Yukawa [3 Dirac - 8 scalar] interaction lagrangian

[Tenens]

Homework Statement

Given an interaction lagrangian

$$L = i \, g \, \bar \psi(x)_i (\lambda^a)_{ij} \gamma_5 \, \psi(x)_j \phi(x)_a$$

where $$\psi_i$$ are three Dirac fermions with mass M and $$\phi_a$$ are eight real scalar fields of mass m and $$\lambda_a$$ are the generators of SU(3).

I have to find all internal symmetries of the Lagrangian and their conserved currents.

Homework Equations

Representations of SU(3)

The Attempt at a Solution

I suspect that there is an SU(3) transformation whose fundamental representation (dimension = 3) acts on the Dirac fermions and whose adjoint representation (dimension = 8) acts on the real scalar fields.

This is suggested to me by the fact that the number of Dirac fermions and real scalar fields corresponds to the dimension of the two representations of SU(3).

I have tried to write down the infinitesimal transformations according to their corresponding representations, then attempting to substitute them in the Lagrangian to show that it remains indeed invariant.

I have figured out that the Dirac fermions $\psi_i$ transform as
$$\begin{cases} \psi_i \rightarrow \psi_i (\delta_{ij} - i \theta_a \lambda^a_{ij})\psi_j \\ \bar \psi_i \rightarrow \psi_i (\delta_{ij} + i \theta_a \lambda^a_{ij})\bar \psi_j \end{cases}$$

I don't know how to represent the infinitesimal adjoint representation in order to go on with my calculation. With it I would try to insert the trasformed fields into the Lagrangian and show that it remains the same after calculations.

Could someone help me?

 

[mark726]

One possible approach to finding the internal symmetries of this Lagrangian would be to start by looking at the generators of SU(3) and how they act on the fields in the Lagrangian. As you noted, the Dirac fermions transform under the fundamental representation of SU(3), while the real scalar fields transform under the adjoint representation.

To find the infinitesimal transformations for the adjoint representation, you can use the fact that the generators of SU(3) satisfy the commutation relations [λa, λb] = ifabcλc, where fabc are the structure constants of the group. This means that the infinitesimal transformation for the fields in the adjoint representation will involve a linear combination of the generators with coefficients given by the structure constants.

Once you have the infinitesimal transformations for both representations, you can substitute them into the Lagrangian and show that it is invariant under these transformations. This will then give you the conserved currents associated with each symmetry.

It may also be helpful to look at the conserved charges associated with these symmetries and see if they have any physical meaning or interpretations in terms of the fields in the Lagrangian. This can give you a deeper understanding of the underlying symmetries and their implications.

I hope this helps and good luck with your calculations!