Griffiths Introduction to Elementary Particles · Problem 6.10

Quantum Electrodynamics — David J. Griffiths

Physics — QFT · scattering-cross-section

Problem

(a) Analyze the problem of elastic scattering (

m_3 = m_1

m_4 = m_2

) in the lab frame (particle 2 at rest). Derive the formula for the differential cross section.

\left[Answer: \frac{d\sigma}{d\Omega} = \left(\frac{\hbar}{8\pi}\right)^2 \frac{\mathbf{p}_3^2 S |\mathcal{M}|^2}{m_2 |\mathbf{p}_1| \left[ |\mathbf{p}_3| c (E_1 + m_2 c^2) - |\mathbf{p}_1| E_3 \cos\theta \right]}\right]

(b) If the incident particle is massless (

m_1 = 0

), show that the result in part (a) simplifies to

\left(\frac{d\sigma}{d\Omega}\right) = S\left(\frac{\hbar E_3}{8\pi m_2 c E_1}\right)^2 |\mathcal{M}|^2