A Diagrammatic Analysis of Two-Body Charmed Baryon Decays with Flavor Symmetry

We study the two-body anti-triplet charmed baryon decays based on the diagrammatic approach with SU(3) flavor symmetry. We extract the two W -exchange effects as EB and E' that contribute to the Λc+→Ξ0K+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\Lambda}_c^{+}\to {\Xi}^0{K}^{+} $$\end{document} decay, together with the relative phases, where EB gives the main contribution. Besides, we find that BΛc+→pπ0=0.8−0.8+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}$$ \mathcal{B}\left({\Lambda}_c^{+}\to p{\pi}^0\right)=\left({0.8}_{-0.8}^{+0.9}\right)\times {10}^{-4} $$\end{document},which is within the experimental upper bound. Particularly, we obtain BΞc+→Ξ0π+=9.3±3.6×10−3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\Xi}_c^{+}\to {\Xi}^0{\pi}^{+}\right)=\left(9.3\pm 3.6\right)\times {10}^{-3} $$\end{document}, BΞc0→Ξ−π+Λ0K¯0=19.3±2.88.3±5.0×10−2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\Xi}_c^0\to {\Xi}^{-}{\pi}^{+},{\Lambda}^0{\overline{K}}^0\right)=\left(19.3\pm 2.8,8.3\pm 5.0\right)\times {10}^{-2} $$\end{document} and BΞc0→Ξ−K+=5.6±0.8×10−4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\Xi}_c^0\to {\Xi}^{-}{K}^{+}\right)=\left(5.6\pm 0.8\right)\times {10}^{-4} $$\end{document}, which all agree with the data. For the singly Cabibbo suppressed Λc+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\Lambda}_c^{+} $$\end{document} decays, we predict that BΛc+→nπ+pη'Σ+K0=7.7±2.07.1±1.419.1±4.8×10−4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B}\left({\Lambda}_c^{+}\to n{\pi}^{+}, p\eta \hbox{'},{\Sigma}^{+}{K}^0\right)=\left(7.7\pm 2.0,7.1\pm 1.4,19.1\pm 4.8\right)\times 10-4 $$\end{document}, which are accessible to the experiments at BESIII, BELLEII and LHCb.

By using H eff in eq. (2.1), we draw different topological diagrams in B c → BM [32][33][34], where the quark lines should be in accordance with the operators in eq. (2.2). As seen in figure 1, we obtain six topological diagrams. The external and internal W -emission diagrams in figures 1a and b can be parameterized as the topological amplitudes T and C, respectively. Since one can factorize T and C as A ∝ M |(q 1 q 2 )|0 B|(q 3 c)|B c [4], which consists of two calculable matrix elements, the T and C are regarded as the factorizable amplitudes [7,22,32]. The other internal W -emission diagram in figure 1c has no factorizable JHEP02(2020)165 form, parameterized as C . In figures 1(d,e,f), the W -exchange amplitudes of (E , E B , E M ) need an additional gluon to relate M and B. Besides, E B (E M ) has the W -boson to connect B and M , with the c-quark transition to be a valence quark in B(M ), whereas M in the E amplitude is unable to connect to the W -boson. In addition to C , (E , E B , E M ) are the non-factorizable amplitudes according to the factorization approach [32,35]. As a result, we clearly identify each (non-)factorizable effect that contribute to the two-body B c → BM decays.
To present the amplitudes of B c → BM with (T, C) and (C , E , E B , E M ), we need the suitable insertions of the final states to match the quark lines, such as π 0 = 1/2(uū−dd), which adds a pre-factor of ± 1/2 to the topological parameters. Likewise, the (η, η ) meson states that mix with η q = 1/2(uū + dd) and η s = ss lead to the other pre-factors. Specifically, the mixing matrix is presented as [45] with the mixing angle φ = (39.3 ± 1.0) • . We hence obtain the amplitudes of B c → BM , given in table 1. The topological amplitudes are in fact complex, presented as 11 parameters: with T set to be relatively real. To obtain the decay widths, we depend on the integration of the phase space for the two-body decays, given by [43] Γ with A(B c → BM ) from table 1.

Discussions and conclusions
Since χ 2 /n.d.f = 0.5 presents a good fit, we demonstrate that the diagrammatic approach can explain the data well. Moreover, the non-factorizable effects depicted in figure 1 now have clear information. The fit of |E B | (|T |, |C|) 0.4 and |C | 2(|E M |, |E |) 0.3 shows that the non-factorizable effects can be as significant as the factorizable ones; nonetheless, neglected in the factorization approach. While the theoretical computations are unable to explain B(Λ + c → Ξ 0 K + ) [8][9][10][11][12], we explicitly present the two W -exchange effects as E B and E that contribute to Λ + c → Ξ 0 K + , together with the relative phases. Besides, we point out that E B has the main contribution. The E M term as the rarely studied W -exchange process is found to have a significant interference with C in Λ + c → pK 0 . In contrast, the SU(3) f parameters cannot distinguish the three W -exchange contributions. As seen in table 1, the Λ + c → Σ + η ( ) , Σ + K 0 decays only receive the non-factorizable effects. Particularly, C and E M in Λ + c → Σ + K 0 give a constructive interference, leading to B(Λ + c → Σ + K 0 ) = (19.1 ± 4.8) × 10 −4 accessible to the BESIII experiment. We hence have a better understanding for the non-factorizable effects.
In the SU(3) f symmetry, the B(Λ + c → pπ 0 ) was once overestimated as two times larger than the experimental upper bound [22]. By recovering one of the previously neglected parameters, which gives the destructive interference, the number has been reduced to agree with the data [30,31]. It is interesting to note that the recovered parameter is recognized as a factorizable effect, which corresponds to our C term in Λ [22,23,50]. This causes 0s 2 c to be 4σ away from the observation. According to the topological diagrams in figure 2, we find that   [31,35] and the calculation with the pole model, current algebra and MIT bag model [32,35]. The data of B(Ξ 0 c with the value of (0.54±0.04)s 2 c to agree with the data. Indeed, the effects of the SU(3) f symmetry breaking can give rise to the new parameters added to Ξ 0 [25]. Here, our interpretation for R 1 (Ξ 0 c ) relies on the additional diagram in figure 2 without invoking the SU(3) f symmetry breaking.
By relating the topological diagrams in figure 1 to the symmetry properties of the baryon wave functions, such as the (anti-)symmetric quark ordering of Σ 0 (Λ 0 ) ∼ (ud±du)s or the irreducible forms in the SU(3) f and SU(2) spin symmetries, one can derive the more restrict parameterization of the topological amplitudes [33,34]. This leads to R 1 (Ξ 0 c ) 1.0s 2 c inconsistent with the data. Moreover, the T term, which contributes to Λ + c → Σ 0 M + with M + = (π + , K + ) in table 1, becomes forbidden in refs. [33,34]. We hence turn off the T term in Λ + c → Σ 0 M + as a test fit, and obtain χ 2 ∼ 30. Clearly, the more restrict representations cannot explain the data well. Without considering the symmetry properties of the baryon wave functions, our topological amplitudes present the most general forms, which are able to receive the short and long-distance contributions both. The long-distance effect has been proposed in the pole model to contribute to the W -exchange process [32,35]. In the Ξ 0 c → Ξ − M + decay, Ξ 0 c transforms as the Λ 0 (Σ 0 ) pole followed by the strong decay Λ 0 (Σ 0 ) → Ξ − M + , which contributes to figures 2b and c; nonetheless, the latter diagram is forbidden with the more restrict representations. Another example comes from the rescattering effect. With Λ * denoting the higher-wave Λ state, the Λ + c → Λ * ρ + decay has a T amplitude (T * ). Through the π 0 exchange, Λ * ρ + rescatter into Σ 0 π + , such that T * contributes to T in Λ + c → Σ 0 π + . In fact, the symmetry properties of the meson wave functions are not involved in the D → M M decays, such that the long-distance contributions have been absorbed into the topological amplitudes [39,40,42,51].
In table 3, the SU(3) f parameters and topological amplitudes that respect the SU(3) f symmetry are found to explain the data of B(Λ + c → BM ) well, indicating that the SU(3) f symmetry in Λ + c → BM has no sizeable broken effects. In the D → KK decays, the Wexchange processes with V cs V us and V cd V ud are parameterized as E (d) and E (s) , respectively. One needs the sizeable broken effect of |E (s) | > |E (d) | to explain B(D 0 → K + K − )/B(D 0 → π + π − ) and B(D 0 → K 0K0 ) [39][40][41][42]. Likewise, since we can present A(Ξ 0 c → Ξ − K + ) as B , eq. (4.1), whether ∆E B is equal to zero or not can be a test of the broken SU(3) f symmetry. This requires more accurate measurements from BELLEII and LHCb.