Gell-Mann–Okubo mass formula

In physics, the Gell-Mann–Okubo mass formula provides a sum rule for baryon and meson masses within a specific multiplet determined by particle spin. The rule was first formulated by Murray Gell-Mann in 1961[1] and independently proposed by Susumu Okubo in 1962[2]. Spin, in its simplest generalization is generated by SU(3), which can be represented by eight unitary and traceless matrices corresponding to the "components" of spin. Three of the matrices correspond to the three components of spin, while four correspond to flavor change, and the final to hypercharge.

For the baryon octet, the final matrix for hypercharge can be transformed to provide two mass difference matrices relating the masses of the baryons within the octet. Using these matrices the Gell-Mann-Okubo formula can be written for the baryon octet,

2(m_N%2Bm_\Xi) = 3m_\Lambda %2B m_\Sigma \,

where mN is the average mass of the proton and neutron, mΞ the average mass of the Ξ0 and Ξ−, and mΣ the average mass of Σ0, Σ+, and Σ−.

The same mass relation can be found for the meson octet, but now the masses are squared[3].

\frac{1}{2}\left((m_{K^-}%2Bm_{\bar{K}^0})^2 %2B (m_{K^%2B}%2Bm_{K^0})^2\right) = 3m_\eta^2 %2B \frac{1}{9}(m_{\pi^0}%2Bm_{\pi^%2B}%2Bm_{\pi^-})^2

For the baryon decuplet, the mass formula is given by three relations,

m_\Delta-m_{\Sigma^*} = m_{\Sigma^*}-m_{\Xi^*} = m_{\Xi^*}-m_\Omega \,

where mX is the average mass between the neutral and charged instances of particle X, just as is done in the baryon octet formulation. The baryon decuplet formula allowed Gell-Mann to successfully predict the mass of the yet undiscovered Ω−.

References

  1. ^ M. Gell-Mann, "The Eightfold Way: A Theory of Strong Interaction Symmetry," California Institute of Technology Synchrotron Laboratory Report CTSL-20 (1961), unpublished.
  2. ^ S. Okubo, Note on Unitary Symmetry in Strong Interactions, Prog. Theor. Phys. 27 (1962)
  3. ^ Griffiths, David (1987). Introduction to Elementary Particles. New York: John Wiley & Sons. ISBN 0-471-60386-4