Measured differential cross section with associated uncertainties as a function of $N$($b$-jets). Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate. For the meaning of the bins please refer to the figure.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{j1}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma} (N_\mathrm{jets}^{p_T>30\ \mathrm{GeV}}=0)$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Measured differential cross section with associated uncertainties as a function of $m_{jj}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Measured differential cross section with associated uncertainties as a function of $\Delta\phi_{jj}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma}$ in bins of $|y_{\gamma\gamma}|$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Measured differential cross section with associated uncertainties as a function of $\Delta\phi_{jj}$ in the VBF fiducial region. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Observed statistical correlations, evaluated with a bootstrapping technique, between $p_\mathrm{T}^{\gamma\gamma}$, $N_\mathrm{jets}$, $m_{jj}$, $\Delta\phi_{jj}$, and $p_\mathrm{T}^{j1}$

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma}$.

Post-fit correlation matrix for the differential cross section measured as a function of $N_\mathrm{jets}$.

Post-fit correlation matrix for the differential cross section measured as a function of $N$($b$-jets).

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{j1}$.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma} (N_\mathrm{jets}^{p_T>30\ \mathrm{GeV}}=0)$.

Post-fit correlation matrix for the differential cross section measured as a function of $m_{jj}$.

Post-fit correlation matrix for the differential cross section measured as a function of $\Delta\phi_{jj}$.

Post-fit correlation matrix for the differential cross section measured as a function of $\Delta\phi_{jj}$ in the VBF fiducial region.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma}$ in bins of $|y_{\gamma\gamma}|$. The bins are the same as in the corresponding plot of the differential cross section.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma1}/m_{\gamma\gamma}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma1}/m_{\gamma\gamma}$.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma2}/m_{\gamma\gamma}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma2}/m_{\gamma\gamma}$.

Measured differential cross section with associated uncertainties as a function of $|y_{\gamma\gamma}|$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $|y_{\gamma\gamma}|$.

Measured differential cross section with associated uncertainties as a function of $m_{\gamma\gamma j}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $m_{\gamma\gamma j}$.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma j}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma j}$.

Measured differential cross section with associated uncertainties as a function of $H_{T}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $H_{T}$.

Measured differential cross section with associated uncertainties as a function of $\tau_{C,j1}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $\tau_{C,j1}$.

Measured differential cross section with associated uncertainties as a function of $\sum\tau_{C,j}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $\sum\tau_{C,j}$.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma} (N_\mathrm{jets}^{p_T>40\ \mathrm{GeV}}=0)$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma} (N_\mathrm{jets}^{p_T>40\ \mathrm{GeV}}=0)$.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma} (N_\mathrm{jets}^{p_T>50\ \mathrm{GeV}}=0)$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma} (N_\mathrm{jets}^{p_T>50\ \mathrm{GeV}}=0)$.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma} (N_\mathrm{jets}^{p_T>60\ \mathrm{GeV}}=0)$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma} (N_\mathrm{jets}^{p_T>60\ \mathrm{GeV}}=0)$.

Measured differential cross section with associated uncertainties as a function of $\pi - |\Delta\phi_{\gamma\gamma,jj}|$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $\pi - |\Delta\phi_{\gamma\gamma,jj}|$.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma jj}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma jj}$.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma}$ in bins of $p_{T}^{\gamma\gamma j}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma}$ in bins of $p_{T}^{\gamma\gamma j}$. The bins are the same as in the corresponding plot of the differential cross section.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma}$ in bins of $\tau_{C,j1}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma}$ in bins of $\tau_{C,j1}$. The bins are the same as in the corresponding plot of the differential cross section.

Measured differential cross section with associated uncertainties as a function of $(p_{T}^{\gamma1}-p_{T}^{\gamma2})/m_{\gamma\gamma}$ in bins of $(p_{T}^{\gamma1}+p_{T}^{\gamma2})/m_{\gamma\gamma}$. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $(p_{T}^{\gamma1}-p_{T}^{\gamma2})/m_{\gamma\gamma}$ in bins of $(p_{T}^{\gamma1}+p_{T}^{\gamma2})/m_{\gamma\gamma}$. The bins are the same as in the corresponding plot of the differential cross section.

Measured differential cross section with associated uncertainties as a function of $|\eta^*|$ in the VBF fiducial region. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $|\eta^*|$ in the VBF fiducial region.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{\gamma\gamma jj}$ in the VBF fiducial region. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{\gamma\gamma jj}$ in the VBF fiducial region.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{j1}$ in the VBF fiducial region. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{j1}$ in the VBF fiducial region.

Measured differential cross section with associated uncertainties as a function of $p_{T}^{j1}$ in bins of $\Delta\phi_{jj}$ in the VBF fiducial region. Each systematic uncertainty source is fully uncorrelated with the other sources and fully correlated across bins, except for the background modelling systematics for which an uncorrelated treatment across bins is more appropriate.

Post-fit correlation matrix for the differential cross section measured as a function of $p_{T}^{j1}$ in bins of $\Delta\phi_{jj}$ in the VBF fiducial region. The bins are the same as in the corresponding plot of the differential cross section.

Observed yields in the boost_1_ge2J signal region category of the hh channel.

Observed yields in the boost_2 signal region category of the hh channel.

Observed yields in the boost_3 signal region category of the hh channel.

Observed yields in the ttH_0 signal region category of the hh channel.

Observed yields in the ttH_1 signal region category of the hh channel.

Observed yields in the VH_0 signal region category of the hh channel.

Observed yields in the VH_1 signal region category of the hh channel.

Observed yields in the VBF_0 signal region category of the hh channel.

Observed yields in the VBF_1 signal region category of the hh channel.

Observed yields in the boost_0_1J signal region category of the lh channel.

Observed yields in the boost_0_ge2J signal region category of the lh channel.

Observed yields in the boost_1_1J signal region category of the lh channel.

Observed yields in the boost_1_ge2J signal region category of the lh channel.

Observed yields in the boost_2 signal region category of the lh channel.

Observed yields in the boost_3 signal region category of the lh channel.

Observed yields in the VH_0 signal region category of the lh channel.

Observed yields in the VH_1 signal region category of the lh channel.

Observed yields in the VBF_0 signal region category of the lh channel.

Observed yields in the VBF_1 signal region category of the lh channel.

Observed yields in the boost_0_1J signal region category of the ll channel.

Observed yields in the boost_0_ge2J signal region category of the ll channel.

Observed yields in the boost_1_1J signal region category of the ll channel.

Observed yields in the boost_1_ge2J signal region category of the ll channel.

Observed yields in the boost_2 signal region category of the ll channel.

Observed yields in the boost_3 signal region category of the ll channel.

Observed yields in the VH_0 signal region category of the ll channel.

Observed yields in the VH_1 signal region category of the ll channel.

Observed yields in the VBF_0 signal region category of the ll channel.

Observed yields in the VBF_1 signal region category of the ll channel.

Best-fit values and uncertainties for the pp $\rightarrow H\rightarrow\tau\tau$ cross-section measurement and the measurement in the four dominant production modes. All measurements include the branching ratio of $H\rightarrow\tau\tau$ and are performed with true Higgs boson rapidity $|y_{H}|<2.5$. The SM predictions for each region, computed using the inclusive cross-section calculations and the simulated event samples are also shown. The contributions to the total uncertainty in the measurements from statistical (Stat. unc.) or systematic uncertainties (Syst. unc.) are given separately.

The measured correlations between each parameter of interest in the measurement of the cross-sections per production mode

Best-fit values and uncertainties for the $H\rightarrow\tau\tau$ cross-sections, in the reduced stage 1.2 STXS scheme. The EW production mode includes vector-boson fusion and qq$\rightarrow$V($\rightarrow$qq)H processes. All measurements include the branching ratio of $H\rightarrow\tau\tau$ and are performed with true Higgs boson rapidity $|y_{H}|<2.5$. The SM predictions for each region, computed using the inclusive cross-section calculations and the simulated event samples are also shown. The contributions to the total uncertainty in the measurements from statistical (Stat. unc.) or systematic uncertainties (Syst. unc.) are given separately. The asterisk symbol (*) indicates that the criteria for m$_{jj}$ only apply to events with at least two reconstructed jets.

The measured correlations between each pair of parameters of interest in the STXS measurement. The asterisk symbol (*) indicates that the criteria for mjj only apply to events with at least two reconstructed jets.

Observed and expected 95% CL lower limits on the $T$-quark mass as a function of the $T$-quark width-to-mass ratio and the branching fraction of the $T \rightarrow Ht$ decay ($\Gamma_{T}$ is the $T$-quark width).

Cutflow table listing the number of events passing each criterion for a $T$-quark hypothesis with a mass of 1.6 TeV and $\kappa_{T} = 0.5$. The initial signal event yield is the predicted number of $T$-quark events inclusive in the Higgs-boson and top-quark decays for 139 fb$^{-1}$.

Observed 95% CL upper limits on the single $T$-quark production cross-section as a function of the $T$-quark coupling $\kappa_{T}$ and $m_{T}$.

Expected 95% CL upper limits on the single $T$-quark production cross-section as a function of the $T$-quark coupling $\kappa_{T}$ and $m_{T}$.

Observed and expected 95% CL lower limits on the $T$-quark mass as a function of the $T$-quark width-to-mass ratio and the branching fraction of the $T \rightarrow Wb$ decay ($\Gamma_{T}$ is the $T$-quark width).

The BDT distribution (with x-axis zoomed in) of the di-Higgs ggF signal for two different values of $\kappa_{\lambda}$ and the main backgrounds in the high mass region ($m^{*}_{b\bar{b}\gamma\gamma} > 350$ GeV). Distributions are normalized to unit area. The dotted lines denote the category boundaries. Events with a BDT score below 0.857 in the high mass region are discarded. The range of BDT scores is from 0.8 to 1.

The $BDT_{tot}$ score for the benchmark signal $m_{X}$ = 300 GeV and for the main backgrounds. Distributions are normalized to unit area. The dotted line denotes the event selection threshold. Events with a $BDT_{tot}$ score below 0.85 for $m_{X}$ = 300 GeV are discarded.

The $BDT_{tot}$ score for the benchmark signal $m_{X}$ = 500 GeV and for the main backgrounds. Distributions are normalized to unit area. The dotted line denotes the event selection threshold. Events with a $BDT_{tot}$ score below 0.75 for $m_{X}$ = 500 GeV are discarded.

Distributions of $m_{\gamma\gamma}$ in high mass BDT tight category for the nonresonant $HH$ search. The data-derived fractions of nonresonant $\gamma\gamma$, $\gamma$-jet or jet-$\gamma$, and dijet background are applied and the total background is normalized to the data sideband.

Distributions of $m_{\gamma\gamma}$ in high mass BDT loose category for the nonresonant $HH$ search. The data-derived fractions of nonresonant $\gamma\gamma$, $\gamma$-jet or jet-$\gamma$, and dijet background are applied and the total background is normalized to the data sideband.

Distributions of $m_{\gamma\gamma}$ in low mass BDT tight category for the nonresonant $HH$ search. The data-derived fractions of nonresonant $\gamma\gamma$, $\gamma$-jet or jet-$\gamma$, and dijet background are applied and the total background is normalized to the data sideband.

Distributions of $m_{\gamma\gamma}$ in low mass BDT loose category for the nonresonant $HH$ search. The data-derived fractions of nonresonant $\gamma\gamma$, $\gamma$-jet or jet-$\gamma$, and dijet background are applied and the total background is normalized to the data sideband.

Distributions of $m_{\gamma\gamma}$ for the selections used for the resonance mass point $m_{X}$ = 300 GeV for the resonant search. The data-derived fractions of nonresonant $\gamma\gamma$, $\gamma$-jet or jet-$\gamma$, and dijet background are applied and the total background is normalized to the data sideband. The scalar resonance signal is scaled to a total production cross section $\sigma(pp \rightarrow X \rightarrow HH)$ = 370 fb.

Distributions of $m_{\gamma\gamma}$ for the selections used for the resonance mass point $m_{X}$ = 500 GeV for the resonant search. The data-derived fractions of nonresonant $\gamma\gamma$, $\gamma$-jet or jet-$\gamma$, and dijet background are applied and the total background is normalized to the data sideband. The scalar resonance signal is scaled to a total production cross section $\sigma(pp \rightarrow X \rightarrow HH)$ = 67 fb.

The number of data events observed in the 120 GeV < $m_{\gamma\gamma}$ < 130 GeV window, the number of $HH$ signal events expected for $\kappa_{\lambda}$ = 1 and for $\kappa_{\lambda}$ = 10, and events expected for single Higgs boson production (estimated using MC simulation), as well as for continuum background. For the single Higgs boson, Rest includes VBF, $WH$, $tHqb$, and $tHW$. The values are obtained from a fit of the Asimov data set generated under the SM signal-plus-background hypothesis, $\kappa_{\lambda}$ = 1. The continuum background component of the Asimov data set is obtained from the fit of the data sideband. The uncertainties in $HH$ signals and single Higgs boson background include the systematic uncertainties discussed in Section 6. 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 and expected limits at 95% CL on the cross section of nonresonant Higgs boson pair production as a function of the Higgs boson self-coupling modifier $\kappa_{\lambda}= \lambda_{HHH}/\lambda^{\textrm{SM}}_{HHH}$. The expected constraints on $\kappa_{\lambda}$ are obtained with a background hypothesis excluding $pp \rightarrow HH$ production. The $\pm 1\sigma$ and $\pm 2\sigma$ variations about the expected limit due to statistical and systematic uncertainties are also shown. The theory prediction curve represents the scenario where all parameters and couplings are set to their SM values except for $\kappa_{\lambda}$. The uncertainty band of the theory prediction curve shows the cross-section uncertainty.

Values of the negative log-profile-likelihood ratio ($-2ln\Lambda$) as a function of $\kappa_{\lambda}$ evaluated for the combination of all the categories of the nonresonant search. The coupling of the Higgs boson to fermions and gauge bosons is set to SM values in the profile likelihood calculation. The Asimov data set is generated under the SM signal-plus-background hypothesis, $\kappa_{\lambda}$= 1. All systematic uncertainties, including the theoretical uncertainties in the di-Higgs boson production cross section, are included. The intersections of the solid curves and the horizontal dashed lines indicate the 1$\sigma$ and 2$\sigma$ confidence-level intervals.

The number of events observed in the 120 < $m_{\gamma\gamma}$ < 130 GeV window in data, the number of events expected for scalar resonance signals of masses $m_{X}$ = 300 GeV and $m_{X}$ = 500 GeV assuming a total production cross section $\sigma(pp \rightarrow X \rightarrow HH)$ equal to the observed exclusion limits of Figure 15, and events expected for SM $HH$ and single Higgs boson production (estimated using MC simulation), as well as for continuum background. The values are obtained from a fit of the Asimov data set generated under the signal-plus-background hypothesis. The continuum background component of the Asimov data set is obtained from the fit of the data sideband. The uncertainties in the resonant signals and the SM $HH$ and single-Higgs-boson backgrounds include the systematic uncertainties discussed in Section 6. 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 and expected limits at 95% CL on the production cross section of a narrow-width scalar resonance $X$ as a function of the mass $m_{X}$ of the hypothetical scalar particle. The black solid line represents the observed upper limits. The dashed line represents the expected upper limits. The $\pm 1\sigma$ and $\pm 2\sigma$ variations about the expected limit due to statistical and systematic uncertainties are also shown.

Breakdown of the dominant systematic uncertainties. The impact of the uncertainties is defined according to the statistical analysis described in Section 7. It corresponds to the relative variation of the expected upper limit on the cross section when re-evaluating the profile likelihood ratio after fixing the nuisance parameter in question to its best-fit value, while all remaining nuisance parameters remain free to float. The impact is shown in %. Only systematic uncertainties with an impact of at least 0.2% are shown. Uncertainties of the "Norm. + Shape" type affect both the normalization and the parameters of the functional form. The rest of the uncertainties affect only the yields.

Cutflow for nonresonant di-Higgs ggF signal sample, yields are normalized to 139 $fb^{-1}$.

Cutflow for resonant signal sample, with $m_{X}$ = 300 GeV, yields are normalized to 139 $fb^{-1}$.

Cutflow for resonant signal sample, with $m_{X}$ = 500 GeV, yields are normalized to 139 $fb^{-1}$.

Comparison of $m_{b\bar{b}}$ distributions when applying the specific b-jet energy calibration and the nominal jet energy calibration. The distributions are fitted using a Bukin function, and the values of the peak position, resolution and the relative improvement are reported in the legend.

Fit results of $m_{b\bar{b}}$ distributions when applying the specific b-jet energy calibration and the nominal jet energy calibration. The distributions are fitted using a Bukin function, and the values of the peak position, resolution and the relative improvement are reported in the legend.

The relative amount (purity) of expected events from SM $HH$ and single Higgs boson production processes for each of the four categories of the nonresonant search. The Higgs boson pair production with $k_{\lambda} = 1$ is considered as signal in (a), while the case with $k_{\lambda} = 10$ is considered as signal in (b).

The expected significance in each of the four categories of the nonresonant search. The Higgs boson pair production with $k_{\lambda} = 1$ is considered as signal in (a), while the case with $k_{\lambda} = 10$ is considered as signal in (b). The single Higgs boson processes and the di-photon continuum spectrum are considered as background.

Spurious signal result for the exponential pdf for the various ggF nonresonant di-Higgs categories. In each category, the spurious signal value ($n_{sp}$) and its ratio to the expected statistical error ($Z_{spur}$) from data are shown.

Spurious signal result for the exponential pdf as function of the resonant di-Higgs signal mass. The spurious signal value and its ratio to the expected statistical error from data are shown.

Distributions of the signal efficiency as a function of $\kappa_{\lambda}$, for the di-Higgs boson ggF nonresonant production mode. The range of $\kappa_{\lambda}$ in the table is from -10 to 10.

Distributions of the signal efficiency as a function of $\kappa_{\lambda}$, for the di-Higgs boson VBF nonresonant production mode. The range of $\kappa_{\lambda}$ in the table is from -10 to 10.

Values of the negative log-profile-likelihood ($-2ln\Lambda$) as a function of $\kappa_{\lambda}$ evaluated for the combination of all the categories of the nonresonant search. The coupling of the Higgs boson to fermions and gauge bosons is set to SM values in the profile likelihood calculation. The Asimov data set is generated under the SM signal-plus-background hypothesis, $\kappa_{\lambda}$= 1. All systematic uncertainties, including the theoretical uncertainties on the di-Higgs boson production cross section, are included. The intersections of the solid curves and the horizontal dashed lines indicate the 1$\sigma$ and 2$\sigma$ confidence level intervals. The best fit value corresponds to $\kappa_{\lambda}$ = $2.8^{+2.0}_{-2.2}(^{+3.8}_{-4.3})$ for the 1$\sigma$(2$\sigma$) confidence interval. The expected value corresponds to $\kappa_{\lambda}$ = $1.0^{+5.5}_{-2.4}(^{+7.3}_{-4.2})$ for the 1$\sigma$(2$\sigma$) confidence interval. The dashed curves represent values of the negative log-profile-likelihood where the Higgs boson branching fractions and the cross section of the production modes are varied as a function of $\kappa_{\lambda}$. In this case,the best fit value corresponds to $\kappa_{\lambda}$ = $2.7^{+2.0}_{-2.2}(^{+3.8}_{-4.3})$ and the expected value corresponds to $\kappa_{\lambda}$ = $1.0^{+5.4}_{-2.5}(^{+7.3}_{-4.3})$ for the 1$\sigma$(2$\sigma$) confidence interval.

Minimum BDT value of the events passing the selection criteria of the resonant search. The combined BDT score is formed using as coefficients $C_{1}$ = 0.65 and $C_{2}$ = 1 − $C_{1}$. The selection efficiency for the resonant $X \rightarrow HH$ signal is also shown.

The $1\sigma_{\text{exp}}$ variation of the expected exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=1.0$.

The observed exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.1$.

The expected exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.1$.

The observed exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.15$.

The expected exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.15$.

The observed exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.2$.

The expected exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.2$.

The observed exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.4$.

The expected exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.4$.

The observed exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0.6$.

The expected exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=0p6$.

The observed exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=1.5$.

The expected exclusion contour at 95% CL as a function of the smuon and neutralino masses, for $\lambda_{231}^{'}=1.5$.

The observed exclusion contour at 95% CL as a function of the leptoquark mass and coupling strength.

The expected exclusion contour at 95% CL as a function of the leptoquark mass and coupling strength.

The minus $1\sigma_{\text{theory}}$ variation of the observed exclusion contour at 95% CL as a function of the leptoquark mass and coupling strength.

The plus $1\sigma_{\text{theory}}$ variation of the observed exclusion contour at 95% CL as a function of the leptoquark mass and coupling strength.

The $1\sigma_{\text{exp}}$ variation of the expected exclusion contour at 95% CL as a function of the leptoquark mass and coupling strength.

Observed yields, and fake lepton background yields in the $e^{+}\mu^{-}$ and $e^{-}\mu^{+}$ channels of SR-MET, along with the results of the $e^{+}\mu^{-}/e^{-}\mu^{+}$ ratio measurement and 1-sided p-value in SR-MET, binned in $M_{T2}$.

Observed yields, and fake lepton background yields in the $e^{+}\mu^{-}$ and $e^{-}\mu^{+}$ channels of SR-JET, along with the results of the $e^{+}\mu^{-}/e^{-}\mu^{+}$ ratio measurement and 1-sided p-value in SR-JET, binned in $H_{\text{P}}$.

Observed and expected 95% CL upper limits on the total number of signal events entering the $e^{+}\mu^{-}$ and $e^{-}\mu^{+}$ channels of each bin of SR-MET. The regions are binned in the same way as the ratio $\rho$ measurement. The limits are shown for a selection of 'z' values, where 'z' is the fraction of the total signal events entering the $e^{+}\mu^{-}$ channel.

Observed and expected 95% CL upper limits on the total number of signal events entering the $e^{+}\mu^{-}$ and $e^{-}\mu^{+}$ channels of each bin of SR-JET. The regions are binned in the same way as the ratio $\rho$ measurement. The limits are shown for a selection of 'z' values, where 'z' is the fraction of the total signal events entering the $e^{+}\mu^{-}$ channel.

Signal yields following each cut in the analysis, for representative $R$-parity-violating supersymmetry and leptoquark signals. All yields are MC generator-weighted and normalised to $139~\text{fb}^{-1}$. The cut labelled `Preselection' includes trigger requirements, and requires exactly one Baseline electron and one Baseline muon. At this point, the muon charge-bias correction weights are also applied. The $R$-parity-violating supersymmetry models were generated by specifying a top-quark in the final state and applying a two-lepton filter, hence the first row also includes events where the top quark decays to an electron.

Final state radiation correction used in the $\phi_{\eta}^{*}$ cross-section measurement. The first uncertainty is statistical and the second is systematic.

Final state radiation correction used in the $y^{Z}-p_{T}^{Z}$ cross-section measurement. The first uncertainty is statistical and the second is systematic.

Final state radiation correction used in the $y^{Z}-\phi_{\eta}^{*}$ cross-section measurement. The first uncertainty is statistical and the second is systematic.

Correlation matrix of statistical uncertainty for one-dimensional $y^Z$ measurement.

Correlation matrix of statistical uncertainty for one-dimensional $p_{T}^{Z}$ measurement.

Correlation matrix of statistical uncertainty for one-dimensional $\phi_{\eta}^{*}$ measurement.

Correlation matrix of statistical uncertainty for two-dimensional $y^Z-p_{T}^{Z}$ measurement.

Correlation matrix of statistical uncertainty for two-dimensional $y^Z-\phi_{\eta}^{*}$ measurement.

Correlation matrix of efficiency uncertainty for one-dimensional $y^Z$ measurement.

Correlation matrix of efficiency uncertainty for one-dimensional $p_{T}^{Z}$ measurement.

Correlation matrix of efficiency uncertainty for one-dimensional $\phi_{\eta}^{*}$ measurement.

Correlation matrix of efficiency uncertainty for two-dimensional $y^Z-p_{T}^{Z}$ measurement.

Correlation matrix of efficiency uncertainty for two-dimensional $y^Z-\phi_{\eta}^{*}$ measurement.

Measured total $Z$-boson cross-section for different datasets. The first uncertainty is statistical, the second systematic, and the third is due to the luminosity.

Measured single differential cross-sections in interval regions of $y^{Z}$. The first uncertainty is statistical, the second systematic, and the third is due to the luminosity.

Measured single differential cross-sections in interval regions of $p_{T}^{Z}$. The first uncertainty is statistical, the second systematic, and the third is due to the luminosity.

Measured single differential cross-sections in interval regions of $\phi_{\eta}^{*}$. The first uncertainty is statistical, the second systematic, and the third is due to the luminosity.

Measured double differential cross-sections in interval regions of $y^{Z}$ and $p_{T}^{Z}$. The first uncertainty is statistical, the second systematic, and the third is due to the luminosity.

Measured double differential cross-sections in interval regions of $y^{Z}$ and $\phi_{\eta}^{*}$. The first uncertainty is statistical, the second systematic, and the third is due to the luminosity.

Systematic uncertainties in the single differential cross-sections in interval regions of $y^{Z}$, presented in percentage. The contributions from efficiency (Eff), background (BKG), final state radiation (FSR), closure test (Closure), and alignment and calibration (Alignment) are shown.

Systematic uncertainties in the single differential cross-sections in interval regions of $p_{T}^{Z}$, presented in percentage. The contributions from efficiency (Eff), background (BKG), final state radiation (FSR), closure test (Closure), and alignment and calibration (Alignment) are shown.

Systematic uncertainties in the single differential cross-sections in interval regions of $\phi_{\eta}^{*}$, presented in percentage. The contributions from efficiency (Eff), background (BKG), final state radiation (FSR), closure test (Closure), and alignment and calibration (Alignment) are shown.

Systematic uncertainties in the double differential cross-sections in interval regions of $y^{Z}$ and $p_{T}^{Z}$, presented in percentage. The contributions from efficiency (Eff), background (BKG), final state radiation (FSR), closure test (Closure), and alignment and calibration (Alignment) are shown.

Systematic uncertainties in the double differential cross-sections in interval regions of $y^{Z}$ and $\phi_{\eta}^{*}$, presented in percentage. The contributions from efficiency (Eff), background (BKG), final state radiation (FSR), closure test (Closure), and alignment and calibration (Alignment) are shown.

The $p_{T}$ dependencies of $v_{1}$ within $-0.4<y<-0.1$ for $t$ in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity and $p_{T}$ dependencies of $v_{1}$ for $^{3}$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The $p_{T}$ dependencies of $v_{1}$ within $-0.1<y<0$ for $^4$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The $p_{T}$ dependencies of $v_{1}$ within $-0.4<y<-0.1$ for $^{4}$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity and $p_{T}$ dependencies of $v_{2}$ for $p$ in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity and $p_{T}$ dependencies of $v_{2}$ for $d$ in 10-40% mid-central Au+Au collisions at 3 GeV.

The $p_{T}$ dependencies of $v_{2}$ within $-0.1<y<0$ for $t$ in 10-40% mid-central Au+Au collisions at 3 GeV.

The $p_{T}$ dependencies of $v_{2}$ within $-0.4<y<-0.1$ for $t$ in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity and $p_{T}$ dependencies of $v_{2}$ for $^{3}$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The $p_{T}$ dependencies of $v_{2}$ within $-0.1<y<0$ for $^4$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The $p_{T}$ dependencies of $v_{2}$ within $-0.4<y<-0.1$ for $^{4}$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity dependencies of $v_{1}$ for $p$, $d$, and $^{3}$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity dependencies of $v_{1}$ for $t$ and $^{4}$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity dependencies of $v_{1}$ from JAM plus coalescence calculations in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity dependencies of $v_{2}$ for $p$, $d$, and $^{3}$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity dependencies of $v_{2}$ for $t$ and $^{4}$He in 10-40% mid-central Au+Au collisions at 3 GeV.

The rapidity dependencies of $v_{2}$ from JAM plus coalescence calculations in 10-40% mid-central Au+Au collisions at 3 GeV.

Light nucleus scaled $v_{1}$ slopes as a function os collision energy in 10-40 mid-cantral Au+Au collisions.

Light nucleus scaled $v_{1}$ slopes from JAM plus coalescence in 10-40 mid-cantral Au+Au collisions at 3 GeV.

Centrality dependence of the proton cumulant ratios for Au+Au collisions at $\sqrt{s_{NN}}$ = 3.0 GeV. Protons are from $-0.5 < y < 0$ and $0.4 < p_{T} < 2.0$ GeV/$c$. Systematic uncertainties are represented by gray bars. Statistical uncertainties are smaller than marker size. CBWC is applied to all cumulant ratios. While open squares represent the data without the VFC correction, blue triangles and red circles are the results with VFC using the $\langle N_{\rm{part}} \rangle$ distributions from the UrQMD and Glauber models, respectively. UrQMD model results are represented as gold dashed line.

Centrality dependence of the proton cumulant ratios for Au+Au collisions at $\sqrt{s_{NN}}$ = 3.0 GeV. Protons are from $-0.5 < y < 0$ and $0.4 < p_{T} < 2.0$ GeV/$c$. Systematic uncertainties are represented by gray bars. Statistical uncertainties are smaller than marker size. CBWC is applied to all cumulant ratios. While open squares represent the data without the VFC correction, blue triangles and red circles are the results with VFC using the $\langle N_{\rm{part}} \rangle$ distributions from the UrQMD and Glauber models, respectively. UrQMD model results are represented as gold dashed line.

Rapidity and transverse momentum dependence of the proton cumulant ratios for 0–5% central collisions. Black-squares, red-dots and blue-triangles stand for data without and with the VFC using Glauber and UrQMD, respectively.

Rapidity and transverse momentum dependence of the proton cumulant ratios for 0–5% central collisions. Black-squares, red-dots and blue-triangles stand for data without and with the VFC using Glauber and UrQMD, respectively.

Rapidity and transverse momentum dependence of the proton cumulant ratios for 0–5% central collisions. Black-squares, red-dots and blue-triangles stand for data without and with the VFC using Glauber and UrQMD, respectively.

Rapidity and transverse momentum dependence of the proton cumulant ratios for 0–5% central collisions. Black-squares, red-dots and blue-triangles stand for data without and with the VFC using Glauber and UrQMD, respectively.

Collision energy dependence of the ratios of cumulants, $C_4/C_2$, for proton (squares) and net-proton (red circles) from top 0-5% Au+Au collisions at RHIC. The points for protons are shifted horizontally for clarity. The new result for proton from $\sqrt{s_{NN}}$ = 3.0 GeV collisions is shown as a filled square. HADES data of $\sqrt{s_{NN}}$ = 2.4 GeV 0-10% collisions is also shown. The vertical black and gray bars are the statistical and systematic uncertainties, respectively. In addition, results from the HRG model, based on both Canonical Ensemble (CE) and Grand-Canonical Ensemble (GCE), and transport model UrQMD are presented.

Fraction of bottom hadron decayed electrons in 0-80% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV.

Fraction of bottom hadron decayed electrons in 0-20% and 20-40% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV.

Fraction of bottom hadron decayed electrons in 40-80% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV.

Nuclear modification factor $R_{\rm AA}$ of inclusive heavy flavor decayed electrons in 0-80% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV. The 8% global uncertainty from the $p$+$p$ reference and a 8% gloabl uncertainty from $N_{\rm coll.}$ are not included in the table values.

Nuclear modification factor $R_{\rm AA}$ of bottom- and charm-decay electrons in 0-80% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV. The 8% global uncertainty from the $p$+$p$ reference and a 8% gloabl uncertainty from $N_{\rm coll.}$ are not included in the table values.

Nuclear modification factor $R_{\rm AA}$ ratio of bottom- to charm-decay electrons in 0-80% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV.

Null hypothesis for the nuclear modification factor $R_{\rm AA}$ ratio of bottom- to charm-decay electrons in 0-80% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV.

Nuclear modification factor $R_{\rm CP}$ ratio of bottom- and charm-decay electrons in 0-80% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV.

Null hypotheses for the nuclear modification factor $R_{\rm CP}$ ratio of bottom- and charm-decay electrons in 0-80% Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV.