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The precision of the CP phases $2\beta$ and $2\beta_s$ determined from themixing-induced CP asymmetries in $B_d\to J/\psi K_S$ and $B_s\to J/\psi \phi$,respectively, is limited by the unknown long-distance contribution of a penguindiagram involving up quarks. The penguin contribution is expected to becomparable in size to the precision of the LHCb and Belle II experiments andtherefore limits the sensitivity of the measured quantities to new physics. Weanalyze the infrared QCD structure of this contribution and find that all softand collinear divergences either cancel between different diagrams or factorizeinto matrix elements of local four-quark operators up to terms suppressed by$\Lambda_{QCD}/m_{\psi}$, where $m_{\psi}$ denotes the $J/\psi$ mass. Ourresults, which are based on an operator product expansion, allow us tocalculate the penguin-to-tree ratio $P/T$ in terms of the matrix elements ofthese operators and to constrain the penguin contribution to the phase $2\beta$as $|\Delta \phi_d|\leq {0.68}^\circ$. The penguin contribution to $2\beta_s$is bounded as $|\Delta \phi_s^{0}|\leq {0.97}^\circ$, $|\Delta\phi_s^{\parallel}|\leq {1.22}^\circ$, and $|\Delta \phi_s^{\perp}|\leq{0.99}^\circ$ for the case of longitudinal, parallel, and perpendicular $\phi$and $J/\psi$ polarizations, respectively. We further place bounds on $|\Delta\phi_d|$ for $B_d\to \psi(2S) K_S$ and the polarization amplitudes in $B_d\toJ/\psi K^*$. In our approach it is further possible to constrain $P/T$ fordecays in which $P/T$ is Cabibbo-unsuppressed and we derive upper limits on thepenguin contribution to the mixing-induced CP asymmetries in $B_d\to J/\psi\pi^0$, $B_d\to J/\psi \rho^0$, $B_s\to J/\psi K_S$, and $B_s\to J/\psi K^*$.For all studied decay modes we also constrain the sizes of the direct CPasymmetries.

Penguin contributions to CP phases in $B_{d,s}$ decays to charmonium