Resolving asymmetries in B⁰ and B⁰_s oscillations

In the Standard Model (SM), CP violation originates from a single complex phase in the 3 × 3 Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix. The unitarity condition of the CKM matrix (V_ud V^*_ub + V_cd V^*_cb + V_td V^*_tb = 0, where V_ij are the CKM matrix elements) can be represented as a triangle in the complex plane, with an area proportional to the amount of CP violation in the quark sector. One angle of this triangle, γ = arg (–V_ud V^*_ub/ V_cd V^*_cb), is of particular interest as it can be probed both indirectly under the assumption of unitarity and in tree-level processes that make no such assumption. Its most sensitive direct experimental determination is currently given by a combination of LHCb measurements of B⁺, B⁰, B⁰_s decays to final states containing a D_(s) meson and one or more light mesons. Decay-time-dependent analyses of tree-level B⁰_s → D^∓_s K^± and B⁰ → D^∓π^± decays are sensitive to the angle γ through CP violation in the interference between mixing and decay amplitudes. Thus, comparing the value of γ obtained from tree-level processes with indirect measurements of γ and other unitary triangle parameters in loop-level processes provides an important consistency check of the SM.

Measurements using neutral B⁰ and B⁰_s mesons are particularly powerful because they resolve ambiguities that other measurements cannot. Due to the interference between B⁰_(s) – B⁰_(S) mixing and decay amplitudes, the physical CP-violating parameters in these decays are functions of a combination of γ and the relevant mixing phase, namely γ + 2β in the B⁰ system, where β = arg(–V_cd V^*_cb/ V_td V^*_tb), and γ–2β_s in the B⁰_s system, where β_s = arg(–V_ts V^*_tb/ V_td V^*_tb). Measurements of these physical quantities can therefore be interpreted in terms of the angles γ and β_(s), and γ can be derived using independent determinations of the other parameter as input.

The LHCb collaboration recently presented a new measurement of B⁰_s → D^∓_s K^± decays collected during Run 2. This is a challenging analysis, as it requires a decay time-dependent fit to extract the CP-violating observables expressed as amplitudes of the four different decay paths that arise from B⁰_s and – B⁰_s to D^∓_s K^± final states. Previously, LHCb measured γ in this decay using the Run 1 dataset, obtaining γ = 128 ⁺¹⁷_–22°. The B⁰_s – B⁰_s oscillation frequency ∆m_s must be precisely constrained in order to determine the phase differences between the amplitudes. In the Run 2 measurement, the established uncertainty on ∆m_s would have been a limiting systematic uncertainty, which motivated the recent LHCb measurement of ∆m_s using the flavour-specific B⁰_s → D^–_s π⁺ decays from the same dataset. Combined with Run 1 measurements of ∆m_s, this has led to the most precise contribution to the world average and has greatly improved the precision on γ in the B⁰_s → D^∓_s K^± analysis. Indeed, for the first time the four amplitudes are resolved with sufficient precision to show the decay rates separately (see figure 1).

The angle γ is determined using inputs from other LHCb measurements of the CP-violating weak phase –2β_s, along with measurements of the decay width and decay-width difference. The final result, γ = 74 ± 11°, is compatible with the SM and is the most precise determination of γ using B⁰_s meson decays to date.