Hadron Formation

According to the assumptions discussed at the beginning of the previous section, reinteractions below the critical point affect only local sets of partons. The minimal groups of partons which may form hadrons independently are color singlets. A color singlet system is defined to transform according to the trivial representation of . Even if a system has zero eigenvalues of the two commuting generators of , it will not in general be a color singlet (c.f. a state with need not have total angular momentum ). To determine, for example, whether a system is a color singlet, one must know not only its total ``color magnetic quantum numbers'' (, ), but also the amplitudes for the possible arrangements of the quark ``color magnetic quantum numbers'' (c.f. the state has while has ; both have ). In the free emission approximation for parton production, the ``color magnetic quantum numbers'' are conserved at each vertex; the phase of the amplitude for each emission is random, so that the final partons are statistically distributed among the possible representations. Hence, for example, a color neutral (i.e., with zero color magnetic quantum numbers) pair produced has probability 1/2 to be in an singlet or an octet, respectively. Similarly, a color neutral pair has probability 1/6 to be a color singlet (in the limit , fixed, a vanishingly small fraction of pairs are color singlets). One might perhaps imagine that final state interactions would mould the amplitudes for different parton states so as to produce particular color representation configurations: such effects would violate the locality assumed, and will therefore be ignored here.

There are several distinct classes of color singlet parton systems which may be considered. First, one might collect all color singlet systems at delimited by a quark and an antiquark [22]. Unfortunately, the very low multiplicity of secondary pairs at realistic (evident in Fig. 2) causes such systems to have masses (although at truly asymptotic , their masses should become [22]); in this case, the final hadron production would not be local. A second scheme for color singlet identification consists in forming the minimum invariant mass color singlet or clusters at . The gluon systems in this case often have rather large masses, because of the low probabilities for color neutral gluon systems to be color singlets. In a third scheme, on which I concentrate here, each gluon at is forcibly split into a pair. Each quark carries one of the spinor color indices of the gluon; every quark is connected by a group theoretical string to its color conjugate antiquark, so as to form a color neutral pair. The formation of color singlet systems from these color neutral may be estimated by combining with probability 1/2 pairs of color neutral systems which arise from splitting of a common gluon ancestor (i.e., the pairs , where originate from forcible splitting of a ``final'' gluon are amalgamated into a single color singlet system with probability 1/2). The prescription of splitting each gluon to a pair may perhaps be regarded as the assumption that any gluonium mesons formed decay infinitely quickly into mesons. (This assumption contradicts the indication of narrow gluonium states, but is supported by the experimental absence of narrow gluonium states.) With this prescription, the mass spectrum of color singlet parton systems is very strongly damped ( or ), and very nearly independent of [2] (except at where the spectrum is usually infinitesimal), yielding a mean color singlet mass . (Recall that in the parton production model of [2], the ``final'' partons produced from parents with were taken to be on their mass shells; if this assumption were relaxed, then , yielding rather massive clusters. If clusters were required to be color neutral, but not color singlet, then .) Of course, while these clusters represent essentially the minimal parton systems which can form hadrons independently, it is certainly possible that several such clusters may often act cooperatively, for example, if their joint invariant mass is below some fixed . In splitting gluons into pairs, I arbitrarily choose the momenta of the quarks to be uniformly distributed over the allowed range (no results are sensitive to this choice) and to have flavors , , with equal probabilities. Just as the color representations of the parton clusters can be determined only statistically, so also their total angular momenta are not determined (the for each parton could be traced, but the orbital angular momentum is entirely undetermined). Nevertheless, I shall below approximate the clusters to decay isotropically in their rest frames, thereby implicitly assuming zero total angular momenta.

If indeed the local color singlet parton clusters described above are formed by reinteractions below , one must then determine how each cluster should decay into hadrons. The discussion above assumes that the clusters may have arbitrary masses. Perhaps, however, each cluster may instead represent a definite meson resonance, with discrete mass. The energy levels of ``atoms'' bound by nonconfining potentials always become dense close to ionization. A ``cooling'' pair may thus be treated classically until it lies in the energy band just below the ionization limit; then the atom cascades by quantum mechanical radiation into the ground state of definite discrete energy. For a confining potential, all the energy levels are discrete, suggesting that ``atoms'' must be directed immediately into discrete levels. This phenomenon must be described by quantum mechanics and would contravene the locality assumption made above. Nevertheless, in a second quantized treatment, the higher levels may decay to lower ones: then the widths of the excited levels may increase faster than their spacings, so that the higher levels merge, effectively yielding a continuous energy band. (This phenomenon occurs for a hydrogen atom in 2-dimensional QED.) This possibility may well be realized for meson resonances: in a constituent model, their level density rises , while phenomenologically their widths . In this case, the available meson (or cluster) masses essentially represent a continuous band: clusters formed in the band may then decay to light mesons with definite masses (25). (The smoothness of the cross-section above suggests that the band of allowed cluster masses extends down to .) The decay properties of the clusters may to some extent be estimated from measured meson resonances, together with low-energy annihilation final states (26). All evidence suggests that quasi-two-body decays are universally dominant. For clusters below , an adequate model is to allow decay into pairs of the lowest-lying mesons, with equal matrix elements for each final spin state (so that the decay branching ratios are determined by the available phase space). This scheme yields a roughly linear increase of multiplicity with mass (for ), apparently as an accidental consequence of the properties of the low-lying mesons. Strange meson production is suppressed simply by the larger mass and by the larger number of than produced in decays of low-lying meson resonances. The approximate constancy of the total multiplicity in annihilation from up to suggests that clusters with masses in this range decay directly to pairs of light mesons, without cascading through clusters of intermediate mass. (Quantum numbers usually leave only one quasi-two-body decay channel open to the known meson resonances, preventing determination of the mass distributions for their decay products (27).) The decay products of low mass clusters are thus taken to have masses comparable to their parents. As the cluster masses increase, the product masses remain unchanged, so that the decay momenta increase. This behavior provides a rather smooth transition to the parton decays at larger invariant masses, where daughter partons have much smaller masses than their parents (see eqs. (1, 2)). In as far as the free emission approximation describes the decay of clusters with sufficiently large mass to lighter clusters, the value of the parameter should be irrelevant: changes in over a certain range would simply assign a different fraction of the hadron production process to the free emission stage and to the phenomenological cluster decay stage, leaving the results unchanged. In practice, however, only rather small changes in exhibit this behavior.

The model defined above purports to describe all features of annihilation final states. A comprehensive investigation will be reported elsewhere [3]; here I make only a few very brief comments. As discussed above, a crucial feature of the model is that it exhibits the locality of hadron formation necessary to justify use of QCD perturbation theory at early times. Previous models (e.g., [15,25]) have failed dismally in this respect: they typically take each produced parton with energy in the c.m.s. to decay into hadrons like one jet of an event with in the original c.m.s. (usually as parametrized by the Field-Feynman model [28]). In this way, the invariant mass of the hadron jet resulting from the parton decay is : the energy of the parton in the c.m.s. of the complete event is crucial in determining its decay to hadrons, and the locality postulate is totally violated. For this reason, any detailed agreement between such models and experimental data should in no way be construed as support for QCD.

The measured mean charged multiplicity in annihilation is roughly constant at small Q, increasing from at to at . At higher , increases rapidly, becoming at and by . (26) This increase presumably reflects the rapid rise in parton multiplicity at high evident in Fig. 2. Given that the quasi-two-body cluster decay model described above reproduces the observed for , the obtained at higher agrees with data to within about 30% for any in the range 1-2 GeV (with ). Almost any plausible cluster decay model suggests . The hadron multiplicity distributions should roughly follow the parton ones, and be broader than Poisson at high . Single hadron energy spectra at are also in adequate agreement with data so long as (a 10% softening in between and is expected). Whereas in the original Field-Feynman model, the charge-weighted distributions for each jet were essentially monotonic, they exhibit considerable oscillations in the present model, especially at small , although the charges of very high hadrons still reflect those of the original quark (c.f. [27]). The transverse momentum spectra of single hadrons obtained are roughly in accord with the experimental data so long as (the increase slowly with as for partons). (Note that, for example, at , measured with respect to the sphericity axis is about 10% larger than that with respect to the original direction; in the former case, consideration of charged hadrons alone effects an reduction in .) Note that the rises slowly with increasing , in contrast with the roughly constant behavior implicit in the Field-Feynman model. Shape parameters, which measure the large-scale angular distribution of energy in the final state, provide an important probe of the processes of hadron production. Recall that successive partons emitted tend to be progressively more collinear, so that only the first few emissions can have a significant effect on the ``shapes'' of the events. If is small, then the hadron clusters formed typically have small masses, and release little transverse momentum in their decays; hence the final hadrons are concentrated along the directions of the first few emitted partons, and the shape parameters for the final states are close to those obtained in low-order perturbation theory. Experimental results for shape parameters indicate that actual final states are much less lumpy, strongly suggesting (note that an increased effective resulting from large higher-order corrections could account for the observed but not the distributions). With such values for , : the hadrons from decays of different clusters thus overlap considerably in phase space (hence analysis methods used to extract properties of clusters in multiparticle production in low hadron collisions [28] could not discern these ``superclusters'', but only the lighter clusters resulting from their decays).

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