Download V.Andreev, N.Maksimenko, O.Deryuzhkova, Polarizability of the

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.atomph/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
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. phys.-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.
3
Effective field Lagrangians describing the interaction of low-energy
electromagnetic field with nucleons based on expansion in powers of inverse
mass of the nucleon M1 4 have been widely used recently [R.J.Hill, G.Lee,
G.Paz, M.P.Solon The NRQED lagrangian at order- 2012. - : http:// hepph/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. D2-11707, 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
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. D2-11707, 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.
5
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
1  
1

L   F F   i D m    i D m .


4
2 
2 


(1)
The following notations were introduced:


 
ie 
D     
 F  ie A,
4m
(2)

 ie 

 
D     
 F  ie A,
4m

  g

(3)

2
~ ~


F F   F F  ..
m
(4)
6
If we substitute expressions (2)-(4) into (1), the effective Lagrangian will have the
form:



1
e
 i
L   F F      m  e A     F  K   ,
(5)
4
2
4m
  

2
~ ~
i


 
K 
F F   F F  ,     ,      ,
m
2


We separate the part related to nucleon polarizabilities in the
Lagrangian (5)
i  

 




2
    ,


2
~ ~
L

F F   F F    ,
m
1
~ ~
F F   F F      F F  ,
2
 ,  
(6)
2 
 

 , 
 
 


L




F
F



F
F

 


.
m 
2

(7)
7
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


2
2

 , 
H  L
 2 E  H ,
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.
8
In order to obtain the equations for interaction of the electromagnetic
field with nucleons, we use the effective Lagrangian (1) and EulerLagrange equations:
 L
 
    A



 L
 L

 0,   
 A
   




 L


 0,   L

 

  


 L

 0.
 

As a result we get:
 e

  F   e      
    G  ,
 2m

 

 
  


 

 i   m   e A   i  K     K      e   F ,


2
4m


(8)
(9)

 i
 

e 






 i   m   e A   K     K  
 F . (10)


2
4m


9

D
 

 

 i D m   0,   i D m   0,









D
G 
L ,  4
 ~ 
 ~ 
    F   F      F  ,
G 

   A
m
~  1 
      .
2









1

 , 
L
  F G  .
4
10

To identify the physical meaning of tensor G
let’s use Gordon decomposition
[Itzykson, C. Quantum field theory / C. Itzykson, J.-B. Zuber. – McGraw-Hill, 1984. – Vol.

II. – 400 p.]. Current density j of Dirac particles with the help of Gordon decomposition
can be represented as follows:

e
 e

j  e  
i    
   ,
2m
 2m


where


G0
e

  ,
2m

je


e

i .
2m
The components of G0 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
jm    G0 .
In the rest frame of the particle, we have the following relations:
i
mo i   ijk Gojk , d 0 i  G i 0 .
2
11
Components of 4-dimensional current can be defined through the dipole moments
 
 

 0   d 0 ,



j0   t d 0  m0 ..
The Lagrangian describing the interaction of electromagnetic field with a charged particle
with a static dipole moment has the form
1 
(12)
LI   je A  G0 F .
2
1
Using Eq. (12), the Lagrangian L   F F  , and the Euler-Lagrange equations, we
4
get:

  F   je    G0 .
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 G  in the following way:


j   G ,
d   G  U , m   1   G U .
2
(13)
12


G   d U   U  d    mU
Tensor
(13).
satisfies to relations
In order to switch to quantum description of the structural particles with induced
dipole moments let’s use operator form [Maksimenko, N.V. Covariant gaugeinvariant Lagrangian formalism with the polarizabilities of the particles / N.V.
Maksimenko, O.M. Deryuzhkova // Vestsi NAS Belarus 2011. – № 2. – P. 27–
30.]:

  
  

i   
 

G 
m    ,
 d  d  


2m 






  
i       

G 
 m   ,
  d  d 


2m 







m
and
where d
are operators of the induced dipole moments, which are
dependent on the electromagnetic field tensor.
13
If one requires that the low-energy theorem for Compton scattering holds true, then these
operators can be defined as


~
d   4  F    , m   4 F    .
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:
1
1
LI   je  A  G0  F  G  F ,
2
4
which implies Maxwell equations in the form
 F   je    G0   G .
14
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:
x    0  x    S F x  x V  ,  xdx,
V  ,  x  
 


i 
 K x  x  K x   x ..
2
(12)
where
15
Let’s define the matrix element S fi of the photons scattering on a nucleon. To do this,
r 
we contract (12) with  p 2 x at t   and use the relation
2

r2 
3






x
S
x

x
d
x
 p2
F
where
p2
r2 

1
2 
3/ 2
  i  p2 r2  x,
t 
m r2  
U  p2   eip2 x .
E2
S fi   i  p2
 r2 
As a result, we get:
xV  ,  xd 4 x.
(13)
Using boundary conditions and the crossing symmetry, the expression (13) can be
represented as:
S fi
 
  
x d

 1
      p2 r2  x   K  21 x    p1 r1  x  
 2
 K  21 x     p1
r1
4
x .
(14)
16
After using the definition of the electromagnetic field tensor in (14) and integrating we get:
2i
( 2 ) 
(1) 

( 2 ) 

4
S fi 
(



)
F
F

F
F



F
F

d
x.(15)

21



21





(1)
( 2)
(1)

m






If we consider the wave functions of the nucleon and the photons in the initial and final
states, the expression (15) takes the form:
S fi 
im k1  p1  k 2  p2 
2 
2
41 2 E1 E2
M,
(16)
where M has the following form:

  2  
2 r2   

M
U  p2 k 2 e2   k 2  e   k1 e 1  P  k1 P e 1  
m


 
 



 1  

 k 2 e 2  P  k 2 P e 2   k 1 e1   k1 e      





 m k 2  e2   k 2 e2   k1 e 1   k1 e 1  U r1   p1 .


  


(17)

17
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]:
M  4 
2
r2 

 


 ( 2 )  ( 1 )
  ( 2 )   ( 1 ) r1 
e e
  n2 e
 n1 e
 .
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:




2


 ( )  ( )
  ( )   ( )
e

r2  
2 
2
M 
   4   e 2 e 1  4  n2 e 2  n1 e 1
 m


 r 
 1 .

(18)
Differential cross section of the Compton scattering for   0  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]:

d   e 
    2 e (   ) 2 .
d  m 
m
2
18
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
2
 direction (   
direction (   0  ) and the backward
) up to
terms can be
written as:
  
 
T
   
 
T
(  0)  8 m f  2  E  M        
(   )  8 m f  2  E   M        
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  QED
(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. D6. – P. 14281438; Denner, A. Complete O(alpha) QED corrections to polarized Compton scattering /
A. Denner, S. Dittmaier // Nucl. Phys. – 1999. – Vol. B540. – P. 58-86]).
19
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:
 Eq  s   Mq  s
 Eq s   Mq s
where

2
2
 QED
11 8 QED  2 


ln 
 
3
3

3 m f 6 3 m f  m f 
2
2
QED
59 4QED  2 


ln 

3
3
3 m f 6 3 m f   
(19)
(20)
parameter is an infinitely small mass of the photon.
20
As follows from (19) and (20), quasi-static polarizabilities include non-analytic
~ ln  , which diverge in the Thomson limit   0 . This property was the
terms
reason that in papers [Llanta, E. Polarizability sum rules in QED / E. Llanta, R. Tarrach //
Phys. Lett. – 1978. – Vol. B78. – 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. D72 . – № 9. – P. 094014]
structures (19) and (20) were called quasi-static polarizabilities.
qs
From (19) and (20) it is easy to find the electric (  Eq  s ) and magnetic (  M
) quasi-static
polarizabilities and assess their contribution to the polarizability of the "Dirac" proton
(point-like fermion with zero anomalous magnetic moment)
𝛼 𝑞−𝑠 + 𝛽 𝑞−𝑠 ≈ −5,8 × 10−7 Fm3.
The experimental values [Review of Particle Physics / K. Nakamura [et al.] // Journal of
Physics G. – 2010. – Vol. 37. – P. 075021]:
𝛼 + 𝛽 ≈ 13,8 ± 0.4 × 10−4 Fm3
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. С74. – P. 589-592].
21
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
 Eq  s   Mq  s
the
combination of quasi-static polarizabilities in QED
framework was obtained. New expression for
is derived.
 Eq  s   Mq  s
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.
22
23