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At leading order in the naive parton model, dimensionless deep-inelastic structure functions depend only on Bjorken x. Quantum chromodynamics evolution introduces scale dependence.

For massless spin-one-half partons at leading order, the unpolarized electromagnetic deep-inelastic structure functions obey this relation. Radiative Quantum chromodynamics corrections and target-mass effects can modify it.

For an unpolarized electromagnetic target, Lorentz covariance and parity symmetry in quantum field theory permit a symmetric hadronic tensor built from $g^{\mu \nu }$, $P^{\mu }$ and $q^{\mu }$, where $P=P_H$. Its surviving general form isThe totally antisymmetric tensor structure ${ϵ}^{\mu \nu \rho \sigma }{P}_{\rho }{q}_{\sigma }$ is excluded by electromagnetic parity symmetry in quantum field theory for this unpolarized target. The other possible antisymmetric structure, $P^{\mu }{q}^{\nu }-q^{\mu }{P}^{\nu }$, cannot be transverse for generic scattering kinematics and hence is excluded by current conservation. Current conservation gives the Ward identities $q_{\mu }{W}^{\mu \nu }=q_{\nu }{W}^{\mu \nu }=0$, imposingTwo scalar functions remain. Taking ${\tilde{P}}^{\mu }=P^{\mu }-\nu q^{\mu }/q^2$ gives the transverse basisThese are the deep-inelastic structure functions; $x$ is Bjorken x. At fixed target mass, the two independent Lorentz scalar invariants can equivalently be chosen as $x,q^2$. In this paper $\nu =P\cdot q$ has dimensions of mass squared, rather than the alternative convention $\nu =P\cdot q/M$ used for energy transfer.

On the physical positive-energy branch, put $\omega =q^0+\xi E_P$. The massless outgoing parton obeysSince $E_{k^{\prime }}=\omega >0$ on the support and $\nu >0$, the positive-energy on-shell delta function identity givesInsert this into the parton model sum, and writeThe parton distribution functions $q_f,{\bar{q}}_f$ are number densities in momentum fraction. The contributing tensor becomesComparing with part (b), under the same leptonic tensor contraction, givesThis is the Callan-Gross relation for massless partons of spin angular momentum $1/2$ at leading order. The longitudinal deep-inelastic structure function is $F_L=F_2-2x{F}_1=0$ in this approximation. Target-mass effects and radiative Quantum chromodynamics corrections can change the relation. The leading parton model also gives Bjorken scaling: at this level the deep-inelastic structure functions depend on $x$ alone.