Charmed baryon decays in $SU(3)_F$ symmetry

In the recent years, fruitful results on charmed baryons are obtained by BESIII, Belle and LHCb. We investigate the two-body non-leptonic decays of charmed baryons in the flavor $SU(3)$ symmetry. Hundreds of amplitude relations are clearly provided, and are classified according to the $I$-, $U$- and $V$-spin symmetries. Among them, some amplitude relations are tested by the experimental data, or used to predict the branching fractions based on the exact flavor symmetry without any other approximation. Some relations of $K^0_S-K^0_L$ asymmetries and $CP$ asymmetries are obtained under the $U$-spin symmetry in the modes of charmed baryon decaying into neutral kaons. Besides, the $U$-spin breaking effect is explored in the $\Lambda_c^+\to \Sigma^+K^{*0}$ and $\Xi_c^+\to p\bar{K}^{*0}$ modes.

fractions based on the exact flavor symmetry without any other approximation. Some relations of K 0 S − K 0 L asymmetries and CP asymmetries are obtained under the U -spin symmetry in the modes of charmed baryon decaying into neutral kaons. Besides, the U -spin breaking effect is explored in the Λ + c → Σ + K * 0 and Ξ + c → pK * 0 modes.

I. INTRODUCTION
Charmed baryon decays have attracted great attentions recently. Many new measurements were performed by the BESIII [1][2][3][4][5][6][7][8][9][10][11][12][13], Belle [14][15][16][17][18][19], and LHCb [20][21][22][23][24] experiments, with a lot of properties firstly determined in the recent five years when more than thirty years after the observation of the charmed baryons. For instance, the absolute branching fractions of two-body non-leptonic charmed baryon decays are measured and collected shown in Table I. Especially, the measurements on the absolute branching fractions of the Ξ 0 c and Ξ + c decays by the Belle collaboration [18,19] are important extensions from the studies of Λ + c decays. Brilliant prospects of charmed baryon decays are expected at BESIII [26], Belle II [27] and LHCb [28] in the near future. Motivated by the experimental progress, many theoretical efforts are devoted to charmed baryon decays since 2016 . It is worthwhile to investigate the charmed baryon decays continuously since they provide a platform to study the weak and strong interactions.
However, the potential of flavor SU (3) symmetry analysis has not been fully explored. The SU (3) amplitude relations of charmed baryon decays were firstly studied in 1990 [69] and have to be updated nowadays. Besides, due to the limited available data and the large number of free parameters in the SU (3) irreducible representation amplitudes, some assumptions are introduced in the global fitting, either by neglecting the 15-dimensional representation which is small compared to the 6-dimensional representation, or considering the factorization hypothesis for the 15-dimensional representation [41][42][43][44][45][46][47][48][49][50]. In this work, we use the exact flavor SU (3) symmetry without any other assumptions to find more and accurate amplitude relations.
The analysis includes the modes charmed baryons decaying into one octet or decuplet light baryon and one pseudoscalar or vector meson, covering almost all the available two-body nonleptonic charm-baryon decays. Some branching fractions of charmed baryon decays are predicted using the SU (3) amplitude relations. In order to test the I-, U -, V -spin symmetries and their breaking effects, the amplitude relations are classified according to the SU (2) subgroups of SU (3) group. We discuss the Λ + c → Ξ 0 K + , Λ + c → Σ 0 K + , Λ + c → Σ 0 π + modes for testing the U -spin symmetry, and the Λ + c → Σ + K * 0 and Ξ + c → pK * 0 modes for the implications of U -spin breaking.
The rest of this paper is organized as follows. In Sec. II, we introduce the SU (3) irreducible representation amplitude approach and derive the amplitude relations under the SU (3) F limit.
The Phenomenological analysis is presented in Sec. III. Sec. IV is a short summary. The decay amplitudes of charmed baryon decays and a series of amplitude relations under I-, U -, V -spin symmetries are listed in Appendixes A and B, respectively.

II. AMPLITUDE RELATIONS IN THE FLAVOR SYMMETRY
In this Section, we introduce the SU (3) irreducible representation amplitude (IRA) approach.
The tree level effective Hamiltonian in charm quark weak decay in the Standard Model (SM) is in which α, β are color indices, q 1,2 are d and s quarks. The non-leptonic decays of charmed hadrons are classified into three types: Cabibbo-favored (CF), singly Cabibbo-suppressed (SCS) and doubly Cabibbo-suppressed (DCS) decays, In the SU (3) picture, the four-quark operators in charm decays embed into an effective Hamiltonian, O ij k can be seen as a tensor representation of SU (3) Operator O ij k can be decomposed into four irreducible representations of SU (3) group: 3⊗3⊗3 = 3 ⊕ 3 ⊕ 6 ⊕ 15. The explicit decomposition is [54,71] All components of the irreducible representations can be found in [54]. The non-zero components corresponding to tree operators in the SU (3) decomposition, under the approximation of V * In the comparison with most literatures [37][38][39][40][41][42][43][44][45][46][47][48][49][50]69], the difference is only one common factor of The pseudoscalar and vector mesons form two nonets (octet + singlet), The mass eigenstates η and η are mixing of η 8 and η 1 , The mixing angle ξ has large uncertainty in literatures. We are not going to discuss the value of ξ because the decay modes involving η and η mesons are not used and predicted in this work. The mass eigenstates ω and φ are mixing of ω 8 and ω 1 , The ideal mixing indicates that sin ξ = 1/ √ 3 and cos ξ = 2/3. The singly charmed baryons form an anti-triplet and a sextet. The anti-triplet reads as or contracted by the Levi-Civita tensor, The light baryon octet reads as The light baryon decuplet is symmetric under the interchange of any two quarks, which can be To obtain the SU (3) irreducible representation amplitude of the B c → B 8 P decay, one can contract all indices in the following manner: Similarly, the decay amplitude of B c → B 10 P is constructed to be The decay amplitudes of a, b, c, e, f, g, h, r and α, β, γ, δ, λ are complex free parameters. For the decay modes involving vector mesons, their amplitudes have the same forms as Eqs. (16) and (17).
For distinguishing, we label superscript for each parameter of vector modes. With Eqs. (16) and (17) or the mixing of V -spin triplet and V -spin singlet, With the decay amplitudes listed in Appendix A, we derive some amplitude relations between different modes. Here we only list those relations which will be used later. The others are listed in Appendix B.
Isospin relations: U -spin relations: One can derive more amplitude relations that are only valid in the flavor SU In the U -spin relations, one type of them, which is relevant to a complete interchange of d and s quarks in the initial and final states in two decay channels, is simplest. For example, under the complete interchange of d ↔ s, the initial and final state particles in Eq. (24) are interchanged as The relations associated with the complete interchange of d and s quarks are very interesting because they can be gotten from their initial and final states without writing down the amplitude

A. Test flavor symmetry
In this Section, we discuss physical applications of the amplitude relations in the SU (3) F limit.
The first application is to test the validity of Isospin, U -spin and V -spin symmetries with available data. For the two-body decay, the partial decay width Γ is expressed as where M is the mass of initial particle, m 1 and m 2 are the masses of final particles, and A is the decay amplitude. According to Eq. (48) and the Isospin symmetry relation (20), the ratio of The experimental data of Br(Λ + c → Σ + π 0 ) and Br(Λ + c → Σ 0 π + ) imply that [25] Br One can find the theoretical prediction is well consistent with the experimental data. It demonstrates that the isospin symmetry is fairly accurate even in the charmed baryon decays.
Other testable equation is the U -spin relation (23). The amplitude magnitudes of Λ + c → Σ 0 π + , Λ + c → Σ 0 K + and Λ + c → Ξ 0 K + modes obtained from available data are If the U -spin symmetry is relatively precise, amplitudes of the three modes should form a triangle in the complex plane. The triangle is shown in Fig. 1. It is found the amplitudes of Λ + c → Σ 0 π + , Λ + c → Σ 0 K + and Λ + c → Ξ 0 K + modes form a triangle within the 1σ error. Using the triangle in Fig. 1, we extract the relative strong phases between the three decay modes from data. The three angles of the triangle are expected to be in which θ( 1 , 2 , 3 ) present the opposite angles of the three sides of the triangle in Fig. 1. These values could provide some guides for the theoretical studies.  [72,73], is not large enough. For comparison, we list the results given in [42][43][44][45][46] in Table II. From Table II, one can find our results, except for Λ + c → ∆ ++ π − , are consistent with the ones given in [42][43][44][45][46] within the range of the errors. It indicates the assumption that the main contributions to charmed baryon decays come from O(6) operators is reasonable. In Table II, the branching fraction of Λ + c → ∆ + K 0 is the most precise prediction because it is derived from Isospin symmetry. Our prediction of Br(Λ + c → pπ 0 ) is given in a large range which satisfies the upper limit by BESIII Collaboration [6], Br(Λ + c → pπ 0 ) < 2.7 × 10 −4 in 90% confidence level. There are some branching fraction ratios relative to Br(Ξ + c → Ξ − π + π + ) given by PDG [25]. For example, the ratio between Br(Ξ + c → Σ * + K 0 ) and Br(Ξ + c → Ξ − π + π + ) is The branching fraction of Ξ + c → Ξ − π + π + is taken from [19]. And then the branching fractions, such as Br(Ξ + c → Σ * + K 0 ), can be predicted using these ratios. The results are presented in Table III. With the results listed in Table. III, one can also predict some branching fractions via amplitude relations. The results are presented in Table IV. From Table II and Table III, Ξ + c → ∆ ++ π − (6.7 ± 1.6) × 10 −5 DCS (6.2 ± 1.5) × 10 −5 [42] Ξ + c → ∆ 0 π + (4.7 ± 1.2) × 10 −5 DCS (8.5 ± 1.5) × 10 −5 [46] (1.9 ± 0.9) × 10 −3 [45] C. K 0 S − K 0 L asymmetry and CP asymmetry in B c → BK 0 S,L decays Flavor SU (3) symmetry can give some interesting arguments for the K 0 S − K 0 L asymmetry and CP asymmetry in charm hadron decays into neutral kaons. For convenience to the analysis below, we first review the key points about K 0 S − K 0 L asymmetry and CP asymmetry in charm hadron decays into neutral kaons. More details can be found in [40,74,75]. The K 0 S − K 0 L asymmetry, which is induced by the interference between CF and DCS amplitudes, is defined by If the ratio between the DCS and CF amplitudes is defined as with the magnitude r f , the relative strong phase δ f , and the weak phase φ ≡ The time-dependent CP asymmetry in the decay chain of B c → BK(t)(→ π + π − ) is defined as with Γ ππ (t) ≡ Γ(B c → BK(t)(→ π + π − )) and Γ ππ (t) ≡ Γ(B c → BK(t)(→ π + π − )). The intermediate state K(t) is recognized as a time-evolved neutral kaon K 0 (t) or K 0 (t), and t is the time interval between the charmed baryon decay and the neutral kaon decay in the kaon rest frame [75,76]. As pointed out in [75], there exist three CP -violation effects, i.e., the indirect CP violation in K 0 −K 0 mixing A K 0 CP , the direct CP asymmetry in charm decays A dir CP , and the effect from the interference between two tree (CF and DCS) amplitudes with neutral kaon mixing A int CP , in which with the parameter characterizing the indirect CP asymmetry in the K 0 − K 0 mixing, the mass m S (m L ) and the width Γ S (Γ L ) of the K 0 S (K 0 L ) meson and Γ ≡ (Γ S + Γ L )/2, ∆m ≡ m L − m S . In the B c → B 8 P and B c → B 10 P decays, we can define seven K 0 S − K 0 L asymmetries which are associated with the decay modes of Λ + c → pK 0 Let us analyze the relation between R(Λ + c → pK 0 S,L ) and R(Ξ + c → Σ + K 0 S,L ) in the U -spin symmetry. The ratio of decay amplitudes of Λ + c → pK 0 and Λ + c → pK 0 is The ratio of decay amplitudes of Ξ + c → Σ Eqs. (63) and (64) show that the magnitude of ratios r p and r Σ + and strong phases δ p and δ Σ + have following relations in the U -spin limit: where θ is the Cabibbo angle. For convenience, we define r p / tan 2 θ = tan 2 θ/r Σ + = −r, δ p = −δ Σ + =δ. Then the K 0 S − K 0 L asymmetries in Λ + c → pK 0 S,L and Ξ + c → Σ + K 0 S,L modes can be written as The following relation between R(Λ + c → pK 0 S,L ) and R(Ξ + c → Σ + K 0 S,L ) is gotten: According to Eqs. (60) and (61), the direct CP asymmetry A dir CP and the interference between charm decay and neutral kaon mixing A int CP are proportional to sin δ f . Since the relative strong phase between DCS and CF amplitudes δ f is opposite in Λ + c → pK 0 S and Ξ + c → Σ + K 0 S modes, A dir CP and A int CP have opposite sign too. Furthermore, the magnitudes of K 0 S − K 0 L asymmetries and CP asymmetries in Λ + c → pK 0 S,L and Ξ + c → Σ + K 0 S,L modes have following relations: 1. If both R(Λ + c → pK 0 S,L ) and R(Ξ + c → Σ + K 0 S,L ) are large, the strong phaseδ is close to zero. The CP asymmetries A dir CP and A int CP are small in Λ + c → pK 0 S and Ξ + c → Σ + K 0 S decays.
2. If both R(Λ + c → pK 0 S,L ) and R(Ξ + c → Σ + K 0 S,L ) are small, the strong phaseδ is close to π/2. The CP asymmetries A dir CP and A int CP are large in Λ + c → pK 0 S and Ξ + c → Σ + K 0 S decays.
3. If R(Λ + c → pK 0 S,L ) is large while R(Ξ + c → Σ + K 0 S,L ) is small, the parameterr is large. The CP asymmetries A dir CP and A int CP in Λ + c → pK 0 S decay are large and the ones in Ξ + c → Σ + K 0 S decay are small.
is large, the parameterr is small. The CP asymmetries A dir CP and A int CP in Λ + c → pK 0 S decay are small and the ones in Ξ + c → Σ + K 0 S decay are large.
Let us take a closer look on Eqs. (63) and (64). The decay modes Λ + c → pK 0 and Ξ + c → Σ + K 0 are connected by a complete interchange of d and s quarks in initial and final states: The decay amplitudes of Λ + c → pK 0 and Ξ + c → Σ + K 0 are associated with an interchange of the CKM matrix elements: The same situation appears in Λ + c → pK 0 and Ξ + c → Σ + K 0 modes. In fact, it is an universal law that if two decay modes connected by the interchange of d ↔ s in the initial and final states, their decay amplitudes are the same under the flavor U -spin symmetry except for an interchange of the CKM matrix elements.
The detailed analysis can be found in our previous work [54]. Even the decay amplitudes are not written down, Eq. (65) can still be obtained. Eq. (65) is valid for any charmed decay modes involving K 0 S,L if other initial-and final-state particles are connected by a complete interchange of d and s quarks, no matter the charmed meson decays, singly and doubly charmed baryon decays, or two-and multi-body decays. All the analysis on Λ + c → pK 0 S,L and Ξ + c → Σ + K 0 S,L can be applied to the modes that satisfy this condition. As examples, one can check decay modes such as D + → K 0 S,L π + and D + s → K 0 S,L K + , Λ + c → ∆ + K 0 S,L and Ξ + c → Σ * + K 0 S,L , Ξ + cc → Λ + c K 0 S,L and Ω + cc → Ξ + c K 0 S,L , Ξ + cc → Σ + c K 0 S,L and Ω + cc → Ξ * + c K 0 S,L , and so on.

D. U -spin breaking
As is well known the SU (3) breaking effects are significantly large in the charmed meson decays [77,78]. It deserves to investigate the SU (3) breaking effects in charmed baryon decays. In charm decays, U -spin breaking is usually studied. A perturbative method to deal with U -spin breaking was proposed in [79,80]. In this method, the corrections of arbitrary order to decay amplitude f |H eff |i are obtained by introducing an s − d spurion mass operator M U brk into the Hamiltonian and initial and final states. It is the U 3 = 0 component in U -spin triplet. Using the s − d spurion mass operator, the author of [80] derives the n-th order U -spin breaking corrections for D 0 → K − π + , D 0 → K + K − , D 0 → π + π − and D 0 → K + π − decays. In this subsection, we try to apply this perturbative method to charmed baryon decays to analyze the singly Cabibbo-suppressed modes Λ + c → Σ + K * 0 and Ξ + c → pK * 0 .
Under the U -spin symmetry limit, the decay amplitudes of Λ + c → Σ + K * 0 and Ξ + c → pK * 0 are equal, as shown in Eq. (24). The first order U -spin breaking is obtained by multiplying an s − d spurion mass operator M U brk ∝ (ss) − (dd) with Hamiltonian and initial and final states, The effective Hamiltonian for SCS decays at first order has an additional penguin term P s+d due to the s − d mass difference [80,81] Hence the first order correction of U -spin breaking is rewritten as in which For Λ + c → Σ + K * 0 and Ξ + c → pK * 0 decays, According to the coupling law of angular momenta and the following property of Clebsch-Gordan coefficients, the relation between the decay amplitudes of Λ + c → Σ + K * 0 and Ξ + c → pK * 0 decays at first order correction of U -spin breaking can be derived. For instance, the expressions of the first term in Eq. (73) in Λ + c → Σ + K * 0 and Ξ + c → pK * 0 modes are written as One can find f |H W M U brk |i term in Λ + c → Σ + K * 0 and Ξ + c → pK * 0 has opposite sign. Similar conclusions are also deduced in other terms of Eq. (73). Thereby, the relation between ratios of the first order U -spin breaking amplitude and the U -spin symmetry amplitude of these two decay modes is So the amplitudes of Λ + c → Σ + K * 0 and Ξ + c → pK * 0 decays in the first-order U -spin breaking are in which the V CKM involves V cs V us sin θ cos θ and V cd V ud − sin θ cos θ. Neglecting the high order contributions, the ratio of A(Λ + c → Σ + K * 0 ) and A(Ξ + c → pK * 0 ) is With the experimental data of branching fractions [19,25], the U -spin breaking parameter Re(ε) in Λ + c → Σ + K * 0 and Ξ + c → pK * 0 modes is extracted to be Re(ε B ) = 0.57 ± 0.38.
The corresponding Re(ε) is given by [80], It is plausible that U -spin breaking in the singly Cabibbo-suppressed transitions is larger than the Cabibbo-flavored and doubly Cabibbo-suppressed transitions and some non-pertubative dynamics enhance the U -spin breaking in both charmed meson and baryon decays.

IV. SUMMARY
In summary, we study the two-body non-leptonic decays of charmed baryons based on the flavor SU (3) symmetry. Hundreds of I-, U -and V -spin relations between different decay channels of charmed baryons are found. Some of them can be used to test the breaking of I-, U -and V -spins. Using these amplitude relations, some branching fractions of charmed baryon decays are predicted, which could provide guides for the future experiments. Several U -spin relations for K 0 S − K 0 L asymmetries and CP asymmetries in the B c → BK 0 S,L modes are proposed. And a possible abnormal U -spin breaking is found in Λ + c → Σ + K * 0 and Ξ + c → pK * 0 modes.