11institutetext: IFPU, Institute for Fundamental Physics of the Universe, via
Beirut 2, I-34014 Trieste, Italy 22institutetext: SISSA, via Bonomea 265,
I-34132 Trieste, Italy 33institutetext: INFN, Sezione di Trieste, SISSA, Via
Bonomea 265, 34136, Trieste, Italy 44institutetext: Instituut-Lorentz, Leiden
University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands 55institutetext:
European Organization for Nuclear Research (CERN), Geneva, Switzerland
66institutetext: Institut für Astroteilchen Physik, Karlsruher Institut für
Technologie (KIT), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-
Leopoldshafen, Germany

# Towards the optimal beam dump experiment to search for feebly interacting
particles

Kyrylo Bondarenko 112233 Alexey Boyarsky 44 Richard Jacobsson 55 Oleksii
Mikulenko 44 Maksym Ovchynnikov 6644

###### Abstract

Future searches for new physics beyond the Standard Model are without doubt in
need of a diverse approach and experiments with complementary sensitivities to
different types of classes of models. One of the directions that should be
explored is feebly interacting particles (FIPs) with masses below the
electroweak scale. The interest in FIPs has significantly increased in the
last ten years. Searches for FIPs at colliders have intrinsic limitations in
the region they may probe, significantly restricting exploration of the mass
range $m_{\text{FIP}}<5-10$ GeV/c2. Beam dump-like experiments, characterized
by the possibility of extremely high luminosity at relatively high energies
and the effective coverage of the production and decay acceptance, are the
perfect option to generically explore the “coupling frontier” of the light
FIPs. Several proposals for beam-dump detectors are currently being considered
by CERN for implementation at the SPS ECN3 beam facility. In this paper, we
analyse in depth how the characteristic geometric parameters of a beam dump
experiment influence the signal yield. We apply an inclusive approach by
considering the phenomenology of different types of FIPs. From the various
production modes and kinematics, we demonstrate that the optimal layout that
maximises the production and decay acceptance consists of a detector located
on the beam-axis, at the shortest possible distance from the target defined by
the systems required to suppress the beam-induced backgrounds.

##  1 Introduction

Despite the success of the Standard Model (SM) of particle physics, evidence
for the existence of new physics beyond the Standard Model is already well
established because the origin of neutrino oscillations, dark matter, and the
baryon asymmetry of the Universe is not known. However, we have no solid
predictions of where to search for it. New particles capable of resolving
these problems can have masses from sub-eV to Planck scale and coupling
constants with SM particles ranging many orders of magnitude. At this
crossroad point of particle physics, it is essential to use efficiently
available or planned experimental facilities to push forward different
frontiers of physics, probing whole classes of models simultaneously.

If the mass of a new particle is below the EW scale, it may be produced at
accelerators not only as a resonance but also in decays of SM particles, such
as heavy bosons $W,Z,h$, as well as mesons $\pi,D,B$. This makes this range of
masses of new particles especially interesting from an experimental point of
view. In this mass range, new particles would have escaped detection, not
because of the limit on available accelerator energy, but because their
creation is extremely rare. Numerous searches at past experiments as well as
at the LHC constrain large values of coupling constants, which is why new
particles of this type are often called feebly-interacting particles, or FIPs
(see e.g. Beacham:2019nyx ; Agrawal:2021dbo ).

FIPs can play a direct role in the beyond SM phenomena, like e.g. heavy
neutral leptons or HNLs in the sub-EW mass range explain neutrino masses via
sea-saw mechanism and matter-antimatter asymmetry via their out-of-equilibrium
kinetics in the early Universe at a temperature above 100 GeV. They can also
be a ”portal” that connects the SM sector with a Dark Sector, i.e. the case in
which the Dark Sector particles only interact with ordinary matter via the FIP
mediator (see e.g. Alekhin:2015byh ). FIPs, with their tiny coupling
constants, form a ”coupling frontier” of particle physics, and are part of a
whole class of SM extensions.

If there is no particular reason (e.g. symmetry) for making these new
particles stable, their lifetime scales with the coupling constant $g$ and
mass $m_{\text{FIP}}$ as $\tau_{\text{FIP}}\propto
g^{-2}m_{\text{FIP}}^{-\alpha}$ where $\alpha=1-5$ Alekhin:2015byh . Depending
on the lifetime, different search strategies should be used to probe the FIP
parameter space efficiently. In particular, particles with lifetimes
$c\tau_{\text{FIP}}\gtrsim 1\text{ mm}$ can be searched for via displaced-
vertices schemes at the LHC and future colliders CMS:2022fut ;
Boiarska:2019jcw ; Blondel:2022qqo ; Abdullahi:2022jlv . The limitation on
$c\tau_{\text{FIP}}$ from below comes from numerous backgrounds caused by
events with SM particles, which occur at small displacements. The limitation
from above is FIP-dependent. It is caused by two factors. First, the collider
detectors have short decay volumes of the order of $\mathcal{O}(1\text{ m})$,
typically defined by the dimensions of their inner trackers CMS:2022fut . As a
result, long-lived FIPs mainly decay well beyond the fiducial volume. Second,
to reduce SM backgrounds, one must impose severe selection criteria on
candidate events with FIPs, such as kinematic properties and specific final
states, resulting in low signal efficiency. For instance, a typical selection
efficiency of recent searches for HNLs at CMS CMS:2022fut  was of the order of
1%. Therefore, even if a long-lived FIP decays inside the tracker, the event
will likely be outside the selection acceptance.

Typically, the FIP lifetime and its production rate are controlled by the same
coupling, i.e. a significantly long FIP lifetime also means a small production
rate.111A counter-example is the model of dark scalars with the mixing and
quartic coupling to the Higgs bosons, $\mathcal{L}=\theta m_{h}^{2}Sh+\alpha
hS^{2}$. Depending on the value of $\alpha$, the production may be dominated
by the Higgs boson decay $h\to SS$, while the decay is mediated by $\theta$.
Together with typically low signal efficiencies coming from the trigger and
the event-versus-background selection, it is possible that the FIP production
rate within the acceptance of the searches is insufficient to provide any
sensitivity. This is the case of dark scalars with the coupling through
mixing, dark photons, and axion-like particles Beacham:2019nyx . Even for FIPs
where the production is in principle sufficient (e.g., heavy neutral leptons
Boiarska:2019jcw ), future collider searches have a limited potential to probe
the parameter space of GeV-scale FIPs. This is explained by the behavior of
the FIP decay length $c\tau_{\text{FIP}}\gamma_{\text{FIP}}\propto
g^{-2}m_{\text{FIP}}^{-\alpha-1}$ – for a given coupling $g$, the lifetime
rapidly increases with decreasing $m_{\text{FIP}}$, in other words, quickly
reducing the FIP decay probability within the fiducial volume and hence the
sensitivity. We illustrate these points in Fig. 1, where we show the parameter
space of heavy neutral leptons (HNLs) and dark scalars mixing with Higgs
bosons. Future development of new trackers Alimena:2021mdu ; LHCP:2022  at
ATLAS, CMS, and LHCb may improve the LHC reach for light FIPs, but the
remaining explorable region in the parameter space will still be large (see
Sec. 6). Altogether, this suggests that we need a special experiment to search
for FIPs in the GeV range.

Figure 1: 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. 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 Asaka:2005an . LHC in high luminosity phase
Boiarska:2019jcw ; Abdullahi:2022jlv  and lepton colliders 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 $m_{N}\lesssim 10\text{ GeV}$. 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 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.

In this paper, we argue that the most suitable experimental setup to search
for FIPs with mass below 5 GeV is a beam dump experiment, where an extracted
proton beam hits a dense target, and where the search is performed in a
displaced decay volume. Although having much lower centre-of-mass energy of
collisions than at colliders, beam dumps can deliver extremely high luminosity
by operating with a more intensive proton beam combined with a high-A/Z
target. This means in particular that they are capable of delivering a large
number of mesons within a relatively small forward solid angle, in particular
$B,D$, that may further decay into FIPs in the mass range of interest. There
is no limitation on the decay volume length. It can easily be several tens of
metres to cover a much larger lifetime acceptance than at collider detectors.
Finally, backgrounds can be significantly reduced by placing the decay volume
behind a chain of components designed to suppress beam-induced particle
backgrounds, such as a hadron absorber and a muon deflector, considerably
reducing the need to impose strict signal selection criteria. The best
placement of such an experiment is the SPS accelerator at CERN, operating with
a high-intensity proton beam of energy $E_{p}=400\text{ GeV}$.

Several proposals of beam dump experiments at the SPS have been made
Aberle:2839677 ; CortinaGil:2839661 ; Alviggi:2839484 . They differ in
geometric parameters, including the placement with respect to the beam axis,
the decay volume size, and the detector angular coverage, both in terms of
production acceptance and decay acceptance. The experiments also differ in the
choice of material for the proton target. Maximum production of FIPs, and
simultaneously maximum suppression of background from pion and kaon decays to
muons and neutrinos, are achieved with a target of the highest possible atomic
mass and atomic number, as well as minimized internal cooling for density.
However, to discuss the optimal experimental layout, we assume in this paper
the same target material for all experiments.

$l_{\text{min}}$ | $S_{\text{det}}$ | $l_{\text{fid}}$ | $l_{\text{det}}$ | $r_{\text{displ}}$
---|---|---|---|---
38 m | $4\times 6\text{ m}^{2}$ | 50 m | 15 m | 0 m
Table 1: Parameters of the hypothetical experiment used as a reference
experiment in our estimates: the longitudinal distance from the target to the
beginning of the decay volume; the transverse dimensions of the decay volume
and the detector, the longitudinal length of the decay volume; the
longitudinal length of the detector; the distance from the centre of the
detector in the transverse plane to the beamline. Here and below, we assume
that the decay volume is oriented parallel to the beamline, which is motivated
by the typical constraints from available space and infrastructure.

In this paper, we perform an analysis of the sensitivity to FIPs with respect
to the geometric parameters of a beam dump experiment, in a maximally model-
independent way, in order to find the optimal configuration. We start with an
on-axis experiment specified in Table 1. Its close analog is the SHiP
experiment SHiP:2015vad ; Aberle:2839677 . We then study how the FIP
sensitivity is affected by changing the parameters from the table. To be
model-independent, we consider a few FIP models covering a wide class of
production mechanisms and decay modes.

Our main results are summarized in Figures 5, 6, 7, 8. We demonstrate that the
setup in Table 1 is optimal for searching for FIPs independently of their
lifetime, being also compatible with the absence of backgrounds. One of the
main reasons for providing the largest signal acceptance is the on-axis
placement. The off-axis location leads to a substantial loss of acceptance,
and significantly worsens the ability to reconstruct properties of FIPs such
as mass, spin, and decay modes, see Figure 8.

The paper is organized as follows. In Sec. 2, we start from the expression for
the number of events and discuss the FIP phenomenology (subsection 3),
describing, in particular, their production and decay modes. In Sec. 4, we
analyze how the number of events at the lower bound of the sensitivity varies
with the experimental configuration, considering separately its placements on-
axis (Sec. 4.1) and off-axis (Sec. 4.2) relatively to the beamline. In Sec.
4.3, we study the impact of the configuration on the potential to probe FIPs
at the upper bound of the sensitivity. In Sec. 5, we apply our findings to
compare the sensitivity of the experiments proposed at SPS, SHiP, SHADOWS, and
HIKE. Finally, in Sec. 6, we make conclusions. Appendices contain all the
relevant technical information on the phenomenology of FIPs and the
calculations.

##  2 Number of signal events

Let us start with the expression for the number of events in the regime of
large lifetimes, where the typical decay length of FIPs,
$c\tau_{\text{FIP}}\langle\gamma_{\text{FIP}}\rangle$, is much larger than the
characteristic scale of the experiments $\simeq 100\text{ m}$ (the opposite
cases of short lifetimes
$c\tau_{\text{FIP}}\langle\gamma_{\text{FIP}}\rangle\lesssim l_{\text{min}}$
is discussed in Sec. 4.3). In this case, the number of events may be
represented in the following schematic form (see Appendix A):

$N_{\text{events}}\approx N_{X,\text{prod}}\times\frac{l_{\text{fid}}\langle
p_{\text{FIP}}^{-1}\rangle}{c\tau_{\text{FIP}}m_{\text{FIP}}}\times\epsilon_{\text{geom}}\times\epsilon_{\text{rec}}\times\text{Br}_{\text{vis}}$
(1)

Here, $N_{X,\text{prod}}=N_{\text{PoT}}\times\text{Br}(pp\to X)$ is the total
number of the FIPs produced in the collisions, with $N_{\text{PoT}}$ being the
number of protons on target, and $\text{Br}(pp\to X)$ the probability of
producing of FIPs by any mechanism; the second factor is the decay probability
in the regime of large lifetimes, with $\langle p_{\text{FIP}}^{-1}\rangle$
being the mean inverse momentum for the FIP at the experiment;
$\epsilon_{\text{geom}}$ is the overall geometric acceptance, following from
geometric limitations of the decay volume and detectors (discussed in details
below); $\text{Br}_{\text{vis}}$ is the branching ratio of the decay of FIPs
into states visible at the given experiment; finally, $\epsilon_{\text{rec}}$
is the total reconstruction efficiency – the fraction of events within the
geometric acceptance that may be reconstructed.

Independently of the configuration, we will assume $\epsilon_{\text{rec}}=1$,
and that $\text{Br}_{\text{vis}}$ includes all decays with at least two
electrically charged particles or two photons.

Considering two experiments located at the same facility (such that the beam
energy and target configuration are the same), for the ratio of the number of
events (1) one has

$\frac{N_{\text{events,1}}}{N_{\text{events,2}}}\approx\frac{l_{\text{fid,1}}}{l_{\text{fid,2}}}\times\frac{\epsilon_{\text{geom,1}}}{\epsilon_{\text{geom,2}}}\times\frac{\langle
E_{\text{FIP}}^{-1}\rangle_{1}}{\langle E_{\text{FIP}}^{-1}\rangle_{2}}$ (2)

Therefore, to compare the lower bounds of sensitivity of two experiments, we
need to understand the behavior of $\epsilon_{\text{geom}}$ and $\langle
E_{\text{FIP}}^{-1}\rangle$.

###  2.1 Geometric acceptance

Schematically, the geometric acceptance is given by

$\epsilon_{\text{geom}}\simeq\epsilon_{\text{FIP}}\times
l_{\text{fid}}^{\text{eff}}/l_{\text{fid}}\times\epsilon_{\text{dec}},$ (3)

see Fig. 2.

Figure 2: Illustration of the impact of different contributions to the
geometric acceptance defined by Eq. (3). 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
$\epsilon_{\text{dec}}$.

The first factor is the FIP acceptance, i.e., the fraction of FIPs with
trajectories pointing to the cross-section of the end of the
detector.222Naively, the mentioned definition of the FIP acceptance looks too
restrictive, as it does not take into account the FIPs decaying inside the
decay volume but not pointing to the detector. However, because of the
4-momentum conservation, FIPs that do not point to the detector typically
cannot decay into particles pointing to the detector. Therefore, instead of
considering FIPs decaying in any direction inside the decay volume, we
considered only the FIPs that already point to the detector.
$l_{\text{fid}}^{\text{eff}}$ is effective fiducial volume length – the mean
length inside the decay volume passed by FIPs pointing to the end of the
detector. If the considered experiment is located on-axis, then
$l_{\text{fid}}^{\text{eff}}\approx l_{\text{fid}}$. However, in case of the
off-axis placement, parallel to the beamline, $l_{\text{fid}}^{\text{eff}}$
gets effectively reduced. Finally, the last factor is the decay product
acceptance, i.e., the fraction of decays of FIPs within
$\epsilon_{\text{FIP}}$ with at least two of their decay products pointing to
the end of the detector. Roughly, the decay products of FIPs with the gamma
factor $\gamma_{\text{FIP}}$ have the opening angle333For simplicity, we
considered here 2-body decays; however, for 3-body decays, the situation is
qualitatively similar.

$\Delta\theta_{\text{dec}}\sim 2/\gamma_{\text{FIP}}$ (4)

If this angle becomes comparable with the angle covered by the detector as
seen from the FIP’s decay point, then $\epsilon_{\text{dec}}$ would
significantly reduce the event rate.

##  3 Kinematic distributions of FIPs

Figure 3: 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).

To understand the behavior of $\epsilon_{\text{FIP}}$ and
$\epsilon_{\text{dec}}$, we need to study in a model-independent way how FIPs
may be produced in proton-target collisions and how they decay. To this
extent, we consider different types of FIPs: dark photons, dark scalars, HNLs,
and ALPs with photon coupling Beacham:2019nyx . By considering all of them, we
can perform the analysis in a maximally model-independent fashion. The
dominant production mechanisms and decay modes are shown in Fig. 3 and in
Table 2. The production channels are decays of mesons (light unflavoured
mesons $\pi,\eta,\eta^{\prime},\rho^{0}$, as well as heavy flavored mesons
$B,D$), or the direct production through proton bremsstrahlung, Drell-Yan
process, coherent proton-nucleus and photon-nucleus scattering. We generate
the distribution of the light mesons at the SPS using the approach of
Dobrich:2019dxc  (see also Jerhot:2022chi ), and use the distribution of $B,D$
mesons from CERN-SHiP-NOTE-2015-009 . We follow the description of the
bremsstrahlung process from SHiP:2020vbd  and the coherent scattering from
Dobrich:2019dxc . For the details on the derivation of the distributions of
FIPs from these production channels, see Appendix A.

FIP | Prod. modes | Decay modes
---|---|---
DP $V$ | $\begin{cases}\pi^{0}/\eta\to V,\ m_{V}<m_{\eta}\\\ \text{Brem/DIS},\ m_{V}>m_{\eta}\end{cases}$ |  $V\to ll$ $V\to 2\pi,3\pi,KK,m_{V}\lesssim 1\text{ GeV}$ $V\to qq,m_{V}\gtrsim 1\text{ GeV}$
ALPγ $a$ |  $\gamma+Z\to a+Z$ $p+Z\to p+Z+a$ | $a\to\gamma\gamma$
Scalar $S$ | $\begin{cases}K\to S+\pi,\ m_{S}<m_{K}-m_{\pi}\\\ B\to S+X,\ m_{S}>m_{K}+m_{\pi}\end{cases}$ |  $S\to ll$ $S<\pi\pi/KK,m_{S}<2\text{ GeV}$ $S\to qq,m_{S}>2\text{ GeV}$
HNL $N$ | $\begin{cases}K\to N+X,\ m_{N}<m_{K},\\\ D\to N+X,m_{N}<m_{D_{s}},\\\ B\to N+X,\ m_{N}>m_{D_{s}}\end{cases}$ |  $N\to ll\nu$ $N\to\pi^{0}\nu,\eta\nu,\pi l,m_{N}\lesssim 1\text{ GeV}$ $N\to qq\nu,qql,m_{N}\gtrsim 1\text{ GeV}$
Table 2: Dominant production and decay modes of GeV-scale FIPs in proton-
proton collisions at the SPS. We consider the dark photon, the ALP with the
photon coupling, the Higgs-like scalar with mixing coupling, and HNLs with the
dominant coupling to electron neutrinos. For more details, see
Ovchynnikov:2023cry .

Figure 4: Kinematics of FIPs produced in proton-target collisions at the SPS.
Molybdenum target is considered. 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 2). 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 1
is indicated with arrows and the vertical dashed line. 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.
Ovchynnikov:2023cry  for details.

The solid angle distributions

$df_{\text{FIP}}/d\Omega_{\text{FIP}}\sim
df_{\text{FIP}}/d\cos(\theta_{\text{FIP}}))$ (5)

of the FIPs produced by these mechanisms, as well as their energy
distributions are shown in Fig. 4. Because of the kinematics of the collisions
with a fixed target, the bulk of the distributions of FIPs are contained
within a relatively small forward solid angle around the beam axis, being flat
up to and quickly dropping at large angles $\theta>\theta_{\text{flat}}$. The
direct production processes are characterized by very small typical transverse
momentum compared to the momentum of the incoming proton. In the case of
Drell-Yan and proton bremsstrahlung, it is of the order of $m_{p}\sim 1\text{
GeV}$ Gorbunov:2014wqa . Given very large typical energies of the FIPs
produced by these mechanisms, $E_{\text{FIP}}\sim 100-200\text{ GeV}$, the
angle is $\theta_{\text{flat}}\simeq 10\text{ mrad}$. For the Primakov
production off nuclei and nucleons, $\theta_{\text{flat}}$ is determined by
typical transverse momenta carried by virtual photons, being
$\theta_{\text{flat}}\simeq m_{p}/E_{\text{beam}}=2.5\cdot 10^{-3}\text{ rad}$
Dobrich:2015jyk .

Light mesons produced in collisions have characteristic $p_{T}$ of the order
of $\Lambda_{\text{QCD}}$, and relatively small mean energy of order of
$\langle E\rangle\simeq 20-30\text{ GeV}$, which leads to
$\theta_{\text{flat}}$ being of the order of a few tenths of mrad
Dobrich:2019dxc .

Heavy mesons $D,B$ have large $p_{T}$ of the order of their mass, but, at the
same time, much larger characteristic energies. As a result, their
distributions start dropping at even smaller angles
$\theta_{\text{flat}}\lesssim 10\text{ mrad}$. For lighter masses, the FIPs
produced by decays of $B,D$ may have broader angular distribution. The reason
is an additional transverse momentum of the order of the energy of the FIP at
the rest frame of the decaying meson, which may be as large as the meson mass
for FIPs with $m_{\text{FIP}}\ll m_{\text{meson}}$.

The main uncertainty for the FIP production (which translates to the main
uncertainty in the signal rate estimates) comes from the thickness of the
target, which gives rise to a cascade production of the SM particles that may
potentially decay into FIPs. The cascade population affects both the total
yield of the particles and the shape of the angle-energy distribution: the
angular distribution of these extra particles is broader, while the energy
distribution is shifted to lower energies.

While for the thin target and SPS beam energies the fluxes of mesons have been
measured Adamovich:1992cv ; Aguilar-Benitez:1991hzq , observing the cascade
flux in the case of thick targets is more complicated (as the mesons or their
decay products may be absorbed inside the target) and requires a dedicated
experiment. Therefore, we may only estimate it via simulations. Such estimates
have already been performed by the SHiP collaboration SHiP:2020vbd ; CERN-
SHiP-NOTE-2015-009 . They suggest that for heavy flavor production, the
cascade enhancement gives a factor of 2 to the total meson yield and is
important for both on-axis and off-axis experiments (see also Dobrich:2018jyi
); we account for these cascade mesons in our calculations. For lighter
mesons, the enhancement may be a factor of 10 or larger SHiP:2020vbd .
However, these mesons are very soft, with energies of order of 1 GeV, and
decay into a very low-energy FIP produced in a broad range of polar angles.
Even if such FIPs would decay inside the decay volume, the products would not
be properly reconstructed because of low energies.444For instance, selection
criterion at SHiP for the decay products is $E>1\text{ GeV}$. As a result,
only a tiny fraction of such cascade mesons contributes to the signal yield.
Therefore, we may safely neglect them.

The decay modes of FIPs may differ by the number of the decay products and
their phase space if the number of products is fixed, which comes from the
ratio of the FIP-to-decay product masses and the matrix element of the decay.
The main decay modes of the FIPs are provided in Table 2.

Another uncertainty in calculating the number of events may come from
theoretical uncertainties in the description of the phenomenology of FIPs.
Namely, for the particles such as Higgs-like scalars, various ways to describe
hadronic decays may change the decay width by 1-2 orders of magnitude (see,
e.g., Boiarska:2019jym ; Gorbunov:2023lga  and references therein), which
affects both the lower and upper bounds of the sensitivity. While it may be
important for the full sensitivity region, it is not relevant when computing
the ratio of the number of events in the regime of long lifetimes, as we do in
this paper – just the parameter space gets shifted.

##  4 Effect on the number of signal events from varying parameters of the
experiment

In this section, we study the impact of varying the geometric parameters of
the experiment on the number of events. To illustrate the effect of the decay
products acceptance and the FIP angular distribution, we consider the
following FIPs and properties:

  * –

HNLs with masses $0.5\text{ GeV}$, $1.5\text{ GeV}$ produced by decays of $D$
mesons, and $4\text{ GeV}$ produced by decays of $B$.

  * –

Dark scalars with masses $0.5\text{ GeV}$ and $4\text{ GeV}$ produced by
decays of $B$.

  * –

Dark photons with masses $m_{V}=300\text{ MeV}$ produced by decays of $\eta$
and $m_{V}=1\text{ GeV}$ by proton bremsstrahlung.

  * –

ALPs with photon coupling with masses $m_{a}=300\text{ MeV}$ and 1 GeV.

###  4.1 On-axis location

Let us first assume the on-axis placement of the experiment. We analyze how
the number of long-lived FIPs is affected by changing the distance to the
decay volume $l_{\text{min}}$, and its length $l_{\text{fid}}$ (the case of
short-lived FIPs with $c\tau_{\text{FIP}}\gamma_{\text{FIP}}\lesssim
l_{\text{min}}$ is discussed in Sec. 4.3).

Figure 5: 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. 2 for details) under change of the distance to the decay volume
$l_{\text{min}}$ (top panels) and its length $l_{\text{fid}}$ (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 1. For convenience, we normalize the
number of events to the one for the configuration from Table 1.

The dependence of the number of events on $l_{\text{min}}$ is shown in Fig. 5
(top panels). We normalize all the values to the corresponding values of the
SHiP-like experiment from Table 1. The main impact of $l_{\text{min}}$ is in
defining the solid angle covered by the detector as seen from the target,

$\Omega_{\text{det-
target}}=S_{\text{det}}/(l_{\text{min}}+l_{\text{fid}}+l_{\text{det}})^{2},$
(6)

and hence the fraction of FIPs $\epsilon_{\text{FIP}}$ pointing to the
detector. If the FIP’s angular distribution $df_{\text{FIP}}/d\Omega$ (Fig. 4)
is flat within the angles covered by the detector (the case of light HNLs,
dark scalars, and ALPs), then $\epsilon_{\text{FIP}}\propto\Omega_{\text{det-
target}}$. From Eq. (6), it follows that to increase $N_{\text{events}}$ for
the SHiP-like configuration by a factor of two, it is necessary to decrease
$l_{\text{min}}$ from $l_{\text{min}}=38\text{ m}$ to $l_{\text{min}}\approx
8\text{ m}$. At the same time, if $df_{\text{FIP}}/d\Omega$ is collimated and
falls at the boundaries (dark photon, ALPs, and heavy HNLs/dark scalars), the
effect on $\epsilon_{\text{FIP}}$ is even smaller. On the other hand, the
price of such a close placement would be a significant increase in the SM
background.

The effect of $l_{\text{fid}}$ is less trivial, see the bottom panels of
Figure 5. It affects the product $\epsilon_{\text{FIP}}\times
l_{\text{fid}}\times\epsilon_{\text{dec}}$. The impact on
$\epsilon_{\text{FIP}}$ is similar to the $l_{\text{min}}$ case (Eq. (6)). The
second factor comes from the decay probability of long-lived FIPs. In
addition, by increasing $l_{\text{fid}}$ and maintaining the aperture of the
detector constant, $\epsilon_{\text{dec}}$ decreases, and vice versa. Indeed,
$l_{\text{fid}}$ enters the solid angle covered by the detector as seen from
the beginning of the decay volume,

$\Omega_{\text{det-fid}}=S_{\text{det}}/(l_{\text{fid}}+l_{\text{det}})^{2}$
(7)

If $l_{\text{fid}}$ is too large, the opening angle between the FIP’s decay
products $\Delta\theta_{\text{dec}}$ (Eq. (4)) becomes comparable with the
detector size, and such events do not contribute. Hence,
$l_{\text{fid}}\times\epsilon_{\text{dec}}$ remains constant under further
increase of $l_{\text{fid}}$.

As a result, the product
$\epsilon_{\text{FIP}}\times\epsilon_{\text{dec}}\times l_{\text{fid}}$ first
scales linearly with $l_{\text{fid}}$, then reaches maximum where
$\Delta\theta_{\text{dec}}$ becomes comparable with the detector size, and
then decreases as a result of decreasing $\Omega_{\text{det-target}}$.

For the FIPs produced by decays of heavy mesons, the maximum is around
$l_{\text{fid}}=50\text{ m}$. The situation is different for the FIPs produced
by decays of $\pi,\eta$, bremsstrahlung, or Primakov process. In these cases,
either the FIPs have small masses or very large energies (Fig. 4), which leads
to a very large $\gamma_{\text{FIP}}$. Hence, the suppression by
$\epsilon_{\text{dec}}$ is not so severe, meaning that the number of events
increases with $l_{\text{fid}}$ over a very large range.

###  4.2 Off-axis location

Let us now analyze the impact on the number of signal events when displacing
the detector off-axis. As in the previous subsection, we first consider the
configuration with the same dimensions and distance from the beam dump as in
Table 1. We will consider only the parallel orientation of the decay volume
and detector relative to the beamline, motivated by the limitations typically
imposed by the infrastructure. Let us start with increasing the transverse
displacement of the centre of the detector relative to the beamline,
$r_{\text{displ}}$. We vary $r_{\text{displ}}$ from 0 to $5$ m. Note that
$r_{\text{displ}}$ is not the same as the off-axis displacement of the side of
the decay volume. In particular, for the considered configuration, the latter
is $>0$ only if $r_{\text{displ}}>2\text{ m}$. For $r_{\text{displ}}=3\text{
m}$, the gap between the side of the decay volume and the beam axis is 1 m.

Figure 6: 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. 2) 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. 4). Depending on the FIP, the decrease may be an order of magnitude or
larger (top panel). It is impossible to compensate for this decrease by
placing the experiment closer to the target (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 1. 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 1.

The dependence of the number of events is shown in Fig. 6, top panels. For all
FIP models considered, the number of events decreases with $r_{\text{displ}}$.
The reason for this is that the FIP angular distribution decreases at the
larger polar angles (Fig. 4) covered with the off-axis displacement
$r_{\text{displ}}$. In the case that the whole detector is placed off-axis
($r_{\text{displ}}>2\text{ m}$, such that the side of the decay volume is
entirely away from the beam axis), we would not only shift the detector in the
domain of large $\theta$, but also decrease the azimuthal coverage in the
domain of small polar angles, which further reduces the FIP acceptance.
Finally, the acceptance gets further suppressed by the shortening of the
effective decay volume length for FIPs that enter from the side (Eq. (3)).

For heavy FIPs, the drop is more significant, which is explained by more
forward-pointing angular distributions and smaller decay acceptance. The
decrease in $\epsilon_{\text{dec}}$ is due to the softer energy spectrum at
large polar angles. It leads to an increase of the typical opening angle
between the decay products $\theta\simeq 1/\gamma_{\text{FIP}}$ (Eq. (4)), and
hence a decrease in $\epsilon_{\text{dec}}$.

It is also interesting to compare the effect of shortening $l_{\text{min}}$ in
the case of off-axis and on-axis placements (Fig. 5 and discussion therein).
The behavior of the number of events is illustrated in Fig. 6, bottom panels.
Unlike the on-axis case, the number of events tends to grow if increasing
$l_{\text{min}}$, which is again a result of the very forward-pointing angular
distributions and decreasing $\epsilon_{\text{dec}}$ at the off-axis
locations.

###  4.3 Upper bound of the sensitivity for on-axis and off-axis

Finally, let us also examine the role of the geometric parameters in
determining the potential of the experiment to search for FIPs with large
couplings, for which the typical FIP decay length is smaller than the distance
to the decay volume,
$c\tau_{\text{FIP}}\langle\gamma_{\text{FIP}}\rangle\lesssim l_{\text{min}}$.

At the upper bound, the number of events is proportional to the following
integral (see Appendix A):

$N_{\text{events}}^{\text{upper bound}}\propto\epsilon=\int dL\int
dE_{\text{FIP}}\\\
\frac{1}{c\tau_{\text{FIP}}\gamma_{\text{FIP}}}\exp\left[-\frac{L}{c\tau_{\text{FIP}}\gamma_{\text{FIP}}}\right]f^{l_{\text{min}}}_{E_{\text{FIP}},L},$
(8)

where $L$ is the modulus of the FIP decay position, and
$f^{l_{\text{min}}}_{E_{\text{FIP}},L}\equiv\bigg{\langle}\frac{df_{\text{FIP}}}{dE_{\text{FIP}}}\epsilon_{\text{dec}}\bigg{\rangle}_{\theta}$
is the FIP distribution averaged over the angular coverage of the detector.
The integral (8) effectively plays the role of geometric acceptance in the
case of short-lived FIPs. It is sensitive to the high-energy tail of the FIP
distribution ($E_{\text{FIP}}>\langle E_{\text{FIP}}\rangle$).

We will start with the simpler case of the on-axis configuration. In this
case, $f^{l_{\text{min}}}_{E_{\text{FIP}},L}$ depends weakly on
$l_{\text{min}}$ and on $L$: $f^{l_{\text{min}}}_{E_{\text{FIP}},L}\approx
f_{E_{\text{FIP}}}$. Indeed, independently of $l_{\text{min}}$, the detector
covers the far-forward domain, which determines the high-energy tail of the
distribution function. As a result, the only impact of decreasing
$l_{\text{min}}$ comes from decreasing $L_{\text{min}}\approx l_{\text{min}}$.
Namely, the whole integral is saturated around its value $L_{\text{min}}$:

$\epsilon\approx\int
dE_{\text{FIP}}\exp\left[-\frac{l_{\text{min}}}{c\tau_{\text{FIP}}\gamma_{\text{FIP}}}\right]f_{E_{\text{FIP}}}$
(9)

Up to logarithmic corrections, at a fixed mass, the upper bound of the
sensitivity, i.e., the smallest lifetimes that may be probed, scales as
$\tau_{\text{FIP}}^{\text{upper}}\propto l_{\text{min}}^{-1}$.

Let us now consider the off-axis case, concentrating on the case where the
whole detector lies off-axis. For the detector dimensions in Table 1, this
case would correspond to the transverse displacement of the centre of the
detector $r_{\text{displ}}>3\text{ m}$. In this case, the situation is less
trivial. First, the FIP energy spectrum becomes softer (remind Fig. 4) at
large polar angles. As a result, the value of the $E_{\text{FIP}}$ integral in
Eq. (8) at fixed $L$ decreases compared to the off-axis case. Second, the FIPs
pointing to the closest (to the beamline) part of the detector enter the decay
volume from the side (Fig. 2). This also decreases
$f^{l_{\text{min}}}_{E_{\text{FIP}},L}$. This behavior is in tension with the
exponent in Eq. (8): small values of $L$ close to $L_{\text{min}}$ correspond
to maximal polar angle and hence smaller energies/acceptance. This destructive
interplay destroys the positive impact of decreasing $l_{\text{min}}$ on the
potential to probe short FIP lifetimes. As a result, the fully off-axis
configurations have a lower number of signal events than the on-axis, even
given much smaller $l_{\text{min}}$.

Figure 7: The energy distribution of the function (10) 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 1; 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.

To illustrate these qualitative arguments, let us consider three setups: the
on-axis experiment from Table 1, the same on-axis experiment but with
$l_{\text{min}}=10\text{ m}$, and finally an off-axis experiment with
$l_{\text{min}}=10\text{ m}$ and $r_{\text{displ}}=3\text{ m}$. For the FIPs,
we will consider dark scalars with mass $m_{S}=3\text{ GeV}$ and lifetime
$c\tau_{S}=5\text{ cm}$, which corresponds to the upper bound of the
sensitivity.

The behavior of the integrand of (8),

$\frac{d\epsilon}{dE_{\text{FIP}}}=\int
dL\frac{1}{c\tau_{\text{FIP}}\gamma_{\text{FIP}}}\exp\left[-\frac{L}{c\tau_{\text{FIP}}\gamma_{\text{FIP}}}\right]f^{l_{\text{min}}}_{E_{\text{FIP}},L},$
(10)

for the three setups is shown in Fig. 7. Obviously, the on-axis setup with
$l_{\text{min}}=10\text{ m}$ has the largest flux of FIPs. However, despite
the fact that the off-axis decay volume is located $\sim 4$ times closer to
the beam dump than the on-axis setup with $l_{\text{min}}=38\text{ m}$, the
off-axis setup has a $\simeq 30$ times lower value of the total integral (8),
following from the factors described above.

##  5 Comparison between the experiment proposals under study at the CERN SPS

In this section, we make a comparison of the physics yields between the three
experiment proposals that are currently being considered for implementation in
the ECN3 beam facility at CERN’s SPS accelerator, HIKE CortinaGil:2839661 ,
SHADOWS Alviggi:2839484 , SHiP Aberle:2839677 . All proposals are based on a
similar detector setup in that they conceptually consist of large decay
volumes followed by spectrometers and particle identification, together with
various veto systems. HIKE is primarily a kaon experiment located on-axis. It
requires a specialised beam setup with a kaon target, a secondary beam line
for the kaon selection, and an absorber of copper/iron for the remaining
proton beam and secondary hadrons from the kaon target. HIKE’s distance to the
kaon target is defined by the optimisation for the kaon physics programme,
resulting in a relatively large distance between HIKE’s decay volume and the
absorber. The kaon physics optimisation imposes limitations on the maximum
beam intensity that is due to both the secondary beam line setup and the
detector, effectively four times lower than it would be for SHiP. HIKE is also
proposed to partially operate in beam-dump mode for FIP physics. In this mode,
the kaon target is moved aside to let the proton beam of the same intensity be
directly dumped on the absorber. With the very small solid angle coverage,
HIKE has mainly sensitivity to dark photons and ALPs with photon coupling.
SHADOWS is an off-axis experiment that would be located alongside HIKE’s kaon
beam, downstream of the absorber, with the detector covering angles
$\theta\gtrsim 30\text{ mrad}$ (Fig. 4). SHADOWS’ distance to the proton
absorber is defined by the infrastructure around the absorber, shielding
requirements, a muon sweeper, and the subsequent beam line elements. SHADOWS
would operate together with HIKE in beam-dump mode, with SHADOWS searching for
the FIPs flying off-axis and HIKE for those produced in the far-forward
region. In this respect, it is expected that the beam time for HIKE and
SHADOWS is split between periods of kaon physics and beam-dump physics. SHiP
is instead a dedicated on-axis experiment with the detector located as close
as possible to a compact target station housing a target of
molybdenum/tungsten that is optimised for FIP physics. SHIP’s distance to the
target is defined by a hadron stopper with minimum depth, and a specialised
magnetic muon deflector that sweeps the muon flux away from the fiducial
volume. SHIP’s location allows it to cover all the FIP production modes at the
SPS.

In Fig. 8 we show the 90% CL sensitivities of HIKE${}_{\text{dump}}$ \+
SHADOWS and of SHIP to HNLs, dark scalars, dark photons, and ALPs with the
coupling to photons. In addition, for SHiP, we include the iso-contours
corresponding to $N_{\text{events}}=100$. Such a large number of events allows
not only to establish the existence of a new particle but also identify its
properties such as branching ratios of various decay channels, precise mass,
etc. Details of the sensitivity estimates are described in Appendix A. We see
that the lower bound of the sensitivity of SHADOWS+HIKE${}_{\text{dump}}$ is
close to the 100 events line of SHiP. This may be easily understood if using
Eq. (1). Namely, for SHADOWS, the product $N_{\text{PoT}}\times
l_{\text{fid}}$ is 10 times smaller than at SHiP. The rest of the suppression
in the number of events at SHADOWS comes from the off-axis placement parallel
to the beamline, significantly decreasing $\epsilon_{\text{FIP}}\times\langle
p^{-1}_{\text{FIP}}\rangle$ – by a factor $1/3-1/20$ depending on the FIP
mass. For HIKE, the product $N_{\text{PoT}}\times l_{\text{fid}}$ is only
twice smaller than at SHiP. However, its detector covers $\sim 20$ times
smaller solid angle as seen from the target and the beginning of the decay
volume, which results in a much lower overall acceptance.

Moreover, despite the fact that SHADOWS is placed much closer to the target,
which could naively mean that it should be able to probe larger couplings of
FIPs, Fig. 8 shows that this is not the case. One of the reasons for this is
that FIPs flying off-axis have a much softer energy spectrum and hence shorter
typical decay length (see Fig. 6 and discussions therein). An additional
reason is that the most energetic FIPs enter the decay volume from the side,
resulting in a shorter flight path within the decay volume and thus
suppressing the fraction of their decays inside the decay volume.

Figure 8: 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 (top left panel), dark scalars with the mixing
coupling and with the quartic coupling fixed by $\text{Br}(h\to SS)=10^{-3}$
(top right panel), dark photons (bottom left), and ALPs with the coupling to
photons (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 Alviggi:2839484 ; CortinaGil:2839661 , while for dark scalars we report
our own estimates based on the SHADOWS configuration specified in the LoI (see
Appendix A 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 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.

##  6 Conclusions

Future searches for new physics beyond the Standard Model are without a doubt
in need of a diverse approach and experiments with complementary sensitivities
to different classes of models, ranging both mass and coupling scales. A
theoretically and experimentally attractive case for new physics that is
largely unexplored but within reach at current facilities, consists of
particles with masses below the electroweak scale that may be produced at
accelerators in decays of SM particles. Their couplings should be small in
order to avoid existing experimental constraints. Hence, they are often called
feebly-interacting particles, or FIPs.

FIPs are now being actively searched for at the LHC and will be searched for
at future collider experiments. However, this type of search has limitations
in probing the parameter space of long-lived FIPs in the mass range
$m_{\text{FIP}}\lesssim 5-10\text{ GeV}$, see Fig. 1, mainly because of strong
backgrounds and too short decay volumes as compared to the typical decay
length of light FIPs. Beam dump-like experiments, characterized by the
possibility of extremely high luminosity at relatively high energies, and
effective coverage of the production and decay acceptance, are the perfect
setup to generically explore the ”coupling frontier” and the case of FIPs with
mass below the B meson mass. Beam dump experiments can be equipped with long
decay volumes and be located at some distance from the target to accommodate
absorbers and deflectors of SM particles, as well as veto systems, to reduce
the backgrounds. The best placement for such a FIP facility is the SPS
accelerator at CERN, where the existing infrastructure and the currently
available proton yield of up to $4\times 10^{19}$ protons per year at 400 GeV
make it possible to implement and operate a-state-of-the-art experiment at
relatively low costs. There are currently three experiment proposals being
considered for implementation in the ECN3 beam facility at CERN’s SPS
accelerator, HIKE CortinaGil:2839661 , SHADOWS Alviggi:2839484 , SHiP
Aberle:2839677 . Their respective objectives and layouts are briefly
summarised in Sec. 5.

In order to determine the optimal experimental geometry, we have made an in-
depth study of the dependence of the number of FIP signal events at the lower
(Secs. 4.1, 4.2) and the upper (Sec. 4.3) bounds of the sensitivity, as a
function of the length of the decay volume, the distance from the target, and
the transverse displacement with respect to the beamline, and other
parameters. We have performed the analysis for several classes of models of
FIPs (“portals”) with different production channels and decay modes, see Sec.
3. Given that all the proposals claim to reach a background-free regime, we do
not consider backgrounds specifically other than the constraints that such
assumptions impose on the geometric parameters.

In particular, we analyzed the effect of displacing the detector off-axis for
different portals. Generically, it leads to a decrease in the number of events
for two specific reasons: first, the angular flux of FIPs decreases at large
angles (Fig. 4, and second, it causes geometric shortening of the effective
length of the decay volume along trajectories pointing to the detector (Fig.
2). The impact of the off-axis placement depends on the dominant FIP
production channel, i.e., by the decays of mesons (such as for HNLs, ALPs, and
dark scalars), or directly in proton-target collisions (dark photons). In the
former case, shifting the decay volume entirely to the side of the beam axis
such that it covers polar angles $\theta\gtrsim 10\text{ mrad}$, leads to a
loss of up to a factor of five in the number of events at the lower bound,
depending on the FIP mass, as compared to the reference configuration in Table
1. In the latter case, the same decrease in yield is seen as soon as the side
of the decay volume wall gets off-axis. Practically, this means that the off-
axis configurations have no sensitivity to dark photons.

In contrast, we have seen that changing the distance from the target to the
decay volume, and its length, for the on-axis configuration does not affect
the number of events at the lower bound by more than a factor of two (Fig. 5)
over a broad range of parameters. On the other hand, significantly decreasing
the distance to the target affects the complexity of the experiment and
background suppression, and hence the cost. Therefore, optimizing the on-axis
configuration is rather a subject of minimizing background and cost.

Finally, we have applied the analysis to the three ECN3 proposals (Sec. 5). In
Fig. 8, we show the 90% CL sensitivities of HIKE${}_{\text{dump}}$ \+ SHADOWS
and of SHIP to HNLs, dark scalars, dark photons, and ALPs with the photon
coupling. We also include the projection of the sensitivity of the LHC
experiments, taking LHCb as an example, for the case that new triggers will be
developed, allowing, e.g., the use of the muon chambers as trackers for FIPs
independently of their production channel LHCP:2022 . In this case, the
sensitivity of SHADOWS+HIKE may be limited in the domain of heavy FIPs with
$m\gtrsim 2\text{ GeV}$.

Figure 9: 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. The right panel: the same
figure but with the reach of colliders included.

For SHiP, we also include the iso-contours corresponding to
$N_{\text{events}}=100$ in Fig. 8. Such a large number of events allows not
only to establish the existence of a new particle but also identify its
properties, such as branching ratios of various decay channels, precise mass,
etc. We conclude that the combination of lower beam intensity and non-optimal
geometric placement for HIKE and SHADOWS worsens their potential to explore
the “coupling frontier” compared to SHiP. In particular, assuming that all the
experiments operate in background-free regime, in the domain of small
couplings, where SHADOWS and HIKE may detect 1 event at 90% CL, SHiP would be
able to reconstruct their parameters, such as mass, spin, and the probability
of various decays.

To stress the importance of the maximization of the FIP event yield, let us
look closer at the model of HNLs. The main motivation for the HNLs is that
they provide a very simple explanation of the observed active neutrino masses,
in a way very similar to other fermions of the SM, i.e., by the mixing of
left-handed and right-handed states via a Dirac mass term. For this, however,
(i) at least two HNLs are required in order to explain to observed neutrino
mass differences $\Delta m_{\text{solar}}^{2},\Delta m_{\text{atm}}^{2}$; (ii)
their coupling constants should be at the level presented in Fig. 1 as the
“sea-saw line”. As we see, such couplings are very far from what will be
accessed by the current experiments. It is, however, possible that the
couplings of each of the HNLs are orders of magnitude larger, but that their
contributions to the active neutrino masses cancel with high precision, due to
a fine-tuning or a symmetry. The fine-tuning scale associated with this
cancellation is $\xi=U^{2}_{\text{seesaw}}/U^{2}$, where
$U^{2}_{\text{seesaw}}\sim 5\cdot 10^{-11}(1\text{ GeV}/m_{N})$ is the see-saw
line which does not require the fine-tuning. As shown in Fig. 9, beam dump
experiments allow probing the part of the parameter space that is unreachable
to collider experiments. In addition, we find that the layout of the SHiP
experiment at the SPS ECN3 beam facility is close to optimal, assuming that
the background is negligibly small.

## Acknowledgements

AB is supported by the European Research Council (ERC) Advanced Grant “NuBSM”
(694896). KB is partly funded by the INFN PD51 INDARK grant. OM is supported
by the NWO Physics Vrij Programme “The Hidden Universe of Weakly Interacting
Particles” with project number 680.92.18.03 (NWO Vrije Programma), which is
(partly) financed by the Dutch Research Council (NWO). MO received support
from the European Union’s Horizon 2020 research and innovation program under
the Marie Sklodowska-Curie grant agreement No. 860881-HIDDeN.

## References

  * (1) J. Beacham et al., “Physics Beyond Colliders at CERN: Beyond the Standard Model Working Group Report,” J. Phys. G 47 no. 1, (2020) 010501, arXiv:1901.09966 [hep-ex].
  * (2) P. Agrawal et al., “Feebly-interacting particles: FIPs 2020 workshop report,” Eur. Phys. J. C 81 no. 11, (2021) 1015, arXiv:2102.12143 [hep-ph].
  * (3) S. Alekhin et al., “A facility to Search for Hidden Particles at the CERN SPS: the SHiP physics case,” Rept. Prog. Phys. 79 no. 12, (2016) 124201, arXiv:1504.04855 [hep-ph].
  * (4) CMS Collaboration, A. Tumasyan et al., “Search for long-lived heavy neutral leptons with displaced vertices in proton-proton collisions at $\sqrt{\mathrm{s}}$ =13 TeV,” JHEP 07 (2022) 081, arXiv:2201.05578 [hep-ex].
  * (5) I. Boiarska, K. Bondarenko, A. Boyarsky, S. Eijima, M. Ovchynnikov, O. Ruchayskiy, and I. Timiryasov, “Probing baryon asymmetry of the Universe at LHC and SHiP,” arXiv:1902.04535 [hep-ph].
  * (6) A. Blondel et al., “Searches for long-lived particles at the future FCC-ee,” Front. in Phys. 10 (2022) 967881, arXiv:2203.05502 [hep-ex].
  * (7) A. M. Abdullahi et al., “The present and future status of heavy neutral leptons,” J. Phys. G 50 no. 2, (2023) 020501, arXiv:2203.08039 [hep-ph].
  * (8) D. Acosta et al., “Review of opportunities for new long-lived particle triggers in Run 3 of the Large Hadron Collider,” arXiv:2110.14675 [hep-ex].
  * (9) L. Shchutska and M. Ovchynnikov, “Prospects for LLP searches at the LHC in Run 3 and HL-LHC,”  2022\.
  * (10) T. Asaka, S. Blanchet, and M. Shaposhnikov, “The nuMSM, dark matter and neutrino masses,” Phys. Lett. B 631 (2005) 151–156, arXiv:hep-ph/0503065.
  * (11) LHCb Collaboration, R. Aaij et al., “Search for long-lived scalar particles in $B^{+}\to K^{+}\chi(\mu^{+}\mu^{-})$ decays,” Phys. Rev. D 95 no. 7, (2017) 071101, arXiv:1612.07818 [hep-ex].
  * (12) SHiP Collaboration, O. Aberle et al., “BDF/SHiP at the ECN3 high-intensity beam facility,” Tech. Rep. CERN-SPSC-2022-032, SPSC-I-258, CERN, Geneva, 2022.  http://cds.cern.ch/record/2839677.
  * (13) HIKE Collaboration, E. Cortina Gil et al., “HIKE, High Intensity Kaon Experiments at the CERN SPS: Letter of Intent,” Tech. Rep. CERN-SPSC-2022-031, SPSC-I-257, SPSC-I-257, CERN, Geneva, 2022.  arXiv:2211.16586.  https://cds.cern.ch/record/2839661.  Letter of Intent submitted to CERN SPSC. Address all correspondence to<EMAIL_ADDRESS>
  * (14) SHADOWS Collaboration, M. Alviggi et al., “SHADOWS Letter of Intent,” Tech. Rep. CERN-SPSC-2022-030, SPSC-I-256, CERN, Geneva, 2022.  https://cds.cern.ch/record/2839484.
  * (15) SHiP Collaboration, M. Anelli et al., “A facility to Search for Hidden Particles (SHiP) at the CERN SPS,” arXiv:1504.04956 [physics.ins-det].
  * (16) B. Döbrich, J. Jaeckel, and T. Spadaro, “Light in the beam dump - ALP production from decay photons in proton beam-dumps,” JHEP 05 (2019) 213, arXiv:1904.02091 [hep-ph]. [Erratum: JHEP 10, 046 (2020)].
  * (17) J. Jerhot, B. Döbrich, F. Ertas, F. Kahlhoefer, and T. Spadaro, “ALPINIST: Axion-Like Particles In Numerous Interactions Simulated and Tabulated,” JHEP 07 no. TTK-22-04, CP3-22-02, (2022) 094, arXiv:2201.05170 [hep-ph].
  * (18) SHiP Collaboration, H. Dijkstra and T. Ruf, “Heavy Flavour Cascade Production in a Beam Dump.” Dec, 2015.  https://cds.cern.ch/record/2115534. CERN-SHiP-NOTE-2015-009.
  * (19) SHiP Collaboration, C. Ahdida et al., “Sensitivity of the SHiP experiment to dark photons decaying to a pair of charged particles,” Eur. Phys. J. C 81 no. 5, (2021) 451, arXiv:2011.05115 [hep-ex].
  * (20) M. Ovchynnikov, J.-L. Tastet, O. Mikulenko, and K. Bondarenko, “Sensitivities to feebly interacting particles: public and unified calculations,” arXiv:2305.13383 [hep-ph].
  * (21) D. Gorbunov, A. Makarov, and I. Timiryasov, “Decaying light particles in the SHiP experiment: Signal rate estimates for hidden photons,” Phys. Rev. D 91 no. 3, (2015) 035027, arXiv:1411.4007 [hep-ph].
  * (22) B. Döbrich, J. Jaeckel, F. Kahlhoefer, A. Ringwald, and K. Schmidt-Hoberg, “ALPtraum: ALP production in proton beam dump experiments,” JHEP 02 (2016) 018, arXiv:1512.03069 [hep-ph].
  * (23) M. Adamovich et al., “Results on charm physics from WA82,” Nucl. Phys. B Proc. Suppl. 27 (1992) 212–218.
  * (24) M. Aguilar-Benitez et al., “Inclusive particle production in 400-GeV/c p p interactions,” Z. Phys. C 50 (1991) 405–426.
  * (25) B. Döbrich, F. Ertas, F. Kahlhoefer, and T. Spadaro, “Model-independent bounds on light pseudoscalars from rare B-meson decays,” Phys. Lett. B 790 (2019) 537–544, arXiv:1810.11336 [hep-ph].
  * (26) I. Boiarska, K. Bondarenko, A. Boyarsky, V. Gorkavenko, M. Ovchynnikov, and A. Sokolenko, “Phenomenology of GeV-scale scalar portal,” JHEP 11 (2019) 162, arXiv:1904.10447 [hep-ph].
  * (27) D. Gorbunov, E. Kriukova, and O. Teryaev, “Scalar decay into pions via Higgs portal,” arXiv:2303.12847 [hep-ph].
  * (28) I. Boiarska, A. Boyarsky, O. Mikulenko, and M. Ovchynnikov, “Constraints from the CHARM experiment on heavy neutral leptons with tau mixing,” Phys. Rev. D 104 no. 9, (2021) 095019, arXiv:2107.14685 [hep-ph].
  * (29) A. Boyarsky, O. Mikulenko, M. Ovchynnikov, and L. Shchutska, “Exploring the potential of FCC-hh to search for particles from B mesons,” JHEP 01 (2023) 042, arXiv:2204.01622 [hep-ph].
  * (30) M. Ovchynnikov, V. Kryshtal, and K. Bondarenko, “Sensitivity of the FACET experiment to Heavy Neutral Leptons and Dark Scalars,” JHEP 02 (2023) 056, arXiv:2209.14870 [hep-ph].

##  Appendix A Technical details of the sensitivity estimates

###  A.1 The number of events

The expression for the number of events that we use looks as follows:

$N_{\text{events}}=\sum_{\text{mother}}N_{X,\text{prod}}\times\int
dE_{X}\int\limits_{\text{decay volume}}d\theta_{X}dz_{X}\times\\\ \times
f_{X}(\theta_{X},E_{X})\times\epsilon_{\text{az}}(\theta_{X},z_{X})\times\\\
\times\frac{dP_{\text{dec}}}{dz_{X}}\times\epsilon_{\text{dec}}(\theta_{X},E_{X},z_{X})\times\text{Br}_{\text{vis}}(m_{X})$
(11)

Here, $N_{X,\text{prod}}=N_{\text{mother}}\times P_{\text{mother}\to X}$ is
the total number of particles $X$ produced by a mother particle, with
$P_{\text{mother}\to X}$ being the probability of production.
$z_{X},\theta_{X},E_{X}$ are, correspondingly, the longitudinal position,
polar angle, and energy of the decaying FIP of type $X$,
$f_{X}(\theta_{X},E_{X})$ is the differential distribution of FIPs in the
polar angle and energy, and $\epsilon_{\text{az}}(\theta_{X},z_{X})$ is the
azimuthal acceptance:

$\epsilon_{\text{az}}=\frac{1}{2\pi}\text{min}\big{[}\Delta\phi_{\text{decay
volume}}(\theta_{X},z_{X}),\\\ \Delta\phi_{\text{end of
spectrometer}}(\theta_{X}))\big{]}$ (12)

i.e. fraction of azimuthal angle for which FIPs 1) decays inside the decay
volume and 2) have a trajectory intersecting the whole spectrometer. The
differential decay probability $\frac{dP_{\text{dec}}}{dz_{X}}$ is:

$\frac{dP_{\text{dec}}}{dz_{X}}=\frac{\exp[-l(z,\theta)/l_{\text{X,decay}}]}{l_{\text{X,decay}}}\frac{dl(z_{X},\theta_{X})}{dz_{X}},$
(13)

with $l=z_{X}/\cos(\theta_{X})$ being the displacement. The decay product
acceptance $\epsilon_{\text{dec}}(\theta_{X},E_{X},z_{X})$ is the fraction of
FIPs within the azimuthal acceptance whose decay products point to the
detector. Finally, $\text{Br}_{\text{vis}}(m_{X})$ is the branching ratio of
the FIP decay into final states visible at the given experiment.

Further details may be found in the works Boiarska:2021yho ; Boyarsky:2022epg
; Ovchynnikov:2022its , where the approach has been used for various
experiments located at different facilities. In addition, the publicly
available code SensCalc, which allows computing sensitivities using this
approach, will be provided in a paper Ovchynnikov:2023cry  soon to appear.

####  A.1.1 Lower bound of the sensitivity

In the regime of long FIP lifetimes, the exponent in the expression for the
decay probability is

$\exp[-l/l_{\text{X,decay}}]\approx 1,$ (14)

and the number of events becomes

$N_{\text{events}}\approx
N_{X,\text{prod}}\times\frac{l_{\text{fid}}}{c\tau_{X}\langle
E_{X}^{-1}\rangle^{-1}/m_{X}}\times\epsilon,$ (15)

where $\epsilon$ is the overall acceptance:

$\epsilon\equiv\frac{1}{l_{\text{fid}}}\int dE_{X}\int\limits_{\text{decay
volume}}d\theta_{X}dz_{X}\times f_{X}(\theta_{X},E_{X})\times\\\
\times\frac{\langle
E^{-1}_{X}\rangle^{-1}}{E_{X}}\times\epsilon_{\text{dec}}(\theta_{X},E_{X},z_{X})\times\epsilon_{\text{az}}(\theta_{X},z_{X})$
(16)

###  A.2 Sensitivity to dark scalars

In Fig. 8, we report the 90% CL sensitivity of SHADOWS to dark scalars. The
sensitivity has been obtained assuming the setup described in the SHADOWS LoI
Alviggi:2839484  and requiring $N_{\text{events}}>2.3$.

Let us discuss the setup in detail. The distance of the decay volume from the
target is considered to be $l_{\text{min}}=10\text{ m}$555Note that
$l_{\text{min}}$ reported in the LoI in the section about experiment
specifications is different, being 14 m. The reason for this discrepancy is
unclear to us., and the decay volume is defined to be located entirely off-
axis with the closest decay volume wall being parallel to the beam-axis at a
distance of 1 m. We consider the given box geometry of the decay volume with
length 20 m and the cross-section equal to $2.5\times 2.5\text{ m}^{2}$, and
use the detector cross-section equal to $2.5\times 2.5\text{ m}^{2}$. In
addition, we require at least two charged decay products that point to the
beginning of the dipole magnet, located $\simeq 2.5$ m downward the end of the
decay volume. We note that the last specification may significantly
overestimate the sensitivity of SHADOWS as the LoI specifies that the
spectrometer and its downstream particle identification system, consisting of
a calorimeter and muon system, have a total length of 12-14 m. For proper
identification of the decay products and reconstruction of their kinematics,
their trajectories have to be contained within the acceptance of the entire
detector, also requiring that the spectrometer magnet field is accounted for.

As for the scalar phenomenology, we follow Boiarska:2019jym . For the
probability of producing scalars, we use

$\text{P}(B\to S)\approx\sum_{B_{x}=B^{0,+}}f_{b\to B_{x}}\times\\\
\times\text{max}\left[\text{Br}_{\text{incl}}(B\to
S),\text{Br}_{\text{excl}}(B\to S+X_{d/s})\right]$ (17)

Here, $f_{b\to B_{x}}$ is the fragmentation fraction ($f_{b\to B^{+}}\approx
f_{b\to B^{0}}\approx 0.411$). $\text{Br}_{\text{incl}}(B\to S)$ is the
inclusive branching ratio, which is equal to $\approx 6.6$ in the domain
$m_{S}\ll m_{b}$. $\text{Br}_{\text{excl}}$ is the sum over all exclusive
channels considered in Boiarska:2019jym , and $X_{d/s}$ is the final state
appearing from the hadronization of $d$ or $s$ quarks. The inclusive
description breaks down at large scalar masses Boiarska:2019jym ; in addition,
PBC recommended using the exclusive description Beacham:2019nyx .
Nevertheless, in Alviggi:2839484 , a purely inclusive estimate has been used.
As a result, the branching ratio (17) matches the value used in
Alviggi:2839484  for scalars with masses $m_{S}\lesssim 2\text{ GeV}$, but is
larger for larger masses.

For the distribution of mother particles, $B$ mesons, we use the angle-energy
distribution and the production probability $\chi_{b\bar{b}}=2.7\cdot 10^{-7}$
from CERN-SHiP-NOTE-2015-009 , which also includes the cascade enhancement
from the amount of $B$s produced in secondary interactions in the thick
target.

Finally, we assume that the detectable decays of scalars are those containing
at least two “visible” particles (photons or charged particles). In this way,
we allow not only fully reconstructable states (where the kinematics may be
fully reconstructed) but also partially reconstructed decays, containing
particles such as neutrinos among the other decay products. Examples of the
latter are decays
$S\to\tau\bar{\tau}\to\nu_{\tau}\bar{\nu}_{\tau}+\text{visible}$, $S\to
D\bar{D}\to\nu+\bar{\nu}+\text{visible}$, which dominate the scalar decays
above di-$\tau$ mass threshold. In contrast, in the LoI, only fully
reconstructable states have been required. We do not require any other cut,
such as the minimum opening angle or the energy cuts.

Therefore, given the discussion above, our sensitivity estimate must match the
sensitivity from the LoI in the mass range $m_{S}\lesssim 2\text{ GeV}$. In
contrast, for larger masses, it should be more optimistic.

The comparison of the sensitivities of SHADOWS obtained by our calculations
and by the calculations in the LoI is shown in Fig. 10.

Figure 10: Sensitivity of SHADOWS to dark scalars with the mixing coupling.
The red line shows the sensitivity obtained from our computation (Eq. (11))
for the experimental setup used in the SHADOWS LoI 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).

We notice a significant difference at the lower and upper bounds of the
sensitivity. Namely, at small masses, the number of events at the lower bound
reported by the LoI is larger by up to a factor of 15 compared to our
predictions. The situation is similar for large masses, where the sensitivity
reported in the LoI should be much more conservative.

While we cannot identify the reason for such discrepancy, we can present a
simple qualitative argument suggesting that the sensitivity presented in the
LoI could be overestimated. Namely, let us consider an experiment with the
projective decay volume along the lines pointing from the target to the
SHADOWS detector window. For the latter, we choose the same plane with cross-
section $2.5\times 2.5\text{ m}^{2}$ located 2.5 m downward the end of the
decay volume.

It would not be possible to build such an experiment because of the
limitations imposed by the infrastructure. However, what is important for us
is that it must have better sensitivity than the real SHADOWS configuration.
Indeed, its detector covers the same polar angles as the detector of SHADOWS,
but, there would be no geometric shortening of the length of the decay volume
for the scalars pointing to the detector, and there would be less azimuthal
suppression.

Further, let us consider the scalar mass $m_{S}\lesssim 1\text{ GeV}$, for
which the decay products fly roughly in the same direction as the decaying
scalar, and we may approximate the decay products acceptance by its upper
bound $\epsilon_{\text{dec}}\approx 1$. Finally, we may consider the lower
bound of the sensitivity only. Under these assumptions, the number of decays
is approximately

$N_{\text{events}}\approx N_{\text{PoT}}\times\chi_{b\bar{b}}\times 2(f_{b\to
B^{+}}+f_{b\to B^{0}})\times\text{Br}_{B\to S}\times\\\
\times\epsilon_{\text{geom}}\times\frac{l_{\text{fid}}\cdot m_{S}\langle
p^{-1}_{S}\rangle}{c\tau_{S}},$ (18)

where $\epsilon_{\text{geom}}$ is the fraction of the scalars pointing to the
detector, and $\langle p^{-1}_{S}\rangle$ is the mean inverse momentum of such
scalars. These quantities may be obtained just from knowing the scalar
distribution function, which may be easily derived given the simple kinematics
of the scalar production in 2-body decays $B\to S+X_{s}$, and the azimuthal
coverage of the SHADOWS detector window, which is shown in Fig. 11.

Figure 11: 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 $S_{\text{det}}/L_{\text{to det}}^{2}\approx
4.6\cdot 10^{-3}\text{ sr}$.

The values of all the relevant parameters for the particular point
$m_{S}=150\text{ MeV}$ are reported in Table 3.

Parameter | $\text{Br}(B\to S)$ | $\epsilon_{S}$ | $m_{S}\langle p_{S}^{-1}\rangle$ | $c\tau_{S},\text{ m}$ | $N_{\text{events}}$
---|---|---|---|---|---
Value | $\approx 6.6$ | $0.09$ | $3.7\cdot 10^{-3}$ | $7.7\cdot 10^{-3}/\theta^{2}$ | $2.3\left(\frac{\theta^{2}}{1.4\cdot 10^{-7}}\right)^{2}$
Table 3: The values of the parameters entering Eq. (18) for the particular
scalar mass $m_{S}=150\text{ MeV}$.

Comparing the lower bound predicted by this formula with the sensitivity from
the SHADOWS LoI, we find that the sensitivity of the projective experiment
lies well above the latter, see Fig. 10. In contrast, it lies below our
estimate. At small masses, it is very close to our estimate. The increasing
difference at large masses is explained by the decay products acceptance which
is assumed to be 1 for the projective setup.