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IL NUOVO CIMENTO VOL. XXX, N. 6 16 Dicembre 1963
K-Nucleon Interactions and Y~-Resonance.
R A ~ E S H C H A N D
Department o] Physics, Syracuse University - Syracuse, N. Y.
(ricevuto il 14 Giugno 1963)
S u m m a r y . - - A theoretical method is given for distinguishing between Watson's two sets of zero effective-range ]~A~ scattering parameter solutions, called solution A and solution B, which fit the K--p data quite well in the K--laboratory momentum range of (350--450) MeV/c. These two sets of solutions, which are quite similar in both the s- and d-waves but differ considerably in p-wave, are combined with the parameters, obtained by requiring Yl*-resonance at 1385 MeV with width 50 MeV as an I ~ 1, pFwave, bound state of the ]~-nucleon system, so as to give two new sets of energy-dependent K~N ) scattering parameters in I ~ 1, p~-state. I t is found that the zero effective-range expansion for p-wave KA" phase shift and the scattering parameters of Watson's solution B are incon- sistent with the requirements of causality and of the positive definiteness of transition probabilities.
1 . - I n t r o d u c t i o n .
Recen t ly , WATSOn" (1) has m a d e a fit to the K - - p r o t o n d a t a for K - - l a b o r -
g t o r y m o m e n t u m kL in the range of (350 - -450)} [eV/c . I n this analysis, Z -A
rela t ive p a r i t y is considered even and the Y o -resonance at 1520 ~ e V is t aken
to have Bre i t -Wigne r shape ~ i t h a w id th of 50 5IeV. This inves t iga t ion gives
two sets of zero effect ive-range K_-nucleon sca t t e r ing p a r a m e t e r solutions,
called solut ion A and solut ion B, which p rov ide good fits to the K - - p d a t a
in the ent i re energy range studied. These two sets of pa ramete rs , which are
de te rmined wi th re la t ive ly small errors, are v e r y similar in bo th the s- and
d-wuves, bu t differ cons iderably in the p-wave . Therefore, in principle, it is
possible to dis t inguish exper imenta l ly be tween these two sets of ampl i tudes
(1) M. B. WATSON: Report, UCRL-10175 (1962).
91 - II Nuovo Cimento.
1 4 4 6 RAM~SH CHAND
by ext rac t ing the values of the p-wave cross-sections from the measured cross- sections. However, due to large s-wave effects in K- -p collisions at low energies (kL~<400 MeV/c), the determinat ion of the ra ther small p-wave cross-sections requires considerably more experimental da ta than available at the present time. Therefore, i t is quite desirable to provide a theoretical means of di- stinguishing between Watson 's two sets of solutions.
The experimental da ta on Y*-resonance, obtained by SttAFER et al. (~), strongly suggest tha t the I = 1 , 7:-A resonance at 1385 MeV with a width of
50 MeV is in a p c s t a t e . Due to the product ion of Y~-resonance in K-nucleon collisions and its subsequent decay into ~:Y channels, it is clear tha t this resonance is coupled to the K ~ , roY channels. Therefore, the existence of Y* should exhibit itself as poles in the amplitudes for KJT --~ K3~ and K ~ ---> 7:Y processes. We propose to use this fact to resolve the ambigui ty between Watson 's solution A and solution B.
2. - K - n u c l e o n in terac t ions .
I t is well known tha t the expressions for scattering (KJV --> K~f', i.e. both elastic and inelastic scattering) and absorption (K3~-->~Y) cross-sections for KJT collisions in a pure, isotopic-spin state I , orbital angular -momentum state l, and total angular-momentum state J are given by
(1.1)
(1.2)
(~.c~tt(I1J, k) = ~ J + 11 -- S(I1J, k)[ ~ ,
~u~(Iij , ~) = -~ j + [1 - 1 s ( z zJ , k) I ~] ,
where k is the center-of-mass momentum of the K ~ system and corresponds to K- labora tory momentum kL. In terms of the K ~ phase shift (~(IIJ, k)~ the amplitudes S can be wri t ten as
(2) S(I IJ , k) = exp [2i6(I1J, k)].
I t is convenient to express the phase shift 6 in terms of the energy-dependent
scattering parameter A ( I I J , k), defined by
(3) k 2t+1 ctg 5(I1J, k) = 1/A(I1J , k ) .
(2) j . B. SHA•v,R, J. J. MURRAY and D. O. Huw~: Phys. Rev. Lett., 10, 179 (1963),
K-NUCLEON 1NTERACTIONS AND Y~-RESONANCE 1447
Due to the presence of s trong absorpt ive reactions leading to the produc t ion of hyperons, the phase shift 5 and the scat ter ing pa rame te r A are complex
numbers . The complex scat ter ing p a r a m e t e r can be wri t ten as
A(I1J , k) -- a( I IJ , k) @ ib(I1J, k ) ,
where a and b are real numbers . In te rms of the scat ter ing paramete r , the
expressions for ]~A " cross-sections in a pure I = 1 , l = 1, and J = 23 s tate are
given by
(4.1)
(4.2)
where
(4.3)
2:n cr~b~(k) = ~-2-[1 - - ] (1 4- i k a A ( k ) ) T ( k ) ] '~j
T(k) = [ 1 - ik3A(k)] -~ .
In te rms of T(lc), the expression for the I = 1 , p~-wave scat ter ing ampl i tude
](k) can be wr i t ten as
(5) ](k) - - sin 6(k) exp [ib(k)]/k = k2A(k) T(k) .
In these expressions, the values of the var iables I , l, and J are not specified explicit ly so as to keep our nota t ion simple, e.g. A(1 , 1, ~, k) has been denoted
by A(k) . Here, we assume charge-independence for nuclear interact ions and therefore, neglect Coulomb interact ions and the mass differences betwen particles belonging to the same mult iplet .
In order to obtain a t~-nucleon bound-s ta te descript ion of the 1385 5IeV Y*- resonance in a p~-state, we replace k by i K in expression (5); where
' ~ E t ~-- m K 4 - ~'to~, , K ~ ~/2tt(E t - E ) ,
ff is the reduced mass trod E the totM energy of the KA ~ system. Therefore, in
the unphysical region of negat ive kinetic energy of the KA ~ system, the expression
for the scat ter ing ampl i tude is of the fo rm
(6) [ ](K) = - - K 2 A ( K ) [ 1 - - K 3 A ( K ) ] -1 ,
= - - K : A ( K ) [ ( 1 - - K3a(K)) - - iK3b(K)] -1 .
Expressing the Brei t -Wigner resonance fo rm ( E - - Er 4- iF~2) -1 in the form of the
sea.ttering ampl i tude ](K) leads us to the conclusion tha t the resonance energy
1448 RAM'~.SH CHAND
E, corresponds to
(7.1) K3, = [2/~(Et - - Er)] ~ = "l/a(K,),
and the width F is given by
(7.2) F = -- 2KS, b(K,)//~ .
In this discussion, we assume the experimental ly de termined (8) values
of Er and F as known quantit ies and would therefore determine the value of the scat ter ing ampli tude A(Kr), necessary for the existence of the pole in the K2~ ampli tude at the position of Yl-resonance. Using E r = 1 3 8 5 MeV (which corresponds to Kr -~ 0.9033 fermi -1) and I ' = 50 MeV in expressions (7), we get
(8) A(K~) = (1.357 - - 0.346 i) fermi s ,
as the value of the I----1, p+-wave K2~ scat ter ing parameter at the position of the Y*-resonance. Expression (8) clearly shows tha t the KJV bound-s ta te de- scription of the Y*-resonance (causality condition) requires the ampli tude b to be negative at the position of this resonance. However , the positive definiteness of t ransi t ion probabilities and hence of absorption cross-section nab. requires the ampli tude b to be positive definite at energies above the KJN" threshold. These two requirements are incompatible with the requirement of the energy- independence of the scattering ampli tude A(k) for p-wave interactions. There- fore, the use of the expression (3) for the I~JV phase shift ~ in the limit
k s ctg 5(k) --= A-l(k) ~ A-I(O),
is not allowed. But , since k 3 ctg ~(k) is an analyt ic funct ion of k s, one can write
(9) k 3 ctg 5(k) ---- A-l(k) ~_ A-I(O) + �89 2 ,
as the simplest possible energy-dependent form for the scat ter ing parameter A(k).
The detailed analysis which WATSON (1) has carried out for all available da ta on K - - p r o t o n interact ions for kr in the range of (350--450)MeV/c has led to two sets of equally acceptable zero effective-range K3~(' scattering am- plitudes. These amplitudes in an I = 1 , p+-state are listed in Table I. Using
the effective range expansion (9) and combining Watson 's da ta at k~. ---- 400 MeV/c
(8) B. P. GREGORY: Proc. 1962 Ann. Intern. Con]. High Energy Phys. at CERN (Geneva, 1962), p. 779.
K-NUCLEON INTERACTIONS AND Y~-RESONANCE 1449
TABLE I. -- Two sets o] I = 1 , p]-wave, zero eJJective-range K-A~ scattering parameters, determined by Watson, which best ]it the K--p data in the region o] (350--450)MeV/c
K--laboratory momentum.
Solution i a (fermi a) b (fermi 3)
A -- 0.016~=0.012 0.0004 • B 0.066 ~= 0.011 0.00003 ~ 0.00010
wi th the va lue of the sca t t e r ing a m p l i t u d e a t k-- - -0 .9033i fermi-l~ g iven b y
express ion (8) ( o b t a i n e d b y the K S ' b o u n d s ta te descr ip t ion of Y*-resonance) ,
gives us two new sets of e n e r g y - d e p e n d e n t K A ~ sca t t e r ing a m p l i t u d e solut ions.
These sets of solut ions in the I = 1, p v s t a t e of the K A p sys t em are called so lu t ion
A ' and so lu t ion B ' , a n d cor respond to W a t s o n ' s so lu t ion A a nd so lu t ion B,
respect ive ly . The va lue of the p a r a m e t e r R, so ob t a i ne d , is g iven b y
(10.1) R = - - (53.16 + 1.463 i) fermi -1,
for so lu t ion A ' and
(10.2) R =- (12.17 - - 0.154i) fermi -1,
for so lu t ion B ' . These two sets of e n e r g y - d e p e n d e n t K A ~ sca t t e r ing p a r a m e t e r s
are l is ted in Tab le I I , for several va lues of kL. Tab le I I shows t h a t the
i m a g i n a r y p a r t of the s ca t t e r ing a m p l i t u d e A(k) , n a m e l y b(k), is nega t i ve for
TABLE II. Two sets o] energy-dependent K~" scattering parameters A(k) (a) in an I = 1, pi-state as a ]unction o] K-laboratory momentum kL.
kL(SIeV/c) - -
!
0 50
100 150 200 250 300 350 400
A (k)=a(k)+ib(k) (fermi a)
Solution A ~
- - 0.048+ 0.0010 i - - 0.046+ 0.0009 i - - 0.042+0.0009 i - - 0.036+ 0.0008 i - - 0 . 0 3 1 + 0 . 0 0 0 7 i
- - 0.026+0.0006 i - - 0.022 + 0.0005 i - - 0.019+0.0005 i -- 0.016+0.0004 i
(a) Here k refers to the c.m. m o m e n t u m of sponding to K- l abo ra to ry m o m e n t u m k L.
Solution B'
0.177--0.0035 0.172--0.0033 0.158--0.0026 i 0.140--0.0019 i 0.12I--0.0012 i 0.104--0.0007 i 0.089--0.0003 i 0.076--0.0001 i 0.066+0.00003 i
the K ~ system corre-
J
i i
1450 RAMESH CHAND
solution B ' for certain real values of kL. But the posit ive definiteness of the absorpt ion cross-sections requires b(k) to be posit ive for all energies which lie above the K2V threshold. This requi rement is being violated by the pa ramete r s
of solution B' , and therefore, the in terpola ted solution B' and hence Watson ' s solution B are inadequate to represent K-nucleon interact ions. Pa rame te r s of
solution A ' are quite consistent with the requirements of causal i ty and posit ive
definiteness of t ransi t ion probabili t ies. Also, we find tha t our conclusions are
unaffected ei ther by the inclusion of uncer ta int ies in Watson ' s ampl i tudes or
by the choice of K m o m e n t u m used in the effective-range expansion (9) to
de te rmine R and A(0), as long as it lies in the range of the va l id i ty of Watson ' s ampli tudes , namely ( 3 5 0 - 450) MeV/c. Thus, we conclude tha t the pa ramete r s of
Watson ' s solution A and not of solution B are the appropr ia te ones to describe K - - p r o t o n scat ter ing and absorpt ion processes around 400 ~ e V / e K - - l a b o r a t o r y
m o m e n t u m . We also note t ha t in principle, one should include higher-order energy-dependent correction te rms in the effective-range expansion (9) for
K ~ " phase shift. However , due to insufficiency of exper imenta l data , this is not possible a t the present t ime.
Scat ter ing and absorpt ion cross-sections have been calculated, using the scat ter ing pa ramete r s of solution A ' and solution B ' and are given in Table I I I .
TABLE I[I . -- Cross-sections ]or processes Kj~'->Kd~? (a,r and K3~-->r:Y ((~b~) in I : 1, p~-state vs. K-laboratory momentum kL, calculated ]or energy-dependent interpolated
solutions A ' and B'.
Solution A' Solution B' k~(MeV/c) i $
a~r (mb) nab8 (mb) asoat~ (mb) ~a~ (mb)
0 50
100 150 200 250 300 350 400
0.0 0.0004 0.005 0.02 0.04 0.07 0.10 0.13 0.16
0.0 0.04 0.07 0.10 0.11 0.12 0.13 0.13 0.13
0.0 0.006 0.07 0.29 0.67 1.15 1.67 2.17 2.62
--0.0 --0.14 --0.22 --0.23 --0.19 --0.14 --0.08 - - 0.03 + 0.01
We first note t ha t due to the requi rement of posi t ive definiteness of (~ab~ for
energies above the K ~ threshold, even though these calculations are not
s t r ic t ly meaningful for solution B' , ye t they show very clearly t ha t due to
the smallness of these I----1, p~-wave ] ~ cross-sections together with the
known fact of large uncertaint ies in the expe r imen ta l measurements (1.4) i t
(4) R. Ross and W. tI~TMPnREY: Reports, UCRL-9749 and UCRL-9752 (1961).
K-NUCLEON INTERACTIONS AND ~r~-RESONANCE 1451
would be quite difficult to decide experimentally between these two sets of
amplitudes. The extract ion of these rather small p-wave effects from the
measured values of the total K,N ) cross-seetions would require considerably
more experimental data than available at the present time. However, a check
on our calculations will be provided by detailed experiments for the deter-
ruination of K-nucleon cross-sections in the I = 1 , p+-state. The smallness of a~b,
and a.~tt in the momentum ran~'e studied, ensures tha t our calculated results
are not in eontradict ion with the experimental results (4) on low-energy
(k~<~300 MeV/c) K- -p ro ton intera.ctions. Here, we have a rather nice example
of the advantages of theoretical predictions over experimental means.
Table I I I shows tha t the values of the scattering and absorption cross-
sections increase monotonieally with increasing momentum, the increase being
more rapid for a~r thun for a~b., as is expected for p-wave interactions. The
smallness of these I-----1, p~-wave KA ~ cross-sections reflect the fact tha t
(a) at low-energies (kL~400 5{eV/e), I~-nucleon interaction is dominant ly an
s-interaction, and (b) the strength of ]~2V interaction in an I = 1 channel is
significantly weaker than the strength in I = 0 channel (4).
The above considerations concerning the existence of a 1385 MeV Yl-reso-
nanee as poles in Kgg' and ,=Y amplitudes for K-nucleon collisions, follow from $
the existence of ~ significant coupling between Yvresonance and KA ~, ~Y
channels, and therefore, all quant i ta t ive understandings about Y*-decay, e t c ,
remain unaffected.
I t is a pleasure to express our indebtedness to Professor E. C. G. SUDA~-
S~IA~" for several useful su~'~'estions and for reading the manuscript .
R I A S S U N T O (*)
Si espone un metodo teorico per distinguere fra i due gruppi di soluzioni dei para- metri dello scattering Kj\" di portata effettiva nulla, dati da Watson e chiamati solu- zione =4 e soluzione B, che si adattano abbastanza bene ai dati K--p nell'intervallo di impulsi del K- nel sistema del laboratorio fra 350 e 450 MeV/c. Si eombinano questi due gruppi di soluzioni, che sono abbastanza simili in onda s e d ma differiseono eonsiderevolmente in onda p, con i parametri ottenuti esigendo una risonanza Y~ di 1385 MeV con ampiezza 50 MeV come stato legato del sistema ~_-nucleone, con I = 1, in onda p~, in modo da ottenere due nuovi gruppi di parametri dello scattering ]~A ~ dipendenti dall'energia hello stato I = 1, p+, Si trova che lo sviluppo della portata effet- tiva nulla per lo spostamento di fase KA" in onda p e i parametri di scattering delle soluzioni B di Watson non concordano con le esigenze di causalit~t e di definizione posi- tiva delle probabilith di transizione.
(*) T r a d u z i o n e a cura del la Redaz ione .
