Study of two-body doubly charmful baryonic B decays with SU(3) flavor symmetry

Within the framework of SU(3) flavor symmetry, we investigate two-body doubly charmful baryonic B→BcB¯c′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ B\to {\textbf{B}}_c{\overline{\textbf{B}}}_c^{\prime } $$\end{document} decays, where BcB¯c′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textbf{B}}_c{\overline{\textbf{B}}}_c^{\prime } $$\end{document} represents the anti-triplet charmed dibaryon. We determine the SU(3)f amplitudes and calculate BB−→Ξc0Ξ¯c−=3.4−0.9+1.0×10−5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({B}^{-}\to {\Xi}_c^0{\overline{\Xi}}_c^{-}\right)=\left({3.4}_{-0.9}^{+1.0}\right)\times {10}^{-5} $$\end{document} and BB¯s0→Λc+Ξ¯c−=3.9−1.0+1.2×10−5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\overline{B}}_s^0\to {\Lambda}_c^{+}{\overline{\Xi}}_c^{-}\right)=\left({3.9}_{-1.0}^{+1.2}\right)\times {10}^{-5} $$\end{document} induced by the single W-emission configuration. We find that the W-exchange amplitude, previously neglected in studies, needs to be taken into account. It can cause a destructive interfering effect with the W-emission amplitude, alleviating the significant discrepancy between the theoretical estimation and experimental data for BB¯0→Λc+Λ¯c−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\overline{B}}^0\to {\Lambda}_c^{+}{\overline{\Lambda}}_c^{-}\right) $$\end{document}. To test other interfering decay channels, we calculate BB¯s0→Ξc0+Ξ¯c0+=3.0−1.1+1.4×10−4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\overline{B}}_s^0\to {\Xi}_c^{0\left(+\right)}{\overline{\Xi}}_c^{0\left(+\right)}\right)=\left({3.0}_{-1.1}^{+1.4}\right)\times {10}^{-4} $$\end{document} and BB¯0→Ξc0Ξ¯c0=1.5−0.6+0.7×10−5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\overline{B}}^0\to {\Xi}_c^0{\overline{\Xi}}_c^0\right)=\left({1.5}_{-0.6}^{+0.7}\right)\times {10}^{-5} $$\end{document}. We estimate non-zero branching fractions for the pure W-exchange decay channels, specifically BB¯s0→Λc+Λ¯c−=8.1−1.5+1.7×10−5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\overline{B}}_s^0\to {\Lambda}_c^{+}{\overline{\Lambda}}_c^{-}\right)=\left({8.1}_{-1.5}^{+1.7}\right)\times {10}^{-5} $$\end{document} and BB¯0→Ξc+Ξ¯c−=3.0±0.6×10−6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\overline{B}}^0\to {\Xi}_c^{+}{\overline{\Xi}}_c^{-}\right)=\left(3.0\pm 0.6\right)\times {10}^{-6} $$\end{document}. Additionally, we predict BBc+→Ξc+Ξ¯c0=2.8−0.7+0.9×10−4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({B}_c^{+}\to {\Xi}_c^{+}{\overline{\Xi}}_c^0\right)=\left({2.8}_{-0.7}^{+0.9}\right)\times {10}^{-4} $$\end{document} and BBc+→Λc+Ξ¯c0=1.6−0.4+0.5×10−5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({B}_c^{+}\to {\Lambda}_c^{+}{\overline{\Xi}}_c^0\right)=\left({1.6}_{-0.4}^{+0.5}\right)\times {10}^{-5} $$\end{document}, which are accessible to experimental facilities such as LHCb.

In the study of singly charmful baryonic B → B c B′ decays, the W ex(an) amplitude was also neglected [11,12].Nonetheless, it has been found that M wex(wan Initially, it was considered that B → B c B′ c receives a single contribution from the W em topology [16,17].In Eq. (1),

II. FORMALISM
To study the two-body doubly charmful baryonic B (c) → B c B′ c decays with B c denoting B + c (bc), the quark-level effective Hamiltonians for the b → cqq ′ weak transitions are required, given by [49,50] where G F is the Fermi constant, V cb and V qq ′ with q = (u, c) and q ′ = (s, d) the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.In Eq. ( 2), we define (q and the subscripts (α, β) denote the color indices; moreover, c 1,2 are the scale (µ)-dependent Wilson coefficients with µ = m b for the b decays.In the SU(3) f representation, H b→ccq ′ ef f and by omitting Lorentz structure are reduced as H i and H i j , respectively, where i and j run from 1 to 3 to represent the flavor indices.Explicitly, the nonzero entries are given by [27] with Accordingly, we present the B meson and B c baryon in the SU(3) f forms: whereas B c is a singlet.By connecting the flavor indices of the initial state to those of the effective Hamiltonian and final states, the SU(3) f approach yields the amplitudes to be where the parameter e and c ′ (c ′ ) correspond to the W ex and W em configurations in Fig. 1a and Fig. 1b(c), respectively.For a later numerical analysis, we use the equation [14]: to compute the branching fractions, where  I.

III. NUMERICAL RESULTS
In the numerical analysis, the CKM matrix elements are adopted from PDG [14]: where A = 0.826 and λ = 0.225 in the Wolfenstein parameterization.In Eq. ( 5), the parameters e and c ′ are complex numbers, which we present as with δ e a relative phase.By using the experimental data in Table II, we solve the parameters as Moreover, we assume c′ = c ′ due to the similarity of the Feynman diagrams in Figs.1b and   1c.Subsequently, we calculate the branching fractions as provided in Table II using the determination in Eq. ( 9).

IV. DISCUSSION AND CONCLUSIONS
The SU(3) f approach enables us to explore all possible B → B c B′ c decays, as summarized in Table I.Furthermore, it helps in deriving constraints on SU(3) f relations, facilitating the decomposition of amplitudes into e and c ′ terms.These terms parameterize the W ex and W em topologies depicted in Fig. 1a and Fig. 1b(c), respectively.
In b → ccs induced decays, the SU(3) f symmetry unequivocally establishes that both f B is the B meson decay constant, q µ the momentum transfer, and the matrix elements present the vacuum (0) to B c B′ c production.As information on the 0 → B c B′ c production is lacking, a model calculation is currently unavailable.For a calculation on M wem , one proposes a meson propagator to provide an additional quark pair, resulting in the branching fractions to be a few times 10 which are also contributed by the single W em amplitude, promising to be measured by experimental facilities such as LHCb.
The previous studies have assumed that B0 → Λ + c Λ− c receives the single W em contribution [16,17].Consequently, the estimated branching fraction B( B0 → Λ + formation, it is reasonable to assume that QCD effects cannot distinguish between the topology in Fig. 1b and the one in Fig. 1c in the hadronization process.Hence, we assume c′ = c ′ , and predict the following branching fractions: which can be probed by the LHCb experiment. In q|0 with m c ≫ m q can alleviate the helicity suppression[13].This resultsin B wex ( B0 s → Λ + c p)and B wex ( B0 → Ξ + c Σ− ) being predicted to be of order 10 −5 , much more reachable than B(B → B B′ ) ∼ 10 −8 − 10 −7 for the test of a non-negligible W ex(an) contribution.However, until very recently, these observations have not been reported.It is worth noting that two-body doubly charmful baryonic B decays, B → B c B′ c , have provided a possible experimental indication of a non-negligible contribution from the W ex term.The measured branching fractions for B → B c B′ c are reported as follows: p cm is the three-momentum of the B c baryon in the B (c) meson rest frame, and τ B (c) stands for the B (c) lifetime.The amplitude M(B (c) → B c B′ c ) can be found in Table −3 for B0 → Ξ + c Λ− c and B − → Ξ 0 c Λ− c [17].Additionally, a theoretical attempt incorporating final state interactions yields B ≃ O(10 −3 ) [51].It appears that the above approaches may overestimate the W em contribution.Without invoking the model calculations, we determine e and c ′ with the experimental data based on the SU(3) f symmetry.Explicitly, we use the experimental results for B( B0 → Ξ + c Λ− c ) and B(B − → Ξ 0 c Λ− c ) to fit |c ′ |.On the other hand, the experimental data have not been sufficient and accurate enough to simultaneously determine |e| and the relative phase δ e , as indicted in Table II.For a practical determination, we fix δ e = 180 • to cause a maximum destructive interference, where the fitted |c ′ | value has been used.As a consequence, the experimental upper bounds of B( B0 s → Λ + c Λ− c ) and B( B0 → Λ + c Λ− c ) can sandwich an allowed range for |e|, as given in Eq. (9).As the numerical results, we obtain B( B0 → Ξ + c Λ− c ) = (7.2+2.1 −1.9 ) × 10 −4 and B(B − → Ξ 0 c Λ− c ) = (7.8+2.3 −2.0 ) × 10 −4 utilizing |c ′ | = (1.29 ± 0.18) GeV 3 , in agreement with the experimental inputs.This demonstrates that c ′ can estimate the W em contribution.Thus, we predict
summary, we have explored the two-body doubly charmful baryonic B → B c B′ c decays.Here, the W ex and W em amplitudes have been parametrized as e and c ′ , respectively, using the SU(3) f approach.With the determination of the SU(3) f parameters, we have calculated the branching fractions B(B − → Ξ 0 c Ξ− c ) = (3.4+1.0 −0.9 ) × 10 −5 and B( B0 s → Λ + c Ξ− c ) = (3.9+1.2 −1.0 ) × 10 −5 .Considering that the single W em contribution to B0 → Λ + c Λ− c has caused the branching fraction to significantly exceed the experimental upper bound, we have added the W ex amplitude (or the SU(3) f parameter e), overlooked in previous studies, to account for a destructive interfering effect.Subsequently, we have alleviated the discrepancy.To further test the interfering decay channels, we have predicted B( B0 s → Ξ 0(+)