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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 Murray Gell-Mann in 1961 and independently proposed by Susumu Okubo in 1962. 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.

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), , 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.


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