Newly observed two-body decays of B mesons in a hybrid perspective

In consistency with B̄ → D(∗)π, J/ψK̄ and J/ψπ decays, recently observed B0 → D+ s π− and B̄0 → D+ s K− are studied in a hybrid perspective in which their amplitude is given by a sum of factorizable and non-factorizable ones. (Quasi) two-body decays of B mesons have been studied extensively by using the factorization [1,2]. However, recently measured rates [3] for the color mismatched spectator (CMS) decays, B̄ d → D(∗)0π0, are much larger than the expectation by the factorization. It suggests that non-factorizable contributions can play an important role in these decays. In addition, very recently, B̄ → D s K− and B → D s π− have been observed [4]. The rate for the former is again much larger than the expectation by the factorization, i.e., it has been expected to be strongly suppressed (the helicity suppression) since it is described by an annihilation diagram in the weak boson mass mW → ∞ limit. It means that the non-factorizable contribution is dominant in this decay. The latter is a pure spectator decay, b̄→ ū + (cs̄), but does not satisfy the kinematical condition of color transparency [5], so that it is not very clear if the factorization works well in this decay. Therefore, it is meaningful to study a possible role of non-factorizable contributions in the newly observed B̄ d → D(∗)0π0, B̄ → D s K− and B → D s π− in consistency with the b→ c type of decays, B̄ → D(∗)π, J/ψK̄ and J/ψπ. We, first, review briefly our (hybrid) perspective (see Ref. [6] for more details). Our starting point is to assume that the amplitude can be decomposed into a sum of factorizable and non-factorizable ones, (MFA and MNF, respectively). MFA is estimated by using the factorization while MNF is assumed to be dominated by dynamical contributions of various hadron states and calculated by using a hard pion (or kaon) approximation in the infinite momentum frame (IMF) [7,8]. In this approximation, MNF is given by a sum of the surface term (MS) which is given by a sum of all possible pole amplitudes and the equal-time commutator term (METC) which arises from the contribution of non-resonant (multi-hadron) intermediate states [9]. Corresponding to the above decomposition of the amplitude, the effective weak Hamiltonian, Hw ' (GF/ √ 2){c1O1 + c2O2} + h.c., (where c1 and c2 are the Wilson coefficients), is decomposed into a sum of the BSW Hamiltonian [1], H (BSW) w , and an extra term, H̃w, i.e., Hw → H w + H̃w, by using the Fierz reshuffling, where H w is given by a sum of products of colorless currents and might provide the factorizable amplitude. However, the “external” hadron states which sandwich H (BSW) w might interact sometimes with each other through hadron dynamics (like a re-scattering, etc.). In this case, corresponding part of the amplitude is non-factorizable and should be included in MNF, so that the values of the coefficients, a1 and a2, in MFA arising from H (BSW) w might not be the same as the original a (BSW) 1 = c1 + c2/Nc and a (BSW) 2 = c2 + c1/Nc in H (BSW) w , where Nc is the color degree of freedom. Therefore, we will treat a1 and a2 as adjustable parameters later. The extra term H̃w which is given by a color singlet sum of colored current products provides non-factorizable amplitudes in the present perspective, although, in Ref. [2], contributions from H̃w have been included in the factorized amplitudes by considering the effective colors. Explicit expression of factorized and non-factorizable amplitudes for the B̄ → D(∗)π, J/ψK̄ and J/ψπ decays have already been given in Ref. [6] in which METC and MS with contributions of low lying meson poles are taken into account. In the same way, we can calculate the amplitude for the b̄ → ū + (cs̄) decays, B → D(∗)+ s π−. These amplitudes, however, include many parameters, i.e., form factors, decay constants of heavy mesons, asymptotic matrix elements of H̃w (matrix elements of H̃w taken between single hadron states with infinite momentum), phases, δ̃I , of M (I) ETC(B̄ → Dπ), (I = 12 and 32), relative to MS and the relative phase (∆) between MFA and MNF which has not been considered in our previous studies. The other parameters involved are known or can be estimated by using related experimental data and asymptotic flavor symmetries [10]. To obtain improved values of the above amplitudes, we update values of parameters involved. Asymptotic matrix elements of axial charges are estimated as follows, i.e., |〈ρ|Aπ+ |π−〉| ' 1.0 from Γ(ρ→ ππ)exp ' 150 MeV [11]. Here we take 〈ρ|Aπ+ |π−〉 = 1.0 and the other ones can be related to it by using related asymptotic flavor symmetries, for example, √ 2〈D∗+|Aπ+ |D0〉 = −〈ρ|Aπ+ |π−〉, etc., as in our previous study [6]. As the values of the CKM matrix elements [12] and the decay constants, we take Vcs ' Vud ' 0.98, Vcd ' −0.22, Vcb ' 0.040, |Vub/Vcb| ' 0.090 and fπ ' 130.7 MeV, fK ' 160 MeV from Ref. [11]. The decay constant, fJ/ψ ' 406 MeV, can be obtained from Γ(J/ψ → e+e−)exp = 5.26± 0.37 keV [11]. The updated values of the decay constants of heavy mesons, fD ' 0.226 GeV, fDs ' 0.250 GeV and fB ' 0.198 GeV, are taken from the lattice QCD [13], and fD∗ ' fD and fD∗ s ' fDs are assumed as expected by the heavy quark effective theory (HQET) [14]. The form factors, F (DB̄) 0 (m 2 π) and A (D∗B̄) 0 (m 2 π), are estimated by using the HQET and the data on the semileptonic decays of B mesons [11] as F (DB̄) 0 (m 2 π) ' 0.74 and A ∗B̄) 0 (m 2 π) ' 0.65. The form factors, F (πB̄) 0 (q ) and F (πB̄) 1 (q ), are estimated by using extrapolation formulas based on the lattice QCD [15]. We here take F (πB) 0 (m 2 D) ' 0.28, F (πB) 0 (m2Ds) ' 0.32, F (πB) 1 (mD∗) ' 0.34, F (πB) 1 (m 2 ψ) ' 0.50 and F (KB) 1 (mψ) ' 0.59. The annihilation amplitudes which contain F (Dπ) 0 (m 2 B) and A (D∗π) 0 (m 2 B) will be small and neglected because of the helicity suppression. The asymptotic matrix element of H̃w is parameterized by 〈D0|H̃(ud;cb) w |B̄0 d〉 VcbVudfπ = BH × 10−5 (GeV), (1) where H̃ w is a component of H̃w which is given by a sum of Õ (ud;cb) 1 = VudVcb{2 ∑a(d̄tau)L(c̄tab)L} and Õ 2 = VudVcb{2 ∑a(c̄tau)L(d̄tab)L} with the color SUc(3) generator t. To evaluate the B̄ → D∗π amplitudes, we assume