title: A natural interpretation on the data of $Λ_c\toΣπ$
authors: Hong-Wei Ke Xue-Qian Li
text: 
  6.5in 9.0in
  
  
  
  
  
  
  
      Λ_ QCD/v/p⟨⟩M_Σ_bM_Σ_c
      ^1 School of Science, Tianjin University,
      Tianjin 300072, China
      
      ^2 School of Physics, Nankai University, Tianjin 300071, China 
      
      
      Even though the Standard Model (SM) has achieved great success, its application to the field of low energies still lacks
      solid foundation due to our limited knowledge on non-perturbative QCD.
      Practically, all theoretical calculations of
      the hadronic transition matrix elements are based various phenomenological models.
      There indeed exist some anomalies in the field which are waiting
      for interpretations. The goal of this work is trying to solve
      one of the anomalies: the discrepancy between the theoretical prediction on the sign of the up-down asymmetry
      parameter of Λ_c→Σπ and the experimental measurement.
      In the literatures several authors calculated the rate and
      determined the asymmetry
      parameter within various schemes, but there exist obvious loopholes in those adopted scenarios. To solve the discrepancy between theory and data,
      we suggest that
      not only the direct transition process contributes to the observed Λ_c→Σπ, but also other portals such as Λ_c→Λρ
      also play a substantial role via an isospin-conserving re-scattering Λρ→Σπ. Taking into account
      of the effects induced by the final state interaction, we re-evaluate the relevant quantities. Our numerical results indicate
      that the new theoretical prediction based on this scenario involving an interference between the direct transition of Λ_c→Σπ
      and the portal Λ_c→Λρ→Σπ can make both the decay rate and sign of the asymmetry parameter to be consistent with
      data.
      
      
      
      
      
      
      
      
      13.30.-a,14.20.Lq, 12.40.-yA natural interpretation on the data of Λ_c→Σπ
           Hong-Wei Ke^1[khw020056@tju.edu.cn]
      and Xue-Qian Li^2[lixq@nankai.edu.cn]
          December 30, 2023
      ==============================================================================
      
      
      
      § INTRODUCTION
      
      
      Even though the Standard Model (SM) has achieved great success, its application to the field of low energies still lacks
      solid foundation due to our limited knowledge on non-perturbative QCD.
      Practically, all theoretical calculations of
      the hadronic transition matrix elements are based various phenomenological models. There
      indeed exist some anomalies in the field which are waiting
      for interpretations.
      One of the anomalies is the discrepancy between the theoretical prediction on the sign of the up-down asymmetry
      parameter of Λ_c→Σπ and the experimental measurement. In fact, besides the meson case, for
      baryons which contain three ingredients, the complexity makes a thorough study on them more difficult than on
      mesons.  However from another aspect, the involved physics in the transitions between baryons is also richer and
      by the research one can get better understanding of the governing mechanisms. An advantage of studying decays of baryons involving heavy flavors
      is obvious just as one does on the heavy mesons (structure, production and decay).
      
      Especially, the charmed hadrons are of special significance because charm quark is heavier than the light quarks (u,d,s), but
      at the same time is not as heavy as the bottom quark, so that relativistic effects are not negligible at all. The issue
      that the lifetimes of B^±, B^0 and Λ_b are close, however, the lifetimes of D^±, D^0
      and Λ_c are quite apart, has been warmly discussed. It is believed that the Pauli interference induces the lifetime differences of D^± and D^0<cit.> which is suppressed for the B-hadrons, but for Λ_c the question still exists.
      
      
      Since 1990s, many decay channels
      of Λ_c have successively been measured by  experimental collaborations<cit.>,
      and the field  has attracted attentions of theorists. Its weak decays have been carefully explored with different approaches<cit.>.
      In this work, we would re-visit the old topic because much larger data-sets with higher precision are available at BESIII, Belle and even LHCb which make
      us to hope
      getting better understanding on the charmed baryons.
      
      
      
      Among the previous theoretical studies on the decay rate of Λ_c→Σπ and the corresponding up-down asymmetry parameter α, the pole model
      was adopted because of its advantage. The pole model is simple and the relevant parameters are adopted by fitting data, therefore one can trust its effectiveness.
      A naive
      conjecture would expect that the prediction obtained with this model should be consistent with data even though an error is unavoidable  in
      this relatively rough picture. However, it is noticed that the prediction on the up-down asymmetry parameter α is positive while the measured value is negative.
      This apparent discrepancy which is not a tolerable deviation, indicates that there must be something wrong.
      Thus to solve this “anomaly" there are two routes. One is that the method adopted for the calculation should be modified whereas another possibility is that besides
      the direct transition, there exist other contributions to the observed data on Λ_c→Σπ. As the first route,
      Cheng and his collaborators <cit.> went on to use the current algebra
      calculating the transition matrix element ⟨Σ| H_eff|Λ_c⟩ and obtained negative α. However, the pre-condition of
      using the current algebra<cit.> is properly extracting the pion field out from the matrix element ⟨πΣ|H'_eff  |Λ_c⟩ under the
      soft-pion approximation. By contrast for the process Λ_c→Σπ, the 3-momentum of the pion is not small to be “soft", thus
      the whole scenario is questionable.
      
      Instead, we follow the second route i.e. the obvious loophole may suggest that there could be another mechanism.
      We propose that other channels of Λ_c decays would contribute to the observed Λ_c→Σπ via final state interactions.
      The interference between the direct transition and the new contribution may lead to the results consistent with data.
      Considering the effective interaction and the decay rate of Λ_c decays, the most possible
      two-step process is that Λ_c first transits into Λρ then, by a re-scattering
      Λρ turns into Σπ and its contribution would interfere with the amplitude of direct transition Λ_c→Σπ.
      
      
      As is well known, the weak decays of heavy hadrons mainly occur
      via an emission of virtual W or Z bosons  which later turn into lepton or quark pairs, from the heavy quark(antiquark). In our case,
      Λ_c is an iso-spin singlet, therefore the u-d subsystem (one may call it as a diquark) in Λ_c exists in an iso-spin singlet (I=0),
      and it is noted that the u-d subsystem in
      Λ is also an isospin singlet, but in Σ^0 the subsystem is an isospin-triplet (I=1). During the transition of Λ_c→Λ
      the u-d subsystem serves generally as a spectator and retains its isospin unchanged, whereas for Λ_c→Σ^0, the isospin of the u-d  subsystem
      is forced to change from a singlet into a triplet. Practically, for the Λ_c→Σ^0π^+ transition,
      the W-boson emitted by the charm quark must be connected to the u-d subsystem which is no longer a real spectator.
      Although the weak interaction
      does not conserve isospin, the isospin analysis may help us to get an insight into what happens during the transition.
      
      
      According to the analysis given in literature <cit.>,
      the rate of Λ_c→Λρ  should be about twice larger than that of Λ_c→Σπ,
      therefore
      a two-steps process: Λ_c→Λρ→Σπ may substantiate and change the picture (especially the sign of the asymmetry parameter).
      This is not a surprise to notice the role of final interaction, in our earlier work,
      we studied the case of D^0→ K^0K̅^0 which has the same rate as D^0→ K^+K^-. In fact the former is strongly suppressed, nevertheless  the later is
      favored. The result is fully understood as D^0→ K^0K̅^0 is realized via a re-scattering K^+K^-→ K^0K̅^0<cit.>.
      Therefore in this work we include the contributions from both the direct transition and that induced by the final state interaction, and their
      interference leads to the final result which is experimentally measured. For a comparison, in the following table, we list the results given in literature.
      
      
      
      
      
      Concretely, all the coupled channels of Λ_c  should contribute  to the observed
      Λ_c→Σπ via final state interactions (re-scattering). In terms of the effective interaction,
      the corresponding coupling constants  the first step  of the sequential
      Λ_c decays could be formulated according to the scenario presented in the literature.
      The re-scattering mechanism has been  successfully applied to  explain some
      anomalies existing in low energy experiments, such as  the decays of Υ and bottomed
      mesons<cit.>, thus we have a full confidence
      that the mechanism also works well  here.
      
      
      This paper is organized as follows: after this introduction we
      will consider the contribution to Λ_c→Σπ by including the final state interactions. In section III  we present our numerical results. Section IV is
      devoted to a brief summary.
      
      
      
      § THE SEQUENTIAL DECAY  Λ_C→ΛΡ→ΣΠ
      
      
      
      The amplitude of  baryon decays ℬ_i→ℬ_f P can be written as<cit.>
          ℳ(ℬ_i→ℬ_f P)=iU̅_ℬ_f [A-Bγ_5] U_ℬ_i,
      
      where ℬ_i (ℬ_f) is the initial (final) baryon and P is a pseudoscalar meson.
      
      
      For the transition ℬ_i→ℬ_f V, the amplitude is
      
          ℳ(ℬ_i→ℬ_f V)=iU̅_ℬ_f ε^*μ[-A_1γ_μγ_5-A_2p_fμγ_5+B_1γ_μ+B_2p_fμ] U_ℬ_i,
      
      where V is a vector meson with polarization ε and p_f is the momentum of ℬ_f. It is noted that the sign of the γ_5 term
      in Ref.<cit.> is opposite to the convention adopted in <cit.>. Here with this minus sign in front of every item involving γ_5 (conventional definition), all the values of B, A_1 and A_2 given in Ref.<cit.> do not need to be  changed and we  directly adopt their formulas. Indeed. this provides us a great convenience to derive relevant quantities.
      
      
      For the decay Λ_c→Σπ the factors A and B should include the contributions of all relevant Feynman diagrams. At the quark level the transition does not occur
      via factorizable  Feynman diagrams but those of non-factorizable ones<cit.>. In Ref.<cit.> the authors employed the simple pole-model to calculate the contributions of non-factorizable  Feynman diagrams. However the sign of the up-down asymmetry α gained in this way is opposite to data.
      
      It is easy to conjecture that in this case, the contribution from the re-scattering of final state in some decays of Λ_c might play an important role. The goal of this
      work is just to check if the re-scattering of the final products can change the scenario, namely simultaneously results in the required production rate for Λ_c→Σπ and a correct up-down asymmetry α.
      In principle  many coupled channels would jointly contribute to the decay Λ_c→Σπ, for example Λ_c→Λρ and Λ_c→Λπ etc.
      Considering the coupling constants and the rates of the first step decay of Λ_c one can decide that the main process is Λ_c→Λρ→Σπ
      where the second step is the isospin conserving re-scattering Λρ→Σπ.
      
      The total amplitude of the practical transition Λ_c→Σπ is
      
          M_(Λ_c→Σπ)      = M^ DIR_(Λ_c→Σπ)+M^ FSI_(Λ_c→Λρ→Σπ)
                =iU̅_Σ[A^ DIR-B^ DIRγ_5] U_Λ_c+iU̅_Σ[A^ FSI-B^ FSIγ_5] U_Λ_c
                =iU̅_Σ[A-Bγ_5] U_Λ_c,
      
      where M^ DIR_(Λ_c→Σπ) and M^ FSI_(Λ_c→Λρ→Σπ)
      correspond to the contributions of the direct transition Λ_c→Σπ and the two-step process Λ_c→Λρ→Σπ respectively, A=A^ DIR+A^ FSI and B=B^ DIR+B^ FSI.
      The amplitude of the direct transition M^ DIR_(Λ_c→Σπ) was calculated in terms of the pole model <cit.> and we will use their numerical results directly.
      
      Now let us begin to study the processes whose corresponding Feynmen diagrams are depicted in Fig.<ref>.
      
      
      The relevant effective interactions are
      <cit.>
          ℒ__ΛΣπ=ig__ΛΣπψ̅__Σγ_5ψ__Λπ
          ℒ_ρππ=      g__ρππ(∂_μπ^0 π^+ρ^-μ-∂_μπ^+π^0ρ^-μ ),
      
      
      Generally, the absorptive part overwhelmingly dominates and the contribution of
       the dispersive one can be ignored, so that we only need to calculate the absorptive part of the Feynmen diagrams in Fig.<ref>.
      where the intermediate states ρ and Λ are on shell and the transition amplitude of Λ_c→Λρ can also
      be written in terms of the pole model. By the Cutkosky rule, one can factorize the transition into two parts as
      M(Λ→Λρ)M(Λρ→Σπ). Let us first
      calculate the amplitude corresponding to diagram Fig 1 (a):
      
      
          M^ FSI_(Λ_c→Λρ^+→Σ^0π^+)=      1/2∫d𝐩_1/(2π)^32E_1d𝐩_2/(2π)^32E_2(2π)^4δ(p-p_1-p_2)ℳ[Λ_c→Λρ]ℳ[Λρ→Σπ]
          =      1/2∫d𝐩_1/(2π)^32E_1d𝐩_2/(2π)^32E_2(2π)^4δ(p-p_1-p_2)iU̅_Λ[-A_1γ_μγ_5-A_2p_1μγ_5γ_μ
                +B_1+B_2p_1μ]
          U_Λ_c g__ΣΛπU̅_Σiγ_5
          U_Λg__ρππi(p_4+q)_ν(-g^μν+p_2^μ
          p_2^ν/m_ρ^2)i/q^2
          -m^2_πF^2(q^2)
          =      ∫|𝐩_1|dΩ/32π^2EU̅_Λ[-A_1γ_μγ_5-A_2p_1μγ_5+B_1γ_μ+B_2p_1μ]
          U_Λ_cU̅_Σγ_5 U_Λ
                
          g__ΣΛπg__ρππ(p_4+q)_ν(-g^μν+p_2^μ
          p_2^ν/m_ρ^2)F^2(q^2)/q^2-m^2_π
          =      ∫|𝐩_1|dΩ/32π^2EU̅_Σγ_5
          (p_1+m_Λ)[-A_1γ_μγ_5-A_2p_1μγ_5+B_1γ_μ+B_2p_1μ]
          U_Λ_c
                
          g__ΣΛπg__ρππ(p_4+q)_ν(-g^μν+p_2^μ
          p_2^ν/m_ρ^2)F^2(q^2)/q^2-m^2_π,
      
      where E, E_1 and E_2 are the energies of Λ_c, Λ and ρ, with p, p_1, p_2, p_3 and p_4 being the momenta of  Λ_c, Λ, ρ, Σ and π, q is the momentum of the exchanged intermediate pions. Since in practice meson and baryon are not point particles,  a form
      factor at each effective vertex should be introduced. The form factor
      suggested by many researchers  is in the form:
      
          F(q,m_ P^2 ) = Λ_1^2 -
          m_P^2 Λ_1^2 -q^2 ,
      
      where Λ_1 is a cutoff parameter and m_ P is equal to m_π. Since the form factor is
      not derived from a fundamental principle, and the concerned cutoff
      parameter is neither determined theoretically, actually so far we
      know little about the cutoff parameter Λ_1. In some
      Refs.<cit.> the form
      factor is parameterized as Λ_1=λΛ_QCD+m_P with
      Λ_QCD≈ 220 MeV and the dimensionless parameter λ
      is of order of unit.
      
      Using the four-momentum relations p_4=p-p_3, q=p_1-p_3, p_2=p-p_1 and contracting the indices μ and ν, these notations p, p_1, p_3
      , p^2, p_1^2, p_3^2, p· p_1, p· p_3 and p_1· p_3 appear in the expression of M^ FSI_(Λ_c→Λρ^+→Σ^0π^+). One can employ Dirac equations p U_Λ_c=mU_Λ_c and p_3 U_Σ=m_3U_Σ to simply the expression. Since Λ_c, Σ and Λ are on-shell, p^2, p_1^2, p_3^2, p· p_1, p· p_3 and p_1· p_3 can be expressed in terms of observable physical quantities. At last, one needs to deal with p_1. In our calculation we choose 𝐩_3  in  z-direction and the
      angle spanned between 𝐩_1 and 𝐩_3 is θ. Since there exists an integration over azimuth one can find p_1=C_1p+C_2p_3 with C_1=E_1|𝐩_3|-|𝐩_1|E_3 cosθ/m|𝐩_3| and C_2=|𝐩_1| cosθ/|𝐩_3|.
      Finally we obtain
      
          M^ FSI_(Λ_c→Λρ^+→Σ^0π^+)=iU̅_Σ[A^ FSI-B^ FSIγ_5] U_Λ_c,
      
      with
      A^ FSI=g__ΣΛπg__ρππ∫C_a|𝐩_1| sinθ F^2(q^2)
      dθ/16π E and B^ FSI=g__ΣΛπg__ρππ∫C_b|𝐩_1| sinθ F^2(q^2) dθ/16π E.
      The detailed expressions of C_a and C_b are
      
          C_a=      -A_1 [ m^3 m_1 + m^2 m_1^2 - m_1^4 -
          2 E_3 m m_1 ( m + m_1 )  +
                 m_1^2 m_2^2 + 2 m_1 m_2^2 m_3
                 -
                 m m_1 ( m_1^2 + m_2^2 - 2 p_1· p_3 )  + 2 m_1^2 p_1· p_3 -
                 4 m_2^2 p_1· p_3 ]/[m_2^2 ( m_π^2 - q^2 ) 
                 - A_2 m_1 
               [ -2 E_3 m m_1^2 + m^2 m_1^2 - m_1^4 + m_1^2 m_2^2 +
                 E_1 m ( 2 E_3 m - m^2 + m_1^2 
                + m_2^2 - 2 p_1· p_3 )  +
                 2 m_1^2 p_1· p_3 - 2 m_2^2 p_1· p_3 ] /[m_2^2 ( m_π^2 - q^2 )
                 ]
                - (m C_1+m_3 C_2)A_1 [ -m^3 - m^2 m_1 + m_1^3 +
          2 E_3 m ( m + m_1 )  -
                   m_1 m_2^2 + 2 m_2^2 m_3
                + m ( m_1^2 + m_2^2 - 2 p_1· p_3 )  -
                   2 m_1 p_1· p_3 ]/[m_2^2 
              ( m_π^2 - q^2 ) ] 
                 + (m c_1+m_3 c_2)A_2 
                 [ -2 E_3 m m_1^2 + m^2 m_1^2 - m_1^4 + m_1^2 m_2^2 +
                   E_1 m ( 2 E_3 m 
                - m^2 + m_1^2 + m_2^2 - 2 p_1· p_3 )  +
                   2 m_1^2 p_1· p_3 - 2 m_2^2 p_1· p_3 ]   /[m_2^2 
              ( m_π^2 - q^2 ) ],
          
          C_b=      B_1  [2  E_3 m  m_1 (m- m_1)-m^3  m_1+m^2  m_1^2+m
          m_1  ( m_1^2+ m_2^2-2
              p_1· p_3 )- m_1^4+ 
                m_1^2  m_2^2+2  m_1^2  p_1· p_3+2  m_1m_2^2  m_3-4  m_2^2  p_1· p_3 ]/[ m_2^2  ( m_π^2- q^2 )]
                + B_2m_1  [ E_1 m  (2  E_3 m-m^2+ m_1^2+ m_2^2-2  p_1· p_3 )-2  E_3 m  m_1^2+m^2  m_1^2- m_1^4
                + m_1^2
              m_2^2+2  m_1^2  p_1· p_3-2  m_2^2  p_1· p_3]/[ m_2^2  ( m_π^2- q^2 )]
          
                + (m c_1-m_3 c_2)B_1  [2  E_3 m (m-
          m_1)-m^3+m^2 m_1+m ( m_1^2+ m_2^2-2  p_1· p_3
          )
                - m_1^3+ m_1m_2^2+2  m_1p_1· p_3-2  m_2^2  m_3 ]/[ m_2^2
              ( m_π^2- q^2 )]
                +(m C_1-m_3 C_2)B_2  [ E_1 m  (2  E_3 m-m^2+ m_1^2+ m_2^2-2  p_1· p_3 )-2
              E_3 m  m_1^2+m^2  m_1^2
                - m_1^4+ m_1^2  m_2^2+2  m_1^2  p_1· p_3-2  m_2^2  p_1· p_3 ] /[ m_2^2
              ( m_π^2- q^2 )],
      
      where m, m_1, m_2, m_3 and m_4 are the masses of Λ_c, Λ, ρ, Σ and π respectively, E_3 is the energy of Σ.
      
      By the same process one obtains the absorptive part of the Feynman diagram (b) on the right panel of Fig.<ref>
      and it is noted that
      
          M^ FSI_(Λ_c→Λρ^+→Σ^+π^0)=-M^ FSI_(Λ_c→Λρ^+→Σ^0π^+),
      
      where the minus sign comes from the effective interaction ℒ__ρππ i.e. the sign of the absorptive part of Λ_c→Σ^0π^+ is opposite to that of Λ_c→Σ^+π^0. The minus sign is necessary for determining the symbol of the asymmetry parameter of the decay Λ_c→Σ^+π^0.
      
      The decay rates of ℬ_i→ℬ_fP
      and up-down asymmetries are <cit.>
          Γ   =   |𝐩_c|/8π[(m_i+m_f)^2-m_P^2/m_i^2|A|^2+
            (m_i-m_f)^2-m_P^2/m_i^2|B|^2], 
          α   =   2κ Re(A^*B)/|A|^2+κ^2|B|^2,
      
      where 𝐩_c is the three-momentum of ℬ_f in the rest frame of
      ℬ_i and κ= |𝐩_c|/E_f+m_f.
      
      The amplitude of the direct transition Λ_c→Σπ is straightforward calculated in terms of the pole model which was shown in
      the early works, so that we omit the details of the calculations in this section.
      
      
      
      § THE THEORETICAL PREDICTIONS AS THE CONTRIBUTION FROM FINAL STATE INTERACTION IS TAKEN INTO ACCOUNT
      
      
      In a new work, the authors restudied the Cabibbo-suppressed decays of Λ_c<cit.> where the 30 years-old work
      <cit.> was cited. It indicates, the fundamental framework does not change and therefore we can employ their numerical results directly. Moreover,
      Λ_c→Λρ has also been investigated
      <cit.>, thus we just directly use their results about the first step transitions M(Λ_c→Λρ). Then we
      concentrate our attention on exploring the
      re-scattering process Λρ→Σπ and analyze the consequences especially how inclusion of the final state interaction  results in
      an opposite sign for the up-down asymmetry parameter from that determined by the direct transition.
      
      
      
      
      
      
      
      
      
      
      
      We list the theoretical predictions given by the authors of Ref.<cit.> in Tab. <ref> and <ref> for a clear reference.
      It is noted that the authors corrected their values about Λ_c→Λρ later and we employ the new ones for  our numerical computations
      (line (a) in Tab. <ref>). Since the effective color-favored Wilson coefficient (∼ 1.315) used in Ref.<cit.> is larger than the present values (∼ 1.26)<cit.> we set it to be 1.26 and repeat the calculations. The values we obtained are listed in line (b) of Tab. <ref> .
      
      
      
      
      In order to perform the numerical computations we need to determine the coupling constant g__ΣΛπ and g__ρππ. Using the data in
      particle data book<cit.> we fix g__ρππ=6.01 and another factor g__ΣΛπ=11.8 was given in Ref.<cit.>.
      Generally the cut-off parameter Λ_1 is about 1 GeV for a light exchanged meson. In our calculation we set it to be 0.8 GeV and 1 GeV respectively to make more sense.
      Using the formula derived above A^ FSI and B^ FSI are calculated with A_1=A^ fac_1+A^ pole_1, A_2=A^ fac_2+A^ pole_2, B_1=B^ fac_1+B^ pole_1 and B_2=B^ fac_2+B^ pole_2 and their numerical values are presented in Tab.<ref>.
      Using the values in table <ref> one can obtain A^ DIR=A^ fac+A^ pole and B^ DIR=B^ fac+B^ pole.
      Summing up the contributions of the direct transition and that involving final state interaction we have A and B in the total amplitude M_(Λ_c→Σπ)
      and our theoretical results are presented in Tab. <ref> where the values in front of (or between) the parentheses are corresponding to those in Tab. <ref> (a) (or (b)). The experimental results on Γ(Λ_c→Σ^+ π^0) and α are (4.08±0.33)×10^-14GeV and -0.45±0.32.
      
      In this scenario, we have made a theoretical prediction on the up-down asymmetry α whose  value
      resides within the error tolerance of the data and its sign is consistent with the experimental measurement.
      The predicted decay width of Γ(Λ_c→Σπ)  is also closer to data than that made
      in Ref.<cit.>. Apparently final state interaction changes the naive results of Ref.<cit.>. From table <ref> one can find that the signs of A^ pole and B^ pole are the same so the sign of α calculated in pole model is positive. As  the final state interaction are taken into account
      the interference between the direct transition and the final state interaction would induce a conversion of the sign of the asymmetry parameter. Namely
      A and B (table <ref>) possess  opposite signs, so the sign of α in is negative.
        It is noticed that in this work, only the contribution from the decay Λ_c→Λρ as an intermediate state is accounted, certainly in principle,
      other decay portals of Λ_c
      should also contribute to the same process via re-scattering. A careful analysis indicates that those contributions are not as important as that of
      Λ_c→Λρ→Σπ, therefore, we ignore those coupled channels in this work.
      
      
      
      
      
      § SUMMARY
      
      At the quark level the decay Λ_c→Σπ receives only the non-factorizable W-exchange and internal W-emission contributions. Based on the valence quark model<cit.> or the pole-model<cit.> these  non-factorizable diagrams were calculated while the resultant sign of the up-down asymmetry conflicts with data. Employing current algebra the authors<cit.> obtained a negative up-down asymmetry as required by data. Generally current algebra can be applied to study the decays where a soft pseudoscalar meson is emitted. However the pion in Λ_c→Σπ is far from being soft so it is not natural to explain the data by using the current algebra. Following the approach in the references<cit.> we suggest that a final state interaction (or re-scattering) in the decays of Λ_c can contribute to the observed Λ_c→Σπ. In terms of the effective interactions, coupling constants
      we calculate the contribution of the subprocess Λ_c→Λρ→Σπ to the observed Λ_c→Σπ. We notice (see the Table <ref>), with the pole model the contribution to both A^ DIR and B^ DIR in
      Eq.(1) are positive while the the contributions originating from the re-scattering to A^ FSI destructively interferes with that of  A^ DIR  of the direct transition whereas
      B^ FSI constructively interferes with  B^ DIR, thus as a consequence, the sign of the asymmetry parameter is reversed due to the destructive interference.
      
      For our concrete calculations, generally considering, if there exists an absorptive part, it should dominate the rate. Thus we only calculate the absorptive part of the triangle (see the Feynman diagrams) where in the intermediate step
      Λ and ρ are on-shell and we can factorize the two steps Λ_c→Λρ and Λρ→Σπ. Including the contribution of the direct transition Λ_c→Σπ calculated using pole model and the subprocess Λ_c→Λρ→Σπ which involves re-scattering effects, we obtain a negative up-down asymmetry α and the resultant decay width of Γ(Λ_c→Σπ) is also closer to data than the original results of Ref.<cit.>. It is also noted that the re-scattering of
      Λρ to other products may reduce the observed  rate of Λ_c→Λρ. We predict the observed width of Λ_c→Λρ would be somehow smaller than the theoretically predicted value given in Tab. <ref>.
      
      In this paper we  study the contribution of final state interaction to the transition Λ_c→Σπ and confirm that the final state interaction plays an important role in many hadronic transitions. In fact there still exist some discrepancies between theoretical estimations and data for other decays of Λ_c, and we hope the mechanism can also be applied to studying those “anomalies".
      
      
      
      § ACKNOWLEDGEMENT
      
      This work is supported by the National Natural Science Foundation
      of China (NNSFC) under Contract No. 11375128, 11675082 and 11735010.
      
      
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