The measured branching fractions are ,

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

The measured branching fractions are B(KL→π+π−)=1.967×10−3\mathcal B(K_L\to\pi^+\pi^-)=1.967\times10^{-3}, B(KS→π+π−)=0.6920\mathcal B(K_S\to\pi^+\pi^-)=0.6920, B(KL→π0π0)=8.64×10−4\mathcal B(K_L\to\pi^0\pi^0)=8.64\times10^{-4} and B(KS→π0π0)=0.3069\mathcal B(K_S\to\pi^0\pi^0)=0.3069, with τS=0.8954×10−10 s\tau_S=0.8954\times10^{-10}\,\mathrm{s} and τL=5.116×10−8 s\tau_L=5.116\times10^{-8}\,\mathrm{s}. ((a)) Show that the double ratio ∣η00/η+−∣2|\eta_{00}/\eta_{+-}|^2 can be written with the four branching fractions alone, with no lifetime in it, and evaluate it. ((b)) Use ∣η00/η+−∣2≈1−6 Re(ϵ′/ϵ)|\eta_{00}/\eta_{+-}|^2\approx1-6\,\mathrm{Re}(\epsilon'/\epsilon) to extract Re(ϵ′/ϵ)\mathrm{Re}(\epsilon'/\epsilon) from your result. ((c)) Suppose the two KSK_S branching fractions were exact and the two KLK_L ones were each known independently to the same fractional uncertainty δ\delta. How small must δ\delta be for a double ratio of the size found in (a) to differ from 11 by three standard deviations? Use the answer to explain why experiments measured the double ratio directly, counting the π+π−\pi^+\pi^- and π0π0\pi^0\pi^0 decays of KLK_L and KSK_S side by side, rather than building it from separately measured branching fractions.

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