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The potential of future collider searches to probe the parameter space of feebly-interacting particles in the plane FIP mass - FIP coupling to SM particles. The figures demonstrate that colliders cannot efficiently explore the parameter space of FIPs with mass of the order of GeV. \textbf{Left panel:} parameter space of heavy neutral leptons (HNLs) that mix with electron neutrinos. The lower bound (seesaw) is defined by their ability to generate masses for active neutrinos~\cite{Asaka:2005an}. LHC in high luminosity phase~\cite{Boiarska:2019jcw,Abdullahi:2022jlv} and lepton colliders~\cite{Blondel:2022qqo} are mainly sensitive to short-lived HNLs, with the typical lifetimes $c\tau_{N}\lesssim \mathcal{O}(100\text{ m})$, see text for details. The scaling of the HNL lifetime with the mass is $\tau_{N}\propto m_{N}^{-5}U_{e}^{-2}$. As a result, colliders have poor sensitivity to HNL masses $m_{N}\lesssim 10\text{ GeV}$. \textbf{Right panel}: dark scalars mixing with Higgs bosons. Given the strict event selection to cope with the backgrounds and the available triggers, the event rate with displaced vertices at colliders is insufficient to provide a competitive sensitivity. Instead, scalars may be searched for with prompt events at LHCb~\cite{LHCb:2016awg}, even though only relatively large couplings are within reach. The parameter space of these examples and other feebly-interacting particles in the GeV range can instead be more efficiently explored in the coming years with beam dump experiments.

Illustration of the impact of different contributions to the geometric acceptance defined by Eq.~\eqref{eq:acceptance-qualitative}. First, the FIPs produced by collisions of the proton beam with the fixed target must point to the detector (the red arrow). The fraction of such events is given by $\epsilon_{\text{FIP}}$. The effective length inside the decay volume passed by decaying FIPs (the dashed blue line) may differ significantly from the nominal decay volume length $l_{\text{fid}}$. This results in the factor $l_{\text{fid,eff}}/l_{\text{fid}}$. Finally, the decay products of FIPs (the green arrows) also have to point to the detector, which is incorporated by $\epsilon_{\text{dec}}$.

Examples of production processes for various FIPs: (a) proton bremsstrahlung (dark photon $V$), (b) coherent scattering off nuclei (ALP with the photon coupling $a$), (c) decays of $B$ mesons (HNLs $N$, dark scalars).

Kinematics of FIPs produced in proton-target collisions at the SPS. Molybdenum target is considered. \textbf{Top panels}: solid angle distributions $df_{\text{FIP}}/d\Omega_{\text{FIP}}\sim df_{\text{FIP}}/d\cos(\theta_{\text{FIP}})$ of various FIPs. Different masses are considered, corresponding to different production channels (Table~\ref{tab:fip-model-channels}). Note that the distribution of heavy HNLs with $m_{N}>3\text{ GeV}$ is very similar to the distribution of scalars, because of the same mother particle and decay kinematics. The polar angle coverage of the detector of the reference setup~\ref{tab:hypothetical-experiment-parameters} is indicated with arrows and the vertical dashed line. \textbf{Bottom panels}: energy spectra of the mesons producing FIPs, dark photons produced by the proton bremsstrahlung, and ALPs with photon coupling. For the case of heavy mesons $B,D$, the distribution is shown assuming two different angular coverage: ``on-axis'' $\theta < 0.05\text{ rad}$, and ``off-axis'', $\theta >0.05\text{ rad}$, to demonstrate how the spectrum gets softer off-axis. See text and Ref.~\cite{Ovchynnikov:2023cry} for details.

The behavior of the number of signal events of a beam dump on-axis experiment at the SPS at the lower bound of the sensitivity ($c\tau_{\text{FIP}}\langle\gamma_{\text{FIP}}\rangle\gg 100\text{ m}$, see Sec.~\ref{sec:qualitative-analysis} for details) under change of the distance to the decay volume $l_{\text{min}}$ (\textbf{top panels}) and its length $l_{\text{fid}}$ (\textbf{bottom panels}) for different models of FIPs. On the one hand, changing these parameters may have a significant impact for the backgrounds to be removed, the complexity of the setup, and costs. On the other hand, the maximal impact of these parameters on the number of events is small, $< \mathcal{O}(2)$, see text for details. Therefore, we conclude that the optimization of these parameters should be a subject of background considerations and costs rather than the maximization of the number of FIP events. The other parameters defining the experimental setup -- the transverse size of the decay volume and the detector dimensions -- are summarized in Table~\ref{tab:hypothetical-experiment-parameters}. For convenience, we normalize the number of events to the one for the configuration from Table~\ref{tab:hypothetical-experiment-parameters}.

The behavior of the number of signal events of a beam dump experiment at the SPS at the lower bound of the sensitivity ($c\tau_{\text{FIP}}\langle\gamma_{\text{FIP}}\rangle\gg 100\text{ m}$, see Sec.~\ref{sec:qualitative-analysis}) assuming an off-axis placement of the centre of its decay volume parametrized in terms of the displacement $r_{\text{displ}}$. From the figures, we see that independently on the FIP type, by increasing $r_{\text{displ}}$, the number of events decreases. This results from the very forward-pointing FIP angular distribution that falls at large polar angles (Fig.~\ref{fig:FIPs-distributions}). Depending on the FIP, the decrease may be an order of magnitude or larger (\textbf{top panel}). It is impossible to compensate for this decrease by placing the experiment closer to the target (\textbf{bottom panel}): despite the increase of the solid angle covered by the detector, the minimal covered polar angle increases, which again results in a decrease of the FIP flux. The other parameters defining the experiment -- transverse dimensions of the decay volume, and detector dimensions -- are fixed as specified in Table~\ref{tab:hypothetical-experiment-parameters}. Note that $r_{\text{displ}}$ does not equal the off-axis displacement of the side of the decay volume. In particular, for the configuration considered, this displacement becomes non-zero only if $r_{\text{displ}} >2\text{ m}$. The displacement $r_{\text{displ}} = 3\text{ m}$ corresponds to 1 m gap between the side of the decay volume and beam axis. For convenience, we normalize the number of events to the one for the configuration specified in Table~\ref{tab:hypothetical-experiment-parameters}.

The energy distribution of the function~\eqref{eq:energy-spectrum-upper-bound} determining the number of events at the upper bound of the sensitivity ($c\tau_{\text{FIP}}\langle\gamma_{\text{FIP}}\rangle\lesssim l_{\text{min}}$, where $l_{\text{min}}$ is the distance from target to the decay volume). Its shape and normalization depend on the value of the distance from the target to the decay volume $l_{\text{min}}$ and the high-energy tail of FIPs within the acceptance for the given experiment. If decreasing $l_{\text{min}}$ for the on-axis placement of the experiment, the high-energy tail would remain unchanged; as a result, we would increase the event rate. This may not be the case for an off-axis placement: the energy spectrum may become much softer and compensate for the decrease of $l_{\text{min}}$ (see text for details). To illustrate these points, we consider a dark scalar with mass $m_{S} = 3\text{ GeV}$ and lifetime $c\tau_{S} = 0.05\text{ m}$ at three various experimental setups at SPS: the configuration from Table~\ref{tab:hypothetical-experiment-parameters}; the same configuration but with $l_{\text{min}} = 10\text{ m}$; and the off-axis experiment with $l_{\text{min}} = 10\text{ m}$, the displacement of the lower edge of its decay volume from the beamline of $1\text{ m}$, and the decay volume length $l_{\text{fid}} = 20\text{ m}$. The number of events at the closer on-axis experiment is larger, while at the off-axis experiment, it is smaller since scalars have smaller energies.

Comparison of the potential of the HIKE, SHADOWS, and SHiP proposals to explore the parameter space of FIPs in beam-dump mode. The FIP physics is here illustrated with HNLs (\textbf{top left panel}), dark scalars with the mixing coupling and with the quartic coupling fixed by $\text{Br}(h\to SS) = 10^{-3}$ (\textbf{top right panel}), dark photons (\textbf{bottom left}), and ALPs with the coupling to photons (\textbf{bottom right}). For all the models except for dark scalars, we use the 90\% CL combined sensitivity of SHADOWS and HIKE$_{\text{dump}}$ from the LoIs~\cite{Alviggi:2839484,CortinaGil:2839661}, while for dark scalars we report our own estimates based on the SHADOWS configuration specified in the LoI (see Appendix~\ref{app:numerics} for detail). For SHiP, we show two curves: the 90\% CL sensitivity, and the domain $N_{\text{events}} >100$, where it may be possible to determine properties of FIPs such as their mass and decay branching ratios. The combined impact of the lower number of protons on the target and the non-optimal placement for SHADOWS and HIKE lead to a significant limitation on their physics potential. Their exclusion domain lies within the FIP identification domain of SHiP. Moreover, in the case of FIPs produced by decays of $B$ mesons (such as e.g. HNLs, dark scalars, and ALPs coupled to fermions), the reach of SHADOWS and HIKE may be overcome by future searches at LHCb~\cite{LHCP:2022} with triggers allowing the use of the muon stations as trackers. We show the sensitivity of such types of searches in the case of HNLs.

\textbf{The left panel}: the HNL parameter space reach of SHiP and SHADOWS in the plane $\xi = U^{2}_{\text{seesaw}}/U^{2}$-HNL mass. The departure of $\xi$ from the seesaw line $\xi = 1$ shows the scale of fine-tuning needed to explain neutrino masses by two HNLs with large couplings. \textbf{The right panel:} the same figure but with the reach of colliders included.

Sensitivity of SHADOWS to dark scalars with the mixing coupling. The red line shows the sensitivity obtained from our computation (Eq.~\eqref{eq:Nevents}) for the experimental setup used in the SHADOWS LoI~\cite{Alviggi:2839484}. The black dots show the sensitivity reported in the LoI. Finally, the green line shows the sensitivity of the projective setup where scalars are required to point to the SHADOWS detector acceptance window and with the decay products acceptance set to 1. The sensitivity of SHADOWS cannot be better than the sensitivity of the projective setup (see text for details).

The fraction of the azimuthal angle covered by the SHADOWS detector window located 2.5 m downwards the end of the decay volume. The solid angle covered by the detector, $\Omega_{\text{det}} = \int \sin(\theta)\Delta \phi_{\text{det,SHADOWS}}d\theta$, matches the simple formula $S_{\text{det}}/L_{\text{to det}}^{2}\approx 4.6\cdot 10^{-3}\text{ sr}$.