L E T T E R E A L L A R E 1 ) A Z I O N E (La responsabilitd scientifica degli scvitli inseritl in qaesla rubrica ~ completamente lasciala

dalla Direzione del periodico ai singoli aulori)

P h o t o p r o d u c t i o n o f P i o n s o n A - H y p e r o n s .

A L L A D I R A M A K R I S H N A N , T. K. R A D H A , ~R. T I IUNGA and A. t ' . ~}ALACHANDRAN

The Ins t i tu te o/ Mathemat ica l Scie~ces - Madras

(r icevuto i | 27 Aprile 1962)

In this le t ter we wish to discuss tile l)rocess - , , ÷ A ~ A + ~ assuming as in the previous cont r ibut ion (1) tha t the low energy A~ scat ter ing proceeds mainly th rough S waves. This process will be of impor tance in the s tudy of reactions like

"l" + A ' -> A ± K + ~ .

In the no ta t ion of CGLN (2) a s t ra ight forward calculat ion leads to the following Born terms for the individual mul t ipole ampl i t ude , in the case of odd EA rela t ive pa r i ty

, _ JY - - ~m A 'm:,~ F~ /~(! JO+ : - -

E~+ : O,

(1) ]1, ,,g q IV- 'n~\

where, as in 1he rest of the calculat ion, we have re ta ined only those mul t ipole ampl i tudes leading to final s tates with 1 = 0 or 1. In eq. (1), we have denoted the to ta l centre-of-mass energy by W. We notice tha t in a stat ic approximat ion , the leading B o r n term is E0~÷, the others being ~:~f the order of l /m A compared to it. Therefore. keeping only E0., we may write

cc

1

(t) ALLADI ]~AMAKRI~]?INAN, T. ]~. RADHA, R . THUN(;A a n d A, P. BALACHANDRAN: NHOVO Cimento, 25, 723 (1962).

(2) (~, F. (-!tIE,V. ,~[. L. GOLDBERGER, F. ~E. 1,o~v ar id ~1-. NAMBU: ])h?/8. Re(L, 106, 1315 (1957).

940 ALLAD1 RA,MAKI4ISHNAN, T. K. I~ADIIA, It . T t I U N G A a n d A. P. B A L A C H A N D R A N

where o) W - - m A and we have set the pion mass equal to unity. Im E0+(~o) can as usual be calculated using the uni tar i ty condition in which only the AT: inter- mediate state is retained. We find, in a static approximation

(3) lm Eo+(W ) = exp [-- i6o] sin 6oEo+ (w) ,

where we have explicitly assumed that the .,,~.~z interaction proceeds dominantly through S-wave and denoted the S-wave scattering phase shift by 3 o. Equation (2) now reduces to the familiar mapping problem discussed by OMN~S (a) whose solution reads

(4) E0+(~o ) =

2z ~o .... 6m

where

(5)

co

~ e x p [ o ( ~ ) ] p : ~ S i n 6 ( ° ; ) e x p [ ° ~ ( ( ° l ) ] d o J l e x p [ i 6 ( o y ) ] " cos ~(o~) + 7~ (~ ' - - 6m) (~ ' - - ~)

1

co

~Op; 6((o')

1

where 6m = m ~ : - m A and EoB+ has also been replaced by its static form. For even ZA relative parity it is found that the contributions from the crossed

channels are of the same order of magnitude as that from the direct channel. Bearing in mind that under crossing A-> A; C - ~ - C and D-+ D, the Born terms for the multipoles in the static approximation turn out ~o be

(6)

= 2 - ~ ,

~B El+ 0 ,

M~+ - - /~g kq 6~ ~o + (~m '

M~_ - - l~g 1 2~ ~ + 6 m

Since we are retaining only the S-wave At. scattering amplitude in the uni tar i ty condition, it is easy to see that the different multipole amplitudes do not get coupled by it so that for each of them we have a relation of the form (3). Final ly the solutions of the dispersion integrals for these amplitudes read

co

(7) E°+(°~) = [2~ oJ cos 6((o)+ ~ exp [o(w)] P - z ( ~ o ' - - o~)

1

d~o' 1 exp[i 6(w)],

(s) I~. OM.'Z]~S: N u o v o Cimento , 8, 316 (1957).

P I I O T O P R O D U C T 1 O N OF P I O N S ()N A - t I Y P E R O N S 94I

(8 ) E I + ( ( o ) = O ,

(9)

fin ~ + ,~m

co

cos . . . . . . ~ .... d ~ ' exp[ i ~/(~)]

1

(10) 311 ( ~ ) = co

. . . . cos <$(o~) [~(m)] . . . . . . . . . d ~ ' exp [ 7 6 ( @ ] . [2~ m-r-6m v (o'(o.l'+ 6 m ) ( w ' - - m )

1

where o(o~) is aga in g iven b y (5). Since we h a v e a l r eady p resen ted effect ive range fo rmulae for S -wave A~ s c a t t e r i n g

in a p rev ious c, o n t r i b u t i o n (1), we can easily ca lcu la te t h e ac tua l va lues of t h e a p p r o p r i a t e cross-sect ions f rom these formulae . W e hope to s t u d y t he inf luence of t he process y + A ~ A + ~ on processes l ike y + A ' - + A + K + ~ on t he basis of our effect ive r ange t h e o r y in a l a t e r c o n t r i b u t i o n .