Weak phases from topological-amplitude parametrization

We propose a parametrization for two-body nonleptonic B meson decays, in which the various topologies of amplitudes are counted in terms of powers of the Wolfenstein parameter λ ∼ 0.22. The weak phases and the amplitudes are determined by comparing this parametrization with available measurements. It is possible to obtain the phase φ3 from the B → Kπ data up to theoretical uncertainty of O(λ ) ∼ 5%. The recently measured B d → π π branching ratio implies a large color-suppressed or penguin amplitude, and that the extraction of the phase φ2 from the B → ππ data may suffer theoretical uncertainty more than the expected one, O(λ) ∼ 5%. Email: [email protected] Email: [email protected] One of the major missions in B physics is to determine the weak phases in the Kobayashi-Maskawa ansatz for CP violation [1] . The phase φ1 can be extracted from the CP asymmetry in the B → J/ψKS decays in an almost model-independent way, which arises from the B-B̄ mixing. The application of the isospin symmetry to the B → ππ decays [2] and to the B → ρπ decays [3] has been considered as giving a model-independent determination of the phase φ2. However, this strategy in fact suffers the theoretical uncertainty from the electroweak penguin, which is expected to be about 5-10% . The phase φ3 can be extracted in a theoretically clean way from the modes involving only tree amplitudes, such as B → πD [4] and B → KD [5, 6]. The difficulty is that one of the modes, such as B d → πD or B → KD, has a very small branching ratio and is not experimentally feasible [7]. The alternative modes B → KD [8] and Bc → DsD [9] improve the feasibility only a bit. It has been pointed out that the B → K(D → f) and B → K(D̄ → f) amplitudes, with D̄ → f being a doubly-Cabibbo suppressed decay, exhibits a strong interference [7, 10, 11]. For this strategy, the strong phase difference between D → f and D̄ → f is a necessary input. Another possibility is to measure the B → DV decays for the vector meson V = ρ, K, · · ·, since an angular analysis involves many observables, which are sufficient for extracting φ3 model-independently [12]. Instead of resorting to theoretically clean modes, which are usually experimentally difficult, one considers the modes with higher feasibility and tries to constrain the decay amplitudes and the weak phases. The problem is that available measurements are usually insufficient to make the constraint, and theoretical inputs are unavoidable. For example, one adopts the (imaginary) tree-over-penguin ratio obtained from the perturbative QCD (PQCD) formalism [13, 14, 15, 16, 17] or from the QCD-improved factorization (QCDF) [18], so that the phase φ2 can be extracted from the CP asymmetries of the B 0 d → ππ decays. One may also employ symmetries to relate the amplitudes of the relevant modes, such as SU(3) [19] and U -spin [20], in order to reduce the number of free parameters. However, the theoretical calculations are subject to subleading corrections, and the symmetry relations are broken with unknown symmetry breaking effects. For these strategies to work, the theoretical uncertainty must be under control. In this paper we shall propose counting rules for the various topologies of amplitudes [21] in two-body nonleptonic B meson decays in terms of powers of the Wolfenstein parameter λ ∼ 0.22 [22]. The relative importance among the topological amplitudes has been known from some physical principles: helicity suppression (color transparency) implies that tree annihilation (nonfactorizable) contributions are smaller than leading factorizable emission contributions. Here we shall assign an explicit power of λ to each topology, such that the relative importance becomes quantitative. This assignment is supported by the known QCD theories [16, 18, 23, 24], and differs from that assumed in [22]. We drop the topologies with higher powers of λ until the number of free parameters are equal to the number of available measurements. The weak phases and the decay amplitudes can then be solved by comparing the resultant parametrization with experimental data. Afterwards, it should be examined whether the solved amplitudes obey the power counting rules. If they do, the extracted weak phases suffer only the theoretical uncertainty from the neglected topologies. If not, the inconsistency could be regarded as a warning to QCD theories for two-body nonleptonic B meson decays. For example, the long-distance rescattering effect has been neglected in PQCD and in QCDF. If this effect is important, the hierarchy among the various topological amplitudes will be destroyed [25]. The comparison of our parametrization with data can tell whether the above assumption is reliable [26]. As shown below, dropping the electroweak penguin amplitude, the phase φ2 can be extracted from the B → ππ data. In principle, the theoretical uncertainly of the ignored amplitudes is around O(λ) ∼ 5%, the same as in the extraction based on the isospin symmetry [2]. Similarly, the phase φ3 can be best determined from the B → Kπ data up to the uncertainty from the neglect of the O(λ) ∼ 1% tree annihilation and color-suppressed electroweak amplitudes. Note that the determination of the phase φ1 from the B → J/ψK decays also bears about 1% theoretical uncertainty. Certainly, a CP asymmetry is an O(λ) quantity itself. Precisely speaking, the above determination of φ2 and φ3, involving the data of CP asymmetries, in fact carries the uncertainly of O(λ) ∼ 20% and O(λ) ∼ 5%, respectively. Because the B → ππ, Kπ measurements are not yet complete, we shall drop more topologies in order to match the currently available data. In this simple demonstration, we observe that the amplitudes solved from the B → Kπ data more or less obey the hierarchy in λ. That is, an almost model-independent determination of φ3 is promising. The solution from the B → ππ analysis is, unfortunately, not consistent with the power counting rules, indicating that the extraction of φ2 may suffer theoretical uncertainty larger than stated above. Hence, our work casts a doubt to the strategy based on the isospin symmetry [2] and gives a warning to the QCD calculations of the B → ππ modes [17, 18, 27].