The Hydrogen Bond of QCD

Using the Born-Oppenheimer approximation, we show that exotic resonances, X and Z, may emerge as QCD molecular objects made of colored two-quark lumps, states with heavy-light diquarks spatially separated from antidiquarks. With the same method we confirm that doubly heavy tetraquarks are stable against strong decays. Tetraquarks described here provide a new picture of exotic hadrons, as formed by the QCD analog of the hydrogen bond of molecular physics.

V (x A , x B ) is the interaction between the heavy particles, e.g. the electrostatic repulsion, and E(x A , x B ) is the lowest energy eigenvalue of the light particles at fixed heavy particles coordinates. The approximation improves with m q /M Q → 0. The application of the Born-Oppenheimer method to doubly heavy tetraquarks in lattice QCD has been suggested recently in [6,7], both for hidden flavor tetraquarks, [cq][cq], i.e. the exotic resonances X, Z [3,[8][9][10], and for double beauty open flavor tetraquarks, bbqq, introduced in [11,12] and, more recently, studied in [13][14][15][16].
We fix the QQ pair to be in color 8 and we consider both possibilites,3 and 6, for QQ. Had we taken QQ in color singlet, the interaction with the light quark pair would be mediated by color singlet exchanges, as in the hadroquarkonium model proposed in [17].
For hidden flavour tetraquarks, we obtain color repulsion within the heavy QQ and the light qq quark pairs, and mutual attraction between heavy and light quarks or antiquarks. Thus, in the [Qq] − [Qq] color singlet molecule, repulsions and attractions among constituents are distributed in the same way as for protons and electrons in the hydrogen molecule. Assuming one-gluon exchange forces, Fig 1(a) describes a configuration of a tight QQ similar to the "quarkonium adjoint meson" discussed in [18]. Increasing the repulsion between light quarks beyond the naive one-gluon exchange force, we obtain a configuration of the potential which separates the diquarks from each other (see Fig. 1(b)), as envisaged in [19], with the phenomenological implications discussed in [10] and [20]. The most compelling one is that decays of X, Z particles into quarkonia+mesons are suppressed with respect to decays into open charm mesons: the tunneling of heavy quark pairs through the barrier gets a larger suppression factor. At difference from what was done originally in [3,8,10], the two lumps of two-quark states Qq +Qq are found in a superposition of diquarkantidiquark in the3 ⊗ 3 and 6 ⊗6 color configurations.
The two light particles are not equal and there are two different heavy-light orbitals: in addition to Qq +Qq, we examine the Qq +Qq case. In the latter, Qq andQq orbitals have a large color singlet component. At large separations between heavy quarks, the lowest state will correspond to a pair of charmed mesons. A minimum of the BO potential is not guaranteed. If there is such a minimum, as in Fig. 2(a), it would correspond to a configuration similar to the quarkonium adjoint meson of the previous case. If repulsion in the qq pair prevails, there is no minimum at all, Fig. 2 The BO potential for (QQ)3 is presented in Fig. 3. The unperturbed orbitals correspond to Qq andQq. Forces among constituents are all attractive and the potential vanishes at large QQ separation. This allows a new, independent estimate of the extra binding of QQ with respect to the naive constituent quark model. We confirm the result in [13,14,16] that the lowest bb tetraquark and possibly bc are stable under strong decays, while cc is borderline, see Tab. I.
(QQ) 6 repel each other. However, with the constraint of an overall color singlet, we find both attractive and repulsive forces and the BO potential may admit a second QQ tetraquark. With the perturbative one-gluonexchange couplings, a shallow bound state is indeed found.
In summary, the BO approximation gives a new insight on tetraquark structure, even with the limitations of our arXiv:1903.10253v1 [hep-ph] 25 Mar 2019 perturbative treatment. A non-perturbative treatment of this problem should be provided by lattice QCD [6].
Hidden Charm. We indicate with x A and x B the coordinates of c andc, and x 1,2 the coordinates of q and q. Both cc and qq are taken in the 8 color representation.
If we restrict to one-gluon exchange we find the interactions between the different pairs in terms of the quadratic Casimir operators q 1,2 are the 3 or3 irreducible representations of the color group depending on wether q 1,2 are quarks or antiquarks, and R is the color representation of the q 1 q 2 pair 1 .
If we find the pair q 1 q 2 in the tetraquark T (q i q j q k q l ) in a superposition of two SU(3) c representations with amplitudes a and b Since both cc and qq are in color octet we have λ cc = λ qq = +1/6 α S . The couplings of the other pairs are found using the Fierz rearrangement formulae for SU(3) c to bring the desired pair in the same quark bilinear. We get The pattern of repulsions and attractions implied by previous formulae is the same as in the hydrogen molecule, substituting electrons with light quarks and protons with heavy ones. We shall take a perturbative approach similar to the one followed in the H 2 case in [5]. For fixed coordinates of the heavy particles, x A and x B , we describe the unperturbed state as the product of two orbitals, i.e. the wave functions of the bound states of one heavy and one light particle around x A and x B , and treat the interactions not included in the orbitals as perturbations.
The cq orbital. In the H 2 molecule, the orbital is just the hydrogen atom wave function in the ground state. In our case, we take the Coulombic interaction given by λ cq in (5) with the addition of a confining linear potential 1 We recall the results: We assume a radial wave-function R(r) of the form and determine A by minimizing the Schroedinger functional We use a costituent light quark mass 2 M q = 0.31 GeV estimated from the meson spectrum [1,3], k = 0.15 GeV 2 from [21] and α S = 0.30 at the charm mass scale. We find A = 0.43 GeV, H min = 0.73 GeV. We write the wave function of the qq state The unperturbed energy of Ψ(1, 2) is given by the quark constituent masses plus the energy of each orbital The perturbation Hamiltonian using the values for λ cc = λ qq and λ cq = λc q found above, is To first order in H pert and with r AB = |x A − x B |, the BO potential is where δE = Ψ(1, 2), H pert Ψ(1, 2) evaluates to The functions I 1,4 are given in [5] for hydrogen wave functions, and may be computed numerically for any given orbital (7) where the vector ξ originates from A and |x B | = r AB . Similarly In addition, we take into account the confinement of the colored diquarks by adding a linearly rising potential that starts at r AB ∼ 10 GeV −1 The potential V (r) computed on the basis of Eqs. (7) is given in Fig. 1(a). Also reported are the wave function and the eigenvalue obtained by solving numerically the radial Schrödinger equation [22].
As it is customary for confined system like charmonia, we fix V 0 to reproduce the mass of the tetraquark, so the eigenvalue is not interesting. However, the eigenfunction gives us information on the internal configuration of the tetraquark.
In Fig. 1(a), with one-gluon exchange couplings, a configuration with c close toc and the light quarks around is obtained, much like the quarkonium adjoint meson described in [18]. Fig. 1(b) is obtained by increasing the repulsion in the qq interaction: +1/6 α S ∼ 0.11 → 2.4. The corresponding cc wave function clearly displays the separation of the diquark from the antidiquark.
The barrier that c has to overcome to reachc, apparent in Fig. 1(b), was suggested in [10], and further considered in [20], to explain the suppression of the J/ψ + ρ/ω decay modes of X(3872), otherwise favoured by phase space with respect to the DD * modes. Indeed, with the parameters in Fig. 1(b), we find |R(0)| 2 = 10 −3 with respect to |R(0)| 2 = 10 −1 with the parameters of Fig. 1(a). The tetraquark state T = |(cc) 8 (qq) 8 1 can be written as The tetraquark picture of X(3872) and the related Z(3900) and Z(4020) have been originally formulated in terms of pure3 ⊗ 3 → 1 diquark-antidiquark states [3,8,10]. The 6 ⊗6 → 1 component results in the opposite sign of the qq hyperfine interactions vs the dominant cq andcq one, and it could be the reason why X(3872) is lighter than Z(3900).
The cq orbital. One obtains the new orbital by replacing −1/3 α S → −7/6 α S in Eq. (6). Correspondingly A = 0.50 GeV, H min = 0.47 GeV. The perturbation Hamiltonian appropriate to this case is and The cq and qc orbitals have a large component of color singlet and the lowest state will correspond, at large |x A − x B |, to a pair of color singlet charmed mesons.
There is no confining potential and V BO → H min +V 0 for r AB → ∞. Including constituent quark masses, the energy of the state at r AB = ∞ is E ∞ = 2(M c + M q + H min + V 0 ) and it must coincide with the mass of a pair of non-interacting charmed mesons, with spin-spin interaction subtracted. Therefore we impose A minimum of the BO potential is not guaranteed. If there is such a minimum, as in Fig. 2(a), it would correspond to a configuration similar to the quarkonium adjoint meson in Fig. 1(a). If repulsion is increased above the perturbative value, e.g. changing +1/6 α S ∼ 0.11 to a coupling ≥ 1 in analogy with Fig. 1(b), the BO potential has no minimum at all, Fig. 2(a). Double beauty tetraquarks: bb in3. We start with the color antisymmetric state. The lowest energy state corresponds to bb in spin one and light antiquarks in spin and isospin zero. The tetraquark state T = |(bb)3, (qq) 3 1 can be Fierz transformed into with all attractive couplings There is only one possible orbital, namely bq, but the unperturbed state now is the superposition of two states withq bound to one or to the other b The denominator is needed to normalize Ψ(1, 2) and it arises because ψ(1) and φ(1) are not orthogonal, being eigenfunctions of different hamiltonians, one centered in x A and the other in x B . The overlap S is defined as The perturbation Hamiltonian is and where δE = Ψ(1, 2), H pert Ψ(1, 2) evaluates to I 1,4 were defined previously whereas [5]  value for the bb pair in color3 and the one-gluon exchange couplings are reported in Fig. 3. There is a bound tetraquark with a tight bb diquark, of the kind expected in the constituent quark model [13,14,16].
The BO potential in the origin is Coulomb-like and, for large r AB , it tends to zero, due to our normalization condition (21). The (negative) eigenvalue E of the Schrödinger equation is the binding energy associated with the BO potential. The mass of the lowest tetraquark with (bb) S=1 , (qq) S=0 and of the B mesons are where κ bb = 15 MeV, κ qq = 98 MeV and κ bq = 23 MeV [3] are the hyperfine interactions in the diquark and in the B meson and E = −84 MeV is the eigenvalue shown in Fig 3(a).  The Q-value for the decay T → 2B + γ is then Results for Q cc,bc are reported in Tab. I. Eq. (33) underscores the result obtained by Eichten and Quigg [14] that the Q-value goes to a negative constant limit for M Q → ∞: Q < −150 MeV+O(1/M Q ). Double beauty tetraquarks: bb in 6. We start from T = |(bb) 6 , (qq)6 , also considered in [16], to find The situation is entirely analogous to the H 2 molecule, with two identical, repelling light particles. For the orbital bq, we find A = 0.43 GeV and H min = 0.72 GeV. The BO potential with the one-gluon exchange parameters admits a very shallow bound state with E = −32 MeV, quantum numbers: (bb) 6,S=0 and (qq)6 ,S=0,I=1 , J P C = 0 ++ , and charges −2, −1, 0. The Q-value for the decay T → 2B is then The picture of diquark-antidiquark states segregated in space by a potential barrier is compatible with the existence of charged partners of the X 0 (3872) to be found in X ± → ρ ± J/ψ final states, with branching fractions considerably smaller than in the neutral channel. This requires to push way further on the available experimental bounds. It also gives an independent thrust to the idea of stable bbqq tetraquarks, still awaiting an experimental assessment.
We are grateful for hospitality by the T.D. Lee Institute and Shanghai Jiao Tong University where this work was initiated. We acknowledge interesting discussions with A. Ali, A. Esposito, R. Lebed and W. Wang.