Comparison of the $m_{miss}$ shapes for the simulated signal events within the fiducial region and those outside it, after including the effect of PU protons as describe in the text, for a generated $m_{X}$ mass of 1000 GeV. The distributions are shown for single(+z)-single(−z) proton reconstruction categories.

Product of the acceptance times the combined reconstruction and identification efficiency, as a function of $m_{X}$ , for events generated inside the fiducial volume defined in Table 1. The curves shown panel display the different final states. The definition of the fiducial region and signal model used to estimate the acceptance is provided in the text.

Product of the acceptance times the combined reconstruction and identification efficiency, as a function of $m_{X}$ , for different proton reconstruction categories in the $Z \rightarrow \mu\mu $ analysis, and for events generated inside the fiducial volume defined in Table 1. The curves shown panel display the different final states. The definition of the fiducial region and signal model used to estimate the acceptance is provided in the text.

Distributions of the reconstructed proton $\xi$ in the negative arm from the proton mixing procedure with simulated MC events, are compared to data. The ee final states are shown.The ee events are displayed without the Z boson $p_T$ requirement.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Distributions of the reconstructed proton $\xi$ in the positive arm from the proton mixing procedure with simulated MC events, are compared to data. The ee final states are shown.The ee events are displayed without the Z boson $p_T$ requirement.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Distributions of the reconstructed di-proton rapidity from the proton mixing procedure with simulated MC events, are compared to data. The ee final states are shown.The ee events are displayed without the Z boson $p_T$ requirement.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Distributions of the reconstructed proton $\xi$ in the negative arm from the proton mixing procedure with simulated MC events, are compared to data. The $\mu\mu$ final states are shown.The $\mu\mu$ events are displayed without the Z boson $p_T$ requirement.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Distributions of the reconstructed proton $\xi$ in the positive arm from the proton mixing procedure with simulated MC events, are compared to data. The $\mu\mu$ final states are shown.The $\mu\mu$ events are displayed without the Z boson $p_T$ requirement.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Distributions of the reconstructed di-proton rapidity from the proton mixing procedure with simulated MC events, are compared to data. The $\mu\mu$ final states are shown.The $\mu\mu$ events are displayed without the Z boson $p_T$ requirement.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Distributions of the reconstructed proton $\xi$ in the negative arm from the proton mixing procedure with simulated MC events, are compared to data. The $\gamma$ final states are shown.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Distributions of the reconstructed proton $\xi$ in the positive arm from the proton mixing procedure with simulated MC events, are compared to data. The $\gamma$ final states are shown.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Distributions of the reconstructed di-proton rapidity from the proton mixing procedure with simulated MC events, are compared to data. The $\gamma$ final states are shown.For illustration, the simulated signal distributions are superimposed for various choices of $m_{X}$ , normalised to a generated fiducial cross section of 100 pb.

Validation of the background modelling method, using the $e\mu$ control sample. Selected $e\mu$ events are mixed with protons from $Z\rightarrow \mu \mu$ events with $p_{T}(Z)<10$ GeV to simulate the combinatorial background shape, while the data points are unaltered $e\mu$ events. The proton $\xi$ distribution for the positive CT-PPS arm is shown.

Validation of the background modelling method, using the $e\mu$ control sample. Selected $e\mu$ events are mixed with protons from $Z\rightarrow \mu \mu$ events with $p_{T}(Z)<10$ GeV to simulate the combinatorial background shape, while the data points are unaltered $e\mu$ events. The proton $\xi$ distribution for the negative CT-PPS arm is shown.

Validation of the background modelling method, using the $e\mu$ control sample. Selected $e\mu$ events are mixed with protons from $Z\rightarrow \mu \mu$ events with $p_{T}(Z)<10$ GeV to simulate the combinatorial background shape, while the data points are unaltered $e\mu$ events. The di-proton invariant mass is shown.

Validation of the background modelling method, using the $e\mu$ control sample. Selected $e\mu$ events are mixed with protons from $Z\rightarrow \mu \mu$ events with $p_{T}(Z)<10$ GeV to simulate the combinatorial background shape, while the data points are unaltered $e\mu$ events. The $m_{miss}$ is shown.

$m_{miss}$ distributions in the $pp\rightarrow ppZeeX$, with the protons reconstructed with the multi(+z)-multi(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 1 pb.

$m_{miss}$ distributions in the $pp\rightarrow ppZeeX$, with the protons reconstructed with the multi(+z)-single(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 1 pb.

$m_{miss}$ distributions in the $pp\rightarrow ppZeeX$, with the protons reconstructed with the single(+z)-multi(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 1 pb.

$m_{miss}$ distributions in the $pp\rightarrow ppZeeX$, with the protons reconstructed with the single(+z)-single(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 1 pb.

$m_{miss}$ distributions in the $pp\rightarrow ppZ\mu\mu X$, with the protons reconstructed with the multi(+z)-multi(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 1 pb.

$m_{miss}$ distributions in the $pp\rightarrow ppZ\mu\mu X$, with the protons reconstructed with the multi(+z)-single(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 1 pb.

$m_{miss}$ distributions in the $pp\rightarrow ppZ\mu\mu X$, with the protons reconstructed with the single(+z)-multi(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 1 pb.

$m_{miss}$ distributions in the $pp\rightarrow ppZ\mu\mu X$, with the protons reconstructed with the single(+z)-single(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 1 pb.

$m_{miss}$ distributions in the $pp\rightarrow pp\gamma X$, with the protons reconstructed with the multi(+z)-multi(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 10 pb.

$m_{miss}$ distributions in the $pp\rightarrow pp\gamma X$, with the protons reconstructed with the multi(+z)-single(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 10 pb.

$m_{miss}$ distributions in the $pp\rightarrow pp\gamma X$, with the protons reconstructed with the single(+z)-multi(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 10 pb.

$m_{miss}$ distributions in the $pp\rightarrow pp\gamma X$, with the protons reconstructed with the single(+z)-single(-z)method. The background distributions are shown after the fit. The expectations for a signal with $m_{X} = 1000$ GeV are superimposed, where the fiducial production cross section is normalised to 10 pb.

Upper limits on the $pp\rightarrow ppZ/\gamma X$ cross section at 95% CL, as a function of $m_X$. The 68 and 95% central intervals of the expected limits are represented by the green and yellow bands, respectively, while the observed limit is superimposed as a curve. The plot corresponds to the $Z\rightarrow ee$ analysis.

Upper limits on the $pp\rightarrow ppZ/\gamma X$ cross section at 95% CL, as a function of $m_X$. The 68 and 95% central intervals of the expected limits are represented by the green and yellow bands, respectively, while the observed limit is superimposed as a curve. The plot corresponds to the $Z\rightarrow \mu\mu$ analysis.

Upper limits on the $pp\rightarrow ppZ/\gamma X$ cross section at 95% CL, as a function of $m_X$. The 68 and 95% central intervals of the expected limits are represented by the green and yellow bands, respectively, while the observed limit is superimposed as a curve. The plot corresponds to the $Z$ analysis.

Upper limits on the $pp\rightarrow ppZ/\gamma X$ cross section at 95% CL, as a function of $m_X$. The 68 and 95% central intervals of the expected limits are represented by the green and yellow bands, respectively, while the observed limit is superimposed as a curve. The plot corresponds to the $\gamma$ analysis.

Expected and observed upper limits on the AQGC operators $a^Z_C/\Lambda^2$, with no unitarization. The $y$ axis shows the limit on the ratio of the observed cross section to the cross section predicted for each anomalous coupling value ($\sigma_\mathrm{AQGC}$).

Limits on LEP-like dimension-6 anomalous quartic gauge coupling parameters, with and without unitarization via a clipping procedure.

Conversion of limits on $a^W_0$ to dimension-8 $f_{M,i}$ operators, using the assumption of vanishing $WWZ\gamma$ couplings to eliminate some parameters. When quoting limits on one of the operators, the other is fixed to zero. The results for $|f_{M,0}/\Lambda^{4}|$ and $|f_{M,4}/\Lambda^{4}|$ are shown with and without clipping of the signal model at 1.4 TeV, when the other parameter is fixed to the SM value of zero.

Conversion of limits on $a^W_0$ and $a^W_C$ to dimension-8 $f_{M,i}$ operators, using the assumption that all $f_{M,i}$ except one are equal to zero. The results are shown with and without clipping of the signal model at 1.4 TeV.

{Expected and observed limits in the two-dimensional plane of $a^W_0/\Lambda^2$ vs. $a^W_C/\Lambda^2$. The limits are described by analytical ellipses of equation $(x-x0)^2/a^2 + (y-y0)^2/b^2 = 1$ and rotated counter-clockwise by $\theta$ degrees, where $x$ and $y$ in the equation correspond to the $a_0^W$ and $a_C^W$ couplings, respectively.

{Expected and observed limits in the two-dimensional plane of $a^W_0/\Lambda^2$ vs. $a^W_C/\Lambda^2$ with unitarization imposed by clipping the signal model at 1.4 TeV. The limits are described by analytical ellipses of equation $(x-x0)^2/a^2 + (y-y0)^2/b^2 = 1$ and rotated counter-clockwise by $\theta$ degrees, where $x$ and $y$ in the equation correspond to the $a_0^W$ and $a_C^W$ couplings, respectively.

{Expected and observed limits in the two-dimensional plane of $a^Z_0/\Lambda^2$ vs. $a^Z_C/\Lambda^2$. The limits are described by analytical ellipses of equation $(x-x0)^2/a^2 + (y-y0)^2/b^2 = 1$ and rotated counter-clockwise by $\theta$ degrees, where $x$ and $y$ in the equation correspond to the $a_0^Z$ and $a_C^Z$ couplings, respectively.

Invariant mass distribution of the diphoton pairs for the elastic selection region with events satisfying a < 0.005 (signal contribution is magnified by a factor 5000).

Predicted number of events having an elastic diphoton pair in association with a pair of protons observed within the range where the proton detectors have a radiation inefficiency less than 10%. The yields where the two-photon and two-proton systems mass and rapidity are matching at 2 and 3$\sigma$ are also quoted. This corresponds to a search region of $m_{\gamma\gamma} > 350$ GeV, $0.070 < \xi^+ < 0.111$, and $0.070 < \xi^- < 0.138$.

Predicted number of events having an elastic diphoton pair in association with a pair of protons observed within the range where the proton detectors have a radiation inefficiency less than 10%. The yields where the two-photon and two-proton systems mass and rapidity are matching at 2 and 3$\sigma$ are also quoted. This corresponds to a search region of $m_{\gamma\gamma} > 350$ GeV, $0.070 < \xi^+ < 0.111$, and $0.070 < \xi^- < 0.138$.

95% CL expected and observed frequentist upper limit for the LbL cross section within the $m_{\gamma\gamma} > 350$ GeV, $0.070 < \xi^+ < 0.111$, and $0.070 < \xi^- < 0.138$ search region.

95% CL expected and observed frequentist upper limit for the LbL cross section within the $m_{\gamma\gamma} > 350$ GeV, $0.070 < \xi^+ < 0.111$, and $0.070 < \xi^- < 0.138$ search region.

95% expected and observed one-dimensional limits on $\zeta_1$ and $\zeta_2$ anomalous LbyL production parameters, when the other parameter is set to zero. This corresponds to a search region of $m_{\gamma\gamma} > 350$ GeV, $0.070 < \xi^+ < 0.111$, and $0.070 < \xi^- < 0.138$.

95% expected and observed one-dimensional limits on $\zeta_1$ and $\zeta_2$ anomalous LbyL production parameters, when the other parameter is set to zero. This corresponds to a search region of $m_{\gamma\gamma} > 350$ GeV, $0.070 < \xi^+ < 0.111$, and $0.070 < \xi^- < 0.138$.

95% expected and observed limits on $\zeta_1$ and $\zeta_2$ anomalous LbyL production parameters. The parametric elliptic form is assumed: $a_0\zeta_1^2+a_1\zeta_1\zeta_2+a_2\zeta_2^2=1$. This corresponds to a search region of $m_{\gamma\gamma} > 350$ GeV, $0.070 < \xi^+ < 0.111$, and $0.070 < \xi^- < 0.138$.

95% expected and observed limits on $\zeta_1$ and $\zeta_2$ anomalous LbyL production parameters. The parametric elliptic form is assumed: $a_0\zeta_1^2+a_1\zeta_1\zeta_2+a_2\zeta_2^2=1$. This corresponds to a search region of $m_{\gamma\gamma} > 350$ GeV, $0.070 < \xi^+ < 0.111$, and $0.070 < \xi^- < 0.138$.