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Relative importance of dual loops
Volume 36B, number 6
PHYSICS
RELATIVE
IMPORTANCE
LETTERS
18 October 1971
OF
DUAL
LOOPS
P.H. F R A M P T O N
CERN, Geneva, Switzerland Received 12 August 1971 We suggest that the size of a dual loop term is determined by the number of quark lines, rather than the power of g. This could explain the smallness of the t r i p l e - P o m e r o n vertex.
In t h i s l e t t e r , it i s s u g g e s t e d that it is u s e f u l in e s t i m a t i n g t h e m a g n i t u d e s of t e r m s in a dual loop e x p a n s i o n to count t h e n u m b e r (m) of d i f f e r ent q u a r k l i n e s o c c u r r i n g in t h e c o r r e s p o n d i n g q u a r k d i a g r a m ; t h i s i s an a l t e r n a t i v e to t h e m o r e u s u a l e x p a n s i o n (useful f o r c o n s i d e r a t i o n s of p e r t u r b a t i v e u n i t a r i t y [ 1]) in t e r m s of t h e n u m b e r of h a d r o n l o o p s , o r e q u i v a l e n t l y of p o w e r s of the dual c o u p l i n g c o n s t a n t (g). E x p a n s i o n of t h e h a d r o n i c s c a t t e r i n g a m p l i t u d e into dual loop t e r m s i m p l i e s t h e i m p o r t a n t r e q u i r e m e n t that the r e n o r m a l i z a t i o n c o n t r i b u t i o n s a r e , in s o m e s e n s e , s m a l l so that t h e B o r n t e r m r e m a i n s a good a p p r o x i m a t i o n to t h e final s o l u t i o n . L e t u s now g i v e a d e f i n i t i o n of what we m e a n by s a y i n g that t h e r e n o r m a l i z a t i o n c o r r e c t i o n s a r e s m a l l . S u p p o s e that t h e s e a r e N e x t e r nal h a d r o n s and m = (mminimu m + m ' ) q u a r k l i n e s w h e r e m minimum i s the n u m b e r of q u a r k s (or a n t i q u a r k s ) c o n t a i n e d in t h e e x t e r n a l l e g s . T h e n , we s h o u l d e x p a n d t h e a m p l i t u d e a s cO
A:
~ Am, m ~=0
,
(1)
e a c h t e r m of w h i c h m a y be e x p a n d e d into t h e n u m b e r (L) of h a d r o n l o o p s v~
A m , =J - 2
~ g2L A m , ' L=m' L "
(2)
All t e r m s with m' = 1 a r e r e n o r m a l i z a t i o n e f f e c t s and the f u n d a m e n t a l a s s u m p t i o n of t h e dual loop w o r k i s that t h e s e r i e s , eq. (1), is v e r y rapidly convergent (much more convergent, for e x a m p l e , than an e x p a n s i o n in p o w e r s of g, s u c h a s eq. (2), i s e x p e c t e d to be). As a z e r o t h o r d e r a p p r o x i m a t i o n , we s h o u l d t a k e only m' = 0, that i s only t e r m s w i t h the m i n i m u m p o s s i b l e n u m b e r of q u a r k l i n e s .
a)
b)
c)
Fig. 1. Orientable primitive graphs. (a) Non-planar loop. {b) Planar tadpole. (c) Non-planar tadpole. L e t u s now c o n s i d e r in t u r n s o m e of the c o n sequences. i) F o r t h e o r i e n t a b l e p r i m i t i v e g r a p h s [2] of t h e dual r e s o n a n c e t h e o r y (fig. 1), we s e e that only t h e p l a n a r t a d p o l e (fig. lb) h a s an a d d i t i o n a l q u a r k l i n e (6m = +1). Both t h e n o n - p l a n a r loop (fig. la) and t h e n o n - p l a n a r t a d p o l e (fig. lc) h a v e no e x t r a q u a r k l i n e ( 6 m = 0). S i m i l a r l y t h e p o w e r s of t h e dual c o u p l i n g c o n s t a n t a r e c l e a r ; we m a y then s u m m a r i z e the p r o p e r t i e s of t h e s e graphs: n o n - p l a n a r loop
6m = 0
g2 ,
planar tadpole
6rn = 1
g2 ,
non-planar tadpole
5m = 0
g4
T o z e r o t h o r d e r , in eq. (1), we i n c l u d e a l l l o o p c o n t r i b u t i o n s without p l a n a r t a d p o l e s . T h u s we i n c l u d e in z e r o t h o r d e r t h e P o m e r o n P (fig. l a ) and i t s i t e r a t i o n s (fig. l a with any n u m b e r of fig. l c ) , w h i c h a r e dual to m u l t i - R e g g e e x c h a n g e cuts. Also included are the Pomeron-Regge cuts (fig. l c ) . ii) T h e a s s u m e d s m a l l h i g h e r o r d e r s a r e r e n o r m a l i z a t i o n e f f e c t s . F o r e x a m p l e , the p l a n a r t a d p o l e g i v e s r i s e to m a s s r e n o r m a l i z a t i o n s a s i n d i c a t e d in fig. 2a. It i s c r u c i a l to the s u c c e s of t h e dual loop p r o g r a m m e that s u c h r e n o r m a l i z a t i o n s be s m a l l , in o r d e r that the B o r n t e r m r e m a i n a good a p p r o x i m a t i o n . 591
PHYSICS
Volume 36B, number 6
o)
b)
Fig. 2. Renormalizationdiagrams. (a) Mass renorma[ization. (b) Vertex renormalization. iii) S i m i l a r l y , r e n o r m a l i z a t i o n c o r r e c t i o n s a p p l y to v e r t i c e s which a r e f o r b i d d e n at t h e m' = 0 l e v e l . F o r e x a m p l e , the c o u p l i n g of an e x o t i c m e s o n (qqct(l) to two n o n - e x o t i c m e s o n s (q~) m a y p r o c e e d t h r o u g h t h e r e n o r m a l i z a t i o n g r a p h (m' = 1) of fig. 2b. C l e a r l y , it is a g a i n c r u c i a l that s u c h c o r r e c t i o n s be s m a l l . F o r e x a m p l e , if the v e r t e x we h a v e c h o s e n to i l l u s t r a t e w e r e a p p r e c i a b l e , we would no l o n g e r e a s i l y u n d e r s t a n d t h e e x p e r i m e n t a l v e r i f i c a t i o n of e x c h a n g e d e g e n e r a c y p r e d i c t i o n s d e r i v e d f r o m the a b s e n c e of e x o t i c s t a t e s in, f o r e x a m p l e , Tr+Tr*----,Tr+Tr+. iv) Now, we t u r n to the t h r e e - p o i n t v e r t i c e s i n v o l v i n g the P o m e r o n . F i r s t , we look at the t r i p l e - P o m e r o n v e r t e x , which is i n d i c a t e d in fig. 3a, both a s a q u a r k d i a g r a m and in t e r m s of i n t e g r a t i o n c o n t o u r s on a R i e m a n n s u r f a c e . N o t e that f o u r q u a r k l i n e s (or i n t e g r a t i o n b o u n d a r i e s ) a r e h e r e n e e d e d to c o u p l e t h r e e (qci) s y s t e m s so that the P P P v e r t e x h a s 5m = I. In fact, we may r e g a r d the P P P v e r t e x a s a
o)
LETTERS
18 October 1971
r e n o r m a l i z a t i o n c o r r e c t i o n to t h e P P R v e r t e x (6m= 0) shown in fig. 3b, s i n c e it i s o b t a i n e d f r o m it by a d d i n g a p l a n a r t a d p o l e *. T h e o b s e r v e d s m a l l n e s s ** of the s t r e n g t h G ( P P P) t h u s c o i n c i d e s with the f a c t that for d u a l - l o o p P o m e r o n s this v e r t e x is only a r e n o r m a l i z a t i o n e f f e c t ***, e x p e c t e d to be v e r y s m a l l a s a r e the c o r r e s p o n d i n g e f f e c t s d i s c u s s e d a b o v e and d e p i c t e d in fig. 2. v) F i n a l l y , we t u r n to the P P R v e r t e x , w h i c h i s a l s o of c o n s i d e r a b l e i n t e r e s t e x p e r i m e n t a l l y . T h i s v e r t e x is shown in fig. 3b, and h a s 6m = 0. C o n s e q u e n t l y , t h e r e is no r e a s o n f o r b e l i e v i n g that i t s s t r e n g t h G(P P R) should be v e r y s m a l l . On the c o n t r a r y , t h i s v e r t e x h a s to be of n o r m a l s t r e n g t h , that is not v e r y d i f f e r ent f r o m G ( P R R ) and G(RRR), s i n c e it can be r e a c h e d by duality t r a n s f o r m a t i o n s f r o m the l a t t e r two v e r t i c e s (fig. 3c). By duality and f a c t o r i z a t i o n at the i n t e r m e d i a t e s t a t e p o l e s of t h e g r a p h s in fig. 3c, we find that G(RRR) G ( P P R ) ~ G ( P R R ) 2 . Note that we cannot s i m i l a r l y c o n s t r a i n G ( P P P) b e c a u s e duality t r a n s f o r m a t i o n s p r e s e r v e t h e n u m b e r of q u a r k l i n e s . N o r m a l s t r e n g t h f o r G( P P R) is t h u s p r e d i c t e d , and t h i s s e e m s to h a v e s o m e s u p p o r t f r o m the e n e r g y d e p e n d e n c e of r e s o n a n c e p r o ductionin p+p p+ N* (1400, 1520, 1690) [5, fig. 4]. C o r r e s p o n d i n g l y , one should e x p e c t to s e e e v i d e n c e of d o u b l e P o m e r o n - e x c h a n g e at (very) high e n e r g y , f o r e x a m p l e by o b s e r v i n g an e n e r g y i n d e p e n d e n t c r o s s - s e c t i o n f o r p r o d u c t i o n of i s o s c a l a r (~+~-) s y s t e m s in the c e n t r e of m a s s of p+p-p+p+(n+~-). This is noted independently by Brower and Waltz. CERN Preprint TH.1335 (1971). ** We refer to the rapid fall-off in energy (like l / s ) of the background in m i s s i n g - m a s s experiments [3]. *** The smallness has alternatively been discussed in terms of the nearness of the Pomeron intercept to one, by Abarbanel e t a [ . [4}.
References b)
N
c} Fig. 3. Pomeron couplings. (a)P P P vertex. (b) P P R vertex. (c) Duality transformation for P P R vertex. 592
[I] d. Scherk, Orsay Preprint, LPTHE 71/17 (1971). [2] D.J.Gross, A. Neveu, J. Scherk and J.H. Sehwarz, Phys. Letters 31B (1970) 592. [3] J.M.Wang and L.L. Wang, BNL Preprint (1971): P.D. Tingand H.J. Yesian, Phys. Letters 35B (1971) 321; G. Pancheri-Srivastava and Y. Srivastava, Northeastern University Preprint (1971); R. Rajamaran, Trieste Preprint IC/71/25 (1971); R. M. Edelstein, V. Rittenberg and H. R. Rubinstein, Phys. Letters 35B (1971) 408. [4] I|. D. I. Abarbanel, G. F. Chew, M. L. Goldberger and L.M. Saunders, Phys. Rev. Letters 26 (1971) 937. [5] Anderson et al., Phys. Rev. Letters 16 (1966) 855.
PHYSICS
RELATIVE
IMPORTANCE
LETTERS
18 October 1971
OF
DUAL
LOOPS
P.H. F R A M P T O N
CERN, Geneva, Switzerland Received 12 August 1971 We suggest that the size of a dual loop term is determined by the number of quark lines, rather than the power of g. This could explain the smallness of the t r i p l e - P o m e r o n vertex.
In t h i s l e t t e r , it i s s u g g e s t e d that it is u s e f u l in e s t i m a t i n g t h e m a g n i t u d e s of t e r m s in a dual loop e x p a n s i o n to count t h e n u m b e r (m) of d i f f e r ent q u a r k l i n e s o c c u r r i n g in t h e c o r r e s p o n d i n g q u a r k d i a g r a m ; t h i s i s an a l t e r n a t i v e to t h e m o r e u s u a l e x p a n s i o n (useful f o r c o n s i d e r a t i o n s of p e r t u r b a t i v e u n i t a r i t y [ 1]) in t e r m s of t h e n u m b e r of h a d r o n l o o p s , o r e q u i v a l e n t l y of p o w e r s of the dual c o u p l i n g c o n s t a n t (g). E x p a n s i o n of t h e h a d r o n i c s c a t t e r i n g a m p l i t u d e into dual loop t e r m s i m p l i e s t h e i m p o r t a n t r e q u i r e m e n t that the r e n o r m a l i z a t i o n c o n t r i b u t i o n s a r e , in s o m e s e n s e , s m a l l so that t h e B o r n t e r m r e m a i n s a good a p p r o x i m a t i o n to t h e final s o l u t i o n . L e t u s now g i v e a d e f i n i t i o n of what we m e a n by s a y i n g that t h e r e n o r m a l i z a t i o n c o r r e c t i o n s a r e s m a l l . S u p p o s e that t h e s e a r e N e x t e r nal h a d r o n s and m = (mminimu m + m ' ) q u a r k l i n e s w h e r e m minimum i s the n u m b e r of q u a r k s (or a n t i q u a r k s ) c o n t a i n e d in t h e e x t e r n a l l e g s . T h e n , we s h o u l d e x p a n d t h e a m p l i t u d e a s cO
A:
~ Am, m ~=0
,
(1)
e a c h t e r m of w h i c h m a y be e x p a n d e d into t h e n u m b e r (L) of h a d r o n l o o p s v~
A m , =J - 2
~ g2L A m , ' L=m' L "
(2)
All t e r m s with m' = 1 a r e r e n o r m a l i z a t i o n e f f e c t s and the f u n d a m e n t a l a s s u m p t i o n of t h e dual loop w o r k i s that t h e s e r i e s , eq. (1), is v e r y rapidly convergent (much more convergent, for e x a m p l e , than an e x p a n s i o n in p o w e r s of g, s u c h a s eq. (2), i s e x p e c t e d to be). As a z e r o t h o r d e r a p p r o x i m a t i o n , we s h o u l d t a k e only m' = 0, that i s only t e r m s w i t h the m i n i m u m p o s s i b l e n u m b e r of q u a r k l i n e s .
a)
b)
c)
Fig. 1. Orientable primitive graphs. (a) Non-planar loop. {b) Planar tadpole. (c) Non-planar tadpole. L e t u s now c o n s i d e r in t u r n s o m e of the c o n sequences. i) F o r t h e o r i e n t a b l e p r i m i t i v e g r a p h s [2] of t h e dual r e s o n a n c e t h e o r y (fig. 1), we s e e that only t h e p l a n a r t a d p o l e (fig. lb) h a s an a d d i t i o n a l q u a r k l i n e (6m = +1). Both t h e n o n - p l a n a r loop (fig. la) and t h e n o n - p l a n a r t a d p o l e (fig. lc) h a v e no e x t r a q u a r k l i n e ( 6 m = 0). S i m i l a r l y t h e p o w e r s of t h e dual c o u p l i n g c o n s t a n t a r e c l e a r ; we m a y then s u m m a r i z e the p r o p e r t i e s of t h e s e graphs: n o n - p l a n a r loop
6m = 0
g2 ,
planar tadpole
6rn = 1
g2 ,
non-planar tadpole
5m = 0
g4
T o z e r o t h o r d e r , in eq. (1), we i n c l u d e a l l l o o p c o n t r i b u t i o n s without p l a n a r t a d p o l e s . T h u s we i n c l u d e in z e r o t h o r d e r t h e P o m e r o n P (fig. l a ) and i t s i t e r a t i o n s (fig. l a with any n u m b e r of fig. l c ) , w h i c h a r e dual to m u l t i - R e g g e e x c h a n g e cuts. Also included are the Pomeron-Regge cuts (fig. l c ) . ii) T h e a s s u m e d s m a l l h i g h e r o r d e r s a r e r e n o r m a l i z a t i o n e f f e c t s . F o r e x a m p l e , the p l a n a r t a d p o l e g i v e s r i s e to m a s s r e n o r m a l i z a t i o n s a s i n d i c a t e d in fig. 2a. It i s c r u c i a l to the s u c c e s of t h e dual loop p r o g r a m m e that s u c h r e n o r m a l i z a t i o n s be s m a l l , in o r d e r that the B o r n t e r m r e m a i n a good a p p r o x i m a t i o n . 591
PHYSICS
Volume 36B, number 6
o)
b)
Fig. 2. Renormalizationdiagrams. (a) Mass renorma[ization. (b) Vertex renormalization. iii) S i m i l a r l y , r e n o r m a l i z a t i o n c o r r e c t i o n s a p p l y to v e r t i c e s which a r e f o r b i d d e n at t h e m' = 0 l e v e l . F o r e x a m p l e , the c o u p l i n g of an e x o t i c m e s o n (qqct(l) to two n o n - e x o t i c m e s o n s (q~) m a y p r o c e e d t h r o u g h t h e r e n o r m a l i z a t i o n g r a p h (m' = 1) of fig. 2b. C l e a r l y , it is a g a i n c r u c i a l that s u c h c o r r e c t i o n s be s m a l l . F o r e x a m p l e , if the v e r t e x we h a v e c h o s e n to i l l u s t r a t e w e r e a p p r e c i a b l e , we would no l o n g e r e a s i l y u n d e r s t a n d t h e e x p e r i m e n t a l v e r i f i c a t i o n of e x c h a n g e d e g e n e r a c y p r e d i c t i o n s d e r i v e d f r o m the a b s e n c e of e x o t i c s t a t e s in, f o r e x a m p l e , Tr+Tr*----,Tr+Tr+. iv) Now, we t u r n to the t h r e e - p o i n t v e r t i c e s i n v o l v i n g the P o m e r o n . F i r s t , we look at the t r i p l e - P o m e r o n v e r t e x , which is i n d i c a t e d in fig. 3a, both a s a q u a r k d i a g r a m and in t e r m s of i n t e g r a t i o n c o n t o u r s on a R i e m a n n s u r f a c e . N o t e that f o u r q u a r k l i n e s (or i n t e g r a t i o n b o u n d a r i e s ) a r e h e r e n e e d e d to c o u p l e t h r e e (qci) s y s t e m s so that the P P P v e r t e x h a s 5m = I. In fact, we may r e g a r d the P P P v e r t e x a s a
o)
LETTERS
18 October 1971
r e n o r m a l i z a t i o n c o r r e c t i o n to t h e P P R v e r t e x (6m= 0) shown in fig. 3b, s i n c e it i s o b t a i n e d f r o m it by a d d i n g a p l a n a r t a d p o l e *. T h e o b s e r v e d s m a l l n e s s ** of the s t r e n g t h G ( P P P) t h u s c o i n c i d e s with the f a c t that for d u a l - l o o p P o m e r o n s this v e r t e x is only a r e n o r m a l i z a t i o n e f f e c t ***, e x p e c t e d to be v e r y s m a l l a s a r e the c o r r e s p o n d i n g e f f e c t s d i s c u s s e d a b o v e and d e p i c t e d in fig. 2. v) F i n a l l y , we t u r n to the P P R v e r t e x , w h i c h i s a l s o of c o n s i d e r a b l e i n t e r e s t e x p e r i m e n t a l l y . T h i s v e r t e x is shown in fig. 3b, and h a s 6m = 0. C o n s e q u e n t l y , t h e r e is no r e a s o n f o r b e l i e v i n g that i t s s t r e n g t h G(P P R) should be v e r y s m a l l . On the c o n t r a r y , t h i s v e r t e x h a s to be of n o r m a l s t r e n g t h , that is not v e r y d i f f e r ent f r o m G ( P R R ) and G(RRR), s i n c e it can be r e a c h e d by duality t r a n s f o r m a t i o n s f r o m the l a t t e r two v e r t i c e s (fig. 3c). By duality and f a c t o r i z a t i o n at the i n t e r m e d i a t e s t a t e p o l e s of t h e g r a p h s in fig. 3c, we find that G(RRR) G ( P P R ) ~ G ( P R R ) 2 . Note that we cannot s i m i l a r l y c o n s t r a i n G ( P P P) b e c a u s e duality t r a n s f o r m a t i o n s p r e s e r v e t h e n u m b e r of q u a r k l i n e s . N o r m a l s t r e n g t h f o r G( P P R) is t h u s p r e d i c t e d , and t h i s s e e m s to h a v e s o m e s u p p o r t f r o m the e n e r g y d e p e n d e n c e of r e s o n a n c e p r o ductionin p+p p+ N* (1400, 1520, 1690) [5, fig. 4]. C o r r e s p o n d i n g l y , one should e x p e c t to s e e e v i d e n c e of d o u b l e P o m e r o n - e x c h a n g e at (very) high e n e r g y , f o r e x a m p l e by o b s e r v i n g an e n e r g y i n d e p e n d e n t c r o s s - s e c t i o n f o r p r o d u c t i o n of i s o s c a l a r (~+~-) s y s t e m s in the c e n t r e of m a s s of p+p-p+p+(n+~-). This is noted independently by Brower and Waltz. CERN Preprint TH.1335 (1971). ** We refer to the rapid fall-off in energy (like l / s ) of the background in m i s s i n g - m a s s experiments [3]. *** The smallness has alternatively been discussed in terms of the nearness of the Pomeron intercept to one, by Abarbanel e t a [ . [4}.
References b)
N
c} Fig. 3. Pomeron couplings. (a)P P P vertex. (b) P P R vertex. (c) Duality transformation for P P R vertex. 592
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