 Solution

ID: past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2015/2/b/solution

Assume unpolarized beams,

s ≫ m_{μ}^{2}

, a single exchanged photon, full azimuthal coverage and no radiative corrections. Take

c_{*} = cos θ^{*}

between incoming

e^{-}

and outgoing

μ^{-}

. The four nonzero helicity amplitudes have magnitudes

e^{2} (1 + c_{*})

for equal incoming/outgoing negative-particle helicities, and

e^{2} (1 - c_{*})

for opposite ones. Averaging four initial spin states and summing the final ones gives

\overline{∣ M ∣^{2}} = \frac{e ^{4}}{4} {2 (1 + c_{*})^{2} + 2 (1 - c_{*})^{2}} = e^{4} (1 + c_{*}^{2}) .

The massless two-body phase space gives

\frac{d σ}{d Ω _{*}} = \frac{α ^{2}}{4 s} (1 + c_{*}^{2}), \frac{d σ}{d c _{*}} = \frac{π α ^{2}}{2 s} (1 + c_{*}^{2}), σ = \frac{4 π α ^{2}}{3 s} .

The even distribution reflects the equal left/right photon couplings and is symmetric under interchange of forward and backward directions.

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