Measurement of $W^{\pm}Z$ production cross sections and gauge boson polarisation in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector

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Abstract

This paper presents measurements of $W^{\pm}Z$ production cross sections in $pp$ collisions at a centre-of-mass energy of 13 TeV. The data were collected in 2015 and 2016 by the ATLAS experiment at the Large Hadron Collider, and correspond to an integrated luminosity of 36.1 fb$^{-1}$. The $W^{\pm}Z$ candidate events are reconstructed using leptonic decay modes of the gauge bosons into electrons and muons. The measured inclusive cross section in the detector fiducial region for a single leptonic decay mode is $\sigma_{W^\pm Z \rightarrow \ell^{'} \nu \ell \ell}^{\textrm{fid.}} = 63.7 \pm 1.0$ (stat.) $\pm 2.3$ (syst.) $\pm 1.4$ (lumi.) fb, reproduced by the next-to-next-to-leading-order Standard Model prediction of $61.5^{+1.4}_{-1.3}$ fb. Cross sections for $W^+Z$ and $W^-Z$ production and their ratio are presented as well as differential cross sections for several kinematic observables. An analysis of angular distributions of leptons from decays of $W$ and $Z$ bosons is performed for the first time in pair-produced events in hadronic collisions, and integrated helicity fractions in the detector fiducial region are measured for the $W$ and $Z$ bosons separately. Of particular interest, the longitudinal helicity fraction of pair-produced vector bosons is also measured.

Figures

Tables

Table 1. Table 1: Observed and expected numbers of events after the W ± ​ Z superscript 𝑊 plus-or-minus 𝑍 W^{\pm}Z  inclusive selection described in Section  5 in each of the considered channels and for the sum of all channels. The expected number of W ± ​ Z superscript 𝑊 plus-or-minus 𝑍 W^{\pm}Z  events from Powheg+Pythia and the estimated number of background events from other processes are detailed. The Powheg+Pythia MC prediction is scaled by a global factor of 1.18 1.18 1.18 to match the N NLO cross section predicted by MATRIX . The sum of background events containing misidentified leptons is labelled “Misid. leptons”. Only statistical uncertainties are reported.

Channel	$eee$		$\mu ee$		$e\mu \mu$		$\mu \mu \mu$		All
Data	$1279$		$1281$		$1671$		$1929$		$6160$
Total Expected	$1221$$\pm$	$7$	$1281$$\pm$	$6$	$1653$$\pm$	$8$	$1830$$\pm$	$7$	$5986$$\pm$	$14$
$WZ$	$922$$\pm$	$5$	$1077$$\pm$	$6$	$1256$$\pm$	$6$	$1523$$\pm$	$7$	$4778$$\pm$	$12$
Misid. leptons	$138$$\pm$	$5$	$34$$\pm$	$2$	$193$$\pm$	$5$	$71$$\pm$	$2$	$436$$\pm$	$8$
$ZZ$	$86$$\pm$	$1$	$89$$\pm$	$1$	$117$$\pm$	$1$	$135$$\pm$	$1$	$426$$\pm$	$3$
$t\overline{t}$+V	$50.0$$\pm$	$0.7$	$54.0$$\pm$	$0.7$	$56.1$$\pm$	$0.7$	$63.8$$\pm$	$0.8$	$225$$\pm$	$1$
$tZ$	$23.1$$\pm$	$0.4$	$24.8$$\pm$	$0.4$	$28.8$$\pm$	$0.4$	$33.5$$\pm$	$0.5$	$110$$\pm$	$1$
$VVV$	$2.5$$\pm$	$0.1$	$2.8$$\pm$	$0.1$	$3.2$$\pm$	$0.1$	$3.6$$\pm$	$0.1$	$12.0$$\pm$	$0.2$

Table 2. Table 2: The C W ​ Z subscript 𝐶 𝑊 𝑍 C_{WZ} factors for each of the e ​ e ​ e 𝑒 𝑒 𝑒 eee , μ ​ e ​ e 𝜇 𝑒 𝑒 \mu{ee} , e ​ μ ​ μ 𝑒 𝜇 𝜇 {e}\mu\mu , and μ ​ μ ​ μ 𝜇 𝜇 𝜇 \mu\mu\mu inclusive channels. The Powheg+Pythia MC event sample with the “resonant shape” lepton assignment algorithm at particle level is used. Only statistical uncertainties are reported.

Channel	$C_{W^{-}Z}$	$C_{W^{+}Z}$	$C_{W^{\pm }Z}$
$eee$	$0.399$ $\pm$ $0.003$	$0.394$ $\pm$ $0.003$	$0.396$ $\pm$ $0.002$
$\mu ee$	$0.470$ $\pm$ $0.004$	$0.469$ $\pm$ $0.003$	$0.469$ $\pm$ $0.002$
$e\mu \mu$	$0.548$ $\pm$ $0.004$	$0.541$ $\pm$ $0.003$	$0.544$ $\pm$ $0.003$
$\mu \mu \mu$	$0.660$ $\pm$ $0.005$	$0.663$ $\pm$ $0.004$	$0.662$ $\pm$ $0.003$

Table 3. Table 3: Summary of the relative uncertainties on the measured fiducial cross section σ W ± ​ Z fid. subscript superscript 𝜎 fid. superscript 𝑊 plus-or-minus 𝑍 \sigma^{\textrm{fid.}}_{W^{\pm}Z} for each channel and for their combination. The uncertainties are reported as percentages. The first rows indicate the main sources of systematic uncertainty for each channel and their combination, which are treated as correlated between channels. A row with uncorrelated uncertainties follows, which comprise all uncertainties of statistical origin including MC statistics as well as statistical uncertainties in the fake-factors calculation, which are uncorrelated between channels.

	$eee$	$\mu ee$	$e\mu \mu$	$\mu \mu \mu$	Combined
Relative uncertainties [%]
$e$ energy scale	$0.2$	$0.1$	$0.1$		$0.1$
$e$ id. efficiency	$2.8$	$1.8$	$1.0$		$1.1$
$\mu$ momentum scale
$\mu$ id. efficiency		$1.3$	$1.6$	$2.8$	$1.5$
$E_T^{\mathrm{miss}}$ and jets	$0.2$	$0.2$	$0.3$	$0.5$	$0.3$
Trigger			$0.2$	$0.3$	$0.2$
Pile-up	$1.0$	$1.5$	$1.2$	$1.5$	$1.3$
Misid. leptons background	$4.7$	$1.1$	$4.5$	$1.6$	$1.9$
$ZZ$ background	$1.0$	$1.0$	$1.1$	$1.0$	$1.0$
Other backgrounds	$1.6$	$1.5$	$1.4$	$1.2$	$1.4$
Uncorrelated	$0.7$	$0.6$	$0.7$	$0.5$	$0.3$
Total systematic uncertainty	$6.0$	$3.5$	$5.4$	$4.1$	$3.6$
Luminosity	$2.2$	$2.2$	$2.2$	$2.2$	$2.2$
Theoretical modelling	$0.5$	$0.5$	$0.5$	$0.5$	$0.5$
Statistics	$3.6$	$3.3$	$3.2$	$2.7$	$1.6$
Total	$7.3$	$5.3$	$6.6$	$5.3$	$4.5$

Table 4. Table 4: Fiducial integrated cross section in fb, for W ± ​ Z superscript 𝑊 plus-or-minus 𝑍 W^{\pm}Z , W + ​ Z superscript 𝑊 𝑍 W^{+}Z and W − ​ Z superscript 𝑊 𝑍 W^{-}Z production, measured in each of the channels e ​ e ​ e 𝑒 𝑒 𝑒 eee , μ ​ e ​ e 𝜇 𝑒 𝑒 \mu{ee} , e ​ μ ​ μ 𝑒 𝜇 𝜇 {e}\mu\mu , and μ ​ μ ​ μ 𝜇 𝜇 𝜇 \mu\mu\mu and for all four channels combined. The statistical ( δ stat . subscript 𝛿 stat \delta_{\mathrm{stat.}} ), experimental systematic ( δ exp . syst . subscript 𝛿 formulae-sequence exp syst \delta_{\mathrm{exp.\,syst.}} ), modelling systematic ( δ mod . syst . subscript 𝛿 formulae-sequence mod syst \delta_{\mathrm{mod.\,syst.}} ), luminosity ( δ lumi . subscript 𝛿 lumi \delta_{\mathrm{lumi.}} ) and total ( δ tot . subscript 𝛿 tot \delta_{\mathrm{tot.}} ) uncertainties are given in percent. The NNLO SM predictions from MATRIX using the NNDPF3.0nnlo set are also reported.

Channel	${\sigma }^{\mathrm{fid}.}$	${\delta }_{\mathrm{stat}.}$	${\delta }_{\exp .\mathrm{syst}.}$	${\delta }_{\mathrm{mod}.\mathrm{syst}.}$	${\delta }_{\mathrm{lumi}.}$	${\delta }_{\mathrm{tot}.}$
	[fb]	[%]	[%]	[%]	[%]	[%]
${\sigma }_{W^{\pm }Z\to {\ell }^{{}^{\prime }}\nu \ell \ell }^{\mathrm{fid}.}$
$e^{\pm }ee$	65.8	3.6	6.0	0.5	2.2	7.3
${\mu }^{\pm }ee$	61.2	3.3	3.5	0.5	2.2	5.3
$e^{\pm }\mu \mu$	62.4	3.2	5.4	0.5	2.2	6.6
${\mu }^{\pm }\mu \mu$	65.3	2.7	4.1	0.5	2.2	5.3
Combined	63.7	1.6	3.6	0.5	2.2	4.5
SM prediction	61.5	—	—	—	—	${}_{2.1}{}^{}{}_{}{}^{2.3}$
${\sigma }_{W^{+}Z\to {\ell }^{{}^{\prime }}\nu \ell \ell }^{\mathrm{fid}.}$
$e^{+}ee$	40.8	4.6	5.4	0.5	2.2	7.4
${\mu }^{+}ee$	36.5	4.3	3.3	0.5	2.2	5.8
$e^{+}\mu \mu$	36.7	4.1	5.0	0.5	2.2	6.8
${\mu }^{+}\mu \mu$	38.2	3.5	4.0	0.5	2.2	5.7
Combined	37.9	2.0	3.4	0.5	2.2	4.5
SM prediction	36.3	—	—	—	—	${}_{2.0}{}^{}{}_{}{}^{2.2}$
${\sigma }_{W^{-}Z\to {\ell }^{{}^{\prime }}\nu \ell \ell }^{\mathrm{fid}.}$
$e^{-}ee$	24.9	6.1	7.1	0.5	2.2	9.6
${\mu }^{-}ee$	24.8	5.3	4.0	0.5	2.2	7.0
$e^{-}\mu \mu$	25.7	5.1	6.2	0.5	2.2	8.3
${\mu }^{-}\mu \mu$	27.1	4.3	4.3	0.5	2.2	6.4
Combined	25.9	8.1	4.0	0.5	2.2	5.2
SM prediction	25.2	—	—	—	—	${}_{2.1}{}^{}{}_{}{}^{2.3}$

Table 5. Table 5: Measured helicity fractions in the fiducial phase space with Born-level leptons, for W ± ​ Z superscript 𝑊 plus-or-minus 𝑍 W^{\pm}Z   W + ​ Z superscript 𝑊 𝑍 W^{+}Z and W − ​ Z superscript 𝑊 𝑍 W^{-}Z events. The total uncertainties in the measurements are reported. The measurements are compared with predictions at electroweak LO from Powheg+Pythia and MATRIX corrected to sin 2 ⁡ θ W eff = 0.23152 superscript 2 superscript subscript 𝜃 W eff 0.23152 \sin^{2}{\theta_{\mathrm{W}}^{\mathrm{eff}}}=0.23152 . The uncertainties on the Powheg+Pythia prediction include QCD scale and PDF uncertainties; the uncertainties in the MATRIX prediction include QCD scale uncertainties.

	$f_0$						$f_L-f_R$
	Data		Powheg+Pythia		MATRIX		Data		Powheg+Pythia		MATRIX
$W^{+}$ in $W^{+}Z$	$0.26$ $\pm$	$0.08$	$0.233$ $\pm$	$0.004$	$0.2448$ $\pm$	$0.0010$	$-0.02$ $\pm$	$0.04$	$0.091$ $\pm$	$0.004$	$0.0868$ $\pm$	$0.0014$
$W^{-}$ in $W^{-}Z$	$0.32$ $\pm$	$0.09$	$0.245$ $\pm$	$0.005$	$0.2651$ $\pm$	$0.0015$	$-0.05$ $\pm$	$0.05$	$-0.063$ $\pm$	$0.006$	$-0.034$ $\pm$	$0.004$
$W^{\pm }$ in $W^{\pm }Z$	$0.26$ $\pm$	$0.06$	$0.2376$ $\pm$	$0.0031$	$0.2506$ $\pm$	$0.0006$	$-0.024$ $\pm$	$0.033$	$0.0289$ $\pm$	$0.0022$	$0.0375$ $\pm$	$0.0011$
$Z$ in $W^{+}Z$	$0.27$ $\pm$	$0.05$	$0.225$ $\pm$	$0.004$	$0.2401$ $\pm$	$0.0014$	$-0.32$ $\pm$	$0.21$	$-0.297$ $\pm$	$0.021$	$-0.262$ $\pm$	$0.009$
$Z$ in $W^{-}Z$	$0.21$ $\pm$	$0.06$	$0.235$ $\pm$	$0.005$	$0.2389$ $\pm$	$0.0015$	$-0.46$ $\pm$	$0.25$	$0.052$ $\pm$	$0.023$	$0.0468$ $\pm$	$0.0034$
$Z$ in $W^{\pm }Z$	$0.24$ $\pm$	$0.04$	$0.2294$ $\pm$	$0.0033$	$0.2398$ $\pm$	$0.0014$	$-0.39$ $\pm$	$0.16$	$-0.156$ $\pm$	$0.016$	$-0.135$ $\pm$	$0.006$

Table 6. Table 6: Summary of the absolute uncertainties in the helicity fractions f 0 subscript 𝑓 0 f_{\mathrm{0}} and f L − f R subscript 𝑓 L subscript 𝑓 R f_{\mathrm{L}}-f_{\mathrm{R}} measured in W ± ​ Z superscript 𝑊 plus-or-minus 𝑍 W^{\pm}Z  events for W 𝑊 W and Z 𝑍 Z bosons.

	$W^{\pm }$ in $W^{\pm }Z$		$Z$ in $W^{\pm }Z$
	$f_0$	$f_L-f_R$	$f_0$	$f_L-f_R$
$e$ energy scale and id. efficiency	0.0024	0.0004	0.005	0.0021
$\mu$ momentum scale and id. efficiency	0.0013	0.0027	0.0018	0.008
$E_T^{\mathrm{miss}}$ and jets	0.0024	0.0010	0.0017	0.005
Pile-up	0.005	0.00009	0.0014	0.005
Misid. lepton background	0.031	0.001	0.007	0.019
$ZZ$ background	0.009	0.0004	0.0007	0.0012
Other backgrounds	0.0012	0.0005	0.0018	0.005
QCD scale	0.0008	0.0013	0.0004	0.008
PDF	0.0011	0.0009	0.00004	0.00001
Modelling	0.004	0.007	0.0015	0.0028
Total systematic uncertainty	0.033	0.008	0.009	0.024
Luminosity	0.0015	0.0001	0.0001	0.0008
Statistics	0.06	0.032	0.04	0.15
Total	0.06	0.033	0.04	0.16

Equations

$$P={\left|\frac{1}{m_{({\ell }^{+},{\ell }^{-})}^2-{\left(m_Z^{\text{PDG}}\right)}^2+\text{i}{\Gamma }_Z^{\text{PDG}}{m}_Z^{\text{PDG}}}\right|}^2\times {\left|\frac{1}{m_{({\ell }^{\prime },{\nu }_{{\ell }^{\prime }})}^2-{\left(m_W^{\text{PDG}}\right)}^2+\text{i}{\Gamma }_W^{\text{PDG}}{m}_W^{\text{PDG}}}\right|}^2,$$

$$m_{\mathrm{T}}^{WZ}=\sqrt{{\left(\sum _{\ell =1}^3{p}_{\mathrm{T}}^{\ell }+E_{\mathrm{T}}^{\mathrm{miss}}\right)}^2-\left[{\left(\sum _{\ell =1}^3{p}_x^{\ell }+E_x^{\mathrm{miss}}\right)}^2+{\left(\sum _{\ell =1}^3{p}_y^{\ell }+E_y^{\mathrm{miss}}\right)}^2\right]}.$$

$${\sigma }_{W^{\pm }Z\to {\ell }^{{}^{\prime }}\nu \ell \ell }^{fid.}=\frac{N_{\text{data}}-N_{\text{bkg}}}{\mathcal{L}\cdot C_{WZ}}\times \left(1-\frac{N_{\tau }}{N_{\text{all}}}\right),$$

$${\sigma }_{W^{\pm }Z}^{tot.}=\frac{{\sigma }_{W^{\pm }Z\to {\ell }^{{}^{\prime }}\nu \ell \ell }^{fid.}}{{\mathcal{B}}_W{\mathcal{B}}_Z{A}_{WZ}},$$

$${\sigma }_{W^{\pm }Z\to {\ell }^{{}^{\prime }}\nu \ell \ell }^{fid.}=63.7\text{leavevmode}\pm \text{leavevmode}1.0\text{(stat.)}\text{leavevmode}\pm \text{leavevmode}2.3\text{(exp. syst.)}\text{leavevmode}\pm \text{leavevmode}0.3\text{(mod. syst)}\text{leavevmode}\pm 1.4\text{(lumi.) fb},$$

$$\frac{{\sigma }_{W^{+}Z\to {\ell }^{{}^{\prime }}\nu \ell \ell }^{\text{fid.}}}{{\sigma }_{W^{-}Z\to {\ell }^{{}^{\prime }}\nu \ell \ell }^{\text{fid.}}}=1.47\pm 0.05\text{(stat.)}\pm 0.02\text{(syst.)}.$$

$${\sigma }_{W^{\pm }Z}^{tot.}=51.0\pm 0.8(stat.)\pm 1.8(exp.syst.)\pm 0.9(mod.syst.)\pm 1.1(lumi.)\mathrm{pb},$$

$$\frac{1}{{\sigma }_{W^{\pm }Z}}\frac{\mathrm{d}{\sigma }_{W^{\pm }Z}}{\mathrm{d}\cos {\theta }_{\ell ,W}}=\frac{3}{8}{f}_{\text{L}}[(1\mp \cos {\theta }_{\ell ,W}{)}^2]+\frac{3}{8}{f}_{\text{R}}[(1\pm \cos {\theta }_{\ell ,W}{)}^2]+\frac{3}{4}{f}_0{\sin }^2{\theta }_{\ell ,W},$$

$$\frac{1}{{\sigma }_{W^{\pm }Z}}\frac{\mathrm{d}{\sigma }_{W^{\pm }Z}}{\mathrm{d}\cos {\theta }_{\ell ,Z}}$$

$$\frac{{\frac{1}{{\sigma }_{W^{\pm }Z}}\frac{\mathrm{d}{\sigma }_{W^{\pm }Z}}{\mathrm{d}\cos {\theta }_{\ell ,W}}|}_{\mathrm{L}/0/\mathrm{R}}}{\frac{3}{8}{f}_{\mathrm{L}}^{gen.}(1\mp \cos {\theta }_{\ell ,W}{)}^2+\frac{3}{8}{f}_{\mathrm{R}}^{gen.}(1\pm \cos {\theta }_{\ell ,W}{)}^2+\frac{3}{4}{f}_0^{gen.}{\sin }^2{\theta }_{\ell ,W}},$$

$$\left.\frac{1}{\sigma_{W^{\pm}Z}}\frac{\mathrm{d}\sigma_{W^{\pm}Z}}{\mathrm{d}\cos\theta_{\ell,W}}\right|\begin{array}[]{r}\mathrm{L}\\ \mathrm{0}\\ \mathrm{R}\end{array}=\frac{3}{8}\left\{\begin{array}[]{r}(1\mp\cos\theta_{\ell,W})^{2}\\ 2\sin^{2}\theta_{\ell,W}\\ (1\pm\cos\theta_{\ell,W})^{2}\end{array}\right.\,,$$

Full text

\AtlasTitle

Measurement of $W^{\pm}Z$ production cross sections and gauge boson polarisation in $pp$ collisions at $\sqrt{s}=13\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}$ with the ATLAS detector \AtlasAbstractThis paper presents measurements of $W^{\pm}Z$ production cross sections in $pp$ collisions at a centre-of-mass energy of $13$ TeV. The data were collected in $2015$ and $2016$ by the ATLAS experiment at the Large Hadron Collider, and correspond to an integrated luminosity of $36.1$ fb*-1*. The $W^{\pm}Z$ candidate events are reconstructed using leptonic decay modes of the gauge bosons into electrons and muons. The measured inclusive cross section in the detector fiducial region for a single leptonic decay mode is $\sigma_{W^{\pm}Z\rightarrow\ell^{{}^{\prime}}\nu\ell\ell}^{\textrm{fid.}}=63.7\,\pm\leavevmode\nobreak\ 1.0\leavevmode\nobreak\ \textrm{(stat.)}\,\pm\leavevmode\nobreak\ 2.3\leavevmode\nobreak\ \textrm{(syst.)}\,\pm\leavevmode\nobreak\ 1.4\leavevmode\nobreak\ \textrm{(lumi.)}$ fb, reproduced by the next-to-next-to-leading-order Standard Model prediction of $61.5^{+1.4}_{-1.3}$ fb. Cross sections for $W^{+}Z$ and $W^{-}Z$ production and their ratio are presented as well as differential cross sections for several kinematic observables. An analysis of angular distributions of leptons from decays of $W$ and $Z$ bosons is performed for the first time in pair-produced events in hadronic collisions, and integrated helicity fractions in the detector fiducial region are measured for the $W$ and $Z$ bosons separately. Of particular interest, the longitudinal helicity fraction of pair-produced vector bosons is also measured.

\AtlasRefCodeSTDM-2018-03 \PreprintIdNumberCERN-EP-2018-327 \AtlasJournalRefEur. Phys. J. C 79 (2019) 535 \AtlasDOI10.1140/epjc/s10052-019-7027-6

1 Introduction

The study of $W^{\pm}Z$ diboson production is an important test of the Standard Model (SM) for its sensitivity to gauge boson self-interactions, related to the non-Abelian structure of the electroweak interaction. It provides the means to directly probe the triple gauge boson couplings (TGC), in particular the $WWZ$ gauge coupling. Improved constraints from precise measurements can potentially probe scales of new physics in the multi-TeV range and provide a way to look for signals of new physics in a model-independent way. Previous measurements have concentrated on the inclusive and differential production cross sections. In addition to more precise measurements of these cross sections that include new data, this paper presents measurements of the three helicity fractions of the $W$ and $Z$ bosons. The existence of the third state, the longitudinally polarised state, is a consequence of the non-vanishing mass of the bosons generated by the electroweak symmetry breaking mechanism. The measurement of the polarisation in diboson production therefore tests both the SM innermost gauge symmetry structure, through the existence of the triple gauge coupling, and the particular way this symmetry is spontaneously broken, via the longitudinal helicity state. Angular observables can be used to look for new interactions that can lead to different polarisation behaviour than predicted by the SM, to which the $W^{\pm}Z$ final state would be particularly sensitive [1, 2]. Precise calculations, at the next-to-leading order (NLO) in QCD, of SM polarisation observables in $W^{\pm}Z$ production as well as electroweak corrections have recently appeared [3]. Polarisation measurements for each charge of the $W$ boson might be helpful in the investigation of $CP$ violation effects in the interaction between gauge bosons [4, 5]. In the longer term, measuring the scattering of longitudinally polarised vector bosons will be a fundamental test of electroweak symmetry breaking [6].

Measurements of the $W^{\pm}Z$ production cross section in proton–antiproton collisions at a centre-of-mass energy of $\sqrt{s}=1.96$ TeV were published by the CDF and DØ collaborations [7, 8] using integrated luminosities of $7.1$ fb*-1* and $8.6$ fb*-1*, respectively. At the Large Hadron Collider (LHC), the most precise measurement of $W^{\pm}Z$ production was reported by the ATLAS Collaboration [9] using $20.1$ fb*-1* of data collected at a centre-of-mass energy of $8$ TeV. Measurements of $W^{\pm}Z$ production at $\sqrt{s}=13$ TeV were reported by the ATLAS [10] and CMS [11] collaborations using integrated luminosities of $3.2$ fb*-1* and $35.9$ fb*-1*, respectively. Other $W^{\pm}Z$ measurements in $pp$ collisions, at centre-of-mass energies of $7$ TeV and $8$ TeV, were reported previously by ATLAS and CMS [12, 13].

At hadron colliders, the polarisation of the $W$ boson was previously measured in the decay of the top quark by the CDF and DØ [14, 15, 16] collaborations and the ATLAS [17] and CMS [18] collaborations, as well as in association with jets by ATLAS [19] and CMS [20]. Polarisation and several other related angular coefficient measurements of a singly produced $Z$ boson were published by the CDF [21], CMS [22] and ATLAS [23] collaborations. The polarisation of $W$ bosons was also measured in $ep$ collisions by the H1 Collaboration [24]. Finally, for dibosons, first measurements of the $W$ polarisation were performed by LEP experiments in $W$ pair production in $e^{+}e^{-}$ collisions [25, 26] and were used to set limits on anomalous triple gauge couplings (aTGC) in Ref. [27].

This paper presents results obtained using proton–proton ($pp$) collisions recorded by the ATLAS detector at a centre-of-mass energy of $\sqrt{s}=13$ TeV in $2015$ and $2016$, corresponding to an integrated luminosity of 36.1\leavevmode\nobreak\ \mbox{fb{}^{-1}}. The $W$ and $Z$ bosons are reconstructed using their decay modes into electrons or muons. The production cross section is measured within a fiducial phase space both inclusively and differentially as a function of several individual variables related to the kinematics of the $W^{\pm}Z$ system and to the jet activity in the event. The reported measurements are compared with the SM cross-section predictions at NLO in QCD [28, 29] and with the recent calculations at next-to-next-to-leading order (NNLO) in QCD [30, 31]. The ratio of the $W^{+}Z$ cross section to the $W^{-}Z$ cross section, which is sensitive to the parton distribution functions (PDF) is also measured. Finally, an analysis of angular distributions of leptons from decays of $W$ and $Z$ bosons is performed and integrated helicity fractions in the detector fiducial region are measured for the $W$ and $Z$ bosons separately.

2 ATLAS detector

The ATLAS detector [32, 33, 34] is a multipurpose particle detector with a cylindrical geometry111ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the $z$-axis along the beam direction. The $x$-axis points from the IP to the centre of the LHC ring, and the y-axis points upward. Cylindrical coordinates $(r,\phi)$ are used in the transverse $(x,y)$ plane, $\phi$ being the azimuthal angle around the beam direction. The pseudorapidity is defined in terms of the polar angle $\theta$ as $\eta=-\textrm{ln}[\textrm{tan}(\theta/2)]$. and nearly $4\pi$ coverage in solid angle. A set of tracking detectors around the collision point (collectively referred to as the inner detector) is located within a superconducting solenoid providing a $2$ T axial magnetic field, and is surrounded by a calorimeter system and a muon spectrometer. The inner detector (ID) consists of a silicon pixel detector and a silicon microstrip tracker, together providing precision tracking in the pseudorapidity range $|\eta|<2.5$, complemented by a straw-tube transition radiation tracker providing tracking and electron identification information for $|\eta|<2.0$. The electromagnetic calorimeter covers the region $|\eta|<3.2$ and is based on a lead/liquid-argon (LAr) sampling technology. The hadronic calorimeter uses a steel/scintillator-tile sampling detector in the region $|\eta|<1.7$ and a copper/LAr detector in the region $1.5<|\eta|<3.2$. The most forward region of ATLAS, $3.1<|\eta|<4.9$, is equipped with a forward calorimeter, measuring electromagnetic and hadronic energies in copper/LAr and tungsten/LAr modules. The muon spectrometer (MS) comprises separate trigger and high-precision tracking chambers to measure the deflection of muons in a magnetic field generated by three large superconducting toroids with coils arranged with an eightfold azimuthal symmetry around the calorimeters. The high-precision chambers cover the range of $|\eta|<2.7$ with three layers of monitored drift tubes, complemented by cathode strip chambers in the forward region, where the particle flux is highest. The muon trigger system covers the range $|\eta|<2.4$ with resistive-plate chambers in the barrel and thin-gap chambers in the endcap regions. A two-level trigger system [35] is used to select events in real time. It consists of a hardware-based first-level trigger that uses a subset of detector information to reduce the event rate to approximately 100 kHz, and a software-based high-level trigger system that reduces the event rate to about 1 kHz. The latter employs algorithms similar to those used offline to identify electrons, muons, photons and jets.

3 Phase space for cross-section measurement

The fiducial $W^{\pm}Z$ cross section is measured in a phase space chosen to follow closely the event selection criteria described in Section 5. It is based on the kinematics of particle-level objects as defined in Ref. [36]. These are final-state prompt222A prompt lepton is a lepton that is not produced in the decay of a hadron, a $\tau$-lepton, or their descendants. leptons associated with the $W$ and $Z$ boson decays. Charged leptons after QED final-state radiation are “dressed” by adding to the lepton four-momentum the contributions from photons with an angular distance $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}<0.1$ from the lepton. Dressed leptons, and final-state neutrinos that do not originate from hadron or $\tau$-lepton decays, are matched to the $W$ and $Z$ boson decay products using an algorithm that does not depend on details of the Monte Carlo (MC) generator, called the “resonant shape” algorithm. This algorithm is based on the value of an estimator expressing the product of the nominal line-shapes of the $W$ and $Z$ resonances

[TABLE]

where $m_{Z}^{\textrm{PDG}}$ ($m_{W}^{\textrm{PDG}}$) and $\Gamma_{Z}^{\textrm{PDG}}$ ($\Gamma_{W}^{\textrm{PDG}}$) are the world average mass and total width of the $Z$ ($W$) boson, respectively, as reported by the Particle Data Group [37]. The input to the estimator is the invariant mass $m$ of all possible pairs ($\ell^{+},\ell^{-}$) and ($\ell^{\prime},\nu_{\ell^{\prime}}$) satisfying the fiducial selection requirements defined in the next paragraph. The final choice of which leptons are assigned to the $W$ or $Z$ bosons corresponds to the configuration exhibiting the largest value of the estimator. Using this specific association algorithm, the gauge boson kinematics can be computed using the kinematics of the associated leptons independently of any internal MC generator details.

The reported cross sections are measured in a fiducial phase space defined at particle level as follows. The dressed leptons from $Z$ and $W$ boson decay must have $|\eta|<2.5$ and transverse momentum $p_{\text{T}}$ above $15$ GeV and $20$ GeV, respectively; the invariant mass of the two leptons from the $Z$ boson decay differs by at most $10$ GeV from the world average value of the $Z$ boson mass $m_{Z}^{\textrm{PDG}}$. The $W$ transverse mass, defined as $m_{\textrm{T}}^{W}=\sqrt{2\cdot p_{\mathrm{T}}^{\nu}\cdot p_{\mathrm{T}}^{\ell}\cdot[1-\cos{\Delta\phi(\ell,\nu)}]}$, where $\Delta\phi(\ell,\nu)$ is the angle between the lepton and the neutrino in the transverse plane, and $p_{\text{T}}^{\ell}$ and $p_{\text{T}}^{\nu}$ are the transverse momenta of the lepton from $W$ boson decay and of the neutrino, respectively, must be greater than $30$ GeV. In addition, it is required that the angular distance $\Delta R$ between the charged leptons from the $W$ and $Z$ decay is larger than $0.3$, and that $\Delta R$ between the two leptons from the $Z$ decay is larger than $0.2$. A requirement that the transverse momentum of the leading lepton be above $27$ GeV reduces the acceptance of the fiducial phase space by less than $0.5\%$. This criterion is therefore not added to the definition of the fiducial phase space, while it is present in the selection at the detector level.

The fiducial cross section is extrapolated to the total phase space and corrected for the leptonic branching fractions of the $W$ and $Z$ bosons, $(10.86\pm 0.09)\%$ and $(3.3658\pm 0.0023)\%$ [37], respectively. The total phase space is defined by requiring the invariant mass of the lepton pair associated with the $Z$ boson to be in the range $66<m_{\ell\ell}<116$ GeV.

For the differential measurements related to jets, particle-level jets are reconstructed from stable particles with a lifetime of $\tau>30$ ps in the simulation. Stable particles are taken after parton showering, hadronisation, and the decay of particles with $\tau<30$ ps. Muons, electrons, neutrinos and photons associated with $W$ and $Z$ decays are excluded from the jet collection. The particle-level jets are reconstructed with the anti-$k_{t}$ algorithm [38] with a radius parameter $R=0.4$ and are required to have a $p_{\text{T}}$ above $25$ GeV and an absolute value of pseudorapidity below $4.5$.

4 Signal and background simulation

A sample of simulated $W^{\pm}Z$ events is used to correct the signal yield for detector effects, to extrapolate from the fiducial to the total phase space, and to compare the measurements with the theoretical predictions. The production of $W^{\pm}Z$ pairs and the subsequent leptonic decays of the vector bosons were simulated at NLO in QCD using the Powheg-Box v2 [39, 40, 41, 42] generator, interfaced to Pythia 8.210 [43] for simulation of parton showering, hadronisation and the underlying event. Final-state radiation resulting from QED interactions is simulated using Pythia 8.210 and the AZNLO [44] set of tuned parameters. The CT10 [45] PDF set was used for the hard-scattering process, while the CTEQ6L1 [46] PDF set was used for the parton shower. The sample was generated with dynamic renormalisation and factorisation QCD scales, $\mu_{\textrm{R}}$ and $\mu_{\textrm{F}}$, equal to $m_{WZ}/2$, where $m_{WZ}$ is the invariant mass of the $WZ$ system. An additional $W^{\pm}Z$ sample was generated by interfacing Powheg-Box v2 matrix elements to the Herwig++ 2.7.1 [47] fragmentation model and is used to estimate the uncertainty due to the fragmentation modelling. Also for this sample, the CT10 and CTEQ6L1 PDF sets are used for the matrix elements and the parton showers, respectively, while QED final-state radiation is internally modelled in Herwig. An alternative signal sample was generated at NLO QCD using the Sherpa 2.2.2 generator [48]. Matrix elements contain all diagrams with four electroweak vertices. They were calculated for up to one parton at NLO and up to three partons at LO using Comix [49] and OpenLoops [50], and merged with the Sherpa parton shower [51] according to the ME+PS$@$NLO prescription [52]. The NNPDF3.0nnlo [53] PDF set was used in conjunction with the dedicated parton shower tuning developed by the Sherpa authors. A calculation using Sherpa 2.1 with one to three partons at LO is also used for comparisons to measured jet observables. Finally, the NLO QCD predictions from the MC@NLO v4.0 [54] MC generator interfaced to the Herwig fragmentation model, using the CT10 PDF set, are also used to estimate signal modelling uncertainties.

NNLO QCD cross sections for $W^{\pm}Z$ production in the fiducial and total phase spaces are provided by the MATRIX computational framework [30, 31, 55, 50, 56, 57, 58, 59]. The calculation includes contributions from off-shell electroweak bosons and all relevant interference effects. The renormalisation and factorisation scales were fixed to $(m_{Z}+m_{W})/2$, chosen following Ref. [30], and the NNPDF3.0nnlo PDF set was used. The predictions from the Powheg+Pythia sample were rescaled by a global factor of $1.18$ to match the NNLO cross section predicted by MATRIX.

The background sources in this analysis include processes with two or more electroweak gauge bosons, namely $ZZ$, $WW$ and $VVV$ ($V=W,Z$); processes with top quarks, such as $t\bar{t}$ and $t\bar{t}V$, single top and $tZ$; and processes with gauge bosons associated with jets or photons ($Z+j$ and $Z\gamma$). MC simulation is used to estimate the contribution from background processes with three or more prompt leptons. Background processes with at least one misidentified lepton are evaluated using data-driven techniques and simulated events are used to assess the systematic uncertainties of these backgrounds (see Section 6).

The Sherpa 2.2.2 event generator was used to simulate $q\bar{q}$-initiated $ZZ$ processes with up to one parton at NLO and up to three partons at LO and using the NNPDF3.0nnlo PDF set. A Sherpa 2.1.1 $ZZ$ sample was generated with the loop-induced $gg$-initiated process simulated at LO using the CT10 PDF, with up to one additional parton. A $K$-factor of $1.67\pm 0.25$ was applied to the cross section of the loop-induced $gg$-initiated process to account for the NLO corrections [60]. Triboson events were simulated at LO with the Sherpa 2.1.1 generator using the CT10 PDF set. The $t\bar{t}V$ processes were generated at NLO with the MadGraph5_aMC@NLO [61] MC generator using the NNPDF3.0nlo PDF set interfaced to the Pythia 8.186 [62] parton shower model. Finally, the $tZ$ events were generated at LO with the MadGraph5_aMC@NLO using the NNPDF2.3lo [63] PDF set interfaced with Pythia 6.428 [64].

All generated MC events were passed through the ATLAS detector simulation [65], based on GEANT4 [66], and processed using the same reconstruction software as used for the data. The event samples include the simulation of additional proton–proton interactions (pile-up) generated with Pythia 8.186 using the MSTW2008LO [67] PDF set and the A2 [68] set of tuned parameters for the underlying event and parton fragmentation. Simulated events were reweighted to match the pile-up conditions observed in the data. Scale factors are applied to simulated events to correct for small differences in the efficiencies observed in data and predicted by MC simulation for the trigger, reconstruction, identification, isolation and impact parameter requirements of electrons and muons [69, 70]. Furthermore, the electron energy and muon momentum in simulated events are smeared to account for small differences in resolution between data and MC events [71, 70].

5 Event selection

Only data recorded with stable beam conditions and with all relevant detector subsystems operational are considered. Candidate events are selected using triggers [35] that require at least one electron or muon. The transverse momentum threshold applied to leptons by the triggers in $2015$ was $24$ GeV for electrons and $20$ GeV for muons satisfying a loose isolation requirement based only on ID track information. Due to the higher instantaneous luminosity in $2016$ the trigger threshold was increased to $26$ GeV for both the electrons and muons. Furthermore, tighter lepton isolation and tighter electron identification requirements were applied in $2016$. Possible inefficiencies for leptons with large transverse momentum are reduced by using additional triggers with tighter thresholds, $p_{\text{T}}=60$ GeV and $50$ GeV for electrons and muons respectively, and no isolation requirements. Finally, a single-electron trigger requiring $p_{\text{T}}>120$ GeV (in $2015$) and $p_{\text{T}}>140$ GeV (in $2016$) with less restrictive electron identification criteria was used to increase the selection efficiency for high-$p_{\text{T}}$ electrons. The combined efficiency of these triggers is close to $100\%$ for $W^{\pm}Z$ events passing the offline selection criteria.

Events are required to have a primary vertex compatible with the luminous region of the LHC. The primary vertex is defined as the reconstructed vertex with at least two charged particle tracks, that has the largest sum of the $p_{\text{T}}^{2}$ for the associated tracks.

All final states with three charged leptons (electrons $e$ or muons $\mu$) and missing transverse momentum ($E_{\text{T}}^{\text{miss}}$) from $W^{\pm}Z$ leptonic decays are considered. In the following, the different final states are referred to as $\mu^{\pm}\mu^{+}\mu^{-}$, $e^{\pm}\mu^{+}\mu^{-}$, $\mu^{\pm}e^{+}e^{-}$ and $e^{\pm}e^{+}e^{-}$, where the first label is from the charged lepton of the $W$ decay, and the last two labels are for the $Z$ decay. No requirement on the number of jets is applied.

Muon candidates are identified by tracks reconstructed in the muon spectrometer (MS) and matched to tracks reconstructed in the inner detector (ID). Muons are required to pass a “medium” identification selection, which is based on requirements on the number of hits in the ID and the MS [70]. The efficiency of this selection averaged over $p_{\text{T}}$ and $\eta$ is larger than $98\%$. The muon momentum is calculated by combining the MS measurement, corrected for the energy deposited in the calorimeters, and the ID measurement. The $p_{\text{T}}$ of the muon must be greater than $15$ GeV and its pseudorapidity must satisfy $|\eta|<2.5$.

Electron candidates are reconstructed from energy clusters in the electromagnetic calorimeter matched to ID tracks. Electrons are identified using a discriminant that is the value of a likelihood function constructed with information about the shape of the electromagnetic showers in the calorimeter, the track properties, and the quality of the track-to-cluster matching for the candidate [69]. Electrons must satisfy a “medium” likelihood requirement, which provides an overall identification efficiency of $90\%$. The electron momentum is computed from the cluster energy and the direction of the track. The $p_{\text{T}}$ of the electron must be greater than $15$ GeV and the pseudorapidity of the cluster must satisfy $|\eta|<1.37$ or $1.52<|\eta|<2.47$ to be within the tracking system, excluding the transition region between the barrel and endcap sections of the electromagnetic calorimeter.

Electron and muon candidates are required to originate from the primary vertex. Thus, the significance of the track’s transverse impact parameter calculated relative to the beam line, $|{d_{0}/\sigma_{d_{0}}}|$, must be smaller than $3.0$ for muons and less than $5.0$ for electrons. Furthermore, the longitudinal impact parameter, $z_{0}$ (the difference between the value of $z$ of the point on the track at which $d_{0}$ is defined and the longitudinal position of the primary vertex), is required to satisfy $|z_{0}\cdot\sin(\theta)|<0.5$ mm.

Electrons and muons are required to be isolated from other particles using both calorimeter-cluster and ID-track information. The isolation requirement for electrons is tuned for an efficiency of at least $90\%$ for $p_{\text{T}}>25$ GeV and at least $99\%$ for $p_{\text{T}}>60$ GeV [69], while fixed requirements on the isolation variables are used for muons, providing an efficiency above $90\%$ for $p_{\text{T}}>15$ GeV and at least $99\%$ for $p_{\text{T}}>60$ GeV [70].

Jets are reconstructed from clusters of energy deposition in the calorimeter [72] using the anti-$k_{t}$ algorithm [38] with a radius parameter $R=0.4$. The energy of jets is calibrated using a jet energy correction derived from both simulation and in situ methods using data [73]. Jets with $p_{\text{T}}$ below $60$ GeV and with $|\eta|<2.4$ have to pass a requirement on the jet vertex tagger [74], a likelihood discriminant that uses a combination of track and vertex information to suppress jets originating from pile-up activity. All jets must have $p_{\text{T}}>25$ GeV and be reconstructed in the pseudorapidity range $|\eta|<4.5$. Jets in the ID acceptance containing a $b$-hadron are identified with a multivariate algorithm [75, 76] that uses the impact parameter and reconstructed secondary vertex information of the tracks contained in the jets. Jets initiated by $b$-quarks are selected by setting the algorithm’s output threshold such that a $70\%$ $b$-jet selection efficiency is achieved in simulated $t\bar{t}$ events. The corresponding light-flavour ($u$,$d$,$s$-quark and gluon) and $c$-jet misidentification efficiencies are $0.3$% and $8.2$%, respectively. Corrections to the flavour-tagging efficiencies are based on data-driven calibration analyses.

The transverse momentum of the neutrino is estimated from the missing transverse momentum in the event, $E_{\text{T}}^{\text{miss}}$, calculated as the negative vector sum of the transverse momentum of all identified hard physics objects (electrons, muons, jets), with a contribution from an additional soft term. This soft term is calculated from ID tracks matched to the primary vertex and not assigned to any of the hard objects (electrons, muons and jets) [77].

To avoid cases where the detector response to a single physical object is reconstructed as two different final-state objects, e.g. an electron reconstructed as both an electron and a jet, several steps are followed to remove such overlaps, as described in Ref. [78].

Events are required to contain exactly three lepton candidates satisfying the selection criteria described above. To ensure that the trigger efficiency is well determined, at least one of the candidate leptons is required to have $p_{\text{T}}>25$ GeV for $2015$ and $p_{\text{T}}>27$ GeV for $2016$ data, as well as being geometrically matched to a lepton that was selected by the trigger.

To suppress background processes with at least four prompt leptons, events with a fourth lepton candidate satisfying looser selection criteria are rejected. For this looser selection, the lepton $p_{\text{T}}$ requirement is lowered to $p_{\text{T}}>5$ GeV, electrons are allowed to be reconstructed in the barrel-endcap calorimeter gap ($1.37<|\eta|<1.52$), and “loose” identification requirements [70, 69] are used for both the electrons and muons. A less stringent requirement is applied for electron isolation and is based only on ID track information.

Candidate events are required to have at least one pair of leptons with the same flavour and opposite charge, with an invariant mass that is consistent with the nominal $Z$ boson mass [37] to within $10$ GeV. This pair is considered to be the $Z$ boson candidate. If more than one pair can be formed, the pair whose invariant mass is closest to the nominal $Z$ boson mass is taken as the $Z$ boson candidate. The remaining third lepton is assigned to the $W$ boson decay. The transverse mass of the $W$ candidate, computed using $E_{\text{T}}^{\text{miss}}$ and the $p_{\text{T}}$ of the associated lepton, is required to be greater than $30$ GeV.

Backgrounds originating from misidentified leptons are suppressed by requiring the lepton associated with the $W$ boson to satisfy more stringent selection criteria. Thus, the transverse momentum of these leptons is required to be greater than $20$ GeV. Furthermore, charged leptons associated with the $W$ boson decay are required to pass the “tight” identification requirements, which results in an efficiency between $90\%$ and $98\%$ for muons and an overall efficiency of $85\%$ for electrons. Finally, muons associated to the $W$ boson must also pass a tighter isolation requirement, tuned for an efficiency of at least $90\%$ ($99\%$) for $p_{\text{T}}>25\leavevmode\nobreak\ (60)$ GeV.

6 Background estimation

The background sources are classified into two groups: events where at least one of the candidate leptons is not a prompt lepton (reducible background) and events where all candidates are prompt leptons or are produced in the decay of a $\tau$-lepton (irreducible background). Candidates that are not prompt leptons are also called “misidentified” or “fake” leptons.

Events in the first group originate from $Z+j$, $Z\gamma$, $t\bar{t}$, and $WW$ production processes and constitute about $40\%$ of the total backgrounds. This reducible background is estimated with a data-driven method based on the inversion of a matrix containing the efficiencies and the misidentification probabilities for prompt and fake leptons [9, 79]. The method exploits the classification of the leptons as loose or tight candidates and the probability that a fake lepton is misidentified as a loose or tight lepton. Tight leptons are signal leptons as defined in Section 5. Loose leptons are leptons that do not meet the isolation and identification criteria of signal leptons but satisfy only looser criteria. The misidentification probabilities for fake leptons are determined from data using dedicated control samples enriched in misidentified leptons from light- or heavy-flavour jets and from photon conversions. The lepton efficiencies and misidentification probabilities are combined with event rates in data samples of $W^{\pm}Z$ candidate events where at least one and up to three of the leptons are loose. Then, solving the system of linear equations, the number of events with at least one misidentified lepton, which represents the amount of reducible background in the $W^{\pm}Z$ sample, is obtained. About $2\%$ of this background contribution arises from events with two fake leptons. The background from events with three fake leptons, e.g., from multijet processes, is negligible. The method allows the shape of any kinematic distribution of reducible background events to be estimated. Another independent method to assess the reducible background was also considered. This method estimates the amount of reducible background using MC simulations scaled to data by process-dependent factors determined from the data-to-MC comparison in dedicated control regions. Good agreement with the matrix method estimate is obtained at the level of $4$% in the yield and $40$% in the shape of the distributions of irreducible background events.

The events contributing to the second group of background processes originate from $ZZ$, $t\bar{t}+V$, $VVV$ (where $V$ = $Z$ or $W$) and $tZ(j)$ events. The amount of irreducible background is estimated using MC simulations because processes with a small cross section and signal leptons can be simulated with a better statistical accuracy than an estimate obtained with data-driven methods. Events from $VH$ production processes with leptonic decays of the bosons can also contribute. This contribution was estimated using MC simulations to be of the order of $0.1$% and was therefore neglected. The dominant contribution in this second group is from $ZZ$ production, where one of the leptons from the $ZZ$ decay falls outside the detector acceptance. It represents about $70\%$ of the irreducible background. The MC-based estimates of the $ZZ$ and $t\bar{t}+V$ backgrounds are validated by comparing the MC expectations with the event yield and several kinematic distributions of a data sample enriched in $ZZ$ and $t\bar{t}+V$ events, respectively.

The $ZZ$ control sample is selected by requiring a $Z$ candidate that meets all the analysis selection criteria and is accompanied by two additional leptons, satisfying the lepton criteria described in Section 5. The $ZZ$ MC expectation needs to be rescaled by a factor of $1.12$ in order to match the observed event yield of data in this control region. This scaling factor relative to Sherpa predictions is in agreement with the $ZZ$ cross-section measurements performed at $\sqrt{s}=13$ TeV [80]. The shapes of distributions of the main kinematic variables are found to be well described by the MC predictions.

The $t\bar{t}+V$ control sample is selected by requiring $W^{\pm}Z$ events to have at least two reconstructed $b$-jets. The $t\bar{t}+V$ MC calculation needs to be rescaled by a factor of $1.3$ in order to match the observed event yield in data. This scaling factor relative to predictions is in line with the $t\bar{t}V$ cross-section measurements performed at $\sqrt{s}=13$ TeV [81]. Here again, the distributions of the main kinematic variables are found to be well described by the MC predictions.

7 Detector-level results

Table 1 summarises the predicted and observed numbers of events together with the estimated background contributions. Only statistical uncertainties are quoted. Figure 1 shows the measured distributions of the transverse momentum and the invariant mass of the $Z$ candidate, the transverse mass of the $W$ candidate, and for the $WZ$ system the variable $m_{\mathrm{T}}^{WZ}$, defined as follows:

[TABLE]

The Powheg+Pythia MC prediction is used for the $W^{\pm}Z$ signal contribution. Figure 1 indicates that the MC predictions provide a fair description of the shapes of the data distributions.

8 Corrections for detector effects and acceptance

For a given channel $W^{\pm}Z$ $\rightarrow\ell^{{}^{\prime}\pm}\nu\ell^{+}\ell^{-}$, where $\ell$ and $\ell^{{}^{\prime}}$ indicates each type of lepton ($e$ or $\mu$), the integrated fiducial cross section that includes the leptonic branching fractions of the $W$ and $Z$ bosons is calculated as

[TABLE]

where $N_{\textrm{data}}$ and $N_{\textrm{bkg}}$ are the number of observed events and the estimated number of background events, respectively, $\mathcal{L}$ is the integrated luminosity, and $C_{WZ}$, obtained from simulation, is the ratio of the number of selected signal events at detector level to the number of events at particle level in the fiducial phase space. This factor corrects for detector efficiencies and for QED final-state radiation effects. The contribution from $\tau$-lepton decays, amounting approximately to $4\%$, is removed from the cross-section definition by introducing the term in parentheses. This term is computed using simulation, where $N_{\tau}$ is the number of selected events at detector level in which at least one of the bosons decays into a $\tau$-lepton and $N_{\textrm{all}}$ is the number of selected $WZ$ events with decays into any lepton.

The $C_{WZ}$ factors for $W^{-}Z$, $W^{+}Z$, and $W^{\pm}Z$ inclusive processes computed with Powheg+Pythia for each of the four leptonic channels are shown in Table 2.

The total cross section is calculated as

[TABLE]

where $\mathcal{B}_{W}=(10.86\pm 0.09)\%$ and $\mathcal{B}_{Z}=(3.3658\pm 0.0023)\%$ are the $W$ and $Z$ leptonic branching fractions [37], respectively, and $A_{WZ}$ is the acceptance factor calculated at particle level as the ratio of the number of events in the fiducial phase space to the number of events in the total phase space as defined in Section 3.

A single acceptance factor of $A_{WZ}$ = $0.343$ $\pm$ $0.002$ (stat.), obtained by averaging the acceptance factors computed in the $\mu ee$ and $e\mu\mu$ channels, is used since it was verified for the fiducial phase space used that interference effects related to the presence of identical leptons in the final state, as in the $eee$ and $\mu\mu\mu$ channels, are below $1\%$ of the cross section. The use of only the $\mu{ee}$ and ${e}\mu\mu$ channels for the computation of $A_{WZ}$ avoids the ambiguity arising from the assignment of particle-level final-state leptons to the $W$ and $Z$ bosons.

The differential detector-level distributions within the fiducial phase space are corrected for detector resolution and for QED final-state radiation effects using simulated signal events and an iterative Bayesian unfolding method [82], as implemented in the RooUnfold toolkit [83]. The number of iterations used ranges from two to four, depending on the resolution in the unfolded variable. The width of the bins in each distribution is chosen according to the experimental resolution and to the statistical significance of the expected number of events in each bin. The fraction of signal MC events generated in a bin that are reconstructed in the same bin is around $70\%$ on average and always greater than $50\%$, except for the jet multiplicity distribution, where it can decrease to $40\%$ for $N_{\mathrm{jets}}=4$.

In the inclusive measurements, the Powheg+Pythia signal sample is used for unfolding since it provides a fair description of the data distributions. For differential measurements with jets, the Sherpa 2.2.2 signal sample is used for the computation of the response matrix since this sample includes up to three partons in the matrix element calculation and therefore better describes the jet multiplicity in data.

9 Systematic uncertainties

The systematic uncertainties in the measured cross sections are due to uncertainties of experimental and theoretical nature in the acceptance, in the correction procedure for detector effects, in the background estimate and in the luminosity.

The theoretical modelling systematic uncertainties in the $A_{WZ}$ and $C_{WZ}$ factors are due to the choice of PDF set, QCD renormalisation $\mu_{\text{R}}$ and factorisation $\mu_{\text{F}}$ scales, and the parton showering simulation. The uncertainties due to the choice of PDF are computed using the CT10 eigenvectors and the envelope of the differences between the CT10 and CT14 [84], MMHT2014 [85] and NNPDF 3.0 [53] PDF sets, according to the PDF4LHC recommendations [86]. The effects of QCD scale uncertainties are estimated by varying $\mu_{\text{R}}$ and $\mu_{\text{F}}$ by factors of two around the nominal scale $m_{WZ}/2$ with the constraint $0.5\leq\mu_{\text{R}}/\mu_{\text{F}}\leq 2$, where $m_{WZ}$ is the invariant mass of the $WZ$ system. Uncertainties arising from the choice of parton shower model are estimated by interfacing Powheg with Pythia or Herwig and comparing the results. Among these three sources of theoretical uncertainty, only the choice of parton shower model has an effect on the $C_{WZ}$ factors, of $0.5\%$. The uncertainty of the acceptance factor $A_{WZ}$ is less than $0.5\%$ due to the PDF choice, less than $0.7\%$ due to the QCD scale choice, and about $0.5\%$ for the choice of parton shower model.

Uncertainties in the unfolding from detector to particle level due to imperfect description of the data by the MC simulation are evaluated using a data-driven method [87]. The MC differential distribution is corrected to match the data distribution and the resulting weighted MC distribution at detector level is unfolded with the response matrix used in the actual data unfolding. The new unfolded distribution is compared with the weighted MC distribution at particle level and the difference is taken as the systematic uncertainty. Uncertainties in the unfolding are typically of the order of $2$% but can vary from $0.1$% to $10$% depending on the considered observable and bin. For the inclusive measurements, differences in the unfolded results if the Powheg+Pythia or Sherpa 2.2.2 MC signal samples are used for the unfolding are found to be covered by these unfolding uncertainties.

The experimental systematic uncertainty on the $C_{WZ}$ factors and on the unfolding procedure includes uncertainties on the scale and resolution of the electron energy, muon momentum, jet energy and $E_{\text{T}}^{\text{miss}}$, as well as uncertainties on the scale factors applied to the simulation in order to reproduce the trigger, reconstruction, identification and isolation efficiencies measured in data. The systematic uncertainties on the measured cross sections are determined by repeating the analysis after applying appropriate variations for each source of systematic uncertainty to the simulated samples. The uncertainties on the jet energy scale and resolution are based on their respective measurements in data [73]. The uncertainty on $E_{\text{T}}^{\text{miss}}$ is estimated by propagating the uncertainties on the transverse momenta of reconstructed objects and by applying momentum scale and resolution uncertainties to the track-based soft term [77]. A variation in the pileup reweighting of the MC is included to cover the uncertainty on the ratio between the predicted and measured inelastic cross-section in the fiducial volume defined by $M_{X}>13$ GeV where $M_{X}$ is the mass of the hadronic system [88]. For the measurements of the $W$ charge-dependent cross sections, an uncertainty arising from the charge misidentification of leptons is also considered. It affects only electrons and leads to an uncertainty of less than $0.15\%$ in the ratio of $W^{+}Z$ to $W^{-}Z$ integrated cross sections determined by combining the four decay channels.

The dominant contribution among the experimental systematic uncertainties in the $eee$ and $\mu ee$ channels is due to the uncertainty on the electron identification efficiency, contributing at most a $2.8\%$ uncertainty to the integrated cross section, while in the $e\mu\mu$ and $\mu\mu\mu$ channels it originates from the muon reconstruction efficiency and is at most $2.8\%$.

The uncertainty on the amount of background from misidentified leptons takes into account the limited number of events in the control regions as well as the differences in background composition between the control region used to determine the lepton misidentification rate and the control regions used to estimate the yield in the signal region. This results in a total uncertainty of $30\%$ on the misidentified-leptons background yield for the integrated cross-section measurements and of $40\%$ when the shape of the differential distributions of the reducible background events is also considered.

A global uncertainty of $\pm 12\%$ is assigned to the amount of $ZZ$ background predicted by the MC simulation, based on the comparison with data in the $ZZ$ control region. Similarly, a global uncertainty of $\pm 30\%$ is assigned to the $t\bar{t}+V$ background.

The uncertainty due to other irreducible background sources is evaluated by propagating the uncertainty in their MC cross sections. These are $20\%$ for $VVV$ [89] and $15\%$ for $tZ$ [9].

The uncertainty on the combined $2015$+$2016$ integrated luminosity is $2.1\%$. It is derived from a calibration of the luminosity scale using $x$–$y$ beam-separation scans, following a methodology similar to that detailed in Ref. [90], and using the LUCID-2 detector for the baseline luminosity measurements [91]. It is applied to the signal normalisation as well as to all background contributions that are estimated using only MC simulations and has an effect of $2.4\%$ on the measured cross sections.

The total systematic uncertainty on the $W^{\pm}Z$ fiducial cross section, excluding the luminosity uncertainty, varies between $4\%$ and $6\%$ for the four different measurement channels, and is dominated by the uncertainty on the reducible background estimate. Table 3 shows the statistical uncertainty and the main sources of systematic uncertainty on the $W^{\pm}Z$ fiducial cross section for each of the four channels and for their combination. The modelling uncertainty on the measurements is dominated by the modelling of the fragmentation.

10 Cross-section measurements

10.1 Integrated cross sections

The measured fiducial cross sections for the four channels are combined using a $\chi^{2}$ minimisation method that accounts for correlations between the sources of systematic uncertainty affecting each channel [92, 93, 94]. The combination of the $W^{\pm}Z$ cross sections in the fiducial phase space for the four channels yields a $\chi^{2}$ per degree of freedom ($\mathrm{dof}$) of $\chi^{2}/n_{\mathrm{dof}}=3.3/3$. The combinations of the $W^{+}Z$ and $W^{-}Z$ cross sections separately yield $\chi^{2}/n_{\mathrm{dof}}=3.7/3$ and $1.5/3$, respectively.

The $W^{\pm}Z$ production cross section in the fiducial phase space resulting from the combination of the four channels including the $W$ and $Z$ branching ratio in a single leptonic channel with muons or electrons is

[TABLE]

where the uncertainties correspond to statistical, experimental systematic, modelling systematic and luminosity uncertainties, respectively. The corresponding SM NNLO QCD prediction from MATRIX is $61.5^{+1.4}_{-1.3}\,\mathrm{fb}$, where the uncertainty corresponds to the QCD scale uncertainty estimated conventionally by varying the scales $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$ by factors of two around the nominal value of $(m_{W}+m_{Z})/2$ with the constraint $0.5\leq\mu_{\mathrm{R}}/\mu_{\mathrm{F}}\leq 2$. This prediction is obtained by correcting the result in Ref. [31] for Born level leptons to dressed leptons by a factor of $0.96$, which is estimated in the fiducial phase space using Powheg+Pythia. Changing the PDF set used from NNPDF3.0nnlo to MMHT2014 or CT14 affects the MATRIX prediction by $+2\%$ and $+1\%$, respectively. The uncertainty due to varying the $\alpha_{\text{S}}$ coupling constant value used in the PDF determination is $0.6\%$ and $1.0\%$ for $W^{+}Z$ and $W^{-}Z$ production, respectively. The measured $W^{\pm}Z$ production cross sections are compared with the SM NNLO prediction from MATRIX using three different PDF sets, NNDPF3.0nnlo, MMHT2014 and CT14, as well as with NLO predictions from Sherpa 2.2.2 in Figure 2. All results for $W^{\pm}Z$, $W^{+}Z$ and $W^{-}Z$ final states are reported in Table 4. The NNLO SM calculations reproduce the measured cross sections well. The production of $W^{\pm}Z$ in association with two jets produced as a result of electroweak processes is not included in the NNLO SM prediction and amounts to $1.2\%$ of the measured cross section, as estimated using Sherpa 2.2.2.

The ratio of the $W^{+}Z$ to $W^{-}Z$ production cross sections is

[TABLE]

Most of the systematic uncertainties, especially the luminosity uncertainty, almost cancel out in the ratio, so that the measurement is dominated by the statistical uncertainty. The measured cross-section ratios, for each channel and for their combination, are compared in Figure 3 with the SM prediction of $1.44^{+0.03}_{-0.06}$, calculated with MATRIX [31] and the NNDPF3.0nnlo PDF set. The uncertainties correspond to PDF uncertainties estimated at NLO with Powheg+Pythia using the CT10 eigenvectors and the envelope of the differences between the CT10 and CT14, MMHT2014 and NNPDF 3.0nnlo PDF sets. The effects of QCD scale uncertainties on the predicted cross-section ratio are negligible. The cross-section ratio is also calculated with MATRIX using the MMHT2014 and CT14 PDF sets, yielding values of $1.42$ and $1.44$, respectively, as shown in Figure 3.

The combined fiducial cross section is extrapolated to the total phase space. The result is

[TABLE]

where the modelling uncertainty accounts for the uncertainties in the $A_{WZ}$ factor due to the choice of PDF set, QCD scales and the fragmentation model. The NNLO SM prediction calculated with MATRIX [30] is $49.1^{+1.1}_{-1.0}\,\mathrm{(scale)}\;\mathrm{pb}$, which is in good agreement with the present measurement. As the MATRIX calculation does not include effects of QED final-state radiation, a correction factor of $0.99$, as estimated from Powheg+Pythia in the total phase space, is applied to it to obtain the above prediction.

10.2 Differential cross sections

For the measurements of the differential distributions, all four decay channels, $eee$, $e\mu\mu$, $\mu ee$, and $\mu\mu\mu$, are added together. The resulting distributions are unfolded with a response matrix computed using a Powheg+Pythia MC signal sample that includes all four topologies and is divided by four, such that cross sections refer to final states where the $W$ and $Z$ bosons decay in a single leptonic channel with muons or electrons.

The $W^{\pm}Z$ production cross section is measured as a function of several variables: the transverse momenta of the $Z$ and $W$ bosons, $p_{\textrm{T}}^{Z}$ and $p_{\textrm{T}}^{W}$, the transverse mass of the $W^{\pm}Z$ system $m_{\textrm{T}}^{WZ}$ and the azimuthal angle between the $W$ and $Z$ bosons in Figure 4; as a function of the $p_{\text{T}}$ of the neutrino associated with the decay of the $W$ boson, $p_{\textrm{T}}^{\nu}$, and the absolute difference between the rapidities of the $Z$ boson and the charged lepton from the decay of the $W$ boson, $|y_{Z}-y_{\ell,W}|$ in Figure 5.

In order to derive $p_{\textrm{T}}^{W}$ and $p_{\textrm{T}}^{\nu}$ from data events, it is assumed that the whole $E_{\textrm{T}}^{\textrm{miss}}$ of each event arises from the neutrino of the $W$ boson decay. The validity of this assumption was verified for SM $WZ$ events using MC samples at the level of precision of the present results.

The measured differential cross sections in Figures 4 and 5 are compared with the predictions at NNLO in QCD from the MATRIX computational framework. The predictions from MATRIX are corrected from Born-level leptons to dressed leptons using binned correction factors determined using Powheg+Pythia. The correction factors are found to be mostly constant over the ranges of all differential distributions, with a mean value of $0.96$. The predicted and measured cross sections are in good agreement. The measurements are also compared with NLO MC predictions from Powheg+Pythia, after a rescaling of its predicted integrated fiducial cross section to the NNLO cross section, and to Sherpa 2.2.2 without rescaling its prediction. Good agreement of the shapes of the measured distributions with the predictions of Powheg+Pythia and Sherpa 2.2.2 is observed. The $\Delta\phi(W,Z)$ distribution, which is sensitive to QCD higher-order perturbative effects, is better described by MATRIX than by Powheg+Pythia or Sherpa 2.2.2.

As shown in previous publications, the high energy tails of the $p_{\mathrm{T}}^{Z}$ [12] and $m_{\mathrm{T}}^{WZ}$ [9] observables are sensitive to aTGC, $p_{\mathrm{T}}^{Z}$ having the disadvantage of being more subject to higher-order perturbative effects in QCD [95] and electroweak theory [96]. This is seen also here with larger NNLO QCD scale uncertainties predicted by MATRIX for $p_{\mathrm{T}}^{Z}$ than for $m_{\mathrm{T}}^{WZ}$. No excess of data events in the tails of these distributions is observed.

The exclusive multiplicity of jets above a $p_{\text{T}}$ threshold of $25$ GeV unfolded at particle level is presented in Figure 6(a). The measurements are compared with predictions from Sherpa 2.2.2, Sherpa 2.1 and Powheg+Pythia. The Sherpa predictions provide a better description of the ratio of [math]-jet to $1$-jet event cross sections than Powheg+Pythia. However, the Sherpa 2.2.2 prediction, which models up to one parton at NLO, tends to overestimate the cross section of events with two or more jets, while Sherpa 2.1 agrees better with data for $N_{\text{jets}}$ up to three. Yields of events with higher jet multiplicities are described by the parton shower modelling of the Powheg+Pythia MC. Finally, the measured $W^{\pm}Z$ differential cross section as a function of the invariant mass, $m_{jj}$, of the two leading jets with $p_{\text{T}}>25$ GeV is presented in Figure 6(b). The measurement is better described by the Sherpa predictions. The production of $W^{\pm}Z$ in association with two jets produced as a result of electroweak processes is not included in the SM predictions presented in the figure. In the last $m_{jj}$ bin it amounts to $17\%$ of the measured cross section, as estimated using Sherpa 2.2.2.

11 Polarisation measurement

11.1 Formalism and analysis principle

The polarisation of a gauge boson can be determined from the angular distribution of its decay products. At the Born level, the expected angular distribution for massless fermions in the rest frame of the parent $W$ boson is given in terms of the diagonal elements $f_{\mathrm{0}}$, $f_{\mathrm{L}}$ and $f_{\mathrm{R}}$ of the spin density matrix [97, 98, 99, 100] by

[TABLE]

where $\theta_{\ell,W}$ is defined using the helicity frame, as the decay angle of the charged lepton in the $W$ rest frame relative to the $W$ direction in the $WZ$ centre-of-mass frame, as shown in Figure 7. The terms $f_{\mathrm{0}}$, $f_{\mathrm{L}}$ and $f_{\mathrm{R}}$ refer to the longitudinal, transverse left-handed and transverse right-handed helicity fractions, respectively, and the normalisation is chosen such that $f_{\mathrm{0}}+f_{\mathrm{L}}+f_{\mathrm{R}}=1$. In the equation, the upper and lower signs correspond to $W^{+}$ and $W^{-}$ bosons, respectively. All dependencies on the azimuthal angle are integrated over.

The expected angular distribution of the lepton decay products of the $Z$ boson is described by the generalisation of Equation (1) [97, 98, 99]:

[TABLE]

where $\theta_{\ell,Z}$ is defined using the helicity frame, as the decay angle of the negatively charged lepton in the $Z$ rest frame relative to the $Z$ direction in the $WZ$ centre-of-mass frame. The parameter $\alpha=(2c_{v}c_{a})/(c_{v}^{2}+c_{a}^{2})$ is expressed in terms of the vector $c_{v}=-\frac{1}{2}+2\sin^{2}{\theta_{\mathrm{W}}^{\mathrm{eff}}}$ and axial-vector $c_{a}=-\frac{1}{2}$ couplings of the $Z$ boson to leptons, respectively, where the effective value of the Weinberg angle $\sin^{2}{\theta_{\mathrm{W}}^{\mathrm{eff}}}=0.23152$ [37] is used. Equation (2) also holds for the contribution from $\gamma^{*}$ and its interference with the $Z$ boson, with appropriate $c_{v}$ and $c_{a}$ coefficients. The tight invariant mass window of $\pm 10$ GeV around the nominal $Z$ boson mass minimises the contribution from $\gamma^{*}$, although all the helicity fractions presented here are effective fractions, containing the small contribution from $\gamma^{*}$.

Equations (1) and (2) are valid only when the full phase space of the leptonic decays of the gauge bosons is accessible. Restrictions on the $p_{\text{T}}$ and $\eta$ values of the charged decay lepton or of the neutrino suppress events at $\left|\cos\theta_{\ell,W(Z)}\right|\sim 1$, as shown in Figure 8, and the analytic expressions of Equations (1) and (2) cannot be used to extract the helicity fractions. Simulated templates therefore must be used.

Another major difficulty arises for the $W$ boson from incomplete knowledge of the neutrino momentum. The large angular coverage of the ATLAS detector enables measurement of the missing transverse momentum, which can be identified as the transverse momentum of the neutrino. The neutrino longitudinal momentum $p_{z}^{\nu}$ is obtained using the $W$ mass constraint. Solving the corresponding equation leads to a twofold ambiguity, which is resolved by choosing the solution with the smaller $|p_{z}^{\nu}|$. If the measured transverse mass is larger than the nominal $W$ mass, no real solutions exist for $p_{z}^{\nu}$. The most likely cause is that the measured $E_{\text{T}}^{\text{miss}}$ is larger than the actual neutrino $p_{\text{T}}$. In this case, the best estimate is obtained by choosing the real part of the complex solutions. As an alternative to the $\cos\theta_{\ell,W}$ observable using this reconstruction of the neutrino momentum, a “transverse helicity” observable introduced in Ref. [19] was tested, but a similar or lower sensitivity for the measurement of the $f_{\mathrm{0}}$ helicity fraction for $W$ bosons was obtained, so it was not pursued further.

For the polarisation measurements, all four decay channels, $eee$, $e\mu\mu$, $\mu ee$, and $\mu\mu\mu$, are added together. The measurements of $W$ and $Z$ boson polarisation are performed separately for $W^{+}Z$, $W^{-}Z$ and $W^{\pm}Z$ events. To allow the datasets of both $W$ boson charges to be combined for the measurement in $W^{\pm}Z$ events, $\cos\theta_{\ell,W}$ is multiplied by the sign of the lepton charge $q_{\ell}$. Figures 9(a), (b) present the reconstructed distributions for $W^{\pm}Z$ events of $q_{\ell}\cdot\cos\theta_{\ell,W}$ for the $W$ bosons and of $\cos\theta_{\ell,Z}$ for $Z$ bosons. The MC predictions provide a good description of the shapes of the data distributions.

The helicity parameters $f_{\mathrm{0}}$ and $f_{\mathrm{L}}-f_{\mathrm{R}}$ are measured in $W^{\pm}Z$ events separately for $W$ and $Z$ bosons using a binned profile-likelihood fit [101] of templates of the three helicity states to the $q_{\ell}\cdot\cos\theta_{\ell,W}$ and $\cos\theta_{\ell,Z}$ distributions. The equation $f_{\mathrm{0}}+f_{\mathrm{R}}+f_{\mathrm{L}}=1$ is used to constrain the independent parameters of the fit to $f_{\mathrm{0}}$, $f_{\mathrm{L}}-f_{\mathrm{R}}$ and the integrated fiducial cross section. The templates of $q_{\ell}\cdot\cos\theta_{\ell,W}$ and $\cos\theta_{\ell,Z}$ distributions for each of the three helicity states of the $W$ and $Z$ bosons are extracted from the Powheg+Pythia MC sample [19]. For each of the gauge bosons, generically denoted as $V$, the predicted helicity fractions of Powheg+Pythia MC events are determined as a function of $p_{\mathrm{T}}^{V}$ and $y_{V}$ by fitting the analytic functions of equations (1) and (2) to the predicted $\cos\theta_{\ell,V}$ distributions in the total phase space. Two dimensional bins as a function of $p_{\mathrm{T}}^{V}$ and $y_{V}$ are used. The bin boundaries are optimised such that possible bias on the evolution of the extracted helicity fractions is minimised. The MC templates at detector level representing longitudinal, left- and right-handed states of the $W$ boson are then obtained by reweighting of Powheg+Pythia MC events according to

[TABLE]

where

[TABLE]

and where $f_{\mathrm{L/0/R}}^{\mathrm{gen.}}$ are the helicity fractions at generator level, extracted by the fit, of Powheg+Pythia MC events. Similar equations hold for the polarisation of the $Z$ boson. The procedural uncertainty of the reweighting method for the generation of MC templates was estimated to be below $0.5$%. Helicity fractions are extracted by the template fit at detector level. To be expressed in a fiducial phase space at particle level, each helicity fraction is then corrected independently for detector efficiencies and QED final-state radiation effects using factors obtained from the simulation. Measured helicity fractions are thus reported at particle level for a fiducial phase space which follows the definition of Section 3 with the difference that leptons with kinematics defined before QED final state radiation (“Born leptons”) are used instead of dressed leptons. Experimental systematic uncertainties detailed in Section 9 are considered and treated as nuisance parameters with an assumed Gaussian distribution. Theoretical systematic uncertainties due to the modelling in the event generator used to evaluate the helicity templates are considered. The effects of PDF and QCD scale uncertainties are estimated as detailed in Section 9. An additional modelling uncertainty is considered and estimated by comparing predictions from the Powheg+Pythia and MC@NLO MC event generators for the shape of helicity template distributions.

11.2 Results

The measurements of $f_{\mathrm{0}}$ and $f_{\mathrm{L}}-f_{\mathrm{R}}$ are summarised in Table 5, where they are compared with the predictions from Powheg+Pythia. The Powheg+Pythia MC sample was generated at LO in the electroweak formalism using the $G_{\mu}$ scheme with $\sin^{2}{\theta_{\mathrm{W}}}=0.2229$. This choice impacts the predicted $f_{\mathrm{L}}-f_{\mathrm{R}}$ values which depend on the chosen value of the Weinberg angle via the angular coefficient $A_{4}$ [102, 97]. The impact of the value of $\sin^{2}{\theta_{\mathrm{W}}}$ on $f_{\mathrm{L}}-f_{\mathrm{R}}$ is estimated using MCFM [103, 104, 105] calculations with two electroweak schemes, the $G_{\mu}$ scheme and a scheme where the value $\sin^{2}{\theta_{\mathrm{W}}^{\mathrm{eff}}}=0.23152$ is imposed. The difference between the two calculations is used to correct the $f_{\mathrm{L}}-f_{\mathrm{R}}$ values predicted by Powheg+Pythia.

The longitudinal helicity fraction $f_{\mathrm{0}}$ of the $Z$ boson is measured with an observed significance of $6.5\,\sigma$, compared to $6.1\,\sigma$ expected. The longitudinal helicity fraction $f_{\mathrm{0}}$ of the $W$ boson is more difficult to extract than for the $Z$ boson and has a larger uncertainty. This measurement establishes the presence of longitudinally polarised $W$ bosons with an observed significance of $4.2\,\sigma$, compared to $3.8\,\sigma$ expected. Table 6 shows the main sources of uncertainty in the measurement of the helicity fractions. The measurements are dominated by statistical uncertainties. Good agreement of the measured helicity fractions of both the $W$ and $Z$ bosons with the predictions from Powheg+Pythia and MATRIX is observed. Measured $f_{\mathrm{0}}$ values agree within $1\sigma$ with the prediction, while $f_{\mathrm{L}}-f_{\mathrm{R}}$ values agree within $2\,\sigma$. The Powheg+Pythia and MATRIX predictions are only at NLO and NNLO in QCD, respectively, but, more importantly for polarisation, both calculations use only LO electroweak matrix elements. Therefore, and also because of the still large statistical uncertainties in the measurements, no stringent constraints nor clear inconsistencies between measurements and predictions can be deduced. The values of $f_{\mathrm{0}}$ and $f_{\mathrm{L}}-f_{\mathrm{R}}$ measured in $W^{\pm}Z$ events are shown in Figure 10 for the $W$ and $Z$ bosons, respectively.

12 Conclusion

Measurements of $W^{\pm}Z$ production cross sections in $\sqrt{s}=13$ TeV $pp$ collisions at the LHC are presented. The data analysed were collected with the ATLAS detector in $2015$ and $2016$ and correspond to an integrated luminosity of 36.1\leavevmode\nobreak\ \mbox{fb{}^{-1}}. The measurements use leptonic decay modes of the gauge bosons to electrons or muons and are performed in a fiducial phase space closely matching the detector acceptance. The measured inclusive cross section in the fiducial region for leptonic decay modes (electrons or muons) is $\sigma_{W^{\pm}Z\rightarrow\ell^{{}^{\prime}}\nu\ell\ell}^{\textrm{fid.}}=63.7\,\pm\leavevmode\nobreak\ 1.0\leavevmode\nobreak\ \textrm{(stat.)}\,\pm\leavevmode\nobreak\ 2.3\leavevmode\nobreak\ \textrm{(syst.)}\,\pm\leavevmode\nobreak\ 1.4\leavevmode\nobreak\ \textrm{(lumi.)}$ fb, in agreement with the NNLO Standard Model expectation of $61.5^{+1.4}_{-1.3}$ fb. The ratio of the cross sections for $W^{+}Z$ and $W^{-}Z$ production is also measured. The $W^{\pm}Z$ production cross section is measured as a function of several kinematic variables and compared with SM predictions at NNLO from the MATRIX calculation and at NLO from the Powheg+Pythia and Sherpa MC event generators. The differential cross-section distributions are fairly well described by the theory predictions, with the exception of the jet multiplicity. The MATRIX calculations show the best agreement with the data.

Furthermore, an analysis of angular distributions of leptons from decays of $W$ and $Z$ bosons has been performed. Helicity fractions of pair-produced vector bosons are measured for the first time in hadronic collisions. Integrated over the fiducial region, the longitudinal polarisation fractions of the $W$ and $Z$ bosons in $W^{\pm}Z$ events are measured to be $f_{0}^{W}=0.26\pm 0.06$ and $f_{0}^{Z}=0.24\pm 0.04$, respectively, in agreement with the SM predictions at NLO in QCD and at LO for electroweak corrections, of $0.238\pm 0.003$ and $0.230\pm 0.003$, respectively. The differences of the left and right transverse polarisations are also measured.The measured values agree with the SM predictions within less than one and two standard deviations of their uncertainties for $f_{\mathrm{0}}$ and $f_{\mathrm{L}}-f_{\mathrm{R}}$, respectively.

These polarisation measurements represent a step towards further new constraints on the electroweak symmetry breaking mechanism of the Standard Model, in particular by polarisation measurements in vector boson scattering.

Acknowledgements

We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.

We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF and DNSRC, Denmark; IN2P3-CNRS, CEA-DRF/IRFU, France; SRNSFG, Georgia; BMBF, HGF, and MPG, Germany; GSRT, Greece; RGC, Hong Kong SAR, China; ISF and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; NWO, Netherlands; RCN, Norway; MNiSW and NCN, Poland; FCT, Portugal; MNE/IFA, Romania; MES of Russia and NRC KI, Russian Federation; JINR; MESTD, Serbia; MSSR, Slovakia; ARRS and MIZŠ, Slovenia; DST/NRF, South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SERI, SNSF and Cantons of Bern and Geneva, Switzerland; MOST, Taiwan; TAEK, Turkey; STFC, United Kingdom; DOE and NSF, United States of America. In addition, individual groups and members have received support from BCKDF, CANARIE, CRC and Compute Canada, Canada; COST, ERC, ERDF, Horizon 2020, and Marie Skłodowska-Curie Actions, European Union; Investissements d’ Avenir Labex and Idex, ANR, France; DFG and AvH Foundation, Germany; Herakleitos, Thales and Aristeia programmes co-financed by EU-ESF and the Greek NSRF, Greece; BSF-NSF and GIF, Israel; CERCA Programme Generalitat de Catalunya, Spain; The Royal Society and Leverhulme Trust, United Kingdom.

The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN, the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA), the Tier-2 facilities worldwide and large non-WLCG resource providers. Major contributors of computing resources are listed in Ref. [106].