Study of nonleptonic $B_{q}^{\ast}$ ${\to}$ $D_{q}V$ and $P_{q} D^*$ weak decays

Motivated by the powerful capability of measurement for the $b$-flavored hadron rare decays at LHC and SuperKEKB/Belle-II, the nonleptonic $\bar{B}^{\ast}$ ${\to}$ $D\bar{D}^{\ast}$, $D{\rho^-}$, $DK^{\ast-}$, ${\pi}D^{\ast}$ and $KD^{\ast}$ weak decays are studied in detail. With the amplitudes calculated with factorization approach and the form factors of $B^{\ast}$ transition into pseudoscalar meson evaluated with the BSW model, branching fractions and polarization fractions are firstly presented. Numerically, the CKM-favored $\bar{B}_{q}^{\ast}$ ${\to}$ $D_{q}D_{s}^{{\ast}-}$ and $D_{q}{\rho}^{-}$ decays have large branching fractions, $\sim$ $10^{-8}$, which should be sought for with priority and firstly observed by LHC and Belle-II experiments. The $\bar{B}^{\ast}_q$ ${\to}$ $D_qK^{\ast}$ and $D_q{\rho}$ decays are dominated by the longitudinal polarization states. While, the parallel polarization fractions of $\bar{B}^{\ast}_q$ ${\to}$ $D_q\bar{D}^{\ast}$ decays are comparable with the longitudinal ones, numerically, $f_{\parallel}$ $+$ $f_{L}$ ${\simeq}$ 95\% and $f_{L}:f_{\parallel}$ $\simeq$ $5:4$. Some comparisons between $\bar{B}^{*0}_q$ $\to$ $D_q V$ and their corresponding $\bar{B}^{0}_q$ $\to$ $D^*_q V$ decays are performed, and the relation $ f_{L,\parallel}(\bar{B}^{\ast 0}\to D V)\simeq f_{L,\parallel}(\bar{B}^0\to D^{\ast +} V^-) $ is presented. Besides, with the implication of $SU(3)$ flavor symmetry, some useful ratios $ R_{\rm du}$ and $ R_{\rm ds}$ are discussed in detail, and suggested to be verified experimentally.

in the past years, many B u,d,s meson decays have been well measured. Thanks to the ongoing LHCb experiment [1] at LHC and forthcoming Belle-II experiment [2] at SuperKEKB, experimental analysis of B meson decays is entering a new frontier of precision. By then, besides B u,d,s mesons, the rare decays of some other b-flavored hadrons are hopefully to be observed, which may provide much more extensive space for b physics.
The excited states B * u,d,s with quantum number of n 2s+1 L J = 1 3 S 1 and J P = 1 − ( n, L, s, J and P are the quantum numbers of radial, orbital, spin, total angular momenta and parity, resptctively), which will be referred as B * in this paper, had been observed by CLEO, Belle, LHCb and so on [3]. However, except for their masses, there is no more experimental information due to the fact that the production of B * mesons are mainly through Υ(5S) decays at e + e − colliders and the integrated luminosity is not high enough for probing the B * rare decays. Moreover, B * decays are dominated by the radiative processes B * → Bγ, and the other decay modes are too rare to be measured easily. Fortunately, with annual integrated luminosity ∼ 13 ab −1 [2] and the cross section of Υ(5S) production in e + e − collisions σ(e + e − →Υ(5S)) = (0.301±0.002±0.039) nb [4], it is expected that about 4×10 9 Υ(5S) samples could be produced per year at the forthcoming super-B factory SuperKEKB/Belle-II, which implies that the B * rare decays with branching fractions > ∼ 10 −9 are possible to be observed. Besides, due to the much larger production cross section of pp collisions, experiments at LHC [5,6] also possibly provide some experimental information for B * decays.
With the rapid development of experiment, accordingly, the theoretical evaluations for where p = d or s, V qb V * q ′ p is the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements; C i are Wilson coefficients, which describe the short-distance contributions and are calculated perturbatively; The explicit expressions of local four-quark operators O i are where (q 1 q 2 ) V ±A =q 1 γ µ (1±γ 5 )q 2 , α and β are color indices, Q p ′ is the electric charge of the quark p ′ in the unit of |e|, and p ′ denotes the active quark at the scale µ ∼ O(m b ), i.e., p ′ = u, d, c, s, b.
To obtain the decay amplitudes, the remaining and also the most intricate work is how to calculate hadronic matrix elements P V |O i |B * . With the factorization approach [8][9][10][11] based on the color transparency mechanism [12,13], in principle, the hadronic matrix element could be factorized as Due to the unnecessary complexity of hadronic matrix element V |J µ |B * and power suppression of annihilation contributions, we only consider one simple scenario where pseudoscalar meson pick up the spectator quark in B * meson, i.e., a = 1, b = 0 and c = 0 in Eq.(7) for the moment. Two current matrix elements can be further parameterized by decay constants and transition form factors, where ǫ and η are the polarization vector, f V is the decay constant of vector meson, V and A 0,1,2 are transition form factors, q = p B * − p P and the sign convention ǫ 0123 = 1. Even though some improved approaches, such as the QCD factorization [14,15], the perturbative QCD scheme [16,17] and the soft-collinear effective theory [18][19][20][21], are presented to evaluate higher order QCD corrections and reduce the renormalization scale dependence, the naive factorization (NF) approximation is a useful tool of theoretical estimation. Because there is no available experimental measurement for now, the NF approach is good enough to give a preliminary analysis, and so adopted in our evaluation.
With the above definitions, the hadronic matrix elements considered here can be decomposed into three scalar invariant amplitudes S 1,2,3 , where the amplitudes S 1,2,3 describe the s, d, p wave contributions, respectively, and are explicitly written as Alternatively, one can choose the helicity amplitudes H λ (λ = 0, +, −), Now, with the formulae given above and the effective coefficients α i defined as we present the amplitudes of nonleptonic two-bodyB * decays as follows: • ForB * q → D qD * decays (the spectator q = u, d and s), • ForB * 0 q → D q V decays (the spectator q = d and s, the V = ρ − and K * − ), • ForB * → πD * decays, • ForB * → KD * decays, In the rest frame ofB * meson, the branching fraction can be written as where the momentum of final states is The longitudinal, parallel and perpendicular polarization fractions are defined as where A and A ⊥ are parallel and perpendicular amplitudes gotten through forB * decays.

III. NUMERICAL RESULTS AND DISCUSSION
Firstly, we would like to clarify the input parameters used in our numerical evaluations.
The decay constants of light vector mesons are [24] For the decay constants of D * (s) mesons, we will take [25] which agree well with the results of the other QCD sum rules [26,27] and lattice QCD with N f = 2 [28].  Besides the decay constants, the B * → P transition form factors are also essential inputs to estimate branching ratios for nonleptonic B * → P V decay. In this paper, the Bauer-Stech-Wirbel (BSW) model [10] is employed to evaluate the form factors A 1 (0), A 2 (0) and V (0), which could be written as the overlap integrals of wave functions of mesons [10], where p ⊥ is the transverse quark momentum, σ y,z are the Pauli matrix acting on the spin indices of the decaying quark, and m q represents the mass of nonspectator quark of pseudoscalar meson. With the meson wave function ϕ M ( p ⊥ , x) as solution of a relativistic scalar harmonic oscillator potential [10], and ω = 0.4 GeV which determines the average transverse quark momentum through p 2 ⊥ = ω 2 , we get the numerical results of the transition form factors summarized in Table I. In our following evaluation, these numbers and 15% of them are used as default inputs and uncertainties, respectively.
To evaluate the branching fractions, the total decay widths (or lifetimes) Γ tot (B * ) are necessary. However, there is no available experimental or theoretical information for Γ tot (B * ) until now. Because of the fact that the QED radiative processes B * → Bγ dominate the decays of B * mesons, we will take the approximation Γ tot (B * ) ≃ Γ(B * →Bγ). The theoretical predictions on Γ(B * →Bγ) have been widely evaluated in various theoretical models, such as relativistic quark model [29,30], QCD sum rules [31], light cone QCD sum rules [32], light front quark model [33], heavy quark effective theory with vector meson dominance hypothesis [34] or covariant model [35]. In this paper, the most recent results [33,35] Tables II and III. In Table II, the first, second and third theoretical errors are caused by uncertainties of the CKM parameters, hadronic parameters (decay constants and form factors) and total decay widths, respectively. From Tables II and III, it could be found that: (1) The hierarchy of branching fractions is clear. (i) The branching fractions ofB * → πD * and KD * decays are much smaller than the ones ofB * → DD * , Dρ and DK * decays, which is caused by that the form factors ofB * → D transition are much larger than those ofB * → π andB * → K transitions. (ii) ForB * → DD * , Dρ and DK * decays,  (2) Besides small form factors, theB * → πD * , KD * decays are either color suppressed or Decay modes the CKM factors suppressed, hence have very small branching fractions (see TableII) to be hardly measured soon. Most of the CKM favored and tree-dominatedB * → DD * , Dρ, DK * decays, enhanced by the relatively largeB * → D transition form factors, have large branching fractions, > ∼ 10 −9 , and thus could be measured in the near future. In particular, branching ratios forB * q → D qD * − s , D q ρ decays can reach up to 10 −8 , and hence should be sought for with priority and firstly observed at the high statistics LHC and Belle-II experiments.
The numerical results and above analyses are based on the NF, in which the QCD corrections are not included. Fortunately, for the color-allowed tree amplitude α 1 , the NF estimate is stable due to the relatively small QCD corrections [15]. For instance, in B → ππ and B → D * L decays, the results α 1 (ππ) = (1.020) LO + (0.018 + 0.018i) N LO [14] and α 1 (D * L) = (1.025) LO + (0.019 + 0.013i) N LO [15] indicate clearly that the O(α s ) correction is only about 2% and thus trivial numerically. For the colorsuppressed decay modes listed in Tables II, even though the NF estimates would suffer significant O(α s ) correction (about 46% in B → ππ decays for instance [36] ), they still escape the experimental scope due to their small branching factions < 10 −9 , and thus will not be discussed further. In the following analyses, we will pay our attention only to the color allowed tree-dominatedB * → DD * , Dρ, DK * decays.
(4) Besides of branching fraction, the polarization fractions f L, ,⊥ are also important observables. For the potentially detectable decay modes with branching fractions > ∼ 10 −9 , our numerical results of f L, are summarized in Table III. For the helicity amplitudes A λ , the formal hierarchy pattern is naively expected. Hence,B * → P V decays are generally dominated by the longitudinal polarization state and satisfy f L ∼ 1 − 1/m 2 B * [37]. ForB * → DV (V = K * , ρ) decays, in the heavy-quark limit, the helicity amplitudes H λ given by Eqs. (15) and (16) could be simplified as The transversity amplitudes could be gotten easily through Eq. (40). Obviously, due to the helicity suppression factor 2m  Table III for numerical results).
It should be noted that above analyses and Eqs. (62) and (63) are based on the case of m 2 V ≪ m 2 B * , and thus possibly no longer satisfied byB * → DD * decays because of the un-negligible vector mass m D * . In fact, for theB * → DD * decays, Eqs. (15) and (16) are simplified as Because the so-called helicity suppression factor 2m D * /m B * ∼ 0.8 is not small, which is different from the case ofB * → DV decays, it could be easily found that the relation of Eq.(61) doesn't follow. Further considering that H ± P V are dominated by the term of A 1 in Eq. (65) due to its large coefficient, the relation f L (DD * ) ∼ f (DD * ) ≫ f ⊥ (DD * ) could be easily gotten. Above analyses and findings are confirmed by our numerical results in Table III, which will be tested by future experiments.
(5) As known, there are many interesting phenomena in B meson decays, so it is worthy to explore the possible correlation between B and B * decays. TakingB * 0 → D + ρ − andB 0 → D * + ρ − decays as example, we find that the expressions of their helicity amplitudes (the former one have be given by Eqs. (62) and (63)