Non-leptonic two-body weak decays of $\Lambda_c(2286)$

We study the non-leptonic two-body weak decays of $\Lambda_c^+(2286)\to {\bf B}_n M$ with ${\bf B}_n$ ($M$) representing as the baryon (meson) states. Based on the $SU(3)$ flavor symmetry, we can describe most of the data reexamined by the BESIII Collaboration with higher precisions. However, our result of ${\cal B}(\Lambda_c^+ \to p\pi^0)=(5.6\pm 1.5)\times 10^{-4}$ is larger than the current experimental limit of $3\times10^{-4}$ (90\% C.L.) by BESIII. In addition, we find that ${\cal B}(\Lambda_c^+ \to \Sigma^+ K^0)=(8.0\pm 1.6)\times 10^{-4}$, ${\cal B}(\Lambda_c^+ \to \Sigma^+ \eta^\prime)=(1.0^{+1.6}_{-0.8})\times 10^{-2}$, and ${\cal B}(\Lambda_c^+ \to p \eta^\prime)=(12.2^{+14.3}_{-\,\,\,8.7})\times 10^{-4}$, which are accessible to the BESIII experiments.

Theoretically, the factorization approach is demonstrated to well explain the B and bbaryon decays [5][6][7], such that it is also applied to the two-body Λ + c → B n M decays [8], of which the amplitudes are derived as the combination of the two computable matrix elements for the Λ + c → B n transition and the meson (M) production. However, the factorization approach does not work for most of the two-body Λ + c → B n M ones. For example, the decays of Λ + c → Σ + π 0 and Ξ 0 K + are forbidden in the factorization approach [9], but their branching ratios turn out to be measured. As a result, several theoretical attempts to improve the factorization by taking into account the nonfactorizable effects have been made [10][11][12][13][14]. In contrast with the QCD-based models, the SU(3) symmetry approach is independent of the detailed dynamics, which has been widely used in the B meson [15][16][17], b-baryon [18,19] and Λ + c (Ξ c ) [9,[20][21][22][23] decays. With this advantage, the two-body Λ + c → B n M decays can be related by the SU(3) parameters, which receive possible non-perturbative and nonfactorizable contributions [9,[20][21][22][23][24], despite of the unknown sources. The minimum χ 2 fit with the p-value estimation [3] can statistically test if the SU(3) flavor symmetry agrees with the data. Being determined from the fitting also, the SU(3) parameters are taken to predict the not-yet-measured modes for the future experimental tests. However, the global fit was once unachievable without the sufficient data and the use of the symmetry for Λ + c → B n M. Clearly, the reexamination with the global fit to match the currently more accurate data is needed. Note that, to study the Λ + c → B n η (′) decays, the singlet state of η 1 should be included [16,17]. In this report, we will extract the SU(3) parameters in the global fit, and predict the branching fractions to be compared with the future BESIII experimental measurements.

II. FORMALISM
From Fig. 1, there are four types of diagrams for the non-leptonic charm quark decays, where Figs. 1a−1c with the W -boson emissions directly connected to quark pairs are the so-called tree-level processes, while Fig. 1d with the W -boson in the loop corresponds to the penguin-level ones. In Fig. 1c, the c → dus transition that proceeds through |V cd V us | ≃ sin 2 θ c is the doubly Cabibbo-suppressed one, with θ c being the well-known Cabibbo angle.
Meanwhile, the c → uqq(ss) transitions in Fig. 1d have the higher-order contributions from the quark loops, with the effective Wilson coefficients [25] calculated to be smaller than the tree-level ones by one order of magnitude. As a result, the decay processes in Figs. 1c and 1d are both excluded in the present study. Accordingly, the effective Hamiltonian for the c → sud and c → uqq transitions with q = (d, s) in Figs. 1a and 1b, respectively, is given by [26] with the current-current operators O (q) 1,2 , written as where G F is the Fermi constant, V ij are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and (q 1 q 2 ) V −A stands forq 1 γ µ (1−γ 5 )q 2 . The operators O 1,2 and O q 1,2 in Eq. (3) lead to the so-called Cabibbo-allowed and Cabibbo-suppressed decay modes due to the factor of In the NDR scheme [25,27], one has that (c 1 , For the four-quark operator (q i q k )(q j c) from the effective Hamiltonian in Eq.
which are formed as the tensor notations of H(6) ij and H(15) i jk , respectively. Note that the Cabibbo-suppressed operatorsÔ − andÔ + have similar irreducible forms, leading to their ownĤ(6) ij andĤ (15) i jk [20,21]. As a result, the effective Hamiltonian in Eq. (3) under the SU (3) representation becomes where the non-zero entries are To proceed, we take the amplitudes of B c → B n M under the SU (3) representations.
First, the B c state acts as3 under the SU(3) flavor symmetry, written as by which one defines T ij = ǫ ijk (B c ) k . Second, B n is the baryon octet, given by To include the octet (π, K, η 8 ) and singlet η 1 , M is presented as the nonet, given by where (η, η ′ ) are the mixtures of (η 1 , η 8 ), decomposed as η 1 = 2/3η q + 1/3η s and η 8 = 1/3η q − 2/3η s with η q = 1/2(uū + dd) and η s = ss. Explicitly, the η − η ′ mixing matrix is given by [28]  with the mixing angle φ = (39 where T (B c → B n M) = T (O 15 ) + T (O 6 ) are given by [9] T  Table I. Note that, although we follow the approach in Ref. [21], the h 1,2 terms are newly added for the singlet η 1 . Due to c − /c + ≃ 2.4, the contribution of O 6 (Ô 6 ) to the decay branching ratio can be 5. to simplify the amplitudes. Since e, f , g and h 2 are complex numbers, we have 7 real independent parameters to be determined by the data, given by where e is set to be real, while an overall phase can be removed without losing generality.
To calculate the decay widths, we use [3]: where , with the integratedover variables of the phase spaces in the two-body decays.

III. NUMERICAL RESULTS AND DISCUSSIONS
For the numerical analysis, we use the minimum χ 2 fit to find the SU(3) parameters in Eq. (14). The theoretical inputs for the CKM matrix elements are given by [3] (V cs , V ud , V us , V cd ) = (1 − λ 2 /2, 1 − λ 2 /2, λ, −λ) , with λ = 0.225 in the Wolfenstein parameterization. There are 9 branching ratios of Λ c → B n M, which are the data inputs, given in the last column of Table II. The equation of the χ 2 fit is given by where B i th and B i ex stand for the branching ratios from the theoretical SU(3) amplitudes in Table I and experimental data inputs in Table II, with σ i ex as the 1σ experimental errors, while i = 1, 2, ..., 9 denote the 9 observed decay modes involved in the global fit, respectively.
where d.o.f stands for the degrees of freedom. The statistical p-value to be smaller than 0.05 will show the inconsistency between the theory and data [3], which is equivalent to  Table II, where the results based on the heavy quark effective theory (HQET) [24], Sharma and Verma (SV) in Ref. [23], pole model (PM) [11] and current algebra (CA) [11] are also listed. In our fit, the SU(3) amplitudes in Eq. (18) have considerable imaginary parts, being included in δ f,g,h 2 . Nonetheless, the studies in Refs. [12,24] depend on real ones. For a test, we turn off δ f,g,h 2 , which causes an unsatisfactory fit to the data with χ 2 /d.o.f ≈ 14 ≫ 2.4 in Eq. (18), suggesting that the imaginary parts are necessary to fit the nine data well. It is similar that, in the D → MM decays, the imaginary parts with the SU(3) flavor symmetry are also considerable, which correspond to the strong phases calculated from the on-shell quark loops in the next-leadingorder QCD models [29]. We hence conclude that the Λ + c → B n M decays are like the D → MM ones, where the phases are in accordance with the higher order contributions in the QCD models, which have not been well developed yet.
In Eq. (21), the simple estimation based on the data inputs gives that B(Λ + c → pπ 0 ) = (5.1 ± 0.7) × 10 −4 , which agrees with our numerical fitting result of B(Λ + c → pπ 0 ) = (5.6 ± 1.5) × 10 −4 , but is larger than the experimental upper bound of 3 × 10 −4 (90%C.L.) in Eq. (2) by BESIII. To check if there is a discrepancy here, we have taken the original data of B(Λ + c → pπ 0 ) = (7.95 ± 13.61) × 10 −5 [4] by BESIII as the input. In this case, we get χ 2 /d.o.f = 4.7, which is two times larger than the value in Eq. (18), showing that the fitting cannot accommodate the present data of B(Λ + c → pπ 0 ). Apart from the SU(3) flavor symmetry, we estimate that B(Λ + c → pπ 0 ) ≃ 5 × 10 −4 in the approach of the factorization, which is also larger than the experimental upper bound. It is clear that a dedicated search for this mode with a more precise measurement should be done. An improved sensitivity to measure Λ + c → pπ 0 will clarify if the currently unmovable discrepancy exists or not. It is also interesting to see that B(Λ + c → Σ + η ′ ) and B(Λ + c → pη ′ ) fitted to be (1.0 +1.6 −0.8 ) × 10 −2 and (12.2 +14.3 − 8.7 ) × 10 −4 are as large as their η counterparts, respectively, while B(Λ b → Λη) ≃ B(Λ b → Λη ′ ) [31]. We note that there is a similar term in Ref. [32] as the h 1,2 terms, which relates Λ + c → Σ + η ′ to Ξ 0 c → Ξ 0 η ′ . In contrast, the theoretical approach in Ref. [23] is based on the SU(3) flavor symmetry also, but without the h 1,2 terms to include the singlet η 1 , such that it leads to B(Λ + c → Σ + (p)η ′ ) < B(Λ + c → Σ + (p)η) [23]. Finally, we remark that, with the SU(3) symmetry, we can extend our study to the two-body Ξ +,0 b decays, which are also accessible to the current experiments. Since the two-body Λ + c → B n V with V the vector meson and three-body Λ + c decays are observed, which require the interpretations, the approach of the SU(3) symmetry can be useful.

IV. CONCLUSIONS
We have studied the two-body Λ + c → B n M decays, which have been recently reanalyzed or newly measured by BESIII. With the SU(3) flavor symmetry, we can describe the data except that for Λ + c → pπ 0 . We have found that B(Λ + c → pπ 0 ) = (5.6 ± 1.5) × 10 −4 , which is almost 2σ above the experimental upper bound of 3 × 10 −4 . We hope that the future experimental measurement of B(Λ + c → pπ 0 ) can resolve this discrepancy. Unlike the previous results, we have predicted that B(Λ + c → Σ + η ′ ) = (1.0 +1.6 −0.8 ) × 10 −2 and B(Λ + c → pη ′ ) = (12.2 +14.3 − 8.7 ) × 10 −4 which are as large as their η counterparts, due to the newly added h 1,2 terms with the singlet η 1 in the SU(3) flavor symmetry. With the SU(3) symmetry, one is able to study Λ + c → B n V and the three-body B c decays, which have been observed but barely interpreted. Moreover, the extensions to study the Ξ +,0 b decays are possible, which are also accessible to the current experiments.