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

Latest revision as of 19:41, 5 June 2020


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
How to Cite This Entry:
Gell-Mann-Okubo formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gell-Mann-Okubo_formula&oldid=16570
This article was adapted from an original article by P. Stovicek (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article