A bstract The non-leptonic D 0 → K − K + and D 0 → π − π + decays are powerful probes of the Standard Model and are related to each other through the U -spin symmetry of the strong interaction. Using lattice QCD inputs we calculate the corresponding colour-allowed tree amplitudes in factorisation and demonstrate that non-factorisable contributions and U -spin-breaking effects at the level of 50% allow us to accommodate the measured branching ratios in the Standard Model. An exciting direct probe of such non-factorisable and U -spin breaking effects is provided by the $$ {D}^0\to {K}_{\mathrm{S}}^0{K}_{\mathrm{S}}^0 $$ D 0 → K S 0 K S 0 channel. This decay is governed by non-factorisable exchange topologies and essentially vanishes in the U -spin limit, although it is experimentally well established with a prominent branching ratio. Extrapolating our D 0 → K − K + results using the isospin symmetry, we find a consistent benchmark picture. Specifically, we can accommodate the measured $$ {D}^0\to {K}_{\mathrm{S}}^0{K}_{\mathrm{S}}^0 $$ D 0 → K S 0 K S 0 branching ratio with U -spin-breaking effects at the 50% level and exchange amplitudes at the level of 50% of the colour-allowed D 0 → K − K + , D 0 → π − π + tree contributions. Finally, we explore the resulting range for direct CP violation in $$ {D}^0\to {K}_{\mathrm{S}}^0{K}_{\mathrm{S}}^0 $$ D 0 → K S 0 K S 0 , obtaining upper bounds in our benchmark scenarios of a few per mille, offering an exciting target for future measurements.

Abstract The complex phases $$\phi _d$$ ϕ d and $$\phi _s$$ ϕ s , associated with the mixing between neutral $$B_q^0$$ B q 0 and $$\bar{B}_q^0$$ B ¯ q 0 mesons ( $$q\in \{d,s\}$$ q ∈ { 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 $$B_d^0\rightarrow J/\psi K^0$$ B d 0 → J / ψ K 0 , $$B_s^0\rightarrow J/\psi \phi $$ B s 0 → J / ψ ϕ and $$B_s^0\rightarrow D_s^+D_s^-$$ B s 0 → D s + D s - . To reach the highest possible precision on $$\phi _d$$ ϕ d and $$\phi _s$$ ϕ 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 $$B_s^0\rightarrow D_s^+D_s^-$$ B s 0 → D s + D s - , $$B_d^0\rightarrow D^+D^-$$ B d 0 → D + D - , $$B^+\rightarrow J/\psi K^+$$ B + → J / ψ K + and $$B^+\rightarrow J/\psi \pi ^+$$ B + → J / ψ π + , as well as from Belle-II on the decay $$B_d^0\rightarrow J/\psi \pi ^0$$ B d 0 → J / ψ π 0 , we present the current state-of-the-art picture on controlling the penguin contributions and extract $$\phi _d$$ ϕ d and $$\phi _s$$ ϕ 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.

The penguin loop-suppressed B decays are highly sensitive to contributions of hypothetical heavy new particles. Particularly interesting probes for testing the Standard Model and revealing such phenomena are provided by CP violation in the B^0_d_ S decay. Standard-Model estimates for the corresponding CP-violating observables are theoretically limited by doubly Cabibbo-suppressed penguin contributions. We study these effects using a factorization approach, and provide predictions for CP asymmetries, to be contrasted with future measurements. To gain additional insight into these hadronic effects, we propose the B_s^0_ S decay as a new channel. We predict the observables for this decay for which currently no measurements exist. By comparing B^0_d_ S and B^+ → ϕK^+, we further derive state-of-the-art constraints on isospin observables within the Standard Model. The same framework enables probing of possible New-Physics contributions, including general effects and those with non-trivial isospin structure. Interesting prospects arise for the high-precision era of flavour physics ahead.

We investigate the Standard Model (SM) description of the singly Cabibbo-suppressed charm decays D^0→ π^-π^+, D^0→ K^-K^+ and D^0→ K_ S^0K_ S^0. Using factorisation together with isospin symmetry, we constrain non-factorisable effects and the associated U-spin breaking required by the measured branching fractions. We find that corrections of order 50% are sufficient to describe all modes, which although sizeable, are not unexpected for hadronic charm decays. Using the constraints from the branching fractions, we derive SM predictions for the direct CP asymmetry in D^0→ K_ S^0K_ S^0, finding it to be at most at the per-mille level in our benchmark scenario, providing motivation for future precision measurements.

The non-leptonic D0 → K−K+ and D0 → π−π+ decays are powerful probes of the Standard Model and are related to each other through the U -spin symmetry of the strong interaction. Using lattice QCD inputs we calculate the corresponding colour-allowed tree amplitudes in factorisation and demonstrate that non-factorisable contributions and U -spin-breaking effects at the level of 50 D^0→K_S^0K_S^0 channel. This decay is governed by non-factorisable exchange topologies and essentially vanishes in the U -spin limit, although it is experimentally well established with a prominent branching ratio. Extrapolating our D0 → K−K+ results using the isospin symmetry, we find a consistent benchmark picture. Specifically, we can accommodate the measured D^0→K_S^0K_S^0 branching ratio with U -spin-breaking effects at the 50 D^0→K_S^0K_S^0 , obtaining upper bounds in our benchmark scenarios of a few per mille, offering an exciting target for future measurements.

Abstract The non-leptonic D 0 → K − K + and D 0 → π − π + decays are powerful probes of the Standard Model and are related to each other through the U -spin symmetry of the strong interaction. Using lattice QCD inputs we calculate the corresponding colour-allowed tree amplitudes in factorisation and demonstrate that non-factorisable contributions and U -spin-breaking effects at the level of 50% allow us to accommodate the measured branching ratios in the Standard Model. An exciting direct probe of such non-factorisable and U -spin breaking effects is provided by the D 0 → K S 0 K S 0 $$ {D}^0\to {K}_{\mathrm{S}}^0{K}_{\mathrm{S}}^0 $$ channel. This decay is governed by non-factorisable exchange topologies and essentially vanishes in the U -spin limit, although it is experimentally well established with a prominent branching ratio. Extrapolating our D 0 → K − K + results using the isospin symmetry, we find a consistent benchmark picture. Specifically, we can accommodate the measured D 0 → K S 0 K S 0 $$ {D}^0\to {K}_{\mathrm{S}}^0{K}_{\mathrm{S}}^0 $$ branching ratio with U -spin-breaking effects at the 50% level and exchange amplitudes at the level of 50% of the colour-allowed D 0 → K − K +, D 0 → π − π + tree contributions. Finally, we explore the resulting range for direct CP violation in D 0 → K S 0 K S 0 $$ {D}^0\to {K}_{\mathrm{S}}^0{K}_{\mathrm{S}}^0 $$ , obtaining upper bounds in our benchmark scenarios of a few per mille, offering an exciting target for future measurements.

Rare decays of the kind B → K∗μ+μ− and Bs → ϕμ+μ− are key players for testing the Standard Model. The current experimental data for their decay rates and angular observables show tensions with the theoretical predictions that may be indications of New Physics. We present a strategy to extract the relevant short-distance coefficients in the presence of new sources of CP violation from a minimal set of observables, utilizing the synergy between the rare modes with vector meson final states and B → Kμ+μ− decays. Using the current data as a guideline, we illustrate the new method to determine the complex coefficients C_9^(') and C_10^(') using only four angular observables. Interestingly, the current experimental picture leaves significant room for CP-violating New Physics. We discuss also the link to leptonic B_s^0→μ^+μ^- decays. Our strategy complements global fit efforts and could also help to disentangle long-distance QCD dynamics from possible new CP-violating interactions.

The complex phases ϕ _d and ϕ _s , associated with the mixing between neutral B_q^0 and B̅_q^0 mesons ( q∈{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→ J/ψ K^0 , B_s^0→ J/ψϕ and B_s^0→ D_s^+D_s^- . To reach the highest possible precision on ϕ _d and ϕ _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→ D_s^+D_s^- , B_d^0→ D^+D^- , B^+→ J/ψ K^+ and B^+→ J/ψπ ^+ , as well as from Belle-II on the decay B_d^0→ J/ψπ ^0 , we present the current state-of-the-art picture on controlling the penguin contributions and extract ϕ _d and ϕ _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.

CP violation offers powerful probes to explore the quark-flavour sector, where decays of B mesons have been key players since decades. I discuss a variety of probes ranging from non-leptonic to rare B decays, offering exciting opportunities at the FCC in the era after the HL-LHC and Belle II.

The complex phases phi d and phi s, associated with the mixing between neutral Bq0 and B & strns;q0 mesons (q is an element of{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 Bd0 -> J/psi K0, Bs0 -> J/psi phi and Bs0 -> Ds+Ds-. To reach the highest possible precision on phi d and phi 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 Bs0 -> Ds+Ds-, Bd0 -> D+D-, B+-> J/psi K+ and B+-> J/psi pi+, as well as from Belle-II on the decay Bd0 -> J/psi pi 0, we present the current state-of-the-art picture on controlling the penguin contributions and extract phi d and phi 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.

The complex phases ϕ_d and ϕ_s, associated with the mixing between neutral B_q^0 and B̅_q^0 mesons (q∈{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→ J/ψ K^0, B_s^0→ J/ψϕ and B_s^0→ D_s^+D_s^-. To reach the highest possible precision on ϕ_d and ϕ_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→ D_s^+D_s^-, B_d^0→ D^+D^-, B^+→ J/ψ K^+ and B^+→ J/ψπ^+, as well as from Belle-II on the decay B_d^0→ J/ψπ^0, we present the current state-of-the-art picture on controlling the penguin contributions and extract ϕ_d and ϕ_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.

The kaon physics programme, long heralded as a cutting-edge frontier by the European Strategy for Particle Physics, continues to stand at the intersection of discovery and innovation in high-energy physics (HEP). With its unparalleled capacity to explore new physics at the multi-TeV scale, kaon research is poised to unveil phenomena that could reshape our understanding of the Universe. This document highlights the compelling physics case, with emphasis on exciting new opportunities for advancing kaon physics not only in Europe but also on a global stage. As an important player in the future of HEP, the kaon programme promises to drive transformative breakthroughs, inviting exploration at the forefront of scientific discovery.

Experimentally, the phases $\phi_d$ and $\phi_s$ are determined from CP asymmetry measurements in the "golden modes" $B_d^0\to J/\psi K_{\mathrm{S}}^0$ and $B_s^0\to J/\psi\phi$. At leading order, the theoretical interpretation of these measurements is straightforward. However, to reach high precision determinations of $\phi_d$ and $\phi_s$, which is essential in view of the searches for signs of beyond the SM physics, corrections from next-to-leading order effects need to be accounted for. These corrections primarily originate from so-called penguin topologies. Using the $SU(3)$ flavour symmetry, these corrections can be determined using suitably chosen control modes. Recent new CP asymmetry measurements from LHCb in $B\to DD$ and Belle-II in $B_d^0\to J/\psi\pi^0$ decays greatly improve our knowledge on the parameters describing the contribution from penguin topologies. These proceedings will discuss the current constraints on the penguin parameters in $B\to J/\psi X$ and $B\to DD$ decays, provide corrected determinations for $\phi_d$ and $\phi_s$, and highlight what can be expected at the end of the HL-LHC and Belle-II programmes.

The non-leptonic D^0→ K^- K^+ and D^0→ π^-π^+ decays are powerful probes of the Standard Model and are related to each other through the U-spin symmetry of the strong interaction. Using lattice QCD inputs we calculate the corresponding colour-allowed tree amplitudes in factorisation and demonstrate that non-factorisable contributions and U-spin-breaking effects at the level of 50

Experimentally, the phases ϕ_d and ϕ_s are determined from CP asymmetry measurements in the "golden modes" B_d^0→ J/ψ K_S^0 and B_s^0→ J/ψϕ. At leading order, the theoretical interpretation of these measurements is straightforward. However, to reach high precision determinations of ϕ_d and ϕ_s, which is essential in view of the searches for signs of beyond the SM physics, corrections from next-to-leading order effects need to be accounted for. These corrections primarily originate from so-called penguin topologies. Using the SU(3) flavour symmetry, these corrections can be determined using suitably chosen control modes. Recent new CP asymmetry measurements from LHCb in B→ DD and Belle-II in B_d^0→ J/ψπ^0 decays greatly improve our knowledge on the parameters describing the contribution from penguin topologies. These proceedings will discuss the current constraints on the penguin parameters in B→ J/ψ X and B→ DD decays, provide corrected determinations for ϕ_d and ϕ_s, and highlight what can be expected at the end of the HL-LHC and Belle-II programmes.

In the pursuit of physics beyond the Standard Model, a promising path is the study of B-meson decays caused by the transition $b \to s\ell^+\ell^-$. A key observable in such decays is the ratio $R_K$, which measures electron--muon universality in $B \to K μ^+μ^-/e^+e^-$. At first sight, the recent LHCb measurement of $R_K \sim 1$ may seem to largely constrain deviations from universality in these decays. However, we show that this is actually not the case: new sources of CP violation allow for significant universality violation consistent with $R_K \sim 1$. This provides an exciting new opportunity to search for New Physics by measuring differences between CP asymmetries in $B \to Kμ^+μ^-$ and $B \to K e^+e^-$.

Experimental data on rare B-meson decays indicate deviations from Standard Model predictions. In studies of these decays, possible new sources of CP violation are often neglected. We discuss CP violation in the rare B-meson decays B→ Kℓ^+ℓ^- (ℓ = μ,e) and point to two phenomena that arise when new sources of CP violation are included. First, the Wilson coefficients C_9ℓ and C_10ℓ become complex, and we show how we can extract their values from measurements of direct and mixing-induced CP asymmetries. Second, new sources of CP violation can generate nontrivial lepton flavour universality violation. Such a violation is usually measured through ratios like R_K and R_K^*, but we show that measuring only these ratios leaves a large parameter space unexplored. These results bring exciting opportunities to reveal New Physics effects in the high-precision era.

The leptonic decay Bs → μ+μ− is both rare and theoretically clean, making it an excellent probe for New Physics searches. Due to its helicity suppression in the Standard Model, this decay is particularly sensitive to new (pseudo)-scalar contributions. We present a new strategy for detecting CP-violating New Physics contributions of this kind, exploiting two observables: 𝒜_Δ_s^μμ , which is accessible due to the sizeable decay width difference of the Bs system, and the mixing-induced CP asymmetry 𝒮_μμ . The strategy also uses information from B → K(*)μ+μ− and Bs → ϕμ+μ− decays. We find remarkably constrained regions in the 𝒜_ΔΓ_s^μμ - 𝒮_μμ plane that serve as promising targets for future measurements.

The phenomenon of $B^0_q$-$\bar{B}^0_q$ mixing ($q=d,s$) provides a sensitive probe for physics beyond the Standard Model. We have a careful look at the determination of the Unitarity Triangle apex, which is needed for the Standard Model predictions of the $B_q$ mixing parameters, and explore how much space for New Physics is left through the current data. We study the impact of tensions between inclusive and exclusive determinations of the CKM matrix elements $|V_{ub}|$ and $|V_{cb}|$, and focus on the $\gamma$ angle extraction. We present various future scenarios and discuss the application of these results for leptonic rare $B$ decays, which allows us to minimise the CKM parameter impact in the New Physics searches. Performing future projections, we explore and illustrate the impact of increased precision on key input quantities. It will be exciting to see how more precise data in the future high-precision era of flavour physics can lead to a much sharper picture.