II. The Model

A. Charged-lepton Scattering

The lowest-order diagram for charged-lepton-nucleon scattering is shown in Fig. 1. The lowest-order electromagnetic radiative corrections come from the diagrams of Figs. 2, 3, and 4 [diagrams not involving real photon emission contribute to the cross section at through interference with Fig. 1]. Self-energy diagrams for external lines are not drawn here or in later diagrams. If the cross section for Fig. 1 is known, the corrections due to the diagrams of Fig. 2 may be calculated in a model-independent manner; the processes depicted in Figs. 3 and 4 are usually ignored on the grounds that the lepton is more violently accelerated than the hadrons and, therefore, radiates more strongly. This assumption is undoubtedly valid for scattering; we discuss its validity for the case below. Since the graphs of Figs. 1 and 2 form a gauge-invariant set, their sum does not contain infrared divergences, though any individual diagram is of course in general infrared divergent.

The radiative process may be considered as having four effects:

(1) The outgoing particle's momentum may be altered, moving events from one kinematical configuration to another.

(2) The incoming particle's momentum may be altered, thereby changing the kinematic configuration for the event, and also the probability for the interaction (though the energy dependence of the cross section).

(3) The overall cross section for the process in a particular kinematical configuration will change.

(4) In experimental studies of exclusive processes, events may be discarded because particles radiate energy and momentum, so that the experimental energy-momentum constraints cannot be satisfied. If the energy-momentum lost in this way is small (and all particles will inevitably radiate to some extent), then the imbalance will not be noticed and the event will be accepted. The evaluation of this effect requires knowledge of the experimental acceptance, resolution, and fitting procedure. In practice all these details are combined into the energy resolution of the experiment. may also be considered as the minimum detectable photon energy.) To lowest order these effects are typically proportional to . The effect (4) is not relevant for the inclusive processes that we discuss.

[ Figure 1 ] The lowest-order diagram for charged-lepton-nucleon scattering.

[ Figure 2 ] First-order electromagnetic radiative corrections to Fig. 1: ``leptonic'' diagrams.

[ Figure 3 ] First-order electromagnetic radiative corrections to Fig. 1: ``hadronic'' diagrams.

[ Figure 4 ] First-order electromagnetic radiative corrections to Fig.1: ``two-photon exchange'' diagrams.

The evaluation of the diagrams for Figs. 3 and 4 would require a model for the hadron current. The parton model is an obvious possibility. It has been used (3) to calculate due to two-photon-exchange diagrams. A more complete calculation has recently been done by Bardin and Shumeiko. (4) They find that the hadronic radiative corrections can be as large as , but are always in regions where the cross section is sizable. In addition one might expect hadronic radiative corrections to give . No such inequality has yet been observed, (5) in agreement with complete theoretical predictions. (4)

The parton model has proved adequate in explaining gross features of differential cross sections for deep-inelastic scattering, but it does not attempt to describe the evolution of the multihadron final state at large distances. It may, however, be photon interactions in this hadronic final state which form the main part of the hadronic electromagnetic radiative correction.

B. Neutrino Scattering

The basic diagram for charged-current neutrino-nucleon scattering is shown in Fig. 5, and the lowest-order electromagnetic-radiative corrections to this process are given in Fig. 6 [diagrams in which photons couple to the are typically suppressed by a factor and are consequently ignored]. Unlike the diagrams for charged-lepton scattering in Figs. 2, 3, and 4, those of Fig. 6 cannot be separated into gauge-invariant sets. They must therefore all be included in order to obtain an infrared-finite result. A model for the hadronic current is hence obligatory in these calculations. We use the parton model, and so the radiative-correction diagrams are those of Fig. 7. We take the mass to be infinite (so that the weak interactions are pointlike), since the finiteness of its mass will not have appreciable effects at the energies we consider. The diagrams of Fig. 7 have been calculated in the deep-inelastic limit by Kiskis. (2) He finds that some of the radiative correction may be absorbed into renormalization of the weak coupling constant. The effective mass of the final parton is a free parameter (the mass of the initial parton is completely constrained by parton-model kinematics; if no photon is emitted it is approximately , where, as usual, ), as is the initial parton distribution. In Fig. 8 we show the correction factor by which theoretical predictions should be changed

as a function of and for various possible choices of the free parameters in the model, with an incoming antineutrino beam of energy 100 GeV. Figure 8(a) shows the corrections computed with the values of these free parameters as used in the rest of the paper (Field-Feynman quark distribution functions (6) and outgoing parton mass 0.3 GeV). Figures 8(b) and 8(c) are for outgoing parton masses of 0.001 and 0.9 GeV, respectively, and Fig. 8(d) shows the correction obtained using the quark distribution functions of Barger and Phillips. (7) Finally in Fig. 8(e) we show the results of using integer charged partons, although the success of the parton model essentially requires the partons to have the usual quark quantum numbers. These figures indicate that changes in the parameters in the model we use do not have any significant effect on our results.

[ Figure 5 ] The lowest-order diagram for charged-current scattering.

[ Figure 6 ] First-order electromagnetic radiative corrections to Fig. 5.

[ Figure 7 ] Parton-model diagrams for electromagnetic radiative corrections to charged-current scattering.

[ Figure 8 ] The radiative-correction factors for deep-inelastic scattering with an incoming neutrino energy of 100 GeV for various choices of the free parameters in the model. (a) Outgoing parton mass ; Field-Feynman quark distribution functions (standard set of parameters). (b) Outgoing parton mass ; Field-Feynman quark distribution functions. (c) Outgoing parton mass ; Field-Feynman quark distribution functions. (d) Outgoing parton mass ; Barger-Phillips quark distribution functions. (e) Outgoing parton mass ; Field-Feynman quark distribution functions; integer-charged quarks.

It should be noted that the appearance of terms proportional to the logarithm of the parton mass......consequence of the fact that it has been......that the momentum of the outgoing muon......measured with arbitrary precision. If one allows for finite resolution in the detection of the muon, then the mass singularities disappear, and the logarithms of the parton mass are replaced by logarithms of the resolution.

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