Volume 99B, number 2 PHYSICS LETTERS 12 February 1981

ON THE POSSIBLE EXISTENCE OF STABLE FOUR-QUARK SCALAR MESONS

WITH CHARM AND STRANGENESS ~

Nathan ISGUR Department of Physics, University of Toronto, Toronto, Canada

and

Harry J. LIPKIN Argonne National Laboratory, Argonne, 1L 60439, USA Fermi National Accelerator Laboratory, Batavia, 1L 60510, USA and Weizmann Institute of Science, Rehovoth, Israel

Received 4 September 1980 Revised manuscript received 20 November 1980

we observe that the existence of the S* and 6 just below KK threshold has a natural interpretation in the four-quark model for these states, and we apply this interpretation to predict the possible existence of stable four-quark scalar mesons with charm and strangeness.

The possibility that four-quark scalar mesons with the composition (cs~a) or (cu-gff) might exist and be stable against strong decay has been pointed out [1,2]. In this paper we present additional arguments for the existence of such states and give a rough quantitative calculation o f binding effects in the four-quark system.

The four-quark scalar states can be considered as bound states of two ordinary pseudoscalar mesons, with binding provided by the hyperfine interaction as first suggested by Jaffe [3] ,1 . The basic physics of this model can be seen as follows: consider a state of two pseudoscalar mesons placed very close together to form a four-particle qq qcl cluster. In the original co lo r - spin coupling each meson is a quark-ant iquark pair in a spin-singlet-color-singlet state and there is no force between the quarks in one meson and the quarks in the other. But suppose the colors and spins of the four particles are recoupled to introduce color octet

¢' Research supported in part by the Natural Sciences and Engineering Research Council of Canada.

,1 G, Gelmini has recently applied such considerations to the qqc~system with some results analogous to those presented here [4].

and spin triplet components into each pair while keeping the overall four-particle state a color singlet and spin singlet. The color-electric interaction is not changed by this recoupling, since it is the same for any spatially symmetric color singlet state, but the spin-dependent hyperfine interaction can change. Some energy is lost in the hyperfine energy of each original pair since the state which is the singlet in both color and spin has the lowest energy, but there are four new pairs involving a quark or antiquark in one of the old pairs and a quark or antiquark in the other. Before the recoupling there was no interaction energy in these four pairs. The ques- tion arises whether a recoupling is possible which gains more binding energy in the forces between the four new pairs than it loses in the forces between the two old pairs.

In order to determine whether a four-particle bound state exists the possible gain in potential energy due to color and spin recoupling must be balanced against the increase in kinetic energy required to keep the four- body system together rather than allowing it to sepa- rate into two mesons. The gain in potential energy can easily be calculated, using Jaffe's color-spin force and

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Volume 99B, number 2 PHYSICS LETTERS 12 February 1981

experimental values for observed hyperfine splittings, for a four-particle wave function which has a spatial dependence between each pair which is the same as any other pair (e.g., like the four nucleons in an alpha particle) and is the same as in ordinary quark-ant i - quark mesons where the values of the hyperfine inter- action matrix elements are known from experimental hyperfine splittings. We shall see below that in the cases of interest, an appreciable gain in potential energy is obtained by such a recoupling of spins. However, it is not clear if this is sufficient to overcome the effect of the kinetic energy. The question of how to modify the wave function from this simple a-particle structure in a way which minimizes the energy has no simple model- independent answer, since it depends upon how the color charge and color hyperfine interactions change when the four-particle wave function is scaled up in size or takes on a two-center molecular type configura- tion rather than that of an alpha particle. Such asym- metric wavefunctions can also lead to effects from color electricity as well as from the confinement po- tential if this latter is color dependent [5,6].

We argue that the existence of the 6 and S* just below the KK threshold indicates that such binding occurs for the four-quark KK system. Jaffe has pointed out that the degeneracy of the isovector 6 and the isoscalar S* which couples much more strongly to KK than to zrTr is simply explained in the four quark model and not in the stan- dard q~t model and that the masses are in the right ball park. We note that the description of these states as just barely bound states of the KK system provides a natural explanation for the occurrence of these states right at the KK threshold. There is no simple explana- tion for this striking experimental fact if the 6 and S* are ordinary q~ mesons.

We therefore suggest that similar bound states of DK and DK should exist near and possibly below the DK threshold. The isoscalar states of these two configu- rations, denoted by 1~ s and F x would then be stable against strong decay.

The increase in potential energy from color-spin re- coupling can be calculated for the alpha particle con- figuration using Jaffe's expressions for the hyperfine interaction. We consider the four-quark SU(6) scalar state with the color-spin classification (21,21 *) in the SU(6) X SU(6) classification, where the two SU(6) groups are the color-spin groups for the quarks and antiquarks, respectively. (For illustrative purposes we

will neglect here the effect o f [1] ~ [405] SU(6) color- spin mixing, and will simply quote the result of the exact calculation below * 2 .) The expectation values of the hyperfine interaction in this wave function for a quark-ant iquark pair and a quark-quark pair, res- pectively, are found to be:

Mq~(21,21 *) = - ~ (M v - M p ) , ( la)

Mqq(21,21 *) = M ~ ( 2 1 , 2 1 *) = - 3 (M v _ Mp), ( lb)

where M v - Mp is the hyperfine splitting for the con- ventional q~ mesons, given by the experimental value of the mass difference between the vector and pseudo- scalar mesons. The hyperfine interaction in a conven- tional pseudoscalar meson is just :

mq~(P) = - 3 (m V _ m p ) . (2a)

This is greater than the value of the hyperfine inter- action ( la ) for a quark-ant iquark pair in the (21,21 *) state, as expected. The change in energy o f a q~ pair in recoupling its spin from the pseudoscalar color- singlet-spin-singlet state to the (21,21 *) state is given by the difference between ( la ) and (2a):

Mq~(21,21 *) - Mqg(P) = 9 (M v _ M p ) . (2b)

The change in binding energy of the alpha particle configuration in recoupling the colors and spins from the two pseudoscalar configuration to the (21,21 *) configuration is seen to contain three components. Eq. (2b) gives the loss in binding energy for each of the two qcl pairs that were originally coupled to pseudo- scalar mesons. Eq. ( la) gives the gain in binding energy for each of the two q~ pairs which were initially not in the same meson and had no initial hyperfine interac- tion. Eq. ( lb ) gives the gain in the binding energy for qq and ~ pairs which also had no initial hyperfine in- teraction. The net gain in binding for the alpha-particle configuration over the 2P configuration is then given by:

M ( a ) - 2M(P) = - 2 ( I + ~ - 9 ) (M v _ Mp)

= - ~ (M V - M p ) . (3)

,2 This calculation may be made using Jaffe's result for [1 ] - [405] mixing in refs. [3] if one takes into account that in eq. (3.19) of the second article the sign of the 10+9 [405 ]> state should be reversed. See also ref. [6] for a technique not involving the use of color-spin.

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The exact calculation gives -0 .53 (M v - Mp), about 25% larger.

Since vector-pseudoscalar splittings are typically several hundreds of MeV, we see that the gain in poten- tial energy from color-spin recoupling in the alpha particle configuration will also be several hundreds of MeV. This is sufficient to be taken seriously as a source for binding. However, the zero point kinetic energies per degree of freedom are of the same order of magni- tude. As mentioned above, whether the binding is suf- ficient is a dynamical question requiring detailed study [6] ; for now, we assume from the KK system that bind- ing does indeed occur. (Note, for example, that the usual 6-function potential used for the hyperfine inter- action is valid only in perturbation theory, and that it is too singular to give a sensible result in the Schr6dinger equation.)

The expression (3) assumes SU(4) flavor symmetry, in which the hyperfine interaction is flavor-independent. However, it is very easy to take flavor dependence into account if we keep the (21,21 *) wave function. This will give an upper bound on the hyperfine energy, since it will be possible to lower the energy by slight changes in color-spin recoupling from the symmetric (21,21 *) configuration if the hyperfine couplings of the quarks are different. For this purpose the expres- sions (1) and (2b) are convenient since each can have a different flavor dependence.

Let us consider the DK and DK systems which could bind to produce the Fs and F x states, respec- tively. These contain six pairs with flavors (cs), (cu), (cd), (su), (sd) and (ud). Each pair gives a contribution to the binding which has the form (la), ( Ib) or (2b) depending upon whether it is a quark-antiquark or quark-quark pair not in the original mesons or whether it is a pair which was in an original meson state. For each pair the relevant value o f M V - Mp is the hyper- fine splitting corresponding to the flavors of the parti- cular pair. We thus obtain:

M{o~; Fx(CUg0.)} - M(D) - M(K) 3

= _ ~ (M o - Mrr +MF* - MF)

+ ~ (MK, - M K + M D , - M D ) = 2 0 5 MeV, (3)

M { a ; Fs(CSgd)} - M(D) - M(K )

= -- 3 ( M p - M . + M F , - M F

+ M K , - M K + M D , - M D ) = I 4 0 M e V , (4)

where M{e; ...} denotes the potential energy in the alpha-particle configuration for the quantum numbers indicated (recall we have isovector-isoscalar degeneracy)

For the case of the KK system which can bind to produce the ~ and S* states, expressions entirely in terms of experimental hyperfine splittings like (4) cannol be obtained, since the hyperfine splitting for an (s~) pair is obscured by mixing and not directly available from experiment. If we assume that hyperflne splittings are inversely proportional to quark masses, we obtain the result

3 M(a; ¢5) - 2M(K) = - q (MK, - MK)

X [ l+(ms/m u + m u / m s - 2)1 ~ - 2 0 0 M e V , (5)

where m s and m u are the masses of the strange and up quarks and we have set m u / m s = 2/3 to obtain the numerical result. Note that this result is very insensitive to the value of m u /ms .

These energies are all in the right ball park to suggest that spatial configurations exist in which these potential energies can barely win over the kinetic energies to pro- duce a bound state. One would expect kinetic energy effects to be smaller for the charmed system because of the increased mass; thus if the S* and 8 are bound KK states, the F x should also be bound and the FS borderline. The same argument applied to the lrrr and KTr systems could explain the absence of bound e and t¢ states.

Note that the RoRo decay mode of the Fs provides a useful signature for its detection. An even more strik- ing signature would be three neutral kaons produced in the cascade decay B 0 --> Fs + K 0 -* R0 + R0 + K 0" Even though the branching ratio for this B 0 decay is probably very small, significant evidence for the existence of the Fs would be obtained from a few events containing three kaons at the B mass with a unique mass just be- low DK threshold for one kaon pair.

N.I. would like to acknowledge continuing conver- sations on the subject of multiquark hadrons with Steve Godfrey, Kim Maltman and John Weinstein.

[1] H.J. Lipkin, Phys. Lett. 70B (1977) 113. [2] H.J. Lipkin, in: The whys of subnuclear physics, Subnuclear

• Series, Vol. 15, ed. H. Zichichi (Plenum, New York, i979). [3] R.L. Jaffe, Phys. Rev. D15 (1977) 267,281. [4] G. Gelmini, Trieste preprint IC/79/163. [5] H.J. Lipkin, Phys. Lett. 45B (1973) 267. [6] N. Isgur, in preparation.

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