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Comparison of data and SM prediction for the \met distribution in (a) the \ttbar and (b) \Wjets control regions; and for the transverse mass of the top-tagged large-$R$ jet and \met system, $m_{\text{T}}(\met,J)$, distribution in the (c) \ttbar and (d) \Wboson/\Zboson+jet control regions used for the dark-matter search ((a) and (b)) and vector-like $T$-quark search ((c) and (d)). Other backgrounds in the 1L regions include multi-jet, $Z$+jets and diboson contributions, while in the 0L regions it is composed of diboson, \ttbar~+$X$ and multi-jet contributions. The expectations in the leptonic (hadronic) channel are obtained from a fit of the background-only hypothesis to data in the 1L (0L) control regions, where the normalisations of the \ttbar and \Wjets (\ttbar and \Wboson/\Zboson+\,jets) processes are treated as nuisance parameters in the fit. The error bands include statistical and systematic uncertainties. The last bin contains the overflow events.

Comparison of data and SM prediction for the \met distribution in (a) the \ttbar and (b) \Wjets control regions; and for the transverse mass of the top-tagged large-$R$ jet and \met system, $m_{\text{T}}(\met,J)$, distribution in the (c) \ttbar and (d) \Wboson/\Zboson+jet control regions used for the dark-matter search ((a) and (b)) and vector-like $T$-quark search ((c) and (d)). Other backgrounds in the 1L regions include multi-jet, $Z$+jets and diboson contributions, while in the 0L regions it is composed of diboson, \ttbar~+$X$ and multi-jet contributions. The expectations in the leptonic (hadronic) channel are obtained from a fit of the background-only hypothesis to data in the 1L (0L) control regions, where the normalisations of the \ttbar and \Wjets (\ttbar and \Wboson/\Zboson+\,jets) processes are treated as nuisance parameters in the fit. The error bands include statistical and systematic uncertainties. The last bin contains the overflow events.

95\%{} CL upper limits on the signal cross-section as a function of (a) the $V$ mass in the non-resonant (NR) model, (b) the mass of the scalar particle $\phi$ in the resonant (R) model and (c) the VLT mass. LO values for the production cross-section were computed for the non-resonant (resonant) DM production modes assuming $m_\chi=1$~\GeV, $a=0.5$ and $g_\chi=1$ ($m_\chi=10$~\GeV, $\lambda = 0.2$ and $y=0.4$).

Representative leading-order diagrams corresponding to the signals sought in this paper: non-resonant (a) $t$-channel and (b) $s$-channel production of a top-quark in association with a vector boson~\textit{V} which decays into two DM particles; (c) resonant production of a coloured scalar~$\phi$ that decays into a DM particle and a top-quark; and (d) single production of a vector-like $T$ quark decaying into $Zt\, (\to \nu\bar \nu b W)$.

Representative leading-order diagrams corresponding to the signals sought in this paper: non-resonant (a) $t$-channel and (b) $s$-channel production of a top-quark in association with a vector boson~\textit{V} which decays into two DM particles; (c) resonant production of a coloured scalar~$\phi$ that decays into a DM particle and a top-quark; and (d) single production of a vector-like $T$ quark decaying into $Zt\, (\to \nu\bar \nu b W)$.

Comparison of data and fitted expectations for the \met and the transverse mass of the top-tagged large-$R$ jet and \met system, $m_{\text{T}}(\met,J)$, distributions in the signal regions. Other backgrounds in the 1L regions include multi-jet, $Z$+jets and diboson contributions, while in the 0L regions it is composed of diboson and \ttbar~+$X$ contributions. The background-only hypothesis is used in the fit: (a) and (b) including the 1L and 0L DM signal regions as well as the 1L and 0L control regions; (c) 0L DM signal and control regions; (d) 0L VLT signal and control regions. The error bands include statistical and systematic uncertainties. The expected shape of a benchmark signal normalised to the theoretical prediction is added on top of the SM prediction. The benchmark signals correspond to: the non-resonant (NR) DM model with $m_{V}=1$~\TeV\ and 2~\TeV, $m_\chi=1$~\GeV, $a=0.5$ and $g_\chi=1$; the resonant (R) DM model with $m_{\phi}=1$~\TeV\ and 2~\TeV, $m_\chi=10$~\GeV, $\lambda = 0.2$ and $y=0.4$; and a VLT with a mass of 0.9~\TeV.

Comparison of data and SM prediction for the \met distribution in (a) the \ttbar and (b) \Wjets control regions; and for the transverse mass of the top-tagged large-$R$ jet and \met system, $m_{\text{T}}(\met,J)$, distribution in the (c) \ttbar and (d) \Wboson/\Zboson+jet control regions used for the dark-matter search ((a) and (b)) and vector-like $T$-quark search ((c) and (d)). Other backgrounds in the 1L regions include multi-jet, $Z$+jets and diboson contributions, while in the 0L regions it is composed of diboson, \ttbar~+$X$ and multi-jet contributions. The expectations in the leptonic (hadronic) channel are obtained from a fit of the background-only hypothesis to data in the 1L (0L) control regions, where the normalisations of the \ttbar and \Wjets (\ttbar and \Wboson/\Zboson+\,jets) processes are treated as nuisance parameters in the fit. The error bands include statistical and systematic uncertainties. The last bin contains the overflow events.

Representative leading-order diagrams corresponding to the signals sought in this paper: non-resonant (a) $t$-channel and (b) $s$-channel production of a top-quark in association with a vector boson~\textit{V} which decays into two DM particles; (c) resonant production of a coloured scalar~$\phi$ that decays into a DM particle and a top-quark; and (d) single production of a vector-like $T$ quark decaying into $Zt\, (\to \nu\bar \nu b W)$.

Representative leading-order diagrams corresponding to the signals sought in this paper: non-resonant (a) $t$-channel and (b) $s$-channel production of a top-quark in association with a vector boson~\textit{V} which decays into two DM particles; (c) resonant production of a coloured scalar~$\phi$ that decays into a DM particle and a top-quark; and (d) single production of a vector-like $T$ quark decaying into $Zt\, (\to \nu\bar \nu b W)$.

The 95\% CL upper limit contours on the signal strength $\sigma/\sigma_\text{theory}$ are shown for the non-resonant (NR) and resonant (R) DM production models. Non-resonant model: (a) $V$ mass vs $a$; (b) $V$ mass vs $g_\chi$ and (c) $V$ mass vs mass of the DM candidate $\chi$. Resonant model: (d) mass of the scalar $\phi$ vs $\lambda$; (e) mass of the scalar $\phi$ vs $y$. The solid (dashed) lines correspond to the observed (median expected and corresponding $\pm 1 \sigma$ and $\pm 2 \sigma$ bands) limits for $\sigma/\sigma_\text{theory}=1$. The predicted cross-sections were computed with \MGMCatNLO.

Expected and observed 95\% CL limits from the combination of the single-production channels on (a) the coupling of the $T$ quark to SM particles, $c_W = \sqrt{c^2_{{\mathrm{L}},W} + c^2_{{\mathrm{R}},W}}$ assuming a singlet $T$, corresponding to a $\mathcal{B}$ of $\approx 25$\%{}; and (b) the absolute value of $\sin\theta_{\mathrm{L}}$, with $\theta_{\mathrm{L}}$ being the mixing angle of a singlet $T$ with the SM top-quark.

Expected and observed 95\% CL limits from the combination of the single-production channels on (a) the coupling of the $T$ quark to SM particles, $c_W = \sqrt{c^2_{{\mathrm{L}},W} + c^2_{{\mathrm{R}},W}}$ assuming a singlet $T$, corresponding to a $\mathcal{B}$ of $\approx 25$\%{}; and (b) the absolute value of $\sin\theta_{\mathrm{L}}$, with $\theta_{\mathrm{L}}$ being the mixing angle of a singlet $T$ with the SM top-quark.

Comparison of data and fitted expectations for the \met and the transverse mass of the top-tagged large-$R$ jet and \met system, $m_{\text{T}}(\met,J)$, distributions in the signal regions. Other backgrounds in the 1L regions include multi-jet, $Z$+jets and diboson contributions, while in the 0L regions it is composed of diboson and \ttbar~+$X$ contributions. The background-only hypothesis is used in the fit: (a) and (b) including the 1L and 0L DM signal regions as well as the 1L and 0L control regions; (c) 0L DM signal and control regions; (d) 0L VLT signal and control regions. The error bands include statistical and systematic uncertainties. The expected shape of a benchmark signal normalised to the theoretical prediction is added on top of the SM prediction. The benchmark signals correspond to: the non-resonant (NR) DM model with $m_{V}=1$~\TeV\ and 2~\TeV, $m_\chi=1$~\GeV, $a=0.5$ and $g_\chi=1$; the resonant (R) DM model with $m_{\phi}=1$~\TeV\ and 2~\TeV, $m_\chi=10$~\GeV, $\lambda = 0.2$ and $y=0.4$; and a VLT with a mass of 0.9~\TeV.

Comparison of data and fitted expectations for the \met and the transverse mass of the top-tagged large-$R$ jet and \met system, $m_{\text{T}}(\met,J)$, distributions in the signal regions. Other backgrounds in the 1L regions include multi-jet, $Z$+jets and diboson contributions, while in the 0L regions it is composed of diboson and \ttbar~+$X$ contributions. The background-only hypothesis is used in the fit: (a) and (b) including the 1L and 0L DM signal regions as well as the 1L and 0L control regions; (c) 0L DM signal and control regions; (d) 0L VLT signal and control regions. The error bands include statistical and systematic uncertainties. The expected shape of a benchmark signal normalised to the theoretical prediction is added on top of the SM prediction. The benchmark signals correspond to: the non-resonant (NR) DM model with $m_{V}=1$~\TeV\ and 2~\TeV, $m_\chi=1$~\GeV, $a=0.5$ and $g_\chi=1$; the resonant (R) DM model with $m_{\phi}=1$~\TeV\ and 2~\TeV, $m_\chi=10$~\GeV, $\lambda = 0.2$ and $y=0.4$; and a VLT with a mass of 0.9~\TeV.

Comparison of data and fitted expectations for the \met and the transverse mass of the top-tagged large-$R$ jet and \met system, $m_{\text{T}}(\met,J)$, distributions in the signal regions. Other backgrounds in the 1L regions include multi-jet, $Z$+jets and diboson contributions, while in the 0L regions it is composed of diboson and \ttbar~+$X$ contributions. The background-only hypothesis is used in the fit: (a) and (b) including the 1L and 0L DM signal regions as well as the 1L and 0L control regions; (c) 0L DM signal and control regions; (d) 0L VLT signal and control regions. The error bands include statistical and systematic uncertainties. The expected shape of a benchmark signal normalised to the theoretical prediction is added on top of the SM prediction. The benchmark signals correspond to: the non-resonant (NR) DM model with $m_{V}=1$~\TeV\ and 2~\TeV, $m_\chi=1$~\GeV, $a=0.5$ and $g_\chi=1$; the resonant (R) DM model with $m_{\phi}=1$~\TeV\ and 2~\TeV, $m_\chi=10$~\GeV, $\lambda = 0.2$ and $y=0.4$; and a VLT with a mass of 0.9~\TeV.

The 95\% CL upper limit contours on the signal strength $\sigma/\sigma_\text{theory}$ are shown for the non-resonant (NR) and resonant (R) DM production models. Non-resonant model: (a) $V$ mass vs $a$; (b) $V$ mass vs $g_\chi$ and (c) $V$ mass vs mass of the DM candidate $\chi$. Resonant model: (d) mass of the scalar $\phi$ vs $\lambda$; (e) mass of the scalar $\phi$ vs $y$. The solid (dashed) lines correspond to the observed (median expected and corresponding $\pm 1 \sigma$ and $\pm 2 \sigma$ bands) limits for $\sigma/\sigma_\text{theory}=1$. The predicted cross-sections were computed with \MGMCatNLO.

The 95\% CL upper limit contours on the signal strength $\sigma/\sigma_\text{theory}$ are shown for the non-resonant (NR) and resonant (R) DM production models. Non-resonant model: (a) $V$ mass vs $a$; (b) $V$ mass vs $g_\chi$ and (c) $V$ mass vs mass of the DM candidate $\chi$. Resonant model: (d) mass of the scalar $\phi$ vs $\lambda$; (e) mass of the scalar $\phi$ vs $y$. The solid (dashed) lines correspond to the observed (median expected and corresponding $\pm 1 \sigma$ and $\pm 2 \sigma$ bands) limits for $\sigma/\sigma_\text{theory}=1$. The predicted cross-sections were computed with \MGMCatNLO.

Comparison of data and SM prediction for the \met distribution in (a) the \ttbar and (b) \Wjets control regions; and for the transverse mass of the top-tagged large-$R$ jet and \met system, $m_{\text{T}}(\met,J)$, distribution in the (c) \ttbar and (d) \Wboson/\Zboson+jet control regions used for the dark-matter search ((a) and (b)) and vector-like $T$-quark search ((c) and (d)). Other backgrounds in the 1L regions include multi-jet, $Z$+jets and diboson contributions, while in the 0L regions it is composed of diboson, \ttbar~+$X$ and multi-jet contributions. The expectations in the leptonic (hadronic) channel are obtained from a fit of the background-only hypothesis to data in the 1L (0L) control regions, where the normalisations of the \ttbar and \Wjets (\ttbar and \Wboson/\Zboson+\,jets) processes are treated as nuisance parameters in the fit. The error bands include statistical and systematic uncertainties. The last bin contains the overflow events.

95\%{} CL upper limits on the signal cross-section as a function of (a) the $V$ mass in the non-resonant (NR) model, (b) the mass of the scalar particle $\phi$ in the resonant (R) model and (c) the VLT mass. LO values for the production cross-section were computed for the non-resonant (resonant) DM production modes assuming $m_\chi=1$~\GeV, $a=0.5$ and $g_\chi=1$ ($m_\chi=10$~\GeV, $\lambda = 0.2$ and $y=0.4$).

95\%{} CL upper limits on the signal cross-section as a function of (a) the $V$ mass in the non-resonant (NR) model, (b) the mass of the scalar particle $\phi$ in the resonant (R) model and (c) the VLT mass. LO values for the production cross-section were computed for the non-resonant (resonant) DM production modes assuming $m_\chi=1$~\GeV, $a=0.5$ and $g_\chi=1$ ($m_\chi=10$~\GeV, $\lambda = 0.2$ and $y=0.4$).

The 95\% CL upper limit contours on the signal strength $\sigma/\sigma_\text{theory}$ are shown for the non-resonant (NR) and resonant (R) DM production models. Non-resonant model: (a) $V$ mass vs $a$; (b) $V$ mass vs $g_\chi$ and (c) $V$ mass vs mass of the DM candidate $\chi$. Resonant model: (d) mass of the scalar $\phi$ vs $\lambda$; (e) mass of the scalar $\phi$ vs $y$. The solid (dashed) lines correspond to the observed (median expected and corresponding $\pm 1 \sigma$ and $\pm 2 \sigma$ bands) limits for $\sigma/\sigma_\text{theory}=1$. The predicted cross-sections were computed with \MGMCatNLO.

The 95\% CL upper limit contours on the signal strength $\sigma/\sigma_\text{theory}$ are shown for the non-resonant (NR) and resonant (R) DM production models. Non-resonant model: (a) $V$ mass vs $a$; (b) $V$ mass vs $g_\chi$ and (c) $V$ mass vs mass of the DM candidate $\chi$. Resonant model: (d) mass of the scalar $\phi$ vs $\lambda$; (e) mass of the scalar $\phi$ vs $y$. The solid (dashed) lines correspond to the observed (median expected and corresponding $\pm 1 \sigma$ and $\pm 2 \sigma$ bands) limits for $\sigma/\sigma_\text{theory}=1$. The predicted cross-sections were computed with \MGMCatNLO.