Nonleptonic three-body charmed baryon weak decays with H(15)

We study the nonleptonic three-body charmed baryon weak decays of $\mathbf{B}_{c}\rightarrow\mathbf{B}_{n}PP^{\prime}$ under the $SU(3)_{F}$ flavor symmetry, where $\mathbf{B}_{c}$ denotes the anti-triplet charmed baryon, comprising $(\Xi^{0}_{c},-\Xi^{+}_{c},\Lambda^{+}_{c})$, and $\mathbf{B}_{n}$ and $P(P^{\prime})$ represent octet baryon and pseudoscalar meson states, respectively. In addition to 12 parameters from the contributions of the color-antisymmetric part of the effective Hamiltonian, denoted as $H(\bar{\mathbf{6}})$, there are 4 parameters from the color-symmetric one, $H(\mathbf{15})$, which were not included in the previous study. With 16 parameters in total and 28 experimental data points, we obtain the minimal $\chi^2$ over degree of freedom of $\chi^{2}/d.o.f=1.5$, which is a great improvement comparing to that without $H(\mathbf{15})$. With the better fitting values, we evaluate the branching ratios and up-down asymmetries of $\mathbf{B}_{c}\rightarrow\mathbf{B}_{n}PP^{\prime}$, which present some interesting results such as $\mathcal{B}\,(\Lambda^{+}_{c}\rightarrow(\Xi(1690)^{0}\rightarrow\Sigma^{+}K^{-})\,K^{+})\equiv(1.5\pm0.4)\times10^{-3}$ and potential $SU(3)$ breaking effects in $\Xi^{+}_{c}\rightarrow p\pi^{+}K^{-}$ and $\Lambda^{+}_{c}\rightarrow \Sigma^{+}\pi^{-}K^{+}$ to be verified by the experiments at BESIII, Belle-II and LHCb.

In contrast to the wealth of observational data, calculating charm quark three-body weak decays into light quarks has proven challenging.This is attributed to the large mass of the charm quark, making the SU(4) F flavor symmetry ineffective, and the failure of the heavy quark expansion due to the insufficiently large m c .The increasing complexity of these decays further causes the ineffectiveness of factorization methods [15].To overcome these challenges, alternative approaches for charmed hadron decays have been explored in various studies [16][17][18][19][20][21][22].These approaches recognize the necessity of considering non-factorizable effects.On the other hand, the SU(3) F flavor symmetry method has been tested as a useful tool both in the beauty and charmed hadron decays.Its feasibility has been established in charmed baryon two-body and three-body semileptonic charmed baryon weak decays [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].
To investigate the non-resonant weak decays of B c → B n P P ′ , we make the assumption that the final state configurations of the pseudoscalar meson-pairs are predominantly characterized by S-wave ones.We express the decay amplitudes in terms of parity-conserving and violating components under SU(3) F , following a similar framework as outlined in Ref. [38].Building upon a thorough discussion [34,40] of the contribution from the color-antisymmetric part of the effective Hamiltonian associated with the irreducible representation 6 under SU(3) f and incorporating additional experimental data, we extend our analysis to consider complete effective Hamiltonian contributions related to both 6 and 15 representations, resulting in 16 real parameters to be fitted with 28 available experiment data points.Furthermore, we discuss the possible error sources and some interesting findings of the new fit.
Our paper is organized as follows.We interpret the formalism and give the explicit amplitudes of all decay channels of B c → B n P P ′ under the SU(3) F flavor symmetry in Sec.II.In Sec.III, we present our numerical fitting results and discussions.Our conclusion is given in Sec.IV.

II. FORMALISM
The nonleptonic three-body charmed baryon weak decays of B c → B n P P ′ can be proceeded through the charmed quark decays of c → su d, c → ud d (uss) and c → dus.Accordingly, the effective Hamiltonian at tree level is given by [41] with the four-quark operators written as: where G F is Fermi constant and c i represent the Wilson coefficients.The four quark op- ± are classified into so-called Cabibbo-favored (CF), singly Cabibbosuppressed (CS) and doubly Cabibbo-suppressed (DCS) processes, respectively.
The three modes of charmed quark decays can be written as c → q i q j q k with q i = (u, d, s) is the triplet of light quarks under the SU(3) F flavor symmetry.The form of q i q j q k can be decomposed as the irreducible representations of 3 ⊗ 3 ⊗ 3 = 15 ⊕ 6 ⊕ 3 ⊕ 3, in which 15 and 6 correspond to the color-symmetric operator O q 1 q 2 + and the color-antisymmetric operator O q 1 q 2 − [23,24], respectively.Consequently, the effective Hamiltonian can be divided into the symmetric part H(15) and antisymmetric part H( 6), defined as Under SU(3) F , the three lowest-lying charmed baryon states of B c form anti-triplet charmed baryon states, and B n and P belong to octet baryon and pseudoscalar meson states.In this work, we adopt the same convention for the SU(3) F tensors as those in Refs.[34,40].
We assume that S-wave (L = 0) pseudoscalar meson pairs dominate in the non-resonant amplitudes.The decay amplitude can be written as where u Bc,n are Dirac spinors of baryons, and A and B represent the parity conserving and parity violating parts, respectively.Assuming the dominance of H(6) over H (15), contains six SU(3) F parameters in A and B amplitudes to be fitted with data [40].Due to the limitation of data points, we assume that final-state interactions (FSIs) are negligible between non-resonance states, and the parameters A and B are considered to be relatively real.
However, the H( 6) fitting [40] has presented large deviations with updated experiment data, indicating the H(15) contribution is of great significance in the SU(3) F fitting.In this study, we first incorporate H(15) into the analysis of charmed baryon three-body weak decay.We list all possible topological diagrams contributed by H(15) before integrating the W boson in Fig. 1.Focusing on H(15), the quarks represented by q c and i of the W-exchange part in Fig. 1e are symmetric in color.Meanwhile, q c and i originate from B c , where the color of the quarks is totally antisymmetric.We conclude that the topology of W-exchange processes like Fig. 1e does not contribute to H (15), owing to the Körner-Pati-Woo theorem [42,43].From these diagrams we can get complete A and B SU(3) F amplitudes, given by We observe that in Figs.1a and 1b, B n is composed of the two spectator quarks from B c , whereas in Figs.1c 1 and 1c 2 , it consists of only one spectator quark.Given that the light quarks in B c and B n have similar wave functions in the quark model, it is reasonable to assume that the contributions from (c 1 ) and (c 2 ) are smaller than those from (a) and (b).Hence, we take a 9,10 and b 9,10 , which come from the topology of Figs.1c 1 and 1c 2 , to be zero in the following.

The explicit full expansions of
As we only consider the physical quantities after integrating over the phase space, we assume the amplitudes of a i and b i to be independent of m 2  23 , which can be justified in the limit of the SU(3) f flavor symmetry 1 .By introducing the kinematic correction κ (m 2 23 ), the differential decay width and averaged up-down asymmetry of B c → B n P P ′ can be derived as [40]  respectively.The kinematic correction κ (m 2  23 ) is defined as with m 23 is the sum of the 3-momentum of the two pseudoscalar mesons in the rest frame.

III. NUMERICAL RESULTS
We make use of the minimum χ 2 fit in the numerical analysis to obtain the values of 16 parameters a i and b i in Eq. ( 5) under SU(3) F for B c → B n P P ′ .The validity can be tested via χ 2 /d.o.f .
The minimum χ 2 fit approach is given by in which B i SU (3) is the i−th decay branching ratio from SU(3) F fitting predictions, B i data represents the i−th experiment data, and σ i data stands for the i−th experiment error, while i = 1, 2, ..., 28 for 28 experiment measured channels in Table I.We now discuss the data input in Table I.Most importantly, the contributions from two-body resonances are excluded from these 28 data in the table.Aside from the usage of nonresonant experimental data, we also take into consideration for subtracting the resonant contributions.To obtain absolute branching ratios, we incorporate specific branching ratio measurements, B (Λ for the degree of freedom.In Table I, the branching ratios of 28 input data have been reproduced, and we can see that the SU(3) F fittings are in good agreement with the data.We list our all numerical fitting results for the branching ratios of Λ + c → B n P P ′ , Ξ + c → B n P P ′ and Ξ 0 c → B n P P ′ in Tables III, IV, V and VI, respectively.

Our predictions include the branching ratio of B (Λ
, which is notably smaller than the one of (2.0 ± 0.4) × 10 −3 observed by BESIII [9] but aligns with the upper limit prediction of 8 × 10 −4 from Belle [53].This suggests that resonant contributions still exist.Working backwards, it suggests that B (Λ + c → (Ξ(1690) 0 → Σ + K − )K + ) is about (1.5 ±0.4) × 10 −3 .Moreover, the predicted branching ratios for B (Λ + c → Σ 0 π + π 0 ) and B (Λ + c → Ξ − π + K + ) are approximately half the magnitude of their experimental measurements, hinting the presence of other resonant contributions from excited state particles.We emphasize the need for further experimental investigations to confirm their existences and extract the branch fractions of these states.
Notably, our prediction of B (Ξ + c → pπ + K − ) = (28.3±2.6)×10−3 significantly exceeds the results of (11 ± 4) × 10 −3 from LHCb [14] and (4.5 ± 2.2) × 10 −3 from Belle [1].It is interesting to point out in the exact SU(3) F symmetry, but their released energies are in great difference of 0.9 GeV and 0.5 GeV for the former and latter, respectively2 , leading to the hierarchy of Γ (Ξ However, this analysis is in sharp contrast to the experimental data, indicating that their decay widths are approximately the same.This implies a large SU(3) F flavor symmetry breaking effect, which may come from resonant hadrons.We strongly suggest that future experiments revise these two channels.Λ 0 π 0 K0 1.18 ± 0.30 Ξ 0 π 0 η 0 0.33 ± 0.09 Λ 0 K0 η 0 0.28 ± 0.05 Ξ 0 η 0 η 0 (7.50 ± 5.18) × −3 We note that the χ 2 fit suffers a Z 2 ambiguity of B (A, B) = B (A, −B).The ambiguity can be broken with an input of α as α (A, B) = α (A, −B).Here, we choose the α of Λ + c → Ξ 0 π 0 K + and Λ + c → Ξ 0 K + η 0 to be negative as their amplitudes are contributed by Fig. 2 mainly, where x and y have a negative helicity in the chiral limit.Consequently, the helicity of B n is negative also, leading to a negative averaged up-down asymmetry.We list the predictions for the up-down asymmetries of α (Λ + c , Ξ + c , Ξ 0 c → B n P P ′ ) in Tables VII, VIII and IX.Meanwhile, without considering CP violation of physical particles K 0 S and K 0 L , we can also give branching ratios and up-down asymmetries of three-body decay channels involving K 0 S and K 0 L of mixed-modes, which are presented in Table X.These results are acquired under the assumption of S-wave meson pairs in the final states, neglecting the contributions from pseudoscalar meson exchanges, which can be used to assess the dominance of S-wave meson pairs in nonleptonic three-body decays.

TABLE II :
Fitting values for a i and b i in unit of G F

TABLE III :
Numerical results for B(Λ + c → B n P P ′ )

TABLE IV :
Numerical results for B(Ξ + c → B n P P ′ )

TABLE V :
Numerical results for B(Ξ 0 c → B n P P ′ )

TABLE VI :
(Continued)Numerical results for B(Ξ 0 c → B n P P ′ )

TABLE VII :
Numerical results for α (Λ + c → B n P P ′ )

TABLE VIII :
Numerical results for α (Ξ + c → B n P P ′ )

TABLE IX :
Numerical results for α (Ξ 0 c → B n P P ′ )

TABLE X :
Decay branching ratios and averaged up-down asymmetries for CF and DCS mixed-mode processes involving K 0 S and K