Figure 1

Sample eight-point loop diagrams for the processes $qg\to W{q}^{\prime }gggg$, $q{\overline{Q}}_1\to W{q}^{\prime }ggg{\overline{Q}}_1$ and $q{\overline{Q}}_1\to W{q}^{\prime }{\overline{Q}}_2{Q}_{2}g{\overline{Q}}_1$, followed by the decay of the $W$ boson to leptons.Reuse & Permissions

Figure 2

Sample nine-point real-emission diagrams for the processes $qg\to W{q}^{\prime }ggggg$ and $q{\overline{q}}^{\prime }\to W{Q}_{1}gg{Q}_2{\overline{Q}}_2{\overline{Q}}_1$, followed by the decay of the $W$ boson to leptons.Reuse & Permissions

Figure 3

The distribution of the relative error in the virtual cross section for three subprocesses, $gd\to e^{-}{\overline{\nu }}_{e}ggggu$, $ud\to e^{-}{\overline{\nu }}_{e}ggguu$, and $ud\to e^{-}{\overline{\nu }}_{e}guus\overline{s}$, reading from left to right. The horizontal axis is the logarithm of the relative error (2.6) between an evaluation by BlackHat, running in production mode, and a target expression evaluated at higher precision. The vertical axis shows the number of phase-space points having that relative error. The dark (black) line labeled by “double” shows the $1/{ϵ}^2$ term; the darkly shaded (red) curve labeled by “single”, the $1/ϵ$ term; and the lightly shaded (green) curve labeled by “finite”, the finite (${ϵ}^0$) term. Each plot is based on approximately 10 000 points in phase space, distributed as in the actual calculation.Reuse & Permissions

Figure 4

The renormalization-scale dependence of the total cross section using a fixed reference scale of ${\mu }_0=2M_W$. The top four panels give the renormalization-scale dependence at both LO and NLO for $W+2$-jets through $W+5$-jets. The bottom panel shows the $K$ factors for these cases, with the top curve for $W+2$-jets and the bottom one for $W+5$-jets. The factorization scale is held fixed.Reuse & Permissions

Figure 5

The improvement in the renormalization and factorization scale dependence of the differential cross section as a function of the hadronic total transverse energy $H_{\mathrm{T}}^{\mathrm{jets}}$, comparing LO to NLO at the LHC at $\sqrt{s}=7\mathrm{TeV}$. In the upper panels, the NLO predictions are shown as solid (black) lines, while the LO predictions are shown as dashed (blue) lines. The thin vertical line in the center of each bin (where visible) gives its numerical integration error, corresponding to the fluctuations in the plots. The lower panels show the predictions for the LO distribution and scale-dependence bands, normalized to the NLO prediction at the scale $\mu ={\widehat{H}}_{\mathrm{T}}^{\prime }/2$. The LO distribution is the dashed (blue) line, and the scale-dependence bands are shaded (gray) for NLO and cross-hatched (brown) for LO.Reuse & Permissions

Figure 6

The $p_{\mathrm{T}}$ distributions of the leading five jets in $W^{-}+5$-jet production at the LHC at $\sqrt{s}=7\mathrm{TeV}$. In the upper panels, the NLO predictions are shown as solid (black) lines, while the LO predictions are shown as dashed (blue) lines. The lower panels show the predictions for the LO distribution and scale-dependence bands normalized to the NLO prediction (at the scale $\mu ={\widehat{H}}_{\mathrm{T}}^{\prime }/2$). The LO distribution is the dashed (blue) line, and the scale-dependence bands are shaded (gray) for NLO and cross-hatched (brown) for LO.Reuse & Permissions

Figure 7

The $p_{\mathrm{T}}$ distributions of the leading five jets in $W^{+}+5$-jet production at the LHC at $\sqrt{s}=7\mathrm{TeV}$.Reuse & Permissions