Singly Cabibbo suppressed decays of $\Lambda_{c}^+$ with SU(3) flavor symmetry

We analyze the weak processes of anti-triplet charmed baryons decaying to octet baryons and mesons with the SU(3) flavor symmetry and topological quark diagram scheme. We study the decay branching ratios without neglecting the contributions from ${\cal O}(\overline{15})$ for the first time in the SU(3) flavor symmetry approach. The fitting results for the Cabibbo allowed and suppressed decays of $\Lambda_{c}^+$ are all consistent with the experimental data. We predict all singly Cabibbo suppressed decays. In particular, we find that ${\cal B}(\Lambda_c^+\to p \pi^0)=(1.3\pm0.7)\times 10^{-4}$, which is slightly below the current experimental upper limit of $2.7\times 10^{-4}$ and can be tested by the ongoing experiment at BESIII as well as the future one at Belle-II.

Recently, the study of the charmed baryons has been receiving increasing attention both theoretically and experimentally. The main reason for this is the recent measurement of the absolute branching fraction of the golden channel Λ + c → pK − π + by the Belle Collaboration [1]. This mode and many other Λ + c ones have also been observed by the BESIII Collaboration [2][3][4][5][6][7][8][9][10] with using Λ + cΛ − c pairs produced by e + e − collisions at a center-of-mass energy of √ s = 4.6 GeV, which provides a uniquely clean background to study charmed baryons. Consequently, the Particle Data Group (PDG) [11] has given a new average of B(Λ + c → pK − π + ) = (6.23 ± 0.33)%. The precision measurement on this mode is very important as it can be used to determine the absolute branching fractions of other Λ + c decays [11] as well as processes involving Λ + c , such as the extractions of the CKM element from Λ b → Λ + c µ −ν µ [12,13]. It is clear that a new era of physics for charmed baryons has begun. For a review on the theoretical progress of charmed baryons, please see Ref. [14].
It is known that it is difficult to make reliable predictions on the charmed baryon decay rates due to the lack of theoretical understanding of underlined dynamics for the charmed baryon structure. Since the Cabbibo allowed decays of Λ + c → Σ 0 π + and Λ + c → Σ + π 0 do not receive any factorizable contributions, the nonzero experimental observed values of their branching fractions imply that the factorization approach is not working in charmed baryon decays. Without the use of a dynamical model, it is clear that the most reliable way to analyze charmed baryon processes is to impose SU(3) F [18][19][20][21][22][23][24][25][26]. In fact, it has been demonstrated [20][21][22][23] [23,25]. On the other hand, the irreducible amplitude in the later approach is more intuitive and gives an insight on dynamics [32]. In particular, it could shed light for us on distinguishing the nonfactorizable and factorizable contributions in the processes. It is expected that these two approaches should give the same results under SU(3) F . The close connections between the two have been recently examined in Ref. [33].
To study the two-body anti-triplet of the lowest-lying charmed baryon decays of B c → and B n and M are the baryon and pseudoscalar octet states, given by Here, we have assumed that the physical state of η is solely made of η 8 due to the small mixing between the weak eigenstates of η 0 and η 8 [11] to reduce our fitting parameters.
We start with the effective Hamiltonian responsible for the tree-level c → sud, c → uqq and c → dus transitions, given by [34] with where corresponding to the scale-dependent Wilson coefficients with the QCD corrections. By [11] representing the well-known Cabbibo angle θ c , the decays associated with O ds ± , O qq ± and O sd ± are the so-called Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed processes, respectively.
Now, we can write down the SU(3) irreducible amplitude for B c → B n M as [20,24] where Here, the Wilson coefficients have been absorbed in the parameters a i .
In order to reduce the fitting parameters for the processes based on the amplitudes in Eqs. (7) and (8) To evaluate A F (O(15)), we need the help of topological quark diagrams. In other words, we have to find out the terms in T (O 15 ) of Eq. (8), which can be factorizable. In Figs. 2a and 2b, we illustrate the factorizable contributions for the color allowed and suppressed processes in the topological diagram approach, 1 respectively. Note that the quark indices 1 It is clear that we have ignored the soft gluon interactions whenever the factorization problem is discussed. represent the light quark lines of hadrons or operators with q i = (u, d, s). From Fig. 2, we obtain that and T (C) represents the color allowed (suppressed) amplitude. By using Eq. (6) and the tensor identity ǫ njk ǫ mjk = 2δ m n , we find that where 2 In general, the term associated with a 6 also contribute the non-factorizable part.
canceled out by the corresponding term in A F (O(6)), resulting in the process to be either color allowed or color suppressed. This can be explicitly demonstrated by the recent work in Ref. [33] on the connection between the topological and SU(3) F approaches.
To illustrate the effect of the only a 6 term from O(15), we show the decay amplitudes of Λ + c → pπ 0 and Λ + c → nπ + , given by [23] A It is clear that the relation of [19] is violated with the contributions from a 6 . This violation has been explicitly pointed out in Ref. [17] based on a dynamical model. On the other hand, some direct relations still exist in some modes. For example, one has that Future experimental searches for these decays will confirm if the discussions based on SU(3) F are right or not.
In Table 2, we list our fitting results for the branching ratios of the Cabibbo allowed and singly Cabibbo suppressed Λ + c decays. In the table, we have also included the previous results based on SU(3) F [22] without O(15) along with the data as well as those from the dynamical model calculations by CKX [17]. As seen in Table 2, our results for the Cabibbo 12.6 ± 2.1 12.9 ± 0.7 12.8 ± 2.3 - 5.9 ± 1.0 5.9 ± 0.9 5.5 ± 1.4 - allowed Λ + c decays with the consideration of O(15) are slightly better than those without O(15), but they all fit the data well. On the other hand, the decay branching ratios for singly Cabibbo suppressed modes of Λ + c with and without O(15) are quite different. In particular, we predict that B(Λ + c → pπ 0 ) = (1.3 ± 0.7) × 10 −4 , which is consistent with the experiments upper limit of 2.7 × 10 −4 as well as the result of 0.8 × 10 −4 calculated by the pole model with current algebra in Ref. [17]. It is clear that the inconsistent branching ratio of (5.7 ± 1.5) × 10 −4 in the previous study with SU(3) F [22] results from the ignorance of O(15), in which a large destructive interference occurs between O(15) and O (6). It is also interesting to note that B(Λ + c → nπ + ) is found to be (6.1 ± 2.0) × 10 −4 , which is reduced by almost a factor 2 in comparing with that in Ref. [22]. Although the signs for the contributions from a 6 to Λ + c → pπ 0 and Λ + c → nπ + in Eq. (13) are opposite, the resulting values are both reduced due to the complex numbers of a 2,3 and a 6 in Eq. (15).
In addition, from Table 2, we have that B(Λ + c → Σ + K 0 S ) = (5.7 ± 1.0) × 10 −4 , which agrees with the experimental value of B(Λ + c → Σ 0 K + ) = (5.2 ± 0.8) × 10 −4 [11]. The future search for Λ + c → Σ + K 0 S is a good test for SU(3) F . Finally, we remark that we are unable to discuss the SU(3) F breaking effects after including the contributions of O(15) in the fit due to the insufficient experimental data points.
Once more experimental data are available in the future, the studies of these effects along with the η ′ channels would be possible.
In sum, we have studied the two-body decays of Λ + c → B n M based on the approach with the SU(3) F flavor symmetry, which is a powerful tool to examine charmed baryon physics and allows us to connect the physical quantities without knowing the underlined dynamics.
We have successfully fitted all the existing experimental data from the Cabibbo allowed and suppressed decays of Λ + c . By considering the approach with the topological quark diagrams, for the first time, the contributions from O (15) have been included in the calculations with the SU(3) F method. As a result, we have predicted all singly Cabibbo suppressed decays.
In particular, we have found that B(Λ + c → pπ 0 ) = (1.3 ± 0.7) × 10 −4 , which is slightly below the current experimental upper limit of 2.7 × 10 −4 . This result can be tested by the experiments at BESIII and Belle-II.