title: Triple-X and beyond: hadronic systems of three and more X(3872)
authors: Lorenzo Contessi Johannes Kirscher Manuel Pavon Valderrama
text: 
  
  
  lorenzo@contessi.net
  
    IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
  
  
  
  
    Theoretical Physics Division, School of Physics and Astronomy,
  
    The University of Manchester, Manchester, M13 9PL, UK
    
    mpavon@buaa.edu.cn
  School of Physics,
  International Research Center for Nuclei and Particles in the Cosmos and 
  
  Beijing Key Laboratory of Advanced Nuclear Materials and Physics, 
  
  Beihang University, Beijing 100191, China 
  
  
  
  
  
  
   
    0ex3ex
    The X(3872) resonance has been conjectured to be a J^PC = 1^++
    charm meson-antimeson two-body molecule.
    
    Meanwhile, there is no experimental evidence for larger, few-body
    compounds of multiple charm meson-antimeson pairs which would resemble larger
    molecules or nuclei.
    
    Here, we investigate such multi-meson states to
    the extent of what can be deduced theoretically from essentials of the interaction
    between uncharged D^0 and D^*0 mesons.
    
    From a molecular X(3872),
    we predict a 4X (4^++) octamer with 
    a binding energy 
    assuming a D^*0D̅^0 system close to
    the unitary limit (as suggested by the mass of the X(3872)).
    
    If we consider heavy-quark spin symmetry explicitly,
    the D^*0D̅^*0 (2^++) system is close to unitarity, too.
    
    In this case, we predict a bound 3X (3^++) hexamer
    with B_3X > 2.29  MeV
    and a more deeply bound 4X octamer 
    with B_4X > 11.21  MeV.
    
    These results exemplify with hadronic molecules a more general phenomenon of
    equal-mass two-species Bose systems comprised of equal number of either type:
    the emergence of unbound four- and six-boson clusters in
    the limit of a short-range two-body interaction which
    acts only between bosons of different species.
    
    Finally, we also study the conditions under which a 2X (2^++) tetramer
    might form.
  
  
  Triple-X and beyond: hadronic systems of three and more X(3872)
  
      Manuel Pavon Valderrama
      December 30, 2023
  ================================================================
  
  
  
  Systems of particles with a two-body scattering length a
  significantly larger than the interaction range R 
  share a series of common/universal properties, which encompass a multitude of phenomena in atomic, nuclear,
  and particle physics <cit.>.
  
  This invariance with respect to a continuous scale transformation, however, holds strictly only
  in the two-body sector.
  
  In the few-body spectrum, this continuous
  scale invariance 
  survives only partially 
  in a discrete version.
  
  An example of this is the Efimov effect <cit.>,
   the appearance of a geometric bound-state spectrum
  of three-boson systems in the unitary limit (a/R →∞).
  
  This effect was found for the first time a decade ago
  in experiments with caesium atoms <cit.>,
  and it is now known to extend to systems of non-identical particles <cit.>
  as well as systems of more than three particles <cit.>.
  
  In nuclear physics, the Efimov effect plays a role
  in the description of the triton <cit.> and
  ^4 He <cit.>, halo nuclei <cit.>,
  and the Hoyle state <cit.>.
  
  In bosonic systems with more than three particles, the same effect realizes
  stable clusters (see , <cit.> where up to 60 bosons where analyzed).
  
  Compared with atoms and nucleons,
  it is more difficult to find instances of universality in hadronic physics
  
  where
  the X(3872) resonance <cit.>
  might qualify as a hadronic system close to the unitary limit.
  
  The X has been conjectured to be a hadronic molecule <cit.>, more precisely, a 
  relatively shallow bound state of two hadrons
  
  because of its proximity to the D^*0D̅^0 threshold
  (∼0.01  MeV) and its narrow width.
  
  This shallowness, in particular, is a signature of universal
  behaviour <cit.>.
  
  To explore the consequences of universality, we will describe X and multi-X systems with
  a contact-range theory <cit.> and use D (D̅) mesons (antimesons) as fundamental degrees of freedom.
  With this approach, we aim to expose characteristic features of composite systems with the minimal assumptions and data on the constituents.
  Alternative descriptions may improve on accuracy if the coupling to other channels, meson
  exchanges, etc. <cit.>
  is considered. The bulk properties of
  the systems we analyse below, however, will not be affected by these refinements.
  
  The identification of universal properties in systems composed of
  more than two charm mesons is an intriguing open question because
  charm meson-antimeson interactions produce qualitatively new features
  that are absent in systems of identical particles.
  
  For instance, both three-body systems D^0 D^0 D̅^*0 and
  D^*0 D^*0D̅^0 do neither form trimers nor do they display
  the Efimov effect <cit.>.
  
  Along with heavy-quark spin symmetry (HQSS) <cit.> and the associated
  more tightly constrained
  charm meson-antimeson potential enter new features.
  
  In the two-body sector, we expect from HQSS
  the interaction in the D^*0D̅^0 (J^PC = 1^++) X-channel
  to be identical to the one in the D^*0D̅^*0 (2^++)
  channel, suggesting the existence of a partner
  molecule of the X <cit.>. 
  
  Like the X(3872), this partner is expected to be shallow but its survival
  as a bound or virtual state, or as a resonance depends on a number of
  uncertainties <cit.>.
  
  HQSS challenges
  the original expectation of an unbound D^*0 D^*0D̅^0 J=2 three-body system, and could thus facilitate the Efimov effect <cit.>.
  
  In four-meson systems and beyond, we expect to find
  new universal phenomena different from the ones known to emerge in 
  atomic and nuclear composites <cit.>.
  
  We will consider, in particular, systems of N=2,3,4 D^0 D̅^*0 pairs
  with maximum spin,  J=2,3,4, respectively.
  
  Bound “polymers” of this kind
  exhibit a characteristic
  scaling inversely proportional to the square of
  the interaction range,  B_2 N∝ 1/R^2.
  
  We infer from this scaling the Thomas collapse <cit.> of
  these systems along with the implied Efimov effect.
  
  This collapse is eventually avoided owing to short-range effects,  the finite interaction range. Conversely, the Efimov effect is impaired by long-range deviations from unitarity,  a finite scattering length.
  Specifically, D^*0D̅^*0 pairs can decay strongly to D^(*)0D̅^0/D^(*)+D̅^- via a short-range D-wave
  operator <cit.> inducing such a finite interaction width.
  
  Using data on the X in support of the assumption of an
  infinite D^*0D̅^0 scattering length (zero binding energy of
  the X molecule) and disregarding HQSS,
  we predict a bound state of four X's: an octamer.
  
  As the tetramer and the hexamer are unbound under these circumstances, this
  octamer resembles a so-called Brunnian <cit.> state:
  a generalization of a Borromean structure.
  
  Finally, we predict that a D^*0D̅^*0
  interaction close to the unitary limit will stabilize the hexamer and thus
  induce the transition from a Brunnian to a Borromean system
  (a still unbound tetramer with a hexamer resembling a Borromean
  bound state of X's).
  
  
  
  
  Theory and calculation method: 
  
  
  
  
  We treat the above-mentioned “polymers” as a non-relativistic few-body problem.
  
  The charm meson and antimesons comprising these “polymers”
  have a ground (D/D̅) and excited (D^*/D̅^*)
  state.
  
  Their isospin I=1/2 discriminates between neutral and charged states.
  Because of their mass difference, we will only consider
  the neutral mesons which dominate the 
  X wave function tail.
  
  In the unitary limit,
  we have to consider only resonant[Two particles interact resonantly if their S-wave scattering length diverges and
  a two-body bound state with zero energy at threshold exists.] two-body interactions.
  
  Mass and quantum numbers of the X from the molecular perspective 
  hint towards such a resonant behaviour
  in the D^*0D̅^0 (1^++) channel
  (  the X channel) <cit.>,
  while HQSS lets us expect the D^*0D̅^*0 (2^++)
  channel to be resonant, too <cit.>.
  
  All other combinations are assumed to be non-resonant and
  set to zero. 
  
  Non-resonant interaction pairs would
  increase the total binding of the systems slightly. 
  Thus they do not alter our conclusions.
  
  To describe the resonant pairs, we employ a contact
  two-body potential 
  
      V(r; R_c) = C(R_c) δ^(3)(r; R_c)   ,
  
  regularized with the Gaussian cutoff function
  
      δ^(3)(r; R_c) = e^-(r/R_c)^2/π^3/2 R_c^3 .
  
  The cutoff-radius R_c and coupling constant C(R_c) are
  calibrated (renormalized) to the mass of the X(3872):
  
      M_X =  m(D^0) + m(D^*0) - B_X   .
  
  Here, m(D^0), m(D^*0) denote the masses of D^0 and D^*0 mesons, respectively,
  and the binding energy B_X is positive for a stable state.
  
  As we are interested in the universal properties of charmed meson clusters,
  we will set B_X = 0 in accordance
  with the current experimental value B_X = -0.01 ± 0.18   MeV
  putting it slightly above threshold <cit.>.
  
  The potential underlying this threshold state is attractive,  C(R_c) < 0, and
  would bind the system if increased by an infinitesimal amount.
  
  The contact-range potential is visualized in fig. <ref>
  for R_c = 1.0, 1.5, 2.0  fm, where the relation between an
  interaction of shorter range with a larger coupling ( a deeper potential),
  as to ensure that the D^*0D̅^0 bound state is always
  at threshold, is apparent.
  
  When solving the Schrödinger equation with this potential (<ref>),
  we will consider all the mesons to have an identical mass of m=1933.29 MeV,
   twice the reduced mass of the D^*0D̅^0 pair within the X
  because it is
  the most important resonant interaction
  in the multi-X systems (see derivation
  of (<ref>)-(<ref>) below). 
  Corrections are deemed to be subleading and without impact
  on the qualitative description of the many-particle states.
  
  Renormalized predictions, in principle, require
  that observables are cutoff independent.
  
  We will show below that hexamer and octamer binding energies do not
  conform with this demand as they Thomas-collapse if R_c → 0.
  
  In few-body systems, this type of divergence can be renormalized via
  an additional three-boson datum <cit.> which, 
  as of now, is unavailable in the few-X sector.
  
  Despite this obstacle, we can obtain information about the existence of
  bound states and estimates of their binding energies.
  
  To this end, we choose a cutoff range near the theory's
  breakdown scale which is determined by the longest omitted
  short-range component of the interaction.
  
  This missing component is in case of the X the charged channel[
  Pion effects are naïvely expected to enter perturbatively at subleading orders <cit.>, suggesting leading-order predictions which are
  indistinguishable in pionfull and pionless treatments. Analogous to the conjectured effect of the charged components of the D's,
  the inclusion of pions is expected to change the breakdown scale of the theory and
  generates a finite-range interaction which
  adds to the attraction in larger clusters instead of their disintegration.],  
  the D^*+D^- component of the X wave
  function <cit.>.
  
  The characteristic momentum scale of the charged channel is
  M_ ch≃ 125  MeV.
  
  It is sensible to expect a cutoff in the vicinity of
  M_ ch R_c ∼ 1 for which the three-body counterterm vanishes,
  and that it remains numerically small within some interval around it.
  
  This smallness suffices to foresee that their inclusion would have no
  effect on the character of a state: bound will remain bound,
  resonance will remain resonance, etc..
  Hence, a bound state found within a cutoff
  range around M_ ch R_c ∼ 1 (specifically, we chose R_c = 1.0-2.0  fm)
  can reliably be considered a renormalized prediction which will not change character
  even with the proper calibration of a collapse-preventing counterterm.
  
  
  
  Interaction between meson pairs: 
  We exemplify the few-X calculations with a detailed discussion
  of the four-body,  two-X problem.
  
  First, we treat the X as a pure D^*0D̅^0 two-body system.
  
  This approximation disregards the shorter-range D^*+D̅^- component
  and assumes the X wave function to be
  
  
      Ψ_X = ϕ_X (r) 1/√(2) [ 
        | D^0 D̅^*0⟩ + | D^*0D̅^0 ⟩ ]  ,
  
  with the spatial two-body wave function ϕ_X(r).
  
  The charm meson-antimeson potential in the X channel is defined
  for the linear combination D^*0D̅^0 + D^0 D̅^*0
  (the positive C-parity combination).
  
  It is practical to use a Fock representation of the potential:
  
      V_X(r; R_c) =
      V_D(r; R_c)/2 [
          | D^0 D̅^*0⟩ ⟨ D^0 D̅^*0 | +
          | D^*0D̅^0⟩ ⟨ D^*0D̅^0 |
          ]  
      
        + V_E(r; R_c)/2 [
          | D^0 D̅^*0⟩ ⟨ D^*0D̅^0 | +
          | D^*0D̅^0⟩ ⟨ D^0D̅^*0 |
          ]   ,
  
  with a direct (V_D) and an exchange term (V_E)
  which combine to the potential in the X channel,
  V_X = V_D + V_E.
  
  As no negative C-parity partner of the X has been found yet,
  we assume |V_D + V_E| ≫ |V_D - V_E|.
  
  Moreover, the isospin-breaking decays of the X <cit.>
  allow access to V_E and corroborate this inequality <cit.>.
  
  Hence, we use the approximation V_D = V_E = 1/2 V and 
  
      V_X =
        V/2[|D^0D̅^*0⟩+|D^*0D̅^0⟩][⟨ D^0D̅^*0|+⟨ D^*0D̅^0|]   ,
  
  where the (r⃗, R_c) dependence of the potential
  has been dropped to improve readability.
  
  
  The two-X tetramer contains in principle the six possible permutations of
  the | D^0 D^0D̅^*0D̅^*0⟩ state that result
  from exchanging ground- and excited-state mesons
  (we assume the spins of all the D^*0/D̅^*0 mesons/antimesons
  to point in the same direction).
  
  However, these permutations are further constrained by symmetries, as we require
  (i) positive C-parity (, invariance wrt. the exchange of particles and antiparticles), and (ii)
  D^0 and D^*0 to obey Bose statistics which we realize with
  symmetric internal and spatial wave-function components as they are expected
  to provide the majority of the attraction
  (, symmetric combinations of D^0 D^*0 and D^*0 D^0 [Antisymmetric combinations – the nuclear analog are proton-proton or neutron-neutron spin-1 contributions to,  ^4He – demand an odd angular momentum with a perturbatively
  small effect in the leading-order framework employed in this work.]).
  
  
  
  This reduces the number of relevant states from six to two:
  
      | 1 ⟩   =   | D^0 D^*0D̅^*0D̅^0⟩ + | D^*0 
          D^0 D̅^0 D̅^*0⟩/√(2)  , 
      
       | 2 ⟩   =   | D^0 D^*0⟩ + | D^*0D^0 ⟩/√(2) | D̅^*0D̅^0⟩
           + | D̅^0 D̅^*0⟩/√(2)  .
  
  In this basis, the potential reads (insert (<ref>) and (<ref>) in (<ref>))
  
      ∑_ij V_X(r_ij; R_c)
        [ | 1 ⟩; | 2 ⟩ ] =
        [   2 V̅ √(2)V̅; √(2)V̅     V̅ ] [ | 1 ⟩; | 2 ⟩ ]  ,
  
  where V̅ represents the average of the potential for all resonant pairs.
  
  Considering that V_X involves particle-antiparticle interactions only,
  and the same ordering as in |1⟩ and |2⟩ (,
  indexing particles before antiparticles):
  
      V̅ = 1/4 [V(r_13) + V(r_14) + V(r_23) + V(r_24)] .
  
  
  The diagonalization of (<ref>) yields
  
      ∑_ij V_X(r_ij; R_c)  | X_2 ⟩  = 3 V̅ | X_2 ⟩  ,
  
  as the most attractive configuration, with  being
  a four-meson eigenstate of ∑ V_X.
  
  The original coupled-channel problem has thereby been recast
  into a single-channel form.
  
  The steps detailed above for the tetramer can be straightforwardly applied
  to the hexamer and octamer.
  
  The six-body case comprises, in principle, 20 possible permutations of
  the | D^0 D^0 D^0D̅^*0D̅^*0D̅^*0⟩ state,
  which are reduced to two states by symmetry constraints.
  
  In the eight-body case, there are 70 possible permutations of the
  | D^0 D^0 D^0 D^0D̅^*0D̅^*0D̅^*0D̅^*0⟩ state, which are reduced to three symmetric ones.
  
  The potential can be diagonalized, as before in the four-body case, leading to a
  series of eigenvalues and eigenvectors of which
  the most attractive configurations are
  
      ∑_ij V_X(r; R_c)  | X_3 ⟩   =    6 V̅ | X_3 ⟩  ,
        
      ∑_ij V_X(r; R_c)  | X_4 ⟩   =    10 V̅ | X_4 ⟩ .
  
  Again, V̅ represents the average of the potential experienced
  by the interacting pairs, while | X_3 ⟩, | X_4 ⟩ are
  the eigenvectors in the internal space of the interaction
  that correspond to the most attractive configuration.
  
  
  In order to analyze the effect of HQSS on our predictions,
  we modify the above-derived interaction.
  First, we infer from HQSS a potential
  in the D^*0D̅^*0 (2^++) channel
  identical to that in the X channel.
  
  Note the approximate character of this symmetry and
  the resulting hypothetical nature of the 2^++
  partner of the X.
  
  The HQSS extension of the two-body potential
  V_X of (<ref>)  is
  
      V_X^ HQSS   =    V_X + V 
          | D^*0D̅^*0⟩ ⟨ D^*0D̅^*0 |  .
  
  Coupling between the 1^++ and the 2^++ channels is precluded in the
  two-body sector but, nevertheless, these transitions become possible in
  the few-X sector, where these states appear as intermediate structures
  in the wave function.
  
  We use the four-body case once again to exemplify how the additional interaction term leads to more attraction
  than expected earlier. In the basis (<ref>), (<ref>), the potential (<ref>)  now reads
  
      ∑_ij V_X^ HQSS(r; R_c)
        [ | 1 ⟩; | 2 ⟩ ] =
        [   2 V̅ √(2)V̅; √(2)V̅   2 V̅ ] [ | 1 ⟩; | 2 ⟩ ]  ,
  
  whose diagonalization gives
  
      ∑_ij V_X^HQSS(r; R_c)  | X_2' ⟩  =
        (2+√(2)) V̅ | X_2' ⟩  ,
  
  with the more attractive eigenvalue (2+√(2)) V̅ and eigenvector |X_2' ⟩ = (| 1 ⟩ + | 2 ⟩) / √(2).
  
  For the six- and eight-body systems, the same assumptions and symmetries result in
  
      ∑_ij V_X^ HQSS(r; R_c)  | X_3' ⟩   =   1/2(11+√(13)) V̅ | X_3' ⟩  , 
      ∑_ij V_X^ HQSS(r; R_c)  | X_4' ⟩   =   (8+√(22)) V̅ | X_4' ⟩  ,
  
  with eigenvectors |X_3' ⟩ and |X_4' ⟩ corresponding
  to configurations in which the potential is most attractive.
  
  In both cases, with and without HQSS, the spectrum of a system
  composed of N/2 X particles is given by the Schrödinger equation 
  
      (- ∑^N_i<jħ^2/2 m(∇_ij)^2
      +η∑_1 ≤ i ≤ N/2   (N/2 +1) ≤ j≤N
        V(r_ij) ) ϕ_N(r)
      
      = E ϕ_N(r)   ,
  
  with ϕ_N(r)=⟨r_1,r_2,⋯,r_N| (N/2)X⟩, η the eigenvalues of (<ref>)  (η= {3/4, 6/9, 10/16} respectively for 2X, 3X, and 4X),
  r_ij=r_i-r_j with the index i(j) representing a charm
  meson (antimeson), where we have indexed the particles first and
  then the antiparticles, and with m being twice the reduced
  mass of the D^0 D^*0 system,  m≈1933MeV.
  
  Finally, E refers to the energy of the N-body system, where we are interested in bound states (B=-E > 0).
  
  In practice, we solve the Schrödinger equation
  with the Stochastic-Variational Method (SVM <cit.>).
  In our implementation, this method expands the wave function in correlated Gaussian functions ((N-1)×(N-1) relative Jacobi coordinates),
  with a non-zero interaction between the relevant pairs (meson-antimeson).
  We abstain from an explicit symmetrization of the spatial wave function, , we do not project onto L=0 and assume
  that the central and parity-preserving character of the potential will produce the energetically favourable symmetric ground states
  in the course of the variational optimization.
  
  
  
  Results and conclusions: 
  
  
  
  
  
  Assuming the charm meson-antimeson interaction in the X-channel
  to dominate, , with the average interactions
  (<ref>), (<ref>), and (<ref>),
  we find solutions to (<ref>)  of the four-body (2X) and six-body (3X)
  systems to be unbound.
  
  Adding another X, we predict the eight-body (4X) system bound
  with B_4X > 2.08  MeV.
  
  Including the attraction induced by HQSS, , the interactions in (<ref>), (<ref>), and (<ref>),  the eight-body binding energy
  increases to 
  
  Furthermore, the six-body system becomes bound with B_3X > 2.29  MeV.
  
  These results represent sensible lower bounds for the binding energies of
  the respective systems obtained
  at a regularization scale of about 2  fm,
  a value deemed soft enough for an attractive three-body
  counterterm.
  
  Furthermore, any attraction from the non-resonant mesonic interactions
  (set to zero in our calculations) is expected to increase binding energies.
  
  As alluded to in the introduction, this appearance of few-body clusters bound by a few MeV 
  as a result of a resonant two-body system with close-to zero binding energy is not unprecedented.
  In comparison with those
  universal A = 4,5,6 bosonic clusters found <cit.>
  attached to A=3 Efimov states, the constraint of a resonant interaction in the X-channel, only,
  amounts to an N-body problem with each particle interacting only with N/2 particles.
  Furthermore, the strength of this interaction is reduced and is no longer
  resonant. To be explicit, instead of 6(28) resonant interaction pairs in a system of
  4(8) bosons, the limitation to resonant interaction only in the 1^++ channel 
  yields a 4(8) equal-mass boson problem with 4(16) non-resonant interaction pairs.
  Multi-X boundstates are thus not expected to expose molecular-X behaviour.
  Therefore, they should be approached,
  as done in this study, as multi-D/D̅ systems.
  We also notice that three X bosons do not show a Efimov spectrum.
  
  In figure <ref>, we show the
  regulator dependence of
  the binding energies as a signature of the Thomas collapse.
  Originally this collapse is expected for one-channel systems of
  identical particles in the zero-range limit <cit.>.
  Here, we demonstrate the occurrence of the collapse for a more complex system
  with more than one channel and where a certain number of
  interaction pairs have been removed.
  
  A range of cutoffs over which the effect of the unenforced renormalization
  condition (, the canonical three-body counterterm) is expected to vanish
  is marked in the figure (gray area).
  
  Another effect of the reduction of resonantly interacting pairs found here
  is the cutoff-independent ratio between 4X and 3X energies, B_4/B_3∼4.9.
  
  Compared with the ratio found in <cit.>
  and <cit.>, B_4/B_3∼4.6,
  we conclude that reducing the number of interacting pairs
  widens the gap between the energy of N- and (N+1)-boson systems. 
  
  However, a single counterterm should still suffice to renormalize both systems.
  
  Interpreted more generally, this study hints towards new universal systems in which part of the resonant interactions
  are amplified by the presence of more two-body channels or totally removed by symmetry effects.
  Consequences of deviations from universality and the effect of multiple open two-body channels on universal ratios
  are problems beyond the scope of this work.
  
  
  
  
  Finally, we revisit the X_2 and assess the conditions for its binding.
  
  The heavy-quark content of the X_2 is ccc̅c̅, a
  double charm-anticharm content which was first observed
  
  when detecting Ξ_cc^++ <cit.>
  and more recently the fully charm tetraquark <cit.>.
  These measurements indicate a possible X_2 discovery in the near future.
  
  In this regard, it is interesting to notice that the D^0D̅^0 interaction
  have been theorized to be attractive and strong enough as to even support
  a bound state <cit.>.
  
  As this interaction is not connected to the V_X potential via HQSS,
  its strength is unknown. Its structure, however, can be included it our
  framework by refining (<ref>) as follows:
  
      Ṽ^ HQSS_X = V^ HQSS_X +
        λ  V | D^0 D̅^0 ⟩⟨ D^0 D̅^0 |
          .
  
  Here, λ parametrizes the relative strength of the interaction
  in the D^0 D̅^0 (0^++) channel with respect to the X.
  
  The diagonalization of this modified potential (see (<ref>)) is
  
      ∑_ijṼ_X^HQSS(r; R_c)  | X_2”⟩   =    
        (2 + λ/2 +
        √(2 + λ^2/4)) V̅ | X_2”⟩
         =    4  η(λ) V̅ | X_2”⟩  .
  
  Compared with (<ref>), the strength of the potential η is now
  a function of the additional attraction parameter λ.
  
  We find numerically that if η≥η^*=0.876(1) the four-body (2X) system
  binds (see Fig. <ref>). This critical strength is approximately
  constant for all considered cutoffs R_c = 1.0, 1.5, 2.0  fm.
  An analysis and explanation of this phenomenon remains beyond this article.
  
  The condition η≥η^* is equivalent to λ≥λ^* = 0.174(8),  ,
  the 2X will bind if an additional (>20%) D^0 D̅^0 attraction is provided.
  
  As bound states of the
  D^0 D̅^0 system are a conceivable scenario <cit.>, 
  the enhanced attraction and the ensuing bound 2X tetramer are plausible.
  
  
  In summary, we have shown how the substructure of a unitary dimer
  – the X – affects the spectrum of its cluster states.
  
  This spectrum differs from the one predicted 
  for point-like bosons in the unitary limit <cit.>
  in an intriguing aspect.
  
  Namely, under certain assumptions about the meson-antimeson interaction,
  the X cluster states realize a novel generalization of
  Borromean/Brunnian systems.
  
  Regardless of the enormous practical difficulties which hamper
  an experimental (or numerical, in the lattice) verification of
  our conjectures (double charm-anticharm production has only been
  recently achieved <cit.>),
  we deem the exposition of the mechanism which “delays”
  the formation of bound structures – the onset of binding with 4X and 3X,
  but not necessarily with 2X under the assumptions we made –
  as a noteworthy result of the above.
  
  Yet in this later case the 2X tetramer will bind provided we include
    a weakly attractive D^0 D̅^0 interaction in our calculations
    of about 20% the strength of the X-channel potential.
    
    In view of previous conjectures about a possible D^0 D̅^0
    molecule <cit.>,
    this condition might very well be met in the real world.
    
    If this is to be the case, the 2X tetramer will be experimentally
    accessible in the near future.
  
  
  
  Acknowledgments: 
  
  We thank Feng-Kun Guo (Institute of Theoretical Physics in Beijing, CAS) and Michael C. Birse (The University of Manchester) for valuable comments and discussions.
  This work is partly supported by the National Natural Science Foundation
  of China under Grants No.  11735003, No. 11975041 and the Fundamental
  Research Funds for the Central Universities.
  L. Contessi acknowledges the support and the framework of the "Espace de Structure et de réactions Nucléaires Théorique" (ESNT, http://esnt.cea.fr ) at CEA.
  M.P.V. is supported by the Thousand Talents Plan for Young Professionals and would like to thank the IJCLab of Orsay for its long-term hospitality.
  
  
  
  
  