Classical Electrodynamics · Problem 15.15

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bremsstrahlung-virtual-quanta-radiative-beta

Problem

One of the decay modes of a K+\text{K}^+ meson is the three-pion decay, K+→π+π+π−\text{K}^+ \rightarrow \pi^+\pi^+\pi^-. The energy release is 75 MeV, small enough that the pions can be treated nonrelativistically in rough approximation. (a) Show that the differential spectrum of radiated intensity at low frequencies in the K meson rest frame is approximately d2Idω dΩ≈2e2π2c⋅T−mπc2⋅sin⁡2θ\frac{d^2I}{d\omega\, d\Omega} \approx \frac{2e^2}{\pi^2 c} \cdot \frac{T_-}{m_\pi c^2} \cdot \sin^2\theta where T−T_- is the kinetic energy of the negative pion and θ\theta is the angle of emission of the photon relative to the momentum of the negative pion. (b) Estimate the branching ratio for emission of a photon of energy greater than Δ\Delta relative to the nonradiative three-pion decay. What is its numerical value for Δ=1\Delta = 1 MeV? 10 MeV? Compare with experiment (∼2×10−3\sim 2 \times 10^{-3} for Δ=11\Delta = 11 MeV).

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