We have measured the inclusive deep inelastic scattering of muons on nucleons in iron using beams of 93 and 215GeV muons. To perform this measurement, we have built and operated the Multimuon Spectrometer (MMS) in the muon beam at Fermilab. Using the known form of the radiativelycorrected electromagnetic cross section, we extract the struc- A. KinematicsThe Feynman diagram for deep inelastic lepton-hadron scattering in the lowest order of QED is shown in Fig. 1, together with a summary of our kinematical notation. To this order, the process is described as the exchange of one virtual photon. QED allows us to calculate unambiguously what happens at the leptonic (upper) vertex. The goal of our experiment is to uncover what happens in the region surrounding the hadronic vertex.We will be studying inclusive scattering, p.N -+ p.X, with no reference to any particular hadronic final state. This means that the only relevant 4-vectors at the hadronic vertex are p and q, and the only Lorentz-invariant quantities are q 2 and p · q (and p 2 = MJ.). While isolating the hadronic vertex is a productive move toward understanding the scattering process, experimentally it is imp_ortant to note that these quantities can be measured 3 ' < using only the initial and final muons:Q 2 = -q 2 = 4EE'sin 2 ~. where E, E', and 9 are the initial and final muon energies and the muon scattering angle as measured in the laboratory frame (the target rest frame). Terms containing the lepton mass have been neglected. Another useful quantity is the invariant mass of the hadronic final state: W 2 = (p + q) 2 = MJ. + 2MNv-Q 2 • The elastic limit is W 2 = MJ. or Q 2 = 2MNV. Resonances appear at fixed W 2 near this limit. Figure 2 shows the region of the Q 2 -v plane accessible to inelastic scattering at fixed incident energy E.It is convenient to describe another set of variables that are Q 2 and v scaled by their maximum values, neglecting lepton masses:v = Q 2 j2MNE = xy.
B. Cross section and structure functionsBy demanding Lorentz covariance and gauge invariance we can translate 7 • 8 the diagram of Fig. 1 directly into an expression for the spin-averaged inclusive cross section:dQ2dv-~ 2Wl(Q ,v)sm 2 + W2 (Q 2 ,v)cos 2 2 ], (3) in terms of the two unknown "structure functions" wl and w2. scalar functions that describe the hadronic electromagnetic current. We treat the incident muon as a source of virtual photons and write the cross section as where T and L refer to transversely and longitudinally polarized virtual ph'?tons and K is a flux factor. Defining R = uL/uT we can eliminate W 1 in favor of R in the cross section Eq. 3. The advantages of this substitution will be described in the next section where it is shown that for some cases of interest R is expected to be small. For now we record the cross section in its new form, The approximate form comes from taking the Bjorken limit where energies (E,Q 2 ,v)-+ oo with x and y-finite. 10 In the kinematic region covered by our data, making such an approximation has a maximum effect of < ! % on our ...