IL NUOVO CIMENTO Voz. XXIX, N. 4 16 Agosto 1963

K-Nucleon Interactions and the Possibility of Yo-Resonance

as a Bound State of the K-~ System.

~:~ A~C[E SI-I CttA]~D

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

(ricevuto il 29 Marzo 1963)

Summary. - - In the zero-range boundary conditions method of Dalitz and Tuan for coupled ]~AP-r:Y channels, interactions of i~-meson with nucleon are discussed. The low-energy K--p data are described in terms of three sets of scattering parameters, called solutions I, I I and I IA. Using these parameters, the possibility of an I = 0 , ~-E resonance at 1405 MeV as an s-wave bound state of l~-nucleon system is investigated.

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

F o r low values of K - - m o m e n t u m ( K - - l a b o r a t o r y m o m e n t u m k~, below 300 MeV/c), the processes which occur for K-- l? collisions are:

(1.1) K - + l? -+ K--}- l? ,

(1.2) -+ K ~ + n ,

(1.3) -->~+ + E - ,

(1.4) ~ 7:- + Z +

(1.5) -> 7:0 + Zo,

(1.6) -+ n ~ § A ,

toge the r w i th a small f rac t ion ( < 1%) of events .

X ~

K - § ~ A § 2 4 7 - , A + 7 : ~ T h e exl?erimental da t a on K--l? l?rocesses for k~=<300 MeV/c has been

968 RAMESH CHAND

obtained by Ross and I-IU)~PltlCEY (L2). The outs tunding features of these da ta are (i) the angular distr ibution for K - - p elastic scat ter ing is isotropic (after subtract ing out the Coulomb-scat ter ing contr ibut ion which is dominan t ly in

the forwurd direction), (ii) the hyperons product ion cross-sections are isotropic and (iii) the product k ~ a ( K - + p - ~ a l l T c § approaches a (;oustant value in

the l imit of kL upproaching zero. These characteris t ics s t rongly suggest t ha t below 300 MeV/c l abora to ry m o m e n t u m of the K - - m e s o n , l(-A ~ interact ion

is dominant ly an s- interact ion U). This can be e~sily unders tood in the clas-

sical sense b y realizing tha t in order to conserve bo th p~ r i t y ~md to ta l angular

m o m e n t u m in strong interactions, the lowest mass which can be exchanged

between a K-meson and a nucleon is t ha t of two pions which corresponds to

a rnnge of about 0.7 fermi. This value of r~nge gives k~ ~ 220 MeV/c for

s-wave interactions of the I~-J~ system. Another impor t an t fe~ture of the K - - p

da ta is tha t the cross-section for the product ion of all hyperons is ~ large fr~ction of the to ta l cross-section and is quite comparable wi th the geometr ic

l imit given by 7~/k ~, k being the center-of-mass m o m e n t u m of the K-A ~ system.

For example, a t k L = 200 5~eV/c,

a ( K - + p -~ all TO+ Y) _~_ a ( K - + p -~ K - § p) __~ 55 m b ,

as compared with the value ~/k'2~ 74 rob. This clearly indicates the inade- quacy of pe r tu rba t ive methods for describing the K-A r processes i rrespect ive of

the s trengths of the coupling constants involved. This is due to the fact t h a t the absorpt ive effects appear only in the higher-order te rms of the pe r tu rba t ion

theory and never in the lowest-order te rm. DALITZ ~nd TUAN have discussed (4) in considerable detail the use of per turbut ion theory and the s t rengths of the coupling constants for describing K-A ~ processes, in tu i t ive ly , one would expect t h a t the s t rength of these coupling constants is large, of the order of uni ty .

The exper imenta l da ta (~.~) which show tha t for k~<300 5~eV/c, K-A~inter - act ion is dominant ly an s-interactionl led JACKSON et al. (5) to propose the use

of effective range expansion for K-A ~ scat ter ing phase shifts which can be

wri t ten as 1 1 1

(2.1) ]c ctg($~ - - A~(k) - - A, + -2 R'k~'

(1) R. Ross: Elastic and charge-exchange scattering o/ K--mesons in hydrogen, Lawrence Radiation Laboratory Report, UCRL-97t9 (June 1961).

(~) W. HVMPHR]~Y: Hyperon product,,on by K--,~uesons on hydrogen, Lawrence Ra- diation Laboratory Report, UCRL-9752 (Jnne 1961).

(~) I t has been rather well established experimentally that relative to A-parity, parities of E-hyperons and K-mesons are even and odd respectively. See, e.g.. R. H. DA- LITZ: Prec. o] the 1962 Anuuai Intern. Con]. on High-Energy Physics at CERN, p. 391, (Geneva, 1962).

(4) R. It. DALITZ and S. F. TUAN: Ann. o] Phys., 3, 307 (1960). (a) j . D. JACKSON, D. C~. RAVENHALL ~nd H. W. WYLD : Nuovo Cimento, 9, 834 (1958).

F'~-NUCLEON INTER,trCTIONS AND THE POSSIBILITY OF Y~--ICESONANCE ETC. 969

for the isotopic-spin I of the K-JT system. Due to s t rong react ive processes

leading to the product ion of hyperons, the phase shift ~ , the scat ter ing length

A 1 and the effective range Rx are all complex numbers . For K - - p scattering,

two complex phase shifts 8o and ~1 describing the I = 0 and I = 1 scat ter ing interact ions are necessary. Therefore, the use of (2.1) requires the specification

of eight real parameters . Due to the incompleteness of the exper imenta l da ta on K - - p scat ter ing and react ion processes, i t is ra ther difficult to determine

so m a n y parameters . Therefore, in order to describe thephenomenon , JACKSON et al. suggested the use of the effective range expansion (2.1) in the l imit of

zero-range, t ha t is, with the approx imat ion

1 1 (2.2) k ergO, ~ _~ ~ - .

A,(k)

The complex scat ter ing length A, for each isotopic-spin s ta te can be wri t ten

as Az= az~-ibx, where a~ and b~ are real quantit ies. ])ALITZ and TUAI~ (4)

have reformulated this zero-effective-range expansion in te rms of boundary,

conditions for coupled KAY-ToY channels by using the energy-independent

K-matr ices . The use of (2.2) allows us to write down the following expressions for the s-wave K - - p collision cross-sections:

(3.1)

(3.2)

(3.3)

(3 .4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.1o)

a (K- -k p ---~ K - + p) =

o,(K-q- p --~K ~ q-n) =- 7~

a ( K - + p ---~ z: + q- E-) ----

a(K-+ p ---~zc-+ E ' ) =

a ( K - + p -+zc ~ + E ~ :

a ( K - q - p --~z: ~ + A ~ =

~o = ITo I ~ -

a~ = a~(E) + a~(A) =

~ ( r , ) = J T~ I ~, ~ ( A ) =

8 --

Ao -k A1 ' 1-- ikAo 1 -- ikA1 2

Ao A1 2 1-- ikAo 1-- ikA1

To T1 2

To T1 2, x/~ 2

To t 2

ZA 2 7

4~bo/ K

l l_ikAoI~ '

4zbl/K l _ i k A l [ 2 '

TA?, ~(A)

(71

970 RAMESH C}IANI)

I t is convenient to define To, :T1, and T A as follows:

~o ( 4 . ] ) T O - -

1 - - i kAo '

M~ .N'A (~ .2 ) ~1 - - - - ~ . / ~ - -

I - - i k A l ' 1 - - i k A 1 "

In order to describe the hyperon production processes, we need an additional

quant i ty F, which is defined by

To TTo (5.1) T-~ = exp [ i ~ .

Using (4), we can write ~ as

(5.2) = ~o + arg [(1 - - i k A ~ ) / ( 1 - - i k A o ) ] ,

where ~o = arg M o - - a r g M1. In this description of K-A r processes, we assume

charge-independence for the nuclear interactions and therefore neglect the

effects due to Coulomb interactions and the mass differences between particles

belonging to the same multiplet. Therefore, the six real parameters repre-

sented by ao, bo, a~, bx, e and ~o constitute a description of the K - - p scattering

and absorption processes.

A complete analysis of all data on K - - p collisions for K- - l abora to ry mo- mentum /% below 300 MeV/c has been made by Ross %nd ttvM1)H~EY (~,~).

Ross has obtained cross-sections for elastic and charge-exchange scattering pro-

cesses, and H u ~ P ~ R ] ~ for the absorptive reactions leading to the product ion

of Z and A-hyperons. Their investigations give two sets of s-wave K-A ~ inter-

action parameters, called solution I and solution I I which fit the experimental

data rather well over the complet e K - - m o m e n t u m range and are listed in

Table I. The most impor tant thing worth noticing about these two solutions

TABLE I. -- The two sets o/ s.wave zero.e//ective ~ange K-~N ~ interaction paramete~'s deter-

mined by Ross and H v M P ~ Y , which bes t / i t the K--p data up to 300 MeV/e laboratory

momentum. The parameters ao, b o, a 1 and b 1 are given in fermi units.

Solution

II

a o

- - 0.22 1.07

-- 0.59 4- 0.46

bo

2.74 4- 0.31

0.96 4-0.17

a l

0:02 i 0.33

1 . 2 0

t= 0.06

bl

0.38 ~= 0.08

0.56 =~ 0.15

0 . 4 0

i 0 .03

0.39 • 0.02

~o

60.3 ~

_ 6 3 . 2 ~

]~-NUCLEON INTERACTIONS AND THE POSSIBILITY OF ~*--RESONANCE ETC. 971

is tha t solution ~ is character ized b y a posi t ive value of ~0 whereas solution I I

has ~o negative. As has been discussed earlier (6), the pa ramete r s of solution I

provide a be t t e r fit to the exper imenta l da ta on K - - p , K~-p, and K - - d processes than those of solution I I .

ICecently, WATSON (~) has made a fit to the K - - p da ta i n K - - l a b o r a t o r y

m o m e n t u m range of (350--450) 5IeV/c. I n this analysis, Z-A relat ive pa r i t y is considered even and the I - - 0, ~-Z-dg resonance (Y** a t 1520 lVieV) is t aken of Brei t -Wigner form. This invest igat ion gives two sets of s ,wave zero effective range

K - . ~ interact ion pa ramete r s called solutions A and B and are listed in TaMe I I .

TABLE II. - The two sets o/ s-wave zero e/]ective range K-2~ ~ interaction parameters deter- mined by WATSON which best/ i t he K--p data in the region o/(350+450) MeV/c K--labo-

ratory m~me~tum. The parameters %, b0, a I and b~ are given in fermi units.

Solution a o b o a~ b~ e 9

A - - 0.90 2.50 - - 0.03 0.41 0.31 - - 119 ~ 0.25 =~ 0.20 • 0.06 • 0.03 ! 0 . 0 3 =~ 3 ~

B - - 0.96 -4-0.14

1 . 7 1

• 0.14

~= 0.05 0.42

~0.03 0.31

• 0.03 --109 ~ ~ 2 ~

I t has been poin ted out by AI{IBA and CAPPS (s) t ha t an appeal to the con-

t inu i ty in the energy-dependence of the Z - / Z + rat io f rom zero-energy to the

energy of the ** Y0 -resonance requires the phase angle q~ to be negat ive in the

entire energy range. This is possible with the pa rame te r s of solution I I and not with those of solution I. Therefore, the possibil i ty of solution I describing the s-wave K-J~ processes is complete ly ruled out. Now, we are left with solution I I whose pa rame te r s do not provide a good fit to the low-energy da ta

on K - - p , K~ and K - - d processes, giving too high values of the cross-sections for the last two processes. This clearly suggests tha t we re-examiue the para- meters of solution I I in comparison with the s-wave scat ter ing paramete rs of

solutions A and B. The most surprising thing abou t Wat son ' s solutions A

and B, which are quite s imi l a r in character , is t h a t the real pa r t of I = 1

scat ter ing length, namely el , is ra ther small, whereas l~oss and H u m p h r e y ' s

solution I I is character ized b y a large value of el, I n fact , the disagreement

between the exper imenta l da ta and the calculated values of the cross-sections

for K~-p and K - - d collisions using the pa ramete r s of solution I I is p robab ly

(~) RA~IESII CIIAND: Ann. o/ Phys.. 3, (39 (1963). (7) 1~. B. WATSON: K--proton interactions near 400 MeV/c, Lawrence Radiation

Laboratory Report, UCRL-10175 (September 1962). (s) T. AKIBA and R. H. CAPP$: Phys. Rev. fJett., 8, 457 (1962).

972 R A M E S H C H A N D

due to this large v a l u e of al. Therefore, it is quite na tura l to expect t ha t

new energy-dependent scat ter ing lengths solution, obta ined by in terpola t ion between l~oss and H u m p h r e y ' s solution I I and Watson ' s solution A (or solu-

0 t ion B) would explain the exper imenta l da ta on K - - p , K~-p, and K - - d inte-

ract ions adequately. In order to pe r fo rm the in terpola t ion for the sca t te r ing lengths A0 a.nd A~, we have made use of the linear expansion (2.1), t ak ing

solution I I to represent the K3~' da ta at kL = 0 MeV/e and solution A at.

k~ = 400 h~eV/c. This gdves us

(6.~) Ro ----- (0.43 ~- 0.51 i) fermi ,

(6.2) R~ ~ - - (1.10 + 2.70 i) f e rmi .

This new set of s-wave energy-depenci[ent scat ter ing lengths solution called solution I I A (as it is obta ined by combining solutions I I and A) is given in

Table I I I for few selected values of k~. The pa rame te r s of this new solu-

TABLE IH. - ~'he s-wave energy.dependent scattering length parameters o] solution I I A as a function o] K--laboratory momentum lcx.. The parameters Ao(k ) and Al(k) are in

Fermi unit ~.

kr (MoV/c) A0(k) Al(l~)

0 100 200 300 400

- - 0.59 4- o.96 i - - 0 . 6 1 4- 1 . 0 1 i

--0.67 4- 1.16i - - 0 . 7 8 + 1.53i - - 0.9O -}- 2.50 i

1.20 4- 0.56 i 1.03 -[- 0.77 i 0.45 4- 0.89 i 0.07 4- 0.63 i

- - 0.03 4- 0.41i

t ion provide a good fit to the exper imenta l da ta on K - - p and K - - d collisions

in the region of k n = (200+300)MeV/c , but not too good an agreement for

k r < 200 MeV/c. Of course, a t kn = 0, solution I I A is identical with Ross and

H u m p h r e y ' s solution I L The da ta on K~ interact ions are also fit ted reason-

ably well b y the pa ramete r s of this solution.

2 . - Y ~ - r e s o n a n c e .

m

In this section, we would use the scat ter ing pa ramete r s of the K-nucleon sys-

t em in order to invest igate the possibil i ty of the I = 0, ~-E r,Jsonance a t 140~) MeV

as an s-wave bound s ta te of the K - ~ system. The possibil i ty of the existence

K-NUCLEON INT]~RACTIONS AND TIf]~5 1)OSSIBILITY OF Y~ RESONANCE ]~TC. 973

of this resonance was first postulated theoret ical ly by CAPES and SHV~T (~) while considering the reaction

(7) K--~ d --> r~+ Z + ~ .

Using impulse approximat ion and the sum rule method in terms of the isotopic- spin states of the Z - ~ system, CAPPS and SHVLT calculated the reaction rates / ~ and R�89 for Z-hyperons production. Here ~ and �89 stand for the isotopic- spin of the Z-2V system. For K--absorp t ion ~t rest, Capps and Shul t found big discrepancy between the experimental data (~0) and the calculated value

of the ratio /~ = R~/R i , the exper imental value being smaller than the cal- culated value by ~ factor of about 2. The calculations have been repeated for these rates with the inclusion of multiple scattering and the corrections from charge-independence due to the mass differences between particles be- longing to the same multiplet . I t is found (6.~) t ha t t h e above discrepancy

still persists, e.g., we huve found R = 2.54 for solution I I (e) us compared with the experimental value R = ]..18 for K- -cap tu re at rest. In order to explain this discrepancy, Capps und Shul t have postulated the existence of an I = 0, ::-Z resonance called Y* just below the threshold of K - -p system (at ~bout - -10MeV) . The existence of such u resonunce would invalidate the assumption of constancy of the magni tude and phase of 3/o for the cal- culations of K- -deu te r ium absorpt ion processes and in fact the relative phase ~o between M0 and M~ would then vury rupidly as the K - - p energy becomes negative. The observed Y* is much fur ther below the K - - p threshold (-- 27 MeV)

and it is ra ther difficult to believe tha t this resonance would muke such a big difference in the calculutions of R~ and R~ absorptive rates as has been claimed by CAeeS and Sn~LT. Therefore, fur ther investigations concerning this dis- erepancy are desirable.

5Tow, we would like to s tudy the possibility of Y*-resonance as an s-wave bound state of K - - p system. For this purpose, we would consider the amplitude for the process K- -~p->7 :o-~Z~ where the final-state particles are in a pure 1 = 0 stute. After dropping ~ constant factor of V/6, the ampli tude for this process can be wri t teu as

i o (8) To -- 1 - - ikAo "

(9) R. L. SHULT and R. H. CxPPS: Phys. l~ev., 122, 1659 (1961). (lo) O. I. DA~, N. Ho~wI~z, D. MILLE~, J. MURRAY and M. B. WATSON: Proc.

o/ the 1960 Annual Intern. Con/. on High-Energy Physics at Rochester (Now York, 1960), p. 417.

(11) Rx~ESH C~AN1) and R. H. DALiTZ: Ann. Phys,, 20, 1 (1962).

974 gAMJ~ STd CnAND

In terms of real K-matrices, Mo is given by (9

(9) Mo __ 1 3o , 1 - - i7o q

where (fwlKl J:), 70 = a n d q is t h e eenter-oi-m ss m o

mentum of the ~~176 system. The reality of K-matrices, namely of rio and yo

and the fact that the Y*-resonance lies above the threshold of the ~-E system into which it decays, implies that the amplitude Mo in expression (9) cannot have any dynamical singularity in the energy range of interest to us. Also, the assumption of constancy of the K-matrices prevents Mo to have any intrinsic singularity either. We must remember that the low-energy K--p data are con- sistent (~.2) with the constancy of Mo. Hence the singularity of To-matrix is completely determined by the kinematical factor of 1--ikAo. Therefore, Y*-resonance as a bound state of K--p system is obtained by simply replacing l~ by i K in expression (8), where

K ~_ ~/2#r:(E, - - E) , E t = MK + M x ,

l% is the reduced mass and E the total energ3r of the K--p system. This gives US

Mo(K) Mo(K) (10) To(K) -- 1 + KAo -- (1 + Kao) + iKbo "

Expressing the Breit-Wigner resonance form ( E - - E , + iI ' /2) -1 into the form [(1 + Kao) + iKbo] -~ of the denominator of expression (10) leads us to the con- clusion that the resonance energy E, corresponds to

(11.1)

and that the width of this resonance is given by

(11.2) i ~ ~_ 2b~ . ~

In this discussion, we assume the experimentally determined values of E~ and F as known quantities and would therefore determine the values of the scat- tering amplitudes which are needed for the explanation of Y~-resonance as bound state of K--proton system. E~-=1405 MeV gives K--=132.4 MeV/c

I~-NUCLEON INTERACTIONS AND THE POSSIBILITY OF Y*-RESONANCE ETC. 975

0.67 fermi -~. At present, there is considerable uncer ta in ty as to the correct width of Y*. There are two exper imental groups which give completely differ- ent values of F, one group giving /7_~ 50 h~[eV, and the other group giving F < 2 1V[eV (~). Using these values in (11), we get

(12.1) a0 ~ - 1.5 f e rmi ,

(12.2) bo ~ 0.69 fe rmi , for F ~ 50 MeV,

(12.3) b0 ~< 0.03 fe rmi , for / ' ~ 2 MeV.

These values are to be compared with those given by the energy-dependent solution I I A at k ~ 0.67i fermi -~ and by the energy-independent solution II . These values are as follows:

(12.4) ao , . o _ (0.59 =]=0.40) f e rmi .

(12.5) b0 --~ (0.96=k0.17) fermi

for solution I I and

(12.6) a0 ~ - - 0.52 fe rmi ,

(12.7) bo ~ 0.81 fe rmi ,

for solution IIA. On the basis of this comparison, i t is ra ther difficult to decide between solutions I I and I IA, both solutions providing a ra ther poor fit to the values of ao and bo required by the existence of Y*-resonanee as an s-w~ve K_Ae bound state. The small value of bo required for the ease of F~<2 ~ e V indicates tha t most probably the correct width of u is about 50 ]VfeV.

F rom these investigations, we conclude tha t an entirely new fit to the low-energy K- - p data, using the energy-dependent scattering p~rameters would

be of great value. Also, it would be very desirable to require tha t below the energy of K- - p threshold, these ~mplitudes extrapolate to the v~lues required

by Y*-resonance. This constraint will be of no value, if later on it is found

(12) B. P. G~]~GORY: Proc. o] the 1962 An~wual Intern. Con/. on High-Energy Physics at C E R N (Geneva, 1962), p. 779.

(18) j . j . Mu~aAY, J. B. SHAFER and D. O. L[UwE: Bull. Amer. Phys. Soc., 8, 22 (1963).

976 RAMESH CHAND

t h a t Y* is n o t an s -wave re sonance , e . g , i t is now r a t h e r wel l e s t a b l i s h e d t h a t

t h e 1385 M e u ~ - A r e s o n a n c e is a p~- resonance (18) a n d t he r e fo re is to be ex-

u m i n e d as a p - w a v e K - A e b o u n d s t a t e . This r equ i r e s t h e ~o rmu la t i on of

p - w a v e b o u n d a r y cond i t i ons for coup led I~AP-T:Y channels .

R I A S S U N T O (')

Si discutono le interazioni del mesone F~ col nucleone entro i l metodo delle con- dizioni ai l imit i di range nullo di Dalitz e Tuan per canali KAP-uY accoppiati . Si descri- vono i dat i K- -p di bassa energia in funzione di t re gruppi di parametr i di scattering, chiamat i soluzioni I, H e I Ia . Servendosi di questi parametr i , si s tudia la possibilit~ di una risonanza n-E, con I ~ 0 , di 1405 MeV come uno stato legato in onda s del sistema ]~-nucleone.

(*) Traduzione a cura della Redazione.