Four-body baryonic decays of $B\to p \bar{p} \pi^+\pi^-(\pi^+K^-)$ and $\Lambda \bar{p} \pi^+\pi^-(K^+K^-)$

TL;DR

This paper analyzes four-body baryonic B decays, providing theoretical predictions for branching ratios that align with recent experimental observations and suggesting new decay modes accessible to current experiments.

Contribution

It offers a theoretical framework for baryonic B decays involving mesons and predicts branching ratios for various decay channels, extending understanding of baryonic decay mechanisms.

Findings

Predicted branching ratios match experimental data.

Identified new decay modes accessible to LHCb and Belle.

Provided theoretical insights into baryonic decay processes.

Abstract

We study the four-body baryonic $B\to {\bf B_1 \bar B_2}M_1 M_2$ decays with $\bf B_{1,2}$ ($M_{1,2}$) being charmless baryons (mesons). In accordance with the recent LHCb observations, each decay is considered to proceed through the $B\to M_1 M_2$ transition together with the production of a baryon pair. We obtain that ${\cal B}(B^-\to \Lambda\bar p \pi^+\pi^-)=(3.7^{+1.5}_{-1.0} )\times 10^{-6}$ and ${\cal B}(\bar B^0\to p\bar p \pi^+\pi^-,p\bar p \pi^+ K^-)=(3.0\pm 0.9,6.6\pm 2.4)\times 10^{-6}$, in agreement with the data. We also predict ${\cal B}(B^-\to\Lambda\bar p K^+ K^-)=(3.0^{+1.3}_{-0.9})\times 10^{-6}$, which is accessible to the LHCb and BELLE experiments.

Tables

Table 1. Table 1: The parameters a i subscript 𝑎 𝑖 a_{i} with N c e ​ f ​ f = 2 ,  3 superscript subscript 𝑁 𝑐 𝑒 𝑓 𝑓 2 3 N_{c}^{eff}=2,\,3 , and ∞ \infty to estimate the non-factorizable effects in the generalized factorization.

	($b\to d$ transition)			($b\to s$ transition)
$a_i$	$N_c^{eff}=2$	$3$	$\infty$	$N_c^{eff}=2$	$3$	$\infty$
$a_1$	—–	—–	—–	$0.98$	$1.05$	$1.17$
$a_2$	$0.22$	$0.02$	$-0.37$	$0.22$	$0.02$	$-0.37$
${10}^4{a}_3$	$-10.4-6.9i$	$72.4$	$237.9+13.9i$	$-13.1-15.6i$	$72.4$	$243.2+31.2i$
${10}^4{a}_4$	$-377.6-34.7i$	$-417.2-37.0i$	$-496.5-41.6i$	$-391.0-77.9i$	$-431.6-83.1i$	$-512.6-93.5i$
${10}^4{a}_5$	$-171.4-6.9i$	$-65.8$	$145.3+13.9i$	$-174.1-15.6i$	$-65.8$	$150.7+31.2i$
${10}^4{a}_6$	$-560.7-34.7i$	$-584.9-37.0i$	$-633.4-41.6i$	$-574.1-77.9i$	$-599.3-83.1i$	$-649.5-93.5i$
${10}^4{a}_9$	$-93.3-1.4i$	$-99.5-1.4i$	$-112.0-1.4i$	$-93.5-2.2i$	$-99.8-2.2i$	$-112.3-2.2i$
${10}^4{a}_{10}$	$-18.5-0.7i$	$0.18-0.46i$	$37.5$	—–	—–	—–

Table 2. Table 2: The branching ratios of B → 𝐁 𝟏 ​ 𝐁 ¯ 𝟐 ​ M 1 ​ M 2 → 𝐵 subscript 𝐁 1 subscript ¯ 𝐁 2 subscript 𝑀 1 subscript 𝑀 2 B\to{\bf B_{1}\bar{B}_{2}}M_{1}M_{2} , where the errors come from the non-factorizable effects, CKM matrix elements, and form factors, respectively.

branching ratios	our results	data
${10}^6ℬ\left(B^{-}\to \Lambda \overline{p}{\pi }^{+}{\pi }^{-}\right)$	${3.7}_{-0.5}^{+1.2}\pm 0.1\pm 0.9$	$5.9\pm 1.1$
${10}^6ℬ\left(B^{-}\to \Lambda \overline{p}{K}^{+}{K}^{-}\right)$	${3.0}_{-0.5}^{+1.1}\pm 0.1\pm 0.7$	—–
${10}^6ℬ\left({\overline{B}}^0\to p\overline{p}{\pi }^{+}{\pi }^{-}\right)$	${3.0}_{-0.3}^{+0.5}\pm 0.3\pm 0.7$	$3.0\pm 0.3$
${10}^6ℬ\left({\overline{B}}^0\to p\overline{p}{\pi }^{\pm }{K}^{\mp }\right)$	$6.6\pm 0.5\pm 0.0\pm 2.3$	$6.6\pm 0.5$

Equations

$$\mathcal{B}({\bar{B}}^0\to p\bar{p}{\pi }^{+}{\pi }^{-})$$

$$\mathcal{B}({\bar{B}}^0\to p\bar{p}{K}^{\mp }{\pi }^{\pm })$$

$$\mathcal{B}(B^{-}\to \Lambda \bar{p}{\pi }^{+}{\pi }^{-})$$

$$\displaystyle{\cal A}_{1}(\bar{B}^{0}_{(s)}\to p\bar{p}M_{1}M_{2})=\frac{G_{F}}{\sqrt{2}}\bigg{\{}\bigg{[}\langle p\bar{p}|\alpha_{+}^{q}(\bar{u}u)_{V}-\alpha_{-}^{q}(\bar{u}u)_{A}|0\rangle+\langle p\bar{p}|\beta_{+}^{q}(\bar{d}d)_{V}-\beta_{-}^{q}(\bar{d}d)_{A}|0\rangle$$

$$\displaystyle+(\alpha_{4}^{q}-\alpha_{10}^{q}/2)\langle p\bar{p}|(\bar{q}q)_{V-A}|0\rangle\bigg{]}\langle M_{1}M_{2}|(\bar{q}b)_{V-A}|\bar{B}^{0}_{(s)}\rangle$$

$$\displaystyle+\alpha_{6}^{q}\langle p\bar{p}|(\bar{q}q)_{S+P}|0\rangle\langle M_{1}M_{2}|(\bar{q}b)_{S-P}|\bar{B}^{0}_{(s)}\rangle\bigg{\}}\,,$$

$$\displaystyle{\cal A}_{2}(B^{-}\to\Lambda\bar{p}M_{1}M_{2})=\frac{G_{F}}{\sqrt{2}}\bigg{\{}(\alpha_{1}^{s}+\alpha_{4}^{s})\langle\Lambda\bar{p}|(\bar{s}u)_{V-A}|0\rangle\langle M_{1}M_{2}|(\bar{u}b)_{V-A}|B^{-}\rangle$$

$$\displaystyle+\alpha_{6}^{s}\langle\Lambda\bar{p}|(\bar{s}u)_{S+P}|0\rangle\langle M_{1}M_{2}|(\bar{u}b)_{S-P}|B^{-}\rangle\bigg{\}}\,,$$

$${\alpha }_{\pm }^q$$

$${\alpha }_{1,2}^q$$

$${\bar{B}}^0\to p\bar{p}{\pi }^{+}{\pi }^{-},{\bar{B}}_s^0\to p\bar{p}{K}^{+}{\pi }^{-},\text{(q=d)}$$

$${\bar{B}}^0\to p\bar{p}{\pi }^{+}{K}^{-},{\bar{B}}_s^0\to p\bar{p}{K}^{+}{K}^{-},\text{(q=s)}$$

$$B^{-}\to \Lambda \bar{p}{\pi }^{+}{\pi }^{-},B^{-}\to \Lambda \bar{p}{K}^{+}{K}^{-}.$$

$$⟨{\mathbf{B}}_1{\bar{\mathbf{B}}}_2|{\bar{q}}_1{\gamma }_{\mu }{q}_2|0⟩$$

$$⟨{\mathbf{B}}_1{\bar{\mathbf{B}}}_2|{\bar{q}}_1{\gamma }_{\mu }{\gamma }_5{q}_2|0⟩$$

$$⟨{\mathbf{B}}_1{\bar{\mathbf{B}}}_2|{\bar{q}}_1{q}_2|0⟩$$

$$⟨M_1{M}_2|{\bar{q}}_1{\gamma }_{\mu }(1-{\gamma }_5)b|B⟩$$

$$F_1=\frac{{\bar{C}}_{F_1}}{t^2},g_A=\frac{{\bar{C}}_{g_A}}{t^2},f_S=\frac{{\bar{C}}_{f_S}}{t^2},g_P=\frac{{\bar{C}}_{g_P}}{t^2},$$

$$d\Gamma =\frac{|\bar{\mathcal{A}}{|}^2}{4(4\pi {)}^6{m}_B^3}X{\alpha }_{\mathbf{B}}{\alpha }_{\mathbf{M}}dsdtd\text{cos}{\theta }_{\mathbf{B}}d\text{cos}{\theta }_{\mathbf{M}}d\varphi ,$$

$$X$$

$${\alpha }_{\mathbf{B}}$$

$${\alpha }_{\mathbf{M}}$$

$$(m_{M_1}+m_{M_2}{)}^2\le s\le (m_B-\sqrt{t}{)}^2,(m_{{\mathbf{B}}_1}+m_{{\bar{\mathbf{B}}}_2}{)}^2\le t\le (m_B-m_{M_1}-m_{M_2}{)}^2,$$

$$0\le {\theta }_{\mathbf{B}},{\theta }_{\mathbf{M}}\le \pi ,0\le \varphi \le 2\pi .$$

$$(V_{ub},V_{tb})=(A{\lambda }^3(\rho -i\eta ),1),$$

$$(V_{ud},V_{td})=(1-{\lambda }^2/2,A{\lambda }^3),$$

$$(V_{us},V_{ts})=(\lambda ,-A{\lambda }^2),$$

$$(C_h,C_{w_{-}}){|}_{B\to \pi \pi }$$

$$(C_h,C_{w_{-}}){|}_{B\to KK(K\pi )}$$

$$C_{F_1}=\frac{5}{3}{C}_{||}+\frac{1}{3}{C}_{\overline{||}},C_{g_A}=\frac{5}{3}{C}_{||}^{\ast }-\frac{1}{3}{C}_{\overline{||}}^{\ast },\text{(for ⟨ppˉ∣uˉγμ(γ5)u∣0⟩)}$$

$$C_{F_1}=\frac{1}{3}{C}_{||}+\frac{2}{3}{C}_{\overline{||}},C_{g_A}=\frac{1}{3}{C}_{||}^{\ast }-\frac{2}{3}{C}_{\overline{||}}^{\ast },\text{(for ⟨ppˉ∣dˉγμ(γ5)d∣0⟩)}$$

$$C_{f_S}=\frac{1}{3}{\bar{C}}_{||},C_{g_P}=\frac{1}{3}{\bar{C}}_{||}^{\ast },\text{(for ⟨ppˉ∣dˉ(γ5)d∣0⟩)}$$

$$C_{F_1}=\sqrt{\frac{3}{2}}{C}_{||},C_{g_A}=\sqrt{\frac{3}{2}}{C}_{||}^{\ast },\text{(for ⟨Λpˉ∣sˉγμ(γ5)u∣0⟩)}$$

$$C_{f_S}=-\sqrt{\frac{3}{2}}{\bar{C}}_{||},C_{g_P}=-\sqrt{\frac{3}{2}}{\bar{C}}_{||}^{\ast },\text{(for ⟨Λpˉ∣sˉ(γ5)u∣0⟩)}$$

$$(C_{||},\delta C_{||})=(154.4\pm 12.1,19.3\pm 21.6){\mathrm{GeV}}^4,$$

$$(C_{\overline{||}},\delta C_{\overline{||}})=(18.1\pm 72.2,-477.4\pm 99.0){\mathrm{GeV}}^4,$$

$$({\bar{C}}_{||},\delta {\bar{C}}_{||})=(537.6\pm 28.7,-342.3\pm 61.4){\mathrm{GeV}}^4,$$

Full text

I introduction

One of the main purposes of the $B$ factories and current LHCb is to study CP violation (CPV), which is important for us to understand the puzzle of the matter-antimatter asymmetry in the Universe. As the observables, the (in)direct CP-violating asymmetries (CPAs) require both weak and strong phases Geng:2016kjv ; Hsiao:2014mua ; Geng:2006jt , whereas the T-violating triple momentum product correlations (TPCs), such as $\vec{p}_{1}\cdot(\vec{p}_{2}\times\vec{p}_{3})$ in a four-body decay, do not necessarily need a strong phase Bensalem ; Geng:2005wt . For example, the LHCb Collaboration has provided the first evidence for CPV from the TPCs in $\Lambda_{b}\to p\pi^{-}\pi^{+}\pi^{-}$ Aaij:2016cla , and measured TPCs in $\Lambda_{b}\to pK^{-}\mu^{+}\mu^{-}$ Aaij:2017mib . As the similar baryonic cases, the four-body baryonic $B$ decays can also provide TPCs.

For a long time, the $B^{-}\to\Lambda\bar{p}\pi^{+}\pi^{-}$ decay was the only observed decay mode in $B\to{\bf B_{1}\bar{B}_{2}}M_{1}M_{2}$ Chen:2009xg . Until very recently, more four-body baryonic B decays have been observed by the LHCb BtoBBMM_LHCb , which motivate us to give theoretical estimations on the corresponding decay branching ratios. The experimental measurements for the branching ratios of $\bar{B}^{0}/B^{-}\to{\bf B_{1}\bar{B}_{2}}M_{1}M_{2}$ at the level of $10^{-6}$ are given by Chen:2009xg ; BtoBBMM_LHCb

[TABLE]

where the resonant ${\cal B}(B^{-}\to\Lambda\bar{p}$ $(\rho^{0},f_{2}(1270)\to)\pi^{+}\pi^{-})$ have been excluded from the data Chen:2009xg . In comparison with ${\cal B}(\bar{B}^{0}\to p\bar{p}K^{+}K^{-})\simeq(1.3\pm 0.3)\times 10^{-7}$ and ${\cal B}(\bar{B}^{0}_{s}\to p\bar{p}\pi^{+}\pi^{-})<7.3\times 10^{-7}$ (90% C.L.) BtoBBMM_LHCb , the decays with ${\cal B}\sim 10^{-6}$ in Eq. (I) are recognized to have the same theoretical correspondence, where $\bar{B}^{0}/B^{-}\to{\bf B_{1}\bar{B}_{2}}M_{1}M_{2}$ proceed through the $B\to M_{1}M_{2}$ transition along with the $\bf B_{1}\bar{B}_{2}$ production, as depicted in Fig. 1. Note that the $\bar{B}^{0}_{s}$ decays of $\bar{B}^{0}_{s}\to p\bar{p}K^{\pm}\pi^{\mp}$ and $p\bar{p}K^{+}K^{-}$ with $\bar{s}$ being replaced by $\bar{d}$ in $\bar{B}^{0}\to p\bar{p}\pi^{+}\pi^{-}$ and $p\bar{p}K^{\mp}\pi^{\pm}$ have also been found with the branching ratios of order $10^{-6}$ BtoBBMM_LHCb , respectively.

In this report, we will calculate the four-body baryonic $B$ decays in accordance with the decaying processes in Fig. 1, with the extraction of the $B\to M_{1}M_{2}$ transition form factors from the $B\to D^{(*)}M_{1}M_{2}$ and $B\to M_{1}M_{2}M_{3}$ decays and the adoption of the timelike baryonic form factors from the two-body and three-body baryonic $B$ decays. Our theoretical approach will be useful for the estimations of TPCs in $B\to{\bf B_{1}\bar{B}_{2}}M_{1}M_{2}$ to be compared to future measurements by the LHCb.

II Formalism

In terms of the quark-level effective Hamilontion for the charmless $b\to q_{1}\bar{q}_{2}q_{3}$ transition, the amplitudes of the four-body baryonic $B$ decays by the generalized factorization approach are derived as ali

[TABLE]

where $G_{F}$ is the Fermi constant, $V_{ij}$ are the CKM matrix elements, and $(\bar{q}_{1}q_{2})_{V(A)}$ and $(\bar{q}_{1}q_{2})_{S(P)}$ stand for $\bar{q}_{1}\gamma_{\mu}(\gamma_{5})q_{2}$ and $\bar{q}_{1}(\gamma_{5})q_{2}$, respectively. The parameters $\alpha^{q}_{\xi}$ and $\beta^{q}_{\eta}$ in Eq. (II) are given by

[TABLE]

with $q=(d,s)$ and $j=(3,4,5,9,10)$, where $a_{i}\equiv c^{eff}_{i}+c^{eff}_{i\pm 1}/N_{c}^{eff}$ for $i=$ odd (even) with the effective color number $N_{c}^{eff}$ and Wilson coefficients $c_{i}^{eff}$ in Ref. ali . From ${\cal A}_{1}(\bar{B}^{0}_{(s)}\to p\bar{p}M_{1}M_{2})$ and ${\cal A}_{2}(B^{-}\to\Lambda\bar{p}M_{1}M_{2})$ in Eq. (II), the allowed decays are

[TABLE]

Note that the $\bar{B}^{0}\to p\bar{p}\pi^{+}K^{-}$ and $\bar{B}^{0}_{s}\to p\bar{p}K^{+}K^{-}$ decays have the matrix elements of $\langle p\bar{p}|(\bar{s}s)_{V,A,S,P}|0\rangle$ with the $\bar{s}s$ quark currents, which eventually cause the terms of $\alpha_{4,6,10}^{s}$ to give nearly zero contributions due to the OZI suppression of $\bar{s}s\to p\bar{p}$ Hsiao:2014tda .

For the matrix elements in Eq. (II), the baryon-pair productions from the quark currents are given by Chua:2002yd ; Geng:2005wt

[TABLE]

where $q=p_{\bf B_{1}}+p_{\bf\bar{B}_{2}}$, $t\equiv q^{2}$, $u$($v$) is the (anti-)baryon spinor, and $(F_{1,2},g_{A},h_{A},f_{S},g_{P})$ are the timelike baryonic form factors. On the other hand, the $B\to M_{1}M_{2}$ transition matrix elements are parameterized as Lee:1992ih

[TABLE]

where $p=p_{M_{2}}+p_{M_{1}}$ and $(h,r,w_{\pm})$ are the form factors. Subsequently, one can also get $\langle M_{1}M_{2}|\bar{q}_{1}(\gamma_{5})b|B\rangle$ from Eq. (6) based on equations of motion. In terms of the approach of pQCD counting rules, the momentum dependences for the $0\to{\bf B_{1}\bar{B}_{2}}$ and $B\to M_{1}M_{2}$ transition form factors are given by Brodsky:1973kr ; Brodsky:2003gs ; Chua:2002pi ; Chua:2004mi

[TABLE]

where $\bar{C}_{i}=C_{i}[\text{ln}({t}/{\Lambda_{0}^{2}})]^{-\gamma}$ with $\gamma=2.148$ and $\Lambda_{0}=0.3$ GeV. We note that since $F_{2}$ is derived to be $F_{2}=F_{1}/(t\text{ln}[t/\Lambda_{0}^{2}])$ Belitsky:2002kj , which is much less than $F_{1}$, while the small value of ${\cal B}(\bar{B}^{0}\to p\bar{p})=(1.5^{+0.7}_{-0.5})\times 10^{-8}$ pdg ; Aaij:2013fta causes a tiny $C_{h_{A}}$ Hsiao:2014zza in $h_{A}=C_{h_{A}}/t^{2}$, we may not consider the effects from $F_{2}$ and $h_{A}$. In addition, by following Ref. Chua:2002pi , we have neglected the terms related to $r$ and $w_{+}$ in Eq. (6) due to the wrong parity Drutskoy:2002ib .

The integration over the phase space of the four-body $B(p_{B})\to{\bf B_{1}}(p_{\bf B_{1}}){\bf\bar{B}_{2}}(p_{\bf\bar{B}_{2}})$ $M_{1}(p_{M_{1}})M_{2}(p_{M_{2}})$ decay relies on the five kinematic variables, that is, $s\equiv p^{2}$, $t$ and the three angles of $\theta_{\bf B}$, $\theta_{\bf M}$ and $\phi$. In Fig. 2, the angle $\theta_{\bf B(M)}$ is between $\vec{p}_{\bf B_{1}}$ ($\vec{p}_{M_{1}}$) of the $\bf B_{1}\bar{B}_{2}$ ($M_{1}M_{2}$) rest frame and the line of flight of the $\bf B_{1}\bar{B}_{2}$ ($M_{1}M_{2}$) system in the $B$ meson rest frame, while the angle $\phi$ is from the $\bf B_{1}\bar{B}_{2}$ plane to the $M_{1}M_{2}$ plane, defined by the momenta of the $\bf B_{1}\bar{B}_{2}$ and $M_{1}M_{2}$ pairs in the $B$ rest frame, respectively. The partial decay width reads Geng:2011tr ; Geng:2012qn

[TABLE]

where $X$, $\alpha_{\bf B}$ and $\alpha_{\bf M}$ are given by

[TABLE]

respectively, with $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2bc-2ca$, while the allowed ranges of the five variables are given by

[TABLE]

III Numerical Results and Discussions

For the numerical analysis, the CKM matrix elements in the Wolfenstein parameterization are presented as

[TABLE]

with $(\lambda,\,A,\,\rho,\,\eta)=(0.225,\,0.814,\,0.120\pm 0.022,\,0.362\pm 0.013)$ pdg . To estimate the non-factorizable effects in the generalized factorization approach ali , $N_{c}^{eff}$ ranges from 2 to $\infty$. In Table 1, we show the values of $a_{i}$ for the $b\to d$ and $b\to s$ transitions with $N_{c}^{eff}=(2,3,\infty)$, respectively.

According to the extractions of $(C_{h},C_{w_{-}})$ in Refs. Chua:2002pi ; Chua:2004mi , we fit the $B\to\pi\pi$ transition form factors with the branching ratios of $\bar{B}^{0}\to D^{(*)0}\pi^{+}\pi^{-}$, $B^{-}\to\pi^{-}\pi^{+}\pi^{-}$ and $B^{-}\to K^{*-}\pi^{+}\pi^{-}$, and the $B\to(K\pi,KK)$ ones with those of $B^{-}\to D^{(*)0}K^{-}K^{0}$, $\bar{B}^{0}\to D^{0}K^{-}\pi^{+}$ and $B^{-}\to K^{*-}K^{+}K^{-}$. Note that the contributions from the resonant $B\to D^{(*)}(M_{0}\to)M_{1}M_{2}$, and $B^{-}\to K^{*-}(M_{0}\to)M_{1}M_{2}$ decays with $\rho^{0},f_{2}(1270)\to\pi^{+}\pi^{-}$ or $\phi\to K^{+}K^{-}$ have been excluded from the data. Unfortunately, the current observations of ${\cal B}(\bar{B}^{0}_{s}\to M_{1}M_{2}M_{3})$ are not sufficient for us to extract the $\bar{B}^{0}_{s}\to M_{1}M_{2}$ transition form factors. As a result, we obtain

[TABLE]

The timelike baryonic form factors in Eq. (II) can be related with the $SU(3)$ flavor and $SU(2)$ spin symmetries, such that $(C_{F_{1}},C_{g_{A}},C_{f_{S}},C_{g_{P}})$ are recombined by a new set of constant parameters as Brodsky:1973kr ; Chua:2002yd ; Geng:2016fdw ; Hsiao:2016amt ; ang_BtopLpi

[TABLE]

with $C_{||(\overline{||})}^{*}\equiv C_{||(\overline{||})}+\delta C_{||(\overline{||})}$ and $\bar{C}_{||}^{*}\equiv\bar{C}_{||}+\delta\bar{C}_{||}$, in which $\delta C_{||(\overline{||})}$ and $\delta\bar{C}_{||}$ have been added to explain the large and unexpected angular distributions in $\bar{B}^{0}\to\Lambda\bar{p}\pi^{+}$ and $B^{-}\to\Lambda\bar{p}\pi^{0}$ ang_BtopLpi ; Wang:2007as , to account for the fact that the $SU(3)$ flavor and $SU(2)$ spin symmetries at large $t$ ($t\to\infty$)  Brodsky:1973kr should be broken at $t\simeq m_{B}^{2}$ ang_BtopLpi . The extractions of the form factors by the data of $\bar{B}^{0}\to n\bar{p}D^{*+}$, $\bar{B}^{0}\to\Lambda\bar{p}D^{(*)+}$, $\bar{B}^{0}\to\Lambda\bar{p}\pi^{+}$, $B^{-}\to\Lambda\bar{p}(\pi^{0},\rho^{0})$, $\bar{B}^{0}_{(s)}\to p\bar{p}$ and $B^{-}\to\Lambda\bar{p}$ give Geng:2016fdw

[TABLE]

where the added constants for the broken effects have been approved by the excellent agreement for ${\cal B}(\bar{B}^{0}_{s}\to\Lambda\bar{p}K^{+}+\bar{\Lambda}pK^{-})$ LHCb:2017khw . Subsequently, we evaluate the branching ratios of $B\to{\bf B_{1}\bar{B}_{2}}M_{1}M_{2}$ as shown in Table 2, and draw the distributions vs. $m_{\bf B_{1}\bar{B}_{2}}$ in Fig. 3.

As seen in Table 2, although the predicted result of ${\cal B}(B^{-}\to\Lambda\bar{p}\pi^{+}\pi^{-})=(3.7^{+1.5}_{-1.0})\times 10^{-6}$ is a little lower, it is consistent with the data in Eq. (I) by taking the uncertainties into account. With the replacement of $B^{-}\to\pi^{+}\pi^{-}$ by $B^{-}\to K^{+}K^{-}$, the $B^{-}\to\Lambda\bar{p}\pi^{+}\pi^{-}$ and $\Lambda\bar{p}K^{+}K^{-}$ decays share the same decaying configuration. We hence predict that ${\cal B}(B^{-}\to\Lambda\bar{p}K^{+}K^{-})=(3.0^{+1.3}_{-0.9})\times 10^{-6}$, which is accessible to the LHCb and BELLE experiments. Unlike the $B^{-}\to\Lambda\bar{p}M_{1}M_{2}$ decays, where $a_{1,4,6}$ are stable by ranging $N_{c}^{eff}$ from 2 to $\infty$, the tree-level dominant $\bar{B}^{0}\to p\bar{p}\pi^{+}\pi^{-}$ decay has $\alpha^{d}_{\pm}\simeq V_{ub}V_{ud}^{*}a_{2}$ in Eq. (II) to be sensitive to the non-factorizable effects. Since the non-factorizable effects are uncomputable, according to the data of ${\cal B}(\bar{B}^{0}\to p\bar{p}\pi^{+}\pi^{-})=(3.0\pm 0.3)\times 10^{-6}$ BtoBBMM_LHCb in Eq. (I), we obtain ${\cal B}(\bar{B}^{0}\to p\bar{p}\pi^{+}\pi^{-})=(3.0\pm 0.9)\times 10^{-6}$, where $a_{2}=0.26\pm 0.01$ with the tiny value of $\delta a_{2}=0.01$ from the new data is compatible to ${\cal O}(0.2-0.3)$ from the two-body $B$ and $\Lambda_{b}$ and three-body baryonic $B$ decays Neubert:2001sj ; Hsiao:2015cda ; Hsiao:2016amt . For the measured branching ratio of $\bar{B}^{0}\to p\bar{p}\pi^{+}K^{-}+p\bar{p}\pi^{-}K^{+}$, it is found that the contribution is mainly from the penguin-level dominant $\bar{B}^{0}\to p\bar{p}\pi^{+}K^{-}$ mode. Note that $a_{3,5}$ from $\alpha^{s}_{\pm}\simeq\beta^{s}_{\pm}=-V_{tb}V_{ts}^{*}(a_{3}\pm a_{5}+a_{9})$ are also sensitive to the non-factorizable effects. With $N_{c}^{eff}=3$, we obtain ${\cal B}(\bar{B}^{0}\to p\bar{p}\pi^{+}K^{-})=(6.6\pm 2.4)\times 10^{-6}$, which suggests that the decay is free from the non-factorizable effects. In Table 2 we have included the data to constrain the non-factorizable effects, which results in $\delta N_{c}^{eff}=0.06$. We note that the two spectra in Fig. 3 for $B^{-}\to\Lambda\bar{p}M_{1}M_{2}$ and $\bar{B}^{0}\to p\bar{p}M_{1}M_{2}$ present the threshold effects as the peaks around the threshold areas of $m_{\Lambda\bar{p}}\simeq m_{\Lambda}+m_{\bar{p}}$ and $m_{p\bar{p}}\simeq m_{p}+m_{\bar{p}}$, respectively, which are commonly observed in the three and four-body baryonic $B$ decays BtoBBMM_LHCb ; Wang:2007as .

Finally, we remark that we cannot explain the data of ${\cal B}(\bar{B}^{0}_{s}\to p\bar{p}K^{\pm}\pi^{\mp},p\bar{p}K^{+}K^{-})=(1.5\pm 0.7,4.6\pm 0.6)\times 10^{-6}$ measured by the LHCb BtoBBMM_LHCb due to the lack of the information for the transition form factors of $\bar{B}^{0}_{s}\to(K^{+}\pi^{-},K^{+}K^{-})$. This calls for the theoretical and experimental studies of the three-body mesonic $\bar{B}^{0}_{s}$ decays that could proceed with the $\bar{B}^{0}_{s}\to M_{1}M_{2}$ transitions, such as the $\bar{B}^{0}_{s}\to D_{s}^{*-}\pi^{+}K^{0}$, $\bar{B}^{0}_{s}\to D^{*0}\pi^{+}K^{-}(K^{+}K^{-})$ and $\bar{B}^{0}_{s}\to\rho^{-}\pi^{+}K^{0}$ decays with one of the mesons to be a vector one, in order to extract both $(h,w_{-})$ in Eq. (6). On the other hand, the observed $\bar{B}^{0}_{s}\to D^{0}K^{+}\pi^{-}$ and $\bar{B}^{0}_{s}\to D^{0}K^{+}K^{-}$ decays pdg are also important as they relate to $w_{-}$.

IV Conclusions

In sum, we have studied the charmless four-body baryonic $B\to{\bf B_{1}\bar{B}_{2}}M_{1}M_{2}$ decays, where the primary decaying processes are regarded as the $B\to M_{1}M_{2}$ transitions along with the baryon-pair productions. According to the new extractions of the $B\to M_{1}M_{2}$ transition form factors from the three-body $B\to D^{(*)}M_{1}M_{2}$ and $B\to M_{1}M_{2}M_{3}$ decays, we have shown that ${\cal B}(B^{-}\to\Lambda\bar{p}\pi^{+}\pi^{-})=(3.7^{+1.5}_{-1.0})\times 10^{-6}$ and ${\cal B}(\bar{B}^{0}\to p\bar{p}\pi^{+}\pi^{-},p\bar{p}\pi^{+}K^{-})=(3.0\pm 0.9,6.6\pm 2.4)\times 10^{-6}$, which agree with the data. We have also predicted ${\cal B}(B^{-}\to\Lambda\bar{p}K^{+}K^{-})=(3.0^{+1.3}_{-0.9})\times 10^{-6}$ to be accessible to the LHCb and BELLE experiments. The study of $B\to{\bf B_{1}\bar{B}_{2}}M_{1}M_{2}$ benefits the future test of T violation, as the T-odd triple momentum product correlation of $\vec{p}_{1}\cdot(\vec{p}_{2}\times\vec{p}_{3})$ can be directly constructed.

ACKNOWLEDGMENTS

We would like to thank Dr. Eduardo Rodrigues for useful discussions. This work was supported in part by National Center for Theoretical Sciences, MoST (MoST-104-2112-M-007-003-MY3), and National Science Foundation of China (11675030).