 Solution

ID: past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2018/3/b/solution

The two indistinguishable paths to the same final state must be added as probability amplitudes:

A_{f} (t) = A_{f} g_{+} (t) + \frac{q}{p} \overset{ˉ}{A}_{f} g_{-} (t) = A_{f} [g_{+} (t) + λ g_{-} (t)] .

Taking the squared modulus gives the time-dependent decay rate, with the common final-state phase-space normalization absorbed into the probability amplitudes:

Γ (K (t) \to f) = ∣ A_{f} ∣^{2} [∣ g_{+} ∣^{2} + ∣ λ ∣^{2} ∣ g_{-} ∣^{2} + 2 Re (λ g_{-} g_{+}^{*})] .

The cross term is the interference between direct decay and decay after flavour conversion. For reference, diagonal propagation gives

g_{\pm} = (e_{ℓ} \pm e_{h}) /2

, where

e_{j} = e^{- i m_{j} t - Γ_{j} t /2}

and

t

is proper time. The unfactored expression remains usable if

A_{f} = 0

, when the ratio

λ

itself is undefined.

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