The forward-backward charge asymmetry for the process $q\bar{q} \rightarrow Z/\gamma^{\ast} \rightarrow \mu^{+}\mu^{-}$ is measured as a function of the invariant mass of the dimuon system. Measurements are performed using proton proton collision data collected with the LHCb detector at $\sqrt{s} = 7$ and 8\tev, corresponding to integrated luminosities of $1$fb$^{-1}$ and $2$fb$^{-1}$ respectively. Within the Standard Model the results constrain the effective electroweak mixing angle to be $$sin^{2}\theta_{W}^{eff} = 0.23142 \pm 0.00073 \pm 0.00052 \pm 0.00056 $$ where the first uncertainty is statistical, the second systematic and the third theoretical. This result is in agreement with the current world average, and is one of the most precise determinations at hadron colliders to date.
Values for $A_{\rm{FB}}$ with the statistical and positive and negative systematic uncertainties for $\sqrt{s}$ = 7 TeV data. The theoretical uncertainties presented in this table, corresponding to the PDF, scale and FSR uncertainties described in Sec. 5, affect only the predictions of $A_{\rm{FB}}$ and the sin$^2\theta^{\rm{eff}}_{\rm W}$ determination, and do not apply to the uncertainties on the measured $A_{\rm{FB}}$.
Values for $A_{\rm{FB}}$ with the statistical and positive and negative systematic uncertainties for $\sqrt{s}$ = 8 TeV data. The theoretical uncertainties presented in this table, corresponding to the PDF, scale and FSR uncertainties described in Sec. 5, affect only the predictions of $A_{\rm{FB}}$ and the sin$^2\theta^{\rm{eff}}_{\rm W}$ determination, and do not apply to the uncertainties on the measured $A_{\rm{FB}}$.