N N annihilation into two pseudoscalar mesons at high energy

Volume

PHYSICS

25B, number 6

RN

ANNIHILATION

Physics

Department,

LETTERS

2 October

INTO TWO PSEUDOSCALAR AT HIGH ENERGY* V. BARGER and D. CLINE University of Wisconsin. Madison, Received

1967

MESONS

Wisconsin.

USA

7 Gugust 1967

The amplitudes for the NN -MM annihilation channel are related to the amplitudes for backward MN elastic scattering using a Reggeized baryon exchange model. The angular distributions for RN -+ iis at high energy are qualitatively predicted from data on nN elastic scattering at large angles. Several cross-section relations are obtained for the EN-+ KK reactions that should be subject to experiment test.

Baryon exchange presumably accounts for the rather sharp backward peaks which have been observed in angular distributions for mesonbaryon scattering processes. Within the framework of the Regge-pole model, the amplitudes for baryon exchange in the pseudoscalar meson-nucleon scattering process Ml+Nf-N2+M2

(1)

fl(G

s) =

E,iM =--

du

1

r(J+g)

cos 7rCY

7#%s) (2)

for e(R2M2) = 180° [for illustration, see fig. l]**. This leads us to expect forward and backward peripheral peaks in RN - MM at high energy. A simultaneous study of reactions (1) and (2) at high energy and large angles should therefore constitute a critical test of the baryon exchange mechanism. In this letter we point out some general qualitative features to be expected in process (2) based on present understanding of reaction (1). The formalism presented below is restricted to pseudoscalar mesons of equal mass. The u channel amplitude for a single Reggeized baryon exchange in meson-nucleon scattering is given by ***

I

(s/so)(y -T ,

(3)

where the trajectory cy and the residue y are functions of the variable (-rP6). The corresponding u channel amplitude for the annihilation process is given by

for scattering angles 0 (Ml M2) N 180’ also describe the annihilation process R2+N1-a1+M2

+,e-in[@ -iI

= Tfl(JX

s).

(4)

The complete u-channel amplitudes are obtained by summing over the contributing trajectories in eqs. (3) and (4). For convenience, we define

u

Ju “/+(%S)= 3 E+M [

Y_(%S) =A

Es [ u

fl(~,S)

u

Ju

fl(-6,s)

+E_M

fl(J%S)

1,

(5)

-E&,,(-Gr,,)-

u-

1

Analo ous definitions apply in relating F*(u,s) to rl(& $ U, s). In terms of Y* and V* the differential cross sections for reactions (1) and (2), respectively, are given by

sq;g= lr+12[s+u+2D2] + +

IT-j2[u(-s+4iG)+D4]+ 4MRe

(6) (Y: Y_) [u+D~],

* Work supported in part by the University

of Wisconsin Research Commitee with funds granted by the Wisconsin Alumni Research Foundation, and in part by the U.S. Atomic Energy Commission under contract AT (ll-l)-881 #COO-881-114. ** Henceforth we use an adaptation of the notation of nuclear physics: Nl(M1, N2)M2 for reaction (1) with B(MlM2) N 180’ [i.e. fJ(Ml N2) = O”j and Nl(N2, -Ml)M2 for reaction (2) with 0 (R2 M2) = 180’ [i.e. (3 (N2 Ml) N 09.

.

*** The notation in eq. (3) is described in particular, 7 denotes the signature of and P the parity. Cont,ributions of the leading trajectory s a-5 and contributions from daughter are not included in the results of eqs.

ref. 1. In the trajectory of order trajectories (3) and (4). 415

Volume 25B, number 6

]JHYSICS L E T T E R S

Mr/ /

i

\

2 October 1967

\ \ ~,

M=/

I

/

/

/

X

/

M,

!

!

S

S

M,+N,-" M=+ N= 8 (M,M=)= 180° N, (M,, N=) M=

N= + N , ~ M, +M= O(NtM=) =180 ° N, (Nt,M,) M=

Fig. 1. Schematic illustration of line reversal for the baryon exchange mechanism of the processes NI(M1, N1)M 2 and NI(N2, M1)M2.

2sp2

dff= I t + ] [ s - 4 M 2 ] +

lr

12[-us-(D2-u)2]+

- 4MRe (~*r_)[u+D2],

(7)

w h e r e D 2 = M 2 - u 2 a n d q~, p ~ a r e the a p p r o p r i a t e c . m . m o m e n t a . W h e n e v e r the e x c h a n g e of a s i n g l e t r a j e c t o r y d o m i n a t e s , the r e l a t i o n I r + ] = = Ir:~l o b t a i n s a n d the a n a l y s i s s i m p l i f i e s . As s --* ~o at a fixed v a l u e of u, t h e s e c r o s s s e c t i o n s a p p r o a c h the a s y m p t o t i c v a l u e s d~

4

12

2]

and

2ddu =4~[]~+ 12 - u ir _12 1. F o r the e x c h a n g e of s e v e r a l t r a j e c t o r i e s , with the s a m e v a l u e of the s i g n a t u r e T,

all

dff

d~ du

~ dq 2 du

for

s ~ ¢¢. *

, ( d a V i s - ( M + u ) 2] [ s _ - ( M - ~ ) 2] s [s + u + 2 0 2] '

d--u = 5 d u /

and h e n c e - - =

F o r c o n c r e t e a p p l i c a t i o n of the above r e s u l t s , we f i r s t c o n s i d e r the r e l a t i o n s h i p b e t w e e n the r e a c t i o n s p(Tr±, p)~± and p(~, 7r=~)~:~. F o r high e n e r g y s c a t t e r i n g , it now s e e m s r e a s o n a b l y well e s t a b l i s h e d that the a m p l i t u d e for p(n +, p)~+ is d o m i n a t e d by the e x c h a n g e of the Not t r a j e c t o r y [4] a n d the a m p l i t u d e for p0r-, p)Tr- is d o m i n a t e d by the e x c h a n g e of the A5 t r a j e c t o r y [1]. O u r p r e s e n t d i s c u s s i o n is l i m i t e d to a s i m p l i f i e d m o del for t h e s e Regge e x c h a n g e s with I P t = = I r - I = 0 in eqs. (5) and (6). (Such an a p p r o x i m a t i o n is e q u i v a l e n t to the a s s u m p t i o n that the t r a j e c t o r y can be r e p r e s e n t e d a s ~ = a + bu a n d the Regge r e s i d u e y is c o n s t a n t [1]). T h i s m o d e l r e p r e s e n t s the i m p o r t a n t q u a l i t a t i v e f e a t u r e s of eqs. (6) and (7) in the r e g i o n u ~ 0. With this s i m p l i f i c a t i o n , we i m m e d i a t e l y find f r o m eqs. (6) a n d (7), the following r e l a t i o n s h i p

(8)

At i n t e r m e d i a t e e n e r g y the c o r r e c t i o n s to f o r m u l a (8) a s given by eqs. (6) a n d (7) m a y modify this r a t i o . • In the asymptotic limit, s --~ oo, such results follow from more general considerations as has been s t r e s sed by L. Van Hove [2]. Specific applications in the asymptotic region have been discussed in ref. 3. 416

(9)

for c r o s s s e c t i o n s e v a l u a t e d at the s a m e v a l u e s of s a n d u. Eq. (9) has b e e n u s e d to p r e d i c t the p(~, ~=)~+ 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 a s shown in fig. 2a a n d 2b. The d a t a p o i n t s in the f i g u r e r e p r e s e n t e x p e r i m e n t a l m e a s u r e m e n t s f r o m p(~:~, p)~:~ at s m a l l u [5] s u i t a b l y c o r r e c t e d u s i n g eq. (9). The s o l i d c u r v e s a r e r e p r e s e n t a t i v e Regge pole fits [6] to a l l e x i s t i n g high e n e r g y data for b a c k w a r d ~N s c a t t e r i n g at s m a l l u . The m a j o r d i f f e r e n c e in the c h a r a c t e r i s t i c f e a t u r e s of d ~ / d u and d~/du

PHYSICS

Volume 25B, number 6

1

2 October 1967

LETTERS

PREDICTED CROSS SECTION FOR pQJ,n*)r-

PREDICTED CROSS SECTION FOR p(p, n-)a* Na DOMINANT REGGE EXCHANGE

\

A8

REGGE

EXCHANGE . FRISKEN et al

. FRISKEN et

d KINEMATIC

1 KINEMATIC

.o

II

-3

LIMIT

-.5 ”

FOR

LIMIT

FOR

P( .-,

P)r-

P&r-)w+

-.7

-.9

-1.1

-1.3

.I

.o

-.I

-.3

-.5 ”

(GeV/cl’

-.7

-.9

-1.1

-I .3

(G&UC)*

Fig. 2. Differential cross section predictions for the reactions (a) p(p,a-)lr’ and (b) p(p,a+)n-. The solid curves are qualitative predictions for p(p,?t)n based on Regge pole fits (ref. 6) to all available data on p(n,p)n near u = 0. The dashed curves represent tentative extrapolations of the fits away from the region where u N 0. The data points shown have been taken from ref. 6 and suitably modified by eq. (9) to obtain the pp - IM cross sections. The kinematic limits for 180’ scattering are shown by cross-hatched lines for both the pp - 1~71and uppi processes. arises from kinematic constraints. For 180' scattering the kinematic limit for p(n, p)n is UR = = +(M2 -~~)~/s whereas for p(P,ii)n the limit is

Using these qualitative features and eq. (9), we predict that the cross section ratio

gf [P(P, ++I

(10) of this difference in kinematic limits, the annihilation channel sees a narrower range of u values than the nN scattering channel. The distinctive features of high energy backward nN elastic cross sections are: Because

(i)$u[p(n+,p)?r’]

(ii) g

= 5g[p(a-,p)77‘]

[p(n+, p).rr+] has an extremely

slope near u = uB followed fixed u = -0.15. (iii) %[~(a-,

sharp

by a pronounced

p)n-] has a remarkably

slope near u = UB.

at U=UB,

dip at

small

/&

[P(P, n+)r-l ,

at u = GR will be approximately 1 to 1.5 at pre sently available energies but will increase with s and ultimately reach the limiting value of

at u = UR. The suppression of this annihilation cross section ratio at present energies relative to the TN backward scattering ratio is directly associated with the zero in the Regge amplitude for the Nor exchange at (Y = -i and the kinematic limit of eq. (10). Consequently, verification of these predictions will serve as a valuable test of Reggeized baryon exchange. On the basis of the calculated curves in fig. 1 we crudely estimate the integrated cross section for Ot(p[p, n*]rs) for / u 1 < 1 (GeV/c)% is = 1.5 pb at J.s = 2.9 GeV and = 0.33pb at Js = 4 GeV. 417

Volume

25B, number 6

PHYSICS

These estimates are probably reliable lower limits on the annihilation cross section. A recent measurement [7] at 4s = 2.9 GeV gave Ft = 8 * f 3tib (on the basis of 3 events) which is of the expected order of magnitude. A number of interesting qualitative predictions also follow in this model for the pp - m processes. It is known that the K-p + pK cross section is at least a factor of 20 smaller than Kfp -+ - pK+ at 3.5 GeV/c backward angles [8]. This experimental result is interpreted as nonexistence of Y = +2 states or as a dynamical suppression of Y = +2 baryon exchanges. We correspondingly expect da[p(P, K+)K-] = 0 and d5[p(p, K”)K-] = 0. The cross section equalities dD[p(p,Kl)Kl] = = de[p(F, Kx)Kl] = da(p(P, Kl)K2) = da[p(P, Kz)Kz] are likewise predicted. The prediction for dB[p(p, Kf)K-] is well verified at J.s = 2.3 GeV [9] where the pp - K+K- cross section shows a strong peripheral peak for Q(pI(+) N 180°, but no events for e(pK-) N 180°. In backward nN scattering at intermediate energies the interference of direct channel resonances with a Reggeized baryon exchange contribution has been useful for quantum number determinations of the resonances [l]. Similarly we anticipate that the interference of direct channel boson resonances with the baryon exchange amplitude will eventually serve as a valuable tool for the study of high mass meson resonances. As a first step it will be necessary to experimentally verify the validity of the baryon exchange amplitude at high energies through such tests as those suggested here. We therefore urge early experi-

*****

418

LETTERS

2 October

1967

mental effort in this direction *. The discussion presented here for the two pseudoscalar channel can also be extended to other dimeson production channels such as ps, on, K* K etc. which are related to backward scattering in the inelastic reactions TN -+pN, nN - wN and KN - K*N. We thank Professor C. Goebel for reading the manuscript of this paper.

References 1. V. Barger and D. Cline, Phys. Rev. 155 (1967)

1792. 2. L.Van Hove, Phys. Letters 5 (1963) 252.

3. A.A.Logunov et al., Phys. Letters 7 (1963) 69; A. Bialas and 0. Czyzewski, Phys. Letters 13 (1964) 337. 4. C. B. Chiu and J. D.Stack, Phys. Rev. 153 (1967) 1575. 5. W. R. Frisken et al., Phys. Rev. Letters 15 (1965) 313. 6. V. Barger and D. Cline, to be published. 7. W. M.Katz, T. Ferbel, B. Forman and W. Metzger, Bull. Am. Phys. Sot. 12 (1967) 470. 8. J. Banaigs et al., Phys. Letters 24B (1967) 31’7; D. Cline, C. Moore and D. Reeder, University of Wisconsin Report (to be published). 9. G. R. Lynch et al., Phys. Rev. 131 (1963) 1287.

* We have been informed that a Carnegie Institute of Technology - Brookhaven National Laboratory collaboration will attempt a measurement of the p[P,%]a differential cross section at 8 GeV/c. This experiment was brought to our attention by Dr. R. Edelstein.