Calculation of mass and width of unstable molecular state using the developed Bethe-Salpeter theory

Applying the developed Bethe-Salpeter theory for dealing with resonance, we investigate the time evolution of molecular state composed of two vector mesons as determined by the total Hamiltonian. Then exotic meson resonance $\chi_{c0}(3915)$ is considered as a mixed state of two unstable molecular states $D^{*0}\bar{D}^{*0}$ and $D^{*+}D^{*-}$, and the mass and width for physical resonance $\chi_{c0}(3915)$ are calculated in the framework of relativistic quantum field theory. In this actual calculation, we minutely show how to obtain the correction for energy level of resonance and to exhibit the key features of dispersion relation in an extended Feynman diagram. The numerical results are consistent with the experimental values.


I. INTRODUCTION
Hadronic molecule structure has been proposed to interpret the internal structure of exotic meson resonance for many years [1,2].In the previous works [1][2][3][4][5], molecular states were considered as meson-meson bound states and homogeneous Bethe-Salpeter (BS) equation was frequently used to investigate molecular states.Solving homogeneous BS equations for meson-meson bound states, the authors of these works obtained the masses and BS wave functions.The mass of meson-meson bound state was regarded as mass of exotic meson resonance.However, all decay channels of resonance should contribute to its physical mass and the correction for energy level of molecular state due to decay channels has seldom been considered [1][2][3][4][5][6][7].Fortunately, recent fundamental research [8] has noticed that hadron resonance should be regarded as an unstable two-body system, and developed BS theory for dealing with the dynamics of coupled channels in the framework of relativistic quantum field theory.Though Ref. [8] illuminated the physical meaning of the developed Bethe-Salpeter theory for dealing with resonance, many details in computational process have not been presented.In this paper, we will comprehensively and systematically show the theoretical approach about unstable molecular state composed of two heavy vector mesons, and this approach is applied to investigate exotic meson resonance χ c0 (3915) [9], once named X(3915), which is considered as a mixed state of two unstable molecular states D * 0 D * 0 and D * + D * − .Since resonance is an unstable state which decays spontaneously into other particles, the molecular state composed of two heavy vector mesons should not be a stationary vectorvector bound state.To investigate this unstable two-body system, we suppose that at some given time this unstable state has been prepared to decay and then study the time evolution of this system as determined by the total Hamiltonian.This prepared state can be described by the ground-state BS wave function for vector-vector bound state at the times t 1 = 0 and t 2 = 0.In our previous works [4,7], the most general form of BS wave functions for the bound states created by two vector fields with arbitrary spin and definite parity has been given.
According to the effective theory at low energy QCD, we have investigated the light meson interaction with quarks in heavy vector mesons and obtained the interaction kernel between two heavy vector mesons derived from one light meson (σ, ω, ρ, φ) exchange [4,10].Solving BS equation with this interaction kernel, we have obtained the mass and BS wave function for bound state composed of two vector mesons [4,11].After providing the description for the prepared state, we can study the time evolution of BS wave function and obtain the pole corresponding to resonance through the scattering matrix element.
The crucial point of our resonance theory is that the scattering matrix element between bound states is calculated in the framework of relativistic quantum field theory.According to dispersion relation, the total matrix element between a final state and an initial bound state should be calculated with respect to arbitrary value of the final state energy [8].It is necessary to note that the total energy of the final state extends over the real interval while the initial state energy is specified.For the initial bound state composed of two heavy vector mesons, we have given the generalized Bethe-Salpeter (GBS) amplitude for four-quark state describing this meson-meson structure [6,7], which should be specified.Because the value of the final state energy is an arbitrary real number over the real interval, we may obtain several closed channels derived from the interaction Lagrangian and all open and closed channels should contribute to the mass of physical resonance.For exotic resonance χ c0 (3915), we consider three open decay channels J/ψω, D + D − , D 0 D0 and one closed channel D * D * from the effective interaction Lagrangian at low energy QCD.Mandelstam's approach is applied to calculate the matrix element between bound states with respect to arbitrary value of the final state energy, which are exhibited by extended Feynman diagrams.Finally, we obtain the correction for energy level of resonance χ c0 (3915) and the physical mass is used to calculate the decay width of physical resonance χ c0 (3915).
The structure of this article is as follows.In Sec.II we give the revised general form of GBS wave functions for meson-meson bound states as four-quark states.The mass and GBS wave function for the mixed state of two bound states D * 0 D * 0 and D * + D * − is obtained in instantaneous approximation.Section III gives the traditional technique to calculate the matrix element with mass of meson-meson bound state, which is applied to investigate the decay modes χ c0 (3915) → J/ψω, χ c0 (3915) → D + D − and χ c0 (3915) → D 0 D0 .Section IV gives the developed Bethe-Salpeter theory.In Sec.V we emphatically introduce the matrix element between bound states with respect to arbitrary value of the final state energy.Three open decay channels J/ψω, D + D − , D 0 D0 and one closed channel D * D * are considered.In Sec.VI we obtain the physical mass and width for unstable molecular state.Our numerical results are presented in Sec.VII and we make some concluding remarks in Sec.VIII.

QUARK STATE
In this paper, we investigate the light meson interaction with the light quarks in heavy mesons.As in effective theory at low energy QCD, the interaction Lagrangian for the coupling of light quark fields to light meson fields is [6] From this effective interaction Lagrangian at low energy QCD, we have to consider that the heavy meson is a bound state composed of a quark and an antiquark and investigate the interaction of light meson with quarks in heavy meson.The quark current J µ coupling with light vector meson and the quark scalar density J coupling with σ meson can be obtained.
In this section, our attention is only focused on the bound state composed of two vector mesons and some errors in previous works are revised.

A. BS wave function for bound state composed of two vector mesons
If a bound state with spin j and parity η P is created by two Heisenberg vector fields with masses M 1 and M 2 , respectively, its BS wave function is defined as where P is the momentum of the bound state, . Making the Fourier transformation, we obtain the BS wave function in the momentum representation where p is the relative momentum of two vector fields and we have 1 and p ′ 2 are the momenta carried by two vector fields, respectively.The polarization tensor of the bound state η µ 1 µ 2 •••µ j can be separated, where the subscripts λ and τ are derived from these two vector fields.The polarization tensor η µ 1 µ 2 •••µ j describes the spin of the bound state, which is totally symmetric, transverse and traceless: From Lorentz covariance, we have where In fact, the relative momenta p µ 1 , • • • , p µ j , p λ , p τ represent the orbital angular momenta.There should be 20 20) in Eq. ( 6).In Ref. [7], three tensor structures are omitted.In this paper, these missing terms are added as the last three terms in Eq. (6).
Using the transversality condition [4,10] p ′ 1λ χ j λτ (P, p) = p ′ 2τ χ j λτ (P, p) = 0 (7) and considering the properties of BS wave function under space reflection, we obtain the revised general form of BS wave functions for the bound states created by two massive vector fields with arbitrary spin and definite parity (see details in [4]), for η P = (−1) j , for η P = (−1) j+1 , where N j is normalization, the independent tensor structures T i λτ are given in Appendix A, Φ i (P • p, p 2 ) and Φ ′ i (P • p, p 2 ) are independent scalar functions.Scalar functions f i in Eq. ( 6) are the linear combinations of Φ i and Φ ′ i .

B. Kernel between two heavy vector mesons
In this paper, we assume that the isoscalar χ c0 (3915) is a mixed state of two unstable where and P becomes the total momentum for the mixed state of two meson-meson bound states, 2 )⊗| 1 2 , − 1 2 are the isospin wave functions of pure bound states D * 0 D * 0 and D * + D * − , respectively; χ D * 0 D * 0 ,j λτ and χ D * + D * − ,j λτ represent BS wave functions for the bound states of two vector mesons, which are the eigenstates of Hamiltonian without considering the coupled-channel terms.These eigenstates have the same quantum numbers.The error in Ref. [7] has been revised.As usual the momentum for the mixed state of two bound states is set as P = (0, 0, 0, iM 0 ) in the rest frame.
P D D represents the momentum of pure bound state in the rest frame, whose fourth component is different from the one of P .This BS wave function should satisfy the equation where V θθ ′ ,κ ′ κ is the interaction kernel, ) 2 We emphasize that the kernel V is defined in two-body channel so V is not complete interaction.
The kernel in homogeneous BS equation ( 13) plays a central role for making two-body system to be a stable bound state, and the solution of homogeneous BS equation ( 13) should only describe bound state.
To construct the interaction kernel between D * l and D * l , we consider that the effective interaction is derived from one light meson (σ, ρ 0 , V 1 and V 8 ) exchange [4,10,11], shown as Fig. 1.The flavor-SU(3) singlet V 1 and octet V 8 states of vector mesons mix to form the physical ω and φ mesons as where the mixing angle θ = 38.58• was obtained by KLOE [12].Then the exchanged mesons should be the octet V 8 and singlet V 1 states, and the relation of the octet-quark coupling constant g 8 and the singlet-quark coupling constant g 1 has the form where the meson-quark coupling constants g 2 ω = 2.42/2 and g 2 φ = 13.0 were determined by QCD sum rules approach [13].
In Fig. 1, V  matrix elements of quark scalar density J and quark current J α can be expressed as where 2 is the momentum of the light meson and w = p ′ − p; h(w 2 ) and h(w 2 ) are scalar functions, the four-vector ε(p) is the polarization vector of heavy vector meson with momentum p Taking away the external lines including normalizations and , we obtain the interaction kernel from one light meson (σ, ρ 0 , V 1 and V 8 ) exchange [4,10] where g σ = B(Mσ ) fσ = 299 60 [14,15], g 2 ρ = 2.42 [13], and these terms containing M 1,2 are neglected because the masses of heavy mesons are large.Using the method above, we can obtain the interaction kernels from one-ρ ± exchange [11].

The extended Bethe-Salpeter equation
Substituting BS wave function given by Eq. ( 12) and the kernel (17) into BS equation (13), we find that the integral of one term on the right-hand side of (12) has contribution to the one of itself and the other term.Ignoring the cross terms, one can obtain two individual equations: where ). Comparing the tensor structures in both sides of Eqs. ( 18) and (19), respectively, we obtain where V 0 + 1 (p, p ′ ; P D D) and V 0 + 2 (p, p ′ ; P D D) are derived from the interaction kernel between D * l and D * l .In instantaneous approximation, we set the momentum of exchanged meson as w = (w, 0).Then Eqs. ( 20) and ( 21) become two relativistic Schrödinger-like equations (see details in Refs.[4,11]) where The potentials between D * l and D * l up to the second order of the p/M D * l expansion are Solving these two equations ( 22) and ( 23), respectively, one can obtain the eigenvalues and the corresponding eigenfunctions Ψ 0 + 1 (p) and Ψ 0 + 2 (p).From Ψ 0 + 1 and Ψ 0 + 2 , it is easy to obtain F 1 and F 2 , respectively.Because the cross terms are small, we can take the ground state BS wave function to be a linear combination of two eigenstates F 10 λτ (P D D • p, p 2 ) and F 20 λτ (P D D • p, p 2 ) corresponding to lowest energy in Eqs.(18) and (19).Then in the basis provided by Substituting ( 26) into BS equation ( 13) and comparing the tensor structures in both sides, we obtain an eigenvalue equation in instantaneous approximation where we have the matrix elements  22) and ( 23), respectively; Ψ 0 + 10 and Ψ 0 + 20 are the corresponding eigenfunctions.From this equation, we can obtain the eigenvalues and eigenfunctions which contain the contribution from the cross terms.Some errors in our previous works have been revised.

Form factors of heavy meson
To calculate these heavy vector meson form factors h(w 2 ) describing the heavy meson structure, we have to know the wave function of heavy vector meson D * l in instantaneous approximation.For heavy vector mesons, the authors of Refs.[16][17][18][19] have obtained their BS amplitudes in Euclidean space: where K is the momentum of heavy meson, k denotes the relative momentum between quark and antiquark in heavy meson, M V is heavy vector meson mass, Γ ) is scalar function fixed by providing fits to observables.The charmed meson D * l is composed of c-quark and l-antiquark.As in heavy-quark effective theory (HQET) [20], we consider that the heaviest quark carries all the heavy-meson momentum and obtain BS wave function of D * l meson where K is set as the momentum of heavy meson in the rest frame, k becomes the relative momentum between c-quark and l-antiquark, m c,l are the constituent quark masses, and ω D * =1.50 GeV [19].The components of this BS wave function are 4 × 4 matrices, which can be written as [21] and the coefficient corresponding to γ µ is Substituting Eq. ( 30) into (32), we can obtain the heavy vector meson wave function in instantaneous approximation In the previous works [4,10,11], we have obtained the form factors for the vertices of heavy vector meson D * l coupling to scalar meson (σ) and to vector meson (ρ, V 1 and V 8 ) where E c,l (p) = p 2 + m 2 c,l and Ψ D * l is the wave function of heavy vector meson expressed as Eq.(33).In this paper, some errors in our previous works have been revised.Equations ( 22) and ( 23) can be solved numerically with these form factors, and then the eigenvalue equation ( 27) can be solved.The masses M D D and wave functions of pure bound states D * 0 D * 0 and D * + D * − with spin-parity quantum numbers 0 + can be obtained.
Considering the interaction kernels from one-ρ ± exchange and using the coupled-channel approach (see details in Ref. [11]), we can calculate the mass M 0 of the mixed state of two pure bound states D * 0 D * 0 and D * + D * − with 0 + .Since the mixing of component wave functions causes the change of energy, the fourth component of P D D in the original BS wave function becomes the total energy of mixed state, and χ 0 + λτ (P D D, p) in Eq. ( 26) becomes We emphasize that the mass M 0 of meson-meson bound state should not be the mass of physical resonance.Substituting Eq. ( 36) into (10), we obtain BS wave function χ D * D * ,0 + λτ (P, p) for the mixed state of two bound states D * 0 D * 0 and D * + D * − with 0 + .

D. GBS wave function for four-quark state
The heavy meson is a bound state consisting of a quark and an antiquark and the mesonmeson bound state is actually composed of four quarks.We have to give GBS wave function of meson-meson bound state as a four-quark state.If a bound state with spin j and parity η P is composed of four quarks, its GBS wave function can be defined as [6] where P is the momentum of the four-quark bound state, Q is the quark operator and its superscript is a flavor label.From translational invariance, this GBS wave function can be written as where ) and m A,B,C,D are the quark masses.In the momentum representation, GBS wave function of four-quark bound state becomes where p 1 , p 3 , p 4 , p 2 are the momenta carried by the fields Q C , QA , Q B , QD ; p, k, k ′ are the conjugate variables to X ′ , x, x ′ , respectively; and p = η 2 (p In the hadronic molecule structure, p is the relative momentum between two mesons in molecular state, k and k ′ are the relative momenta between quark and antiquark in these two mesons, respectively, shown as Fig. 2.This work is aimed to investigate the bound state composed of two vector mesons.In Fig. 2, V M represents the vector meson with mass M 1 , V M ′ represents the anti-particle of vector meson V M ′ with mass M 2 , and  In Fig. 2, there are three two-body systems: a meson-meson bound state and two quark-antiquark bound states.We define BS wave functions of these two-body systems as ), respectively.BS wave function for the bound state of two vector mesons has been given by Eq. ( 3) and BS wave functions of two vector mesons are where p ′ 1 and p ′ 2 are the momenta of two vector mesons, respectively, Applying the Feynman rules and comparing with Eq.
(39), we obtain the revised GBS wave function for four-quark state describing the bound state composed of two vector mesons with arbitrary spin and definite parity [6,7] From Eq. ( 30), we obtain BS wave functions of vector mesons In this section, we consider a mixed state of two bound states D * 0 D * 0 and D * + D * − with spin-parity quantum numbers 0 + .In Fig. 2, V M and V M ′ become D * l and D * l , respectively, and in Eq. (37) the flavor labels C = D and A = B represent c-quark and l-quark, respectively.From Eqs. ( 10), ( 42) and (43), we obtain the GBS wave function for meson-meson bound state as a four-quark state where E. Normalization of BS wave function

Heavy vector meson
Here, we determine normalizations N D * 0 and N D * + .The authors of Refs.[18,19] employed the ladder approximation to solve the BS equation for quark-antiquark state, and the reduced normalization condition for the BS wave function of D * l meson given by Eq. ( 30) is where S F (p) −1 is the inverse propagator for quark field and the factor 1/3 appears because of the sum of three transverse directions.

Molecular state
The reduced normalization condition for χ 0 + λτ (P, p) expressed as Eq. ( 36) is where ∆ F βα ′ (p) −1 is the inverse propagator for the vector field with mass m, ∆ F βα [6].After determining normalization N 0 + , we automatically obtain the normalized BS wave function for the mixed state of two components D * 0 D * 0 and D * + D * − given by Eq. (10).Immediately, the normalized GBS wave function for meson-meson bound state as a four-quark state expressed as Eq. ( 44) is obtained.

III. SCATTERING MATRIX ELEMENT FROM FOUR-QUARK STATE TO FI-NAL STATE
In experiments two strong decay modes of χ c0 (3915) have been observed: J/ψω and D + D − .The narrow state χ c0 (3915) was discovered in 2005 [22] by the Belle collaboration and for a long time a series of experiments [23][24][25][26] only observed one strong decay mode of χ c0 (3915): J/ψω denoted as c ′ 1 .In 2020 the LHCb Collaboration observed another decay channel D + D − [27] denoted as c ′ 2 .Though the neutral channel D 0 D0 still has not been observed, this neutral channel should exist for the isospin conservation, which is denoted as In this section, we present the traditional technique to calculate decay width for these processes and revise some errors in previous works [6,7].
A. Decay channel J/ψω with respect to mass of bound state Mandelstam's approach is a technique based on BS wave function for evaluating the general matrix element between bound states [28].Applying Mandelstam's approach, we have obtained the scattering matrix element from a four-quark state to a heavy meson plus a light meson [6] in the momentum representation, as shown in Fig. 3.In this work, we retain only the lowest order term of the two-particle irreducible Green's function.In Fig. 3, V M and V M ′ still represent D * l and D * l , respectively; HM represents J/ψ with momentum ).The momentum of the initial state is set as P = (0, 0, 0, iM 0 ) in the rest frame, and M 0 is the mass of the mixed state of two pure bound states D * 0 D * 0 and D * + D * − , which should not be the physical mass of resonance.It is necessary to emphasize that the momenta in the final state satisfy Q + Q ′ = P in this section.Here, we consider that in the final state the light vector meson ω is an elementary particle and the heavy vector meson J/ψ is a bound state of cc.From Eq. (43), we obtain the BS wave function of heavy vector meson J/ψ where ϕ J/ψ (q 2 ) = exp(−q 2 /ω 2 J/ψ ) and ω J/ψ =0.826 GeV was obtained from lattice QCD (see details in Ref. [6]).The reduced normalization condition for BS wave function of J/ψ meson expressed as Eq. ( 48) is where the factor 1/3 appears for the three transverse directions are summed.Normalization N J/ψ can be determined.These momenta in Fig. 3 become Using the Heisenberg picture, we obtain the total matrix element from the initial state where T (c ′ 1 ;b)a (M 0 ) is the T -matrix element with mass M 0 for channel c ′ 1 .According to Mandelstam's approach, we obtain where ε ̺=1,2,3 ν (Q) and ε ̺ ′ =1,2,3 µ (Q ′ ) are the polarization vectors of J/ψ and ω, respectively, Here χ 0 + λτ (P, p) is expressed as Eq. ( 36).In Eq. ( 53) the trace of the product of 8 γ-matrices contains 105 terms and the resulting expression has been given in Appendix B of Ref. [6].In our approach, the p integral is computed in instantaneous approximation.To calculate this , we have given a simple method in Ref. [6].It is obvious that the tensor where ) are scalar functions.The above expression is multiplied by these tensor structures respectively; and a set of equations is obtained where U ′ i are numbers.Subsequently, we numerically calculate U ′ i and solve this set of equations.The values of U i can be obtained.
Then we can obtain the decay width with mass of meson-meson bound state for channel where 0 ).To calculate the decay width Γ 1 (M 0 ), we use the transverse condition 0 and the completeness relation.It is necessary to emphasize that the decay width Γ 1 (M 0 ) expressed as Eq. ( 56) is not the decay width of physical resonance.To simplify the computational process, we use the vertex function for the exchanged light meson, heavy pseudoscalar and vector mesons; and then Fig. 4 can be reduced to Fig.

5.
From the Lorentz-structure, we obtain the matrix elements of quark scalar density J and quark current J α between heavy pseudoscalar and vector mesons × {h where where K is the momentum of heavy meson, k denotes the relative momentum between quark and antiquark in heavy meson, N P is normalization, and ϕ P (k 2 ) is scalar function fixed by providing fits to observables.Using the approach introduced in Sec.II C 2, we can obtain the heavy pseudoscalar meson wave function in instantaneous approximation where m c,d are the constituent quark masses, ) and ω D =1.50 GeV [19].Then we can apply the method given in Refs.[4,10,11] to obtain the explicit forms of the vertex function for heavy pseudoscalar meson D and vector meson D * l coupling to scalar meson (σ) and to vector meson (ρ, V 1 and V 8 ) where E d (p) = p 2 + m 2 d , Ψ D + and Ψ D * l are the wave functions of heavy pseudoscalar and vector mesons expressed as Eqs.( 59) and (33), respectively.

Taking away the external lines including normalizations and polarization vectors ε
2 ) in Eq. ( 57), we obtain the interaction from one light meson (σ, ρ 0 , V 1 and V 8 ) VM PM exchange where E 1 = E 2 = M 0 /2, and w = (w, 0).The interaction from one-ρ ± exchange becomes These momenta in Fig. 5 become where P = (0, 0, 0, iM 0 ), For decay channel D + D − , we obtain the total matrix element where T (c ′ 2 ;b)a (M 0 ) is the T -matrix element with mass M 0 for channel c ′ 2 .From Fig. 5, we obtain where Here χ 0 + λτ (P, p) is expressed as Eq.(36).The p integral is also computed in instantaneous approximation.Then the decay width with mass of meson-meson bound state for channel The decay width Γ 2 (M 0 ) also is not the width of physical resonance.
C. Decay channel D 0 D0 with respect to mass of bound state Since the χ c0 (3915) state is a isoscalar, there should exist the neutral channel D 0 D0 .
In Figs. 4, 5 and 6, P M and P M ′ represent pseudoscalar mesons D 0 and D0 , respectively.
Following the same procedure as for charged channel D + D − , we can obtain the T -matrix element T (c ′ 3 ;b)a (M 0 ) and the decay width Γ 3 (M 0 ) with mass M 0 for neutral channel c ′ 3 .The decay width Γ 3 (M 0 ) should not be the width of physical resonance.

IV. THE DEVELOPED BETHE-SALPETER THEORY
Secs.II and III give the traditional technique to deal with molecular state in present particle physics.These masses of meson-meson bound states were regarded as masses of resonances [1][2][3][4][5] and used to calculate decay widths of resonances [6,7], which should not be impeccable.To deal with resonance in the framework of relativistic quantum field theory, we considered the time evolution of molecular state as determined by the total Hamiltonian and provided the developed Bethe-Salpeter theory in Ref. [8].
Because the time evolution of molecular state is determined by the total Hamiltonian, exotic meson resonance should be considered as an unstable meson-meson molecular state.
According to the developed Bethe-Salpeter theory for dealing with resonance [8], this unstable state has been prepared to decay at given time, and the prepared state can be regarded as a bound state with ground-state energy.Solving BS equation for arbitrary meson-meson bound state, one can obtain the mass M 0 and BS wave function χ P (x ′ 1 , x ′ 2 ) for this bound state with momentum P = (P, i P 2 + M 2 0 ).Setting t 1 = 0 and t 2 = 0 in the ground-state BS wave function, we obtain a description for the prepared state (ps) Now it is necessary to consider the total Hamiltonian where K I represents the interaction responsible for the formation of stationary bound state and V I stands for the interaction responsible for the decay of resonance.Then the time evolution of this system determined by the total Hamiltonian H has the explicit form where (ǫ − H) −1 is the Green's function and the contour C 2 runs from ic r + ∞ to ic r − ∞ in energy-plane.The positive constant c r is sufficiently large that no singularity of (ǫ − H) −1 lies above C 2 .The time-dependent wave function X (t) provides a complete description of the system for t > 0. Since H = K I , this system should not remain in the prepared state X ps a .Then at arbitrary time t the probability amplitude of finding the system in the state X ps a is In field theory the operator T (ǫ) is just the scattering matrix with energy ǫ, and T aa (ǫ) is the T -matrix element between two bound states, which is defined as Because of the analyticity of T aa (ǫ), we define where ǫ approaches the real axis from above, D and I are the real and imaginary parts, respectively.In experiments, many exotic particles are narrow states and their decay widths are very small compared with their energy levels, i.e., (2π) 3 I(M 0 ) ≪ M 0 .This situation is ordinarily interpreted as implying that both (2π) 3 |D(ǫ)| and (2π) 3 I(ǫ) are also very small quantities, as compared to M 0 .Therefore, we can expect that [ǫ − M 0 − (2π) 3 T aa (ǫ)] −1 has a pole on the second Riemann sheet where ∆M = (2π) 3 D(M 0 ) is the correction for energy level of resonance and M = M 0 + (2π) 3 D(M 0 ) is the physical mass for resonance.This pole at ǫ pole describes the resonance.
The mass M 0 of two-body bound state is obtained by solving homogeneous BS equation, which should not be the mass of physical resonance.Γ(M 0 ) with mass M 0 also should not be the width of physical resonance, which should depend on its physical mass M. We will minutely show the computational process of T -matrix element between two bound states T aa (ǫ) in the next section.
When there is only one decay channel, we can use the unitarity of T aa (ǫ) to obtain [29] 2I where the symbol P means that this integral is a principal value integral and the variable of integration is the total energy ǫ ′ of the final state.To calculate the real part, we need calculate the function I(ǫ ′ ) of value of the final state energy ǫ ′ , which is an arbitrary real number over the real interval ǫ M < ǫ ′ < ∞.As usual the momentum of initial bound state a is set as P = (0, 0, 0, iM 0 ) in the rest frame and ǫ M denotes the sum of all particle masses in the final state.We suppose that the final state b may contain n composite particles and n ′ elementary particles in decay channel c ′ .From Eq. (76), we have where the momenta of final elementary and composite particles, respectively; P ǫ ′ = (0, 0, 0, iǫ ′ ), T (c ′ ;b)a (ǫ ′ ) is the T -matrix element with respect to ǫ ′ , and spins represents summing over spins of all particles in the final state.In Eq. ( 78) the energy in scattering matrix is equal to the total energy ǫ ′ of the final state b, which is an arbitrary real number over the real interval ǫ M < ǫ ′ < ∞.The mass M 0 and BS amplitude of initial bound state a have been specified and the value of the initial state energy in the rest frame is a specified value M 0 .From Eq. (78), we have I(ǫ ′ ) > 0 for ǫ ′ > ǫ M and I(ǫ ′ ) = 0 for ǫ ′ ǫ M , which is the reason that the integration in dispersion relation (77) ranges from ǫ M to +∞.
If there are several decay channels, we should write instead A. Channel J/ψω with respect to arbitrary value of the final state energy From Eq. (51), we obtain the total matrix element between the final state J/ψ(Q), ω(Q ′ ) out| and the specified initial four-quark state |P in where the total energy ǫ ′ of the final state extends from ǫ c ′ 1 ,M to +∞, i.e., ) is the bound state matrix element with respect to ǫ ′ for channel c ′ 1 , shown as Fig. 7.It is necessary to emphasize that the energy in the twoparticle irreducible Green's function is equal to the final state energy ǫ ′ while the mass M 0 and BS amplitude of initial bound state is specified.We have introduced extended Feynman diagram in Ref. [8] to represent arbitrary value of the final state energy.In Fig. 7, the quark momenta in left-hand side of crosses depend on the final state energy and the momenta in right-hand side depend on the initial state energy, i.e., p 1 − p 2 − p 3 + p 4 = Q + Q ′ = P ǫ ′ and p ′ 1 − p ′ 2 = P .When ǫ ′ = M 0 , the crosses in Fig. 7 disappear and then Fig. 7 becomes Fig. 3; T (c ′ 1 ;b)a (ǫ ′ = M 0 ) is the T -matrix element with mass M 0 for channel c ′ 1 expressed as Eq. ( 52).Though the T -matrix element T (c ′ 1 ;b)a (ǫ ′ ) has the same form expressed as Eq. ( 52), these momenta should become where P = (0, 0, 0, iM 0 ), ).The initial state is considered as a four-quark state, so the specified GBS amplitude of initial state should be where k ′ depends on P .Then we obtain the function I 1 (ǫ ′ ) for channel J/ψω B. Channel D + D − with respect to arbitrary value of the final state energy The T -matrix element with respect to ǫ ′ for channel c ′ 2 can be represented graphically by Fig. 8.The total energy ǫ ′ of the final state extends from ǫ c ′ 2 ,M to +∞, i.e., ǫ c ′ 2 ,M < ǫ ′ < ∞ and ǫ c ′ 2 ,M = M D + + M D − .In Fig. 8, the crosses mean that the momenta of quark propagators and the momentum w of the exchanged light meson depend on Q 1 and Q 2 , i.e., We still use the vertex function to calculate the T -matrix element with respect to ǫ ′ for channel c ′ 2 .However, different from the ordinary vertex function, we should introduce the vertex function with respect to ǫ ′ , which is shown as Fig. 9.In Fig. 9, Q 1 depends on P ǫ ′ , p ′ 1 depends on P and the crosses mean that the momenta of quark propagators and the momentum w of the exchanged light meson depend on the final state energy ǫ ′ .
Using the approach introduced in Sec.III B, we can obtain the explicit forms for the vertex functions with respect to ǫ ′ , and then Fig. 8 can be reduced to Fig. 10.In Fig. 10, we have From Eq. (65), we obtain the total matrix element between the final state where T (c ′ 2 ;b)a (ǫ ′ ) is the bound state matrix element with respect to ǫ ′ for channel c ′ 2 , shown as Fig. 10.When ǫ ′ = M 0 , the crosses in Figs. 8, 9 and 10 disappear and then these three extended Feynman diagrams become Figs.4, 6 and 5, respectively; T (c ′ 2 ;b)a (ǫ ′ = M 0 ) is the T -matrix element with mass M 0 for channel c ′ 2 expressed as Eq.(66).Though the T -matrix element T (c ′ 2 ;b)a (ǫ ′ ) has the same form expressed as Eq. ( 66), these momenta should become where P = (0, 0, 0, iM 0 ), P ǫ ′ = (0, 0, 0, iǫ ′ ), Q 1 = (Q D (ǫ ′ ), iǫ ′ /2), Q 2 = (−Q D (ǫ ′ ), iǫ ′ /2) and The coefficients E 1 and E 2 in interaction V λτ (Q 1 , Q 2 , p) given by Eq. (62) should become E 1 (ǫ ′ ) = E 2 (ǫ ′ ) = √ ǫ ′ M 0 /2.Then we obtain the function ; the crosses mean that the momenta of quark propagators and the momentum w of the exchanged light meson depend on Q 1 and Q 2 , i.e., p 1 − p 2 − p 3 + p 4 = p 1 − p 2 − q 3 + q 4 = Q 1 + Q 2 = P ǫ ′ , and p ′ 1 − p ′ 2 = P .To calculate the T -matrix element with respect to ǫ ′ for channel c ′ 4 , we also introduce the form factor of heavy meson with respect to ǫ ′ , which is shown as Fig. 12.Using the approach introduced in Sec.II C 2, we can obtain the explicit forms for the heavy meson form factors h(w 2 ) with respect to ǫ ′ , and then Fig. 11 can be reduced to Fig. 13.In Fig. 13, we have Q 1 + Q 2 = P ǫ ′ , p ′ 1 − p ′ 2 = P , and the crosses lie on the right-hand side of numerical results from meson mass M σ are also very small in our previous works [4,6,7,11] and Refs.[5,33].Therefore, in our approach the calculated mass and decay width are uniquely determined.
Up to now, a theoretical approach from QCD to investigate resonance which is regarded as an unstable two-body system has been established.In this paper, we only explore exotic meson resonance which is considered as an unstable molecular state composed of two heavy vector mesons.The extension of our approach to more general resonances is straightforward, while the interaction Lagrangian may be modified.More importantly, it is most reasonable and fascinating to investigate resonance as far as possible from QCD.In the framework of quantum field theory, the nonperturbative contribution from the vacuum condensates can be introduced into BS wave function [11] and the two-particle irreducible Green's function, and then the calculated mass and decay width of resonance will contain more inspiration of QCD.
molecular states D * 0 D * 0 and D * + D * − with spin-parity quantum numbers 0 + .In this section, we only investigate the mixed state of two stable bound states D * 0 D * 0 and D * + D * − , and BS wave function for this system is a linear combination of two components as χ D * D * Let D * l denote one of D * 0 and D * + , and l = u, d represents the u or d antiquark in heavy vector meson D * 0 or D * + , respectively; D * l denotes the antiparticle of D * l .From Eq. (8), we can obtain BS wave function describing pure bound state D * l D * l
and b 2 10 (M D D)/(2µ R ) and b 2 20 (M D D)/(2µ R ) are the eigenvalues corresponding to lowest energy in Eqs. (

FIG. 2 :
FIG. 2: Generalized Bethe-Salpeter wave function for four-quark state in the momentum representation.The solid lines denote quark propagators, and the unfilled ellipses represent Bethe-Salpeter amplitudes.

FIG. 3 :
FIG.3:The lowest order matrix element between bound states in the momentum representation.
Q and Q ′ , so in Minkowski space M

FIG. 4 :
FIG. 4: Matrix element for decay channel D + D − .The momenta in the final state satisfy Q 1 +Q 2 = P .w represents the momentum of the exchanged light meson.

FIG. 6 :
FIG.6: Vertex function for the exchanged light meson, heavy pseudoscalar and vector mesons.
ǫ) = b (2π) 4 δ (3) (P b − P)δ(E b − ǫ)|T ba (ǫ)| 2 , (76) where P b = (P b , iE b ) is the total energy-momentum vector of all particles in the final state and the T -matrix element T ba (ǫ) is defined as b out|a in = −i(2π) 4 δ (3) (P b − P)δ(E b − ǫ)T ba (ǫ).The delta-function in Eq. (76) means that the energy ǫ in scattering matrix is equal to the total energy E b of the final state, and b represents summing over momenta and spins of all particles in the final state.For E b = ǫ, we also denote the total energy of the final state by ǫ and I(ǫ) becomes a function of the final state energy.Using dispersion relation for the function T aa (ǫ), we obtain where c ′ represents summing over all open and closed channels.Because the total energy ǫ ′ of the final state extends from ǫ M to +∞, we may obtain several closed channels derived from the interaction Lagrangian.Assuming that resonance χ c0 (3915) is a mixed state of two components D * 0 D * 0 and D * + D * − , we obtain one closed channel D * D * derived from the interaction Lagrangian (1), denoted as c ′ 4 .Since bound state lies below the threshold, i.e., M 0 < M D * + M D * , the closed channel c ′ 4 can not occur inside the physical world.

FIG. 7 :FIG. 8 :
FIG.7: Matrix element with respect to ǫ ′ for channel J/ψω.The momenta in the final state satisfyQ + Q ′ = P ǫ ′and the momentum of the initial state is P .The final state energy extends from ǫ M to +∞ while the initial state energy is specified, and the crosses mean that the momenta of quark propagators depend on the final state energy ǫ ′ .

FIG. 9 :FIG. 10 :
FIG.9: Vertex function for the exchanged light meson, heavy pseudoscalar and vector mesons with respect to ǫ ′ .Q 1 depends on P ǫ ′ and p ′ 1 depends on P .The crosses mean that the momenta of quark propagators and the momentum w of the exchanged light meson depend on the final state energy ǫ ′ .
out| and the mixed state of two pure bound states D * 0 D * 0 and D

2 D 2 D 1 = −M 2 D * l and Q 2 2 =
) C. Channel D 0 D0 with respect to arbitrary value of the final state energy In Figs. 8, 9 and 10, P M and P M ′ represent pseudoscalar mesons D 0 and D0 , respectively.Following the same procedure as for charged channel D + D − , we can obtain the T -matrix element T (c ′ 3 ;b)a (ǫ ′ ) with respect to ǫ ′ and the function I 3 (ǫ ′ ) for neutral channel c ′ 3 .Here, the total energy ǫ ′ of the final state extends fromǫ c ′ 3 ,M to +∞, i.e., ǫ c ′ 3 ,M < ǫ ′ < ∞ and ǫ c ′ 3 ,M = M D 0 + M D0 .D. Closed channel D * D *The final state D * , D * out| can be written asD * , D * out| = 1 √ * 0 , D * 0 out| + 1 √ * + , D * − out|.(87)The total energy ǫ ′ of the final state extends fromǫ c ′ 4 ,M to +∞, i.e., ǫ c ′ 4 ,M < ǫ ′ < ∞ and ǫ c ′ 4 ,M = M D * l + M D * l .Considering the lowest order term of the two-particle irreducible Green's function, we can obtain the T -matrix element between the final state D * l , D * l out| and the initial four-quark state, which can be represented graphically by Fig.11.In Fig.11, V M and V M ′ still represent D * l and D * l , respectively; Q 1 and Q 2 still represent the momenta of final particles, but Q 2 Exotic resonance χ c0 (3915) is considered as a mixed state of two unstable molecular states D * 0 D * 0 and D * + D * − , and we investigate the time evolution of meson-meson molecular state as determined by the total Hamiltonian.According to the developed Bethe-Salpeter theory, the total matrix elements for all decay channels should be calculated with respect to arbitrary value of the final state energy.Because the total energy of the final state extends from ǫ M to +∞, we consider three open decay channels J/ψω, D + D − , D 0 D0 and one closed channel D * D * from the effective interaction Lagrangian at low energy QCD, which are exhibited by extended Feynman diagrams.Using the developed Bethe-Salpeter theory, we calculate the mass M and full width Γ of physical resonance χ c0 (3915), which are in good agreement with the experimental data.Obviously, our work can be extended to more general resonances.dation, China under Grants No. ZR2016AQ19 and ZR2016AM31; and SDUST Research Fund under Grant No. 2018TDJH101.T 8 µ 1 represents the meson-meson bound state.