Isospin analysis of charmless B-meson decays

Abstract

We discuss the determination of the CKM angle α using the non-leptonic two-body decays B → ππ, B → ρρ and B → ρπ using the latest data available. We illustrate the methods used in each case and extract the corresponding value of α. Combining all these elements, we obtain the determination α dir = (86.2• . We assess the uncertainties associated to the breakdown of the isospin hypothesis and the choice of the statistical framework in detail. We also determine the hadronic amplitudes (tree and penguin) describing the QCD dynamics involved in these decays, briefly comparing our results with theoretical expectations. For each observable of interest in the B → ππ, B → ρρ and B → ρπ systems, we perform an indirect determination based on the constraints from all the other observables available and we discuss the compatibility between indirect and direct determinations. Finally, we review the impact of future improved measurements on the determination of α.

Resolving the ϕ2 (α) ambiguity in B → ρρ

“…In this paper, I employ the frequentist approach adopted by the CKMfitter Group [22] where a χ 2 is constructed comparing theoretical forms for relations between parameters of interest and the physical observables with the experimentally measured values of those observables. The value of ∆χ 2 across the range of φ 2 can then be converted into a p-value scan, assuming it is distributed with one degree of freedom, from which confidence intervals can be derived.…”

“…Decays that proceed predominantly through theb →ūud tree transition (figure 1a) in the presence of B 0 -B 0 mixing are sensitive to the interior angle of the Unitarity Triangle φ 2 = α ≡ arg(−V td V * tb )/(V ud V * ub ), which can be accessed through mixing-induced CP violation observables measured from time-dependent, flavour-tagged analyses. This quark process manifests itself in multiple systems, including B → ππ [3][4][5][6][7][8], ρπ [9,10], ρρ [11][12][13][14][15][16][17] and a ± 1 π ∓ [18][19][20], where the angle φ 2 has so far been constrained with an overall uncertainty of around 4 • [21][22][23]. However, one of the salient features of the overall φ 2 combination is the persistence of degenerate solutions within the range [0, π]: up to the 2σ level, two solutions currently remain, while beyond this further solutions emerge.…”

Measurement of CP asymmetries in two-body B(s)0 -meson decays to charged p

“…More recently, it has been proposed to combine the CP asymmetries of the B 0 → π þ π − and B 0 s → K þ K − decays with information provided by the semileptonic decays B 0 → π − l þ ν and B 0 s → K − l þ ν, in order to achieve a substantial reduction of the theoretical uncertainty on the determination of −2β s [12,13]. The CP asymmetry in the B 0 → π þ π − decay is also a relevant input to the determination of the CKM angle α, when combined with other measurements from the isospin-related decays B 0 → π 0 π 0 and B þ → π þ π 0 [14][15][16]. In this paper, measurements of the time-dependent CP asymmetries in B 0 → π þ π − and B 0 s → K þ K − decays and of the time-integrated CP asymmetries in…”

The CKM Parameters

“…The present average of these constraints yields α meas = (88 Figure 3 depicts the different constraints and shows the discrete symmetries present in the ππ and ρρ cases, as well as the fact that two solutions are allowed by the combination of the measurements. In addition to the statistical uncertainties of the measurements, the accuracy is limited by two main hypotheses: ∆I = 3/2 contributions coming from electroweak penguins are neglected, and isospin symmetry in strong interactions is not perfect [129][130][131] followed by carefully chosen D decay processes. This is the least precisely known angle of the UT, and its determination from tree decays is considered to be free from contributions beyond the SM and unaffected by hadronic uncertainties, contrary to the angles α and β [132].…”

“…Constraints on the CKM angle α from B → ππ, B → ρπ, and B → ρρ. The combination of the constraints and the outcome of the global fit are also represented[129].…”