Investigating new type of doubly-charmed molecular tetraquarks composed of the charmed meson in $H$-doublet and the charmed meson in $T$-doublet

Stimulated by the newly reported doubly-charmed tetraquark state $T_{cc}^+$ by LHCb, we carry out a systematic investigation of the $S$-wave interactions between the charmed meson ($D,\,D^{*}$) in $H$-doublet and the charmed meson ($D_{1},\,D_{2}^{*}$) in $T$-doublet by adopting the one-boson-exchange model. Both the $S$-$D$ wave mixing effect and the coupled channel effect are taken into account. By performing a quantitative calculation, we suggest that the $S$-wave $D^{*} D_{1}$ states with $I(J^{P})=0(0^{-},\,1^{-})$ and the $S$-wave $D^{*}D_{2}^{*}$ state with $I(J^{P})=0(1^{-})$ should be viewed as the most promising candidates of the doubly-charmed molecular tetraquark states, and the $S$-wave $DD_{1}$ state with $I(J^{P})=0(1^{-})$, the $S$-wave $DD_{2}^{*}$ state with $I(J^{P})=0(2^{-})$, and the $S$-wave $D^{*}D_{2}^{*}$ state with $I(J^{P})=0(2^{-})$ are the possible doubly-charmed molecular tetraquark candidates. With the accumulation of experimental data at Run III and after High-Luminosity-LHC upgrade, these predicted doubly-charmed molecular tetraquark states can be accessible at LHCb in the near future.


I. INTRODUCTION
The study of the exotic hadronic states, which cannot fit into the conventional meson and baryon [1,2], has become one of the central topics at the precision frontier of the hadron physics. As an important part of the hadron spectroscopy, these exotic hadronic states include the hybrid mesons, glueballs, multiquark states, and so on [3,4]. Studying the exotic hadronic states can provide the valuable hints to shed light on the non-perturbative behavior of the quantum chromodynamics (QCD). Since the observation of the X(3872) in 2003 [5], a series of charmonium-like XYZ states and several P c states had been discovered experimentally [3,4,[6][7][8][9][10]. Since the masses of these X/Y/Z/P c states are close to the corresponding thresholds of two hadrons, the explanations of the hiddencharm molecular tetraquark and pentaquark states were extensively proposed and studied in the past decades [3,4].
Very recently, the LHCb Collaboration reported an important observation of the doubly-charmed tetraquark state T + cc in the D 0 D 0 π + invariant mass spectrum via the proton-proton collisions [11]. The Breit-Wigner parameters of this doublycharmed tetraquark state are δm = −273 ± 61 ± 5 +11 −14 keV/c 2 and Γ = 410 ± 165 ± 43 +18 −38 keV, where the mass difference is with respect to the D 0 D * + threshold [11]. We should emphasize that the observation of the doubly-charmed tetraquark T + cc is a great breakthrough for the hadron physics, and it is the firstly observed doubly-charmed tetraquark state with the typical quark configuration ccūd, which can be assigned as the isoscalar DD * molecular state with J P = 1 + [12][13][14][15][16].
In 2003, the charmoniumlike state X(3872) was observed by the Belle Collaboration [5], and more theoretical groups joined the discussion of whether the X(3872) can be viewed as the DD * molecular state [17][18][19][20][21][22][23][24][25][26][27][28][29], which is involved in the / * / * / * → / * FIG. 1: With a simple replacement of the charmed mesons, studying the interactions between the charmed meson in H-doublet and the charmed meson in T -doublet is the natural extension of former investigation of the interactions involving two S -wave charmed mesons [12][13][14]. depiction of the interactions between an S -wave charmed meson and an S -wave anti-charmed meson. Later, the experimental collaborations reported the more and more observations of charmonium-like states with higher mass [10], which results in the discussion of whether they can also be regarded as the hidden-charm molecular tetraquark states by checking the interactions of an S -wave charmed meson and a P-wave anti-charmed meson [3,4]. For example, the Z(4430) [30] can be interpreted as the D * D ( ) 1 molecular state [31,32], the Y(4260) [33] as the DD 1 molecular state was given in Refs. [34][35][36], the Y(4360) [37] was interpreted as the 2S bound state in the D * D 1 system [38,39], and so on. If borrowing the research experience of the hidden-charm molecular tetraquark states [3,4], it is time to make realistic study of the doublycharmed molecular tetraquark candidates composed of the charmed meson in H-doublet and the charmed meson in Tdoublet. Meanwhile, the study of the interactions between the charmed meson in H-doublet and the charmed meson in Tdoublet is the natural extension of former study of the interactions involved two S -wave charmed mesons [12][13][14] (see Fig.  1), which has inspired the observation of the doubly-charmed T + cc state to some extent [11]. Recently, these possible doublycharmed molecular tetraquark states were considered with the light vector meson exchange in Ref. [40].
Considering future experimental potential, we believe that the doubly-charmed molecular tetraquark states can be accessible through the proton-proton collisions. In 2018, the LHC Collaboration announced a white paper on the future physics program [41], Run III will be carried out with higher statistic data accumulation in near future, and the High-Luminosity-LHC will be upgraded for the physics reach. Thus, it is possible that more doubly-charmed molecular tetraquark states will be found by LHCb in near future. As an area full of opportunities and challenges, exploring various types of the doublycharmed molecular tetraquark states should be paid more attention by both theorists and experimentalists.
In this work, we perform a quantitative calculation of the Swave interactions between the charmed meson in H-doublet and the charmed meson in T -doublet. For achieving this goal, we adopt the one-boson-exchange (OBE) model, which was extensively applied to study the hadronic molecular states [3,4], to deduce the effective potentials for these discussed Swave HT systems. In our calculation, the S -D wave mixing effect and the coupled channel effect are taken into account. Based on the obtained effective potentials, we may get the bound state properties of these discussed S -wave HT systems, and further judge whether these discussed new type of the doubly-charmed molecular tetraquark states exist or not, by which we hope that these crucial information can encourage experimental colleagues to focus on these predicted doublycharmed molecular tetraquark states in near future. In addition, we hope that the join effort from both experimentalist and theorist makes the family of the doubly-charmed molecular tetraquark states become more complete [12][13][14].
This paper is organized as follows. In Sec. II, we present the detailed deduction of the S -wave interactions between the charmed meson in H-doublet and the charmed meson in Tdoublet by adopting the OBE model, and the corresponding bound state properties of the S -wave HT systems will be presented in Sec. III. Finally, we will give a summary in Sec. IV.

II. THE DETAILED DEDUCTION OF THE HT INTERACTIONS
In the present work, we mainly want to answer whether or not the charmed meson in H-doublet and the charmed meson in T -doublet can be bound together to form new type of the doubly-charmed molecular tetraquark states. Firstly, we study the S -wave effective interactions between the charmed meson in H-doublet and the charmed meson in T -doublet within the OBE model [3,4], including the contribution from the σ, π, η, ρ, and ω exchange interactions in our concrete calculation, which is inspired by the experience of the nuclear force [3,4]. Here, the π, σ/η, and ρ/ω exchanges provide the long-range, intermediate-range, and short-range interactions, respectively.

A. The scattering amplitude
In the framework of the OBE model, we firstly write out the scattering amplitude M(h 1 h 2 → h 3 h 4 ) for the scattering process h 1 h 2 → h 3 h 4 by exchanging allowed light mesons. At the hadronic level, we usually adopt the effective Lagrangian approach to express the concrete scattering amplitude quantitatively. For the scattering process HT → HT , the relevant Feynman diagrams are given in Fig. 2. Here, we need to construct the corresponding effective Lagrangians to describe these related interactive vertexes.
The relevant Feynman diagrams for the scattering process HT → HT . Here, H and T stand for the D(D * ) and D 1 (D * 2 ), respectively.
By considering the heavy quark symmetry, the chiral symmetry, and the hidden local symmetry [49][50][51][52][53], one constructs the effective Lagrangians relevant to the charmed mesons in the H/T -doublet and the light scalar, pseudoscalar, or vector mesons, where the compact effective Lagrangians can be explicitly written as [34] for the light scalar meson exchange, for the light pseudoscalar meson exchange, and for the light vector meson exchange. In the above effective Lagrangians, the superfield H (Q) a is defined as a linear combination of the pseudoscalar charmed meson D with I(J P ) = 1/2(0 − ) and vector charmed meson D * with I(J P ) = 1/2(1 − ). And, the superfield T (Q)µ a is expressed as a combination of the axial-vector charmed meson D 1 with I(J P ) = 1/2(1 + ) and tensor charmed meson D * 2 with I(J P ) = 1/2(2 + ) in the heavy quark limit [54], which are defined by [34] H (Q) respectively. Here, v µ = (1, 0) denotes the four velocity under the non-relativistic approximation, and their conjugate fields satisfy H [34]. In addition, the covariant derivative D µ , the axial current A µ , the vector current V µ , the pseudo-Goldstone field ξ, the vector meson field ρ µ , and the vector meson field strength tensor F µν (ρ) can be defined as and F µν (ρ) = ∂ µ ρ ν − ∂ ν ρ µ + ρ µ , ρ ν , respectively. Here, the matrices for pseudoscalar mesons P and vector mesons V µ are expressed as [42,46,47] respectively. At the leading order of the pseudo-Goldstone field ξ, the axial current A µ and the vector current V µ read as [44] A µ = i f π ∂ µ P, and V µ = 0, (2.6) respectively. By expanding the compact effective Lagrangians, we can obtain the detailed effective Lagrangians to describe the coupling involving the charmed mesons in the H/T -doublet and the light scalar, pseudoscalar, or vector mesons, which can be found in Refs. [43][44][45].
As the important input parameters, the information about the coupling constants is crucial when studying the existence possibilities of the hadronic molecular states. These coupling constants can be extracted from the experimental data or calculated by the theoretical model [43-45, 48, 50, 55-60], and the corresponding signs for several coupling constants can be determined by the quark model [61]. For example, the coupling constant g is extracted from the experimental width of the D * + → D 0 π + process [56], the coupling constant β can be determined from the vector meson dominance mechanism [56], and the coupling constant k can be estimated with the quark model [61]. Here, we need to indicate that these coupling constants are widely used to discuss the strong interactions [43-45, 48, 57-60]. In addition, the masses of these involved hadrons are taken from the Particle Data Group [62]. In Table I, we collect these values of the coupling constants and hadron masses. Units of the coupling constants h = (h 1 + h 2 )/Λ χ , λ, λ , and µ 1 are GeV −1 , the coupling constant f π is given in unit of GeV, and the masses of the hadrons are given in unit of MeV. In addition, we need the normalization relations for these discussed charmed mesons to write out the concrete scattering amplitude quantitatively, which include (2.7) Here, the polarization vector µ m (m = 0, ±1) describing spin-1 field can be expressed as µ 0 = (0, 0, 0, −1) and µ ± = (0, ±1, i, 0) / √ 2 in the static limit, and the polarization tensor ζ µν m (m = 0, ±1, ±2) describing spin-2 field can be constructed as ζ µν m = m1,m2 1, m 1 ; 1, m 2 |2, m µ m 1 ν m 2 [63].
B. The effective potential in the momentum (coordinate) space For the effective potential in the momentum space in terms of the Breit approximation [64,65] and the nonrelativistic normalization. In order to solve the coupled channel Schrödinger equation in the coordinate space, the effective potential in the coordinate space V h 1 h 2 →h 3 h 4 E (r) should be given, which can be obtained by performing Fourier transformation for the effective potential in the momentum space V h 1 h 2 →h 3 h 4 E (q). The general relation can be explicitly expressed as In the above Fourier transformation, q and m E are the fourmomentum and mass of the exchanged light mesons, respectively. Generally speaking, the monopole type form factor is introduced in each interactive vertex phenomenologically [66,67], which describes the finite size effect of the discussed hadrons and compensates the off-shell effect of the exchanged light mesons. Here, Λ is the cutoff of the OBE potential [3]. The more explicit forms of the relevant Fourier transformation can be referred to Ref. [45].

C. The flavor and spin-orbital wave functions
For the doubly-charmed molecular tetraquark states composed of the charmed meson in H-doublet and the charmed meson in T -doublet, their isospin quantum numbers are either I = 0 or 1. When discussing the concrete effective potentials of these investigated doubly-charmed tetraquark systems within the OBE model, we take the same form for the flavor wave functions as that given in the former work [12,14,16]. In Table II, we summarize the flavor wave functions |I, I 3 for these discussed doubly-charmed tetraquark HT systems.
2 ), I and I 3 represent their isospin and the third component of these discussed doubly-charmed tetraquark HT systems, respectively.
And then, the general expressions for the spin-orbital wave functions | 2S +1 L J are written as In the above expressions, the notations A 0 , A 1 , and A 2 denote the charmed mesons D, D * (D 1 ), and D * 2 , respectively, while C e, f ab,cd and |Y L,m L stand for the Clebsch-Gordan coefficient and the spherical harmonics function, respectively.
As is known well, the deuteron is a loosely bound state composed of a proton and a neutron, which may be regarded as the well-established molecular state [43,[66][67][68]. The study of the deuteron shows that the S -D wave mixing effect may play important role in forming the loosely bound states [43].
Thus, we consider the S -D wave mixing effect in our calculation, where the relevant channels are summarized in Table III. Here, we use the notation | 2S +1 L J to simply denote the total spin S , relative angular momentum L, and total angular momentum J for the corresponding doubly-charmed tetraquark systems, and the detailed information of finding out the bound state solutions for these S -wave HT systems can be found in Ref. [43] when considering the S -D wave mixing effect and the coupled channel effect. The quantum numbers J P and possible channels | 2S +1 L J for the S -wave HT systems. Here, "..." means that the S -wave components in the corresponding states do not exist.
Through the above procedure, we can obtain the general expressions of the effective potentials in the coordinate space for these investigated doubly-charmed molecular tetraquark HT systems, which are collected in Appendix A.

III. BOUND STATE SOLUTIONS FOR THESE DISCUSSED S -WAVE HT SYSTEMS
Based on the obtained effective potentials in the coordinate space for these discussed doubly-charmed tetraquark systems, we solve the coupled channel Schrödinger equation, where the obtained bound state solutions include the binding energy E, the root-mean-square radius r RMS , and the probabilities for different components P i , which may provide us the critical information to judge whether these discussed doubly-charmed molecular tetraquark states exist or not.
Before producing the numerical calculation, we want to specify three points: 1. Since there exists the repulsive centrifugal potential ( + 1)/2µr 2 , a hadronic state with the higher partial wave 1 is less likely to generate the bound state [3,7]. Thus, we mainly focus on the S -wave HT systems in the current work.
2. Only the cutoff is a free parameter in the OBE model [3], and we attempt to find the loosely bound state solutions by changing the cutoff parameter in the range of 0.8-3.0 GeV in the following numerical analysis [44,47]. According to the experience of studying the deuteron, a loosely bound state with the cutoff parameter around 1.0 GeV can be suggested as the possible hadronic molecular candidate [43,[66][67][68], which is widely regarded as a reasonable input parameter to study the hadronic molecular states [3,4]. Although the bound state properties of these investigated S -wave HT systems depend on the cutoff parameter, we can predict possible doubly-charmed molecular tetraquark candidates at the qualitative level.
3. In addition, when judging whether the loosely bound state is an ideal hadronic molecular candidate, we expect that the reasonable binding energy should be at most tens of MeV, and the typical root-mean-square radius should be larger than the size of all the included component hadrons [3,69]. This is mainly because that the hadronic molecular state is a loosely bound state [3,4].
In the following, we analyze the bound state properties of these investigated S -wave HT systems by performing the single channel, S -D wave mixing, and coupled channel analysis.
A. The S -wave DD 1 system For the DD 1 system, we notice that there does not exist the long-range π exchange, but the σ, ρ, and ω exchanges are allowed within the OBE model because of the spin-parity conservation. In Table IV, we present the bound state properties for the S -wave DD 1 system. In our numerical analysis, we firstly get the bound state solutions with the single channel analysis. Furthermore, we consider the contribution of the S -D wave mixing effect and the coupled channel effect, and repeat the whole procedure of finding the bound state solutions of the S -wave DD 1 system.
For the S -wave DD 1 state with I(J P ) = 1(1 − ), we cannot find the bound state solution when scanning the cutoff parameter Λ = 0.8-3.0 GeV, even if we consider the contribution from the S -D wave mixing effect and the coupled channel effect. Therefore, there does not exist the S -wave DD 1 molecular state with I(J P ) = 1(1 − ).
In contrast, we can obtain the bound state solution for the S -wave DD 1 state with I(J P ) = 0(1 − ) by setting the cutoff parameter Λ around 2.30 GeV when only considering the contribution of the S -wave channel, and the bound state solution can also be found when the cutoff parameter is fixed to be larger than 2.25 GeV after adding the contribution of the D-wave channel. Besides, we further include the coupled channel effect for the S -wave DD 1 coupled system with I(J P ) = 0(1 − ). It is possible to find the bound state solution when we tune the cutoff parameter to be around 1.53 GeV, where the contribution of the D * D * 2 channel becomes obvious with increasing the cutoff value. Here, we notice that this state is the mixture of the DD 1 and D * D * 2 channels, where their probabilities are all over several tens of percents. Thus, our numerical results indicate that the contribution from the coupled channel effect may play an essential role in generating the S -wave DD 1 bound state with I(J P ) = 0(1 − ). If the existence of the S -wave DD 1 molecular state with I(J P ) = 0(1 − ) is possible, it can decay into the DD, DD * , and D * D * channels through the P-wave interaction. In addition, the three-body strong decay channels DDπ, DD * π, and D * D * π are also allowed. IV: Bound state solutions for the S -wave DD 1 system. The cutoff Λ, the binding energy E, and the root-mean-square radius r RMS are in units of GeV, MeV, and fm, respectively. Here, "×" indicates no bound state solutions when scanning the cutoff parameter 0.8-3.0 GeV, and we label the major probability for the corresponding channels in a bold manner.
Single channel analysis S -D wave mixing analysis In this subsection, we present the cutoff parameter Λ, the binding energy E, the root-mean-square radius r RMS , and the probabilities of the individual channel P i for the S -wave DD * 2 system in Table V. For the S -wave DD * 2 state with I(J P ) = 0(2 − ), there exist the loosely bound state solution when the cutoff parameter Λ is fixed to be around 1.66 GeV or even larger value when performing the single channel analysis. If making comparison of the bound state solutions with and without the S -D wave mixing effect, it is obvious that the S -D wave mixing effect is not obvious when generating the S -wave DD * 2 bound state with I(J P ) = 0(2 − ), where the D-wave contribution is less than 1%. After including the contribution of the coupled channel effect, we can obtain the bound state solution with the cutoff parameter Λ above 1.55 GeV for the S -wave DD * 2 coupled system with I(J P ) = 0(2 − ), where the dominant channel is DD * 2 with probability 97 percent and the remaining channels have very small probabilities. By including the S -D mixing effect and the coupled channel effect step by step, the cutoff value becomes smaller when reproducing the same binding energy. Thus, the coupled channel effect plays positive role in the formation of the S -wave DD * 2 bound state with S -D wave mixing analysis Coupled channel analysis If the cutoff parameter Λ smaller than 1.6 GeV is a reasonable input, we may conclude that the S -wave DD * 2 state with I(J P ) = 0(2 − ) is the possible doubly-charmed molecular tetraquark candidate.
For the S -wave DD * 2 state with I(J P ) = 1(2 − ), there do not exist the bound state solution until we increase the cutoff parameter to be around 3.0 GeV, even if we take into account the contribution of the S -D wave mixing effect and the coupled channel effect. Therefore, we conclude that our quantitative calculation does not support the existence of the S -wave DD * 2 molecular state with I(J P ) = 1(2 − ).
In the following, we also discuss the strong decay channels for the possible S -wave DD * 2 molecular candidate with I(J P ) = 0(2 − ). This state can decay into the DD * and D * D * channels through the P-wave interaction, and the DD 1 decay channel can be suppressed since it is a typical D-wave decay process. Additionally, the three-body strong decay modes are the DDπ, DD * π, and D * D * π channels.
C. The S -wave D * D 1 system Different from the former S -wave DD 1 and DD * 2 states, the S -wave D * D 1 states are more abundant due to the possibility of different quantum number combination. In Table VI, we collect the bound state properties for the S -wave D * D 1 system. In particular, we consider the coupled channel effect for the S -wave D * D 1 system in the present work, and find that the coupled channel effect provides slight contribution in forming the S -wave D * D 1 bound states, which is similar to the S -wave D * D 1 system [48] mainly determined by the non-diagonal matrix elements of the effective interactions [70].
For the S -wave D * D 1 states with I(J P ) = 0(0 − ) and 0(1 − ), we can obtain the bound state solutions when choosing the cutoff parameters around 0.96 GeV and 1.27 GeV, respectively. Additionally, after considering the contribution of the S -D wave mixing effect, the bound state solutions can also be found with the cutoff parameters around 0.94 GeV and 1.23 GeV, respectively. Since they have shallow binding energy and suitable root-mean-square radius under the reasonable range of the cutoff parameter, we conclude that the Swave D * D 1 states with I(J P ) = 0(0 − ) and 0(1 − ) should be viewed as the ideal doubly-charmed molecular tetraquark candidates according to the experience of studying the deuteron [43,[66][67][68], where the π exchange provides strongly attractive force at the long-range for the S -wave D * D 1 states with I(J P ) = 0(0 − , 1 − ). For the S -wave D * D 1 state with I(J P ) = 0(2 − ), there exist weakly bound state solution when the cutoff parameter is taken around 2.30 GeV if we only consider the contribution of the S -wave channel. And then, we can also find weakly bound state solution with the cutoff parameter around 2.05 GeV when adding the contribution of the D-wave channels. However, such cutoff parameter is away from the reasonable range around 1.0 GeV [43,[66][67][68]. By comparing the bound state properties of the S -wave D * D 1 molecular candidates with I(J P ) = 0(0 − , 1 − ), it is obvious that the Swave D * D 1 state with I(J P ) = 0(2 − ) as the doubly-charmed molecular tetraquark candidate is no priority. Based on the analysis mentioned above, it is clear that the S -wave isoscalar D * D 1 states with the lower spin may be bound more tightly compared to those higher spin states, which is related to the obtained operator values O ( ) k for the S -wave D * D 1 states with the difference total angular momentum J [71].
Contrary to the above situation, when only considering the contribution of the S -wave channel, there do not exist the bound state solution with the cutoff parameter restricted to be below 3.0 GeV for the S -wave isovector D * D 1 states. This situation almost keeps the same when the S -D wave mixing effect is included. Thus, we exclude the possibilities of the existence of the S -wave isovector D * D 1 molecular states. This can be easily understood since the π exchange potential provides the repulsive force for the S -wave isovector D * D 1 states with the lower spin.
To summarize, we can predict that the S -wave D * D 1 states with I(J P ) = 0(0 − ) and 0(1 − ) can be viewed as the ideal candidates of the doubly-charmed molecular tetraquark states for their reasonable cutoff value, binding energy, and root-meansquare radius [43,[66][67][68]. In the following, we also discuss their possible strong decay behaviors. For the S -wave D * D 1 molecular candidates with I(J P ) = 0(0 − ) and 0(1 − ), the allowed two-body and three-body strong decay channels include respectively. Thus, we suggest the experiments should firstly carry out the search for these suggested doubly-charmed molecular tetraquark candidates in the future. For the S -wave D * D * 2 system, the corresponding bound state properties are collected in Table VII for all possible spinparity configurations. In order to check the specific roles of the S -D wave mixing effect, we present the numerical results without and with considering the S -D wave mixing effect.
From analyzing the bound state solutions for the S -wave D * D * 2 system presented in Table VII, we summarize the following points: • For the S -wave D * D * 2 state with I(J P ) = 0(1 − ), we can obtain the binding energies around a few MeV and the root-mean-square radii around a few fm with the cutoff parameter Λ larger than 1.06 GeV, when only considering the contribution of the S -wave channel. Besides, we can also get the bound state solution when the cutoff parameter Λ is tuned larger than 1.04 GeV after considering the S -D wave mixing effect. However, the contribution of the D-wave channels is very small. Borrowing the experience of studying deuteron [43,[66][67][68], we may regard the S -wave D * D * 2 state with I(J P ) = 0(1 − ) as the possible doubly-charmed molecular tetraquark candidate. For the S -wave D * D * 2 state with I(J P ) = 1(1 − ), we fail to find the bound state solution with the cutoff parameter less than 3.0 GeV, even if    the contribution of the S -D wave mixing effect is introduced. Thus, the S -wave D * D * 2 state with I(J P ) = 1(1 − ) cannot be bound together to form the hadronic molecular state.
• For the S -wave D * D * 2 state with I(J P ) = 0(2 − ), when the cutoff parameter Λ is slightly bigger than 1.60 GeV, we can obtain the bound state solution with shallow binding energy and suitable root-mean-square radius by performing single channel analysis. When we further consider the S -D wave mixing effect, the loosely bound state solution can also be obtained if we tune the cutoff parameter Λ larger than 1.58 GeV. Thus, the S -wave D * D * 2 state with I(J P ) = 0(2 − ) seems to be the possible doubly-charmed molecular tetraquark candidate. For the S -wave D * D * 2 state with I(J P ) = 1(2 − ), we cannot find the bound state solution corresponding to the cutoff parameter 0.8 < Λ < 3.0 GeV, even if we consider the contribution of the S -D wave mixing effect. Therefore, we can exclude the possibilities of the S -wave D * D * 2 state with I(J P ) = 1(2 − ) as the doubly-charmed molecular tetraquark.
• For the S -wave D * D * 2 state with I(J P ) = 0(3 − ), if the contribution of the D-wave channels does not considered, we should mention that the interactions are not sufficient to form the bound state until we increase the cutoff parameter Λ to be around 3.0 GeV. For the S -wave D * D * 2 state with I(J P ) = 1(3 − ), we find that the binding energy can reach up to several MeV when taking the cutoff parameter Λ to be around 2.00 GeV with the single channel analysis. However, when we further consider the contribution of the S -D wave mixing effect, we can find that there exist the loosely bound state solutions for the S -wave D * D * 2 states with I(J P ) = 0(3 − ) and 1(3 − ) if setting the cutoff parameters Λ larger than 2.66 GeV and 1.98 GeV, respectively. Comparing the numerical results with and without considering the S -D wave mixing effect, we notice that the S -D wave mixing effect is important in the formation of the S -wave D * D * 2 bound state with I(J P ) = 0(3 − ). However, such cutoff parameters deviate from the reasonable range around 1.0 GeV [43,[66][67][68]. Thus, it seems that the S -wave D * D * 2 states with I(J P ) = 0(3 − ) and 1(3 − ) as the doubly-charmed molecular tetraquark candidates are no priority.
Additionally, we also provide the allowed two-body and three-body strong decay channels for these predicted doublycharmed molecular tetraquark candidates. For the most promising S -wave D * D * 2 molecular candidate with I(J P ) = 0(1 − ), the allowed two-body decay modes include the DD, DD * , D * D * , DD 1 , DD * 2 , and D * D 1 channels, and the possible three-body decay modes are the DDπ, DD * π, and D * D * π channels. For the possible S -wave D * D * 2 molecular candidate with I(J P ) = 0(2 − ), the allowed two-body and three-body decay modes are the DD * , D * D * , DD 1 , DD * 2 , D * D 1 , DDπ, DD * π, and D * D * π channels.

IV. SUMMARY
Exploring the exotic hadronic state is an interesting and important research topic of the hadron physics. Very recently, the LHCb Collaboration reported a new structure T + cc when analyzing the D 0 D 0 π + invariant mass spectrum [11]. This doubly-charmed tetraquark T + cc can be assigned as the isoscalar DD * molecular state with J P = 1 + [12,13,15,16], which is due to the mass of this doubly-charmed tetraquark state T + cc locates just below the DD * threshold. To some extent, this new observation makes the study of the interactions between two charmed mesons becomes an intriguing research issue.
In this work, we attempt to explore new type of the doublycharmed molecular tetraquark candidates composed of the charmed meson in H-doublet and the charmed meson in Tdoublet. In our concrete calculation, the effective potentials can be obtained by the OBE model when including the contribution from the σ, π, η, ρ, and ω exchanges, and we consider both the S -D wave mixing effect and the coupled channel effect. Our numerical results show that the S -wave D * D 1 states with I(J P ) = 0(0 − , 1 − ) and the S -wave D * D * 2 state with I(J P ) = 0(1 − ) should be viewed as the most promising doubly-charmed molecular tetraquark candidates, the S -wave DD 1 state with I(J P ) = 0(1 − ), the S -wave DD * 2 state with I(J P ) = 0(2 − ), and the S -wave D * D * 2 state with I(J P ) = 0(2 − ) are the possible candidates of the doubly-charmed molecular tetraquark states, and other investigated S -wave HT states as the doubly-charmed molecular tetraquark candidates are no priority (see Fig. 3). Here, we need to emphasize that these possible doubly-charmed molecular tetraquark candidates have the typical exotic quark configuration ccqq different from the conventional meson states, which gives us a good opportunity to identify the tetraquark hadronic states [48].
[3735] Similarly to the observation of the T + cc in the D 0 D 0 π + invariant mass spectrum [11], these predicted doubly-charmed molecular tetraquark states can be searched in these allowed two-body or three-body strong decay channels, since these predicted S -wave HT molecular candidates can decay into the DD, DD * , D * D * , DD 1 , DD * 2 , D * D 1 , DDπ, DD * π, and D * D * π channels if the kinetically allowed (see Fig. 3 for more details). These two-body or three-body strong decay modes can provide valuable information when searching for these predicted doubly-charmed molecular tetraquark states experimentally. As a potential experimental research issue, we strongly suggest the LHCb Collaboration to pay attention to these possible two-body or three-body strong decay channels from the proton-proton collisions with the accumulation of experimental data. With more jigsaws of new types of the T cc states will be observed in future experiment, the zoo of the doubly-charmed molecular tetraquark becomes booming.
Appendix A: The effective potentials Before presenting the effective potentials for these investigated doubly-charmed tetraquark systems, we firstly define the function Y(Λ i , m i , r), i.e., (1.1) Here, H(I) and G(I) are the isospin factors for these investigated doubly-charmed tetraquark systems, and I denotes the isospin quantum number.
With the above preparation, the expressions of the effective potentials in the coordinate space for these investigated doubly-charmed tetraquark systems are given by • For the process DD 1 → DD 1 : • For the process DD 1 → D 1 D: • For the process DD * 2 → DD * 2 : • For the process DD * 2 → D * 2 D: • For the process D * D 1 → D * D 1 : (1.10) • For the process D * D 1 → D 1 D * : (1.11) • For the process D * D * 2 → D * D * 2 : (1.12) • For the process D * D * 2 → D * 2 D * : In the above expressions, the variables q i (i = 1, · · ·, 4) are written as q 1 = m D 1 − m D , q 2 = m D * 2 − m D , q 3 = m D 1 − m D * , and q 4 = m D * 2 − m D * . In the above OBE effective potentials, we also introduce several operators O ( ) k involved in this work [45,48], i.e., Here, we define = m,n,a,b C 2,m+n 1m,1n C 2,a+b 1a,1b , and T (x, y) = 3 (r · x) (r · y) − x · y withr = r/|r| is the tensor force operator. For the operators O k , we just need to make the change with the subscripts of the polarization vector i for the operators O k , i.e., 1 ↔ 2 and 3 ↔ 4. In our calculation, the corresponding matrices elements f |O ( ) k |i are obtained by sandwiched these operators O ( ) k between the relevant spin-orbit wave functions of the initial and final states. In Tables VIII-IX, we collect the obtained operator matrix elements O ( ) k [J] (k = 1, · · ·, 21), which will be used in the calculation [45,48].  13).