-nucleon interactions-nucleon interactions

IL NUOVO CIMENT0 VOL. XXXII , N. 5 1 o Giugno 1964=

J(-Nucleon Interactions (*).

RA~IESH CHARD

Syracuse University, Department o] Physics - Syracuse, N .Y .

(ricevuto 1'11 Novembre 1963)

S u m m a r y . - - Total scattering and absorption cross-sections for K-nucleon collisions in I ~ 1, p~-channel are given as functions of the two sets of energy-dependent KA" scattering parameters solutions, called solution A' and solution B'. These scattering parameters are obtained by linear interpolations between Watson's amplitudes around 400 MeV/c and the amplitude at the position of the pole in the KJ~ c' scattering amplitude corresponding to the p~-wave 1385 MeV Y~ resonance with 50MeV width. The zero-range expansion for p-wave K-nucleon phase shift and the scattering parameters of Watson's solution B are found to be in violation of the requirements of causality and of positive definiteness of transition probabilities.

1 . - I n t r o d u c t i o n .

Recently, WATSO~ (1) has analysed the K - - p r o t o n da ta in t e rms of two sets

of zero-range K-nucleon scat ter ing pa ramete r s solutions, culled solution A and

solution B, which provide good fits to the K - - p da ta for K - - l a b o r a t o r y momen-

t u m k L in the range of ( 350 - -450 )~eV/c . These two sets of ampl i tudes are

very similar except in the p -wave and therefore, in principle, i t is possible

to distinguish between these two solutions by ex t rac t ing the values of p-wave

cross-sections f rom the measured values. However , due to large s-wave effects

for low-energy K - - p collisions, this requires considerably more exper imenta l

da ta than available at the present t ime. Since the ]385 MeV Y*-resonance

(') Supported by National Science Foundation grant. (1) M. WATSON: Lawrence Radiation Laboratory Report, UCRL-10175 (1962).

~ - N U C L E O N I X~TEI~ A~.;TION S 1235

with 50 3[eV width is now ra ther well established to be a p~-resonance and the fact t ha t Y* is significantly coupled to the ]~.V and ¢Y cannels, the existence

of this resonance should exhibi t itself as poles in the KA; scgttering and

react ion ampl i tudes in I = 1 , p~-state. Basing on this ~act, we (2) gave a theo-

retical means of dist inguishing between the two sets of Watson ' s ampli tudes .

_Now, we plan to give detai led results on the i = 1, pCw~ve K-nucleon a) sc~t-

terin~ paramete r , and b) sea.ttering and absorpt ion cross-sections.

2. - K-nucleon interactions in I = 1, p~-state.

The K-nucleon scat ter ing ampl i tude in I ~ 1. 1 ~ 1 , and J = ~ , s tate can be wri t ten as

( 1 ) I T(k) ~_ - - sin 5(k) exp [i 6(k)]/k,

[ = - t . ' - A ( k ) [ 1 - - i ~ : ~ A ( k ) ] - ~ ,

where the ]£~V scat ter ing p a r a m e t e r A(k) for ]~ c.m. m o m e n t u m k is defined

in te rms of the phase shif t 3(k) by the relation

(2) k a ctg ~(k) ~ 1/A(k) .

Due to the presence of s trong regctions leading to the product ion of hyperons ,

the phase shift 5(k) and the scat ter ing p a r a m e t e r A(k) are complex numbers .

The complex scat ter ing p a r a m e t e r can be w~itten as

A(k) = a(~') i b(~'),

where a and b are real numbers .

:Now, in order to obta in the K-nucleon bound-s ta te description of the $

Yl-resonance, we replace k by iK in expression (1), where

K _~ [2#(E t - E)] "~ , Et = m K -- r ex ,

# is the reduced mass and E the to ta l energy of the K ~ ~ system. Therefore,

in the unphysieal region of negat ive kinetic energy of the K ~ ) system, the

(2) RA)I~SH CH.iYD: Nuoco Uimento, 30, 1445 (1963).

1236 R A~[ESI-I C H A N D

expression for the scat ter ing ampl i tude can be wr i t t en as

(3) T(K) = K ~ A ( K ) [ 1 - - K 3 A ( K ) ] -* = K ~ A ( K ) [ ( 1 - - K 3 a(K)) - - i K ~ b(K)]-* .

Comparison of the Brei t -Wigner resonance fo rm ( E - - E , + i I ' / 2 ) -~ to the fo rm

of the scat ter ing ampl i tude T ( K ) leads us to the conclusion t ha t the resonance

energy E+ of Y* corresponds to

(4a) K~ = [ 2 / t ( E , - E~)] ~ = 1/a(K~),

and the full width at half m a x i m u m is given by

(4b) F = - - 2 K ~ b ( K , ) / # .

Since K~, kt and F are posi t ive quant i t ies , expression (4b) clearly shows tha t

the value of the ampl i tude b at the posit ion of the * ¥ , - resonance is necessarily

negat ive. However , the posi t ive definiteness of the t ransi t ion ra te and hence of

the absorpt ion cross-section a~b , (expression (Tb)) requires the ampl i tude b to be posit ive at energies above the I~A ° threshold. These two requi rements are

inconsistent wi th the requi rement of energy-independence of the scat ter ing

p a r a m e t e r A(k) for p -wave K-nucleon interact ions. Therefore, the use of the expansion (2) for the K~V phase shift ~(k) in the l imit of zero-range, namely

k a ctg ~(k) == A-~(k) ~ A-l (0)

is not permissible. But , since k 3 ctg 3(k) is an analy t ic funct ion of k 2, one

can write

(5) k ~ ctg ~(k) _-= A-I(~:) ~ A-x(0) ~- ½R~ -u

as the simplest possible energy-dependent fo rm for the scat ter ing p a r a m e t e r A(k). In this paper , we use the exper imenta l ly known values of E , and /7 in

order to determine the value of tile ampl i tude A ( K ~ ) ( = a ( K ~ ) + i b ( K ~ ) ) , ne-

cessary for the existence of the pole in the K S scat ter ing ampl i tude at the

posi t ion of the ¥*-resonance. Using E~ = 1385 MeV (corresponds to K ~ =

= 0 . 9 0 3 3 fermi-*) and F = 5 0 MeV in (4), we get

(6) A(Kr) = ( l . 3 5 7 - 0.346 i) fermi 3 .

The detai led analysis on K - - p r o t o n interact ions, carried out by WATSOn',

for k L in the range of (350- -450)MeV/c , has led to two sets of equally ac- ceptable zero-range K,W scat ter ing pa ramete r s which are l isted in Table I

F - ~ - N I ! ( ' I A : ; ( ) N 1 N T E R A ( ' T I O N S 1237

TABLE I. -- 7'WO sets O/ I = 1, pi-wace zero-range K-&" scattering parameters, determined by WATSON, which best ]it the K--p data in the region, o] (350--450) MeV/c K - laboratory

momentum k z.

Solution a (fermi a) b (fermi a)

i A -- 0.016±0.012 0.0004 _~0.0004 B 0.066 ± 0.011 0.000 03 ~ 0.00010

in I - - l , p~-s ta te . Using' t he e f fec t ive - range e x p a n s i o n (5) a n d c o m b i n i n g

W a t s o n ' s a m p l i t u d e s to r ep re sen t K - n u c l e o n processes ~t a) k w = 350 MeV/e,

b) k w - 400 MeV/c a n d e) k w = 150 M e V / c , t he v a l u e of t h e s c a t t e r i n g p a r a -

m e t e r a t t he p o s i t i o n of the Y~-resonan(.e (],'= 0 . 9 0 3 3 i fermi-~)~ ~'iven b y (6),

/ i r e s us two new sets of e n e r g y - d e p e n d e n t K , V s c a t t e r i n g p a r a m e t e r s so lu t ions ,

ca l l ed so lu t ion A ' a n d s<)lution B', a.nd <'orrespond to W a t s o n ' s so lu t ions A

a n d B, res t )ee t ive ly . The va lues of t he ran(ge p a r a m e t e r R, so o b t a i n e d a re

~'iven in Tab le I1 for t he t h r e e va lues of k w. The va lues of t he ene r l zy -dependen t

TABLE II . - Values o/ the range parameter R /or F~,\' collisions i~ 1 - -1 , pivstate r,~. the mome~tum k w o] Watson's ar~plitude.~ ]or the two .~ets o] energy.dependent K~N ~ seatterin 9

ptlram, eter.~ solutiow~, solution A ' and solution B'.

Solut, ion A ' Solution B'

R e R (fermi -1) ] Im R (fermi -I) Re L' (fermi -~) " Im R (fermi -~)

k w -- 350 MeV/c

, + 26.2 , la . ~ + 3 . o ---0.18±0.03 ~ ) 1 . 9 as3. 9 1 + 1.5 - - - " 7 - 4 5 . 6 . . . . . 2 . 1

k w - 400 MeV/c

-+ 1.a 1") .)+2.o I - -0 .15±0 .03 - 1 . 0 _ 3 9 . 2 i . . . . 1 . 8 i

k w -- 450 MeV/c

• + 1.1 1 0 . 6 + ~ . ~ 1 . 3 _ 3 ~ . o _ .

- , ~ . ~ + 2 2 . 5 - - .~ ) , J .Z 1 5 7 . 9

I

+ 1 9 . 5 - . 4 6 . 1 la6.9 - -0 .13~0 .03

s c a t t e r i n g p a r a m e t e r s A ( k ) are ~ iven m Ta,ble 1I I as func t ions of K - l a b o r a t o r y

m o m e n t u m k z a n d the m o m e n t u m k w for W a t s o n ' s a m p l i t u d e s , for the in te r -

p o l a t e d so lu t ions A ' a n d B ' c o r r e s p o n d i n g to W a t s o n ' s so lu t ions A a n d B,

r e s p e c t i v e l y . Th is t a b l e shows t h a t t i le w d u e s of b(k) are n e g a t i v e in the case

of s o l u t i o n B ' for p h y s i c a l va lues of k L i r r e s p e e t i v e of t he va lue s of k W. How-

ever , t he p o s i t i v e n a t u r e of t h e a b s o r p t i o n ( ' ross-sect ion for ] ~ , N ' - + ~ Y pro-

cesses a b o v e the K . V t h r e s h o h l r equ i r e s b(k) to be p o s i t i v e (express ion (Tb))

1238 n~&.~IESH CItAN D

TABLE I I I . - Values o] the scattering parameter A(k) = a(k)+ib(k) /or KA" collisions in I = 1 , pvstate as a ]unction o] K laboratory momentum k L ]or several values o] the mome~tum k~v ]or Watson's amplitudes are given ]or the two sets o] energy-dependent Kz~"

scattering parameters solutions, solution A ' and solution B'.

i Solution A ' I kL ! i

(MeV/c)' I

Solution B'

! a(k) (fermi a) b(k) (fermi 3) ] a(k) (fermi 3) b(k) (fermi 3)

0 100 200 300 400

0 100 200 300 400

0 100 200 300 400

- - 0 . 0 4 1 4 - 0 . 0 3 1

--0.0364-0.027 - -0 .026±0.020 --0.0194-0.014 --0.0144-0.010

- - 0.0484-0.037 - - 0.042 4-0.032 - - 0.031 4- 0.024 --0.0224-0.017 - - 0.0164-0.012

- - 0.055 4- 0.043 - - 0.048 4- 0.038 --0.0364-0.027 --0.0254-0.019 --0.0184-0.014

kw = 350 MeV/c

0,0009 4-0.0015 0.154±0.024 0.00084-0.0012 0.1384-0.022 0,0006 4- 0.0008 0.105 =~0.017 0.00054-0.0005 0.077 ±0.013 0.00034-0.0003 0.0574-0.010

kw = 400 MeV/c

0.00104-0.0018 0.1774-0.028 0.00094-0.0015 0.1584-0.025 0.00074-0.0010 0.121±0.020 0.00054-0.0006 0.0894-0.015 0.00044-0.0004 0.0664-0.011

kw = 450 MeV/c

0.0010--0.0022 0.200~0.030 0,00104-0.0019 0,179±0.028 0.00084-0.0012 0.1384-0.022 0.0006=[_0.0007 0.1014-0.017 0.00054-0.0005 0.076±0.013

- - 0.0025 4- 0.0010 - - 0.00184-0.0008 - - 0.0007 -4- 0.0004 - - 0.0001 4-0.0002

0.0001 4-0.0001

- - 0.0035 j : 0.0015 - - 0.0026 ± 0.0011 - -0 .0012±0.0006 - - 0.0003 ~= 0.0003

0.0003 :J: 0.0001

- - 0.0049 ~ 0.0018 - -0 .0037±0.0015 - - 0.0018 ~= 0.0008 - - 0.0006 ~ 0.0004 - - 0.0001 -V 0.0002

for a l l rea.1 va`lues of k z. Th is r e q u i r e m e n t is b e i n g viola, t e d b y t h e ene rgy -

d e p e n d e n t sca` t ter ing p a r a m e t e r s of our i n t e r p o l a t e d s o l u t i o n B ' a n d n o t b y

t h o s e of s o l u t i o n A ' . The re fo re , t he p a r a m e t e r s of s o l u t i o n A ' und n o t t h o s e of

so lu t i on B ' a`re t he a d e q u a t e ones to d e s c r i b e K - n u c l e o n i n t e r a c t i o n s in I----1,

p~-s ta te . Th i s i m p l i e s t h a t t he i = l , p:~-wave K ~ " p a r a m e t e r s of Wa`tson 's

so lu t i on A (and n o t t hose of so lu t i on B) are t he u p p r o p r i u t e ones for t h e

d i scuss ion of K - - p sca t te r in? : a n d a b s o r p t i o n processes u r o u n d 400 MeV/c K - -

la `borutory m o m e n t u m .

The expres s ions for t he t o t a l scat tering" c ross - sec t ion a,o~t~(K,V-~ K ~ ) a n d

t h e a b s o r p t i o n c ross - sec t ion a~b~(K,V --~ ,~Y) in I = 1, p~.-st~te ea`n be w r i t t e n a`s

(7(,) %~tt = snL T(k)L 2 ,

(7b) a~b ~ = 8 , ~ k b ( k ) I t ( k ) 1 2 ,

KINUCLE(}N INT]~RACTIONS 1239

where

(7c) t(~') = [ l --i~'3A(~')] 1

Sca t t e r ing a n d abso rp t i on cross-sections, ca lcu la ted us ing the energy-de-

p e n d e n t K ~ scattering" p a r a m e t e r s of solut ions A ' a n d B ' are p resen ted in

! t a )

0 . 4 -

0.2 F 0 F,

t b) 0.6~

E 0'4L~ ~:~o.2 ~ ~o i

c)

0.6 ~ I

0.~-

0.2=

0 200 400

0 ~

4! r) i

:! 0 200 400

kL(MeV/c)

Fig. 1 . - Maximum, mean (corresponding to mean values of theF~ ~" scattering parameters) and minimum values of the total scattering cross-section a~ t t for I ~ - + K ~ V processes in I = 1 , p~-state vs. I~-laboratory momentum k~, calculated using the energy-dependent ~ scattering parameters of solutions A' and B', as a func- t ion of the momentum k~ for Watson's amplitudes, a) Solution A', k w-350MeV/c; b) solution A', kw=400 MeW/e; c) solu- t ion A', kw--450 MeV/c; d) solution B ' , kw =350MeW/e; e) solution B',

kw=400 MeW/c; ]) solution B', kw = 450 MeV/c.

0 , 3 7 -

0.2 °>/1---

0.3~

b~ o

_o.1[ ~ , ~ ~ i , i i

0.4 i c) 0 1 I -o.

0 . 2 , / ~ -0.2

0 F ~ _ ~ -o.4

-o

--0.1 --0.2

-0.1

-0.2

-0.3

i i i t i I i

200 400 k L (MeV/c)

Fig. 2. - Maximum, mean (corresponding to mean values of tile I~h" scattering parameters) and minimum values of the total absorption cross-section nab ~ for K ~ ' - + n Y processes in I - - l , p}-state vs. I~-laboratory momentum k z, calculated using the energy-dependent K..N ~ scattering parameters of solutions A' and B', as a func- tion of the momentum t'~ for Watson's amplitudes, a) Solution A', k w = 350 MeV/c ; b) solution A', kw--400 MeV/c; c) solution A' , kw--450MeV/c ; d) solution B', kw--350MeV/c; e) solution B',

kw--400 MeW/c; ]) solution B', kw = 450 MeV/c.

F ig . 1 a n d 2. F i r s t of all we no te that. due to nega t ive values of the ampl i -

t u d e b(k) for ~olut ion B ' , these eah, u la t ions are no t s t r ic t ly mean ing fu l for

1240 RA_~ESH CHAND

paramete rs of solution B' . However , these results clearly demons t ra te the fact tha t due to the smallness of these I = 1 , p t -wave KJ~' cross-sections to-

gether with the known fact of large uncertaint ies in the exper imenta l measure-

ments , i t would be ex t remely difficult exper imenta l ly to resolve the ambigu i ty of Watson ' s solutions. Even though the scattering" cross-sections ob ta ined for

solution B ' are about 6 t imes larger than for solution A', yet they lie within the uncertaint ies of exper imenta l measurements . The absorpt ion cross-sections

calculated for solution B' are negat ive in the i ( - m o m e n t u m rnnge of

(0--300) MeV/e. In case of solution A' , the m a x i m u m allowed values of a~b .

together wi th the mean values (corresponding to mean values of Watson ' s am-

plitudes) are posit ive in the entire energy range. This clearly shows tha t the para-

meters of solution A' and not those of B ' are the appropr ia te ones to describe

I = 1 , pCwave K-nucleon interact ions up to 400 5[eV/c. However , an im-

por t an t check on our calculations requires t ha t detai led exper iments be per-

formed on K - - p r o t o n interact ions for k L in the range of (0+400)5{eV/c , in

order to determine the values of the p -wave cross-sections. The smNluess of

these I = ] , p -wave ](-nucleon cross-sections is due to the known fact thu t at low energies (kz<400 MeV/c), a) K-nucleon interact ion is dominan t ly an s-interaction and b) the s t rength of K&" interact ion in the 1 = 1 channel is

significantly weaker than in the I = 0 channel.

3 . - C o n c l u s i o n s .

F r o m these investigations, we conclude tha t the use of the effective-range expansion for p-wave ](~'~ phase shift in the l imi t of zero-range violates the

requi rements of causali ty and of posi t ive definiteness of t ransi t ion probabi l i t ies for physical values of ]~-momentum. This means t ha t in principle, one can not use the p-wave K-nucleon phase-shif t expansion in the l imi t of zero-range. We also note t ha t in principle one should include higher-order energy-dependent

t e rms in the effective-range expansion (5). However , due to the lack of exper-

imenta l data , this is not possible at the present t ime. I t is found tha t the

energy-dependent pa ramete r s of our in terpola ted solution B' are inconsis tent

with the requi rement of positiveness of the absorpt ion cross-section. This im-

plies tha t the ]~2V sca, t te r ing ampl i tudes of Watson ' s solution A and not those

of solution B u r e the appropr ia te ones for describing K - - p r o t o n processes

around 400 ~eV/c . T*

The above considerations requir ing the existence of the 1385 MeV Y 1-resonance with 50 5IeV width as poles in I ~ 1 , p~-chunnel for K~V K.~N and K2V-+ r~¥ ampl i tudes follow f rom the existence of a significant coupling between

YI* and K~N,-- ~ ~Y channels. Therefore, all quan t i t a t ive unders tandings about

the Yl-decay etc., renmin unaffected by our invest igat ions.

K - N U C L E O N 1 NT~RACTIONS 1241

W e a re p l ea sed to t h a n k the stuff of I B M 7094 c o m p u t e r a t Gene ra l E l e c t r i c

C o m p a n y in S y r a c u s e for t h e i r co -ope ra t i on dm~in~ " n u m e r i c a l c o m p u t u t i o n s .

] ~ I A S S I : N T ~ ) (')

Si dhnno le sezioni d 'ur to total i di scattering ed assorbimento per le collisioni K-nucleone nel canale I = l , p+ in funzione di due ~ 'uppi di soluzioni dei parametr i dell() scattering K:~' dipendenti dall 'energia, chiamate sohtzioni A ' e soluzione B'. Questi parametr i di scattering si ottengono per interpolazione lineare fra le ampiezze di Watson attorno ai 400 MeV/c e l 'ampiezza nel punto del polo della ampiezza di scattering K~" corrispondente alla risonanza Y* di 1385 MeV in onda p+, di ampiezza di 50 MeV. Si t rova che lo sviluppo di rango zero per lo spostamento di fase K-nucleone in onda p e i parametr i di scattering della soluzione B di Watson violano le esigenze di eausalith e di definizione posit iva delle pr~)babilit'~ di transizione.

(°) Traduz ione a cura della Redazione.

-nucleon interactions-nucleon interactions

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