The spin-summed

PHYSICS · Multi-part · 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

The spin-summed, spin-averaged squared amplitude for Z→μ+μ−Z\to\mu^+\mu^-, quoted from electroweak theory (label: asserted), is ∣M∣2‾=385 GeV2\overline{|\mathcal{M}|^2} = 385\,\mathrm{GeV^2}, treating the muon as massless compared with MZ=91.19 GeVM_Z=91.19\,\mathrm{GeV}. ((a)) Use the two-body width formula Γ=p∗∣M∣2‾/8πM2\Gamma = p^*\overline{|\mathcal{M}|^2}/8\pi M^2, with p∗=MZ/2p^*=M_Z/2, to calculate Γ(Z→μ+μ−)\Gamma(Z\to\mu^+\mu^-) in MeV. ((b)) Given the total width ΓZ=2495.5 MeV\Gamma_Z=2495.5\,\mathrm{MeV} (PDG 2024), estimate the branching ratio B(Z→μ+μ−)\mathcal{B}(Z\to\mu^+\mu^-).

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