Doubly CKM-suppressed corrections to CP asymmetries in $B^0\to J\mathrm{/}\psi K^0$

A doubly CKM-suppressed amplitude in $B^0\to J\mathrm{/}\psi K_S$ leads to corrections in CP asymmetries $S=\sin 2\beta ,C=0$, which may be enhanced by long-distance rescattering. It has been suggested that this enhancement may lead to several percent corrections. We calculate an upper bound of order ${10}^{-3}$ on rescattering corrections using measured branching ratios for charmless $\mathrm{\mid }\mathrm{\Delta }S\mathrm{\mid }=1$ $B^0$ decays.