Soft-pion production and algebra of currents

[ 8.A.1

I

i

I

Nuclear Physics 87 (1967) 581--591; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by p h o t o p r i n t or microfilm without written permission from the publisher

SOFT-PION

PRODUCTION

AND ALGEBRA OF CURRENTS s. 1WAd

lnstitut fiir theoretisehe Physik der Universitiit Bern, Bern, Switzerland t

Received 3 August 1966 Abstract: The production of two soft pions in collision processes are here investigated under the

hypotheses of the algebra of currents and the partially-conservedaxial-vector current. The condition of softness is made to meet with the experimental situation. A general formula to estimate the two soft-pion production is derived. A scalar part as well as the normal isospin part of the reduction amplitude are included in this formula. The calculated cross section is large enough to be measured experimentally. The baryon-baryon collision and the production of double soft pions are studied in detail. Assuming the SU(3) symmetry for baryon systems, we get many sum rules and selection rules. The non-go selection rules are classified into three classes according to the forbiddenness of the scalar and isospin part of the reaction amplitude, viz. (i) no scalar part and forbidden isospin part, (ii) allowed scalar part but forbidden by the isospin part, (iii) completely forbidden by both scalar and isospin part.

1. IntroducUon

T h e idea o f an algebra o f currents 1) which manifest the s y m m e t r y o f elementary particles has been d e v e l o p e d in the language o f q u a n t u m field t h e o r y 2), leading to a successful d e r i v a t i o n o f the axial-vector c o u p l i n g - c o n s t a n t r e n o r m a l i z a t i o n 3) in the n e u t r o n b e t a decay. In this d e r i v a t i o n the hypothesis o f the partially-conserved axialvector current 4) ( P C A C ) a n d a suitable limiting procedure, so-called soft-pion s) limit, were utilized. U n d e r the similar hypotheses it was f o u n d a n u m b e r o f sum rules lbr all three types o f interactions 6) (strong, weak a n d electromagnetic ones) *t. W e expect even m o r e useful relations by this a p p r o a c h . Recently W e i n b e r g 7) has discussed multiple s o f t - p i o n p r o d u c t i o n by assuming exactly conserved axial-vector current ( C A C ) . H e gave some criteria for the softness o f the pions a n d some selection rules involved in the equal t i m e c o m m u t a t o r o f the currents. However, we feel that his discussion is t o o general to be a p p l i e d to the real physical situation. W e shall confine ourselves to the p h e n o m e n a where the softness c o n d i t i o n can easily be a p p l i e d a n d seek results which could be tested experimentally. F o r this p u r p o s e we restrict ourselves to the process in which only two soft pions are involved. G o o d examples in this c a t e g o r y will be the d o u b l e p i o n p r o d u c t i o n in a b a r y o n - b a r y o n collision, BB -~ BB2n (where B represents a b a r y o n ) , further N N ~ 4n a n d K K 2n in which two o f the pions are soft ones, etc. *) This work is supported in part by the Swiss National Science Foundation. t+) Here we cite only the papers related to the hadronic decay of hyperons. 581

582

s. IWAO

Some years ago Shrauner 8) had discussed the single soft-pion production in the pion-nucleon collision at 1.3 GeV lab pion kinetic energy. He obtained a good explanation of the angular distribution of the fast pion but the total cross section seemed to be underestimated by a factor of 7 under the assumptions he made *). Weinberg 7) has interpreted this as well as the work of Nambu et al. 5) as a sign of the CAC. Shrauner argued at that time that this large discrepancy might be caused by the neglected symmetry-beaking effect, such as an S-wave pion-pion interaction. If instead we consider double soft-pion production, Shrauner's difficulty will be avoided since we can replace the transition-amplitude operator by the isopin operator by making use of the equal-time commutator of two chiral currents. This is another reason to study double soft-pion production. The kinematical situation becomes simple only if two soft pions are produced. Experimentally it is difficult to know how to select soft pions especially in view of the multiple hard pions associated. In order to get a good selection of soft pions, Shrauner 8) picked up the low-energy pion relative to the lab as well as the outgoing nucleon. Looking for the similarity of the experimental angular distribution of fast pions in the two systems, he concluded that he could pick up the desired soft pions. Under similar conditions, we can apply his criterion to the K N collision with double soft-pion production since kaon energies can be measured in any desired system. We discuss a more stringent condition than those of Weinberg and Shrauner in the case of the multiple pion production. It has the advantage that the general expression of the cross section becomes simple and may be applied to various collision processes. In our derivation we start from the reduction formula for two reduced soft pions of Kawarabayashi and Suzuki 9) (hereafter referred as KS); it includes an additional term proportional to the scalar quantity t o) ,, in the soft-pion limit. One could drop this term by assuming the CAC as Weinberg does 7). If one of the soft pions is neutral and the other charged, the scalar term does not appear due to the vanishing value of the structure constant. If one pion is positive and the other negative, the scalar part becomes proportional to the zeroth or eighth component of the SU(3) generator, so it consists of the symmetry-non-breaking and -breaking effect. If finally they are identically charged n°n ° and/or n+-n +-, the former contributes to the scalar part but the latter contributes neither to the scalar nor to the isospin operator. If we cancel the scalar KS part, the result becomes simpler. A successful result is obtained by neglecting the scalar KS part, for instance, for the elastic S-wave pion-nucleon scattering 11, 1o). It automatically vanishes for the chargeexchange scattering 12). However there is no a priori reason to cancel it from the beginning for other processes, so we include it in our discussion. The general formula t) He feels that the large discrepancy in his paper is most likely due to the fact that the experimental data available were too far removed from where his calculations should be applied (private c o m m u n i cation f r o m E. Shrauner). ~t) The first extensive discussion on the scalar quantity was given in this reference. Unfortunately it was published after completing the present work and the scalar part from ref. a) was used here.

SOFT-PION PRODUCTION

583

for the double soft-pion-production cross section is derived and estimated numerically. It is large enough to be measured. Under this observation we proceed to analyse the production of double soft pions in the baryon-baryon collision. The energy of two-baryon states for this process will be high enough that the prediction under the SU(3) symmetry for two-baryon states may be better than that in the low-energy elastic scattering 13). Many selection rules in addition to sum rules are derived. These rules are so closely related to our assumptions that they will supply a good test of the soft-pion hypothesis. The selection rules are classified into three groups (i) missing scalar part and forbidden isospin part, (ii) allowed scalar part but forbidden by the isospin part and (iii) completely forbidden by both scalar and isospin part of the amplitudes. In sect. 2 we shall discuss the condition for softness; sect. 3 establishes the general lbrmula; and in sect. 4 the sum and selection relus ['or the BB ~ BB2~ reaction are discussed. Sect. 5 is a summary of our results, in the appendix we discuss the differential spectra of the spectator particle for four final particle states. 2. Two Soft Pions in Production

As a strict condition of softness, we require that the physical amplitude with soft pions ,,ill be close to the one without soft pions. For definiteness let us consider the collision of particles A and B and the production of C, D and two soft pions: A+B ~ C+D+2

soft pions.

(1)

Here C and/or D can be many-particle states so that the reaction (1) is a completely general collision process. The soft-pion general rule can be expressed by A o2 ,

Bg, Co2 , D o2 > . SA+B--Sc+ D ~ (211rr)2

(2)

where Ao, Bo, Co and D O are the energies of particles A, B, C, and D in the lab as well as the c.m. frame, SA+8 and Sc+D the squares of the total energies of systems A + B and C + D, respectively, and p,~ the pion mass. Our expression is equivalent to that of Weinberg 7). We shall confine ourselves to the physical region close to the right-hand equality in eq. (2). I f we split further, say, D into D = E + n hard pions, and if we assume C and E are one-particle states, we must replace Do2 in eq. (2) by n + 1 terms E~ and q/Zo (i = 1, 2 , . . . , n). The above statement is yet not clear enough to apply to the experimental analysis. We shall therefore adopt an additional condition for softness, viz. total energy of any pair of soft and hard pions as well as that of the soft pion and baryon should be much different from their resonant energies. It may require considerable work in order to satisfy this condition experimentally. More explicitly we require that the two soft pions do not affect the kinematics of A + B C + D. This may not be true even for BB -~ BB2~ near the threshold. I f not, we shall have to modify our expression of the cross section in this limit (see the appendix) but not the selection and sum rules.

584

s. I W A O

3. Cross Section for Two Soft-Pion Production

Let us start from the reduced S-matrix element of two soft pions for reaction (1) (CDn~rtoJtlABi.)

=

l x/ & f V

2 d 4x d 4y e -~k~-~k'y.(#~_

z

2

-

× (CDo°dT{q~'(x)~pJ(y)}IAB,,), (3) where ~i and rtj are the soft-pion states with isospin component i and j, k and k' their four-momenta, ~oi(x) and ¢J(y) the corresponding field operators and V the quantization volume. The PCAC 4) is defined by

q)i(x) = c

OJaui(x) OXu

'

c =

g~KNN"(o) 2

.

(4)

g A mN tlTt

Here Ja~(x) is the axial-vector current density and c the isospin-independent constant defined by the rationalized and renormalized pion-nucleon coupling constant gr, its vertex correction KNN"(k2) (normalized to 1 on the pion mass shell), the axial-vector coupling-constant renormalization gA, the nucleon mass m N and the square of the pion mass /z2,. Substituting (4) into (3), performing partial integration twice with respect to x and y, keeping terms up to first order with respect to pion momenta in the limit of k ~ 0 and k' --, 0 (in the sense of Nambu and Luri6 5)) and finally making use of the equal-time commutation relation of the chiral currents and of the chiral current and pseudoscalar quantity, we get 1

2

4

lim (CDrt'rC~utlAB~.) = --27rf(EA+a--Ec+D) 2V c /~ k~0 k'~0

i u ,~0

1

//r~

where dijk and fijk a r e the generalized Kronecker symbols introduced by Gell-Mann, iJVok(x) the fourth component of the vector current density and uko(x) the scalar density defined by 1) the equal-time commutator of the axial-vector current density and the pseudoscalar density rio(X) = jaoJ(x)

f d3 y[j~i(x), VJo(Y)]xo=,o = idlik uko(x).

(9)

Eq. (5) is nothing but the generalized expression of KS. Let us define Uk by

lim i|d3xu~(x)/" d

= Uk.

(7)

SOFT- PION PRODUCTION

585

Substituting eq. (7) and the definition of the isospin operator into eq. (5), we get a simple expression lim 4,CDrc ' i7Zout]ABin J ) k~O

=

--27rc~(EA+B--Ec+D)

k'~0

1 2V

x lim c 2p~4 [Ljk(CDo~tI/~IABi,,) - dii~(CDo.,I UkIAB~,)].

(8)

~0

We shall call the first and the second term in the squared bracket an isospin and a scalar part of the amplitude, respectively. We consider now the process A+B +

C+E+nTr+ni+Trj,

(D = E+mr),

(9)

in which 7r~ and M are soft pions. We have then

q~+pC+PE--pA--pB)NAB~CD(.Ik)(n,CEITIAB), (10)

(CDoutllk[ABin) = i(27r)4~4( ~ i

where NAU+CD is the kinematical factor, AB~CEnTz and (Ik)

(lk)

----- ( / C / D ,

If, If3,

(n, CEITIAB) the T-matrix element for

fl]lkllAIB, li, 1i3'

~)"

Here I, are the isospins of the corresponding states (s = A, B, C, D), I~ and If the total isospins of the initial and final states, Ii3 and /f3 their third components, and and fl the additional quantum numbers such as the hypercharges and the unitary spins. Similarly we define (CDomi UkIAB~,,) = i(2~z)464( ~

qi+Pc+pE--pA--pB)NAB+cD(Uk)(n,CEITiAB),

i

where

(1 I ) (Uk)

= (/C/D,

I f , I f 3 , fllUk]IAIB,

li, Ii3, ~)"

Substituting (10) and (11) into (9), we get g j

lim (CD= ZrootlABin) = i(2~)'6"( ~

k+O k'~O

i

x (27r)c3(EA+

_ Ec+o)

1

qi+pc"l-pE--

PA - -

PB) 27" N~u~cD

l i m C2 fl~(fijk(lk/~ 4 - -- dijk(, U k ) ) ( f l , C E I TiAB)-

(12)

~0

This will be compared with i

lim (final~n]Slinitial~.) = i(2rr)4c~4( ~ qi + Pc + PE-- PA -- PB) lim -- NAB ~CD k~0 i ~,,~o 2Vp~

k'~0

x (n, CE, 2ITIAB),

(13)

where S is the S-matrix and (n, CE, 2]TIAB) is the T-matrix element for process (9).

586

s. IWAO

Comparing (12) and (13), we find (n, CE, 2[ TIAB) ~ (2rc)6(EA+ B - - Ec+D)C2/.tn5 ( n , C E ] T [ A B ) ( f i j k ( l k ) -- dijk(Uk) ). (14) In this relation the zero soft-pion energy arising from the kinematical factor eq. (13) was extrapolated to the pion mass. Maing use of (14), the total cross section a for process (9) can be obtained by the standard procedure 14), viz. 4 8

1

G ~, C ,Un (2~)2 [fljk(lk)--dij k(Uk)12 1

2

f d4k6(k 2 + p~)O(K)6(EA +~ -- EC .D-- ko -- k'o)

x fd4k'6(k 'z + p2)O(k')6(EA+,- Ec+ D - ko - k~))aAa-*CO,

(15)

where aAa.CD is the cross section for the process AB--*CD. If we assume that the kinematical condition of this reaction is not affected by the production of two soft pions, we may take aA~.CO out of the integrand and find a O'AB~CÜ

g 64 (KNN~(0)f ) 4 \

~JA

/

((ko-p~)(ko2 2

t2 __~12))½ ]fijk(lk)--dijk (Uk)[2'

(16)

]j2

w h e r e f 2 = (p,J2mN)2(gZ/4rc). Makinguse o f f 2 = 0.082 (ref. 15)), 1OAI = 1.17 (ref. 12)) KNN~(0) = 1, the kinematical factor in (16) becomes 0.048, 0.10, 0.16, 0.22, and 0.29 for ko = k~ = 1.1, 1.2, 1.3, 1.4 and 1.5/~,, respectively. The matrix element for the internal spin part is of the order of 1 so that the production cross section of double soft pions is a quantity, which can be easily measured. One can introduce the correlation between two soft pions into eq. (15) if it becomes important. Moreover if the approximation (16) is not valid, we can modify it by taking the differential spectra in eq. (15). This modification may complicate the problem except for the simplest case where only four particles are present in the final state including the soft pions (see the appendix). It is worthwhile summarizing the possible soft-pion states and relevant numerical factors arising from the commutators of the currents. As we mentioned before, the identical charge state rt°Tz° contributes only to the scalar part but ~z-+rt -+ give no contribution at all. The remaining three combinations 7r-+7t° and ~+ n - contribute to the isospin part of the amplitudes, viz.

[fJ'~(+-'(x)dZx,fJ'~Z(y)dZy]x° ro = I'-+',

(17)

Note that this part of the current algebra holds for both chiral U(2)®U(2) and U(3)

587

SOFT-PION PRODUCTION

® U ( 3 ) . The eigenvalues o f the isospin operators I (-+) and 13 are normalized as those in the appendix o f ref. 14). The pion fields corresponding to their charge states are defined by 7z+ __+ x/T((;ol+__icD2), %o ..+ (p3 so that the insertion of these states into the left-hand side o f eq. (4) requires the factor x/½ and 1 for the charged and the neutral pion, respectively. Normalization o f our current operators is the same as that o f Gell-Mann 1). Therefore the overall normalization gives the factor ,v/~ and ½ for n + n ° and n + n - , respectively, on the right-hand side o f eq. (12) by the corresponding insertion o f pions on its left-hand side. These remarks will suffice for the understanding o f the discussion in the next section. 4. Two Soft-Pion Production in BB Collision The c o m p u t a t i o n made in the preceding section will allow us to prepare a table for the double soft-pion production in the b a r y o n - b a r y o n collision. For this, we calculate from eqs. (17) and (18) the matrix elements of the three isospin operators between the initial and final b a r y o n states. N o w the amplitudes for the production o f soft n + and n - are obtained by multiplying the eigenvalues o f 13 by those found in table 3 o f re['. 13) and taking into account the last remark in the previous section. Therefore we have only to evaluate the contribution arising from eq. (17). Matrix elements which do not vanish are given in table 1. By elaborating it, we have assumed the SU(3) symmetry as before 13). The notation used here is slightly different f r o m that in ref. 13), i.e., the internal symmetry of baryons is specified by overall suffices S and A on the T-matrix and the order o f initial and final states are interchanged according to the convention o f this paper. There are only a few independent amplitudes in partial waves, so we may expect to find a n u m b e r of sum rules. Their explicit forms will follow from the modified table 3 of ref. 13) as just discussed and can be seen in our table l. F r o m this table we find that the allowed space-spin states between baryons in pp --+ pn 7r ~ n ° and pn ~ nn n + n o are odd while that for Y - p ~ S +s o n - n o is even. We come now to the discussion o f non-go selection rules. Since the scalar does not contribute to n -+n o production, this kind of selection rule is important. F r o m the fact that there is no generator connecting I -- ½ with I = 3 states, a first class o f rules take the form A ( S + p ~ Apn°n +) = 0. (19) A similar relation holds for the inverse reaction with charge-conjugated pions. A n d still three other relations o f the same class hold, viz. A ( ~ ° p ~ A A n ° n +)

= 0,

I = 1 --+ 0 forbidden,

(20)

A(-=°p --* Z+Z+n°n -) = 0,

I = 1 ~ 2 forbidden,

(21)

A(~E°p~Z°Z°n°n

+) = 0,

no/=

l s t a t e i n Z ° Z °.

(22)

588

S. I W A O

The second class of rules which m a y or may not be zero for the scalar part c o m e s from the vanishing value o f
A ( p n ~ pn)

= 0,

(23)

A ( A A --+ A A )

= 0,

(24)

A ( E - p --+ .z---p)

= 0,

(25)

A ( E - p --+ E ° n )

= 0,

(26)

A ( S - p --+ A A )

= 0,

(27)

A ( E - p --+ S+ S - ) = 0.

(28)

Here we omitted the final pions rt°rr ° or rc +Tr-. TABLE

1

Matrix element for changed and neutral pion production

Coefficients of Amplitude (pnll(-)[pp>s s

(An[I¢-'[Ap>s A (~'° nile ~[Ap>s

A (27)

A (8~)

A(10)

v'2 9 10

l 10

.x/ 3

,v/ 3

1

...... 10

I

2

2

,,/'3 6

",/3 6

10

1

(Z+n[ll-~IZ+p> a (Z'°p]I~-'[Z+ p>s < Z ° p f ~[Z+p>A s

1

~/2

%/2 7~/2 10

3V'2 I0

x./2

A


6 x/6 -10

x/2 6

x/6

\'6

V6

6--

6-

2

-5 1

1

1

~,'6 6

\'6 6

\."2

\'2

,~/2

3

6

6

< 3 - n l I c )13-p>A s

2-~/2 3

- 16

A s

A(10*)

x/2

A (Z'+ nJlHl~'+ p> s

A(8~)

x/6 5

x/6 5

A

A(1)

SOFT- PION

PRODUCTION

589

The third class of rules derives from the fact that they are not only forbidden by the scalar and isospin but part also by the matrix elements for baryon states, viz. A(pp -* nn~+Tr +)

= 0,

I~ = 1 ~ - 1 forbidden,

(29)

A(S+p ~ An~+~ +) = 0,

I~ = } ~ - ½ forbidden,

(30)

A ( X + p - - * Z ° n ~ + ~ +) = O,

I~ = 3--* - ½ f o r b i d d e n .

(3[)

These rules are abundant enough to be tested experimentally. It is interesting to add that the modified table 3 of ref. ~3). will give quantitative information on the scalar part from the ratio (=°=°/7~+~-), since the numerator cancels for the isospin part. If the strict condition of softness stipulated at the ending of sect. 2 does not apply, we have to take the differential spectra per momentum and per solid angle of the spectator particle in the equation corresponding to eq. (15) (see the appendix). This will imply a more complicated experimental analysis. It may be useful to remark that the two-pion production threshold in the nucleonnucleon collision is around 600 MeV in kinetic energies. This gives some idea as to what should be reasonably assumed in the course of an experimental study.

5. Discussion Summarizing, we have studied the double soft-pion production in a scheme combining algebra of currents, PCAC and quantum field theory. As a result, we have found in a fairly good approximation the estimate of the cross section which is large enough to suggest the importance of the study in the understanding of the production amplitudes, This prediction should be tested in various production processes. We have pointed out a few specific processes well-suited for such tests. A baryon-baryon collision accompanied by the production of two soft pions has been worked out in detail. The existence of many open channels allows us to establish more sum rules than in the elastic scattering. An experimental confirmation of the selection rules (19)-(31) would be the best criterion for the existence or the vanishing of the scalar part. If this conclusion of our theory turns out to be correct, it would be advisable to work out a variety of processes left unstudied, such as double soft-pion production in meson-nucleon collision, which includes an additional amplitude known as the octetoctet interference ~6). Therefore we hope for an opportunity to compare our theory with the eventual experimental study of two soft-pion production. The author is indebted to Professor A. Mercier for reading the manuscript and for his encouragement. He also wants to express his thanks to the ICTP, to the SLAC. to the NAVIS and to his friends for sending him their preprints before publication.

590

s. IWAO Appendix

FINAL FOUR-PARTICLE STATES We shall confine to the case that D in eq. (9) is a one-particle state. In this case the cross section is l

4

10

c fin a ~ 2X/2(S, m 2, m 2) (27z)6

[fijk(Ik)--di'ik(Uk)[2 /i 1 ' 2 2 .... 2 2x / d r 2 j x¢x/(ko--/2,~)~.Ko --/2,0d k__df2k'' J, (A.I)

where s is the square of the total energy, rn A and m~ the masses of particles A and B, 2(a, b, c) = a2+b2+c2-2ab-2bc-2ca, Ok and Ok. the solid angles of two soft pions, and J is defined by

d = f d*pc~(p 2 + rnE)O(pc) f d 4pDt~(p2 + m~)O(pD)t~3(pA + PB - -

PC-

PD -- k - k')

x 6(EA+B--Ec+D) Z I ( C D I T I A B ) I 2, (A.2) spin

where pi(i = A, B, C, D) are f o u r - m o m e n t a of particles A, B, C, and D and m c and m D the masses of particles C and D, respectively. Choosing the coordinate system PA + Ps--PC = 0 after integrating over PD, we find d2j dlpcldf2c 1

,

d

k k ) , 4 ..~. ,....d~°~l~l(CDl T I A B ) I2. - 8s x/2(s, m 2, m 2 ) - 2 ( s + m ~ - m 2 ) ( k + k ' ) 2 + ( + (A.3) Here f2c is the solid angle for particle C. Substituting (A.3) into (A.1), we find (

)

2

2

,2

dEa - 64 KNN~(0)f * ~/(ko--/2~)(ko dlPcldQc \ gA / /22 X

1 2 ~df2gdl2k ' x/2(s, (4re) d

2

--/2~)[fijk(ik)_dljk(Uk)i2

m 2, mE)-2(s+mE-m2)(k+k')2+(k+k') * dEtrAB_~CO x/2(s, m 2 , m 2)

dlPcIdf2c (A.4)

T h e differential spectra derived here can be c o m p a r e d with the experimental study.

References

1) M. Gell-Mann, Phys. Rev. 125 (1962) 1067; Physics 1 (1964) 63 2) S. Fubini and G. Furlan, Physics 1 (1965) 229; S. Fubini, G. Furlan and C. Rossetti, Nuovo Cim. 40 (1965) 1171

SOFT-PION PRODUCTION

59 ]

3) S. L. Adler, Phys. Rev. Lett. 14 (1965) 1051; W. I. Weisberger, ibid 14 (1965) 1047 4) M. Gell-Mann and M. L6vy, Nuov. Cim. 16 (1960) 705 5) Y. Nambu and D. Luri6, Phys. Rev. 125 (1962) 1429; Y. Nambu and E. Shrauner, ibid 128 (1962) 862 6) M. Suzuki, Phys. Rev. Lett. 15 (1965) 986; H. Sugawara, ibid 15 (1965) 870; Y. Hara, Y. Nambu and J. Schechter, ibid 16 (1966) 380; L. S. Brown and C. M. Sommerfield, ibid. 16 (1966) 751; Y. T. Chiu and J. Schechter, ibid. 16 (1966) 1022 7) S. Weinberg, Phys. Rev. Lett. 16 (1966) 879 8) E. Shrauner, Phys. Rev. 131 (1963) 1847 9} K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16 (1966) 255 10) K. Kawarabayashi and W. W. Wada, Phys. Rev. 146 (1966) 1209 11} Y. Tornozawa, Princeton preprint (April, 1966) 121} S. lwao, University of Bern preprint (June, 1966); K. Raman and E. C. G. Sudarshan, Syracuse University preprint, NYO-3399-66 (April 1966) 13) S. lwao, Nuovo Cim. 34 (1964) 1167; Phys. Lett. 11 (1964) 188 14) G. K/ill6n, Elementary particle physics (Addison-Wesley Publ. Co., Reading, Mass., 1964) 15) V. K. Samaranayake and W. S. Woolkock, Phys. Rev. Lett. 15 (1965) 936 16) C. A. Levinson, H. J, Lipkin and S. Meshkov, Phys. Lett. 1 (1962) 44