Griffiths Introduction to Elementary Particles · Problem 7.44

Electrodynamics of Quarks and Hadrons — David J. Griffiths

Physics — QFT · scalar-photon-pair-annihilation

Problem

(a) Find the amplitude,

\mathcal{M}

, for pair annihilation,

e^+ + e^- \to \gamma + \gamma

, in this theory. (b) Determine

\langle|\mathcal{M}|^2\rangle

, assuming the energy is high enough that we can ignore both the electron and the ``photon'' mass (

m_e, m_\gamma \to 0

). (c) Evaluate your result, in (b), in the

CM

system. Express your answer in terms of the incident electron energy,

E

, and the scattering angle,

\theta

. (d) Find the differential cross section for pair annihilation, in the

CM

system, still assuming

m_e = m_\gamma = 0

. Is the total cross section finite? [Note: This problem pertains to the massive scalar photon model described before Problem 7.42.]