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Treat a nucleon as incoherent collinear spin-half quarks and antiquarks carrying fractions $x$ of its momentum. Neglect their masses and transverse motion, use a charged-current four-fermion interaction, and take $Q^2\ll m_W^2$. The subprocess energy is $\hat{s}\simeq xs$. Antineutrino scattering on a quark has the same selected-current helicity factor $(1-y{)}^2$ as in part a; crossing to an antiquark changes the allowed spin pairing and gives a flat distribution. Multiplying the elementary strength by the respective parton densities givesHere $q,\bar{q}$ are the effective weak flavour combinations with the isoscalar proton-neutron average and CKM weights absorbed into their definitions. With only light flavours and charge symmetry, the relevant up and down combinations coincide on an isoscalar target. Rescaling those density definitions requires rescaling the prefactor consistently.

The equation is a leading-order scaling result: it neglects logarithmic QCD evolution, heavy-quark thresholds, higher-twist and target-mass corrections, nuclear modifications, and weak-propagator suppression. The current produces ${\mu }^{+}$ for an incident muon antineutrino. At higher momentum transfer the weak propagator supplies a factor approximately $[m_W^2/(m_W^2+Q^2){]}^2$, and at low energy resonance and quasielastic scattering invalidate a pure DIS description.

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.

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.