SU(3) symmetry breaking in charmed baryon decays

We explore the breaking effects of the $SU(3)$ flavor symmetry in the singly Cabibbo-suppressed anti-triplet charmed baryon decays of ${\bf B}_c\to {\bf B}_n M$, with ${\bf B}_c=(\Xi_c^0,\Xi_c^+,\Lambda_c^+)$ and ${\bf B}_n(M)$ the baryon (pseudo-scalar) octets. We find that these breaking effects can be used to account for the experimental data on the decay branching ratios of ${\cal B}(\Lambda_c^+\to \Sigma^{0} K^{+},\Lambda^{0} K^{+})$ and $R'_{K/\pi}$=${\cal B}(\Xi^0_c \to \Xi^- K^+)$/${\cal B}(\Xi^0_c \to \Xi^- \pi^+)$. In addition, we obtain that ${\cal B}(\Xi_{c}^{0} \to \Xi^{-} K^{+},\Sigma^{-} \pi^{+})=(4.6 \pm 1.7,12.8 \pm 3.1)\times 10^{-4}$, ${\cal B}(\Xi_c^0\to pK^-,\Sigma^+\pi^-)=(3.0 \pm 1.0, 5.2 \pm 1.6)\times 10^{-4}$ and ${\cal B}(\Xi_c^+\to \Sigma^{0(+)} \pi^{+(0)})=(10.3 \pm 1.7)\times 10^{-4}$, which all receive significant contributions from the breaking effects, and can be tested by the BESIII and LHCb experiments.


I. INTRODUCTION
It is known that the theoretical approach based on the factorization and quantum chromodynamics (QCD) barely explains the charmed hadron decays [1]. This is due to the fact that the mass of the charm quark, m c ≃ 1.5 GeV, is not as heavy as that of the bottom one, m b ≃ 4.8 GeV, resulting in an undetermined correction to the heavy quark expansion, such that the alternative models have to take place for this correction [2][3][4][5][6][7]. On the other hand, the SU(3) flavor (SU(3) f ) symmetry that works in the b-hadron decays [8][9][10][11][12] can be well applied to D → MM and B c → B n M [13][14][15][16][17][18][19][20][21][22][23][24], where B c = (Ξ 0 c , −Ξ + c , Λ + c ) are the lowestlying anti-triplet charmed baryon states, while B n and M represent baryon and pseduscalar meson states, respectively. Particularly, the SU(3) f symmetry has been extended to investigate the singly charmed baryon sextet states as well as the doubly and triply charmed baryon ones [22,23]. For D → MM decays, the measurements produce [25] in comparison with (R D 0 (K/π) , B D 0 (2K 0 s ) ) ≃ (1, 0) given by the theoretical calculations based on the SU(3) f symmetry. The disagreements between the theory and experiment imply that the breaking effects of the SU(3) f symmetry cannot be ignored in the singly Cabibbosuppressed (SCS) processes. We note that, in the literature, the SU(3) f breaking effects were used to relate R K/π to the possible large difference of the CP violating asymmetries of [14,26,27], which is recently measured to be (−0.10 ± 0.08 ± 0.03)% by LHCb [28].
For the two-body B c → B n M decays, both Cabibbo flavored (CF) and SCS decays are not well explained. In particular, the experimental measurements show that [29,30] , [25] , where s c ≡ sin θ c = 0.2248 [25] with θ c the well-known Cabbibo angle. However, theoretical evaluations based on the SU(3) f symmetry lead to B pπ 0 = (5.7 ± 1.5) × 10 −4 and R ′ K/π ≃ 1.0s 2 c [20], and those in the factorization approach give where we have used the data of B(Λ + c → pK 0 ) = (3.16 ± 0.16) × 10 −2 [25]. In addition, the fitted results of B(Λ + c → Λ 0 K + , Σ 0 K + ) = (4.6 ± 0.9, 4.0 ± 0.8) × 10 −4 [21] are (1.3 − 1.6)σ away from the data in Eq. (2). In this study, we will consider the breaking effects of the SU(3) f symmetry due to the fact of m s ≫ m u,d in the B c → B n M decays, particularly, the SCS processes, in accordance with the D → MM ones. Our goal is to find out whether the data in Eq. (2) can be understood by introducing the breaking effects.
The paper is organized as follows. In Sec. II, we provide the formalism, in which the amplitudes of the B c → B n M decays with and without the breaking effects of SU(3) f symmetry are presented. The numerical analysis is performed in Sec. III. In Sec. IV, we discuss our results and give the conclusions.

II. FORMALISM
The two-body charmed baryon weak decays, such as Ξ 0 c → Ξ − π + (Ξ − K + ) and Λ + c → pπ 0 , proceed through the quark-level transitions of c → sud, c → udd and c → uss, with the effective Hamiltonian given by [31] where G F is the Fermi constant, c ± are the scale-dependent Wilson coefficients, and the CKM matrix elements V cs V ud ≃ 1 and V cs V us ≃ −V cd V ud ≃ s c correspond to the Cabibbofavored (CF) and singly Cabibbo-suppressed (SCS) charmed hadron decays, respectively, are the four-quark operators, written as where q = (d, s) and (q 1 q 2 ) =q 1 γ µ (1 − γ 5 )q 2 . With q i = (u, d, s) as the triplet of 3, the operator of (q i q kq j )c can be decomposed as the irreducible forms, that is, (3 × 3 ×3)c = (3 +3 ′ + 6 + 15)c. Accordingly, the operators O given by [16] O ∓ ≃O 6(15) = 1 2 (ūds ∓sdū)c , such that the effective Hamiltonian in Eq. (3) can be transformed into the tensor form of with the non-zero entries: where the notations of (i, j, k) are quark indices, to be connected to the initial and final states in the amplitudes. Note that H 23 (6) and H 32 (6) are derived from O s 6 and O d 6 , respectively. The lowest-lying charmed baryon states B c are an anti-triplet of3 to consist of (ds − sd)c, together with the baryon and meson octets, given by where we have removed the octet η 8 and singlet η 1 meson states to simplify our discussions.
There remain two measurements to be explained. In Eq. (2), the prediction for B pπ 0 has the 2σ gap to reach the edge of the experimental upper bound. However, with R(15/6) to be small, it is nearly impossible that, by restoring a 4,5,6,7 that have been ignored in the literature [19][20][21][22]24], one can accommodate the data of B pπ 0 but without having impacts on the other decay modes, which are correlated with the same sets of parameters. Moreover, as seen from Eq. (12), there is no room for R ′ K/π as it is fixed to be (1.0)s 2 c . On the other hand, the results for D → MM decays in Eq. (1) suggest some possible corrections from the breaking effects of the SU(3) f symmetry in the SCS processes. In the charm baryon decays, we consider the similar effects. Due to m s ≫ m u,d , we present the matrix of M s = ǫ(λ s ) i j [13] to break SU(3) f , where ǫ ∼ 0.2 − 0.3 and λ s is given by [13,14,17] which transforms as an octet of 8, such that its coupling to H(6) is in the form of 8 ×6 =3 + 6+15+24, and3 is for the simplest break effects to be confined in the SCS processes [17,34].
Note that from

CF mode
T -amp )sc the new T -amplitudes, given by where  Table I, to be used to calculate the decay widths, given by [25] where

III. NUMERICAL ANALYSIS
In the numerical analysis, we examine B(Λ + c → Λ 0 K + , Σ 0 K + , pπ 0 ) and R ′ K/π by including the breaking effects of the SU(3) f symmetry to see if one can explain their data in Eq. (2).
The theoretical inputs for the CKM matrix elements are given by [25] with λ = 0.2248 in the Wolfenstein parameterization. We perform the minimum χ 2 fit, in terms of the equation of [21] , where the subscripts th and ex are denoted as the theoretical inputs from the amplitudes in Table I and the   experimental data points in Table II, respectively, while σ i,j correspond to the 1σ errors.
By following Refs. [20][21][22], we extract the parameters, which are in fact complex numbers, given by where a 4,5,...,7 have been ignored as discussed in Sec. III. Since only the relative phases contribute to the branching ratios, a 1 is set to be real without losing generality. However, we take v i to be real numbers in order to fit 8 parameters with the 9 data points in Table II.
Subsequently, the fit with the breaking effects in the SU (3) Table III.

IV. DISCUSSIONS AND CONCLUSIONS
As seen from Tables II and III, the breaking effects associated with v 2 and v 3 on the branching ratios of the SCS Λ + c → B n M decays are at most around 30%, which is close to the naive estimation of (f K /f π ) 2 ≃ 40%. In particular, we get B(Λ + c → Λ 0 K + , Σ 0 K + ) = (6.1±0.9, 5.2±0.7)×10 −4 , which explain the data in Eq. (2) well and alleviate the (1.3−1.6)σ deviations by the fit with the exact SU(3) f symmetry [21]. Meanwhile, the branching ratios for the CF modes are fitted to be the same as those without the breaking except B(Λ + c → Ξ 0 K + ), which is slightly different in order to account for the recent observational value given by BESIII [35].

SCS mode
Exact [21] Broken with v 1 in the T amplitudes are also projected in the Ξ 0,+ c modes, particularly, Ξ 0 c → Ξ − K + and Ξ 0 c → Σ 0 π 0 . Clearly, these SCS Ξ c decays all contain sizable SU(3) f breaking effects, and can be treated as golden modes to test the SU(3) f symmetry.