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Regge pole dominance at high energies; an experimental test
Volume 21, number 3
PHYSICS LETTERS
Table 1 Predictions for the coefficient of q2 in eq. (2). Partial wave ~-plane s-plane dispersion relations W-plane Cini-Fubini relation Chew-Low theory SU6 (G.V. interpretation) Current algebra Experiment
15 May 1966
(iii) The p a r t i a l wave d i s p e r s i o n r e l a t i o n in the s - p l a n e gives a good r e s u l t , s u g g e s t i n g that the left hand cut i n t e g r a l i s m o r e r a p i d l y c o n v e r g e n t h e r e than in the w- or W-plane; t h e o r e t i c a l a r g u m e n t s why this might happen a r e given in ref. 10.
1.25 0.69 0.91 1.5 1.33 0.96 = 0.73 0.68
I would like to thank Dr. H. Burkhaxdt for u s e ful suggestions and long d i s c u s s i o n s c o n c e r n i n g this work. 1. R.Gatto and G.Veneziano, Physics Letters 19 (1965) 512 and 20 (1966) 439. 2. G.Furlan, R.Jengo and E.Remiddi, Trieste P r e print 1C/66/13 (1966). 3. D. Amati and S. Fubtni, Ann. Rev. Nuclear Science 12 (1962) 359. 4. J. Hamilton and W. S. Woolcock, Rev. Mod. Phys. 35 (1963) 737; table 3. 5. A.Donnachie, A.T.Lea and C.Lovelace, CERN preprint 65/1320/5 - TH. 599 (1965). 6. A.Dcmmchie, J.Hamilton and A.T.Lea, Phys.Rev. 135 (1964) B515. 7. Sum rules like this have been derived and used for the lrN amplitudes by A.Donnachie and J.Hamilton, Phys.Rev. 138 (1965) B678. 8. G.F.Chew and F.E.Low, Phys.Rev. 101 (1956) 1570. 9. D.H.Lyth, University of Birmingham preprint. 10. D.H.Lyth, Consequences of pure absorption, etc. (University of Birminghnrn preprint, revised version in preparation).
gives in the n a r r o w r e s o n a n c e a p p r o x i m a t i o n the value { f 2 for (1), independently of cor (i.e. inde-
pendently of the value of the Chew-Low cutoff parameter). T h r e e r e m a r k s a r e worth making. (i) The value of the i n t e g r a l (1) follows f r o m v e r y g e n e r a l t h e o r e t i c a l a r g u m e n t s , w h e r e a s the i n t e g r a n d depends on the details of the d y n a m i c s (e.g. high e n e r g y i n e l a s t i c i t y in the N / D method, o r the value of the Chew-Low cutoff p a r a m e t e r ) . In the n a r r o w r e s o n a n c e a p p r o x i m a t i o n this a m o u n t s to s a y i n g that the width I" and the position q r depend on the d e t a i l s f, but the quantity (I'/q3r) does not. (if) In view of this, the suggestion of Gatto and Veneziano that the N* d y n a m i c s can be i m p r o v e d by i m p o s i n g the ' s y m m e t r y r e s u l t ' (1) is not r e a s o n a b l e ; the suggestion a m o u n t s to i n s e r t i n g into the d y n a m i c s a l m o s t the only thing it can predict.
f The precise extent of the uncertainty in the N* mass (given the left hand cut) is e~,aluated in: A simple graph ical method for predicting resonances [9].
*****
REGGE
POLE DOMINANCE AT HIGH ENERGIES; AN E X P E R I M E N T A L TEST
E. ABERS *
H. BURKHARDT
University of California, Los Angeles
University of Birmingham, England and
V. T E P L I T Z
C. WILKIN
Massachusetts Institute of Technology, Cambridge, Mass.
Broohhaven National Laboratory, Upton, New York Received 12 April 1966
The energy dependence of the shadow effect in the scattering of high energy particles from deuterons is shown to provide a sensitive check of the dominance of moving singularities in the angular momentum plane in the high energy scattering from protons and.neutrons. Regge theory predicts that the shadowing should decrease as an inverse power of the energy.
We wish to propose a new type of e x p e r i m e n t a l check of the famous c o n j e c t u r e [1] that high e n e r * Partially supported by National Science Foundation.
gy e l a s t i c s c a t t e r i n g a m p l i t u d e s can be d e s c r i b e d in s o m e a p p r o x i m a t i o n by the exchange of a s e t of Regge t r a j e c t o r i e s . Our p r o c e d u r e i s l e s s d i r e c t , 339
Volume 21, number3
PHYSICS LETTERS
d
15 May 1966
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Fig. 2. Diagram for the eclipse correction to x-d scattering.
Fig. 1. Diagram for the impulse approximation to x-d scattering. t h e o r e t i c a l l y , than a t t e m p t s to p a r a m e t r i z e highe n e r g y c r o s s s e c t i o n s by t r a j e c t o r i e s , but m a y n e v e r t h e l e s s be s u b j e c t to f e w e r a m b i g u i t i e s . We do not a s k the e x p e r i m e n t s to fit a d e t a i l e d equation but only to a n s w e r a y e s - n o type of question, n a m e l y , d o e s the e c l i p s e c o r r e c t i o n to the s c a t t e r i n g of h i g h - e n e r g y p a r t i c l e s on d e u t e r o n s v a n i s h like a power of the incident e n e r g y , o r only l o g a r i t h m i c a l l y ? The q u e s t i o n of h i g h - e n e r g y d e u t e r o n s c a t t e r ing has a long h i s t o r y . C o n s i d e r the s c a t t e r i n g of a s c a l a r p a r t i c l e , x, on a d e u t e r o n , d, which, to s i m p l i f y the a r g u m e n t , we c o n s i d e r to be a s c a l a r bound s t a t e of two s c a l a r nucleons. In p r a c t i c e , the p r o j e c t i l e s x about which we have s o m e i n f o r m a t i o n a r e n u c l e o n s and pions and p e r h a p s kaons. In f i r s t a p p r o x i m a t i o n the total xd c r o s s s e c t i o n (~xd i s j u s t axn + axp'. But, even i g n o r i n g t h r e e - b o d y i n t e r a c t i o n s , axd i s d i m i n i s h e d f r o m this s u m b e c a u s e e a c h nucleon h a s a finite p r o b a b i l i t y of b e i n g in the " s h a d o w " of the other. Using a s e m i - c l a s s i c a l a p p r o x i m a t i o n , G l a u b e r [3] h a s given a f o r m u l a for this c o r r e c tion. Let t be the i n v a r i a n t m o m e n t u m t r a n s f e r v a r i a b l e for e i t h e r x - N o r x - d s c a t t e r i n g , and w the incident l a b o r a t o r y - f r a m e e n e r g y . L e t F ( w , t) be the s c a t t e r i n g a m p l i t u d e n o r m a l i z e d so that the d i f f e r e n t i a l c r o s s s e c t i o n is
340
d~/dt= IF(~,t)[ 2 Then G l a u b e r ' s f o r m u l a r e a d s : (~xd = gxn + gxp +
(1)
+ -2- R e f G ( p 2 ) F x p ( w , - p2) Fxn (w , - p 2 ) d 2 p , ~7
w h e r e G(t) is the d e u t e r o n f o r m f a c t o r , n o r m a l i z e d so that G(0) = 1. T h e r e is an analogous f o r m u l a for the e l a s t i c a m p l i t u d e Fxd(W , t). Udgaonkar and G e l l - M a n n [4] e x a m i n e d what h a p p e n s when the F x N a r e given by the exchange of the P o m e r a n c h u k t r a j e c t o r y , i.e., when FxN(W , t) = ~iJ-~C~xn(¢O) BN(t)Cs/so) a (t)- 1 ,
(2)
w h e r e BN(0) = a(0) = 1. They pointed out that if eq. (2) i s i n s e r t e d into eq. (1), then the c o r r e c tion has the t y p i c a l f o r m of a Regge cut, with a b r a n c h point at 5(t) = 2 a ( ¼ t ) - 1. It i s w e l l known [5] that the f i r s t two t e r m s on the r i g h t hand s i d e of eq. (1) c o r r e s p o n d to d i a g r a m s like fig. 1, in the s e n s e that they contain the a n o m a l o u s cut contribution, which, b e c a u s e of the d e u t e r o n ' s s m a l l binding e n e r g y , dominate., the d i a g r a m in the p h y s i c a l r e g i o n . T h i s "impulse a p p r o x i m a t i o n " m a y be obtained by taking only th( m a s s - s h e l l p a r t of the p r o p a g a t o r belonging to th~ s p e c t a t o r nucleon, and identifying the r e m a i n i n g p r o p a g a t o r s with the d e u t e r o n wave-function. By an analogous a r g u m e n t it has been shown [6, 7]
Volume 21, number 3
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that the e c l i p s e c o r r e c t i o n c o r r e s p o n d s to an a p p r o x i m a t i o n to the d i a g r a m of fig. 2. Being a "Regge cut" t e r m , the e c l i p s e c o r r e c tion a s w r i t t e n in eq. (1) v a n i s h e s like I / l o g f o r l a r g e e n e r g i e s . At m o r e m o d e r a t e e n e r g i e s it m a y be a p p r o x i m a t e d by [3] (1/47r) ~xn" ~xd)~~< r - 2 > ,
(3)
w h e r e r is the s e p a r a t i o n of the n u c l e o n s in the deuteron. However, M a n d e l s t a m [8] h a s shown that at high e n e r g i e s c a n c e l l a t i o n s m a y o c c u r in d i a g r a m s like that of fig. 2. In that c a s e , the Re~ge cut b e h a v i o u r a p p e a r s on the s e c o n d s h e e t of the e n e r g y v a r i a b l e and thus does not d o m i n a t e the p h y s i c a l s c a t t e r i n g for l a r g e e n e r g i e s . We have shown in r e f . 7 that t h e r e i s indeed no Regge cut in fig. 2. It t h e r e f o r e follows that the c o r r e c t i o n is in fact p r o p o r t i o n a l to o~2~(°°)-2 , w h e r e ~(oo) i s the l e a d ing s i n g u l a r i t y in the a n g u l a r m o m e n t u m plane for t ~ _oo. This conclusion holds only if ~(oo) < 1. If 9(°o) = 1, the l o g a r i t h m i c b e h a v i o u r s t i l l obt a i n s . The c a n c e l l a t i o n , in the c a s e 9(00) < 1, c o m e s f r o m the c o n t r i b u t i o n of m a n y p a r t i c l e i n t e r m e d i a t e s - c h a n n e l s t a t e s . In p h y s i c a l t e r m s t h i s c o r r e s p o n d s to the c a n c e l l a t i o n of the G l a u b e r shadowing of one nucleon by a n o t h e r , by r a d i a t i o n f r o m the f i r s t n u c l e o n ' s pion cloud. Thus if the u s u a l P o m e r a n c h u k t r a j e c t o r y p i c t u r e of x - N s c a t t e r i n g i s c o r r e c t , the e c l i p s e effect, ~ x d - ~rxp- ~xn, will v a n i s h like s o m e p o w e r of the e n e r g y ; in s i m p l e m o d e l s such a s the l a d d e r a p p r o x i m a t i o n [9], a s f a s t a s co-4. It is not excluded that x - N s c a t t e r i n g c o n t a i n s a fixed cut with b r a n c h point at cr = 1 f o r a l l t. Such a b e h a v four, unhappy though it be for a s i m p l e p i c t u r e of high e n e r g y s c a t t e r i n g , does not v i o l a t e any known bound [e.g. 10]. F u r t h e r m o r e the p o s s i b i l i t y exists that the sequence of ladder diagrams in which the "rungs" are themselves Pomeranchuk trajectories gives a sequence of cuts whose branch points approach l = 1 for aUt. Dominance of high energy scattering by this sequence cannot at present be excluded. To the extent that x-N scattering allows description by the exchange of simple Regge trajectories without the fixed I = 1 cut, the eclipse effect should drop off from the value given by eq. (3) like some power of the energy, rather than the relatively constant [1/log 0J] behaviour cf Glauber's formula. In s u m m a r y our a i m i s to t e s t the t - b e h a v i o u r of the l e a d i n g lplane s i n g u l a r i t y ~(t) by e x a m i n i n g a quantity s e n s i t i v e to the a m p l i t u d e F(s, t) at m o d e r a t e l y high ( a c c e s s i b l e ) s and l a r g e n e g a t i v e t in cont r a s t to e l a s t i c s c a t t e r i n g e x p e r i m e n t s which look a t F(s, t) i t s e l f at v e r y high (and c u r r e n t l y i n a c c e s s i b l e ) s and s m a l l n e g a t i v e t.
LETTERS
15 May 1966
F i n a l l y , we should point out that t h e r e m a y be 1 / l o g w c o n t r i b u t i o n s to x - d s c a t t e r i n g f r o m c o m p l i c a t e d d i a g r a m s with i n e l a s t i c i n t e r m e d i a t e s t a t e s , which do have moving cuts in the /-plane. N u m e r i c a l e s t i m a t e s i n d i c a t e that t h e s e a r e s u b s t a n t i a U y s m a l l e r than the c o r r e c t i o n due to fig. 2 in the r e g i o n w h e r e that i s i m p o r t a n t , and w h e r e i t s magnitude can be e s t i m a t e d f r o m eq. (3), so that p o w e r - l a w f a l l off will be o b s e r v e d o v e r a l a r g e e n e r g y r a n g e , leaving a s m a l l r e s i d u a l shadow effect. What is the p r e s e n t e x p e r i m e n t a l s i t u a t i o n ? At this t i m e r e l e v a n t m e a s u r e m e n t s of sufficient a c c u r a c y s e e m to us to be too s c a n t y to check the b e h a v i o u r of the e c l i p s e t e r m with confidence. T h i s is due p r i n c i p a l l y to u n c e r t a i n t i e s involved in e x t r a c t i n g e l e c t r o m a g n e t i c e f f e c t s and the a b s e n c e of d i r e c t ~ - n and p - n m e a s u r e m e n t s at v e r y high e n e r g i e s . A v a i l a b l e e v i d e n c e on pion c r o s s s e c t i o n s [11], h o w e v e r , s e e m s to indicate a p e r s i s t e n t , r o u g h l y c o n s t a n t c o r r e c t i o n up to about 20 GeV. But the quoted u n c e r t a i n t i e s a r e l a r g e (we need to know with s o m e p r e c i s i o n a s m a l l 5% to 8 ~ - d f f f e r e n c e between c r o s s s e c tions) and, a s a l w a y s , we can n e v e r be s u r e we a r e y e t in the " a s y m p t o t i c " r e g i o n *. W h e r e a s o b s e r v a n c e of a p o w e r - l a w f a l l off would be s t r i k i n g evidence f o r Regge pole d o m i nance, we should c l e a r l y be m o r e c a r e f u l in d r a w i n g the o p p o s i t e conclusion f r o m p r e s e n t e x p e r i m e n t s b e c a u s e of t h e s e e x p e r i m e n t a l and t h e o r e t i c a l loopholes. It s e e m s to us, t h e r e f o r e , that it would b e v e r y u s e f u l to have e x p e r i m e n t s on p - d and p - n s c a t t e r i n g (the l a t t e r r e q u i r e s a neutron b e a m ) at v e r y high e n e r g i e s of sufficient a c c u r a c y to m a k e a s t a t e m e n t with confidence about the high e n e r g y b e h a v i o u r O f ~ p d - ~pn- ~pp"
References 1. R. Blankenbecler and M. L. Goldberger, Phys. Rev. 126 (1962) 766;
G. F. Chew and S. C. Frautschi, Phys. Rev. Letters 8 (1962) 41; G.F.Chew, S.C.Frautechi and S.Mandelstam, Phys.Rev. 126 (1962) 1202; S. C. Frautschi, M. Gell-Mann and F. Zachariasen, Phys.Rev. 126 (1962) 2204. 2. R.J.N.Phillips and W.Rarita, Phys.Rev. 138 (1965) B723 and 139 (1965) B1336. We wish to emphasize that we are testing the leading singularity in the t--channel/-plane for large negative t (of the order of -s/10, say). We do not expect detailed canc¢llation mechanisms, which at small negative t might conspire to make the leading l-plane singularity appear fixed, to operate identically at large negative t. 341
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3. R . J . G l a u b e r , P h y s . Rev. 100 (1955) 242~. 4. B . M . Udgaonkar and M. Gell-Mann, Phys. Rev. L e t t e r s 8 (1962) 346. 5. R . E . C u t k o w s k y , P r o c . 1960 High energy c o n f e r e n c e , R o c h e s t e r (1960) p.236; J . J . B r e h m and J: Sucher, Ann. P h y s i c s (New York) 25 (1963) 1; F . G r o s s , P h y s . R e v . 140 (1965) B410. 6. H . N . P e n d l e t e n , P h y s . R e v . 131 (1963) 1833. 7. E. A b e r s , H. B u r k h a r d t , V. Teplitz and C. Wilkin, Nuovo Cimento, to be published. 8. S . M a n d e l s t a m , Nuovo Cimento 30 (1963) 1127 and 1148.
342
LETTERS 9 . B . W . L e e and R . F . S a w y e r ,
15 May 1966
P h y s . R e v . 127 (1962) 2266. 10. A. Martin, in Scottish U n i v e r s i t i e s S u m m e r School, ed. R. G. Moorehouse (Oliver and Boyd, E d i n b u r g , 1963) p. 105. 11. W. Galbraith, E . W . J e n k i n s , T . F . O . K y c i a , B . A . Leontic, R.H. Phillips, A . L . Read and R. Rubenstein, P h y s . R e v . 138 (1965) B913; W . F . B a k e r , E . W . J e n k i n s , T . F . O . K y c i a , R.H. Phillips, A . L . Read, K. F. Riley and H. Ruderman, P r o c . Sienna Conf. on E l e m e n t a r y p a r t i c l e s , 1963 (Societa Italiana di F i s i c a , Bologna, 1963).
PHYSICS LETTERS
Table 1 Predictions for the coefficient of q2 in eq. (2). Partial wave ~-plane s-plane dispersion relations W-plane Cini-Fubini relation Chew-Low theory SU6 (G.V. interpretation) Current algebra Experiment
15 May 1966
(iii) The p a r t i a l wave d i s p e r s i o n r e l a t i o n in the s - p l a n e gives a good r e s u l t , s u g g e s t i n g that the left hand cut i n t e g r a l i s m o r e r a p i d l y c o n v e r g e n t h e r e than in the w- or W-plane; t h e o r e t i c a l a r g u m e n t s why this might happen a r e given in ref. 10.
1.25 0.69 0.91 1.5 1.33 0.96 = 0.73 0.68
I would like to thank Dr. H. Burkhaxdt for u s e ful suggestions and long d i s c u s s i o n s c o n c e r n i n g this work. 1. R.Gatto and G.Veneziano, Physics Letters 19 (1965) 512 and 20 (1966) 439. 2. G.Furlan, R.Jengo and E.Remiddi, Trieste P r e print 1C/66/13 (1966). 3. D. Amati and S. Fubtni, Ann. Rev. Nuclear Science 12 (1962) 359. 4. J. Hamilton and W. S. Woolcock, Rev. Mod. Phys. 35 (1963) 737; table 3. 5. A.Donnachie, A.T.Lea and C.Lovelace, CERN preprint 65/1320/5 - TH. 599 (1965). 6. A.Dcmmchie, J.Hamilton and A.T.Lea, Phys.Rev. 135 (1964) B515. 7. Sum rules like this have been derived and used for the lrN amplitudes by A.Donnachie and J.Hamilton, Phys.Rev. 138 (1965) B678. 8. G.F.Chew and F.E.Low, Phys.Rev. 101 (1956) 1570. 9. D.H.Lyth, University of Birmingham preprint. 10. D.H.Lyth, Consequences of pure absorption, etc. (University of Birminghnrn preprint, revised version in preparation).
gives in the n a r r o w r e s o n a n c e a p p r o x i m a t i o n the value { f 2 for (1), independently of cor (i.e. inde-
pendently of the value of the Chew-Low cutoff parameter). T h r e e r e m a r k s a r e worth making. (i) The value of the i n t e g r a l (1) follows f r o m v e r y g e n e r a l t h e o r e t i c a l a r g u m e n t s , w h e r e a s the i n t e g r a n d depends on the details of the d y n a m i c s (e.g. high e n e r g y i n e l a s t i c i t y in the N / D method, o r the value of the Chew-Low cutoff p a r a m e t e r ) . In the n a r r o w r e s o n a n c e a p p r o x i m a t i o n this a m o u n t s to s a y i n g that the width I" and the position q r depend on the d e t a i l s f, but the quantity (I'/q3r) does not. (if) In view of this, the suggestion of Gatto and Veneziano that the N* d y n a m i c s can be i m p r o v e d by i m p o s i n g the ' s y m m e t r y r e s u l t ' (1) is not r e a s o n a b l e ; the suggestion a m o u n t s to i n s e r t i n g into the d y n a m i c s a l m o s t the only thing it can predict.
f The precise extent of the uncertainty in the N* mass (given the left hand cut) is e~,aluated in: A simple graph ical method for predicting resonances [9].
*****
REGGE
POLE DOMINANCE AT HIGH ENERGIES; AN E X P E R I M E N T A L TEST
E. ABERS *
H. BURKHARDT
University of California, Los Angeles
University of Birmingham, England and
V. T E P L I T Z
C. WILKIN
Massachusetts Institute of Technology, Cambridge, Mass.
Broohhaven National Laboratory, Upton, New York Received 12 April 1966
The energy dependence of the shadow effect in the scattering of high energy particles from deuterons is shown to provide a sensitive check of the dominance of moving singularities in the angular momentum plane in the high energy scattering from protons and.neutrons. Regge theory predicts that the shadowing should decrease as an inverse power of the energy.
We wish to propose a new type of e x p e r i m e n t a l check of the famous c o n j e c t u r e [1] that high e n e r * Partially supported by National Science Foundation.
gy e l a s t i c s c a t t e r i n g a m p l i t u d e s can be d e s c r i b e d in s o m e a p p r o x i m a t i o n by the exchange of a s e t of Regge t r a j e c t o r i e s . Our p r o c e d u r e i s l e s s d i r e c t , 339
Volume 21, number3
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15 May 1966
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d
Fig. 2. Diagram for the eclipse correction to x-d scattering.
Fig. 1. Diagram for the impulse approximation to x-d scattering. t h e o r e t i c a l l y , than a t t e m p t s to p a r a m e t r i z e highe n e r g y c r o s s s e c t i o n s by t r a j e c t o r i e s , but m a y n e v e r t h e l e s s be s u b j e c t to f e w e r a m b i g u i t i e s . We do not a s k the e x p e r i m e n t s to fit a d e t a i l e d equation but only to a n s w e r a y e s - n o type of question, n a m e l y , d o e s the e c l i p s e c o r r e c t i o n to the s c a t t e r i n g of h i g h - e n e r g y p a r t i c l e s on d e u t e r o n s v a n i s h like a power of the incident e n e r g y , o r only l o g a r i t h m i c a l l y ? The q u e s t i o n of h i g h - e n e r g y d e u t e r o n s c a t t e r ing has a long h i s t o r y . C o n s i d e r the s c a t t e r i n g of a s c a l a r p a r t i c l e , x, on a d e u t e r o n , d, which, to s i m p l i f y the a r g u m e n t , we c o n s i d e r to be a s c a l a r bound s t a t e of two s c a l a r nucleons. In p r a c t i c e , the p r o j e c t i l e s x about which we have s o m e i n f o r m a t i o n a r e n u c l e o n s and pions and p e r h a p s kaons. In f i r s t a p p r o x i m a t i o n the total xd c r o s s s e c t i o n (~xd i s j u s t axn + axp'. But, even i g n o r i n g t h r e e - b o d y i n t e r a c t i o n s , axd i s d i m i n i s h e d f r o m this s u m b e c a u s e e a c h nucleon h a s a finite p r o b a b i l i t y of b e i n g in the " s h a d o w " of the other. Using a s e m i - c l a s s i c a l a p p r o x i m a t i o n , G l a u b e r [3] h a s given a f o r m u l a for this c o r r e c tion. Let t be the i n v a r i a n t m o m e n t u m t r a n s f e r v a r i a b l e for e i t h e r x - N o r x - d s c a t t e r i n g , and w the incident l a b o r a t o r y - f r a m e e n e r g y . L e t F ( w , t) be the s c a t t e r i n g a m p l i t u d e n o r m a l i z e d so that the d i f f e r e n t i a l c r o s s s e c t i o n is
340
d~/dt= IF(~,t)[ 2 Then G l a u b e r ' s f o r m u l a r e a d s : (~xd = gxn + gxp +
(1)
+ -2- R e f G ( p 2 ) F x p ( w , - p2) Fxn (w , - p 2 ) d 2 p , ~7
w h e r e G(t) is the d e u t e r o n f o r m f a c t o r , n o r m a l i z e d so that G(0) = 1. T h e r e is an analogous f o r m u l a for the e l a s t i c a m p l i t u d e Fxd(W , t). Udgaonkar and G e l l - M a n n [4] e x a m i n e d what h a p p e n s when the F x N a r e given by the exchange of the P o m e r a n c h u k t r a j e c t o r y , i.e., when FxN(W , t) = ~iJ-~C~xn(¢O) BN(t)Cs/so) a (t)- 1 ,
(2)
w h e r e BN(0) = a(0) = 1. They pointed out that if eq. (2) i s i n s e r t e d into eq. (1), then the c o r r e c tion has the t y p i c a l f o r m of a Regge cut, with a b r a n c h point at 5(t) = 2 a ( ¼ t ) - 1. It i s w e l l known [5] that the f i r s t two t e r m s on the r i g h t hand s i d e of eq. (1) c o r r e s p o n d to d i a g r a m s like fig. 1, in the s e n s e that they contain the a n o m a l o u s cut contribution, which, b e c a u s e of the d e u t e r o n ' s s m a l l binding e n e r g y , dominate., the d i a g r a m in the p h y s i c a l r e g i o n . T h i s "impulse a p p r o x i m a t i o n " m a y be obtained by taking only th( m a s s - s h e l l p a r t of the p r o p a g a t o r belonging to th~ s p e c t a t o r nucleon, and identifying the r e m a i n i n g p r o p a g a t o r s with the d e u t e r o n wave-function. By an analogous a r g u m e n t it has been shown [6, 7]
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that the e c l i p s e c o r r e c t i o n c o r r e s p o n d s to an a p p r o x i m a t i o n to the d i a g r a m of fig. 2. Being a "Regge cut" t e r m , the e c l i p s e c o r r e c tion a s w r i t t e n in eq. (1) v a n i s h e s like I / l o g f o r l a r g e e n e r g i e s . At m o r e m o d e r a t e e n e r g i e s it m a y be a p p r o x i m a t e d by [3] (1/47r) ~xn" ~xd)~~< r - 2 > ,
(3)
w h e r e r is the s e p a r a t i o n of the n u c l e o n s in the deuteron. However, M a n d e l s t a m [8] h a s shown that at high e n e r g i e s c a n c e l l a t i o n s m a y o c c u r in d i a g r a m s like that of fig. 2. In that c a s e , the Re~ge cut b e h a v i o u r a p p e a r s on the s e c o n d s h e e t of the e n e r g y v a r i a b l e and thus does not d o m i n a t e the p h y s i c a l s c a t t e r i n g for l a r g e e n e r g i e s . We have shown in r e f . 7 that t h e r e i s indeed no Regge cut in fig. 2. It t h e r e f o r e follows that the c o r r e c t i o n is in fact p r o p o r t i o n a l to o~2~(°°)-2 , w h e r e ~(oo) i s the l e a d ing s i n g u l a r i t y in the a n g u l a r m o m e n t u m plane for t ~ _oo. This conclusion holds only if ~(oo) < 1. If 9(°o) = 1, the l o g a r i t h m i c b e h a v i o u r s t i l l obt a i n s . The c a n c e l l a t i o n , in the c a s e 9(00) < 1, c o m e s f r o m the c o n t r i b u t i o n of m a n y p a r t i c l e i n t e r m e d i a t e s - c h a n n e l s t a t e s . In p h y s i c a l t e r m s t h i s c o r r e s p o n d s to the c a n c e l l a t i o n of the G l a u b e r shadowing of one nucleon by a n o t h e r , by r a d i a t i o n f r o m the f i r s t n u c l e o n ' s pion cloud. Thus if the u s u a l P o m e r a n c h u k t r a j e c t o r y p i c t u r e of x - N s c a t t e r i n g i s c o r r e c t , the e c l i p s e effect, ~ x d - ~rxp- ~xn, will v a n i s h like s o m e p o w e r of the e n e r g y ; in s i m p l e m o d e l s such a s the l a d d e r a p p r o x i m a t i o n [9], a s f a s t a s co-4. It is not excluded that x - N s c a t t e r i n g c o n t a i n s a fixed cut with b r a n c h point at cr = 1 f o r a l l t. Such a b e h a v four, unhappy though it be for a s i m p l e p i c t u r e of high e n e r g y s c a t t e r i n g , does not v i o l a t e any known bound [e.g. 10]. F u r t h e r m o r e the p o s s i b i l i t y exists that the sequence of ladder diagrams in which the "rungs" are themselves Pomeranchuk trajectories gives a sequence of cuts whose branch points approach l = 1 for aUt. Dominance of high energy scattering by this sequence cannot at present be excluded. To the extent that x-N scattering allows description by the exchange of simple Regge trajectories without the fixed I = 1 cut, the eclipse effect should drop off from the value given by eq. (3) like some power of the energy, rather than the relatively constant [1/log 0J] behaviour cf Glauber's formula. In s u m m a r y our a i m i s to t e s t the t - b e h a v i o u r of the l e a d i n g lplane s i n g u l a r i t y ~(t) by e x a m i n i n g a quantity s e n s i t i v e to the a m p l i t u d e F(s, t) at m o d e r a t e l y high ( a c c e s s i b l e ) s and l a r g e n e g a t i v e t in cont r a s t to e l a s t i c s c a t t e r i n g e x p e r i m e n t s which look a t F(s, t) i t s e l f at v e r y high (and c u r r e n t l y i n a c c e s s i b l e ) s and s m a l l n e g a t i v e t.
LETTERS
15 May 1966
F i n a l l y , we should point out that t h e r e m a y be 1 / l o g w c o n t r i b u t i o n s to x - d s c a t t e r i n g f r o m c o m p l i c a t e d d i a g r a m s with i n e l a s t i c i n t e r m e d i a t e s t a t e s , which do have moving cuts in the /-plane. N u m e r i c a l e s t i m a t e s i n d i c a t e that t h e s e a r e s u b s t a n t i a U y s m a l l e r than the c o r r e c t i o n due to fig. 2 in the r e g i o n w h e r e that i s i m p o r t a n t , and w h e r e i t s magnitude can be e s t i m a t e d f r o m eq. (3), so that p o w e r - l a w f a l l off will be o b s e r v e d o v e r a l a r g e e n e r g y r a n g e , leaving a s m a l l r e s i d u a l shadow effect. What is the p r e s e n t e x p e r i m e n t a l s i t u a t i o n ? At this t i m e r e l e v a n t m e a s u r e m e n t s of sufficient a c c u r a c y s e e m to us to be too s c a n t y to check the b e h a v i o u r of the e c l i p s e t e r m with confidence. T h i s is due p r i n c i p a l l y to u n c e r t a i n t i e s involved in e x t r a c t i n g e l e c t r o m a g n e t i c e f f e c t s and the a b s e n c e of d i r e c t ~ - n and p - n m e a s u r e m e n t s at v e r y high e n e r g i e s . A v a i l a b l e e v i d e n c e on pion c r o s s s e c t i o n s [11], h o w e v e r , s e e m s to indicate a p e r s i s t e n t , r o u g h l y c o n s t a n t c o r r e c t i o n up to about 20 GeV. But the quoted u n c e r t a i n t i e s a r e l a r g e (we need to know with s o m e p r e c i s i o n a s m a l l 5% to 8 ~ - d f f f e r e n c e between c r o s s s e c tions) and, a s a l w a y s , we can n e v e r be s u r e we a r e y e t in the " a s y m p t o t i c " r e g i o n *. W h e r e a s o b s e r v a n c e of a p o w e r - l a w f a l l off would be s t r i k i n g evidence f o r Regge pole d o m i nance, we should c l e a r l y be m o r e c a r e f u l in d r a w i n g the o p p o s i t e conclusion f r o m p r e s e n t e x p e r i m e n t s b e c a u s e of t h e s e e x p e r i m e n t a l and t h e o r e t i c a l loopholes. It s e e m s to us, t h e r e f o r e , that it would b e v e r y u s e f u l to have e x p e r i m e n t s on p - d and p - n s c a t t e r i n g (the l a t t e r r e q u i r e s a neutron b e a m ) at v e r y high e n e r g i e s of sufficient a c c u r a c y to m a k e a s t a t e m e n t with confidence about the high e n e r g y b e h a v i o u r O f ~ p d - ~pn- ~pp"
References 1. R. Blankenbecler and M. L. Goldberger, Phys. Rev. 126 (1962) 766;
G. F. Chew and S. C. Frautschi, Phys. Rev. Letters 8 (1962) 41; G.F.Chew, S.C.Frautechi and S.Mandelstam, Phys.Rev. 126 (1962) 1202; S. C. Frautschi, M. Gell-Mann and F. Zachariasen, Phys.Rev. 126 (1962) 2204. 2. R.J.N.Phillips and W.Rarita, Phys.Rev. 138 (1965) B723 and 139 (1965) B1336. We wish to emphasize that we are testing the leading singularity in the t--channel/-plane for large negative t (of the order of -s/10, say). We do not expect detailed canc¢llation mechanisms, which at small negative t might conspire to make the leading l-plane singularity appear fixed, to operate identically at large negative t. 341
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3. R . J . G l a u b e r , P h y s . Rev. 100 (1955) 242~. 4. B . M . Udgaonkar and M. Gell-Mann, Phys. Rev. L e t t e r s 8 (1962) 346. 5. R . E . C u t k o w s k y , P r o c . 1960 High energy c o n f e r e n c e , R o c h e s t e r (1960) p.236; J . J . B r e h m and J: Sucher, Ann. P h y s i c s (New York) 25 (1963) 1; F . G r o s s , P h y s . R e v . 140 (1965) B410. 6. H . N . P e n d l e t e n , P h y s . R e v . 131 (1963) 1833. 7. E. A b e r s , H. B u r k h a r d t , V. Teplitz and C. Wilkin, Nuovo Cimento, to be published. 8. S . M a n d e l s t a m , Nuovo Cimento 30 (1963) 1127 and 1148.
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LETTERS 9 . B . W . L e e and R . F . S a w y e r ,
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P h y s . R e v . 127 (1962) 2266. 10. A. Martin, in Scottish U n i v e r s i t i e s S u m m e r School, ed. R. G. Moorehouse (Oliver and Boyd, E d i n b u r g , 1963) p. 105. 11. W. Galbraith, E . W . J e n k i n s , T . F . O . K y c i a , B . A . Leontic, R.H. Phillips, A . L . Read and R. Rubenstein, P h y s . R e v . 138 (1965) B913; W . F . B a k e r , E . W . J e n k i n s , T . F . O . K y c i a , R.H. Phillips, A . L . Read, K. F. Riley and H. Ruderman, P r o c . Sienna Conf. on E l e m e n t a r y p a r t i c l e s , 1963 (Societa Italiana di F i s i c a , Bologna, 1963).