Doubly heavy tetraquarks in an extended chromomagnetic model

Using an extended chromomagnetic model, we perform a systematic study of the masses of the doubly heavy tetraquarks. We find that the ground states of the doubly heavy tetraquarks are dominated by color-triplet $\ket{(qq)^{\bar{3}_{c}}(\bar{Q}\bar{Q})^{3_{c}}}$ configuration, which is opposite to that of the fully heavy tetraquarks. The combined results suggest that the color-triplet configuration becomes more important when the mass difference between the quarks and antiquarks increases. We find three stable states which lie below the thresholds of two pseudoscalar mesons. They are the $IJ^{P}=01^{+}$ $nn\bar{b}\bar{b}$ tetraquark, the $IJ^{P}=00^{+}$ $nn\bar{c}\bar{b}$ tetraquark and the $J^{P}=1^{+}$ $ns\bar{b}\bar{b}$ tetraquark.

In the quark model [114][115][116][117][118], the mass of hadron can be decomposed into the quark masses, the kinetic energy and the potentials which include the color-independent Coulomb and confinement interactions, and the hyperfine interactions like the spin-spin, spin-orbit, and tensor terms. If we restrict to the S-wave states, the spin-orbit and tensor interactions do not contribute. We can use the extended chromomagnetic model [1,58,[118][119][120][121][122][123][124][125][126][127]. In this model, the masses of S-wave hadrons consist of effective quark masses, the color interaction and the chromomagnetic interaction. This simplified model gives good account of all S-wave mesons and baryons [125]. In this work, we use the extended chromomagnetic model to study the S-wave doubly heavy tetraquarks. With the wave function obtained, we further use a simple method to estimate the partial decay ratios of the tetraquark states. In Sec. II we introduce the methods of present work, The numerical results are presented and discussed in Sec. III. We conclude in Sec. IV.

A. Hamiltonian
In the chromomagnetic model, the Hamiltonian of the S-wave hadron reads [123,[125][126][127][128][129][130][131] where m i is the effective mass of ith quark, H CE is the chromoelectric (CE) interaction [123,[125][126][127] and H CM is the chromomagnetic (CM) interaction [1,26,[120][121][122] Here, a ij and v ij ∝ α s (r ij )δ(r ij ) /m i m j are effective coupling constants which depend on the constituent quark masses and the spatial wave function. S i = σ i /2 and F i = λ i /2 are the quark spin and color operators. For the antiquark, and the total color operator i F i nullifies any colorsinglet physical state, we can rewrite the Hamiltonian as [125][126][127] where is the quark pair mass parameter. V C ij = F i · F j and V CM ij = S i · S j F i · F j are the color and CM interactions between quarks.

B. Wave function
To investigate the masses of the tetraquarks, we need to construct the wave functions. The total wave function is a direct product of the orbital, color, spin and flavor wave functions. Here, the orbital wave function is symmetric since we only consider the S-wave states. Since the Hamiltonian does not contain a flavor operator explicitly, we first construct the color-spin wave function, and then incorporate the flavor wave function to account for the Pauli principle.
The spins of the tetraquarks can be 0, 1 and 2. In the qq⊗qq configuration, the possible color-spin wave functions {α J i } are listed as follows, 2. J P = 1 + : 3. J P = 2 + : where the superscript 3,3, 6 or6 denotes the color, and the subscript 0, 1 or 2 denotes the spin.

C. Partial decay rates
Next we consider the strong decay properties of the tetraquarks. There are various methods for studying the tetraquark decays, such as the dimeson decay through the quark interchange model [132][133][134][135] and the dibaryon decay through the 3 P 0 model [136][137][138][139]. These models require the dynamical structure of the hadrons, which is beyond the power of the chromomagnetic model. Here we adopt a simple method to estimate the partial decay ratios of the tetraquark states.
In Sec. II B we have constructed the wave function in the qq⊗qq configuration, the tetraquark states are superposition of the bases. The tetraquark states can also be written as the linear superposition of the bases in the qq⊗qq configuration (see Appendix A). Normally, the qq component in the tetraquark can be either of colorsinglet or of color-octet. The former one can easily dissociate into two S-wave mesons in relative S wave, which is called "Okubo-Zweig-Iizuka-(OZI-)superallowed" decays. The recoupling coefficient tell us the overlap between the tetraquark and a particular meson × meson state. Then we can determine the decay amplitude of the tetraquark into that particular meson × meson channel. The latter one can only fall apart through the gluon exchange [120,140]. In this work, we will focus on the "OZI-superallowed" decays.
For each decay mode, the branching fraction is proportional to the square of the coefficient c i of the corresponding component in the eigenvectors, and also depends on the phase space. For two body decay through L-wave, the partial decay width reads [126,141] where m is the mass of the initial state, k is the momentum of the final states in the rest frame of the initial state, α is an effective coupling constant, and γ i is a quantity determined by the decay dynamics. Generally, γ i is determined by the spatial wave functions of both initial and final states, which are different for each decay process. In the quark model, the spatial wave functions of the pseudoscalar and vector mesons are the same. Thus for each tetraquark, we have where M i and M * i are pseudoscalar and vector mesons respectively. Then we can estimate the partial decay width ratios of the tetraquark states.

A. Parameters
To calculate the tetraquark masses, one needs to estimate the parameters {m t ij , v t ij }. In Ref. [125] we used the meson and baryon masses to extract the parameters } between two heavy quarks cannot be fitted from baryons because of the lack of experimental data. For this reason, we adopted the assumptions and R bm q1q2 ≡v b q1q2 /v m q1q2 = 2/3 ± 0.30 (30) to estimate them from the meson parameters {m m Q1Q2 , v m Q1Q2 }. The resulting parameters are listed in Table I. Since the CM interaction strength v ij 's are inversely proportional to the quark masses, the meson parameters {v cc , v cb , v bb } between heavy flavors are quite small. Thus the large uncertainty of the ratio R bm q1q2 does no have much effects on the baryon parameters {v cc , v cb , v bb } and the mass spectrum of the doubly heavy tetraquarks. As shown in Ref. [125], the introduction of the first assumption makes the difference δm bm q1q2 ≡m b q1q2 − m m q1q2 separable over the two quarks where δm bm q ≡m b q − m m q is the difference of the effective quark mass extracted from the baryon and meson. In this way, the ten δm bm q1q2 's reduce to four δm bm q 's. Actually, such property can be achieved by a weaker assumption. Namely, we assume that the difference a b q1q2 − a m q1q2 is separable over the two quarks  (34) which includes the quark mass difference and the differences between color interactions. We again reduce the ten δm bm q1q2 's into four degrees of freedom. All results are unchanged except that we reinterpret the δm bm q of Ref. [125] as δm bm q (see Table II or Table VI of Ref. [125]). Now we consider the tetraquarks. In Ref. [127], We used the following scheme to estimate the masses of the fully heavy tetraquarks Within this scheme, we found that the ground states of the fully heavy tetraquarks are dominated by color-sextet configurations, which is consistent with the dynamical calculations [142,143]. Nonetheless, this scheme ignores the difference of the spatial configurations between the tetraquarks and the normal hadrons, which will evidently cause large uncertainties [1,143,144]. To appreciate the uncertainty, we introduce three additional schemes for comparison (see Table III). The scheme III (IV) differs from the scheme I (II) by Due to the smallness of v b qQ and v m qQ , the results in scheme I (II) are very similar to those in scheme III (IV). Thus we will focus on the scheme I and scheme II.
B. The nnQQ systems

The nncc and nnbb tetraquarks
Inserting the parameters into the Hamiltonian, we can determine the tetraquark masses. The masses and eigenvectors of the nnQQ tetraquarks are listed in Table IV.
Here, we assume that the SU(2) flavor symmetry is exact and denote u, d quarks collectively as n. In the following, we will use T i (nnQQ, m, I, J P ) to represent the nnQQ tetraquarks, where the subscript i denotes the particular scheme of the parameters. In Figs 1-2, we plot the relative position of the nnQQ tetraquarks and their meson-meson thresholds.
We first consider the nncc tetraquarks. The quantum number of its lightest state is IJ P = 01 + , namely the Parameter The other isoscalar state is T I (nncc, 3976.1, 0, 1 + ) or T II (nncc, 4230.8, 0, 1 + ). We find that the scheme II always gives larger masses than the scheme I. The reason is that the two schemes choose different value of m t qiqj , which results in different values of the color interaction. More precisely, the difference of the color interaction between the two schemes is where in the last line we have ignored the terms proportional to i F i . Note that both (q 1 q 2 ) 6c (q 3q4 )6 c and (q 1 q 2 )3 c (q 3q4 ) 3c are eigenstates of (F 1 + F 2 ) 2 , with eigenvalues 10/3 and 4/3 respectively. In other words, For the nncc system, i δm bm i = 212.9 MeV. The ground state T I (nncc, 3749.8, 0, 1 + ) is dominated by the color-triplet configuration, and its mass is increased by about 118.9 MeV. While the mass of the color-sextet configuration dominated state T I (nncc, 3976.1, 0, 1 + ) is increased by 254.7 MeV. The deviation from Eq. (41) is caused by the color mixing. In the isovector sector, we have four tetraquark states. They are all above the corresponding S-wave decay channels. It is interesting to note that the T II (nncc, 3686.7, 0, 1 + ) in scheme II is quite close to the newly observed T + cc state. The nnbb tetraquarks is very similar to the nncc tetraquarks. Its lightest state also have quantum number IJ P = 01 + , namely the T I (nnbb, 10291.6, 0, 1 + ) or T II (nnbb, 10390.9, 0, 1 + ). In both schemes, this state lies below the BB threshold and is stable against strong decays. In scheme I, the T I (nnbb, 10468.9, 1, 0 + ), T I (nnbb, 10485.3, 1, 1 + ) and T I (nnbb, 10507.9, 1, 2 + ) also lie below the the BB threshold. But they are not stable in scheme II. Thus we cannot draw a definite conclusion.
Besides the masses, the eigenvectors also help understand the nature of the tetraquarks. Within the four possible quantum numbers, the IJ P = 10 + one and the IJ P = 01 + one are of particular interest because they both have two possible color configurations, namely the color-sextet |(qq) 6c ⊗(QQ)6 c and colortriplet |(qq)3 c ⊗(QQ) 3c . For simplicity, we denote them as 6 c ⊗6 c and3 c ⊗3 c . As pointed out by Wang et al. [142], there are two competing effects in determining whether the 6 c ⊗6 c or3 c ⊗3 c dominates the tetraquark's ground state. In the one-gluon-exchange (OGE) model, the color interactions in color-triplet diquark are attractive, while those in color-sextet diquark are repulsive. On the other hand, the attractions between 6 c diquark and6 c antidiquark and between the3 c ⊗3 c counterpart are both attractive, and the former one is much stronger. The authors of Refs. [127,142,143] found that the colorsextet configuration has more net attractions for most fully heavy tetraquarks. Thus the ground states contain more color-sextet components than the color-triplet one. The only exception is the ccbb tetraquark in model II of Ref [142], whose ground state has 53% of the3 c ⊗3 c component. It is also interesting to note that, when the mass ratio between quarks and antiquarks deviates from one, the color-triplet configuration becomes more important in the ground states. For example, Ref. [127] found that the T (bbbb, 18836.1, 0 ++ ) and T (cccc, 6044.9, 0 ++ ) have 18.5% and 30.5% of the3 c ⊗3 c components, while the T (ccbb, 12596.3, 0 ++ ) has 48.4%. This tendency also exists in the doubly heavy tetraquarks. As shown in Table IV, the3 c ⊗3 c components become dominant in ground states of the nnQQ tetraquarks. This phenomenon can also be explained by the color interaction Hamiltonian, Taking scheme I as an example, we have while for the fully heavy tetraquarks δm ccbb = +15.15 MeV .
As the ratios mq/m q 's increase, the3 c ⊗3 c components become more important in the ground states. Another interesting conclusion from the Hamiltonian is that the color interaction does not mix the 6 c ⊗6 c and 3 c ⊗3 c configurations. Actually, this conclusion applies for all S-wave tetraquarks with q 1 = q 2 orq 3 =q 4 . Let's consider the matrix element of color interaction α | H CE | β . Note that the color interaction is independent of the spin operator, and thus is a rank-0 tensor in the q 1 q 2 spin space. Its matrix elements over different q 1 q 2 spin states always vanish. If q 1 = q 2 , the Pauli principle further renders the matrix elements vanish unless the bases α and β possess the same color symmetry over q 1 q 2 . The same argument works forq 3q4 as well. In summary, the color interaction does no mix the 6 c ⊗6 c and 3 c ⊗3 c color configurations if q 1 = q 2 orq 3 =q 4 .
Next we consider their decay properties. For the decays of the tetraquark states in this work, the (k/m) 2 's are all of O(10 −2 ) or even smaller. All higher wave decays are suppressed. Thus we will only consider the S-wave decays in this work. First we transform the wave function of nnQQ tetraquarks into the nQ⊗nQ configuration. Then we can calculate the k · |c i | 2 's and partial decay width ratios. The corresponding results are listed in Tables V-X. Note that the two schemes give very similar results, we will mainly focus on the scheme I in the following. In the isovector sector, the nncc tetraquarks are mostly above the S-wave decay channels, thus are wide states. Depending on the schemes, the J P = 2 + state may lie on or above thē D * D * threshold. Namely the T I (nncc, 4017.1, 1, 2 + ) or T II (nncc, 4123.8, 1, 2 + ). A firm conclusion requires more detailed studies. The T I (nncc, 4127.4, 1, 0 + ) can decay into bothDD andD * D * channels, with partial decay width ratio ΓD * D * : ΓDD ∼ 37.5 .

The nncb tetraquark
Next we consider the nncb tetraquark. We list the masses and eigenvectors of these states in Table IV. Their relative position and possible decay channels are plotted in Fig. 2. There are two possible stable nncb tetraquark states. The first state is T I (nncb, 7003.4, 0, 0 + ), which lies below theDB threshold by more than 100 MeV. Even in scheme II, this state is about 20 MeV below the threshold. The second state is T I (nncb, 7046.2, 0, 1 + ). It is about 100 MeV lighter than theDB threshold. However, this state lies above the threshold in scheme II. Nonetheless, it lies below its S-wave decay modeDB * , thus should be a narrow state.
Since the two antiquarks do not have to obey the Pauli principle, we have much bigger number of states than the nncc/nnbb cases. For each isospin, we have two 0 + states, three 1 + states and one 2 + state. From Table IV, we see that for each possible quantum number, the lower mass states are dominated by color-triplet configurations, while the color-sextet configurations are more important in the higher mass states. For example, the two stable states T I (nncb, 7003.4, 0, 0 + ) and T I (nncb, 7046.2, 0, 1 + ) have 80.6% and 90.0% of3 c ⊗3 c components respectively. This can be explained by the color interaction Note that both 6 c ⊗6 c and3 c ⊗ 3 c configurations are eigenstates of V C 12 + V C 34 , with eigenvalues 2/3 and −4/3 respectively. The negative value of δm indicates that the color interaction favors the3 c ⊗ 3 c configuration.
In Table XI, we transform the nncb tetraquarks into the nc⊗nb configuration. Then we calculate the values of k · |c i | 2 and relative partial decay widths, as shown in Tables XII-XIII. Besides the two stable states discussed above, two heavier isoscalar states T I (nncb, 7220.3, 0, 0 + ) and T I (nncb, 7232.9, 0, 1 + ) are above theDB * , while T I (nncb, 7329.3, 0, 1 + ) can decay into bothD * B and DB * modes, with relative width ΓD * B : ΓD B * ∼ 11.2 : 1 (54) In the isovector sector, the lower 0 + state can only decay intoDB mode, while the higher one can also decay intō D * B * mode. All 1 + states can decay intoDB * in Swave, while only the highest one can decay intoD * B andD * B * modes, with partial decay rates ΓD * B * : ΓD * B : ΓD B ∼ 4.6 : 2.0 : 1 (55) There is no doubt that the current results rely on the mass estimation. In scheme II, the higher masses allow the states to have more decay modes. Yet we find that the partial decay width ratios are quite stable in the two schemes.

C. The ssQQ systems
We list the numerical results of the ssQQ in Table XIV-XXIII. We also plot the relative position and possible decay channels in Figs. 3-4. The pattern of the mass spectrum is very similar to that of the nnQQ tetraquarks with isospin I = 1.
First we focus on the sscc tetraquarks. The ground state is T I (sscc, 4043.7, 0 + ). It can decay intoD sDs in S-wave, and thus might be a wide state. The most heavy state T I (sscc, 4311.1, 0 + ) lies above theD * Thus theD * sD * s mode is dominant. The other two state T I (sscc, 4192.6, 1 + ) and T I (sscc, 4264.5, 2 + ) can decay intoD sD * s andD * sD * s modes respectively. Next we turn to the ssbb tetraquarks. In scheme I, the T I (ssbb, 10697.1, 0 + ) and T I (ssbb, 10718.2, 1 + ) lie below the B s B s threshold, and the T I (ssbb, 10742.5, 2 + ) lies just above the B s B s threshold, which suggests that they are stable states. However, they become heavier than their S-wave decay channels in scheme II. A detailed study with dynamical model, or experimental researches, is required to distinguish which of the two schemes gives better description of the ssbb tetraquarks. In both schemes, the three states are dominated by3 c ⊗3 c configuration. Actually, their wave functions are nearly the same, except that they have different total spin, which is the reason for their different masses.
From Fig. 4, we see that the sscb tetraquarks are all above the S-wave decay channels and are probably broad states. Among them, the T I (sscb, 7534.3, 2 + ) is slightly above theD * s B * s . Its decay width may be relatively narrow. We also calculate the partial decay width ratios of the sscb tetraquarks. It is interesting that some of the ratios are different in the two schemes. For example, in scheme I and in scheme II which can be used to distinguish the two schemes.

D. The nsQQ systems
We list the masses and wave functions of the nsQQ in Table XXIV. The ground states of the nscc and nsbb tetraquarks are both of 1 + . They are strange counterparts of the IJ P = 01 + nnQQ tetraquarks. Among them, the T I (nscc, 3919.0, 1 + ) lies above theDD s threshold, while the T I (nsbb, 10473.1, 1 + ) lies deeply below the BB s threshold. In scheme II, the former one lies above its S-wave decay channelsD * D s andDD * s , while the latter one is still stable. We hope the future experiment can reach for this state.
The last class of the doubly heavy tetraquarks is the nscb system. It is composed of four different quarks. Similar to the nncb tetraquarks, the ground state of the nscb tetraquarks has quantum number 0 + . Depending on the scheme used, it may be a stable state. A full dynamical quark model study is needed to have a better understanding of these states.
We also study the decay properties of the nsQQ tetraquarks, which can be found in Tables XXV-XXXIV

IV. CONCLUSIONS
In this work, we systematically study the mass spectrum of the doubly heavy qqQQ tetraquarks in the frame-work of an extended chromomagnetic model. In addition to the chromomagnetic interaction, the effect of color interaction is also considered in this model. The model pa- rameters are fitted from the mesons and baryons. Since the spatial configurations of the qq (qq) and qq pairs are different in the conventional hadrons and the tetraquarks, applying these parameters to the tetraquarks may cause errors. To appreciate this uncertainty, we adopt two schemes of parameters to study the qqQQ tetraquarks. As indicated in Eq. (41), the scheme II gives larger masses than the scheme I. However, the wave functions and decay properties of the two schemes are very similar for the qqQQ tetraquarks. We find three states which are stable in both schemes. They are the nnbb tetraquark with quantum number IJ P = 01 + , the nncb tetraquark with quantum number IJ P = 00 + and the nsbb tetraquark with quantum number J P = 1 + . They all lie below the thresholds of two pseudoscalar mesons, which can only decay through weak processes. Meanwhile, many narrow states which lie below S-wave decay channels are also found. It shall be interesting to search for these states.
The tetraquarks have two possible color configurations, namely the color-sextet configuration |(qq) 6c (QQ)6 c and the color-triplet one |(qq)3 c (QQ) 3c . Unlike the fully heavy tetraquarks, the ground states of the doubly heavy tetraquarks favor the color-triplet configurations. Combining the results of fully and doubly heavy tetraquarks, we can clearly see the trend that the color-triplet config-uration is more and more important when the mass ratio between the quarks and antiquarks increases.
Besides the mass spectrum, we also estimate the decay properties of the tetraquarks. We hope these states can be searched for by future experiments. To calculate the partial decay rates, we need to construct the tetraquark wave functions in the qq⊗qq configuration. The possible color-spin wave functions {β J i } are listed as follows, where the superscript 1 or 8 denotes the color, and the subscript 0, 1 or 2 denotes the spin. Among them, the