Past exam of the particle physics course of the University of Oxford / 2018 / 3 / d / Solution

Introduce all four physical probability amplitudes: $A_f,{\bar{A}}_f,A_{\bar{f}},{\bar{A}}_{\bar{f}}$. Under exchange of the flavour and final-state labels, defineThis CP-conjugate decay-amplitude ratio gives the conjugate-channel ratesCP is a unitary discrete transformation, so exchanging state labels does not mean complex-conjugating every strong phase by fiat. Nor should one assume that CP-conjugate probability amplitudes are equal while studying CP violation.

The equality $\bar{\lambda }={\lambda }^{-1}$ holds whenin particular when $\bar{f}$ and $f$ are the same CP eigenstate ray, with a consistent state phase. It is not automatic for two distinct CP-conjugate final states. For example, $p=q$, $A_f={\bar{A}}_{\bar{f}}=2$ and ${\bar{A}}_f=A_{\bar{f}}=1$ conserve the paired decay probability amplitudes but give $\lambda =\bar{\lambda }=1/2$, not inverse values.