Singly Cabibbo-suppressed hadronic decays of $\Lambda_c^+$

We study singly Cabibbo-suppressed two-body hadronic decays of the charmed baryon $\Lambda_c^+$, namely, $\Lambda_c^+\to \Lambda K^+, p\pi^0, p\eta, n\pi^+,\Sigma^0K^+,\Sigma^+ K^0$. We use the measured rate of $\Lambda_c^+\to p\phi$ to fix the effective Wilson coefficient $a_2$ for naive color-suppressed modes and the effective number of color $N_c^{\rm eff}$. We rely on the current-algebra approach to evaluate $W$-exchange and nonfactorizable internal $W$-emission amplitudes, that is, the commutator terms for the $S$-wave and the pole terms for the $P$-wave. Our prediction for $\Lambda_c^+\to p\eta$ is in excellent agreement with the BESIII measurement. The $p\eta$ ($p\pi^0$) mode has a large (small) rate because of a large constructive (destructive) interference between the factorizable and nonfactorizable amplitudes for both $S$- and $P$-waves. Some of the SU(3) relations such as $M(\Lambda_c^+\to n\pi^+)=\sqrt{2}M(\Lambda_c^+\to p\pi^0)$ derived under the assumption of sextet dominance are not valid for decays with factorizable terms. Our calculation indicates that the branching fraction of $\Lambda_c^+\to n\pi^+$ is about 3.5 times larger than that of $\Lambda_c^+\to p\pi^0$. Decay asymmetries are found to be negative for all singly Cabibbo-suppressed modes and range from $-0.56$ to $-0.96$.


I. INTRODUCTION
The study of hadronic decays of charmed baryons is an old subject (for a review, see [1,2]). For a long time, both experimental and theoretical progresses in this arena were very slow. Almost all the model calculations of two-body nonleptonic decays of charmed baryons were done before millennium and most of the experimental measurements were older ones. Theoretical interest in hadronic weak decays of charmed baryons peaked around the early 1990s and then faded away. To date, we still do not have a good and reliable phenomenological model, not mentioning the QCD-inspired approach as in heavy meson decays, to describe the complicated physics of charmed baryon decays. 1 From the theoretical point of view, baryons being made out of three quarks, in contrast to two quarks for mesons, bring along several essential complications. First of all, the factorization approximation that the hadronic matrix element is factorized into the product of two matrix elements of single currents and that the nonfactorizable term such as the W -exchange contribution is negligible relative to the factorizable one is known empirically to be working reasonably well for describing the nonleptonic weak decays of heavy mesons. However, this approximation is a priori not directly applicable to the charmed baryon case as W -exchange there, manifested as pole diagrams, is no longer subject to helicity and color suppression. This is different from the naive color suppression of internal W -emission. It is known in the heavy meson case that nonfactorizable contributions will render the color suppression of internal W -emission ineffective. However, the W -exchange in baryon decays is not subject to color suppression even in the absence of nonfactorizable terms. The experimental measurements of the decays Λ + c → Σ 0 π + , Σ + π 0 and Λ + c → Ξ 0 K + , which do not receive any factorizable contributions, 2 indicate that W -exchange and nonfactorizable internal W -emission indeed play an essential role in charmed baryon decays.
Recently, there are two major breakthroughs in charmed-baryon experiments in regard to hadronic weak decays. First of all, it is concerned with the absolute branching fraction of Λ + c → pK − π + . Experimentally, nearly all the branching fractions of the Λ + c were measured relative to the pK − π + mode. On the basis of ARGUS and CLEO data, Particle Data Group (PDG) had made a model-dependent determination of the absolute branching fraction, B(Λ + c → pK − π + ) = (5.0±1.3)% [4]. Recently, Belle reported a value of (6.84 ± 0.24 +0. 21 −0.27 )% [5] from the reconstruction of D * pπ recoiling against the Λ + c production in e + e − annihilation. Hence, the uncertainties are much smaller, and, most importantly, this measurement is model independent! More recently, BESIII has also measured this mode directly with the result B(Λ + c → pK − π + ) = (5.84 ± 0.27 ± 0.23)% [6]. Its precision is comparable to the Belle's result. A new average of (6.35 ± 0.33)% for this benchmark mode is quoted by the PDG [7].
Second, in 2015 BESIII has measured the absolute branching fractions for more than a dozen of decay modes directly for the first time [6]. Not only the central values are substantially different 1 An exception is the heavy-flavor-conserving hadronic decay of the heavy baryon, for example, Ξ c → Λ c π, which can be reliably studied within the framework that incorporates both heavy-quark and chiral symmetries [3]. 2 At first sight, it appears that the decay modes such as Λ + c → Σ 0 π + , Σ 0 K + can proceed through the external W -emission process. However, the spectator diquark ud of the Λ + c is antisymmetric in flavor, while the same diquark in Σ 0 is symmetric in flavor. Hence, the external W -emission is prohibited.
from the PDG ones (versions before 2016), but also the uncertainties are significantly improved. For example, B(Λ + c → Σ + ω) = (2.7 ± 1.0)% quoted in 2014 PDG [4] now becomes (1.74 ± 0.21)% in 2016 PDG [7] due to the new measurement of BESIII. In other words, all the PDG values before the 2016 version for the branching fractions of charmed baryon decays become obsolete.
The decay amplitude of the charmed baryon generally consists of factorizable and nonfactorizable contributions. The study of nonfactorizable effects arising from W -exchange and internal W -emission conventionally relies on the pole model. Under the pole approximation, one usually concentrates on the most important low-lying 1/2 + and 1/2 − pole states. Consider the charmed baryon decay with a pseudoscalar meson in the final state, B c → B + P . In general, its nonfactorizable S-and P -wave amplitudes are dominated by 1 2 − low-lying baryon resonances and 1 2 + ground-state baryon poles, respectively. It is known that the pole model is reduced to current algebra in the soft pseudoscalar-meson limit. The great advantage of current algebra is that the evaluation of the S-wave amplitude does not require the information of the troublesome negative-parity baryon resonances which are not well understood in the quark model. Nevertheless, the use of the pole model is very general and is not limited to the soft meson limit and to the pseudoscalar-meson final state. For example, current algebra is not applicable to the decays B c → B + V . However, the estimation of pole amplitudes is a difficult and nontrivial task since it involves weak baryon matrix elements and strong coupling constants of 1 2 + and 1 2 − baryon states. As a consequence, the evaluation of pole diagrams is far more uncertain than the factorizable terms.
In Table I we show various model calculations of branching fractions and up-down decay asymmetries of Cabibbo-allowed Λ + c → B + P decays. Two explicit pole-model calculations were carried out in [9] and [10,14] and a variant of the pole model was considered in [12]. In [13], the S-wave amplitude was calculated using current algebra. Similar calculations based on current algebra also can be found in [10] (denoted by CA in Table I). Authors of [8] chose to use the covariant quark model to tackle the three-body transition amplitudes (rather than two-body transitions) directly. This work was further developed in [11]. We see from Table I that the predicted rates by most of the models except current algebra are generally below experiment. Moreover, the pole model, the covariant quark model and its variant all predict a positive decay asymmetry α for both Λ + c → Σ + π 0 and Σ 0 π + , while it is measured to be −0.45 ± 0.31 ± 0.06 for Σ + π 0 by CLEO [15]. In contrast, current algebra always leads to a negative decay asymmetry for aforementioned two modes: −0.49 in [10], −0.31 in [13], −0.76 in [16] and −0.47 in [17]. BESIII will measure decay asymmetry parameters for Λ + c → Λπ + , Σ 0 π + , Σ + π 0 and pK 0 and the sensitivity for measuring α Σ + π 0 is estimated to be (10 ∼ 77)% [18]. It will be of great interest to see if the negative sign of α Σ + π 0 measured by CLEO is confirmed.
Writing the nonfactorizable S-wave amplitude as the term (A − A CA ) can be regarded as an on-shell correction to the current-algebra result. It turns out that in the existing pole model calculations [9,10,14], the on-shell correction (A − A CA ) always has a sign opposite to that of A CA . Moreover, its magnitude is sometimes even bigger than |A CA | for some of the decays such as Λ + c → Σ 0 π + , Σ + π 0 . That is, the on-shell correction is large enough to flip the sign of the parity-violating (PV) amplitudes. This explains the smaller calculated rate in the pole model and the sign difference of α Σ + π 0 ,Σ 0 π + between the pole model and current algebra. If the negative sign of α Σ + π 0 is confirmed, this means that the on-shell correction (A − A CA ) has been overestimated in previous pole model calculations probably owing to our poor knowledge of the negative-parity baryon resonances. The empiric fact that current algebra seems to work reasonably well for Λ + c → B + P is a bit surprising and annoying since the pseudoscalar meson produced in Λ + c decays is generally far from being soft. We plan to examine this important issue and the pole model in a separate work.
In this work we will focus on singly Cabibbo-suppressed hadronic decays of the Λ + c , specifically, Λ + c → ΛK + , pπ 0 , pη, nπ + , Σ 0 K + , Σ + K 0 . Among them, evidence of Λ + c → pη was found by BESIII recently [19], while a stringent upper limit on Λ + c → pπ 0 was also set. Besides dynamical model calculations, two-body nonleptonic decays of charmed baryons have been analyzed in terms of SU(3)-irreducible-representation amplitudes [20,21]. However, the quark-diagram scheme (i.e., analyzing the decays in terms of topological quark-diagram amplitudes) has the advantage that it is more intuitive and easier for implementing model calculations. A general formulation of the quark-diagram scheme for charmed baryons is given in [22] (see also [23]). Analysis of Cabibbosuppressed decays using SU(3) flavor symmetry was first carried out in [24]. This approach became popular recently [25][26][27][28]. Nevertheless, we shall perform dynamical model calculations based on current algebra. This work is organized as follows. In Sec. II we set up the formalism for analyzing factorizable and nonfactorizable contributions to singly Cabibbo-suppressed decays of the charmed baryon Λ + c . Numerical model calculations and discussions are presented in Sec. III. Sec. IV gives our conclusion. Appendix A is devoted to the study of the decay Λ + c → pφ to fix the relevant Wilson coefficient. The MIT bag model evaluation of baryon matrix elements is sketched in Appendix B. Axial-vector form factors and baryon wave functions relevant to the present work are summarized in Appendices C and D, respectively.

II. FORMALISM
The effective weak Hamiltonian for singly Cabibbo-suppressed decays at the scale µ = m c reads [29] with q = d, s and the four-quark operators are given by  Tables  VI and VII of [29]). Because in this work we will not consider effects of CP violation, we shall assume real CKM matrix elements for simplicity thereafter.
The general amplitude for B i → B f + P is given by where A and B are the S-and P -wave amplitudes, respectively. Note that if we write While the factorizable amplitude vanishes in the soft meson limit, the nonfactorizable one is not.

A. Factorizable contributions
We first consider the factorizable amplitudes for some of singly Cabibbo-suppressed modes: where a 1 = c 1 + c 2 Nc for the external (color-allowed) W -emission amplitude and a 2 = c 2 + c 1 Nc for internal (color-suppressed) W -emission in nave factorization. In terms of the decay constants and form factors defined by 3 There is a sign ambiguity for the one-body matrix element. We define Eq. (2.6) in such a way that a correct relative sign between the factorizable and nonfactorizable amplitudes, e.g. between Eqs. (2.8) and (2.13), is ensured.
where we have neglected contributions from the form factors f 3 and g 3 . We have learned from charmed meson decays that naive factorization does not work for color-suppressed decay modes. Empirically, it was realized in the 1980s that if the Fierz-transformed terms characterized by 1/N c are dropped, the discrepancy between theory and experiment will be greatly improved [30]. This leads to the so-called large-N c approach for describing hadronic D decays [31]. As the discrepancy between theory and experiment for charmed meson decays gets much improved in the 1/N c expansion method, it is natural to ask if this scenario also works in the baryon sector. This issue can be settled down by the experimental measurement of the Cabibbo-suppressed mode Λ + c → pφ, which receives contributions only from the factorizable diagrams [14]. Using the recent BESIII measurement of Λ + c → pφ [32], we obtain |a 2 | = 0.45 ± 0.03, corresponding to N eff c ≈ 7 (see Appendix A below). Recall that a 2 = −0.19 for N c = 3. Hence, color suppression in the factorizable amplitude is not operative.

B. Nonfactorizable contributions
Besides factorizable terms, there exist nonfactorizable contributions arising from W -exchange (see e.g. diagrams E 1,2,3 in Fig. 1 below) or nonfactorizable internal W -emission (e.g. diagram C 2 in Fig. 1). How do we tackle with the nonfactorizable contributions? One popular approach is to consider the contributions from all possible intermediate states. Among all possible pole contributions, including resonances and continuum states, one usually focuses on the most important poles such as the low-lying 1/2 + and 1/2 − states, known as pole approximation. More specifically, the S-wave amplitude is dominated by the low-lying 1/2 − resonances and the P -wave one governed by the ground-state poles. The nonfactorizable S-and P -wave amplitudes for the process B i → B f +M are then given by [14] A pole = − respectively. Ellipses in the above equation denote other pole contributions which are negligible for our purposes, 4 and the baryon-baryon matrix elements are defined by [14] B (2.14) When M = P , one can apply the Goldberger-Treiman relation for the strong coupling g B ′ BP and its generalization for g B * BP to express Eq. (2.13) as with the decay constant normalized to f P 3 = f π = 132 MeV. In the soft pseudoscalar-meson limit, p f = p i and hence the S-wave amplitude can be recast to the form The above expression for A com is precisely the well-known soft-pion theorem in the current-algebra approach. Using the relation [Q a 5 , H PV eff ] = [Q a , H PC eff ], we see that in the soft meson limit, the parityviolating amplitude is reduced to a simple commutator term expressed in terms of parity-conserving matrix elements. Therefore, the great advantage of current algebra is that the evaluation of the parity-violating S-wave amplitude does not require the information of the negative-parity 1/2 − poles.
To apply the soft-meson theorem, we notice that where I ± , U ± and V ± are isospin, U -spin and V -spin ladder operators, respectively, with The use of the hypercharge Y = 2 √ 3 Q 8 has been made in the last line of Eq. (2.19). In the SU(3) case, the hypercharge is given by the well-known relation Y = B + S. However, its generalization to the SU(4) case depends on the generalized definition of the hypercharge. For example, Y = B+S−C is derived in the textbook of [37], while the relation Y = B+S+C also can be found in the literature. For our purpose, we will adopt the first one, so that Y (p) = 1 and Y (Λ + c ) = 0. We will come back to this point in Sec. III.
Applying Eq. (2.20) to the commutator terms for singly Cabibbo-suppressed modes: Λ + c → ΛK + , pπ 0 , pη, nπ + , Σ 0 K + , Σ + K 0 , we obtain where the superscript π 0 of g A(π 0 ) pp implies that the form factor g A pp is evaluated using the the axialvector current corresponding to P 3 = π 0 , and likewise for the superscript η 8 of g . For the singlet component η 0 , the soft pseudoscalar meson theorem is not applicable. Hence, we will not consider the S-wave amplitude of Λ + c → pη 0 within the currentalgebra framework. As shown in Appendix C, the axial-vector form factor vanishes for antitripletantitriplet heavy baryon transitions, i.e. g A B3B3 = 0. Hence, in the P -wave amplitudes we can drop those terms with g A ΛcΛc or g A ΞcΛc .

C. Baryon matrix elements
To evaluate the nonfactorizable amplitudes we need to know the baryon matrix elements and the axial-vector form factor at q 2 = 0, g A B ′ B . For the matrix elements, we write with O q ± = O q 1 ± O q 2 = (qc)(ūq) ± (qq)(ūc) and c ± = c 1 ± c 2 . Since the four-quark operator O + is symmetric in color indices while O − is antisymmetric, the former does not contribute to the baryon transition matrix element since the baryon wave function is totally antisymmetric in color. Hence, We shall evaluate the matrix elements using the MIT bag model (see Appendix B). The relevant PC matrix elements are where X q 1 and X q 2 with q = d, s are the bag integrals defined in Eq. (B8). The numerical values of the bag integrals can be found in Eq. (B10). It should be stressed that the relative signs of matrix elements are fixed by the baryon wave functions given in Appendix D.
For the q 2 dependence of the form factors defined in Eq. (2.7), we follow the conventional practice to assume a pole dominance  [14,[38][39][40]. Presumably, the SU(3) relation (2.28) should be respected at zero recoil q 2 = (m i − m f ) 2 . For our purpose, we shall follow [38] to use for Λ c − p transition. Form factors for Λ c − Λ transition will be discussed in Sec. III below. As for the axial-vector form factors g A B ′ B , they are discussed in Appendix C.

III. RESULTS AND DISCUSSIONS
In terms of the decay amplitude of B i → B f + P given in Eq. (2.3), its decay rate reads The predicted S-and P -wave amplitudes of singly Cabibbo-suppressed decays Λ + c → B + P in units of G F 10 −2 GeV 2 . Branching fractions and the asymmetry parameter α are shown in the last three columns. Experimental results are taken from [7,19]. with p c being the c.m. three-momentum in the rest frame of B i , and the up-down asymmetry α is given by If the parent baryon B i is unpolarized, the produced baryon B f is longitudinally polarized by an amount of α. The predicted S-and P -wave amplitudes of singly Cabibbo-suppressed decays Λ + c → ΛK + , pπ 0 , pη, nπ + , Σ 0 K + , Σ + K 0 , their branching fractions and decay asymmetries are shown in Table II.
We first discuss the two modes Λ + c → pπ 0 and pη. In the topological quark-diagram approach for charmed baryon decays [22], the relevant quark diagrams for Λ + c → pη, pπ 0 are depicted in Fig.  1. There are two internal W -emission diagrams C 1 and C 2 and three W -exchange ones E 1 , E 2 and E 3 . Symmetry properties of the baryon wave function are taken into account in the analysis of [22]. Among these diagrams, only C 1 is factorizable. Since the CKM matrix elements V cs V us and V cd V ud are similar in magnitude but opposite in sign and since the decay constants f s η and f q η also have opposite signs (see Eq. (2.12)), it is obvious that factorizable amplitude of pη is significantly larger than pπ 0 in magnitude owing to the constructive interference in the former (see Table II). Considering the factorizable contributions alone, we already have B(Λ + c → pη) fac = 4.0×10 −4 , while B(Λ + c → pπ 0 ) fac = 0.93×10 −4 . We rely on the current-algebra approach to evaluate nonfactorizable W -exchange amplitudes, namely, the commutator terms for the S-wave and the current-algebra pole terms for the P -wave.
Various other model predictions for the singly Cabibbo-suppressed decays Λ + c → B + P are summarized in Table III. Except for the dynamic calculation in [41] and the consideration of  a The P -wave amplitude of Λ + c → Ξ 0 K + is assumed to be positive. b The P -wave amplitude of Λ + c → Ξ 0 K + is assumed to be negative. factorizable contributions in [42], all other predictions are based on the SU(3) symmetry argument. A global fit of the SU(3) amplitudes of Λ + c → B + P to the data of branching fractions of Cabibboallowed decays Λ + c → pK 0 , Λπ + , Σ + π 0 , Σ 0 π + , Σ + η, Ξ 0 K + , and singly-Cabibbo-suppressed decays [26] yields B(Λ + c → pπ 0 ) = (5.6± 1.5)× 10 −4 , which is too large compared to the experimental limit of 2.7 × 10 −4 [19]. Assuming the sextet 6 dominance over 15 (i.e. c − O − ≫ c + O + ), the authors of [25] obtained the relation 6 and the sum rule derived from the relations [26] √ The current PDG values for branching fractions [7] lead to B(Λ + c → nπ + ) ∼ 0.97 × 10 −3 and hence B(Λ + c → pπ 0 ) ∼ 0.48 × 10 −3 . The prediction of the latter is consistent with the SU(3) global fit of [26]. The discrepancy between the SU(3) approach and experiment for Λ + c → pπ 0 is ascribed to the SU(3) relations given by Eqs. (3.3) and (3.4). First of all, the relation (3.3) does not hold in the general quark diagram approach owing to the presence of factorizable contributions [22]. Since the factorizable amplitude of Λ + c → nπ + (Λ + c → pπ 0 ) is governed by the external (internal) W -emission, we have (see also Table II) Hence, the factorizable amplitudes alone strongly violate the SU (3)  should be respected by A com and B ca terms, but not by A fac,tot and B fac,tot (see Table II). By the same token, the first line of Eq. (3.5) does not hold as the factorizable amplitudes of Λ + c → Λπ + and Λ + c → pK 0 are of different types, governed by a 1 and a 2 , respectively. Hence, we conclude that the rates of nπ + and pπ 0 cannot be extracted from experiment through the invalid SU(3) relations (3.4) and (3.3). In our work, both nπ + and pπ 0 are suppressed owing to the destructive interference between factorizable and nonfactorizable terms. Experimentally, the Cabibbo-allowed decay Λ + c → nK S π + involving a neutron was observed by BESIII recently [43]. It is conceivable that the Cabibbo-suppressed mode Λ + c → nπ + can be reached in the near future. Only factorizable contributions to Λ + c → nπ + and pπ 0 were considered in [42]. In the naive factorization with N eff c = 3, the branching ratio of Λ + c → pπ 0 of order 10 −6 is smaller than that of Λ + c → nπ + by a factor of order 50. It was argued in [42] that final-state rescattering effects through Λ + c → {nπ + , nρ + , ΛK + , ΛK + * } → pπ 0 will enhance the former so that B(Λ + c → pπ 0 ) > ∼ B(Λ + c → nπ + ) (see Tables 2 and 3 of [42]). We would like to make two remarks: (i) In order to enhance the rate of pπ 0 to the order of 10 −4 , a common wisdom is that the branching fraction of the intermediate states, e.g. Λ + c → nρ + , ΛK + , should be at least two orders of magnitude larger than 10 −4 [44]. (ii) We find that even in the absence of final-state rescattering, the nonfactorizable contributions denoted by A com and B ca in Table II, which were neglected in [42], will yield B(Λ + c → pπ 0 ) nf = 3.3 × 10 −4 . Therefore, it is mandatory to take into account the nonfactorizable contributions from internal W -emission (denoted by C 2 in Fig. 1) and W -exchange (E 1 , E 2 , E 3 ) in the study.
As for Λ + c → ΣK decays, we see from Eqs. (2.25) and (2.26) that a Ξ + to the smallness of the bag integrals X d,s 1 compared to X s 2 (see Eq. (B10)). It follows from Eq.

IV. CONCLUSIONS
We have studied singly Cabibbo-suppressed two-body hadronic decays of the charmed baryon Λ + c . We use the measured rate of Λ + c → pφ to fix the effective Wilson coefficient a 2 for naive color-suppressed modes and the effective number of color N eff c . We rely on the current-algebra method to evaluate W -exchange and nonfactorizable internal W -emission amplitudes, that is, the commutator terms for the S-wave and the pole terms for the P -wave. Our prediction for Λ + c → pη is in excellent agreement with the BESIII measurement. The pη (pπ 0 ) mode has a large (small) rate because of a large constructive (destructive) interference between the factorizable and nonfactorizable amplitudes for both S-and P -waves. Some of the SU(3) relations such as M (Λ + c → nπ + ) = √ 2M (Λ + c → pπ 0 ) and Eq. (3.4) derived under the assumption of sextet dominance are not valid for decays with factorizable contributions. Sextet dominance is justified for nonfactorizable terms as the baryon matrix elements a B ′ B are governed by the four quark operator O − , but not for factorizable amplitudes as both O − and O + operators contribute. Our calculation indicates that the branching fraction of Λ + c → nπ + is about 3.5 times larger than that of Λ + c → pπ 0 . Decay asymmetries are found to be negative for all singly Cabibbo-suppressed modes and range from −0.56 to −0.96. asymmetry then read 7 with the S-, P -and D-waves given by [47,48] S = −A 1 , Using the data B(Λ + c → pφ) = (1.04 ± 0.21) × 10 −3 [19], we obtain |a 2 | = 0.45 ± 0.03 and hence N eff c ≈ 7 for c 1 = 1.346 and c 2 = −0.636 and f φ = 215 MeV. This leads to a 1 = 1.26 ± 0.01 .
where R is the radius of the bag and u(r), v(r) are the large and small components of the quark wave function, respectively, defined by ψ = iu(r)χ v(r)σ ·rχ . (B3) As an example for illustration, we consider the matrix element a pΛc given by Applying the relation and the wave functions (see Appendix D) with obvious notation for permutation of quarks, it is straightforward to show that with for q = d, s. Hence, Numerically, we obtain