💼 Case: consider fermions scattering with some exchange particle $X$

• Left: we have our incoming particles we treat this as $t=-\infin$
• Right: we have our outgoing particles which we treat as $t=\infin$
• 🧽 Assume: interaction is instant at $t=0$

🗒️ Note: Feynman diagrams can be used to find the probability of a decay(the scattering amplitude)

Particles in Feynman diagrams

Interactions in Feynman Diagrams

• In red is a vertex
• Here we have a fermion emitting an electroweak boson
• The fermion lines have to be continuous
• Typically we give $g_X$ which is the coupling strength of the interaction

• With that we give $\alpha_X=\frac{g_X^2}{4\pi}$ which we recognise from the Yukawa potential

💃 Example: for EM the coupling strength is $g_\text{EM}\equiv \text{electric charge} \,(e)$ and thus $\alpha_\text{EM}\sim \frac{1}{137}$ is the fine structure constant

💃 Example: The Feynman Diagram for $e^- \mu^- \to e^- \mu^-$ scattering via the $\rm EM$

💡 Tips:

• Draw initial and final states
• Then draw the interaction

Rules

Conserved quantities

• Charge conservation (all types of charges not just electric)

  $$ \left [ \sum_i Q_i \right ]^\text{initial}=\left [ \sum_{i}Q_i \right ]^\text{final} $$

• Generational lepton number conservation

  $$ L_e^\text{initial} =L_e^\text{final} $$

  where $L_e=N(e^-,\nu_e)-N(e^+,\overline \nu_e)$ note that the 3 generations $L_\mu$ and $L_\tau$ also conserved

• Baryon number conservation

  $$ B^\text{initial}=B^\text{final} $$

  where $B=\frac{1}{3} [N(q)-N(\overline q)]$ here $q$ is any type of quark

Time-ordering

We done represent time orderings:

❌ No good

❌ No good

✅ Fantastic

Supremacy of lowest-order diagram

💼 Case: if we consider the process $e^- \mu^- \to e^- \mu ^-$