Suppl. Fig. 7. The reconstructed shower angle for all single photon events. The individual signal and background event type categories added together form the unconstrained prediction.

Suppl. Fig. 7. The constrained covariance matrix for the reconstructed shower angle for all single-photon events. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Suppl. Fig. 7. The unconstrained covariance matrix for reconstructed shower angle for all single-photon events. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Fig. 3. The reconstructed number of protons, with a proton KE energy threshold of 35 MeV, for all single-photon events. The individual signal and background event type categories added together form the unconstrained prediction.

Fig. 4, Suppl. Fig. 8. The reconstructed shower energy for zero proton events binned from 0 to 600 MeV. The individual signal and background event type categories added together form the unconstrained prediction.

Fig. 4. The constrained covariance matrix for the reconstructed shower energy for zero proton events binned from 0 to 600 MeV. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Fig. 4. The unconstrained covariance matrix for reconstructed shower energy for zero proton events binned from 0 to 600 MeV. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Fig. 6. The reconstructed shower energy for zero proton events binned from 150 to 1250 MeV, with LEE prediction. The individual signal and background event type categories added together form the unconstrained prediction. The "LEE" category is the unfolded MiniBooNE LEE model prediction scaled under the neutrino-target nucleon interaction assumption.

Fig. 6. The constrained covariance matrix for the reconstructed shower energy for zero proton events binned from 150 to 1250 MeV. Both with and without the LEE model added. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Fig. 6. The unconstrained covariance matrix for reconstructed shower energy for zero proton events binned from 150 to 1250 MeV. Both with and without the LEE model added. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Suppl. Fig. 10. The reconstructed shower backwards projected distance for zero proton events. The individual signal and background event type categories added together form the unconstrained prediction.

Suppl. Fig. 9. The reconstructed shower angle for zero proton events. Includes LEE model prediction. The individual signal and background event type categories added together form the unconstrained prediction. The "LEE" category is the unfolded MiniBooNE LEE model prediction scaled under the neutrino-target nucleon interaction assumption.

Suppl. Fig. 9. The constrained covariance matrix for the reconstructed shower angle for zero proton events. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

Suppl. Fig. 9. The unconstrained covariance matrix for reconstructed shower angle for zero proton events. The matrix shows uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Data statistical uncertainties are not included. An example of how to add Pearson data statistical uncertainties can be found in the example code repository.

1$\gamma$Xp shower energy and angle selection efficiencies for true 1$\gamma$Xp events. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied. Energy and Angle values represent bin lower edges.

1$\gamma$0p shower energy and angle selection efficiencies for true single-photon events with no other true particles of any energy. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied. Energy and Angle values represent bin lower edges.

Covariance matrix showing uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Pearson data statistical uncertainties have been included, and include small correlations due to events which can be selected by both WC and Pandora. The first four bins are the WC $1\gamma Np$, WC $1\gamma 0p$, Pandora $1\gamma 1p$, and Pandora $1\gamma 0p$ channels, and the next four sets of 16 bins are the NC $\pi^0 Np$, NC $\pi^0 0p$, $\nu_\mu$CC $Np$, and $\nu_\mu$CC $0p$ channels. This corresponds to Fig. 1 of the paper and Fig. 6 of the Supplemental Material.

Constrained WC $1\gamma Np$ energy distribution. There are 6 bins from 0 to 600 MeV of reconstructed shower energy, with an additional overflow bin. Constrained systematic uncertainties on the predictions are illustrated, and a more detailed covariance matrix is included in the Constrained Energy Distributions Covariance Matrix tab. This corresponds to Fig. 2 of the paper.

Constrained WC $1\gamma 0p$ energy distribution. There are 14 bins from 0 to 700 MeV of reconstructed shower energy, with an additional overflow bin. Constrained systematic uncertainties on the predictions are illustrated, and a more detailed covariance matrix is included in the Constrained Energy Distributions Covariance Matrix tab. This corresponds to Fig. 2 of the paper.

Constrained Pandora $1\gamma 1p$ energy distribution. There are 6 bins from 0 to 600 MeV of reconstructed shower energy, with an additional overflow bin. Constrained systematic uncertainties on the predictions are illustrated, and a more detailed covariance matrix is included in the Constrained Energy Distributions Covariance Matrix tab. This corresponds to Fig. 2 of the paper.

Constrained Pandora $1\gamma 0p$ energy distribution. There are 14 bins from 0 to 700 MeV of reconstructed shower energy, with an additional overflow bin. Constrained systematic uncertainties on the predictions are illustrated, and a more detailed covariance matrix is included in the Constrained Energy Distributions Covariance Matrix tab. This corresponds to Fig. 2 of the paper.

Constrained covariance matrix showing constrained uncertainties and correlations between bins due to flux uncertainties, cross-section uncertainties, hadron reinteraction uncertainties, detector systematic uncertainties, Monte-Carlo statistical uncertainties, and dirt (outside cryostat) uncertainties. Pearson data statistical uncertainties have been included, and include small correlations due to events which can be selected by both WC and Pandora. The first seven bins are the WC $1\gamma Np$ reconstructed shower energy from 0-600 MeV with an overflow bin, the next 15 bins are the WC $1\gamma 0p$ reconstructed shower energy from 0-700 MeV with an overflow bin, the next seven bins are the Pandora $1\gamma 1p$ reconstructed shower energy from 0-600 MeV with an overflow bin, and the last 15 bins are the Pandora $1\gamma 0p$ reconstructed shower energy from 0-700 MeV with an overflow bin. This corresponds to Fig. 2 of the paper.

2D efficiency matrix showing the efficiency of the WC $1\gamma Np$ selection with respect to all true NC Delta Radiative Decay events which interact in the fiducial volume. The efficiency is shown as a function of true photon energy and angle. There are 10 bins from -1 to 1 in true photon $\cos \theta$ and 15 bins from 0 to 1500 MeV in true photon energy. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied. This corresponds to Fig. 9 of the Supplemental Material.

2D efficiency matrix showing the efficiency of the WC $1\gamma 0p$ selection with respect to all true NC Delta Radiative Decay events which interact in the fiducial volume. The efficiency is shown as a function of true photon energy and angle. There are 10 bins from -1 to 1 in true photon $\cos \theta$ and 30 bins from 0 to 1500 MeV in true photon energy. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied. This corresponds to Fig. 9 of the Supplemental Material.

2D efficiency matrix showing the efficiency of the Pandora $1\gamma 1p$ selection with respect to all true NC Delta Radiative Decay events which interact in the fiducial volume. The efficiency is shown as a function of true photon energy and angle. There are 10 bins from -1 to 1 in true photon $\cos \theta$ and 15 bins from 0 to 1500 MeV in true photon energy, with lower bin edges indicated in the table. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied. This corresponds to Fig. 9 of the Supplemental Material.

2D efficiency matrix showing the efficiency of the Pandora $1\gamma 0p$ selection with respect to all true NC Delta Radiative Decay events which interact in the fiducial volume. The efficiency is shown as a function of true photon energy and angle. There are 10 bins from -1 to 1 in true photon $\cos \theta$ and 30 bins from 0 to 1500 MeV in true photon energy, with lower bin edges indicated in the table. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied. This corresponds to Fig. 9 of the Supplemental Material.

2D efficiency matrix showing the efficiency of the WC $1\gamma 0p$ selection with respect to all true zero-primary-final-state-proton NC Delta Radiative Decay events which interact in the fiducial volume. The efficiency is shown as a function of true photon energy and angle. There are 10 bins from -1 to 1 in true photon $\cos \theta$ and 30 bins from 0 to 1500 MeV in true photon energy, with lower bin edges indicated in the table. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied.

2D efficiency matrix showing the efficiency of the Pandora $1\gamma 0p$ selection with respect to all true zero-primary-final-state-proton NC Delta Radiative Decay events which interact in the fiducial volume. The efficiency is shown as a function of true photon energy and angle. There are 10 bins from -1 to 1 in true photon $\cos \theta$ and 30 bins from 0 to 1500 MeV in true photon energy, with lower bin edges indicated in the table. NaN values indicate a "divide by zero" case where zero events were present in the bin before any selection was applied.

$a_2(1320)$ differential cross section in the m=-2 projection split into different reflectivity components, $\frac{d\sigma^+_{-2}}{dt}$ and $\frac{d\sigma^-_{-2}}{dt}$, in bins of four-momentum transfer. The first uncertainty is statistical, the second systematic.

$a_2(1320)$ differential cross section in the m=0 projection split into different reflectivity components, $\frac{d\sigma^+_{0}}{dt}$ and $\frac{d\sigma^-_{0}}{dt}$, in bins of four-momentum transfer. The first uncertainty is statistical, the second systematic.

$a_2(1320)$ differential cross section in the m=+1 projection split into different reflectivity components, $\frac{d\sigma^+_{+1}}{dt}$ and $\frac{d\sigma^-_{+1}}{dt}$, in bins of four-momentum transfer. The first uncertainty is statistical, the second systematic.

$a_2(1320)$ differential cross section in the m=+2 projection split into different reflectivity components, $\frac{d\sigma^+_{+2}}{dt}$ and $\frac{d\sigma^-_{+2}}{dt}$, in bins of four-momentum transfer. The first uncertainty is statistical, the second systematic.

Observed and expected 95 % CL exclusion limits on $\tan\beta$ as a function of $m_{H^{\pm}}$, shown in the context of the mh125 scenario, for $m_{H^{\pm}}>150$ GeV and $(1 \leq \tan\beta \leq 60)$. The surrounding shaded bands correspond to the 1$\sigma$ and 2$\sigma$ confidence intervals around the expected limit.

Observed and expected 95 % CL exclusions on $\tan\beta$ as a function of $m_{H^{\pm}}$ derived from exclusion limits on $\mathrm{\cal{B}}(t\to bH^+)\times \mathrm{\cal{B}}(H^+ \to \tau \nu)$, for Type I 2HDM low mass scenario. The surrounding shaded bands correspond to the 1$\sigma$ and 2$\sigma$ confidence intervals around the expected limit.

Observed and expected 95 % CL exclusions on $\tan\beta$ as a function of $m_{H^{\pm}}$ derived from exclusion limits on $\mathrm{\cal{B}}(t\to bH^+)\times \mathrm{\cal{B}}(H^+ \to \tau \nu)$, for Lepton Specific 2HDM low mass scenario. The surrounding shaded bands correspond to the 1$\sigma$ and 2$\sigma$ confidence intervals around the expected limit.

The distribution of the $p_\mathrm{T}{}_{\mu\mathrm{-had}}^\mathrm{col}$ in the VR. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top-quark, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic sources.

The distributions of $m_{\mu\mathrm{-had}}^\mathrm{col}$ corresponding to the SR selection criteria defined in the text. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top-quark, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic sources.

The distributions of $m_{\mu\mathrm{-had}}^\mathrm{col}$ corresponding to the SR_\text{std} selection criteria defined in the text. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top-quark, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic sources.

The distributions of $m_{\mu\mathrm{-had}}^\mathrm{col}$ corresponding to the SR_\text{tight} selection criteria defined in the text. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top-quark, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic sources.

The distributions of $m_{\mu\mathrm{-had}}^\mathrm{col}$ corresponding to the SR_\text{tight}^\text{std} selection criteria defined in the text. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top-quark, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic sources.

The distribution of $\Delta\phi^\mathrm{signed}_{\mu-MET}$ with all other event selection criteria applied in the SR; the straight lines indicate the positions of the selection requirements. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of $\Delta\phi^\mathrm{signed}_{\mu-MET}$ with all other event selection criteria applied in the VR; the straight lines indicate the positions of the selection requirements. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $m_{\mu\mathrm{-had}}^\mathrm{col}$ in the SR. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $m_{\mu\mathrm{-had}}^\mathrm{col}$ in the VR. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $p_\mathrm{T}$ of the $\tau_\mathrm{had}^{\mu\mkern-10mu\backslash}$ in the SR. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $p_\mathrm{T}$ of the $\tau_\mathrm{had}^{\mu\mkern-10mu\backslash}$ in the VR. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $p_\mathrm{T}$ of the muon removed from the $\tau_\mathrm{had}^{\mu\mkern-10mu\backslash}$ in the SR `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $p_\mathrm{T}$ of the muon removed from the $\tau_\mathrm{had}^{\mu\mkern-10mu\backslash}$ in the VR `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $p_\mathrm{T}$ of the leading jet in the SR. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $p_\mathrm{T}$ of the leading jet in the VR. `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $\Delta{R}_{\tau\tau}^\mathrm{reco}$ between the muon and the $\tau_\mathrm{had}^{\mu\mkern-10mu\backslash}$ in the SR `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the $\Delta{R}_{\tau\tau}^\mathrm{reco}$ between the muon and the $\tau_\mathrm{had}^{\mu\mkern-10mu\backslash}$ in the VR `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the reconstructed $E_\mathrm{T}^\mathrm{miss}$ in the SR `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

The distribution of the reconstructed $E_\mathrm{T}^\mathrm{miss}$ in the VR `$Z(\rightarrow\tau\tau)+\text{jets}$' represents the contributions from the signal process. `Diboson' indicates the contributions from $WW$, $WZ$, and $ZZ$ processes. `Top' represents the predicted contributions from the $t\bar{t}$, single-top, and $tW$ processes. `Other' includes the contributions from the $Z(\rightarrow\ell\ell)$+jets, $W$+jets, and Higgs boson processes. The uncertainties shown include both statistical and systematic errors.

Differential cross sections for exclusive dielectron production, in the fiducial phase space of Table 1, as functions of the pair invariant mass. Data are compared with SUPERCHIC + FSR(PHOTOS++), STARLIGHT + FSR(PY8), and gamma-UPC + FSR(PY8) predictions.

Differential cross sections for exclusive dielectron production, in the fiducial phase space of Table 1, as functions of the pair |cos $\theta$*|. Data are compared with SUPERCHIC + FSR(PHOTOS++), STARLIGHT + FSR(PY8), and gamma-UPC + FSR(PY8) predictions.

Differential exclusive diphoton cross sections in the fiducial phase space of Table 1 as a function of the diphoton rapidity measured in data compared with SUPERCHIC and gamma-UPC@NLO predictions.

Differential exclusive diphoton cross sections in the fiducial phase space of Table 1 as a function of the diphoton invariant mass measured in data compared with SUPERCHIC and gamma-UPC@NLO predictions.

Observed and expected 95% CL limits on the production cross section $\sigma(\gamma\gamma \rightarrow a \rightarrow \gamma\gamma)$ as a function of the ALP mass $m_{a}$. The uncertainties indicate regions containing 68 and 95% of the distribution of limits expected under the background-only hypothesis. The additional datasets indicate the expected ALP cross sections as a function of $m_{a}$ for decreasing photon couplings ($g_{a\gamma} = 0.3, 0.1, 0.05 TeV^{-1}$).

Exclusion limits at 95% CL in the axion-photon coupling $g_{a\gamma}$ versus axion mass $m_a$ plane, for the operator $\frac{1}{4\Lambda}aF\widetilde{F}$ (assuming ALPs coupled only to photons) extracted in this analysis.

The observed and expected ($\pm2\sigma$) upper limits at 95% CL on the branching ratio for $H\rightarrow aa\rightarrow \gamma\gamma\tau\tau$ as a function of the resonance mass hypothesis $m_{a}$.

The polynomial parameterisation of the total signal selection efficiency as a function of $m_{a}$, defined as the fraction of the reconstructed and selected events out of the total number of events in the signal simulated samples.

Values of the test statistic $t_{\kappa_{g,\mathrm{off-shell}}}$ as a function of $\kappa_{g,\mathrm{off-shell}}$ while profiling $\kappa_{V,\mathrm{off-shell}}$ obtained with an Asimov dataset and with data. The values with all nuisance parameters fixed at their best-fit values (stat-only) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{\kappa_{V,\mathrm{off-shell}}}$ as a function of $\kappa_{V,\mathrm{off-shell}}$ while profiling $\kappa_{g,\mathrm{off-shell}}$ obtained with an Asimov dataset and with data. The values with all nuisance parameters fixed at their best-fit values (stat-only) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{\kappa_{H}}$ as a function of $\kappa_{H}=\Gamma_H/\Gamma_H^{\mathrm{SM}}$ obtained with an Asimov dataset and with data. The values with all nuisance parameters fixed at their best-fit values (stat-only) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{R_{gg}}$ as a function of $R_{gg}=\kappa_{g,\mathrm{on-shell}}^2/\kappa_{g,\mathrm{off-shell}}^2$ while profiling $R_{VV}=\kappa_{V,\mathrm{on-shell}}^2/\kappa_{V,\mathrm{off-shell}}^2$ and fixing $\kappa_H=1$ obtained with an Asimov dataset and with data. The values with all nuisance parameters fixed at their best-fit values (stat-only) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{R_{VV}}$ as a function of $R_{VV}=\kappa_{V,\mathrm{on-shell}}^2/\kappa_{V,\mathrm{off-shell}}^2$ while profiling $R_{gg}=\kappa_{g,\mathrm{on-shell}}^2/\kappa_{g,\mathrm{off-shell}}^2$ and fixing $\kappa_H=1$ obtained with an Asimov dataset and with data. The values with all nuisance parameters fixed at their best-fit values (stat-only) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{\mu_{\mathrm{off-shell}}}$ assuming a single parameter of interest $\mu_{\mathrm{off-shell}}$ obtained with an Asimov dataset and with data when combining the $H^*\rightarrow ZZ\rightarrow 4\ell$ and $H^*\rightarrow ZZ\rightarrow 2\ell 2\nu$ decay channels. The values from the histogram-based analysis (Phys. Lett. B 846 (2023) 138223) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{\kappa_{H}}$ as a function of $\kappa_{H}=\Gamma_H/\Gamma_H^{\mathrm{SM}}$ obtained with an Asimov dataset and with data in the $H^*\rightarrow ZZ\rightarrow 4\ell$ decay channel. The values from the histogram-based analysis (Phys. Lett. B 846 (2023) 138223) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{\kappa_{H}}$ as a function of $\kappa_{H}=\Gamma_H/\Gamma_H^{\mathrm{SM}}$ obtained with an Asimov dataset and with data when combining the $H^*\rightarrow ZZ\rightarrow 4\ell$ and $H^*\rightarrow ZZ\rightarrow 2\ell 2\nu$ decay channels. The values from the histogram-based analysis (Phys. Lett. B 846 (2023) 138223) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{R_{gg}}$ as a function of $R_{gg}$ while profiling $R_{VV}$ and fixing $\kappa_H=1$ obtained with an Asimov dataset and with data. The values from the histogram-based analysis (Phys. Lett. B 846 (2023) 138223) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Values of the test statistic $t_{R_{VV}}$ as a function of $R_{VV}$ while profiling $R_{gg}$ and fixing $\kappa_H=1$ obtained with an Asimov dataset and with data. The values from the histogram-based analysis (Phys. Lett. B 846 (2023) 138223) are added for comparison. The 68% and 95% confidence intervals obtained from the Neyman construction are also added.

Invariant mass of the lepton pair plus jet system, for data, background (pre- and post-reweighting) and three $H\rightarrow Za$ signal hypotheses (for $a\rightarrow q\bar{q}/gg$ inclusively). Events are required to pass the complete event selection but not the classification NN requirement. The background normalization is set equal to that of the data for events passing the preselection and being in the $m_{\ell\ell j}$ 100-180 GeV region. The signal normalization assumes the SM Higgs boson inclusive production cross-section, $\mathcal{B}(H\to Za)=100\%$, and it is scaled up by a factor of 100. The error bars (hatched regions) represent the data (MC) sample's statistical uncertainty in the histograms and the ratio plots. Vertical arrows indicate data points that fall outside the displayed $y$-axis range.

Regression NN output variable, for data, reweighted background, and three $H\rightarrow Za$ signal hypotheses ($a\rightarrow q\bar{q}/gg$ inclusively). Events are required to pass the complete event selection, including a $120<m_{\ell\ell j}<140$ GeV requirement, but not the classification NN output variable requirement. The background normalization is set equal to that of the data for events passing the preselection and being in the $m_{\ell\ell j}$ 100-180 GeV region. The signal normalization assumes the SM Higgs boson inclusive production cross-section, $\mathcal{B}(H\to Za)=100\%$, and it is scaled up by a factor of 100. The error bars (hatched regions) represent the data (MC) sample's statistical uncertainty in the histograms and the ratio plots. Vertical arrows indicate data points that fall outside the displayed $y$-axis range.

Classification NN output variable, for data, reweighted background, and three $H\rightarrow Za$ signal hypotheses ($a\rightarrow q\bar{q}/gg$ inclusively). Events are required to pass the complete event selection, including a $120<m_{\ell\ell j}<140$ GeV requirement, but not the classification NN output variable requirement. The background normalization is set equal to that of the data for events passing the preselection and being in the $m_{\ell\ell j}$ 100-180 GeV region. The signal normalization assumes the SM Higgs boson inclusive production cross-section, $\mathcal{B}(H\to Za)=100\%$, and it is scaled up by a factor of 100. The error bars (hatched regions) represent the data (MC) sample's statistical uncertainty in the histograms and the ratio plots. Vertical arrows indicate data points that fall outside the displayed $y$-axis range.

Invariant mass of the lepton pair plus jet system for data, reweighted background, and various signal hypotheses. Events are required to pass the complete event selection (preselection and classification NN requirement). The pre-fit background distribution is shown along with three $H\rightarrow Za$ signal hypotheses coming from the pure MC sample ($a\rightarrow q\bar{q}/gg$ inclusively). The background normalization is set equal to that of the data for events passing the preselection and being in the region 100-180 GeV. The error bars represent the data sample's statistical uncertainty and the grey bands represent the full background model uncertainty before the fit (assuming all the uncertainties are uncorrelated). The lower panel shows the ratio of the data to the pre-fit background prediction.

Invariant mass of the lepton pair plus jet system for data, reweighted background, and 2.0 GeV signal hypotheses. Events are required to pass the complete event selection (preselection and classification NN requirement). The background shape and normalization come from the fit to the data. The signal includes only $gg$ decays, its shape comes from the fit, and it is normalized to $\mathcal{B}(H\rightarrow Za)=100\%$. The error bars represent the data sample's statistical uncertainty and the grey bands represent the full background model uncertainty after the fit. The lower panel shows the per-bin signal significance, $(N_\text{data}-N_\text{MC})\,\text{\Large{/}}\sqrt{\sigma_\text{data}^2+\sigma_\text{MC}^2}$. The post-fit background uncertainty is calculated from a fit under the background-only hypothesis. Vertical arrows indicate data points that fall outside the displayed $y$-axis range.

Observed 95$\%$ CL upper limits on $\mathcal{B}(H \to Za)=100\%$ for $\mathcal{B}(a \to gg)=100\%$, and the expectation under the background-only hypothesis together with its $\pm1\sigma$ and $\pm2\sigma$ intervals. The weaker limits from the previous version of the analysis (PRL 125(2020) 221802) are also shown. Linear interpolation is used to set limits between the mass points for which MC signal samples were generated.

Observed 95$\%$ confidence level upper limits on $\mathcal{B}(H \to Za)=100\%$ for $\mathcal{B}(a\rightarrow q\bar{q})=100\%$, and the expectation under the background-only hypothesis together with its $\pm1\sigma$ and $\pm2\sigma$ intervals. The weaker limits from the previous version of the analysis (PRL 125(2020) 221802) are also shown. Linear interpolation is used to set limits between the mass points for which MC signal samples were generated.

Invariant mass distribution of final state particles in SR1 HF category ($\text{H}\to\text{J}/\psi\gamma$ signal)

Invariant mass distribution of final state particles in SR1 Z-LP category ($\text{Z}\to\text{J}/\psi\gamma$ signal)

Invariant mass distribution of final state particles in SR1 Z-HP category ($\text{Z}\to\text{J}/\psi\gamma$ signal)

Invariant mass distribution of final state particles in SR2 category ($\text{H}\to\psi(\text{2S})\gamma$ signal)

Invariant mass distribution of final state particles in SR2 category ($\text{H}\to\psi(\text{2S})\gamma$ signal)

Invariant mass distribution of final state particles in control region category ($\text{H}\to\mu\mu\gamma$ background)

Observed and expected (with $\pm 1$, $\pm 2$ standard deviation bands) exclusion limits at 95% C.L. on the branching fraction $\mathcal{B}$ of the $(\text{H,Z}) \to \psi(\text{nS})\gamma$ decays

Observed (expected) upper limits at 95% C.L. on the normalized values with respect to the SM expectation, denoted as the signal strength parameter $\mu$, of the product of the cross section $\sigma$ and the branching fraction $\mathcal{B}$ of the $(\text{H,Z}) \to \psi(\text{nS})\gamma$ decays, and the branching fraction, assuming a SM Z and H boson cross section.