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

The precision of the CP phases $2\beta$ and $2{\beta }_s$ determined from the mixing-induced CP asymmetries in $B_d\to J\mathrm{/}\psi K_S$ and $B_s\to J\mathrm{/}\psi \varphi$, respectively, is limited by the unknown long-distance contribution of a penguin diagram involving up quarks. The penguin contribution is expected to be comparable in size to the precision of the LHCb and Belle II experiments and therefore limits the sensitivity of the measured quantities to new physics. We analyze the infrared QCD structure of this contribution and find that all soft and collinear divergences either cancel between different diagrams or factorize into matrix elements of local four-quark operators up to terms suppressed by ${\mathrm{\Lambda }}_{QCD}\mathrm{/}{m}_{\psi }$, where $m_{\psi }$ denotes the $J\mathrm{/}\psi$ mass. Our results, which are based on an operator product expansion, allow us to calculate the penguin-to-tree ratio $P\mathrm{/}T$ in terms of the matrix elements of these operators and to constrain the penguin contribution to the phase $2\beta$ as $\mathrm{\mid }\mathrm{\Delta }{\varphi }_d\mathrm{\mid }\le {0.68}^{\circ }$. The penguin contribution to $2{\beta }_s$ is bounded as $\mathrm{\mid }\mathrm{\Delta }{\varphi }_s^0\mathrm{\mid }\le {0.97}^{\circ }$, $|\Delta \phi_s^{\parallel}|\leq {1.22}^\circ$,and$|\Delta \phi_s^{\perp}|\leq {0.99}^\circ$forthecaseoflongitudinal,parallel,andperpendicular$\phi$and$J/\psi$polarizations,respectively\mathrm{.}Wefurtherplaceboundson$|\Delta \phi_d|$for$B_d\to \psi(2S) K_S$andthepolarizationamplitudesin$B_d\to J/\psi K^$\mathrm{.}Inourapproachitisfurtherpossibletoconstrain$P/T$ for decays in which $P\mathrm{/}T$ is Cabibbo-unsuppressed and we derive upper limits on the penguin 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 CP asymmetries.