Main Text

Quarks and gluons generated by QCD processes involving large momentum transfers initially have large invariant masses ; as they travel outwards from their production point, they radiate other partons, which in turn radiate, until each has a small invariant mass and the original has been converted into transverse momenta between partons in the jet. During most of this cascading process, one may neglect interference between the amplitudes for successive emissions: the spectra of radiated partons are adequately described by independent probability distributions, and the evolution of the jet may conveniently be simulated by an iterative Monte Carlo method. The model described below uses this technique to generate parton jets which exhibit essentially all known features of QCD final states, and yields direct and complete QCD predictions for suitably inclusive measurements at sufficiently high energies. To obtain further details, one must venture beyond perturbative QCD, and adopt a model for the formation of hadrons from parton jets. A simple (but phenomenologically accurate) model will be presented in a forthcoming paper [1], and provides complete predictions for hadronic final states (giving multijet events from large angle emissions, and scaling violations in jet energy spectra from multiple small angle emissions). We consider here primarily annihilation, although our methods may be applied to any other processes, including those with partons in the initial state (e.g., deep inelastic scattering or hadroproduction; high-energy collisions on hadron or nuclear targets involving only small momentum transfers may perhaps also be treated).

Fig. 1 illustrates the spacetime development of typical annihilation events at c.m. energies and generated by our Monte Carlo method. Any emissions at distances (e.g., the first gluon radiated in the upper jet of fig. 1a) are typically at large angles to the off-shell , so that they generate separated parton jets (fig. 1a would probably be classified as a ``three-jet event''): the probability for such emissions is and thus small. (The wavelengths of any large angle radiations in one jet typically encompass the other jet, so that interferences may occur; to account for this [in practice small] effect, we use the complete explicit differential cross section for to describe the first gluon emission in an annihilation event.) As discussed in refs. [2,3,4], the rate of radiation from an off-shell parton at distances decreases roughly inversely with time (1) (so that emissions are very roughly equally spaced on the logarithmic longitudinal distance scale of fig. 1), apart from a logarithmic rise from the effective coupling . (2) Here is the invariant mass of the radiating parton and , with the number of light quark flavors. The partons are typically emitted at progressively smaller angles: their energies remain , but the typical c.m. distance between emissions approaches , so that interferences are small, and the radiations may be treated independently and iteratively.

[ Figure 1 ] A projection of two typical events evolved to a cutoff mass GeV. Note that the semi-logarithmic scale causes both a suppressed origin and the curved paths for the evolution of both free and virtual partons.

Perturbation theory suggests that an off-shell parton should always continue to emit progressively softer and more collinear radiation. However, an actual parton propagates only for a finite time before confinement acts and its jet condenses into hadrons: the perturbative evolution treated by our model is curtailed by hadronization when the parton invariant masses fall below a critical value , probably a few times (3). (The development in fig. 1 has been cut off with : the parton jets therefore extend a longitudinal c.m.s. distance , where is the mean ``final'' parton momentum.) With some models for hadronization, earlier emissions (with ) may nevertheless be too soft or collinear to affect the structure of the hadron final state. Only emissions which will lead to distinguishable hadron states need be included in the Monte Carlo simulation , the evolution of coarse-grained entropy in statistical mechanics). The cuts on elementary parton processes which censor irrelevant emissions depend in detail on the model for hadronization: we discuss several procedures below. Suitably coarse measurements on the final state (which, for example, consider only its energy distribution lumped into large angular bins) are, however, unaffected by the cuts (this insensitivity is the essential characteristic of an ``infrared-stable'' observable).

The development of a QCD jet from a large invariant-mass quark or gluon by a cascade of independent emissions is analogous to the formation of an electromagnetic shower in matter from a high energy electron or photon. An electromagnetic shower develops by successive independent bremsstrahlung and pair production until the and produced fall below some critical energy (governed by inverse atomic sizes) when ionization losses dominate, and the cascade is absorbed (e.g., producing scintillation light). The longitudinal development of QED showers is conventionally treated by moments or master equations [5] just as for their QCD counterparts; the transverse profile of a QED shower is, however, dominated by multiple Coulomb scattering, which has no QCD analogue (4).

The leading logarithm approximation (LLA) of independent small angle emissions is best investigated using axial gauges for the gluon propagator (so that gluon polarization vectors are orthogonal to the fixed vector ). In these gauges, the dominant Feynman diagrams have a simple iterative ladder structure (the squared amplitudes correspond to planar diagrams) as illustrated in fig. 2a: the total probability is to LLA a product (with suitable kinematic convolutions) of the probabilities for each individual emission. In axial gauges, the LLA probability for a parton of type with invariant mass to propagate from its production and ``decay'' into approximately massless partons of type and carrying (roughly) fractions, and of its longitudinal momentum may be calculated directly (e.g., from the diagram of fig. 2b), yielding

where ( labels the flavors of quarks, which are for now taken effectively massless) (5)

[ Figure 2 ] A typical diagram contributing to the LLA in an axial gauge. (b) The diagram corresponding to eq. (2a). (c) The section of a parton shower discussed in the text.

The choice of determines the precise interpretation of : different choices extract probabilities (1) which typically differ by subleading log terms rather than , where is the invariant mass of the preceding decay [ for the first emission]). Taking along the direction of one quark in annihilation is convenient in analytical calculations since it causes all radiation to originate from the other quark. This choice identifies as the relative longitudinal Sudakov variable, or fraction of each emitted parton with respect to its parent (which travels along the 3-axis). For the Monte Carlo model, we choose symmetrically (e.g., along the direction) yielding equal radiation from each quark, and identifying as the fraction (6). In the ``decay'' shown in fig. 2b, therefore, . (7) As mentioned above, avoiding ultimately irrelevant emissions may give tighter constraints on (e.g., if ``final'' partons are to have , then always).

In addition to collinear divergences, the probabilities (1) exhibit further divergences when emitted gluons become soft. As usual, such soft divergences are canceled by virtual processes (for which the discontinuity in, e.g., fig. 2b is taken through rather than a real final state), which add to a divergent term such as to give finite results for smoothly weighted integrals over . The running coupling constant in (1) arises in axial gauges from summing the LLA effects of higher-order corrections on each external parton leg. For example, the corrections to the process in fig. 2b arise from the sum of vacuum polarization insertions on the outgoing gluon leg and diagrams in which the produced gluon subsequently decays into two real gluons [or a pair]. When integrated over the available phase space, the infrared divergences cancel between the real and virtual diagrams, but the upper limit on the invariant mass of the real pair prevents complete cancellation, and introduces a term multiplied by the lowest-order cross section. In higher orders, the gluon undergoes self-energy corrections before decaying. The sum of such terms is accounted for by use of an effective in the lowest order form. In our case, , but numerical results are close to those with .

We now describe the iterative procedure for generating LLA parton showers, which has been implemented in a FORTRAN computer program (available from us). For simplicity, we here first assume that the only resolvable emissions are those which pass a cut much more stringent than any kinematical constraint (8); softer emissions serve only to degrade the parent invariant mass, and not to initiate independent jets. We consider a section of shower development illustrated in fig. 2c. The probability that the parton (which has maximum invariant mass ) should evolve until it has ) (at which mass we terminate its radiation) emitting only unresolvably soft ( or ) partons is given to LLA by a formula directly analogous to the structure function moment evolution equation for deep inelastic scattering:

where we have used

the crucial arises from the nesting of the integrations. The effects of virtual diagrams have been included in (3a) by using (3c), which simply states that the total probability for all possible decays of i is one (to LLA). The first step in our Monte Carlo procedure is to determine whether will emit resolvable radiation: if so, its is chosen according to the distribution (1) modified by possible subsequent emissions (the probability for these is given by (3a) with replaced by ): the resultant distribution is given by

which satisfies

This step is achieved by generating a random number uniformly between 0 and 1, and then solving for in

if the resulting , emits no resolvable radiation. Having determined that is to radiate, and selected its , the type and momenta of its ``decay'' products and are chosen according to (2); their fractions satisfy , and their momenta are distributed uniformly in azimuth. and are then evolved as was , and the cascade continues until all partons have chosen to generate no further resolvable radiation with .

The practical implementation of this procedure involves several complications. First, the actual range of for visible emissions depends on , and is not fixed, as assumed above. This is accounted for by initially choosing in the procedure above to be the minimum over . Then if the generated lie outside the kinematic region, the procedure is repeated, but with replaced by the first generated [thus correcting the form (3)]. In the LLA, the products of each parton ``decay'' are formally approximated as massless (since strictly the in LLA are strongly ordered ). Although the implied by the LLA are usually small, one must, in practice, account for the effects of on the kinematic regions for emissions; the LLA provides no guidance as to the correct procedure (the problem does not yet exist for at : to investigate it, one must evaluate in an axial gauge, and it is quite possible that the optimal prescription may differ in higher orders and other processes). Here we use an ad hoc but physically reasonable prescription: results which are sensitive to the details of the procedure cannot in any case be trusted. We first assign , and then allow to run only up to the corresponding kinematic limits ). (Another, but less satisfactory, procedure would be to take , but to reassign if the generated are kinematically forbidden.)

The parton showers in fig. 1 were generated according to the scheme described above (with ); virtual partons were taken to survive for a time , while ``final'' partons were drawn until the point at which hadrons form in the model of ref. [1].

Fig. 3 shows the distribution of parton final states from annihilation generated according to our procedure in the shape parameter , where [8] (the sum runs over all pairs of final partons, including )

For two parton events, , while for isotropic events, . The curves of fig. 3 are insensitive to the value of chosen (as expected from the infrared stability of ), except very close to . The main effect of including multiple gluon emissions on the distribution is to soften the unphysical singularity obtained at . Replacement of the exact cross section for the first gluon emission by the LLA makes an indiscernible change to the distribution (and thus inspires hope as to the accuracy of higher-order LLA). The obtained from the distributions of fig. 3 are all very close to the result . Results for the thrust observable [9] will be very similar to those obtained for .

[ Figure 3 ] The distributions calculated by our Monte Carlo method for cutoff mass . and various energies . Also shown is the result for .

Three-parton final states must have : to obtain smaller requires emission of at least two gluons. A direct measure of such four-parton final states is afforded by deviations from coplanarity, which give [10]

( for an isotropic event). For continuum annihilation , while on a heavy resonance decaying to three gluon jets, .

A standard application of the LLA gives the evolution of parton energy spectra ( distributions) in jets as a function of . At , the quark energy spectrum is simply . As increases, progressively more emissions occur before , and the quark energy spectrum is correspondingly softened. The usual results for these ``scaling violations'' are recovered in our model by constraining all partons to be collinear; the use of realistic kinematics is most important for the first few emissions, and typically changes by . The extensive LLA results for multiparton spectra derived in ref. [14] are in most cases significantly modified (at accessible ) by inclusion of exact kinematics.

To make complete predictions for observed jets, one must treat the formation of hadrons from parton showers at large times. The details of this process are yet unknown, but there are some theoretical and many phenomenological indications that it should be universal: sets of partons with particular quantum numbers and masses should fragment into hadrons in a manner independent of the process by which they were made. If the perturbative evolution of a parton shower is curtailed at comparatively short distances by a large cut (say, ), then each ``final'' quark should presumably form a jet of hadrons roughly like those in annihilation events at (and analogously for gluons). For our model to be consistent, the final results for hadronic events must be independent of the value of chosen (so long as it is neither too close to the original nor to . (This behavior is familiar from LLA investigations of single hadron distributions, where is the ``reference mass'' .) In attempting a less phenomenological approach to hadron production, it is interesting to reduce the value of . Although perturbation theory cannot precisely describe the final stages of hadron formation, some properties of the parton system which it prepares may be relevant for hadronization. In particular, one may guess that sufficiently localized color singlet systems of partons should evolve and form hadrons independently [11] from the rest of the final state. A convenient method for identifying such systems is to associate a ``string'' with each spinor color index (hence two strings per gluon). Then the final state may be specified as a collection of two-ended strings representing color singlet systems of partons. Fig. 4 shows the distributions of invariant masses of such strings in parton final states from annihilation, together with the invariant-mass distributions for the nearest (but not necessarily color singlet) pairs of partons in the final state. The mass distributions are essentially independent of the original , and are power-law damped above , so that most strings have . (The distributions are, of course, sensitive to infrared cutoffs: other simply scale them, but other forms of cut lead to different damping powers.) If is a few GeV an attractively simple model for hadron formation would be to take each string to condense into hadrons isotropically in its c.m.s., with the final hadrons distributed according to the available phase-space volumes. In this model, the mean final hadron multiplicity is proportional to the string multiplicity: using for the string decay parameters those determined from low-energy multiparticle production (cf., ref. [12]) gives total multiplicities agreeing with annihilation measurements over the range to within about 20%. Fig. 5 shows the transverse momentum distributions for strings produced at various . The grows only slowly with but the tails on the distributions increase rapidly. The distributions are sensitive to details of cuts only for . The results for partons track those for strings at large with a constant difference in overall normalization. Detailed predictions of this model for hadron formation will be given in ref. [1].

[ Figure 4 ] The mass distributions for both color singlet strings and parton pairs from annihilation at and 200 GeV.

[ Figure 5 ] The transverse momentum distributions calculated by our Monte Carlo method for color singlet strings as a function of energy .

Finally, we discuss some extensions of our model. We have considered only emissions from outgoing partons: radiation from incoming partons approaching a collision may also be treated by our Monte Carlo methods. In that case, a radiating parton begins nearly on-shell, but its invariant mass becomes progressively more spacelike because of emissions. To leading order, the energy of the incoming parton is shared by emissions just as for outgoing partons in eqs. (1), (2).

We have assumed that all partons are unpolarized: polarizations of quarks and gluons may be treated by using spin-dependent kernels in (2). We have considered only light quarks, with effective masses . Heavy quarks (with masses ) may be included by using massless kernels (1), (2) but allowing the quarks to evolve only until (it is the mass of the heavy quarks, rather than finite propagation time, which regularizes small infrared divergences).

Emission of hard may be included by adding suitable kernels to (2); collinear divergences in emissions from quarks are cut off by hadron masses, so that (a larger would require inclusion of collinear in fragmentation functions), whereas for emission from leptons, . Many QED radiative corrections to cross sections may be obtained by using a LLA (equivalent approximation) for soft photon emissions from leptons and hadrons: our Monte Carlo method may also be used directly to simulate both QED and QCD corrections, in, for example, deep inelastic scattering. In this way, one may clarify the effects of QED radiative corrections on scaling violations, previously treated in a less satisfactory manner (e.g., [13]).

We are grateful to R.D. Field for discussions.

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