II. Heavy-particle Production

A. Charged-heavy-lepton Production Cross Sections

Charged heavy leptons () which cannot come from decays of weakly decaying heavier hadrons should be produced dominantly in the process

(The reaction should not contribute significantly to cosmic-ray production.) The Drell-Yan model (3) for (1) depicted in Fig. 1 provides a very adequate description of muon pair production at accelerator energies. In view of the scaling properties of this model, the extrapolation to cosmic-ray energies is not very drastic. The cross section for the production of with mass by hadrons of energy incident on nucleons at rest is given in this model by

where is the probability that the quark (with charge ) occurs in the hadron with a fraction of its momentum. We use quark distributions which give a satisfactory description of recent data (4) on , but our conclusions do not depend on their detailed form or dependence. (5) Note that since (but not ) contains valence antiquarks the cross section near threshold for is somewhat larger than that for .

[ Figure 1 ] The Drell-Yan picture for massive-lepton-pair production

B. Heavy-hadron Production Cross Sections

There is no universally accepted model for heavy-hadron () production. However, a simple model (6) based on lowest-order quantum-chromodynamics (QCD) perturbation theory probably provides an adequate estimate. This model is consistent with present limits on the production of charm by protons and photons. The subprocesses contributing to heavy-hadron production in the model are shown in Fig. 2; the most important is ( is a vector gluon, and the new quark presumably associated with any new type of hadron). Writing the differential cross section for these subprocesses as , one finds (6) that the invariant production cross section for by a hadron incident with energy on a nucleon at rest given by

where is the probability that the final carries a fraction of the momentum of the Q from which it evolved. We take

Since this ``fragmentation function'' is normalized to unity, the total -production cross section becomes

Our results are not sensitive to the form of the gluon momentum distribution assumed. (5) Note the similarity of Eq. (4) to the (successful) Drell-Yan model cross section for leptons given in Eq. (2). However, because it is dominantly gluons rather than quarks and antiquarks which react to produce the pair, the cross sections for this process are almost identical when it is initiated by protons and by pions.

[ Figure 2 ] QCD subprocesses contributing to heavy-quark production.

C. The Cosmic-ray Flux

The last two sections were devoted to a discussion of the production cross sections for heavy leptons and hadrons; to find their cosmic-ray production rates we must convolute these cross sections with the cosmic-ray flux through the atmosphere. An approximate empirical formula for the flux of hadrons of energy at a depth (in kg m---sea level corresponds to ) is (7)

This flux contains both pions and nucleons; the presence of pions will increase the heavy-lepton production rate by a factor of about 2. Secondary hadron production in the atmosphere [which is accounted for by the form of the exponential in Eq. (5)] increases all production rates by a factor of about 1.3. The total flux of a particle produced with cross section by a cosmic ray of energy is then simply

where is the total cross section.

D. Production Rates

In Fig. 3 we have plotted the expected direct pair-production cross section for heavy leptons of several masses as a function of the energy of the incident hadron. (8) Figure 4 shows our prediction for heavy hadrons. It is clear from Fig. 3 that the flux of heavy leptons would probably be very small, unless they could be generated in a hadron decay, in which case their flux would be simply the flux for that hadron multiplied by its branching ratio to the heavy lepton. When these cross sections are convoluted with the total cosmic-ray flux according to Eq. (6), we find the total flux of heavy hadrons and leptons as a function of their mass given in Fig. 5. Tolerable fits to these fluxes are given by

These results are not affected drastically by requiring that the incident hadron energy exceed some minimum. For example, with a cutoff of 10 TeV, the hadron flux is reduced significantly only for masses below about 5 GeV.

[ Figure 3 ] Cross section for proton-proton collisions to produce a pair of heavy charged leptons with various masses as a function of the incoming proton energy .

[ Figure 4 ] Cross section for proton-proton collisions to produce heavy hadrons of various masses as a function of the incoming proton energy .

Far above threshold our predictions for heavy-hadron production (Fig. 4) are rather insensitive to the details of our model. Close to threshold, however, the predicted cross section depends strongly on, for example, the assumed form of the gluon distribution. Since it is in this threshold region that most accelerator searches for heavy hadrons have been performed, we can make no meaningful comparison of their limits with the predictions of our model.

[ Figure 5 ] Total flux of stable heavy hadrons and directly produced charged leptons generated by cosmic-ray particles in the atmosphere, based on a logarithmic extrapolation of the observed rise in .

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