Fiducial differential cross section of the electroweak $W^\pm W^\pm jj$ production as a function of $N_{\mathrm{gap}\,\mathrm{jets}}$. The correlation of uncertainties of the measured cross section across bins is presented in Table 14.

Fiducial differential cross section of the electroweak $W^\pm W^\pm jj$ production as a function of $\xi_{\mathrm{j}3}$. The correlation of uncertainties of the measured cross section across bins is presented in Table 15.

Fiducial differential cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $m_{\ell\ell}$. The correlation of uncertainties of the measured cross section across bins is presented in Table 16.

Fiducial differential cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $m_{\mathrm{T}}$. The correlation of uncertainties of the measured cross section across bins is presented in Table 17.

Fiducial differential cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $m_{\mathrm{jj}}$. The correlation of uncertainties of the measured cross section across bins is presented in Table 18.

Fiducial differential cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $N_{\mathrm{gap}\,\mathrm{jets}}$. The correlation of uncertainties of the measured cross section across bins is presented in Table 19.

Fiducial differential cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $\xi_{\mathrm{j}3}$. The correlation of uncertainties of the measured cross section across bins is presented in Table 20.

Observed correlations between the bins of the LH-unfolded cross section of the electroweak $W^\pm W^\pm jj$ production as a function of $m_{\ell\ell}$. See Table 1.

Observed correlations between the bins of the LH-unfolded cross section of the electroweak $W^\pm W^\pm jj$ production as a function of $m_{\mathrm{T}}$. See Table 2.

Observed correlations between the bins of the LH-unfolded cross section of the electroweak $W^\pm W^\pm jj$ production as a function of $m_{\mathrm{jj}}$. See Table 3.

Observed correlations between the bins of the LH-unfolded cross section of the electroweak $W^\pm W^\pm jj$ production as a function of $N_{\mathrm{gap}\,\mathrm{jets}}$. See Table 4.

Observed correlations between the bins of the LH-unfolded cross section of the electroweak $W^\pm W^\pm jj$ production as a function of $\xi_{\mathrm{j}3}$. See Table 5.

Observed correlations between the bins of the LH-unfolded cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $m_{\ell\ell}$. See Table 6.

Observed correlations between the bins of the LH-unfolded cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $m_{\mathrm{T}}$. See Table 7.

Observed correlations between the bins of the LH-unfolded cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $m_{\mathrm{jj}}$. See Table 8.

Observed correlations between the bins of the LH-unfolded cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $N_{\mathrm{gap}\,\mathrm{jets}}$. See Table 9.

Observed correlations between the bins of the LH-unfolded cross section of the inclusive $W^\pm W^\pm jj$ production as a function of $\xi_{\mathrm{j}3}$. See Table 10.

Evolution of the one-dimensional expected and observed limits at 95% CL on the parameters corresponding to the quartic operators with label M0 as a function of the cut-off scale. The unitarity bounds as a function of the cut-off scale are defined for one non-zero Wilson coefficient.

Evolution of the one-dimensional expected and observed limits at 95% CL on the parameters corresponding to the quartic operators with label M1 as a function of the cut-off scale. The unitarity bounds as a function of the cut-off scale are defined for one non-zero Wilson coefficient.

Evolution of the one-dimensional expected and observed limits at 95% CL on the parameters corresponding to the quartic operators with label M7 as a function of the cut-off scale. The unitarity bounds as a function of the cut-off scale are defined for one non-zero Wilson coefficient. The limits on M7 were obtained without taking into account the SM-EFT interference for the EW W$^\pm$Zjj final state.

Evolution of the one-dimensional expected and observed limits at 95% CL on the parameters corresponding to the quartic operators with label S02 as a function of the cut-off scale. The unitarity bounds as a function of the cut-off scale are defined for one non-zero Wilson coefficient.

Evolution of the one-dimensional expected and observed limits at 95% CL on the parameters corresponding to the quartic operators with label S1 as a function of the cut-off scale. The unitarity bounds as a function of the cut-off scale are defined for one non-zero Wilson coefficient.

Evolution of the one-dimensional expected and observed limits at 95% CL on the parameters corresponding to the quartic operators with label T0 as a function of the cut-off scale. The unitarity bounds as a function of the cut-off scale are defined for one non-zero Wilson coefficient.

Evolution of the one-dimensional expected and observed limits at 95% CL on the parameters corresponding to the quartic operators with label T1 as a function of the cut-off scale. The unitarity bounds as a function of the cut-off scale are defined for one non-zero Wilson coefficient.

Evolution of the one-dimensional expected and observed limits at 95% CL on the parameters corresponding to the quartic operators with label T2 as a function of the cut-off scale. The unitarity bounds as a function of the cut-off scale are defined for one non-zero Wilson coefficient.

Expected and observed exclusion limits at 95% CL for $\sigma_{\mathrm{VBF}}(\mathrm{H}^{\pm\pm}_5) \times \mathcal{B}(\mathrm{H}^{\pm\pm}_5 \rightarrow W^{\pm}W^{\pm})$ as a function of the doubly-charged Higgs mass.

Expected and observed exclusion limits at 95% CL for $\sin\theta_{\mathrm{H}}$ as a function of the doubly-charged Higgs mass in the Georgi-Machacek model.

Distribution of ${E}_{T}^{miss}$ after the fit of the multijet backgrounds, in the electron channel, in the $t\bar{t}$ 3-jets VR, without applying the cut on ${E}_{T}^{miss}$. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit. The last bin includes the overflow. The uncertainty band indicates the simulation's statistical uncertainty, the normalisation uncertainties for different processes ($40$ % for $W$+jets production, $30$ % for multijet background and $6$ % for top-quark processes) and the multijet background shape uncertainty in each bin, summed in quadrature. The lower panel of the figure shows the ratio of the data to the prediction.

Distribution of ${E}_{T}^{miss}$ after the fit of the multijet backgrounds, in the electron channel, in the $t\bar{t}$ 4-jets VR, without applying the cut on ${E}_{T}^{miss}$. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit. The last bin includes the overflow. The uncertainty band indicates the simulation's statistical uncertainty, the normalisation uncertainties for different processes ($40$ % for $W$+jets production, $30$ % for multijet background and $6$ % for top-quark processes) and the multijet background shape uncertainty in each bin, summed in quadrature. The lower panel of the figure shows the ratio of the data to the prediction.

Distribution of $m_{T}^{W}$ after the fit of the multijet backgrounds, in the muon channel, in the signal region, without applying the cut on $m_{T}^{W}$. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit. The last bin includes the overflow. The uncertainty band indicates the simulation's statistical uncertainty, the normalisation uncertainties for different processes ($40$ % for $W$+jets production, $30$ % for multijet background and $6$ % for top-quark processes) and the multijet background shape uncertainty in each bin, summed in quadrature. The lower panel of the figure shows the ratio of the data to the prediction.

Distribution of $m_{T}^{W}$ after the fit of the multijet backgrounds, in the muon channel, in the $W$+jets VR, without applying the cut on $m_{T}^{W}$. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit. The last bin includes the overflow. The uncertainty band indicates the simulation's statistical uncertainty, the normalisation uncertainties for different processes ($40$ % for $W$+jets production, $30$ % for multijet background and $6$ % for top-quark processes) and the multijet background shape uncertainty in each bin, summed in quadrature. The lower panel of the figure shows the ratio of the data to the prediction.

Distribution of $m_{T}^{W}$ after the fit of the multijet backgrounds, in the muon channel, in the $t\bar{t}$ 3-jets VR, without applying the cut on $m_{T}^{W}$. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit. The last bin includes the overflow. The uncertainty band indicates the simulation's statistical uncertainty, the normalisation uncertainties for different processes ($40$ % for $W$+jets production, $30$ % for multijet background and $6$ % for top-quark processes) and the multijet background shape uncertainty in each bin, summed in quadrature. The lower panel of the figure shows the ratio of the data to the prediction.

Distribution of $m_{T}^{W}$ after the fit of the multijet backgrounds, in the muon channel, in the $t\bar{t}$ 4-jets VR, without applying the cut on $m_{T}^{W}$. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit. The last bin includes the overflow. The uncertainty band indicates the simulation's statistical uncertainty, the normalisation uncertainties for different processes ($40$ % for $W$+jets production, $30$ % for multijet background and $6$ % for top-quark processes) and the multijet background shape uncertainty in each bin, summed in quadrature. The lower panel of the figure shows the ratio of the data to the prediction.

Expected distributions of the MEM discriminant $P(S|X)$ in the SR, for the s-channel single-top signal, and for the $t\bar{t}$ and $W$+jets backgrounds, for MEM discriminant values larger than $2.0\times10^{-4}$. Each distribution is normalised to unity. The binning is the same as the optimised binning used in the signal extraction fit, resulting in a non-linear horizontal scale.

Distribution of the MEM discriminant $P(S|X)$ in the $W$+jets VR. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit presented in Section 5 in the paper. The uncertainty band indicates the simulation's statistical uncertainty and the normalisation uncertainties for the various processes in each bin, summed in quadrature. The ratio of the observed number to the predicted number of events in each bin is shown in the lower panel of the figure, with different vertical axis ranges. The binning is the same as the optimised binning used in the signal extraction fit described in Section 8 in the paper, resulting in a non-linear horizontal scale.

Distribution of the MEM discriminant $P(S|X)$ in the $t\bar{t}$ 3-jets VR. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit presented in Section 5 in the paper. The uncertainty band indicates the simulation's statistical uncertainty and the normalisation uncertainties for the various processes in each bin, summed in quadrature. The ratio of the observed number to the predicted number of events in each bin is shown in the lower panel of the figure, with different vertical axis ranges. The binning is the same as the optimised binning used in the signal extraction fit described in Section 8 in the paper, resulting in a non-linear horizontal scale.

Distribution of the MEM discriminant $P(S|X)$ in the $t\bar{t}$ 4-jets VR. Simulated events are normalised to the expected number of events given the integrated luminosity, after applying the normalisation factors obtained in the multijet fit presented in Section 5 in the paper. The uncertainty bands indicate the simulation's statistical uncertainty and the normalisation uncertainties for the various processes in each bin, summed in quadrature. The ratio of the observed number to the predicted number of events in each bin is shown in the lower panel of the figure, with different vertical axis ranges. The binning is the same as the optimised binning used in the signal extraction fit described in Section 8 in the paper, resulting in a non-linear horizontal scale.

Distribution of the MEM discriminant $P(S|X)$ in the SR before the fit to data, for MEM discriminant values larger than $2.0\times10^{-4}$. The lower panel of the figure shows the ratio of the data to the prediction, with different vertical axis ranges. The uncertainty band indicates the total uncertainties and their correlations in each bin. The uncertainties in the $t\bar{t}$ and $W$+jets normalisation factors, as well as in the s-channel signal cross-section, are not defined pre-fit and therefore not included. The binning is the same as the optimised binning used in the fit, resulting in a non-linear horizontal scale.

Distribution of the MEM discriminant $P(S|X)$ in the SR after the fit to data, for MEM discriminant values larger than $2.0\times10^{-4}$. The lower panel of the figure shows the ratio of the data to the prediction, with different vertical axis ranges. The uncertainty band indicates the total uncertainties and their correlations in each bin. The binning is the same as the optimised binning used in the fit, resulting in a non-linear horizontal scale.

Distribution of the MEM discriminant $P(S|X)$ in the SR after the fit to data, for MEM discriminant values larger than $2.0\times10^{-4}$, after subtraction of all backgrounds. The fitted distribution for the simulation of the signal is shown together with the post-fit uncertainty in the backgrounds. The binning is the same as the optimised binning used in the fit, resulting in a non-linear horizontal scale.

Pre-fit and post-fit event yields in the SR, for MEM discriminant values larger than $2.0\times10^{-4}$. The central value of the event yield for each process is calculated by summing the values of the discriminant bin contents, using the nominal expected yield for the pre-fit value, and the best-fit estimate for the post-fit value. The error includes statistical and systematic uncertainties summed in quadrature. All sources of systematic uncertainties are included, taking into account correlations and anti-correlations in the post-fit case. The uncertainties in the $t\bar{t}$ and $W$+jets normalisation factors, as well as in the s-channel signal cross-section, are not defined pre-fit and therefore only included in the post-fit uncertainties.

Observed impact of the different sources of uncertainty on the measured s-channel signal cross-section, grouped by categories. The impact of each category is obtained by repeating the fit after having fixed the set of nuisance parameters corresponding to that category, subtracting the square of the resulting uncertainty from the square of the uncertainty found in the full fit, and calculating the square root. The 'Systematic uncertainties' category combines all sources of systematic uncertainties. The statistical uncertainty is obtained by repeating the fit after having fixed all nuisance parameters, including the $t\bar{t}$ and $W$+jets normalisation factors. 'Total' gives the total uncertainty on the measurement.

Observed impact of the different sources of $t\bar{t}$ modelling uncertainty on the measured s-channel signal cross-section. The impact of each category is obtained by repeating the fit after having fixed the set of nuisance parameters corresponding to that category, subtracting the square of the resulting uncertainty from the square of the uncertainty found in the full fit, and calculating the square root. 'PS & had.' refers to the parton shower and hadronisation model, and 'ME/PS matching' to the matching of the ME to the parton shower.

Observed impact of the different sources of s-channel modelling uncertainty on the measured s-channel signal cross-section. The impact of each category is obtained by repeating the fit after having fixed the set of nuisance parameters corresponding to that category, subtracting the square of the resulting uncertainty from the square of the uncertainty found in the full fit, and calculating the square root. 'PS & had.' refers to the parton shower and hadronisation model, as described in Section 7 in the paper.

Observed impact of the different sources of t-channel modelling uncertainty on the measured s-channel signal cross-section. The impact of each category is obtained by repeating the fit after having fixed the set of nuisance parameters corresponding to that category, subtracting the square of the resulting uncertainty from the square of the uncertainty found in the full fit, and calculating the square root. 'PS & had.' refers to the parton shower and hadronisation model, as described in Section 7 in the paper.

Observed impact of the different sources of $tW$ modelling uncertainty on the measured s-channel signal cross-section, grouped by categories. The impact of each category is obtained by repeating the fit after having fixed the set of nuisance parameters corresponding to that category, subtracting the square of the resulting uncertainty from the square of the uncertainty found in the full fit, and calculating the square root. 'PS & had.' refers to the parton shower and hadronisation model, and '$t\bar{t}$ overlap' to the algorithm removing the overlap between $tW$ and $t\bar{t}$ production at NLO, as described in Section 7 in the paper.

Observed impact of the different sources of PDF uncertainties on the measured s-channel signal cross-section, grouped by categories. The impact of each category is obtained by repeating the fit after having fixed the set of nuisance parameters corresponding to that category, subtracting the square of the resulting uncertainty from the square of the uncertainty found in the full fit, and calculating the square root.

Comparison between data and prediction after the fit to data in the signal region for the leading-jet $p_{T}$. The last bin includes the overflow. The uncertainty band includes all uncertainties and their correlations. The lower panel of the figure shows the ratio of the data to the prediction.

Comparison between data and prediction after the fit to data in the signal region for the leading-jet $\eta$. The uncertainty band includes all uncertainties and their correlations. The lower panel of the figure shows the ratio of the data to the prediction.

Comparison between data and prediction after the fit to data in the signal region for the subleading-jet $p_{T}$. The last bin includes the overflow. The uncertainty band includes all uncertainties and their correlations. The lower panel of the figure shows the ratio of the data to the prediction.

Comparison between data and prediction after the fit to data in the signal region for the subleading-jet $\eta$. The uncertainty band includes all uncertainties and their correlations. The lower panel of the figure shows the ratio of the data to the prediction.

Comparison between data and prediction after the fit to data in the signal region for the lepton $p_{T}$. The last bin includes the overflow. The uncertainty band includes all uncertainties and their correlations. The lower panel of the figure shows the ratio of the data to the prediction.

Comparison between data and prediction after the fit to data in the signal region for the lepton $\eta$. The uncertainty band includes all uncertainties and their correlations. The lower panel of the figure shows the ratio of the data to the prediction.

Comparison between data and prediction after the fit to data in the signal region for the ${E}_{T}^{miss}$. The last bin includes the overflow. The uncertainty band includes all uncertainties and their correlations. The lower panel of the figure shows the ratio of the data to the prediction.

Comparison between data and prediction after the fit to data in the signal region for the $m_{T}^{W}$. The last bin includes the overflow. The uncertainty band includes all uncertainties and their correlations. The lower panel of the figure shows the ratio of the data to the prediction.

Nuisance parameters ranked according to their post-fit impacts on the best-fit value of the ratio $\mu$ of the measured cross-section to the predicted cross-section. In the figure, only the 20 nuisance parameters with the largest post-fit impacts are shown. The empty (solid) blue rectangles illustrate the pre-fit (post-fit) impact on $\mu$, corresponding to the upper axis. The pre-fit (post-fit) impact of each nuisance parameter, $\Delta\mu$, is calculated as the difference in the fitted value of $\mu$ between the nominal fit and the fit when fixing the corresponding nuisance parameter to $\hat{\theta}\pm\Delta\theta$ ($\hat{\theta}\pm\Delta\hat{\theta}$), where $\hat{\theta}$ is the best-fit value of the nuisance parameter and $\Delta\theta$ ($\Delta\hat{\theta}$) is its pre-fit (post-fit) uncertainty. Several systematic uncertainties are split into different nuisance parameters, which are indicated by NP. JES (JER) indicates jet energy scale (resolution), and $\gamma$ indicates a nuisance parameter associated to the MC statistics in one of the 18 bins numbered from 0 to 17. The black points show the best-fit values of the nuisance parameters, with the error bars representing the post-fit uncertainties. Each nuisance parameter is shown wrt. its nominal value, $\theta_0$, and in units of its pre-fit uncertainty, except the free-floating normalisation factors of the $t\bar{t}$ and $W$+jets backgrounds, and the parameters associated to the MC statistics in each bin, for which the post-fit values and uncertainties are shown.

Correlation matrix of the nuisance parameters and of the ratio $\mu$ of the measured cross-section to the predicted cross-section. The correlations are given after the fit to data. In the figure, only the parameters which have a correlation of at least 0.2 with any other parameter are shown.

Distribution of the MEM discriminant $P(S|X)$ in the SR for MEM discriminant values larger than $2.0\times10^{-4}$, for the collision data used for the measurement, and for 1000 pseudo-data replicas, generated using a bootstrapping technique, in order to assess the statistical correlations between this measurement and others, for the purpose of combinations. The replicas are obtained by reweighting each observed data event by a random integer generated according to Poisson statistics, using the <a href="https://zenodo.org/record/5361038">BootstrapGenerator</a> software package , which implements a technique described in <a href="https://cds.cern.ch/record/2759945/">ATL-PHYS-PUB-2021-011</a>. The ATLAS event number and run number of each event are used as seed to uniquely but reproducibly initialise the random number generator for each event. Each pseudo-data replica is assigned an index, ranging from 0 to 999, corresponding to the random number index used consistently for each observed data event.

Measured values of the signal cross-section and of the $t\bar{t}$ and $W$+jets normalisation factors, obtained by statistical-only fits to the collision data used for the measurement, and to 1000 pseudo-data replicas, generated using a bootstrapping technique, in order to assess the statistical correlations between this measurement and others, for the purpose of combinations. The central values and their statistical uncertainties are obtained by repeating the fit after having fixed all nuisance parameters, except the $t\bar{t}$ and $W$+jets normalisation factors, which are let free-floating (unlike for the statistical uncertainty on the cross-section quoted in the paper). The replicas are obtained by reweighting each observed data event by a random integer generated according to Poisson statistics, using the <a href="https://zenodo.org/record/5361038">BootstrapGenerator</a> software package , which implements a technique described in <a href="https://cds.cern.ch/record/2759945/">ATL-PHYS-PUB-2021-011</a>. The ATLAS event number and run number of each event are used as seed to uniquely but reproducibly initialise the random number generator for each event. Each pseudo-data replica is assigned an index, ranging from 0 to 999, corresponding to the random number index used consistently for each observed data event.