(a) A hypothetical fourth-generation charged lepton has mass and decays through the…

PHYSICS · Subjective · Fermi University Physics · Particle Physics & Quantum Fields · Particles, interactions and experiment · Electroweak theory and the Standard Model

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

Problem

(a) A hypothetical fourth-generation charged lepton L−L^- has mass mL=20.0 GeVm_L = 20.0\,\mathrm{GeV} and decays through the same point-like weak interaction, with coupling GFG_F, as the muon, with the same five-channel counting used for the τ\tau, so that B(L−→e−νˉeνL)=1/5\mathcal{B}(L^-\to e^-\bar\nu_e\nu_L)=1/5. Using the measured muon lifetime τμ=2.1970 μs\tau_\mu=2.1970\,\mu\mathrm{s}, mμ=105.658 MeVm_\mu=105.658\,\mathrm{MeV}, and the fifth-power mass scaling of the point-like decay rate, predict the lifetime of L−L^-. (b) The asserted squared amplitude for π+→ℓ+ν\pi^+\to\ell^+\nu gives Γ(π→ℓν)=Cπmℓ2mπ(1−mℓ2/mπ2)2\Gamma(\pi\to\ell\nu)=C_\pi m_\ell^2 m_\pi(1-m_\ell^2/m_\pi^2)^2; the same functional form applies to any charged pseudoscalar meson decaying through the same point-like weak interaction, with a constant CPC_P that cancels in a same-meson ratio. Using mK+=493.677 MeVm_{K^+}=493.677\,\mathrm{MeV}, me=0.511 MeVm_e=0.511\,\mathrm{MeV}, mμ=105.658 MeVm_\mu=105.658\,\mathrm{MeV}, predict RK≡Γ(K+→e+ν)/Γ(K+→μ+ν)R_K \equiv \Gamma(K^+\to e^+\nu)/\Gamma(K^+\to\mu^+\nu). Split the prediction into a phase-space factor pe∗/pμ∗p_e^*/p_\mu^* (with pℓ∗=(mK2−mℓ2)/2mKp_\ell^*=(m_K^2-m_\ell^2)/2m_K for the massless neutrino) and a squared-amplitude factor, and compare with the measured value (2.488±0.009)×10−5(2.488\pm0.009)\times10^{-5} (PDG 2024).

Browse the problem library