Semileptonic $B-\to p\bar{p} \ell-\bar{\nu}_\ell$ decays

We study the four-body exclusive semileptonic baryonic $\bar B$ decays of $B^-\to p\bar p \ell^- \bar\nu_{\ell}$ ($\ell=e,\mu,\tau$) in the standard model. We find that their decay branching ratios are about $(1.0, 1.0,0.5)\times 10^{-4}$, respectively. In particular, the electron mode is close to the corresponding CLEO's upper limit of $5.2\times 10^{-3}$, while all results are about one or two orders of magnitude larger than the previous estimated values for the inclusive modes of $\bar B\to {\bf B\bar B'}\ell \bar \nu$. Clearly, both B-factories of Belle and BaBar should be able to observe these exclusive four-body modes.


I. INTRODUCTION
In the semileptonic B → Mℓν ℓ decay with a meson M and a charged lepton ℓ, the ℓν ℓ pair involves no direct QCD interaction so that the theoretical description of the amplitude can be reduced to a simple form with the B → M transition.For example, the rate for B0 → π + e − νe is proportional to |V ub f + (q 2 )| 2 , where the form factor f + (q 2 ) for the B0 → π + transition depends on the momentum transfer squared, q 2 .This benefits the precision measurement of |V ub |, where |V ub | is one of the least known Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [1,2] in the Standard Model (SM).As long as we choose a point q 2 = q 2 i in the decay spectrum, the corresponding data point with other parameters can be fixed to extract the value of |V ub |.However, f + (q 2 ) relies on the calculations in the QCD models, such as quark models [3], lattice QCD [4], and Light Cone Sum Rules [5].Starting with q 2 i and |V ub |, one is allowed to inversely extract the q 2 dependence of f + (q 2 ) in different q 2 intervals from the measured data [6][7][8][9][10].The extraction compared with various theoretical models hence improves the knowledge of f + (q 2 ).Moreover, such extraction also provides crosschecks for the B → ρ and B → η (′) transition form factors [8][9][10].In particular, the size of the gluonic singlet contribution [11][12][13] to the B → η ′ transition to explain the unexpectedly large two-body hadronic B → Kη ′ decay rate has been constrained by measuring B → η (′) ℓν decays [8][9][10].Similar to the mesonic cases, it should be interesting to extend the study to baryonic decay modes, such as B → B B′ ℓν with B B′ being a baryon pair, to investigate the B → B B′ transition form factors, which have been used as theoretical inputs in the three-body B → ppM decays.
The factorizable amplitudes for the three-body baryonic B → B B′ M decays are normally classified into current and transition parts, given by respectively, where (q 1 q 2 ) and (q 3 b) stand for the weak currents.The matrix elements of 0 → B B′ in A C are presented as the timelike baryonic form factors, for which the theoretical calculations are available, such as the approach of the pQCD counting rules [14][15][16].
Moreover, we expect that the measurements of the angular distributions in B − → ppe − νe will provide some information to understand the unexpectedly large angular distribution At present, the CLEO Collaboration has given an experimental upper limit: [43] while the theoretical estimation has only been done for the inclusive B → B B′ ℓν decays with charmless dibaryons, given by [44] In this paper, we concentrate on the exclusive four-body semileptonic baryonic decay of In particular, we will study its decay branching ratio in the SM.
The paper is organized as follows.In Sec.II, we provide the formalism, in which we show the decay amplitude and rate of B − → ppℓν ℓ along with the definitions of the transition form factors of B → B B′ .We give our numerical results and discussions in Sec.III.In Sec.
IV, we present the conclusions.

II. FORMALISM
In terms of the effective Hamiltonian, given by for the b → u transition with the W boson emission to ℓν at the quark level, we easily factorize the amplitude for the B − → ppℓν ℓ decay to be where we have parameterized the amplitude in terms of the transition matrix element of B → pp.With Lorentz invariance, the most general forms of the B → B B′ transition form factors can be written as [31] B B′ |q with p = p B − p B − pB′ for the vector and axial-vector quark currents, respectively.For the momentum dependences of f i and g i , we can rely on the results in the B → ppM decays as they share the same B → B B′ transition form factors. Since the pp invariant mass distributions for B → ppM have been observed to peak near the threshold area and flatten out at the large energy region, inspired by the pQCD counting rules [14][15][16]27], we simply take the form factors as [40] with n = 3 and t ≡ (p p + p p) 2 ≡ m 2 pp where D f i and D g i are constants determined by the B → ppM data.Note that the number of n = 3 is for three hard gluons as the propagators to form a baryon pair in the approach of the pQCD counting rules, where two of them attach to valence quarks in pp, while the third one kicks and speeds up the spectator quark in B.
We then need the kinematics for the four-body B(p B ) → B(p B ) B′ (pB′)ℓ(p ℓ )ν(p ν ) decay to integrate over the phase space.As the formalisms in K l4 , D l4 , and B l4 decays given in Refs.[45][46][47], we use five kinematic variables, s ≡ (p ℓ + p ν ) 2 ≡ m 2 ℓν , t, θ B , θ L , and φ to describe the decay.The three angles θ B , θ L , and φ are depicted in Fig. 2, where the angle where X, β B , and β L are given by

III. NUMERICAL RESULTS AND DISCUSSIONS
In our numerical analysis, we take |V ub | = (3.89± 0.44) × 10 −3 from the PDG [48].To deal with D g i and D f i in Eq. ( 7), it is helpful to use the approach of the pQCD counting rules again, where with SU(3) flavor and SU(2) spin symmetries the vector and axial-vector currents are incorporated as two chiral currents in the large t limit [27,31,40].Consequently, D g i and D f i from the vector currents are related by another set of constants D || and D || from the chiral currents.Explicitly, for the B − → pp transition form factors we have [31,40]  ( Thus, the total branching ratios of B − → ppℓν ℓ are found to be where the two errors in Eq. ( 13) are from those in Eq. ( 12) and |V ub |, respectively.The invariant mass spectra and angular distributions for B − → ppe − νe are shown in Fig. 3.The

integrated angular distribution asymmetries, defined by
are obtained to be where the errors are from those in Eq. ( 12).
Since our result on B(B − → ppe − νe ) in Eq. ( 13) is around 1.0 × 10 −4 , which is the same order of magnitude as those of the well measured mesonic B decays at Belle and BaBar, such as B( B0 → π + (ρ + )ℓ − νℓ ) and B(B − → ρ 0 ℓ − νℓ ), this four-body mode should be observed at these B-factories [49].Moreover, as seen from Fig. 3a, the B − → ppe − νe decay inherits the same threshold enhancement as those in the three-body baryonic B decays, resulting from the adoption of 1/t 3 for the momentum dependence in the B − → pp transition form factors.
The spectrum in Fig. 3b reflects the fact that in the helicity structure the amplitude of the e − νe pair is proportional to (E e + E νe ).
It is interesting to note that our study of B − → ppℓν ℓ is similar to that of Finally, it is interesting to point out that the angular distribution asymmetries in Eq. (14) in the B − → ppℓν ℓ decay are sensitive to new physics, such as the currents of (V + A) and (S ± P ) beyond the SM.Note that A θ L = 0.59 in Eq. (15) (see also Fig. 3c) is caused by the ℓ − νℓ pair of (V − A) in the SM, which forms a polarization vector ε µ − (p) to couple to the left-handed helicity state of the virtual weak boson W * − .Therefore, a new physics with the (V + A) current, which lets the ℓν ℓ pair to be another polarization state ε µ + (p), must result in the deviation of A θ L in Eq. (15).Apart from a direct CP violation [41]
, with λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2bc − 2ca.The regions for the five variables of the phase space are given by

FIG. 3 .
FIG. 3. Invariant mass spectra as functions of the invariant masses m pp and m eνe and angular distributions as functions of θ i (i = B, L) for B − → ppe − νe , respectively.
B − → ppK * − [40-42].The terms related to g 2 and f 2 in the B − → pp transition form factors give the main contributions to B − → ppℓ − νℓ .Since the pair of the left-handed electron and the right-handed anti-neutrino in the helicity structure behaves as one of the polarization vector ε µ − (p) with p = p ℓ + p νℓ , leading to ε • p = 0, the contributions from f 3 and g 3 disappear.Those from f 4 and g 4 are effectively small due to the tiny |D 4 || | ≃ 10 GeV 4 .As the branching ratio receives the most contribution near the threshold area, the g 5 (f 5 )-accompanied term (p p − p p ) = (E p − E p , p p − p p ) → (0, 0) is suppressed.Moreover, since the terms of g 2 and f 2 contain σ µν p ν , we have the relation |D g 2 (f 2 ) p| ≃ 300 |p| GeV 5 > |D f 1 | ≃ 200 GeV 5 ≫ |D g 1 | ≃ 20 GeV 5 , which explains why g 2 and f 2 prevail over other terms in the B − → ppℓν ℓ decay.
, B − → ppℓν ℓ can easily create T -odd triple product correlations (TPC's) to test direct T violation effects.Since the three-momenta of pp and those of ℓν ℓ are not in the same plane, p ℓ • ( p p × p p) can be a nonzero TPC observable.Like the case of s Λ • ( p Λ × p p) in B0 → Λpπ − [50] with s Λ denoting the Λ spin, there are other TPC observables s ℓ • ( p p × p p) and s p • ( p p × p p) in B − → ppℓν ℓ .These rich TPC observables are expected to be useful to test new physics in the advantage of B of order 10 −4 much larger than the sensitivity of 10 −7 in the B factories.IV.CONCLUSIONS We have examined the four-body semileptonic baryonic B decay of B − → ppℓν ℓ in the SM, which proceeds via b → uℓν ℓ at the quark level.The transition form factors of B − → pp, which are well studied in the three-body baryonic B → ppM decays, play the key role in the theoretical calculation.We have found that B(B − → ppℓν ℓ ) = (1.04,1.04, 0.46) × 10 −4 for ℓ = e, µ, τ , respectively, which are just a little below the CLEO's upper limit of 5.2 × 10 −3 for B(B − → ppe − νe ) but much larger than the previous estimations of 10 −5 − 10 −6 for the inclusive modes of B → B B′ ℓν.It is clear that the four-body decays of B − → ppℓν ℓ , in particular the light charged leoton modes, should be observed by the B-factories of Belle and BaBar as well as future B-factories, such as Super-Belle and LHCb.