Forest Bhabha Scattering

Bhabha (electron -positron) Scattering

Bhabha scattering identifies the scatterng of an electron and positron (particle and anti-particle). There are two processes that can occur

1.) scattering via the exchange of a virtual photon

2.) annihilation in which the e+ and e- spend some time as a photon which then reconverts back to an e+e- pair

variables

Let:

$p_1 \equiv$ initial electron 4-momentum

u_1 \equiv initial electron spinor

p_2 \equiv final electron 4-momentum

u_2 \equiv final electron spinor

p_3 \equiv initial positron 4-momentum

\bar{u}_3 \equiv initial positron spinor

p_4 \equiv finial positron 4-momentum

$\bar{u}_4 \equiv$ finial positron spinor

Matrix element for scattering

According to the Feynman RUles for QED:

the term

$ig_e \gamma^{\mu}$

is used at the vertex to describe the Quantum electrodynamic (electromagneticc) interaction between the two fermion spinor states entering the vertex and forming a photon which will "connect" this vertex with the next one.

The QED interaction Lagrangian is

$-eA_{\mu} \bar{\Psi} \gamma^{\mu} \Psi$

$\mathcal{M}_s = \,$ $e^2 \left( \bar{u}_{3} \gamma^\nu u_4 \right) \frac{1}{(p_1+p_2)^2} \left( \bar{u}_{2} \gamma_\nu u_{1} \right)$

Matrix element for annihilation

$\mathcal{M}_a = \,$ $-e^2 \left( \bar{u}_{3} \gamma^\mu u_{1} \right) \frac{1}{(p_1-p_3)^2} \left( \bar{u}_{2} \gamma_\mu u_4 \right)$

Forest_QMII