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Deep inelastic e-p scattering in a Veneziano-like model including diffraction
~
Nuclear Physics B31 (1971) 429-442. Nor~:h-Holland Publishing Company
DEEP INELASTIC e-p SCATTERING IN A VENEZIANO-LIKE MODEL INCLUDING DIFFRACTION D. ATKINSON Physikalisches Institut der Universitttt Bonn, Bonn, Ger~nany and A. P. CONTOGOURIS * Department of Physics, McGill University, Montreal, Canada Received 23 March 1971 Abstract: The Veneziano-inspired approach to deep inelastic e-p scattering is extended to include pomeron-dominated (diffractive) contributions. A four-point amplitude is presented in which the Pomeranchuk singularity, olp(t), is dual to a background contribution in s. This background is characteristic of a Reggecut, as seen in the large [ t asymptotics. When the ampl'itude is extended to virtual photon-nucleon scattering, it scales in the Bjorken limit. When it is combined with the dual, non-diffractive amplitude of Landshoff and Polkinghorne, it accounts well for the proton inelastic form factor in the whole scaling region, and leads to a total pN cross-section in order-of-magnitude agreement with experiment.
1. INTRODUCTION Among v a r i o u s efforts to c o n s t r u c t a V e n e z i a n o - l i k e model f o r two c u r r e n t s (virtual p h o t o n - h a d r o n s c a t t e r i n g ) , that of Landshoff and Polkinghorne (LP) is probably the m o s t s u c c e s s f u l [1, 2]. T h e s e a u t h o r s have c o n s t r u c t e d a dual, c r o s s i n g - s y m m e t r i c , R e g g e - behaved amplitude, which s a t i s f i e s the F u b i n i - G e l l - M a n n (FGM) sum rule, and which embodies s e v e r a l i m p o r tant f e a t u r e s of e l a s t i c hadron f o r m f a c t o r s . The model s a t i s f i e s s c a l i n g p r o p e r t i e s in the Bjorken limit [3] and, what is m o r e striking, it is in o r d e r - o f - m a g n i t u d e a g r e e m e n t with the e x p e r i m e n t a l data for the inelastic proton f o r m f a c t o r , vW2, even though t h e r e a r e e s s e n t i a l l y no f r e e p a r a m e t e r s . F r o m the p h y s i c a l point of view, the L P model is the f i r s t p r o p o s a l that a dual, n o n - d i f f r a c t i v e m e c h a n i s m m a y account f o r a significant p a r t of the inelastic l e p t o n - h a d r o n s c a t t e r i n g , even in the scaling region. This p r o p o s a l is now r e c e i v i n g d i r e c t support in the data a n a l y s i s of Bloom and Gilman [4]. N e v e r t h e l e s s , both f r o m a t h e o r e t i c a l and f r o m a phenomenological point of view, the complete a b s e n c e of a diffractive, p o m e r o n - d o m i n a t e d * Also supported by the National Research Council of Canada.
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D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
contribution is probably the m o s t i m p o r t a n t s h o r t c o m i n g of the L P ansatz. The p r e s e n t work is an attempt to c o n s t r u c t a model for the p o m e r o n contribution to e-p deep inelastic s c a t t e r i n g , and c e r t a i n r e l a t e d p r o c e s s e s , and is intended to be added to the L P f o r m u l a . This new model for the p o m e r o n , besides supplementing the L P a n s a t z , can also be useful in the a n a l y s i s of purely hadronic p r o c e s s e s , in conjunction with o r d i n a r y Veneziano t e r m s . The b a s i c f e a t u r e s of our model may be s u m m a r i z e d as follows: (i) In a c c o r d with c u r r e n t ideas, the p o m e r o n t r a j e c t o r y , a p ( t ) , is dual to a background contribution a s s o c i a t e d with b r a n c h - c u t s t r u c t u r e in the v a riable s of the c r o s s e d channel. (ii) The a s y m p t o t i c behaviour as It] -* ~o (s fixed) c o r r e s p o n d s to a m o v ing b r a n c h point in the s - c h a n n e l complex angular m o m e n t u m plane (J): this can well be taken to r e p r e s e n t a M a n d e l s t a m - R e g g e cut, due to exchange of the nucleon t r a j e c t o r y plus the p o m e r o n (i.e. an absorptive c o r r e c t i o n to nucleon exchange). (iii) In the Bjorken limit, our t w o - c u r r e n t amplitude s a t i s f i e s scaling. When it is p r o p e r l y n o r m a l i z e d and added to the L P contribution to vW2, the a g r e e m e n t with e x p e r i m e n t is significantly i m p r o v e d (see fig. 2). In our picture, the e x p e r i m e n t a l value of vW2 f o r the proton is accounted for mostly by the n o n - d i f f r a c t i v e contribution when co <5 (to ~ - 2 M y / q 2 ) : in this domain, the p o m e r o n exchange is r e l a t i v e l y unimportant. As ¢0 i n c r e a s e s , the r e l a t i v e i m p o r t a n c e of the diffractive contribution i n c r e a s e s ; and, as w -~ 0% the inelastic f o r m f a c t o r tends to a finite value, v W 2 -~ 0.1 (which is about one third of its m a x i m u m value). (iv) In the deep inelastic region, simple v e c t o r - d o m i n a n c e does not hold in our model. Each c u r r e n t is a s s o c i a t e d , not with a single v e c t o r meson, but r a t h e r with a s e r i e s of v e c t o r m e s o n s : we might call this " g e n e r a l i z e d v e c t o r - d o m i n a n c e " . However, n e a r the two-fold p - m e s o n pole (q 2 = q2 = m z, where ql and q2 a r e the f o u r - m o m e n t a of the c u r r e n t s ) , our ~nod~l ddes satisfy simple p-dominance. The c o r r e s p o n d i n g r e s i d u e is d i r e c t l y r e l a t e d to the pN elastic s c a t t e r i n g amplitude, and the n o r m a l i z a t i o n i m posed by the diffractive contribution to v W 2 leads to an a s y m p t o t i c total pN c r o s s section in o r d e r - o f - m a g n i t u d e a g r e e m e n t with experiment. The plan of the paper is as follows: In sect. 2 we r e v i e w briefly the L P model and d i s c u s s some of its m o s t i m p o r t a n t physical a s p e c t s and s h o r t comings. In sect. 3, we p r e s e n t our diffractive amplitude f o r the f o u r - p o i n t function, and d e m o n s t r a t e its a s y m p t o t i c and analytic p r o p e r t i e s in s and t. In sect. 4, we extend our model to v i r t u a l photon-nucleon s c a t t e r i n g ; and, in sect. 5, we d e t e r m i n e the diffractive contribution to v W 2 , and to the a s y m p t o t i c total pN c r o s s section.
2. THE LANDSHOFF-POLKINGHORNE MODEL
We c o n s i d e r the amplitude of fig. 1, which may be written T~flv = f d 4 e iq2x( p 2 ] [ J.a .~),
j~(O)]1pl)0(Xo)
(2.1)
D.Atkinson and A.P. Contogouris, Inelastic e-p scattering
431
S
Fig. 1. Virtual photon-nucleon scattering. H e r e j ~ is an i s o v e c t o r - v e c t o r c u r r e n t , which is s c a t t e r e d f r o m a "nucleon" of spin O and isospin ½. The suffix, /~, is the s p a c e - t i m e index, and the s u p e r s c r i p t , a , is the isospin index. T h e p r o c e s s may be thought of as the v i r t u a l Compton effect 7a(ql)+N(Pl) yfl(q2)+N(P2). We define
s = ( q l + P l )2, t = ( p l - P 2 )2' u = ( P l - q 2 ) 2 ,
(2.2)
and we shall u s e the n o r m a l i z a t i o n of r e f . [1]. The t e n s o r Tt~ u has a decomposition in t e r m s of ten invariant a m p l i t u d e s (see, f o r instance, r e f . [2], t h i r d item):
T# uotl3= Aaflpu Pu + B°tflP ql, v +C°tflP q2, v +"" '
(2.3)
where P =Pl +P'~' and w h e r e the amplitudes A, B, C, . . . . , a r e functions of s, t, q2, q2. ~ / e shall c o n c e n t r a t e exclusively upon A~fl(s, t, q2, q2). Let a(t)
= a
O
+at,
with a o ~ ½, be the P - w - f o -A2 d e g e n e r a t e Regge t r a j e c t o r y , and let
fl(s) = ~t(-M 2 + s ) , be the t r a j e c t o r y c o r r e s p o n d i n g to a s c a l a r nucleon of m a s s M. Landshoff_ and Polkinghorne p r o p o s e the following ansatz for the amplitude, A(s, t,q'~, q2), that c o r r e s p o n d s to or, fl = +, -:
A =kN 2 ~ dUldU2dz
o
"1 2
"i, "2, × f(u 1, u 2, z)g(u 1, u2, z; s)h(ul, u 2, z),
where *
jot_ .1 ..2 _ .+ ~-31.t +OP.=J~ ,
~ .1 ..2_ .J = Jr- zJv=Ju"
(2.4)
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D.Atkinson and A.P. Contogouris, Inelastic e-p scattering
and **
z(u 1 +u 2 +UlU2) ] f=
!1+
~ - ; - ~ T ~ - I ~2
m
J
[ , g=
z
]Z(s),
l+l+UlU2/(l+Z+Ul+U2) j h=
l+z 1 + z + N ( u 1 + u2) .
(2.4")
T h i s a m p l i t u d e w a s c o n s t r u c t e d to h a v e a n u m b e r of d e s i r a b l e p r o p e r t i e s . It h a s a s y m p t o t i c R e g g e b e h a v i o u r and r e s o n a n c e p o l e s in s and t. M o r e o v e r , the r e s i d u e s of the p o l e s on the leading t r a j e c t o r y in s a r e f a c t o r i z able. T h e r e is an infinite s e q u e n c e of p o l e s in q/2 (i = 1,2), c o r r e s p o n d i n g to (~(q~) = 1, 2, 3, . . . , which thus a s s o c i a t e s a s e r i e s of v e c t o r m e s o n s with each c u r r e n t ( g e n e r a l i z e d v e c t o r d o m i n a n c e ) . In the l i m i t s -~ - % t fixed, A(s, t, q2, q2) h a s , in addition to the R e g g e t e r m , a f i x e d pole c o n t r i b u t i o n
-F(t)/s , w h e r e F(t) is the e l a s t i c n u c l e o n f o r m f a c t o r . T h i s m e a n s t h a t the w e l l known F G M s u m r u l e , oO
1
"IT
is s a t i s f i e d .
f d s I m A ( s , t, q21, q ~ ) = F ( t )
T h e r e q u i r e m e n t F ( o ) = 1 f i x e s the n o r m a l i z i n g c o n s t a n t , N :
N -1 = B(m, 1 - Oto).
(2.5)
F i n a l l y , the c o n s t a n t , rn, can be d e t e r m i n e d by m a t c h i n g the l a r g e - t b e h a v i o u r of F(t), s i n c e
F(t ) ~ t - m .
(2.6)
T h e r e a r e then no f r e e p a r a m e t e r s left. C o n s i d e r now q2 = q2 = q2, and i n t r o d u c e the s c a l i n g v a r i a b l e ¢o= 1+ 8 2" -q
(2.7)
In the B j o r k e n l i m i t , s -~ ~ , w fixed (and t fixed), the a m p l i t u d e (2.4) h a s the r e m a r k a b l e p r o p e r t y
A(s, t, q~, q 2 2 ) - s - l dp(t, co), w h e r e 4) is a c a l c u l a b l e function of t and ¢o only. On the a s s u m p t i o n that i s o s c a l a r and i s o v e c t o r p h o t o n s c o n t r i b u t e equally in this l i m i t , one finds, f o r the i n e l a s t i c f o r m f a c t o r , ** This form is slightly simpler than that of ref. [1], and corresponds to that proposed by Landshoff, Phys. Letters 32B (1970) 57.
D.A tkinson and A. P. Contogouris, Inelastic e-p scattering 0 50
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Fig. 2. Inelastic proton form factor in the scaling region: (a) The LP dual non-diffractive contribution (eq. {2.8)). (b) The diffractive contribution by itself (eq. (5.2). (c) The total inelastic form factor, being the sum of (a) and (b). The experimental data are as in ref. [10]. 2
2
2
a
vW2(v, q21 : 2M~'Im{A(s'lr o, q21, q2)+A(u, o, ql' q2 )} =N¢o o
-m
(w_ 1)rn-1
(2.8) where
v = Pl" ql/M" In fig. 2, the b r o k e n c u r v e (a) i s a plot of t h i s e x p r e s s i o n f o r rn = 2.5 a n d a o = 0.5. F r o m the p h y s i c a l p o i n t of v i e w , the m a i n i m p o r t a n c e of the L P w o r k is i n p r o v i d i n g a p a r a m e t e r - f r e e m o d e l , in which a s i g n i f i c a n t p a r t of the i n elastic f o r m factor c o m e s f r o m a dual, n o n - d i f f r a c t i v e m e c h a n i s m . Recently, i m p o r t a n t e v i d e n c e in s u p p o r t of s u c h a m e c h a n i s m h a s b e e n s u p p l i e d by d i r e c t a n a l y s i s of e x p e r i m e n t a l data [4]. In t h i s d a t a , it is found that, a s _ q2 i n c r e a s e s , e a c h s - c h a n n e l r e s o n a n c e c o n t r i b u t i o n d e c r e a s e s r a p i d l y , b u t in s u c h a way t h a t , f o r s m a l l co, the s u m of the s - c h a n n e l r e s o n a n c e s b u i l d s up a s m o o t h ~W 2. T h e s a m e t h i n g h a p p e n s in the L P m o d e l . M o r e o v e r , by v i r t u e of the s e q u e n c e of v e c t o r m e s o n p o l e s in q2, i = 1, 2, t h e m o d e l o f f e r s a s i m p l e d e s c r i p t i o n of the deep i n e l a s t i c f o r m f a c t o r that a m o u n t s to a g e n e r a l i z a t i o n of the n o t i o n of v e c t o r d o m i n a n c e . W i t h i n the V e n e z i a n o a p p r o a c h , the s a m e d e s c r i p t i o n s u c c e s s f u l l y a c c o u n t s a l s o f o r the e l a s t i c f o r m f a c t o r d a t a [5]. N e v e r t h e l e s s , t h e c o m p l e t e a b s e n c e of a p o m e r o n c o n t r i b u t i o n m a y b e the m o s t s e r i o u s d e f e c t of t h e a n s a t z (2.4) ¢. F o r one thing, the r e s i d u e of ¢ The LP model, as well as that of sect. 4 of the present work, also contains no Atrajectory in the baryon channel, but a A-contribution could be added in a straightforward manner, and this would not affect the above conclusions. The question of c u r r e n t conservation is discussed in sect. 5 of ref. [1]; but we have nothing to add to this discussion.
D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
434
A(s, t, q2, q2)at the two-fold pole_ cz(q2) = 1 - cz(q2) is R(s, t) =XN2B[2 - ol(t), -fl(s)] s ~"'J~ (- Xs)°t(t)-2 "
(2.9)
H e r e R(s, t) is p r o p o r t i o n a l to the a m p l i t u d e f o r e l a s t i c p N s c a t t e r i n g . T h i s would m e a n that the s -~ % t - f i x e d l i m i t of ON -~ p N is d o m i n a t e d by p - f o exchange, w h e r e a s e x p e r i m e n t shows a c l e a r diffraction c h a r a c t e r . T h i s s u g g e s t s that a p o m e r o n contribution is needed. The total c r o s s section f o r the a b s o r p t i o n of r e a l photons by p r o t o n s is p r a c t i c a l l y constant between 5 and 16 GeV. T h i s would indicate p o m e r o n d o m i n a n c e in r e a l photon s c a t t e r i n g , and then the c o m p l e t e a b s e n c e of the p o m e r o n in v i r t u a l photon s c a t t e r i n g could not e a s i l y be u n d e r s t o o d . Finally, although the l a r g e - w b e h a v i o u r of the e x p e r i m e n t a l vW2 is not well e s t a b l i s h e d , the existing data (fig. 2) a r e c e r t a i n l y s u g g e s t i v e of s o m e contribution, e s p e c i a l l y f o r l a r g e ¢o, in addition to that of eq. (2.8).
3. A F O U R - P O I N T AMI,LITUDE WITH I,OMERON EXCHANGE Our c o n s t r u c t i o n of a d i f f r a c t i v e f o u r - p o i n t amplitude will be b a s e d upon the H a r a r i - F r e u n d p i c t u r e , which a s s o c i a t e s o r d i n a r y t - c h a n n e l t r a j e c t o r i e s (like o~(t) of s e c t . 2) with s - c h a n n e l r e s o n a n c e s , but the p o m e r o n t r a j e c t o r y , ~i,(t), with s o m e s - c h a n n e l background. We need an i n t e g r a l r e p r e s e n t a t i o n s i m i l a r to that of the B-function, f o r this will m a k e it r e latively e a s y to set up an amplitude for v i r t u a l photon-nucleon s c a t t e r i n g (sect. 4) that h a s the r e q u i r e d p r o p e r t i e s , in p a r t i c u l a r Bjorken scaling. Our f o u r - p o i n t a n s a t z is &(s,
t)
Fj zd Z ( l + } ) ~ ( t ) , , "l +z'v(s)
:
1 - ( l + z ) -v log(1 + z)
(3.1)
o
Here ~(t) -
ai,(t) -2,
n(s) - no+XoS ,
w h e r e ~?o, Xo and V a r e constants to be fixed l a t e r . In the l i n e a r a p p r o x i m a tion, the p o m e r o n t r a j e c t o r y is ai,(t) = 1 +Xi,t. The p r o p e r t i e s of A(s,t ) m a y be e s t a b l i s h e d as follows: C o n s i d e r a in eq. (3.1) to be a function of ~ and 77 and r e s t r i c t attention initially to g a n d ~ r e a l and negative. It is then e a s y to justify differentiating (3.1) u n d e r the integral: 07 o
D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
435
F r o m this e x p r e s s i o n , it f o l l o w s that
:
/ld~'B(-~, -V'), ~l-~
o r , by a s i m p l e c h a n g e of v a r i a b l e ,
A(S, t) = f o
d x B [ - ~ ( t ) , x-~/(s)].
(3.2)
T h i s e x p r e s s i o n c a n now be c o n t i n u e d f r o m the n e g a t i v e r e a l a x e s of ~ and ~? to t h e e n t i r e c o m p l e x p l a n e s (i.e. to all v a l u e s of s and t).
For Is] -~oo ( a r g [ s ] ¢
O) andt fixed, O~p(t)-2
A(s, t) ~ v r (- ~)(-n)~ - v r [ 2 - ep(t)] (- Xos) T h i s c o r r e s p o n d s to a s i m p l e R e g g e pole s t r u c t u r e , and s f i x e d at a r e a l , n e g a t i v e v a l u e , we find A ~ (_~)7/ f o
(3.3)
F o r It [ -~ ~ ( a r g t ¢ 0),
dxF(x_~?)(_~)-x.
(3.4)
Since the p o w e r of ( - ¢ ) h a s b e e n s m e a r e d f r o m 7/to 7 / - 7 , this c o r r e s p o n d s to a R e g g e cut, r a t h e r than a R e g g e pole. Indeed, s i n c e F ( x - 7 / ) is a s m o o t h ly v a r y i n g function, f o r 7? r e a l and n e g a t i v e , we can e x p e c t the l e a d i n g b e h a v i o u r of A to be o b t a i n e d by r e p l a c i n g F(x - 7/) by F ( - ~?) in eq. (3.4) *: A(s, t) ~ F ( - ~ ) (- ~)~ - (- ~) ~)rl-7 ~ log(-
r(- no- XoS)[-aP(t)]v°+ xoS
log[-ap(t)], (3.5)
which is indeed the kind of behaviour expected from a crossed-channel Regge cut. The singularities of eq. (3.2) in the variable t are easily seen to be simplepoles at ~(t) = m, m =0, 1, 2 , . . . , with residues
(-1)
dx(x - 77 - 1 ) ( x - ~7- 2) . . . . . ( x - ~ - m ) .
m
(3.6)
o T h i s e x p r e s s i o n is a p o l y n o m i a l of d e g r e e m in 77 (or s), so that t h e r e a r e no a n c e s t o r s . To d e t e r m i n e the s i n g u l a r i t i e s in s, one can w r i t e eq. (3.2) in the f o r m
* Clearly this step is not rigorous, since (_~)-x is a rapidly oscillating function, but the general conclusion that a Regge pole has been smeared into a Regge cut follows directly from eq. (3.4), and is certainly eorrect.
436
D.Atkinson and A .P. Contogouris, Inelastic e-p scattering
(- 1)/ F(-~) I = 0 I ! F ( - ~ -l) l - ~ + x
o
:
F(-~)
/=0
l!r(-~
-l) (_ 1)/log ( I _ A ) .
(3.7)
(3.8)
This gives a series of branch cuts I < ~(s) < l + y ,
(3.9)
or
l-~/° --~< O
l +7 -~7o s ~ < - ~,
(3.10)
O
l = 0 , 1, 2 , . . . The a m p l i t u d e (3.1) contains t h r e e p a r a m e t e r s , 77o, ~o and ~, which we now need to fix. According to eq. (2.9), the Veneziano t e r m of pN s c a t t e r i n g , R(s, 1), has the following l a r g e - t behaviour:
n(s, t) - x ~ v 2 r [ - t~(s)] (- ~t) ~(s) , which c o r r e s p o n d s , of c o u r s e , to the nucleon Regge pole at J = ~(s) = ~(-M 2 +s) in the s - c h a n n e l a n g u l a r m o m e n t u m plane. Since A (S, t) is e v e n tually to be added to R(s, t ), it would be nice to m a k e its b r a n c h - p o i n t , at J = ~?(s) =- ~?o + ~o s, c o r r e s p o n d to a M a n d e l s t a m moving b r a n c h - p o i n t f o r m e d by the exchange of the nucleon and the p o m e r o n (i.e. an a b s o r p t i v e Regge cut a s s o c i a t e d with the nucleon). T h i s r e q u i r e s that the s = o i n t e r c e p t of ~?(s) should be the s a m e a s that of ~(s), the nucleon t r a j e c t o r y , and that the slope of 77(s) should be the h a r m o n i c m e a n of those of/3(s) and Otp(S). That is,
77o = -~M2' ~o- ~ + ~ p "
(3.11)
F r o m (3.10) we s e e that the lowest b r a n c h - p o i n t o c c u r s at
s o
=M 2 --~+~P kp
(3.11')
For any reasonable ~tp, this is well above the ~ N normal threshold, so we may take the cuts (3.10) to be a contribution to the inelastic right-hand cut. We will choose T = 2, for in this way we will have a cut all the way along the right-hand cut, starting from s o (3.11'). However, it is to be emphasized that this is not the elastic cut, but only part of the inelastic cut. Our ansatz (3.1) belongs to the class of "smeared" or "smoothed" Veneziano models, and so is similar to that first proposed by Martin [6], and then discussed by several authors [7]. However, in our case, the pole in s has been completely smoothed away, and no pinching mechanism has
D.A tkinson and A.P. Contogouris, Inelastic e-p scattering
437
b e e n i n t r o d u c e d to r e - i n s t a t e it on t h e s e c o n d s h e e t , a s in r e f . [ 6 ] . T h e P o m e r a n c h u k s i n g u l a r i t y , ap(t), on t h e o t h e r h a n d , i s a s i m p l e m o v i n g p o l e in c o m p l e x J . S e v e r a l e x p e r i m e n t a l a n d p h e n o m e n o l o g i c a l a n a l y s e s of
elastic hadron reactions establish ~tp ~ 0.4~, implving that, if ther~ arebosons on the t r a j e c t o r y ,
t h e y a r e of v e r y h i g h m a s s * (m 2 >~ ~ 1
~ 2.5GeV").
4. A P O M E R O N - D O M I N A T E D A M P L I T U D E F O R TWO C U R R E N T S C o n s i d e r now t h e a m p l i t u d e T~.fi. with a = fl = 3 a n d i t s t e n . s o r d e c o m p o s i t i o n (2.3). F o r t h e corresponding~La~mplitude A aft =- Ap(s, t, q2, q2), we s h a l l w r i t e an i n t e g r a l r e p r e s e n t a t i o n s i m i l a r to (2.4), but a p p r o p r i a t e to p o m e r o n d o m i n a n c e , a n d i n c o r p o r a t i n g t h e f o u r - p o i n t f u n c t i o n of s e c t . 3. Our representation is
Ap(S, t, q21, q22, ×/p(ul,
:a N
u2,
o
dUldU_dZ Ul~2 z ~[ol(q~,, ol(q~,,
olp(t,;ul,
u2, Z]
Z)gp(Ul, u2, z;s)hp(Ul, u 2, z)P(u 1, u 2, z ; t ) .
(4.1)
H e r e ;~N i s a n o r m a l i z i n g c o n s t a n t (with d i m e n s i o n G e V - 2 ) , a n d ~P i s t h e s a m e a s in (2.4'), e x c e p t t h a t ~ p ( t ) r e p l a c e s ~ ( t ) a s t h e t h i r d a r g u m e n t . Further,
f p = I 1+ Z_UlU~ 2 _l-I t z + UlU2J '
z gp =
1+ 1+
l'q(s)
UlU2/(u 1 + u2)J
'
l+z
h p = l+Ul +U2 +Z,
(4.2)
w h e r e it i s a c o n s t a n t to b e f i x e d l a t e r . T h e f a c t o r s (4.2) a r e s i m i l a r to t h o s e of (2.4"), but with s o m e s i m p l i f i c a t i o n s m a d e p o s s i b l e by t h e a b s e n c e of s - c h a n n e l p o l e s , a n d b y t h e f a c t t h a t A p d o e s not n e e d to s a t i s f y a n F G M * Another possibility would be to smoothe away the poles in t also, with an ansatz such as
f
"
and then there would be no p a r t i c l e s on the Pomeranehuk t r a j e c t o r y . The nature of the branch points can also be changed by generalizing the s m e a r i n g function. These m o r e general p o s s i b i l i t i e s will be considered in a forthcoming work: for the p r e sent, we shall be satisfied with eq. (3.1) as a simple formula for a diffractive contribution.
438
D.Atkinson and A .P. Contogouris, Inelastic e-p scattering
sum rule. Finally,
w(z) +UlU 2 [1 P =
l+UlU 2
~p(t)
]-1
l+UlU2(Z+Z-1)]
,
(4.3)
where
w(z) = 1- ( 1 + z ) - 7 log(1 +z) " The additional factor P(Ul, u2, z; t), in the integrand of (4.1), establishes the n e c e s s a r y diffractive properties of Ap. The properties of Ap a r e as follows:
(a) There are poles in q21 and q~ occurring at ~(q~) : 1, 2, 3 , . . . ,
i : 1,
2, which come f r o m end-point singularities atu i = 0, i = 1, 2, just as in eq. (2.4). (b) The residue at the two-fold pole at a(q 2) = a(q 2) = 1 is easily seen to be R e s A p ( s , t, q21, q22)=~N[1-ap(t)]'l A(s, t ) ,
(4.4)
with A(s, t) as in (3.1). By adding the s ~-~u c r o s s i n g - s y m m e t r i c t e r m , we obtain the complete pomeron contribution for the amplitude pN--. pN:
Rv(s, t) : X N [ 1 - a p ( t ) ] - i [A(s, t ) + A ( u , t ) ] .
(4.5)
For s -- ~ and t fixed,
Rp(s, t ) ~ XN[1 - ap(t ) ]-lT {[_ ~?(s)]~(t ) +[_ ~?(u)]~(t) 1 + e - i ~ a P (t) . ~ XNWF[ap(t)]
sin~ap(t)
ap(t)-2 [XoS] .
(4.5')
This has the c o r r e c t form for a diffractive contribution to pN-~ pN. (c) The branch points in s a r i s e as end-point singularities at z = ~. The corresponding contribution to Ap is 2
2
2
2
Ap(S, t, ql' q2 ) ~ XN~P(ql' q2 )A(s' t ) ,
(4.6)
where
and where A(s, t) was given in eq. (3.1). Hence A p has the branch-point s t r u c t u r e (3.10) in the variable s. (d) In a s i m i l a r way, one can show that there a r e poles in t, at ~ p ( t ) = 2, 3, 4, . . . . corresponding to the poles of A(s, t), as discussed in sect. 3.
D.Atkinson and A.P. Contogouris, Inelastic e-p scattering
439
(e) So f a r a s the a s y m p t o t i c b e h a v i o u r in s is c o n c e r n e d , we s h a l l be s a t i s f i e d with a d i s c u s s i o n of the l i m i t f o r r e a l s - ~ - % just a s in r e f . [1]. M a k e the c h a n g e of v a r i a b l e X'
(s)'
and let s ~ - ~ . In this l i m i t , x' gP = 1-1+.1.2/(u l +u2) ~ 1
rl(S)
-l+.1.2/(u1+u2)
exp
so that *
2
22
Ap(s, t, ql'
~'N
q2 )
l-(~p(t) [-~(s)]~(t) f/o 1
x
1 u2J
dUldU2
11~(q1)-1
UlU2 [ 1 + UlJ
F dx', ,,-~
xul+U2o
~7(x)
x' exp-l+UlU2/(Ul+U2)
] J"
With the c h a n g e of v a r i a b l e Xt =
1+
ulu2 ix u I +u2J
'
this e x p r e s s i o n m a y be w r i t t e n
Ap(s, t, q21, q22)~
~,NF[I - a p ( t ) ] A(q21, q22, t)[-XoS]
aP(t)-2 ,
(4.7)
where
x [ul+"2]-I[: + ulu2 ]_:(t). L
(4.7')
u I +u2J
T h i s is R e g g e b e h a v i o u r , c o r r e s p o n d i n g to p o m e r o n e x c h a n g e . T h e f o r m (4.7) and (4.7') is a c t u a l l y only c o r r e c t f o r t > 0: f o r t < 0 t h e r e i s a l s o a f i x e d pole at J = 1, which, h o w e v e r , is c a n c e l l e d when the s ~ u c r o s s i n g s y m m e t r i c t e r m is added. (f) In the l i m i t t ~ _oo, t h e b e h a v i o u r is d e t e r m i n e d by the l a r g e - z c o n t r i b u t i o n to (4.1), which is given by (4.6). T h e r e is t h e r e f o r e a m o v i n g b r a n c h point in the c o m p l e x J - p l a n e . * The asymptotic properties of eq. (4.1) have been established in certain sectors only of the variables in question, as in ref. [lj. We stress, however, that the limits eqs. (3.3) and (3.4) for A itself are uniform, except for arg s = 0, arg t = 0.
D.Atkinson and A. P. Contogouris, Inelastic e-p scattering
440
(g) The Bjorken limit is defined as in sect. 2. We f i r s t make the change of variables u 1 =-7?(s)x 1, u2=-~(s)x 2. (4.8) In deriving our scaling form, we observe that convergence of the integral representation (4.1) r e q u i r e s that we s t a r t with co < 1. Eventually we shall continue the l a r g e - s asymptotic expression to co > 1 just as in ref. [1], since this corresponds to the physical region for e-p s c a t t e r i n g * . Thus we have (1+ eta(q2)-1 I 1 l~o-l+}'s/(1-co) = 1 -~12 u- -I
[ ~ s~-~ _oo exp -
1 ~P[c~(q21), ~(q22), a ( t ) ; u l ,
j
,
Xl+X2 XlX2]"
u2, z ] ~ i x + l l ~ e x p I - l . c °
Thus
(,+
S
x (Xl~2x2)exp[-(Z+l~)Xl+X~21. XlX 2 A The XlX 2 double integral can be p e r f o r m e d explicitly, and this gives
2
2
Ap(s, t, ql' q2 )
X
N 1 ~p(t, co),
X
s
O
where 1
-1
o so one has Bjorken scaling, for any fixed t.
5. INELASTIC PROTON FORM FACTOR AND TOTAL pN CROSS SECTION To d e t e r m i n e our d i f f r a c t i v e contribution to the e l a s t i c proton f o r m f a c t o r , we need to continue (4.9) f r o m co < I to co > 1 (much a s in ref. [I]). The function (4.10) has a b r a n c h cut 1 < co < oo, on which i t s i m a g i n a r y p a r t is
* The procedure of continuing an asymptotic expression in a subsidiary variable is highly non-rigorous and fraught with danger, as we realize no less than do Landshoff and Polkinghorne. We can only hope that it works in this case.
D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
Im ~p(t,
¢o) .
441
0 (.¢ o - 1 ) . ~ - k N.( ~ ° ~ l ) ~ ~oa p ( t ) - i
(5.1)
O
The diffractive contribution to W2 is
W2p(v, q2) = 2-M-MTIrm {Ap(s, o, q2, q 2 ) + A p ( u , F r o m eq. (2.7), we s e e that, f o r l a r g e s, s ~ u finally
o, q2, q2)} .
m e a n s w -~ -w, so we obtain
vW2p(v ' q2) = ~ - \ - ~ /
(5.2)
O
C l e a r l y , a s w ~ ~o, vW2p t e n d s to a c o n s t a n t , ~tN/~to, which is a t y p i c a l f e a t u r e of d i f f r a c t i v e m o d e l s [8]. H we add (5.2) to (2.8), we can get good a g r e e m e n t with e x p e r i m e n t if we c h o o s e ~tN/~t o in the r a n g e 0.1 to 0.15. It t u r n s out t h a t vW2p is r a t h e r i n s e n s i t i v e to the p a r a m e t e r , p, e x c e p t f o r ¢0 n e a r 1, but in this r e g i o n the n o n - d i f f r a c t i v e c o n t r i b u t i o n d o m i n a t e s a n y w a y . C u r v e (a) of fig. 2 s h o w s the L P dual c o n t r i b u t i o n (2.8), which a c c o u n t s f o r m o s t of the e x p e r i m e n t a l d a t a at s m a l l w. T h i s is in a c c o r d with the a n a l y s i s of B l o o m and G i l m a n [4], and s u g g e s t s a r a t h e r l a r g e v a l u e of tz *. In fig. 2, the d i f f r a c t i v e c o n t r i b u t i o n is p l o t t e d by i t s e l f in c u r v e (b), and the s u m of the n o n - d i f f r a c t i v e p a r t (a), and the d i f f r a c t i v e p a r t (b) is g i v e n in c u r v e (c), s h o w i n g a good fit to the e x p e r i m e n t a l p o i n t s . F o r c u r v e s (b) and (c), the v a l u e s t~ = 5 and )tN/h ° = 0 . 1 1 5 ,
(5.3)
have been chosen. F r o m (4.5') and t h e v a l u e (5.3) f o r hN/)to, we can obtain an e s t i m a t e f o r the p N t o t a l c r o s s s e c t i o n . N e a r the t w o - f o l d pole a(q 2) = a(q 2) = 1, pd o m i n a n c e s h o u l d hold, and it is e a s y to a p p l y the o p t i c a l t h e o r e m to o b tain in the l i m i t s ~ 0%
~'N
2
~f p
~pN(S) ~ ~/~o 3~t2rn4 " P
(5.4)
With.y = 2, a s in s e c t . 3, f 2 / 4 ~ = 2.5 ref[9] and ~t = 0.85 GeV -2, this g i v e s a-N 15 m b , which is in o i ~ d e r - o f - m a g n i t u d e a g r e e m e n t with the e x p e r i mPental v a l u e of 24 m b . It is a p l e a s u r e to t h a n k M r . R. G a s k e l l and P r o f e s s o r s yon G e h l e n f o r u s e f u l d i s c u s s i o n s .
V. M t l l l e r and G.
* The exact value o f ~ might be detormined by some additional requirements, as, for example, the q~dependence of the process e-p --. e - p p (ref. [10]).
442
D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
REFERENCES [1] P. V. Landshoff and J. C. Polkinghorne, Nuel. Phys. B19 (1970) 432. [2] M. Bander, Nucl. Phys. B13 (1969) 587; R. C. Brower and J. H. Weis, Phys. Rev. 188 (1969) 2486, 2495; 1R. C. Brower, A. Rabl and J. H. Weis, Nuovo Cimento 65A (1970) 654; M. Ademollo and E. Del Giudice, Nuovo Cimento 63A (1969) 639; D. Z. Freedman, Phys. Rev., to be published. [3] J. D. Bjorken, Phys. Rev. 179 (1969) 1547. [4] E.D. Bloom and F . J . Gilman, Phys.Rev. Letters 25 (1970) 1140; C. H. Llewellyn Smith, Invited paper at the Austin Meeting of P a r t i c l e s and Fields, November 1970. [5] P. Di Vecchia and F. Drago, F r a s c a t i preprint LNF-69/25; P. H. ~ rampton, University of Chicago preprint (1970). [6] A. M art i n , Phys. Letters 29B (1969) 431. [7] N. F. Bali, D.D. Coon and J. W. Dash, Phys. Rev. Letters 23 (1969) 900; K. Huang, Phys. Rev. Letters 23 (1969) 903. [8] H. H a r a r i , Phys. Rev. Letters 22 (1969) 1078; H. Abarbanel, M. L. Goldberger and S. B. T r e i m a n , Phys,Rev. Letters 22 (1969) 500. [9] J. J. Sakurai, Currents and mesons (University of Chicago P r e s s , 1969) ch. III. [10] M. Briedenbach et al., P h y s , R e v . Letters 23 (1969) 935; see also E. D. Bloom et al., SLAC-PUB 796 (1970) and C. L. Jordan, invited paper at the Austin Meeting of P a r t i c l e s and Fields, November 1970.
Nuclear Physics B31 (1971) 429-442. Nor~:h-Holland Publishing Company
DEEP INELASTIC e-p SCATTERING IN A VENEZIANO-LIKE MODEL INCLUDING DIFFRACTION D. ATKINSON Physikalisches Institut der Universitttt Bonn, Bonn, Ger~nany and A. P. CONTOGOURIS * Department of Physics, McGill University, Montreal, Canada Received 23 March 1971 Abstract: The Veneziano-inspired approach to deep inelastic e-p scattering is extended to include pomeron-dominated (diffractive) contributions. A four-point amplitude is presented in which the Pomeranchuk singularity, olp(t), is dual to a background contribution in s. This background is characteristic of a Reggecut, as seen in the large [ t asymptotics. When the ampl'itude is extended to virtual photon-nucleon scattering, it scales in the Bjorken limit. When it is combined with the dual, non-diffractive amplitude of Landshoff and Polkinghorne, it accounts well for the proton inelastic form factor in the whole scaling region, and leads to a total pN cross-section in order-of-magnitude agreement with experiment.
1. INTRODUCTION Among v a r i o u s efforts to c o n s t r u c t a V e n e z i a n o - l i k e model f o r two c u r r e n t s (virtual p h o t o n - h a d r o n s c a t t e r i n g ) , that of Landshoff and Polkinghorne (LP) is probably the m o s t s u c c e s s f u l [1, 2]. T h e s e a u t h o r s have c o n s t r u c t e d a dual, c r o s s i n g - s y m m e t r i c , R e g g e - behaved amplitude, which s a t i s f i e s the F u b i n i - G e l l - M a n n (FGM) sum rule, and which embodies s e v e r a l i m p o r tant f e a t u r e s of e l a s t i c hadron f o r m f a c t o r s . The model s a t i s f i e s s c a l i n g p r o p e r t i e s in the Bjorken limit [3] and, what is m o r e striking, it is in o r d e r - o f - m a g n i t u d e a g r e e m e n t with the e x p e r i m e n t a l data for the inelastic proton f o r m f a c t o r , vW2, even though t h e r e a r e e s s e n t i a l l y no f r e e p a r a m e t e r s . F r o m the p h y s i c a l point of view, the L P model is the f i r s t p r o p o s a l that a dual, n o n - d i f f r a c t i v e m e c h a n i s m m a y account f o r a significant p a r t of the inelastic l e p t o n - h a d r o n s c a t t e r i n g , even in the scaling region. This p r o p o s a l is now r e c e i v i n g d i r e c t support in the data a n a l y s i s of Bloom and Gilman [4]. N e v e r t h e l e s s , both f r o m a t h e o r e t i c a l and f r o m a phenomenological point of view, the complete a b s e n c e of a diffractive, p o m e r o n - d o m i n a t e d * Also supported by the National Research Council of Canada.
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D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
contribution is probably the m o s t i m p o r t a n t s h o r t c o m i n g of the L P ansatz. The p r e s e n t work is an attempt to c o n s t r u c t a model for the p o m e r o n contribution to e-p deep inelastic s c a t t e r i n g , and c e r t a i n r e l a t e d p r o c e s s e s , and is intended to be added to the L P f o r m u l a . This new model for the p o m e r o n , besides supplementing the L P a n s a t z , can also be useful in the a n a l y s i s of purely hadronic p r o c e s s e s , in conjunction with o r d i n a r y Veneziano t e r m s . The b a s i c f e a t u r e s of our model may be s u m m a r i z e d as follows: (i) In a c c o r d with c u r r e n t ideas, the p o m e r o n t r a j e c t o r y , a p ( t ) , is dual to a background contribution a s s o c i a t e d with b r a n c h - c u t s t r u c t u r e in the v a riable s of the c r o s s e d channel. (ii) The a s y m p t o t i c behaviour as It] -* ~o (s fixed) c o r r e s p o n d s to a m o v ing b r a n c h point in the s - c h a n n e l complex angular m o m e n t u m plane (J): this can well be taken to r e p r e s e n t a M a n d e l s t a m - R e g g e cut, due to exchange of the nucleon t r a j e c t o r y plus the p o m e r o n (i.e. an absorptive c o r r e c t i o n to nucleon exchange). (iii) In the Bjorken limit, our t w o - c u r r e n t amplitude s a t i s f i e s scaling. When it is p r o p e r l y n o r m a l i z e d and added to the L P contribution to vW2, the a g r e e m e n t with e x p e r i m e n t is significantly i m p r o v e d (see fig. 2). In our picture, the e x p e r i m e n t a l value of vW2 f o r the proton is accounted for mostly by the n o n - d i f f r a c t i v e contribution when co <5 (to ~ - 2 M y / q 2 ) : in this domain, the p o m e r o n exchange is r e l a t i v e l y unimportant. As ¢0 i n c r e a s e s , the r e l a t i v e i m p o r t a n c e of the diffractive contribution i n c r e a s e s ; and, as w -~ 0% the inelastic f o r m f a c t o r tends to a finite value, v W 2 -~ 0.1 (which is about one third of its m a x i m u m value). (iv) In the deep inelastic region, simple v e c t o r - d o m i n a n c e does not hold in our model. Each c u r r e n t is a s s o c i a t e d , not with a single v e c t o r meson, but r a t h e r with a s e r i e s of v e c t o r m e s o n s : we might call this " g e n e r a l i z e d v e c t o r - d o m i n a n c e " . However, n e a r the two-fold p - m e s o n pole (q 2 = q2 = m z, where ql and q2 a r e the f o u r - m o m e n t a of the c u r r e n t s ) , our ~nod~l ddes satisfy simple p-dominance. The c o r r e s p o n d i n g r e s i d u e is d i r e c t l y r e l a t e d to the pN elastic s c a t t e r i n g amplitude, and the n o r m a l i z a t i o n i m posed by the diffractive contribution to v W 2 leads to an a s y m p t o t i c total pN c r o s s section in o r d e r - o f - m a g n i t u d e a g r e e m e n t with experiment. The plan of the paper is as follows: In sect. 2 we r e v i e w briefly the L P model and d i s c u s s some of its m o s t i m p o r t a n t physical a s p e c t s and s h o r t comings. In sect. 3, we p r e s e n t our diffractive amplitude f o r the f o u r - p o i n t function, and d e m o n s t r a t e its a s y m p t o t i c and analytic p r o p e r t i e s in s and t. In sect. 4, we extend our model to v i r t u a l photon-nucleon s c a t t e r i n g ; and, in sect. 5, we d e t e r m i n e the diffractive contribution to v W 2 , and to the a s y m p t o t i c total pN c r o s s section.
2. THE LANDSHOFF-POLKINGHORNE MODEL
We c o n s i d e r the amplitude of fig. 1, which may be written T~flv = f d 4 e iq2x( p 2 ] [ J.a .~),
j~(O)]1pl)0(Xo)
(2.1)
D.Atkinson and A.P. Contogouris, Inelastic e-p scattering
431
S
Fig. 1. Virtual photon-nucleon scattering. H e r e j ~ is an i s o v e c t o r - v e c t o r c u r r e n t , which is s c a t t e r e d f r o m a "nucleon" of spin O and isospin ½. The suffix, /~, is the s p a c e - t i m e index, and the s u p e r s c r i p t , a , is the isospin index. T h e p r o c e s s may be thought of as the v i r t u a l Compton effect 7a(ql)+N(Pl) yfl(q2)+N(P2). We define
s = ( q l + P l )2, t = ( p l - P 2 )2' u = ( P l - q 2 ) 2 ,
(2.2)
and we shall u s e the n o r m a l i z a t i o n of r e f . [1]. The t e n s o r Tt~ u has a decomposition in t e r m s of ten invariant a m p l i t u d e s (see, f o r instance, r e f . [2], t h i r d item):
T# uotl3= Aaflpu Pu + B°tflP ql, v +C°tflP q2, v +"" '
(2.3)
where P =Pl +P'~' and w h e r e the amplitudes A, B, C, . . . . , a r e functions of s, t, q2, q2. ~ / e shall c o n c e n t r a t e exclusively upon A~fl(s, t, q2, q2). Let a(t)
= a
O
+at,
with a o ~ ½, be the P - w - f o -A2 d e g e n e r a t e Regge t r a j e c t o r y , and let
fl(s) = ~t(-M 2 + s ) , be the t r a j e c t o r y c o r r e s p o n d i n g to a s c a l a r nucleon of m a s s M. Landshoff_ and Polkinghorne p r o p o s e the following ansatz for the amplitude, A(s, t,q'~, q2), that c o r r e s p o n d s to or, fl = +, -:
A =kN 2 ~ dUldU2dz
o
"1 2
"i, "2, × f(u 1, u 2, z)g(u 1, u2, z; s)h(ul, u 2, z),
where *
jot_ .1 ..2 _ .+ ~-31.t +OP.=J~ ,
~ .1 ..2_ .J = Jr- zJv=Ju"
(2.4)
432
D.Atkinson and A.P. Contogouris, Inelastic e-p scattering
and **
z(u 1 +u 2 +UlU2) ] f=
!1+
~ - ; - ~ T ~ - I ~2
m
J
[ , g=
z
]Z(s),
l+l+UlU2/(l+Z+Ul+U2) j h=
l+z 1 + z + N ( u 1 + u2) .
(2.4")
T h i s a m p l i t u d e w a s c o n s t r u c t e d to h a v e a n u m b e r of d e s i r a b l e p r o p e r t i e s . It h a s a s y m p t o t i c R e g g e b e h a v i o u r and r e s o n a n c e p o l e s in s and t. M o r e o v e r , the r e s i d u e s of the p o l e s on the leading t r a j e c t o r y in s a r e f a c t o r i z able. T h e r e is an infinite s e q u e n c e of p o l e s in q/2 (i = 1,2), c o r r e s p o n d i n g to (~(q~) = 1, 2, 3, . . . , which thus a s s o c i a t e s a s e r i e s of v e c t o r m e s o n s with each c u r r e n t ( g e n e r a l i z e d v e c t o r d o m i n a n c e ) . In the l i m i t s -~ - % t fixed, A(s, t, q2, q2) h a s , in addition to the R e g g e t e r m , a f i x e d pole c o n t r i b u t i o n
-F(t)/s , w h e r e F(t) is the e l a s t i c n u c l e o n f o r m f a c t o r . T h i s m e a n s t h a t the w e l l known F G M s u m r u l e , oO
1
"IT
is s a t i s f i e d .
f d s I m A ( s , t, q21, q ~ ) = F ( t )
T h e r e q u i r e m e n t F ( o ) = 1 f i x e s the n o r m a l i z i n g c o n s t a n t , N :
N -1 = B(m, 1 - Oto).
(2.5)
F i n a l l y , the c o n s t a n t , rn, can be d e t e r m i n e d by m a t c h i n g the l a r g e - t b e h a v i o u r of F(t), s i n c e
F(t ) ~ t - m .
(2.6)
T h e r e a r e then no f r e e p a r a m e t e r s left. C o n s i d e r now q2 = q2 = q2, and i n t r o d u c e the s c a l i n g v a r i a b l e ¢o= 1+ 8 2" -q
(2.7)
In the B j o r k e n l i m i t , s -~ ~ , w fixed (and t fixed), the a m p l i t u d e (2.4) h a s the r e m a r k a b l e p r o p e r t y
A(s, t, q~, q 2 2 ) - s - l dp(t, co), w h e r e 4) is a c a l c u l a b l e function of t and ¢o only. On the a s s u m p t i o n that i s o s c a l a r and i s o v e c t o r p h o t o n s c o n t r i b u t e equally in this l i m i t , one finds, f o r the i n e l a s t i c f o r m f a c t o r , ** This form is slightly simpler than that of ref. [1], and corresponds to that proposed by Landshoff, Phys. Letters 32B (1970) 57.
D.A tkinson and A. P. Contogouris, Inelastic e-p scattering 0 50
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Fig. 2. Inelastic proton form factor in the scaling region: (a) The LP dual non-diffractive contribution (eq. {2.8)). (b) The diffractive contribution by itself (eq. (5.2). (c) The total inelastic form factor, being the sum of (a) and (b). The experimental data are as in ref. [10]. 2
2
2
a
vW2(v, q21 : 2M~'Im{A(s'lr o, q21, q2)+A(u, o, ql' q2 )} =N¢o o
-m
(w_ 1)rn-1
(2.8) where
v = Pl" ql/M" In fig. 2, the b r o k e n c u r v e (a) i s a plot of t h i s e x p r e s s i o n f o r rn = 2.5 a n d a o = 0.5. F r o m the p h y s i c a l p o i n t of v i e w , the m a i n i m p o r t a n c e of the L P w o r k is i n p r o v i d i n g a p a r a m e t e r - f r e e m o d e l , in which a s i g n i f i c a n t p a r t of the i n elastic f o r m factor c o m e s f r o m a dual, n o n - d i f f r a c t i v e m e c h a n i s m . Recently, i m p o r t a n t e v i d e n c e in s u p p o r t of s u c h a m e c h a n i s m h a s b e e n s u p p l i e d by d i r e c t a n a l y s i s of e x p e r i m e n t a l data [4]. In t h i s d a t a , it is found that, a s _ q2 i n c r e a s e s , e a c h s - c h a n n e l r e s o n a n c e c o n t r i b u t i o n d e c r e a s e s r a p i d l y , b u t in s u c h a way t h a t , f o r s m a l l co, the s u m of the s - c h a n n e l r e s o n a n c e s b u i l d s up a s m o o t h ~W 2. T h e s a m e t h i n g h a p p e n s in the L P m o d e l . M o r e o v e r , by v i r t u e of the s e q u e n c e of v e c t o r m e s o n p o l e s in q2, i = 1, 2, t h e m o d e l o f f e r s a s i m p l e d e s c r i p t i o n of the deep i n e l a s t i c f o r m f a c t o r that a m o u n t s to a g e n e r a l i z a t i o n of the n o t i o n of v e c t o r d o m i n a n c e . W i t h i n the V e n e z i a n o a p p r o a c h , the s a m e d e s c r i p t i o n s u c c e s s f u l l y a c c o u n t s a l s o f o r the e l a s t i c f o r m f a c t o r d a t a [5]. N e v e r t h e l e s s , t h e c o m p l e t e a b s e n c e of a p o m e r o n c o n t r i b u t i o n m a y b e the m o s t s e r i o u s d e f e c t of t h e a n s a t z (2.4) ¢. F o r one thing, the r e s i d u e of ¢ The LP model, as well as that of sect. 4 of the present work, also contains no Atrajectory in the baryon channel, but a A-contribution could be added in a straightforward manner, and this would not affect the above conclusions. The question of c u r r e n t conservation is discussed in sect. 5 of ref. [1]; but we have nothing to add to this discussion.
D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
434
A(s, t, q2, q2)at the two-fold pole_ cz(q2) = 1 - cz(q2) is R(s, t) =XN2B[2 - ol(t), -fl(s)] s ~"'J~ (- Xs)°t(t)-2 "
(2.9)
H e r e R(s, t) is p r o p o r t i o n a l to the a m p l i t u d e f o r e l a s t i c p N s c a t t e r i n g . T h i s would m e a n that the s -~ % t - f i x e d l i m i t of ON -~ p N is d o m i n a t e d by p - f o exchange, w h e r e a s e x p e r i m e n t shows a c l e a r diffraction c h a r a c t e r . T h i s s u g g e s t s that a p o m e r o n contribution is needed. The total c r o s s section f o r the a b s o r p t i o n of r e a l photons by p r o t o n s is p r a c t i c a l l y constant between 5 and 16 GeV. T h i s would indicate p o m e r o n d o m i n a n c e in r e a l photon s c a t t e r i n g , and then the c o m p l e t e a b s e n c e of the p o m e r o n in v i r t u a l photon s c a t t e r i n g could not e a s i l y be u n d e r s t o o d . Finally, although the l a r g e - w b e h a v i o u r of the e x p e r i m e n t a l vW2 is not well e s t a b l i s h e d , the existing data (fig. 2) a r e c e r t a i n l y s u g g e s t i v e of s o m e contribution, e s p e c i a l l y f o r l a r g e ¢o, in addition to that of eq. (2.8).
3. A F O U R - P O I N T AMI,LITUDE WITH I,OMERON EXCHANGE Our c o n s t r u c t i o n of a d i f f r a c t i v e f o u r - p o i n t amplitude will be b a s e d upon the H a r a r i - F r e u n d p i c t u r e , which a s s o c i a t e s o r d i n a r y t - c h a n n e l t r a j e c t o r i e s (like o~(t) of s e c t . 2) with s - c h a n n e l r e s o n a n c e s , but the p o m e r o n t r a j e c t o r y , ~i,(t), with s o m e s - c h a n n e l background. We need an i n t e g r a l r e p r e s e n t a t i o n s i m i l a r to that of the B-function, f o r this will m a k e it r e latively e a s y to set up an amplitude for v i r t u a l photon-nucleon s c a t t e r i n g (sect. 4) that h a s the r e q u i r e d p r o p e r t i e s , in p a r t i c u l a r Bjorken scaling. Our f o u r - p o i n t a n s a t z is &(s,
t)
Fj zd Z ( l + } ) ~ ( t ) , , "l +z'v(s)
:
1 - ( l + z ) -v log(1 + z)
(3.1)
o
Here ~(t) -
ai,(t) -2,
n(s) - no+XoS ,
w h e r e ~?o, Xo and V a r e constants to be fixed l a t e r . In the l i n e a r a p p r o x i m a tion, the p o m e r o n t r a j e c t o r y is ai,(t) = 1 +Xi,t. The p r o p e r t i e s of A(s,t ) m a y be e s t a b l i s h e d as follows: C o n s i d e r a in eq. (3.1) to be a function of ~ and 77 and r e s t r i c t attention initially to g a n d ~ r e a l and negative. It is then e a s y to justify differentiating (3.1) u n d e r the integral: 07 o
D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
435
F r o m this e x p r e s s i o n , it f o l l o w s that
:
/ld~'B(-~, -V'), ~l-~
o r , by a s i m p l e c h a n g e of v a r i a b l e ,
A(S, t) = f o
d x B [ - ~ ( t ) , x-~/(s)].
(3.2)
T h i s e x p r e s s i o n c a n now be c o n t i n u e d f r o m the n e g a t i v e r e a l a x e s of ~ and ~? to t h e e n t i r e c o m p l e x p l a n e s (i.e. to all v a l u e s of s and t).
For Is] -~oo ( a r g [ s ] ¢
O) andt fixed, O~p(t)-2
A(s, t) ~ v r (- ~)(-n)~ - v r [ 2 - ep(t)] (- Xos) T h i s c o r r e s p o n d s to a s i m p l e R e g g e pole s t r u c t u r e , and s f i x e d at a r e a l , n e g a t i v e v a l u e , we find A ~ (_~)7/ f o
(3.3)
F o r It [ -~ ~ ( a r g t ¢ 0),
dxF(x_~?)(_~)-x.
(3.4)
Since the p o w e r of ( - ¢ ) h a s b e e n s m e a r e d f r o m 7/to 7 / - 7 , this c o r r e s p o n d s to a R e g g e cut, r a t h e r than a R e g g e pole. Indeed, s i n c e F ( x - 7 / ) is a s m o o t h ly v a r y i n g function, f o r 7? r e a l and n e g a t i v e , we can e x p e c t the l e a d i n g b e h a v i o u r of A to be o b t a i n e d by r e p l a c i n g F(x - 7/) by F ( - ~?) in eq. (3.4) *: A(s, t) ~ F ( - ~ ) (- ~)~ - (- ~) ~)rl-7 ~ log(-
r(- no- XoS)[-aP(t)]v°+ xoS
log[-ap(t)], (3.5)
which is indeed the kind of behaviour expected from a crossed-channel Regge cut. The singularities of eq. (3.2) in the variable t are easily seen to be simplepoles at ~(t) = m, m =0, 1, 2 , . . . , with residues
(-1)
dx(x - 77 - 1 ) ( x - ~7- 2) . . . . . ( x - ~ - m ) .
m
(3.6)
o T h i s e x p r e s s i o n is a p o l y n o m i a l of d e g r e e m in 77 (or s), so that t h e r e a r e no a n c e s t o r s . To d e t e r m i n e the s i n g u l a r i t i e s in s, one can w r i t e eq. (3.2) in the f o r m
* Clearly this step is not rigorous, since (_~)-x is a rapidly oscillating function, but the general conclusion that a Regge pole has been smeared into a Regge cut follows directly from eq. (3.4), and is certainly eorrect.
436
D.Atkinson and A .P. Contogouris, Inelastic e-p scattering
(- 1)/ F(-~) I = 0 I ! F ( - ~ -l) l - ~ + x
o
:
F(-~)
/=0
l!r(-~
-l) (_ 1)/log ( I _ A ) .
(3.7)
(3.8)
This gives a series of branch cuts I < ~(s) < l + y ,
(3.9)
or
l-~/° --~< O
l +7 -~7o s ~ < - ~,
(3.10)
O
l = 0 , 1, 2 , . . . The a m p l i t u d e (3.1) contains t h r e e p a r a m e t e r s , 77o, ~o and ~, which we now need to fix. According to eq. (2.9), the Veneziano t e r m of pN s c a t t e r i n g , R(s, 1), has the following l a r g e - t behaviour:
n(s, t) - x ~ v 2 r [ - t~(s)] (- ~t) ~(s) , which c o r r e s p o n d s , of c o u r s e , to the nucleon Regge pole at J = ~(s) = ~(-M 2 +s) in the s - c h a n n e l a n g u l a r m o m e n t u m plane. Since A (S, t) is e v e n tually to be added to R(s, t ), it would be nice to m a k e its b r a n c h - p o i n t , at J = ~?(s) =- ~?o + ~o s, c o r r e s p o n d to a M a n d e l s t a m moving b r a n c h - p o i n t f o r m e d by the exchange of the nucleon and the p o m e r o n (i.e. an a b s o r p t i v e Regge cut a s s o c i a t e d with the nucleon). T h i s r e q u i r e s that the s = o i n t e r c e p t of ~?(s) should be the s a m e a s that of ~(s), the nucleon t r a j e c t o r y , and that the slope of 77(s) should be the h a r m o n i c m e a n of those of/3(s) and Otp(S). That is,
77o = -~M2' ~o- ~ + ~ p "
(3.11)
F r o m (3.10) we s e e that the lowest b r a n c h - p o i n t o c c u r s at
s o
=M 2 --~+~P kp
(3.11')
For any reasonable ~tp, this is well above the ~ N normal threshold, so we may take the cuts (3.10) to be a contribution to the inelastic right-hand cut. We will choose T = 2, for in this way we will have a cut all the way along the right-hand cut, starting from s o (3.11'). However, it is to be emphasized that this is not the elastic cut, but only part of the inelastic cut. Our ansatz (3.1) belongs to the class of "smeared" or "smoothed" Veneziano models, and so is similar to that first proposed by Martin [6], and then discussed by several authors [7]. However, in our case, the pole in s has been completely smoothed away, and no pinching mechanism has
D.A tkinson and A.P. Contogouris, Inelastic e-p scattering
437
b e e n i n t r o d u c e d to r e - i n s t a t e it on t h e s e c o n d s h e e t , a s in r e f . [ 6 ] . T h e P o m e r a n c h u k s i n g u l a r i t y , ap(t), on t h e o t h e r h a n d , i s a s i m p l e m o v i n g p o l e in c o m p l e x J . S e v e r a l e x p e r i m e n t a l a n d p h e n o m e n o l o g i c a l a n a l y s e s of
elastic hadron reactions establish ~tp ~ 0.4~, implving that, if ther~ arebosons on the t r a j e c t o r y ,
t h e y a r e of v e r y h i g h m a s s * (m 2 >~ ~ 1
~ 2.5GeV").
4. A P O M E R O N - D O M I N A T E D A M P L I T U D E F O R TWO C U R R E N T S C o n s i d e r now t h e a m p l i t u d e T~.fi. with a = fl = 3 a n d i t s t e n . s o r d e c o m p o s i t i o n (2.3). F o r t h e corresponding~La~mplitude A aft =- Ap(s, t, q2, q2), we s h a l l w r i t e an i n t e g r a l r e p r e s e n t a t i o n s i m i l a r to (2.4), but a p p r o p r i a t e to p o m e r o n d o m i n a n c e , a n d i n c o r p o r a t i n g t h e f o u r - p o i n t f u n c t i o n of s e c t . 3. Our representation is
Ap(S, t, q21, q22, ×/p(ul,
:a N
u2,
o
dUldU_dZ Ul~2 z ~[ol(q~,, ol(q~,,
olp(t,;ul,
u2, Z]
Z)gp(Ul, u2, z;s)hp(Ul, u 2, z)P(u 1, u 2, z ; t ) .
(4.1)
H e r e ;~N i s a n o r m a l i z i n g c o n s t a n t (with d i m e n s i o n G e V - 2 ) , a n d ~P i s t h e s a m e a s in (2.4'), e x c e p t t h a t ~ p ( t ) r e p l a c e s ~ ( t ) a s t h e t h i r d a r g u m e n t . Further,
f p = I 1+ Z_UlU~ 2 _l-I t z + UlU2J '
z gp =
1+ 1+
l'q(s)
UlU2/(u 1 + u2)J
'
l+z
h p = l+Ul +U2 +Z,
(4.2)
w h e r e it i s a c o n s t a n t to b e f i x e d l a t e r . T h e f a c t o r s (4.2) a r e s i m i l a r to t h o s e of (2.4"), but with s o m e s i m p l i f i c a t i o n s m a d e p o s s i b l e by t h e a b s e n c e of s - c h a n n e l p o l e s , a n d b y t h e f a c t t h a t A p d o e s not n e e d to s a t i s f y a n F G M * Another possibility would be to smoothe away the poles in t also, with an ansatz such as
f
"
and then there would be no p a r t i c l e s on the Pomeranehuk t r a j e c t o r y . The nature of the branch points can also be changed by generalizing the s m e a r i n g function. These m o r e general p o s s i b i l i t i e s will be considered in a forthcoming work: for the p r e sent, we shall be satisfied with eq. (3.1) as a simple formula for a diffractive contribution.
438
D.Atkinson and A .P. Contogouris, Inelastic e-p scattering
sum rule. Finally,
w(z) +UlU 2 [1 P =
l+UlU 2
~p(t)
]-1
l+UlU2(Z+Z-1)]
,
(4.3)
where
w(z) = 1- ( 1 + z ) - 7 log(1 +z) " The additional factor P(Ul, u2, z; t), in the integrand of (4.1), establishes the n e c e s s a r y diffractive properties of Ap. The properties of Ap a r e as follows:
(a) There are poles in q21 and q~ occurring at ~(q~) : 1, 2, 3 , . . . ,
i : 1,
2, which come f r o m end-point singularities atu i = 0, i = 1, 2, just as in eq. (2.4). (b) The residue at the two-fold pole at a(q 2) = a(q 2) = 1 is easily seen to be R e s A p ( s , t, q21, q22)=~N[1-ap(t)]'l A(s, t ) ,
(4.4)
with A(s, t) as in (3.1). By adding the s ~-~u c r o s s i n g - s y m m e t r i c t e r m , we obtain the complete pomeron contribution for the amplitude pN--. pN:
Rv(s, t) : X N [ 1 - a p ( t ) ] - i [A(s, t ) + A ( u , t ) ] .
(4.5)
For s -- ~ and t fixed,
Rp(s, t ) ~ XN[1 - ap(t ) ]-lT {[_ ~?(s)]~(t ) +[_ ~?(u)]~(t) 1 + e - i ~ a P (t) . ~ XNWF[ap(t)]
sin~ap(t)
ap(t)-2 [XoS] .
(4.5')
This has the c o r r e c t form for a diffractive contribution to pN-~ pN. (c) The branch points in s a r i s e as end-point singularities at z = ~. The corresponding contribution to Ap is 2
2
2
2
Ap(S, t, ql' q2 ) ~ XN~P(ql' q2 )A(s' t ) ,
(4.6)
where
and where A(s, t) was given in eq. (3.1). Hence A p has the branch-point s t r u c t u r e (3.10) in the variable s. (d) In a s i m i l a r way, one can show that there a r e poles in t, at ~ p ( t ) = 2, 3, 4, . . . . corresponding to the poles of A(s, t), as discussed in sect. 3.
D.Atkinson and A.P. Contogouris, Inelastic e-p scattering
439
(e) So f a r a s the a s y m p t o t i c b e h a v i o u r in s is c o n c e r n e d , we s h a l l be s a t i s f i e d with a d i s c u s s i o n of the l i m i t f o r r e a l s - ~ - % just a s in r e f . [1]. M a k e the c h a n g e of v a r i a b l e X'
(s)'
and let s ~ - ~ . In this l i m i t , x' gP = 1-1+.1.2/(u l +u2) ~ 1
rl(S)
-l+.1.2/(u1+u2)
exp
so that *
2
22
Ap(s, t, ql'
~'N
q2 )
l-(~p(t) [-~(s)]~(t) f/o 1
x
1 u2J
dUldU2
11~(q1)-1
UlU2 [ 1 + UlJ
F dx', ,,-~
xul+U2o
~7(x)
x' exp-l+UlU2/(Ul+U2)
] J"
With the c h a n g e of v a r i a b l e Xt =
1+
ulu2 ix u I +u2J
'
this e x p r e s s i o n m a y be w r i t t e n
Ap(s, t, q21, q22)~
~,NF[I - a p ( t ) ] A(q21, q22, t)[-XoS]
aP(t)-2 ,
(4.7)
where
x [ul+"2]-I[: + ulu2 ]_:(t). L
(4.7')
u I +u2J
T h i s is R e g g e b e h a v i o u r , c o r r e s p o n d i n g to p o m e r o n e x c h a n g e . T h e f o r m (4.7) and (4.7') is a c t u a l l y only c o r r e c t f o r t > 0: f o r t < 0 t h e r e i s a l s o a f i x e d pole at J = 1, which, h o w e v e r , is c a n c e l l e d when the s ~ u c r o s s i n g s y m m e t r i c t e r m is added. (f) In the l i m i t t ~ _oo, t h e b e h a v i o u r is d e t e r m i n e d by the l a r g e - z c o n t r i b u t i o n to (4.1), which is given by (4.6). T h e r e is t h e r e f o r e a m o v i n g b r a n c h point in the c o m p l e x J - p l a n e . * The asymptotic properties of eq. (4.1) have been established in certain sectors only of the variables in question, as in ref. [lj. We stress, however, that the limits eqs. (3.3) and (3.4) for A itself are uniform, except for arg s = 0, arg t = 0.
D.Atkinson and A. P. Contogouris, Inelastic e-p scattering
440
(g) The Bjorken limit is defined as in sect. 2. We f i r s t make the change of variables u 1 =-7?(s)x 1, u2=-~(s)x 2. (4.8) In deriving our scaling form, we observe that convergence of the integral representation (4.1) r e q u i r e s that we s t a r t with co < 1. Eventually we shall continue the l a r g e - s asymptotic expression to co > 1 just as in ref. [1], since this corresponds to the physical region for e-p s c a t t e r i n g * . Thus we have (1+ eta(q2)-1 I 1 l~o-l+}'s/(1-co) = 1 -~12 u- -I
[ ~ s~-~ _oo exp -
1 ~P[c~(q21), ~(q22), a ( t ) ; u l ,
j
,
Xl+X2 XlX2]"
u2, z ] ~ i x + l l ~ e x p I - l . c °
Thus
(,+
S
x (Xl~2x2)exp[-(Z+l~)Xl+X~21. XlX 2 A The XlX 2 double integral can be p e r f o r m e d explicitly, and this gives
2
2
Ap(s, t, ql' q2 )
X
N 1 ~p(t, co),
X
s
O
where 1
-1
o so one has Bjorken scaling, for any fixed t.
5. INELASTIC PROTON FORM FACTOR AND TOTAL pN CROSS SECTION To d e t e r m i n e our d i f f r a c t i v e contribution to the e l a s t i c proton f o r m f a c t o r , we need to continue (4.9) f r o m co < I to co > 1 (much a s in ref. [I]). The function (4.10) has a b r a n c h cut 1 < co < oo, on which i t s i m a g i n a r y p a r t is
* The procedure of continuing an asymptotic expression in a subsidiary variable is highly non-rigorous and fraught with danger, as we realize no less than do Landshoff and Polkinghorne. We can only hope that it works in this case.
D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
Im ~p(t,
¢o) .
441
0 (.¢ o - 1 ) . ~ - k N.( ~ ° ~ l ) ~ ~oa p ( t ) - i
(5.1)
O
The diffractive contribution to W2 is
W2p(v, q2) = 2-M-MTIrm {Ap(s, o, q2, q 2 ) + A p ( u , F r o m eq. (2.7), we s e e that, f o r l a r g e s, s ~ u finally
o, q2, q2)} .
m e a n s w -~ -w, so we obtain
vW2p(v ' q2) = ~ - \ - ~ /
(5.2)
O
C l e a r l y , a s w ~ ~o, vW2p t e n d s to a c o n s t a n t , ~tN/~to, which is a t y p i c a l f e a t u r e of d i f f r a c t i v e m o d e l s [8]. H we add (5.2) to (2.8), we can get good a g r e e m e n t with e x p e r i m e n t if we c h o o s e ~tN/~t o in the r a n g e 0.1 to 0.15. It t u r n s out t h a t vW2p is r a t h e r i n s e n s i t i v e to the p a r a m e t e r , p, e x c e p t f o r ¢0 n e a r 1, but in this r e g i o n the n o n - d i f f r a c t i v e c o n t r i b u t i o n d o m i n a t e s a n y w a y . C u r v e (a) of fig. 2 s h o w s the L P dual c o n t r i b u t i o n (2.8), which a c c o u n t s f o r m o s t of the e x p e r i m e n t a l d a t a at s m a l l w. T h i s is in a c c o r d with the a n a l y s i s of B l o o m and G i l m a n [4], and s u g g e s t s a r a t h e r l a r g e v a l u e of tz *. In fig. 2, the d i f f r a c t i v e c o n t r i b u t i o n is p l o t t e d by i t s e l f in c u r v e (b), and the s u m of the n o n - d i f f r a c t i v e p a r t (a), and the d i f f r a c t i v e p a r t (b) is g i v e n in c u r v e (c), s h o w i n g a good fit to the e x p e r i m e n t a l p o i n t s . F o r c u r v e s (b) and (c), the v a l u e s t~ = 5 and )tN/h ° = 0 . 1 1 5 ,
(5.3)
have been chosen. F r o m (4.5') and t h e v a l u e (5.3) f o r hN/)to, we can obtain an e s t i m a t e f o r the p N t o t a l c r o s s s e c t i o n . N e a r the t w o - f o l d pole a(q 2) = a(q 2) = 1, pd o m i n a n c e s h o u l d hold, and it is e a s y to a p p l y the o p t i c a l t h e o r e m to o b tain in the l i m i t s ~ 0%
~'N
2
~f p
~pN(S) ~ ~/~o 3~t2rn4 " P
(5.4)
With.y = 2, a s in s e c t . 3, f 2 / 4 ~ = 2.5 ref[9] and ~t = 0.85 GeV -2, this g i v e s a-N 15 m b , which is in o i ~ d e r - o f - m a g n i t u d e a g r e e m e n t with the e x p e r i mPental v a l u e of 24 m b . It is a p l e a s u r e to t h a n k M r . R. G a s k e l l and P r o f e s s o r s yon G e h l e n f o r u s e f u l d i s c u s s i o n s .
V. M t l l l e r and G.
* The exact value o f ~ might be detormined by some additional requirements, as, for example, the q~dependence of the process e-p --. e - p p (ref. [10]).
442
D.Atkinson and A . P . Contogouris, Inelastic e-p scattering
REFERENCES [1] P. V. Landshoff and J. C. Polkinghorne, Nuel. Phys. B19 (1970) 432. [2] M. Bander, Nucl. Phys. B13 (1969) 587; R. C. Brower and J. H. Weis, Phys. Rev. 188 (1969) 2486, 2495; 1R. C. Brower, A. Rabl and J. H. Weis, Nuovo Cimento 65A (1970) 654; M. Ademollo and E. Del Giudice, Nuovo Cimento 63A (1969) 639; D. Z. Freedman, Phys. Rev., to be published. [3] J. D. Bjorken, Phys. Rev. 179 (1969) 1547. [4] E.D. Bloom and F . J . Gilman, Phys.Rev. Letters 25 (1970) 1140; C. H. Llewellyn Smith, Invited paper at the Austin Meeting of P a r t i c l e s and Fields, November 1970. [5] P. Di Vecchia and F. Drago, F r a s c a t i preprint LNF-69/25; P. H. ~ rampton, University of Chicago preprint (1970). [6] A. M art i n , Phys. Letters 29B (1969) 431. [7] N. F. Bali, D.D. Coon and J. W. Dash, Phys. Rev. Letters 23 (1969) 900; K. Huang, Phys. Rev. Letters 23 (1969) 903. [8] H. H a r a r i , Phys. Rev. Letters 22 (1969) 1078; H. Abarbanel, M. L. Goldberger and S. B. T r e i m a n , Phys,Rev. Letters 22 (1969) 500. [9] J. J. Sakurai, Currents and mesons (University of Chicago P r e s s , 1969) ch. III. [10] M. Briedenbach et al., P h y s , R e v . Letters 23 (1969) 935; see also E. D. Bloom et al., SLAC-PUB 796 (1970) and C. L. Jordan, invited paper at the Austin Meeting of P a r t i c l e s and Fields, November 1970.