Difference in Sign for 6.4 Equation

[touqra]

I have a hard time trying to derive the equation in 6.4 . This is what I've done, and as you will see I have a sign difference compared to his.
http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PRVDAQ000070000001015012000001&idtype=cvips&prog=normal
Starting with page 8, equation 6.2, considering the second term in the action, i.e.

$$D_{\mu}\overline{\psi}\gamma^{\mu}\psi$$

note: the author used the four-component Dirac spinor as $$\left( \begin{array}{cc} \chi\\ \overline{\psi} \end{array} \right)$$

I also calculated some identities:
$$\psi* \gamma^0 \gamma^{\mu} = \gamma^0 \gamma^{\mu} \psi*$$
$$\gamma^{\mu}\gamma_{\mu} = 4$$
$$\{\gamma^{\mu} , \gamma^{\nu}\} = 2g^{\mu\nu}$$

With these,
I obtain for $$\frac{1}{4z}\gamma_{\mu}\gamma_{5}\overline{\Psi}\gamma^{\mu}\Psi = \frac{-2}{z}(\chi\psi - \overline{\chi}\overline{\psi})$$

I have a sign difference with the author's in equation 6.4
I calculated for the first term too. Still a sign difference. In fact, I get something like $$\frac{3}{z} (\chi\psi - \overline{\chi}\overline{\psi})$$

 

[quasar987]

you might want to provide the name of the book you're using?