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 Incoming links: Semileptonic wrong-sign mixing asymmetry

Past exam of the particle physics course of the University of Oxford / 2018 / 3 / e / Solution 2026-10-25

Assume the usual semileptonic flavour selection rule

Δ S = Δ Q

, negligible wrong-sign decay probability amplitudes, and equal magnitudes for CP-conjugate allowed probability amplitudes. Then

f_{+} = π^{-} ℓ^{+} ν

is reached directly only from

K^{0}

, and

f_{-} = π^{+} ℓ^{-} \overset{ν}{ˉ}

only from

\overset{ˉ}{K}^{0}

. Put

∣ A_{f_{+}} ∣ = ∣ \overset{ˉ}{A}_{f_{-}} ∣ = ∣ A ∣

. The four rates are

Γ (K (t) \to f_{+}) Γ (K (t) \to f_{-}) = ∣ A ∣^{2} ∣ g_{+} ∣^{2}, = ∣ A ∣^{2} ∣ q / p ∣^{2} ∣ g_{-} ∣^{2}, Γ (\overset{ˉ}{K} (t) \to f_{-}) Γ (\overset{ˉ}{K} (t) \to f_{+}) = ∣ A ∣^{2} ∣ g_{+} ∣^{2}, = ∣ A ∣^{2} ∣ p / q ∣^{2} ∣ g_{-} ∣^{2} .

There is only one path to each tagged semileptonic channel, so no direct/mixed decay interference survives. The semileptonic wrong-sign mixing asymmetry is

a_{SL} = \frac{Γ ( K ˉ ( t ) \to f _{+} ) - Γ ( K ( t ) \to f _{-} )}{Γ ( K ˉ ( t ) \to f _{+} ) + Γ ( K ( t ) \to f _{-} )} = \frac{1 - ∣ q / p ∣ ^{4}}{1 + ∣ q / p ∣ ^{4}} .

It isolates CP violation in mixing under the stated assumptions. Allowing direct semileptonic CP violation or wrong-sign probability amplitudes would spoil that isolation.^[ref]

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