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PHYSICAL REVIEW D 97, 094015 (2018)

Heavy-flavored tetraquark states with the QQQ¯ Q¯ configuration

† ‡ Jing Wu,1 Yan-Rui Liu,1,* Kan Chen,2,3 Xiang Liu,2,3, and Shi-Lin Zhu4,5,6, 1School of Physics and Key Laboratory of Physics and Particle Irradiation (MOE), Shandong University, Jinan 250100, China 2School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China 3Research Center for and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China 4School of Physics and State Key Laboratory of and Technology, Peking University, Beijing 100871, China 5Collaborative Innovation Center of Quantum , Beijing 100871, China 6Center of High Energy Physics, Peking University, Beijing 100871, China

(Received 7 May 2016; published 17 May 2018)

In the framework of the color-magnetic interaction, we systematically investigate the mass spectrum of the tetraquark states composed of four heavy with the QQQ¯ Q¯ configuration in this work. We also show their strong decay patterns. Stable or narrow states in the bbb¯c¯ and bcb¯c¯ systems are found to be possible. We hope the studies shall be helpful to the experimental search for heavy-full exotic tetraquark states.

DOI: 10.1103/PhysRevD.97.094015

I. INTRODUCTION reported the hidden-charm Pcð4380Þ and 1 Pcð4450Þ in the invariant mass of ψp. They are consistent Searching for exotic states is an interesting research field with the structures [29–37]. With a model- full of challenges and opportunities. Among the various independent analysis, the LHCb confirms their previous exotic candidates, the multiquark state is a very popular model-dependent evidence for these states [38]. Very hadronic configuration. In fact, the concept of the multi- recently, a structure Xð5568Þ was announced by the D0 state was proposed a long time ago in Refs. [1–3].In Collaboration [39]. This narrow state is about 200 MeV the past decades, experimentalists have been trying to find ¯ 0 below the BK threshold and decays into Bsπ . Therefore, them. The observations of the charmoniumlike/bottomo- Xð5568Þ niumlike XYZ states [4–11] have provided valuable hints the may be a typical tetraquark state composed of four different flavors [40–52] while the state of the existence of the exotic states. Especially, these Xð5568Þ charged charmoniumlike or bottomoniumlike states like assignment to is not favored [53]. However, the LHCb Collaboration did not confirm the existence of the Zð4430Þ [12–15], the Z1ð4050Þ [16], the Z2ð4250Þ the Xð5568Þ [54], which makes some theorists consider [16], the Zcð3900Þ [17–20], the Zcð3885Þ [21–23], the the difficulty of explaining the Xð5568Þ as a genuine Zcð4020Þ [24,25], the Zcð4025Þ [26,27], the Zcð4200Þ resonance [55–63]. Although there exist different opinions [14], the Zbð10610Þ [28], and the Zbð10650Þ [28] have of the Xð5568Þ, the observation of the Xð5568Þ again forced us to consider the existence of the multiquark matter ’ very seriously. So many observations in the heavy quark ignites a theorist s enthusiasm of exploring exotic tetra- sector in recent years are surprising, which provides new quark states. As discussed above, there exist possible candidates of the opportunities for us to understand the nature of the strong hidden-charm tetraquark states and the tetraquark with a interaction. Experimentalists continue to surprise us after the obser- heavy flavor quark and three light quarks. If the multiquark vations of XYZ states. In 2015, the LHCb Collaboration states indeed exist in nature, we have a strong reason to believe that there are more tetraquark states with other flavor configurations. In this work, we focus on the heavy-flavor *[email protected] Q Q Q¯ Q¯ † tetraquark states with the 1 2 3 4 configuration, where [email protected] ‡ Qi is a c,orb quark. Although these tetraquark states are still [email protected] missing in experiment, it is time to carry out a systematic ¯ ¯ Published by the American Physical Society under the terms of investigation of the mass spectrum of the Q1Q2Q3Q4 the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1For convenience, here and in the following, ψ means J=ψ and DOI. Funded by SCOAP3. once the symbol is adopted.

2470-0010=2018=97(9)=094015(13) 094015-1 Published by the American Physical Society WU, LIU, CHEN, LIU, and ZHU PHYS. REV. D 97, 094015 (2018) tetraquark system, which may provide important information where λi’s are the Gell-Mann matrices and σi’s are the Pauli to further experimental exploration. matrices. The above Hamiltonian was deduced from the In Ref. [64], Iwasaki studied the hidden-charm tetraqark one--exchange interaction [85]. The effective cou- state composed of a pair of charmed quark and antiquark as pling constants Cij incorporate effects from the spatial well as a pair of light quark and antiquark, which has the wave function and the quark masses, which depend on 2 2 ccq¯ q¯ configuration. The dimeson configuration (Q q¯ )is the system. We determine their values in the next section. stable against the strong decay into two [65]. Chao This Hamiltonian leads to a mass formula for the studied performed a systematical investigation of the ccc¯ c¯ tetra- system quark system in a quark-gluon model for the first time X in Ref. [66]. In Ref. [67], the authors used the Born- M ¼ m þhH i; ð Þ i CM 2 Oppenheimer approximation for heavy quarks in the MIT i bag and found that the heavy-quark system c2c¯2 is stable cc¯ against a breakup into two pairs. But in a potential where mi is the effective mass of the ith constituent quark, model calculation [68], the authors suggested that for which includes the constituent quark mass and effects from ¯ ¯ identical quarks, there is no stable QQQ Q state. Similar other terms such as the color-electric interaction and color opinions were shared by the authors of Ref. [69]. However, confinement. in Ref. [70], Lloyd and Vary obtained bound tetraquark To calculate the matrix elements for the color-spin states by adopting a parametrized Hamiltonian to compute interaction, one may construct the color ⊗ spin wave the spectrum of the ccc¯ c¯ tetraquark states. The tetraquark functions explicitly and calculate them by definition. spectrum was also studied with a generalization of the In studying multiquark systems, a simpler way was hyperspherical harmonic formalism in Ref. [71]. In addi- used in Refs. [86–88]. One just calculates the matrix tion, the calculation of the chromomagnetic interaction for elements in color space and inP spin space separately the Q2Q¯ 2 system was performed in Ref. [72]. In under- H ¼ − C λ λ H ¼ withP the Hamiltonians C i

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6 ¯ ¯ 6¯ ϕ1χ1 ¼jðQ1Q2Þ1ðQ3Q4Þ1i2δ12δ34; quark-antiquark interaction is related with the G-parity transformation, the CMI for this J ¼ 1 state is expected to ϕ χ ¼jðQ Q Þ3¯ ðQ¯ Q¯ Þ3i ; 2 1 1 2 1 3 4 1 2 be weak. For the case J ¼ 0, there are two possible wave ¯ 6 ¯ ¯ 6 ϕ2χ3 ϕ1χ6 ϕ1χ2 ¼jðQ1Q2Þ1ðQ3Q4Þ1i1δ12δ34; functions and . Although the color-spin struc- tures are different, their mixing is allowed by the color- ϕ χ ¼jðQ Q Þ3¯ ðQ¯ Q¯ Þ3i ; 2 2 1 2 1 3 4 1 1 magnetic interaction. The obtained symmetric CMI matrix ¯ T 6 ¯ ¯ 6 in the ðϕ2χ3; ϕ1χ6Þ base is ϕ1χ3 ¼jðQ1Q2Þ1ðQ3Q4Þ1i0δ12δ34; pffiffiffi  3¯ ¯ ¯ 3 16 ϕ χ ¼jðQ Q Þ ðQ Q Þ i ; ðCbb − 2Cbb¯ Þ 8 6Cbb¯ 2 3 1 2 1 3 4 1 0 hH i¼ 3 : ð6Þ ¯ CM 6 ¯ ¯ 6 8Cbb ϕ1χ4 ¼jðQ1Q2Þ1ðQ3Q4Þ0i1δ12; ϕ χ ¼jðQ Q Þ3¯ ðQ¯ Q¯ Þ3i δ ; The ccc¯ c¯ system has similar expressions. For the case 2 4 1 2 1 3 4 0 1 34 16 J ¼ 2, hH i¼ ðCcc þ Ccc¯ Þ with the color-spin wave ϕ χ ¼jðQ Q Þ6ðQ¯ Q¯ Þ6¯ i δ ; CM 3 1 5 1 2 0 3 4 1 1 34 ϕ χ J ¼ 1 hH i¼16 ðC − C Þ function 2 1. For the case , CM 3 cc cc¯ 3¯ ¯ ¯ 3 ϕ2χ5 ¼jðQ1Q2Þ0ðQ3Q4Þ1i1δ12; with the wave function ϕ2χ2. For J ¼ 0, the CMI matrix is pffiffiffi  6 ¯ ¯ 6¯ 16 ϕ1χ6 ¼jðQ1Q2Þ0ðQ3Q4Þ0i0; ðCcc − 2Ccc¯ Þ 8 6Ccc¯ hH i¼ 3 : ð7Þ 3¯ ¯ ¯ 3 CM 8C ϕ2χ6 ¼jðQ1Q2Þ0ðQ3Q4Þ0i0δ12δ34; ð5Þ cc

color ¯ ¯ color The above compact tetraquark system has the same where the used notation is jðQ1Q2Þ ðQ3Q4Þ i .We spin spin spin quark content as the molecular states composed of two δ insert a symbol ij in the wave functions to reflect the bottomonium (charmonium) mesons. However, the inter- constraint from the Pauli principle. When the ith quark and action between heavy quarks is dominantly through the the jth quark are the same, δij ¼ 0. Otherwise, δij ¼ 1. short-range gluon exchange. Once the interaction is strong Replacing Qi with a c or b quark, one gets nine enough to bind the two mesons, the resulting object very possibilities for the flavor content, six of which need to probably tends to form a compact structure instead of a ¯ ¯ ¯ ¯ ¯ be studied: bbbb, ccc¯ c¯, bbc¯ c¯, bbbc¯, ccc¯ b, and bcbc¯. The loosely bound molecule. other three cases ccb¯ b¯, cbb¯ b¯, and bcc¯ c¯ correspond to the In the extreme case that the exist, the of bbc¯ c¯, bbb¯ c¯, and ccc¯ b¯, respectively. So configuration mixing is possible. For the S-wave ϒϒ their formulas are not independent. Here, considering the (ψψ) state, the allowed quantum numbers are IGðJPCÞ¼ Pauli principle, one may categorize the six systems into 0þð2þþÞ or 0þð0þþÞ since the wave function in spin space ¯ ¯ three sets: (1) bbbb, ccc¯ c¯, and bbc¯ c¯, where δ12 ¼ should be symmetric for identical mesons. The quantum ¯ ¯ S η η δ34 ¼ 0; (2) bbbc¯ and ccbc¯, where δ12 ¼ 0, δ34 ¼ 1; numbers for the -wave state composed of two b ( c) ¯ 0þð0þþÞ η and (3) bcb c¯ where δ12 ¼ δ34 ¼ 1. The number of inde- mesons are only . For the state composed of one b (ηc) and one ϒ (ψ), the quantum numbers are just pendent wave functions for them is 4, 6, and 12, respec- G PC − þ− PC tively. We present the resulting CMI matrix elements I ðJ Þ¼0 ð1 Þ. Therefore, the allowed J appears system by system. By the way, the usually mentioned both in the molecule and diquark-antidiquark pictures. The ¯ number of states is also equivalent. Considering that the “good” diquark (spin ¼ 0, color ¼ 3c) exists only in the systems containing the ðbcÞ substructure since the most interaction between heavy quarks is through the gluon 3¯ 3¯ exchange force, one does not expect a large mass difference ðbbÞ c ðccÞ c attractive spin-0 and spin-0 objects are forbidden by between these two configurations. The compact structure the Pauli principle. may contribute significantly to the properties of the molecules with the same quantum numbers. A. The bbb¯ b¯ and ccc¯ c¯ systems If the tetraquark states have larger masses, the quark The color-spin structures of these two systems are the rearrangements into the meson-meson channels with the same. The only difference lies in the quark mass. So we put same quantum numbers may happen. We will discuss such them together for discussions. possible decay patterns after their masses are estimated. The bbb¯ b¯ system is a neutral state. Its possible quantum ¯ ¯ numbers are IGðJPCÞ¼0þð2þþÞ, 0−ð1þ−Þ,or0þð0þþÞ. B. The bbc¯c¯ and ccbb systems The number of states is constrained by the Pauli principle. The two systems are related through the C-parity trans- For the case J ¼ 2, the wave function is ϕ2χ1 and the formation, and they have the same color-spin matrix hH i 16 ðC þ C Þ obtained CM is given by 3 bb bb¯ . The color- elements. The possible quantum numbers of them are magnetic interaction is certainly repulsive. For the case IðJPÞ¼0ð2þÞ, 0ð1þÞ,or0ð0þÞ. Again the Pauli principle J ¼ 1 ϕ χ hH i¼ ¯ ¯ , the wave function is 2 2 and CM results in four states for the bbc¯ c¯ (or ccbb) system. 16 3 ðCbb − Cbb¯ Þ. Since the quark-quark interaction and the The color-spin wave functions are ϕ2χ1 and ϕ2χ2 for the

094015-3 WU, LIU, CHEN, LIU, and ZHU PHYS. REV. D 97, 094015 (2018) case of J ¼ 2 and J ¼ 1, respectively. The corresponding C. The bbb¯ c¯ and cbb¯ b¯ systems hH i’ 8ðC þC þ2C Þ 8ðC þC −2C Þ CM s are 3 bb cc bc¯ and 3 bb cc bc¯ . The two systems have the same matrix elements. For the case J ¼ 0, the allowed wave functions are the Their quantum numbers are also IðJPÞ¼0ð2þÞ, 0ð1þÞ, ϕ χ ϕ χ same as those of the previous systems, 2 3 and 1 6. or 0ð0þÞ. We here focus on the bbb¯ c¯ system. Now Considering their mixing, one gets the number of the vector states is 3 and that of the pffiffiffi  8 scalar states is 2. For the J ¼ 2 case, the color-spin 3 ðCbb þ Ccc − 4Cbc¯ Þ 8 6Cbc¯ hH i¼ : ð8Þ wave function is again ϕ2χ1. The resulting color- CM 4ðC þ C Þ bb cc hH i¼8 ðC þ C þ magnetic matrix element is CM 3 bb bc These systems have the same quark content as the Cbb¯ þ Cbc¯ Þ. For the case J ¼ 1, three possible wave ðÞ ðÞ ϕ χ ϕ χ ϕ χ Bc Bc meson-meson states. The quantum numbers of functions 2 2, 2 4, and 1 5 are allowed. The last − − þ þ P þ þ Bc Bc (Bc Bc ) are IðJ Þ¼0ð2 Þ or 0ð0 Þ, those of one has a the different color structure from the other two. − − þ þ þ − − Bc Bc (Bc Bc ) are 0ð0 Þ. There is only one Bc Bc Although their mixing occurs, from the obtained matrix þ þ P þ T (Bc Bc ) state with IðJ Þ¼0ð1 Þ. [base: ðϕ2χ2; ϕ2χ4; ϕ1χ5Þ ]

0 pffiffi 1 8 8 2 ðCbb þ Cbc − Cbb¯ − Cbc¯ Þ ðCbb¯ − Cbc¯ Þ 8ðCbb¯ − Cbc¯ Þ B 3 3 pffiffiffi C hH i¼B 8 C; ð Þ CM @ 3 ðCbb − 3CbcÞ −4 2ðCbc¯ þ Cbb¯ Þ A 9 4 3 ð3Cbb − CbcÞ one observes that the mixing strength for ϕ2χ2 and ϕ2χ4 and that for ϕ2χ2 and ϕ1χ5 may be both small. The remaining case is for J ¼ 0, where the possible wave functions are ϕ2χ3 and ϕ1χ6. Now, one has pffiffiffi  8 ðC þ C − 2C − 2C Þ 4 6ðC þ C Þ hH i¼ 3 bb bc bb¯ bc¯ bb¯ bc¯ : ð Þ CM 10 4ðCbb þ CbcÞ

¯ − − − − The meson-meson systems with the quark content bbb c¯ are ϒBc , ϒBc , ηbBc , and ηbBc . Their quantum numbers are IðJPÞ¼0ð1þÞ, 0ð½2; 1; 0þÞ, 0ð0þÞ, and 0ð1þÞ, respectively.

D. The ccc¯ b¯ and bcc¯ c¯ systems The situation is similar to the bbb¯ c¯ and cbb¯ b¯ systems. By exchanging b and c there, one easily gets relevant matrix ccc¯ b¯ J ¼ 2 hH i¼8 ðC þ C þ C þ C Þ elements. For comparison, we focus on the system. For the case , one has CM 3 cc bc bc¯ cc¯ . For the case J ¼ 1, the matrix for the color-spin interaction reads pffiffi 0 8 8 2 1 3 ðCcc þ Cbc − Cbc¯ − Ccc¯ Þ 3 ðCcc¯ − Cbc¯ Þ 8ðCcc¯ − Cbc¯ Þ B pffiffiffi C hH i¼@ 8 A: ð Þ CM 3 ðCcc − 3CbcÞ −4 2ðCcc¯ þ Cbc¯ Þ 11 4 3 ð3Ccc − CbcÞ For the case J ¼ 0, the matrix is pffiffiffi  8 ðC þ C − 2C − 2C Þ 4 6ðC þ C Þ hH i¼ 3 cc bc bc¯ cc¯ bc¯ cc¯ : ð Þ CM 12 4ðCcc þ CbcÞ The signs for the nondiagonal matrix elements seem to be inconsistent with the previous systems after the replacements b → c and c → b. Actually they do not affect the final results. For a detailed argument, one may consult Eq. (2) of Ref. [89] and relevant explanations there. The meson-meson states that these tetraquarks can rearrange into are ψBc, ψBc, ηcBc, and ηcBc.

E. The bcb¯ c¯ system The Pauli principle does not give any constraints on the wave functions now. The two wave functions for J ¼ 2 and the four wave functions for J ¼ 0 will mix, respectively. However, one should be careful in discussing the mixing with the six wave functions for J ¼ 1 because the system is neutral and may have C parity.

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If J ¼ 2, both the diquark and the antidiquark have an angular momentum 1. The state should have definite C parity þ G PC þ þþ T and the quantum numbers are I ðJ Þ¼0 ð2 Þ. With the base ðϕ1χ1; ϕ2χ1Þ , one may get the CMI matrix pffiffiffi 2  − ð4C − 5C ¯ − 10C − 5C Þ 2 2ðC ¯ − 2C þ C Þ hH i¼ 3 bc bb bc¯ cc¯ bb bc¯ cc¯ : ð Þ CM 4 13 3 ð4Cbc þ Cbb¯ þ 2Cbc¯ þ Ccc¯ Þ If J ¼ 0, both the diquark and the antidiquark have the same angular momentum. The quantum numbers for the system are IGðJPCÞ¼0þð0þþÞ. The obtained CMI matrix is 0  1 2C þ 5C pffiffiffi pffiffiffi 4 bc bb¯ 10 − − pffiffi ðC ¯ − 2C þ C Þ 4 2ðC ¯ − 2C þ C Þ 2 6ðC ¯ þ 2C þ C Þ B 3 3 bb bc¯ cc¯ bb bc¯ cc¯ bb bc¯ cc¯ C B þ10Cbc¯ þ 5Ccc¯ C B pffiffiffi C hH i¼B 8C 2 6ðC þ 2C þ C Þ 0 C; CM B bc bb¯ bc¯ cc¯ C B 8 4 C @ ð2C − C ¯ − 2C − C Þ − pffiffi ðC ¯ − 2C þ C Þ A 3 bc bb bc¯ cc¯ 3 bb bc¯ cc¯

−16Cbc ð14Þ

T where the base is ðϕ1χ3; ϕ1χ6; ϕ2χ3; ϕ2χ6Þ . If J ¼ 1, the states ϕ1χ2 and ϕ2χ2 have negative C parities. All the other four wave functions ϕ1χ4, ϕ2χ4, ϕ1χ5, and ϕ2χ5 are not invariant under C-parity transformation. But we may construct four states which are invariant under C-parity transformations. The basic procedure is similar to that given in Ref. [90]. Explicitly, the two C ¼þstates are

66¯ 1 33¯ 1 ½ϕχ ¼ pffiffiffi ðϕ1χ4 þ ϕ1χ5Þ; ½ϕχ ¼ pffiffiffi ðϕ2χ4 þ ϕ2χ5Þ; ð15Þ þ 2 þ 2 and the two C ¼ − states are

66¯ 1 33¯ 1 ½ϕχ ¼ pffiffiffi ðϕ1χ4 − ϕ1χ5Þ; ½ϕχ ¼ pffiffiffi ðϕ2χ4 − ϕ2χ5Þ: ð16Þ − 2 − 2 Only states with the same quantum numbers may mix. So we have two color-spin matrices. For the states with IGðJPCÞ¼0þð1þþÞ, the matrix is pffiffiffi 2  ð4C þ 5C ¯ þ 5C − 10C Þ −2 2ðC ¯ þ C þ 2C Þ hH i¼ 3 bc bb cc¯ bc¯ bb cc¯ bc¯ ; ð Þ CM 4 17 3 ð−4Cbc þ Cbb¯ þ Ccc¯ − 2Cbc¯ Þ

66¯ 33¯ T G PC − þ− with the base ð½ϕχþ ; ½ϕχþ Þ . For the states with I ðJ Þ¼0 ð1 Þ, the matrix reads 0  1 4C þ 5C ¯ pffiffiffi pffiffiffi − 2 bc bb 2 2ðC þ C − 2C Þ 20 ðC − C Þ −4 2ðC − C Þ B 3 bb¯ cc¯ bc¯ 3 bb¯ cc¯ bb¯ cc¯ C B þ5Ccc¯ þ 10Cbc¯ C B  C B 4C − C ¯ pffiffiffi C B 4 bc bb −4 2ðC − C Þ 8 ðC − C Þ C B 3 bb¯ cc¯ 3 bb¯ cc¯ C B −Ccc¯ − 2Cbc¯ C hH i¼B  C; CM B 4C − 5C pffiffiffi C B 2 bc bb¯ C B 3 2 2ðCbb¯ þ Ccc¯ þ 2Cbc¯ Þ C B −5C ¯ þ 10C ¯ C B cc bc  C @ 4C þ C A 4 bc bb¯ − 3 þCcc¯ − 2Cbc¯ ð18Þ

66¯ 33¯ T where the base is ðϕ1χ2; ϕ2χ2; ½ϕχ− ; ½ϕχ− Þ . There are two kinds of molecule configurations for the bcb¯ c¯ system. In the bottomonium þ charmonium case, the G PC þ þþ − þ− − þ− allowed quantum numbers are I ðJ Þ¼0 ð0 Þ for the ηbηc system, 0 ð1 Þ for the ηbψ system, 0 ð1 Þ for the ϒηc, þ þþ − þ þ þþ − þ and 0 ð½2; 1; 0 Þ for the ϒψ system. In the meson-antimeson case, those for Bc Bc are 0 ð0 Þ, those for Bc Bc − þ þ − þ þ þþ − þ− Bc Bc are 0 ð1 Þ, and those for Bc Bc are 0 ð½2; 0 Þ or 0 ð1 Þ.

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C III. NUMERICAL RESULTS effectively attractive (repulsive), the approximation QQ ≈ 2 CQQ¯ 3 A. Parameters should result in heavier (lighter) tetraquark states. Here, the effective interaction within diquarks reflects the repulsion We need to determine six coefficients Cbb¯ , Ccc¯ , Cbc¯ , Cbb, or attraction effect from the enhancement or cancellation Ccc, and Cbc in discussing the mass splittings for various ¯ ¯ between the quark-quark (and antiquark-antiquark) inter- Q1Q2Q3Q4 systems. Their masses may be further esti- mated with the Hamiltonian in Eq. (2) once the effective action in the case of channel coupling. For comparison, we use these two approximations in our estimation. The values quark masses mc and mb are used. ϒ η m − m ¼ of the quark-quark coupling parameters estimated with From the mass splitting between and b, ϒ ηb 16 them are listed in Table I. ½ 3 Cbb¯ − ½−16Cbb¯ ¼61 MeV [91], one extracts Cbb¯ ¼ 2 9 C ¼ 5 3 To determine the masses of the tetraquarks, we adopt two . MeV. Similarly, the value of cc¯ . MeV is approaches in the present work: 1) one estimates the meson obtained from the mass splitting mJ=ψ − mη ¼ c masses with the effective heavy quark masses mb ¼ 114 B MeV. Since the excited c meson has not been 5052.9 MeV and mc ¼ 1724.8 MeV. These values were C observed yet, we just estimate the value of bc¯ to be adopted in understanding the strange properties of the m − m ¼ 70 3.3 MeV from Bc Bc MeV calculated with a tetraquark states [98,99] and the pentaquark states [100];2) þþ quark model [92]. Although the Ξcc was confirmed one calculates the masses from a meson-meson threshold, recently by the LHCb Collaboration [93] after the first M ¼ M − hH i þhH i where the mass formula is th CM th CM , observation at SELEX [94,95], the available heavy baryon and the relevant meson masses are mϒ ¼ 9460.3 MeV, masses are still not enough for us to extract the value of Ccc. m ¼ 3096 9 m ¼ 6275 1 ψ . MeV, and Bc . MeV [91]. The Here, we perform our calculation with the approximation latter method has been used in estimating the mass of an C ¼ C ¯ Q ¼ c b QQ QQ ( , ). Since there are no dynamics in the exotic Tcc [101,102]. present model, the choice of the approximation to deter- ¯ ¯ mine CQQ is not unique. However, the results induced by B. The bbb b and ccc¯ c¯ systems C the change of QQ should not be large [96,97].For It is easy to get the numerical results for the CMI matrix C C comparison, we also adopt the approximation QQ ¼ nn ≈ CQQ¯ Cnn¯ elements with the above two sets of parameters. Adopting 2 the approximation CQQ ¼ CQQ¯ , we obtain the CMI matri- 3 and check the extreme case CQQ ¼ 0, where Cnn ¼ 18.4 MeV is extracted from the light baryon masses [98]. ces, their eigenvalues and corresponding eigenvectors, By using the mass difference between ρ and π, one gets and estimated masses with two different approaches. These results are presented in Table II. In the approxima- Cnn¯ ¼ 29.8 MeV. The latter approximation certainly gives Cnn tion CQQ ¼ CQQ¯ , the estimated masses of the tetraquark a smaller CQQ. If the interactions within the diquarks are Cnn¯ states are slightly different from those in the former approximation. In the following discussions, we mainly TABLE I. Two sets of effective coupling parameters in units of use the masses estimated with the parameters in set I. MeV with different approximations. Since the hadron masses estimated with the effective quarks   – CQQ Cnn are usually like an upper limit [98 100,103], we focus on ðC ¼ C ¯ Þ Set II C ¼ C Set I QQ QQ QQ¯ nn¯ the results estimated with reference thresholds. We assume

Ccc 5.3 3.3 that these masses are all reasonable values. To have an Cbb 2.9 1.9 impression for the spectrum, we plot relative positions for ¯ ¯ Cbc 3.3 2.0 the bbb b (ccc¯ c¯) states in Fig. 1(a) [Fig. 1(b)]. The solid/ black lines are for the approximation CQQ ¼ CQQ¯ and the

¯ ¯ TABLE II. Results for the bbbb and ccc¯ c¯ systems in units of MeV with the approximation CQQ ¼ CQQ¯ . The masses in the fifth column are estimated with mb ¼ 5052.9 MeV and mc ¼ 1724.8 MeV. The last column lists masses estimated from the ðϒϒÞ or T ðJ=ψJ=ψÞ threshold. The base for the J ¼ 0 case is ðϕ2χ3; ϕ1χ6Þ .

PC System J hHCMi Eigenvalue Eigenvector Mass ðϒϒÞ=ðψψÞ ðbbb¯b¯Þ 2þþ 30.9 30.9 1 20243 18921 þ− 1 0.0 0.0h 1ih 20212ih 18890i þþ  hi 0 −15.5 56.8 63.9 ð0.58;0.81Þ 20275 18954 56.8 23.2 −56.2 ð−0.81;0.58Þ 20155 18834 ðccc¯c¯Þ 2þþ 56.5 56.5 1 6956 6194 þ− 1  0.0h 0.0ih 1ih 6899ih 6137i þþ 0 −28.3 103.9 116.8 ð0.58;0.81Þ 7016 6254 103.9 42.4 −102.6 ð−0.81;0.58Þ 6797 6035

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(a) (b)

(c) (d)

(e) (f)

FIG. 1. Relative positions for the considered tetraquark states. The solid (black) and dashdotted (blue) lines correspond to masses Cnn estimated with the approximations CQQ ¼ CQQ¯ and CQQ ¼ CQQ¯ , respectively. The dashed (red) lines are for the case CQQ ¼ 0. The Cnn¯ dotted lines indicate various meson-meson thresholds. When the quantum numbers in the subscript of the symbol for a meson-meson state are equal to the JPC (or JP) of an initial state, the decay for the initial state into that meson-meson channel through a S or D wave is allowed. We adopt the masses estimated with the reference thresholds of (a) ϒϒ, (b) ψψ, (c) BcBc, (d) ϒBc, (e) ψBc, and (f) BcBc. The masses are all in units of MeV. dashdotted/blue lines are for the approximation the charmed system. One finds that the mixing between Cnn CQQ ¼ C CQQ¯ . We also show the results in the extreme different color structures is important here, which enlarges nn¯ the mass difference between these two states. If one does case CQQ ¼ 0 with the dashed/red lines. The uncertainty not consider the mixing, the masses for the bottom case are C caused by the change of QQ is less than 20 (37) MeV in 18913 MeV and 18875 MeV. Both states are below the ϒϒ ¯ ¯ the bbb b (ccc¯ c¯) case. threshold and above the ηbϒ threshold. Once the mixing is The mass splitting between the scalar tetraquarks is considered, the higher state (6c bb dominates) becomes a ¯ around 120 MeV for the bottom system and 220 MeV for state above the ϒϒ threshold while the lower one (3c bb

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¯ ¯ dominates) becomes a state below the ηbϒ threshold. TABLE III. Results for the bbc¯ c¯ and ccb b systems in units of Certainly the mass shift affects decay properties. The MeV with the approximation CQQ ¼ CQQ¯ . The masses in the charmed case is similar. fifth column are estimated with mb ¼ 5052.9 MeV and From the diagrams (a) and (b) in Fig. 1, the estimated mc ¼ 1724.8 MeV. The last column lists masses estimated from tetraquark masses are all above the lowest meson-meson the ðBcBcÞ threshold. The base for the J ¼ 0 case is ðϕ χ ; ϕ χ ÞT threshold. This observation is consistent with those in 2 3 1 6 . ¯ ¯ Refs. [74,79,82]. The lowest bbb b mass in [79] and ours P J hH i Eigenvalue Eigenvector Mass ðBcBcÞ are similar. From these diagrams, the masses obtained with CM 2þ 39.5 39.5 1 13595 12695 parameters in set II are all lower than those in set I. This þ 1  4.3h 4.3ih 1ih 13560ih 12660i means that the interactions within the diquarks are effectively þ 0 −13.3 64.7 78.4 ð0.58;0.82Þ 13634 12734 repulsive, which can be verified from the Hamiltonian 64.7 32.8 −58.9 ð−0.82;0.58Þ 13496 12597 expressions. If stable multiquark states need attractive diquarks, these bbb¯ b¯ and ccc¯ c¯ compact tetraquarks would tend to become meson-meson states because the quark- ccb¯ b¯ antiquark interaction is usually attractive (see the diagonal system and relevant meson-meson thresholds in matrix elements in the Hamiltonian expressions), and these Fig. 1(c). tetraquarks should not be stable. Basically, the behaviors for the rearrangement decays are bbb¯ b¯ ccc¯ c¯ If the studied states do exist, finding out their decay similar to those for the and systems. The main difference lies in the C parity. The former states have properties are helpful to the search at experiments. Possible ¯ ¯ rearrangement decay modes are easy to be understood from definite C parities while the ccb b states not. Without the þ ¯ ¯ Fig. 1. Since the feature for the ccc¯ c¯ system is very similar to condition of C parity conservation, the 2 ccb b tetraquark þ þ the bottom case, we here only concentrate on the latter can also decay into the Bc Bc channel through a D wave. one. For the J ¼ 2 tetraquark, the present model estimation Up to now, experiments confirm only the ground Bc gives a mass around the ϒϒ threshold. If the approximation meson. It means that the axial vector tetraquark decaying 2 B B CQQ ≈ CQQ¯ is more appropriate, the state is blow the into c c may not be observed in the near future. In the 3 B threshold, and it should have a relatively narrow width case that the c meson is confirmed with enough data, 1þ ccb¯ b¯ bbc¯ c¯ through D-wave decay into ηbηb. It is a basic feature that the search for the (or ) tetraquark is also high spin multiquark states have dominantly D-wave decay possible. However, the interactions within the diquarks are modes and should not be very broad [99,100,103]. For the effectively repulsive, and these tetraquarks should not be J ¼ 1 tetraquark, its mass is 30 MeV above the ηbϒ stable. threshold. From the quantum numbers, its rearrangement ¯ ¯ ¯ ¯ decay channel is only this ηbϒ. For the J ¼ 0 tetraquarks, D. The bbb c and cbb b systems the higher one can decay into both ϒϒ and ηbηb through We show the results for the bbb¯ c¯ (and cbb¯ b¯) states in S D both -and -wave interactions, while the lower one Table IV and the spectrum for bbb¯ c¯ in Fig. 1(d). The η η decays only into b b. If we use their masses to denote maximum mass splitting is around 130 MeV. Comparing 18954 > these states, probably the ordering of the widths is with the former systems, the Pauli principle works only for 18890 ∼ 18834 > 18921 . one diquark now, which leads to one more 1þ tetraquark. From Figs. 1(a) and 1(b), the feature that all the states From Fig. 1(d), we may easily understand the rearrange- can decay is consistent with the feature that the effective ment decay behaviors for the bbb¯ c¯ states. The two scalar interaction within the diquarks is repulsive. If we want to states have similar properties to the states in the former find relatively stable compact tetraquarks in Fig. 1, good systems. For the J ¼ 2 tetraquark, there is one more candidates should be those states for which dashed/red D ccb¯ b¯ bbb¯ b¯ -wave decay channel compared to the case because lines are above solid/black lines. Although the and the violation of the heavy quark spin symmetry results in ccc¯ c¯ systems do not satisfy this condition, we will see that the nondegeneracy for the thresholds of ηbBc and ϒBc. The such systems exist. P þ interesting observation appears for the J ¼ 1 states. The color-spin mixing affects the masses of the states relatively C. The bbc¯ c¯ and ccb¯ b¯ systems largely. The resulting observation is the highest and the ¯ ¯ We present the numerical results for the ccb b intermediate states are kinematically allowed to decay into ¯ ¯ states in Table III. The bbc c states are antiparticles of ϒBc and ηbBc channels, while the lowest state has no ¯ ¯ the ccb b states and have the same results. Now the rearrangement decay channel. From the relative positions mass splitting between different spins is less than for the solid/black, dashdotted/blue, and dashed/red lines, 140 MeV. This number lies between the splittings for the interactions within the diquarks are effectively attractive the bbb¯ b¯ case and the ccc¯ c¯ case, and thus, the mixing for the lowest 1þ tetraquark, repulsive for the intermediate effect is in the middle of the two. To understand the 1þ state, and also attractive for the highest 1þ state. This decay properties easily, we plot the spectrum for the feature is a result of balance between attraction/repulsion in

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¯ ¯ ¯ TABLE IV. Results for the bbb c¯ and cbb b systems in units of MeV with the approximation CQQ ¼ CQQ¯ . The masses in the fifth column are estimated with mb ¼ 5052.9 MeV and mc ¼ 1724.8 MeV. The last column lists masses estimated from the ðϒBcÞ T T threshold. The base for the J ¼ 1 case is ðϕ2χ2; ϕ2χ4; ϕ1χ5Þ and that for the J ¼ 0 case is ðϕ2χ3; ϕ1χ6Þ . JP hH i ðϒB Þ CM Eigenvalue Eigenvector Mass c 2þ 33.1!" 33.1#" 1#" 16917#" 15806 # 1þ 0.0 −1.5 −3.2 −43.3 ð−0.07; −0.82; −0.57Þ 16840 15729 −1.5 −18.7 −35.1 31.7 ð−0.06; −0.57; 0.82Þ 16915 15804 −3.2 −35.17h.2 0.1 ihð1.00; −0.09; 0ih.01Þ 16884ih15773i þ 0 −16.5 60.7 68.3 ð0.58;0.81Þ 16952 15841 60.7 24.8 −60.0 ð−0.81;0.58Þ 16823 15713

bc, repulsion in bb, attraction between quarks and anti- only one decay channel ηcBc, the intermediate one has two quarks, and channel coupling. From the effective inter- ηcBc and ψBc, and the high mass state has one more ¯ action within the diquarks and the estimated mass, the channel ψBc. Contrary to the bbb c¯ case, the effective þ lowest 1 state is a good candidate of stable tetraquarks. interaction within the diquarks in the lowest 1þ state is ¯ Apparently, this state looks like an excited Bc where a bb repulsive, which is a result that the cc interaction is stronger pair is excited (see discussions in Ref. [99] for other than the bb interaction. If this state does exist, it should be tetraquarks). By checking the CMI matrix elements, one less stable than the lowest 1þ bbb¯ c¯. finds that the stable state is possible mainly because of the Early experimental investigations on possible resonances ¯ attraction within the bc diquark. The situation is similar to in the ccc¯ b system should be through the channel ψBc, the Tcc state (ccq¯ q¯) where the attractive ud diquark which means that four tetraquarks, a high mass scalar, two contributes dominantly [101,102]. axial vectors, and one tensor, could be observed first. With the observed bottomonium and Bc states, in η B principle, resonance structures above the b c threshold F. The bcb¯ c¯ system can all be investigated. Once experiments collect enough Bc data, interesting states in the ϒBc channel would We show the results for the twelve states in Table VI. probably be observed first. Now the maximum mass splitting (240 MeV) again occurs between the scalar tetraquarks. In estimating the tetraquark ¯ masses, there are two reference thresholds we can use, ϒψ E. The ccc¯ b and bcc¯ c¯ systems þ − and Bc Bc . As the investigations in other systems [99,100], We may call such tetraquarks as “mirror” partners of the the former threshold leads to lighter masses that can be previous states. The color-spin structure is the same as the treated as a lower limit on the theoretical side. We here ¯ bbb c¯ system but the decay feature relies on masses and adopt the masses estimated with the latter threshold. Future may be different. We present the numerical results in measurements may answer whether the scheme is reason- Table V and draw the spectrum in Fig. 1(e). The maximum able or not. Seven meson-meson channels are involved in mass splitting between the two scalar tetraquarks is around discussing decay properties with the obtained results. We 180 MeV. Rearrangement decays for these tetraquarks are show the spectrum and these channels in Fig. 1(f). possible only when the mass is high enough. Since there is no constraint from the Pauli principle, the The rearrangement decay properties for the J ¼ 2 and obtained spectrum is more complicated than other systems. J ¼ 0 tetraqaurks are similar to those for the bbb¯ c¯ states. All the masses are above the lowest meson-meson thresh- For the three axial vector tetraquarks, the low mass one has old, and they should decay. At least four rearrangement

¯ TABLE V. Results for the ccc¯ b and bcc¯ c¯ systems in units of MeV with the approximation CQQ ¼ CQQ¯ . The masses in the fifth column are estimated with mb ¼ 5052.9 MeV and mc ¼ 1724.8 MeV. The last column lists masses estimated from the ðψBcÞ T T threshold. The base for the J ¼ 1 case is ðϕ2χ2; ϕ2χ4; ϕ1χ5Þ and that for the J ¼ 0 case is ðϕ2χ3; ϕ1χ6Þ .

P J hHCMi Eigenvalue Eigenvector Mass ðψBcÞ 2þ 0 45.91 " 45.9#" 1#" 10273#" 9442 # 1þ 54 4 ð0 16; −0 57; 0 81Þ 10282 9451 0.07.516.0 . . . . B C −53 1 ð0 28; −0 76; −0 59Þ 10174 9343 @ 7.5 −12.3 −48.6 A . . . . 3 3 ð0 95; 0 32; 0 04Þ 10231 9400 16.0 −48.616.8 . . . .  hihihihi þ 0 −22.9 84.3 94.7 ð0.58;0.81Þ 10322 9491 84.3 34.4 −83.3 ð−0.81;0.58Þ 10144 9313

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¯ TABLE VI. Results for the bcb c¯ system in units of MeV with the approximation CQQ ¼ CQQ¯ .The masses in the fifth column are estimated with mb ¼ 5052.9 MeV and mc ¼ 1724.8 MeV. The last two columns list masses estimated from the ðϒψÞ and the ðBcBcÞ T T thresholds. The bases for the J ¼ 2 and J ¼ 0 cases are ðϕ1χ1; ϕ2χ1Þ and ðϕ1χ3; ϕ1χ6; ϕ2χ3; ϕ2χ6Þ , respectively. If J ¼ 1, the bases 66¯ 33¯ T 66¯ 33¯ T for the cases C ¼þand C ¼ − are ð½ϕχþ ; ½ϕχþ Þ and ðϕ1χ2; ϕ2χ2; ½ϕχ− ; ½ϕχ− Þ , respectively. JPC hH i ðϒψÞðB B Þ CM Eigenvalue Eigenvector Mass c c 2þþ ð40.5 4.5 Þ ð43.7Þ ½ ð−0.82;−0.58Þ ð13599Þð12557Þ ð12700Þ 0 4.5 37.3 1 0 34.1 1 2 ð0.58;−0.82Þ 3 0 13590 1 0 12548 1 0 12690 1 0þþ −159 4 ð0 82; 0 08; −0 11; −0 56Þ 13396 12354 12496 −107.5 −9.29.172.5 . . . . . B C B 78 8 C 6 ð−0 02; 0 81; 0 58; −0 03Þ 7 B 13634 C B 12592 C B 12735 C B −9.226.472.50.0 C B . C 6 . . . . 7 B C B C B C B C @ −72 7 A 4 ð0 08; −0 58; 0 80; −0 14Þ 5 @ 13483 A @ 12441 A @ 12583 A @ 9.172.5 −21.9 −3.7 A . . . . . −2 4 ð0 57; −0 01; 0 08; 0 82Þ 13553 12511 12653 72.50.0 −3.7 −52.8 . . . . . 1þþ ð 14.1−41.9 Þ ð−45.1Þ ½ ð0.58;0.82Þ ð13510Þ ð12468Þð12611Þ 0 −41.9−15.5 1 0 43.7 1 2 ð−0.82;0.58Þ 3 0 13599 1 0 12557 1 0 12700 1 1þ− −58.14.5 −16.013.6 −77.7 ð−0.73; 0.16; −0.43; 0.50Þ 13478 12436 12578 B C B C 6 7 B C B C B C B 4.5 −2.113.6 −6.4 C B 36.8 C 6 ð0.05; −0.18; −0.80; −0.56Þ 7 B 13592 C B 12550 C B 12693 C @ −16.013.63.541.9 A @ −35.5 A 4 ð0.68; 0.19; −0.41; 0.58Þ 5 @ 13520 A @ 12478 A @ 12620 A 13.6 −6.441.9 −19.7 −0.1 ð0.00; 0.95; 0.00; −0.31Þ 13555 12513 12656 decay channels for the 2þþ states are D-wave channels. If tetraquark states with the simple color-magnetic interaction þþ þ − the 2 state is above the Bc Bc threshold, the S-wave in this work. PC þþ ¯ channel is also opened. For the channels with J ¼ 1 , For the case (bcb c¯ system) without the constraint from the S-wave ϒψ is the lowest one. The decays of the two the Pauli principle, the number of color-singlet tetra- þþ ¯ 1 tetraquarks into this channel are both allowed. For quarks with a 6c diquark is equal to that with a 3c the four 1þ− states, they all have S-wave decay channels diquark. The two color structures may couple through the ϒηc and ηbψ. There is at least one allowed rearrangement color-magnetic interaction. From Table VI, the coupling þþ þþ decay channel, ηbηc, for the four 0 tetraquarks through for the 2 states is weak while that for other states is an S-wave interaction. It seems that all the bcb¯ c¯ tetra- stronger. For the case with a constraint from the Pauli quarks are not stable. principle, the number of color-singlet tetraquarks with the ¯ On the other hand, one finds that the interactions 3c diquark is bigger than that with the 6c diquark. Their within the diquarks in several states are effectively mixing is generally significant (see Tables II–V). In both attractive. This indicates that in the competition between cases, the tetraquarks with a 6c diquark do not exist the interactions of QQ and QQ¯ , the stronger attraction in independently. QQ may make such states relatively stable. The lowest After the configuration mixing effects are considered, 0þþ state satisfies this condition, and it has only one both the heaviest tetraquark and the lightest tetraquark rearrangement decay channel. Maybe this state has a for a system are the scalar states. The maximum mass relatively narrow width. The lower 1þþ state has also differences are around 120, 220, 140, 130, 180, and ¯ ¯ ¯ similar features. Its S-wave decay channel is ϒψ, and this 240 MeV for the bbb c¯, ccc¯ c¯, bbc¯ c¯, bbb c¯, ccc¯ b, and tetraquark is a worthwhile study. Because of the existence bcb¯ c¯ systems, respectively. Other states fall in these mass of possible tetraquark states, searching for exotic phe- difference ranges. Whether the tetraquarks decay through nomena with ϒψ may help us to understand the strong quark rearrangements and how many thresholds fall in interactions between heavy quarks. In Ref. [82], Richard these ranges rely on the tetraquark masses. For comparison, et al. also observed that bound bcb¯ c¯ might be more we estimate the masses with two approaches. We mainly favorable than bbb¯ b¯ and ccc¯ c¯. discuss the tetraquark properties by using the masses estimated with the threshold of some meson-meson state, the second approach. IV. DISCUSSIONS AND SUMMARY Since the parameter CQQ cannot be extracted with In the chiral quark model, the interaction between the experimental data, we use two approximations to determine light quarks may also arise from the exchange of Goldstone them. We also show results in the extreme case CQQ ¼ 0. . For the pure heavy systems, such an interaction is From the comparison study, we find that (1) the estimated absent. Therefore, one needs to consider only the gluon- tetraquark masses are affected slightly and (2) the plotted ¯ ¯ exchange interaction for the present Q1Q2Q3Q4 systems. spectra can be easily used to judge which states contain As a short range force, it helps to form compact tetraquarks effectively attractive diquarks. From the viewpoint that rather than meson-meson molecules if bound four-quark narrow compact tetraquarks should have attractive diquarks states do exist. We have studied the mass splittings of these inside them and have as few S-wave decay channels as

094015-10 HEAVY-FLAVORED TETRAQUARK STATES WITH THE QQ … PHYS. REV. D 97, 094015 (2018) possible, we find that most QQQ¯ Q¯ states are not stable. considered. We have estimated their masses with two However, a stable 1þ bbb¯ c¯ state is observed, and relatively approaches and discussed possible rearrangement decay ¯ narrow bcb¯ c¯ tetraquarks are also possible in the present channels. Stable or narrow tetraquarks in the bbb c¯ and ¯ model investigations. The latter system is also proposed to bcb c¯ systems are a worthwhile study in heavy meson- have bound states in Ref. [82]. Although the model we use meson channels such as ϒψ. Hopefully, the exotic tetra- is simple and it does not involve dynamics, the basic quark states composed of four heavy quarks may be features of the obtained spectra might be somehow rea- observed at LHCb in the future. sonable. To give more reliable results, the present model needs to be improved. ACKNOWLEDGMENTS If experiments could observe one resonant state in the channel of two heavy quarkonia, its nature as a Y. R. L. thanks F. Huang (UCAS, Beijing) for a dis- tetraquark is favored. More tetraquarks should also exist cussion and people at Tokyo Institute of Technology and searches for them are strongly called for. We hope the for their hospitality where the updated draft was decay channels discussed in this paper are helpful for the finalized. This project is supported by National Natural experimental search. Science Foundation of China under Grants No. 11775132, To summarize, we have explored the mass splittings No. 11222547, No. 11175073, No. 11261130311, and 973 ¯ ¯ between the Q1Q2Q3Q4 tetraquarks with a simple program. X. L. is also supported by the National Program model. The mixing between different color structures is for Support of Top-notch Young Professionals.

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