A beam of mesons is produced with momentum in the laboratory

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Particle Physics & Quantum Fields · Quantum dynamics and its formulations

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

Problem

A beam of K0K^0 mesons is produced with momentum p=5.00 GeV/cp=5.00\,\mathrm{GeV}/c in the laboratory. Take mK=497.6 MeV/c2m_K=497.6\,\mathrm{MeV}/c^2, τS=0.8954×10−10 s\tau_S=0.8954\times10^{-10}\,\mathrm{s} (cτS=2.68 cmc\tau_S=2.68\,\mathrm{cm}), τL=5.116×10−8 s\tau_L=5.116\times10^{-8}\,\mathrm{s}, and ΔmKτS=0.474\Delta m_K\tau_S=0.474. ((a)) Find βγ\beta\gamma for this beam and the laboratory decay lengths of the short- and long-lived components. ((b)) Using P(K0→Kˉ0;t)=14[e−ΓSt+e−ΓLt−2e−Γˉtcos⁡(Δm t)],P(K^0\to\bar K^0;t)=\tfrac14\big[e^{-\Gamma_St}+e^{-\Gamma_Lt}-2e^{-\bar\Gamma t}\cos(\Delta m\,t)\big], with tt the kaon's proper time, find the probabilities that a kaon born as K0K^0 is found as K0K^0 or as Kˉ0\bar K^0 after a proper time t=2τSt=2\tau_S, and give the corresponding laboratory distance for this beam. ((c)) At that point the beam enters a block of matter. Explain, without further calculation, why the block adds to the KSK_S amplitude still present in the beam a new coherent KSK_S amplitude proportional to f−fˉf-\bar f, where ff and fˉ\bar f are the forward scattering amplitudes of K0K^0 and Kˉ0\bar K^0 on a nucleon, and name the strong-interaction process that makes the forward scattering amplitudes of K0K^0 and Kˉ0\bar K^0 on nucleons different.

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