Distribution of the NN output in the $\geq$1fj$\,$SR in data and the expected contribution of the signal and background processes after the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions considering the correlations of the uncertainties as obtained by the fit.

Distribution of the NN output in the $t\bar{t}\gamma\,$CR in data and the expected contribution of the signal and background processes after the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions considering the correlations of the uncertainties as obtained by the fit.

Total event yield in the $W\gamma\,$CR in data and the expected contribution of the signal and background processes after the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions considering the correlations of the uncertainties as obtained by the fit.

Distribution of the scalar sum of the jet transverse momenta in the 0fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions. The first and last bins include the underflow and overflow, respectively.

Distribution of the $\eta$ of the $b$-tagged jet in the 0fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions.

Distribution of the reconstructed top-quark mass in the 0fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions. The first and last bins include the underflow and overflow, respectively.

Distribution of the $p_{\mathrm{T}}$ of the top-quark+photon system in the 0fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions and the last bin includes the overflow.

Distribution of the photon $p_{\mathrm{T}}$ in the 0fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions and the last bin includes the overflow.

Distribution of the photon $\eta$ in the 0fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions.

Distribution of the scalar sum of the jet transverse momenta in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions and the last bin includes the overflow.

Distribution of the invariant mass of the $b$-tagged jet and the highest-$p_{\mathrm{T}}$ forward jet in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions and the last bin includes the overflow.

Distribution of $p_{\mathrm{T}}$ of the highest-$p_{\mathrm{T}}$ forward jet in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions and the last bin includes the overflow.

Distribution of the difference in $\eta$ between the highest-$p_{\mathrm{T}}$ forward jet and the photon in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions and the last bin includes the overflow.

Distribution of the energy of the system formed by the highest-$p_{\mathrm{T}}$ forward jet and the photon in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions. The first and last bins include the underflow and overflow, respectively.

Distribution of the $\eta$ of the $b$-tagged jet in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions.

Distribution of the reconstructed top-quark mass in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions. The first and last bins include the underflow and overflow, respectively.

Distribution of the $p_{\mathrm{T}}$ of the top-quark and photon system in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions and the last bin includes the overflow.

Distribution of the photon $p_{\mathrm{T}}$ in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions and the last bin includes the overflow.

Distribution of the photon $\eta$ in the $\geq$1fj$\,$SR in data and for the sum of all processes expectations before the profile-likelihood fit. The "Total" column corresponds to the sum of the expected contributions from the signal and background processes. The uncertainty represents the sum of statistical and systematic uncertainties in the signal and background predictions.

Ordered list of the 30 systematic uncertainties with the largest impact on the measured signal normalisation in the fit to data in the parton-level measurement considered as nuisance parameters (NPs) in the profile-likelihood fit. The column "NP value, error" corresponds to the nominal best-fit values and the corresponding uncertainties. The impact of each NP, $\Delta\sigma$/$\sigma_{\mathrm{pred}}$, is computed by comparing the nominal best-fit value of the POI ($\sigma$/$\sigma_{\mathrm{pred}}$) with the result of the fits when fixing the considered NP to its best-fit value shifted by its pre-fit and post-fit uncertainties. The corresponding impacts are listed in the "POI impact prefit high/low" and "POI impact high/low" columns, respectively. The "MC stat." NPs represent the MC statistical uncertainty and they enter the likelihood with a Poisson term, while all the other NPs enter the likelihood via a Gaussian term.

Ordered list of the 30 systematic uncertainties with the largest impact on the measured signal normalisation in the fit to data in the particle-level measurement considered as nuisance parameters (NPs) in the profile-likelihood fit. The column "NP value, error" corresponds to the nominal best-fit values and the corresponding uncertainties. The impact of each NP, $\Delta\sigma$/$\sigma_{\mathrm{pred}}$, is computed by comparing the nominal best-fit value of the POI ($\sigma$/$\sigma_{\mathrm{pred}}$) with the result of the fits when fixing the considered NP to its best-fit value shifted by its pre-fit and post-fit uncertainties. The corresponding impacts are listed in the "POI impact prefit high/low" and "POI impact high/low" columns, respectively. The "MC stat." NPs represent the MC statistical uncertainty and they enter the likelihood with a Poisson term, while all the other NPs enter the likelihood via a Gaussian term.

This table lists the kinematic requirements on parton-level objects used to define of the fiducial phase space for the parton-level measurement. Frixione isolation ($\href{https://arxiv.org/abs/hep-ph/9801442}{\text{hep-ph/9801442}}$) with a chosen radius of $\Delta R = 0.2$ is applied to photons ($\gamma$). The measured fiducial parton-level cross section is $\sigma_{tq\gamma}\times\mathcal{B}(t\rightarrow l\nu b) = 688\pm 23(\text{stat.})^{+75}_{-71}(\text{syst.})\,$fb.

This table lists the kinematic requirements on particle-level objects used to define of the fiducial phase space for the particle-level measurement. The particle level objects are photons ($\gamma$) not from a hadron decay, neutrinos not from a hadron decay ($\nu$), prompt electrons and muons ($\ell$) "dressed" by adding close-by ($\Delta R < 0.1$) photons, and anti-$k_t$ $R = 0.4$ jets built from stable particles ($\tau > 30\,$ps) and tau leptons excluding neutrinos and prompt dressed muons. Jets are $b$-tagged ($b$-jet) using ghost-matched $b$-hadrons with $p_{\text{T}} > 5\,$GeV. Apart from the kinematic requirements, isolation and overlap removal criteria are applied. Jets within $\Delta R = 0.4$ of a photon are removed if the $p_{\text{T}}$ of charged particles within $\Delta R = 0.3$ of the photon is smaller than $10\,\%$ of its $p_{\text{T}}$. Jets within $\Delta R = 0.4$ of a lepton are removed. Events are removed where a photon is close ($\Delta R < 0.4$) to a lepton or a surviving jet. The measured fiducial particle-level cross section is $\sigma_{tq\gamma}\times\mathcal{B}(t\rightarrow l\nu b)+\sigma_{t(\rightarrow l\nu b\gamma)q} = 303\pm 9(\text{stat.})^{+33}_{-32}(\text{syst.})\,$fb.

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