A hypothetical neutral meson pair

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Flavour, neutrinos and CP violation · Flavour, neutrinos and CP violation

Symmetry and Flavour, Ch Neutral kaons: mixing, strangeness oscillation and CP violation

Problem

A hypothetical neutral meson pair P0P^0, Pˉ0\bar P^0 obeys the equal-diagonal effective Hamiltonian H=M−i2ΓH = M - \tfrac{i}{2}\Gamma with H11=H22=m0−i2γH_{11} = H_{22} = m_0 - \tfrac{i}{2}\gamma and H12=H21=μ−i2νH_{12} = H_{21} = \mu - \tfrac{i}{2}\nu, where m0=500.000000 MeVm_0 = 500.000000\,\mathrm{MeV}, μ=3.5×10−6 MeV\mu = 3.5\times10^{-6}\,\mathrm{MeV}, the average lifetime corresponding to γ\gamma is τ0=ℏ/γ=1.00×10−10 s\tau_0 = \hbar/\gamma = 1.00\times10^{-10}\,\mathrm{s}, and ν=0.6γ\nu = 0.6\gamma. Take ℏ=6.582×10−22 MeV s\hbar = 6.582\times10^{-22}\,\mathrm{MeV\,s}. ((a)) Find the mass splitting Δm=2μ\Delta m = 2\mu of the two eigenstates, in MeV. ((b)) Find the widths Γ1=γ+ν\Gamma_1 = \gamma+\nu and Γ2=γ−ν\Gamma_2 = \gamma-\nu of the two eigenstates, in MeV. ((c)) Convert these widths to lifetimes τ1\tau_1 and τ2\tau_2 in seconds, and check that their average width reproduces γ\gamma.

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