As we know the algebra of SU(3) consist of two Cartan generators and 6 raising and lowering operators. We define the eigenstates of the Cartan operators as u,d,s, correspoding to the three lightest quarks.(adsbygoogle = window.adsbygoogle || []).push({});

Now when we study the [itex] 3\otimes 3[/itex] tensor product we can show that the Hilbert space of these states decompose as

[itex]3\otimes 3 = 8\oplus 1[/itex]

Which means that we can devide this Hilbert into two invariant subspaces.

And similar for baryons

[itex]3\otimes 3\otimes 3=10\oplus 8 \oplus 8\oplus 1[/itex]

My question is following. We can categorize the mesons in two multiplets of 8 particles and 1 singlet, and the baryons in one multiplet of 10 particles, one of 8 particles and one singlet. So is not obvious to me which is the correspondence between these groups of particles and the above Hilbert spaces. If the decomposition of [itex] 3\otimes 3[/itex] has one 8-plet why we have two mesons diagrams and for baryons why we have only one 8-plet of particles?

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# Decomposition of SU(3) and particles

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