

Schumacher

//

[Electronic

resource]
.



2013
.


http
:
//

hep
-
ph
/
1301
.
1567
.

pdf
]
.




In

the

case

of

a

variable

electromagnetic

field

the

signs

of

polarizabilities

in

the

Lagrangian

(in

the

non
-
relativistic

approximation)

will

change

[
Detmold,

W
.

Electromagnetic

and

spin

polarisabilities

in

lattice

QCD

/

W
.
Detmold
,

B
.
C
.

Tiburzi
,

A
.

Walker
-
Loud

//

[Electronic

resource]
.



2003
.


http
:
//
hep
-
lat/
0603026
.

However,

the

structure

of

tensor

contraction

in

(
7
)

does

not

change
.





,
2
2
2
,
H
E
L
H












9

In

order

to

obtain

the

equations

for

interaction

of

the

electromagnetic

field

with

nucleons,

we

use

the

effective

Lagrangian

(
1
)

and

Euler
-
Lagrange

equations
:







As

a

result

we

get
:


(8)



(9)



(10)





,
0




















A
L
A
L


,
0


















L
L




.
0


















L
L


,
2
























G
m
e
e
F




,
4
2



































F
m
e
K
K
i
A
e
m
i







.
4
2












F
m
e
K
K
i
A
e
m
i

























10



.
4
1
,




G
F
L




.
2
1
~


















,
~
~
4
,




















F
F
F
m
A
L
G










G

D


D

,
0












m
D
i

,
0












m
D
i

11



To

identify

the

physical

meaning

of

tensor

let’s

use

Gordon

decomposition

[
Itzykson
,

C
.

Quantum

field

theory

/

C
.

Itzykson
,

J
.
-
B
.

Zuber
.



McGraw
-
Hill
,

198
4
.



Vol
.

II
.



400

p
.
]
.

Current

density

of

Dirac

particles

with

the

help

of

Gordon

decomposition

can

be

represented

as

follows
:






where



The

components

of

tensor,

which

is

called

anti
-
symmetric

dipole

tensor,

are

static

dipole

moments

of

point
-
like

particles
.

With

the

help

of

this

tensor

we

can

define

the

current




In

the

rest

frame

of

the

particle,

we

have

the

following

relations
:





G

j
,
2
2
























m
e
i
m
e
e
j

,
2
0






m
e
G
.
2







i
m
e
j
e

0
G
.
0



G
j
m


.
0
0
i
i
G
d

,
2
ojk
ijk
i
o
G
i
m


12

Components

of

4
-
dimensional

current

can

be

defined

through

the

dipole

moments



The

Lagrangian

describing

the

interaction

of

electromagnetic

field

with

a

charged

particle

with

a

static

dipole

moment

has

the

form


(12)



Using

Eq
.

(
12
),

the

Lagrangian

and

the

Euler
-
Lagrange

equations,

we

get
:




In

relativistic

electrodynamics

a

similar

tensor

with

induced

dipole

moments

is

introduced

[
De

Groot,

S
.
R
.

Electrodynamics

/

S
.
R
.

de

Groot,

L
.
G
.

Sattorp
.



M
.:

Nauka
,

1982
.

-

560

p
.
]
.

The

current

density

and

moments

are

expressed

through

in

the

following

way
:



(13)



,
4
1


F
F
L


.
2
1
0




F
G
A
j
L
e
I





,
0
0
d








..
0
0
0
m
d
j
t








.
0





G
j
F
e




,



G
j


,



U
G
d

.
2
1





U
G
m


G
13

Tensor

satisfies

to

relations

(
13
)
.


In

order

to

switch

to

quantum

description

of

the

structural

particles

with

induced

dipole

moments

let’s

use

operator

form

[
Maksimenko,

N
.
V
.

Covariant

gauge
-
invariant

Lagrangian

formalism

with

the

polarizabilities

of

the

particles

/

N
.
V
.

Maksimenko,

O
.
M
.

Deryuzhkova

//

Vestsi

NAS

Belarus

2011
.





2
.



P
.

27

30
.
]
:











where

and

are

operators

of

the

induced

dipole

moments,

which

are

dependent

on

the

electromagnetic

field

tensor
.












U
m
d
U
U
d
G



,
2








































m
d
d
m
i
G
,
2



































m
d
d
m
i
G






d


m
14

If

one

requires

that

the

low
-
energy

theorem

for

Compton

scattering

holds

true,

then

these

operators

can

be

defined

as





Thus,

the

expression

(
11
)

is

the

anti
-
symmetric

tensor

of

the

induced

dipole

moments

of

the

nucleon
.

In

this

case,

the

interaction

Lagrangian

is

defined

as

follows
:





which

implies

Maxwell

equations

in

the

form





.
~

4





F
m


,


4






F
d


,
4
1
2
1
0






F
G
F
G
A
j
L
e
I




.
0







G
G
j
F
e








We

define

the

contribution

of

electric

and

magnetic

polarizabilities

to

the

amplitude

of

Compton

scattering
.

To

do

this,

we

use

the

Green’s

function

method

given

in

[
Bogush
,

A
.
A
.

Introduction

to

the

theory

of

classic

fields

/

А
.
А
.

Bogush
,

L
.
G
.

Moroz
.

-

Minsk
:

Science

and

technology,

1968
.

-

387

p
.;

Bogush
,

A
.
A
.

Introduction

to

the

gauge

field

theory

of

electroweak

interactions

/

А
.
А
.

Bogush
.

-

Minsk
:

Science

and

technology,

1987
.

-

359

p
.;

Bjorken
,

J
.
D
.

Relativistic

quantum

field

theory

/

J
.
D
.

Bjorken
,

E
.
D
.

Drell
.

-

M
:

Science
.

-

1978
.



Vol
.

1
.

-

295

p
.
]
.

We

present

the

differential

equation

(
9
),

which

takes

into

account

only

the

contribution

of

polarizabilities
,

in

the

integral

form
:





(12)




where

15













,
,
0
x
d
x
V
x
x
S
x
x
F




























..
2
,
x
x
K
x
x
K
i
x
V






















16



Let’s define the matrix element of the photons scattering on a nucleon. To do this,
we contract (12) with at and use the relation





where



As a result, we get:




(13)



Using boundary conditions and the crossing symmetry, the expression (13) can be
represented as:



(14)



fi
S




x
r
p
2
2



t












,
2
2
2
2
3
x
i
x
d
x
x
S
x
r
p
t
F
r
p


















.
2
1
2
2
2
2
2
2
2
/
3
x
ip
r
r
p
e
p
U
E
m
















.
4
,
2
2
x
d
x
V
x
i
S
r
p
fi





























.
2
1
4
21
21
1
1
1
1
2
2
x
d
x
x
K
x
x
K
x
S
r
p
r
p
r
p
fi






























17

After

using

the

definition

of

the

electromagnetic

field

tensor

in

(
14
)

and

integrating

we

get
:



(
15)



If

we

consider

the

wave

functions

of

the

nucleon

and

the

photons

in

the

initial

and

final

states,

the

expression

(
15
)

takes

the

form
:



(16)

where

M

has

the

following

form
:






(17)















.


)
(
2
4
21
)
1
(
)
2
(
21
)
2
(
)
1
(
)
1
(
)
2
(
x
d
F
F
F
F
F
F
m
i
S
fi


























,
4
2
2
1
2
1
2
2
2
1
1
M
E
E
p
k
p
k
im
S
fi



























































.
2
1
1
1
2
2
1
1
2
2
1
1
2
2
2
1
1
1
2
2
1
1
2
2
1
1
2
2
2
p
U
e
k
e
k
e
k
e
k
m
e
k
e
k
e
P
k
P
e
k
e
P
k
P
e
k
e
k
e
k
p
U
m
M
r
r






































































18

We

now

define

the

amplitude

(
17
)

in

the

rest

frame

of

the

target

and

limit

M

up

to

the

second
-
order

terms
.

In

this

case,

we

have

[
Petrunkin
,

V
.
A
.

Electrical

and

magnetic

polarizabilities

of

hadrons

/

V
.
A
.

Petrunkin

/

/

Fiz
.

-

1981
.

-

Vol
.

12
.

-

P
.

692
-
753
]
:





If

in

the

amplitude

M

along

with

the

contribution

of

polarizabilities

and

we

take

into

account

the

contribution

of

the

electric

charge,

then

M

can

be

represented

as
:




(18)


Differential cross section of the Compton scattering for computed using equation
(18) has the form [
Petrunkin
, V.A.
Electrical and magnetic
polarizabilities

of hadrons /
V.A.
Petrunkin

/ /
Fiz
.
-

1981.
-

Vol. 12.
-

P. 692
-
753
]:



















.
4
1
1
2
1
2
2
)
(
1
)
(
2
)
(
)
(
2
r
r
e
n
e
n
e
e
M

































.
4
4
1
1
2
1
2
2
)
(
1
)
(
2
2
)
(
)
(
2
2
r
r
e
n
e
n
e
e
m
e
M






































.
)
(
2
2
2
















m
m
d
d
e
e

0



Let’s

find

the

quasi
-
static

polarizabilities

of

point
-
like

fermions,

which

appear

in

the

Compton

scattering

due

to

higher

order

terms
.

In

general,

the

ACS

T

in

the

forward

direction

(

)

and

the

backward

direction

(

)

up

to

terms

can

be

written

as
:







On

the

other

hand,

it

is

possible

to

calculate

matrix

elements

and

the

amplitude

of

Compton

scattering

in

the

framework

of

QED,

including

next
-
to
-
the
-
Born

order

of

perturbation

theory

over


(see,

e
.
g
.
,

[
Tsai,

W
.
-
Y
.

Compton

scattering
.

ii
.

differential

cross
-
sections

and

left
-
right

asymmetry

/

W
.
-
Y
.

Tsai,

L
.
L
.

Deraad
,

K
.
A
.

Milton

//

Phys
.

Rev
.



1972
.



Vol
.

D
6
.



P
.

1428
-
1438
;

Denner
,

A
.

Complete

O(alpha)

QED

corrections

to

polarized

Compton

scattering

/

A
.

Denner
,

S
.

Dittmaier

//

Nucl
.

Phys
.



1999
.



Vol
.

B
540
.



P
.

58
-
86
])
.



19


0





2



2
( 0) 8
f E M
T m


 
      
 
 


 
   


2
( ) 8
f E M
T m


  
       
 
 


  
   
QED

20




In [
Andreev, V.
The invariant amplitudes of Compton scattering in QED / V. Andreev, A.M.
Seitliev
/
/
Vesti

NAS Belarus.
Ser.fiz
.
-
mat.
navuk
.
-

2011.
-

№ 3.
-

P. 60
-
65
] a method of
calculating fermions
polarizabilities

in the framework of quantum field models and theories
was developed by comparing the corresponding matrix elements. As a result, the following
equation take place:



(19)



(20)



where parameter is an infinitely small mass of the photon.

















2 2
3 3
8
11 2
3 6 3
QED QED
q s q s
E M
f f f
ln
m m m
 

 
 
 
 
   
 
 
 
2 2
3 3
4
59 2
3 6 3
QED QED
q s q s
E M
f f
ln
m m
 

 
  
 
 
    
 
 

21



As

follows

from

(
19
)

and

(
20
),

quasi
-
static

polarizabilities

include

non
-
analytic


terms

,

which

diverge

in

the

Thomson

limit

.

This

property

was

the

reason

that

in

papers

[
Llanta
,

E
.

Polarizability

sum

rules

in

QED

/

E
.

Llanta
,

R
.

Tarrach

//

Phys
.

Lett
.



1978
.



Vol
.

B
78
.



P
.

586
-
589
;

Holstein,

B
.
R
.

Sum

rules

for

magnetic

moments

and

polarizabilities

in

QED

and

chiral

effective
-
field

theory

/

B
.
R
.

Holstein,

V
.

Pascalutsa
,

M
.

Vanderhaeghen

//

Phys
.

Rev
.



2005
.



Vol
.

D
72

.






9
.



P
.

094014
]

structures

(
19
)

and

(
20
)

were

called

quasi
-
static

polarizabilities
.



From

(
19
)

and

(
20
)

it

is

easy

to

find

the

electric

(

)

and

magnetic

(

)

quasi
-
static

polarizabilities

and

assess

their

contribution

to

the

polarizability

of

the

"Dirac"

proton

(point
-
like

fermion

with

zero

anomalous

magnetic

moment
)



𝑞

𝑠
+

𝑞

𝑠


5
,
8
×
10

7
Fm
3
.


The

experimental

values

[
Review

of

Particle

Physics

/

K
.

Nakamura

[et

al
.
]

//

Journal

of

Physics

G
.



2010
.



Vol
.

37
.



P
.

075021
]
:


+


13
,
8
±
0
.
4
×
10

4
Fm
3



Numerical

estimates

are

consistent

with

the

estimates

given

in

[
Gerasimov
,

S
.
B
.

Scattering

of

light

of

low

frequency

and

charged

particle

polarizability

/

S
.
B
.

Gerasimov
,

L
.
D
.

Soloviev

//

Nucl
.

Phys
.



1965
.



Vol
.

С
74
.



P
.

589
-
592
]
.






ln
~
0


s
q
E


s
q
M


22

s
q
M
s
q
E





s
q
M
s
q
E







In

the

framework

of

gauge
-
invariant

approach

we

obtain

the

covariant

equations

of

motion

of

a

nucleon

in

the

electromagnetic

field

with

account

for

its

electric

and

magnetic

polarizabilities
.


Based

on

the

solutions

of

electrodynamic

equations

of

nucleon

motion

obtained

using

the

Green’s

function

method,

it

was

shown

that

the

developed

covariant

Lagrange

formalism

for

interaction

of

photons

with

nucleons

is

consistent

with

the

low
-
energy

theorem

of

Compton

scattering
.


On

the

basis

of

the

original

technique

a

well
-
known

result

for


the

combination

of

quasi
-
static

polarizabilities

in

QED

framework

was

obtained
.

New

expression

for

is

derived
.



The

apparent

advantage

of

the

method

for

defining

"
polarizabilities
"

mentioned

in

Section

3

is

its

relative

simplicity
.

This

approach

gives

wider

opportunities

for

the

study

of

the

internal

structure

of

nucleons

and

can

be

applied

in

various

quantum

field

theories

and

models
.




23