Quark Model Families

Question

I cannot understand what we mean when we say that we organise baryons and mesons in families. In other words, i have seen that a lot of books mention the baryon J=1/2 family. What do we mean by that? Also, how can we find how many particles are in a given family?

Answer 1

First, $$J = L +S.$$ Given that the mesons/baryons are in the ground state, we have $ L =0$. What left is to compute $S$.

We also know that baryons are composed of three quarks. Quarks have spin $1/2$, so that based on spin addition rule, you can get $S = 1/2, 3/2$.

For each of this quantum number, you group them to a given multiplet of $SU(3)_f$. To count how many particles are there in a given "family", you have to know in which representation the particle belong. First notice that these baryons/mesons made from quarks $u, d, s$ that form a fundamental representation $3$ of $SU(3)_f$. Again, because baryons are composed of three quarks, from
$$3 \times 3 \times 3 = 10 + 8 + 8 +1,$$ we can group the baryons into singlet (1), octet (8), and decouplet (10) representation of $SU(3)_f$. For example, neutron with $J=1/2$ belong to octet representation, so its family composed of 8 members.