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Multi-reggeon effects for inclusive reactions in the pionization region
r~Nuclear
L
Physics B42 (1972) 5_89-606. North-Holland Publishing Company
MULTI-REGGEON EFFECTS FOR INCLUSIVE REACTIONS IN THE PIONIZATION REGION J. C. BOTKE * Dcparl~zc~ll qf Applic, d 3Iathc~alics a~d Theorelica[ Physics. Univ~,rsilv qf Cambridge. Etlgla~zd
Received
6 January
1972
Abstract: We determine the non-leading behaviour of the inclusive differentia] cross section (IDCS) in the pionization region by considering a number of GribovFeynman graphs which are related to the IDCS through Mueller's generalized optical theorem. We find in addition to "normal" regge cut dependence, (ln s) -1, a dependence of the form S (c°nst)p2± (ln s) -1 which has no analogy in two-body scattering.
1. I N T R O D U C T I O N In a r e c e n t p a p e r by M u e l l e r [1], a g e n e r a l i z e d o p t i c a l t h e o r e m was p r o p o s e d which r e l a t e s the i n c l u s i v e d i f f e r e n t i a l c r o s s - s e c t i o n (IDCS) for the p r o c e s s 1 + 3 - - 2 + X to the M 2 - ( p l + P 3 - p,2)2 d i s c o n t i n u i t y of the c o n n e c t e d p a r t of the t o t a l l y f o r w a r d , u n p h y s i c a l a m p l i t u d e of fig. 1. In this p a p e r we want to show that this p i c t u r e is c o n s i s t e n t with the r e s u l t o b t a i n e d by a s u m m a t i o n of g r a p h s of the G r i b o v - F e y n m a n [2] type. In p a r t i c u l a r , the c l a s h i n g c o n t i n u a t i o n p r e s c r i p t i o n s a s s o c i a t e d with p h y s i c a l r e g i o n L a n d a u s i n g u l a r i t i e s [3], which a r e n e c e s s a r y in o r d e r t o p r e s e r v e the r e a l i t y of the c r o s s - s e c t i o n , do not c a u s e a n y t r o u b l e . A f t e r d i s c u s s i n g s o m e g e n e r a l f e a t u r e s of the p r o b l e m in s e c t . 2, we w i l l in the f o l l o w i n g s e c t i o n s , c o n s i d e r the a s y m p t o t i c b e h a v i o u r of a n u m b e r of g r a p h s in o r d e r to e x t r a c t the n o n - l e a d i n g b e h a v i o u r of the IDCS in the p i o n i z a t i o n r e g i o n . One i n t e r e s t i n g f e a t u r e of this w o r k is that in a d d i t i o n to the e x p e c t e d " n o r m a l " R e g g e c u t s , which b e h a v e a s y m p t o t i c a l l y as ( l n s ) -1 we find a c l a s s of g r a p h s which a p p r o a c h s(COnst)p2 a ins * Work supported by the Science Research Council.
590
J. C. Botke, Alulti-Reggeon effects
,M 2
3
•
1'
Fig. i. in e x a c t a n a l o g y with the r e s u l t of D r u m m o n d [4] for the 2 - 3 p a r t i c l e p r o c e s s . T h i s is to be c o m p a r e d with an i n v e r s e - s q u a r e r o o t b e h a v i o u r a s s o c i a t e d with n o r m a l m e s o n t r a j e c t o r y e x c h a n g e .
2. K I N E M A T I C S AND G E N E R A L DISCUSSION Our discussion will be concerned with amplitude F defined to be the connected part of the complete amplitude for the process 1 + 2 + 3 ~ 1' + 2' + 3' We defined the quantities,
sij : (pi+Pj) 2~- 2Pi.P j ,
(1)
q i = P'i - P i ,
(2)
M 2 = (pl+P2+P3)2~- s12+s23+s13,
(3)
and s12s23 -
s13
(4)
T h u s , for the i n c l u s i v e p r o c e s s 1 + 3 ~ 2 + X , s13 is the t o t a l e n e r g y and s12 , s23 a r e m o m e n t u m t r a n s f e r s . In a l l that f o l l o w s , we will c o n f i n e o u r a t t e n t i o n to the p i o n i z a t i o n r e g i o n d e f i n e d by Sl 3 ~ oo ,
(5)
s12, s23 ~ _ oo ,
(6)
with
s13
1,
(7)
fixed .
(8)
M2 and
O u r p r o g r a m will be to s t u d y the a s y m p t o t i c b e h a v i o u r of the G r i b o v F e y n m a n g r a p h s c o r r e s p o n d i n g to the e x c h a n g e of 2, 3 a n d 4 r e g g e o n s .
J. C.Botke, Mulli-Reggeo~ cJfec'ls
591
To t h i s end, it w i l l b e u s e f u l to d e c o m p o s e a l l v e c t o r s into two v e c t o r s l y i n g in and n o r m a l to the p l a n e of P l and P3 [51. We f i r s t d e f i n e the v e c t o r s m2 P l - +g13 P3 ,
Pl
m2 03 = P 3 - s l ~ P l
(9) (10)
,
s u c h t h a t p2 _~ p32 _~ 0. T h e n P2, f o r i n s t a n c e , c a n be w r i t t e n s23 s12 P2 : S l 3 p l + s i 3 p 3 + p 2 ~ w h e r e P 2 ± i s a s p a c e l i k e v e c t o r n o r m a l to the p l a n e o f P l p o s e t h e m a s s s h e l l c o n s t r a i n t on t h i s r e l a t i o n , we find
(11) andP3.
2 2 2 P2 : ?7 + P 2 1 = m
If we i m -
(12)
In g e n e r a l , the c o m p l e t e 3 - 3 a m p l i t u d e c a n b e c o n s i d e r e d a s an i n f i n i t e s u m of F e y n m a n g r a p h s . G r i b o v ' s a p p r o a c h is to c o n s i d e r s e l e c t i v e i n f i n i t e s u m s of g r a p h s w h i c h p r o d u c e g e n e r a l i z e d F e y n m a n g r a p h s c o n t a i n i n g complete off-shell two-bodyamplitudes, A ( s , l , m~). It is then a s s u m e d t h a t t h e d y n a m i c s is s u c h t h a t t h e s e t w o - b o d y a m p l i t u d e s a r e m a x i m a l in t h e l i m i t !s I ~ ~, t / s ~ 0 and m 2 / s ~ O. In t h i s l i m i t , t h e R e g g e h y p o t h e s i s implies 2
A ( s , t, m i) = g(l, m3,
(13)
where for s ~ +% 1 f d l ~ l G ( l , l) si G(t, s) : - 4~
(14)
with the signature factor
e - i ~ l +.r ~l-
s i n ?rl
(15)
T h e l i n t e g r a t i o n l i e s e n t i r e l y to the r i g h t of the s i n g u l a r i t i e s of G(l, l). F o r t h e l i m i t s ~ - ~ , we r e p l a c e
s l~
"r(-s) l
(16)
in eq. (14). T h e f u n c t i o n s g and g ' a r e the f a c t o r i z e d o f f - s h e l l r e s i d u e f u n c t i o n s w h i c h a r e a s s u m e d to h a v e o n l y r i g h t - h a n d c u t s and to f a l l off a s 1m21 ~ oo. In t h e s u b s e q u e n t s e c t i o n s , we w i l l d e f i n e m o r ? p r e c i s e l y t h e b e h a v i o u r r e q u i r e d . A l s o , we w i l l n e e d the d i s c o n t i n u i t y of A a c r o s s the s cut w h i c h is obtained by making the replacement ~l ~ ~l(1 - ~-e i~l) = - 2i in eq. (14).
(17)
d. C. Bolkc, Multi-Reggco~ effccts
592 If t h e r i g h t - m o s t
s i n g u l a r i t y of G(l, t) i s a R e g g e p o l e ,
G
1
l- a(t)
(18)
the G r i b o v - F e y n m a n g r a p h r e d u c e s to the u s u a l r e g g e o n e x c h a n g e d i a g r a m . W e c a n , h o w e v e r , i n c l u d e m o r e c o m p l e x d e p e n d e n c e in G to i n c o r p o r a t e cuts as well. F i n a l l y , s i n c e we a r e c o m p u t i n g a c r o s s s e c t i o n , it i s n e c e s s a r y to get t h e p h a s e of e a c h g r a p h c o r r e c t . To s a v e the r e a d e r m a n y f r u s t r a t i n g h o u r s w e w i l l s t a t e t h e w e l l known but e a s i l y f o r g o t t e n r u l e [6] w h i c h is to a s s o c i a t e a (+i) with e a c h A and e a c h p r o p a g a t o r , a ( - i ~ ) with e a c h v e r t e x , and a n o v e r a l l ( - i ) with e a c h g r a p h . We now p r o c e e d to the g e n e r a l i z e d o p t i c a l t h e o r e m , da - ---c~ P20 d3p~
1 r(++-)l 2 i s 1 3 AM2
(19)
iqi=0
w h e r e A , I 2 d e n o t e s the M2 d i s c o n t i n u i t y of the u n p h y s i c a l a m p l i t u d e F w h o s e a r g u m e n t s r e f e r to the d i s p o s i t i o n s of s13 , M 2 and s l , 3, r e l a t i v e to t h e i r cuts. The other invariantssl2, s 2 3 a r e both n e g a t i v e and thus r e m o v e d f r o m t h e i r c u t s . R a t h e r than w o r k i n g d i r e c t l y with t h i s a m p l i t u d e , h o w e v e r , w e w i l l find it c o n v e n i e n t to c o n s i d e r t h e p h y s i c a l a m p l i t u d e F ( + + + ) w h i c h i s r e l a t e d to the r e q u i r e d d i s c o n t i n u i t y by the i d e n t i t y , AM2 F ( + + - ) = (1 - A s l , 3 ,) AM2 F ( + + + )
(20a)
: A~I 2 ( 1 - A S l , 3 ,) F ( + + + ) .
(20b)
In p r a c t i c e , the M 2 d e p e n d e n c e of a g i v e n g r a p h wit[ not be e x p l i c i t , n o r f o r that m a t t e r w i l l the S l , 3, d e p e n d e n c e , w h i c h p r e v e n t s us f r o m e v a l u a t i n g t h e d i s c o n t i n u i t y v i a an e x p l i c i t a n a l y t i c c o n t i n u a t i o n . I n s t e a d , we u s e the f o r m a l p r e s c r i p t i o n [6] of s l i c i n g the g r a p h in a l l p o s s i b l e w a y s and r e p l a c i n g e a c h cut a m p l i t u d e o r p r o p a g a t o r by i t s d i s c o n t i n u i t y c o n s i s t e n t with the two rules: (i) T h e two a m p l i t u d e s w h i c h r e s u l t f r o m a p a r t i c u l a r p a r i t i o n of F m u s t b o t h be c o n n e c t e d s i n c e t h e y a r e i n d i v i d u a l l y s u p p o s e d to be a m p l i t u d e s f o r the i n c l u s i v e p r o c e s s . (ii) If m o r e than one s l i c e is p o s s i b l e , the t o t a l d i s c o n t i n u i t y i s found by a d d i n g the c o n t r i b u t i o n of t h e v a r i o u s p a r t i t i o n s (P) a c c o r d i n g to the r u l e , F(+++) :
l E fL(+,+,+)(AM2f) p=l
pfR(+,-,+)
,
(21)
w h e r e f L and f R r e f e r to the a m p l i t u d e s w h i c h r e s u l t on the l e f t and r i g h t of a g i v e n p a r t i t i o n d e n o t e d by (AM2f) p. We can m a k e one f u r t h e r s i m p l i f i c a t i o n by n o t i n g t h a t the p a r t i t i o n s of F i s o l a t e the d e p e n d e n c e on s13 and Sl, 3, and in f a c t f L ( f R ) d o e s not d e p e n d on Sl,3,(s13). T h e t o t a l d i s c o n t i n u i t y , eq. (20), i s then
J. C. Bolke. MuIti-Reggeon e.ffecls
593
l AM2F(++-)
:
~ fL(+)(AM2f)pfR(+) p:l
(22)
w h e r e we have u s e d fR(-) = fR(+) .
(23)
We can now e a s i l y s e e the p h y s i c a l i n t e r p r e t a t i o n of this r e l a t i o n . If we w e r e to e v a l u a t e the IDCS d i r e c t l y , we would c o m p u t e
~ f d(~nF(+)F(-)
(24)
n w h e r e F i s the c o m p l e t e a m p l i t u d e for the p r o c e s s 1 + 3 ~ 2 + X n. If we exp a n d F in t e r m s of a s u m of c o n t r i b u t i o n s f r o m v a r i o u s g r a p h s f , a t y p i c a l c o n t r i b u t i o n will be of the f o r m f dg)n f L (+) J R ( - ) Now, s i n c e
(AM2f) p
(25)
of eq. (22) c o n v e r t s loop i n t e g r a t i o n s into p h a s e s p a c e
i n t e g r a t i o n s , eq. (22) has j u s t the f o r m of eq. (25). In p a r t i c u l a r , s i n c e p h y s i c a l r e g i o n L a n d a u s i n g u l a r i t i e s a r i s e in the s a m e m a n n e r in both r e l a t i o n s , they c a u s e no t r o u b l e in a s s o c i a t i n g t h e m 2 d i s c o n t i n u i t y of F ( + + - ) with the IDCS. We now p r o c e e d with the e v a l u a t i o n of s p e c i f i c g r a p h s . F o r t u n a t e l y , a l t h o u g h we c a n n o t c o m p u t e a n u m e r i c a l r e s u l t for e a c h g r a p h , it is p o s s i b l e to d e t e r m i n e w h e t h e r or not a s p e c i f i c g r a p h will v a n i s h . T h i s v a n i s h i n g o c c u r s a s a c o n s e q u e n c e of one o r m o r e of the f o l l o w i n g t h r e e m e c h a n i s m s : (i) T a k i n g AM2 v i o l a t e s r u l e (i) a b o v e . (ii) T h e a m p l i t u d e h a s a v a n i s h i n g M 2 d i s c o n t i n u i t y which will be r e f e r r e d to a s a k i n e m a t i c z e r o (KZ). (iii) T h e a m p l i t u d e F(+ + +) v a n i s h e s in the s a m e m a n n e r as the w e l l k n o w n A F S d i a g r a m [6]. T h i s m e c h a n i s m will be d i s c u s s e d in d e t a i l in s e c t . 4 w h e r e it will he shown that t h i s is d e p e n d e n t on the a s s u m p t i o n c o n c e r n i n g the m a s s d e p e n d e n c e of the o f f - s h e l l a m p l i t u d e s . F o r want of a b e t t e r n a m e , we will r e f e r to this as a c l o s e d c o n t o u r z e r o (CCZ).
3. T W O - R E G G E O N GRAPHS A l t h o u g h the t w o - r e g g e o n c o n t r i b u t i o n h a s b e e n c o n s i d e r e d p r e v i o u s l y [7], we will d i s c u s s it ip s o m e d e t a i l f o r the i n s i g h t it a f f o r d s into the t e c h n i c a l a s p e c t s of this s o r t of c a l c u l a t i o n . In p a r t i c u l a r , we want to e l a b o r a t e on the m e c h a n i s m s d i s c u s s e d in s e c t . 2 by w h i c h c e r t a i n g r a p h s vanish. T h e r e l e v a n t g r a p h s a r e i l l u s t r a t e d in fig. 2 w h e r e the l a b e l s (abc) c o r r e s p o n d to p e r m u t a t i o n s of (12 3). C o n s i d e r i n g fig. 2b f i r s t , we s e e that t h e s e v a n i s h as a c o n s e q u e n c e of r u l e (i). Of t h o s e r e m a i n i n g , only the c a s e
J. C. Bolke, Multi-Reggeon effects
594
hlvla
b
b
¢
c
"-t
a b
"
~
c
a b c
I
(a)
(b)
Fig. 2. T w o - r e g g e o n g r a p h s .
Fig. 3.
b = 2 is non-vanishing; the remaining possibilities vanishing as a result of kinematic r e s t r i c t i o n s (rule (ii)). We will consider first the case b ¢ 2. The Gribov-Feynman graph for this situation is given in fig. 3 where of course the choice b = 3 instead of b = 1 is a r b i t r a r y . If we define Sudakov p a r a m e t e r s [5] for the loop momentum k we have k = otPl+fip3+g,
(26)
k 2 -~ c~/3s13 + K2 ,
(27)
where
and d 4 k = ½ I s 1 3 I do~ d / 3 d 2 ~ .
(28)
(½P3±k)2-m2+ic=K2+(}m2+~Sl3)(}±fi)-rn2+iE,
(29)
The propagators
become
and the energies S l = 2 P l " (½P3 + k ) = ~~s 1 3 + c~rn2 + $ s 1 3 ,
(30a)
s 2 = 2P2 ' ( ½ P 3 - k) = ½s23 - c~s12 - $ s 2 3 - 2P2 ± . K .
(30b)
Keeping the energies determines
large
and the "masses"
~ O (1) .
f i n i t e a s s 1 3 - - % 77 f i x e d
(32)
Making the change of variables x = s13 a y = [5 , we have
,
(33) (34)
595
d. C. Botke, Multi-Reggeon effects
(½P3 ± k ) 2 - r n 2 + i e
~- K2+ (½rn2 ±x)(½
±y)-rn2+ic
and
s 1 : s13 (½+Y) ,
(36a)
s2 = s23 (½-Y) • The amplitude for this process is
(36b)
F(+++) =-i~ 2 f d4k AIA2 (2v)4 [(½P3 + k)2- m2 +i~]2[(½p3- k)2- m2+ie] 2
(37)
which becomes 2
F(+++) = - i •
~ dlj {J}~jGj(lj,
0)~(0, m 2, m2)}
j=l
(38)
× (s13)/1 (- s23 )/2 M l l l 2 , where
Mhl2 : ½x2 (½~)2f dxdyd2K (2~) 4 g l g 2 ( Y + ½ ) l l ( y _ ½ ) / 2
× {[K 2-m2+(~m2+x)(½+y)+iE]2[K 2-m2+(½m2-x)(½-y)+ic]2}-I
(39)
T h e functions gl and g2 are a s s u m e d to have only right-hand cuts which are disposed in the s a m e m a n n e r as the propagators with regard to displacements off the real axis of integration. It is then a simple matter to show I that if IYl -" ~, all the singularities of the integrand as a function of x lie on one side of the r e a l x a x i s . If, as assumed, t h e g ' s d e c r e a s e sufficiently rapid as a function of the " m a s s e s " , we can close the x integration in the 1 appropriate half-plane to get a zero result. Thus, we must have l Yl < ~. However, from eq. (36), we see that this implies s 2 < 0 and consequently the M 2 discontinuity of the amplitude is zero. We now turn to the situation indicated in fig. 4, and proceed as before. The propagators become
(½P2 ± k ) 2 - m2 + ie = (K ± sp2_L) - m
~ ½±
½±
+ie , (40)
and the energies 1
l
1
1
s I = 2P I . (~P2 +k) = ~s12 +~m 2 +fist3 ,
s 2 : 2P3"(~p2- k) = ~s23- ~ s 1 3 - t i m 2
•
(41a)
(41b)
Again, we r e q u i r e that the energies a r e l a r g e and the " m a s s e s " finite as s --~ 0% q fixed with the result
J. C. Botke. Mulli-Regge'on ei~tecls
596
I
I
Fig. 4. (42)
t'~
0
(43)
~?
(s23) •
D e f i n i n g new v a r i a b l e s
(44)
~lx = a S l 2 , 77y = - 3 s 2 3
(45)
,
we find (½P2 + k ) 2 - m 2 + i c
= ( K + ~ p 2 ± ) 2 - rn2 +~(~1 + x)(½ :~ y ) + i ~ ,
(46)
and s I = (-Sl2)(y-
~) ,
s 2 = (-s23)(x-½) The amptitude,
2
.
(47a) (47b)
in this c a s e , b e c o m e s
dlj
F(~++) : -i [l f. / 2~ ~J Gj(~, 0)~)(0, m2, .~2/} j=l × (-s12 )11 (-s23)12N1112(V) ,
(48)
where
N l l l 2 = }~)~2 (½v)2 f d x d y d 2 g g l g2(Y- ½)/1 ( x - ½)12 (27r)4
× {[(K+½P2±) 2 - m 2 + ~ ? ( ½ + x ) ( ~ - y ) + i e ] 2 × [(K- ~p2±) 2 - m 2 + r/(~ - x) (~ + y) + i~]2}-1
(49)
597
J. C. Bolke. Mulli-Reg$eo~z eJfecls
1
With r e g a r d to j u s t the p r o p a g a t o r s and the g ' s we find a g a i n that if lit'l ~, the s i n g u l a r i t i e s a r e alt on one s i d e of the r e a l x a x i s as b e f o r e . H o w e v e r in this c a s e , in view of eqs. (47), t a k i n g the M 2 d i s c o n t i n u i t y i m p o s e s c o n s t r a i n t s on both x and y which p r e v e n t s the c l o s i n g of e i t h e r c o n t o u r and the r e s u l t is n o n - z e r o . If we a s s u m e that the r i g h t - m o s t s i n g u l a r i t i e s of the G ' s a r e s i m p l e p o l e s we find a c o n t r i b u t i o n to the IDCS of the f o r m
g l g2
(_ s12) ~1(0) (_ s23) c~2(0) s13 --
A~ lC~207)
,
(50)
w h i c h if we a s s u m e p o m e r o n d o m i n a n c e b e c o m e s g2)?Nc~l:Ot2=l(r]) .
(51)
T h e f u n c t i o n .~ is a p o s i t i v e , r e a l f u n c t i o n of p2.c : m 2 - ~ d e f i n e d by eq. (49) 1 1 with the r e s t r i c t i o n x > s a n d y : 5. We thus o b t a i n the e x p e c t e d s e a l i n g result.
4. T H R E E - R E G G E O N
GRAPHS
T h e r e is a l a r g e n u m b e r of s u c h g r a p h s to be c o n s i d e r e d and to p r o v i d e s o m e o r d e r , we will c l a s s i f y t h e m a c c o r d i n g to the n u m b e r of r e g g e o n -
i
-If__ _ I_
X
(a)
X I-
i (b)
m
(c)
-%
l!i
(d)
(¢)
(f)
Fig. 5. Three-reggeon graphs with no reggeon-reggeon couplings.
Pl PPa
iA,
iA~ ~
-k ~
Fig. 6.
_
-
pf
~L
J. C. Botke, Multi-Reggeon effects
598
ii
"V
--
(a)
--
(b)
A_-
A
(d)
t
(c)
-N
(e)
(f)
D~ _!I (g)
Fig. 7. Three-reggeon graphs with one reggeon-reggeon coupling. I
F i g . 8.
oi
-
(a)
/J
/~_/}
(b)
(c)
U) (d)
Fig. 9. Three-reggeon graphs with two reggeon-reggeon couplings,
d. C. Botke, Multi-Reggeon effects 11
599
el
I
Fig. i0. Gribov-Feynman
graph for fig. 9b (i 2 3) with two of the six possible M 2 partitions.
f
\ "
Pt
(a) •
'Pz
P3
(b)
Fig. ii. Three-reggeon
y (a)
graph with three reggeon-reggeon couplings. partitions are indicated in (b).
Three
@ 1
2
2
(b)
3
3
Fig. 12. Three-reggeon graphs with a triple-reggeon coupling.
Fig. 13.
typical
J. C. Bolke. Mzdli-Reggc'on e:l'Jects
500
r e g g e o n v e r t i c e s . T h e s e a r e r e p r e s e n t e d s c h e m a t i c a l l y in figs. 5, 7, 9, 11 a n d 12 w h e r e e a c h g r a p h is m e a n t to r e p r e s e n t all s i m i l a r g r a p h s o b t a i n e d by p e r m u t i n g p a r t i c l e s and v e r t i c e s . B e c a u s e t h e r e is s u c h a l a r g e n u m b e r of g r a p h s , we will not p r e s e n t a d e t a i l e d a n a l y s i s but only quote the e s s e n tial results. C o n s i d e r i n g fig. 5 f i r s t , we s e e i m m e d i a t e l y that, as a c o n s e q u e n c e of r u l e (i), 5a and 5b do not c o n t r i b u t e . In fact 5c and 5d a l s o fail to c o n t r i b u t e f o r the s a m e r e a s o n a l t h o u g h it is p e r h a p s n e c e s s a r y to c o n s t r u c t the a p p r o p r i a t e G r i b o v - F e y n m a n g r a p h s to s e e t h i s . T h e r e m a i n i n g g r a p h s 5e a n d 5f both v a n i s h as a r e s u l t of c l o s e d c o n t o u r z e r o s . To i l l u s t r a t e m o r e p r e c i s e l y what is m e a n t , we will c o n s i d e r in d e t a i l a p a r t i c u l a r g r a p h of the s e t 5e, n a m e l y that i l l u s t r a t e d in fig. 6. P r o c e e d i n g as b e f o r e , the p r o p a gators are ( P l +/e)2 - m 2 + ie = g2 + at3Sl3 + c~m2 +t3s13 + ie ,
(52a)
( P2 - k)2 - DI2 + I'E =
(52b)
K2+Ceg3SI3-C~SI2-t3S23-2P2±'K+iE,
a n d the e n e r g i e s , s t = s 3
:
s12
(53a)
,
(535)
s2 = 2P3 ' ( P 2 - k) = s 2 3 - C~Sl3- i3rn2 w h i c h d e t e r m i n e s the p a r a m e t e r s ,
(55)
(56) M a k i n g the c h a n g e of v a r i a b l e s x = c~s 12 ,
(56a)
y = i3s 13 ,
(56b)
gives
(Pl+k)2-m2+ic
~ g2+y+ie ,
(P2-k)2-m2+ie
~ K2 - 2 p 2 ± ' , ~ - x + i e
(57a) ,
(57b)
and s2 = - s 2 3 ( ' ; - 1 )
•
(58)
T h e i m p o r t a n t p o i n t to n o t i c e is that the y d e p e n d e n c e is c o n t a i n e d e n t i r e l y in eq. (57a) and the c o r r e s p o n d i n g g f u n c t i o n s . S i n c e the s i n g u l a r i t i e s in y a r e thus a l w a y s b e l o w the r e a l a x i s , and p r o v i d e d the i n f i n i t e s e m i - c i r c l e g i v e s no c o n t r i b u t i o n to the i n t e g r a l , we can c l o s e the c o n t o u r in the u p p e r h a l f p l a n e to get a z e r o r e s u l t . T h i s . in fact, p r o v i d e s the q u a n t a t i v e u p p e r
J. C. B o t t e c . A l ~ d l i - R c g g c o , e
601
Table i Fig.
Non-zero
Disconnected
5(a-d) 5e, f
Kinenmtie
zero
Closed contour zero
all all all
7a
7(b-d) 7e 7f 7g
(i 23)
9a 9b 9c
(I 23)
all
(13 2)
(2 13) all (213)
(123), (132) all
11
(I 3 2) (132)
(2 13) (123), (213)
all
12a
all
(12 3)
12b
(13 2), (2 13)
bound on t h e g ' s m e n t i o n e d in s e c t . 2. In t a b l e 1, we h a v e s u m m a r i z e d these results. T h e g r a p h s with o n e r e g g e o n - r e g g e o n v e r t e x a r e i l l u s t r a t e d in fig. 7. A s i n d i c a t e d in t a b l e 1, a ll v a n i s h with the e x c e p t i o n of t h e (1 2 S) (and (S 2 1)) o r d e r i n g of fig. 7e. T h e G r i b o v - F e y n m a n g r a p h is i n d i c a t e d in fig. 8 w h e r e the d a s h e d l i n e s d e n o t e the p o s s i b l e M 2 p a r t i t i o n s . T h e r e is one p o i n t i n t r o d u c e d by t h i s g r a p h w h i c h we want to d i s c u s s b r i e f l y . T h e e n e r g i e s of A 1 l a nd A 3 a r e s 1 = s 2 = e S l 2 and as s12 ~ - ~ , we e n c o u n t e r t h e l e f t - h a n d c u t s of the t w o - b o d y a m p l i t u d e s w h i c h s e e m s to c o n t r a d i c t the s u p p o s i t i o n ( s e c t . 2) that the t o t a l a m p l i t u d e h a s c u t s in a p a r t i c u l a r s i j only f o r p o s i t i v e v a l u e s of that v a r i a b l e . T h e r e s o l u t i o n of t h i s d i f f i c u l t y is that as s12 ~ a c o r r e s p o n d i n g e n e r g y s, d e f i n e d by s = ~ m 2 - l - s 1 2 , is a p p r o a c h i n g p o s i t i v e i n f i n i t y and the cut we e n c o u n t e r is the p o s i t i v e e n e r g y cut a s s o c i a t e d with t h i s e n e r g y . T h i s is, in f a c t , j u s t a p a r t i c u l a r f i n al s t a t e s u b e n e r g y of the i n c l u s i v e a m p l i t u d e and c o n s e q u e n t l y , is e v a l u a t e d a b o v e o r b e l o w its cut in the s a m e s e n s e as M 2. T h e g e n e r a l f o r m of t h i s c o n t r i b u t i o n is
(-s23)~2(0) f d2K !~I(K) 2
(-sI2)2~I(K2)-IN(~2(0)
s13
7, K)
(59)
'
w h e r e we h a v e a s s u m e d c~1 = c~3. T h e f u n c t i o n N c a n n o t be e v a l u a t e d in c l o s e d f o r m but it d o e s h a v e the u s e f u l p r o p e r t i e s that it is r e a l and p o s i t i v e We can a p p r o x i m a t e the a s y m p t o t i c b e h a t r i o u r by s e t t i n g K = 0 e v e r y w h e r e e x c e p t in the e x p o n e n t with the r e s u l t , (_ s23 )c~2(0) (_ s12)2c~1 (0)-1
s13 ln(- s12 )
15c~1(o) i2 2~"0)i~
N(c~2(0)' 77, 0) ,
(60)
J. C. Botke, Multi-Rcggeo~t effects
602
which
if we assume
pomeron
dominance
becomes
(const)~?N0?, K = 0)
(61)
In(- s12) Including the (3 2 i) ordering, the total contribution is
(const)~N(~) [
L__+
kln(-s 12)
L ~
In(- s23)J "
(62)
It i s i n t e r e s t i n g to note t h a t a l t h o u g h t h e s e e f f e c t s r e s u l t f r o m a r e g g e o n r e g g e o n cut of the a m p l i t u d e F ( + + + ) , the i n c l u s i v e a m p l i t u d e s , in t h i s c a s e , do not c o n t a i n t h e s e c u t s s i n c e t h e y i n d i v i d u a l l y c o n t a i n o n l y s i n g l e r e g g e o n exchanges with, however, variable momentum transfer. T h e next g r o u p a r e t h o s e with two r e g g e o n - r e g g e o n v e r t i c e s w h i c h a r e i l l u s t r a t e d in fig. 9 a n d s u m m a r i z e d in t a b l e 1. T h e g r a p h of fig. 9d is not a l l o w e d b e c a u s e the " e n e r g y " a s s o c i a t e d with t h e r c e n t e r r e g g e o n in n e v e r l a r g e . T h e G r i b o v - F e y n r n a n g r a p h f o r fig. 9b (1 2 3) i s g i v e n in fig. 10 w h e r e two of the s i x p o s s i b l e M 2 p a r t i t i o n s a r e i n d i c a t e d . A s y m p t o t i c a l l y , t h i s h a s a f o r m s i m i l a r to eq. (59), n a m e l y ,
(-
s23)a3(°)f
d2t~(-
s12) al(K2)+a2(K2)-I N ( a l , c~2, ~, K) .
(63)
s13
In t h i s c a s e N, w h i c h is a s u m of t e r m s c o r r e s p o n d i n g to the p o s s i b l e p a r t i t i o n s , i s c o m p l e x r a t h e r t h a n r e a l a s a c o n s e q u e n c e of p h y s i c a l r e g i o n L a n d a u s i n g u l a r i t i e s . T h e s e a r i s e in the t e r m c o r r e s p o n d i n g to p a r t i t i o n 6 in a s s o c i a t i o n with the p r o p a g a t o r s i n d i c a t e d by the a r r o w s . T h e s e n s e of continuation the s i n g u l a r i t y i s , of c o u r s e , s p e c i f i e d by the i e ' s a t t a c h e d to the p r o p a g a t o r s . T h e r e a l i t y of t h e c r o s s s e c t i o n i s r e c o v e r e d by a d d i n g t h e g r a p h in w h i c h the s i n g l e r e g g e o n c o u p l i n g with p a r t i c l e 2 f o l l o w s r a t h e r t h a n p r e c e e d s the two r e g g e o n c o u p l i n g w h i c h , a s a c o n s e q u e n c e of eq. (22), i s the c o m p l e x c o n j u g a t e of eq. (63). In the p o m e r o n l i m i t , the t o t a l c o n t r i b u t i o n h a s the f o r m of eq. (62). T h e only g r a p h with t h r e e r e g g e o n - r e g g e o n v e r t i c i e s is i l l u s t r a t e d in fig. 11 w h e r e t h r e e t y p i c a l p a r t i t i o n s a r e a l s o i n d i c a t e d . T h e l e a d i n g b e h a v i o u r of t h i s g r a p h h a s the f o r m
past
l_fd2 S13
K (_S12)al(K2)+a3(K2)-1
(_S23)a2(K2)+a3(K2)-1N(ai,~, K) (64)
w h e r e N in t h i s c a s e i s r e a l d e s p i t e t h e e x i s t e n c e of n u m e r o u s p h y s i c a l r e g i o n L a n d a u s i n g u l a r i t i e s ; the r e a l i t y b e i n g p r e s e r v e d by the l e f t - r i g h t s y m m e t r y of t h e g r a p h . We h a v e the p o s s i b i l i t y , in t h i s c a s e of a p i n c h b e t w e e n two L a n d a u s i n g u l a r i t i e s w h i c h a r i s e f r o m o p p o s i t e s i d e s of a p a r t i t i o n and thus a p p r o a c h the i n t e g r a t i o n c o n t o u r f r o m o p p o s i t e s i d e s of t h e r e a l a x i s . S i n c e the i n t e g r a t i o n c o n t o u r c a n n o t be d e f o r m e d to a v o i d the p i n c h , we d e f i n e the i n t e g r a l by r e p l a c i n g t h e r e l e v a n t p r o p a g a t o r s on o n e s i d e of the p a r t i t i o n by t h e i r c o m p l e x c o n j u g a t e s p l u s d i s c o n t i n u i t i e s w h i c h l e a d s to a p i e c e w h i c h i s a n a l y t i c p l u s a n o n - a n a l y t i c c o r r e c t i o n w h i c h p r e s e r v e s the r e a l i t y of the c r o s s s e c t i o n .
J. C. Botke , Multi-Reggeo~z ej)k,c'ts
603
In general, as discussed in ref. [4], this contribution contains two reggeon- reggeon cuts corresponding to c~1 + c~3 and c~2 + a 3 and the full traingle singularity corresponding to the pair ot 1 + c~3 and c~2 + c~3" This will be discussed more fully in sect. 5. In this case, however, since the argument of all the c~i i s K2, the reggeon-reggeon cuts are coincident with the triangle singularity. Again, assuming pomeron dominance, the leading behaviour is
(const)r/N(r/) In s13 -
(65)
The remaining graphs of this group are those of fig. 12. As one might expect, those of fig. 12a vanish and those of fig. 12b lead to a contribution of the form eq. (62) except that the sign of the contribution is negative.
5. F O U R - R E G G E O N G R A P H S We do not i n t e n d to w o r k o u r way t h r o u g h the e n t i r e s e t of f o u r - r e g g e o n g r a p h s s i n c e the m a j o r i t y w i l l l e a d to h i g h e r o r d e r c u t s e f f e c t s of j u s t the s o r t a l r e a d y d i s c u s s e d . T h e r e i s h o w e v e r , one c l a s s of g r a p h s w h i c h l e a d to a n e w t y p e of b e h a v i o u r u n l i k e any found in t w o - b o d y s c a t t e r i n g . T h e r e m a i n d e r of t h i s s e c t i o n w i l l be s p e n t d i s c u s s i n g t h e s e i d e a s . T h e g r a p h with w h i c h we a r e c o n c e r n e d , i s i l l u s t r a t e d in fig. 13. T h e n o v e l f e a t u r e i s t h a t p a r t i c l e 2 b r e a k s into a r e g g e o n - r e g g e o n p a i r r a t h e r t h a n p r o c e e d i n g d i r e c t l y t h r o u g h the d i a g r a m a s in the p r e v i o u s e x a m p l e s . A s a c o n s e q u e n c e of t h i s , the t r a n s v e r s e m o m e n t u m of p a r t i c l e 2, P2 , g e t s m i x e d up with the m o m e n t u m t r a n s f e r a s s o c i a t e d with the r e g g e o n s w h i c h u l t i m a t e l y c o n t r o l the a s y m p t o t i c b e h a v i o u r of the g r a p h . A g a i n , we w i l l d i s p e n s e with a d e t a i l e d d e r i v a t i o n and s i m p l y s t a t e the r e s u l t t h a t t h e c o n t r i b u t i o n to the IDCS h a s the f o r m ,
I= ( c o n s t ) f d2~ ( - s 1 2 ) a l + a 3 - 1 w h e r e the a r g u m e n t s
( - s 2 3 ) c ~ 2 + a 4 - 1 N ( a i , •,p2±) ,
(66)
of the a i a r e (K + ½P2±)2 .
i = 1, 3 ,
(67a)
(K - ½p2±) 2 •
i = 2, 4 .
(67b)
S i n c e the b e h a v i o u r of t h i s t y p e of f u n c t i o n h a s b e e n d i s c u s s e d in d e t a i l by D r u m m o n d [4], we w i l l o n l y o u t l i n e the a r g u m e n t . We f i r s t d e f i n e the function
IjlJ2
=S
d(-Sl2)(-Sl2)
SO
-jl-1
f~ d(- s 2 3 ) (- s 2 3 ) - J12 -
I(s12, s23) ,
(68)
SO
which has the inverse d j l dJ2 I " " I = f (2~ j lJ2(-s12)31 (-s23))2 "
(69)
J. C.Botke. Multi-Reggeon cftk'cls
604 In a d d i t i o n ,
where
we d e f i n e the p a r a m e t e r s (- s12) = X(qs13 )x ,
(70a)
(- s23) = x - l ( r l S l 3 )y ,
(70b)
eq. (4) i m p l i e s x+y
-- 1 .
(70c)
Then we have (_ S l 2 ) J l (_ s23)J2 = ~Jl-J2 (~lSl3)Xjl+YJ2
.
(71)
Defining new variables J = xJl+YJ2 '
(72a)
J = J l - J2 ,
(72b)
eq. (69) b e c o m e s I : f 2~i dJ '
(73)
(VSl3)J
with
~ ( j ) : f d ~j i aj I j l J 2
"
(74)
T h e a s y m p t o t i c b e h a v i o u r i s t h u s d e t e r m i n e d by the r i g h t - m o s t s i n g u l a r i t y of qp(d) w h i c h in t u r n d e p e n d s on the s i n g u l a r i t i e s of Ij lJ2" By i n s e r t i n g eq. (66) i n t o eq. (68), we find N ( a , t~, p2±) I j l J 2 = (const)f d2K ( j l _ o 1 1 _ a t 3 + 1 ) ( J 2 _ ~ 2 _ c~4+1 ) . (75) T h i s f u n c t i o n h a s two t y p e s of j p l a n e s i n g u l a r i t i e s . T h e f i r s t a r e the t w o r e g g e o n c u t s a s s o c i a t e d w i t h the v a n i s h i n g of the two f a c t o r s in t h e d e n o m i nator separately. The branch points are at Jl = a13 =
al+a3-
J2 = c~24 = a 2 + a 4 -
1 ,
(76a)
1 ,
(76b)
w h e r e we h a v e a s s u m e d eli(l) = cli+bil. T h e s e c o n d t y p e o c c u r s f o r j l ' c~13 , J2 • a 2 4 w h e n b o t h f a c t o r s v a n i s h s i m u l t a n e o u s l y . T h e e q u a t i o n f o r the b r a n c h p o i n t is the p a r a b o l a
p2
(77)
( r l ± ;~2 )2 = _ 2 - ' where
2 al +a3--]1rI : bl+b 3
1
,
(78)
J. C. Botkc , Multi-Rcggeon ql?}'cls
605
and 2
(12 + a 4 - J2 - 1
r2 :
b ~ 1,4
(79)
T h e s i n g u l a r i t i e s of O(J) a r e d e t e r m i n e d by the p i n c h e s of the j i n t e g r a t i o n of eq. (74) w h i c h o c c u r at (1)
J = xc~13+YCi24 ,
(80)
(2)
J : XJlL+ YJ2L '
(81)
and
a n d J2L a r e d e t e r m i n e d by t h e i n t e r s e c t i o n of the c u r v e with the b r a n c h cut eq. (77). A s shown by D r u m m o n d [4], the f i r s t t y p e of p i n c h i s i n e f f e c t i v e with the r e s u l t that the l e a d i n g b e h a v i o u r i s g i v e n by the s e c o n d w h i c h if we a s s u m e b 1+ b 3 = b 2 + 54 g i v e s
where)lL
J=Xjl+YJ2
N(p 21) ,
Sl 3 I n s 13
(82)
where d : xc~13 + yc~24 +xy (b 1 + b3)P~ ± .
(83)
If t h e q u a n t u m n u m b e r s of p a r t i c l e 2 a r e s u c h that it c a n c o u p l e to two p o m e r o n s , t h e n t h e l e a d i n g b e h a v i o u r is
S l~Y3(bl +b 3 ) p2~ lnsl 3
N(P2±) .
(84)
A c t u a l l y , t h e r e h a s b e e n n o t h i n g in the a r g u m e n t to p r e v e n t us f r o m a s s u m i n g p a r t i c l e 2 i s a s y s t e m of p a r t i c l e s which c o u p l e s to the two p o m e r o n s in w h i c h c a s e (84) s t i l l a p p l i e s w i t h P 2 ± , the t o t a l t r a n s v e r s e momentum. In the p i o n i z a t i o n r e g i o n xy ~ ~ and with b i ~ 1 GeV - 2 the e x p o n e n t of eq. (84) i s a p p r o x i m a t e l y ½P22±_which thus r e s e m b l e s a r e g g e p o l e with i n t e r s e p t 1 1 a n d s l o p e ~ GeV - 2 . In s u m m a r y , t h e t o t a l IDCS h a s the f o r m dcr
i
i
P20 d3p2 = Fl(r/)+ F2(r/) [in (- s12) +ln (-s23) ] 2
2x3 P2± F 3 (r/) s13 +in(sl3 ) + F 4 0 7 ) i n s 1 3
+-..
,
where F407) vanishes if particle 2, or system 2, cannot couple to two pomerons. All the Fi(~l) are real and FI(~/) is positive. The signs of the remaining F i are less well determined and may be negative.
(85)
506
d. C. B o t k e . M u l l i - R e g g e o n ¢~'fects
T h e a u t h o r w o u l d l i k e to t h a n k P r o f e s s o r J. C. P o i k i n g h o r n e f o r s u g g e s t ing t h i s t o p i c and f o r h is c o n t i n u e d i n t e r e s t . I a m a l s o i n d e b t e d to I. T. D r u m m o n d for useful c o n v e r s a t i o n s c o n c e r n i n g this work.
RE FERENCES [1] [2] [3] [4] [5] [6]
A. tt. Mueller, Phys. Hey. D2 (1970) 2963. V. N. Gribov, ZhETF 53 (1967) 654; [JETP (Soy. Phys.) 26 (1968) 414]. J. C. Polkinghorne, Nuovo Cimento 7A (1972) 555. I.T. Drummond, Phys. Rev. 176 (1968) 2003. V.V. Sudakov, ZhETF 30 (1956) 97; [JETP (Soy. Phys.) 3 (1956) 65]. R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The analytic Smatrix, Cambridge University P r e s s 1966. [7] I. G. Haliiday and G. W. P e r r y , Imperial College preprint, ICTP/70/33.
L
Physics B42 (1972) 5_89-606. North-Holland Publishing Company
MULTI-REGGEON EFFECTS FOR INCLUSIVE REACTIONS IN THE PIONIZATION REGION J. C. BOTKE * Dcparl~zc~ll qf Applic, d 3Iathc~alics a~d Theorelica[ Physics. Univ~,rsilv qf Cambridge. Etlgla~zd
Received
6 January
1972
Abstract: We determine the non-leading behaviour of the inclusive differentia] cross section (IDCS) in the pionization region by considering a number of GribovFeynman graphs which are related to the IDCS through Mueller's generalized optical theorem. We find in addition to "normal" regge cut dependence, (ln s) -1, a dependence of the form S (c°nst)p2± (ln s) -1 which has no analogy in two-body scattering.
1. I N T R O D U C T I O N In a r e c e n t p a p e r by M u e l l e r [1], a g e n e r a l i z e d o p t i c a l t h e o r e m was p r o p o s e d which r e l a t e s the i n c l u s i v e d i f f e r e n t i a l c r o s s - s e c t i o n (IDCS) for the p r o c e s s 1 + 3 - - 2 + X to the M 2 - ( p l + P 3 - p,2)2 d i s c o n t i n u i t y of the c o n n e c t e d p a r t of the t o t a l l y f o r w a r d , u n p h y s i c a l a m p l i t u d e of fig. 1. In this p a p e r we want to show that this p i c t u r e is c o n s i s t e n t with the r e s u l t o b t a i n e d by a s u m m a t i o n of g r a p h s of the G r i b o v - F e y n m a n [2] type. In p a r t i c u l a r , the c l a s h i n g c o n t i n u a t i o n p r e s c r i p t i o n s a s s o c i a t e d with p h y s i c a l r e g i o n L a n d a u s i n g u l a r i t i e s [3], which a r e n e c e s s a r y in o r d e r t o p r e s e r v e the r e a l i t y of the c r o s s - s e c t i o n , do not c a u s e a n y t r o u b l e . A f t e r d i s c u s s i n g s o m e g e n e r a l f e a t u r e s of the p r o b l e m in s e c t . 2, we w i l l in the f o l l o w i n g s e c t i o n s , c o n s i d e r the a s y m p t o t i c b e h a v i o u r of a n u m b e r of g r a p h s in o r d e r to e x t r a c t the n o n - l e a d i n g b e h a v i o u r of the IDCS in the p i o n i z a t i o n r e g i o n . One i n t e r e s t i n g f e a t u r e of this w o r k is that in a d d i t i o n to the e x p e c t e d " n o r m a l " R e g g e c u t s , which b e h a v e a s y m p t o t i c a l l y as ( l n s ) -1 we find a c l a s s of g r a p h s which a p p r o a c h s(COnst)p2 a ins * Work supported by the Science Research Council.
590
J. C. Botke, Alulti-Reggeon effects
,M 2
3
•
1'
Fig. i. in e x a c t a n a l o g y with the r e s u l t of D r u m m o n d [4] for the 2 - 3 p a r t i c l e p r o c e s s . T h i s is to be c o m p a r e d with an i n v e r s e - s q u a r e r o o t b e h a v i o u r a s s o c i a t e d with n o r m a l m e s o n t r a j e c t o r y e x c h a n g e .
2. K I N E M A T I C S AND G E N E R A L DISCUSSION Our discussion will be concerned with amplitude F defined to be the connected part of the complete amplitude for the process 1 + 2 + 3 ~ 1' + 2' + 3' We defined the quantities,
sij : (pi+Pj) 2~- 2Pi.P j ,
(1)
q i = P'i - P i ,
(2)
M 2 = (pl+P2+P3)2~- s12+s23+s13,
(3)
and s12s23 -
s13
(4)
T h u s , for the i n c l u s i v e p r o c e s s 1 + 3 ~ 2 + X , s13 is the t o t a l e n e r g y and s12 , s23 a r e m o m e n t u m t r a n s f e r s . In a l l that f o l l o w s , we will c o n f i n e o u r a t t e n t i o n to the p i o n i z a t i o n r e g i o n d e f i n e d by Sl 3 ~ oo ,
(5)
s12, s23 ~ _ oo ,
(6)
with
s13
1,
(7)
fixed .
(8)
M2 and
O u r p r o g r a m will be to s t u d y the a s y m p t o t i c b e h a v i o u r of the G r i b o v F e y n m a n g r a p h s c o r r e s p o n d i n g to the e x c h a n g e of 2, 3 a n d 4 r e g g e o n s .
J. C.Botke, Mulli-Reggeo~ cJfec'ls
591
To t h i s end, it w i l l b e u s e f u l to d e c o m p o s e a l l v e c t o r s into two v e c t o r s l y i n g in and n o r m a l to the p l a n e of P l and P3 [51. We f i r s t d e f i n e the v e c t o r s m2 P l - +g13 P3 ,
Pl
m2 03 = P 3 - s l ~ P l
(9) (10)
,
s u c h t h a t p2 _~ p32 _~ 0. T h e n P2, f o r i n s t a n c e , c a n be w r i t t e n s23 s12 P2 : S l 3 p l + s i 3 p 3 + p 2 ~ w h e r e P 2 ± i s a s p a c e l i k e v e c t o r n o r m a l to the p l a n e o f P l p o s e t h e m a s s s h e l l c o n s t r a i n t on t h i s r e l a t i o n , we find
(11) andP3.
2 2 2 P2 : ?7 + P 2 1 = m
If we i m -
(12)
In g e n e r a l , the c o m p l e t e 3 - 3 a m p l i t u d e c a n b e c o n s i d e r e d a s an i n f i n i t e s u m of F e y n m a n g r a p h s . G r i b o v ' s a p p r o a c h is to c o n s i d e r s e l e c t i v e i n f i n i t e s u m s of g r a p h s w h i c h p r o d u c e g e n e r a l i z e d F e y n m a n g r a p h s c o n t a i n i n g complete off-shell two-bodyamplitudes, A ( s , l , m~). It is then a s s u m e d t h a t t h e d y n a m i c s is s u c h t h a t t h e s e t w o - b o d y a m p l i t u d e s a r e m a x i m a l in t h e l i m i t !s I ~ ~, t / s ~ 0 and m 2 / s ~ O. In t h i s l i m i t , t h e R e g g e h y p o t h e s i s implies 2
A ( s , t, m i) = g(l, m3,
(13)
where for s ~ +% 1 f d l ~ l G ( l , l) si G(t, s) : - 4~
(14)
with the signature factor
e - i ~ l +.r ~l-
s i n ?rl
(15)
T h e l i n t e g r a t i o n l i e s e n t i r e l y to the r i g h t of the s i n g u l a r i t i e s of G(l, l). F o r t h e l i m i t s ~ - ~ , we r e p l a c e
s l~
"r(-s) l
(16)
in eq. (14). T h e f u n c t i o n s g and g ' a r e the f a c t o r i z e d o f f - s h e l l r e s i d u e f u n c t i o n s w h i c h a r e a s s u m e d to h a v e o n l y r i g h t - h a n d c u t s and to f a l l off a s 1m21 ~ oo. In t h e s u b s e q u e n t s e c t i o n s , we w i l l d e f i n e m o r ? p r e c i s e l y t h e b e h a v i o u r r e q u i r e d . A l s o , we w i l l n e e d the d i s c o n t i n u i t y of A a c r o s s the s cut w h i c h is obtained by making the replacement ~l ~ ~l(1 - ~-e i~l) = - 2i in eq. (14).
(17)
d. C. Bolkc, Multi-Reggco~ effccts
592 If t h e r i g h t - m o s t
s i n g u l a r i t y of G(l, t) i s a R e g g e p o l e ,
G
1
l- a(t)
(18)
the G r i b o v - F e y n m a n g r a p h r e d u c e s to the u s u a l r e g g e o n e x c h a n g e d i a g r a m . W e c a n , h o w e v e r , i n c l u d e m o r e c o m p l e x d e p e n d e n c e in G to i n c o r p o r a t e cuts as well. F i n a l l y , s i n c e we a r e c o m p u t i n g a c r o s s s e c t i o n , it i s n e c e s s a r y to get t h e p h a s e of e a c h g r a p h c o r r e c t . To s a v e the r e a d e r m a n y f r u s t r a t i n g h o u r s w e w i l l s t a t e t h e w e l l known but e a s i l y f o r g o t t e n r u l e [6] w h i c h is to a s s o c i a t e a (+i) with e a c h A and e a c h p r o p a g a t o r , a ( - i ~ ) with e a c h v e r t e x , and a n o v e r a l l ( - i ) with e a c h g r a p h . We now p r o c e e d to the g e n e r a l i z e d o p t i c a l t h e o r e m , da - ---c~ P20 d3p~
1 r(++-)l 2 i s 1 3 AM2
(19)
iqi=0
w h e r e A , I 2 d e n o t e s the M2 d i s c o n t i n u i t y of the u n p h y s i c a l a m p l i t u d e F w h o s e a r g u m e n t s r e f e r to the d i s p o s i t i o n s of s13 , M 2 and s l , 3, r e l a t i v e to t h e i r cuts. The other invariantssl2, s 2 3 a r e both n e g a t i v e and thus r e m o v e d f r o m t h e i r c u t s . R a t h e r than w o r k i n g d i r e c t l y with t h i s a m p l i t u d e , h o w e v e r , w e w i l l find it c o n v e n i e n t to c o n s i d e r t h e p h y s i c a l a m p l i t u d e F ( + + + ) w h i c h i s r e l a t e d to the r e q u i r e d d i s c o n t i n u i t y by the i d e n t i t y , AM2 F ( + + - ) = (1 - A s l , 3 ,) AM2 F ( + + + )
(20a)
: A~I 2 ( 1 - A S l , 3 ,) F ( + + + ) .
(20b)
In p r a c t i c e , the M 2 d e p e n d e n c e of a g i v e n g r a p h wit[ not be e x p l i c i t , n o r f o r that m a t t e r w i l l the S l , 3, d e p e n d e n c e , w h i c h p r e v e n t s us f r o m e v a l u a t i n g t h e d i s c o n t i n u i t y v i a an e x p l i c i t a n a l y t i c c o n t i n u a t i o n . I n s t e a d , we u s e the f o r m a l p r e s c r i p t i o n [6] of s l i c i n g the g r a p h in a l l p o s s i b l e w a y s and r e p l a c i n g e a c h cut a m p l i t u d e o r p r o p a g a t o r by i t s d i s c o n t i n u i t y c o n s i s t e n t with the two rules: (i) T h e two a m p l i t u d e s w h i c h r e s u l t f r o m a p a r t i c u l a r p a r i t i o n of F m u s t b o t h be c o n n e c t e d s i n c e t h e y a r e i n d i v i d u a l l y s u p p o s e d to be a m p l i t u d e s f o r the i n c l u s i v e p r o c e s s . (ii) If m o r e than one s l i c e is p o s s i b l e , the t o t a l d i s c o n t i n u i t y i s found by a d d i n g the c o n t r i b u t i o n of t h e v a r i o u s p a r t i t i o n s (P) a c c o r d i n g to the r u l e , F(+++) :
l E fL(+,+,+)(AM2f) p=l
pfR(+,-,+)
,
(21)
w h e r e f L and f R r e f e r to the a m p l i t u d e s w h i c h r e s u l t on the l e f t and r i g h t of a g i v e n p a r t i t i o n d e n o t e d by (AM2f) p. We can m a k e one f u r t h e r s i m p l i f i c a t i o n by n o t i n g t h a t the p a r t i t i o n s of F i s o l a t e the d e p e n d e n c e on s13 and Sl, 3, and in f a c t f L ( f R ) d o e s not d e p e n d on Sl,3,(s13). T h e t o t a l d i s c o n t i n u i t y , eq. (20), i s then
J. C. Bolke. MuIti-Reggeon e.ffecls
593
l AM2F(++-)
:
~ fL(+)(AM2f)pfR(+) p:l
(22)
w h e r e we have u s e d fR(-) = fR(+) .
(23)
We can now e a s i l y s e e the p h y s i c a l i n t e r p r e t a t i o n of this r e l a t i o n . If we w e r e to e v a l u a t e the IDCS d i r e c t l y , we would c o m p u t e
~ f d(~nF(+)F(-)
(24)
n w h e r e F i s the c o m p l e t e a m p l i t u d e for the p r o c e s s 1 + 3 ~ 2 + X n. If we exp a n d F in t e r m s of a s u m of c o n t r i b u t i o n s f r o m v a r i o u s g r a p h s f , a t y p i c a l c o n t r i b u t i o n will be of the f o r m f dg)n f L (+) J R ( - ) Now, s i n c e
(AM2f) p
(25)
of eq. (22) c o n v e r t s loop i n t e g r a t i o n s into p h a s e s p a c e
i n t e g r a t i o n s , eq. (22) has j u s t the f o r m of eq. (25). In p a r t i c u l a r , s i n c e p h y s i c a l r e g i o n L a n d a u s i n g u l a r i t i e s a r i s e in the s a m e m a n n e r in both r e l a t i o n s , they c a u s e no t r o u b l e in a s s o c i a t i n g t h e m 2 d i s c o n t i n u i t y of F ( + + - ) with the IDCS. We now p r o c e e d with the e v a l u a t i o n of s p e c i f i c g r a p h s . F o r t u n a t e l y , a l t h o u g h we c a n n o t c o m p u t e a n u m e r i c a l r e s u l t for e a c h g r a p h , it is p o s s i b l e to d e t e r m i n e w h e t h e r or not a s p e c i f i c g r a p h will v a n i s h . T h i s v a n i s h i n g o c c u r s a s a c o n s e q u e n c e of one o r m o r e of the f o l l o w i n g t h r e e m e c h a n i s m s : (i) T a k i n g AM2 v i o l a t e s r u l e (i) a b o v e . (ii) T h e a m p l i t u d e h a s a v a n i s h i n g M 2 d i s c o n t i n u i t y which will be r e f e r r e d to a s a k i n e m a t i c z e r o (KZ). (iii) T h e a m p l i t u d e F(+ + +) v a n i s h e s in the s a m e m a n n e r as the w e l l k n o w n A F S d i a g r a m [6]. T h i s m e c h a n i s m will be d i s c u s s e d in d e t a i l in s e c t . 4 w h e r e it will he shown that t h i s is d e p e n d e n t on the a s s u m p t i o n c o n c e r n i n g the m a s s d e p e n d e n c e of the o f f - s h e l l a m p l i t u d e s . F o r want of a b e t t e r n a m e , we will r e f e r to this as a c l o s e d c o n t o u r z e r o (CCZ).
3. T W O - R E G G E O N GRAPHS A l t h o u g h the t w o - r e g g e o n c o n t r i b u t i o n h a s b e e n c o n s i d e r e d p r e v i o u s l y [7], we will d i s c u s s it ip s o m e d e t a i l f o r the i n s i g h t it a f f o r d s into the t e c h n i c a l a s p e c t s of this s o r t of c a l c u l a t i o n . In p a r t i c u l a r , we want to e l a b o r a t e on the m e c h a n i s m s d i s c u s s e d in s e c t . 2 by w h i c h c e r t a i n g r a p h s vanish. T h e r e l e v a n t g r a p h s a r e i l l u s t r a t e d in fig. 2 w h e r e the l a b e l s (abc) c o r r e s p o n d to p e r m u t a t i o n s of (12 3). C o n s i d e r i n g fig. 2b f i r s t , we s e e that t h e s e v a n i s h as a c o n s e q u e n c e of r u l e (i). Of t h o s e r e m a i n i n g , only the c a s e
J. C. Bolke, Multi-Reggeon effects
594
hlvla
b
b
¢
c
"-t
a b
"
~
c
a b c
I
(a)
(b)
Fig. 2. T w o - r e g g e o n g r a p h s .
Fig. 3.
b = 2 is non-vanishing; the remaining possibilities vanishing as a result of kinematic r e s t r i c t i o n s (rule (ii)). We will consider first the case b ¢ 2. The Gribov-Feynman graph for this situation is given in fig. 3 where of course the choice b = 3 instead of b = 1 is a r b i t r a r y . If we define Sudakov p a r a m e t e r s [5] for the loop momentum k we have k = otPl+fip3+g,
(26)
k 2 -~ c~/3s13 + K2 ,
(27)
where
and d 4 k = ½ I s 1 3 I do~ d / 3 d 2 ~ .
(28)
(½P3±k)2-m2+ic=K2+(}m2+~Sl3)(}±fi)-rn2+iE,
(29)
The propagators
become
and the energies S l = 2 P l " (½P3 + k ) = ~~s 1 3 + c~rn2 + $ s 1 3 ,
(30a)
s 2 = 2P2 ' ( ½ P 3 - k) = ½s23 - c~s12 - $ s 2 3 - 2P2 ± . K .
(30b)
Keeping the energies determines
large
and the "masses"
~ O (1) .
f i n i t e a s s 1 3 - - % 77 f i x e d
(32)
Making the change of variables x = s13 a y = [5 , we have
,
(33) (34)
595
d. C. Botke, Multi-Reggeon effects
(½P3 ± k ) 2 - r n 2 + i e
~- K2+ (½rn2 ±x)(½
±y)-rn2+ic
and
s 1 : s13 (½+Y) ,
(36a)
s2 = s23 (½-Y) • The amplitude for this process is
(36b)
F(+++) =-i~ 2 f d4k AIA2 (2v)4 [(½P3 + k)2- m2 +i~]2[(½p3- k)2- m2+ie] 2
(37)
which becomes 2
F(+++) = - i •
~ dlj {J}~jGj(lj,
0)~(0, m 2, m2)}
j=l
(38)
× (s13)/1 (- s23 )/2 M l l l 2 , where
Mhl2 : ½x2 (½~)2f dxdyd2K (2~) 4 g l g 2 ( Y + ½ ) l l ( y _ ½ ) / 2
× {[K 2-m2+(~m2+x)(½+y)+iE]2[K 2-m2+(½m2-x)(½-y)+ic]2}-I
(39)
T h e functions gl and g2 are a s s u m e d to have only right-hand cuts which are disposed in the s a m e m a n n e r as the propagators with regard to displacements off the real axis of integration. It is then a simple matter to show I that if IYl -" ~, all the singularities of the integrand as a function of x lie on one side of the r e a l x a x i s . If, as assumed, t h e g ' s d e c r e a s e sufficiently rapid as a function of the " m a s s e s " , we can close the x integration in the 1 appropriate half-plane to get a zero result. Thus, we must have l Yl < ~. However, from eq. (36), we see that this implies s 2 < 0 and consequently the M 2 discontinuity of the amplitude is zero. We now turn to the situation indicated in fig. 4, and proceed as before. The propagators become
(½P2 ± k ) 2 - m2 + ie = (K ± sp2_L) - m
~ ½±
½±
+ie , (40)
and the energies 1
l
1
1
s I = 2P I . (~P2 +k) = ~s12 +~m 2 +fist3 ,
s 2 : 2P3"(~p2- k) = ~s23- ~ s 1 3 - t i m 2
•
(41a)
(41b)
Again, we r e q u i r e that the energies a r e l a r g e and the " m a s s e s " finite as s --~ 0% q fixed with the result
J. C. Botke. Mulli-Regge'on ei~tecls
596
I
I
Fig. 4. (42)
t'~
0
(43)
~?
(s23) •
D e f i n i n g new v a r i a b l e s
(44)
~lx = a S l 2 , 77y = - 3 s 2 3
(45)
,
we find (½P2 + k ) 2 - m 2 + i c
= ( K + ~ p 2 ± ) 2 - rn2 +~(~1 + x)(½ :~ y ) + i ~ ,
(46)
and s I = (-Sl2)(y-
~) ,
s 2 = (-s23)(x-½) The amptitude,
2
.
(47a) (47b)
in this c a s e , b e c o m e s
dlj
F(~++) : -i [l f. / 2~ ~J Gj(~, 0)~)(0, m2, .~2/} j=l × (-s12 )11 (-s23)12N1112(V) ,
(48)
where
N l l l 2 = }~)~2 (½v)2 f d x d y d 2 g g l g2(Y- ½)/1 ( x - ½)12 (27r)4
× {[(K+½P2±) 2 - m 2 + ~ ? ( ½ + x ) ( ~ - y ) + i e ] 2 × [(K- ~p2±) 2 - m 2 + r/(~ - x) (~ + y) + i~]2}-1
(49)
597
J. C. Bolke. Mulli-Reg$eo~z eJfecls
1
With r e g a r d to j u s t the p r o p a g a t o r s and the g ' s we find a g a i n that if lit'l ~, the s i n g u l a r i t i e s a r e alt on one s i d e of the r e a l x a x i s as b e f o r e . H o w e v e r in this c a s e , in view of eqs. (47), t a k i n g the M 2 d i s c o n t i n u i t y i m p o s e s c o n s t r a i n t s on both x and y which p r e v e n t s the c l o s i n g of e i t h e r c o n t o u r and the r e s u l t is n o n - z e r o . If we a s s u m e that the r i g h t - m o s t s i n g u l a r i t i e s of the G ' s a r e s i m p l e p o l e s we find a c o n t r i b u t i o n to the IDCS of the f o r m
g l g2
(_ s12) ~1(0) (_ s23) c~2(0) s13 --
A~ lC~207)
,
(50)
w h i c h if we a s s u m e p o m e r o n d o m i n a n c e b e c o m e s g2)?Nc~l:Ot2=l(r]) .
(51)
T h e f u n c t i o n .~ is a p o s i t i v e , r e a l f u n c t i o n of p2.c : m 2 - ~ d e f i n e d by eq. (49) 1 1 with the r e s t r i c t i o n x > s a n d y : 5. We thus o b t a i n the e x p e c t e d s e a l i n g result.
4. T H R E E - R E G G E O N
GRAPHS
T h e r e is a l a r g e n u m b e r of s u c h g r a p h s to be c o n s i d e r e d and to p r o v i d e s o m e o r d e r , we will c l a s s i f y t h e m a c c o r d i n g to the n u m b e r of r e g g e o n -
i
-If__ _ I_
X
(a)
X I-
i (b)
m
(c)
-%
l!i
(d)
(¢)
(f)
Fig. 5. Three-reggeon graphs with no reggeon-reggeon couplings.
Pl PPa
iA,
iA~ ~
-k ~
Fig. 6.
_
-
pf
~L
J. C. Botke, Multi-Reggeon effects
598
ii
"V
--
(a)
--
(b)
A_-
A
(d)
t
(c)
-N
(e)
(f)
D~ _!I (g)
Fig. 7. Three-reggeon graphs with one reggeon-reggeon coupling. I
F i g . 8.
oi
-
(a)
/J
/~_/}
(b)
(c)
U) (d)
Fig. 9. Three-reggeon graphs with two reggeon-reggeon couplings,
d. C. Botke, Multi-Reggeon effects 11
599
el
I
Fig. i0. Gribov-Feynman
graph for fig. 9b (i 2 3) with two of the six possible M 2 partitions.
f
\ "
Pt
(a) •
'Pz
P3
(b)
Fig. ii. Three-reggeon
y (a)
graph with three reggeon-reggeon couplings. partitions are indicated in (b).
Three
@ 1
2
2
(b)
3
3
Fig. 12. Three-reggeon graphs with a triple-reggeon coupling.
Fig. 13.
typical
J. C. Bolke. Mzdli-Reggc'on e:l'Jects
500
r e g g e o n v e r t i c e s . T h e s e a r e r e p r e s e n t e d s c h e m a t i c a l l y in figs. 5, 7, 9, 11 a n d 12 w h e r e e a c h g r a p h is m e a n t to r e p r e s e n t all s i m i l a r g r a p h s o b t a i n e d by p e r m u t i n g p a r t i c l e s and v e r t i c e s . B e c a u s e t h e r e is s u c h a l a r g e n u m b e r of g r a p h s , we will not p r e s e n t a d e t a i l e d a n a l y s i s but only quote the e s s e n tial results. C o n s i d e r i n g fig. 5 f i r s t , we s e e i m m e d i a t e l y that, as a c o n s e q u e n c e of r u l e (i), 5a and 5b do not c o n t r i b u t e . In fact 5c and 5d a l s o fail to c o n t r i b u t e f o r the s a m e r e a s o n a l t h o u g h it is p e r h a p s n e c e s s a r y to c o n s t r u c t the a p p r o p r i a t e G r i b o v - F e y n m a n g r a p h s to s e e t h i s . T h e r e m a i n i n g g r a p h s 5e a n d 5f both v a n i s h as a r e s u l t of c l o s e d c o n t o u r z e r o s . To i l l u s t r a t e m o r e p r e c i s e l y what is m e a n t , we will c o n s i d e r in d e t a i l a p a r t i c u l a r g r a p h of the s e t 5e, n a m e l y that i l l u s t r a t e d in fig. 6. P r o c e e d i n g as b e f o r e , the p r o p a gators are ( P l +/e)2 - m 2 + ie = g2 + at3Sl3 + c~m2 +t3s13 + ie ,
(52a)
( P2 - k)2 - DI2 + I'E =
(52b)
K2+Ceg3SI3-C~SI2-t3S23-2P2±'K+iE,
a n d the e n e r g i e s , s t = s 3
:
s12
(53a)
,
(535)
s2 = 2P3 ' ( P 2 - k) = s 2 3 - C~Sl3- i3rn2 w h i c h d e t e r m i n e s the p a r a m e t e r s ,
(55)
(56) M a k i n g the c h a n g e of v a r i a b l e s x = c~s 12 ,
(56a)
y = i3s 13 ,
(56b)
gives
(Pl+k)2-m2+ic
~ g2+y+ie ,
(P2-k)2-m2+ie
~ K2 - 2 p 2 ± ' , ~ - x + i e
(57a) ,
(57b)
and s2 = - s 2 3 ( ' ; - 1 )
•
(58)
T h e i m p o r t a n t p o i n t to n o t i c e is that the y d e p e n d e n c e is c o n t a i n e d e n t i r e l y in eq. (57a) and the c o r r e s p o n d i n g g f u n c t i o n s . S i n c e the s i n g u l a r i t i e s in y a r e thus a l w a y s b e l o w the r e a l a x i s , and p r o v i d e d the i n f i n i t e s e m i - c i r c l e g i v e s no c o n t r i b u t i o n to the i n t e g r a l , we can c l o s e the c o n t o u r in the u p p e r h a l f p l a n e to get a z e r o r e s u l t . T h i s . in fact, p r o v i d e s the q u a n t a t i v e u p p e r
J. C. B o t t e c . A l ~ d l i - R c g g c o , e
601
Table i Fig.
Non-zero
Disconnected
5(a-d) 5e, f
Kinenmtie
zero
Closed contour zero
all all all
7a
7(b-d) 7e 7f 7g
(i 23)
9a 9b 9c
(I 23)
all
(13 2)
(2 13) all (213)
(123), (132) all
11
(I 3 2) (132)
(2 13) (123), (213)
all
12a
all
(12 3)
12b
(13 2), (2 13)
bound on t h e g ' s m e n t i o n e d in s e c t . 2. In t a b l e 1, we h a v e s u m m a r i z e d these results. T h e g r a p h s with o n e r e g g e o n - r e g g e o n v e r t e x a r e i l l u s t r a t e d in fig. 7. A s i n d i c a t e d in t a b l e 1, a ll v a n i s h with the e x c e p t i o n of t h e (1 2 S) (and (S 2 1)) o r d e r i n g of fig. 7e. T h e G r i b o v - F e y n m a n g r a p h is i n d i c a t e d in fig. 8 w h e r e the d a s h e d l i n e s d e n o t e the p o s s i b l e M 2 p a r t i t i o n s . T h e r e is one p o i n t i n t r o d u c e d by t h i s g r a p h w h i c h we want to d i s c u s s b r i e f l y . T h e e n e r g i e s of A 1 l a nd A 3 a r e s 1 = s 2 = e S l 2 and as s12 ~ - ~ , we e n c o u n t e r t h e l e f t - h a n d c u t s of the t w o - b o d y a m p l i t u d e s w h i c h s e e m s to c o n t r a d i c t the s u p p o s i t i o n ( s e c t . 2) that the t o t a l a m p l i t u d e h a s c u t s in a p a r t i c u l a r s i j only f o r p o s i t i v e v a l u e s of that v a r i a b l e . T h e r e s o l u t i o n of t h i s d i f f i c u l t y is that as s12 ~ a c o r r e s p o n d i n g e n e r g y s, d e f i n e d by s = ~ m 2 - l - s 1 2 , is a p p r o a c h i n g p o s i t i v e i n f i n i t y and the cut we e n c o u n t e r is the p o s i t i v e e n e r g y cut a s s o c i a t e d with t h i s e n e r g y . T h i s is, in f a c t , j u s t a p a r t i c u l a r f i n al s t a t e s u b e n e r g y of the i n c l u s i v e a m p l i t u d e and c o n s e q u e n t l y , is e v a l u a t e d a b o v e o r b e l o w its cut in the s a m e s e n s e as M 2. T h e g e n e r a l f o r m of t h i s c o n t r i b u t i o n is
(-s23)~2(0) f d2K !~I(K) 2
(-sI2)2~I(K2)-IN(~2(0)
s13
7, K)
(59)
'
w h e r e we h a v e a s s u m e d c~1 = c~3. T h e f u n c t i o n N c a n n o t be e v a l u a t e d in c l o s e d f o r m but it d o e s h a v e the u s e f u l p r o p e r t i e s that it is r e a l and p o s i t i v e We can a p p r o x i m a t e the a s y m p t o t i c b e h a t r i o u r by s e t t i n g K = 0 e v e r y w h e r e e x c e p t in the e x p o n e n t with the r e s u l t , (_ s23 )c~2(0) (_ s12)2c~1 (0)-1
s13 ln(- s12 )
15c~1(o) i2 2~"0)i~
N(c~2(0)' 77, 0) ,
(60)
J. C. Botke, Multi-Rcggeo~t effects
602
which
if we assume
pomeron
dominance
becomes
(const)~?N0?, K = 0)
(61)
In(- s12) Including the (3 2 i) ordering, the total contribution is
(const)~N(~) [
L__+
kln(-s 12)
L ~
In(- s23)J "
(62)
It i s i n t e r e s t i n g to note t h a t a l t h o u g h t h e s e e f f e c t s r e s u l t f r o m a r e g g e o n r e g g e o n cut of the a m p l i t u d e F ( + + + ) , the i n c l u s i v e a m p l i t u d e s , in t h i s c a s e , do not c o n t a i n t h e s e c u t s s i n c e t h e y i n d i v i d u a l l y c o n t a i n o n l y s i n g l e r e g g e o n exchanges with, however, variable momentum transfer. T h e next g r o u p a r e t h o s e with two r e g g e o n - r e g g e o n v e r t i c e s w h i c h a r e i l l u s t r a t e d in fig. 9 a n d s u m m a r i z e d in t a b l e 1. T h e g r a p h of fig. 9d is not a l l o w e d b e c a u s e the " e n e r g y " a s s o c i a t e d with t h e r c e n t e r r e g g e o n in n e v e r l a r g e . T h e G r i b o v - F e y n r n a n g r a p h f o r fig. 9b (1 2 3) i s g i v e n in fig. 10 w h e r e two of the s i x p o s s i b l e M 2 p a r t i t i o n s a r e i n d i c a t e d . A s y m p t o t i c a l l y , t h i s h a s a f o r m s i m i l a r to eq. (59), n a m e l y ,
(-
s23)a3(°)f
d2t~(-
s12) al(K2)+a2(K2)-I N ( a l , c~2, ~, K) .
(63)
s13
In t h i s c a s e N, w h i c h is a s u m of t e r m s c o r r e s p o n d i n g to the p o s s i b l e p a r t i t i o n s , i s c o m p l e x r a t h e r t h a n r e a l a s a c o n s e q u e n c e of p h y s i c a l r e g i o n L a n d a u s i n g u l a r i t i e s . T h e s e a r i s e in the t e r m c o r r e s p o n d i n g to p a r t i t i o n 6 in a s s o c i a t i o n with the p r o p a g a t o r s i n d i c a t e d by the a r r o w s . T h e s e n s e of continuation the s i n g u l a r i t y i s , of c o u r s e , s p e c i f i e d by the i e ' s a t t a c h e d to the p r o p a g a t o r s . T h e r e a l i t y of t h e c r o s s s e c t i o n i s r e c o v e r e d by a d d i n g t h e g r a p h in w h i c h the s i n g l e r e g g e o n c o u p l i n g with p a r t i c l e 2 f o l l o w s r a t h e r t h a n p r e c e e d s the two r e g g e o n c o u p l i n g w h i c h , a s a c o n s e q u e n c e of eq. (22), i s the c o m p l e x c o n j u g a t e of eq. (63). In the p o m e r o n l i m i t , the t o t a l c o n t r i b u t i o n h a s the f o r m of eq. (62). T h e only g r a p h with t h r e e r e g g e o n - r e g g e o n v e r t i c i e s is i l l u s t r a t e d in fig. 11 w h e r e t h r e e t y p i c a l p a r t i t i o n s a r e a l s o i n d i c a t e d . T h e l e a d i n g b e h a v i o u r of t h i s g r a p h h a s the f o r m
past
l_fd2 S13
K (_S12)al(K2)+a3(K2)-1
(_S23)a2(K2)+a3(K2)-1N(ai,~, K) (64)
w h e r e N in t h i s c a s e i s r e a l d e s p i t e t h e e x i s t e n c e of n u m e r o u s p h y s i c a l r e g i o n L a n d a u s i n g u l a r i t i e s ; the r e a l i t y b e i n g p r e s e r v e d by the l e f t - r i g h t s y m m e t r y of t h e g r a p h . We h a v e the p o s s i b i l i t y , in t h i s c a s e of a p i n c h b e t w e e n two L a n d a u s i n g u l a r i t i e s w h i c h a r i s e f r o m o p p o s i t e s i d e s of a p a r t i t i o n and thus a p p r o a c h the i n t e g r a t i o n c o n t o u r f r o m o p p o s i t e s i d e s of t h e r e a l a x i s . S i n c e the i n t e g r a t i o n c o n t o u r c a n n o t be d e f o r m e d to a v o i d the p i n c h , we d e f i n e the i n t e g r a l by r e p l a c i n g t h e r e l e v a n t p r o p a g a t o r s on o n e s i d e of the p a r t i t i o n by t h e i r c o m p l e x c o n j u g a t e s p l u s d i s c o n t i n u i t i e s w h i c h l e a d s to a p i e c e w h i c h i s a n a l y t i c p l u s a n o n - a n a l y t i c c o r r e c t i o n w h i c h p r e s e r v e s the r e a l i t y of the c r o s s s e c t i o n .
J. C. Botke , Multi-Reggeo~z ej)k,c'ts
603
In general, as discussed in ref. [4], this contribution contains two reggeon- reggeon cuts corresponding to c~1 + c~3 and c~2 + a 3 and the full traingle singularity corresponding to the pair ot 1 + c~3 and c~2 + c~3" This will be discussed more fully in sect. 5. In this case, however, since the argument of all the c~i i s K2, the reggeon-reggeon cuts are coincident with the triangle singularity. Again, assuming pomeron dominance, the leading behaviour is
(const)r/N(r/) In s13 -
(65)
The remaining graphs of this group are those of fig. 12. As one might expect, those of fig. 12a vanish and those of fig. 12b lead to a contribution of the form eq. (62) except that the sign of the contribution is negative.
5. F O U R - R E G G E O N G R A P H S We do not i n t e n d to w o r k o u r way t h r o u g h the e n t i r e s e t of f o u r - r e g g e o n g r a p h s s i n c e the m a j o r i t y w i l l l e a d to h i g h e r o r d e r c u t s e f f e c t s of j u s t the s o r t a l r e a d y d i s c u s s e d . T h e r e i s h o w e v e r , one c l a s s of g r a p h s w h i c h l e a d to a n e w t y p e of b e h a v i o u r u n l i k e any found in t w o - b o d y s c a t t e r i n g . T h e r e m a i n d e r of t h i s s e c t i o n w i l l be s p e n t d i s c u s s i n g t h e s e i d e a s . T h e g r a p h with w h i c h we a r e c o n c e r n e d , i s i l l u s t r a t e d in fig. 13. T h e n o v e l f e a t u r e i s t h a t p a r t i c l e 2 b r e a k s into a r e g g e o n - r e g g e o n p a i r r a t h e r t h a n p r o c e e d i n g d i r e c t l y t h r o u g h the d i a g r a m a s in the p r e v i o u s e x a m p l e s . A s a c o n s e q u e n c e of t h i s , the t r a n s v e r s e m o m e n t u m of p a r t i c l e 2, P2 , g e t s m i x e d up with the m o m e n t u m t r a n s f e r a s s o c i a t e d with the r e g g e o n s w h i c h u l t i m a t e l y c o n t r o l the a s y m p t o t i c b e h a v i o u r of the g r a p h . A g a i n , we w i l l d i s p e n s e with a d e t a i l e d d e r i v a t i o n and s i m p l y s t a t e the r e s u l t t h a t t h e c o n t r i b u t i o n to the IDCS h a s the f o r m ,
I= ( c o n s t ) f d2~ ( - s 1 2 ) a l + a 3 - 1 w h e r e the a r g u m e n t s
( - s 2 3 ) c ~ 2 + a 4 - 1 N ( a i , •,p2±) ,
(66)
of the a i a r e (K + ½P2±)2 .
i = 1, 3 ,
(67a)
(K - ½p2±) 2 •
i = 2, 4 .
(67b)
S i n c e the b e h a v i o u r of t h i s t y p e of f u n c t i o n h a s b e e n d i s c u s s e d in d e t a i l by D r u m m o n d [4], we w i l l o n l y o u t l i n e the a r g u m e n t . We f i r s t d e f i n e the function
IjlJ2
=S
d(-Sl2)(-Sl2)
SO
-jl-1
f~ d(- s 2 3 ) (- s 2 3 ) - J12 -
I(s12, s23) ,
(68)
SO
which has the inverse d j l dJ2 I " " I = f (2~ j lJ2(-s12)31 (-s23))2 "
(69)
J. C.Botke. Multi-Reggeon cftk'cls
604 In a d d i t i o n ,
where
we d e f i n e the p a r a m e t e r s (- s12) = X(qs13 )x ,
(70a)
(- s23) = x - l ( r l S l 3 )y ,
(70b)
eq. (4) i m p l i e s x+y
-- 1 .
(70c)
Then we have (_ S l 2 ) J l (_ s23)J2 = ~Jl-J2 (~lSl3)Xjl+YJ2
.
(71)
Defining new variables J = xJl+YJ2 '
(72a)
J = J l - J2 ,
(72b)
eq. (69) b e c o m e s I : f 2~i dJ '
(73)
(VSl3)J
with
~ ( j ) : f d ~j i aj I j l J 2
"
(74)
T h e a s y m p t o t i c b e h a v i o u r i s t h u s d e t e r m i n e d by the r i g h t - m o s t s i n g u l a r i t y of qp(d) w h i c h in t u r n d e p e n d s on the s i n g u l a r i t i e s of Ij lJ2" By i n s e r t i n g eq. (66) i n t o eq. (68), we find N ( a , t~, p2±) I j l J 2 = (const)f d2K ( j l _ o 1 1 _ a t 3 + 1 ) ( J 2 _ ~ 2 _ c~4+1 ) . (75) T h i s f u n c t i o n h a s two t y p e s of j p l a n e s i n g u l a r i t i e s . T h e f i r s t a r e the t w o r e g g e o n c u t s a s s o c i a t e d w i t h the v a n i s h i n g of the two f a c t o r s in t h e d e n o m i nator separately. The branch points are at Jl = a13 =
al+a3-
J2 = c~24 = a 2 + a 4 -
1 ,
(76a)
1 ,
(76b)
w h e r e we h a v e a s s u m e d eli(l) = cli+bil. T h e s e c o n d t y p e o c c u r s f o r j l ' c~13 , J2 • a 2 4 w h e n b o t h f a c t o r s v a n i s h s i m u l t a n e o u s l y . T h e e q u a t i o n f o r the b r a n c h p o i n t is the p a r a b o l a
p2
(77)
( r l ± ;~2 )2 = _ 2 - ' where
2 al +a3--]1rI : bl+b 3
1
,
(78)
J. C. Botkc , Multi-Rcggeon ql?}'cls
605
and 2
(12 + a 4 - J2 - 1
r2 :
b ~ 1,4
(79)
T h e s i n g u l a r i t i e s of O(J) a r e d e t e r m i n e d by the p i n c h e s of the j i n t e g r a t i o n of eq. (74) w h i c h o c c u r at (1)
J = xc~13+YCi24 ,
(80)
(2)
J : XJlL+ YJ2L '
(81)
and
a n d J2L a r e d e t e r m i n e d by t h e i n t e r s e c t i o n of the c u r v e with the b r a n c h cut eq. (77). A s shown by D r u m m o n d [4], the f i r s t t y p e of p i n c h i s i n e f f e c t i v e with the r e s u l t that the l e a d i n g b e h a v i o u r i s g i v e n by the s e c o n d w h i c h if we a s s u m e b 1+ b 3 = b 2 + 54 g i v e s
where)lL
J=Xjl+YJ2
N(p 21) ,
Sl 3 I n s 13
(82)
where d : xc~13 + yc~24 +xy (b 1 + b3)P~ ± .
(83)
If t h e q u a n t u m n u m b e r s of p a r t i c l e 2 a r e s u c h that it c a n c o u p l e to two p o m e r o n s , t h e n t h e l e a d i n g b e h a v i o u r is
S l~Y3(bl +b 3 ) p2~ lnsl 3
N(P2±) .
(84)
A c t u a l l y , t h e r e h a s b e e n n o t h i n g in the a r g u m e n t to p r e v e n t us f r o m a s s u m i n g p a r t i c l e 2 i s a s y s t e m of p a r t i c l e s which c o u p l e s to the two p o m e r o n s in w h i c h c a s e (84) s t i l l a p p l i e s w i t h P 2 ± , the t o t a l t r a n s v e r s e momentum. In the p i o n i z a t i o n r e g i o n xy ~ ~ and with b i ~ 1 GeV - 2 the e x p o n e n t of eq. (84) i s a p p r o x i m a t e l y ½P22±_which thus r e s e m b l e s a r e g g e p o l e with i n t e r s e p t 1 1 a n d s l o p e ~ GeV - 2 . In s u m m a r y , t h e t o t a l IDCS h a s the f o r m dcr
i
i
P20 d3p2 = Fl(r/)+ F2(r/) [in (- s12) +ln (-s23) ] 2
2x3 P2± F 3 (r/) s13 +in(sl3 ) + F 4 0 7 ) i n s 1 3
+-..
,
where F407) vanishes if particle 2, or system 2, cannot couple to two pomerons. All the Fi(~l) are real and FI(~/) is positive. The signs of the remaining F i are less well determined and may be negative.
(85)
506
d. C. B o t k e . M u l l i - R e g g e o n ¢~'fects
T h e a u t h o r w o u l d l i k e to t h a n k P r o f e s s o r J. C. P o i k i n g h o r n e f o r s u g g e s t ing t h i s t o p i c and f o r h is c o n t i n u e d i n t e r e s t . I a m a l s o i n d e b t e d to I. T. D r u m m o n d for useful c o n v e r s a t i o n s c o n c e r n i n g this work.
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