The ratio of branching ratios of the $W$ boson to muons and electrons, $R^{\mu/e}_W=B(W\rightarrow \mu\nu)/B(W\rightarrow e\nu)$, has been measured using 140 fb$^{-1}$ of $pp$ collision data at $\sqrt{s}=13$ TeV collected with the ATLAS detector at the LHC, probing the universality of lepton couplings. The ratio is obtained from measurements of the $t\bar{t}$ production cross-section in the $ee$, $e\mu$ and $\mu\mu$ dilepton final states. To reduce systematic uncertainties, it is normalised by the square root of the corresponding ratio $R^{\mu\mu/ee}_Z$ for the $Z$ boson measured in inclusive $Z\rightarrow ee$ and $Z\rightarrow\mu\mu$ events. By using the precise value of $R^{\mu\mu/ee}_Z$ determined from $e^+e^-$ colliders, the ratio $R^{\mu/e}_W$ is determined to be $R^{\mu/e}_W = 0.9995 \pm 0.0022 \pm 0.0036 \pm 0.0014$. The three uncertainties correspond to data statistics, experimental systematics and the external measurement of $R^{\mu\mu/ee}_Z$, giving a total uncertainty of 0.0045, and confirming the Standard Model assumption of lepton flavour universality in $W$-boson decays at the 0.5% level.
The measurement of the ratio of the rate of decay of W bosons to muons and electrons, $R(\mu/e)=B(W\rightarrow\mu\nu)/B(W\rightarrow e\nu)$.
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