Volume 34B, number 4 P H Y S I C S L E T T E R S 1 M a r c h 197:

I N V E S T I G A T I O N O F T H E ( b - M E S O N R E S O N A N C E B Y

E L E C T R O N - P O S I T R O N C O L L I D I N G B E A M S

V. E. BALAKIN, G . I . BUDKER, E . V . PAKHTUSOVA, V . A . S I D O R O V A. N. SKRINSKY, G . M . TUMAIKIN and A. G. K H A B A K H P A S H E V '

Institute of Nuclear Physics of the Siberian Division of the USSR Academy of Sciences, Novosibirsk, USSR

Received 20 January 1971

Using the colliding beam machine , VEPP-2 , the excitat ion curve of the (b-meson resonance was inves t i - gated in the three main decay modes. The resonance p a r a m e t e r s obtained are the width, the total c r o s s - sect ion and the branching rat ios . Namely:

F - (4.67 ± 0.42)MeV

(~o - (3.96 ± 0.35) ~B

B(e+e -) = (2,81 ± 0.25) x 10 -4

B(K*K-) = (54.0 ± 3,4) %

o O B(KsK~) - (25.7 ± 3,0) c/r:

B@r+T;~ °) = (20.3 ± 4,2) %

In addition, the pion form factor at 2E :: 1020 MeV is obtained together with an upper l imit for branching ra t io of the (b-meson decay into a pion pair :

i FTr 12 2.9 • 1,2 B(TT+Tr -) < 0.8 ~ .

U s i n g two c o l l i d i n g - b e a m i n s t a l l a t i o n s (Novo- s i b r i s k and O r s a y ) a n u m b e r of n e a t r a l v e c t o r m e s o n i n v e s t i g a t i o n s h a v e b e e n p e r f o r m e d [1-8] . T h e p r e s e n t w o r k on the ( / ) -meson r e s o n a n c e i s an e x t e n s i o n of t h e s e r e s e a r c h e s . I t s d i f f e r e n c e f r o m the s i m i l a r w o r k [6, 7] by t he O r s a y g r o u p c o n s i s t s in the s i m u l t a n e o u s d e t e c t i o n of t he t h r e e m a i n ~ - m e s o n d e c a y m o d e s : K+K ", K~K~ ( th rough K~ ~ ~+~-) and 7r+~-n ° , a s we l l a s t he :r+~ - p a i r s . T h e p r e l i m i n a r y r e s u l t s of t he w o r k a r e a l r e a d y p u b l i s h e d [8].

T h e e x p e r i m e n t s h a v e b e e n m a d e u s i n g the e l e c t r o n - p o s i t r o n s t o r a g e r i n g V E P P - 2 . T h e w o r k i n g c o n d i t i o n s of t he s t o r a g e r i n g d u r i n g the e x p e r i m e n t s (in t h e s u m m e r of 1969) a r e s p e c i f i e d by the f o l l o w i n g p a r a m e t e r s , n a m e l y : t he a v e r a g e l u m i n o s i t y i s 1028 c m -2 s e c -1 , t he i n i t i a l c u r r e n t s a r e 60 m A and 40 mA, f o r e l e c - t r o n s and p o s i t r o n s , r e s p e c t i v e l y , and the e f - f e c t i v e l i f e t i m e of t he b e a m s i s 5 h o u r s .

T h e s p a r k c h a m b e r s y s t e m u s e d in t he e x - p e r i m e n t c o n s i s t s of two i d e n t i a l p a r t s (upper and l o w e r ) c o v e r i n g t he s o l i d a n g l e 2 × 0.9 s t e r a d i a n s n e a r t he v e r t i c a l a x i s . T h e l ayou t of t h e s p a r k c h a m b e r s y s t e m i s shown in f ig . 1. T h i n p l a t e c h a m b e r s a r e u s e d to d e t e r m i n e t he a n g l e s of o u t c o m i n g p a r t i c l e s a s we l l a s t he c o -

o r d i n a t e s of t he i n t e r a c t i o n po in t . The p a r t i c l e i d e n t i f i c a t i o n i s a c h i e v e d by o b s e r v i n g t h e i r i n - t e r a c t i o n wi th the m a t e r i a l p l a t e s in the " s h o w e r " - and t h e " r a n g e " - s p a r k c h a m b e r s . T h e " s h o w e r " c h a m b e r c o n s i s t s of 11 l e a d p l a t e s of 2 m m t h i c k each . T h e " r a n g e " c h a m - b e r c o n s i s t s of 21 p l a t e s e a c h of 8 m m t h i c k s t a i n l e s s s t e e l . An e l a b o r a t e s y s t e m of m i r r o r s a l ] owed t he u s e of a s i n g l e c a m e r a fo r r e c o r d i n g

T h e s p a r k c h a m b e r s y s t e m i s t r i g g e r e d by fou r s c i n t i l l a t i o n c o u n t e r s in c o i n c i d e n c e wi th t h e r e s o l v i n g t i m e of 20 n s e c . T h e f o u r - f o l d c o - i n c i d e n c e d e t e c t e d the m i n i m u m i o n i z i n g p a r t i c - l e s wi th an e f f i c i e n c y of 97%. T h e t w o - f o l d c o - i n c i d e n c e b e t w e e n t he i n n e r s c i n t i l l a t i o n c o u n - t e r s , c l o s e to the v a c u u m c h a m b e r , d e t e c t e d c h a r g e d K - m e s o n s wi th an e f f i c i e n c y of 100%. T h e k i n e t i c e n e r g y of K - m e s o n s a t the ( b - r e s o - n a n c e m a x i m u m i s only 16 MeV, and h e n c e t he l i gh t y i e l d in the s c i n t i l l a t o r i s c o n s i d e r a b l y g r e a t e r t h a n t h a t of r e l a t i v i s t i c p a r t i c l e s . To f a c i l i t a t e t he l a t t e r a n a l y s i s the m a g n i t u d e of t h e s e p u l s e s f o r e a c h even t was r e c o r d e d on t h e f i l m of t he s p a r k c h a m b e r p i c t u r e s .

An a n t i c o i n c i d e n c e c o u n t e r of d i m e n s i o n s 160 × 160 × 5 c m 3 was v i e w e d by a s i n g l e p h o t o - m u l t i p l i e r F E U - 6 5 and i s u s e d fo r v e t o i n g c o s -

328

Volume 34B, number 4 P H Y S I C S L E T T E R S 1 M a r c h [971

II \ "",,,\ . . ,, i ¸

p

i l l i , I L - ~ - -

_'2 ! • ~'~ ---

- i e / e - - £ : -

.- j .~ ~ - ~ _ A

I

. - . , _ _ ~ : : = =

i l ~ , i I I

~ ' ,

i '

Fig. 1. Spark Chamber system, 1) anticoincidence sc int i l la t ion counter , 2) lead abso rbe r , 200 mm thick, 3} range spa rk chamber , 4) shower spark chamber , 5), 7) scint i l la t ion counters , 6) duraluminium abso rbe r , 20mm thick, 8), t0) thin-plate spa rk chamber , 9) s t o r - age r ing magnet , 11) in teract ion region, 12), 13) inner

and outer vacuum chambers .

m i c r a y s . B e t w e e n t h e v e t o - c o u n t e r and t he c h a m b e r s a 20 c m t h i c k l e a d a b s o r b e r i s p l a c e d to p r e v e n t t h e p a r t i c l e s u n d e r i n v e s t i g a t i o n f r o m r e a c h i n g t h i s c o u n t e r . T h e a n t i c o i n c i d e n c e c o u n - t e r r e d u c e d t he c o s m i c - r a y t r i g g e r i n g r a t e by m o r e t h a n a f a c t o r of 50. An a d d i t i o n a l r e d u c t i o n of a f a c t o r 4 was o b t a i n e d when t h e c o u n t e r s w e r e g a t e d by the r f s i g n a l . U n d e r t h e s e c o n d i - t i o n s t h e t r i g g e r i n g r a t e due to c o s m i c - r a y s i s 40 c o u n t s p e r h o u r w h i l e t he t r i g g e r i n g r a t e due to t he p a r t i c l e s f r o m the c o l l i d i n g b e a m s w as m u c h h i g h e r .

T h e a c c u r a c y of t h e a b s o l u t e e n e r g y d e t e r - m i n a t i o n by m e a n s of m a g n e t i c f i e l d m e a s u r e - m e n t s i s a b o u t 1%. T h e s t a b i l i t y in e n e r g y a n d i t s r e p r o d u c i b i l i t y o v e r v a r i o u s r u n s w as c h e c k e d by t he n u c l e a r r e s o n a n c e m e t h o d w i t h a n a c c u r a c y of 0.01%. T h e a b s o l u t e e n e r g y c a l i b r a - t i on w a s m a d e by u s i n g t he t a b u l a t e d e n e r g y v a l u e , 1019.5 MeV of t h e ( ~ - m e s o n r e s o n a n c e [9].

C h e c k i n g the c o n d i t i o n s of t he b e a m c o l l i s i o n and t he l u m i n o s i t y m o n i t o r i n g ( i n t e g r a t i o n ) w e r e

Table 1 Summarizes the resu l t s of measu remen t s .

effect background cosmic

Measurement t ime (hour) 222 67 68

2 y - m o n i t o r (103) 1292 2.3

Number of p ic tures (103 ) 86.1 25.6 10.2

Luminosity integral (1033 crn_2) 8.5

c a r r i e d out by r e c o r d i n g t he doub le b r e m s s t r a h - lung e v e n t s [16]. T h i s p r o c e s s was r e c o r d e d by t he two NaI s c i n t i l l a t i o n c o u n t e r s in c o i n c i d e n c e (2 v - m o n i t o r ) . T h e e n e r g y t h r e s h o l d of t h e s e c o u n t e r s was s e t a t 5 MeV.

T h e m a i n bu lk of r e c o r d i n g took 1.5 m o n t h s of 24 h o u r s p e r day o p e r a t i o n of t he s t o r a g e r i n g . T h e e x p e r i m e n t s w e r e p e r f o r m e d a t n i n e e n e r g y v a l u e s of t he c o l l i d i n g b e a m s , f r o m 508 MeV u p to 514 MeV. O n e - f o u r t h of t he w h o l e p e r i o d was s p e n t on " b a c k g r o u n d " m e a s u r e m e n t s wi th t he b e a m s d i s p l a c e d v e r t i c a l l y f r o m e a c h o t h e r a t a d i s t a n c e of 2 m m . In a d d i t i o n t h e b a c k g r o u n d of c o s m i c - r a y s was m e a s u r e d .

T h r o u g h the a n a l y s i s of t he p h o t o g r a p h e d e v e n t s t he f o l l o w i n g s ix t y p e s of r e a c t i o n w e r e s e l e c t e d :

1) e+e - -* e+e - . T h e c o l l i n e a r i t y a n d the p r e - s e n c e of c h a r a c t e r i s t i c s h o w e r s in b o t h u p p e r a n d l o w e r c h a m b e r s y s t e m s a l l o w u s to s e l e c t t h i s t ype of p r o c e s s c l e a r l y w i t h o u t b a c k g r o u n d c o n t a m i n a t i o n . T h e d e v i a t i o n , /x~o, f r o m c o l - l i n e a r i t y w a s r e q u i r e d to s a t i s f y AW < 10 o.

2) e+e - -* K+K - . T h e m a i n c r i t e r i a f o r s e l e c - t i on w e r e c o l l i n e a r i t y (A~o < 10 °) a n d p u l s e s of h i g h a m p l i t u d e in a p a i r of t r i g g e r i n g s c i n t i l l a - t i on c o u n t e r s . In " s h o w e r " and " r a n g e " c h a m - b e r s a b o u t o n e - t h i r d of t he r e g i s t e r e d e v e n t s of t h i s p r o c e s s s h o w e d a c h a r a c t e r i s t i c p i c t u r e of t h e d e c a y of a K - m e s o n s t o p p e d in t he f i r s t s c i n - t i l l a t i o n c o u n t e r . T h e l a t t e r e v e n t s h a v e no b a c k g r o u n d .

o o 3) e+e - ~ KsK L. T h e d e c a y K~ ~ ~+~- was

r e c o r d e d . T h e b r a n c h i n g r a t i o of t h i s m o d e i s 68.7 + 0.6% [9]. T h e a n g l e of n o n - c o l l i n e a r i t y , Aw, was r e q u i r e d to b e f r o m 10 ° to 34 ° . T h e u p p e r l i m i t i s s e t by the K ~ - m e s o n m o m e n t u m . T h e r a n g e of the ~ - m e s o n s f r o m t h i s p r o c e s s d o e s not a l l o w t h e m to p e n e t r a t e b e y o n d the f o u r t h gap of the " r a n g e " c h a m b e r . T h i s c o n d i c t i o n was u s e d a s an a d d i t i o n a l c r i t e r i o n fo r s e l e c t i o n . T h e

329

Volume 34B. number 4 P H Y S I C S L E T T E R S

T a b l e 2

1 M a r c h 1971

Channel

e + e -

K+K-

O O KsK L

77-+ T/~-T/-O

~+~-

Number of events detected

back- effect cosmics ground

530 0 0

633 20 2

95 0 0

21 0 0

40 3 15

Normalizat ion factor 1 0.298 1.32

Pure effect Efficiency of (background detection substracted)

530 • 23 (0.0625gb)

565 ± 29 7.1 :~ 0.1

95 • 10 3.4 + 0.2

21 ± 4.6 0.71 :e 0.08

28.6± 7 6.0 ±0 .7

e v e n t s wi th a z i m u t h a l c o m p o n e n t , A(p, of A¢0 s m a l l e r than 4 ° w e r e no t a c c e p t e d , s i n c e they a r e c o n t i m a t e d by the p r o c e s s e+e - ~ ~+Tr-).

4) e+e - ~ ~+~r-ro. To t h i s p r o c e s s w e r e a s - s i g n e d t h o s e e v e n t s in w h i c h t h e a n g l e of n o n - c o l l i n e a r i t y o r the r a n g e of one of t h e p i o n s e x - c e e d e d t he v a l u e s e x p e c t e d f o r t he p r e c e d i n g p r o c e s s . T h e c r i t e r i o n A~ < 4 ° i s a l s o r e t a i n e d h e r e . An a d d i t i o n a l c o n d i t i o n of A ~ < 65 ° i s f u r t h e r i m p o s e d in o r d e r to d e c r e a s e t he b a c k - g r o u n d c o n t a m i n a t i o n .

5) e+e - -* ~r+Tr - . To t h i s p r o c e s s w e r e a s - s i g n e d t h o s e e v e n t s in w h i c h b o t h ends of the c o l - l i n e a r t r a c k , (Aw < 2.5 o) w e r e e i t h e r in t h e " s h o w e r " - o r in t he " r a n g e " c h a m b e r .

6) e+e - ~ ~+ g - . To t h i s p r o c e s s w e r e a s - s i g n e d t h o s e e v e n t s in w h i c h t he c o l l i n e a r t r a c k (Aw < 2.5 °) e x t e n d e d in b o t h d i r e c t i o n s b e y o n d t he " r a n g e " c h a m b e r s . T h e r e s u l t s of t h i s p r o - c e s s a r e g i v e n e l s e w h e r e [10] w h e r e t he d o m a i n of a p p l i c a b i l i t y of QED i s i n v e s t i g a t e d .

T h e r e s u l t s of the e x p e r i m e n t a r e s u m m a - r i z e d in t a b l e 2.

T h e n u m b e r of b a c k g r o u n d e v e n t s a m o n g the a b o v e a c c e p t e d e v e n t s w as n o r m a l i z e d to t h a t in the " b a c k g r o u n d " r u n by u s i n g t he t o t a l n u m b e r s of p i c t u r e s t a k e n in t he r e s p e c t i v e r u n s . In e s t i - m a t i n g t he b a c k g r o u n d f o r the p r o c e s s ~+~r- we u s e d t he " c o s m i c " r u n i n s t e a d of " b a c k g r o u n d " r u n b e c a u s e of t he b e t t e r s t a t i s t i c a l a c c u r a c y .

D u r i n g t he c o u r s e of t he e x p e r i m e n t a l r u n s the r e l a t i v e i n t e g r a l l u m i n o s i t y w as c h e c k e d by the 2 ~ - m o n i t o r . T h e a b s o l u t e v a l u e of t he l u m i - n o s i t y was d e t e r m i n e d f r o m t he l a r g e a n g l e e l e c - t r o n - p o s i t r o n e l a s t i c s c a t t e r i n g .

T h e g e o m e t r i c a l d e t e c t i o n e f f i c i e n c y of t he s p a r k c h a m b e r s y s t e m was c a l c u l a t e d by a

330

M o n t e C a r l o m e t h o d . T h e d e t e c t i o n p r o b a b i l i t i e s g i v e n in t a b l e 2 i n c l u d e c o r r e c t i o n s f o r t he e f - f i c i e n c y of t he c o i n c i d e n c e c i r c u i t , a n d fo r the p r o b a b i l i t y of ~ - m e s o n s s a t i s f y i n g t he r a n g e r e - q u i r e m e n t , e t c . T h e e l a s t i c s c a t t e r i n g c r o s s - s e c t i o n g iven in t a b l e 2 f o r t h e p r e s e n t d e t e c t o r g e o m e t r y a l r e a d y i n c l u d e s t h e r a d i a t i v e c o r r e c - t i on [11]. In c a l c u l a t i n g t h e I~K 0 mode c r o s s - s e c t i o n i t was t a k e n in to a c c o u n t ~ h a t a p a r t of t he e v e n t s 7r+Tr-Tr o would s a t i s f y t he s e l e c t i o n r e - q u i r e m e n t s f o r K~K~. T h e p r o b a b i l i t y f o r a n 1r+~-~ ° - e v e n t to be m i s t a k e n f o r a K ~ K ~ - e v e n t i s (0.14 + 0.02) %.

T h e r a d i a t i v e c o r r e c t i o n s a r e r a t h e r i m p o r - t a n t f o r the a n a l y s i s of t he r e s o n a n c e p r o c e s s u n d e r i n v e s t i g a t i o n [12]. T h e m e a s u r e d c r o s s - s e c t i o n , a m , i s r e l a t e d by t he fo l l owing f o r m u l a to the " i d e a l " c r o s s - s e c t i o n , ~, w h i c h we a r e l ook ing f o r , a m = a(1 + 5R). At the p e a k of the ~ - r e s o n a n c e , 6 R = - 0 . 2 4 .

T h e d u r a t i o n of the r u n s i s f r o m 2 to 5 h o u r s . S i n c e the t i m e of r a d i a t i v e p o l a r i z a t i o n of the b e a m s a t t he e n e r g y 510 MeV in s t o r a g e r i n g V E P P - 2 i s t h r e e h o u r s [13], in the a b s e n c e of d e p o l a r i z a t i o n e f f e c t s the p o l a r i z a t i o n of t he b e a m wi l l i n f l u e n c e t he r e s u l t s of c r o s s - s e c t i o n m e a s u r e m e n t s in t h e K+K - c h a n n e l . In t h i s c a s e t he c o r r e c t i o n f o r t h e c r o s s - s e c t i o n wi l l b e 5 R = 0 .06. In o t h e r w o r d s , i f t h e r e a r e no d e p o l a r i z a - t i on e f f e c t s , o u r r e s u l t s f o r t he t o t a l c r o s s - s e c - t i on of t h i s m o d e s h o u l d b e r e d u c e d by 6%. I t i s p o s s i b l e to e s t i m a t e t h i s c o r r e c t i o n f r o m t h e o b - s e r v a t i o n of t h e r a t e of o c c u r e n c e of K+K - - e v e n t s a s a f u n c t i o n of t i m e a f t e r the b e a m s w e r e i n j e c t e d in to t he s t o r a g e r i n g . T h i s m e t h o d g a v e 5p = 0.02 + 0.03 and we t ook t he v a l u e 5p = 0 in the p r e s e n t a n a l y s i s .

Volume 34B, number 4 P H Y S I C S

i

I E, AC KGR OL~t~ ! 'J

0 ~ I0~'6 ~0'20 1024 2 ( , N e V

Fig. 2. e+e ~ -~ <b -*K+K --reaction. At the chosen nor- malization the ordinates of the points correspond to

the number of detected events.

40

] e e-*-K,L

t046 4090 *0~. ~(, f, leV

÷ - O O Fig. 3. e e -~ <b ~KsK T. The admixture of the 7r+Tr-Tr ° channel's shown by the dotted line.

~. e+e-_ ¢_~i+1 i -~ o

qOI6 t020 102k '2( NeV

Fig. 4. e+e - -,d) --* ~r+~-~r°-reaction.

Figs. 2-4 show the results of the experimen- tal data analysis of the three main decay modes of the <b-meson. Fitting the experimental r e - sults to the Breit-Wigner formula using the maxi- mum likely-hood method gave the following values for the resonance width and for the par - tial resonance cross-sec t ions of the three reac- tion modes:

L E T T E R S I March 1971

F = (4.67 ± 0.42) MeV

~(K+K -) = (2.13 + 0.17) b a r n

~(K~K~) = (I.01 + 0.15) barn

~(~+~-~o) = (0.81 ± 0.21) barn.

The X2-value of this fit is 15.5 which corre- sponds to 84% confidence level for 28-6 = 22 de- grees of f r e e d o m .

A s s u m i n g t h e a b s e n c e of t he o t h e r d e c a y m o d e s of t h e <b-meson , t he t o t a l c r o s s - s e c t i o n f o r 0 - m e s o n p r o d u c t i o n was found to be

~o = (3.96 a: 0 . 3 5 ) ~ b a r n

a n d t he b r a n c h i n g r a t i o s f o r the m a i n m o d e s of d e c a y a r e

B(K+K -) = (54.0 + 3.4) %

B(I~K~) = (25.7 + 3.0) %

B(~+~-~ o) = (20.3 * 4.2) %.

U s i n g t h e s e r e s u l t s we can o b t a i n t he b r a n c h i n g r a t i o a s we l l a s t he wid th of the e+e - - d e c a y of t h e <b-meson . N a m e l y :

B(e+e -) = (2.81 • 0 .25) × 10 -4

F (e+e -) = (1.31 + 0.12) keV.

T h e c o u p l i n g c o n s t a n t of t he <b-meson i s

g2 ~ 2 m (b a <b 4~ - 3 F ~ = 11.7 • 1.1,

w h e r e a<b = 0 .85 t a k e s c a r e of t he e f f e c t of t he f i n i t e w i d t h [14].

T h e r e s u l t s o b t a i n e d h e r e a r e in s a t i s f a c t o r y a g r e e m e n t w i t h t h o s e of t h e O r s a y g r o u p in F r a n c e .

T h e a n a l y s i s of ~+~- p r o c e s s a l l o w e d t he p r e - s e n c e of a n o t i c e a b l e a m o u n t of n o n - r e s o n a n t p r o d u c t i o n in t he e n e r g y r a n g e i n v e s t i g a t e d . T h e f o r m f a c t o r of t he ~ - m e s o n i s t h e n e s t i m a t e d to b e (at 1020 MeV)

12 -- 3.2 • 0.8.

T h i s v a l u e i s h i g h e r t h a n t he v a l u e 1.6 + 0 .3 e s t i - m a t e d f r o m t h e e x t r a p o l a t i o n of the B r e i g t - W i g - n e r r e l a t i o n f o r t he p - m e s o n r e s o n a n c e [2, 4].

O u r a t t e m p t to o b s e r v e t h e i n t e r f e r e n c e b e - t w e e n t h i s n o n - r e s o n a n t t a i l and t h e r e s o n a n t <b ~ ~+~- d e c a y was u n s u c c e s f u l . We o b t a i n e d a n u p p e r l i m i t f o r t he b r a n c h i n g r a t i o of the ~+Tr-

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decay m o d e of the S - m e s o n . Th i s i s

B(~+n -) < 0,8 %

with the c o n f i d e n c e l eve l of 95%. Inc lud ing the p o s s i b l e i n t e r f e r e n c e e f f ec t t he m a x i m u m l i k e l y - hood e s t i m a t e g i v e s

iF~ i 2 = 2,9 ± 1,2.

The a u t h o r s a r e g r a t e f u l to V. N. B a y e r , A . I . V a i n s h t e i n and I. B. K h r i p l o v i c h f o r u se fu l d i s - c u s s i o n s and to the l a r g e g r o u p of c o - w o r k e r s t ak ing p a r t in da ta p r o c e s s i n g .

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433. [2] V. L. Auslender et al., Yadern. Fiz. 9 (1969}

[3] I. E. Augustin et al.. Phys. Rev. Let ters 20 (1968) 126.

[4] I. E. Augustin et al., Phys. Let ters 28B (1969) 508. [5] I. E. Augustin et al., Phys. Let ters 28B (1969) 513. [6] I. E. Augustin et al., Phys. Let ters 28B ~l .q69) 517. [71 I. P e r e z - y - Y o r b a . Intern. Symposium (Daresbury.

(1969). [8] V. A. Sidorov, Intern. Symposium (Daresbury,

1969). [9] A. N. Rosenfeld, Tables (1970).

[10] V. E. Balakin et al. , Submitted to the Intern. Con- ference (Kiev, 1970).

[11] S. M. Sukhanov, V. S. Fadin and V. A. Khoze, Sov. Phys. Dokl., 178 (1968) 822.

[12] V. N. Bayer and V. S. Fadin, Phys. Let ters 27B (1968) 223.

[13] V. N. Bayer. International School of Physics "Enrico Fermi" XLVI (Academic P r e s s , 1970).

[14] F .Renard . Nucl. Phys. B15 (1970) 267. [15] L. D. Landau and E. M. Lifshi ts . , Phys. 6 (1950)

615. [16] P. I. Golubnichiy et al., Proc. Int. Symp. on e l ec -

tron and positron storage rings (Saclay, 1966).

332