Past exam of the particle physics course of the University of Oxford / 2016 / 5 / 3

Lepton universality means that the weak charged current couples equally to the three charged lepton families after accounting for masses and known radiative corrections. In the ratio of two tau lepton leptonic decay widths, the parent coupling constant, mean lifetime and $m_{\tau }^5$ cancel:The universal prediction is $B_{\mu }/B_e=0.97256/0.9999993=0.9725607$. The measured ratio is $17.41/17.83=0.97644$, givingif the quoted errors are treated as independent. This is consistent with unity at about $1.2$ standard deviations. An actual precision fit must retain the measured covariance and the small radiative corrections; the rounded table alone does not supply those correlations.

The Cabibbo-Kobayashi-Maskawa matrix is the unitary change of basis between down-type quark mass eigenstates and the combinations appearing in the weak charged current. In particular, $u\bar{d}$ and $u\bar{s}$ production carry $V_{ud}$ and $V_{us}$. Exclusive tau lepton decays into a charged pseudoscalar meson giveup to small radiative corrections. The pseudoscalar decay constants specify the vacuum-to-meson axial-current matrix elements, $⟨0|\bar{q}{\gamma }^{\mu }{\gamma }^{5}u|P(p)⟩=i{f}_P{p}^{\mu }$.

The observed branching fraction ratio is $0.70/10.83=0.06464$, and the kinematic factor is approximately $0.8622$. Hence the directly determined combination isWith an independent, representative lattice input $f_K/f_{\pi }\simeq 1.193$, this becomes $|V_{us}/V_{ud}|\simeq 0.2295$, close to the source's rounded expectation $0.224/0.975=0.2297$. Conversely that expectation and the hadronic input predict a branching fraction ratio near $0.0648$. Without the decay-constant input the table does not isolate a pure Cabibbo-Kobayashi-Maskawa matrix ratio. Setting $f_K=f_{\pi }$ would give approximately $0.0455$, demonstrating the importance of flavor symmetry breaking.^[ref]

Assuming that the two listed leptonic decay categories exhaust the leptonic modes, the inclusive hadronic branching fraction is $100-17.83-17.41=64.76\%$. ThusAt the simplest massless quark level, the W boson couples to $u\bar{d}$ and $u\bar{s}$ in three color charge states, so equal quark and lepton weak coupling constant values predictThis is a rough check of color multiplicity in a decay width and universal weak-current normalization. It is much less precise than the two-lepton ratio because hadronic Quantum chromodynamics and electroweak interaction corrections are sizeable: the data require a net correction factor near $3.632/3.0024\simeq 1.21$. They must be included before extracting a quark-to-lepton coupling constant ratio. It would be incorrect to interpret the excess over three directly as a failure of weak universality. The displayed exclusive hadronic modes alone would underestimate the inclusive sum.