Gell-Mann–Okubo mass formula

In physics, the Gell-Mann–Okubo mass formula provides a sum rule for the masses of hadrons within a specific multiplet, determined by their isospin (I) and strangeness (or alternatively, hypercharge)

${\displaystyle M=a_{0}+a_{1}Y+a_{2}\left[I\left(I+1\right)-{\frac {1}{4}}Y^{2}\right],}$

where a₀, a₁, and a₂ are free parameters.

The rule was first formulated by Murray Gell-Mann in 1961[1] and independently proposed by Susumu Okubo in 1962.[2][3] Isospin and hypercharge are generated by SU(3), which can be represented by 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 representations of the Lie algebra su(3). In particular, the meson octet corresponds to the root system of the 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 quarks 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), ${\displaystyle Y={\tfrac {2}{\sqrt {3}}}F_{8}=\operatorname {diag} (1,1,-2)/3~}$, 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.[4]

This formula is phenomenological, describing an approximate relation between meson and baryon masses, and has been superseded as theoretical work in quantum chromodynamics advances, notably chiral perturbation theory.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

Baryon properties[5]

Octet
	Name	Symbol	Isospin	Strangeness	Mass (MeV/c^2)
	Nucleons	N	​^1⁄2	0	939
	Lambda baryons	Λ	0	−1	1116
	Sigma baryons	Σ	1	−1	1193
	Xi baryons	Ξ	​^1⁄2	−2	1318
Decuplet
	Delta baryons	Δ	​^3⁄2	0	1232
	Sigma baryons	Σ^*	1	−1	1385
	Xi baryons	Ξ^*	​^1⁄2	−2	1533
	Omega baryon	Ω	0	−3	1672

Using the values of relevant I and S for baryons, the Gell-Mann–Okubo formula can be rewritten for the baryon octet,

${\displaystyle {\frac {N+\Xi }{2}}={\frac {3\Lambda +\Sigma }{4}}\,}$

where N, Λ, Σ, and Ξ represent the average mass of corresponding baryons. Using the current mass of baryons,[5] this yields:

${\displaystyle {\frac {N+\Xi }{2}}=1128.5~\mathrm {MeV} /c^{2}}$

and

${\displaystyle {\frac {3\Lambda +\Sigma }{4}}=1135.25~\mathrm {MeV} /c^{2}}$

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

${\displaystyle \Delta -\Sigma ^{*}=\Sigma ^{*}-\Xi ^{*}=\Xi ^{*}-\Omega =a_{1}+2a_{2}\approx \,-147~\mathrm {MeV} /c^{2}}$

where Δ, Σ^*, Ξ^*, and Ω represent the average mass of corresponding baryons.

The baryon decuplet formula famously allowed Gell-Mann to predict the mass of the then undiscovered Ω⁻.[6][7]

Mesons

The same mass relation can be found for the meson octet,

${\displaystyle {\frac {1}{2}}\left({\frac {K^{-}+{\bar {K}}^{0}}{2}}+{\frac {K^{+}+K^{0}}{2}}\right)={\frac {3\eta +\pi }{4}}}$

Using the current mass of mesons,[5] this yields

${\displaystyle {\frac {1}{2}}\left({\frac {K^{-}+{\bar {K}}^{0}}{2}}+{\frac {K^{+}+K^{0}}{2}}\right)=496~\mathrm {MeV} /c^{2}}$

and

${\displaystyle {\frac {3\eta +\pi }{4}}=445~\mathrm {MeV} /c^{2}}$

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:[8]

${\displaystyle {\frac {1}{2}}\left[\left({\frac {K^{-}+{\bar {K}}^{0}}{2}}\right)^{2}+\left({\frac {K^{+}+K^{0}}{2}}\right)^{2}\right]={\frac {3\eta ^{2}+\pi ^{2}}{4}}}$

This now yields

${\displaystyle {\frac {1}{2}}\left[\left({\frac {K^{-}+{\bar {K}}^{0}}{2}}\right)^{2}+\left({\frac {K^{+}+K^{0}}{2}}\right)^{2}\right]=246\times 10^{3}~\mathrm {MeV^{2}} /c^{4}}$

and

${\displaystyle {\frac {3\eta ^{2}+\pi ^{2}}{4}}=230\times 10^{3}~\mathrm {MeV^{2}} /c^{4}}$

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[9][10] in the context of chiral perturbation theory, just for pseudoscalar mesons, since these are the pseudogoldstone bosons of dynamically broken 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
• Quark model
• SU(3)

References

1. M. Gell-Mann (1961). "The Eightfold Way: A Theory of Strong Interaction Symmetry" (PDF). Synchrotron Laboratory Report CTSL-20. California Institute of Technology. doi:10.2172/4008239. OSTI 4008239. Cite journal requires |journal= (help)
2. S. Okubo (1962). "Note on Unitary Symmetry in Strong Interactions". Progress of Theoretical Physics. 27 (5): 949–966. Bibcode:1962PThPh..27..949O. doi:10.1143/PTP.27.949.
3. S. Okubo (1962). "Note on Unitary Symmetry in Strong Interactions. II —Excited States of Baryons—". Progress of Theoretical Physics. 28 (1): 24–32. Bibcode:1962PThPh..28...24O. doi:10.1143/PTP.28.24.
4. Sidney Coleman (1988). Aspects of Symmetry. Cambridge University Press. ISBN 978-0-521-31827-3.
5. J. Beringer et al. (Particle Data Group) (2012). "Review of Particle Physics". Physical Review D. 86 (1): 010001. Bibcode:2012PhRvD..86a0001B. doi:10.1103/PhysRevD.86.010001. hdl:1854/LU-3822071. and 2013 partial update for the 2014 edition.
6. Gell-Mann, M. (1962). "Strange Particle Physics. Strong Interactions". In J. Prentki (ed.). Proceedings of the International Conference on High-Energy Physics at CERN, Geneva, 1962. p. 805.
7. V. E. Barnes; et al. (1964). "Observation of a Hyperon with Strangeness Number Three" (PDF). Physical Review Letters. 12 (8): 204. Bibcode:1964PhRvL..12..204B. doi:10.1103/PhysRevLett.12.204.
8. D. J. Griffiths (1987). Introduction to Elementary Particles. John Wiley & Sons. ISBN 978-0-471-60386-3.
9. J. F. Donoghue; E. Golowich; B. R. Holstein (1992). Dynamics of the Standard Model. Cambridge University Press. pp. 188–191. ISBN 978-0-521-47652-2.
10. S. Weinberg (1996). The Quantum Theory of Fields, Volume 2. Cambridge University Press. pp. 225–233. ISBN 978-0-521-55002-4.

Further reading

The following book contains most (if not all) historical papers on the Eightfold Way and related topics, including the Gell-Mann–Okubo mass formula.

• M. Gell-Mann; Y. Ne'eman, eds. (1964). The Eightfold Way (PDF). W. A. Benjamin. LCCN 65013009.