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A measurement of the cross-section for $W \to e\nu$ 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 oftransverse momentum and to lie between 2.00 and 4.25 in pseudorapidity. Theinclusive $W$ production cross-sections, where the $W$ decays to $e\nu$, aremeasured to be \begin{equation*} \sigma_{W^{+} \to e^{+}\nu_{e}}=1124.4\pm2.1\pm 21.5\pm 11.2\pm 13.0\,\mathrm{pb}, \end{equation*} \begin{equation*}\sigma_{W^{-} \to e^{-}\bar{\nu}_{e}}=\,\,\,809.0\pm 1.9\pm 18.1\pm\,\,\,7.0\pm\phantom{0}9.4\,\mathrm{pb}, \end{equation*} where the first uncertainties arestatistical, the second are systematic, the third are due to the knowledge ofthe LHC beam energy and the fourth are due to the luminosity determination.Differential cross-sections as a function of the electron pseudorapidity aremeasured. The $W^{+}/W^{-}$ cross-section ratio and production charge asymmetryare also reported. Results are compared with theoretical predictions atnext-to-next-to-leading order in perturbative quantum chromodynamics. Finally,in a precise test of lepton universality, the ratio of $W$ boson branchingfractions is determined to be \begin{equation*} \mathcal{B}(W \toe\nu)/\mathcal{B}(W \to \mu\nu)=1.020\pm 0.002\pm 0.019, \end{equation*} wherethe first uncertainty is statistical and the second is systematic.

Measurement of forward $W\to e\nu$ production in $pp$ collisions at $\sqrt{s}=8\,$TeV