The Leading Pole Approximation

In the classical approximation discussed above, the development of a shower (or jet) of partons is described by a sequence of independent ``decays'' of off-shell partons into partons with smaller invariant masses. To leading order in , it is sufficient to consider only (quasi-)twobody ``decays''. Then the probability for a parton of type to have an invariant mass squared in the range and to ``decay'' into partons of types , carrying (roughly) fractions and of its longitudinal momentum is given by

if the invariant masses of the final partons are sufficiently small that the ``decay'' is kinematically allowed, and otherwise. The (Altarelli-Parisi [4]) distributions for the various possible decays are (2)

where the color ``charges'' are given by , . The probability for each decay is uniform in azimuthal angle. The which appears as the argument of the effective coupling in (1) is : its exact value will be discussed below. As explained below, the in (1, 2) is identified as the fraction of each daughter parton with respect to its parent (as measured in the rest frame (3) ). In the rest frame of the decaying parton, is roughly the cosine of the angle between the spin direction of the decaying parton and the (oppositely-directed) momenta of its decay products: the parton decays are not isotropic. (The spin of the decaying parton depends on its momentum with respect to its parent, and ultimately with respect to the original .)

The differential cross-sections for multiparton production obtained by suitable application of the probabilities (1, 2) to all possible decay chains could, in principle, also be found by explicit evaluation of all the contributing high order Feynman diagrams. The results of the latter exact (but intractably complicated beyond ) approach must agree with the approximation, at least to leading order in for each emission. However, the probabilistic interpretation of the resulting differential cross-section as a sequence of independent ``decays'' will not in general be manifest: individual diagrams apparently involving interference between different decay chains may appear to be important. To establish the probabilistic interpretation directly from individual diagrams, one must use particular gauges, in which the gluon spin is explicitly constrained to be orthogonal to its momentum, at least when the gluon approaches its mass shell, thereby preventing propagation of unphysical gluon polarization states. This is achieved in an axial gauge, for which the gluon polarization tensor is , where is a fixed four-vector, to which the gluon spin is approximately orthogonal. Then the decay probabilities may be calculated from explicit diagrams involving ``incoming'' off-shell partons (but, in the leading pole approximation, with on-shell outgoing partons): interference diagrams are explicitly relegated to nonleading order in . (Corrections to the leading pole approximation probe details of the off-shell extrapolation; in calculating these, one must consider explicitly the ``decay'' by which the off-shell parton was generated.) Different choices for the gauge vector share the leading pole contribution differently among the various radiating partons: for example, if is chosen nearly along the momentum of some parton, then the diagrams involving radiation from that parton will give no leading pole contribution (4). To obtain directly the probabilities (1, 2), one must choose away from the momenta of radiating partons: the appearing will then be fractions evaluated in the rest frame of . Of course, the sum of all diagrams regardless of gauge reproduces leading pole results obtained by applying the probabilities (1, 2) for radiation from all possible partons.

As discussed above, the ``leading pole approximation'' (1, 2) undoubtedly becomes inadequate when : the later development must be treated by other means. This limitation also affects the distributions in , which become inaccurate when approaches 0 or 1 too closely. Consider the decay of a parton 0 into two partons, 1 and 2. For the of the final partons to be maximal, they must have zero invariant mass. In this case, parton 1 has . By energy conservation, , and hence , where is the fraction for parton 0 with respect to the original momentum. Since we require , the minimum `` loss'' for (1, 2) to remain accurate is : the soft divergences in for gluon emission are always avoided.

The classical and iterative nature of parton evolution for makes this phase of jet development eminently suited to investigation by Monte Carlo methods. The parton showers shown in Fig. 1 were generated using a Monte Carlo computer program [2], taking , , . Virtual partons were drawn to travel for a proper time before radiating. Note that all parton trajectories are curved by the semilogarithmic scales-used to display the events.

The parton showers in Fig. 1 resulting from the ``decay'' of a high invariant mass parton are in many respects analogous to electromagnetic showers, initiated by the entry of a high-energy electron or photon into matter (5). In the latter case, the initial particle is on its mass shell, but is repeatedly ``poked'' off shell by interactions with nuclei and generates a shower by successive Bremsstrahlung radiation and pair production. Eventually, when the energies of produced fall below some fixed critical value, interactions with ambient atomic electrons (ionization losses) become important, so that free radiation, as described by probabilities analogous to (1, 2), no longer dominates. The radiation from an accelerated, and hence off shell, electron may be found directly by solution of classical electrodynamics equations with suitable boundary conditions. The explicit simulation of photon emissions may be considered as a Monte Carlo solution of these equations. For the QCD case, however, the classical equations are much less tractable, and Monte Carlo methods become almost obligatory. The primary reason for this is that gluon decays cause parton showers to have a much more dendritic structure than their QED counterparts: this extra cascading means, for example, that the gluon potential at a point samples sources not only on the surface, but also throughout the volume of its past light cone (hence a ``pulse'' of gluon radiation will become dispersed, even propagating through the vacuum).

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