Eur. Phys. J. C (2026) 86: 215
https://doi.org/10.1140/epjc/s10052-026-15401-z

Regular Article - Theoretical Physics

How to tame penguins: advancing to high-precision measurements of ${\varphi }_d$ and ${\varphi }_s$

¹ Nikhef, Science Park 105, 1098 XG, Amsterdam, The Netherlands
² Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, 9747, Groningen, The Netherlands
³ Faculty of Science, Vrije Universiteit Amsterdam, 1081 HV, Amsterdam, The Netherlands
⁴ Center for Particle Physics Siegen, University of Siegen, 57068, Siegen, Germany

^a This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 16 May 2025
Accepted: 2 February 2026
Published online: 3 March 2026

Abstract

The complex phases ${\varphi }_d$ and ${\varphi }_s$, associated with the mixing between neutral $B_q^0$ and ${\overline{B}}_q^0$ mesons ($q\in \{d,s\}$), are key observables to test the Standard Model and search for contributions from new physics. They are conventionally determined from the measurements of mixing-induced CP violation in the decays $B_d^0\to J/\psi K^0$, $B_s^0\to J/\psi \varphi$ and $B_s^0\to D_s^{+}{D}_s^{-}$. To reach the highest possible precision on ${\varphi }_d$ and ${\varphi }_s$, it is crucial that corrections from next-to-leading order effects – primarily associated with penguin decay topologies – are accounted for. The strategy adopted in this paper uses the SU(3) flavour symmetry of QCD to relate the unknown contributions from penguin topologies to their counterparts in suitably-chosen control modes, where their effects are enhanced. Utilising new CP asymmetry measurements from LHCb on the decays $B_s^0\to D_s^{+}{D}_s^{-}$, $B_d^0\to D^{+}{D}^{-}$, $B^{+}\to J/\psi K^{+}$ and $B^{+}\to J/\psi {\pi }^{+}$, as well as from Belle-II on the decay $B_d^0\to J/\psi {\pi }^0$, we present the current state-of-the-art picture on controlling the penguin contributions and extract ${\varphi }_d$ and ${\varphi }_s$ from the corresponding observables. We explore the prospects for the end of the Belle-II and HL-LHC flavour physics programmes, and demonstrate the importance of measuring the control modes with future data.