Electroweak gauge boson

In the $(A^1,A^2,A^3,B)$ basis with gauge transformations $U=e^{i\alpha Q}$ and $Q=T_3+Y$, the non-Abelian connection transforms by conjugation. The Lie brackets $[T_3,T_1]=i{T}_2$ and $[T_3,T_2]=-i{T}_1$ give $\delta A^1=\alpha A^2$, $\delta A^2=-\alpha A^1$, with neutral $A^3,B$. Thus $\delta A=i\alpha Q_{\mathrm{gauge}}A$ gives the displayed Hermitian matrix. The fields $W^{\pm }=(A^1\mp i{A}^2)/\sqrt{2}$ transform as $e^{\pm i\alpha }{W}^{\pm }$, while both neutral combinations are uncharged. This separates the Adjoint representation on the gauge fields from the doublet charge matrix $\mathrm{diag}(1,0)$ on the Higgs field.

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 $gv/2$.

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.

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.

The neutral massive electroweak gauge field is proportional to $g{W}_{\mu }^3-g^{\prime }{B}_{\mu }$ and has mass $v\sqrt{g^2+g^{\prime 2}}/2$.

For a vertex $(g/c_W){\gamma }^{\mu }(c_V-c_A{\gamma }^5)$ into one massless color-singlet particle-antiparticle pair, the massive-vector spin average and two-body Lorentz-invariant phase space give $\Gamma =g^2{m}_Z(c_V^2+c_A^2)/(12\pi c_W^2)$. With Standard Model neutral-current vector and axial couplings, one neutrino flavor has ${\Gamma }_{\nu }=g^2{m}_Z/(96\pi c_W^2)$ and one charged-lepton flavor has ${\Gamma }_{\ell }={\Gamma }_{\nu }(1-4s_W^2+8s_W^4)$. Chiral projectors already remove inactive neutrino helicities. These are tree-level massless results.

Each effectively massless active neutrino species contributes the same invisible Z boson partial decay width. Subtracting calculated or measured visible widths from the total width therefore measures the displayed number, assuming no additional invisible channels. The inference concerns particles lighter than $M_Z/2$ with ordinary weak couplings. It does not exclude heavier neutrinos or right-handed neutrinos that are gauge singlets. Early LEP resonance measurements were consistent with three light active species, with statistical uncertainty. See the OPAL report on its 1989 data for the original resonance-width comparison.

For the Z boson vertex $g{\gamma }^{\mu }(c_v-c_a{\gamma }^5)/(2c_W)$, sum over the three initial polarizations and final fermion spins. The symmetric Dirac trace is $4(c_v^2+c_a^2)(k_1^{\mu }{k}_2^{\nu }+k_1^{\nu }{k}_2^{\mu }-g^{\mu \nu }{k}_1\cdot k_2)$. The vector-axial interference is antisymmetric and drops out of the massive vector polarization sum. The longitudinal term drops out by current conservation for massless final particles. Contracting the remaining trace gives the displayed result. The massive-vector spin average divides it by three; two-body Lorentz-invariant phase space then gives $\Gamma =g^2{M}_Z(c_v^2+c_a^2)/(48\pi c_W^2)$ per color. These coefficients are twice those used with a vertex prefactor $g/c_W$.