Four - Quark Model for the Mesons

A four-quark mechanism is proposed which generates virtually the entire known spectrum of (non-charm) meson states to remarkable accuracy. The ground state Oand lnonets are taken as input; exclusive of octetsinglet mixing there are no free parameters. (Submitted to Phys. Rev. Lett.) This work was supported by the National Science Foundation, Grant No. PHY76-24629 AOl. tWork supported by the Department of Energy. -2Previous attempts to derive the meson spectrum on the basis of the quark model have focused almost entirely on the qq system. Such approaches ~ invariably involve the introduction of a potential (and/or a bag), with associated free parameters. Unfortunately, our experience indicates that the potential must be very complex (and the number of parameters large) in order to generate a detailed description (e.g., at the level of this paper). Recently, several authors have studied 4-quark (4499) systems with regard to a few select conventional states (Ott), and O-exotics. 1 However, this approach is again within the potential/bag framework, and is not (at present) quantitative. In contrast, taking the ground state Oand lnonets as input, the 4-quark mechanism proposed in this Letter involves no free parameters, and generates every well-determined meson state to remarkable accuracy. This implies that the mechanism responsible for confining the q; s-waves, whatever its nature, does not directly pro. duce low mass states with R F 1. Specifically, I propose that the excitations commonly associated with R 2 1 arise as the consequence of a simultaneity condition involving the pairwise masses of all four q; combinations. In order to understand this condition, we first consider Fig. la, which depicts three mesons ml, m2, m3 resonating in pairs to produce particles A (mlm2) and B (mlm3) simultaneously. This can occur only if the invariant three-body mass takes on a particular value MO determined by the masses of A, B and the three mesons. Almost a generation back, Peierls noted the sharp energy-dependence of this effect, and proposed that it could be responsible for generating the N*(1512).2 Others extended Peierls' treatment to produce excellent predictions for the masses of the Al, Q and E mesons. Physically, there -3is nothing strange about this effective "force"; for example, in the singly ionized hydrogen molecule, in which a single electron can be bound to ~' either&f two protons, the molecular binding is produced by exactly such an (exchange) mechanism. At the technical level, however, subsequent analysis by several authors singularity is on the wrong a resonance. 3 demonstrated that the associated S-matrix Riemann sheet, and hence cannot be related to On the other hand, I recently discussed an alternative mechanism in which particles m 1' m3 again form resonance B, but particles m2, m3 are taken to be relatively at rest. The related singularity is now on the correct sheet, and closely similar values are generated for M 0 in many cases (e.g., Al, Q, D, E). 4 This is easy to see, since particles m2 and m3 must be identical in the case of a single type of resonance (A=B), and hence the pairwise masses Ml2 and Ml3 are equal; M12=M13=mA. In the Al, . for example, A, B are the p, and ml, m2, m3 are all pions. I have shown that for broad states A(characteristic of physical mesons) this effect will be important only under rather special circumstances, but in the limit that A has zero width it is guaranteed to be strong (i.e., at the level of the quark model). For our purposes it is important to note that nothing has changed physically with respect to the Peierls idea; this reformulation is aimed at the mathematical subtleties. We now turn to Fig. lb, representing a 4-quark system which we regard as arising from the excitation of an original 2-quark state (qlq2). We thus suppose that ql, y2 are moving originally as free particles in the center of a bag or very deep potential well, and that a fluctuation occurs in which an additional pair y3, q4 is created (also in the bag/well). The -4energy for this process will be minimal if the pair c3, q4 is relatively at rest; to conserve charge they must both clearly be of the same flavor ( uu, d;L, S-G), and form an I=0 singlet. In analogy with the above, we shall assume that the extremely strong forces responsible for confinement will attempt to force as many qq,pairs as possible into.the unique energy levels observed in the O-, lnonets. We would then expect the favored energy levels to correspond to the pairs (q1<3), (c2q4), and (qly2) forming meson states a, b, and c, respectively. Applying the conditions (p1+p3)2=mi , etc. to the 4-body system, it is simple to derive the expression M2 = mz+2(m:-m2 41 ) -!2(4-m2 92 ) (1) for the mass M of the excited state, where mq i is the mass of quark qi. Below I demonstrate that this simple formula correctly generates the observed spectrum of meson states. Of course, it is not sufficient just to calculate a sequence of masses; one must show that each excitation has the right quantum numbers to be identified with the corresponding physical state. Fortunately, given our picture of the excitation mechanism, we have a number of facts to work with. In the first place, the isospin of the system must be identical with that of meson c (I=Ic). Secondly, for neutral states the effect of the charge conjugation operator C is to rotate the system about the horizontal axis bisecting the diagram of Fig. lb. Thus, if mesons a and b have relative angular momentum L, we associate the state with C=(-)L. Together, these facts determine the g-parity; g=(-)'+I. If we then make the reasonable assumption that the particular combination of a and b is only important if their quantum numbers allow them to be emitted as -5actual decay products, we deduce the additional constraint g=gagb; this is useful if IaS;%, I,#%. From our identification of a, b, c as either ~ vectors,or pseudoscalars, we can also determine the most likely value of the total spin S of the excited state. Thus, if a, b, c are all O-, the individual quark spins are alternately up or-down, and we infer S=O (other choices suggest S=O,1,2). Given L and S, we severely limit the possible -. values of total angular momentum J. The parity P is simply (-)L PaPb. Finally, it is natural to assume that the purely pseudoscalar combinations a, b, c with L=O generate the lowest-lying excitations (J "=O*), and that successively replacing each with a vector leads (in general) to a state of higher J. Taking all these rules together, it is possible to achieve an almost totally unambiguous classification of states (with an exception discussed below). As a specific example, let us first consider the possible excitations 1 involving just 7~'s and p's. In doing so, it is useful to introduce the notation (ab)c for the particles involved, and to represent the common mass of the u and d quarks by m. According to Eq. (l), (ITT)V leads to the relation M2=5m2-4m2. ?r At this state we clearly need a value for m, which we fix by requiring that this lowest level excitation leads simply back to the pion; i.e., m=m TrWhile this is certainly not the only choice one might make a priori, it turns out to be the only acceptable one in our scheme, and has some very intesting consequences. Thus (ITT)~ with L=O has 1=1, C=+, g=-i it hence cannot decay into two pions a and b (whatever the value of m). On the other hand, (rn)p yields an acceptable vector (l--) with L=l, and predicts M=m if m=m p7rr' Therefore, the pion is stable against the lowest excitation, but the p has an acceptable mechanism for decay; p -+ (ITIT)P -f VT. To go beyond the r and p, the next -6level is (p~)7~. For this the possibility of pr decay requires g=-; with I=1 we need L even. Since S=l the lowest choice L=O yields 1 ii, with a -. -. ~ predictzd mass M2=2m2-m2. Numerically, M(ltc)=1080 MeV, which may be P 71 compared with A,(l100).5 The next state is (pn)p; again S=l and L=even. Since this is a higher level we take L=2, yielding possibilities 1 *, 2*, 3++ Having a (lower) 1 * +i. already, the most likely choice is 2 ; we then predict M(2*)=1319 MeV, to be compared with A2(1310). Finally, (PP)P has S=2, I=l, g=+ (implying L=odd); for L=l one may have l--, 2--, 3--. Having a l-(and a better candidate for 2-below), we predict M(3--)=1699 MeV, to be compared with g(1680). Thus we achieve the sequence r, p, Al, A2, g2 with the correct quantum numbers, decay modes, and masses (to the level of 1% error). In order to generate the I = $ states, we next consider the diagram in which ql is taken to be the s quark. In precisely the same fashion as . the above, we obtain the K and K* from (KT)K and (Kr)K*, providing that the s mass is equal to mK. The next excitation (L=O) has two modes (K*a)K and (Kp)K, leading to the respective values 1161 and 1179 for M(l*). This splitting arises from the fact that rn:-rnt is not precisely equal to m&-G empirically; at the level of our argument it seems reasonable to neglect this and predict a single state at the average value 1170. This agrees nicely with the value Q,(l180> extracted from a recent analysis of (l+) Knn data. 6 A continuation of this sequence leads to M(2*)=1388, M(3--)=1745; the experimental states are K*(1420) and K*(1780). Having established the rules of our game, it is straightforward to PC generate the I=0 partners of these states, and other J combinations. For this reason I shall defer noncritical details to a subsequent (more -7lengthy) article, and concentrate on some points of particular interest. (1) Results are displayed in Tables I and II for the lowest lying J PC ~' -c, states. For reference purposes, an index N has been introduced to characterize the particular 4-quark configuration involved. Using the symbol m to represent either d or u, the diagrams (Fig. lb) with q q 34 = &n and 9142 = mm, sm, ss are labeled by N=1,2,3,.respectively. The diagrams with G3q4 = ss and q1q2 = %n, is, gs are labeled by N=4,5,6. (2) The I=0 6 and ss configurations are regarded as projections onto the pure octet and singlet states. At the level of unmixed SU(3), the excitations are thus computed using the appropriate octet and singlet masses. In order to compare with experiment, however, one must allow for some mixing of states such as (an,)c and (anl)c. The simplest assumption is that mixing occurs precisely as it appears empirically in the O-, l1 ground states. Due to the nature of our mass formula, this means that one may simply employ the physical masses of w, +, n, n' in Eq. (1). With a few exceptions, this postulate leads to generally excellent results. The overall quality can be considerably improved, however, by allowing for slightly different mixing of n, n'. The values quoted have thus been computed using the effective masses m n=617 MeV, m ,=920 Mev. n (3) The Ott ground states are crucial as input for generating certain higher J PC excitations. For example, the first I=1 entry in Table II shows a state I denoted as "b" at 861 MeV; this arises from-(nx)r. It has not been seen experimentally, but may be difficult to disentangle from the 6. It is presumably of comparably narrow width, and I have therefore assumed that it can be employed as one of the a,b,c states (communicating via the qi p-wave). Correspondingly, an I=% partner (b') -8is generated at 983 MeV by (Kn)K. This can decay to Kr and KITTW, but is also unobserved. For our purposes, the predicted b, b' states are vital 4 in generating well-determined states such as B, Q,, h, U (in addition to A3, L, etc.); their experimental verification would provide a crucial test of this theory. The use of relatively broad excitations as input is perhaps questionable; as an example I have listed 4 +Istates arising from Al and Q,. The results are clearly encouraging, yielding h, U and X(1900). (4) In Table II I have listed four nonets of O* states; there are additional I=0 scalars of higher mass. Among the former are good candidates for relatively narrow states corresponding to 6(980), S*(980), ~(1400);~ and to 6'(1260) and S*'(1310).7 Others may be associated with broad enhancements ~(1250), ~(1300) in the K~,wT s-waves. In addition, states degenerate in mass with n and w which couple strongly to TIT are predicted ~ (labeled nk, w*); possible evidence for such states has been noted in the decays I/J -+ $I,+,and J, -t WIT+IT-. 8 (5) In addition to those states normally attributed to qi, it is possible to construct exotics such as the O-nonet shown. These (lightest) excitations are all close to the busy 1400 MeV region, and will not be easy to sort out. (6) The l+ I=% system is particularly striking. A recent analysis by this author suggested two Q states; Q,(1180) + K*?r, pK and Ql(1290) + pK, 6 whereas previous analyses found the same Q,, but a higher mass state Q2(1400) -t KNIT, EK. 9 The present theory suggests that all of these exist, with Q, occurring in C = +, and Q,, Q, associated with C = combinations of K+7;t'lrand K-r+,-. Note that the L=l (Kb) and (K6) configurations may well prefer to decay into the s-wave Kp or K*v configurations (i.e., -9horizontally, in the sense of Fig. lb). This Q triplet is quite unexpected"from the standpoint of qq dynamics, and is an important signature of this mechanism. In summary,-1 feel that this model must be regarded as a seerious alternative to the simple picture of qs excitations. For those states which are best determined experimentally and uncomplicated by mixing ambiguities (e.g., 6, B, A2, E, Q,, g, u, h, ~*(1420), L, K*(1780)) the predictions are essentially exact. With slight differences in mixing, and considering experimental errors, this could conceivably be true for the entire spectrum. In addition, new effects such as the Q splitting, the b, b' particles, and the low mass -frr states are predicted, and could provide a definitive test of the theory. A generalization to the baryon isobars, and (comparable) results for charm particles, will be reported in subsequent articles. -lOREFERENCES 1. -'