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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.

At $y=0$, both fractions equal $x=0.0402$. Reading the original plot near this value gives representative momentum densitiesThese are approximate graphical readings, not a precision PDF fit. Undo the sea-curve scale and divide by eight flavours/antiflavours:All products have the same $1/x^2$ factor, which cancels in the fractions. The three relative weights areTheir sum is $0.3128$, givingReading the central curves more finely changes these percentages by about a percentage point, well within what the plot supports. Omitting its factor $0.05$ would substantially underestimate the sea contribution.

Use the leading-order parton model: scatter incoherently from an approximately on-shell collinear quark carrying nucleon momentum fraction $x$, neglect quark and lepton masses, use an unpolarized target, and neglect nuclear binding and higher-twist corrections. The condition $Q^2\gg m_p^2$ suppresses target-mass effects, while $Q^2\ll M_W^2$ permits the local Fermi interaction instead of the full W propagator. CKM-weighted accessible final flavours are approximated by their leading light-quark sums. The struck-quark centre-of-mass energy is $\hat{s}\simeq xs$, with $s=(k+p{)}^2\simeq 2m_N{E}_{\nu }$.

A neutrino scatters through $\nu d\to {\mu }^{-}u$ or $\nu s\to {\mu }^{-}c$, and through the corresponding crossed antiquark channels such as $\nu \bar{u}\to {\mu }^{-}\bar{d}$. The chiral charged current makes the massless quark contribution independent of $y$ and the antiquark contribution proportional to $(1-y{)}^2$. The scattering probability is weighted by the parton distribution functions; the explicit factor $x$ comes from $\hat{s}=xs$. In the usual convention $q(x)dx$ is a number probability density, while $xq(x)$ is a momentum density. Approximate Bjorken scaling leaves mainly $x$ dependence; QCD evolution adds a logarithmic $Q^2$ dependence.

The printed proton-density formula is not an isoscalar formula in the standard PDF convention. For a proton, neglecting incoming charm and taking accessible CKM sums as unity, the conventional leading-order result isFor an isoscalar target per nucleon, average proton and neutron and use charge symmetry of nucleon parton distributions: $d_n=u_p$, ${\bar{u}}_n={\bar{d}}_p$, $s_n=s_p$. This givesThe standard normalization follows equivalently by inserting $F_2^{\nu p}=2x(d+s+\bar{u})$ and $x{F}_3^{\nu p}=2x(d+s-\bar{u})$ into the charged-current structure-function formula.^[ref]

A concrete discrepancy remains even with a simple valence model: take $u_p=2h(x)$, $d_p=h(x)$ and all sea densities zero. The correctly averaged isoscalar bracket is $3h(x)$, while the printed bracket is only $h(x)$ with the same prefactor. Thus the intended helicity and $x$ dependence can be explained, but the literal printed isoscalar claim cannot be derived. If the symbols were instead defined as momentum densities, the extra explicit $x$ would also need removal.

For a valence-dominated isoscalar target, $u_v^N(x)=d_v^N(x)=q_v(x)$. A left-chiral incident neutrino has unsuppressed scattering on a left-chiral quark and the $(1-y{)}^2$ weight for a right-chiral quark. Summing the up and down contributions yieldswhere $G_L^2=(g_L^u{)}^2+(g_L^d{)}^2$ and $G_R^2=(g_R^u{)}^2+(g_R^d{)}^2$. The common normalization here uses the conventional Fermi coupling and PDFs in a; the requested $y$ shape is independent of an overall normalization convention.

Put $w={\sin }^2{\theta }_W$. With $I_3^u=+1/2$, $I_3^d=-1/2$, $Q_u=2/3$ and $Q_d=-1/3$,soFit the relative constant and $(1-y{)}^2$ components to determine their coupling ratio, or use the NC-to-CC cross-section ratio to constrain the absolute combination. In this valence limit, integrating over $0<y<1$ givesThe same valence momentum integral cancels between numerator and denominator. Experimentally, NC events have hadronic activity without the charged-current outgoing muon; known fluxes or CC normalization, muon identification and acceptance corrections are needed. Sea-quark contributions, charm thresholds, QCD corrections and departures from isoscalarity must be included for a precision weak-mixing-angle measurement.

The charged-current expressions in Section 18.1 specify the conventions used to audit a; the normalization is per target nucleon with standard number-density PDFs.

Assume approximate charge symmetry of nucleon parton distributions, the interchange of up quarks and down quarks under the isospin rotation carrying the proton into the neutron:Corrections from the up quark-down quark mass difference and the electromagnetic interaction are neglected. With the proton-only shorthand in d,If charm is active, add the same $4x(c^p+{\bar{c}}^p)/9$ to each; heavier flavors are treated similarly. These formulae use the same $Q^2$ and factorization convention for all parton distribution functions.

Under the stipulated peak-energy approximation, set $E=m_V$ separately for $V=W,Z$ in the supplied Breit-Wigner resonance cross section. For a spin-one resonance,Define the measured ratio as $R=\sigma (W\to \ell \nu )/\sigma (Z\to \ell \ell )$. With the known production prefactor ratio neglected as instructed, the spin factors cancel andSolving for the positive physical width givesIf the production prefactors are retained as $C_W,C_Z$, multiply the quantity under the square root by $C_W/C_Z$. The partial decay widths in this extraction are calculated inputs, while the total decay width being inferred need not be fixed to its Standard Model value.

The peak and integrated resonance width dependence explains why the square root is essential to the specified peak approximation. An energy-integrated narrow-width cross section contains one inverse power of the total decay width, because integrating over the resonance supplies a factor proportional to its width. That different observable leads to a different extraction formula. Real hadron-collider analyses also include parton distribution functions and the energy convolution; the present result follows the explicit approximation in the source.

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.

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.

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.

A quark or antiquark in the nonvalence component of a hadron's parton distribution functions. Such pairs do not change the net valence flavor numbers. Approximately equal light-flavor sea distributions can dominate small-x deep inelastic scattering, but exact equality is not required by their definition.

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.