Revisiting semileptonic $B^-\to p\bar{p} \ell^- \bar{\nu}_{\ell}$ decays

We systematically revisit the baryonic four-body semileptonic decays of $B^- \to {\bf B}\bar{\bf B}'\ell^- \bar{\nu}_{\ell}$ by the perturbative QCD counting rules with ${\bf B}$ representing octet baryons and $\ell=e,\mu$. We study the transition form factors of $B^- \to {\bf B} \bar{\bf B}'$ in the limit of $(p_{\bf B}+p_{\bar{\bf B}'})^2 \to \infty $ with the three-body $\bar{B}\to {\bf B}\bar{\bf B}' M$ and $B^- \to p\bar{p} \mu^- \bar{\nu}_{\mu}$ data along with $SU(3)_f$ flavor symmetry. We calculate the decay branching ratios and angular asymmetries as well as the differential decay branching fractions of $B^- \to p \bar{p} \ell^- \bar{\nu}_{\ell}$. In particular, we find that our new result of ${\cal B}( B^- \to p \bar{p} \ell^- \bar{\nu}_{\ell})=(5.21\pm0.34)\times 10^{-6}$, which is about one order of magnitude lower than the previous theoretical prediction of $(10.4\pm2.9)\times 10^{-5}$, agrees well with both experimental measurements of $(5.8^{+2.6}_{-2.3})\times 10^{-6}$ and $(5.3\pm0.4)\times 10^{-6}$ by the Belle and LHCb Collaborations, respectively. We also evaluate the branching ratios and angular asymmetries in other channels of $B^- \to {\bf B}\bar{\bf B}\ell^- \bar{\nu}_{\ell}$, which can be tested by the ongoing experiments at LHCb and BelleII.

These decay modes are useful to determine the value of |V ub | as the works in the other baryonic modes [4] as well as the underlying new CP/T violating effects. The main difficulty in both extracting |V ub | from B − → ppℓν ℓ and constraining the new CP/T violating effects is how to obtain their hadronic transition amplitude of B − → pp as it is hard to be calculated via the usual QCD methods, such as the factorizations and sum rules, which have been widely used in the mesonic decays of B − → π + π − ℓ −ν ℓ (B ℓ4 ) [5][6][7]. Nevertheless, these modes should be considered in the fit for the extraction of V ub . Qualitatively speaking, to reduce the theoretical values for the decay branching ratios of B − → ppℓν ℓ , a smaller value of |V ub | is needed besides the form factors. It is similar to the extractions from the exclusive B and Λ b decays, but lower than that from the inclusive B decays. Clearly, as a baryonic complementary version of B ℓ4 decays, both theoretical and experimental studies of ℓ may shed light on the baryonic transition amplitude of B − → pp, uncover the nature of the QCD dynamics, and improve the measurement of |V ub |.
Because of the rareness of four-body B − → BB ′ ℓ −ν ℓ decays with B (′) representing octet baryons, people have concentrated on the three-bodyB → BB ′ M decays to extract the baryonic transition from factors in the BB ′ transitions, where B(B ′ ) and M are octet (anti-)baryons and pseudoscalar or vector mesons, respectively. There have been several theoretical analyses on the baryonic three-body B → BB ′ M decays based on the factorization assumptions [8][9][10][11][12][13][14]. These baryonic B decays can be basically classified into current production C, transition T and hybrid C + T types [14], with the quark flow diagrams shown in Fig. 1. Among them, the transition one is the only channel directly related tō LHCb [3]. The main reason for such a large prediction is that there were short of relevant data as well as lack of the understanding of the underlined QCD dynamics for the baryonic transition of B − → pp. In this work, we would like to reanalyze the semi-leptonic decays of B − → ppℓ −ν ℓ with the same strategy as that in Ref. [1] with the updated data. In addition, we shall use the flavor symmetry to extend our results to other B − → BBℓ −ν ℓ decays. This work is the first step to know the property ofB → BB ′ transition matrix elements. After getting a better understanding of these elements, we can use them for not only improving the measurement of |V ub | but probing or constraining the new physics effects, such as the T violating triple momentum correlations due to the rich kinematic structure in the four-body decays ofB → BB ′ ℓν.
This paper is organized as follows. In Sec. II, we present our formalism, which contains the effective Hamiltonians and generalized transition form factors. In Sec. III, we show our numerical results of the form factors fitted by three-bodyB → BB ′ M processes and the latest B − → ppµ −ν µ result, and present our predictions of the branching ratios and angular asymmetries in B − → BBℓ −ν ℓ . We also compare our results of pp invariant mass spectrum in the B − → ppµ −ν µ decay with the one measured by the LHCb. We give our conclusions in Sec. IV.

II. FORMALISM
The effective Hamiltonian forB → BB ′ ℓ −ν ℓ at the quark level is given by where G F is the Fermi constant and V ub represents the element of the CKM matrix. The transition amplitude ofB → BB ′ ℓ −ν ℓ can be easily factorized into hadronic and leptonic parts, written as wherel and ν ℓ are the usual Dirac spinors and BB ′ |ūγ µ (1−γ 5 )|B is the unknown hadronic transition amplitude. The most general Lorentz invariant forms of the hadronic transitions for the vector and axial-vector currents can be parametrized by [1,14] BB ′ |ūγ µ b|B = iū(p B ) [g 1 γ µ + ig 2 σ µν p ν + g 3 p µ + g 4 (p B + pB′) µ + g 5 (p B − pB′) µ ] γ 5 v(pB′), respectively, where f i and g i (i = 1, 2, · · · , 5) are the form factors and Inspiring from the threshold effects [15], which have been observed in three-bodyB → BB ′ M decays [16][17][18], and the pQCD counting rules [19][20][21], the momentum dependences of f i and g i can be assumed to be with n = 3, where C f i ,g i are constants determined by the branching ratios of the input channels. Note that n relates to the number of hard-gluon propagators as shown in Fig. 2.
In theB → BB ′ transition, two hard gluons produce the valance quarks in the BB ′ pair separately as well as one more hard gluon is needed to speed up the spectator quark in B [14]. As a result, we can use C f i and C g i to describe the hadronic form factors in both transition type three-body and semileptonic four-body decays. With the help of SU(3) flavor and SU(2) spin symmetries in t → ∞ and heavy quark limit, C f i and C g i are related by only two chiral-conserving parameters C RR and C LL and one chiral-flipping parameter where e RR , e LL and e LR are the electroweak coefficients determined by the spin-flavor structure ofB and BB ′ , and m B,B ′ correspond to baryon and anti-baryon masses, respectively.
The detail derivations of Eq. 5 are presented in Appendix. We list the coefficients of relevant channels in Table I.
Following the same formalism in the literature of B ℓ4 , D ℓ4 and K ℓ4 analyses [22,23], we examine theB → BB ′ ℓ −ν ℓ system in theB rest frame with five kinematic variables, s = (p ℓ + pν ℓ ) 2 , t, θ B , θ ℓ and φ, where √ s and √ t are the invariant masses of lepton and BB ′ pairs, respectively, and three kinematic angles are shown in Fig. 3. The differential decay width is given by where |Ā| 2 is the spin-averaged amplitude and X, β B , β L are given by We can also define the integrated θ B and θ ℓ asymmetries of BB ′ and lepton pairs as follows: with f = B and ℓ, respectively.
leading to |V ub | = 3.8×10 −3 . To extract the form factors, we use the factorization assumption and follow the formula in Ref. [14] to calculate the branching ratios ofB → BB ′ M. The full analysis of three-body kinematics and detail derivations ofB → BB ′ M factorization amplitudes can be found in Ref. [14]. Based on Refs. [14,25,26], the effective Wilson respectively, with χ 2 /d.o.f = 0.28. Our fitting results along with the input data for the transition-type three-body decays ofB → BB ′ M and B − → ppµ −ν µ are presented in Table. III. In Table IV, we show our predictions of other four-body B − → BB ′ ℓ −ν ℓ decays. In Tables III and IV,  Moreover, these momentum behaviors can match the newest B − → ppµ + ν µ differential decay width measured by the LHCb. It is interesting to see that the SU(3) f flavor symmetry guarantees that all observables in B − → Λ 0Σ0 ℓ −ν ℓ are the same as those in B − → Σ 0Λ0 ℓ −ν ℓ . We note that the angular distribution asymmetries in B − → BB ′ ℓ −ν ℓ mainly depend on the electroweak coefficients, which are associated with the spin-flavor structures of the BB ′ pairs. Interestingly, the angular asymmetries of B − → ΛΛℓ −ν ℓ vanish because only the chiral-conserving interaction participates in B − → ΛΛ. As a result, the physical observables in B − → ΛΛℓ −ν ℓ are sensitive to test the availability of pQCD counting rules as well as the asymptotic relations in the limit of t → ∞.
In Table V, we summarize our results of B − → ppℓ −ν ℓ along with the previous theoretical ones [1] as well as the experimental data [2,3]. We note that the theoretical calculations are insensitive to the lepton mass for the ℓ = e and µ channels. As seen from Table V, our new result of B(B − → ppℓ −ν ℓ ) = (5.21 ± 0.34) × 10 −6 is about one order of magnitude lower than the previous theoretical prediction of (10.4 ± 2.9) × 10 −5 in Ref. [1], but the  [2] and agrees well with the combined one of (5.8 +2.6 −2.3 ) × 10 −6 by Belle [2] as well as the recent µ-channel data of (5.3 ± 0.4) × 10 −6 by LHCb [3] which is one of our input channels. Clearly, more precise modeling and explanation are needed to find the QCD origin of the effective color number being N ef f c = 0.51 ± 0.03, indicating that the non-peturbative effects in three-bodyB → BB ′ M channels is much stronger than those in two-bodyB → M 1 M 2 ones. In Fig. 4, we plot the pp invariant mass spectrum in B − → ppℓ −ν ℓ . By comparing our results with the LHCb measurement [3], we find that our spectrum is consistent with the observed one. We further show the differential branching fractions of B − → ppℓ −ν ℓ as functions of √ s ≡ m ℓν , cos θ B and cos θ ℓ in Fig. 5, respectively, which can provide us not only the information of the leptonic sector but also the the spin-flavor relations in the t → ∞ asymptotic limit. These differential branching fractions could be tested by the ongoing experiments. and cos θ ℓ , respectively.

IV. CONCLUSIONS
We have systematically revisited the baryonic four-body semileptonic decays of B − → BB ′ ℓ −ν ℓ with ℓ = e, µ. We have reduced the ten form factors in the hadronic transition of B − → BB ′ into three free parameters in the heavy quark limit and t → ∞. We have performed the minimum χ 2 method to fit the three parameters and the effective color number of N ef f c with χ 2 /d.o.f = 0.28 by using five three-body decays ofB → BB ′ M along with the B − → ppµ −ν µ measurement. We have obtained a consistent fitting result of B(B − → ppℓ −ν ℓ ) = (5.21 ± 0.34) × 10 −6 as well as other input channels. Our B − → ppℓ −ν ℓ decay branching ratio is about one order of magnitude lower than the previous theoretical prediction of (10.4 ± 2.9) × 10 −5 in Ref. [1], and agrees well with the experimental data of (5.8 +2.6 −2.3 ) × 10 −6 and (5.3 ± 0.4) × 10 −6 by Belle [2] and LHCb [3], respectively. In addition, our evaluation of the m pp invariant mass spectrum is also consistent with that by the LHCb measurement [3], demonstrating that the threshold effect and the t −3 dependence of the form factors from the QCD counting rules is still dominant in the baryonic four-body semileptonic decays, while the other Lorentz invariant variables, such as (pB +p B ) 2 , as well as the resonant states are highly suppressed. Furthermore, we have plotted the differential branching fractions with respect to the kinematic variables of m ℓν and cos θ B,ℓ in B − → ppℓ −ν ℓ to provide the information in the lepton sector and angular distributions, respectively. We have also used the flavor symmetry to explore the physical observables in other B − → BB ′ ℓ −ν ℓ decays. In particular, we have found that that the angular asymmetries of B − → ΛΛℓ −ν ℓ vanish due to the absence of the chiral-flipping interaction (e LR = 0). On the theoretical side, the non observed transition modes of the three-bodyB → BB ′ M and four-bodȳ B → BB ′ ℓν decays can help us to relax the assumption of the heavy quark limit once they are measured. Otherwise, the lattice simulation would be currently the most trustworthy theoretical method to reliably extract the hadronic form factors. On the experimental side, some of our results in B − → BB ′ ℓ −ν ℓ can be tested by the ongoing experiments at Belle-II and LHCb. Finally, we remark that the theoretical determination of the four-body decays ofB → BB ′ ℓν would provide a valuable opportunity to search for T violating effects from the triple momentum correlations and improve the measurement of |V ub | as the works in exclusive B and Λ b decays.

APPENDIX
Starting with transition matrix elements of BB ′ |J µ V − J µ A |B with J µ V (A) =qγ µ (γ 5 )b, we assume that theB meson state can be approximately expressed by the field operator of free quarks |B ∼bγ 5 q ′ |0 . Therefore, the matrix elements become where J ′µ = 2q L γ µ / p b q ′ R andJ µ = 2m bqL γ µ q ′ L with q L(R) = (1∓γ 5 )/2q andq L(R) =q(1±γ 5 )/2. Note that, by inserting QCD (gluon-quark-antiquark) vertices in the corresponding diagrams, the Dirac structure in Eq. (11) could be altered. However, because of the asymptotic freedom in QCD, we can treat these alterations from QCD vertices as small perturbations, which are negligible in the limit of (p B + p B ′ ) 2 → ∞. In terms of the crossing-symmetry (c.s.), the final state anti-baryon (B ′ ) is transformed as the initial baryon (B ′ ) in the initial state with opposite four-momentump B ′ = −pB′, resulting in According to Refs. [11,21], the amplitude can be parameterized as In the asymptotic limit of (p B −p B ′ ) 2 → ∞, the helicity of a particle can be approximately treated as its chirality, so that the amplitudes with a specific chirality can be written as where e LR = B, L|(a L q ) † a R q ′ |B ′ , R , e RR = B, R|(a L q ) † a L q ′ |B ′ , R , e LL = B, L|(a L q ) † a L q ′ |B ′ , L , with the corresponding particle creation and annihilation operators (a s q ) † and a s q , and other combinations are zero due to the angular-momentum conservation. From Eqs. (13), (14) and (15), we find that F ′+ = e LR F LR , F ′− = 0,F + = e RR F RR ,F − = e LL F LL .
Consequently, the transition amplitude in (p B −p B ′ ) 2 → ∞ is given by After applying the crossing-symmetry again, we get that where we have used the approximations of p b ≃ pB and m b ≃ mB in the heavy quark limit.