I want to show that the free Wess-Zumino Lagrangian is invariant under a SUSY transformation, e.g. following this reference (section 3.1).
However, I have a hard time understanding the daggers and stars on the fields. In particular, with the fermionic fields. The fermion Lagrangian looks like this: $$ \mathcal L_\text{fermion}=\text{i} \psi^\dagger \bar\sigma^\mu \partial_\mu \psi. \tag{3.1.2} $$ In index notation, this should be $\text{i} \bar \psi_{\dot a} (\bar\sigma^\mu)^{\dot aa} \partial_\mu \psi_a$. If we start with $$ \delta\psi_a = -\text{i} (\sigma^\mu \epsilon^\dagger)_a \partial_\mu\phi+\epsilon_aF = -\text{i} (\sigma^\mu)_{a\dot a} \bar\epsilon^{\dot a} \partial_\mu\phi+\epsilon_aF \tag{3.1.15}, $$ then my guess for the conjugate transformation $\delta\bar\psi_{\dot a}$ would be: $$ \begin{align}\delta\bar\psi_{\dot a} &= \text{i} \big((\sigma^\mu)_{a\dot a} \bar\epsilon^{\dot a}\big )^* \partial_\mu\phi^* +\bar\epsilon_{\dot a}F^* \\&= \text{i} (\sigma^\mu)_{\dot aa} \epsilon^{a} \partial_\mu\phi^* +\bar\epsilon_{\dot a}F^* \\&= \text{i} \epsilon^{a}(\sigma^\mu)^T_{a\dot a} \partial_\mu\phi^* +\bar\epsilon_{\dot a}F^* \\& = \text{i} (\epsilon \sigma^{\mu T})_{\dot a}\partial_\mu\phi^* +\bar\epsilon_{\dot a}F^* \end{align}$$ where I used the fact that the Pauli matrices are hermitian (therefore, complex conjugation becomes a transpose). However, it should actually be $$ \delta\bar\psi_{\dot a} = \text{i} (\epsilon \sigma^{\mu})_{\dot a}\partial_\mu\phi^* +\bar\epsilon_{\dot a}F^* \tag{3.1.15}$$ i.e. without the transpose on the $\sigma^\mu$ matrix.
Where's my mistake? I feel like I don't really understand the spinor index notation.
For what it's worth, I'm using these assignments in order to use index notation, $$ \begin{align} \psi &\sim \psi_a \\ \bar\psi = \psi^* &\sim \bar\psi_{\dot a} \\ \psi^T &\sim \psi^a \\ \bar\psi^T=\psi^\dagger &\sim \psi^{\dot a} \end{align} $$ as well as contracting indices like ${}^a{}_a$ and ${}_{\dot a}{}^{\dot a}$.
I have already considered these questions [1, 2, 3, 4], but did not find a solution to my problem.