Inphysics, theeightfold way is an organizational scheme for a class of subatomic particles known ashadrons that led to the development of thequark model. The American physicistMurray Gell-Mann and the Israeli physicistYuval Ne'eman independently and simultaneously proposed the idea in 1961.^[1][2][a]The name comes from Gell-Mann's (1961) paper, "The Eightfold Way: A theory of strong interaction symmetry." It is an allusion to theNoble Eightfold Path ofBuddhism andwas meant to be a joke.^[3]

By 1947, physicists believed that they had a good understanding of what the smallest bits of matter were. There wereelectrons,protons,neutrons, andphotons (the components that make up the vast part of everyday experience such asvisible matter and light) along with a handful of unstable (i.e., they undergoradioactive decay) exotic particles needed to explaincosmic rays observations such aspions,muons and the hypothesizedneutrinos. In addition, the discovery of thepositron suggested there could beanti-particles for each of them. It was known a "strong interaction" must exist to overcomeelectrostatic repulsion in atomic nuclei. Not all particles are influenced by this strong force; but those that are, are dubbed "hadrons"; these are now further classified asmesons (from the Greek for "intermediate") andbaryons (from the Greek for "heavy").

But the discovery of the neutralkaon in late 1947 and the subsequent discovery of a positively charged kaon in 1949 extended the meson family in an unexpected way, and in 1950 thelambda particle did the same thing for the baryon family. These particles decay much more slowly than they are produced, a hint that there are two different physical processes involved. This was first suggested byAbraham Pais in 1952. In 1953,Murray Gell-Mann and a collaboration in Japan, Tadao Nakano withKazuhiko Nishijima, independently suggested a new conserved value now known as "strangeness" during their attempts to understand the growing collection of known particles.^[4][5][b]The discovery of new mesons and baryons continued through the 1950s; the number of known "elementary" particles ballooned. Physicists were interested in understanding hadron-hadron interactions via the strong interaction. The concept ofisospin, introduced in 1932 byWerner Heisenberg shortly after the discovery of the neutron, was used to group some hadrons together into "multiplets" but no successful scientific theory as yet covered the hadrons as a whole. This was the beginning of a chaotic period in particle physics that has become known as the "particle zoo" era. The eightfold way represented a step out of this confusion and towards thequark model, which proved to be the solution.

Group representation theory is the mathematical underpinning of the eightfold way, but technical mathematics is not needed to understand how it helps organize particles. Particles are sorted into groups as mesons or baryons. Within each group, they are further separated by theirspin angular momentum. Symmetrical patterns appear when these groups of particles have theirstrangeness plotted against theirelectric charge. (This is the most common way to make these plots today, but originally physicists used an equivalent pair of properties calledhypercharge andisotopic spin, the latter of which is now known asisospin.) The symmetry in these patterns is a hint of the underlying symmetry of thestrong interaction between the particles themselves. In the plots below, points representing particles that lie along the same horizontal line share the same strangeness,s, while those on the same left-leaning diagonals share the same electric charge,q (given as multiples of theelementary charge).

In the original eightfold way, the mesons were organized into octets and singlets. This is one of the finer points of differences between the eightfold way and the quark model it inspired, which suggests the mesons should be grouped into nonets (groups of nine).

The eightfold way organizes eight of the lowestspin-0mesons into an octet.^[1][6] They are:

• K⁰
  ,K⁺
  ,K⁻
  andK⁰
  kaons
• π⁺
  ,π⁰
  , andπ⁻
  pions
• η, theeta meson

Diametrically opposite particles in the diagram areanti-particles of one another, while particles in the center are their own anti-particle.

The chargeless, strangeless eta prime meson was originally classified by itself as a singlet:

• η′

Under the quark model later developed, it is better viewed as part of a meson nonet, as previously mentioned.

The eightfold way organizes thespin-⁠1/ 2 ⁠baryons into an octet. They consist of

• neutron (n) andproton (p)
• Σ⁻
  ,Σ⁰
  , andΣ⁺
  sigma baryons
• Λ⁰
  , thestrange lambda baryon
• Ξ⁻
  andΞ⁰
  xi baryons

Theorganizational principles of the eightfold way also apply to the spin-⁠3/2⁠ baryons, forming adecuplet.

• Δ⁻
  ,Δ⁰
  ,Δ⁺
  , andΔ⁺⁺
  delta baryons
• Σ^∗−
  ,Σ^∗0
  , andΣ^∗+
  sigma baryons
• Ξ^∗−
  andΞ^∗0
  xi baryons
• Ω⁻
  omega baryon

However, one of the particles of this decuplet had never been previously observed when the eightfold way was proposed. Gell-Mann called this particle theΩ⁻
and predicted in 1962 that it would have astrangeness −3,electric charge −1 and a mass near1680 MeV/c². In 1964, a particle closely matching these predictions was discovered^[7] by aparticle accelerator group atBrookhaven. Gell-Mann received the 1969Nobel Prize in Physics for his work on the theory ofelementary particles.

Historically, quarks were motivated by an understanding of flavour symmetry. First, it was noticed (1961) that groups of particles were related to each other in a way that matched therepresentation theory of SU(3). From that, it was inferred that there is an approximate symmetry of the universe which is represented by the group SU(3). Finally (1964), this led to the discovery of three light quarks (up, down, and strange) interchanged by these SU(3) transformations.

The eightfold way may be understood in modern terms as a consequence offlavor symmetries between various kinds ofquarks. Since thestrong nuclear force affects quarks the same way regardless of their flavor, replacing one flavor of quark with another in a hadron should not alter its mass very much, provided the respective quark masses are smaller than the strong interaction scale—which holds for the three light quarks. Mathematically, this replacement may be described by elements of theSU(3) group. The octets and other hadron arrangements arerepresentations of this group.

There is an abstract three-dimensional vector space:$${\displaystyle \text{up quark}\to \left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\text{down quark}\to \left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),\text{strange quark}\to \left(\begin{array}{c}0\\ 0\\ 1\end{array}\right),}$$and the laws of physics areapproximately invariant under a determinant-1unitary transformation to this space (sometimes called aflavour rotation):$${\displaystyle \left(\begin{array}{c}x\\ y\\ z\end{array}\right)\mapsto A\left(\begin{array}{c}x\\ y\\ z\end{array}\right),\text{where}A\text{is in}SU(3).}$$

Here,SU(3) refers to theLie group of 3×3 unitary matrices with determinant 1 (special unitary group). For example, the flavour rotationis a transformation that simultaneously turns all the up quarks in the universe into down quarks and conversely. More specifically, these flavour rotations are exact symmetries ifonlystrong force interactions are looked at, but they are not truly exact symmetries of the universe because the three quarks have different masses and different electroweak interactions.

This approximate symmetry is calledflavour symmetry, or more specificallyflavour SU(3) symmetry.

Assume we have a certain particle—for example, a proton—in a quantum state${\displaystyle |\psi ⟩}$. If we apply one of the flavour rotationsA to our particle, it enters a new quantum state which we can call${\displaystyle A|\psi ⟩}$. Depending onA, this new state might be a proton, or a neutron, or a superposition of a proton and a neutron, or various other possibilities. The set of all possible quantum states spans a vector space.

Representation theory is a mathematical theory that describes the situation where elements of a group (here, the flavour rotationsA in the group SU(3)) areautomorphisms of a vector space (here, the set of all possible quantum states that you get from flavour-rotating a proton). Therefore, by studying the representation theory of SU(3), we can learn the possibilities for what the vector space is and how it is affected by flavour symmetry.

Since the flavour rotationsA are approximate, not exact, symmetries, each orthogonal state in the vector space corresponds to a different particle species. In the example above, when a proton is transformed by every possible flavour rotationA, it turns out that it moves around an 8 dimensional vector space. Those 8 dimensions correspond to the 8 particles in the so-called "baryon octet" (proton, neutron,Σ⁺
,Σ⁰
,Σ⁻
,Ξ⁻
,Ξ⁰
,Λ). This corresponds to an 8-dimensional ("octet") representation of the group SU(3). SinceA is an approximate symmetry, all the particles in this octet have similar mass.^[8]

EveryLie group has a correspondingLie algebra, and eachgroup representation of the Lie group can be mapped to a correspondingLie algebra representation on the same vector space. The Lie algebra${\displaystyle \mathfrak{s}\mathfrak{u}}$(3) can be written as the set of 3×3 tracelessHermitian matrices. Physicists generally discuss the representation theory of the Lie algebra${\displaystyle \mathfrak{s}\mathfrak{u}}$(3) instead of the Lie group SU(3), since the former is simpler and the two are ultimately equivalent.

• M. Gell-Mann; Y. Ne'eman, eds. (1964).The Eightfold Way.W. A. Benjamin.LCCN 65013009. (contains most historical papers on the eightfold way and related topics, including theGell-Mann–Okubo mass formula.)

• Particle physics
• Hadrons
• Quarks
• Concepts in physics
• Murray Gell-Mann

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