ON QUARK ATOMS, MOLECULES AND CRYSTALS * 3. Department of Nuclear Physics. Weizmann Institute of Science. Abstract

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1 thhtiiibeeutg WIS-20/75-Ph ON QUARK ATOMS, MOLECULES AND CRYSTALS * 3 Harry J. Lipkin Department of Nuclear Physics Weizmann Institute of Science Rehovot, Israel Abstract The existence of exotic states in the nonrelativistic quark model with three triplets and an octet of colored gluons is shown to depend on the radial dependence of the potential. Quasi-atomic potentials with smoothraonotonicradial dependence do not produce exotics. Quasi-molecular potentials with repulsive cores can produce not only exotics but also bound crystal lattices. A color dependent repulsive core becomes attractive in some states and could produce a new spectrum of "collapsed" states. Supported in part by the Israel Commission for Basic Research.

2 -1- A nonrelativistic quark model with three triplets and an octet of colored gluons was recently shown to lead to a "quasi-atomic*' model for hadrons which predicts the existence only of bound states having just the quantum numbers observed in the hadron spectrum. The same force binds both mesons and baryons. The quark-antiquark and three-quark systems behave as neutral atoms which do not bind with additional quarks to form large systems with exotic quantum numbers. Although certain peculiar types of forces could produce exotic bound states, it was shown that exotics would not be bound by reasonable quasi-atomic potentials such as the Coulomb, Yukawa or harmonic oscillator commonly considered in quark models. 21 These conclusions have been challenged by Dolgov et al., who present a model in which exotic states are more strongly bound than the non-exotic states. They argue that the model requires existence of exotic particles. The purpose of this note is to point out that the results of Dolgov et al. are not in disagreement with the results of rc-f. 1. The essential difference between the two treatments is in the spatial dependences of the potentials used. The quasi-atomic potential of ref. lf which gives the desired result of saturation at the quark-antiquark and threequark levels has a singularity or an extremum at the origin and a smooth radial dependence which decreases monotonically in magnitude. Dolgov et ai. use a "quasi-molecular" potential with a repulsion at small radius which keeps the quarks separated by a finite distance. In contrast to the quasi-atomic case where the spatial size or" the bound state is detftrmined by the uncertainty principle, the quasi-molecular model

3 -2- neglects the kinetic energy and assumes that the spatial dimensions of the bound state are determined entirely by the features of the potential. Vie show below that this type of force tends to produce a crystal lattice and does not saturate. Both treatments consider the two-quark - two-antiquark system with the four particles arranged at the corners of a square, with the identical particles at opposite diagonals so that the distances separating quark-antiquark pairs are equal to the side of the square while the distances between quark-quark and antiquark-antiquark pairs are equal to the diagonal. For a smooth quasi-atomic potential this four-particle configuration will dissociate into two quark-antiquark bound states. However, for rapidly varying potentials 3uch as a square well or the molecular type potential, a stronger binding than that of two separated quark-antiquark pairs is obtainable by choosing a configuration in which the dominant force is that between nearest neighbors and adjusting the couplings in the color degree of freedom to make the quark-antiquark forces attractive while the quark-quark and antiquark-antiquark forces are repulsive. To show that forces dominated by nearest neighbor interactions lead to stable lattices in the colored-quark-gluon model, we consider a. system of n quarks and n anti^uarks, interacting with the potential used in ref. 1. U = i J u.. J g. g. (1) where u.- describes the dependence of the potential on all the noncolor variables of particles i and j and g. (as 1...8) denote the 8 generators of the group SUC3), acting on a single quark or antiquark i. The summation over i and j goes over all the quarks and antiquarks in the system.

4 Let us now consider the particular color singlet state of n quarks and n antiquarks in which the n quarks are coupled to the totally symmetric state in SU(3), : i.e. the representation with dimension r.(n+l) (n+2)/6,and similarly color for the n antiquarks. These two totally symmetric states are then coupled to a color singlet. For this state the interaction energy (1) can be shown by a little algebra to be given by U(Sym, Sym, 0) = i "H * J I u u ' <f + H") I J J i=q i=q i=q j=q j=q j=q U ii J (2) where the first two terms on the right hand side come from the quark-quark and antiquark-antiquark interactions and the third terra from the. quark -ant iquark interaction. For the case where the interaction u.. is the same for all pairs, ij the interaction (2) reduces to u.. = u (independent of i and j)t (3a) U(Sym, Sym, Q) = - n u (3b) This is the result obtained in refersnces 1 and 2 for any color singlet state of 2n particles when till interactions are equal. It is exactly equal to the interaction energy of n separated color singlet quark-antiquark clusters; e.g. of an n-aeson scattering state. Let us now examine the case of a potential whose spatial dependence is either like the square well of ref. 1 or the molecular potential of ref. 2 where the possibility exists of configurations in which the interaction is large between nearest neighbors and drops off very rapidly with distance so that all other interactions are negligible. Consider a very large nuabsr

5 -4- of quark-antiqusrk pairs arranged in a body-centered cubic Lattice with each quark at the center of a cube formed by eight antiquarks and vice versa. If the potential and the lattice constant are such that the potential between nearest neighbors is large and all other potentials are negligible, the first two terms on the right hand side of eq. (2) do not contribute since quarkquark and antiquark-antiquark pairs are never nearest neighbors. In the third term there are contributions for each quark i from 8 antiquarks j which are nearest neighbors. Thus uisy..ny.0) m.-%nu m (4) where the subscript nn denotes that we are only considering the interactions between nearest neighbors. Comparison of eq. (4) with eq. (3b] shows that the interaction energy of the body-centered cubic lattice is twice as large as the interaction energy of n free quark-anti quark pairs. Thus;, in any model where the spatial dependence of the potential is such that there can be dominance of the nearest neighbor interaction the state of lowest energy will be a large crystal. A simple picture of the coupling scheme in color space is seen in a two color model where the su ( 2 )coi0_ group can be called "color-spin". The color coupling scheme used is one in which the color spins of all the quarks are coupled to the largest possible spin, namely n/2, and the same for the antiquarks, and these two spins of n/2 are then coupled to a total spin of zero. Then any quark pair or any antiquark pair is in the color-symmetric state of spin 1. However, the quark-antiquark pairs are not in color-spin eigenstates but in a mixture of singlet and triplet states. For the case of large n we can use a classical picture in which the antiquark spin is always antiparallel to th* quark spin. A given qutrk-antiquark pair is thus in a mixture of the

6 -5- states quark-up antiquark-down and quark-down antiquark-i^p, but the mixture is incoherent since the relative phase of these two components depends upon the spin variables of all the other quarks. This state is just an equal mixture of singlet and triplet spins. Extending this picture to the analogous state in the case of three colors we see that all quark pairs and antiquark pairs are in the totally symmetric color sextet state, while the quark-antiquark pairs are not in a color eigenstate but in a mixture of the singlet and octet states. For large n the analogy with spin holds and any quark-antiquark pair is a mixture of states in which both the quark and the antiquark have the same color but the relative phases are incoherent. It is thus 1/3 singlet and 2/3 octet. The coefficients of equation (2) are obtained from the values of the twe-body interaction tabulated in Table 1 of ref. 1. The coefficients of the first two terms in eq, (2) are obtained from the value +2/3 for the repulsive interaction in the color sextet state. The leading term in the coefficient of the third term in eq. (2) is obtained from the values of -8/3 for the singlet state and +1/3 for the octet state and taking 1/3 of the singlet and 2/3 of the octet. The use of static potentials with exchange character is questionable. There must be corrections for retardation effects and higher order exchanges. But one can conclude qualitatively that molecular type forces which do not allow quarks to come very close together do not lead to bound systems resembling the observed hadrons but tend to form crystal lattices. On the other hand quasi-atomic potentials which have no repulsive'cores and lead to systems with a size determined by the uncertainty principle can give a hadron spectrum with the observed quantum numbers and no bound exotic states.

7 -6- Dolgov et al J give no indication of the origin of their effective repulsion, nor how it fits into their formalism which is equivalent to our equation (1, Such a repulsive core cannot be put into the spatial dependence of the potential u.. since the sign of the potential depends on the color variables. If the core ir, repulsive in those channels where the long range interaction is attractive the core will be attractive in those channels where the long range interaction is repulsive and give rise to very strong binding at short distances which seems peculiar. There does not seem to be any simple way to obtain an interaction which is repulsive at short distances for all possible color states of the quark-quark and quark-antiquark systems. In nuclear physics the ^-exchange interaction which is believed to be responsible for the repulsive core in the nucleon-nucleon interaction gives a strong attraction at; small distances in the nucleon-antinucleon interactions. The suggestion has been made that such short range attractive interactions 31 night give rise to a new kind of "collapsed" nuclear bound state. In the colored quark model with a core which is repulsive in color singlet states and attractive in color octet states, such "collapsed" states would have color octet quantus nunbers, be stable against decay by color-conserving strong interactions and have suppressed matrix elements for simple radiative decays. Perhaps such peculiar properties deserve further investigation. ^

8 7 References 1) li.j. Lipkin, Phys. Lett. 45_B (1973) ) A.D. Dolgov, L.B. Okun and V.I. Zakharov, Phys. Lett. 49J! (1974) 453.,i) Y. Ne'eman, in Symmetry Principles at High Energy, Proceedings of the Fifth Coral Gables Conference, edited by A. Perlmutter, C. Hurst and B. Kursunoglu, K'.A. Benjamin, New York (1968) p A.R. Bodmer, Phys. Rev. D4_ (1971) ) H.J. Lipkin, Are the New Resonances Superexotic or Collapsed Han Nambu States, WIS-75/18-Ph.