Extensions

For simplicity, I have considered above only quarks with vanishing rest masses (all partons nevertheless receive effective masses from their finite propagation). Small quark masses introduce (or possibly mass corrections to the quark decay probabilities in eq. (1, 2)). (These are a species of higher twist corrections; the mass insertions responsible for them are analogous to the insertions of extra collinear gluons responsible for higher twist terms.) When , the quark decay probabilities are kinematically constrained to vanish. For the process ( denotes a heavy quark), a very good approximation to the exact differential cross-section is provided by using the usual for but setting for ( is, as always, interpreted as the fraction, thus accounting for Nachtmann scaling corrections). It seems likely that this prescription will also be satisfactory for multiple gluon emissions. Note that in the effective coupling , the available is now , rather than . For close to , the product in a ``decay'' will have small velocity relative to its parent, so that reinteractions will be important. Thus the free emission approximation in this case should fail when : the effective cutoff for emissions from heavy quarks should therefore be . For , the production of a pair from or should be suppressed relative to production by a factor where is the relative velocity of the outgoing . and should thus be reduced by this factor for , and set to zero below threshold (as decreases, the angular, or equivalently, distribution for the decay flattens slowly, becoming isotropic (i.e., flat) at threshold). Just above threshold, the produced have small relative velocity, and thus undergo extensive final state interactions. At , gluon exchange in gives . In higher orders, the cross-section exhibits isolated peaks at the positions of resonances below threshold for meson production, and further resonant peaks just above threshold. Note that such effects are probably less important for than for because in the former case, the cannot bind into a hadron because of their color. Nevertheless, in all cases, the peaks and valleys in the cross-sections should average out when smeared over a range around , and the lowest order result should suffice. In the prescription for forming color singlet parton systems described above, each gluon at is forcibly split into a light pair: this process, if it occurs at all, is undoubtedly not of perturbative origin, and one may guess that heavy production by it would be suppressed by . Hence secondary pairs should be generated only at the perturbative stage, and thus be rare.

Heavy quarks should eventually combine with light quarks and gluons to form color singlet clusters with masses . These clusters should then decay into mesons and light hadrons, with branching ratios determined approximately by available phase space (just as for light mesons, ``'' as well as ``D'' production should probably be included explicitly, thereby accounting correctly for decays: for very heavy , meson masses and branching ratios may presumably be estimated from potential models). The ground state (presumably pseudoscalar) mesons will then decay weakly (the lifetimes for these weak decays are much longer than the time necessary for the decaying meson to form, and they should therefore be treated separately). The decay may proceed either through separate decay of the heavy quark, or for purely hadronic modes, by exchange with the spectator . With the first mechanism, the is produced roughly uniformly (isotropic) in ; its subsequent decay to a fermion pair may be approximated as independent, and described by the distribution (for ) , with the various flavors of quarks and leptons weighted with mixing angles according to their appearance in the weak current. When , real production is permitted; the relevant decay probabilities are analogous to those for (or ) production. The second decay mechanism may be effective for purely hadronic modes; its relative importance may depend on the charge of the decaying meson. The final pair generated by this mechanism is isotropic in the rest frame: the various possible quark flavors are weighted by the requisite mixing angles. Other decay modes, such as or probably have very small branching ratios. The partons emitted in decays may be off-shell, and thus radiate, producing hadrons as in decay. Note that heavy lepton decays may be treated analogously to the first mechanism for decays.

Below threshold for production, resonances (e.g., , ; denoted generically ) should be produced in decay. The lightest such resonance presumably decays mainly to , , or ; radiation from these partons may be treated as described above. (Note that the gluons in the decay are distributed almost uniformly in the available phase space.) Excited ) mesons may decay either to lower-lying states, or directly to lighter partons; the branching ratios may be estimated from potential models. For , a cutoff permits almost no radiation from the produced, and usually combines them into just one hadron cluster; this then decays identically to the single hadron cluster produced by decay with . Of course, in the model described here, both quark and gluon ``fragmentation functions'' are completely determined.

The hadron clusters discussed in the previous section are taken to decay into light meson pairs according to available phase space. For clusters with masses above , baryon pair production is also possible. To be consistent with other assumptions, no suppression of baryon pair production beyond phase space restrictions should be introduced. Then, at asymptotic , should tend slowly to a constant determined by .

In addition to gluons, virtual quarks produced in decay may also radiate real or virtual photons, and, if they have sufficiently large masses, real or virtual , , and perhaps Higgs () particles. The probabilities for photon emission are just as for gluon emission (after the replacement ). Direct photon production in the decays of the hadron clusters may probably be ignored. Just as outgoing or incoming quarks may emit gluons, so also incoming may emit photons. Most of the resulting electromagnetic radiative corrections may be treated by the direct Monte Carlo methods discussed above and in [2].

In the discussion above, the polarizations of quarks and gluons have not been traced explicitly. It is simple to include spin-dependent decay probabilities [4], but the likely presence of orbital angular momentum in cluster formation makes deduction of final hadron polarizations difficult.

In this paper, I have concentrated on hadron production in annihilation. Processes which involve partons in the initial state may be treated by largely analogous methods. The valence quarks in incoming hadrons may plausibly be taken to be distributed in the hadron rest frames, for example, Gaussian (c.f. nonrelativistic harmonic oscillator wavefunctions) in all four momentum components so as to have a mean invariant mass . (There is no difficulty in boosting this momentum probability distribution.) Prior to interactions involving high invariant masses (e.g., absorption of a highly spacelike virtual photon (deep inelastic scattering), or production of a highly timelike (Drell-Yan process)), these nearly on-shell partons may radiate (small timelike invariant mass) partons, and themselves attain progressively more spacelike invariant masses (up to ). The probabilities for these decays are, in the free independent emission approximation, essentially the same as those for the decays of timelike invariant mass partons considered above, with suitable reinterpretation of . (The optimal argument of becomes rather than , as in the timelike case; hence contributions to the decay probabilities in the two cases differ by terms.) The emitted gluons and ``sea quarks'' may be considered to provide extra constituents of the incoming hadrons: their momentum distributions will as usual ``evolve'' as increases. Note that with this model, the ``constituent'' partons will exhibit a distribution of transverse momenta with respect to the incoming hadron direction. After the hard scattering, remaining partons may be off-shell with timelike invariant masses up to , and may radiate just as in decay. Partons from the initial hadrons which do not participate in the hard scattering should preserve their original momenta (roughly collinear with the incoming hadrons) until they are combined with other partons in the formation of color singlet hadron clusters. The model outlined here should allow discussion of all high-transverse momentum hadron processes [29]. Comparison with results from triggered experiments will, however, require nontrivial importance sampling in theoretical Monte Carlo calculations. One may speculate that the methods used to describe high transverse momentum scatterings between incoming partons could also be applied to low transverse momentum multiparticle production processes. The incoming valence partons would exchange a small momentum, with a cross-section given by one-gluon exchange, but with its Coulomb singularity at small regularized for , by the presence of additional partons from the incoming hadrons, which shield the color of the valence quarks at distances . This small momentum transfer, say , causes the incoming high-energy (but nearly on-shell) partons to be ``poked'' off shell to invariant masses , where is the c.m.s. energy in the parton collision. These off-shell partons would then radiate just like partons with the same invariant masses produced in decay. At sufficiently high , the transverse momentum distribution of the final hadrons should therefore broaden.

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