# Baryon in terms of quark fields – spinor index structure

What is the most general way to write down a current describing a baryon made from quarks $\psi_i^\alpha$? Let's say we suppress flavour indices but want to write colour $(i,j)$ as well as spinor $(\alpha,\beta)$ indices. Then I suppose the colour structure would be something like $$\psi_i \psi_j \psi_k \varepsilon^{ijk}$$ in order to be antisymmetric. How can one explicitly add spinor indices to this expression and how would one then decide if this describes a baryon of spin 1/2 or 3/2?

Edit To be more clear, I stumbled upon this reading Witten's Baryons in the 1/N expansion. There he investigates a 2d model of $SU(N)$ QCD in section 9. In equation (39), page 109, he introduces a current "with the quantum numbers to create a baryon": $$J(x) = \psi_1(x) \psi_2(x) \cdots \psi_N(x)$$ He then goes on to say

"Since we are not keeping track of Dirac indices, the Dirac indices have not been written in (39). If one wishes to keep track of Dirac indices, one should choose in (39) the same Dirac component for each of the $N$ quark fields. For instance, one may consider the positive chirality component of each quark field."

I do not understand why this choice is useful and if it is only true due to peculiarities of spinor representation in two dimensions. I thought I would understand if I saw the explicit structure as far as SU(3) in 4d is concerned.

• First thoughts - That would be independent spaces. Direct product implied? – 299792458 Oct 16 '14 at 15:38
• Ok, I guess one would just write $\psi_\alpha(x)\psi_\beta(x) \psi_\gamma (x) \varepsilon_{ijk}$ and then pick out the right components to get baryons of e.g. spin 1/2. I am still not sure about the piece in the paper by Witten. Maybe someone can help? – PassfishSwordword Oct 21 '14 at 15:07

Anyways, to be complete, a current describing the isobar $\Delta^{++}$ can be found in equation (13) there and has the structure $$\eta_\mu (x) = \left(u^i(x) C\gamma_\mu u^j(x)\right) u^k(x) \varepsilon^{ijk}.$$
A possible current for a proton can be seen in equation (46): $$\eta(x) =\left(u^i(x) C \gamma_\mu u^j (x) \right) \gamma_5\gamma_\mu d^k (x) \varepsilon^{ijk}.$$ So whereas a spin 1/2 baryon only carries one spinor index, a spin 3/2 one has an additional Lorentz index.