II. Real and Virtual Photoproduction

In this section we will discuss the associated production of heavy hadrons carrying new flavors by real and virtual photons. Some experimental information on the photoproduction of charm already exists. The observation of a charmed antibaryon (3) provides direct evidence for the production of charm by real photons although the rate is not yet known. Indirect evidence for a charm signal comes from the observed energy dependence of near the threshold for charmed-hadron production. (4) Arguments based on generalized vector-meson dominance allow an estimate for the cross section (10) while unitarity and the Okubo-Zweig-Iizuka (OZI) (11) rule provide a lower bound on this quantity. (12)

Virtual photons should also be effective in producing charm or other heavy flavors and there exists some evidence for the production of charm in this manner from the observation of events of the type . (1) A knowledge of the size and dependence of this cross section is important in determining whether charm production is an important background at small Bjorken to the scaling violations in expected in field theories of the strong interactions.

The Basic Calculation

We shall discuss our calculation of the cross section for the photoproduction of heavy particles in some detail in order to illustrate the techniques and assumptions used. Our starting point involves the diagrams of Fig. 1, where is a vector gluon and denotes any heavy quark.

[ Figure 1 ] Diagrams for in QCD perturbation theory.

We shall tend to refer to this below as charm, but our results should also apply to the production of heavier quarks. These diagrams constitute the first-order perturbation-theory approximation in QCD to the associated production of free charmed quarks. In order to relate this to an observable cross section we assume that the outgoing charmed quarks are dressed to form charmed hadrons with unit probability, independent of their momenta. We also assume that no other flavor of quark may dress to form a charmed hadron. We further assume the mechanism by which quarks are forged into hadrons is sufficiently soft that the invariant mass of the pair is approximately the effective mass of the charmed hadron system containing them. The threshold invariant mass can then be taken to be either or (corresponding roughly to the associated production process where is the lightest charmed baryon) rather than being given by the effective quark masses appearing in the matrix element. For charm production we take

If we define the probability that a gluon carries a fraction momentum of the nucleon which contains it to be we can write the cross section defined by the diagrams in Fig. 1 as

where

and

For simplicity we consider the photoproduction of heavy quarks only from transverse photons. In the deep-inelastic region this corresponds to calculating the effect of charm production on the structure function . In the spirit of the parton model we assume that the gluon is on its mass shell. The calculation of the spin-averaged, color-averaged for the subprocess is given in Appendix A. The result is

For , the expression (2.4) reduces to that given by Jones and Wyld. (13) Except for the overall factors , (due to color averaging) and the cross section is the same as that for the process .

To evaluate the cross section given by (2.2) we must choose a value for , the QCD effective coupling constant and a form for . Motivated by a study of the application of QCD to similar processes, we use a running coupling constant. By allowing the coupling to be a function of the invariant mass of the exchanged quark or gluon in a given diagram we are including some contributions from a set of higher-order diagrams in our nominally lowest-order calculation. We shall allow a range of possible values for which reflects the uncertainties in its determination, (14)

Our results are fairly insensitive to the dependence of the coupling constant. For example, in real photoproduction with , the value for the charm cross section with a running coupling constant is less than 1% lower than that with a coupling constant fixed at its threshold value, . The main situation in which the dependence given by (2.5) is important is thus in comparing the production of different flavors.

Gluon Distributions

The most important input to our calculation (2.2) is the gluon momentum distribution, . In fact, we need to know the probability for a gluon to be seen by a photon of but we will defer a discussion of the scaling violations associated with the dependence of this function until Sec. IV. From the amount of missing momentum in deep-inelastic scattering one can conclude

but there are no other experimental constraints on the gluon distribution. Since we are forced to make theoretical models for this distribution we will present results for a variety of models representing different schools of thought. The first possible choice for is

where the behavior near is motivated by constituent counting rules (15) and the behavior near by a correspondence with Regge theory. (16)

It is also possible to calculate in the spirit of the scale-invariant parton model the gluon distributions which arise from a given valence quark distribution from processes of the form shown in Fig. 2.

[ Figure 2 ] Bremsstrahlung of gluons.

We then have

where is the probability that a quark emits a gluon carrying a fraction of its momentum. Some authors would omit the factor which is included here to mimic the phase-space integration. A massive particle with momentum decaying into two massless particles, gives a particle with momentum with a probability independent of . That is, it has . If instead, a prescription based on constituent-counting rules which ignores the complications due to the spin of quarks and gluons is used, we have . There is some uncertainty, therefore, in the exact relation between valence quark and gluon distributions in this approach. One estimate of this relation gives (14)

The overall normalization of (2.8) is determined by (2.6) while the behavior near is determined by the relation between gluons and antiquarks implied by diagrams such as those in Fig. 2.

The softest gluon distribution we consider is obtained by assuming the gluons to be confined within a rigid bag of radius 0.7 fm, with normalization fixed by (2.6):

To obtain a significantly softer distribution would require cooperative effects among the gluons. More frequently we will use a distribution which combines the idea of nonperturbative effects with bremsstrahlung from valence quarks (which is smaller by a factor ): (14)

This is only slightly different than (2.7) and represents a plausible soft distribution of gluons. The gluon distributions (2.7), (2.9), and (2.11) are shown in Fig. 3 where they are compared with a parametrization due to Field and Feynman (17) of quark and antiquark distributions.

[ Figure 3 ] The gluon distributions used in our calculations are compared to the Field-Feynman parameterization of and . This figure is taken from Ref. 8 where there is more discussion of gluon distributions.

The Results for Real Photoproduction

Our results for the energy dependence of the cross section for the production of charm by real photons are shown in Fig. 4.

[ Figure 4 ] Energy dependence for the cross section calculated in our model. The calculation with the bag-brems gluon distribution (2.11) is shown as a band of values corresponding to the range of in (2.5). For the brems (2.9) and the naive (2.5) distributions we calculate only with . Also shown are the range for which might be inferred from the data on of Ref. 4, a prediction of the GVMD model (Ref. 10) and a lower bound (Ref. 12) from unitarity.

For the bag-brems gluon distribution (2.11) we show the range of predictions corresponding to our range of estimates for the strong coupling constant (2.5). For the other two gluon distributions we show only the curves corresponding to , and the range can be inferred by eye.

Also shown on the graph are some phenomenological constraints which can be used to judge the reliability of the model. Using an argument based on unitarity and the OZI rule indicated schematically in Fig. 5., one can derive the rigorous inequality (12)

[ Figure 5 ] Bound on charm photoproduction from unitarity and the OZI rule in photoproduction.

where is a parameter which measures OZI-rule violations and should be quite small, where is the diffractive amplitude for , and

Using data (18) on and the nuclear absorption measurement (19)

Eq. (2.12) yields an empirical lower bound on the charm-photoproduction cross section. This is shown in Fig. 4.

Also shown in Fig. 4 is a band of 2--5 b which is due to a careful experiment (4) which measures for 20--100 GeV and compares it with an extrapolation of low-energy data. Depending slightly on the parametrization used they find a surplus in . One interpretation of this surplus is that it is due to charm production. Also, the generalized vector-meson-dominance (GVMD) model (11) predicts

It is significant, therefore, that our QCD calculation gives results approximately a factor of 10 lower than this GVMD estimate or the surplus of the cross section observed experimentally. Our calculation is, in fact, consistent with charm production approximately saturating the lower bound derived from unitarity. The saturation of this lower bound for charm production was argued for on the basis of the photoproduction of strange particles and it is interesting that we have a calculation completely independent of which gives this result. Our results in Fig. 4 agree with those of Jones and Wyld (13) and we find approximate agreement with the sum rule of Shifman, Vainshtein, and Zakharov. (20)

The experimental measurement of the charm signal in collisions at Fermilab or the CERN SPS should therefore distinguish quite easily between the GVMD result and the range calculated here. If experiments find the lower values predicted here it would be significant support for the validity of simple QCD perturbation theory for this type of calculation. If the results of QCD are born out at high energy, we can see from Fig. 4 that a measurement of the charm signal at would be able to discriminate between the gluon distributions (2.7), (2.19), and (2.11). Since gluon distributions are not very accessible experimentally, this is a potentially valuable capability.

It is important to note that the cross section (2.6) is peaked near threshold where the outgoing charmed quarks do not have a large relative velocity. There is, therefore, little justification for the assumption (which we are forced to make) of ignoring the final-state interactions between the quarks. We will return to this problem in Sec. V.

Photoproduction of a Flavor Heavier Than Charm

If we assume the existence of a new flavor of quark (b) associated with the (9.5) we can use our formula (2.2) to calculate the production of hadrons carrying this new flavor by photon beams. In what follows we shall assume that this quark has charge , but our curves can be adjusted trivially to take into account other possibilities. We take and . Figure 6 shows our prediction for the energy dependence of the cross section for the photoproduction of this new flavor. The curve given is for the bag-bremsstrahlung gluon distribution (2.11). We may again compare this calculation with a simple prediction based on the vector-meson-dominance model:

If we assume that and this gives

[ Figure 6 ] The energy dependence for the cross section calculated using (2.2)--(2.5) with and charge .

In our QCD calculation using Eq. (2.2), we observe that the internal cross section for peaks near threshold. This means that we can approximately evaluate charm production in the high-energy limit where the detailed shape of the gluon distribution is not important to obtain

where and are the masses of the quarks. This gives

for , , and . This is somewhat similar to (2.17), but it is important to note that specific GVMD models make much larger predictions for . An estimate due to Margolis (21) is

Virtual Photoproduction and Deep-inelastic Scattering

We now turn to the cross section for the production of charm by virtual photons. For simplicity, we calculate only the production of charm by transversely polarized photons. The transverse cross section is related to the deep-inelastic scattering structure functions by

In the deep-inelastic region it is possible to interpret our results in terms of the absorption of a photon by a charmed quark or antiquark in the sea of the nucleon. We believe that for some range of length scales the sum of the first few terms in the series represented by the diagram in Fig. 7 may give a reasonable estimate for the momentum distribution of heavy quarks in the sea. Note that, for example, diagrams of the form of Fig. 8 are already taken into account through the choice of the gluon distribution in (2.2). Our results for are plotted at fixed as a function of in Fig. 9. The cross section shows some evidence for scaling [ a function of only] when . As in the case of real photoproduction we can compare our prediction with ones based on GVMD. According to the generalized vector-meson-dominance model, the cross section for the production of a heavy flavor is (10)

[ Figure 7 ] Diagrams for pairs in the proton.

[ Figure 8 ] Diagram for pair production absorbable into given momentum distribution.

[ Figure 9 ] The cross section for for transversely polarized virtual photons at fixed The value for QCD perturbation theory is shown with two different gluon distributions. The GVMD curve is obtained from Ref. 10.

where is the lowest vector meson containing a quark and antiquark carrying the new flavor, is the splitting between vector-meson radial recurrences, and is the generalized Riemann function. The series for the function is truncated when ; vector mesons with larger masses should not be excited.

The magnitude of the charm-production cross section predicted by the GVMD-model calculation (2.22) is significantly larger than that implied by the present QCD calculation (2.2). This difference between the predictions of QCD and GVMD is relevant to the question of the contribution of charmed hadron production to the observed rise of the deep-inelastic structure functions with at small . If the GVMD estimate for charm production is correct, then a significant fraction of the apparent scaling violations observed at small can be attributed to this process. However, our QCD calculation gives a rate which is a factor of 5--10 smaller, suggesting that it may be reasonable to interpret the experimental results for the rise in at small in terms of nonscaling effects not associated with hadronic thresholds. The size of the nonscaling effects is then in rough agreement with predictions (22) based on the short-distance behavior of QCD.

Preliminary results from an experiment at Fermilab on events of the form appear to indicate that the charm production cross section is smaller than that predicted by the GVMD model and smaller than would be needed to explain the observed rise in the structure function. (1) However, a quantitative comparison of experimental results with our predictions for the charm-photoproduction cross section based on QCD is not yet possible.

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