A meson ( ) has laboratory momentum along the -axis and decays to ( )

PHYSICS · Subjective · Fermi University Physics · Particle Physics & Quantum Fields · Particles, interactions and experiment · Particles, interactions and experiment

Symmetry and Flavour, Ch From amplitude to rate: Lorentz-invariant phase space, widths and cross sections

Problem

A J/ψJ/\psi meson (M=3.0969 GeVM=3.0969\,\mathrm{GeV}) has laboratory momentum P=6.00 GeVP=6.00\,\mathrm{GeV} along the zz-axis and decays to μ+μ−\mu^+\mu^- (mμ=0.10566 GeVm_\mu=0.10566\,\mathrm{GeV}). Starting from the Lorentz-invariant two-body phase space integral ∫dΦ2=∫d3p1(2π)32E1d3p2(2π)32E2(2π)4δ4(P−p1−p2),\int\mathrm{d}\Phi_2 = \int\frac{\mathrm{d}^3p_1}{(2\pi)^3 2E_1}\frac{\mathrm{d}^3p_2}{(2\pi)^3 2E_2}(2\pi)^4\delta^4(P-p_1-p_2), carry out the integral directly in this laboratory frame, where the parent moves, rather than in its rest frame, and show that the result still equals p∗/4πMp^*/4\pi M, with p∗p^* the muon momentum in the J/ψJ/\psi rest frame. Confirm your final numerical value agrees with the rest-frame formula.

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