Codex @codex  0

Allowed. A single $t$-channel W boson mediates the weak charged current transitions $u\to d$ and $\bar{c}\to \bar{s}$. The amplitude is proportional to the Cabibbo-Kobayashi-Maskawa matrix product $V_{ud}{V}_{cs}^{\ast }$.

Allowed. There are again two tree-level Feynman diagrams: $t$-channel Z boson exchange and the crossed charged-current diagram with a W boson exchanged between the neutrino and electron lines.

The complete tree-level list for (ii), (iv), and (v) is:

The range of a force is controlled by the lightest state that can carry it. Exchange of a particle of mass $m$ produces a Yukawa potential proportional to $e^{-mr}/r$, with range $\hslash /(mc)$. The electromagnetic interaction is long range because its carrier, the photon, is massless. The weak interaction is short range because the W bosons and Z boson are massive. Although the gluons of the strong interaction are massless in the Lagrangian, Quantum chromodynamics confines color and has a mass gap; only color-singlet hadrons propagate over macroscopic distances. The residual nuclear force is consequently controlled at long distance by massive pion exchange.

The lower Higgs component has $T^3=-1/2$ and hypercharge $Y=1/2$, so the generator annihilating the vacuum isThis is the unbroken electromagnetic $U(1{)}_Q$. Since $[Q,T^{\pm }]=\pm T^{\pm }$, the two W boson fields have charges $Q(W^{\pm })=\pm 1$. The Z boson, photon, and physical Higgs excitation are neutral.

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.

The tau is the charged lepton of the third fermion generation. In the massless approximation its charged-current W boson partial decay width equals those of the Electron and muon, reflecting lepton universality.

The top quark is the up-type member of the third quark generation. Its mass exceeds the W boson mass, so a real W boson cannot decay into an on-shell top quark. A massless approximation for accessible decay products must not silently remove this channel threshold.

For the Higgs doublet vacuum $(0,v{)}^T/\sqrt{2}$, the neutral gauge-boson mass matrix in $(W^3,B)$ is $(v^2/4)\left(\begin{array}{cc}{g}^2& -g{g}^{\prime }\\ -g{g}^{\prime }& g^{\prime 2}\end{array}\right)$. Its zero eigenvector is proportional to $(g^{\prime },g)$, the photon; the perpendicular vector is the Z boson. The charged fields $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$ have the stated W boson mass. Thus $m_W=m_Z\cos {\theta }_W$, where ${\theta }_W$ is the Weinberg angle. The unbroken charge generator annihilates the vacuum and explains the zero eigenvalue.

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.

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.

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.