Singly Cabibbo-suppressed hadronic decays of $Λ_c^+$
Hai-Yang Cheng Xian-Wei Kang Fanrong Xu
hep-ph hep-ex


    ⟨⟩1 2 B Phys. Rev.  Phys. Rev. Lett.  Phys. Lett.  Nucl. Phys.  Z. Phys. ↑↓ -1.2pt<5pt∼  -1.2pt>5pt∼ =cmbx10 scaled 2 March, 2018 1.5 cm
    
     Singly Cabibbo-suppressed hadronic decays of Λ_c^+ Hai-Yang Cheng^1, Xian-Wei Kang^1, 2, Fanrong Xu^3^1Institute of Physics, Academia SinicaTaipei, Taiwan 115, Republic of China^2College of Nuclear Science and Technology, Beijing Normal UniversityBeijing 100875, People's Republic of China^3 Department of Physics, Jinan UniversityGuangzhou 510632, People's Republic of China Abstract
    We study singly Cabibbo-suppressed two-body hadronic decays of the charmed baryon Λ_c^+, namely, Λ_c^+→Λ K^+, pπ^0, pη, nπ^+,Σ^0K^+,Σ^+ K^0. 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_c^ 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 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^+)=√(2)M(Λ_c^+→ pπ^0) derived under the assumption of sextet dominance are not valid for decays with factorizable terms. 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.
    
    
    
    
    § INTRODUCTION
    
    The study of hadronic decays of charmed baryons is an old subject (for a review, see <cit.>). 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.
    [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 <cit.>.]
    
    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^0π^+, Σ^+π^0 and
    Λ^+_c→Ξ^0K^+, which do not receive any factorizable
    contributions,
    [At first sight, it appears that the decay modes such as Λ_c^+→Σ^0π^+,Σ^0K^+ 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.]
    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)%<cit.>. Recently, Belle reported a value of (6.84±0.24^+0.21_-0.27)%<cit.> 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)%<cit.>. 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 <cit.>.
    
    
    Second, in 2015 BESIII has measured the absolute branching fractions for more than a dozen of  decay modes directly for the first time <cit.>.  Not only the central values are substantially different from the PDG ones (versions before 2016), but also the uncertainties are significantly improved. For example,  (Λ_c^+→Σ^+ω)=(2.7±1.0)% quoted in 2014 PDG <cit.> now becomes (1.74±0.21)%  in 2016 PDG <cit.> 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, _c→+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  _c→+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 <ref> 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 <cit.> and <cit.> and a variant of the pole model was
    considered in <cit.>. In <cit.>, the S-wave amplitude was calculated using current algebra. Similar calculations based on current algebra also can be found in <cit.> (denoted by CA in Table <ref>).
    Authors of <cit.> 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 <cit.>. We see from Table <ref> 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^0π^+, while it is measured to be -0.45±0.31±0.06 for Σ^+π^0 by CLEO <cit.>. In contrast, current algebra always leads to a negative decay asymmetry for aforementioned two modes: -0.49 in <cit.>, -0.31 in <cit.>, -0.76 in <cit.> and -0.47 in <cit.>.  BESIII will measure decay asymmetry parameters for Λ_c^+→Λπ^+,Σ^0π^+,Σ^+π^0 and pK̅^0 and the sensitivity for measuring α_Σ^+π^0 is estimated to be (10∼ 77)%<cit.>. 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
    
    A=A^ CA+(A-A^ CA),
    
    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
    <cit.>, 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^0π^+,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^0π^+ 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^+→+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^+→Λ K^+, pπ^0, pη, nπ^+,Σ^0K^+,Σ^+ K^0. Among them, evidence of Λ_c^+→ pη
    was found by BESIII recently <cit.>, 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 <cit.>. 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 <cit.> (see also <cit.>).  Analysis of  Cabibbo-suppressed decays using SU(3) flavor symmetry was first carried out in <cit.>. This approach became popular recently <cit.>. 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.
    
    
    
    
    § FORMALISM
    
    The effective weak Hamiltonian for singly Cabibbo-suppressed decays at the scale μ=m_c reads <cit.>
    H_ eff=G_F√(2)∑_q=d,sV^*_uqV_cq(c_1O_1^q+c_2O_2^q)+h.c.,
    
    with q=d,s and the four-quark operators are given by
    
    O_1^q=(q̅c)(u̅q),      O_2^q=(q̅q)(u̅c),
    
    with (q̅_1q_2)≡q̅_1γ_μ(1-γ_5)q_2. For the Wilson coefficients, we shall use the lowest order values c_1=1.346 and c_2=-0.636 obtained at the scale μ=1.25 GeV with Λ^(4)_ MS=325 MeV (see Tables VI and VII of <cit.>). 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 _i→_f+P is given by
    
        M(_i→_f+P)=iu̅_f(A-Bγ_5)u_i,
    
    where A and B are the S- and P-wave amplitudes, respectively. Note that
    if we write M(_i→_f+P)=iu̅_f(A+Bγ_5)u_i, the P-wave amplitudes given in Eqs. (<ref>) and (<ref>) below and the decay asymmetry α  in Eq. (<ref>) will be flipped in sign. The decay amplitude
    generally consists of factorizable and nonfactorizable ones
    
    M(_i→_f+P)=M(_i→_f+P)^ fac+M(_i→_f+P)^ nf.
    
    While the factorizable amplitude vanishes in the soft meson limit, the nonfactorizable one is not.
    
    
    
     §.§ Factorizable contributions
    
    We first consider the factorizable amplitudes for some of singly Cabibbo-suppressed modes:
    
    M(Λ_c^+→Λ K^+)^ fac   =   G_F√(2)V_csV_usa_1 K^+|(u̅s)|0Λ|(s̅ c)|Λ_c^+, 
    
    M(Λ_c^+→ pπ^0)^ fac   =   G_F√(2)V_cdV_uda_2π^0|(d̅d)|0 p|(u̅ c)|Λ_c^+,
    
    where a_1=c_1+c_2 N_c for
    the external (color-allowed) W-emission amplitude and a_2=c_2+c_1 N_c for internal (color-suppressed) W-emission in nave factorization.
    In terms of the decay constants and form factors defined by
    [There is a sign ambiguity for the one-body matrix element. We define Eq. (<ref>) in such a way that a correct relative sign between the factorizable and nonfactorizable amplitudes, e.g. between Eqs. (<ref>) and (<ref>), is ensured.] K^+(q)|(u̅s)|0=-if_K q_μ,     π^0(q)|(d̅d)|0=i√(2)f_π q_μ,
    
    and
    Λ(p_Λ)|(s̅c)|Λ_c^+(p_Λ_c)   =   u̅_Λ[f_1^Λ_cΛ(q^2)γ_μ-f_2^Λ_cΛ(q^2)iσ_μνq^ν m_Λ_c+
    f_3^Λ_cΛ(q^2)q_μ m_Λ_c
           -( g_1^Λ_cΛ(q^2)γ_μ-g_2^Λ_cΛ(q^2)iσ_μνq^ν m_Λ_c
    +g_3^Λ_cΛ(q^2) q_μ m_Λ_c)γ_5 ]u_Λ_c,
    
    with q=p_Λ_c-p_Λ, we obtain
    
    M^ fac(Λ_c^+→Λ K^+)    =    -iG_F√(2)V_csV_us a_1f_K [(m_Λ_c-m_Λ)f_1^Λ_cΛ(m_K^2)
     +(m_Λ_c+m_Λ)g_1^Λ_cΛ(m_K^2)γ_5], 
    
    M^ fac(Λ_c^+→ pπ^0)    =    iG_F 2V_cdV_ud a_2f_π[(m_Λ_c-m_p)f_1^Λ_c p(m_π^2)
     +(m_Λ_c+m_p)g_1^Λ_c p(m_π^2)γ_5],
    
    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 <cit.>. This leads
    to the so-called large-N_c approach for describing hadronic D
    decays <cit.>.
    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 <cit.>.
    Using the recent BESIII measurement of Λ_c^+→ pϕ<cit.>, we obtain |a_2|=0.45±0.03, corresponding to N_c^ eff≈ 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.
    
    For Λ_c^+→ pη^(') decays, we need to consider the η-η' mixing parametrized by
    
        |η⟩   =   cosϕ|η_q⟩-sinϕ|η_s⟩, 
        
        |η'⟩   =   sinϕ|η_q⟩+cosϕ|η_s⟩,
    
    where the flavor states qq̅=(uu̅+dd̅)/√(2) and ss̅ are labeled as η_q and η_s, respectively.
    The mixing angle ϕ is determined to be 39.3^∘± 1.0^∘ in the
    Feldmann-Kroll-Stech mixing scheme <cit.>,  which is consistent
    with the recent result ϕ=42^∘±2.8^∘ extracted from
    the CLEO data <cit.>. The factorizable amplitudes then read
    
    A^ fac(Λ_c^+→ pη^('))    =    -G_F√(2) a_2 (V_csV_usf_η^(')^s+1√(2)V_cdV_udf_η^(')^q) (m_Λ_c-m_p)f_1^Λ_c p(m_η^2),  
    
    B^ fac(Λ_c^+→ pη^('))    =   G_F√(2) a_2 (V_csV_usf_η^(')^s+1√(2)V_cdV_udf_η^(')^q) (m_Λ_c+m_p)g_1^Λ_c p(m_η^2),
    
    where the decay constants are defined by
    η^(')|q̅γ_μγ_5 q|0=i1√(2)f_η^(')^q q_μ,     η^(')|s̅γ_μγ_5 s|0=if_η^(')^s q_μ.
    
    We follow <cit.> to use
    
    f_η^q= 107  MeV,      f^s_η= -112  MeV,     
    f_η'^q= 89  MeV,     f^s_η'=137  MeV
    for ϕ=39.3^∘.
    
    
    
    
     §.§ Nonfactorizable contributions
    
    
    Besides factorizable terms, there exist nonfactorizable contributions arising from W-exchange (see e.g. diagrams E_1,2,3 in Fig. <ref> below) or nonfactorizable internal W-emission (e.g. diagram C_2 in Fig. <ref>). 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 _i→_f+M are then given by  <cit.>
    A^ pole   =    -∑__n^*(1/2^-)[ g___f_n^*Mb_n^*i m_i-m_n^*+ b_fn^*g___n^*_iM m_f-m_n^*]+⋯, 
    
    B^ pole   =    -∑__n[ g___f_nM a_ni m_i-m_n+ a_fng___n_iM m_f-m_n]+⋯
    respectively. Ellipses in the above equation denote other pole contributions which are negligible for our purposes,
    [For example, contributions to the S-wave amplitude from the parity-violating matrix elements b_ij defined
    in Eq. (<ref>) are much smaller than the parity-conserving ones a_ij, which have been shown explicitly in <cit.>.]
    and the baryon-baryon matrix elements are defined by <cit.>_i|H_ eff|_j=u̅_i(a_ij-b_ijγ_5)u_j,     _i^*(1/2^-)|H_ eff^ PV|_j = ib_i^*ju̅_i u_j.
    
    When M=P, one can apply the Goldberger-Treiman relation for the strong coupling g__' P and its generalization for g__^* P
    g__' P^a= √(2) f_P^a(m_'+m_)g^A_',     
    g__^* P^a= √(2) f_P^a(m_^*-m_)g^A_^*,
    
    to express Eq. (<ref>) as
    
    A^ pole   =    - √(2) f_P^a∑__n^*(1/2^-)[ g__f_n^*^A p/_f -m_n^* p/_i-m_n^*b_n^*i - b_fn^*p/_i -m_n^* p/_f-m_n^* g__n^*_i^A  ], 
    
    B^ pole   =    -√(2) f_P^a∑__n[ g__f_n^A m_f+m_n m_i-m_n a_ni+ a_fnm_i+m_n m_f-m_n g__n_i^A],
    
    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
    
    A^ com   =     - √(2) f_P^a∑__n^*(1/2^-)[ _f|Q_5^a|_n^*_n^*|H_ eff^ PV|_i-
    _f|H_ eff^ PV|_n^*_n^*|Q_5^a|_i]  
       =    -√(2) f_P^a_f| [Q_5^a, H_ eff^ PV]|_i,
    
    with
    
    Q^a=∫ d^3x q̅γ_0λ^a 2q,       Q^a_5=∫ d^3x q̅γ_0γ_5λ^a 2q.
    
    The above expression for A^ com  is precisely the well-known soft-pion theorem in the current-algebra approach. Using the relation [Q_5^a, H_ eff^ PV]=[Q^a, H_ eff^ PC], we see that in the soft meson limit, the parity-violating 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
           A^ com(_i→_fπ^0)=-√(2) f_π_f| [I_3, H_ eff^ PC]|_i,      A^ com(_i→_fπ^±)=-1 f_π_f| [I_∓, H_ eff^ PC]|_i,  
           A^ com(_i→_f K^±)=-1 f_K_f| [V_∓, H_ eff^ PC]|_i,     A^ com(_i→_f (-)K^0)=-1 f_π_f| [U_∓, H_ eff^ PC]|_i, 
           A^ com(_i→_f η_8)= -√(3 2)1 f_η_8_f|[Y, H_ eff^ PC]|_i,
    
    where I_±,U_± and V_± are isospin, U-spin and V-spin ladder operators, respectively, with
    
    I_+|d=|u,    I_-|u=|d,    U_+|s=|d,    U_-|d=|s,    V_+|s=|u,    V_-|u=|s.
    
    The use of the hypercharge Y=2√(3)Q^8 has been made in the last line of  Eq. (<ref>). 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 <cit.>, 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. (<ref>) to the commutator terms for singly Cabibbo-suppressed modes:
    Λ_c^+→Λ K^+, pπ^0, pη, nπ^+,Σ^0K^+,Σ^+ K^0, we obtain
           A^ com(Λ_c^+→Λ K^+) = 1 f_K(√(3/2) a_pΛ_c+a_ΛΞ_c^0),     
    A^ com(Λ_c^+→ pπ^0) = -1√(2)f_πa_pΛ_c, 
           A^ com(Λ_c^+→Σ^0 K^+) = 1√(2)f_K(a_pΛ_c+√(2)a_Σ^0Ξ_c^0 ),    A^ com(Λ_c^+→ n π^+) = -1 f_πa_pΛ_c,
     
           A^ com(Λ_c^+→Σ^+ K^0) = 1 f_K(a_pΛ_c-a_Σ^+Ξ_c^+ ),        
    A^ com(Λ_c^+→ pη_8) =  -√(3 2)1 f_η_8a_pΛ_c, 
    for S-wave amplitudes with a___c≡|H_ eff^^ PC|_c. For P-wave amplitudes, we have
    
    B^ ca(Λ_c^+→ pπ^0)    =    -√(2) f_π(g_pp^A(π^0)m_p+m_p m_Λ_c-m_p a_pΛ_c+ a_pΣ_c^+m_Λ_c+m_Σ_c m_p-m_Σ_c g_Σ_c^+Λ_c^A(π^0) + a_pΛ_cm_Λ_c+m_Λ_c m_p-m_Λ_c g_Λ_cΛ_c^A(π^0)), 
    
    B^ ca(Λ_c^+→ n π^+)    =    -1 f_π(g_np^Am_n+m_p m_Λ_c-m_p a_pΛ_c+ a_nΣ_c^0m_Λ_c+m_Σ_c m_n-m_Σ_c g_Σ_c^0Λ_c^A ),  
    
    B^ ca(Λ_c^+→ pη_8)    =    -√(2) f_η_8(g_pp^A(η_8)m_p+m_p m_Λ_c-m_p a_pΛ_c+ a_pΣ_c^+m_Λ_c+m_Σ_c m_p-m_Σ_c g_Σ_c^+Λ_c^A(η_8) + a_pΛ_cm_Λ_c+m_Λ_c m_p-m_Λ_c g_Λ_cΛ_c^A(η_8)), 
    
    B^ ca(Λ_c^+→Λ K^+)    =    -1 f_K(g_Λ p^Am_Λ+m_p m_Λ_c-m_p a_pΛ_c + a_ΛΞ_c^0m_Λ_c+m_Ξ_c m_Λ-m_Ξ_c g_Ξ_c^0Λ_c^A + a_ΛΞ_c^'0m_Λ_c+m_Ξ_c' m_Λ-m_Ξ_c' g_Ξ_c^'0Λ_c^A), 
    
    B^ ca(Λ_c^+→Σ^0 K^+)    =     -1 f_K(g_Σ^0 p^Am_Σ+m_p m_Λ_c-m_p a_pΛ_c+ a_Σ^0Ξ_c^0 m_Λ_c+m_Ξ_c m_Σ-m_Ξ_c g_Ξ_c^0Λ_c^A  
               +  a_Σ^0Ξ_c^'0m_Λ_c+m_Ξ_c' m_Σ-m_Ξ_c' g_Ξ_c^'0Λ_c^A), 
    
    B^ ca(Λ_c^+→Σ^+ K^0)    =     -1 f_K(g_Σ^+p^Am_Σ+m_p m_Λ_c-m_p a_pΛ_c+ a_Σ^+Ξ_c^+ m_Λ_c+m_Ξ_c m_Σ-m_Ξ_c g_Ξ_c^+Λ_c^A 
               +a_Σ^+Ξ_c^'+m_Λ_c+m_Ξ_c' m_Σ-m_Ξ_c' g_Ξ_c^'+Λ_c^A ), 
    where the superscript π^0 of g_pp^A(π^0) implies that the form factor g_pp^A is evaluated using the the axial-vector current corresponding to P^3=π^0, and likewise for  the superscript η_8 of g_pp^A(η_8).
    In Eqs. (<ref>) and (<ref>), η_8 is the octet component of the η and η'η=cosθη_8-sinθη_0,     η'=sinθη_8+cosθη_0,
    
    with θ=-15.4^∘<cit.>. 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 current-algebra framework. As shown in Appendix C, the axial-vector form factor vanishes for antitriplet-antitriplet heavy baryon transitions, i.e. g__3̅_3̅^A=0. Hence, in the P-wave amplitudes we can drop those terms with g^A_Λ_cΛ_c or g^A_Ξ_cΛ_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_'.
    For the matrix elements, we write
    
    a__c≡|H_ eff^ PC|_c  = G_F2√(2)∑_q=d,sV_cqV_uq|c_+O_+^q+c_-O_-^q|_c ,
    
    with O_±^q=O_1^q± O_2^q=(q̅c)(u̅q)±(q̅q)(u̅c) 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,
    
    a__c = G_F2√(2)∑_q=d,sV_cqV_uq(c_1-c_2)|O_-^q|_c .
    
    We shall evaluate the matrix elements using the MIT bag model (see Appendix B).
    The relevant PC matrix elements are
           p|O_-^d|Σ_c^+=2√(2) 3(-X_1^d+9X_2^d)(4π),    Σ^+|O_-^d|Σ_c^+=-2√(2) 3(-X_1^d+9X_2^d)(4π),
           p|O_-^d|Λ_c^+ =4√(6)(X_1^d+3X_2^d)(4π),      n|O_-^d|Σ_c^0 = 5(X_1^d+X_2^d)(4π), 
          Λ|O_-^d|Ξ_c^0=-4X_2^d(4π),             Λ|O_-^s|Ξ_c^0= 2(-X_1^s+X_2^s)(4π), 
          Λ|O_-^d|Ξ_c^'0=-4√(3)X_2^d(4π),           Λ|O_-^s|Ξ_c^'0=-2√(3)(X_1^s+3X_2^s)(4π) ,  
          Σ^0|O_-^d|Ξ_c^0= -4√(3)X_1^d(4π),           Σ^0|O_-^s|Ξ_c^0= -2√(3)(X_1^s+3X_2^s)(4π), 
          Σ^0|O_-^d|Ξ_c^'0=4 3X_1^d(4π),             Σ^0|O_-^s|Ξ_c^'0=-2 3(X_1^s-9X_2^s)(4π) , 
          Σ^+|O_-^s|Ξ_c^+= 2√(6)X_2^s(4π),           Σ^+|O_-^s|Ξ_c^'+= 4 3√(2)(X_1^s-9X_2^s)(4π), 
    where X_1^q and X_2^q with q=d,s are the bag integrals defined in Eq. (<ref>). The numerical values of the bag integrals can be found in Eq. (<ref>). 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. (<ref>), we follow the conventional practice to assume a pole dominance
    
    f_i(q^2)=f_i(0) (1-q^2/m_V^2)^n ,      g_i(q^2)=g_i(0) (1-q^2/m_A^2)^n ,
    
    with n=2 or 1,
    where m_V(m_A) is the pole mass of the vector (axial-vector) meson with the same quantum number under consideration, for example, m_V=m_D^*_s and m_A=m_D_s1(2536) for Λ_c→Λ transition.
    Form factors f_i and g_i for Λ_c→Λ and Λ_c→ p transitions at zero recoil and at maximal recoil q^2=0 have been calculated in the literature <cit.>. Presumably, the SU(3) relation
    
    f_i^Λ_c p(q^2)=-√(3 2)f_i^Λ_cΛ(q^2),     
    g_i^Λ_c p(q^2)=-√(3 2)g_i^Λ_cΛ(q^2),
    
    should be respected at zero recoil q^2=(m_i-m_f)^2. For our purpose, we shall follow <cit.> to use
    
    f_1^Λ_c p(0)=-0.470,      g_1^Λ_c p(0)=-0.414 .
    
    for Λ_c-p transition.
    Form factors for Λ_c-Λ transition will be discussed in Sec. III below.
