= Time-dependent neutral-B CP asymmetry
{title2=$A_{\rm tag}(\Delta t)=S\sin(\Delta m\Delta t)-C\cos(\Delta m\Delta t)$}
With <CPT symmetry>, write the propagating states of <neutral meson mixing> as $p|B^0\rangle\pm q|\bar B^0\rangle$, defining the coefficient ratio $q/p$. Let $\Delta m$ be their mass splitting and use units $\hbar=1$.
For rates $f_\pm$ tagged as $B^0,\bar B^0$ and $\Delta t=t_f-t_{\rm tag}$, neglect the <decay width> difference and take $|q/p|=1$. Define
$$
\lambda_f=\frac qp\frac{\bar A_f}{A_f},\qquad
S=\frac{2\operatorname{Im}\lambda_f}{1+|\lambda_f|^2},\qquad
C=\frac{1-|\lambda_f|^2}{1+|\lambda_f|^2}.
$$
Here $A_f,\bar A_f$ are <decay amplitudes>. The tagged asymmetry is $(f_+-f_-)/(f_++f_-)=S\sin(\Delta m\Delta t)-C\cos(\Delta m\Delta t)$ in this convention.<Observation of CP violation in the B0 meson system by BABAR Collaboration>[PDF p. 4, Eq. (1), tag signs and time convention]{ref}
For a single <weak phase> in a <charmonium>-<kaon> <CP eigenstate> with eigenvalue $\eta_f$, $\lambda_f\simeq\eta_f e^{-2i\beta}$ gives $S=-\eta_f\sin2\beta$, $C\simeq0$, with
$$
\beta=\arg\!\left(-\frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*}\right).
$$
This connects the <CKM unitarity triangle> to <quantum interference> between direct decay and decay after <neutral meson mixing>.<Observation of CP violation in the B0 meson system by BABAR Collaboration>[PDF p. 4, Eq. (1) and charmonium weak phase]{ref}
For an independent teaching example choose $x=\Delta m\Delta t$, $\lambda_f=4i/3$ and an arbitrary common positive envelope $F(\Delta t)$. Then $S=24/25$, $C=-7/25$ and
$$
f_\pm=F(\Delta t)\left[1\pm\left(\frac{24}{25}\sin x+\frac7{25}\cos x\right)\right].
$$
Because $(24/25)^2+(7/25)^2=1$, both rates remain nonnegative. At $x=0$, the asymmetry is $7/25$; at $x=\pi/2$, it is $24/25$. These chosen values illustrate the sign convention and are not experimental estimates. More generally $S^2+C^2=1-[2\operatorname{Re}\lambda_f/(1+|\lambda_f|^2)]^2\le1$.
If a tag flips independently with probability $w$, exchanging $f_+$ and $f_-$ for that fraction of events gives $A_{\rm observed}=(1-2w)A_{\rm tag}$ by direct subtraction. The plotted toy choice $w=1/5$ reduces the amplitude to $3/5$ of its ideal value; at $w=1/2$ the tag contains no flavour information. Finite time resolution additionally averages over nearby oscillation phases. Extracting a <weak phase> consequently requires calibrated readout as well as a <CKM matrix> prediction.
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