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The leading subprocess is $u\bar{d}\to W^{+}$. There are two beam assignments, and charge conjugation relates antiproton PDFs to proton PDFs. ThereforeIn a narrow-width leading-order normalization the prefactor is $\pi g^2/(12s)$ when $|V_{ud}|=1$; only the PDF dependence is needed for the fractions below.

Write the common sea PDF as $S=\bar{u}=\bar{d}=\bar{s}=\bar{c}=\Sigma /8$. The Valence and sea decomposition of proton-antiproton W production gives the three parton luminositiesThe factor two in the sea-sea term includes both beam assignments. The antiproton's valence antiquarks make valence-valence Annihilation possible.

With the four-momenta in a, $q^2=-2E{E}^{\prime }(1-\cos \theta )$, $p_2\cdot q=m_p(E-E^{\prime })$, and $p_2\cdot p_1=m_{p}E$. Thus the measured beam energy $E$, scattered electron energy $E^{\prime }$ and scattering angle $\theta$ determineHere $Q^2$ measures the spacelike momentum transfer, $y$ is the fractional energy transfer, and the Bjorken scaling variable $x$ is the struck parton's longitudinal momentum fraction in the leading collinear description. The hadronic invariant mass satisfies $W^2=(p_2+q{)}^2=m_p^2+Q^2(1/x-1)$; large $Q^2$ and $W^2$ select deep inelastic scattering.

A beam spectrometer measures $E$ before a thin stationary hydrogen target. Tracking layers on both sides of a magnetic spectrometer measure the outgoing electron direction, charge and curvature; an electromagnetic calorimeter measures its energy. Matching the track to its particle shower, with downstream muon rejection where useful, separates the scattered electron from hadronic tracks. A downstream hadronic detector measures the hadronic activity and supports event-quality checks. The sketch in a shows these separate measurements.

The scattered electron provides a clean, single-particle reconstruction. Reconstructing the entire hadronic system requires summing many charged and neutral fragments, correcting unequal particle calorimeter response and losses through beam holes, and sometimes accounting for escaping neutrinos. Thus the leptonic definition is experimentally convenient. It is not a fundamental requirement of Lorentz invariance: exact reconstruction of $p_X$ gives the same $q=p_X-p_2$, and hadronic or combined reconstructions can be preferable in some kinematic regions.

At leading order in QED, the electron exchanges one spacelike virtual photon with a charged parton inside the proton.

The two Feynman vertices contribute $e$ and $e{e}_q$, while the photon propagator contributes $1/q^2=-1/Q^2$. Hence the scattering amplitude scales as $e^2{e}_q/Q^2$ and its squared absolute value as $e^4{e}_q^2/Q^4$. Since the fine-structure constant is $\alpha =e^2/(4\pi )$, the electromagnetic coupling dependence is ${\alpha }^2/Q^4$. The numerical $4\pi$ in the stated differential scattering cross-section additionally uses the flux, Lorentz-invariant phase space, azimuthal integration and the convention defining the deep-inelastic structure function; it cannot be deduced from the propagator alone.

For each measured $x,Q^2$ bin form the reduced quantityPlot this against $Q^2$ at several fixed $x$ values, normally using a logarithmic $Q^2$ axis. Bjorken scaling predicts an approximately horizontal curve at each $x$, or equivalently collapse of different-$Q^2$ data onto one $F_2(x)$ curve. Raw cross sections do not collapse because their explicit kinematic factors vary. QCD predicts logarithmic scaling violations, so exact independence is only the ideal parton model limit. The stated extraction also assumes negligible longitudinal deep-inelastic structure function, as in the leading massless spin-half parton approximation.

This labelled part supplies a definition rather than a separate task. Its notation is interpreted as an incoherent charge-squared sum: the photon couples to a quark with electric charge $e{e}_q$, and the squared scattering amplitude weights its contribution by $e_q^2$. To make the antiquark convention explicit, letThen, restricting the displayed example to three active light quark flavors, the deep-inelastic structure function isThus the source's sum must either include antiquarks as separate partons or define each $q^p$ to include its antiquark distribution. It must not silently omit the sea quarks, especially in the small-$x$ limit used below.

Resolving a length requires momentum transfer $q$ of order $\hslash /\ell$. Taking a hard relativistic scale $qc\sim 4\mathrm{TeV}$ givesThe full de Broglie wavelength convention gives $hc/E\simeq 3.1\times 10^{-19}\mathrm{m}$, differing by $2\pi$; the intended scale is roughly $10^{-19}\mathrm{m}$, far below a proton radius. Two opposing beams have a larger total collision energy, but proton energy is shared among partons: the actual resolution depends on the hard scattering momentum transfer, not simply on beam energy alone.

An ideal Bjorken scaling model has $F_2(x,Q^2)=f(x)$: varying $Q^2$ at fixed Bjorken x leaves this dimensionless deep-inelastic structure function unchanged. The comparison requires extracting a deep-inelastic structure function, rather than plotting raw event counts. Quantum chromodynamics predicts departures from exact Bjorken scaling; finite hadron masses also limit the idealization.

For an original shape comparison, choose $f(x)=x(1-x{)}^3$ and define $S(Q^2)=F_2(0.3,Q^2)/F_2(0.3,Q_0^2)$ with $Q_0^2=2{\mathrm{GeV}}^2$. The ideal model has $S=1$. To display a contrasting size-dependent suppression, define an illustrative elastic-form-factor shape $G(Q^2)=(1+Q^2/{\Lambda }^2{)}^{-2}$ with the freely chosen ${\Lambda }^2=1{\mathrm{GeV}}^2$. Its normalized squared magnitude isThus $A(5{\mathrm{GeV}}^2)=1/16$, whereas $S(5{\mathrm{GeV}}^2)=1$. The elastic-form-factor shape is a comparison of suppression mechanisms, not an elastic differential scattering cross-section assigned to $x=0.3$: actual elastic scattering from a proton has $W=M$ and $x=1$. Neither curve is measured data or a fit to the SLAC-MIT experiment.

The experimental interpretation required more than a flat curve. The angular dependence and small longitudinal response supported spin one-half partons, expressed at leading order by the Callan-Gross relation $F_2=2x{F}_1$. Identification with fractionally charged quarks also drew on later electron, muon and neutrino comparisons. A scaling plot alone does not determine the individual electric charges of constituents.^[ref]

The leading parton model formula already shows the ambiguity: $F_2=x{\sum }_f{Q}_f^2(q_f+{\bar{q}}_f)$ combines squared electric charges with unknown parton distribution functions. One measured function cannot independently fix all these quantities. Confinement also explains why an inclusive final hadron state need not contain an isolated outgoing quark.