= Solution
The middle panel shows the <Feynman diagram> for the leptonic <W boson> decay. With the charged-current <Lagrangian density> sign in the question, its vertex <Feynman rule> is
$$
\boxed{-\frac{ig}{2\sqrt2}\gamma^\mu(1-\gamma_5).}
$$
An overall field-phase convention can change the overall amplitude sign but cannot change the <decay width>. With the standard <Dirac field> current ordering, the amplitude is
$$
\mathcal M=-\frac{g}{2\sqrt2}\epsilon_\mu(p)\,
\overline u(q_1)\gamma^\mu(1-\gamma_5)v(q_2).
$$
The unusual order of barred fields in the printed description should be interpreted as this physical <weak charged current> matrix element.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-54-processes.png]
{title=Higgsstrahlung, leptonic W decay and photon-mediated deep inelastic scattering}
For massless final <fermions>, the <fermion spin sums> are $\sum_su_s\overline u_s=\not q_1$ and $\sum_sv_s\overline v_s=\not q_2$. The <massive-vector spin average> divides by three. Thus
$$
\overline{|\mathcal M|^2}=\frac{g^2}{24}K_{\mu\nu}T^{\mu\nu},\qquad
K_{\mu\nu}=-g_{\mu\nu}+p_\mu p_\nu/p^2,
$$
with the chiral trace
$$
T^{\mu\nu}=\operatorname{Tr}[\not q_1\gamma^\mu(1-\gamma_5)
\not q_2\gamma^\nu(1-\gamma_5)].
$$
Anticommuting $\gamma_5$ through the two following gamma matrices and using $(1-\gamma_5)^2=2(1-\gamma_5)$ gives its symmetric part
$$
T_S^{\mu\nu}=8(q_1^\mu q_2^\nu+q_1^\nu q_2^\mu-g^{\mu\nu}q_1\cdot q_2).
$$
Its remaining part is an antisymmetric epsilon tensor from the <gamma matrix trace identities>; it vanishes against the symmetric polarization tensor $K$. Direct contraction gives $-g_{\mu\nu}T_S^{\mu\nu}=16q_1\cdot q_2$ and
$$
\frac{p_\mu p_\nu}{p^2}T_S^{\mu\nu}
=8\left[\frac{2(p\cdot q_1)(p\cdot q_2)}{p^2}-q_1\cdot q_2\right].
$$
Hence
$$
\boxed{\overline{|\mathcal M|^2}=\frac{g^2}{3}
\left[q_1\cdot q_2+\frac{2(p\cdot q_1)(p\cdot q_2)}{p^2}\right].}
$$
No additional averaging over hypothetical right-handed neutrino states is appropriate: the <chiral projector> already enforces the physical helicity coupling.
On shell, $p=q_1+q_2$, $p^2=M_W^2$ and $q_i^2=0$, so $q_1\cdot q_2=M_W^2/2$ and $p\cdot q_i=M_W^2/2$. The squared <matrix element> reduces to $g^2M_W^2/3$. In the parent rest frame, integrate the spatial momentum delta first in <two-body Lorentz-invariant phase space>. The remaining radial integral is
$$
\int d\Phi_2=\frac1{(2\pi)^2}\int\frac{k^2dk\,d\Omega}{4k^2}\delta(M_W-2k)
=\frac1{8\pi}.
$$
Consequently the <massless leptonic W decay width> is
$$
\boxed{\Gamma_e=\frac1{2M_W}\frac{g^2M_W^2}{3}\frac1{8\pi}
=\frac{g^2M_W}{48\pi}\equiv\Gamma_\ell.}
$$
Lepton universality gives $\Gamma(W^-\to\mu^-\overline\nu_\mu)=\Gamma(W^-\to\tau^-\overline\nu_\tau)=\Gamma_\ell$ in the same massless approximation. The <muon> and <tau lepton> masses produce small corrections if retained.
With an identity <CKM matrix>, the allowed quark channels are $d\overline u$ and $s\overline c$. The corresponding third-generation channel $b\overline t$ is closed by the top threshold. Each open channel has three orthogonal matching-color final states, so the <color multiplicity in a decay width> gives \b[$\Gamma(d\overline u)=\Gamma(s\overline c)=3\Gamma_\ell$]. There is no initial color average because the <W boson> is colorless. The <tree-level W branching fractions> are therefore:
|| Final channel
|| Partial width
|| Branching fraction, rounded
| $e^-\overline\nu_e$
| $\Gamma_\ell$
| $1/9\simeq11\%$
| $\mu^-\overline\nu_\mu$
| $\Gamma_\ell$
| $1/9\simeq11\%$
| $\tau^-\overline\nu_\tau$
| $\Gamma_\ell$
| $1/9\simeq11\%$
| $d\overline u$
| $3\Gamma_\ell$
| $1/3\simeq33\%$
| $s\overline c$
| $3\Gamma_\ell$
| $1/3\simeq33\%$
The rounded percentages sum to 99% because of rounding. The exact fractions sum to one, with total leptonic fraction $1/3$ and hadronic fraction $2/3$. The total <decay width> is
$$
\boxed{\Gamma_W=9\Gamma_\ell=\frac{3g^2M_W}{16\pi}
=\frac{3G_FM_W^3}{2\pi\sqrt2}.}
$$
As an illustrative numerical estimate, taking $g=0.65$ and $M_W=80\,\mathrm{GeV}$ gives $\Gamma_\ell\simeq0.224\,\mathrm{GeV}$, each open quark width $\simeq0.672\,\mathrm{GeV}$ and $\Gamma_W\simeq2.02\,\mathrm{GeV}$. These are tree-level model estimates with the stated illustrative inputs, not a fit to a measured width. Strong radiative corrections increase the inclusive quark widths.
The produced <quark> and <antiquark> radiate <gluons>, generating a parton shower. As their virtualities fall towards the strong-interaction scale, perturbation theory gives way to <hadronization>: confinement converts the colored partons into color-neutral <hadrons>, typically seen as two <particle jets>, with additional radiation sometimes producing resolved extra jets. Unstable hadrons subsequently decay; the primary quarks are not observed as free particles.
Including the <CKM matrix>, the charged current couples an accessible up-type flavour $u_i$ to every kinematically accessible down-type flavour $d_j$. Therefore
$$
\boxed{\Gamma(W^-\to d_j\overline u_i)=3\Gamma_\ell|V_{ij}|^2,\qquad
u_i=u,c,\quad d_j=d,s,b,}
$$
The sums over $j$ remain $3\Gamma_\ell$ by row <unitarity>, so the massless tree-level total remains $9\Gamma_\ell$. Quark mixing redistributes the hadronic <branching fractions>; it does not create a new total color factor. If a family is above threshold, only its accessible channels enter, and the unitarity sum must not be applied to an incomplete set of final flavours.
For completeness, <CKM parameter counting> starts with $N^2$ real degrees of freedom of an $N\times N$ <unitary matrix>. This follows by counting $2N^2$ real entries and $N^2$ independent Hermitian constraints in $V^\dagger V=I$. The real-orthogonal subclass has $N(N-1)/2$ mixing angles; the remaining $N(N+1)/2$ parameters can be viewed as phases. Rephasing each up-type and down-type mass eigenfield changes $V_{ij}$ by $e^{-i\alpha_i}e^{i\beta_j}$. Of the $2N$ phases, the common phase does nothing, so $2N-1$ are removable. Thus
$$
\boxed{N_{\mathrm{physical}}=(N-1)^2,\qquad
N_{\mathrm{angles}}=\frac{N(N-1)}2,\qquad
N_{\mathrm{CP\ phases}}=\frac{(N-1)(N-2)}2.}
$$
For three generic nondegenerate families this is three angles and one <CP-violating phase>. Degenerate masses can allow additional changes of basis and reduce the count.
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