Volume 58B, number 1 PHYSICS LETTERS 18 August 1975

ON Q U A R K ATOMS , M O L E C U L E S A N D C R Y S T A L S *

H.J. LIPKIN Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel

Received 23 June 1975

The existence of exotic states in the nortrelativistic quark model with three triplets and an octet of colored gluons is shown to depend on the radial dependence of the potential. Quaff-atomic potentials with smooth monotonic radial 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.

A nonrelativistic quark model with three triplets and an octet of colored gluons [1 ] 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 spec- trum. 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.

These conclusions have been challenged by Dolgov et al. [2], 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 ref. [1]. The essential difference between the two treatments is in the spatial depen- dences of the potentials used. The quasi-atomic poten- tial of ref. [ 1 ], which gives the desired result of satura- tion at the quark-antiquark and three-quark levels has a singularity or an extremum at the origin and a smooth radial dependence which decreases monoton- ically in magnitude. Dolgov et al. use a "quasi-molecu- lar" potential with a repulsion at small radius which

* Supported in part by the Israel Commission for Basic Research.

keeps the quarks separated by a finite distance. In con- trast to the quasi-atomic case where the spatial size of the bound state is determined by the uncertainty prin- ciple, the quasi-molecular model neglects the kinetic energy and assumes that the spatial dimensions of the bound state are determined entirely by the features of the potential. We 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 di- agonals so that the distances separating quark-anti- quark pairs are equal to the side of the square while the distances between quark-quark and antiquark-anti- quark 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 such 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 de- gree of freedom to make the quark-antiquark forces attractive while the quark-quark and antiquark-anti- quark 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 antiquarks, interacting with the potential used in ref. [1].

v : l: uii g, og/o (1) 2 i . /

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Volume 58B, number 1 PHYSICS LETTERS 18 August 1975

where ui] describes the dependence of the potential on all the noncolor variables of particles i and] and gio (o = 1 ... 8) denote the 8 generators of the group SU(3)color acting on a single quark or antiquark i. The summation over i and j goes over all the quarks and antiquarks in the system. Let us now consider the par- ticular color singlet state o fn quarks and n antiquarks in which the n quarks are coupled to the totally sym- metric state in SU(3)color; i.e. the representation with dimension n(n + 1)(n + 2)/6, and similarly for the n antiquarks. These two totally symmetric states are then coupled to a color singlet. For this state the inter- action energy (1) can be shown by a little algebra to be given by

U(Sym'Sym'O)=~i~=qUij 3 - \ 3 nli=q ui] i=q ]=~l ]=~l i¢/ i¢/ (2)

where the first two terms on the right hand side come from the quark-quark and antiquark-antiquark inter- actions and the third term from the quark-antiquark interaction. For the case where the interaction ui/is the same for all pairs,

ui] = u (independent o f i and]), (3a)

the interaction (2) reduced to

U(Sym, Sym, 0) = 8 - ~ n u . ( 3 b )

This is the result obtained in refs. [1,2] for any color singlet state of 2n particles when all interactions are equal. It is exactly equal to the interaction energy of n separated color singlet quark-antiquark clusters; e.g. of an n-meson 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 inter- action is large between nearest neighbors and drops off very rapidly with distance so that all other inter- actions are negligible. Consider a very large number of quark-antiquark pairs arranged in a body-centered cubic lattice with each quark at the center of a cube formed by eight antiquarks and vice versa. I f the po- tential and the lattice constant are such that the po- tential 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

quark-quark and antiquark-antiquark pairs are never nearest neighbors. In the third term there are contribu- tions for each quark i from 8 antiquarks] which are nearest neighbors. Thus

U(Sym, Sym, 0)n n = - ~ n Unn (4)

where the subscript nn denotes that we are only con- sidering the interactions between nearest neighbors. Comparison ofeq. (4) with eq. (3b) shows that the in- teraction energy of the body-centered cubic lattice is twice as large as the interaction energy o fn free quark- antiquark 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)color

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 anti- quark spin is always antiparallel to the quark spin. A given quark-antiquark pair is thus in a mixture of the states quark-up antiquark-down and quark-down anti- quark-up, but the mixture is incoherent since the rela- tive 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 quarkpairs and anti- quark 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 eq. (2) are obtained from the Values of the two-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

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Volume 58B, number 1 PHYSICS LETI'ERS 18 August 1975

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 charac- ter is questionable. There must be corrections for re- tardation 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 not 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.

Dolgov et al. [2] give no indication of the origin of their effective repulsion, nor how it fits into their formalism which is equivalent to our eq. (1). Such a repulsive core cannot be put into the spatial depen- dence of the potential uii since the sign of the poten- tial depends on the color variables. I f the core is repul- sive 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 co-exchange interaction which is believed to be responsible for the repulsive core in the nucleon- nucleon interactiqn gives a strong attraction at small distances in the nucleon-antinucleon interactions.

The suggestion has been made that such short range attractive interactions might give rise to a new kind of "collapsed" nuclear bound state [3]. 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 quantum numbers, be stable against decay by color-conserving strong interactions and have suppressed matrix ele- ments for simple radiative decays. Perhaps such pecu- liar properties deserve further investigation [4].

References

[1] H.J. Lipkin, Phys. Lett. 45B (1973) 267. [2] A.D. Dolgov, L.B. Okun and V.I. Zakharov, Phys. Lett.

49B (1974) 453. [3] Y. Ne'eman, in Symmetry principles at high energy, Proc.

Fifth Coral Gables ConL, eds. A. Perlmutter, C. Hurst and B. Kursunoglu, W.A. Benjamin, New York (1968)p. 149. A.R. Bodmer, Phys. Rev. I)4 (1971) 1601.

[4 ] HJ . Lipkin, Ate the new resonances superexotic or collapsed Han Nambu states, WIS-75/18-ph.

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