Measurements of inclusive $W$ and $Z$ boson production cross-sections in $pp$ collisions at $sqrt{s}=7$ TeV using final states containing muons are presented. The data sample corresponds to an integrated luminosity of 37 pb$^{-1}$ collected with the LHCb detector. The $W$ and $Z$ bosons are reconstructed from muons with a transverse momentum above 20 GeV$/c$ and pseudorapidity between 2.0 and 4.5, and, in the case of the $Z$ cross-section, a dimuon invariant mass between 60 and 120 GeV$/c^2$. The cross-sections are measured to be {$831pm9pm27pm 29$ pb} for $W^+$, {$656pm8pm19pm 23$ pb} for $W^-$ and {$76.7pm1.7pm3.3pm2.7$ pb} for $Z$, where the first uncertainty is statistical, the second is systematic and the third is due to the luminosity. Differential cross-sections, $W$ and $Z$ cross-section ratios and the lepton charge asymmetry are also measured in the same kinematic region. The ratios are determined to be $sigma_{W^+rightarrowmu^+ u} / sigma_{W^-rightarrowmu^-bar{ u}}= ${$1.27pm0.02pm0.01$}, and $sigma_{W^+rightarrowmu^+ u} + sigma_{W^-rightarrowmu^-bar{ u}})/sigma_{Zrightarrowmumu}= ${$19.4pm0.5pm0.9$}. The results are in general agreement with theoretical predictions, performed at next-to-next-to-leading order in QCD using recently calculated parton distribution functions.
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