$D D^{*}$ potentials in chiral perturbation theory and possible molecular states

The $DD^{*}$ potentials are studied within the framework of heavy meson chiral effective field theory. We obtain the effective potentials of the $DD^{*}$ system up to $O(\epsilon^2)$ at the one-loop level. In addition to the one-pion exchange contribution, the contact and two-pion exchange interactions are also investigated in detail. Furthermore, we search for the possible molecular states by solving the Schr\"odinger equation with the potentials. We notice that the contact and two-pion exchange potentials are numerically non-negligible and important for the existence of a bound state. In our results, no bound state is found in the $I=1$ channel within a wide range of cutoff parameter, while there exists a bound state in the $I=0$ channel as the cutoff is near $m_\rho$ in our approach.


I. INTRODUCTION
Chiral effective field theory (ChEFT) is an effective field theory respecting the chiral symmetry of Quantum chromodynamics (QCD) at low momenta. A prominent feature of ChEFT is that the results are expanded as a power series of small momenta rather than small coupling constants, which enables us to systematically study into the non-perturbative regime of the strong interaction. Pseudo-Goldstone bosons such as pion and kaon, with light masses, play very important roles for the low energy processes. Chiral symmetry constrains the form of the interaction quite strongly. Owing to the clear power counting scheme, ChEFT is very powerful to investigate the properties of light pseudoscalar bosons [1][2][3].
The situation becomes complicated when heavy hadrons involve. The power counting rule is broken because of large hadron masses. However, for the system with single heavy hadron and few light pseudoscalar bosons, the power counting scheme can be easily rebuilt, and many approaches of ChEFT have been developed to deal with the relevant scattering, interaction, electromagnetic moments, and other properties of such system. Heavy hadron chiral perturbation theory, the infrared regularization, and the extended-on-mass-shell scheme are frequently used in one heavy hadron sector [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Unfortunately, these approaches cannot be directly extended to study the properties about few heavy hadrons, like nuclear force.
Two-nucleon interaction bears another power counting problem. Two approximately on-shell nucleons in loop diagrams cause extra enhancement compared to the naive power counting, which prevents us from calculating scattering matrix directly. Weinberg proposed a framework to deal with the issue [19,20]. One can first calculate an effective potential, i.e., sum of all two-particle irreducible (2PI) diagrams, and then iterate it with equations, such as Lippmann-Schwinger * Electronic address: liuzhanwei@lzu.edu.cn † Electronic address: xiangliu@lzu.edu.cn and Schrödinger equation, to retrieve two-particle reducible (2PR) contributions. The Weinberg's formalism has been further extended and developed [21-24, 27, 29-35]. For example, a unitary transformation is presented to remove the energy dependence of the potential in Refs. [23,24]. The renormalization of potentials are carefully studied in Refs. [25][26][27][28][29]. The authors in Ref. [30] revisit the nucleon-nucleon potential up to NNNLO within ChEFT. In Refs. [31,32], the nucleonantinucleon potential is investigated within ChEFT. Very recently, a covariant formalism of the N-N interaction is proposed in Ref. [33]. Three body and even four body nuclear forces have been systematically studied within ChEFT, see Refs. [34,35] for a review. The application of ChEFT has been definitely advancing our understanding of the nuclear force [36].
With successes in the study of nuclear force, one may wonder whether ChEFT can help us to comprehend the interactions of heavy (charmed, bottomed) meson systems. Obviously, since heavy meson is heavier, we can make some assumptions such as the heavy quark limit without worries, and thus heavy hadron ChEFT is even more suitable than that in the nucleon system.
The XYZ and similar exotic states have attracted a lot of interest in the hadron physics, and it is well known the interaction between heavy mesons is quite responsible for the strange behavior at close threshold in charmonium and bottomonium spectra (see Ref. [37] for a review). This starts with the discovery of the famous X(3872), which was observed by the Belle Collaboration in B decay process B +− → K +− π + π − J/ψ in 2003 [38]. X(3872) is extremely close to the threshold of D 0D * 0 . Its mass is much smaller than quark model (such as the Godfrey-Isgur model [39]) predictions if it is regarded as χ ′ c1 (2P) charmonium, and moreover it has a large decay width for the isospin violation process X(3872) → J/ψρ. After that, more and more XYZ and other exotic states candidates were discovered, such as recent observed pentaquark P c (4380) + and P c (4450) + [40] and still debated X(5568) [41].
There are many models dealing with these states, such as the one-boson-exchange molecular model, some underlying multiquark models, kinematical effect, and so on (see the re-view [37]). For example, in Refs. [42,43] D * (s)D * (s) and DD * systems are studied within the local hidden gauge formalism to dynamically generate Y(3940), Z(3930), X(4160) and Z c (3900). In Ref. [44], the authors have investigated the DD * system and its relation to Z c (3900) using the covariant spectator theory. Z c (3900) is also studied from the pole counting rule [45]. The authors in Ref. [46] have discussed D ( * )D( * ) with the constituent quark models, and solved the four-body Schrödinger equation with the Gaussian expansion method. The contact interaction of DD * (BB * ) is specially investigated in Ref. [47] with the effective field theory, which is implemented with the heavy quark symmetry. The DD * system is also intensively studied with different kinds of effective field theories, see Refs. [48][49][50][51][52][53][54][55][56][57][58] and many other works citied therein. For example, in Ref. [48], the authors studied the DD * with XEFT using perturbative pions. In Ref [50], the authors studied X(3872) and DD * using non-perturbative pions. Moreover, the authors in Ref. [58] further included the effects of the D * width. In Ref. [59], the study of hadronic molecules with effective field theories are reviewed.
As mentioned above, there are many models dealing with heavy meson systems. Among them, the one-boson-exchange model has interpreted many exotic phenomenas and made some predictions which have been verified by the later discoveries of new particles at experiment. This model can provide the dynamical potentials of hadron systems, and then one can solve the Schrödinger equation to see if there is a bound state. The model has been widely used to study the interaction of the two-heavy-hadron systems and related exotic states. The research of the charmed-anticharmed system and X(3872) experiences a long progress. It starts from the pion and σ exchanges in early Ref. [60], directly extends to the multi-state exchanges [61], and then includes more complicated effects from S -D mixing [62], isospin violation [63], and so on. For the investigation of nuclear force, after the boson exchange model develops for decades (see the discussion in Ref. [64]), ChEFT is applied at last and help us systematically build the modern system of knowledge. Following their steps, it is natural to introduce ChEFT into the study of heavy meson systems after the one-boson-exchange model.
There exist many works on heavy mesons system with oneboson-exchange model and effective field theories, as mentioned above. It is interesting to investigate their higher order effects in chiral effective field theory, and then discuss the potential in coordinate space and search for the bound state by solving Schrödinger equation. We will also compare the results with one-boson-exchange model.
In this work, we focus on the doubly charmed-meson system DD * , which is clearer than the hidden charmed system for the absence of annihilation channels. It provides us another insight to understand the heavy-flavor dynamics and nonperturbative QCD. Furthermore, it is analogous to deuteron since they both have contact, one-pion exchange (OPE), and two-pion exchange (TPE) contributions without annihilation channels in our framework.
Till now, the only observed doubly heavy-flavor system is the Ξ ++ cc baryon which was first discovered by SELEX collaboration [65]. Systems like ccu and ccd have been discussed a lot, and their properties such as masses and electromagnetic moments still need more efforts to get clarified [69][70][71][72][73][74][75][76]. Very recently, LHCb group confirmed the existence of Ξ ++ cc but disfavored the mass measured at SELEX [68]. With the technique and apparatus well developed nowadays, it is also possible to search for the doubly charmed boson made of DD * at experiment.
In Ref. [77], the authors studied D ( * ) D ( * ) (B ( * ) B ( * ) ) system to search for bound and resonant states, and they used pion and vector meson exchange potentials which are constrained by heavy quark symmetry and chiral symmetry. They found that in isospin 0 channel there exists a bound state in S -wave with binding energy 62.3 MeV, and no bound state is found in isospin 1 with S -wave. In Ref. [78] the authors have studied D ( * ) D ( * ) system using the one-boson-exchange model, and found that there exists a bound state consisting of DD * with binding energy 5 ∼ 43 MeV in the isospin 0 channel. The authors in Ref. [79] investigate deuteron-like molecules with both open charm and bottom using the heavy-meson effective theory. In Ref. [80], charm-beauty meson bound states are dynamically generated from the B ( * ) D ( * ) and B ( * )D( * ) interactions, and they also give the informations of the scattering lengths. There also exist lattice studies on BB and BB * interaction [81][82][83]. Especially in Ref. [83], the authors considered both diquark-antidiquark and meson-meson configuration. In Ref. [84], we have investigatedBB interaction within heavy meson chiral effective field theory (HMChEFT). We obtain the potentials of theBB system at one loop level, and have discussed the contact and two-pion exchange contributions in momentum space.
We investigate the DD * system in this work. As we mentioned before, we need to study the potentials first, and then access physical observables indirectly. Furthermore, the potential in coordinate space can give us more intuitive information about interaction between mesons, and we can further solve a dynamic equation to see whether there exists a hadronic molecule. This paper is organized as follows. After introduction, we elucidate the framework in Sec. II. In Sec. III, we give results of potentials in momentum space. In Sec. IV, we study the potential in coordinate space to search possible molecules. At last, we summarize our conclusions.

II. LAGRANGIANS AND WEINBERG SCHEME
To study the DD * system under HMChEFT, we need to show Lagrangians and provide results systematically in a strict power counting scheme. Our results are arranged order by order with the small parameter ǫ = p/Λ χ , where p can be the momentum of pion, the residual momentum of heavy mesons, or the D-D * mass splitting, and Λ χ represents either the chiral breaking scale or the mass of the heavy mesons. In this work, flavor SU(2) symmetry is always kept.
A. Lagragians at the leading order At the leading order O(ǫ 0 ), both OPE diagrams and contact diagrams contribute to the amplitudes, and thus we should first build the Langrangians for DD * π interaction vertices, the corresponding contact vertices, and so on.
The DD * π Lagrangian at leading order [85][86][87] is given by In the above, H field represents the (D, D * ) doublet in the heavy quark limit v = (1, 0, 0, 0) stands for the 4-velocity of the H field. The last term in Eq.(1) is included to account for D-D * mass shift which is not zero in the chiral limit, and δ is the mass difference in (D, D * ) doublet. The axial vector field u and chiral connection Γ are expressed as where ξ = exp(iφ/2 f ), f is the bare constant for pion decay, and The contact Lagrangian at O(ǫ 0 ) is constructed as follows [47,51,84] where D a , D b , E a , E b are four independent low energy constants (LECs).
B. Lagrangians at the next to leading order At chiral order O(ǫ 2 ), the total amplitudes consists of the contact corrections, OPE corrections, and TPE amplitudes. These one-loop amplitudes must be renormalized with the help of O(ǫ 2 ) Lagrangians. The divergences in the one-loop amplitudes are canceled by the infinite parts of the LECs in the following lagrangians [84], whereχ Note that, the term L (2,d) 4H in Ref. [84] vanishes in our SU(2) case.
In addition to canceling the divergences of the loop diagrams, the above Lagrangians also contain finite parts that contribute to tree-level diagrams at O(ǫ 2 ). They are governed by a large amount of LECs appearing in Eqs. (6)-(8).

C. Weinberg scheme
In this work, we adopt the power counting scheme from Weinberg to study the DD * systems [19,20]. This framework has been widely applied to nucleon-nucleon system as mentioned in the introduction. Let us start with a nucleonnucleon TPE box diagram depicted in Fig. 1. As illustrated in Ref. [84], the amplitude can be written under the heavy hadron formalism: where m N is the mass of the nucleon, q 1 = P+ l and q 2 = P− l. Naive power counting gives the l 0 integral O(| P| −1 ), while we notice from Eq. (10) that the l 0 integral should be of O(| P| −2 ), i.e. the true order is enhanced by | P| −1 . Such enhancement definitely violates the power counting rule, which would invalidate the chiral expansion. As pointed out in Ref. [19,20] , the origin of such a contradiction comes from double poles in Eq. (10) which relates to two-particle-reducible (2PR) part of the box diagram in Fig. 1. With the above analysis in mind, we just fall into the same situation when studying the interaction of the doubly-charmed meson pair, and thus can not directly calculate the scattering amplitude. Alternatively, we apply the Weinberg's power counting scheme. First, with the usual power counting rule, we compute the 2PI contributions of all diagrams, and this leads to effective potentials. Then we substitute the potentials into iterated equations such as Lippmann-Schwinger equation or Schrödinger equation to recover the 2PR contributions. Finally, we would obtain the desired scattering amplitudes or energy levels.

III. EFFECTIVE POTENTIALS OF DD * SYSTEM
The effective potentials of DD * system receive contributions from the contact and OPE diagrams at the leading order O(ǫ 0 ). At the next to leading order O(ǫ 2 ), there are both tree and one-loop corrections. The effective potentials V are related to the Feynman amplitudes M of 2PI diagrams which follows from the one-boson-exchange model despite some differences in conventions [90,91]. At the lowest order O(ǫ 0 ), there are two diagrams at tree level illustrated in Fig. 2. They represent the contact and OPE contributions, individually. The contact terms mainly affect the short range interaction between particles while the OPE contribution determines the behavior of the long range interaction. With Lagrangians (1) and (5), the corresponding amplitudes can be easily computed. For the process D(p 1 )D * (p 2 ) → D(p 3 )D * (p 4 ) with isospin I = 1, the amplitudes for diagrams (a) and (b) in Fig. 2 read For the process D(p 1 )D * (p 2 ) → D(p 3 )D * (p 4 ) with I = 0, the amplitudes are In above equations, momentum p = p 1 − p 4 , the superscript (0) denotes the order O(ǫ 0 ), and the subscripts "I = 0, 1" stand for the process DD * → DD * with isospin 0, 1, respectively. At O(ǫ 2 ), a number of diagrams emerge. The tree diagrams at O(ǫ 2 ) are similar to Fig. 2 (a), but the vertices should be replaced with those from Lagrangians (6-8). There are additionally three sets of one-loop diagrams.
The diagrams in the first set are for one-loop corrections to the contact terms. They are depicted in Fig. 3. Diagrams (a12)-(a12) represents contributions from the wave function renormalization of external legs.
We show the second set of diagrams in Fig. 4. They represent one-loop corrections to the OPE diagrams. The diagrams (b1)-(b6) and (b8)-(b9) in Fig. 4 contribute to the renormalization of the DD * π vertex. Therefore, we must use the value for the bare coupling g in M (0) at O(ǫ 0 ) to avoid double counting. We show the relation between the bare coupling g and the experiment coupling g (2) in Eq. (B1) in Appendix B. Similarly, the bare decay constant f should be used in Eqs. (13,15), too.
Final set is for the TPE diagrams which are illustrated in Fig. 5. They are important for the medium range interaction.
As discussed in the previous section, some diagrams, such as the box diagrams in Fig. 5, contain a 2PR part that should be subtracted. If there exists a loop function of a box diagram like  following Ref. [84,88], we can separate the 2PR and 2PI parts by The term proportional to Dirac δ function is just the 2PR part which should be dropped in potentials. All the one-loop amplitudes of the diagrams Figs. 3-5 for the processes DD * → DD * are shown in Appendix A. The divergences of the loop functions are regularized with the dimensional regularization, and subtracted by the modified minimal subtraction scheme. Also, we list the definitions of the loop functions in Appendix C. The finite parts of the high order lagrangians should also contribute to tree level diagrams at O(ǫ 2 ), and they are governed by a large number of LECs. However, it needs plenty of data for DD * (or other channels such as DD, DD) scattering in different partial waves to fit these LECs, but there is still lack now. Therefore in the present work, we only focus on the loop contributions at O(ǫ 2 ). We can easily obtain the potentials V DD * I=1 and V DD * I=0 from the Feynman amplitudes by multiplying a factor −1/4. The polarized vectors in the potentials are delicately dealt with in Ref. [89]. In this work, we only consider the S -wave interaction, which leads to the following substitutions in Eqs. (12)- where we follow from the one-boson exchange model in Ref. [90,91]. After all these procedures, the effective potentials V DD * I=1 and V DD * I=0 in the momentum space can be obtained. But, the potentials are energy dependent. A solution to this problem is proposed in Refs. [23,24], where they apply a unitary transformation to get rid of the energy dependence. While in this work, we just take the transfered energies equal to zero, i.e. p 0 = 0 and q 0 = 0 for simplicity, as in the one-boson exchange model [60]. Also, we take the residual energies of the heavy mesons equal to zero, too.
First, we list the results for the contact contributions. For V DD * I=1 in the channel of isospin 1, the effective potential at O(ǫ 0 ) and O(ǫ 2 ) is as follows And for the channel of isospin 0, we obtain Obviously, the contact contributions are just constants, and they result in δ(r) potentials in coordinate space, which describes short distance effect. From Eqs. (20)- (23), we see the convergence of the series expansion is good. From Eqs. (21) and (23), the contact coupling constant D a does not appear in the effective potential at O(ǫ 2 ) because the contributions from D a term are canceled among various diagrams in Fig. 3. Next, we focus on the properties of the OPE and TPE contributions. We illustrate the corresponding potentials for channels with isospin 0 and 1 in Figs. 6 and 7, respectively, ranging from q = |q| = 0 to 300 MeV. those at O(ǫ 0 ). The sums of OPE contributions are negative from Figs. 6 and 7, which means the OPE interaction is attractive in both the I = 0 and I = 1 channels. We also notice that the OPE interaction in I = 0 is more attractive than I = 1.
The situation for the TPE potentials is more complicated. The TPE contributions behave differently in I = 0 and I = 1 channels. In Fig. 6, the TPE interaction for I = 0 channel is attractive in the range 0 ∼ 300 MeV, and it tends to grow beyond 300 MeV. The TPE potential at O(ǫ 2 ) is larger than the OPE one at O(ǫ 0 ) in the range 0 ∼ 60 MeV, while the OPE contribution exceeds that of TPE rapidly when q is larger than 60 MeV, and becomes dominant. We can say that the convergence of the chiral series is good. Looking at Fig. 7, we see the TPE potential is repulsive in the range 0 ∼ 120 MeV, while it becomes attractive as q is beyond the range. The TPE potential at O(ǫ 2 ) is smaller than the OPE contribution at O(ǫ 0 ) in the lower range of the momentum, and becomes comparable in large momenta. It seems to indicate the convergence of the chiral series would be spoiled at larger transfered momenta. From the blue dot-dashed lines in Figs. 6 and 7, we see the TPE interaction in I = 1 is more attractive than that in I = 0.
Let us turn to the sum of these three contributions. The total contribution in Fig. 6 for V DD * I=0 is attractive, while in Fig. 7 for V DD * I=1 it is less attractive and tends to repulsive as q becomes smaller than 50 MeV because of the repulsive TPE contribution. It makes us wonder whether there could form a bound state in the DD * system with the inclusion of contact contributions.

V. POTENTIALS IN COORDINATE SPACE AND POSSIBLE MOLECULAR STATE
Although the pion exchange interaction is attractive at most momenta, there can still be no bound states if not attractive enough. Moreover, the contact interaction might be repulsive and furthermore decrease the possibility for the existence of a bound state. Thus the contact potentials must be first obtained numerically by the determination of the LECs. After that, we can investigate the effective potentials in coordinate space, and then solve the Schrödinger equation to search for possible molecular states.

B. Potentials in coordinate space
After the determination of the LECs, we are ready to transfer the potentials into coordinate space: However, since V(q) in ChEFT is proportional to the power series of q, the higher order terms diverge worse. The evaluation of V(r) is essentially a non-perturbative problem, and it originates from the resummation of the 2PI potentials. We have to regularize Eq. (30) non-perturbatively. Enormous efforts have been made to explore the non-perturbative renormalization, such as Refs. [21,23,[109][110][111][112][113][114]. Here we resort to a simple Gaussian cutoff exp(− p 2n /Λ 2n ) to suppress the higher momentum contributions, as in Ref. [22,24,33]. We use n = 2 as in Ref. [33]. In the nucleon-nucleon ChEFT, the value of cutoff parameter is commonly below the ρ meson mass [30], and therefore we adopt Λ = 0.7 GeV in our work. The resulting full potentials are shown in Figs. 8 and 9, where we set Λ = 0.7 GeV. From Figs. 8 and 9, we find the OPE and TPE interactions are attractive in both cases, and the contact terms lead to the attractive interaction in the I = 0 channel while repulsive interaction in the I = 1 channel. Obviously, this difference brings more opportunity to form a bound state in the I = 0 channel than in the I = 1 channel. Let us focus on the total results. The total potential in the short distance for the I = 1 channel is repulsive but small while that for the I = 0 channel is attractive and large.

C. Possible bound states
With the potentials in hand, we are finally able to solve Schrödinger equation. We find a bound state with the bind- The radial wave function for the I = 0 channel is plotted in Fig. 10. It extends to quite large distance, which means the constituents D and D * are separated.
It is worth noticing that in pion and vector-meson exchange potential model [77] a bound state was found with a binding energy of 62.3 MeV in I = 0 channel, while no state was found in I = 1 channel. In the one-boson exchange model, there is also a bound state in the I = 0 channel, and the binding energy is about 5 ∼ 43 MeV with a reasonable cutoff [78]. No bound state was found in the I = 1 channel in that model, either [78]. Our results are consistent.
From Fig. 9, we notice the contact interaction is repulsive at short distance. However, we cannot still find a bound state even if dropping the contact interaction in the I = 1 chan-nel, which states the pion exchange interaction is not attractive enough for binding DD * . If we repeat and turn off the contact potential in the I = 0 channel, the shallow bound state will disappear. We cannot obtain a reasonable energy eigenvalue of the Schrödinger equations, either, if keeping the OPE potentials themselves for two channels. The attractive contact and TPE interactions are important for the existence of the molecule in the I = 0 channel.
Theoretically, the obtained observable (such as binding energy) is independent of the regularization procedure in Eq. (30). The formal dependence on the cutoff Λ in Eq. (30) can be compensated by the Λ dependence of the LECs. However, the results are sometimes sensitive with different choices of Λ in practice. Here we investigate the influence of the cutoff with the LECs fixed. We plot the full potentials with different cutoffs in Fig. 11. From the figure, we notice that the potential becomes deeper and steeper in the short range as the cutoff increases. After solving Schrödinger equation, we obtain the binding energy 1.1 MeV, 17.5 MeV and 53.1 MeV with Λ = 0.6, 0.7 GeV, and m ρ , respectively. The binding energy is sensitive to the cutoff. However, bound state solution exists as cutoff is near m ρ . Furthermore, as we stressed earlier, the cutoff dependence can be compensated if readjusting the LECs at different cutoffs. There also exist other sources of uncertainties. Firstly, we discuss the uncertainty from the resonance saturation model which is utilized to determine LECs of contact terms. From the numerical values of D a and E a in Eq. 29, we can see the contributions from f 0 and a 0 are small, and ρ, ω and σ exchanges dominate D a and E a . For D b and E b in Eq. 29, the uncertainty brought by axial-vector exchanges are not small, and therefore they have considerable effects on binding energy. However the estimation of the axial-vector contributions is quite rough, we hope we can obtain much more reliable input for g HHV A in the future. In general, the uncertainty in Eq (29) gives the binding energy at Λ = 0.7 GeV: 17.5 +4.1+18.3 Secondly, the uncertainty can come from that of axial coupling g. When we include the experimental error [103] (width and branching fraction), we obtain the bare coupling g = 0.65 +0.02 −0.01 , and the binding energy in I = 0 channel with Λ = 0.7 GeV is 17.5 +9.6 −3.9 MeV. We can see the binding energy is sensitive to the coupling g, but not much sensitive as cutoff Λ. Including the uncertainty of D a(b) , E a(b) discussed above, we obtain the binding energy 17.5 +21.1 −15.0 at Λ = 0.7 GeV. This uncertainty is largely brought by axial-vector mesons, the uncertainty from g is moderate, and the uncertainty from f 0 and a 0 is smallest.
The third uncertainty comes from truncation error. Here we partially estimate few loop diagrams of contact contribution at O(ǫ 4 ) to show how large the truncation error is. For O(ǫ 4 ) contact loop contribution, there exist many Feynmann diagrams. We pick some diagrams and plot them in Fig. 12.
In the first four diagrams of Fig. 12, each one contains two separated loops, and the sum of these reads: We can see they are generally O( 1 100 ) relative to those at O(ǫ 2 ) by comparing with Eqs. (21) and (23). The last three set of diagrams in Fig. 12 indicate the wave function renormalization of the O(ǫ 2 ) diagrams in Fig. 3, and the sum of these reads: They are O( 1 10 ) relative to those at O(ǫ 2 ) from Eqs. (21) and (23). Therefore we expect when all the contact O(ǫ 4 ) diagrams are included, the convergence may not be bad.

VI. SUMMARY
In this work, we have systematically studied the DD * system with ChEFT. Due to the intrinsic difficulty of the ChEFT, we cannot obtain the physical observables directly from the Feynman diagrams. We alternatively calculate the potentials, i.e., the sum of all the 2PI diagrams, and then iterate them into Lippmann-Schwinger or Schrödinger equation to recover the 2PR contributions.
We have investigated the DD * effective potentials in ChEFT with Weinberg scheme. With the effective potentials obtained in momentum space, we have analyzed the contact, OPE and TPE contribution in detail. The OPE and TPE contributions are free of many LECs, and thus they are more model independent than the contact interaction since the LECs are determined with the resonance saturation model in this work. The OPE contribution at O(ǫ 2 ) is smaller than that at O(ǫ 0 ). The potential from TPE at O(ǫ 2 ) is relatively large compared to that from OPE at O(ǫ 0 ) in the I = 1 channel, while it shows a good convergence in the I = 0 channel. The TPE interaction is important and non-negligible.
We have determined the LECs in contact contributions with the resonance saturation model, and further explored the full potentials in coordinate space, which are regularized with a simple Gaussian cutoff. The roles of each contributions have been discussed, and the total potentials are very different in two channels. We have also discussed the importance of the contact contribution and the infulence of the cutoff in detail. Furthermore, we discuss the uncertainties of our approach, which comes from axial coupling g, LECs and truncation error. We find that the TPE contribution is non-negligible and attractive in general, while the contact contributions is an important element to compete the π-exchange contributions and cause quite different behavior in each channels. Despite the roughly estimated LECs, we notice that no bound state exists in the I = 1 channel in a wide range of cutoff parameter, while there is a bound state in the I = 0 channel as the cutoff is near m ρ in our approach. The binding energy is sensitive to the cutoff. Our results are consistent with those in the one-bosonexchange model [78].
In this work, we have ignored many other sub-leading effects from the isospin violation, S -D mixing, recoiling, and so on. These effects can be investigated in future, and our framework shall be proved to be elegant.
We point out that the DD * molecule may be discovered at experiments through various processes. Since at Tevatron and LHCb there are a number of B c events, the DD * molecule can be produced via B c weak decay: singly Cabibbo-supressed process B c → X(DD * )K, doubly Cabibbo-supressed processes B c → X(DD * )π and B c → X(DD * )D. Moreover we hope the e + e − process such as e + e − → X(DD * )DD at BelleII can be studied to observe the state. The molecular states may be constructed through DD final states. We also expect the lattice simulations to test our results.
Our exploration of the DD * system can help to make more profound understanding of the heavy meson system and nonperturbative QCD. We expect our results can be tested by future LHCb and BelleII experiments, and help the extrapolations of future lattice simulations.
We first list the amplitudes of the process D(p 1 )D * (p 2 ) → D(p 3 )D * (p 4 ). The difference between the amplitudes for the I = 0 and I = 1 channels is just a factor.
For the one-loop corrections to the contact terms in Fig. 3, the Feynman amplitudes are (A1) (A7) For the one-loop corrections to the OPE potentials in Fig. 4, the Feynman amplitudes are ×ε µ (p 2 )ε * ν (p 4 ); (A20) For the TPE potentials in Fig. 5, the Feynman amplitudes are In above, J F i j is the short notation for J F i j (m 1 , m 2 , q), J S i j and J T i j are J S i j (m 1 , m 2 , ω, q) and J T i j (m 1 , m 2 , ω, q), respectively. J B i j and J R i j are J B i j (m 1 , m 2 , ω 1 , ω 2 , q) and J R i j (m 1 , m 2 , ω 1 , ω 2 , q), respectively. These loop functions like J g are defined in the Appendix C.
In Eqs. (A1)-(A33), the constants A are different with different isospin. We list them in Tables I, II and III. The remaining constants are: for I = 1. And I: The coefficients for the contact amplitudes in the processes DD * → DD * .
In Eqs. (A1)-(A33), M is the D meson mass, δ is the mass difference between D * and D, m, m 1 , m 2 are all pion masses, p = p 1 − p 4 , q = p 1 − p 3 , µ is the renormalization scale in the dimensional regularization, and Appendix B: Renormalized and bare couplings We provide the relation between the renormalized coupling g (2) at experiment and the bare coupling g in the Lagrangian The expression relating the renormalized f (2) and the bared f is well known We use f (2) = f π = 0.092 GeV.
and L is defined in Eq. (A36). One should notice that in Eqs. (C1)-(C5), if the form of the integral Eq. (16) is encountered, the 2PR part must be subtracted using Eq. (17).
However, the evaluations of above loop integrals are not complete since the kinetic energy terms in the propagators are not included. Here, we further illustrate the calculations con-sidering the kinetic energy terms q 2 2M . We choose J b 0 as an example, We first apply the Feynman parametrization to Eq. (C16):