Stable double-heavy tetraquarks: spectrum and structure

Bound states of double-heavy tetraquarks are studied in a constituent quark model. Two bound states are found for isospin and spin-parity I(J^P ) = 0(1^+) in the bb\bar{u}\bar{d} channel. One is deeply bound and compact made of colored diquarks, while the other is shallow and extended as a BB^* molecule. The former agrees well with lattice QCD results. A systematic decrease in the binding energy is seen by replacing one of the heavy quarks to a lighter one. Altogether we find ten bound states. It is shown for the first time that hadrons with totally different natures emerge from a single Hamiltonian.

This paper addresses the spectrum and structure of bound tetra-quark states with two heavy quarks.This is a key problem of quantum chromodynamics (QCD) that will lead to resolving confusion in interpreting the nature of exotic resonances observed recently at high energy accelerator facilities such as LHC, KEK and BEP.
Hadron spectroscopy has turned to a new phase in the past 15 years by successive discoveries of new hadron resonances, such as X(3872), P c and others [1][2][3].As they do not fit into the conventional meson (q q) and baryon (qqq) pictures, their structure and dynamics must be different from the ordinary hadrons.Another notable feature of some of the newly found resonances is their narrow widths, in spite of sizable phase space open to hadronic decay channels.
Various interpretations have been given for the observed exotic resonances [4,5].Some are consistent with loosely bound states of hadrons, forming hadronic molecules.For instance, X(3872) was suggested to be a molecular bound state of D and D * [6].Another interpretation is a threshold cusp, i.e., a kinematical effect, as it is located just at the threshold of D plus D * mesons [7,8].
Yet it has been also claimed that X(3872) is a superposition of a compact cc state and a D-D * molecular component [5,9].This example shows that, in many cases, the interpretations are not conclusive due to uncertainties in hadron interactions and to the presence of many open channels.
An alternative picture for exotic resonances is a compact multi-quark (tetra-, or penta-quark) state.QCD does not forbid such color-singlet multi-quark configurations.Indeed, many theoretical works predicted compact tetraquarks [10][11][12], pentaquarks [13] and dibaryons [14,15].Yet, none of them has so far been confirmed experimentally, because the predicted states, that are above some two-hadron thresholds, become resonances with often a large fall-apart decay width.
Recently, with experimental developments in heavy hadron spectroscopy, possibilities of stable multi-quark states are being discussed frequently.Let us focus on the simplest one, tetraquarks formed by two quarks and two antiquarks.Compact tetraquarks may be composed of correlated colored diquarks generated by the strong color Coulomb attraction.It was suggested that this effect becomes critically important for systems with two heavy quarks, QQ q q in Refs.[16,17], where Q ( ) and q ( ) denote heavy (c and b) and light (u, d, s) quarks, respectively.Unlike the Q Q q q system, QQ q q is more likely to have a bound state that is stable against strong decays, mainly because the threshold energy for the latter, Qq + Q q , is larger than the former, Q Q + q q .In fact, there have been many theoretical studies about this possibility over the years (see, for instance, Ref. [18,19]), which however remained inconclusive.Meanwhile, the existence of the doubly charmed baryon Ξ cc has been experimentally established [20].This made a semi-quantitative discussion for double heavy teraquarks possible, giving large binding energies from an empirical mass formula [16,21].
The purpose of this paper is to systematically study stable QQ q q tetraquark states with various flavor combinations in the non-relativistic quark model.We find several stable states, one of which is a strongly bound bbq q with isospin and spin-parity I(J P ) = 0(1 + ), having a binding energy of almost 200 MeV.This confirms the earlier discussions [16,21] and is also consistent with the predictions of lattice QCD [22][23][24][25].We have also found a shallow state for the same I(J P ) = 0(1 + ) channel.By computing density distributions, it is shown that the deep one is a compact tetraquark state, while the shallow one is regarded as a loosely-bound molecule of two color singlet mesons, B and B * .This is a hadronic analogue of the cluster formation in light nuclei [26], the first example that hadrons with totally different nature emerge from a single Hamiltonian.It is a universal feature of quantum many-body systems which will clarify unsolved problems of colored QCD dynamics.
For the quark model Hamiltonian, we employ the form of AP1 of Ref. [27] (See Eq. ( 2) of [27]), which is composed of a power-law confinement term and a gluonexchange potential with non-relativistic kinetic energy.This Hamiltonian has been also employed for our former studies of pentaquarks of qqqcc and ssscc [28,29].For determining the existence of bound states, it is important for the calculation to treat the relevant threshold energies consistently.In order to improve the fit to the threshold meson masses, we have tuned the potential parameters.In Table I, we compile the values of the Hamiltonian parameters and the calculated masses of the heavy mesons relevant to the present study of tetraquarks.Compared with the experimental values, the meson masses are reproduced within the errors of at most 30 MeV or much less.The errors of the binding energies are expected to be less, as large part of errors will be cancelled by taking the mass differences of the tetraquark and threshold mesons.
One missing element here is hadron dynamics, in particular meson-exchange interactions at long distances.There are reasons, however, important features of our present discussions are robust.For deeply bound compact states such dynamics can be negligible.Whether or not shallow states exist may be modified, while their molecular structure remains unchanged as long as binding energies are small.
To solve the four-body problem accurately, we employ the Gaussian expansion method [30].The variational wave function of a tetraquark, Ψ I,JM , with isospin I and total spin (J, M ) is formed as follows: where ξ 1 stands for the color singlet (indicated by the lower index 1) wave function, η for the isospin of light quarks, χ for the spin of each quark, and φ, ψ, φ denote spatial wave functions.The label (C) specifies a set of Jacobi coordinates shown in Fig. 1, which are to coincide with the color combinations of quarks.When two quarks are connected by a line, they form a color 3, while a quark and an antiquark will be connected to form a color singlet state.For example, the color wave functions, ξ  1) includes all quantum numbers needed for the expansion, γ ≡ {s, Σ, K, n, N, ν, , L, λ, G}.
The expansion coefficients, or the variational parameters, B (C) γ , are determined by matrix diagonalization.Details of the method and its validity and accuracy are discussed in Ref [30].It should be noted that the precision is very important in the present analysis because the bound states are often close to the two-body thresholds, where the system becomes very dilute, making it much harder to obtain accurate wave functions and eigenenergies.
Bound tetraquark states, QQ q q , are searched for various flavor combinations from light to heavy quarks with spin and parity J P = 0 + , 1 + and 2 + .In the presence of light quarks, flavor combinations are expressed by isospin I.We have found altogether ten bound tetraquarks as shown in Fig. 2, six for J P = 1 + (red bars), two for 0 + and two for 2 + (blue bars).Other combinations, such as the one with all heavy quarks, do not accommodate stable states due to the relatively low threshold masses of fall apart mesons.We therefore conclude that the combination of heavy and light quarks is the key to generate stable bound states.
In Fig. 2, the resulting energies −E B (E B : binding energy) are shown in units of MeV together with their quantum numbers I(J P ).In the figure, dashed bars stand for fall-apart two meson thresholds as indicated beside the bars.The columns are drawn relative to the threshold energies of the pseudoscalar (0 − ) plus vector (1 − ) meson masses such as BB * , DB * for each quantum number.Let us discuss the nature of these bound states.J P = 1 + : For bbū d (I = 0), we have obtained two bound states; one is deeply bound with a binding energy of 173 MeV, and the other shallow one with a binding energy of 4 MeV.As we will discuss shortly, these two states have very different internal structures.If we change the bottom quarks to charm or strange quarks for the deeply bound state, its binding energy decreases; specifically, in the order of the reduced masses of the quark pairs bb, bc, cc, bs, it decreases systematically as 173, 40, 23 and 5 MeV, respectively.This behavior is explained by the color electric force between heavy quarks, as emphasized in Refs.[16,17].For color 3 states, it provides half of the attraction strength of the color singlet quark and antiquark pair.Moreover, due to its 1/r behavior at short distances the attraction increases proportional to the reduced mass of the two quarks.To demonstrate this explicitly, we plot the expectation values of the Coulomb (1/r) term of the color-electric potential for the bQ pair in a bQq q tetraquark (red line) and for the QQ pair in a QQq q tetraquark (blue line) as functions of m Q in Fig. 3.When m Q = m b , the two results agree, with the large attraction energy of ∼ −200 MeV.As m Q decreases down to ∼ 1 GeV, where a bound state still exists, the absolute values of both the QQ and bQ energies decrease monotonically.The Coulomb energy for bQ is more attractive than for QQ, because the reduced mass of bQ is larger than that of QQ.The increase in the attractive energy is also understood intuitively by the decrease in the size of the bQ pair as shown in Table III.
There is another bound state for bbsq : I(J P ) = 1/2(1 + ) with a binding energy of 59 MeV.This is the strange analogue of the deeply bound state of 173 MeV.The difference between the two energies is partly due 3. Coulomb energies of the bQ pair in the bQq q tetraquark (red line) and that of the QQ pair in the QQq q tetraquark (blue line), as functions of mQ.
to the the spin-spin interaction, which is weaker for the strange quark than for the up and down quarks.
Other J P 's: We have found two bound states with I(J P ) = 0(0 + ) for bcq q bound below the BD threshold by 37 MeV, and for bsq q by 7 MeV.Their Q and Q are in symmetric configurations, so that their siblings in the bbq q or ccq q channels are forbidden by the Pauli principle.This is realized in a lattice QCD calculation as well [23].
Lastly, we have also found two more states with J P = 2 + .The one in the bbq q channel of I = 1 is located only 3 MeV below the B * B * threshold.This state is formed by the bad anti-diquark q q of (I(J + ) = 1(1 + )) bound to the heavy vector diquark bb.The mass difference from the state of 0(1 + ) with 173 MeV binding energy can mostly be explained by the spitting between the good (I(J P ) = 0(0 + )) and bad anti-diquarks.The other 2 + bound state appears in a bcq q configuration with a small binding energy of 5 MeV below the D * B * threshold.
TABLE II.The energies of stable tetraquarks −EB in comparison with recent lattice QCD calculations in units of MeV [22][23][24][25].N stands for "no bound state".Refs.[22,23,25] report binding energies directly, while only the meson and tetraquark energies are given in Ref. [24].The shown errors for Ref. [24] here are estimated by combining the errors of the individual hadron masses in quadrature.
Next, we compare our results with those of recent lattice QCD calculations [22][23][24][25] in Table II.We see that for the channels containing either cc or bb heavy quarks, the agreement between the lattice and our quark model results is rather good.Especially for the deeply bound bbq q and bbsq cases with I(J P ) = 0(1 + ) for which calculations of multiple lattice QCD collaborations are available, the quark model states lie within an energy range of at most 40 MeV of the lattice results.For all other states with cc or bb heavy quarks, the bound states, if any, are only rather shallow both for the quark model and the lattice calculations.Conversely, for the channels with bc and bs quarks which have been studied in Ref. [24], there is some disagreement between the lattice and the quark model results.Specifically, we find bound states for all of them in our work, while on the lattice no such bound state is obtained.
We continue by discussing the two-body density distributions for quark pairs in the tetraquarks, which will help revealing their spatial structure.The two-body density distribution of a qq pair, where q or q indicates any quark or anti-quark in the tetraquark, is defined by where r qq = |r qq | is the distance between q and q , rqq is the angular part of the relative q-q coordinate, and x 1 and x 2 denote the other Jacobi coordinates.
In Fig. 4, we show r 2 ρ qq (r) for various qq pairs in the two bbq q tetraquarks of I(J P ) = 0(1 + ).For the deeply bound state (a), we see a very compact structure for the bb pair, while the bq and q q pairs have extended density distributions.This is what we expect; the bb pair is strongly attracted due to the color-electric force, while this effect is smaller for the bq and qq pairs as the attraction is proportional to their reduced masses.Turning to the shallow bound state (b), all diquark pairs are extended and furthermore, the bb distribution shows a node-like structure.This implies that this state is a nodal excitation of the bb pair.
To understand these features more quantitatively, we summarize in Table III, the mean distances, R qq ≡ r 2 ρ qq (r) r 2 dr/ ρ qq (r) r 2 dr 1/2 , of various pairs of quarks (and antiquarks).One sees clear tendencies that the density distributions depend on the types of quark pairs and their binding energies.Namely, the deep bound states have a smaller R Qq−Q q , the distance between the centers of mass of Qq and Q q, compared to the shallow ones, for which R Qq < R Qq−Q q .This indicates that the shallow states are loosely bound (molecular) states of color singlet mesons, (Qq) 1 + (Q q) 1 , where the index 1 denote color singlet.In particular, the node-like structure of bb may transfer to the similar structure for the mesons.It is very interesting to see two extreme cases of bound states, one deep and compact, the other shallow and molecular, simultaneously in the spectrum of the single quark model Hamiltonian.This is the first example of a hadronic analogue of cluster formation in spectra of light nuclei, where cluster structures made of α particles are developed around the α emission thresholds [26], while the lower bound states are compact shell-model-like states.QQ q q I(J P ) −EB R QQ RQq R Q q Rqq R Qq−Q q bbq q 0(1 + ) −173 0.34 0.84 0.74 0.32 bbq q 0(1 + ) −4  The states that we have discussed so far are stable against the strong decay, while they decay through the electro-magnetic or weak interactions.For example, the I(J P ) = 0(1 + ) state of bcū d with binding energy 40 MeV will decay radiatively into D + B + γ(M 1).Similarly all the J P = 1 + states above the two 0 − meson thresholds, and 2 + states above the 0 − and 1 − meson thresholds, are subject to such decays.The two deeply bound states, the bbū d (0(1 + )) and bcū d (0(0 + )) states, on the other hand, can decay only via the weak interaction.
Summarizing, we have found a few stable bound states in QQ q q tetra quark systems in the quark model.The deep compact bound state in bbū d (and also in ccū d) with I(J P ) = 0(1 + ) agrees well with the lattice QCD prediction.A shallow bbū d (0(1 + )) bound state is also found, whose wave function is consistent with a moleculetype loosely bound state of B and B * mesons.This is the first hadronic example of a set of a deep and shallow bound states in the same channel.

FIG. 2 .
FIG.2.Bound tetraquarks with their energies −EB (MeV) measured from the thresholds for various flavor contents.The labels beside each bar indicate isospin and spin-parity quantum numbers I(J P ).The hatch pattern in the bcq q sector indicates that the distance between the DB * -D * B * thresholds does not reflect the actual scale.
FIG.3.Coulomb energies of the bQ pair in the bQq q tetraquark (red line) and that of the QQ pair in the QQq q tetraquark (blue line), as functions of mQ.

FIG. 4 .
FIG. 4. Density distibutions for various quark pairs in the deep (a) and shallow (b) bbq q tetraquarks of J P = 1 + .

TABLE I .
The parameters of the Hamiltonian and the calculated masses (Cal) of heavy mesons compared with their experimental values (Exp).

TABLE III .
Mean distance R qq [fm] for various tetraquarks.Binding energies EB are in units of MeV.