Past exam of the sub-atomic physics course of the University of Oxford / 2016 / 4 / a / Solution

For the first generation the flavor-diagonal weak neutral current permitsEach quark channel has three possible matching color charges. The weak charged current permitsThe 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.

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: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.