The Stages of an Annihilation Event

In QCD perturbation theory, an annihilation event is initiated by the decay of the virtual photon () into a quark and an antiquark. If the QCD coupling constant were zero, then the would propagate freely to infinity, and would therefore be produced on their mass shells. In fact, the need propagate only for a finite time before interacting or radiating, and may therefore be produced with a distribution of invariant masses usually peaked at low values, but with a power-law tail extending up to the kinematic limit imposed by the mass of the . Large initial quark invariant masses should be dissipated predominantly by radiation of gluons: the outgoing should emit gluons at a rate decreasing roughly inversely with (proper) time, thereby converting their invariant masses into transverse momenta of the produced gluons, and spreading their energy and color into a cone of finite aperture. The emitted gluons may also have invariant masses up to those of their parent quarks, and hence may themselves radiate more gluons (and occasionally, another pair), generating a cascade or shower of partons, as illustrated in Fig. 1. However, even in perturbation theory, such free emissions cannot continue unchecked forever: as the invariant masses of the partons become small, back reactions in which emitted partons reinteract with their parents or other ambient partons should become increasingly important. Eventually, these reactions, together, perhaps, with qualitatively new phenomena not visible in perturbation theory, should cause the system of partons to condense into color singlet hadrons. The magnitude of the critical invariant mass , below which free perturbative emissions no longer dominate is presumably determined by the masses of hadrons, and by the renormalization group invariant mass (which gives the position of the infrared Landau divergence in the leading log effective coupling constant . Phenomenological comparisons (mentioned below) suggest that is probably of order 1-2 GeV, and give some hints on the transformation of a system of partons below this critical mass into hadrons. As the mass Q of the original (c.m.s. energy in the collision) is increased, the extent of the period during which free perturbative emissions dominate should correspondingly increase: at sufficiently high energies phenomena occurring below should become irrelevant. However, it will turn out that for most purposes, the residual effects decrease slower than , and are by no means negligible compared, for example, with hard gluon emissions at presently available energies .

[ Figure 1 ] Spacetime development of typical parton showers initiated by decay of a virtual photon with invariant mass , traced until each parton has invariant mass below the critical , and with , generated using the Monte Carlo computer program of [2].

A parton off-shell by an amount will typically survive without radiating (``decaying'') for a proper time (1). The transverse momentum between its ``decay'' products is kinematically bounded by . At very early times, gluons may be emitted with large transverse momenta , leading to clearly separated additional parton ``jets''. As discussed in the next section, the partons produced in each ``decay'' tend to have much smaller invariant masses than their parents, and thus tend to survive for much longer times before decaying themselves. Partons emitted at later times must therefore be progressively much more closely collinear with their parent partons, and their existence should thus affect the final angular distribution of energy over smaller and smaller regions.

In so far as the partons emitted in each decay tend to have much smaller masses than the decaying parton, their energies may remain of the same order as the energy (mass) of the decaying parton. Typically, the energies of partons emitted by the decays of partons with invariant masses decrease only logarithmically with (c.f. ``scaling violations'' in ); their average wavelength therefore remains for a long time . On the other hand, the distance traveled by the decaying partons . Hence, the distance between successive emissions soon typically becomes much larger than the wavelengths of the decaying or emitted partons: thus the amplitudes for successive emissions should not interfere appreciably. Hence, so long as the energy of an emitted parton is sufficiently large, but its invariant mass is not too close to the mass of its parent, its decay should be independent of its production, and the spectrum of its decay products well described by an independent classical probability distribution. This point is crucial in simplifying the discussion on the development of parton final states below. Any gluons with must be emitted very soon after the decay and must arise from very short-lived virtual quarks, typically with . The wavelengths of these virtual quarks (and the gluons they radiate) are therefore no longer than their decay paths: interference between amplitudes for successive gluon emissions, or radiations from and , could thus be important. Explicit calculation of the first gluon emission to in perturbation theory suggests, however, that such effects are numerically insignificant. On the other hand, at very large times, quantum mechanical interferences are undoubtedly crucial: the organization of the final state into color singlet hadrons with definite masses may be viewed as resulting from destructive interference between the amplitudes for producing illegal final states.

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