The Structure of Parton Final States

By using the leading pole approximation discussed above, and assuming that higher order corrections may alter the overall normalization but not the kinematic structure of the ``decay'' probabilities , one may trace the development of parton final states in annihilation until the invariant masses of the partons are degraded below the critical mass . As mentioned above, the use of Monte Carlo methods [2] allows exact account of kinematic constraints to be taken, yielding in many cases important corrections to results found by ``asymptotic'' analytical techniques (e.g., [12,1]).

Figure 2 shows the mean total multiplicity of partons produced before the cutoff as a function of , with taken as 0.5 GeV. For smaller , more partons are radiated before the evolution is curtailed. Note that the detailed quantitative behavior of the parton multiplicity is somewhat sensitive to the details of the imposition of the cut (15) qualitative features are, however, entirely insensitive. Nearly all the partons are gluons: the curve for mean multiplicity with given in Fig. 2 indicates that in this case, an average of one secondary quark per event is achieved only at . (Note that only three quark flavors are considered, since only these may be excited by the momentum transfers which dominate the development of the parton final state.) The results in Fig. 2 approach slowly the asymptotic form [11,12,1] when . at asymptotic should lag only by a power of : at accessible energies, however, the suppression is numerically large. In QED, the asymptotic multiplicity distributions are of the Poisson form (so that ): a sequence of photons is emitted independently from an off-shell electron line, typically with energies much lower than the electron energy so that kinematic correlations are absent. In QCD, however, each emission changes the color of the radiating parton, destroying the independence. Moreover, the much larger total multiplicity is dominated by radiation of gluons from low energy gluons: kinematic correlations are therefore significant, and the multiplicity distributions deviate from the Poisson form (asymptotically [12,14], with , and the constant depending on the cutoff prescription. The dispersion of the distribution remains roughly constant over the range shown in Fig. 2 (with a value for ), rather than decreasing as for a Poisson distribution.

[ Figure 2 ] Mean multiplicity of partons produced in the decay of a virtual photon with invariant mass by radiation from virtual partons with invariant masses The dashed line gives the multiplicity of quarks and antiquarks. The parton production cross-sections were estimated by a leading pole approximation, with .

For , no gluons are emitted, so that the quark energy distribution is simply , where . As increases, progressively more gluons are emitted, and the quark energy spectra soften. Figure 3 shows the mean fractional energy carried by gluons as a function of . The percentage of events in which no gluons are emitted above the critical invariant mass is also marked, and decreases rapidly with increasing . The standard leading log approximation often used to estimate the evolution of moments of the distributions is obtained from the leading pole approximation (1, 2) by making the kinematic approximation that all emitted partons are collinear with their parents, but nevertheless formally allowing the invariant mass of each emitted parton to run up to the mass of its parent. The leading log approximation for the evolution of a quark energy spectrum at up to gives

this form is compared with the complete result (for ) in Fig. 3. The effects of the kinematic approximations fall off only rather slowly with , mainly because of the importance of multigluon emissions, in which each gluon has only a small fraction of the total available energy . At asymptotically large , tends logarithmically from below to the ``equilibrium'' value ; the final limit is independent of or but is approached more rapidly for smaller . For most of the curves in Fig. 3, the in the decay probability (1) was approximated by , with . However, as discussed above, the choice should account for subleading log higher order corrections (with a momentum-independent subtraction scheme used for renormalization): the result for with this form shown in Fig. 3 suggests that these corrections are quite insignificant. Changes in the overall normalization of due to higher order corrections alter the in roughly the same manner as changes in .

[ Figure 3 ] Mean fraction of total energy in decay of a virtual photon with mass carried by gluons. The production of partons by leading pole approximation cross-sections has been truncated when their invariant masses fall below the cutoff . The percentage of events in which no gluons were emitted above this cutoff is marked. Results obtained by using approximate collinear kinematics (the usual ``leading log approximation'') are also shown. The consequences of a modification of the effective coupling constant as for parton ``decays'' discussed in the text are shown.

Each off-shell parton ``decay'' imparts a relative transverse momentum between its products. If the transverse momentum distributions in the individual ``decays'' had finite variance, then the central limit theorem implies that the resulting total distribution should be roughly Gaussian. In fact, the power law distributions in each decay implied by the factor in the decay probability give rise to a power law tail in the average single parton distributions measured with respect to the initial directions, as illustrated in Fig. 4. (This is mathematically analogous to the tail of the (Molière) multiple Coulomb scattering transverse momentum distribution for electrons traversing matter.) The results in Fig. 4 are all for : a larger removes small partons, but does not affect ``hard'' partons with and thus increases the .

[ Figure 4 ] Transverse momentum distributions for single partons produced in the decay of a virtual photon with mass with respect to the primary pair direction. Parton emissions were truncated by the cut .

As discussed above, partons emitted at early times typically have large transverse momenta with respect to their parents, but because of the form of the decay probabilities (1, 2), partons emitted later are progressively more collinear with their parents. This ordering leads at sufficiently high to considerable clustering in the angular distribution of energy for the parton final state (16). Figure 5 shows the distribution of partons in the northern hemisphere of a reasonably typical simulated event at , with , . (The original quark was directed towards the north pole. The event displayed in Fig. 5 is somewhat more isotropic than the average.) The angular clumping of the partons is evident. Observables which are sensitive to the angular distribution of energy in the final state only at large angular scales should probe only the total momenta of the clumps, and be insensitive to the precise distribution of the momenta between their individual constituent partons. (17) Since typically the angle between a pair of partons produced by the decay of a parent with invariant mass is (where is the fraction of the original energy carried by the parent), observables which probe the final state energy distribution in angular bins of width should be sensitive only to decays with . In as far as the final formation of hadrons affects only invariant masses , so the distribution of hadronic energy over angular scales should reflect the structure of the parton system. A convenient set of observables for measuring the final state angular energy distribution is given by (18) [15]

where the sum runs over all final particles, including the case . For a final state consisting of just two massless particles (e.g., ), , while for an isotropic final state, . The are the coefficients in a Legendre expansion of the energy correlation [15,17] between two point detectors as a function of their relative angle. For high , the Legendre polynomials may roughly be approximated by , so that the lump together systems of partons subtending angles , and probe the evolution of the final state only at . Figure 6 shows as a function of for various with . At low , no emissions are possible above and . As increases, the effects of the cutoff decrease (this linear behavior reflects the linearity of in final particle momenta necessary to ensure infrared finiteness). At high , approaches the result (19) obtained by treating only the first emission in perturbation theory. (As discussed above, however, higher order terms may modify the normalization of .) A small deviation remains even at high . Note that reaches its asymptotic form at much lower than did , mainly because multigluon effects are suppressed by rather than . The distributions for at various are given in Ref. 2. The divergence at exponentiates to provide ``radiation damping'' at when higher order emissions are summed. Figure 6 also shows the behavior of when , and the form of at in perturbation theory. is more sensitive to later emissions than . Typically, at large , the effects of successive emissions on decrease as , while the cutoff provides corrections. As .

[ Figure 5 ] Distribution of partons in the northern hemisphere of a reasonably typical decay final state with , . The sizes of the dots are roughly proportional to the energies of the partons they represent. The original quark produced in the decay was directed towards the north pole; its final position is marked with a cross. All other partons are gluons. The lines drawn between the parton directions describe the parton color indices. The event shown is somewhat more isotropic than the average; the values of some shape parameters for it are given.

[ Figure 6 ] Mean values of the shape parameters and (defined by eq. 6) for parton final states from the decay of a virtual photon with mass , and with cutoff . Results obtained by considering only a single gluon emission at with are also shown.

The emission of a single gluon deflects energy in an event from the line. Emission of two or more gluons spreads energy outside a plane. The deviation of a final state from coplanarity may be measured, for example, by [15] . (20)

If the quarks produced in the decay were exactly on-shell and massless, then their angular distribution with respect to the original direction would be . When the quarks are off-shell, their original angular distribution becomes instead roughly ; the gluons radiated remain roughly uniform in azimuth with respect to the direction [15].

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