Two-body charmed baryon decays involving decuplet baryon in the quark-diagram scheme

In the quark-diagram scheme, we study the charmed baryon decays of ${\bf B}_c\to {\bf B}^* M$, where ${\bf B}_c$ is $\Lambda_c^+$ or $\Xi_c^{+(0)}$, together with ${\bf B}^*$ ($M$) the decuplet baryon (pseudoscalar meson). It is found that only two $W$-exchange processes are allowed to contribute to ${\bf B}_c\to {\bf B}^* M$. Particularly, we predict ${\cal B}(\Lambda_c^+ \to \Sigma^{*0(+)} \pi^{+(0)})=(2.8\pm 0.4)\times 10^{-3}$, which respects the isospin symmetry. Besides, we take into account the $SU(3)$ flavor symmetry breaking, in order to explain the observation of ${\cal B}(\Lambda_c^+\to \Sigma^{*+}\eta)$. For the decays involving $\Delta^{++}(uuu)$, we predict ${\cal B}(\Lambda_c^+\to \Delta^{++} \pi^-,\Xi_c^+ \to \Delta^{++} K^-) =(7.0\pm 1.4,13.5\pm 2.7)\times 10^{-4}$ as the largest branching fractions in the singly Cabibbo-suppressed $\Lambda_c^+,\Xi_c^+\to{\bf B}^*M$ decay channels, respectively, which are accessible to the LHCb, BELLEII and BESIII experiments.

For a better understanding of the hadronization in B c → B * M, there have been some theoretical attempts, which are in terms of the pole model, quark model and irreducible [4][5][6][7][22][23][24]. Particularly, the quark-diagram scheme with the topological SU(3) f symmetry provides a clear picture for the decay processes [17,[25][26][27]. Due to the fact that B * is a spin-3/2 baryon with totally symmetric quark contents, it can be shown that the topological diagrams involving the flavor anti-symmetric quark pair in B * are forbidden or suppressed. Therefore, we purpose to use the quark-diagram scheme to relate all possible B c → B * M decay channels. With the existing data, we will perform the numerical analysis, and determine different topological contributions. We can hence test the validity of the topological scheme, which involves the SU(3) f symmetry and its broken effect. Furthermore, we will give predictions for B(B c → B * M) to be compared to the future measurements, which can help to clarify how B c → B * M, B * → BM ′ mixes with B c → BV, V → MM ′ and the non-resonant configuration in B c → BMM ′ .

A. Effective Hamiltonian in the flavor symmetry
To study the two-body charmed baryon decays, the corresponding quark-level effective Hamiltonian is given by [28] where (q 1 q 2 ) =q 1 γ µ (1 − γ 5 )q 2 , and the subscripts (α, β) denote the color indices. With For the lowest-lying anti-triplet charmed baryon states Ξ 0 c , Ξ + c and Λ + c that consist of (ds − sd)c, (su − us)c and (ud − du)c, respectively, we present them as The pseudoscalar meson states are given by where (η, η ′ ) mix with η q = 1/2(uū + dd) and η s = ss. The mixing angle φ = (39.3 ± 1.0) • in (sφ, cφ) ≡ (sin φ, cos φ) comes from the mixing matrix, given by [29]  The decuplet baryons are written as By neglecting the Lorentz indices, H ef f for the c → q iqj q k transition can be presented with the tensor notation, H ki j , and the nonzero entries are given by [27] B. The quark-diagram scheme Furthermore, it is found in Figs. 1(a,b) where q a q b q k(i) are totally symmetric, such that (T, C) give no contributions to B c → B * M. Thus, the B c → B * M decays are purely non-factorizable processes. In , which is in accordance with the Körner-Pati-Woo theorem [31]. With the current-current structure of (q (2), q i and q k are color anti-symmetric. When combined as the constituents of the baryon, q i,k are flavor anti-symmetric, such that the topological diagrams (C ′ , E ′ ) in To proceed, we derive the amplitudes as Explicitly, the T amplitudes (T -amps) read [17,25,27] T where the parameters E  Tables I, II and III, with m ± = m B * ± m M , where τ Bc stands for the B c baryon lifetime.
We use two scenarios for the global fit. In the first scenario (S1), we take E s B(M ) = E B(M ) under the exact SU(3) f symmetry. Since E B and E M are complex numbers, it leads to three independent parameters, given by where E B is set to be real, and δ E M is a relative strong phase. Using the χ 2 -fit, we extract that (|E B |, |E M |) = (0.41 ± 0.03, 0.34 ± 0.03) GeV 3 ,   shown to be in tension with the observation of (9.1 ± 2.0) × 10 −3 . Sizeably, it adds 3.6 to the total χ 2 value.
Since Λ + c → Σ * + η is in association with |E s M |, the tension hints the broken SU(3) f symmetry, where |E s M | is not equal to |E M |. On the other hand, B(Ξ 0 c → Ω − K + ) is fitted to agree with the data, indicating that |E s B | is not deviating from |E B |. Currently, the data points are not sufficient for an independent extraction of |E s M |. We hence adopt the numerical results from the two-body D meson decays, where the similar W EX contributions have been found to induce the severe SU(3) f symmetry breaking [18][19][20][21]. In the second scenario (S2), we take |E s M | = n q × |E M | and |E s B | ≃ |E B |, with n q = 1.4 adopted from [21]. Consequently, we obtain where χ 2 /n.d.f is reduced as 1.3. As the demonstration, we obtain B(Λ + c → Σ * + η) = (7.3 ± 1.5) × 10 −3 , which alleviates the deviation from the observation. With the fit values of (|E B |, |E M |, δ E M ) in S1 and S2, we present the branching ratios of the B c → B * M decays in Tables I, II and III, along with the recent theoretical results for comparison.
Uniquely, the decuplet baryon can contain three identical quarks, denoted by B * (qqq), which leads to an additional weight factor of √ 3 among the decuplet baryons in Eq. (6).
The factor can be considered as the main reason why Λ + c → ∆ ++ K − and Ξ 0 c → Ω − K + are measured with the largest branching fractions in the CA decay channels of Λ + c , Ξ 0 c → B * M, respectively. Accordingly, the T -amps with B * (qqq) are listed as While B(Λ + c → ∆ ++ K − ) and B(Ξ 0 c → Ω − K + ) have been observed, the other branching fractions are given by which are predicted as the largest branching fractions in the SCS Λ + c (Ξ + c ) and DCS Ξ +(0) c decay channels, respectively. Here, we present our predictions of S2, which is favored by the χ 2 -fit. The equality of T (Λ + c → Σ * 0 π + ) = T (Λ + c → Σ * + π 0 ) corresponds to the isospin symmetry. The branching fraction, given by can be used to test the broken effect. The decays B c → B * η (′) , Λ + c → Σ * + η, Ξ 0 c → Ξ * 0 η and Ξ + c → Σ * + η have sizeable branching fractions, which is due to the constructive interferences between E B and E M . However, the other branching fractions of B c → B * η (′) are typically small with the destructive interferences. Moreover, we find that B(Λ + c → Σ * + η ′ ) = 0 with m Λ + c < m Σ * + + m η ′ .
The approach of the irreducible SU(3) f symmetry has been widely used in the hadron weak decays [4][5][6][7][8][9][10][11][12][13][14][15][16]. For B c → B * M, there exist four parameters a 8 and a 9,10,11 [4,6], which correspond to the decomposition of H ef f = H(6) + H(15) in the SU(3) f representation of 6 and 15, respectively. By comparison, we derive that such that a i are found to correspond to the topologies. Since (E ′ , C ′ ) have been the vanishing topological parameters, one has a 9 = −a 10 and a 11 = 0. Moreover, our global fits for E B,M indicate that a 9(10) from H(15) has a non-zero value. By contrast, the numerical analysis performed with the irreducible SU(3) f symmetry neglects the contributions from H(15) [7], whose results are given in the tables. In the physical mass scenario (S pm ) for the global fit in Ref. [7], where m Bc , m B * and m M are taken from the physical values in Ref. [1], B(Λ + c → Σ * + η, Ξ * 0 K + ) and B(Ξ 0 c → Ω − K + ) are fitted to be a few times smaller than the observations. Instead of considering the SU(3) f symmetry breaking, one performs another fit in the equal mass scenario (S em ), where m Λc = m Ξc , m ∆ = m Σ * = m Ω and m π = m η = m K , resulting in the raised values of the above branching fractions.