⇦Chinese Physics Letters, 2021, Vol. 38, No. 9, Article code 092001Views & Comments Perfect $DD^*$ Molecular Prediction Matching the $T_{\rm cc}$ Observation at LHCb Ning Li1*, Zhi-Feng Sun2,3,4*, Xiang Liu2,3,4*, and Shi-Lin Zhu5* Affiliations 1School of Physics, Sun Yat-Sen University, Guangzhou 510275, China 2School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China 3Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China 4Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China 5School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China Received 1 August 2021; accepted 19 August 2021; published online 2 September 2021 Supported by the China National Funds for Distinguished Young Scientists (Grant No. 11825503), the National Key Research and Development Program of China (Grant No. 2020YFA0406400), the 111 Project (Grant No. B20063), and the National Natural Science Foundation of China (Grant No. 11975033, 12070131001, and 12047501).
*Corresponding authors. Email: lining59@mail.sysu.edu.cn; sunzf@lzu.edu.cn; xiangliu@lzu.edu.cn; zhusl@pku.edu.cn Citation Text: Li N, Sun Z F, Liu X, and Zhu S L 2021 Chin. Phys. Lett. 38 092001    Abstract In 2012, we investigated the possible molecular states composed of two charmed mesons [Phys. Rev. D 88 (2013) 114008; 2012 arXiv:1211.5007 [hep-ph]]. The $D^*D$ system with the quantum numbers of $I(J^P)=0(1^+)$ was found to be a good candidate of the loosely bound molecular state. This state is very close to the $D^*D$ threshold with a binding energy around 0.47 MeV. This prediction was confirmed by the new LHCb observation of $T_{\rm cc}^+$ [see Franz Muheim's talk at the European Physical Society conference on high energy physics 2021]. DOI:10.1088/0256-307X/38/9/092001 © 2021 Chinese Physics Society Article Text Since the Belle Collaboration discovered the $X(3872)$ in 2003,[1] a series of charmonium-like $XYZ$ states which are close to the thresholds of two hadrons have been observed. There exists difficulty to describe these states in the traditional hadron picture, which has posed a big challenge and brought valuable opportunities to the whole community. The study of these new exotic hadron states has become the hot new frontier of the hadron physics. Especially, in the last three years, the LHCb Collaboration has been playing a key role in finding new hadronic states. In 2019, the characteristic mass spectrum of three $P_{\rm c}$ states, the $P_{\rm c}(4312)^+$, $P_{\rm c}(4440)^+$ and $P_{\rm c}(4457)^+$[2] provide strong evidence of the existence of the hidden-charm molecular pentaquark. Additionally, LHCb reported the evidence of the $P_{\rm cs}(4459)$[3] in the $J/\psi\varLambda$ invariant mass distribution. And then, several hidden-charm tetraquark candidates [the $Z_{\rm cs}(4000)^+$ and $Z_{\rm cs}(4220)^+$[4]] and possible single-charm tetraquarks, the $d\bar{c}u\bar{s}$ tetraquarks [$X_0(2900)$ and $X_1(2900)$[5,6]] were observed. All these measurements have attracted much attention from both theorists and experimentalists.[7,8] Although so many multiquark state candidates have been reported in experiment, it is not the end of whole story. At the European Physical Society conference on high energy physics 2021, Franz Muheim gave a talk on the “LHCb Highlights”, which shows that LHCb observed the $T_{\rm cc}^+$ by analyzing the $D^0D^0\pi^+$ invariant mass spectrum.[9] The difference between mass of this state and the $D^{*+}D^0$ threshold is $-273 \pm 61 \pm 5 ^{+11}_{-14}$ keV, and its width is $410\pm 165 \pm 43 ^{+18}_{-38}$ keV.[9] Obviously, as the first observation of the double-charm tetraquark, the $T_{\rm cc}$ has the $cc\bar u\bar d$ configuration. In this note we shall briefly introduce our previous work[10] on this topic. An extensive review of the $T_{\rm cc}$ system can be found in Ref. [11]. In 2012, we investigated the system of the $D^{(*)}D^{(*)}[I(J^P) = 0(1^+)]$ very carefully by taking into account the coupled channel effect and the $S$–$D$ mixing effect, which play an important role in the formation of the loosely bound deuteron.[12] With the one-pion-exchange force only, we found a loosely $D^{*}D[I(J^P)=0(1^+)]$ bound state with the binding energy $1.24$ MeV for a reasonable cutoff $1.05$ GeV, which is introduced to suppress the very high-momentum or short-range contribution. The corresponding root-mean-square radius is $3.11$ fm, which is comparable to the size of the well-known deuteron (about $2.0$ fm). There exists a small contribution around $2.79\%$ from the $D^*D^*(^{3}\!S_1)$ channel because of the large mass gap (about $140$ MeV) between the thresholds of $DD^*$ and $D^*D^*$. However, the probability of the $D$-wave interaction is tiny, $0.73\%$ for the $DD^*({}^{3}\!D_1)$ and $0.08\%$ for the $D^*D^*(^{3}\!D_1)$, which indicates that the $D^{*}D[I(J^P) = 0(1^+)]$ molecular state is almost dominated by the s-wave interaction. We should mention that for the pion exchange in the crossed channel, the potential is complex because the exchanged pion is on-shell due to the large mass difference between $D$ and $D^*$. With the same approach in the study of $X(3872)$, we keep the real part of the potential. Such a formalism has no effects on the spectrum of the $DD^*$ system. When the heavier $\rho$ and $\omega$ exchanges are included, the binding becomes deeper for the same cutoff as that with the one-pion-exchange potential only. With a smaller cutoff $0.95$ GeV, there exists a loosely $D^{(*)}D^{(*)}[I(J^P) = 0(1^+)]$ state with a binding energy $0.47$ MeV and hence with the mass $3875.38$ MeV, which can be identified as the newly observed $T_{\rm cc}$ by the LHCb Collaboration. In such a case, the heavier $\rho$ and $\omega$ exchanges cancel each other significantly since for the isospin-zero system the isospin factor of the $\rho$ meson exchange is $-3$ while that of the $\omega$ meson exchange is $1$, and the residual force is helpful to strengthen the binding. Additionally, the contributions from the $\eta$ and $\sigma$ exchanges are very small. The $T_{\rm cc}$ and $X(3872)$ shares the same one-pion-exchange potential. Their long-range dynamics is similar and correlated to each other. If the $X(3872)$ is a loosely bound molecular state, the existence of the $X(3872)$ implies the existence of the $T_{\rm cc}$. Numerically, the binding energy of the $T_{\rm cc}$ depends weakly on the cutoff, which is similar to the case of the $X(3872)$. For example, the binding energy of the $T_{\rm cc}$ is $0.47$ MeV for the cutoff = 0.95 GeV and $18.72$ MeV for cutoff = 1.05 GeV.
cpl-38-9-092001-fig1.png
Fig. 1. (The wave function [$\chi(r) = r\psi(r)$] of the $D^{(*)}D^{(*)}I(J^P)=0(1^+)$. Here 1 stands for the channel $[DD^*(^{3}\!S_1)]$, 2 for $[DD^*(^{3}\!D_1)]$, 3 for $[D^*D^*(^{3}\!S_1)]$ and 4 for $D^*D^*(^{3}\!D_1)]$.
The effects of the $\sigma$, $\rho$ and $\omega$ exchanges are also analyzed by turning off the contribution of the $\pi$ and $\eta$ exchanges. A loosely bound state with a binding energy $0.78$ MeV and root-mean-square radius $3.74$ fm was obtained when the cutoff parameter is tuned to be $1.44$ GeV, which is larger than $1.05$ GeV used for the one-pion-exchange case. The contribution of the long-range pion exchange is larger than that of the heavier vector meson exchange in the formation of the loosely bound $D^{(*)}D^{(*)}[I(J^P)=0(1+)]$ state. The wave function is shown in Fig. 1, from which one can see that there is no node except the origin. In other words, it is really a ground state. In summary, in 2012 we predicted a loosely bound state of the $DD^*[I(J^P) = 0(1^+)]$ which can be perfectly identified as the newly observed $T_{\rm cc}$ by the LHCb Collaboration. For this state, the probability of the $D$-wave interaction is very small, and it arises from the s-wave interaction. There is small contribution from the $D^*D^*$ channel. The long-range pion exchange is strong enough to form the loosely bound state, and the medium-range $\eta$ and $\sigma$ exchanges and the short-range $\rho$ and $\omega$ exchanges are helpful to strengthen the binding. In Ref. [13], the authors also studied the doubly charmed systems within the hidden gauge formalism in a coupled-channel unitary approach. For the $D^*D^*$ system with $C=2$, $S=0$ and $I=0$, they only obtained a bound state with quantum number $I(J^P)=0(1^+)$. However, the pole appeared at 3969 MeV, which is about 100 MeV larger than our result. In contrast, we considered the coupled channel effect between the $DD^*$ and $D^*D^*$. Actually, what we obtained is a $DD^*$ bound state instead of a $D^*D^*$ bound state.
cpl-38-9-092001-fig2.png
Fig. 2. The allowed strong and radiative decays of $T_{\rm cc}$.
The state $DD^*[0(1^+)]$ cannot decay into a double charm baryon plus a light baryon. The masses of the lightest doubly-charmed baryon and light baryon are 3518 MeV and 938 MeV, respectively, corresponding to $\varXi_{\rm cc}^+$ and proton as listed in Particle Data Group.[14] The mass of the molecular state is around 3875 MeV, much smaller than the sum of the masses of a doubly-charmed baryon and a light baryon. Therefore such a decay is kinematically forbidden. However, the heavy vector meson $D^*$ within the exotic molecular state mainly decays to $D\pi$ via the strong interaction as shown in Figs. 2(a) and 2(b). It also decays into $D\gamma$, which is illustrated in Figs. 2(c) and 2(d). The main decay modes should be $DD\gamma$ and $DD\pi$, and the $D^{(*)}$ meson may also decay via the weak interaction. The above typical decay modes provide important information to experimental investigation of the properties of $T_{\rm cc}$. With further theoretical and experimental progress, we shall gain new insights into this structure. The observation of this doubly charmed structure has opened a new window for the exotic hadron states in this exciting era, which is beyond the hidden charm “exotic” states which have been widely investigated since the $X(3872)$ was first observed in 2003 by the Belle Collaboration. References Observation of a Narrow Charmoniumlike State in Exclusive B ± → K ± π + π − J / ψ DecaysObservation of a Narrow Pentaquark State, P c ( 4312 ) + , and of the Two-Peak Structure of the P c ( 4450 ) + Evidence of a J / ψ Λ structure and observation of excited Ξ - states in the Ξ b - → J / ψ Λ K - decayObservation of new resonances decaying to $J/ψK^+$ and $J/ψϕ$Model-Independent Study of Structure in B + → D + D − K + DecaysAmplitude analysis of the B + → D + D − K + decayProduction of P cs (4459) from Ξ b DecayX 0 (2900) and X 1 (2900): Hadronic Molecules or Compact TetraquarksCoupled-channel analysis of the possible D ( * ) D ( * ) , B ¯ ( * ) B ¯ ( * ) and D ( * ) B ¯ ( * ) molecular statesPentaquark and Tetraquark StatesHigh-precision, charge-dependent Bonn nucleon-nucleon potentialNew interpretation for the D s 2 * ( 2573 ) and the prediction of novel exotic charmed mesonsReview of Particle Physics
[1] Choi S K et al. [Belle Collaboration] 2003 Phys. Rev. Lett. 91 262001
[2] Aaij R et al. [LHCb Collaboration] 2019 Phys. Rev. Lett. 122 222001
[3] Aaij R et al. [LHCb Collaboration] 2021 Sci. Bull. 66 1278
[4] Aaij R et al. [LHCb Collaboration] 2021 arXiv:2103.01803 [hep-ex]
[5] Aaij R et al. [LHCb Collaboration] 2020 Phys. Rev. Lett. 125 242001
[6] Aaij R et al. [LHCb Collaboration] 2020 Phys. Rev. D 102 112003
[7] Wu Q, Chen D Y, and Ji R 2021 Chin. Phys. Lett. 38 071301
[8] Chen H X, Chen W, Dong R R, and Su N 2020 Chin. Phys. Lett. 37 101201
[9]See the talk by Franz Muheim at the European Physical Society conference on high energy physics 2021 on July 28, https://indico.desy.de/event/28202/contributions/102717/
[10] Li N, Sun Z F, Liu X, and Zhu S L 2013 Phys. Rev. D 88 114008
[11] Liu Y R, Chen H X, Chen W, Liu X, and Zhu S L 2019 Prog. Part. Nucl. Phys. 107 237
[12] Machleidt R 2001 Phys. Rev. C 63 024001
[13] Molina R, Branz T, and Oset E 2010 Phys. Rev. D 82 014010
[14] Group P D and Tanabashi M et al. 2018 Phys. Rev. D 98 030001