Local duality in the invariant amplitudes for πN scattering

7.B.1

Nuclear Physics B24 (1970) 490-508. North-Holland Publishing Company

Ls-A-sI LOCAL DUALITY IN THE I N V A R I A N T AMPLITUDES FOR zrN SCATTERING Kunio SHIGA and K i s e i KINOSHITA Department of P h y s i c s , liyushu University, Futeuoka and F u m i h i k o TOYODA The Second Department, Kinki University, lizuka Received 2 June 1970

Abstract: Local duality for vN scattering in the low- and intermediate-encrgy regions is investigated from comparisons of the invariant amplitudes constructed from phase-shift data with the interpolated Regge ones. The following results are obtained: (i) Local duality is satisfied with the order A '(+), B(-), A '(--) and B(+). In particular, local duality for the A '(~) amplitude is valid even for the c.m. energy down to 1.5 GeV and finite t-region. (ii) The c r o s sing-even amplitudes having excellent local duality are made up constructively by the prominent resonances which lie on the leading trajectories and have the regularity of the Kycia-Riley rule. (iii) The F r e u n d - I l a r a r i conjecture is examined for isospin nonflip amplitudes A ' ( ' ) and B(+). The conjecture works well for the imaginary part but not for the real part. Moreover, the modified interference model proposed by Coulter, Ma and Shaw is also examined for the B(-) amplitude: The real part of the background term fills up thc dip near t = -0.6, in contradiction with the observations.

i. INTRODUCTION In the r e c e n t p r o g r e s s of h a d r o n p h y s i c s , r e s o n a n c e - R e g g e d u a l i t y is now o n e of the m o s t i n t e r e s t i n g p r o b l e m and is i n v e s t i g a t e d by m a n y a u t h o r s [1] f r o m the v a r i o u s p o i n t of view. The d u a l i t y c o n c e p t has v a r i o u s p o s s i b l e v e r s i o n s how to c o r r e l a t e r e s o n a n c e s with Regge p o l e s . A few y e a r s ago, B a r g e r and C l i n e [21 p r o p o s e d the i n t e r f e r e n c e m o d e l (IFM) p r o v i d i n g an e x p l i c i t f o r m f o r s c a t t e r i n g a m p l i t u d e s in the i n t e r m e d i a t e - e n e r g y r e g i o n . The I F M , h o w e v e r , i s shown to have a s e r i o u s d o u b l e c o u n t i n g d i f f i c u l t y a s e x t e n s i v e l y s t u d i e d by Chiu and S t i r l i n g [3]. A f t e r t h a t , m a n y a u t h o r s [4] s t u d i e d the m o d i f i c a t i o n s of the o r i g i n a l IFM. On the o t h e r h a n d , the u s e of F E S R [5] l e a d s us to " a v e r a g e d u a l i t y " of r e s o n a n c e s and Regge p o l e s a s b e i n g d i s c u s s e d i n t e n s i v e l y by D o l e n , H o r n and S c h m i d [6]. F u r t h e r m o r e , S c h m i d [7] d e c o m p o s e d the R e g g e - e x c h a n g e a m p l i t u d e s into s - c h a n n e l p a r t i a l w a v e s , found the c i r c l e s in the A r g a n d d i a g r a m s and a r g u e d that they c o r r e s p o n d to s - c h a n n e l r e s o n a n c e s . How-

LOCAL DUALITY

491

e v e r , the a r g u m e n t of S c h m i d [7] d e p e n d s so g r e a t l y on the d e t a i l s of R e g g e parametrizations that many authors may obtain various results, sometimes, contradicting each other * S i n c e the R e g g e a m p l i t u d e s do not e x a c t l y r e p r o d u c e the d e t a i l s of the s c a t t e r i n g a m p l i t u d e s in the i n t e r m e d i a t e - and l o w - e n e r g y r e g i o n s , it i s r a t h e r h a r d to o b t a i n d e f i n i t e c o n c l u s i o n s on r e s o n a n c e - R e g g e d u a l i t y p r o b l e m t h r o u g h the s t u d i e s of the p a r t i a l w a v e d e c o m p o s i t i o n of the R e g g e a m p l i t u d e s s u f f e r i n g f r o m the a m b i g u i t y of the p a r a m e t r i z a t i o n s . On the c o n t r a r y , we m a y s t u d y the v a l i d i t y of the l o c a l d u a l i t y f o r the R e g g e a m p l i tude t h r o u g h the c o n s t r u c t i o n of the i n v a r i a n t a m p l i t u d e f r o m the p h a s e s h i f t d a t a . In t h i s p a p e r we p u r s u e t h i s l i n e of a p p r o a c h and i n v e s t i g a t e l o c a l d u a l i t y f o r r,N s c a t t e r i n g in the l o w - and i n t e r m e d i a t e - e n e r g y r e g i o n s w h e r e the p h a s e - s h i f t d a t a a r e a v a i l a b l e [11]. U s i n g the i n v a r i a n t a m p l i t u d e , we s e e how the p r o m i n e n t r e s o n a n c e s o b s e r v e d in the e x p e r i m e n t c o n s t r u c t the Regge-exchange amplitude. R e g g e p a r a m e t r i z a t i o n s f o r nN s c a t t e r i n g a r e t a k e n f r o m the one of B a r g e r and P h i l l i p s [12] and i n t e r p o l a t e d into the e n e r g y r e g i o n w h e r e a lot of i n f o r m a t i o n i s a v a i l a b l e f o r p h a s e - s h i f t a n a l y s i s [11]. We o b t a i n the f o l l o w i n g r e s u l t s . F o r c r o s s i n g - e v e n a m p l i t u d e s , the p r o m i n e n t r e s o n a n c e s b e l o n g i n g to Na, Ny and 4 6 t r a j e c t o r i e s , w h i c h h a v e the r e g u l a r i t y of the K y c i a - R i l e y r u l e [13], b u i l d up the R e g g e - e x c h a n g e a m p l i t u d e so w e l l t h a t e x c e l l e n t l o c a l d u a l i t y i s s a t i s f i e d even f o r the l o w e n e r g y and f i n i t e l - r e g i o n . On the o t h e r h a n d , the p r o m i n e n t r e s o n a n c e s c o n t r i b u t e to c r o s s i n g - o d d a m p l i t u d e s with a l t e r n a t i n g s i g n . Then l o c a l d u a l i t y i s p o o r f o r c r o s s i n g odd a m p l i t u d e s . In the i n v e s t i g a t i o n s of d u a l i t y , the s p e c i a l r o l e of p o m e r o n i s an i m p o r t a n t p r o b l e m . F r e u n d [141 and H a r a r i [15] p r o p o s e d the m o d e l that the o r d i n a r y R e g g e p o l e s a r e c o n n e c t e d by F E S R [5] to the d i r e c t - c h a n n e l r e s o n a n c e s and a l s o p o m e r o n to the l o w energ~y b a c k g r o u n d . F u r t h e r , H a r a r i and Z a r m i [16] s t u d i e d the m o d e l q u a n t i t a t i v e l y u s i n g the p a r t i a l - w a v e a m p l i t u d e s . H o w e v e r , the p a r t i a l w a v e a m p l i t u d e s d e p e n d s e n s i t i v e l y on the l a r g e - / a m b i g u i t i e s . In t h i s p a p e r , the F r e u n d - H a r a r i (FH) c o n j e c t u r e [14, 15] i s e x a m i n e d f o r A ' ( + ) and B(+) a m p l i t u d e s i n c l u d i n g p o m e r o n . T h e M I F M p r o p o s e d by C o u l t e r et al. [17] a r e a l s o e x a m i n e d . T h e r e a l p a r t of the a m p l i t u d e , h o w e v e r , d o e s not w o r k w e l l a t a l l f o r both F H c o n j e c t u r e and the M I F M . In s e c t . 2, the f o r m a l i s m of the i n v a r i a n t a m p l i t u d e and the r e s u l t s of c a l c u l a t i o n s a r e shown. A n a l y s i s of l o c a l d u a l i t y is g i v e n in s e c t . 3 and the c h e e k of F H c o n j e c t u r e and the M I F M i s shown in s e c t . 4. S u m m a r y and d i s c u s s i o n s a r e g i v e n in the final s e c t i o n . * Collins et al. [8] agree with Schmid that p a r t i a l - w a v e projections of Regge-pole a m plitudes result in loops in the Argand d i a g r a m s , but argue that such loops should not be associated with physical resonances. F u r t h e r , Kreps et al. [9] have made partial analysis of Hegge-pole amplitude for ~'N charge-exchange scattering and concluded that there is a lack of correspondence between Schmid loops and m e a s ured resonances. This conclusion is in strong contrast to the results of Lipshutz [10], who also analyzed nN charge-exchange data and obtained good correspondences between Schmid loops and physical resonances.

492

K. SHIGA et al.

2. FORMALISM AND THE CALCULATED RESULTS The invariant amplitude A and B a r e c o n s t r u c t e d from the partial wave amplitude fl+ as follows: .4 = 4 ~

E+~-tl

I1

B = 47r ~ T - M - f l

- N~-~f2

1

+ ~5-_M-S2

,

],

(1)

and f l =~-~ (f(/-1)+ - f ( l + l ) - ) P ' ( c ° s 0),

f2 = ~ ( f l - - f l+ ) P ' ( c o s 0),

(2)

where W is the total e n e r g y of the s y s t e m and M and E are the m a s s and the kinetic e n e r g y of the nucleon, r e s p e c t i v e l y . Using the c r o s s i n g m a t r i x we c o n s t r u c t t - c h a n n e l isospin 0(F +) and I ( F - ) amplitudes as follows.

/,-+!

/1

Where F stands for .4- a n d / ? - a m p l i t u d e s . For the invariant amplitudes, we adopt A'(±) and B(±) r a t h e r than A (±) and B (±). The A'(e) amplitude is d e fined as follows

A,(± ) =.4(±) + w + t / 4 M B(±) ' 1 - t/4M 2

(4)

w h e r e w is the l a b o r a t o r y energy of the pion. In the following our r e s u l t s from c o m p a r i s o n s of the e x p e r i m e n t with the interpolated Regge amplitude a r e shown in the real and i m a g i n a r y p a r t s of the invariant amplitudes s e p a r a t e l y . F i r s t , the e n e r g y dependence of the f o r w a r d invariant amplitudes and secondly, the m o m e n t u m - t r a n s f e r dependence for fixed energy. The duality for the i m a g i n a r y part of s c a t t e r i n g amplitude has been d i s c u s s e d by many authors [1]. It is also worthwhile to examine local duality for the real part of amplitude. We investigate local duality not only for the i m a g i n a r y p a r t but also for the real part using the invariant amplitude. (1) The energy dependence of the f o r w a r d invarianl amplitudes F r o m figs. la and lb, we see that the c r o s s i n g - e v e n amplitudes A'(+) and B(-) have good local duality, but the c r o s s i n g - o d d amplitudes A ' ( - ) and B(+) have poor local duality. To be m o r e p r e c i s e , it is c l e a r that local duality is satisfied with the o r d e r A '(+), B(-), A ' ( - ) and B(+). These r e sults a r e also supported by the t-dependence for fixed energy.

_* l-

_*--

Re

.-

:L._-___---’ -m

..-._

-

--._

-Al ‘*. ‘.

_‘._

O-O1.5

A’

(-’ _.-

_,-

A’(_’

“i,

- -----.-3>4 c-

I’

‘-L.a_---=3Y--.__Hj’

Ja5/_-4,7

__c’

Irn

\

k_ -

2.1

e

2.2

W(G.5')

_-_--mm,

*.

Re*, ;g..........--0.1,

Fig.

la.

The energy ,ct,

c-17

dependence x

of the forward

A’(*) invariant

amplitudes.

10-O

Me”

-Fteqqe .......

Fig.

lb.

The energy

dependence

of the forward

Experiment

B(*)

invariant

amplitudes

K. SHIGA et al.

494

I

-

Etegge

. . . . . .

Experiment

i: t

(GeV21

Fig. 2a. The t-dependence

:!

of A’(+) amplitude for the fixed energy 1.5 GeV A’ (+IW=2.18

Re

-0.04

-0.

0

,o -

'1 '\ '\ '\ : '\ : I '1 \ : :

-0.:

1

, :

-1.0

Ge")

0 .oo

‘I i

! 1 -1.5

I

I I

:

I

Fig. 2b. The t-dependence of A’(+) amplitude for the top energy of phase-shift data 2.18 GeV.

495

LOCAL DUALITY B (-)

(W:l. 8

GeY)

Im

Re

0.~

0.0

0.0

0.~l

0.~

0.8

0.4

0.6

/ !

-0.5 J

i

-1.0 I s

i

/ /

iJ

Fig. 3a. The t-dependence of B(-) amplitude for the lowest energy 1.8 GeV satisfying good local duality. (2) The t-dependence of the invariant amplitudes f o r fixed energy F r o m figs. 2 to 5, we note following. A,((a)~.), L o c a l dualiW is good f o r low e n e r g y and f i n i t e - t r e g i o n with the o r d e r B ( - ) , A ' ( - ) and B(+). This r e s u l t is c o n s i s t e n t with the o b s e r v a t i o n s in (i). (b) At the top energy about 2.2 GeV given by phase-shift analysis in the present stage, all the invariant amplitudesA'~+), B(-), A'(-) and B(+) have good local duality. (c) In particular, the A'(+) amplitude has very good local duality even down to the energy 1.5 GeV. In addition to the results (a) to (c), it is interesting that local duality is good even for large t beyond the dip position near -0.6 without u-channel Regge contributions. In order to see the overall fit of the interpolated Regge amplitude to the invariant amplitude constructed from the phase-shift data, we plot in the (s,i) plane the real and imaginary parts of the crossing even amplitude, separately *. Clearly, the above observations (a), (b) and (c) are confirmed. * Footnote see next page.

496

K. SHIGA et al. B (-)

(W=2.18

GeV)

Im

Re 0.0

0.2

0.4

0.6

0.0

0.2

0.4

0.6

0.0

~

J

-0.5

I

10~/

I

t

-1.5

I I Fig. 3b. The t-dependence of B(-) amplitude for the top energy of phase-shift data 2.18 GeV.

3. L O C A L D U A L I T Y AND THE R E G U L A R I T Y O F THE P R O M I N E N T RESONANCES The r e s u l t s o b t a i n e d in the p r e v i o u s s e c t i o n a r e e x p l a i n e d by d e c o m p o s i n g the i n v a r i a n t a m p l i t u d e into the p a r t i a l - w a v e a m p l i t u d e s as follows. F o r small-t regions 6M

8~wA'(+)(s,t)

~

_1 { ~-~ (l + 1)(.f~+ + 2 f l + ) + )--'~ l ( f ~ _ + 2 f ~ l

l

), -

6(w-~?B(-)(~,t/~ z(z+ll(j~JL-/~-+~ 1

87r W

86M . w -".

l

,(_>,~s, ~ / ~ :cz (~ + ~ ( / L

1

-

f]+),

-/i\~

+ :cz z (/~ - - / z - ~,

-

* Chiu, Eden and Tan [18] stressed the importance of the phase contour of the invariant amplitudes on the (s,t) plane. However, the phase of the amplitude sometimes depends sensitively on the detailed behaviour of its real and imaginary parts.

LOCAL

DUALITY

A”-)

iW=2.18

497

GeV)

InI 3

0.02 ,

0.04

0.06

/’ /’

,,*’

Fig.

4. The t-dependence

of A’(-)

amplitude

for the fixed

energy

2.18 GeV.

(5) In the calculations of B(*) amplitudes, a term proportional to 1/(W+M)2 is neglected comparing with one proportional to l/(W-l)2. It is well known that the prominent resonances which lie on leading trajectories and have 9 1 L large elasticity are included in f/+ and fF_ partial waves. Other _fi _ and fi+ partial waves are supposed to give non-resonating backgrounds or rather inelastic resonances, We represent the contributions of the prominent resoas {NJ, (NY} and {A6} symnances belonging to N,, N and A6 trajectories bolically. These {NJ, (N 1 and {A5} contributions to the invariant amplitudes are written as follo&: A’(f)

Sz

B(-) =

498

K. SHIGA et al. B (+)

(W-2.18

3eV)

0

0.2

Re -0.4

0.0

Im -0.2

0.0

0.[

0_6

Il

-0.5

-i.0

\ ,

-1.5

11

\ Fig. 5. The t - d e p e n d e n c e of B(+) a m p l i t u d e f o r the fixed e n e r g y 2.18 GeV.

A '(-) ~ {Na} + {Ny} - {A6},

B(+) 3

+

- 2 {%}.

(6)

l

The resonating wave//+(//_) satisfying the Kycia-Riley rule [131 has A/3 and A5 (Na and N),) trajectories; but the A.~ ~Jtrajectory has shaky evidence for $31(1640) and D35(1950). The Na, N~. and Ab-trajectories have many prominent resonances on it. In the A'(+)"amplitude, the factor 2 of {A6} contributions compensates for almost missing contributions from A trajectory a rathand, as a result, the total contribution of {Na}, {NF}and 2 {A6} gives ~ er smooth behaviour to the A'(+) amplitude. Furthermore, the A'(+) amplitude has additive contributions from all the partial waves as shown in eq. (5). On the other hand, B(-) has equal weight of contributions from Na, NF and A6 trajectories as shown in eq. (6). Therefore, it is also possible to consider that the B(-) Regge-exchange amplitudes are built up by the promi-

LOCAL DUALITY t

499

(GeV2) -3.0

-2.01

-0.02

f 1.00 -I.0

-0.02

-0.0

1.5

1.6

~ 1.7

0.02

1.8

1.9

2.0

2.1

2.2 W(GeV)

Fig. 6a. The s - and t-dependence of the real part of the invariant amplitude A'(+) constructed from the phase-shift data.

nent r e s o n a n c e s lying on the Na, Ny and A 5 t r a j e c t o r i e s having the r e g u l a r i ty of the K y c i a - R i l e y r u l e [13]. In s u m m a r y , it is c l a r i f i e d that the p r o m i nent r e s o n a n c e s having the r e g u l a r i t y of the K y c i a - R i l e y r u l e r e p r o d u c e the c r o s s i n g - e v e n R e g g e - e x c h a n g e a m p l i t u d e s A '(+) and B(-) with c o n s i d e r a b l e amount. This fact is v e r y i n t e r e s t i n g and p r o m i s i n g one in an a p p r o a c h of i n v e s t i g a t i o n s of r e s o n a n c e - R e g g e local d u a l i t y . On the o t h e r hand, the c r o s s i n g - o d d a m p l i t u d e s A ' ( - ) and B (+) have a l t e r n a t i n g c o n t r i b u t i o n s f r o m the p r o m i n e n t r e s o n a n c e s as shown in eq. (7). Then, this s i t u a t i o n l e a d s to p o o r l o c a l d u a l i t y of the c r o s s i n g - o d d a m p l i t u d e s . T h e s e q u a l i t a t i v e a r g u m e n t a r e s u p p o r t e d by e x p e r i m e n t a l s i t u a t i o n s shown in sect. 2.

4. WHAT IS THE BACKGROUND TERM ? When we c o n s i d e r a s c a t t e r i n g a m p l i t u d e in one channel which c o n t a i n s r e s o n a n c e s , it is a v e r y i m p o r t a n t p r o b l e m to s e p a r a t e a r e s o n a n c e p a r t f r o m n o n - r e s o n a t i n g b a c k g r o u n d p a r t [191 . At high e n e r g i e s what c a r r i e s the r o l e of l o w - e n e r g y b a c k g r o u n d p a r t is a v e r y i m p o r t a n t p r o b l e m f o r

500

K. SHIGA et al.

-3.0

.0

-2.0 - -

~0.0

-I.0 0.0

---0.02

~ - 0 . 0 4 -0.0 1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

Fig. 6b. The s - and /-dependence of the real part of the interpolated Regge amplitude A' (+). t h e i n v e s t i g a t i o n s of r e s o n a n c e - R e g g e d u a l i t y . In t h i s s e c t i o n , w e e x a m i n e t h e F r e u n d - H a r a r i c o n j e c t u r e [14, 15] f o r t h e b a c k g r o u n d t e r m and the m o d i f i e d i n t e r f e r e n c e m o d e l of C o u l t e r , M a and Shaw [17]. (1) A test of F H conjecture using A ' ( + ) and t~+~ invariant amplitudes F r e u n d [14] a n d H a r a r i [151 p r o p o s e d a m o d e l in w h i c h a l o w - e n e r g y background is connected with a high-energy pomeron-exchange term through F E S R [5]. H o w e v e r , the q u o t e d b a c k g r o u n d in t h i s m o d e l i s o n l y an i m a g i n a r y o n e . H e r e , we e x a m i n e t h e i r i d e a not o n l y f o r t h e i m a g i n a r y p a r t b u t a l s o f o r the r e a l o n e in o r d e r to r e v e a l the p r o p e r t i e s of l o w - e n e r g y b a c k ground. F r o m f i g s . 10 and 11 , i t i s o b v i o u s t h a t the c o n j e c t u r e i s v a l i d f o r the o n l y i m a g i n a r y p a r t b u t n o t f o r the r e a l one. T h a t i s , l o c a l d u a l i t y f o r the i m a g i n a r y p a r t i s good in c o n t r a s t to the b a d n e s s of the r e a l p a r t . (2) A tesl of the M I F M by Coulter el al. C o u l t e r et al. [17] p r o p o s e d the M I F M in w h i c h the b a c k g r o u n d t e r m c o r r e s p o n d s to the n o n - s i g n a t u r e d R e g g e e x c h a n g e a m p l i t u d e . W e e x a m i n e their model for the B(-) amplitude. The invariant amplitudes constructed f r o m the p h a s e - s h i f t d a t a a r e a l m o s t a n a l o g o u s to the o n e f r o m R e g g e - e x * The curves marked pomeron (plus P ' and P'~ in figs. 10 to 11 a r e obtained by isolating the pomeron p a r t from other Regge poles using B a r g e r and P h i l l i p s ' Regge p a r a m e t r i z a t i o n [12].

LOCAL DUALITY

501

-3.0 0.0

.0 -2.0

0.02

--0.02 0.04 -0.06 008 '"0.io

I /

o. o Fig. 7a.

1.s

• 1.6

1.7

-

1.8

~

-

1.9

-

-

2.0

~

:

2.1

~

2.~

T h e s - and t-dependence of t h e i m a g i n a r y part of the invariant a m p l i t u d e A'(~) constructed from t h e p h a s e - s h i f t d a t a .

0.0

-3.0

-2.0

-1.0

~'-"~--0.02

.0.04 --0.06 ~ 0 . I0 0.0

/ 1.5

--1.6

1.7

1.8

~

~ 1.9

F i g . 7b. T h e s - a n d t - d e p e n d e n c e of t h e i m a g i n a r y R e g g e a m p l i t u d e .4' (+).

2.0

'

"0.12 2.1

2.2

p a r t of t h e i n t e r p o l a t e d

502

K. SHIGA et al.

-3.0

-2.0

0.0 -0.2

-i.0 0.0 o

~

~0.2 0.4

0.0

1.5

1.6

1.7

1.8

1.9

2.0

2.1

0"62.2

Fig. 8a. T h e s - and t - d e p e n d e n c e of the r e a l p a r t of the i n v a r i a n t a m p l i t u d e B(-) c o n s t r u c t e d f r o m the p h a s e - s h i f t data.

-3.0

0.0

0.2 -2.0 ¸

-1.0

I

~

0.2 0.4

0.0" 1.5

0.6 1.6

1.7

1.8

1.9

2.0

2.1

Fig. 8b. The s - and t - d e p e n d e n c e of the r e a l p a r t of the i n t e r p o l a t e d Regge a m p l i t u d e B ( - ) .

2.2

LOCAL DUALITY

503

-3.0

-2.0

-I.0

0.0 1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

Fig. 9a. The s - and t-dependence of the imaginary part of the invariant amplitude B(-) constructed from the phase-shift data.

-3.0

~

0

0'2.0

0.2 -2.0

0.0

-0.2 -1.0 -0.4

.0.0

L 1.5

,0.2 0.4 1.6

l.?

1.8

1.9

2.0

0.6 2.1 2.2

Fig. 9b. The s - and t-dependence of the imaginary part of the interpolated Regge amplitude Bt-).

504

K. StIIGA et al. A '(+)

(W-l.7

tier)

Imaginary

Q 1 • 0.i0( M e V ) '~Exp .'% R e g g e (p÷p, } Pomeron

)

~ERes

i

l -0.05

(

i MeV

)

~al

Fig. 10a. A test of the FH conjecture for A '(+) amplitude for the fixed energy 1.7 GeV. ( a p = 1+0.37 rand a p , = 0.56+0.86t). Here ~Res stands for the sum of the contributions from F15 , F37 , D15 , D13 and P l l resonances. A l s o O , • and • indicate t = ,0.0, -0.5 and -1.0 respectively. c h a n g e a m p l i t u d e f o r b o t h r e a l and i m a g i n a r y p a r t s . H o w e v e r , when w e c o n s i d e r the p r o p o s e d f o r m of R e g g e a m p l i t u d e w i t h o u t s i g n a t u r e f a c t o r , l o c a l d u a l i t y i s so b a d f o r the r e a l p a r t of the a m p l i t u d e t h a t the d i p n e a r - 0 . 6 f o r ~N s c a t t e r i n g can n o t b e r e p r o d u c e d c o r r e c t l y .

5. DISCUSSIONS I ~ c a l d u a l i t y f o r nN s c a t t e r i n g h a s b e e n i n v e s t i g a t e d t h r o u g h the i n v a r i a n t a m p l i t u d e in t h i s p a p e r in o r d e r to c l a r i f y how the p r o m i n e n t r e s o n a n c e s o b s e r v e d in t h e e x p e r i m e n t c o n s t r u c t the R e g g e - e x c h a n g e a m p l i t u d e . In the f o l l o w i n g , we d i s c u s s o u r r e s u l t s , f u r t h e r p r o b l e m s and the e x t e n d e d a n a l y s i s f o r l~N s c a t t e r i n g . (1) F o r the c r o s s i n g e v e n a m p l i t u d e s of nN s c a t t e r i n g , the p r o m i n e n t r e s o n a n c e s h a v i n g the r e g u l a r i t y of the K y c i a - R i l e y r u l e [13] b u i l d up R e g g e e x c h a n g e a m p l i t u d e s w i t h c o n s i d e r a b l e e f f e c t . W h e n we e x t e n d the a b o v e a n a l y s i s to KN s c a t t e r i n g , w h e t h e r the a r g u m e n t s s i m i l a r to the ,~N c a s e a r e a p p l i c a b l e o r not d e p e n d on the f o l l o w i n g p r o b l e m s . F i r s t , the c r o s s i n g m a t r i x i s not e q u a l to the o n e f o r r,N c a s e . S e c o n d l y , w h e t h e r t h e h y p e r o n r e s o n a n c e s o b s e r v e d in e x p e r i m e n t h a v e the r e g u l a r i t y o r not. T h i r d l y , t h e r e i s the d i s c r e p a n c y of s t r e n g t h b e t w e e n g Y ~ R N and g Y { K N c o u p l i n g

LOCAL DUALITY

505

Imaginary A'(+)(W: 2.0

GeV)

0.15

1

( MeV )

'~lI ~ Regge (P+P ')

a~,

,,

0.10

Exp-::Res \ •

t

\): ~Pomeron 0.05

j. -0.05

Real

1 ( MeV )

Fig. 10b. A test of the FH conjecture for A'(~) amplitude for the fixed energy 2.0 GcV. ( a p = 1 ~0.37t and a p , = 0.56+0.86t). Here ~ R e s s t a n d s for the sum of the contributions from F37 and G17 resonances. Also O, • and • indicate t = +0.0, -0.5 and -1.0 respectively. c o n s t a n t s [20]. F o r t u n a t e l y p r o b l e m (1) and (3) a r e c o m p e n s a t e d by a d i f f e r e n c e of f a c t o r in the c r o s s i n g m a t r i x . M o r e o v e r , a s shown in t a b l e 1, the p r o m i n e n t and l e a d i n g h y p e r o n r e s o n a n c e s h a v e the r e g u l a r i t y of the K y c i a R i l e y r u l e a s n u c l e o n - and 33 r e s o n a n c e s e r i e s . H o w e v e r , the p r o m i n e n t h y p e r o n r e s o n a n c e s h a v e d i f f e r e n t i n t e r n a l s p i n and SU(3) d i m e n s i o n a l i t y f r o m n u c l e o n and 33 r e s o n a n c e s e r i e s . T h i s d i f f e r e n c e , e v e n if the s i m i l a r a r g u m e n t s of l o c a l d u a l i t y a r e a p p l i c a b l e to I~N s c a t t e r i n g , m a y l e a d u s to v e r y i n t e r e s t i n g s t r u c t u r e of h a d r o n s p e c t r u m . Such p r o b l e m s w i l l be i n v e s t i g a t e d e x t e n s i v e l y in a f o l l o w i n g p a p e r . (2) In t h i s p a p e r we c o n s i d e r e d the only t - c h a n n e l R e g g e c o n t r i b u t i o n s p a r a m e t r i z e d b y B a r g e r a n d P h i l l i p s [12]. In s p i t e of t h i s l i m i t a t i o n , l o c a l duality even for finite-t region is good for the crossing-even amplitudes. L o c a l d u a l i t y in b a c k w a r d r e g i o n s h o u l d b e s t u d i e d f u r t h e r , i n c l u d i n g t h e u channel Regge poles. (3) T h e r e m a y a r i s e s o m e q u e s t i o n s w h e t h e r the i n t e r p o l a t e d R e g g e a m p l i t u d e i s v a l i d o r not a t s u c h a low e n e r g y . A few d a u g h t e r t r a j e c t o r i e s a r e i n c l u d e d in the p a r a m e t r i z a t i o n a d o p t e d h e r e [12]. T h e p r o b l e m w h e t h e r the d i s c r e p a n c y b e t w e e n t h e e x p e r i m e n t a l s i t u a t i o n and the i n t e r p o l a t e d R e g g e

K. SHIGA et 3/.

506

B (÷) (W=2.18 GeV)

Imaglnar} ~

e(P*P'+P")

(_~,2~xla McV

-4 )

0.4 .%./pomeron

Exp

"" 0.2

EX

s

-0.4

"~.

-0.2

Real

'

Fig. 11. A t e s t of the FH 2.18 GeV. ( a p = 1 + 0 . 3 6 t of the contributions f r o m t=

(---!.~x I 0 -4 ) MeV

c o n j e c t u r e for B(+) amplitude for the fixed e n e r g y and a p , = 0 . 5 6 + 0 . 8 6 t ) . H e r e ~ R e s s t a n d s for the sum F37 and G17 r e s o n a n c e s . Also O, @ and • indicate +0.0, -0.5 and -1.0 r e s p e c t i v e l y .

Table 1 P r o m i n e n t 7rN and I~N r e s o n a n c e s e r i e s s a t i s f y i n g the K y c i a - R i l e y rule and its internal spin and SU(3) d i m e n s i o n a l i t y . Trajectory

Resonance series

Internal spin

N•

N( 940)½+,N(1688){ + , . . .

s=~

SU (3) dimensionality

8

I~

N(1520)~-,N(2190)~-,...

s=½

~

a(n36)~+,A(1950)½+,..,

s =~

10

A~

A(1640)½-,A(1950)~-,...

s =½

10

1÷

1

Kycia-Riley rule

8

Aa

A ( I 1 1 5 ) ~, A(1815)~ +. . . .

s =~

8

Ay

A(1520)~-, A(2100)~-,...

s:½

I

~5

~(1385){ +, ~(2030){ +. . . .

s =~2

I0

J-L=I-1

J-L=I-~

LOCAL DUALITY B [-)

(W=2.18

507

GeV)

Real 0.0

0~0

%2 /

~

0_a j

J)

-0.5,

-1.0 t (GeV 2 )

-I. ~.

i ~J~ .-----gxp-rRes

!i:

Fig. 12. A test of the MIFM for the real p a r t of B(-) amplitude for the top energy of the p h a s e - s h i f t data 2.18 GeV. Here ~ R e s stands for the sum of the contributions from F37 and G17 resonances.

b e h a v i o u r in the i n t e r m e d i a t e - e n e r g y r e g i o n i s to be r e m e d i e d o r not by the i n t r o d u c t i o n of d a u g h t e r t r a j e c t o r i e s s h o u l d w a i t f u r t h e r i n v e s t i g a t i o n s . H o w e v e r , a l m o s t o n l y the l e a d i n g R e g g e p o l e s l e a d s to ,good l o c a l d u a l i t y f o r the c r o s s i n g - e v e n a m p l i t u d e s . A m o n g t h e m , the A , t + ) a m p l i t u d e h a s p a r t i c u l a r l y e x c e l l e n t l o c a l d u a l i t y down to the low e n e r g y 1.5 GeV w i t h o u t d a u g h t e r t r a j e c t o r i e s *. T h u s , o u r a r g u m e n t s do not d e p e n d s e r i o u s l y on t h e i n t r o d u c t i o n of d a u g h t e r t r a j e c t o r i e s P " and p ' . (4) F r o m the s t a n d - p o i n t of h a d r o n s p e c t r u m , n o n - l e a d i n g and i n e l a s t i c r e s o n a n c e s [21] with r e p u l s i v e b a c k g r o u n d s m u s t be c l a r i f i e d in c o n n e c t i o n w i t h R e g g e p o l e s w h e n e v e r we p u r s u e r e s o n a n c e - R e g g e d u a l i t y . T h e s t u d y of t h e s e n o n - l e a d i n g r e s o n a n c e s m a y b r i n g a k e y of r e v e a l i n g the d y n a m i c s of h a d r o n s p e c t r u m . T h e a u t h o r s w o u l d l i k e to t h a n k P r o f e s s o r

M. U e h a r a f o r h i s c r u c i a l

* Here we take the viewpoint that pomeron is e x t r a o r d i n a r y [1-1, 15] and P' is the leading ordinary Regge pole for the isospin nonflip amplitudes.

508

K. SHIGA et al.

c o m m e n t s and d i s c u s s i o n s . T h a n k s a r e a l s o due to P r o f e s s o r S. O t s u k i and D r . M. I m a c h i f o r t h e i r d i s c u s s i o n s and to the o t h e r m e m b e r s of r e s e a r c h g r o u p of K y u s h u U n i v e r s i t y .

REFERENCES [1] R.Dolen, D. Horn and C.Schmid, Phys. Rev. 166 {1968) 1768; C.Schmid, Phys. Rev. L e t te r s 20 (1968) 628; G. F. Chew and A. Pignotti, Phys. Rev. Letters 20 (1968) 1078; G. Veneziano, Nuovo Cimento 57A (1968) 190. [2] V. B a r g e r and M.Olsson, Phys. Rev. 151 {1966} 1123; V. B a r g e r and D. Cline, Phys. Rev. 155 (1967) 1792. [3] C. B. Chiu and A . V . S t i r l i n g , Phys. Letters 26B (1968) 236. [4] M. Bando and T. }tattori, Prog. Theor. Phys. 43 (1970) 463; Y. Kosaka, O. Miyamura, F. Takagi and K. Itabashi, Nuovo Cim. Let t er s 1 {1969} 404; P. W. Coulter, E.S. Ma and G. L. Shaw, Phys. Rev. Letters 23 (1969) 106. [5] K. Igi and S. Matsuda, Phys. Rev. Letters 18 (1967) 625; A. A. Lognov, L. D. Soloviev and A. N. Tavkhelidze, Phys. Let t er s 24B {1967) 181 [6] loc. cit. [i]. [7] loc. cit. [I]. [8] P. D. B. Collins, R. C. Johnson and E. J. Squires, Phys. Let t er s 27B (1968) 23; P . D . B . Collins, R . C . J o h n s o n and G.G. Ross, Phys. Rev. 176 (1969) 1952. [9] R. E. Kreps and R. K. Logan, Phys. Rev. 177 (1969) 2328. [10] N.R. Lipshutz, Phys. Rev. 181 (1969) 1972. [11] A. Donnachie, R.G. Kirsopp and C. Lovelace, Phys. Letters 26B (1968) 161, and its addendum. [12] V. B a r g e r and R. J. N. Phillips, Phys. Rev. 187 (1969) 2210. [13] T. F. Kycia and K. F. Riley, Phys. Rev. L e tt e r s 10 {1963) 266. [14] P . G . O . Freund, Phys. Rev. L e t t e r s 20 {1968) 235. [15] H . H a r a r i , Phys. Rev. L e t te r s 20 {1968) 1395. [16] }l. Harari and Y. Zarmi, Phys. Rev. 187 (1969) 2230. [17] loc. cit. [4]. [18] C.B. Chiu, R . J . Eden and C.I. Tan, Phys. Rev. 170 {1968} 1490. [19] K. Kinoshita and Y. Kinoshita, Nucl. Phys. B7 (1968) 121. [20] J . W . Kim, Phys. Rev. L e t t e r s 19 (1967) 1074, 1079; N. Zovko, Phys. L e tt e r s 23 (1966) 143; R . L . W a r n o c k and G. F r y e , Phys. Rev. 138 {1965) 947. [21] Y. Kinoshita, P r o g r . Theor. Phys. 42 (1969) 629.