Diffraction shrinking and regge trajectories

Volume 5, number 2

PHYSICS LETTERS

DIFFRACTION

SHRINKING

AND

REGGE

15 June 1963

TRAJECTORIES

R. J. N. P H I L L I P S

A. E. R. E., Harwell, England Received 16 April 1963

If a single R e g g e pole d o m i n a t e s h i g h - e n e r g y s c a t t e r i n g , its t r a j e c t o r y ~(t) can be found f r o m the e n e r g y dependence of differential c r o s s s e c t i o n s 1) (-t being the s q u a r e d m o m e n t u m t r a n s f e r ) . Now the e n e r g y dependence of total c r o s s s e c t i o n s , with the optical t h e o r e m , show that m o r e than one pole p l a y s a p a r t at t = 0, f o r p r e s e n t a c c e l e r a t o r e n e r g i e s , but s u g g e s t that one pole d o m i n a t e s above say 10 GeV. If this is t r u e f o r t < 0 a l s o , a s i n g l e - p o l e i n t e r p r e t a t i o n of data g i v e s the leading t r a j e c t o r y a p p r o x i m a t e l y 2). The o b s e r v e d shrinking of the p - p d i f f r a c t i o n peak as e n e r g y i n c r e a s e s then sugg e s t s 3) that f o r t < 0, a(t) falls well below the value ~(0) = 1. H o w e v e r , s e c o n d a r y poles m a y be quite i m p o r t a n t in the r e g i o n t < 0, even at 30 GeV, in which c a s e a s i n g l e - p o l e a n a l y s i s could be m o s t misleading. The p r e s e n t note d i s c u s s e s this p o s s i b i l i t y , and in p a r t i c u l a r the c a s e s w h e r e two poles dominate. A s s u m e f i r s t that s p i n - d e p e n d e n t s c a t t e r i n g is negligible in the d i f f r a c t i o n peak ; this is r e a s o n a b l e at l e a s t f o r the v a c u u m , ~ and ~ t r a j e c t o r i e s 1,2) and l e a v e s a s c a l a r p r o b l e m . Then the a s y m p t o t i c contribution to the s c a t t e r i n g amplitude f r o m each Regge pole has the f o r m

Aj = ~j(t)

1 + cj exp (- i ~ ~j(t))( E L ~ ~j(t) (1) sin ~r c~j(t) \ Eoj/ "

H e r e j labels the R e g g e pole ; a~(t) is the t r a j e c t o r y and B~(t) is a r e s i d u e function (IJoth supposedly r e a l in t h 6 r e g i o n of interest) ; cj = ± 1 is the s i g n a t u r e ; EOj is an e n e r g y s c a l e and E L is the total l a b o r a t o r y b o m b a r d i n g e n e r g y . C l e a r l y Bj and Eo ~ a r e not •

.

T::7~dlittudP:r:d~'~r:d

.

~

.

2 th:t%a~ ~fEer0~:ntilGeV

c r o s s s e c t i o n is d

/dt

= IZ. #I 2

2 •

(2)

] If a single pole d o m i n a t e s c o m p l e t e l y , then at fixed t In(dcr/dt) v a r i e s l i n e a r l y with In E L and the slope is 2~(t) - 2 ; this l o g - l o g plot is b o t h the s t a n d a r d test of s i n g l e - p o l e d o m i n a n c e and a m e a n s of deducing a(t). However, if the e n e r g y r a n g e and e x p e r i m e n t a l a c c u r a c y a r e limited, a c u r v e d line

m a y a p p e a r e s s e n t i a l l y s t r a i g h t ; thus two o r m o r e poles (or b r a n c h cuts 4, 5)) can m a s q u e r a d e as a single pole and m i s l e a d the o b s e r v e r . This is the situation to be d i s c u s s e d . C o n s i d e r just two t e r m s (j = 1,2) at s o m e fixed t, with ~1 > a2- Then at high enough e n e r g y the upper pole d o m i n a t e s and the plot of ln(dc;/dt) v e r s u s In E L is l i n e a r with slope 2'v 1 - 2 ; at low enough e n e r g y the lower pole d o m i n a t e s and the slope is 2~ 2 - 2. (The low e n e r g y l i m i t has no p h y s i c a l s i g n i f i c a n c e , s i n c e n e i t h e r the a s y m p t o t i c f o r m in eq. (1) nor the neglect of other poles a r e justified h e r e , but it helps to show the t r e n d of e n e r g y variation. ) At i n t e r m e d i a t e e n e r g i e s t h e r e is a m o r e o r l e s s s m o o t h change of s l o p e ; it is helpful to mention t h r e e s p e c i a l c a s e s . 1. If the two R e g g e pole t e r m s a r e out of p h a s e ( e . g . , ¢1 = ¢2, ~1 = ~2 + 1), their contributions to d ~ / d t do not i n t e r f e r e and t h e r e is a s m o o t h t r a n sition as in fig. 1, c u r v e (a). 2. If the two t e r m s i n t e r f e r e c o n s t r u c t i v e l y , the t r a n s i t i o n is m o r e g r a d u a l and the c u r v a t u r e is l e s s , as in fig. 1, c u r v e (b). 3. If the two t e r m s a r e in p h a s e and i n t e r f e r e d e s t r u c t i v e l y , d c / d t v a n i s h e s at s o m e e n e r g y , as in fig. 1, c u r v e (c). If they a r e s o m e w h a t out of p h a s e , d ~ / d t does not vanish exactly and the dip is p a r t l y filled. F o r a limited e n e r g y r a n g e , each c u r v e in fig. 1 is roughly l i n e a r and s u g g e s t s a single Regge pole with s o m e a p p a r e n t t r a j e c t o r y a. In c a s e s 1 and 2 a lies between a l and a2 ; in c a s e 3 it lies above a l o~" below a2 ; in each c a s e the value at any p a r ticular e n e r g y depends on the r a t i o of E1 to B2" Thus two suitable poles can s i m u l a t e any value of a. But the question r e m a i n s : o v e r what e n e r g y r a n g e and to what a c c u r a c y does the log-lug plot a p p e a r l i n e a r ? F o r an a n s w e r , c o n s i d e r two examples. E x a m p l e I : two flat t r a j e c t o r i e s fake a sloping one. Take ~l(t) ---1, e 1 = + l ; a 2 ( t ) = 0 , ¢2 = - 1. Then a suitable choice of B2(t)/Bl(t) s i m u l a t e s a sloping " P o m e r a n c h u k " t r a j e c t o r y a(t). The plots of l n ( d a / d t ) v e r s u s in E L a r e shown in fig. 2, f o r v a r i o u s values of the a p p a r e n t t r a j e c t o r y . F o r a 159

Volume 5, number 2

PHYSICS LETTERS

15 June 1963 6

10

18

i

i

i

30 EL GeV i

0

~=I0

~

a=08

-2

v c -3

-4

~

L.

I

I

I

Ill

I I I In E L (0rb~trory Scc]le)

i

.... I

Fig. 1. The dependence of ln(d~/dt) on l n E L , for two Regge poles. Three particular cas6-s (a), (b) and (c) are shown. f a c t o r 5 in e n e r g y r a n g e , f r o m 6 to 30 GeV, the c u r v e s a r e l i n e a r within 10% (The v a l u e of t and the a b s o l u t e m a g n i t u d e of the ~j f o r any p a r t i c u l a r are freely adjustable). E x a m p l e I I : two s l o p i n g t r a j e c t o r i e s fake a f l a t one. T a k e ~ l ( t ) and ~ ( t L t o b e p a r a l l e l s t r a i g h t l i n e s of slope ~ ( G e V / c ) ' ' , p a s s i n g t h r o u g h ¢Xl(0) = 1 and c~(0) = 0.2, with e I = + 1 and e 2 = - 1. T h e n a s u i t a b l e c h o i c e of fl2(t)TBl(t) s i m u l a t e s a f l a t " P o m e r a n c h u k " t r a j e c t o r y , f o r a f a i r r a n g e of t. P l o t s of I n ( d a / d t ) v e r s u s in E L a r e shown in fig. 3, f o r E L b e t w e e n 6 and 30 GeV. The c u r v a t u r e i s s e e n to b e c o m e s e r i o u s as 1tl i n c r e a s e s . But i n the s m a l l e r e n e r g y r a n g e 1 0 - 30 GeV t h e r e i s s t i l l a good s i m u l a t i o n of a s i n g l e t r a j e c t o r y with s m a l l slope. F r o m the d i s c u s s i o n and e x a m p l e s a b o v e , the following p o i n t s e m e r g e . 1. Two Regge poles can s i m u l a t e - within l i m i t s a s i n g l e one. 2. T h i s a p p a r e n t t r a j e c t o r y m a y l i e f a r f r o m the t r u e l e a d i n g t r a j e c t o r y . In c e r t a i n c a s e s it c a n lie a n y w h e r e . 3. T h e r e s e m b l a n c e to a s i n g l e pole is m o r e c o n v i n c i n g when the a p p a r e n t t r a j e c t o r y l i e s b e t w e e n the t r u e t r a j e c t o r i e s , o r i s c l o s e to one of t h e m . 4. The d i f f r a c t i o n p a t t e r n c a n s h r i n k o v e r a f i n i t e e n e r g y r a n g e even when n e i t h e r t r a j e c t o r y has a slope. V a r i o u s r e m a r k s m a y b e added. 5. F u r t h e r p o l e s could m a k e the l o g - l o g p l o t s m o r e

160

=

t

10%

-5 15

2=0

215

310 In EL

315

Fig. 2. Two Hat trajectories fake a sloping one. ln(d~/dt) is plotted against In ET; the apparent trajectory a is indicated for each c~rve. + 10% uncertainty is illustrated, for comparison.

6 i

10

18 i

30 EL GeV i

~i=I0

cq=09

0(,=08

i

(X~=07

J

~3 J

~

o(t=0 6

-4

= 10% -5 15

210

215

310 In EL

315

Fig. 3. Two sloping trajectories fake a fiat one. ln(dT/dt) is plotted against In EL: the value of the true leading trajectory a 1 is shown for each curve. + 10% uncertainty is illustrated, for comparison.

Volume 5, number 2

PHYSICS

LETTERS

15 June 1963

n e a r l y l i n e a r ~ but s e e m to add nothing r e a l l y new 9. An a p p l i c a t i o n of p o i n t 3 a b o v e . When a s i n g l e ( B r a n c h c u t s p r e s u m a b l y a c t l i k e s e t s of p o l e s ) . pole analysis suggests a sloping trajectory, it is 6. S p i n - d e p e n d e n t a m p l i t u d e s would c o n t r i b u t e e x t r a e a s y to b e l i e v e t h a t the t r u e l e a d i n g t r a j e c t o r y t e r m s to d e / d t , e a c h of w h i c h c o u l d h a v e the b e m a y h a v e a s m a l l e r s l o p e , b u t not t h a t i t h a s a haviour discussed above. much bigger slope. 7. Eq. (1) t a k e s o n l y t h e h i g h e s t p o w e r of EL, f o r each Regge pole. For a true scalar prob~m, this I a m g r a t e f u l to D r . P . G . B u r k e f o r u s e f u l c o n d e p e n d s on a p p r o x i m a t i n g a L e g e n d r e f u n c t i o n and v e r s a t i o n s . is reasonable for the examples considered ( ~ ( t ) >/- 0.2, E L >16 GeV, a s s u m i n g n u c l e o n o r plon m a s s e s ) . But in g e n e r a l the i m p o r t a n c e of l o w e r t e r m s c a n a l s o d e p e n d on t h e r a t i o s of i n dependent residue functions (see for instance the References 1) E .g. : S C. Frautsehi, M.Gell-Mann and F. Zaehariasen, - N s • i n - f l i p a m p l i t u d e in eq. (4.6) of F r a u t s c h i Phys. Rev. 126 (1962) 2204; Proc. 1962 Int. Conf. on et al. 1)). Such c o r r e c t i o n s m a y n o t a l l b e n e g High-Energy Physics (CERN, Geneva}. l i g i b l e at a c c e l e r a t o r e n e r g i e s (as p o i n t e d out by F.Hadjioannou, R. J.N. Phillips and W.Rarita, Phys. G. F . Chew 6)). 2) Rev. Letters 9 (1962} 183. 8. R e c e n t l y N a m b u and S u g a w a r a 7) h a v e s t r e s s e d 3) A.N.Diddens et a l . , Phys.Rev. Letters 9 (1962} 108, that diffraction shrinking over a finite energy 111. r a n g e d o e s not n e c e s s a r i l y i m p l y a s y m p t o t i c 4) D.Amati, S. Fubini and A.Stanghellini, Physics Letters 1 (1962) 29. s h r i n k i n g . E x a m p l e I i l l u s t r a t e s t h i s point. I. R. Gatland and J.W. Moffat, Phys. Rev., in p r e s s . E x a m p l e II i l l u s t r a t e s the c o n v e r s e , t h a t a b s e n c e 5) 6) R. J. Eden, private communication. of s h r i n k i n g o v e r a f i n i t e r a n g e m a y a l s o b e i l 7) Y.Nambu and M.Sugawara, Phys. Rev. Letters 10 (1963} lusory. 304.

INTERNAL

CONVERSION

COEFFICIENT

OF

14.4

keV

F e 57

TRANSITION

*

A. H. MUIR, J r .

North American Aviation Science Center, Canoga Park, California and E. K A N K E L E I T and F . BOEHM

California Institule of Technology, Pasadena, California Received 6 May 1963

R e c e n t l y , T h o m a s et al. 1) r e p o r t e d a new v a l u e s T = 9.94 + 0.60 f o r the t o t a l i n t e r n a l c o n v e r s i o n c o e f f i c i e n t of t h e 14.4 keV T - r a y in F e 57. B e c a u s e of the s u b s t a n t i a l d i s a g r e e m e n t of t h i s r e s u l t with the g e n e r a l l y a c c e ~ t e d p r e v i o u s v a l u e c~T = 15 :L 1 of L e m m e r et al. ) and b e c a u s e of the i m p o r t a n c e of t h i s c o n v e r s i o n c o e f f i c i e n t f o r c e r t a i n F e 57 M ~ s s b a u e r effect i n v e s t i g a t i o n s **, w e d e c i d e d to m a k e a c a r e f u l i n d e p e n d e n t m e a s u r e m e n t of t h i s q u a n t i t y . It m a y b e r e m a r k e d that s o m e w o r k e r s 3 , 4 ) h a v e s u g g e s t e d that the v a l u e ,~ = 15 i s too l a r g e on * P a r t of the support for this work was provided by the United States Atomic Energy Commission. ** Since in any case c is considerably greater than unity, the resonance c r o s s - s e c t i o n is nearly inverseley p r o portional to c..

t h e b a s i s of Mt~ssbauer e f f e c t m e a s u r e m e n t s . T h e d e c a y of Co57 and t h e p r o p e r t i e s of the F e 57 l e v e l s h a v e b e e n s t u d i e d by a n u m b e r of w o r k e r s , and the d e c a y s c h e m e a s shown in fig. 1 i s now w e l l e s t a b l i s h e d 5). P r e v i o u s v a l u e s of a T h a v e b e e n d e r i v e d p r i m a r i l y f r o m m e a s u r e m e n t s of r e l a t i o n s h i p s b e t w e e n t h e T - r a y s and X - r a y s in the Co 57 d e c a y . T h o m a s et al. 6) d e t e r m i n e d a T by c o m p a r i n g t h e t o t a l i n t e n s i t y of 72 + 73 with the c o i n c i d e n c e i n t e n s i t y of 71, 72. Both s c i n t i l l a t i o n and g a s p r o p o r t i o n a l c o u n t e r s w e r e u s e d . L e m m e r et a l . 2) m a d e two d i f f e r e n t d e t e r m i n a t i o n s of a T. In one c a s e t h e y c o m p a r e d t h e i n t e n s i t y of T1 m e a s u r e d with a k r y p ton p r o p o r t i o n a l c o u n t e r to the t o t a l i n t e n s i t y of 72 and 73 m e a s u r e d with a s c i n t i l l a t i o n c o u n t e r and

161