- Lagrangian: Where the 1st term corresponds to the Photon, 2nd to the Fermion, 3rd to the Photon-Fermion interaction and the 4th fixes the otherwise arbitrary gauge as
\begin{cases} \frac{1}{2\xi}(\partial_\mu A^\mu)^2,& \text{covariant gauges}\ n_\mu A^\mu, ;n^2=1, & \text{axial gauges} \end{cases}
- A general form of a physical #cross-section is $$d\sigma=\frac{1}{flux}|\mathcal{M}_{i\rightarrow f}|.\text{phase space}$$ We use a Lorentz Invariant Phase Space element, aka LIPS: $$\text{LIPS}=(2\pi)^4\delta^4(\sum_{f=1}^{n_f}{}p_f-\sum_{f=1}^{n_i}p_i) {\prod_{f=1}^{n}\frac{d^3\vec{p_f}}{(2\pi)^32E_f}}$$ - A 2 -> n #cross-section $$d\sigma=\frac{1}{4\sqrt{(p_1.p_2)-m_1^2m_2^2}}d\text{LIPS}<|\mathcal{M}_{i\rightarrow f}|^2>$$ where the final term is an average over final state spins and polarisations and a sum over the initial state ones. - 1 -> n processes #decay-rates $$d\Gamma=\frac{1}{2m}d\text{LIPS}<|\mathcal{M}_{i\rightarrow f}|^2>$$ 2. Feynman Diagram Rules $$\alpha,\beta \text{ are Dirac Indices; }\mu,\nu \text{ are Lorentz Indices; }\lambda,\kappa \text{ are Polarisation Indices}$$ $$\text{and } s,s' \text{are Spin Indices}$$ 1. External Wavefunctions ![[External Wavefunctions]] 2. Propagators: Photon $$\frac{-i(g^{\mu\nu}+(\xi-1)\frac{q^\mu q^\nu}{q^2})}{q^2+i\epsilon}$$ Fermion $$\frac{i(\cancel q + m)_{\alpha\beta}}{q^2-m^2+i\epsilon}$$ 3. Vertex: $$iQe\gamma^\mu_{\alpha\beta}$$ 3. Amplitude construction: - Draw all topologically distinct Feynman Diagrams. - Only connected diagrams - No disconnected parts - No vacuum bubbles in the external legs - Assign momenta to all external lines. - Assign momenta to internal lines using momentum conservation at each vertex. - Integrate over all undefined internal momenta - $$\int \frac{d^4k}{(2\pi)^4}$$ for each internal momentum k. - At tree level, there will be no undefined internal momenta. - At n loop level, there will be n undefined internal momenta. - Include a factor of (-1) for every closed fermion loop. Include another factor of (-1) for diagrams differing only by exchange of identical fermions. - Finally divide by the respective symmetry factor.