A measurement of the cross-section for $W to e u$ production in $pp$ collisions is presented using data corresponding to an integrated luminosity of $2,$fb$^{-1}$ collected by the LHCb experiment at a centre-of-mass energy of $sqrt{s}=8,$TeV. The electrons are required to have more than $20,$GeV of transverse momentum and to lie between 2.00 and 4.25 in pseudorapidity. The inclusive $W$ production cross-sections, where the $W$ decays to $e u$, are measured to be begin{equation*} sigma_{W^{+} to e^{+} u_{e}}=1124.4pm 2.1pm 21.5pm 11.2pm 13.0,mathrm{pb}, end{equation*} begin{equation*} sigma_{W^{-} to e^{-}bar{ u}_{e}}=,,,809.0pm 1.9pm 18.1pm,,,7.0pm phantom{0}9.4,mathrm{pb}, end{equation*} where the first uncertainties are statistical, the second are systematic, the third are due to the knowledge of the LHC beam energy and the fourth are due to the luminosity determination. Differential cross-sections as a function of the electron pseudorapidity are measured. The $W^{+}/W^{-}$ cross-section ratio and production charge asymmetry are also reported. Results are compared with theoretical predictions at next-to-next-to-leading order in perturbative quantum chromodynamics. Finally, in a precise test of lepton universality, the ratio of $W$ boson branching fractions is determined to be begin{equation*} mathcal{B}(W to e u)/mathcal{B}(W to mu u)=1.020pm 0.002pm 0.019, end{equation*} where the first uncertainty is statistical and the second is systematic.
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