Three-body B⁺ decays violate CP

New sources of CP violation (CPV) are needed to explain the absence of antimatter in our matter-dominated universe. The LHCb collaboration has reported new results describing CPV in B⁺ → π⁺K⁺K^– and B⁺ → π⁺π⁺π^– decays. Until very recently, all observations of CPV in B mesons were made in two-body and quasi-two-body decays; however, it has long been conjectured that the complex dynamics of multi-body decays could give rise to other manifestations. For CPV to occur in B decays, competing decay amplitudes with different weak phases (which change sign under CP) and strong phases (which do not) are required. The weak phase differences are tied to fundamental parameters of the Standard Model (SM), but the strong phase difference can arise from loop-diagram contributions, final-state re-scattering effects, and phases associated with intermediate resonant structure.

The three-body B decays under study proceed mainly via various intermediate resonances – effectively, a cascade of two-body decays – but also include contributions from non-resonant three-body interactions. The phase space is two-dimensional (it can be fully described by two kinematic variables) and its size allows a rich tapestry of resonant structures to emerge, bringing quantum-mechanical interference into play. Much as in Young’s double-slit experiment, the total amplitude comprises the sum of all possible decay paths. The interference pattern and its phase variation could contribute to CPV in regions where resonances overlap.

One of the most intriguing LHCb results was the 2014 observation of large CPV effects in certain phase-space regions of B⁺ → π⁺K⁺K^– and B⁺ → π⁺π⁺π^– decays. In the new analysis, these effects are described with explicit amplitude models for the first time (figure 1). A crucial step in the phenomenological description of these amplitudes is to include unitarity-conserving couplings between final states, most notably ππ and KK. Accounting for these is essential to accurately model the complex S-wave component of the decays, which is the configuration where there is no relative angular momentum between a pair of oppositely-charged final-state particles, and which contains broad resonances that are difficult to model. Three complementary approaches were deployed to describe the complicated spin-0 S-wave component of the B⁺ → π⁺π⁺π^– decay: the classical isobar model, which explicitly associates a line-shape with a clear physical interpretation to each contribution in the phase space; the K-matrix method, which takes data from scattering experiments as an input; and finally a quasi-model-independent approach, in which the S-wave magnitude and phase are extracted directly from the data.

LHCb’s amplitude analyses of these decays are based on data from Run 1 of the LHC and contain several groundbreaking results, including the largest CP asymmetry in a single component of an amplitude analysis, found in the ππ ↔ KK re-scattering amplitude; the first observation of CPV in the interference between intermediate states, seen in the overlap between the dominant spin-1 ρ(770)⁰ resonance and the π⁺π⁺ S-wave; and the first observation of CPV involving a spin-2 resonance of any kind, found in the decay B⁺ → f₂(1270)π⁺. These results provide significant new insights into how CPV in the SM manifests in practice, and motivate further study, particularly into the strong-phase-generating QCD processes that govern CP violation.