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Ratio of the cross-sections computed in the WW approximation and at ETL, $\sigma_{2\rightarrow3}^{Z'}\big|_\text{WW}/\sigma_{2\rightarrow3}^{Z'}\big|_\text{ETL}$, as a function of the muon beam energy, $E_0$, for different $Z'$ masses, $m_{Z'}$. More details can be found in \cite{Kirpichnikov:2021jev}.

Comparison of the single-differential cross-section as a function of the fractional energy of the emitted $\gamma$ and $Z'$ through respectively SM muon-bremsstrahlung and dark bremsstrahlung, $\mu N\rightarrow\mu NZ'$, in the mass limit $m_{Z'}\rightarrow0$. These results are obtained both at ETL and in the WW and IWW approaches, with mixing strength $\epsilon_{Z'}=g_{Z'}/e=1$. More details can be found in Ref. \cite{Kirpichnikov:2021jev}.

Experimental set-up schematic overview of the search for $Z^\prime\rightarrow\text{invisible}$ \cite{Andreev:2024sgn}. Top: The \emph{upstream} experimental region for the reconstruction of the incoming muon momentum through the MS1 (BEND6) magnet spectrometer using MM$_{1-4}$, ST$_{5,4}$ and BMS$_{1-6}$. The beam-defining optics quadrupoles, QPL$_{29-32}$ and QPL$_{33}$, part of the FODO scheme, are shown. For completeness, the distances between the different detector elements are given in cm. See text for more details.

The \emph{downstream} region of the experiment for final-state muon identification through detector response and momentum reconstruction in MS2. The distances between the detector elements are given in cm. See text for more details.

Left: Distributions of the fractional energy, $x$, for repsectively SM muon bremsstrahlung, $\mu N\rightarrow\mu N\gamma$ and dark bremsstrahlung in the limit $m_{Z'}\rightarrow0$. The events are obtained from a minimal \texttt{GEANT4} simulation of the NA64$\mu$ target, assuming a fixed muon beam energy $E_0=160$ GeV. Right: Production cross-sections $\sigma_{2\rightarrow3}$ as a function of the muon beam energy, $E_0$, extracted from a realistic \texttt{GEANT4} simulation of both SM muon bremsstrahlung, $\mu N\rightarrow\mu N\gamma$, and dark bremsstrahlung in the limit $m_{Z'}\rightarrow0$, within the NA64$\mu$ ECAL.

Left: Distributions of the fractional energy, $x$, for repsectively SM muon bremsstrahlung, $\mu N\rightarrow\mu N\gamma$ and dark bremsstrahlung in the limit $m_{Z'}\rightarrow0$. The events are obtained from a minimal \texttt{GEANT4} simulation of the NA64$\mu$ target, assuming a fixed muon beam energy $E_0=160$ GeV. Right: Production cross-sections $\sigma_{2\rightarrow3}$ as a function of the muon beam energy, $E_0$, extracted from a realistic \texttt{GEANT4} simulation of both SM muon bremsstrahlung, $\mu N\rightarrow\mu N\gamma$, and dark bremsstrahlung in the limit $m_{Z'}\rightarrow0$, within the NA64$\mu$ ECAL.

The muon emission angles $\psi_{\mu}^\prime$ for kinematical regimes with $m_{Z'}=10,500$ MeV. The normalized sampled angle (crosses) and the normalized target partial distribution function (PDF, lines) are shown for comparison. Small deviations from the target PDF at larger $\psi_\mu^\prime$ are due to fewer statistics in the binned sample distribution.

Calibration trigger acceptance effects (blue solid line) on the initial-state beam muons before the ECAL target.

Final-state muons energy distribution from the SM process $\mu N\rightarrow\mu N\gamma$ using the calibration trigger configuration (green line) and different physical trigger configurations with the S$_4$ counter shifted respectively -40 mm (black line) and -65 mm (blue line) along the deflection axis past MS2. For completeness, bremsstrahlung-like events $\mu N\rightarrow\mu NZ'$ in the $m_{Z'}=1$ MeV scenario (solid red dots) are shown for S$_4$ at -65 mm.

Energy distributions for both HCAL and ECAL from secondary products from hard muon bremsstrahlung and nuclear interactions in the target. A final-state muon with energy less than $E_\mu^\prime\leq100$ GeV is required. Both HCAL transverse sizes with 60$\times$60 cm$^2$ (blue) and $120\times60$ cm$^2$ (red) are shown. The upper bound of the signal box (black dashed line) is shown (see Sec. \ref{sec:data-analysis}).

Event distribution in the hermeticity plane defined by the reconstructed momentum after the ECAL target, $p_\text{out}$, and the sum of energy deposit in the calorimeters, $E_\text{CAL}=E_\text{ECAL}+E_\text{VHCAL}+E_\text{HCAL}$ \cite{Andreev:2024sgn}. The sample of events corresponds to the sum of both physical trigger configurations 1+2, with $N_\text{MOT}=(1.98\pm0.02)\times10^{10}$. The signal region is blinded (green right-hashed box). Left: Events distribution after applying selection criteria (i-ii) to select a single-track-compatible event with a muon traversing the whole set-up. For completeness, different regions of the phase space are highlighted, with regions $A$ and $B$ used as \emph{control} regions for background extrapolation (see text). Right: Event distribution after additionally requiring a MIP in the calorimeters and no activity in the VHCAL and VETO (criteria (i-iii)).

Event distribution in the hermeticity plane defined by the reconstructed momentum after the ECAL target, $p_\text{out}$, and the sum of energy deposit in the calorimeters, $E_\text{CAL}=E_\text{ECAL}+E_\text{VHCAL}+E_\text{HCAL}$ \cite{Andreev:2024sgn}. The sample of events corresponds to the sum of both physical trigger configurations 1+2, with $N_\text{MOT}=(1.98\pm0.02)\times10^{10}$. The signal region is blinded (green right-hashed box). Left: Events distribution after applying selection criteria (i-ii) to select a single-track-compatible event with a muon traversing the whole set-up. For completeness, different regions of the phase space are highlighted, with regions $A$ and $B$ used as \emph{control} regions for background extrapolation (see text). Right: Event distribution after additionally requiring a MIP in the calorimeters and no activity in the VHCAL and VETO (criteria (i-iii)).

\texttt{GenFit}-based \cite{Rauch:2014wta} track reconstruction through the whole experimental set-up.

Distribution of single-track muon events deflection past MS2, defined as $\delta x=(\text{MM}_7)_{x}-(\text{GEM}_1)_x$, for both data (blue triangle) and MC (solid magenta line) in the calibration trigger configuration.

Distributions of energy deposited around the MIP peak compatible with a muon for both data (blue triangle) and MC (solid magenta line) in the calibration trigger configuration. Left: The ECAL module. Right: The whole HCAL module (first and second HCAL modules). The spectra are normalized to a similar number of events.

Distributions of energy deposited around the MIP peak compatible with a muon for both data (blue triangle) and MC (solid magenta line) in the calibration trigger configuration. Left: The ECAL module. Right: The whole HCAL module (first and second HCAL modules). The spectra are normalized to a similar number of events.

First HCAL module energy distribution for a sample of muon interacting in the ECAL, $\mu N\rightarrow\mu+X$, with high energetic secondaries escaping the detector and propagating through MS2. Both data (blue triangle) and MC (solid magenta line) correspond to the physics trigger configuration 1. The spectra are normalized to a similar number of events. See text for more details.

Final-state muon energy distributions for both pions (green line) and kaons (blue line) decays, $\pi,K\rightarrow\mu+X$, as extracted from a dedicated \texttt{GEANT4} simulation of the process within the experimental set-up.

Exponential (dashed magenta line) and Crystal Ball (dashed green line) fits of the low-energy tails of the scattered muon momentum distribution, after applying selection criteria (iii) and (i), with $p_\text{in}=160\pm10$ GeV/c, on a sample of events extracted from the calibration trigger configuration.

Effect of the geometrical acceptance associated with trigger 1 (S$_4$ and S$_\mu$ shifted respectively 65 mm and 152 mm along the magnetic deflection direction) on the signal efficiency computed as a function of the mass. The track reconstruction efficiency is also considered.

The NA64$\mu$ 90\% C.L. exclusion limits in the parameter space compatible with a light boson as an explanation for the muon $(g-2)_\mu$. Left: The $Z'$ vector boson parameter space $(m_{Z'},\ g_{Z'})$ together with existing constraints from neutrino experiments such as BOREXINO \cite{Kamada:2015era,Kaneta:2016uyt,Gninenko:2020xys} and CCFR \cite{Altmannshofer:2014pba,CCFR:1991lpl}, visible searches in electron-positron annihilation with BaBar \cite{Capdevilla:2021kcf}, Belle II constraints \cite{Belle-II:2019qfb} and the NA64 electron program limits \cite{Andreev:2024lps}. Projections for the pre-LS3, pre-LS4, and post-LS4 phases of the muon program are shown together with the $M^{3}$ missing momentum searches \cite{Kahn:2018cqs}. Right: The $S$ scalar boson parameter space $(m_{S},\ g_{S})$ together with existing constraints from BaBar and projections for the pre-LS3, pre-LS4, and post-LS4 phases of the muon program, as well as ATLAS HL-LHC \cite{Galon:2019owl} and $M^{3}$.

The NA64$\mu$ 90\% C.L. exclusion limits in the parameter space compatible with a light boson as an explanation for the muon $(g-2)_\mu$. Left: The $Z'$ vector boson parameter space $(m_{Z'},\ g_{Z'})$ together with existing constraints from neutrino experiments such as BOREXINO \cite{Kamada:2015era,Kaneta:2016uyt,Gninenko:2020xys} and CCFR \cite{Altmannshofer:2014pba,CCFR:1991lpl}, visible searches in electron-positron annihilation with BaBar \cite{Capdevilla:2021kcf}, Belle II constraints \cite{Belle-II:2019qfb} and the NA64 electron program limits \cite{Andreev:2024lps}. Projections for the pre-LS3, pre-LS4, and post-LS4 phases of the muon program are shown together with the $M^{3}$ missing momentum searches \cite{Kahn:2018cqs}. Right: The $S$ scalar boson parameter space $(m_{S},\ g_{S})$ together with existing constraints from BaBar and projections for the pre-LS3, pre-LS4, and post-LS4 phases of the muon program, as well as ATLAS HL-LHC \cite{Galon:2019owl} and $M^{3}$.

The NA64$\mu$ 90\% C.L. exclusion limits in the LTDM parameter space $y-m_{\chi}$ compatible with an invisibly decaying $Z'\rightarrow\text{DM}$ with (left) $g_\chi=5\times10^{-2}$ and (right) $g_\chi=1$ and mass ratio $m_\chi/m_{Z'}=3$. The existing constraints from the CCFR experiment \cite{Altmannshofer:2014pba,CCFR:1991lpl} are compared, and the thermal targets for complex scalar, (pseudo-)Dirac and Majorana thermal relics plotted \cite{Berlin:2018bsc}. Projections for the pre-LS3, pre-LS4, and post-LS4 periods of the muon program are shown together with the $M^{3}$ missing momentum searches \cite{Kahn:2018cqs}. The NA64 electron program limits are plotted for completeness \cite{Andreev:2024lps}.

The NA64$\mu$ 90\% C.L. exclusion limits in the LTDM parameter space $y-m_{\chi}$ compatible with an invisibly decaying $Z'\rightarrow\text{DM}$ with (left) $g_\chi=5\times10^{-2}$ and (right) $g_\chi=1$ and mass ratio $m_\chi/m_{Z'}=3$. The existing constraints from the CCFR experiment \cite{Altmannshofer:2014pba,CCFR:1991lpl} are compared, and the thermal targets for complex scalar, (pseudo-)Dirac and Majorana thermal relics plotted \cite{Berlin:2018bsc}. Projections for the pre-LS3, pre-LS4, and post-LS4 periods of the muon program are shown together with the $M^{3}$ missing momentum searches \cite{Kahn:2018cqs}. The NA64 electron program limits are plotted for completeness \cite{Andreev:2024lps}.

The NA64$\mu$ 90\% C.L. exclusion limits on the dark photon scenario, $A'\rightarrow\text{invisible}$. The $(m_{A'},\ \epsilon)$ parameter space is shown, together with the latest results from NA64$e^{-}$ \cite{NA64:2023wbi} and NA64$e^{+}$ \cite{NA64:2023ehh} and the existing limits from BaBar \cite{BaBar:2017tiz}. The peak is related to fermionic DM assuming $\alpha_D=0.1$. Projections for the pre-LS3, pre-LS4, and post-LS4 periods are shown.

The NA64$\mu$ 90\% C.L. exclusion limits on the dark photon scenario, $A'\rightarrow\text{invisible}$ in the $(m_{\chi},\ y)$ parameter space, together with the DM target relic abundance for scalar, (pseudo-)Dirac and Majorana scenarios \cite{Berlin:2018bsc}. Left: Scenario with $\alpha_D=0.1$. Right: Scenario with $\alpha_D=0.5$. Projections for the pre-LS3, pre-LS4, and post-LS4 periods are shown for completeness. The combined projected limits (green dashed curve) for NA64$e^{-},e^{+},\mu$ are plotted, using the projections for $10^{13}$ EOT and $10^{11}$ $e^{+}$OT.

The NA64$\mu$ 90\% C.L. exclusion limits on the dark photon scenario, $A'\rightarrow\text{invisible}$ in the $(m_{\chi},\ y)$ parameter space, together with the DM target relic abundance for scalar, (pseudo-)Dirac and Majorana scenarios \cite{Berlin:2018bsc}. Left: Scenario with $\alpha_D=0.1$. Right: Scenario with $\alpha_D=0.5$. Projections for the pre-LS3, pre-LS4, and post-LS4 periods are shown for completeness. The combined projected limits (green dashed curve) for NA64$e^{-},e^{+},\mu$ are plotted, using the projections for $10^{13}$ EOT and $10^{11}$ $e^{+}$OT.

The NA64$\mu$ 90\% C.L. exclusion limits on invisibly decaying spin-0 scalar mediator, $S\rightarrow\text{invisible}$. The thermal targets for light DM are shown for respectively $m_S/m_\chi=3$ and $m_S/m_\chi=2.1$ with $g_\chi=1$, and extracted from \cite{Chen:2018vkr}. Projections for $3\times10^{11}$, $2\times10^{13}$ and $10^{14}$ MOT are plotted.