POLARIZABILITY OF NUCLEON IN QUANTUM-FIELD APPROACH

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

Introduction

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. 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.

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. 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

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. 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.

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

1. The covariant equations describing the electromagnetic field interaction with nucleon taking

into account polarizabilities.

.2

1

2

1

4

1

mDimDiFFL�

,4

AieFm

ieD

,4

AieFm

ieD

��

..~~2

FFFF

mg

7

,424

1

KFm

eAem

iFFL

�

,~~2

FFFFm

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,

FFFFm

L

,2

1~~

FFFFFF

.2

2,

FFFFm

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 22, HELH

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

LL

.0

LL

,2

Gm

eeF

,42

F

m

eKK

iAemi

.42

Fm

eKK

iAemi

�

10

 

.4

1,

GFL

.2

1~

,~~4,

FFFmA

LG

G

D

D�

,0

mDi

,0

mDi�

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, 1984. – 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

,22

m

ei

m

eej

�

,20

m

eG .

2 �

im

eje

0G

.0

Gjm

.00

ii Gd ,2 ojk

ijkio G

im

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 FFL

.2

10

FGAjL eI

,00 d

..000 mdj t

.0

GjF e

,

Gj , UGd .

2

1

UGm

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.

UmdUUdG

,2

mdd

m

iG

,2

mdd

m

iG

����

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 Fm

, 4 Fd

,4

1

2

10

FGFGAjL eI

.0

GGjF 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

2. Covariant representation of the amplitude of Compton scattering on the nucleon with polarizabilities

,,0 xdxVxxSxx F

..2

, xxKxxKi

xV

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)

fiS xr

p2

2 t

,2

2

2

2

3 xixdxxSx rp

tF

rp

.2

1222

2 22

2/3xiprr

p epUE

m

.4,2

2xdxVxiS r

pfi

.

2

1

421

21

1

1

1

1

2

2

xdxxK

xxKxS

rp

rp

rpfi

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( xdFFFFFF

m

iS fi

,42 2121

22211 M

EE

pkpkimS fi

.

2

11122

1122

11222

11122

1

122

11

2

22

pUekekekekm

ekekePkPek

ePkPekekekpUm

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 112122 )(1

)(2

)()(2 rr eneneeM

.44 112122 )(1

)(2

2)()(22

rr eneneem

eM

.)(2 22

mmd

d ee

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. D6. – P. 1428-1438; Denner, A. Complete O(alpha) QED corrections to polarized Compton scattering / A. Denner, S. Dittmaier // Nucl. Phys. – 1999. – Vol. B540. – P. 58-86]).

19

3. Quasi-static polarizability of particles spin ½ in QED

0 2

2( 0) 8 f E MT m

2( ) 8 f E MT 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

811 2

3 6 3QED QEDq s q s

E Mf f f

lnm m m

2 2

3 3

459 2

3 6 3QED QEDq s q s

E Mf f

lnm 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. 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.

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)

.

The experimental values [Review of Particle Physics / K. Nakamura [et al.] // Journal of Physics G. – 2010. – Vol. 37. – P. 075021]:

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].

ln~ 0

sqE

sqM

22

 Conclusion

sqM

sqE

sq

Msq

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.

Thanks for your attention

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