Gell-Mann-Okubo formula

A perturbative formula for the mass spectrum of strongly interacting particles, baryons and mesons. In 1961, M. Gell-Mann and Y. Ne'eman classified baryons and mesons and grouped them into multiplets, labeled by irreducible representations of the Lie algebra $ \mathfrak s \mathfrak u ( 3 ) $, with each particle in a multiplet being represented by a normalized weight vector (the number of particles in the multiplet equals the dimension of the representation) and with weights giving values of observable quantities: the isotopic spin $ I _ {3} $ and the hypercharge $ Y $[a1]. To explain the variation of masses of particles belonging to the same multiplet, a mass formula was suggested by Gell-Mann and S. Okubo [a2]: $ m _ {f} = ( Tf,f ) $, where $ f $ is the normalized weight vector representing a particle and

$$ T = m _ {0} 1 + aY + b \left [ I ( I + 1 ) - { \frac{1}{4} } Y ^ {2} \right ] . $$

Here, $ m _ {0} $, $ a $ and $ b $ are empirical constants related to a given multiplet, $ Y $ and $ I ( I + 1 ) $ are the representatives of the two elements from the universal enveloping algebra of $ \mathfrak s \mathfrak u ( 3 ) $ that are expressed in terms of the Gell-Mann matrices as $ {1 / {\sqrt 3 } } \lambda _ {8} $ and $ {1 / 4 } ( \lambda _ {1} ^ {2} + \lambda _ {2} ^ {2} + \lambda _ {3} ^ {2} ) $, respectively.

References

[a1]	M. Gell-Mann, Y. Ne'eman, "The eightfold way" , Benjamin (1964)
[a2]	S. Okubo, "Note on unitary symmetry in strong interactions" Progress Theor. Phys. , 27 (1962) pp. 949–969