144 Progress of Theoretical Physics, Vol. 32, No. 1, July 1964

Quartet Scheme for Elementary Particles Ziro MAKI and Y oshio OHNUKI

Institute for Theoretical Physics Nagoya University, Nagoya (Received February 3, 1964)

A possible unified model for elementary particles is developed by generalizing the viewpoint of Sakata-Nagoya model. Instead of the Sakata triplet (p, nand A), we work with the quartet of new baryons Xo and X CX1, Xz and Xs) of spin 1/2, of which Xo belongs to U(1) whereas the other three to U(3). Strong interactions are subjected to a "broken U(1) X U(3)" symmetry. Baryon octet (N, 1:,

A and 3) and meson octets (K, 11:, 71, K; etc.) are interpreted, respectively, as the three (XXXo)or two (XX) -body composite systems both belonging to 8-dimensional representations of SU(3). One of the crucial test of the model is the prediction of triplet mesons (XXo), the iso-doublet of which can be identified with /C.-mesons. The baryon quartet, Xo and X, corresponds to four leptons in the sense of "modified" baryon-lepton symmetry. This enables us to explain Cabibbo's phenomenological weak interaction as a consequence of the present scheme.

§I. Introduction and summary Recent developments of particle physics are characterized by rapid mcrease of experimental knowledge about new "particles" and "resonances" of strongly interacting systems and also properties of weak interactions of both hadrons (strongly interacting particles) and leptons. It seems to us, therefore, that one of the most exciting problems in the theory of elementary particles is to explore possible ways of unified description of elementary particles aiming to find out a more profound basis of symmetry among them. Many attempts have been made so far along such a standpoint. Among them, the group-theoretical approach to the properties of strong interactions has been successfully developed based on the U(3)- or SU(3)-symmetry. Of these, there are two typical ways to describe baryon- and meson-families. The one is the "full symmetry" theoryl) developed by several au~hors 2 ) in close relation to the Sakata modelP and the other is the "eightfold way" invented by Yamaguchi, Gell-Mann4 ) and N e' em an 5 ) (Y. G. N.) . A mar ked difference between these two lies in the assignment of baryon octet (N, Z, A and 8). According to the Y. G. N.-scheme, empirical mass relation among baryon octet can be explained by Gell-Mann and Okubo's mass formula 6) in a· natural way. Therefore, if the spin-parity fo ahyperon is expected to be determined as (1/2)+, the Y. G. N.-approach to the baryon octet may have to be taken as the better alternative than that of the full symmetry -scheme. 2 ) However, we should remember the fact that there remain some problems,

Quartet Scheme for Elementary Particles

145

e.g. for the Y.G.N.-scheme, of clarifying physical basis of SU(3)-symmetry, because the group-theoretical approach is merely a phenomenology to be located at the first step of the recognition of inter-relation among baryons and mesons.*> Moreover, once we could find some underlying physical ground for the symmetry properties of hadrons, we had to further explore the relation between hadrons and leptons in order to get a unified understanding of existing particles. It is, however, rather difficult at first sight to construct such a scheme after the discovery of the second neutrino, since the number of "fundamental" baryons, which are supposed to generate U(3) or SU(3) symmetry of hadrons, is never four but three contrary to the number of the kind of leptons. In spite of such an apparent obstacle, it has been suggested by one of the present authors (Z.M.) 7 ) that a crucial step for this purpose could be taken by postulating the existence of a new baryon (the "fourth" baryon), and at the same time, by re-defining the fundamental baryon "triplet" not as p, n and A in the original sense of the Sakata model but as hypothetical ones whose existence should be searched for in the course of development of high energy experiments-. These points will be recapitulated in §2, 2·1. We may call this model "the quartet scheme".**) (See a diagram given in the text.) The aim of the present paper is to construct a unified scheme of elementary particles by adopting this new point of view. The selection rules in strong interaction from the four baryon scheme: viz. U(l) X U(3), is obviously more stringent than the strangeness rule requires. This is, however, not a drawback of the present model. In fact, it will be pointed out that the violation of the strict-symmetry by a moderate strong interaction can reproduce the strangeness rule on the one hand, and account for the breakdown of exact U(3)-invariance on the other, thus providing a raison d'etre of "broken symmetry". (§2. 2 · 2.) In §3, we shall give a tentative classification of hadrons with a special emphasis em identification of newly predicted particles with those whkh have been reported by experimental physicists, including the ones not as yet been well established. Also to be noted is that, according to our standpoint, the baryon octet (N, I;, A, 8) and meson octets (K, n:, r;, K, etc.) are interpreted, respectively, as the three- or two-body composite systems in a unified way. This seems rather natural frorn the view of the Regge pole, 8 > since, according to it, the eight baryons are supposed to be composite systems. One of the remarkable features of the present scheme is the possible existence of "triplet" mesons which have never been predicted by the octet model. In §4, we shall examine their properties in some detail and point out a possibility of identifying the iso-doublet of them with K-mesons which were recently reported. *) This viewpoint has been stressed by Sakata in the research meeting at Hiroshima University (March, 1963) ; see "Soryushiron Kenkyu" 28 (19.63), llO. **) A similar idea was once suggested by Y. Yamaguchi in ·1960 (private communication).

146

Z ..l'vfaki and Y. Ohnuki

A natural, but important, consequence from the present scheme is the theoretical. justification of the phenomenological model of baryonic weak interaction proposed recently by Cabibbo (§5) .9 ) This is not surprising, because in the present scheme the fundamental quartet of baryons, from which all hadrons are composed, is chosen in such a way that they correspond to the quartet of leptons in the sense of "modified" baryon-lepton symmetry. 10)

§2.

Model

The fourth baryon and the modified baryon-lepton symmetry

2 ·1.

One of the basic ideas upon which we will construct a new scheme of elementary particles is the particle-mixture theory of neutrinos proposed previously by Sakata, Nakagawa and one of the present authors (Z.M.), 10 ) and independently also by the Kyoto group_ll) They postulated that the correspondence between "fundamental" baryons (p, n and A) and four leptons should be realized through the B-matter 12) as follows:

(2·1) where v 1 1s a particle state of the neutrino defined by J.11

=ve coso+ J.l~'- sino.

(2 ·2a)

If there 1s no baryon corresponding to "another" neutrino state vo, orthogonal to v1; J.lo

= - J.le

sino+ J.l~'- coso,

(2·2b)

then the baryonic current JA may be expressed as

which 1s obtained from the leptonic weakcurrent:

(2·4) It was pointed out that the slowness of leptonic decays of baryons with 1 and the difference between Gv' s of /3- and p~e decays would be explained from the structure of JA (2 · 3), if sino=l/5.**> But, the relation something like

ILIS I =

Nothing or V+ = ..

'"d 0

...0

(Mesons)

I

Name (MeV) (JPG)

\ I

s

XoXo

0

0

XXo

1/2 0

1 0

0

0

roAna(317)(0 )? ~

1/2 1

1

K(494)

Configurations

1e

c\1 c~

0 1/2

I Note

(725) (0 or 1)

p1°(720) ? or ro (781) +

c~

151

+

n-(140)

0

:

(549) } (0).. K(494)

0 -1

7J

I 1

..

'"d 0

...0

Cl 0

K3*(1630)?

17)

-

..,j.

others

I

means a probable alternative. In the assignment A or B, the former (latter) corresponds to the case in which the spin of is taken to be 0 (1). ,Other conventions are the same with that of Table II. ••·-?

1e

C1XoX 0 • On the other hand Y~'(.-.....1680MeV ?, 1=0, S= -1, J=l/2- ?) resonance, the possible existence of which has been examined by Ioffe/ 8 ) may well be regarded as a member of C1Xo Xo or X. It should be noted that at present we have no candidates which correspond to any configurations C§ 5 , C~ and Ct Table III shows a classification of mesons. In this case, there are no mesons to be classified into the four-body configurations, except K 3 * (1630 MeV?, I= 3/2, S= 1, JPG = ?) , whose existence is yet questionable. One of the most remarkable feature of the present scheme would be the possible existence of a meson triplet, XXo and Xo X which will be denoted by

respectively. In the next section, we shall discuss 1n detail the behaviour of such new mesons. §4.

Possible existence of triplet mesons

As was mentioned in the previous sections, it is possible in the present model to make correspondence any of representations of SU(3) to baryons, mesons or their resonances. This is a different feature from the octet model. In this respect, it is of interest to examine a possibility of a meson triplet fh ( = XiXo)), which is the simplest example of "composite systems" not appearing in the octet model.

Z. Maki and Y. Ohnuki

152

In recent experiments a K-rc resonance (K) 22 ) with mass 725 MeV has been reported, whose isotopic spin is 1/2. The spin is supposed to be 1 or 0. Obviously the K cannot be regarded as a member of an octet, since the reality of (, which would be an isovector partner in the octet, is now quite questionable. In our model, however, it will be identified with a member of a triplet (XXo). In this case there exists a neutral member of isospin zero, 03 . In the symmetry limit of the strong interaction, this triplet cannot decay into the lower mass mesons owing to the X0 -number conservation; the decay should occur through the moderately strong interaction. In the following, we shall examine behaviors of 03 in some detail. Since the antiparticle of 03 is not equal . to itself, the decay should occur from the state

(03+03)/v2 or (0 3-03)/v2 just like the K 0 -meson, where (}3 stands for antiparticle of fls . First, let us assume the spin of the triplet to be one, and write

(4 ·1) both of which represent real fields because of (}3 + = - 0 3 . Then the quantum numbers I, G and P of the V1 are identified with those of cp, which belong to the vector octet in the symmetrical limit. Accordingly the broken symmetry interaction (2 · 8) gives rise to a mixing between them in its first order,

"""' vl =

-qcp+p v1,

(4·2)

p2+q2=1. Here rnv 1 , the decay

(4·13) occurs. Accordingly, if we assume 1nv2>mv 1, the V2 will decay into V1 and r in main but not into 4n, because ( 4 ·13) is the two-body decay with r emission not through the moderately strong interaction.*) Therefore it is rather natural that has not yet been obs~rved. Next we shall consider the case in which ~e is scalar. In this case we must identify (J) and rrtsl).

(4·16c)

sl

may be identified to the resonance Pl (720 "Nle V)' though its existence is now in question. As for S2 it is not clear which of a, b or c is the most probable. But if ms2 I.e I, the strength of decay interaction (4 · 12) should be weaker than that merely through the moderately strong interaction. In fact, since in virtue of the charge conjugation invariance Fz cannot make a transition to the 3S1 or the 3 D1 state through (2·8), its decay should occur through the {ourbody component.

Quartet Scheme for Elementary Particles

155

1s realized m our scheme by tansformations (4 ·17) Consequently, if our theory were invariant under R in the symmetry limit, the meson (Xi Xo), whose mass is the same as (Xi Xo), would exist. It may be, however, impossible to make its mass more than 2 Bev by virtue of the broken symmetry interaction.

§5. Weak interactions 5 ·1.

Structure of the weak interactions

Let us now discuss the structure of weak interactions of the present scheme. Starting with the leptonic current

we have, like (1 · 3), the fundamental baryonic weak-current JA: (5 ·1) Note that Xo does not appear. in JA. This comes from the postulate that B 0 and B+ are separately conserved. Or, equivalently, if there exists only charged weak bosons w±, we are led automatically to the .form, (5 ·2) because the "neutral current"

(5·3) cannot couple to w±. It _is easy to show that the vector part of JA behaves as the conserved vector current (CVC). Namely, noticing that the total isospin current TA (x) (fh TA (x) = O) in our system is expressed as with

(5 ·4)

we find the L1S = 0 leptonic interaction of the physical nucleon to be of the form,

(N' I T);+)(x) IN)jA coso, where N = ( ~) .

Then one ·obtains the relation

M2 )112 - (N'l TA(x) (E'E

i IN>= ---u'rrA.u 2

156

Z. Maki and Y. Ohnuki

in the limit of zero-momentum transfer. Thus, as far as the nucleon N are treated as if it were "elerr.entary", we have to supplement the term (i/2) N-rr;.. N to (5 · 4) in order to obtain the "effective" conserved current, (5·5) By generalizing this argument, we easily observe that the effective vector part of baryonic weak current Jf.·eff is to be introduced as

(5·6) where ji(x) (i= 1, ···, 8) are the conserved octet current connected with the generators Fi (i= 1, ···, 8) of SU(3)-transformation group,

(5·7) being a totally anti-symmetric tensor by which the algebra of generators defined:

fikl

IS

(5 ·8) As is well-known, Cj1 +ij2) and (j4+ij 5) represent LlS=O and L1S=1 (vector) current, respectively. The axial-vector currents will appear corresponding to each ji in virtue of the (V-A) -structure of the fundamental baryonic current (5 ·1). Thus, by writing

(jl + ij2)A + (axial-vector part)= J~O) (j4 +ij5)>. +(axial-vector part)

=J~l)'

we have

J1ff = [ Cx1 x2)>- + J~ 0 )] cos8

+ [ Cx1 x3) >- + JJl)] sino.

(5 ·9)

In this expression, the induced term J~O) coso+J~l) sino is nothing but the baryonic current introduced recently by Cabibbo. 9 ) It should be stressed that the "octet structure" of the baryonic current is an immediate ~onsequence from the present model unlike an ad hoc assumption made by him. 5 · 2.

On the axial-vector parts

Let us concentrate our attention to the LJS = 0 axial-vector part of the baryonic current. From the experimental informations we know the relation for a-decay:

On the other hand, as was shown in §3, the nucleons belong to the three cody configuration C~Xo. Therefore, in order to interpret the relation GA=:::::..-1.25 Gv

Quartet Scheme for Elementary Particles

157

we must inquire the dynamical structure of such a composite system. It is to be noted that, although p and n may be described as the "bound" state of K+ and K 0 with Xo, respectively,

p == [K+,

Xo],

n== [K 0 ' X] Q_

(5 "10) '

we cannot accept this structure in a naive sense. For example, if these bound states are in the S-state, GA vanishes for the relation (K+ I xl rA r5 x21 K 0 ) = 0. So, we had better describe nucleons as the bound states such as

P= [Os,

X1J,

(5 ·11)

n= [Os, X2J, by using the triplet mesons 0, the properties of which were discussed in the previous section (§4).

§6.

Concluding remarks

The model we have discussed in this paper is a natural generalizaton of the Sakata- modelS) for hadrons and of the Nagoya model. 12) It should, however, be emphasized that the present scheme obeys transitionary character in essence even if it will succeed to explain a number of experimental materials, since many fundamental problems seem to remain to be elucidated m future. In this connection, we would like to raise a conjecture that the origin of entrance of neutrino-mixture, p.-e mass difference and of symmetry violating interaction might well be correlated to each other to be interpreted from a more profound theoretical basis.

Acknowledgement We would like to express sincere gratitude to Prof. S. Sakata for his encouragement and fruitful comments. We are also indebted to Dr. M. Nakagawa and other members of Nagoya University for valuable discussion~. Added Notes In a private communication with Y. Hara, we were recently informed of a neutral resonance of 930 MeV. If it really exists, the mixing interaction of w-