Transforming [product] with Fierz Transformations

[Lester]

If I have a product like

[tex]
\bar\ psi\gamma^\mu\psi\bar\psi\gamma_\mu\psi
[tex]

how can i rearrange with Fierz transformations?

 

[samalkhaiat]

I assume, you meant to write

$$V(4,2;3,1) = \bar{\psi}(4) \gamma_{a} \psi(2) \bar{\psi}(3) \gamma^{a} \psi(1)$$

The Fietz transformation is

$$V(4,2;3,1) = \left( - S + \frac{1}{2} V - \frac{1}{2} A + P \right) (4,1;3,2)$$

where

$$S(4,1;3,2) = \bar{\psi}(4) \psi(1) \bar{\psi}(3) \psi(2)$$

$$A(4,1;3,2) = \bar{\psi}(4) \gamma_{5} \gamma_{a} \psi(1) \bar{\psi}(3) \gamma^{a}\gamma_{5} \psi(2)$$

$$P(4,1;3,2) = \bar{\psi}(4) \gamma_{5} \psi(1) \bar{\psi}(3) \gamma_{5} \psi(2)$$

and similar one for V(4,1;3,2). See page 160-162 in Itzykson & Zuber; QFT.

regards

sam

 

[Lester]

Thanks a lot, Sam. This is what I was looking for.

Jon