Use the Feynman rules to obtain the invariant amplitudes for...

Use the Feynman rules to obtain the invariant amplitudes for the "spinless" processes $\mathrm{e}^{-} \mu^{+} \rightarrow \mathrm{e}^{-} \mu^{+}$and $\mathrm{e}^{-} \mathrm{e}^{+} \rightarrow \mu^{-} \mu^{+}$. Check your answers by appropriately crossing the $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$amplitude of Section 4.2.

Key Concepts

Scattering Processes

Scattering processes are interactions in which initial particles collide and transform into final particles. In quantum field theory, these processes are analyzed using perturbation theory, and the outcomes are predicted using the invariant amplitudes derived from Feynman diagrams. Understanding these processes is essential for relating theoretical predictions to experimental measurements.

Feynman Rules

Feynman rules are a set of prescriptions in quantum field theory that translate interaction Lagrangians into diagrams and corresponding mathematical expressions (amplitudes) for particle processes. They provide the vertices, propagators, and external line factors needed for computing scattering amplitudes, ensuring that all contributions are accounted for according to the theory's dynamics.

Invariant Amplitude

The invariant amplitude is a Lorentz-invariant quantity derived from Feynman diagrams that encapsulates the dynamical information of a scattering process. It is independent of the specific reference frame and combines all contributions from the relevant diagrams, serving as the core element in calculating cross sections and probabilities for transitions between different quantum states.

Crossing Symmetry

Crossing symmetry is a fundamental property in quantum field theory that relates scattering amplitudes of different processes by interchanging particles between the initial and final states. This principle allows one to obtain amplitudes for certain processes by analytic continuation of those from a related process, thereby providing a consistency check and unifying different reactions under a common framework.

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