EIGHTFOLD WAY

The eightfold way is the term coined by Murray Gell-Mann in 1961 to describe a classification scheme for elementary particles that he and Yuval Ne'eman had devised. The name, adopted from the Eightfold Path of Buddhism, refers to the eight-member families to which many sets of particles belong.

In the 1950s Gell-Mann and Kazuo Nishijima invented a scheme to explain a "strange" feature of certain particles; they appeared to be easily produced in cosmic-ray and accelerator reactions but decayed slowly, as if something were hindering their decays. These particles were assumed to carry a property known as strangeness that would be preserved in production but could be changed in decays. Two examples of plots of electric charge Q (in units of the fundamental charge |e |) versus strangeness for certain particles, most of which were known in the late 1950s, are given in Figure 1.

Mesons include the π particles, known as pions, whose existence was proposed by Hideki Yukawa in 1935 to explain the strong nuclear force, and the K particles (also known as kaons), discovered in cosmic radiation in the 1940s. Pions and kaons weigh about one-seventh and one-half as much as protons, respectively. Baryons (the prefix bary - is Greek for heavy )

FIGURE 1

include the proton p , the neutron n , and heavier relatives Λ (lambda), Σ (sigma), and Ξ (xi), collectively known as hyperons and discovered in the 1940s and 1950s. The rationale for these families was sought through symmetries of the strong interactions.

According to the Gell-Mann–Nishijima scheme, reactions in which these particles are produced must have equal total strangeness on each side. For example, Κ⁰ and λ can be produced by the reaction

π^¯(S = 0) + ρ (S = 0)→ Κ⁰(S = 1) + Λ(S = -1).

This scheme thus explained another curious feature of the "strange particles": they never appeared to be produced singly in reactions caused by protons, neutrons, and π mesons.

In the 1930s Werner Heisenberg and others had recognized that the similarities of the proton and neutron with respect to their nuclear interactions and masses could be described by a quantity known as isotopic spin. This quantity, called isospin for short, is analogous to ordinary spin with the proton's isospin pointing "up" and the neutron's pointing "down." Mathematically, isospin is described by a symmetry group that is a set of transformations that leaves interactions unchanged, known as SU(2). The 2 refers to the proton and neutron.

Families whose members are related to one another by SU(2) transformations can have any number of members, including the two-member family to which the proton and neutron belong. Collectively, ρ and n are known as nucleons and denoted by the symbol Ν . Isospin predicts that certain sets of particles with different charges (e.g., Κ or Σ) should have similar masses and strong interactions, as is observed.

In 1956 Shoichi Sakata proposed that mesons were composed of the proton ρ , the neutron n , the lambda Λ, and corresponding antiparticles, with binding forces so large as to overcome most of their masses. Thus, for example, the Κ⁺ would be ρ ƛ̄. (The bar over a symbol denotes its antiparticle; the electric charges and strangeness of antiparticles are opposite to those of the corresponding particles.) The remaining known baryons (the Σ and Ξ) had to be accounted for in more complicated ways. The Sakata model had the symmetry known as SU(3), where 3 referred to p , n , and Λ.

Gell-Mann and Ne'eman recognized that if electric charge were to be part of the SU(3) description, particles whose electric charges were integer multiples of |e | could belong only to certain families. The simplest of these contained one, eight, and ten members. Other families, such as those containing three and six members, would have fractionally charged members, and fractional charges had never been seen in nature. Both the mesons and baryons mentioned above would then have to belong to eight-member families. The baryons fit such a family exactly, leading Gell-Mann to call his scheme the eightfold way. In addition to the known Κ and π mesons shown in the meson octet of Figure 1, there would have to be an eighth meson, which was neutral and had zero strangeness. This particle, now called the η (eta), was discovered in 1961 by Pevsner et al.

A consequence of the eightfold way for describing mesons and baryons was that their masses Μ could be related to one another by formulae proposed by Gell-Mann and Susumu Okubo in 1962:

Mesons: 4Μ (Κ ) = Μ(π) + Μ(μ)

Baryons: 2[Μ (N ) + Μ (Ξ)] = Μ (Σ) + 3Μ (Λ).

These formulae, particularly the one for baryons, were obeyed quite well. More evidence for SU(3) soon materialized as the result of another experimental discovery.

Certain baryons known as Δ (delta), Σ* (sigma-star), and Ξ* (xi-star) appeared to fit into a ten-member family (a decuplet, which would be completed by a not yet observed particle known as the Ω¯[omega-minus]). The mass of the Ω¯ could be anticipated within a few percent because the Gell-Mann–Okubo mass formula for these particles predicted

Μ (Ω¯) - Μ (Ξ*) = Μ (Ξ*) - Μ (Σ*)= Μ (Σ*) - Μ (Δ).

An experiment at Brookhaven National Laboratory (BNL), discussed by Barnes et al., detected this particle with the predicted mass through a decay that left no doubt as to its nature. Figure 2 shows its place in the baryon decuplet.

FIGURE 2

An early application of the eightfold way, building on suggestions by Gell-Mann and Maurice Levy in 1960 and by Gell-Mann again in 1962, was made by Nicola Cabibbo in 1963. Applying the concept to certain decays of baryons, Cabibbo showed that SU(3) symmetry could be used to describe not only the existence and masses of particles but also their interactions.

Underlying the success of the eightfold way and the symmetry group SU(3) is the existence of fundamental subunits of matter, called quarks by Gell-Mann in 1964 and aces by their coinventor George Zweig in 1964. These objects can belong to a family of three fractionally charged members u (up), d (down), and s (strange), as shown in Figure 3.

The fact that fractionally charged objects have not been seen in nature requires quarks to combine with one another in such a way as to produced only integrally charged particles. This is one successful prediction of the theory of the strong interactions, quantum chromodynamics (QCD). Baryons are made of three quarks, whereas mesons are made of a quark and an antiquark (with reversed charge and strangeness). For example, the Δ⁺⁺ is made of uuu ; the Δ¯ is made of ddd ; the Ω¯ is made of sss ; and the Κ⁺ is made of us̄ .

The SU(3) symmetry described here refers to the flavor of quarks (u, d, s ). A separate SU(3), associated with quantum chromodynamic degrees of freedom, describes the colors of quarks. Each flavor of quark can exist in three colors. Other flavors of quarks—charm (c ), bottom (b ), and top (t )—were

FIGURE 3

discovered subsequently. They are much heavier than u, d, and s and so do not fit easily into a generalization SU(n ) of SU(3) with n > 3. The approximate SU(3) symmetry of particles containing u, d, and s quarks remains a useful guide to properties of the strong interactions.

See also:Flavor Symmetry; Particle Physics, Elementary; Quarks, Discovery of; SU(3)

Bibliography

Barnes, V. E. "Observation of a Hyperon with Strangeness Minus Three." Physical Review Letters12 , 204–206 (1964).

Cabibbo, N. "Unitary Symmetry and Leptonic Decays." Physical Review Letters10, 531–533 (1963).

Gell-Mann, M. "The Eightfold Way: A Theory of Strong Interaction Symmetry." California Institute of Technology Report. CTSL-20 (1961).

Gell-Mann, M. "Symmetries of Baryons and Mesons." Physical Review125 , 1067–1084 (1962).

Gell-Mann, M. "A Schematic Model of Baryons and Mesons." Physical Review Letters8 , 214–215 (1964).

Gell-Mann, M., and Levy, M. "The Axial Vector Current in

Beta Decay." Nuovo Cimento16 , 705 (1960).

Gell-Mann, M., and Ne'eman, Y. The Eightfold Way (W. A. Benjamin, New York, 1964).

Ne'eman, Y. "Derivation of Strong Interactions from a Gauge Invariance." Nuclear Physics26 , 222–229 (1961).

Okubo, S. "Note on Unitary Symmetry in Strong Interactions." Progress of Theoretical Physics (Kyoto) 27 , 949–966 (1962).

Pevsner, A., et al. "Evidence for a Three Pion Resonance Near 550 MeV." Physical Review Letters7 , 421–423 (1961).

Sakata, S. "On a Composite Model for the New Particles." Progress of Theoretical Physics (Kyoto) 16 , 686–688 (1956).

Zweig, G. "An SU(3) Model for Strong Interaction Symmetry and its Breaking" in Developments in the Quark Theory of Hadrons, Vol. 1, edited by D. B. Lichtenberg and S. P. Rosen (Hadronic Press, Nonantum, MA, 1980), pp. 22–101.

Jonathan L. Rosner