Fierz identity for chiral fermions

Question

First of all I define the convention I use.

The matrices $\bar{\sigma}^\mu$ I will use are $\{ Id, \sigma^i \}$ where $\sigma^i$ are the Pauli matrices and $Id$ is the 2x2 identity matrix. I will use the Chiral Fierz Identity $$(\bar{\sigma}^\mu)[\bar{\sigma}^\nu] = (\bar{\sigma}^\mu][\bar{\sigma}^\nu) + (\bar{\sigma}^\nu][\bar{\sigma}^\mu) - \eta^{\mu\nu}(\bar{\sigma}^\lambda][\bar{\sigma}_\lambda) + i\epsilon^{\mu\nu\rho\lambda}(\bar{\sigma}_\lambda][\bar{\sigma}_\rho)$$ where I used the Takashi notation.

Let us consider the left-handed component $\chi$ of a massless fermion field $\psi$ and the operator defined as $$\mathcal{O} = \chi^\dagger \bar{\sigma}^\mu\chi (\partial_\mu\partial_\nu\chi^\dagger)\bar{\sigma}^\nu\chi.$$

If I have use the Chiral Fierz identity I get $\mathcal{O} = 2\mathcal{O}$ where I used $\partial_\mu\partial^\mu \chi = 0$. So, I get $\mathcal{O}=0$.

This equality, if true, suggests me there is another way to show that this operator is null for massless fermions. Is there any way? Do you suggest anything?

This post imported from StackExchange Physics at 2016-05-07 13:22 (UTC), posted by SE-user FrancescoS