Srednicki — Quantum Field Theory · Problem 11.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · compton-scattering-klein-nishina

Problem

Consider Compton scattering, in which a massless photon is scattered by an electron, initially at rest. (This is the FT frame.) In problem 59.1, we will compute ∣T∣2|\mathcal{T}|^2 for this process (summed over the possible spin states of the scattered photon and electron, and averaged over the possible spin states of the initial photon and electron), with the result ∣T∣2=32π2α2[m4+m2(3s+u)−su(m2−s)2+m4+m2(3u+s)−su(m2−u)2+2m2(s+u+2m2)(m2−s)(m2−u)]+O(α4)(11.50)|\mathcal{T}|^2 = 32\pi^2\alpha^2\left[\frac{m^4 + m^2(3s+u) - su}{(m^2-s)^2} + \frac{m^4 + m^2(3u+s) - su}{(m^2-u)^2} + \frac{2m^2(s+u+2m^2)}{(m^2-s)(m^2-u)}\right] + O(\alpha^4) \tag{11.50} where α=1/137.036\alpha = 1/137.036 is the fine-structure constant. a) Express the Mandelstam variables ss and uu in terms of the initial and final photon energies ω\omega and ω′\omega'. b) Express the scattering angle θFT\theta_{\mathrm{FT}} between the initial and final photon three-momenta in terms of ω\omega and ω′\omega'. c) Express the differential scattering cross section dσ/dΩFTd\sigma/d\Omega_{\mathrm{FT}} in terms of ω\omega and ω′\omega'. Show that your result is equivalent to the Klein-Nishina formula dσdΩFT=α22m2ω′2ω2[ωω′+ω′ω−sin⁡2θFT].(11.51)\frac{d\sigma}{d\Omega_{\mathrm{FT}}} = \frac{\alpha^2}{2m^2}\frac{\omega'^2}{\omega^2}\left[\frac{\omega}{\omega'} + \frac{\omega'}{\omega} - \sin^2\theta_{\mathrm{FT}}\right]. \tag{11.51}

Related theory

Browse the problem library