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Charged pion photoproduction and finite energy sum rules
Volume 29H, number 4
PHYSICS LETTERS
CHARGED PION PHOTOPRODUCTION AND FINITE ENERGY SUM RULES
12 May 1969
*
J. D. JACKSON and C. QUIGG Department of Physics and Lawrence Radiation Laboratory, University of California, Berkeley, California 94720, USA Received 24 March 1969 The large s, small t pion photoproduction data and the corresponding continuous moment sum rules are fitted with evasive Regge poles and associated Regge cuts, instead of the commonly accepted pion conspiracy. The FESR actually yield the real parts of the photoproduction amplitudes, essentially independent of the model. A further assumption permits the prediction (in good agreement with existing data) of high energy cross sections directly from the FESR, without commitment to any particular model.
In this note we c o n s i d e r the photoproduction of charged pions at high e n e r g i e s and s m a l l m o m e n t u m t r a n s f e r s : a) We c o n s t r u c t a model with e v a s i v e Regge poles and a s s o i c a t e d Regge cuts that fits both the high e n e r g y d i f f e r e n t i a l c r o s s s e c t i o n s (including the data with l i n e a r l y p o l a r ized photons) and the continuous m o m e n t finite e n e r g y s u m r u l e s , b) We show that the n o r m a l finite e n e r g y s u m r u l e s in fact d e t e r m i n e a p p r o x i m a t e l y the r e a l p a r t s of the photoproduction a m p l i t u d e s at the u p p e r l i m i t of i n t e g r a t i o n . c) We develop a p s e u d o m o d e l which d i r e c t l y r e l a t e s the finite e n e r g y sum r u l e s to the high e n ergy data without the need for c o m m i t m e n t to any p a r t i c u l a r t h e o r e t i c a l model. Our most i m p o r t a n t c o n c l u s i o n is that, although finite e n e r g y s u m r u l e s for photoproduction allow an i m m e diate connection between the low e n e r g y and high e n e r g y r e g i o n s , they make no d i s t i n c t i o n among a l a r g e c l a s s of m o d e l s , and in p a r t i c u l a r , do not e s t a b l i s h the e x i s t e n c e of a pion c o n s p i r a c y in photoproduction of pions. The s h a r p f o r w a r d peaks in n - p c h a r g e exchange s c a t t e r i n g [ 1] and in charged pion photop r o d u c t i o n [e.g., 2] a r e m o s t c o m m o n l y cited as e v i d e n c e for a p a r i t y - d o u b l e t c o n s p i r a c y of the pion (class III L o r e n t z pole with M= 1). Ball et al. [3] and Henyey [4] have given v e r y s a t i s f a c t o r y fits to the p o s i t i v e pion photoproduction data at s m a l l m o m e n t u m t r a n s f e r s with such a c o n s p i r a c y model. A p p a r e n t l y independent support for the idea of a pion c o n s p i r a c y c a m e f r o m the g e n e r a l r e s u l t s of D r e l l and Sullivan [5], f r o m the application of n o r m a l finite e n e r g y s u m * This work was supported in part by the U.S. Atomic Energy Commission. 236
r u l e s (FESR) to the photoproduction a m p l i t u d e s by Bietti et al. [6] and by Roy and Chu [7], and f r o m the subsequent use of continuous m o m e n t finite energy s u m r u l e s (CMSR) to d e t e r m i n e the t r a j e c t o r i e s and r e s i d u e s of the pion and its cons p i r a t o r [8, 9]. While the pion c o n s p i r a c y is a t t r a c t i v e l y e c o n o m i c a l in fitting photoproduction, it is c e r t a i n l y not unique [10, 11] a n d it c a u s e s difficulties in other p r o c e s s e s via f a c t o r i z a t i o n [ 12]. N u m e r o u s models of hadronic p r o c e s s e s have been made u s i n g Regge poles and ad hoc cuts [13, 14] or cuts obtained f r o m the a b s o r p t i v e optical model [15-20]. F o r photoproduction, m o d els with poles plus cuts have been d i s c u s s e d by A m a t i et al. [11], FrdYland and Gordon [21] and Henyey et al. [22]. All of this work shows that models with poles and cuts can avoid the f a c t o r ization difficulties r e s u l t i n g f r o m the c o n s p i r a c y hypothesis, and provide explanations for the c r o s s - o v e r p h e n o m e n a [ 18], p o l a r i z a t i o n [ 17], and p r o b a b l y also the dip [20] in p i o n - n u c l e o n charge exchange. The r e m a i n i n g question conc e r n i n g the pion c o n s p i r a c y , or lack of it, is whether the evidence deduced f r o m CMSR can be r e g a r d e d as conclusive. We show, by explicit c o n s t r u c t i o n of a c o u n t e r - e x a m p l e , that the CMSR r e s u l t s cannot be so r e g a r d e d ; in the p r o c e s s we show what can actually be l e a r n e d f r o m the sum r u l e s in photoproduction. Our model is s i m i l a r to those of r e f s . 16-20 in that it has e v a s i v e Regge poles and cuts g e n e r a t e d by the r e c i p e of the a b s o r p t i v e model. It d i f f e r s in detail f r o m all of them in its f o r m for the Regge poles and in the a s s u m p t i o n that the k e r n e l in the c u t - g e n e r a t i n g i n t e g r a l is e n e r g y -
Volume 29B, number 4
PHYSICS LETTERS
independent. B e c a u s e the a b s o r p t i v e c o r r e c t i o n s a r e e x p r e s s e d in t e r m s of s - c h a n n e l a m p l i t u d e s while the R e g g e e x c h a n g e s and the a m p l i t u d e s u s e d in the s u m r u l e s a r e in the t - c h a n n e l , it is n e c e s s a r y to s p e c i f y a m p l i t u d e s in both channels. In the t - c h a n n e l we u s e theFi(i = 1, 2, 3,4) of r e f . 3. In the s - c h a n n e l we c h o o s e the four a m p l i t u d e s gj, w h e r e j = 1, 2, 3, 4 s t a n d s for the photon, i n i t i a l nucleon, and final nucleon h e l i c i -
ties (ky, ~N ;kN') = (+,+ ; -), (+, - ; -), (+,+ ;+), (-,+ ; -), respectively. The net helicity flips values n~j) ~ [Zy - kN+ XN' [ are 0, I, 1, 2, respectively. At high energies and small momentum transfers the two sets of amplitudes are related b y g = ( u / ~ ) A F , w h e r e u = (s - rn2)/2m i s the photon e n e r g y in the l a b o r a t o r y and the c r o s s i n g matrix A is
/-7/}m-2m/("2+7)
ll
0 ~,
[(7)~
0
(7)~/2rn
2rn(7)
A =/(7)~
0
(7)½/2rn
2rn0(7)~"
2rn/(g2 + 7)
1
\-7/2m
(1)
In eq. (1) we have i n t r o d u c e d 7 = -t, and have n e g l e c t e d t e r m s of r e l a t i v e o r d e r (T/4m2), a s w e l l a s nonleading p o w e r s of u. The r e c i p e of the a b s o r p t i v e m o d e l l e a d s [ 19, 22] to s - c h a n n e l a m p l i t u d e s of the f o r m : oo
gj(u, 7) = gjRegge (g, 7) - ~ o f
dT' exp{-½ (7+7')} ×
(2) × In(j ) [a(77,)½ ] g R e g g e (v, 7 ' ) , w h e r e a and C a r e c h a r a c t e r i s t i c of the a b s o r b ing r e g i o n (or of the P o m e r a n c h u k pole, if the cut i s viewed a s a r i s i n g f r o m two -R egge - p o l e exchange), In(X) i s a m o d i f i e d B e s s e l function of the f i r s t kind of o r d e r n, and gl~egge i s the a m p l i t u d e m a d e f r o m one o r m o r e R e g g e p o l e s . Eq. (2) can b e w r i t t e n in the c o m p a c t o p e r a t o r form:
gj = Kj ® g ~ e g g e .
(3)
S p e c i f i c a t i o n of the Regge a m p l i t u d e s and the p a r a m e t e r s a and C d e t e r m i n e s the model. F o r our p r e s e n t p u r p o s e s , w h e r e we c o n c e n t r a t e on the v e r y s m a l l z b e h a v i o r , we i g n o r e the p t r a j e c t o r y and k e e p only the pion and the A2, c o n t r i b u t i n g r e s p e c t i v e l y to F 2 and to F 1 and F 3. E x p l i c i t l y the s - c h a n n e l Regge a m p l i t u d e s a r e g~egge=g
_gA,
g~egge =g~3egge=hA ,
(4) w h e r [ ~ e g g e = -gz~- g A '
12May 1969
g~ = ega~7 F ( - a ~ ( 7 ) ) × ×
[ 1 + exp { - i ~ n ( z ) }] (u/uO) °t~(7) '
gA = eg~-Z2 (~1+~2) F ( 1 - OtA(7)) × 4m
× [1 + exp ( - i ~ a A ( 7 ) }] (U/Uo)°tA(7), h A = ~z ~
(41 - T~2/4rn2) r ( 1 - ~A(7)) × × [ 1 + exp { - i ~ A ( 7 ) } ] (u/u0)~A(7).
(5) The pion a m p l i t u d e g i v e s the c o r r e c t r e s i d u e in g l and g4 at the pion pole; it h a s the T - d e p e n d ence n o r m a l l y a s s o c i a t e d with a R e g g e p o l e p l u s a n e c e s s a r y f a c t o r of 7. The A 2 a m p l i t u d e s c o r r e s p o n d to the G e l l - M a n n m e c h a n i s m at the nons e n s e point ~ =0 [23]; the r e s i d u e s ~i a r e chosen a s c o n s t a n t s for s i m p l i c i t y . In the f o r w a r d d i r e c t i o1 n the s - c h a n n e l h e l i c i t y a m p l i t u d e s v a n i s h a s (7)~ n. Thus only g l will s u r v i v e at 7 = 0. Since both the pion and A 2 R e g g e a m p l i t u d e s in (5) vanish a s 7, the e n t i r e value of g l in the f o r w a r d d i r e c t i o n c o m e s f r o m the cut contribution. The m e c h a n i s m f o r the s h a r p f o r w a r d peak in the c r o s s s e c t i o n i s now c l e a r . The cut contribution v a r i e s r e l a t i v e l y slowly in the f o r w a r d d i r e c t i o n . The pion pole t e r m , i n c r e a s i n g l i n e a r l y f r o m z e r o at 7 = 0, i n t e r f e r e s d e s t r u c t i v e l y with the cut c o n t r i b u tion, c a u s i n g a s h a r p d r o p in the c r o s s section. B e f o r e p r e s e n t i n g a fit to the high e n e r g y c r o s s s e c t i o n we t u r n to the finite e n e r g y s u m r u l e s . We u s e the g e n e r a l notation of r e f s . 6 and 8. The r e l e v a n t sum r u l e s a r e t h o s e for the i s o v e c t o r photon a m p l i t u d e s F 2 ( - ) and F 3 ( - ) , which r e c e i v e c o n t r i b u t i o n s f r o m the pion and f r o m the pion c o n s p i r a t o r and the A 2 in m o d e l s u s i n g only p o l e s [23]. In t e r m s of the s - c h a n n e l a m p l i t u d e s (3), we have
F2(_ ) . = (~2 + 7) 4my [(KI+K4) ®gn - (K1-K4)®gA]' F 3 ( - ) : ~ u [ - ( K 1 -K4) ® gu +
(6) 1
+ ( K I + K 4 ) ® g A _ _(~-_(g
2®hA)]
.
The a m p l i t u d e F 3 ( - ) d o e s not contain the pion pole, but d o e s get a plon cut contribution. It is b a s i c a l l y t h i s cut a m p l i t u d e that p l a y s the r o l e taken by the pion c o n s p i r a t o r in o t h e r m o d e l s . It i s e a s y to v e r i f y f r o m the b e h a v i o r of Kj a s 23"/
Volume 29B, n u m b e r 4
PHYSICS
LETTERS
~ 0 t h a t F 2 ( - ) a n d F 3 ( - ) in (6) s a t i s f y t h e conspiracy relation, F2(-) = at i-=0. In t h e C M S R i t i s n e c e s s a r y to e v a l u a t e a n i n t e g r a l of t h e f o r m : ~ i =
500 ,
-(~2/2m)F3(-) ,
=
-(./n)JoVmaxdv(v/Vmax) ~ Im[exp(-ir~,)Fi(v, r)],
o r to do s o m e t h i n g e q u i v a l e n t [18]. If w e a s s u m e that in eq. (3) i s i n d e p e n d e n t of v t h e i n t e g r a t i o n o v e r v of t h e R e g g e c u t a m p l i t u d e b e c o m e s t h e s a m e a s f o r a R e g g e p o l e [24]. It i s t h e n s t r a i g h t f o r w a r d t o s h o w t h a t to go f r o m in (6) to ~ i o n e m e r e l y s u b s t i t u t e s i n (6) a c c o r d i n g to
Kj
1 2 M a y 1969
;
I
~
II
/_,
I
..... 1.0
1
IE 2 0 0
0.4
(n
I00
"'"~
0.2
Fi(-)
O 0
I 0.1
I 0.2
I 0.3 r I,'2
1-g(v,'c')--" C : v _ . sin [½~(~ + ~(~'))] ~[7 + ol(v')]
(7) exp { ½ i ~ a ( v ' ) } g ( u m a x , T') .
Here g stands for g~, gA or hA, and a(T') is me a p p r o p r i a t e ol. T h e h i g h e n e r g y s i d e s of t h e C M S R , [ ( l a ) a n d ( l b ) of r e f . 8] a r e t h e n ~ 2 ( ~ , Vmax, ~") = (~2+T) = 4m [ ( K I + K 4 ) ® G~ - (K 1 - K 4 ) ® G A ] , (8) 1
~P3(~, Vmax, T) = ~ [ - ( K 1 - K 4 ) ® G~ + + ( K I + K 4) ® G A - ~(~")½ - - (K 2 ® HA)] T h e s i m u l t a n e o u s f i t t i n g of t h e C M S R a n d t h e high energy cross section for small T values was d o n e b y t r i a l a n d e r r o r . No a t t e m p t w a s m a d e to u s e e l a b o r a t e r e s i d u e s in o r d e r to o b t a i n a p r e c i s e f i t o v e r a w i d e r a n g e of ~" v a l u e s s i n c e w e a r e i n t e r e s t e d m a i n l y i n I r J < 0. ] The
(GeV/c)2.
5
(GeV)/¢
I 0.4
..... 0 0.5
d(r/dt =
Fig. 1. C r o s s sections f o r y p ~?r+n : ( s - m 2 ) 2 ~ . ~ ~Y, with unpolarized photons (LH s c a l e , solid symbols), ~ ~ C~± - ~ , )/(~± + ~ , ) for l i n e a r l y p o l a r ized photons (RH scale, open symbols). Data points a r e from SLAC [2] (O, 5 G e V ; . , 8 GeV; i , 11 GeV; &, 16 GeV) and f r o m DESY [26] ([:], 3.4 GeV; C), 5 GeV). The dashed curve is the prediction of our model * (actually evaluated at 11 GeV, but v i r t u a l l y e n e r g y independent). The dotted c u r v e is the pseudomodel prediction [10]. The solid c u r v e for ~ is the model r e sult at 5 GeV; the pseudomodel r e s u l t is e s s e n t i a l l y indistinguishable. r e s u l t s of o n e f i t * a r e s h o w n i n f i g s . 1 t h r o u g h 3. The cross sections for both unpolarized and line a r l y p o l a r i z e d [25] p h o t o n s a r e g i v e n in fig. 1. F i g . 2 i l l u s t r a t e s t h e f i t to t h e ~, d e p e n d e n c e of • The fit we display was obtained with the p a r a m e t e r s : a = 8 ( G e V / c ) - 2 , C =1.19, ~1 = - 0 . 7 5 , ~2 = - 4 - 9 2 , o~/r(7") = - 0 . 6 5 ( T + ~ 2 ), O/A~(~") = 0 . 4 5 - ' r , So-= 2mY0= = 1 (GeV) 2. The nonunique-ness of this fit is i l l u s t r a t e d by the fact that an equally good fit to the FESR (fig. 3) and an even b e t t e r fit to the high energy data (fig. 1) is obtained with just an e l e m e n t a r y pion (ot~ ~ 0 ) and cut Ca = 8 , C = 1 . 3 ) .
-
-
"~'-o....
4'~ x 10 3
~ O"" ",,O,,.."
O,,,.
o
,,
e~'e""e~"e"ee"
•
~2 x 10 3
-5 Fig. 2. CMSR, evaluated at ~ = 0, as a function of the m o m e n t p a r a m e t e r 7. The solid (dashed) line is ¢P2C/~2(p3/2m) from Fox [26]. The c i r c l e s a r e from our model * 238
Volume 29B, number 4 0.02
P H Y SI C S L E T T E R S
I''''I''''
0.01 -
''''I ....
I
y=O
-~2
-0.01 -0.02
-0.03
/;
12 May 1969
and t h e h i g h - e n e r g y d a t a without an i n t e r v e n i n g m o d e l . W e n e e d one m o r e a s s u m p t i o n o r e m p i r i c a l f a c t . F i g . 1 s h o w s t h a t the p o w e r l a w b e h a v i o r of t h e p h o t o p r o d u c t i o n a m p l i t u d e s c o r r e s p o n d s to vc~eff w i t h Oteff ~ 0. I n d e p e n d e n t of t h e p o s s i b l e p r e s e n c e of l o g a r i t h m i c v a r i a t i o n in t h e P h r a g m ~ n - L i n d e l 6 f f t h e o r e m a l l o w s u s to c o n c l u d e that t h e p h a s e s of t h e a m p l i t u d e s a r e g i v e n by 1 + exp(-irrot). With ot ~ 0, t h e a m p l i t u d e s a r e p r e d o m i n a n t l y r e a l . T h e high e n e r g y cross sections for linearly polarized photons can t h u s b e w r i t t e n f o r s m a l l T d i r e c t l y in t e r m s of the F E S R qg/: F (~2) 2
J,
[lO]
-0.04 I
-0.2
27
~,,-: ½(s-m2) 2 -~ ~ (m2/~T~.2)L ~ + T ( ~ 4 )
! !
2 2 dcr. -O.l
0 v
O.l
Fig. 3. FESR (CMSR, 9/= 0) v e r s u s "r. The solid (dashed) curves are from Fox [26] (our model, *). Agreement between c i r c l e s (RHS of eq. (9)) and dashed curves indicates the validity of (9) for our particular model. t h e C M S R a t T = 0 , Vma x = 1 . 2 8 G e V . T h e low e n e r g y s i d e s of t h e CMS1R a r e t h o s e of F o x [26], b a s e d on t h e fit of W a l k e r [27] to the low e n e r g y d a t a on p h o t o p r o d u c t i o n . T h e c o m p a r i s o n of F o x ' s r e s u l t s w i t h o u r m o d e l a s a f u n c t i o n of T (for ~ = 0) is shown in fig. 3. T h e o v e r a l l fit to b o t h high e n e r g y d a t a and CMSR i s s e e n to b e s a t i s f a c t o r y f o r I~-I < 0.1 ( G e V / c ) 2. T h i s e s t a b l i s h e s that t h e CMS1R f o r F 2 ( - ) and F 3 ( - ) a r e s a t i s f i e d e q u a l l y by a m o d e l w i t h an e v a s i v e pion p l u s a l~egge cut and m o d e l s w i t h a p i o n c o n spiracy. W e now e x a m i n e t h e c o n n e c t i o n b e t w e e n t h e t c h a n n e l a m p l i t u d e s g i v e n by eq. (6) and the u s u a l FES1R [6, 7] g i v e n b y eq. (8) w i t h the G ' s , of eq. (7), h a v i n g 7 = 0. T o t h e e x t e n t t h a t ½{1 + + COST:O~(T')} ~ sinT:c~(~-')/7:o~(~'),the FESI~ and the amplitudes are related directly by
F "l-)(Vmax,T), (9)
i n d e p e n d e n t of t h e p a r t i c u l a r f o r m of K j in eq. (3). In p r a c t i c e t h i s m e a n s t h a t if t h e m a j o r c o n t r i b u t i o n s to t h e i n t e g r a l s o v e r r ' c o m e f r o m regions where all the Regge poles have I ot(r')l < 1, eq. (9) i s an e s s e n t i a l l y m o d e l - i n d e p e n d e n t r e s u l t . V e r i f i c a t i o n of eq. (9) f o r t h e p a r t i c u l a r m o d e l d e s c r i b e d a b o v e i s shown in f i g . 3. T h e d i r e c t c o n n e c t i o n b e t w e e n the F E S R ~0i and the t - c h a n n e l a m p l i t u d e s F i a f f o r d e d by eq. (9) a l l o w s u s to c o n n e c t t h e l o w - e n e r g y d a t a * See footnote previous page.
2
0.2
IGeV/c) 2
~i(Y=O, u m a x , ~) ~ ~~v m a x R e
/17r" 2
Strictly speaking these are only the contributions f r o m i s o v e c t o r p h o t o n s , but t h e d a t a on t h e ~ - / ~ + r a t i o in d e u t e r i u m [28] i n d i c a t e t h a t f o r 7 < 0.1 (GeV/c) 2 t h e s e c o n t r i b u t i o n s d o m i n a t e . T h e F E S R r e s u l t s [26] show t h a t f o r s m a l l ~- the c r o s s s e c t i o n i s g i v e n e n t i r e l y by t h e f i r s t t e r m s in eq. (10). C o m p a r i s o n of t h e p s e u d o m o d e l g i v e n by eq. (10) with the d a t a at s m a l l ~" i s s h o w n in fig. 1 by t h e d o t t e d c u r v e . T h e a g r e e m e n t is s u r p r i s i n g l y good, in v i e w of the r a t h e r s m a l l v a l u e of Umax n e c e s s a r i l y u s e d in t h e F E S R $. But o u r p o i n t h a s b e e n m a d e , n a m e l y f i n i t e e n e r g y s u m r u l e s r e l a t e low e n e r g y and high e n e r gy d a t a , u s u a l l y by m e a n s of m o d e l s f o r t h e h i g h e n e r g y b e h a v i o r , but t h e y do not d i s c r i m i n a t e a m o n g m o d e l s . In the p r e s e n t e x a m p l e , e s s e n t i a l l y no m o d e l i s n e c e s s a r y to g e n e r a t e t h e high energy cross sections. We w i s h to thank D r . G e o f f r e y C. F o x f o r k i n d l y s u p p l y i n g u s with his e v a l u a t i o n s of t h e CMSR. $ A possible reason is that, near "r = 0 at least, the Born t e r m s almost saturate the FESR. Harari [29] has emphasized this point as an explanation of why the gauge-invariant electric Born t e r m s give roughly the correct high energy cross section near T = 0 [2]. The pseudomodel continues to fit the c r o s s s e c tion for T at least as large as 0.5 (GeV/c)2, while the Born c r o s s section fails beyond T = 0.1 (GeV/c}2.
References 1. G. Manning et al., Nuovo Cimento 41A (1966) 2. A.M. Boyarski et al., Phys. Rev. Letters 20 300; Abstract 430, Proc. XIVth Intern. Conf. High-energy physics, Vienna, 1968 (see fig.
167. (1968) on 2 p.3). 239
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3. J. S. Ball, W.R. Frazer and M. Jacob, Phys. Rev. Letters 29 (1968) 518. 4. F.S. Henyey, Phys. Rev. 170 (1968) 1619. 5. S.D. Drell and J. D. Sullivan, Phys. Rev. Letters 19 (1967) 268. 6. A. Bietti et al., Phys. Letters 26B (1968) 457; 736(E). 7. D. P. Roy and S.-Y. Chu, Phys. Rev. 171 (1968) 1762. 8. P. DiVecchia et al., Phys. Letters 27B (1968) 296, 521. 9. K. Raman and K. V. Vasavada, Phys. Rev. 175 (1968) 2191. I0. K. Dtetz and W. Korth, Phys. Letters 26B ([968) 394. ii. D. Amati et al., Phys. Letters 26B (1968) 510. 12. M. LeBellac, Phys. Letters 25B (1967) 524; M. Aderholz et al., Phys. Letters 27B (1968) 174. 13. C.B. Chiu and J. Finkelstein, Nuovo Cimento 48A (1967) 820. 14. K. Huang and I, J. Muzinich, Phys. Rev. 164 (1967) 1726. 15. R. C. Arnold, Phys. Rev. 153 (1967) 1523.
240
LETTERS
12 May 1969
16. G. Cohen-Tannoudji et al., Nuovo Cimento 48A (1967) 1075. 17. J.N.J. White, Phys. Letters 27B (1968) 92. 18. C. Michael, Nuel, Phys. B8 (1968) 431. 19. F. Schrempp, Nucl. Phys0 B6 (1968) 487. 20. F.S. Henyey et al., Phys. Rev. Letters 21 (1968) 946. 21. J. Frdyland and D. Gordon, Phys. Rev., to be published. 22. F.S. Henyey et al., Phys. Rev., to be published. 23. R. C. Brower and J. W. Dash, Phys. Rev. 175 (1968) 2014. 24. T.W. Rogers and J. H, Schwarz, Phys. Rev. 172 (1968) 1595. 25. Chr. Geweniger et al., Abstract 335, Proc. XIVth Intern. Conf. on High-energy physics, Vienna, 1968 (see fig. 9 p. 7). 26. G.C. Fox, private communication. 27. R.L.Walker, Phys. Rev., to be published. 28. A.M. Boyarski et al., Phys. Rev. Letters 21 (1968) 1767. 29. H. Harari, Proc. 1967 Intern. Syrup. on Electron and photon interactions at high energies, Stanford, 1967 (IN.]3. S. publication CONF.-670923, UC-34Physics (T. I.D.-4500, 51st Edition)), pp. 357-359.
PHYSICS LETTERS
CHARGED PION PHOTOPRODUCTION AND FINITE ENERGY SUM RULES
12 May 1969
*
J. D. JACKSON and C. QUIGG Department of Physics and Lawrence Radiation Laboratory, University of California, Berkeley, California 94720, USA Received 24 March 1969 The large s, small t pion photoproduction data and the corresponding continuous moment sum rules are fitted with evasive Regge poles and associated Regge cuts, instead of the commonly accepted pion conspiracy. The FESR actually yield the real parts of the photoproduction amplitudes, essentially independent of the model. A further assumption permits the prediction (in good agreement with existing data) of high energy cross sections directly from the FESR, without commitment to any particular model.
In this note we c o n s i d e r the photoproduction of charged pions at high e n e r g i e s and s m a l l m o m e n t u m t r a n s f e r s : a) We c o n s t r u c t a model with e v a s i v e Regge poles and a s s o i c a t e d Regge cuts that fits both the high e n e r g y d i f f e r e n t i a l c r o s s s e c t i o n s (including the data with l i n e a r l y p o l a r ized photons) and the continuous m o m e n t finite e n e r g y s u m r u l e s , b) We show that the n o r m a l finite e n e r g y s u m r u l e s in fact d e t e r m i n e a p p r o x i m a t e l y the r e a l p a r t s of the photoproduction a m p l i t u d e s at the u p p e r l i m i t of i n t e g r a t i o n . c) We develop a p s e u d o m o d e l which d i r e c t l y r e l a t e s the finite e n e r g y sum r u l e s to the high e n ergy data without the need for c o m m i t m e n t to any p a r t i c u l a r t h e o r e t i c a l model. Our most i m p o r t a n t c o n c l u s i o n is that, although finite e n e r g y s u m r u l e s for photoproduction allow an i m m e diate connection between the low e n e r g y and high e n e r g y r e g i o n s , they make no d i s t i n c t i o n among a l a r g e c l a s s of m o d e l s , and in p a r t i c u l a r , do not e s t a b l i s h the e x i s t e n c e of a pion c o n s p i r a c y in photoproduction of pions. The s h a r p f o r w a r d peaks in n - p c h a r g e exchange s c a t t e r i n g [ 1] and in charged pion photop r o d u c t i o n [e.g., 2] a r e m o s t c o m m o n l y cited as e v i d e n c e for a p a r i t y - d o u b l e t c o n s p i r a c y of the pion (class III L o r e n t z pole with M= 1). Ball et al. [3] and Henyey [4] have given v e r y s a t i s f a c t o r y fits to the p o s i t i v e pion photoproduction data at s m a l l m o m e n t u m t r a n s f e r s with such a c o n s p i r a c y model. A p p a r e n t l y independent support for the idea of a pion c o n s p i r a c y c a m e f r o m the g e n e r a l r e s u l t s of D r e l l and Sullivan [5], f r o m the application of n o r m a l finite e n e r g y s u m * This work was supported in part by the U.S. Atomic Energy Commission. 236
r u l e s (FESR) to the photoproduction a m p l i t u d e s by Bietti et al. [6] and by Roy and Chu [7], and f r o m the subsequent use of continuous m o m e n t finite energy s u m r u l e s (CMSR) to d e t e r m i n e the t r a j e c t o r i e s and r e s i d u e s of the pion and its cons p i r a t o r [8, 9]. While the pion c o n s p i r a c y is a t t r a c t i v e l y e c o n o m i c a l in fitting photoproduction, it is c e r t a i n l y not unique [10, 11] a n d it c a u s e s difficulties in other p r o c e s s e s via f a c t o r i z a t i o n [ 12]. N u m e r o u s models of hadronic p r o c e s s e s have been made u s i n g Regge poles and ad hoc cuts [13, 14] or cuts obtained f r o m the a b s o r p t i v e optical model [15-20]. F o r photoproduction, m o d els with poles plus cuts have been d i s c u s s e d by A m a t i et al. [11], FrdYland and Gordon [21] and Henyey et al. [22]. All of this work shows that models with poles and cuts can avoid the f a c t o r ization difficulties r e s u l t i n g f r o m the c o n s p i r a c y hypothesis, and provide explanations for the c r o s s - o v e r p h e n o m e n a [ 18], p o l a r i z a t i o n [ 17], and p r o b a b l y also the dip [20] in p i o n - n u c l e o n charge exchange. The r e m a i n i n g question conc e r n i n g the pion c o n s p i r a c y , or lack of it, is whether the evidence deduced f r o m CMSR can be r e g a r d e d as conclusive. We show, by explicit c o n s t r u c t i o n of a c o u n t e r - e x a m p l e , that the CMSR r e s u l t s cannot be so r e g a r d e d ; in the p r o c e s s we show what can actually be l e a r n e d f r o m the sum r u l e s in photoproduction. Our model is s i m i l a r to those of r e f s . 16-20 in that it has e v a s i v e Regge poles and cuts g e n e r a t e d by the r e c i p e of the a b s o r p t i v e model. It d i f f e r s in detail f r o m all of them in its f o r m for the Regge poles and in the a s s u m p t i o n that the k e r n e l in the c u t - g e n e r a t i n g i n t e g r a l is e n e r g y -
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PHYSICS LETTERS
independent. B e c a u s e the a b s o r p t i v e c o r r e c t i o n s a r e e x p r e s s e d in t e r m s of s - c h a n n e l a m p l i t u d e s while the R e g g e e x c h a n g e s and the a m p l i t u d e s u s e d in the s u m r u l e s a r e in the t - c h a n n e l , it is n e c e s s a r y to s p e c i f y a m p l i t u d e s in both channels. In the t - c h a n n e l we u s e theFi(i = 1, 2, 3,4) of r e f . 3. In the s - c h a n n e l we c h o o s e the four a m p l i t u d e s gj, w h e r e j = 1, 2, 3, 4 s t a n d s for the photon, i n i t i a l nucleon, and final nucleon h e l i c i -
ties (ky, ~N ;kN') = (+,+ ; -), (+, - ; -), (+,+ ;+), (-,+ ; -), respectively. The net helicity flips values n~j) ~ [Zy - kN+ XN' [ are 0, I, 1, 2, respectively. At high energies and small momentum transfers the two sets of amplitudes are related b y g = ( u / ~ ) A F , w h e r e u = (s - rn2)/2m i s the photon e n e r g y in the l a b o r a t o r y and the c r o s s i n g matrix A is
/-7/}m-2m/("2+7)
ll
0 ~,
[(7)~
0
(7)~/2rn
2rn(7)
A =/(7)~
0
(7)½/2rn
2rn0(7)~"
2rn/(g2 + 7)
1
\-7/2m
(1)
In eq. (1) we have i n t r o d u c e d 7 = -t, and have n e g l e c t e d t e r m s of r e l a t i v e o r d e r (T/4m2), a s w e l l a s nonleading p o w e r s of u. The r e c i p e of the a b s o r p t i v e m o d e l l e a d s [ 19, 22] to s - c h a n n e l a m p l i t u d e s of the f o r m : oo
gj(u, 7) = gjRegge (g, 7) - ~ o f
dT' exp{-½ (7+7')} ×
(2) × In(j ) [a(77,)½ ] g R e g g e (v, 7 ' ) , w h e r e a and C a r e c h a r a c t e r i s t i c of the a b s o r b ing r e g i o n (or of the P o m e r a n c h u k pole, if the cut i s viewed a s a r i s i n g f r o m two -R egge - p o l e exchange), In(X) i s a m o d i f i e d B e s s e l function of the f i r s t kind of o r d e r n, and gl~egge i s the a m p l i t u d e m a d e f r o m one o r m o r e R e g g e p o l e s . Eq. (2) can b e w r i t t e n in the c o m p a c t o p e r a t o r form:
gj = Kj ® g ~ e g g e .
(3)
S p e c i f i c a t i o n of the Regge a m p l i t u d e s and the p a r a m e t e r s a and C d e t e r m i n e s the model. F o r our p r e s e n t p u r p o s e s , w h e r e we c o n c e n t r a t e on the v e r y s m a l l z b e h a v i o r , we i g n o r e the p t r a j e c t o r y and k e e p only the pion and the A2, c o n t r i b u t i n g r e s p e c t i v e l y to F 2 and to F 1 and F 3. E x p l i c i t l y the s - c h a n n e l Regge a m p l i t u d e s a r e g~egge=g
_gA,
g~egge =g~3egge=hA ,
(4) w h e r [ ~ e g g e = -gz~- g A '
12May 1969
g~ = ega~7 F ( - a ~ ( 7 ) ) × ×
[ 1 + exp { - i ~ n ( z ) }] (u/uO) °t~(7) '
gA = eg~-Z2 (~1+~2) F ( 1 - OtA(7)) × 4m
× [1 + exp ( - i ~ a A ( 7 ) }] (U/Uo)°tA(7), h A = ~z ~
(41 - T~2/4rn2) r ( 1 - ~A(7)) × × [ 1 + exp { - i ~ A ( 7 ) } ] (u/u0)~A(7).
(5) The pion a m p l i t u d e g i v e s the c o r r e c t r e s i d u e in g l and g4 at the pion pole; it h a s the T - d e p e n d ence n o r m a l l y a s s o c i a t e d with a R e g g e p o l e p l u s a n e c e s s a r y f a c t o r of 7. The A 2 a m p l i t u d e s c o r r e s p o n d to the G e l l - M a n n m e c h a n i s m at the nons e n s e point ~ =0 [23]; the r e s i d u e s ~i a r e chosen a s c o n s t a n t s for s i m p l i c i t y . In the f o r w a r d d i r e c t i o1 n the s - c h a n n e l h e l i c i t y a m p l i t u d e s v a n i s h a s (7)~ n. Thus only g l will s u r v i v e at 7 = 0. Since both the pion and A 2 R e g g e a m p l i t u d e s in (5) vanish a s 7, the e n t i r e value of g l in the f o r w a r d d i r e c t i o n c o m e s f r o m the cut contribution. The m e c h a n i s m f o r the s h a r p f o r w a r d peak in the c r o s s s e c t i o n i s now c l e a r . The cut contribution v a r i e s r e l a t i v e l y slowly in the f o r w a r d d i r e c t i o n . The pion pole t e r m , i n c r e a s i n g l i n e a r l y f r o m z e r o at 7 = 0, i n t e r f e r e s d e s t r u c t i v e l y with the cut c o n t r i b u tion, c a u s i n g a s h a r p d r o p in the c r o s s section. B e f o r e p r e s e n t i n g a fit to the high e n e r g y c r o s s s e c t i o n we t u r n to the finite e n e r g y s u m r u l e s . We u s e the g e n e r a l notation of r e f s . 6 and 8. The r e l e v a n t sum r u l e s a r e t h o s e for the i s o v e c t o r photon a m p l i t u d e s F 2 ( - ) and F 3 ( - ) , which r e c e i v e c o n t r i b u t i o n s f r o m the pion and f r o m the pion c o n s p i r a t o r and the A 2 in m o d e l s u s i n g only p o l e s [23]. In t e r m s of the s - c h a n n e l a m p l i t u d e s (3), we have
F2(_ ) . = (~2 + 7) 4my [(KI+K4) ®gn - (K1-K4)®gA]' F 3 ( - ) : ~ u [ - ( K 1 -K4) ® gu +
(6) 1
+ ( K I + K 4 ) ® g A _ _(~-_(g
2®hA)]
.
The a m p l i t u d e F 3 ( - ) d o e s not contain the pion pole, but d o e s get a plon cut contribution. It is b a s i c a l l y t h i s cut a m p l i t u d e that p l a y s the r o l e taken by the pion c o n s p i r a t o r in o t h e r m o d e l s . It i s e a s y to v e r i f y f r o m the b e h a v i o r of Kj a s 23"/
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PHYSICS
LETTERS
~ 0 t h a t F 2 ( - ) a n d F 3 ( - ) in (6) s a t i s f y t h e conspiracy relation, F2(-) = at i-=0. In t h e C M S R i t i s n e c e s s a r y to e v a l u a t e a n i n t e g r a l of t h e f o r m : ~ i =
500 ,
-(~2/2m)F3(-) ,
=
-(./n)JoVmaxdv(v/Vmax) ~ Im[exp(-ir~,)Fi(v, r)],
o r to do s o m e t h i n g e q u i v a l e n t [18]. If w e a s s u m e that in eq. (3) i s i n d e p e n d e n t of v t h e i n t e g r a t i o n o v e r v of t h e R e g g e c u t a m p l i t u d e b e c o m e s t h e s a m e a s f o r a R e g g e p o l e [24]. It i s t h e n s t r a i g h t f o r w a r d t o s h o w t h a t to go f r o m in (6) to ~ i o n e m e r e l y s u b s t i t u t e s i n (6) a c c o r d i n g to
Kj
1 2 M a y 1969
;
I
~
II
/_,
I
..... 1.0
1
IE 2 0 0
0.4
(n
I00
"'"~
0.2
Fi(-)
O 0
I 0.1
I 0.2
I 0.3 r I,'2
1-g(v,'c')--" C : v _ . sin [½~(~ + ~(~'))] ~[7 + ol(v')]
(7) exp { ½ i ~ a ( v ' ) } g ( u m a x , T') .
Here g stands for g~, gA or hA, and a(T') is me a p p r o p r i a t e ol. T h e h i g h e n e r g y s i d e s of t h e C M S R , [ ( l a ) a n d ( l b ) of r e f . 8] a r e t h e n ~ 2 ( ~ , Vmax, ~") = (~2+T) = 4m [ ( K I + K 4 ) ® G~ - (K 1 - K 4 ) ® G A ] , (8) 1
~P3(~, Vmax, T) = ~ [ - ( K 1 - K 4 ) ® G~ + + ( K I + K 4) ® G A - ~(~")½ - - (K 2 ® HA)] T h e s i m u l t a n e o u s f i t t i n g of t h e C M S R a n d t h e high energy cross section for small T values was d o n e b y t r i a l a n d e r r o r . No a t t e m p t w a s m a d e to u s e e l a b o r a t e r e s i d u e s in o r d e r to o b t a i n a p r e c i s e f i t o v e r a w i d e r a n g e of ~" v a l u e s s i n c e w e a r e i n t e r e s t e d m a i n l y i n I r J < 0. ] The
(GeV/c)2.
5
(GeV)/¢
I 0.4
..... 0 0.5
d(r/dt =
Fig. 1. C r o s s sections f o r y p ~?r+n : ( s - m 2 ) 2 ~ . ~ ~Y, with unpolarized photons (LH s c a l e , solid symbols), ~ ~ C~± - ~ , )/(~± + ~ , ) for l i n e a r l y p o l a r ized photons (RH scale, open symbols). Data points a r e from SLAC [2] (O, 5 G e V ; . , 8 GeV; i , 11 GeV; &, 16 GeV) and f r o m DESY [26] ([:], 3.4 GeV; C), 5 GeV). The dashed curve is the prediction of our model * (actually evaluated at 11 GeV, but v i r t u a l l y e n e r g y independent). The dotted c u r v e is the pseudomodel prediction [10]. The solid c u r v e for ~ is the model r e sult at 5 GeV; the pseudomodel r e s u l t is e s s e n t i a l l y indistinguishable. r e s u l t s of o n e f i t * a r e s h o w n i n f i g s . 1 t h r o u g h 3. The cross sections for both unpolarized and line a r l y p o l a r i z e d [25] p h o t o n s a r e g i v e n in fig. 1. F i g . 2 i l l u s t r a t e s t h e f i t to t h e ~, d e p e n d e n c e of • The fit we display was obtained with the p a r a m e t e r s : a = 8 ( G e V / c ) - 2 , C =1.19, ~1 = - 0 . 7 5 , ~2 = - 4 - 9 2 , o~/r(7") = - 0 . 6 5 ( T + ~ 2 ), O/A~(~") = 0 . 4 5 - ' r , So-= 2mY0= = 1 (GeV) 2. The nonunique-ness of this fit is i l l u s t r a t e d by the fact that an equally good fit to the FESR (fig. 3) and an even b e t t e r fit to the high energy data (fig. 1) is obtained with just an e l e m e n t a r y pion (ot~ ~ 0 ) and cut Ca = 8 , C = 1 . 3 ) .
-
-
"~'-o....
4'~ x 10 3
~ O"" ",,O,,.."
O,,,.
o
,,
e~'e""e~"e"ee"
•
~2 x 10 3
-5 Fig. 2. CMSR, evaluated at ~ = 0, as a function of the m o m e n t p a r a m e t e r 7. The solid (dashed) line is ¢P2C/~2(p3/2m) from Fox [26]. The c i r c l e s a r e from our model * 238
Volume 29B, number 4 0.02
P H Y SI C S L E T T E R S
I''''I''''
0.01 -
''''I ....
I
y=O
-~2
-0.01 -0.02
-0.03
/;
12 May 1969
and t h e h i g h - e n e r g y d a t a without an i n t e r v e n i n g m o d e l . W e n e e d one m o r e a s s u m p t i o n o r e m p i r i c a l f a c t . F i g . 1 s h o w s t h a t the p o w e r l a w b e h a v i o r of t h e p h o t o p r o d u c t i o n a m p l i t u d e s c o r r e s p o n d s to vc~eff w i t h Oteff ~ 0. I n d e p e n d e n t of t h e p o s s i b l e p r e s e n c e of l o g a r i t h m i c v a r i a t i o n in t h e P h r a g m ~ n - L i n d e l 6 f f t h e o r e m a l l o w s u s to c o n c l u d e that t h e p h a s e s of t h e a m p l i t u d e s a r e g i v e n by 1 + exp(-irrot). With ot ~ 0, t h e a m p l i t u d e s a r e p r e d o m i n a n t l y r e a l . T h e high e n e r g y cross sections for linearly polarized photons can t h u s b e w r i t t e n f o r s m a l l T d i r e c t l y in t e r m s of the F E S R qg/: F (~2) 2
J,
[lO]
-0.04 I
-0.2
27
~,,-: ½(s-m2) 2 -~ ~ (m2/~T~.2)L ~ + T ( ~ 4 )
! !
2 2 dcr. -O.l
0 v
O.l
Fig. 3. FESR (CMSR, 9/= 0) v e r s u s "r. The solid (dashed) curves are from Fox [26] (our model, *). Agreement between c i r c l e s (RHS of eq. (9)) and dashed curves indicates the validity of (9) for our particular model. t h e C M S R a t T = 0 , Vma x = 1 . 2 8 G e V . T h e low e n e r g y s i d e s of t h e CMS1R a r e t h o s e of F o x [26], b a s e d on t h e fit of W a l k e r [27] to the low e n e r g y d a t a on p h o t o p r o d u c t i o n . T h e c o m p a r i s o n of F o x ' s r e s u l t s w i t h o u r m o d e l a s a f u n c t i o n of T (for ~ = 0) is shown in fig. 3. T h e o v e r a l l fit to b o t h high e n e r g y d a t a and CMSR i s s e e n to b e s a t i s f a c t o r y f o r I~-I < 0.1 ( G e V / c ) 2. T h i s e s t a b l i s h e s that t h e CMS1R f o r F 2 ( - ) and F 3 ( - ) a r e s a t i s f i e d e q u a l l y by a m o d e l w i t h an e v a s i v e pion p l u s a l~egge cut and m o d e l s w i t h a p i o n c o n spiracy. W e now e x a m i n e t h e c o n n e c t i o n b e t w e e n t h e t c h a n n e l a m p l i t u d e s g i v e n by eq. (6) and the u s u a l FES1R [6, 7] g i v e n b y eq. (8) w i t h the G ' s , of eq. (7), h a v i n g 7 = 0. T o t h e e x t e n t t h a t ½{1 + + COST:O~(T')} ~ sinT:c~(~-')/7:o~(~'),the FESI~ and the amplitudes are related directly by
F "l-)(Vmax,T), (9)
i n d e p e n d e n t of t h e p a r t i c u l a r f o r m of K j in eq. (3). In p r a c t i c e t h i s m e a n s t h a t if t h e m a j o r c o n t r i b u t i o n s to t h e i n t e g r a l s o v e r r ' c o m e f r o m regions where all the Regge poles have I ot(r')l < 1, eq. (9) i s an e s s e n t i a l l y m o d e l - i n d e p e n d e n t r e s u l t . V e r i f i c a t i o n of eq. (9) f o r t h e p a r t i c u l a r m o d e l d e s c r i b e d a b o v e i s shown in f i g . 3. T h e d i r e c t c o n n e c t i o n b e t w e e n the F E S R ~0i and the t - c h a n n e l a m p l i t u d e s F i a f f o r d e d by eq. (9) a l l o w s u s to c o n n e c t t h e l o w - e n e r g y d a t a * See footnote previous page.
2
0.2
IGeV/c) 2
~i(Y=O, u m a x , ~) ~ ~~v m a x R e
/17r" 2
Strictly speaking these are only the contributions f r o m i s o v e c t o r p h o t o n s , but t h e d a t a on t h e ~ - / ~ + r a t i o in d e u t e r i u m [28] i n d i c a t e t h a t f o r 7 < 0.1 (GeV/c) 2 t h e s e c o n t r i b u t i o n s d o m i n a t e . T h e F E S R r e s u l t s [26] show t h a t f o r s m a l l ~- the c r o s s s e c t i o n i s g i v e n e n t i r e l y by t h e f i r s t t e r m s in eq. (10). C o m p a r i s o n of t h e p s e u d o m o d e l g i v e n by eq. (10) with the d a t a at s m a l l ~" i s s h o w n in fig. 1 by t h e d o t t e d c u r v e . T h e a g r e e m e n t is s u r p r i s i n g l y good, in v i e w of the r a t h e r s m a l l v a l u e of Umax n e c e s s a r i l y u s e d in t h e F E S R $. But o u r p o i n t h a s b e e n m a d e , n a m e l y f i n i t e e n e r g y s u m r u l e s r e l a t e low e n e r g y and high e n e r gy d a t a , u s u a l l y by m e a n s of m o d e l s f o r t h e h i g h e n e r g y b e h a v i o r , but t h e y do not d i s c r i m i n a t e a m o n g m o d e l s . In the p r e s e n t e x a m p l e , e s s e n t i a l l y no m o d e l i s n e c e s s a r y to g e n e r a t e t h e high energy cross sections. We w i s h to thank D r . G e o f f r e y C. F o x f o r k i n d l y s u p p l y i n g u s with his e v a l u a t i o n s of t h e CMSR. $ A possible reason is that, near "r = 0 at least, the Born t e r m s almost saturate the FESR. Harari [29] has emphasized this point as an explanation of why the gauge-invariant electric Born t e r m s give roughly the correct high energy cross section near T = 0 [2]. The pseudomodel continues to fit the c r o s s s e c tion for T at least as large as 0.5 (GeV/c)2, while the Born c r o s s section fails beyond T = 0.1 (GeV/c}2.
References 1. G. Manning et al., Nuovo Cimento 41A (1966) 2. A.M. Boyarski et al., Phys. Rev. Letters 20 300; Abstract 430, Proc. XIVth Intern. Conf. High-energy physics, Vienna, 1968 (see fig.
167. (1968) on 2 p.3). 239
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3. J. S. Ball, W.R. Frazer and M. Jacob, Phys. Rev. Letters 29 (1968) 518. 4. F.S. Henyey, Phys. Rev. 170 (1968) 1619. 5. S.D. Drell and J. D. Sullivan, Phys. Rev. Letters 19 (1967) 268. 6. A. Bietti et al., Phys. Letters 26B (1968) 457; 736(E). 7. D. P. Roy and S.-Y. Chu, Phys. Rev. 171 (1968) 1762. 8. P. DiVecchia et al., Phys. Letters 27B (1968) 296, 521. 9. K. Raman and K. V. Vasavada, Phys. Rev. 175 (1968) 2191. I0. K. Dtetz and W. Korth, Phys. Letters 26B ([968) 394. ii. D. Amati et al., Phys. Letters 26B (1968) 510. 12. M. LeBellac, Phys. Letters 25B (1967) 524; M. Aderholz et al., Phys. Letters 27B (1968) 174. 13. C.B. Chiu and J. Finkelstein, Nuovo Cimento 48A (1967) 820. 14. K. Huang and I, J. Muzinich, Phys. Rev. 164 (1967) 1726. 15. R. C. Arnold, Phys. Rev. 153 (1967) 1523.
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16. G. Cohen-Tannoudji et al., Nuovo Cimento 48A (1967) 1075. 17. J.N.J. White, Phys. Letters 27B (1968) 92. 18. C. Michael, Nuel, Phys. B8 (1968) 431. 19. F. Schrempp, Nucl. Phys0 B6 (1968) 487. 20. F.S. Henyey et al., Phys. Rev. Letters 21 (1968) 946. 21. J. Frdyland and D. Gordon, Phys. Rev., to be published. 22. F.S. Henyey et al., Phys. Rev., to be published. 23. R. C. Brower and J. W. Dash, Phys. Rev. 175 (1968) 2014. 24. T.W. Rogers and J. H, Schwarz, Phys. Rev. 172 (1968) 1595. 25. Chr. Geweniger et al., Abstract 335, Proc. XIVth Intern. Conf. on High-energy physics, Vienna, 1968 (see fig. 9 p. 7). 26. G.C. Fox, private communication. 27. R.L.Walker, Phys. Rev., to be published. 28. A.M. Boyarski et al., Phys. Rev. Letters 21 (1968) 1767. 29. H. Harari, Proc. 1967 Intern. Syrup. on Electron and photon interactions at high energies, Stanford, 1967 (IN.]3. S. publication CONF.-670923, UC-34Physics (T. I.D.-4500, 51st Edition)), pp. 357-359.