- #1

philipke

- 5

- 0

## Homework Statement

We are asked to calculate the amplitude for the annihilation of a positron electron pair into two photons in the high-energy limit. The high-energy limit is assumed by taking massless Weyl spinor for the electron and positron.

## Homework Equations

I get in my amplitude a factor that looks like this.

[tex]

\bar{u_R}(p_2)\frac{-\gamma^{\nu}k_2\!\!\!/\gamma^{\mu} + 2\gamma^{\nu}p_1^{\mu}}{-2p_1k_2}u_R(p_1) = C

[/tex]

I call it C to use it below. I am supposed to apply the Fierz identiy (from problem 5.3)

[tex]

\bar{u}_L(p_1)\gamma^{\mu}u_L(p_2)[\gamma_{\mu}]_{ab} = 2[u_L(p_2)\bar{u}_L(p_1) + u_R(p_1)\bar{u}_R(p_2)]_{ab}

[/tex]

I have not really an idea how to apply this identity on the first term in the factor of my amplitude. The complete amplitude looks like

[tex]

A(\gamma_{\nu})*B(\gamma_{\mu})*(C + C(\mu <-> \nu[/tex] and [tex] k_1 <-> k_2))

[/tex]

where A and B are sums of bilinears of u_R and u_L (The polarization vectors from the problems's description)

Thanks in advance!