Understanding Fierz Identity Transformations

[Luca_Mantani]

Hi,
I was calculating some amplitudes and I end up with an expression like this:
$$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_L u)$$

In the solution of the exercise they say that, from the Fierz identity:
$$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_Lu)=16(\bar{c}\gamma^\mu P_L b)(\bar{d}\gamma_\mu P_L u)$$

How can I show that?

 

[eys_physics]

Hi,
I was calculating some amplitudes and I end up with an expression like this:
$$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_L u)$$

In the solution of the exercise they say that, from the Fierz identity:
$$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_Lu)=16(\bar{c}\gamma^\mu P_L b)(\bar{d}\gamma_\mu P_L u)$$

How can I show that?

Probably, it is useful for you to read this reference:
https://arxiv.org/pdf/hep-ph/0306087.pdf