A measurement of the ratio of the branching fractions, $R_{\tau/e} = B(W \to \tau \nu)/ B(W \to e \nu)$, is performed using a sample of $W$ bosons originating from top-quark decays to final states containing $\tau$-leptons or electrons. This measurement uses $pp$ collisions at $\sqrt{s}=13$ TeV, collected by the ATLAS experiment at the Large Hadron Collider during Run 2, corresponding to an integrated luminosity of 140 fb$^{-1}$. The $W \to \tau \nu_\tau$ (with $\tau \to e \nu_e \nu_\tau$) and $W \to e \nu_e$ decays are distinguished using the differences in the impact parameter distributions and transverse momentum spectra of the electrons. The measured ratio of branching fractions $R_{\tau/e} = 0.975 \pm 0.012 \textrm{(stat.)} \pm 0.020 \textrm{(syst.)}$, is consistent with the Standard Model assumption of lepton flavour universality in $W$-boson decays.
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Number of events in the $\mu e$ channel from different sources, as estimated by the fit to the data, compared with the observed yield. Uncertainties include the statistical and systematic contribution. The uncertainty in the total expected number of events can be smaller than the uncertainties of the individual contributions because of correlations between them.
Number of events in the $e e$ channel from different sources, as estimated by the fit to the data, compared with the observed yield. Uncertainties include the statistical and systematic contribution. The uncertainty in the total expected number of events can be smaller than the uncertainties of the individual contributions because of correlations between them.
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