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Homework Statement:

The charge conjugation for the complex KleinGordon scalar field ##\phi(x)## is defined by the following transformation
$$\phi \rightarrow U \phi U^{1} = \eta_c \phi^{\dagger}$$
Where ##U## is a unitary operator which leaves the vacuum invariant (i.e. ##U  \ 0 > =  \ 0>##) and ##\eta_c## is a phase factor.
a) Show that ##\mathcal{L} = \partial_{\mu} \phi^{\dagger} \partial^{\mu} \phi m^2 \phi^{\dagger} \phi## is invariant under such a transformation.
b) Show that the charge current ##s^{\alpha} = \frac{iq}{\hbar} ((\partial^{\alpha} \phi^{\dagger}) \phi  (\partial^{\alpha} \phi) \phi^{\dagger})## picks up a change of sign under such a transformation.
Source: (part of) Ex. 3.5 QFT book by Franz Mandl and Graham Shaw.
Relevant Equations:
 $$\phi \rightarrow U \phi U^{1} = \eta_c \phi^{\dagger}$$
a)
Alright, I think that the trick here is to consider ##\phi^{\dagger}## and ##\phi## as independent scalar fields.
I've read that the unitary matrices read as follows
$$U = e^{i \epsilon}$$
Thus here we have to consider two separate transformations
$$\phi \rightarrow \phi' = e^{i \epsilon} \phi \sim (1 + i \epsilon) \phi$$
$$\phi^{\dagger} \rightarrow \phi'^{\dagger} = e^{i \epsilon} \phi^{\dagger} \sim (1  i \epsilon) \phi^{\dagger}$$
Both are infinitesimal transformations
$$\phi \rightarrow \phi' = \phi + \delta \phi = \phi + i \epsilon \phi$$
$$\phi^{\dagger} \rightarrow \phi'^{\dagger} = \phi^{\dagger} + \delta \phi^{\dagger} = \phi i \epsilon \phi^{\dagger}$$
OK this is all the theory I've learned but how can I actually prove invariance of the Lagrangian ##\mathcal{L} = \partial_{\mu} \phi^{\dagger} \partial^{\mu} \phi m^2 \phi^{\dagger} \phi## under such transformations? Could you please give me a hint?
Thank you.
Alright, I think that the trick here is to consider ##\phi^{\dagger}## and ##\phi## as independent scalar fields.
I've read that the unitary matrices read as follows
$$U = e^{i \epsilon}$$
Thus here we have to consider two separate transformations
$$\phi \rightarrow \phi' = e^{i \epsilon} \phi \sim (1 + i \epsilon) \phi$$
$$\phi^{\dagger} \rightarrow \phi'^{\dagger} = e^{i \epsilon} \phi^{\dagger} \sim (1  i \epsilon) \phi^{\dagger}$$
Both are infinitesimal transformations
$$\phi \rightarrow \phi' = \phi + \delta \phi = \phi + i \epsilon \phi$$
$$\phi^{\dagger} \rightarrow \phi'^{\dagger} = \phi^{\dagger} + \delta \phi^{\dagger} = \phi i \epsilon \phi^{\dagger}$$
OK this is all the theory I've learned but how can I actually prove invariance of the Lagrangian ##\mathcal{L} = \partial_{\mu} \phi^{\dagger} \partial^{\mu} \phi m^2 \phi^{\dagger} \phi## under such transformations? Could you please give me a hint?
Thank you.