# My Struggle with Fierz Identity

I am following BUSSTEPP Lectures on Supersymmetry to learn SUSY.

The Lagrangian of a interacting Wess-Zumino model in 4D is given by

$$\mathcal{L}=-\frac{1}{2}(\partial_{\mu}S)(\partial^{\mu}S)-\frac{1}{2}(\partial_{\mu}P)(\partial^{\mu}P)-\frac{1}{2}\bar{\psi}\partial\!\!\!/\psi-\frac{1}{2}m^{2}S^{2}-\frac{1}{2}m^{2}P^{2}-\frac{1}{2}m\bar{\psi}\psi-\lambda[\bar{\psi}(S-P\gamma_{5})\psi+\frac{1}{2}\lambda(S^{2}+P^{2})^{2}+mS(S^{2}+P^{2})],$$

where $$\psi$$ is a Grassmann-valued Majorana spinor, $$S$$ is a scalar and $$P$$ is a pseudo-scalar.

Here $$\bar{\psi}$$ is the Majorana conjugate adjoint spinor, which is defined as

$$\bar{\psi}=\psi^{T}\mathcal{C},$$

where $$\mathcal{C}$$ is the charge conjugation matrix. In my notations, the gamma matrices and the charge conjugation matrix satisfy

$$\mathcal{C}\gamma_{\mu}=-(\gamma_{\mu})^{T}\mathcal{C},\quad(\gamma_{5})^{2}=-1,\quad\mathcal{C}\gamma_{5}=(\gamma_{5})^{T}\mathcal{C},\quad\mathcal{C}^{T}=-\mathcal{C}$$

with the metric given by $$\mathrm{diag}\left\{-1,+1,+1,+1\right\}$$, and $$M^{T}$$ is the transpose of $$M$$, i.e. $$M_{ab}=(M^{T})_{ba}$$.

In what follows, the latin indices $$a,\cdots,b$$ are referred to as 4-spinor indices, and run through the dotted and undotted 2-spinor indices. The raising and lowering rules are given by

$$\psi^{a}=\mathcal{C}^{ab}\psi_{b},\quad\psi_{a}=\psi^{b}\mathcal{C}_{ba}.$$

The SUSY transformations are given by

$$Q_{a}\cdot S=\psi_{a},\quad Q_{a}\cdot P=-(\gamma_{5})_{a}^{\,\,\,b}\psi_{b}$$

and

$$(Q_{a}\cdot\psi)_{b}=-\partial_{\mu}S(\gamma^{\mu})_{ab}+\partial_{\mu}P(\gamma^{\mu}\gamma_{5})_{ab}-mS\mathcal{C}_{ab}-mP(\gamma_{5})_{ab}-\lambda(S^{2}-P^{2})\mathcal{C}_{ab}-2\lambda SP(\gamma_{5})_{ab},$$

where the SUSY charge $$Q$$ is Grassmann-odd.

I want to check the on-shell closure of the SUSY algebra, and so I compute the symmetric commutator

$$(\left\{Q^{a},Q^{b}\right\}\cdot\psi)^{c}.$$

It is more convenient to directly compute the anti-symmetric commutator

$$[\delta_{1},\delta_{2}]\psi_{c}\equiv[\epsilon_{1a}Q^{a},\epsilon_{2b}Q^{b}]\psi_{c},$$

where $$\epsilon_{1}$$ and $$\epsilon_{2}$$ are two arbitrary Grassmann-valued constant Majorana spinors.

Then, I encounter the following term

$$(\epsilon_{2b}\partial_{\mu}\psi^{b})\epsilon_{1a}(\gamma^{\mu})^{ac}-(\epsilon_{1a}\partial_{\mu}\psi^{a})\epsilon_{2b}(\gamma^{\mu})^{bc}.$$

I want to use the Fierz identity to rearrange the positions of spinors to compute the above thing. The Fierz identity is given as follows.

$$(\bar{\alpha}\beta)\chi_{a}=-\frac{1}{4}(\bar{\alpha}\chi)\beta_{a}-\frac{1}{4}(\bar{\alpha}\gamma_{\mu}\chi)(\gamma^{\mu}\beta)_{a}-\frac{1}{4}(\bar{\alpha}\gamma_{5}\chi)(\gamma_{5}\beta)_{a}+\frac{1}{4}(\bar{\alpha}\gamma_{\mu}\gamma_{5}\chi)(\gamma^{\mu}\gamma_{5}\beta)_{a}+\frac{1}{8}(\bar{\alpha}\gamma_{\mu\nu}\chi)(\gamma^{\mu\nu}\beta)_{a}.$$

Now I want to set

$$\alpha\equiv(\epsilon_{1/2})_{a},\quad\beta=\partial_{\mu}\psi.$$

Should I set $$\chi^{c}=(\bar{\epsilon}_{2}\gamma^{\mu})^{c}$$? This is weird because it has a bar and carries upper index.