JP = 12± baryon resonances and nonleptonic hyperon decays

I fJ> = aw

(2.20)

I u> = kc3cp> Y54-4

Y544,

(2.21)

where the g,, are WB pseudoscalar coupling constants. Henceforth we take SU, symmetric values for the g,, and fPo which means that f: = 0 so that the pole terms contribute to only the parity-conserving amplitudes (8) whereas the equal-time commutator term (2.5) contributes to only the parity-violating amplitudes (9). Then the pole terms for the scattering process 01+ q + /3 + 7~are Tpole = T L?s&W3

s - MB2 [ (Mf + MB)(s - ME2 + k) +

s - iili2k+

ir

1y5u(a)

NONLEPTONIC

HYPERON

151

DECAYS

Finally, when we set q = 0 we get

as the contribution from the octet baryon poles to the parity-conserving amplitude of the decay 01+ /3 + n-. The equal-time commutator and octet baryon pole terms for the decay (Y-+ jI + rr are listed in Table I. We have expressed the SU, symmetric pseudoTABLE

I

EQUAL-TIMECOMMUTATORAND OCTET BARYONPOLE TERMS

Decay

AE.T.C.

A-0

q

x

(--D

B Polk? x

t-f;“)

-

3F)

q

(-D

-

%'zg,NN

MA-I-MN

3F)

-

~MN(MA

+ q z++

0

CD -

?(D

- F)

(MA

+

-

2M~cM.r

z--

D-F

(’ -

‘jcD

v)(D ~(-0

-

+ d\/g ~(-0 --3-

+(-D

+ 3F)

d$

T(D + F) + ”

-

ME)

-

MN) Mz+MN

F)

- Mz) ME+MN

=&(MN

+ d$

MN

+ Mz)(MN

Mz+MN

0

+ (1 -

MN)

MI

(1 -

-

3F)

(ME

Mz+MN F)2Mz,(MN -

+

M&MN

-

MA)

-

MA)

Mz) Mz+MN

3F)

(ME

+ MA)(MN

ME + MA (MA + M&ME - ML) 27x0

-

3F)

ME ~Ms&'/I

+ MA - ME)

scalar coupling constants in terms of gnNN and 77 (our 77 corresponds to GellMann’s (10) 01)and the fg in terms of the SU, parameters D and F. The unlisted amplitudes can be obtained from the 1LIZ / = + sum rules y.0 A00 + (1-O = 0, .z-

+ 1/2 ‘x0:0’ = Z+f, 2/T Eo” + E- = 0.

(2.24) (2.25) (2.26)

152

TROFIMENKOFF

111.PION-OCTET BARYON INTERMEDIATE

STATES

The P-B intermediate states corresponding to diagrams (a) and (b) of Fig. 2 arise in the s and u channels, respectively. In general, there is more than one possible intermediate state for each mode 01+ 4 + p + ?r. For the present, we pretend that there is only one such state; later, we perform the relevant isospin decomposition. In order to simplify our calculation we consider the pion in the intermediate state to be massless and take the mass of the intermediate octet baryon in the S(U) channel to be equal to the mass of the final (initial) baryon.

I 6 I1’71 S

, I I IV I

p W

3

a

I

*.....q ...

(a)

lb)

FIG. 2. Diagrams for pion-octet baryon intermediate states in (a) the s channel and (b) the I( channel.

Consider the s channel. As may be inferred from diagram (a) of Fig. 2 or from Eq. (2.14) the strong vertex corresponds to the T-B scattering process p + X’ -+ /3 + z-; the weak vertex corresponds to the weak nonleptonic process 01+ q + p + T/, that is, to the type of process for which we are calculating a matrix element. This means that the dispersion relations (2.12) and (2.13) yield a set of coupled integral equations for the weak nonleptonic amplitudes. We do not attempt to solve these integral equations exactly; instead, we make an approximation for the weak amplitudes of 01+ q + p + rr’ appearing in the imaginary parts of the amplitudes for 01+ q - /3 + z- in terms of the equal-time commutator and octet baryon pole terms found in Section II. The approximation that we make is more reliable than taking a first iteration of the integral equations. Using Eq. (2.14), we now find expressions for the imaginary parts of the amplitudes in the s channel. To explicitly maintain the reality of the Im Ti(s, t) we use the trick due to Goldberger and Treiman (II) “out” and an “in” intermediate state / n>.

of taking one-half the sum over an

NONLEPTONIC

HYPERON

153

DECAYS

Then Im T(s, t) = - q

Re

c s4(pB + k - P,, - U P,.k’

spin p

x (/3 / @‘c’(o) j p + d; out)(p

In the notation

+ 7r’; out I --4&L(O)

I a:.

(3.1)

of Gasiorowicz (12) we have2 I p + T’; out) = @)[-a*

where B and B are Z--B scattering amplitudes. into Eq. (3.1) we get

+ jir * (k + k’) i?*] u(p) Substituting

g

(3.2)

Eq. (3.2) and (2.8)

S4(ps t k - p,, - k’)

(3.3)

where (B4, (Pp), and (pa) indicate the baryons and consequently the momenta involved in each amplitude. The sum over the spin of p and the integrations over momenta can be easily performed after going to the r-/3 center of mass system and expressing all the amplitudes in terms of the partial-wave expansions given in the appendix. The result is

Imfi,(b4 = I k I Re[=~@p)hJp41,

(3.5)

where

=z*=&

eisz* sin 6,* ,

are the T-B partial-wave scattering amplitudes and gL+(fiJ are the weak nonleptonic parity-conserving (parity-violating) partial-wave amplitudes. In order to evaluate the Im gl* and Im& exactly we would need to know the explicit dependence of the phase shifts a,+ on the center of mass energy W. Since 2 We use the symbol N in A and 17 only to distinguish the n - B scattering amplitudes from the weak amplitudes A and B of Eq. (2.16).

154

TROFIMENKOFF

these phase shifts are not well known and since we are specifically interested in baryon resonances with .P = g*, we make the following approximations: (1) We neglect all inelastic effects in T-B scattering so that we can take the phase shifts to be real. If we had used only “out” intermediate states to evaluate Eq. (2.14) we would have obtained

(3.8) instead of Eq. (3.4) and (3.5). This means that when the phase shifts are real we can take g&4

= eis%?&4

(3.9)

f&4

= eis”4+(P4,

(3.10)

where tl, and jr, are both real (this is just the usual final-state interaction theorem). Therefore, we can write Im s&W

= sin(h+) &,(P)~

(3.11)

Im.h,(/W

= sin@t,) f&d

(3.12)

(2) We make the resonance approximation sin 6,* = 1

for

MI, - ir,*

sin 6,* = 0 otherwise,

< W < Ml* + Br,*

(3.13) (3.14)

where Mz+ and r,* are the masses and widths of resonances in the corresponding n-B partial-wave scattering amplitudes. More realistic forms for sin 6,+ at resonances, say, Breit-Wigner forms, could be used; however, this choice greatly simplifies our calculations and, furthermore, should not seriously affect our conclusions. (3) We consider resonances in only the Of and 1 - partial-wave amplitudes, that is, only resonances with J p - +*. The reason for neglecting resonances in the other partial waves will be discussed in Section V. We now approximate the parity-conserving amplitudes g!,(pol) by the octet baryon pole terms (2.22) and the parity-violating amplitudes fz,(pol) by the equaltime commutator term (2.5). In making these approximations we take the pseudoscalar coupling constants g and thef” to be independent of momenta. Furthermore, in the pole terms (2.22) we take Mt = M, and M, = M, and set s = Me2 and

NONLEPTONIC

HYPERON

DECAYS

155

u = Mo2. Denoting by T&XX) the amplitudes which correspond to the partial-wave amplitudes &,(pol) and jr,&), we now have the following approximations:

U(p)T&.4 44 = [E.T.C. (~41,

(3.15)

(3.16)

T2(P4 = 0,

(3.17) (3.18)

Note that approximating gl* and fi, rather than & and j,* by the equal-time commutator and octet baryon pole terms would correspond to taking a first iteration of the integral equations resulting from the dispersion relations. However, such a first iteration is unlikely to yield a good approximation to the solution of the integral equations because the equal-time commutator and octet baryon pole terms do not have the correct phases shown in Eq. (3.9) and (3.10) (this is particularly obvious at a resonance where the real parts of gl, andf,* should be zero). Our procedure is an improvement over simply taking a first iteration because it corresponds to first assigning the correct phase to the equal-time commutator and octet baryon pole terms and then taking a first iteration of the integral equations. Using the approximations discussed above, explicit expressions for the Im g,*(Pa) and Irnfr,(j&x) can now be found. The partial-wave decompositions given in the appendix can then be used to assemble the Im g,,(j?ol) and Imf&(,%) into expressions for the Im TJs’, 6 = 0) which are to be used in the dispersion relations (2.12) and (2.13). By this procedure we get the following expressions for the Im T&s’, t = 0): Im Tdb)

= &w [( W + MB) sin so+ f ( W - M,) sin 6,-l T&cu),

Im T2(@) = &

(3.19)

[(W + MB) sin 6,+ + (W - MB) sin 6,-l T2(pcy)

i- +w [--(W2

- MB2) sin a,+ + (W’ - MB2) sin a,-] F&XX),

Im T,@ol) = GW [- sin 6,+ + sin a,-] ZF1(pn),

(3.20) (3.21)

Im T&%X) = 2w -!- [- sin S,, + sin S,-] T2(pcy) + ew[(W-

M,)sinS

o++ (W + MB)sin h-1 ~h4,

(3.22)

156

TROFIMENKOFF

where we have introduced the notation W = 1/Z. The expressions (3.19)-(3.22) can also be inferred from the absorptive amplitudes used by Okubo et al. (13) in a final-state interaction calculation. In the final instance we are interested in obtaining the real parts of the hyperon decay amplitudes (the imaginary parts are small) at 4 = 0. Substituting the Im Ti of Eq. (3.19)-(3.22) into the dispersion relations (2.12) and (2.13) and setting q = 0, we find the following contributions to the amplitudes A and B of Eq. (2.17) and (2.18) from baryon resonances in the s channel: AR = A:+,(R) T(p)

+ &WI

T&.x4,

(3.23)

(3.24)

where

(3.25)

The limits of integration

in the A,+ , Bt_ , and C,, are Ml& -

tr,&

< W < M,, + &,

(3.26)

and R denotes the resonance with mass Ml* and width r,+ . To obtain Eq. (3.23)(3.26) we have taken into account the approximations (3.13) and (3.14) and the fact that the T&CL) of Eq. (3.15)43.18) are independent of W. We now consider T-B intermediate states in the u channel which corresponds to diagram (b) of Fig. 2. Since Im T(u, t) of Eq. (2.15) is in the unphysical region we follow the standard procedure of using a crossing relation. Crossing gives

NONLEPTONIC

the following relation a + (-q) -+ /3 + +(--k)

HYPERON

between the matrix and p + q--f 01+ I:

out j --iHp&O)

1CL>= -(a

157

DECAYS

elements

+ x(-C)(k);

for

out ) -i&~(o)

the

processes / 8)“. (3.27)

In order to evaluate the u channel contribution to 01+ q---f /3 + T(C) we first calculate the corresponding s channel contribution to /3 + q + 01+ T+) and then use the crossing relation (3.27). Using this procedure and approximations for the strong and weak amplitudes similar to those made previously for the s channel, we find the following contributions to the amplitudes A and B of Eq. (2.17) and (2.18) from Jp = +* baryon resonances in the u channel:

+ C,U,{R) ~@u> + C,U_{R’, T,(/~u).

(3.29)

The A”, B”, and C” can be found from the corresponding A”, B”, and C” given in Eq. (3.25) by making the interchange M, tf M,. The T((j3u) in Eq. (3.28) and (3.29) are defined by ii(fl) 7=&b) u(u) = [E.T.C. (/3u,],

(3.30) (3.31)

T,(Bo) = 0,

(3.32) (3.33)

Since the phase shifts S,+ in Eqs. (3.19)-(3.22) and the corresponding u channel expressions refer to T-B scattering amplitudes with specific values of isospin we must perform an isospin decomposition on the weak amplitudes. The resonances that we include in the s channel are, in the notation of Rosenfeld et aE. (14), N(1550) and N’(1710) with Jp = &- and Z = j-, A(1640) with Jp = a- and Z = #, and N’(1470) with Jp = 3+ and Z = + . In the u channel we include only A(1405) with Jp = t- and Z = 0 in V-Z scattering. Performing the isospin decomposition on the weak amplitudes by the standard methods, we finally get the following

158

TROFIMENKOFF

contributions p = +*:

to the hyperon decay amplitudes

from baryon resonances with

A&Lo)

= [&+{N(1550)}

+ A;+{N’(1710)}

+ &{N’(1470)}]

B&Lo)

= [B;+{N(1550)}

+ B~+{iv’(1710)} + B;-{N’(1470)}]

T&4-“), T&Lo)

+ {B' -+ C", T2 --+ i=..},

-4G++) = 4i+@wwwl(~-->1 + L4,8,W(l550)]

+ L4~+;(~‘(1710)]

+ A~-{W(147O)}][T~(Z+‘)

- $T&z--)I

+ ~0”,(~(1405)][!&(~++)

+ BT,(L-)l,

B&2'++) = B~{~l(1640)}[9~&-)] + [B,+W(1550)}

- +T2(Z--)]

+ Bo”,Wl405)H~~G++)

+ @zG--)I

(3.34)

C", B"--+ C", i'2 -+ T4),

= A,“+{A(1640)} T.(Z--) + ~;+:,(~(1405)>Wl(~++)

B&m-)

1O)}

+ B;-{iV’(147O)}][~~(.c,“)

+ (B8+ A&E-)

+ B;+W’(l7

= B;+{41640)}

+ B~lG--)l,

T,(zm-)

+ Bo”,Wl405)>[t~0++) + {B* -+ C', B"-+

+ iiT@--> C", T2 -+ F4}.

The decays K- and Eoo do not receive any contributions from the resonances considered here and the other unlisted amplitudes can be found from the 1Al 1 = 4 sum rules (2.24) and (2.25). IV.

NUMERICAL

RESULTS

The total S-wave amplitudes now consist of the equal-time commutator terms listed in TabIe I and the contributions AR from resonances given in Eq. (3.34). The total P-wave amplitudes consist of the octet baryon pole terms in Table I and the BR of Eq. (3.34). Note that in including the resonances we have introduced

NONLEPTONIC

HYPERON TABLE

A COMPARISON

OF EQUAL-TIME

COMMUTATOR EXPERIMENTAL

159

DECAYS

II AND OCTET AMPLITUDES

BARYON

POLE TERMS WITH

S

P

-_-Decay A-Q z++ .?-Em-

__Experimental +1.ooo +0.005 t1.200 -1.304

* i + &

[E.T.C.] 0.015 0.022 0.011 0.019

f1.00 0 +2.10 -1.78

Experimental +0.380 +1.201 -0.010 +0.265

i + i f

Pole 0.016 0.022 0.025 0.023

f0.299 +0.925 0.000 to.209

no parameters, except the masses and widths of resonances, that do not appear in the equal-time commutator and octet baryon pole terms. The evaluation of the AR and BR is now a simple, although a tedious task; therefore, we give only numerical results. In order to show the effect of including the resonances with .Jp = $* we make two comparisons with the experimental amplitudes. In Table II we neglect the contribution from the resonances and retain only the equal-time commutator and octet baryon pole terms (these results are almost the same as those given by Itzykson and Jacob (3)). In Tables III and IV we include the resonances and also display the contribution from each individual resonance. The S- and P-wave amplitudes in Tables II-IV are defined by S = A,

(4.1)

’

(4.2)

and we have normalized the experimental amplitudes which are taken from Cabibbo (1.5) such that the S-wave amplitude of (1-O is unity. Also, we have normalized S(L”) to unity in both sets of calculated amplitudes. To obtain the results in Tables II to IV we have taken (12)

with (16) GA = 1.253

(4.4)

160

TROFIMENKOFF

and masses and widths of resonances from Rosenfeld et al. (14) and have chosen the following values for the unknown parameters: D - = -0.845, F

(4.5)

7 = 0.635.

(4.6)

These values of D/F and 7 give about the best fit to the experimental amplitudes; the fit gets worse very rapidly as D/F or 7 is varied. A comparison of Tables II, III, and IV shows that including baryon resonances with Jp = Q* does not substantially affect the fit to the experimental amplitudes. However, interesting features become evident from an examination of the contribution from individual resonances displayed in Tables III and IV. TABLE

III

CALCULATED S-WAVE AMPLITUDES INCLUDING BARYON RESONANCES‘WITH JP = -&*

Decay

[E.T.C.]

N(1550)

N’(1710)

N’(1470)

A(1640)

A(1405)

A-0 -T+ z.e

+0.940 0 t-1.969 - 1.676

+0.026 -0.032 0 0

+0.036

-0.002 +0.002 0 0

0 +0.031 +0.093 0

0 -0.039 -0.039 0

-0.041 0 0

Total +1.000 -0.079 +2.023 -1.676

The dominant feature of the S-wave amplitudes given in Table III is that individual resonances with Jp = +- yield contributions which are an order of magnitude larger than those from N’(1470) which has Jp = &+. Perhaps it is best to compare the contribution from N’(1470) (F = 210) with that from N(1550) (r = 130) and N’(1710) (F = 300). Then it is evident that the larger contributions from the resonances with Jp = &- arise even though these resonances have larger mass than N’(1470) and comparable widths. That the feature that the contribution from Jp = +- resonances should be substantially larger than the contribution from Jp = ++ resonances should arise can be inferred from an examination of Eq. (3.23) and the structure of the A,, and A,- given in Eq. (3.25). On the other hand, for P-wave decays, Table IV shows that when there is a substantial contribution it comes from the Jp = ++ resonance N’(1470) (in L’++ decay) and it is an order of magnitude larger than the contribution from individual Jp = +- resonances. As may be expected, the major portion of this contribution comes from the term in Eq. (3.24) which involves C,- .

NONLEPTONIC

HYPERON

TABLE CALCULATED

Decay A-0

z++ z---P-

P-WAVE

Pole +0.2so5 f0.8690 -0.0001 +0.1966

AMPLITUDES

161

DECAYS

IV

INCLUDING

BARYON

RESONANCES

WITH

Jp = i=

N( 1550)

N’(1710)

N’(1470)

A(1640)

A(1405)

Total

-0.0067 -0.0102 0 0

-0.0121 -0.0138 0 0

+0.0131 +0.2426 0 0

0 +0.001s +0.0055 0

0 -0.0085 -0.0085 0

+0.2748 + 1.0809 -0.0031 +0.1966

V. CONCLUSION

An investigation was made of baryon resonances with Jp = i* in nonleptonic hyperon decays with a view to establishing a criterion for neglecting or retaining particular resonances. The investigation was carried out within a dispersion relation formulation of the CA-PCAC method which has been developed by Okubo et al. (4) and applied to nonleptonic hyperon decays by Okubo (4). In previous calculations resonances have been treated in pole approximations (4, 5). Here, equal-time commutator and octet baryon pole terms were first calculated and then pion-octet baryon intermediate states were included in the calculation of the imaginary parts of amplitudes for the dispersion relations. A resonance approximation was then used for the pion-octet baryon scattering amplitudes, and the equal-time commutator and octet baryon pole terms were used in an approximation for the weak nonleptonic amplitudes which appear in the imaginary parts of the amplitudes when pion-octet baryon intermediate states are included. The resuhing dispersion integrals were evaluated and the contribution to the hyperon decay amplitudes from the known baryon resonances with Jp = 4’ was given numerically. What conclusions on the relative importance of Jp = +* baryon resonances in nonleptonic hyperon decays can be reached on the basis of our results ? Firstly, it is evident from Tables III and IV and the discussion in Section IV that, in general, the contribution to S-wave (P-wave) decays from a resonance with Jp = &(Jp = ++) is an order of magnitude larger than the contribution from a resonance with Jp = &+ (Jp = &-). Therefore, if the contribution from baryon resonances is calculated in a pole approximation it is safe to neglect Jp = ++ (Jp = &-) resonances in S-wave (P-wave) decays. Secondly, several calculations (.5,18) have been made in which /1(1405) is retained but the other Jp = $* resonances are neglected. Our results show no evidence whatever that the contribution from (1(1405) dominates that from the other Jp = 8* resonances and, 595/55/I-11

162

TROFIMENKOFF

therefore, retaining only fl(l405) cannot be supported. Thirdly, in our results the contribution from each individual resonance is quite small except from N’(1470) in Z++ P-wave decay. It has been pointed out (6) that in the case of “universal spurion coupling” (19) the octet baryon pole terms for Z++ decay are much too small to account for the experimental P-wave amplitude. Our results suggest that N’( 1470) may enhance the P-wave amplitude of .Z++ sufficiently to overcome this difficulty. The reader may well ask if our conclusions on the relative importance of resonances with Jp = Q* are any more reliable than those that could be reached by examining the structure of a pole approximation. In pole approximations the coupling constants at the weak vertices associated with the baryon resonances are unknown and, consequently, the relative importance of resonances can be judged only if a reasonable guess at the magnitudes of these coupling constants can be made. The corresponding unknown quantities in our calculation are the weak nonleptonic amplitudes which appear in the imaginary parts of the amplitudes; corresponding to the guess at the magnitudes of the coupling constants is our replacement of the weak amplitudes in the imaginary parts by the equal-time commutator and octet baryon pole terms. At present there is no method of estimating the weak coupling constants for the pole approximation and, therefore, our calculation would yield information which is more reliable if our method of using the equal-time commutator and octet baryon pole terms provides a reasonable approximation to the weak amplitudes even at the resonances. An examination of the typical structure of amplitudes obtained in calculations of final-state interactions (13, 17) leads us to believe that, although our method of using these terms may not give the correct absolute magnitudes, it gives roughly the correct relative magnitudes at the resonances and, therefore, our method of estimating the relative importance of baryon resonances with Jp = &* is more reliable than examining the structure of a pole approximation. Because of the numerous approximations made in our calculations the resonance contributions listed in Tables III and IV should be considered as rough estimates. Therefore, it may be worthwhile to carry out a final-state interaction calculation of the type performed by Okubo et al. (13). With recent data on pion-nucleon scattering it should be possible to calculate more realistically the effects of finalstate interactions and, at the same time, to check on the relative importance of resonances with Jp = 4’ in A and Z decays. Furthermore, such a calculation may yield evidence that the Z+ + P-wave amplitude is in fact enhanced by N’(1470). Throughout this article an important class of low-lying baryon resonances, namely, the decuplet resonances, was neglected. With our approximations for the Ti , Eq. (3.15H3.18), there is no contribution from the decuplet resonances because there is no dependence on angles in the Ti . Even if u was not set equal

NONLEPTONIC

163

HYPERON DECAYS

to MO2 to obtain Eq. (3.18) the contribution from the decuplet resonances would be extremely small because at energies near these resonances the weak nonleptonic partial-wave amplitudes &+ would be small. In order to get any substantial contribution from the decuplet resonances (or any other low-lying baryon resonances with Jp # 8’) the f,,, would have to be assigned some momentum dependence. The correct momentum dependence is beyond our present knowledge. APPENDIX PARTIAL-WAVE

DECOMPOSITIONS

For the T-B scattering amplitudes A, B, and aI., found in Eq. (3.1)-(3.6) we use the partial-wave decomposition provided by Gasiorowicz (12). For the weak amplitude T of Eq. (2.8) we write

T = GW’l +

T2y5

d&M-

f iy * W3 + T4y5)144

x*[F’,

+

F2

=

- h

= - i

+

G,

Q * $ +

G,

Q1f;l x,

(A.l)

B a where A denotes a unit vector, in the center of mass system. Then for the parityconserving amplitudes we have

(W- MdG,

-(W + M,) G1

T,=--l[__

2 w d(Es F%i,)(E,--M,)

+ “?EB - MW,

7,=1 [ ---- Gl 2137 +o + M,WL - Ma)+

1 1’

+ MJ ’

G2 d(E8 - MB)(EoI+x

(A-2)

(A*3)

and the inverse relations

G, = h% + M,Wa - Mm)l--T2 + (W - MA TJ,

(A.4)

G, = +B - W&E, + Ma) P”2+ W’ + MJ TA.

(A.9

The T, and Gj are functions of the total center of mass energy W and of h with x =

came

(A.@

where 8 is the angle between k and q. We can expand G1 and G, as GdK

4 = f’ a+( W f’;+l@> l=O

G2VK

f gz-W’) K-d@,

4 = f k-t W>- g~+W’)lP,‘GV, Z=l

(A.7)

2=2

WV

164

TROFIMENKOFF

where gl,( W) are the partial wave amplitudes,

(A-9) The corresponding relations for the parity-violating amplitudes by making the following changes in Eq. (A.2) to (A.9):

can be obtained

(A. 10)

ACKNOWLEDGMENTS

The author thanks Dr M. Nakagawa and Dr S. Fumi for conversations on this work, and Professor K. J. Le Couteur for his hospitality. RECEIVED: April

4, 1969 REFERENCES

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