|
EVENT SHAPES IN e+e- ANNIHILATION (1979) |
|
5.1. The Model for Jet Development
At present, QCD can give little guidance as to the way in which quarks and gluons `fragment' into hadrons. We use the explicit model for the development of hadronic jets due to Field and Feynman [18]. The basic assumption of this model is that hadrons are emitted from a `fragmenting' quark or gluon independently and with limited transverse momenta. Strong and electromagnetically decaying hadrons such as the 
are taken to be among those produced, and their decays are treated exactly. The fragmentation of a particular type of quark or gluon is specified by giving the probability function 
, where 
is the fraction of the jet's total momentum that is carried off by the first hadron emitted. If one chooses
then the distribution of the fractional momenta 
of all hadrons in the jet is given approximately by
A precise treatment requires the inclusion of transverse momenta and the possibility of producing many species of hadrons [18]. Field and Feynman adjusted the forms of 
so as to agree with experimental estimates of 
for quark fragmentation. Their final choice was
We also need the corresponding function for gluons. In the absence of good experimental constraints, we use the simple form of eq. (5.1). We usually take 
, but we also present some results using 
and 
.
Our calculations of the expected distributions in the 
for genuine hadronic events require a model for the complete structure of a hadron jet, rather than just one-particle momentum distributions . In using the complete jets provided by the Field-Feynman model [18], we are relying more heavily on its basic assumptions (especially independent emission) than do predictions for one-particle distributions. Limited experimental tests [19] on the detailed structure of jets predicted by the model have, however, proved successful. When further experimental data become available, the model for jet development must be refined accordingly.
The parameters in the Field-Feynman model were determined by fitting data from various reactions at comparatively low Q
assuming that all events contained only the minimum number of jets (two for 
annihilation). In reality, some of the events will have contained extra jets. On the other hand, we use the model to simulate the fragmentation of a single jet, and sometimes include explicitly the fragmentation of the extra jets. It therefore appears that the contribution of extra jets has been counted twice. However, our results show that for the low Q at which the jet-model parameters were determined, the distributions of hadrons produced in two jets and in three jets of the same total energy are almost identical, so it seems that roughly the correct parameters to describe single-jet fragmentation at high energies were found.
Note that the Field-Feynman jet model gives rise to hadron transverse and longitudinal momentum distributions which asymptotically become independent of the total jet energy, in contradiction to the predictions of QCD. In addition, as discussed in sect. 3, it does not include the `mixing' of jets generated from quarks and from gluons implied by QCD.
The present formulation of the Field-Feynman jet model does not conserve energy and momentum exactly, as discussed in detail in ref. [18]. The violations are worst for low jet energies, and the model becomes unreliable for jets with energies below about 2 GeV. This inadequacy of the model for small jet energies prevents any useful comparison of its predictions with existing data on hadron production in 5.2. Momentum Distributions and Multiplicities for Two- and Three-jet Processes In this and subsects. 5.3, 5.4, we discuss hadronic final states arising from the three basic processes
Of course, the process Fig. 31 shows the single-hadron fractional momentum Fig. 32 shows the mean hadron multiplicity due to the processes
if we use our standard choice of 5.3. Shapes of Two- and Three-jet Events We now discuss the
Note that the Figs. 33--35 show the distributions annihilation. Moreover, there is some inconsistency between the data from different experiments [1].
[ Figure 21 ]
Distributions in fractional momentum for single hadrons produced by 
, (G) and 
GGG as a function of the c.m. energy 
.
G cannot be observed in isolation, but only in combination with . We denote the combined processes by (G). As discussed in sect. 3, the method of combination is not entirely unambiguous. Events arising from the process G are divided into two classes according to whether the value of 
calculated for the quarks and gluons is above or below a cut-off 
. The events below the cut-off are considered as true three-jet events, and the fragmentations of the quark, antiquark and gluon into hadrons are treated separately. Events above the cut-off are taken to contain two, rather than three, jets and to be indistinguishable from . The total cross section is only finite when these events are combined with the genuine term. We simply generate two-jet (quark and antiquark) final states for the combined G (
) and 
(
) terms. An example may make this prescription clearer. For 
, the cross section for G for 
is 0.41 times the point cross section. In the same units, the total cross section to order 
is 
. At this energy, therefore, the ratio of three- to two-jet final states is 
. We shall usually take 
, although, as we shall discuss below; this choice is unreasonable in certain cases.
distributions (longitudinal plus transverse momentum) for hadronic final states resulting from the three basic processes , (G) and GGG (
is a heavy vector meson). Note that the turnover in the distributions at small 
moves closer to zero as increases. The presence of such a turnover is a consequence of the finite transverse momenta 
of the hadrons in the jets. The Field-Feynman model is such that 
and 
are roughly constant with energy, so that as increases, the transverse momenta of the hadrons become insignificant compared to their longitudinal momenta, and the turnover disappears. At very high energies, the hadron momentum distributions tend to a limit given for gluon jets by eq. (5.2) and for quark jets by ref. [18]. Notice that at all energies, the distributions for hadrons from the process GGG are significantly steeper than those from the processes and (G), while results for the latter two processes never differ by more than about 20%. The model we use predicts that even at 
, the process 
GGG should give a steeper hadron distribution than . One would, therefore, expect a difference between the distributions in 
decays and in the surrounding continuum. This prediction seems difficult to avoid as it also occurs in simpler models not involving the generation of complete hadronic final states, but formulated solely in terms of fragmentation functions into single hadrons. It is, therefore, surprising that the experimental single-hadron distributions seem identical on and off the resonance [1]. Nevertheless, the correct inclusion of scaling violations in the fragmentation functions (which is complicated by the need to consider 
) could remove the discrepancy.
, (G) and GGG as a function of the c.m. energy . At sufficiently high energies (
), the multiplicities rise like 
, with coefficients given by the heights of the rapidity plateaus for the various processes (or equivalently, the limits of 
as 
). Our choice for the gluon fragmentation function (eq. (5.2)) leads to a higher rapidity plateau (hence, higher hadron multiplicity) for gluon jets than for quark jets at high energies. At low energies, however, the rapidity plateau is not fully developed, and it is a matter of detailed calculation to determine whether quark or gluon jets have higher hadron multiplicities. We find, in fact, that below 
, the process 
GGG gives a lower mean hadron multiplicity than . The details are sensitive to the gluon fragmentation function. For example, in the 
region (
), we find
for the gluon fragmentation function. If we take instead 
or 
, then the mean multiplicity for GGG becomes 12.8 or 11.2, respectively.
[ Figure 22 ]
The mean multiplicity of hadrons from the processes
, (G) and GGG as a function of .
distributions for realistic hadronic events resulting from the processes , (G) and GGG. It should be emphasized that, if QCD is correct, then the pure two-jet process should never be observed; only the combination of two- and three-jet processes which we denote by (G) should be present. (Of course, higher-order processes involving the production of more than three jets should also occur, but we do not expect their inclusion to modify our results appreciably.)
could also be used to distinguish the approximate two-jet structure expected from QCD from purely isotropic production of particles [17]. For an exactly isotropic event, 
for all 
.
, 
and 
for simulated hadronic events resulting from , (G) and GGG, at various center of mass energies . Table 7 gives 
, 
and 
for these and other cases. Perhaps the most striking feature of these figures and the table is the large difference between the results for realistic hadronic final states, and those obtained using the free quark and gluon approximation (fig. 23 and table 6). We believe that large modification of free quark and gluon results for realistic final states is not a phenomenon peculiar to the ; rather, it will occur for all the other shape parameters previously investigated [5,6]. As we shall discuss in subsect. 5.4, use of only the higher momentum hadrons in each event leads to distributions which are closer to the free quark and gluon approximations.
[ Figure 23 ]
The distributions for hadronic events resulting from the processes , (G) and GGG, at various c.m. energies, . All
hadrons in the simulated events were used, no momentum cut 
being
imposed. On some of the graphs for high values of , we also show the
distributions obtained for final states of free quarks and gluons (see fig. 23). The parameter defined in subsect. 5.1 was taken to be 0.8 at all energies except 
, for which 
was used.
[ Figure 24 ]
The distributions 
for hadronic events resulting from the processes , (G) and GGG, at various c.m. energies, . See
fig. 33 for other details.
for hadronic events resulting from the processes , (G) and GGG, at various c.m. energies, . See
fig. 33 for other details.
for realistic hadronic events resulting from
the processes , (G) (combination of and G calculated through 
) and GGG, for various c.m. energies
In some cases, a momentum cut 
has been applied so that only hadrons with momenta above the cut are used in the calculation of the for each event. `Charged only' means that for that case, the 
were computed using only the charged hadrons in the final state. 
is a parameter which divides (G) processes into two and three-jet events. is the gluon fragmentation function discussed in subsect. 5.1. The defaults are discussed in sect. 5.
As discussed above, the model for jet fragmentation which we use is unreliable for total jet energies below about 2 GeV. Our results for three-jet production at 
should, therefore, not be taken too seriously while at , even the results for two-jet production should be considered somewhat suspect. However, we expect the similarity between the distributions for the various processes at these energies to survive the use of a more adequate model.
At 
, the distribution for hadronic events resulting from the process GGG is very different from that due to or (G). We find that at this energy (see table 7)
A difference between the distributions and on and off resonance in the region should, therefore, be easily measurable, and would provide an important test of the mechanism for the decays of heavy vector mesons. We usually take 
for the gluon fragmentation function. Fig. 36 shows the dependence of the distribution at from (G) and GGG on the form of the gluon fragmentation function . We consider the choices 
, (standard choice) and 
. Very little change is effected in the distribution for (G), but the distribution for GGG changes appreciably when different forms for the gluon fragmentation function are used. Since, however, the single-hadron momentum distributions from decay should allow the gluon fragmentation function to be determined, the form of the distribution for the decay remains an important test of its basic mechanism. Note that, as discussed in sect. 4, even on the resonance, there should be (in reality (G)) as well as GGG final states. The contribution of the former must be subtracted before a study of the process GGG may be made. We do not expect the change to be very large.
for hadronic events resulting from the processes (G) and GGG for three possible forms of
the gluon fragmentation function .
As shown by fig. 35 and table 7, the differences between the 
distributions and their means also provide a good method for identifying the possible two- and three-jet processes at . and are particularly effective at discriminating between GGG and (G). On the other hand, the 
distributions at for the processes (G) and GGG are rather similar, but are very different from those for .
For , the , and distributions from , (G) and GGG events are all very different. Once again, the distribution is the best measure of any contributions from pure processes, while the and especially distributions exhibit the largest differences between the processes GGG and (G) expected on and off heavy vector meson resonances.
At 
, the distributions for GGG and (G) show marked similarity to the distributions obtained for these processes in the approximation of free final quarks and gluons. Note the appearance of a kink in for (G) events around 
. Below the kink, most events of this type will have three-jet final states, while above, most will give only two jets. The distribution in for pure events is, of course, very similar to that for (G) when 
. The and distributions at bear less resemblance to the free quark and gluon results than the distribution, but they still offer ample discrimination between events of different types.
It is interesting that 
is required for even the distribution for realistic hadron final states to be similar to the free quark and gluon approximation. This phenomenon should be universal to all parametrizations of the `shapes' of final hadron states. Previous estimates of its importance for spherocity and thrust based on very simple models for quark and gluon fragmentation [6] were probably too optimistic. Nevertheless, it seems likely that our results for realistic hadronic events do not depend significantly on the details of the model for quark and gluon fragmentation used, and so remain firm predictions of QCD.
Finally, at , all the distributions are very close to the free quark and gluon predictions. The cross sections for the simulated hadronic events drop away very quickly at the boundaries of the kinematic region for the quark-gluon subprocesses (
, 
, 
). Four-jet events may give important contributions to the cross sections outside the physical regions for three-particle subprocesses.
In the distribution for (G) events at , a dip around 
is clearly visible. This dip signals the boundary between two- and three-jet processes, and is an artifact of our model. It has no physical significance.
As discussed above, it is necessary to separate (G) processes into two- and three-jet events in order to treat their fragmentation into hadrons. This separation is achieved by dividing the events according to the value of corresponding to the fragmenting quarks and gluons. For , (G) events are treated as being of the form , while for 
the event is assumed to contain three jets, and the fragmentations of the 
, 
and G into hadrons are treated separately. The magnitude of the two-jet contribution is determined by demanding that the sum of the two- and three-jet pieces reproduce the correct total cross section for (G) to . For most of the distributions shown in figs. 33--35 we used , but at , we took instead . The value of should be chosen so that G processes with are indistinguishable from ones as far as their fragmentation to hadrons is concerned. may be thought of as a parameter which determines the `resolution' of the final hadron state to different configurations of the quarks and gluons produced in the primary interaction. The closer is to one, the lower the energy (and angle) of the gluon in the process G must be for the hadronic final state not to reflect its presence, and to consist only of two jets. The choice of the `resolution parameter' for the hadronic final state depends on the model for quark and gluon fragmentation used. Even in the context of a particular model, its value must, at present, essentially be guessed. We have simply tried to choose it so that our results are not obviously unphysical. The signal for too small a choice of 
is the appearance of separated peaks in corresponding to two- and three-jet events. In this case, the `resolution' of the simulated hadron final state is really finer than the division into two- and three-jet processes chosen; the hadron final state can `see' the arbitrary division between configurations of the quarks and gluons which are supposed to give two- and three jet final states. If a very large value of is chosen then negative amounts of have to be added in order to obtain the correct total cross section for (G) events, which seems undesirable.
In the Field-Feynman model for quark and gluon fragmentation, we find that, for most values of , the choice 
avoids the pathologies mentioned above. At , however, there is no value of which avoids both of them. This is undoubtedly a signal of the incompatibility of the Field-Feynman model with QCD. Inclusion of the rise of for the hadrons from quark and gluon fragmentation with increasing implied by QCD would probably serve to alleviate the difficulty. We used at . Fig. 37 shows the dependence of for (G) events on the choice of . It seems clear from these results that is adequate for predictions at c.m. energies which will be attained by accelerators in the immediate future (
), and that at these energies the dependence of the results on the value of the unknown parameter is very slight.
for hadronic events resulting from the process (G) for various choices of the parameter which divides two- from three-jet final states.
5.4. Results for Incomplete Final States
The distributions for realistic hadronic events presented in sect. 4 exhibited rather large deviations from the free quark and gluon approximation. The discrepancy is, in part, due to the presence of many soft hadrons in the final state, and may be decreased by using only those particles whose momenta are above some lower cut-off in the calculation of the (16). For sufficiently large , such a cut leaves essentially only those hadrons moving along the jet direction, and removes the roughly isotropic background of low-momentum hadrons. The latter particles typically have momenta comparable with the transverse momentum of the jet fragmentation. If is taken too large, however, then all the hadrons in a particular jet may be ignored, and the resulting will once again not reflect the free quark and gluon results. We shall consider 
and 1 GeV. Of course, our previous results used all the hadrons in each event, corresponding to 
.
Since we always work in the virtual photon rest frame, the three-momenta of all particles in an event should sum to zero, so that, for each event, 
. In addition, if all hadrons in an event are massless, then energy conservation requires that 
. The non-zero masses of the final hadrons serve to decrease the value of 
from one. The effect is small, and in the results of sect. 4 we always removed it by dividing by for each event. We followed the same procedure when considering incomplete final states. Here the deviation of from 
was, of course, much larger. The total momentum measured in an `incomplete' event will also be non-zero, so that 
. Fig. 38 shows the distributions 
for (G) and GGG events at , with various choices for . corresponds to measuring all particles in each event. That 
is not always zero even in this case is a reflection of the fact that the model we use to generate final states does not conserve momentum exactly. The narrowness of distribution in for this case shows that the discrepancy is, in fact, very small.
for simulated hadronic final states computed using only hadrons with momenta greater
than . The non-zero values of even when all hadrons in each event are included are a reflection of the fact that the
model for jet fragmentation used does not conserve momentum exactly. In one of the curves, only the charged particles
in each event have been used for the calculation of 
.
Fig. 39 shows the 
distributions for hadronic events resulting from the processes , (G) and GGG computed using only hadrons with momenta greater than in each event. Note that larger values of improve the agreement between the results for realistic hadronic final states and those obtained using the approximation of free quarks and gluons. However if is increased significantly above 1 GeV at , the agreement begins to deteriorate again; the apparent values of in some events are too small because all the particles in a jet have momenta below the cut-off . In no case does the use of a momentum cut significantly improve the discrimination between different types of hadronic events provided by the ; it merely serves to make some of the distributions more similar to those obtained in the free quark and gluon approximation. These trends are reflected in the values of 
, 
and 
given in table 7.
for simulated hadronic events calculated using only hadrons with momenta greater than .
In actual experiments, not all the hadrons in each event will usually be detected. Some will be lost through imperfect angular acceptance, while others will simply not trigger the detectors. Note that the rotational invariance of the renders them independent of the orientation of events, so that their values should be insensitive to gaps in angular acceptance. As an illustration of the consequences of these effects, we show in fig. 40 the distributions in , 
and 
for hadronic events at 
and 20 GeV calculated using only the charged hadrons in the final state. The discrimination between different processes provided by the is not unduly affected by this cut. The enhancement of events at larger if only charged particles are considered is a consequence of the large apparent violations of momentum conservation in this case. This phenomenon is reflected in the very broad distribution for these events shown in fig. 37. This artificial behaviour can be removed by considering only events with (charged) multiplicity greater than 3. [17][29]. Many of the new generation of detectors can measure neutral particles, and so will not encounter these problems. Of course, we cannot estimate the distributions which should be observed by actual experiments; that would require a knowledge of the details of the apparatus used.
and 
for simulated hadronic events calculated using only charged hadrons.
It is possible that cuts other than on hadron momenta may serve to `clean up' the distributions. For example, one might consider only events with a certain range of total multiplicities.
5.5. Heavy Quark and Lepton Production
In this subsection, we present results for realistic events involving the production and decay of pairs of heavy quarks and leptons. We use the models for these processes discussed in subsect. 4.5, which are adequate only close to threshold.
Fig. 41 shows the fractional momentum distributions for single hadrons from events involving heavy quark or lepton production (at ). The distributions are all very similar, but are much softer than those for events of the (G) type.
distributions for simulated hadronic events involving the production and decay of heavy quarks and leptons, according to the models discussed in subsect. 4.5.
Figs. 42-44 show the distributions 
for realistic hadronic final states resulting from the production and decay of heavy quarks and leptons, and table 8 gives the 
for these processes at various c.m. energies. The fact that our models for heavy quark and lepton decay give rise to many jets of hadrons means that higher c.m. energies than for two- or three-jet production are necessary to obtain well-collimated jets, or for the distributions for realistic hadronic events to approach the free quark and gluon approximation. The decays of heavy quark and lepton pairs into four hadronic jets give similar and distributions, but the heavy lepton gives a significantly harder distribution. The different mechanisms for heavy quark decay give slightly different 
distributions, even at , and it seems, therefore, that an analysis based on the should be able to give some indication as to the mechanism for the weak decay of mesons with masses around 5 GeV. Mesons containing 
quarks should have masses in this region, and it seems probable that they will decay through weak interactions.
for simulated hadronic events involving the production and decay of heavy quarks and leptons.
for simulated hadronic events
involving the production and decay of heavy quarks and leptons.
for simulated hadronic events
involving the production and decay of heavy quarks and leptons.
for realistic hadronic events resulting from heavy quark and lepton production and decay by the various mechanisms discussed in subsect. 4.5
At , the two-jet mechanisms for heavy quark decay give rise to definitely harder and distributions than the three-jet mechanism. In addition, because (in our model) quark jets are better collimated than gluon jets the decay mode gives a harder distribution than the qG mode.
Finally, at , the distributions for realistic hadronic events become very similar to those obtained in the free quark and gluon approximation (fig. 30), although they remain somewhat softer at large . This is due to the large number of jets produced in each event; even at each jet does not have sufficient energy to appear like a free quark or gluon.
5.6. The 
for Large 
The forms of the distributions for high 
in the free quark and gluon approximation for the processes (G) and GGG were presented in fig. 23. The in these cases were given in table 6. In fig. 45, we plot the for realistic hadronic events resulting from the processes , (G) and GGG for the range of c.m. energies which will be attained by the next generation of accelerators. It is clear that the power of the to discriminate between the various processes diminishes as increases. For large , the reflect only the small-scale angular structure of each event, which is determined more by the structure of the hadron jets than by the nature of the subprocess. Note also that the approach of the for realistic hadronic events to the obtained in the free quark and gluon approximation is less rapid for high . Little new information may be gleaned from a measurement of the for high , and we believe that an experimental measurement of the distributions in and should be sufficient. The clearly have less power to discriminate between different types of events than do the complete distributions in the .
for realistic two- and three-jet events as a function of the c.m. energy over the range accessible to the next generation of accelerators.
From the for realistic events, one may in principle construct an average `correlation function' 
(see sect. 2), using the series formula (2.10). Fig. 46 shows the computed from the for realistic events of various types at . For most of the curves, the series (2.10) was truncated at 
, but for events, we also show the form for obtained by truncating the series at 
. The two results differ significantly, illustrating the very slow convergence of the series for . Note that cannot be computed directly from real events, as the formula (2.7) involves products of delta functions. Rather, one must find the function 
(defined in eq. (2.22)) for some small, but finite, detector size 
. This indicates that although in principle the and contain the same information (see subsect. 2.1), in practice they are not equivalent.
calculated from the for realistic two- and three-jet events at using the series (2.10). The series was truncated at 
.
