W boson ($W^{\pm }$)

The charged electroweak gauge fields are $W_{\mu }^{\pm }=(W_{\mu }^1\mp i{W}_{\mu }^2)/\sqrt{2}$. After electroweak symmetry breaking they have electric charges $\pm 1$ and mass $gv/2$.

Neglect final-state masses and QCD corrections. Each of the three leptonic W boson decay channels contributes one common unit of width. Each open quark channel contributes $3|V_{ij}{|}^2$, with the factor three counting final colours. The top quark is kinematically inaccessible in an on-shell W decay because $m_t>m_W$. Thus $B(W\to \ell \nu )=1/D\simeq 1/9$ and $B(W^{-}\to b\bar{u},b\bar{c})=3(|V_{ub}{|}^2+|V_{cb}{|}^2)/D$. The large $V_{tb}$ does not open a forbidden top-quark channel.

In unitary gauge, the free massive W boson has the displayed momentum-space propagator. At external momentum small compared with $M_W$, its leading term is $i{g}_{\mu \nu }/M_W^2$. Inserting it between two weak charged current vertices produces the local Fermi interaction, with $G_F/\sqrt{2}=g^2/(8M_W^2)$ when each vertex is written with $1-{\gamma }_5$. Higher momentum terms give higher-dimensional operators; other gauges distribute the longitudinal contribution differently while giving the same physical amplitudes.

With three massless lepton families, two accessible up-type quarks, three colors, and no Quantum chromodynamics radiative correction, a W boson has total decay width $9{\Gamma }_{\ell }$. With an identity CKM matrix, each leptonic channel has branching fraction $1/9$ and each of the two allowed quark-flavour channels has $1/3$. With mixing, each accessible quark channel is weighted by $|V_{ij}{|}^2$, while row unitarity preserves the total at this order if all down-type channels are accessible.

The chiral weak charged current vertex $-ig{\gamma }^{\mu }(1-{\gamma }_5)/(2\sqrt{2})$ gives the spin-averaged squared matrix element $g^2{M}_W^2/3$ for an on-shell W boson decaying to a massless charged lepton and antineutrino. Integrating two-body Lorentz-invariant phase space gives ${\Gamma }_{\ell }=g^2{M}_W/(48\pi )$. The massive-vector spin average divides by three; an extra neutrino-helicity factor would incorrectly double count the chiral projector.

For a resting W boson with real spatial polarization vector along $z$, the fermion spin sum gives $\sum |\mathcal{M}{|}^2=g^2{M}_W^2{\sin }^2\theta /2$ for massless charged lepton and antineutrino momenta. The two-body Lorentz-invariant phase space gives the displayed angular decay rate. Since $\int {\sin }^2\theta d\Omega =8\pi /3$, the integrated width is $g^2{M}_W/(48\pi )$, equal to the unpolarized width. The massive-vector spin average changes the angular distribution, not the total width. No initial spin average is taken for a specified pure polarization.

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.

Ignoring accessible quark masses and strong corrections, the W boson flavor width is $\Gamma (W^{+}\to q{\bar{q}}^{\prime })=N_c{G}_F{M}_W^3|V_{q{q}^{\prime }}{|}^2/(6\pi \sqrt{2})$. The CKM matrix fixes the flavor factor, while the color multiplicity in a decay width supplies $N_c=3$. Only the up and charm rows are accessible for an on-shell W boson, giving a total hadronic width $G_F{M}_W^3/(\pi \sqrt{2})$ by row unitarity.