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Double logarithmic asymptotics of quantum electrodynamics
Volume 22, number 5
PHYSICS LETTERS
pothesis would be valid to a good approximation - thereby making practical reductions in the n u m ber of parameters in Regge pole phenomenology. Therefore, should the present communication encourage m e a s u r e m e n t of the above reactions, it has already served its purpose. I would like to thank P r o f e s s o r G. F. Chew for his i n t e r e s t and e n c o u r a g e m e n t . I would a l s o like to thank D r s . V. B a r g e r , C.H. Chan, J. K i r z and W. R a r i t a for d i s c u s s i o n s .
DOUBLE
LOGARITHMIC
ASYMPTOTICS
G. V. F R O L O V ,
V.G. G O R S H K O V ,
15 September 1966
References 1. A.Ahmadzadeh, Phys.Rev. Letters 16 (1966) 952. 2. R.C.Arnold, Phys. Rev. Letters 14 (1965) 657. 3. A.Ahmadzadeh and C.H. Chan, Sum rules for high energy scattering phenomena, University of California, San Diego, Report UCSD-10PI0-6, June, 1966 (to be published). 4. I.Mannelli, A. Bigi, R. Carrara, M. Wahlig and L. Sodickson, Phys.Rev. 14 (1965) 408. 5. A.V.Sterling et al., Phys. Rev. Letters 14 (1965) 763. 6. A.Ahrnadzadeh. University of California, San Diego, Report UCSD-10P10-1, May 1966, Phys. Letters 22 (1966) 96. See this paper also for references to the quark decomposition hypothesis.
OF
QUANTUM
V.N. G R I B O V
ELECTRODYNAMICS and L. N. L I P A T O V
Ioffle Physical Technical Institute, LeningLrad, USSR Received 8 August 1966
The double logarithmic asymptotics of various two-particle processes in quantum electrodynamics are investigated at high energy and at fixed momentum transfer.
The asymptotics of the two-particle process at high energies s = (Pl +#2) 2 and at fixed t = = (Pl -p~)z in the case of spirfless particles with ,/~ ¢'3 interaction has been found for a 2 In s << 1 by Polkinghorne [1]. In this case the sequence of ladder diagrams is dominant, their s u m being cx (a i n s K(t)) n-1 = ~ exp(a K(t) In s). (n- 1) ! s
(1)
n=l s
These a s y m p t o t i c s may be called single l o g a r i t h m i c , b e c a u s e t h e r e i s one power of In s for each O/.
In o r d e r to obtain the a s y m p t o t i c s of the l a d d e r d i a g r a m it i s c o n v e n i e n t to decompose the i n t e g r a t i o n f o u r - m o m e n t a k into the l o n g i t u d i n a l and the t r a n s v e r s e p a r t s k = UPl +vP2 +kx [2]. Then du, dv i n t e g r a t i o n gives In s~. and K(t) i s the twod i m e n s i o n a l i n t e g r a l over d k± c o r r e s p o n d i n g to one loop of the c o n t r a c t e d d i a g r a m (fig. la). However, if one of the p a r t i c l e s in the loop i s m a s s l e s s (photon), the c o r r e s p o n d i n g i n t e g r a l K(t) d i v e r g e s at s m a l l k ~. F u r t h e r m o r e , if both p a r t i c l e s of the loop have spin ½ ( e l e c t r o n s ) , K(t) d i v e r g e s at l a r g e k£ also. As the i n i t i a l d i a g r a m (fig. 1) does not d i v e r g e both at s m a l l and at
l a r g e kx, this m e a n s that the a s y m p t o t i c f o r m (1) f a i l s to hold. The m o r e d e t a i l e d a n a l y s i s shows that in the i n t e g r a l K(t) t h e r e i s in fact a cut-off at s "1 f o r s m a l l k± and at s at l a r g e k±. T h e r e fore the a p p a r e n t l o g a r i t h m i c d i v e r g e n c e of the i n t e g r a l r e s u l t s actually in the o c c u r r e n c e of an additional power of In s in each loop. Thus we have l n 2 s for each c~ and the r e s u l t a n t a s y m p t o t i c s may be c a l l e d double l o g a r i t h m i c . The doubl~ l o g a r i t h m i c t e r m s connected with the divergen-'~e at s m a l l k~. c o r r e s p o n d always to real internat photons k 2 = k2+k~ = 0. These photons are similar to real Bremsstrahlung photons which, as is known, give double logarithmic terms in the cross section. Such internal Bremsstrahlung photons are shown by the wavy lines in the figures. Let us consider all possible two-particle processes of quantum electrodynamics. It is convenient to characterize the processes in question by the c h a r g e Z of the t - c h a n n e l s t a t e s . If Z = 0, then t h e r e a r e two photons or two f e r m i o n s in the i n i t i a l and the final s t a t e s , the d i r e c t i o n s of the f e r m i o n l i n e s being opposite to each other. The charge in this c a s e is e i t h e r a b s e n t (photon-photon s c a t t e r i n g ) , or does not 6'/1
Volume 22, number 5
PHYSICS LETTERS
I t t I t
I I I i I
L
!
Fig. 1.
change i t s d i r e c t i o n of motion a f t e r s c a t t e r i n g . The r e l e v a n t p r o c e s s e s a r e the f o r w a r d a n n i h i l a t i o n of the p a i r e+e - into the p a i r # + ~ - (I) *; e+e - and e - e - f o r w a r d s c a t t e r i n g (H); f o r w a r d Compton s c a t t e r i n g (I]]). A s the c h a r g e d o e s not change d i r e c t i o n of motion, one m a y e x p e c t that B r e m s s t r a h l u n g photons do not c o n t r i b u t e to the c r o s s s e c t i o n . It t u r n s out that double l o g a r i t h m i c c o n t r i b u t i o n s of the B r e m s s t r a h l u n g photons shown (e.g. in fig. 2), r e a l l y c a n c e l out. Thus the double l o g a r i t h m i c c o n t r i b u t i o n s o c c u r only in d i a g r a m s which d i v e r g e at l a r g e k± b e c a u s e of the p r e s ence of s p i n o r n u m e r a t o r s (fig. l a ) . A s well a s in the s p i n l e s s c a s e , the s i g n i f i c a n t d i a g r a m s t u r n out to be l a d d e r ones. S u m m i n g up a l l t h e s e * We are indebted to I.A.Malkin who has drawn our attention to this reaction.
15 September 1966
d i a g r a m s we obtain the s a m e r e s u l t f o r a l l p r o cesses: ~ ( ~ ) = fo~
Ii(~) ;
~2 _ 2_~a~in 2 s .
w h e r e f o i s the B o r n s c a t t e r i n g a m p l i t u d e f o r a given p r o c e s s , II(~) i s the B e s s e l function of the i m a g i n a r y a r g u m e n t . The a s y m p t o t i c s ~ _ c o r r e spond to the s q u a r e - r o o t b r a n c h p o i n t , / l 2 - 4 a / ~ in the t - c h a n n e l p a r t i a l wave a m p l i t u d e . S i m i l a r a s y m p t o t i c s w e r e found f o r the f i r s t t i m e by B j o r k e n and Wu [3], who c o n s i d e r e d a s p e c i a l model. F o r p r o c e s s (I) the a s y m p t o t i c s (2) a r e dominant. At the s a m e t i m e , f o r p r o c e s s e s (II) and (IH) involving two i d e n t i c a l f e r m i o n s , d i a g r a m s of the type of f i g s . 2a and 3a with photons in i n t e r m e d i a t e s t a t e s a r e r e l e v a n t . T h e s e d i a g r a m s a r e p r o p o r t i o n a l to s and d o m i n a t e at v e r y high s. However, fig. 3a c o n t a i n s an a d d i t i o n a l f a c t o r a s c o m p a r e d with fig. 3, b e i n g at ~ In 2 s / ~ ~ 1 of the s a m e o r d e r of m a g n i t u d e a s the double l o g a r i t h m i c a s y m p t o t i c s of the p r o c e s s (III). The s a m e f a c t o r ~ a p p e a r s in the c o r r e s p o n d i n g d i a g r a m f o r photon-photon s c a t t e r i n g . F o r p r o c e s s (II) the double l o g a r i t h m i c a s y m p t o t i c s a r e i n s i g nificant, f o r in t h i s c a s e d i a g r a m s of the type of fig. 2a p r e v a i l . The double l o g a r i t h m i c a s y m p t o t i c s of the p r o c e s s e s (I), (II) and (III) have been c o n s i d e r e d by A b r i k o s o v [4] and M i l e k h i n and F r a d k i n [5], but t h e i r r e s u l t s s e e m to be i n c o r rect. If Z = 1, then one of the e x t e r n a l l i n e s of fig. 1 i s the photon line. The c o r r e s p o n d i n g p r o c e s s i s the b a c k w a r d Compton s c a t t e r i n g . In this c a s e only B r e m s s t r a h l u n g p h o t o n s give r i s e to the
,/ I I I
Fig. 2
672
(2)
Fig. 3.
f
Volume 22, number 5
PHYSICS
double l o g a r i t h m i c a s y m p t o t i c s , which we c o n s i d e r e d e a r l i e r [6]. If Z = 2, then t h e d i r e c t i o n of m o t i o n of the c h a r g e a f t e r s c a t t e r i n g i s o p p o s i t e to the i n i t i a l one. In t h i s c a s e both l a r g e and s m a l l k± y i e l d the double l o g a r i t h m i c c o n t r i b u t i o n s . The c o r r e sponding p r o c e s s e s a r e the b a c k w a r d s c a t t e r i n g e ' e +, /I-~+ and the backward annihilation c~ the pair e-e + into the pair ~t-~+. The significant diagrams are now analogous to the diagrams of fig. 2, but the directions of the fermion lines a r e the s a m e , r a t h e r than opposite. The double l o g a r i t h m i c c o n t r i b u t i o n s of the individual d i a g r a m s due to the B r e m s s t r a h l u n g photons, which w e r e s u b t r a c t e d f o r Z = 0, a r e now added. The d i a g r a m s of the type of fig. 2a a r e a b s e n t in t h i s c a s e , s o double l o g a r i t h m i c a s y m p t o t i c s a r e d o m i n a n t . We have found that the double l o g a r i t h m i c a s y m p t o t i c s f o r t h e s e processes are given by
c+i°°
d
gf(~) = f o 2i~ exp(_½}2 ) f d/exp(/~) ~ c-i~
_¼(l)
(3)
where ~)-¼(1) is the parabolic cylinder function. The exponential factor standing in front of the integral in 'eq. (3) is caused by the contribution of the Bremsstraldung photons with Ik±l 2 << 1. For large ~ the leading contribution to asymptotics yields the rightmost pair of complex conjugate zeroes of the function ~_¼(l). The asymptotic behavior of the amplitude is oscillating in this case.
LETTERS
15 September 1966 -8
exp(-2.261~) cos(1.843~) + . . .
s~(}) = exp (_½~2) JI
~>>I
(4)
!
1 - ~ 2 +i.~2~4 + . . .
,~<< 1
The d i f f e r e n t i a l c r o s s s e c t i o n f o r p r o c e s s e s with Z = 0 i s given by eq. (2) s q u a r e d . F o r Z = 2 the s u m of the e l a s t i c c r o s s s e c t i o n and the c r o s s s e c t i o n due to the B r e m s s t r a h l u n g of photons with [k± 12 << 1 i s given by eq. (3) s q u a r e d e x c e p t f o r the exponential f a c t o r in f r o n t of the i n t e g r a l . The double l o g a r i t h m i c a s y m p t o t i c s a t l a r g e s , t, u have b e e n obtained e a r l i e r by s e v e r a l a u t h o r s [7-9]. Thus the double l o g a r i t h m i c a s y m p t o t i c s of t w o - p a r t i c l e p r o c e s s e s in quantum e l e c t r o d y n a m i c s a r e now i n v e s t i g a t e d in a l l c a s e s . We w i s h to thank I. A. Malkin, I. Ya. P o m e r a n chuk and E. S. F r a d k i n f o r fruitful d i s c u s s i o n s . 1. 2. 3. 4. 5. 6. 7. 8. 9.
J.C.Polkinghorne, J.Math. Phys. 4 (1963)503. V.V.Sudakov, Zh. Eksp. i Teor. Fiz. 30 (1956) 87. J.D.Bjorken and T.T.Wu, Phys.Rev. 130 (1963) 2566. A.A.Abrikosov, Zh.Eksp. iTeor.Fiz.30 (1965) 386,544. G.A. Milekhin and E.S. Fradkin, Zh. Eksperim. i Teor. Fiz. 45 (1963) 1926. G°V. Frolov, V.G.Gorshkov and V.N.Gribov, Phys. Letters 20 (1966) 544. V.N. Bajer and S. A. Kheffetz, Zh. Eksperim. i Teor. Fiz.40 (1961) 613. D.R.Yennie, S.C.Frautschi and H.Suura, Ann.of Phys. 13 (1961) 379. I.A.Malkin, JETP Letters, to be published.
* * * *
673
PHYSICS LETTERS
pothesis would be valid to a good approximation - thereby making practical reductions in the n u m ber of parameters in Regge pole phenomenology. Therefore, should the present communication encourage m e a s u r e m e n t of the above reactions, it has already served its purpose. I would like to thank P r o f e s s o r G. F. Chew for his i n t e r e s t and e n c o u r a g e m e n t . I would a l s o like to thank D r s . V. B a r g e r , C.H. Chan, J. K i r z and W. R a r i t a for d i s c u s s i o n s .
DOUBLE
LOGARITHMIC
ASYMPTOTICS
G. V. F R O L O V ,
V.G. G O R S H K O V ,
15 September 1966
References 1. A.Ahmadzadeh, Phys.Rev. Letters 16 (1966) 952. 2. R.C.Arnold, Phys. Rev. Letters 14 (1965) 657. 3. A.Ahmadzadeh and C.H. Chan, Sum rules for high energy scattering phenomena, University of California, San Diego, Report UCSD-10PI0-6, June, 1966 (to be published). 4. I.Mannelli, A. Bigi, R. Carrara, M. Wahlig and L. Sodickson, Phys.Rev. 14 (1965) 408. 5. A.V.Sterling et al., Phys. Rev. Letters 14 (1965) 763. 6. A.Ahrnadzadeh. University of California, San Diego, Report UCSD-10P10-1, May 1966, Phys. Letters 22 (1966) 96. See this paper also for references to the quark decomposition hypothesis.
OF
QUANTUM
V.N. G R I B O V
ELECTRODYNAMICS and L. N. L I P A T O V
Ioffle Physical Technical Institute, LeningLrad, USSR Received 8 August 1966
The double logarithmic asymptotics of various two-particle processes in quantum electrodynamics are investigated at high energy and at fixed momentum transfer.
The asymptotics of the two-particle process at high energies s = (Pl +#2) 2 and at fixed t = = (Pl -p~)z in the case of spirfless particles with ,/~ ¢'3 interaction has been found for a 2 In s << 1 by Polkinghorne [1]. In this case the sequence of ladder diagrams is dominant, their s u m being cx (a i n s K(t)) n-1 = ~ exp(a K(t) In s). (n- 1) ! s
(1)
n=l s
These a s y m p t o t i c s may be called single l o g a r i t h m i c , b e c a u s e t h e r e i s one power of In s for each O/.
In o r d e r to obtain the a s y m p t o t i c s of the l a d d e r d i a g r a m it i s c o n v e n i e n t to decompose the i n t e g r a t i o n f o u r - m o m e n t a k into the l o n g i t u d i n a l and the t r a n s v e r s e p a r t s k = UPl +vP2 +kx [2]. Then du, dv i n t e g r a t i o n gives In s~. and K(t) i s the twod i m e n s i o n a l i n t e g r a l over d k± c o r r e s p o n d i n g to one loop of the c o n t r a c t e d d i a g r a m (fig. la). However, if one of the p a r t i c l e s in the loop i s m a s s l e s s (photon), the c o r r e s p o n d i n g i n t e g r a l K(t) d i v e r g e s at s m a l l k ~. F u r t h e r m o r e , if both p a r t i c l e s of the loop have spin ½ ( e l e c t r o n s ) , K(t) d i v e r g e s at l a r g e k£ also. As the i n i t i a l d i a g r a m (fig. 1) does not d i v e r g e both at s m a l l and at
l a r g e kx, this m e a n s that the a s y m p t o t i c f o r m (1) f a i l s to hold. The m o r e d e t a i l e d a n a l y s i s shows that in the i n t e g r a l K(t) t h e r e i s in fact a cut-off at s "1 f o r s m a l l k± and at s at l a r g e k±. T h e r e fore the a p p a r e n t l o g a r i t h m i c d i v e r g e n c e of the i n t e g r a l r e s u l t s actually in the o c c u r r e n c e of an additional power of In s in each loop. Thus we have l n 2 s for each c~ and the r e s u l t a n t a s y m p t o t i c s may be c a l l e d double l o g a r i t h m i c . The doubl~ l o g a r i t h m i c t e r m s connected with the divergen-'~e at s m a l l k~. c o r r e s p o n d always to real internat photons k 2 = k2+k~ = 0. These photons are similar to real Bremsstrahlung photons which, as is known, give double logarithmic terms in the cross section. Such internal Bremsstrahlung photons are shown by the wavy lines in the figures. Let us consider all possible two-particle processes of quantum electrodynamics. It is convenient to characterize the processes in question by the c h a r g e Z of the t - c h a n n e l s t a t e s . If Z = 0, then t h e r e a r e two photons or two f e r m i o n s in the i n i t i a l and the final s t a t e s , the d i r e c t i o n s of the f e r m i o n l i n e s being opposite to each other. The charge in this c a s e is e i t h e r a b s e n t (photon-photon s c a t t e r i n g ) , or does not 6'/1
Volume 22, number 5
PHYSICS LETTERS
I t t I t
I I I i I
L
!
Fig. 1.
change i t s d i r e c t i o n of motion a f t e r s c a t t e r i n g . The r e l e v a n t p r o c e s s e s a r e the f o r w a r d a n n i h i l a t i o n of the p a i r e+e - into the p a i r # + ~ - (I) *; e+e - and e - e - f o r w a r d s c a t t e r i n g (H); f o r w a r d Compton s c a t t e r i n g (I]]). A s the c h a r g e d o e s not change d i r e c t i o n of motion, one m a y e x p e c t that B r e m s s t r a h l u n g photons do not c o n t r i b u t e to the c r o s s s e c t i o n . It t u r n s out that double l o g a r i t h m i c c o n t r i b u t i o n s of the B r e m s s t r a h l u n g photons shown (e.g. in fig. 2), r e a l l y c a n c e l out. Thus the double l o g a r i t h m i c c o n t r i b u t i o n s o c c u r only in d i a g r a m s which d i v e r g e at l a r g e k± b e c a u s e of the p r e s ence of s p i n o r n u m e r a t o r s (fig. l a ) . A s well a s in the s p i n l e s s c a s e , the s i g n i f i c a n t d i a g r a m s t u r n out to be l a d d e r ones. S u m m i n g up a l l t h e s e * We are indebted to I.A.Malkin who has drawn our attention to this reaction.
15 September 1966
d i a g r a m s we obtain the s a m e r e s u l t f o r a l l p r o cesses: ~ ( ~ ) = fo~
Ii(~) ;
~2 _ 2_~a~in 2 s .
w h e r e f o i s the B o r n s c a t t e r i n g a m p l i t u d e f o r a given p r o c e s s , II(~) i s the B e s s e l function of the i m a g i n a r y a r g u m e n t . The a s y m p t o t i c s ~ _ c o r r e spond to the s q u a r e - r o o t b r a n c h p o i n t , / l 2 - 4 a / ~ in the t - c h a n n e l p a r t i a l wave a m p l i t u d e . S i m i l a r a s y m p t o t i c s w e r e found f o r the f i r s t t i m e by B j o r k e n and Wu [3], who c o n s i d e r e d a s p e c i a l model. F o r p r o c e s s (I) the a s y m p t o t i c s (2) a r e dominant. At the s a m e t i m e , f o r p r o c e s s e s (II) and (IH) involving two i d e n t i c a l f e r m i o n s , d i a g r a m s of the type of f i g s . 2a and 3a with photons in i n t e r m e d i a t e s t a t e s a r e r e l e v a n t . T h e s e d i a g r a m s a r e p r o p o r t i o n a l to s and d o m i n a t e at v e r y high s. However, fig. 3a c o n t a i n s an a d d i t i o n a l f a c t o r a s c o m p a r e d with fig. 3, b e i n g at ~ In 2 s / ~ ~ 1 of the s a m e o r d e r of m a g n i t u d e a s the double l o g a r i t h m i c a s y m p t o t i c s of the p r o c e s s (III). The s a m e f a c t o r ~ a p p e a r s in the c o r r e s p o n d i n g d i a g r a m f o r photon-photon s c a t t e r i n g . F o r p r o c e s s (II) the double l o g a r i t h m i c a s y m p t o t i c s a r e i n s i g nificant, f o r in t h i s c a s e d i a g r a m s of the type of fig. 2a p r e v a i l . The double l o g a r i t h m i c a s y m p t o t i c s of the p r o c e s s e s (I), (II) and (III) have been c o n s i d e r e d by A b r i k o s o v [4] and M i l e k h i n and F r a d k i n [5], but t h e i r r e s u l t s s e e m to be i n c o r rect. If Z = 1, then one of the e x t e r n a l l i n e s of fig. 1 i s the photon line. The c o r r e s p o n d i n g p r o c e s s i s the b a c k w a r d Compton s c a t t e r i n g . In this c a s e only B r e m s s t r a h l u n g p h o t o n s give r i s e to the
,/ I I I
Fig. 2
672
(2)
Fig. 3.
f
Volume 22, number 5
PHYSICS
double l o g a r i t h m i c a s y m p t o t i c s , which we c o n s i d e r e d e a r l i e r [6]. If Z = 2, then t h e d i r e c t i o n of m o t i o n of the c h a r g e a f t e r s c a t t e r i n g i s o p p o s i t e to the i n i t i a l one. In t h i s c a s e both l a r g e and s m a l l k± y i e l d the double l o g a r i t h m i c c o n t r i b u t i o n s . The c o r r e sponding p r o c e s s e s a r e the b a c k w a r d s c a t t e r i n g e ' e +, /I-~+ and the backward annihilation c~ the pair e-e + into the pair ~t-~+. The significant diagrams are now analogous to the diagrams of fig. 2, but the directions of the fermion lines a r e the s a m e , r a t h e r than opposite. The double l o g a r i t h m i c c o n t r i b u t i o n s of the individual d i a g r a m s due to the B r e m s s t r a h l u n g photons, which w e r e s u b t r a c t e d f o r Z = 0, a r e now added. The d i a g r a m s of the type of fig. 2a a r e a b s e n t in t h i s c a s e , s o double l o g a r i t h m i c a s y m p t o t i c s a r e d o m i n a n t . We have found that the double l o g a r i t h m i c a s y m p t o t i c s f o r t h e s e processes are given by
c+i°°
d
gf(~) = f o 2i~ exp(_½}2 ) f d/exp(/~) ~ c-i~
_¼(l)
(3)
where ~)-¼(1) is the parabolic cylinder function. The exponential factor standing in front of the integral in 'eq. (3) is caused by the contribution of the Bremsstraldung photons with Ik±l 2 << 1. For large ~ the leading contribution to asymptotics yields the rightmost pair of complex conjugate zeroes of the function ~_¼(l). The asymptotic behavior of the amplitude is oscillating in this case.
LETTERS
15 September 1966 -8
exp(-2.261~) cos(1.843~) + . . .
s~(}) = exp (_½~2) JI
~>>I
(4)
!
1 - ~ 2 +i.~2~4 + . . .
,~<< 1
The d i f f e r e n t i a l c r o s s s e c t i o n f o r p r o c e s s e s with Z = 0 i s given by eq. (2) s q u a r e d . F o r Z = 2 the s u m of the e l a s t i c c r o s s s e c t i o n and the c r o s s s e c t i o n due to the B r e m s s t r a h l u n g of photons with [k± 12 << 1 i s given by eq. (3) s q u a r e d e x c e p t f o r the exponential f a c t o r in f r o n t of the i n t e g r a l . The double l o g a r i t h m i c a s y m p t o t i c s a t l a r g e s , t, u have b e e n obtained e a r l i e r by s e v e r a l a u t h o r s [7-9]. Thus the double l o g a r i t h m i c a s y m p t o t i c s of t w o - p a r t i c l e p r o c e s s e s in quantum e l e c t r o d y n a m i c s a r e now i n v e s t i g a t e d in a l l c a s e s . We w i s h to thank I. A. Malkin, I. Ya. P o m e r a n chuk and E. S. F r a d k i n f o r fruitful d i s c u s s i o n s . 1. 2. 3. 4. 5. 6. 7. 8. 9.
J.C.Polkinghorne, J.Math. Phys. 4 (1963)503. V.V.Sudakov, Zh. Eksp. i Teor. Fiz. 30 (1956) 87. J.D.Bjorken and T.T.Wu, Phys.Rev. 130 (1963) 2566. A.A.Abrikosov, Zh.Eksp. iTeor.Fiz.30 (1965) 386,544. G.A. Milekhin and E.S. Fradkin, Zh. Eksperim. i Teor. Fiz. 45 (1963) 1926. G°V. Frolov, V.G.Gorshkov and V.N.Gribov, Phys. Letters 20 (1966) 544. V.N. Bajer and S. A. Kheffetz, Zh. Eksperim. i Teor. Fiz.40 (1961) 613. D.R.Yennie, S.C.Frautschi and H.Suura, Ann.of Phys. 13 (1961) 379. I.A.Malkin, JETP Letters, to be published.
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