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A particle with zero invariant rest mass. A positive-energy massless state has $E=|\mathbf{p}|$ in natural units and has no rest frame. Its finite-helicity representations differ from massive spin representations.

For an isoscalar valence target and massless leading-order scattering, $R_{\nu }=G_L^2+G_R^2/3$, where $G_{L/R}^2=(g_{L/R}^u{)}^2+(g_{L/R}^d{)}^2$. The factor $1/3$ is ${\int }_0^1(1-y{)}^{2}dy$. With Standard Model quark couplings and $w={\sin }^2{\theta }_W$, this becomes $1/2-w+20w^2/27$. Sea quarks and threshold, nuclear and QCD corrections matter beyond this limit.

Treat the electron and electron antineutrino as effectively massless particles in the centre-of-momentum frame, retain only the charged current, and use the Fermi interaction: $m_e^2\ll s\ll M_W^2$. Take ${\theta }^{\ast }$ between incoming and outgoing electron antineutrino directions. The V-A interaction selects a right-helicity electron antineutrino and a left-helicity electron. Their opposite incoming directions give total angular momentum projection one. The outgoing pair has the same projection along its own axis. The Wigner D-matrix element is $d_{11}^1({\theta }^{\ast })=(1+\cos {\theta }^{\ast })/2$, so the squared helicity amplitude is proportional to $(1+\cos {\theta }^{\ast }{)}^2$. The inactive initial electron helicity is included in the unpolarized normalization supplied by the paper.

The boost from the centre-of-momentum frame to the laboratory frame has $\beta =E/(E+m_e)$, while the incoming electron antineutrino has $E_{\ast }=(s-m_e^2)/(2\sqrt{s})=m_{e}E/\sqrt{s}$ and $s=m_e^2+2m_{e}E$. ThusThe requested identity therefore uses $E\gg m_e$; it is not exact at finite target mass. With $c_{\ast }=\cos {\theta }^{\ast }$, the absolute Jacobian determinant is $|dy/d{c}_{\ast }|=1/2$, givingThe suppression at backward scattering follows from angular momentum conservation: rotating projection $+1$ through $\pi$ cannot leave projection $+1$ along the original axis. Replacing $1+c_{\ast }$ by $2(1-y)$ gives the supplied $(1-y{)}^2$ dependence.

Neglect the participating quark and lepton masses, so that the weak charged current selects a left-helicity $u$ and a right-helicity $\bar{d}$. Choose $+z$ along the proton beam in the partonic centre-of-momentum frame. Then the up quark moves along $+z$ with spin projection $-1/2$, while the down antiquark moves along $-z$ with helicity $+1/2$ and also has spin projection $-1/2$ along $+z$. ThusThe collinear two-body initial state has no orbital projection about its collision axis. In this approximation the produced W boson therefore has the indicated transverse polarization; its longitudinal spin state is absent. The unpolarized hadron beams contain other constituent spin states, but these do not participate in this massless charged-current subprocess.

In the final state the Positron has helicity $+1/2$ and the electron neutrino has helicity $-1/2$. Along the outgoing charged-lepton direction their combined angular-momentum projection is $+1$, because the oppositely moving neutrino has spin projection $+1/2$ on that axis. Finite masses, noncollinear production and sea contributions modify the ideal configuration.

The ordering of Weyl components depends on the supplied basis transformation. In this paper use $u_W=U{u}_D=(\xi ,\eta {)}^{\mathsf{T}}$. Applying the printed $U$ to the printed Dirac matrices givesThus $\xi$ is right chiral and $\eta$ left chiral in this ordering. The free massless Weyl equations areThey describe spin-one-half massless fields in the two inequivalent chiral spinor representations of the Lorentz group. Their rotation generators agree, but their boost generators have opposite signs; spatial parity interchanges the two. A positive-energy particle in the $\xi$ field has helicity $+1/2$, and one in the $\eta$ field has helicity $-1/2$. Their antiparticles have the opposite helicities, so a Weyl field is not a field containing only one helicity for both particles and antiparticles.

Neutrinos provide the observed ultrarelativistic approximation to a left-chiral particle and right-helicity antiparticle. However, neutrino oscillation establishes nonzero neutrino mass differences, so physical neutrinos are not all exactly massless Weyl particles. Chiral Weyl fields occur in the Standard Model, but an isolated exactly massless elementary Weyl particle is not established by neutrino observations. Whether neutrino mass eigenstates are Dirac or Majorana remains a separate question.^[ref]

With the gamma matrices derived in a, the momentum-space Dirac equation readsFor a positive-energy particle at rest, $E=m>0$ and $\mathbf{p}=0$, so $\xi =\eta$. The Dirac adjoint in this basis is $\bar{u}=({\eta }^{†},{\xi }^{†})$, and the specified normalization givesTherefore, for any normalized two-component spinor $\chi$,Equivalently, choosing ${\chi }_{+}=(1,0{)}^{\mathsf{T}}$ and ${\chi }_{-}=(0,1{)}^{\mathsf{T}}$ gives the two independent rest-frame spin states. A strictly massless particle has no rest frame; this step requires $m>0$.

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 the stationary target, $p=(m_N,0)$. The invariant definition immediately givesNo massless approximation is needed for this laboratory identity when binding and target recoil conventions are neglected.

For the angular relation, specify the neutrino-parton centre-of-momentum frame and neglect masses there. The incoming and outgoing leptons then each have energy $\sqrt{\hat{s}}/2$. Their momentum transfer satisfiesSince $Q^2=xys$ and $\hat{s}=xs$, cancellation givesThis identifies the antiquark weight $(1-y{)}^2$ with the corresponding squared angular factor. The relevant massless condition is sufficient energy in the lepton-parton subprocess; $E_{\nu }\gg m_{\mu }$ alone does not control every endpoint configuration.

The frame qualifier matters: the complete neutrino-nucleon centre-of-mass final state contains a variable-mass hadronic system. In that frame, still neglecting masses but retaining $x$, direct two-body kinematics for $\mu +X$ gives insteadFor example $x=1/2$ and ${\theta }_{\nu N}=\pi /2$ give $y=2/3$, not $1/2$. Thus the simple angle-only relation implicitly uses the partonic frame.

Sections 14.2, 14.4 and 14.9.2 distinguish massless Standard Model fields, massive neutrino propagation and the Dirac–Majorana question used in a.

Use coordinates $X^{\mu }=(ct,\mathbf{x})$ and the mostly-plus Minkowski metric ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$. Along a future-directed timelike world line, proper time satisfiesThe four-velocity and four-acceleration areThe Minkowski norm is fixed: $U^{\mu }{U}_{\mu }=-c^2$. Differentiating with respect to proper time, using the constant Minkowski metric, givesIn the instantaneous rest reference frame, $U^{\mu }=(c,0,0,0)$, so the orthogonality says $A^0=0$ there. This construction uses a timelike world line; a massless particle has no rest-frame proper time parameter.

For an on-shell W with zero transverse recoil and massless leptons, $p_T=(M_W/2)\sin \theta$. Mapping both angular roots to one transverse momentum produces an integrable inverse-square-root edge. For a normalized polar distribution $3(1-\cos \theta {)}^2/8$, the density in $x=2p_T/M_W$ is $3x(2-x^2)/(4\sqrt{1-x^2})$. Finite width, transverse recoil and detector resolution smear the edge.

The free massless relativistic field equation for one chiral two-component spinor. With mostly-minus metric, the left-chiral equation has the minus sign and the right-chiral equation the plus sign. The positive-energy particle helicities are respectively $-1/2$ and $+1/2$; antiparticle helicities are opposite.