Measurement of the forward $W$ boson cross-section in $pp$ collisions at $\sqrt{s} = 7\mathrm{\,Te\kern -0.1em V}$

A measurement of the inclusive $W \to \mu\nu$ production cross-section using data from $pp$ collisions at a centre-of-mass energy of $\sqrt{s} = 7\mathrm{\,Te\kern -0.1em V}$ is presented. The analysis is based on an integrated luminosity of about 1.0 fb$^{-1}$ recorded with the LHCb detector. Results are reported for muons with a transverse momentum greater than $20\mathrm{\,Ge\kern -0.1em V\!/}c$ and pseudorapidity between 2.0 and 4.5. The $W^{+}$ and $W^{-}$ production cross-sections are measured to be \begin{align*} \sigma_{W^{+} \to \mu^{+}\nu} &= 846.9 \pm 2.0 \pm 10.9 \pm 29.9 {\rm \,pb}, \\ \sigma_{W^{-} \to \mu^{-}\bar{\nu}} &= 664.6 \pm 1.8 \pm 8.7 \pm 23.5 {\rm \,pb}, \end{align*} \noindent where the first uncertainty is statistical, the second is systematic and the third is due to the luminosity determination. Cross-section ratios and differential distributions as functions of the muon pseudorapidity are also presented. The ratio of $W^{+}$ to $W^{-}$ cross-sections in the same fiducial kinematic region is determined to be \begin{equation*} \frac{\sigma_{W^{+} \to \mu^{+}\nu}}{\sigma_{W^{-} \to \mu^{-}\bar{\nu}}} = 1.274 \pm 0.005 \pm 0.009, \end{equation*} where the uncertainties are statistical and systematic, respectively. Results are in good agreement with theoretical predictions at next-to-next-to-leading order in perturbative quantum chromodynamics.