fulltext

The electron scattering and graviton photoproduction

in external electromagnetic fields

D. TATOMIR(1), D. RADU(1) and M. AGOP(2)

(1_{) Department of Theoretical Physics, Faculty of Physics, “Al. I. Cuza” University}

Iasi 6600, Romania

(2_{) Department of Physics, “Gh. Asachi” Technical University}

Iasi 6600, Romania

(ricevuto il 22 Aprile 1996; approvato il 30 Agosto 1996)

Summary. — Starting from the principle of minimal coupling in Quantum Electrodynamics and Quantum Gravity and using the Feynman diagram technique, the processes of Dirac particle scattering and photon-graviton transformation in external electromagnetic fields (homogeneous electrostatic or magnetostatic fields and electric- or magnetic-dipole fields) are studied. A remarkable analogy between the cross-sections of Dirac-particle scattering and graviton photoproduction is noticed. The results for the latter type of processes are in agreement in some peculiar cases with those obtained by other authors in a different manner. According to our knowledge, the previous papers in this field did not studied the process of photoproduction of gravitons in the field of an electric dipole. We believe that this process exists, while the emission of gravitational waves appears in this case only as a quantum effect.

PACS 04.20 – Classical general relativity.

1. – Introduction

As is well known, the interaction problem of an electric charged particle (e.g., an electron) with the electromagnetic field has been solved in a complete and consistent manner only in a few cases from a quantum point of view. In this paper, we first intend to approach this problem by using the Feynman diagram technique applied to a covariant perturbative-like problem within the frame of quantum field theory (QFT) formalism. More exactly, we shall study in the above-mentioned manner the electron scattering process for the following four field types, considered as exterior electro-magnetic fields:

a) the electrostatic field of a flat condenser; b) the homogeneous magnetostatic field; c) the electric-dipole field;

d) the magnetic-dipole field.

The second purpose of this paper is to approach the process of production of gravitational waves by turning photons into gravitons in the field of an electric dipole. Finally, we generalize some of the results already obtained by us [1-3] and also by other authors [4-7] in a different manner, concerning the photoproduction of gravitons in other types of exterior electromagnetic fields (i.e. the electrostatic and magneto-static fields and the magnetic-dipole field). The remarkable analogy of the results obtained for the two types of processes (i.e. the electron scattering process and the graviton photoproduction process) concerning the expressions of differential cross-sections is worth mentioning.

All the processes studied in this paper are treated in the first order of approximation (linear processes in coupling constants), a very credible hypothesis for the diffusion of the electrons in electromagnetic exterior fields and especially for the photoproduction of the gravitons in exterior electromagnetic fields. In this last case, because of the smallness of the gravitational constant k 4k16 pG , where G is Newton’s gravitational constant, the cross-sections are generally too small to be of practical interest in the laboratory experiments. But in astrophysics things are different. So, as was shown by Papini and Valluri [4], very interesting astrophysical objects have recently been observed and a reassessment of the role of gravitons in this context may

a priori be interesting quite independently of the related problem of the existence of

gravitational waves. This is also suggested by Boccaletti [8] who showed that for astrophysical situations the rate of production of gravitons in quantum processes can be at least as high as the classical one.

The recent results concerning the graviton photoproduction processes have shown that the scattering cross-sections may have detectable values even in the laboratory conditions. Thus, Long et al. [6] have shown that in their scenario (respectively, the conversion of photons into gravitons in the electric field of a flat condenser and the magnetic field of a solenoid) the scattering cross-sections may be increased as much as possible on the basis of the field intensity (or of the volume containing the field) increase. This is possible because the external electromagnetic field is considered as a classical field. Besides, the same authors have shown that the reverse phenomenon is also possible. Thus, the conversion of gravitons into photons can take place both in the exterior static electromagnetic fields and in the periodic external electromagnetic fields. In the latter case they calculated the differential cross-sections in the quasi-static limit [9].

The problem of the scattering of a massive Dirac field “propagating” in a domain occupied by the electromagnetic field may be treated by quantizing both the electro-magnetic background and the scattered field. In this scenario the two fields couple according to the Feynman vertex rules. However, since our interest is restricted to an electromagnetic-background configuration generated by classical energy-momentum distributions which are not affected considerably by the scattering process, we may replace the virtual photon by an external field. In particular, we considered only static electromagnetic fields.

As previously shown by other authors, we have found the Feynman-diagram technique to be far more powerful than partial-wave analyses or other working methods for studying these processes. Historically, the Feynman technique was first used in conjunction with quantum-electrodynamical processes, but its efficiency as a problem-solving tool soon led to its widespread use in many other aspects of quantum interactions and not only. Since a classical process is in fact the

long-wavelength limit of a quantum one, one can perfectly well use this technique to solve classical problems too [10].

Concerning the second process studied in this paper (i.e. the graviton photo-production in external electromagnetic fields), we mention that far from considering graviton emission from strong sources like black holes, whose existence is strongly hinted and which may require use of the full mathematical apparatus of Einstein’s theory, we have undertaken the study of this process with objects like pulsars, neutron stars and quasars in mind. These have the common characteristic of being either strong emitters of electromagnetic radiation or sources of intense magnetic fields, as shown by Papini and Valluri [4]. Thus, a process in which gravitons are produced in the interaction of photons with static electromagnetic fields would look a priori promising. Our paper comprises six sections. The second section offers the general working frame and the calculus and notation conventions as well. The two following sections are dedicated to the study of the scattering process of the electrons in the electromagnetic static fields and to the process of photoproduction of gravitons in the same exterior fields. Section 5 is concerned with some observations and astrophysical considerations, while the last section contains a final overview on the result presented in the paper as well as some concluding remarks.

2. – General conventions and notations

By gmn_{, h}mn _{and y}mn _{we denoted the metric tensor, the Minkowski tensor—}

diag (11, 212121)—and the tensor of the weak gravitational field, respectively. Then, Gupta’s linear approximation reads k_{2g g}mn_{4 h}mn_{2 ky}mn, where g 4det(gmn). The notations for electrostatic and magnetostatic fields are E and H, while for the electric- and magnetic-dipole fields we use D and M. For the former type of classical field the potential of the static Maxwell field is

Aext

m (x) 4idm4V(x) , V(x) 4

D Q x

4 pNxN3 , E(x) 42˜V(x) ,

(1)

and in the latter case we have

Aext

m (x) 4dmiAi(x) , A(x) 4 M 3x

4 pNxN3 , H(x) 42˜3A(x) , ˜Q A(x) 40 ,

(2)

or, for the tensor of the classical Maxwell field,

Fext

mn (x) 4i(dm4dnj2 dmjdn4) Ej(x) , (3)

and

Fmnext(x) 4idmidnjeijkHk(x) , (4)

completely antisymmetric Levi-Civita tensor (1_{). Components of 3-vectors are labeled}

by Roman indices, while components of 4-vectors carry Greek indices.

3. – The Dirac-particle scattering in static electromagnetic fields

Starting from the principle of minimal coupling in QED [11] the well-known expression for the interaction Lagrangian between the Maxwell and Dirac fields is

LMD4 iecgmcAm. (5)

Here c and c–_{(4c* g}4) are the operators corresponding to massive spinor field, gmare the usual Dirac matrices, the asterisk signifies Hermitian conjugation and e is the electromagnetic coupling constant. Using the S-matrix formalism, the rules of Feynman type for diagrams in the external-electromagnetic-field approximation are deduced, by means of which the corresponding differential cross-sections are obtained. The process is described by the Feynman diagram in fig. 1, where k and (r), p and (s) are the four-momenta and polarization (spin) states of the incoming and outgoing particles—for which r , s 41, 2—respectively, and q is the four-momentum of the virtual photon (q 4p2k and q04 p02 k04 0—the conservation of energy).

According to the standard quantum field theory the part of (5) Lagrangian— casted into the normal form—which describes the interaction of massive charged Dirac particles with the external Maxwell field is

N[LMD(x) ] 4iec(2)(x) gmc(1)(x) Amext(x) , (6) p ( )s k ( )r q e gm E D H M, , ,

Fig. 1. – The electron-photon-electron diagram. The wavy line represents a photon. The dash-dotted lines represent the Dirac particles.

(1_{) The notations corresponding to complex Euclidean space x}

where (1) and (2) denote the positive and negative frequency parts, corresponding to the annihilation and creation of spinor particles in x, respectively. The matrix element in the external-electromagnetic-field approximation, corresponding to the diagram in fig. 1, is (7) _{apNSNkb 42} me ( 2 pk0p0)1 O2



_d3 qu(s)(p) gmu(r)(k) Amext(q) d(p 2k2q) d(p02 k0) 4 4 F(k , p) d(p02 k0) ,

where u, u–are the positive-energy Dirac spinors and m is the mass of spinor particles. After integration we obtained

F(k , p) 42 me

( 2 p)1 O2k0

Q(k , p) , _{Q(k , p) 4u}(s)_{(p) u}(r)_{(k) A}ext

m (q) , q 4p2k ,

(8)

where the Fourier transform of the classical electromagnetic potential

Aext m (q) 4 1 ( 2 p)3 O2



_d3_{x e}2iq Q x_Aext m (x) (9)

must be introduced. The differential cross-section results by averaging over the initial and summing over the final spin (polarization) states of particles [12]:

ds 4 (2p)2_k2 0

o

!

f , spNF(k , p) N 2

p

i , sp dV , (10)

where dV 4sin u du dW is the infinitesimal solid-angle element in the direction of emerging particles, u being the scattering angle (between k and p). Thus we have

o

!

f , spNF(k , p) N 2

p

i , sp4 m2_e2 2 pk02 1 2

!

spNQ(k , p) N 2_. (11)

Using the covariant form of the projection operator

P_{1(k) 4}

!

r 41 2 u(r)_{(k) u}(r) (k) 4 gmkm1 im 2 im (12)

and the reality properties of Amext(x), one obtains

(13) 1 2

!

spNQ(k , p) N 2 4 21 2 A ext m (q) Anext(2q) Sp [gmP1(k) gnP1(p) ] 4 4 1 2 m

g

kmpn1 pmkn1 2 dmnk 2_sin u 2

h

A ext m (q) Anext(2q) , so that the differential cross-section has the form

ds dV 4 pe 2

g

_k mpn1 pmkn1 2 dmnk2sin u 2

h

A ext m (q) Anext(2q) , q 4p2k . (14)

a

q

j

E D H M, , , k z p y x

Fig. 2. – The spatial orientation of the incident (k) and emergent (p) particle directions relative to the direction of the electromagnetic fields (E, D, H, M).

This general result can be specialized for different forms of the external Maxwell fields. Finally, in order to obtain the Fourier transforms of classical electromagnetic fields F f E , D, H, M, we suppose the F vector situated in the (x , O , z)-plane and a the angle between F and the incoming-electron movement direction considered along the

Oz-axis, i.e.

.

/

´

F f (F sin a , 0 , F cos a) , _{k f ( 0 , 0 , NkN) ,} p f (NkNsin u cos W, NkNsin u sin W, NkNcos u) ,

q f

(

_{NkNsin u cos W , NkNsin u sin W , 2NkN( 1 2 cos u)}

)

, (15)

W being the angle between the (F , k) and (p , k) planes, as in fig. 2.

3.1. The electric-field case. – In this case, using the potential of Maxwell field (1) and the Fourier transform (9), we have

Aext

m (q) 4idm4Vext(q) , Vext(q) 4iV(q) , (16)

obtaining for the homogeneous electrostatic or electric-dipole fields the expressions

V(q) 4 q Q E(q)

q2 , V(q) 42 D Q q

( 2 p)3 O2_q2 ,

respectively, and the differential cross-section becomes ds dV 4 2 pe 2

g

_k2 02 k2sin2 u 2

h

[V(q) ] 2_, q 4p2k . (18)

Thus, for the scattering of the electrons with k0 energy in the homogeneous

electrostatic field E(x) of a plane capacitor in a domain of A 3B3C dimensions, using the Fourier transform

E(q) 4 E ( 2 p)3 O2



2AO2 AO2 dx e2ixNkNsin u cos W



2BO2 BO2 dy e2iyNkNsin u sin W



2CO2 CO2 dz e2izNkN( 1 2 cos u)_, (19) we find ds dV 4 e2E2

g

k022 k 2 sin2 u 2

h

p2_k8_sin4_{u sin}6 u 2sin 2_{W cos}2_W

g

sin a cos u

2 cos W 2cos a sin

u 2

h

2 3 (20) 3sin2

g

1

2ANkN sinu cosW

h

sin

2

g

1

2 BNkNsin u sin W

h

sin

2

g

1

2CNkN(12cos u)

h

, and for the electron scattering in the electric-dipole field of D moment one obtains ds dV 4 e2D2

g

k022 k 2 sin2 u 2

h

( 4 p)2_k2_sin2 u 2

g

sin a cosu

2 cos W 2cos a sin

u

2

h

2

. (21)

We shall notice that, on the one hand, in the limit case W 4a4pO2 the differen-tial cross-section (20) vanishes and, on the other hand, in the ultrarelativistic limit

(

_{NkN K k}0

)

, for the case when u 4a4pO2, W40, the differential cross-section (20)

becomes ds dV 4 e2E2B2 2 p2_k4 0 sin2

g

1 2Ak0

h

sin 2

g

1 2Ck0

h

. (22)

3.2. The magnetic-field case. – In this case, using again relation (1) and the Fourier transform (9), we have

Aext

m (q) 4dmiAiext(q) (i 41, 2, 3) , Aext(q) 4iA(q) , (23)

obtaining for the homogeneous magnetostatic or magnetic-dipole fields the expressions

A(q) 4 q 3H(q)

q2 , q Q A(q) 40 , A(q) 42

M 3q

( 2 p)3 O2_q2 ,

respectively, and the differential cross-section (14) becomes ds

dV 4 2 pe

2

m

[k Q A(q) ][p Q A(q) ] 1k2_sin2 u

2[A(q) ]

2

n

_,

q 4p2k .

(25)

Thus, for the scattering of the electrons with k0 energy in the homogeneous

magnetostatic field (2

) H(x) in a domain of A 3B3C dimensions, using the Fourier transform similar to (19) we find

ds

dV 4

e2H2

k

sin2a sin2_{W 1sin}2 u

2

g

sin a sin

u

2 cos W 1cos a cos

u

2

h

2

l

p2k6sin4u sin6_{(uO2) sin}2W cos2W 3

(26)

3sin2

g

1

2ANkN sinu cosW

h

sin

2

g

1

2 BNkNsin u sin W

h

sin

2

g

1

2CNkN(12cos u)

h

, and for the electron scattering in the magnetic dipole of M moment one obtains

ds dV 4 e2_M2

k

_sin2_{a sin}2 W 1sin2 u 2

g

sin a sin u

2 cos W 1cos a cos

u 2

h

2

l

( 4 p)2_sin2 u 2 . (27)

We notice that in the limit case u 4W4a4pO2, for ultrarelativistic electrons, the differential cross-section (26) becomes

ds dV 4 2 e2H2A2 p2_k4 0 sin2

g

1 2 Bk0

h

sin 2

g

1 2 Ck0

h

, (28)

and in the other one, when u 4a4pO2, W40, the differential cross-section is ds dV 4 e2_H2_B2 2 p2_k4 0 sin2

g

1 2Ak0

h

sin 2

g

1 2Ck0

h

. (29)

4. – The graviton photoproduction in static electromagnetic fields

In order to describe the interaction between the gravitational and electromagnetic fields, using the principle of minimal coupling in quantum gravity [13], we must add to the expression of Einstein Lagrangian the Maxwell field’s Lagrangian written in the curved space: LM4 2 1 4k2g g ma_gnb_F mnFab. (30)

(2) This has nothing to do with the synchrotron radiation which is generated by ultrarelativistic electrons moving in a magnetic field (e.g., moving in synchrotrons or storage rings), this being a second-order process.

Developing all quantities in series in terms of k, i.e

.

`

/

`

´

k2g 4 1 2 1 2ky 1O(k 2_{) ,} y 4ya a , gmn 4 hmn2 khmn1 O(k2) , hmn_{4 y}mn₂ 1 2h mn y , (31)

and passing to the flat space, the first-order interaction Lagrangian between the Einstein and Maxwell fields has the form

L( 1 ) MG(k) 42 1 2kFmnFmauna, (32) where umn4 ymn2 1 4dmny , y 4y a a . (33)

Using the S-matrix formalism the rules of Feynman type for diagrams in the

external-electromagnetic-field approximation are deduced, by means of which the

corresponding differential cross-sections are obtained. The process is described by the Feynman diagram in fig. 3, where k and em(a)(k), p and emn(b)(p) are the four-momenta and polarization vector or tensor of the incoming and outgoing particles, respectively

k q E D H M, , , k p e( )b( )p e( )a( )k mn m

Fig. 3. – The photon-photon-graviton diagram. The dash-dotted line represents a graviton. The wavy lines represent the photons.

(a , b 41, 2 for real photons and gravitons) and q is the four-momentum of the virtual photon (q 4p2k and q04 p02 k04 0—the conservation of energy).

Taking into account that for real gravitons y 40, the part of Lagrangian (32)—casted into the normal form—is

N[L( 1 ) MG(x) ] 42 1 2kF ext mn (x) yna(2)(x) Fma(1)(x) . (34)

In order to obtain the Fourier transforms of the external electromagnetic fields

F f E , D , H , M : Fext mn (q) 4 1 ( 2 p)3 O2



_d3_{x e}2iq Q x_Fext mn (x) , (35)

we suppose again—as in (15)—the F vector situated in the (x , O , z)-plane and a the angle between F and the incoming-photon movement direction considered along the

Oz-axis, i.e.

.

/

´

F f (F sin a , 0 , F cos a) , k f ( 0 , 0 , k0) ,

p f (k0sin u cos W , k0sin u sin W , k0cos u) ,

q f

(

k0sin u cos W , k0sin u sin W , 2k0( 1 2cos u)

)

,

(36)

W being the angle between the (F , k) and (p , k) planes.

4.1. The electric-field case. – In this case using relation (3) and the Fourier transform (35) of the external electromagnetic tensor field we have for the homo-geneous electrostatic or electric-dipole fields

Fext mn (q) 4 i ( 2 p)3 O2(dm4dnj2 dmjdn4) Ej(q) , Ej(q) 42 D Q q (2p)3 O2 qj q2 , (37)

respectively, and the matrix element in the external-electromagnetic-field approximation is apNSNkb 4 (38) 4 ik 4( 2 pk0p0)1 O2



_d3

q(dm4dnj2 dmjdn4) Ej(q) ena(b)(p)[ea(a)(k) km2 em(a)(k) ka] 3 3d(p 2 k 2 q) d(p02 k0) 4F(k, p) d(p02 k0) .

Thus, for the scattering of the photons with k0energy in the homogeneous electrostatic

field E(x) of a plane capacitor, in a domain of A 3B3C dimensions, choosing for photons and gravitons the gauge in which e4(a)(k) 40, e4 a(b)(p) 40, respectively, one

obtains F(k , p) 42 k 4( 2 p)1 O2Q(k , p) , Q(k , p) 4e (a) i (k) eij(b)(p) Ej(q) , (39)

where

E(q) 4 E

( 2 p)3 O2



2AO2

AO2

dx e2ixk0sin u cos W



2BO2

BO2

dy e2iyk0sin u sin W



2CO2

CO2

dz e2izk0( 1 2cos u)_.

(40)

Averaging over the initial spin states (polarization) of photons and summing over the final gravitons ones, i.e.

o

_{f , sp}

!

NF(k , p) N2

p

i , sp 4 (41) 4 k 2 32 p 1 2

!

polNQ(k , p) N 2 4 k 2 64 p

y

1 1 (k Q p)2 k2_p2

z

{

[E(q) ] 2 2 [p Q E(q) ] 2 p2

}

,

and using for photons and gravitons the completeness relations

.

`

/

`

´

!

a 41 2 ei(a)(k) ej(a)(k) 4dij2 kikj k2 (i , j 41, 2, 3) ,

!

b41 2 eij(b)(p) ekl(b)(p)4dik(p) djl(p)1dil(p) djk(p)2dij(p) dkl(p) (i , j , k , l41, 2, 3) , (42)

the differential cross-section has the form ds dV 4 (43) 4 1 2

g

kE 2 pk2

0sin2u sin2(uO2) sin W cos W

h

2

[ 1 2 ( sin a sin u cos W1cos a cos u)2] 3

3( 1 1 cos2u) sin2

g

1

2Ak0sin u cos W

h

sin

2

g

1

2 Bk0sin u sin W

h

sin

2

g

1

2 Ck0( 1 2cos u)

h

, generalizing the results obtained before us by Mitskevich [7]. We notice that in the limit case u 4a4pO2, W40 the differential cross-section vanishes, and in another limit case when u 4a4W4pO2 the differential cross-section becomes

ds dV 4 k2_E2_A2 2( 2 p)2k02 sin2

g

1 2Bk0

h

sin 2

g

1 2Ck0

h

, (44)

in agreement with Mitskevich’s result, obtained in a different manner.

For the electric dipole of D moment, taking into account the transversality condition pieij(b)(p)40 and choosing the same gauge as above, i.e. e4(a)(k)40, e4 a(b)(p)40,

the matrix element in the external-electromagnetic-field approximation has the form (45) _apNSNkb4 2ik ( 4 p)2_(k 0p0)1 O2



_d3_q(d m4dnj2dmjdn4) ena(b)(p) [ea(a)(k) km2em(a)(k) ka]3 3D Q q q2 qjd(p 2k2q)(p02 k0) 4F(k, p) d(p02 k0) , and finally F(k , p) 4 kD 2( 4 p)2

g

cos a 2ctg u 2 cos W sin a

h

Q(k , p) , Q(k , p) 4e (a) i (k) ei3(b)(p) . (46)

Averaging over the polarization states of photons and summing over the gravitons ones, the differential cross-section

(

using the completeness relations (42)

)

is

ds dV 4 k2_D2_k2 0 2( 8 p)2 ( 1 1cos 2 u)

g

sin a cos u

2 cos W 2cos a sin

u 2

h

2 cos2 u 2 . (47)

Since this process has not been approached in the literature till now, we look into it more than other ones. Thus, after u-integration of relation (47) one obtains the following W-differential cross-section:

f1(a , W) 4 1 K ds dW(a , W) 4 1 K



0 p ds dV(a , u , W) sin u du 4 (48) 4 1

30

m

13 2cos 2a17 cos 2W2

7

2[ cos ( 2 a 22W)1cos (2a12W) ]2

75

16p sin 2 a cosW

n

, and after W-integration of the same relation one obtains the following u-differential cross-section: f2(a , u) 4 1 K ds dv(a , u) 4 (49) 4 1 K



0 2 p ds dV(a , u , W) dW 4 p 16cos 3 u

2 [ 6 12 cos 2a22 cos u2

2 3 cos ( 2 a 2 u) 2 3 cos ( 2 a 1 u) ]

g

6 sinu 2 2 sin

3 u

2 1 sin

5 u

2

h

, dv 4sin u du ,

whose graphical representations are given in fig. 4 and fig. 5, respectively.

1.0 0.5 0.0 0 0 1 2 2 3 4 6 f₁( , )a j j a

Fig. 4. – The viariation of W-differential cross-section f1(a , W) 4 (1OK)( dsOdW)(a, W) with respect

to the angles a and W .

performed easily. The result is

s *tot(a) 4 1 Kstot(a) 4 1 K



0 2 p

u



0 p ds dV(a , u , W) sin u du

v

dW 4 1 15( 13 2cos 2a) , (50)

and represents the total cross-section corresponding to the photon-graviton transfor-mation process in an exterior static electric-dipole field. In relations (48), (49) and (50)

K 4k 2_D2_k2

0

2( 8 p)2 . The variation of s *tot(a) with respect to the angle a is given in fig. 6. From

eq. (47) we see that the differential cross-section for the photoproduction process of gravitons in the electric-dipole field of D moment depends quadratically on the magnitude D and the photon energy k0.

4.2. The magnetic-field case. – In this case using relation (4) and the Fourier transform (9) we have for the homogeneous magnetostatic or magnetic-dipole fields

Fext mn (q) 4 1 ( 2 p)3 O2dmidnjeijkHk(q) , Hk(q) 4 1 ( 2 p)3 O2

g

Mk2 qk M Q q q2

h

, (51)

respectively, and the matrix element in the external-electromagnetic-field approxima-tion is

apNSNkb 4 k

4( 2 pk0p0)1 O2



_d3_qd

midnjHk(q) ena(b)(p)[ea(a)(k) km2 em(a)(k) ka] 3 (52)

2 0 0 0 1 ₁ 2 2 3 3 6 f2( , )a q q a 1

Fig. 5. – The variation of u-differential cross-section f2(a , u) 4 (1OK)( dsOdv)(a, u) with respect

to the angles a and u .

a 0.0 0.5 1.0 1.5 2.0 2.5 3.0 * ( ) tot a s 2.9 2.8 2.7 2.6

Fig. 6. – The variation of s *tot(a) with respect to the angle a.

Thus, for the scattering of the photons with k0 energy in the homogeneous

magnetostatic field H(x) in a domain of A 3B3C dimensions, choosing for photons and gravitons the same gauge, i.e. e4(a)(k) 40, e4 a(b)(p) 40, respectively, one

obtains

.

/

´

F(k , p) 4 k 4( 2 p)1 O2_k 0 Q(k , p) , Q(k , p) 4eijkHk(q)[e (a) n (k) ki2 e (a) i (k) kn] e (b) jn (q) , (53)

where the Fourier transform H(q) is given by an expression similar to (40). Averaging and summing over the initial and final polarization states of photons and gravitons,

i.e.

o

_{f , sp}

!

NF(k , p) N2

p

i , sp4 (54) 4 k 2 32 pk2 0 1 2

!

polNQ(k , p) N 2 4 k 2 64 pk2 0

y

k2 1 (k Q p) 2 p2

z

{

[H(q) ] 2 2 [p Q H(q) ] 2 p2

}

,

and using the completeness relations (42), the differential cross-section has the form ds dV4 1 2

u

kH 2 pk2 0sin2u sin2 u 2 sin W cos W

v

2

[ 12( sin a sin u cos W1cos a cos u)2

]3 (55)

3( 1 1 cos2u) sin2

g

1

2Ak0sin u cos W

h

sin

2

g

1

2 Bk0sin u sin W

h

sin

2

g

1

2 Ck0( 1 2cos u)

h

, generalizing the results obtained by Mitskevich [7]. In the limit case u 4a4pO2,

W 40 the differential cross-section vanishes and for u4a4W4pO2 the differential

cross-section becomes ds dV 4 k2H2A2 2( 2 p)2_k2 0 sin2

g

1 2Bk0

h

sin 2

g

1 2Ck0

h

, (56)

in agreement with Mitskevich’s result, obtained in a different manner. Finally, for the scattering of photons in the static magnetic-dipole field H(x) with the moment M, the matrix element in the external-electromagnetic-field approximation is

apNSNkb 4 k ( 4 p)2_(k 0p0)1 O2



_d3 qdmidnjeijk

g

Mk2 qk M Q q q2

h

e (b) na(p) 3 (57)

and choosing for photons and gravitons the same gauge as above, one obtains

.

`

/

`

´

F(k , p) 4 k ( 4 p)2_k 0 Q(k , p) , Q(k , p) 4eijk

y

Mk2 (pk2 kk) M Q (p 2k) 4 k2 0sin2(uO2)

z

[e(a) n (k) ki2 ei(a)(k) kn] ejn(b)(p) . (58)

Averaging and summing again over both the initial polarization states of photons and the final states of gravitons, respectively, and taking into account the completeness relations (42), the differential cross-section has the form

ds dV4 k2_M2_k2 0 2( 16 p)2 ( 11cos 2_u)[f2

1(a , u , W) sin2a1f22(a , u , W) cos2a2f32(a , u , W) ] ,

(59) where

.

`

/

`

´

f1(a , u , W) 421g(a, u, W) ctg a sin u cos W ,

f2(a , u , W) 4

m

g2(a , u , W) sin2u sin W 14

k

1 2g(a, u, W) sin2

u

2

l

2

n

1 O2

,

f3(a , u , W) 4sin a sin u cos W1cos a(11cos u) ,

g(a , u, W) 412tg a ctgu

2 cos W . (60)

This result has the most general form and contains all the particular cases studied in the literature concerning this process. Thus, if the direction of motion of the photon is along M (a 40), the differential cross-section becomes

dsll dV 4 k2M2k02 2( 16 p)2 sin 2 u( 1 1cos2_{u) .} (61)

If it is perpendicular to M (a 4pO2), one obtains ds» dV 4 k2M2k02 2( 8 p)2

g

1 1cos 2_{u)( sin}2 W 1cos2_{W sin}4 u 2

h

, (62)

in agreement with Papini and Valluri’s [4, 14] results, obtained by other means. From both eq. (61) and eq. (62) we see that the differential cross-sections depend quadratically on the magnitude M of the magnetic-dipole moment and the photon energy k0.

We notice that there is a remarkable analogy between the cross-sections of

electromagnetic electron scattering given by (20), (21), (26), (27) and the

“corresponding” graviton photoproduction cross-sections given by (43), (47), (55) and (59), specifying, however, that in different peculiar cases the spatial distribution of emerging particles is different.

5. – Observations. Some astrophysical applications

As one knows, several processes in which gravitons interact with other elementary particles are studied in the literature. As we mentioned above, because of the weakness of the gravitational constant, only processes linear in this constant are usually considered. Even if the cross-sections are too small to be of any practical interest in laboratory experiments, this may not be so, however, in astrophysics. Thus:

a) For the scattering process of photons with k0 energy in the homogeneous

electrostatic field E of a plane capacitor or in the homogeneous magnetostatic field H in a domain of A 3B3C dimensions, the differential cross-section is given by

ds dV 4 k2_F2 »A 2 2( 2 p)2k02 sin2

g

1 2Bk0

h

sin 2

g

1 2Ck0

h

, (63)

where F»4 E , H is the intensity of the field oriented perpendicular to the photon

movement direction, the photons traveling along the A dimension. In the limit of scatterers with linear dimensions all large compared with the wavelength of the incident photon (l b V1 O3_{) the total cross-section is s 4 (1O8) k}2_LF2

»V , where L is the

distance of the photon traveling through the field, this being contained in the volume V. The cross-section vanishes when the photon traveling is along the field.

According to Mitskevich’s estimations, in the laboratory one can create conditions such that the total cross-section is about s B10230_cm2

. In this case the field intensity must be F»B 10

10

OAl , where l is the photon’s wavelength traveling the field. Also, this effect can take place in cosmic space for travel of photons through the magnetic fields of the galaxies, namely, a magnetic field about H B1025_{Oe is necessary in order}

that it is enough at the considered value of s . In this case—according to Mitskevich—the reverse phenomenon can be observed, i.e. the transformation of gravitational waves into electromagnetic ones, when they arrive in a region where there is a magnetic field. Thus, because of irradiation by free gravitational waves, an amplified magnitude of the spiral galactic branches, in the domains of large wavelength, should be observed.

b) For the scattering process of photons with k0 energy in the static

magnetic-dipole field H of moment m, when the photon moves along the m direction, the total cross-section is s C (3O5) k2_m2_k2

0 and if the photon direction is perpendicular

to m, one obtains s C (1O4) k2m2k02. According to Poznanin’s estimation, for the

scattering of photons with k0B 109eV in the magnetic field of an electron, the total

cross-section is s B10260_cm2_{. Supposing that in the laboratory it is possible to create a}

magnetic dipole with the moment m C104 _{(in C.G.S.e.m. units), then for visible light}

(k0C 1 eV ) one obtains s B 10232cm2. According to De Sabbata et al.’s ideas [15], the

reverse process has some importance in astrophysical situations if we explore the possibility of electromagnetic wave emission when a gravitational wave interacts with a dipole magnetic field, as we can find then fields in peculiar astrophysical objects as neutron stars and black holes. If we apply these considerations to a model of galactic center consisting of a cluster of neutron stars, we may consider the possibility of the interaction of gravitational waves produced in the cluster itself with the magnetic dipoles of the neutron stars. The gravitational waves are produced in the cluster during the collisions of neutron star pair and one obtains for the whole cluster a power of about

P B1031_–1032

Finally, if we apply these results to the magnetic field of the Earth, with the assumption that a flux of gravitational waves arrives on the Earth from the galactic center, we have an emission of electromagnetic waves about P B1024_–1023_{ergOs at the}

same frequency.

6. – Summary and conclusions

The electron scattering processes in four types of static exterior electromagnetic fields are considered in a thorough and unitary manner, and the corresponding cross-sections are calculated. The particular results concerning the photon-graviton

transformation in electrostatic, magnetostatic and magnetic-dipole fields are

generalized. The process of graviton photoproduction in an exterior electric-dipole field is studied. This process has not been approached in the literature till now and we believe that it exists only as a quantum effect. We note that a remarkable analogy between the cross-sections of Dirac-particle scattering and the photon-graviton transformation is worthy to be pointed out. As an application of the results we obtained in this paper, we make some observations and astrophysical evaluations, which are consistent with those obtained by other authors.

As a final comment, we note that a more interesting but much more difficult problem would be the approach of the processes studied in this paper in the presence of strong sources, like black holes, for instance. We intend to do some future research in this field.

* * *

We thank Prof. Dr. I. GOTTLIEB for some stimulating discussions and useful

suggestions. We are also grateful to Prof. Dr. I. MERCHESfor pointers regarding our literary style.

R E F E R E N C E S

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