Noether current of a Lagrangian with interactions

Question

I am a bit confused regarding Noether Current. The Lagrangian of two complex scalar fields is $$ \mathcal{L}=\partial^\mu\phi_i^*\partial_\mu\phi_i-m_i^2|\phi_i|^2+\lambda(\phi_2^3\phi_1+\text{h.c.}). $$ It has a symmetry given by $$ \phi_1\rightarrow e^{-3i\alpha}\phi_1, $$ $$ \phi_2\rightarrow e^{i\alpha}\phi_2. $$ I am trying to find the Noether current, and was unsure regarding the interaction term. My instinct was to do something similar to what I know can be done for an interaction-less field, i.e., $$ J^\mu=\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi_i)}\delta\phi_i+\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi_i^*)}\delta\phi_i^*. $$

But shouldn't I take into account the interaction term when calculating the Noether current?

Answer 1

Since you will be differentiating the Lagrangian with interactions, you are already taking the interactions into account. The expression for the Noether current is derived assuming a fairly general Lagrangian, which can present interactions. In this particular case, it seems that the interactions don't affect the current.