Two-body charmed baryon decays involving vector meson with $SU(3)$ flavor symmetry

We study the two-body anti-triplet charmed baryon decays of ${\bf B}_c\to {\bf B}_n V$, with ${\bf B}_c=(\Xi_c^{0},-\Xi_c^{+},\Lambda_c^+)$ and ${\bf B}_n(V)$ the baryon (vector meson) states. Based on the $SU(3)$ flavor symmetry, we predict that ${\cal B}(\Lambda^{+}_{c}\to \Sigma^{+}\rho^{0},\Lambda^0 \rho^+)=(0.61\pm 0.46,0.74\pm 0.34)\%$, in agreement with the experimental upper bounds of $(1.7,6)\%$, respectively. We also find ${\cal B}(\Lambda^+_c \to \Xi^0 K^{*+},\Sigma^0 K^{*+},\Lambda^0 K^{*+}) =(8.7 \pm 2.7,1.2\pm 0.3,2.0\pm 0.5)\times 10^{-3}$ to be compatible with the pseudoscalar counterparts. For the doubly Cabibbo-suppressed decay $\Xi^+_c \to p\phi$, measured for the first time, we predict its branching ratio to be $(1.5\pm 0.7)\times 10^{-4}$, together with ${\cal B}(\Xi^+_c \to p \bar K^{*0},\Sigma^+ \phi) =(7.8 \pm 2.2,1.9\pm0.9)\times 10^{-3}$. The ${\bf B}_c\to{\bf B}_n V$ decays with ${\cal B}\simeq {\cal O}(10^{-4}-10^{-3})$ are accessible to the BESIII, BELLEII and LHCb experiments.


I. INTRODUCTION
The two-body B c → B n V decays have not been abundantly measured as the B c → B n M counterparts, where B c = (Ξ 0 c , −Ξ + c , Λ + c ) are the anti-triplet charmed baryon states, together with B n and V (M) the baryon and vector (pseudo-scalar) meson states, respectively. For example, all Cabibbo-favored (CF) Λ + c → B n M decays have been measured [1], including the recent BESIII observation for Λ + c → Σ + η ′ [2], whereas for the CF vector modes only Λ + c → pK * 0 , Σ + ω, Σ + φ have absolute branching fractions [1]. In addition, the first absolute branching ratio for the Ξ 0 c decays is Ξ 0 c → Ξ − π + [3], instead of any Ξ 0 c → B n V decays.
Nevertheless, the B c → B n V decays are not less important than the B c → B n M counterparts. First, the participations of BESIII, BELLEII and LHCb Collaborations are expected to make more accurate measurements for B c → B n V , such as Λ + c → Σ + ρ 0 , Λ 0 ρ + , presented as B(Λ + c → Σ + ρ 0 , Λ 0 ρ + ) < (1.7, 6)% due to the previous measurements [1]. Second, in the three-body B c → B n MM ′ decays, the MM ′ meson pair is assumed to be mainly in the S-wave state [4]. However, the resonant B c → B n V, V → MM ′ decay causes MM ′ to be in the P-wave state, of which the contribution to the total B(B c → B n MM ′ ) needs clarification. Note that (S,P) denote L = (0, 1) as the quantum numbers for the orbital angular momentum between M and M ′ . Third, the three-body Ξ + c decays can be measured as the ratios of B(Ξ + c → B n V )/B(Ξ + c → B n MM ′ ). Particularly, the doubly Cabibbo-suppressed Ξ + c → pφ decay is observed for the first time, with B(Ξ + c → pφ)/B(Ξ + c → pK − π + ) = (19.8 ± 0.7 ± 0.9 ± 0.2) × 10 −3 [5]. The information of Since the study of B c → B n V is necessary, it is important to provide a corresponding theoretical approach. The factorization approach for the heavy hadron decays [6][7][8] [19][20][21][22][23][24][25], such as the purely non-factorizable Λ + c → Ξ 0 K + decay [31]. In addition, the predicted values of B(Λ + c → Σ + η ′ ) and B(Ξ 0 c → Ξ − π + ) are in agreement with the recent observations [2,3,22,23]. Therefore, we propose to extend the SU(3) f symmetry to B c → B n V , while the existing observations have been sufficient for the numerical analysis. In this report, we will extract the SU(3) f amplitudes, and predict the to-be-measured B c → B n V branching fractions.

II. FORMALISM
To obtain the amplitudes for the two-body B c → B n V decays, where B c(n) is the singly charmed (charmless) baryon state and V the vector meson, we present the relevant effective Hamiltonian (H ef f ) for the tree-level c quark decays, given by [26] with (q 1 q 2 ) ≡q 1 γ µ (1 − γ 5 )q 2 . By neglecting the Lorentz indices, the operator of (q 1 q 2 )(q 3 c) transforms as (q i q kq j )c under the SU(3) f symmetry, where q i = (u, d, s) represent the triplet of 3. The operator can be decomposed as irreducible forms, which is accordance with (3 × 3 ×3)c = (3 +3 ′ + 6 + 15)c. One hence has [15,16] O −(+) ∼ O 6(15) = 1 2 (ūds ∓sdū)c , with the subscripts (6, 15) denoting the two irreducible SU where the tensor notations of 1/2ǫ ijl H(6) lk and H(15) k ij contain O (q,′) 6 and O (q,′) 15 , respectively. In terms of (  (15) = −s 2 c as the non-zero entries [15]. Note that n = 0, 1 and 2 in s n c correspond to the Cabibbo-flavored (CF), singly Cabibbo-suppressed (SCS), and doubly Cabibbo-suppressed (DCS) decays, respectively. We also need B c and B n (V ) to be in the irreducible representation of the SU(3) f symmetry, given by Subsequently, H ef f in Eq. (4) is enabled to be connected to the initial and final states in instead of introducing the details of the QCD calculations for the hadronization. Explicitly, the T amplitudes (T -amps) are given by [15,16] T where T ij ≡ (B c ) k ǫ ijk , and (c − , c + ) have been absorbed into the SU(3) parameters (ā i ,h (′) ).
With the full expansion of T -amps in Table I, the two-body B c → B n V decays are presented with the SU(3) f parameters. Since ω = (uū + dd)/ √ 2 and φ = ss actually mix with In terms of the we can compute the branching ratio with the SU(3) f amplitudes, where τ Bc denotes the B c lifetime. The SU(3) f amplitudes are accounted to be 9 complex numbers, leading to 17 independent parameters to be extracted, whereas there exist 10 data points for the numerical analysis. To have a practical fit, we follow Refs. [19,22,23,25] to reduce the parameters.
On the other hand, (ā 1,2,3 , h) from H(6) are kept for the fit, represented as with the phases δā 2,3 ,h , andā 1 set to be relatively real.

III. NUMERICAL ANALYSIS
For the numerical analysis, we collect (the ratios of) the branching fractions for the observed B c → BV decays in Table II, where B(Ξ + c → pK * 0 , Σ + φ, Σ +K * 0 ) are in fact measured to be relative to B(Ξ + c → Ξ − π + π + ) [1,29], recombined as R 1,2 (Ξ + c ). We obtain , with the input of B(Ξ 0 c → Ξ − π + ) measured by BELLE [3]. In addition, the ratio of R(Λ + c ) = (Λ + c → Σ + ρ 0 )/B(Λ + c → Σ + ω) comes from the data events in Ref. [32]. Besides, s c = 0.22453 ± 0.00044 [1] is the theoretical input for the CKM matrix elements. By using the equation of [9] we are able to obtain the minimum χ 2 value, such that the SU(3) f parameters can be extracted with the best fit. Note that B i (R j ) represents (the ratio of) the branching fraction, with the subscript th (ex) denoting the theoretical (experimental) input, while σ i(j) ex stands for the experimental error. As the inputs in Eq. (10), B(R) th come from the T -amps in Table I, while B(R) ex and σ ex the data points in Table II. Subsequently, the global fit gives where n.d.f represents the number of the degree of freedom. With the fit results in Eq. (11), we calculate the branching ratios, R(Λ + c ) and R 1,2 (Ξ + c ) to be compared to their data inputs in Table II. Moreover, we predict the branching fractions for the B c → B n V decays, given in Table III.

ACKNOWLEDGMENTS
This work was supported by National Science Foundation of China (11675030).