Time-dependent neutral-B CP asymmetry ($A_{\mathrm{tag}}(\Delta t)=S\sin (\Delta m\Delta t)-C\cos (\Delta m\Delta t)$)

With CPT symmetry, write the propagating states of neutral meson mixing as $p|B^0⟩\pm q|{\bar{B}}^0⟩$, defining the coefficient ratio $q/p$. Let $\Delta m$ be their mass splitting and use units $\hslash =1$.

For rates $f_{\pm }$ tagged as $B^0,{\bar{B}}^0$ and $\Delta t=t_f-t_{\mathrm{tag}}$, neglect the decay width difference and take $|q/p|=1$. DefineHere $A_f,{\bar{A}}_f$ 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.^[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 $S=-{\eta }_f\sin 2\beta$, $C\simeq 0$, withThis connects the CKM unitarity triangle to quantum interference between direct decay and decay after neutral meson mixing.^[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 $S=24/25$, $C=-7/25$ andBecause $(24/25{)}^2+(7/25{)}^2=1$, 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 $S^2+C^2=1-[2\mathrm{Re}{\lambda }_f/(1+|{\lambda }_f{|}^2){]}^2\le 1$.

If a tag flips independently with probability $w$, exchanging $f_{+}$ and $f_{-}$ for that fraction of events gives $A_{\mathrm{observed}}=(1-2w)A_{\mathrm{tag}}$ 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.