DOI: 10.17182/hepdata.135074.v2/t4
title: "Table 4" of "Observation of gauge boson joint-polarisation states in $W^{\pm}Z$ production from $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector"

description: Correlation matrix for the unfolded cross section.
description: CERN-LHC. Measurements of joint-polarisation states of $W$ and $Z$ gauge bosons in $W^{\pm}Z$ production are presented.
The data set used corresponds to an integrated luminosity of \mbox{$139$~fb$^{-1}$} of proton–proton collisions at a centre-of-mass energy of $13$~TeV recorded by the ATLAS detector at the CERN Large Hadron Collider.
The $W^{\pm}Z$ candidate events are reconstructed using leptonic decay modes of the gauge bosons into electrons and muons.
The simultaneous pair-production of longitudinally polarised vector bosons is measured for the first time with a significance of $7.1$ standard deviations.
The measured joint helicity fractions integrated over the fiducial region are $f_{\mathrm{00}} = 0.067 \pm 0.010$, $f_{\mathrm{0T}} = 0.110 \pm 0.029$, $f_{\mathrm{T0}} = 0.179 \pm 0.023$ and $f_{\mathrm{TT}} = 0.644 \pm 0.032$, in agreement with the next-to-leading-order Standard Model predictions.
Individual helicity fractions of the $W$ and $Z$ bosons are also measured and found to be consistent with joint helicity fractions within
the expected amounts of correlation.
Both the joint and individual helicity fractions are also measured separately in $W^+Z$ and $W^-Z$ events.
Inclusive and differential cross sections for several kinematic observables sensitive to polarisation are presented.

Measured differential cross sections are reported in this HEPdata entry.

Integrated cross sections over bins of different kinematic observables are reported. 
The reported cross sections are not divided by the bin width.

The cross sections are measured in a fiducial phase space reflecting the detector acceptance, defined by:
- $p_{\mathrm{T}}$ of electrons and muons from Z0 decays &gt; 15 GeV
- $p_{\mathrm{T}}$ of electrons and muons from the $W^{\pm}$ decays &gt; 20 GeV
- $|\eta|$ of muons and electrons &lt; 2.5
- Leptons from the Z0 boson are separated by $\Delta R(\ell,\ell) &gt; 0.2$ from each other
- Leptons from the Z0 and W bosons are separated by $\Delta R(\ell,\ell) &gt; 0.3$ from each other
- |dilepton mass - Z0 mass| &lt; 10 GeV
- $m_{\mathrm{T}}$ of $W^{\pm}$ &gt; 30 GeV.


Cross sections are reported using kinematics of final-state prompt leptons defined before QED final-state radiation (FSR),
i.e. Born leptons, or using dressed leptons.

At particle level, the kinematics of final-state prompt dressed electrons and muons is computed including the contributions from final-state radiated photons within a distance in the ($\eta,\phi$) plane of 
$\Delta R = \sqrt{(\Delta\eta)^2 + (\Delta\phi)^2} = 0.1$ around the direction of the charged lepton. 
These dressed leptons and the final-state neutrinos that do not originate from hadron or $\tau$ decays are 
associated with the $W$ and $Z$ boson decay products with an algorithmic approach, called ``resonant shape''. 
This algorithm is based on the value of an estimator expressing the product of the nominal line shapes of 
the $W$ and $Z$ resonances

 $P = \left| \frac{1}{ m^2_{(\ell^+,\ell^-)} - \left(m_Z^{\textrm{PDG}}\right)^2 + i \; \Gamma_Z^{\textrm{PDG}} \; m_Z^{\textrm{PDG}} } \right|^2 \times \; \left| \frac {1} { m^2_{(\ell',\nu_{\ell'})} - \left(m_W^{\textrm{PDG}}\right)^2 + i \; \Gamma_W^{\textrm{PDG}} \; m_W^{\textrm{PDG}} } \right|^2$

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. 
The input to the estimator is the invariant mass $m$ of all possible pairs ($\ell^+,\ell^-$) and
($\ell',\nu_{\ell'}$) satisfying the fiducial selection requirements. 
The final choice of which leptons are assigned to the $W$ or $Z$ bosons corresponds to the configuration 
exhibiting the highest value of the estimator.

For cross sections using Born leptons, the association of leptons to $W$ or $Z$ bosons is performed using the Powheg Monte Carlo generator.