Selection efficiencies for the barrel region at object level. Definitions used in this table: Track-Match efficiency (barrel / endcap) is the fraction of reconstructed superclusters matched to at least two tracks (within a ΔR < 0.15 cone) with $p_T$>25GeV, normalized to the number of generated electron pairs. Merged-electron-pair (MEP) reconstruction efficiency (barrel / endcap) is the fraction of events in which an MEP is successfully reconstructed, using the same normalization. Iso-cuts efficiency (barrel / endcap) is the fraction of reconstructed MEPs that pass the isolation requirement. BDT efficiency (barrel / endcap) is the fraction of MEPs that pass the BDT selection among those passing the isolation requirement. Merged-electron-pair ID efficiency (barrel / endcap) is the fraction of MEPs that satisfy all signal-selection criteria, again normalized to the number of generated electron pairs.

Selection efficiencies for the endcap region at object level. Definitions as above.

Trigger-level and overall event-level efficiency.Definitions used in this table: Trigger efficiency is defined as the number of events passing both the trigger and the HAA4e offline selection, divided by the number passing the offline selection alone. Number of normalized events passing all selections gives the luminosity-scaled event yield after applying all cuts. Total event efficiency is the fraction of generated events in which a four-electron invariant mass is successfully reconstructed within the 80-250 GeV range, with event weights applied.

Selection efficiencies for the barrel region at object level. Definitions used in this table: Track-Match efficiency (barrel / endcap) is the fraction of reconstructed superclusters matched to at least two tracks (within a ΔR < 0.15 cone) with $p_T$>25GeV, normalized to the number of generated electron pairs. Merged-electron-pair (MEP) reconstruction efficiency (barrel / endcap) is the fraction of events in which an MEP is successfully reconstructed, using the same normalization. Iso-cuts efficiency (barrel / endcap) is the fraction of reconstructed MEPs that pass the isolation requirement. BDT efficiency (barrel / endcap) is the fraction of MEPs that pass the BDT selection among those passing the isolation requirement. Merged-electron-pair ID efficiency (barrel / endcap) is the fraction of MEPs that satisfy all signal-selection criteria, again normalized to the number of generated electron pairs.

Selection efficiencies for the endcap region at object level. Definitions as above.

Trigger-level and overall event-level efficiency.Definitions used in this table: Trigger efficiency is defined as the number of events passing both the trigger and the HAA4e offline selection, divided by the number passing the offline selection alone. Number of normalized events passing all selections gives the luminosity-scaled event yield after applying all cuts. Total event efficiency is the fraction of generated events in which a four-electron invariant mass is successfully reconstructed within the 80-250 GeV range, with event weights applied.

Selection efficiencies for the barrel region at object level. Definitions used in this table: Track-Match efficiency (barrel / endcap) is the fraction of reconstructed superclusters matched to at least two tracks (within a ΔR < 0.15 cone) with $p_T$>25GeV, normalized to the number of generated electron pairs. Merged-electron-pair (MEP) reconstruction efficiency (barrel / endcap) is the fraction of events in which an MEP is successfully reconstructed, using the same normalization. Iso-cuts efficiency (barrel / endcap) is the fraction of reconstructed MEPs that pass the isolation requirement. BDT efficiency (barrel / endcap) is the fraction of MEPs that pass the BDT selection among those passing the isolation requirement. Merged-electron-pair ID efficiency (barrel / endcap) is the fraction of MEPs that satisfy all signal-selection criteria, again normalized to the number of generated electron pairs.

Selection efficiencies for the endcap region at object level. Definitions as above.

Trigger-level and overall event-level efficiency.Definitions used in this table: Trigger efficiency is defined as the number of events passing both the trigger and the HAA4e offline selection, divided by the number passing the offline selection alone. Number of normalized events passing all selections gives the luminosity-scaled event yield after applying all cuts. Total event efficiency is the fraction of generated events in which a four-electron invariant mass is successfully reconstructed within the 80-250 GeV range, with event weights applied.

Selection efficiencies for the barrel region at object level. Definitions used in this table: Track-Match efficiency (barrel / endcap) is the fraction of reconstructed superclusters matched to at least two tracks (within a ΔR < 0.15 cone) with $p_T$>25GeV, normalized to the number of generated electron pairs. Merged-electron-pair (MEP) reconstruction efficiency (barrel / endcap) is the fraction of events in which an MEP is successfully reconstructed, using the same normalization. Iso-cuts efficiency (barrel / endcap) is the fraction of reconstructed MEPs that pass the isolation requirement. BDT efficiency (barrel / endcap) is the fraction of MEPs that pass the BDT selection among those passing the isolation requirement. Merged-electron-pair ID efficiency (barrel / endcap) is the fraction of MEPs that satisfy all signal-selection criteria, again normalized to the number of generated electron pairs.

Selection efficiencies for the endcap region at object level. Definitions as above.

Trigger-level and overall event-level efficiency.Definitions used in this table: Trigger efficiency is defined as the number of events passing both the trigger and the HAA4e offline selection, divided by the number passing the offline selection alone. Number of normalized events passing all selections gives the luminosity-scaled event yield after applying all cuts. Total event efficiency is the fraction of generated events in which a four-electron invariant mass is successfully reconstructed within the 80-250 GeV range, with event weights applied.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\omega}$ meson $c\tau$ for $m_{\tilde{\omega}}=5~\textrm{GeV}$ and $\mathcal{B}(\tilde{\omega}\to\mu\mu)=0.162$ in the vector portal model. It is assumed that $m_{\tilde{\omega}}=\tilde{\Lambda}=m_{\tilde{\eta}}$, where $m_{\tilde{\omega}}$, $\tilde{\Lambda}$, and $m_{\tilde{\eta}}$ are parameters of the dark sector: the mass of the spin-one meson, confinement scale, and the mass of the spin-zero meson, respectively.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\omega}$ meson $c\tau$ for $m_{\tilde{\omega}}=15~\textrm{GeV}$ and $\mathcal{B}(\tilde{\omega}\to\mu\mu)=0.15$ in the vector portal model. It is assumed that $m_{\tilde{\omega}}=\tilde{\Lambda}=m_{\tilde{\eta}}$, where $m_{\tilde{\omega}}$, $\tilde{\Lambda}$, and $m_{\tilde{\eta}}$ are parameters of the dark sector: the mass of the spin-one meson, confinement scale, and the mass of the spin-zero meson, respectively.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\omega}$ meson $c\tau$ for $m_{\tilde{\omega}}=20~\textrm{GeV}$ and $\mathcal{B}(\tilde{\omega}\to\mu\mu)=0.15$ in the vector portal model. It is assumed that $m_{\tilde{\omega}}=\tilde{\Lambda}=m_{\tilde{\eta}}$, where $m_{\tilde{\omega}}$, $\tilde{\Lambda}$, and $m_{\tilde{\eta}}$ are parameters of the dark sector: the mass of the spin-one meson, confinement scale, and the mass of the spin-zero meson, respectively.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\textrm{A}'$ $c\tau$ for $m_{\tilde{\pi}_{3}}=1~\textrm{GeV}$ and $m_{\textrm{A}'}=0.33~\textrm{GeV}$ , and $\mathcal{B}(\textrm{A}'\to\mu\mu)=0.458$ in Scenario A. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\textrm{A}'$ $c\tau$ for $m_{\tilde{\pi}_{3}}=4~\textrm{GeV}$ and $m_{\textrm{A}'}=0.4~\textrm{GeV}$ , and $\mathcal{B}(\textrm{A}'\to\mu\mu)=0.436$ in Scenario A. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\textrm{A}'$ $c\tau$ for $m_{\tilde{\pi}_{3}}=2~\textrm{GeV}$ and $m_{\textrm{A}'}=0.67~\textrm{GeV}$, and $\mathcal{B}(\textrm{A}'\to\mu\mu)=0.17$ in Scenario A. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\textrm{A}'$ $c\tau$ for $m_{\tilde{\pi}_{3}}=4~\textrm{GeV}$ and $m_{\textrm{A}'}=1.33~\textrm{GeV}$, and $\mathcal{B}(\textrm{A}'\to\mu\mu)=0.305$ in Scenario A. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\pi}_{3}$ meson $c\tau$ for $m_{\tilde{\pi}_{3}}=1~\textrm{GeV}$ and $m_{\textrm{A}'}=0.33~\textrm{GeV}$ , and $\mathcal{B}(\textrm{A}'\to\mu\mu)=0.458$ in Scenario B1. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\pi}_{3}$ meson $c\tau$ for $m_{\tilde{\pi}_{3}}=2~\textrm{GeV}$ and $m_{\textrm{A}'}=0.4~\textrm{GeV}$ , and $\mathcal{B}(\textrm{A}'\to\mu\mu)=0.436$ in Scenario B1. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\pi}_{3}$ meson $c\tau$ for $m_{\tilde{\pi}_{3}}=2~\textrm{GeV}$ and $m_{\textrm{A}'}=0.67~\textrm{GeV}$ , and $\mathcal{B}(\textrm{A}'\to\mu\mu)=0.17$ in Scenario B1. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\pi}_{3}$ meson $c\tau$ for $m_{\tilde{\pi}_{3}}=4~\textrm{GeV}$ and $m_{\textrm{A}'}=1.33~\textrm{GeV}$, and $\mathcal{B}(\textrm{A}'\to\mu\mu)=0.305$ in Scenario B1. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\omega}$ meson mass for $c\tau=10~\textrm{mm}$ in the vector portal model. It is assumed that $m_{\tilde{\omega}}=\tilde{\Lambda}=m_{\tilde{\eta}}$, where $m_{\tilde{\omega}}$, $\tilde{\Lambda}$, and $m_{\tilde{\eta}}$ are parameters of the dark sector: the mass of the spin-one meson, confinement scale, and the mass of the spin-zero meson, respectively. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\omega}$ meson mass for $c\tau=100~\textrm{mm}$ in the vector portal model. It is assumed that $m_{\tilde{\omega}}=\tilde{\Lambda}=m_{\tilde{\eta}}$, where $m_{\tilde{\omega}}$, $\tilde{\Lambda}$, and $m_{\tilde{\eta}}$ are parameters of the dark sector: the mass of the spin-one meson, confinement scale, and the mass of the spin-zero meson, respectively. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\to\psi\overline{\psi}\right)$ as a function of the $\mathrm{A}'$ mass in Scenario A, for $\mathrm{A}'$ $c\tau=10~\textrm{mm}$, and $m_{\tilde{\pi}_{3}}=10m_{\textrm{A}'}$. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and ${\sin\theta=0.1}$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\to \mathrm{A}'\mathrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\to\psi\overline{\psi}\right)$ as a function of the $\mathrm{A}'$ mass in Scenario A, for $\mathrm{A}'$ $c\tau=10~\textrm{mm}$, and $m_{\tilde{\pi}_{3}}=3m_{\textrm{A}'}$. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and ${\sin\theta=0.1}$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\to \mathrm{A}'\mathrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\to\psi\overline{\psi}\right)$ as a function of the $\mathrm{A}'$ mass in Scenario A, for $\mathrm{A}'$ $c\tau=100~\textrm{mm}$, and $m_{\tilde{\pi}_{3}}=10m_{\textrm{A}'}$. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and ${\sin\theta=0.1}$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\to \mathrm{A}'\mathrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\to\psi\overline{\psi}\right)$ as a function of the $\mathrm{A}'$ mass in Scenario A, for $\mathrm{A}'$ $c\tau=100~\textrm{mm}$, and $m_{\tilde{\pi}_{3}}=3m_{\textrm{A}'}$. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and ${\sin\theta=0.1}$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\to \mathrm{A}'\mathrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\to\psi\overline{\psi}\right)$ as a function of the $\mathrm{A}'$ mass in Scenario B1, for $\tilde{\pi}_{3}$ $c\tau=10~\textrm{mm}$, and $m_{\tilde{\pi}_{3}}=10m_{\textrm{A}'}$. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and ${\sin\theta=0.1}$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\to \mathrm{A}'\mathrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\to\psi\overline{\psi}\right)$ as a function of the $\mathrm{A}'$ mass in Scenario B1, for $\tilde{\pi}_{3}$ $c\tau=10~\textrm{mm}$, and $m_{\tilde{\pi}_{3}}=3m_{\textrm{A}'}$. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and ${\sin\theta=0.1}$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\to \mathrm{A}'\mathrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\to\psi\overline{\psi}\right)$ as a function of the $\mathrm{A}'$ mass in Scenario B1, for $\tilde{\pi}_{3}$ $c\tau=100~\textrm{mm}$, and $m_{\tilde{\pi}_{3}}=10m_{\textrm{A}'}$. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and ${\sin\theta=0.1}$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\to \mathrm{A}'\mathrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\to\psi\overline{\psi}\right)$ as a function of the $\mathrm{A}'$ mass in Scenario B1, for $\tilde{\pi}_{3}$ $c\tau=100~\textrm{mm}$, and $m_{\tilde{\pi}_{3}}=3m_{\textrm{A}'}$. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and ${\sin\theta=0.1}$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\to \mathrm{A}'\mathrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Observed upper limits at 95% CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as a function of the $\tilde{\omega}$ meson mass and $c\tau$ in the vector portal model. The phase space region that is left out is kinematically forbidden in the model. It is assumed that $m_{\tilde{\omega}}=\tilde{\Lambda}=m_{\tilde{\eta}}$, where $m_{\tilde{\omega}}$, $\tilde{\Lambda}$, and $m_{\tilde{\eta}}$ are parameters of the dark sector: the mass of the spin-one meson, confinement scale, and the mass of the spin-zero meson, respectively. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Observed upper limits at $95\%$ CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as functions of the $\textrm{A}'$ mass and $c\tau$ for $m_{\tilde{\pi}_{3}}=10m_{A'}$ in Scenario A. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Observed upper limits at $95\%$ CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as functions of the $\textrm{A}'$ mass and $c\tau$ for $m_{\tilde{\pi}_{3}}=3m_{A'}$ in Scenario A. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Observed upper limits at $95\%$ CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as functions of the $\textrm{A}'$ mass and $c\tau$ for $m_{\tilde{\pi}_{3}}=10m_{A'}$ in Scenario B1. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Observed upper limits at $95\%$ CL on the branching fraction $\mathcal{B}\left(\textrm{H}\rightarrow\psi\overline{\psi}\right)$ as functions of the $\textrm{A}'$ mass and $c\tau$ for $m_{\tilde{\pi}_{3}}=3m_{A'}$ in Scenario B1. It is assumed that $m_{\tilde{\eta}}=\tilde{\Lambda}=4m_{\tilde{\pi}_{2}}$ and $\sin\theta=0.1$, where $m_{\tilde{\eta}}$ is the mass of the dark-sector pseudoscalar meson and $\theta$ is the mixing angle parametrising the isospin violation. The branching fraction $\mathcal{B}(\tilde{\pi}_{3}\rightarrow \textrm{A}'\textrm{A}')$ is assumed to be 1. The grey bands correspond to mass regions of known SM resonances, which are masked in the search.

Cutflow efficiencies for representative signal model points in the vector portal.

Cutflow efficiencies for representative signal model points in Scenario A.

Cutflow efficiencies for representative signal model points in Scenario B1.

Normalized differential distributions of the $\Delta R (\gamma,\mathrm{tt})$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process.

Absolute differential distributions of the min. $\Delta R (\gamma,\mathrm{t})$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process.

Normalized differential distributions of the min. $\Delta R (\gamma,\mathrm{t})$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process.

Absolute differential distributions of the $m(\mathrm{tt})$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process.

Normalized differential distributions of the $m(\mathrm{tt})$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process.

Absolute differential distributions of the leading lepton $p_{\mathrm{T}}$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process. The theory predictions are computed at fixed order, at NLO in QCD, as described in Section 9.2 of the paper.

Normalized differential distributions of the leading lepton $p_{\mathrm{T}}$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process. The theory predictions are computed at fixed order, at NLO in QCD, as described in Section 9.2 of the paper.

Absolute differential distributions of the photon $p_{\mathrm{T}}$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process. The theory predictions are computed at fixed order, at NLO in QCD, as described in Section 9.2 of the paper.

Normalized differential distributions of the photon $p_{\mathrm{T}}$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process. The theory predictions are computed at fixed order, at NLO in QCD, as described in Section 9.2 of the paper.

Absolute differential distributions of the $\Delta \phi (\ell,\ell)$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process. The theory predictions are computed at fixed order, at NLO in QCD, as described in Section 9.2 of the paper.

Normalized differential distributions of the $\Delta \phi (\ell,\ell)$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process. The theory predictions are computed at fixed order, at NLO in QCD, as described in Section 9.2 of the paper.

Absolute differential distributions of the ratio between the cross sections of $\mathrm{tt\gamma}$ and $\mathrm{tt}$, as a function of the leading top quark $p_{\mathrm{T}}$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process.

Absolute differential distributions of the ratio between the cross sections of $\mathrm{tt\gamma}$ and $\mathrm{tt}$, as a function of the leading lepton $p_{\mathrm{T}}$. The nominal MC prediction used to compare the experimental results to is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and Madgraph5 at LO in QCD for photons from the decay part of the process. The alternative prediction is obtained with Madgraph5 at NLO in QCD for photons from the production part of the process and a POWHEG+Pythia $\mathrm{tt}$ simulation at NLO in QCD for photons from the decay part of the process. The theory predictions are computed at fixed order, at NLO in QCD, as described in Section 9.2 of the paper.

Correlation matrix of the total uncertainty in the absolute differential cross section as a function of the leading top quark $p_{\mathrm{T}}$.

Correlation matrix of the statistical uncertainty in the absolute differential cross section as a function of the leading top quark $p_{\mathrm{T}}$.

Correlation matrix of the total uncertainty in the absolute differential cross section as a function of the $\Delta R (\gamma,\mathrm{tt})$.

Correlation matrix of the statistical uncertainty in the absolute differential cross section as a function of the $\Delta R (\gamma,\mathrm{tt})$.

Correlation matrix of the total uncertainty in the absolute differential cross section as a function of the min. $\Delta R (\gamma,\mathrm{t})$.

Correlation matrix of the statistical uncertainty in the absolute differential cross section as a function of the min. $\Delta R (\gamma,\mathrm{t})$.

Correlation matrix of the total uncertainty in the absolute differential cross section as a function of the $m(\mathrm{tt})$.

Correlation matrix of the statistical uncertainty in the absolute differential cross section as a function of the $m(\mathrm{tt})$.

Correlation matrix of the total uncertainty in the absolute differential cross section as a function of the leading lepton $p_{\mathrm{T}}$.

Correlation matrix of the statistical uncertainty in the absolute differential cross section as a function of the leading lepton $p_{\mathrm{T}}$.

Correlation matrix of the total uncertainty in the absolute differential cross section as a function of the photon $p_{\mathrm{T}}$.

Correlation matrix of the statistical uncertainty in the absolute differential cross section as a function of the photon $p_{\mathrm{T}}$.

Correlation matrix of the total uncertainty in the absolute differential cross section as a function of the $\Delta \phi (\ell,\ell)$.

Correlation matrix of the statistical uncertainty in the absolute differential cross section as a function of the $\Delta \phi (\ell,\ell)$.

Correlation matrix of the total uncertainty in the absolute differential $\mathrm{tt\gamma}$ / $\mathrm{tt}$ cross section ratio as a function of the leading top $p_{\mathrm{T}}$.

Correlation matrix of the statistical uncertainty in the absolute differential $\mathrm{tt\gamma}$ / $\mathrm{tt}$ cross section ratio as a function of the leading top $p_{\mathrm{T}}$.

Correlation matrix of the total uncertainty in the absolute differential $\mathrm{tt\gamma}$ / $\mathrm{tt}$ cross section ratio as a function of the leading lepton $p_{\mathrm{T}}$.

Correlation matrix of the statistical uncertainty in the absolute differential $\mathrm{tt\gamma}$ / $\mathrm{tt}$ cross section ratio as a function of the leading lepton $p_{\mathrm{T}}$.

The $v_{3}\{2,\lvert\Delta\eta\rvert>2\}$ ratios for charged particles as functions of centrality in NeNe to OO collisions at 5.36 TeV.

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.

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.

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-ej and (c, d) SR-2L-ej of the e+light-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_de) = (2.0&nbsp;TeV, 1.0) and S&#771;₁ (m, y_de) = (3.0&nbsp;TeV, 1.0), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-&mu;j and (c, d) SR-2L-&mu;j of the &mu;+light-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_s&mu;) = (2.0&nbsp;TeV, 1.5) and S&#771;₁ (m, y_s&mu;) = (3.0&nbsp;TeV, 1.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-&mu;j and (c, d) SR-2L-&mu;j of the &mu;+light-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_s&mu;) = (2.0&nbsp;TeV, 1.5) and S&#771;₁ (m, y_s&mu;) = (3.0&nbsp;TeV, 1.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-&mu;j and (c, d) SR-2L-&mu;j of the &mu;+light-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_s&mu;) = (2.0&nbsp;TeV, 1.5) and S&#771;₁ (m, y_s&mu;) = (3.0&nbsp;TeV, 1.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-&mu;j and (c, d) SR-2L-&mu;j of the &mu;+light-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_s&mu;) = (2.0&nbsp;TeV, 1.5) and S&#771;₁ (m, y_s&mu;) = (3.0&nbsp;TeV, 1.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-eb and (c, d) SR-2L-eb of the e+b-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_be) = (1.5&nbsp;TeV, 2.5) and S&#771;₁ (m, y_be) = (2.0&nbsp;TeV, 2.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-eb and (c, d) SR-2L-eb of the e+b-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_be) = (1.5&nbsp;TeV, 2.5) and S&#771;₁ (m, y_be) = (2.0&nbsp;TeV, 2.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-eb and (c, d) SR-2L-eb of the e+b-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_be) = (1.5&nbsp;TeV, 2.5) and S&#771;₁ (m, y_be) = (2.0&nbsp;TeV, 2.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-eb and (c, d) SR-2L-eb of the e+b-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_be) = (1.5&nbsp;TeV, 2.5) and S&#771;₁ (m, y_be) = (2.0&nbsp;TeV, 2.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-&mu;b and (c, d) SR-2L-&mu;b of the &mu;+b-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_b&mu;) = (1.5&nbsp;TeV, 1.5) and S&#771;₁ (m, y_b&mu;) = (2.0&nbsp;TeV, 1.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-&mu;b and (c, d) SR-2L-&mu;b of the &mu;+b-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_b&mu;) = (1.5&nbsp;TeV, 1.5) and S&#771;₁ (m, y_b&mu;) = (2.0&nbsp;TeV, 1.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-&mu;b and (c, d) SR-2L-&mu;b of the &mu;+b-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_b&mu;) = (1.5&nbsp;TeV, 1.5) and S&#771;₁ (m, y_b&mu;) = (2.0&nbsp;TeV, 1.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Data (dots) and post-fit SM distribution (histograms) of m_&#8467;j in (a, b) SR-1L-&mu;b and (c, d) SR-2L-&mu;b of the &mu;+b-jet channel obtained by a CR+SR background-only fit for Run&nbsp;2 and Run&nbsp;3, respectively. The lower panel shows the ratio of observed data to the total post- and pre-fit SM prediction. The last bin includes the overflow. Uncertainties in the background estimates include both the statistical and systematic uncertainties, with correlations between uncertainties taken into account. The dashed lines show the predicted yields for two benchmark signal models corresponding to S&#771;₁ (m, y_b&mu;) = (1.5&nbsp;TeV, 1.5) and S&#771;₁ (m, y_b&mu;) = (2.0&nbsp;TeV, 1.5), respectively. Note: the values in the table are normalized by the width of corresponding bin

Exclusion limits for minimal models of S&#771;₁ production with only y_de being non-zero obtained from a simultaneous fit to SR-1L and SR-2L of the e+light-jet channel combining Run-2 and Run-3 data. All limits are computed at 95&percnt; CL and the observed (red solid lines) and expected (black dashed lines) exclusion limits are shown in the m(S&#771;₁) &ndash; y_de plane. The observed exclusion should be interpreted as the region above the red line. The yellow inner (green outer) shaded band around the expected limits corresponds to the &plusmn;1 &sigma; (&plusmn;2 &sigma;) variations of the expected limit, accounting for all uncertainties. The observed limit obtained from ATLAS searches for LQ pair production is also shown as dark blue line [13]. Constraints from weak charge measurements of protons and nuclei on y_de couplings derived by Ref. [10] are shown as light magenta line.

Exclusion limits for minimal models of S&#771;₁ production with only y_s&mu; being non-zero obtained from a simultaneous fit to SR-1L and SR-2L of the &mu;+light-jet channel combining Run-2 and Run-3 data. All limits are computed at 95&percnt; CL and the observed (red solid lines) and expected (black dashed lines) exclusion limits are shown in the m(S&#771;₁) &ndash; y_s&mu; plane. The observed exclusion should be interpreted as the region above the red line. The yellow inner (green outer) shaded band around the expected limits corresponds to the &plusmn;1 &sigma; (&plusmn;2 &sigma;) variations of the expected limit, accounting for all uncertainties. The observed limit obtained from ATLAS searches for LQ pair production is also shown as dark blue line [13].

Exclusion limits for minimal models of S&#771;₁ production with only y_be being non-zero obtained from a simultaneous fit to SR-1L and SR-2L of the e+b-jet channel combining Run-2 and Run-3 data. All limits are computed at 95&percnt; CL and the observed (red solid lines) and expected (black dashed lines) exclusion limits are shown in the m(S&#771;₁) &ndash; y_be plane. The observed exclusion should be interpreted as the region above the red line. The yellow inner (green outer) shaded band around the expected limits corresponds to the &plusmn;1 &sigma; (&plusmn;2 &sigma;) variations of the expected limit, accounting for all uncertainties. The observed limit obtained from ATLAS searches for LQ pair production is also shown as dark blue line [13].

Exclusion limits for minimal models of S&#771;₁ production with only y_b&mu; being non-zero obtained from a simultaneous fit to SR-1L and SR-2L of the &mu;+b-jet channel combining Run-2 and Run-3 data. All limits are computed at 95&percnt; CL and the observed (red solid lines) and expected (black dashed lines) exclusion limits are shown in the m(S&#771;₁) &ndash; y_b&mu; plane. The observed exclusion should be interpreted as the region above the red line. The yellow inner (green outer) shaded band around the expected limits corresponds to the &plusmn;1 &sigma; (&plusmn;2 &sigma;) variations of the expected limit, accounting for all uncertainties. The observed limit obtained from ATLAS searches for LQ pair production is also shown as dark blue line [13].

Exclusion limits for minimal models of S&#771;₁ production obtained from a fit to SR-1L, SR-2L and their combination of (a) the e+light-jet, (b) the &mu;+light-jet, (c) the e+b-jet and (d) the &mu;+b-jet channel using Run-2 and Run-3 data. All limits are computed at 95&percnt; CL and the observed (solid lines) and expected (dashed lines) exclusion limits are shown in the m(S&#771;₁) &ndash; y plane. The observed exclusion regions should be interpreted as the region above the solid lines. Constraints from weak charge measurements of protons and nuclei&nbsp; and ATLAS searches for LQ pair production&nbsp; are shown as light magenta and dark blue areas, respectively.

Exclusion limits for minimal models of S&#771;₁ production obtained from a fit to SR-1L, SR-2L and their combination of (a) the e+light-jet, (b) the &mu;+light-jet, (c) the e+b-jet and (d) the &mu;+b-jet channel using Run-2 and Run-3 data. All limits are computed at 95&percnt; CL and the observed (solid lines) and expected (dashed lines) exclusion limits are shown in the m(S&#771;₁) &ndash; y plane. The observed exclusion regions should be interpreted as the region above the solid lines. Constraints from weak charge measurements of protons and nuclei&nbsp; and ATLAS searches for LQ pair production&nbsp; are shown as light magenta and dark blue areas, respectively.

Exclusion limits for minimal models of S&#771;₁ production obtained from a fit to SR-1L, SR-2L and their combination of (a) the e+light-jet, (b) the &mu;+light-jet, (c) the e+b-jet and (d) the &mu;+b-jet channel using Run-2 and Run-3 data. All limits are computed at 95&percnt; CL and the observed (solid lines) and expected (dashed lines) exclusion limits are shown in the m(S&#771;₁) &ndash; y plane. The observed exclusion regions should be interpreted as the region above the solid lines. Constraints from weak charge measurements of protons and nuclei&nbsp; and ATLAS searches for LQ pair production&nbsp; are shown as light magenta and dark blue areas, respectively.

Exclusion limits for minimal models of S&#771;₁ production obtained from a fit to SR-1L, SR-2L and their combination of (a) the e+light-jet, (b) the &mu;+light-jet, (c) the e+b-jet and (d) the &mu;+b-jet channel using Run-2 and Run-3 data. All limits are computed at 95&percnt; CL and the observed (solid lines) and expected (dashed lines) exclusion limits are shown in the m(S&#771;₁) &ndash; y plane. The observed exclusion regions should be interpreted as the region above the solid lines. Constraints from weak charge measurements of protons and nuclei&nbsp; and ATLAS searches for LQ pair production&nbsp; are shown as light magenta and dark blue areas, respectively.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$ej$ and (b) Run 2 SR-2L-$ej$ and (c) Run 3 SR-1-$ej$ and (d) Run 3 SR-2-$ej$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$ej$ and (b) Run 2 SR-2L-$ej$ and (c) Run 3 SR-1-$ej$ and (d) Run 3 SR-2-$ej$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$ej$ and (b) Run 2 SR-2L-$ej$ and (c) Run 3 SR-1-$ej$ and (d) Run 3 SR-2-$ej$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$ej$ and (b) Run 2 SR-2L-$ej$ and (c) Run 3 SR-1-$ej$ and (d) Run 3 SR-2-$ej$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$ej$ and (b) Run 2 SR-2L-$ej$ and (c) Run 3 SR-1-$ej$ and (d) Run 3 SR-2-$ej$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$ej$ and (b) Run 2 SR-2L-$ej$ and (c) Run 3 SR-1-$ej$ and (d) Run 3 SR-2-$ej$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$ej$ and (b) Run 2 SR-2L-$ej$ and (c) Run 3 SR-1-$ej$ and (d) Run 3 SR-2-$ej$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$ej$ and (b) Run 2 SR-2L-$ej$ and (c) Run 3 SR-1-$ej$ and (d) Run 3 SR-2-$ej$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu j$ and (b) Run 2 SR-2L-$\mu j$ and (c) Run 3 SR-1L-$\mu j$ and (d) Run 3 SR-2L-$\mu j$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu j$ and (b) Run 2 SR-2L-$\mu j$ and (c) Run 3 SR-1L-$\mu j$ and (d) Run 3 SR-2L-$\mu j$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu j$ and (b) Run 2 SR-2L-$\mu j$ and (c) Run 3 SR-1L-$\mu j$ and (d) Run 3 SR-2L-$\mu j$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu j$ and (b) Run 2 SR-2L-$\mu j$ and (c) Run 3 SR-1L-$\mu j$ and (d) Run 3 SR-2L-$\mu j$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu j$ and (b) Run 2 SR-2L-$\mu j$ and (c) Run 3 SR-1L-$\mu j$ and (d) Run 3 SR-2L-$\mu j$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu j$ and (b) Run 2 SR-2L-$\mu j$ and (c) Run 3 SR-1L-$\mu j$ and (d) Run 3 SR-2L-$\mu j$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu j$ and (b) Run 2 SR-2L-$\mu j$ and (c) Run 3 SR-1L-$\mu j$ and (d) Run 3 SR-2L-$\mu j$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu j$ and (b) Run 2 SR-2L-$\mu j$ and (c) Run 3 SR-1L-$\mu j$ and (d) Run 3 SR-2L-$\mu j$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$eb$ and (b) Run 2 SR-2L-$eb$ and (c) Run 3 SR-1L-$eb$ and (d) Run 3 SR-2L-$eb$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$eb$ and (b) Run 2 SR-2L-$eb$ and (c) Run 3 SR-1L-$eb$ and (d) Run 3 SR-2L-$eb$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$eb$ and (b) Run 2 SR-2L-$eb$ and (c) Run 3 SR-1L-$eb$ and (d) Run 3 SR-2L-$eb$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$eb$ and (b) Run 2 SR-2L-$eb$ and (c) Run 3 SR-1L-$eb$ and (d) Run 3 SR-2L-$eb$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP of $\tilde{S}$ in (a) Run 2 SR-1L-$eb$ and (b) Run 2 SR-2L-$eb$ and (c) Run 3 SR-1L-$eb$ and (d) Run 3 SR-2L-$eb$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP of $\tilde{S}$ in (a) Run 2 SR-1L-$eb$ and (b) Run 2 SR-2L-$eb$ and (c) Run 3 SR-1L-$eb$ and (d) Run 3 SR-2L-$eb$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP of $\tilde{S}$ in (a) Run 2 SR-1L-$eb$ and (b) Run 2 SR-2L-$eb$ and (c) Run 3 SR-1L-$eb$ and (d) Run 3 SR-2L-$eb$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP of $\tilde{S}$ in (a) Run 2 SR-1L-$eb$ and (b) Run 2 SR-2L-$eb$ and (c) Run 3 SR-1L-$eb$ and (d) Run 3 SR-2L-$eb$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu b$ and (b) Run 2 SR-2L-$\mu b$ and (c) Run 3 SR-1L-$\mu b$ and (d) Run 3 SR-2L-$\mu b$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu b$ and (b) Run 2 SR-2L-$\mu b$ and (c) Run 3 SR-1L-$\mu b$ and (d) Run 3 SR-2L-$\mu b$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu b$ and (b) Run 2 SR-2L-$\mu b$ and (c) Run 3 SR-1L-$\mu b$ and (d) Run 3 SR-2L-$\mu b$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for resonant production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu b$ and (b) Run 2 SR-2L-$\mu b$ and (c) Run 3 SR-1L-$\mu b$ and (d) Run 3 SR-2L-$\mu b$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu b$ and (b) Run 2 SR-2L-$\mu b$ and (c) Run 3 SR-1L-$\mu b$ and (d) Run 3 SR-2L-$\mu b$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu b$ and (b) Run 2 SR-2L-$\mu b$ and (c) Run 3 SR-1L-$\mu b$ and (d) Run 3 SR-2L-$\mu b$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu b$ and (b) Run 2 SR-2L-$\mu b$ and (c) Run 3 SR-1L-$\mu b$ and (d) Run 3 SR-2L-$\mu b$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Values of the signal acceptance multiplied by the selection efficiency ($A\times\epsilon$) for DY+SP production of $\tilde{S}$ in (a) Run 2 SR-1L-$\mu b$ and (b) Run 2 SR-2L-$\mu b$ and (c) Run 3 SR-1L-$\mu b$ and (d) Run 3 SR-2L-$\mu b$. The $A \times \epsilon$ values are calculated as the number of events of reconstructed-level signal simulation divided by the total cross-section of the signal process times the integrated luminosity.

Event selection cutflows of SR-1L-$eb$ for the signal sample with $m(\tilde{S}) = 2$ TeV and $y_{be} = 1.0$ individually for resonant production (RP) and combined Drell-Yan plus single production (DY+SP). The first number in the column for each mass points corresponds to the signal event yield after applying the associated cut while the second number in parentheses represents the efficiency with respect to the previous cut.

Event selection cutflows of SR-2L-$eb$ for the signal sample with $m(\tilde{S}) = 2$ TeV and $y_{be} = 1.0$ individually for resonant production (RP) and combined Drell-Yan plus single production (DY+SP). The first number in the column for each mass points corresponds to the signal event yield after applying the associated cut while the second number in parentheses represents the efficiency with respect to the previous cut.

Event selection cutflows of SR-1L-$ej$ for the signal sample with $m(\tilde{S}) = 2$ TeV and $y_{de} = 1.0$ individually for resonant production (RP) and combined Drell-Yan plus single production (DY+SP). The first number in the column for each mass points corresponds to the signal event yield after applying the associated cut while the second number in parentheses represents the efficiency with respect to the previous cut.

Event selection cutflows of SR-2L-$ej$ for the signal sample with $m(\tilde{S}) = 2$ TeV and $y_{de} = 1.0$ individually for resonant production (RP) and combined Drell-Yan plus single production (DY+SP). The first number in the column for each mass points corresponds to the signal event yield after applying the associated cut while the second number in parentheses represents the efficiency with respect to the previous cut.

Event selection cutflows of SR-1L-$\mu b$ for the signal sample with $m(\tilde{S}) = 2$ TeV and $y_{b\mu} = 1.0$ individually for resonant production (RP) and combined Drell-Yan plus single production (DY+SP). The first number in the column for each mass points corresponds to the signal event yield after applying the associated cut while the second number in parentheses represents the efficiency with respect to the previous cut.

Event selection cutflows of SR-2L-$\mu b$ for the signal sample with $m(\tilde{S}) = 2$ TeV and $y_{b\mu} = 1.0$ individually for resonant production (RP) and combined Drell-Yan plus single production (DY+SP). The first number in the column for each mass points corresponds to the signal event yield after applying the associated cut while the second number in parentheses represents the efficiency with respect to the previous cut.

Event selection cutflows of SR-1L-$\mu j$ for the signal sample with $m(\tilde{S}) = 2$ TeV and $y_{s\mu} = 1.0$ individually for resonant production (RP) and combined Drell-Yan plus single production (DY+SP). The first number in the column for each mass points corresponds to the signal event yield after applying the associated cut while the second number in parentheses represents the efficiency with respect to the previous cut.

Event selection cutflows of SR-2L-$\mu j$ for the signal sample with $m(\tilde{S}) = 2$ TeV and $y_{s\mu} = 1.0$ individually for resonant production (RP) and combined Drell-Yan plus single production (DY+SP). The first number in the column for each mass points corresponds to the signal event yield after applying the associated cut while the second number in parentheses represents the efficiency with respect to the previous cut.

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.

Distributions of the VLQ-candidate mass, m_VLQ, in the (a&ndash;c) SRs, (d&ndash;f) W+jets CRs and (g&ndash;i) tt&#772; 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&ndash;c) SRs, (d&ndash;f) W+jets CRs and (g&ndash;i) tt&#772; 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&ndash;c) SRs, (d&ndash;f) W+jets CRs and (g&ndash;i) tt&#772; 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&ndash;c) SRs, (d&ndash;f) W+jets CRs and (g&ndash;i) tt&#772; 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&ndash;c) SRs, (d&ndash;f) W+jets CRs and (g&ndash;i) tt&#772; 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&ndash;c) SRs, (d&ndash;f) W+jets CRs and (g&ndash;i) tt&#772; 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 tt&#772; 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 tt&#772; 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 tt&#772; 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 tt&#772; 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 tt&#772; 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 tt&#772; 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&percnt;&nbsp;CL upper limits on the VLQ coupling &kappa; 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&nbsp; is overlaid; this includes the decay modes T&rarr;H t and T&rarr;Z t. These interpretations are limited to parameter combinations for which the model corrections are valid, by restricting the VLQ relative width to &Gamma;_Q/m_Q &lt; 0.5, as indicated by the grey dashed line.

Expected (background-only) and observed 95&percnt;&nbsp;CL upper limits on the VLQ coupling &kappa; 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&nbsp; is overlaid; this includes the decay modes T&rarr;H t and T&rarr;Z t. These interpretations are limited to parameter combinations for which the model corrections are valid, by restricting the VLQ relative width to &Gamma;_Q/m_Q &lt; 0.5, as indicated by the grey dashed line.

Exclusion limit (at 95&percnt; CL) for (a) T-singlet and (b) Y-triplet, expressed in terms of an upper limit on the VLQ relative width &Gamma;_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 &kappa; in Figure&nbsp;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&percnt; CL) for (a) T-singlet and (b) Y-triplet, expressed in terms of an upper limit on the VLQ relative width &Gamma;_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 &kappa; in Figure&nbsp;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.