Physics beyond colliders at CERN: beyond the Standard Model working group report

A large class of BSM models includes interactions with light new vector particles. Such particles could result from extra gauge symmetries of BSM physics. New vector states can mediate interactions both with the SM fields, and extra fields in the dark sector that e.g. may represent the DM states.

The most minimal vector portal interaction can be written as

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{vector}}={{ \mathcal L }}_{\mathrm{SM}}+{{ \mathcal L }}_{\mathrm{DS}}-\displaystyle \frac{\epsilon }{2\cos {\theta }_{W}}{F}_{\mu \nu }^{{\prime} }{B}_{\mu \nu },\end{eqnarray} \tag{ 2.2 }$$

where ${{ \mathcal L }}_{\mathrm{SM}}$ is the SM Lagrangian, B_μν and ${F}_{\mu \nu }^{{\prime} }$ are the field stengths of hypercharge and new $U(1)^{\prime}$ gauge groups, is the so-called kinetic mixing parameter [24], and ${{ \mathcal L }}_{\mathrm{DS}}$ stands for the dark sector Lagrangian that may include new matter fields χ charged under $U^{\prime} (1)$

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{DS}}=-\displaystyle \frac{1}{4}{\left({F}_{\mu \nu }^{{\prime} }\right)}^{2}+\displaystyle \frac{1}{2}{m}_{A^{\prime} }^{2}{\left({A}_{\mu }^{{\prime} }\right)}^{2}+| ({\partial }_{\mu }+{{ig}}_{D}{A}_{\mu }^{{\prime} })\chi {| }^{2}+...\,.\end{eqnarray} \tag{ 2.3 }$$

If χ is stable or long-lived it may constitute a fraction of entirety of DM. At low energy this theory contains a new massive vector particle, a dark photon state, coupled to the electromagnetic current with -proportional strength, ${A}_{\mu }^{{\prime} }\times \epsilon {J}_{{EM}}^{\mu }$.

We define the following important benchmark cases (BC1–BC3) for the vector portal models.

• BC1, Minimal dark photon model: in this case the SM is augmented by a single new state $A^{\prime}$. DM is assumed to be either heavy or contained in a different sector. In that case, once produced, the dark photon decays back to the SM states. The parameter space of this model is $\{{m}_{A^{\prime} },\epsilon \}$.
• BC2, Light DM coupled to a dark photon: here a minimally coupled WIMP DM model can be constructed [14, 15]. The preferred values of dark coupling ${\alpha }_{D}={g}_{D}^{2}/(4\pi )$ are such that the decay of $A^{\prime}$ occurs predominantly into $\chi {\chi }^{* }$ states. These states can further rescatter on electrons and nuclei due to -proportional interaction between SM and DS states mediated by the mixed ${AA}^{\prime}$ propagator [22, 25].The parameter space for this model is $\{{m}_{A^{\prime} },\epsilon ,{m}_{\chi },{\alpha }_{D}\}$ with further model-dependence associated to the properties of χ (boson or fermion). The suggested choices for the PBC evaluation are 1. versus ${m}_{A^{\prime} }$ with ${\alpha }_{D}\gg {\epsilon }^{2}\alpha$ and $2{m}_{\chi }\lt {m}_{A^{\prime} }$, 2. y versus m_χ plot where the yield variable y, $y={\alpha }_{D}{\epsilon }^{2}{\left({m}_{\chi }/{m}_{A^{\prime} }\right)}^{4}$, is argued [26] to contain a combination of parameters relevant for the freeze-out and DM–SM particles scattering cross section. One possible choice is α_D = 0.1 and ${m}_{A^{\prime} }/{m}_{\chi }=3$.
• BC3, Millicharged particles: this is the limit ${m}_{A^{\prime} }\to 0$, in which case χ has an effective electric charge of $| {Q}_{\chi }| =| \epsilon {g}_{D}e|$ [24, 27]. The suggested choice of parameter space is $\{{m}_{\chi },{Q}_{\chi }/e\}$, and χ can be taken to be a fermion.

The kinetic mixing coupling of $A^{\prime}$ to matter is the simplest and most generic, but not the only possible vector portal. Other cases considered in the literature include gauged B − L and ${L}_{\mu }-{L}_{\tau }$ models, and somewhat less motivated leptophylic and leptophobic cases, when $A^{\prime}$ is assumed to be coupled to either total lepton current, or total baryon current with a small coupling $g^{\prime}$.

Such other exotic vector mesons however, generically mix with the SM photon at one loop, which is often enhanced by the number of flavors and/or colors of the quarks/leptons running in the loop. This means that the kinetically mixed dark photon benchmarks outlined above also cover these scenarios, to some extent (see, e.g. [28]).

The 2012 discovery of the BEH mechanism, and the Higgs boson h, prompts to investigate the so called scalar or Higgs portal, that couples the dark sector to the Higgs boson via the bilinear ${H}^{\dagger }H$ operator of the SM. The minimal scalar portal model operates with one extra singlet field S and two types of couplings, μ and λ [29],

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{scalar}}={{ \mathcal L }}_{\mathrm{SM}}+{{ \mathcal L }}_{\mathrm{DS}}-(\mu S+\lambda {S}^{2}){H}^{\dagger }H.\end{eqnarray} \tag{ 2.4 }$$

The dark sector Lagrangian may include the interaction with DM $\chi ,{{ \mathcal L }}_{\mathrm{DS}}=S\bar{\chi }\chi +...$. Most viable DM models in the sub-EW scale range imply $2\cdot {m}_{\chi }\gt {m}_{S}$ [30].

At low energy, the Higgs field can be substituted for $H=(v+h)/\sqrt{2}$, where v = 246 GeV is the EW vacuum expectation value, and h is the field corresponding to the physical 125 GeV Higgs boson. The non-zero μ leads to the mixing of h and S states. In the limit of small mixing it can be written as

$$\begin{eqnarray}&&\theta =\displaystyle \frac{\mu v}{{m}_{h}^{2}-{m}_{S}^{2}}.\end{eqnarray} \tag{ 2.5 }$$

Therefore the linear coupling of S to SM particles can be written as ${\theta }_{S}\times {\sum }_{\mathrm{SM}}{O}_{h}$, where O_h is a SM operator to which Higgs boson is coupled and the sum goes over all type of SM operators coupled to the Higgs field.

The coupling constant λ leads to the coupling of h to a pair of S particles, λ S². It can lead to pair-production of S but cannot induce its decay. An important property of the scalar portal is that at loop level it can induce flavor-changing transitions, and in particular lead to decays $K\to \pi S,B\to {K}^{(* )}S$ etc [29, 31, 32] and similarly for the hS² coupling [33]. We define the following benchmark cases for the scalar portal models:

• BC4, Higgs-mixed scalar: in this model we assume λ = 0, and all production and decay are controlled by the same parameter θ. Therefore, the parameter space for this model is {θ, m_S}.
• BC5, Higgs-mixed scalar with large pair-production channel: in this model the parameter space is {λ, θ, m_S}, and λ is assumed to dominate the production via e.g. $h\to {SS},B\to {K}^{(* )}{SS},{B}^{0}\to {SS}$ etc. In the sensitivity plots shown in section 9.2 a value of the branching fraction ${BR}(h\to {SS})$ close to 10⁻² is assumed in order to be complementary to the LHC searches for the Higgs to invisible channels.

We also note that while the 125 GeV Higgs-like resonance has properties of the SM Higgs boson within errors, the structure of the Higgs sector can be more complicated and include e.g. several scalar doublets. In the two-Higgs doublet model the number of possible couplings grows by a factor of three, as S can couple to 3 combinations of Higgs field bilinears, ${H}_{1}^{\dagger }{H}_{1},{H}_{2}^{\dagger }{H}_{2}$ and ${H}_{1}{H}_{2}$. Therefore, the experiments could investigate their sensitivity to a more complicated set of the Higgs portal couplings that are, however, beyond the present document.

The neutrino portal extension of the SM is very motivated by the fact that it can be tightly related with the neutrino mass generation mechanism. The neutrino portal operates with one or several dark fermions N, that can be also called heavy neutral leptons or HNLs. The general form of the neutrino portal can be written as

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{vector}}={{ \mathcal L }}_{\mathrm{SM}}+{{ \mathcal L }}_{\mathrm{DS}}+\sum {F}_{\alpha I}({\bar{L}}_{\alpha }H){N}_{I},\end{eqnarray} \tag{ 2.6 }$$

where the summation goes over the flavor of lepton doublets L_α, and the number of available HNLs, N_I. The ${F}_{\alpha I}$ are the corresponding Yukawa couplings. The dark sector Lagrangian should include the mass terms for HNLs, that can be both Majorana or Dirac type. For a more extended review, see [23, 34]. Setting the Higgs field to its v.e.v., and diagonalizing mass terms for neutral fermions, one arrives at ${\nu }_{i}-{N}_{J}$ mixing, that is usually parametrized by a matrix called U. Therefore, in order to obtain interactions of HNLs, inside the SM interaction terms, one can replace ${\nu }_{\alpha }\to {\sum }_{I}{U}_{\alpha I}{N}_{I}$. In the minimal HNL models, both the production and decay of an HNL are controlled by the elements of matrix U.

PBC has defined the following benchmark cases:

• BC6, Single HNL, electron dominance: assuming one Majorana HNL state N, and predominantly mixing with electron neutrinos, all production and decay can be determined as function of parameter space $({m}_{N},| {U}_{e}{| }^{2})$.
• BC7, Single HNL, muon dominance: assuming one Majorana HNL state N, and predominantly mixing with muon neutrinos, all production and decay can be determined as function of parameter space $({m}_{N},| {U}_{\mu }{| }^{2})$.
• BC8, Single HNL, tau dominance: one Majorana HNL state with predominantly mixing to tau neutrinos. Parameter space is $({m}_{N},| {U}_{\tau }{| }^{2})$.

These are representative cases which do not exhaust all possibilities. Multiple HNL states, and presence of comparable couplings to different flavors can be even better motivated than the above choices. The current choice of benchmark cases is motivated by simplicity.

QCD axions are an important idea in particle physics [35–37] that allows for a natural solution to the strong CP problem, or apparent lack of CP violation in strong interactions. Current QCD axion models are restricted to the sub-eV range of axions. However, a generalization of the minimal model to ALPs can be made [27]. Taking a single pseudoscalar field a one can write a set of its couplings to photons, quarks, leptons and other fields of the SM. In principle, the set of possible couplings is very large and in this study we consider only the flavor-diagonal subset

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{\mathrm{axion}} & = & {{ \mathcal L }}_{\mathrm{SM}}+{{ \mathcal L }}_{\mathrm{DS}}+\displaystyle \frac{a}{4{f}_{\gamma }}{F}_{\mu \nu }{\tilde{F}}_{\mu \nu }+\displaystyle \frac{a}{4{f}_{G}}\mathrm{Tr}{G}_{\mu \nu }{\tilde{G}}_{\mu \nu }\\ & & +\displaystyle \frac{{\partial }_{\mu }a}{{f}_{l}}\displaystyle \sum _{\alpha }{\bar{l}}_{\alpha }{\gamma }_{\mu }{\gamma }_{5}{l}_{\alpha }+\displaystyle \frac{{\partial }_{\mu }a}{{f}_{q}}\displaystyle \sum _{\beta }{\bar{q}}_{\beta }{\gamma }_{\mu }{\gamma }_{5}{q}_{\beta }.\end{array}\end{eqnarray} \tag{ 2.7 }$$

The DS Lagrangian may contain new states that provide a UV completion to this model (for the case of the QCD axion they are called the Peccei–Quinn sector). All of these interactions do not lead to large additive renormalization of m_a, making this model technically natural. Note, however, that the coupling to gluons does lead to a non-perturbative contribution to m_a.

The PBC proposals have considered the following benchmark cases:

• BC9, photon dominance: assuming a single ALP state a, and predominant coupling to photons, all phenomenology (production, decay, oscillation in the magnetic field) can be determined as functions on $\{{m}_{a},{g}_{a\gamma \gamma }\}$ parameter space, where ${g}_{a\gamma \gamma }={f}_{\gamma }^{-1}$ notation is used.
• BC10, fermion dominance: assuming a single ALP state a, and predominant coupling to fermions, all phenomenology (production and decay) can be determined as functions on $\{{m}_{a},{f}_{l}^{-1},{f}_{q}^{-1}\}$. Furthermore, for the sake of simplicity, we take f_q = f_l.
• BC11, gluon dominance: this case assumes an ALP coupled to gluons. The parameter space is $\{{m}_{a},{f}_{G}^{-1}\}$. Notice that in this case the limit of ${m}_{a}\lt {m}_{a,{\rm{QCD}}}{| }_{{f}_{a}={f}_{G}}$ is unnatural as it requires fine tuning and therefore is less motivated.

The ALP portals, ${BC}9-{BC}11$, are effective interactions, and would typically require UV completion at or below the f_i scales. This is fundamentally different from vector, scalar and neutrino portals that do not require external UV completion. Moreover, the renormalization group (RG) evolution is capable of inducing new couplings. All the sensitivity plots shown in section 7 assume a cut-off scale of Λ = 1 TeV. Details about approximations and assumptions assumed in computing sensitivities for the BC10 and BC11 cases are reported in appendices A and B.

The IAXO is a new generation axion helioscope [44], aiming at the detection of solar axions with sensitivities to the axion-photon coupling ${g}_{a\gamma }$ down to a few 10⁻¹² GeV⁻¹, a factor of 20 better than the current best limit from CAST (a factor of more than 10⁴ improvement in signal-to-noise ratio). Its physics reach is highly complementary to all other initiatives in the field, with unparalleled sensitivity to highly motivated parts of the axion parameter space that no other experimental technique can probe. The proposed baseline configuration of IAXO includes a large-scale superconducting multi-bore magnet, specifically built for axion physics, together with the extensive use of x-ray focusing based on cost-effective slumped glass optics and ultra-low background x-ray detectors. The unique physics potential of IAXO can be summarized by the following statements:

• (1)  
  IAXO follows the only proposed technique able to probe a large fraction of QCD axion models in the meV to eV mass band. This region is the only one where astrophysical, cosmological (DM) and theoretical (strong CP problem) motivations overlap.
• (2)  
  IAXO will fully probe the ALP region invoked to solve the transparency anomaly, and will largely probe the axion region invoked to solve observed stellar cooling anomalies.
• (3)  
  IAXO will partially explore viable QCD axion DM models, and largely explore a subset of predictive ALP models (dubbed ALP miracle) recently studied to simultaneously solve both DM and inflation.
• (4)  
  The above sensitivity goals do not depend on the hypothesis of axion being the DM, i.e. in case of non-detection, IAXO will robustly exclude the corresponding range of parameters for the axion/ALP.
• (5)  
  IAXO relies on detection concepts that have been tested in the CAST experiment at CERN. Risks associated with the scaling up of the different subsystems will be mitigated by the realization of small scale prototype BabyIAXO.
• (6)  
  IAXO will also constitute a generic infrastructure for axion/ALP physics with potential for additional search strategies (e.g. the option of implementing RF cavities to search for DM axions).

The main element of IAXO is a new dedicated large-scale magnet, designed to maximize the helioscope figure of merit. The IAXO magnet will be a superconducting magnet following a large multi-bore toroidal configuration, to efficiently produce an intense magnetic field over a large volume. The design is inspired by the ATLAS barrel and end-cap toroids, the largest superconducting toroids ever built and presently in operation at CERN. Indeed the experience of CERN in the design, construction and operation of large superconducting magnets is crucial for the project. IAXO will also make extensive use of novel detection concepts pioneered at a small scale in CAST. This includes x-ray focussing and low background detectors. The former relies on the fact that, at grazing incident angles, it is possible to realize x-ray mirrors with high reflectivity. IAXO envisions newly-built optics similar to those used onboard NASA's NuSTAR satellite mission, but optimized for the energies of the solar axion spectrum. Each of the eight ∼60 cm diameter magnet bores will be equipped with such optics. For BabyIAXO, using existing optics from the ESA's XMM mission is being considered. At the focal plane of each of the optics, IAXO will have low-background x-ray detectors. Several technologies are under consideration, but the most developed one are small gaseous chambers read by pixelised microbulk Micromegas planes. They involve low-background techniques typically developed in underground laboratories, like the use of radiopure detector components, appropriate shielding, and the use of offline discrimination algorithms. Alternative or additional x-ray detection technologies are also considered for IAXO, like GridPix detectors, magnetic metallic calorimeters, transition edge sensors, or silicon drift detectors. All of them show promising prospects to outperform the baseline Micromegas detectors in aspects like energy threshold or resolution, which are of interest, for example, to search for solar axions via the axion-electron coupling, a process featuring both lower energies that the standard Primakoff ones, and monochromatic peaks in the spectrum.

As a first step the Collaboration pursues the construction of BabyIAXO, an intermediate scale experimental infrastructure. BabyIAXO will test magnet, optics and detectors at a technically representative scale for the full IAXO, and, at the same time, it will be operated and will take data as a fully-fledged helioscope experiment, with sensitivity beyond CAST and potential for discovery.

After a few years of preparatory phase, project socialization and interaction with funding bodies, the IAXO Collaboration was eventually formalized in July 2017. A Collaboration agreement document (bylaws) was signed by 17 institutions from Croatia, France, Germany, Italy, Russia, Spain, South Africa, USA, as well as CERN. They include about  75 physicists at the moment. It is likely that this list will increase with new members in the near future. A Collaboration management is already defined and actively implementing steps towards the BabyIAXO design and construction. The experiment will likely be sited at DESY, and it is expected to be built in 2–3 years, entering into data taking in 3–4 years.

The Collaboration already nicely encompasses all the know-how to cover BabyIAXO needs, and therefore a distribution of responsibilities in the construction of the experiment exists already. The magnet of (Baby)IAXO is of a size and field strength comparable to that of large detector magnets typically built in high energy physics. For this IAXO relies on the unique expertize of CERN in large superconducting magnets. The CERN magnet detector group has led all magnet design work so far in the IAXO CDR. The technical design of the BabyIAXO magnet, for which CERN has allocated one Applied Fellow, has started in January 2018. Further CERN participation is expected in terms of, at the least, allocation of expert personnel to oversee the construction of the magnet, as well as the use of existing CERN infrastructure. Other groups with magnet expertize in the Collaboration are CEA-Irfu and INR. The groups of LLNL, MIT and INAF are experts in the development and construction of x-ray optics, in particular in the technology chosen for the IAXO optics. Detector expertize exists in many of the Collaboration groups, encompassing the technologies mentioned above. Experience in general engineering, large infrastructure operation and management is present in several groups and in particular in centers like CERN or DESY. Many of the groups have experience in axion phenomenology and the connection with experiment, and more specifically experience with running the CAST experiment. Following these guidelines the Collaboration board is in the process of defining a Collaboration agreement (MoU) to organize the distribution of efforts and commitments among the collaborating institutes.

IAXO has also submitted a separate document³⁶ to be considered in the update of the European Strategy for Particle Physics (ESPP).

The pioneer LSW experiment was conducted in Brookhaven by the BFRT Collaboration [45], and the two most recent results are those of the experiments ALPS [46] and OSQAR [47]. ALPS is DESY based and used a decommissioned HERA magnet. ALPS is currently performing a major improvement to phase II, where a set of 10 + 10 HERA magnets will be coupled to two 100 m long Fabry–Perot cavities. ALPS II [48] will in fact take advantage of a resonant regeneration apparatus [49, 50], thus expecting a major improvement of the current limit on LSW experiment given by OSQAR. ALPS II will represent the current state of the art LSW experiment, and for this reason its activities are monitored with interest by the PBC since they will give key elements to judge the proposal Joint Undertaking on the Research for ALPs (JURA).

ALPS II aims to improve the sensitivity on ALP-photon couplings by three orders of magnitude compared to existing exclusion limits from laboratory experiments in the sub-meV mass region. ALPS II will inject a 30 W laser field into the 100 m long production cavity (PC) which is immersed in a 5.3 T magnetic field. The circulating power inside the PC is expected to reach 150 kW. The 100 m long regeneration cavity (RC) on the other side of the wall will have a finesse of 120 000. The RC is also placed inside a similar 5.3 T magnetic field. The employed two different photon detection concepts are expected to be able to measure fields with a photon rate as low as ∼10⁻⁴ photons per second. A next generation experiment for a LSW techniques will mainly rely on improved magnetic field structure, since from the optical part only limited improvements seems to be feasible. The project JURA basically combines the optics and detector development at ALPS II with dipole magnets for future accelerators under development at CERN.

The sensitivity of ALPS II in the search for ALPs is mainly limited by the magnetic field strength and the aperture (which limits the length of the cavities) of the HERA dipole magnets. JURA assumes the usage of magnets under development for an energy upgrade of LHC or a future FCC.

Several variants of these future magnets are of interest to the JURA initiative. In one of them the inner high temperature superconductor part would be omitted, so that magnets with a field of about 13 T and 100 mm aperture would be available (the modified HERA dipoles provide 5.3 T and 50 mm). In table 3 experimental parameters of ALPS II and this option of JURA are compared. They follow from assuming the installation of optical cavities inside the magnet bore in a (nearly) confocal configuration.

Table 3.  Comparison of experimental parameters of ALPS II at DESY and the JURA proposal.

Parameter	Sensitivity	ALPS II	JURA	Rel. sensitivity
				JURA/ALPS II
Magnet aperture		50 mm	100 mm
Magnetic field amplitude B	${g}_{a\gamma }\propto {B}^{-1}$	5.3 T	13 T	2.5
Magnetic field length L	${g}_{a\gamma }\propto {L}^{-1}$	189 m	960 m	5.1
Effective laser power P	${g}_{a\gamma }\propto {P}^{-1/4}$	0.15 MW	2.5 MW	2.0
Regeneration build up
(finesse F)	${g}_{a\gamma }\propto {F}^{-1/4}$	40 k	100 k	1.3
Detector noise rate R	${g}_{a\gamma }\propto {R}^{1/8}$	10−4 Hz	10−6 Hz	1.8
Total gain				56

The project JURA is a long term development, for which the experiment ALPS II can be considered as a feasibility study, especially for the resonant regeneration scheme. There are in fact some open questions: for example, the possibility of running mutually resonant cavities of very high finesses for distances of the order of several hundreds meters is still open. The linewidth of such cavities is in fact of the order of a few Hz, about one order of magnitude smaller than current state of the art. Another issue is the detector noise, however recent developments using coherent detection schemes seem to be very promising. Of course, the development of new magnets at CERN is not related to JURA, and thus this project will just rely on other projects' results.

JURA in the abovementioned configuration would surpass IAXO by about a factor of 2 in the photon-ALP coupling. It would allow to determine the photon-coupling of a lightweight ALP discovered by IAXO unambiguously and in a model-independent fashion or probe a large fraction of the IAXO parameter space model independently in case IAXO does not see anything new.

ALPS II is currently being constructed at DESY in the HERA tunnels. The tunnels and hall are currently being cleared and magnet installation will begin in 2019. The optics installation will begin at the end of 2019 and first data run is scheduled for 2020. About two years of operation is then expected. The time schedule for JURA is foreseen to be for a 2024–2026 starting time by using a LHC dipole magnet in a first phase. At the moment there is no real Collaboration and JURA might be considered an idea for a possible experiment which should grow within the years to come.

Brief presentation, unique features

Beam, beam time

Key requirements for detector

The REDTOP detector is being designed to sustain a maximum inelastic interaction rate of about 5 × 10⁸ evt s⁻¹. These capabilities exceed the event rate expected at CERN by about one order of magnitude and could portend to running the detector at future, high-intensity proton facility (for example, PIP-II at Fermilab). In order to sustain such an event rate, the detector must be: (a) very fast; (b) blind to baryons. The latter are produced within the target with a multiplicity of about 5/event and could easily pile-up if detected. On the other hand, since BSM physics is being searched for mostly in channels with charged leptons in the final state, the detector must have good efficiency to electrons and muons and excellent PID capabilities. The above requirements are fullfilled by adopting an Optical-TPC [52] for the tracking systems and a high-granularity, dual-readout ADRIANO calorimeter [53].

A fiber tracker, with identical features as that under construction for the LHCb experiment [54], has been recently included in the detector layout.

The schematic layout of the detector is shown in figure 3.

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Open questions, feasibility studies

Timeline

Status of the Collaboration

The NA64 is a hermetic general purpose detector to search for dark sector particles in missing energy events from high-energy (∼100 GeV) electrons, muons, and hadrons scattering off nuclei in an active dump. A high energy electron beam, for example, can be used to produce a vector mediator, e.g. dark photon $A^{\prime}$, via the reaction ${e}^{-}+Z\to {e}^{-}+Z+A^{\prime} \ A^{\prime} \to \chi \overline{\chi }$ where Z is the atomic number of the nuclei and $A^{\prime}$ is produced via kinetic mixing with bremsstrahlung photons and then decay promptly and invisibly into light (sub-GeV) DM particles [55, 56] in a hermetic detector [57, 58]. The signature of possible $A^{\prime}$ would appear as a single isolated electromagnetic shower in the active dump with detectable energy accompanied by missing energy in the rest of the detector.

The advantage of this technique compared to traditional beam dump experiments is that the sensitivity to $A^{\prime}$ scales as ² instead of ⁴, being the kinetic mixing strength, as the $A^{\prime}$ is required to be produced but not detected in the far apparatus. Another advantage of the NA64 approach is the high energy of the incident beam, that boosts the center-of-mass system relative to the laboratory system: this results in an enhanced hermeticity of the detector which provides a nearly full solid angle coverage.

The missing energy technique can be used also with high energy muon and hadron beams. The reaction of muon scattering off nuclei $\mu +Z\to \mu +Z+{Z}_{\mu }^{{\prime} }$ is sensitive to dark sector particles predominantly coupled to muons [59, 60], and, as such, is fully complementary to the dark photon searches. This search is quite appealing and very timely in particular in connection to the g_μ − 2 anomaly [61], and will be competing with another proposal at Fermilab [62] and elsewhere (see, e.g. [63] for a review). A Z_μ model gauging the L_μ–L_τ lepton number could also explain the hints of LFU violations in R_K and ${R}_{{K}^{* }}$ ratios observed by LHCb [64, 65]. The sensitivity to a Z_μ particle compatible with the observed B-anomalies and other constraints is currently under study by the Collaboration.

High energy hadron beams can be used to search for dark sector particles in the decays ${K}_{L},{K}_{S},{\pi }^{0},\eta ,\eta ^{\prime} \to$ invisible, where the neutral mesons M⁰ are produced via the charge-exchange reactions $\pi (K)+p\to {M}^{0}\,+\,n+{E}_{\mathrm{miss}}$ [66–68]. This type of search with neutral kaons is also quite complementary, see e.g. [68, 69] to the current CERN and the proposed PBC program in the kaon sector.

NA64 is currenly taking data at the H4 beam line of the SPS [70–72]. The beam line is a 100 GeV electron beam with a maximum intensity of $\sim {10}^{7}\,{e}^{-}$ per SPS spill. Beam intensity and transverse size have been optimized to guarantee an efficient detection of the synchrotron radiation during NA64 operation. The detection of synchrotron radiation is necessary to reach the required beam purity.

NA64 has collected about 3 × 10¹¹ eot before LS2, and aims at reaching 5 × 10¹² eot during Run 3.

The NA64 detector is a spectrometer with a low material budget tracker, micro-MEGAS, GEM and straw-tubes based, followed by an active target, which is a hodoscopic electromagnetic calorimeter (ECAL), shashlik-type, for the measurement of the energy of the recoil electrons. A high-efficient veto counter and a massive hermetic hadronic calorimeter are positioned just after the ECAL to efficiently detect muons or hadronic secondaries produced in the ${e}^{-}$-nucleus interactions in the active target.

The key feature of the NA64 apparatus is the detection of the synchrotron radiation from 100 GeV electrons in the magnetic field to significantly enhance electron identification and suppress background from the hadron contamination in the beam. The addition of a compact tungsten calorimeter after the syncrotron radiation detector as an active target for the production of energetic $A^{\prime}$ or X-boson explaining the ${}^{* }{Be}$ anomaly [73, 74], enables the search of $A^{\prime} \to {e}^{+}{e}^{-}$ visible decays. The first results obtained in 2016–2017 for the both $A^{\prime} \to \chi \overline{\chi }$ and $A^{\prime} \to {e}^{+}{e}^{-}$ decay modes [70–72] confirm the validity and sensitivity of the NA64 technique for searching for dark sector physics.

The NA64⁺⁺ experiment proposed in the PBC context aims at using high-energy electron, muon and hadron beams extracted at the SPS and currently available at the CERN North Area, starting in Run 3.

• (1)  
  NA64${}^{++}(e)$: NA64 plans to continue the data taking after LS2 with the main goal to integrate up to 5 × 10¹² eot at the H4 line in about $(6\mbox{--}8)$ months. The preparation of an area able to host a quasi-permanent installation of NA64 began in 2018.An upgrade of the detector is also needed in order to cope with a high intensity beam: this includes the replacement of the electromagnetic calorimeter electronics, the addition of a zero-degree hadron calorimeter, and the upgrade of the data acquisition system.
• (2)  
  NA64⁺⁺(μ):A new detector served by the M2 beam line and located in the EHN2 experimental hall in the CERN North Area is proposed to be built after LS2 to investigate dark sector predominantly coupled to the second and third generation and LFV μ–τ conversion with a high energy muon beam. The M2 line, currently serving the COMPASS experiment, is able to provide muons with momentum of ≃(100–160) GeV/c, and intensity up to ∼10⁸ μ/spill.The detector setup follows closely the one currently operating with e⁻ beam: an active (muon) target followed by a large hadron calorimeter located in a magnetic field, which is used both to measure the outgoing muon momentum and to veto events with associated hadrons. The signal consists of a muon with outgoing momentum significantly lower than the incoming one and no energy deposition in the rest of the detector.A key issue is the purity of the incoming muon beam: a background study performed in 2017 shows that the level of the hadron contamination in the muon beam can be reduced down to the negligible level ≤10⁻⁶ by using nine Be absorbers. Another key issue is the precise measurement of the momentum of the outgoing muon and its identification with high purity.Some modification of the M2 upstream part are also foreseen, as described in the PBC Conventional Beams WG Report [75]. Assuming a muon beam intensity of ∼3 × 10⁷μ/spill, with ∼4 × 10³ spills per day, about 1.5 years are necessary to accumulate ∼5 × 10¹³ mot. NA64 has submitted in October the addendum for the SPSC⁴⁰ for the Phase 1 of NA64⁺⁺(μ), which requires 10⁶ muons s⁻¹ at 100 GeV.
• (3)  
  NA64⁺⁺(h):The NA64 studies with hadron beams are less advanced and will continue during the coming years. Integrated luminosities of 10¹³ pions-on-target, 10¹² kaons-on-target, and 10¹² protons-on-target could investigate dark sector models complementary to the dark photon one. These searches would require (20–50) GeV hadron beams that could be provided by the H4 line without modification. The Collaboration aims to start data taking with hadron beams after LS3.

The main open question for NA64⁺⁺(e) is the detector ability to cope with the higher beam intensity, which is already available, and hence increased pile-up: this has already been positively answered based on the preliminary analysis of the data sample collected during the 2018 run where an intensity close to $\sim {10}^{7}e$/spill has been reached. With such an intensity and 4000/spills/day, about four months will be required to collect 4 × 10¹² eot. Upgrades of the detector and data acquisition system are planned during LS2. As for NA64⁺⁺(e), key issues for NA64⁺⁺(μ) are the beam purity and beam momentum measurement, and detector hermeticity. In both cases, the time sharing in the two (highly demanded) beam lines (H4 and M2) with other potential users (eg. COMPASS, MUonE, etc) is an issue and will require a careful planning and prioritization of the operations.

The Collaboration currently consists of ≃50 participants representing 14 institutions from Chile, Germany, Greece, Russia, Switzerland, and USA. An updated list of authors and institutions can be found at: https://na64.web.cern.ch. The NA64 Collaboration has also submitted a separate document⁴¹ for the next update of the ESPP.

NA62 [76] is a fixed target experiment at the CERN SPS with the main goal of measuring the BR of the ultra-rare decay ${K}^{+}\to {\pi }^{+}\nu \overline{\nu }$ with 10% precision. It is currently taking data at the K12 beam line at the CERN SPS. The NA62 long decay volume, hermetic coverage, low material budget, full PID capability and excellent tracking performance, make NA62 a suitable detector for the search for hidden particles (SHiP). The possibility of dedicating part of Run 3 to this physics case is timely, since the projected sensitivity surpasses that of competitive experiments in the same time range. NA62 proposes to integrate ∼10¹⁸ pot operating the detector in dump mode for few months during Run 3 [77].

NA62 is currently operating at the K12 beam line in the North Area. At full intensity, a beam of $3\times {10}^{12}$ protons-per-pulse (ppp), 400 GeV momentum, in 3.5 s long effective spills from the SPS hit a beryllium target to produce a 75 GeV momentum-selected 750 MHz intense secondary beam of positive particles, 6% of which are charged kaons. The beryllium target used by NA62 is followed by two 1.6 m long, water-cooled, beam-defining copper collimators (TAX) which can act also as a dump of ∼10.7 nuclear interaction lengths each. In the standard NA62 operation, roughly 50% of the beam protons punch through the beryllium target and are absorbed by the TAX collimators.

At the NA62 nominal beam intensity, 10¹⁸ pot can be acquired in ${ \mathcal O }(3)$ months of data taking. The dump-mode operation can be obtained by lifting the NA62 Beryllium target away from the beam line and by closing the first TAX collimator, placed ∼22 m downstream of the target. The muon halo emerging from the dump is partially swept away by the existing muon clearing system. The switching from the standard beam mode to the beam-dump mode takes a few minutes and it is already done regularly. About 3 × 10¹⁶ pot in dump mode have already been collected and are being analyzed for background studies.

The NA62 Collaboration is preparing a thorough plan for running after the end of LS2 with a fraction of the beam time in dump mode during Run 3 (2021–2023). A possible sharing could be two years in beam mode to complete the measurement of the branching fraction (BR) of the ${K}^{+}\to {\pi }^{+}\nu \overline{\nu }$ mode and ${ \mathcal O }(1)$ year in beam dump mode. The proposal will take into account the results obtained on the measurement of the ${BR}({K}^{+}\to {\pi }^{+}\nu \overline{\nu })$ based on the analysis of data taken in recent (2016–2018) run.

A schematic layout of the NA62 detector is shown in figure 4.

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The secondary positively charged hadron beam of 75 GeV/c momentum reaches the 120 m long, 2 m diameter, in-vacuum decay volume, placed 100 m downstream of the target. A Cherenkov counter (KTAG) filled with N₂ along the beam line identifies and timestamps kaons, which are about 6% of the hadron beam. Three silicon pixel stations (Gigatracker, GTK) measure the momentum and the time of all the particles in the beam at a rate of 750 MHz. A guard ring detector (CHANTI) tags hadronic interactions in the last GTK station at the entrance of the decay volume. Large angle electromagnetic calorimeters (LAV) made of lead glass blocks surround the decay vessel can be used to veto particles up to 50 mrad. A magnetic spectrometer made of straw tubes in vacuum measures the momentum of the charged particles.

A 17 m long RICH counter filled with Neon separates π, μ and e up to 40 GeV/c. The time of charged particles is measured both by the RICH and by scintillator hodoscopes (CHOD and NA48-CHOD) placed downstream to the RICH. The electromagnetic calorimeter filled with liquid krypton (LKr) covers the forward region and complements the RICH for the particle identification. A shashlik small-angle calorimeter (IRC) in front of LKr detects γ directed on the inner edges of the LKr hole around the beam axis. The hadronic calorimeter made of two modules of iron-scintillator sandwiches (MUV1 and MUV2) provides further π–μ separation based on hadronic energy. A fast scintillator array (MUV3) identifies muons with sub-nanosecond time resolution. A shashlik calorimeter (SAC) placed on the beam axis downstream of a dipole magnet bending off-axis the beam at the end of the NA62 detector, detects γ down to zero angle. A multi-level trigger architecture is used when operated in beam mode. The hardware-based level-0 trigger uses timing information from CHOD, RICH and MUV3, and calorimetric variables from electromagnetic and hadronic calorimeters. Higher-level software-based trigger requirements are based on variables from KTAG, LAV and magnetic spectrometer.

Such a setup is perfectly suited to perform a comprehensive SHiP in a large variety of visible final states.

The operation of NA62 in dump mode does not pose particular problems and no show-stoppers have been identified. The analysis of ${ \mathcal O }({10}^{16})$ pot collected in dump mode shows that the background can be kept under control for hidden particles decaying to final states that are then fully reconstructed. The addition of an Upstream Veto at the front of the fiducial volume is currently being studied: this detector should be able to further reduce the combinatorial di-muon background coming from random combinations of halo muons and to open the possibility of detecting also partially reconstructed final states. In normal operation mode half of the protons do not interact with the Be target and impinge upon the TAXes: these data are used for some specific background studies, namely for the di-muon background.

Minor modifications to the beam line are possible, too, aimed at reducing the upstream-produced background (again mainly halo muons). A full Geant4-based simulation of the beam line has been implemented and is being used to study optimized settings of the existing magnetic elements of the line and possibly an optimized new layout for the beam-dump operation. Preliminary studies show that the component of the muon flux above 20 GeV can be reduced by two orders of magnitude with an appropriate setting of the magnetic elements of the beam line. The maximum intensity achievable is under study, as well, with some prospects of increase beyond the present nominal one. These aspects are under study within the PBC Conventional Beams working group.

The NA62 Collaboration is made of 213 authors from 31 institutions. An updated list of authors and institutions can be found at: https://na62.web.cern.ch.

Brief presentation, unique features

Key requirements for detector, beam, beam time, timeline

Reaching the full potential of the missing-momentum technique places demanding constraints on the experiment and the beamline supporting it. A high repetition rate of electrons is required (as much as ∼10⁹ electrons-on-target (eot) per second) in order to reach the envisaged integrated luminosities of 10¹⁴–10¹⁶ eot, while keeping an extremely low electron density per bunch ($1-5\ {e}^{-}$/bunch).

This requires a fast detector that can individually resolve the energies and angles of incident electrons, while simultaneously rejecting a variety of potential background processes that vary in rate over many orders of magnitude. The LDMX design makes use of a low-mass, silicon-based tracking system in a 1.5 T dipole magnet to measure the momentum of the incoming electrons, and to cleanly reconstruct electron recoils, thereby providing a measure of missing momentum. A high-speed, high-granularity SiW calorimeter with MIP sensitivity is used to reject potential high rate bremsstrahlung background at trigger level, and to work in tandem with a scintillator-based hadron calorimeter to veto rare photo-nuclear reactions. The design leverages new and existing calorimeter technology under development for the HL-LHC, as well as existing tracking technology and experience from the HPS experiment [80]. The experiment is fairly small-scale for HEP standards. Thus it could be built, commissioned and run over the course of a few years. A rendering of the proposed experimental design is shown in figure 5.

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The scenario for a CERN SPS beam outlined below envisages a beam energy between 3.5 and 16 GeV [79]. Further requirements for the beam are a low current and large beam-spot to ease the identification of individual electrons, paired with a high duty factor for large integrated luminosity.

All of this can be provided at CERN in three basic steps: a new LINAC providing electrons with 3.5 GeV, injecting into the SPS where the electrons are accelerated to up to 16 GeV, followed by a slow extraction of electrons to be delivered to the experiment. The bunch spacing can be any multiple of 5 ns up to 40 ns, the average number of electrons per bunch can range from <1 up to anything that can be tolerated by the experiment, and there is a high flexibility in the beam size, such that for example a beam spot of 2 cm × 30 cm is perfectly feasible. To achieve 10¹⁶ eot in one year would require approximately one third of the time currently used by the SPS to accelerate protons.

The Collaboration anticipates the beamline could be available conservatively in 2025 (or even a few years earlier depending on CERN priorities) and that this would accommodate comfortably the time needed for the final design and construction of the detector. Hence, data taking could start in 2025 (or earlier), and be completed within a few years, as little as 1–2 years for the most optimistic luminosity scenarios. In addition to the LDMX experiment itself, the main construction needs are the electron linac as injector to the SPS, a 50 m tunnel for last path of the extracted beam, and a small experimental hall. The potential of such a primary electron beam facility goes beyond LDMX: (i) It also opens for a beam-dump search for visibly decaying dark photons, (ii) gives a Jefferson laboratory type facility with extended energy range for Nuclear Physics, and (iii) would be a significant Accelerator Physics R&D-asset at CERN.

Open questions, planned feasibility studies

Status of the Collaboration

Brief presentation, unique features

Key requirements for detector, beam, beam time, timeline

The dark photons decay in a decay volume of order 10 m long, and the decay products are detected in three micromegas trackers MM1, MM2, MM3 as well as a tungsten plastic shashlik calorimeter, ECAL and the further possible addition of a HCAL. A downstream magnet separates decay products and allows the momentum reconstruction. A schematic layout of the experiment is shown in figure 6.

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The advantage of this experimental setup is the luminosity gain provided by deploying bunches of electrons. This enables a larger eot in a shorter time frame and results in an extended coverage of the sensitivity parameter space. Taking into account the LIU-SPS with upgraded extraction kickers and a 12 week experimental run with a 70% SPS duty cycle, AWAKE/NA64 expects to integrate 10¹⁶ eot in one year of operation. This is more than three orders of magnitude larger than the expected integrated eot by NA64 in Run 3.

The proposed experiment requires a location accessible to SPS protons that drive the AWAKE accelerator and tunnel length long enough to accommodate a 50–100 m long plasma cell as well as 20 m of dump, drift volume and detectors.

A possible location is in the former CNGS target hall and decay tunnel. This project relies on the successful implementation of the AWAKE acceleration concept and could be installed at earliest during LS3.

Open questions, planned feasibility studies

Status of the Collaboration

The main goal of the KLEVER experiment is to look for New Physics in the multi-TeV mass range via a measurement of the rare decay ${K}_{L}\to {\pi }^{0}\nu \overline{\nu }$ and is discussed in section 6. However, the experiment may also be sensitive to specific signatures of hidden sector physics at the MeV–GeV scale, as discussed in section 9.

Brief presentation, unique features

Detector description, key requirements for detector

The main experimental challenge concerns the requirement of highly efficient reduction of beam-induced backgrounds to below 0.1 events in the projected sample of 2 × 10²⁰ protons on target. To this end, the experimental configuration includes a long target made of heavy material to stop pions and kaons before their decay, a decay volume in vacuum, a muon shield based on magnetic deflection able to reduce the flux of muons emerging from the target by six orders of magnitude in the detector acceptance, and a hermetic veto system surrounding the whole decay volume.

The SHiP experiment incorporates two complementary apparatuses. The first detector immediately downstream of the muon shield consists of an emulsion based spectrometer optimized for recoil signatures of hidden sector particles and τ neutrino physics. The second detector system aims at measuring the decays of Hidden Sector particles to fully reconstructible final states as well as partially reconstructible final states that involve neutrinos. The spectrometer is designed to accurately reconstruct the decay vertex, the mass, and the impact parameter of the hidden particle trajectory at the proton target. A set of calorimeters and muon stations provide particle identification. A dedicated timing detector with ∼100 ps resolution provides a measure of coincidence in order to reject combinatorial backgrounds. The decay volume is surrounded by background taggers to tag neutrino and muon interactions in the vacuum vessel walls and in the surrounding infrastructure.

A schematic diagram of the detector layout is shown in figure 7.

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Beam requirements, beam time, timeline The BDF facility is described in a separate report⁴⁵ . It consists of a 400 GeV momentum primary proton beam line slowly extracted from the SPS in 1 s long spills per 7.2 s long cycle. It is able to provide up to 4.0 × 10¹³ protons per cycle. The SHiP operational scenario is based on a similar fraction of beam time as the past CERN Neutrinos to Gran Sasso (CNGS) program. In the baseline scenario, the beam sharing delivers an annual yield of 4 × 10¹⁹ protons to the SHiP experimental facility and a total of 10¹⁹ to the other physics programs at the CERN North Area, while respecting the beam delivery required by the LHC and HL-LHC. The physics sensitivities are based on acquiring a total of 2 × 10²⁰ protons on target, which may thus be achieved in five years of nominal operation.

CERN's North Area has a large space next to the SPS beam transfer lines which is largely free of structures and underground galleries, and is entirely located on the current CERN territory. The proposed implementation is based on minimal modification to the SPS complex and maximum use of the existing beam lines. The design foresees space for future extensions. SHiP profits from the unique feature in the SPS of slow extraction of a de-bunched beam over a timescale of around a second. It allows tight control of combinatorial background, and allows diluting the large beam power deposited on the proton target both spatially and temporally. Should an observation require consolidation, a second mode of operation with slow extraction of bunched beam is also foreseen in order to further increase the discrimination between the signature of a light dark matter object, by measuring their different times of flight, and background induced by neutrino interactions.

The schedule for the SHiP experiment and the experimental facility is largely driven by the CERN long-term accelerator schedule. Accordingly, the schedule aims at profiting as much as possible from data taking during Run 4 (currently 2027–2029). Most of the experimental facility can be constructed in parallel to operating the North Area beam facilities. The connection to the SPS has been linked to Long Shutdown 3 (i.e. for LHC 2024–2026) but requires that the stop of the North Area is extended by one year (2025–2026). The schedule requires preparation of final prototypes and the TDRs for both the detector and the facility by beginning 2022, and construction and installation between 2023 and beginning 2027.

Background and feasibility studies

Open questions:

Status of the collaboration

FASER (ForwArd SEarch expeRiment at the LHC) is a proposed small and inexpensive experiment designed to search for light, weakly-interacting particles at the LHC. Such particles are dominantly produced along the beam collision axis and are typically LPs, traveling hundreds of meters before decaying. To exploit both of these properties, FASER is to be located along the beam collision axis, 480 m downstream from the ATLAS interaction point (IP). At this location, FASER and a larger successor, FASER2, will enhance the LHC's discovery potential by providing sensitivity to dark photons, dark Higgs bosons, HNLs, ALPs, and many other proposed new particles.

The FASER signal is LLPs that are produced at or close to the IP, travel along the beam collision axis, and decay visibly in FASER:

$$\begin{eqnarray}&&p+p\to \mathrm{LLP}+X,\ \ \mathrm{LLP}\ \mathrm{travels}\sim 480\,{\rm{m}},\ \ \mathrm{LLP}\to \mathrm{charged}\ \mathrm{tracks}+X\ (\mathrm{or}\ \gamma \gamma +X).\,\end{eqnarray} \tag{ 5.1 }$$

These signals are striking: two oppositely charged tracks (or two photons) with very high energy (∼TeV) that emanate from a common vertex inside the detector and which have a combined momentum that points back through 100 m of rock and concrete to the IP.

The sensitivity reach of FASER has been investigated for a large number of new physics scenarios [28, 82–92]. FASER will have the potential to discover a broad array of new particles, including dark photons, other light gauge bosons, HNLs with dominantly τ couplings, and ALPs. FASER2 will extend FASER's physics reach in these models to larger masses and also probe currently uncharted territory for dark Higgs bosons, other types of HNLs, and many other possibilities.

FASER will be located 480 m downstream from the ATLAS IP in service tunnel TI12 as shown in figure 8. TI12 was formerly used to connect the SPS to the LEP tunnel, but is currently empty and unused.

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The planned timeline for FASER is to be installed in TI12 during Long Shutdown 2 (LS2), in time to collect data during Run 3 of the 14 TeV LHC from 2021 to 2023. FASER's cylindrical active decay volume has a radius R = 10 cm and length L = 1.5 m, and the detector's total length is under 5 m. To allow FASER to maximally intersect the beam collision axis, the floor of TI12 should be lowered by 45 cm. This will not disrupt essential services, and no other excavation is required. FASER will run concurrently with the LHC and requires no beam modifications. Its interactions with existing experiments are limited only to requiring bunch crossing timing and luminosity information from ATLAS.

If FASER is successful, a larger version, FASER2, with an active decay volume with R = 1 m and L = 5 m, could be installed during LS3 and take data in the 14 TeV HL-LHC era. FASER2 would require extending TI12 or widening the staging area UJ12 adjacent to TI12.

The layout of the FASER detector is illustrated in figure 9. At the entrance to the detector on the left is a double layer of scintillators (gray) to veto charged particles coming through the cavern wall from the IP, primarily high-energy muons. The veto layer is followed by a 1.5 m long, 0.6 T permanent dipole magnet (red) with a 20 cm aperture. This serves as the decay volume for LLPs decaying into a pair of charged particles, with the magnet separating these to a detectable distance. Next is a spectrometer consisting of two 1 m long, 0.6 T dipole magnets with three tracking stations (blue), each composed of layers of precision silicon strip detectors located at either end and in between the magnets. Scintillator planes (gray) for triggering and precision time measurements are located at the entrance and exit of the spectrometer. The final component is an electromagnetic calorimeter (purple) to identify high energy electrons and photons and measure the total electromagnetic energy.

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The FASER signals are two extremely energetic (∼TeV) coincident tracks or photons that start at a common vertex and point back to the ATLAS IP. Muons and neutrinos are the only known particles that can transport such energies through 100 m of rock and concrete between the IP and FASER. Preliminary estimates show that muon-associated radiative processes and neutrino-induced backgrounds may be reduced to negligible levels.

Recently a FLUKA study [93–95] from the CERN Sources, Targets and Interactions group has been carried out to assess possible backgrounds and the radiation level in the FASER location. The study shows that no high energy (>100 GeV) particles are expected to enter FASER from proton showers in the dispersion suppressor or from beam-gas interactions. In addition, the radiation level expected at the FASER location is very low due to the dispersion function in the LHC cell closest to FASER.

Emulsion detectors and battery-operated radiation monitors were installed in TI12 and TI18 during Technical Stops in 2018. The results from these in situ measurements have validated the FLUKA estimates, confirming that the high-energy particle background is highly suppressed and radiation levels are also very low and not expected to be problematic for detector electronics. Additional work is ongoing to refine background estimates, evaluate signal efficiencies, and optimize the detector.

FASER submitted a Letter of Intent [96] to the LHCC in July 2018. At its September meeting, the LHCC reviewed the LoI favorably and encouraged the FASER Collaboration to submit a Technical Proposal. This was submitted to the LHCC in November 2018, and based on a positive review, the LHCC has approved FASER in March 2019. A working group has also been created within the PBC activities to study the interplay between the detector, the civil engineering, the backgrounds and radiation levels at the FASER installation point. Two private foundations contribute to support FASER's construction and operation costs.

The FASER group currently (December 2018) consists of 27 collaborators (22 experimentalists and 5 theorists) from 16 institutions in China, Germany, Israel, Japan, Poland, Switzerland, the United Kingdom, and the United States. The list of institutes is the following: Tsinghua University (China), University of Mainz (Germany), Technion (Israel), Weizmann Institute (Israel), KEK (Japan), Kyushu University (Japan), Nagoya University (Japan), National Centre for Nuclear Research (Poland), University of Bern (Switzerland), CERN (Switzerland), University of Geneva (Switzerland), University of Sheffield (United Kingdom) Rutgers University (United States), University of California (United States), University of Oregon (United States), University of Washington (United States).

The updated status of the Collaboration and experiment are available at: https://twiki.cern.ch/twiki/bin/viewauth/FASER/WebHome. FASER has submitted also a separate document⁴⁹ for the next ESPP update.

The basic motivation for the MATHUSLA detector (MAssive Timing Hodoscope for Ultra-Stable neutraL pArticles) [97] is the search for LLPs produced in $\sqrt{s}=14$ TeV HL-LHC collisions, with lifetimes much greater than the size of the main detectors and up to the BBN constraint of ∼0.1 s, with the peak sensitivity near βcτ ∼ 100 m. MATHUSLA also has a secondary physics case as a cosmic ray telescope.

This proposal has been the subject of several studies [86, 98–108], and the physics motivation from both a bottom-up and top-down point of view, including connections to naturalness, dark matter, baryogenesis and neutrino masses, has been explored in a comprehensive white paper [109]. The MATHUSLA Collaboration has also recently presented its Letter of Intent [110] to the LHCC. Given that some overlap exists between the MATHUSLA physics case and the PBC framework, the LHC Committee recommended MATHUSLA to be discussed within the PBC framework as well.

The size of the detector and the corresponding location is not yet finalized. All sensitivity estimates in this document assume the MATHUSLA200 benchmark geometry from the Letter of Intent [110], which was also the original layout proposed in [97, 109]. This geometry assumes a very large (200 × 200 m²) area detector built on the surface, situated 100 m horizontally and vertically away from a LHC interaction point IP (either ATLAS or CMS IP), and a decay volume height of 20 m above the ground.

It is very unlikely that a detector with these large dimensions can be implemented at CERN, hence the sensitivity plots shown in this document should be properly rescaled once the final geometry will be finalized and the exact distance from the ATLAS/CMS IP points determined. An integrated luminosity of 3 ab⁻¹ corresponding to the full HL-LHC period is assumed, with a hypothetical start of the data taking during Run 4. The timeline for the construction of the detector is under study.

The MATHUSLA detector is essentially a large tracker, situated above an air-filled decay fiducial volume on the surface above ATLAS or CMS, that is able to robustly reconstruct displaced vertices (DVs) from the decay of neutral LLPs into two or more charged particles. The tracker should have on the order of five planes to provide robust tracking with ∼ ns timing and cm spatial resolution. This is vital for rejecting cosmic ray (CR) and other backgrounds, and allows for the reconstruction of multi-pronged DV for LLPs with boost up to ∼10³, corresponding to minimum LLP mass of ${ \mathcal O }(10$ MeV) if the LLP is produced in exotic B-meson decays and ${ \mathcal O }(0.1-1\,\mathrm{GeV})$ for weak or TeV scale production [109]. Analyzing the geometry and multiplicity of the DV final states also allows the LLP decay mode and mass to be determined in many scenarios [102]. A layer of detectors in the floor is also considered, since this will improve LLP reconstruction and provide additional veto capabilities that may be necessary to reject upwards-going backgrounds like high-energy muons from the HL-LHC.

For the current MATHUSLA design, the focus is on proven and relatively cheap technologies to allow for MATHUSLA's construction in time for the HL-LHC upgrade. Therefore, the trackers are envisioned to be implemented with Resistive Plate Chambers (RPCs), which have been used for very large area experiments in the past [111, 112], or extruded scintillators which have also been used extensively [113, 114].

Assuming the baseline dimensions of 200 × 200 m², with five active layers, this would correspond to 200 000 m² of active detectors that have to provide time and space coordinates with ∼ ns time resolution and ∼ cm space resolution.

The main open questions for MATHUSLA are related to its large dimensions and to its capacity of controlling the backgrounds mostly coming from the tens of MHz of cosmic rays crossing the detector in all directions, with a total integrated rate of ∼10¹⁵ charged particle trajectories over the whole HL-LHC run.

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  Cost: The Collaboration has not provided an official estimate of the cost of the detector because of ongoing design optimizations. MATHUSLA requirements on resolution and rate are significantly lower compared to past experiments using similar detector technologies. The scale of the detector area is a further opportunity for cost optimization by employing mass production techniques. The detector size and location are currently being optimized to take into account land constraints and opportunities, with the hope to be able to reduce the size while keeping similar sensitivity. The detector design is modular for a staged implementation. The total cost will be driven by civil engineering and the large area tracking detectors. The Collaboration is investigating low-cost solutions with the challenging goal to keep the overall cost of the full size detector below 100 MCHF.
• (2)  
  Background: As was argued in detail in [97], it is crucial for the projected sensitivities that MATHUSLA can search for LLP decays without backgrounds. The surface location shields MATHUSLA from ubiquitous QCD backgrounds from the LHC collision. It was quantitatively demonstrated that muon and neutrino backgrounds from the IP can be sufficiently rejected. Extremely stringent signal requirements and 4-dimensional DV reconstruction would limit the probability of cosmic rays to fake the hadronic or even leptonic LLP decays. Background estimates using a combination of detailed Monte Carlo studies with full detector simulation, the known cosmic ray spectrum, and empirical measurements at the LHC using a test stand detector, are currently in progress. The outcome of these studies will quantitatively determine whether the proposed background rejection strategies are sufficiently effective to reach the zero-background regime. However, to date, no quantitative analysis based on the full Geant4 simulation of the detector with large Monte Carlo samples has been shown, and, as such, the assumption that MATHUSLA200 is a zero-background experiment is still to be demonstrated.

The Collaboration is currently studying a modular detector design, evaluating possible experimental sites at CERN and developing simulations of background and signal acceptance. Crucial to this endeavor is the data from the MATHUSLA test stand, a ∼(3 × 3 × 5) m³ MATHUSLA-type detector that is currently taking data on local cosmic rays and LHC muon backgrounds at CERN Point 1, allowing simulation frameworks to be calibrated and reconstruction strategies to be verified.

Precise timelines for the full detector proposal are still being established, but the aim is to have the full detector operating roughly by the time the HL-LHC goes online, around 2025 or shortly thereafter. The MATHUSLA Collaboration has also prepared a separate stand-alone submission⁵⁰ to the ESPP update.

A snapshot of the MATHUSLA Collaboration is provided by the author list of the Letter of Intent [110]. It includes 64 authors, of which 48 are experimentalists and 16 are theorists. The institutes of the 48 experimentalists are the following: Universidad Mayor de San Andrés (Bolivia), University of Campinas (Brazil), Institute of High Energy Physics (Beijing, China) Tel Aviv University (Israel), Politecnico di Bari (Italy), INFN, sezione di Roma Tor Vergata (Italy), Università degli Studi di Roma La Sapienza (Italy), Benemérita Universidad Autónoma de Puebla (Mexico), Universidad Michoacana de San Nicolás de Hidalgo (Mexico), Universidad Autónoma de Chiapas (Mexico), CERN, Boston University (US), NYU (US), Ohio State University(US), Rutgers (US), SLAC (US), University of Arizona (US), University of Maryland (US), University of Washington (US).

The NA62 experiment at the CERN SPS is expected to measure ${BR}({K}^{+}\to {\pi }^{+}\nu \overline{\nu })$ to within 10% by the end of LHC Run 3. In order to fully constrain the CKM matrix, or possibly, distinguish between different NP scenarios, it is necessary to measure ${BR}({K}_{L}\to {\pi }^{0}\nu \overline{\nu }$) as well. The KOTO experiment at J-PARC, should have enough data for the first observation of the ${K}_{L}\to {\pi }^{0}\nu \overline{\nu }$ decay by the late 2020s⁵¹ , but a next-generation experiment is needed in order to measure the BR.

As far as KOTO is concerned, a new detector and an upgraded beam line would be required to go to ${ \mathcal O }(100)$ events sensitivity: an extension of the J-PARC hadron hall is currently being considered by the Japan Science Council with KOTO⁺⁺ as a priority.

The KLEVER experiment aims to measure ${BR}({K}_{L}\to {\pi }^{0}\nu \overline{\nu })$ to ∼20% accuracy assuming the SM branching fraction, corresponding to the collection of 60 SM events with an S/B ratio of ∼1 using a high-energy neutral beam at the CERN SPS starting in Run 4.

Relative to KOTO, which uses a neutral beam with a mean momentum of about 2 GeV, the boost from the high-energy beam in KLEVER facilitates the rejection of background channels such as ${K}_{L}\to {\pi }^{0}{\pi }^{0}$ by detection of the additional photons in the final state. On the other hand, the layout poses particular challenges for the design of the small angle vetoes, which must reject photons from K_L decays escaping through the beam pipe amidst an intense background from soft photons and neutrons in the beam. Background from ${\rm{\Lambda }}\to n{\pi }^{0}$ decays in the beam must also be kept under control.

KLEVER would make use of the 400 GeV SPS proton beam to produce a neutral secondary beam with a mean K_L momentum of 40 GeV, leading to a fiducial volume acceptance of 4%, and a K_L yield of 2 × 10⁻⁵ K_L/pot. With a selection efficiency of 5%, collection of 60 SM events would require a total primary flux of 5 × 10¹⁹ pot, corresponding to an intensity of 2 × 10¹³ ppp under NA62-like slow-extraction conditions. This is a six-fold increase in the primary intensity relative to NA62. The feasibility of an upgrade to provide this intensity on the T10 target is under study in the Conventional Beams working group [75]. Preliminary indications are positive: there is general progress on issues related to the slow extraction of the needed intensity to the North Area (including duty cycle optimization); a workable solution for T4-to-T10 bypass has been identified. The ventilation in the TCC8 cavern appears to be reasonably hermetic, obviating the need for potentially expensive upgrades. A four-collimator neutral beamline layout for ECN3 has been developed and simulation studies with FLUKA and Geant4 are in progress to quantify the extent and composition of beam halo, muon backgrounds, and sweeping requirements.

KLEVER would aim to start data taking in LHC Run 4 (2026). Assuming a delivered proton intensity of 10¹⁹ pot yr⁻¹, collection of 60 SM events would require five years of data taking. To be ready for the 2026 start date, detector construction would have to begin by 2021 and be substantially concluded by 2025, leaving three years from the present for design consolidation and R&D.

A schematic layout of the experiment is shown in figure 11. Most of the subdetector systems for KLEVER will have to be newly constructed. Early studies indicated that the NA48 liquid-krypton calorimeter (LKr) could be reused as the Main Electromagnetic Calorimeter (MEC), and indeed, the efficiency and energy resolution of the LKr appear to be satisfactory for KLEVER.

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However, the LKr timing resolution would be a major liability. The LKr would measure the event time in KLEVER with 500 ps resolution, while the total rate of accidental vetoes from the SAC could be 100 MHz. The LKr time resolution might be improved via a comprehensive readout upgrade, but concerns about the service life of the LKr would remain, and the size of the inner bore would limit the beam solid angle (and hence kaon flux). The Collaboration is investigating the possibility of replacing the LKr with a shashlyk-based MEC patterned on the PANDA FS calorimeter (in turn, based on the KOPIO calorimeter [129]). This is a shashlyk design incorporating 'spy tiles' for longitudinal sampling of the shower development, resulting in additional information for γ/n separation. A first test of this concept was carried out with a prototype detector at Protvino in April 2018.

The upstream veto (UV), which rejects ${K}_{L}\to {\pi }^{0}{\pi }^{0}$ decays upstream of the fiducial volume, would use the same shashlyk technology as the MEC. The active final collimator (AFC), inserted into the hole in the UV for passage of the beam, is a LYSO collar counter with angled inner surfaces. This provides the last stage of beam collimation while vetoing photons from K_L that decay in transit through the collimator itself. Because of the boost from the high-energy beam, it is sufficient for the large-angle photon vetoes (LAVs) to cover polar angles out to 100 mrad. The LAVs are lead/scintillating-tile detectors based on the CKM VVS [130]. Extensive experience with this type of detector (including in prototype tests for NA62) demonstrates that the low-energy photon detection efficiency will be sufficient for KLEVER [131, 132].

As far as the rejection of charged particles is concerned, simulations indicate that the needed rejection can be achieved with two staggered planes of charged-particle veto (CPV) each providing 99.5% detection efficiency, supplemented by the μ and π recognition capabilities of the MEC (assumed in this case to be equal to those of the LKr) and the current NA62 hadronic calorimeters and muon vetoes.

Finally, a pre-shower detector featuring a 0.5 X₀ converter and two planes of tracking with σ_x,y ∼ 100 μm (assumed to be large-area MPGDs) would allow angular reconstruction of at least one γ from ${K}_{L}\to {\pi }^{0}{\pi }^{0}$ events with two lost γ's to be reconstructed in 50% of cases.

Simulations of the experiment carried out with fast-simulation techniques (idealized geometry, parameterized detector response, etc) show that the target sensitivity is achievable (60 SM events with S/B = 1). Background channels considered at high simulation statistics include ${K}_{L}\to {\pi }^{0}{\pi }^{0}$ (including events with reconstructed photons from different ${\pi }^{0}{\rm{s}}$ and events with overlapping photons on the MEC), ${K}_{L}\to 3{\pi }^{0}$ and ${K}_{L}\to \gamma \gamma$.

Background from ${\rm{\Lambda }}\to n{\pi }^{0}$ and from decays with charged particles is assumed to be eliminated on the basis of studies with more limited statistics. An effort is underway to develop a comprehensive simulation and use it to validate the results obtained so far. Of particular note, backgrounds from radiative K_L decays, cascading hyperon decays, and beam-gas interactions remain to be studied, and the neutral-beam halo from more detailed FLUKA simulations needs to be incorporated into the simulation of the experiment. Preliminary studies indicate that the hit and event rates are similar to those in NA62, with the notable exception of the SAC, which will require an innovative readout solution. Offline computing resources required are similarly expected to be on the scale of NA62.

A PBC concern is related to the overall cost of the project if compared to the current strength of the Collaboration. The Collaboration is well aware that success in carrying out the KLEVER experimental program will require the involvement of new institutions and groups, with resources to contribute to the project, and initiatives to seek new collaborators are a major focus at present.

The proton sharing with existing or potential users in the North Area (as, e.g. SHiP), is also a concern: this will require a proper schedule and prioritization among the proposals.

About 13 institutions currently participating in NA62 have expressed support for and interest in the KLEVER project. These are: University of Sofia (Bulgaria), Charles University (Czech Republic), Mainz (Germany), University and INFN Ferrara (Italy), University and INFN Firenze (Italy), University and INFN Naples (Italy), University and INFN Pisa (Italy), University and INFN Tor Vergata (Italy), University and INFN Torino (Italy), INR (Moscow, Russia), IHEP (Protvino, Russia), George Mason University (US).

Individuals from UK institutions participating in NA62 have indicated an interest in the KLEVER project and are exploring the possibility of joining. In addition 5 people from CERN EN-EA are currently dedicated to the study of the KLEVER beam line.

In addition to direct KLEVER input for the European Strategy update⁵² , an Expression of Interest to the SPSC is in preparation and will serve as an opportunity to consolidate project membership.

The TauFV Collaboration aims at exploiting the high intensity of the BDF at CERN and install a detector, upstream of the proposed SHiP beam-dump target, which will have world-leading sensitivity to many LFV decay modes, for example probing for $\tau \to \mu \mu \mu$ decays down into the 10⁻¹⁰ regime. For the $\tau \to \mu \mu \mu$ mode, a limit of $2.1\times {10}^{-8}$ at the 90% confidence level has been set by the Belle Collaboration [133]. Results of similar sensitivity have been obtained by BaBar [134] and LHCb [135]. The Belle-II experiment expects to reach a sensitivity of 1 × 10⁻⁹ [136], but may be able to go lower if all background is suppressed.

TauFV will be well suited to other LFV studies in tau decays, for example ${\tau }^{-}\to {e}^{-}{e}^{+}{e}^{-}$, ${\tau }^{-}\to {\mu }^{-}{e}^{+}{e}^{-},{\tau }^{-}\to {e}^{-}{\mu }^{+}{\mu }^{-},{\tau }^{-}\to {\mu }^{+}{e}^{-}{e}^{-}$ and ${\tau }^{-}\to {\mu }^{+}{e}^{-}{e}^{-}$. Particularly high sensitivity is expected for the latter two modes, where the initial level of contamination will be lower. Lepton number violation (LNV) searches will be performed with decays such as ${\tau }^{-}\to {h}^{-}{h}^{-}{{\ell }}^{+}$ (h = any hadron, ℓ = e or μ). The experiment will also have access to an enormous number of charm decays (e.g. $5\times {10}^{15}$ D⁰ mesons), which will allow a parallel program of LFV and LNV study with modes such as $D\to h{\mu }^{-}{e}^{+}$ and $D\to h{{\ell }}^{-}{{\ell }}^{-}$. World-leading measurements will be possible in the field of charm physics, many enabled by the excellent calorimetry of the experiment, including CP-violation measurements and searches for suppressed decays such as ${D}^{0}\to {\mu }^{+}{\mu }^{-}$ and ${D}^{0}\to \gamma \gamma$.

The baseline scenario is to use 2% of the protons currently intended for the SHiP experiment, which could be achieved with an integrated target thickness of 2 mm of tungsten. A five year period of operation would produce 4 × 10¹⁸ protons-on-target, which would result in 8 × 10¹³ ${D}_{s}^{-}\to {\tau }^{-}\overline{\nu }$ decays. This enormous yield is two orders of magnitude larger than the number of τ leptons so far produced at the LHCb interaction point, and five orders of magnitude larger than that produced at Belle.

The timescale for installing and operating TauFV is dictated both by the construction of the BDF, and by the development of the challenging sub-detector technology, in particular the front-end ASICs. The TauFV experimental hall could be prepared in 2026–27, in parallel with the installation of SHiP. If the project proceeds rapidly it would be possible to deploy the full detector at this time. Alternatively, a first-stage experiment, capable of demonstrating the possibility of performing high-precision flavor physics in this new environment, could be installed instead. The full scale experiment would then be assembled in LS4, currently foreseen for 2030. An attractive feature of TauFV is that the physics reach is not limited by the intensity of the available beam. Therefore, it is conceivable that future upgrades could be planned, integrating significantly more pot, depending on how both the detector technology and the physics landscape evolve.

From the beam optics point of view, several locations can provide the required beam conditions and the beam drift space to accommodate the detector along the new 200 m transfer line between the TDC2 switch-yard cavern and the BDF target station, without either affecting the location of the BDF experimental area or requiring significant changes to the beam-line configuration. The choice is instead driven by considerations related to the civil engineering in the vicinity of the existing installations, radiological protection, and access and transport requirements, both above ground and underground. Lateral space is required on both sides for shielding in order to limit the radiation exposure of the surrounding underground area to levels typical for the rest of the beam line. The currently preferred location is situated 100 m upstream of the BDF target bunker. An access and service complex for the transfer line is already foreseen at this location. This complex will be extended and reconfigured to include a bypass tunnel, the detector bunker, service cavern and the required surface infrastructure.

The target system of TauFV will consist of a set of thin tungsten blades, matched to an elliptical beam profile of vertical size ∼1 mm, each separated by ∼2 cm and distributed over a length of 10–20 cm (figure 12, left). This layout will ensure that interactions will be well spread both longitudinally and transversally, which is desirable for background rejection. Furthermore, the majority of the τ leptons will decay in free space, and there will be a low probability of a decay track passing through a downstream target.

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The spectrometer design (figure 12, right) has an acceptance in polar angle between 20 and 260 mrad, and length of around 7 m. A Vertex Locator (VELO), comprising planes of silicon-pixel detectors broadly similar to the LHCb VELO, interleaves the target system, and continues downstream of it. Bending of charged tracks is provided by a dipole of integrated field of ∼2.5 Tm, which is followed by a tracker, a TORCH detector [137, 138], a high performance ECAL and a muon system. All detector components will have fast-timing capabilities, good radiation hardness and high granularity. The TORCH detector provides time resolution for charged tracks of <20 ps, which is a key weapon in the suppression of combinatoric background, and also brings hadron identification capabilities, which will enhance the charm-physics program of the experiment.

R&D is starting on the most critical elements of the detector, in particular the VELO and the ECAL. Here there is very close synergy with the requirements of Upgrade II of LHCb [139]. The VELO stations will be built from hybrid pixel sensors, and discussions have begun with the MediPix Collaboration concerning the requirements of the ASIC. A promising solution for the ECAL would be to employ crystal modules, based on YAG or GAGG crystal as a scintillator, and using the leading edge of the light pulse, or alternatively a silicon preshower, to provide the fast-timing information. Crystal samples have already been acquired, with the aim of constructing and evaluating a prototype module in 2019.

Evaluation of the physics reach of the TauFV experiment has so far focused on the benchmark mode ${\tau }^{-}\to {\mu }^{-}{\mu }^{+}{\mu }^{-}$. Studies are still ongoing, but preliminary results suggest that excellent control of combinatoric background will be achievable, mainly due to the distributed target system, which suppresses the likelihood of fake combinations, and the fast-timing provided by the TORCH and other sub-detectors. Hence, combinatorics will certainly be the sub-dominant source of background, and will not significantly impact upon the measurement down to signal branching ratios of 1 × 10⁻¹⁰, and maybe lower. Background from same topology decays of ${D}^{+}$ and D^s mesons involving three leptons will be a greater concern, but will be controlled through good mass resolution, kinematic requirements involving the direction between the interaction and decay vertices, and the possibility to tag the soft photon from ${D}_{s}^{* }\to {D}_{s}\gamma$ decays, thereby rejecting backgrounds from ${D}^{+}$ mesons. Restrictions on the invariant mass of each di-muon pair can isolate ultra-pure regions of phase space, but at the expense of introducing model-dependence into the interpretation of the results. All these methods will be combined in a multivariate analysis to obtain maximum discrimination. Although final results are not yet available, it seems probable that sensitivity to branching ratios of a few 10⁻¹⁰ will be attainable. The physics reach in modes like ${\tau }^{-}\to {e}^{+}{\mu }^{-}{\mu }^{-}$, which are afflicted by combinatoric background alone, will be even better by an order of magnitude.

The potential of TauFV in charm physics can be assessed by a direct comparison with LHCb, as the sources of background are the same, and are generally dominated by combinatorics, that TauFV can suppress effectively. The ECAL, optimized for soft photons, will give TauFV exciting possibilities in radiative decays. TauFV will therefore provide charm measurements of similar or higher precision to those of the proposed LHCb Upgrade II across a wide range of decay topologies, including modes that are complementary to the collider experiment.

The project is currently at a very early stage: no results from simulation are available, and the evaluation of the background is currently ongoing. This will be a mandatory step to address a definitive estimate of the physics reach. In order to pursue the proposed physics program, a strengthening of the Collaboration is also necessary.

The current TauFV Collaboration consists of nine physicists from the University of Bristol (UK), CERN, Imperial College London (UK), the University of Oxford (UK), the University of Zurich (Switzerland) and ETH Zurich (Switzerland). Before the end of this year these groups will complete the initial optimization of the layout and determine the physics reach for benchmark channels. In parallel, discussions will take place with additional potential collaborators.

There has been a substantial number of dedicated search experiments for permanent EDMs in a variety of systems over the past 60 years. All of them were well motivated and with a clear potential to discover new physics. They can be distinguished in four types depending on the particles studied. I.e. there are experiments on:

• (1)  
  Free elementary particles, such as electron, muon, tau, but also approximately free elementary particles as the proton and the neutron;
• (2)  
  Atoms, such as Hg, Xe, Tl, Cs;
• (3)  
  Molecules such as YbF, ThO, BaF, HfF⁺;
• (4)  
  Condensed matter samples, such as ferroelectric materials, liquid Xe.

Each of these lines of research has its own merits. Since a finite value of the not yet fully understood Θ_QCD parameter could cause EDM in hadrons, only an EDM found in a lepton would immediately indicate non-SM physics.

Any discovered EDM would call for further experiments to unravel the potentially different sources of the underlying new process of CP violation. Several hadronic EDMs could be used to demonstrate or disprove a Θ_QCD explanation, the combination with leptons will be indispensable to disentangling new physics. Because of the known CP-violation in the SM, permanent EDMs of fundamental particles are predicted which arise, e.g. for neutrons from three loop processes and for leptons from at least four loop processes. The SM EDM values are of order 10⁻³² ecm for neutrons and 10⁻⁴⁰ ecm for electrons [140]. Such small values are orders of magnitude below present experimental possibilities and they therefore open large windows of opportunity for observing New Physics. For almost all particles speculative models exist which can provide for EDMs almost as large as the present experimental limits [141].

Motivation to carry out experiments to search for EDMs in one or another system therefore require judgment calls on the viability of such speculative models. Independent of this, the non-observation of any EDM has ruled out more speculative theories than any other known experimental approach⁵³ . As one example of power of future EDM experiments, searches for EDMs on baryons and light nuclei, i.e. neutron, proton, deuteron and ³He, have particular potential to unravel different models of CP violation [142]. Below, we briefly present the main techniques currently used to search (directly or indirectly) for EDM in elementary particles as neutrons, protons, electrons and muons.

• (1)  
  Neutron EDM using ultra-cold neutrons Experiments to search for the EDM of the free neutron (d_n) have been conducted since the 1950s [143]. A long chain of experiments with ever increasing sensitivity, first with neutron beams and later with stored ultracold neutrons (UCN), has yielded the present best limit of $| {d}_{n}| \lt 3\times {10}^{-26}\,\mathrm{ecm}$ [144]. Presently there are at least five different sizeable efforts⁵⁴ aiming at improving the sensitivity to the neutron EDM in steps to 10⁻²⁷ ecm and then to 10⁻²⁸ ecm over the next 5–10 years. Several efforts (at PSI, ILL, LANL, TRIUMF) will use improved intensities of UCN stored in vacuum and at room temperature. One effort (at SNS) aims at conducting the measurement with UCN inside cold superfluid He (SFHe). The SFHe environment offers advantages [145] of potentially larger numbers of UCN exposed to larger electric fields, however, at the cost of considerable complication of the setup and handling. A first measurement in the cryogenic environment has still to be demonstrated. Beyond 10 years, some proposed or ongoing R&D efforts might succeed with cryogenic setups. Alternatively, also an experiment at a pulsed cold neutron beam of the ESS has been proposed [146].
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  Neutral atoms as probe of neutron and proton EDMsEDMs have also been searched for in neutral atoms. From EDM searches in diamagnetic atoms numerous limits on parameters describing physics within the SM or beyond could be extracted. The most recent table top experiment on ¹⁹⁹Hg has established $| {d}_{{\rm{Hg}}}| \lt 7.4\times {10}^{-30}\,\mathrm{ecm}$ (95% C.L.) [147]. From this value various other limits have been derived when assuming that there was for each case only one process that causes an EDM in ¹⁹⁹Hg. Amongst others a best neutron EDM limit of $| {d}_{n}| \lt 1.6\times {10}^{-26}\,\mathrm{ecm}$ [148] and a proton EDM limit of $| {d}_{p}| \lt 2.0\times {10}^{-25}\,\mathrm{ecm}$ [148] have been established as well as a limit on the QCD Θ parameter at $| {{\rm{\Theta }}}_{{\rm{QCD}}}| \lt 1.1\times {10}^{-10}$ [149].
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  Paramagnetic atoms and molecules as probe of an electron EDMEDM searches in paramagnetic atoms have yielded limits primarily on the electron EDM. Those early limits have been superseded since by searches in molecules and in molecular ions, where internal electric fields in these molecules give rise to some 10⁵→ 10⁹ fold enhancement for an electron EDM for example by using excited states of ThO in a molecular beam [150] or the ground state HfF⁺ in an rf-particle trap [151] Bounds could be established at $| {d}_{e}| \lt 1.1\times {10}^{-29}\,\mathrm{ecm}$ and $| {d}_{e}| \lt 1.3\times {10}^{-28}\,\mathrm{ecm}$ (90% C.L.), respectively, with these experiments exploiting significantly different techniques. Further improvements are expected soon from projects using these and also further molecules, such as YbF [152] and BaF [153]. It is highly realistic to expect that within the coming decade sensitivities better than 10⁻³⁰ ecm will be achieved for the EDM on the electron.
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  Muon EDMThe most sensitive EDM search experiments so far have been conducted on systems involving particles from the first particle generation. Yet limits on higher generation particles could be established as well. Along with measurements of the muon magnetic anomaly (muon g-2) EDM values could be obtained, the best limit currently being $| {d}_{\mu }| \lt 1.8\times {10}^{-19}\,\mathrm{ecm}$ (95% C.L.) [154]. The series of muon g-2 experiments since the 1960s has exploited the strong motional magnetic fields muons experience when moving at high velocities (close to the speed of light) through static magnetic fields. This basic concept underlies a muon EDM experiment proposed for the Paul Scherrer Institute (PSI) [155]. As a major improvement in the experimental concept a radial electric field is installed in the storage volume to compensate the particle's g-2 value related spin precession. An EDM on the muon manifests itself as an out of orbit plane precession of the muon spin, which can be detected via the time evolution of spatial distribution of decay electrons.At existing muon facilities a statistics limited sensitivity of $| {d}_{\mu }| \approx 7\times {10}^{-23}\,\mathrm{ecm}$ can be achieved within 1 year of data taking. At this precision the viability of the technique to directly search for an EDM on even short-lived charged particles can be demonstrated. Further, already at this sensitivity a number of SM extensions can be tested [155] which in particular account for the fact that the muon is a second generation particle.

Further limits on higher generation particles have been established. Figure 13 displays limits on the EDMs of fundamental particles. Muon and neutron limits have been deduced from measuring directly on these free particles, while e.g. the limit on the electron EDM results from the ACME experiment on ThO [150] assuming the electron EDM as the sole CP violating source.

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The limits on the neutrino EDMs are together with limits on magnetic moments deduced from cross sections which would be affected by electromagnetic couplings. The experimental limits are displayed as red bars from the top. From below come the SM estimates from CKM CP violation and Θ_QCD, taken to be the maximal value consistent with the neutron EDM limits. White regions indicate safe BSM discovery territory for the experiments. The range of ongoing or proposed experimental projects is indicated in orange.

Improved sensitivities can in several cases be obtained with the projects proposed for CERN within this PBC study: the proton EDM is the topic of the CPEDM Collaboration, and the strange and charm baryons might be improved or measured for the first time at all [156, 157] with the experiment proposed by the LHC-FT group:

• (1)  
  LHC-FT: measurement of EDMs in charmed and strange baryons Interest in hadronic EDM of second and even third generation quarks comes, e.g. from the fact that the indirect constrains on the charm EDM are rather weak, of order $4\times {10}^{-17}\,\mathrm{ecm}$ [158] only. As no finite EDM has been observed so far and no source of BSM CP violation is known yet, experimental efforts covering uncharted territory are necessary. The charm quark as well as the muon might via unexpectedly large EDM give clues on specific flavor structure of new physics.The experiment concept relies on a bent crystal to extract protons from the LHC beam halo. These protons will then hit a dense target and produce charged heavy and strange baryons that will then be channeled in bent crystals positioned in front of the detector. The intense electric field between the crystal atomic planes is able to induce a sizeable spin precession during the lifetime of the particle. The EDM, along with the magnetic dipole moment [159] can be determined by analyzing the angular distribution of the decay particles. Recently, the possibility to use the same technology for measuring the EDM (and MDM) of the tau lepton has been discussed in [160, 161].The LHC interaction point IP8, where the LHCb detector [162, 163] sits, has been identified as a suitable location of the experiment. A main challenge is represented by the limited coverage of the detector in the very forward region, requiring a secondary crystal with a large bending exceeding 15 mrad. A W target of ≈2 cm thickness hit by a proton flux of ≈10⁷ protons s⁻¹ is the upper limit for a parallel detector operation. R&D is ongoing to assess the feasibility of the secondary crystal along other challenges of the proposal, which include the compatibility with the machine, its operation mode, maximum reachable proton flux and the design of the absorber downstream the detector.About $2.4\times {10}^{14}$ protons on target could be reached in Run 3 with three years of data taking after the installation during an LHC technical stop, either with two weeks per year of a dedicated detector running at 10⁸ proton s⁻¹ or with a parallel detector operation at 10⁷ proton s⁻¹. This would lead to EDM sensitivities of about 10⁻¹⁷ ecm for charm baryons. Extending the detector coverage down to 10 mrad along with an increase of the proton flux during LHC Run 4 and Run 5, either at LHCb or at a dedicated experiment would improve sensitivity by about one order of magnitude.Figure 14 shows the EDM sensitivity for different baryons in two different scenarios, scenario 1 (S1) corresponds to data collected at the LHCb interaction point in a first phase at low luminosity (about 2 × 10¹⁴ pot); scenario 2 (S2) corresponds to data collected at a possible next-generation experiment at higher luminosity (∼10¹⁷) and enhanced coverage.
• (2)  
  CPEDM: measurement of proton and deuteron EDMsThe same experimental concept as for muons, i.e. exploiting a magnetic storage ring and motional electromagnetic fields, underlies the proposed deuteron EDM experiment for CERN. The spin analysis in this case is achieved by a newly developed deuteron polarimeter. Numerous preparations including polarimetry and spin manipulation are already being studied by the JEDI Collaboration. The COSY experiment is an indispensable proof of principle at ∼10⁻²⁴ ecm sensitivity for a ring experiment using a dedicated magnetic storage ring for deuterons (or protons) at CERN, which is needed to observe or establish a limit on the deuteron EDM at the level of 10⁻²⁹ ecm.For a deuteron experiment at CERN a new magnetic storage ring is required with some 80 m circumference to store polarized deuterons and observe the time evolution of their polarization. The precursor experiment at COSY is expected to develop all required detectors with sufficient sensitivity and to test the viability of the approach for hadrons. Both the muon and deuteron EDM experiment concepts take advantage of the fact that the magnetic anomaly is rather small and therefore magnetic spin precession in a magnetic storage ring can be compensated effectively by radial electrostatic fields.The proton EDM experiment proposed by the CPEDM Collaboration for CERN uses as in contrast a purely electrostatic dedicated storage ring of some 400–500 m circumference and with alternating field gradients, since the magnetic anomaly is much larger than for deuteron amd muon. For sensitivity 10⁻²⁹ ecm it exploits a proton beam of 233 MeV energy. The device needs provision for clockwise and counter-clockwise particle injection to minimize systematics. External magnetic fields at the experimental site need to be compensated to some 10 nT all over the particle storage volume and through the experimental running time. The substantial necessary expenses require a full structured program of stepwise testing of all essential concepts and necessary devices. The proton EDM project at CERN is a joint effort of the proton and deuteron EDM communities. It appears that a small size proof of principle experiment would be indispensable.An experiment on the proton EDM tests to a large part the same speculative models as the neutron EDM, except for those constructed with isospin dependence. Therefore a proton EDM experiment will need to exceed the prospected future sensitivity values expected for neutron experiments in order to justify the expenditures. Here, one expects some 10⁻²⁷ ecm by 2025 and 10⁻²⁸ ecm by 2030. Note that, for the deuteron an EDM can arise from either a proton or a neutron EDM (or both) and in addition an EDM may be due to CP violating parts in the proton-neutron interaction of the deuteron binding. Both experiments, once they have proven sufficient sensitivity, are therefore strongly motivated and they have robust discovery potential. Yet, the speed of progress in the area of molecular EDM searches and the significantly lower costs of table top experiments need to weighted against those of the storage ring approaches. The CeNTREX experiment at Yale⁵⁵ aims for a 30 times improvement for the proton EDM (and 100 fold improvement on ${{\rm{\Theta }}}_{{\rm{QCD}}}$) as compared to limits established for the proton to date⁵⁶ .

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The most updated review on the laboratory searches for axions and ALPs has been given in the recent paper by Irastorza and Redondo [39]. Figure 15 shows the current constraints for the axion-photon coupling ${g}_{a\gamma }$ versus axion mass m_a in the sub-eV mass range. The figure has been updated with the recent result of ADMX [164].

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The figure follows a color scheme to present results obtained with different methods: black/gray for laboratory results, bluish colors for helioscope searches and bounds related to stellar physics, greenish for haloscopes or cosmology dependent arguments. Hinted regions, like the QCD axion, are in yellow/orange.

Laboratory limits (dark gray area in figure 15) are essentially due to the results of OSQAR (region below 1 meV), and PVLAS (region above 1 meV). OSQAR [165] is a CERN based light shining through a wall experiment based on a protoype LHC magnet. PVLAS [166] is a sensitive polarimeter employing two rotating 2.5 T permanent magnets and an ultra high finesse Fabry–Perot cavity to search for the magnetic birefringence of the vacuum [167]. A possible next generation magnetic polarimeter to study this effect is under discussion within the PBC Technology Group [168] under the name VMB@CERN.

The bounds from helioscopes and haloscopes experiments are mostly driven by CAST [169] and ADMX [164, 170] results, respectively.

• (1)  
  CASTCAST is an helioscope, searching from axions/ALPs with photon-coupling produced in the Sun through Primakoff conversion of plasma photons in the electrostatic field of charged particles.The most efficient way to detect solar axions is through their reconversion into photons in the presence of a static electromagnetic field (usually magnetic dipole field) [164]. Reconverted photons are then detected by using low background x-ray devices. The achievable sensitivity in terms of the axion photon coupling constant is proportional to

  $$\begin{eqnarray}&&\mathrm{sens}({g}_{a\gamma \gamma })\propto \displaystyle \frac{{b}^{1/8}}{{B}^{1/2}{L}^{1/2}{A}^{1/4}{t}^{1/8}},\end{eqnarray} \tag{ 8.1 }$$

  where L is the length of the magnetic field of amplitude $B,A$ is the area of the useful bore, b the background rate and t the integration time. Large volume magnets are therefore a primary ingredient for such a research.By using a prototype LHC dipole magnet with 9 T magnetic field along 9.3 m length, CAST for the first time was able to explore solar axion in the QCD model range, in the mass region 0.1–1 eV. To maximize observing efficiency, the magnet was supported by a structure capable to track the Sun for a sizeable fraction of the day. The last CAST result [169] set the current best limit on the axion-photon coupling strength (0.66 × 10⁻¹⁰ GeV⁻¹ at 95% confidence level), thus competing with the most stringent limits from astrophysics on this coupling.CAST has also searched for other axion production channels in the Sun, enabled by the axion-electron or the axion-nucleon couplings. The project is now over, and the magnet may be utilized for building a haloscope (project RADES [171]). Most of the CAST Collaboration will be entering IAXO. Among the key elements of the CAST apparatus were the use of x-ray focusing optics and very low background micromegas detector.
• (2)  
  ADMXAxion or ALPs can be the main component of the dark matter halo of our galaxy and produce measurable signals in a suitable terrestrial detector. Such a detector usually exploits the long coherence length of these low mass particles, inside the galactic halo, in such a way to obtain detectability in spite of their very weak interactions with ordinary matter. Under the assumption that the searched for particle is the only constituent of the DM halo, limits on the coupling can be obtained in the absence of a detected signal. Strictly speaking, the limit is on the product between the coupling and the fraction of the local DM density in the case of a subdominant component. The oldest strategy to search for axions is the Sikivie or Primakoff haloscope [40], which has given almost all current limits for direct detection of dark matter in the sub eV range.In a Sikivie type detector, a high Q tunable microwave resonator is immersed in a strong static magnetic field. DM axions can be converted into real photons via a Primakoff process and deposit energy into the resonant mode of the cavity. In the last two decades the Axion dark matter experiment—ADMX—has implemented this method for cavities in the GHz range. Under the assumption of dominant DM component for the axion, ADMX has excluded the KSVZ axion in the 1.91–3.69 μeV mass range [170], and very recently the DFSZ one in the 2.66–2.81 μeV range [164]. The apparatus is based on a large volume high Q tunable copper cavity, operated in the sub-K temperature range and read by a SQUID based detection chain. Coverage of masses up to 40 μeV (10 GHz) is envisioned for the near future by combining the outputs of multiple co-tuned cavity resonators in the current 8 T superconducting magnet.

For the stellar and cosmology limits shown in figure 15 the acronyms are as follows (see [172, 173]):

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  HB, Sun, SN1987a: limits from stellar evolution obtained by studying the ratio of horizontal branch (HB) to red giants in globular clusters (GCs) [174], by a combined fit of solar data (Sun) [175], and by the study of the SN1987A neutrino pulse duration [176];
• (2)  
  Telescopes, x-rays, γ-rays: photons produced from axions decays inside galaxies show up as a peak in galactic spectra that must not exceed the observed background;
• (3)  
  x_ion: the ionization of primordial hydrogen caused by the decay photons of axions must not contribute significantly to the optical depth after recombination;
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  EBL: photons produced in ALP decays when the Universe is transparent must not exceed the extragalactic background light;
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  CMB: axions decay photons must not cause spectral distortions in the CMB spectrum;
• (6)  
  BBN: the decay of high mass ALPs produces electromagnetic and hadronic showers that must not spoil the agreement of big bang nucleosynthesis (BBN) with observations of primordial nuclei.

Figure 16 shows the physics reach of the proposed PBC experiments, BabyIAXO, IAXO and JURA compared with other experiments currently proposed and/or planned in the world. Both IAXO and JURA projects could be operated on a ${ \mathcal O }(10)$ year timescale. Table 4 shows the list of the relevant parameters of the IAXO project, together with the BabyIAXO setup and other past or competing experiments. Table 5 shows the key parameters for the JURA proposal.

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Table 4.  List of the relevant parameters of the IAXO project, together with the Baby-IAXO setup and other past or competing experiments. For the meaning of the parameters see equation (8.1).

Experiment	Status	B (T)	L (m)	A (cm2)	N (counts keV−1 cm−2 s−1)	t (years)
BNL E840 [177]	End	2.2	1.8	130
SUMICO [178]	End	4	2.5	18
CAST [169]	Running	9	9.3	30	10−6	1.1
TASTE [179]	Proposal	3.5	12	$2.8\times {10}^{3}$	5 × 10−7	3
BabyIAXO	Design	2.5	10	$2.8\times {10}^{3}$	1 × 10−7	1
IAXO	Design	2.5	22	2.3 × 104	1 × 10−8	3 + 3(gas)

The experiments planned or proposed in the world that could be able to produce results earlier or on the same timescale of the PBC projects are listed below.

• (1)  
  ALPS IIIn a photon regeneration experiment axions are produced by an electromagnetic beam (laser or microwave) traversing an external dipolar magnetic field. These axions are then reconverted into photons after a wall and can be detected with very sensitive detector combatting only technical and thermal noise.The pioneer experiment was conducted in Brookhaven by the BFRT Collaboration [45], and the two most recent results are those of the experiments ALPS [46] and OSQAR [47]. ALPS is DESY based and used a decommissioned HERA magnet. It is currently performing a major improvement to phase II, where a set of 10 + 10 HERA magnets will be coupled to two 100 m long Fabry–Perot cavities. ALPS II [48] will in fact take advantage of a resonant regeneration apparatus, thus expecting a major improvement over the current limit on LSW experiment given by OSQAR. ALPS II will represent the current state of the art LSW experiment, and for this reason its activities are monitored with interest by the PBC since they will give key elements to judge the proposal JURA.

• (1)  
  HAYSTACHAYSTAC is a high frequency version of the Sikivie detector, planned by a group that was Collaborating with ADMX. Its most notable feature is the use of a Josephson parametric amplifier with very low noise temperature, allowing the experiment to reach cosmological sensitivity in the mass region around 20 μeV. [180].
• (2)  
  KLASHWISPDMX and KLASH proposals aim at studying the low mass region (0.1–1 μeV), by employing large resonator and refurbished magnets from high energy physics experiments [181].
• (3)  
  CAPP Activities on axion searches are also pushed forward by the South Korean Center for Axion and Precision Physics-CAPP. The initiative CULTASK [182] is a CAPP based standard haloscope whose strength is the development of very high field large bore magnets, with fields up to 35 T and above. A CAPP-CAST Collaboration [183] is also ongoing to use rectangular cavities embedded inside the CAST magnet, while the CAPP initiative ACTION [184] studies the use of toroidal geometry.
• (4)  
  ORGANORGAN plans to study the higher mass region in the 50–200 μeV range, with specially designed resonant systems [185].

The search for axions with masses above tens of μeV is very challenging when using resonant cavity detectors. Typically the useable cavity volume is reduced but also other factors like a decrease in the technically achievable resonant enhancement are a challenge. In view of this, new initiatives are being developed where the detectors are broadband and instrumenting large volumes. The explored coupling is still the one with the photon, and again there is the need for large volume of high static magnetic field.

• (1)  
  BRASS and MADMAX By exploiting the axion induced electric field on a boundary immersed in a static magnetic field [186], the BRASS experiment will use a magnetized 8 m radius disk immersed in a 1 T static magnetic field to study the mass region 10 μeV–10 meV simultaneously. At the moment it is at very preliminary stage. The same concept of radiating disk is at the heart of MADMAX experiment [187], where, however, a multiple disk configuration is used to obtain resonant enhancement, albeit with a relatively large width of the resonance. This Collaboration is already being developed and it is in the R&D phase.
• (2)  
  DM Radio and ABRACADABRA Another method for ultra low mass dark matter axion detection is the use of a lumped element LC resonator inside a strong magnet, where an alternating current is induced by the axion field. Studies are underway to implement such idea within the ADMX magnet for a detector with sensitivity in the mass region below 1 μeV. The DM radio experiment [188] is based on the same idea but uses a tunable LC resonator shielded by a superconducting structure and read by a SQUID. ABRACADABRA [189] is a 1m scale broadband detector based on a toroidal magnet equipped with with a superconducting pick up loop read by SQUID. Again, the best sensitivity is obtained for masses below 1 μeV. All these efforts are just finalizing their R&D phase and should come out with first data in a few years.

In the minimal dark photon model, the SM is augmented by a single new state $A^{\prime}$. DM is assumed to be either heavy or contained in a different sector. In that case, once produced, the dark photon decays back to the SM states. The parameter space of this model is then (${m}_{A^{\prime} },\epsilon$) where ${m}_{A^{\prime} }$ is the mass of the dark photon and the coupling parameter of the Dark Photon with the standard photon.

Current bounds

Visible decays of vector mediators are mostly constrained from searches for di-electron or di-muon resonances [190–192] and from the re-interpretation of data from fixed target or neutrino experiments in the low (<1 GeV) mass region [193–195]. NA48/2 [191], A1 [192] and BaBar [190] experiments put the strongest bounds for  > 10⁻³ in the 0.01–10 GeV mass range. They search for a bump in the e⁺e⁻ or μ⁺μ⁻ invariant mass distribution over a smooth background. These experiments consider a variety of dark photon production mechanisms, such as meson decays (NA48/2 [191]), bremsstrahlung (A1 [192]), and annihilation (BaBar [190], KLOE [196–199], LHCb [200]). These results are complemented by those from beam dump experiments, such as E141 [193] and E137 [194, 201] at SLAC, E774 at Fermilab [195], CHARM [202, 203] and NuCal [204]. The KOTO experiment has also set recently a limit on the ${BR}({K}_{L}\to {\pi }^{0}X)\lt 2.4\times {10}^{-9}$ (90% CL) [205] that could partially fill of the hole of the E949 coverage around $m={m}_{{\pi }^{0}}$.

Existing limits in the plane mixing strength versus mass of the dark photon are shown in figure 17. In the following we briefly detail the contributions by classifying them as a function of the experimental technique used.

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• 1. Searches for dilepton resonances:

  • (1)  
    NA48/2 @ CERN: searches for dark photons decays to e⁺e⁻ final state in the decay chain ${\pi }^{0}\to \gamma A^{\prime}$ using $\sim 2\times {10}^{7}$ fully reconstructed ${\pi }^{0}\to \gamma {e}^{+}{e}^{-}$ decays collected in 2003–2004 [191].
  • (2)  
    BaBar @ KEK: searches for a dark photon in the reaction ${e}^{+}+{e}^{-}\to \gamma \,+A^{\prime} ,A^{\prime} \,\to {e}^{+}{e}^{-},{\mu }^{+}{\mu }^{-}$ using 514 fb⁻¹ of data [190].
  • (3)  
    KLOE @ DAFNE: searches for dark photon in visible final states using a large variety of production modes, such as meson decay ($\phi \to \eta A^{\prime}$), and annihilation (${e}^{+}+{e}^{-}\,\to \gamma +A^{\prime}$) [196–199].
  • (4)  
    LHCb: inclusive di-muon search in pp collisions at $\sqrt{s}=13$ TeV performed with the current Run 2 LHCb data above the di-muon threshold [200].

• 2. Reinterpretation of data of fixed target experiments:

  • (1)  
    E137 @ SLAC (electron beam dump): E137 was an experiment conducted at SLAC in 1980–1982 where a 20 GeV electron beam was dumped on a target. Dark matter interacting with electrons (e.g. via a dark photon) could have been produced in the electron-target collisions and scattered off electrons in the E137 detector, producing the striking, zero-background signature of a high-energy electromagnetic shower that points back to the beam dump [194, 201].
  • (2)  
    CHARM @ CERN (proton beam dump): the CHARM Collaboration performed a search for ALPs decaying to photon, electron or muon pairs using the 400 GeV, 2.4 × 10¹⁸ protons-on-target (pot) dumped on a thick copper target distant 480 m from the 35 m long decay volume [202].
  • (3)  
    E141 @ SLAC (electron beam dump): the E141 Collaboration searched for high-energy positron signals from a hypothetical ${X}^{0}\to {e}^{+}{e}^{-}$ decay, produced in the interactions of 2 × 10¹⁵ 9 GeV electrons dumped on a 10 and 12 cm long W-target [193].
  • (4)  
    E774 @ Fermilab (electron beam dump): the E774 Collaboration used 5.2 × 10⁹ eot from an electron beam of 275 GeV at FNAL. A hypothetical X⁰ particle could have been produced by bremsstrahlung in the dump and then decays to e⁺e⁻ pairs in the ∼2 m long decay volume [195].

Future experimental landscape

Several experiments and proposals not considered in the PBC activity will search for dark photons in decays to visible final states using different types of beams and experimental techniques. The projections of their sensitivity in the near future are shown in figure 18. The status of these projects is briefly reported in the following.

• (1)  
  Belle-II @ KEK: will search for visible dark photon decays $A^{\prime} \to {e}^{+}{e}^{-},{\mu }^{+}{\mu }^{-}$ where $A^{\prime}$ is produced in the process ${e}^{+}+{e}^{-}\to A^{\prime} +\gamma$. Projections are based on 50 ab⁻¹. Timeline: data taking started in 2018, expected about 50 ab⁻¹ by 2025 [206].
• (2)  
  LHCb @ CERN: LHCb will search for dark photon in visible final states both using the inclusive di-muon production [207] and the ${D}^{* 0}\to {D}^{0}{e}^{+}{e}^{-}$ decays [208]. The ${D}^{0* }$ search will cover dark photon masses from the di-electron mass threshold up to 1.9 GeV. The ${D}^{* 0}$ search requires the upgrade of the current LHCb trigger system, currently scheduled during 2019–2020. The projections are based on 15 fb⁻¹, 3 years data taking with 5 fb⁻¹/yr⁻¹ with an upgraded detector after Long Shutdown 2.
• (3)  
  HPS @ JLAB: electron beam-dump at CEBAF electron beam (2.2–6.6 GeV, up to 500 nA), search for visible ($A^{\prime} \to {e}^{+}{e}^{-}$) dark photon (prompt and displaced) decays produced via bremsstrahlung production in a thin W target.The experiment makes use of the 200 nA electron beam available in Hall-B at Jefferson Lab. Of the 180 PAC-days⁶⁰ granted, HPS collected data for 1.7 PAC-days in 2015 (engineering run) at 1.06 GeV beam energy and 5.4 PAC-days for a physics run in 2016 at 2.3 GeV. Results of the 2015 analysis are reported in [209]. A 28 PAC-days data taking at 4.55GeV beam energy is expected in Summer 2019 [210].
• (4)  
  APEX @ JLAB: electron beam dump at CEBAF electron beam, search for visible dark photon decays. Status: planned one-month physics run in 2018–2019 [211, 212].
• (5)  
  SeaQuest @ FNAL: will search for visible dark photon decays $A^{\prime} \to {e}^{+}{e}^{-}$ at the 120 GeV Main Injector beamline at FNAL. SeaQuest plans to install a refurbished electromagnetic calorimeter (ECAL) from the PHENIX detector at Brookhaven National Laboratory within the next few years. The Collaboration plans to submit an official proposal to the Fermilab physics advisory committee in 2019 to install the ECAL at the end of the next polarization target run in 2021 and acquire $\sim {10}^{18}$ (10²⁰) pot by 2024 (2030s). The 10²⁰ pot yield could be collected as a result of the Fermilab Proton Improvement Plan [105, 213].
• (6)  
  VEPP3 at BINP: missing mass method and visible decay searches at BINP at Novosibirsk. Dark photons are produced by colliding a 500 MeV positron beam on an internal gaseous hydrogen target, and both visible and invisible (via the missing mass mode) final state are identified. Timeline: First run is anticipated for 2019–2020 [214].
• (7)  
  Mu3e @ PSI: Search for $\mu \to {eee}$ decay at PSI. Phase I: sensitivity $2\times {10}^{-15}$ with the existing muon line, from proton cyclotron of 2.4 mA protons at 590 MeV. Phase II: sensitivity of 10⁻¹⁶ with upgraded muon line.
• (8)  
  MAGIX at MESA (Mainz, Germany): is a step beyond the traditional visible dark photon decay searches with a dipole spectrometer at the 105 MeV polarized electron beam at A1/MAMI. The MESA accelerator has ${E}_{\max }\,=$ 155 MeV energy, and up to 1 mA current. The MAGIX detector is a twin arm dipole spectrometer placed around a gas target. Production mechanism similar to HPS and identification through a di-electron resonance. The possibility of a beam dump setup similar to BDX is under study. Timeline: Proposal in 2017 with targeted operations in 2021–2022 and 2 years of data taking [215].

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Physics reach of PBC projects on 5 and 10–15 year timescales

Figure 19 shows the 90% CL exclusion limits for searches for dark photons decaying to visible final states performed by PBC proposals that might produce results on ∼5 year timescale: NA64${}^{++}(e)$, NA62⁺⁺ and FASER with 150 fb⁻¹. These projects will be competing with other initiatives in the same timescale, as for example SeaQuest, HPS and LHCb, as shown in figure 18.

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The physics reach of PBC projects on a 10–15 year timescale is shown in figure 20. On this timescale several projects could be ready and operated, such as REDTOP, SHiP, FASER2, MATHUSLA200, AWAKE, and LDMX. The sensitivity for dark photons decaying in visible final states will be dominated by SHiP, while FASER2, LDMX and AWAKE will be directly competing with SeaQuest, LHCb, HPS, and others as shown in figure 18. MATHUSLA200 in this scenario is however not competitive, mostly due to the fact that the dark photon is produced in the forward direction.

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This is the model where minimally coupled viable WIMP dark matter model can be constructed with a dark photon as light mediator. Preferred values of the dark coupling ${\alpha }_{D}={g}^{2}D/4\pi$ are such that the decay of $A^{\prime}$ occurs predominantly into DM $\chi \bar{\chi }$ states. These states can further rescatter on electrons and nuclei due to -proportional interaction between SM and DM states mediated by the mixed ${AA}^{\prime}$ propagator. The parameter space for this model is $({m}_{A^{\prime} },\epsilon ;{m}_{\chi };{\alpha }_{D})$ with further model-dependence associated with the properties of the dark matter candidate χ (boson or fermion).

The sensitivity plots for this benchmark case can be shown in two ways:

• (a)  
  the plane versus ${m}_{A^{\prime} }$ where ${\alpha }_{D}^{2}\gg \epsilon {\alpha }_{D}$ and ${m}_{A^{\prime} }\gt 2{m}_{\chi };$
• (b)  
  the plane y versus ${m}_{\chi }$ plot where the 'yield' variable y, $y={\alpha }_{D}{\epsilon }^{2}{\left({m}_{\chi }/{m}_{A^{\prime} }\right)}^{4}$ , is argued to contain a combination of parameters relevant for the freeze-out and DM–SM particles scattering cross section. Here α_D is the dark fine structure constant that describes the interactions between dark photon and dark matter. The coupling of the dark photon to SM particles happens via the millicharge e. The choice adopted by the PBC is ${\alpha }_{D}=0.1$ and ${m}_{A^{\prime} }/{m}_{\chi }=3$.

In case (b), the yield variable y can be put in direct connection to the DM thermal relic abundance. In fact, the direct DM annihilation responsible of the thermal relic abundance, is driven by the same couplings that define the direct DM scattering, leading to rather well defined predictions:

$$\begin{eqnarray*}&&\langle \sigma \cdot v\rangle \sim \displaystyle \frac{y}{{m}_{\chi }},\end{eqnarray*}$$

where v is the velocity. The measured dark matter abundance imposes a minimum bound on this cross-section, $\langle \sigma \cdot v\rangle \gt \langle \sigma \cdot v{\rangle }_{\mathrm{relic}}$. This lower bound can be translated in turn into a lower bound on the strength of the SM-mediator and DM-mediator couplings, and, as a consequence, opens up the possibility to link results obtained at accelerator-based experiments to those coming from DM direct detection experiments, depending on the nature of the DM candidate. Two cases considered in this study are Elastic Scalar and Pseudo-Dirac fermion dark matter.

Current bounds and future experimental landscape

In case of dark photons with invisible decays, the strongest limits on the coupling strength of DM with light vector mediator for DM and mediator masses in the MeV–GeV range are provided by the NA64 experiment [70], and from a recent result on mono-photon search from BaBar [216]. Limits in the low mass range come from a re-interpretation by theorists of old results from the liquid scintillator near detector (LSND) [217] and E137 [201] experiments, and as such, should be taken with many caveats. A re-analysis of electron-scattering data from direct detection experiments has led to constraints in the sub-GeV DM region [218, 219]

(a) Plane versus ${m}_{A^{\prime} }$ Figure 21 shows the current 90% CL upper limits in the plane versus ${m}_{A^{\prime} }$ from BaBar [216], E787/E949 [220, 221] and NA64 [70] as filled areas and future perspectives from projects not PBC related as solid or dashed curves. The region preferred by the ${(g-2)}_{\mu }$ puzzle [222] is also shown in the plot. Most of the future projections come from experiments using the missing momentum/missing mass techniques, as explained below.

• (1)  
  Belle II @ KEK: search for dark photons in the process ${e}^{+}+{e}^{-}\to A^{\prime} +\gamma ,A^{\prime} \to$ invisible relies on a L1 trigger sensitive to mono-energetic ISR photon with energy $E=({E}_{{CM}}^{2}-{M}_{A^{\prime} }^{2})/2{E}_{{cm}}$. A trigger threshold as low as 1.2 GeV is anticipated to be applied for the 2018–2019 dataset, corresponding to ∼20 fb⁻¹ of data. [206].
• (2)  
  MMAPS @ Cornell: MMAPS aims at searching for dark photons in the process ${e}^{+}+{e}^{-}\to A^{\prime} +\gamma$ using the interactions of a 5.3 GeV positron beam extracted from the Cornell synchrotron with a fixed Be target. The measure of the outgoing photon kinematics with a CsI calorimeter allows to infer the $A^{\prime}$ mass. This method provides sensitivity to all possible decay modes. The main limitations arise from the detector resolution and QED backgrounds, such as ${e}^{+}+{e}^{-}\to \gamma +\gamma$ or ${e}^{+}+{e}^{-}\to {e}^{+}+{e}^{-}\gamma$ where charged final particle(s) sometimes escape undetected.Timeline: proposal stage, no starting date (>2020).
• (3)  
  PADME @ LNF: missing momentum searches at the Beam Test Facility (BTF) in LNF. The principle is similar to the MMAPS experiment, using a 550 MeV positron beam on a diamond target. In addition to invisible $A^{\prime}$ decays, PADME is studying its sensitivity to di-photon decays of ALPs and dark Higgs decays. Timeline: Expected to collect 10¹³ positron on target by end of 2019.

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We do expect NA62 will be able to produce results in the next few years as a by-product of the ${K}^{+}\to {\pi }^{+}\nu \overline{\nu }$ analysis, but no sensitivity curves have been provided by the Collaboration so far.

Figure 22 show the projections from the PBC experiments. NA64${}^{++}(e)$ with $5\times {10}^{12}$ eot will be able to explore a large part of the parameter space on a 5 year timescale. KLEVER could further push the search for dark photons in the invisible final states in the mass region between 100 and 200 MeV as a by-product of the analysis of the ${K}_{L}\to {\pi }^{0}\nu \overline{\nu }$ rare decay. The ultimate sensitivity can be reached by LDMX, either at the DASEL facility with 8 GeV electron beam and even further at the eSPS facility with 16 GeV electron beam at CERN.

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(b) Plane y versus ${m}_{\chi }$ The current bounds and future perspectives in the plane y versus DM mass are shown in figure 23 for two different hypotheses on the dark matter nature: Elastic Scalar and Pseudo-Dirac fermion.

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In this plot, the lower limits for the thermal relic targets are also shown, under that hypothesis that a single DM candidate is responsible for the whole DM abundance. Under the hypothesis of an elastic scalar DM candidate, limits from direct detection DM experiments, as CRESST-II [223], XENON 10/100 [224, 225] and Super-CDMS [226] can be used, which is not the case for a DM as Pseudo-Dirac fermion hypothesis.

In contrast, results from accelerator-based experiments, are largely independent of the assumptions on the specific DM nature as in this case it is produced in relativistic regime and the strength of the interactions with light mediators and SM particles is only fixed by thermal freeze-out.

Future initiatives that could explore still uncovered parameter space in the plane y versus DM mass for DM masses below 1 GeV are all those that have sensitivity in the plane versus dark photon mass and, in addition, experiments exploiting DM scattering with nucleons and/or electrons, both accelerator-based and from direct detection searches. Among the accelerator-based experiments, there are BDX at JLab [227], MiniBooNE at FNAL [228] and COHERENT at ORNL [229], as explained below.

• (1)  
  BDX at JLab (electron beam-dump): the Beam Dump eXperiment (BDX) is aiming to detect light dark matter produced in the interaction of an intense electron beam with the dump. The experiment is sensitive to elastic DM scattering ${e}^{-}+\chi \to {e}^{-}+\chi$ in the detector after production in ${e}^{-}+Z\to {e}^{-}+Z(+A^{\prime} ,A^{\prime} \to \chi \chi )$. A detector placed ∼20 m downstream of the Hall-A beam-dump at Jefferson lab is expected to identify a dark matter scattering by measuring a electromagnetic shower produced by the DM interaction with atomic electrons of the detector. The BDX detector is composed by a ∼GeV electromagnetic calorimeter surrounded by two layers of active plastic scintillator vetos. The calorimeter re-uses ∼800 CsI(Tl) crystals formerly used in the BaBar EM Cal, upgraded with SiPM-readout and triggerless data acquisition. The experiment makes use of the high energy (∼10 GeV) and high intensity (∼100 μA) electron beam available in Hall-A running in parallel (parasitically) with the scheduled hadron physics program. BDX has been approved with the maximum scientific rating (A) by the JLab PAC and granted with 285 PAC-days of data taking, corresponding to an integrated yield of 10²² eot. The BDX Collaboration is currently seeking for funds to build the new experimental hall that will host the BDX detector. The experiment is expected to be deployed and take data in 2–3 years from now [227].
• (2)  
  MiniBooNE @ FNAL (proton beam dump): neutrino detector at the 8 GeV Booster Neutrino Beamline at FNAL. MiniBooNE is a 800 ton mineral oil Cherenkov detector situated 490 m downstream of the beam dump. The DM is searched for via the chain $p+p\to X+{\pi }^{0}/\eta ,{\pi }^{0}/\eta \to \gamma +A^{\prime}$ and $A^{\prime} \to \chi \chi$. The results are based on $1.8\times {10}^{20}$ pot and have been published for DM-nucleon and electron-elastic scattering [228].
• (3)  
  COHERENT (proton beam dump): the COHERENT Collaboration aims to measure Coherent Elastic Neutrino-Nucleus Scattering using the high-quality pion-decay-at-rest neutrino source at the Spallation Neutron Source in Oak Ridge, Tennessee. The SNS provides an intense flux of neutrinos in the energy range of few tens-of-MeV. The beam has a sharply-pulsed timing structure that is beneficial for background rejection. The current experimental appartus includes ${ \mathcal O }$(10 kg) NaI(Tl) and CsI(Tl) detectors, and a 35 kg single-phase LAr scintillation detector. The same apparatus can be used to search for dark matter mainly produced via the process ${\pi }^{0}/\eta \to \gamma +A^{\prime}$ with $A^{\prime} \to \chi {\chi }^{* }$, where the ${\pi }^{0}/\eta$ are produced out of collisions from the primary proton beam. The DM candidates are identified through coherent scattering leading to a detectable nuclear recoil. Timeline: currently taking data, upgrade after 2019 [229]⁶¹ .
• (4)  
  SBN @ FNAL (proton beam-dump): the SBN program consists of three LAr-based detectors of 112 ton (SBND), 89 ton (microBooNE), and 476 ton (ICARUS-T600) situated at 110 m, 470 m and 600 m downstream the beam dump, respectively, of the 8 GeV primary proton beam of the Booster Neutrino Beamline at FNAL. Dark matter could be primarily produced via pion decays created in the collisions of the protons with the dump and scatter in LAr TPC detectors. SBND is expected to yield the most sensitive results and could improve upon MiniBooNE by more than an order of magnitude with $6\times {10}^{20}$ pot. Projections shown in figure 23 are based on 2 × 10²⁰ pot.

Several experiments designed to perform direct detection DM searches will be able to put bounds.

• (1)  
  SENSEI: is a direct detection experiment that will be able to explore DM candidates with masses in the MeV–GeV range, by detecting the signal released in DM-electron scattering interactions in a fully depleted silicon CCD. For the first time, it has been demonstrated that the charge in each pixel of a CCD—in a detector consisting of millions of pixels—can be measured with sub-electron noise. A 1 g detector is already operating in the NUMI access tunnel. A larger project (100 g) can be deployed at a deeper site on a 1–2 year timescale if funding is obtained [230].
• (2)  
  CRESST-II: uses cryogenic detectors to search for nuclear recoil events induced by elastic scattering of DM particles in CaWO₄ crystals. Because of its low-energy threshold, the sensitivity to DM was extended in the sub-GeV region. Current bounds are derived from a dataset corresponding to 52 kg live days [223].
• (3)  
  XENON 10/100: DM-electron scattering searches have already illustrated their potential, probing down to m_χ > 5 MeV [218, 219] using XENON10 data [224] sensitive to single electrons and down to m_χ > 35 MeV [219] using XENON100 data [225].

Physics reach of PBC projects on 5 and 10–15 years timescales PBC projects able to put bounds on the y versus m_χ plane are NA64⁺⁺(e) on a 5 year timescale and LDMX and SHiP on a 10–15 year timescale, as shown in figure 24. NA64⁺⁺(e) and LDMX will use the missing energy/missing momentum techniques, respectively. SHiP, instead, will exploit the elastic scattering of DM candidates with the electrons in the medium of the emulsion-based neutrino detector. As such, SHiP is fully complementary to the other two.

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Milli-charged particles (mCP) can be seen as a specific limit of the vector portal when ${m}_{A^{\prime} }$ goes to zero and the parameter space simplifies to the mass (m_χ) and effective charge ($| Q| =| \epsilon {g}_{D}e|$) of milli-charged particles. The suggested choice of parameter space is (m_χ, Q_χ/e) and χ can be taken to be a fermion. The searches for millicharged particles can be performed either through missing energy techniques or through minimum ionizing (milli-charged) signals.

A recent review [231] showed the potential of the existing data from MiniBooNE [232] and the LSND [233], and the soon to be released data from MicroBooNE, the ongoing SBN program [234], the Deep Underground Neutrino Experiment (DUNE) [235], beyond the standard electron beam-dump experiments [236, 237] to probe the milli-charged particles model. In the following sections we stick on experimental published bounds and official projections of current experiments and future proposals.

Current bounds and future experimental landscape The most stringent current experimental bounds on millicharged particles arise from the SLAC milliQ experiment [236], the EDGES experiment [238] and from colliders [237].

• (1)  
  SLAC milliQ experiment A dedicated search for mQ's has been carried out at SLAC in the late 90s. This search was sensitive to particles with electric charge in the range (10⁻¹–10⁻⁵)e and masses between 0.1 and 1000 MeV. The experiment, located near the positron-production target of the SLC beam, looked for extremely feebly ionization and/or excitation signals in scintillators counters (down to a single scintillation photon) that might arise from millicharged particles surviving the 110 m sandstone filling the distance between the detector and the positron source [236].
• (2)  
  CollidersIn the mass region above 100 MeV the strongest direct bounds arise from colliders, mainly from the constraint from the invisible width of the Z, as well as direct searches for fractionally charged particles at LEP [237].
• (3)  
  EDGES experimentThe unexpected strength of 21 cm absorption signal measured by the Experiment to Detect the Global Epoch of Reionization (EDGES) could be naturally explained [239, 240] if even only a fraction (less than 0.4%) of the dark matter due to CMB constraints [241]⁶² is in form of milli-charged particles. Data from [238], interpretation from [241].
• (4)  
  SuperNovae 1987A bounds The number of neutrinos detected on Earth during the explosion of the SN 1987A agree roughly with the theoretical expectations. This allows us to use the generic stellar energy-loss argument that if other particles were contributing to the cooling of the proto neutron star these would have reduced the neutrino fluxes and duration of the neutrino signal.
• (5)  
  milliQanFuture experimental bounds outside the PBC projects will be set by the milliQan experiment. milliQan has been proposed to be sited in the PX56 Observation and Drainage gallery above the CMS underground experimental cavern (UXC). It consists of one or more scintillator detector layers of roughly 1 m³ each. The experimental signature would consist of a few photo-electrons arising from the small ionization produced by the mCPs that travel without interacting through material after escaping the CMS detector. milliQan plans to integrate ∼150 fb⁻¹ during Run 3 and 3 ab⁻¹ in the HL-LHC era [245].
• (6)  
  FerMINIThe FerMINI project [246] is a milliQan-type detector proposed to be installed downstream of the proton target of a neutrino experiment, either in MINOS near detector hall and/or the proposed DUNE near detector hall, both at Fermilab. FerMINI can achieve unprecedented sensitivity for milli-charged particles with mass in the MeV–GeV range and fractional charge Q_χ/e in the 10⁻⁴–10⁻¹ range.

PBC projects on 5 and 10–15 years timescale Three PBC projects have sensitivity to search for milli-charged particles, as shown in figure 25: NA64${}^{++}(e)$ and NA64⁺⁺(μ) on 5–10 year timescale and LDMX on a ∼10 to 15 year timescale. NA64⁺⁺ (e) with 5 × 10¹² eot collected during Run 3 can explore the region with masses between 100 and 1000 MeV and fractional charge Q/e = 10⁻³–10⁻¹; NA64⁺⁺(μ) with 5 × 10¹³ mot can improve over NA64⁺⁺(e) by pushing down the limit of the fractional charge by almost an order of magnitude. LDMX, with an electron beam of 16 GeV momentum and a collected yield of 10¹⁶ eot will further improve the search in particular in the intermediate (100–1000 MeV) mass region.

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A light scalar particle mixing with the Higgs with angle θ can be a mediator between DM and SM particles. The Langrangian to be added to the SM one is of the form:

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{scalar}}={{ \mathcal L }}_{\mathrm{SM}}+{{ \mathcal L }}_{\mathrm{DS}}-(\mu S+\lambda {S}^{2}){H}^{\dagger }H.\end{eqnarray} \tag{ 9.1 }$$

The minimal scenario (BC4) assumes for simplicity that λ = 0 and all production and decay processes of the dark scalars are controlled by the same parameter μ = sinθ. Therefore, the parameter space for this model is (θ, m_S). A more general approach (BC5) consists in having both λ and μ being different from zero: in this case, the parameter space is {λ, θ, m_S}, and λ is assumed to dominate the production via e.g. $h\to {SS},B\to {K}^{(* )}{SS},{B}^{0}\to {SS}$ etc. In the following we will assume the branching fraction ${BR}(h\to {SS})\sim {10}^{-2}$ in order to be complementary to the LHC searches for the Higgs to invisible channels.

A key feature of the scalar portal is that its production is often proportional to one of the larger Yukawa couplings, y_t, in the case of the EW penguin, while its decay is controlled by one of the smaller Yukawa's or the induced gluon coupling. This means that it is natural for dark scalars to be both long-lived and be produced at a relatively large rate, which makes them an excellent target for the proposals discussed in this study.

Current bounds and future experimental landscape Figure 26 shows the current bounds on the mixing parameter ${\sin }^{2}\theta$ versus the mass of the dark scalar m_S. Bounds on this scenario come from recast of data from old beam dump experiments [247, 248], bump hunt in visible B meson decays [249–251] and cosmological and astrophysical arguments, as explained below.

• (1)  
  CHARMThe CHARM Collaboration has put bounds on light ALPs using a 400 GeV proton beam impinging on a copper target [202]. Figure 26 shows the reinterpretation of the CHARM data from [248] as yellow shaded region.
• (2)  
  Visible Meson DecaysA visibly decaying scalar mediator can contribute to the processes ${B}^{+}\to {K}^{+}{\mu }^{+}{\mu }^{-}$ and ${B}^{0}\to {K}^{* 0}{\mu }^{+}{\mu }^{-}$, which are tightly constrained by LHCb [249, 250] and Belle [251] measurements. In the same parameter space, we also show bounds computed by us⁶³ based on the measurement of the ${K}^{+}\to {\pi }^{+}\nu \overline{\nu }$ branching fraction from E949 experiment [252].
• (3)  
  BBNA sufficiently light (M <10 MeV), weakly coupled scalar particle with a thermal number density can decay appreciably during BBN and spoil the successful predictions of light element yields accumulated in the early universe.
• (4)  
  SupernovaeA light, weakly coupled scalar mediator can be produced on shell during a supernova (SN) explosion and significantly contribute to its energy loss, thereby shortening the duration of the observable neutrino pulse emitted during core collapse. The most significant constraint arises from SN1987a which has been used to constrain the parameter space for axions and ALPs [253–256].

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Searches in the near (∼5 years) future will be performed by: SeaQuest at FNAL [105], using the same dataset for the search for a dark photon into e⁺e⁻ final state as explained in section 9.1; LHCb will update the bump hunt in $B\to K{{\ell }}^{+}{{\ell }}^{-}$ decays with an integrated luminosity of ∼15fb⁻¹ which is expected to be collected during Run 3. NA62 in kaon-mode will be able to explore the mass range below the kaon mass, as a side product of the measurement at ${ \mathcal O }(10 \% )$ accuracy of the rare decay ${K}^{+}\to {\pi }^{+}\nu \overline{\nu }$, by interpreting it as ${K}^{+}\to {\pi }^{+}S$. The NA62 search should be able to push down the limit currently set by the E949 experiment by, at least, an order of magnitude, even if official projections have not yet been made by the Collaboration.

Physics reach of PBC projects on 5 and 10–15 year timescale Figure 27 shows the sensitivity of FASER during its first phase of data taking during Run 3, and NA62⁺⁺ in dump mode with 10¹⁸pot collected in about 100 days of data taking during Run 3. These results could be obtained on a ∼5 year timescale. NA62⁺⁺ in dump mode should be able to improve the limit between the di-muon mass and ∼1 GeV range and will compete with SeaQuest on the same timescale; FASER, in its first phase, is instead not competitive.

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On a longer timescale (10–15 years) the explored parameter space will be significantly extended by bigger PBC projects, as SHiP with 2 × 10²⁰ pot, KLEVER with 5 × 10¹⁹ pot delivered, REDTOP with 10¹⁷ pot collected at the η threshold, MATHUSLA200 and FASER2 with 3 ab⁻¹, running parasitically at the ATLAS or CMS interaction points, and CODEX-b with 300 fb⁻¹, if the LHCb phase-II upgrade will be approved during the HL-LHC era. Above the di-muon threshold SHiP, FASER2, MATHUSLA200 and CODEX-b have comparable sensitivity. Below the kaon mass, KLEVER will be able to close the gap between the recasted limit from the data of the E949 experiment (and possible future result from NA62) and the Super Novae bound. REDTOP with 10¹⁷ pot at the η threshold instead is not competitive with respect to the others. This is shown in figure 28.

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The extended version of the minimal Higgs-Dark Scalar model, with both couplings μ and λ different from zero, allows to cover a larger fraction of the parameters space, as shown in figure 29, due to the new contributions arising from a virtual (as in the $B\to {KSS}$ mode) or real (as in the case $h\to {SS}$) Higgs in the chain. Also in this case the larger impact is provided by the bigger experiments, MATHUSLA200, SHiP, FASER2 and CODEX-b which will be able to explore the region well above the GeV mass scale in a fully uncharted range of couplings. The experiments at a central location near the LHC interaction points, MATHUSLA and CODEX-b, will have sensitivity all the way up to ∼60 GeV, if the assumption of zero background is valid.

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In this section we consider the case in which HNLs couple only to first SM generation and the sensitivity plots are shown in the plane {$| {U}_{e}{| }^{2},{m}_{N}$}.

Current bounds, experimental landscape and PBC projects on 5 year timescale Current bounds and the future experimental landscape in the next ∼5 years, including some PBC projects, are shown in figure 30 for the case of HNLs with couplings only to the first lepton generation and masses in the MeV–GeV range.

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Existing bounds, shown as filled colored areas, for masses below the charm mass arise mostly from beam dump experiments (PS191 [275] and CHARM [276]) while those above the charm mass from LEP data, dominated by the DELPHI result [274], from Belle [291] and more recently from CMS [292]. The allowed range of couplings is bounded from below by the BBN constraint [270], and the see-saw limit [293].

• (1)  
  PS 191 @ CERN: the PS191 CERN experiment was specifically designed to search for neutrino decays in a low-energy neutrino beam. The apparatus consisted of 10 m long nearly empty decay volume instrumented by flash chambers, calorimeter and scintillator hodoscope [275].
• (2)  
  CHARM @ CERN: a search for heavy neutrinos was performed by the CHARM Collaboration by dumping ${ \mathcal O }(2\cdot {10}^{18})$ 400 GeV protons on a thick Copper beam dump and looking for visible decays with electrons in the final state in the 35 m long decay volume with a spectrometer of 3 × 3 m² cross section [276].
• (3)  
  Belle @ KEK: Belle performed a search for heavy neutrinos with 772 M of $B\overline{B}$ pairs using leptonic and semileptonic B mesons decays, $B\to {Xl}{\nu }_{R}$, where ℓ = e, μ and X was a charmed meson ${D}^{(* )}$, a light meson $(\pi ,\rho ,\eta ,\mathrm{etc}.)$ or nothing (purely leptonic decays), in a range of masses between the kaon and the B mass [291].
• (4)  
  LEP data: the most stringent limits above the B meson mass have been put by DELPHI [274]. HNLs have been searched for using data collected by the DELPHI detector corresponding to $3.3\times {10}^{6}$ hadronic Z⁰ decays at LEP1.
• (5)  
  CMS @ LHC: CMS searched for HNLs in three prompt charged leptons sample in any combination of electrons and muons collected at a center-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 35.9 fb⁻¹. The search is performed in the HNL mass range between 1 GeV and 1.2 TeV [292].
• (6)  
  BBN constraint: a HNL with parameters to the left of the BBN line would live sufficiently long in the early Universe to result in an overproduction of primordial Helium-4 [270].
• (7)  
  See-saw limit: below the see-saw limit, the mixing of the HNL with active neutrinos becomes too weak to produce the observed pattern of neutrino flavor oscillations [293].

Only two PBC projects with a ∼5 year timescale can contribute to this benchmark case: FASER with 150 fb⁻¹, which unfortunately is not competitive with the current bounds set by CHARM, and more interestingly, NA62⁺⁺ that can push down the CHARM limits by about one order of magnitude in the same mass range by collecting ∼10¹⁸ pot in dump mode. The NA62⁺⁺ projections correspond to the 90% CL exclusion limits in case all the final states with at least two charged tracks are considered [294]. In the projections the zero background hypothesis is assumed. Studies performed with the already acquired 3 × 10¹⁶ pot dataset in dump mode show that the background can be reduced to zero with the current setup for fully reconstructed final states, while for open final states the addition of an Upstream Veto in front of the decay volume is required. The addition of this detector is currently under study by the Collaboration.

Physics reach of PBC projects on 10–15 year timescale On a 10–15 year timescale many PBC projects can contribute to this benchmark case, as shown in figure 31: MATHUSLA200, FASER2, CODEX-b and SHiP. For MATHUSLA200 we show separately the contributions from heavy mesons and gauge bosons decays. The SHiP sensitivity curve is obtained without (solid curve) and with (dashed curve) a particular assumption for the contribution from B_c. This is because the σ(B_c)/σ(B) fraction at the SPS energies is not reliably known. We therefore show an upper sensitivity limit provided by assuming the fraction measured at the LHC energy.

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Also in this case the plot shows the 90% CL exclusion limits under the hypothesis of zero background. This hypothesis is a strong assumption that has been properly validated only by SHiP so far, using a full Geant4 simulation of the detector, including digitization and reconstruction, and large samples of Monte Carlo data. The background evaluation for MATHUSLA200, CODEX-b and FASER2 is still work in progress and will be carried on in the coming years. Figure 31 shows also projections from the LBNE near detector as 5-year sensitivity corresponding to an exposure of 5 × 10²¹ protons on target for a detector length of 30 m and assuming a normal hierarchy of neutrinos masses [295] and from FCC-ee with 10¹² Z⁰ decays and HNLs decaying between 10 and 100 cm from the interaction vertex [296].

In this section we consider the case in which HNLs couple only to second SM generation and the sensitivity plots are shown in the plane {$| {U}_{\mu }{| }^{2},{m}_{N}$}.

Current bounds, experimental landscape and PBC projects on 5 year timescale Current bounds and the future experimental landscape in the next ∼5 years, including some PBC projects, are shown in figure 32 for the case of HNL with couplings only to the second lepton generation and masses in the MeV–GeV range.

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Also in this case the allowed range of couplings is bounded from below by the BBN constraint [270], and the see-saw limit [293]. Existing experimental limits are shown as filled colored areas: for masses below the charm mass they arise mostly from the same beam dump experiments contributing to the sensitivity for electron-flavor dominance (PS191 [275] and CHARM [276], as explained in the previous subsection) with the addition of NuTeV [277], and E949 [278]:

• (1)  
  NuTeV @ Fermilab: a search for HNLs decaying in muonic final states has been performed at the neutrino detector NuTeV at Fermilab in 1996–1997, using 2 × 10¹⁸ 800 GeV protons interacting with a Beryllium-oxide target and a proton dump. [277].
• (2)  
  E949 @ BNL: Evidence of a heavy neutrino, in the process ${K}^{+}\to {\mu }^{+}{\nu }_{R}$ was searched for by the E949 Collaboration using 1.7 × 10¹² stopped kaons.

Above the charm mass, current bounds are set by DELPHI [274], Belle [291], CMS [292] with the same analysis used to set bounds for electron-dominance, and by LHCb with a dedicated analysis to search for prompt and displaced ${\pi }^{-}{\mu }^{+}$ vertices in ${B}^{+}\to {\pi }^{-}{\mu }^{+}{\mu }^{+}$ LNV decays [297].

In this scenario, NA62⁺⁺ in dump mode will be able to perform this search with competitive physics reach in a time scale of ∼5 years.

Physics reach of PBC projects on 10–15 year timescale Figure 33 shows the 90% CL exclusion limits from MATHUSLA200, FASER2, CODEX-b and SHiP in a 10–15 years time scale. Also in this case the curves are obtained under the assumption of zero background, for which the same considerations drawn in the previous subsection hold.

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In this section we consider the case in which HNLs couple only to third SM generation and the sensitivity plots are shown in the plane {$| {U}_{\tau }{| }^{2},{m}_{N}$}.

Current bounds and experimental landscape Current bounds and future experimental landscape in the next ∼5 years, including some PBC projects, are shown in figure 34 for the case of HNL coupling only to the third lepton generation and masses in the MeV–GeV range. Also in this case the allowed range of couplings is bounded from below by the BBN constraints [270], and the see-saw limit [293].

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Main bounds in this benchmark case arise from CHARM [298], NOMAD [299], and again the same data from DELPHI [274] used for the other two benchmark cases (BC6 and BC7).

• (1)  
  CHARM: limits on the square mixing strength $| {U}_{\tau }{| }^{2}$ in a mass range 10–290 MeV were set by re-interpreting the null result of a search for events produced by the decay of neutral particles into two electrons performed by the CHARM experiment using the neutrino flux produced by ${ \mathcal O }(2\times {10}^{18})$ 400 GeV protons on a solid copper target [298].
• (2)  
  NOMAD: a search for heavy neutrinos was performed using 4.1 × 10¹⁹ 450 GeV protons on target at the WANF facility at CERN in 1996–1998. The HNLs were searched in the process ${D}_{s}\to \tau {\nu }_{R}$ followed by the decay ${\nu }_{R}\to {\nu }_{\tau }{e}^{+}{e}^{-}$ in the NOMAD detector. This allowed to derive an upper limit on the mixing strength between the heavy neutrino and the tau neutrino in the ν_R mass range from 10 to 190 MeV [299].

Physics reach of PBC projects on 5 and 10–15 year timescale Among the PBC projects the only two contributing on a 5 year timescale are, again, FASER with 150 fb⁻¹ and NA62⁺⁺, as shown in figure 34. Figure 35 shows the 90% CL exclusion limits from MATHUSLA200, FASER2, CODEX-b and SHiP in a 10–15 years time scale. The physics reach from FCC(ee) with 10¹² Z⁰ is also shown.

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Assuming a single ALP state a, and the predominant coupling to photons, all phenomenology (production, decay, oscillation in the magnetic field) can be determined as functions of the (${m}_{a};{g}_{a\gamma \gamma }={f}_{\gamma }^{-1}$) parameter space.

Current bounds and near future experimental landscape The current bounds for ALPs with photon coupling are shown in figure 36, left. A zoom on the region of interest for experiments at accelerators is shown in the right panel and covers a range of masses between MeV and GeV. We note that this is also the mass region of interest in models where ALPs serve as mediators to a DM sector.

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Searches for ALPs with photon coupling have been performed using monophoton searches at LEP, ${e}^{+}{e}^{-}\to {\gamma }^{* }\to a\gamma$, mono-photon searches at BaBar (${e}^{+}{e}^{-}\to \gamma a,a\to$ invisible), radiative ϒ decays (${\rm{\Upsilon }}({nS})\to {\gamma }^{* }\to \gamma a$), radiative Z-boson decays, and electron- and proton beam dump experiments, where the ALPs are produced mainly via the Primakoff effect, i.e. the conversion of a photon into an ALP in the vicinity of a nucleus [310].

• (1)  
  E141 @ SLAC (electron beam dump): primarily searched for LLPs decaying into the ${e}^{+}{e}^{-}$ final state [193] but the addition of a photon converter in front of the detector for certain period of data taking opened the possibility to be sensitive also to photons from an ALP decay. Reinterpretation of the E141 results was performed in [312, 313], leading to somehow more conservative bound with respect previous interpretations [314].
• (2)  
  E137 @ SLAC (electron beam dump): dedicated search for ALPs coupling only to photons [194]. However the exclusion limits presented in the paper did not contain the turnover towards large couplings due to the exponential suppression of the number of ALPs that reach the detector, this has been added in [311].
• (3)  
  CHARM (proton beam dump): a search for a ALP in 400 GeV proton interactions with a thick copper target was performed with the CHARM detector [202]. The target was placed 480 m apart from the 35 m long decay volume and the hypothetical decay $a\to \gamma \gamma$ has been sought using a fine-grained calorimeter of 9 m² active area.
• (4)  
  NuCal (proton beam dump): the production and decay of a light scalar and pseudoscalar particle has been investigated in a proton-iron beam dump experiment at the 70 GeV Serpukhov accelerator [204].
• (5)  
  LEP: limits from LEP data in the ALP mass range MeV–GeV arise from a reinterpretation [315] of the LEP ${Z}^{0}\to \gamma \gamma$ data [316–319] where one of the two neutral clusters was considered the result of a merging of two highly collimated photons from the ALP decay in the process ${Z}^{0}\to a\gamma ,a\to \gamma \gamma$. Mono-photon searches, i.e. searches for highly-energetic photons in association with missing energy resulting from the process ${e}^{+}{e}^{-}\to \gamma +a(a\to$ invisible) have been performed as well [320] but they are not sensitive to ALPs with mass in the sub-GeV range [311].
• (6)  
  Bound from Astrophysics: Supernova 1987A. Weakly coupled particles such as axions or ALPs with masses up to about 100 MeV can be copiously produced in the hot core of a supernova. Because of their weak couplings these particles stream out of the core and thereby constitute a new energy loss mechanism. In the absence of such new particles the main cooling mechanism is due to neutrino emission. The corresponding neutrino signal has been observed in the case of SN 1987A, placing a bound on possible exotic energy loss mechanisms, which should not exceed the energy loss via neutrino emission [311].

We note that mono-photon searches have also been carried out at the LHC (for the most recent analyses see e.g. [321]), but their sensitivity does not significantly improve on the bound from LEP.

Near future bounds will come from Belle-II where the ALP can be searched in the invisible and 3γ decay modes. A phenomenological study has been performed in [311] where the authors consider ALP decays into dark matter (invisible) and two (resolved) photons. The bounds have been shown in figure 36. However the $a\to$ invisible sensitivity heavily relies on the possibility to use the single-photon trigger with a low threshold (1.8 GeV), that in not guaranteed during the nominal Belle-II luminosity regime.

PBC projects on 5 and 10–15 years timescale Three PBC experiments can perform searches of ALPs with photon coupling on a 5 year timescale: NA62⁺⁺ in dump mode and FASER, will look for visible ALP decays, $a\to \gamma \gamma$, while NA64${}^{++}(e)$ will be able to perform a search into visible and invisible decays. The contour plots are shown in figure 37. On a longer timescale the big PBC projects can enter in the game further extending the physics reach in a still uncharted parameter space. In this respect SHiP and LDMX will be fully complementary, the first covering larger masses and smaller coupling, the latter filling the uncovered phase space between the old beam-dump experiments and the colliders, in the 10–300 MeV mass range.

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Assuming a single ALP state a, and the predominant coupling to fermions, all phenomenology (production and decay) can be determined as functions of (${m}_{a};{g}_{Y}=2{{vf}}_{{\ell }}^{-1},2{{vf}}_{q}^{-1}$), with v the vev of the Higgs. Furthermore, for the sake of simplicity, we take ${f}_{q}={f}_{{\ell }}$. Details about approximations and assumptions in computing sensitivities for this benchmark case are reported in appendices A and B.

The effective Yukawa coupling ALP-SM fermions is proportional to the mass of the SM fermions. Hence: a possible ALP with fermion coupling is mostly originated from meson decays and only very rarely from electrons. Heavy mesons can be produced in ${e}^{+}{e}^{-}$ colliders, pp colliders and in the interactions of a proton beam with a target.

Searches for ALPs with fermion coupling are being pursued at the LHC, namely in the analysis of rare B decays as for example ${B}^{+}\to {K}^{* 0}{\mu }^{+}{\mu }^{-}$ [250]. The geometry of the LHC experiments, on the other hand, is such that a search can be performed only if the ALP decays more or less instantaneously, hence has large couplings. For ALPs with smaller couplings and longer lifetimes these searches are much less effective even though ALPs may still be produced in abundance.

Beam-dump experiments in contrast are particularly sensitive to long-lived and very weakly coupled light new states, which can travel through the hadron absorber before decaying. Several constraints already exist, mostly coming from old beam dump experiments as CHARM [202], NuCal [204], and E613 [322]. Other constraints are derived from K and B mesons experiments, as explained below.

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Current bounds and near future prospects, including PBC projects The current status of the exclusion limits for ALPS with fermion coupling in the MeV–GeV range is shown in figure 39, as filled colored areas.

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Most of the current bounds arise from a re-interpretation of experimental results from CHARM [202], E949 [221, 323], KTeV [324] performed by theorists [325, 326]. As such, these bounds should be taken with many caveats. A few searches are instead coming directly from experiments, as for example BaBar [327] and LHCb [249, 250].

• (1)  
  ${K}^{+}\to {\pi }^{+}+X:$ the ${K}_{\mu }2$ experiment has studied the momentum distribution of charge pions produced in the decay ${K}^{+}\to {\pi }^{+}$ [328]. In presence of a light pseudoscalar, the decay channel ${K}^{+}\to {\pi }^{+}a$ would lead to a bump in the spectrum.
• (2)  
  ${K}^{+}\to {\pi }^{+}+$ invisible: reinterpretation of the E949 results [221, 323] as an upper bound on the process ${K}^{+}\to {\pi }^{+}a$ performed in [325], and cross-checked by the KLEVER Collaboration. The curve assumes that the ALP escapes the decay volume.
• (3)  
  ${B}^{0}\to {K}_{S}$ + invisible: This search is the analogous as the ${K}^{+}\to {\pi }^{+}$ + invisible search. The strongest bounds come from CLEO [329].
• (4)  
  ${K}_{L}\to {\pi }^{0}{{\ell }}^{+}{{\ell }}^{-}$: in the mass range $210\lt m\lt 420\,\mathrm{MeV}$ the pseudoscalar will decay predominantly to muon pairs. The KTeV/E749 Collaboration has set an upper limit on the ${K}_{L}\to {\pi }^{0}{{\ell }}^{+}{{\ell }}^{-}$ decay [324] and this result has been converted into an upper limit for the branching fraction of the decay ${K}_{L}\to {\pi }^{0}a$ under the hypothesis that the ALP decays instantaneously [325].
• (5)  
  Radiative ϒ decays: pseudo-scalar particles have been sought by the BaBar Collaboration in radiative ϒ decays ${\rm{\Upsilon }}\to a\gamma$, with $a\to {\mu }^{+}{\mu }^{-}$ for $a\lt 2{m}_{\tau }$ [327] and $a\to {\tau }^{+}{\tau }^{-}$ for m_a above threshold [330].
• (6)  
  $B\to K{\mu }^{+}{\mu }^{-}$: the measurement of the branching fraction of the ${B}^{+}\to {K}^{+}{\mu }^{+}{\mu }^{-}$ decay as a function of the di-muon mass performed by LHCb [331] has been interpreted in [325] as an upper bound for the process ${B}^{+}\to {K}^{+}a,a\to {\mu }^{+}{\mu }^{-}$ in each ${\mu }^{+}{\mu }^{-}$ mass bin, under the hypothesis that a decays instantaneously. Dedicated searches have been instead performed by the Collaboration, allowing also for displaced di-muon vertices in the ${B}^{0}\to {K}^{0* }{\mu }^{+}{\mu }^{-}$ and ${B}^{+}\to {K}^{+}{\mu }^{+}{\mu }^{-}$ processes, and the results reported in [250] and [249], respectively⁶⁴ .

Near (∼5 years) and medium-far (10-15 years) prospects for PBC experiments are shown as solid lines in figures 39 and 40, respectively. This benchmark case considers a scenario in which the ALP a only couples to the gluon field at a scale Λ = 1 T eV. One can write down the corresponding low-energy Lagrangian at the tree level as

$$\begin{eqnarray}&&{ \mathcal L }={{ \mathcal L }}_{{\rm{SM}}}+{{ \mathcal L }}_{{\rm{DS}}}+a\displaystyle \frac{{g}_{s}^{2}}{8{f}_{G}}{G}_{\mu \nu }^{b}\tilde{{G}^{b\mu \nu }}.\end{eqnarray} \tag{ 9.2 }$$

Because the ALP mixes with the neutral pseudoscalar mesons, it is produced in any process that produces such mesons. Moreover it can be produced also in B mesons decays, as explained in section 2.1.4. Details about approximations and assumptions assumed in computing sensitivities for this benchmark case are reported in appendices A and B.

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Figure 41 shows the current bounds (as colored filled areas) and the prospects for PBC projects (solid lines) both on 5 year (FASER) and 10–15 year (CODEX-b, MATHUSLA200, FASER2) timescale. Below the three pion threshold, the CODEX-b and MATHUSLA200 reach for this benchmark is conditional upon the eventual detectors being sensitive to the di-photon final state. Production from K and B decays depend on UV completion and the results shown assume $\approx [\mathrm{log}{{\rm{\Lambda }}}_{{\rm{UV}}}^{2}/{m}_{t}^{2}\pm { \mathcal O }(1)]\Rightarrow 1$. The CODEX-b curve has been obtained considering B-decays only, hence it is conservative. NA62⁺⁺ and SHiP are also expected to be sensitive to this benchmark case but they did not provide the sensitivity curves on the timescale of this paper.

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Current bounds arise from flavor physics, old beam-dump experiments and LEP data. A comprehensive reinterpretation of these data has been performed in [332] in the ${m}_{\pi }\,\lt {m}_{a}\lt 3\,\mathrm{GeV}$ mass region, namely:

• (1)  
  Data from LEP [333, 334] and old beam dump experiments, E137 [194] and NuCal [204], have been used to recast limit on the $a\gamma \gamma$ vertex and translated into a limit in the ${BR}(a\to \gamma \gamma );$
• (2)  
  The limits on the branching fractions of the decays $\phi \to \pi \pi \gamma \gamma$ and $\eta ^{\prime} \to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}{\pi }^{0}$ [335] are used to set a limit on the rate of the processes $\phi \to \gamma a(\pi \pi \gamma )$ and $\eta ^{\prime} \to {\pi }^{+}{\pi }^{-}a({\pi }^{+}{\pi }^{-}{\pi }^{0})$, assuming that the full rate is due to ALPs;
• (3)  
  Decays driven by the $b\to {sa}$ penguin diagram are considered and a recast of results is performed while analyzing:

  • –  
    the ${m}_{\eta \pi \pi }$ spectrum of the decay ${B}^{\pm }\to {K}^{\pm }\eta {\pi }^{+}{\pi }^{-}$, interpreted as ${B}^{\pm }\to {K}^{\pm }a(\eta {\pi }^{+}{\pi }^{-})$, from [336];
  • –  
    the ${m}_{{K}^{* }K}$ spectrum of the decay ${B}^{\pm }\to {K}^{\pm }{K}^{\pm }{K}_{{\rm{S}}}{\pi }^{\mp }$, interpreted as ${B}^{\pm }\,={K}^{\pm }a({K}^{\pm }{K}_{{\rm{S}}}{\pi }^{\mp })$, from [336];
  • –  
    the measurement of the two decay rates, ${BR}({B}^{0}\to {K}^{0}\phi \phi )$ [337] and ${BR}({B}^{\pm }\to {K}^{\pm }\omega (3\pi ))$ [338], to put a constraints on the processes ${B}^{0}\to {K}^{0}a(\phi \phi )$ and ${B}^{\pm }\to {K}^{\pm }a({\pi }^{+}{\pi }^{-}{\pi }^{0})$, respectively.
• (4)  
  Measurements on processes driven by the $s\to d$ penguin diagram, as ${K}^{\pm }\to {\pi }^{\pm }\gamma \gamma$ [339] and ${K}_{{\rm{L}}}\to {\pi }^{0}\gamma \gamma$ [340], are used to recast limits on ALPs.

For cases (2) and (3) listed above, at one loop, the ${agg}$ vertex generates an axial-vector ${att}$ coupling [341] which enhances the rate for $B\to {K}^{(* )}a$ decays [31, 342–344]. Following [332] the UV-dependent factor contained in the loop, $\approx [\mathrm{log}{{\rm{\Lambda }}}_{{\rm{UV}}}^{2}/{m}_{t}^{2}\pm { \mathcal O }(1)]$, is approximated to unity (which corresponds to a UV scale ∼TeV).

In case of the TauFV experiment, the physics goal is to observe and measure, or alternatively set the upper bound on, the several LFV τ or D-meson decays. To estimate the sensitivity to a multi-TeV New Physics scale, the LFV process ${\tau }^{\pm }\to {\mu }^{+}{\mu }^{\pm }{\mu }^{-}$ is considered. This process is almost entirely forbidden in the SM, and any attempt to measure this small branching will automatically probe the New Physics that violates approximate τ and μ flavor conservation.

To quantify the New Physics reach, one can introduce a series of effective operators that mediate such transitions. For this particular decay process, at lowest order, one can have

$$\begin{eqnarray}&&{ \mathcal L }=\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\phi }}{{{\rm{\Lambda }}}_{\mu \tau }^{2}}\times (\bar{\mu }{\gamma }_{\alpha }\mu )(\bar{\mu }{\gamma }_{\alpha }\tau )\,+({\rm{h}}.{\rm{c}}.)+\,\mathrm{other}\,\mathrm{Lorentz}\,\mathrm{structures}.\end{eqnarray} \tag{ 10.3 }$$

In this expression, all coupling constants have been subsumed into the definition of the energy scale ${{\rm{\Lambda }}}_{\mu \tau }$, apart from a possible phase ϕ.

The resulting branching ratio for the τ → 3μ decay in the m_μ ≪ m_τ limit is given by

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\tau \to 3\mu }=\displaystyle \frac{{m}_{\tau }^{5}}{256{\pi }^{3}{{\rm{\Lambda }}}_{\mu \tau }^{4}}.\end{eqnarray} \tag{ 10.4 }$$

Given the stated goal of the TauFV experiment (in the absence of positive signal) is to reach the exclusion level of ${\mathrm{BR}}_{\tau \to 3\mu }\sim {10}^{-10}$, one can translate the above formula to the ${{\rm{\Lambda }}}_{\mu \tau }$ sensitivity,

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\mu \tau }\gt 55\,\mathrm{TeV}\times {\left(\displaystyle \frac{{10}^{-10}}{{\mathrm{Br}}_{\tau \to 3\mu }}\right)}^{1/4}.\end{eqnarray} \tag{ 10.5 }$$

The KLEVER proposal seeks to complement the NA62 experiment by measuring ${K}_{L}\to {\pi }^{0}\,+\,\text{missing energy}$. In the SM, the missing energy is carried by neutrinos, ${K}_{L}\to {\pi }^{0}\nu \bar{\nu }$, and the corresponding branching ratio is predicted to be ${\mathrm{BR}}_{{\rm{SM}}}=(3.4\pm 0.6)\,\times {10}^{-11}$ [345].

The branching ratios for the decays $K\to \pi \nu \bar{\nu }$ are among the observables in the quark-flavor sector most sensitive to NP. Because the SM decay amplitudes are strongly suppressed by the GIM mechanism and the CKM hierarchy and dominated by short-distance physics, the SM rates are small and predicted very precisely, making the $K\to \pi \nu \bar{\nu }$ BRs potentially sensitive to NP at mass scales of hundreds of TeV, in general surpassing the sensitivity of B decays in SM extensions [117]. Observations of lepton-flavor-universality-violating phenomena are mounting in the B sector. Most explanations for such phenomena predict strong third-generation couplings and thus significant changes to the $K\to \pi \nu \bar{\nu }$ BRs through couplings to final states with tau neutrinos [119].

The BR for the decay ${K}_{L}\to {\pi }^{0}\nu \bar{\nu }$ has never been measured. The current experimental result, $\mathrm{BR}({K}^{+}\to {\pi }^{+}\nu \bar{\nu })={1.73}_{-1.05}^{+1.15}\times {10}^{-10}$, obtained at Brookhaven from ${K}^{+}$ decays at rest with seven candidate events [221], together with considerations of isospin symmetry [346], leads to the model-independent bound $\mathrm{BR}({K}_{L}\to {\pi }^{0}\nu \bar{\nu })\lt 1.4\times {10}^{-9}$. This limit has to be compared to the direct limit set by the KOTO experiment, $\mathrm{BR}({K}_{L}\to {\pi }^{0}\nu \bar{\nu })\lt 3.0\,\times {10}^{-9}$ at 90% CL [205]. Because the amplitude for ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ has both real and imaginary parts while the amplitude for ${K}_{L}\to {\pi }^{0}\nu \bar{\nu }$ is purely imaginary, the decays have different sensitivity to new sources of CP violation.

In general, the measurement of the ${\rm{BR}}({K}_{L}\to {\pi }^{0}\nu \overline{\nu }$) is sensitive to additional sources of flavor violation coming from NP at, or above, the TeV scale. Parametrizing the effective Lagrangian for new physics in terms of effective operators as before, and taking one flavor of neutrinos for simplicity,

$$\begin{eqnarray}&&{ \mathcal L }=\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\phi }}{{{\rm{\Lambda }}}_{{ds}}^{2}}\times (\bar{\nu }{\gamma }_{\alpha }(1-{\gamma }_{5})\nu )(\bar{d}{\gamma }_{\alpha }(1-{\gamma }_{5})s)\,+({\rm{h}}.{\rm{c}}.)+\,\mathrm{other}\,\mathrm{Lorentz}\,\mathrm{structures},\end{eqnarray} \tag{ 10.6 }$$

one can quantify the sensitivty of KLEVER to NP. The decay ${K}_{L}\to {\pi }^{0}\nu \bar{\nu }$ is ${CP}-$ violating, and therefore the amplitude is proportional to $\sin (\phi ),\phi$ being the phase of NP contributions. In contrast, the ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ branching fraction is phase-independent, so it can also be used as a probe of TeV physics independently from ${K}_{L}\to {\pi }^{0}\nu \bar{\nu }$. Should NA62 discover deviations from the SM, a KLEVER-type of measurement would be required to further investigate their origins.

If the sensitivity of KLEVER can reach the SM branching ratio level, it would also entail sensitivity to New Physics at approximately

$$\begin{eqnarray}&&\displaystyle \frac{| \sin \phi | }{{{\rm{\Lambda }}}_{{ds}}^{2}}\sim \displaystyle \frac{1}{{\left(500\mathrm{TeV}\right)}^{2}}.\end{eqnarray} \tag{ 10.7 }$$

Figure 42, reproduced from [347], illustrates a general scheme for the expected correlation between the charged and neutral decays under various scenarios.

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If the NP has a CKM-like structure of flavor interactions, the K_L and ${K}^{+}$ BRs will lie along the band of correlation shown in green. In models with only left-handed or only right-handed couplings to the quark currents (e.g. models with modified Z couplings or littlest-Higgs models with T parity), because of constraints from ${\epsilon }_{K}^{{\prime} }$, the BRs must lie along one of the branches shown in blue. If the NP has an arbitrary flavor structure and both left-handed and right-handed couplings (e.g. in Randall–Sundrum models), there is little correlation, as illustrated in red.

In a recent breakthrough, the RBC-UKQCD Collaboration obtained the first result for $\mathrm{Re}\,{\epsilon }_{K}^{{\prime} }/{\epsilon }_{K}$ from a lattice calculation thought to have reliable systematics: $\mathrm{Re}\,{\epsilon }_{K}^{{\prime} }/{\epsilon }_{K}\,=(1.38\pm 5.15\pm 4.59)\times {10}^{-4}$, 2.1σ less than the experimental value [348]. Estimates from large-Nc dual QCD support the lattice result [349].

With this result for $\mathrm{Re}\,{\epsilon }_{K}^{{\prime} }/{\epsilon }_{K}$, the correlation between ${\epsilon }_{K}$ and $\mathrm{BR}({K}_{L}\to {\pi }^{0}\nu \bar{\nu })$ has been examined in various SM extensions at energy scales Λ in the neighborhood of 1–10 TeV by several authors, in many cases, with constraints from ${\epsilon }_{K}^{{\prime} },{\rm{\Delta }}{m}_{K}$, and $\mathrm{BR}({K}_{L}\to \mu \mu )$ considered as well. The results of these studies are summarized in table 7. In general, an observed value of ${\epsilon }_{K}$ that is larger than expected in the SM implies a suppression of $\mathrm{BR}({K}_{L}\to {\pi }^{0}\nu \bar{\nu })$ to below the SM value. However, it is possible to construct models in which ${\epsilon }_{K}$ and $\mathrm{BR}({K}_{L}\to {\pi }^{0}\nu \bar{\nu })$ are simultaneously enhanced. With moderate parameter tuning (e.g., cancellation among SM and NP interference terms to the 10%–20% level), $\mathrm{BR}({K}_{L}\to {\pi }^{0}\nu \bar{\nu })$ may be enhanced by up to an order of magnitude.

Table 7.  Effects on BRs for $K\to \pi \nu \bar{\nu }$ decays in various SM extensions, with constraints from other kaon observables, including in particular $\mathrm{Re}\,{\epsilon }_{K}^{{\prime} }/{\epsilon }_{K}$.

Model	Λ (TeV)	Effect on $\mathrm{BR}({K}^{+}\to {\pi }^{+}\nu \bar{\nu })$	Effect on $\mathrm{BR}({K}_{L}\to {\pi }^{0}\nu \bar{\nu })$	References
Leptoquarks, most models	1–20	Very large enhancements; mainly ruled out		[350]
Leptoquarks, U1	1–20	+10% to +60%	+100% to +800%	[350]
Vector-like quarks	1–10	−90% to +60%	−100% to +30%	[351]
Vector-like quarks + $Z^{\prime}$	10	−80% to +400%	−100% to 0%	[351]
Simplified modified Z, no tuning	1	−100% to +80%	−100% to −50%	[352]
General modified Z, cancellation to 20%	1	−100% to +400%	−100% to +500%	[352]
SUSY, chargino Z penguin	4–6 TeV		−100% to −40%	[353]
SUSY, gluino Z penguin	3–5.5 TeV	0% to +60%	−20% to +60%	[354]
SUSY, gluino Z penguin	10	Small effect	0% to +300%	[355]
SUSY, gluino box, tuning to 10%	1.5–3	±10%	±20%	[356]
LHT	1	±20%	−10% to −100%	[357]

The KLEVER experiment aims to use a high-energy neutral beam at the CERN SPS to achieve 60-event sensitivity for the decay ${K}_{L}\to {\pi }^{0}\nu \bar{\nu }$ at the SM BR with an S/B ratio of 1. At the SM BR, this would correspond to a relative uncertainty of about 20%, demonstrating a discrepancy with 5σ significance if the observed rate is a bit more than twice or less than one-quarter of the SM rate, or with 3σ significance if the observed rate is less than half of the SM rate. These scenarios are consistent with the rates predicted for many different SM extensions, as seen in table 7.