IL NUOV() (!IMENTO VoL. XIV, N. 5 lo Dicen~bre 1959

K-Hyperon Production Threshold Phenomena (*)(**).

L. FONDA (***) and R. G. ~-~:WT()N

Physics Department, Indiana University - Bloomington, Ind.

(ricevuto il 25 Luglio 1959)

S u m m a r y . - - The energy dependence of pion proton scattering and K-meson hyperon production cross sections is studied on the basis of a simple model. I t is found that the threshold effects can be big enough to be experimentally detectable for the purpose of determining relative hyperon and K-meson parities.

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

E n e r g y anomal ies a t thresholds for h y p e r o n p roduc t ion provide us wi th

inst~mces where the (~ Wigner cusps ~)(~) m a y be of considerable exper imenta l

impor tance . The mos t p romis ing of these effects is t h a t in the A - K associated

p roduc t ion near the threshold for the p roduc t ion of E-hyperons . Tha t the

re la t ive A E pa r i t y m a y be exper imen ta l ly de t e rmined b y observing the energy

dependence of the reac t ion = - ÷ p --~ A ÷ K ° or of ,~, -f- p -~ A ~- K + in the vici-

n i ty of the th reshold where the E - K associated p roduc t i on becomes possible was first suggested b y AD.&m (2) and i ndependen t ly b y BAz and O r ~ N ' (3). The physics invo lved is the fol lowing (***).

(*) Supported in part by the National Science Foundation. (**) A preliminary report of this work was presented at the Spring Meeting of

the American Physical Society, Washington D.C., April 3 0 - M a y 2, 1959. (***) On leave of absence from the University of Trieste, Italy.

(1) E. P. WIdeNER: Phys. Rev., 73, 1002 (1948). (2) R. K. ADA~I~: Phys. Rev., 111, 632 (1958). (5) A. N. BAZ' and L. B. OKUN': Soy. Phys. J.E.T.P., 8, 526 (1959); 2uric.

f;ksper. Teor. Fiz., 35, 757 (1958). (.**) Tile following is a condensation of the general argument given in ref. (4); see

also ref. (5~7). (4) R. G. NEWTON: Pltys. Rev., 114, 1611 (1959). (5) R. G. NEWTOn': Aw~. Phys., 4. 29 (1958). (~) L. F0~-I)A: ,¥uovo Cimet~to. 13. 956 (1959). (7) A. 1. BAz: Soy. Phys. J.E.T.P., 6, 709 (1958); ~urtt. F, ksper. Teor. Fiz., 33,

923 (1959).

1028 L. FONDA and a. G. NEWTON

Owing to the fact tha t the Y,K S-wave product ion cross-section starts with an infinite slope at threshold, all o ther cross-sections for processes connected by non-vanishing S-matr ix elements to tha t E K S-wave can be expected to have infinite energy derivatives there also. Consequently there will be a (~ cusp ~ or (~ rounded step ~) in the AK production cross-section at the E K pro- duct ion threshold, and it will appear in the A K S-wave if the AE par i ty is even, in the AK _P-wave if it is odd. Observation of the energy dependence of the AK angular distribution in the vicinity of the ZK threshold ma y there- fore lead to an unambiguous determinat ion of the relative AZ pari ty. More- over, if an anomaly at the Y,K product ion threshold is found in the A K pro-

duction S-wave then only a Z spin of ½ is compatible with a K spin of zero

and a A spin of ½. Similarly, there will be a cusp or rounded step in the pion proton elastic

scattering at the thresholds for AK and Z K product ion and they will appear in the S or P-wave depending on the respective parities PAKq~ and PzKq~ (*)"

The present paper is a s tudy of these threshold effects. Our purpose is to see if, on the basis of a simple model, one could expect the phenomena in question to be big enough for experimental detection. I t is clear f rom quite general considerations (4), and was pointed out part icularly by BAz and OKU~' (~), tha t the size of the effect in the A K production cross-section is largely determined by the size of the cross-section for Z K product ion b y AK collisions near threshold. Tha t however leads to no simple explicit s ta tement about the (( size of the cusp ~) in its over-all shape and therefore its experimental observability. Fur thermore , we want to determine how strongly the threshold

effects depend on the parameters in the model chosen. We shall investigate all four of the possible relative par i ty cases for the

(=p), (AK) and (ZK) systems. Since the (AK) system, however, is in a T = ½ isotopic spin state, only tha t state has all three systems coupled. Hence we are interested in the (rcp) and (ZK) systems in the T = ½ state only. The T = ~ states of the la t ter systems are coupled separately to one another, thus

producing another cusp or rounded step in the ~p scattering at the ZK pro-

duction threshold. We shall leave tha t simpler problem aside. Assuming, as we shall, tha t the spins of the A and Z are ½ and tha t of the

K is zero, only the J = ½ state contains the S-wave of the newly produced channel. Therefore only the partial cross-sections referring to tha t state exhibit

threshold anomalies. We shall consequently confine ourselves to the J = ½

state and then talk about S and P waves only. The fact t ha t we are looking at the T ~- ½, J -~ ½ state only, makes it difficult,

of course, to compare numbers to experiments. The experimental data we

(*) The AK production threshold lies at 762 MeV and the ZK production thresh- old at 892 MeV pion laboratory energy.

K-HYP~RON PRODUCTION THRESHOLD PHENOMENA 1029

w~nt to fit lie ut the end of our cross-section curves, at 9_10 to 960 MeV pion

laboratory energy. Wha t is known are the A K product ion S und P-waves

at 910 and 960 MeV (~), the EK production S-wave (P-wuve small) at 960 MeV (s)

and the total : :p elastic scattering cross-section at 950 MeV (9). We consi-

dered these data as upper limits for our purposes and made generous allowances

for the other states present.

Fur ther restrictions in the model we chose are the neglect of all electro-

magnetic effects including the E ° - E - and K ° - K + mass difference (*), and of

the multiple production of pions. The inclusion of multiple pion production

may very well appreciably alter the absolute cross-sections obtuined. How-

ever, there is no good reason to expect tha t it would very much change the

relative size of the threshold effects we ure interested in. The problem we are

studying is therefore tha t of three coupled channels: no. 1, the (~-p) channel;

no. 2, the (AK) channel; no. 3, the (EK) channel; each in the T = ½ , J = ½

state.

2 . - M o d e l a n d c a l c u l a t i o n .

The model we w~nt to describe the coupling of the three two-particle

systems (~:p), (AK) und (EK) in the center of mass frame, should be simple

enough for ready calculutions and yet contuin the essential elements tha t le~d

to the threshold effects we are looking for. At the same time we want to

work a little closer to the customary picture of (( forces ~) than, for example,

simply writing down ~n R-m~trix. We therefore chose ~ potential description

in a Schr6dingcr-type equation. Each channel, of course, satisfies ~ separate

wave equation with a different mass and energy. The three equations arc

coupled by given functions of the relative distance which we refer to as the

production potentials (**). There are, then the three ordinary potentials,

~ = V(~p --~ r:p), V22 = V(AK I+ AK), V33 = V(EK --> EK), and the three pro-

(s) r . E1SLi~R, R. PLAN(), A. PRt)DELL. N. SAMIOS, ~[. SCH~ARTZ, J . STEINBERGER, P. BAssi. V. BOREI.LI. G. PUPPy, H. TANAKA. P. WALOSCHEK, V. ZOBOLI, ~'~. CONVEI~,SI, P. FRANZINI, [. MANNELLI, R. ~ANTANGELO and V. ~ILVESTRINI: N~OVO Cime~to. 10. 468 (1958).

(9) A. R. ERWIN and J . K. KoPP: Phys. Rev., 109, 1364 (1958). (*) Since the (EK) channel of strangeness zero and total charge zero consists either

of E - ÷ K + or of E°÷K °, the two mass differences ~lmost cancel; according to ref. (~o), M E - - M~0 ~ 6 MeV, ~nd according to ref. (11), MK0-- ~]/K + ~ 5 MeV.

(") For a general discussion of such systems of equations for several channels, see ref. (~.6).

(10) W. H. BAI~KAS and A. H. ROSENFELD: University o/ Cali/ornia Rad. Lab., Report 8030 (1958).

(11) F. S. CRAWFORD, M. CRESTI, M. L. GOOD, M. L. STEVENSON and H. K. TICHO : Phys. Rev. Lett.. 2, 112 0959).

11030 L. )'ONDA and ~. o. N~WTON

duction potentials V,I=V(::p -+ AK), I '~t=V(=p -+ ZK) , and Vs.,=V(AK - + Z K ) . Since we shall neglect possible spin dependence, the three product ion poten-

tials, as well as the diagonal ones, mus t be real and those for reciprocal re-

actions mus t be equal due to t i m e reversal invariance. ~Ve combine all of

t h e m into one real symmetr ic 3 × 3 potent ia l matr ix . A fur ther word is necessary concerning the equat ion to be used. Since the

Z K product ion threshold lies a t a pion labora tory energy of about 892 MeV, which is not only high with respect to the pion rest mass bu t comparable to

the pro ton mass, nei ther relativist ic nor recoil effects can be neglected. E v e n

the simple pion nucleon scat tering alone therefore confronts us with a rela-

t ivist ic two body problem. A simple potent ia l description then certainly no

longer suffices. I f nevertheless we simulate the real physics b y an (( effective ))

potential , we still have a relat ively complicated free Hamfl tonian, i . e . , the

~um of the two free particle relativistic energies. Such an equat ion would be

r a the r difficult to solve even for the simplest potentials . We therefore con-

t en ted ourselves with introducing the (( potent ia l ~) into the equat ion af ter it has been conver ted into a second order differential equat ion (*). In other

words, the equations (**)

(1) W = (p2 + m~>)½ + (p2 + ~ m~<) , i = 1 , 2 , 3

for each channel are squared twice and then lead to the equations

p* = 4 W ~ i = 1, 2, 3 .

(2)

where

I f we conver t t hem into wave equations at this stage we have second order differential equations of the Schr6dinger type. We then add a potent ia l t e rm 2V#, where # has the dimensions of a mass, and have a manageable model.

The equations are 3

- - 1 n - l k - ' 7 2 , ~t~i ~ ~,~ + ~ V/j~:j = ci~,':i i = 1 , 2, 3, j = l

(3) ei = ( W " - m ~ > - ~n~<)2 _ 4~ni>~n?< _ k~

8 ~ t i W 2 - 2[Zi

The only remaining question is wha t should #i be. In the non-relativist ic

case, of course,/~i is the reduced mass in the i - th channel. There is, however,

one difficulty.

(') See the Appendix for a discussion of the relation between the two equations. (*') We use natural units, i .e . , ~ = c = l .

K - I [ Y P E R O N P R O D U C T I O N T H R E S H O L D P H E N O M E N A . 1031

We can derive a un i t a ry S-mat r ix f rom (2). The numbers #~ and kl will en te r in it as though k,/#~ were the relative veloci ty in the i - th channel. On the other hand, if we derive the form of the un i ta ry S-mat r ix f rom the asynlp-

tot ie requi rement of conservation of energy relativistically, then the ratio of outgoing and incoming relativistic relat ive velocities enters, which is the rat io

of two k~/#~ with #~ equal to the reduced energy:

(k~ + m~>)½. (k~ + m]<)}

#~ = (k~ + m~>)½ ~- (k~ + m~<)½ "

We therefore took these values oi #~ ra ther than the reduced masses (*). Equat ions (2) are not mean t to be any th ing bu t a simple easily manageable

model which m a y simulate the real physics within a re la t ively narrow energy

interval. They will certainly break down at low energies, since they contain

no mechanism for excluding the negat ive energy solutions when the to ta l energy W gets low enough. When the sum of the masses in one channel is

smaller than the difference of the masses in another , t hey lead to an unphysicM

threshold a t low energies (see Appendix for a more detailed discussion). In

the present ease there is no reason to believe tha t they cannot reproduce all the essential features exper imenta l ly observed.

For simplicity we m a y write (2) in ma t r ix form:

(5)

where V is the 3 x 3 potent ia l matr ix , e and K are the diagonal matr ices with

elements ~ and ki given by (3), M is the diagonal ma t r ix whose elements

are (4), and F is a 3 × 3 ma t r ix wave function whose rows indicate the three

(.hannel components and whose columns differ by their boundary conditions, i.e., the first column has an incoming wave only in the np channel, the second

only in the A K channel, the third only in the E K channel ('*). Equa t ion (5) is analyzed in te rms of par t ia l waves just like the ordinary

SehrSdinger equation. We assume tha t V is invar ian t under rotat ions. I f the intrinsic parities in the three channels are equal, then V will therefore couple only equal orbi ta l angula r m o m e n t a ; if the three par i t ies are not all equal,

i t will couple the l ----J--½ s ta te of one pa r i t y to the l ' = J + ½ s ta te of the

(') Owing to the structure of (2), there is ~ simple correspondence between any two choices of ltd. From the solution of an equation with one set of /~ and potentials V~

t ! ! ~ ! m~e can immediately obtain that of an equation with/~ and V~j= K~j(/~t//~) (/~j//@}. since the /ti given by (4} are energy dependent and the reduced masses are not, a .description with (4) and energy independent potentials corresponds to one with reduced masses and e~tergy dependent potentiMs, and vicevers~.

('*) Our notation differs from that in ref. (5) by transposition of all matrices.

1032 L. FONDA and R. (~. N]~WTON

other. Hence the radial equations for the J = ½ state are

(6) - - ½M-I~O " g- V~, + r-~Py+ = ½ K ~ M - 1~ ,

where P is a diagonal matrix with a one for the channels in the P-s ta te and

a zero for those in the S-state. In other words the equation for the pion pro ton

p --

S-wave has

(i°i)oO (i ° 1

0

(i ° 0

0

i) , if P a ~ = ~ - 1 , PAKq~ -~ - - I

!) , if P A = = q - l , PAK~ = -~ 1 ,

(i°i)ol , if PAX = - - 1 , PAKC~ = - - 1

, if P A ~ - - 1 , P a K q ~ = - ~ I ,

while the equations, for the pion proton P-wave are obtained by subtract ing

the above P from the unit matrix.

The type of potential we choose in (6) is a square well and for the sake

of simplicity we take the radii of all the elements to be the same. i n other

words, for r < a the matr ix V is a constant and for r > a it vanishes. We

then have seven parameters at our disposal: the common radius a, the three

diagonal potential depths, and the three production potential strengths.

I f the parities are all the same so that P is a multiple of the unit matr ix,

then eq. (6) can be solved explicitly and the S-matrix is calculated in a straight-

forward manner. I f the parities are not all the same, an explicit solution is

no longer possible. We then calculate the (( inside ~) solution as a power series

in the radial distance. I t is readily seen that the expansion is of the form

o a

~p(r) = ~. r2n+l(b~ + rcn) ,

K-HYP~ROS PRODUCTION THRESHOLD PIIENOMEI~'A

where the ma t r ix coefficients b~ and c,, satisfy the recursion relations

G)

1_/) P ] b n : b,~-I(2VM-- K 2) 2 n ~ +1) + ( 2 n - ] - 2 ) ( 2 n - - ] ) '

lO33

and if they are unequal

The actual computa t ion of the cross-sections was done with an IBM 650

electronic computer . We shall present three different sets of potentials for each of the four pa r i ty cases. Since the exper imenta l da ta even in the form

l 1--P P 1 c~-= c, ,_I(2VM--K ~) (2n ~- 1)(2n ~-2) ~- 2n(2n ~-3) "

This function is then fit ted in the usual manner to the (~ outside ~> solution

a n d the ~ -mat r ix is obta ined f rom the asympto t i c form.

The scattering and product ion ampl i tude ma t r ix for to ta l angular too- m e n t u m J, leading f rom the orbital angular m o m e n t u m component 1 of the

incident beam in the fi channel of to ta l spin j and spin z-component v, and

travelling in the z-direction, to the orbital angular m o m e n t u m component l'

of the outgoing wave in the 7 channel of spin j ' and spin z-component v', seen

ill the direction k ' is

k ' i ~-z'~l o'/,>~.,:,,,~.,( ) = [:~(:~ + ] ) ] ~ k ; ' y ' i ; " ( k ' ) - r ', , ' ~ ( J ) • C~j(J, v; O, v) C~,~,(J, v; ~ , - -v ' , v jzt~.~,r~,,

where the C's are the Clebsch-Gordan coefficients and T = I - - S . F r o m this

we obtain the in tegra ted part ial cross-sections for our case:

1 f d ' (~) ,~") (P ~ 7)=~Y-,,,,,,.,I .o I 0,~,, ~,,~,.,(k )'[~ ,

where j = l is understood. I f the intrinsic parities of the incoming and out- .,om~o'." ,, channels are equal, then

1034 L. FONDA and R. G. ~]~WTON

of limitations exist at best for six of the cross-sections at about 960 MeV pion laboratory energy, we have more parameters than points and hence aI~

. infinity of fits. The twelve sets of curves are to be considered simply as ex- amples of what variations are possible within the sparse experimental limits.

There are two additional points we want to illustrate in passing. One is the possibility, if the threshold cusp for example in the A K product ion at the E K product ion threshold is reasonably large, of obtaining from it the product ion cross-section for E's by AK collisions. The other point is the pos- sibility of obtaining, for example, the total cross-section for Z K collisions near zero energy from the E K product ion cross-section curve. Assuming tha t mul- tiple pion product ion is negligible, this total cross-section must equal the sum of those for pion proton and A K product ion (since the elastic scattering at zero energy is finite), both of which can be obtained also b y reciproci ty

f rom other observations (the first direct ly from the E K product ion by rip col- lisions and the second indirectly as described above). A comparison of the results will then give a measure of the multiple pion production. A similar argument applies, of course, to obtaining the total A K collision cross-section

near zero energy. The possibility of obtaining the A ÷ K - ~ E ÷ K product ion cross-section

near threshold follows directly from the general equation (14) of ref. (4). I f we apply it to the present case, using cross-sections integrated over angles~

then we obtain

(s) 2

O ' ~ A K b :

1 - 1

The notat ion on the left hand side refers to the derivative of the entire cross- section for r: ~ -p -~ A~-K, evaluated once as the E K threshold is approached from above, and once, f rom below; on the right hand side each parenthesis refers to the limit at the E K threshold. The states referred to in the subscripts

on the right hand side must all agree; tha t is to say, since in 62]K_~A K at threshold only the J = ½ state enters, all three cross-sections refer only to J = ½; since the AK system is in a T = ½ state, only the T = ½ part ial cross-section is

meant by a ~ _ ~ ; since, depending on the relative AE pari ty, it is either the

S or the P state in the A K system which is produced in E - ~ K -> A ÷ K near zero energy, it is the same S or P state, respectively, which is meant in the

final state in a=~_~.~; finally on similar grounds, it is the S or P state in the ~p system which enters bo th cross-sections on the right, depending on the re- lat ive par i ty PAKq~" Hence we can obtain the A + K - + E ~-K cross-section near threshold from (8) only if we know the J = ½ and T ~ - 1 partial cross-sections on the right, as well as the left hand side. If only the sum of the cross-sections

K - H Y P E R O N PRODUCTION THRESHOLD PH]gNOMENA 1 0 3 5

v , K for J , T = ½, ~ is known, we may obtain upper limits on the A + K --~ - --

cross-section. The total EK collision cross-section near zero energy is obtMned from

eq. (16) of ref. (~):

1 (9) -~ (k:~k a~,-.zK) 1 - - (kZKa~K)

where the parentheses on the right hand side again indicate the values at

threshold. One can therefore obtain a~K*°~ from the slope and intercept of the 1 curve of kEK am,__>~K near k~K--O.

3 . - D i s c u s s i o n .

First a remark on the potentials. I t is clear tha t all physical results are

invariant under a simultaneous sign change of all off-diagonM potentials. A

sinmltaneous si~n change of any two off-diagonal potentials is also of no phy-

sicM consequence. We may therefore fix the sign of two of them to be always

positive. We are giving our results in terms of positive V ( ~ p - ~ A K ) and

V(~p --> ZK).

Whenever in the following the pion proton elastic cross-sections are uninte-

resting, i.e., exhibit no marked structure, they are not shown. Table I gives

their values for each case at 950 ~[eV pion laboratory energy (*). The

T: ~-p ~ F~+ K product ion Pl-Wave below 960 ~ e V is in many cases too small

to be visible on our scale. In those cases we have not included it in the graphs.

Table I Mso lists the value of the A + K -~ V + K cross-section at 910 MeV for the purpose of correlation with the size of the EK threshold effects. I t is to be

noticed tha t there is no clear correlation between the sizes of the potential

V(AK -~ EK) and of the cross-section ~A~:-~K" In Table I I we list the values

of k=,, ! kA~: i and I kzK I in the region of interest for the purpose of easy deter- ruination of what constitutes a (( large )) or a (~ small )) cross-section.

In most cases the graphs speak for themselves. We may simply add a few conlments.

In comparing cases A and C, notice tha t the anomaly is bigger in C than

in A in spite of the smaller V ( A K - * EK) in the former. There is a marked

difference in radii for the two eases. Case B has a small cusp despite a large

V(AK --> EK). The AK S-wave is aeeidentMly rising very slowly.

(*) All our results are given in terms of the pion l~boratory energy. For that it is merely necessary to express the C.M. energy W in (3) in terms of the pion laboratory k i n e t i c e n e r g y , Ela b

I'V 2 = (m= + n%,) "2 + 2 ,n%El~ b .

TA

BL

E I

. --

Tlt

e va

riou

s ca

ses

cons

ider

ed i

n th

e te

xt.

Lis

ted

are

the

rela

tive

par

itie

s,

the

wel

l ra

dius

, th

e po

tent

ial

dept

hs,

the

pion

pr

oton

par

tial

S

and

P~

elas

t@ s

catt

erin

g cr

oss-

sect

ion

at

950

MeV

pio

n la

bora

tory

en

ergy

, an

d tl

~e p

arti

al l

=0

E

K

prod

ucti

on

cros

s-

sect

ion

by A

K

coll

isio

ns a

t 85

MeV

C

.M.

kine

tic

ener

gy (

kA

K=

l.2

4"

1() ':

3 cm

-').

PA

r. :

= q

- 1,

PA

KC

/~=

- 1

A

PA

~ =

+

1, P

AK

0Z=

+

1

51"75

ioo,

i:or

_

....

__

P

A~

=

--I:

PA

Kq

/--+

I P

A~

.---

-1,

PaR

r//=

--

1

a (i

n I0

-la

cm)

.575

.7

5 .7

.8

--

'i0

0

..

..

..

..

..

..

..

V(g

p

-+~Z

la )

(in

MeV

) --

2(

100

-- 2

00

v

500

, --

.

..

..

..

..

..

..

..

..

O

V(~

p -+

AK

) 22

5 '

175

175

200

3(

I(

125

I 75

> v

V(~

p -+

ZK

) .

..

..

..

..

..

.

100

200

200

50L

20

0 3(

1(

~

-Vo

o

_ o

o

_ o

o

..

..

..

..

..

..

..

..

..

..

..

..

.

V(A

K -

+ Z

K)

150

-- 3

( 2(

V(E

K-+

EK

) 50

50

50

0

--2

00

--

4(

--4

( 10

0 L

..

..

..

..

..

..

..

1

a (°)

~_+~

p (at

95

0M

eV,

inm

b)

1.57

!

3.29

4.

21

3.49

/

.62

I .1

3.

~ 4.

99

I

J K

~

i .7

--2

00

40

50

50

100

100

--IO

0

--5

00

50

50

1.66

3.

34

4

a(1)

nv-+

~p (a

t 95

0 ~¢

[eV

, in

mb)

*-

~ 0

~(o)

1.

74

AK

--+~

.K

(at

85 M

eV

C.M

.,

in

mb)

1.68

.2

1 .3

0 .3

5 1.

'76

1.67

.3

5 2.

25

1.51

5.53

•

I .

10.4

[

.' .

. .2

8 8.

67

K - H Y P E R O N P R O D U C T I O N T H R E S H O L D P H E N O M E N A 1 0 3 7

0.6

0.5

0.4

0.3

0.2

0.1

AK S-Wave

.(3 E

ZK S-Wove

750 800

AK P,- Wove ~ ~ / ZK PcWave

850 900 950 MeV

Fig. 1. - A K and E K par t i a l p roduc t ion cross sections for case A (see Tab le I) against pion l abo ra to ry energy.

0.5

0A

0.3 £

02

0 T i 750 800 850 900 950

MeV

Fig. 2. - AK and EK part ial production cross sections for case B.

0.1

67 - II N~oco ( ' imento .

1038 L. FONDA and R. G. N~WTON

0.8 AK 5-Wave / ~ 8

/ \ .-^, / \

0.6

o., ~ " - - _ _ _ _ < ~ r~

E

0.2 1 2

AK P1 - Wave ~ I ~'~ p - wnve . . . . 1 - -~ . I ' " ~ I ~.K Pl- Wave

o ~ o ~oo ~ o = ~ o HeV

Fig. 3. - A K and Z K par t i a l p roduc t ion cross sections for ease C; the par t i a l p ion p ro ton elast ic sca t te r ing cross section for l = 0 is g iven by the dashed line and refers

to a different scale ind ica ted on the r ight .

04

[13

J ~

02 E

0 / L ~ I I 0 I ~ I

?50 800 850 900 950

HeY Fig. 4. - A K and ~.K pa r t i a l p roduc t ion cross sections for case D. The Z K P~-wave

is too small to be visible.

K - H Y P ] ~ R O N PRODUCTION THRESHOLD PHENOM:ENA 1039

TABLE I I . Absolute values o] the wave numbers in the three cha~nels at various pio~ laboratory energies.

E~2b k~ I kAK I I k~K I (MeV) (10 la cm -1) (1013 cm -1) (1013 cm 1)

96[) 950 930 910 895 894 893 892 891 890 885 880 860 840 820 800 780 770 764 762 761 759 750 740 710 65O 60O

3.01 2.99 2.96 2.92 2.89 2.89 2.89 2.89 2.88 2.88 2.87 2.86 2.83 2.79 2.75 2.71 2.67 2.65 2.63 2.63 2.63 2.62 2.61 2.59 2.52 2.39 2.28

1.44 1.40 1.33 1.24 1.18 1.17 1.17 1.16 1.16 1.16 1.13 1.11 1.01

.901

.776

.629

.434

.293

.155

.0599

.0816

.165

.345

.471

.725 1.06 1.27

.822

.758

.610

.413

.143

.103

.0236

.O971

.139

.172

.282

.359

.572

.725

.849

.957 1.05 1.10 1.12 1.13 1.14 1.15 1.18 1.22 1.33 1.53 1.67

Case D, F ig . 5, shows a s izeble d o w n w a r d cusp in t h e p ion p r o t o n scat-

t e r i ng P l - w a v e ut t h e E K th re sho ld . The effect a t t he A K p r o d u c t i o n th re sho ld

is not a d o w n w a r d cusp, as t he in se r t shows. I t fol lows a t once f rom t h e me-

c h a n i s m t h a t p r o d u c e s t he inf in i te d e r i v a t i v e a t t h e u p p e r s ide of t h e t h r e s h o l d

b y conse rva t i on of f lux (~), t h a t a b o v e t h e f i rs t t h r e s h o l d t h e d e r i v a t i v e m u s t

b e n e g a t i v e (5). I n cases E a n d F (see F ig . 7 a n d 8), wh ich differ m a i n l y in

t h e size of t h e p o t e n t i a l V ( E K -~ EK) , t he re is a r e sonance in t h e p ion p r o t o n

e las t i c P l - w a v e j u s t be low t h e t h r e s h o l d for t h e A K channe l . I n b o t h cases

t h e d i a g o n a l p o t e n t i a l s a re t o o w e a k to b i n d t h e A K a n d E K s y s t e m s b y

t h e m s e l v e s . The coup led A K a n d E K sys t ems , u n c o u p l e d f rom t h e p ion

p r o t o n channe l , howeve r , p r e s u m a b l y h a v e a b o u n d s t a t e of smal l b i n d i n g

e n e r g y in these cases. No t i ce t h e sha rp r ise of t h e A K p r o d u c t i o n c ross -sec t ion

in b o t h cases nea r t h r e s h o l d (Fig. 6 a n d 8) owing to these resonances . The

1 0 4 0 L . F O N D A a n d R . G. N E W T O N

/ 1245~

F

0.1 5

0.10

~P P~ - Wave

005

..o E

0 I I I I #

750 800 850 900 950 MeV

Fig. 5. - P i o n p r o t o n p a r t i a l / ) 1 e las t ic s c a t t e r i n g cross sec t ion for case D. T h e i n s e r t shows t h e de ta i l a t t h e A K p r o d u c t i o n t h r e s h o l d .

1.2

1.0

G8

0.6

0.4

02 ~,K 5 - W a v e

AK O-Wave ~ l I )

75 0 800 850 900 9 50 MeV

Fig. 6. - A K a n d Z K p a r t i a l p r o d u c t i o n cross sec t ions for case E ; t h e Z K P1 w a v e is too smal l to be visible.

14_-ItYPERON PRODUCTION THRESHOLD PHENOMENA 1041

I

2.5 ! i

2.0 !

0.5 ~ p P-Wave

I i i I

0 750 800 850 900 MeV

95O

Fig. 7. Pion pro ton par t i a l P1 elastic sca t te r ing cross section for case E.

3 i I I l J l l~ ~ 5- Wave

2 I .D I I l\\\ F_ /

I / ! \ \

// \,~np Pl-Wave "~

/

, AK P~ Wave I I

0 750 800 850 900 950 MeV

Fig. 8. - A K and E K par t i a l p roduc t ion cross sections as well as t i le pion p ro ton par t i a l P1 elastic sca t te r ing cross section for case F . Tile E K P~ w a v e is too small

to be visible.

0.8

L. F O N D A a n d R. G. N E W T O N

750

0.6

t~

E

OA

0.2

800

1042

1.0

~ AK P~-Wave

850 900 950 MeV

Fig. 9. - A K a n d Z K p a r t i a l p r o d u c t i o n cross sec t ions for case G. The Z K P1 w a v e is too smal l to be visible.

1.6

1.2

E

np 3-Wave

0 i i J

750 800 950 850 900 M e V

Fig. 10. - P i o n p r o t o n p a r t i a l e las t ic s c a t t e r i n g cross sect ion, l=O, for case G.

K - H Y P E R O N P R O D U C T I O N T H R E S H O L D P H E N O M E N A 1 0 4 3

25

2ff

15

10

~P

0 74O

I I

750 760 MeV

F I

770

Fig. II. Pion proton partial P1 elastic scat- tering cross section for

case H.

1,2

10

0.8

o6L [

04

02

~ A K 5-Wgve

0 I L 75O 80O

ZK S-Wave

AK ~ ~ I

850 900 950 MeV

Fig. I2. - AK and EK partial production cross sections for case H. The EK P1 wave is too small to be visible.

1 0 4 4

0.6

L. F O N D A and R . G. : N E W T O N

0.5

0.~

0.3

0.2

0.1

JD

E

0 I 1 750 800

AK ~- Wo~/~ ZK ~-Wave

900 950 850 HeY

Fig. 13. - A K and E K pa r t i a l p roduc t i on cross sect ions for case I .

1.2

1.0

O.8

O.0

0.~-

0 750

i I I I 800 850 go0 950

HeV

Fig. 14. - P ion p r o t o n pa r t i a l elast ic sca t t e r ing cross sect ions for case I .

K - H Y P E R O N PRODUCT ION THRESHOLD P H E N O M E N A 1045;

0,6

0./. AK 3-Wove JD

0 I I . i 750 800 850 900 950

MeV

F i g . 15. - A K a n d ~ K p a r t i a l p r o d u c t i o n c r o s s s e c t i o n s fo r c a s e J . T h e E K P1 w a v e -

is t o o s m a l l to b e v i s i b l e .

0.6

0./,

0.2

,1o E

~K 5-Wave

800 850 0 I I I

750 900 950

MeV

F i g . 16. - A K a n d E K partia,1 p r o d u c t i o n c r o s s s e c t i o n s fo r c a s e K . T h e E K P1 w a v e

is t o o s m a l l t o b e v i s i b l e .

2046 L. FONDA and R. G. NEWTON

J 3 2 E

~P P,-Wove/ 1 1

i i I i

0 ?50 800 850 900 950 MeV

Fig. 17. - Pion p ro ton pa r t i a l Pz elastic sca t t e r ing cross sect ion for case K.

3

2 A K P-Wa

gK S-Wave I I 0

750 800 8 5 0 900 950 MeV

Fig. 18. - AK and ZK part ial production cross seetions for case L. The ZK PI wave is too smal l to be visible.

K - I I Y P E R O N P R O D U C T I O N T H R E S H O L D P H E N O M E N A ] 0 4 7

8 - I

4

2

I I I i L

750 8~ 850 900 950 M e V

Fig. I9. - Pion proton partial elas- tic scattering cross sections for case L.

ELab ( M e V ) EL. b ( M e V ) 8 8 0 8 9 0

7621 7641 770, 780E I 5 I 2 ? 910

1.3 ~-

1.2

1.1

1.0

0 9 - - L ± I I l 0 1 2 3 4 5

RAK( i012 c m -1 )

Fig. 20. The, AK production cross section divided by kAK plotted against ~;AK n e a r threshold for case A. ] 'he corresponding pion laboratory energy is

given in the upper scale.

0.50

045 c~

E

0.40

I I I I .._ 2 0 2

I k~ l (10~2crn -~ ) Fig. 2 1 . - AK production cross section plotted against I kEEl near tile ZK threshold. The cor- responding pion laboratory energy is given in tlle upper scale. The scale on the left refers only to the partial cross section f o r / = 0 ; adding tile P1 wave does not alter the shape of the curve.

1048 L. FONDA a.nd R. G. NEWTON

mechanism for tha t is, of course, quite different f rom the familiar resonance.

near zero energy in an elastic cross-section if there is a weakly bound state.

in the same channel (*). Cases A to F all had equal parit ies for the A and E hyperons. The fol-

lowing cases are those of unequal parit ies for those two particles. Case G has a resonance in the pion pro ton S-wave (Fig. 10) and in t h e

A K P l -wave (Fig. 9) owing to a bound state at 850 MeV of 24 M e ¥ b ind ing

energy in the E K system. The m a x i m a in the elastic pion proton scattering.

and in the A K product ion are shifted relat ive to each other, with the A K

m a x i m u m closer to the bound state. Case H has a sharp resonance (Fig. 11),

jus t below the A K threshold and a concomitant very sharp rise in the A K

product ion near threshold (Fig. 12) in spite of the fact t ha t the diagonal

V ( A K - + AK) potent ia l does not bind. In case I is a similar bu t less sharp,

effect, a l though the V(AK--> AK) is much too weak to bind. The cusps in

the A K P l -wave (Fig. 13) and in the ~p S-wave (downward; see Fig. 14) a r e quite marked . Notice tha t case I has a relat ively large radius, a----0.8" 10 -13 cm.

Case J shows ve ry small cusps a t the Y,K threshold (see Fig. 15; the ~p, wave is not shown). Case K, with a smaller radius and a bigger V(AK -+ EK),

has ra ther large cusps in the A K Pl -wave (Fig. 16) and in the rcp P l -wave (Fig. 17). I n ease L the threshold effects are dominated b y a resonance below the E K threshold bu t considerably above the A K threshold. This resonance

is produced in spite of the fact t ha t the diagonal potent ia l V ( E K - + EK) is

repulsive (**). Fig. 20 shows the A K product ion cross-section divided by kAK near kAK= 0 ~

aS a funct ion of kAK for case A. I t s curvature is small over a considerable range. of pion l abora to ry energy so tha t the to ta l A K collision cross-section coulff be obta ined f rom an equat ion like (9). Nevertheless it mus t be recognizeff t ha t the errors introduced b y exper imenta l uncertainties in a curve like Fig. 20 '

are ve ry large. Fig. 21 shows the A K product ion S-wave on bo th sides of the E K threshold

as ~ function of I/~:z~] for ease A. The lines being very nearly s t raight o v e r

appreciable pion labora tory energies, use of eq. (8) is pract ical ly quite feasible~

in this case.

The authors are indebted to Professors R. B. CURTIS and M. H. Ross for'

some fruitful conversations and to the Ind iana Univers i ty Research Computing:

Center for the use of their facilities.

(*) For a detailed discussion of such a phenomenon see ref. (12). (12) IJ. FONDA and R. G. N~wTo~: Ann. Phys., to be publi~,hed. (**) See ref. (1~) for an explanation.

K - H Y P E R . O N P R O D U C T I O N T H R E S H O L D P H E N O M E N A 1049

A P P E N D I X

This appendix consti tutes a more detailed analysis of the break down of eq. (2) at low energies.

I t is clear tha t a more proper way of t reat ing our relativistic two body problem would have been to s tar t f rom eq. (1) and after having converted it into a wave equation, to introduce into it a potent ia l mat r ix V:

(A.1) m ~ )½_ [(p ~ + m~>)~ + (p~ + ,< W:]~,, -- - X V,v,,. J

Since this equat ion still neglects re tardat ion effects it is, of course, not really correct; bu t it has more reali ty than (2).

The integral equat ion corresponding to eq. (A.1) and the outgoing wave boundary condition is

(A.2) V, -- ~v~- ~+> ~ v , ~ ,

where the Green's funct ion _~(v(*) is given by ( s > 0):

(A.3) ~¢(~+'(W, l r - - r ' l ) - -

+co

--i fd pexp[iplr-r'l] (2z)2]r r,i, P (p2+ my>)½ + (p~+m]<)½--W--is"

'The integral is carried out on tha t sheet of the Riemann surface of the inte- grand where, on the real axis, the square roots are bo th positive. The inte- grand in (A.3) has branch points on the imaginary axis as well as a simple pole in the upper half plane. After adding two quar ter circles in the upper half plane at infinity to the pa th of integration, the integral therefore natural ly :splits into the sum of two integrals:

(A.4) ~+). = ~i(¢(1) _+_ ~12) ,

where ~1) is the integral over the contour C~ 1) in Fig. 22 enclosing the pole Pi of the integrand in (A.3), and ~2) is the integral .over the contour C(~ 2) around the branch lines.

~(1) is easily calculated: i

(A.5) 2~[r--r'[

Fig. 22. - Contours of integration in tile p-plane for eq. (A.4).

-m > (p)

cO)

®

1050 L. FONDA and R. G. ~]~WTON

with k~ and #~ defined by (3) ~nd (4). We note t ha t ~(~'~ is jus t the p roper Green's funct ion required for solving eq. (2). f¢~*) cannot be eva lua ted in closed fo rm; it vanishes asympto t ica l ly for large r exponentially, essentially like e x p [ - - m , : < - -

Fig. 23 shows the p a t h

f~ l l l !

m~ (P)

( -

described b y the pole P , as W decreases. When W ~ m~> ÷ m~<, P~ moves along the r ight of the posit ive imaginary axis, the i - th channel now be ing

m 2 closed. When W becomes less t h a n ( ~> --m~<)½, the pole moves across a b ranch line onto ano the r sheet of the R iemann surface, where on the rea l axis (p~÷m~<)½ is negative. F r o m then on the re is no residue contr ibut ion to the Green's funct ion ~(~+', which will be given by :

(A.6) f¢'+' ca(2' for W < (m~> m~<)½

Fig. 23. - The path described by the pole of the integrand in (A.3) as the total energy W decreases. The dashed line represents the continuation on another sheet of the

Riemann surface.

This shows the mos t essential difference be tween eqs. (A.1) and (2). The ]a t ter contains no mechan i sm for avoiding a negat ive energy for the l ighter pa i r tc le when W ~ (m~> - - m~<)½. When W ~ m~> - - m/< eq. (2) leads to an outgoing wave in the i - th channe~ and hence to an unphysical threshold. Eq. (A.1), on the contrary , au tomat ica l ly excludes nega t ive energy effects and the unphysical outgoing wave.

Asympto t i ca l ly as ei ther r or W goes to infinity, the Green 's funct ion for eq. (A.1) equals the Green 's funct ion for eq. (2) since then f¢~' vanishes relat ive to _,(vcl) (except in the energy region where f¢~I) vanishes identically).

Operat ion of (A.1) b y [ ~ ' ] - ~ = (p~/2/t,) - - e, on the left leads to an equatior~ of the t ype (2):

wi th an equivalent ~ potent ia l ,) t e rm on the r ight hand side which is b o t h non local and energy dependent (the first t e rm on the r ight hand side will be absent for all i for which W ~ * ( m , > -

R I A S S U N T O

In questo lavoro ~ stato eseguito lo studio delle sezioni d'urto di scattering pione- protone e di produzione mesone K-iperone in funzione dell'energia del pione incidente. Dato che gli effetti dl soglia risultano cospicui per molti dei casi considerati, esiste la possibilit~ di determinare sperimentalmente le relative parit~ PAX e PAKq~.