Study on possible molecular states composed of $Λ_c\bar D$ ($Λ_b B$) and $Σ_c\bar D$ ($Σ_b B$) within the Bethe-Salpeter framework
Hong-Wei Ke Mei Li Xiao-Hai Liu Xue-Qian Li

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 
    P_c(4312) observed by the LHCb collaboration is confirmed as a
    pentaquark and its structure, production, and decay behaviors
    attract great attention from theorists and experimentalists. Since
    its mass is very close to sum of Σ_c and D̅ masses,
    it is naturally tempted to be considered as a molecular state
    composed of Σ_c and D̅. Moreover, P_c(4312) is
    observed in the channel with J/ψ p final state, requiring
    that isospin conservation P_c(4312) is an isospin-1/2
    eigenstate. In literature, several groups used various models to
    estimate its spectrum. We systematically study the pentaquarks
    within the framework of the Bethe-Salpeter equation; thus
    P_c(4312) is an excellent target because of the available data.
    We calculate the spectrum of P_c(4312) in terms of the
    Bethe-Salpter equations and further study its decay modes. Some
    predictions on other possible pentaquark states that can be tested
    in future experiments are made.
    
    
    
    12.39.Mk, 12.40.-y ,14.40.NdStudy on possible molecular states composed of Λ_cD̅ (Λ_b B) and Σ_cD̅ (Σ_b B)   within the Bethe-Salpeter framework 
         Hong-Wei Ke^1[Corresponding author, khw020056@tju.edu.cn], Mei Li^1, Xiao-Hai Liu^1[xiaohai.liu@tju.edu.cn]and
            Xue-Qian Li^2[lixq@nankai.edu.cn],
       
        December 30, 2023
    ==================================================================================================================================================================
    
    
    
    § INTRODUCTION
    
    
    Due to the innovation of experimental techniques and facilities as
    well as the advances in theory of recent years several exotic
    states have been experimentally observed and theoretically
    studied. Indeed, more constituents would cause more ambiguities,
    unlike the simplest qq̅ for mesons and qqq for baryons.
    The inner structures of the exotic states are still not clear yet,
    those discoveries stir up large numbers of
    discussions<cit.>. Indeed the theoretical exploration
    is crucial for getting a better understanding of the quark model
    and obtaining valuable information about non-perturbative physics.
    Definitely, to complete the theoretical job achieving more
    accurate data would compose the key.
    
    Some hidden charm or bottom states were measured in two-meson
    final
    states<cit.>.
    They are regarded as tetraquark states or meson-meson molecular
    states. In 2003 a baryon was measured by LEPS<cit.>
    which was conjectured as a pentaquark, however later the
    allegation was negated by further more accurate experiments.
    Breaking the frustration on existence of pentaquark which was
    predicted by Gell-Mann in his first paper on quark model, the LHCb
    collaboration reported two pentaquark states observed in
    Λ_b decays where peaks appear at the J/ψ p final
    states<cit.>.
    
    Recently another narrow pentaquark state
    P_c(4312)<cit.> has also been observed in the
    J/ψ p mass spectrum.  Its mass and width are
    4311.9±0.7^+6.8_-0.6 MeV and 9.8± 2.7^+3.7_-4.5
    MeV respectively. Since its mass is very close to the sum of
    Σ_c and D̅ masses, it is natural to regard it as a
    molecular state of Σ_c D̅<cit.>. Furthermore its width is
    rather wide in accordance with the property of molecular states,
    so the phenomenon further supports the proposal of its molecular
    structure. Some other theorists conjecture P_c(4312) as a
    compact pentaquark.<cit.> instead. In
    Ref. <cit.> the authors think the
    interaction between Σ_c and D̅ is too weak to bind
    them into a bound state. It is worth deeper explorations about
    whether the molecule picture is reasonable. In this work we will
    calculate the mass spectrum of P_c(4312) based on the assumption
    that it is a stable bound state of Σ_c and D̅.
    Additionally we also study other possible bound states of
    Λ_cD̅, Λ_b B and Σ_b B and see if they
    can be formed.
    
    We will employ the Bethe-Salpeter (B-S) equation to study the
    possible bound state which consists of a baryon and a meson. The
    B-S equation is a relativistic equation to deal with the bound
    state and established on the basis of quantum field
    theory<cit.>. Initially, people used the B-S
    equation to study the bound state of two
    fermions<cit.> and the system of
    one-fermion-one-boson<cit.>. In
    Ref.<cit.> the authors employed the
    Bethe-Salpeter equation to study the KK̅ and BK̅
    molecular states and their decays. With the same approach we
    studied the molecular state of Bπ<cit.>,
    D^(*)D^(*) and B^(*)B^(*)<cit.>. Recently
    the approach is extended to explore double charmed
    baryons<cit.> and pentaquarks which are
    assumed to be two-body bound systems. In Ref.<cit.>
    the authors studied possible bound states of Λ (Σ)
    and K̅. In this work we will employ a similar approach to
    study the possible bound states of Σ_cD̅,
    Λ_cD̅, Λ_b B and Σ_b B.
    
    At present, pentaquark states P_c(4312), P_c(4380),
    P_c(4440) and P_c(4457) have been measured in decays of
    Λ_b where the pentaquark states peak up at the invariant
    mass spectrum of J/ψ p, so their isospin is 1/2
    because of isospin conservation. Thus we require that the two
    hadron constituents reside in an isospin eigenstate. Instead, for
    the Λ_cD̅ (as well Λ_b B) system  its isospin
    must be 1/2 but the Σ_cD̅ ( or Σ_b B)
    system may reside in either isospin 1/2 or
    3/2. Certainly, for  a bound system with  spin-parity
    1/2^- the two constituents are in the Swave.
    
    For carrying on our calculation the interactions between two
    constituents are needed. According to the quantum field theory two
    particles interact via exchanging certain mediate particles. Since
    two constituents in a pentaquark are color-singlet hadrons the
    exchanged particles are some light hadrons such as ρ or (and)
    ω etc.. The effective interactions are deduced from the
    chiral Lagrangian<cit.>
    which we list in Appendix A. With the effective interactions we
    obtain the kernel and establish the corresponding B-S equation.
    
    
    With a reasonable parameter set, the B-S equation is solved. For a
    spin-isospin eigenstate, if the equation does not possess a
    solution, then we would conclude that the corresponding bound
    state should not exist in nature; on the contrary, a solution of
    the B-S equation implies the bound state being formed. At the same
    time the B-S wave function is obtained and  we are able to use the
    corresponding formula for calculating the rates of strong decay
    P_c(4312)→ proton+𝒱 (vector) which can be
    compared with the data.
    
    This paper is organized as follows: after this introduction we
    will derive the B-S equations related to  possible bound states
    composed of a baryon and a meson and the formula for its strong decays.
    Then in section III we will solve the B-S equation numerically and
    present our  results by figures and tables. Section IV is devoted to a brief
    summary.
    
    
    
    § THE BOUND STATES OF Λ_CD̅  AND
    Σ_CD̅
    
    
    Since the newly found pentaquarks  P_c(4312), P_c(4380),
    P_c(4440) and P_c(4457) are all  hadrons containing hidden
    charms(or hidden bottoms) and their masses are close to the sums
    of the masses of several real hadrons, we will focus on the
    molecular structures composed of one charmed (bottomed) baryon and
    an anti-charmed(anti-bottomed) meson. Concretely, in this paper we
    study  Λ_cD̅, Σ_cD̅, Λ_b B and
    Σ_b B systems whose spin-parity is 1/2^- i.e. the
    spatial wave function is in Swave. In this section as an example
    we only formulate the corresponding quantities for Λ_cD̅ and Σ_cD̅ systems. These formulas can be equally
    applied to Λ_b B and Σ_b B systems.
    
    
    
     §.§ The isospin states of Λ_cD̅ and Σ_cD̅
    
    
    The isospin structure of the possible bound state of
    Λ_cD̅ is
    
        |1/2,1/2⟩=|Λ_cD̅^0⟩.
    
    We will use P'_c(1/2,1/2) to denote this resonance.
    
    Instead, the possible bound states of
    Σ_cD̅ should be in three isospin assignments i.e.
    |I,I_3⟩ are |1/2,±1/2⟩|3/2,±1/2⟩ and |3/2,±3/2⟩. Let us work out the explicit
    isospin states
    
        |1/2,1/2⟩=√(2/3)|Σ_c^++
        D^-⟩-√(1/3)|Σ_c^+D̅^0⟩,
    
        |3/2,1/2⟩=√(1/3)|Σ_c^++
        D^-⟩+√(2/3)|Σ_c^+D̅^0⟩,
    
    and
    
        |3/2,3/2⟩=|Σ_c^++D̅^0⟩.
    
    The states |1/2,-1/2⟩,
    |3/2,-1/2⟩  and
    |3/2,-3/2⟩ are just the charge conjugate
    states of  |1/2,1/2⟩,
    |3/2,1/2⟩  and
    |3/2,1/2⟩, therefore their hadronic
    properties are the same. We use P_c(1/2,1/2),
    P_c(3/2,1/2) and
    P_c(3/2,3/2) to denote the three isospin
    states of Σ_cD̅: |1/2,1/2⟩,
    |3/2,1/2⟩  and
    |3/2,3/2⟩  respectively for latter
    discussions.
    
    In order to discuss the Isospin factors in the B-S equation  we
    define the fields of baryons and mesons  in the
    expressions<cit.>:
    
        ℬ_1(x)=∫d^4q(2π)^4√(2m_ℬ^++)
        (a_ℬ^–e^-iqx+a^†_ℬ^++e^iqx),
              ℬ_2(x)=∫d^4q(2π)^4√(2m_ℬ^+)
        (a_ℬ^-e^-iqx+a^†_ℬ^+e^iqx),
              ℳ_1(x)=∫d^4q(2π)^4√(2m_ℳ^+)
        (a_ℳ^+e^-iqx+a^†_ℳ^-e^iqx),
              ℳ_2(x)=∫d^4q(2π)^4√(2m_ℳ^0)
        (a_ℳ^0e^-iqx+a^†_M^0e^iqx),
    
    where ℬ represents Λ_c or Σ_c and
    ℳ denotes D.
    
    
    
     §.§ The Bethe-Salpeter (B-S) equation for 1/2^- molecular state
    
    In the effective theory a meson and a baryon can interact via
    exchanging hadrons. The Feynman diagram at the leading order is
    depicted in Fig. <ref> (It is noted, the diagram where the
    exchanged hadron is a heavy baryon, is ignored at the leading
    order). The relative and total momenta of the bound state in the
    equations are defined as
    
        p = η_2p_1 -
        η_1p_2 ,   q = η_2q_1 - η_1q_2 ,   P = p_1 + p_2
        =q_1 + q_2  ,
    
    where p and q are the relative momenta at the two sides of the
    effective vertex, p_1 (q_1) and p_2 (q_2) are those
    momenta of the constituents, P is the total momentum of the
    bound state, k is the momentum of the exchanged meson, η_i =
    m_i/(m_1+m_2) and m_i  (i=1,2) is the mass of the ith
    constituent meson.
    
    The bound state composed of a baryon and a meson can be written as
    
        χ_P(x_1,x_2)
        =⟨0|Tℬ(x_1)ℳ(x_2)|P⟩.
    
    
    The B-S wave function is a Fourier transformation of that in momentum
    space
    
        χ_P(x_1,x_2) =e^-iPX∫d^4q/(2π)^4χ_P(p).
    
    
    By the so-called ladder approximation the corresponding B-S
    equation was deduced in earlier references as
    
        χ_P( p)
        =S_B(p_1)∫d^4q(2π)^4 K(P,p,q)χ__P(q)S_M(p_2) ,
    
    where S_B(p_1) is the propagator of the baryon (Λ_c or
    Σ_c), S_M(p_2) is that of the meson ( D̅) and
    K(P,p,q) is the kernel which can be obtained by calculating the
    Feynman diagram in Fig. 1. For later convenience the relative
    momentum p is decomposed into the longitudinal p_l (≡
    p· v) and transverse projection p^μ_t (≡
    p^μ-p_lv^μ)=(0, 𝐩_T) according to  the momentum of
    the bound state P (v=P/M).
    
    
    
        S_B(η_1
        P+p)=i[(η_1M+p_l)v+p_t+m_1]/(η_1M+p_l+ω_l-iϵ)(η_1M+p_l-ω_l+iϵ),
    
        S_M(η_2
        P-p)=i/(η_2M-p_l+ω_2-iϵ)(η_2M-p_l-ω_2+iϵ),
    
    where M is the total energy of the bound state, ω_i =
    √( p_l^2 + m_i^2) and m_1 (m_2) is the mass of the
    baryon (meson).
    
    By the Feynman diagram the kernel K(P,p,q) is
    written as
    
        K(P,p,q)=-C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱(γ^α-
        κ_ℬℬρ/2m_ℬσ^αβk_β)
        (p_2+q_2)^μΔ_αμ(k,m_V)F^2(k),
    
    where m_V is the mass of the exchanged meson, g_MMV,
    g_BBV and κ_BBρ are the concerned coupling
    constants, C_I,I_z is the isospin coefficient which is given
    in Appendix B and
    Δ_αμ(k,m_V)=(-g_αμ+k_αk_μ/m_v^2)/(k^2-m_v^2).
    Apparently the contribution of the tensor term is much smaller
    than that of the first term, thus we can ignore it in practical
    computations. Indeed, a numerical estimate verifies this
    allegation.
    
    Since the constituents of the molecule (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 of
    the form:
    
        F( k,m_ V^2 ) = Λ^2 -
        m_ V^2 Λ^2 +  k^2 ,   k = 
        p- p'  ,
    
    where Λ is a cutoff parameter. Since the form factor is
    not derived from a fundamental principle, the concerned cutoff
    parameter is neither determined theoretically, thus until now we
    know little about the cutoff parameter Λ. In some
    Refs.<cit.> the form
    factor is parameterized as λΛ_QCD+m_V with
    Λ_QCD=220 MeV and the dimensionless parameter λ
    is of order of unit. We will employ the expression
    Λ=λΛ_QCD+m_V in our calculation.
    
    
    
    
    The three-dimensional B-S wave function is obtained after
    integrating over p_l
        χ_P( p_t) =∫dp_l/2πχ_P( p).
    
    
    For the Swave system, the spatial wave function can be easily
    derived <cit.>
        χ_P( p_t)
        =[f_1(|𝐩_T|)+f_2(|𝐩_T|)p_t]u(v,s),
    
    where f_1(|𝐩_T|) and f_2(|𝐩_T|) are the
    radial wave functions, u(v,s), v and s are the spinor,
    velocity and total spin of the pentaquark respectively.
    
    
    
    Substituting Eq. (<ref>) into Eq. (9) and employing the
    so-called covariant instantaneous approximation where q_l=p_l
    i.e. p_l takes the place of q_l in the kernel K(P,p,q),
    K(P,p,q)  no longer depends on q_1. Then we are performing a
    series of manipulations: integrate over q_l on the right side of
    Eq. (<ref>); multiply ∫dp_l/(2π) on the
    both sides of Eq. (<ref>), and integrate over p_l on
    the left side using Eq. (<ref>). Finally, substituting
    Eq. (<ref>)  we obtain
    
        [f_1(|𝐩_T|)+f_2(|𝐩_T|)p_t]u(v,s)=-∫dp_l/(2π)∫d^3𝐪_T/(2π)^3iC_I,I_zg_MMVg_BBV
        [(η_1M+p_l)v+p_t+m_1]/[(η_1M+p_l)^2-ω^2_l+iϵ][(η_1M-p_l)^2-ω^2_2+iϵ)]
              {-κ[2(η_2M-p_l)
        v-p_t-q_t](p_t-q_t)-(p_t-q_t)
        [2(η_2M-p_l)v-p_t-q_t]/4m_B[-(𝐩_T-𝐪_T)^2-m_V^2]+
        
              2(η_2M-p_l)v-p_t-q_t-
        (p_t-q_t)(𝐩_T^2-𝐪_T^2)/m_V^2/-(𝐩_T-𝐪_T)^2-m_V^2}
        F^2(k,m_V)[f_1(|𝐪_T|)+f_2(|𝐪_T|)q_t]u(v,s).
    
    
    Now let us finally fix the expressions of f_1(|𝐩_T|)
    and f_2(|𝐩_T|). Multiplying u̅(v,s) on both sides
    of Eq.(<ref>), we get an expression which only
    contains f_1 whereas multiplying u̅(v,s)p_t to
    the expression, f_2 is obtained, then by taking a trace, the
    resultant formulas are
    
        f_1(|𝐩_T|)=-∫dp_l/(2π)∫d^3𝐪_T/(2π)^3iC_I,I_zg_MMVg_BBV
        F^2(k,m_V)/[(η_1M+p_l)^2-ω^2_l+iϵ][(η_1M-p_l)^2-ω^2_2+iϵ]
              {𝐩_T·𝐪_T + 𝐩_T^2 + 2( m_1 + p_l + Mη_1) (
        Mη_2-p_l
         ) +(𝐩_T^2 -𝐩_T·𝐪_T  )
        ( 𝐩_T^2 -𝐪_T^2) 
         / m_V^2/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)+
              ( 𝐩_T^2 -𝐪_T^2)
           (𝐩_T·𝐪_T - 𝐪_T^2 )
            ( m_1 + p_l + Mη_1 ) /m_V^2
          + (m_1+Mη_1+p_l)(
        𝐩_T·𝐪_T + 𝐪_T^2 )  +
        2(Mη_2-p_l)𝐩_T·𝐪_T/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)
              +κ/m_B[
        𝐩_T·𝐪_T^2 -
        𝐩_T^2𝐪_T^2 + (m_1+p_l+Mη_1)(
        𝐪_T^2 -𝐩_T·𝐪_T )(p_l-Mη_2)
            ]f_2(|𝐪_T|)/[-(𝐩_T-𝐪_T)^2-m_V^2]
              -κ/m_B(
        𝐩_T·𝐪_T - 𝐩_T^2 ) ( p_l -
        Mη_2
        )f_1(|𝐪_T|)/[-(𝐩_T-𝐪_T)^2-m_V^2]}.
    
        f_2(|𝐩_T|)𝐩_T^2=-∫dp_l/(2π)∫d𝐪_T/(2π)^3-iC_I,I_zg_MMVg_BBV
        F^2(k,m_V)/[(η_1M+p_l)^2-ω^2_l+iϵ][(η_1M-p_l)^2-ω^2_2+iϵ]
              {-𝐩_T^2(𝐩_T·𝐪_T + 𝐪_T^2) + 2𝐩_T·𝐪_T( m_1-p_l - Mη_1) ( Mη_2-p_l
         ) +𝐩_T^2(𝐪_T^2 -𝐩_T·𝐪_T  )
        ( 𝐩_T^2 -𝐪_T^2) /m_V^2/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)+
              ( 𝐩_T^2 -𝐪_T^2) (𝐩_T·𝐪_T - 𝐩_T^2 )
            ( -m_1 + p_l + Mη_1 ) /m_V^2
          + (m_1-Mη_1+p_l)(
        𝐩_T·𝐪_T + 𝐩_T^2 )  -
        2Mη_2𝐩_T^2-2p_l𝐩_T·𝐪_T/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)
              -κ/m_B[(
        Mη_2 - pl ) 𝐩_T·𝐪_T𝐩_T^2 -
         𝐩_T·𝐪_T^2( m_1 - p_l- Mη_1 )  +
          𝐩_T^2𝐪_T^2( m_1 - M
             )]f_2(|𝐪_T|)/[-(𝐩_T-𝐪_T)^2-m_V^2]
              -κ/m_B(
        𝐩_T·𝐪_T - 𝐩_T^2 ) ( p_l -
        Mη_2
        )(m_1-p_l-Mη_1)f_1(|𝐪_T|)/[-(𝐩_T-𝐪_T)^2-m_V^2]}.
    
    To extract f_1(|𝐩_T|) and f_2(|𝐩_T|) from the
    above equations, instead of the procedure adopted in earlier works,
    we multiply u̅(v) from the right side of the equation and sum
    over the spin projections of u(v), then taking a trace of the
    modified equation, the job is done. The advantage of this procedure
    is to keep the equation of motion v u(v,s)=u(v,s).
    
    
    
    Now we perform an integral over p_l on the right side of Eqs.
    (<ref>) and (<ref>) where four
    poles exist at -η_1M-ω_1+iϵ,
    -η_1M+ω_1-iϵ, η_2M+ω_2-iϵ and
    η_2M-ω_2+iϵ. By choosing an appropriate contour
    (<ref>) and (<ref>) we calculate
    the residuals at p_l=-η_1M-ω_1+iϵ and
    p_l=η_2M-ω_2+iϵ. The coupled equations after the
    contour integrations are collected in the appendix [Eqs.
    (<ref>) and (<ref>) ]. Then
    one can carry out the azimuthal integration and reduce Eqs.
    (<ref>) and (<ref>) to
    one-dimensional integral equations
    
        f_1(|𝐩_T|)=∫d|𝐪_T|[A_11(|𝐪_T|,|𝐩_T|)
        f_1(|𝐪_T|)+A_12(|𝐪_T|,|𝐩_T|)f_2(|𝐪_T|)]
              
        f_2(|𝐩_T|)=∫d|𝐪_T|[A_21(|𝐪_T|,|𝐩_T|)
        f_1(|𝐪_T|)+A_22(|𝐪_T|,|𝐩_T|)f_2(|𝐪_T|)],
    
    where A_11, A_12, A_21 and A_22 are presented in
    the appendix [see Eqs. (<ref>), (<ref>), (<ref>) and (<ref>)].
    
    
    
    
    
     §.§ The normalization condition for the B-S wave function
    
    The normalization condition for the B-S wave function of a bound
    state is<cit.>
        i∫d^4pd^4q/(2π)^8χ̅_P(p)∂/∂
        P_0[I(P,p,q)+K(P,p,q)]χ_P(q)=1,
    
    where P_0 is the energy of the bound state and the spinor relation
    ∑_s u(v,s)u̅(v,s)=v+1/2 is used.
    I(P,p,q) is the reciprocal of the four-point propagator
    
        I(P,p,q)=δ^4(p-q)/(2π)^4[S_ℬ(p_1)]^-1[S_ℳ(p_2)]^-1.
    
    
    For the molecular sates composed of two mesons the second term in
    the normalization condition is several orders smaller than the
    first term <cit.>; thus we have all every
    reason to believe that the rule also applies to the case where the
    molecule is composed of a baryon and a meson, consequently the
    term ∂/∂ P_0K(P,p,q)  can be ignored and
    then
    
        -∫d^4p/(2π)^4χ̅_P(p)η_1v[S_ℳ(p_2)]^-1χ_P(q)-∫d^4p/(2π)^4χ̅_P(p)2η_2p_2·
        v[S_ℬ(p_1)]^-1χ_P(q)=1.
    
    Let us define the transverse projections of the B-S wave function
    as follows:
    
        α_P(p)=-i[S_ℬ(p_1)]^-1χ_P(q)[S_ℳ(p_2)]^-1,
              β_P(p)=-i[S_ℳ(p_2)]^-1χ̅_P(q)[S_ℬ(p_1)]^-1,
    
    the normalization condition is
    
        -∫d^4p/(2π)^4
        Tr[α_P(p)β_P(p)S_ℬ(p_1)η_1v
        S_ℬ(p_1)S_ℳ(p_2)]
              -∫d^4p/(2π)^4 Tr[α_P(p)β_P(p)2η_2p_2·
        v S_ℬ(p_1) S_ℳ(p_1) S_ℳ(p_1)]=1.
    
    
    Substituting the expression χ̅_P(p) [Eq.
    (<ref>)] into Eqs. (<ref>) under the covariant
    instantaneous approximation one can obtain the expressions of
    α_P(p) and β_P(p), for example
    
        α_P(p)=-∫d^3𝐪_T/(2π)^3C_I,I_zg_ℳℳ𝒱
        g_ℬℬ𝒱{2(η_2M-p_l)v-p_t-q_t
        -(p_t-q_t)
        (𝐩_T^2-𝐪_T^2)/m_V^2/-(𝐩_T-𝐪_T)^2-m_V^2
              ­κ[2(η_2M-p_l)
        v-p_t-q_t](p_t-q_t)-(p_t-q_t)
        [2(η_2M-p_l)v-p_t-q_t]/4m_B[-(𝐩_T-𝐪_T)^2-m_V^2]
              }
        F^2(k,m_V)[f_1(|𝐪_T|)+f_2(|𝐪_T|)q_t]u(v,s).
    
    and α_P(p) and β_P(p) can be parameterized into
    
        α_P(p)=[h_1(|𝐩_T|)+h_2(|𝐩_T|)p_t]u(v,s),
              β_P(p)=u̅(v,s)[h_1(|𝐩_T|)+h_2(|𝐩_T|)p_t],
    
    with
    
        h_1(|𝐩_T|)=-∫d^3𝐪_T/(2π)^3{
        2f_1(𝐪_T)(Mη_2-p_l )+f_2(|𝐪_T|)[
        𝐪_T^2+𝐩_T·𝐪_T + (
        𝐩_T^2 - 𝐪_T^2)
        (𝐩_T·𝐪_T -𝐪_T^2 ) /m_V^2 ]
        /-(𝐩_T-𝐪_T)^2-m_V^2
              +4κ[f_2(|𝐪_T|)(
        (Mη_2-p_l)(𝐩_T·𝐪_T- 𝐪_T^2
        )]/4m_B[-(𝐩_T-𝐪_T)^2-m_V^2]}C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱F^2(k,m_V),
              h_2(|𝐩_T|)=-∫d^3𝐪_T/(2π)^3{f_1(𝐪_T)(
        (𝐩_T·𝐪_T -𝐩_T^2 )
                (𝐩_T^2 - 𝐪_T^2 ) /m_V^2-𝐩_T·𝐪_T -𝐩_T^2 )  +
        2f_2(𝐪_T)( p_l -
        Mη_2)𝐩_T·𝐪_T/𝐩_T^2[-(𝐩_T-𝐪_T)^2-m_V^2]
              +4κ(
        [f_2(𝐪_T)(
        𝐩_T^2𝐪_T^2-𝐩_T·𝐪_T^2
        )  +
          f_1(𝐪_T)(𝐩_T·𝐪_T -𝐩_T^2 ) (p_l - Mη_2 )
        ]/4m_B𝐩_T^2[-(𝐩_T-𝐪_T)^2-m_V^2]}
              C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱
        F^2(k,m_V),
    
    Substituting Eqs. (<ref>), (<ref>) and
    equation group (<ref>) into Eq. (<ref>) we
    obtain
    
        i∫d^4p/(2π)^42{h_1^2[ m_1^2 + p_l^2 + 𝐩_T^2 + 2Mp_lη_1 + M^2η_1^2 +
               2m_1( p_l + Mη_1 )  ]+
              h_2^2𝐩_T^2( m_1^2 + p_l^2 + 𝐩_T^2 + 2Mp_lη_1 +
               M^2η_1^2 - 2m_1( p_l + Mη_1 )  )  -4h_1h_2𝐩_T^2( p_l
        + Mη_1 )
        }
              /{[(η_1M+p_l)^2-ω^2_l+iϵ]^2[(λ_1M-p_l)^2-ω^2_2+iϵ)]}
              +i∫d^4p/(2π)^42[
         h_1^2( m_1 + p_l + Mη_1
        )-2h_1h_2𝐩_T^2 + h_2^2𝐩_T^2
             (  p_l-m_1  + Mη_1 )   ]
              /{[(η_1M+p_l)^2-ω^2_l+
        iϵ]^2[(λ_1M-p_l)^2-ω^2_2+iϵ)]}=1.
    
    After the contour integration on p_l and the azimuthal
    integration the normalization condition can be calculated
    numerically and the values of f_1(|𝐩_T|) and
    f_2(|𝐩_T|) are fixed at the same time.
    
    
    
    
    
     §.§ the decay of P_c→𝒱+proton
    
    
    Now we investigate the strong decays of P_c in terms of the
    framework formulated above.
    
    
    
    The amplitudes corresponding to the two diagrams in Fig. <ref>
    are,
    
        𝒜_a=C_Ig_ℬℬ'
        Dg_DD𝒱∫d^4p/(2π)^4U̅_ℬ'γ^5
        χ_P(p)(k-p_2)_νϵ^ν1/k^2-M^2_DF^2(k,m_D),
    
        𝒜_b      =2C_Ig_ℬℬ'
        D^*g_DD^*𝒱∫d^4p/(2π)^4U̅_ℬ'(γ^σ-κ_BBD^*/2m_Bσ^σωk_ω)
        χ_P(p)ε^αβμνk_μq_2_αϵ_ν
              g_σβ-k_β
        k_σ/M_D^*^2/k^2-M^2_D^*F^2(k,m_D^*),
    
    where C_I is the isospin coefficient of the transition,
    k=p-(η_2 q_1-η_1 q_2), ℬ denotes the charmed baryon in the molecular
    state: Σ_c or Λ_c; ϵ is the polarization
    vector of 𝒱 and ℬ' represents proton. We
    still take the approximation k_0=0 to carry out the calculation.
    
    The total amplitude is
    
        𝒜=𝒜_a+𝒜_b= u̅_ℬ'[γ^5g_1γ^μ+iγ^5g_2σ^μν
        q_2ν+ig_3γ_νε^μναβP_α
        q_2β] u(v) ϵ_μ.
    
    The factors g_1, g_2 and g_3 can be extracted from the
    expressions of 𝒜_1 and 𝒜_2.
    
     Then the partial width is expressed as
    
        dΓ=1/32π^2|𝒜|^2|q_2|/M^2dΩ.
    
    
    § NUMERICAL RESULTS
    
    
     §.§ the numerical results 
    
    
    
    
    In order to solve the B-S equation numerically some parameters are
    needed. The mass m_Λ_c, m_Σ_c, m_D,
    m_ω, m_ρ is taken from the databook<cit.>.
    Following Ref.<cit.>, we set the
    coupling constants g_DDω=g_DDρ=3.02,
    g_Λ_cΛ_cω=8.125,
    g_Σ_cΣ_cω=g_Σ_cΣ_cρ=7.475,
    f_Σ_cΣ_cω=κ
    g_Σ_cΣ_cω=9.9125,
    f_Σ_cΣ_cρ=κ g_Σ_cΣ_cρ=9.9125
    .
    
    
    
    With these parameters and the corresponding isospin factors a
    complete B-S equation [the coupled equations (<ref>)] is established. These coupled equations are
    complicated integral equations, thus to numerically solve them,
    the standard way is to discretize them, namely we would convert
    them into algebraic  equations. Concretely,  we set a reasonable
    finite range for |p_T| and |q_T|, and let the
    variables take n ( n=129 in our calculation) discrete values
    Q_1, Q_2,...Q_n which distribute with equal gap from
    Q_1=0.001 GeV to Q_n=2 GeV . The gap between two adjacent
    values is Δ |p_T|=(1.999/128) GeV. For clarity, we let
    n values of f_1( |p_T|) and n values of f_2(
    |p_T|) constitute a column matrix with 2n rows and the 2n
    elements f_1( |q_T|), f_2( |q_T|) construct another
    column matrix residing on the right side of the equation as shown
    below. The column matrix composed of f_1( |p_T|) and
    f_2( |p_T|) is associated with the right column matrix of
    f_1( |q_T|) and f_2( |q_T|) by a 2n× 2n
    matrix whose elements are the coefficients given in Eq. (<ref>). The standard way to treat the equation is to
    let |p_T| and |q_T| take the same sequential values
    Q_1, Q_2,...Q_n for discretizing the integral equation.
    
        ([   f_1(Q_1);        ...; f_1(Q_129);   f_2(Q_1);        ...; f_2(Q_129) ])=A(Δ E, λ)([   f_1(Q_1);        ...; f_1(Q_129);   f_2(Q_1);        ...; f_2(Q_129) ]).
    
    As a matter of fact, it is a homogeneous linear equation group. If
    it possesses non-trivial solutions, the necessary and sufficient
    condition is the coefficient determinant to be zero. In our case,
    it is |A(Δ E,λ)-I|=0 (I is the unit matrix) where
    A(Δ E,λ). Now we calculate the determinant of
    |A(Δ E,λ)-I|  is a function of the binding energy
    Δ E=m_1+m_2-M and parameter λ. Our strategy is
    following: we arbitrarily vary Δ E within a possible range,
    by requiring |A(Δ E,λ)-I|=0, we obtain a
    corresponding λ. In Ref.<cit.>λ was
    fixed to be 3. In our earlier paper<cit.> we change the
    value of λ from 1 to 3 to explore possible dependence of
    the results on it,  it seems that a value of λ within the
    range of 0∼ 4 is reasonable for forming a bound state of two
    hadrons. Consequently, if the obtained λ is much beyond
    the range, we would conclude that the resonance cannot exist.
    
    To get the wavefunction
    T(f_1(Q_1),f_1(Q-2)...,f_2(Q_1)...f_2(Q_129), we adopt a
    special method. Namely, we suppose a matrix equation (A(Δ
    E,λ)_ij)(f(j))=β (f(i)) which just is an eigenequation. In
    terms of the standard software, we can find all the possible
    “eigenvalues" β, and among them only β=1 is the solution
    we expect, then the corresponding wavefunction is gained which
    just is the solution of the B-S equation.
    
    For |A(Δ E,λ)-I|=0, inputting some  binding energies,
    we would check whether we can obtain reasonable values for
    λ. If yes, we substitute the values of λ and the
    binding energy into the matrix equation to obtain the B-S
    wavefunctions. With this strategy, we investigate the molecular
    structure of Λ_c and D̅ as well as that of
    Σ_c and D̅.
    
    
    
    
    
    If the exchanging particles are limited to light vector meson,
    only ω and ρ can be exchanged between charmed baryons
    and D. Of course, exchanging two ρ mesons between
    Λ_c (Σ_c) and D̅ can also induce a potential,
    but it undergoes a loop  suppression, therefore, we do not
    consider that contribution.
    
    
    
    As the first trial, let us study a simple compound, namely
    we explore the possible bound states
    of Λ_c and D̅ which is an I=1/2 state, therefore
    only ω can be exchanged between Λ_c and D̅.
    We find that there is no solution for the B-S equation, therefore we
    would conclude that the interaction induced by the single ω exchange
    is repulsive.
    
    With the same procedure, we study a molecule composed of
    Σ_c and D̅ whose isospin could be either 1/2 or
    3/2 and the coefficient is C_1/2,1/2=1.
    Since P_C(4312) is observed in the J/ψ p portal, it is
    confirmed to be a state of I=1/2. In this case both ω and
    and ρ exchanges between the two ingredients are allowed. The
    isospin factor for the ρ exchange is -2, namely plays an
    opposite role to the ω exchange. We try to solve the
    equation |A(Δ E,Λ)-I|=0 for some chosen Δ E
    and find a solution for Σ_c D̅ with the quantum number
    I(J)=1 2(1 2) where the factor λ can span
    a large range.
    
    
    
    
    The result indicates that although, the ω exchange
    contributes a repulsive interaction, for Σ_c D̅
    molecule, the total interaction can be attractive due to a larger
    contribution from the ρ exchange. Numerically, the obtained
    values of λ and corresponding Δ E for Σ_cD̅ system are presented in Table <ref>. Our numerical
    computation also confirms that the tensor coupling in the
    ℒ_ℬℬ𝒱 has little effect on the results.
    For example setting Δ E=8 MeV one can fix λ=3.77
    and 3.88  with and without the tensor contribution the obtained
    wave functions are very close to each other so we can safely
    ignore the tensor coupling in the vertex
    ℒ_ℬℬ𝒱. Apparently when Δ E is very
    small the obtained λ is smaller than 4, so Σ_c and
    D̅ should form a weak bound state. At present  the
    pentaquark P_c(4312) has been experimentally observed in
    Λ_b→ J/ψ pK portal, which is peaked at the invariant
    mass spectrum of J/ψ p and has the invariant mass of about
    4312 MeV. Apparently its isospin is 1/2, and the
    majority of authors
    <cit.>
    regarded this pentaquark as a bound state of Σ_c and D̅ and we agree with it.
    
    
    
    Using the normalized wave functions the transition
    P_c(1/2,1/2)→ J/ψ+p is calculable. The
    form factors defined in Eq. (<ref>) with the coupling constants
    are evaluated: g_ℬℬ'D=2.7,
    g_ℬℬ'D^*=3.0, g_DDψ=7.4,
    g_DD^*ψ=2.5 GeV^-1<cit.>. We obtain
    g_1=0.396 GeV, g_2=0.270, g_3=0.00632 GeV^-1 and the
    decay width Γ[P_c(1/2,1/2)→ J/ψ
    p]=3.66 MeV. If the binding energy is 20 MeV, g_1=0.412 GeV,
    g_2=0.282, g_3=0.00923 GeV^-1 can be obtained and the
    estimated decay width is Γ[P_c(1/2,1/2)→ J/ψ
    p]=3.90 MeV.  We notice that our results are close to that of
    Ref.<cit.>, but the results given in
    Ref.<cit.> are 1-3 orders smaller than ours where
    different ultraviolet regulators are employed.
    
    
    By our observation given above, for the state with I=3/2
    the isospin factor is 1 for  exchanging
    either  ω or
    ρ, therefore the total interaction is repulsive, it means that
    Σ_c and D̅ cannot form a bound state
    with I=3/2.
    
    
    
     §.§ predictions about pentaquark P_b
    
    The isospin of the Λ_b B^+ system is
    
        |1/2,1/2⟩=|Λ_b B^+⟩.
    
    
    The isospin of the Σ_b B system can be
    |1/2,±1/2⟩|3/2,±1/2⟩ and |3/2,±3/2⟩. Let us work on the isospin states
    
        |1/2,1/2⟩=√(2/3)|Σ_b^+
        B^0⟩-√(1/3)|Σ_b^0 B^+⟩,
    
        |3/2,1/2⟩=√(1/3)|Σ_b^+
        B^0⟩+√(2/3)|Σ_b^0 B^+⟩,
    
    and
    
        |3/2,3/2⟩=|Σ_b^+ B^+⟩.
    
    
    
    
    Using the masses of Λ_b, Σ_b and B presented in
    Ref.<cit.> and other parameters listed in previous sections,
    we solve those B-S equations. It is found that only the equation
    for the Σ_b  B system with I=1/2 has a solution.
    The binding energy and corresponding λ values are
    presented in Table <ref>. That implies that the bound
    state with I=1/2 can exist in the nature. Under the
    heavy quark symmetry we suppose that the couplings are unchanged
    when b-hadrons replace c-hadrons. We turn to study the
    transition P_b(1/2,1/2)→Υ p. We
    obtain g_1=0.00346 GeV, g_2=0.252, g_3=0.0000911 GeV^-1
    and predict the decay width
    Γ[P_b(1/2,1/2)→Υ p]=0.690 keV
    as the binding energy is 10 MeV. If Δ E=20 MeV the decay
    width Γ[P_b(1/2,1/2)→Υ p]=1.09
    keV and g_1=0.00435 GeV, g_2=0.318, g_3=0.000149 GeV^-1.
    
    
    
    § CONCLUSION AND DISCUSSION
    
    
    Within the B-S framework we explore several bound states which are
    composed of a baryon and a meson. Their total spin and parity is
    1/2^- i.e. the orbital angular momentum L=0
    (S-wave). We try to solve  the B-S equation for getting possible
    spatial wave functions for Λ_c D̅, Σ_c D̅,
    Λ_b  B and Σ_b  B systems. If the B-S equation for
    a supposed molecular structure has a stable solution, we would
    conclude that the concerned pentaquark could exist in the nature,
    oppositely, no-solution means the supposed pentaquark cannot
    appear as a resonance or the molecular state is not an appropriate
    structure. The solution can apply as a criterion for the
    structures of the pentaquark states which have already been or
    will be experimentally observed. In this scenario, the two
    constituents interact by exchanging light vector mesons. For the
    Λ_cD̅ (Λ_b B) system only ω is the
    exchanged mediate meson, while for the Σ_cD̅ system
    (Σ_b B) both ω and ρ contribute. The chiral
    interaction determines if those molecular states can be formed.
    
    
    For  1/2^- baryon (S-wave), the B-S wave function
    possesses two scalar functions f_1(|𝐩_𝐓|) and
    f_2(|𝐩_𝐓|) which should be solved numerically.
    Discretizing the integral equations, we simplify the B-S equation
    into two coupled algebraic equations about f_1(|𝐩_𝐓|)
    and f_2(|𝐩_𝐓|).
    
    As |𝐩_𝐓|   (i=1,2) takes n discrete values the two
    coupled equations are converted into a matrix equation which can
    be easily solved numerically in terms of available softwares. When
    all known parameters are input there still is one undetermined
    parameter λ. Our strategy is inputting binding energies
    within a range and then fixing λ by solving the matrix
    equation. If λ is located in a reasonable range one can
    expect the bound state to exist. We find the B-S equation of  the
    state Λ_c D̅ system has no solution for λ when
    the binding energy takes experimentally allowed values. For the
    Σ_c D̅ system there are three isospin eigenstates. Due
    to the isospin factors, the B-S equations for
    P_c(1/2,1/2), P_c(3/2,1/2)
    and P_c(3/2,3/2) are set. We find the equation
    for |1/2,1/2⟩ has a solution for λ
    falling into the reasonable range. It means that P_c(4312)is
    maybe a molecular state of Σ_c D̅. The decay width of
    P_c(1/2,1/2)→ J/ψ p is calculated within
    this framework and we obtain it as about 3.66 MeV.
    
    It is noted, we ignore the couple
    channel effects in the Bethe-Salpeter. We also note that if the couple channel
    interaction between Λ_c D̅ and Σ_c D̅ is taken
    into account, just as
    the authors of Ref.<cit.> did, a bound state of
    Λ_c D̅ may exist via the coupled channel with
    Σ_c D̅ (I=1/2). In other words, there
    is a Λ_c D̅ component in the physical state of
    P_c(1/2,1/2).
    
    In this work, we
    study  Λ_b  B and Σ_b  B systems and
    solve the B-S equations for  Σ_b  B and
    Λ_b  B.  Our conclusion is that the bound state
    P_b(1/2,1/2) is still a promising pentaquark.
    The partial width P_b(1/2,1/2)→Υ p is about 1.06 keV at Δ E=20 MeV, which will be
    checked by the future experiments.
    
    
    
    Within the B-S framework, we systematically investigate the
    molecular structure of pentaquarks. We pay a special attention to
    P_c(4312) because it is experimentally well measured. From that
    study, we have accumulated valuable knowledge on probable
    molecular structure of pentaquarks which can be applied to the
    future research.  Definitely, the discovery of pentaquarks opens a
    window for understanding the quark model established by Gell-Mann
    and several other predecessors. Deeper study on their structure
    and concerned effective interaction which binds the ingredients to
    form a molecule would greatly enrich our theoretical asset. So we
    will continue to do research along the line.
    
    
    
    
    
    
    
    
    § ACKNOWLEDGEMENT
    
    This work is supported by the National Natural Science Foundation
    of China (NNSFC) under Contract No. 11375128, , 11675082, 11735010
    and 11975165.. We would like to
     thank professors Bing-Song Zou, Xiang Liu and Yu-Ming Wang, as well as Dr. Zhen-Yang Wang for their
     suggestions and useful discussions.
    
    
    
    
    
    
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    § THE EFFECTIVE INTERACTIONS
    
    
    The effective interactions can be found
    in<cit.>
        ℒ_𝒫𝒫ρ=ig_𝒫𝒫ρϕ_𝒫ρ^μ·τ∂_μϕ_𝒫+c.c.,
              ℒ_𝒫𝒫𝒱=ig_𝒫𝒫𝒱ϕ(x)_𝒫∂_μϕ(x)_𝒫ϕ(x)_𝒱^μ+c.c.,
        
              ℒ_ℬℬρ=-g_ℬℬρψ̅_ℬ(γ^μ-κ_ℬℬρ/2m_ℬσ^μν∂_ν)ρ^μ·τψ_ℬ,
              ℒ_ℬℬω=-g_ℬℬωψ̅_ℬ(γ^μ-κ_ℬℬω/2m_ℬσ^μν∂_ν)ω^μψ_ℬ,
        
              ℒ_𝒱𝒱𝒫=ig_𝒱𝒱𝒫ε_μ,ν,α,β∂^μϕ_𝒱(x)^ν∂_αϕ_𝒱(x)^βϕ_𝒫(x)+c.c.,
        
              ℒ_ℬℬ𝒫=ig_ℬℬ𝒫ψ̅_ℬγ^5ψ_ℬ,
    
    where c.c. is the complex conjugate term, τ is pauli matrix for I=1/2 and τ_x=
    1/√(2)([ 0 1 0; 1 0 1; 0 1 0 ]),
    τ_y=1/√(2)([  0 -i  0;  i  0 -i;  0  i  0 ]) and τ_z=1/√(2)([  1  0  0;  0  0  0;  0  0 -1 ]) for I=1. when
    ϕ_𝒫=([ D^0; D^+ ]) and ψ_ℬ=([ Σ_c^++;  Σ_c^+;  Σ_c^0 ]) the effective
    interactions are consistent with those in Ref<cit.>.
    
    
    
    § THE ISOSPIN FACTORS IN THE KERNEL
    
    
    To gain the characteristic hadronic property of the pentaquark, one needs to project the bound states on the vacuum via the field operators
    ℬ_1, ℬ_2, ℳ_1 and
    ℳ_2 and
    
        ⟨0|Tℬ_i(x_1)ℳ_j(x_2)|P⟩_I,I_3=C^ij_(I,I_3)χ^I_P(x_1,x_2)
        ,
     where χ^I_P(x_1,x_2) is the B-S wave function
    for the bound state with isospin I. The isospin coefficients
    C^22_(1/2,1/2) for Λ_cD bound state
    is 1, the isospin coefficients for Σ_cD bound states are
    
        C^11_(1/2,1/2)=√(2/3),  C^22_(1/2,1/2)=-√(1/3)
        ,  C^11_(3/2,1/2)=√(2/3),  C^22_(3/2,1/2)=√(1/3)
        ,  C^12_(3/2,3/2)=1.
    
    
    
    Then corresponding B-S equation was deduced in
    Ref.<cit.> as
    
        C^ij_(I,I_3)χ^I_P( p)
        =S_ℬ(p_1)∫d^4q(2π)^3 K^ij,lk(P,p,q)C^lk_(I,I_3)χ^I__P(q)S_ℳ(p_2) ,
    
    where K^ij,lk(P,p,q) is still the kernel and its superscripts
    ij and lk denote the initial and final  components.
    
    For Λ_cD̅
        χ_P( p)
        =S_ℬ(p_1)∫d^4q(2π)^3 K^22,22χ__P(q)S_ℳ(p_2) ,
    
    For I=1/2 ,I_z=1/2 state Σ_cD̅  if
    the components are Σ_c^++D^-
        χ_P( p)
        =S_ℬ(p_1)∫d^4q(2π)^3 (-K^11,11-1/√(2)K^11,22)χ__P(q)S_ℳ(p_2) ,
    
    if the components are Σ_c^+D̅^0
        χ_P( p)
        =S_ℬ(p_1)∫d^4q(2π)^3 (K^22,22-√(2)K^22,11)χ__P(q)S_ℳ(p_2) ,
    
    so 
        χ_P( p)
        =S_ℬ(p_1)∫d^4q(2π)^3 (-2/3K^11,11-√(2)/3K^11,22+
        1/3K^22,22-√(2)/3K^22,11)χ__P(q)S_ℳ(p_2) ,
    
    
    For I=3/2 ,I_z=1/2Σ_cD̅ state if
    the components are Σ_c^++D^-
        χ_P( p)
        =S_ℬ(p_1)∫d^4q(2π)^3 (-K^11,11+√(2)K^11,22)χ__P(q)S_ℳ(p_2) ,
    
    if the components are Σ_c^+D̅^0
        χ_P( p)
        =S_ℬ(p_1)∫d^4q(2π)^3 (K^22,22+1/√(2)K^22,11)χ__P(q)S_ℳ(p_2) ,
    
    so 
        χ_P( p)
        =S_ℬ(p_1)∫d^4q(2π)^3 (-1/3K^11,11+√(2)/3K^11,22+
        2/3K^22,22+√(2)/3K^22,11)χ__P(q)S_ℳ(p_2) .
    
    
    The sign “-" before K^11,11 in Eq. (<ref>) and
    (<ref>) comes from the interactions in Appendix A.  For
    Λ_cD̅ state the two components interact only by
    exchanging ω. However ω and ρ can contribute to
    the Σ_cD̅ state. One also has
    K^11,11(ω)=K^22,22(ω),
    K^11,22(ω)=K^22,11(ω)=0,
    K^11,22(ρ)=K^22,11(ρ)=√(2)K^11,11(ρ) and
    K^22,22(ρ)=0. In the Eqs. (<ref>) and
    (<ref>) K^11,11, K^11,22 and K^22,11 can be
    changed into K^11,11 and then the coefficient of K^11,11
    is just the isospin factor C_I,I_z:C_1/2,1/2=1,-2 for ω and ρ,
    C_3/2,1/2=C_3/2,3/2=1,1 for
    ω and ρ.
    
    
    
    
    § THE COUPLED EQUATION OF F_1(|𝐩_T|) AND
    F_2(|𝐩_T|) AFTER INTEGRATING OVER P_L AND SOME FORMULAS FOR AZIMUTHAL INTEGRATION
    
        f_1(|𝐩_T|)=-∫d^3𝐪_T/(2π)^3C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱
        F^2(k,m_V)/2ω_1(M+ω_1+ω_2)(M+ω_1-ω_2)
              {𝐩_T·𝐪_T + 𝐩_T^2 + 2( m_1 - ω_1 ) (
        M+ω_1
         ) +(𝐩_T^2 -𝐩_T·𝐪_T  )
        ( 𝐩_T^2 -𝐪_T^2) 
         / m_V^2/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)+
              ( 𝐩_T^2 -𝐪_T^2)
           (𝐩_T·𝐪_T - 𝐪_T^2 )
            ( m_1 - ω_l ) /m_V^2
          + (m_1-ω_1)(
        𝐩_T·𝐪_T + 𝐪_T^2 )  +
        2(M+ω_1)𝐩_T·𝐪_T/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)
              -κ/4m_B-4[
        𝐩_T·𝐪_T^2 -
        𝐩_T^2𝐪_T^2 +
        (m_1-ω_1)(𝐩_T·𝐪_T- 𝐪_T^2
        )(ω_1+M)
            ]/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)
              -κ/4m_B4(
        𝐩_T^2 -𝐩_T·𝐪_T) ( M+ω_1
        )/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)}+
              ∫d^3𝐪_T/(2π)^3C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱
        F^2(k,m_V)/2ω_2(M+ω_1-ω_2)(M-ω_1-ω_2)
              {𝐩_T·𝐪_T + 𝐩_T^2 + 2( m_1- ω_2 +
        M)ω_2 +(𝐩_T^2 -𝐩_T·𝐪_T )
        ( 𝐩_T^2 -𝐪_T^2) 
         / m_V^2/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)+
              ( 𝐩_T^2 -𝐪_T^2)
           (𝐩_T·𝐪_T - 𝐪_T^2 )
            ( m_1 - ω_2 + M ) /m_V^2
          + (m_1+M-ω_2)(
        𝐩_T·𝐪_T + 𝐪_T^2 )  +
        2ω_2𝐩_T·𝐪_T/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)
              -κ/4m_B-4[
        𝐩_T·𝐪_T^2 -
        𝐩_T^2𝐪_T^2 + (m_1-ω_2+M)(
        𝐪_T^2 -𝐩_T·𝐪_T )(-ω_2)
            ]/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)
              -κ/4m_B4(
        𝐩_T·𝐪_T - 𝐩_T^2 )
        (-ω_2)/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)}.
    
        f_2(|𝐩_T|)𝐩_T^2=-∫d^3𝐪_T/(2π)^3-C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱
        F^2(k,m_V)/2ω_1(M+ω_1+ω_2)(M+ω_1-ω_2)
              {-𝐩_T^2(𝐩_T·𝐪_T + 𝐪_T^2) + 2𝐩_T·𝐪_T( m_1+ω_1) ( M+ω_1
         ) +𝐩_T^2(𝐪_T^2 -𝐩_T·𝐪_T  )
        ( 𝐩_T^2 -𝐪_T^2) /m_V^2/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)+
              (
           𝐩_T^2 -𝐪_T^2) (𝐩_T·𝐪_T - 𝐩_T^2 )
            ( -m_1- ω_1  ) /m_V^2
          + (m_1-ω_1)(
        𝐩_T·𝐪_T + 𝐩_T^2 )  -
        2M𝐩_T^2+2ω_1𝐩_T·𝐪_T/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)
              -κ/4m_B4[(
        M +ω_1 ) 𝐩_T·𝐪_T𝐩_T^2
        -
         𝐩_T·𝐪_T^2( m_1 + ω_1 )  +
          𝐩_T^2𝐪_T^2( m_1 - M
             )]/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)
              -κ/4m_B4(
        𝐩_T·𝐪_T - 𝐩_T^2 ) (-ω_1 -
        M
        )(m_1+ω_1)/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)}
              +∫d^3𝐪_T/(2π)^3-C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱
        F^2(k,m_V)/2ω_2(M+ω_1-ω_2)(M-ω_1-ω_2)
              {-𝐩_T^2(𝐩_T·𝐪_T + 𝐪_T^2) + 2𝐩_T·𝐪_T( m_1+ω_2 - M)  ω_2
          +𝐩_T^2(𝐪_T^2 -𝐩_T·𝐪_T  )
        ( 𝐩_T^2 -𝐪_T^2) /m_V^2/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)+
              ( 𝐩_T^2 -𝐪_T^2) (𝐩_T·𝐪_T - 𝐩_T^2 )
            (  M-m_1 - ω_2  ) /m_V^2
          + (m_1-M-ω_2)(
        𝐩_T·𝐪_T + 𝐩_T^2 )
        +2ω_2𝐩_T·𝐪_T/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)
              -κ/4m_B4[
        ω_2 𝐩_T·𝐪_T𝐩_T^2 -
         𝐩_T·𝐪_T^2( m_1 + ω_2- M )  +
          𝐩_T^2𝐪_T^2( m_1 - M
             )]/-(𝐩_T-𝐪_T)^2-m_V^2f_2(|𝐪_T|)
              -κ/4m_B4(
        𝐩_T·𝐪_T - 𝐩_T^2 ) ω_2
        (M-m_1-ω_2)/-(𝐩_T-𝐪_T)^2-m_V^2f_1(|𝐪_T|)}.
    
    
    Since d^3𝐪_T=𝐪_T^2
    sin(θ)d|𝐪_T|dθ dϕ and 𝐩_T·𝐪_T=|𝐩_T||𝐪_T| cos(θ) one
    can carry out the azimuthal integration for Eqs. (<ref>) and (<ref>) analytically. Some
    useful integrations are defined as follow
    
    
        J_0≡∫_0^π sin(θ)dθ1/-(𝐩_T-𝐪_T)^2-m_V^2[Λ^2-m^2_V/Λ^2-(𝐩_T-𝐪_T)^2]^2
        
              =∫_0^π
        sin(θ)dθ/-[𝐩_T^2+𝐪_T^2-2|𝐩_T||𝐪_T|
        cos(θ)]-m_V^2{Λ^2-m^2_V/Λ^2-[𝐩_T^2+𝐪_T^2-2|𝐩_T||𝐪_T|
        cos(θ)]}^2
              =-2(m_V^2-Λ^2)/[(|𝐩_T|-|𝐪_T|)^2+Λ^2][(|𝐩_T|+|𝐪_T|)^2
        +Λ^2]
              +1/2|𝐩_T||𝐪_T|{
        Ln[(|𝐩_T|+|𝐪_T|)^2+Λ^2/(|𝐩_T|-|𝐪_T|)^2+Λ^2]-
        Ln[(|𝐩_T|+
        |𝐪_T|)^2+m_V^2/(|𝐩_T|-|𝐪_T|)^2+m_V^2]},
    
        J_1≡∫_0^π
        sin(θ)dθ𝐩_T·𝐪_T/-(𝐩_T-𝐪_T)^2-m_V^2[Λ^2-m^2_V/Λ^2-(𝐩_T-𝐪_T)^2]^2
        
              =∫_0^π|𝐩_T||𝐪_T|
        cos(θ)
        sin(θ)dθ/-[𝐩_T^2+𝐪_T^2-2|𝐩_T||𝐪_T|
        cos(θ)]-m_V^2{Λ^2-m^2_V/Λ^2-[𝐩_T^2+𝐪_T^2-2|𝐩_T||𝐪_T|
        cos(θ)]}^2
              =-(m_V^2-Λ^2)(|𝐩_T|^2+|𝐪_T|^2
        +Λ^2)/[(|𝐩_T|-|𝐪_T|)^2+Λ^2][(|𝐩_T|+|𝐪_T|)^2
        +Λ^2]
              +(|𝐩_T|^2+|𝐪_T|^2
        +m_V^2)/4|𝐩_T||𝐪_T|{Ln[(|𝐩_T|+|𝐪_T|)^2+Λ^2/(|𝐩_T|-|𝐪_T|)^2+Λ^2]-Ln[(|𝐩_T|+
        |𝐪_T|)^2+m_V^2/(|𝐩_T|-|𝐪_T|)^2+m_V^2]},
    
        J_2≡∫_0^π
        sin(θ)dθ(𝐩_T·𝐪_T)^2/-(𝐩_T-𝐪_T)^2-m_V^2[Λ^2-m^2_V/Λ^2-(𝐩_T-𝐪_T)^2]^2
        
              =∫_0^π|𝐩_T|^2|𝐪_T|^2 cos^2(θ)
        sin(θ)dθ/-[𝐩_T^2+𝐪_T^2-2|𝐩_T||𝐪_T|
        cos(θ)]-m_V^2{Λ^2-m^2_V/Λ^2-[𝐩_T^2+𝐪_T^2-2|𝐩_T||𝐪_T|
        cos(θ)]}^2
              =-(m_V^2-Λ^2)(|𝐩_T|^2+|𝐪_T|^2
        +Λ^2)^2/2[(|𝐩_T|-|𝐪_T|)^2+Λ^2][(|𝐩_T|+|𝐪_T|)^2
        +Λ^2]
              +1/8|𝐩_T||𝐪_T|{(|𝐩_T|^2+|𝐪_T|^2
        +2m_V^2-Λ^2)(|𝐩_T|^2+|𝐪_T|^2
        +Λ^2)
        Ln[(|𝐩_T|+|𝐪_T|)^2+Λ^2/(|𝐩_T|-|𝐪_T|)^2+Λ^2]
              -(|𝐩_T|^2+|𝐪_T|^2
        +m_V^2)^2 Ln[(|𝐩_T|+
        |𝐪_T|)^2+m_V^2/(|𝐩_T|-|𝐪_T|)^2+m_V^2]}.
    
        A_11(𝐩_𝐓,𝐪_𝐓)=-𝐪_T^2/(2π)^2C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱/2ω_1(M+ω_1+ω_2)(M+ω_1-ω_2)
              {[J_1
        + 𝐩_T^2J_0 + 2( m_1 - ω_1 ) ( M+ω_1
         )J_0 +(𝐩_T^2J_0 -J_1  )
        ( 𝐩_T^2 -𝐪_T^2) 
         / m_V^2]f_1(|𝐪_T|)
              -κ/4m_B[4(
        𝐩_T^2J_0 -J_1) ( Mη_2+ω_1
        )]f_1(|𝐪_T|)}+
              𝐪_T^2/(2π)^2C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱/2ω_2(M+ω_1-ω_2)(M-ω_1-ω_2)
              {[J_1
        + 𝐩_T^2J_0 + 2( m_1- ω_2 + M)ω_2J_0
        +(𝐩_T^2J_0 -J_1 ) ( 𝐩_T^2
        -𝐪_T^2) 
         / m_V^2]f_1(|𝐪_T|)
              -κ/4m_B[4(
        J_1 - 𝐩_T^2J_0 ) (-ω_2)]f_1(|𝐪_T|)}.
    
        A_12(𝐩_𝐓,𝐪_𝐓)=-𝐪_T^2/(2π)^2C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱/2ω_1(M+ω_1+ω_2)(M+ω_1-ω_2)
              {[(
        𝐩_T^2 -𝐪_T^2)
           (J_1 - 𝐪_T^2J_0 )
            ( m_1 - ω_l ) /m_V^2
          + (m_1-ω_1)(
        J_1 + 𝐪_T^2J_0 )  +
        2(M+ω_1)J_1]f_2(|𝐪_T|)
              +κ/m_B[
        J_2 - 𝐩_T^2𝐪_T^2J_0 + (m_1-ω_1)(J_1-
        𝐪_T^2J_0 )(ω_1+M)
            ]f_2(|𝐪_T|)}+
              𝐪_T^2/(2π)^2C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱/2ω_2(M+ω_1-ω_2)(M-ω_1-ω_2)
              {[(
        𝐩_T^2 -𝐪_T^2)
           (J_1 - 𝐪_T^2J_0 )
            ( m_1 - ω_2 + M ) /m_V^2
          + (m_1+M-ω_2)(
        J_1 + 𝐪_T^2J_0 )  +
        2ω_2J_1]f_2(|𝐪_T|)
              +κ/m_B[
        J_2 - 𝐩_T^2𝐪_T^2J_0 + (m_1-ω_2+M)(
        𝐪_T^2J_0 -J_1 )(-ω_2)
            ]f_2(|𝐪_T|)}.
    
        A_21(𝐩_𝐓,𝐪_𝐓)𝐩_T^2=𝐪_T^2/(2π)^2C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱/2ω_1(M+ω_1+ω_2)(M+ω_1-ω_2)
              {[(
           𝐩_T^2 -𝐪_T^2) (J_1 - 𝐩_T^2J_0 )
            ( -m_1- ω_1  ) /m_V^2
          + (m_1-ω_1)(
        J_1 + 𝐩_T^2J_0 )  -
        2M𝐩_T^2J_0+2ω_1J_1]f_1(|𝐪_T|)
              -κ/4m_B[4(
        J_1 - 𝐩_T^2J_0 ) (-ω_1 - M
        )(m_1+ω_1)/-(𝐩_T-𝐪_T)^2-m_V^2]f_1(|𝐪_T|)}
              +𝐪_T^2/(2π)^2C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱/2ω_2(M+ω_1-ω_2)(M-ω_1-ω_2)
              {[(
        𝐩_T^2 -𝐪_T^2) (J_1 - 𝐩_T^2J_0 )
            (  M-m_1 - ω_2  ) /m_V^2
          + (m_1-M-ω_2)(
        J_1 + 𝐩_T^2J_0 )
        +2ω_2J_1]f_1(|𝐪_T|)
              -κ/4m_B[4(
        J_1 - 𝐩_T^2J_0 ) ω_2
        (M-m_1-ω_2)]f_1(|𝐪_T|)}.
    
        A_22(𝐩_𝐓,𝐪_𝐓)𝐩_T^2=𝐪_T^2/(2π)^2C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱/2ω_1(M+ω_1+ω_2)(M+ω_1-ω_2)
              {[-𝐩_T^2(J_1
        + 𝐪_T^2J_0) + 2J_1( m_1+ω_1) ( M+ω_1
         ) +𝐩_T^2(𝐪_T^2J_0 -J_1  )
        ( 𝐩_T^2 -𝐪_T^2) /m_V^2]f_2(|𝐪_T|)
              -κ/4m_B4[(
        M +ω_1 )𝐩_T^2 J_1 - ( m_1 + ω_1 )J_2
        +
          𝐩_T^2𝐪_T^2( m_1 - M
             )J_0]f_2(|𝐪_T|)}
              +𝐪_T^2/(2π)^2C_I,I_zg_ℳℳ𝒱g_ℬℬ𝒱/2ω_2(M+ω_1-ω_2)(M-ω_1-ω_2)
              {[-𝐩_T^2(J_1
        + 𝐪_T^2J_0) + 2( m_1+ω_2 - M)  ω_2J_1
          +𝐩_T^2(𝐪_T^2J_0 -J_1  )
        ( 𝐩_T^2 -𝐪_T^2) /m_V^2]f_2(|𝐪_T|)
              -κ/4m_B4[
        ω_2 𝐩_T^2J_1 - ( m_1 + ω_2- M )J_2  +
          𝐩_T^2𝐪_T^2( m_1 - M
             )J_0]f_2(|𝐪_T|)}.
