An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 5.2

Elementary Processes of Quantum Electrodynamics — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · qed-bhabha-scattering

Problem

Compute the differential cross section

d\sigma/d\cos\theta

for Bhabha scattering,

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

. You may work in the limit

E_{\text{cm}} \gg m_e

, in which it is permissible to ignore the electron mass. There are two Feynman diagrams; these must be added in the invariant matrix element before squaring. Be sure that you have the correct relative sign between these diagrams. The intermediate steps are complicated, but the final result is quite simple. In particular, you may find it useful to introduce the Mandelstam variables

s

t

, and

u

. Note that, if we ignore the electron mass,

s + t + u = 0

. You should be able to cast the differential cross section into the form

\frac{d\sigma}{d\cos\theta} = \frac{\pi\alpha^2}{s} \left[ u^2 \left(\frac{1}{s} + \frac{1}{t}\right)^2 + \left(\frac{t}{s}\right)^2 + \left(\frac{s}{t}\right)^2 \right].

Rewrite this formula in terms of

\cos\theta

and graph it. What feature of the diagrams causes the differential cross section to diverge as

\theta \to 0

Related theory

Qed Scattering Amplitudes

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