Infer proton structure from inclusive lepton scattering

Let P be the target momentum and q the incoming minus outgoing lepton momentum. For spacelike exchange, Q squared is minus q squared and is positive. Bjorken x is Q squared divided by twice P dot q; inelasticity y compares P dot q with P dot k for the incoming lepton. In the target rest frame, the energy transfer nu is the lepton energy loss and y is nu divided by its initial energy.

The inclusive hadronic system has momentum P plus q. Expanding its square gives a useful consistency check relating its mass to x and Q squared. Elastic scattering has that mass equal to the target mass and x equal to one. A large Q alone does not ensure an inclusive continuum if x is too near one. Deep inelastic reasoning needs both adequate virtuality and adequate final-state mass, with target-mass and resonance effects checked for the selected kinematics.

At leading order with massless spin-half partons and electromagnetic exchange, the transverse response counts quarks and antiquarks weighted by their electric charges squared. The density q of x is defined per unit momentum fraction, not as the probability that the whole proton consists of only that quark. Up-type flavors have weight four ninths and down-type flavors one ninth. A photon cannot separately determine every flavor combination from a single proton structure function, because several flavors share the same charge.

In this simplest limit the Callan–Gross relation sets F two equal to twice x F one, corresponding to zero longitudinal structure function after neglecting target-mass effects. QCD radiation produces a nonzero longitudinal response and logarithmic scaling violations. Weak charged- and neutral-current measurements supply other flavor and parity combinations. A neutron comparison can help isolate up and down content, but extracting a neutron from nuclear data introduces nuclear corrections that are not part of the free-proton leading-order identity.

The photon-exchange double differential cross section contains a known leptonic weight multiplying F two and another multiplying F longitudinal. Changing the beam energy at fixed x and Q squared changes y and can separate these responses. At higher order the structure functions are convolutions of coefficient functions with PDFs. The physical response is independent of the factorization prescription when all terms are included consistently, whereas the individual fitted densities and coefficients depend on scheme and scale.

Approximate Bjorken scaling follows when the hard probe resolves effectively pointlike partons, but exact scale independence is not expected in QCD. Evolution accounts for leading-twist logarithms; corrections suppressed by powers of the hard scale describe additional long-distance correlations. Heavy-flavor thresholds and small-x logarithms may require specialized treatment. A toy charge-weighted calculation tests the algebra, not a global PDF extraction. Likewise the projection lab can illustrate weights and components, but contains no hadronic tensor, lepton detector or proton model.