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)
where a0, a1, and a2 are free parameters.
The rule was first formulated by
Theory
The mass formula was obtained by considering the
The formula is underlain by the octet enhancement hypothesis, which ascribes dominance of SU(3) breaking to the hypercharge generator of SU(3), , and, in modern terms, the relatively higher mass of the strange quark.[4][5]
This formula is phenomenological, describing an approximate relation between meson and baryon masses, and has been superseded as theoretical work in quantum chromodynamics advances, notably chiral perturbation theory.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
Octet | ||||
---|---|---|---|---|
Name | Symbol | Isospin | Strangeness | Mass (MeV/c2) |
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,
where N, Λ, Σ, and Ξ represent the average mass of corresponding baryons. Using the current mass of baryons,[6] this yields:
and
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
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−.[7][8]
Mesons
The same mass relation can be found for the meson octet,
Using the current mass of mesons,[6] this yields
and
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:[9]
This now yields
and
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[10][11] 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
- ^ M. Gell-Mann (1961). "The Eightfold Way: A Theory of Strong Interaction Symmetry" (PDF). Synchrotron Laboratory Report CTSL-20. )
- ^ S. Okubo (1962). "Note on Unitary Symmetry in Strong Interactions". .
- ^ S. Okubo (1962). "Note on Unitary Symmetry in Strong Interactions. II —Excited States of Baryons—". .
- ISBN 978-0-521-31827-3.
- .
- ^ a b c
J. Beringer; et al. (hdl:1854/LU-3822071. and 2013 partial updatefor the 2014 edition.
- ^ M. Gell-Mann (1962). "Strange Particle Physics. Strong Interactions". In J. Prentki (ed.). Proceedings of the International Conference on High-Energy Physics at CERN, Geneva, 1962. CERN. p. 805.
- OSTI 12491965.
- ^
D. J. Griffiths (1987). Introduction to Elementary Particles. ISBN 978-0-471-60386-3.
- ^
J. F. Donoghue; E. Golowich; B. R. Holstein (1992). Dynamics of the Standard Model. ISBN 978-0-521-47652-2.
- ^
S. Weinberg (1996). The Quantum Theory of Fields, Volume 2. 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). LCCN 65013009.