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 Bc→BnM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{B}_c\rightarrow \mathbf{B}_n M$$\end{document}, with Bc=(Ξc0,Ξc+,Λc+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{B}_c=(\Xi _c^0,\Xi _c^+,\Lambda _c^+)$$\end{document} and Bn(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{B}_n(M)$$\end{document} 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 B(Λc+→Σ0K+,Λ0K+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Lambda _c^+\rightarrow \Sigma ^{0} K^{+},\Lambda ^{0} K^{+})$$\end{document} and RK/π′=B(Ξc0→Ξ-K+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R'_{K/\pi }={\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- K^+)$$\end{document}/B(Ξc0→Ξ-π+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- \pi ^+)$$\end{document}. In addition, we obtain that B(Ξc0→Ξ-K+,Σ-π+)=(4.6±1.7,12.8±3.1)×10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Xi _{c}^{0} \rightarrow \Xi ^{-} K^{+},\Sigma ^{-} \pi ^{+})=(4.6 \pm 1.7,12.8 \pm 3.1)\times 10^{-4}$$\end{document}, B(Ξc0→pK-,Σ+π-)=(3.0±1.0,5.2±1.6)×10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Xi _c^0\rightarrow pK^-,\Sigma ^+\pi ^-)=(3.0 \pm 1.0, 5.2 \pm 1.6)\times 10^{-4}$$\end{document} and B(Ξc+→Σ0(+)π+(0))=(10.3±1.7)×10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Xi _c^+\rightarrow \Sigma ^{0(+)} \pi ^{+(0)})=(10.3 \pm 1.7)\times 10^{-4}$$\end{document}, which all receive significant contributions from the breaking effects, and can be tested by the BESIII and LHCb experiments.


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][8]. On the other hand, the SU (3) flavor (SU (3) f ) symmetry that works in the b-hadron decays [9][10][11][12][13] can be well applied to D → M M and B c → B n M [14][15][16][17][18][19][20][21][22][23][24][25], where B c = ( 0 c , − + c , + c ) are the lowest-lying antitriplet charmed baryon states, while B n and M represent baryon and pseudoscalar 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 [23,24]. For D → M M decays, the measurements produce [26] R D 0 (K /π ) ≡ B(D 0 → K + K − ) B(D 0 → π + π − ) = 2.82 ± 0.07, a e-mail: geng@phys.nthu.edu.tw 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 Cabibbo-suppressed (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 C P violating asymmetries of [15,27,28], which is recently measured to be (−0.10 ± 0.08 ± 0.03)% by LHCb [29].
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 [30,31], where s c ≡ sin θ c = 0.2248 [26] 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 [21], and those in the factorization approach give [26]. In addition, the fitted results of B( + c → 0 K + , 0 K + ) = (4.6 ± 0.9, 4.0 ± 0.8) × 10 −4 [22] 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 → M M 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 Sect. 2, 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 Sect. 3. In Sect. 4, we discuss our results and give the conclusions.

Formalism
The two-body charmed baryon weak decays, such as 0 c → − π + ( − K + ) and + c → pπ 0 , proceed through the quark-level transitions of c → sud, c → udd and c → uss, with the effective Hamiltonian given by [32] 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, while O (d,s) ± 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 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, (us − su)c and (ud − du)c, presented as 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. Subsequently, the amplitudes of B c → B n M can be derived as with with T i j ≡ (B c ) k i jk . In Eq. (11), a 1,2,3 and a 4,5,6,7 are the SU (3) parameters from H (6) and H (15), respectively, in which c ∓ have been absorbed. We hence obtain [22] based on the exact SU (3) f symmetry. According to Refs. [21,22], the numerical analysis with the minimum χ 2 fit has well explained the ten observed B c → B n M decays by neglecting the terms associated with a 4,5,6,7 to reduce the parameters [20][21][22][23]25]. This reduction is due to the fact that the contributions to the branching rates from H (15) and H (6) lead to a small ratio of R (15/6) 1.78) calculated at the scale μ = 1 GeV in the NDR scheme [33,34]. 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 [20][21][22][23]25], 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 3 and λ s is given by [14,15,18] 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, and 3 is for the simplest break effects to be confined in the SCS processes [18,35]. Note that from 1 8 (6) jn and the nonzero entry of H (3) 1 = s c from the coupling of H (6) 23 and H (6) 32 [14], one can trace back to the break effect between SCS c → uss and c → udd transitions. As a result, the SU (3) f symmetry breaking gives rise to the new T -amplitudes, given by where  Table 1, to be used to calculate the decay widths, given by [26] where

Numerical analysis
In the numerical analysis, we examine B( + c → 0 K + , 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 [26] with λ = 0.2248 in the Wolfenstein parameterization. We perform the minimum χ 2 fit, in terms of the equation of [22] where the subscripts th and ex are denoted as the theoretical inputs from the amplitudes in Table 1 and the experimental data points in Table 2, respectively, while σ i, j correspond to the 1σ errors. By following Refs. [21][22][23], we extract the parameters, which are in fact complex numbers, given by where a 4,5,...,7 have been ignored as discussed in Sect. 3. 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 2. In the calculation, δ a i (i=2 or 3) from a i e δ a i is a fitting parameter, which can absorb the phase of δ v i from the interference in the data fitting. Note that δ a i (i = 2, 3) have been fitted with the imaginary parts [22]. As a result, we may set δ v 2,3 along with the overall phase of δ v 1 to be zero for the estimations of the decay branching ratios due to the SU (3) breaking effects. We will follow Ref. [25] to test our assumption, where a similar global fit in the approach of the SU (3) f symmetry has been done to extract a 2 e δ a 2 by freely rotating the angle of δ a 2 from −180 • to 180 • to estimate the uncertainties of the branching ratios. Subsequently, the fit with the breaking effects in the SU (3) f symmetry yields   [22]. With the parameters in Eq. (19), we obtain the branching ratios of the CF and SCS B c → B n M decays, shown in Table 3.
2 s c

Discussions and conclusions
As seen from Tables 2 and 3, the breaking effects associated with v 2 and v 3 on the branching ratios of the SCS + 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( + c → 0 K + , 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 [22]. Meanwhile, the branching ratios for the CF modes are fitted to be the same as those without the break-ing except B( + c → 0 K + ), which is slightly different in order to account for the recent observational value given by BESIII [36].
In our calculation, we treat v 3 as the norm in T ( Table I, such that δ v 3 is allowed to rotate from −90 • to 50 • without letting B( + c → 0 K + ) exceed the data. Since the allowed range for δ v 3 is large, it is clear that its value is insensitive to the data. On the other hand, in order to explain the experimental data of R K /π with the smallest corrections from v i e δ v i , we assume maximumly destructive interferences between a i e δ a i and v i e δ v i .