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Tree-level Feynman diagrams for the processes $e^+ e^-\to X a$ with $X=\gamma, Z, h$.

Tree-level Feynman diagrams for the processes $e^+ e^-\to X a$ with $X=\gamma, Z, h$.

Production cross sections of ALPs produced in the decays of Higgs and $Z$ bosons at the LHC ($\sqrt{s} = 14\,$TeV) versus the new-physics scale $\Lambda$. We set $m_a=0$ and fix the relevant Wilson coefficients to 1. For the green contour in the left plot, we fix $C_{Zh}^{(5)}=0$ and only consider the dimension-7 coupling in \eqref{LeffD>5}. The grey regions in the two plots are excluded by Higgs coupling measurements and the measurement of the total $Z$ width, respectively.

Leading order Higgs production cross sections at $e^+e^-$ colliders as a function of the center-of-mass energy, produced with \texttt{MadGraph5} \cite{Alwall:2014hca}.

Left: Summary plot of constraints on the parameter space spanned by the ALP mass and ALP-photon coupling. Right: Enlarged display of the constraints from collider searches: LEP (light blue and blue), CDF (purple), LHC from associated production and $Z$ decays (orange), LHC from photon fusion (light orange), and from heavy-ion collisions at the LHC (green).

Left: Existing constraints on the ALP mass and coupling to gluons by mono-jet searches at the LHC (light blue), rare kaon decays (light red) and three-jet events (purple). Right: Constraints on the ALP mass and coupling to leptons from searches for solar axions (purple), the evolution of red giants (light red), beam dump searches for ALP decays into electrons (blue) and BaBar searches for $e^+e^-\to 4 \mu$.

Left: Existing constraints on the ALP mass and coupling to gluons by mono-jet searches at the LHC (light blue), rare kaon decays (light red) and three-jet events (purple). Right: Constraints on the ALP mass and coupling to leptons from searches for solar axions (purple), the evolution of red giants (light red), beam dump searches for ALP decays into electrons (blue) and BaBar searches for $e^+e^-\to 4 \mu$.

Projected sensitivity regions for searches for $e^+e^-\to \gamma a \to 3\gamma$ (left) and $e^+e^-\to Z a \to Z_\text{vis}\gamma \gamma$ (right) at future $e^+e^-$ colliders for $\text{Br}(a\to\gamma \gamma)=1$. The constraints from Figure~\ref{fig:Cggbounds} are shown in the background. The sensitivity regions are based on 4 expected signal events.

Left: Projected sensitivity regions for searches for $e^+e^-\to ha \to b\bar b \gamma\gamma$ (upper panels) and $e^+e^-\to ha \to b\bar b \ell^+\ell^-$ (lower panels) for future $e^+e^-$ colliders, assuming that $|C_{Zh}^\text{eff}|=0.72\,\Lambda/\text{TeV}$ and $\text{Br}(a\to\gamma \gamma)=1$ (upper panels) and $\text{Br}(a\to\ell^+\ell^-)=1$ (lower panels). Right: Sensitivity regions for the example of the FCC-ee with $|C_{Zh}^\text{eff}|=0.72\,\Lambda/\text{TeV}$ (solid contour), $|C_{Zh}^\text{eff}|=0.1\,\Lambda/\text{TeV}$ (dashed contour), and $|C_{Zh}^\text{eff}|=0.015\,\Lambda/\text{TeV}$ (dotted contour), which corresponds to Br$(h \to Za)=34\%, 1\% $ and Br$(h \to Za)=0.02\% $, respectively. The constraints from Figure~\ref{fig:Cggbounds} are shown in the background. The sensitivity regions are based on 4 expected signal events.

Left: Projected sensitivity regions for searches for $e^+e^-\to ha \to b\bar b \gamma\gamma$ (upper panels) and $e^+e^-\to ha \to b\bar b \ell^+\ell^-$ (lower panels) for future $e^+e^-$ colliders, assuming that $|C_{Zh}^\text{eff}|=0.72\,\Lambda/\text{TeV}$ and $\text{Br}(a\to\gamma \gamma)=1$ (upper panels) and $\text{Br}(a\to\ell^+\ell^-)=1$ (lower panels). Right: Sensitivity regions for the example of the FCC-ee with $|C_{Zh}^\text{eff}|=0.72\,\Lambda/\text{TeV}$ (solid contour), $|C_{Zh}^\text{eff}|=0.1\,\Lambda/\text{TeV}$ (dashed contour), and $|C_{Zh}^\text{eff}|=0.015\,\Lambda/\text{TeV}$ (dotted contour), which corresponds to Br$(h \to Za)=34\%, 1\% $ and Br$(h \to Za)=0.02\% $, respectively. The constraints from Figure~\ref{fig:Cggbounds} are shown in the background. The sensitivity regions are based on 4 expected signal events.

Parameter regions which can be probed for $e^+e^-\to ha \to b\bar b \gamma\gamma$ (upper panels) and $e^+e^-\to ha \to b\bar b \ell^+\ell^-$ (lower panels) at future $e^+e^-$ colliders. The grey shaded area is excluded by LHC Higgs measurements. The parameter space to the right of the dotted contours corresponds to the sensitivity reach of the FCC-ee with the indicated ALP branching ratios. The sensitivity regions are based on 4 expected signal events.

Parameter regions which can be probed for $e^+e^-\to ha \to b\bar b \gamma\gamma$ (upper panels) and $e^+e^-\to ha \to b\bar b \ell^+\ell^-$ (lower panels) at future $e^+e^-$ colliders. The grey shaded area is excluded by LHC Higgs measurements. The parameter space to the right of the dotted contours corresponds to the sensitivity reach of the FCC-ee with the indicated ALP branching ratios. The sensitivity regions are based on 4 expected signal events.

Projected exclusion contours for searches for $e^+e^-\to \gamma a \to \gamma \ell^+\ell^-$ (left) and $e^+e^-\to Z a \to Z_\text{vis}\ell^+ \ell^-$ (right) for future $e^+e^-$ colliders, and $\text{Br}(a\to\ell^+ \ell^-)=1$. The constraints from Figure~\ref{fig:Cllbounds} are in the background. The sensitivity regions are based on 4 expected signal events.

Parameter regions which can be probed in the decay $h \to Z a$ with $a \to \gamma \gamma$ (upper row) and $h \to a a$ with $a \to \gamma \gamma$ (lower row) at future $e^+e^-$ colliders. The grey shaded area is excluded by LHC Higgs measurements. The dotted contours correspond to the sensitivity region of the FCC-ee for ALP branching ratios smaller than 1. The sensitivity regions are based on 4 expected signal events.

Parameter regions which can be probed in the decay $h \to Z a$ with $a \to \gamma \gamma$ (upper row) and $h \to a a$ with $a \to \gamma \gamma$ (lower row) at future $e^+e^-$ colliders. The grey shaded area is excluded by LHC Higgs measurements. The dotted contours correspond to the sensitivity region of the FCC-ee for ALP branching ratios smaller than 1. The sensitivity regions are based on 4 expected signal events.

Parameter regions which can be probed in the decay $h \to Z a$ with $a \to \ell^- \ell^+$ (upper row) and $h \to a a$ with $a \to \ell^+ \ell^-$ (lower row) at future $e^+e^-$ colliders. The grey shaded area is excluded by LHC Higgs measurements. The dotted contours correspond to the sensitivity region of the FCC-ee for ALP branching ratios smaller than 1. The sensitivity regions are based on 4 expected signal events.

Projected exclusion contours for searches for $pp\to h \to Z a $ (left) and $pp\to h \to aa $ (right) with the subsequent ALP decay $a \to gg$ and $\text{Br}(a\to gg)=1$ with the MATHUSLA detector. The different contours correspond to different values of $C_{Zh}^\text{eff}$ and $C_{ah}^\text{eff}$.

Parameter regions which can be probed in the decay $h \to Z a$ with $a \to \ell^- \ell^+$ (upper row) and $h \to a a$ with $a \to \ell^+ \ell^-$ (lower row) at future $e^+e^-$ colliders. The grey shaded area is excluded by LHC Higgs measurements. The dotted contours correspond to the sensitivity region of the FCC-ee for ALP branching ratios smaller than 1. The sensitivity regions are based on 4 expected signal events.

Allowed regions in the parameters space of the Wilson coefficients $C_{\gamma\gamma}^{\rm eff}-C_{\gamma Z}^{\rm eff}$ obtained from projections for the two-parameter global electroweak fit at 68\% CL, 95\% CL and 99\% CL at FCC-ee (violet) and at 95\% CL for the LHC at $\sqrt{s}=14\,$TeV (green). For the parameter space within the dashed black contour, the FCC-ee measurement of $\alpha(m_Z)$ is within its projected errors at 95\% CL \cite{Janot:2015gjr}. The red dots represent the best fit points based on the current electroweak fit.

Projected reach in searches for $pp \to a \to \gamma\gamma $ with the LHC (green), HE-LHC (light green) and a $100\,$TeV collider (blue). Contours of constant branching ratios $\text{Br}(a\to\gamma\gamma)$ are shown as dotted lines. The sensitivity regions are based on 100 expected signal events.

Parameter regions which can be probed in the decay $Z \to \gamma a$ with $a \to \gamma \gamma$ at hadron colliders . The projected reach is coloured green (LHC), light green (HE-LHC) and turquoise (FCC-hh). We assume $\text{Br}(a\to\gamma\gamma)=1$. The sensitivity regions are based on 100 expected signal events.

Projected reach in searches for $h \to Za \to \ell^+\ell^-+2\gamma $ and $h \to aa \to 4\gamma $ decays with the LHC (green), HE-LHC (light green) and a $100\,$TeV collider (blue). The parameter region with the solid contours correspond to a branching ratio of $\text{Br}(a\to \gamma\gamma)=1$, and the contours showing the reach for smaller branching ratios are dotted. The sensitivity regions are based on 100 expected signal events.

Projected reach in searches for $h \to Za \to \ell^+\ell^-+2\gamma $ and $h \to aa \to 4\gamma $ decays with the LHC (green), HE-LHC (light green) and a $100\,$TeV collider (blue). The parameter region with the solid contours correspond to a branching ratio of $\text{Br}(a\to \gamma\gamma)=1$, and the contours showing the reach for smaller branching ratios are dotted. The sensitivity regions are based on 100 expected signal events.

Projected reach in searches for $h \to Za \to \ell^+\ell^-+\ell^+\ell^- $ and $h \to aa \to 4\ell $ decays with the LHC (green), HE-LHC (light green) and a $100\,$TeV collider (blue). The parameter region with the solid contours correspond to a branching ratio of $\text{Br}(a\to \ell^+\ell^-)=1$, and the contours showing the reach for smaller branching ratios are dotted. The sensitivity regions are based on 100 expected signal events.

Projected reach in searches for $h \to Za \to \ell^+\ell^-+\ell^+\ell^- $ and $h \to aa \to 4\ell $ decays with the LHC (green), HE-LHC (light green) and a $100\,$TeV collider (blue). The parameter region with the solid contours correspond to a branching ratio of $\text{Br}(a\to \ell^+\ell^-)=1$, and the contours showing the reach for smaller branching ratios are dotted. The sensitivity regions are based on 100 expected signal events.

Left: Geometric setup of the MATHUSLA surface detector above the ATLAS/CMS cavern together with a sketch of the $pp\to h\to aZ$ process with a subsequent decay of the ALP in the MATHUSLA detector volume. Right: Total percentage of ALPs decaying within the ATLAS or CMS detector per ALPs produced in the Higgs decay $h\to aZ$ (green), fraction of ALPs produced decaying in ATLAS/CMS together with a leptonically decaying $Z$ (dashed green), and the percentage of ALPs decaying within the MATHUSLA detector volume (red). The grey area shows the distance between the interaction point and the electromagnetic calorimeter.

Projected reach in searches for $h \to Za \to \ell^+\ell^-+2\gamma$ (top) and $h \to aa \to4\gamma$ (bottom) decays at the LHC (green) and MATHUSLA (red) with $\sqrt{s}=14\,$TeV center-of-mass energy and $3\,$ab$^{-1}$ integrated luminosity. The parameter region with solid contours correspond to a branching ratio of $\text{Br}(a\to \gamma\gamma)=1$, and contours showing the reach for smaller branching ratios are dotted. The sensitivity regions are based on 4 (MATHUSLA) and 100 (LHC) expected signal events, respectively .

Projected reach in searches for $h \to Za \to \ell^+\ell^-+2\gamma$ (top) and $h \to aa \to4\gamma$ (bottom) decays at the LHC (green) and MATHUSLA (red) with $\sqrt{s}=14\,$TeV center-of-mass energy and $3\,$ab$^{-1}$ integrated luminosity. The parameter region with solid contours correspond to a branching ratio of $\text{Br}(a\to \gamma\gamma)=1$, and contours showing the reach for smaller branching ratios are dotted. The sensitivity regions are based on 4 (MATHUSLA) and 100 (LHC) expected signal events, respectively .

Projected reach in searches for $h \to Za \to \ell^+\ell^-+\mu^+\mu^- $ (left) and $h \to aa \to \mu^+\mu^-+\mu^+\mu^- $ (right) decays with ATLAS/CMS (green) and MATHUSLA (red) with $\sqrt{s}=14\,$TeV center-of-mass energy and $3\,$ab$^{-1}$ integrated luminosity. The parameter region with solid contours correspond to a branching ratio of $\text{Br}(a\to \mu^+\mu^-)=1$, and contours showing the reach for smaller branching ratios are dotted. The sensitivity regions are based on 4 (MATHUSLA) and 100 (LHC) expected signal events, respectively.

Projected reach in searches for $Z \to \gamma a \to 3\gamma$ with MATHUSLA for $\sqrt{s}=14\,$TeV center-of-mass energy, $3\,$ab$^{-1}$ integrated luminosity and $\text{Br}(a\to \gamma\gamma)=1$, together with the expected sensitivity of FASER taken from \cite{Feng:2018pew} and SHiP \cite{Alekhin:2015byh}. The sensitivity regions are based on 4 (MATHUSLA) and 100 (LHC) expected signal events, respectively.

Left: Existing constraints on the ALP mass and coupling to gluons by mono-jet searches at the LHC (light blue), rare kaon decays (light red) and three-jet events (purple). Right: Constraints on the ALP mass and coupling to leptons from searches for solar axions (purple), the evolution of red giants (light red), beam dump searches for ALP decays into muons (blue) and BaBar searches for $e^+e^-\to 4 \mu$.

Projected exclusion contours for searches for $pp\to h \to Z a $ (left) and $pp\to h \to aa $ (right) with the subsequent ALP decay $a \to gg$ and $\text{Br}(a\to gg)=1$ with the MATHUSLA detector. The different contours correspond to different values of $C_{Zh}^\text{eff}$ and $C_{ah}^\text{eff}$. The sensitivity regions are based on 4 expected signal events, respectively.