I L N U O V O C I M E N T O VOL. X X X V , N. 2 16 G e n n a i o 1965

Final-State Interactions in K--+37: Decays.

R. I'RASAD (*)

U ~ i v e r s i t y Col lege L o m l o ~ - L o m l m ~

( r i c e v u t o i l 3 N o v e m b r e 1964)

T h a t t h e d e n s i t y of p o i n t s in a D a l i t z - p l o t fo r t h e t h r e e K - d e c a y m o d e s :

K + - + T:+ + ~ + +T: - ,

K + __+ ~o + ~o + ~+ ,

K o _+~++~-+~o,

c a n b e r e p r e s e n t e d a s a f u n c t i o n w i t h l i n e a r d e p e n d e n c e on t i l e y - v a r i a b l e s e e m s

n o w to be w e l l e s t a b l i s h e d (~). I f t h e v a r i a b l e s a r e c h o s e n as

(l)

w h e r e

a n d

a n d

l 2

y = 2 t 3 - 1 ,

I i ~ T i / r f ,

T i - - ( m - - #)~ - - s i , ~ _ ( m - - #)2 - - 4,u 2

2 m 2 m

.% = (K - - ki) 2 = - - (k, -V kk) z ,

(*) Present address: I s t i t u t Henri Poincar~, Rue Pierre Curie, Paris, 5 e. (1) For z-decay see L. T. S.~ITH, D. J. PROWSE and D. H. STORK: Phys . Left., 2, 204 (1962);

for v'-deeay see G. E. K A y : S , A. KEI~NAN, ~[:~. T. PU, ~V. ~I. ~)OWE~ and R. DOWD : Phys . Rev. Left., i3, 99 (1964); for v~-decay see D. LUERS, I. S. MYF~RA, ~V. J. WILLIS ~1(:]. S. S. YAMA~OTO; Phys . Rev., 133, B 1276 (1964); R. K. ADAIR and L. P. LEIPUNER: P h y s . L e t t . , | 2 , 67 (1964).

FINAL-STATE INTERACTIONS IN K - - - 3 ~ DECAYS 6S3

t h e n t h e d e n s i t y of p o i n t s is g i v e n b y

d 2 w (2) = i m,(z, y)!2 = ~ + / ~ y

dx dy

and the slopes of the energy spectra (fil~) are (~)

~r -- ~ 0 .23~ : 0 . 0 4 ,

~.~, = - - 0 . 6 5 ~ 0 . 0 5 ,

~T~-- - - 0.68 ~ 0.15 .

T h e a b s o l u t e d e c a y r a t e is g iven b y (2)

(3) o ) = 2 z ~ f f f dkl dk2 dk3 j j j i2~) ~ (2g) 3 (2z) 3 ' (2z)464( K - - / " 1 - - ]~2--- k3)"

1 1 . 1 f f ~(~'1, ~'2) d:rdy4~t[4ztM(a~,Y)[ 2 2m 2'w V 2w z. 2ws (2z~) ~ j j O(x, y)

w h e r e x a n d y a re c o n s t r a i n e d b y t h e c o n d i t i o n s

x 2 + y?~< 1 , x > 0 .

T h e e x p e r i m e n t a l v a l u e s for t h e r a t e s a r e (3)

w~ = (4.66 ~: 0.25)" lO ~ s -1 ,

w~, = (1.39 ~ 0 .17) '106 s -1 ,

w~ = (2.90 :~ 0.72)" l06 s -1 .

T h e p r e s e n t c a l c u l a t i o n is b a s e d on t h e a s s u m p t i o n t h a t p i o n - p i o n f i n a l - s t a t e i n t e r a c t i o n s p r o d u c e t h e o b s e r v e d s p e c t r a a n d t h a t t h e d e c a y r a t e s a re d e t e r m i n e d b y t h e d e n s i t y of s t a t e s in b o t h (I)- a n d (J)-Slfin spaces .

FADDEEV (4) h a s p r o v e d t h e e x i s t e n c e of a se t of t h r e e c o u p l e d i n t e g r a l e q u a t i o n s w h i c h r e p r e s e n t t h e t h r e e - p a r t i c l e i n t e r a c t i o n w a v e - f u n c t i o n :

(4)

I T(1) ]

T = ~o ÷ T<I)÷ T(2)+ T ¢ ~ ,

o3 o, [o 23E1 1 ~b(12> __ ~bo] T12(E) T12(E) [ t//,a>j

(~) See M. GELL-MANN and A. It. ROSENFELD: Ann. ~e~'. Nttel. Scf., 7, 455 (1957). (~) These figures are taken ~rom R. I[. Dt~LITZ: Brookhaven Conlerertce on Weak Interactions

[~Iso D. STERN, T. O. ]~]NFORD, V. O. LIND, J. ilk. ANDERSON, F. S. CRAWFORD jr. and R. L. GOLDEN: Phys. Rev. LetL, 12, 459 (1964)].

(a) L. D. •ADDEEV: Soviet Physics JETP, 12, 1014 (1962).

684 R. PRASAD

where

and

wi th

(D (iD O (+) r5

" ( i D : 1 ( H o + ~' i j - - l¢~o i v ) - l l ' / ~

¢o := exp [i{k~ "r~ + k 2 "r.,. + k3. r.~}]

and where the following nota t ion has been used

3 1 3 Ho Y ~ i = ~ 1

i=l i=l 2m-~i A i , G o : (H o - E o - i s ) - 1 ,

and the T-matr ices T~j sat isfy

Ti~(E) | ~ j - - V~.iGo(E) T,-j(E) .

I t is a simple m a t t e r to show tha t the second inhomogeneous t e rm of eq. (4) represents the to ta l effect of mul t ip le sca t te r ing and resca t te r ing processes. If, therefore, the fur ther assumpt ion is made tha t these processes have only a small effect on the decay processes, the second t e rm m a y be neglected.

The above equat ions are best handled by means of a ~, k inemat ic ro ta t ion ,~ of the co-ordinate axes:

[ kij : (m¢ki-- mikj)/('m,i + m~) , (5) ~ Pk = {mAk~ + k ~ ) - - ( m ~ + ?~,~)k~.}/(m: + 'm.~+ 're.k), i ' j ; k = J, 2; 3;

using these co-ordinates, the kinet ic energy opera tor in the c.m. frame, T¢, is invar ian t

where

2/Qi 2/tL i, ,1 ; t: = 1, 2 3

1 t 1 1 1 1 - - + + J~ij ~l'i "ll~,j /Ik ~ll~: ~Tb i -~ ~H,j

and each wave funct ion T¢i) can be expressed in its (, o w n , co-ordinate system

(6) T ( i )= T(i)(kjk, p , ) , i ; j , k = 1; 2, 3 .

However , this me thod also entai ls the corresponding ro ta t ion in isotopic-spin space, bu t has the advan tage tha t the three pions are all t r ea ted on an equal footing.

The ampl i tudes for t ransi t ion to a s ta te wi th given s y m m e t r y (i.e. [3] or [2, 1]) and given m o m e n t u m dis t r ibut ion (i.e. P l , P ~ , P 3 with Pl>~P2>~P3) are g iven by

FINAL-STATE INTERACTIONS IN K-~3= DECAYS 6 8 5

DALITZ (5) for ~ and ~' decays, and the same analysis can be carried through for the neutral decays. (The symmetry s tate ([1, 1, 1]) which can occur in this case is assumed to make a negligible contribution.)

I t is necessary to use all three co-ordinate systems since, making the usual assumption tha t the plane wave is modified into spherical outgoing waves, the r~-~ scattering ampli tudes pick-up a numerical coefficient corresponding to the co-ordinate system and symmetry s tate being considered.

Assuming the predominance of s-wave scattering the various ampli tudes can be expressed in terms of isospin states by symmetrizing out the T- -1 states from the two-pion wave functions. A complication tha t occurs, however, is tha t the charge-exchange scattering ampli tude corresponding to

x + + x- ~ ~o + ~o

is of the same order of magnitude as the noncharge-exehange amplitudes. This process great ly complicates the problem since it couples each of the two possible decay channels ( + + - - ) and (00 ÷ ) or ( ÷ - - 0 ) and (000) . However it is pos- sible to separate out the terms corresponding to charge-exchange scattering and put them in the correct channel, account being taken of the phase-space change.

The usual scattering-length-effective-range parameter izat ion of the ampli tudes is made :

(7)

exp [i@] sin ~r

1 , 1 2

qi~ctg(~r ~ 72r rqo ,

T = 0 , 2 ; i , j - : 1,2 ,3 .

All terms to be evaluated are of the form

(S) l all~ + hi13 + cf.~3 j'~

and eventually lead to a decay energy spectrum of the form

d~w 2 ~ 2 2 ( 9 ) - - : A - - Bql 2 - C(qx 3 -~ q 2 3 ) •

dx dy

(It is impor tant to distinguish between the numbering of the pions as observed and the numbering of the pious in various co-ordinate systems.)

The slope of the energy spectrum in terms of tile Dalitz variables is

(10)

where

8 - - ~ 2 ( B - C)

~A - e 2 { B + C(5 - - 6to)} '

(5) R. H. DALITZ: Prec. Phys. Soc., A 69, 527 (1956).

686 ~. PnASAD

a~ld

m - - 3/~ 1 to (tl ~ t2÷ t3) .

3T 3

A i/~-fit can therefore be made to the three slopes and, independent ly , the three rates by vary ing s imul taneously a o, a 2, r 0, r 2 th rough all physical ly reasonable values.

A number of different solutions for the parameters have been proposed. As examples we can give the Chew-Mandels tam analysis (~) of ~-r: sca t ter ing which gave a t t r ac t i ve forces in bo th the ~/'--0 and T = 2 s tates wi th 2a 0~ 5a 2. The analysis by Hami l ton ' s group (7) of the cont r ibut ion of ~-r~ sca t te r ing to ~-~) sca t ter ing agrees wi th this resul t and gives a 0 ~ l . 3 ± 0 . 4 . The t au -decay analyses give confl ict ing results however . KnuRl and TREIMAN (s) and SAWYER and WALI (9) obta ined repulsive T = 0 and T = 2 states wi th a 0 ~ - - 1 . 0 and a ~ - - 0 . 4 . These analyses where confirmed in their conclusions by the work of B.~RTON and KACSER (10) who also included P - w a v e effects. BROWN and StNt)ER (H) propose a T = 0 resonance a t 400 MeV to ta l energy, and MITRA (12) has considered be th T - - 0 and T = 2 resonant phases. TnO~'LAS and HOLLABAY (13) o n the o ther hand suggested a T = 2 resonant s tate .

The present analysis is in agreement w i th the result of Brown and Singer. The values of the paramete rs which minimize Z z for the slopes are a 0 = + 1 . 4 ~ : 0 . 1 , a2= ÷ 0 . 7 ± 0 . 1 , r 0 = - - 2.2 ± 0.1, r 2 - + 2 . 0 ± 0 . 1 ; and the values which minimise Z'~ for the rates are a o = + l . l ± 0 . 1 , a 2 _ + 0 . 8 ~ 0 . i , r u = - - 2 . 0 ± 0 . 1 , r 2 = + 2 . 0 ~ 0 . t . In the case of bo th the slopes and the rates the 7. ̀'~ dropped to ~0 .1 . The ranges of the paramete rs inves t iga ted were as follows:

a) no resonances or bound states in e i ther channel (i .e. all parameters posi t ive) ;

b) a resonance in e i ther 7' 0 or T - - 2 channels (i .e. r r negat ive and <% posit ive) ;

c) a bound s ta te in e i ther T - - 0 or T 2 channels (i .e. ~ negat ive and r r posi t ive) .

The possibi l i ty of all parameters being nega t ive is of course ident ical to the case a) and need no t be inves t iga ted separately.

In all ranges of the parameters inves t iga ted Z -~ was found to be a monotonica l ly increasing or decreasing function.

The present m e t h o d does not insert a resonant form into the ampli tudes , ne i ther does i t inser t the re la t ive sign or order of magn i tude into the slopes. I t does, however , insert the dens i ty of s tates in isotopic-spin space (assumption of a T l three-pion state) and the re la t ive phase-space factors for the different channels (which arc caused by 7:±_,~0 mass differences).

(6) (]. CIIEW a n d S. MANDEI.,STAM: Phys. ltev., | 1 9 , 467 (196o). (7) j . IIAMILTON, P. 3[ENOTTI, G. OADES a n d L. VXCK: Phys. ]~'('t'., 128, ISSl (1962). (a) N. K.~URI a n d S, B. TREII~IAN: Phys. Rev., 1i9 , 1115 (1960). (g) R. F. SAWYER a n d K . C. XV~LI: Phys. Rev., 119, 1429 (1960).

(I0) G. BARTON a n d C. KAOSEI~: Phys. Rev. Lett., 8, 226, 353 (E) (1962); a n d C l a r e n d o n L a b o r a - t o r y R e p o r t 95/62 (unpubl i shed) .

(H) L. M. BROWN a n d P. SINOER: Phys. Rev. Left., 8, 460 (1962); Phys. Rev., 133, 13 812 (1964). (1~) A. N. MITRA: Nucl. Phys., 6, 404 (1958); 18, 502 (1960). (ts) B. S. THOI~IAS a n d XV. G. HOLLADAY: Phys. ReV., ][15, 1329 (1959).

FINAL-STATE INTERACTIONS IN K-~ 3~ DECAYS 687

The agreement of the slopes requires only that the relative orders of magnitude of the four parameters take on their physical values. The agreement of the rates requires, in addition, that their absolute values are the physical ones. In the cases of both the rates and the spectra the numerical values obtained were found to be sensitive functions of the parameter variation.

The conclusion is, therefore, that the linear matrix element squared is compat- ible with a resonant =-T: phase in the T = 0 channel with roughly the position and width assigned to it by BRowx and SI~ER.

The ~--> 3u Dalitz plot when the statistics are more reliable will provide a crucial test of the method.

I should like to thank Professor J. HAMILTON for originally suggesting the problem for investigation and for continued guidance while the work was carried out.

I should also like to thank members of the Physics Department at University College, London, for st imulating discussions and, in particular, Dr. G. C. OADES for information on the pion-pion interactions and Mr. M. A. HFNNELL for discussions of the Faddeev equations. I am grateful also to the London Univelsi ty Computer Unit for the use of their Atlas Computer and to the Department of Scientific and Industrial Research for the award of a Research Studentship.