Editing Eightfold way (physics)

{{Short description|Classification scheme for hadrons}}
[[Image:meson octet.png|thumb|The [[pseudoscalar meson]] octet. Particles along the same horizontal line share the same [[Strangeness (particle physics)|strangeness]], {{mvar|s}}, while those on the same left-leaning diagonals share the same [[electric charge|charge]], {{mvar|q}} (given as multiples of the [[elementary charge]]).]]
In [[physics]], the '''eightfold way''' is an organizational scheme for a class of subatomic particles known as [[hadron]]s that led to the development of the [[quark model]]. Both the American physicist [[Murray Gell-Mann]] and the Israeli physicist [[Yuval Ne'eman]] independently and simultaneously proposed the idea in 1961.<ref name=Gell-Mann-1961-TID-12608/><ref name=Ne-eman-1961-08/>{{efn|
In Gell-Mann's 1961 paper, Reference&nbsp;6 says
{{blockquote|After the circulation of the preliminary version of this work (January&nbsp;1961) the author has learned of a similar theory put forward independently and simultaneously by [[Yuval Ne'eman|Y.&nbsp;Ne'eman]] ([[Nuclear Physics (journal)|Nuclear Physics]], to be published). Earlier uses of the 3&nbsp;dimensional unitary group in connection with the [[Sakata model]] are reported by Y.&nbsp;Ohnuki at the 1960 Rochester Conference on High Energy Physics. [[Abdus Salam|A.&nbsp;Salam]] and J.&nbsp;Ward ([[Nuovo Cimento]], to be published) have considered related questions. The author would like to thank Dr.&nbsp;Ne'eman and Professor Salam for communicating their results to him.}}
while the very end of Ne'eman's (1961) paper reads,
{{blockquote|I am indebted to Prof. A.&nbsp;Salam for discussions on this problem. In fact, when I presented this paper to him, he showed me a study he had done on the unitary theory of the Sakata model, treated as a gauge, and thus producing a similar set of vector bosons. Shortly after the present paper was written, a further version, utilizing the 8&nbsp;representation for baryons, as in this paper, reached us in a [[preprint]] by Prof. [[Murray Gell-Mann|M. Gell Mann]].}}
}}
The name comes from Gell-Mann's (1961) paper and is an allusion to the [[Noble Eightfold Path]] of [[Buddhism]].<ref name=Young-Freedman-2004-Sears-Zemansky/>

==Background==
By 1947, physicists believed that they had a good understanding of what the smallest bits of matter were. There were [[electron]]s, [[proton]]s, [[neutron]]s, and [[photon]]s (the components that make up the vast part of everyday experience such as [[macroscopic scale|visible matter]] and light) along with a handful of unstable (i.e., they undergo [[radioactive decay]]) exotic particles needed to explain [[cosmic ray]]s observations such as [[pion]]s, [[muon]]s and the hypothesized [[neutrino]]s. In addition, the discovery of the [[positron]] suggested there could be [[anti-particle]]s for each of them. It was known a "[[strong interaction]]" must exist to overcome [[Coulomb's law|electrostatic 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 as [[meson]]s (from the Greek for "intermediate") and [[baryon]]s (from the Greek for "heavy").

But the discovery of the neutral [[kaon]] 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 the [[lambda 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 by [[Abraham Pais]] in 1952. In 1953, [[Murray Gell-Mann]] and a collaboration in Japan, Tadao Nakano with [[Kazuhiko Nishijima]], independently suggested a new conserved value now known as "[[strangeness]]" during their attempts to understand the growing collection of known particles.<ref name=Gell-Mann-1953-11/><ref name=Nakano-Nishijima-1953-11/>{{efn|
A footnote in Nakano and Nishijima's paper says
{{blockquote|After the completion of this work, the authors knew in a private letter from Prof.&nbsp;Nambu to Prof.&nbsp;Hayakawa that Dr.&nbsp;Gell-Mann has also developed a similar theory.}}
}}
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 of [[isospin]], introduced in 1932 by [[Werner 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 the [[quark model]], which proved to be the solution.

==Organization==
[[Group representation theory]] is the mathematical underpinning of the eightfold way, but that rather 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 their [[Spin (physics)|spin]] angular momentum. Symmetrical patterns appear when these groups of particles have their [[Strangeness (particle physics)|strangeness]] plotted against their [[electric charge]]. (This is the most common way to make these plots today, but originally physicists used an equivalent pair of properties called ''hypercharge'' and ''isotopic spin'', the latter of which is now known as ''isospin''.) The symmetry in these patterns is a hint of the underlying symmetry of the [[strong interaction]] between the particles themselves. In the plots below, points representing particles that lie along the same horizontal line share the same strangeness, {{mvar|s}}, while those on the same left-leaning diagonals share the same electric charge, {{mvar|q}} (given as multiples of the [[elementary charge]]).

===Mesons===
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).

====Meson octet====
[[Image:meson octet.png|thumb|The [[pseudoscalar meson]] octet]]
The eightfold way organizes eight of the lowest [[Spin (physics)|spin]]-0 [[meson]]s into an octet.<ref name=Gell-Mann-1961-TID-12608/><ref name=Gell-Mann-1962/> They are:
* {{SubatomicParticle|Kaon0}}, {{SubatomicParticle|Kaon+}}, {{SubatomicParticle|Kaon-}} and {{SubatomicParticle|AntiKaon0}} [[kaon]]s
* {{SubatomicParticle|pion+}}, {{SubatomicParticle|pion0}}, and {{SubatomicParticle|pion-}} [[pion]]s
* {{SubatomicParticle|eta}}, the [[eta meson]]

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

====Meson singlet====
The chargeless, strangeless eta prime meson was originally classified by itself as a singlet:
* [[Eta meson|{{SubatomicParticle|eta prime}}]]

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

===Baryons===

====Baryon octet====
[[Image:Baryon octet.png|thumb|The {{mvar|J}} = {{sfrac|1|2}} [[baryon]] octet]]
The eightfold way organizes the [[Spin (physics)|spin]]-{{sfrac|1| 2 }} [[baryon]]s into an octet. They consist of
* [[neutron]] (n) and [[proton]] (p) 
* {{SubatomicParticle|sigma-}}, {{SubatomicParticle|sigma0}}, and {{SubatomicParticle|sigma+}} [[sigma baryon]]s
* {{Subatomic particle|Lambda0}}, the [[lambda baryon|strange lambda baryon]]
* {{SubatomicParticle|xi-}} and {{SubatomicParticle|xi0}} [[xi baryon]]s

====Baryon decuplet====
[[Image:Baryon decuplet.png|thumb|The {{mvar|J}} = {{sfrac|3|2}} [[baryon decuplet]]]]
The [[Special unitary group#Lie algebra 3|organizational principles of the eightfold way]] also apply to the spin-{{sfrac|3|2}} baryons, forming a [[Baryon#Isospin and charge|decuplet]].
* {{SubatomicParticle|delta-}}, {{SubatomicParticle|delta0}}, {{SubatomicParticle|delta+}}, and {{SubatomicParticle|delta++}} [[delta baryon]]s
* {{SubatomicParticle|sigma*-}}, {{SubatomicParticle|sigma*0}}, and {{SubatomicParticle|sigma*+}} [[sigma baryon]]s
* {{SubatomicParticle|xi*-}} and {{SubatomicParticle|xi*0}} [[xi baryon]]s
* {{SubatomicParticle|omega-}} [[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 the {{SubatomicParticle|link=yes|Omega-}} and predicted in 1962 that it would have a [[strangeness]] &minus;3, [[electric charge]] &minus;1 and a mass near {{val|1680|u=MeV/c2}}. In 1964, a particle closely matching these predictions was discovered<ref name=Barnes-Connolly-etal-1964/> by a [[particle accelerator]] group at [[Brookhaven National Laboratory|Brookhaven]]. Gell-Mann received the 1969 [[Nobel Prize in Physics]] for his work on the theory of [[elementary particle]]s.

==Historical development==

===Development===

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 the [[Clebsch–Gordan coefficients for SU(3)|representation 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.

===Modern interpretation===

The eightfold way may be understood in modern terms as a consequence of [[flavour (particle physics)|flavor]] symmetries between various kinds of [[quark]]s. Since the [[strong interaction|strong 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 the [[special unitary group|SU(3) group]]. The octets and other hadron arrangements are [[representation theory|representations]] of this group.

==Flavor symmetry==
{{main|Flavour (particle physics)}}

===SU(3)===

There is an abstract three-dimensional vector space:
<math display="block">
 \text{up quark} \to \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \qquad
 \text{down quark} \to \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad
 \text{strange quark} \to \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix},
</math>
and the laws of physics are ''approximately'' invariant under a determinant-1 [[unitary transformation]] to this space (sometimes called a ''flavour rotation''):
<math display="block">
 \begin{pmatrix} x \\ y \\ z \end{pmatrix} \mapsto
 A \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad
 \text{where}\ A\ \text{is in}\ SU(3).</math>

Here, [[SU(3)]] refers to the [[Lie group]] of 3×3 unitary matrices with determinant 1 ([[special unitary group]]). For example, the flavour rotation
<math display="block">
 A = \begin{pmatrix}
 \phantom- 0 & 1 & 0 \\
          -1 & 0 & 0 \\
 \phantom- 0 & 0 & 1
\end{pmatrix}</math>
is 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 if ''only'' [[strong 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 called ''[[flavour symmetry]]'', or more specifically ''flavour SU(3) symmetry''.

{{see also|Clebsch–Gordan coefficients for SU(3)#Representations of the SU(3) group}}

===Connection to representation theory===
{{main|Particle physics and representation theory}}
{{see also|Compact group#Representation theory of a connected compact Lie group}}Assume we have a certain particle—for example, a proton—in a quantum state <math>|\psi\rangle</math>. If we apply one of the flavour rotations ''A'' to our particle, it enters a new quantum state which we can call <math>A|\psi\rangle</math>. Depending on ''A'', 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 rotations ''A'' in the group SU(3)) are [[automorphism]]s 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 rotations ''A'' 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 rotation ''A'', it turns out that it moves around an 8&nbsp;dimensional vector space. Those 8&nbsp;dimensions correspond to the 8&nbsp;particles in the so-called "baryon octet" (proton, neutron, [[Sigma baryon|{{SubatomicParticle|Sigma+}}, {{SubatomicParticle|Sigma0}}, {{SubatomicParticle|Sigma-}}]], [[Xi baryon|{{SubatomicParticle|Xi-}}, {{SubatomicParticle|Xi0}}]], [[Lambda baryon|{{SubatomicParticle|Lambda}}]]). This corresponds to an 8-dimensional ("octet") representation of the group SU(3). Since ''A'' is an approximate symmetry, all the particles in this octet have similar mass.<ref name=Griffiths-2008/>

Every [[Lie group]] has a corresponding [[Lie algebra]], and each [[group representation]] of the Lie group can be mapped to a corresponding [[Lie algebra representation]] on the same vector space. The Lie algebra <math>\mathfrak{su}</math>(3) can be written as the set of 3×3 traceless [[Hermitian matrices]]. Physicists generally discuss the representation theory of the Lie algebra <math>\mathfrak{su}</math>(3) instead of the Lie group SU(3), since the former is simpler and the two are ultimately equivalent.

==Notes==
{{notelist}}

==References==
{{reflist|25em|refs=

<ref name=Barnes-Connolly-etal-1964>
{{cite journal
 |author1 =Barnes, V.E.    |author2 =Connolly, P.L.   |author3 =Crennell, D.J.
 |author4 =Culwick, B.B.   |author5 =Delaney, W.C.    |author6 =Fowler, W.B.
 |author7 =Hagerty, P.E.   |author8 =Hart, E.L.       |author9 =Horwitz, N.
 |author10=Hough, P.V.C.   |author11=Jensen, J.E.     |author12=Kopp, J.K.
 |author13=Lai, K.W.       |author14=Leitner, J.      |author15=Lloyd, J.L.
 |author16=London, G.W.    |author17=Morris, T.W.     |author18=Oren, Y.
 |author19=Palmer, R.B.    |author20=Prodell, A.G.    |author21=Radojičić, D.
 |author22=Rahm, D.C.      |author23=Richardson, C.R. |author24=Samios, N.P.
 |author25=Sanford, J.R.   |author26=Shutt, R.P.      |author27=Smith, J.R.
 |author28=Stonehill, D.L. |author29=Strand, R.C.     |author30=Thorndike, A.M.
 |author31=Webster, M.S.   |author32=Willis, W.J.     |author33=Yamamoto, S.S. 
 |display-authors=6
 |year=1964
 |title=Observation of a hyperon with strangeness minus three
 |journal=[[Physical Review Letters]]
 |volume=12  |issue=8  |page=204
 |bibcode=1964PhRvL..12..204B
 |doi=10.1103/PhysRevLett.12.204
 |osti =12491965 |url=http://teachers.web.cern.ch/teachers/archiv/HST2001/bubblechambers/omegaminus.pdf
}}
</ref>

<ref name=Gell-Mann-1953-11>
{{cite journal
 |last=Gell-Mann |first=M. |author-link=Murray Gell-Mann
 |date=November 1953
 |title=Isotopic spin and new unstable particles
 |journal=Phys. Rev.
 |volume=92  |issue=3  |pages=833–834
 |doi=10.1103/PhysRev.92.833  |bibcode=1953PhRv...92..833G 
 |url=https://authors.library.caltech.edu/60471/1/PhysRev.92.833.pdf
}}
</ref>

<ref name=Gell-Mann-1961-TID-12608>
{{cite report
 |last=Gell-Mann |first=Murray |author-link=Murray Gell-Mann
 |date=15 March 1961
 |title=The Eightfold Way: A theory of strong interaction symmetry
 |publisher=Office of Scientific and Technical Information (OSTI)
 |doi=10.2172/4008239
 | doi-access=free
 |url=https://www.osti.gov/scitech/servlets/purl/4008239
}}
</ref>

<ref name=Gell-Mann-1962>
{{cite journal
 | last = Gell-Mann | first = M. | author-link = Murray Gell-Mann
 | year = 1962
 | title = Symmetries of baryons and mesons
 | journal = Physical Review
 | volume = 125 | issue = 3 | page = 1067
 | bibcode = 1962PhRv..125.1067G
 | doi = 10.1103/physrev.125.1067 | doi-access = free
}}
</ref>

<ref name=Griffiths-2008>
{{cite book
 |author=Griffiths, D. 
 |year=2008
 |title=Introduction to Elementary Particles
 |edition=2nd.
 |publisher=[[Wiley-VCH]]
 |isbn=978-3527406012 
}}
</ref>

<ref name=Nakano-Nishijima-1953-11>
{{cite journal
 |last1=Nakano    |first1=Tadao    |author-link1=Tadao Nakano 
 |last2=Nishijima |first2=Kazuhiko |author-link2=Kazuhiko Nishijima
 |date=November 1953
 |title=Charge independence for ''V''-particles
 |journal=Progress of Theoretical Physics
 |volume=10 |issue=5 |pages=581–582
 |doi=10.1143/PTP.10.581 |doi-access=free
 |bibcode=1953PThPh..10..581N
}}
</ref>

<ref name=Ne-eman-1961-08>
{{cite journal
 |last=Ne'eman |first=Y. |author-link=Yuval Ne'eman
 |date=August 1961
 |title=Derivation of strong interactions from a gauge invariance
 |journal=[[Nuclear Physics (journal)|Nuclear Physics]]
 |publisher=North-Holland Publishing Co.
 |location=Amsterdam
 |doi=10.1016/0029-5582(61)90134-1
 |volume=26 |issue=2 |pages=222–229
 |bibcode=1961NucPh..26..222N
}}
</ref>

<ref name=Young-Freedman-2004-Sears-Zemansky>
{{cite book
 |last1=Young |first1=Hugh D.
 |last2=Freedman |first2=Roger A.
 |others=contributions by A. Lewis Ford
 |year=2004
 |title=Sears and Zemansky's University Physics with Modern Physics
 |edition=11th International
 |publisher=Pearson/Addison Wesley
 |location=San Francisco, CA
 |isbn=0-8053-8684-X
 |page=1689
 |quote=The name is a slightly irreverent reference to the ''[[Noble Eightfold Path]]'', a set of principles for right living in [[Buddhism]].
}}
</ref>

<!--  -----------------------
 park <ref name=...> ... </ref> references here and link
 by substituting the into the text, just the label
   <ref name=.../>    for the
 body of the reference (kept here).
  -----------------------   -->

}}  <!-- end "refs=" -->

==Further reading==
* {{cite book
 |editor1=M. Gell-Mann
 |editor2=Y. Ne'eman
 |year=1964
 |title=The Eightfold Way
 |url=http://bookzz.org/book/1271076/8ff905
 |publisher=[[W. A. Benjamin]]
 |lccn=65013009
}} (contains most historical papers on the eightfold way and related topics, including the [[Gell-Mann–Okubo mass formula]].)

{{Particles}}

[[Category:Particle physics]]
[[Category:Hadrons]]
[[Category:Quarks]]
[[Category:Concepts in physics]]
[[Category:Murray Gell-Mann]]