Abstract
Motivated by the latest discovery of a new tetraquark with two charm quarks and two light antiquarks by LHCb Collaboration, we investigated the hadronic molecule interpretation of . By calculation, the mass and the decay width of this new structure can be understood in one-meson exchange potential model. The binding energies for these hadronic molecules with are around 1 MeV. Besides, we also studied the possible beauty partners of hadronic molecule , which may be feasible in future LHCb experiments.
1. Introduction
Hadron spectroscopy provides a unique window for us to understand the fundamental strong interactions. In naive quark model, hadrons are established by quark-antiquark pair or three quark objects. However, the quantum chromodynamics (QCD) theory tells us that some exotic states such as multiquark states or gluon-participated states, which are apart from the conventional configurations, may also be confined into a color-singlet hadron. The earliest evidence of exotic states is the X(3872) discovered by the Belle Collaboration in 2003, which lies above the two open charm meson threshold but has a very narrow decay width ( MeV) [1]. Interpretation and verification of the special properties of exotic states attracted a lot of attempt from both theoretical and experimental aspects (see the reviews [2–4]). The studies of exotic states are not only to gradually filling in the period table of hadrons but also to enriching our knowledge of QCD color-confining principle.
Very recently, the LHCb Collaboration has reported the first discovery of a new tetraquark with two charm quarks and two light antiquarks in the mass spectrum using a proton-proton collision data set corresponding to an integrated luminosity of [5], where the mass and decay width of are determined as
And the spin-parity is determined as . Later, the LHCb Collaboration has released a more profound decay analysis [6]; then, the mass and decay width of are updated as
The LHCb exotic state has an electrical charge and two charm quantum numbers and thus leads to a strong evidence of least quark content . This exotic system has interesting points. First, the two heavy quarks inside the system have small relative velocities due to its large masses compared to the QCD typical energy scale (). There exists an attractive color force between the color antitriplet heavy quark pair. Similar attraction is produced for the light antiquark pair. From these arguments, diquark models were employed [7–15]. On the other hand, a lower bound state may be produced between two heavy hadrons by exchanging light mesons. Hadronic molecules are also popular choices for the system of two heavy quarks and two light antiquarks [16–22]. In addition, there are other proposals to explain the doubly heavy tetraquarks: compact tetraquarks [23–50], chiral quark model [51–53], constituent quark model [54], and hydrogen-like molecules [55]. The production properties of doubly heavy tetraquarks have been studied in literatures, for example, Refs. [56–59], while the decay properties of doubly heavy tetraquark have been studied in literatures [60–68].
Considering that the LHCb exotic state is near threshold of a pair of charm mesons, we will investigate the hadronic molecule interpretation of in this work. From the mass of , it is extremely close to the threshold. The binding energy is less than 1 MeV. We will study the mass and decays of doubly charm tetraquarks in one-meson exchange potential (OMEP) model. The effective coupling constants among light mesons and charm mesons are revisited. By power counting, we only consider the leading order contribution from the lightest mesons, i.e., pseudoscalar mesons. Then, the number of parameters is further reduced in OMEP model. By the investigation of the decay channels, it is also possible to hunting for the charge-partners and states. As a by-product, we also study the mass spectra of the possible doubly bottomed tetraquarks and discuss their golden decay channels.
This paper is organized as follows. We give the low energy effective Lagrangian and the effective potential after the Introduction. The OMEP model is employed to extract the effective potential. In Section 3, we present the calculation detail of the mass of the possible ground states of doubly heavy tetraquarks below the heavy meson pair threshold. In Section 4, we give the decay amplitude and decay width of the process . We conclude in the end.
2. Low Momentum Interaction Effective Theory
In the heavy quark limit , the heavy quark behaves like a static point, and the heavy meson dynamics is determined by the degree of freedom of the light quark. In heavy quark spin symmetry, the spin-0 and spin-1 heavy-light mesons are combined into a matrix where the pseudoscalar and vector heavy-light meson fields are explicitly expressed as and for charm sector and and for bottom sector; is the velocity of the heavy quark with the constraint . Here, is a triplet in SU(3) flavor symmetry when considering the fact that the masses of light quarks , , and can be ignored compared with the heavy quark mass .
When one considers the exchanging of low energy light mesons between heavy hadrons, it is required to employ the chiral perturbation theory. Using this low energy theory, it becomes easy to separate the long and short range dynamics. The doubly charm tetraquark observed in LHCb experiment is very close to the threshold of ; the binding energy is less than 1 MeV if we treat the as a bound state. We expect that the low energy expansion converges well. For the decay channel , the final particles will have small velocities and can be treated as nonrelativistic objects. For example, the maximum energy of in is MeV, which is only 2.84 MeV above the mass of meson with MeV. Similarly, the maximum energy of in is MeV, which is only 5.279 MeV above the mass of meson with MeV. Thus, we also use the low-energy effective theory to study the decay properties.
The low momentum interaction effective theory Lagrangian at leading order is written as [69] where and and . is exponentially related to the light pseudoscalar mesons with
In principle, the vector and scalar light mesons may also bring new effects in the binding and decay properties of the near-threshold doubly charm tetraquark states in OMEP model, but these effects are expected to be suppressed as according to the power counting rules. Compared to the long distance interaction from pseudoscalar light mesons, the interactions from scalar and vector light mesons are medium and short ranges. On the other hand, one needs to introduce more parameters in the effective theory, and some of them are not well investigated currently. We will address these points in future works.
Using the above Lagrangian, the two-body potential between a vector and a pseudoscalar heavy mesons becomes as [4, 16]
3. Doubly Heavy Tetraquark Spectra from System
In this section, we will investigate the spectra of doubly heavy tetraquark from the possible bound states in OMEP model. After implementing the Fourier transformation on the potential in momentum space, the potential in coordinate space can be obtained. Considering the size of the exchanged light meson, the Fourier transformation on the potential with the dipole form factors becomes where a UV cut-off is introduced to regularize the short-distance effects.
In coordinate space, the one-meson exchange potential is then written as [4, 16] where
In the potential, the scale is chose as due to the recoil effect from unequal heavy mesons. For the interactions, , and we should replace as and take the real parts in the integrals [4]. The form factor parameter is chose as usual GeV. A higher UV cut-off may be employed if one includes the medium and short dynamics. When interpreting the doubly charm tetraquark as the possible bound states with , the state can be rewritten as
For the system with , the may be in the S-wave state with orbital angular momentum or D-wave state with orbital angular momentum . The parameter is expressed as with isospin quantum number of the hadrons. Consider the above mixing between S-wave and D-wave states, one has the matrix [16]
In nonrelativistic approximation, the binding energy can be solved by Schrödinger equation where is the reduced mass [70]. We only focus on the stable bound states with binding energy .
In the calculation, one needs to input the parameter values. The hadron masses are adopted from PDG [71]: MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, MeV, and MeV. The effective coupling constants are needed to extract from experiments or lattice QCD calculations. In flavor SU(3) symmetry, they are approximated to be equal to . For the decays, the Feynman amplitude can be written as
The data of the decay widths and branching ratios can be inputted from PDG [71]: keV, MeV, , , and . Use the channel, the effective coupling constant is estimated as . Use the channel, the effective coupling constant is estimated as . While use the channel and keV, one can get . And the effective coupling constant is estimated from as
We list the numerical results for the bound states in Table 1, where the isospin, strange, and beauty partners of hadronic molecule are also given. The binding energies for and with are near to 0.6 MeV. The binding energies for strange partners with are close to 1 MeV, while the binding energies for these hadronic molecules without strange quantum numbers are around 6 MeV, and then, states are more stable.
4. Decays
In this section, we will study the decays of . Here, we only focus on the stable hadronic molecules with spin-parity given in Table 1. Consider the fact that the mass splitting between and mesons is larger than the pion mass, there have decay channels. is also allowed when is close to or above threshold. However, the mass splitting between and mesons is small and less than the pion mass. Thus, channel is forbidden due to the lack of phase space when is below or close to threshold. The LHCb Collaboration has employed the golden channel in the discovery of the first doubly charm tetraquark. The typical Feynman diagram is plotted in Figure 1.
For and processes, the leading-order Feynman amplitudes are similar and can be written as where is the effective coupling constant for the vertex.
In general, the three-body partial decay width is written as [72] where represents the invariant mass of two heavy charm mesons and represents the invariant mass of one heavy charm meson and the pion. For the process , the phase space constraints then read as [73]
The energies in the rest frame are
The decay widths of can be estimated as
Currently, it is not easy to determine the value of the effective coupling . If we choose and , the decay width of is estimated as , which is consistent to the first round data at LHCb experiment. While we employ the second round LHCb data, the effective coupling is extracted as . In this case, the decay width of is then estimated as with the choice of the effective coupling . Due to limited phase space, does not decay into . For the strange partners of , we will discuss its decay properties in future works due to the lack of information of the effective couplings. For the beauty partners , one can use the electromagnetic decay channels to detect due to the limited phase space. The beauty partners shall be more stable than the states in turn.
5. Conclusion
In this paper, we investigated the mass spectrum and the decay properties of the state first observed at LHCb experiment. Numerical results indicate that the state can be well-understood in the hadronic molecule model. As a by-product, the mass spectra of the doubly charm tetraquarks and doubly bottomed tetraquarks in heavy hadronic molecule framework are studied, some of which may be hunted in future LHCb experiments. Especially, it is worthwhile to notice the stable doubly bottomed tetraquarks , , and from the decay channels.
Data Availability
The data used to support the findings of this study are available from the authors upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the NSFC under grant Nos. 12075124 and 11975127 and the Youth Talent Support Program of Nanjing Normal University.