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

The precision of the CP phases $2β$ and $2β_s$ determined from the mixing-induced CP asymmetries in $B_d\to J/ψK_S$ and $B_s\to J/ψϕ$, 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 $Λ_{QCD}/m_ψ$, where $m_ψ$ denotes the $J/ψ$ mass. Our results, which are based on an operator product expansion, allow us to calculate the penguin-to-tree ratio $P/T$ in terms of the matrix elements of these operators and to constrain the penguin contribution to the phase $2β$ as $|Δϕ_d|\leq {0.68}^\circ$. The penguin contribution to $2β_s$ is bounded as $|Δϕ_s^{0}|\leq {0.97}^\circ$, $|Δϕ_s^{\parallel}|\leq {1.22}^\circ$, and $|Δϕ_s^{\perp}|\leq {0.99}^\circ$ for the case of longitudinal, parallel, and perpendicular $ϕ$ and $J/ψ$ polarizations, respectively. We further place bounds on $|Δϕ_d|$ for $B_d\to ψ(2S) K_S$ and the polarization amplitudes in $B_d\to J/ψK^*$. In our approach it is further possible to constrain $P/T$ for decays in which $P/T$ is Cabibbo-unsuppressed and we derive upper limits on the penguin contribution to the mixing-induced CP asymmetries in $B_d\to J/ψπ^0$, $B_d\to J/ψρ^0$, $B_s\to J/ψK_S$, and $B_s\to J/ψK^*$. For all studied decay modes we also constrain the sizes of the direct CP asymmetries.