Introduction

We attempt once more, as in the global symmetry scheme, to treat the eight known baryons as a supermultiplet, degenerate in the limit of a certain symmetry but split into isotopic spin multiplets by a symmetry-breaking term. Here we do not try to describe the symmetry violation in detail, but we ascribe it phenomenologically to the mass differences themselves, supposing that there is some analogy to the (\mu)-e mass difference.

The symmetry is called unitary symmetry and corresponds to the “unitary group” in three dimensions in the same way that charge independence corresponds to the “unitary group” in two dimensions. The eight infinitesimal generators of the group form a simple Lie algebra, just like the three components of isotopic spin. In this important sense, unitary symmetry is the simplest generalization of charge independence.

The baryons then correspond naturally to an eight-dimensional irreducible representation of the group; when the mass differences are turned on, the familiar multiplets appear. The pion and K meson fit into a similar set of eight particles, along with a predicted pseudoscalar meson (\chi^0) having (I = 0). The pattern of Yukawa couplings of (\pi), K and (\chi) is then nearly determined, in the limit of unitary symmetry.

The most attractive feature of the scheme is that it permits the description of eight vector mesons by a unified theory of the Yang-Mills type (with a mass term). Like Sakurai, we have a triplet

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(\rho) of vector mesons coupled to the isotopic spin current and a singlet vector meson (\omega^0) coupled to the hypercharge current. We also have a pair of doublets M and (\overline{\text{M}}), strange vector mesons coupled to strangeness-changing currents that are conserved when the mass differences are turned off. There is only one coupling constant, in the symmetric limit, for the system of eight vector mesons. There is some experimental evidence for the existence of (\omega^0) and M, while (\rho) is presumably the famous (I = 1), (J = 1), (\pi)-(\pi) resonance.

A ninth vector meson coupled to the baryon current can be accommodated naturally in the scheme.

The most important prediction is the qualitative one that the 8 baryons should all have the same spin and parity and that the pseudoscalar and vector mesons should form “octets”, with possible additional “singlets”.

If the symmetry is not too badly broken in the case of the renormalized coupling constants of the eight vector mesons, then numerous detailed predictions can be made of experimental results.

The mathematics of the unitary group is described by considering three fictitious “leptons”, (\nu), e, and (\mu), which may or may not have something to do with real leptons. If there is a connection, then it may throw light on the structure of the weak interactions.

1. Introduction

The strongly interacting particles were grouped into isotopic multiplets. It seemed that they would show traces of a higher symmetry that is somehow broken.

Under the higher symmetry, the 8 familiar baryons would be degenerate and form a supermultiplet.

As the higher symmetry is broken, the (\Xi), (\Lambda), (\Sigma), and N would split apart, leaving inviolate only the conservation of isotopic spin, of strangeness, and of baryons.

Of these 3:

• the first is partially broken by electromagnetism
• the second is broken by the weak interactions

Only the conservation of baryons and of electric charge are absolute.

An attempt(^{1,2)}) to incorporate these ideas in a concrete model was the scheme of “global symmetry”, in which the higher symmetry was valid for the interactions of the (\pi) meson, but broken by those of the K.

The mass differences of the baryons were thus attributed to the K couplings, the symmetry of which was unspecified, and the strength of which was supposed to be significantly less than that of the (\pi) couplings.

The theory of global symmetry has:

• failed in predicting experimental results
• many defects

The peculiar distribution of isotopic multiplets among the observed mesons and baryons is left unexplained. The arbitrary K couplings (which are not really particularly weak) bring in several adjustable constants.

Furthermore, as admitted in Reference 1 and reemphasized recently by Sakurai,(^{3,4)}) in his remarkable articles predicting vector mesons, the global model makes no direct connection between physical couplings and the currents of the conserved symmetry operators.

I replace it with a new model of the higher symmetry of elementary particles which has none of these faults and a number of virtues.

The isotopic spin group is the same as the group of all unitary 2x2 matrices with unit determinant.

Each of these matrices can be written as exp(iA), where A is a hermitian 2x2 matrix. Since there are three independent hermitian 2x2 matrices (say those of Pauli), there are three components of the isotopic spin.

Our higher symmetry group is the simplest generalization of isotopic spin, namely the group of all unitary 3x3 matrices with unit determinant.

There are 8 independent traceless 3x3 matrices and consequently the new “unitary spin” has 8 components.

The first three are just the components of the isotopic spin, the eighth is proportional to the hypercharge Y (which is +1 for N and K, -1 for (\Xi) and (\overline{\text{K}}), 0 for (\Lambda), (\Sigma), (\pi), etc.), and the remaining four are strangeness-changing operators.

Just as isotopic spin possesses a three-dimensional representation (spin 1), so the “unitary spin” group has an eight-dimensional irreducible representation, which we shall call simply 8. In our theory, the baryon supermultiplet corresponds to this representation. When the symmetry is reduced, then (\underline{I}) and Y are still conserved but the four other components of unitary spin are not; the supermultiplet then breaks up into (\Xi), (\Sigma), (\Lambda), and N. Thus the distribution of multiplets and the nature of strangeness or hypercharge are to some extent explained.

The pseudoscalar mesons are also assigned to the representation 8. When the symmetry is reduced, they become the multiplets K, (\overline{\text{K}}), (\pi), and (\chi), where (\chi) is a neutral isotopic singlet meson the existence of which we predict. Whether the PS mesons are regarded as fundamental or as bound states, their Yukawa couplings in the limit of “unitary” symmetry are describable in terms of only two coupling parameters.

The vector mesons are introduced in a very natural way, by an extension of the gauge principle of Yang and Mills(^{5)}). Here too we have a supermultiplet of eight mesons, corresponding to the representation 8. In the limit of unitary symmetry and with the mass of these vector mesons “turned off”, we have a completely gauge-invariant and minimal theory, just like electromagnetism. When the mass is turned on, the gauge invariance is reduced (the gauge function may no longer be space-time-dependent) but the conservation of unitary spin remains exact. The sources of the vector mesons are the conserved currents of the eight components of the unitary spin(^{6)}).

When the symmetry is reduced, the eight vector mesons break up into a triplet (\rho) (coupled to the still-conserved isotopic spin current), a singlet (\omega) (coupled to the still-conserved hypercharge current), and a pair of doublets M and (\overline{\text{M}}) (coupled to a strangeness-changing current that is no longer conserved). The particles (\rho) and (\omega) were both discussed by Sakurai. The (\rho) meson is presumably identical to the I = 1, J = 1, (\pi)-(\pi) resonance postulated by Frazer and Fulco(^{7)}) in order to explain the isovector electromagnetic form factors of the nucleon. The (\omega) meson is no doubt the same as the I = 1, J = 0 particle or 3(\pi) resonance predicted by Nambu(^{8)}) and later by Chew(^{9)}) and others in order to explain the isoscalar form factors of the nucleon. The strange meson M may be the same as the K(^*) particle observed by Alston et al.(^{10)}).

Thus we predict that the eight baryons have the same spin and parity, that K is pseudoscalar and that (\chi) exists, that (\rho) and (\omega) exist with the properties assigned to them by Sakurai, and that M exists. But besides these qualitative predictions there are also the many symmetry rules associated with the unitary spin. All of these are broken, though, by whatever destroys the unitary symmetry, and it is a delicate matter to find ways in which these effects of a broken symmetry can be explored experimentally.

Besides the eight vector mesons coupled to the unitary spin, there can be a ninth, which is invariant under unitary spin and is thus not degenerate with the other eight, even in the limit of unitary symmetry. We call this meson B(^0). Presumably it exists too and is coupled to the baryon current. It is the meson predicted by Teller(^{11)}) and later by Sakurai(^{3)}) and explains most of the hard-core repulsion between nucleons and the attraction between nucleons and antinucleons at short distances.

We begin our exposition of the “eightfold way” in the next Section by discussing unitary symmetry using fictitious “leptons” which may have nothing to do with real leptons but help to fix the physical ideas in a rather graphic way. If there is a parallel between these “leptons” and the real ones, that would throw some light on the weak interactions, as discussed briefly in Section VI.

Section 3 is devoted to the 8 representation and the baryons and Section IV to the pseudoscalar mesons. In Section V we present the theory of the vector mesons.

The physical properties to be expected of the predicted mesons are discussed in Section VII, along with a number of experiments that bear on those properties.

In Section VIII we take up the vexed question of the broken symmetry, how badly it is broken, and how we might succeed in testing it.

2. The “Leptons” as a Model for Unitary Symmetry**

Unitary symmetry begins with “leptons”, although our theory really concerns the baryons and mesons and the strong interactions.

The particles we consider here for mathematical purposes do not necessarily have anything to do with real leptons, but there are some suggestive parallels. We consider three leptons, (\nu), e(^-), and (\mu^-), and their antiparticles.

The neutrino is treated on the same footing as the other two, although experience suggests that if it is treated as a four-component Dirac field, only two of the components have physical interaction. (Furthermore, there may exist two neutrinos, one coupled to the electron and the other to the muon.)

The electrical and weak interactions are absolutely symmetrical between e(^-) and (\mu^-), which are distinguished, however from (\nu). The charged particles e(^-) and (\mu^-) are separated by the mysterious difference in their masses.

We shall not necessarily attribute this difference to any interaction, nor shall we explain it in any way. (If one insists on connecting it to an interaction, one might have to consider a coupling that becomes important only at exceedingly high energies and is, for the time being, only of academic interest.) We do, however, guess that the (\mu)-e mass splitting is related to the equally mysterious mechanism that breaks the unitary symmetry of the baryons and mesons and splits the supermulacademic interest.)

We do, however, guess that the μ-e mass splitting is related to the equally mysterious mechanism that breaks the unitary symmetry of the baryons and mesons and splits the supermultiplets into isotopic multiplets. For practical purposes, we shall put all of these splittings into the mechanical masses of the particles involved.

It is well known that in present quantum electrodynamics, no one has succeeded in explaining the e-ν mass difference as an electro- magnetic effect. Without prejudice to the question of its physical origin, we shall proceed with our discussion as if that mass difference were “turned on” along with the charge of the electron.

If we now “turn off” the μ-e mass difference, electromagnetism, and the weak interactions we are left with a physically vacuous theory of three exactly similar Dirac particles with no rest mass and no known couplings. This empty model is ideal for our mathematical purposes, however, and is physically motivated by the analogy with the strongly interacting particles, because it is at the corresponding stage of total unitary symmetry that we shall introduce the basic baryon masses and the strong interactions of baryons and mesons.

The symmetric model is, of course, invariant under all unitary transformations on the three states, ν, e¯, and μ¯.

Let us first suppose for simplicity that we had only two par- ticles ν and e¯. We can factor each unitary transformation uniquely into one which multiplies both particles by the same phase factor and one (with determinant unity) which leaves invariant the product of the phase factors of ν and e¯. Invariance under the first kind of transformation corresponds to conservation of leptons ν and e¯. It may be considered separately from invariance under the class of transformations of the second kind (called by mathematicians the unitary unimodular group in two dimensions).

Each transformation of the first kind can be written as a matrix e^(iφ)1, where 1 is the unit 2x2 matrix. The infinitesimal transformation is 1 + i(δφ)1 and so the unit matrix is the infini- tesimal generator of these transformations. The transformations of the second kind are generated in the same way by the three indepen- dent traceless 2x2 matrices, which may be taken to be the three Pauli isotopic spin matrices τ₁, τ₂, τ₃. We thus have

1 + i ∑_{k=1}^{3} δφ_k (τ_k / 2) (2.1)

Symmetry under all the transformations of the second kind is the same as symmetry under τ₁, τ₂, τ₃, in other words charge indepen- dence or isotopic spin symmetry. The whole formalism of isotopic spin theory can then be constructed by considering the transforma- tion properties of this doublet or spinor (ν, e⁻) and of more com- plicated objects that transform like combinations of two or more such leptons.

The Pauli matrices τₖ are hermitian and obey the rules

Tr τᵢ τⱼ = 2δᵢⱼ [τᵢ, τⱼ] = 2iεᵢⱼₖ τₖ {τᵢ, τⱼ} = 2δᵢⱼ 1 (2.2)

We now generalize the idea of isotopic spin by including the third object μ⁻. Again we factor the unitary transformations on the leptons into those which are generated by the 3x3 unit matrix 1 (and which correspond to lepton conservation) and those that are generated by the eight independent traceless 3x3 matrices (and which form the “unitary unimodular group” in three dimensions). We may construct a typical set of eight such matrices by analogy with the 2x2 matrices of Pauli. We call them λ₁……λ₈ and list them in Table I. They are hermitian and have the properties

Tr λᵢ λⱼ = 2δᵢⱼ [λᵢ, λⱼ] = 2ifᵢⱼₖ λₖ {λᵢ, λⱼ} = 4/3 δᵢⱼ 1 + 2dᵢⱼₖ λₖ , (2.3)

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where the fᵢⱼₖ are real and totally antisymmetric like the Kronecker symbols εᵢⱼₖ of Eq. (2.2), while the dᵢⱼₖ are real and totally sym- metric. These properties follow from the equations

Tr λₖ [λᵢ, λⱼ] = 4ifᵢⱼₖ Tr λₖ {λᵢ, λⱼ} = 4dᵢⱼₖ (2.4)

derived from (2.3).

The non-zero elements of fᵢⱼₖ and dᵢⱼₖ are given in Table II for our choice of λᵢ. Even and odd permutations of the listed in- dices correspond to multiplication of fᵢⱼₖ by +1 respectively and of dᵢⱼₖ by +1.

The general infinitesimal transformation of the second kind is

…

by analogy with (2.1). Together with conservation of leptons, in- variance under the eight λᵢ corresponds to complete “unitary sym- metry” of the three leptons.

It will be noticed that λ₁, λ₂, and λ₃ correspond to τ₁, τ₂, and τ₃ for ν and e⁻ and nothing for the muon. Thus, if we ignore symmetry between (ν, e⁻) and the muon, we still have conservation of isotopic spin. We also have conservation of λ₈, which commutes with λ₁, λ₂, and λ₃ and is diagonal in our representation. We can diagonalize at most two λ’s at the same time and we have chosen them to be λ₃ (the third component of the ordinary isotopic spin) and λ₈, which is like strangeness or hypercharge, since it distinguishes the isotopic singlet μ⁻ from the isotopic doublet (ν, e⁻) and commutes with the isotopic spin.

Now the turning-on of the muon mass destroys the symmetry under λ₄, λ₅, λ₆, and λ₇ (i.e., under the “strangeness-changing” components of the “unitary spin”) and leaves lepton number, “iso- topic spin,” and “strangeness” conserved. The electromagnetic interactions (along with the electron mass) then break the conser- vation of λ₁ and λ₂, leaving lepton number λ₃, and strangeness con- served. Finally the weak interactions allow the strangeness to be changed (in muon decay) but continue to conserve the lepton number n_L and the electric charge

Q = n_Q (λ₃ + λ₈/√3 - 4/3 n_L) (2.6)

where n_L is the number of leptons minus the number of antileptons and equals 1 for ν, e⁻, and μ⁻ (i.e., the matrix 1).

We see that the situation is just what is needed for the baryons and mesons. We transfer the symmetry under unitary spin to them and assign them strong couplings and basic symmetrical masses. Then we turn on the mass splittings, and the symmetry under the 4th, 5th, 6th, and 7th components of the unitary spin is lifted, leaving baryon number, strangeness, and isotopic spin conserved. Electro- magnetism destroys the symmetry under the 1st and 2nd components of the spin, and the weak interactions destroy strangeness conservation. Finally, only charge and baryon number are conserved.

3. Mathematical Description of the Baryons

In the case of isotopic spin $\underline{\text{I}}$ , we know that the various pos- sible charge multiplets correspond to “irreducible representations” of the simple 2x2 matrix algebra described above for ( $\nu$ , e⁻). Each multiplet has 2I + 1 components, where the quantum number I distin- guishes one representation from another and tells us the eigenvalue I(I + 1) of the operator $\sum_{i=1}^{3} \text{I}_i^2$ , which commutes with all the ele- ments of the isotopic spin group and in particular with all the infinitesimal group elements $1 + i \sum_{i=1}^{3} \delta\varphi_i \text{I}_i$ . The operators $\text{I}_i$ are represented, within the multiplet, by hermitian (2I + 1) x (2I + 1) matrices having the same commutation rules

$[\text{I}_i, \text{I}_j] = i\varepsilon_{ijk} \text{I}_k$ (3.1)

as the 2x2 matrices $\tau_i/2$ . For the case of I = 1/2, we have just $\text{I}_i = \tau_i/2$ within the doublet.

If we start with the doublet representation, we can build up all the others by considering superpositions of particles that trans- form like the original doublet. Thus, the antiparticles e⁺, -ν also form a doublet. (Notice the minus sign on the antineutrino state or field.) Taking $(e^+ - \bar{\nu})/\sqrt{2}$ , we obtain a singlet, that is, a one- dimensional representation for which all the $\text{I}_i$ are zero. Calling the neutrino and electron $e_\alpha$ with $\alpha = 1, 2$ , we can describe the singlet by $\frac{1}{\sqrt{2}} \bar{e}_\alpha e_\alpha$ or, more concisely, $\frac{1}{\sqrt{2}} \bar{e}e$ . The three components of a triplet can be formed by taking $e^+\nu = \frac{1}{\sqrt{2}} \bar{e}(\tau_1 - i\tau_2)e$ , $\frac{e^+e^- - \nu\nu}{\sqrt{2}} = \frac{1}{\sqrt{2}} \bar{e} \tau_3 e$ , and $\nu^-e^- = \frac{1}{\sqrt{2}} \bar{e}(\tau_1 + i\tau_2)e$ . Rearranging

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these, we have just $\frac{1}{\sqrt{2}} \bar{e} \tau_j e$ with j = 1, 2, 3. Among these three states, the 3x3 matrices $\text{I}_j^{jk}$ of the three components of $\underline{\text{I}}$ are given by

$\text{I}_j^{jk} = -i\varepsilon_{jjk}$ (3.2)

Now let us generalize these familiar results to the set of three states ν, e⁻, and μ⁻. Call them $\ell_\alpha$ with $\alpha = 1, 2, 3$ and use $\bar{\underline{\ell}}\underline{\text{I}}\ell$ to mean $\bar{\ell}_\alpha \text{I}_{\alpha\beta} \ell_\beta$ , etc. For this system we define $\text{F}_i = \frac{\lambda_i}{2}$ with i = 1, 2,…,8, just as $\text{I}_i = \frac{\tau_i}{2}$ for isotopic spin. The $\text{F}_i$ are the 8 components of the unitary spin operator $\underline{\text{F}}$ in this case and we shall use the same notation in all representations. The first three components of $\underline{\text{F}}$ are identical with the three components of the isotopic spin $\underline{\text{I}}$ in all cases, while $\text{F}_8$ will always be $\sqrt{3}$ times

the hypercharge Y (linearly related to the strangeness). In all representations, then, the components of $\underline{\text{F}}$ will have the same com- mutation rules

$[\text{F}_i, \text{F}_j] = if_{ijk} \text{F}_k$ (3.3)

that they do in the simple lepton representation for which $\text{F}_i = \lambda_i/2$ . (Compare the commutation rules in Eq. (2.3).) The trace properties and anticommutation properties will not be the same in all representations any more than they are for $\underline{\text{I}}$ . We see that the rules (3.1) are just a special case of (3.3) with indices 1, 2, 3, since the f’s equal the ε’s for these values of the indices.

We must call attention at this point to an important difference between unitary or $\underline{\text{F}}$ spin and isotopic or $\underline{\text{I}}$ spin. Whereas, with a simple change of sign on $\bar{\nu}$ , we were able to construct from $\bar{e}_\alpha$

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a doublet transforming under $\underline{\text{I}}$ just like $e_\alpha$ , we are not able to do the same thing for the $\underline{\text{F}}$ spin when we consider the three antileptons $\bar{\ell}_\alpha$ compared to the three leptons $\ell_\alpha$ . True, the antileptons do give a representation for $\underline{\text{F}}$ , but it is, in mathematical language, inequi- valent to the lepton representation, even though it also has three dimensions. The reason is easy to see: when we go from leptons to antileptons the eigenvalues of the electric charge, the third com- ponent of $\underline{\text{I}}$ , and the lepton number all change sign, and thus the eigenvalues of $\text{F}_8$ change sign. But they were $1/2\sqrt{3}$ , $-1/2\sqrt{3}$ , and $-1/\sqrt{3}$ for leptons and so they are a different set for antileptons and no similarity transformation can change one representation into the other. We shall refer to the lepton representation as $\underline{3}$ and the antilepton representation as $\underline{\bar{3}}$ .

Now let us consider another set of “particles” $\text{L}_\alpha$ transfor- ming exactly like the leptons $\ell_\alpha$ under unitary spin and take their antiparticles $\bar{\text{L}}_\alpha$ . We follow the same procedure used above for the isotopic spin and the doublet e. We first construct the state $\frac{1}{\sqrt{3}} \bar{\text{L}}_\alpha \ell_\alpha$ or $\frac{1}{\sqrt{3}} \bar{\text{L}}\ell$ . Just as $\frac{\bar{e}e}{\sqrt{2}}$ gave a one-dimensional represen- tation of $\underline{\text{I}}$ for which all the $\text{I}_i$ were zero, so $\frac{\bar{\text{L}}\ell}{\sqrt{3}}$ gives a one- dimensional representation of $\underline{\text{F}}$ for which all the $\text{F}_i$ are zero. Call this one-dimensional representation $\underline{1}$ .

Now, by analogy with $\frac{\bar{e} \tau_i e}{\sqrt{2}}$ with i = 1, 2, 3, we form $\frac{\bar{\text{L}} \lambda_i \ell}{\sqrt{2}}$ with i = 1, 2,……8. These states transform under unitary spin $\underline{\text{F}}$ like an irreducible representation of dimension 8, which we shall call $\underline{8}$ . In this representation, the 8x8 matrices $\text{F}_i^{jk}$ of the

eight components $\text{F}_i$ of the unitary spin are given by the relation

$\text{F}_i^{jk} = -if_{ijk}$ , (3.4)

analogous to Eq. (3.2).

When we formed an isotopic triplet from two isotopic doublets, in the discussion preceding Eq. (3.2), we had to consider linear com- binations of the $\frac{\bar{e} \tau_i e}{\sqrt{2}}$ in order to get simple states with definite electric charges, etc. We must do the same here. Using the symbol ~ for “transforms like”, we define

$\Sigma^+ \sim \frac{1}{\sqrt{2}} \bar{\text{L}}(\lambda_1 - i\lambda_2)\ell$ ~ $\text{D}^0\nu$ $\Sigma^- \sim \frac{1}{\sqrt{2}} \bar{\text{L}}(\lambda_1 + i\lambda_2)\ell$ ~ $\text{D}^0e^-$ $\Sigma^0 \sim \frac{1}{\sqrt{2}} \bar{\text{L}} \lambda_3 \ell$ ~ $\frac{\text{D}^0\nu - \text{D}^+e^-}{\sqrt{2}}$ p ~ $\frac{1}{\sqrt{2}} \bar{\text{L}}(\lambda_4 - i\lambda_5)\ell$ ~ $\text{S}^+\nu$ n ~ $\frac{1}{\sqrt{2}} \bar{\text{L}}(\lambda_6 - i\lambda_7)\ell$ ~ $\text{S}^+e^-$ $\Xi^0 \sim \frac{1}{\sqrt{2}} \bar{\text{L}}(\lambda_6 + i\lambda_7)\ell$ ~ $\text{D}^+\mu^-$ $\Xi^- \sim \frac{1}{\sqrt{2}} \bar{\text{L}}(\lambda_4 + i\lambda_5)\ell$ ~ $\text{D}^0\mu^-$ $\Lambda \sim \frac{1}{\sqrt{3}} \bar{\text{L}} \lambda_8 \ell$ ~ $(\text{D}^0\nu + \text{D}^+e^- - 2\text{S}^+\mu^-)/\sqrt{6}$ (3.5)

The most graphic description of what we are doing is given in the last column, where we have introduced the notation $\text{D}^0$ , $\text{D}^+$ , and $\text{S}^+$ for the $\bar{\text{L}}$ particles analogous to the $\bar{\ell}$ particles $\bar{\nu}$ , $e^+$ , and $\mu^+$ respectively. D stands for doublet and S for singlet with respect to isotopic spin. Using the last column, it is easy to see that the

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isotopic spins, electric charges, and hypercharges of the multiplets are exactly as we are accustomed to think of them for the baryons listed.

We say, therefore, that the eight known baryons form one degenerate supermultiplet with respect to unitary spin. When we introduce a perturbation that transforms like the μ-e mass difference, the supermultiplet will break up into exactly the known multiplets. (Of course, D will split from S at the same time as e⁻, ν from μ⁻.)

Of course, another type of baryon is possible, namely a sin- glet neutral one that transforms like $\frac{1}{\sqrt{3}} \bar{\text{L}}\text{L}$ . If such a particle exists, it may be very heavy and highly unstable. At the moment, there is no evidence for it.

We shall attach no physical significance to the $\ell$ and $\bar{\text{L}}$ “particles” out of which we have constructed the baryons. The dis-

cussion up to this point is really just a mathematical introduction to the properties of unitary spin.

4. Pseudoscalar Mesons

The baryon fields $\text{N}_j$ transform like an octet $\underline{8}$ under $\underline{\text{F}}$ , so that the matrices of $\underline{\text{F}}$ for the baryon fields are given by Eq. (3.4). We now demand that all mesons transform under $\underline{\text{F}}$ in such a way as to have $\underline{\text{F}}$ -invariant strong couplings. If the 8 mesons $\pi_i$ are to have Yukawa couplings, they must be coupled to $\bar{\text{N}}\text{O}_i\text{N}$ for some matrices $\text{O}_i$ , and we must investigate how such bilinear forms transform under $\underline{\text{F}}$ .

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In mathematical language, what we have done in Section III is to look at the direct product $\underline{3} \times \underline{3}$ of the representations $\underline{3}$ and $\underline{3}$ and to find that it reduces to the direct sum of $\underline{8}$ and $\underline{1}$ . We identi- fied $\underline{8}$ with the baryons and, for the time being, dismissed $\underline{1}$ . What we must now do is to look at $\underline{8} \times \underline{8}$ . Now it is easy to show that actually $\underline{\bar{3}}$ is equivalent to $\underline{3}$ ; this is unlike the situation for $\underline{3}$ and $\underline{\bar{3}}$ . (We note that the values of Y, I₃, Q, etc. are symmetrically disposed about zero in the $\underline{8}$ representation.) So the antibaryons transform essentially like the baryons and we must reduce out the direct product $\underline{8} \times \underline{8}$ . Standard group theory gives the result

$\underline{8} \times \underline{8} = \underline{1} + \underline{8} + \underline{8} + \underline{10} + \underline{\overline{10}} + \underline{27}$ , (4.1)

where $\underline{\overline{27}} = \underline{27}$ (this can happen only when the dimension is the cube of an integer). The representation $\underline{27}$ breaks up, when mass differ- ences are turned on, into an isotopic singlet, triplet, and quintet with Y = 0, a doublet and a quartet with Y = 1, a doublet and a quartet with Y = -1, a triplet with Y = 2, and a triplet with Y = -2. The representation $\underline{10}$ breaks up, under the same conditions, into a triplet with Y = 0, a doublet with Y = -1, a quartet with Y = +1, and a singlet with Y = +2. The conjugate representation $\underline{\overline{10}}$ looks the same, of course, but with equal and opposite values of Y. None of these much resembles the pattern of the known mesons.

The $\underline{8}$ representation, occurring twice, looks just the same for mesons as for baryons and is very suggestive of the known $\pi$ , K, and $\bar{\text{K}}$ mesons plus one more neutral pseudoscalar meson with I = 0,

Y = 0, which corresponds to Λ in the baryon case. Let us call this meson χ⁰ and suppose it exists, with a fairly low mass. Then we have identified the known pseudoscalar mesons with an octet under unitary symmetry, just like the baryons. The representations $\underline{1}$ , $\underline{10}$ , $\underline{\overline{10}}$ , and $\underline{27}$ may also correspond to mesons, even pseudoscalar ones, but presumably they lie higher in mass, some or all of them perhaps so high as to be physically meaningless.

To describe the eight pseudoscalar mesons as belonging to $\underline{8}$ , we put (very much as in (3.5))

$\chi^0 = \pi_8$ $\pi^+ = (\pi_1 - i\pi_2)/\sqrt{2}$ $\pi^- = (\pi_1 + i\pi_2)/\sqrt{2}$ $\pi^0 = \pi_3$ $\text{K}^+ = (\pi_4 - i\pi_5)/\sqrt{2}$ $\text{K}^0 = (\pi_6 - i\pi_7)/\sqrt{2}$ $\bar{\text{K}}^0 = (\pi_6 + i\pi_7)/\sqrt{2}$ $\text{K}^- = (\pi_4 + i\pi_5)/\sqrt{2}$ (4.2)

and we know then that the matrices of $\underline{\text{F}}$ connecting the $\pi_j$ are just the same as those connecting the $\text{N}_j$ , namely $\text{F}_i^{jk} = -if_{ijk}$ .

To couple the 8 mesons invariantly to 8 baryons (say by $\gamma_5$ ), we must have a coupling

$2ig_0 \bar{\text{N}} \gamma_5 \text{O}_i \text{N} \pi_i$ (4.3)

for which the relation

$[\text{F}_i, \text{O}_j] = if_{ijk} \text{O}_k$ (4.4)

holds. Now the double occurrence of $\underline{8}$ in Eq. (4.1) assures us that there are two independent sets of eight 8x8 matrices $\text{O}_i$ obey- ing (4.4). One of these sets evidently consists of the $\text{F}_i$ them- selves. It is not hard to find the other set if we go back to the commutators and anticommutators of the λ matrices in the $\underline{3}$ repres- entation (Eq. (2.3)). Just as we formed $\text{F}_i^{jk} = -if_{ijk}$ , we define

$\text{D}_i^{jk} = d_{ijk}$ (4.5)

and it is easy to show that the D’s also satisfy Eq. (4.4). We recall that where the F matrices are imaginary and antisymmetric with respect to the basis we have chosen, the D’s are real and symmetric.

Now what is the physical difference between coupling the pseudoscalar mesons $\pi_i$ by means of $\text{D}_i$ and by means of $\text{F}_i$ ? It lies in the symmetry under the operation

R: p $\leftrightarrow \Xi^-$ , n $\leftrightarrow \Xi^0$ , $\Sigma^+ \leftrightarrow \Sigma^-$ , $\Sigma^0 \leftrightarrow \Sigma^0$ , $\Lambda \leftrightarrow \Lambda$ $\text{K}^+ \leftrightarrow \text{K}^-$ , $\text{K}^0 \leftrightarrow \bar{\text{K}}^0$ , $\pi^+ \leftrightarrow \pi^-$ , $\pi^0 \leftrightarrow \pi^0$ , $\chi^0 \leftrightarrow \pm \chi^0$ , (4.6)

which is not a member of the unitary group, but a kind of reflec- tion. In the language of $\text{N}_i$ , we may say that R changes the sign of the second, fifth, and seventh particles; we note that $\lambda_2$ , $\lambda_5$ , and $\lambda_7$ are imaginary while the others are real. From Table II we can see that under these sign changes $f_{ijk}$ is odd and $d_{ijk}$ even.

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It may be that in the limit of unitary symmetry the coupling of the pseudoscalar mesons is invariant under R as well as the uni- tary group. In that case, we choose either the plus sign in (4.6) and the D coupling or else the minus sign and the F coupling. The two possible coupling patterns are listed in Table III.

If only one of the patterns is picked out (case of R-invariance) it is presumably the D coupling, since that gives a large $\Lambda\pi\Sigma$ inter- action (while the F coupling gives none) and the $\Lambda\pi\Sigma$ interaction is the best way of explaining the binding of $\Lambda$ particles in hypernuclei.

In general, we may write the Yukawa coupling (whether funda- mental or phenomenological, depending on whether the $\pi_i$ are elemen- tary or not) in the form

$\text{L}_{\text{int}} = 2ig_0 \bar{\text{N}} \gamma_5 [\alpha \text{D}_i + (1 - \alpha) \text{F}_i] \text{N} \pi_i$ . (4.7)

We note that in no case is it possible to make the couplings $\Lambda\text{K}\text{N}$ and $\Sigma\text{K}\text{N}$ both much smaller than the $\text{N}\pi\text{N}$ coupling. Since the evidence from photo-K production seems to indicate smaller effective coupling constants for $\Lambda\text{K}\text{N}$ and $\Sigma\text{K}\text{N}$ than for $\text{N}\pi\text{N}$ (indeed that was the basis of the global symmetry scheme) we must conclude that our sym- metry is fairly badly broken. We shall return to that question in Section VII.

A simple way to read off the numerical factors in Table III, as well as those in Table IV for the vector mesons, is to refer to the chart in Table V, which gives the transformation properties of mesons and baryons in terms of the conceptual “leptons” and “L particles” of Section 3.

An interesting remark about the baryon mass differences may be added at this point. If we assume that they transform like the μ-e mass difference, that is, like the 8th component of the unitary spin, then there are only two possible mass-difference matrices, $\text{F}_8$ and $\text{D}_8$ . That gives rise to a sum rule for baryon masses:

$1/2 (\text{m}_{\text{n}} + \text{m}_{\Xi^0}) = 3/4 \text{m}_{\Lambda} + 1/4 \text{m}_{\Sigma}$ , (4.8)

which is very well satisfied by the observed masses, much better than the corresponding sum rule for global symmetry.

There is no particular reason to believe, however, that the analogous sum rules for mesons are obeyed.

V. Vector Mesons

The possible transformation properties of the vector mesons under $\underline{\text{F}}$ are the same as those we have already examined in the pseudoscalar case. Again it seems that for low mass states we can safely ignore the representations $\underline{27}$ , $\underline{10}$ , and $\underline{\overline{10}}$ . We are left with $\underline{1}$ and the two cases of $\underline{8}$ .

A vector meson transforming according to $\underline{1}$ would have Q = 0, I = 0, Y = 0 and would be coupled to the total baryon current $i\bar{\text{N}}\gamma_\mu\text{N}$ , which is exactly conserved. Such a meson may well exist and be of great importance. The possibility of its existence has been envisaged for a long time.

We recall that the conservation of baryons is associated with the invariance of the theory under infinitesimal transformations

$\text{N} \rightarrow (1 + i \varepsilon)\text{N}$ , (5.1)

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where ε is a constant. This is gauge-invariance of the first kind. We may, however, consider the possibility that there is also gauge invariance of the second kind, as discussed by Yang and Lee $^{12)}$ . Then we could make ε a function of space-time. In the free baryon Lagrangian

$\text{L}_{\text{N}} = -\bar{\text{N}}(\gamma_\alpha \partial_\alpha + \text{m}_0)\text{N}$ (5.2)

this would produce a new term

$\text{L}_{\text{N}} \rightarrow \text{L}_{\text{N}} - i\bar{\text{N}}\gamma_\alpha \text{N} \partial_\alpha \varepsilon$ (5.3)

which can be cancelled only if there exists a neutral vector meson field $\text{B}_\alpha$ coupled to the current $\bar{\text{N}}\gamma_\alpha\text{N}$ :

$\text{L}_{\text{B}} = -1/4 (\partial_\alpha \text{B}_\beta - \partial_\beta \text{B}_\alpha)^2$ … $- i g \bar{\text{N}}\gamma_\alpha \text{N}\text{B}_\alpha$ (5.4)

and which undergoes the gauge transformation

$\text{B}_\alpha \rightarrow \text{B}_\alpha + 1/f_0 \partial_\alpha \varepsilon$ . (5.5)

As Yang and Lee pointed out, such a vector meson is massless and if it existed with any appreciable coupling constant, it would simulate a kind of anti-gravity, for baryons but not leptons, that is contradicted by experiment.

We may, however, take the point of view that there are vector mesons associated with a gauge-invariant Lagrangian plus a mass term, which breaks the gauge invariance of the second kind while leaving inviolate the gauge invariance of the first kind and the conservation

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law. Such situations have been treated by Glashow $^{13)}$ , Salam and Ward $^{14)}$ , and others, but particularly in this connection by Sakurai $^{3)}$ .

The vector meson transforming according to $\underline{1}$ would then be of such a kind. Teller $^{11)}$ , Sakurai $^{3)}$ , and others have discussed the notion that such a meson may be quite heavy and very strongly coupled, binding baryons and anti-baryons together to make the pseudoscalar mesons according to the compound model of Fermi and Yang $^{15)}$ . We shall leave this possibility open, but not consider it further here. If it is right, then the Yukawa couplings (4.7) must be treated as phenomenological rather than fundamental; from an immediate practical point of view, it may not make much difference.

We go on to consider the $\underline{8}$ representation. An octet of vector mesons would break up into an isotopic doublet with Y = 1, which we shall call M (by analogy with K – the symbol L is already used to mean $\pi$ or $\mu$ ); the corresponding doublet $\bar{\text{M}}$ analogous to $\bar{\text{K}}$ ; a triplet $\rho$ with Y = 0 analogous to $\pi$ ; and a singlet $\omega^0$ with Y = 0 analogous to $\chi^0$ .

We may tentatively identify M with the $\text{K}^*$ reported by Alston et al. $^{10)}$ at 884 Mev with a width $\Gamma \simeq 15$ Mev for break-up into $\pi$ + K. Such a narrow width certainly points to a vector rather than a scalar state. The vector meson $\rho$ may be identified, as Sakurai has proposed, with the I = 1, J = 1, $\pi$ - $\pi$ resonance discussed by Frazer and Fulco $^{7)}$ in connection with the electromagnetic structure of the nucleon. The existence of $\omega^0$ has been postulated for simi- lar reasons by Nambu $^{8)}$ , Chew $^{9)}$ , and others.

In principle, we have a choice again between couplings of the D and the F type for the vector meson octet. But there is no ques- tion which is the more reasonable theory. The current $i\bar{\text{N}}\gamma_\alpha\text{F}_i\text{N}$ is the current of the F-spin for baryons and in the limit of unitary sym- metry the total F spin current is exactly conserved. (The conserva- tion of the strangeness-changing currents, those of $\text{F}_4$ , $\text{F}_5$ , $\text{F}_6$ , and $\text{F}_7$ , is broken by the mass differences, the conservation of $\text{F}_2$ and $\text{F}_3$ by electromagnetism, and that of $\text{F}_3$ and $\text{F}_8$ separately by the weak interactions. Of course, the current of the electric charge

$Q = e(\text{F}_3 + \frac{\text{F}_8}{\sqrt{3}})$ (5.6)

is exactly conserved.)

Sakurai has already suggested that $\rho$ is coupled to the iso- topic spin current and $\omega$ to the hypercharge current. We propose in addition that the strange vector mesons M are coupled to the strange- ness-changing components of the F spin current and that the whole system is completely invariant under $\underline{\text{F}}$ before the mass-differences have been turned on, so that the three coupling constants (suitably defined) are approximately equal even in the presence of the mass differences.

Now the vector mesons themselves carry F spin and therefore contribute to the current which is their source. The problem of con- structing a nonlinear theory of this kind has been completely solved in the case of isotopic spin by Yang and Mills $^{5)}$ and by Shaw $^{5)}$ . We have only to generalize their result (for three vector mesons) to the case of F spin and eight vector mesons.

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We may remark parenthetically that the Yang-Mills theory is irreducible, in the sense that all the 3 vector mesons are coupled to one another inextricably. We may always make a “reducible” theory by adjoining other, independent vector mesons like the field $\text{B}_\alpha$ dis- cussed earlier in connection with the baryon current. It is an interesting mathematical problem to find the set of all irreducible Yang-Mills tricks. Glashow and the author $^{16)}$ have shown that the problem is the same as that of finding all the simple Lie algebras, one that was solved long ago by the mathematicians. The possible dimensions are 3, 8, 10, 14, 15, 21, and so forth. Our generali- zation of the Yang-Mills trick is the simplest one possible.

But let us “return to our sheep”, in this case the 8 vector mesons. We first construct a completely gauge-invariant theory and then add a mass term for the mesons. Let us call the eight fields