Probing the pole origin of $X(3872)$ with the coupled-channel dynamics
Jun-Zhang Wang Zi-Yang Lin Yan-Ke Chen Lu Meng Shi-Lin Zhu





wangjzh2022@pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China

lzy_15@pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China

chenyanke@stu.pku.edu.cn
School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China

lu.meng@rub.de
Institut für Theoretische Physik II, Ruhr-Universität Bochum,  D-44780 Bochum, Germany 

zhusl@pku.edu.cn

School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China









The X(3872), as the first and the most crucial member in the exotic charmoniumlike XYZ family, has been studied for a long time. However, its dynamical origin, whether stemming from a DD̅^* hadronic molecule or the first excited P-wave charmonium χ_c1(2P), remains controversial. In this Letter, we demonstrate that the X(3872) definitely does not result from the mass shift of the  higher bare χ_c1(2P) resonance pole  in the coupled-channel dynamics involving a short-distance cc̅ core and the long-distance DD̅^* channels. Instead, it originates from either the DD̅^* molecular pole or the shadow pole associated with the P-wave charmonium, which depends on the concrete coupling mode between the cc̅ and DD̅^*. In order to further exploit the nature of X(3872), we carefully investigate potential mechanisms that contribute to its pole width, which suggests that the coupled-channel dynamics plays a critical role in causing a noticeable discrepancy between the pole widths of X(3872) and T_cc^+. Interestingly, we bridge the quantitative connection among the dynamics origin of X(3872), its pole width and the properties of the predicted new resonance. The precise measurement of the pole width of X(3872) and the search for the new charmoniumlike resonance become highly significant and can be anticipated in future LHCb, BESIII and Belle II experiments.




Probing the pole origin of X(3872) with the coupled-channel dynamics
    Shi-Lin Zhu 0000-0002-4055-6906
    April 29, 2024
====================================================================







Introduction.—
During the past two decades, one of the most important topics in the physics of the strong interaction is to understand a large number of charmoniumlike XYZ resonances observed by high energy experiments including BaBar, Belle, BESIII and LHCb, and so on (see Refs. <cit.> for relevant progresses). As a superstar among them, the nature of X(3872) is the most mysterious, which is mainly manifested in that its mass exactly coincides to the threshold of D^0D̅^*0/D^*0D̅^0 and its decay pattern is strongly coupled with D^0D̅^0π^0 <cit.>. These behaviors imply the central role of DD̅^* (DD̅^* refers to DD̅^*/D̅D^* and similarly hereinafter) in the generation of X(3872).  As a result, the hadronic molecular interpretation of X(3872) was naturally proposed <cit.>.  Intriguingly, the LHCb Collaboration recently discovered the doubly charmed tetraquark T_cc^+ in the prompt production
of the pp̅ collision <cit.>, which has been widely acknowledged as an ideal candidate for the shallow DD^* bound state <cit.>. 

Compared with the doubly charmed tetraquark T_cc^+, a more complicated aspect of X(3872) is that its DD̅^* component inevitably interacts with the conventional P-wave charmonium states with the same quantum number J^PC=1^++ <cit.>, especially the nearby χ_c1(2P) state (hereinafter denoted as χ_c1^'). Therefore, it can be expected that the realistic wave function of X(3872) should behave as a compact charmonium core in the short-distance region and a DD̅^* structure in the long-distance region <cit.>. In this coupled-channel framework, a prevalent opinion suggests that the X(3872) arises from the mass shift of the higher χ_c1^' state due to the unquenched DD̅^* loop correction to the self-energy of the P-wave charmonium <cit.>. Thus, whether the X(3872) is a hadronic molecule or an excited charmonium remains controversial. Regarding this long-standing view, in this Letter, we demonstrate that the X(3872) definitely does not originate from the bare χ_c1^' resonance pole in the coupled-channel dynamics involving the cc̅ core and the DD̅^* channels. 
The X(3872) stems from either the DD̅^* pole in the so-called weak-coupling mode or the shadow pole associated with χ_c1^' in the strong-coupling mode.  

To further decipher the nature of X(3872), we systematically investigate the pole width of X(3872). We establish the DD̅^* interaction with the chiral effective field theory (ChEFT), and the transition amplitude between the P-wave charmonium and DD̅^* with the quark-pair-creation (QPC) model. We explore three kinds of potential mechanisms contributing to the pole width, namely the DD̅π three-body cut from the one-pion exchange (OPE) in the DD̅^* scattering,  the dynamical width of D̅^* and the non-open-charm decays of the bare χ_c1^' state. We find that the coupled-channel dynamics plays a crucial role in generating a significantly larger pole width of X(3872) compared with that of T_cc^+. Furthermore, we establish a quantitative connection among the dynamical origin of X(3872), its pole width and the properties of the predicted new resonance, which underscores the importance of precise measurements of the pole width of X(3872) in future high-energy experiments.
These findings should constitute an important step towards thoroughly unraveling the mystery of X(3872) and constructing the P-wave charmonium family.









Pole evolution of X(3872) in the coupled-channel dynamics.—
We adopt a complete coupled-channel framework to study the properties of X(3872), in which the DD̅^* scattering dynamics is also included.
The hadronic fock state can be written as

    | Φ⟩=c_0 | Φ_0 ⟩+∑_i∫d^3q/(2π)^3ϕ_i(q)| Φ_i ⟩_q,

where | Φ_0 ⟩ and | Φ_i ⟩_q correspond to the bare fock state and the i-channel hadron-hadron continuum state associated with the relative momentum q, respectively. Then the coupled-channel Schrödinger equation can be written as 

    [  ℋ_0 ℋ_01 ℋ_02    …; ℋ_10  ℋ_1 ℋ_12    …; ℋ_20 ℋ_21  ℋ_2    …;    ⋮    ⋮    ⋮    ⋱;      ][ c_0 | Φ_0 ⟩;     | Φ_1 ⟩;     | Φ_2 ⟩;           ⋮;             ]   =E [ c_0 | Φ_0 ⟩;     | Φ_1 ⟩;     | Φ_2 ⟩;           ⋮;             ]

with 

    | Φ_i ⟩=∫d^3q/(2π)^3ϕ_i(q)| Φ_i ⟩_q,   (i=1,2,⋯).

After expanding the above matrix equation and applying the relation ⟨Φ_0| ℋ_0 | Φ_0 ⟩=M_0, ⟨Φ_j |_qℋ_j0 | Φ_0 ⟩=𝒱_0j(q) and ⟨Φ_j |_q^'ℋ_ji | Φ_i  ⟩ _q=δ_ijE_k + V_ij(q,q^'), one can obtain

    ∑_i ∫ϕ_i(q) 𝒱_i0(q)  d^3q/(2π)^3=(E-M_0)c_0,  
           c_0𝒱_0j(q)+∑_i∫(δ_ijE_k + V_ij(q,q^'))ϕ_i(q^') d^3q^'/(2π)^3
           =E ϕ_j(q)   (j=1,2,3 …),

respectively, where M_0 is the bare mass of | Φ_0 ⟩, the 𝒱_0j corresponds to the transition amplitude between the bare state and the j-th channel, and the E_k and V_ij stand for the kinetic term q^2/(2μ) and the potential of the corresponding hadron-hadron scattering, respectively. Combining  Eqs. (<ref>) and (<ref>), we have

    ∑_i(δ_ijE_k +∫( V_ij(q,q^')+ 𝒱_ij(q,q^'))ϕ_i(q^') d^3q^'/(2π)^3) 
           =Eϕ_j(q)   (j=1,2,3 …),

where 𝒱_ij(q,q^')=𝒱_i0(q^')𝒱_0j(q)/E-M_0+iϵ. Equation (<ref>) is a standard coupled-channel Schrödinger equation in momentum space. The coupled-channel problem between the bare state and the hadronic continuum is converted to a hadron-hadron scattering problem including an extra s-channel effective potential  𝒱_ij(q,q^').  



If expanding the above matrix equation, one can obtain 

    ℋ_0c_0| Φ_0 ⟩+∑_i ∫ϕ_i(q)ℋ_0i| Φ_i ⟩_q d^3q=E c_0| Φ_0 ⟩,  
          ℋ_j0c_0| Φ_0 ⟩+∑_i∫ϕ_i(q)ℋ_ji| Φ_i ⟩_q d^3q= E  ∫ϕ_j(q) 
          ×| Φ_j ⟩_q d^3q   (j=1,2,3 …).

By applying for the relation of ⟨Φ_0| ℋ_0 | Φ_0 ⟩=M_0, ⟨Φ_j |_qℋ_j0 | Φ_0 ⟩=𝒱_0j(q) and ⟨Φ_j |_q^'ℋ_ji | Φ_i  ⟩ _q=δ_ijE_k + V_ij(q,q^'),  the Eqs. (<ref>) and (<ref>) are  equivalent to




We consider a two-channel calculation involving the [D^0D̅^*0] and [D^+D̅^*-], where the square brackets are the shorthands of the C=+ states such as [D^0D̅^*0]=1/√(2)(D^0D̅^*0-D̅^0D^*0). Because the mass of X(3872) lies close to the threshold of D^0D̅^*0, the DD̅^* → DD̅^* scattering should be governed by the leading-order  contact and OPE interactions  in the effective field theory <cit.>.  The total effective potential can be written as

    V_Total(q,q^')=
    [    C_t-V_π^0+𝒱_11 C_t^'-2V_π^±+𝒱_12; C_t^'-2V_π^±+𝒱_21    C_t-V_π^0+𝒱_22;                   ],
 
where C_t and C_t^'  are the contact terms, V_π is the OPE potential and 𝒱_11=𝒱_12=𝒱_21=𝒱_22. Since the ChEFT only works at small momentum, we introduce a Gaussian cutoff to regularize the contact and OPE potential, i.e., 

    ℱ(q,q^')=exp(-(q^2+q^' 2)/Λ^2),

where the cutoff Λ=0.5 GeV is usually taken for the DD^* or DD̅^* system <cit.>.
In order to explore the trajectory of the pole generated in  the DD̅^* → DD̅^* scattering in the coupled-channel dynamics, we first ignore the OPE contribution and adopt the momentum-dependent form of 𝒱=g^2/2M_0 e^-(q^2+q^' 2)/α^2/(E-M_0+iϵ), where  α and g characterize the coupling range and strength between χ_c1^' and DD̅^*, respectively. Then the effective potential in Eq. (<ref>) becomes separable, and it would be very convenient to analytically solve the Lippmann-Schwinger equation (LSE) and search for the poles of the T matrix <cit.>.


The general coupled-channel LSE is given by

    T^α_β(q,q^')   =   V^α_β(q,q^')+∑_γ∫_0^∞dk k^2/(2π)^3V^α_γ(q,k)T^γ_β(k,q^')/E-k^2/(2μ_γ),

where V^α_β(q,q^') is the partial-wave-projected potential from the α channel to the β channel, and q and q^' correspond to the initial and final momentum, respectively. The reduced mass μ_γ is defined by

    μ_γ=m_1m_2/m_1+m_2.









With the above preparations, we investigate the pole origin of X(3872). We first focus on the situation with the strict isospin symmetry, in which the two-channel calculation in Eq. (<ref>) is reduced to a single-channel one. The pole trajectories in the momentum k_0-plane are presented in Fig. <ref>.  Here, k_0 is defined by k_0=√(2μ(E-m_D-m_D^*)). k_0 with positive and negative imaginary parts correspond to the first and second Riemann sheets of the T matrix, respectively. The  bare mass M_0=3.96 GeV is adopted according to the quark model estimations <cit.>. The subfigures 1, 2, 3 represent the cases with no contact interaction (C_t=0 GeV^-2), an attractive contact interaction (C_t=-10 GeV^-2) and a repulsive contact interaction (C_t=10 GeV^-2), respectively.  In Fig. <ref>(a) and (b), we show the pole trajectories of the DD̅^*  scattering when increasing the coupling strength g  in the so-called weak coupling mode with a typical value α=1.0 GeV and in the strong coupling mode with α=1.4 GeV, respectively, which indicates a smaller and larger overlap range between the wave functions of  the cc̅ core and DD̅^* channel, respectively. 




In Fig. <ref>(a-1) with the weak coupling mode, there appear three poles of the T matrix in the complex plane, although  only the s-channel interaction 𝒱 is  considered.  It is easy to identify the nature of these three poles by setting g^2 → 0. The red, blue and green poles correspond to the bare χ_c1^' resonance, the shadow pole associated with χ_c1^' resonance  and a virtual state of DD̅^* at the infinity, respectively. With gradually increasing g^2,  the pole of χ_c1^' resonance moves first towards and then away from the threshold of DD̅^*, while the DD̅^* virtual state continuously moves towards the threshold and finally crosses the unitary branch cut of DD̅^* and becomes a bound state, which corresponds to the observed X(3872) structure. For the strong coupling mode shown in Fig. <ref>(b-1), the situation is completely different. When g^2 increases to a critical value, the χ_c1^' pole and its shadow pole will meet at the same point as virtual states and then continue to move along the negative imaginary axis of the k_0-plane in opposite directions. Subsequently, the χ_c1^' pole will meet the DD̅^* virtual state pole arising from the infinity and then they evolve into a pair of resonance and anti-resonance.  In contrast, the shadow pole associated with χ_c1^' will enter the first Riemann sheet and be related to a X(3872) structure. 

It is important to highlight that this pole separation behavior associated with the χ_c1^' state is directly attributed to the tiny imaginary part iϵ of the effective potential 𝒱∝ (m_th+k^2_0/(2μ)-M_0+iϵ)^-1.
Thus, such a phenomenon is still universal when including the intrinsic  DD̅^* dynamics, as shown in Fig. <ref>(a-2), (a-3), (b-2) and (b-3).   It can be seen that an intrinsic attractive force between D and D̅^* prompts the pole evolution behaviors  to favor the weak coupling mode and a repulsive force causes poles to evolve in favor of the strong coupling mode.
Interestingly, due to the influence of the repulsive interaction, the χ_c1^' pole might continuously emerge as a virtual state by a strong enough threshold coupling.
To sum up, whatever the dynamics origin of X(3872) is, 

X(3872) does not originate from the mass shift of the bare χ_c1^' resonance pole in the coupled-channel dynamics.


In the following, we qualitatively understand the underlying reason of the evolution trends of the χ_c1^' pole and its shadow pole.
The pole of the dressed χ_c1^' state is determined by effective potential 𝒱_ij(q,q^'), which is proportional to (m_th+k^2_0/(2μ)-M_0+iϵ_2/2)^-1 (ϵ_2 are small quantity). When considering k_0=ϵ_1-iκ near the negative imaginary axis of k_0 plane, then

    𝒱_ij∝(m_th-M_0+-κ^2+ϵ_1^2/2μ+i(ϵ_2/2-κϵ_1/μ))^-1
          ≈1/𝒜_0^2(𝒜_0-i(ϵ_2/2-κϵ_1/μ))

with 𝒜_0=(m_th-M_0+-κ^2+ϵ_1^2/2μ). It can be seen that positive (negative) ϵ_1 will relatively weaken (strengthen) the attractive interaction, which determines that the χ_c1^' pole must move away from the positive real axis of the k_0 plane and its shadow state will be closer to the origin point so that it can move into the first Riemann sheet. 









 §.§ a. Explaining the evolution trend of χ_c1^' pole


The pole of the dressed χ_c1^' is determined by effective potential 𝒱_ij(q,q^'), which is proportional to (m_th+k^2_0/(2μ)-M_0+iM_0ϵ_2)^-1 (ϵ_2 are small quantity). When considering k_0=ϵ_1-iκ near the negative imaginary axis of k_0 plane, then

    𝒱_ij∝(m_th^2-M_0^2+κ^4-6κ^2ϵ_1^2/4μ^2+-(κ^2+ϵ_1^2)m_th/μ+  
          i(M_0ϵ_2+κ^3ϵ_1/μ^2-2κ/μm_thϵ_1))^-1≈
          1/𝒜_0^2(𝒜_0-i(M_0ϵ_2+κ^3ϵ_1/μ^2-2κ/μm_thϵ_1))

with 𝒜_0=(m_th^2-M_0^2+κ^4-6κ^2ϵ_1^2/4μ^2+-(κ^2+ϵ_1^2)m_th/μ). It can be seen that positive (negative) ϵ_1 will induce a relatively weaker (stronger) attractive interaction for κ≪1, which determines that the χ_c1^' pole must move away from the positive real axis of the k_0 plane and its shadow state will be closer to the origin point so that it can move into the first Riemann sheet. 







For X(3872), the isospin breaking effect is significant since the mass difference between the neutral and charged channels is up to  8 MeV <cit.>. Thus, we further study the pole evolution of DD̅^* → DD̅^* in the two-channel scattering, which is presented in Fig. <ref>. Here, in order to make the T-matrix a single-valued function in the locally flat surface, a uniformization variable z is introduced to achieve a mapping from energy E to z <cit.>, whose definition is summarized in Appendix.
In the z/E-plane of Fig. <ref>, the orange and black solid lines stand for the unitary branch cuts related to the [D^0D̅^*0] and [D^+D̅^*-], respectively. The dashed lines represent the real axis of the E-plane unoccupied by the branch cut. The four Riemann sheets (+,+), (-,+), (+,-) and (-,-) for the two-channel system correspond to the upper half z-plane outside and inside the circle, and the lower half z-plane inside and  outside the circle, respectively, where + and - represent the first and the second Riemann sheets, respectively. 











Since the impact of the intrinsic DD̅^* dynamics on the pole evolution in the two-channel case is very similar to the lesson learned from the single-channel analysis, here we only show the two-channel calculations  with C_t=C_t^'=0 GeV^-2. In Fig. <ref>(a-1) with a typical value α=1.0 GeV, a large number of new poles appear due to the isospin symmetry violation, and it can be found that the evolution behaviors of the dressed χ_c1^' resonance and its shadow pole in the (-,-) sheet are very similar to those of the weak coupling mode in the single-channel case. However, the X(3872) pole no longer arises from the virtual DD̅^* state in the (-,-) sheet. Instead, it arises from the shadow resonance of χ_c1^' in the (-,+) sheet. In Fig. <ref>(a-2), we show the corresponding pole trajectories  in the complex energy plane. It can be seen that the dressed χ_c1^' resonance is always higher than the [D^+D̅^*-] threshold. Its shadow pole in the (-,+) sheet first moves into the energy region below the [D^0D̅^*0] threshold and then returns and crosses the unitary cut of [D^0D̅^*0] into the (+,+) sheet and becomes a bound state X(3872). This trajectory is still maintained when employing a stronger coupling mode such as α= 1.4 GeV. However, when adopting a weaker coupling mode (α= 0.8 GeV), as presented in  Fig. <ref>(b-1) and (b-2), the shadow pole of χ_c1^' in the (-,+) sheet will not be pulled to the imaginary axis of the z-plane and the X(3872) originates from a DD̅^* virtual state pole in the (-,+) sheet. 
















The pole width of X(3872) and the relevant implications.—
Recently, the LHCb Collaboration and BESIII Collaboration extracted the pole position of X(3872) by analyzing the line shape, whose imaginary part is extracted to be -(130^+320_-180)i keV <cit.>  and -(190^+206_-161)i keV <cit.>, respectively. Although the experimental errors are large, both of the center values are obviously larger than that of T_cc^+, which is extracted to be -(48±2)i keV   <cit.>. The pole width of T_cc^+ has been reproduced well in theoretical calculations and can be attributed to the dominant three-body DDπ decay <cit.>.  However, the DD̅π half decay width of X(3872) from the three-body cut of the OPE potential is estimated to be only 15∼34 keV in Refs. <cit.>. The possible large pole width of X(3872) has not been understood well in the present theoretical studies.

Here, we conjecture that the coupled-channel dynamics plays a critical role in producing the larger pole width of X(3872), which is completely inaccessible to the T_cc^+. The motivation bases on the evidence that the experimental total width of the ground state P-wave charmonium χ_c1(1P), whose open-charm decay channels are not opened yet, can reach 0.88±0.05 MeV <cit.>.  Hence, it can be expected that the non-open-charm decay widths of the bare χ_c1^' state possess the same order of magnitude as that of the ground state. In order to verify this point, we employ a quark-model-independent scheme to estimate the decay behaviors of the bare χ_c1^' state. The main non-open-charm modes of the bare χ_c1^' include the radiative and light hadron decays, which is usually related to the spatial wave function of the charmonium. The radial wave function of χ_c1^' can be described by a simple harmonic oscillator function <cit.>, i.e., 

    R_nL(r,β)=β^3/2√(2n!/Γ(n+L+3/2))(β r)^Le^-r^2β^2/2L_n^L+1/2(r^2β^2),

where L_n^L+1/2 is the associated Laguerre polynomial with the orbital angular momentum L and the radial quantum number n.  Under the heavy quark symmetry, it can be concluded that the P-wave charmonium triplets share the same radial wave function, so the unique unknown parameter β=0.7∼0.9 GeV <cit.> can be reliably determined by the experimental total width of χ_c2^' and χ_c0^' <cit.>. Subsequently, the decays into light hadrons or a charmonium with a photon can be calculated. The coupling amplitude 𝒱_0j(q) and 𝒱_i0(q^') in Eq. (<ref>) can also be obtained in the quark pair creation (QPC) model without extra parameter dependence. The relevant details can be found in Appendix. Now, the s-channel effective potential should be modified as

    𝒱_ij(q,q^')=𝒱_i0(q^')𝒱_0j(q)/E-M_0+iϵ→𝒱_i0(q^')𝒱_0j(q)e^-λ^2(q^2+q^' 2)/E-M_0+i1/2(Γ_a+Γ_b),

where Γ_a=3488 keV and Γ_b=220 keV  correspond to the light hadron decays and radiative decays of the bare χ_c1^' when adopting β=0.75 GeV, respectively. According to suggestions from Refs. <cit.>, an extra cutoff parameter λ is required to adjust the large momentum suppression in 𝒱_i0 from the input of the QPC amplitude. Additionally, we also simultaneously take into account the  three-body DD̅π threshold effect from the OPE interaction <cit.> and the dynamical width of D̅^* from its strong decay D̅π and electromagnetic decay D̅γ (see Appendix for relevant formalisms). The complete scattering amplitudes of DD̅^* → DD̅^* involving these three potential mechanisms contributing to the pole width are presented in Fig. <ref>.









The inclusion of these decay dynamics makes the system Hamiltonian no longer Hermitian. For this situation, a more convenient approach to extract the pole positions of the full scattering amplitudes in Fig. <ref> is the complex scaling method (CSM) based on the Schrödinger equation <cit.>. 
In our previous studies on T_cc^+ in ChEFT <cit.>, it is found that the pole width is not sensitive to the cutoff Λ within a large range of Λ=0.4-1.0  GeV, and then the  Λ cutoff dependence can be safely renormalized by the contact interaction. Thus, the Λ is fixed to 0.5 GeV and the undetermined parameters are λ, C_t and C_t^'. The last two can be related to two new contact constants V_ct^I=0 and V_ct^I=1 (see Appendix for their definitions). 

Firstly we do not consider the isospin violation of the contact interaction, i.e., V_ct^I=1=0 GeV^-2. The solved observable pole structures and the corresponding component possibilities are listed in Table <ref>.  For different cutoff λ, one can adjust the V_ct^I=0 to reproduce the small binding energy of X(3872), and it can be seen that the pole width of X(3872) is still small if the non-open-charm decays of the bare χ_c1^' are not considered. When involving the width Γ_a+Γ_b from the bare state, its decay dynamics can be transferred into other poles by the coupled-channel mechanism, which significantly enlarges the pole width of X(3872) to the order of 𝒪(10^-1) MeV. Therefore, the accurate pole width of X(3872) can actually be treated as a novel indicator to convincingly probe the role of the coupled-channel dynamics in the DD̅^* scattering.


As a prediction, there is indeed a  higher resonance pole around 4 GeV, which could correspond to the genuine dressed charmonium χ_c1^' resonance as argued in Refs. <cit.> or a distorted DD̅^* resonance, in the case that the bare χ_c1^' resonance has evolved into a virtual pole far from the physical area in the (-,-) sheet.  Furthermore, we find that the pole width of X(3872) and the properties of the new resonance significantly depend on the cutoff λ, which embodies the strong or weak coupling modes mentioned in the above section. For example,  it can be identified that the X(3872) pole originates from the shadow resonance of χ_c1^' in the (-,+) sheet for λ=0.5 GeV^-1 or the DD̅^* pole in the (-,+) sheet for λ=2.5 GeV^-1, which correspond to the strong or weak coupling mode, respectively.  Thus, the precise measurement of X(3872) and the search for the missing higher charmoniumlike resonance might provide a strong limitation and reveal the dynamical origin and nature of X(3872). The results assuming an evident isospin violation of the contact interaction are summarized in Appendix, in which the similar conclusions are obtained.















    𝒱_ij(q,q^')=ℳ_0i^†(q^')ℳ_0j^†(q)/E-M_0→ℳ_0i^†(q^')ℳ_0j^†(q)e^-α^2(q^2+q^' 2)/E-M_0




















Summary.—
We have studied the pole origin of X(3872) with the coupled-channel dynamics involving the interaction between the first excited P-wave charmonium core and the DD̅^* continuum channels and definitely demonstrated that the X(3872) does not originate from the mass shift of the bare χ_c1^' resonance pole. Accordingly, 
it stems from either the DD̅^* virtual state pole in the weak-coupling mode or the shadow pole associated with χ_c1^' in the strong-coupling mode. Furthermore, we focused on the pole width of X(3872) by considering the complete scattering dynamics, which is implemented by the complex scaling method.  Here, three important dynamical mechanisms including the three-body threshold effect from the OPE potential, the dynamical width of D̅^* and the non-open-charm decays of the bare  χ_c1^' state are considered to produce the pole width. Our findings indicate that  the coupled-channel dynamics will result in an apparently larger pole width of X(3872). In addition, we have established a quantitative connection among the dynamical origin of X(3872), its pole width and the properties of the predicted new resonance. We emphasize that the precise measurement of the pole width of X(3872) and the search for the higher charmoniumlike resonance should be very significant for thoroughly solving the puzzle of X(3872), which shall hopefully be achieved with the forthcoming upgrade of LHCb, BESIII and Belle II.







§ ACKNOWLEDGEMENTS


This work is supported by the National Science Foundation of China under Grants No. 11975033, No. 12070131001 and No. 12147168. This project was also funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation, Project ID 196253076-TRR 110). J.Z.W. is also supported by the National Postdoctoral Program for Innovative Talent.






§ APPENDIX





 §.§ The uniformization in the two-channel problem



For the two-channel threshold m_th0 and m_th±, we define two momentum-like  variables 

    p_0^2=E^2-m_th0^2,    p_±^2=E^2-m_th±^2,

and p_±^2-p_0^2=Δ^2. The operation mapping E to z is given by the following relations,

    p_±+p_0=Δ z,     p_0-p_±=Δ/z.

Based on the definition of the variable z, the same energy E in four different Riemann sheets can be mapped onto different regions in the z-plane, which is the so-called uniformization process. Here, the origin of the z-plane is mapped to the infinity of the (-,+) and (+,-) Riemann sheets, so the molecular pole and the χ_c1^' state stem from the origin and the real axis, respectively.





 §.§ The non-open charm and open charm decay behaviors of bare P-wave charmonium χ_c1^'


The important non-open charm decays of the cc̅ state include the light hadron decay and the radiative transition into the lower charmoniums. For a n^3P_1 charmonium, its inclusive decays to various light hadron final states mainly occur through the annihilation process n^3P_1 → qq̅g, whose width depends on the first-order derivative of its radial wave function at the origin <cit.>, 

    Γ(n^3P_1 → qq̅g)=32α_s^3/9π m_c^4|R^'_nP(0)|^2 ln(m_c  < R  >),

where < R  > is the average radius of the n^3P_1 state, m_c=1.65 GeV and α_s=0.26.  For the radiative decays of the charmonium, the partial width of the E1 transition n^2S+1L_J → n^'2S+1L^'_J^'γ is given by <cit.>,

    Γ_E1=4/3α e_c^2 ω^3δ_L,L'±1C_if|⟨ψ_f|r|ψ_i⟩|^2,

with

    C_if=max(L,L')(2J'+1)  L'   J'   S
     J   L   1 ^2,

where the e_c is the charm quark charge in units of |e|, α is the fine-structure constant, ω is the emitted photon energy and ⟨ψ_f|r|ψ_i⟩=∫_0^∞R_n'L'(r)rR_nL(r)r^2dr is the transition matrix element. 
The partial width of the M1 radiative transition n^2S+1L_J →n'^2S'+1L_J'γ with the spin flip can be written as <cit.>

    Γ_M1=4α e_c^2ω^3/3m_c^2δ_S,S'±12J'+1/2L+1|< ψ_f|j_0(ω r/2)|ψ_i>|^2,

with

    < ψ_f|j_0(ω r/2)|ψ_i>=∫_0^∞R_n'L'(r)j_0(ω r/2)R_nL(r)r^2dr,

where  j_0(ω r/2) is the spherical Bessel function. For the first excited P-wave charmonium χ_c1^', the kinematically allowed final states of E1 transition include J/ψγ, ψ(3686)γ, ψ(3770)γ and ψ_2(3823) γ, and the corresponding M1 transition process is χ_c1^'→ h_c γ.


The quark pair creation (QPC) model is a very successful phenomenological approach in depicting the two-body OZI-allowed strong decay behaviors covering the light meson to heavy quarkonium system <cit.>. The QPC model assumes that the quark-antiquark pair qq̅ created from the vacuum is a ^3P_0 state with the spin-parity J^PC=0^++, and the transition operators 𝒯 can be expressed as

    𝒯    =    -3γ∑_m⟨ 1m;1-m|00⟩∫ d p_3dp_4 δ ^3 (p_3+p_4) 
          ×𝒴_1m( p_3-p_4/2) χ _1,-m^34ϕ _0^34ω_0^34 b_3i^† (p_3) d_4j^†(p_4),

where 𝒴_lm( p)=p^lY_lm(θ_p,ϕ_p), b_3^†(d_4^†) is the quark (antiquark) creation operator, and ϕ _0^34=(uu̅+dd̅+ss̅)/√(3) and ω_0^34 are SU(3) flavor and color wave function of vacuum quark pair, respectively. The dimensionless parameter γ^2, reflecting the strength of the vacuum quark pair creation, has been determined as 40.9±8.2 in Ref. <cit.> for the charmonium family. The transition amplitude of χ_c1^'→ DD̅^* is

    ℳ=< DD̅^* |𝒯 | χ_c1^'>,

which is proportional to the overlap integral of the wave functions in momentum space

    I(q)   =   ∫ d^3pΨ_n_AL_AM_L_A(q+p) Ψ^∗ _n_BL_BM_L_B(m_q q/m_c+m_q+p)  
          ×𝒴_lm(p)
    Ψ^∗ _n_CL_CM_L_C(m_q q/m_c̅+m_q+p),

where q denotes the momentum of either outgoing charmed meson, and Ψ_nLM_L(p) is the spatial wave function, while the notation A, B and C refer to the χ_c1^', D and D̅^* states, respectively. It is worth emphasizing that the overlap integral I(q) provides the momentum-dependent part of the 𝒱_0j(q) and 𝒱_i0(q^') in Eq. (<ref>).








The numerical coupled-channel interaction 𝒱_0j(q) between the bare χ_c1^' and the D^0D̅^*0/D^+D̅^*- continuum  estimated by the QPC model is shown in Fig. <ref> with β=0.7, 0.8 and 0.9 GeV. β is a quantity reflecting the wave function distribution of χ_c1^'.  We also illustrate the variation of the total decay width of χ_c1^'→ DD̅^*+c.c. with β, which is estimated to be 250∼320 MeV. Such a large decay width also implies that the s-channel coupling with the DD̅^* may be too strong to be ignored.  Based on the same range of  β, the estimated widths of the radiative transitions and inclusive light hadron decays of the bare χ_c1^' are presented in  Fig. <ref>. It can be seen that the ψ(3686)γ is the dominant radiative decay mode although its phase space is far smaller than that of J/ψγ, which mainly benefits from the node effect from the wave function overlap. Additionally, the ψ(3770)γ and ψ_2(3823) γ are also important radiative modes, whose widths are 10∼20 keV.  The Γ(χ_c1^'→ qq̅g) can reach 2.6∼7.6 MeV, which should govern the non-open-charm decays of the bare χ_c1^'.
If assuming the DD̅^* component of X(3872) does not decay to ψ(3686)γ and J/ψγ, the theoretical value of the ratio R=Γ(ψ(3686)γ)/Γ(J/ψγ) including the phase space correction coincides with the corresponding experimental values from the Belle and LHCb Collaboration <cit.> within a narrower range of β=0.73∼0.766 GeV.




 §.§ The three-body DD̅π threshold dynamics and the self-energy effect of D̅^* in the DD̅^* scattering



A typical OPE potential V_π of the [DD̅^*]_i → [DD̅^*]_j can be written as

    V_π^ij(q,q^',z)=g^2/4f_π^2(ε· p)(ε^'· p)/p_0^2-(q^2+q^' 2-2qq^' z)-m_π^2+iϵ,

where i(j)=1, 2 stands for [D^0D̅^*0] and [D^+D̅^*-], respectively, and the p is defined as the four-momentum (p_1-p_4) of the exchanged pion, and p_1, p_2, p_3 and p_4 correspond to the four-momentum of the initial charmed meson D, D̅^* and final charmed meson D̅ and D^*, respectively. The q(q^') and z denote the magnitudes of the three-momentum p_1(p_4) and the cosine of the angle between p_1 and p_4, respectively. The coupling constant g=0.53 <cit.> and pion decay constant f_π=0.086 <cit.>.  Since the p_0∼(m_D^*-m_D)>m_π^phy will lead to an on-shell pion exchange, the DD̅^* system couples to the three-body channel of DD̅π. In order to include this three-body threshold effect, the OPE potential can be modified as <cit.>

    V_π^ij(q,q^',z)= 
          g^2/4f_π^2(ε· p)(ε^'· p)/(E^'+δ_ij)^2-(q^2+q^' 2-2qq^' z)-m_π^2+iϵ,

where the system energy E is the sum of the center-of-mass kinetic energy E^'=k_0^2/(2μ) and the lowest two-body threshold m_D^0+m_D̅^*0, and δ_11=m_D^*0-m_D^0, δ_12=m_D^*0-m_D^+, δ_21=m_D^*0-m_D^+ and δ_11=m_D^*0+m_D^0-2m_D^+.   Here, for the concerned poles near the two-body threshold, ignoring the kinetic energy of the heavy meson is a good approximation. When considering the kinetic energy term of heavy meson, Eq. (<ref>) can be further revised by the following replacement of δ→δ-(p^2+p^'2)/(2m_D).  More discussions and applications of the three-body threshold effects can be found in Refs. <cit.>.

The matrix form of the contact interaction of the DD̅^* scattering reads

    V_contact=[   C_t C_t^'; C_t^'   C_t;       ].

We define two new leading-order low-energy coupling constant V_ct^I=0 and V_ct^I=1, which can be related to C_t and C_t^' by 

    C_t   =   1/2(V_ct^I=0+V_ct^I=1), 
    
    C_t^'   =   1/2(V_ct^I=0-V_ct^I=1).

Here, the I represents the isospin of the DD̅^* system.


Because the D^*0 and D^*±  have comparable widths with the pole width of X(3872), it is necessary to take into account the width for the propagator of D^* <cit.>, which includes both the intermediate D̅π and D̅γ self-energy diagrams and  is different from the mechanism of the revised OPE potential with the three-body threshold cut. The real part of the self-energy of D^* is approximately absorbed into the physical mass of the D^* meson. The imaginary part of the self-energy of D^* is given by

    Γ_D^*+(E)   =   g^2m_D^0/12π f_π^2m_D^*+k^3_D^0π^++g^2m_D^+/24π f_π^2m_D^*+k^3_D^+π^0
          +Γ(D^*+→ D^+γ), 
    Γ_D^*0(E)   =   g^2m_D^0/24π f_π^2m_D^*0k^3_D^0π^0,

where the D^* width is treated as a function of the center-of-mass energy E rather than a constant since the D^* is not always on-shell in the complex energy space. Consequently, the the width of D^* can be introduced into the scattering dynamics by modifying the Schrödinger equation as <cit.>

    Eϕ(p)=(p^2/2μ-iΓ(E)/2)ϕ(p)+∫d^3k/(2π)^3 V(p,k)ϕ(k),

where p^2/2μ-iΓ(E)/2 means a modified kinetic energy term for the unstable system.





























 §.§ The pole behaviors of the full DD̅^* scattering dynamics by considering the isospin violation of the contact potential 



Considering the full DD̅^* scattering dynamics and the evident isospin violation of the contact interaction, the pole behaviors of the X(3872) and the new resonance are summarized in Table <ref> and Table <ref>, which correspond to V_ct^I=1=30 GeV^-2 and V_ct^I=1=-30 GeV^-2, respectively. Interestingly, there almost exists a one-to-one correspondence between the cutoff λ and the pole imaginary of the new charmoniumlike resonance, which almost does not depend on the contact potential V_ct^I=1.

Furthermore, it can be seen that the isospin violation impact on X(3872) is obviously greater than that on the dressed χ_c1^' resonance or the distorted DD̅^* resonance. This feature can explain the absence of the experimental signal of this higher charmoniumlike resonance in the final states of J/ψπ^+π^- <cit.>, which is a typical isospin violation channel of discovering the X(3872). Additionally, if adopting the center value of the pole width of X(3872) measured by the BESIII and LHCb Collaborations <cit.> as inputs, the pole width of  the new charmoniumlike resonance is predicted to be around 100 MeV, and such a large width might explain why this state is still missing in experiments. 














































































































































































































































































































































































































































































































































































































































































































































































































































