Resonant state $D_0^\ast(2400)$ in the quasi-two-body $B$ meson decays

We study four quasi-two-body decay processes including $D^\ast_0(2400)$ as the intermediate state in perturbative QCD (PQCD) approach. The branching fraction predicted in this work for the decay mode $B^-\to D^{*}_0(2400)^0\pi^-\to D^+\pi^-\pi^-$ agree with the data from Belle, BaBar and LHCb Collaborations. The PQCD prediction of the branching ratio for the decay $\bar B^0\to D^{*+}_0K^-\to D^0\pi^+K^-$ is consistent with the value given by LHCb. For the decays $\bar B^0\to D^{*+}_0\pi^-\to D^0\pi^+\pi^-$ and $B^-\to D^{*0}_0K^-\to D^+\pi^-K^-$, the PQCD predicted branch ratios are $2.85^{+1.23}_{-0.80}(\omega_B)^{+1.05}_{-0.81}(\omega_{D\pi})^{+0.33}_{-0.31}(a_{D\pi})^{+0.06}_{-0.05}(\Gamma_{D^{*+}_0}) \times 10^{-4}$ and $4.65^{+1.89}_{-1.30}(\omega_B)^{+1.51}_{-1.24}(\omega_{D\pi})^{+0.40}_{-0.38}(a_{D\pi})^{+0.22}_{-0.18}(\Gamma_{D^{*0}_0}) \times 10^{-5}$, respectively. We analyze the experimental branching fractions using the ratios $R_{D^{*0}_0}$ and $R_{D^{*+}_0}$ which are related to the decays with the neutral and charged $D^\ast_0(2400)$, respectively. The available experimental results for the quasi-two-body decays including $D^\ast_0(2400)$ are not in agreement with the isospin relation and $SU(3)$ flavor symmetry.

We study four quasi-two-body decay processes including D * 0 (2400) as the intermediate state in the perturbative QCD (PQCD) approach.
[1] for a review. One of them, the p-wave orbitally excited state D * 0 (2400), with the light degree of freedom j p q = 1 2 + [2-4] and quantum number J P = 0 + [5], was first discovered by Belle Collaboration in the three-body decays B − → D + π − π − , with the mass m D * 0 0 = 2308 ± 17 ± 15 ± 28 MeV and width Γ D * 0 0 = 276 ± 21 ± 18 ± 60 MeV [6]. For simplicity, we adopt D * 0 to denote the D * 0 (2400) state and the inclusion of charge-conjugate processes is implied throughout this work. The neutral resonant state D * 0 0 has been confirmed by BaBar Collaboration in the same decay processes in [7], with the close but preciser values for its mass and width. While in the wideband photoproduction experiment, differ from those three-body decay processes, FOCUS Collaboration provided quite different values for the broad structure D * 0 0 in [8]. One has m D * 0 0 = 2407 ± 21 ± 35 MeV and Γ D * 0 0 = 240 ± 55 ± 59 MeV in company with m D * + 0 = 2403 ± 14 ± 35 MeV and Γ D * + 0 = 283 ± 24 ± 34 MeV for a charged state D * + 0 from Ref. [8]. Unlike the charmed-strange state D * s0 (2317) [9][10][11], which lies just below DK threshold and mainly decays into the isospin breaking channel D s π, the state D * 0 is expected to decay rapidly through the s-wave pion emission, the conservation of its angular momentum implies this resonance primarily couple to Dπ and has a broad decay width [12] as revealed by experiments. While the discrepancy of its properties between the experimental results [6,7] and the predictions from the quark model [13,14] has triggered many studies on its true nature. The strong decays, radiative decays and/or the spectra have been studied extensively in Refs. [3,4,[15][16][17][18][19][20][21] to explore the exact feature of the resonant state D * 0 . In Refs. [22,23], the possible four-quark structure of D * 0 was investigated, the authors pointed out that the four-quark structure is acceptable for the resonant state observed by Belle [6] and BaBar [7], but not for the cases observed by FOCUS [8]. While in Refs. [24][25][26], it was claimed that there exist two poles in D * 0 energy region. And a pole near the Dπ threshold was obtained from lattice QCD in [27], which was said to share the similarities with the experimental resonance D * 0 . The resonant state D * 0 has also been explained as a mixture of two-and four-quark state [28] or the bound state of Dπ [29].
Semileptonic or hadronic B meson decays including a resonant state D * 0 shall yield clues to its properties. Employing constituent quark model or the light-cone sum rules to evaluate the B → D * 0 transition form factors, the decays of B → D * 0 lν have been studied in Refs. [30,31]. With the help of a chiral unitarity model, the ratio between the decay widths ofB 0 s → D * + s0 (2317)lν l andB 0 → D * + 0 lν l was calculated in [32]. The model independent studies of B → D * * lν have been performed within the standard model in [33] and beyond the standard model in [34] based on heavy quark symmetry. In [35], the branching fraction for the semileptonic decay B → D * 0 lν was predicted assuming the conventional quark-antiquark configuration for D * 0 state. And the hadronic matrix elements were evaluated in the Bethe-Salpeter approach for B c → D * 0 µμ decays in [36]. Since Belle's announcement [6], many works have been done on two-body hadronic B meson decays including the state D * 0 . For example, within the covariant light-front approach, the branching ratio of B − → D * 0 0 π − was predicted to be 7.3 × 10 −4 in [37]. In [38], the information on the Isgur-Wise functions at zero recoil was extracted from theB 0 → D * + 0 π − decay process. Using the improved version of the Isgur-Scora-Grinstein-Wise quark model for the B → D * 0 transition form factors, the decays B − → D * 0 0 π − and B 0 → D * − 0 π + have their branching ratios as 7.7 × 10 −4 and 2.6 × 10 −4 , respectively, in Ref. [39]. Two-body decays B → D * 0 π have also been discussed within factorization framework [40] and the perturbative QCD (PQCD) approach [41].
a Total S-wave D + π − contribution b Isobar model c K-matrix model In Table I, we have the data for the four quasi-two-body hadronic B meson decay modes including D * 0 from Belle, BaBar and LHCb Collaborations. In the processes B → D * 0 h → Dπh, where h is a charged pion or kaon, the weak interaction point accompany the birth of the bachelor particle h, the intermediate state D * 0(+) 0 generated from the hadronization of c-quark plusū(d)-quark as demonstrated in Fig. 1. We stress that the resonance D * 0 is not necessary to be conventional quart-antiquark structure. In this letter, we shall analyze those four decay processes in a quasi-two-body framework based on the PQCD factorization approach [47][48][49][50]. In Refs. [51][52][53][54][55], the PQCD approach has been employed in the studies of the three-body B meson decays. With the help of the two-pion distribution amplitudes [56][57][58] and the experimental inputs for the time-like pion form factors, in Ref. [59], we calculated the decays B → Kρ(770), Kρ ′ (1450) → Kππ in the quasi-two-body framework. The method used in [59] have been adopted for some other quasi-two-body B decays in Refs. [60][61][62]. In this work, we extend the previous studies to thē Refer to the Kπ system in Refs. [63,64], we define the scalar form factor F Dπ 0 (s) for the final state with the constant where the m D (m π ) is the mass of D(π) meson, the m c = 1.275 GeV and m u = 2.2 MeV (which could be neglected safely) for the mass of c and u quarks are adopted from [5]. Then we have and Π BW One has different values from 78 MeV [65] to 148 +40 −46 MeV [40] in different works for this decay constant, see [37,40,41,[65][66][67][68], we support the moderate one f D * 0 = 0.13 GeV which was adopted in the PQCD approach in [41]. The denominator D BW = m 2 0 − s − im 0 Γ(s), the mass-dependent decay width Γ(s) has its definition as Γ(s) = Γ 0 q q0 m0 √ s , m 0 and Γ 0 are the pole mass and width of the resonant state D * 0 and s is the invariant mass square for the Dπ pair in the final state. In the rest frame of the resonance D * 0 , its daughter D + or π − has the magnitude of the momentum as and q 0 is the value of q at s = m 2 0 . The coupling constant g D * 0 Dπ has its value from the relation [32,39] We define then we have F Dπ 0 (s) = C Dπ · F Dπ (s), with the parameter In the rest frame of the B meson, with m B being its mass, we define the momentum p = mB √ 2 (1, η, 0) in the light-cone coordinates for the resonant state D * 0 and the Dπ pair coming out from the resonance. Its easy to see η = s/m 2 B with s = p 2 . The light spectator quark comes from B meson and goes into D * 0 in the hadronization processes in Fig. 1 (a) got the momentum k = ( mB √ 2 z, 0, k T ), z is the momentum fraction. The momenta p B , p 3 , k B and k 3 for the B meson, bachelor meson h and the associated spectator quarks for B and h have their definitions as where x B and x 3 are the corresponding momentum fractions. The S-wave Dπ system distribution amplitude could be collected into [41,54,59] Φ S−wave and the distribution amplitude the a Dπ = 0.40 ± 0.10 and ω Dπ = 0.40 ± 0.10 GeV are adopted in the calculation in this work by catering to our numerical results to the data in Table I and considering the related parameters for the D and D * mesons in the literature. The numbers for ω Dπ and a Dπ in Ref. [41] have been considered as the references in this work, but we don't use the same values because of the different framework of the two-body and quasi-two-body decays and the different definitions of the distribution amplitudes. The distribution amplitudes for the pion, kaon and B meson are the same as those widely adopted in the PQCD approach to hadronic B meson decays, one can find their expressions and the relevant parameters in Ref. [69]. The decay amplitude A for the quasi-two-body decay processesB in the PQCD approach is given by [51,52,59] where the symbol ⊗ means convolutions in parton momenta, the hard kernel H contains only one hard gluon exchange at leading order in the strong coupling α s as in the two-body formalism and the distribution amplitude φ B (φ h , φ Dπ ) absorbs nonperturbative dynamics in the decay processes. We then have the differential branching fraction (B) [5] τ B being the B meson mean lifetime, the magnitude momentum for bachelor h, in the center-of-mass frame of the Dπ pair, as The m h is the mass of the bachelor meson pion or kaon.
In the numerical calculation, we adopt Λ The decay constant f B = 0.19 GeV for B meson comes from lattice QCD [70]. The masses and the mean lifetimes for the neutral and charged B meson, the pole masses and the widths of the neutral and charged D * 0 state, the Wolfenstein parameters, the masses of pion, kaon and D meson are all come from the Particle Data Group [5]. Utilizing the the differential branching fraction Eq. (13) and the decay amplitudes collected in Appendix A, we obtain the branching fractions in Table II for the concerned quasi-two-body decay processes. The shape parameter uncertainty of the B meson, ω B = 0.40 ± 0.04 GeV, contributes the largest error for the branching fractions in Table II Table II. There are other errors, which come from the uncertainties of the parameters in the distribution amplitudes for bachelor pion(kaon) [69] and the Wolfenstein parameters [5], are small and have been neglected.  The distributions of those four branching ratios in Table II in the Dπ pair invariant mass m Dπ are shown in Fig. 2, with the curves for B − → D * 0 0 π − → D + π − π − (the dash line) andB 0 → D * + 0 π − → D 0 π + π − (the solid line) on the left, and the curves for B − → D * 0 0 K − → D + π − K − (the dash line) andB 0 → D * + 0 K − → D 0 π + K − (the solid line) at the right. The small mass difference of the charged and the neutral D * 0 exhibit the different peaks of the m Dπ dependence for the different decay modes. The main portion of each branching ratio lies obviously in the region around the pole mass of the resonant state D * 0 in the Fig. 2, the contributions from the energy region m Dπ > 3 GeV can be safely omitted.
Assuming the D * 0 state decays essentially into two-body modes, from the Clebsch-Gordan coefficients, we have  [40] within errors, but it's smaller than the corresponding results in Ref. [41].
Compare our numerical results in Table II with the corresponding data in Table I, we find that the PQCD value of the branching fraction for the quasi-two-body decay process B − → D * 0 0 π − → D + π − π − in this work agree well with the values (6.1 ± 0.6 ± 0.9 ± 1.6) × 10 −4 taken from Belle [6] and (6.8 ± 0.3 ± 0.4 ± 2.0) × 10 −4 picked up from BaBar [7]. In Ref. [42], LHCb Collaboration presented a result (5.78 ± 0.08 ± 0.06 ± 0.09 ± 0.39) × 10 −4 for the total S-wave Dπ system, which should be supposed to mainly contributed by the D * 0 state, in the B − → D + π − π − decays. For the decayB 0 → D * + 0 K − → D 0 π + K − , the result 2.38 +0.95+0.85+0.30+0.04 −0.65−0.68−0.28−0.03 × 10 −5 in Table II is consistent with the data (1.77 ± 0.26 ± 0.19 ± 0.67 ± 0.20) × 10 −5 given by LHCb [46]. While for the other two decay modes, there are apparent inconsistencies for the branching ratios between the PQCD predictions and the results from Belle and LHCb Collaborations. The Belle's branching fraction [43] for the decayB 0 → D * + 0 π − → D 0 π + π − is only about 21% of the PQCD prediction in this work, other two values from LHCb [44] in Table I for this process are some larger, but still less than 30% of our result when considering only the central values. The data for the decay [45] is probably worse than theB 0 → D * + 0 π − → D 0 π + π − case, the branching fraction B = (6.1 ± 1.9 ± 0.5 ± 1.4 ± 0.4) × 10 −6 is about one order of magnitude smaller than the predicted value in Table II. The B → D * 0 (→ Dπ)π processes can be decomposed in terms of two isospin amplitudes, A 1/2 and A 3/2 , as have been done for theB 0 → D + π − and B − → D 0 π − decays in Ref. [72]. With the absolute value |A 1/2 / √ 2A 3/2 | and the relative strong phase between A 1/2 and A 3/2 in [72], we have the ratio R ≈ 0.59, which is close to the result R ≈ 0.54 from Table III, between the branching fractions of decaysB 0 → D + π − and B − → D 0 π − . It is reasonable to expect that the ratio between the branching fractions of the decaysB 0 → D * + 0 (→ D 0 π + )π − and B − → D * 0 0 (→ D + π − )π − is not far from the 0.54. Our results in this work provide R ≈ 0.48 for the corresponding two decays with pion as the bachelor particle, while the value for R from the data in Table I is just slightly larger than 0.1. So one could conclude that the data in Table I for the decaysB 0 → D * + 0 (→ D 0 π + )π − and B − → D * 0 0 (→ D + π − )π − are probably inconsistent with the isospin relation.
For the decay processes B − → D * 0 0 π − → D + π − π − and B − → D * 0 0 K − → D + π − K − , we have an identical step D * 0 0 → D + π − , the difference of the two decay modes originated from the bachelor particles pion and kaon. Within the SU (3) flavor symmetry, we have a straightforward ratio R D * 0 0 for the branching fractions of these two decays as from Review of Particle Physics [5], then we have R D * 0 0 ≈ 0.076. It's easy to obtain a similar ratio for the decay modesB 0 → D * + 0 K − → D 0 π + K − andB 0 → D * + 0 π − → D 0 π + π − . The energy dependent curves of the R D * 0 0 and R D * + 0 predicted by PQCD are shown in Fig. 3, from which one can find that there is little variation for the R D * 0 0 or R D * + 0 as m Dπ runs from its threshold to 3.5 GeV. There are similar patterns for the ratios of the related branching fractions for the decay modes including a pseudoscalar D or a vector D * rather than D * 0 as listed in the Table III. If we accept the average value B(B − → D * 0 0 π − ) × B(D * 0 0 → D + π − ) = (6.4 ± 1.4) × 10 −4 in Ref. [5], the branching fraction B = (4.86 ± 1.06) × 10 −5 , which agree well with the PQCD prediction in Table II, for the decay process B − → D * 0 0 K − → D + π − K − could be derived from Eq. (15). If we believe the result B = (1.77 ± 0.77) × 10 −5 given by LHCb [46] for the decay B 0 → D * − 0 K + →D 0 π − K + , then the three values listed in Table I for the decaȳ B 0 → D * + 0 π − → D 0 π + π − announced by Belle and LHCb are simply not credible when considering the Eq (17). In fact, there is a preliminary result from the Dalitz plot analysis of the B 0 →D 0 π + π − decay processes in Ref. [71] announced by BaBar as This result is consistent with the prediction 2.85 +1.23+1.05+0.33+0.06 −0.80−0.81−0.31−0.05 × 10 −4 within errors. To sum up, we studied the quasi-two-body decays The concerned quasi-two-body decay amplitudes are given, in the PQCD approach, by in which G F is the Fermi coupling constant, V 's are the CKM matrix elements. And it should be understood that the Wilson coefficients c 1 and c 2 appear in convolutions in momentum fractions and impact parameters b. The amplitudes from Fig. 1 are written as The evolution factors in the above factorization formulas are given by in which S (B,C,P ) (t) are in the Appendix of [73], the hard functions h, h a , h have their explicit expressions in the Ref. [73]. We need to stress that, because of the different definitions of the momenta for the initial and final states, the concerned expressions in [73] could be employed in this work only after the replacements {x 1 → x B , b 1 → b B , x 2 → z, b 2 → b, r 2 → η}. The parameter c in the Eq. (A1) of [73] is adopt to be 0.4 in this work according to the Refs. [74,75].