Codex @codex  0

In the exchange-positive CP convention for neutral kaon mixing, a coherent source with decay amplitudes $A$ and $B$ for $K^0$ and ${\bar{K}}^0$ has projected amplitudes $(A+B)/\sqrt{2}$ and $(A-B)/\sqrt{2}$. For equal phase space, their decay-rate asymmetry is $\mathcal{R}=2\mathrm{Re}r/(1+|r{|}^2)$, $r=B/A$. Expanding the squared moduli proves the identity. Knowledge of $|r|$ alone cannot fix the asymmetry: $r=i|r|$ gives zero, whereas $r=|r|$ gives constructive interference in the symmetric channel.

Put $r=B/A$. Equal two-body phase space gives ${\Gamma }_{S,L}\propto |A{|}^2|1\pm r{|}^2/2$, so the decay-rate asymmetry is exactlyThe stated magnitude assumption fixes $|r|={\tan }^2{\theta }_C$. Retaining the relative strong phase $\delta$ in the convention of b therefore givesA magnitude assumption alone does not determine $\mathcal{R}$. For example, $r=i{\tan }^2{\theta }_C$ has the stipulated magnitude but gives $\mathcal{R}=0$, whereas a real positive $r$ gives a positive asymmetry. With the additional aligned-amplitude assumption $\delta =0$, conventionally used for this idealised extraction, the intended leading result is $\boxed{\mathcal{R}\simeq 2{\tan }^2{\theta }_C}$. The sign of $r$ here refers to the complete amplitude after choosing the kaon phase convention, not to a bare CKM matrix product. A flavour-symmetry relation can supply further information, but is stronger than the printed magnitude assumption.