Trace from Casimir's trick (muon neutrino- electron inelastic scattering)

Question

Griffiths defines Feynman amplitude for the CC process $\nu_\mu + e \rightarrow \mu + \nu_e$ as

$$\mathscr{M} = \frac{ g_w^2}{8M_W^2} \left[\bar \nu_e \gamma^\mu (\mathbb{1} -\gamma^5) e\right] \left[\bar \mu \gamma_\mu (\mathbb{1}-\gamma^5) \nu_\mu\right]$$

When we want $|\mathscr{M}|^2$ the complex conjugate is multiplied as

$$\sum_\text{spins} |\mathscr{M}|^2 = \frac{ g_w^4}{64M_W^4} \left[\bar \nu_e \gamma^\mu (\mathbb{1} -\gamma^5) e\right] \left[\bar \mu \gamma_\mu (\mathbb{1}-\gamma^5) \nu_\mu\right] \left[\bar \nu_e \gamma^\mu (\mathbb{1} -\gamma^5) e\right]^* \left[\bar \mu \gamma_\mu (\mathbb{1}-\gamma^5) \nu_\mu\right]^*$$

where the particle symbols are spinors. My question starts now: using Casimir's trick I found

\begin{equation} \begin{split} \left<|\mathscr{M}|^2 \right> = \left(\frac{g_w}{4 M_W}\right)^4 &Tr[\gamma^\mu (\mathbb{1}-\gamma^5) (\displaystyle{\not}{p_e} + \mathbb{m_e}) {\gamma^0}(\mathbb{1} -\gamma^5) {\gamma^\nu}^{\dagger} {\gamma^0} \displaystyle{\not}{p}_{\nu_e}]\\ &Tr[\gamma_\mu (\mathbb{1}-\gamma^5) \displaystyle{\not} {p}_{\nu_\mu} {\gamma^0}(\mathbb{1} -\gamma^5) {\gamma_\nu}^\dagger{\gamma^0} (\displaystyle{\not}{p}_{\mu} + \mathbb{m_\mu})] \end{split} \end{equation}

but Griffths's solution is

\begin{equation} \begin{split} \left<|\mathscr{M}|^2 \right> = \left(\frac{g_w}{4 M_W}\right)^4 &Tr[\gamma^\mu (\mathbb{1}-\gamma^5) (\displaystyle{\not}{p_e} + \mathbb{m_e}) {\gamma^\nu}(\mathbb{1} -\gamma^5) \displaystyle{\not}{p}_{\nu_e}]\\ &Tr[\gamma_\mu (\mathbb{1}-\gamma^5) \displaystyle{\not} {p}_{\nu_\mu}{\gamma_\nu}(\mathbb{1} -\gamma^5) (\displaystyle{\not}{p}_{\mu} + \mathbb{m_\mu})] \end{split} \end{equation}

Can anybody help me to understand the possible mistake? Thanks in advance.

Answer 1

The results are the same. Since

$$\big\{ \gamma^{\mu\dagger},\gamma^5 \big\} = 0$$

and

$$\gamma^0\gamma^{\mu\dagger}\gamma^\mu = \gamma^\mu$$

you obtain

$$\gamma^0(1 - \gamma^5)\gamma^{\mu\dagger}\gamma^0 = \gamma^0\gamma^{\mu\dagger}(1 + \gamma^5)\gamma^0 = \gamma^0\gamma^{\mu\dagger}\gamma^0(1 - \gamma^5) = \gamma^{\mu}(1 - \gamma^5).$$