View source for Gell-Mann–Okubo mass formula

In [[physics]], the '''Gell-Mann–Okubo mass formula''' provides a [[sum rule in quantum mechanics|sum rule]] for the masses of [[hadron]]s within a specific multiplet, determined by their [[isospin]] (''I'') and [[strangeness]] (or alternatively, [[hypercharge]])
:<math>M = a_0 + a_1 Y + a_2 \left[ I \left( I + 1 \right)-\frac{1}{4} Y^2 \right] ,</math>
where ''a''<sub>0</sub>, ''a''<sub>1</sub>, and ''a''<sub>2</sub> are [[free parameter]]s.

The rule was first formulated by [[Murray Gell-Mann]] in 1961<ref>
{{cite paper
 |author=M. Gell-Mann
 |year=1961
 |title=The Eightfold Way: A Theory of Strong Interaction Symmetry
 |publisher=[[California Institute of Technology]]
 |series=Synchrotron Laboratory Report CTSL-20
 |doi=10.2172/4008239
 |osti=4008239
}}</ref> and independently proposed by [[Susumu Okubo]] in 1962.<ref>
{{cite journal
 |author=S. Okubo
 |year=1962
 |title=Note on Unitary Symmetry in Strong Interactions
 |journal=[[Progress of Theoretical Physics]]
 |volume=27 |issue=5 |pages=949–966
 |bibcode=1962PThPh..27..949O
 |doi=10.1143/PTP.27.949
}}</ref><ref>
{{cite journal
 |author=S. Okubo
 |year=1962
 |title=Note on Unitary Symmetry in Strong Interactions. II —Excited States of Baryons—
 |journal=[[Progress of Theoretical Physics]]
 |volume=28 |issue=1 |pages=24–32
 |bibcode=1962PThPh..28...24O
 |doi=10.1143/PTP.28.24
}}</ref> Isospin and hypercharge are generated by [[SU(3)]], which can be represented by [[Gell-Mann matrices|eight hermitian and traceless matrices]] corresponding to the "components" of isospin and hypercharge. Six of the matrices correspond to flavor change, and the final two correspond to the third-component of isospin projection, and hypercharge.

==Theory==
The mass formula was obtained by considering the [[Lie algebra representation|representations]] of the [[Lie algebra]] [[su(3)]]. In particular, the meson octet corresponds to the [[root system]] of the [[adjoint endomorphism|adjoint representation]]. However, the simplest, lowest-dimensional representation of su(3) is the [[fundamental representation]], which is three-dimensional, and is now understood to describe the approximate [[flavor symmetry]] of the three [[quark]]s ''u'', ''d'', and ''s''. Thus, the discovery of not only an su(3) symmetry, but also of this workable formula for the [[mass spectrum]] was one of the earliest indicators for the existence of quarks.

The formula is underlain by the ''octet enhancement hypothesis'', which ascribes dominance of SU(3) breaking to the hypercharge generator  of SU(3), 
<math>Y=\tfrac{2}{\sqrt{3}}F_8=\operatorname{diag}(1,1,-2)/3~</math>,
and, in modern terms, the relatively higher mass of the strange quark. An elegant abstract derivation of it is available  in Ch. 1.3.5 and  1.4 of S. Coleman's text.<ref>{{cite book |author= Sidney Coleman
|title= Aspects of Symmetry
|year= 1988
|publisher= Cambridge University Press
| url= https://books.google.com/books?id=PX2Al8LE9FkC
|isbn= 978-0-521-31827-3}}</ref>

This formula is ''phenomenological'', describing an approximate relation between meson and baryon masses, and has been superseded as theoretical work in [[quantum chromodynamics]] advances, notably chiral perturbation theory.

==Baryons==
{| class="wikitable" style="float:right; text-align:center; margin: 0em 0em 0em 1em;"
|+ Baryon properties<ref name=PDG>
{{cite journal
 |author=J. Beringer ''et al.'' ([[Particle Data Group]])
 |coauthors=<!-- -->
 |year=2012
 |title=Review of Particle Physics
 |journal=[[Physical Review D]]
 |volume=86 |issue=1 |page=010001
 |bibcode=2012PhRvD..86a0001B
 |doi=10.1103/PhysRevD.86.010001
 |postscript=<!-- -->
|hdl=1854/LU-3822071}} and [http://pdg.lbl.gov/ 2013 partial update] for the 2014 edition.</ref>
|-
!rowspan="6"| Octet
|-
! Name || Symbol || [[Isospin]] || [[Strangeness]] || Mass (MeV/''c''<sup>2</sup>)
|-
| [[Nucleon]]s || N || {{frac|1|2}} || 0 || 939
|-
| [[Lambda baryon]]s || &Lambda; || 0 || &minus;1 || 1116
|-
| [[Sigma baryon]]s || &Sigma; || 1 || &minus;1 || 1193
|-
| [[Xi baryon]]s || &Xi; || {{frac|1|2}} || &minus;2 || 1318
|-
!rowspan="5"| Decuplet
|-
| [[Delta baryon]]s || &Delta; || {{frac|3|2}} || 0 || 1232
|-
| [[Sigma baryon]]s || &Sigma;<sup>*</sup> || 1 || &minus;1 || 1385
|-
| [[Xi baryon]]s || &Xi;<sup>*</sup> || {{frac|1|2}} || &minus;2 || 1533
|-
| [[Omega baryon]] || &Omega; || 0 || &minus;3 || 1672
|-
|}

Using the values of relevant ''I'' and ''S'' for baryons, the Gell-Mann–Okubo formula can be rewritten for the baryon octet,
:<math>\frac{N + \Xi}{2} = \frac{3 \Lambda + \Sigma}{4} \, </math>
where ''N'', &Lambda;, &Sigma;, and &Xi; represent the average mass of corresponding baryons. Using the current mass of baryons,<ref name=PDG/> this yields:
:<math> \frac{N + \Xi}{2} = 1128.5~\mathrm{MeV}/c^2 </math>
and
:<math> \frac{3 \Lambda + \Sigma}{4} = 1135.25~\mathrm{MeV}/c^2</math>
meaning that the Gell-Mann–Okubo formula reproduces the mass of octet baryons within ~0.5% of measured values.

For the baryon decuplet, the Gell-Mann–Okubo formula can be rewritten as the "equal-spacing" rule
:<math>\Delta -\Sigma^* = \Sigma^* - \Xi^* = \Xi^* - \Omega = a_1 + 2a_2 \approx \, -147 ~\mathrm{MeV}/c^2</math>
where &Delta;, &Sigma;<sup>*</sup>, &Xi;<sup>*</sup>, and &Omega; represent the average mass of corresponding baryons.

The baryon decuplet formula famously allowed Gell-Mann to predict the mass of the then undiscovered &Omega;<sup>&minus;</sup>.<ref>
{{cite conference
 |last=Gell-Mann |first=M.
 |year=1962
 |title=Strange Particle Physics. Strong Interactions
 |editor=J. Prentki
 |booktitle=Proceedings of the International Conference on High-Energy Physics at CERN, Geneva,  1962
 |page=805
}}</ref><ref>{{cite journal
 |author=V. E. Barnes|year=1964
 |title=Observation of a Hyperon with Strangeness Number Three
 |url=http://teachers.web.cern.ch/teachers/archiv/HST2001/bubblechambers/omegaminus.pdf
 |journal=[[Physical Review Letters]]
 |volume=12 |issue=8 |page=204
 |bibcode=1964PhRvL..12..204B
 |doi=10.1103/PhysRevLett.12.204
|display-authors=etal}}</ref>

== Mesons ==
The same mass relation can be found for the meson octet,
:<math>\frac{1}{2}\left(\frac{ K^- + \bar{K}^0 }{2} + \frac{ K^+ + K^0}{2}\right) = \frac{3\eta + \pi}{4}</math>
Using the current mass of mesons,<ref name=PDG/> this yields
:<math> \frac{1}{2}\left(\frac{ K^- + \bar{K}^0 }{2} + \frac{ K^+ + K^0}{2}\right)= 496~\mathrm{MeV}/c^2</math>
and
:<math> \frac{3\eta + \pi}{4} = 445~\mathrm{MeV}/c^2</math>

Because of this large discrepancy, several people attempted to find a way to understand the failure of the GMO formula in mesons, when it worked so well in baryons. In particular, people noticed that using the square of the average masses yielded much better results:<ref>
{{cite book
 |author=D. J. Griffiths
 |year=1987
 |title=Introduction to Elementary Particles
 |publisher=[[John Wiley & Sons]]
 |isbn=978-0-471-60386-3
}}</ref>
:<math>\frac{1}{2}\left[ \left( \frac{ K^- + \bar{K}^0 }{2} \right)^2 + \left( \frac{ K^+ + K^0}{2} \right)^2\right] = \frac{3\eta^2 + \pi^2}{4}</math>
This now yields
:<math>\frac{1}{2}\left[ \left( \frac{ K^- + \bar{K}^0 }{2} \right)^2 + \left( \frac{ K^+ + K^0}{2} \right)^2\right]= 246\times10^3~\mathrm{MeV^2}/c^4</math>
and
:<math>\frac{3\eta^2 + \pi^2}{4} = 230\times10^3~\mathrm{MeV^2}/c^4</math>
which fall within 5% of each other.

For a while, the GMO formula involving the square of masses was simply an [[empirical relationship]]; but later a justification for using the square of masses was found<ref>
{{cite book
 |author1=J. F. Donoghue
 |author2=E. Golowich
 |author3=B. R. Holstein
 |year=1992
 |title=Dynamics of the Standard Model
 |pages=188–191
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-47652-2
}}</ref><ref>
{{cite book
 |author=S. Weinberg
 |year=1996
 |title=The Quantum Theory of Fields, Volume 2
 |pages=225–233
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-55002-4
}}</ref>  in the context of [[chiral perturbation theory]], just for pseudoscalar mesons, since these are the pseudogoldstone bosons of dynamically broken [[chiral symmetry breaking|chiral symmetry]], and, as such, obey Dashen's mass formula. Other, mesons, such as vector ones, need no squaring for the GMO formula to work.

==See also==
*[[Gell-Mann–Nishijima formula]]
*[[Eightfold Way (physics)|Eightfold Way]]
*[[Quark model]]
*[[SU(3)]]

==References==
{{reflist}}

==Further reading==
The following book contains most (if not all) historical papers on the Eightfold Way and related topics, including the Gell-Mann&ndash;Okubo mass formula.
*{{cite book
 |editor1=M. Gell-Mann
 |editor2=Y. Ne'eman
 |year=1964
 |title=The Eightfold Way
 |url=http://www.imamu.edu.sa/Scientific_selections/abstracts/Physics/THE%20EIGHTFOLD%20WAY.pdf
 |publisher=[[W. A. Benjamin]]
 |lccn=65013009
}}

{{DEFAULTSORT:Gell-Mann-Okubo mass formula}}
[[Category:Hadrons]]
[[Category:Quantum chromodynamics]]