= Neutral B meson mixing
{title2=$B^0\leftrightarrow\bar B^0$}
<Neutral B meson mixing> converts the <flavor eigenstates> $B^0=\bar bd$ and $\bar B^0=b\bar d$ into one another through the <weak interaction>. The propagating <mass eigenstates> are $|B_L\rangle=p|B^0\rangle+q|\bar B^0\rangle$ and $|B_H\rangle=p|B^0\rangle-q|\bar B^0\rangle$, with $|p|^2+|q|^2=1$.
In units $\hbar=c=1$, neglect their width difference and take $|q/p|=1$. Evolution of an initial $B^0$ at <proper time> $t$ gives the undecayed flavour <probabilities>
$$
P_{B^0}(t)=e^{-\Gamma t}\cos^2\left(\frac{\Delta m\,t}{2}\right),\qquad
P_{\bar B^0}(t)=e^{-\Gamma t}\sin^2\left(\frac{\Delta m\,t}{2}\right),
$$
where $\Delta m=m_H-m_L$ and $\Gamma$ is the common <decay width>. Their sum is the survival <probability>, not one.
For a common final state, mixing and direct decay interfere through $\lambda_f=(q/p)(\bar A_f/A_f)$. In the <Standard Model>, $f=J/\psi\,K_S$ combines a <J psi meson> with a <short-lived neutral kaon>. Neglecting subleading weak amplitudes and small kaon <CP violation>,
$$
\lambda_f\simeq-e^{-2i\beta},\qquad
\beta=\arg\left[-\frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*}\right].
$$
Here the $V_{ij}$ are <CKM matrix> elements; the minus sign reflects the approximate negative <CP> parity of this specified final state. A different charmonium-kaon final state need not have that same sign.<Review of Particle Physics by Particle Data Group (2022)>[CP Violation in the Quark Sector, equation 13.82 and preceding paragraph: the J/psi K_S channel]{ref}
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