Massive gauge boson pair production at the LHC: A next-to-leading order story

Electroweak gauge boson pair production is one of the most important Standard Model processes at the LHC, not only because it is a benchmark process but also by its ability to probe the electroweak interaction directly. We present full next-to-leading order predictions for the production cross sections and distributions of on-shell massive gauge boson pair production in the Standard Model. This includes the QCD and electroweak (EW) corrections. We study the hierarchy between the different channels when looking at the size of the QCD gluon-quark-induced processes and the EW photon-quark-induced processes and provide the first comprehensive explanation of this hierarchy thanks to analytical leading-logarithmic results. We also provide a detailed study of the theoretical uncertainties affecting the total cross section predictions that stem from scale variation, parton distribution function and ${\alpha }_s$ errors. We then compare with the present LHC data.

Figure 1

Representative tree-level diagrams for $V{V}^{\prime }$ production processes.

Figure 2

Representative diagrams for $V{V}^{\prime }$ production via gluon-gluon fusion.

Figure 3

Selected fermion-loop diagrams with ${\gamma }_5$.

Figure 4

$Z$ transverse momentum $p_T$ (in GeV) distribution (left) and $Z$ pair invariant mass $M_{ZZ}$ (in GeV) distribution (right) of $pp\to ZZ$ cross section at the LHC (in pb/GeV), including NLO QCD and EW corrections calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a. In the $M_{ZZ}$ distribution the $q\overline{q}$ LO (dotted red) and $q\overline{q}$ NLO EW (dashed blue) curves nearly coincide.

Figure 5

$Z$ transverse momentum $p_T$ (in TeV) distribution (left, using $\Delta K$) and $Z$ pair invariant mass $M_{ZZ}$ (in GeV) distribution (right, in %) of the NLO QCD corrections to $pp\to ZZ$ cross section at the LHC, calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a.

Figure 6

Same as Fig. 5 but for EW corrections (in %). The $q\overline{q}$ EW NLO and $q\overline{q}$ EW virtual curves nearly coincide as well as the photon-quark-induced and photon-radiated curves.

Figure 7

$Z$ pair rapidity $y_{ZZ}$ distribution of the NLO QCD corrections and NLO EW corrections (upper left and right, respectively, in %), as well as the distribution of the $pp\to ZZ$ cross section (lower, in pb) at the LHC, calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a. The $q\overline{q}$ LO (dotted red) and $q\overline{q}$ NLO EW (dashed blue) curves nearly coincide in the lower panel.

Figure 8

$W$ and $Z$ transverse momentum $p_T$ (in GeV) distributions of $pp\to W^{+}Z$ (upper left and right, respectively) and $pp\to W^{-}Z$ (lower left and right, respectively) cross sections at the LHC (in pb/GeV), including NLO QCD and EW corrections calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a. The $q\overline{q}$ LO (dotted red) and $q\overline{q}$ NLO EW (dashed blue) curves nearly coincide.

Figure 9

$W$ and $Z$ transverse momentum $p_T$ (in TeV) distributions of the NLO QCD corrections (using $\Delta K$) to $pp\to W^{+}Z$ (upper left and right, respectively) and $pp\to W^{-}Z$ (lower left and right, respectively) cross sections at the LHC, calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a.

Figure 10

Same as Fig. 9 but for EW corrections (in %).

Figure 11

$M_{WZ}$ invariant mass (in GeV) distribution of $pp\to W^{+}Z$ (upper left) and $pp\to W^{-}Z$ (upper right) cross sections at the LHC (in pb/GeV), including NLO QCD and EW corrections calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a. The $q\overline{q}$ LO (dotted red) and $q\overline{q}$ NLO EW (dashed blue) curves nearly coincide. Middle left (right) panels: $M_{WZ}$ invariant mass distribution of the NLO QCD corrections to $pp\to W^{+}Z$ ($pp\to W^{-}Z$) cross section at the LHC (in %); lower left and right: the same but for the NLO EW corrections.

Figure 12

$WZ$ pair rapidity $y_{WZ}$ distributions of $pp\to W^{+}Z$ (upper left, in pb) and $pp\to W^{-}Z$ (upper right, in pb) as well as the $y_{WZ}$ distributions of the NLO QCD corrections and the NLO EW corrections in $pp\to W^{+}Z$ channel (middle left and lower left, respectively, in %) and in $pp\to W^{-}Z$ channel (middle right and lower right, respectively), at the LHC, calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a. The $q\overline{q}$ LO (dotted red) and $q\overline{q}$ NLO EW (dashed blue) curves nearly coincide in the upper panels.

Figure 13

$W^{+}$ transverse momentum $p_T$ (in GeV) distribution (left) and $W$ pair invariant mass $M_{WW}$ (in GeV) distribution (right) of $pp\to WW$ cross section at the LHC (in pb/GeV), including NLO QCD and EW corrections calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a. The LO $q\overline{q}$ (dotted red) and NLO EW $q\overline{q}$ (dashed blue) curves nearly coincide, as well as the LO $\gamma \gamma$ (solid light blue) and NLO $\gamma \gamma$ (dotted violet) in the $M_{WW}$ distribution.

Figure 14

$W^{+}$ transverse momentum $p_T$ (in TeV) distribution (left, using $\Delta K$) and $W$ pair invariant mass $M_{WW}$ (in GeV) distribution (right, in %) of the NLO QCD corrections to $pp\to WW$ cross section at the LHC, calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a.

Figure 15

Same as Fig. 14 but for EW corrections (in %). The LO $\gamma \gamma$ (solid light blue) and NLO $\gamma \gamma$ (dotted violet) curves coincide in the $M_{WW}$ distribution.

Figure 16

$W$ pair rapidity $y_{WW}$ distribution of the NLO QCD corrections and NLO EW corrections (upper left and right, respectively, in %), as well as the distribution of the $pp\to WW$ cross section (lower, in pb) at the LHC, calculated with MRST2004QED PDF set and with the input parameters described in Sec. 3a. The LO $\gamma \gamma$ (solid light blue) and NLO $\gamma \gamma$ (dotted violet) curves coincide as well as the LO $q\overline{q}$ (dotted red) and NLO EW $q\overline{q}$ (dashed blue) in the lower panel.

Figure 17

Comparison between the full results (plain black line) and the analytical leading-logarithmic approximation (dashed red line) for gluon-quark-induced processes in $ZZ$ (left), $W^{+}Z$ (middle) and $WW$ (right) $p_T$ differential distributions.

Figure 18

Same as Fig. 17 but for photon-quark-induced processes.

Figure 19

Representative $t$-channel diagrams for $bg\to W^{+}{W}^{-}b$ production cross section.

Figure 20

Scale uncertainty for a scale variation in the interval $\frac{1}{2}{\mu }_0\le {\mu }_R={\mu }_F\le 2{\mu }_0$ in $\sigma (pp\to WW,ZZ,WZ)$ (in pb) at the LHC as a function of $\sqrt{s}$ (in TeV). Upper left: $WW$ cross section. Upper right: $ZZ$ cross section. Lower: $W^{+}Z$ and $W^{-}Z$ cross sections. In the inserts the relative deviation from the central cross section obtained with ${\mu }_R={\mu }_F={\mu }_0=M_{V_1}+M_{V_2}$ is shown, $V_i=W/Z$.

Figure 21

Total cross sections (in pb) as a function of the c.m. energy (in TeV) for $WW$ (upper left), $ZZ$ (upper right), $W^{+}Z$ (lower left) and $W^{-}Z$ (lower right) production channels at the LHC when using different PDF sets. In the inserts the relative deviation from the central cross section obtained with MSTW2008 PDF set is shown.

Figure 22

PDF and $\mathrm{PDF}+{\alpha }_s$ uncertainties using MSTW2008 PDF set in $\sigma (pp\to WW,ZZ,WZ)$ (in pb) at the LHC as a function of the c.m. energy $\sqrt{s}$ (in TeV). Upper left: $WW$ cross section. Upper right: $ZZ$ cross section. Lower: $W^{+}Z$ and $W^{-}Z$ cross sections. In the inserts the relative deviation from the central cross section is shown.

Figure 23

The NLO $\mathrm{QCD}+\mathrm{EW}$ total cross section (black/full, in pb) of the processes $pp\to WW$ (upper left), $pp\to ZZ$ (upper right) and $pp\to W^{+}Z+W^{-}Z$ (lower) at the LHC as a function of the c.m. energy $\sqrt{s}$ (in TeV) including the total theoretical uncertainty (red/dashed) as discussed in the text. The insert shows the relative deviation from the central cross sections, and the experimental data points are also displayed on the main figures.

Figure 24

Diagrams for $u\gamma \to W^{+}Zd$ production cross section in the case of the differential $p_T^Z$ distribution. The soft $W^{+}$ is displayed in red and the diagrams are classified in four categories depending on the $2\to 2$ hard $Z$ subprocess: (a) is for $d\gamma \to Zd$, (b) is for $u\gamma \to Zu$, (c) is for $u\gamma \to W^{+}d$ and (d) is for $u{W}^{-}\to Zd$.