Three-body charmed baryon Decays with SU(3) flavor symmetry

We study the three-body anti-triplet ${\bf B_c}\to {\bf B_n}MM'$ decays with the $SU(3)$ flavor ($SU(3)_f$) symmetry, where ${\bf B_c}$ denotes the charmed baryon anti-triplet of $(\Xi_c^0,-\Xi_c^+,\Lambda_c^+)$, and ${\bf B_n}$ and $M(M')$ represent baryon and meson octets, respectively. By considering only the S-wave $MM'$-pair contributions without resonance effects, the decays of ${\bf B_c}\to {\bf B_n}MM'$ can be decomposed into irreducible forms with 11 parameters under $SU(3)_f$, which are fitted by the 14 existing data, resulting in a reasonable value of $\chi^2/d.o.f=2.8$ for the fit. Consequently, we find that the triangle sum rule of ${\cal A}(\Lambda_c^+\to n\bar K^0 \pi^+)-{\cal A}(\Lambda_c^+\to pK^- \pi^+)-\sqrt 2 {\cal A}(\Lambda_c^+\to p\bar K^0 \pi^0)=0$ given by the isospin symmetry holds under $SU(3)_f$, where ${\cal A}$ stands for the decay amplitude. In addition, we predict that ${\cal B}(\Lambda_c^+\to n \pi^{+} \bar{K}^{0})=(0.9\pm 0.8)\times 10^{-2}$, which is $3-4$ times smaller than the BESIII observation, indicating the existence of the resonant states. For the to-be-observed ${\bf B_c}\to {\bf B_n}MM'$ decays, we compute the branching fractions with the $SU(3)_f$ amplitudes to be compared to the BESIII and LHCb measurements in the future.


I. INTRODUCTION
The three-body charmed baryon B c → B n MM ′ decays have been recently searched by the experimental Collaborations of BELLE, BESIII and LHCb, where B c ≡ (Ξ 0 c , −Ξ + c , Λ + c ) denotes the charmed baryon anti-triplet, and B n and M ( ′ ) correspond to the baryon and meson octets, respectively. For example, the decay of Λ + c → pK − π + has been observed with high precision by BELLE and BESIII [1,2], which improves the accuracy of the Λ b decays with Λ + c as one of the final states [5]. Besides, the crucial information on the higher wave baryon resonances like Λ(1405) has been extracted from the Σπ invariant mass spectra of the Λ + c → Σππ decays [3]. The another interest comes from the test of the theoretical approach. For example, the first observation of Λ + c → nK 0 s π + has been used to examine the isospin relation [4], that is, R(∆) ≡ A(Λ + c → nK 0 π + ) + A(Λ + c → pK − π + ) + √ 2A(Λ + c → pK 0 π 0 ) = 0 [6, 7] 1 Since the Λ + c → pK − π + decay shares the similar diagrams as the doubly Cabibbosuppressed Λ + c → pK + π − one, we have the ratio of B(Λ + c → pK + π − )/B(Λ + c → pK − π + ) = R Kπ tan 4 θ c with R Kπ ≃ 1.0 and θ c the Cabibbo angle should hold. Nonetheless, the values of R Kπ = 0.82 ± 0.12 [9] and 0.58 ± 0.06 [10] have been measured by BELLE and LHCb, respectively, showing a possible deviation caused by an additional W -exchange amplitude for Λ + c → pK − π + . As a result, the B c → B n MM ′ decays are important for achieving a deeper insight into the hadronization of particle interactions.
In contrast with the abundant observations, there rarely exist systematic theoretical studies on the B c → B n MM ′ decays, apart from those based on the isospin symmetry [6,7]. This is due to the fact that the scale of the charm quark mass (m c ) is too large for the flavor SU(3) (SU(3) f ) symmetry, but the theories based on the heavy quark expansion may not be valid as m c is not large enough. In addition, the factorization fails to work well in the charmed hadron decays [6,11], whereas it is successfully used in the beauty hadron ones [12][13][14]. The alternative approaches for the charmed hadron decays have been shown in Refs. [15][16][17][18][19][20], which take into account the non-factorizable effects. On the other hand, the SU(3) f symmetry has been tested as a useful tool both in the beauty and charmed 1 To calculate the decay amplitude of A, we use the conventions of |π + = −|11 and |K 0 = −| 1 2 1 2 , whereas |π + = |11 and |K 0 = | 1 2 1 2 are taken in Refs. [6,7], resulting in the relation to be R(∆) ≡ A(Λ + c → nK 0 π + ) − A(Λ + c → pK − π + ) − √ 2A(Λ + c → pK 0 π 0 ) = 0. However, the different signs in R(∆) and other similar relations do not affect the physical consequences of these relations due to the arbitrariness of the phase of the amplitude. hadron decays [21][22][23][24][25][26][27][28], particularly, the two-body B c → B n M decays [6,[29][30][31][32][33][34][35][36][37][38]. It is hence expected that the same symmetry can be applied to the three-body B c → B n MM ′ decays.
In this paper, we will relate the possible B c → B n MM ′ decay processes with the SU(3) f parameters [29], by which the systematic numerical analysis can be performed for the first time. Under the SU(3) f symmetry, we will derive the relation of R(∆) = 0, and examine the value of R Kπ from the ratio of B(Λ + c → pK + π − )/B(Λ + c → pK − π + ). Our paper is organized as follows. We give the formalism in Sec. II, where the amplitudes for the three-body charmed baryon decays under the SU(3) f symmetry are presented. In Sec. III, we show our numerical results and discussions. Our conclusions are in Sec. IV.

II. FORMALISM
The three-body B c → B n MM ′ decays can proceed through the charm quark decays of c → sud, c → udd (uss) and c → dus, where B c,n and M ( ′ ) denote the baryon and meson states, respectively. Accordingly, the tree-level effective Hamiltonian is given by [39] has been used for O † ∓ to combine the c → udd and c → uss transitions. By means of the Cabibbo angle θ c , it is given that In Eq. (2), (q 1 q 2 )(q 3 c) can be rewritten as (q i q kq j )c with q i = (u, d, s) the triplet of 3 under the SU(3) f symmetry, by suppressing the Dirac and Lorentz indices. Furthermore, since (q i q kq j )c can be decomposed as the irreducible forms of (3 × 3 ×3)c = (3 +3 ′ + 6 + 15)c, one derives that [29] O −(+) ≃O 6(15) = 1 2 (ūds ∓sdū)c , Subsequently, the effective Hamiltonian in Eq. (1) has the expression under the SU(3) f symmetry, given by [32][33][34][35]] where H(6, 15) are presented as the tensor forms of (O with (i, j, k) for the quark indices. Correspondingly, the three lowest-lying charmed baryon states of B c form an anti-triplet of3 to consist of (ds − sd)c, (us − su)c and (ud − du)c, and B n (M) belongs to the baryon (meson) octet of 8, which are written as respectively. Now, one is able to connect the octets of (B n , M) i j and anti-triplet of (B c ) i to (ǫ ijl H(6) lk , H(15) ij k ) in H ef f of Eq. (4) to get the SU(3) f amplitudes. Since the Wilson coefficients are scaledependent, in the NDR scheme it is calculated that (c − , c + ) = (1.78, 0.76) at the scale µ = 1 GeV [40,41]. The value of (c − /c + ) 2 ≃ 5.5 implies the suppressed branching ratios associated with H (15). Hence, we follow Refs. [6,32,37] to ignore the amplitudes from with T ij = (B c ) a ǫ aij , where c 2 c and c − in H ef f have been absorbed into the SU(3) f parameters a i (i = 1, 2, ..., 6). While there exists the relative orbital angular momentum L between the two-meson states, we have assumed the S-wave MM ′ -pair (L = 0) in the dominant amplitudes in Eq. (7), whereas the P-wave one (L = 1) is neglected. However, there are some cases in which the S-wave contributions vanish, but P-wave ones are dominant, resulting in the another set of amplitudes to be studied elsewhere. For example, the decay of Λ + c → Λπ + π 0 with the measured branching ratio around 7.1% is mainly from the P-wave contribution.
The integration over the phase space of the three-body decay relies on the equation of [5] where . In Tables I, II and III, we show the full expansions , respectively. In general, the SU(3) f parameters depend on m 12 and m 23 . However, all structures in the Dalitz plots come from the dynamical effects, such as those from the resonant states. Clearly, the squared amplitude in the Dalitz plot is almost structureless for the decay without the resonance. As a result, we treat our decay amplitudes as constants without energy dependences so that they can be factored out from the integrals as an approximation. Σ + π 0 π 0 , pK + π − ) [3,10]. data our results

III. NUMERICAL RESULTS AND DISCUSSIONS
In the numerical analysis, we perform the minimum χ 2 fit to examine if the SU(3) f symmetry is valid in the B c → B n MM ′ decays. The equation of the χ 2 fit is given by where B th as B(B c → B n MM ′ ) is calculated by the SU(3) f parameters, and B ex the experimental value in Table IV, with σ the experimental error. With sin θ c = 0.2248 [5], one obtains that t c = 0.2307 as the input in Eq. (5). The SU(3) f parameters are written as a 1 , a 2 e iδa 2 , a 3 e iδa 3 , a 4 e iδa 4 , a 5 e iδa 5 , a 6 e iδa 6 , where the phases δ a 2,3,...,6 are due to the nature of complex numbers associated with a i , while a 1 can be relatively real. This leads to the reduced 11 parameters to be extracted with 14 data inputs in Table IV, where the fitting values of a i and δ a i are shown in Table V To determine the SU(3) f parameters, we use the non-resonant parts of Λ + c → pK − π + from the PDG [5]. Note that the resonant Λ + c → p(K * 0 →)K − π + , K − (∆(1232) ++ →)pπ + and π + (Λ(1520) →)pK − contributions are separated from its total branching ratio. In addition, the decay of Λ + c → pK − K + is free from the resonant one of Λ + c → p(φ →)K − K + . For the other Λ + c decays in Table IV, some of their resonant parts might be present, but taken to be small, such as B(Λ + c → Σ + (ρ 0 →)π + π − ) < 1.7% [5], which should be insensitive to the fit. We hence use their total branching ratios, instead of excluding the resonant contributions. The Ξ 0,+ c → B n MM ′ decays are partially observed, such that we can barely use their data. Nonetheless, in terms of T (Λ + c → Ξ − K + π + ) = 1/(−2t c )T (Ξ + c → Σ − π + π + ) = −2a 6 and the data of B(Λ + c → Ξ − K + π + ), we obtain B(Ξ + c → Σ − π + π + ) = (1.1 ± 0.1) × 10 −2 , by which the observed ratio of B(Ξ + c → Σ − π + π + )/B(Ξ + c → Ξ − π + π + ) = 0.18 ± 0.09 and it leads to B(Ξ + c → Ξ − π + π + ) = (6.1 ± 3.1) × 10 −2 as given in Table IV. With χ 2 /d.o.f being 2.8 in Eq. (V), it turns out to be a reasonable fit, so that the SU(3) f symmetry with the reduced parameters can be used to explain the three-body which agrees with our numerical analysis. Note that the calculation of B(Λ + c → Σ − π + π + ) needs an additional pre-factor of 1/2 to T (Λ + c → Σ − π + π + ) due to the fact that the π + π + meson-pair involves two identical bosons.
There exist the sum rules for the T-amplitudes in Table I. In particular, by taking the CF Λ + c decay modes as an example, we obtain Note that the first relation of R(∆) in Eq. (12), which has been used in Ref. [4] to reveal the broken isospin symmetry, is also derived by the isospin symmetry in Refs. [6,7] with some different signs in the relation due to the conventions of the π + andK 0 states. In addition, the second relation in Eq. (12) can be identified as the special case in Ref. [7], given by with the symmetrized amplitude of where T ′ (Λ + c → Σ + π ± π ∓ ) are the amplitudes calculated by the isospin analysis in Ref. [7]. Likewise, one can take the relations in Eq. (12) to explore the broken SU(3) f symmetry.
There are other relations and sum rules obtained from the U-spin symmetry, which is also The not-yet-observed B(Λ + c → B n MM ′ ) can be calculated by the SU(3) f parameters, which are given in Table VI Tables VII and VIII, respectively, to be compared to the upcoming data.