A group of charged particles with charges ej and coordinates rj(t) undergo interactions and are accelerated only during a time −τ/2<t<τ/2, during which their velocities change from cβj to cβj′.
(a) Show that for ωτ≪1 the intensity of radiation emitted with polarization ϵ per unit solid angle and unit frequency interval is
dωdΩd2I=4π2c1∣ϵ∗⋅E∣2
where
E=j∑ej(1−n⋅βj′βj′−1−n⋅βjβj)e−iωn⋅rj(0)/c
(b) An ω0 meson of mass 784 MeV decays into π+π− and e+e− with branching ratios of 1.3×10−2 and 8×10−5, respectively. Show that for both decay modes the frequency spectrum of radiated energy at low frequencies is
dωdI=πce2[(β1+β2)ln(1−β1+β)−2]≈πc4e2[ln(mMω)−21]
where Mω is the mass of the ω0 meson and m is the mass of one of the decay products. Evaluate approximately the total energy radiated in each decay by integrating the spectrum up to the maximum frequency allowed kinematically. What fraction of the rest energy of the ω0 is it in each decay?