Volume 26B. number 8 P H Y S I C S L E T T E R S 18 March 1968

F I N I T E E N E R G Y S U M R U L E S A N D T H E C R O S S O V E R P R O B L E M I N K - N S C A T T E R I N G

P. DI V E C C H I A , F. DRAGO Laboratori Nazionali del C . N . E . N . , Frascati , Italy

and

M. L. P A C I E L L O Istituto di Fisica dell 'UniversitY, Roma, Italy

Received 6 February 1968

We study the behaviour of the cO-residue functions in kaon-nueleon elastic scattering using the "finite energy sum rules" technique. We find that both spin flip and non spin flip co-residues show a zero near t ~ -0.1 (GeV/c)2: this is in agreement with the usual explanation of the crossover phenomena in KN and NN elastic scattering, but in contradiction with the behaviour of the CO-residue functions deduced from other reactions.

Recently the Regge pole model experienced some difficulties in connection with the so called "cross- over phenomena" [1-3] in the elastic scattering ~±p, Kip, pp and pp.

In fact these phenomena have been explained by Phillips [5,6] and Rarita assuming that the ¢o and p Regge pole residue functions vanish at the cross-over point t = t c. However it has been shown that the amplitudes O~k} characterized by Regge poles with odd change conjugation in the crossed channel [3], do not vanish-at- t = t c in the r e a c t i o n s ~ + N ~ p + N [7] and Y + N --*~ + N [4]. M o r e o v e r fo r the pho to - p roduc t i on r e a c t i o n i t ha s been shown tha t the co e x c h a n g e g i v e s the only i m p o r t a n t con t r i bu t ion to O{~}(I=O). T h i s i m p l i e s that , if the i n t e r p r e t a t i o n of the c r o s s o v e r p h e n o m e n a i s c o r r e c t ( i .e. v a n i s h - ing of O{k}) and if the co e x c h a n g e g i v e s the only i m p o r t a n t con t r ibu t ion to O{~} in n u c l e o n - n u c l e o n s c a t t e r i f i g , the f a c t o r i z a t i o n b r e a k s down and, wi th i t , the p r e d i c t i v i t y of the R e g g e po le t heo ry . In o r - d e r to avo id th is unp l ea san t s i tua t ion , B a r g e r and Durand [2] h a v e s p e c u l a t e d about the p o s s i b l e e x i s - t e n c e of add i t iona l ~ c o n t r i b u t i o n s .

In th is l e t t e r we sha l l s tudy the c r o s s o v e r p r o b l e m in K - N s c a t t e r i n g , u s ing the " f in i t e e n e r g y s u m r u l e " a p p r o a c h [8-10]. T h i s t e c h n i q u e e n a b l e s us to s tudy the spin f l ip and non s p i n - f l i p r e s i d u e f u n c - t ions as a func t ion of t, r e l a t i n g t h e m to low e n e r g y i n t e g r a l s .

The s a m e t e c h n i q u e has been u s e d by the a u t h o r s [3] in o r d e r to s tudy the b e h a v i o u r of the w r e s i d u e func t ions in no pho toproduc t ion : t h e s e func t ions do not p r e s e n t any z e r o e s at t = tc, c o n t r a d i c t i n g , in a s t r i k i n g m a n n e r , the u sua l exp lana t ion of the c r o s s o v e r p h e n o m e n a .

Only the ca e x c h a n g e in the c r o s s e d t - c h a n n e l c o n t r i b u t e s to the a m p l i t u d e

Mco (v, t) = ¼ [ M ( K - p ~ K - p ) + M(K+p-*K+p) - M ( K - n - * K - n ) - M(K+n ~ K + n ) ] ,

w h e r e v = co + t / 4 m N , co be ing the e n e r g y of the inc iden t K m e s o n in the l a b o r a t o r y s y s t e m . At a s y m p - to t i c e n e r g i e s we a s s u m e that

I m Aco(v, t) = (2~co + 1)SA ((~co) (C( t ) /m K) (V/mK) ~co(t) , (1)

ImBco(v , t) = SB(aco) (D(t) /m 2) ( v / m K ) a w ( t ) - I . (2)

$ A(p,t) is the non spinflip amplitude, which Singh call A' [II]. S A and S B depend on the behaviour of the co trajec- tory near the nonsense point otco = 0: if the co "chooses sense" $4((~) ~ I and SB(OI) ~Ol, if it chooses nonsense, as recently suggested [3], SA(OD ~ SB(Ot) ~Ol.

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Volume 26B, number 8 P H Y S I C S L E T T E R S

Taking into account the c ross ing proper t i es one has the following sum rules

(2C~co + 1)SA(aw) C(t) ( c°A + t/4mN ] aw(t)+l 1 f A ImAw(~, t)dw = 2.2mN(C~co + 1) \ mK / 2"2mN o

18 March 1968

, (3)

mNfWA u ImBw(u , t)dw 27r2 o

mNSB(aw) D(t) ( ¢°A + t/4 rrlq) aw(t)+l

2 .2(aw + 1) mK (4)

Extract ing the pole t e rms and assuming that the amplitudes a re dominated by the Y;(1405) and Y~(1385) f rom the (An) to the physical threshold, one has:

2 .

2 rn~q (~-~-~2 4 ~-2°~ m21 X+(Yo,t)+~(__~_~ ) mN + I ( g A A ) ~ x (A,t) 3 gE 1 _ _ 3 "gY1 \ A Z ( Y ~ ' t ) (5)

+ _ _ 1 ~o = (2a~o+l)SA(a) (WA + t/4m N a w ( t ) + l A ImAw(u,t)dw C(t) . mK / ,

2.2raN m K 2n2mN(aw + 1)

and

1 (.g2 3 g2 1 (g2;~

2 . 2

m N f A mNSB(a) w A + t/4mN)aW(t)+l + u ImBw(u ,t)dw - D(t) ( ~n K /

27r 2 ~n K 2.2(otw + 1) '

where

(6)

2 2 2 my - m N- m K t v y

~y(t) - 2m N + 4m~ and Xi(Y, t) = ± m y + m N + 1- t /

and Z(Y1, t) can be found in ref. 12 where sum rules s imi la r to ours have been used to investigate the behaviour of the A 2 t r a j ec to ry near the nonsense right s ignature point a A 2 = 0. The integrals have been evaluated in the na r row width approximation, that gives

Fel. 1 ~ A 1 ±clie l±

2,2-mN~K ImAw(v' t )d¢° = 8m 2 I,I±2 --q3± x (7)

2 , × L L~X+(MI±,t)[(MI±-mN )2 mK]Pli±l(COSOl±)-X (Ml±,t)r(Ml±+mN)2t 21~, , 0 '~ - _ -mKJ~l±tCOS le)J ,

COjA v lmBw( v,t)dw = N 2n 2 m K

(8) Fel.

=_1 ~ . , ~ l± /4- t 2 2 ' ±c - - ~ \_ 4mN/(~,,l.+--'l{[(Ml±-mN) _mKlPl±.l(CosOl±) 2 2 ' 8 I,l± - [(Ml± + raN) - mK]Pl±(COS Ol±) }

ql±

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V o l u m e 2 6 B , n u m b e r 8 P H Y S I C S L E T T E R S 18 M a r c h 1968

w h e r e the usua l no ta t ion has been used . The c o e f f i c i e n t C is equal to ~ fo r I = 0 and to 3 fo r I = 1. The d e r i v a t i v e s of the L e g e n d r e p o l y n o m i a l s a r e to be e v a l u a t e d at c o s S / ~ = 1 + t / 2 q ~ , ql± be ing the c . m . m o m e n t u m at the r e s o n a n c e e n e r g y .

F o r c o n v e n i e n c e the c h o i c e ¢~A = 2 GeV has been made : s o m e a r g u m e n t s which show that th i s can be a r e s o n a b l e v a l u e h a v e been g iven in r e f . 12.

V a r i o u s a m b i g u i t i e s a r i s e in the eva lua t i on of the s u m r u l e s (5) and (6). L e t us d i s c u s s t h e m s e p a - r a t e l y .

1) T h e coup l ing cons t an t gA and g x . Al though many i n v e s t i g a t i o n s [13] h a v e been devo ted to t hem, they a r e s t i l l p r a c t i c a l l y unknown. A r e c e n t a n a l y s i s by K i m [14,15] a p p e a r e d to c l a r i f y the s i tua t ion , showing tha t both gA and g ~ a r e in a g r e e m e n t wi th t he SU(3) p r e d i c t i o n s . H o w e v e r v e r y r e c e n t l y [16] s o m e a r g u m e n t s h a v e been p r e s e n t e d to show tha t both the c o n s t a n t s c a t t e r i n g l eng th [17] and the k - m a - t r i x a p p r o a c h [14] g ive v a l u e s i n c o m p a t i b l e wi th SU(3) i n v a r i a n c e . In v i e w of th is u n c e r t a i n t y we e v a l u - a t ed ou r s u m r u l e s in the two c a s e s

a) Zovko va lues [13]:2g /4 : 6 9 • 2.9 andg /4 : 2.1 0 .9 b) K i m v a l u e s [15]: g / ~ / 4 ~ = 16.0 ± 2.5 and g ~ / - 4 ~ = 0.3 + 0.5. 2 * . . . . . ) T h e Yo(1405) c o u p h n g has been e v a l u a t e d u s ing the cons t an t s c a t t e r i n g l eng th a p p r o x i m a t i o n , o b -

t a in ing g 2 ~ / 4 ~ = 0.32 [18]. F o r t u n a t e l y h o w e v e r the Yo con t r i bu t i on i s so s m a l l tha t a v a r i a t i o n • 100% of i t s coup l ing cons t an t would not p r o d u c e s i g n i f i c a n t v a r i a t i o n s of the r e s u l t s .

3) T h e Y~(1385) has been shown [14] to be ma in ly a ~A s c a t t e r i n g r e s o n a n c e and i t s coup l ing to the kN channe l i s e x t r e m e l y weak. In the fo l l owing we sha l l t ake gy~ = 0.

4) The r e s o n a n c e s p a r a m e t e r s h a v e been t aken f r o m the R o s e n f e l d t a b l e s [19]: only t he f i r m l y e s - t a b l i s h e d r e s o n a n c e s h a v e been inc luded : th i s po in t wi l l be d i s c u s s e d be low.

T h e r e s u l t s ob ta ined with t h e s e da ta a r e shown in f ig s . 1 and 2 fo r both c a s e a) (Zovko va lue s ) and b) (Kim v a l u e s ) . F o r t u n a t e l y the q u a l i t a t i v e f e a t u r e s of the r e s u l t s a r e not too d i f f e r e n t in the two c a s e s . In c a s e a the l e f t hand s i d e of the s u m r u l e (5) changes s ign n e a r t ~ -0 .09 (GeV/c) 2 and t ~ - 0 . 9 (GeV/c) 2 and of the s u m r u l e (6) n e a r t ~ -0 .11 (GeV/c) 2 and t ~ -1 .2 (GeV/c) 2. F o r c a s e b t h e s e z e r o e s r e s p e c - t i v e l y a t the po in t s t ~ -0 .1 (GeV/c) 2, t ~ -0 .8 (GeV//c) 2 fo r the s u m r u l e (5) and t ~ -0 .11 (GeV//c) 2, t ~ -1 .2 (GeV//c) ~ fo r the s u m r u l e (6). M o r e o v e r f r o m f ig . 1 we s e e that the p r e d i c t i o n s ob ta ined f r o m the s u m r u l e (5) a r e in q u a l i t a t i v e a g r e e m e n t , fo r t ~> -0 .7 (GeV/c) 2, wi th the r e s u l t s ob ta ined u s ing the nonf l ip p a r a m e t e r s of the ¢o po le g iven by the high e n e r g y f i t by P h i l l i p s and R a r i t a [5]. As g e n e r a l f e a t u r e of t h e s e f i n i t e e n e r g y s u m r u l e s we w i s h to r e m a r k the e x c e l l e n t a g r e e m e n t be tween the high e n e r g y p a r a m e t e r s f i t t ed by high e n e r g y da ta and t h o s e ob ta ined by the low e n e r g y s t r u c t u r e . T h e r e f o r e the f i n i t e e n e r g y s u m r u l e s s e e m to be a v e r y p o w e r f u l a p p r o a c h in o r d e r to s tudy the h igh e n e r g y s t r u c t u r e of the s t r o n g i n t e r a c t i o n s when p h a s e sh i f t a n a l y s i s in t he 2 to 4 GeV/c r e g i o n b e c o m e a v a i l a b l e [e .g . 20].

F r o m fig. 2 we s e e a l s o tha t the w e x c h a n g e g i v e s r i s e to a non n e g l i g i b l e sp in f l ip amp l i t ude : up to now the high e n e r g y p h e n o m e n o l o g i c a l f i t s w e r e unab le to e s t a b l i s h th i s po in t [e .g. 20]: in fac t the w- e x c h a n g e sp in f l ip a m p l i t u d e was n e g l e c t e d in r e f . 5.

L e t us a t f i r s t c o n s i d e r the z e r o e s in the A w and Bo) a m p l i t u d e s n e a r t = -0.1 (GeV/c) 2 under the u sua l a s s u m p t i o n that t he w e x c h a n g e g i v e s the only s i g n i f i c a n t C = -1 con t r i bu t ion . T h e z e r o in A w is in p e r f e c t a g r e e m e n t wi th the exp lana t ion of the c r o s s o v e r in kN and ~N s c a t t e r i n g . A z e r o in B w i s to be e x p e c t e d if the exp lana t ion of the c r o s s o v e r in pp and pp i s c o r r e c t and the f a c t o r i z a t i o n t h e o r e m ho lds . H o w e v e r t h e s e r e s u l t s a r e in c o n t r a d i c t i o n , v i a the f a c t o r i z a t i o n t h e o r e m , wi th the b e h a v i o u r of the w r e s i d u e func t ions that has been found in o t h e r r e a c t i o n s [3,7].

We c o n s i d e r now the z e r o e s n e a r t ~ -1 (GeV/c)2: such z e r o e s could be a s s o c i a t e d wi th the v a n i s h i n g of t he t r a j e c t o r y , p r o v i d e d tha t the w i s n o n s e n s e choos ing [3] a t the w r o n g s i g n a t u r e po in t ~ = 0. H o w - e v e r t h e s e z e r o e s o c c u r at v a l u e s of t r a t h e r d i f f e r e n t fo r the A and B a m p l i t u d e s and t h e s e v a l u e s of / t t a r e r a t h e r l a r g e to be a s s o c i a t e d with t he v a n i s h i n g of t he w - t r a j e c t o r y . If th is d i s p l a c e m e n t i s not m e r e l y a t h i rd double s p e c t r a l func t ion e f fec t [21], and if t h e s e z e r o e s a t l a r g e v a l u e s of I t I can be t aken s e r i o u s l y , they could s u g g e s t tha t , b e s i d e t he w exchange , o t h e r c o n t r i b u t i o n s , p o l e s o r b r a n c h cu t s , a r e i m p o r t a n t .

F i n a l l y , we t r i e d to check how t h e s e r e s u l t s depend upon the p o s s i b l e n e g l e c t of s o m e s t i l l u n d i s - c o v e r e d d i r e c t - c h a n n e l r e s o n a n c e s . To th i s end we h a v e inc luded t h r e e m o r e r e s o n a n c e s $ w h o s e p a r a m -

i+ $ We included two resonances ~ , respect ively with I = 0 and I = 1, which are supposed to belong to an unitary octet and one resonace I - with I = 1 supposed to belong to a decuplet.

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Volume 26B, number 8 P H Y S I C S L E T T E R S 18 March 1968

2 ; G.v

/ in(GeV)~ It[ .~ - .

T , f •

0.5 1 1.5

, / / , 0.5 1

itl in (GeV~' k-E--/

1.5

-1

Fig. 1. The curves Ia and Ib are the plots of the lef t- hand side of eq. (5) using for gAand gN Zovko's and Kim's values respect ively . Dashed curve r ep resen t the expression:

(20t co+ 1) / t \ a co(t)+1 2~.2mN(ffw+l) C(t) k wA +4-~-N)

fitted by Phillips and Rari ta in ref. 5.

_21 Fig. 2. The curves Ia and I b a r e the plots of the left- hand side of eq. (6) using for gA and gN Zovko's and

Kim's values respect ively.

e t e r s have been r o u g h l y e x t i m a t e d a s s u m i n g SU(3) i n v a r i a n c e . T h e q u a l i t a t i v e f e a t u r e s of our r e s u l t s do not c h a n g e much: t h i s s e e m s to s u g g e s t tha t ou r r e s u l t s a r e of g e n e r a l va l i d i t y , a l though the s a t u r a t i o n p r o c e d u r e we u s e d i s a d m i t t e n l y r a t h e r r ough .

The c o n c l u s i o n s of t h i s w o r k a r e thus r a t h e r u n p l e a s a n t . The v a r i o u s open p o s s i b i l i t i e s a p p e a r to be the fo l lowing:

1) a d d i t i o n a l s i n g u l a r i t i e s a r e p r e s e n t in the c o m p l e x J - p l a n e . S o m e new ~ [2] c o n t r i b u t i o n s , in the f o r m of a new po le o r of a cut , a r e n e e d e d a t l e a s t in k a o n - n u c l e o n and n u c l e o n - n u c l e o n s c a t t e r i n g .

2) the f a c t o r i z a t i o n t h e o r e m d o e s not hold , p e r h a p s b e c a u s e a b s o r p t i v e c o r r e c t i o n s a r e i m p o r t a n t .

It i s a p l e a s u r e to thank p ro f . L. B e r t o c c h i f o r the c r i t i c a l r e a d i n g of the m a n u s c r i p t and p ro f . N, Cab ibbo f o r s o m e u se fu l s u g g e s t i o n s and d i s c u s s i o n s .

References

1. R . J . N . Phil l ips, in: Proc. of the Heidelberg Intern. Conf. on Elementary Par t ic les (North-Holland, Amsterdam. 1968) p. 442.

2. V. Barger and L.Durand, Phys. Le t te r s 19 (1967) 1295. 3. P . D i V e c c h i a , F .Drago a n d M . L . P a e i e l l o . Universi ty of Rome. preprint no. 150 (1968). 4. M.Braunschweig. W. Braunschweig, D. Husmann, K. Liibelsmeyer and D. Schmitz, Single photoproduetion of

neutral ?r-mesons on hydrogen at small angles between 4 and 5.8 GeV: prepr int presented to: Heidelberg Intern. Conf. on Elementary par t ic le physics, October 1967.

5. R . J . N . Phil l ips and W. Rari ta . Phys. Rev. 139 (1965) B1336.

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6. W. R a r i t a . R . J . R idde l , C . B . Chiu and R . J . N . P h i l l i p s , U n i v e r s i t y of Ca l i f o rn i a Rad ia t ion L a b o r a t o r y Repo r t , U C R L - 1 7 5 2 3 (1967). Addi t ional r e f e r e n c e s a r e g iven in t h i s p a p e r .

7. A . P . C o n t o g o u r i s , H . J . Luba t t i and J . T r a n T hanh Van. P h y s . Hev. L e t t e r s 19 (1967} 1352. 8. K. I g i a n d S . M a t s u d a , P h y s . Rev . L e t t e r s 18 {1967) 625. 9. A . A . Log~nov, L . D . Soloviev and A. N. T a v k h e l i d z e . P h y s . L e t t e r s 24B (1967) 181.

10. R . D o l e n , D . H o r n a n d C . S c h m i d . P h y s . Rev . L e t t e r s 19 (1967} 402; R . D o l e n , D . H o r n a n d C . S c h m i d , P r e p r i n t C A L T - 6 8 - 1 4 3 (1967}.

11. V .S ingh , P h y s . Rev . 129 (1963) 1889. 12. S. M a t s u d a a n d K. lgi , P h y s . R ev . L e t t e r s 19 (1967} 928. 13. M . L u s i g n o l i . M . R e s t i g n o l i , G . A . S n o w a n d G . V i o l i n i , P h y s . L e t t e r s 21 (1966)229; Nuovo C i m e n t o 4 5 A (1966)

792; N. Zovko. P h y s . L e t t e r s 23 (1966) 143. And m a n y o the r .

14. J . K . K i m , P h y s . Rev . L e t t e r s 19 (1967) 1074. 15. J . K . Kim. P h y s . Rev . L e t t e r s 19 (1967} 1079. 16. A . D . M a r t i n and G . G . R o s s , "A p r e d i c t i o n fo r the AKN and ~ ' K N coupl ing c o n s t a n t s f r o m a f in i te e n e r g y s u m

r u l e " . U n i v e r s i t y of D u r h a m p r e p r i n t . 17. J . K . K im. P h y s . Re v . L e t t e r s 14 (1965) 29;

M. Sakitt e t a l . , P h y s . R ev . 139 (1965) B719. 18. R . L . W a r n o c k a n d G . F r y e , P h y s . Rev . 138 (1965) B947. 19. A . H . R o s e n f e l d et a l . , Rev . Mod. P h y s . 39 (1967) 1. 20. F o r e x a m p l e s ee the V. B a r g e r t a lk at the "Top ica l C o n f e r e n c e on High E n e r g y C o l l i s i o n s of H a d r o n s ~, CERN,

J a n u a r y 1968. 21. S. M a n d e l s t a m and L . L. Wang, P h y s . Rev . 160 (1967) 1490.

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