Show that the cross section for elastic scattering of...

Show that the cross section for elastic scattering of unpolarized electrons from spinless point-like particles is $$ \left.\frac{d \sigma}{d \Omega}\right|_{\text {Lab, }}=\left(\frac{\alpha^2}{4 E^2 \sin ^4 \frac{\theta}{2}}\right) \frac{E^{\prime}}{E} \cos ^2 \frac{\theta}{2}, $$ where as before we neglect the mass of the electron. Justify using (6.18) with $L_{\mu \nu}^{\text {muon }}$ replaced by $\left(p+p^{\prime}\right)_\mu\left(p+p^{\prime}\right)_v$. Comparing the cross section with that for $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$, we see that the $\sin ^2(\theta / 2)$ in $(6.50)$ is due to scattering from the magnetic moment of the muon.

Show that the cross section for elastic scattering of unpolarized electrons from spinless point-like particles is
$$
\left.\frac{d \sigma}{d \Omega}\right|_{\text {Lab, }}=\left(\frac{\alpha^2}{4 E^2 \sin ^4 \frac{\theta}{2}}\right) \frac{E^{\prime}}{E} \cos ^2 \frac{\theta}{2},
$$
where as before we neglect the mass of the electron. Justify using (6.18) with $L_{\mu \nu}^{\text {muon }}$ replaced by $\left(p+p^{\prime}\right)_\mu\left(p+p^{\prime}\right)_v$. Comparing the cross section with that for $\mathrm{e}^{-} \mu^{-} \rightarrow \mathrm{e}^{-} \mu^{-}$, we see that the $\sin ^2(\theta / 2)$ in $(6.50)$ is due to scattering from the magnetic moment of the muon.

Key Concepts

Elastic Scattering in QED

This concept involves the interaction of charged particles via the exchange of a virtual photon within the framework of quantum electrodynamics. Elastic scattering refers to processes where the internal structure of the particles remains unchanged, and energy–momentum is conserved. The analysis often makes use of Feynman diagrams to represent the virtual photon exchange that mediates the electromagnetic force between the scattering particles.

Differential Cross Section

The differential cross section quantifies the likelihood of scattering into a specific solid angle. It is derived from the square of the scattering amplitude and provides essential information about the angular distribution of the scattered particles. In quantum field theory, it is computed using the matrix elements of the interaction, incorporating factors from both the propagator and vertex terms.

Electromagnetic Vertex Structure

Referring to the coupling between particles and the electromagnetic field, the vertex structure embodies how charge and current distributions interact with the photon. For spinless point-like particles, the electromagnetic vertex is simpler—involving the combination (p + p')—compared to particles with spin, where additional components such as magnetic moment interactions contribute to the vertex.

Spin Effects and Magnetic Moments

The magnetic moment arises from intrinsic spin and its associated current distribution. In scattering processes, particles like the muon possess a magnetic moment that modifies the angular dependence of the cross section, as evidenced by the presence of additional angular factors (e.g., sin²(?/2)) in the cross section formula. Comparing scattering off a spinless particle with that off a particle with spin highlights the impact of these magnetic interactions.

Kinematic Simplifications for Massless Electrons

Neglecting the electron mass simplifies the kinematics, which is a common approximation in high-energy scattering analyses. This leads to clearer expressions for energy and momentum conservation, and simplifies the scattering amplitude by reducing complications that arise from mass-dependent terms. The energy ratios and angular dependences in the differential cross section become more transparent under this assumption.

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