Relativistic Collisions in Field Theory. Feynman Rules. Decays

Abstract

The initial and final states for this process are,

$$ [\left| i \right\rangle {\text{ = }}{{\hat b}^ + }\left( {{p_1},{\lambda _1}} \right){{\hat d}^ + }\left( {{p_{2,}},{\lambda _2}} \right)\left| 0 \right\rangle ,\langle f| = \langle 0|\hat \alpha \left( {{k_1},{\eta _1}} \right)\hat \alpha \left( {{k_2},{\eta _2}} \right) $$

(11.1.1)

where the η are the helicities of the photons and we define photon states without a factor $1/\sqrt {n!}$, as we did for electron states. The interaction Hamiltonian is still given by (10.4.1). In the Born approximation,

$$ \begin{gathered} \langle f|\hat S{\left| i \right\rangle _{Born}} = \frac{{{i^2}{e^2}}}{{2!}}\int {{d^4}{x_1}{d^4}{x_2}\sum\limits_{\mu \upsilon } {{g_\mu }_\mu {g_\upsilon }_\upsilon \langle 0|\hat \alpha \left( {{k_1},{\eta _1}} \right)\hat \alpha } } \left( {{k_2},{\eta _2}} \right) \hfill \\ \times T(:\hat \bar \psi \left( {{x_1}} \right){\gamma _\upsilon }\hat \psi :{{\hat A}_\mu }\left( {{x_1}} \right){{\hat A}_\nu }\left( {{x_2}} \right):\hat \bar \psi \left( {{x_2}} \right){\gamma _\nu }\hat \psi \left( {{x_2}} \right)\left. : \right) \hfill \\ x{{\hat b}^ + }\left( {{p_1},{\lambda _1}} \right){{\hat d}^ + }\left( {{p_2},{\lambda _2}} \right)\left| {\left. 0 \right\rangle .} \right. \hfill \\ \end{gathered} $$

(11.1.2)