- Email: [email protected]
Regge analysis of electroproduction and photoproduction
Volume 34B. number 4
REGGE
PttYSICS
ANALYSIS
OF
LETTERS
ELECTROPRODUCTION
1 March 1971
AND
PHOTOPRODUCTION
$
H. R. P A G E L S
The Rockefeller University, New York. New York 10021. USA Received 7 January 1971
We establish the analytic properties of the Regge residue functions in electroproduction and hvpothesize that all Regge poles have scaling behavior. We fit the proton structure function with F-2P(p) -~:0.11 + 0.48 p-l/2 and using photoproduction data predict FP(D) - F~(p) = 0.15 p -1/2 for p = v / - q 2 > 4.
T h e o b j e c t of t h i s l e t t e r is to d e m o n s t r a t e the a n a l y t i c p r o p e r t i e s of the p a r t i a l w a v e a m p l i t u d e in a v i r t u a l m a s s v a r i a o l e w h i c h f o l l o w f r o m the F r o i s s a r t - G r i b o v d e f i n i t i o n s of this a m p l i t u d e and the DGS [1] r e p r e s e n t a t i o n . W e then w i l l e x p l o i t t h i s r e p r e s e n t a t i o n and the i n f o r m a t i o n it i m p l i e s about R e g g e r e s i d u e s to e x t r a p o l a t e the high e n e r g y b e h a v i o r of the t o t a l p h o t o n - p r o t o n c r o s s - s e c t i o n to the electroproduction region. We b e g i n with the d e f i n i t i o n of the f o r w a r d v i r t u a l C o m p t o n a m p l i t u d e F = e ; T~v(p,q)e v f o r photons of m a s s q2 and p o l a r i z a t i o n e~ s c a t t e r i n g on a t a r g e t h a d r o n with m a s s n o r m a l i z e d a c c o r d i n g to p2 = 1, where
T~v(P,q) qLzqv • = Tl(q2,v)(-g~v +~--~_~) q2 + T2(q2,v!(P ~ - ~p'q - q ~ ) ( P v ' ~ p'q qv ) and v = p.q. In the f o l l o w i n g we w i l l d i s c u s s only the a m p l i t u d e T l ( q 2 , v) although with m o d i f i c a t i o n s t h i s a n a l y s i s can be a l s o a p p l i e d to T2(q2 , v). T h e a b s o r p t i v e p a r t of T l ( q 2 , v) d e f i n e s the t o t a l t r a n s v e r s e c r o s s - s e c t i o n a c c o r d i n g to the o p t i c a l t h e o r e m . W l ( q 2 v) =--1 I m T l ( q 2
v) = ( v - ½ 1 q 2 1 ) a t ( q 2 , v ) / 4 ~ 2 a
At q2 = 0 at(0 , v) = at(v) is j u s t the t o t a l p h o t o n - h a d r o n c r o s s - s e c t i o n . limit exists lim
q2~-oo ¢o -q2/v fixed
(1) We w i l l a l s o a s s u m e the s c a l i n g
W l ( q 2 , v) = F l ( W ) .
N e x t we d e f i n e the s i g n a t u r e d a m p l i t u d e F(+)(q 2, v)
F(+)(q2,v) = T l ( q 2 , 0 )
+
2v~ dv'ImTl(q2,v') 7I YO
~'~:
V))
'
(2)
so that Tl(q2 , v) = ½[F(+)(q2, v) + F(+)(q2, -v)]. A subtraction term has been inserted in the dispersion relation for F(+)(q2, v) as required by the Regge analysis [2] as v ~ ~ for proton and neutron targets. Further it can be shown that Tl(q2, v) is a crossed channel helicity-nonflip amplitude so we may decompose F(+)(q2, v) into partial waves according to Work supported in part by the US Atomic Energy Commission under Contract number AT(30-1)-4204.
299
Volume 34B, n u m b e r 4
PHYSICS
LETTERS
1 March 1971
z : v/'~q2
F(+)(q2. v) = ~ ( 2 J + l ) A ( + ) ( q 2 . j ) P j ( z ) J=0
(3)
F r o m (2) and (3) we have the F r o i s s a r t - G r i b o v d e f i n i t i o n of the p a r t i a l wave
A(+)(q2.j) =~,2 f dzqj(z)fmTl(q2,z)
ReJ:
1.
(4)
Z o
w h e r e , for the m o m e n t , we r e s t r i c t o u r s e l v e s to ReJ :> 1 b e c a u s e of the p r e s e n c e of the s u b t r a c t i o n t e r m . L a t e r we a s s u m e this r e p r e s e n t a t i o n d e f i n e s the a n a l y t i c p r o p e r t i e s of A (+)(q2,j) in J for R e J -~. It is f u r t h e r p o s s i b l e to e s t a b l i s h the a n a l y t i c p r o p e r t i e s of A(+)(q2,j) in the q 2 - p l a n e if we m a k e u s e of the DGS r e p r e s e n t a t i o n [1] a p p r o p r i a t e to an a m p l i t u d e with one s u b t r a c t i o n which r e a d s Tl(q2" P) = f ~ d;~2h(X2 )
09
+1 (q ÷
and 09
~v _ i m) T l l( q 2
= vf
+1 d X 2 f dfig(X2,fi) 0 -1
5[(q+fip)2
- 421.
(5)
w h e r e the s p e c t r a l f u n c t i o n g ( ~ 2 fi) is odd in ft. We a l s o a s s u m e g(~2. fl) is s t r o n g l y d a m p e d a s 4 2 -~ 09. I n s e r t i n g (5) in (4) we o b t a i n 1
d(+)(q2.j) =f d ~ 2 f d~g(x2.~)qz(v)y; 0
y : (X2-fl 2
-q2)/2fiv~2
(6)
0
as o u r r e p r e s e n t a t i o n for the p a r t i a l wave a m p l i t u d e e m b r a c i n g the c o r r e c t J - p l a n e a n d q 2 _ p l a n e a n a lytic p r o p e r t i e s . In the s c a l i n g l i m i t we can e a s i l y e x t r a c t the b e h a v i o r of the p a r t i a l w a v e s . U s i n g eq. (4) and t a k i n g the l i m i t ]ql = J - - ~ ,o% o) = _q2/v fixed we find lim
iql ~,~ where
A(+)(q2.j) = 2h(J)f(J)/(ilql)J; f(J)
h(J) =~l/2r(J+l)/2J+lr(J+~).
(7)
is the M e l l i n t r a n s f o r m of the i n e l a s t i c s t r u c t u r e function
2 f(J)
-
f dww J - 1 F 1 (a~). 0
(8) 09
U s i n g eq. (6) we a l s o obtain t a k i n g this s a m e l i m i t the r e l a t i o n 2Fl(~V) = f ~ dh2g(~ 2, ½w). Knowing the b e h a v i o r of A(+)(q2,j) as t ql ~ 09 we can e x t r a c t the full c o n t e n t of the s ~t r u c t u r e f u n c t i o n F1(¢o). We now a s s u m e that A (+)(q2 j ) has Regge p o l e s in the J p l a n e at J = c~/ and that t h e s e a r e the l e a d ing s i n g u l a r i t i e s for R e J :~ 0. F r o m the r e p r e s e n t a t i o n (6) the r e q u i r e m e n t that A(÷)(q2,j) have p o l e s at J = ~ i i m p l i e s the b e h a v i o r of the s p e c t r a l function to be l i m g(~2 fl) =__ -~Ri(~2)fi-~i
~0
(9)
which d e f i n e s the f u n c t i o n s Ri(X2). F r o m eqs. (6) and (9) one o b t a i n s for the Regge r e s i d u e fl~+)(q2," c~i) = lira (J- ~i)A(+)(q 2. J)
J~ai fll+)(q 2, c~i) =
09
-h(~i) f
d;~2Ri(;~ 2) [2 ~ / ( ; t 2 _q2)] (10) 0 T h i s r e s u l t (10) s p e c i f i e s c o m p l e t e l y the a n a l y t i c p r o p e r t i e s of the r e s i d u e f u n c t i o n in q2 and we
300
Volume 34B. number 4
PHYSICS
LETTERS
1 March 1971
l e a r n that -/3(+)(q 2, c~)/(2 q 4 ~ ) a h ( a ) is a g e n e r a l i z e d S t i e l t j e s t r a n s f o r m of o r d e r a T. We m a y now e x p l o i t t h i s r e p r e s e n t a t i o n f o r the r e s i d u e f u n c t i o n u s i n g the R e g g e r e p r e s e n t a t i o n . T h e R e g g e r e p r e s e n t a t i o n f o r the a m p l i t u d e T l ( q 2, v) is o b t a i n e d f r o m the s t a n d a r d a s s u m p t i o n that A ( + ) ( q 2 , j ) is a n a l y t i c f o r R e J > - ½ and we find
T l ( q 2, v) = - ½ v ~ ( 2 a i + 1) i
fi(+)(q2 a . , i = ' z [l+exp (ivai)]Pai(z) + background, s m va i
w h e r e the s u m on the R e g g e p o l e s ~ i > -31 and the b a c k g r o u n d i n t e g r a l w i l l be r e l a t i v e l y s u p p r e s s e d . T a k i n g the i m a g i n a r y p a r t of T l ( q 2 , u) and the u -~ ~o l i m i t we h a v e , d r o p p i n g the b a c k g r o u n d i n t e g r a l , 1
1
l i m ~-Im T l ( q 2 , u ) = W l ( q 2 , u) = - 5 ~ . [/3 +)(q2,c~i)/h(oei)]z°ti V~
co
z = v/~
(11)
l
P a r a m e t r i z i n g the t o t a l p h o t o p r o d u c t i o n c r o s s - s e c t i o n ucrT(u)/47r2c~ = WI(0, u)
a c c o r d i n g to the l e a d i n g R e g g e p o l e s
l i m WI(0 , u) = ~ a i v ai u~o i
(12a)
and the i n e l a s t i c s t r u c t u r e f u n c t i o n by l i m Fl(¢O) = ~. b i co-ai
(12b)
w e obtain f r o m eq. (II) and the representation for the residue function (i0) as q2 spectively oo
a i = 2ai-lf
~
dX2Ri(X2)(h2)-ai; 0
~ 0 and _q2
~ ~ re-
2
b i : 2 a i - 1 f dX2Ri( x ) 0
SO co
ai/b i = f
oo
d~2Ri(~2).
dX 2 Ri(X2)(X 2 ) - a i / f
0
(13)
0
One i m m e d i a t e c o n c l u s i o n that f o l l o w s f r o m eq. (13) is that if a R e g g e p o l e c o u p l e s to the p h o t o p r o d u c t i o n a m p l i t u d e at q2 = 0 so a i ¢ 0 then b a r r i n g a v e r y unusual b e h a v i o r of Ri(X2) r e q u i r i n g the z e r o t h m o m e n t to v a n i s h we m u s t a l s o c o u p l e the ~ o l e to e l e c t r o p r o d u c t i o n b i ¢ O. S i n c e the R e g g e r e s i d u e s ~/(+) (q2, a i ) / ( ~ ) a i a r e the m a t r i x e l e m e n t s of the c o u p l i n g of two p h o t o n s of m a s s q2 to the R e g g e p o l e t h e y m i g h t be e x p e c t e d to b e h a v e l i k e the s q u a r e of an e l e c t r o m a g n e t i c f o r m f a c t o r . T h i s s u g g e s t s that the R e g g e r e s i d u e s do not c h a n g e s i g n f o r s p a c e l i k e q2 0 < _q2 < ~o and if we a s s u m e that t h i s is the c a s e then we c o n c l u d e f r o m e q s . (10) and (13) that a i / b i > 0 so the r e l a t i v e s i g n of the c o u p l i n g s is d e t e r m i n e d . F o r p o m e r o n c o n t r i b u t i o n at a p = 1, the l e a d i n g s i n g u l a r i t y , the p o s i t i v i t y of the c r o s s s e c t i o n s i s a l o n e s u f f i c i e n t to r e q u i r e a p / b p > O. We c o n c l u d e by e x a m i n i n g the i m p l i c a t i o n s of t h i s a n a l y s i s f o r the i n t e r p r e t a t i o n of r e c e n t e x p e r i m e n t s . T h e t o t a l c r o s s - s e c t i o n d a t a f o r p h o t o p r o d u c t i o n on p r o t o n s and n e u t r o n s [4] i n d i c a t e s the c o u 1 p l i n g of the p o m e r o n , the P ' and the A~ m e s o n a s the l e a d i n g t r a j e c t o r i e s w i t h a p = 1, u p , ~ ~ A ~ ~ 3. A s u = photon e n e r g y / G e V ~ ~o the d a t a can be p a r a m e t r i z e d a c c o r d i n g to aTP(V) = 94 + 79 v-1/2(/~b) ,
(yTP(V) - C~T(V) = 24.6 v-1/2(/xb)
(14)
and c o m p a r i n g w i t h eq. (12a) and u s i n g the f a c t that P and P ' a r e I = 0 and the A~ 1 = 1 we find a p = 0.61, a p , + a A o = + 0 . 5 1 , 2aAo = + 0 16 2 " " To m a k e c o n t a c t with e l e e t r o p r o d u e t i o n d a t a we a s s u m e , as is c o n s i s t e n t with t h e s e e x p e r i m e n t s [5] that the l o n g i t u d i n a l a m p l i t u d e FL(W ) = F 2 ( w ) / w - F l ( w ) ~ 0 so F2(w) = WFl(W). I n t r o d u c i n g p = 1 / w = - v / q 2 we h a v e the r e s u l t f r o m eq. (12b) a s p ~ ~o. This result appears in the paper of Brandt [3]. F o r a related discussion see ref. [13]. 301
Volume 34B. number 4
F~(p)
= 1,ppetp-1 +
F~(p) - F~(p)
PHYSICS
bp,pC~p,-1 + /,A.~P~A~_I+
...
LETTERS
~ bp +
(bp, + b A ~ ) p - 1 / 2
1 March 1971 ;
: 2hA,~ p ~ A ~ - l , ~ 2bA,~ p-1/'2
(15)
On the a s s u m p t i o n that the r e l a t i v e sign is s p e c i f i e d by ai/b i :: 0 a s d i s c u s s e d p r e v i o u s l y we c o n clude that the p r o t o n s t r u c t u r e f u n c t i o n FP(p) is a p p r o a c h i n g the c o n s t a n t p o m e r o n l i m i t f r o m above by f a l l i n g like p - l / 2 and that F ~ ( p ) - F~(p) d e c r e a s e s to z e r o through p o s i t i v e v a l u e s like p - l / 2 . Both of t h e s e q u a l i t a t i v e f e a t u r e s a p p e a r to be c o n s i s t e n t with the p r e s e n t data. In fig. 1 we have plotted the data for p • 4 and given fits in a c c o r d a n c e with the b e h a v i o r p r e s c r i b e d by eq. (15). The l a r g e p data for the p r o t o n a p p e a r s to be f a l l i n g h o w e v e r the r a t e of d e c r e a s e is d e p e n d e n t on the r a t i o R = Crs/~t which is in any event s m a l l [5 t. C u r v e B with F2(p) = 0.11 + 0.48p -1/2 r e p r e s e n t s the data [6] at l a r g e p for R = 0 and is taken f r o m the 6 ° data in fig. 2(a) of ref. [6] (at a t i m e when a f a l l i n g F2(P) was t h e o r e t i c a l l y u n f a s h i o n a b l e ) . C u r v e A with F2(P) = 0.275 + 0.13p -1/2 is the m o r e r e c e n t a n a l y s i s [7] r e p o r t e d at Kiev with R = 0.18 and is taken f r o m fig. 18 of ref. [7 I. In e i t h e r c a s e the s i g n i f i c a n t p o i n t , if o u r i n t e r p r e t a t i o n is c o r r e c t , is that the p o m e r o n l i m i t is a p p r o a c h e d f r o m above and a f a l l i n g FP(p) i n d i c a t e s the n e e d for o t h e r t r a j e c t o r i e s . o4~ !
A~
~
AFz(P) :0 275 + 013P ½
i
~-~ 0 2 " o
o~-
~"
l
¢o
F
C
0~
C2(p):O 15p-½
5
I© D
=
2'0
- y/Q2
Fig. i. ]nel~Isl.ic structure funetions: A: proton, R- 0.18; 13: proton. R
0: C: proton-neutron.
If we a s s u m e that the f u n c t i o n s Ri(~ 2) d e p e n d only on the pole p o s i t i o n and ),2 so R i ( ~ 2) = R(;~2, ~i ) then (13) i m p l i e s with c~D,~ = ~A o2 that a•...../bD, = aao/b~o2 ~ 2 h e n c e (b~,~+b^o)/2b^o~2 "~2 = ( a p ' + a A ~ ) / 2 a A ~ " F r o m eq. (14) and the fit to c u r v e B. b p , + bAQ = 0.48 we have 2bAO = 0.15 so that F2P(p) - F~(p) = Z . . . . Z 0 . 1 5 p - 1 / 2 ( p ,~o). T h i s is plotted on c u r v e C of fig. 1 and is c o n s i s t e n t with the v e r y r o u g h data r e p o r t e d . H a r a r i [81 and A b a r b a n e l . G o l d b e r g e r and T r e p a n [9] have m a d e the o b s e r v a t i o n that if C~TT(~) = c o n s t a n t then Regge b e h a v i o r and s c a l i n g i m p l y F~ (~) = c o n s t a n t . T h e i r w o r k is a s p e c i a l i z a t i o n to j u s t the p o m e r o n c o u p l i n g of the c o n c l u s i o n s r e a c h e d h e r e . The work of B r a n d t [10] u t i l i z i n g the DGS r e p r e s e n t a t i o n a t t e m p t e d to e s t a b l i s h a q u a n t i t a t i v e c o n n e c t i o n b e t w e e n aTP(Oo) and FP(~o). The p r e s e n t w o r k 2 g e n e r a l i z e d t h e s e r e s u l t s to a r b i t r a r y c~i and s u g g e s t s that the s e c o n d a r y t r a j e c t o r i e s c~p, and C~AO m a y . . . . 2 play an i m p o r t a n t r o l e m m t e r p r e t m g the b e h a v i o r of the i n e l a s t i c s t r u c t u r e f u n c t i o n s F2(P) a s p -- oo. F i n a l l y t h e r e is the i n t r i g u i n g p r o b l e m of the i n t e r p r e t a t i o n of the s t r u c t u r e f u n c t i o n F-P(p) in the low , 2 p r e g i o n ~ p 4. A clue is given by the r e c e n t w o r k of B l o o m and G i l m a n [11] who showed how this r e g i o n e m e r g e s a s an a v e r a g e o v e r the r e s o n a n c e b e h a v i o r a s _q2 a d v a n c e s f r o m low - 2 to t a r e - 2 > I(GeV) 2. T h i s s u g g e s t s we c o n s i d e r an a m p l i t u d e Wl(q 2, u) which is c o n s t r u c t e d fromqwl(q 2. gel b-yq e x t r a p o l a t i o n of the Regge b e h a v i o r at l a r g e u, w h e r e t h e s e f u n c t i o n s a g r e e , to low r, by a v e r a g i n g t h r o u g h the r e s o n a n c e s . So ~PT(U) = 4~2 ~ 1 ( 0 . u) ~-1 is the Tp total c r o s s - s e c t i o n a v e r a g i n g the Regge b e h a v i o r through the r e s o n a n c e s . R e p r e s e n t i n g V/l(q2, u) by a Regge r e p r e s e n t a t i o n and k e e p i n g only the Regge p o l e s we o b t a i n ¢x)
Wl(q 2. u) = ~1 . h(oti)( 2 ~ i + 1) Pc~i(v/#q) f d ~2Ri(~2)[2 q~q2//(~2 -q2)] c~i . t 0 302
Volume 34B. n u m b e r 4
PHYSICS
A t l a r g e v :, 4 G e V we a r e in t h e t r a n s r e s o n a n c e obtain FP(p ) = ~2c~i-1
p
ai-1
i
LETTERS
1 March 1971
r e g i o n f o r w h i c h W1 = W1 a n d t a k i n g t h e s c a l i n g l i m i t
f ~ d~t2 Ri()t 2) , 0
w h i l e a t q2 = 0 ~TP(U)/'4~2(~
=~ 2 ~ i
If we n o w a s s u m e
• -1
oei-1
~
f dX2Ri(X2)/(~t2)c~i. 0
that the ratio
ri(~i) =J d~2Ri(~2)(h2)-~i//f
d ~ 2 R i ( i t 2) w i t h r i ( 0 ) = 1 i s n o t a n e x /0 tremely~ v a r y i n g f u n c t i o n of t h e t r a j e c t o r y a i ( t h i s s e e m s to b e t r u e of t h e P . P ' a n d A~ t e r m s ) t h e n FP(p) a n d ~,~(~,)/47r 2 (~ s h o u l d h a v e r o u g h l y t h e s a m e s h a p e s i n c e t h e e x p a n s i o n c o e f f i c i e n t s a r e a p p r o x i m a t e l y the same, except near threshold where these considerations must fail. From the y-p total cross-sect i o n d a t a [12] e x t r a p o l a t i n g t h e R e g g e b e h a v i o r s m o o t h l y t h r o u g h t h e r e s o n a n c e s to z e r o a t t h r e s h o l d a n d c o m p a r i n g w i t h t h e i n e l a s t i c s t r u c t u r e f u n c t i o n we l e a r n t h i s i s a p p r o x i m a t e l y t h e c a s e . B o t h FP(p) a n d @TT(~) r i s e f r o m z e r o a t t h e i r r e s p e c t i v e t h r e s h o l d s to a m a x i m u m a n d t h e n f a l l l i k e p-l~2 a n d ~ - 1 / 2 - t o t h e c o n s t a n t p o m e r o n l i m i t . T h e q u a n t i t a t i v e d e t a i l s d e p e n d on Ri(~t2) b u t t h e q u a l i t a t i v e f e a t u r e s d i s c u s s e d h e r e t e n d to s u p p o r t t h e a n a l y s i s of B l o o m a n d G i l m a n [11]. 0
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
S. D e s e r . W. Gilbert and E. C. G. Sudarshan. Phys. Rev. 115 (1959) 731. H, H a r a r i . Phys. Rev. L e t t e r s 17 (1966) 1303. R , A . B r a n d t . Phys. Rev. D1 (1970) 2808 (Appendix). W.P. Hesse et ai.. Phys. Rev. L e t t e r s 25 (1970) 613. R . E . T a y l o r in Fourth Intern. Symp. on Electron and photon interaction at high energies• 1969 (Daresbury. England). M. Breidenbach et ai.. Phys. Rev. L e t t e r s 23 (1969) 935. M . I . T . - S . L . A . C . collaboration. SLAC-PUB-796 Sept. (1970). H . H a r a r i , Phys. Rev. L e t t e r s 22 (1969) 1078. H. D . I . A b e r b a n e i , M. Goldberger and S . T r e i m a n . Phys. Rev. L e t t e r s 22 (1969) 500. R . A . B r a n d t . Phys. Rev. L e t t e r s 22 (1969) 1149. E. Bloom and F. Gilman, to be published. R. Diebold. High e n e r ~ , photoproduction. Proe. of the Boulder conference on High energ3' physics {Aug. 1969. Boulder) fig, 1, p. 39. N. Nakanishi, Phys. Rev. 133 (1964) B214; J. M. Cornwall. D. C o r r i g a n and R. E. Norton. Phys. Rev. D. to be published.
303
REGGE
PttYSICS
ANALYSIS
OF
LETTERS
ELECTROPRODUCTION
1 March 1971
AND
PHOTOPRODUCTION
$
H. R. P A G E L S
The Rockefeller University, New York. New York 10021. USA Received 7 January 1971
We establish the analytic properties of the Regge residue functions in electroproduction and hvpothesize that all Regge poles have scaling behavior. We fit the proton structure function with F-2P(p) -~:0.11 + 0.48 p-l/2 and using photoproduction data predict FP(D) - F~(p) = 0.15 p -1/2 for p = v / - q 2 > 4.
T h e o b j e c t of t h i s l e t t e r is to d e m o n s t r a t e the a n a l y t i c p r o p e r t i e s of the p a r t i a l w a v e a m p l i t u d e in a v i r t u a l m a s s v a r i a o l e w h i c h f o l l o w f r o m the F r o i s s a r t - G r i b o v d e f i n i t i o n s of this a m p l i t u d e and the DGS [1] r e p r e s e n t a t i o n . W e then w i l l e x p l o i t t h i s r e p r e s e n t a t i o n and the i n f o r m a t i o n it i m p l i e s about R e g g e r e s i d u e s to e x t r a p o l a t e the high e n e r g y b e h a v i o r of the t o t a l p h o t o n - p r o t o n c r o s s - s e c t i o n to the electroproduction region. We b e g i n with the d e f i n i t i o n of the f o r w a r d v i r t u a l C o m p t o n a m p l i t u d e F = e ; T~v(p,q)e v f o r photons of m a s s q2 and p o l a r i z a t i o n e~ s c a t t e r i n g on a t a r g e t h a d r o n with m a s s n o r m a l i z e d a c c o r d i n g to p2 = 1, where
T~v(P,q) qLzqv • = Tl(q2,v)(-g~v +~--~_~) q2 + T2(q2,v!(P ~ - ~p'q - q ~ ) ( P v ' ~ p'q qv ) and v = p.q. In the f o l l o w i n g we w i l l d i s c u s s only the a m p l i t u d e T l ( q 2 , v) although with m o d i f i c a t i o n s t h i s a n a l y s i s can be a l s o a p p l i e d to T2(q2 , v). T h e a b s o r p t i v e p a r t of T l ( q 2 , v) d e f i n e s the t o t a l t r a n s v e r s e c r o s s - s e c t i o n a c c o r d i n g to the o p t i c a l t h e o r e m . W l ( q 2 v) =--1 I m T l ( q 2
v) = ( v - ½ 1 q 2 1 ) a t ( q 2 , v ) / 4 ~ 2 a
At q2 = 0 at(0 , v) = at(v) is j u s t the t o t a l p h o t o n - h a d r o n c r o s s - s e c t i o n . limit exists lim
q2~-oo ¢o -q2/v fixed
(1) We w i l l a l s o a s s u m e the s c a l i n g
W l ( q 2 , v) = F l ( W ) .
N e x t we d e f i n e the s i g n a t u r e d a m p l i t u d e F(+)(q 2, v)
F(+)(q2,v) = T l ( q 2 , 0 )
+
2v~ dv'ImTl(q2,v') 7I YO
~'~:
V))
'
(2)
so that Tl(q2 , v) = ½[F(+)(q2, v) + F(+)(q2, -v)]. A subtraction term has been inserted in the dispersion relation for F(+)(q2, v) as required by the Regge analysis [2] as v ~ ~ for proton and neutron targets. Further it can be shown that Tl(q2, v) is a crossed channel helicity-nonflip amplitude so we may decompose F(+)(q2, v) into partial waves according to Work supported in part by the US Atomic Energy Commission under Contract number AT(30-1)-4204.
299
Volume 34B, n u m b e r 4
PHYSICS
LETTERS
1 March 1971
z : v/'~q2
F(+)(q2. v) = ~ ( 2 J + l ) A ( + ) ( q 2 . j ) P j ( z ) J=0
(3)
F r o m (2) and (3) we have the F r o i s s a r t - G r i b o v d e f i n i t i o n of the p a r t i a l wave
A(+)(q2.j) =~,2 f dzqj(z)fmTl(q2,z)
ReJ:
1.
(4)
Z o
w h e r e , for the m o m e n t , we r e s t r i c t o u r s e l v e s to ReJ :> 1 b e c a u s e of the p r e s e n c e of the s u b t r a c t i o n t e r m . L a t e r we a s s u m e this r e p r e s e n t a t i o n d e f i n e s the a n a l y t i c p r o p e r t i e s of A (+)(q2,j) in J for R e J -~. It is f u r t h e r p o s s i b l e to e s t a b l i s h the a n a l y t i c p r o p e r t i e s of A(+)(q2,j) in the q 2 - p l a n e if we m a k e u s e of the DGS r e p r e s e n t a t i o n [1] a p p r o p r i a t e to an a m p l i t u d e with one s u b t r a c t i o n which r e a d s Tl(q2" P) = f ~ d;~2h(X2 )
09
+1 (q ÷
and 09
~v _ i m) T l l( q 2
= vf
+1 d X 2 f dfig(X2,fi) 0 -1
5[(q+fip)2
- 421.
(5)
w h e r e the s p e c t r a l f u n c t i o n g ( ~ 2 fi) is odd in ft. We a l s o a s s u m e g(~2. fl) is s t r o n g l y d a m p e d a s 4 2 -~ 09. I n s e r t i n g (5) in (4) we o b t a i n 1
d(+)(q2.j) =f d ~ 2 f d~g(x2.~)qz(v)y; 0
y : (X2-fl 2
-q2)/2fiv~2
(6)
0
as o u r r e p r e s e n t a t i o n for the p a r t i a l wave a m p l i t u d e e m b r a c i n g the c o r r e c t J - p l a n e a n d q 2 _ p l a n e a n a lytic p r o p e r t i e s . In the s c a l i n g l i m i t we can e a s i l y e x t r a c t the b e h a v i o r of the p a r t i a l w a v e s . U s i n g eq. (4) and t a k i n g the l i m i t ]ql = J - - ~ ,o% o) = _q2/v fixed we find lim
iql ~,~ where
A(+)(q2.j) = 2h(J)f(J)/(ilql)J; f(J)
h(J) =~l/2r(J+l)/2J+lr(J+~).
(7)
is the M e l l i n t r a n s f o r m of the i n e l a s t i c s t r u c t u r e function
2 f(J)
-
f dww J - 1 F 1 (a~). 0
(8) 09
U s i n g eq. (6) we a l s o obtain t a k i n g this s a m e l i m i t the r e l a t i o n 2Fl(~V) = f ~ dh2g(~ 2, ½w). Knowing the b e h a v i o r of A(+)(q2,j) as t ql ~ 09 we can e x t r a c t the full c o n t e n t of the s ~t r u c t u r e f u n c t i o n F1(¢o). We now a s s u m e that A (+)(q2 j ) has Regge p o l e s in the J p l a n e at J = c~/ and that t h e s e a r e the l e a d ing s i n g u l a r i t i e s for R e J :~ 0. F r o m the r e p r e s e n t a t i o n (6) the r e q u i r e m e n t that A(÷)(q2,j) have p o l e s at J = ~ i i m p l i e s the b e h a v i o r of the s p e c t r a l function to be l i m g(~2 fl) =__ -~Ri(~2)fi-~i
~0
(9)
which d e f i n e s the f u n c t i o n s Ri(X2). F r o m eqs. (6) and (9) one o b t a i n s for the Regge r e s i d u e fl~+)(q2," c~i) = lira (J- ~i)A(+)(q 2. J)
J~ai fll+)(q 2, c~i) =
09
-h(~i) f
d;~2Ri(;~ 2) [2 ~ / ( ; t 2 _q2)] (10) 0 T h i s r e s u l t (10) s p e c i f i e s c o m p l e t e l y the a n a l y t i c p r o p e r t i e s of the r e s i d u e f u n c t i o n in q2 and we
300
Volume 34B. number 4
PHYSICS
LETTERS
1 March 1971
l e a r n that -/3(+)(q 2, c~)/(2 q 4 ~ ) a h ( a ) is a g e n e r a l i z e d S t i e l t j e s t r a n s f o r m of o r d e r a T. We m a y now e x p l o i t t h i s r e p r e s e n t a t i o n f o r the r e s i d u e f u n c t i o n u s i n g the R e g g e r e p r e s e n t a t i o n . T h e R e g g e r e p r e s e n t a t i o n f o r the a m p l i t u d e T l ( q 2, v) is o b t a i n e d f r o m the s t a n d a r d a s s u m p t i o n that A ( + ) ( q 2 , j ) is a n a l y t i c f o r R e J > - ½ and we find
T l ( q 2, v) = - ½ v ~ ( 2 a i + 1) i
fi(+)(q2 a . , i = ' z [l+exp (ivai)]Pai(z) + background, s m va i
w h e r e the s u m on the R e g g e p o l e s ~ i > -31 and the b a c k g r o u n d i n t e g r a l w i l l be r e l a t i v e l y s u p p r e s s e d . T a k i n g the i m a g i n a r y p a r t of T l ( q 2 , u) and the u -~ ~o l i m i t we h a v e , d r o p p i n g the b a c k g r o u n d i n t e g r a l , 1
1
l i m ~-Im T l ( q 2 , u ) = W l ( q 2 , u) = - 5 ~ . [/3 +)(q2,c~i)/h(oei)]z°ti V~
co
z = v/~
(11)
l
P a r a m e t r i z i n g the t o t a l p h o t o p r o d u c t i o n c r o s s - s e c t i o n ucrT(u)/47r2c~ = WI(0, u)
a c c o r d i n g to the l e a d i n g R e g g e p o l e s
l i m WI(0 , u) = ~ a i v ai u~o i
(12a)
and the i n e l a s t i c s t r u c t u r e f u n c t i o n by l i m Fl(¢O) = ~. b i co-ai
(12b)
w e obtain f r o m eq. (II) and the representation for the residue function (i0) as q2 spectively oo
a i = 2ai-lf
~
dX2Ri(X2)(h2)-ai; 0
~ 0 and _q2
~ ~ re-
2
b i : 2 a i - 1 f dX2Ri( x ) 0
SO co
ai/b i = f
oo
d~2Ri(~2).
dX 2 Ri(X2)(X 2 ) - a i / f
0
(13)
0
One i m m e d i a t e c o n c l u s i o n that f o l l o w s f r o m eq. (13) is that if a R e g g e p o l e c o u p l e s to the p h o t o p r o d u c t i o n a m p l i t u d e at q2 = 0 so a i ¢ 0 then b a r r i n g a v e r y unusual b e h a v i o r of Ri(X2) r e q u i r i n g the z e r o t h m o m e n t to v a n i s h we m u s t a l s o c o u p l e the ~ o l e to e l e c t r o p r o d u c t i o n b i ¢ O. S i n c e the R e g g e r e s i d u e s ~/(+) (q2, a i ) / ( ~ ) a i a r e the m a t r i x e l e m e n t s of the c o u p l i n g of two p h o t o n s of m a s s q2 to the R e g g e p o l e t h e y m i g h t be e x p e c t e d to b e h a v e l i k e the s q u a r e of an e l e c t r o m a g n e t i c f o r m f a c t o r . T h i s s u g g e s t s that the R e g g e r e s i d u e s do not c h a n g e s i g n f o r s p a c e l i k e q2 0 < _q2 < ~o and if we a s s u m e that t h i s is the c a s e then we c o n c l u d e f r o m e q s . (10) and (13) that a i / b i > 0 so the r e l a t i v e s i g n of the c o u p l i n g s is d e t e r m i n e d . F o r p o m e r o n c o n t r i b u t i o n at a p = 1, the l e a d i n g s i n g u l a r i t y , the p o s i t i v i t y of the c r o s s s e c t i o n s i s a l o n e s u f f i c i e n t to r e q u i r e a p / b p > O. We c o n c l u d e by e x a m i n i n g the i m p l i c a t i o n s of t h i s a n a l y s i s f o r the i n t e r p r e t a t i o n of r e c e n t e x p e r i m e n t s . T h e t o t a l c r o s s - s e c t i o n d a t a f o r p h o t o p r o d u c t i o n on p r o t o n s and n e u t r o n s [4] i n d i c a t e s the c o u 1 p l i n g of the p o m e r o n , the P ' and the A~ m e s o n a s the l e a d i n g t r a j e c t o r i e s w i t h a p = 1, u p , ~ ~ A ~ ~ 3. A s u = photon e n e r g y / G e V ~ ~o the d a t a can be p a r a m e t r i z e d a c c o r d i n g to aTP(V) = 94 + 79 v-1/2(/~b) ,
(yTP(V) - C~T(V) = 24.6 v-1/2(/xb)
(14)
and c o m p a r i n g w i t h eq. (12a) and u s i n g the f a c t that P and P ' a r e I = 0 and the A~ 1 = 1 we find a p = 0.61, a p , + a A o = + 0 . 5 1 , 2aAo = + 0 16 2 " " To m a k e c o n t a c t with e l e e t r o p r o d u e t i o n d a t a we a s s u m e , as is c o n s i s t e n t with t h e s e e x p e r i m e n t s [5] that the l o n g i t u d i n a l a m p l i t u d e FL(W ) = F 2 ( w ) / w - F l ( w ) ~ 0 so F2(w) = WFl(W). I n t r o d u c i n g p = 1 / w = - v / q 2 we h a v e the r e s u l t f r o m eq. (12b) a s p ~ ~o. This result appears in the paper of Brandt [3]. F o r a related discussion see ref. [13]. 301
Volume 34B. number 4
F~(p)
= 1,ppetp-1 +
F~(p) - F~(p)
PHYSICS
bp,pC~p,-1 + /,A.~P~A~_I+
...
LETTERS
~ bp +
(bp, + b A ~ ) p - 1 / 2
1 March 1971 ;
: 2hA,~ p ~ A ~ - l , ~ 2bA,~ p-1/'2
(15)
On the a s s u m p t i o n that the r e l a t i v e sign is s p e c i f i e d by ai/b i :: 0 a s d i s c u s s e d p r e v i o u s l y we c o n clude that the p r o t o n s t r u c t u r e f u n c t i o n FP(p) is a p p r o a c h i n g the c o n s t a n t p o m e r o n l i m i t f r o m above by f a l l i n g like p - l / 2 and that F ~ ( p ) - F~(p) d e c r e a s e s to z e r o through p o s i t i v e v a l u e s like p - l / 2 . Both of t h e s e q u a l i t a t i v e f e a t u r e s a p p e a r to be c o n s i s t e n t with the p r e s e n t data. In fig. 1 we have plotted the data for p • 4 and given fits in a c c o r d a n c e with the b e h a v i o r p r e s c r i b e d by eq. (15). The l a r g e p data for the p r o t o n a p p e a r s to be f a l l i n g h o w e v e r the r a t e of d e c r e a s e is d e p e n d e n t on the r a t i o R = Crs/~t which is in any event s m a l l [5 t. C u r v e B with F2(p) = 0.11 + 0.48p -1/2 r e p r e s e n t s the data [6] at l a r g e p for R = 0 and is taken f r o m the 6 ° data in fig. 2(a) of ref. [6] (at a t i m e when a f a l l i n g F2(P) was t h e o r e t i c a l l y u n f a s h i o n a b l e ) . C u r v e A with F2(P) = 0.275 + 0.13p -1/2 is the m o r e r e c e n t a n a l y s i s [7] r e p o r t e d at Kiev with R = 0.18 and is taken f r o m fig. 18 of ref. [7 I. In e i t h e r c a s e the s i g n i f i c a n t p o i n t , if o u r i n t e r p r e t a t i o n is c o r r e c t , is that the p o m e r o n l i m i t is a p p r o a c h e d f r o m above and a f a l l i n g FP(p) i n d i c a t e s the n e e d for o t h e r t r a j e c t o r i e s . o4~ !
A~
~
AFz(P) :0 275 + 013P ½
i
~-~ 0 2 " o
o~-
~"
l
¢o
F
C
0~
C2(p):O 15p-½
5
I© D
=
2'0
- y/Q2
Fig. i. ]nel~Isl.ic structure funetions: A: proton, R- 0.18; 13: proton. R
0: C: proton-neutron.
If we a s s u m e that the f u n c t i o n s Ri(~ 2) d e p e n d only on the pole p o s i t i o n and ),2 so R i ( ~ 2) = R(;~2, ~i ) then (13) i m p l i e s with c~D,~ = ~A o2 that a•...../bD, = aao/b~o2 ~ 2 h e n c e (b~,~+b^o)/2b^o~2 "~2 = ( a p ' + a A ~ ) / 2 a A ~ " F r o m eq. (14) and the fit to c u r v e B. b p , + bAQ = 0.48 we have 2bAO = 0.15 so that F2P(p) - F~(p) = Z . . . . Z 0 . 1 5 p - 1 / 2 ( p ,~o). T h i s is plotted on c u r v e C of fig. 1 and is c o n s i s t e n t with the v e r y r o u g h data r e p o r t e d . H a r a r i [81 and A b a r b a n e l . G o l d b e r g e r and T r e p a n [9] have m a d e the o b s e r v a t i o n that if C~TT(~) = c o n s t a n t then Regge b e h a v i o r and s c a l i n g i m p l y F~ (~) = c o n s t a n t . T h e i r w o r k is a s p e c i a l i z a t i o n to j u s t the p o m e r o n c o u p l i n g of the c o n c l u s i o n s r e a c h e d h e r e . The work of B r a n d t [10] u t i l i z i n g the DGS r e p r e s e n t a t i o n a t t e m p t e d to e s t a b l i s h a q u a n t i t a t i v e c o n n e c t i o n b e t w e e n aTP(Oo) and FP(~o). The p r e s e n t w o r k 2 g e n e r a l i z e d t h e s e r e s u l t s to a r b i t r a r y c~i and s u g g e s t s that the s e c o n d a r y t r a j e c t o r i e s c~p, and C~AO m a y . . . . 2 play an i m p o r t a n t r o l e m m t e r p r e t m g the b e h a v i o r of the i n e l a s t i c s t r u c t u r e f u n c t i o n s F2(P) a s p -- oo. F i n a l l y t h e r e is the i n t r i g u i n g p r o b l e m of the i n t e r p r e t a t i o n of the s t r u c t u r e f u n c t i o n F-P(p) in the low , 2 p r e g i o n ~ p 4. A clue is given by the r e c e n t w o r k of B l o o m and G i l m a n [11] who showed how this r e g i o n e m e r g e s a s an a v e r a g e o v e r the r e s o n a n c e b e h a v i o r a s _q2 a d v a n c e s f r o m low - 2 to t a r e - 2 > I(GeV) 2. T h i s s u g g e s t s we c o n s i d e r an a m p l i t u d e Wl(q 2, u) which is c o n s t r u c t e d fromqwl(q 2. gel b-yq e x t r a p o l a t i o n of the Regge b e h a v i o r at l a r g e u, w h e r e t h e s e f u n c t i o n s a g r e e , to low r, by a v e r a g i n g t h r o u g h the r e s o n a n c e s . So ~PT(U) = 4~2 ~ 1 ( 0 . u) ~-1 is the Tp total c r o s s - s e c t i o n a v e r a g i n g the Regge b e h a v i o r through the r e s o n a n c e s . R e p r e s e n t i n g V/l(q2, u) by a Regge r e p r e s e n t a t i o n and k e e p i n g only the Regge p o l e s we o b t a i n ¢x)
Wl(q 2. u) = ~1 . h(oti)( 2 ~ i + 1) Pc~i(v/#q) f d ~2Ri(~2)[2 q~q2//(~2 -q2)] c~i . t 0 302
Volume 34B. n u m b e r 4
PHYSICS
A t l a r g e v :, 4 G e V we a r e in t h e t r a n s r e s o n a n c e obtain FP(p ) = ~2c~i-1
p
ai-1
i
LETTERS
1 March 1971
r e g i o n f o r w h i c h W1 = W1 a n d t a k i n g t h e s c a l i n g l i m i t
f ~ d~t2 Ri()t 2) , 0
w h i l e a t q2 = 0 ~TP(U)/'4~2(~
=~ 2 ~ i
If we n o w a s s u m e
• -1
oei-1
~
f dX2Ri(X2)/(~t2)c~i. 0
that the ratio
ri(~i) =J d~2Ri(~2)(h2)-~i//f
d ~ 2 R i ( i t 2) w i t h r i ( 0 ) = 1 i s n o t a n e x /0 tremely~ v a r y i n g f u n c t i o n of t h e t r a j e c t o r y a i ( t h i s s e e m s to b e t r u e of t h e P . P ' a n d A~ t e r m s ) t h e n FP(p) a n d ~,~(~,)/47r 2 (~ s h o u l d h a v e r o u g h l y t h e s a m e s h a p e s i n c e t h e e x p a n s i o n c o e f f i c i e n t s a r e a p p r o x i m a t e l y the same, except near threshold where these considerations must fail. From the y-p total cross-sect i o n d a t a [12] e x t r a p o l a t i n g t h e R e g g e b e h a v i o r s m o o t h l y t h r o u g h t h e r e s o n a n c e s to z e r o a t t h r e s h o l d a n d c o m p a r i n g w i t h t h e i n e l a s t i c s t r u c t u r e f u n c t i o n we l e a r n t h i s i s a p p r o x i m a t e l y t h e c a s e . B o t h FP(p) a n d @TT(~) r i s e f r o m z e r o a t t h e i r r e s p e c t i v e t h r e s h o l d s to a m a x i m u m a n d t h e n f a l l l i k e p-l~2 a n d ~ - 1 / 2 - t o t h e c o n s t a n t p o m e r o n l i m i t . T h e q u a n t i t a t i v e d e t a i l s d e p e n d on Ri(~t2) b u t t h e q u a l i t a t i v e f e a t u r e s d i s c u s s e d h e r e t e n d to s u p p o r t t h e a n a l y s i s of B l o o m a n d G i l m a n [11]. 0
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
S. D e s e r . W. Gilbert and E. C. G. Sudarshan. Phys. Rev. 115 (1959) 731. H, H a r a r i . Phys. Rev. L e t t e r s 17 (1966) 1303. R , A . B r a n d t . Phys. Rev. D1 (1970) 2808 (Appendix). W.P. Hesse et ai.. Phys. Rev. L e t t e r s 25 (1970) 613. R . E . T a y l o r in Fourth Intern. Symp. on Electron and photon interaction at high energies• 1969 (Daresbury. England). M. Breidenbach et ai.. Phys. Rev. L e t t e r s 23 (1969) 935. M . I . T . - S . L . A . C . collaboration. SLAC-PUB-796 Sept. (1970). H . H a r a r i , Phys. Rev. L e t t e r s 22 (1969) 1078. H. D . I . A b e r b a n e i , M. Goldberger and S . T r e i m a n . Phys. Rev. L e t t e r s 22 (1969) 500. R . A . B r a n d t . Phys. Rev. L e t t e r s 22 (1969) 1149. E. Bloom and F. Gilman, to be published. R. Diebold. High e n e r ~ , photoproduction. Proe. of the Boulder conference on High energ3' physics {Aug. 1969. Boulder) fig, 1, p. 39. N. Nakanishi, Phys. Rev. 133 (1964) B214; J. M. Cornwall. D. C o r r i g a n and R. E. Norton. Phys. Rev. D. to be published.
303