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

The local Fermi interaction gives a three-body leptonic decay width of the format tree level with negligible neutrino masses and universal weak coupling constant values. The fifth power follows from the dimension of $G_F$ and the three-body phase space; $f$ results from integrating the V-A interaction matrix element over the reduced phase space when the charged daughter has nonzero mass. It is more than a purely geometric volume correction because the spin-dependent matrix element is included.

As $z\to 0$, $f(z)\to 1$; increasing daughter mass reduces the available energy and suppresses the rate. The largest ratio is $m_{\mu }/m_{\tau }$, so the $\tau \to \mu$ correction is visibly below one. The electron is extremely light relative to either parent. Using the lepton masses givesThese reproduce the supplied factors to their stated accuracy. Radiative and finite-W boson corrections are separate small corrections when a precision lepton universality test is performed.

At the Z boson resonance, a tau lepton pair produces two approximately opposite narrow hemispheres, usually with one or three charged tracks per decay, and missing energy from neutrinos. Multihadronic Z boson decays have much higher track multiplicity. Direct $e^{+}{e}^{-}$ and ${\mu }^{+}{\mu }^{-}$ events have two tracks but almost the full beam energy in an electromagnetic calorimeter or muon detector; the tau lepton daughters typically have smaller visible energy. Track directions, calorimetric energy, missing transverse momentum and particle identification together control these backgrounds.

Use a tag-and-probe method: select one hemisphere with an independently understood tau lepton tag, then classify the opposite hemisphere into mutually exclusive decay categories. Correct the category counts for efficiency, migration and background, including an explicit remainder for modes absent from the displayed table. Conditional on a genuine tag, the opposite-hemisphere probabilities sum to one. This avoids assuming that a raw event count equals the number of decaying tau leptons.

An absolute cross-check uses single and double tags. If $S_i$ counts tagged hemispheres and $D_{ii}$ counts events with both hemispheres in mode $i$, thenThe double-tag efficiency need not factorize. Fit all categories with their efficiencies and correlations, imposing the inclusive normalization. The nine displayed entries sum to $96.18\%$, not $100\%$. Renormalizing those entries alone to $100\%$ would bias every quoted branching fraction.

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.

The isospin coupling $1\otimes 1\to 1$, with third component $-1$, has the normalized stateThe two Clebsch-Gordan coefficients have equal squared magnitude. Neglecting electromagnetic effects and charged-neutral mass differences, the two rho meson charge channels have equal rates. Their subsequent strong decays are ${\rho }^0\to {\pi }^{+}{\pi }^{-}$ and ${\rho }^{-}\to {\pi }^{-}{\pi }^0$, with nearly unit probabilities. Consequentlyin the ideal isospin model.

Each final state contains one pair of identical pions. Summing the alternative resonant-pair assignments coherently, with the corresponding identical-particle phase space factors, does not generate an extra factor of two for just one channel. In the narrow-rho meson approximation the equality is already evident from equal $\rho \pi$ rates; the consistent isospin amplitudes extend it to the integrated three-body distributions.

The tabulated ratio $9.02/9.30=0.970$ is close to this expectation but is not exactly one. Treating the printed errors as independent gives an uncertainty about $0.012$, so a percent-level comparison requires isospin breaking, radiative corrections, realistic resonant dynamics and experimental correlations. The idealized argument predicts the leading pattern rather than an exact identity for the rounded measurements.