Isospin doublet and quark content from contraction of quarks

Question

Let's introduce a quark $SU(2)$ doublet. We are in the $m_u \approx m_d$ limit. So we have $$ q = \begin{pmatrix} u\\ d \end{pmatrix}. $$ Then we can construct the Nucleonic field $$ N := q q q = \begin{pmatrix} p\\ n \end{pmatrix}. $$

I expect this to transform as a $SU(2)$ doublet, which corresponds to the Isospin limit. Also, I expect to find that the proton $p$ is made up of two ups and one down., while the neutron is made of one up and two downs.

But how do we see this from contracting the indices of $q$?

Answer 1

Apart from strong and weak isospin mixup as mentioned in the comments, let's also be careful not to mix up color SU(3) and flavor SU(3). We can leave the color out of it entirely. Also full flavor SU(3) is not needed, only the (strong) isospin subgroup SU(2). And there you just add three isospin $\frac12$ states to a total isospin $\frac12$.

This is not the only thing you can do to combine three $q$'s, you could also have combined them into a total isospin $\frac32$ state. That would give you the $\Delta$'s. It's a choice you make, in the two cases you contract the indices in a different way. All described of course by Clebsch-Gordan coefficients, which technically speaking should answer your question.

But you can also say that to combine them to $p$ and $n$, you'll need terms in the summation that always have the isospin $z$-component add up to $\pm\frac12$. If, on the other hand, you combine them to $\Delta$'s, you'll usually start creating the combinations with isospin $z$-component $\pm\frac32$ (which is easy to do) and then raising and lowering operators can do the rest. That will then help you to find the $n$ and $p$ states, which should be orthogonal to the two "middle states" of the $\Delta$'s, the ones that have isospin $z$-component $\pm\frac12$. (To pin down the exact states you'll also have to use total symmetry for spin+isospin, since the color part of the state is already totally antisymmetric).

NB: to define everything with the right phase factor it might be wise to check the Condon & Shortley phase convention.