Parton model

A high-energy hadron is treated as constituents scattering incoherently. In electromagnetic deep inelastic scattering, the charged constituents are quarks and antiquarks.

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.

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

Quantum chromodynamics makes parton distribution functions depend on resolution scale through gluon radiation and parton splitting. Thus Bjorken scaling and the linear-energy neutrino cross section are approximate leading-order patterns rather than exact laws.

Dependence of a deep-inelastic structure function on $Q^2$ at fixed Bjorken scaling variable. In QCD, radiation and gluon splitting evolve the parton distribution functions with the resolution scale, producing logarithmic departures from the simplest parton model scaling limit.

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.

The leading high-energy approximation neglects quark and target masses and transverse constituent momentum. A lightlike reference momentum is used for the hadron; this does not describe a massless particle at rest.

In the leading massless collinear parton approximation, $P^2=0$ and $q^2=-Q^2$ imply $(\xi P+q{)}^2=2\xi P\cdot q-Q^2$. With Bjorken x $x=Q^2/(2P\cdot q)$ and $P\cdot q>0$, the Dirac delta change-of-variable formula yields the displayed identity. The positive-energy branch of the final parton also gives $\delta (E_q+\xi E_P-E_k)/E_k=2\delta ((E_q+\xi E_P{)}^2-E_k^2)$ locally on physical support. Keeping target mass instead introduces ${\xi }^2{M}^2$ and changes the root and Jacobian.

A density of partons with a given flavour and momentum fraction at a specified parton factorization scale. It is a number density, not generally a distribution normalized to one.

A nuclear parton distribution describes parton momentum fractions in a bound nuclear target. Its normalization must specify whether it is per nucleus or per nucleon and how $x$ is defined. Nuclear binding, shadowing and other nuclear modifications mean it need not equal a simple sum of free-nucleon distributions.

In the collinear approximation, a parton carries a fraction $0<x<1$ of its parent hadron's longitudinal momentum. A parton distribution function $f_i(x,Q^2)$ gives its distribution at the specified factorization scale. For production of one on-shell particle with no transverse recoil, $x_{+}{x}_{-}s=M^2$ and $x_{\pm }=(M/\sqrt{s})e^{\pm y}$, where $y$ is its rapidity.

For charged current production $u\bar{d}\to W^{+}$, charge conjugation of the antiproton distributions gives $u(x_{+})d(x_{-})+\bar{d}(x_{+})\bar{u}(x_{-})$. With a common sea quark distribution $S$, the valence quark terms are $u_v{d}_v$, the mixed terms are $u_{v}S+S{d}_v$, and the two sea assignments give $2S(x_{+})S(x_{-})$. Omitting the second beam assignment undercounts the pure sea contribution by two.

The approximate equality of proton and neutron parton distribution functions after interchanging up quarks and down quarks, including their antiquarks. It follows from the corresponding isospin symmetry when electromagnetic and light-quark mass-difference effects are neglected. Isospin-singlet sea distributions are unchanged.

The parton distribution function is a number density. Integrating the quark distribution minus the antiquark distribution counts the net flavour charge of the hadron, since every sea quark-antiquark pair contributes zero. The corresponding flavour-vector conserved current counts particles minus antiparticles. For a proton, the net up/down numbers are two and one; for a neutron they are one and two. The rule fixes the difference of distributions, not their sum, and does not imply absence of a sea.

With only up/down valence quarks and no antiquarks, the quark number sum rule and leading parton model expression give the charge-squared number moment $\int F_2(x)dx/x={\sum }_f{Q}_f^2{N}_f$. The squared charges are $4/9$ for up and $1/9$ for down, so the proton has $8/9+1/9=1$ and the neutron has $4/9+2/9=2/3$. Restricting the flavour list to up/down does not itself remove antiquarks; the no-sea assumption is essential for these individual moments.

The scale at which collinear contributions are assigned to parton distribution functions rather than the hard scattering coefficient. The separate factors depend on this scale, while a consistently calculated physical differential scattering cross-section does not.