Testing the W-exchange mechanism with two-body baryonic B decays

TL;DR

This paper investigates the significance of W-exchange diagrams in baryonic B decays, demonstrating their non-negligible role through theoretical predictions and aligning some with experimental bounds, thus providing a new perspective on decay mechanisms.

Contribution

The study provides the first detailed analysis of W-exchange contributions in baryonic B decays, offering predictions for branching ratios that can be tested experimentally.

Findings

W-exchange diagrams are significant in baryonic B decays.

Predicted branching ratio for _c ar{\u1EF9} decays aligns with experimental bounds.

Future measurements by LHCb and Belle II can test these predictions.

Abstract

The role of $W$-exchange diagrams in baryonic $B$ decays is poorly understood, and often taken as insignificant and neglected. We show that charmful two-body baryonic $B\to {\bf B}_c \bar {\bf B}'$ decays provide a good test-bed for the study of the $W$-exchange topology, whose contribution is found to be non-negligible; here ${\bf B}_c$ is an anti-triplet or a sextet charmed baryon, and ${\bf \bar B}'$ an octet charmless (anti-)baryon. In particular, we calculate that ${\cal B}(\bar B^0\to\Sigma_c^{+}\bar p)=(2.9^{+0.8}_{-0.9})\times 10^{-6}$ in good agreement with the experimental upper bound. Its cousin $\bar B_s^0$ mode, $\bar B_s^0\to\Lambda^+_c\bar p$, is a purely $W$-exchange decay, hence is naturally suited for the study of the role of the $W$-exchange topology. We predict ${\cal B}(\bar B_s^0\to\Lambda^+_c\bar p)=(0.8\pm 0.3)\times 10^{-6}$, a relatively large branching ratio to be tested with a future measurement by the LHCb collaboration. Other predictions, such as ${\cal B}(\bar B^0\to\Xi_c^+\bar\Sigma^-)=(1.1\pm 0.4)\times 10^{-5}$, can be tested with future Belle II measurements.

Tables

Table 1. Table 1: Classification for B → 𝐁 c ​ 𝐁 ¯ ′ → 𝐵 subscript 𝐁 𝑐 superscript ¯ 𝐁 ′ B\to{\bf B}_{c}\bar{\bf B}^{\prime} decays, where 𝒜 ex subscript 𝒜 ex {\cal A}_{\rm ex} and 𝒜 em subscript 𝒜 em {\cal A}_{\rm em} denote the amplitudes through the W 𝑊 W -exchange and W 𝑊 W -emission diagrams, respectively.

${\overline{B}}^0\to {𝐁}_c{\overline{𝐁}}^{\prime }$	Amplitudes	${\overline{B}}^0\to {𝐁}_c{\overline{𝐁}}^{\prime }$	Amplitudes
${\Lambda }_c^{+}\overline{p},{\Lambda }_c^{+}{\overline{\Delta }}^{-}$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$	${\Sigma }_c^{++}{\overline{\Delta }}^{--}$	${𝒜}_{\mathrm{ex}}$
${\Xi }_c^{+}{\overline{\Sigma }}^{-}$	${𝒜}_{\mathrm{ex}}$	${\Sigma }_c^{+}\overline{p},{\Sigma }_c^{+}{\overline{\Delta }}^{-}$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$
${\Xi }_c^0\overline{\Lambda },{\Xi }_c^0{\overline{\Sigma }}^0$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$	${\Sigma }_c^0\overline{n},{\Sigma }_c^0{\overline{\Delta }}^0$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$
		${\Xi }_c^{{}_{}{}^{\prime }+}{\overline{\Sigma }}^{-}$	${𝒜}_{\mathrm{ex}}$
		${\Xi }_c^{{}_{}{}^{\prime }0}\overline{\Lambda },{\Xi }_c^{{}_{}{}^{\prime }0}{\overline{\Sigma }}^0$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$
		${\Omega }_c^0{\overline{\Xi }}^0$	${𝒜}_{\mathrm{ex}}$
${\overline{B}}_s^0\to {𝐁}_c{\overline{𝐁}}^{\prime }$	Amplitudes	${\overline{B}}_s^0\to {𝐁}_c{\overline{𝐁}}^{\prime }$	Amplitudes
${\Lambda }_c^{+}\overline{p},{\Lambda }_c^{+}{\overline{\Delta }}^{-}$	${𝒜}_{\mathrm{ex}}$	${\Sigma }_c^{++}{\overline{\Delta }}^{--}$	${𝒜}_{\mathrm{ex}}$
${\Xi }_c^{+}{\overline{\Sigma }}^{-}$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$	${\Sigma }_c^{+}\overline{p},{\Sigma }_c^{+}{\overline{\Delta }}^{-}$	${𝒜}_{\mathrm{ex}}$
${\Xi }_c^0\overline{\Lambda },{\Xi }_c^0{\overline{\Sigma }}^0$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$	${\Sigma }_c^0\overline{n},{\Sigma }_c^0{\overline{\Delta }}^0$	${𝒜}_{\mathrm{ex}}$
		${\Xi }_c^{{}_{}{}^{\prime }+}{\overline{\Sigma }}^{-}$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$
		${\Xi }_c^{{}_{}{}^{\prime }0}\overline{\Lambda },{\Xi }_c^{{}_{}{}^{\prime }0}{\overline{\Sigma }}^0$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$
		${\Omega }_c^0{\overline{\Xi }}^0$	${𝒜}_{\mathrm{ex}}+{𝒜}_{\mathrm{em}}$

Table 2. Table 2: The relations between different timelike baryon form factors. The 𝟑 ¯ c subscript ¯ 3 𝑐 \bar{\mathbf{3}}_{c} and 𝟔 c subscript 6 𝑐 \mathbf{6}_{c} denote the anti-triplet and the sextet charmed baryons, respectively, and the 𝟖 8 \mathbf{8} denotes the octet charmless baryons.

${\overline{\mathrm{𝟑}}}_c\otimes \mathrm{𝟖}$	$\left|\xi \right|$		${\mathrm{𝟔}}_c\otimes \mathrm{𝟖}$	$\left|\zeta \right|$
${\Lambda }_c^{+}\overline{p}$	$1$		${\Sigma }_c^{+}\overline{p}$	1
${\Xi }_c^{+}{\overline{\Sigma }}^{-}$	$1$		${\Sigma }_c^0\overline{n}$	$\sqrt{2}$
${\Xi }_c^0\overline{\Lambda }$	$\frac{1}{\sqrt{6}}$		${\Xi }_c^{{}_{}{}^{\prime }+}{\overline{\Sigma }}^{-}$	1
${\Xi }_c^0{\overline{\Sigma }}^0$	$\frac{1}{\sqrt{2}}$		${\Xi }_c^{{}_{}{}^{\prime }0}{\overline{\Sigma }}^0$	$\frac{1}{\sqrt{2}}$
			${\Xi }_c^{{}_{}{}^{\prime }0}\overline{\Lambda }$	$\sqrt{\frac{3}{2}}$
			${\Omega }_c^0{\overline{\Xi }}^0$	$\sqrt{2}$

Table 3. Table 3: Branching ratios of the B ¯ ( s ) 0 → 𝐁 c ​ 𝐁 ¯ ′ → subscript superscript ¯ 𝐵 0 𝑠 subscript 𝐁 𝑐 superscript ¯ 𝐁 ′ \bar{B}^{0}_{(s)}\to{\bf B}_{c}\bar{\bf B}^{\prime} decays from the W 𝑊 W -exchange diagrams.

Decay modes	$ℬ\times {10}^5$		Data		Decay modes	$ℬ\times {10}^6$		Data
${\overline{B}}^0\to {\Lambda }_c^{+}\overline{p}$	${1.0}_{-0.3}^{+0.4}$		$1.54\pm 0.18$ pdg		${\overline{B}}^0\to {\Sigma }_c^{+}\overline{p}$	${2.9}_{-0.9}^{+0.8}$		pdg
${\overline{B}}^0\to {\Xi }_c^{+}{\overline{\Sigma }}^{-}$	$1.1\pm 0.4$				${\overline{B}}^0\to {\Sigma }_c^0\overline{n}$	${5.8}_{-1.8}^{+1.5}$
${\overline{B}}^0\to {\Xi }_c^0{\overline{\Sigma }}^0$	$0.6\pm 0.2$				${\overline{B}}^0\to {\Xi }_c^{{}_{}{}^{\prime }+}{\overline{\Sigma }}^{-}$	${3.1}_{-1.0}^{+0.9}$
${\overline{B}}^0\to {\Xi }_c^0\overline{\Lambda }$	$0.2\pm 0.1$				${\overline{B}}^0\to {\Xi }_c^{{}_{}{}^{\prime }0}{\overline{\Sigma }}^0$	${1.6}_{-0.5}^{+0.4}$
					${\overline{B}}^0\to {\Xi }_c^{{}_{}{}^{\prime }0}\overline{\Lambda }$	${4.6}_{-1.5}^{+1.3}$
					${\overline{B}}^0\to {\Omega }_c^0{\overline{\Xi }}^0$	${6.4}_{-2.1}^{+1.8}$
Decay modes	$ℬ\times {10}^6$				Decay modes	$ℬ\times {10}^7$
${\overline{B}}_s^0\to {\Lambda }_c^{+}\overline{p}$	$0.8\pm 0.3$				${\overline{B}}_s^0\to {\Sigma }_c^{+}\overline{p}$	${2.3}_{-0.7}^{+0.6}$
${\overline{B}}_s^0\to {\Xi }_c^{+}{\overline{\Sigma }}^{-}$	$0.9\pm 0.3$				${\overline{B}}_s^0\to {\Sigma }_c^0\overline{n}$	${4.5}_{-1.4}^{+1.1}$
${\overline{B}}_s^0\to {\Xi }_c^0{\overline{\Sigma }}^0$	$0.4\pm 0.2$				${\overline{B}}_s^0\to {\Xi }_c^{{}_{}{}^{\prime }+}{\overline{\Sigma }}^{-}$	${2.5}_{-0.8}^{+0.6}$
${\overline{B}}_s^0\to {\Xi }_c^0\overline{\Lambda }$	$0.2\pm 0.1$				${\overline{B}}_s^0\to {\Xi }_c^{{}_{}{}^{\prime }0}{\overline{\Sigma }}^0$	${1.2}_{-0.4}^{+0.3}$
					${\overline{B}}_s^0\to {\Xi }_c^{{}_{}{}^{\prime }0}\overline{\Lambda }$	${3.6}_{-1.2}^{+0.9}$
					${\overline{B}}_s^0\to {\Omega }_c^0{\overline{\Xi }}^0$	${5.1}_{-1.7}^{+1.4}$

Equations

$$\mathcal{B}({\bar{B}}^0\to {\Lambda }_c^{+}\bar{p})=(1.54\pm 0.18)\times 10^{-5},$$

$$\mathcal{B}({\bar{B}}^0\to {\Sigma }_c^{+}\bar{p})<2.4\times 10^{-5},$$

$$\mathcal{B}(B^{-}\to {\Sigma }_c^0\bar{p})=(2.9\pm 0.7)\times 10^{-5}.$$

$$\mathcal{A}(B\to \ell {\bar{\nu }}_{\ell })$$

$$\mathcal{A}(B\to {\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime })$$

$$\mathcal{B}({\bar{B}}^0\to {\Xi }_c^{+}{\bar{\Sigma }}^{-},{\Sigma }_c^{++}{\bar{\Delta }}^{--},{\Xi }_c^{\prime +}{\bar{\Sigma }}^{-},{\Omega }_c^0{\bar{\Xi }}^0)=0,$$

$$\mathcal{B}({\bar{B}}_s^0\to {\Lambda }_c^{+}\bar{p},{\Lambda }_c^{+}{\bar{\Delta }}^{-},{\Sigma }_c^{++}{\bar{\Delta }}^{--},{\Sigma }_c^{+}\bar{p},{\Sigma }_c^{+}{\bar{\Delta }}^{-},{\Sigma }_c^0\bar{n},{\Sigma }_c^0{\bar{\Delta }}^0)=0.$$

$${\mathcal{H}}_{\mathrm{eff}}=\frac{G_F}{\sqrt{2}}\sum _{q=d,s}{V}_{cb}{V}_{uq}^{\ast }\left[c_1^{\mathrm{eff}}(\bar{q}u)(\bar{c}b)+c_2^{\mathrm{eff}}(\bar{c}u)(\bar{q}b)\right],$$

$$\mathcal{A}({\bar{B}}_{(s)}^0\to {\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime })$$

$$⟨0|\bar{q}{\gamma }^{\mu }(1-{\gamma }_5)b|B⟩$$

$$⟨{\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime }|\bar{c}{\gamma }^{\mu }u|0⟩$$

$$⟨{\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime }|\bar{c}{\gamma }^{\mu }{\gamma }_{5}u|0⟩$$

$$\mathcal{A}(B\to {\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime })$$

$$A_{S}=C_{S}m_{-}\bigg{[}f_{1}+\left(\frac{m_{B}^{2}}{m_{+}m_{-}}\right)f_{3}\bigg{]}\,,\,\,\,\,\,\,\,\,\,A_{P}=C_{P}m_{+}\bigg{[}g_{1}+\left(\frac{m_{B}^{2}}{m_{+}^{2}}\right)g_{3}\bigg{]}\,,$$

$$F(t)=\frac{F(0)}{1-a\left(t/m_B^2\right)+b\left(t^2/m_B^4\right)},$$

$$(C_f^{{\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime }},C_g^{{\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime }})=\xi (C_f^{{\Lambda }_c^{+}\bar{p}},C_g^{{\Lambda }_c^{+}\bar{p}}),(C_f^{{\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime }},C_g^{{\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime }})=\zeta (C_f^{{\Sigma }_c^{+}\bar{p}},C_g^{{\Sigma }_c^{+}\bar{p}}),$$

$$\mathcal{B}(B\to {\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime })=\frac{|{\vec{p}}_{{\mathbf{B}}_c}|{\tau }_B}{8\pi m_B^2}|\mathcal{A}(B\to {\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime }){|}^2,$$

$$|{\vec{p}}_{{\mathbf{B}}_c}|=\frac{\sqrt{(m_B^2-m_{+}^2)(m_B^2-m_{-}^2)}}{2m_B},$$

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

$$(C_f^{{\Lambda }_c^{+}\bar{p}},a,b)=(0.41_{-0.09}^{+0.07},0.75,0.45),$$

$$(C_g^{{\Lambda }_c^{+}\bar{p}},a,b)=(0.37_{-0.08}^{+0.07},0.90,0.50),$$

$$(C_f^{{\Sigma }_c^{+}\bar{p}},a,b)=(-0.25\pm 0.04,0.70,0.40),$$

$$(C_g^{{\Sigma }_c^{+}\bar{p}},a,b)=(-0.21\pm 0.04,0.84,0.52),$$

$$\mathcal{B}(B\to \ell {\bar{\nu }}_{\ell })$$

$$\mathcal{B}(B\to {\mathbf{B}}_c{\bar{\mathbf{B}}}^{\prime })$$

$$\mathcal{B}({\bar{B}}^0\to {\Sigma }_c^{+}\bar{p})$$

$$\mathcal{B}({\bar{B}}^0\to {\Sigma }_c^0\bar{n})$$

$$\mathcal{B}({\bar{B}}^0\to {\Xi }_c^0{\bar{\Sigma }}^0,{\Xi }_c^0\bar{\Lambda })=(0.6\pm 0.2,0.2\pm 0.1)\times 10^{-5},$$

$$\mathcal{B}({\bar{B}}_s^0\to {\Xi }_c^0{\bar{\Sigma }}^0,{\Xi }_c^0\bar{\Lambda })={\mathcal{R}}_{B_s}\times \mathcal{B}({\bar{B}}^0\to {\Xi }_c^0{\bar{\Sigma }}^0,{\Xi }_c^0{\bar{\Lambda }}^0),$$

$$\mathcal{B}({\bar{B}}^0\to {\Xi }_c^{+}{\bar{\Sigma }}^{-})$$

$$\mathcal{B}({\bar{B}}_s^0\to {\Lambda }_c^{+}\bar{p})$$

$$\mathcal{B}({\bar{B}}^0\to {\Xi }_c^{{}^{\prime }+}{\bar{\Sigma }}^{-},{\Omega }_c^0{\bar{\Xi }}^0)$$

Full text

I Introduction

Decays of $B$ mesons to multi-body baryonic final states, such as $B\to{\bf B\bar{B}^{\prime}}M$ and ${\bf B\bar{B}^{\prime}}MM^{\prime}$, where ${\bf B}$ ($M$) represents a baryon (meson), have been richly studied. Their branching ratios are typically at the level of $10^{-6}$ pdg . These relatively large branching ratios are due to the fact that the baryon-pair production tends to occur in the threshold region of $m_{\bf B\bar{B}^{\prime}}\simeq m_{\bf B}+m_{\bf\bar{B}^{\prime}}$, where the threshold effect with a sharply raising peak can enhance the branching ratio Wei:2007fg ; Aaij:2017vnw ; Aaij:2017pgn ; Lu:2018qbw . On the other hand, without the recoiled meson(s) to carry away the large energy release, the two-body $B\to{\bf B\bar{B}^{\prime}}$ decays proceed at the $m_{B}$ scale, several GeV away from the threshold region, resulting in the suppression of their decay rates Hou:2000bz ; Suzuki:2006nn . So far, only three charmless modes have been seen experimentally: $\bar{B}^{0}\to p\bar{p}$ (${\cal B}\simeq{\cal O}(10^{-8})$) and $B^{-}\to\Lambda^{(*)}\bar{p}$ (${\cal B}\simeq{\cal O}(10^{-7})$), with $\Lambda^{*}\equiv\Lambda(1520)$ Aaij:2017gum ; Aaij:2016xfa ; Aaij:2013fla .

The tree-level dominated $B\to{\bf B\bar{B}^{\prime}}$ decays can proceed through the $W$-exchange, emission and annihilation diagrams, depicted in Figs. 1(a,b,c), respectively. However, the $W$-exchange (annihilation) process is regarded as helicity suppressed Chen:2008pf ; Bevan:2014iga , and hence neglected in theoretical studies Chernyak:1990ag ; Ball:1990fw ; Chang:2001jt ; Cheng:2001tr ; Chua:2013zga ; He:2006vz . Moreover, one also neglects the penguin-level gluon-exchange (annihilation) contributions, which leads to ${\cal B}(\bar{B}_{s}^{0}\to p\bar{p})\simeq 0$ Chua:2013zga . Consequently, the observation of the $\bar{B}_{s}^{0}\to p\bar{p}$ decay would provide valuable information on whether contributions from the exchange (annihilation) processes play a significant role. The smallness of the current upper bound on its branching ratio, $\mathcal{B}(\bar{B}_{s}^{0}\to p\bar{p})<1.5\times 10^{-8}$ Aaij:2017gum , indicates an experimentally difficult decay mode to study the role of exchange (annihilation) diagrams.

On the other hand, experimental data show that ${\cal B}(B\to{\bf B}_{c}{\bf\bar{B}}^{\prime})\simeq(10^{2}-10^{3}){\cal B}(B\to{\bf B\bar{B}^{\prime}})$, where ${\bf B}_{c}$ denotes a charmed baryon pdg . With significantly larger decay rates, charmful two-body baryonic $B$ decays offer an interesting and suitable environment in which to study and test the role of the $W$-exchange (annihilation) mechanism. The set of measured $B\to{\bf B}_{c}{\bf\bar{B}}^{\prime}$ branching ratios is nevertheless scarce pdg ; Gabyshev:2002dt ; Aubert:2008ax :

[TABLE]

As in the case of the charmless final states considered above, the $B\to{\bf B}_{c}{\bf\bar{B}}^{\prime}$ decays can proceed through both the $W$-exchange and $W$-emission diagrams. Again, theoretical studies regard the $W$-exchange diagram as helicity-suppressed, which is in analogy with the leptonic $B\to\ell\bar{\nu}_{\ell}$ decays, and take the $W$-emission diagram as the dominant contribution Chernyak:1990ag ; Ball:1990fw ; Cheng:2001ub ; Cheng:2002sa ; He:2006vz . For clarification, we present the amplitudes of $B\to\ell\bar{\nu}_{\ell}$ and $B\to{\bf B}_{c}\bar{\bf B}^{\prime}$ with $W$-exchange contribution in Fig. 1a as

[TABLE]

where ${\cal A}(B\to{\bf B}_{c}\bar{\bf B}^{\prime})$ by equation of motion is presented as a reduced form with the quark mass $m_{c}$ Hsiao:2014zza . In Eq. (I), the small $m_{\ell}$, with $\ell=(e,\mu)$, is responsible for the helicity suppression in $B\to\ell\bar{\nu}_{\ell}$. Nonetheless, $m_{c}\sim 1.3$ GeV pdg in ${\cal A}(B\to{\bf B}_{c}\bar{\bf B}^{\prime})$ is clearly helicity allowed, indicating that neglecting its contribution may not be a valid assumption to make. For completeness, we note that the theoretical studies in Refs. Bediaga:1991eu ; Pham:1980dc ; Pham:1980xe ; Hsiao:2014zza also considered the exchange and annihilation contributions in $B\to{\bf B\bar{B}^{\prime}}$ and $D_{s}^{+}\to p\bar{n}$.

We therefore propose the study of the family of charmful two-body baryonic $B\to{\bf B}_{c}\bar{\bf B}^{\prime}$ decays to improve our knowledge of the role of $W$-exchange diagrams in $B$ decays to baryonic final states. Since modes such as $\bar{B}^{0}\to\Xi_{c}^{+}\bar{\Sigma}^{-}$ and $\bar{B}_{s}^{0}\to\Lambda^{+}_{c}\bar{p}$ can only proceed via the $W$-exchange diagram, measurements of their branching ratios are direct tests of the $W$-exchange mechanism.

II Formalism

Besides the $W$-emission (${\cal A}_{\rm em}$) amplitudes studied elsewhere Cheng:2001ub ; Cheng:2002sa ; He:2006vz , we consider the often neglected $W$-exchange (${\cal A}_{\rm ex}$) amplitudes for the charmful two-body baryonic $\bar{B}^{0}_{(s)}\to{\bf B}_{c}{\bf\bar{B}}^{\prime}$ decays, where ${\bf B}_{c}$ denotes the anti-triplet and the sextet charmed baryon states, $(\Xi_{c}^{+,0},\Lambda_{c}^{+})$ and $(\Sigma_{c}^{++,+,0},\Xi_{c}^{\prime+,0},\Omega_{c}^{0})$, respectively, and ${\bf\bar{B}}^{\prime}$ an octet charmless (anti-)baryon. The decays with the decuplet charmless (anti-)baryons are excluded from the calculations in this paper due to the lack of the corresponding timelike baryon form factors.

We show in Table 1 the amplitudes involved in the interested $\bar{B}^{0}_{(s)}\to{\bf B}_{c}\bar{\bf B}^{\prime}$ modes. The decay rate of modes that can only occur through the $W$-exchange diagram would be vanishingly small by construction if the importance of these diagrams was to be insignificant:

[TABLE]

None of these relations has yet been verified experimentally.

The relevant part of the Hamiltonian for the $\bar{B}^{0}_{(s)}\to{\bf B}_{c}\bar{\bf B}^{\prime}$ decays has the following form Buras:1998raa :

[TABLE]

where $G_{F}$ is the Fermi constant, $V_{ij}$ stand for the CKM matrix elements, and $(\bar{q}_{1}q_{2})_{V-A}\equiv\bar{q}_{1}\gamma^{\mu}(1-\gamma_{5})q_{2}$. In the factorization approach, the $W$-exchange amplitude of $\bar{B}_{(s)}^{0}\to{\bf B}_{c}\bar{\bf B}^{\prime}$ is given by Cheng:2001ub ; Hsiao:2014zza

[TABLE]

where $q=d(s)$ for $\bar{B}^{0}_{(s)}$, and $a_{2}=c^{\rm eff}_{2}+c^{\rm eff}_{1}/N_{c}$ consists of the effective Wilson coefficients $(c_{1}^{\rm eff},c_{2}^{\rm eff})=(1.168,-0.365)$ and the color number $N_{c}$.

The matrix elements in Eq. (5) are defined as pdg ; Cheng:2002sa

[TABLE]

where $f_{B}$ is the $B$ meson decay constant, $q^{\mu}=(p_{{\bf B}_{c}}+p_{\bar{\bf B}^{\prime}})^{\mu}$ the momentum transfer, and $f_{i}$ and $g_{i}$ ($i=1,\,2,\,3$) are the timelike baryonic ($0\to{\bf B}_{c}\bar{\bf B}^{\prime}$) form factors. The decay amplitude of $B\to{\bf B}_{c}\bar{\bf B}^{\prime}$ in the general form is written as

[TABLE]

with $A_{S,P}$ standing for the $(S,P)$-wave amplitudes. By only receiving the $W$-exchange contributions in Eq. (5), the $A_{S,P}$ are given by

[TABLE]

where $C_{S,P}=\mp i{G_{F}}\,a_{2}V_{cb}V^{*}_{uq}f_{B}/{\sqrt{2}}$, $m_{\pm}=m_{{\bf B}_{c}}\pm m_{\bar{\bf B}^{\prime}}$, and $(f_{2},g_{2})$ vanish due to the contraction of $\sigma_{\mu\nu}q^{\mu}q^{\nu}=0$. We use the timelike baryonic form factors with the light-front quark model Lih_Lcpbar , which have been widely applied to the $b$ and $c$-hadron decays Bakker:2003up ; Ji:2000rd ; Bakker:2002aw ; Choi:2013ira ; Cheng:2004cc ; Cheng:2003sm ; Schlumpf:1992vq . In the light-front frame, since the momentum transfer $q^{\mu}$ is presented as $q^{+}=q^{0}+q^{3}=0$, it simply indicates the non-contributions from $(f_{3},g_{3})$ as in the case of the $B\to M$ transition Cheng:2003sm ; Schlumpf:1992vq . This is in accordance with lattice QCD calculations, where the derivations of $(f_{3},g_{3})\propto(f_{1},g_{1})/t$ show the suppression in the spacelike region Meinel:2016dqj , even though the corrections beyond leading order have been considered. Therefore, $(f_{3},g_{3})$ are often neglected in the literature Cheng:2002sa ; Khodjamirian:2011jp .

By the double-pole parameterization, $F(t)\equiv(f_{1}(t),g_{1}(t))$ is presented as Khodjamirian:2011jp

[TABLE]

with $t\equiv q^{2}$. The $SU(3)$ flavor symmetry can relate $F(0)$ in the different decays, given by

[TABLE]

with $(C_{f},C_{g})\equiv(f_{1}(0),g_{1}(0))$ and $|\xi|$ ($|\zeta|$) for the anti-triplet (sextet) charmed baryons listed in Table 2.

We compute the branching ratios from the decay-rate equation for two-body decays, given by pdg

[TABLE]

where $\tau_{B}$ denotes the $B$ meson lifetime.

III Numerical analysis

In the numerical analysis, we adopt the Wolfenstein parameterization for the CKM matrix elements, given by pdg

[TABLE]

where $\lambda=0.22453\pm 0.00044$ and $A=0.836\pm 0.015$, together with the $B$ meson decay constants $(f_{B},f_{B_{s}})=(0.19,0.23)$ GeV pdg . The light-front quark model provides

[TABLE]

where $C_{f,g}^{\Lambda^{+}_{c}\bar{p}}$ and $C_{f,g}^{\Sigma^{+}_{c}\bar{p}}$ relate the different decays with $(|\xi|,|\zeta|)$ in Table 2.

By considering the non-factorizable $W$-emission contributions only, the pole model relates $B^{-}\to\Sigma_{c}^{0}\bar{p}$ and $\bar{B}^{0}\to\Lambda_{c}^{+}\bar{p}$ with the same sets of strong coupling constants Cheng:2002sa ; Bevan:2014iga . We follow Ref. Cheng:2002sa to re-extract the strong coupling constants with the most current data of ${\cal B}(B^{-}\to\Sigma_{c}^{0}\bar{p})=2.9\times 10^{-5}$ pdg , which leads to ${\cal B}(\bar{B}^{0}\to\Lambda_{c}^{+}\bar{p})=0.4\times 10^{-5}$ in comparison with the data of $(1.55\pm 0.18)\times 10^{-5}$. In order to explain the data, the often neglected $W$-exchange contribution to ${\cal B}(\bar{B}^{0}\to\Lambda_{c}^{+}\bar{p})$ should be at the level of $10^{-5}$. With $N_{c}=2.9$ ($a_{2}=0.04$), we are able to get ${\cal B}(\bar{B}^{0}\to\Lambda_{c}^{+}\bar{p})=(1.0^{+0.4}_{-0.3})\times 10^{-5}$ with the errors from the uncertainties of the form factors. In the generalized factorization, one allows $N_{c}$ to shift from 2 to $\infty$ as an effective number to account for the non-factorizable strong interaction. Particularly, $N_{c}\simeq 3$ implies a mild correction ali . Accordingly, we estimate the previously neglected $W$-exchange contributions to ${\cal B}(B\to{\bf B}_{c}\bar{\bf B}^{\prime})$, given in Table 3.

IV Discussions and Conclusions

Several suppression factors have been proposed to support the neglecting of the $W$-exchange (annihilation) contributions. The first one is in analogy with the semileptonic $B\to\ell\bar{\nu}_{\ell}$ decays Bevan:2014iga . In fact, we obtain

[TABLE]

with ${\cal R}_{m}\equiv(m_{-}/m_{+})^{2}$. While $m_{\mu}^{2}$ is responsible for the helicity suppressed ${\cal B}(B^{-}\to\mu^{-}\bar{\nu}_{\mu})\simeq 10^{-7}$ Hou:2019uxa , $B\to{\bf B}_{c}\bar{\bf B}^{\prime}$ with $m_{+}^{2}=(m_{{\bf B}_{c}}+m_{\bar{\bf B}^{\prime}})^{2}$ obviously allows for helicity-flip. The second one is the decay constant $f_{B}=0.19$ GeV in Eq. (IV), regarded as small and suppressing the $W$-exchange contribution Jarfi:1990ej . However, it is found that ${\cal B}(B^{-}\to\tau^{-}\bar{\nu}_{\tau})$ with $f_{B}$ can still be as large as $(1.09\pm 0.24)\times 10^{-4}$ pdg . Besides, the ratio of $m_{\mu}^{2}/m_{\tau}^{2}\sim{\cal O}(10^{-3})$ implies the smallness of ${\cal B}(B^{-}\to\mu\bar{\nu}_{\mu})/{\cal B}(B^{-}\to\tau\bar{\nu}_{\tau})\sim{\cal O}(10^{-3})$. Hence, it is reasonable to infer that $f_{B}$ cannot be the main cause of the suppression. Third, in the charmless $B\to{\bf B}\bar{\bf B}^{\prime}$ decays, one of the suppression factors for ${\cal B}(\bar{B}^{0}\to p\bar{p})\sim{\cal O}(10^{-8})$ comes from $(f_{1},g_{1})\propto(\alpha_{s}/t)^{2}$ with $t=m_{B}^{2}$, which corresponds to the hard gluons that transfer the energy of $m_{B}$ Hsiao:2014zza ; Hsiao:2018umx . In the $B\to{\bf B}_{c}\bar{\bf B}^{\prime}$ decays, however, $(f_{1},g_{1})$ with a charmed baryon are not small at $t=m_{B}^{2}$, as shown in Eqs. (9) and (III). Note that $(f_{1},g_{1})$ in the timelike and spacelike regions can be associated with the crossing symmetry, and $(f_{1},g_{1})$ with the light-front quark model agree with those in lattice QCD calculations Lih_Lcpbar ; Meinel:2016dqj . By contrast, ${\cal B}(\bar{B}^{0}\to\Lambda_{c}^{+}\bar{p})$ was once calculated as small as $4.6\times 10^{-7}$ Korner:1988mx , taken as the another theoretical support for the neglect of the $W$-exchange diagram Cheng:2002sa ; Bevan:2014iga . However, the estimation was done with the baryonic form factors in the dipole form of $F(0)/(1-t/m_{D^{*}}^{2})^{2}$ adopted from the $\Lambda_{c}^{+}\to p$ transition Korner:1988mx , such that the momentum transfer at $m_{B}$ exceeds the $D^{*}$ meson pole, causing the suppression, whereas the validity of the $D^{*}$ meson pole has never been tested in the timelike region. Besides, $F(0)$ was not clearly given, due to the lack of the studies on the quark and QCD models at that time.

With the $W$-exchange contributions, we obtain

[TABLE]

where ${\cal B}(\bar{B}^{0}\to\Sigma_{c}^{+}\bar{p})$ is consistent with the current data in Eq. (I). In particular, ${\cal B}(\bar{B}^{0}\to\Sigma_{c}^{0}\bar{n})$ is predicted to be ten times larger than the pole model calculation Cheng:2002sa . On the other hand, by only considering the $W$-emission contribution, the pQCD approach gives that ${\cal B}(\bar{B}^{0}\to\Lambda_{c}^{+}\bar{p})=(2.3-5.1)\times 10^{-5}$ He:2006vz , which is more than 4 standard deviations away from the measured central value. It is interesting to know if the pQCD approach would overestimate ${\cal B}(B^{-}\to\Sigma_{c}^{0}\bar{p})$ as well. One way of comparing the $W$-emission and exchange contributions is by studying decays other than $\bar{B}^{0}\to\Lambda_{c}^{+}\bar{p},\Sigma_{c}^{+}\bar{p}$, which receive contributions from both ${\cal A}_{\rm{ex}}$ and ${\cal A}_{em}$. By taking ${\cal A}_{\rm{ex}}$ as the primary contribution, we predict that

[TABLE]

with ${\cal R}_{B_{s}}\equiv|(V_{us}f_{B_{s}})/(V_{ud}f_{B})|^{2}=0.075$. In future measurements the deviation of ${\cal R}_{B_{s}}$ from the theoretical prediction can be used to evaluate the size of ${\cal A}_{em}$ in the decays. Being pure $W$-exchange processes, $\bar{B}^{0}\to\Xi_{c}^{+}\bar{\Sigma}^{-},\Xi_{c}^{{}^{\prime}+}\bar{\Sigma}^{-},\Omega_{c}^{0}\bar{\Xi}^{0}$ and $\bar{B}^{0}_{s}\to\Sigma_{c}^{0}\bar{n},\Lambda_{c}^{+}\bar{p},\Sigma_{c}^{+}\bar{p}$ can be excellent probes for the examination of the $W$-exchange mechanism. Therefore, we predict that

[TABLE]

which are all within the capability of the current $B$ factories.

In summary, we have studied the charmful two-body baryonic $B_{(s)}^{0}\to{\bf B}_{c}\bar{\bf B}^{\prime}$ decays, with ${\bf B}_{c}=(\Xi_{c}^{+,0},\Lambda_{c}^{+})$ or $(\Sigma_{c}^{++,+,0},\Xi_{c}^{\prime+,0},\Omega_{c}^{0})$. We have found that the often neglected $W$-exchange contribution is in fact not negligible. Particularly, we obtain that ${\cal B}(\bar{B}^{0}\to\Sigma_{c}^{+}\bar{p})=(2.9^{+0.8}_{-0.9})\times 10^{-6}$, agreeing with the current data. For decays that only proceed via the $W$-exchange diagram, we predict ${\cal B}(\bar{B}_{s}^{0}\to\Lambda^{+}_{c}\bar{p})=(0.8\pm 0.3)\times 10^{-6}$ and ${\cal B}(\bar{B}^{0}\to\Xi_{c}^{+}\bar{\Sigma}^{-})=(1.1\pm 0.4)\times 10^{-5}$, which provide an excellent window for the LHCb and Belle II experiments to look into the long-ignored $W$-exchange mechanism.

ACKNOWLEDGMENTS

This work was supported in part by National Science Foundation of China (11675030) and U. S. National Science Foundation award ACI-1450319.