Distributions of the VLQ-candidate mass, m_VLQ, in the (a–c) SRs, (d–f) W+jets CRs and (g–i) tt̄ CRs after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or that contain two W/Z bosons. The last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the (a–c) SRs, (d–f) W+jets CRs and (g–i) tt̄ CRs after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or that contain two W/Z bosons. The last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the (a–c) SRs, (d–f) W+jets CRs and (g–i) tt̄ CRs after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or that contain two W/Z bosons. The last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the (a–c) SRs, (d–f) W+jets CRs and (g–i) tt̄ CRs after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or that contain two W/Z bosons. The last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the (a–c) SRs, (d–f) W+jets CRs and (g–i) tt̄ CRs after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or that contain two W/Z bosons. The last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the (a–c) SRs, (d–f) W+jets CRs and (g–i) tt̄ CRs after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or that contain two W/Z bosons. The last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the W+jets VRs (a-f) and tt̄ VRs (g-i) after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or containing combinations having two W/Z bosons. Last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the W+jets VRs (a-f) and tt̄ VRs (g-i) after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or containing combinations having two W/Z bosons. Last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the W+jets VRs (a-f) and tt̄ VRs (g-i) after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or containing combinations having two W/Z bosons. Last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the W+jets VRs (a-f) and tt̄ VRs (g-i) after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or containing combinations having two W/Z bosons. Last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the W+jets VRs (a-f) and tt̄ VRs (g-i) after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or containing combinations having two W/Z bosons. Last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Distributions of the VLQ-candidate mass, m_VLQ, in the W+jets VRs (a-f) and tt̄ VRs (g-i) after the fit to the background-only hypothesis. The columns correspond from left to right to the low-, middle-, and high-p_T^W bins in each region. Other includes remaining backgrounds from top quarks or containing combinations having two W/Z bosons. Last bin includes overflow. Note: the 'Data' values in the table are normalized by the width of the bin to correspond to the number of events per 100 GeV

Expected (background-only) and observed 95% CL upper limits on the VLQ coupling κ as functions of the VLQ mass m_Q, for T-singlet (a) and Y-triplet (b) vector-like quarks. The green and yellow bands indicate the systematic uncertainties which are profiled in the fit for the observed upper limits. For the T-singlet model, the result of the latest ATLAS combination of searches for single vector-like top-quarks  is overlaid; this includes the decay modes T→H t and T→Z t. These interpretations are limited to parameter combinations for which the model corrections are valid, by restricting the VLQ relative width to Γ_Q/m_Q < 0.5, as indicated by the grey dashed line.

Expected (background-only) and observed 95% CL upper limits on the VLQ coupling κ as functions of the VLQ mass m_Q, for T-singlet (a) and Y-triplet (b) vector-like quarks. The green and yellow bands indicate the systematic uncertainties which are profiled in the fit for the observed upper limits. For the T-singlet model, the result of the latest ATLAS combination of searches for single vector-like top-quarks  is overlaid; this includes the decay modes T→H t and T→Z t. These interpretations are limited to parameter combinations for which the model corrections are valid, by restricting the VLQ relative width to Γ_Q/m_Q < 0.5, as indicated by the grey dashed line.

Exclusion limit (at 95% CL) for (a) T-singlet and (b) Y-triplet, expressed in terms of an upper limit on the VLQ relative width Γ_Q/m_Q, within its validity range for the modelling used, as a function of the VLQ mass m_Q. This is an alternative presentation of the upper limit set in terms of the coupling κ in Figure 6. As can be seen by the fact the interference-included and signal-only exclusion curves overlap almost entirely, the effect of neglecting the interference is negligible for the vector-like Y-quark.

Exclusion limit (at 95% CL) for (a) T-singlet and (b) Y-triplet, expressed in terms of an upper limit on the VLQ relative width Γ_Q/m_Q, within its validity range for the modelling used, as a function of the VLQ mass m_Q. This is an alternative presentation of the upper limit set in terms of the coupling κ in Figure 6. As can be seen by the fact the interference-included and signal-only exclusion curves overlap almost entirely, the effect of neglecting the interference is negligible for the vector-like Y-quark.

Unfolded $R(\rho_{s})$ observable in the 2-to-3 measurement (normalized and divided by bin width). The parton level is defined with two stable top-quarks and a jet with $p_{T}>50$ GeV and $|\eta|<2.5$.

Covariance matrix for statistical effects of the measured $R(\rho_{s})$ observable after unfolding, for the 2-to-3 measurement (normalized)

Covariance matrix for statistical and systematic effects of the measured $R(\rho_{s})$ observable after unfolding, for the 2-to-3 measurement (normalized)

Impact of systematic uncertainties on the 2-to-3 unfolded observable. Values are given in percentage of bin content.

Central value and breakdown of the uncertainties affecting the top-quark pole mass extraction from the 2-to-3 unfolded observable.

Unfolded number of events in the 2-to-7measurement (not normalized). The parton level is defined with two neutrinos, one electron and one muon of opposite electric charges, two $b$-jets and an additional jet (extrajet). The four-momentum of the sum of neutrinos has transverse component larger than 30 GeV. The $p_{T}$-leading lepton has $p_{T}>28$ GeV, while the sub-leading has $p_{T}>20$ GeV. The two $b$-jets with have $p_{T}>30$ GeV and the extrajet has $p^\text{extrajet}_{T}>60$ GeV. All the leptons and jets are separated by $\Delta R >0.4$ and have $|\eta|<2.5$.

Covariance matrix for statistical effects of the measured number of events after unfolding, for the 2-to-7 measurement (not normalized)

Covariance matrix for statistical and systematic effects of the measured number of events after unfolding, for the 2-to-7 measurement (not normalized)

Unfolded $R(\rho_{s})$ observable in the 2-to-7 measurement (normalized and divided by bin width). The parton level is defined with two neutrinos, one electron and one muon of opposite electric charges, two $b$-jets and an additional jet (extrajet). The four-momentum of the sum of neutrinos has transverse component larger than 30 GeV. The $p_{T}$-leading lepton has $p_{T}>28$ GeV, while the sub-leading has $p_{T}>20$ GeV. The two $b$-jets with have $p_{T}>30$ GeV and the extrajet has $p^\text{extrajet}_{T}>60$ GeV. All the leptons and jets are separated by $\Delta R >0.4$ and have $|\eta|<2.5$.

Covariance matrix for statistical effects of the measured $R(\rho_{s})$ observable after unfolding, for the 2-to-7 measurement (normalized)

Covariance matrix for statistical and systematic effects of the measured $R(\rho_{s})$ observable after unfolding, for the 2-to-7 measurement (normalized)

Impact of systematic uncertainties on the 2-to-7 unfolded observable. Values are given in percentage of bin content.

Central value and breakdown of the uncertainties affecting the top-quark pole mass extraction from the 2-to-7 unfolded observable.

Observed profile likelihood scans of $\kappa_\lambda$. The scans are performed by varying only the coupling modifier of interest, while all other relevant coupling modifiers are fixed to unity.

Expected profile likelihood scans of $\kappa_\lambda$. The scans are performed by varying only the coupling modifier of interest, while all other relevant coupling modifiers are fixed to unity.

Observed profile likelihood scans of $\kappa_{2V}$. The scans are performed by varying only the coupling modifier of interest, while all other relevant coupling modifiers are fixed to unity.

Expected profile likelihood scans of $\kappa_{2V}$. The scans are performed by varying only the coupling modifier of interest, while all other relevant coupling modifiers are fixed to unity.

Confidence level contours at 68% (solid line) and 95% (dashed line) in the $(\kappa_\lambda, \kappa_{2V})$ parameter space, when all other coupling modifiers are fixed to their SM predictions. The corresponding expected contours are shown by the inner and outer shaded regions The SM prediction is indicated by the star, while the best-fit value is denoted by the cross.

Confidence level contours at 68% (solid line) and 95% (dashed line) in the $(\kappa_\lambda, \kappa_{2V})$ parameter space, when all other coupling modifiers are fixed to their SM predictions. The corresponding expected contours are shown by the inner and outer shaded regions The SM prediction is indicated by the star, while the best-fit value is denoted by the cross.

Observed $m_{\gamma\gamma}$ distributions for the high-mass categories in Run 2. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the high-mass categories in Run 3. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the high-mass categories in Run 2. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the high-mass categories in Run 3. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the high-mass categories in Run 2. 3he lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the high-mass categories in Run 3. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the low-mass categories in Run 2. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the low-mass categories in Run 3. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the low-mass categories in Run 2. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the low-mass categories in Run 3. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the low-mass categories in Run 2. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the low-mass categories in Run 3. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the low-mass categories in Run 2. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed $m_{\gamma\gamma}$ distributions for the low-mass categories in Run 3. The lines show the fit results for the continuum background only (light dotted), adding the single Higgs boson background (black dotted) and the full fit (solid).

Observed and expected profiled likelihood scans on $\kappa_\lambda$ for the low-mass (LM) and high-mass (HM) regions separately and for their combination. The scan is performed by floating the indicated parameter of interest while fixing all the others to their SM value.

Observed and expected profiled likelihood scans on $\kappa_\lambda$ for the low-mass (LM) and high-mass (HM) regions separately and for their combination. The scan is performed by floating the indicated parameter of interest while fixing all the others to their SM value.

Observed and expected profiled likelihood scans on $\kappa_\lambda$ for the low-mass (LM) and high-mass (HM) regions separately and for their combination. The scan is performed by floating the indicated parameter of interest while fixing all the others to their SM value.

Observed and expected profiled likelihood scans on $\kappa_\lambda$ for the low-mass (LM) and high-mass (HM) regions separately and for their combination. The scan is performed by floating the indicated parameter of interest while fixing all the others to their SM value.

Observed and expected profiled likelihood scans on $\kappa_\lambda$ for the low-mass (LM) and high-mass (HM) regions separately and for their combination. The scan is performed by floating the indicated parameter of interest while fixing all the others to their SM value.

Observed and expected profiled likelihood scans on $\kappa_\lambda$ for the low-mass (LM) and high-mass (HM) regions separately and for their combination. The scan is performed by floating the indicated parameter of interest while fixing all the others to their SM value.

The expected number of events (estimated by using simulation) from the SM HH signals and single Higgs boson production, and the expected number of events from the continuum background, evaluated in the 120 GeV < $m_{\gamma\gamma}$ < 130 GeV window in Run 3 for the different low-mass (LM) and high-mass (HM) categories. For comparison, the number of data events is also shown. The uncertainties in the HH signals and single Higgs boson backgrounds include the systematic uncertainties discussed in Section 6. Asymmetric uncertainties arise primarily from the theory calculation of the SM ggF HH cross section and the large uncertainty in the yield of single Higgs bosons produced in ggF events in association with heavy-flavour jets. The uncertainty in the continuum background is given by the sum in quadrature of the statistical uncertainty from the fit to the data and the spurious signal uncertainty.

Observed 95% $\text{CL}_\text{S}$ upper limits on the production cross-section times acceptance times branching ratio to jets, $\sigma \cdot A \cdot \text{BR}$, of Gaussian-shaped signals of 5%, 10%, and 15% width relative to their peak mass, $m_G$. Also included are the corresponding expected upper limits predicted for the case the $m_{jj}$ distribution is observed to be identical to the background prediction in each bin and the $1\sigma$ and $2\sigma$ envelopes of outcomes expected for Poisson fluctuations around the background expectation. Limits are derived from the J100 signal region.

Observed 95% $\text{CL}_\text{S}$ upper limits on the the value of the coupling to quarks, $g_q$, of the $Z^\prime$ model as a function of the resonance mass $m_{Z^\prime}$. Also included are the corresponding expected upper limits predicted for the case the $m_{jj}$ distribution is observed to be identical to the background prediction in each bin and the $1\sigma$ and $2\sigma$ envelopes of outcomes expected for Poisson fluctuations around the background expectation. Limits are derived from the J50 signal region.

Observed 95% $\text{CL}_\text{S}$ upper limits on the the value of the coupling to quarks, $g_q$, of the $Z^\prime$ model as a function of the resonance mass $m_{Z^\prime}$. Also included are the corresponding expected upper limits predicted for the case the $m_{jj}$ distribution is observed to be identical to the background prediction in each bin and the $1\sigma$ and $2\sigma$ envelopes of outcomes expected for Poisson fluctuations around the background expectation. Limits are derived from the J100 signal region.

Acceptance of simulated $Z^\prime$ signal events given the analysis event selection as a function of $m_{Z^\prime}$. The solid black line corresponds to the acceptance without a lower $m_{jj}$ requirement applied, the solid blue line includes the $m_{jj} > 344$ GeV requirement of the J50 signal region, and the solid red line includes the $m_{jj} > 481$ GeV requirement of the J100 signal region.

Numbers of events in each of the two considered datasets after consecutively applying the analysis selections.