Quasi-two-body decays $B_{(s)} \to P D_0^*(2400) \to P D \pi$ in the perturbative QCD approach

TL;DR

This paper calculates branching ratios for specific B meson decays involving D_0^*(2400) resonances using perturbative QCD, providing predictions that can be tested in future experiments.

Contribution

It presents the first detailed perturbative QCD analysis of quasi-two-body B decays involving D_0^*(2400), including predictions for branching ratios and their ratios, considering CKM suppression effects.

Findings

Branching ratios range from 10^{-9} to 10^{-4}.

Predicted ratios agree with flavor-SU(3) symmetry expectations.

Results are consistent with current theoretical frameworks and can be tested experimentally.

Abstract

We study the quasi-two-body decays $B\to P D^{\ast}_0(2400) \to P D\pi $ with $P=(\pi, K, \eta, \eta^{\prime})$ in the perturbative QCD factorization approach. The predicted branching fractions for the considered decays are in the range of $10^{-9}$-$10^{-4}$. The strong Cabibbo-Kobayashi-Maskawa (CKM) suppression factor $R_{CKM}\approx \lambda^4 (\bar{\rho}^2 + \bar{\eta}^2) \approx 3\times 10^{-4}$ results in the great difference of the branching ratios for the decays with $D_0^*$ and $\bar{D}_0^*$ as the intermediate states. The ratio $R_{\bar{D}_0^{*0}}$ between the decays $B^0 \to \bar{D}_0^{*0} K^0\to D^-\pi^+K^0$ and $B^0 \to \bar{D}_0^{*0}\pi^0 \to D^-\pi^+\pi^0$ is about $0.091^{+0.003}_{-0.005}$, consistent with the flavour-$SU$(3) symmetry result. The ratio for the branching fractions is found to be $1.10^{+0.05}_{-0.02}$ between $\mathcal{B}(B_s^0\to D_0^{*+}K^-\to D^0\pi^+K^-)$ and $\mathcal{B}(B^0\to D_0^{*+} \pi^-\to D^0\pi^+ \pi^-)$ and to be $1.03^{+0.06}_{-0.07}$ between $\mathcal{B}(B_s^0\to\bar{D}_0^{*0} \bar{K}^0\to D^-\pi^+ \bar{K}^0)$ and $2\mathcal{B}(B^0\to \bar{D}_0^{*0}\pi^0\to D^-\pi^+\pi^0)$. The predictions in this work can be tested by the future experiments.

Tables

Table 1. Table 1: PQCD predictions for branching fractions of the quasi-two-body decays B + → D 0 ∗ ​ P → D ​ π ​ P → superscript 𝐵 superscript subscript 𝐷 0 𝑃 → 𝐷 𝜋 𝑃 B^{+}\to D_{0}^{*}P\to D\pi P together with the available experimental data.

Mode	Unit	$ℬ$	Data
$B^{+}\to D_0^{\ast 0}{\pi }^{+}\to D^{+}{\pi }^{-}{\pi }^{+}$	(${10}^{-8}$)	${1.13}_{-0.26}^{+0.36}{\left({\omega }_B\right)}_{-0.14}^{+0.13}{\left({\omega }_{D\pi }\right)}_{-0.05}^{+0.03}{\left(a_{D\pi }\right)}_{-0.05}^{+0.06}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^{+}\to {\overline{D}}_0^{\ast 0}{\pi }^{+}\to D^{-}{\pi }^{+}{\pi }^{+}$	(${10}^{-4}$)	${5.95}_{-1.64}^{+2.37}{\left({\omega }_B\right)}_{-1.55}^{+1.97}{\left({\omega }_{D\pi }\right)}_{-0.49}^{+0.54}{\left(a_{D\pi }\right)}_{-0.21}^{+0.29}\left({\Gamma }_{D_0^{\ast 0}}\right)$	RPPTanabashi:2018oca : $6.4\pm 1.4$
			BelleAbe:2003zm : $6.1\pm 0.6\pm 0.9\pm 1.6$
			BaBarAubert:2009wg : $6.8\pm 0.3\pm 0.4\pm 2.0$
			LHCbAaij:2016fma : $5.78\pm 0.08\pm 0.06\pm 0.09\pm 0.39$
$B^{+}\to D_0^{\ast 0}{K}^{+}\to D^{+}{\pi }^{-}{K}^{+}$	(${10}^{-7}$)	${3.56}_{-0.78}^{+1.02}{\left({\omega }_B\right)}_{-0.52}^{+0.46}{\left({\omega }_{D\pi }\right)}_{-0.15}^{+0.09}{\left(a_{D\pi }\right)}_{-0.12}^{+0.16}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^{+}\to {\overline{D}}_0^{\ast 0}{K}^{+}\to D^{-}{\pi }^{+}{K}^{+}$	(${10}^{-5}$)	${4.65}_{-1.30}^{+1.89}{\left({\omega }_B\right)}_{-1.24}^{+1.51}{\left({\omega }_{D\pi }\right)}_{-0.38}^{+0.40}{\left(a_{D\pi }\right)}_{-0.18}^{+0.22}\left({\Gamma }_{D_0^{\ast 0}}\right)$	LHCbAaij:2015vea : $0.61\pm 0.19\pm 0.05\pm 0.14\pm 0.04$
$B^{+}\to D_0^{\ast +}{\pi }^0\to D^0{\pi }^{+}{\pi }^0$	(${10}^{-7}$)	${1.40}_{-0.34}^{+0.48}{\left({\omega }_B\right)}_{-0.01}^{+0.02}{\left({\omega }_{D\pi }\right)}_{-0.00}^{+0.01}{\left(a_{D\pi }\right)}_{-0.02}^{+0.03}\left({\Gamma }_{D_0^{\ast +}}\right)$	-
$B^{+}\to D_0^{\ast +}{K}^0\to D^0{\pi }^{+}{K}^0$	(${10}^{-9}$)	${5.52}_{-0.21}^{+0.15}{\left({\omega }_B\right)}_{-1.42}^{+1.73}{\left({\omega }_{D\pi }\right)}_{-0.36}^{+0.41}{\left(a_{D\pi }\right)}_{-0.12}^{+0.13}\left({\Gamma }_{D_0^{\ast +}}\right)$	-
$B^{+}\to D_0^{\ast +}\eta \to D^0{\pi }^{+}\eta$	(${10}^{-8}$)	${6.26}_{-1.49}^{+2.11}{\left({\omega }_B\right)}_{-0.03}^{+0.04}{\left({\omega }_{D\pi }\right)}_{-0.02}^{+0.03}{\left(a_{D\pi }\right)}_{-0.10}^{+0.14}\left({\Gamma }_{D_0^{\ast +}}\right)$	-
$B^{+}\to D_0^{\ast +}{\eta }^{\prime }\to D^0{\pi }^{+}{\eta }^{\prime }$	(${10}^{-8}$)	${4.01}_{-0.96}^{+1.34}{\left({\omega }_B\right)}_{-0.03}^{+0.02}{\left({\omega }_{D\pi }\right)}_{-0.01}^{+0.02}{\left(a_{D\pi }\right)}_{-0.06}^{+0.07}\left({\Gamma }_{D_0^{\ast +}}\right)$	-

Table 2. Table 2: PQCD prediction of branching fraction for the quasi-two-body decays B 0 → D 0 ∗ ​ P → D ​ π ​ P → superscript 𝐵 0 superscript subscript 𝐷 0 𝑃 → 𝐷 𝜋 𝑃 B^{0}\to D_{0}^{*}P\to D\pi P together with the available experimental data.

Mode	Unit	$ℬ$	Data
$B^0\to D_0^{\ast -}{\pi }^{+}\to {\overline{D}}^0{\pi }^{-}{\pi }^{+}$	(${10}^{-4}$)	${2.85}_{-0.80}^{+1.23}{\left({\omega }_B\right)}_{-0.81}^{+1.05}{\left({\omega }_{D\pi }\right)}_{-0.31}^{+0.33}{\left(a_{D\pi }\right)}_{-0.05}^{+0.06}\left({\Gamma }_{D_0^{\ast +}}\right)$	RPPTanabashi:2018oca : $0.76\pm 0.08$
			BelleKuzmin:2006mw : $0.60\pm 0.13\pm 0.15\pm 0.22$
			LHCbAaij:2015sqa : $0.77\pm 0.05\pm 0.03\pm 0.03\pm 0.04$111 Isobar model
			LHCbAaij:2015sqa : $0.80\pm 0.05\pm 0.08\pm 0.04\pm 0.04$222 K-matrix model
$B^0\to D_0^{\ast +}{\pi }^{-}\to D^0{\pi }^{+}{\pi }^{-}$	(${10}^{-7}$)	${2.56}_{-0.65}^{+0.85}{\left({\omega }_B\right)}_{-0.02}^{+0.01}{\left({\omega }_{D\pi }\right)}_{-0.03}^{+0.02}{\left(a_{D\pi }\right)}_{-0.06}^{+0.03}\left({\Gamma }_{D_0^{\ast +}}\right)$	-
$B^0\to D_0^{\ast -}{K}^{+}\to {\overline{D}}^0{\pi }^{-}{K}^{+}$	(${10}^{-5}$)	${2.38}_{-0.65}^{+0.95}{\left({\omega }_B\right)}_{-0.68}^{+0.85}{\left({\omega }_{D\pi }\right)}_{-0.28}^{+0.30}{\left(a_{D\pi }\right)}_{-0.03}^{+0.04}\left({\Gamma }_{D_0^{\ast +}}\right)$	LHCbAaij:2015kqa : $1.77\pm 0.26\pm 0.19\pm 0.67\pm 0.20$
$B^0\to D_0^{\ast 0}{\pi }^0\to D^{+}{\pi }^{-}{\pi }^0$	(${10}^{-9}$)	${4.20}_{-1.07}^{+1.62}{\left({\omega }_B\right)}_{-0.48}^{+0.44}{\left({\omega }_{D\pi }\right)}_{-0.07}^{+0.09}{\left(a_{D\pi }\right)}_{-0.12}^{+0.07}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^0\to {\overline{D}}_0^{\ast 0}{\pi }^0\to D^{-}{\pi }^{+}{\pi }^0$	(${10}^{-5}$)	${2.29}_{-0.61}^{+0.87}{\left({\omega }_B\right)}_{-0.43}^{+0.51}{\left({\omega }_{D\pi }\right)}_{-0.06}^{+0.09}{\left(a_{D\pi }\right)}_{-0.04}^{+0.12}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^0\to D_0^{\ast 0}{K}^0\to D^{+}{\pi }^{-}{K}^0$	(${10}^{-7}$)	${2.69}_{-0.66}^{+0.91}{\left({\omega }_B\right)}_{-0.32}^{+0.30}{\left({\omega }_{D\pi }\right)}_{-0.08}^{+0.09}{\left(a_{D\pi }\right)}_{-0.11}^{+0.12}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^0\to {\overline{D}}_0^{\ast 0}{K}^0\to D^{-}{\pi }^{+}{K}^0$	(${10}^{-6}$)	${4.15}_{-1.09}^{+1.54}{\left({\omega }_B\right)}_{-0.72}^{+0.74}{\left({\omega }_{D\pi }\right)}_{-0.03}^{+0.03}{\left(a_{D\pi }\right)}_{-0.14}^{+0.19}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^0\to D_0^{\ast 0}\eta \to D^{+}{\pi }^{-}\eta$	(${10}^{-9}$)	${2.81}_{-0.58}^{+0.78}{\left({\omega }_B\right)}_{-0.33}^{+0.30}{\left({\omega }_{D\pi }\right)}_{-0.14}^{+0.11}{\left(a_{D\pi }\right)}_{-0.09}^{+0.13}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^0\to D_0^{\ast 0}{\eta }^{\prime }\to D^{+}{\pi }^{-}{\eta }^{\prime }$	(${10}^{-9}$)	${1.80}_{-0.37}^{+0.49}{\left({\omega }_B\right)}_{-0.21}^{+0.19}{\left({\omega }_{D\pi }\right)}_{-0.09}^{+0.07}{\left(a_{D\pi }\right)}_{-0.06}^{+0.08}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^0\to {\overline{D}}_0^{\ast 0}\eta \to D^{-}{\pi }^{+}\eta$	(${10}^{-5}$)	${1.79}_{-0.41}^{+0.60}{\left({\omega }_B\right)}_{-0.28}^{+0.30}{\left({\omega }_{D\pi }\right)}_{-0.03}^{+0.07}{\left(a_{D\pi }\right)}_{-0.06}^{+0.09}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-
$B^0\to {\overline{D}}_0^{\ast 0}{\eta }^{\prime }\to D^{-}{\pi }^{+}{\eta }^{\prime }$	(${10}^{-5}$)	${1.15}_{-0.27}^{+0.38}{\left({\omega }_B\right)}_{-0.18}^{+0.19}{\left({\omega }_{D\pi }\right)}_{-0.02}^{+0.04}{\left(a_{D\pi }\right)}_{-0.04}^{+0.06}\left({\Gamma }_{D_0^{\ast 0}}\right)$	-

Table 3. Table 3: PQCD prediction of branching fraction for the quasi-two-body decays B s 0 → D 0 ∗ ​ P → D ​ π ​ P → subscript superscript 𝐵 0 𝑠 superscript subscript 𝐷 0 𝑃 → 𝐷 𝜋 𝑃 B^{0}_{s}\to D_{0}^{*}P\to D\pi P .

Mode	Unit	$ℬ$
$B_s^0\to D_0^{\ast -}{\pi }^{+}\to {\overline{D}}^0{\pi }^{-}{\pi }^{+}$	(${10}^{-7}$)	${2.70}_{-0.36}^{+0.29}{\left({\omega }_B\right)}_{-0.58}^{+0.60}{\left({\omega }_{D\pi }\right)}_{-0.31}^{+0.43}{\left(a_{D\pi }\right)}_{-0.01}^{+0.06}\left({\Gamma }_{D_0^{\ast +}}\right)$
$B_s^0\to D_0^{\ast +}{\pi }^{-}\to D^0{\pi }^{+}{\pi }^{-}$	(${10}^{-9}$)	${2.90}_{-0.15}^{+0.08}{\left({\omega }_B\right)}_{-0.83}^{+0.95}{\left({\omega }_{D\pi }\right)}_{-0.23}^{+0.26}{\left(a_{D\pi }\right)}_{-0.06}^{+0.07}\left({\Gamma }_{D_0^{\ast +}}\right)$
$B_s^0\to D_0^{\ast +}{K}^{-}\to D^0{\pi }^{+}{K}^{-}$	(${10}^{-7}$)	${2.82}_{-0.74}^{+1.09}{\left({\omega }_B\right)}_{-0.01}^{+0.02}{\left({\omega }_{D\pi }\right)}_{-0.00}^{+0.01}{\left(a_{D\pi }\right)}_{-0.04}^{+0.06}\left({\Gamma }_{D_0^{\ast +}}\right)$
$B_s^0\to D_0^{\ast 0}{\pi }^0\to D^{+}{\pi }^{-}{\pi }^0$	(${10}^{-9}$)	${1.48}_{-0.04}^{+0.03}{\left({\omega }_B\right)}_{-0.42}^{+0.46}{\left({\omega }_{D\pi }\right)}_{-0.13}^{+0.12}{\left(a_{D\pi }\right)}_{-0.07}^{+0.08}\left({\Gamma }_{D_0^{\ast 0}}\right)$
$B_s^0\to {\overline{D}}_0^{\ast 0}{\pi }^0\to D^{-}{\pi }^{+}{\pi }^0$	(${10}^{-7}$)	${1.38}_{-0.19}^{+0.24}{\left({\omega }_B\right)}_{-0.33}^{+0.48}{\left({\omega }_{D\pi }\right)}_{-0.16}^{+0.22}{\left(a_{D\pi }\right)}_{-0.04}^{+0.07}\left({\Gamma }_{D_0^{\ast 0}}\right)$
$B_s^0\to D_0^{\ast 0}{\overline{K}}^0\to D^{+}{\pi }^{-}{\overline{K}}^0$	(${10}^{-9}$)	${9.09}_{-2.38}^{+3.65}{\left({\omega }_B\right)}_{-0.95}^{+0.84}{\left({\omega }_{D\pi }\right)}_{-0.23}^{+0.38}{\left(a_{D\pi }\right)}_{-0.31}^{+0.41}\left({\Gamma }_{D_0^{\ast 0}}\right)$
$B_s^0\to {\overline{D}}_0^{\ast 0}{\overline{K}}^0\to D^{-}{\pi }^{+}{\overline{K}}^0$	(${10}^{-5}$)	${4.70}_{-1.39}^{+2.05}{\left({\omega }_B\right)}_{-0.75}^{+0.76}{\left({\omega }_{D\pi }\right)}_{-0.05}^{+0.04}{\left(a_{D\pi }\right)}_{-0.16}^{+0.21}\left({\Gamma }_{D_0^{\ast 0}}\right)$
$B_s^0\to D_0^{\ast 0}\eta \to D^{+}{\pi }^{-}\eta$	(${10}^{-8}$)	${9.37}_{-2.70}^{+4.31}{\left({\omega }_B\right)}_{-0.77}^{+0.67}{\left({\omega }_{D\pi }\right)}_{-0.15}^{+0.21}{\left(a_{D\pi }\right)}_{-0.30}^{+0.43}\left({\Gamma }_{D_0^{\ast 0}}\right)$
$B_s^0\to D_0^{\ast 0}{\eta }^{\prime }\to D^{+}{\pi }^{-}{\eta }^{\prime }$	(${10}^{-7}$)	${1.62}_{-0.43}^{+0.65}{\left({\omega }_B\right)}_{-0.15}^{+0.16}{\left({\omega }_{D\pi }\right)}_{-0.05}^{+0.06}{\left(a_{D\pi }\right)}_{-0.05}^{+0.09}\left({\Gamma }_{D_0^{\ast 0}}\right)$
$B_s^0\to {\overline{D}}_0^{\ast 0}\eta \to D^{-}{\pi }^{+}\eta$	(${10}^{-6}$)	${1.27}_{-0.39}^{+0.55}{\left({\omega }_B\right)}_{-0.20}^{+0.18}{\left({\omega }_{D\pi }\right)}_{-0.03}^{+0.04}{\left(a_{D\pi }\right)}_{-0.04}^{+0.05}\left({\Gamma }_{D_0^{\ast 0}}\right)$
$B_s^0\to {\overline{D}}_0^{\ast 0}{\eta }^{\prime }\to D^{-}{\pi }^{+}{\eta }^{\prime }$	(${10}^{-6}$)	${2.24}_{-0.64}^{+0.93}{\left({\omega }_B\right)}_{-0.30}^{+0.29}{\left({\omega }_{D\pi }\right)}_{-0.04}^{+0.02}{\left(a_{D\pi }\right)}_{-0.08}^{+0.11}\left({\Gamma }_{D_0^{\ast 0}}\right)$

Equations

$$\mathcal{A}={\varphi }_B\otimes H\otimes {\varphi }_P\otimes {\varphi }_{D\pi }^{\text{S-wave}},$$

$$\displaystyle\left(\begin{array}[]{c}|\eta\rangle\\ |\eta^{\prime}\rangle\end{array}\right)=\left(\begin{array}[]{cc}\text{cos}\phi&-\text{sin}\phi\\ \text{sin}\phi&\text{cos}\phi\end{array}\right)\left(\begin{array}[]{c}|\eta_{q}\rangle\\ |\eta_{s}\rangle\end{array}\right)\;,$$

$${\Phi }_P(p,z)=\frac{i}{\sqrt{2N_c}}{\gamma }_5\left[p/{\varphi }^A(z)+m_0{\varphi }^P(z)+m_0(v/n/-1){\varphi }^T(z)\right],$$

$${\varphi }^A(z)$$

$${\varphi }^P(z)$$

$${\varphi }^T(z)$$

$$C_1^{\frac{3}{2}}(t)=3t,C_2^{\frac{1}{2}}(t)=\frac{1}{2}(3t^2-1),C_2^{\frac{3}{2}}(t)=\frac{3}{2}(5t^2-1),$$

$$C_4^{\frac{1}{2}}(t)=\frac{1}{8}(3-30t^2+35t^4),C_4^{\frac{3}{2}}(t)=\frac{15}{8}(1-14t^2+21t^4),$$

$$m_{B^{\pm ,0}}$$

$$m_{D_0^{\ast 0}}$$

$$m_{{\pi }^{\pm }}$$

$$f_{\pi }$$

$$\displaystyle R_{CKM}=\bigg{|}\frac{V_{ub}^{*}V_{cd}}{V_{cb}^{*}V_{ud}}\bigg{|}^{2}\approx\lambda^{4}(\bar{\rho}^{2}+\bar{\eta}^{2})\approx 3\times 10^{-4}\;.$$

$$R_1$$

$$R_2$$

$$R_3$$

$$R_4$$

$$R_5$$

$$R_{{\bar{D}}_0^{\ast 0}}$$

$${\left|\frac{V_{us}}{V_{ud}}\right|}^2\cdot \frac{f_K^2}{f_{\pi }^2}=0.076$$

$$\frac{\mathcal{B}(B_s^0\to D_0^{\ast +}{K}^{-}\to D^0{\pi }^{+}{K}^{-})}{\mathcal{B}(B^0\to D_0^{\ast +}{\pi }^{-}\to D^0{\pi }^{+}{\pi }^{-})}=1.10_{-0.02}^{+0.05}.$$

$$\frac{\mathcal{B}(B_s^0\to {\bar{D}}_0^{\ast 0}{\bar{K}}^0\to D^{-}{\pi }^{+}{\bar{K}}^0)}{2\mathcal{B}(B^0\to {\bar{D}}_0^{\ast 0}{\pi }^0\to D^{-}{\pi }^{+}{\pi }^0)}=1.03_{-0.07}^{+0.06}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to\pi^{+}[D_{0}^{*0}\to]D^{+}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to\pi^{+}[\bar{D}_{0}^{*0}\to]D^{-}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to K^{+}[D_{0}^{*0}\to]D^{+}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to K^{+}[\bar{D}_{0}^{*0}\to]D^{-}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to\pi^{0}[D_{0}^{*+}\to]D^{0}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to K^{0}[D_{0}^{*+}\to]D^{0}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to\eta_{q}[D_{0}^{*+}\to]D^{0}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to\eta[D_{0}^{*+}\to]D^{0}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{+}\to\eta^{\prime}[D_{0}^{*+}\to]D^{0}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\pi^{+}[D_{0}^{*-}\to]\bar{D}^{0}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\pi^{-}[D_{0}^{*+}\to]D^{0}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to K^{+}[D_{0}^{*-}\to]\bar{D}^{0}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\pi^{0}[D_{0}^{*0}\to]D^{+}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\pi^{0}[\bar{D}_{0}^{*0}\to]D^{-}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to K^{0}[D_{0}^{*0}\to]D^{+}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to K^{0}[\bar{D}_{0}^{*0}\to]D^{-}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\eta_{q}[D_{0}^{*0}\to]D^{+}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\eta[D_{0}^{*0}\to]D^{+}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\eta^{\prime}[D_{0}^{*0}\to]D^{+}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\eta_{q}[\bar{D}_{0}^{*0}\to]D^{-}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\eta[\bar{D}_{0}^{*0}\to]D^{-}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B^{0}\to\eta^{\prime}[\bar{D}_{0}^{*0}\to]D^{-}\pi^{+}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B_{s}^{0}\to\pi^{+}[D_{0}^{*-}\to]\bar{D}^{0}\pi^{-}\big{)}$$

$$\displaystyle{\mathcal{A}}\big{(}B_{s}^{0}\to\pi^{-}[D_{0}^{*+}\to]D^{0}\pi^{+}\big{)}$$

Full text

I INTRODUCTION

The strong dynamics contained in the three-body hadronic $B$ meson decays is much more complicated than that in the two-body cases. There are resonant and nonresonant contributions, final-state interactions Bediaga:2015mia ; Bediaga:2017axw , and complex interplay between the weak processes and the low-energy strong interactions Charles:2017ptc in the three-body $B$ meson decays. The traditional approaches for the two-body decays are no longer satisfactory in the three-body processes Amato:2016xjv . In order to extract the most information from the experimental data of those three-body processes, different methods have been adopted abundantly in theoretical works Wang:2018dfq . Three-body hadronic $B$ decays are known, in most cases, to be dominated by the low-energy scalar, vector, and tensor resonant states. In this situation, for the numerous three-body $B$ meson processes, it is urgent to study the resonance contributions, which could be handled in the quasi-two-body framework where the factorization procedure can be applied Amato:2016xjv ; Boito:2017jav .

The $p$-wave orbitally excited state $D_{0}^{*}$ 111For the sake of convenience, we employ $D_{0}^{*}$ to denote $D_{0}^{*}(2400)$ in this work., with its $j_{q}=1/2$ Datta:2003re ; Godfrey:2005ww ; Godfrey:2015dva and $J^{P}=0^{+}$ Tanabashi:2018oca , decays rapidly through $S$-wave pion emission. It was thought to be the $c\bar{q}$ state in the traditional quark model Godfrey:1985xj ; Godfrey:1986wj ; DiPierro:2001dwf , but the mass observed in experiments Abe:2003zm ; Aubert:2009wg is lower than the quark model predictions. One possible explanation is that the self-energy hadronic loop could pull down the mass of the heavy scalar Guo:2007up supported by Cheng:2017oqh within the framework of heavy meson chiral perturbation theory. The tetraquark structure for $D_{0}^{*}$ was investigated in Bracco:2005kt with the help of the QCD sum rule, and the authors of Bracco:2005kt suggested that the charmed scalar meson $D_{0}^{*0}(2308)$ observed by the Belle Collaboration Abe:2003zm and $D_{0}^{*0(+)}(2405)$ observed by the FOCUS Collaboration Link:2003bd are different resonances. It was claimed that two poles exist in the $D_{0}^{*}$ energy region Albaladejo:2016lbb , which has been supported by the lattice QCD analysis Moir:2016srx . The resonant state $D_{0}^{*}$ has also been explained as a mixture of $c\bar{q}$ and tetraquarks Vijande:2006hj or a meson-meson bound state Gamermann:2006nm . Since the Belle Collaboration’s announcement Abe:2003zm , much work Cheng:2003sm ; Jugeau:2005yr ; Cheng:2003id ; Cheng:2006dm ; Chen:2003rt has emerged for the two-body hadronic $B$ decays involving $D_{0}^{*}$.

By studying the three-body hadronic $B$ meson decays involving $D_{0}^{*}$, one could provide the constraint on the unitary triangle Aaij:2016bqv ; Gershon:2009qc ; Craik:2017dpc ; Bondar:2018gpb and probe the inner structure of the intermediate resonances. In Ref Wang:2018fai , four quasi-two-body decay processes involving $D_{0}^{*}$ have been studied in the perturbative QCD (PQCD) approach Keum:2000ph ; Keum:2000wi ; Lu:2000em ; Li:2003yj . In this work, we extend the study to the quasi-two-body decays $B_{(s)}\to PD_{0}^{*}\to PD\pi$, with the bachelor particle $P$ which denotes the light pseudoscalar $\pi$, $K$, $\eta$, or $\eta^{\prime}$. Typical diagrams for the $B_{(s)}\to PD^{*}_{0}\to PD\pi$ decays’ processes are shown in Fig. 1. Inspired by the generalized parton distribution in hard exclusive two pion production Diehl:1998dk ; Mueller:1998fv ; Polyakov:1998ze ; Hagler:2002nh , the two-meson distribution amplitude was introduced in three-body hadronic $B$ decays in Chen:2002th ; Chen:2004az as the universal nonperturbative input within the PQCD approach. The PQCD approach has been employed in Chen:2002th ; Chen:2004az ; Wang:2014ira ; Wang:2015uea ; Wang:2017hao for the three-body and in Wang:2016rlo ; Li:2016tpn ; Li:2017mao ; Li:2018qrm ; Ma:2016csn ; Ma:2017kec for the quasi-two-body $B$ meson decays. The decay amplitude for a three-body or quasi-two-body $B$ decay can be expressed as the convolution of the nonperturbative wave function and hard kernel Chen:2002th ; Chen:2004az ; Wang:2016rlo . Taking $B\to PD_{0}^{*}\to PD\pi$ as an example, we have the decay amplitude

[TABLE]

where hard kernel $H$ is calculated at leading order which contains one hard gluon, and the distribution amplitudes $\phi_{B},\phi_{P}$ and $\phi_{D\pi}^{\text{S-wave}}$ absorb the nonperturbative dynamics in the decay processes.

The layout of this paper is as follows. We give a brief introduction of the theoretical framework in Sec. II. Then the numerical results, a discussion and conclusions are given in Sec. III and IV. The relevant factorization formulas for the decay amplitudes are collected in the Appendix.

II FRAMEWORK

The definitions of the momenta for the $B_{(s)}$ meson, $S$-wave $D\pi$ system, and the bachelor meson are the same as those in Ref. Wang:2018fai . The distribution amplitude and the parameters for the $S$-wave $D\pi$ system employed in this work as the same as those in Wang:2018fai . The wave functions for $B_{(s)}$ and the relevant parameters can be found in Wang:2012ab . The decay constants $f_{B^{0,\pm}}=0.190$ GeV for $B^{0,\pm}$ and $f_{B_{s}^{0}}=0.230$ GeV for $B_{s}^{0}$ were adopted from recent lattice QCD updated results with $N_{f}=2+1+1$ Aoki:2019cca . The physical states $\eta$ and $\eta^{\prime}$ are related to the flavor states $\eta_{q}$ and $\eta_{s}$ via Thomas:2007uy ; Feldmann:1998vh ; Feldmann:1998sh

[TABLE]

with the decay constants $f_{q}=(1.07\pm 0.02)f_{\pi}$ and $f_{s}=(1.34\pm 0.06)f_{\pi}$ for $\eta_{q}$ and $\eta_{s}$, respectively, and the mixing angle $\phi=39.3^{\circ}\pm 1.0^{\circ}$, which is close to the recent measurement $\phi=(40.1\pm 1.4_{\mathrm{stat}}\pm 0.5_{\mathrm{syst}})^{\circ}$ by the BESIII Collaboration Ablikim:2019rjz . The wave functions for the states $\pi,K,\eta_{q}$ and $\eta_{s}$ in this work are written as

[TABLE]

where $m_{0}$ is the chiral mass, $n=(1,0,\mathbf{0}_{T})$ and $v=(0,1,\mathbf{0}_{T})$ are the dimensionless lightlike unit vectors, $p$ and $z$ are, respectively, the momentum and corresponding momentum fraction of states $\pi,K,\eta_{q}$, and $\eta_{s}$. The distribution amplitudes $\phi^{A}(z),\phi^{P}(z),\phi^{T}(z)$ can be written as Ball:1998tj ; Ball:1998je ; Ball:2004ye ; Ball:2006wn

[TABLE]

where the Gegenbauer moments are $a_{1}^{\pi,\eta_{q,s}}=0,\ a_{1}^{K}=0.06,\ a_{2}^{\pi,K}=0.25,\ a_{2}^{\eta_{q,s}}=0.115,\ a_{4}^{\pi,\eta_{q,s}}=-0.015$, and the parameters are $\rho_{\pi}=m_{\pi}/m_{0}^{\pi},\ \rho_{K}=m_{K}/m_{0}^{K},\ \rho_{\eta_{q}}=2m_{q}/m_{qq},\ \rho_{\eta_{s}}=2m_{s}/m_{ss}$, $\eta_{3}=0.015,\ \omega_{3}=-3$. Where $m_{q}$ is the mass of the up or down quark, $m_{s}$ is the mass of the strange quark, $m_{qq,ss}$ are related to $m_{0}^{\eta_{q},\eta_{s}}$ by $m_{0}^{\eta_{q}}=m^{2}_{qq}/(m_{u}+m_{d})$ and $m_{0}^{\eta_{s}}=m^{2}_{ss}/2m_{s}$, respectively. We adopt $m_{0}^{\pi}=(1.4\pm 0.1)$ GeV, $m_{0}^{K}=(1.6\pm 0.1)$ GeV, $m_{0}^{\eta_{q}}=1.07$ GeV, and $m_{0}^{\eta_{s}}=1.92$ GeV in the numerical calculation. The Gegenbauer polynomials are defined as

[TABLE]

where the variable $t=2z-1$.

III RESULTS

For the numerical calculations, we adopt from Tanabashi:2018oca the masses and mean lifetimes for the $B^{0,\pm}$ and $B_{s}^{0}$ mesons, the pole masses and width for $D_{0}^{*0,\pm}$, the masses and decay constants for the light pseudoscalar mesons pion and kaon, and the Wolfenstein parameters as:

[TABLE]

where the masses, decay constants and widths are in units of GeV and lifetimes in units of $ps$.

By using the decay amplitudes for the decays $B_{(s)}\to PD_{0}^{*}\to PD\pi$ in the Appendix and the differential branching fraction ($\mathcal{B}$), Eq. (13) in Wang:2018fai , we obtain the branching fractions for the decays involving $B^{+}$ in Table LABEL:tableB+, the results for the processes including $B^{0}$ in Table LABEL:tableB0, and the values for the $B_{s}^{0}$ decay modes in Table LABEL:tableBs with the existing data from Abe:2003zm ; Aubert:2009wg ; Aaij:2016fma ; Aaij:2015vea ; Kuzmin:2006mw ; Aaij:2015sqa ; Aaij:2015kqa . The first error of these results in Tables LABEL:tableB0-LABEL:tableBs comes from the shape parameters $\omega_{B^{0,\pm}}=0.40\pm 0.04$ GeV for $B^{0,\pm}$ and $\omega_{B_{s}^{0}}=0.5\pm 0.05$ GeV for $B_{s}^{0}$ Wang:2012ab . The second error comes from the shape parameter $\omega_{D\pi}=0.40\pm 0.10$ GeV for the $D\pi$ system, and the Gegenbauer moment $a_{D\pi}=0.40\pm 0.10$ produces the third one Wang:2018fai . The last one comes from the uncertainty of decay width $\Gamma_{D_{0}^{*0}}=267\pm 40$ MeV or $\Gamma_{D_{0}^{*+}}=230\pm 17$ MeV Tanabashi:2018oca . We have neglected the errors induced by the uncertainties of the parameters in the distribution amplitudes of the light pseudoscalar mesons and the Wolfenstein parameters since they are very small.

The four quasi-two-body decays $B^{+}\to\bar{D}_{0}^{*0}\pi^{+}\to D^{-}\pi^{+}\pi^{+}$, $B^{+}\to\bar{D}_{0}^{*0}K^{+}\to D^{-}\pi^{+}K^{+}$, $B^{0}\to D_{0}^{*-}\pi^{+}\to\bar{D}^{0}\pi^{-}\pi^{+}$ and $B^{0}\to D_{0}^{*-}K^{+}\to\bar{D}^{0}\pi^{-}K^{+}$ have been discussed in Ref. Wang:2018fai . For completeness, we keep their branching ratios in Tables LABEL:tableB+ and LABEL:tableB0. In Fig. 2, we show the $D\pi$ invariant mass-dependent differential branching fraction for the quasi-two-body decay $B^{0}\to\bar{D}_{0}^{*0}\pi^{0}\to D^{-}\pi^{+}\pi^{0}$. One can find that the main portion of branching fraction for $B^{0}\to\bar{D}_{0}^{*0}\pi^{0}\to D^{-}\pi^{+}\pi^{0}$ comes from the region around the pole mass of the resonant state $D_{0}^{*}$. The contributions from the $m_{D\pi}$ mass region larger than $3$ GeV can be neglected safely as argued in Ref. Wang:2018fai .

For the CKM suppressed decay modes $B\to D_{0}^{*}\pi\to D\pi\pi$ and $B_{s}\to D_{0}^{*}\bar{K}\to D\pi\bar{K}$, their branching ratios are much smaller than the corresponding results of $B\to\bar{D}_{0}^{*}\pi\to D\pi\pi$ and $B_{s}\to\bar{D}_{0}^{*}\bar{K}\to D\pi\bar{K}$ decays as predicted by PQCD in this work. The major reason comes from the strong CKM suppression factor Ma:2016csn

[TABLE]

For the CKM suppressed and CKM favored decay modes concerned in this work, we define the following ratios of the branching fractions for the the corresponding decays as

[TABLE]

The ratios $R_{1},R_{2},R_{3}$, and $R_{4}$ are close to each other, because all four decay pairs in these four ratios decay through the same colour suppressed emission topologies, and the nonfactorizable diagrams in Fig. 1 play the dominant role. The nonvanishing charm quark mass in the fermion propagator generates the main differences between the $R_{CKM}$ and $R_{1,2,3,4}$. For the decay process $B^{+}\to\bar{D}_{0}^{*0}\pi^{+}\to D^{-}\pi^{+}\pi^{+}$, one has the contributions from both the $B\to\bar{D}_{0}^{*0}$ transition and the $B\to\pi$ transition, while for $B^{+}\to D_{0}^{*0}\pi^{+}\to D^{+}\pi^{-}\pi^{+}$, one has only the colour suppressed transition $B\to\pi$. So it is not surprising to have a quite small value for $R_{5}$.

Assuming factorization and flavour-$SU$(3) symmetry, the ratio between the two decays $B^{0}\to D_{0}^{*-}K^{+}\to\bar{D}^{0}\pi^{-}K^{+}$ and $B^{0}\to D_{0}^{*-}\pi^{+}\to\bar{D}^{0}\pi^{-}\pi^{+}$ will not very far from 0.076, as discussed in Ref. Wang:2018fai . The same situation should happen to the decays $B^{0}\to\bar{D}_{0}^{*0}K^{0}\to D^{-}\pi^{+}K^{0}$ and $B^{0}\to\bar{D}_{0}^{*0}\pi^{0}\to D^{-}\pi^{+}\pi^{0}$. With the PQCD predictions in Table LABEL:tableB0, we have

[TABLE]

The deviation between the $R_{\bar{D}_{0}^{*0}}$ and

[TABLE]

could be due to the violation of the flavour-$SU$(3) symmetry and the contributions from annihilation diagrams in the $B^{0}\to\bar{D}_{0}^{*0}\pi^{0}\to D^{-}\pi^{+}\pi^{0}$ process.

The ratio of branching fractions with topologically similar decay processes $B_{s}^{0}\to D_{0}^{*+}K^{-}\to D^{0}\pi^{+}K^{-}$ and $B^{0}\to D_{0}^{*+}\pi^{-}\to D^{0}\pi^{+}\pi^{-}$ is expected to be close to 1 in the naïve factorization because of the close values for the $B_{s}^{0}\to K^{-}$ and $B^{0}\to\pi^{-}$ transition form factors Wang:2012ab . With the predictions in Tables LABEL:tableB0 and LABEL:tableBs, we have

[TABLE]

A similar relation for $B_{s}^{0}\to\bar{D}_{0}^{*0}\bar{K}^{0}\to D^{-}\pi^{+}\bar{K}^{0}$ and $B^{0}\to\bar{D}_{0}^{*0}\pi^{0}\to D^{-}\pi^{+}\pi^{0}$ is

[TABLE]

induced from Tables LABEL:tableB0 and LABEL:tableBs.

IV CONCLUSION

We have studied the quasi-two-body decays $B_{(s)}\to PD_{0}^{*}\to PD\pi$, where the bachelor particle $P$ denotes $\pi$, $K$, $\eta$, or $\eta^{\prime}$ in the PQCD approach. The predicted branching fractions for the considered decays are in the range of $10^{-9}-10^{-4}$. For the decays $B\to D_{0}^{*}\pi\to D\pi\pi$ and $B\to\bar{D}_{0}^{*}\pi\to D\pi\pi$ as well as $B_{s}\to D_{0}^{*}\to D\pi\bar{K}$ and $B_{s}\to\bar{D}_{0}^{*}\bar{K}\to D\pi\bar{K}$, the great difference in their corresponding branching fractions can be understood by a strong CKM suppression factor $R_{CKM}\approx\lambda^{4}(\bar{\rho}^{2}+\bar{\eta}^{2})\approx 3\times 10^{-4}$. The flavour-$SU$(3) symmetry can be employed to analyse the quasi-two-body decays with the same topologies, such as $B^{0}\to\bar{D}_{0}^{*0}K^{0}\to D^{-}\pi^{+}K^{0}$ and $B^{0}\to\bar{D}_{0}^{*0}\pi^{0}\to D^{-}\pi^{+}\pi^{0}$, while $R_{\bar{D}_{0}^{*0}}$ was predicted to be $0.091^{+0.003}_{-0.005}$ for their branching ratios. The ratio for the branching fractions was found to be $1.10^{+0.05}_{-0.02}$ between $\mathcal{B}(B_{s}^{0}\to D_{0}^{*+}K^{-}\to D^{0}\pi^{+}K^{-})$ and $\mathcal{B}(B^{0}\to D_{0}^{*+}\pi^{-}\to D^{0}\pi^{+}\pi^{-})$ and to be $1.03^{+0.06}_{-0.07}$ between $\mathcal{B}(B_{s}^{0}\to\bar{D}_{0}^{*0}\bar{K}^{0}\to D^{-}\pi^{+}\bar{K}^{0})$ and $2\mathcal{B}(B^{0}\to\bar{D}_{0}^{*0}\pi^{0}\to D^{-}\pi^{+}\pi^{0})$, which can be tested by the precise data from the future experiments.

Acknowledgements.

We are grateful to Muhammad Waqas for helpful comments. This work is supported in part by the National Natural Science Foundation of China under Grants No. 11547038, No. 11505148, and No. 11575103. *

Appendix A Decay Amplitudes

The amplitudes from Fig. 1 are written as

[TABLE]

[TABLE]

where $G_{F}$ is the Fermi constant, $V$’s are the CKM matrix elements, $C_{1}$ and $C_{2}$ are Wilson coefficients and $a_{1}=C_{1}/3+C_{2}$ and $a_{2}=C_{2}/{3}+C_{1}$. The factorization formulas for decay amplitudes from Fig. 1 are collected below:

[TABLE]

where $x_{B}$, $x_{3}$, and $z$ are momentum fractions of the corresponding spectator quarks, as defined in Ref. Wang:2018fai . $b_{B}$, $b_{3}$, and $b$ are the conjugate variables of transverse momenta $P_{B}$, $P_{3}$, and $P$, respectively. Variable $\eta$ is defined as $\eta=m^{2}_{D\pi}/m^{2}_{B}$. The ratio $r_{0}=m_{0}/m_{B}$, where $m_{0}$ is the chiral mass of light pseudoscalars. $r_{c}=m_{c}/m_{B}$ is the ratio of the charm quark mass to the $B$ meson mass. The functions $E_{1mn}$ and $E_{2mn}$($m=a,c,e,g$ and $n=b,d,f,h$) are the evolution factors, which are given by

[TABLE]

in which Sudakov exponents $S_{(B,D,P)}(t)$ are defined as

[TABLE]

where the quark anomalous dimension $\gamma_{q}=-\alpha_{s}/\pi$. The explicit form for $s(Q,b)$ at one loop can be found in Ali:2007ff . $t_{1x}$ and $t_{2x}$($x=a,b\cdots h$) are hard scales which are chosen to be the maximum of the virtuality of the internal momentum transition in the hard amplitudes as

[TABLE]

The hard functions can be written as

[TABLE]

[TABLE]

where $K_{0}$, $I_{0}$, and $H_{0}=J_{0}+iY_{0}$ are Bessel functions. The function $S_{t}(x)$ can be parametrized as

[TABLE]

with $c=0.4$ for numerical calculation Kurimoto:2001zj ; Li:2009pr .