Notations and Conventions

A four component Dirac Fermion with mass M is described by the Lagrangian ${\mathcal{L}}_{Dirac}=i{\bar{\Psi }}_D{\gamma }^{\mu }{\partial }_{\mu }{\Psi }_D-M{\bar{\Psi }}_D{\Psi }_D$ We use the specific representation of the 4x4 gamma matrices in 2x2 blocks by ${\gamma }^{\mu }=\left(\begin{array}{cc}0& {\sigma }^{\mu }\\ -{\sigma }^{\mu }& 0\end{array}\right){\gamma }_5=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$ where ${\sigma }^0={\bar{\sigma }}^0=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right){\sigma }^1=-{\bar{\sigma }}^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right){\sigma }^2=-{\bar{\sigma }}^2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right){\sigma }^3=-{\bar{\sigma }}^3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ In this representation, a four-component Dirac spinor is written in terms of 2 two-component complex, anti-commuting objects; ${\Psi }_D=\left(\begin{array}{c}{\xi }_{\alpha }\\ {\chi }^{†\dot{\alpha }}\end{array}\right)$