The Hydrogen Bond of QCD

TL;DR

This paper proposes a novel QCD molecular model for exotic hadrons, using the Born-Oppenheimer approximation, revealing stable doubly heavy tetraquarks and new insights into the structure of exotic resonances.

Contribution

It introduces a QCD analog of the hydrogen bond to describe exotic hadrons and confirms the stability of doubly heavy tetraquarks against strong decays.

Findings

Exotic resonances X and Z may be QCD molecular states.

Doubly heavy tetraquarks are stable against strong decays.

A new molecular picture of exotic hadrons is proposed.

Abstract

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.

Tables

Table 1. Table 1: Q 𝑄 Q values in MeV for decays into meson+meson+ γ 𝛾 \gamma . The models in  Karliner:2017qjm ; Eichten:2017ffp ; Luo:2017eub are different elaborations of the constituent quark model we use throughout this paper. More details can be found in the original references. We also refer the reader to the lattice QCD literature providing alternate conclusions on these states  latts . Results in parentheses are obtained with a string tension k = 1 / 4 × 0.15 𝑘 1 4 0.15 k=1/4\times 0.15  GeV 2 in Eq. ( 6 ).

$Q{Q}^{\prime }\overline{u}\overline{d}$	This work	K&R Karliner:2017qjm	E&Q Eichten:2017ffp	Luo et al. Luo:2017eub
$cc\overline{u}\overline{d}$	$-10\left(+7\right)$	$+140$	$+102$	$+39$
$cb\overline{u}\overline{d}$	$-73\left(-58\right)$	$\sim 0$	$+83$	$-108$
$bb\overline{u}\overline{d}$	$-154\left(-137\right)$	$-170$	$-121$	$-75$

Equations

$$V_{\mathrm{BO}}({\mathbf{x}}_A,{\mathbf{x}}_B)=V({\mathbf{x}}_A,{\mathbf{x}}_B)+\mathcal{E}({\mathbf{x}}_A,{\mathbf{x}}_B)$$

$$\lambda_{q_{1}q_{2}}({\bm{R}})=\alpha_{S}\frac{1}{2}\Big{(}C_{2}({\bm{R}})-C_{2}(\bm{q}_{1})-C_{2}(\bm{q}_{2})\Big{)}$$

$$T=a|(q_1{q}_2{)}_{{\mathbf{R}}_1}\cdots {⟩}_1+b|(q_1{q}_2{)}_{{\mathbf{R}}_2}\cdots {⟩}_1$$

$${\lambda }_{q_1{q}_2}=a^2{\lambda }_{q_1{q}_2}({\mathbf{R}}_1)+b^2{\lambda }_{q_1{q}_2}({\mathbf{R}}_2)$$

$${\lambda }_{cq}={\lambda }_{\bar{c}\bar{q}}=-\frac{1}{3}{\alpha }_S{\lambda }_{c\bar{q}}={\lambda }_{\bar{c}q}=-\frac{7}{6}{\alpha }_S$$

$$V_{cq}=-\frac{1}{3}\frac{{\alpha }_S}{r}+kr+V_0$$

$$R(r)=\frac{A^{3/2}}{\sqrt{4\pi }}{e}^{-Ar}$$

$$\langle H(A)\rangle=\frac{\Big{(}R(r),(-\frac{1}{2M_{q}}\Delta+V_{cq}-V_{0})R(r)\Big{)}}{(R(r),R(r))}$$

$$\Psi (1,2)=\psi (1)\varphi (2)=R(|{\mathbf{x}}_1-{\mathbf{x}}_A|)R(|{\mathbf{x}}_2-{\mathbf{x}}_B|)$$

$$H_{\mathrm{pert}}$$

$$V_{\mathrm{BO}}(r_{AB})=+\frac{1}{6}{\alpha }_S\frac{1}{r_{AB}}+\delta E$$

$$\delta E=-\frac{7}{6}{\alpha }_{S}2I_1(r_{AB})+\frac{1}{6}{\alpha }_S{I}_4(r_{AB})$$

$$I_1(r_{AB})=\int d^3\xi |\psi (\xi ){|}^2\frac{1}{|\mathbf{\xi }-{\mathbf{x}}_B|}$$

$$I_4(r_{AB})=\int d^3\xi d^3\eta |\psi (\xi ){|}^2|\varphi (\eta ){|}^2\frac{1}{|\mathbf{\xi }-\mathbf{\eta }|}$$

$$V_{\mathrm{conf}}(r)=k_T\times (r-R_0)\times \theta (r-R_0)$$

$$V(r)=V_{\mathrm{BO}}(r)+V_{\mathrm{conf}}(r)$$

$$T=\sqrt{\frac{2}{3}}|(cq{)}_{\bar{3}}(\bar{c}\bar{q}{)}_3{⟩}_1-\sqrt{\frac{1}{3}}|(cq{)}_6(\bar{c}\bar{q}{)}_{\bar{6}}{⟩}_1$$

$$k_T=\left(\frac{2}{3}+\frac{1}{3}\frac{C_2(6)}{C_2(3)}\right)k=1.5k$$

$$H_{\mathrm{pert}}$$

$$V_{\mathrm{BO}}=+\frac{1}{6}{\alpha }_S\frac{1}{r_{AB}}+\delta E$$

$$T=\sqrt{\frac{8}{9}}|(\bar{c}q{)}_1(\bar{q}c{)}_1{⟩}_1-\frac{1}{\sqrt{9}}|(\bar{c}q{)}_8(\bar{q}c{)}_8{⟩}_1$$

$$⟨H{⟩}_{\min }+V_0=0$$

$$T=\sqrt{\frac{1}{3}}|(\bar{q}b{)}_1,(\bar{q}b{)}_1{⟩}_1-\sqrt{\frac{2}{3}}|(\bar{q}b{)}_8,(\bar{q}b{)}_8{⟩}_1$$

$${\lambda }_{bb}={\lambda }_{\bar{q}\bar{q}}=-\frac{2}{3}{\alpha }_S{\lambda }_{b\bar{q}}=-\frac{1}{3}{\alpha }_S$$

$$\Psi (1,2)=\frac{\psi (1)\varphi (2)+\varphi (1)\psi (2)}{\sqrt{2\left(1+S^2\right)}}$$

$$S=\int d^3\xi \psi (\xi )\varphi (\xi )$$

$$H_{\mathrm{pert}}$$

$$V_{\mathrm{BO}}(r_{AB})=2(⟨H{⟩}_{\min }+V_0)-\frac{2}{3}{\alpha }_S\frac{1}{r_{AB}}+\delta E$$

$$\delta E=\frac{1}{1+S^2}\left[-\frac{2}{3}{\alpha }_S(I_1+S{I}_2)-\frac{2}{3}{\alpha }_S(I_4+I_6)\right]$$

$$I_2(r_{AB})=\int d^3\xi \psi (\xi )\varphi (\xi )\frac{1}{|\mathbf{\xi }-{\mathbf{x}}_B|}$$

$$I_6(r_{AB})=\int d^3\xi d^3\eta \psi (\xi )\varphi (\xi )\psi (\eta )\varphi (\eta )\frac{1}{|\mathbf{\xi }-\mathbf{\eta }|}$$

Full text

Introduction. In this letter we present a description of tetraquarks book ; Esposito:2016noz ; Maiani:2004vq in terms of color molecules: two lumps of two-quark (colored atoms) held together by color forces. The variety of tetraquarks decribed here identifies a new way of looking at multiquark hadrons, as formed by the QCD analog of the hydrogen bond of molecular physics.

We restrict to heavy-light systems, $Q\bar{Q}q\bar{q}$ or $QQ\bar{q}\bar{q}$, and apply the Born-Oppenheimer (BO) approximation, see e.g. weinbergQM , the method used for the hydrogen molecule, see pauling . The method consists in solving the eigenvalue problem for the light particles with fixed coordinates of the heavy ones, ${\bm{x}}_{A},{\bm{x}}_{B}$, and then solve the Schrödinger equation of the heavy particles in the BO potential

[TABLE]

$V({\bm{x}}_{A},{\bm{x}}_{B})$ is the interaction between the heavy particles, e.g. the electrostatic repulsion, and ${\cal E}({\bm{x}}_{A},{\bm{x}}_{B})$ is the lowest energy eigenvalue of the light particles at fixed heavy particles coordinates. The approximation improves with $m_{q}/M_{Q}\to 0$.

The application of the Born-Oppenheimer method to doubly heavy tetraquarks in lattice QCD has been suggested recently in bicudo ; lebed , both for hidden flavor tetraquarks, $[cq][\bar{c}\bar{q}]$, i.e. the exotic resonances $X,Z$ Maiani:2004vq ; Maiani:2014aja ; Maiani:2017kyi ; Ali:2017wsf , and for double beauty open flavor tetraquarks, $bb\bar{q}\bar{q}$, introduced in Esposito:2013fma ; Guerrieri:2014nxa and, more recently, studied in Karliner:2017qjm ; Eichten:2017ffp ; Eichten:2017ual ; Luo:2017eub .

We fix the $Q\bar{Q}$ pair to be in color ${\bm{8}}$ and we consider both possibilites, ${\bar{\bm{3}}}$ and ${\bm{6}}$, for $QQ$. Had we taken $Q\bar{Q}$ in color singlet, the interaction with the light quark pair would be mediated by color singlet exchanges, as in the hadroquarkonium model proposed in Dubynskiy:2008mq .

For hidden flavour tetraquarks, we obtain color repulsion within the heavy $Q\bar{Q}$ and the light $q\bar{q}$ quark pairs, and mutual attraction between heavy and light quarks or antiquarks. Thus, in the $[Qq]-[\bar{Q}\bar{q}]$ 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 $Q\bar{Q}$ similar to the “quarkonium adjoint meson” discussed in braatenBO , see also Brambilla:2017uyf . 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, Fig. 1(b), as envisaged in Selem:2006nd , with the phenomenological implications discussed in Maiani:2017kyi and Esposito:2018cwh . 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 Maiani:2004vq ; Maiani:2014aja ; Maiani:2017kyi , the two lumps of two-quark states $Qq+\bar{Q}\bar{q}$ are found in a superposition of diquark-antidiquark in the $\bar{\bm{3}}\otimes\bm{3}$ and $\bm{6}\otimes\bar{\bm{6}}$ color configurations.

The two light particles are not equal and there are two different heavy-light orbitals: in addition to $Qq+\bar{Q}\bar{q}$, we examine the $Q\bar{q}+\bar{Q}q$ case. In the latter, $Q\bar{q}$ and $\bar{Q}q$ orbitals have a color octet component. As we dhall see, however, at large separations between heavy quarks the lowest state will correspond to a pair of color singlet 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 $q\bar{q}$ pair prevails, there is no minimum at all, Fig. 2(b).

The BO potential for $(QQ)_{\bar{\bm{3}}}$ is presented in Fig. 3. The unperturbed orbitals correspond to $Q\bar{q}$ and $\bar{Q}q$. 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 $QQ$. We confirm the result obtained in Karliner:2017qjm ; Eichten:2017ffp ; Luo:2017eub with different variants of the naive constituent quark model, that the lowest $bb$ tetraquark and possibly $bc$ are stable under strong decays, while $cc$ is borderline, see Tab. 1.

$(QQ)_{{\bm{6}}}$ 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-gluon-exchange couplings, a shallow bound state is indeed found.

In conclusion, the BO approximation, even with the limitations of our perturbative treatment, gives a new insight on the tetraquark structure and provides new opportunities in the intricate field of exotic resonances properties. We hope that our approach may be the basis of further investigations on the internal structure of multiquark hadrons and the phenomenology of their decays. Non-perturbative investigations along these lines should be provided by lattice QCD (see for example bicudo ), following the growing interest shown for doubly heavy tetraquarks latts .

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 $X^{\pm}\to\rho^{\pm}\,J/\psi$ 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 $bb\bar{q}\bar{q}$ tetraquarks, still awaiting an experimental confirmation.

Hidden Charm. We indicate with ${\bm{x}}_{A}$ and ${\bm{x}}_{B}$ the coordinates of $c$ and $\bar{c}$, and ${\bm{x}}_{1,2}$ the coordinates of $q$ and $\bar{q}$. Both $c\bar{c}$ and $q\bar{q}$ are taken in the $\bm{8}$ color representation.

Suppressing coordinates $T=(\bar{c}\lambda^{a}c)(\bar{q}\lambda^{a}q)$ with the sum over $a=1,\dots,8$ understood.

If we restrict to one-gluon exchange we find the interactions between the different pairs in terms of the quadratic Casimir operators

[TABLE]

$\bm{q}_{1,2}$ are the $\bm{3}$ or $\bar{\bm{3}}$ irreducible representations of the color group depending on wether $q_{1,2}$ are quarks or antiquarks, and ${\bm{R}}$ is the color representation of the $q_{1}q_{2}$ pair 111We recall the results: $C_{2}({\bm{1}})=0$, $C_{2}({\bm{R}})=C_{2}({\bar{\bm{R}}})$, $C_{2}({\bm{3}})=4/3$, $C_{2}({\bm{6}})=10/3$, $C_{2}({\bm{8}})=3$..

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 $b$

[TABLE]

then we use

[TABLE]

Since both $c\bar{c}$ and $q\bar{q}$ are in color octet we have $\lambda_{c\bar{c}}=\lambda_{q\bar{q}}=+1/6\,\alpha_{S}$. 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

[TABLE]

The pattern of repulsions and attractions in (5) is the same as in the hydrogen molecule, substituting electrons with light and protons with heavy quarks. We take a perturbative approach similar to the one in the $H_{2}$ case pauling . For fixed coordinates of the heavy particles, ${\bm{x}}_{A}$ and ${\bm{x}}_{B}$, 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 ${\bm{x}}_{A}$ and ${\bm{x}}_{B}$, and treat the interactions not included in the orbitals as perturbations.

Two subcases are allowed: $i)$ $cq$ (and $\bar{c}\bar{q}$) or $ii)$ $c\bar{q}$ (and $\bar{c}q$).

The ${\bm{c}}{\bm{q}}$ 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 $\lambda_{cq}$ in (5) with the addition of a confining linear potential

[TABLE]

We assume a radial wave-function $R(r)$ of the form

[TABLE]

and determine $A$ by minimizing the Schroedinger functional

[TABLE]

We use a costituent light quark mass222For heavy quarks we take $M_{c}=1.67$ GeV, $M_{b}=5.0$ GeV book ; Maiani:2004vq . $M_{q}=0.31$ GeV estimated from the meson spectrum book ; Maiani:2004vq , $\alpha_{S}=0.30$ at the charm mass scale and $k=0.15~{}{\rm GeV}^{2}$ from lattice . Another option is that $k$ follows the coefficient of the Coulombic force Bali:2000gf , which leads to $k=1/4\times 0.15$ GeV2. We comment later on this alternative.

We find $A=0.43~{}{\rm GeV},\,\langle H\rangle_{\rm min}=0.73~{}{\rm GeV}$.

We write the wave function of the $q\bar{q}$ state

[TABLE]

The unperturbed energy of $\Psi(1,2)$ is given by the quark constituent masses plus the energy of each orbital $E_{0}=2(M_{c}+M_{q}+\langle H\rangle_{\rm min}+V_{0})$.

The perturbation Hamiltonian using the values for $\lambda_{c\bar{c}}=\lambda_{q\bar{q}}$ and $\lambda_{c\bar{q}}=\lambda_{\bar{c}q}$ found above, is

[TABLE]

To first order in $H_{\rm pert}$ and with $r_{AB}=|{\bm{x}}_{A}-{\bm{x}}_{B}|$, the BO potential is

[TABLE]

where \delta E=\big{(}\Psi(1,2),H_{\rm pert}\Psi(1,2)\big{)} evaluates to

[TABLE]

The functions $I_{1,4}$ are given in pauling for hydrogen wave functions, and may be computed numerically for any given orbital (7)

[TABLE]

where the vector $\bm{\xi}$ originates from $A$ and $|\bm{x}_{B}|=r_{AB}$. Similarly

[TABLE]

In addition, we take into account the confinement of the colored diquarks by adding a linearly rising potential determined by a string tension $k_{T}$ and the onset point, $R_{0}$

[TABLE]

For orientation, we choose $R_{0}=10$ GeV*-1*, greater than $2A^{-1}\sim 5$ GeV*-1*, where the two orbitals start to separate 333$R_{0}$ should be considered a free parameter, to be fixed on the phenomenology of the tetraquark, as we discuss below.. As for $k_{T}$, we note that the tetraquark $T=|(\bar{c}c)_{\bm{8}}(\bar{q}q)_{\bm{8}}\rangle_{\bm{1}}$ can be written as

[TABLE]

At large distances the diquark-antidiquark system is a superposition of ${\bar{\bm{3}}}\otimes{\bm{3}}\to{\bm{1}}$ and ${\bm{6}}\otimes{\bar{\bm{6}}}\to{\bm{1}}$. The hypothesis of Casimir scaling of $k_{T}$ Bali:2000gf and (16) would give

[TABLE]

However, as discussed in Bali:2000gf , gluon screening gives the ${\bm{6}}$ diquark a component over the ${\bar{\bm{3}}}$, which appears in the product ${\bm{6}}\otimes{\bm{8}}$, bringing $k_{T}$ closer to $k$. For simplicity, we adopt $k_{T}=k$.

The potential $V(r)$ computed on the basis of Eqs. (15) is given in Fig. 1(a). Also reported are the wave function and the eigenvalue obtained by solving numerically the radial Schrödinger equation schroed .

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 to $\bar{c}$ and the light quarks around is obtained, much like the quarkonium adjoint meson described in braatenBO .

Fig. 1(b) is obtained by increasing the repulsion in the $q\bar{q}$ interaction: $+1/6\,\alpha_{S}\sim 0.11\to 2.4$. The corresponding $c\bar{c}$ wave function clearly displays the separation of the diquark from the antidiquark. Had we used $k=1/4\times 0.15$ GeV2 in Eq. (6), the required enhancement would be $+1/6\,\alpha_{S}\to 3.3$.

The barrier that $c$ has to overcome to reach $\bar{c}$, apparent in Fig. 1(b), was suggested in Maiani:2017kyi , and further considered in Esposito:2018cwh , to explain the suppression of the $J/\psi+\rho/\omega$ 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 perturbative parameters of Fig. 1(a).

The tetraquark picture of $X(3872)$ and the related $Z(3900)$ and $Z(4020)$ have been originally formulated in terms of pure ${\bar{\bm{3}}}\otimes{\bm{3}}$ diquark-antidiquark states Maiani:2004vq ; Maiani:2014aja ; Maiani:2017kyi . The ${\bm{6}}\otimes{\bar{\bm{6}}}$ component in (16) results in the opposite sign of the $q\bar{q}$ hyperfine interactions vs the dominant $cq$ and $\bar{c}\bar{q}$ one, and it could be the reason why $X(3872)$ is lighter than $Z(3900)$.

The ${\bm{c}}{\bar{\bm{q}}}$ orbital. One obtains the new orbital by replacing $-1/3~{}\alpha_{S}\to-7/6~{}\alpha_{S}$ in Eq. (6). Correspondingly $A=0.50~{}{\rm GeV},\,\langle H\rangle_{\rm min}=0.47~{}{\rm GeV}$. The perturbation Hamiltonian appropriate to this case is

[TABLE]

and

[TABLE]

The tetraquark state is

[TABLE]

At large $|{\bm{x}}_{A}-{\bm{x}}_{B}|$ the lowest energy state (two color singlet mesons) has to prevail, as concluded also in Bali:2000gf on the basis of the screening of octet charges due to gluons.

There is no confining potential and $V_{\rm BO}\to\langle H\rangle_{\rm min}+V_{0}$ for $r_{AB}\to\infty$. Including constituent quark masses, the energy of the state at $r_{AB}=\infty$ is $E_{\infty}=2(M_{c}+M_{q}+\langle H\rangle_{\rm min}+V_{0})$ and it must coincide with the mass of a pair of non-interacting charmed mesons, with spin-spin interaction subtracted. Therefore we impose

[TABLE]

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 $+1/6~{}\alpha_{S}\sim 0.11$ to a coupling $\geq 1$ in analogy with Fig. 1(b), the BO potential has no minimum at all, Fig. 2(a).

Double beauty tetraquarks: $\bm{b}\bm{b}$ in ${\bar{\bm{3}}}$. The lowest energy state corresponds to $bb$ in spin one and light antiquarks in spin and isospin zero. The tetraquark state $T=|(bb)_{\bar{\bm{3}}},(\bar{q}\bar{q})_{{\bm{3}}}\rangle_{\bm{1}}$ can be Fierz transformed into

[TABLE]

with all attractive couplings

[TABLE]

As in Eq. (20), the ${\bm{8}}$ charges are screend by gluons, so at large separations the state in Eq. (22) behaves like the product of two color singlets. There is only one possible orbital, namely $b\bar{q}$, but the unperturbed state now is the superposition of two states with $\bar{q}$ bound to one or to the other $b$

[TABLE]

The denominator is needed to normalize $\Psi(1,2)$ and it arises because $\psi(1)$ and $\phi(1)$ are not orthogonal, with the overlap $S$ defined as

[TABLE]

The perturbation Hamiltonian is

[TABLE]

and

[TABLE]

where \delta E=\big{(}\Psi(1,2),H_{\rm pert}\Psi(1,2)\big{)} evaluates to

[TABLE]

$I_{1,4}$ were defined previously whereas pauling

[TABLE]

For the orbital $b\bar{q}$ we find $A=0.44~{}{\rm GeV},\,\langle H\rangle_{\rm min}=0.75~{}{\rm GeV}$.

The BO potential, wave function and eigenvalue for the $bb$ pair in color ${\bar{\bm{3}}}$ 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 Karliner:2017qjm ; Eichten:2017ffp ; Luo:2017eub .

The BO potential in the origin is Coulomb-like and it tends to zero, for large $r_{AB}$, due to (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 $(bb)_{S=1},~{}(\bar{q}\bar{q})_{S=0}$ and of the $B$ mesons are

[TABLE]

where $\kappa_{bb}=15$ MeV, $\kappa_{qq}=98$ MeV and $\kappa_{b\bar{q}}=23$ MeV Maiani:2004vq are the hyperfine interactions and $E=-84$ MeV is the eigenvalue shown in Fig 3(a) with $\alpha_{s}(m_{b})=0.20$.

The $Q$-value for the decay $T\to 2B+\gamma$ is then

[TABLE]

Results for $Q_{cc,bc}$ are reported in Tab. 1 using $\alpha_{s}((m_{b}+m_{c})/2)=0.23$. Eq. (33) underscores the result obtained by Eichten and Quigg Eichten:2017ffp that the $Q$-value goes to a negative constant limit for $M_{Q}\to\infty$: $Q=-150$ MeV$+{\cal O}(1/M_{Q})$.

Double beauty tetraquarks: $\bm{b}\bm{b}$ in $\bm{6}$. We start from $T=|(bb)_{\bm{6}},(\bar{q}\bar{q})_{{\bar{\bm{6}}}}\rangle$, also considered in Luo:2017eub , to find

[TABLE]

therefore

[TABLE]

The situation is entirely analogous to the $H_{2}$ molecule, with two identical, repelling light particles. For the orbital $b\bar{q}$, we find $A=0.43~{}{\rm GeV}$ and $\langle H\rangle_{\rm min}=0.72$ GeV. The BO potential with the one-gluon exchange parameters admits a very shallow bound state with $E=-32$ MeV, quantum numbers: $(bb)_{{\bm{6}},S=0}$ and $(\bar{q}\bar{q})_{{\bar{\bm{6}}},S=0,I=1}$, $J^{PC}=0^{++}$, and charges $-2,-1,0$. The $Q$-value for the decay $T\to 2B$ is then

[TABLE]

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, N. Brambilla, A. Esposito, R. Lebed and W. Wang.