Abstract
We demonstrate the roles of $D_{(s)}^\ast$ and their contributions in the quasi-two-body decays $B_{(s)} \to D_{(s)} h h^\prime$ ($h, h^\prime = \{\pi, K\}$) in the perturbative QCD approach, stemming from the quark flavour changing $\bar{b} \to \bar{c} \, q_2 \, \bar{q}_1$ and $\bar{b} \to c \, \bar{q_1} \, \bar{q_2}$ with $q_1, q_2 = \{s/d, u\}$. The main motivation of this study is the measurements of significant derivations from the simple phase-space model in the channels $B_{(s)} \to D_{(s)} h h^\prime$ at $B$ factories and LHC, which is now clarified as the Breit-Wigner-tail effects from the corresponding intermediate resonant states $D_{(s)}^\ast$. We confirm that these effect from $D^\ast$ is small ($\sim 5\%$) in the quasi-two-body $B_{(s)} \to D \pi \pi (K)$ decaying channels, and predict the tiny ($< 1 \%$) contributions from $D^\ast$ in the $B_{(s)} \to D_s K \pi (K)$ decaying channels, our result for the $B_s \to D K \pi (K)$ decaying channels contributed only from the Breit-Wigner-tail effect of $D_s^\ast$ is in agreement with the current LHCb measurement. We recommend the Belle-II and the LHCb collaborations to restudy the processes $B^+\to \bar D^{\ast 0}\pi^+(K^+) \to D^-\pi^+\pi^+(K^+)$ to reveal the structure of $D^{\ast 0}$ and the strong decay $D^{\ast 0} \to D^+ \pi^-$.
Highlights
Three-body B decays have a much richer phenomenology with the number of decaying channels being about 10 times larger than the number in two-body decays
We studied systematically the role of DÃðsÞ and their contributions in BðsÞ → DðsÞhh0 (h0 1⁄4 π, K) decays by taking the dimeson light cone distribution amplitude (LCDA) of DðsÞh system in the framework of the perturbative QCD (pQCD) approach
In the local kinematic region where the invariant mass of final DðsÞh system locates in/ around the interval of DÃðsÞ and simultaneously, the other invariant mass approaching zero, the three-body decay can be treated as a quasi-two-body decaying process and divided into two ingredients
Summary
Three-body B decays have a much richer phenomenology with the number of decaying channels being about 10 times larger than the number in two-body decays. We take into account only the leading twist wave functions of the heavy meson and dimeson system, say, B and DðsÞh, and take the light meson wave functions up to a twist three level with the chiral mass mπ0 and mK0 , only the powers Oðmc=mB ≡ rcÞ and Oðmπ0=mB ≡ r0Þ, and the momentum transfer parameter ζ appear in the expressions of charmed two-body B decaying amplitudes In this way, the reliability of pQCD approach can be checked at least at the leading power of OðrcÞ for the charmed B decays [32,33,34,35,36], and in this paper, we are working at this accuracy. 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1⁄2ðm2B − m2h0 Þ2 − 2ðm2B þ m2h0 Þs þ s2=s: ð20Þ
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