View source for Gell-Mann-Okubo formula

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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]]  $  \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 $[[#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]]]:  $  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|universal enveloping algebra]] of  $  \mathfrak s \mathfrak u ( 3 ) $
that are expressed in terms of the [[Gell-Mann matrices|Gell-Mann matrices]] as  $  {1 / {\sqrt 3 } } \lambda _ {8} $
and  $  {1 / 4 } ( \lambda _ {1}  ^ {2} + \lambda _ {2}  ^ {2} + \lambda _ {3}  ^ {2} ) $, 
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>