Virtual contributions from $D^{\ast}(2007)^0$ and $D^{\ast}(2010)^{\pm}$ in the $B\to D\pi h$ decays

We study the quasi-two-body decays $B\to D^*h \to D\pi h$ with $h=(\pi, K)$ in the perturbative QCD approach and focus on the virtual contributions from the off-shell $D^{\ast}(2007)^0$ and $D^{\ast}(2010)^{\pm}$ in the four measured decays $\bar B^0 \to D^0\pi^+\pi^-$, $\bar B^0\to D^0\pi^+K^-$, $B^- \to D^+\pi^-\pi^-$ and $B^- \to D^+\pi^-K^-$. For the $\bar B^0 \to D^{*+}\pi^-\to D^0\pi^+\pi^-$ and $\bar B^0\to D^{*+}K^-\to D^0\pi^+K^-$ decays, their branching fractions concentrate in a very small region of $m_{D^0\pi^+}$ near $D^{*+}$ pole mass, and the virtual contributions from $D^{*+}$, in the region $m_{D^0\pi^+}>2.1$ GeV, are about $5\%$ of the corresponding quasi-two-body results. We define two ratios $R_{D^{*+}}$ and $R_{D^{*0}}$, from which we conclude that the flavor-$SU(3)$ symmetry will be maintained for the $B\to D^* h\to D\pi h$ decays with very small breaking at any physical value of the $m_{D\pi}$. The $B^-\to D^{*0}\pi^-\to D^0\pi^0\pi^-$ and $B^-\to D^{*0}K^-\to D^0\pi^0K^-$ decays can be employed as a constraint for the $D^{*0}$ decay width, with preferred values consistent with previous theoretical predictions for this quantity.

FIG. 1: Typical Feynman diagrams for the decay processes B → D * h → Dπh, h =(π, K). The symbol ⊗ is the weak vertex, × denotes possible attachments of hard gluons and the rectangle represents the vector states D * .
accessible region of the phase space [17,21]. That is to say, although the pole mass of D * 0 is lower than the threshold of D + π − pair, the natural decay tunnel D * 0 → D + π − is blocked, but the resonance tail will contribute to the total branching fractions of the B − → D + π − h − processes, and the off-shell effects were found surprisingly large in [45]. For the decaysB 0 → D 0 π + h − , the portion ofB 0 → D * + h − with the natural decay D * + → D 0 π + were always excluded from the total three-body branching fractions by a cut of the D 0 π + invariant mass, while the necessary off-shell effects were retained in the decay amplitudes [15,18,19].
This work is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework. In Sec. III, we show the numerical results. Discussions and conclusions are given in Sec. IV. The factorization formulas for the relevant quasi-two-body decay amplitudes are collected in the Appendix.

II. FRAMEWORK
In the rest frame of B meson, with m B being its mass, we define momentum p B and the light spectator quark momentum k B for it as in the light-cone coordinates, where x B is the momentum fraction. The momenta p 3 and k 3 for the bachelor final state h and its spectator quark have their definitions as For the state D * and the Dπ pair decays from it in the Feynman diagrams, the Fig. 1, for the quasi-two-body processes B → D * h → Dπh, we define their momentum p = mB √ 2 (1, η, 0). Its easy to see η = s/m 2 B , with the invariant mass square s = p 2 for the Dπ pair. The light spectator quark comes from B meson and goes into intermediate state in the D * hadronization as shown in Fig. 1 (a) has the momentum k = ( mB √ 2 z, 0, k T ). Where x 3 and z are the corresponding momentum fractions and run from 0 to 1.
The distribution amplitudes for the B meson and the bachelor final state pion or kaon in this work are the same as those widely adopted in the PQCD approach in the hadronic B meson decays, one can find their expressions and the relevant parameters in Ref. [86]. For the longitudinal polarization structure of the P -wave Dπ system which including the D * hadronization and the D * → Dπ processes, based on the discussions in Refs. [39,42,75,80,87,88], one could write Φ P -wave with the distribution amplitude where the a Dπ and ω Dπ are the Gegenbauer moment and the shape parameter for the P -wave Dπ system, respectively. The time-like form factor F Dπ (s) has its definition in the matrix elements where p = (p 1 + p 2 ), p 1 (p 2 ) is the momentum for D(π) and m D (m π ) is the mass of D(π) meson. The F 0 (s) is the S-wave form factor for Dπ system. With Eq. (5), by inserting the intermediate state D * , it's easy to have the following expression for the form factor F Dπ (s) The f D * above is the decay constant for D * , one can find its different values in Refs [89][90][91][92][93]. We adopt f D * = (250±11) MeV [89,90] in the numerical calculations. The energy dependent relativistic Breit-Wigner denominator D BW (s) equals to m 2 D * − s − im D * Γ(s), with m D * the pole mass for D * , and the mass dependent decay width defined as where the barrier radius r BW = 4.0 GeV −1 as it in Refs. [17,18,21], the Blatt-Weisskopf barrier factor [94] is and ] /s is the magnitude of the momentum for the daughter state D or π in the rest frame of the D * , q 0 is the value of q at s = m 2 D * . For the virtual contributions from the state D * 0 , the m D * in the q 0 shall be replaced with the m eff 0 , which has its formula in Refs. [17,21,95]. The coupling constant g D * Dπ in Eq. (6) could be related to the decay width Γ 0 in Eq. (7) for D * . For the total decay width of the D * + , which is the sum of the partial widths of the decays D * + → D 0 π + , D * + → D + π 0 and D * + → D + γ, was firstly measured by CLEO Collaboration with Γ(D * + ) = 96 ± 4(stat.) ± 22(syst.) keV [96]. A more precise measurement performed by BaBar Collaboration presented Γ(D * + ) = 83.3 ± 1.2(stat.) ± 1.4(syst.) keV and g D * Dπ = 16.92 ± 0.13 ± 0.14 [97,98] with the isospin relation g D * Dπ = g D * D 0 π + = − √ 2g D * D + π 0 . For the state D * 0 , there is no accurate experimental result for its decay width. In the measurement of three-body decays including virtual D * 0 contributions, the width was fixed to 0.1 MeV by BaBar [16], the experimental upper limit of 2.1 MeV was adopted by LHCb [17,21], while the decay width for D * 0 in the work [14] from Belle Collaboration was calculated from the width of the D * + assuming isospin invariance and HQET.
The Lorentz invariant amplitude A for the quasi-two-body B → D * h → Dπh decay processes in the PQCD approach, according to Fig. 1, is given by [72,73] where the symbol ⊗ means convolutions in parton momenta, the hard kernel H contains one hard gluon exchange as shown in Fig. 1 where τ B being the B meson mean lifetime. The magnitudes of h meson momentum q h , in the rest frame of the D * , is written as The m h is the mass of the bachelor meson pion or kaon. The decay amplitudes for B → D * h → Dπh are collected in the Appendix.

III. RESULTS
In the numerical calculation, we adopt the decay constant f B = 0.19 GeV [100], the mean lifetimes τ B 0 = (1.520 ± 0.004) × 10 −12 s and τ B ± = (1.638 ± 0.004) × 10 −12 s [99] for the B meson. The masses of the neutral and charged B, D, π and K mesons, the pole masses of the neutral and charged D * and the Wolfenstein parameters λ and A are presented in Table I.  [99].
Utilizing the the differential branching fraction the Eq. (10) and the decay amplitudes collected in Appendix A, we obtain the branching fractions for the virtual contributions (B v ) in Table II of the concerned quasi-two-body decay processes B → D * h → Dπh. The invariant mass of the Dπ system has been cut at 2.1 GeV for the results in Table II by following the step of Ref. [19], and the decay width Γ D * 0 = 2.1 MeV which has been adopted by LHCb Collaboration in Refs. [17,21] is employed for the two B − decay modes. The largest error for the branching fractions in Table II comes from the B meson shape parameter uncertainty ω B = 0.40 ± 0.04 GeV, the error induced by the decay constant f D * = (250 ± 11) MeV [89,90] takes the second place, the uncertainty of the Wolfenstein parameter A in Table I contributes the fourth one, while the third error and the last one originated from the D * Gegenbauer moment a Dπ = 0.50 ± 0.10 and shape parameter ω Dπ = 0.10 ± 0.02 [42,101], respectively. There are other errors, which come from the uncertainties of the parameters in the distribution amplitudes for bachelor pion(kaon) [86], the Wolfenstein parameters λ [99], etc. are small and have been neglected. One has the integrated branching ratios for the two-body decaysB 0 → D * + π − andB 0 → D * + K − as from the corresponding quasi-two-body decays by integrating the whole physical region of the D 0 π + invariant mass and considering the data B(D * + → D 0 π + ) = 67.7% [99]. The two results above predicted by PQCD agree well with the branching fractions (2.74 ± 0.13) × 10 −3 and (2.12 ± 0.15) × 10 −4 for the two-body decaysB 0 → D * + π − and B 0 → D * + K − in the Review of Particle Physics [99], respectively.

Mode
Unit 1 GeV even if the D * 0 decay width is 2.1 MeV. As a result, the variation of the r BW from 4.0 GeV −1 to 1.6 GeV −1 [14,19] in Eq. (7) makes the virtual contributions for the B → D * h → Dπh decays in Table II essentially unchanged. The same situation will happen again because of the same reason when one replaces Blatt-Weisskopf barrier factor, the Eq. (8), with the exponential form factor (EFF) F (z) = exp(−(z − z ′ )) for the denominator D BW (s), where z and z ′ have their expressions in Ref. [16]. The EFF R(m 2 (Dπ)) = e −(β1+iβ2)m 2 (Dπ) , with the free parameters β 1 and β 2 , has been used in the experimental Dalitz plot analyses [19] to describe the contributions from the off-shell D * (2010) − and the generalD 0 π − P-wave. We don't tend to employ an EFF to replace the time-like form factor of Eq. (6) because the EFF will bring us an unknown parameter and reduce the ability of theoretical prediction. As a test of the effect of m eff 0 instead of m D * 0 for the virtual contributions in the decays involving D * 0 in this work, we employ the value (m eff 0 + ∆m) to replace the m eff 0 for q 0 in the Eq. (8). When ∆m is ±0.5 GeV or even ±1.0 GeV, the results for are almost the same as they in Table II, the variations are found less than 0.1% for the corresponding values.
. The two small diagrams are for the corresponding virtual contributions in the mDπ region (2.1 ∼ 3.5) GeV.
It's clear that the breaking effects of the flavor-SU (3) symmetry is quite small for R D * + . The small errors induced by the uncertainties of ω B and a Dπ for R D * + are caused by the cancellation, which means the increase or decrease for the values of these parameters will result in nearly identical change of the weight at the same direction for the branching ratios of these two decays. And the errors of R D * + come from the f D * , Wolfenstein parameter A and ω Dπ are zeros for the same reason. The result of Eq. (16) is consistent with the data (7.76 ± 0.34 ± 0.29)% presented by BaBar [102] and (0.074 ± 0.015 ± 0.006) announced by Belle [103]. The energy dependent R D * + is shown as the left diagram in Fig. 4. A similar ratio R D * 0 , which has the definition as for the quasi-two-body decays B − → D * 0 K − → D + π − K − and B − → D * 0 π − → D + π − π − is shown as the right diagram of Fig. 4 in the m Dπ region (2.03 ∼ 3.50) GeV. An interesting conclusion could be made from the R D * lines in Fig. 4 is that the flavor-SU (3) symmetry will be maintained at any physical point of the invariant mass m Dπ for the concerned quasi-two-body B → D * h → Dπh decays. The ratio between the branching fractions of the two-body decays B − → D * 0 K − and B − → D * 0 π − were measured to be (7.930±0.110(stat)±0.560(syst))×10 −2 by LHCb [104] and 0.0813 ± 0.0040(stat) +0.0042 −0.0031 (syst) at BaBar [105], which are close to the result in the region (2.1 ∼ 3.5) GeV deduced from the results in Table II. The PQCD predictions of the virtual contributions in Table II for theB 0 → D * + π − → D 0 π + π − andB 0 → D * + K − → D 0 π + K − decays are 3.4% and 3.5% in the Dπ invariant mass region √ s > 2.1 GeV from the intermediate state D * + , respectively, of the corresponding two-body decay branching ratios, the Eq. (12) and Eq. (13). By considering B(D * + → D 0 π + ) = 67.7% [99], we have about 5% of the total quasi-two-body branching ratios for the virtual contributions of the two decay processes involving D * + . The virtual contributions in Table II for the two B − decay modes are 3.9% and 3.7% of the two-body data for B − → D * 0 π − and B − → D * 0 K − in [99], respectively. Because of the threshold of D + π − , we don't have the integrated quasi-two-body branching fractions for the decays B − → D * 0 π − → D + π − π − and B − → D * 0 K − → D + π − K − . But we can analyse the quasi-two-body processes B − → D * 0 π − → D 0 π 0 π − and B − → D * 0 K − → D 0 π 0 K − from the D 0 π 0 threshold. As an extreme example, if the experimental upper limit of 2.1 MeV is used for the D * 0 width, we have B(B − → D * 0 π − ) = 2.69 × 10 −4 and B(B − → D * 0 K − ) = 2.10 × 10 −5 as the central values for the these two decays after take into the factor B(D * 0 → D 0 π 0 ) = 64.7% [99]. Obviously, the branching fractions for the two-body decays B − → D * 0 π − and B − → D * 0 K − are highly underestimated with Γ D * 0 = 2.1 MeV. Although there is no direct measurement for the Γ D * 0 , we have theoretical results 58 keV [106], 59.6 ± 1.2 keV [107], 55.9 ± 1.6 keV [108] and 53 ± 5 ± 7 keV [109] for it. If we replace the decay width 2.1 MeV with Γ D * 0 = 53 keV [109] and considering the factor B(D * 0 → D 0 π 0 ) = 64.7% [99], we have  Fig. 5, the dash-dot curves are the PQCD predictions, the blue lines and the gray bands are the data with their errors from [99]. The branching ratios in Fig. 5 can be exploited to constrain the D * 0 decay width which could be read as Γ D * 0 ≈ 53 keV from these two diagrams. The detailed discussion including the impacts of different parameter uncertainties about D * 0 decay width in the three-body hadronic B meson decays shall be left for the future study. It must be pointed out that, the changes are tiny for the two virtual contributions involving D * 0 in the Table II when we adopt 53 keV for Γ D * 0 , the reason is that the |m D * Γ(s)| shall be less than 1/10 4 of |m 2 D * − s| even if Γ D * 0 equals to 2.1 MeV when m D + π − is larger than 2.1 GeV. The small branching fractions for the two-body decays B − → D * 0 π − and B − → D * 0 K − with the subprocess D * 0 → D 0 π 0 in this work with Γ D * 0 = 2.1 MeV is caused by the insufficient contributions in the m D 0 π 0 region near the D * 0 pole mass but not by the lower virtual contributions in the region m D 0 π 0 > 2.1 GeV.

IV. CONCLUSION
In this work, we studied the quasi-two-body decays B → D * h → Dπh and focused on the virtual contributions originated from off-shell D * (2007) 0 and D * (2010) ± in the decays of B − → D + π − π − , B − → D + π − K − ,B 0 → D 0 π + π − andB 0 → D 0 π + K − which have been measured by Belle, BaBar and LHCb Collaborations. For theB 0 → D * + π − → D 0 π + π − andB 0 → D * + K − → D 0 π + K − decays, we found that the main portions of their quasi-two-body branching fractions concentrate in a very small region of the D 0 π + invariant mass, the percentage is larger than 90% for the branching ratios in the realm of 2.5 MeV around the pole mass of D * + . And the virtual contributions from D * + , in the region of m D 0 π + > 2.1 GeV, are about 5% of the integrated values for the corresponding quasi-two-body results by considering B(D * + → D 0 π + ) = 67.7%. The virtual contributions in this work for B − → D * 0 π − → D + π − π − and B − → D * 0 K − → D + π − K − decay modes were found to be 3.9% and 3.7% of the two-body data for B − → D * 0 π − and B − → D * 0 K − in Review of Particle Physics, respectively.
From the ratios R D * + and R D * 0 defined between the quasi-two-body decays including D * + and D * 0 as the intermediate states, respectively, we concluded that the flavor-SU (3) symmetry will be maintained with very small breaking at any physical value of the invariant mass m Dπ for the concerned B → D * h → Dπh decays. We found that the decays B − → D * 0 π − → D 0 π 0 π − and B − → D * 0 K − → D 0 π 0 K − have strong dependence on the D * 0 decay width for their branching fractions which could be employed as a constraint for Γ D * 0 . 2.1 MeV for D * 0 decay width will make the branching ratios of the quasi-two-body decays B − → D * 0 π − → D 0 π 0 π − and B − → D * 0 K − → D 0 π 0 K − be highly underestimated because of the insufficient contributions in the m D 0 π 0 region near the D * 0 pole mass. With Γ D * 0 = 53 keV, we predicted as the virtual contributions, which are about 3% of the corresponding quasi-two-body results.
Appendix A: DECAY AMPLITUDES 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, the Wilson coefficients c 1 and c 2 will 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 [42], the hard functions h, h a , h (1,2) (b,d,f ) and the hard scales t (1,2) (e,b,i,d,a,f ) have their explicit expressions in the Ref. [42]. Because of the different definitions of the momenta for the initial and final states, the concerned expressions in [42] 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 [42] is adopt to be 0.35 in this work according to the Refs. [111,112].