I wanted to check the electroweak standard model Lagrangian is invariant under CPT transformation by first checking how the bilinears of $\bar{\psi} \psi$, $\bar{\psi}\gamma_5 \psi$ $\bar{\psi}\gamma^{\mu} \psi$, $\bar{\psi}\gamma^{\mu}\gamma_5 \psi$, $\bar{\psi}\sigma^{\mu \nu} \psi$ transform under discrete symmetries (C, P, T) separately.
I seem to have a problem with the time-reversal symmetry only. I define the T operator as acting
$$\psi \to i\gamma_1 \gamma_3 \psi^*$$ $$\psi^* \to -i\gamma_1^* \gamma_3^* \psi = -i\gamma_1 \gamma_3 \psi$$ $$\bar{\psi} = {\psi^*}^T \gamma^0 = (-i\gamma_1 \gamma_3 \psi)^T \gamma^0$$
Hence, I get
$$\bar{\psi} \psi = (-i\gamma_1 \gamma_3 \psi)^T \gamma^0 (i\gamma_1 \gamma_3 \psi^*)= (\psi^T \gamma_1^T \gamma_3^T \gamma^0 \gamma_1 \gamma_3 \psi^*)$$
Then I use $\gamma_3^T \gamma_1^T = \gamma_3 \gamma_1$ by the definition of gamma matrices and $\gamma^0 \gamma_{1,3} = -\gamma_{1,3} \gamma^0$ from their anticommutation relations and $\gamma_{1,3}^2 = -1$. Hence I got
$$\bar{\psi} \psi = (\psi^T \gamma_1 \gamma_3 \gamma^0 \gamma_1 \gamma_3 \psi^*) = -(\psi^T \gamma^0 \psi^*) = -(\psi^{\alpha} \gamma^0_{\alpha \beta} \psi^{\beta*}) = +\psi^{\beta*}\gamma^0_{\alpha \beta} \psi^{\alpha}$$
now use the fact that $\gamma_0 = \gamma_0^T$ so
$$\bar{\psi} \psi = -\psi^{\beta*}\gamma^0_{\alpha \beta} \psi^{\alpha} = + \bar{\psi}\psi$$
As expected this scalar term under T transform as $\bar{\psi}\psi = \bar{\psi}\psi$. The axial part
$$\bar{\psi} \gamma_5 \psi = (-i\gamma_1 \gamma_3 \psi)^T \gamma^0 \gamma_5 (i\gamma_1 \gamma_3 \psi^*)= (\psi^T \gamma_1^T \gamma_3^T \gamma^0 \gamma_5\gamma_1 \gamma_3 \psi^*) = (\psi^T \gamma_1 \gamma_3 \gamma^0 \gamma_5\gamma_1 \gamma_3 \psi^*) = (\psi^T \gamma_3 \gamma^0 \gamma_5 \gamma_3 \psi^*)= - (\psi^T \gamma^0 \gamma_5 \psi^*)$$
By using the anticommutation of the fields as done for the scalar term $\{{\psi, \psi^*}\}=0$, from the axial term:
$$\bar{\psi} \gamma_5 \psi = - (\psi^T \gamma^0 \gamma_5 \psi^*) = (\bar{\psi} \gamma_5 \psi)$$
However this should be with an opposite sign and I can't figure out where I did wrong.
