 Solution

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.