- Email: [email protected]
The zero in the three Pomeron vertex
Volume 38B, n u m b e r 2
PHYSICS
THE
ZERO
IN
THE
LETTERS
THREE
POMERON
2 4 J a n u a r y 1972
VERTEX
P. GODDARD * a n d A. R. W H I T E
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England Received 20 D e c e m b e r 1971
Multiparticle unitarity is used to show that a reflection of the t w o - P o m e r o n cut in the three P o m e r o n v e r t e x provides a zero which r e m o v e s the analogue of the G r i b o v - P o m e r a n c h u k fixed-pole in the forward limit giving the inclusive c r o s s - s e c t i o n . This c r o s s - s e c t i o n then has a l i n e a r zero in the momentum t r a n s f e r a r i s i n g from the signature factor.
A n u m b e r of r e c e n t p a p e r s [ 1 - 9 ] h a v e d i s c u s s e d the t r i p l e P o m e r o n c o n t r i b u t i o n to t h e i n c l u s i v e p r o c e s s a + b ~ c + a n y t h i n g ( s e e fig. 1). In p a r t i c u l a r it h a s b e e n s h o w n t h a t if t h i s c o n t r i b u t i o n i s to d o m i n a t e in the a p p r o p r i a t e l i m i t ( m i s s i n g m a s s M 2 ~ ~ , S / M 2 ~ oo) t h e n c o n s i s t e n c y w i t h conslanl total cross sections (that is the assumption that total cross-sections are also Pomerondominated) requires azeroin the triple Pomeronvertex. T h e o c c u r r e n c e of s u c h a z e r o h a s b e e n d e m o n s t r a t e d in t h e d u a l m o d e l [3] a n d in a c e r t a i n f i e l d - t h e o r e t i c m o d e l [4]. T h e z e r o c a n be r e g a r d e d a s o c c u r r i n g a t a n o n s e n s e w r o n g - s i g n a t u r e p o i n t (of o n e of the P o m e r o n s r e l a t i v e to t h e h e l i c i t i e s of the o t h e r two). It m i g h t t h e r e f o r e b e e x p e c t e d t h a t the z e r o will b e c a n c e l l e d by a " f i x e d - p o l e " of the G r i b o v - P o m e r a n c h u k t y p e , w h e n t h i r d d o u b l e s p e c t r a l f u n c t i o n e f f e c t s , l a c k i n g in the m o d e l s m e n t i o n e d , a r e i n c l u d e d [7, 8]. A s A b a r b a n e l a n d G r e e n [6] h a v e p o i n t e d out, t h e r e s i d u e of s u c h a f i x e d - p o l e m u s t h a v e a f u r t h e r z e r o . T h i s i s b e c a u s e t h i s r e s i d u e d e t e r m i n e s the d i s c o n t i n u i t y of t h e two P o m e r o n cut in t h e f o u r p o i n t f u n c t i o n a n d u n i t a r i t y r e q u i r e s t h i s d i s c o n t i n u i t y to v a n i s h a t t h e b r a n c h - p o i n t [ 1 0 - 1 2 ] . H o w e v e r , t h e a r g u m e n t of A r b a r b a n e l a n d G r e e n i s not s u f f i c i e n t l y d e t a i l e d to d i s t i n g u i s h in w h i c h v a r i a b l e of t h e three Pomeronvertex f u n c t i o n g ( l l , 12, to) ( s e e fig. 2) t h i s z e r o o c c u r s , o r to d i s c u s s i t s s t r e n g t h . In f a c t t h e s e a r e the c r u c i a l q u e s t i o n s w h i c h d e t e r m i n e c o n s i s t e n c y i s a c h i e v e d . A c o m p l e t e d i s c u s s i o n of t h i s p r o b l e m r e q u i r e s t h e u s e of t h e full u n i t a r i t y e q u a t i o n s c o n t i n u e d to c o m p l e x a n g u l a r m o m e n t u m [10, 12, 13]. B a s i n g o u r d i s c u s s i o n on t h i s , we will t h e n o u t l i n e h o w it c a n be s h o w n t h a t t h e z e r o r e f e r r e d to by A b a r b a n e l a n d G r e e n i s , in f a c t , t h e r e f l e c t i o n of the two P o m e r o n cut in t h e t r i p l e P o m e r o n v e r t e x . It t h e n f o l l o w s t h a t t h e z e r o o c c u r s a t t O = 0 a n d so c o m p l e t e l y r e m o v e s t h e f i x e d - p o l e p a r t of t h e v e r t e x in the i n c l u s i v e a p p l i c a t i o n . R e f s . [2, 3], w h i l e g i v i n g a r g u m e n t s f o r t h e n e c e s s i t y of a z e r o i n g ( l , l, 0) a s ! ~ 0, do n o t d i s c u s s t h e m i n i m u m s t r e n g h t of t h e z e r o r e q u i r e d . In f a c t t h e y p r o d u c e d i f f e r e n t r e q u i r e m e n t s . Ref. [2] i m p l i e s t h a t g(l, t, 0) < ( l o g t ) - 1 - c f o r ~ ~ - ~ w h i l e the a r g u m e n t of r e f . [3], w h i c h i s m o r e d i r e c t in t h a t it only r e l i e s on t h e M u e l l e r d i s c o n t i n u i t y f o r m u l a , i m p l i e s E < 0. Now, a s we s h a l l e l a b o r a t e , the z e r o p r o d u c e d by a t w o - R e g g e o n cut h a s t h e s t r e n g t h of a n i n v e r s e l o g a r i t h m . T h u s if t h i s z e r o o n l y a c t u a l l y o c c u r r e d in t h e l i m i t t - - 0 at t O = 0, a s A b a r b a n e l a n d G r e e n s e e m to i m p l y , t h e a r g u m e n t of r e f . [3] w o u l d s t i l l l e a d to a n i n e o n s i s } e n c y w i t h c o n s t a n t c r o s s - s e c t i o n s . I f g ( t , l, O) ~ ( l o g / ) -1 c r o s s - s e c t i o n s m u s t d i v e r g e a t l e a s t a s f a s t a s log l o g l o g s . No z e r o is r e q u i r e d at a l l if c r o s s s e c t i o n s d i v e r g e l i k e l o g l o g s , and, of c o u r s e , n e i t h e r t h i s o r the p r e v i o u s f o r m of d i v e r g e n c e i s , a s y e t , d e t e e t a b l e e x p e r i m e n t a l l y . B o t h a r e e q u a l l y o b j e c t i o n a b l e f r o m t h e p o i n t of v i e w of a s s u m i n g P o m e r o n d o m i n a n e e f o r total cross sections. T h e s t r u c t u r e of t h e t h r e e R e g g e o n c o n t r i b u t i o n to t h e i n c l u s i v e c r o s s - s e c t i o n in the l i m i t M 2 ~ ~, S / M 2 ~ °0 i s
* CERN, Geneva, Switzerland.
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24 January 1972
S c a / ~ t2
M2 Fig. 1. The inclusive process. dcr
dM2dt
Fig. 2. The three Reggeon vertex.
~r 1 r 2 C~l ~c~ 2 ~3ac~l(t)/3acc~2 (t) sin ~ ( a l ( t ) + a2(t) - ~0(0))
t, O)s cq(l)+c~2(t)-2 (M2) (~O(O)-al(t)-a2(l)
(1)
x 7c~1a2~0(/, w h e r e ~ = (-r + exp (- inn)) and the ~ ' s a r e two a s t ~ 0, the t h r e e P o m e r o n v e r t e x g(t, t, 0) = l i n e a r z e r o a s I ~ 0. F o r 1o < 0, l 1 ~' t 2 the a p p r o p r i a t e c r o s s e d c o n t r i b u t i o n to the a s y m p t o t i c b e h a v i o u r of the i~
A6 (0, ~1, ~2 ~
p a r t i c l e / R e g g e o n c o u p l i n g f u n c t i o n s . If 7 is n o n - s i n g u l a r 7 a p a p a p ( l , t, 0) sin;r (2ap(t) - otp(0)) w i l l h a v e at l e a s t a p a r t i a l - w a v e e x p a n s i o n [14] g i v e s the t r i p l e R e g g e p o l e s i x - p o i n t f u n c t i o n in the f o r m
i~o
- a 0 ~ a l ~a2 q0 gl g2 f d n l f. dn2exp(inlwl) -i~ -1~o
exp(in2c°2) A((~0' ~ 1 ' c ~ 2 ' n l ' n 2 )
(2)
X
w h e r e ~0, ~1, ~2 a r e b o o s t v a r i a b l e s , w 1, w 2 a r e a z i m u t h a l v a r i a b l e s , so that n 1 and n 2 a r e ( c o m p l e x ) • ~2 o/0 . he.lici.ty l a b e l s . A is a k m e m a t m. f a c t o r and F nOtl l n2 is the r e s i d u e of the p a r t i a l - w a v e c o e f f i c i e n t F?17270 at the t r i p l e R e g g e p o l e Ji = a i , i = 0, 1, 2. The a s y m p t o t i c s i n g u l a r i t y s t r u c t u r e o f A 6 (that is ~ln2 no s i n g u l a r i t i e s in o v e r l a p p i n g a s y m p t o t i c i n v a r i a n t s [3], a s r e q u i r e d by the S t e i n m a n n r e l a t i o n s ) is now r e f l e c t e d in the v a r i o u s " k i n e m a t i c h e l i c i t y p o l e s " in A. (In r e f . [15] we d i s c u s s e d the s t r u c t u r e of the two R e g g e o n / p a r t i c l e v e r t e x f r o m this point of v i e w u s i n g the S o m m e r f e l d - W a t s o n t r a n s f o r m a t i o n of the d i r e c t c h a n n e l p a r t i a l - w a v e e x p a n s i o n . The c r o s s - c h a n n e l e x p a n s i o n is p r e f e r r a b l e to the S o m m e r f e l d - W a t s o n t r a n s f o r m b e c a u s e of c o n v e r g e n c e d i f f i c u l t i e s [16]. In any c.a.se, it is l i k e l y that t h e r e ]vYl)2J0 and the F r o i s s a r t is a s i m p l e r e l a t i o n [15] b e t w e e n the c r o s s - c h a n n e l p a r t i a l - w a v e c o e f f i c i e n t s --nln2 G r i b o v c o n t i n u a t i o n s a(j 1, J 2, J O, nl, n2) of the d i r e c t - c h a n n e l p a r t i a l - w a v e c o e f f i c i e n t s [16] , and we s h a l l a s s u m e s u c h a r e l a t i o n in the s u b s e q u e n t d i s c u s s i o n ) , n I and n 2 can be c h o s e n so that the p a r t of (2) with an M 2 d i s c o n t i n u i t y is g i v e n by p o l e s of A a t n I = a 1, a l - 1 , - . . , n 2 = 0~2, ~ 2 = 1 , - . . and so p u l l i n g b a c k the n I and n 2 c o n t o u r s g i v e s A6
0
~2 (cos ¢Ol)(~1 (cos ¢02)
A(e~0' a l '
~
~1~2
~
to)
~1, ~2, ~0 ~ oo c o s ¢Ol, c o s w2 ~ ~¢ + t e r m s with no M 2 d i s c o n t i n u i t y . T h i s e q u a t i o n can now be r e - e x p r e s s e d 94
(3)
in t e r m s of v a r i a n t s and the M 2 d i s c o n t i n u i t y taken. The r e s u l t
Volume 38B, n u m b e r 2
PHYSICS
LETTERS
24 J a n u a r y 1972
c o n t i n u e d to /1 = t 2 a n d t h e n [ 0 0 w i t h ~1 = {2 a n d w I - w 2 w i l l b e (1). T h e s i g n a t u r e f a c t o r s , in p a r t i c u l a r t h e v i t a l s i n ~ ( a l ( / ) + e ~ 2 ( l ) - a 0 ( 9 ) ) c a n b e d e d u c e d f r o m t h e r e a l a n a l y s i c i t y p r o p e r t i e s of A 6. (We will d i s c u s s all t h e s e p o i n t s , a n d s u b t l e t i e s c o n n e c t e d w i t h the f a c t t h a t the l i m i t at l 0 0 is r e a l l y a h e l i c i t y p o l e / R e g g e p o l e l i m i t [1, 31 in a s u b s e q u e n t p a p e r [17].) F C~ 1 (~o Ot0 ~s, . e~crr) of c o u r s e , a s s u m e d to f a c t o r i s . e . i l ) t h e f o r m s h o w n in eq. (1). H o w e v e r , YC~pOtp~p (l, /, 0) m a y b e s i n g u l a r a t ! 0, if F 1! 7210 h a s a n o n s e n s e f i x e d p o l e a t J0 = n 1 + ~z2 - 1 o c c u r r i n g m u l t i p l i c a t i v e l y w i t h t h e t r i p l e R e g g e p o l e . T h i s f i x e d p o l e i s c l o s e l y I i n k e d to t h e t w o - R e g g e o n ( " 1 , c~2) cut at J 0 - c~0(t0)- If AoQot2(Jo, l 0) is the " t w o - R e g g e o n ~ t w o - p a r t i c l e a m p l i t u d e " d e f i n e d by f a c t o r i s a t i o n of the r e s i d u e of the d o u b l e p o l e in a(jl, J2, Jo, Jl, J2) at J l - C~l, J 2 = °e2 t h e n t h e " f i x e d - p o l e " will o c c u r * in A or lC~2(J0, 10) at J 0 a 1 + c~2 - 1. T h e r e s i d u e d e t e r m i n e s the d i s c o n t i n u i t y a c r o s s t h e t w o - R e g g e o n cut in t h e f o u r - p o i n t f u n c t i o n [10, 12]. T h e s t r u c t u r e of t h i s cut c a n be a n a l y s e d in d e t a i l u s i n g m u l t i p a r t i c l e u n i t a r i t y . T h e b a s i c a n a l y s i s h a s b e e n d o n e by G r i b o v , P o m e r a n c h u k and Ter-Martirosyan in r e f , [10] a n d t h e m o d i f i c a t i o n s of t h e i r r e s u l t s r e q u i r e d by r e f . [13] w i l l b e g i v e n in r e f . [12]. T h e v i t a l p o i n t i s t h a t t h e f i x e d - p o l e a t J 0 - c ~ 1 + ~ 2 - 1 m u s t o c c u r m u l t i p l i e d by t h e a s s o c i a t e d t w o - R e g g e o n cut. S i n c e the t w o - R e g g e o n c u t t u r n s out to b e e s s e n t a i l l y a n i n v e r s e l o g a r i t h m [10, 12] we c a n w r i t e * * ~(1) 2 ( .30' / 0 ) / ( j 0 _ C ~ l ( t l ) - c~2( t2)+ Ac~lC~2(J0' to) =JC~lC~
l)[l°g(Jo-C~c(lO ))]+ f(ot2)ol 1 2 (Jo" lO)
w h e r e b o t h f (ffl)1 c~,) a n d ~~(2) o t l c ~ , 2 a r e r e g u l a r at the pole. f(1)lC~, ~ is f i n i t e a t t h e b r a n c h - p o i n t of it w h i c h i s n o n - z e r o i s r e g u l a r . T h e log -1 f a c t o r i s t h e z e r o r e f e r r e d to b y A b a r b a n e l a n d it i s c l e a r f r o m r e f . [10] t h a t it g i v e s the c o n s i s t e n c y w i t h u n i t a r i t y r e q u i r e d by r e f . b-,Otlff2J0 T h e s t r u c t u r e (4) will c a r r y o v e r into _ cq a 2 a n d s o g o i n g to the p o l e at j 0 = co0(/0) s t r u c t u r e f o r y:
),(t 1 , l 2,
~0 ) = ~3(1)(/1 ,
12,
/0) '[c~0(/0) - c~1(/1) -
c~2(12)+ 1] [log
(4) and that part a n d G r e e n [6] [11]. we o b t a i n a
(c~0(t 0) - C~c(/0)) ] + 3(2)(/1, [2" 10) "
(5)
Now q u i t e g e n e r a l l y (~c(0) - c q ( 0 ) + c~2(0 ) - 1 a n d if (~0(0) a l s o h a s t h i s v a l u e , a s i s the c a s e f o r the t h r e e P o m e r o n v e r t e x , the f i r s t t e r m v a n i s h e s in the i n c l u s i v e a p p l i c a t i o n (t O 0, l 1 - 12) a n d Y ~ p c ~ p c ~ p ( / , t, 0) i s i n d e e d n o n - s i n g u l a r a s l ~ 0 . (We a r e , of c o u r s e , a s s u m i n g the P o m e r o n to b e a R e g g e p o l e w i t h u n i t i n t e r c e p t . T h e r e s u l t a l s o h o l d s if only ~ 1 , s a y , i s a P o m e r o n a n d c~2(0) = c~0(0).) It i s p e r h a p s w o r t h r e m a r k i n g t h a t t h e a b o v e a r g u m e n t s a r e b a s e d o n l y on w e l l - j u s t i f i e d a s s u m p t i o n s of a n a l y t i c i t y i n h e l i c i t y a n d a n g u l a r m o m e n t u m a n d a s s u c h a r e e n t i r e l y m o d e l i n d e p e n d e n t . In r e f s . [7, 8] it i s s h o w n t h a t a c e r t a i n c l a s s of n o n - p l a n a r F e y n m a n g r a p h s do n o t h a v e Y O t p e t p ~ p ( t , l, 0) ~ 0 as l~0because of t h e p r e s e n c e of a f i x e d p o l e . H o w e v e r , it is w e l l - k n o w n t h a t to p r o d u c e cut d i s c o n t i n u i t i e s c o n s i s t e n t w i t h u n i t a r i t y it i s n e c e s s a r y to s u m o v e r a n i n f i n i t e n u m b e r of s u c h g r a p h s [ 1 8 - 2 0 ] . W e would t h e r e f o r e e x p e c t the z e r o in the f i x e d - p o l e c o n t r i b u t i o n at l 0 : 0 to a p p e a r only w h e n s u c h a sum is performed. F i n a l l y we n o t e t h a t t h e d e r i v a t i o n of a l o w e r b o u n d f o r yc~pc~pc~p(t, l, 0) g i v e n in r e f . [9] in f a c t r e q u i r e s o u r r e s u l t s to j u s t i f y the n e g l e c t of the m u l t i p l i c a t i v e n a t u r e of t h e f i x e d - p o l e a n d t h e t r i p l e P o m e r o n p o l e . O u r r e s u l t s do not, h o w e v e r , g i v e a n y basi.s f o r n e g l e c t i n g t h e m u l t i p l e P o m e r o n c u t s w h i c h a c c u m u l a t e at t o 0 a n d l : 0 (as i s a l s o n e c e s s a r y f o r the a r g u m e n t of r e f . [9]). If a n y t h i n g o u r r e s u l t s e m p h a s i s e t h e i m p o r t a n c e of c o n s i d e r i n g at l e a s t the t w o - P o m e r o n cut. O n e of u s (P. G.) i s g r a t e f u l to the C E R N T h e o r e t i c a l
Study D i v i s i o n f o r h o s p i t a l i t y .
* The definition of a ( J l , J2, ]0, n l , n2) given in refs. [10, 16] has this fixed-pole removed by a kinematic factor. We shall take it that this is not the case with the amplitudes we use here. ** S t r i c t l y A a l a 2 ( J 0 , to) will h.ave a '~hreshold" singularity at X(tl, t2, to) =- t2+t2+t2-2tlt°-2tlt,-2t, t, = 0 of the .form ~ u to• BOtlOt z •2.u It isz Botla u . . .Ac~ 1o~2 = Bot 1 ~ ~(]0 - ~1- ix2)/2+ . . . . The r esult (4) should really be applied 9 that con~riou~es to (t) since tne s i n g m a r i t y in X d i s a p p e a r s when (3) is r e - e x p r e s s e d in t e r m s of invariants.
95
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24 January
References [1] C. DeTar, C . E . J o n e s , F . E . Low, C.I. Tan, J . H . Weis and J . E . Young, Phys. Rev. Letters 26 (1971) 675. [2] H. D. I. Abarbanel, G. F. Chew, M. L. Goldberger and L. M. Saunders, Phys. Rev. Letters 26 (1971) 937 and Princeton University P r e p r i n t . [3] C . D e T a r and J. H. Weis, MIT P r e p r i n t CTP-218 (1971). [4] S. J. Chang, D. Gordon, F. E. Low and S. B. T r e i m a n , NAL P r e p r i n t THY 16 (1971). [5] D. Gordon and G. Veneziano, Phys. Rev. D3 (1971) 2116. [6] H. D. I. Abarbanel and M. B. Green, Institute for Advanced Study, Princeton P r e p r i n t (1971). [7] D. Gordon, NAL P r e p r i n t THY 23 (1971). [8] A. H. Mueller and T. L. Trueman, Brookhaven P r e p r i n t BNL 16350 (1971). [9] J. Ellis, J. Finkelstein and R. D. P e c c e i , Stanford P r e p r i n t , ITP-396, SLAC-PUB-990 (1971). [10] V. N. Gribov, I. Ya. Pomeranchuk and K. A. T e r - M a r t i r o s y a n , Yad. Fiz. 2 (1965) 361, Soy. J. Nucl. Phys. 2 (1966) 258, Phys. Rev. 139B (1965) 184. [11] J . B . Bronzan and C.E. J o n e s , Phys. Rev. 160 (1967) 1494. [12] A . R . W h i t e , to appear. [13] A. R. White, Cambridge p r e p r i n t DAMTP 71/35 (1971), Nucl. Phys. to be published. [14] P. Goddard and A.R. White, Nucl. Phys. B17 (1970) 45. [15] P. Goddard and A. R. White, Nuovo Cimento 1A (1971) 645. [16] A.R. White, Cambridge p r e p r i n t DAMTP 71/34 (1971), Nucl. Phys. to be published. [17] P. Goddard and A. H. White, in preparation. [18] D. I. Olive and J. C. Polkinghorne, Phys. Rev. 171 (1968) 1475. [19] J. C. Polkinghorne, Nucl. Phys. B6 (1969) 441. [20] P . V . Landshoff and J. C. Polkinghorne, Phys. Rev. 181 (1969) 1989.
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IN
THE
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POMERON
2 4 J a n u a r y 1972
VERTEX
P. GODDARD * a n d A. R. W H I T E
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England Received 20 D e c e m b e r 1971
Multiparticle unitarity is used to show that a reflection of the t w o - P o m e r o n cut in the three P o m e r o n v e r t e x provides a zero which r e m o v e s the analogue of the G r i b o v - P o m e r a n c h u k fixed-pole in the forward limit giving the inclusive c r o s s - s e c t i o n . This c r o s s - s e c t i o n then has a l i n e a r zero in the momentum t r a n s f e r a r i s i n g from the signature factor.
A n u m b e r of r e c e n t p a p e r s [ 1 - 9 ] h a v e d i s c u s s e d the t r i p l e P o m e r o n c o n t r i b u t i o n to t h e i n c l u s i v e p r o c e s s a + b ~ c + a n y t h i n g ( s e e fig. 1). In p a r t i c u l a r it h a s b e e n s h o w n t h a t if t h i s c o n t r i b u t i o n i s to d o m i n a t e in the a p p r o p r i a t e l i m i t ( m i s s i n g m a s s M 2 ~ ~ , S / M 2 ~ oo) t h e n c o n s i s t e n c y w i t h conslanl total cross sections (that is the assumption that total cross-sections are also Pomerondominated) requires azeroin the triple Pomeronvertex. T h e o c c u r r e n c e of s u c h a z e r o h a s b e e n d e m o n s t r a t e d in t h e d u a l m o d e l [3] a n d in a c e r t a i n f i e l d - t h e o r e t i c m o d e l [4]. T h e z e r o c a n be r e g a r d e d a s o c c u r r i n g a t a n o n s e n s e w r o n g - s i g n a t u r e p o i n t (of o n e of the P o m e r o n s r e l a t i v e to t h e h e l i c i t i e s of the o t h e r two). It m i g h t t h e r e f o r e b e e x p e c t e d t h a t the z e r o will b e c a n c e l l e d by a " f i x e d - p o l e " of the G r i b o v - P o m e r a n c h u k t y p e , w h e n t h i r d d o u b l e s p e c t r a l f u n c t i o n e f f e c t s , l a c k i n g in the m o d e l s m e n t i o n e d , a r e i n c l u d e d [7, 8]. A s A b a r b a n e l a n d G r e e n [6] h a v e p o i n t e d out, t h e r e s i d u e of s u c h a f i x e d - p o l e m u s t h a v e a f u r t h e r z e r o . T h i s i s b e c a u s e t h i s r e s i d u e d e t e r m i n e s the d i s c o n t i n u i t y of t h e two P o m e r o n cut in t h e f o u r p o i n t f u n c t i o n a n d u n i t a r i t y r e q u i r e s t h i s d i s c o n t i n u i t y to v a n i s h a t t h e b r a n c h - p o i n t [ 1 0 - 1 2 ] . H o w e v e r , t h e a r g u m e n t of A r b a r b a n e l a n d G r e e n i s not s u f f i c i e n t l y d e t a i l e d to d i s t i n g u i s h in w h i c h v a r i a b l e of t h e three Pomeronvertex f u n c t i o n g ( l l , 12, to) ( s e e fig. 2) t h i s z e r o o c c u r s , o r to d i s c u s s i t s s t r e n g t h . In f a c t t h e s e a r e the c r u c i a l q u e s t i o n s w h i c h d e t e r m i n e c o n s i s t e n c y i s a c h i e v e d . A c o m p l e t e d i s c u s s i o n of t h i s p r o b l e m r e q u i r e s t h e u s e of t h e full u n i t a r i t y e q u a t i o n s c o n t i n u e d to c o m p l e x a n g u l a r m o m e n t u m [10, 12, 13]. B a s i n g o u r d i s c u s s i o n on t h i s , we will t h e n o u t l i n e h o w it c a n be s h o w n t h a t t h e z e r o r e f e r r e d to by A b a r b a n e l a n d G r e e n i s , in f a c t , t h e r e f l e c t i o n of the two P o m e r o n cut in t h e t r i p l e P o m e r o n v e r t e x . It t h e n f o l l o w s t h a t t h e z e r o o c c u r s a t t O = 0 a n d so c o m p l e t e l y r e m o v e s t h e f i x e d - p o l e p a r t of t h e v e r t e x in the i n c l u s i v e a p p l i c a t i o n . R e f s . [2, 3], w h i l e g i v i n g a r g u m e n t s f o r t h e n e c e s s i t y of a z e r o i n g ( l , l, 0) a s ! ~ 0, do n o t d i s c u s s t h e m i n i m u m s t r e n g h t of t h e z e r o r e q u i r e d . In f a c t t h e y p r o d u c e d i f f e r e n t r e q u i r e m e n t s . Ref. [2] i m p l i e s t h a t g(l, t, 0) < ( l o g t ) - 1 - c f o r ~ ~ - ~ w h i l e the a r g u m e n t of r e f . [3], w h i c h i s m o r e d i r e c t in t h a t it only r e l i e s on t h e M u e l l e r d i s c o n t i n u i t y f o r m u l a , i m p l i e s E < 0. Now, a s we s h a l l e l a b o r a t e , the z e r o p r o d u c e d by a t w o - R e g g e o n cut h a s t h e s t r e n g t h of a n i n v e r s e l o g a r i t h m . T h u s if t h i s z e r o o n l y a c t u a l l y o c c u r r e d in t h e l i m i t t - - 0 at t O = 0, a s A b a r b a n e l a n d G r e e n s e e m to i m p l y , t h e a r g u m e n t of r e f . [3] w o u l d s t i l l l e a d to a n i n e o n s i s } e n c y w i t h c o n s t a n t c r o s s - s e c t i o n s . I f g ( t , l, O) ~ ( l o g / ) -1 c r o s s - s e c t i o n s m u s t d i v e r g e a t l e a s t a s f a s t a s log l o g l o g s . No z e r o is r e q u i r e d at a l l if c r o s s s e c t i o n s d i v e r g e l i k e l o g l o g s , and, of c o u r s e , n e i t h e r t h i s o r the p r e v i o u s f o r m of d i v e r g e n c e i s , a s y e t , d e t e e t a b l e e x p e r i m e n t a l l y . B o t h a r e e q u a l l y o b j e c t i o n a b l e f r o m t h e p o i n t of v i e w of a s s u m i n g P o m e r o n d o m i n a n e e f o r total cross sections. T h e s t r u c t u r e of t h e t h r e e R e g g e o n c o n t r i b u t i o n to t h e i n c l u s i v e c r o s s - s e c t i o n in the l i m i t M 2 ~ ~, S / M 2 ~ °0 i s
* CERN, Geneva, Switzerland.
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S c a / ~ t2
M2 Fig. 1. The inclusive process. dcr
dM2dt
Fig. 2. The three Reggeon vertex.
~r 1 r 2 C~l ~c~ 2 ~3ac~l(t)/3acc~2 (t) sin ~ ( a l ( t ) + a2(t) - ~0(0))
t, O)s cq(l)+c~2(t)-2 (M2) (~O(O)-al(t)-a2(l)
(1)
x 7c~1a2~0(/, w h e r e ~ = (-r + exp (- inn)) and the ~ ' s a r e two a s t ~ 0, the t h r e e P o m e r o n v e r t e x g(t, t, 0) = l i n e a r z e r o a s I ~ 0. F o r 1o < 0, l 1 ~' t 2 the a p p r o p r i a t e c r o s s e d c o n t r i b u t i o n to the a s y m p t o t i c b e h a v i o u r of the i~
A6 (0, ~1, ~2 ~
p a r t i c l e / R e g g e o n c o u p l i n g f u n c t i o n s . If 7 is n o n - s i n g u l a r 7 a p a p a p ( l , t, 0) sin;r (2ap(t) - otp(0)) w i l l h a v e at l e a s t a p a r t i a l - w a v e e x p a n s i o n [14] g i v e s the t r i p l e R e g g e p o l e s i x - p o i n t f u n c t i o n in the f o r m
i~o
- a 0 ~ a l ~a2 q0 gl g2 f d n l f. dn2exp(inlwl) -i~ -1~o
exp(in2c°2) A((~0' ~ 1 ' c ~ 2 ' n l ' n 2 )
(2)
X
w h e r e ~0, ~1, ~2 a r e b o o s t v a r i a b l e s , w 1, w 2 a r e a z i m u t h a l v a r i a b l e s , so that n 1 and n 2 a r e ( c o m p l e x ) • ~2 o/0 . he.lici.ty l a b e l s . A is a k m e m a t m. f a c t o r and F nOtl l n2 is the r e s i d u e of the p a r t i a l - w a v e c o e f f i c i e n t F?17270 at the t r i p l e R e g g e p o l e Ji = a i , i = 0, 1, 2. The a s y m p t o t i c s i n g u l a r i t y s t r u c t u r e o f A 6 (that is ~ln2 no s i n g u l a r i t i e s in o v e r l a p p i n g a s y m p t o t i c i n v a r i a n t s [3], a s r e q u i r e d by the S t e i n m a n n r e l a t i o n s ) is now r e f l e c t e d in the v a r i o u s " k i n e m a t i c h e l i c i t y p o l e s " in A. (In r e f . [15] we d i s c u s s e d the s t r u c t u r e of the two R e g g e o n / p a r t i c l e v e r t e x f r o m this point of v i e w u s i n g the S o m m e r f e l d - W a t s o n t r a n s f o r m a t i o n of the d i r e c t c h a n n e l p a r t i a l - w a v e e x p a n s i o n . The c r o s s - c h a n n e l e x p a n s i o n is p r e f e r r a b l e to the S o m m e r f e l d - W a t s o n t r a n s f o r m b e c a u s e of c o n v e r g e n c e d i f f i c u l t i e s [16]. In any c.a.se, it is l i k e l y that t h e r e ]vYl)2J0 and the F r o i s s a r t is a s i m p l e r e l a t i o n [15] b e t w e e n the c r o s s - c h a n n e l p a r t i a l - w a v e c o e f f i c i e n t s --nln2 G r i b o v c o n t i n u a t i o n s a(j 1, J 2, J O, nl, n2) of the d i r e c t - c h a n n e l p a r t i a l - w a v e c o e f f i c i e n t s [16] , and we s h a l l a s s u m e s u c h a r e l a t i o n in the s u b s e q u e n t d i s c u s s i o n ) , n I and n 2 can be c h o s e n so that the p a r t of (2) with an M 2 d i s c o n t i n u i t y is g i v e n by p o l e s of A a t n I = a 1, a l - 1 , - . . , n 2 = 0~2, ~ 2 = 1 , - . . and so p u l l i n g b a c k the n I and n 2 c o n t o u r s g i v e s A6
0
~2 (cos ¢Ol)(~1 (cos ¢02)
A(e~0' a l '
~
~1~2
~
to)
~1, ~2, ~0 ~ oo c o s ¢Ol, c o s w2 ~ ~¢ + t e r m s with no M 2 d i s c o n t i n u i t y . T h i s e q u a t i o n can now be r e - e x p r e s s e d 94
(3)
in t e r m s of v a r i a n t s and the M 2 d i s c o n t i n u i t y taken. The r e s u l t
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c o n t i n u e d to /1 = t 2 a n d t h e n [ 0 0 w i t h ~1 = {2 a n d w I - w 2 w i l l b e (1). T h e s i g n a t u r e f a c t o r s , in p a r t i c u l a r t h e v i t a l s i n ~ ( a l ( / ) + e ~ 2 ( l ) - a 0 ( 9 ) ) c a n b e d e d u c e d f r o m t h e r e a l a n a l y s i c i t y p r o p e r t i e s of A 6. (We will d i s c u s s all t h e s e p o i n t s , a n d s u b t l e t i e s c o n n e c t e d w i t h the f a c t t h a t the l i m i t at l 0 0 is r e a l l y a h e l i c i t y p o l e / R e g g e p o l e l i m i t [1, 31 in a s u b s e q u e n t p a p e r [17].) F C~ 1 (~o Ot0 ~s, . e~crr) of c o u r s e , a s s u m e d to f a c t o r i s . e . i l ) t h e f o r m s h o w n in eq. (1). H o w e v e r , YC~pOtp~p (l, /, 0) m a y b e s i n g u l a r a t ! 0, if F 1! 7210 h a s a n o n s e n s e f i x e d p o l e a t J0 = n 1 + ~z2 - 1 o c c u r r i n g m u l t i p l i c a t i v e l y w i t h t h e t r i p l e R e g g e p o l e . T h i s f i x e d p o l e i s c l o s e l y I i n k e d to t h e t w o - R e g g e o n ( " 1 , c~2) cut at J 0 - c~0(t0)- If AoQot2(Jo, l 0) is the " t w o - R e g g e o n ~ t w o - p a r t i c l e a m p l i t u d e " d e f i n e d by f a c t o r i s a t i o n of the r e s i d u e of the d o u b l e p o l e in a(jl, J2, Jo, Jl, J2) at J l - C~l, J 2 = °e2 t h e n t h e " f i x e d - p o l e " will o c c u r * in A or lC~2(J0, 10) at J 0 a 1 + c~2 - 1. T h e r e s i d u e d e t e r m i n e s the d i s c o n t i n u i t y a c r o s s t h e t w o - R e g g e o n cut in t h e f o u r - p o i n t f u n c t i o n [10, 12]. T h e s t r u c t u r e of t h i s cut c a n be a n a l y s e d in d e t a i l u s i n g m u l t i p a r t i c l e u n i t a r i t y . T h e b a s i c a n a l y s i s h a s b e e n d o n e by G r i b o v , P o m e r a n c h u k and Ter-Martirosyan in r e f , [10] a n d t h e m o d i f i c a t i o n s of t h e i r r e s u l t s r e q u i r e d by r e f . [13] w i l l b e g i v e n in r e f . [12]. T h e v i t a l p o i n t i s t h a t t h e f i x e d - p o l e a t J 0 - c ~ 1 + ~ 2 - 1 m u s t o c c u r m u l t i p l i e d by t h e a s s o c i a t e d t w o - R e g g e o n cut. S i n c e the t w o - R e g g e o n c u t t u r n s out to b e e s s e n t a i l l y a n i n v e r s e l o g a r i t h m [10, 12] we c a n w r i t e * * ~(1) 2 ( .30' / 0 ) / ( j 0 _ C ~ l ( t l ) - c~2( t2)+ Ac~lC~2(J0' to) =JC~lC~
l)[l°g(Jo-C~c(lO ))]+ f(ot2)ol 1 2 (Jo" lO)
w h e r e b o t h f (ffl)1 c~,) a n d ~~(2) o t l c ~ , 2 a r e r e g u l a r at the pole. f(1)lC~, ~ is f i n i t e a t t h e b r a n c h - p o i n t of it w h i c h i s n o n - z e r o i s r e g u l a r . T h e log -1 f a c t o r i s t h e z e r o r e f e r r e d to b y A b a r b a n e l a n d it i s c l e a r f r o m r e f . [10] t h a t it g i v e s the c o n s i s t e n c y w i t h u n i t a r i t y r e q u i r e d by r e f . b-,Otlff2J0 T h e s t r u c t u r e (4) will c a r r y o v e r into _ cq a 2 a n d s o g o i n g to the p o l e at j 0 = co0(/0) s t r u c t u r e f o r y:
),(t 1 , l 2,
~0 ) = ~3(1)(/1 ,
12,
/0) '[c~0(/0) - c~1(/1) -
c~2(12)+ 1] [log
(4) and that part a n d G r e e n [6] [11]. we o b t a i n a
(c~0(t 0) - C~c(/0)) ] + 3(2)(/1, [2" 10) "
(5)
Now q u i t e g e n e r a l l y (~c(0) - c q ( 0 ) + c~2(0 ) - 1 a n d if (~0(0) a l s o h a s t h i s v a l u e , a s i s the c a s e f o r the t h r e e P o m e r o n v e r t e x , the f i r s t t e r m v a n i s h e s in the i n c l u s i v e a p p l i c a t i o n (t O 0, l 1 - 12) a n d Y ~ p c ~ p c ~ p ( / , t, 0) i s i n d e e d n o n - s i n g u l a r a s l ~ 0 . (We a r e , of c o u r s e , a s s u m i n g the P o m e r o n to b e a R e g g e p o l e w i t h u n i t i n t e r c e p t . T h e r e s u l t a l s o h o l d s if only ~ 1 , s a y , i s a P o m e r o n a n d c~2(0) = c~0(0).) It i s p e r h a p s w o r t h r e m a r k i n g t h a t t h e a b o v e a r g u m e n t s a r e b a s e d o n l y on w e l l - j u s t i f i e d a s s u m p t i o n s of a n a l y t i c i t y i n h e l i c i t y a n d a n g u l a r m o m e n t u m a n d a s s u c h a r e e n t i r e l y m o d e l i n d e p e n d e n t . In r e f s . [7, 8] it i s s h o w n t h a t a c e r t a i n c l a s s of n o n - p l a n a r F e y n m a n g r a p h s do n o t h a v e Y O t p e t p ~ p ( t , l, 0) ~ 0 as l~0because of t h e p r e s e n c e of a f i x e d p o l e . H o w e v e r , it is w e l l - k n o w n t h a t to p r o d u c e cut d i s c o n t i n u i t i e s c o n s i s t e n t w i t h u n i t a r i t y it i s n e c e s s a r y to s u m o v e r a n i n f i n i t e n u m b e r of s u c h g r a p h s [ 1 8 - 2 0 ] . W e would t h e r e f o r e e x p e c t the z e r o in the f i x e d - p o l e c o n t r i b u t i o n at l 0 : 0 to a p p e a r only w h e n s u c h a sum is performed. F i n a l l y we n o t e t h a t t h e d e r i v a t i o n of a l o w e r b o u n d f o r yc~pc~pc~p(t, l, 0) g i v e n in r e f . [9] in f a c t r e q u i r e s o u r r e s u l t s to j u s t i f y the n e g l e c t of the m u l t i p l i c a t i v e n a t u r e of t h e f i x e d - p o l e a n d t h e t r i p l e P o m e r o n p o l e . O u r r e s u l t s do not, h o w e v e r , g i v e a n y basi.s f o r n e g l e c t i n g t h e m u l t i p l e P o m e r o n c u t s w h i c h a c c u m u l a t e at t o 0 a n d l : 0 (as i s a l s o n e c e s s a r y f o r the a r g u m e n t of r e f . [9]). If a n y t h i n g o u r r e s u l t s e m p h a s i s e t h e i m p o r t a n c e of c o n s i d e r i n g at l e a s t the t w o - P o m e r o n cut. O n e of u s (P. G.) i s g r a t e f u l to the C E R N T h e o r e t i c a l
Study D i v i s i o n f o r h o s p i t a l i t y .
* The definition of a ( J l , J2, ]0, n l , n2) given in refs. [10, 16] has this fixed-pole removed by a kinematic factor. We shall take it that this is not the case with the amplitudes we use here. ** S t r i c t l y A a l a 2 ( J 0 , to) will h.ave a '~hreshold" singularity at X(tl, t2, to) =- t2+t2+t2-2tlt°-2tlt,-2t, t, = 0 of the .form ~ u to• BOtlOt z •2.u It isz Botla u . . .Ac~ 1o~2 = Bot 1 ~ ~(]0 - ~1- ix2)/2+ . . . . The r esult (4) should really be applied 9 that con~riou~es to (t) since tne s i n g m a r i t y in X d i s a p p e a r s when (3) is r e - e x p r e s s e d in t e r m s of invariants.
95
Volume 38B, number 2
PHYSICS
LETTERS
24 January
References [1] C. DeTar, C . E . J o n e s , F . E . Low, C.I. Tan, J . H . Weis and J . E . Young, Phys. Rev. Letters 26 (1971) 675. [2] H. D. I. Abarbanel, G. F. Chew, M. L. Goldberger and L. M. Saunders, Phys. Rev. Letters 26 (1971) 937 and Princeton University P r e p r i n t . [3] C . D e T a r and J. H. Weis, MIT P r e p r i n t CTP-218 (1971). [4] S. J. Chang, D. Gordon, F. E. Low and S. B. T r e i m a n , NAL P r e p r i n t THY 16 (1971). [5] D. Gordon and G. Veneziano, Phys. Rev. D3 (1971) 2116. [6] H. D. I. Abarbanel and M. B. Green, Institute for Advanced Study, Princeton P r e p r i n t (1971). [7] D. Gordon, NAL P r e p r i n t THY 23 (1971). [8] A. H. Mueller and T. L. Trueman, Brookhaven P r e p r i n t BNL 16350 (1971). [9] J. Ellis, J. Finkelstein and R. D. P e c c e i , Stanford P r e p r i n t , ITP-396, SLAC-PUB-990 (1971). [10] V. N. Gribov, I. Ya. Pomeranchuk and K. A. T e r - M a r t i r o s y a n , Yad. Fiz. 2 (1965) 361, Soy. J. Nucl. Phys. 2 (1966) 258, Phys. Rev. 139B (1965) 184. [11] J . B . Bronzan and C.E. J o n e s , Phys. Rev. 160 (1967) 1494. [12] A . R . W h i t e , to appear. [13] A. R. White, Cambridge p r e p r i n t DAMTP 71/35 (1971), Nucl. Phys. to be published. [14] P. Goddard and A.R. White, Nucl. Phys. B17 (1970) 45. [15] P. Goddard and A. R. White, Nuovo Cimento 1A (1971) 645. [16] A.R. White, Cambridge p r e p r i n t DAMTP 71/34 (1971), Nucl. Phys. to be published. [17] P. Goddard and A. H. White, in preparation. [18] D. I. Olive and J. C. Polkinghorne, Phys. Rev. 171 (1968) 1475. [19] J. C. Polkinghorne, Nucl. Phys. B6 (1969) 441. [20] P . V . Landshoff and J. C. Polkinghorne, Phys. Rev. 181 (1969) 1989.
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