# Hadronic Molecule Interpretation ofTcc+ and Its Beauty Partners

Authors: Huimin Ren; Fan Wu; Ruilin Zhu
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

Motivated by the latest discovery of a new tetraquarkTcc+ with two charm quarks and two light antiquarks by LHCb Collaboration, we investigated the DD∗ hadronic molecule interpretation of Tcc+. By calculation, the mass and the decay width of this new structure Tcc+ can be understood in one-meson exchange potential model. The binding energies for these DD∗ hadronic molecules with JP=1+ are around 1 MeV. Besides, we also studied the possible beauty partners Tbb10598 of hadronic molecule Tcc+, which may be feasible in future LHCb experiments.


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## Body

## 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 (Γ<1.2 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 tetraquarkTcc+ with two charm quarks and two light antiquarks in the D0D0π+ mass spectrum using a proton-proton collision data set corresponding to an integrated luminosity of 9fb−1 [5], where the mass and decay width of Tcc+ are determined as
(1)δm=mTcc+−mD∗++mD0=−273±61±5−14+11keV,ΓTcc+=410±165±43−38+18keV.And the spin-parity is determined asJP=1+. Later, the LHCb Collaboration has released a more profound decay analysis [6]; then, the mass and decay width of Tcc+ are updated as
(2)δm=mTcc+−mD∗++mD0=−361±40keV,ΓTcc+=47.8±1.9keV.The LHCb exotic stateTcc+ has an electrical charge and two charm quantum numbers and thus leads to a strong evidence of least quark content ccu¯d¯. 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 (mQ≫ΛQCD). 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 stateTcc+ is near threshold of a pair of charm mesons, we will investigate the DD∗ hadronic molecule interpretation of Tcc+ in this work. From the mass of Tcc+, it is extremely close to the DD∗ 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 Tcc0 and Tcc++ states. As a by-product, we also study the mass spectra of the possible doubly bottomed tetraquarks Tbb 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 pairHH∗ threshold. In Section 4, we give the decay amplitude and decay width of the process TQQ⟶H+H+π. We conclude in the end.


## 2. Low Momentum Interaction Effective Theory

In the heavy quark limitmQ⟶∞, 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 4×4 matrix
(3)Ha=1+v2Ha∗μγμ+iHaγ5,where the pseudoscalar and vector heavy-light meson fields are explicitly expressed as Ha=D0,D+,Ds+ and Ha∗=D∗0,D∗+,Ds+ for charm sector and Ha=B−,B¯0,B¯s0 and Ha∗=B∗−,B¯∗0,B¯s∗0 for bottom sector; v is the velocity of the heavy quark with the constraint vHa=Ha. Here, Ha∗ is a triplet in SU(3) flavor symmetry when considering the fact that the masses of light quarks u, d, and s can be ignored compared with the heavy quark mass mQ.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 tetraquarkTcc+ observed in LHCb experiment is very close to the threshold of DD∗; the binding energy is less than 1 MeV if we treat the Tcc+ as a DD∗ bound state. We expect that the low energy expansion converges well. For the decay channel Tcc+⟶D0+D0+π+, the final particles D0D0π+ will have small velocities and can be treated as nonrelativistic objects. For example, the maximum energy of D0 in Tcc+⟶D0D0π+ is ED0max=mTcc2+mD2−mD+mπ2/2mTcc=1867.68 MeV, which is only 2.84 MeV above the mass of D0 meson with mD0=1864.84 MeV. Similarly, the maximum energy of π+ in Tcc+⟶D0D0π+ is Eπ+max=mTcc2+mπ2−mD+mD2/2mTcc=144.85 MeV, which is only 5.279 MeV above the mass of π+ meson with mπ+=139.57 MeV. Thus, we also use the low-energy effective theory to study the Tcc+⟶D0+D0+π+ decay properties.The low momentum interaction effective theory Lagrangian at leading order is written as [69]
(4)L0=fP28Tr∂μΣ∂μΣ†−iTrH¯avμδab∂μ+iVabμHb+gPTrH¯aHbγμγ5Abaμ,where Vabμ=1/2ξ†∂μξ+ξ∂μξ†ab and Aabμ=i/2ξ†∂μξ−ξ∂μξ†ab and ξ=Σ. Σ=Exp2iM/fP is exponentially related to the light pseudoscalar mesons with
(5)M=π02+η6π+K+π−−π02+η6K0K−K¯0−2η6.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 asOmπ2/mV,S2 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]
(6)VHH∗q⟶=−gP2fP2τ⟶1·τ⟶2ε⟶1·q⟶ε⟶2∗·q⟶q⟶2+μ2.


## 3. Doubly Heavy Tetraquark Spectra fromHH∗ System

In this section, we will investigate the spectra of doubly heavy tetraquark from the possibleHH∗ 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
(7)VHH∗r=∫d3q⟶2π3eiq⟶·r⟶VHH∗q⟶Λ2−μ2Λ2+q⟶22,where a UV cut-off Λ is introduced to regularize the short-distance effects.In coordinate space, the one-meson exchange potential is then written as [4, 16]
(8)VHH∗r=−gPfP2γIC0Cr+S12r^Tr,where
(9)S12r^=3ε⟶1·r^ε⟶∗2·r^−1,(10)Cr=μ24πe−μrr−e−Λrr−Λ2−μ22Λe−Λr,(11)Tr=14π3+3μr+μ2r2e−μrr3−3+3Λr+Λ2r2e−Λrr3+μ2−Λ221+Λre−Λrr.In the potential, the scale is chose asμ2=mP2−mD∗−mD2 due to the recoil effect from unequal heavy mesons. For the DD∗ interactions, μ2<0, and we should replace μ2 as ∣μ2∣ and take the real parts in the integrals [4]. The form factor parameter Λ is chose as usual Λ≃1.1 GeV. A higher UV cut-off Λ may be employed if one includes the medium and short dynamics. When interpreting the doubly charm tetraquark Tcc+ as the possible DD∗ bound states with JP=1+, the Tcc+ state can be rewritten as
(12)Tcc+=D0D∗+±D+D∗02.For theDD∗ system with JP=1+, the DD∗ may be in the S-wave state with orbital angular momentum ℓ=0 or D-wave state with orbital angular momentum ℓ=2. The parameter γI is expressed as γI=−2II+1−I1I1+1−I2I2+1/3 with isospin quantum number Ii of the hadrons. Consider the above mixing between S-wave and D-wave states, one has the matrix [16]
(13)<C0>=1001,<S12r^>=0−2−21.In nonrelativistic approximation, theDD∗ binding energy E can be solved by Schrödinger equation
(14)−ℏ22μR∇2+VDD∗rψr=Eψr,where μR=mDmD∗/mD+mD∗ is the reduced mass [70]. We only focus on the stable bound states with binding energy E<0.In the calculation, one needs to input the parameter values. The hadron masses are adopted from PDG [71]: mD0=1864.84 MeV, mD±=1869.65 MeV, mD∗0=2006.85 MeV, mD∗±=2010.26 MeV, mDs±=1968.34 MeV, mDs∗±=2112.2 MeV, mB0=5279.65 MeV, mB±=5279.34 MeV, mBs0=5366.88 MeV, mB∗0=5324.65 MeV, mB∗±=5324.65 MeV, mBs∗0=5415.4 MeV, mπ±=139.57 MeV, mπ0=134.977 MeV, mK±=493.677 MeV, mK0=497.611 MeV, and mη=547.862 MeV. The effective coupling constants gP are needed to extract from experiments or lattice QCD calculations. In flavor SU(3) symmetry, they are approximated to be equal to gπ. For the D∗⟶D+π decays, the Feynman amplitude can be written as
(15)iMD∗⟶D+π=iλπpπμεD∗,μ.The data of the decay widths and branching ratios can be inputted from PDG [71]: ΓD∗±=83.4 keV, ΓD∗0<2.1 MeV, BD∗±⟶D0π±=67.7%, BD∗±⟶D±π0=30.7%, and BD∗0⟶D0π0=64.7%. Use the D∗±⟶D0π± channel, the effective coupling constant is estimated as λπ~16.8. Use the D∗±⟶D+π0 channel, the effective coupling constant is estimated as λπ~11.9. While use the BD∗0⟶D0π0=64.7% channel and ΓD∗0≈60 keV, one can get λπ~12.3. And the effective coupling constant gπ is estimated from λπ as
(16)gπ≃λπfπ2mD∗mD~0.4,0.6.We list the numerical results for theDD∗ bound states in Table 1, where the isospin, strange, and beauty partners of hadronic molecule Tcc+ are also given. The binding energies for D0D∗+ and D∗0D+ with JP=1+ are near to 0.6 MeV. The binding energies for strange partners with JP=1+ are close to 1 MeV, while the binding energies for these BB∗ hadronic molecules without strange quantum numbers are around 6 MeV, and then, Tbb states are more stable.Table 1
Predictions of the masses (MeV) ofHH∗ stable hadronic molecules with spin-parity JP=1+. The uncertainty is from the choice of the effective coupling gπ=0.5∓0.1.
Tcc statesIsospinContentsMass (MeV)Tbb statesIsospinContentsMass (MeV)Tcc+0D0D∗+,D∗0D+3875.1−0.3+0.2Tbb−0B¯0B∗−,B¯∗0B−10598−3+2Tcc00D0D∗03871.0−0.3+0.2Tbb00B¯0B¯∗010598−3+2Tcc++0D+D∗+3879.2−0.3+0.2Tbb−−0B−B∗−10598−3+2Tccs,+12D0Ds∗+,D∗0Ds+3974.8−0.5+0.4Tbbs,−1/2B¯s0B∗−,B¯s∗0B−10692.4−0.4+0.3Tccs,++12Ds+D∗+,Ds∗+D+3979.3−0.3+0.2Tbbs,012B¯s0B¯∗0,B¯s∗0B¯010692.4−0.3+0.3Tccss,++0Ds+Ds∗+4079.6−0.2+0.2Tbbss,00B¯s0B¯s∗010781.7−0.2+0.3


## 4.TQQ⟶HHπ Decays

In this section, we will study the decays ofTQQ⟶HHπ. Here, we only focus on the HH∗ stable hadronic molecules with spin-parity JP=1+ given in Table 1. Consider the fact that the mass splitting between D∗ and D mesons is larger than the pion mass, there have D∗⟶D+π decay channels. Tcc⟶DDπ is also allowed when Tcc is close to or above DD∗ threshold. However, the mass splitting between B∗ and B mesons is small and less than the pion mass. Thus, Tbb⟶BBπ channel is forbidden due to the lack of phase space when Tbb is below or close to BB∗ threshold. The LHCb Collaboration has employed the golden channel Tcc+⟶D0D0π+ in the discovery of the first doubly charm tetraquark. The typical Feynman diagram is plotted in Figure 1.Figure 1
Typical Feynman diagram for theTQQ⟶HHπ decay.ForTcc+⟶D0D0π+ and Tcc+⟶D0D+π0 processes, the leading-order Feynman amplitudes are similar and can be written as
(17)iMTcc+⟶DDπ=iλπpπμ−ipD+pπ2−mD∗2−gμν+pD+pπμpD+pπνpD+pπ2imTccλTccεTccν,where λTcc is the effective coupling constant for the Tcc−D−D∗ vertex.In general, the three-body partial decay width is written as [72]
(18)dΓdsdt=12π3132mTcc3M¯2,where s represents the invariant mass of two heavy charm mesons and t represents the invariant mass of one heavy charm meson and the pion. For the process Tcc+⟶D0D+π0, the phase space constraints then read as [73]
(19)tmin=E2+E32−E22−mD+2+E32−mπ022,tmax=E2+E32−E22−mD+2−E32−mπ022,(20)smin=mD++mD02,smax=mTcc−mπ02.The energies in thes rest frame are
(21)E2=s−mD02+mD+22s,E3=mTcc2−s−mπ022s.The decay widths ofTQQ⟶HHπ can be estimated as
(22)ΓTcc+⟶D0D+π0≈λTcc62gπ0.4250keV,(23)ΓTcc+⟶D0D0π+≈λTcc62gπ0.42287keV,(24)ΓTcc0⟶D0D0π0≈λTcc62gπ0.42223keV,(25)ΓTcc++⟶D0D+π+≈λTcc62gπ0.42103keV.Currently, it is not easy to determine the value of the effective couplingλTcc. If we choose λTcc=6 and gπ=0.4, the decay width of Tcc+ is estimated as ΓTcc+~337keV, which is consistent to the first round data at LHCb experiment. While we employ the second round LHCb data, the effective coupling λTcc is extracted as λTcc≃2.26. In this case, the decay width of Tcc+ is then estimated as ΓTcc+~47.8,107.6keV with the choice of the effective coupling gπ=0.4,0.6. Due to limited phase space, Tcc++ does not decay into D+D+π0. For the strange partners of Tcc+, we will discuss its decay properties in future works due to the lack of information of the effective couplings. For the beauty partners Tbb, one can use the Tbb⟶BBγ electromagnetic decay channels to detect due to the limited phase space. The beauty partners Tbb shall be more stable than the Tcc states in turn.


## 5. Conclusion

In this paper, we investigated the mass spectrum and the decay properties of theTcc+ state first observed at LHCb experiment. Numerical results indicate that the Tcc+ state can be well-understood in the DD∗ hadronic molecule model. As a by-product, the mass spectra of the doubly charm tetraquarks and doubly bottomed tetraquarks in heavy HH∗ hadronic molecule framework are studied, some of which may be hunted in future LHCb experiments. Especially, it is worthwhile to notice the stable JP=1+ doubly bottomed tetraquarks Tbb−10598, Tbb010598, and Tbb−−10598 from the Tbb⟶BBγ decay channels.



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