Low-type integral equations for the absorption and production of K mesons

Nuclear Physics 13 (1969) 436-448; Not

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to be reproduced

by photoprint

INTEGRAL AND

@ North-Holland Publishing Co., Amsterdam

or microfilm

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EQUATIONS

PRODUCTION SAUL

FOR

permission

THE

from

tbs

publisher

ABSORPTION

OF R MESONS

BARSHAY

Institute for Thewetical Physics. Received

written

t

University of Copenhagen

26 June

1969

Abstract: A discussion is given of Low-type integral equations for approximate transition amplitudes which describe the processes R+N +n+Y and n+N + K+Y. Isobaric spin and angular momentum decompositions of the equations are performed, and the resulting sets of integral equations for the various amplitudes are discussed with nonrelativistic approximations for the interaction current operators. Certain assumptions are made with ,regard to the as yet unknown relative parities, coupling strengths, and scattering properties of the strange particles, in a preliminary effort to utilize the field equations as a model for correlating certain features of the present experimental data on the absorption of low-energy K- mesons by protons.

1. Introduction

In this paper an attempt is made to derive and to investigate Low-type l) equations for the scattering amplitudes which describe two of the simpler processes involving pions, nucleons, K mesons, and hyperons. These processes, which are related by the crossing symmetry, are a) R+N

+ n+Y,

b) n+N +- K+Y.

(1)

The particle symbols are used to denote the mesons and the symbol Y stands for either a A or a Z hyperon and N for a nucleon. The approximate equations to be considered are linear integral equations for the desired amplitudes, the inhomogeneous terms being “renormalized” Born approximations, and the kernels of the integral terms being related to pion-nucleon and pionhyperon scattering amplitudes. The equations may be useful in going somewhat beyond the Born approximation calculations of the amplitudes, and in this respect they are somewhat similar to the Chew-Low “) development of photopion production. For simplicity, the equations are derived within the static approximation for the baryon fields (that is, the bilinear combinations of baryon field operators in the Hamiltonian are considered as replaced by source functions and virtual antibaryons are removed from consideration). However, the final equations may be generalized to their relativistic analot National Science Foundation

Postdoctoral 436

Fellow.

436

SAUL BARSHAY

gues in the same manner as the Chew-Low equation for photopion production 2, derived within the static model may be generalized to the corresponding Low equation l). In a preliminary discussion of the equations, we shall examine the scattering amplitudes for the processes (la) for K kinetic energies of zero to m 100 MeV, utilizing static model approximations for the current operators under several assumptions about the unknown relative parities, coupling strengths, and scattering properties of the strange particles. This discussion is meant to see whether a semi-phenomenological analysis of the equations may provide any correlation of the accumulating experimental data on K--proton interactions leading to pion-hyperon final states. The discussion is, at present, rather limited by a crude treatment of the strong K-meson interaction current. 2. Derivation We consider the total Hamiltonian to be H = H,+H,+H,, where H, is the free-field Hamiltonian (with the observed masses), H, describes the pion-baryon interactions, and H, describes the K meson-baryon interactions. Suppressing the mass counter-terms which must be subtracted from the interaction Hamiltonians, we may define the pion and K-meson current operators, ji and jZ, by

Hl =

c hA(P)+c,+il+w~

H2=

Iiv,j2(~)+cP+i2+(P)l~

(2)

9

where the c, (C,) and CD+ (CD+) are annihilation and creation operators, respectively, for pions (K mesons) of momentum p and energy w = (p2+ ,~~(rn”)}*. Here the symbol + denotes the hermitian adjoint, ,U is the pion mass, m is the K-meson mass; the isobaric spin indices may be considered to be included in the symbol $. We denote by xa and x,, the initial and final free particle states for the reaction (la); these are eigenstates of H, corresponding to a total energy E = CO,+ E, = CO,,+E,, where w,, tab, E, and E, are, respectively, the total energies of K meson, pion, nucleon and hyperon. We denote by va (+) the exact state vector (with outgoing waves) for the initial state; it satisfies Hya’+’ = Eya’+‘, ya’+’ = Qnc+)x,,

(3)

where AZ+’ = I+ (E-H+iq)-l(H,+H,). Finally, we denote by qb(-’ a state vector for the final state, corresponding

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to the situation with H, = 0. It satisfies (H,+H,)vb(-) a(-)

= E(&)(-‘, = a(-, Xb,

(4

where D-j = 1+ (E--HO--H,--i?j)-rH,. The exact matrix element is given by

(5)

Tab= hlTkJ>

where

T = (H,+H,)Q(+).

This may be transformed into T&b= (P]t,‘-‘jH&&(+)) = (X,,la’-)t H&‘+‘]&.

(6)

The equivalence of the two forms is established simply by noting that the difference, T-i?(-)tH,O’+)

= D WE-H,,

gives a zero contribution to the matrix element, i.e., (&]D]& = 0. we now perform a straightforward reduct.ion of the last form of the matrix element in eq. (6). We write the initial free-particle state containing a K meson of momentum and isobaric spin denoted by a, and a nucleon of specified momentum, spin, and isobaric spin, 1~~) = CILt]XN).For the final freeparticle state containing a pion of momentum and isobaric spin denoted by b, and a hyperon of specified momentum, spin, and isobaric spin, we write, Ix,,) = Cbt]Xy). Then, the IIKhiX element iS given by

Tab = &I

[cb,

[~(-‘%~(+‘P

cat]]

IXN)

=


zl&‘*

(7)

The square brackets denote the commutator. By calculation of the commutator we obtain 2 = [D(-)tH,Q(+), C,t] = D(-)tiz(a)Q(+) --D-)tiz(a){

(E---H+z+)-~--

(E--H--o,+i~)-1}(H1+H2)

+~(-)tH,(E-H+ir)-lj,(a)+~(-)tH,(E-H+i9)-Ij,

(a)

(8)

x (E-H--wa+i~)-l(Hl+H,). The first and second terms of eq. (8) combine to give the following contribution to the matrix element (7): (xylcb~ic-‘ti2(a)(1+

(E,--H+~~)-~(H,+H,)}Ix,)

=
where IN) denotes an eigenstate of H corresponding to the physical nucleon. The last two terms of eq. (8) combine to give the’ following contribution to the matrix element (7):

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BARSHAY

<~~Icb~(-)+H2(E--H+~rl)-1ja(a)(l+(E,--H+~~)-1(Hl+Ha)}I~,) = (cp,,(-)jH2(E-H+iq)-1jalN).

(10)

The complete matrix element is then

(11) This form of the matrix element is particularly convenient if we wish to begin an examination of the problem by treating the K-meson current, iz, as acting only once. If we neglect the K-meson self-field effects in the state vector IN), we have the first term in eq. (11) written in terms of eigenstates of H’ = H,+H,. A perturbation expansion of the second term allows identification of all effects of higher order in jz (including K-meson contributions to vertex renormalizations). We will turn our attention in the next section to an expansion of the first term in eq. (11) in intermediate eigenstates of H’, that is, we start by considering the K-meson current ja, to act once (note, however, the phenomenological modification discussed at the end of this section). We may state this in another way. The exact matrix element is given by (pb(-)IJIN), where J is a new current operator defined by J = jz +H,(E-H+ir])-lj2. We approximate J by jz (as we have, at present, no better representation). The coupling constants that will appear will then be renormalized in the pionic sense, that is, pionic corrections to the vertex function are taken into account by embedding the Heisenberg operators between eigenstates of H’. The term we shall analyze contains many of the effects of the one coupling which we know experimentally to be very strong (as measured by ga/4nM 15), the pion-nucleon coupling. We note, however, that an important contribution from higher order effects in the K-meson interaction could be included phenomenologically in the following manner. Expand the first term of eq. (11) in intermediate eigenstates of the total Hamiltonian and retain the physical baryon and the “one-pion” states, as we shall do explicitly in the next section. Then, approximate all the matrix elements, except those relating to pion-hyperon scattering, by “turning off” the K-meson interactions in the state vectors. This will express all the matrix elements except the above-mentioned ones in terms of the eigenstates of H’. However, matrix elements for pion-hyperon scattering, exact within an approximation which involves incorporating the K-meson self-field effects back into the hyperon eigenstates of H’, will appear in the “one-pion” terms of the integral equation. Since we will resort to a purely phenomenological description of these matrix elements, we can include in the latter description the possible important effects “) of the K-meson interaction on pion-hyperon scattering in the vicinity of the K--p threshold. In section 4, in which we deal in a preliminary manner with the phenomenology of the equations, we shall consider that this approx-

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imation procedure has been carried out and that the pion-hyperon scattering amplitudes that appear in the integral equation can be correlated with the physical pion-hyperon scattering properties t. By standard techniques, we may write (v,,(-)&IN)

= (Ylu,t(E-H’+irl)-lisIN) +
(12)

In this equation, U, and V, refer to the parts of ii corresponding to the interaction of pions with hyperons and nucleons, respectively, IY) denotes a hyperon eigenstate of H’, and we may substitute E = Ey+q., . Eq. (12) satisfies a simple “crossing” symmetry which may be summarized by the following equations: U,t zz ja (in the first term), (13) V,t s ja

(in the second term),

u 1 *Vi# @b -+-c-)b, E N eEEy. The first three equations are, of course, simply more explicit writing of the operation jit * ia. If we denote the operator between the baryon states in eq. (12) by Tx(n) and further, define T,(K) by T,(K)

= iat(E-H’+ir)-lU,+V1(EN--wb--H’)-ljat,

(14

then we have the following unitarity condition: T,,t(K)--x(z) 3. Analysis

= 2ni 2 d(E-E,JT,t(rt)T,(lz). n

of the “One-Pion”

(15)

Approximation

We consider now the approximation of retaining only the baryon and the baryon plus-one-pion states in an expansion of eq. (12) obtained by introducing a complete set of eigenstates of H’ (with incoming waves) between the energy denominator and the current operators. We do this because we would like to make a crude investigation of the effect upon the K absorption t A standard Low equation I), with the exact matrix element written in terms of the operators jr and jr embedded between eigenstates of H, may of course, be written down quite analogously to eq. (12) below. The “renormalized” Born approximations would then contain the completely renormalized couplings. We have explicitly separated off the effects of pion-hyperon couplings induced by the K-meson and pion-nucleon couplings. (However, we have not attempted to calculate such effects in this paper.) The pion-renormalized Born approximations represent the effects of primary pion-baryon couplings. A separation of these effects is possible in a simple, non-relativistic (i.e., cut-off and finite) model. Aside from the model, it is an interesting question to what extent pion-hyperon couplings are induced by the strong pion-nucleon and Kmeson interactions. (However, we have not attempted to calculate such effects in this paper; since we make no explicit use of the separation, the concluding remarks with regard to the pion-renormalized couplings are also applicable if the couplings are considered to have been completely renormalized.)

SAUL

440

BARSHAY

processes of certain assumptions concerning the interaction properties of the pion-baryon system whose scattering amplitudes enter linearly in the “one-pion” terms. The equation for the truncated transition amplitude now reads (~,,~-‘lizlN>

= (Ylu,+lY’>(Y’ll’zlN)(E--E,,+ir)-l

+(Ylj,lN’)(N’l~,+lN)(EN-~~-E~,)-l +(YIU,+lY’,

z(-I)((-)z,

+(Y&IN’,

z(-))((%r,

(16)

Y’linlN)(E-Ey,-~+irl)-l N’IV,+lN)(E,-~,,--w--E,,)-l.

In this equation, hyperon intermediate states (with energy EY,) and nucleon intermediate states (with energy EN,) are denoted by IY’) and IN’), respectively; the pion-hyperon and pion-nucleon intermediate states (with energies E,,+o and E,,+cu) are denoted by IY’, z(-)) and IN’, n(-)), respectively; sums over the various charge combinations, as well as an integration over the intermediate momentum, are to be understood. We perform a decomposition of eq. (16) in isobaric spin space. First consider reaction (la) in which a pion and a Z hyperon are produced; we want the equations for the production amplitudes in the states of isobaric spin zero and one. It is convenient to introduce two auxiliary amplitudes, A and B such that the isobaric dependence of the matrix element we are considering is given by

(P,,(-)lizlW = %A (34,,--t,+r,)+Br,+

4,

(17)

where z, and ty refer to the components of the Pauli isobaric spin operator along the pion and hyperon states appropriate to the reaction (lb) obtained from (la) by “crossing” the pion and K-meson fields. The isobaric spin zero and one amplitudes are then given by A, = $(A-B), We now write the integral (a)

A&,,

b,a) =

P, a)+(

+ (mi,+m)-lA

equations

satisfied

(18)

by the amplitudes

A and B:

(b.a.)A+(1/2n2)/d3p[(d+Wb-W+~~)-1

x~,+(QJ, P, b)D(a X&A

A, =:1/6(2&-B).

P, a)+(Ob-CO+ir)-l((~GCO+(W, $ao+(o,

(-m*,

-_p, a)A(~,

P, b)+&+(w,

P, b))B(w,

P, b)-&+(w

P, b))

P, a)}

P, b)l, (19)

(b)

B(w,,

b, a) =

x (-&)zl+(~,

(b.a.),

+ (l/2n2) ~d3p[(d+w,--w+~~)-1

P, b)D(o,

P, a)+(wb-w+ir)-l((~ao+(w,

++a,+ (WI P, b)) B (0, P, a) + ($a,+ (w P, b) -$q+ xA(o,

P, 4}+(~b+4-1B;(--U)*P

In these equations,

D(o,

-P,

p, a) is an amplitude

a)A(w

P, b)

(m P, b)) P,

b)l.

for the process K+N

+ n+A

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441

(this matrix element is written (vb(-)jjalN) = -D(o, b, a)T,,/l/3); z1 is an amplitude for the scattering of the isobaric spin one part of a n--Z state into a n--,4 state; a,, and a1 are amplitudes for n-Z scattering in the isobaric spin zero and one states, respectively; /?aand #?rare the P-wave amplitudes for pion-nucleon scattering in the states of isobaric spin $ and i, respectively; A is the difference in mass between the Z and A; @.a.) denotes the respective pion-renormalized Born approximations. For the amplitude describing the production of A hyperons we have the following equation:

Dbba b,

a) = @.a.)D+ X +lt(w,

(1~2x2)~d3p[(-d+w,-w+~~)-’ P, b)(B(w, p, a)--A

(W Pa a))+

XE,t(co, P, b)D(w P, a)+ (%+o)-lD(--w*,

x Ah

P,

(~b-cd-id-l

(20)

--PI a)

b)l.

Here, i, is an amplitude for the scattering of a n-A state into the isobaric spin one part of a n--Z state, and 5, is an amplitude for n-A scattering. We now perform an angular momentum decomposition of eqs. (19) and (20) for two possible situations with regard to the relative parities of the strange particles. Case A) The relative Z-A

@a&y is evert and the K meson is psewdoscalar.

We then obtain the following set of integral equations for the production amplitudes “A, YB, ‘D, from an initial P-state K-nucleon system to a final P-state z-- hyperon system (the superscript v = 2j+ 1 (on all of the amplitudes entering into the equations) refers to the state of total angular momentum i) :

(a) lA (mb)= 3(ab/pm){- tab)-lfZnfZK+

(A+%)--%infnd

+(l/n)~~-~d~[(~+~-~+i)-‘(-1/32/8)’21*(w)’D(~~ + (mb-m+ir])-l+{ (2 lao*(~)+lal*(~))lA(~)+(la~*(~)-lal*(~))l~(~)) -(a+,+~)-~&{(

-4183(0)+4383(0))3A(--O*)+(1B3(Lo)+y3~3(0))

xlA(--(0*)}],

(21) (b)

‘A(Ob) = (l/n) ~~~do[(d+~b-Wf~~)-‘(-1/31/6)‘z~*(w)’D(~~) + (ob-++i?j)-‘+{

( 23aO*(~)+3a1*(~))3A(~)+(3a,*(~)-3ar*(~))

442

k)

SAUL

lD(wb)

=

BARSHAY

-~(ab/~m)(3(Wb--))-‘f~,

fx-

(“b)-lffAK)

ob-o+irl)-‘(1/2/32/3)1~~*(o){1B(~)-1A

+Plx)j;PWF+ + (mb--o+i?$‘@i,*

(0))

(0)1D(~)-((Lob+o)-1~((-4181(o)+4%(w))

x”~(-~*)+(181(w)+~381(~))1~(--w*)}], (21) (4

80(ob)

=

+ (lin)~;

2~~(aW,W fidd(

x(3B(t+-3A

(mb)-‘tfAK

--d (m)}+

+ cob-o+ir)-1(2/2/32/3)3~1*(o) (~b-~+i~)-1~1~~*(~)3~(~)

-(~b+~)-‘~{(~‘!%(~)+3i-d(~))3~(--o*)

+(-21B,(w)+2381(w))1D(--w*))l. The equation for lB(cq,) is obtained from (22a) by replacing the Born approximation by

and by the substitutions D--f

-20;

/?a+ @r;

(21a,*+1a1*)1A

(lao*--‘al*)lB

+

(2la,*--la,*)

+

(1a,,*+21a,*)1B; IA.

Similarly, the equation for SB(q,) is obtained from (22b) by replacing the Born apprximation by - 6 (ah/pm) (Wb)-‘ffrk and by the same substitutions as enumerated above, with the superscripts on the amplitudes a, A and B changed from 1 to 3. In these equations, f is the pion-renormalized, unrationalized, pseudovector, pion-nucleon coupling constant, and the other quantities f are similar coupling constants for the interactions indicated by the subscripts. The symbol * denotes complex conjugation. The transition operators are constructed from the amplitudes and standard projection operators appropriate to the transition. For example, for the process K-fp -+ n-+2+, we have, in this case, A(‘%, b, a) = (haWb)-+{%‘i (~b)+

*b U * a

+3A (ob)+(3b . a-u

- b a. a)Xd,,-$+tt,}.

(22)

By the “crossing” symmetry, we may construct from the scalar amplitudes taken at negative values of the energy argument (however, very different in magnitude from those appropriate to R absorption) the transition operators for the associated production processes (1b)t. t An earlier consideration II) of the equations for these high-energy processes came to naught because of the probable necessity of, and the difficulty in, estimating the effects of intermediate states with pairs.

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Case B) The relative Z-A

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443

fiarity is odd ami the relative K-Z

parity is even.

This model has some nice features when applied to K+-nucleon scattering ‘), allowing for both S and P-wave scattering in the non-relativistic limit of the interaction, in a manner that is in qualitative agreement with recent experimental data 6). The set of integral equations for the production amplitudes now reads (a)

lA

kb)

=

-

@/p)

(%)%n&,

~,,-~+i~)-1(-1/,,6)oz~*(~)o~(~)

+W$&W(~+

+(Ob-O+ir)-1~{(21ao*(o)+1ac,*(w))1A(~) -l-(lao*(~)-l~l*(~))

lB(o)}-((ob+o)-l~lB,(w)lA

(-w*)],

wb-0+i~)-1(-1/32/6)1~1*(co)1D(co) (b) OA@I,) = MC)j+;fid43~+ + (‘%--w+ir])-l+{

( 2o ao* (0) +Oa,* (0)) OA(w)

+(“ao*(~)-oa~*(~))oB(~)~l~

(cl “D(ob)= x

-1/3(wb-d)-‘g~~g=,+(l/~)~~~d~

[(--d+~,,--w+i~)-~x’~l~I*(o){lB(ci+-lA(w)}

(23)

+(O~-co+iq)-%i,*(o)OD(w)],

(d) ‘D&b) = X

--d/3(ab/w)(%)-lff,d- (l/n) ~~~d~[(-d+mb-w-t6)-1

%&“~I*(o){oB(o)-oA

(O))+ (wb-~+irl)-“3’~,*(o)‘D(w)

-(~+~)-1&{(-4181(~)+4381(~))3D(--cr)*) +(‘Bl(~)+~3Bl(~))1~(-~*)llJ

(e)

30(mb)

=

2~(ab/v)

(wb)-%K+

(lin)~~

#dw

X [(~b-~+irj)-1~3~~*(~)3~(~)--~b+~)-1~{(8181(~) +3Bl(o))30(-0*)+(--2181(0)+2381(0))1D(--0*)11.

is obtained from (23a) by replacing the Born

The equation for lB(q,) approximation by

2@/p)

(~b)%K{fZn-if)

and by the substitutions D + -20;

B3 -+ &; (lao*-lal*)

(21ao*+1aI*)1A + (lao*+2L1*)lB; lB + 2(lao*-la1*)lA.

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BARSHAY

Finally, the equation for OB(O~) is obtained from (23b) by replacing the Born approximation by (a/m) (d +wr,)-lgnnfAx, and by the above substitutions with the superscripts on the amplitudes a, A and B changed from 1 to 0. The pion-renormalized, unrationalized, scalar couplings are denoted by g with appropriate subscripts. In eqs. (23), the superscripts on the amplitudes A, B and tc, only, refer to the orbital angular momentum of the pion-hyperon state (in the case of a P state, j = $). In eq. (23c), the superscripts on the amplitudes D and & also refer to the pion-hyperon orbital angular momentum. The superscripts on the amplitudes z1 and i, refer to the orbital angular momentum of the pion-hyperon state produced by the amplitude multiplying z1 and zl, respectively. 4. Discussion We limit ourselves to a few, perhaps somewhat academic, remarks, in the light of the present crude approximation J M jz, and also in the light of the somewhat unclear experimental situation with regard to the production processes in low-energy K--p interactions. Let us first outline our picture of the latter situation. We assume that the original data 6, on K--p captures represents predominantly interactions from initial S Bohr orbits, and therefore that the observed branching ratios s), Z- : Z+ : Z” :-A=4 : 2 : 2 : $, will be continuous with those to be observed in K--p interactions at the bottom of the continuum (perhaps with a suitable allowance made for some at-rest captures from P Bohr orbits). We then consider two interesting variations as the K- energy is raised. Perhaps the most striking is that inferred from a number of emulsion studies ‘v8), namely that the “direct” II production, so infrequent in captures at rest (contrary to simple phase space considerations), appears to rise sharply with energy, becoming comparable with the total charged ,Z production at K- kinetic energies of the order of 100 MeV. Secondly, there appears to be a variation over the first 100 MeV energy interval in the Z-/Z+ ratio ‘sg), from a value of w 2 in captures at rest to a value of M 1. Consider now the situation A). The non-relativistic reduction of the pseudoscalar interactions of pions and K mesons leaves us only with equations for the transition amplitudes from initial E-nucleon P states to final pionhyperon P states. We may, under the assumption stated above, take the S-state to S-state branching ratios from experiment (as we have no theoretical feeling for the S-wave effects induced by the operator ys) and attempt to understand the increasing A production as a feature of an increasingly significant P-wave interaction. The variation in the Z-/X+ ratio may be partially due to a suitable energy dependence 10) in the S-state transition amplitudes, but one would also expect the P-state transition amplitudes (if

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446

these are also of increasing importance in ,Z production) to confirm the tendency of this ratio to be M 1. In perhaps the simplest situation (since there is no “a priori” reason Why similar renormalized pseudovector coupling strengths should differ markedly), in which we take IfrJ w Ifn,J w IfI and j/rxI M I/J, one finds that a crude examination of eq. (22) (under, for example, the “global” symmetry hypothesis that the important pion-baryon scattering amplitudes, other than 3/?3,are likely to be the pion-hyperon amplitudes in the states of isotopic spin one and angular momentum $) suggests that, if the P-Wave interaction is indeed responsible for an increasing A production, then a non-negligible P-wave ,Z production is also likely; and that this production strongly favours Z- over L’+, except in the case fnr fzK

= ffm

f&l = -f*

In case B), we have found a slightly better reconciliation of the two variations with energy which we are considering. In this case, the very low energy continuum transitions are from. an. initial $-nucleon S state to a final n-2 P state or a final z---li’S’ state. A crude examination of the eqs. (23), under the assumption that the important’integral term involves. the mixing amplitude zr, suggests that, if fz,,m f, and if the pion-renormalized scalar coupling, gAn, is about an order of magnitude smaller than the pionrenormalized pseudoscalar coupling g, corresponding to f, then the at-rest branching ratios L’- : L’+ : Z” M 2 : 1 : 1 are roughly obtained (from the Born approximation), with strong suppression of A production. Further, if we choose the pion-renormalized scalar K-meson coupling, gz,, to be about an order of magnitude smaller than the pion-renormalized pseudoscalar coupling, g,x, corresponding to fifK, then, at about 60 MeV K- laboratory kinetic energy, the P-state II production can become comparable with the total charged Z production. At this energy, there is sufficient absorption of P-wave K leading to S-wave z-Z systems (with these transitions characterized by a .?-/Z+ ratio < 1) to reduce the overall Z-/Z+ ratio by perhaps about 35 %, in the crude approximation in which we neglect any energy variation in the ratio characteristic of the S-state to P-state transitions. A further remark may be made in case B) with regard to the hyperon angular distributions and polarizations. The above crude estimates of the amplitudes suggest that, either the and Z+ and Z- angular distributions in the centre-of-mass system are of the form IAIf IB[ cos 8, respectively, at those energies where li production has become significant, or, if these angular distributions are largely isotropic, then the charged hyperons must have a significant polarization induced by the production process. Absence of both angular anisotropy and evidence for polarization in charged ,Z production at energies where LI production has increased would weaken an attempt to understand the latter effect as due to significant P wave K-nucleon interactions.

446

SAUL

BARSHAY

In conclusion, we remark that a better representation of the operator J, embodying higher order effects in the strong K-meson interaction current, might make the Low-type field equations discussed in this paper useful models for correlating the accumulating experimental data and the purely phenomenological considerations concerning the absorption of R mesons by nucleons. From the experimental side, it would be useful to know whether the energy variations we have discussed here in a preliminary way are indeed significant, i.e., whether the d production increases markedly over the first 100 MeV in K--p interactions, and whether the Z--/Z+ ratio tends to unity in the same energy interval, and further, whether the branching ratios at the bottom of the continuum are continuous with those at rest. I would like to thank Professor Niels Bohr, Professor Aage Bohr and Professor C. Merller,as well as the other members and guests of the institute for Theoretical Physics for their kind hospitality and stimulating interest during my stay in Copenhagen. I would like to acknowledge my debt to the National Science Foundation for fellowship support. References 1) F. Low, Phys. Rev. 97 (1955) 1392 2) G. F. Chew and F. Low, Phys. Rev. 101 (1956)1679 3) R. H. Dalitz and S. F. Tuan, Annals of Physics (to be published) 4) S. Barshay, Phys. Rev. Letters 1 (1958)97 5) Keefe, Kernan, Montwill, Grilli, Guerriero and Salandin, Nuovo Cimento 3 (1959) 241 6) Alvarez, Bradner, Falk-Variant, Gow, Rosenfeld, Solmitz and Tripp. U.C.R.L. 3775 (unpublished) 7) Eisenberg, Koch, Lohrmann, MikoliC, Schneeberger and Winzeler, Nuovo Cimento 9 (1958) 746 8) Eisenberg, Koch, Mikolic, Schneeberger and Winzeler, Nuovo Cimento (to be published) 9) Preliminary data kindly supplied to the author by Professor A. Rosenfeld of Berkeley 10) R. H. Dalitz, Report to the 1958 Conference on High-Energy Physics at C.E.R.N. 11) S. Barshay, Phys. Rev. 107 (1957) 1464; 108 (1967) 1648