Semileptonic baryonic B decays

We study the semileptonic B→BB′¯LL¯′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}L{\bar{L}}'$$\end{document} decays with BB¯′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}{\bar{\textbf{B}}'}$$\end{document} (LL¯′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L{\bar{L}}'$$\end{document}) representing a baryon (lepton) pair. Using the new determination of the B→BB′¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\rightarrow {\textbf{B}{\bar{\mathbf{B'}}}}$$\end{document} transition form factors, we obtain B(B-→pp¯μ-ν¯μ)=(5.4±2.0)×10-6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}(B^-\rightarrow p{\bar{p}} \mu ^-{\bar{\nu }}_\mu ) =(5.4\pm 2.0)\times 10^{-6}$$\end{document} agreeing with the current data. Besides, B(B-→Λp¯νν¯)=(3.5±1.0)×10-8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}(B^-\rightarrow \Lambda {\bar{p}} \nu {\bar{\nu }})=(3.5\pm 1.0)\times 10^{-8}$$\end{document} is calculated to be 20 times smaller than the previous prediction. In particular, we predict B(B¯s0→pΛ¯e-ν¯e,pΛ¯μ-ν¯μ,pΛ¯τ-ν¯τ)=(2.1±0.6,2.1±0.6,1.7±1.0)×10-6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}({\bar{B}}^0_s\rightarrow p{\bar{\Lambda }} e^- {\bar{\nu }}_e,p{\bar{\Lambda }} \mu ^- {\bar{\nu }}_\mu ,p{\bar{\Lambda }} \tau ^- {\bar{\nu }}_\tau ) =(2.1\pm 0.6,2.1\pm 0.6,1.7\pm 1.0)\times 10^{-6}$$\end{document} and B(B¯s0→ΛΛ¯νν¯)=(0.8±0.2)×10-8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{B}({\bar{B}}^0_s\rightarrow \Lambda {\bar{\Lambda }} \nu {\bar{\nu }})=(0.8\pm 0.2)\times 10^{-8}$$\end{document}, which can be accessible to the LHCb experiment.

The semileptonic B decays of B − → ppℓ − νℓ and B − → Λpν ℓ νℓ with ℓ denoting e, µ or τ can provide another evidence for the B → B B′ transition [8,9].Like the studies of the semileptonic B − → π + π − ℓ − νℓ 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.
Experimentally, it has been measured that [13][14][15][16] B ex (B − → ppe − νe ) = (5.8± 3. (1) The threshold effect commonly observed in B → B B′ M is also observed in B − → ppµ − νe [15], which is drawn as a peak around the threshold area of m B B′ ≃ m B + mB′ in the B B′ invariant mass spectrum.There is no sign that the B to B B′ transition receives a resonant contribution.Nonetheless, it is clearly seen that B ex (B − → ppµ − νµ ) 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 R e/µ ≃ 1 as a test of the lepton universality is not conclusive, and the prediction of B(B − → Λpν ν) is within the experimental upper bound.
In Ref. [6], the B → B B′ transition form factors (F B B′ ) are extracted with the data from B → B B′ M, which cause the overestimation of B(B → ppℓν).With the same theoretical inputs, B(B − → Λpν ν) might be overestimated as well [9].A question is hence raised:  where G F is the Fermi constant, V ub and λ t ≡ V * ts V tb are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and D(x t ) with x t ≡ m 2 t /m 2 W is the top-quark loop function [18][19][20][21].According to H(b → uℓν ℓ , sν ℓ νℓ ), the amplitudes of B → B B′ L L′ with L L′ = (ℓν ℓ , ν ℓ νℓ ) can be derived as [8,9] with (q, q ′ ) = (u, d) for B − → ppπ − and B − → ppρ − , (q, q ′ ) = (u, s) for B − → ppK − and B − → ppK * − , and (q, q ′ ) = (d, s) for B0 → pp K0 and B0 → pp K * 0 .The parameters in Eq. ( 4) result from the factorization approach [30], written as with with where |B q ∼ bγ 5 q|0 has been used.As the chiral charge, Q ≡ R µ=0 annihilates the b quark, and creates a valence quark in B, while the spectator quark in the B meson is transformed as a valence quark (q i ) in B′ .We hence obtain R,L as the B → B B′ transition form factors in the chiral representation.When the chirality states of a spinor (R, L) are taken as the helicity states (↑, ↓), one can see qi with the helicity to be (anti-)parallel [||(||)] to the helicity of B′ , such that the chiral charge acting on qi can be more explicitly defined as where e for pp|(ūb)|B − , pp|( db)| B0 , Λp|(sb)|B − , p Λ|(ūb)| B0 s , and Λ Λ|(sb)| B0 s , respectively.Likewise, we perform a derivation for ḡj ( fj ) through the (pseudo-)scalar current, which leads to [28,29]  for pp|(ūb)|B − and pp|( db)| B0 , respectively.Note that R(L) ∼↑ (↓) is based on the approximation with the large energy transfer, which is conveniently presented as t → ∞.It is also derived that the correction term is of order m q / √ t [31][32][33].In fact, √ 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, √ t > 2 GeV is also sufficient for the holding of the relations in Eqs.(10) and (11). where For integration, the allowed ranges of the five variables are (m where and θ is the angle between the meson and baryon moving directions in the B B′ rest frame.The allowed regions of the variables are −1 < cos θ < 1 and (m B + mB′) For the global fit in the next section, we define the CP asymmetry [4,37], and angular asymmetries of B → B B′ M [3, 26, 28] and B → B B′ L L′ [8,9], written as where B → B B′ M represents the anti-particle decay.
with N ef f c = 2 and ∞ for B → ppM(V ) and B → ppD 0( * ) , respectively.Using the parameters in Eq. ( 18), we calculate the branching fractions and angular asymmetries of B − → ppℓν, Λpν ν and B0 s → p Λℓν, Λ Λν ν, of which the results are compared with the experimental data in Table III.We also draw the pp invariant mass spectrum for B − → ppµ − νµ in Fig. 4.

IV. DISCUSSIONS AND CONCLUSIONS
Since χ 2 /n.d.f = 1.86 presents a reasonable fit, it indicates that the most recent data in Table II can be explained.It is interesting to note that B(B − → ppπ − , ppρ − ) [3,4] were once overestimated [7,43,44], and the relation of was not verified by the measurements [43,44].This is due to F B B′ determined by the  By normalizing the prediction of the pQCD model [8], LHCb draws the m pp spectrum of B − → ppµν 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 pp ∼ 2.5 GeV.Subsequently, the peak should reach 17 × 10 −6 to be above the data point around m pp ∼ 2 GeV for integrating over the partial branching fraction as large as B ≃ 5 × 10 −6 .In comparison, our result prefers to agree with the threshold data points; however, requiring some broadening to give a sufficient branching fraction.
FIG. 1. Feynman diagrams for the B → B B′ L L′ decays, where (a) depicts B − → ppℓ − νℓ and B0 s → p Λℓ − νℓ , while (b, c) B − → Λpν ℓ νℓ and B0 s → Λ Λν ℓ νℓ .whether there exist the universal B → B 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 → B B′ L L′ with L L′ denoting a lepton pair and B → B B′ M. With F B B′ determined from the new global fit, we will re-investigate B − → Λpν ν.Since LHCb hasbeen able to accumulate more events for the B0 s decays, we will study B0 s → p Λℓ − ν and B0 s → Λ Λν ν decays for future measurements.

FIG. 3 .
FIG. 3. The angular variables θ B , θ L and φ depicted for the four-body B → B B′ L L′ decays.
The four-body B(p B ) → B(p B ) B′ (pB′)L(p L ) L′ (pL′) decay involves five kinematic variables in the phase space, that is, s ≡ (p L + pL′) 2 ≡ m 2 L L′ , t, and (θ B , θ L , φ)[34][35][36].As depicted in Fig.3, the angle θ B(L) is between p B ( p L ) in the B B′ (L L′ ) rest frame and the line of flight of the B B′ (L L′ ) system in the B meson rest frame.The angle φ is from the B B′ plane to the L L′ plane defined by the momenta of the B B′ pair and L L′ pair in the B meson rest frame, respectively.The partial decay width then reads[8,9] and 0 ≤ φ ≤ 2π.The partial decay width of B(p B ) → B(p B ) B′ (pB′)M(p M )involves two variables in the phase space, given by[3,28] With 16 experimental inputs from Table II and |V ub | ex , we fit (D || , D || , D 2,3,4,5 ) and ( D|| , D|| , D2,3 ) 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 χ 2 /n.d.f = 1.86 as a measure of the global fit, and extract that TABLE II.Experimental data for the B − → ppℓ − ν ℓ and B → ppM (c) decays, where the notation † for A F B denotes the contribution from m pp < 2.85 GeV, and B(B − → ppµ − νµ ) has combined the Belle and LHCb data in Eq. (1).

B
→ ppK data [3], while B → ppK are in fact the penguin dominated decays with M6 ∝ pp|(S − P ) b |B 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 B(B − → ppK − ), TABLE III.Our calculations for the semileptonic B → B B′ L L′ decays.For B(B → B B′ ℓν ℓ ), the values in the parentheses correspond to ℓ = (e, µ, τ ), where the first and second errors come from |V ub | and the form factors in Eq. (18), respectively.For B(B → B B′ ν ν) = Σ ℓ B(B → B B′ ν ℓ νℓ ) and A F B (B → B B′ L L′ ), the errors take into account the uncertainties of the form factors in Eq. (18).