Quantum field-theoretical electromagnetic and gravitational two-particle potentials

P H Y S I C A L R E V I E W D V O L U M E 2 1 , N U M B E R 8 1 5 A P R I L 1 9 8 0

Quantum field-theoretical electromagnetic and gravitational two-particle potentials

Suraj N. Gupta and Stanley F. Radford Department of Physics, Wayne State University, Detroit, Michigan 48202

(Received 6 November 1979)

The fourth-order static electromagnetic and gravitational two-particle potentials are derived from the scattering operator by using the techniques of the standard field theory. Integrals over four-vector variables are evaluated in a covariant manner, while in earlier treatments such integrals are usually converted to noncovariant forms. Our treatment is more direct, and we are able to carry out regularization and renormalization of divergent integrals unambiguously where necessary. Moreover, calculations of multiple parameter integrals are reduced to elementary integrations in the static and long-range approximations. Our quantum field-theoretical results are in complete agreement with the classical potentials.

I. INTRODUCTION electromagnetic interaction, we sha l l employ dimensional regularization6 for the gravitational in-

The derivation of higher-order two-particle po- teraction. Infrared divergences will be handled by tentials presents an interesting a s well a s a com- introducing a parameter X in field contractions to plicated problem in quantum field theory. We give a nonvanishing m a s s to a photon o r a graviton, sha l l show how the fourth-order s ta t i c e lectro- and letting X - 0 in the final resu l t s . The two- magnetic and gravitational potentials can be de- part ic le potentials a r e expressible in various r ived f rom the sca t te r ing operator by using the equivalent fo rms ,7 but for convenience we shal l techniques of the s tandard field theory. We shal l always obtain them by reducing the scat ter ing m a - consider the interaction of two s c a l a r par t ic les , t r i x elements to the s implest possible fo rm. but our t reatment is a l so applicable to other p a r - t ic les .

11. ELECTROMAGNETIC SCATTERING OF SCALAR The Integration procedure in our derivation will PARTICLES

be quite different f rom that in the e a r l i e r t rea t - ments of the electromagnetic' and gravitational2 F o r the electromagnetic interaction of a complex interactions of two-particle s y ~ t e m s . ~ We shal l s c a l a r fleld U , the coupling t e r m s in the Lagran- evaluate integrals over four-vector variables in a gian density a r e covariant manner , while in e a r l i e r t reatments such integrals a r e usually converted to nonco- L,,, = ( ~ ~ / ~ R ) A , ( ~ ~ u * u - u * ~ , u )

variant fo rms . (3ur covariant t reatment is more - ( ~ / c ~ ~ ~ ~ ) A , , ~ u * u . ( 2 . 1 ) di rec t , and allows us to c a r r y out regularization and renormalizat ion of divergent Integrals un- T o derive the scat ter ing mat r ix elements , we

a l s o requ l re the contractions of the field opera- ambiguously where necessary. This fact is e s - pecially important for the derivation of the fourth- tors in the interaction picture

. .

o r d e r gravitational potential, where a diagram A,(x)'A, (x ' ) ' = - i c t i6 , ,DF(x - x') , requir ing renormalizat ion yields a finite contribu-

u(x)'u*(x')' = - i c t ih , (x - x') , ( 2 . 2 ) tion. Fur ther , we shal l simplify multiple-param- e t e r integrations by making s tat ic and long-range and the Fourier decomposition of the s c a l a r field approximations. These approximations will a l so enable us to d rop nonrenormalizable t e r m s a r i s ing U = V - 1 1 2 ~ ( ~ t i / 2 p , ) ~ / ~ [ a ( j ' 5 ) e ' ~ ' + b * ( f i ) e - i t . 1 , f rom one-loop diagrams in the gravitational inter- ..

P actions of s c a l a r particles. Our quantum field- ( 2 . 3 ) theoret ical resu l t s will be shown to be in complete with agreement with the classical electromagnetic and

Po = ( / 1 2 + F ) 1 / 2 . gravitational potentials. The importance of the fourth-order s ta t i c gravitational potential in the Since we shall re ta in the constants c and t i , it t reatment of the advance of the perihelion of should be noted that 3, Po, and p a r e re la ted to planets i s , of course , well known. the momentum 5 , energy E, and m a s s m of the

We shal l follow a s t raightforward approach4 for s c a l a r par t ic le a s the derivation of the two-particle potential f rom the scat ter ing operator . While regularization by f i =?/ t i , po = E / C E , p = ~ c / E . ( 2 . 5 )

means of auxiliary fields5 is sufficient fo r the Also, we shal l deal with two different s c a l a r

21 - 2213 @ 1980 The American Physical Society

2214 S U R A J N . G U P T A A N D S T A N L E Y F . R A D F O R D 21 -

f ie lds , and use the indices 1 and 2 for them where necessary.

Le t us consider the electromagnetic scat ter ing of two s c a l a r par t ic les of m a s s e s m, and m,, whose propagation four-vectors a r e p and q in the initial s ta te and p' and q' in the final s ta te , and le t

k =p' - P = -(ql - q ) . (2.6)

The diagrams for the fourth-order electromagnetic scat ter ing a r e shown in Fig. 1, where tad- pole diagrams with vanishing contributions have been ignored. F o r the t reatment of the fourth- o rder d iagrams , we shal l use not only the cen te r - of-mass sys tem but a l so the s tat ic approximation by setting

* p + h l = O , 5 + $ = 0 , (2.11)

We shal l use the center-of-mass sys tem, s o that and it follows from (2.7) and (2.11) that * P=-G, $I=-4', P ; = P ~ , q i = q o , .. + - * (2.7) 2 , 2 7 2 7 2 f

p=-L; $ '=LE G='k q e = - L k k ='p' -5 = -('ql -G), ko = O ,

1'2 1 / 2 , (2.12) ~ ~ = p b = ( p : + i z ~ ) ~ ' ~ , q O = q ~ = ( p 2 2 + q k ) The second-order contribution of the scat ter ing

operator for the above process is

S 2 = - V - 2 ( i / ~ f i ) ( 2 n ) 4 6 ( p + q - p ' -9')

x a?(fi1)%*('s')V2(~)azC9)al(3), (2.8)

with

which reduces in the center-of-mass sys tem in the nonrelativistic approximation to

It is understood that we a r e interested here only in the static-long-range interaction, s o that we shal l take I k / << p l and I k I<< p2 , and-retain only t e r m s involving inverse powers of / k l in the sca t - ter ing mat r ix elements.

We shal l now consider the contributions of the various diagrams in Fig. 1, where the external l ines a r e to be labeled with the propagation four- vectors p , p', q , and q' in al l possible ways. Thus, the diagram (c) gives r i s e to two distinct possibilities, the sixth diagram in the s a m e figure gives r i s e to four distinct possibilities, and s o on.

A. Diagrams (a) and (b)

The contribution of the scat ter ing operator for the diagram (a) is expressible in the form (2.8) with v,(G) replaced by

ie4 1 'u, =-

[ ( ~ P I - I). (29' + ~)I[(P' + P - I). (ql + q + 111 4(2n)'efi (poqop~q~)1 /2 / -9' - p - 1)' + h2][(p' - 1)' + p:][(ql + 1)' + p,l] '

By using (2.12), V , can be reduced to

with

a o = 4 p , ~ 2 , ~ , = ~ ( P ~ - C L ~ ) , G = ( P , + c ~ , ) ~ / ~ P , P ~ , % = 2 , (2.15)

where t e r m s of higher o r d e r s in E and 1 have been dropped in the numerator because, a s explained in the Appendix, such t e r m s do not generate inverse powers of in 21,. The integral (2.14) is of the form (Al) , and according to (A23), i t yields

Similar ly, for the diagram (b),

ie4 2, I { d l [(ZP' - 1). (29 - 1)1[(P' + p - 1). (9' +q - 1)l

- 4(2n)4~f i ( ~ ~ q , , P ~ q ~ ) ~ ' ~ (1' +x2)[(p' - p - 1)' +h2][(p' - 1)' +p12][(q - 1)' +p22] '

which can be obtained f rom (2.13) by the replacements q'- -q and q - -q'. With the use of (2.12), i t is possible to reduce 21, to

Q U A N T U M F I E L D - T H E O R E T I C A L E L E C T R O M A G N E T I C ... 2215

with

bo=4~11112, b 1 = - 4 ( ~ 1 + ~ 2 ) , (2.19)

because, as_also explained in the Appendix, the t e r m s dropped i n the numerator do not generate inverse powers of 1 k 1 in O,. Then, according to (A25),

From (2.16) and (2.20), the total contribution of the diagrams (a) and (b) is found to be

B. Diagram ( c )

The contribution of the scat ter ing operator fo r the diagram (c) gives

v, =u: +v: , where

ie4 v1 = I JdZ ( 2 ~ ' - I). (P' +p - 1) ~ ( ~ s ) ~ c E (poqd;q;)1'2 (z2 +xz)[(p' - P - z ) ~ +x21[(p1 - zI2 +pl21 '

and v," can be obtained f rom by interchanging the ro les of the two part ic les . In view of (2.12), u: reduces to

where t e r m s involving powers of G and 1 have been dropped in the numera tor , again for the reason that-such t e r m s do not give r i s e to inverse powers of Ikl in v:. Fur ther , i t follows f r o m (2.24), by v i r tue of (A27), that

FIG. 1. Fourth-order electromagnetic scattering of scalar particles. Solid and broken lines represent scalar particles and photons, respectively.

I

and s ince V: can be obtained f rom V: by replacing C12 by i J . 1 7

C. Remaining diagrams

The remaining d iagrams in Fig. 1 can be t reated in a s imi la r manner. It is well known that a l l the divergences involved in these d iagrams can be renormalized, and we have verified that af ter r e - normalization they do not yield any relevant t e r m s .

111. ELECTROMAGNETIC TWO-PARTlCLE POTENTIAL

The second-order electromagnet ic potential i m - mediately follows f rom (2.10), and with the use of (2.5), we get

However, for the fourth-order potential, i t is necessary to obtain -.

V4(k) =v, + v , +u, +6V4, (3.2)

where V,+V, and V, a r e given by (2.21) and (2.26),


and for any 'p and 5". In the center-of-mass sys tem, p =pZ2 and 5.'

1 6'1: =-- u2(Sf, h")u2(5", h ) =GO2, and the denominator in (3.3) can be expanded 4 ( 2 ~ ) ~ I dh" ( p , +q0)cR - (p: +qz)cli 7 a s

(3.3) (po+qo) - ( p ; +q;)

with (2.10) expressed in the fo rm 1 1

(54-pn4) +. . . e2

o r

Moreover, setting -.I * -. p - 5 = k , h N - 5 = l ,

we have in the s ta t i c approximation -. - * * p = - L k 2 , $h'='s 2 7 $ " = I -$k

With the use of (3.4)-(3.6)) i t is possible to express (3.3) a s

Since the-above integra_l can give r i2e to inverse powers of only when li 1. 121, t e r m s involving powers of k and 1 higher than k2, i 2 , o r H . 1 can be dropped in the numerator , and thus

Integration over the i space leads to

where A is given by (A9), and then upon integration over ul and u, with the use of (A15),

811~2 (Pl +PZ)P P l +P2 PlP2

(3.7)

Substitution of (2.21)) (2.26), and (3.7) into (3.2) gives

s o that the fourth-order s ta t i c electromagnetic potential is

which a g r e e s with the c lass ica l result. '

IV. GRAVITATIONAL SCATTERING OF SCALAR PARTICLES

The gravitational interaction is known to be much more complicated than the electromagnetic interaction. However, s ince our fo rmal i sm is quite general , the electromagnetic and gravitation- a l potentials will be derived in a s i m i l a r manner , and often the s a m e symbol will be used to denote corresponding quantities in the two cases .

F o r a gravitational field interacting with a neu- t r a l s c a l a r field, the coupling t e r m s i n the La- grangian density in n-dimensional space-t ime a r e given by8

with

21 - Q U A N T U M F I E L D - T H E O R E T I C A L E L E C T R O M A G N E T I C ...

where C is the gauge-compensating field.g I t is convenient to rewr i te ( 4 . 3 ) a s

~ y n t = - + K 6 ~ v , ~ p , K o , ~ ~ h p u a a h X p a ~ h K o + O ( K ' ) 9 ( 4 . 5 )

where

The contractions of the field operators in the interact ion picture, required for the derivation of the scat ter ing mat r ix elements , a r e

U(X)'U(X') ' = - i c E h , ( x - x ' ) , h,,(x)'hX,(xl) ' = - icEbp, , X p D F ( ~ - x') , ( 4 . 7 )

c , ( x ) ' c $ ( x l ) ' = - i c E 6 , , D F ( x - X I ) ,

where

6 , U , x p = ~ p x 6 v p + ~ , p 6 " X - ~ P U ~ X P ' ( 4 . 8 )

and the Four ie r decomposition of U takes the form

F o r simplification of the scat ter ing mat r ix e le - ments , i t is useful to note the identity

with

which reduces in the center-of-mass sys tem in the nonrelativistic approximation to

where we have s e t n = 4 . The diagrams for the fourth-order gravitational

scat ter ing a r e shown in Fig. 2 , where some tad- pole and leaf diagrams with vanishing contributions have been ignored. We shal l obtain the contributions of these diagrams to u,(z) in the center-of- m a s s sys tem in the s tat ic approximation by following the s a m e procedure a s used for the electromagnetic case.

A P u 6 y u , ~ p = (A, , +A", - Aaa6pv)6pi6up. ( 4 . 1 0 )

We shall now consider the scat ter ing of two s c a l a r par t ic les a s in Sec. 11, but with the use of the gravitational couplings ( 4 . 1 ) . The second- o rder contribution of the scat ter ing operator fo r the above process is >-o-< >,.,<

FIG. 2 . Fourth-order gravitational scattering of scal- S2 = - v - " ( i / c E ) ( 2 n ) " 6 ( ~ +q - p' - q') ar particles. Solid and wavy lines represent scalar par-

ticles and gravitons, respectively, while dotted lines x ~:($')@(G')V~(~)%("~)U,@), ( 4 . 1 1 ) represent gauge-compensating particles.


A. Diagrams (a) and (b)

The contribution of the scattering operator for the diagram (a ) is expressible in the fo rm (4.11) with 'LIZ(%) replaced by

By using (2.12), retaining only the relevant t e r m s in the numerator as in the electromagnetic case, and setting n = 4 in integrals free f rom ultraviolet divergence, V, can be reduced to

with

ao=p13p23, a l = 4 ~ 1 2 ~ 2 2 ( ~ 1 - p 2 ) , a ~ = Z ~ l ~ 2 ( ~ 1 1 2 + ~ 2 ' ) + ~ 1 2 ~ 1 2 ' , a j = 2 ~ 1 2 1 ~ 2 ~ )

and then, upon application o f (A23),

Similarly, for the diagram ( b ) ,

x (p ' - 1 ) . (q - 1)p. q1 + (8' - 1 ) . q'(q - 1 ) . p - (P' - 1 ) . P(9 - 1 ) . 9' [ - p ; ( q - L ) . q 1 - p ; ( p 1 - ~ ) . ~ - - n - 2 72 P:u22]

(p ' - I ) . (q - 1)p'. q + (p' - 1 ) . q(q - 1 ) . P' - (P' - 1 ) . ~ ' ( 9 - 1 ) . 9

which can be obtained f rom (4.14) by the replacements q'- -q and q - -qt. It is possible to reduce Z1, to

with

bo ' ~ J . ~ ~ P ~ ~ , bl + P A , (4.20)

where only the relevant terms are retained in the numerator. Then, according to (A25),

From (4.17) and (4.21), the total contribution of the diagrams (a ) and (b) is found to be

21 - Q U A N T U M F I E L D - T H E O R E T I C A L E L E C T R O M A G N E T I C . . .

B. Diagram ( c )

The contribution of the scat ter ing operator for the diagram (c) gives

v , = v ; + v ; ,

where

and V: is obtainable from 0; by interchange of the ro les of the two part ic les . By using ( 2 . 1 2 ) again, retaining only the relevant t e r m s i n the numerator , and set t ing n = 4 , 2); can be

reduced to

which yields, with the application of ( A 2 7 ) ,

and consequently

C. Diagram (d)

F r o m the contribution of the scat ter ing operator for the diagram (d), i t is found that

'u, =v: +v;, where

with

~ K ' C K d l ' ~ v ( p ' p f ) = ~ I ( z 2 + x 2 ) [ ( ~ ' - p - + x z ] [ ( p l - z ) ~ + p ; T

x [ ( P I - ~ ) , P ~ + P , ( P ' - 11,- ( P . P' - P . l + p , 2 ) 6 , , 1

x [(PC l ) h ~ b + P ) x ( p ' - Z I P - ( P I 2 - P ' . 1 + p 1 ~ ) 6 ~ p I

X [ l a ( p ' - P - 1 ) ~ ( 6 ~ v , ~ p , ~ o , c i ~ + 6 p u , K o , ~ p , , a )

+ ( p l - p - l)ct(P - P ' ) B ( 6 ~ p , r o . p . a B + 6 h p , p v , ~ o , B a ) + ( P - ~ ' ) a ~ 0 ( ~ K o , ~ v , X p , a 0 ~ 6 ~ o , h p , p v , B u ) l ,

( 4 . 3 0 ) while 2):: can be obtained from 2); by interchanging the ro les of the two part ic les .

According to ( 2 . 1 2 ) , .. q , q : + q / q u - ( 4 . 4 ' + p 2 2 ) ~ , u = - 2 j 1 2 2 ~ ~ 4 6 v 4 + O ( k 2 ) ,

and s ince we a r e interested only in inverse powers of in^;, ( 4 . 2 9 ) reduces to

I t i s now necessary to evaluate A , , ( P , P ' ) up to f i r s t o rder in F/, and perform renormalization. We observe that ( 4 . 2 9 ) coincides with ( 4 . 1 2 ) if A, , (P ,P ' ) is replaced by

2220 S U R A J N . G U P T A A N D S T A N L E Y F . R A D F O R D

and therefore for renormalizat ion we s e t

where C is a renormalization constant, and the physical t e r m ~ b ( p , p ' ) vanishes for p' =p. It follows f rom (4.33) that, up to f i r s t o rder in PI ,

and, on set t ing p' =p,

1 A ~ ~ ( P , P ) = ( ~ ~ - ~ ) P , ~ C ,

s o that

A$4(P,Pf) =&~(P,P ' ) - A44(p,P).

Further , A4,(p,pt), given by (4.30), can be reduced with the use of (2.12) to

where t e r m s of higher o r d e r s in k = (G, 0 ) and I have been dropped in the numerator because, according to the t reatment in the Appendix, such t e r m s do not contribute to AC,(p,pl) up to f i r s t o rder in Fl. We a l so find, f rom (4.6),

SO that A4,(p,pf) becomes

and then the application of (A29) gives fo r n = 4 ,

After dropping the renormalizat ion constant we get , by substituting (4.36) into (4.32),

and therefore

D. Remaining diagrams

The remaining diagrams in Fig. 2 can be t reated in a s i m i l a r manner. In the s ta t i c and long-range approximations, a l l nonrenormalizable divergences in the contributions of these d iagrams a r e found to drop out, and af ter renormalization we do not obtain any relevant t e rms .

I t is interest ing that both electromagnetic and gravitational couplings a r e such that we can s e e

whether a diagram with a loop will yield relevant t e r m s simply by finding out whether the denomina- t o r in the loop contribution can generate inverse powers of I;/. F o r instance, in the contribution of the ver tex loop of the diagram (d), the integral a r i s ing from the denominator is

while corresponding to the ver tex loop of the seventh diagram in the s a m e figure, we have

21 - Q U A N T U M F I E L D - T H E O R E T I C A L E L E C T R O M A G N E T I C ... 2221

o r vanishing, and therefore (4.39) can give r i s e to

1 inverse powers of $1. On the other hand, in

i.211 dul lul d"2 ptu: - P ( b 1 -U,)Z. (4.40) (4.40) the coefficient of 0; cannot vanish without the vanishing of the coefficient of g2 , and there -

In (4.39), i t is possible fo r the co_efficient of pO2 fore this integral cannot give r i s e to inverse to vanish while the coefficient of k2 remains non- powers of $1 .

V. GRAVITATIONAL TWO-PARTICLE POTENTIAL

The second-order gravitational potential, which immediately follows from (4.13), is

where

However, to obtain the fourth-order potential, i t is a l so necessary to derive 6U4, which is given by (3.3) with (4.13) expressed in the form

f o r any "p and 9". By following the procedure of Sec. 111, i t is found that

o r , retaining only the relevant t e r m s ,

Integration over the space leads to

and then upon integration over u, and u, with the use of (A15),

From (4.22), (4.27), (4.38), and (5.4) we obtain

s o that the fourth-order s ta t i c gravitational potential is

which a l s o agrees with the c lass ica l r e s u k 7 Besides the purely electromagnetic and gravitational two-particle potentials, i t i s a l so possible to obtain

a mixed potential of o r d e r Ge2 by quantum field-theoretical methods.1°

S U R A J N . G U P T A A N D S T A N L E Y F . R A D F O R D

ACKNOWLEDGMENT

This work was supported in part by the U . S . Department of Energy under Contract No. EY-76-S-02-2302,

APPENDIX: EVALUATION OF INTEGRALS IN STATIC APPROXIMATION

W e shall describe the procedure for the evaluation o f multiple integrals encountered in our treatment of the fourth-order diagrams, and show how calculations can be reduced to elementary integrations by making the static approximation and dropping terms that cannot give rise to inverse powers of p/ in the scattering matrix elements. W e shall f i rs t treat the most complicated integral, and then discuss the other integrals.

A. Integration over I

Consider the integral

where the a's are independent o f ck and I . By combining the denominators, I , can be expressed as

Then, af ter a shif t o f the origin as * i -i +G(u, - h2) ,

10 - 10 +Po% - P o ~ 3 - 9oU3 2

and integration over the 1 space,

(A2)

with

~ = n , + [ a , +(a, +a4)ul- (a, +Ba4)u2P

- ( POu3 +qo% - P O u , )

+(a, +a,)( POu3 +qou3 - P o u ~ ) ~ , (-43)

D = ( P0u3 + q0u3 - PO%

+G2[(u, -u2)(1 - u l ) - &4221 +k2(1 - u2) . (-44)

B. Integration over u3

For integration over u,, we set

( ~ 0 + 4 0 ) ~ 3 - P 0 ~ 2 = ~ , 645)

so that (A2) takes the fo rm

with

and the X 2 t e rm in the denominators has been dropped for convenience.

We perform integration over s as

("I") = & [. -tan-' - 40uz

where the appearance o f the term n/lcklh is related to the fact that the limits o f integration of s have opposite signs. Similarly, upon integration over s , (A7) yields

f ( ~ 1 , u z ) = f o ( ~ l ) ~ 2 ) + f l ( ~ l ) ~ 2 ) + f2 (~1 , u2) , '(A10)

where

f l(ul , u,) = - +- tan-' - ( 1 ) C14)

where while f,(u,,u,) can be obtained f rom fl(ul , u,) by the replacements go-Po and a,- -a,.


C. Integration over u2 and u ,

According to (A6) and (AlO), I, i s expressible a s

la =Io +I, +I, , where

and I, and I, have s i m i l a r meanings. The integral I, can be readily evaluated by using

where the imaginary par t s have been dropped, and thus

I

Fur ther , when expanded in powers of $1, f,(u,,u,) given by (A12) does not generate any in- v e r s e powers. This shows that f, can possibly give r i s e to inverse powers of (kl only when u, = $J/g0, because then a n expansion of f,(u,, u,) in powers of lG( is not permissible . We therefore s e t

with

A ~ ~ = u ~ ( ~ - u ~ ) , 6 = - ( 1 - ~ ~ ) ~ ~ - & ~ ~ , (-418)

expandf,(u,,u,) in powers of 6 , and re ta in only those t e r m s in f,(u,,u,) which behave a s $I-" with n .)- 2 for u, s $ I/qo. This gives

Then, ca r ry ing out integration over u,, expanding the r e s u l t in powers of $1, and retaining only I2 = - &(PO "4 +go) [ I n ) nZIk2

+ . (A22)

t e r m s involving the inverse powers , we find F r o m (Ale) , (A21), and (A22) we obtain, by using

i a2 I, = -

2qo(Po +go) po = @, +E2/8@, + . . . , ~ a b ( 1 - u ~ ) nu, go = 11, +E2/8@, + . . . , [ldu1Le +8go F~A; 2 $ bo 1. (A20) and again retaining only the relevant t e r m s ,

Since the f i r s t two t e r m s in (A20) diverge upon integration over u,, i t i s necessary h e r e to take into account the X2 t e r m in the denominators, s o that A,' becomes

D. Discussion

and, with X - 0 af ter integration,

Thus, with only the relevant t e r m s retained,

and s imi la r ly i t is found that

The integral (Al) a r i s e s in the t reatment of the box diagram, and we have shown that i t yields the r e s u l t (A23) in the s ta t i c and long-range approximations. Fur ther , by following the above procedure together with regularization where nec- essa ryA it can be shown that t e r m s invzlving pow- e r s of k and I higher than G2, 12, o r T. k in the numerator of (Al) do not generate inverse powers of $ 1 in I , , and therefore they can be dropped. In fac t , according to the final r e s u l t (A23), i t i s sufficient to find the coefficients Q, a,, %, and a, in ( ~ 1 ) for the box diagram.

For the c rossed box diagram we have, c o r r e s - ponding to (Al), the integral

2224 S U R A J N . G U P T A A N D S T A N L E Y F . R A D F O R D 2 1 -

where the denominator can be obtained from the denominator of (Al) by revers ing the sign of go. In this c a s e , integration over u3 can be performed in an analogous manner by introducing the variable

Since both l imits of integration of s' have the s a m e sign, we do not get any t e r m s corresponding to (A16), while the t e r m s corresponding to (A21) and (A22) yield

which shows that it is sufficient to find the coefficients b o and b , i n (A24) f o r the c rossed box diagram. In fact , bo and bl can be obtained from -a, and -al by reversing the s ign of p,.

Other diagrams give r i s e to integrals with l e s s complicated denominators which can be t reated by a s impler version of the procedure described above. One of these integrals is

co 1 c = ! ~ ~ ( ~ ~ + ~ ~ ~ ~ ~ - 2 i . ~ + ~ ~ + 2 ) ( ~ ~ - ~ ~ ~ + 2 l ~ ~ ) ' (A26)

Af te r integration over I , i t becomes

where the integrand is s i m i l a r to but much s impler than (A12), and can be t reated in the s a m e manner . Thus, I, can be reduced to

whence carrying out integration over u,, expanding the resu l t in powers of I;/, and retaining only the r e l e - vant t e r m s , we get

I t can a l so be easily verified that t e r m s involving powers of % and 1 in the numerator of (A26) do not gene- r a t e inverse powers of 61 in I,.

hnother integral required for our investigation is

which h a s to be evaluated up to f i r s t o rder in because it is accompanied by a factor l/i;2 in the s c a t t e r - ing mat r ix element. After integration over 1 with the use of dimensional regularization, it is found that

where the divergent constant q is given by

A s in the t reatment of (A12), by retaining only the relevant t e r m s , I , can be reduced to


and then upon carrying out integration over u,, expanding the resu l t in powers of g/, and again retaining only the relevant t e r m s ,

I t can a l so be verified that inclusion of t e r m s involving higher powers of and I in the numerator of (A28) only affects the constant 77 in (A29).

'see A. Nandy, Phys. Rev. D 5, 1531 (1972), and earlier references therein.

'Y. Iwasaki, Prog. Theor. Phys. 2, 1587 (1971); K . Hiida and H. Okamura, ibid. 4 7 , 1743 (1972).

3 ~ n interesting but unusual treaGnt of the two-particle systqm has been given by C . Fronsdal and R. W. Huff, Phys. Rev. D 1, 3609 (1973).

4 ~ e e S. N. Gupta, Quantum Electrodynamics (Gordon and Breach, New York, 1977) , p. 198.

%. N. Gupta, Proc. Phys. Soc. London=, 129 (1953). 6 ~ . G. Bollini and J. J. Giambiagi, Nuovo Cimento E,

20 (1972); G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972).

'B-M. Barker and R . F. O'connell, J. Math. Phys. 18, 1818 (1977). Our results for the electromagnetic and gravitational potentials correspond to their Eq. (22)

with up= a,= -&M. h he expansion procedure for the gravitational field is

the same as that given by S. N. Gupta, Proc. Phys. Soc. London A65, 608 (1952), except that we now use n dimensions inspace-time for regularization, and we have replaced the symbol y,, by h,,.

or the derivation of the gauge-compensating terms within the framework of the standard field theory with indefinite metric, see S. N. Gupta, Phys. Rev. D 14, 2596 (1976). Also, see references therein for earlier derivations of the gauge-compensating terms by other methods.

'OK. A. Milton, Phys. Rev. D 4, 3579 (1971); 1 5 , 538 (1977); F. A . Berends and R. Gastmans, ~ n ~ P h y s . (N.Y.) 98, 225 (1976).

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Quantum field-theoretical electromagnetic and gravitational two-particle potentials