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Schematic representation of the NA64$-e$ detector, searching for an invisibly decaying $\Zpr$ produced by the interaction of 100 GeV electrons with the material of the active ECAL target. See text for further details.

The relative difference between the $\Pi$ function on-shell value $|\Pi|^2$ and the $q^2$-averaged one $I_{|\Pi|^2}$ as a function of the $\Zpr$ mass, for different $\alpha_D$ values.

Total atomic cross section for the process $e^+e^-\rightarrow \Zpr \rightarrow \nu \bar{\nu}$ on lead as a function of the impinging positron energy, considering (red) or not (black) the motion of atomic electrons.

The new exclusion limits from the NA64$-e$ experiment in the $(g_\Zpr,m_\Zpr)$ parameter space for the $L_\mu-L_\tau$ model, considering the ``vanilla'' scenario without any DS coupling (left) and the full ``dark sector one'' (right), for $\alpha_D=0.1$ (continuous curve) and $\alpha_D=0.02$ (dashed curve). The most competitive bounds reported by other experiments are also shown (see text for further details). The green band correspond to preferred combination of the model parameters that would solve the muon $g-2$ tension. The blue dashed line defines the region excluded by cosmological and astrophysical considerations. Finally, the black curves in the right plot are the preferred combination of the parameters to explain the observed dark matter relic density (so-called ``thermal target'').

The new exclusion limits from the NA64$-e$ experiment in the $(g_\Zpr,m_\Zpr)$ parameter space for the $L_\mu-L_\tau$ model, considering the ``vanilla'' scenario without any DS coupling (left) and the full ``dark sector one'' (right), for $\alpha_D=0.1$ (continuous curve) and $\alpha_D=0.02$ (dashed curve). The most competitive bounds reported by other experiments are also shown (see text for further details). The green band correspond to preferred combination of the model parameters that would solve the muon $g-2$ tension. The blue dashed line defines the region excluded by cosmological and astrophysical considerations. Finally, the black curves in the right plot are the preferred combination of the parameters to explain the observed dark matter relic density (so-called ``thermal target'').

The three curves represent the cross sections of the resonant $\Zpr$ production on a thin diamond target as a function of the positron beam nominal energy $E_{B}$, obtained analytically considering the RHF (orange) and CP (green) model from~\cite{Arias-Aragon:2024qji}, and from a simulation via DMG4 (gray). See the text for more details.