Gell-Mann–Okubo mass formula

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This formula is phenomenological, describing an approximate relation between meson and baryon masses, and has been superseded as theoretical work in quantum chromodynamics advances, notably chiral perturbation theory.

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]

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.

• Gell-Mann–Nishijima formula
• Eightfold Way
• Quark model
• SU(3)

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.