Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the combined fit, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the combined fit, when all other coupling modifiers are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the combined fit, when all other coupling modifiers are fixed to their SM predictions.

Likelihood observed contours at 68% CL (solid line) and 95% CL (dashed line) in the ($\kappa_{\lambda}$, $\kappa_{2V}$) parameter space, when all other coupling modifiers are fixed to their SM predictions. The best-fit value, denoted by a cross in the plot, corresponds to the very first entry in the table.

Likelihood expected contours at 68% CL (teal shaded region) and 95% CL (yellow shaded region) in the ($\kappa_{\lambda}$, $\kappa_{2V}$) parameter space, when all other coupling modifiers are fixed to their SM predictions. In the plot, the SM prediction is indicated by the star.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{hhh}$, $c_{gghh}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross. The best-fit value corresponds to the very first entry in the table.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{hhh}$, $c_{gghh}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{hhh}$, $c_{tthh}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross. The best-fit value corresponds to the very first entry in the table.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{hhh}$, $c_{tthh}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{H}$, $c_{H◻}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross. The best-fit value corresponds to the very first entry in the table.

Likelihood contours at 68% (solid line) and 95% (dashed line) CL in the ($c_{H}$, $c_{H◻}$) parameter space, when all other Wilson coefficients are fixed to their SM values. The corresponding expected contours are shown by the inner and outer shaded regions. The SM prediction is indicated by the star, while the observed best-fit value is denoted by the black cross.

The acceptance times efficiency [in %] for the signal ggF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel. The other coupling modifiers affecting ggF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal ggF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ SLT channel. The other coupling modifiers affecting ggF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal ggF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ LTT channel. The other coupling modifiers affecting ggF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ SLT channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ LTT channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{2V}$ in each analysis category for the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{2V}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ SLT channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

The acceptance times efficiency [in %] for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{2V}$ in each analysis category for the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ LTT channel. The other coupling modifiers affecting VBF $HH$ production are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the fit to the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the fit to the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the fit to the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{\lambda}$ for the fit to the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the fit to the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the fit to the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the fit to the $\tau_{\mathrm{had}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Observed (solid line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$ for the fit to the $\tau_{\mathrm{lep}}\tau_{\mathrm{had}}$ channel, when all other coupling modifiers are fixed to their SM predictions.

Summary of the expected (blue) and observed (orange) one-dimensional constraints at 68% (solid error bars) and 95% (dashed error bars) CL on the coefficients of selected HEFT (top) and SMEFT (bottom) operators describing Higgs boson interactions affecting $HH$ production through gluon-gluon fusion. The results are obtained from one dimensional scans of the profile log-likelihood assuming that all other Wilson coefficients are fixed to their SM values. The contribution from VBF $HH$ production is neglected.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of $\kappa_{2V}$, when all other coupling modifiers are fixed to their SM predictions.

Likelihood observed contours at 68% CL (solid line) and 95% CL (dashed line) in the $\kappa_{\lambda}$, $\kappa_{2V}$ parameter space, when all other coupling modifiers are fixed to their SM predictions. The best-fit value, denoted by a cross in the plot, corresponds to the very first entry in the table.

Likelihood expected contours at 68% CL (teal shaded region) and 95% CL (yellow shaded region) in the $\kappa_{\lambda}$, $\kappa_{2V}$ parameter space, when all other coupling modifiers are fixed to their SM predictions.In the plot, the SM prediction is indicated by the star.

Likelihood observed contours at 68% CL (solid line) and 95% CL (dashed line) in the $c_{gghh}$ versus $c_{hhh}$ HEFT parameter space, with the remaining coefficient fixed to its SM value. The best-fit value, denoted by a cross in the plot, corresponds to the very first entry in the table.

Likelihood expected contours at 68% CL (teal shaded region) 95% CL (yellow shaded region) in the $c_{gghh}$ versus $c_{hhh}$ HEFT parameter space, with the remaining coefficient fixed to its SM value. In the plot, the SM prediction is indicated by the star.

Likelihood observed contours at 68% CL (solid line) and 95% CL (dashed line) in the $c_{tthh}$ versus $c_{hhh}$ HEFT parameter space, with the remaining coefficient fixed to its SM value. The best-fit value, denoted by a cross in the plot, corresponds to the very first entry in the table.

Likelihood exptected contours at 68% CL (teal shaded region) and 95% CL (yellow shaded region) in the $c_{tthh}$ versus $c_{hhh}$ HEFT parameter space, with the remaining coefficient fixed to its SM value. In the plot, the SM prediction is indicated by the star.

The observed (filled circles) and expected (hollow circles) 95% CL upper limits on the $HH$ ggF production cross-section in the Standard Model and for seven HEFT benchmark points. The teal and yellow colored bands indicate the $\pm 1\sigma$ and $\pm 2\sigma$ variations on the expected limit due to statistical and systematic uncertainties. The predicted cross-sections of each of the models under consideration are shown by the red crosses. The contribution from VBF production to the total $HH$ production cross-section is neglected.

Likelihood observed contours at 68% CL (solid line) and 95% CL (dashed line) in the $c_{{H}\boxed{}}$ versus $c_{H}$ SMEFT parameter space, with the remaining coefficient fixed to its SM value. The best-fit value, denoted by a cross in the plot, corresponds to the very first entry in the table.

Likelihood expected contours at 68% CL (teal shaded region) 95% CL (yellow shaded region) in the $c_{{H}\boxed{}}$ versus $c_{H}$ SMEFT parameter space, with the remaining coefficient fixed to its SM value. In the plot, the SM prediction is indicated by the star.

The acceptance times efficiency for the signal ggF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category. The other coupling modifiers affecting ggF $HH$ production are fixed to their SM predictions. The bands indicate the simulation statistical uncertainty.

The acceptance times efficiency for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ in each analysis category. When varying $\kappa_{\lambda}$, the other coupling modifiers affecting the VBF $HH$ production mode are set to their SM predictions. The bands indicate the simulation statistical uncertainty.

The acceptance times efficiency for the signal VBF $HH$ process as a function of the coupling modifier $\kappa_{2V}$ in each analysis category. When varying $\kappa_{2V}$, the other coupling modifiers affecting the VBF $HH$ production mode are set to their SM predictions. The bands indicate the simulation statistical uncertainty.

Expected yields of the signal $HH$ process as a function of the coupling modifier $\kappa_{\lambda}$ after applying the analysis preselection and in each analysis category after the final selection. The $HH$ yield is obtained considering both the $ggF$ and $VBF$ contributions. The other coupling modifiers affecting $HH$ production are fixed to their SM predictions. The bands indicate the simulation statistical uncertainty.

The acceptance times efficiency for the signal ggF $HH$ process in each analysis category as a function of the $c_{hhh}$ HEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The acceptance times efficiency for the signal ggF $HH$ process in each analysis category as a function of the $c_{tthh}$ HEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The acceptance times efficiency for the signal ggF $HH$ process in each analysis category as a function of the $c_{gghh}$ HEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The acceptance times efficiency for the signal ggF $HH$ process in each analysis category as a function of the $c_{H}$ SMEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The acceptance times efficiency for the signal ggF $HH$ process in each analysis category as a function of the $c_{{H}\boxed{}}$ SMEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The yield of the signal ggF $HH$ process in each analysis category as a function of the $c_{hhh}$ HEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The yield of the signal ggF $HH$ process in each analysis category as a function of the $c_{tthh}$ HEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The yield of the signal ggF $HH$ process in each analysis category as a function of the $c_{gghh}$ HEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The yield of the signal ggF $HH$ process in each analysis category as a function of the $c_{H}$ SMEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

The yield of the signal ggF $HH$ process in each analysis category as a function of the $c_{{H}\boxed{}}$ SMEFT coefficients. The dashed lines denote values that are excluded at 95% CL. The bottom panels show the efficiency of the sum of all analysis categories.

Distributions of the diphoton invariant mass for events in data (black dots with error bars) compared to the sum of the expected signal and backgrounds (histograms) in the low mass 1 region.

Distributions of the diphoton invariant mass for events in data (black dots with error bars) compared to the sum of the expected signal and backgrounds (histograms) in the low mass 2 region.

Distributions of the diphoton invariant mass for events in data (black dots with error bars) compared to the sum of the expected signal and backgrounds (histograms) in the low mass 3 region.

Distributions of the diphoton invariant mass for events in data (black dots with error bars) compared to the sum of the expected signal and backgrounds (histograms) in the low mass 4 region.

Distributions of the diphoton invariant mass for events in data (black dots with error bars) compared to the sum of the expected signal and backgrounds (histograms) in the high mass 1 region.

Distributions of the diphoton invariant mass for events in data (black dots with error bars) compared to the sum of the expected signal and backgrounds (histograms) in the high mass 2 region.

Distributions of the diphoton invariant mass for events in data (black dots with error bars) compared to the sum of the expected signal and backgrounds (histograms) in the high mass 3 region.

Observed (solid line) value of $-2\ln\Lambda$ as a function of the $c_{hhh}$ HEFT coefficients when all other coupling modifiers are fixed to the SM value

Expected (dashed line) value of $-2\ln\Lambda$ as a function of the $c_{hhh}$ HEFT coefficients when all other coupling modifiers are fixed to the SM value

Observed (solid line) value of $-2\ln\Lambda$ as a function of the $c_{gghh}$ HEFT coefficients when all other coupling modifiers are fixed to the SM value

Expected (dashed line) value of $-2\ln\Lambda$ as a function of the $c_{gghh}$ HEFT coefficients when all other coupling modifiers are fixed to the SM value

Observed (solid line) value of $-2\ln\Lambda$ as a function of the $c_{tthh}$ HEFT coefficients when all other coupling modifiers are fixed to the SM value

Expected (dashed line) value of $-2\ln\Lambda$ as a function of the $c_{tthh}$ HEFT coefficients when all other coupling modifiers are fixed to the SM value

Observed (solid line) value of $-2\ln\Lambda$ as a function of the $c_{H}$ SMEFT coefficients when all other coupling modifiers are fixed to the SM value.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of the $c_{H}$ SMEFT coefficients when all other coupling modifiers are fixed to the SM value.

Observed (solid line) value of $-2\ln\Lambda$ as a function of the $c_{{H}\boxed{}}$ SMEFT coefficients when all other coupling modifiers are fixed to the SM value.

Expected (dashed line) value of $-2\ln\Lambda$ as a function of the $c_{{H}\boxed{}}$ SMEFT coefficients when all other coupling modifiers are fixed to the SM value.

Summary of the expected (blue) and observed (orange) one-dimensional constraints at 68% (solid error bars) and 95% (dashed error bars) CL on the coefficients of selected HEFT (top) and SMEFT (bottom) operators describing Higgs boson interactions affecting $HH$ production through gluon--gluon fusion. The results are obtained from one dimensional scans of the profile log-likelihood assuming that all other Wilson coefficients are fixed to their SM values. The contribution from VBF $HH$ production is neglected.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy narrow-width pseudoscalar resonance that decays to a 260 GeV scalar resonance and a Z boson, which decays to neutrinos.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy narrow-width pseudoscalar resonance that decays to a 300 GeV scalar resonance and a Z boson, which decays to neutrinos.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy narrow-width pseudoscalar resonance that decays to a 400 GeV scalar resonance and a Z boson, which decays to neutrinos.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy large-width pseudoscalar resonance that decays to a 260 GeV scalar resonance and a Z boson, which decays to neutrinos.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy large-width pseudoscalar resonance that decays to a 300 GeV scalar resonance and a Z boson, which decays to neutrinos.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy large-width pseudoscalar resonance that decays to a 400 GeV scalar resonance and a Z boson, which decays to neutrinos.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy narrow-width pseudoscalar resonance that decays to a 260 GeV scalar resonance and a Z boson, which decays to charged leptons.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy narrow-width pseudoscalar resonance that decays to a 300 GeV scalar resonance and a Z boson, which decays to charged leptons.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy narrow-width pseudoscalar resonance that decays to a 400 GeV scalar resonance and a Z boson, which decays to charged leptons.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy large-width pseudoscalar resonance that decays to a 260 GeV scalar resonance and a Z boson, which decays to charged leptons.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy large-width pseudoscalar resonance that decays to a 300 GeV scalar resonance and a Z boson, which decays to charged leptons.

Acceptance times efficiency as a function of pseudoscalar resonant mass for each event selection step in the search for a neutral heavy large-width pseudoscalar resonance that decays to a 400 GeV scalar resonance and a Z boson, which decays to charged leptons.

Expected and observed 95% CL upper limits on the cross-section of resonant $H\to 4b$ production in association with a W boson. The $\pm 1 \sigma$ and $\pm 2 \sigma$ uncertainty ranges for the expected limits are shown.

Expected and observed 95% CL upper limits on the cross-section of resonant $H\to 4b$ production in association with a Z boson. The $\pm 1 \sigma$ and $\pm 2 \sigma$ uncertainty ranges for the expected limits are shown.

Expected 95% CL upper limits on the cross-section of a heavy narrow-width pseudoscalar resonance decaying to a Z boson and a heavy scalar resonance decaying to $H\to 4b$. The $\pm 1 \sigma$ and $\pm 2 \sigma$ uncertainty ranges for the expected limits are shown.

Observed 95% CL upper limits on the cross-section of a heavy narrow-width pseudoscalar resonance decaying to a Z boson and a heavy scalar resonance decaying to $H\to 4b$. The $\pm 1 \sigma$ and $\pm 2 \sigma$ uncertainty ranges for the expected limits are shown.

Expected 95% CL upper limits on the cross-section of a heavy large-width pseudoscalar resonance decaying to a Z boson and a heavy scalar resonance decaying to $H\to 4b$. The $\pm 1 \sigma$ and $\pm 2 \sigma$ uncertainty ranges for the expected limits are shown.

Observed 95% CL upper limits on the cross-section of a heavy large-width pseudoscalar resonance decaying to a Z boson and a heavy scalar resonance decaying to $H\to 4b$. The $\pm 1 \sigma$ and $\pm 2 \sigma$ uncertainty ranges for the expected limits are shown.

Data and post-fit signal and background from S+B fit for 315 GeV resonant $H\to 4b$ production in association with a W boson.

Data and post-fit signal and background from S+B fit for 400 GeV resonant $H\to 4b$ production in association with a W boson.

Data and post-fit signal and background from S+B fit for 550 GeV resonant $H\to 4b$ production in association with a Z boson.

Data and post-fit signal and background from S+B fit for 400 GeV resonant $H\to 4b$ production in association with a Z boson.

Data and post-fit signal and background from S+B fit for a 790 GeV narrow-width pseudoscalar resonance decaying to a Z boson and a 300 GeV scalar resonance decaying to $H\to 4b$.

Data and post-fit signal and background from S+B fit for a 420 GeV large-width pseudoscalar resonance decaying to a Z boson and a 320 GeV scalar resonance decaying to $H\to 4b$.

Data and post-fit signal and background from S+B fit for a 700 GeV large-width pseudoscalar resonance decaying to a Z boson and a 380 GeV scalar resonance decaying to $H\to 4b$.

Data and post-fit signal and background from S+B fit for SM VHH production, with each Higgs boson decaying to $2b$.

Corrected $m(HH)$ distribution in the resolved $4b$ signal region (dots), after the fit under the background-only hypothesis. The error bars on the $4b$ points represent the Poisson uncertainties corresponding to their event yields. The background model (teal histogram) is shown with its total post-fit uncertainty (gray band). The final bin includes overflow. Representative spin-0 signal hypotheses (dashed, dotted, and dashed-dotted lines) are overlaid, normalized to the overall expected limits on their cross-sections. The bottom panel shows the difference between the $4b$ distribution and the background model, relative to the background model. No significant excess of data relative to the SM background is observed.

Corrected $m(HH)$ distribution in the resolved $4b$ signal region (dots), after the fit under the background-only hypothesis. The error bars on the $4b$ points represent the Poisson uncertainties corresponding to their event yields. The background model (teal histogram) is shown with its total post-fit uncertainty (gray band). The final bin includes overflow. Representative spin-2 signal hypotheses (dashed, dotted, and dashed-dotted lines) are overlaid, normalized to the overall expected limits on their cross-sections. The bottom panel shows the difference between the $4b$ distribution and the background model, relative to the background model. No significant excess of data relative to the SM background is observed.

Cumulative signal acceptance times efficiency as a function of the resonance mass for various selection steps in the boosted channel. The steps up to the $b$-tag categorization are shown for the spin-0.

Cumulative signal acceptance times efficiency as a function of the resonance mass for various selection steps in the boosted channel. The steps up to the $b$-tag categorization are shown for the spin-2.

Cumulative signal acceptance times efficiency as a function of the resonance mass for various selection steps in the boosted channel. The efficiencies of the three b-tag categories are shown for the spin-0 scenario; this efficiency is obtained after the other selection steps including the SR definition. The signal efficiency in the 4b region has a maximum around 1.5 TeV. Above that value the track-jets start to merge together, and for the highest resonance masses the 2b category becomes the most efficient.

Cumulative signal acceptance times efficiency as a function of the resonance mass for various selection steps in the boosted channel. The efficiencies of the three b-tag categories are shown for the spin-2 scenario; this efficiency is obtained after the other selection steps including the SR definition. The signal efficiency in the 4b region has a maximum around 1.5 TeV. Above that value the track-jets start to merge together, and for the highest resonance masses the 2b category becomes the most efficient.

Comparison of the background model (stacked histograms) with data (dots) in the $2b$ validation region. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background uncertainty (gray band) is computed by adding all individual components in quadrature and is not allowed to extend below zero.

Comparison of the background model (stacked histograms) with data (dots) in the $3b$ validation region. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background uncertainty (gray band) is computed by adding all individual components in quadrature and is not allowed to extend below zero.

Comparison of the background model (stacked histograms) with data (dots) in the $4b$ validation region. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background uncertainty (gray band) is computed by adding all individual components in quadrature and is not allowed to extend below zero.

The $m(HH)$ distributions in the boosted $2b$ signal regions (dots), after the fit under the background-only hypothesis. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background model (stacked histogram) is shown with its total post-fit uncertainty (gray band). The uncertainty bands are defined using an ensemble of curves constructed by sampling a multivariate Gaussian probability density function built from the covariance matrix of the fit. Representative spin-0 signal hypotheses (dashed and dashed-dotted lines) are overlaid, normalized to the overall expected limits on their cross-sections. No significant excess of data relative to the SM background is observed.

The $m(HH)$ distributions in the boosted $2b$ signal regions (dots), after the fit under the background-only hypothesis. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background model (stacked histogram) is shown with its total post-fit uncertainty (gray band). The uncertainty bands are defined using an ensemble of curves constructed by sampling a multivariate Gaussian probability density function built from the covariance matrix of the fit. Representative spin-2 signal hypotheses (dashed and dashed-dotted lines) are overlaid, normalized to the overall expected limits on their cross-sections. No significant excess of data relative to the SM background is observed.

The $m(HH)$ distributions in the boosted $3b$ signal regions (dots), after the fit under the background-only hypothesis. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background model (stacked histogram) is shown with its total post-fit uncertainty (gray band). The uncertainty bands are defined using an ensemble of curves constructed by sampling a multivariate Gaussian probability density function built from the covariance matrix of the fit. Representative spin-0 signal hypotheses (dashed and dashed-dotted lines) are overlaid, normalized to the overall expected limits on their cross-sections. No significant excess of data relative to the SM background is observed.

The $m(HH)$ distributions in the boosted $3b$ signal regions (dots), after the fit under the background-only hypothesis. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background model (stacked histogram) is shown with its total post-fit uncertainty (gray band). The uncertainty bands are defined using an ensemble of curves constructed by sampling a multivariate Gaussian probability density function built from the covariance matrix of the fit. Representative spin-2 signal hypotheses (dashed and dashed-dotted lines) are overlaid, normalized to the overall expected limits on their cross-sections. No significant excess of data relative to the SM background is observed.

The $m(HH)$ distributions in the boosted $4b$ signal regions (dots), after the fit under the background-only hypothesis. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background model (stacked histogram) is shown with its total post-fit uncertainty (gray band). The uncertainty bands are defined using an ensemble of curves constructed by sampling a multivariate Gaussian probability density function built from the covariance matrix of the fit. Representative spin-0 signal hypotheses (dashed and dashed-dotted lines) are overlaid, normalized to the overall expected limits on their cross-sections. No significant excess of data relative to the SM background is observed.

The $m(HH)$ distributions in the boosted $4b$ signal regions (dots), after the fit under the background-only hypothesis. The error bars on the data points represent the Poisson uncertainties corresponding to their event yields. The background model (stacked histogram) is shown with its total post-fit uncertainty (gray band). The uncertainty bands are defined using an ensemble of curves constructed by sampling a multivariate Gaussian probability density function built from the covariance matrix of the fit. Representative spin-2 signal hypotheses (dashed and dashed-dotted lines) are overlaid, normalized to the overall expected limits on their cross-sections. No significant excess of data relative to the SM background is observed.

Expected (dashed black lines) and observed (solid black lines) 95% CL upper limits on the cross-section of resonant $HH$ production in the spin-0 signal models. The $\pm 1 \sigma$ and $\pm 2 \sigma$ uncertainty ranges for the expected limits (colored bands) are shown. Expected limits using each of the resolved and boosted channels individually (dashed colored lines) are shown. The nominal $H\rightarrow b\bar{b}$ branching ratio is taken as 0.582.

Expected (dashed black lines) and observed (solid black lines) 95% CL upper limits on the cross-section of resonant $HH$ production in the spin-2 signal models. The $\pm 1 \sigma$ and $\pm 2 \sigma$ uncertainty ranges for the expected limits (colored bands) are shown. Expected limits using each of the resolved and boosted channels individually (dashed colored lines) are shown. The theoretical prediction for the bulk RS model with $k/\bar{M}_{\text{Pl}} = 1$ (solid red line) is shown; the decrease below 350 GeV is due to a sharp reduction in the $G^{*}_{\text{KK}} \rightarrow HH$ branching ratio. The nominal $H\rightarrow b\bar{b}$ branching ratio is taken as 0.582.

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.