View source for 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|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101301.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101302.png" /> and the hypercharge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101303.png" /> [[#References|[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 [[#References|[a2]]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101305.png" /> is the normalized weight vector representing a particle and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101306.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g1101309.png" /> are empirical constants related to a given multiplet, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g11013010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g11013011.png" /> are the representatives of the two elements from the [[Universal enveloping algebra|universal enveloping algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g11013012.png" /> that are expressed in terms of the [[Gell-Mann matrices|Gell-Mann matrices]] as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g11013013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110130/g11013014.png" />, respectively.

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