Abstract
We study the semileptonic \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\) decays with \(\textbf{B}{\bar{\textbf{B}}'}\) (\(L{\bar{L}}'\)) representing a baryon (lepton) pair. Using the new determination of the \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition form factors, we obtain \(\mathcal{B}(B^-\rightarrow p{\bar{p}} \mu ^-{\bar{\nu }}_\mu ) =(5.4\pm 2.0)\times 10^{-6}\) agreeing with the current data. Besides, \(\mathcal{B}(B^-\rightarrow \Lambda {\bar{p}} \nu {\bar{\nu }})=(3.5\pm 1.0)\times 10^{-8}\) is calculated to be 20 times smaller than the previous prediction. In particular, we predict \(\mathcal{B}({\bar{B}}^0_s\rightarrow p{\bar{\Lambda }} e^- {\bar{\nu }}_e,p{\bar{\Lambda }} \mu ^- {\bar{\nu }}_\mu ,p{\bar{\Lambda }} \tau ^- {\bar{\nu }}_\tau ) =(2.1\pm 0.6,2.1\pm 0.6,1.7\pm 1.0)\times 10^{-6}\) and \(\mathcal{B}({\bar{B}}^0_s\rightarrow \Lambda {\bar{\Lambda }} \nu {\bar{\nu }})=(0.8\pm 0.2)\times 10^{-8}\), which can be accessible to the LHCb experiment.
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1 introduction
In the non-leptonic baryonic B decays, the observation of \(B\rightarrow p{\bar{p}} (\pi ,K^{(*)},D^{(*)})\) and \(B^-\rightarrow \Lambda {\bar{p}} (J/\psi ,\gamma )\) suggests the unique existence of the \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition [1,2,3], with which the CP asymmetries of \(B^-\rightarrow p{\bar{p}} (\pi ^-,K^{(*)-})\) [4, 5] and the branching fractions of \(B^-\rightarrow \Lambda {\bar{p}} D^{(*)0}\), \({\bar{B}}^0\rightarrow \Sigma ^0{\bar{\Lambda }} D^0\) [6] have been predicted, and verified by the later measurements [7].
The semileptonic B decays of \(B^-\rightarrow p{\bar{p}} \ell ^-{\bar{\nu }}_\ell \) and \(B^-\rightarrow \Lambda {\bar{p}}\nu _\ell {\bar{\nu }}_\ell \) with \(\ell \) denoting e, \(\mu \) or \(\tau \) can provide another evidence for the \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition [8, 9]. Like the studies of the semileptonic \(B^-\rightarrow \pi ^+\pi ^-\ell ^-\bar{\nu }_\ell \) decays [10, 11], the full dibaryon invariant mass spectrum can be used to test the possible co-existence of the resonant and non-resonant contributions. Therefore, we have predicted \(\mathcal{B}(B^-\rightarrow p{\bar{p}} e^- \bar{\nu }_e) =(1.04\pm 0.26\pm 0.12)\times 10^{-4}\) [8] and \(\mathcal{B}(B^-\rightarrow \Lambda {\bar{p}} \nu {\bar{\nu }}) =(7.9\pm 1.9)\times 10^{-7}\) [9]. We have also predicted \(\mathcal{R}_{e/\mu }\equiv \) \(\mathcal{B}(B^-\rightarrow p{\bar{p}} e^- {\bar{\nu }}_e)/\mathcal{B}(B^-\rightarrow p{\bar{p}} \mu ^- {\bar{\nu }}_\mu )\simeq 1\) [8]. By contrast, the pole model argument leads to the evaluation of \(\mathcal{B}(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\ell \bar{\nu }_\ell )=10^{-5}-10^{-6}\) [12].
Experimentally, it has been measured that [13,14,15,16]
The threshold effect commonly observed in \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}M\) is also observed in \(B^- \rightarrow p{\bar{p}} \mu ^-\bar{\nu }_e\) [15], which is drawn as a peak around the threshold area of \(m_{\textbf{B}{\bar{\mathbf{B'}}}} \simeq m_\textbf{B}+m_{\bar{{ B'}}}\) in the \({\textbf{B}{\bar{\mathbf{B'}}}}\) invariant mass spectrum. There is no sign that the B to \({\textbf{B}{\bar{\mathbf{B'}}}}\) transition receives a resonant contribution. Nonetheless, it is clearly seen that \(\mathcal{B}_{ex}(B^- \rightarrow p{\bar{p}} \mu ^-{\bar{\nu }}_\mu )\) is 20 times smaller than the prediction [8]. This has been pointed out as the theoretical challenge to alleviate the discrepancy [17]. On the other hand, the ratio \(\mathcal{R}_{e/\mu }\simeq 1\) as a test of the lepton universality is not conclusive, and the prediction of \(\mathcal{B}(B^- \rightarrow \Lambda \bar{p} \nu {\bar{\nu }})\) is within the experimental upper bound.
In Ref. [6], the \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition form factors (\(F_{\textbf{B}{\bar{\mathbf{B'}}}}\)) are extracted with the data from \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}M\), which cause the overestimation of \(\mathcal{B}(B\rightarrow p{\bar{p}}\ell {\bar{\nu }})\). With the same theoretical inputs, \(\mathcal{B}(B^- \rightarrow \Lambda {\bar{p}} \nu {\bar{\nu }})\) might be overestimated as well [9]. A question is hence raised: whether there exist the universal \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition form factors to explain the nonleptonic and semileptonic baryonic B decays.
In this paper, we propose to perform a new global fit, in order to accommodate the current data of \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\) with \(L{\bar{L}}'\) denoting a lepton pair and \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}M\). With \(F_{\textbf{B}{\bar{\mathbf{B'}}}}\) determined from the new global fit, we will re-investigate \(B^- \rightarrow \Lambda {\bar{p}} \nu {\bar{\nu }}\). Since LHCb has been able to accumulate more events for the \({\bar{B}}_s^0\) decays, we will study \({\bar{B}}_s^0\rightarrow p{\bar{\Lambda }} \ell ^- {\bar{\nu }}\) and \(\bar{B}_s^0\rightarrow \Lambda {\bar{\Lambda }} \nu {\bar{\nu }}\) decays for future measurements.
2 Formalism
The semileptonic baryonic B decays come from the quark-level \(b\rightarrow u\ell {\bar{\nu }}_\ell \) and \(b\rightarrow s\nu _\ell {\bar{\nu }}_\ell \) processes. In Fig. 1a, \(b\rightarrow u\ell {\bar{\nu }}_\ell \) appear as the tree-level \(b\rightarrow uW,W\rightarrow \ell {\bar{\nu }}_\ell \) decays. Due to the loop contributions from the penguin-level \(b\rightarrow sZ,Z\rightarrow \nu _\ell {\bar{\nu }}_\ell \) and box diagrams in Fig. 1b,c, respectively [18], \(b\rightarrow s\nu _\ell {\bar{\nu }}_\ell \) can be rarer than \(b\rightarrow u\ell {\bar{\nu }}_\ell \). The effective Hamiltonians for the above semileptonic b decays are given by [18, 19]
where \(G_F\) is the Fermi constant, \(V_{ub}\) and \(\lambda _t\equiv V_{ts}^* V_{tb}\) are the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements, and \(D(x_t)\) with \(x_t\equiv m_t^2/m_W^2\) is the top-quark loop function [18,19,20,21]. According to \(\mathcal{H}(b\rightarrow u \ell {\bar{\nu }}_\ell ,s \nu _\ell \bar{\nu }_\ell )\), the amplitudes of \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\) with \(L{\bar{L}}'=(\ell {\bar{\nu }}_\ell ,\nu _\ell {\bar{\nu }}_\ell )\) can be derived as [8, 9]
with \(\langle {\textbf{B}{\bar{\mathbf{B'}}}}|({\bar{q}}b)|B\rangle \) representing the matrix elements of the B meson to \({\textbf{B}{\bar{\mathbf{B'}}}}\) transition. In Fig. 1a–c, \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\) occur as \(B^-\rightarrow p{\bar{p}}\ell {\bar{\nu }}_\ell ,\Lambda {\bar{p}}\nu _\ell {\bar{\nu }}_\ell \) and \({\bar{B}}_s^0\rightarrow p{\bar{\Lambda }}\ell {\bar{\nu }}_\ell ,\Lambda \bar{\Lambda }\nu _\ell {\bar{\nu }}_\ell \) for our study.
The amplitudes of the non-leptonic \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}M\) decays have two forms [1, 22, 23]: \(\mathcal{M}_C\propto \langle {\textbf{B}{\bar{\mathbf{B'}}}}|({\bar{q}} q')|0\rangle \) \(\times \langle M|({\bar{q}} b)|B\rangle \) and \(\mathcal{M}_T\propto \langle M|({\bar{q}} q')|0\rangle \langle {\textbf{B}{\bar{\mathbf{B'}}}}|({\bar{q}} b)|B\rangle \), where \(\mathcal{M}_C\) denotes the current amplitude with \({\textbf{B}{\bar{B'}}}\) produced from the quark current [22,23,24,25,26], and \(\mathcal{M}_T\) the transition amplitude with \({\textbf{B}{\bar{B'}}}\) from the B meson transition [1,2,3]. Clearly, \(\mathcal{M}_T(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}M)\) and \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\) can be related by \(\langle {\textbf{B}{\bar{\mathbf{B'}}}}|({\bar{q}} b)|B\rangle \) [8, 9]. As seen in Fig. 2, \(B\rightarrow p{\bar{p}} M\) with \(M=(\pi ,K)\), \(B\rightarrow p{\bar{p}} V\) with \(V=(\rho ,K^{*})\), and \({\bar{B}}^0\rightarrow p{\bar{p}} D^{0(*)}\) involve the transition amplitudes, given by [1, 27,28,29]
with \((q,q')=(u,d)\) for \(B^-\rightarrow p{\bar{p}} \pi ^-\) and \(B^-\rightarrow p\bar{p}\rho ^-\), \((q,q')=(u,s)\) for \(B^-\rightarrow p{\bar{p}} K^{-}\) and \(B^-\rightarrow p{\bar{p}} K^{*-}\), and \((q,q')=(d,s)\) for \({\bar{B}}^0\rightarrow p{\bar{p}} \bar{K}^0\) and \({\bar{B}}^0\rightarrow p{\bar{p}} {\bar{K}}^{*0}\). The parameters in Eq. (4) result from the factorization approach [30], written as
with \(a_i=c^{eff}_i+c^{eff}_{i\pm 1}/N_c\) for \(i=\)odd (even), where \(c_i^{eff}\) are the effective Wilson coefficients, and \(N_c\) the color number [30].
The matrix elements of the \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition in Eqs. (3) and (4) can be presented as [1, 3, 6, 17, 27,28,29]
with \(V^b_\mu (A^b_\mu )\equiv {\bar{q}}\gamma _\mu (\gamma _5) b\), \(S^b(P^b)\equiv {\bar{q}} b\), and \(p_\mu \equiv (p_B-p_\textbf{B}-p_{\bar{\textbf{B}'}})_\mu \). \(F_{\textbf{B}{\bar{\mathbf{B'}}}}\equiv (g_i,f_i,{\bar{g}}_j,{\bar{f}}_j)\) with \(i=1,2,\ldots ,5\) and \(j=1,2, 3\) are the \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition form factors.
\(F_{\textbf{B}{\bar{\mathbf{B'}}}}\) are momentum dependent. In terms of perturbative QCD (pQCD) counting rules [1, 3, 6, 28, 31,32,33], one parameterizes that
with \(t\equiv (p_\textbf{B}+p_{\bar{\textbf{B}'}})^2\). For \(F_{\textbf{B}{\bar{\mathbf{B'}}}}\propto 1/t^n\), \(n=3\) corresponds to the three gluon propagators, which are drawn in Figs. 1a–c and 2a–c. Since \(V_\mu ^b\) and \(A_\mu ^b\) can be combined as the right-handed chiral current \(R_\mu =(V_\mu ^b+A_\mu ^b)/2\), and the baryon decomposed of the right and left-handed states, that is, , it leads to [6, 17]
where \(|B_q\rangle \sim {\bar{b}}\gamma _5 q|0\rangle \) has been used. As the chiral charge, \(Q\equiv R_{\mu =0}\) annihilates the b quark, and creates a valence quark in \(\textbf{B}\), while the spectator quark in the B meson is transformed as a valence quark (\({\bar{q}}_i\)) in \({\bar{\textbf{B}}}'\). We hence obtain \(G_{R,L}^{(k)}\) as the \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition form factors in the chiral representation. When the chirality states of a spinor (R, L) are taken as the helicity states \((\uparrow ,\downarrow )\), one can see \({\bar{q}}_i\) with the helicity to be (anti-)parallel \([||(\overline{||})]\) to the helicity of \({\bar{\textbf{B}'}}\), such that the chiral charge acting on \({\bar{q}}_i\) can be more explicitly defined as \(Q_{||(\overline{||})}(i)\) (\(i=1,2,3\)). We thus derive that
where \(e_{||}^{R(L)}\) and \(e_{\overline{||}}^{R(L)}\) sum over the weight factors of \(\textbf{B}_{R(L)}{\bar{\textbf{B}'}}_{R(L)}\), and \(\bar{e}_{||}^{R(L)}\) and \({\bar{e}}_{\overline{||}}^{R(L)}\) those of \(\textbf{B}_{L(R)}{\bar{\textbf{B}'}}_{R(L)}\). By defining \(G_{||(\overline{||})}^{(k)}\equiv D_{||(\overline{||})}^{(k)}/t^3\) \((k=2,3,\ldots ,5)\), we relate the two sorts of the form factors as [1, 3, 28]
for \(\langle p{\bar{p}}|({\bar{u}} b)|B^-\rangle \), \(\langle p{\bar{p}}|(\bar{d} b)|{\bar{B}}^0\rangle \), \(\langle \Lambda {\bar{p}}|({\bar{s}} b)|B^-\rangle \), \(\langle p{\bar{\Lambda }}|({\bar{u}} b)|\bar{B}^0_s\rangle \), and \(\langle \Lambda {\bar{\Lambda }}|({\bar{s}} b)|\bar{B}^0_s\rangle \), respectively. Likewise, we perform a derivation for \({\bar{g}}_j\) (\({\bar{f}}_j\)) through the (pseudo-)scalar current, which leads to [28, 29]
for \(\langle p{\bar{p}}|({\bar{u}} b)|B^-\rangle \) and \(\langle p\bar{p}|({\bar{d}} b)|{\bar{B}}^0\rangle \), respectively. Note that \(R(L)\sim \uparrow (\downarrow )\) is based on the approximation with the large energy transfer, which is conveniently presented as \(t\rightarrow \infty \). It is also derived that the correction term is of order \(m_q/\sqrt{t}\) [31,32,33]. In fact, \(\sqrt{t}\) of a few GeV has been large enough to suppress the correction term [33]. Consequently, the relations with the chirality (helicity) symmetry are shown to be able to describe the scattering processes [33]. For the baryonic B decays, \(\sqrt{t} >2\) GeV is also sufficient for the holding of the relations in Eqs. (10) and (11).
The four-body \(B(p_B)\rightarrow \textbf{B}(p_\textbf{B}) {\bar{\textbf{B}'}}(p_{\bar{\textbf{B}'}})L(p_L){\bar{L}}' (p_{{\bar{L}}'})\) decay involves five kinematic variables in the phase space, that is, \(s\equiv (p_{L}+p_{\bar{L}'})^2\equiv m_{L{\bar{L}}'}^2\), t, and \((\theta _\textbf{B},\theta _\textbf{L},\phi )\) [34,35,36]. As depicted in Fig. 3, the angle \(\theta _\mathbf{B(L)}\) is between \(\vec {p}_\textbf{B}\) (\(\vec {p}_{L}\)) in the \({\textbf{B}{\bar{B'}}}\) (\(L\bar{L}'\)) rest frame and the line of flight of the \({\textbf{B}{\bar{B'}}}\) (\(L\bar{L}'\)) system in the B meson rest frame. The angle \(\phi \) is from the \({\textbf{B}{\bar{B'}}}\) plane to the \(L{\bar{L}}'\) plane defined by the momenta of the \({\textbf{B}{\bar{B'}}}\) pair and \(L{\bar{L}}'\) pair in the B meson rest frame, respectively. The partial decay width then reads [8, 9]
where \(X=[(m_B^2-s-t)^2/4-st]^{1/2}\), \(\alpha _\textbf{B}=\lambda ^{1/2}(t,m_\textbf{B}^2,m_{\bar{\textbf{B}'}}^2)/t\), and \(\alpha _\textbf{L}=\lambda ^{1/2}(s,m_{L}^2,m_{{\bar{L}}'}^2)/s\), with \(\lambda (a,b,c)=a^2+b^2+c^2-2ab-2bc-2ca\). For integration, the allowed ranges of the five variables are \((m_L+m_{{\bar{L}}'})^2\le s\le (m_{B}-\sqrt{t})^2\), \((m_\textbf{B}+m_{\bar{\textbf{B}'}})^2\le t\le (m_{B}-m_L-m_{{\bar{L}}'})^2\), \(0\le \theta _\textbf{B,L}\le \pi \), and \(0\le \phi \le 2\pi \). The partial decay width of \(B(p_B)\rightarrow \textbf{B}(p_\textbf{B}){\bar{\textbf{B}'}}(p_{\bar{\textbf{B}'}})M(p_M)\) involves two variables in the phase space, given by [3, 28]
where \(\beta _\textbf{B}=[1-(m_\textbf{B}+m_{\bar{\textbf{B}'}})^2/t][1-(m_\textbf{B}-m_{\bar{\textbf{B}'}})^2/t]\), \(\beta _t=[(m_B+m_M)^2-t][(m_B-m_M)^2-t]\), and \(\theta \) is the angle between the meson and baryon moving directions in the \({\textbf{B}{\bar{B'}}}\) rest frame. The allowed regions of the variables are \(-1<\cos \theta <1\) and \((m_\textbf{B}+m_{\bar{\textbf{B}'}})^2<t<(m_B-m_M)^2\). For the global fit in the next section, we define the CP asymmetry [4, 37], and angular asymmetries of \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}M\) [3, 26, 28] and \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\) [8, 9], written as
where \({\bar{B}}\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}} {\bar{M}}\) represents the anti-particle decay.
3 Numerical results and discussions
In the numerical analysis, the CKM matrix elements in the Wolfenstein parameterization read [7]
with \((\lambda ,A,\rho ,\eta )=(0.225,0.826,0.163\pm 0.010,0.357\pm 0.010)\).
From Refs. [18,19,20,21], we adopt D(x) as
where \(L_2(1-x)\equiv \int ^x_1 ln(t)/(1-t) dt\) and \(\mu =m_b\). For \(B\rightarrow p{\bar{p}} M(V)\) and \({\bar{B}}^0\rightarrow p{\bar{p}} D^{0(*)}\), we present \(c_i^{eff}\) in Table 1, where b and \({\bar{b}}\) decays are both considered, together with the decay constants \((f_\pi ,f_K,f_\rho ,f_{K^*})= (130.2\pm 1.2,155.7\pm 0.3,210.6\pm 0.4,204.7\pm 6.1)\) MeV [7, 38] and \((f_D,f_{D^*})=(208.9\pm 6.5,252.2\pm 22.7)\) MeV [28, 39]. In the generalized edition of the factorization [30, 37], \(N_c\) is taken as the effective color number with \(N_c^{(eff)}=(2,3,\infty )\), in order that the non-factorizable QCD corrections can be estimated.
Using the minimum \(\chi ^2\)-fit of
we test if the observables of non-leptonic and semileptonic baryonic B decays can both be interpreted, where \(\mathcal{O}_{th}^i\) stand for the theoretical calculations of \(\mathcal{B}\), \(\mathcal{A}_{CP}\) and \(\mathcal{A}_{FB}\), while \(\mathcal{O}_{ex}^i\) the experimental inputs in Table 2, together with \(\sigma _{ex}^i\) the experimental errors. Since the \(V_{ub}\) in Eq. (3) is for the exclusive baryonic \(B_{(s)}\) decays, which can be different from that in the inclusive ones [40,41,42], we choose \(|V_{ub}|_{ex}=(3.43\pm 0.32)\times 10^{-3}\) determined from the \({\bar{B}}_s^0\) and baryonic \(\Lambda _b\) decays [7] as our experimental input in Eq. (17).
With 16 experimental inputs from Table 2 and \(|V_{ub}|_{ex}\), we fit \((D_{||},D_{\overline{||}},D_{2,3,4,5})\) and \(({\bar{D}}_{||},{\bar{D}}_{\overline{||}},{\bar{D}}_{2,3})\) in Eqs. (10) and (11), respectively, and \(|V_{ub}|_{th}\), which amount to 11 parameters, such that the number of degrees of freedom denoted by d.n.f is counted as \(d.n.f=16-11=5\). As a result, we obtain \(\chi ^2/n.d.f=1.86\) as a measure of the global fit, and extract that
with \(N_c^{eff}=2\) and \(\infty \) for \(B\rightarrow p{\bar{p}} M(V)\) and \(B\rightarrow p{\bar{p}} D^{0(*)}\), respectively. Using the parameters in Eq. (18), we calculate the branching fractions and angular asymmetries of \(B^-\rightarrow p{\bar{p}}\ell {\bar{\nu }},\Lambda \bar{p}\nu {\bar{\nu }}\) and \({\bar{B}}_s^0\rightarrow p{\bar{\Lambda }}\ell \bar{\nu },\Lambda {\bar{\Lambda }}\nu {\bar{\nu }}\), of which the results are compared with the experimental data in Table 3. We also draw the \({p{\bar{p}}}\) invariant mass spectrum for \(B^-\rightarrow p\bar{p}\mu ^-{\bar{\nu }}_\mu \) in Fig. 4.
4 Discussions and conclusions
Since \(\chi ^2/n.d.f=1.86\) presents a reasonable fit, it indicates that the most recent data in Table 2 can be explained. It is interesting to note that \(\mathcal{B}(B^-\rightarrow p{\bar{p}} \pi ^-,p\bar{p}\rho ^-)\) [3, 4] were once overestimated [7, 43, 44], and the relation of \(\mathcal{A}_{FB}(B^-\rightarrow p{\bar{p}}\pi ^-)\simeq \mathcal{A}_{FB}(B^-\rightarrow p{\bar{p}} K^-)\) [3] was not verified by the measurements [43, 44]. This is due to \(F_{\textbf{B}{\bar{\mathbf{B'}}}}\) determined by the \(B\rightarrow p{\bar{p}} K\) data [3], while \(B\rightarrow p{\bar{p}} K\) are in fact the penguin dominated decays with \(\hat{\mathcal{M}}_6\propto \langle p\bar{p}|(S-P)^b|B\rangle \) to give the main contribution. To avoid the inconsistency unable to be solved at that time, one performed the extraction of Ref. [6] that excluded \(\mathcal{B}(B^-\rightarrow p{\bar{p}} K^-)\), \(\mathcal{B}({\bar{B}}^0\rightarrow p{\bar{p}} {\bar{K}}^0)\), and \(\mathcal{A}_{FB}(B^-\rightarrow p{\bar{p}} K^-)\), in order that the more associated tree dominated decays of \(B\rightarrow p{\bar{p}}(\pi ,\rho )\), \({\bar{B}}^0\rightarrow p{\bar{p}} D^{0(*)}\), and \(B\rightarrow {\textbf{B}{\bar{\textbf{B}}}}L{\bar{L}}'\) can be studied. However, it resulted in an unsatisfactory global fit not to accommodate the all data.
As \(F_{\textbf{B}{\bar{\mathbf{B'}}}}\) determined in this work can be universal for the non-leptonic and semileptonic decay channels, we calculate \(\mathcal{B}(B^-\rightarrow p{\bar{p}} e^-{\bar{\nu }}_e)=(5.3\pm 2.0)\times 10^{-6}\) and \(\mathcal{B}(B^-\rightarrow p{\bar{p}}\mu ^-{\bar{\nu }}_\mu )=(5.4\pm 2.0)\times 10^{-6}\) agreeing with the experimental values. Moreover, we revisit \(B^-\rightarrow \Lambda {\bar{p}}\nu {\bar{\nu }}\), and obtain \(\mathcal{B}(B^-\rightarrow \Lambda {\bar{p}}\nu {\bar{\nu }})=(3.5\pm 1.0)\times 10^{-8}\) 20 times smaller than the number of Ref. [9].
Like the theoretical illustration in \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) and \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}M\) [29, 46], the gluon propagators of Fig. 1a–c play the key role in the \({\textbf{B}{\bar{\mathbf{B'}}}}\) formation of \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\), where two of them provide the valence quarks in \({\textbf{B}{\bar{B'}}}\), while the another one speeds up the spectator quark in B. Accordingly, the approach of pQCD counting rules derives that \(F_{\textbf{B}{\bar{\mathbf{B'}}}}\propto 1/t^3\).
One can test the momentum dependence of \(B^-\rightarrow p{\bar{p}}\mu ^-\bar{\nu }_\mu \), which is by scanning the partial branching fraction as a function of \(\sqrt{t}=m_{p{\bar{p}}}\). In Fig. 4, as we draw the line to agree with the five data points [15]; particularly, those around the area of \(\sqrt{t}\sim m_\textbf{B}+m_{\bar{\textbf{B}'}}\) for the threshold effect, it is shown that \(F_{\textbf{B}{\bar{\mathbf{B'}}}}\) as a function of 1/t can describe \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\).
By normalizing the prediction of the pQCD model [8], LHCb draws the \(m_{p{\bar{p}}}\) spectrum of \(B^-\rightarrow p{\bar{p}}\mu {\bar{\nu }}\) in Fig. 4 of Ref. [15], where the line is higher and narrower than our result. The difference is caused by the fact that the line of of Ref. [15] is chosen to more agree with the two data points around \(m_{p{\bar{p}}}\sim 2.5\) GeV. Subsequently, the peak should reach \(17\times 10^{-6}\) to be above the data point around \(m_{p{\bar{p}}}\sim 2\) GeV for integrating over the partial branching fraction as large as \(\mathcal{B}\simeq 5\times 10^{-6}\). In comparison, our result prefers to agree with the threshold data points; however, requiring some broadening to give a sufficient branching fraction.
The decay channel \({\bar{B}}^0_s\rightarrow \Lambda {\bar{p}} K^+({\bar{\Lambda }} p K^-)\) is the first observation of a baryonic \({\bar{B}}_s^0\) decay [47], whose branching fraction \(\mathcal{B}(\bar{B}^0_s\rightarrow \Lambda {\bar{p}} K^+ +{\bar{\Lambda }} p K^-)=5.46\times 10^{-6}\) is as large as those of the three-body baryonic \(B^-\) (\({\bar{B}}^0\)) decays. Hence, the semileptonic baryonic \({\bar{B}}^0_s\) decay is supposed to be compatible with \(B^-\rightarrow p{\bar{p}}\ell {\bar{\nu }}_\ell \). In our prediction, we present
which are accessible to the LHCb experiment, whereas \(\mathcal{B}(\bar{B}^0_s\rightarrow \Lambda {\bar{\Lambda }} \nu {\bar{\nu }})=(0.8\pm 0.2)\times 10^{-8}\) is relatively small.
Because of \(m_\tau \gg m_{e,\mu }\) that strongly shrinks the phase space, it is anticipated that \(\mathcal{B}(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\tau {\bar{\nu }})\ll \mathcal{B}(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}e{\bar{\nu }},{\textbf{B}{\bar{\mathbf{B'}}}}\mu {\bar{\nu }})\). Nonetheless, the amplitude of Eq. (3) and the matrix elements of Eq. (6) result in
in \(\mathcal{M}(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}} \ell {\bar{\nu }})\), where \(m_\tau \) is able to enhance the decay. We thus obtain
as large as their counterparts. Likewise, the mass effect can be found in \(\mathcal{M}(B\rightarrow \tau {\bar{\nu }})\propto m_\tau \bar{u}_\tau (1+\gamma _5)v_{{\bar{\nu }}}\) [48, 49] and \(\mathcal{M}(B\rightarrow \textbf{B}_c\bar{\textbf{B}}')\propto m_c \langle \textbf{B}_c\bar{\textbf{B}}'|\bar{c}(1+\gamma _5)q|0\rangle \) [50], where \(m_\tau \) and \(m_c\) alleviate the decays from helicity suppression.
We study the angular asymmetries of the semileptonic \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\) decays. While \(\mathcal{A}_{FB,\theta _\textbf{B}}(B^-\rightarrow p{\bar{p}} \ell ^- {\bar{\nu }}_\ell )\) are around several percents, \(\mathcal{A}_{FB,\theta _\textbf{B}}({\bar{B}}^0_s\rightarrow p{\bar{\Lambda }} e^- {\bar{\nu }}_e, p{\bar{\Lambda }} \mu ^- {\bar{\nu }}_\mu )\) and \(\mathcal{A}_{FB,\theta _\textbf{B}}({\bar{B}}^0_s\rightarrow \Lambda {\bar{\Lambda }} \nu \bar{\nu })\) can be around 25%. Like the three-body baryonic B decays [3, 26, 28], this implies a theoretical sensitivity for \(F_{\textbf{B}{\bar{\mathbf{B'}}}}\) to be confirmed by future measurements.
In summary, we have investigated the semiletonic \(B^-(\bar{B}_s^0)\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'\) decays with \(L\bar{L}'=(\ell {\bar{\nu }}_\ell ,\nu {\bar{\nu }})\). We have newly extracted the \(B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}\) transition form factors with the global fit that includes the data of \(B\rightarrow p{\bar{p}} M(V)\), \(\bar{B}^0\rightarrow p{\bar{p}} D^{0(*)}\) and \(B\rightarrow p{\bar{p}} e^-{\bar{\nu }}_e,p\bar{p}\mu ^-{\bar{\nu }}_\mu \) decays. In our demonstration, \(\mathcal{B}(B^-\rightarrow p{\bar{p}} e^-{\bar{\nu }}_e,p{\bar{p}}\mu ^-{\bar{\nu }}_\mu )\) once overestimated to be as large as \(10^{-4}\) has been reduced to be around \(5\times 10^{-6}\), in agreement with the current data. We have also presented \(\mathcal{B}(B^-\rightarrow \Lambda {\bar{p}}\nu {\bar{\nu }})=(3.5\pm 1.0)\times 10^{-8}\). It has been found that \(\mathcal{B}({\bar{B}}^0_s\rightarrow p\bar{\Lambda }e^- {\bar{\nu }}_e,p{\bar{\Lambda }} \mu ^- {\bar{\nu }}_\mu , p\bar{\Lambda }\tau ^- {\bar{\nu }}_\tau ) =(2.1\pm 0.6,2.1\pm 0.6,1.7\pm 1.0)\times 10^{-6}\) and \(\mathcal{B}({\bar{B}}^0_s\rightarrow \Lambda {\bar{\Lambda }} \nu {\bar{\nu }})=(0.8\pm 0.2)\times 10^{-8}\) can be promising for future measurements.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment:The data used in this paper are publicly available and they can be found in the corresponding references.]
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Acknowledgements
The author would like to thank Prof. C. K. Chua for useful discussions. This work was supported by NSFC (Grants No. 11675030 and No. 12175128).
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Hsiao, YK. Semileptonic baryonic B decays. Eur. Phys. J. C 83, 300 (2023). https://doi.org/10.1140/epjc/s10052-023-11489-9
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DOI: https://doi.org/10.1140/epjc/s10052-023-11489-9