III. Production of Heavy Hadrons in Hadron-hadron Collisions

In this section we consider the problem of calculating the cross section for the production of massive hadrons carrying new flavors in hadron-hadron collisions. Unlike photoproduction for which there exist some sketchy experimental data with which to compare our theoretical predictions, there are, as yet, no indications of the hadronic production of charm in accelerator experiments using hadron beams.

The Calculation

The lowest-order diagrams in QCD perturbation theory contributing to the process are shown in Fig. 10. (23) We may write the cross section for the production of heavy quarks in the form

[ Figure 10 ] Lowest-order diagrams for hadronic charm production.

where

We use the quark and antiquark distributions inferred from deep-inelastic scattering data by Field and Feynman. (17) The gluon distributions we use are discussed in Sec. II. The matrix element and its calculation are presented in Appendix B. We find

and

where , , and are the kinematic invariants for the subprocess, is the effective mass of the heavy quark, and . The angular brackets indicate numbers arising from color averaging.

Our remaining assumptions are essentially as described above for photoproduction. We take the charmed quarks to have unit probability to form charmed hadrons and assume that no other quarks may become charmed hadrons in the final state. Further, we assume that the invariant mass of the pair is the same as that of the hadrons which contain them. Figure 11 shows our results on the energy dependence of the cross section for the associated production of charmed hadrons. We show the band of values corresponding to our range of , in (2.7) for the ``bag-bremsstrahlung'' gluon distribution (2.11) while for the ``naive'' (2.7) and ``bremsstrahlung'' (2.19) distributions we given only the curves corresponding to . Also shown is the result obtained by considering only the subprocess ; this corresponds roughly to the estimate for charm production made by Fritzsch. (24) Curves from early model calculations due to Sivers (25) and Bourquin and Gaillard (26) are given for comparison. The experimental upper bound on charm production shown is that obtained from an emulsion exposure at Fermilab. (6) This upper bound,

[ Figure 11 ] The cross section as a function of energy. Early model calculations from Sivers (Ref. 25) and Bourquin and Gaillard (Ref. 26) are shown for comparison. The experimental upper bound is due to Coremans-Bertrand et. al. (Ref. 6). The result using only the mechanism is essentially the same as that of Fritzsch (Ref. 24).

appears to favor small values of the strong coupling constant and/or soft gluon distributions in our model. It is significant that for the value obtained in the calculation is not sensitive to the shape of the gluon distribution. Below the crossover point, the ``bag'' gluon distribution function (2.10) gives the smallest cross section, followed by the ``bag-bremmsstrahlung'' one. The ``naive'' gluon distribution gives a rather larger cross section, and the ``bremmsstrahlung'' one (2.9) gives the largest of all. This ordering is to be expected, since the function given by the last integral in (3.6) tends to increase with and , so that the hardest gluon distributions give the largest cross sections. At some value of and , however, the function begins to decrease again. As increases, this critical value decreases, so that only the small- behavior of the is relevant, and under these circumstances the roles of the different gluon distributions are interchanged. The origin of this damping is the cross section in (3.3), which disfavors the interaction of high-energy gluons. The cross section at CERN ISR energies is seen to be a factor of 10--20 larger than would be obtained with the mechanism alone. Experimental measurements in this energy range will thus be important in determining whether gluons play a role similar to partons in hadron dynamics. Table I gives our predictions for for a variety of inputs at values of corresponding to those available at various experimental facilities.

[ Table 1 ] Cross sections under various assumptions.

Our results also depend on the value for the charmed-quark effective mass used and, since this cannot be extracted directly from experiment, it is important to see how our prediction for the cross section depends on its value. This is shown in Fig. 12 for . Because we are calculating the cross section for the production of quarks by hard scattering, our assumption that the effective mass of the charmed-quark pair is the same as the hadrons which contain them may be questioned. One way to test this assumption is to change the value of the threshold used in the calculation. Even if our assumption were strictly correct, we would still have to consider varying the threshold since we do not know the proportion of 's ), 's ), 's ), and 's which is produced. Probably predominantly 's should be produced, because of their slightly smaller mass. The variation of the cross section with effective at is shown in Fig. 12. In view of the fact we can change our prediction by a factor of 2--3 by reasonable variations of these parameters the question of whether current experimental limits rule out a hard gluon distribution remains open. This range gives some idea of the overall sensitivity of our calculation, and therefore, if the experimental bound on the charm-hadroproduction cross section were improved by a factor of 5--10, we would find it very hard to understand in our model.

[ Figure 12 ] Dependence of the cross section on effective and effective at .

We now turn to predictions for charm-production in and interactions. The quark and gluon distributions in antiprotons are easily obtained from those in protons by charge conjugation,

and inserting these into (3.1) we obtain the cross section shown in Fig. 13. The greater importance of the mechanism in collisions is evident, but above , is still the dominant mechanism in the cross section because it is almost the same as for .

[ Figure 13 ] Energy dependence of with distributions given by (3.6).

In order to calculate the charm-production cross section in collisions we must make some further assumptions regarding the gluon distributions. In the absence of any theoretical or experimental guidance we will assume that the fraction of momentum carried by the gluons is the same in the pion as it is in the nucleon. If we neglect any difference in size between the and the , we can obtain a gluon distribution analogous to (2.11),

The analog of the bremsstrahlung nucleon distribution (2.9) is chosen to be

Our results for for these two gluon distributions are shown in Fig. 14.

[ Figure 14 ] Energy dependence of . Quark distributions are those of Field and Feynman (Ref. 17). Gluon distributions for the are defined in (3.7) and (3.8) and for the proton in (2.11) and (2.9). If we use the ``soft'' bag-brems distribution for the we also use it for the .

We again use the Field-Feynman distributions for quarks and antiquarks.

Production of Flavors Heavier Than Charm in Hadron-hadron Collisions

Using exactly the same methods as for charm production, we may compute the cross section for where denotes a new type of quark with , carrying a new flavor which could be contained in the Our results for the energy dependence of the cross section for the production of particles carrying this new flavor are shown in Fig. 15.

[ Figure 15 ] Energy dependence of and with in our model.

The variation of the cross section with the mass of the produced quark for and is displayed in Fig. 16. For this graph we have taken and used the ``bag-bremsstrahlung'' gluon distribution. From Figs. 15 and 16 one can see that the production cross section for flavors heavier than charm in hadron-hadron collisions should be quite small. Cross sections of this magnitude will be rather difficult to measure unless distinctive triggers can be found.

[ Figure 16 ] Dependence of on the mass of the quark at and 200 GeV in our model. Curves calculated using the bag-brems gluon distribution (2.11).

Large- Production of Charm and Direct Leptons

We can use the basic mechanisms illustrated in Fig. 10 to estimate the production of charmed particles at large transverse momentum (). If our basic model is correct, this calculation should be somewhat more reliable than the estimate of the total charm-production cross section. The reason for this is that any final-state interactions occurring near the subprocess threshold which could be important in the evaluation of the total production cross section should not significantly affect the form of the spectrum for large . We write the differential cross section for the production of a decay product (a charmed meson or a muon arising from semileptonic charmed-particle decay) of one of the charmed quarks in the form

where

are the Mandelstam variables for the hard-scattering subprocesses

and

The form of (3.9) is a simple modification of the usual hard-scattering-model formula which takes into account the masses of the produced quarks. The function gives the probability that the detected particle carries a fraction of the momentum of its parent charmed quark. We assume that one and only one charmed hadron arises from each charmed quark so that

Since we expect the production of charmed hadrons with a small fraction of the charmed quark momentum to be negligible, we should not take

as is commonly done for production. Instead we take the simple form

suggested by constituent-counting rules.

We can also use (3.9) when is a lepton ( or ) which arises from the semileptonic decay of a charmed hadron. In this case we normalize

where is the average semileptonic branching ratio of the produced hadrons. Experimental data from production of charm (27) gives

Folding the probability for producing a with momentum fraction followed by its decay where is a hadronic system (probably a with mass ) we have (28)

where

In the limit this reduces to

which gives a slightly harder lepton spectrum than the expression (3.16). Using (3.18) we calculate the differential cross section for inclusive lepton production. This is displayed in Fig. 17. Also shown for comparison is the fit of Field and Feynman to multiplied by . If this curve is taken to be representative of the cross section attributable to electromagnetic sources of muons, we would predict that between 1 and 10% of the prompt muons observed in collisions at should arise from charm production and consequently not occur in pairs. This exercise verifies the hypothesis that it should be possible to enhance the charm-production signal in high-energy hadron-hadron collisions by triggering on a prompt lepton. If our calculation is correct, the occurrence of coincidences due to charm production can be expected at about times the direct signal at .

[ Figure 17 ] Direct leptons from charmed-particle decay compared to .

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