4. Results for Idealized Jets

In this section, we present our results on the cross sections and moments for the production of free quarks and gluons by a virtual photon. Sect. 5 discusses the results obtained using a realistic model for the fragmentation of the quarks and gluons into hadrons.

4.1. (G)

We consider first the cross section for the process G, which proceeds (in lowest order) through the diagrams of fig. 18. Defining (see the appendix)

where is the total c.m. energy (the virtual photon mass), and taking a finite gluon mass as a regularizer, one finds that the differential cross section for this process (summed over the colors of the final particles) is given by (8)

Here is the lowest-order cross section for the process shown in fig. 19, and is the QCD coupling constant.

[ Figure 8 ] The lowest-order diagrams for the process G.

[ Figure 9 ] The lowest-order diagram for the process .

The cross section for at order resulting from the interference of the diagrams of fig. 20 with the lowest-order diagram (fig. 19.) is [2]

(Note that each `wave-function renormalization' diagram occurs only once in the product of figs. 19 and 20.)

[ Figure 10 ] Contributions to the process at order .

Integrating the differential cross sections (4.2) and (4.3) over the final particles' phase space, one finds that the total cross section for annihilation into free quarks and gluons to order is finite and independent of . It is given by [26]

In the case of this total cross section, there are good arguments based on the renormalization group [11] which suggest that the relevant coupling constant in (4.4) is given by (9)

(We take and , but our results are not sensitive to these choices.) We assume that the appearing in the differential cross sections (4.2) and (4.3) should also be the given in eq. (4.5).

4.2. Decays of Heavy Quark Resonances

It seems reasonable to guess that the decay of a heavy resonance with into hadrons will be initiated by its decay to three gluons, as in fig. 21 [4]. We assume that the quarks and behave as if free at the time of the annihilation. Then the differential cross section for the decay GGG is

where . The cross section (4.6) is identical to that for the decay of positronium into three photons [13]. Note that it is finite throughout the physical region. (It is the only mechanism for GGG at , so that if the total cross section for GGG at is to be finite, then so must the cross section for the process of fig. 21 be.)

[ Figure 11 ] The lowest-order diagram for the decay of a heavy vector meson containing a pair of heavy quarks into three gluons.

The process illustrated in fig. 22 may also contribute to the hadronic decay of the . The ratio of the total decay rates of the due to the diagrams of figs. 22 and 21 may be estimated as ( is the charge of the quark)

where we have assumed (without much justification) that the coupling constant relevant to the process of fig. 21 is . The sensitivity of eq. (4.7) to the value of makes an estimate of for the difficult. It seems likely, however, that

[ Figure 12 ] The lowest-order diagram for the electromagnetic decay of a heavy vector meson containing a pair of heavy quarks (Q) into a pair of light quarks .

A number of decays other than to GGG and may also be considered. The main ones of interest are

The first of these processes might be interesting [14] because the gluon jets will have higher energies than in GGG, and so presumably will be better collimated. The decay

should give somewhat different gluon energy and angular distributions than GGG, but the effect is difficult to estimate reliably.

4.3. The Moments

For a 2-jet event, . For a three-jet event (see the appendix for the definitions of )

This formula realizes the claim made in sect. 3 that may be divided into a constant term and a term which damps the divergence in the 3-jet differential cross section (4.2).

To evaluate the moments of for the process G, we write

The is given to by eq. (4.4). In the case , it is possible to evaluate in closed form. Including the contribution from 2-jet events at , the final result for to is (10)

The coefficients of for some higher moments of are given in table 2. Values of the moments for various c.m. energies are also given there, using . Note the negative values obtained for some of the higher moments of at low . These unphysical results are signals of the breakdown of perturbation theory. As higher moments of are evaluated, so the region in the differential cross section closer and closer to the 2-jet limit is probed. However, as discussed in subsect. 3.2, the lowest order in the perturbation expansion is no longer a good estimate of the differential cross section in this region. Results which depend critically on the behavior of the G cross section for kinematic configurations close to that for cannot, therefore, be determined reliably from an estimate. These difficulties in practice effect only the high-order moments of the for the process G at low . The form of for hadrons resulting from this process will be entirely unaffected by these problems (see sect. 5).

[ Table 1 ] Moments of for the process (G) (the sum of and G calculated through )

The first moment (mean value) of for the process GGG is found to be given by

Higher moments of for this process are given in table 3. The decrease of with in this case is a consequence of the approximate flatness of for GGG. ( for GGG can never be negative, since in this case everywhere.)

[ Table 2 ] Moments of for the process GGG, where is a heavy resonance

In general, . For a three-jet event, however, the value of is more tightly constrained. The form (4.9) for is easily seen to be minimized for (the event is most spherical in this configuration), at which point . It is maximized in the collinear (`two-jet') configuration, for which . For a three-particle final state, therefore

In contrast, a two-jet event gives , while a spherical (`phase-space') event has .

For a two-jet event, . For a three-jet event

The integral contains no infrared divergences since the weight of the kinematic configurations close to the 2-jet one which lead to divergences is zero, For the process G, we find

while for GGG

Notice that both for a spherical and for a 2-jet event. Only for events with non-trivial structure is it non-zero. Tables 4 and 5 give the values of some higher moments of for G and for GGG.

[ Table 3 ] Moments of for the process (G) (the sum of and G calculated through )

[ Table 4 ] Moments of for the process GGG, where is a heavy resonance

For any three-jet process

The maximum value is attained when (the event is most `spiky' in this configuration).

For a 2-jet event, . There is no simple formula for in a three-jet event (11) , although it can be shown that in this case

where the minimum value is realized for . One finds

where by we mean the sum of the processes and G.

4.4. Differential Cross Sections in the

The differential cross sections for the processes (G) and GGG are shown in fig. 23 for and 4. Note that (apart from the dependence of ), all these results are exactly scale invariant (do not depend on the value of ). (The cross section for G is proportional to . was chosen to evaluate for fig. 23. Results for other values of may be obtained by a trivial rescaling.) The differential cross sections for G shown in fig. 23 exhibit infrared divergences, but as discussed above, these divergences cancel when the moments of the are evaluated. Notice the sharp cut-offs in the differential cross sections at the boundaries of the physical region for three-jet processes (, , ).

[ Figure 13 ] The distributions in , and for the processes G and GGG. gives , and . The distributions for GGG are normalized to give unit total cross section. Of course, G yields an infinite total cross section, due to its divergence at . When added to the cross section for calculated through , it gives a finite total cross section of for the complete process (G). The curves in this figure have been calculated using . Results for other values of may be obtained by a trivial rescaling.

As will be discussed in sect. 5, the fragmentation of the quarks and gluons into hadrons serves to make considerable modifications to the . At very high energies, however, such modifications become less important, and at asymptotic energies, the free quark and gluon results of fig. 23 are regained.

We have calculated only to order . In higher orders, processes in which more than three jets are produced will occur, and the two- and three-jet production cross sections will be modified. As discussed above, these higher-order effects are clearly important for those moments of the which probe close to the region of infrared divergence, in some cases, they are even necessary to obtain positive results. Events involving more than three jets will also undoubtedly populate ranges of the outside those allowed for three-jet events by kinematics. Except in these circumstances, we expect and higher corrections to be small.

4.5. Production of Heavy Quark and Lepton Pairs

The production and weak decay of pairs of heavy mesons (D, B, ) carrying new flavors will give rise to events whose shapes are distinct from those of the two- and three-jet configurations discussed above. Three basic mechanisms for the weak decay of a heavy meson may be considered. The first, illustrated in fig. 24, involves the standard weak decay of the heavy quark Q into . If a V--A coupling is assumed, then the differential cross section for the decay in the fractional energies

is

just as in muon decay. The second mechanism, shown in fig. 25, involves the process QqG (12). Note that in these first two mechanisms for heavy meson decay, we have considered only the weak decay of the heavy quark Q. The `spectator' quark will carry only a small fraction of the energy of the decaying meson, and so will not usually generate a jet. A third mechanism for heavy meson decay may also be envisaged. It is illustrated in fig. 26, and involves the exchange of a between the heavy quark Q and the `spectator' . The importance of this third mechanism in strange particle decays is presently unknown; it is probably slightly more effective than the mechanism of fig. 24.

[ Figure 14 ] Diagram for the weak decay of a heavy meson (M) through the decay of the heavy quark (Q) into three light quarks.

[ Figure 15 ] Diagram for the weak decay of a heavy meson (M) through the decay of the heavy quark (Q) into a gluon and a light quark.

[ Figure 16 ] Diagram for the weak decay of a heavy meson (M) in which the heavy quark (Q) undergoes a weak interaction with the spectator quark to produce a pair of light quarks.

It is also possible that the mixing angles between heavy quarks are so arranged that the decay of a heavy quark involves many stages, each consisting of a decay to a quark of slightly lower mass. The cascades generated in this way would lead to values of the very close to zero. The observation of jet structures resulting from the processes of figs. 24--26 would provide definitive evidence as to the mechanism of weak decays of possible very massive mesons carrying new flavors. The rates of these decay modes are modified by radiative corrections, but the energies of the extra final-state particles tend to be very small, so that they should not generate extra jets (13). The radiative corrections to the first decay mode (fig. 24 may be computed from corrections to decay and from eq. (4.4) (color averaging decouples the radiative corrections to the produced quark pair). One finds that for a V--A coupling, , giving a 15% correction for . For the second mechanism (fig. 25), the correction due to the process

(the lowest-order vacuum polarization insertion to the gluon propagator must also be included, and renormalized off the gluon mass shell) simply contributes to replacing by . Note that the pair in this process usually have a small invariant mass (hence small opening angle) and so act as one jet. The rate for the third decay mode (fig. 26) becomes roughly , about a 1% correction for .

The basic mechanism for the inclusive production and decay of new heavy mesons (M) in annihilation is shown schematically in fig. 27. The hadronic shower associated with the primary vertex in fig. 27 will occur on time-scales of . The end-products of the showers will be M mesons with small momenta at least near the M threshold, which will typically live before undergoing weak decays. The two sources of hadrons in these events act at very different times, and so will be effectively decoupled. At very high energies, the decay products of the M mesons will carry a negligible fraction of the total energy of the event; most will be radiated in the hadronic shower. At such energies, therefore, processes like fig. 28 will appear as two-jet events. Just above the threshold for M production, the events should, however, be very different from two-jet ones. We assume that no hadronic shower is generated, and that all the particles in the event come from the weak decay of the M meson. Since these have spin 0, there should be no correlation between the decays of the M and produced in a particular event. Making the approximation that the decays of M and are to free quarks (or gluons) and assuming independent decays at rest through the mechanism of fig. 24, we find that

The distributions are shown in fig. 29. If the decay mechanisms of fig. 25 or fig. 26 are assumed, we find that

where is the angle between the pairs of final particles from the two mesons. The stationary points of the Legendre polynomials in eq. (4.22) are responsible for the spectacular peaks apparent in the distributions for these events. The expression (4.22) for yields

upon integration over . Note that the independence of the decays of the two mesons causes the for the complete process to be proportional to an average of the for the two decays. In the notation of sect. 2, the for the complete process become (for and normalizing the densities so that their sum gives )

The for systems which are randomly rotated with respect to each other therefore obey a linear superposition principle.

[ Figure 17 ] Mechanism for the production and weak decay of heavy meson (M) pairs (containing heavy quarks Q) in annihilation.

[ Figure 18 ] Diagram for the weak decay of a (sequential) heavy lepton (L).

[ Figure 19 ] The distributions in , and for the production and decay of heavy quarks and leptons, through the mechanisms of figs. 24--26. (The decay schemes of figs. 25 and 26 both give rise to final states containing two jets. In the approximation of free final quarks and gluons, these two, therefore, give the same results.)

Some of the weak decays of heavy mesons should produce leptons. Electrons and muons will appear as `jets' containing only one particle, while heavy leptons produced in the decays will decay mostly to hadron jets.

Heavy lepton (L) pairs may also be produced directly in annihilation. If they carry new conserved quantum numbers, then their decays should proceed dominantly through the diagram of fig. 28. (We shall not discuss other possible quantum number assignments and will assume a V--A coupling.) Experimentally, the total visible energy in such events will be significantly less than the total c.m. energy because of the presence of the neutrino. (In an apparatus with complete acceptance, the missing energy will probably, in fact, be the best method for identifying these events.) In a free-quark model for the final state, the average fraction of the total energy visible is . We then find

Note that even at very high energies, heavy lepton production events should retain their four-jet structure although for large the pairs of jets will be boosted to small opening angles in the c.m.s., thereby changing the for the events. The differential cross-sections for idealized (free quark) heavy lepton production events are shown in fig. 29

4.6. The for Large

The for from 0 to 10 are given in table 6 for each of the processes discussed above (, (G), GGG, , G (or ) and ). For (G), the coefficient of in ( even), ( odd) is given. (As discussed above, these means are calculated from the complete cross section for annihilation, which includes one-loop diagram contributions to as well as the process G.) From the formula (2.21), it is possible to compute the limits of the as . As discussed in subsect. 3.4, they are only finite at the lowest order in for each process. In that case, one finds

It is clear from table 6 that these limits are, in fact, approached very quickly as increases (to within 5% at ). For (G), the approach a limit at large , but this limit is not infrared finite and diverges as the artificial gluon mass is taken to zero. This behavior is evident in table 6.

[ Table 5 ] Average values of the for the processes discussed in sect. 4

For (G) (the sum of and G calculated through , the coefficient of is given. The are related to this by ( even) and ( odd).

The distributions are shown in fig. 30 for up to 9. Spikes in the distributions occur when the formula for is stationary with respect to variations of the parameters which specify the configuration of the final state. For production and decay of pairs of heavy quarks into a total of four final particles, the formula (4.22) shows that is stationary with respect to the angle between the directions of the decays when the value of corresponds to a stationary point in . The minima are determined from the minima of the Legendre polynomials, by . In our model, six-particle final states resulting from the production and decay of pairs of heavy quarks or leptons are specified by the values of four parameters. Spikes in the distribution for six-particle final states will occur only if the value of is stationary with respect to variations in all four parameters. This does not occur for .

[ Figure 20 ] The distributions for various processes in the free quark and gluon approximation.

The forms of the for three-particle final states are stationary with respect to the values of and which specify the final-state configuration at values of and within the physical region. For , the only stationary point is at , and yields the minimum (maximum) value of for even (odd) . For , other stationary points develop within the physical region, and spikes appear in the distributions at the values of corresponding to these stationary points. The maximum (minimum) values of for odd (even) always occur when (or , or ). Many stationary points develop along these lines for high , it is always the one nearest to (but not on) the edge of the physical which yields the extremal value of . For example, is stationary when , at which , respectively. The first of these stationary points is at the edge of the physical region, where goes smoothly to zero. The stationary point at is one of three absolute maxima placed symmetrically on the lines , and . There is a local minimum between these three peaks at . Finally, at there is a saddle point between the peaks at and .

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