Pion Nucleon Scattering and Group Theory in Quantum Field Theory

[Physiana]

Pion Nucleon Scattering and a bit about group theory/ representations

Hello everybody,

I am going through Ryder's book about Quantum Field Theory right now. In the chapter about Pion Nucleon scattering he writes the interaction term like

L_{int}=ig \bar{\psi} \gamma_5 \tau_a \psi \phi^a

where the \tau_a are the Pauli matrices. I wonder now; \psi is the nucleon field, \phi the pions
my script of the lecture says that the nucleon is in fundamental representation of SU(2). So we have nucleon dublet corresponding to dimension of the Pauli matrices?
What representation does he use for the Pions? And why does the number \pi^-,\pi^+,\pi^0 correspond to the number of generators and the dimension of the \tau to the number of nucleon States?
I do not exactly understand why I treat the dublet of proton and neutron so differently from the pion fields. Both are Isospin multiplets so I should actually describe them in the same "space"... e.g. regarding the nucleons also as a linear combination of the pauli matrices like we did with the pions, but then actually setting the coefficient of \tau_1 to zero \psi_1 = 0, because we do not have negative nucleon charge... I always thought the number of generators would correspond to the number of multiplets?

Thank you for your help :)

 

[Physiana]

Is there really no one who has an idea?

 

[Meir Achuz]

The pion have Ispin 1.
The index a has the values 1,2,3, with 3 being like the z component.
Linear combos of 1 and 2 are + and -
The nucleon charge is q=(1+\tau_3)/2
In vector notation it reads {\vec \tau}\cdot{\vec \phi}.
That is just like s.L in atomic physics, with L=1.
An older book on particle physics may have more detail than Ryder.