The mixing frequency is

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

Symmetry and Flavour, Ch B mesons: mixing, time-dependent CP asymmetries and the unitarity triangle

Problem

The BsB_s mixing frequency is Δms=17.765 ps−1\Delta m_s = 17.765\,\mathrm{ps^{-1}}. A finite proper-time resolution σt\sigma_t smears an oscillating signal of frequency Δms\Delta m_s, multiplying its amplitude by a factor exp⁡ ⁣[−(Δmsσt)2/2]\exp\!\left[-\left(\Delta m_s\sigma_t\right)^2/2\right]. ((a)) Estimate the largest proper-time resolution σt\sigma_t (in femtoseconds) that keeps this smearing factor at 0.900.90 or above. ((b)) A separate measurement of sin⁡2β\sin2\beta in B→J/ψKSB \to J/\psi K_S decays has an effective tagging power Q=0.30Q = 0.30. Estimate the number of flavour-tagged, reconstructed decays NN needed to reach a statistical precision σS=0.04\sigma_S = 0.04 on sin⁡2β\sin2\beta, using the chapter's estimate N≈1/(0.35 QσS2)N \approx 1/(0.35\,Q\sigma_S^2), where 0.35=⟨sin⁡2Δm Δt⟩0.35=\langle\sin^2\Delta m\,\Delta t\rangle accounts for each event carrying Ssin⁡(Δm Δt)S\sin(\Delta m\,\Delta t) rather than SS itself.

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