Indian Journal of Pure & Applied Physics Vo1.37, March 1999,-pp. 163-169

Scattering of a photon by a dyon Prern Prakash Purohit, V P Pandey & B S Rajput •

Department of Physics, Kumaun University, NainitaI 263 002

Received 31 July 1998; accepted 29 October 1998

Undertaking the study of scattering of photon by monopole and dyon it has been shown that photon associated with monopole and dyon has enormously high energy. It has also been shown that two photons are necessary f~r Compton scattering of dyon through S-matrix expansion. Study of bound state of a dyon and an anti-dyon has also been carried out and it has been shown that this state is very short lived and decays in to four or six photdns depending on the spin statistics of the dyons involved.

1 Introduction

Keeping in view the enormous potential importance 1-7 of monopoles and dyons a self­consistent quantum field theory of generalized electromagnetic fields associated with dyons has been constructed8 by using two four-potentials to avoid the use of arbitrary string variables and assuming the generalized charge, generalized four current and generalized four-potential associated with dyons as complex quantities with their real and imaginary parts as electric and magnetic constituents . Extending this work in the present letter we have undertaken the study of Compton scattering for monopoles and dyons, their S-matrix expansion for Compton scattering and bound state of dyon and anti-dyon i.e. anti-dyonium.

In order to study Compton scattering of a monopole and photon we assume that a monopole is at rest and there is a photon incident on it. It is reasonable to expect that incident photon having very low energy will make little impact on it. In other words, for observing Compton shift in photon­monopole scattering we need a photon of very high energy. When this highly energetic photon collides with monopole which is assumed at rest in the laboratory coordinate system, the photQI1 transfers a

·Chairman, UPState Council of Higher Education, Lucknow *Vice-Chancellor, Kumaun University, Nainital 263 00 I

part of its energy to the monopole which recoils witt

a velocity v in a direction making an angle () with the direction of incident photon. The photon it self with

the reduced energy, is scattered in a direction ¢ with

the original direction . These scattered photon constitute the scattered modified radiation . Applying the laws of conservation of energy and momentum and proceeding in usual way we obtained Compton shift.

... ( I)

where Mrna is the mass of monopole 9. For o

observing Compton shift of 0.02428571 A 10

monopole photon scattering we need a photon of

energy khu{9.332885 xl 0- 12 Joule) , where

k = M rna / M and M is the mass of electron,

which is very high in comparison to the energy

hU{1.989 x 10- 15 Joule) of a photon that makes o

corresponding Compton shift of 0.02428571 A In

electron photon scattering . This seems to be reasonable as the mass of monopole IS

4.69225 x 103 times greater than that of electron.

For dyon-photon scattering we need two photons, one of which is responsible for scattering by electric charge of the dyon while another one is responsible for scattering of magnetic charge of dyon. This

164 INDIAN J PURE APPL PIWS. VOL 37, MARCH 1999

concept of two photons has also been used in S-matrix expansion for dyon-photon scattering. It is just not possible to get Compton scattering for magnetic charge on dyon with the help of ordinary photon and vice-versa. This is shown graphically in Fig. I. Applying the law of conservation of energy, we get

... (2)

incident photons (~scattered photons ----------

Fig. I - Black line denotes ordinary photon, red line denotes highly energetic photon, [This convention of denoting different photons will remain same throughout the letter.]

Now applying the law of conservation of momentum, we get

(1 +k)hu = c

(1 + k)hu m"v coscjl + -( 2 2)1/2 cose,

c 1-·v Ic

along X-axis

and

(1 + k )hu . rna V • o = sm cjI- ( 2 2 )1/2 sm e

c I-v Ic

along Y-axis

... (3)

... (4)

with the help of these equations, we get the change in

wavelength (A" - A) as

(l+k)h( ) ~A = --- 1 - cos~

rn"c .. . (5)

which is Compton shift for the dyon . The direction of scattered dyon and energy are given by the following expressions

and

1 tane = --cot~ 12

1+ a

where

(1 + k)hu a----·

- m c2 o

.. . (6)

.. . (6a)

These results show that one of these two photons IS ordinary one which is responsible for y + e ~ y + e scattering and its energy IS

hU{I.989 x 1O-I~ Joule) while the second is highly

energetic photon responsible for y + g ~ r + g scattering and its energy IS

khu{9.332885 x 10-12 Joule) .

Let us now analyze the S-matrix expansion for dyon-photon scattering which may be written as 10

~ ., (-;)"" '" '" of ~ ( ) ~ () ~ ( )] s= L -- Jdl l JdI2 •••• Jdl,,~ LHI II HI h .... H I In If=U n! -z, -JO -21

... (7)

where H/(/) is the perturbation Hamiltonian In

interaction picture. The Lagrangian density for interaction of relativistic dyons may be written as

i(x)=

- ~(x~y ~ {a~ - i Re(q ·V~)} + m]\V(x)

... (8)

1

y

PUROHIT et al.: seA TIERING OF PHOTON BY OYON 165

where q is the charge of dyon i.e. q = e - ig and VI!

is the potential given as VI! = AI! - iBl! . We may

easily get the interaction Hamiltonian density from the above Lagrangian as

A

HI{X}=

-ieN(~)~et -igN(~gB~Jx ... (9)

Substituting the value of Eq. (9) in to Eq. (7), we get

s = t (- It en fd 4xI fd 4x2 .. .. fd 4xJ'[(vi) r/fe t, n =o n.

(vi) r/fe t, .. .... (vi) r/f. L ] ~ (-It n fd 4 fd 4 fd 4 T' + L..,.-,-g XI X 2 .... X n ·

n=o n.

[(vigBr/fg t, (vigBr/fg t ...... (WgBr/fg t]

The first order S-matrix , S(I) given by

8(1) = - e fd 4xT' k 11 IIF } ~e 'f' e x

- g fd 4xT' ~ gB ~g L

... (10)

... ( II)

gives nse to the eight basic processes (Feynman diagrams) which do not lead to any physical processes as they would violate the conservation laws. The second order S-matrix is given as

2 A(2) e J 4 J 4 ' S = 2! d XI d X2 T

... ( 12)

By the application of Wick's theorem we have

where

2

SP) = ;! f d 4xI f d 4

x2N

[(~) ~, t (~, 11 ~, L] 2

+ ~! f d 4 XI f d 4

x2 N

[(~g B~g t, (~g Bljtg L ...( 13)

.. . ( 14)

(Compton type scattering)

... ( 15)

166 INDIAN J PURE APPL PHYS. VOL n, MARCH 1999

[dyon-dyon scattering (Moller type scattering) and dyon-anti-dyon scattering(Bhabha type scattering)]

... (16)

[self energy of dyons]

... (17)

[photon self energy]

... (18)

[ no transition]

sF) does not lead to any real proc~s. For

undertaking the study of Compton scattering we may

writeS~2) as

and

... (20)

where

- ig2 J d 4xI

J d 4 x2 \V ~ (XI)Y ~SF (X I - x2)Y uB; (xl)i1: (x2)1jJ; (X2) ... (21 )

- ig2 f d 4 XI f d 4

x2 ~; (XI)Y ~ S,,(XI - x2)y v B~(X2 )B~7(X,)IjJ;(X2 )

.. . (22)

[Compton type scattering by dyon]

_ig2 f d4x,f d4x21jJ;(XI)Y ~ISF

(XI -x2)y )}~~(xl)B:(X2)~;(X2) .. . (23)

PUROHIT et aJ.: seA TIERING OF PHOTON BY DYON 167

- ig2 J d4xl J d4X2~;(XI}y ~

SF(XI - x2}y )~~(x2)B:(xJ~;(X2) ... (24)

[Compton type scattering by anti-dyon]

S~(2) - .2 Jd4 Jd4 ~-( )~ 2e - -Ie Xl X 2 'V e Xl Y ~

SF(X\ - X2)Yu ~~(x2)A:(XI)A~(X2)

_ig2 J d4x;J d4X2~;(XI)Y ~SF (Xl - X2)Y v~;(x2)B:(XI)B~(X2)

... (25)

[dyon -anti-dyon pair creation]

- ig2 J d4xl J d4X2B~(XI)B~(X2)Y ~

SF(XI -X2)Yu~;(XI)~;(X2) ... (26)

[dyon-anti-dyon pair annihilation].

This set of equation shows different diagrams of Compton scattering e.g. Eq. (21) and Eq. (22) corresponds to Compton scattering by dyons. This is shown graphically in Fig. (2).

We have further analyzed the bound state of a dyon and an anti-dyon i.e. anti-dyonium and it is interesting to observe that we get four photons or six photons emission instead of usual two or three photons emission of positronium atom of quantum electrodynamics. Let (j 4y be cros~ section for four

photons annihilation of a free pair averaged over the spin directions of both the particles. In non­relativistic limit

... (27)

where v is the relative velocity of the particles. The annihilation probability w4y IS obtained on

multiplying a 4y by the flux density v\'V( 0 )\2. The

normalized wave function 'V(r) , of the anti­dyoniumn ground state is given as

Fig. 2 - Where, A" + and A~' represent absorption and

emission of photon corresponding to electric charge on dyon and

B" + and B~' represent absorption and emission of photon

corresponding to magnetic charge on dyon

_ 1 _% _ 2h2 'V(r)- ( )v q ,a- m{ 2 2) na 3 / 2 e + g

where a is the Bohr radius 11 of anti-dyonium and m is the mass of the dyon. The mean decay probability

W4y is .related to the para-anti-dyonium decay

probability W 0 in the following manner

.. . (28)

so

With the help of these equations, we get

168 INDIAN J PURE APPL PHYS. VOL 37, MARCH 1999

m{e lO + e4 g6 + 3e6 g4 + 3e8 g2) W =

o 21;16C 3

m{glO + g4 e6 + 3g8e2 + 3g6e4) + .

2tz 6c3

The lifetime for para-anti-dyonium is

2tz[1 1] '0 = --2 -5 +-5 ::::~ 2.648963 x 10-14 s me a e a g

and lifetime for ortho-ant i-dyonium is

97tn [I I j -- II tl = 2( 2 ) 6+6 ~ 2.9866 x 10 s. 2me 7t -9 a a

e g

This can be represented digramatical~ as shown in Fig. 3.

I K, K,

~ , .. P-( q)

I I I

L4 - P+(-q) ill K2 K2

EJ ':, p .. (q) -P+(-q) P-(q)

-p.- (-q) (b) S i x - photon d .. cay

(a) Four - photon d~cay

with the results 12-16 , where it has been conjectured that exact gauge group is not

SU(3) x SU(2) x U(I) but it could be

SU(3) x SU(2) x U(I) x U' (1). Moreover,

observation of monopoles and dyoills in heterotic string theory 17 confirms that very high energy is associated with these particles. Recently, Ignatiev and Joshi 18 tried to introduce massive photon to make the existence of magnetic monopole c:ompatible with the finite range electrodynamics. We are, however, not introducing massive photons but we are concluding that photon associated with monopole is not ordinary and it is certainly different from photon associated with electric charge.

The lifetime for anti-dyonium bolth the cases of para and ortho-anti-dyonium is very small in comparison to positronium lifetime. It implies that this bound state is very small lived and it decays in two modes. for para-anti-dyonium, we get almost instantly four photons, two of them will be extra­ordinary and for ortho-anti-dyonium, we get six photons, three of them will be extra-ordinary. Owing to the high energy of extra-ordinary photon it would be suggesting to design experiment to observe monopoles in the concerned energy range.

References

Dirac PAM, Proc R Soc London, A 133 (1931) 60; Phys Rev. 14 (1948) 817.

Fig. 3 - The parameters k I, k2 and k3 are the momenta of 2 Schwinger J, Phys Rev , 144 (1966) 1087; 151 (1966) 1048, Witten E, Phys Lett B, 86 (1979) 283 . photon corresponding to electric charge and kl' , ki and 3' are

the momenta of photon corresponding to magnetic charge 3

4

5 The foregoing analysis shows that the

introduction of monopole and dyon has completely 6

revolutionalized the quantum electrodynamics. For magnetic monopole, we need a photon of extremely 7

high energy and the energy required to pair produce monopole . and dyon is also very high 8

(- 4.803691 x 103 Me V) which is reasonable as

the masses involved are very large. This extra ordinarily energetic photon is inevitable if we wish to incorporate magnetic monopole in quantum 9

electrodynamics. Our results are also in confirmation

Dokos C & Tomaras T, Phys Rev D, 21 (1980) 2940.

Preskill J P, Nucl Pari Sci, 34 (1984).

Mandelstam S, Phys Rep C, 23 (1976) 245;Phys Rev D , 19 (1979) 249.

HooftG t', Nucl Phys B, 138 (1978) I; Nucl Phys B, 190 (1981) 455 ; 153 (1979) 141.

Rubakov Y, JETP Lett, 33 (1981) 645 ;Nucl Phys B, 203 (1982)311.

Rajput B S & Prakash Om, Indian J Pure & Appl Phys, 16 (1978) 593; Rajput B S & Joshi 0 C, Hadronic J , 4 (1 981) 1805; Rajput B S, Kumar S R & Negi 0 P S, Ind J Pure & Appl Phys, 21 ( 1983) 638; Lell Nuovo Cim, 36 (1983) 75; Rajput B S & Bhakuni 0 S, Lell Nuovo Cim, 34 (1982) 509; Nuovo Cim A , 92 (1986) 72.

Aker 0, II Nuovo Cim A, 105 (1992) 935; Inl J Theo Phys , 26 (1987) 613.

10 Rajput B S & Bhakuni D S, PhD thesis, Physics Department. Kumaun University, Nainital, awarded in 1986; Pant P C. Pandey V P & Rajput B S, II Nuovo O m A 110 (1997) 829.

II Pandey V P, Chandola H C & Rajput B S, Can J Phys. 67 (1989) 1002.

12 Olive D I, Nucl Phys B, 113 (1976) 413 ; Comgan E & Oli ve D I, Nucl Phys B, 1 \0 (1976) 237.

13 Goddard P Nuyts J & Olive D, Nucl Phys B, 125 (1977 ) I.

169

14 Rubakov V A, Phys Lett B. 120 (1983) 19 1.

15 Barr S M. Reiss D B & Zee A. Phys Rev Lett, 50 (1983 ) 3 17.

16 Rana J M S. Chandola H C & Raj put B S. Prog Theo Phys, 82 ( 1989) 153 .

17 Sen Ashok. Phys Lell S , 303 ( 1993) 22.

18 Ignatiev A Yu & Joshi G C. Mod Phys Lell A, II ( 1996) 2735.