Beyond Free Emission

The approximation of free (and independent) parton emissions used in the previous section becomes inaccurate when the invariant masses of the partons in an event have fallen so low that the rate of emissions from them no longer dominates over the rate of interactions between them. Above, I have simply truncated all radiation at this critical point, parametrized by a critical parton invariant mass . However, for an accurate description of even the most inclusive experimental measurements at available , it is essential to venture beyond the critical point, and model the final formation of hadrons.

The first simplifying assumption which I shall make is that the cross-section for production of a given final hadron state involves only an incoherent sum over parton states at the critical point. This is in keeping with the free emission approximation for parton production, which provides cross-sections for parton configurations at the critical point: it suggests that interferences between different parton states which are transformed into the same hadron state (21) should vanish. The processes of hadron formation are taken to be sensitive only to the probabilities, and not the amplitudes, for the possible configurations of the parton system. As a simple, but somewhat inappropriate analogue, consider a reaction in which many are produced: the assumption requires that interferences between processes in which a given positronium atom arises from an system and an system should cancel. This occurs in so far as the amplitudes for population of the various states of the interfering systems have random phases (as would follow from classical free emission). A further (but to some extent related) assumption is that the evolution beyond the critical point depends only on the local structure of the parton system over small spacetime volumes at the critical point. Hence, suitable sets of partons at the critical point will evolve to form hadrons independently, and irrespective of the processes by which they were produced. For example, I shall below often assume that low-mass color singlet clusters of partons present at the critical point condense directly into hadrons, independently from each other. If the formation of hadrons does not occur in some such local and universal manner, there seems little hope of obtaining useful predictions from QCD without very detailed knowledge of the structure of hadrons. If, even at high , the whole parton system at the critical point acted cooperatively to generate the final hadrons, then the disposition of partons could be largely irrelevant, and perturbative parton production above the critical point would be rendered invisible. I shall entirely neglect this possibility, and will assume that the processes of hadron formation act universally and locally: the precise constitution of the independent parton systems is, however, unknown; several possibilities will be discussed below.

For some purposes, it may, at sufficiently large , be adequate to make the approximation that each individual parton in the final state (rather than, say, each color singlet cluster of partons) ``decays'' independently into hadrons. (This approximation fails to account for color conservation, and therefore must be violated eventually.) It is conventional, for example, to define ``fragmentation functions'' which describe inclusive hadron spectra in parton decays. In this approximation, the function gives the probability for a hadron of type to carry a fraction of the (roughly) longitudinal momentum of the parton , whose invariant mass is less than . If the perturbative evolution of the final state is truncated when all parton invariant masses fall below , then some inclusive properties of the final hadron state may be found by taking each parton to decay according to a suitable . In this case, changes in affect only the perturbative evolution: the decay of the parton system below may be described by the same , regardless of . Thus the approximation allows the change (``scaling violations'') in single hadron inclusive energy spectra as a function of to be estimated without explicit knowledge of the processes of hadron formation. The approximation fails, however, when and are small enough that many pairs of partons at have invariant masses , and therefore may act cooperatively in forming hadrons, thereby necessitating introduction of a further joint two-parton fragmentation functions . Since only the variation of the single hadron distributions is required, such cooperative hadron formation processes are irrelevant unless they depend on . Hence, the effects of processes such as in which a parton acts cooperatively with the last gluons which it radiated before reaching are accounted for by use of the physical fragmentation functions for the radiating parton. On the other hand, for example, the process leads to -dependent (``higher twist'') corrections. The probability for parton 2 in (8) to be emitted in a kinematic configuration which allows partons 3 and 4 to have an invariant mass decreases , or formally . The higher twist corrections (8) proportional to the joint fragmentation function are therefore roughly of order for hadrons of transverse momentum with respect to the direction of parton 1. The most important higher twist corrections plausibly result when parton 1 is the original or . In this case, the single hadron transverse momentum spectra receive corrections , yielding higher twist corrections to the development of single hadron energy spectra. (Numerically, the suppression may be more than compensated for by the smallness of relative to . Note that higher twist corrections to energy spectra should become more important for lower energy hadrons at a given .) The dominant, leading pole piece of the cross-section for the decay of parton 1 in (8) into 2, 3, 4 corresponds to independent emissions of 1, 2, 3, 4. As discussed in the previous section, there are ``subleading log'' correction terms, some of which depend on the process by which parton 1 was created (e.g., spin 1 or spin 0 ``'' decay): hence higher twist corrections will be no more universal to different processes than are subleading log corrections [18]. Note that in models (such as that discussed below) which describe the complete formation of hadrons from independent parton systems, higher twist terms are automatically present, and their nonuniversal nature accounted for by the form of corrections to leading pole parton decay probabilities. As discussed at the beginning of this section, the apparently plausible assumption is nevertheless made in such a model that processes such as (8) cannot interfere with processes in which the gluon 4 is absent.

In addition to single hadron inclusive spectra, fragmentation functions may also be used to describe multihadron spectra [12,13]. However, the different hadrons may arise either from the decays of separate partons, or as some of the decay products of a single parton. The possibility of r contribution requires introduction of new multiple fragmentation functions which may be determined only from experiment. This contribution may be removed by requiring that no hadron pairs considered have invariant masses , and therefore all must have originated from distinct partons. Nevertheless, the fragmentation function approach to multihadron spectra is increasingly affected by higher twist corrections, and is rendered largely impractical by the prohibitive number of parameters to be determined from experimental data. A more complete dynamical model for hadron formation is therefore required (which implicitly provides estimates for fragmentation functions).

The mechanisms for parton production discussed above were based on perturbation theory. However, it is possible, especially when , that partons may be produced by effects not visible in perturbation theory (e.g., ). (Note that such effects cannot be classified by a twist expansion: their coefficient functions as well as operator matrix elements would not be amenable to perturbative treatment.) In a uniform QED electric field (corresponding to a potential , pairs with transverse momenta with respect to the field direction are generated spontaneously at a rate [19] . An pair in the uniform electric field separated by a distance has a potential energy ; for , this potential energy exceeds the energy necessary to generate a real pair. The energetically favored state containing a real pair is reached by quantum mechanical tunneling, but at an exponentially small rate. The result for this rate exhibits an essential singularity at , and therefore cannot be obtained by any perturbation expansion about . The presence of real pair production eventually results in shielding, so that a uniform electric field cannot be maintained for an infinite time. In QCD, uniform color magnetic as well as electric fields are unstable (22) , because the gluon magnetic moment deviates from the Dirac value. The dominant higher-order perturbative corrections to the decay rate may be accounted for by the replacement of the coupling constant in the exponential by the effective coupling (appropriate for scattering of the pair from the potential). With this form, the rise of the pair production rate at small would be damped for .

The results for spontaneous nonperturbative pair production in uniform fields also hold for separating point charges in -dimensional QED or QCD (where [20], or in -dimensional QCD if as yet unknown effects concentrate color flux into a tube between the separating [19]. If instead, the color electric field is taken to have the perturbative dipole form resulting from a static pair separated by a distance a, then the potential difference between two points for ; again spontaneous pair production should occur, at a rate for (and vanishing with a higher power in the exponent for larger ). If the sources (and hence the field generated by them) were indeed static, then the only source of pairs would be such nonperturbative spontaneous production. However, in practice, the are accelerated at the decay point, and then move rapidly apart, generating a time-dependent field usefully parametrized by elementary gluon excitations and resulting in the perturbative pair production discussed extensively above. Note that pairs produced in the latter manner exhibit power-law, rather than exponential, damping in . Perturbative parton production will modify the field in which nonperturbative tunneling may occur. Typically, at late times, newly-emitted gluon pairs will provide sites for spontaneous production with small and thus high fields. Any spontaneously-produced gluons will be very closely collinear with the separating gluons; they would form a polarization cloud, which would bleach the color of the high momentum gluons, and reduce their reinteraction cross-section (23). Their effects will probably be important, however, only long after the free emission approximation has failed.

The spontaneous nonperturbative production of partons discussed above occurs by tunneling from a state containing just the field to a state containing, in addition, a parton pair, but having the same energy as the original state. The exponent in the tunneling probability is (minus) the action associated with the classical propagation of the partons through the field in imaginary time. As well as those which lead to additional parton production, there may also occur tunneling processes between identical states, which serve to alter the amplitude for the persistence (propagation) of the state. For example, the parton propagators receive corrections from processes in which tunneling occurs by way of an instanton solution to the (sourceless) classical field equations. Such corrections are presumably and therefore almost certainly irrelevant.

It is very difficult to make realistic estimates on the failure of the free parton emission approximation. Certainly the simple cutoff on the invariant mass of individual partons used above is an oversimplification: presumably invariant masses of pairs of partons are also involved. (Since small invariant mass parton pairs often arise from decay of a single small invariant mass parton, these prescriptions are at least roughly equivalent). In addition, the cutoff will not be sharp: the approximation will progressively become more inaccurate. (This behavior could perhaps be parametrized by choosing the value of for each parton from a distribution, rather than taking a fixed value.) Nevertheless, the mere and undoubtedly correct assumption that the parton invariant masses determine the validity of the approximation already has the important consequence that hadrons form after a time in the rest frame of the original and therefore at a distance in the rest frame. Hence, as suggested by Fig. 1, the longitudinal extension of the parton system before hadron formation increases with . The locus of points in the rest frame at which hadrons form is roughly a cylinder of length and breadth . If the high invariant mass were produced by a high interaction in a nucleus, then hadrons should form only far outside the nucleus, and therefore no additional secondary hadron production should occur in the nucleus. The similarity of high jet production from nucleons and nuclei observed experimentally [21] supports this picture. If instead, hadrons formed at a fixed time in the c.m.s., then different structure would be expected. In this case, hadron formation would occur roughly on the surface of a sphere (with ) in the rest frame. Moreover, this alternative (which violates the locality assumption necessary to justify consideration of partons) implies a cutoff on parton invariant masses which asymptotically yields no scaling violations in single hadron spectra.

The first corrections to the free emission approximation presumably arise within perturbation theory from the increasing importance of reinteractions. After many emissions, the density of partons in some regions of phase space will become so high that invariant masses of pairs of partons are often smaller than the invariant masses of other individual partons. In this case, the rate of exchanges between the partons will exceed the rate of radiation from a single parton, and the free emission approximation will fail. This effect occurs to some extent for a final state containing many or pairs in QED. At first, reinteractions result in energy loss through Bremsstrahlung; finally, when the invariant masses of the pairs fall below the masses of the Coulomb bound states, many of the charged particles combine into neutral atoms (c.f. recombination in a cooling plasma, e.g., in the early universe). (Note that the structure of a normal positronium or hydrogen atom depends crucially on the nonzero mass of the electron: when the ``atom'' becomes either infinitely extended (24) or generates pairs until its charge neutralization radius . It is not clear whether the nonzero current masses are crucial in the dynamics of hadron formation; their kinematic effects will be mentioned below: their importance may be gauged by differences between hadron systems produced in decay (from ) and decay at a given .) In QCD, the increase of the effective coupling at large distances presumably leads to reinteractions which ultimately collect all partons into color singlet bound states. The invariant mass below which such effects dominate is perhaps determined by the point at which (c.f. the critical charge for zero radius atoms in massless Q D mentioned above) and is therefore plausibly a few times .

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