Basics of doubly heavy tetraquarks
A. Valcarce J. -M. Richard J. Vijande





^1 Departamento de Física Fundamental e IUFFyM, 
           Universidad de Salamanca, E-37008 Salamanca, Spain

^2 Université de Lyon, Institut de Physique Nucléaire de Lyon,
           IN2P3-CNRS–UCBL,
 4 rue Enrico Fermi, 69622  Villeurbanne, France

^3 Unidad Mixta de Investigación en Radiofísica e Instrumentación Nuclear en Medicina (IRIMED),
           Instituto de Investigación Sanitaria La Fe (IIS-La Fe),
           Universitat de Valencia (UV) and IFIC (UV-CSIC), Valencia, Spain

valcarce@usal.es, j-m.richard@ipnl.in2p3.fr, javier.vijande@uv.es 


We outline the most important results regarding the stability of doubly heavy tetraquarks QQq̅q̅
with an adequate treatment of the four-body dynamics. We consider both color-mixing and spin-dependent effects. 
Our results are straightforwardly applied to the case of all-heavy tetraquarks QQQ̅Q̅. We conclude
that the stability is favored in the limit M_Q/m_q ≫ 1 pointing to the stability of the bbu̅d̅ state
and the instability of all-heavy tetraquarks. 




§ INTRODUCTION

Despite the impression given by the recent flurry of studies dealing with multiquark states,
flavor-exotic multiquarks have already a long history <cit.> and have motivated an abundant literature
(see Ref. <cit.> for a recent compendium). In the pioneering 
work of Ader, Richard and Taxil <cit.> it was shown that
QQq̅q̅ four-quark configurations become more and more
bound when the mass ratio M_Q/m_q increases. The critical value 
of M_Q/m_q for binding is somewhat model dependent.

Currently, a broad theoretical consensus about the existence of a stable
axial vector doubly bottom tetraquark has been reached.
Lattice QCD calculations find unambiguous signals for a stable J^P=1^+ 
bottom-light tetraquark <cit.>. Based on a diquark hypothesis, Ref. <cit.> 
uses the discovery of the Ξ_cc^++ baryon to calibrate the binding energy in a QQ diquark. Assuming that
the same relation is true for the bb binding energy in a tetraquark,
it concludes that the axial vector bbu̅d̅ state is stable.
The Heavy-Quark Symmetry analysis of Ref. <cit.> predicts the existence
of narrow doubly heavy tetraquarks. Using as input for the doubly bottom baryons, not 
yet experimentally measured, the diquark-model calculations
of Ref. <cit.> also leads to a bound axial vector bbu̅d̅ 
tetraquark. Other approaches, using Wilson twisted mass lattice QCD <cit.>, 
also find a bound state. Few-body calculations using quark-quark 
Cornell-like interactions <cit.>,
simple color magnetic models <cit.>, QCD sum rule analysis <cit.>, or
phenomenological studies <cit.> come to similar conclusions. More doubtful has become
the prediction about the stability of all-heavy tetraquarks <cit.>.

In the present note, we stress that a careful treatment of the few-body problem 
is required before drawing any conclusion about the existence of stable states in 
a particular model. There is, indeed, a dramatic spread of strategies: some 
authors use the full machinery of a variational method based on correlated 
Gaussians or hyperspherical expansion, and others use a crude trial wave function 
or a cluster approximation.



§ GENERAL RESULTS BASED ON SYMMETRY BREAKING



The analogy between the stability of few-charge systems
and multiquarks in additive chromoelectric potentials
offers a good guidance for identifying the favorable
configurations. There are, however, some differences
mainly due to the color algebra replacing the simpler
algebra of electric charges.
Unlike in the case of the positronium molecule, the equal-mass
tetraquarks are unstable in the chromoelectric model
with frozen color wave functions <cit.>. 
In both the atom and quark cases, the four-body system and its threshold, 
after simple rescaling, are governed by a generic Hamiltonian


    H=∑_i p⃗_i^ 2/2 m_i- c.o.m. + ∑_i<j g_ij v(r_ij)   ,    ∑_i<j g_ij=2 ,


with v(r)=-1/r in the atomic case, and v(r)=-a/r+b r in the quark case <cit.>.

In quantum mechanics, the minimum of a Hamiltonian containing a symmetric and an antisymmetric 
term is always lower than the minimum of  the symmetric part. From this result, one can analyze the effect 
of symmetry breaking in systems of four-charged particles. Let us first consider the hydrogen molecule, M^+ M^+ m^- m^-.
The Hamiltonian for this system reads,


    H     = p⃗_1^ 2/2   M +
    p⃗_2^ 2/2   M +
    p⃗_3^ 2/2   m +
    p⃗_4^ 2/2   m + V = H_0 + H_1 
       = [∑_ip⃗_i^  2/2  μ  +   V ]   +  (1/4 M  -  1/4 m)(
    p⃗_1^ 2 + p⃗_2^ 2 - p⃗_3^ 2 - p⃗_4^ 2)   ,


where 2 μ^-1=M^-1+m^-1. The C-parity breaking term, H_1, lowers the ground state
energy of H with respect to the C-parity even part, H_0, which is simply a rescaled version of
the Hamiltonian of the positronium molecule. Since H_0 and H
have the same threshold, and since the positronium molecule is stable, the hydrogen molecule
is even more stable, and stability improves when M/m increases. 
Clearly, the Coulomb character of V hardly matters in this reasoning.  
The key property is that the potential does not change when the masses are modified. 

One can use the same reasoning to study the stability of four-charged particles 
when C-parity is preserved but particle symmetry is broken, in other words the
M^+ m^+ M^- m^- configuration. The Hamiltonian is that of Eq. (<ref>) by 
exchanging 2↔ 3.
The same arguments used above lead to the conclusion that this configuration gains 
binding with respect to the threshold (M^+m^-)+(M^-m^+) that it shares with H_0.
However, there is another threshold that lies lower, (M^+M^-)+(m^+m^-). This 
threshold gains more from the symmetry breaking than the four-body molecule, and, 
indeed, it is found  that the molecule becomes unstable for M/m ≥ 2.2. 

The above arguments can be directly translated to four-quark systems:
the QQq̅q̅ configuration  becomes more and more
bound when the mass ratio M_Q/m_q increases. This has been first established 
in Ref. <cit.>, and discussed and confirmed in further studies. 
Arguments based on diquarks, as e.g. <cit.>, might considerably 
overestimate the binding, as analyzed in <cit.>. 
There are  many variants of the so-called diquark model. An extreme point of view is that 
diquarks are almost-elementary objects, with their specific interaction with quarks and 
between them. In the case of doubly heavy baryons QQq there is obviously a QQ clustering 
which makes it tempting to use a two-step approach:  first a (QQ) diquark and then a 
(QQ)q quasi-meson, as the diquark has the same color 3̅ as an antiquark. 
The exercise can be repeated for the QQq̅q̅ states. For simplicity, we consider 
only the case of a frozen 3̅3 color wave function, i.e., the Hamiltonian (<ref>). 
In Fig. <ref>, we compare the exact solution 
of (<ref>) with the approximation consisting of first computing the QQ diquark with 
r_12/2 alone and qq with r_34 alone, and then (QQ)(q̅q̅) as a meson with a 
potential r_12,34 and constituent masses 2 M and 2 m. 



A remaining problem is to understand why the positronium
molecule lies slightly below its dissociation threshold, while a chromoelectric
model associated with the color additive rule does not bind (at least according 
to most computations). This is due to a 
larger disorder in the color coefficients than in the electrostatic
strength factors entering the Coulomb potential <cit.>.  
An alternative proof is based on the so-called Hall-Post inequalities <cit.>.
The principle is rather simple. If a Hamiltonian is decomposed as a sum of Hamiltonians,


    H= h' + h” + ⋯  ,


then for the lowest energy,


    E(H) ≥ E(h') + E(h”) + ⋯  .


With a 3̅ 3 color wave function and a quark mass set to M=1/2 for simplicity, 
the Hamiltonian of the all-heavy tetraquark QQQ̅Q̅ can be written as <cit.>,


    H_4=∑_ip⃗_i^ 2  +  1/2( V_12 + V_34) + 1/4( V_13 + V_14 + V_23 + V_24)   ,


where V_ij=v(r_ij) is the quarkonium potential. Now, we can rewrite this expression as,


    H_4=1/2( h_12 + h_34) + 1/4( h_13 + h_14 + h_23 + h_24)   ,


where h_ij= p⃗_i^  2 + p⃗_j^  2 + V_ij is the quarkonium Hamiltonian. By using Eq.( <ref>)
one gets,


    E_ min(H_4) ≥ 2   E_ min(h_13) = 2   E_ min(QQ̅)   ,


that demonstrates the instability of all-heavy tetraquarks.
The above reasoning on the ground state holds for a single 
color channel. It is observed in explicit computations than 
the mixing of color states does no help much <cit.>. 
The lower bound (<ref>) can even be significantly improved 
if one relates Hamiltonians that are free of center-of-mass motion <cit.>.




§ COLOR DYNAMICS.


In the heavy-quark limit, the lowest lying tetraquark configuration
resembles the helium atom <cit.>, a factorized system with separate dynamics
for the compact color 3̅ QQ nucleus and for the
light quarks bound to the stationary color 3 state, to construct
a QQq̅q̅ color singlet. This argument has been mathematically proved and numerically
checked time ago <cit.>, see the probabilities 
for the axial vector bbu̅d̅ tetraquark shown in Table II
(note that the 66̅ probability
in a compact QQq̅q̅ tetraquark tends to zero for M_Q →∞). 

The ≪ij model of Eq. (<ref>), with a pairwise potential due to 
color-octet exchange, induces mixing between 3̅3 and 66̅ states in the 
QQ-q̅q̅ basis. If one starts from a 3̅3 
state with QQ in a spin triplet, and, for instance q̅q̅=u̅d̅ with spin and 
isospin S=I=0, then its orbital wave function is mainly made of an s-wave in all coordinates. 
It can mix with a color 66̅ with orbital excitations in the x⃗ and y⃗ linking QQ 
and q̅q̅, respectively. A minimal wave function in this sector can be chosen as:


    Ψ_6    ∝   x⃗.y⃗ exp(-a x⃗^  2-b y⃗^  2) ,         or
    Ψ_6    ∝   exp[-a_12 x⃗^  2-a_34 y⃗^  2 -α(r⃗_13^  2+r⃗_24^  2)
     -β(r⃗_14^  2+r⃗_23^  2)]-{α↔β} .


To illustrate the role of color-mixing we use the potential AL1 <cit.>. 
Its central part is a Coulomb-plus-linear potential. Its spin-spin part is a regularized Breit-Fermi 
interaction, with a smearing parameter that depends on the reduced mass.

The energy as a function of M/m without and with color-mixing is shown in the left panel of Fig. <ref>. 
The ground state of the QQu̅d̅, candidate for stability with J^P=1^+, has its main component 
with color 3̅3, and spin {1,0} in the QQ-u̅d̅ basis. The main admixture consists of 66̅ 
with spin {1,0} and an antisymmetric orbital wave function of which (<ref>) is a prototype, and of 
66̅ with spin {0,1} with a symmetric orbital wave function.



Note how the diquark hypothesis and color mixing have opposite effects that tend to cancel in the charm sector.




§ SPIN-DEPENDENT CORRECTIONS


In Ref. <cit.> it was acknowledged that, within current models,  a pure additive interaction such as
(<ref>) will not bind ccq̅q̅, on the sole 
basis that this tetraquark configuration benefits from the strong cc chromoelectric attraction 
that is absent in the Qq̅+Qq̅ threshold. When qq=ud, there is in 
addition a favorable chromomagnetic interaction in the tetraquark, while the threshold experiences 
only heavy-light spin-spin interaction, whose strength is suppressed by a factor m/M.
 
For illustration, we use the again the potential AL1 <cit.>. 
The results are shown in the right panel of Fig. <ref> for QQu̅d̅, as a function of the mass ratio M/m.
The system bbu̅d̅ is barely bound without the spin-spin term, though the mass ratio m_b/m_q is 
very large. It acquires its binding energy of the order of 150 MeV when the spin-spin is restored. 
The system ccu̅d̅ is clearly unbound when the spin-spin interaction is switched off. This 
is shown here for the AL1 model, but this is true for any realistic interaction, including an early 
model by Bhaduri et al. <cit.>. The case of ccu̅d̅ is actually remarkable. 
Here the binding requires both the color mixing of 3̅3 with 66̅, and the spin-spin interaction. 
Moreover, the  binding is so tiny that it cannot be obtained with a simple variational method. One needs 
either a fully converged expansion on a basis of correlated Gaussians, or a hyperspherical expansion up to 
a grand orbital momentum K_max of the order of 12. 
Semay and Silvestre-Brac <cit.>, who used the AL1 potential, missed the binding, but their method of 
systematic expansion on the eigenstates of an harmonic oscillator is not very efficient to account 
for the short-range correlations. 
Janc and Rosina <cit.> were the first to obtain binding with such potentials, and their calculation 
was checked in Ref. <cit.>. The stability of ccu̅d̅ with J^P=1^+ 
is with respect to the nominal DD^* threshold. Depending on its binding energy, 
it decays into DDπ or DDγ. The bb analog decays weakly. 



§ CONCLUSIONS


The four-body problem of tetraquarks is rather delicate, especially for systems at the edge of stability.
The analogy with atomic physics is a good guidance to indicate the most favorable configurations. 
However, unlike the positronium molecule, the all-heavy configuration QQQ̅Q̅ is not 
stable if one adopts a standard quark model and solve the four-body problem correctly. 
The mixing of the 3̅ 3 and 66̅ color configurations is important, especially for states very near the threshold.  
This mixing occurs by  both  the spin-independent and the spin-dependent parts of the potential. 

Approximations are welcome, especially if they shed some light on the four-body dynamics. The diquark-antidiquark approximation 
is not supported by a rigorous solution of the 4-body problem, but benefits of a stroke of luck, as the erroneous extra attraction 
introduced in the color 3̅3 channel is somewhat compensated by the neglect of the coupling to the color 66̅ channel.  

Finally, ccu̅d̅ with J^P=1^+ is at the edge of binding within current quark models. For this state, all contributions 
should be added, in particular the mixing of states with different color structure, and the four-body problem should be 
solved with extreme accuracy. In comparison, achieving the binding of bbu̅d̅ looks easier. Still, with a typical quark model, 
the stability of the ground state cannot be reached if spin-effects and color mixing are both neglected. 

Work funded by Ministerio de Economía, Industria y Competitividad
and EU FEDER under Contract No. FPA2016-77177 and
by Generalitat Valenciana PrometeoII/2014/066.




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