# Octets for baryons

1. Apr 18, 2014

### jono90one

Hi,
I am just learning some materials and struggling to find what the other octet is. I know the following:
$3 \otimes 3 \otimes 3 = 10 \oplus 8 \oplus 8 \oplus 1$

Now I understand the 10 and one of the 8's. But I am a little unsure of what the other octet and singlet is in terms of quarks. Is the other octet just an excited state of the other octet (higher spin)??

http://proj.ncku.edu.tw/research/articles/e/20080523/images/080408014859tzxABW.gif [Broken]
(This is the octet I already know)

Thanks.

Last edited by a moderator: May 6, 2017
2. Apr 18, 2014

### Bill_K

Quarks must obey Fermi-Driac statistics. Their wavefunction is a product of four parts: flavor, color, space and spin. All hadrons are colorless, meaning the color part is totally antisymmetric. In a ground state one assumes L = 0, meaning the space part is totally symmetric. This leaves flavor and spin, which together must be totally symmetric.

An SU(3) singlet is totally antisymmetric, meaning it wants to be combined with a totally antisymmetric spin part.

But three spin-halfs can only be combined in two ways: either as S = 3/2 (totally symmetric) or S = 1/2 (mixed). There is no totally antisymmetric way to combine three spin-halfs.

So the only remaining way to make a baryon which is an SU(3) singlet is to include orbital angular momentum. It's believed that the Λ(1890) is such a particle.