 Parton convolution normalization for a hadronic tensor

In the incoherent massless collinear parton approximation, $f_f(\xi )d\xi$ is the number of quarks in a momentum-fraction interval, not a covariantly normalized external state weight. A single-particle response divided by its external-state normalization is proportional to $w/(2k^0)$; the hadron normalization is $2P^0$. Thus converting a number-weighted parton sum to the target tensor contributes $P^0/k^0=1/\xi$. This gives the displayed convolution, including antiquarks. The spin-averaged electromagnetic parton tensor and $\delta (2\xi P\cdot q-Q^2)=\delta (\xi -x)/(2P\cdot q)$ then give $F_1=\frac{1}{2}{\sum }_f{Q}_f^2(f_f+{\bar{f}}_f)$ and $F_2=x{\sum }_f{Q}_f^2(f_f+{\bar{f}}_f)$, proving the Callan-Gross relation at this order. Target masses, transverse momentum and radiative effects are beyond this leading approximation.