= Solution
For the first generation the flavor-diagonal <weak neutral current> permits
$$
Z\longrightarrow e^-e^+,\quad\nu_e\bar\nu_e,\quad u\bar u,\quad d\bar d.
$$
Each <quark> channel has three possible matching <color charges>. The <weak charged current> permits
$$
W^+\longrightarrow e^+\nu_e,\ u\bar d;\qquad
W^-\longrightarrow e^-\bar\nu_e,\ d\bar u.
$$
The two charges have equal <partial decay widths> to charge-conjugate channels to the accuracy needed here. <Lepton universality> and <color multiplicity in a decay width> give one unit of width for a massless lepton pair and three for a <quark> pair, apart from <quark> mixing and radiative corrections.
\b[If only the first generation is included], the <branching fractions> are $1/(1+3)=1/4$ to the lepton channel and $3/4$ to <quarks>. For the actual accessible <Standard Model> channels, there are three lepton units and two accessible up-type <quark> rows, $u$ and $c$, each with three colors. The top <quark> is too heavy. Unitarity of the <CKM matrix> sums over accessible down-type partners, giving nine units in total:
$$
\boxed{B(W\to\ell\nu)\simeq\frac19\ \text{for one lepton flavor},\quad
B(W\to\text{any leptons})\simeq\frac13,\quad
B(W\to\text{any quarks})\simeq\frac23.}
$$
These fractions concern a charged lepton together with its appropriate <neutrino>; “lepton pair” does not mean two charged leptons in a $W$ decay. The <Z boson> has different vector and axial couplings for different fermions, so the same simple equal-unit rule cannot be transferred to all its <partial decay widths>.
Back to article page
