My professor asked me to find the lagrangian density for a complex scalar field and a lagrangian density for a field that is a doublet of SU(2).

I started from the well established lagrangian density for a scalar field and I found for the complex scalar field:

$$\mathcal{L}_\text{compl scaclar}=(\partial_\mu\phi^\dagger)(\partial^\mu\phi)-m^2\phi^\dagger\phi - \lambda(\phi^\dagger\phi)^2$$

Its symmetry is U(1) I believe.

I had taken only an introductory course on group theory so I know not much about SU(2) and SU(2)xSU(2). So it is kind of cumbersome to write formal arguments to construct the Lagrangian for the SU(2) doublet. Anyway, I tried and my efforts took me to consider the field $\phi_i= (\phi_1\phi_2)^T$

I know that are some arguments of isomorphism between U(1) and SU(2), so I just wonder if I could just start from the complex scalar lagrangian density and write:

first: $$(\partial_\mu\phi^\dagger_i)(\partial^\mu\phi_i) = (\partial_\mu\phi^\dagger_1)(\partial^\mu\phi_1) + (\partial_\mu\phi^\dagger_2)(\partial^\mu\phi_2)$$

second: $$\phi^\dagger_i\phi_i = \phi^\dagger_1\phi_1 + \phi^\dagger_2\phi_2 $$

Third: $$(\phi^\dagger_i\phi_i)^2 = (\phi^\dagger_1\phi_1 + \phi^\dagger_2\phi_2)^2 = (\phi^\dagger_1\phi_1)^2 + (\phi^\dagger_2\phi_2)^2 + 2\phi^\dagger_1\phi_1\phi^\dagger_2\phi_2 $$

So... I just put it all in the lagrangian above and I believe that the answer is right although I am not sure about the formal aspects of what I am really doing. My professor didn't asked for formality anyway, because we will start the formalism after, what I really want here is to know if I am reasoning fairly.

My professor asked me to find the lagrangian density for a complex scalar field and a lagrangian density for a field that is a doublet of $SU(2)$.

I started from the well established lagrangian density for a scalar field and I found for the complex scalar field:

$$\mathcal{L}_\text{compl scaclar}=(\partial_\mu\phi^\dagger)(\partial^\mu\phi)-m^2\phi^\dagger\phi - \lambda(\phi^\dagger\phi)^2$$

Its symmetry is $U(1)$ I believe.

I had taken only an introductory course on group theory so I know not much about $SU(2)$ and $SU(2)\times SU(2)$. So it is kind of cumbersome to write formal arguments to construct the Lagrangian for the $SU(2)$ doublet. Anyway, I tried and my efforts took me to consider the field $\phi_i= (\phi_1\phi_2)^T$

I know that are some arguments of isomorphism between $U(1)$ and $SU(2)$, so I just wonder if I could just start from the complex scalar lagrangian density and write:

first: $$(\partial_\mu\phi^\dagger_i)(\partial^\mu\phi_i) = (\partial_\mu\phi^\dagger_1)(\partial^\mu\phi_1) + (\partial_\mu\phi^\dagger_2)(\partial^\mu\phi_2)$$

second: $$\phi^\dagger_i\phi_i = \phi^\dagger_1\phi_1 + \phi^\dagger_2\phi_2 $$

Third: $$(\phi^\dagger_i\phi_i)^2 = (\phi^\dagger_1\phi_1 + \phi^\dagger_2\phi_2)^2 = (\phi^\dagger_1\phi_1)^2 + (\phi^\dagger_2\phi_2)^2 + 2\phi^\dagger_1\phi_1\phi^\dagger_2\phi_2 $$

So... I just put it all in the lagrangian above and I believe that the answer is right although I am not sure about the formal aspects of what I am really doing. My professor didn't asked for formality anyway, because we will start the formalism after, what I really want here is to know if I am reasoning fairly.