IV. Higher-order Corrections

In the calculation presented above, we have used lowest-order perturbation theory for our calculation of the cross section for the hard-scattering subprocesses. We have therefore implicitly assumed that nonperturbative and higher-order corrections to our expressions are small. We do not have good justifications for either of these assumptions, but must make them if we are to calculate anything.

In Figs. 18 and 19 we indicate the types of diagrams which contribute to and through order in the amplitude. Just as in QED, when calculating a cross section to order cancellations between terms and those of the form (Ref. 29) are known to be essential.

[ Figure 18 ] Typical diagrams for the amplitude classified by order in perturbation theory.

[ Figure 19 ] Typical diagrams for the amplitude classified by order in perturbation theory.

We must, of course, consider the processes represented by the diagrams in Fig. 18 as portions of hadronic processes of the type depicted in Fig. 11, and their interpretation is inexorably tied up with the physical content of expressions such as Eq. (3.2). (30) Consider, for example, the second diagram in Fig. 18 in which a gluon is emitted by an incoming quark. This diagram contributes to deviations from scaling. From the analysis of scaling violations in the framework of the quark-parton model by Altarelli and Parisi (31) we may infer that the contribution of some of this diagram may be absorbed by the choice of at the expense of allowing this to become a function of as well as . gives the distance scale on which the functions are observed, or perhaps the invariant mass of the virtual particle in the scattering subprocess. Similarly, it seems probable that diagrams in which a final-state particle emits a gluon may be at least partially absorbed into the definition of the decay (fragmentation) distribution . We should be wary, however, since many diagrams must be combined in order to obtain a gauge-invariant amplitude. This could be a signal that the scaling violations are process dependent. Nevertheless, for the Drell-Yan process Politzer (32) has shown that the expected violations of scaling are similar to those found in deep-inelastic lepton-hadron scattering.

In view of the above, we consider it important to test the sensitivity of our results to possible scaling violations in the quark and gluon distribution functions. For simplicity we take these to be the same size and type as the scaling violations in leptoproduction. There are some expectations (33) that the quark and gluon distribution (and decay) functions should have slightly different dependences, but we shall ignore this possibility here. We use the empirical parametrization of scaling violations due to Perkins, Schreiner, and Scott, (34)

with for both quarks and gluons. The ensuing violations of the gluon momentum sum rule (2.6) are insignificant. We have explicitly tested the result of calculating using the ``bag-bremsstrahlung'' gluon distribution (2.11) both with and without the scaling violation given in (4.1). At large , the cross section is found to increase due to the softening of the quark and gluon distributions. At low energies the cross section is suppressed as indicated in Table I.

Some of the diagrams in Figs. 18 and 19 contain parts which in the limit contribute to the running coupling constant . The leading (at least for ) terms of these subdiagrams may be approximately obtained simply by making the replacement .

Since the production process we consider is dominated by the subprocess threshold region there may well be significant final-state interactions between the slow, heavy produced quarks. It is unclear whether low-order terms in the perturbation expansion can give a good estimate of these effects. On the whole, we expect final-state interactions to enhance the probability of finding two quarks in a color-singlet state since the color forces tend to be attractive in this channel. Evidence for this is seen in the production of charm in annihilation where resonances and other effects raise the cross section above the value predicted by the naive parton model. In contrast, repulsive final-state interactions between in a color-octet state should depress the cross section. Since our model for produces only color octets we may be overestimating the cross section by neglecting final-state interactions near threshold. In hadronic production the mechanism produces a mixture of singlets and octets and there should be both attractive and repulsive corrections. We have tested for the effect of final-state interactions by changing the effective quark mass (and hence moving the subprocess threshold) as discussed in Sec. III. We have also inserted oscillations in the cross section near threshold representing resonances, but these did not change the overall result by more than a few percent indicating that it is gluon distribution functions which represent the most sensitive part of the calculation for the cross section. One particular class of diagrams deserves further attention. The first of these qualitatively new diagrams appears at and at a large number of this class of diagram occur. Examples are given in Fig. 20. Their distinguishing feature is that it is gluons or light quarks which undergo hard scattering but these generate a pair in the final state. Note that, because of their unusual final states, such diagrams will only occur at in the matrix element squared. Nevertheless it seems possible that the large number of analogous diagrams in higher orders will cause the class as a whole to be not unimportant. A higher-order member of the class would be, for example, that shown in Fig. 21 in which the ``fragments'' into a in the final state. One may only guess that, just as the -integrated probability for a quark to fragment to an quark is less than that for it to fragment to a quark, then so it will be much smaller for it to yield a quark.

[ Figure 20 ] Diagrams for .

[ Figure 21 ] Higher-order diagram for .

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