Double-charm tetraquark under the complex scaling method

The LHCb Collaboration discovered a double-charm tetraquark $T_{cc}^{+}$ with a very small width. We investigate the $T_{cc}^{+}$ as a $DD^{*}$ molecule with $J^{P}=1^{+}$ in the framework of the one-boson-exchange potential model. The isospin breaking effect and $S-D$ wave coupling are taken into account carefully. We adopt the complex scaling method (CSM) to study the $DD^{*}$ system and obtain a quasibound state corresponding to the $T_{cc}^{+}$. Its binding energy relative to the $D^{0}D^{*+}$ and width are $-354$ keV and $61$ keV respectively. The isospin breaking effect is found to be enormous, and the $S-$wave $D^{0}D^{*+}$ and $D^{+}D^{*0}$ components give dominant contributions with the probabilities of $72.1\%$ and $27.1\%$ respectively. In addition, we do not find any resonances in the $DD^{*}$ system. As a by-product, we study the $X(3872)$ as a $(D\bar{D}^*-D^*\bar{D})/\sqrt{2}$ molecule with $J^{PC}=1^{++}$. We also find a quasibound state corresponding to the $X(3872)$. Its binding energy relative to the $D^{0}\bar{D}^{*0}$ threshold and width are $-111$ keV and $26$ keV respectively. The $S-$wave $(D^{0}\bar{D}^{*0}-D^{*0}\bar{D}^{0})/\sqrt{2}$ component dominates this state with the probability of $92.7\%$.


I. INTRODUCTION
Recently, the LHCb Collaboration reported the observation of a double-charm tetraquark T + cc in the D 0 D 0 π + mass distribution [1]. The mass relative to the D * + D 0 threshold and the width extracted from the Breit-Wigner fit are δm BW = −273 ± 61 keV/c 2 and Γ BW = 410 ± 165 keV.
Obviously, the mass is just below the D * + D 0 threshold, and the width is quite narrow, which indicates a DD * molecule structure. Subsequently, the LHCb Collaboration also analyzed the pole mass relative to the D 0 D * + threshold and width in Ref. [2], and gave δm pole = −360 ± 40 +4 −0 keV/c 2 and Γ pole = 48 ± 2 +0 −14 keV.
In the aforementioned theoretical studies, the molecule and compact tetraquark pictures obtain most of the attention. In the compact tetraquark picture [5, 8, 10, 12, 14, 16, 20, 25, 26, 30-33, 40, 51], the double-heavy systems have a heavy diquark-antiquark symmetry, which * jbcheng@pku.edu.cn † lzy 15@pku.edu.cn ‡ zhusl@pku.edu.cn can contribute a deep attractive force. Therefore, this scheme may generate a bound state relative to the corresponding dimeson threshold. On the other hand, when the molecule picture is taken into account, the one-bosonexchange (OBE) potential may also provide an attractive potential to form a bound state. For instance, the work in [21] gives a prediction on the molecule T + cc with I(J P ) = 0(1 + ) quantum number, and the binding energy is −0.47 MeV, which is very close to the experimental value. Besides, the recent work in [34] considered the isospin breaking effect and predicted a new resonance T + cc .
In this work, we investigate the T + cc and X(3872) in the molecule picture. Indeed, these two exotic hadrons have some similar features. Besides the small binding energy and narrow width, one can get a quite similar OBE potential. However, they also have some evident differences. For instance, the X(3872) has some hidden-charm decay patterns, such as the J/ψρ and η c ω. By comparison, the T + cc has only the open-charm decay channels DDπ and DDγ, which provides us convenience when investigating the dimeson molecule.
In fact, compared with the general system like the BB * or BD * , the DD * has a unique behavior due to the decay process D * → Dπ. In the one-pion-exchange (OPE) effective potential, the 0-th component of the transferred momentum can be slightly larger than the pion mass. This difference will induce an effect-the pole location of the OPE potential will move from the imaginary axis (general) to the real axis (DD * ) in the complex transferred momentum plane. In the Lippmann-Schwinger equation (LSE) or the momentum space Schrödinger equation, one will get an imaginary part and reach a pole when integrating the OPE potential along the real momentum axis. In other words, the calculation is divergent. Some works [34,58] take the Cauchy principal (PV) value for the OPE potential to deal with this problem. However, we tend to get over it in a different way.
In our framework, we use the complex scaling method (CSM) [59,60] to study the DD * molecule system. We will retain the imaginary contribution from the OPE potential. After a complex scaling operation, the pole of OPE potential would be rotated from the real axis to the upper half of the momentum plane. The integral along the real momentum axis would bypass the pole and avoid divergence. In this way, we could solve the problem and get the bound states and resonances directly. As a byproduct, we also discuss the X(3872) as a molecule with the J P C = 1 ++ , and consider the isospin breaking effect.
This paper is organized as follows. In Sec. II, we will introduce our framework explicitly. In Sec. III, we present the effective potentials. In Sec. IV, we solve the complex scaled Schrödinger equation and give the results of the T + cc and X(3872) by adopting the OBE potential. The last section V is a brief summary.

II. FRAMEWORK
In this work, we assume that the T + cc is a molecule with the quantum number J P = 1 + . The mass of the T + cc is very close to the threshold of the D 0 D * + /D + D * 0 , and the interactions between these two channels are similar. Thus the isospin breaking effect should be considered. The masses of the D ( * ) mesons and exchanged light mesons are shown in Table I. To deal with the isospin breaking effect, we take into account the channels D 0 D * + ( 3 S 1 , 3 D 1 ) and D + D * 0 ( 3 S 1 , 3 D 1 ), see Table II. We do not consider the channels D * D * ( 3 S 1 , 3 D 1 , 5 D 1 ). Their thresholds are evidently higher than the mass of the T + cc , indicating very small contributions. System  We first consider the OPE potential. As mentioned above, the DD * system is unique in contrast to theBB * orBD * /DB * system-the D * can decay to Dπ, which provides an imaginary part in the OPE potential. To study the influence of the imaginary part, we could adopt one of the following equations: 1) The coordinate space Schrödinger equation 2) The momentum space Schrödinger equation 3 p 2 dp V l,l (p, p )φ l (p ) = Eφ l (p). (4) These two equations are equivalent, and we only consider the first case in this paper.

A. A brief introduction to the CSM
Before discussing the analyticity of the OPE potentials, we give a brief introduction to the CSM herein. This method was proposed by Aguilar, Balslev and Combes [59,60] and the corresponding conclusion is called the ABC theorem. This powerful tool could directly and simultaneously get the solutions of the bound states and resonances. In this method, the resonances can be solved in the same way as the bound states. In the CSM, a simple transformation U (θ) for the radial coordinate r and its conjugate momentum k is introduced as: Then the radial Schrödinger equation is transformed as As explained in the ABC theorem, one could get resonances after making a rotation on the momentum k. If the rotation angle θ is large enough, the resonance pole will cross the branch cut into the first Riemann sheet, as shown in Fig. 1. The details can be seen in Ref. [61,62]. In fact, the authors of Ref. [19] used the CSM to discuss the double-heavy tetraquark system. They took an instantaneous approximation and ignored the q 0 , which is the 0-th component of the transferred momentum. However, this treatment may change the analyticity of the OPE potential. In this framework, we retain the q 0 and get some different features from the situation in the general dimeson systems. We will consider the BB * and DD * cases to interpret these differences. The OPE potentials of the processBB * →B * B and DD * → D * D have the same form as follows where the is the polarization vector of theB * or D * meson, the τ 1 and τ 2 are the isospin operators of theB ( * ) or D ( * ) meson. The denominator above gives For theBB * case, we have q 0 ≈ m B * − m B < m π . Therefore, the poles are located on the imaginary axis of the transferred momentum | q|. When making the Fourier transformation to get the potential V π (r) in coordinate space, we need to take a contour integral in the upper For theBB * /DD * systems, we show the poles of the OPE potential in the momentum space, and the corresponding contour integral (red dashed lines). We plot (a) theBB * case (b) the DD * case in the Cauchy principal (PV) value scheme (c) the DD * case in the Feynman prescription (FP) scheme (d) the DD * case in the complex scaling method.
half of the transferred momentum plane. The pole inside the contour will give a contribution to this integral, as illustrated in Fig. 2(a). In addition, this situation can also occur in the one-σ/η/ρ/ω-exchange potentials for theBB * and DD * systems. However, the OPE potential for the DD * case is different. We take the process D 0 D * + → D * + D 0 as an example. Since the q 0 ≈ m D * + − m D 0 > m π + , one could get a denominator −(q 2 − m 2 ef f ), where the shorthand Obviously, the poles are located on the real transferred momentum axis, and the ambiguity emerges. In this situation, different contour integral schemes may lead to different results. One of the schemes is to take the Cauchy principal (PV) value. The OPE potential in the PV scheme is proportional to The contour integral is illustrated in Fig. 2(b). However, we will adopt another scheme-contour integral under the Feynman prescription (FP), see Fig. 2(c). The corresponding OPE potential is proportional to Obviously, these two schemes will lead to different results. One can get a purely real potential in the PV scheme and a complex potential in the FP scheme. In other words, the OPE potential will contribute an imaginary part in the latter case. Then we explore the analyticity of the OPE potential in the CSM. As introduced in II A, the CSM will rotate the propagator momentum k with the angle θ, and the poles of the transferred momentum become ±e iθ q 2 0 − m 2 π . The pole on the positive real axis can be naturally moved into the contour region under this operation, while the other one is moved outside the region, as illustrated in Fig. 2(d). It can be seen that the CSM coincides with the FP scheme shown in Fig.2(c). In other words, they both choose the pole on the positive real axis, which leads to an extra imaginary potential.
In addition, when using the momentum space Schrödinger equation (3) to handle the OPE case for the DD * system, we will reach a pole located on the real axis of momentum. This pole will lead to the divergence in numerical calculations. In fact, the CSM could also handle this problem. Similar to the situation in the Fourier transformation, the poles will be moved from the real axis to the complex region of the momentum plane under the complex scaling operation. Then the integral along the real momentum axis will bypass the pole, and the divergence disappears.

C. Three-body intermediate coupled channel
As interpreted in the previous subsection, the additional imaginary contribution in the DD * case is from the decay process D * → Dπ. Therefore, the three-body coupled channel effect of DDπ cannot be neglected. In Subsec. II B, we use an approximate expression q 0 ≈ m D * − m D for the DD * case. However, this approximation is kind of rough since the q 0 is quite close to the m π . Therefore, we should make a careful discussion on the q 0 herein. For simplicity, we take the D 0 D * + → D 0 D * + case as an example.
The total energy of the D 0 D * + is E, and the D 0 is assumed to be on shell.
As illustrated in Fig. 3, we denote the total energy of the D 0 D * + as E and assume the D 0 meson to be on shell. Then the can get a similar propagator as shown in Ref. [43]. They use the time-ordered perturbation theory with the non-relativistic propagator. By contrast, we only make a non-relativistic approximation for the charmed meson D/D * propagator, and keep a relativistic form for the π meson propagator.
In fact, we will neglect the kinetic energy terms p 2 /2m D 0 and p 2 /2m D 0 of the charmed mesons, which are much less than the q 0 −m π . Then we make an energy shift E → E + m D 0 + m D * + , and the potential for the D 0 D * + → D 0 D * + process gives Obviously, when solving the Schrödinger equation in coordinate space, one cannot get the solution as usual since the energy E is unknown. However, we can still get a high precision solution by iterating the Schrödinger equation.

III. POTENTIALS
The effective Lagrangians to describe the interactions for the DD * system are constructed in terms of the heavy quark symmetry and chiral symmetry. The concrete expressions of the OBE Lagrangians can be seen from Ref. [21].
The other OPE potentials in momentum space are We add a monopole form factor at each vertex where the q 2 = q 2 0 − q 2 , and the Λ is a cutoff parameter. After the Fourier transformation we can get the coordinate space potentials where the q 0 s of Eqs. (13)(14)(15) are listed in the Eqs. (8)(9)(10) respectively. In the above equations, S(a, b) = a · b and T (a, b) = 3(a·r)(b·r)/r 2 −a·b. The matrix elements are given in Table III. The Y 3 , H 3 functions and relevant Y , Z, H functions are defined as , where the The factor Sign[Re(e iθ m 2 − q 2 0 )] can be deduced from the DD * potentials in Subsec. II B. The CSM rotation angle θ is in the region 0 ≤ θ ≤ π/2. We take the Y (ur) function as an example to discuss the behavior of the potentials. One can get Obviously, the Y (ur) can contribute an imaginary part to the OPE potential when m 2 − q 2 0 < 0.
We also consider the short-and medium-range oneσ, η, ρ, ω-exchange potentials from the OBE interactions. The corresponding potentials are given in Eqs. (18)(19)(20)(21). These potentials have no imaginary contributions due to the large exchanged meson mass, and the isospin breaking effects barely have influence on the one-σ, η, ρ, ωexchange potentials.
So, we ignore the mass difference between the D 0 (D * 0 ) and D + (D * + ) herein. Then the 0−th components of the transferred momenta have simple forms where the q D 0 and q C 0 correspond to the direct and cross diagrams in the processes DD * → DD * .

IV. NUMERICAL RESULTS
A. The OPE potential results for the DD * system We first introduce the OPE potentials only to study the DD * system. The Gaussian expansion method (GEM) [67] is adopted to solve the complex scaled Schrödinger equation, see Eq. (6). In this calculation, to determine the only unknown parameter Λ, we take the mass of the T + cc as an input. Then we get the cutoff Λ = 1602 MeV, and illustrate the CSM eigenvalue distribution in Fig. 4. We plot two lists of eigenvalues obtained from rotational angles θ = 15 • , 25 • . Only one pole is found with the energy E = −364 − 31i keV relative to the threshold of D 0 D * + . This pole could be interpreted as a quasibound state corresponding to the T + cc .
Considering the coupled channels D 0 D * + and D + D * 0 (D 0 D 0 π + and D 0 D + π 0 ), we could find the branch cuts with their branch point at the two-body (three-body) thresholds. Obviously, the quasibound state pole in Fig.  4 is located on the first Riemann sheets (physical sheets) corresponding to the D 0 D * + and D + D * 0 channels and

Im(Energy)
[MeV] the second Riemann sheets (unphysical sheets) corresponding to the D 0 D 0 π + and D 0 D + π 0 channels. In Fig.  5, we choose θ = 15 • and present the real and imaginary part of its wave function. The probabilities of the channels D 0 D * + ( 3 S 1 , 3 D 1 ) and D + D * 0 ( 3 S 1 , 3 D 1 ) are shown in Table IV. Obviously, the S-wave D 0 D * + and D + D * 0 channels dominate this quasibound state, which is similar to the result of Ref. [34]. However, we do not find the resonance predicted in their work.
B. The OBE potential results for the DD * system In this part, we further employ the OBE potential to include the short-and medium-range contribution. When taking the mass of the T + cc as an input, we use the cutoff Λ = 1170 MeV and get the CSM eigenvalue distribution as illustrated in Fig. 6. Similar to the OPE potential case, no resonance is found herein. The only pole of the quasibound state has the energy E = −354 − 31i keV relative to the threshold of the D 0 D * + . The wave functions with θ = 15 • are also shown in Fig. 7. The probabilities for the four channels are also shown in Table IV. One can see that the S-wave D 0 D * + and D + D * 0 channels still dominate this quasibound state. We interpret this pole as the T + cc , and its width also coincides with the experimental data nicely.
Interestingly, after we take the binding energy as input, the OPE and OBE cases give quite similar results, including the width, constituent and size. When we consider the long-range behavior of the T + cc , such a similarity turns out to be reasonable. In our framework, the T + cc is assumed to be a molecule with a very small binding energy, which has a very large size. The RMS in Table IV also supports this speculation. Therefore, the long-range OPE interactions should be dominant in this system, and we should obtain similar results with these two schemes.
In addition, we adjust the cutoff Λ and get a list of eigenvalues to see the correlation between the binding energy and width of this pole. As presented in Fig. 8, the width decreases rapidly as the binding energy becomes deeper. And the width vanishes when the energy is smaller than the thresholds of D 0 D 0 π + and D 0 D + π 0 . This threshold effect is reasonable since we do not consider the electromagnetic decay process in this work. Thus, when the DDπ channel is closed, the imaginary part of the pole disappears, and the pole becomes a stable bound state.  Table II. The RMS is the root-mean-square radius in the CSM, which has been discussed in the Ref. [68]. Its real part is interpreted as an expectation value, and the imaginary part corresponds to a measure of the uncertainty in observation.

C. The OBE potential results for the [DD * ] system
We also discuss the molecule system [DD * ] with J P C = 1 ++ , where the shorthand notation [DD * ] is  Table II. The OPE potentials of the [DD * ] are given in Eq. (24), and the other terms of the OBE potential can be found in Ref. [58]. The matrix elements S( † 3 , 2 ), S( † 4 , 2 ), T ( † 3 , 2 ) can also be  obtained from Table III.
In this work, we need the mass of the X(3872) as input. Recently, the LHCb Collaboration studied the lineshape One can see the mass of the X(3872) is extremely close to the D 0D * 0 threshold, so the accurate binding energy relative to the latter is hard to be determined. We take a binding energy −111 keV relative to the [D 0D * 0 ] as input within the experimental error, and the corresponding cutoff is Λ = 1155 MeV. Then we get the CSM eigenvalue distribution as illustrated in Fig. 9. Similar to the DD * system, we only find a quasibound state with energy E = −111 − 13i keV. The pole is located on the first Riemann sheets (physical sheets) corresponding to the [D 0D * 0 ] and [D +D * − ] channels and the second Riemann sheet (physical sheets) corresponding to the D 0D0 π 0 channel. The probabilities for the [DD * ] channels are shown in Table IV, and the S−wave [D 0D * 0 ] dominates the system. The width value is smaller than the central value in Eq. (23) given by the LHCb Collaboration. However, our width value is still consistent with the new measurement within one standard deviation if one considers the large experimental errors. This possible difference may arise from the neglect of the other channels such as the hidden-charm channels J/ψρ, η c ω and the electromagnetic channel D 0D0 γ, which may provide a considerable contribution.
For the X(3872), the error of the binding energy given by LHCb Collaboration is significant compared with the binding energy value. Thus, we also give a list of eigenvalues with different binding energies corresponding to the pole in Fig. 10. One notices that the width will vanish when the pole energy is less than the D 0D0 π 0 threshold and reach the largest value ≈ 30 keV when the binding energy is around −300 keV.

V. SUMMARY
In 2021, the LHCb Collaboration observed a doublecharm tetraquark T + cc [1] with a very small binding energy relative to the D 0 D * + threshold and a narrow width, which indicates a molecule structure. This discovery encourages us to make a careful analysis of this exotic state. We use the complex scaling method to investigate the possibility of the T + cc as a J P = 1 + molecule with both the OPE and OBE potential. To see the influence of the isospin breaking, we take into account the channels The OPE potentials for the DD * system are unique. They have two poles located on the real transferred momentum axis. Some previous works chose the Cauchy principal value (PV) contour integral scheme, which leads to a real OPE potential. However, as shown in Fig.2(c), we adopt the Feynman prescription (FP) contour integral scheme, which can provide an imaginary contribution. This imaginary part comes from the process D * → Dπ, which can be naturally understood in the framework of the CSM.
When adopting the OBE potential, we get a pole corresponding to the T + cc , whose binding energy relative to D 0 D * + is −354 keV, and the width is 61 keV. The Swave D 0 D * + and D + D * 0 components dominate the system, whose probabilities are 72.1% and 27.1% respectively. One can see that the isospin breaking effect plays an important role in this quasibound state. Interestingly, the OPE and OBE potential cases almost give the same results, which indicates a domination of the long-range OPE dynamics. Since our approach could give the energy and width simultaneously with only one parameter (Λ), one can predict the width once the energy is determined. Therefore, this method can provide a satisfactory description of the T + cc observed in the experiment. The energy-width dependence in the DD * system is presented in Fig. 8. Besides, we do not find a resonance in this framework.
We also study the X(3872) with the same method. Similarly, we find a quasibound state with the binding energy −111 keV relative to the [D 0D * 0 ], and the corresponding width around 26 keV. The dominant channel is the S−wave [D 0D * 0 ] with the probability 92.7%. Apparently, the isospin breaking effect is significant. The numerial results are in Table IV. It is helpful to make a simple analysis of the decay behavior of the T + cc . In this work, the T + cc is interpreted as a molecule with the constituents D 0 D * + and D + D * 0 . Since the channels D + D + π − and D + D − π + are kinetically forbidden, the primary decay channels are the D 0 D 0 π + and D 0 D + π 0 . In addition, the D 0 D 0 π + is expected to be the more important channel due to the larger isospin factor.
On the other hand, the long-range OPE interactions play an important role in both T + cc and X(3872) molecules in our framework. The obtained sizes of the T + cc and X(3872) are around 5 fm and 7 fm respectively.
Therefore, a similar long-range electromagnetic interaction may also provide a non-negligible partial width. Thus, it is very interesting to study the three-body eletromagnetic decay partial widths of the T + cc and X(3872). We may make a detailed investigation of these topics in the coming future.