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 $B_d\to J/\psi K_S$ and $B_s\to J/\psi \varphi$, 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 ${\mathrm{\Lambda }}_{\mathrm{QCD}}/m_{\psi }$, where $m_{\psi }$ denotes the $J/\psi$ 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 $2\beta$ as $|\mathrm{\Delta }{\varphi }_d|\le 0.68°$. The penguin contribution to $2{\beta }_s$ is bounded as $|\mathrm{\Delta }{\varphi }_s^0|\le 0.97°$, $|\mathrm{\Delta }{\varphi }_s^{\parallel }|\le 1.22°$, and $|\mathrm{\Delta }{\varphi }_s^{\perp }|\le 0.99°$ for the case of longitudinal, parallel, and perpendicular $\varphi$ and $J/\psi$ polarizations, respectively. Further, we place bounds on $|\mathrm{\Delta }{\varphi }_d|$ for $B_d\to \psi (2S)K_S$ and the polarization amplitudes in $B_d\to J/\psi K^{*}$. 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 $B_d\to J/\psi {\pi }^0$, $B_d\to J/\psi {\rho }^0$, $B_s\to J/\psi K_S$, and $B_s\to J/\psi K^{*}$. For all studied decay modes, we also constrain the sizes of the direct $CP$ asymmetries.

Figure 1

The LO diagram is shown in (a). The soft IR divergence of the diagram (b) factorizes with the corresponding diagram of the effective-theory side shown in (c). The diagram (d) is an example of a diagram with a collinear IR divergence. In (e) a spectator scattering diagram is given.