Regge-pole model for vector meson production

~7~

Nuclear Physics B8 (1968) 661-685. North-Holland Publ. Comp., Amsterdam

REGGE-POLE MODEL FOR V E C T O R M E S O N P R O D U C T I O N (1): THE R E A C T I O N 7tN -+ oN G u r u V a c h a n DASS a n d C. D. F R O G G A T T

Theoretical Physics Division, A.E.R.E., Harwell, Didcot, Berkshire, U.K. and Rutherford High Energy Laboratory, Chilton, Didcot, Berkshire, U.K.

Received 25 September 1968 Abstract: The Regge-pole model of vector-meson production is discussed. Analyses of the reaction rrN --+pN with an evasive and conspiring Regge pion pole are contrasted, Contributions from the A2 and co Regge poles are included in the p r e sented fits (to data between 2.7 GeV/c and 15 GeV/c lab momenta) which are s i m ilar to those obtained from the absorption model. The decay correlations of the p - m e s o n favour the conspiracy solution. Problems associated with factorisation are pointed out.

1. I N T R O D U C T I O N C o n s i d e r a b l e p r o g r e s s h a s b e e n m a d e r e c e n t l y in e x t e n d i n g R e g g e - p o l e t h e o r y to q u a s i - t w o - b o d y p r o c e s s e s with g e n e r a l s p i n a n d m a s s [1-7]. In t h i s ( p a r t I) a n d a n o t h e r * ( p a r t II) p a p e r [8], we r e p o r t the r e s u l t s of a s t u d y of v e c t o r m e s o n p r o d u c t i o n in m e s o n - n u c l e o n c o l l i s s i o n s u s i n g t h i s f o r m a l i s m . D a t a on the r e a c t i o n s ~N -+ pN i n the r a n g e 2.7 to 15 G e V / c i n c i d e n t lr m o m e n t a h a v e b e e n u s e d . A s i n m o s t i n e l a s t i c c o l l i s i o n s , t h e s e r e a c t i o n s o c c u r with r e l a t i v e l y s m a l l 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 t a n d t h e r e have b e e n m a n y a t t e m p t s to i n t e r p r e t t h e m in t e r m s of the p e r i p h e r a l or e l e m e n t a r y p a r t i c l e e x c h a n g e m o d e l [9, 10]. Any p a r t i c l e e x c h a n g e m o d e l e m p h a s i z e s the s t a t e s in t h e c r o s s e d t - c h a n n e l a n d the d e n s i t y m a t r i x f o r t h e p r o d u c e d r e s o n a n c e h a s a v e r y d e f i n i t e d e p e n d e n c e on the s p i n - p a r i t y of t h e s e s t a t e s [11]. Of c o u r s e , r e s c a t t e r i n g c o r r e c t i o n s [12-15] i n the i n i t i a l a n d f i n a l s t a t e s m i x o p p o s i t e p a r i t y into the s i m p l e o n e - p a r t i c l e e x c h a n g e t h e o r i e s a n d h a v e had c o n s i d e r a b l e s u c c e s s in f i t t i n g the o b s e r v e d d e n s i t y - m a t r i x e l e m e n t s [16-17]. H o w e v e r the a b s o r p t i o n m o d e l s u f f e r s f r o m the f a m i l i a r i n c r e a s e with i n v a r i a n t e n e r g y s of c r o s s s e c t i o n s , ~ ~ s2J-2, f o r high s p i n - J e x c h a n g e . • Part II is devoted to the reaction KN --+ K*N.

662

G.V. DASS and C.D. FROGGATT

It has t h e r e f o r e been s u c c e s s f u l only for p r o c e s s e s in which 7r exchange dominates. R e g g e - p o l e t h e o r y p r o v i d e s a p r a c t i c a l method for r e m o v i n g t h e s e d i v e r g e n c e s and, f r o m the compilation of M o r r i s o n [18], we know that the energy dependences of m o s t two-body r e a c t i o n c r o s s sections a r e in r e a s o n a b l e a c c o r d with its p r e d i c t i o n s . Regge cuts [19-20] introduce mixed parity states in the t-channel in a way s i m i l a r to absorption. We a r e reluctant to introduce the complications of b r a n c h cuts and we attempt to fit the data with the exchange of Regge poles alone. However ultimately it may well be n e c e s s a r y to introduce cuts [21-23]. In the f o r w a r d direction, w h e r e a s y m p t o t i c a l l y the invariant m o m e n t u m t r a n s f e r vanishes, t = 0, f a c t o r i s a t i o n and the exchange of a definite p a r i t y impose i m p o r t a n t c o n s t r a i n t s [5, 24]. As a result, c e r t a i n hellcity amplitudes for Regge pole exchange vanish m o r e rapidly than the gene r a l kinematic behaviour expected f r o m angular m o m e n t u m c o n s e r v a t i o n alone. This is p a r t i c u l a r l y important for pion exchange and, in this case, a c o n s p i r a c y with an opposite parity Regge pole has been suggested [24-29] as an a l t e r n a t i v e m e a n s of satisfying the kinematic conditions at t = 0. The r e a c t i o n s ~rN ~ pN a r e dominated by pion exchange and the a b s o r p tion model fits the data quite well [16]. Our R e g g e - p o l e fits to these r e a c tions include 7r (with and without c o n s p i r a c y ) , A 2 and co exchange and we obtain r e s u l t s s i m i l a r to the a b s o r p t i o n model.

2. FORMALISM In the R e g g e - p o l e model, it is m o s t useful to c o n s i d e r the helicity a m plitudes for the c r o s s e d t-channel. Although invariant amplitudes for v e c t o r - m e s o n production a r e known [30-32], it is m o r e convenient to d i s c u s s the kinematic s i n g u l a r i t i e s for this p r o c e s s in t e r m s of the g e n e r a l methods r e c e n t l y developed [1-7]. We follow closely the a p p r o a c h of Fox [6, 23] who c o n s i d e r s directly the a s y m p t o t i c f a c t o r i s e d amplitudes f12k4,

2tl)t 3

_ 1 +re -irro(t) -½i~(k13-k24 ) (~_~;(t) sin[Tra(t)] e 7)t3kl(t) yk4k2(t)

(1)

for the exchange of a Regge pole a(t) with signature r. This a p p r o a c h and the conventions used in this paper a r e s u m m a r i s e d in the appendix; the n o n - a s y m p t o t i c c o r r e c t i o n s a r e also given there. F o r the production of a v e c t o r m e s o n by s c a t t e r i n g a p s e u d o s c a l a r p a r ticle off a nucleon*, P +N ~ V +N, the kinematic c o n s t r a i n t s on the v e r t e x functions y(t) a r e as follows, where the reduced v e r t e x functions 9(t) a r e in g e n e r a l finite at t h r e s h o l d s , p s e u d o t h r e s h o l d s and t = 0. Type (i): Exchange of an evasive natural parity Regge t r a j e c t o r y , 7_P= +1. (a) The Meson Vertex Y o o ( t ) - O, * We identify the particles 1,2, 3 and 4 with P, N, V and N respectively.

(2)

REGGE-POLE MODEL

663

710(t) = -7_10(/) = ~fi710(t)

(3)

•

(b) T h e N u c l e o n V e r t e x H e r e we have the s e l e c t i o n r u l e GP(-1) I = + 1 f o r a t r a j e c t o r y with i s o spinIandG-parity G. Also, m 2 = m 4 = m. y}t4~2(t) = 7_X4_~t2(t) = ~/[~t24 I ~ X 4 ) t 2 ( t ) / ~

,

7 ~11( t ) - 7 - ~11 ( t) ~ 4 t - 4 m 2

(4)

(5)

T y p e (ii): E x c h a n g e of an e v a s i v e u n n a t u r a l p a r i t y R e g g e t r a j e c t o r y T P = -1.

(a) T h e M e s o n V e r t e x 7~3~ 1 (t) = 7-~3-~tl (t) = 471 ;~131 ~ 3 ~ t l (l)/713 (t) ,

(6)

7oo(t) - 4~71o(t) ~ Tla(t).

(7)

(b) T h e N u c l e o n V e r t e x (a) T r a j e c t o r i e s with GP(-1) / = +1.

7 ~11( ) t ~ o 7-~(

) h a s no k i n e m a t i c c o n s t r a i n t s

(8)

({3) T r a j e c t o r i e s with GP(-1) I = -1.

7½-I(0

=

7-½½(0 ~ o , -7_½_½(t)= 47~½½(t).

(9) (10)

T y p e (iii): E x c h a n g e of c o n s p i r i n g p a r i t y doublet R e g g e t r a j e c t o r i e s , M = 1. We c o n s i d e r the e x c h a n g e of an u n n a t u r a l - p a r i t y pole ~ with G P ( - 1 ) I = -1 and an o p p o s i t e - p a r i t y p a r t n e r , c, with GP(-1) I = +1, w h i c h h a v e t r a j e c t o r i e s c o i n c i d e n t at t = 0, ac(0) = a~(0). (a) T h e M e s o n V e r t e x (m 3 > m l )

--

?

o/T13(o

,

(11) (12)

7 o(t) -- 7 1o(t) :

710(t=O) : "zYlO(t=O) •

(13)

7~o(t ) h a s no k i n e m a t i c f a c t o r s and eqs. (2) and (7) still hold f o r c and n r e spectively. (b) T h e N u c l e o n V e r t e x

7~_,(t)--7 c _1 _~(t) = -,/-t?ci/fft-4m 2 22

-2-2

22

(14)

664

G.V.

DASS and

C.D.

=

_-

C

FROGGATT

(15)

7r

!_~(t=0) = T!_~(t=0) ,

-22

(16)

22

7~_(t) has no k i n e m a t i c f a c t o r s and eqs. (5) and (9) still hold f o r c and n r e 22 . spectlvely. All the a b o v e t r a j e c t o r i e s a r e t a k e n to be l e a d i n g m e m b e r s (or p a r e n t s ) of a f a m i l y of R e g g e p o l e s at t = 0 and f o r e v a s i o n , they have n o n - s i n g u l a r r e s i d u e f u n c t i o n s . In p a r t i c u l a r , an e v a s i v e TP = -1, GP(-1) I = -1 t r a j e c t o r y (e.g. a s i m p l e pion) d e c o u p l e s f r o m the NN s y s t e m at t = 0 and does not c o r r e s p o n d to the u s u a l (refs. [33, 34]) 0(4) d a u g h t e r a s s i g n m e n t . T h e f i r s t d a u g h t e r of s u c h a t r a j e c t o r y a l s o d e c o u p l e s f r o m the NN s y s t e m . F o r c o n s p i r a c y , the c o n s t r a i n t eqs. (13) and (16) e n s u r e that the r e l a t i o n [31, 32,35] 01 is s a t i s f i e d .

+ if~½,

01 ~ , / i

(17)

F o r t y p e (i) and type (iii) e x c h a n g e s , we have the b e h a v i o u r f ; ½ , 00 ~ , / 7 .

(18)

T h e d i f f e r e n t i a l c r o s s s e c t i o n , dcr/dt, c o i n c i d e s with an a n a l y t i c f u n c t i o n [36] of t in the p h y s i c a l r e g i o n and r e m a i n s n o n - s i n g u l a r on c o n t i n u a t i o n to the t h r e s h o l d s t = (m 1+ m3)2 and 4~n2. T h i s r e s u l t is g u a r a n t e e d in the a b o v e f o r m a l i s m by the c o n s t r a i n t eqs. (5) and (7). H o w e v e r the d e n s i t y m a t r i x e l e m e n t s of the v e c t o r m e s o n , eq. (A.7), e v a l u a t e d in the J a c k s o n f r a m e do c o n t a i n s i n g u l a r i t i e s at t h e s e t h r e s h o l d s . It h a s t h e r e f o r e b e e n s u g g e s t e d [6, 37] that e x t r a d y n a m i c a l i n f o r m a t i o n m a y be o b t a i n e d if the d e n s i t y m a t r i x e l e m e n t s of a p r o d u c e d r e s o n a n c e in the s - c h a n n e l h e l i c i t y f r a m e w e r e a l s o given in e x p e r i m e n t a l p a p e r s . T h e r e q u i r e m e n t that all the e i g e n v a l u e s of the spin d e n s i t y m a t r i x f o r a f r e e v e c t o r m e s o n p r o d u c e d in a p a r i t y c o n s e r v i n g r e a c t i o n be p o s i t i v e s e m i - d e f i n i t e [38, 39] g i v e s the following r e s u l t s in the J a c k s o n f r a m e 011 - ]01-1! >~0,

(19)

0oo(011 - 01-1) - 2101012 >/0 .

(20)

F r o m the d e c a y c o r r e l a t i o n s of the v e c t o r m e s o n , the r e a l e l e m e n t s 0oo, 1 011 = ~(1-Poo), P l - 1 and a l s o R e P l 0 a r e d e t e r m i n e d . F o r n a t u r a l p a r i t y e x c h a n g e we have the w e l l - k n o w n r e s u l t [11] Poo = P l 0 = 0 .

(21)

Pl-1 = i Pll

(22)

In l e a d i n g o r d e r we have

f o r n a t u r a l and u n n a t u r a l R e g g e - p o l e e x c h a n g e s r e s p e c t i v e l y , so that the inequality eq. (19) is s a t u r a t e d . It follows that in R e g g e t h e o r y Poo and

REGGE-POLE MODEL

665

( P l l - P l - 1 ) a c t a s p r o j e c t i o n o p e r a t o r s f o r t h e c o n t r i b u t i o n s of u n n a t u r a l p a r i t y e x c h a n g e s when t h e v e c t o r m e s o n h a s s p i n - c o m p o n e n t z e r o and one, r e s p e c t i v e l y . It i s e a s i l y v e r i f i e d t h a t t h e s e r e s u l t s f o l l o w i n d e p e n d e n t l y of t h e n a t u r e of p a r t i c l e s 2 a n d 4. In l e a d i n g o r d e r e x a c t l y a n a l o g o u s p r o j e c t i o n o p e r a t o r s c a n b e f o r m e d in t h e h e l i c i t y f r a m e . F i n a l l y we n o t e t h a t t h e i n e q u a l i t y eq. (20) i s s a t u r a t e d f o r any n u m b e r of R e g g e p o l e s e x c h a n g e s , p r o v i d e d t h a t no m o r e t h a n one h a s u n n a t u r a l p a r i t y , i . e . we h a v e (23)

Poo(Pll - Pl-1) - 2(ReP10) 2 = 0 .

3. REMARKS

ON

PION

EXCHANGE

Due to the closeness of the pion pole at t ~ 0.02 GeV 2 to the physical scattering region, the contribution of the low ranking pion trajectory is enhanced and, with a simple Reggeisation, is similar to that for elementary pion exchange. This predicts strong narrow minima in the forward direction for the reactions np --* pn and ~p ~ ~+n, in contrast with the observed sharp peaks. In order to reproduce the latter results, it is necessary to mix opposite parity exchanges in a correlated way. Regge cuts or absorptive corrections provide such a mechanism. However a simpler model involving only Regge pole exchanges has been suggested [6, 24, 26-28]. This model introduces an opposite parity partner c for the pion having a trajectory which is coincident at E - 0. With an M = 1 conspiracy between ~ and c, the s-channel vertices with unit helicity flip can be finite at E = 0, but then the non-flip vertices must vanish [6, 23,24]. In this way, the observed peaks in np ~ pn and yp -~ ~+n have been fitted [26-28] with the conspirator .choosing nonsense at ~ = 0. However, it seems that a more complicated model of pion exchange is necessary [23]. Here we consider the predictions of the evasive and conspiring Regge pole models for vector meson production. For vector meson production, P+N--~ V+N, the dominant pion amplitude is unaltered by conspiracy and the main effect is in the small t behaviour of the density matrix elements. As the scattering angle 0 tends to zero, the general behaviour of the spin density matrix element Pr~z' for a resonance produced in an inelastic reaction is 6 Im-m' I . In our Regge-pole models for vector-meson production, the asymptotic behaviour at t = 0 of the modified density matrix 2 Pro, rn' = (8~sP12

d~/dt)Pm, m'

= ( _ l ) m _ m,

~ ~2,~4

i s as f o l l o w s

ft

t.'t*

~-9~a,O~Ag~.4,,~ ,.,= 0m'

'

(24)

666

G . V . DASS a n d C. D. F R O G G A T T

with evasive pion 2

with conspiring pion 2

t

constant

11 Pl0

-Pl-1

3

t~/(t-m2)

~2 t2/(t-mv)

t

t

A f t e r n o n - a s y m p t o t i c c o r r e c t i o n s (eq. (A.16)) a r e m a d e , t h e c o n t r i b u t i o n s of s i m p l e R e g g e p o l e s in t h e f o r w a r d d i r e c t i o n to P l l a r e n e g l i g i b l e in c o m p a r i s o n w i t h t h e d o m i n a n t p i o n a m p l i t u d e at p r e s e n t e n e r g i e s . H e n c e f o r e v a s i o n , we h a v e e f f e c t i v e l y t h e p r e d i c t i o n t h a t Poo ~ 1 a s ~ ~ 0 f o r t h e normalised density matrix. However the conspirator c gives a non-vanishing c o n t r i b u t i o n in t h e f o r w a r d d i r e c t i o n a n d t h e g e n e r a l k i n e m a t i c a l l y a l l o w e d b e h a v i o u r 0 < Poo < 1 i s t h e n o b t a i n e d . S i m i l a r l y , f o r c o n s p i r a c y , t h e matrix element/510 has its general kinematic behaviour whereas evasion c a u s e s it to v a n i s h m o r e s t r o n g l y in t h e f o r w a r d d i r e c t i o n . O u r a n a l y s i s n e g l e c t s t h e k i n e m a t i c c o n s t r a i n t s of e q s . (5) a n d (7) at t = 4 m 2 a n d (m 1 + m3)2. H o w e v e r we do c o n s i d e r t h e e f f e c t s of t h e c o n s t r a i n t eq. (7) at t h e p s e u d o t h r e s h o l d t = t_ = (m 3 - m l ) 2 w h i c h i s c l o s e r to t h e p h y s i c a l r e g i o n . Of c o u r s e , f o r e l e m e n t a r y p i o n e x c h a n g e t h i s c o n s t r a i n t i s t r i v i a l l y s a t i s f i e d w i t h Yl0 --- 0 and Yoo h a v i n g t h e l e s s s i n g u l a r b e h a v i o u r at t - . In t h e R e g g e p o l e m o d e l t h e g e n e r a l k i n e m a t i c b e h a v i o u r i s e x p e c t e d at t_. If it i s a s s u m e d t h a t , a p a r t f r o m t h e d y n a m i c a l g h o s t k i l l i n g f a c t o r in 710 a t ~ = 0, t h e r e a r e no z e r o s in t h e r e d u c e d v e r t e x f u n c t i o n s ~rr b e t w e e n t = 0 a n d t - , t h e n t h e s i g n of P l 0 in the p h y s i c a l r e g i o n i s d e t e r m i n e d by t h e c o n s t r a i n t eq. (7). F o r e v a s i o n P l 0 i s p r e d i c t e d to b e p o s i t i v e , w h e r e a s f o r c o n s p i r a c y t h e k i n e m a t i c f a c t o r s a t t = 0 in t h e m e s o n v e r t i c e s Y~oo a n d 7~0 a r e i n t e r c h a n g e d a n d a n e g a t i v e s i g n i s p r e d i c t e d . H e n c e to o b t a i n t h e o b s e r v e d n e g a t i v e v a l u e s f o r R e P l 0 w i t h e v a s i o n it i s n e c e s s a r y to i n s e r t a z e r o into one of t h e m e s o n v e r t e x f u n c t i o n s . In p r a c t i c e ( s e e eq. (33)) a z e r o i s i n s e r t e d into ~ 0 w h e n t h e p s e u d o t h r e s h o l d c o n s t r a i n t i s i m p o s e d . It i s of c o u r s e i m p l i c i t in t h i s d i s c u s s i o n of Re P l 0 t h a t , a p a r t f r o m t h e p i o n , a l l t h e t r a j e c t o r i e s e x c h a n g e s h a v e TP = +1.

4. T H E M O D E L W e now d i s c u s s t h e p o s s i b l e s t a t e s w h i c h m i g h t be e x c h a n g e d in t h e t c h a n n e l a n d t h e R e g g e p o l e s i n c l u d e d in o u r f i t s to t h e d a t a . D e n o t i n g t h e a m p l i t u d e s ( o m i t t i n g h e l i c i t y l a b e l s ) f o r v+p -~ p e p by T~ a n d f o r v - p -~ P°n by To, i s o s p i n c o n s e r v a t i o n g i v e s T_ - T+ = ~/2 T o . T h e r e a r e s i x d i f f e r e n t t y p e s of odd G p a r i t y t r a j e c t o r i e s c h a n g e d in t h e s e r e a c t i o n s :

(25) which can be ex-

REGGE- POLE MODEL (i) I s o v e c t o r a n d i s o s c a l a r w i t h

667

TP = +1:

A2, w.

(ii) I s o v e c t o r s w i t h TP = -1 a n d T = +1: (iii) I s o s c a l a r s

~, A 1.

w i t h TP = -1 a n d T = +1:

H, H l ( s a y ).

T r a j e c t o r i e s w i t h GP(-1) I = +1 (A2, w, A1, H e t c . ) c o u p l e to t h e t r i p l e t NN s t a t e , w h i l e t r a j e c t o r i e s w i t h GP(-1)I = -1 m u s t h a v e u n n a t u r a l p a r i t y (~, H 1 e t c . ) in o r d e r to c o u p l e to t h e NN s y s t e m in t h e s i n g l e t s t a t e . U s i n g s u b s c r i p t s to i n d i c a t e t h e R e g g e p o l e s e x c h a n g e d , we h a v e T+ = + ( T ~ + T A I + T A 2

)+T

w+T H+TH1 .

(26)

Then

IT+f2 +

] 2 - ] T o l 2 : 2LT

+

THI 12 ,

(27)

gives the contribution from trajectories with charge conjugation C = -i. F r o m t h e p r o j e c t i o n o p e r a t o r s Poo a n d P l l + P l - 1 a n d t h e d i f f e r e n t i a l c r o s s s e c t i o n s f o r t h e d i f f e r e n t c h a r g e m o d e s , one c a n in p r i n c i p l e o b t a i n t h e f o l l o w i n g c o n t r i b u t i o n s in l e a d i n g o r d e r ITA212 , ]Twl 2 ,

I T ~ 1 2 + ] T A l l 2 a a d l T H I 2 + I T H 1 ]2 ,

f o r l o n g i t u d i n a l a n d t r a n s v e r s e p - m e s o n p r o d u c t i o n (the n u c l e o n h e l i c i t i e s being averaged over). In p r a c t i c e , t h e d a t a a r e not a c c u r a t e e n o u g h to a p p l y t h e a b o v e p r o c e d u r e . T h e s e r e a c t i o n s a r e d o m i n a t e d by p i o n e x c h a n g e o v e r a r a n g e of e n ergies and the observed density matrix elements suggest that the backg r o u n d c o m e s p r e d o m i n a n t l y f r o m n a t u r a l p a r i t y e x c h a n g e . In p a r t i c u l a r t h e d a t a a r e c o n s i s t e n t w i t h t h e e q u a l i t y , eq. (23), but t h e e r r o r s a n d p o s s i b l e c o r r e l a t i o n s b e t w e e n t h e m m a k e it d i f f i c u l t to t e s t a c c u r a t e l y . So in m o s t of o u r f i t s to t h e d a t a we t o o k ~, A2 a n d w R e g g e - p o l e e x c h a n g e s . F o r a c o n s p i r i n g p i o n , t h e c o n s p i r a t o r c c o n t r i b u t e s to t h e TA2 a m p l i t u d e s . D i f f e r e n c e s b e t w e e n t h e two n o n - c h a r g e - e x c h a n g e r e a c t i o n s , ~+p ~ p+p, t h e n a r i s e f r o : l i n t e r f e r e n c e b e t w e e n t h e TA2 a n d T w a m p l i t u d e s . T h e A 2 a n d w t r a j e c t o r i e s h a v e o p p o s i t e s i g n a t u r e s and s o t h e i n t e r f e r e n c e b e t w e e n t h e m i s s m a l l . E x p e r i m e n t a l l y , t h e c r o s s s e c t i o n s f o r t h e s e two c h a r g e modes are compatible with equality. With the available data, a meaningful s e p a r a t i o n of t h e TP = +1 p o l e s into n u c l e o n h e l i c i t y f l i p a n d n o n - f l i p a m p l i t u d e s i s not p o s s i b l e . So, in o u r f i t s w i t h co and c e x c h a n g e , we r e t a i n only t h o s e n u c l e o n h e l i c i t y v e r t e x f u n c t i o n s w h i c h a r e m o s t s i n g u l a r a t t = 0. T h i s c o r r e s p o n d s to h e l i c i t y n o n - f l i p a n d h e l i c i t y f l i p r e s p e c t i v e l y a n d s o o u r w a n d c c o n t r i b u t i o n s do not i n t e r f e r e . F o r t h e ~, (c), co a n d A 2 e x c h a n g e m o d e l we h a v e [40] ITcsl 2 = ½ ( I T + I 2 + IT_ 12- iTol 2) .

(28)

668

G.V.DASS and C.D.FROGGATT

5. P A R A M E T R I S A T I O N O F T H E A M P L I T U D E S W e c o n s i d e r t h e e x c h a n g e of t h e f o l l o w i n g t r a j e c t o r i e s a(t): t h e R e g g e p o l e s A2, co, 77 a n d , in t h e c a s e of c o n s p i r a c y , i t s o p p o s i t e p a r i t y p a r t n e r c. In t h e c a s e of A2 e x c h a n g e , t h e r a t i o of t h e two n u c l e o n v e r t i c e s i s t a k e n f r o m a R e g g e p o l e a n a l y s i s of ~ p r o d u c t i o n [41]. U n i t s w i t h ~ = c = 1 a r e u s e d a n d a l l m a s s e s a r e m e a s u r e d in u n i t s of 1 GeV. In p a r t i c u l a r , t h e a r b i t r a r y s c a l e p a r a m e t e r s o i s t a k e n to b e 1 GeV 2. T h e d o m i n a n t p i o n e x c h a n g e a m p l i t u d e i s u n a l t e r e d by c o n s p i r a c y and, in t h e n o t a t i o n of eq. (1), h a s t h e f o r m 77 77 2 2 7oo7½½ = ½G77a' ( [ T13(t=rn77 ) e x p [ F o ( t - m 2 ) ] / ' r l 3 ( t

) ,

(29)

w h e r e t h e p i o n t r a j e c t o r y i s t a k e n to b e a(t) = ( t - m 2 ) a '. T h i s a m p l i t u d e i s n o r m a l i s e d at t=rn2 = 0.0196 GeV 2 by t h e v a l u e s of t h e p h y s i c a l p i o n c o u p l i n g c o n s t a n t s . T a k i n g t h e m a s s a n d w i d t h of p to b e [42]

rnp = 0.76 G e V , and the rationalised pseudoscalar b y g2/477 = 14.5, we f i n d t h a t

F p = 0.12 GeV ,

(30)

p i o n n u c l e o n c o u p l i n g c o n s t a n t to b e g i v e n 1

G = 15.2 (mb)~

for

~+p ~ p + p

F o r e v a s i v e p i o n e x c h a n g e we p a r a m e t e r i s e tex as 77

.

(31)

the meson helicity flip ver-

77

710 7½½ = ta(t) e x p ( F l t)H(t) /'rl3 (t) .

(32)

W h e n t h e k i n e m a t i c c o n s t r a i n t at t h e p s e u d o t h r e s h o l d eq. (7) i s i m p o s e d we take

H(t) = H o ( t - t o ) ,

(33)

where

0 < t o < t_ = ( r n 3 - m l ) 2 in o r d e r t o o b t a i n a n e g a t i v e Re

10.

T h e c o n s t a n t H o i s d e t e r m i n e d in t e r m s of t h e o t h e r p a r a m e t e r s

by

2

2

H 0 = G77a' 713(t=rn77) 2 J2[_(t_-to)a(t_)

exp{_Forn2 +t-(Fo-F1)}

(34)

If t h e k i n e m a t i c c o n s t r a i n t at ! = t_ i s n e g l e c t e d in a f i t to t h e d a t a , we s i m p l y t a k e t h e f o r m f a c t o r to be a c o n s t a n t

H(t) = H . F o r t h e e x c h a n g e of a c o n s p i r i n g p i o n w e p a r a m e t e r i s e vertex as 77

7r

Yl0 Y½½ = a~(t)H1 e E l t/'r 13 (t)

(35) the helicity flip

(36)

REGGE-POLE MODEL

669

and the k i n e m a t i c c o n s t r a i n t eq. (7) g i v e s H 1 = Gna' z132 (t=m2) f / e x p { _ F

o r e 2n +t_(Fo_F1)}/(24-~a(t_) ) .

(37)

T h e c o n s p i r a t o r c is a s s u m e d to c h o o s e n o n s e n s e at a c = 0 and we take c yc, YlO - ~

= ac(t)H 2 e F 2 ~ t - ~

2

(38)

w h e r e , f r o m eqs. (13) and (16), H2 : -2mHx/(m2-m2 ) .

(39)

In o u r fits to the d a t a we t a k e the p a r t n e r t r a j e c t o r i e s to be d e g e n e r a t e with a s l o p e a' = I(GeV) -2, ac(t) = an(t) = t - m 2

(40)

.

It is a s s u m e d that the e v e n - s i g n a t u r e pole A 2 c h o o s e s n o n s e n s e at a = 0. F o r w e x c h a n g e , we include only the s e n s e - n o n s e n s e a m p l i t u d e which h a s the s a m e g h o s t - k i l l i n g f a c t o r s f o r the s e n s e o r n o n s e n s e c h o o s i n g m e c h a n i s m . We then p a r a m e t e r i s e the s i m p l e TP = +1 Regge pole c o n t r i b u t i o n s in our model as Yl0 ~y~I,

= 4-[a(t)A e B t / ] t - 4 r n

Yl0Y½½

= 4-[a(t)X(t)/~

2

for

w

for

A2 ,

for

A2 ,

(41)

X(t) = A 2 e B 2 t {0.06 e -0"91t (2m - ½ t / m ) ( 2 a + l ) } .

(42)

Yl0Y-½½ = t a ( t ) A 2 e B 2 t / ] t ~ 4 m 2 w h e r e [41]

A n e g a t i v e s i g n can of c o u r s e be i n s e r t e d into eq. (42) without a f f e c t i n g the r e s u l t s of ref. [41] o r of this p a p e r p r o v i d e d the s i g n of A 2 in eq. (41) is a l s o changed. T h e k i n e m a t i c c o n s t r a i n t eq. (5), at t = 4m2 has b e e n n e g l e c t e d [43] in r e f . [41] and t h i s g i v e s r i s e to the f a c t o r ( 2 m - ½ t / m ) in eq. (42). H o w e v e r this t h r e s h o l d is f a r f r o m the s - c h a n n e l p h y s i c a l r e g i o n and h a s little effect on the n a t u r e of the fit. It s h o u l d be noted h e r e that a z e r o h a s not b e e n i n c l u d e d in the co r e s i d u e function [40, 43-471 n e a r t = - 0 . 1 5 (GeV) 2 n o r in the n r e s i d u e function n e a r t = -0.03 (GeV)~ ( r e f s . [23, 26-28]). T h e t r a j e c t o r y f u n c t i o n s a r e t a k e n [41, 40, 47] to be aA2(t) = -1 + 1 . 9 6 / ( 1 . 4 - 0.65t) , aw(t)

=0.45+0.9/.

We will c o m m e n t f u r t h e r on o u r c h o i c e of p a r a m e t e r i s a t i o n

(43) in s e c t . 7.

670

G.V. DASS and C. D. FROGGATT

6. DATA ANALYSIS Given below a r e the e x p e r i m e n t a l data on dg/dt(s, t) and Prnrn,(s, t) for - t <~ 1 (GeV/c) 2 which w e r e used in our s e a r c h e s . Some r e l e v a n t r e m a r k s follow. (i) Occasionally, the f i r s t bins n e a r the f o r w a r d d i r e c t i o n a r e omitted when it a p p e a r s that scanning b i a s e s make the o b s e r v e d d(r/dt too low; an example is the 8 G e V / c , 7r-p~ p-p e x p e r i m e n t [48]. (ii) When data f r o m different l a b o r a t o r i e s d i s a g r e e at a given energy, we take the one that i n t e r p o l a t e s smoothly with adjacent e n e r g i e s . (iii) In some c a s e s , a p r e l i m i n a r y v e r s i o n of the data (now published) is u s e d in our s e a r c h e s ; data a r e always taken f r o m the f i r s t quoted p a p e r in the a p p r o p r i a t e r e f e r e n c e . (iv) When the e x p e r i m e n t a l e r r o r s on dg/dt a r e not given, we take them to be only the n o r m a l s t a t i s t i c a l e r r o r s . (v) As a model f o r the effects of i n t e r f e r e n c e between the i s o s c a l a r Swave ~Tr state and the po on Prnm' we take [49, 50] a constant value for pSoo = 0.072. The Prnrn' a r e r e - n o r m a l i s e d and the quantities Prnrn,/(1-pSoo ) a r e used as our data points. We t u r n to the data now. (A) Charge exchange: 229 data points

d(~/dt for 7r+n ~ pOp at 6 G e V / c (ref. [51]):

14 data points

dg/dt for ~-p ~ p ° n at 15 G e V / c (ref. [52]), 8 G e V / e (ref. [53]), 7 C e V / e (ref. [54]), 4.2 G e V / c (ref. [55]), 4 G e V / c (ref. [56]) and 2.7 G e V / c (ref. [57]):

110 data points

Prnrn' for ~-p ~ P ° n at 11 G e V / c (ref. [58]), 8 G e V / c (ref. [53]), 7 G e V / c (ref. [54]), 4.2 G e V / c (ref. [55]), 4 G e V / c (ref. [56]), 3 G e V / c (ref. [59]), 2.75 G e V / c (ref. [60])* and 2.7 G e V / c (ref. [57]):

105 data points

Non-charge-exchange: 231 data points dcs/dt for ~+p ~ p+p at 8 G e V / c (ref. [61]) and (B)

4 G e V / e (ref. [62]):

24 data points

dc~/dt for ~-p -~ p-p at 8 G e V / c (ref. [48]), 4.2 G e V / c (ref. [55]), 4 G e V / c (ref. [63]) and 2.7 GeV/c (ref, [57]):

90 data points

Prnrn' for 7r+p-~p+p at 8 G e V / c (ref. [61]) and 4 G e V / c (ref. [62]):

18 data points

* This experiment, in facL, includes da[a in the range 2.05--* 3.22 GeV/c. The experimental Drnrn' for the process 7T-p-~p°n shown in fig. 2 (b) should be renormalised.

R E G G E - P O L E MODEL

671 ~-t(Gev/¢)

2

•2

EOC

4

2 8

I0

6

I0

>

FIG.

& 2

4 6 - t (o,v/c) 2

Ib

!B

i!o

8

I0

-t (Gev/c) 2 2

4

5

I iJ

/9 +

\ \ 8 GeV/c

!

~/

t

"\ \

ol

\ OOI i

~..

,!o ~-t(GcV/c)

2

FIG. I~

0

2

.4

6 -t (ccv/c)

FIG.

8

Ic

I0

2

Fig. 1. D i f f e r e n t i a l c r o s s s e c t i o n s d(7/dt in [mb/(GeV/c) 2] f o r the p r o c e s s e s (a) 7T-p---*/:)On (at l a b m o m e n t a of 2.7, 4, 4.2, 7, 8 and 15 G e V / c ) and 7P~n---~pOp (at a l a b m o m e n t u m of 6 GeV/c), (b) 7T-p--~p-p (at lab m o m e n t a of 2.7, 4, 4.2 and 8 G e V / c ) and (c) 7r+p--~ p+p (at lab m o m e n t a of 4 and 8 G e V / c ) , plotted as f u n c t i o n s of t(GeV/c) 2. T h e dotted and full line c u r v e s (in t h i s and the following f i g u r e s ) a r e o u r r e s u l t s f o r the e v a s i v e and the c o n s p i r a t o r i a l s o l u t i o n s r e s p e c t i v e l y .

672

G . V . DASS and C. D. FROGGATT

~o

Poo

2" 7 0 °

-

-O 5 I-

R'~o 0 / Y::~-

I

J_._i ~ ~

"!f +

2 75,0 °

_ ~ ~ ~-~--+

~_

R+p,o

'-t-' '-~

-O 5h iI

~ - t (G,v/~)

FIG.

i

r

-t (oev/c)2

2

2o

FIG.

2b

i / .$

I0

\\\

/:%o

pooO~I ,OH

i- ~ ~-~-=.~=..~

~0_

5 -Q 3p ° I

-5 I-

P~-i 5 f

~ ~ 4p°

-5 mCP,o

RcP'°_. 5

-t (G+V/c) 2

-t (c+v/~) 2 FIG.

FIG.

2c

F i g . 2. (a)-(d).

2d

___

REGGE - P O L E M O D E L

I

673

O I'0~\

05 J0=e C

&o

~o

P,-I

I

0 -5

42p°

~CP~oo -5

~

~

6

o !,

PI-I _

_

o!2 o!3 ~-t (Gev/¢) ~

o!4

0

, L

:

-

/ .//

........

-.5

b!s -,

FIG. 2¢

.

~

(G,v/c)'

FIG.

IO~x~ x

t"

2f

I

Poo

~o

5

,Or.~ 8p °

-SL "5L

HP°

oi._

Rcp,: 5

6

~

~ FIG. 29

',

"

,!o

~

o ~,

~

~

~

--~

FIG. 2h

F i g . 2. (a)-(h). D e n s i t y m a t r i x e l e m e n t s f o r the p r o c e s s 7r-p - - , p ° n at lab m o m e n t a of 2.7, 2.75, 3, 4, 4.2, 7, 8 and 11 G e V / c r e s p e c t i v e l y , plotted as f u n c t i o n s of t(GeV/c) 2.

674

G . V . DASS and C. D. F R O G G A T T

1.0[-~..

\\ P°° 0.5

~ / 2

°t PI-, PH

t

---~

,

2 7sp-

~ '

!

..~o_iI :~:~-~" -~-~-~-+ -5

•',

6

.'2

.S

~

.~

6

-t(G,v/~)2

',

!2

.'3

's

~

-t (c,¢v/c) 2

P~ 05

P~

0

s

,,\

0

'

~ . . ~

•5 ~

~ _--

P,-,

1 2-8ep-.5

I

t~

3p-

____L

L__L_I-__[

R'e'O~O. 5~

-5

65

•',

I~ - t

.~

~'s

-~

-t CCeV/c)2

(G,v/:) ~ F i g . 3. (a)-(d).

•~

,'.o

R E G G E - P O L E MODEL

675

I 0t~.,. no° o

P~oS i " ~

1~

P,-,0.5[

[.___

,,. . . . . -

P,-, 0

0

-0.5

5

n'P'°-o-s Of''"

o'

/

4 2,0-

4p-

L

n'P--s '°°ir~L~L ~ L_~__

I ',

2 -~

3

,4

il I~S -t (Gev/c) 2

.'s

(G,v/c) 2

2

i

P'° .5 0 7p-

P,~

.5 0 --5

~ c P ' ° - - 5 0 ~ o'2

0'4 o'-6 (c,v/{) 2

-t

o!.

l '0

(~

.',

.2

.3

.~.

"5

, (o,v~c)2

Fig. 3. (a)-(h). D e n s i t y m a t r i x e l e m e n t s f o r the p r o c e s s ~ - p - - ~ p - p at lab m o m e n t a of 2.7, 2.75, 2.88, 3, 4, 4.2, 7 and 8 G e V / c res}~ectively, plotted as f u n c t i o n s of t (GeV/c) z.

6 76

G . V . DASS and C. D. F R O G G A T T

o sJ- 1~-~_

//"

Z

"~

,~

P~

5 0 ~ -05 l

,~

8p*

4P+ ~.5

R'zPI° _ _ ~

I

-, (o,w~)'

'2-

3

•~

,I5"

-t(G,vl~)'

F i g . 4. (a, b). D e n s i t y m a t r i x e l e m e n t s f o r the p r o c e s s ?r+p--*p+p at lab m o m e n t a of 4 and 8 G e V / c r e s p e c t i v e l y , plotted as f u n c t i o n s of t (GeV/c)2.

2.0

1.0

O2

0.~

06 O8 -t (o¢v/¢~

10

.{73

.1 ,15 .2 - t |GcWc) 2

,25

x 0.2,

o~ -0.2'

-0.1 •

Fig. 5. V a r i a t i o n of the q u a n t i t y [40] X(s, t) ---[da(~7+p-~ p+p)/dt + d~(rr-p--~ p-p)/dt d(~(TT-p --~p°n)/dt]as a f u n c t i o n of t at lab m o m e n t a of (a) 4 G e V / c and (b) 8 G e V / c . The r e s u l t s of o u r s o l u t i o n s (a) and (b) a r e i n d i s t i n g u i s h a b l e f o r X(s, t). In o u r m o d e l only ¢0 e x c h a n g e c o n t r i b u t e s to X(s, t). -

REGGE- POLE MODEL

677

Prom' f o r ~ - p - ~ p - p

a t 8 G e V / c ( r e f . [48]), 4.2 G e V / e ( r e f . [55]), 4 G e V / c ( r e f . [63]), 3 G e V / c ( r e f . [59]), 2.88 G e V / c ( r e f . [64]),

2.75 GeV/c (ref. [60])* and 2.7 GeV/c (ref. [57]):

99 d a t a p o i n t s

T h e p a r a m e t e r s of t h e m o d e l a r e a d j u s t e d to m i n i m i z e ×2. D a t a f o r a l l t h e c h a r g e s m o d e s a r e a n a l y s e d t o g e t h e r . O u r r e s u l t s f o r t h e s o l u t i o n s (a) w i t h an e v a s i v e a n d (b) w i t h a c o n s p i r a t o r i a l p i o n R e g g e p o l e a r e s h o w n in f i g s . 1-5 a s d o t t e d a n d f u l l l i n e c u r v e s r e s p e c t i v e l y . T h e p a r a m e t e r i s a t i o n of eq. (35) w a s u s e d f o r t h e e v a s i v e s o l u t i o n (a); s e e . h o w e v e r , s e c t . 7. T h e p a r a m e t e r s q u o t e d b e l o w r e f e r to t h e ~+p ~ p+p a m p l i t u d e . T h e n o n a s y m p t o t i c c o r r e c t i o n s of eq. (A.16) h a v e b e e n i n c l u d e d in o u r f i t s . (a) Evasive solution; ×2 = 970 f o r 7 a d j u s t a b l e p a r a m e t e r s : Fo=

0.13 GeV - 2 ,

A 2 = 117

~r~mb

H=-143 ,

B2 =

~mmb. G e V , 6.2GeV -2,

F1 =

7.1GeV -2,

A = 35

~ m b . GeV

a n d B = - 0 . 3 5 GeV - 2 . (b) Conspiratorial solution; X2 = 751 f o r 7 a d j u s t a b l e p a r a m e t e r s : Fo = A2 =

0.058 G e V - 2 , 15

~-m-mb

,

F1 =

0.59 GeV - 2 ,

B2 =

1.2

GeV - 2 ,

F2 =

2.7 GeV - 2 ,

A = 34

V ~ - b . GeV

a n d B -~ - 0 . 3 1 GeV - 2 . 7. DISCUSSION T h e a b s o r p t i v e p e r i p h e r a l m o d e l h a s b e e n s u c c e s f u l [16~ in r e p r e d u c i n g a l l t h e m a i n f e a t u r e s of t h e r e a c t i o n ~N ~ p N f o r - t ~ 1 5 m ~ on t h e b a s i s of pion exchange alone. There have been previous evasive Regge pole analyses [65-67] of p p r o d u c t i o n . In t h i s p a p e r we h a v e m a d e a d e t a i l e d s t u d y of t h i s r e a c t i o n , i n c l u d i n g a c a r e f u l t r e a t m e n t of t h e k i n e m a t i c s a n d d e n s i t y m a t r i x e l e m e n t s . It w a s not p o s s i b l e to m a k e a u n i q u e d e t e r m i n a t i o n of t h e p a r a m e t e r s in o u r m o d e l . S o m e t y p i c a l e v a s i v e a n d c o n s p i r a t o r i a l f i t s a r e s h o w n in f i g s . 1-5 a s d o t t e d a n d full l i n e c u r v e s r e s p e c t i v e l y . A n u m b e r of t y p e s of fit w e r e m a d e to d i f f e r e n t s u b s e t s of t h e d a t a . A n a l y s i n g t h e t h r e e d i f f e r e n t m o d e s s e p a r a t e l y w i t h ~ a n d A 2 e x c h a n g e , it w a s found t h a t t h e c h a r g e d f i n a l s t a t e s p - p a n d p + p r e q u i r e d s i m i l a r p a r a m e t e r s but pOn r e q u i r e d l e s s n a t u r a l p a r i t y e x c h a n g e . I n c l u s i o n of w e x change provided a natural method for explaining this difference. The par a m e t e r s f o r t h e i s o v e c t o r e x c h a n g e s o b t a i n e d f r o m an a n a l y s i s of t h e p ° n f i n a l s t a t e w e r e t h e n v e r y s i m i l a r to t h o s e o b t a i n e d in t h e c o m b i n e d f i t s to a l l t h e d a t a , c o r r e s p o n d i n g to t h e p l o t t e d c u r v e s . * See

footnote

page

670.

678

G.V.DASS and C.D. FROGGATT

T h e c o n s p i r a c y f i t s g a v e m u c h l o w e r X2 a s t h e y r e p r o d u c e d t h e v a l u e s of Poo and R e P l 0 b e t t e r at s m a l l t v a l u e s . U n f o r t u n a t e l y m o s t of t h e a c c u r a t e d a t a on d e n s i t y m a t r i x e l e m e n t s a r e f r o m t h e l o w e n e r g y r e g i o n w h e r e c o r r e c t i o n s to t h e l e a d i n g o r d e r R e g g e t h e o r y m i g h t b e n e c e s s a r y . F o r i n s t a n c e an A 1 d a u g h t e r t r a j e c t o r y w i t h s i n g u l a r r e s i d u e f u n c t i o n s [33, 68, 69] could interfere with the ~ contribution and destroy the less singular behavio u r of o u r e v a s i v e m o d e l a t [ 6 , 3 1 , 3 2 , 3 5 ] t = 0. In o u r f i t s , o v e r h a l f of t h e d i f f e r e n c e in X2 a r i s e s f r o m t h e pO d e n s i t y m a t r i x d a t a , s e e fig. 2, f o r w h i c h t h e r e i s s o m e a m b i g u i t y d u e to i n t e r f e r e n c e w i t h a n I = 0, ~ b a c k g r o u n d * . It s h o u l d b e r e m e m b e r e d w h e n c o m p a r i n g o u r f i t s w i t h t h e d a t a t h a t , in a v e r a g i n g o v e r a r a n g e of t, t h e d e n s i t y m a t r i x e l e m e n t s a r e w e i g h t e d w i t h t h e m a g n i t u d e of t h e d i f f e r e n t i a l c r o s s s e c t i o n . F o r t h e c o n s p i r a t o r i a l s o l u t i o n , m o s t of t h e i s o v e c t o r n a t u r a l p a r i t y e x c h a n g e i s p r o v i d e d by c a n d i s n o n - v a n i s h i n g at t = 0. T h e A 2 e x c h a n g e i s t h e n not v e r y i m p o r t a n t a n d , c h o o s i n g n o n s e n s e a t ~ = 0, g i v e s a c h a r a c t e r i s t i c a l l y b r o a d c o n t r i b u t i o n [41] to d ~ / d t . S i m p l e ~-P = +1 R e g g e p o l e e x c h a n g e a m p l i t u d e s v a n i s h in t h e f o r w a r d d i r e c t i o n o r a t t = 0. T h e s e c o n t r i b u t i o n s (eq. (41)) c a n only b e e n h a n c e d at s m a l l t by t a k i n g l a r g e v a l u e s f o r t h e p a r a m e t e r s A 2 a n d B 2. C o n s e q u e n t l y in o u r e v a s i v e s o l u t i o n , t h e A 2 R e g g e p o l e g i v e s a s t r o n g n a r r o w c o n t r i b u t i o n to t h e d i f f e r e n t i a l c r o s s s e c t i o n s . T h e m e s o n h e l i c i t y f l i p a m p l i t u d e f o r t h e e v a s i v e p i o n (eq. (32)) i s r e q u i r e d to h a v e a s i m i l a r b e h a v i o u r . It i s t h e r e f o r e f o u n d t h a t in o u r e v a s i v e f i t s f o r - t > 0.5 GeV 2, t h e pO p r o d u c t i o n d i f f e r e n t i a l c r o s s s e c t i o n i s v e r y s m a l l , s e e fig. 1 (a), g i v i n g a s u b s t a n t i a l c o n t r i b u t i o n to X2; a l s o Poo rises for large -t. This can be off-set by introducing extra t-dependence i n t o one o r m o r e of t h e i s o v e c t o r R e g g e p o l e r e s i d u e f u n c t i o n s . I n c l u d i n g t h e p s e u d o t h r e s h o l d c o n s t r a i n t (eqs. (33) a n d (34)), in o u r e v a s i v e f i t s s e e m e d a r t i f i c i a l a s t h e s e a r c h p r o g r a m m e t h e n t e n d e d s l o w l y to a s o l u t i o n w i t h t o v e r y c l o s e to t, g i v i n g r e s u l t s in t h e p h y s i c a l r e g i o n a l m o s t i d e n t i c a l to t h e u n c o n s t r a i n e d f i t s . H e n c e , a s t_ ~ 0.38 GeV 2 i s not t o o c l o s e to t h e p h y s i c a l r e g i o n in t h i s r e a c t i o n , we h a v e p l o t t e d an e v a s i v e s o l u t i o n w i t h t h e u n c o n s t r a i n e d p a r a m e t e r i s a t i o n of eq. (35). T h e d o m i n a n t p i o n a m p l i t u d e h a s b e e n n o r m a l i s e d to g i v e t h e p h y s i c a l c o u p l i n g c o n s t a n t s at t = m 2 a n d it r e p r o d u c e s q u i t e w e l l , w i t h one e x c e p t i o n , t h e n o r m a l i s a t i o n of d~/dt n e a r t h e f o r w a r d d i r e c t i o n , w i t h o u t any r a p i d v a r i a t i o n in t h e r e s i d u e f u n c t i o n . In f a c t t h e d a t a a r e i n c o n s i s t e n t w i t h t h e z e r o n e a r t = - 0 . 0 5 GeV 2 s u g g e s t e d by f a c t o r i s a t i o n f r o m p r e v i o u s p i o n c o n s p i r a c y f i t s [ 2 6 - 2 8 ] . T h e m a g n i t u d e of d~/dt at 15 G e V / c i s u n d e r e s t i m a t e d by o u r f i t s . T h i s d i s c r e p a n c y i s i n c r e a s e d if t h e r e v i s e d p u b l i s h e d d a t a [52] a r e u s e d . T h e s h a p e of t h e d i f f e r e n t i a l c r o s s s e c t i o n s u g g e s t s e x c h a n g e i s s t i l l d o m i n a n t at s m a l l t and, if c o n f i r m e d , t h e c h a n g e of e n e r gy d e p e n d e n c e a t h i g h e r m o m e n t a w i l l b e d i f f i c u l t to e x p l a i n . T h e f i t s a r e r a t h e r i n s e n s i t i v e to v a r i a t i o n s in t h e n a t u r a l p a r i t y t r a j e c t o r y f u n c t i o n s . In t h e p l o t t e d c u r v e t h e co t r a j e c t o r y (eq. (43)) p a s s e s t h r o u g h z e r o n e a r t = - 0 . 5 GeV 2 a s s u g g e s t e d b y an a n a l y s i s of ~o p h o t o p r o * There seems to be a significant difference between the experimental pO density m a t r i x at 2.7 and 2.75 GeV/c.

REGGE- POLE MODEL

679

d u c t i o n [47]. T h e r e a r e f e w a c c u r a t e d a t a at l a r g e t a n d w i t h o u r p a r a m e t e r i s a t i o n , w e x c h a n g e t e n d s to b e t h e m a j o r c o n t r i b u t o r t h e r e ; so, the e f f e c t of t h e n o n s e n s e z e r o i s e n h a n c e d in o u r c u r v e s . T h e e v i d e n c e f o r s u c h a v a n i s h i n g in p p r o d u c t i o n i s r a t h e r w e a k at p r e s e n t a n d c o m e s e s s e n t i a l l y f r o m one e x p e r i m e n t [62] ( s e e fig. 1 (c)). In fig. 5, we s h o w t h e w c o n t r i b u t i o n f r o m o u r f i t s to t h e p r o j e c t i o n s X(s, t) of t h e d i f f e r e n t i a l c r o s s s e c t i o n s g i v e n in r e f . [40] c o r r e s p o n d i n g to i s o s c a l a r e x c h a n g e ( s e e eq. (28)). T h e p r e s e n c e of t h e n o n s e n s e z e r o is a l s o r e f l e c t e d in t h e p r e d i c t e d b e h a v i o u r of t h e p+ d e n s i t y m a t r i x e l e m e n t s n e a r t = - 0 . 5 ( f i g s . 3, 4). T h e r e is no e v i d e n c e f o r a n o n s e n s e z e r o f o r - t < 0.8 G e V 2 f r o m w e x c h a n g e in KN e l a s t i c s c a t t e r i n g [46]. T h e z e r o in w r e s i d u e f u n c t i o n s n e a r l = - 0 . 1 5 GeV 2, s u g g e s t e d to e x p l a i n t h e c r o s s - o v e r e f f e c t [ 4 3 - 4 6 ] , i s not s e e n in p p r o d u c t i o n o r ~o p h o t o p r o d u c t i o n .

8. CONCLUSIONS In g e n e r a l we h a v e o b t a i n e d r e s u l t s s i m i l a r to t h o s e of t h e a b s o r p t i o n m o d e l . T h e c o n s p i r a c y s o l u t i o n g i v e s a g o o d fit to t h e d a t a o v e r t h e i n v e s t i g a t e d r a n g e of e n e r g i e s w i t h r e a s o n a b l e p a r a m e t e r s n a t u r a l l y s a t i s f y i n g t h e k i n e m a t i c c o n s t r a i n t a t t h e p s e u d o t h r e s h o l d . H o w e v e r it i s i n c o n s i s t e n t w i t h f a c t o r i s a t i o n a n d c o n s p i r a c y f i t s to np ~ pn and y p ~ ~+n h a v i n g a z e r o n e a r t = - 0 . 0 5 GeV2. D o u b l e r e s o n a n c e p r o d u c t i o n f a v o u r s e v a s i v e p i o n e x c h a n g e [23, 70], but f o r 7 r N - p N e v a s i v e s o l u t i o n s c o n s i s t e n t l y h a v e d i f f i c u l t y in r e p r o d u c i n g t h e o b s e r v e d d e n s i t y m a t r i x e l e m e n t s . A l s o , t h e p a rameters obtained are less reasonable and the pseudothreshold constraint i s n e g l e c t e d . U n f o r t u n a t e l y , t h e l a r g e i m p r o v e m e n t in ×2 f o r t h e c o n s p i r a c y s o l u t i o n a r i s e s m a i n l y f r o m d a t a on pO p r o d u c t i o n a n d f r o m l o w - e n e r g y d a t a . In f a c t a l l r e a c t i o n s s e e m t o p r e f e r p i o n - e x c h a n g e a m p l i t u d e s to h a v e t h e k i n e m a t i c a l l y m o s t s i n g u l a r b e h a v i o u r a l l o w e d at t = 0, in c o n t r a d i c t i o n w i t h f a c t o r i s a t i o n . T h i s s t r o n g l y s u g g e s t s [23] t h e p r e s e n c e of R e g g e c u t s o r of o t h e r R e g g e p o l e c o n t r i b u t i o n s i n t e r f e r i n g with a n d s t r o n g l y c o r r e l a t e d to t h o s e of t h e p i o n [27]. M o r e a c c u r a t e d a t a on p p r o d u c t i o n a n d o t h e r r e a c t i o n s w i l l s h e d m o r e l i g h t on t h e p r o b l e m s of ~ and w e x c h a n g e . In p a r t i c u l a r if t h e r e p o r t e d c h a n g e of e n e r g y d e p e n d e n c e at h i g h e r m o m e n t a [52] i s c o n f i r m e d t h i s w i l l b e d i f f i c u l t to i n t e r p r e t . We s h o u l d l i k e to t h a n k D r . R. J. N. P h i l l i p s f o r s u g g e s t i n g t h i s i n v e s t i g a t i o n a n d h i s c o n t i n u e d i n t e r e s t in it. A l s o we a r e g r a t e f u l to t h e s o m a n y a u t h o r s who k i n d l y p r o v i d e d u s w i t h d a t a p r i o r to p u b l i c a t i o n . One of u s ( C . D . F . ) w o u l d l i k e to t h a n k D r . G. C. F o x f o r a n u m b e r of i l l u m i n a t i n g d i s cussions.

APPENDIX T h e T - m a t r i x i s d e f i n e d in t e r m s of t h e S - m a t r i x by

680

G.V. DASS and C. D. FROGGATT Sfi = 6 fi + i(27r)454(Pf-Pi)Tfi •

(A. 1)

Following Wang [3] our helicity a m p l i t u d e s a r e defined by taking m a t r i x e l e m e n t s between i n v a r i a n t l y n o r m a l i s e d helicity s t a t e s . ( P a P b k a k b i P a P b k a k b ) = (2~)6 poppa o 63( P a - P a )6 ( P b - P b ) S k a k a 5kty~b .(A.2) T h e p h a s e conventions of J a c o b and Wick [71] a r e u s e d and the c e n t r e - o f m a s s f r a m e of
fx2x 4,xlx 3

=(½(i- cos Ot)-½ IX13-X24[(½(1 + c o s 0t)) -½1x13+x241 f kt2 k 4 , k l k 3 ,

(A.3)

which have nice analyticity p r o p e r t i e s in s [1-7]. H e r e we have k13 = h I - k 3 etc. and 4

cosOt:[2st+t2-t ~ m2+(m~-m2)(m2-m~.)]/(T13~24), i=l

(A.4)

with

r2j(t) = [t- (mi+ mj)2][ t- (m i-

mj) 2] ,

(A.5)

and m i is the m a s s of p a r t i c l e i. Of c o u r s e the k i n e m a t i c c o n s t r a i n t s in the m o m e n t u m t r a n s f e r v a r i a b l e t for the d i r e c t channel p r o c e s s 1 +2 -~ 3 +4 a r e m o r e n a t u r a l l y given by the s - c h a n n e l helicity a m p l i t u d e s f ~ 3 ~ 4 , ~ 1 P 2 " H o w e v e r it is now well u n d e r s t o o d [1-7] how to e x p r e s s the analyticity of f-~3g4,tZlt~ 2 as a function of t in t e r m s of the t - c h a n n e l a m p l i t u d e s fkt2k4,Xlk3 • In our continuation of f/2)~4,~1~ 3 into the s - c h a n n e l p h y s i c a l region,

"rij--. + ]'rij I tion

and ~ ¢ ~ +~Yd) w h e r e ¢ is the p h y s i c a l r e g i o n boundary func-

R E G G E - P O L E MODEL

681

z

_t(m2_m2)(m2_m2)_(m 122m4_m22m3)(m122 + m 4 2 _ r n 2 - m 2 ) . The differential c r o s s section for the s-channel p r o c e s s is ~-d(r: ( ~

[f$~2~t4,)tlX3t 12)/(4~sP~2(2S1

+ I)(2S2 + I)) ,

(A.6)

where P12 is the initial centre-of-mass m o m e n t u m and S i is the spin of particle i. In the usual Jackson frame [11, 73] the density matrix for particle 3 is

Pro, m' = (_l)m_ m,

t

~ klX2h 4

t*

f~2)t4,~tlmf ~2~4,~lrn,//(~ If~2x4,XlX t 3 12) . (A.7)

We have taken the y - a x i s along the direction P l × P3 where P l and P3 are the initial- and f i n a i - s t a t e c e n t r e - o f - m a s s momenta for p a r t i c l e s 1 and 3 respectively. If the s a m e y - a x i s is used for particle 4 there is no (-1) m-re' in the corresponding f o r m u l a for its density m a t r i x elements [6, 23, 73]. Following Fox [6], we write the asymptotic f a c t o r i s e d contributions of a boson Regge pole a(t) of signature T to the t-channel helicity amplitude in the f o r m fxt,asymp 2~4,~tlX3

_ l+'re -i~a(t)

~i~(X13 7t24)TX37tl(t ) sin[~ra(t)] e - 7X4X2(t)

(s_]a(t) . (A.8) \Sol

The kinematic s t r u c t u r e of the vertex functions y(t) at thresholds (and pseudothresholds for unequal m a s s vertices) a r e most readily found [6, 7] by introducing the t r a n s v e r s i t y amplitudes of Kotanski [74]. This gives

d $3

~tlk3 /~3X3

(½n) dS11 (½n) oc (T13)±~I+~3 X1 7~t3~tI ,

(A.9)

where the negative sign is only taken for the lighter particle at t =(m l-m3) 2 and on passing out of the physical region we have taken 4-~ -~i }~bI~. So in the continuation of the vertex functions we have vr~ --~ i v~. Parity conservation gives the useful relation [71]

Y-k3-kl(t) = ~P(-1) S1+$3 ~1~3 7~3~tl (t) ,

(A.10)

where P is the parity of the physical states on the t r a j e c t o r y and 77i is the intrinsic parity of particle i. In the forward direction, which asymptotically is t = 0, the T r u e m a n and Wick [75] c r o s s i n g angles take a simple f o r m and the constraints due to angular momentum conservation can be easily discussed in t e r m s of the leadi n g - o r d e r s - c h a n n e l amplitudes. For a simple or evasive Regge-pole exchange this gives [5, 6, 24]

682

G.V.DASS and C. D. FROGGATT (i) in the u n e q u a l - m a s s c a s e 7X3Xl(t) cc (f2-t-t)l x131 ,

)t13 = Xl-X 3 .

(A.11)

(ii) In the equal m a s s case, we introduce, near t = 0, the functions gtx3~ 1 = X3)t1

d 83 (½v) dSllx I (½7r)(-1) 83+)t3 exp{½i~ (kl+)t3)}Y)t3)t 1 , (A.12) tx3X3

and we have g U 3 ~ l cc (4U7) 1/~13]

"-13 = /~1-/x3 "

(A.13)

However, for an M = 1 c o n s p i r a c y [6, 24, 33], for which the Regge pole a(t) has an opposite p a r i t y p a r t n e r a'(t) of the s a m e signature and satisfying q'(0) = a(0), we have: (i) In the u n e q u a l - m a s s c a s e (say m 3 > ml) at t = 0 y,X3Xl = _iYx3)t 1 cc (~r2~)~13-1

)t13 >/ 1

and y~3~ 1 ~q~2-/,

(A.14)

~13 = 0 .

(ii) In the e q u a l - m a s s c a s e s i m i l a r equations hold for g g 3 P l : '

'

~ (4-~) ~ 1 3 - I

g~3~l =zg~3~l

~13 ~ 1

and g~3~l

cc ( 2 / ,

~13 = 0 .

(A.15)

C o r r e s p o n d i n g equations hold for yX4k2(g~4t~2) with )t13(f~13) r e p l a c e d by X24(tz24). As is well known [68, 69, 76], t h e r e is in general a singularity in the t r a n s f o r m a t i o n (s, t) ~ (cos 0 t, t) at t = 0 and daughter t r a j e c t o r i e s can be introduced to r e m o v e the poles introduced into non-asymptotic t e r m s . Howe v e r in phenomenological fitting the lower ranking daughter t r a j e c t o r i e s a r e probably not important and we make only the s i m p l e s t c o r r e c t i o n s to our a s y m p t o t i c f o r m , eq. (A.8). This involves assuming [3] that the Regge asymptotic behaviour holds in the f o r w a r d d i r e c t i o n for the modified helici-t ty amplitudes f)t2~4,Xl)t3 • Then

/t k2)t4,klX3 .t,asymp [½(1_cos0t)]½1X13-X241[_½(1 0t)]½l ~t13+~t241 (- ~-13~24) xm = }X2)t4,XlX3 + cos st (A(16) where

Am

= Max

(IX131

, 1 x241)"

REGGE-POLE

It f o l l o w s t h a t in p r a c t i c e h e l i c i t y a m p l i t u d e s [1]

MODEL

we are parameterising

683

the parity conserving

t f+ = 3 ~t ~t2X4,;tl;t 3 ~t2;t4,~l~t 3 ± ~f_~t2_~t4,~tl~t3

(A.17)

w h e r e ~ = (-1) X 1 3 + ~ m ~/4 ~2 (-1) $4+$2" T o l e a d i n g o r d e r f ± a r e d o m i n a t e d by R e g g e p o l e s of n a t u r a l a n d u n n a t u r a l p a r i t y r e s p e c t i v e l y .

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REGGE-POLE MODEL

685

N O T E A D D E D IN P R O O F R e c e n t a n a l y s i s of the ~ + p - ~ p + p d a t a at 8 G e V / c [77] h a s p r o d u c e d f u r t h e r e v i d e n c e f o r a dip n e a r t = -0.5 GeV 2 in d ~ / d t f o r this m o d e , as given by an w ' n o n s e n s e z e r o ' f o r both ~±p ~ ~±p. New ~ - p ~ p ° n d a t a [78] at 11.2 G e V / c does not show the c h a n g e in e n e r g y d e p e n d e n c e r e p o r t e d in ref. [52] and is c o n s i s t e n t with o u r fit. H o w e v e r a new difficulty is p o s e d by the s m a l l It I b e h a v i o u r of the ~ - p - - ~ p ° n d a t a at 8 G e V / c [79]. As t --~ O, d a / d t d o e s not s e e m to t u r n o v e r f o r - t < m 2 as p r e d i c t e d by o u r m o d e l and o b s e r v e d f o r ~+p--~p+p at the s a m e e n e r g y [77]; u n f o r t u n a t e l y s c a n n i n g l o s s e s due to s h o r t p r o t o n t r a c k s a r e quite s t r o n g f o r low Itl. We r e c a l l h e r e that the quantity (see fig. 5) X(t) =- 2 d ~ / d t (hOp __~pop) m u s t not be n e g a t i v e . F u r t h e r d a t a on d a / d t and Prom' in this s m a l l It l r e g i o n f o r all c h a r g e m o d e s would be v e r y valuable. A l s o it is s u g g e s t e d in ref. [79] that an i s o t r o p i c ~ b a c k g r o u n d could c o n t r i b u t e s i g n i f i c a n t l y to pSoo (see r e m a r k (v) in sect. 6). T h i s would r e q u i r e a s t r o n g e r r e n o r m a l i s a t i o n of the e x p e r i m e n t a l Prom' f o r pO p r o d u c t i o n and s o m e w h a t r e d u c e the d i f f e r e n c e in ×2 b e t w e e n o u r two s o l u t i o n s .

R E F E R E N C E S TO N O T E A D D E D IN P R O O F [77] Aachen-Berlin-CERN Collaboration, M. Aderholz et al., Nucl. Phys. B8 (1968) 45. [78] B. D, Hyams et al., Nucl. Phys. B7 (1968) 1. [79] W. Selove, F. Forman and H. Yuta, Phys. Rev. Letters 21 (1968) 952.