Equivalent $SU(3)_f$ approaches for two-body anti-triplet charmed baryon decays

For the two-body ${\bf B}_c\to{\bf B}M$ decays, where ${\bf B}_c$ denotes the anti-triplet charm baryon and ${\bf B}(M)$ the octet baryon (meson), there exist two theoretical studies based on the $SU(3)$ flavor [$SU(3)_f$] symmetry. One is the irreducible $SU(3)_f$ approach (IRA). In the irreducible $SU(3)_f$ representation, the effective Hamiltonian related to the initial and final states forms the amplitudes for ${\bf B}_c\to{\bf B}M$. The other is the topological-diagram approach (TDA), where the $W$-boson emission and $W$-boson exchange topologies are drawn and parameterized for the decays. As required by the group theoretical consideration, we present the same number of the IRA and TDA amplitudes. We can hence relate the two kinds of the amplitudes, and demonstrate the equivalence of the two $SU(3)_f$ approaches. We find a specific $W$-boson exchange topology only contributing to $\Xi_c^0\to{\bf B}M$. Denoted by $E_M$, it plays a key role in explaining ${\cal B}(\Xi_c^0\to \Lambda^0 K_S^0,\Sigma^0 K_S^0,\Sigma^+ K^-)$. We consider that $\Lambda_c^+ \to n \pi^+$ and $\Lambda_c^+ \to p\pi^0$ proceed through the constructive and destructive interfering effects, respectively, which leads to ${\cal B}(\Lambda_c^+ \to n \pi^+)\gg{\cal B}(\Lambda_c^+ \to p\pi^0)$ in agreement with the data. With the exact and broken $SU(3)_f$ symmetries, we predict the branching fractions of ${\bf B}_c\to{\bf B}M$ to be tested by future measurements.

It is reasonable to regard IRA and TDA as the equivalent approaches for the heavy hadron decays [37,39,[41][42][43][44]. However, He and Wang first point out that the previous analyses using IRA and TDA could not consistently match [42]. In the two-body D and B decays, one seeks the overlooked TDA amplitudes to solve the mismatch problem [42][43][44].
In the two-body B c → BM decays, the mismatch problem remains unsolved, which is due to the inconclusive TDA amplitudes involved in the decays [37,43].
The equivalence should be in accordance with the equal number of the IRA and TDA amplitudes. For instance, one derives two SU(3) f amplitudes and two topological ones for B c → B * M [39], where B * denotes the decuplet baryon. Without considering the singlet contributions to the formation of η 1 , there can be seven independent IRA amplitudes in the B c → BM decays [18,26,43], whereas TDA leads to six, seven, eight and sixteen topological amplitudes from Refs. [38], [15], [36,37] and [43], respectively. Clearly, the unique unification of the two SU(3) f approaches is unavailable. Therefore, we propose to clarify how many independent topological amplitudes can actually exist, and newly unify the IRA and TDA amplitudes. We will perform the numerical analysis, in order to demonstrate that the new unification can accommodate the new data.

II. FORMALISM
For the anti-triplet charmed baryon decays, we present the effective Hamiltonian for the c → uqq ′ decays as [45,46] where G F is the Fermi constant, and λ q ′ q ≡ V uq ′ V * cq denotes the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Besides, (q 1 q 2 ) =q 1 γ µ (1 − γ 5 )q 2 are quark currents, and (α, β) represent the color indices. As q (′) takes the values d and s, the decays with are regarded as the Cabibbo-allowed (CA), singly Cabibbo suppressed (SCS), and doubly Cabibbo suppressed (DCS) processes, respectively, where θ c in s c ≡ sin θ c ≃ 0.22 is the Cabibbo angle. Moreover, the Wilson coefficient c 1(2) is a scale (µ)-dependent number, and we take µ = m c in the c decays.
By omitting Lorentz structure, H c is seen as H c ∼ (q i q jqk )c, where q i = (u, d, s) is a triplet (3) under the SU(3) f symmetry. In IRA, H c is decomposed as two irreducible SU(3) f forms, 6 H and 15 H , whereas TDA only values the flavor changes of c → q iqj q k in H c . Thus, the effective Hamiltonian can be rewritten as [18,19,26,39,40] where c ∓ = (c 1 ∓c 2 ), and the non-zero entries of H(6) lk , H(15) ij k and H ki j are given by [18,39] As the final states, the octet baryon and meson (8 B and 8 M ) have components where the octet baryon can also be written as B ijk = ǫ ijl B l k . Moreover, the η state mixes η q = 1/2(uū + dd) and η s = ss as η = η q cos φ − η s sin φ, where φ = (39.3 ± 1.0) • is the mixing angle [47]. For the anti-triplet charmed baryon states (3 c ), Ξ 0 c , Ξ + c and Λ + c consist of (ds − sd)c, (su − us)c and (ud − du)c, respectively, and we present them in the two forms, We can hence construct the amplitudes of B c → BM in IRA and TDA, written as [36,[38][39][40]43] M IRA = M 6 + M 15 , where T ij ≡ B c k ǫ ijk , a i (i = 1, 2, ...  are regarded as QCD-(dis)favored parameters. Under the group theoretical consideration, [40], which lead to 3 and 4 SU(3) f singlets to be in accordance with a 1,2,3 and a 4,5,6,7 , respectively. Hence, there can be seven independent IRA amplitudes as those in [18,26,43].
For M TDA , the topological amplitudes correspond to the W -boson emission (W EM ) and W -boson exchange (W EX ) diagrams in Fig. 1(a-c) and Fig. 1(d-h), respectively. More specifically, T and C (′) represent the external and internal W EM processes in Fig. 1a and Fig. 1b Fig. 1d, where the meson receives no quark to relate to the W -boson.
Based on the SU(3) f symmetry, TDA and IRA should have seven independent amplitudes. Nonetheless, we derive eight TDA amplitudes. Since we find the appearance of can be redundant. We thus reduce the TDA amplitudes to seven by choosing to work with the convention of E ′ B = 0. Using M IRA = M TDA , we find the relations as and and a 5 = 0, such that we "topologize" the SU(3) f invariant amplitudes.
The W EX diagrams require g → qq added to BM, where qq can be uū, dd, or ss. Due to m s ≫ m u,d , it is possible that the exchange topologies with g → ss can cause the sizeable M and E ′(s) , respectively, for the B c → BM decays. Note that one has included the SU(3) f breaking effect via the W EX diagrams, and demonstrated that it can be applicable to D → P P (P V ) [49] and B c → B * M [39].
In terms of the equations given by [1]  Tables I, II and III.

III. NUMERICAL ANALYSIS
In the numerical analysis, we adopt the CKM matrix elements and (m B (c) , τ Bc ) from with λ = s c = 0.22453 ± 0.00044 in the Wolfenstein parameterization. Making use of 2 )]sc we perform a minimum χ 2 -fit. As the theoretical input, B th is calculated with the equations in Eq. (9) and M(B c → BM) in Tables I and II. As the experimental input, B ex can be found in Tables V and VI, along with σ ex the experimental error. Until very recently, only the upper limit of B ex (Λ + c → pπ 0 ) > 0.8 × 10 −4 has been reported by Belle in Ref. [5], from which the likelihood distribution in Fig. 7 as a function of B(Λ + c → pπ + ) can be used to estimate that B ex (Λ + c → pπ + ) = (0.3 ± 0.3) × 10 −4 . We perform the global fit in the two scenarios. In the first scenario (S1), we exactly The topological amplitudes as complex numbers can be written as where T has been set as a relatively real number, and E ′ M = E B has been implied in Eq. (7). In the second scenario (S2), we test the SU(3) f symmetry breaking, which is indicated by the ratio of B(Ξ 0 With E s B = E B , we obtain R(Ξ 0 c ) = s 2 c ≃ 0.05 away from the data of 0.03 ± 0.01 by two standard deviation. Note that the IRA amplitudes would cause the same deviation, when 2 )]s 2 c one investigates Ξ 0 c → Ξ − K + and Ξ 0 c → Ξ − π + in IRA without considering the broken SU(3) f symmetry [22,[25][26][27]32]. Since R(Ξ 0 c ) suggests the existence of the broken effect, to the parameters in Eq. (12) for the S2 global fit. We thus determine the parameters in the two scenarios (S1 and S2), given in Table IV

IV. DISCUSSIONS AND CONCLUSIONS
Under the group theoretical consideration, we present the seven SU(3) f singlets for B c → BM, which are in agreement with the seven independent IRA amplitudes in Eq. (6).
Since TDA also relies on the SU(3) f symmetry, there should exist seven independent TDA amplitudes. We draw and parameterize eight TDA amplitudes as those using the topologicaldiagram scheme [36,37]. By finding that either E ′ or E ′ B is redundant, we reduce them to seven. We hence obtain the unique relations in Eqs. (7) and (8), and demonstrate that TDA and IRA are the equivalent SU(3) f approaches.
Confusingly, there can be six, seven and sixteen topological amplitudes from Refs. [38], [15] and [43], respectively. In Ref. [38], the less TDA amplitudes reflects the fact that one disregards the quark orderings for B (c) . It also fails to present the isospin relation: . Although Ref. [15] provides the seven TDA parameters, the equality of M(Ξ 0 is not given, which disagrees with the SU(3) f approaches. In Ref. [43], the sixteen amplitudes are due to that all possible quark orderings of the octet baryon are taken into account, which can be reduced with the identity B ijk + B kij + B jki = 0 [36,40].
The equivalent SU(3) f approaches are able to provide more information on the hadronization in the B c → BM decays. For instance, we derive E ′ M = E B = a 4 to reduce the topological diagrams involved in the decays, and a 1 = (T − C)/2 indicates that a 1 connects the two W -emission topological amplitudes. With a 5 = 0, a 5 has no topological correspondence.
Therefore, it is unlikely that the QCD disfavored parameters are negligible, whereas they were commonly discarded.
In Eq. (13), since R(Ξ 0 c ) has indicated that one cannot explain B ex (Ξ 0 c → Ξ − K + ) with the exact SU(3) f symmetry, it has been excluded in the S1 global fit. As a result, χ 2 /n.d.f = 0.9 presents a reasonable fit. We also include B ex (Ξ 0 c → Ξ − K + ) for a test, which causes χ 2 /n.d.f = 5.5. We hence add E s B as the new parameters in the S2 global fit, in order to accommodate B ex (Ξ 0 c → Ξ − K + ). It turns out to be a reasonable fit with χ 2 /n.d.f = 1.4, and B th (Ξ 0 c → Ξ − K + ) = (4.1 ± 2.8) × 10 −4 can explain the data. Moreover, we extract |n q | = 1.2 ± 0.4 and δ nq = (50.3 ± 11.5) • in E s B = |n q |e iδn q E B as the measure of the SU(3) f symmetry breaking. One can study the broken SU(3) f symmetry with IRA [19,23]. Without the topological indication in Eq. (13), three IRA amplitudes should be introduced for the broken effects in B c → BM [23].
As can be seen in Tables V and VI, the pole model [10,11], IRA with the neglecting of (a 4 , a 5 , a 7 ) [25], and TDA of this work (S1) all lead to R(Ξ 0 c ) ≃ 0.05. It seems that TDA of Ref. [38] gives the consistent R(Ξ 0 c ) ≃ 0.03; it is, however, based on M(Ξ 0 3.28 ± 0.58 5.5 ± 0.7 (6.9 ± 1.0, 2.4 ± 0.9) global fit using IRA without neglecting the QCD-disfavored parameters can present R(Ξ 0 c ) ≃ 0.04 close to the data [31], where the amplitudes are considered to depend on the actual particle masses.
In summary, we have studied the two-body anti-triplet charmed baryon decays using the irreducible SU(3) f approach (IRA) and topological-diagram approach (TDA). Due to the group theoretical consideration, we have presented that there can be the same number of the IRA and TDA amplitudes. We have hence found out the unique relations, and demonstrated that IRA and TDA are the equivalent SU(3) f approaches. We have explained the recently measured branching fractions, that is, B(Ξ 0 c → Λ 0K 0 , Σ 0K 0 , Σ + K − ) and B(Λ + c → nπ + , pπ 0 , pη). Moreover, we have predicted the branching fractions of B c → BM under the exact and broken SU(3) f symmetries, which can be tested by future measurements.