Classical Electrodynamics · Problem 11.19

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · two-body-decay-kinematics

Problem

A particle of mass MM and 4-momentum PP decays into two particles of masses m1m_1 and m2m_2. (a) Use the conservation of energy and momentum in the form, p2=P−p1p_2 = P - p_1, and the invariance of scalar products of 4-vectors to show that the total energy of the first particle in the rest frame of the decaying particle is E1=M2+m12−m222ME_1 = \frac{M^2 + m_1^2 - m_2^2}{2M} and that E2E_2 is obtained by interchanging m1m_1 and m2m_2. (b) Show that the kinetic energy TiT_i of the iith particle in the same frame is Ti=ΔM(1−miM−ΔM2M)T_i = \Delta M\left(1 - \frac{m_i}{M} - \frac{\Delta M}{2M}\right) where ΔM=M−m1−m2\Delta M = M - m_1 - m_2 is the mass excess or QQ value of the process. (c) The charged pi-meson (M=139.6M = 139.6 MeV) decays into a mu-meson (m1=105.7m_1 = 105.7 MeV) and a neutrino (m2=0m_2 = 0). Calculate the kinetic energies of the mu-meson and the neutrino in the pi-meson's rest frame. The unique kinetic energy of the muon is the signature of a two-body decay.

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