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?
Best Answer
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.