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Vector dominance and forward photoproduction of charged pions
~
Nuclear Physics B19 (1970) 259-268. North-Holland Publishing Company
VECTOR DOMINANCE A N D FORWARD PHOTOPRODUCTION OF CHARGED PIONS A. DAR Deutsches Elektronen-Synchrotron DESY, Hamburg and Teehnion, Israel Institute of Technology, Haifa, Israel* Received 2 February 1970 Abstract: The validity of vector-meson dominance (VMD) relations between photoninitiated reactions and vector-meson-initiated reactions is investigated within the framework of realistic models for high-energy exchange reactions. The models suggest that the VMD relations should be valid at high enough energy, in the helicity frame. In particular the narrow forward structure that has been found in photon initiated reactions should be found also in the analogous hadron reactions, at high enough energy.
1. INTRODUCTION The v e c t o r m e s o n dominance (VMD) hypothesis lead to r e l a t i o n s between p h o t o n - i n i t i a t e d r e a c t i o n s and r e a c t i o n s initiated by t r a n s v e r s e l y - p o l a r i z e d v e c t o r m e s o n s . In p a r t i c u l a r the following r e l a t i o n s b a s e d on VMD and t i m e r e v e r s a l s y m m e t r y , have been p r o p o s e d [1] H d~ fgPTr~ t2 d(~ P l l d/(Tr-p ~ p ° n ) = . - ~ - j d-/(YV p ~ ~+n) ,
(1)
H
1
L(da/dt±- ( d a / d t ) , 7V p ~ ~+n
:
LP H l l A7r-p --* p°n
where pH is the density m a t r i x for the v e c t o r m e s o n in the helicity f r a m e , ~¢ denotes i s o v e c t o r photon and ~± and a, a r e the c r o s s s e c t i o n s with photons l i n e a r l y p o l a r i z e d p e r p e n d i c u l a r and p a r a l l e l , r e s p e c t i v e l y , to the p r o d u c t i o n plane. A s is well known, r e l a t i o n (1) is in good a g r e e m e n t with e x p e r i m e n t [2] while the a s y m m e t r y r e l a t i o n (2) and the rotational i n v a r iant relation * Permanent address.
250
A. DAR (Pll + p H 1) d~
aT
_
p - p°n) =
FgpTr7r ]2 d(~± j--at ( r v p -
(a)
a r e badly violated [3]. M o r e o v e r , the relation [4] H d~ I ~ P l l ~ - (~+p -~ P°N*) = L
~
2 dcr dr- (YVp ~ ~-N*) ,
(4)
A
which in addition to the VMD a s s u m p t i o n r e q u i r e s also an a s s u m p t i o n about the behaviour of the hadron c r o s s section under u - s c r o s s i n g , s e e m s to be badly violated [5] at 8 GeV/c. Since the concept of t r a n s v e r s e p o l a r i z a t i o n is not L o r e n t z invariant, it was suggested [6] that one should not abandon v e c t o r dominance b e f o r e one show that t h e r e e x i s t s no f r a m e in which the VMD r e l a t i o n s a r e satisfied. However, in a r e c e n t publication with the s a m e tittle H a r a r i and Horovitz [7] speculated that at high energy and s m a l l m o m e n t u m t r a n s f e r s ( - t £ m 2) the VMD r e l a t i o n s (1) and (2) should be badly violated in any f r a m e . More specifically, in a second publication Avni and H a r a r i [8] p r e d i c t e d that p~t1 d~(~-p ~ p ° n ) / d t will not show the sharp f o r w a r d peak o b s e r v e d [9] in dcr~yp ~ ~ + n ) d t . The question of a suitable f r a m e for VMD r e l a t i o n s is obviously a dyn a m i c a l one and can not be settled by p u r e l y k i n e m a t i c a l c o n s i d e r a t i o n s . We t h e r e f o r e have c a r r i e d out t h e o r e t i c a l t e s t s of VMD r e l a t i o n s within the f r a m e w o r k of v a r i o u s r e a l i s t i c m o d e l s for h i g h - e n e r g y exchange r e a c t i o n s . In this note we intend to show that v a r i o u s m o d e l s that s u c c e s s f u l l y account for the f o r w a r d s t r u c t u r e in c h a r g e d - p i o n photoproduction (the gauge i n v a r iant OPE model [10] the p e r i p h e r a l model of Dar, Watts and Weisskopf [11] and the a b s o r p t i v e R e g g e - p o l e - e x c h a n g e model [12]) support the VMD r e l a tions if c o m p a r i s o n is made in the helicity f r a m e . In p a r t i c u l a r we expect a s h a r p f o r w a r d peak in P~I d~(~-p ~ p ° n ) / d t and a n a r r o w f o r w a r d dip in P~I d¢(7'+P ~ P°N*)/dr at high enough e n e r g y (Plab ~> 10 G e V / c and Plab > 20 GeV c , r e s p e c t i v e l y ) s i m i l a r to the s t r u c t u r e s that have been found in the analogous photon initiated r e a c t i o n s . We also account for the violation of relation (4) at 8 G e V / c , but expect it to be valid at about P l a b Z 20 GeV/c. Finally, we conclude that the e x p e r i m e n t a l violations of eqs. (2) and (3) is only p a r t l y due to the r e l a t i v e low e n e r g i e s where c o m p a r i s o n has been made, but it is m o r e likely to be due to the p o o r l y d e t e r m i n e d p H 1•
2. THE GAUGE-INVARIANT OPE MODEL The VMD r e l a t i o n s (1) and (2) have been studied r e c e n t l y within the f r a m e w o r k of the e l e c t r i c Born model by Cho and Sakurai [13]. In this model, which is a p a r t i c u l a r c a s e of gauge invariant OPE models, the OPE t - c h a n n e l d i a g r a m is supplemented by s - and u - c h a n n e l n u c l e o n - p o l e diag r a m s with a p u r e ~ t coupling for the YvNN and pNN v e r t i c e s . In the limit s ~ ~ and t fixed both eqs. (1) and (2) a r e exactly satisfied, which s u g g e s t s ,
7r+ FORWARD PHOTOPRODUCTION
261
a m o n g o t h e r t h i n g s , t h a t P ~ I dcr (Tr-p -~ p ° n ) / d t h a s a s h a r p f o r w a r d p e a k e v e n t h o u g h t h e f u l l c r o s s s e c t i o n dcr ( ~ - p -~ p ° n ) / d t h a s a f o r w a r d dip ( s e e e x p l i c i t e x p r e s s i o n s in r e f . [13]). H o w e v e r , e q s . (1), (2) a n d (3) a r e b a d l y v i o l a t e d in t h e o t h e r f r a m e s t h a t w e r e p r o p o s e d [14] f o r t e s t i n g VMD. A d d i t i o n of t h e c o n t r i b u t i o n of t h e s p i n - c u r r e n t p a r t of t h e n u c l e o n p o l e t e r m , w h i c h by i t s e l f i s g a u g e i n v a r i a n t , d o e s not a f f e c t t h e s e c o n c l u s i o n s : T h e c o n t r i b u t i o n i s e q u a l f o r b o t h s i d e s of eq. (1). It h a s t h e f o r m 2
i (t~n+2 pp)2
2
L4~JFgP~' [g4--4~-~] ~]
r
s
t
(5)
,
mN
w h e r e p p = 1.79 a n d p n = - 1 . 9 1 a r e t h e a n o m a l o u s m a g n e t i c m o m e n t s of t h e proton and neutron respectively. This contribution is very small for t £ rn 2 a n d t h u s it d o e s not a f f e c t t h e g o o d a g r e e m e n t o b t a i n e d b e t w e e n e x p e r i m e n t a l r e s u l t s on y p ~ ~+n at s m a l l t - v a l u e s a n d t h e p r e d i c t i o n s of the electric Born model. A s i m i l a r g a u g e i n v a r i a n t O P E m o d e l [10] c a n a l s o b e u s e d to t e s t r e l a t i o n (4). O n c e m o r e one c a n s h o w t h a t a t h i g h - s r e l a t i o n (4) i s e x a c t l y s a t i s f i e d in t h e h e l i c i t y f r a m e . T h i s s u g g e s t a m o n g o t h e r t h i n g s t h a t pH 1 d~ (~+p ~ p ° N * ) / d t h a s a n a r r o w f o r w a r d dip e v e n t h o u g h t h e f u l l c r o s s s e c t i o n d~(Tr+p ~ p ° N * ) / d t h a s a s h a r p f o r w a r d p e a k . M o r e o v e r , t h e m o d e l p r e d i c t s c o r r e c t l y t h e s h a p e a n d m a g n i t u d e of t h e f o r w a r d c r o s s s e c t i o n f o r yp ~ ~ - N * . Although the gauge invariant OPE models predict correctly the behavi o u r of y p -~ ~+n a n d yp -~ ~ - N * at s m a l l t - v a l u e s , t h e m o d e l s f a i l f o r larger t-values. Moreover, the models fail completely for any t-value w h e n a p p l i e d to yp -~ K Z a n d yp ~ KA. I t s s u c c e s s f o r yp ~ ~+n a n d y p -~ 7 - N * m a y t h e r e f o r e b e v i e w e d a s an a c c i d e n t . M o r e r e a l i s t i c m o d e l s s h o u l d b e e x a m i n e d b e f o r e any f i n a l c o n c l u s i o n s c a n b e d r a w n . Two s u c h m o d e l s a r e e x a m i n e d b e l o w . W e w i l l s h o w t h a t e s s e n t i a l l y t h e y l e a d to t h e same conslusions. -
3. T H E NEW P E R I P H E R A L
MODEL
In t h e n e w p e r i p h e r a l m o d e l of D a r , W a t t s a n d W e i s s k o p f [11] t h e c o n t r i b u t i o n f r o m i m p a c t p a r a m e t e r b to t h e s c a t t e r i n g a m p l i t u d e f o r a + b c + d with h e l i c i t y s i t u a t i o n [hi -=ha, ~b, h c , h d , i s g i v e n b y an a b s o r p t i o n model type prescription: M[h](b) = 7(5) B[h](b) .
(6a)
H e r e M[h](b) i s d e f i n e d b y t h e r e l a t i o n M[h ] : k 2
fb db
w h e r e zxh = (h a - hc) + (h d - hb) a n d t ' = t - t m i n. ( t m i n i s t h e m i n i m u m m o -
(65)
262
A. DAR
m e n t u m t r a n s f e r a l l o w e d b y t h e k i n e m a t i c s of t h e r e a c t i o n . ) B[;~] i s t h e B o r n a p p r o x i m a t i o n e x p r e s s i o n f o r t h e c o n t r i b u t i o n of i m p a c t t i a i r a m e t e r b to t h e m a t r i x e l e m e n t with h e l i c i t y s i t u a t i o n [~]. T h e B o r n a p p r o x i m a t i o n i s c a l c u l a t e d on t h e b a s i s of a n e x c h a n g e of one o r m o r e s u i t a b l e e x c h a n g e p a r t i c l e s in t h e t - c h a n n e l . B[~](b) f o r t h e e x c h a n g e of a p a r t i c l e with m a s s m e a n d s p i n J e i s g i v e n by [11] B[~](b) ~ C[;~]
s Je-1 K~(P.eb) ,
(7)
w h e r e ~2=m2-tmin. T h e q u a n t i t i e s C[~] d e p e n d o n l y on t h e h e l i c i t y s i t u a t i o n but not on b. F o r o u r p u r p o s e s it i s enough to p o i n t out t h a t t h e a t t e n u a t i o n f u n c t i o n v(b) c a n b e a p p r o x i m a t e d b y a s t e p f u n c t i o n t l ~(b) :
0
b << R(s)
b <<
R(s),
(8)
w i t h a n a r r o w width. T h e r a d i u s R , w h e r e t h e t r a n s i t i o n f r o m 0 to 1 t a k e s p l a c e , i s o n l y s l o w l y v a r y i n g with e n e r g y . F r o m f o r m u l a (6), (7) a n d (8) o n e o b t a i n s [11] f o r s m a l l t oO
M[)~]
C[;~] s Je-1 k 2 f
R(s)
bdb K~,~(~zeb) JA~(b~-t ) C[~] J e
~/tleR(S )
e-t~e R(s) J ~ ( R ( s ) ~ - t ' ) 2 tZe- t'
(9)
F o r m u l a (9) i s o u r k e y to m a n y p r e d i c t i o n s . L e t u s f i r s t e x a m i n e t h e s m a l l - t b e h a v i o u r of t h e h e l i c i t y a m p l i t u d e s . F o r l a r g e s v a l u e s * t m i n -~ 0 s o t h a t t' -- t a n d g e ~ me" F r o m (9) o n e m a y c o n c l u d e t h e n t h a t in t h e high- s limit (i) T h e s m a l l t b e h a v i o u r of t h e h e l i c i t y a m p l i t u d e s i s i n d e p e n d e n t of t h e m a s s of t h e e x t e r n a l v e c t o r m e s o n . (In o r d e r to p r o v e t h e VMD r e l a t i o n s one s t i l l h a s t o s h o w t h a t t h e c o e f f i c i e n t s C[~] a r e i n d e p e n d e n t t o o of t h e m a s s of t h e e x t e r n a l v e c t o r m e s o n . ) (if) F o r r e a c t i o n s d o m i n a t e d by u - e x c h a n g e : (1) h e l i c i t y a m p l i t u d e s w i t h A~ = 0 h a v e a s h a r p f o r w a r d p e a k r e s u l t i n g f r o m t h e p i o n p r o p a g a t o r (t~ 2 - t ' = m 2 - t ) in eq. (9); (2) h e l i c i t y a m p l i t u d e s with ~h ¢ 0 h a v e a n a r r o w f o r w a r d dip. T h e w i d t h of t h e s e s t r u c t u r e s i s a p p r o x i m a t e l y g i v e n by m 2. T h e p r e s e n c e of a s h a r p f o r w a r d p e a k o r a n a r r o w f o r w a r d dip in t h e differential cross section for a particular reaction dominated by u-exc h a n g e t h u s , r e f l e c t s t h e p r e p o n d e r e n c e of a o n e t y p e , ¢;~ = 0 o r A~ ¢ 0 a m p l i t u d e s , r e s p e c t i v e l y , in t h a t c r o s s s e c t i o n . L e t u s now e x a m i n e t h e d e p e n d e n c e of C[h] on t h e e x t e r n a l m a s s e s in a r e a c t i o n . L e t u s c o n c e n t r a t e on u - e x c h a n g e ivhich i s b o t h t h e s i m p l e s t e x * lmin -* 0 like s -1 if both m a ¢ me, m b ¢ rod; tmin ~ 0 like s -2 if m a = m e or mb=m d •
~'+ F O R W A R D P H O T O P R O D U C T I O N
263
c h a n g e and the only one t h a t c a n l e a d to f a s t v a r i a t i o n at s m a l l t - v a l u e s . F o r O P E , the B o r n a p p r o x i m a t i o n c a n be w r i t t e n a s VXcXa(S, t)VXdXb(S, t) 2 '
B [ l ] ( s , t) :
m
77
(10)
-t
w h e r e the V a r e the v e r t e x f u n c t i o n s f o r the c o r r e s p o n d i n g F e y n m a n d i a g r a m . In g e n e r a l V d e p e n d s both on the d y n a m i c s and k i n e m a t i c s at the v e r t e x . It c a n be shown that C[t ] f o r O P E d i a g r a m s is given by 2 C[)t] : V)te)ta(S , m~)VXdxb(S , m ) .
(11)
T h u s a d e p e n d e n c e of C[)t] on the m a s s of the e x t e r n a l v e c t o r m e s o n in ~initiated r e a c t i o n s c a n 6nIy c o m e f r o m the v e r t e x f u n c t i o n
V~c~ (s,t) : 2gcv c~(pc,he)pap ,
(12)
w h e r e ~/~ is the spin one p o l a r i z a t i o n v e c t o r . H o w e v e r in the h e l i c i t y f r a m e in the l i m i t pa ~Pc ~ ~o m~
V I o ( s , rn 2 ) % -gc~Tr ~
(13)
'
i n d e p e n d e n t of m c ( h o w e v e r , 2
Voo(S' m~ ) ~- - gcTr~
m 2 - 2m 2 c
7T
2m c
(14)
s t r o n g l y d e p e n d s on mc). F o r v e c t o r m e s o n s t r a n s v e r s e l y p o l a r i z e d in the h e l i c i t y f r a m e the h e l i c i t y a m p l i t u d e s c a n depend on the m a s s of the e x t e r n a l v e c t o r m e s o n only t h r o u g h the v e r t e x f a c t o r (13). H o w e v e r , in the h e l i c i t y f r a m e (13) d o e s not depend on the m a s s of the e x t e r n a l v e c t o r m e s o n s ! We t h e r e f o r e c o n c l u d e that f o r l a r g e s the VMD r e l a t i o n s (1), (2), (3) and (4) a r e e x a c t l y s a t i s f i e d in the h e l i c i t y f r a m e . A s i m i l a r c a l c u l a tion s h o w s that t h e s e r e l a t i o n s a r e b a d l y v i o l a t e d in the o t h e r f r a m e s that w e r e p r o p o s e d [14] f o r t e s t i n g VMD r e l a t i o n s . What do we e x p e c t f o r the s m a l l - t b e h a v i o u r of d e / d r f o r t h o s e r e a c t i o n s ? In o r d e r to a n s w e r t h i s q u e s t i o n we m u s t e x a m i n e a l s o the ~pn and 7rpN* v e r t e x f a c t o r s . F o r the 7rpn v e r t e x one c a n show that V v a n i s h e s u n less~the n u c l e o n f l i p s its helicity. T h i s h e l i c i t y flip c o m b i n e d with a unit h e l i c i t y c h a n g e at the u p p e r np v e r t e x (which is r e q u i r e d in o r d e r to p r o duce t h e r e a t r a n s v e r s e pO) r e s u l t s in a total h e l i c i t y c h a n g e A)t = 0. We t h e r e f o r e e x p e c t a s h a r p f o r w a r d p e a k in P~I de (n-p ~ p ° n ) / d t and a n a l o g o u s l y in de ( y v p ~ 7r+n)/dt. Note h o w e v e r that f o r the ~p v e r t e x V00 >> V10 ( s e e eqs. (13) And (14)). N a m e l y at the np v e r t e x h e l i c i t y p r e f e r s not to be c h a n g e d . T h i s , c o m b i n e d with h e l i c i t y flip at the n u c l e o n v e r t e x , r e s u l t s
264
A. DAI{
in a d o m i n a n t ~,)t = 1 h e l i c i t y a m p l i t u d e . W e t h e r e f o r e p r e d i c t t h a t at high energy the unpolarized cross section d~(~-p ~ p°n)/dt has a narrow forw a r d dip. I n d i c a t i o n of s u c h a dip h a s b e e n f o u n d in a r e c e n t e x p e r i m e n t [18] 2. T h e s i t u a t i o n f o r t h e r e a c t i o n ~+p ~ p°N* i s c o m p l e t e l y r e v e r s e d . One c a n s h o w t h a t in t h e pN* v e r t e x t h e h e l i c i t y s t r o n g l y p r e f e r s not to b e c h a n g e d . C o n s e q u e n t l y we e x p e c t a n a r r o w f o r w a r d dip in * / d! and analogously * / " P~I dcr (~+P " ~p 0 N~), m• d~(7',rp -~ 7r- N!),,dt. Such a dlp has been observed in the second reaction [18]. VSimilar arguments lead to the prediction of a sharp forward peak in the unpolarized cross section d~ (~+p -~ p°N*)/d[, which also has been observed experimentally [18[. Since experiments are performed at finite energies, it is worth studying the effect of ~JZp ¢0 on the various cross sections and polarizations at finite s-values. The most dramatic effect of ~'Hp ¢0 comes through ~Le2, the distance between the physical I region and the pion pole: From (9) we see that when pe2 increases the forward structure becomes broader and the forward cross section becomes smaller. For larger /-values the magnitude of the cross section is somewhat reduced but the shape of the cross section is not appreciably altered. Since Pe depends on ~J~p through train we expect the VMD relations to be valid for s-values that satisfy AP << 1 , (15) P w h e r e A# i s t h e d i f f e r e n c e b e t w e e n t h e d i s t a n c e s of t h e p h y s i c a l - / r e g i o n f r o m t h e p i o n p o l e in t h e p h o t o n r e a c t i o n a n d in t h e a n a l o g o u s h a d r o n r e a c t i o n , r e s p e c t i v e l y . A s i m p l e c a l c u l a t i o n s h o w s t h a t r e l a t i o n s (1-3) a n d (4) a r e e x p e c t e d to be v a l i d w i t h i n 10% a c c u r a c y , o n l y f o r P l a b ~> 10 GeV a n d P l a b ~ 20 GeV, r e s p e c t i v e l y . H o w e v e r , f o r t h e r e a c t i o n s y p ~ ~Op a n d y p -~ ~Op, w h e r e T , - e x c h a n g e i s f o r b i d d e n , t h e V M D r e l a t i o n s a r e e x p e c t e d to b e v a l i d a l r e a d y at few GeV.
4. S T R O N G A B S O R P T I V E R E G G E - P O L E
EXCHANGE MODELS
For small t and large s evasive reggeized pion exchange behaves like t h e f i x e d O P E . (At t h e p i o n p o l e t h e y c o i n c i d e e x a c t l y . ) T h e l a r g e p a r t i a l w a v e s of t h e two a m p l i t u d e s a r e t h e n p r a c t i c a l l y t h e s a m e w h i l e t h e low partial waves are removed from the amplitudes by the absorption presc r i p t i o n s . W e a r e l e f t t h e r e f o r e p r a c t i c a l l y with t h e s a m e a m p l i t u d e s a s for the previous model. All our conclusions are therefore reproduced. T h e s e c o n c l u s i o n s c a n b e g e n e r a l i z e d to a n y e x c h a n g e [24].
5. CONCLUSIONS Since our conclusions were based on formula (9) which is only approximate, we have carried a detailed numerical study of the effect of mp ¢0 on A detailed study of the large impact parameter behaviour of OPE diagrams, its connection to narrow forward structure have been made by ref. [15].
and
7r+ FORWARD PHOTOPRODUCTION
I°I
I.O
265
I"c- p ~ p ° l l Z GeV
n
I
as "yp --rc+n ~ p ----p° n
/
0
o
I
GeV/c
0!2
Ol 013
200y. p ~1~ +n If p~p:n
& c)
0
d3 F ~ aerie
o'.os
o'.1
o.~s o!1o - t' GeVZlc 2
o ~zs
o3o
Fig. 1. P e r i p h e r a l model calculations of the y-exchange contribution to the reactions 7p --~ Y+n and 7r-p --~ p°n. We used g2yTr/4y= 2.6. g2m/4Y 29.4 and Nb) that was given in ref. [19]. a) Comparison be'tween predietio'fl's--at 5 GeV/c and experimental r e s u l t s [201 for the "parallel" c r o s s sections (oH1 _pH 1) dcr/dt, b) Comparison between predictions at 5 GeV/c and experimental resuTts [211 for the a s y m m e t r y ratios PHi~pHil. c) Comparison between prediction and experimental r e s u l t s [16} at 11.2 GeV/c for the unpolarized c r o s s section dcr(y-p ~
p°n)/dt.
t h e p r e d i c t i o n s of o u r p e r i p h e r a l m o d e l . W e a l s o e x a m i n e d c o n t r i b u t i o n s from vector and tensor meson exchanges. All our conclusions presented b e f o r e w e r e n u m e r i c a l l y v e r i f i e d . A f u l l a c c o u n t of o u r s t u d y w i l l b e g i v e n e l s e w h e r e [19]. S o m e t y p i c a l r e s u l t s a r e shown in f i g s . 1, 2 a n d 3. T h e fact that our peripheral model succesfully reproduces both the photon and t h e h a d r o n c r o s s s e c t i o n s s u g g e s t t h a t it m a y h a v e s o m e t h i n g to do with r e a l i t y . It t h e n r e i n f o r c e s t h e c o n c l u s i o n s of Cho a n d S a k u r a i [13]. N a m e l y : 1) V e c t o r - m e s o n d o m i n a n c e m u s t be t e s t e d in t h e h e l i c i t y f r a m e . 2) T h e e f f e c t of m p ¢ 0 i s i m p o r t a n t f o r t h e p r o d u c t i o n of l o n g i t u d i n a l l y p o l a r i z e d p but d i s a p p e a r s c o m p l e t e l y f o r t r a n s v e r s e l y p o l a r i z e d p a s s~oo. 3) W e e x p e c t a f o r w a r d p e a k in pH 1 d ~ ( y - p ~ p ° n ) / d t a n a l o g o u s to the f o r w a r d p e a k o b s e r v e d in d~ (yp ~ y + n ) / d t . 4) A t low e n e r g i e s r e l a t i o n s (1), (2) a n d (3) m a y be v i o l a t e d , h o w e v e r t h e r e p o r t e d f a i l u r e of r e l a t i o n s (2) a n d (3) i s m o r e l i k e l y due to t h e p o o r l y d e t e r m i n e d pH_l. In a d d i t i o n , t h e f o l i o w i n g c o n c l u s i o n s c a n b e d r a w n :
266
A. DAR
12 ['yp--~ N~+'yn~n; N3/2] ]~ 5 GeV SLAC
TC+p ~
lO-
p° N "K"++
8 GeV
!
% % gl~
O
0.1
b) 0.01
'o!2
o',
& ' o!8 GeV/c
,!o
'oo'2
' 0.0 ~z
' 0.0t6
(108
-t' GeV2/c 2
Fig. 2. Peripheral model predictions for the g-exchange contribution to the r e a c tions TP ~ 7r-N*, r n --~/r+N* and 7r~-p--+ p°N!*. We used g'~rrTr/47r=2.6, g2,nz/47r= 0.37 and r](b) th"at was given2in ref. [19]. a) ~omparison Between prediction~s at 5 GeV and experimental results [22] for ~[d(y{Fp --* 7r-N~)/dt +do{Tn -~ 7r+N_~')/dt]. b) Comparison between predictions and experimental results at 8 GeV/c for the unpolarized cross section dcr(Tr+p --+ p°N*~)/dt. 5) V e c t o r - d o m i n a n c e r e l a t i o n s a r e e x p e c t e d to w o r k only when the e n e r g y i s l a r g e e n o u g h so t h a t c o n d i t i o n (15) i s w e l l s a t i s f i e d . 6) T h e " f a i l u r e " of the v e c t o r d o m i n a n c e r e l a t i o n (4) i s l i k e l y due to t h e r e l a t i v e l y low e n e r g y w h e r e c o m p a r i s o n h a s b e e n m a d e . 7) We e x p e c t a f o r w a r d dip in P ~ I dcr (~+p ~ p°N*)/dt a n a l o g o u s to the f o r w a r d dip in d~ (yp -~ n - N * ) / d t , a f o r w a r d p e a k i n dcr (Tr+p ~ p°N*)/dt a n d a f o r w a r d dip in d(r (rr-p ~ p°n)/dL T h e a u t h o r w o u l d l i k e to t h a n k P r o f s . W. J e n t s c h k e , E. L o h r m a n n , S. C. C. T i n g a n d M. W. T e u c h e r f o r t h e i r w a r m h o s p i t a l i t y d u r i n g the a u t h o r ' s v i s i t at DESY.
7r+ FORWARD P H O T O P R O D U C T I O N
I% \ ,
267
yp--~° p I 4 GeV DESY 4 GeV CEA
q2:0
0
O.l
0.01
000~
-t GeV2/c 2
F i g . 3. C o m p a r i s o n b e t w e e n p e r i p h e r a l m o d e l p r e d i c t i o n s and e x p e r i m e n t a l r e s u l t s [23] at 4 G e V / c f o r dcr(yp --~ 7r°p)/dt. We u s e d the coupling c o n s t a n t s and u(b) of r e f . [19].
REFERENCES [1] D.S. B e d e r , P h y s . Rev. 149 (1966) 1203: A. D a r , V. F. W e i s s k o p f , C . A . L e v i n s o n and H. J. Lipkin, P h y s . Rev. L e t t e r s 20 (1968) 1261; C . I s o and H . Y o s h i i , Ann. of P h y s . 47 (1968) 424: M . K r a m m e r , P h y s . L e t t e r s 26B (1968) 633: R. Diebold and J. A. P o i r i e r , P h y s . Rev. L e t t e r s 20 (1968) 1532: M . K r a m m e r and D. S c h i l d k n e c h t , Nucl. P h y s . B7 (1968) 583. [2] D . S c h i l d k n e c h t , DESY r e p o r t 6 9 / 1 0 . [3] C . G e w e n i g e r et al., P h y s . L e t t e r s 28B (1968) 155: R. Diebold and J. A. P o i r i e r , P h y s . Rev. L e t t e r s 22 (1969) 255: L. J. Gutay et al., P h y s . Rev. L e t t e r s 22 (1969) 424. [4] A . D a r , Nucl. P h y s . B l l (1969) 634; C . I s o and H. Yoshii, Ann. of P h y s . 48 (1968) 237. [5] A . M . B o y a r s k i et al., P h y s . Rev. L e t t e r s 22 {1968) 148. [6] A. B i a l a s and K. Z a l e w s k i , P h y s . L e t t e r s 28B (1969) 436; D . S c h i l d k n e c h t , r e f . [2]; s e e a l s o ref. [14]. [7] H . H a r a r i and B. H o r o v i t z , P h y s . L e t t e r s 29B (1969) 314. [8] Y . A v n i and H . H a r a r i , P h y s . Rev. L e t t e r s 23 (1969) 262.
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[9] A . M . B o y a r s k i et al.. P h y s . Rev. L e t t e r s 20 (1968) 300; 21 (1968) 1767; G . B u s c h h o r n et al.. P h y s . Rev. L e t t e r s 17 (1966) 1027; 18 (1967) 571. [10] P . S t i c h e l . DESY r e p o r t T H . 6 6 / 2 ; P . S t i c h e l and M . S c h o l z , Nuovo C i m e n t o 34 (1964) 1381; G . K r a m e r and P . S t i c h e l , Z. P h y s . 178 (1964) 513; N . N . A c h a s o v . V . I . B e l i n i c h e r and L . M . S a m k o v J E T P L e t t e r s 6 (1967) 103; D . H o r n and M . J a c o b . Nuovo C i m e n t o 56A (1968) 83; H . F r a a s and D . S c h i l d k n e c h t . NucI. P h y s . B6 (1968) 395. [11] A . D a r . T . L . W a t t s and V . F . W e i s s k o p f . Nucl. P h y s . B13 (1969) 477; A . D a r , DESY r e p o r t 6 9 / 3 8 . [12] J . D . J a c k s o n . R e v i e w t a l s in P r o c . of the Lund conf., J u n e , 1969. [13] C . F . C h o and J . J . S a k u r a i , P h y s . L e t t e r s 30B (1969) 119. [14] A . B i a l a s and K. Z a l e w s k i , r e f . [ 6 ] ; Z . G . T . G u i r a g o s s i a n and A . L e v y , P h y s . L e t t e r s 30B (1969) 48. [15] P . R . S t e v e n s . UCLA, to be p u b l i s h e d ( p r i v a t e c o m m u n i c a t i o n t h r o u g h N . B e y e r s ) [16] B . D . H y a m s et al.. Nucl. P h y s . B7 (1967) 1. [17] A . M . B o y a r s k i et al., P h y s . Rev. L e t t e r s 22 (1968) 148. [18] M . A d e r h o l z et al.. (ABC c o l l a b o r a t i o n ) P h y s . L e t t e r s 27B (1968) 174. [19] A . D a r , T . L . W a t t s and V . F . W e i s s k o p f , P h y s . L e t t e r s 30B (1969) 264; A . D a r , to be p u b l i s h e d . [20] C . G e w e n i g e r et al., P h y s . L e t t e r s 29B (1969) 41; s e e a l s o r e f . [ 2 2 ] . [21] H . B . B u r f e i n d t et al., DESY p r e p r i n t . [22] A . M . B o y a r s k i et al., to be p u b l i s h e d ; s e e a l s o r e f . [17]. [23] G . C . B o l o n et al., P h y s . Rev. L e t t e r s 18 (1967) 926; M . B r a u n s c h w e i g c t a l . , P h y s . L e t t e r s 26B (1968) 405. [24] M . L e - B e l l a c and G . P l a u t , P r e p r i n t NICE TH 6 9 / 2 ; G . E i l a m ( T e c h n i o n ) , to be p u b l i s h e d .
Nuclear Physics B19 (1970) 259-268. North-Holland Publishing Company
VECTOR DOMINANCE A N D FORWARD PHOTOPRODUCTION OF CHARGED PIONS A. DAR Deutsches Elektronen-Synchrotron DESY, Hamburg and Teehnion, Israel Institute of Technology, Haifa, Israel* Received 2 February 1970 Abstract: The validity of vector-meson dominance (VMD) relations between photoninitiated reactions and vector-meson-initiated reactions is investigated within the framework of realistic models for high-energy exchange reactions. The models suggest that the VMD relations should be valid at high enough energy, in the helicity frame. In particular the narrow forward structure that has been found in photon initiated reactions should be found also in the analogous hadron reactions, at high enough energy.
1. INTRODUCTION The v e c t o r m e s o n dominance (VMD) hypothesis lead to r e l a t i o n s between p h o t o n - i n i t i a t e d r e a c t i o n s and r e a c t i o n s initiated by t r a n s v e r s e l y - p o l a r i z e d v e c t o r m e s o n s . In p a r t i c u l a r the following r e l a t i o n s b a s e d on VMD and t i m e r e v e r s a l s y m m e t r y , have been p r o p o s e d [1] H d~ fgPTr~ t2 d(~ P l l d/(Tr-p ~ p ° n ) = . - ~ - j d-/(YV p ~ ~+n) ,
(1)
H
1
L(da/dt±- ( d a / d t ) , 7V p ~ ~+n
:
LP H l l A7r-p --* p°n
where pH is the density m a t r i x for the v e c t o r m e s o n in the helicity f r a m e , ~¢ denotes i s o v e c t o r photon and ~± and a, a r e the c r o s s s e c t i o n s with photons l i n e a r l y p o l a r i z e d p e r p e n d i c u l a r and p a r a l l e l , r e s p e c t i v e l y , to the p r o d u c t i o n plane. A s is well known, r e l a t i o n (1) is in good a g r e e m e n t with e x p e r i m e n t [2] while the a s y m m e t r y r e l a t i o n (2) and the rotational i n v a r iant relation * Permanent address.
250
A. DAR (Pll + p H 1) d~
aT
_
p - p°n) =
FgpTr7r ]2 d(~± j--at ( r v p -
(a)
a r e badly violated [3]. M o r e o v e r , the relation [4] H d~ I ~ P l l ~ - (~+p -~ P°N*) = L
~
2 dcr dr- (YVp ~ ~-N*) ,
(4)
A
which in addition to the VMD a s s u m p t i o n r e q u i r e s also an a s s u m p t i o n about the behaviour of the hadron c r o s s section under u - s c r o s s i n g , s e e m s to be badly violated [5] at 8 GeV/c. Since the concept of t r a n s v e r s e p o l a r i z a t i o n is not L o r e n t z invariant, it was suggested [6] that one should not abandon v e c t o r dominance b e f o r e one show that t h e r e e x i s t s no f r a m e in which the VMD r e l a t i o n s a r e satisfied. However, in a r e c e n t publication with the s a m e tittle H a r a r i and Horovitz [7] speculated that at high energy and s m a l l m o m e n t u m t r a n s f e r s ( - t £ m 2) the VMD r e l a t i o n s (1) and (2) should be badly violated in any f r a m e . More specifically, in a second publication Avni and H a r a r i [8] p r e d i c t e d that p~t1 d~(~-p ~ p ° n ) / d t will not show the sharp f o r w a r d peak o b s e r v e d [9] in dcr~yp ~ ~ + n ) d t . The question of a suitable f r a m e for VMD r e l a t i o n s is obviously a dyn a m i c a l one and can not be settled by p u r e l y k i n e m a t i c a l c o n s i d e r a t i o n s . We t h e r e f o r e have c a r r i e d out t h e o r e t i c a l t e s t s of VMD r e l a t i o n s within the f r a m e w o r k of v a r i o u s r e a l i s t i c m o d e l s for h i g h - e n e r g y exchange r e a c t i o n s . In this note we intend to show that v a r i o u s m o d e l s that s u c c e s s f u l l y account for the f o r w a r d s t r u c t u r e in c h a r g e d - p i o n photoproduction (the gauge i n v a r iant OPE model [10] the p e r i p h e r a l model of Dar, Watts and Weisskopf [11] and the a b s o r p t i v e R e g g e - p o l e - e x c h a n g e model [12]) support the VMD r e l a tions if c o m p a r i s o n is made in the helicity f r a m e . In p a r t i c u l a r we expect a s h a r p f o r w a r d peak in P~I d~(~-p ~ p ° n ) / d t and a n a r r o w f o r w a r d dip in P~I d¢(7'+P ~ P°N*)/dr at high enough e n e r g y (Plab ~> 10 G e V / c and Plab > 20 GeV c , r e s p e c t i v e l y ) s i m i l a r to the s t r u c t u r e s that have been found in the analogous photon initiated r e a c t i o n s . We also account for the violation of relation (4) at 8 G e V / c , but expect it to be valid at about P l a b Z 20 GeV/c. Finally, we conclude that the e x p e r i m e n t a l violations of eqs. (2) and (3) is only p a r t l y due to the r e l a t i v e low e n e r g i e s where c o m p a r i s o n has been made, but it is m o r e likely to be due to the p o o r l y d e t e r m i n e d p H 1•
2. THE GAUGE-INVARIANT OPE MODEL The VMD r e l a t i o n s (1) and (2) have been studied r e c e n t l y within the f r a m e w o r k of the e l e c t r i c Born model by Cho and Sakurai [13]. In this model, which is a p a r t i c u l a r c a s e of gauge invariant OPE models, the OPE t - c h a n n e l d i a g r a m is supplemented by s - and u - c h a n n e l n u c l e o n - p o l e diag r a m s with a p u r e ~ t coupling for the YvNN and pNN v e r t i c e s . In the limit s ~ ~ and t fixed both eqs. (1) and (2) a r e exactly satisfied, which s u g g e s t s ,
7r+ FORWARD PHOTOPRODUCTION
261
a m o n g o t h e r t h i n g s , t h a t P ~ I dcr (Tr-p -~ p ° n ) / d t h a s a s h a r p f o r w a r d p e a k e v e n t h o u g h t h e f u l l c r o s s s e c t i o n dcr ( ~ - p -~ p ° n ) / d t h a s a f o r w a r d dip ( s e e e x p l i c i t e x p r e s s i o n s in r e f . [13]). H o w e v e r , e q s . (1), (2) a n d (3) a r e b a d l y v i o l a t e d in t h e o t h e r f r a m e s t h a t w e r e p r o p o s e d [14] f o r t e s t i n g VMD. A d d i t i o n of t h e c o n t r i b u t i o n of t h e s p i n - c u r r e n t p a r t of t h e n u c l e o n p o l e t e r m , w h i c h by i t s e l f i s g a u g e i n v a r i a n t , d o e s not a f f e c t t h e s e c o n c l u s i o n s : T h e c o n t r i b u t i o n i s e q u a l f o r b o t h s i d e s of eq. (1). It h a s t h e f o r m 2
i (t~n+2 pp)2
2
L4~JFgP~' [g4--4~-~] ~]
r
s
t
(5)
,
mN
w h e r e p p = 1.79 a n d p n = - 1 . 9 1 a r e t h e a n o m a l o u s m a g n e t i c m o m e n t s of t h e proton and neutron respectively. This contribution is very small for t £ rn 2 a n d t h u s it d o e s not a f f e c t t h e g o o d a g r e e m e n t o b t a i n e d b e t w e e n e x p e r i m e n t a l r e s u l t s on y p ~ ~+n at s m a l l t - v a l u e s a n d t h e p r e d i c t i o n s of the electric Born model. A s i m i l a r g a u g e i n v a r i a n t O P E m o d e l [10] c a n a l s o b e u s e d to t e s t r e l a t i o n (4). O n c e m o r e one c a n s h o w t h a t a t h i g h - s r e l a t i o n (4) i s e x a c t l y s a t i s f i e d in t h e h e l i c i t y f r a m e . T h i s s u g g e s t a m o n g o t h e r t h i n g s t h a t pH 1 d~ (~+p ~ p ° N * ) / d t h a s a n a r r o w f o r w a r d dip e v e n t h o u g h t h e f u l l c r o s s s e c t i o n d~(Tr+p ~ p ° N * ) / d t h a s a s h a r p f o r w a r d p e a k . M o r e o v e r , t h e m o d e l p r e d i c t s c o r r e c t l y t h e s h a p e a n d m a g n i t u d e of t h e f o r w a r d c r o s s s e c t i o n f o r yp ~ ~ - N * . Although the gauge invariant OPE models predict correctly the behavi o u r of y p -~ ~+n a n d yp -~ ~ - N * at s m a l l t - v a l u e s , t h e m o d e l s f a i l f o r larger t-values. Moreover, the models fail completely for any t-value w h e n a p p l i e d to yp -~ K Z a n d yp ~ KA. I t s s u c c e s s f o r yp ~ ~+n a n d y p -~ 7 - N * m a y t h e r e f o r e b e v i e w e d a s an a c c i d e n t . M o r e r e a l i s t i c m o d e l s s h o u l d b e e x a m i n e d b e f o r e any f i n a l c o n c l u s i o n s c a n b e d r a w n . Two s u c h m o d e l s a r e e x a m i n e d b e l o w . W e w i l l s h o w t h a t e s s e n t i a l l y t h e y l e a d to t h e same conslusions. -
3. T H E NEW P E R I P H E R A L
MODEL
In t h e n e w p e r i p h e r a l m o d e l of D a r , W a t t s a n d W e i s s k o p f [11] t h e c o n t r i b u t i o n f r o m i m p a c t p a r a m e t e r b to t h e s c a t t e r i n g a m p l i t u d e f o r a + b c + d with h e l i c i t y s i t u a t i o n [hi -=ha, ~b, h c , h d , i s g i v e n b y an a b s o r p t i o n model type prescription: M[h](b) = 7(5) B[h](b) .
(6a)
H e r e M[h](b) i s d e f i n e d b y t h e r e l a t i o n M[h ] : k 2
fb db
w h e r e zxh = (h a - hc) + (h d - hb) a n d t ' = t - t m i n. ( t m i n i s t h e m i n i m u m m o -
(65)
262
A. DAR
m e n t u m t r a n s f e r a l l o w e d b y t h e k i n e m a t i c s of t h e r e a c t i o n . ) B[;~] i s t h e B o r n a p p r o x i m a t i o n e x p r e s s i o n f o r t h e c o n t r i b u t i o n of i m p a c t t i a i r a m e t e r b to t h e m a t r i x e l e m e n t with h e l i c i t y s i t u a t i o n [~]. T h e B o r n a p p r o x i m a t i o n i s c a l c u l a t e d on t h e b a s i s of a n e x c h a n g e of one o r m o r e s u i t a b l e e x c h a n g e p a r t i c l e s in t h e t - c h a n n e l . B[~](b) f o r t h e e x c h a n g e of a p a r t i c l e with m a s s m e a n d s p i n J e i s g i v e n by [11] B[~](b) ~ C[;~]
s Je-1 K~(P.eb) ,
(7)
w h e r e ~2=m2-tmin. T h e q u a n t i t i e s C[~] d e p e n d o n l y on t h e h e l i c i t y s i t u a t i o n but not on b. F o r o u r p u r p o s e s it i s enough to p o i n t out t h a t t h e a t t e n u a t i o n f u n c t i o n v(b) c a n b e a p p r o x i m a t e d b y a s t e p f u n c t i o n t l ~(b) :
0
b << R(s)
b <<
R(s),
(8)
w i t h a n a r r o w width. T h e r a d i u s R , w h e r e t h e t r a n s i t i o n f r o m 0 to 1 t a k e s p l a c e , i s o n l y s l o w l y v a r y i n g with e n e r g y . F r o m f o r m u l a (6), (7) a n d (8) o n e o b t a i n s [11] f o r s m a l l t oO
M[)~]
C[;~] s Je-1 k 2 f
R(s)
bdb K~,~(~zeb) JA~(b~-t ) C[~] J e
~/tleR(S )
e-t~e R(s) J ~ ( R ( s ) ~ - t ' ) 2 tZe- t'
(9)
F o r m u l a (9) i s o u r k e y to m a n y p r e d i c t i o n s . L e t u s f i r s t e x a m i n e t h e s m a l l - t b e h a v i o u r of t h e h e l i c i t y a m p l i t u d e s . F o r l a r g e s v a l u e s * t m i n -~ 0 s o t h a t t' -- t a n d g e ~ me" F r o m (9) o n e m a y c o n c l u d e t h e n t h a t in t h e high- s limit (i) T h e s m a l l t b e h a v i o u r of t h e h e l i c i t y a m p l i t u d e s i s i n d e p e n d e n t of t h e m a s s of t h e e x t e r n a l v e c t o r m e s o n . (In o r d e r to p r o v e t h e VMD r e l a t i o n s one s t i l l h a s t o s h o w t h a t t h e c o e f f i c i e n t s C[~] a r e i n d e p e n d e n t t o o of t h e m a s s of t h e e x t e r n a l v e c t o r m e s o n . ) (if) F o r r e a c t i o n s d o m i n a t e d by u - e x c h a n g e : (1) h e l i c i t y a m p l i t u d e s w i t h A~ = 0 h a v e a s h a r p f o r w a r d p e a k r e s u l t i n g f r o m t h e p i o n p r o p a g a t o r (t~ 2 - t ' = m 2 - t ) in eq. (9); (2) h e l i c i t y a m p l i t u d e s with ~h ¢ 0 h a v e a n a r r o w f o r w a r d dip. T h e w i d t h of t h e s e s t r u c t u r e s i s a p p r o x i m a t e l y g i v e n by m 2. T h e p r e s e n c e of a s h a r p f o r w a r d p e a k o r a n a r r o w f o r w a r d dip in t h e differential cross section for a particular reaction dominated by u-exc h a n g e t h u s , r e f l e c t s t h e p r e p o n d e r e n c e of a o n e t y p e , ¢;~ = 0 o r A~ ¢ 0 a m p l i t u d e s , r e s p e c t i v e l y , in t h a t c r o s s s e c t i o n . L e t u s now e x a m i n e t h e d e p e n d e n c e of C[h] on t h e e x t e r n a l m a s s e s in a r e a c t i o n . L e t u s c o n c e n t r a t e on u - e x c h a n g e ivhich i s b o t h t h e s i m p l e s t e x * lmin -* 0 like s -1 if both m a ¢ me, m b ¢ rod; tmin ~ 0 like s -2 if m a = m e or mb=m d •
~'+ F O R W A R D P H O T O P R O D U C T I O N
263
c h a n g e and the only one t h a t c a n l e a d to f a s t v a r i a t i o n at s m a l l t - v a l u e s . F o r O P E , the B o r n a p p r o x i m a t i o n c a n be w r i t t e n a s VXcXa(S, t)VXdXb(S, t) 2 '
B [ l ] ( s , t) :
m
77
(10)
-t
w h e r e the V a r e the v e r t e x f u n c t i o n s f o r the c o r r e s p o n d i n g F e y n m a n d i a g r a m . In g e n e r a l V d e p e n d s both on the d y n a m i c s and k i n e m a t i c s at the v e r t e x . It c a n be shown that C[t ] f o r O P E d i a g r a m s is given by 2 C[)t] : V)te)ta(S , m~)VXdxb(S , m ) .
(11)
T h u s a d e p e n d e n c e of C[)t] on the m a s s of the e x t e r n a l v e c t o r m e s o n in ~initiated r e a c t i o n s c a n 6nIy c o m e f r o m the v e r t e x f u n c t i o n
V~c~ (s,t) : 2gcv c~(pc,he)pap ,
(12)
w h e r e ~/~ is the spin one p o l a r i z a t i o n v e c t o r . H o w e v e r in the h e l i c i t y f r a m e in the l i m i t pa ~Pc ~ ~o m~
V I o ( s , rn 2 ) % -gc~Tr ~
(13)
'
i n d e p e n d e n t of m c ( h o w e v e r , 2
Voo(S' m~ ) ~- - gcTr~
m 2 - 2m 2 c
7T
2m c
(14)
s t r o n g l y d e p e n d s on mc). F o r v e c t o r m e s o n s t r a n s v e r s e l y p o l a r i z e d in the h e l i c i t y f r a m e the h e l i c i t y a m p l i t u d e s c a n depend on the m a s s of the e x t e r n a l v e c t o r m e s o n only t h r o u g h the v e r t e x f a c t o r (13). H o w e v e r , in the h e l i c i t y f r a m e (13) d o e s not depend on the m a s s of the e x t e r n a l v e c t o r m e s o n s ! We t h e r e f o r e c o n c l u d e that f o r l a r g e s the VMD r e l a t i o n s (1), (2), (3) and (4) a r e e x a c t l y s a t i s f i e d in the h e l i c i t y f r a m e . A s i m i l a r c a l c u l a tion s h o w s that t h e s e r e l a t i o n s a r e b a d l y v i o l a t e d in the o t h e r f r a m e s that w e r e p r o p o s e d [14] f o r t e s t i n g VMD r e l a t i o n s . What do we e x p e c t f o r the s m a l l - t b e h a v i o u r of d e / d r f o r t h o s e r e a c t i o n s ? In o r d e r to a n s w e r t h i s q u e s t i o n we m u s t e x a m i n e a l s o the ~pn and 7rpN* v e r t e x f a c t o r s . F o r the 7rpn v e r t e x one c a n show that V v a n i s h e s u n less~the n u c l e o n f l i p s its helicity. T h i s h e l i c i t y flip c o m b i n e d with a unit h e l i c i t y c h a n g e at the u p p e r np v e r t e x (which is r e q u i r e d in o r d e r to p r o duce t h e r e a t r a n s v e r s e pO) r e s u l t s in a total h e l i c i t y c h a n g e A)t = 0. We t h e r e f o r e e x p e c t a s h a r p f o r w a r d p e a k in P~I de (n-p ~ p ° n ) / d t and a n a l o g o u s l y in de ( y v p ~ 7r+n)/dt. Note h o w e v e r that f o r the ~p v e r t e x V00 >> V10 ( s e e eqs. (13) And (14)). N a m e l y at the np v e r t e x h e l i c i t y p r e f e r s not to be c h a n g e d . T h i s , c o m b i n e d with h e l i c i t y flip at the n u c l e o n v e r t e x , r e s u l t s
264
A. DAI{
in a d o m i n a n t ~,)t = 1 h e l i c i t y a m p l i t u d e . W e t h e r e f o r e p r e d i c t t h a t at high energy the unpolarized cross section d~(~-p ~ p°n)/dt has a narrow forw a r d dip. I n d i c a t i o n of s u c h a dip h a s b e e n f o u n d in a r e c e n t e x p e r i m e n t [18] 2. T h e s i t u a t i o n f o r t h e r e a c t i o n ~+p ~ p°N* i s c o m p l e t e l y r e v e r s e d . One c a n s h o w t h a t in t h e pN* v e r t e x t h e h e l i c i t y s t r o n g l y p r e f e r s not to b e c h a n g e d . C o n s e q u e n t l y we e x p e c t a n a r r o w f o r w a r d dip in * / d! and analogously * / " P~I dcr (~+P " ~p 0 N~), m• d~(7',rp -~ 7r- N!),,dt. Such a dlp has been observed in the second reaction [18]. VSimilar arguments lead to the prediction of a sharp forward peak in the unpolarized cross section d~ (~+p -~ p°N*)/d[, which also has been observed experimentally [18[. Since experiments are performed at finite energies, it is worth studying the effect of ~JZp ¢0 on the various cross sections and polarizations at finite s-values. The most dramatic effect of ~'Hp ¢0 comes through ~Le2, the distance between the physical I region and the pion pole: From (9) we see that when pe2 increases the forward structure becomes broader and the forward cross section becomes smaller. For larger /-values the magnitude of the cross section is somewhat reduced but the shape of the cross section is not appreciably altered. Since Pe depends on ~J~p through train we expect the VMD relations to be valid for s-values that satisfy AP << 1 , (15) P w h e r e A# i s t h e d i f f e r e n c e b e t w e e n t h e d i s t a n c e s of t h e p h y s i c a l - / r e g i o n f r o m t h e p i o n p o l e in t h e p h o t o n r e a c t i o n a n d in t h e a n a l o g o u s h a d r o n r e a c t i o n , r e s p e c t i v e l y . A s i m p l e c a l c u l a t i o n s h o w s t h a t r e l a t i o n s (1-3) a n d (4) a r e e x p e c t e d to be v a l i d w i t h i n 10% a c c u r a c y , o n l y f o r P l a b ~> 10 GeV a n d P l a b ~ 20 GeV, r e s p e c t i v e l y . H o w e v e r , f o r t h e r e a c t i o n s y p ~ ~Op a n d y p -~ ~Op, w h e r e T , - e x c h a n g e i s f o r b i d d e n , t h e V M D r e l a t i o n s a r e e x p e c t e d to b e v a l i d a l r e a d y at few GeV.
4. S T R O N G A B S O R P T I V E R E G G E - P O L E
EXCHANGE MODELS
For small t and large s evasive reggeized pion exchange behaves like t h e f i x e d O P E . (At t h e p i o n p o l e t h e y c o i n c i d e e x a c t l y . ) T h e l a r g e p a r t i a l w a v e s of t h e two a m p l i t u d e s a r e t h e n p r a c t i c a l l y t h e s a m e w h i l e t h e low partial waves are removed from the amplitudes by the absorption presc r i p t i o n s . W e a r e l e f t t h e r e f o r e p r a c t i c a l l y with t h e s a m e a m p l i t u d e s a s for the previous model. All our conclusions are therefore reproduced. T h e s e c o n c l u s i o n s c a n b e g e n e r a l i z e d to a n y e x c h a n g e [24].
5. CONCLUSIONS Since our conclusions were based on formula (9) which is only approximate, we have carried a detailed numerical study of the effect of mp ¢0 on A detailed study of the large impact parameter behaviour of OPE diagrams, its connection to narrow forward structure have been made by ref. [15].
and
7r+ FORWARD PHOTOPRODUCTION
I°I
I.O
265
I"c- p ~ p ° l l Z GeV
n
I
as "yp --rc+n ~ p ----p° n
/
0
o
I
GeV/c
0!2
Ol 013
200y. p ~1~ +n If p~p:n
& c)
0
d3 F ~ aerie
o'.os
o'.1
o.~s o!1o - t' GeVZlc 2
o ~zs
o3o
Fig. 1. P e r i p h e r a l model calculations of the y-exchange contribution to the reactions 7p --~ Y+n and 7r-p --~ p°n. We used g2yTr/4y= 2.6. g2m/4Y 29.4 and Nb) that was given in ref. [19]. a) Comparison be'tween predietio'fl's--at 5 GeV/c and experimental r e s u l t s [201 for the "parallel" c r o s s sections (oH1 _pH 1) dcr/dt, b) Comparison between predictions at 5 GeV/c and experimental resuTts [211 for the a s y m m e t r y ratios PHi~pHil. c) Comparison between prediction and experimental r e s u l t s [16} at 11.2 GeV/c for the unpolarized c r o s s section dcr(y-p ~
p°n)/dt.
t h e p r e d i c t i o n s of o u r p e r i p h e r a l m o d e l . W e a l s o e x a m i n e d c o n t r i b u t i o n s from vector and tensor meson exchanges. All our conclusions presented b e f o r e w e r e n u m e r i c a l l y v e r i f i e d . A f u l l a c c o u n t of o u r s t u d y w i l l b e g i v e n e l s e w h e r e [19]. S o m e t y p i c a l r e s u l t s a r e shown in f i g s . 1, 2 a n d 3. T h e fact that our peripheral model succesfully reproduces both the photon and t h e h a d r o n c r o s s s e c t i o n s s u g g e s t t h a t it m a y h a v e s o m e t h i n g to do with r e a l i t y . It t h e n r e i n f o r c e s t h e c o n c l u s i o n s of Cho a n d S a k u r a i [13]. N a m e l y : 1) V e c t o r - m e s o n d o m i n a n c e m u s t be t e s t e d in t h e h e l i c i t y f r a m e . 2) T h e e f f e c t of m p ¢ 0 i s i m p o r t a n t f o r t h e p r o d u c t i o n of l o n g i t u d i n a l l y p o l a r i z e d p but d i s a p p e a r s c o m p l e t e l y f o r t r a n s v e r s e l y p o l a r i z e d p a s s~oo. 3) W e e x p e c t a f o r w a r d p e a k in pH 1 d ~ ( y - p ~ p ° n ) / d t a n a l o g o u s to the f o r w a r d p e a k o b s e r v e d in d~ (yp ~ y + n ) / d t . 4) A t low e n e r g i e s r e l a t i o n s (1), (2) a n d (3) m a y be v i o l a t e d , h o w e v e r t h e r e p o r t e d f a i l u r e of r e l a t i o n s (2) a n d (3) i s m o r e l i k e l y due to t h e p o o r l y d e t e r m i n e d pH_l. In a d d i t i o n , t h e f o l i o w i n g c o n c l u s i o n s c a n b e d r a w n :
266
A. DAR
12 ['yp--~ N~+'yn~n; N3/2] ]~ 5 GeV SLAC
TC+p ~
lO-
p° N "K"++
8 GeV
!
% % gl~
O
0.1
b) 0.01
'o!2
o',
& ' o!8 GeV/c
,!o
'oo'2
' 0.0 ~z
' 0.0t6
(108
-t' GeV2/c 2
Fig. 2. Peripheral model predictions for the g-exchange contribution to the r e a c tions TP ~ 7r-N*, r n --~/r+N* and 7r~-p--+ p°N!*. We used g'~rrTr/47r=2.6, g2,nz/47r= 0.37 and r](b) th"at was given2in ref. [19]. a) ~omparison Between prediction~s at 5 GeV and experimental results [22] for ~[d(y{Fp --* 7r-N~)/dt +do{Tn -~ 7r+N_~')/dt]. b) Comparison between predictions and experimental results at 8 GeV/c for the unpolarized cross section dcr(Tr+p --+ p°N*~)/dt. 5) V e c t o r - d o m i n a n c e r e l a t i o n s a r e e x p e c t e d to w o r k only when the e n e r g y i s l a r g e e n o u g h so t h a t c o n d i t i o n (15) i s w e l l s a t i s f i e d . 6) T h e " f a i l u r e " of the v e c t o r d o m i n a n c e r e l a t i o n (4) i s l i k e l y due to t h e r e l a t i v e l y low e n e r g y w h e r e c o m p a r i s o n h a s b e e n m a d e . 7) We e x p e c t a f o r w a r d dip in P ~ I dcr (~+p ~ p°N*)/dt a n a l o g o u s to the f o r w a r d dip in d~ (yp -~ n - N * ) / d t , a f o r w a r d p e a k i n dcr (Tr+p ~ p°N*)/dt a n d a f o r w a r d dip in d(r (rr-p ~ p°n)/dL T h e a u t h o r w o u l d l i k e to t h a n k P r o f s . W. J e n t s c h k e , E. L o h r m a n n , S. C. C. T i n g a n d M. W. T e u c h e r f o r t h e i r w a r m h o s p i t a l i t y d u r i n g the a u t h o r ' s v i s i t at DESY.
7r+ FORWARD P H O T O P R O D U C T I O N
I% \ ,
267
yp--~° p I 4 GeV DESY 4 GeV CEA
q2:0
0
O.l
0.01
000~
-t GeV2/c 2
F i g . 3. C o m p a r i s o n b e t w e e n p e r i p h e r a l m o d e l p r e d i c t i o n s and e x p e r i m e n t a l r e s u l t s [23] at 4 G e V / c f o r dcr(yp --~ 7r°p)/dt. We u s e d the coupling c o n s t a n t s and u(b) of r e f . [19].
REFERENCES [1] D.S. B e d e r , P h y s . Rev. 149 (1966) 1203: A. D a r , V. F. W e i s s k o p f , C . A . L e v i n s o n and H. J. Lipkin, P h y s . Rev. L e t t e r s 20 (1968) 1261; C . I s o and H . Y o s h i i , Ann. of P h y s . 47 (1968) 424: M . K r a m m e r , P h y s . L e t t e r s 26B (1968) 633: R. Diebold and J. A. P o i r i e r , P h y s . Rev. L e t t e r s 20 (1968) 1532: M . K r a m m e r and D. S c h i l d k n e c h t , Nucl. P h y s . B7 (1968) 583. [2] D . S c h i l d k n e c h t , DESY r e p o r t 6 9 / 1 0 . [3] C . G e w e n i g e r et al., P h y s . L e t t e r s 28B (1968) 155: R. Diebold and J. A. P o i r i e r , P h y s . Rev. L e t t e r s 22 (1969) 255: L. J. Gutay et al., P h y s . Rev. L e t t e r s 22 (1969) 424. [4] A . D a r , Nucl. P h y s . B l l (1969) 634; C . I s o and H. Yoshii, Ann. of P h y s . 48 (1968) 237. [5] A . M . B o y a r s k i et al., P h y s . Rev. L e t t e r s 22 {1968) 148. [6] A. B i a l a s and K. Z a l e w s k i , P h y s . L e t t e r s 28B (1969) 436; D . S c h i l d k n e c h t , r e f . [2]; s e e a l s o ref. [14]. [7] H . H a r a r i and B. H o r o v i t z , P h y s . L e t t e r s 29B (1969) 314. [8] Y . A v n i and H . H a r a r i , P h y s . Rev. L e t t e r s 23 (1969) 262.
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[9] A . M . B o y a r s k i et al.. P h y s . Rev. L e t t e r s 20 (1968) 300; 21 (1968) 1767; G . B u s c h h o r n et al.. P h y s . Rev. L e t t e r s 17 (1966) 1027; 18 (1967) 571. [10] P . S t i c h e l . DESY r e p o r t T H . 6 6 / 2 ; P . S t i c h e l and M . S c h o l z , Nuovo C i m e n t o 34 (1964) 1381; G . K r a m e r and P . S t i c h e l , Z. P h y s . 178 (1964) 513; N . N . A c h a s o v . V . I . B e l i n i c h e r and L . M . S a m k o v J E T P L e t t e r s 6 (1967) 103; D . H o r n and M . J a c o b . Nuovo C i m e n t o 56A (1968) 83; H . F r a a s and D . S c h i l d k n e c h t . NucI. P h y s . B6 (1968) 395. [11] A . D a r . T . L . W a t t s and V . F . W e i s s k o p f . Nucl. P h y s . B13 (1969) 477; A . D a r , DESY r e p o r t 6 9 / 3 8 . [12] J . D . J a c k s o n . R e v i e w t a l s in P r o c . of the Lund conf., J u n e , 1969. [13] C . F . C h o and J . J . S a k u r a i , P h y s . L e t t e r s 30B (1969) 119. [14] A . B i a l a s and K. Z a l e w s k i , r e f . [ 6 ] ; Z . G . T . G u i r a g o s s i a n and A . L e v y , P h y s . L e t t e r s 30B (1969) 48. [15] P . R . S t e v e n s . UCLA, to be p u b l i s h e d ( p r i v a t e c o m m u n i c a t i o n t h r o u g h N . B e y e r s ) [16] B . D . H y a m s et al.. Nucl. P h y s . B7 (1967) 1. [17] A . M . B o y a r s k i et al., P h y s . Rev. L e t t e r s 22 (1968) 148. [18] M . A d e r h o l z et al.. (ABC c o l l a b o r a t i o n ) P h y s . L e t t e r s 27B (1968) 174. [19] A . D a r , T . L . W a t t s and V . F . W e i s s k o p f , P h y s . L e t t e r s 30B (1969) 264; A . D a r , to be p u b l i s h e d . [20] C . G e w e n i g e r et al., P h y s . L e t t e r s 29B (1969) 41; s e e a l s o r e f . [ 2 2 ] . [21] H . B . B u r f e i n d t et al., DESY p r e p r i n t . [22] A . M . B o y a r s k i et al., to be p u b l i s h e d ; s e e a l s o r e f . [17]. [23] G . C . B o l o n et al., P h y s . Rev. L e t t e r s 18 (1967) 926; M . B r a u n s c h w e i g c t a l . , P h y s . L e t t e r s 26B (1968) 405. [24] M . L e - B e l l a c and G . P l a u t , P r e p r i n t NICE TH 6 9 / 2 ; G . E i l a m ( T e c h n i o n ) , to be p u b l i s h e d .