The Forward Physics Facility at the High-Luminosity LHC

To meet the physics requirements, the experimental cavern is designed to be located on the LOS, beginning approximately 617 m from the ATLAS IP1 and 10 m away from the LHC tunnel. The cavern will be 65 m-long and 8.5 m-wide, leaving enough space around the experiments for easy access for transport and installation of the required services, as shown in figure 5. The floor level is set at 1.5 m under the LOS, with a 1.25% fall towards IP1, following the inclination of the LOS.

Zoom In Zoom Out Reset image size

The experiments currently envisioned to be installed at the FPF include FASER2, FASERν2, AdvSND, FLArE, and FORMOSA. Their transverse dimensions and pseudorapidity coverage are given in table 1, and they are described in greater detail in section 3.. The experiments are centralized on the LOS and are served by a crane system along the cavern, as shown in figures 6 and 7. For safety reasons, given the potential of cold gas leakage, a 1 m-deep trench is foreseen under the liquid argon (LAr) detector (FLArE), as shown in figure 7.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The cavern is connected to the surface through an 88 m-deep and 9.1 m-diameter access shaft located on the top of the cavern. It will be equipped with a lift and staircase for access with enough space reserved for transport, as shown in the figure 8.

Zoom In Zoom Out Reset image size

To comply with CERN's safety requirements and avoid any possible dead ends, a safety gallery will connect the experimental cavern to the LHC and serve as a secondary emergency exit, as shown in figure 9.

Zoom In Zoom Out Reset image size

A key virtue of the dedicated facility is that access to the cavern will be possible during LHC operations. This would allow the installation of services and experiments, as well as maintenance and upgrades of experiments, to be possible at any time. A radioprotection (RP) study (discussed in detail in section 2.7) has been carried out to assess the feasibility of this due to the radiation level in the cavern, which shows this is sensitive to the design of the safety gallery connecting the cavern to the LHC tunnel.

Based on the first RP study, the initial layout of the gallery has been modified to further reduce the dose levels in the cavern. As part of the modification, a third chicane wall was added, the thickness of the walls was increased from 40 cm to 80 cm, and a change was made in the location of the walls, as shown in figures 9 and 10. A new RP study will be made to verify the effectiveness of the above-mentioned modifications.

Zoom In Zoom Out Reset image size

The safety gallery will only be used as an emergency escape route from the FPF cavern into the LHC tunnel, with an interlocked access door between the two. In the case that this door is opened, LHC operation will be automatically stopped for safety reasons.

Above ground, an access building and two auxiliary buildings are proposed to house the necessary infrastructure and utilities for the experiments, with their size being based on similar projects at CERN. The proposed layout of the buildings is shown in figures 11 and 12.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The 33 m-long and 21 m-wide access building located over the shaft will provide access from the ground level to the experiments for both personnel and equipment. It is designed as a basic steel portal frame structure with an internal height of 15 m, however the walls on the south and southwest will be part-formed from a retaining wall to support the excavation. The hall will be equipped with a 25 t overhead crane to lower the experiments into the cavern.

The service buildings for the electrical, cooling, and ventilation infrastructure are also characterized as steel portal frame structures, similar to the access building. The cooling and ventilation building will be 20.5 m long, 21 m wide, and 13.5 m high, with the wall on the west side part-formed from a retaining wall to support the excavation. The electrical building will be adjacent to the cooling and ventilation building and will be 20.5 m long, 12 m wide, and 5.5 m high. Both buildings will have a 1.2 m-deep false floor to allow the services to be distributed into the shaft. The proposed design of the surface buildings is shown in figure 13.

Zoom In Zoom Out Reset image size

An access road will be provided, linking to the existing roads and infrastructure of the SM18 buildings at the northeast, as shown in figure 14. A requirement for a maximum gradient of 6% has been respected, in line with the requirements of CERN's transportation services.

Zoom In Zoom Out Reset image size

The volume of earthworks arising from the project is significant, predominantly because of the existing site levels and ground conditions. The proposed location has been previously used as a spoil disposal area, and the ground levels vary between 453–457 m above sea level, 5–8 m above the existing infrastructures in the surrounding area. As a result, to reduce the volume of the excavated material as much as possible, and taking into consideration the ground condition, the finished ground level of the buildings is proposed to be at 450 m, as shown in figure 15.

Zoom In Zoom Out Reset image size

At particle colliders, it is essential to characterize the radiation field to cope with the multiple effects of the interaction of regular and accidental beam losses on machine and detector components. To quantify these effects starting from the relevant loss terms, multipurpose Monte Carlo (MC) codes are a critical tool, enabling the evaluation of macroscopic quantities through the microscopic description of particle transport and interactions in matter. This requires tracking through magnetic fields, as well as accounting for all applicable electromagnetic (EM) and nuclear processes over an extremely wide energy range. The code's reliability is verified through individual benchmarking of physics models against exclusive data. A profitable calculation requires modeling the machine and detector geometry, including material information, to a challenging degree of accuracy. This allows in turn for an inclusive validation against measurements from extended monitor systems (see below).

At CERN, FLUKA [40–42] is the reference tool to assess the machine protection aspects and the complementary radiation protection scope, as well as the machine-induced background to experiments. It is regularly and extensively used for the whole accelerator chain, from the beam dump design of low-energy injectors as Linac4, up to LHC collimation and the HL upgrade of the LHC. For future colliders, FLUKA also plays a crucial role, starting in the early stages of planning, for both accelerator and detector design and for both hadron and lepton machines. This means that simulations have to deal with protons up to some million TeV (the beam energy with a target at rest required to reach 100 TeV CM collisions) down to the lowest transport limit of 100 eV photons (for the study of lepton ring synchrotron radiation). Such a task calls for the continuous improvement of the different interaction models, having in mind that, despite the very high energy of beams of interest, several quantities, such as those related to radionuclide inventory, are extremely sensitive to the low-energy nuclear physics ingredients ruling the reaction fragment de-excitation.

Together with this physics-oriented effort, notable technical developments have made it possible to automatize the construction of consistent geometry models of several-hundred-meter accelerator portions [43, 44].

The operation of the LHC has provided the opportunity to probe the degree of reliability of simulations performed over the long course of the LHC design phase. In particular, the beam loss monitor (BLM) system, consisting of a few thousand ionization chambers all along the 27 km beam line, provides on-line measurements of the energy released by the particle shower originated by beam particle interactions, and it triggers beam aborting if the detected values exceed pre-defined thresholds. Particle shower calculations make it possible to predict BLM signals for different loss scenarios, correlating them at the same time with the energy deposition levels in the most exposed or sensitive elements and the radiation levels, namely differential particle fluences, in areas of interest. Various representative examples of comparisons between BLM measurements and FLUKA predictions are presented in reference [45].

For the FPF, it is particularly important to understand the particle debris generated by colliding beams at the ATLAS IP during HL-LHC running. Given an inelastic cross section of about 80 mb (including diffractive events), the collision products carry almost 7 kW towards each side of IP1 at the ultimate instantaneous luminosity of 7.5 × 10³⁴ cm⁻² s⁻¹ for 7 TeV proton operation. Only a limited fraction of this power ($< 5$%) impacts the detector itself, while 80% of it is transported out of the experiment's cavern by high-energy, forward-angle particles travelling inside the beam vacuum chamber through the accelerator elements. These particles amount on average to 6 per single pp collision out of the 155 particles emerging on average from IP1 after neutral pion decay, and are mostly photons, charged pions, protons, and neutrons. The accelerator elements will be protected by a 1.8 m-long copper absorber, called the TAXS, located near the interface between the cavern and the LHC tunnel about 20 m from IP1, and featuring a 60 mm-diameter cylindrical aperture, considerably larger than the present one. The TAXS absorbs another 8%, while the following 60 m-long string composed of single-bore superconducting quadrupoles (the final focus triplet), corrector magnets and separation dipole (D1) take 22% (as part of the above 80%, which does not include the TAXS fraction, though). In fact, the majority of the energetic pions matching the TAXS aperture are then bent by the magnetic field onto the massive beam screen structure traversing the aforementioned string and embedding tungsten layers to protect the 150 mm-diameter aperture of the magnets.

A new version of the other main absorber (TAXN) will sit 45 m after the D1, incorporating the transition between one single central aperture and two separate symmetrical apertures of 88 mm-diameter and intercepting the LOS of neutral particles coming from IP1. Its effectiveness in protecting downstream elements will be slightly weakened by the adoption of horizontal beam crossing in IP1, since, in this case, the axis of the debris cone, instead of hitting exactly in-between the two TAXN twin apertures (as for vertical crossing), moves closer to the external aperture, as a function of the crossing angle. For the baseline half crossing angle of 250 μrad, the ATLAS TAXN is expected to absorb 20% of the one-side debris power.

An 8 m-long superconducting twin bore recombination dipole will open to the matching section, taking 100 m of the experimental insertion up to the dispersion suppressor (DS). The matching section features four main superconducting quadrupole assemblies (numbered Q4 to Q7) and ends 270 m from IP1. Less than 10% of the one-side debris power, mostly carried by photons, neutrons, and protons, is absorbed in this machine segment, concentrated in the first (TCLPX4) of the three metallic horizontal collimators to be installed on the outgoing beam aperture to further shield the cold magnet coils. Most of the remaining debris power goes to the preceding tunnel walls.

The third TCL collimator, in front of Q6, can also provide an effective cleaning of the initial part of the DS, where the beam lines are bent by the LHC main dipoles and no layout modification is planned for the HL-LHC era. Nevertheless, beyond the TCL6 range, losses take place in the DS odd half-cells, according to the periodicity of the single turn dispersion, as already regularly observed. They consist of protons that underwent diffraction at IP1 and are affected by a magnetic rigidity deficit of the order of 1%, leading them to touch the beam screen of the outgoing beam chamber in the horizontal plane towards the center of the ring.

The implementation in FLUKA of the ATLAS insertion model described above has enabled multiple studies, mainly oriented to HL-LHC design [46], but also serving other purposes, for instance, the evaluation of background and neutrino fluxes for SND@LHC [13]. We present here preliminary results for the calculation of the flux of high-energy muons reaching FPF along the ATLAS LOS, which originate in both primary IP1 collisions and downstream shower development.

Thanks to a dedicated optimization, featuring a suitable transport threshold adjustment and combined biasing techniques, including artificially increasing the decay probability of parent mesons and controlling statistical weight fluctuations, it was possible to produce a meaningful muon sample just upstream of the proposed sweeper magnet location, as shown in figure 22. For both positive and negative muons, the fluence is maximal on the accelerator line, a bit less than 1 m from the ATLAS LOS at x = y = 0. In contrast, a fluence minimum is found at the LOS for positive muons, while negative muons display a concentration on the horizontal plane (y = 0) on the external side of the ring, compatible with the bending action of the DS dipole field. The positive and negative muon energy spectra, averaged over the square of figure 22, are reported in figure 23, indicating that positive muons are predominant above 1 TeV.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The muon samples shown constitute the source term for a second step simulation implementing the sweeper magnet (see section 2.10) and the further propagation of muons to the FPF through the rock. The goal of this second step will be to quantify the muon background, as well as the related dose equivalent contribution (see section 2.7).

In preparation for the FASER experiment, in situ measurements were made during 2018 LHC running in the TI12 and TI18 tunnels in the LHC, 480 m from IP1 and along or close to the LOS. Measurements were made using emulsion-based detectors, which can be used to determine the number of charged particles that traverse the detector while it is installed and very accurately measure the track angles. There is, however, no knowledge of the time the particles cross the detector. In addition to this, measurements were made with a TimePix3 BLM, which is able to measure the rate of charged particles with excellent time resolution, but, given a lack of calibration, could not give an absolute rate. The measurements are discussed in detail in the FASER Technical Proposal [10]. The measurements were used to validate the FLUKA estimate of the muon flux for the 2018 LHC running conditions on the LOS.

The emulsion detector observed a clear peak of charged particles entering the detector with an angle consistent with the direction from IP1. The number of particles observed in this angular peak was (1.2–1.9) × 10⁴ fb cm⁻², when normalized by the luminosity that the detector was exposed to. This can be compared to the estimate from FLUKA of 2.0 × 10⁴ fb cm⁻² with an uncertainty of $\mathcal{O}(50\%)$. The FLUKA simulation result is consistent with the measurement within the expected uncertainties from FLUKA. The measurements with the BLM showed that the observed rate is correlated with the instantaneous luminosity in IP1 during an LHC fill, with the rate falling off during the fill as the luminosity decreases. Again, this is consistent with the FLUKA expectation that the background muon rate is originating from collision debris. With two circulating beams that were not colliding, the measured rate was very close to zero. The agreement between the FLUKA simulations and in situ measurements for the 2018 setup of the LHC gives confidence that FLUKA estimates for the HL-LHC will describe the background with reasonable accuracy. To further validate the simulations of the muon backgrounds in this region, when Run 3 begins in 2022, the FASER collaboration plans to make further measurements using small emulsion detectors placed at various distances from the LOS.

The CERN radiation protection (RP) rules are provided in the so-called 'Safety Code F' [47, 48]. The objective of Safety Code F is to define the rules for the protection of personnel, the population, and the environment from ionizing radiation produced at CERN. Safety Code F is based on and updated to the most advanced standards of European and other relevant international legislations, including the legislation of CERN host states France and Switzerland.

With regard to the design of new facilities, different RP aspects must be taken into account at the design level, including shielding requirements, radiation levels during operation (prompt) and technical stops (residual), area classification, radiation monitoring, and the activation for future disposal of radioactive wastes. Among these, area classification and dose limits are discussed here. These aspects are particularly relevant, as they determine whether access to the new experimental cavern will be possible during LHC beam operation.

Areas inside CERN's perimeter are classified as a function of the effective dose a person would receive during his stay in the area under normal working conditions and routine operation. The potential external, as well as internal, exposures must be taken into account when assessing the effective dose researchers may receive when working in the area considered. The exposure limitation in terms of effective dose is ensured by limiting correspondingly the operational quantity ambient dose equivalent rate ${\dot{H}}^{\ast }(10)$ for exposure from external radiation, and the action levels of specific airborne radioactive material (airborne radioactivity) and specific surface contamination at the corresponding workplaces for exposure from incorporated radionuclides. In addition, exposure of people working on the CERN site, the public, and the environment must remain below the dose limits under normal, as well as abnormal, conditions of operation. Table 3 shows the limits for area classification of non-designated and supervised radiation areas at CERN. These limits are relevant for the experimental FPF cavern, which will be discussed in the following.

The RP-FLUKA simulations aim to determine the prompt radiation levels in the new purpose-built FPF cavern and in the shaft for different scenarios (normal/abnormal LHC operation); verify the accessibility of the experimental cavern during LHC and SPS operation; and check the effectiveness of the chicane in the safety tunnel.

The FLUKA model of the LHC tunnel (figure 24) presented in section 2.6 has been modified to include a detailed model of the new experimental cavern. This model contains $\sim 500$ m of LHC beam line elements (from 250 m to 750 m to the west of IP1) and the experimental cavern and its access shaft, based on technical drawings provided by CERN CE. The safety tunnel connecting the LHC tunnel to FPF includes a chicane made of 2 × 40 cm concrete walls, representing the baseline layout at the time of computation. Simulations were conducted with and without the chicane walls to verify its effectiveness.

Zoom In Zoom Out Reset image size

Several source terms were considered to simulate the operational and accidental scenarios relevant for the RP calculations:

• Beam-gas interactions: this source term is relevant for normal LHC operation, as it originates from inelastic interactions of the 7 TeV proton beam with residual gas within the beam pipes.
• Direct muon component from IP1: this source term is also relevant for normal LHC operation, and originates from muons directly streaming from ATLAS's pp collisions and from the particle showers produced in the long straight section (LSS) by the collision debris.
• Loss of the LHC beam: this source is relevant only for abnormal LHC operation (accidental scenario). It considers the loss of the full 7 TeV proton beam on the MB.B15R1, the superconductive dipole placed in front of the safety tunnel connection to the LHC tunnel.
• Loss of the SPS beam: this source term is relevant for the access of the FPF shaft, due to the vicinity of the two infrastructures, and it considers the loss of the 450 GeV proton beam in the SPS tunnel.

In all LHC simulations, beam 1, the clockwise beam traveling from the ATLAS IP toward the FPF, was simulated. In addition, the HL-LHC beam intensity was simulated using a scaling/normalization factor to consider 2748 bunches and 2.3 × 10¹¹ proton per bunch. With regard to beam-gas interactions, a conservative residual gas-density of 1.0 × 10¹⁵ H₂ m⁻³ was used. As reported in reference [49], this value ensures a 100 h beam lifetime. Recent studies conducted at CERN [50] indirectly determined lower residual gas densities during Run 2 operations (2.25–0.25 × 10¹³ H₂ m⁻³), with higher values registered at the beginning of the physic run.

Finally, the prompt ambient dose equivalent was scored in the LHC tunnel, the safety tunnel, and the new experimental cavern by using a Cartesian XYZ mesh.

Figures 25 and 26 show the prompt ambient dose equivalent rates, in μSv h⁻¹, during normal LHC operation. The particle shower due to beam-gas interactions streams through the safety tunnel into the FPF cavern, but the presence of the chicane lowers the ${\dot{H}}^{\ast }(10)$ at the FPF entrance.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figures 27 and 28 show the prompt ambient dose equivalent in mSv if the full LHC 7 TeV proton beam is accidentally lost on MB.B15R1, the superconductive dipole placed in front of the safety tunnel connection to the LHC tunnel. Similar to the normal operation case, the particle shower generated during this undesired event streams through the safety tunnel and could be potentially harmful for people standing at the entrance of the FPF cavern.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

A quantitative assessment of the radiation levels along the safety tunnel, for LHC normal and abnormal operation, is provided through the 1D profile shown in figures 29 and 30. Figure 29 shows the ${\dot{H}}^{\ast }(10)$ considering different residual gas densities in the LHC beam screen: independently from the shielding configuration (with or without the chicane), even considering a conservative residual gas density of 1.0 × 10¹⁵ H₂ m⁻³, the ${\dot{H}}^{\ast }(10)$ remains below the limit for non-designated radiation areas. On the other hand, the results shown for the accidental scenario in figure 30 show that the cumulative H*(10) exceeds the annual dose limit for classified personnel (20 mSv), even including the positive effect of the chicane.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

With regard to the other source terms to be considered, the loss of the full 450 GeV proton beam in the SPS tunnel, which is most relevant for the FPF shaft, produces a negligible dose, since the distance between the shaft and the SPS tunnel is $> 35$ m. The direct muon contribution coming from IP1/LSS1 to the prompt ambient dose equivalent rate needs to be further investigated and simulations are currently ongoing.

Zoom In Zoom Out Reset image size

The verification of the accessibility of the experimental cavern during LHC operation requires the evaluation of different scenarios/source terms, considering both normal and abnormal operation. At present, the limiting scenario is the possible loss of the full LHC beam on the MB.B15R1 dipole: possible mitigation actions such as adding a turn in the safety tunnel, adding another wall ('triple chicane') and thickening the chicane walls, might be evaluated and integrated into the CE model. However, the missing direct muon contribution might have an impact on the accessibility of the FPF cavern during operation, which needs to be addressed.

For any particle physics experiment in proximity to an accelerator, an understanding of the background sources, their origin, the particle types, and their spectra is crucial at both the design stage as well as during operation. To predict and understand these backgrounds requires the use of MC techniques, as the particle fluence close to an accelerator originates from many different and indirect sources. For the FPF specifically, we focus on the region of the LHC close to IP1, where the ATLAS experiment is located. This entails predicting the particle flux due to:

• (a)  
  pp collisions at IP1 (and potentially also Pb–Pb and Pb-p)
• (b)  
  Inelastic proton interactions with residual vacuum gas in the LHC arcs
• (c)  
  Other beam losses in the arcs due to other-IP physics debris, collimation losses, and other beam losses

The first is expected to be the dominant contribution, as the FPF is on the LOS of the IP where the collisions occur, and they will produce high-energy, penetrating particles, such as muons and neutrinos. The second and third sources are from a significantly different direction, but are expected to be small contributions, since the distance of between 10 m and 16 m from the FPF inside wall to the LHC tunnel is expected to be sufficient to absorb the majority of background particles from these sources.

To simulate the particle fluence, a 3D radiation transport model is required, along with the transport and description of many subatomic particles. Both EM and hadronic interactions with the material of the accelerator, the tunnel, and the surrounding rock must also be simulated, as well as the deflection of charged particles by the many unique magnetic fields of the accelerator magnets.

In contrast to conventional transverse-orientated detectors, the far-forward location of the FPF requires simulation of the accelerator complex including its varied conditions throughout operation. The magnet strengths (the optics) are varied throughout each fill and generally throughout the operation (the Run) for various purposes. The crossing-angle of the colliding beams as well as the beam size (and therefore divergence angle) are varied to maintain, or level, the luminosity at the collision point to best serve the experiments. Throughout the Run, the optics may generally be improved upon depending on the machine performance, machine protection, the upstream injector-chain performance, and collimation performance.

BDSIM [51] is a MC tool based on Geant4 [52–54], ROOT [55], and CLHEP that creates Geant4 models of accelerators from an optical description (one concerned with the magnetic strengths), such as MADX [56], used as the magnetic description of the LHC. It includes a library of geometries for many accelerator components in many styles including those of the LHC. It also includes parameterized magnetic fields for all of the conventional accelerator magnets, as well as the ability to load and interpolate field maps. Crucially, being based on Geant4, it allows the production and tracking of all subatomic particles provided by Geant4, as well as the extensive physics processes for them. Geant4 is a particle physics library commonly used in detector simulation and underpins the MC for countless experiments in particle physics as well as being used in the space and medical physics domains. It is regularly updated with the latest developments from the field. Similar codes also used for this purpose include FLUKA [41, 42], described above in section 2.6, and MARS [57, 58].

Although BDSIM provides a library of approximate and scalable geometries suitable for most applications, it is required in many cases to provide a more accurate geometry for a specific installation, where relevant. In this case, BDSIM provides the ability to load GDML format geometry, which is the geometry persistence format of Geant4 based on XML. The Python library pyg4ometry [59] is used to create, convert, and composite geometry. This allows detailed descriptions in other formats, such as the IR1 tunnel complex at CERN described in FLUKA geometry format, to be converted and incorporated into the model.

An important feature of BDSIM is the ability to filter and store select trajectories of particles in a linked data structure in ROOT format. Therefore, a BDSIM model allows not only fluences to be estimated, but also may be used to provide insight into the origin of background sources and their production mechanisms.

As an input, event generator output in HepMC [60] format can be used for each event. BDSIM is also usable as a C++ class to simulate individual events for a given model. Therefore, it can be integrated into an analysis framework if required, or used to generate a static MC sample. The output of BDSIM is a ROOT-format file that is standard in high-energy physics, and either ROOT itself or the included tools with BDSIM can be used to perform a large scalable analysis, including skimming. BDSIM has been used for studying accelerator beam losses in a variety of machines; the models described here were originally developed for LHC collimation studies [59, 61].

A model of the LHC accelerator from IP1 towards the FPF was created using BDSIM. This model was originally developed for the FASER experiment [62] at both the pilot detector location ('C-side' of IP1 at the TI18 LOS location) and the FASER experiment location ('A-side' of IP1 at the TI12 LOS location). Several data sources are used to prepare the BDSIM input automatically using its associated pybdsim Python library. These include:

• (a)  
  A MADX 'Twiss table' file providing an optical description of the magnets
• (b)  
  A detailed aperture model from the LHC collimation group (BE-ABP-NDC)
• (c)  
  Corresponding collimator settings for the optical configuration
• (d)  
  Tunnel geometry converted from FLUKA format
• (e)  
  Tunnel geometry from BDSIM and pyg4ometry
• (f)  
  Select geometry pieces for various components (e.g., TAN, JSCA(A, B) shielding).

Items (a) and (c) are expected to vary throughout the Run and the model can be easily adjusted to reflect these through automated preparation. The model was originally created for LHC Run 2 (2015–18) and Run 3 (2022–25). However, for the FPF, during HL-LHC [46], a new model is required as the accelerator layout will have changed; see figure 31. There are several key differences from the LHC-era machine, namely:

• New TAXS absorber replaces the TAS absorber with increased aperture
• New TAXN absorber replaces the TAN absorber
• The 'D1' separation dipole is now superconducting
• New collimators for incoming and outgoing beams in the layout

The TAXS absorber (currently TAS in the LHC) is a cylindrical copper absorber approximately 19 m from the IP. The TAXN (currently TAN in the LHC) is an absorber between the two separation dipoles to protect the machine from predominantly neutral physics debris. The layout and geometry in the BDSIM model has been updated according to the optics configuration HL-LHC V1.5.

A 3D view of the complete BDSIM model is shown in figure 32. The proposed FPF cavern is not simulated, as a particle flux on its entrance is desired in this study and will have no effect on that result. In the future, the geometry for the cavern will be added.

Zoom In Zoom Out Reset image size

To generate a MC sample, the CRMC [63] event generator tool is used to generate the $\sqrt{s}=14$ TeV pp collisions. CRMC provides several underlying generators including EPOS-LHC [64], QGSJet [65], and Sibyll [66]. For this study, Sibyll was used. CRMC writes a HepMC format file that is subsequently loaded by BDSIM. BDSIM is set to simulate only loaded particles in the forward direction of the model and with a pseudorapidity η ⩾ 2.3 for simulation efficiency. A minimum kinetic energy cut of 10 GeV was used. Geant4 V10.7.p03 was used with reference physics list FTFP_BERT (a complete physics list including EM, decay, and hadronic processes). HL-LHC V1.5 MADX optics were used with a primary proton beam energy of 7 TeV. The crossing angle was 250 μrad. An aperture model was used from the BE-ABP-NDC collimation group.

Cross section biasing is used to reduce the computational time required to estimate muon fluxes by increasing the cross section of several particles' decay processes. Given the small forward area at a great distance, this biasing scheme is required to estimate the relevant quantities in a reasonable amount of time on an available computer farm. The numerical biasing factors used per particle are shown in table 4.

Passive and invisible planes, called 'samplers' in BDSIM, were placed at several positions in the model to record muons and neutrinos only, as it is foreseen that these are the most relevant SM particles at the FPF for this study. These samplers record the kinematic variables of particles in a 2D plane at a given position with a given square size. Samplers were placed at three locations in the LOS of IP1, as shown in figure 33. These were at distances 370 m, 475 m, and 617 m, which correspond to just into the tunnel after the accelerator starts to bend; the opening into the TI18 tunnel; and in front of the FPF. The data from the samplers can be analyzed individually or together, as well as be re-launched into the model or another for further study.

Zoom In Zoom Out Reset image size

The model preparation was validated by tracking 10 000 primary protons from the IP to the end of the model, analyzing the beam size, mean offset, and calculated optical Twiss functions from the tracks recorded in samplers after every magnet. The tracking showed excellent agreement with the MADX model.

For this study, a proposed sweeper magnet was included. It was 20 cm × 20 cm in area, with a peak field of 1.4 T, and placed in the LOS (X, Y = 0, 0 m) at Z = 370 m. The field was orientated vertically, with the return flux in an iron layer 10 cm wide. The core of the magnet was samarium cobalt. A cross section of it with overlaid magnetic field map is shown in figure 34. The field map is based on previous permanent magnet designs and was provided by the SY-STI-BMI group at CERN. The geometry was created in GDML using pyg4ometry.

Zoom In Zoom Out Reset image size

A MC sample of 100M pp events was generated using the Royal Holloway computing cluster. This sample includes the trajectories of (exclusively) muons and neutrinos that reach the 2 m × 2 m sample plane at the entrance to the FPF. Additionally, the select parent track trajectories linking each particle back to the primary vertex are stored. This will enable a detailed analysis of origins and production mechanisms in the future.

The BDSIM studies provide results for the distribution of muons along the LOS at several distances from the ATLAS IP. The results can be displayed using the coordinate system shown in figure 33. Firstly, the 2D distribution of muons at Z = 370 m (the green line in figure 33) is shown in figure 35, with a magnified view in figure 36. As the primary proton energy is 7 TeV, very high-energy, and therefore penetrating, muons can be produced. A subset of the muons shown in figure 35 is displayed in figure 37, where only muons with E_k > 1 TeV are shown. These are less frequent but are more likely to reach the FPF through the accelerator complex and the surrounding rock.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The 2D muon distribution in front of the entrance to the FPF, Z = 617 m from IP1 (the pink line in figure 33), is shown in figure 38. The spectrum of the muons and neutrinos at the FPF entrance plane is shown in figure 39. It should be noted that the production of τ^± is not implemented in the version of Geant4 used (10.7.p03), but the recently released Geant4 V11.0 can produce them through positron annihilation. Here, we also expect only electron and muon neutrinos to be present in the simulation. The high end of the kinetic energy spectra is still statistically limited, but is expected to continue downwards.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The predicted muon fluxes based on averaging the 2 m × 2 m sample plane at the FPF location are given in table 5. These are integrated across all kinetic energies in the simulation (E_k ⩾ 10 GeV).

Using the trajectory information stored in ROOT-format BDSIM output files, we can visualize the origin of muons that reach the FPF 2 m × 2 m sample plane. The origins are shown in figure 40 as a function of the global Cartesian coordinate Z. The distribution shows that the majority of muons originate from before the accelerator begins to curve at approximately 350 m, and there appears to be significant structure. Many of the peaks (e.g., from Z = 100 m to Z = 250 m) are explained by the TAXN (the absorber at the two-beam separation point), as well as by subsequent collimators that are designed to protect the machine from the high-power forward physics debris. Secondary particles can in turn produce muons that may reach the FPF from these locations. The two broad peaks are initially believed to be from the increasing dispersion (transverse position-energy correlation) in the accelerator as we enter the arcs. The increased dispersion causes greater transverse excursions for off-energy particles resulting in their loss and interaction with the accelerator, eventually producing muons. These results provide an initial insight into where muons originate from and how they can be mitigated with the choice of sweeper magnet location.

Zoom In Zoom Out Reset image size

The model presented is a first step in modelling the FPF muon backgrounds and further improvements are required. Firstly, a simplified geometry based on the aperture can be exported for a RIVET module suitable for testing new and existing generators and their models. This method and tool is described in reference [67]. Such a technique allows rapid iteration to test key quantities and responses of potential experiments to different physics models. The MC sample here required $\sim 80\,000$ cpu-hours to be generated, despite the decay cross section biasing. Whilst the simulation data provided is highly useful in understanding backgrounds, the approach using RIVET by reference [67] using a simplified geometry from this model will allow many more variations to be studied.

Secondly, the geometry can be improved further to include the collimator vacuum tanks and cooling apparatus. The BDSIM-provided LHC magnet geometries will also be improved. The magnetic fields used in the yokes of the magnets are parameterized multipole fields according to the Biot–Savart law. However, more accurate field maps will be used in the future. Additional systematic errors can be investigated, including the varied proton beam optical parameters (e.g., the crossing angle) and the effect of the rock density uncertainty.

Here, only the contribution from collisions at IP1 is presented. The contribution from the inelastic interaction of protons with residual gas in the vacuum can be simulated, as can any physics debris from other IPs that can cause proton losses, albeit at a low rate, throughout the whole accelerator.

The model presented and BDSIM in general present a method that can be used as part of a physics analysis and MC chain to predict quantities at the FPF, as well as estimate the impact of design choices including a sweeper magnet.

Early feasibility studies carried out for the FASER experiment showed that we expect a significant rate of high-energy muons travelling along the LOS; see section 2.6.4 and reference [10]. The rate on the LOS is reduced due to the LHC magnets, particularly the separation/re-combination dipoles, D1 and D2. FLUKA simulations and in situ measurements show that the direction of the muons is consistent with following the LOS from the IP, which means that a sweeper magnet should be placed on the LOS in between the IP and the FPF. The LOS is inside the LHC beam pipe for the long straight section of the LHC machine, and then inside the LHC magnet cryostat volume for the first part of the arc. However, after about 350 m from the IP, the LOS leaves the cryostat and remains in the LHC tunnel for about a further 50 m. This is the location where a sweeper magnet could be installed, and it is shown in figure 41.

Zoom In Zoom Out Reset image size

A preliminary integration study, using the 3D model of the LHC machine in this region, was carried out. This suggested that there would be space for a 7 m-long magnet, of transverse size 20 cm diameter, to be placed on the LOS. This magnet would be about 200 m from the FPF, providing a significant lever arm for the deflected muons to move away from the LOS before they get to the FPF. Figure 42 shows the plan view and side view of the magnet compared to the existing LHC infrastructure from this integration study. The magnet is very close to the QRL cryogenic line, which is a delicate piece of equipment in the LHC tunnel. The study does not take into account the support of the magnet, or the handling equipment needed to install and remove it from this location.

Zoom In Zoom Out Reset image size

Following this study, a laser scan was carried out in this area of the LHC tunnel to validate the 3D integration model used in the integration study. Unfortunately, this revealed a number of components that were not included in the original 3D model used. These components significantly reduce the available space for the sweeper magnet, assuming no modifications to the infrastructure are made. In this case the magnet length would be limited to about 2 m, as shown in figure 43. We are currently investigating if some of the infrastructure in this area of the LHC could be minimally modified, to allow the 7 m magnet to be installed there.

Zoom In Zoom Out Reset image size

The location of the sweeper magnet in the LHC arc has significant radiation when the LHC is operating. It is therefore difficult to reliably operate a power converter in this region, which suggests that a permanent magnet could be a desirable solution for the sweeper magnet. A conceptual design for a possible permanent sweeper magnet is shown in figure 44. The magnet is made up of a central block of permanent magnetic material (e.g., NdFeb or SmCo) of 10 cm by 10 cm transverse dimensions, surrounded by an outer ring of construction steel (making up a 20 cm by 20 cm transverse size magnet). The magnetic field distribution for this configuration is shown in figure 45 with a central field of 1.1 T for a SmCo magnet or 1.4 T for a NdFeb magnet ²³⁷ . The radial stray magnetic field is negligible at the level of $<$0.002 T at 10 cm from the magnet, which is not expected to be problematic for the LHC beam or any equipment in the LHC tunnel. Assuming a field integral of 7 Tm for the sweeper magnet, this would sweep a 100 GeV (500 GeV) muon 4.2 m (0.8 m) from the LOS at the FPF (200 m away). However, it still needs to be studied with simulation if the magnet would also sweep muons into the LOS, and if so what the overall reduction of the background along the LOS would be.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For the conceptual design mentioned above, a 7 m-long magnet would weigh 2.3 tonnes and would likely be constructed and installed in 1 m-long sections. The cost of such a magnet (not taking into account the support and handling infrastructure, nor any modifications that would need to be made to the tunnel infrastructure to create room for the magnet to be installed) is about 150 kCHF.

The existing FASER experiment is already set to probe new parameter space in the search for BSM physics. However, the overall size of FASER, and therefore its possible decay volume, has been heavily constrained, ever since the initial stages of planning, by the available space underground. This directly affects the sensitivity and reach obtainable by FASER, as, for many representative BSM models, the sensitivity is directly related to the length and radius of the decay volume. This strongly motivates the case for an enlarged detector, FASER2, which was already explored in the original FASER proposal [6], the FASER letter of intent [7], technical proposal [10], and physics reach [9] documents.

In previous studies, the nominal FASER2 design is comprised of a decay volume 5 m in length and 2 m in diameter. This results in an angular acceptance of neutral pions that increases from 0.6% in FASER to 10% in FASER2, as shown in figure 5 (left) of reference [9]. In addition, there is a significant improvement in sensitivities to LLPs produced in the decays of heavy mesons, due to the additional acceptance to B-meson production, as shown in figure 5 (right) of reference [9]. The larger decay volume also improves sensitivity to larger LLP masses and longer LLP lifetimes. The combined effect of all these factors is an improvement in reach of 4 orders of magnitude for some models.

There are several key design considerations for FASER2. The larger radius puts less emphasis on the importance of being directly on-axis. The significant improvement in sensitivity to higher mass LLPs has the consequence of exposing FASER2 to a more complicated mixture of decay channels; this strongly motivates the need for particle identification capabilities, for example, between electrons, muons, pions, and kaons. In addition, the factor of 10 increase in decay volume radius corresponds to a factor of 100 increase in area that needs to be instrumented. It therefore becomes much more challenging to accommodate an extended version of the ATLAS SCT tracker module configuration, due to cost considerations and the services required. However, the marked increase in detector length of FASER2 creates the potential to achieve larger decay product separations with different and possibly cheaper technology. The overall increase in detector size will also lead to a larger background rate, which is likely to require more complicated trigger and data analysis techniques.

Given these considerations there is much to be studied in terms of possible detector configurations and technologies. So far studies have focused on general size/layout optimizations. Several possibilities for decay volume sizes and locations have been considered, as shown in table 6. These are based on the constraints imposed from the FPF facility scenarios, which we will refer as the ‘alcoves’ and ‘new cavern’ options.

Figure 46 shows the sensitivity to dark photon (left) and dark Higgs boson (right) models for a selection of the possible FASER2 scenarios shown in table 6. The sensitivity contours have been determined using the FORESEE tool [73], described in section 4.1. As already discussed, the sensitivity to dark photons in the original FASER2 configuration is significantly improved with respect to FASER. However, the alcoves option does not allow for such a large detector, and the figure shows a significant loss of sensitivity with respect to the original configuration. On the other hand, the new cavern option is able to accommodate a detector that can recover and even improve upon the original FASER2 sensitivity, making it the strongly preferred scenario. The only downside to the new cavern scenario is the slight shift in sensitivity to lower couplings, resulting from its increased distance from the ATLAS IP, but this is a rather small effect. Similar conclusions can be drawn for the dark Higgs boson sensitivity, where the effect of the increased radius is even stronger, because of the enhancement in acceptance of B-meson decays, as already discussed.

Zoom In Zoom Out Reset image size

The FASER2 design can be optimized for either the new cavern or alcoves option. The design is not yet strictly defined, but it will be similar to FASER in the general philosophy, modulo changes needed to ameliorate some of the additional challenges already described. A schematic layout of the FASER2 detector, assuming the baseline new cavern option, is given in figure 47. The veto system will be scintillator-based, similar to FASER. The significantly increased area of the active volume makes it impractical to use silicon tracker technology. A silicon-photomultiplier (SiPM) and scintillating fiber tracker technology, such as LHCb’s SciFi detector [74], is a strong candidate to replace the ATLAS SCT modules used in FASER. In addition, monitored drift tube technology, similar to that used in the ATLAS New Small Wheel [75], is also being considered, although this option requires the use of gases in the LHC tunnel that could be problematic for the UJ12 alcoves scenario.

Zoom In Zoom Out Reset image size

For the magnets, superconducting technology would be required to maintain sufficient field strength across the much larger aperture. Suitable technology for this already exists and can be built for FASER2. There are several possibilities for the cooling of such magnets; the use of cryocoolers and the possibility to share a single cryostat across several magnets are being considered. The calorimeter needs to have sufficient spatial resolution to be able to identify particles at $\sim 1\text{\mbox{--}}10$ mm separation; good energy resolution; improved longitudinal separation with respect to FASER; and the capability to perform particle identification, separating, for example, electron and pions. Dual readout calorimetry [76, 77] is a good candidate to satisfy all these requirements. Finally, the ability to identify separately electrons and muons would be very important for signal characterization, background suppression, and for the interface with FASERν2 (see section 3.2) and other detectors. To achieve this, a mass of iron will be placed after the calorimeter, with sufficient depth to absorb pions and other hadrons, followed by a detector for muon identification.

The FASER2 design requirements that must be defined with highest priority are the needs of the spectrometer, both in terms of the magnetic field strength and the tracker resolution. Studies are underway to investigate the characteristic particle separations, starting with dark photons that decay to e⁺ e⁻ pairs. Figure 48 shows a schematic of the Geant4 [52] simulations for the alcoves and new cavern scenarios. In each case, tracker station positions are indicated by vertical dashed lines.

Zoom In Zoom Out Reset image size

In figure 49 (left), the solid contours show the distribution of transverse spatial separation between the e⁺ and e⁻ tracks resulting from a 100 MeV dark photon decaying to e⁺ e⁻ pairs in a 1 T magnetic field. With this magnetic field strength, the track separations are approximately 5 mm at station 1 and 20 mm at station 3 (and at the calorimeter). These lengths determine the spatial resolutions that would be needed to reliably identify separate decay products.

Zoom In Zoom Out Reset image size

Considering the same signal model, and taking into account the almost twice as strong magnetic field assumed here, the particle separations are much larger than might naively be expected based on what was observed in FASER [7] for a similar longitudinal detector layout. This is explained by figure 49 (right), which shows that with a larger-radius decay volume, there is much more acceptance for lower energy dark photons, whose decay products are then easier to separate with a given magnetic field. The dashed contours in figure 49 (left) show the distribution of spatial separations for the e⁺ and e⁻ tracks for the subset of dark photons that are produced in the decay volume of the same transverse size as FASER, that is, within a decay volume radius of R_DV = 10 cm from the LOS. In this case, it can be seen that the particle separations are significantly reduced and more in line with what might be expected from a naive extrapolation from FASER.

In figure 50 (left), the same distributions of transverse track separations are shown, but for the new cavern detector. For the same 1 T magnetic field, the significantly increased detector length results in large separations even at the first station. A tracker resolution of 50 mm would be very efficient for station 1 and much more coarse resolutions would be required for stations 2 and 3 and for the calorimeter. With such large separations it is worth investigating whether a magnet is needed to obtain sufficiently large separations. In the same figure, the dashed lines show the separations without any magnetic field and even then they are comparable to the alcove scenario. However, this is obviously not representative of all energy ranges. Figure 50 (right) shows the same information but for dark photons with an energy of 1 TeV. Here, even with a 1 T magnetic field, the resolutions needed for good track separation are approximately 10 mm at station 1 and approximately 100 mm at station 3 and at the calorimeter.

Zoom In Zoom Out Reset image size

However, the most important consideration is the impact of the different particle separation efficiencies on the sensitivity of FASER2. There is a non-trivial interplay between the LLP mass, coupling, decay product separation, and reach, and so this requires dedicated study.

The effect of different separation cuts on the reach for the new cavern scenario is shown in figure 51 for station 1 (left) and station 3/calorimeter (right). For a requirement of 10 mm separation, the reduction in sensitivity is small enough that the tracker technologies under consideration would be more than sufficient. The particle separations are also large enough at the calorimeter that even a relatively coarse granularity could be sufficient, given the field strength of the proposed magnets.

Zoom In Zoom Out Reset image size

To conclude, the physics potential of a larger-scale successor to FASER is clear. Possible scenarios for this larger detector are being explored and initial studies strongly indicate a preference for a FPF with a dedicated new cavern. Much progress has been made on defining the possible FASER detector designs and identifying detector technologies. Several studies are still required to finalize the design boundaries of FASER2, such as understanding the physics needs and possible detector performance capabilities for LLP mass and pointing reconstruction and particle identification.

As a component of the existing FASER experiment, the FASERν neutrino detector [10, 11] was designed to detect collider neutrinos for the first time and examine their properties at TeV energies. In 2021, the FASER collaboration reported the detection of the first neutrino interaction candidates at the LHC in a 29 kg pilot detector exposed in 2018 [5]. Starting in 2022 at LHC Run 3, FASERν, with a total tungsten target mass of 1.1 tonnes, will measure about 10 000 flavor-tagged, CC neutrino interactions, opening up the new field of neutrino physics at colliders.

So far, no cross section data are available at the TeV-energy scale. For muon neutrinos, the FASERν measurements will probe the gap between current accelerator measurements (E_ν < 360 GeV) [78] and existing IceCube data (E_ν > 6.3 TeV) [79]. For electron and tau neutrinos, the FASERν cross section measurements will extend existing data to much higher energies. In addition to CC interactions, neutral-current (NC) interactions can be measured. Such measurements can provide a new limit on the non-standard interactions (NSIs) of neutrinos to complement existing limits [80]. Furthermore, as discussed in section 6, forward hadron production, which is poorly constrained by other LHC experiments, can be investigated using FASERν. In addition, uncertainties in forward charm production limit the clarification of the atmospheric neutrino background to astrophysical neutrino observations using neutrino telescopes; see section 8. FASERν measurements of high-energy electron neutrinos, which mainly originate from charm decays, can provide the first data on high-energy and large-rapidity charm production, providing vital input from a controlled environment for astrophysical neutrino observations.

In LHC Run 3, FASERν will detect of $\sim \!\mathcal{O}(10)$ tau neutrinos and anti-tau neutrinos. This will be a welcome supplement to the small number of tau neutrinos that have been detected so far, but will be insufficient to probe tau neutrino properties in detail. Given the status of the tau neutrino as the least well-studied SM particle, there is strong motivation to study it more thoroughly with measurements that may be included among the precision probes of the general flavor structure of quarks and leptons.

The FASERν2 detector is designed as a much larger successor to FASERν to greatly extend the physics potential for tau neutrino studies. It will be an emulsion-based detector able to identify heavy flavor particles produced in neutrino interactions, including τ leptons and charm and beauty particles. In the HL-LHC era, FASERν2 will be able to perform precision tau neutrino measurements and heavy flavor physics studies, eventually testing lepton universality in neutrino scattering and new physics effects. Furthermore, FASERν2 can provide extraordinary opportunities for a broad range of physics studies, with additional and important implications for QCD, neutrino physics, and astroparticle physics, as described in sections 6 to 8.

To detect and distinguish the various neutrino flavors, the requirements for the FASERν2 detector are as follows:

On-axis. The detector should be centered along the beam collision axis to maximize the neutrino interaction event rate of all three flavors (ν_e , ν_μ , ν_τ ), and also to maximize the detected neutrino energies.

Large mass. The detector should have a high-density and large-mass target to maximize the neutrino interaction event rate within the space constraints of the underground cavern.

Flavor sensitivity. The detector should be able to identify different lepton flavors: sufficient target material to identify muons; finely sampled detection layers to identify electrons and to distinguish them from gamma rays; and good position and angular resolutions to detect tau and charm decays.

Energy reconstruction. The detector should be able to measure muon and hadron momenta and the energy of EM showers, and be able to estimate neutrino energy.

Tau-neutrino/anti-tau neutrino separation. To distinguish ν_τ and ${\bar{\nu }}_{\tau }$ interactions in the case of tau decays to muons, charge measurement of muons is required. A global analysis that combines information from FASERν2 with the FASER2 spectrometer, described in section 3.1, with the help of an interface detector, is required.

Figure 52 shows a view of the FASERν2 detector. Its ideal location is in front of the FASER2 spectrometer along the beam collision axis. The FASERν2 detector is envisioned to be composed of 3300 emulsion layers interleaved with 2 mm-thick tungsten plates. The total volume of the tungsten target is 40 cm × 40 cm × 6.6 m, and the mass is 20 tonnes. The FASERν2 detector will also include a veto detector and interface detectors to the FASER2 spectrometer, with one interface detector in the middle of the emulsion modules and the other detector downstream of the emulsion modules to make the global analysis and muon charge measurement possible. Similar to FASER2, the veto system will be scintillator-based, and the interface detectors could be based on the SiPM and scintillating fiber tracker technology. The detector length, including the emulsion films and interface detectors, will be ∼8 m. Both the emulsion modules and interface detectors will be situated in a cooling system (not drawn).

Zoom In Zoom Out Reset image size

Tau neutrino CC interactions produce τ leptons, which have one-prong decays 85% of the time. Figure 53 shows the distribution of these events in the (τ flight length, kink angle)-plane [11]. The mean τ flight length is 3 cm. To detect a kink, the τ must cross at least one emulsion layer. In addition, the kink angle should be larger than four times the angular resolution and more than 0.5 mrad, and the flight length should be less than 6 cm, where the last requirement is implemented to reduce hadronic backgrounds. Figure 54 shows event displays of a simulated ν_τ event in FASERν and FASER. The ν_τ interacts and the tau decays into a muon in FASERν, and the muon then passes through the FASER spectrometer. Similar events are expected in FASERν2 and FASER2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The high muon background in the LHC tunnel might be an experimental limitation. The emulsion detector readout and reconstruction work for track densities up to $\sim 1{0}^{6}$ tracks/cm². To keep the detector occupancy low, the possibility of sweeping away such muons with a magnetic field placed upstream of the detector is currently being explored, as described in section 2.10. Considering the expected performance, the emulsion films will be replaced every year during the winter stops.

The emulsion sensitive layers consist of silver bromide micro-crystals, which are semiconductors with a band gap of 2.684 eV, dispersed in a gelatin substrate. The diameter of the crystals which will be used for FASERν2 will be approximately 200 nm. An emulsion detector with 200 nm crystals has a spatial resolution of 50 nm. The two-dimensional intrinsic angular resolution of a double-sided emulsion film with 200 nm-diameter crystals and a base thickness of 210 μm is therefore 0.35 mrad. More details on the emulsion technology are summarized in [81].

The emulsion gel and film production will be performed at a large-scale production facility established at Nagoya University. The sensitivity of the emulsion layers was examined by exposing the produced emulsion to several tens of MeV electrons, measuring ∼45 grains per 100 μm along with the trajectory of particles. This sensitivity is sufficient for detecting minimum ionizing particles by setting an emulsion thickness of larger than 50 μm. The emulsion gel produced can then be used to produce films with two 65 μm emulsion layers deposited on both sides of a 210 μm plastic base by using the coating system shown in figure 55. The capability of the facility to produce emulsion films can reach $\sim 2000\enspace {\text{m}}^{2}$ per year. It is possible to produce 3300 emulsion films, corresponding to a total of $\sim 530\enspace {\text{m}}^{2}$ of emulsion films, for FASERν2 every year. The production of emulsion gel and films will be scheduled half a year before each installation.

Zoom In Zoom Out Reset image size

Analyses of the data collected in the emulsion modules will be based on the readout of the full emulsion volume using the hyper track selector (HTS) system [82]. The readout speed of the HTS system is 0.45 m²/hour/layer, which is a big leap from previous generations. Currently, an upgraded system HTS2, which is about five times faster, is under commissioning and a further upgraded system HTS3 with about 10 m²/hour/layer is being developed. The readout speeds of these systems are summarized in table 7, and a photo of the HTS system is shown in figure 56. The total emulsion film surface to be analyzed in FASERν2 is ∼530 m² yr⁻¹, implying a readout time of ∼2400 h yr⁻¹ with HTS or ∼420 h yr⁻¹ with HTS2. It will be possible to finish reading the data taken in each year within a year using either of the above systems.

Zoom In Zoom Out Reset image size

Reconstruction of the emulsion data will enable the localization of neutrino interaction vertices, identification of muons, the measurement of charged particle momenta by the multiple Coulomb scattering method, and the energy measurement of EM showers. Additionally, with the FASER2 spectrometer using the interface detectors, the charges of muons will be identified.

In the HL-LHC, given the 20 times luminosity and 20 times target mass of FASERν, FASERν2 will collect two orders of magnitude higher statistics than FASERν, allowing precision measurements of neutrino properties for all three flavors. For tau neutrinos, $\sim 2300$ $(\sim 20\,000)$ ν_τ CC interactions are expected, using the event generator Sibyll-2.3d (DPMJet 3.2017), as shown in table 8. As for the uncertainty on the tau neutrino flux, forward charm production is poorly constrained by other experiments, but it can be studied by measuring electron neutrino interactions in FASERν2. Roughly 178k (668k) ν_e CC interactions are expected in FASERν2, using the event generator Sibyll 2.3d (DPMJet 3.2017). Electron neutrinos at high energies above ∼500 GeV, which mainly originate from charm decays, can constrain forward charm production. The major remaining uncertainty could be at the 20% level from the dependency on the charm species, which other experiments or theoretical predictions might further constrain.

Table 8. The expected number of neutrino interactions obtained using two different event generators, Sibyll-2.3d and DPMJet 3.2017, for FASERν [67] and FASERν2 [37].

	Generator	νe +${\bar{\nu }}_{e}$ CC	νμ +${\bar{\nu }}_{\mu }$ CC	ντ +${\bar{\nu }}_{\tau }$ CC
FASERν	Sibyll	0.9k	4.8k	15
	DPMJet	3.5k	7.1k	97
FASERν2	Sibyll	178k	943k	2.3k
	DPMJet	668k	1400k	20k

The Advanced SND project is meant to extend the physics case of the SND@LHC experiment [13]. It will consist of two detectors: one placed in the same η region as SND@LHC, i.e. 7.2 < η < 8.4, hereafter called FAR, and the other one in the region 4 < η < 5, hereafter denoted NEAR. In the first part of this section, we review the way the physics case would be extended, and in the second part, we describe the detector design and layout in more detail. These two detectors are meant to operate during Run 4 of the LHC (i.e., HL-LHC) and beyond. The FPF would host the FAR detector. The NEAR detector, given the higher average angle, would have to be placed more upstream to get a sizeable azimuth angle coverage. Note that the extension of the physics case covered here is related to neutrinos and, more generally, to SM physics, while the BSM physics case is described in section 4.

QCD measurements. Electron neutrinos in the pseudorapidity range of SND@LHC, 7.2 < η < 8.4, are mostly produced by charm decays. Therefore, ν_e s can be used as a probe of charm production in an angular range where the charm yield has a large uncertainty, to a large extent coming from the gluon PDF; see section 6. Electron neutrino measurements can thus constrain the uncertainty on the gluon PDF in the very small (below 10⁻⁵) x region. The interest therein is two-fold: first, the gluon PDF in this x domain will be relevant for Future Circular Collider (FCC) detectors; and second, the measurement will reduce the uncertainty on the flux of very-high-energy (100 PeV) atmospheric neutrinos produced in charm decays, an essential input to the study of neutrinos from astrophysical sources, as discussed in section 8. The charm measurement by SND@LHC in Run 3 will be affected by a systematic uncertainty at the level of 30% and by a statistical uncertainty of 5%. The large systematic uncertainty mostly comes from the procedure linking neutrinos to charm. To reduce this uncertainty, the NEAR detector of AdvSND comes into play, since the charm yield was measured with high precision by LHCb in the 4.0 < η < 4.5 region [83]. The comparison between neutrino measurements and LHCb direct charm measurements will reduce the systematic uncertainties, with the goal of bringing them down to the level of the statistical one.

The operation in Run 4 of the FAR detector will reduce the statistical uncertainty below 1%, as it is clear from table 9. Figure 57 shows the ratio between charm measurements in different η regions normalized to the LHCb measurement: the gluon PDF uncertainty provides the largest contribution. AdvSND will measure charm in two η regions: in the 7.2 < η < 8.4 region with the FAR detector and in the 4.5 < η < 5 range with the NEAR one, where the relative contribution of PDF uncertainty is even larger.

Table 9. The number of neutrinos passing through the FAR detector of AdvSND and the number of CC neutrino interactions in the detector target, assuming 3000 fb⁻¹, as estimated with the Pythia 8 generator.

AdvSND-FAR
Flavor	ν in acceptance		CC DIS
	HardQCD: $c\bar{c}$	HardQCD: $b\bar{b}$	HardQCD: $c\bar{c}$	HardQCD: $b\bar{b}$
νμ + ${\bar{\nu }}_{\mu }$	6.3 × 1012	1.5 × 1011	1.2 × 104	200
νe + ${\bar{\nu }}_{e}$	6.7 × 1012	1.7 × 1011	1.2 × 104	220
ντ + ${\bar{\nu }}_{\tau }$	7.1 × 1011	4.7 × 1010	880	40
Tot	1.4 × 1013		2.5 × 104

Zoom In Zoom Out Reset image size

Neutrino cross section measurements. Figure 58 shows the scatter plots of neutrino energy versus η for neutrinos originating from b and c and from W decays. Neutrinos from leptonic W decays are seen to be kinematically well separated [84]. Note that LHCb has measured charm, beauty, and W production cross sections in the 2 < η < 5 range: 1.5 nb for W, 144 μb for beauty, and 8.6 mb for charm [83]. The W measurement was carried out at 7 TeV while the other two were done at 13 TeV. Accounting for all that, and considering the case of tau neutrinos, which shows a low branching ratio in charm decays (c → ν_τ ∼ 5 × 10⁻³), we expect a factor 10⁵ more charm-induced than W and Z-induced ν_τ s. The role of W and Z decays is therefore marginal in this context and we focus on charm and beauty in the following.

Zoom In Zoom Out Reset image size

Figure 59 shows the neutrino energy spectra for the two η regions, separately for the different neutrino parents. The energy spectra of charm and beauty-induced neutrinos is much softer in the NEAR location, as expected.

Zoom In Zoom Out Reset image size

The large uncertainty on the charm-induced neutrino flux in the large η region prevents SND@LHC from making a neutrino cross section measurement. AdvSND will instead be able to perform this measurement with the NEAR detector, since the neutrino flux from charm and beauty in the 4.0 < η < 4.5 region is very reliable, given the measurements performed by LHCb [83]. This will lead to a neutrino cross section measurement with very small systematic uncertainties of all three neutrino flavors, including tau neutrinos. The expected number of events in the NEAR detector is given in table 10. The lower average energy of neutrinos in the NEAR location results in a lower neutrino cross section, which explains the differences between the neutrino yields in the two detectors, despite the similar flux.

Table 10. The number of neutrinos passing through the NEAR detector of AdvSND and the number of CC neutrino interactions in the detector target, assuming 3000 fb⁻¹, as estimated with the Pythia 8 generator.

AdvSND-NEAR
Flavor	ν in acceptance		CC DIS
	HardQCD: $c\bar{c}$	HardQCD: $b\bar{b}$	HardQCD: $c\bar{c}$	HardQCD: $b\bar{b}$
νμ + ${\bar{\nu }}_{\mu }$	2.1 × 1012	3.3 × 1011	980	200
νe + ${\bar{\nu }}_{e}$	2.2 × 1012	3.3 × 1011	1000	200
ντ + ${\bar{\nu }}_{\tau }$	2.7 × 1011	1.4 × 1011	80	50
Tot	5.4 × 1012		2.5 × 103

Thus, one expects the leading uncertainty to be the statistical one: a few percent for electron and muon neutrinos and about 10% for tau neutrinos as one can derive from table 10. Notice that the yield of muon neutrinos from π and K decays is not included in this table.

Lepton flavor universality with neutrino interactions. In the 7.2 < η < 8.4 region, electron and tau neutrinos come essentially only from charm decays. Therefore, the uncertainty on the flux cancels out in the ratio, which can then be used to test lepton flavor universality with neutrino interactions. The corresponding measurement by SND@LHC is dominated by a 30% statistical uncertainty due to the poor ν_τ statistics. AdvSND will reduce the statistical uncertainty down to less than 5%; see table 9. At this point, the systematic uncertainty due to the charm quark hadronization fraction into D_s mesons, ${f\!}_{{D}_{s}}$, would be leading. This would turn into a measurement of lepton flavor universality at the 20% level.

More constraints on this ratio could come from the NEAR detector, where all charmed hadron species, including D_s , have been identified by the LHCb collaboration. Given the expected number of electron and tau neutrino interactions reported in table 10, lepton flavor universality with electron and tau neutrinos could be tested with an accuracy of 10%.

Lepton flavor universality can also be tested with the electron to muon neutrino ratio. In this case, charm can be considered also as the source of muon neutrinos if an energy cut is applied. A tentative value of 600 GeV is assumed as the energy threshold in the forward location with 7.2 < η < 8.4, and SND@LHC will have a 10% accuracy in this ratio for both systematic and statistical uncertainty. With the FAR detector operating at the FPF, AdvSND can reduce the statistical uncertainty down to a few percent, and the accuracy will be limited by the systematic uncertainty. It has to be underlined that the systematic uncertainty will have to be re-evaluated in this region in the HL regime.

The NEAR detector would have a much smaller systematic uncertainty and therefore the reach is driven by the statistical uncertainty, at the level of a few percent.

Both detectors will be made of three elements. The upstream one is the target region for the vertex reconstruction and the EM energy measurement with a calorimetric approach. It will be followed downstream by a hadronic calorimeter and a muon identification system. The third and most downstream element will be a magnet for the muon charge and momentum measurement, thus allowing for neutrino/anti-neutrino separation for muon neutrinos and for tau neutrinos in the muonic decay channel of the τ lepton.

The target will be made of thin sensitive layers interleaved with tungsten plates, for a total mass of $\sim 5$ tons. The use of nuclear emulsion at the HL-LHC is prohibitive due to the very HL that would make the replacement rate of the target incompatible with technical stops. As discussed in section 2.10, this motivates the study of a sweeper magnet to reduce the muon background. As an alternative approach, the collaboration is investigating the use of compact electronic trackers with high spatial resolution fulfilling both tasks of vertex reconstruction with micrometer accuracy and EM energy measurement. The hadronic calorimeter and the muon identification system will be about 10 λ, which will bring the average length of the hadronic calorimeter to about 12 λ, thus improving the muon identification efficiency and energy resolution. The magnetic field strength is assumed to be about 1 T over about 2 m length. A schematic view of the detector is shown in figure 60.

Zoom In Zoom Out Reset image size

The magnet is a key element in the detector design, because it makes it possible to distinguish ν_μ from ${\bar{\nu }}_{\mu }$ and ν_τ from ${\bar{\nu }}_{\tau }$ when the resulting tau lepton decays to a muon. Figure 61 shows the momentum spectrum of muons induced by the CC interactions of muon neutrinos in the 7.2 < η < 8.4 region. The layout of a spectrometer measuring the bending angle of a track is shown in figure 62 with all the relevant parameters. We describe in the following the design of this kind of spectrometer for AdvSND.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

After traversing a magnet with field strength B and length ℓ, the track of a charged particle with momentum p is bent by an angle $\theta =\frac{\ell }{r}=\frac{eB\ell }{p}$. The bending angle θ is determined by two tracking planes that are located before the magnet and separated by the lever arm a, which measure the track coordinates x₁ and x₂, and two tracking planes that are located behind the magnet and also separated by the lever arm a, measuring the track coordinates x₃ and x₄. From the track coordinates the bending angle is determined to be

$$\begin{equation}\theta =\frac{{x}_{4}-{x}_{3}}{a}-\frac{{x}_{2}-{x}_{1}}{a}.\end{equation} \tag{ 3.1 }$$

Denoting the measurement error of a track coordinate for a tracking station by , the error on the bending angle measurement is ${\Delta}\theta =\frac{2{\epsilon}}{a}$. Hence, the momentum resolution of the spectrometer is

$$\begin{equation}\frac{{\Delta}p}{p}\approx \frac{{\Delta}\theta }{\theta }=\frac{2{\epsilon}p}{eB\ell a}.\end{equation} \tag{ 3.2 }$$

For a given total length L of the spectrometer the choice of the length ℓ of the magnet and of the lever arm a, which results in the best momentum resolution, is defined by $a=\frac{L}{4}=\frac{\ell }{2}$. For the muon charge assignment, the maximum muon momentum, p_max, for which a charge assignment is possible, can be defined by requiring at least a 4σ separation from infinite momentum, corresponding to a momentum resolution of

$$\begin{equation}\frac{{\Delta}p}{p}=\frac{1}{4}\quad \text{for}\enspace p={p}_{\mathrm{max}},\end{equation} \tag{ 3.3 }$$

Thus, the maximum momentum, up to which a muon charge assignment is possible, is obtained:

$$\begin{equation}{p}_{\mathrm{max}}=\frac{eB\ell a}{8{\epsilon}}.\end{equation} \tag{ 3.4 }$$

Assuming a magnetic field of B = 1 T, a spatial resolution of the tracking chambers of = 100 μm, ℓ = 2 m, and a = 1 m, the spectrometer allows for a charge assignment up to 750 GeV/c, thus covering 95% of the momentum spectrum. The overall length of the spectrometer is 4 m.

Table 11 summarizes the main parameters of the two locations and the corresponding detectors.

Because of the spatial, energy, and time resolution of the LArTPC, it is well motivated by the requirements of neutrino detection and the light DM search [87–89]. In particular, the TPC is an excellent choice for the detection and measurement of energetic EM showers. Single muon tracks as well as showers of hadronic tracks also benefit from the superb spatial and charge resolution of this detector. The detector has no insensitive mass and therefore the energy loss and scattering can be measured along a long track. This capability leads to superb particle identification at momenta of ∼1 GeV and also to excellent momentum resolution for high energy muons. The kinematic resolutions in angle and momentum and how they affect various backgrounds for neutrino physics at the TeV scale needs further study.

The detector is expected to measure millions of neutrino interactions, including tau neutrinos. The detector should have sufficient capability to measure these very high energy $( > 100\enspace \text{GeV})$ events, so that the cross section for each flavor can be measured. Identification of tau neutrinos with low backgrounds needs detailed simulations and reconstruction studies. As an approximate estimate, there will be about 50 high-energy neutrino events per tonne per fb⁻¹; this is approximately the integrated luminosity expected to be collected every day during the HL running of the LHC. The majority of this flux will be muon neutrinos, with electron neutrinos forming about 1/5 of the event rate. The tau neutrino rate is expected to be $\sim 0.1$ event/tonne/fb⁻¹ with a very large uncertainty due to QCD modeling in the forward direction. The high energy electron neutrino and the tau neutrino fluxes come from charm meson decays in the forward region, and therefore careful measurements of these event types has broad implications for particle physics, as described in sections 6–8.

Table 12 summarizes the main parameters of a LArTPC for the FPF. A detector with a fiducial mass of approximately 10 tonnes of LAr is envisioned. We are also considering this same detector with a filling of liquid krypton. The dimensions of the TPC are extremely preliminary and could depend considerably on event containment and the energy loss properties of LAr or LKr. For both the LAr and LKr options, the efficiency for well-measured events that happen in the downstream portion of the detector can be enhanced by the addition of a hadron calorimeter, which is included in the table.

Table 12. Detector parameters for a LArTPC for the FPF. The top part of the table shows the nominal geometric parameters for the time projection chamber to be considered for the FPF, and the bottom part shows the additional hadron and muon detectors.

	Value	Remarks
Detector length	7 m	Not including cryostat
TPC drift length	$> 0.5$ m	2 TPC volumes with HV cathode in center
TPC height	$> 1.3\enspace \text{m}$	Dimension dependent on containment
Total LAr mass	$\sim 16$ tonnes	Volume in the cryostat
Total LKr mass	$\sim 27.5$ tonnes	As an option
Fiducial mass LAr/LKr	10/17 tonnes
Charge readout	Wires or pixels	Hybrid approach is possible
Light readout	SiPM array	Needed for neutrino trigger
Background muon rate	$\sim 1$ cm− 2 s−1	Maximum luminosity of 5 × 1034 cm− 2 s−1
Neutrino event rate	$\sim 50/$tonne/fb−1	For all flavors of neutrinos
Cryostat type	Membrane 0.5 m	Thickness of membrane
Hadronic calorimeter	$\sim 6\text{\mbox{--}}10\lambda$	Interactions lengths
Dimension	2 m × 2 m × 1.6 m (depth)	Fe/Scint sandwich
Muon range	To be determined

For 3 ab⁻¹, such a detector will collect as many as millions of muon neutrino/antineutrino CC events, about a hundred thousand electron neutrino events, and thousands of tau neutrino events. These numbers have large uncertainties due to the poorly understood production cross section in the forward region [91, 92]. It is also important to note that this flux of events will have the same time structure as the LHC accelerator with a bunch spacing of 25 ns. At the same time, muons from interactions at the IP will produce a background flux of about $\sim 1$ muon/cm²/s at the nominal maximum luminosity of 5 × 10³⁴ cm⁻² s⁻¹ at the HL-LHC. As the luminosity is increased to $\sim 7.5\times 1{0}^{34}\enspace {\text{cm}}^{-2}\enspace {\text{s}}^{-1}$ in later years of the HL-LHC, the background muon flux will correspondingly increase. As discussed in section 2.10, this muon flux may be reduced by the addition of a sweeper magnet placed between the ATLAS IP and the FPF.

If the TPC can be operated with liquid krypton, several advantages are expected. The radiation length of LKr (4.7 cm) is much shorted than LAr (14 cm) leading to much more compact EM showers [93–95]. This performance naturally leads to much higher event containment for neutrino events. The higher density of LKr should also allow higher event rate. The overall increase of useful event rate is expected to be almost a factor of 2 at energies above 1 TeV. Detailed simulations of event reconstruction need to be performed, but the better resolution from LKr is expected to lead to much better performance for tau neutrinos. The choice of krypton for the TPC is, however, not simple because of intrinsic radioactivity due to ⁸⁵Kr.

The nominal configuration for the LArTPC detector would include a central cathode operating at a large high voltage and two anode planes on two sides of the detector parallel to the beam from the ATLAS IP. The electric field between the cathode and the anode will be at $\sim 500$ V cm⁻¹, providing a drift field for ionization electrons; the drift time for a 0.5 m-long drift will be about 0.3 ms. For a detector with an approximate cross section of 1 m², there will be about 3 muon tracks within a single drift time. Neutrino and dark matter events must be selected out of these overlaying background particle trajectories. For the TPC, a readout using wires or pixels is possible [96]. A readout of the scintillation light is crucial to allow the measurement of the distance along the drift. It is also important for the selection of events that originate in the detector (such as a neutrino or a dark matter event), as well as generating the trigger necessary for acquiring the data. Neutrino events need to be identified at the trigger level as events with tracks or showers that originate from a common vertex within the detector volume. As stated above, the expected event rate is $\sim 50$ per day per ton; the detector scintillation system and the trigger selection must be designed to retain high efficiency for these real events, while maintaining a low ≪1 Hz trigger rate.

For the detection of neutrino and dark matter events at the 10 ton fiducial mass scale, further simulation work is needed to understand event reconstruction and background rejection, especially for tau neutrino events. For detector design, in particular, simulation work is needed to understand neutrino event containment and energy resolution in a 7 m-long detector. Study of kinematic resolution in the case of wire readout versus pixel readout is needed. And finally, the design and performance of the photon detector system needs to be investigated and demonstrated by R&D. Lastly, a LKrTPC would have remarkable resolution for EM showers and the event containment is expected to be excellent. In either the LAr or LKr case, a downstream hadronic calorimeter and a muon range detector is needed to contain particles escaping from the TPC for energetic neutrino events. Looking further to the future, the addition of magnetic field and momentum measurement either with a downstream magnet or as part of the TPC needs to be explored.

FLArE will be deployed in an underground cavern, and so it is necessary to develop compact liquid cryogenic facilities to condense, recirculate, and filter LAr or krypton. The CERN cryogenics group with their enormous experience is expected to take the major responsibility for this facility at CERN. Targeted R&D to understand the differences between LAr and liquid krypton recirculation and purification systems is also important. In particular, the liquid krypton system will have additional requirements because of the higher mass and the need to keep losses very low. There could also be differences due to the needed purity levels.

The LArTPC will be installed in a membrane cryostat with passive insulation and with nominal inner dimensions of 2.1 × 2.1 × 8.2 m³. These cryostat dimensions allow excellent high voltage safety for the TPC dimensions in table 12 using experience from previous designs. However, the cryostat design will need to be closely coordinated with the TPC as the design changes. The membrane cryostat technology allows the cryostat to be constructed underground. The insulation, being passive, ensures reliable and safe long-term performance. The cryogenic system must re-condense the boil-off, keeping the ullage absolute pressure stable to better than 1 mbar, and purify the LAr bath. A standard approach is to re-condense the argon with a heat exchanger with liquid nitrogen. A LAr flow of 500 kg h⁻¹ through the purification circuit is considered sufficient to reach and maintain the required LAr purity.

The total heat input due to the cryostat and the cryogenics system is estimated to be of the order of few kW: $<$ few kW from the cryostat depending on the size, 1 kW from the GAr circuit, 1 kW from the LAr purification and 1 kW from other inefficiencies. To this, the detector electronics should be added. A Turbo-Brayton (8 m × 1.6 m × 2.7 m) TBF-80 unit from air liquid installed in the vicinity of the cryostat provides approximately 10 kW cooling power from ≈100 kW electrical power and 5 kg s⁻¹ of water at ambient temperature. LAr and nitrogen storage tanks are required above ground and connected via piping to the underground cryogenics. Exhaust of gasses will be done to the atmosphere on the surface. For safety reasons, as noted in section 2, the cryostat in the cavern is placed in a trench 1.5 m deep, 6.9 m wide, and 12.6 m long, which collects the argon in case of a leak. Oxygen deficiency is the main risk associated with the LArTPC. A properly designed ventilation system will constantly extract air in the proximity of the cryostat/cryogenics. A detection of low oxygen content by the ODH system in the cavern and in the trench will trigger increased air extraction. The design and technology for the cryosystem is well understood because of the experience from protoDUNE. If LKr is considered as a fill, then the requirements for the cryosystem need to be further examined in detail, but there is also experience with a LKr system at CERN from the NA48 experiment, which has run for a long time [95].

The FLArE detector concept requires a few key R&D efforts. Neutrino events at energies of 1 TeV and above from an accelerator source have not been detected, and simulations of these events require much tuning of software codes and comparison with existing data. This effort will lead to a much better understanding of kinematic resolutions and the design of a hadron calorimeter and a muon rangefinder. In particular, detailed simulations are needed to understand the signal to background ratio achievable for tau neutrinos with event shape and kinematic cuts and other software techniques. The design of the TPC and the cathode and anode readout depends strongly on the spatial and charge resolution that will be needed for particle identification. The three options are: a wire readout, a pixel readout, or a hybrid readout. The wire readout has been analyzed thoroughly [96] and is best suited for very large detectors with very low channel occupancy. On the other hand, a pixel readout could be quite expensive in terms of channel counts. A new type of readout plane is in the process of development for the DUNE vertical drift detector; this design could be appropriate for FLArE. Finally, the design and performance of the photon detector system needs to be investigated and demonstrated by R&D. In particular, the photon system has to serve three functions in order of increasing difficulty: separation of beam related muon and neutrino events, accurate timing performance to measure the location of the neutrino or dark matter events in the TPC, and association of the neutrino or dark matter events with a bunch crossing in the collider detector.

3.4.4.1. Time projection chamber design.

Figure 63 shows the preliminary conceptual design for the FLArE detector, and table 12 summarizes the main parameters. This nominal design is being implemented with Geant4 for detector response and physics reach simulations. A detector with a fiducial mass of approximately 10 tonnes (24 tonnes) of LAr (liquid krypton) is envisioned. During the HL-LHC era, more than 500 neutrino events per day are expected in the detector.

Zoom In Zoom Out Reset image size

The TPC serves to measure the ionization tracks in the liquid with close-to-ideal spatial resolution for an electronic detector. The ionization electrons are drifted over a length of 0.5 to 1 m away from the cathode plane in the center and detected by their induced currents in electrodes on the anode plane. The electrodes (wires or pixels) can be arranged according to the needed spatial resolution down to a few mm. The TPC has three essential components in the design: the high voltage cathode and the field cage, the anode and the electrode design, and the readout electronics.

The cathode and the field cage define the static electric field. High voltage must be provided using a penetration (feed-through) in the cryostat. The design of this system is now very well understood from experience from protoDUNE and ICARUS detectors. Fields of 500 volts cm⁻¹ can now easily be sustained over several meters of drift. Two important challenges that must be confronted for the design of FLArE are the availability of space above the detector for the installation of the HV feedthrough, and providing a cathode that permits scintillation light to reach both sides of the TPC. A particular issue arises for liquid krypton due to the high level of intrinsic radioactivity $(\sim 500\enspace \mathrm{b}\mathrm{q}\enspace \mathrm{c}{\mathrm{m}}^{-\mathrm{3}})$ which creates a cloud of charge that reduces the applied electric field (space charge). This problem needs careful examination and may require smaller TPC gaps or high fields. Both the LAr and krypton options require careful evaluation for space charge due to high rates of muon tracks from the LHC.

The anode and the electrode design determines the spatial resolution for measurement of ionization in 3D. A design using several (more than 3) planes of wires has been used in the protoDUNE test detector. It is also the design for ICARUS, MicroBooNE, and the first module of DUNE. The wires in each plane allow measurement of a single plane projection of the ionization image. Drifting electrons induce currents in each of the planes of wires before being collected on the final wire plane. Combining three 2D images electronically produces a 3D pattern. This technique, however, requires low track multiplicity in the events to reduce reconstruction ambiguities [96] and is otherwise subject to inefficient event recognition. Another option is being examined in the context of the DUNE second module, in which a rigid plane is employed with small holes to allow electrons to pass. Induced currents on metalized electrodes on both sides of the rigid plane (or electronic board) are used to create a projection of the image. This technique could lead to an inexpensive pixelized detector that is not subject to reconstruction ambiguities.

Typical amount of charge from a minimum ionizing particle through LAr (or krypton) is about $\sim 1\text{\mbox{--}}2\times 1{0}^{4}$ electrons for a few mm of track length. The collected charge depends on the electric field strength, purity of the liquid, and losses due to attachment over the drift distance. The liquid must have very high purity ($\sim 0.1$ ppb) to allow charge drift over 1 meter without significant attenuation. Detection of induced current pulses from this small charge requires low noise amplification of the pulses as close to the electrode as possible. Such low noise cryogenic electronics has now been fully developed and tested in the protoDUNE detector. The entire readout chain of amplification, digitization, and data transmission has been designed to work in LAr. The FLArE detector can fully utilize this technology with minimal changes regardless of the detailed design of the electrode.

3.4.4.2. Detector response and physics reach simulation.

Simulations based on various neutrino and DM event generators and Geant4 detector modeling play a critical role in understanding FLArE's response to all relevant physics processes and evaluating its physics research for neutrino studies and DM searches. Preliminary simulations have been performed to compare the development of EM showers in LAr and LKr. These results are shown in figure 64 and demonstrate LKr's remarkable resolution for EM showers and excellent event containment. Further studies will be used to optimize the experiment design and understand event reconstruction and event selection; study the kinematic resolution in the case of wire readout versus pixel readout; study the performance of the proposed photon detection system as well as alternatives; and investigate the potential benefits of a magnetic field, created either by a downstream magnet or as part of the TPC.

Zoom In Zoom Out Reset image size

3.4.4.3. Photon sensor system.

3.4.4.4. AI/machine learning-based trigger studies.

To be sensitive to the small dE/dx of a particle with Q ≲ 0.1e, an mCP detector must contain a sufficient amount of sensitive material in the longitudinal direction pointing to the IP. Plastic scintillator, such as, for example, Eljen EJ-200 [104] or Saint-Gobain BC-408 [105], provides a detection medium with the best combination of photon yield per unit length, response time, and cost. The FORMOSA detector is planned to be a 1 m × 1 m × 5 m array of plastic scintillator. The length of the bars is determined by the desire to efficiently reconstruct a particle of charge $\mathcal{O}(1{0}^{-3})$, however, this may be optimized further. No gain in sensitivity is expected using a larger area thin scintillator detector (as planned for the Run 3 milliQan detector [100]) as the reach in charge is typically below that which such a detector could efficiently reconstruct.

The array will be oriented such that the long axis points at the ATLAS IP and it will be located on the beam collision axis. The array contains four longitudinal 'layers' arranged to facilitate a four-fold coincident signal for feebly interacting particles (FIPs) originating from the ATLAS IP. Each layer in turn contains one hundred 5 cm × 5 cm × 100 cm scintillator 'bars' in a 20 × 20 array. The bars will be held in place by a steel frame. A conceptual design of the FORMOSA detector is shown in figure 65.

Zoom In Zoom Out Reset image size

Although omitted for clarity in figure 65, three additional scintillator 'panels' of 5 cm × 100 cm × 400 cm, placed on each side of the detector, will be used to actively veto cosmic muon shower and beam halo particles. Finally, scintillator panels of 5 cm × 100 cm × 100 cm will be placed on the front and back of the detector to aid in the identification of muons resulting from LHC proton collisions.

To maximize sensitivity to the smallest charges, each scintillator volume will be coupled to a high-gain PMT capable of efficiently reconstructing the waveform produced by a single photoelectron (PE). To reduce random backgrounds, mCP signal candidates will be required to have a quadruple coincidence of hits with N_PE ⩾ 1 within a small time window. The PMTs must therefore measure the timing of the scintillator photon pulse with a resolution of $\leqslant 5$ ns. Three different species of PMT were deployed as part of the milliQan demonstrator: Hamamatsu R878, Hamamatsu R7725 [106], and Electron Tube 9814B [107]. While all three species meet the minimal requirements for the FORMOSA detector, the R7725 and Electron Tube PMTs were found to have the best dark rate, timing, and response performance. The afterpulsing properties of the PMT species used will also have significant impact on the background faced by the detector. To avoid sensitivity to residual magnetic fields in the cavern, each PMT will be wrapped with mu-metal shielding. This shielding consists of two parts: one is directly around the PMT within the mount, and another thin layer is wrapped around the outside of a completed bar covering a region 2 cm on either side of the photo-cathode position.

Waveforms from each PMT will be digitized, read out, and stored for offline analysis. As the pulse rate per PMT is large, a trigger will be used to record only those waveforms during interesting time windows when signal-like activity in the detector is observed with at least three layers in a 2 × 2 × 4 bar region having a pulse above the single PE threshold. To avoid the rate being dominated by through-going muons, large pulses in the front and end panels will be vetoed in the trigger. The digitization can be performed using 25 16-channel CAEN V1743 digitizers [108], operating at 1.6 × 10⁹ samples per second with 12-bit resolution, providing 1024 samples within a 640 ns acquisition window. The 16 channels are arranged into eight trigger groups, each of which can output a trigger bit via LVDS. These trigger bits can then be combined by dedicated electronics to form the trigger decision.

Commercial CAEN HV power supply modules connected to fan-out boards will provide power for up to 12 groups of up to 12 PMTs (144 PMTs total) for each HV supply module. Therefore, three CAEN A1535DN power supply modules can be utilized to power all scintillator bars and panels.

Even though the pointing, four-layered, design will be very effective at reducing background processes, small residual contributions from sources of background that mimic the signal-like quadruple coincidence signature are expected. These include overlapping dark rate pulses, cosmic muon shower particles and beam muon afterpulses. In reference [100], data from the milliQan prototype was used to predict backgrounds from dark rate pulses and cosmic muon shower particles for a closely related detector design and location. Based on these studies, such backgrounds are expected to be negligible for FORMOSA. An additional background is expected from the large through-going muon rate of $\sim 1\enspace \mathrm{H}\mathrm{z}\enspace {\mathrm{c}\mathrm{m}}^{-2}$. Although the large pulses from the muons can easily be distinguished from the signal, the afterpulses caused by ionising in the PMTs can mimic the signal signature. As detailed in reference [99], it is expected that these can be rejected by vetoing a 10 μs time window in the detector following through-going beam muons. To more effectively track the paths of beam muons through the detector, it is possible to segment the veto panels at the front and back of the detector to form hodoscopes. This may allow areas impacted by beam muons to be identified, rather than necessitating deadtime for the full detector. In addition, input on the beam muon paths could be provided to other detectors within the FPF to allow improved background rejection. Finally, the use of a sweeper magnet (detailed in section 2.10) to lessen the rate of beam muons incident on the detector could significantly reduce the deadtime and backgrounds in the detector.

The signal process is simulated from a range of production modes, as detailed in reference [99]. This provides the expected flux for each mass and charge of the signal. This can then be used to determine the expected limits as shown in figure 66. For much of the parameter space, the sensitivity is limited by the efficiency of the scintillator bars to detect through-going mCPs. In this regime, the mCP flux is very high, and so only a small area of higher performance scintillator can allow substantial gains in sensitivity. One possibility is an upgraded design in which an additional 2 × 2 × 4 bars are installed using a higher performance scintillator, such as LaBr₃(Ce). By placing these bars close to the larger plastic scintillator, the active veto background rejection capabilities of the array for sources such as cosmic showers are maintained. Shielding of these would mitigate backgrounds for the larger detector caused by the radioactivity of the LaBr₃(Ce). In this scenario, the optimal charge reach could be lowered by as much as a factor of 5.

Zoom In Zoom Out Reset image size

Finally, the FLArE detector may also have sensitivity to mCPs. It could be possible for FORMOSA to prove ∼10 cm (40 cm) resolution tracking of mCP paths at the front (back) of FLArE. This could help FLArE provide an independent measurement of the properties of the mCP signal. The combination of measurements from multiple detectors in the FPF can provide powerful insights into both background and signal processes.

The most minimal realization of a novel U(1)_X symmetry is one where the SM remains uncharged under the new symmetry and there are no new fermions present. This translates to ${j}_{\alpha }^{X}=0$ in equation (4.1). Hence, the dark photon A' is completely secluded from the SM and the only interaction of the dark photon A' with SM fermions proceeds through kinetic mixing (see, e.g. references [141–143], for reviews of the secluded dark photon). The mixed kinetic terms in equation (4.1) can be removed by a field redefinition of the neutral gauge bosons $({B}_{\mu },{W}_{\mu }^{3},{X}_{\mu })$. Two consecutive orthogonal rotations diagnoalize the neutral gauge boson mass terms. The full transformation translating to the mass basis $({A}_{\mu },{Z}_{\mu },{A}_{\mu }^{\prime })$ then reads

$$\begin{align}\hfill \left(\begin{matrix}\hfill {B}_{\mu }\hfill \\ \hfill {W}_{\mu }^{3}\hfill \\ \hfill {X}_{\mu }\hfill \end{matrix}\right)& =\underset{G({{\epsilon}}_{Y})}{\underbrace{\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill -\frac{{{\epsilon}}_{Y}}{\sqrt{1-{{\epsilon}}_{Y}^{2}}}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\sqrt{1-{{\epsilon}}_{Y}^{2}}}\hfill \end{matrix}\right)}}\underset{{R}_{1}({\theta }_{\text{W}})}{\underbrace{\left(\begin{matrix}\hfill \mathrm{cos}\,{\theta }_{\text{W}}\hfill & \hfill -\mathrm{sin}\,{\theta }_{\text{W}}\hfill & \hfill 0\hfill \\ \hfill \mathrm{sin}\,{\theta }_{\text{W}}\hfill & \hfill \mathrm{cos}\,{\theta }_{\text{W}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right)}}\hfill \\ \hfill & \quad \times \underset{{R}_{2}(\xi )}{\underbrace{\left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathrm{cos}\,\xi \hfill & \hfill -\mathrm{sin}\,\xi \hfill \\ \hfill 0\hfill & \hfill \mathrm{sin}\,\xi \hfill & \hfill \mathrm{cos}\,\xi \hfill \end{matrix}\right)}}\left(\begin{matrix}\hfill {A}_{\mu }\hfill \\ \hfill {Z}_{\mu }\hfill \\ \hfill {A}_{\mu }^{\prime }\hfill \end{matrix}\right),\hfill \end{align} \tag{ 4.2 }$$

where A_μ denotes the SM photon, Z_μ the weak neutral gauge boson and ${A}_{\mu }^{\prime }$ the mass eigenstate of the new U(1)_X bosons, which we refer to as hidden or dark photon. Finally, couplings of the mass eigenstates of the neutral bosons $({A}_{\mu },{Z}_{\mu },{A}_{\mu }^{\prime })$ to the SM fermions are described by the interaction term of the kind,

$$\begin{equation}{\mathcal{L}}_{\text{int}}=-(e{j}_{\alpha }^{\,\text{em}},{g}_{Z}{j}_{\alpha }^{Z},{g}_{x}{j}_{\alpha }^{X})\ K\ \left(\begin{matrix}\hfill {A}^{\alpha }\hfill \\ \hfill {Z}^{\alpha }\hfill \\ \hfill {A}^{\prime \alpha }\hfill \end{matrix}\right),\end{equation} \tag{ 4.3 }$$

where the coupling matrix is given by [144]

$$\begin{equation}K={\left[{R}_{2}(\xi ){R}_{1}({\theta }_{\text{W}}){G}^{-1}({{\epsilon}}_{Y}){R}_{1}{({\theta }_{\text{W}})}^{-1}\right]}^{-1}\approx \left(\begin{matrix}\hfill 1\hfill & \hfill 0\hfill & \hfill -{\epsilon}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\epsilon}\,\mathrm{tan}\,{\theta }_{\text{W}}\hfill & \hfill 1\hfill \end{matrix}\right).\end{equation} \tag{ 4.4 }$$

Here, θ_W denotes the weak mixing angle, e the EM coupling constant, the Z-coupling constant g_Z = e/(sin θ_W cos θ_W) and ${j}_{\mu }^{\text{em}}$ and ${j}_{\mu }^{Z}$ are the EM and weak NCs. Furthermore, we have defined the physical kinetic mixing parameter as = _Y  cos θ_W.

In the domain of low-energy observables, it is sufficient to consider mixing of the new gauge boson with the photon of QED. The interaction of the mass eigenstate dark photon with the SM sector is then parametrized by equation (4.3) and reads explicitly

$$\begin{equation}{\mathcal{L}}_{\text{secl}}={\epsilon}e{j}_{\mu }^{\text{em}}{A}^{\prime \mu }.\end{equation} \tag{ 4.5 }$$

This interaction naturally suggest the name of hidden or dark photon for the new boson A' since it couples to the EM current ${j}_{\mu }^{\text{em}}$ exactly analogous to the SM photon, but suppressed by the kinetic mixing parameter .

Models of purely kinetically-mixed dark photons received a lot of attention in the literature when it was shown that such GeV-scale new mediators can help to explain the cosmic ray positron excess [145, 146]. Especially, after it had been established that such models naturally arise from weak-scale SUSY breaking [147–152], the secluded dark photon model rejoiced from an increased popularity in the literature [153–157].

A consequence of the kinetic mixing interaction in equation (4.5) is that the dark photon can be produced and searched for in a number of EM processes where a SM photon is replaced by a dark photon A'. Natural candidates to search for dark photons are e⁺ e⁻ and hadron colliders, where dark photons can be abundantly produced in Bremsstrahlung processes. For example, in e⁺ e⁻ machines the dark photon can be produced in radiative return (e⁺ e⁻ → γA') [155] or via heavy meson decays and searched for in prompt dilepeton or pion decays. These search strategies have been employed in a number of e⁺ e⁻ collider experiments in the past like, e.g., BaBar [118, 158], Belle [159, 160] and KLOE [161–164]. At hadron colliders like the LHC the dark photon can also be produced via Drell–Yan production $(q\bar{q}\to {A}^{\prime })$ or via the decay of heavy resonances, like e.g. H → ZA' [165], or heavy meson decays, like e.g. D* → DA' [129]. Searches for prompt decays of such produced dark photons have been conducted at ATLAS and CMS [165] and at LHCb [122, 129, 130, 166]. In the left panel of figure 70 these existing collider constraints are shown as gray regions in the (kinetic mixing, mass) plane for a secluded dark photon. As can be seen these experiments cover the region of ≳ 10⁻³ over a very large range of dark photon masses.

Zoom In Zoom Out Reset image size

Nevertheless, the aforementioned collider experiments lose sensitivity for light (M_A' ≲ 1 GeV) and very weakly coupled dark photons. This region, however, can be probed by searching for displaced decays of dark photons. Abundant sources of such light and long-lived dark photons are provided by electron or proton beam dump and fixed target experiments. In these experiments beams of charged particles are dumped onto a block of material where a large number of dark photons can be produced from Bremsstrahlung or secondary meson decays. These dark photons can then be searched for at a macroscopic displacement from the target material. Such searches have been conducted in the past at electron beam dump experiments like SLAC E137 and E141 [120, 121, 157, 167], Fermilab E774 [168], Orsay [169], electron fixed target experiments like APEX [170], A1/MAMI [171, 172], HPS [173], NA64 [174, 175], as well as at proton beam dumps like CHARM [176], LSND [177] and U70/Nu-Cal [178, 179], and proton fixed-target experiments, such as SINDRUM I [180] and NA48/2 [123]. The resulting constraints are also shown in the of figure 70 as gray regions.

At the future FPF, experiments like FASER2 will be able to search for such long-lived dark photons in the very forward direction of LHC proton–proton interactions. As for beam dump and fixed target experiments, dark photons can be produced via proton Bremsstrahlung in the pp collisions at LHC. The total number of dark photons produced can be computed from [6]

$$\begin{align}\hfill N& ={N}_{p}\ \vert {F}_{1}({M}_{{A}^{\prime }}^{2}){\vert }^{2}{\int }_{{M}_{{A}^{\prime }}}^{{E}_{p}-{m}_{p}}\mathrm{d}{E}_{{A}^{\prime }}\frac{1}{{E}_{p}}\frac{{\sigma }_{pp}(2{m}_{p}({E}_{p}-{E}_{{A}^{\prime }}))}{{\sigma }_{pp}(2{m}_{p}{E}_{p})}\hfill \\ \hfill & \quad \times {\int }_{0}^{{p}_{\perp ,\mathrm{max}}^{2}}{\omega }_{{A}^{\prime }p}({p}_{\perp }^{2})\mathrm{d}{p}_{\perp }^{2}{\Theta}({{\Lambda}}_{\text{QCD}}^{2}-{q}^{2}){A}_{\text{geom}}{\mathcal{P}}_{{A}^{\prime }},\hfill \end{align} \tag{ 4.6 }$$

where N_p , E_p , m_p denote the total number of proton collisions, the proton energy and mass, σ_pp denotes the proton–proton cross section, ${F}_{1}({p}_{{A}^{\prime }}^{2})$ denotes the time-like form factor of the proton from the vector meson dominance (VMD) model [181, 182], A_geom is a geometrical acceptance factor, ${\omega }_{{A}^{\prime }p}({p}_{\perp }^{2})$ is a weighting function relating the 2 → 3 process p + p → p + p + A' to p + p → p + p and ${\mathcal{P}}_{{A}^{\prime }}={\text{e}}^{-{L}_{\mathrm{min}}/{\ell }_{{A}^{\prime }}}-{\text{e}}^{-{L}_{\mathrm{max}}/{\ell }_{{A}^{\prime }}}$ is the probability that the dark photon decays within the fiducial detector volume. Details on these expressions can be found in references [6, 179].

Besides Bremsstrahlung, another important source of dark photon is provided for by secondary decays of mesons produced in the proton–proton collisions. The number of expected dark photons from meson decays can be expressed as [178]

$$\begin{equation} N=\frac{{N}_{p}}{\sigma (pp\to X)}{\int }_{0}^{1}\mathrm{d}{x}_{\text{F}}{\int }_{0}^{{p}_{\perp ,\mathrm{max}}^{2}}\,\mathrm{d}{p}_{\perp }^{2}\frac{\mathrm{d}\sigma (pp\to MX)}{\mathrm{d}{x}_{\text{F}}\,\mathrm{d}{p}_{\perp }^{2}}\text{Br}(M\to {A}^{\prime }\gamma ){A}_{\text{geom}}{\mathcal{P}}_{{A}^{\prime }},\end{equation} \tag{ 4.7 }$$

with x_F denoting the Feynman-x variable and $\mathrm{d}\sigma (pp\to MX)/(\mathrm{d}{x}_{\text{F}}\,\mathrm{d}{p}_{\perp }^{2})$ being the differential meson production cross section for the meson M. Here Br(M → A'γ) denotes the branching ratio of the meson M decaying into a photon and a dark photon, which in the case of a secluded dark photon is simply given by

$$\begin{equation}\text{Br}(M\to {A}^{\prime }\gamma )=2{{\epsilon}}^{2}{\left(1-\frac{{M}_{{A}^{\prime }}^{2}}{{M}_{M}^{2}}\right)}^{3}\text{Br}(M\to \gamma \gamma ).\end{equation} \tag{ 4.8 }$$

For dark photon production at LHC the most relevant meson decays are from π⁰, η and η' mesons.

In the left panel of figure 70 the projected sensitivities of FASER and the future FASER2 experiment located at the FPF are shown by the red solid lines. For comparison we also show the projected sensitivity of the future CERN based proton fixed target experiment SHiP [29, 183] by a dashed green line. As can be seen FASER and the future FASER2 experiments will probe regions of dark photon parameter space that has not been tested at any other beam dump or fixed target experiment. In particular, FASER2 will have a significantly increased mass reach, comparable to what can be achieved by SHiP.

The peculiar flavor structure of the SM leads to the presence of the accidental global symmetries U(1)_B , $U{(1)}_{{L}_{e}}$, $U{(1)}_{{L}_{\mu }}$ and $U{(1)}_{{L}_{\tau }}$ in the SM Lagrangian. Under the minimal extension of the SM by three right-handed neutrino fields these global symmetries can be promoted to anomaly-free gauge symmetries by combining them into either a mixed baryo-leptophilic (i) or a purely leptophilic (ii) two-parameter U(1)_X group with [184]

$$\begin{equation}\begin{aligned}\hfill (\text{i})\quad X& =B-{x}_{e}{L}_{e}-{x}_{\mu }{L}_{\mu }-(3-{x}_{e}-{x}_{\mu }){L}_{\tau },\hfill \\ \hfill (\text{ii})\quad X& ={y}_{e}{L}_{e}+{y}_{\mu }{L}_{\mu }-({y}_{e}+{y}_{\mu }){L}_{\tau },\hfill \end{aligned}\end{equation} \tag{ 4.9 }$$

where x_e , x_μ , y_e and y_μ are real numbers. Amongst these groups the combination with x_e = x_μ = 1 plays a special role, since it leads to the flavor-universal U(1)_B−L group under which all generations of quarks carry the same charge, as do all generations of leptons.

In contrast to the secluded dark photon, the B − L gauge boson additionally couples to the B − L current according to equation (4.3), with

$$\begin{equation}{j}_{\mu }^{B-L}=\frac{1}{3}\bar{Q}{\gamma }_{\mu }Q+\frac{1}{3}{\bar{u}}_{\text{R}}{\gamma }_{\mu }{u}_{\text{R}}+\frac{1}{3}{\bar{d}}_{\text{R}}{\gamma }_{\mu }{d}_{\text{R}}-\bar{L}{\gamma }_{\mu }L-{\bar{\ell }}_{\text{R}}{\gamma }_{\mu }{\ell }_{\text{R}}-{\bar{\nu }}_{\text{R}}{\gamma }_{\mu }{\nu }_{\text{R}}.\end{equation} \tag{ 4.10 }$$

As a consequence the B − L boson couples to all SM fermions proportional to the coupling strength g_B−L . Hence, as long as the kinetic mixing parameter is smaller than the new gauge coupling, ≪ g_B−L , kinetic mixing effects are irrelevant for this model. The two major differences of the phenomenology of the B − L boson compared to the secluded dark photon are the different charges of the quarks under the two models, and most importantly, the fact that neutrinos are charged under U(1)_B−L coupling them to the associated gauge boson.

The latter has significant consequences for the employed search strategies. First of all, the B − L boson can decay invisibly into pairs of neutrinos, making searches for invisible final states a sensitive probe for this model. In particular, mono-photon (or even mono-Φ) searches at BaBar [185], NA64 [186] and, in the future, Belle-II provide stringent constraints to this model. Furthermore, a number of neutrino scattering experiments sensitive to extra neutrino interactions provide very stringent bounds on leptonically-coupled dark photons like the B − L gauge boson. Most noticeably, the evaluation [187, 188] of reactor neutrino data from Texono [189] and the neutrino beam experiment Charm-II [190, 191], as well as the recent re-evaluation [192] of solar neutrino scattering at Borexino [193, 194] are probing new regions of parameter space in the sensitivity gap between conventional prompt collider and displaced beam dump and fixed target searches for dark photons. This is shown in the right panel of figure 70.

Since the coupling structure of the B − L gauge boson to charged leptons and hadrons is very similar to the secluded dark photon, the constraints from visible prompt collider and displaced beam dump and fixed target experiments are also quite similar. These limits have been derived for the case of U(1)_B−L for example in references [144, 195] and are shown by the gray areas in the right panel of figure 70. Comparable to the secluded dark photon case, the LHC forward experiments FASER and in particular FASER2 have the potential to cover large untested regions of the B − L parameter space complementary to future searches of prompt invisible decays at Belle-II and NA64μ [144]. The major difference of the FASER(2) projections compared to the secluded dark photon case is a slightly reduced sensitivity to a U(1)_B−L gauge boson, since this has now a sizeable invisible branching fraction into neutrinos.

Among the purely leptophilic anomaly-free U(1) extensions of the SM discussed in section 4.2.2, there are three special subgroups, which are anomaly-free even without the addition of right-handed neutrinos (if Majorana mass terms for the neutrinos are forbidden). These are the three groups $U{(1)}_{{L}_{\mu }-{L}_{e}}$, $U{(1)}_{{L}_{e}-{L}_{\tau }}$ and $U{(1)}_{{L}_{\mu }-{L}_{\tau }}$. Under these groups the associated gauge bosons couple according to equation (4.3) to the gauge currents

$$\begin{equation}{j}_{\mu }^{i-j}={\bar{L}}_{i}{\gamma }_{\mu }{L}_{i}+{\bar{\ell }}_{i}{\gamma }_{\mu }{\ell }_{i}-{\bar{L}}_{j}{\gamma }_{\mu }{L}_{j}-{\bar{\ell }}_{j}{\gamma }_{\mu }{\ell }_{j},\end{equation} \tag{ 4.11 }$$

with i ≠ j = e, μ, τ. The lack of any gauge interactions with hadrons sets these groups quite apart from the previously discussed secluded dark photon and B − L gauge boson.

However, the fact that part of the SM leptons are charged under these leptophilic gauge groups induces a kinetic mixing term at the one-loop level. At energies well below the weak scale, q ≪ v_EW, this mixing is to first order only with the SM photon and the loop-induced mixing parameter can be expressed

$$\begin{equation}{{\epsilon}}_{ij}({q}^{2})=\frac{e{g}_{ij}}{2{\pi }^{2}}{\int }_{0}^{1}\mathrm{d}x\,x(1-x)\mathrm{log}\left(\frac{{m}_{i}^{2}-x(1-x){q}^{2}}{{m}_{j}^{2}-x(1-x){q}^{2}}\right),\end{equation} \tag{ 4.12 }$$

where g_ij denotes the $U{(1)}_{{L}_{i}-{L}_{j}}$ coupling constant and m_i and m_j are the masses of the charged leptons of the ith and jth generation. This irreducible loop-induced kinetic mixing is finite and effectively leads to loop suppressed interactions of the $U{(1)}_{{L}_{i}-{L}_{j}}$ with hadrons. When presenting results below, we assume that there is no bare tree-level kinetic mixing and the dynamics are fully-governed by the loop-induced mixing of equation (4.12). Such a tree-level mixing term can be either forbidden by the underlying UV completion of the model, or it can simply be neglected if it is much smaller than the loop-induced finite mixing.

L_μ − L_e . The phenomenology of the $U{(1)}_{{L}_{\mu }-{L}_{e}}$ gauge boson is governed by its couplings to electrons and muons. Therefore, it is constrained by a large number of e⁺ e⁻ collider, as well as electron beam dump and fixed target dark photon searches, as can be seen in the top left panel of figure 71. Similar to the case of U(1)_B−L discussed in section 4.2.2, its sizeable couplings to neutrinos make it subject to bounds from invisible decays searches and neutrino scattering experiments like Borexino [144] and Texono [187, 188]. The most stringent limit due to neutrino interactions, however, is the bound on neutrino NSIs derived from global fits to neutrino oscillation data [196]. In the future, this bound can be even further pushed by more precise measurements of neutrino oscillation at DUNE [197].

Zoom In Zoom Out Reset image size

The major difference to the U(1)_B−L case is the heavily suppressed couplings of the $U{(1)}_{{L}_{\mu }-{L}_{e}}$ boson to hadrons, which are only induced at the loop level. Consequently, experiments relying on hadronic dark photon production processes are significantly less sensitive to this boson. For example, limits from proton beam dumps like NuCal/U70 or LSND are much less sensitive. Similarly, the limits from LHCb dimuon searches for dark photons with masses of M_A' ≳ 10 GeV are substantially weaker and not even visible on the plot in the top left panel of figure 71. Since the LHC is a proton–proton collider, the production of such a purely leptophilic gauge boson is heavily suppressed. Therefore also FASER and FASER2 will not be able to probe unconstrained parameter space of the $U{(1)}_{{L}_{\mu }-{L}_{e}}$ boson. Noticeably, SHiP still will have some sensitivity to untested parameter space due to its very high statistics, a suitable geometry and a high boost factor. Most progress in searches for this boson can be expected from visible and invisible searches at Belle-II, as well as invisible searches at NA64 μ [198, 199].

L_e − L_τ . The phenomenology of the $U{(1)}_{{L}_{e}-{L}_{\tau }}$ boson is very similar to the case of $U{(1)}_{{L}_{\mu }-{L}_{e}}$, since both bosons have gauge interactions with electrons, but only loop-suppressed interactions with hadrons. The main difference between the two cases are the loop-suppressed interactions with muons of the $U{(1)}_{{L}_{e}-{L}_{\tau }}$ boson. As a consequence the limit from KLOE search for dimuon resonances is absent, and the future muon run of NA64 will not be sensitive to this scenario.

Another difference to the L_μ − L_e case is the slightly larger loop-induced kinetic mixing parameter for L_e − L_τ . The kinetic mixing parameter from equation (4.12) can be approximated as

$$\begin{equation}{{\epsilon}}_{ij}\approx \frac{e{g}_{ij}}{6{\pi }^{2}}\,\mathrm{log}\left(\frac{{m}_{i}}{{m}_{j}}\right),\end{equation} \tag{ 4.13 }$$

yielding

$$\begin{equation}{{\epsilon}}_{\mu e}\approx \frac{{g}_{\mu e}}{37},\quad {{\epsilon}}_{e\tau }\approx -\frac{{g}_{e\tau }}{25},\quad {{\epsilon}}_{\mu \tau }\approx -\frac{{g}_{\mu \tau }}{70}.\end{equation} \tag{ 4.14 }$$

This larger kinetic mixing results in slightly more constraining limits from NuCal/U70 and LSND, and a more constraining projection for SHiP in the top right panel of figure 71. As a consequence, also FASER and FASER2 are more sensitive to the case of $U{(1)}_{{L}_{e}-{L}_{\tau }}$ due to the enhanced gauge boson production in hadronic processes. FASER2 is even able to probe a small region of previously unconstrained parameter space.

L_μ − L_τ . Finally, we consider the phenomenology of an extra $U{(1)}_{{L}_{\mu }-{L}_{\tau }}$ gauge symmetry. This differs most from all previously discussed models, since the associated gauge boson couples to electrons and hadrons only via loop-suppressed kinetic mixing. As ordinary matter is composed of electrons, protons and neutrons—and so are the experimental apparatuses—the $U{(1)}_{{L}_{\mu }-{L}_{\tau }}$ boson is very hard to produce and to detect. As a consequence, we can see in the bottom panel of figure 71 that constraints from proton and electron beam dump and fixed target searches are entirely absent. By the same token there are no constraints from prompt decay searches at collider experiment, with the exception of four-muon final state searches at BaBar [200] and CMS [201].

Instead, the most stringent constraints are due to the neutrino interactions of the $U{(1)}_{{L}_{\mu }-{L}_{\tau }}$ boson, like neutrino trident production at Charm II [202, 203] and CCFR [202] (which is only shown as a gray dotted line due to some dispute over additional background [204]). Moreover, stringent bounds arise from solar neutrino scattering at Borexino [192–194] as well as from white dwarf cooling [144]. At small masses of M_A' ≲ 10 MeV a very strong bound arises from cosmological constraints on extra relativistic degrees of freedom, ΔN_eff, in the early Universe [205].

Most interestingly, a $U{(1)}_{{L}_{\mu }-{L}_{\tau }}$ group is the only of these minimal anomaly-free extra U(1)_X symmetries that can accommodate a solution of the muon (g − 2)_μ [206–209]. The preferred region of parameter space, where the (g − 2)_μ anomaly is resolved is shown by the light green band in the bottom panel of figure 71. In the future, there are a number of experimental searches that can probe this remaining parameter space. For example, one of the soonest searches to test this region will be the missing energy search at NA64μ [198, 199] (or similarly at the proposed Fermilab M³ experiment [210]). With a similar search strategy, the (g − 2)_μ region can also be tested via missing energy searches in kaon decays at NA62 [204]. Furthermore, future DD experiments like LZ and Darwin will be able to test parts of this interesting parameter space via solar neutrino scattering [192], as will spallation source experiments like COHERENT [211] or Coherent CAPTAIN-Mills (CCM) and ESS [212]. However, due to the loop-suppressed hadronic couplings of the $U{(1)}_{{L}_{\mu }-{L}_{\tau }}$ boson, it is very hard to produce at LHC. Therefore, FASER and FASER2 will not be sensitive to this scenario. Only SHiP will be able to test large fractions of unconstrained parameter space due to its very high statistics [144].

In this section we will study the groups $U{(1)}_{B-3{L}_{i}}$ as an example of the minimal anomaly-free baryo-leptophilic U(1) extensions of the SM, discussed in section 4.2.2, that feature family-non-universal couplings in the lepton sector. Their phenomenology will be yet different from the previously discussed examples and searching for the associated gauge bosons requires some dedicated search strategies. To illustrate the maximal effect of flavor-specific couplings in the lepton sector, we will set the kinetic mixing to zero, or assume that it is negligibly small, ≈ 0.

B − 3L_e . The phenomenology of the $U{(1)}_{B-3{L}_{e}}$ gauge boson is still quite similar to the family-universal U(1)_B−L case, with the major difference coming from the absence of muon couplings. Hence, e.g. the search for dimuon resonances at LHCb is not sensitive to this scenario and consequently the LHCb limits are absent in the top left panel of figure 72. In contrast, this model is constrained from prompt and invisible searches at e⁺ e⁻ colliders, like at A1 [171, 172], APEX [170], BaBar [118, 185], KLOE [161, 162, 213] or the fixed target experiment NA64 [186]. Furthermore, very strong constraints arise from proton and electron beam dump and fixed target experiments (like e.g. E137, E141, E774 [157], Orsay [167], NuCal/U70 [178, 179], LSND [214]), which have been derived in reference [215]. As for B − L, the $U{(1)}_{B-3{L}_{e}}$ gauge boson is also subject to constraints from neutrino experiments, like neutrino scattering at COHERENT and Borexino [215] or from searches for NSI with oscillation data [196].

Zoom In Zoom Out Reset image size

In the future, searches for dielectron resonances and missing energy at Belle-II [126, 216] will significantly push the limits from collider searches. However, due to the gauge couplings to hadrons, the $U{(1)}_{B-3{L}_{e}}$ gauge boson can also be abundantly produced in proton–proton collisions at LHC and searched for in electronic and hadronic final states at the FPF. Hence, both FASER and FASER2 will be able to probe previously untested regions of parameter space [215] and improve over current beam dump limits. The FASER and FASER2 reaches are shown as the red lines in the top left panel of figure 72. Finally, also SHiP will be able to probe large areas of previously uncovered parameter space thanks to its large statistics.

B − 3L_μ . In the case of an extra $U{(1)}_{B-3{L}_{\mu }}$ symmetry, dark photon searches at electron beam experiments or e⁺ e⁻ colliders are not sensitive since the associated boson is not coupling to electrons and hence cannot be produced in these kind of experiments. Thus, limits from electron beam dump and fixed targets are entirely absent in the top right panel of figure 72. Below the dimuon threshold, M_A' < 2m_μ , the $U{(1)}_{B-3{L}_{\mu }}$ boson can only decay invisibly into neutrinos. Therefore, in this mass range constraints mostly arise from experiments exploiting its neutrinos interactions, like searches for neutrino coherent scattering at COHERENT [215], NSI in neutrino oscillations [196] or extra relativistic degrees of freedom in the early Universe [205, 217]. Above the dimuon threshold, M_A' ≳ 2m_μ , the $U{(1)}_{B-3{L}_{\mu }}$ gauge boson can be searched for in dimuon resonance searches, like e.g. at LHCb [122], or in four-muon final states at BaBar [200] and CMS [201]. In this region of parameter space, there is also a constraint coming from dark photon searches in muonic final states at the proton beam dump experiment NuCal [215].

In the future, the muon beam experiment NA64μ [198, 199] will be able to improve significantly over current in muon plus missing energy searches. As the $U{(1)}_{B-3{L}_{\mu }}$ boson has gauge interactions with hadrons it can be abundantly produced at the LHC, where it can be searched for at the FPF in displaced muonic and hadronic final states. This will allow FASER2 (and similarly the proton fixed target experiment SHiP) to probe previously unexplored parameter space [215], as is illustrated in the top right panel of figure 72.

B − 3L_τ . Finally, the fact that first and second generation lepton couplings are entirely absent in a $U{(1)}_{B-3{L}_{\tau }}$ model makes it very hard to test. In particular, it is not subject to constraints from past beam dump or fixed target experiment, since the energies have not been high enough to produce dark photons massive enough that they can decay into purely hadronic final states. The most stringent constraints on a $U{(1)}_{B-3{L}_{\tau }}$ boson are due to its neutrino interactions. A rather strong constraint comes from searches for the decay ${\pi }^{0}\to \gamma (X\to \nu \bar{\nu })$ at NA62 [218]. Furthermore, similar to the previous case of $U{(1)}_{B-3{L}_{\mu }}$, the most stringent constraints arise from bounds on NSI in neutrino oscillations [196] or extra relativistic degrees of freedom in the early Universe [205, 217], as can be seen in the bottom panel of figure 72. Furthermore, a previous search for tau neutrino scattering at the DONuT [219] experiment has been used to derive limits to this model [220], which are competitive in a small region around the ω-resonance.

Similarly to the DONuT search, FASERν will be sensitive to scattering of LHC produced neutrinos. It can thus be used to search for extra tau neutrinos originating from the decay of a $U{(1)}_{B-3{L}_{\tau }}$ boson. The corresponding projection for the FASERν sensitivity has been derived in reference [220] and is illustrated by a red line in the bottom panel of figure 72. This will be able to improve over the DONuT and NSI limit in the ω-resonance region. Lastly, the abundant production of $U{(1)}_{B-3{L}_{\tau }}$ bosons in proton collisions combined with the high energies at LHC will allow FASER2 to probe untested parameter space of this model in hadronic decays around the ω-resonance [215]. Similarly, SHiP will be able to search for this boson in hadronic final states and test new regions of parameter space.

All previously mentioned Abelian U(1) mediator models are anomaly-free just by requiring a certain choice of charges and without working in some extra fermion family or embedding it into a UV-complete model. Nevertheless, we might also consider mediators that only couple to lepton or baryon number keeping in mind that it is not per se anomaly-free. In the following, we consider a U(1)_B model, or B model, where the mediator entertains gauge quark couplings but only couples to leptons via kinetic mixing induced by loop effects. The Lagrangian is given by

$$\begin{equation}{\mathcal{L}}_{{Z}_{B},\mathrm{i}\mathrm{n}\mathrm{t}}\supset e{\epsilon}{J}_{\text{em}}^{\mu }{Z}_{B\mu }-{g}_{B}{J}_{B}^{\mu }{Z}_{B\mu },\end{equation} \tag{ 4.15 }$$

where ${J}_{\text{em}}^{\mu }$ is the SM EM current and ${J}_{B}^{\mu }$ a new vector current given by

$$\begin{equation}{J}_{\text{B}}^{\mu }=\frac{1}{3}\sum\limits _{q}\bar{q}{\gamma }^{\mu }q.\end{equation} \tag{ 4.16 }$$

The kinetic mixing parameter is usually assumed to be of typical one-loop size = eg_B /(4π)² and hence, depends on g_B . If direct couplings with g_B , the loop contributions can be neglected. They only become relevant if leptonic decays are the only option since no hadronic decay channels are open.

The B model has been considered for several decades already starting with the early work [227] about MeV bosons in π⁰, or K⁺ and η decays if the gauge boson mass is smaller than the pseudoscalar meson mass. Some of the work even considered vector masses at the order of $\mathcal{O}(1{0}^{2})$ GeV suggesting searches at collision experiments such as the LHC [228, 229]. But soon the focus has been shifted more towards masses below the Z mass, with limits coming from ϒ and Z decays [230, 231], and even further down to 1–10 GeV including rare decays of B or D mesons and quarkonia states [232]. On the theoretical side, various possibilities of model realizations have been discussed as for example in connection to the seesaw-mechanism [233], discussing the mixing with the SM photon [234], anomaly cancellations by adding a single new fermionic generation [235], and other UV-complete models as for example in the context of asymmetric DM [236], usually containing a U(1)_B breaking Higgs field [237]. More recently, the B model has reached the sub-GeV range again where processes have to be described within the framework of VMD [195, 238] for sub-GeV gauge bosons where several signatures have been proposed including revised descriptions of mesonic decays into vector particles or vector mediator decays into hadrons.

Most boson searches assume leptonic couplings when searching for signatures in various experiments. It is often easier to search for a clean leptonic signature at electron facilities such as Belle-II [126] or NA64 [239], instead of hadronic signatures involving jets or mesonic decays which usually are probed at proton facilities like the LHC. If the mediator is coupling to neutrinos, a wide range of neutrino searches can further constrain parts of the parameter space. Moreover, the branching ratio into hadronic decay channels is often smaller than leptonic branching ratios as for example in B − L, or B − 3L_i models as discussed in section 4.2.8 based on [216]. This changes drastically if we consider bosons that predominately couple to quarks as in baryon number gauging U(1)_B models. We show that current and proposed far-forward experiments as proposed to be accommodated by the FPF, can enhance the potential to probe this particular model which remain beyond the reach of experiments focusing on BSM electron couplings.

Without DM. In figure 73, we present several constraints for the model reaching from current bounds from accelerator and collider searches, to rare anomaly-induced decays to new LLP decay signatures that can be probed in the near future. One can see that for vector mediator masses below the pion mass threshold, the only available decay channel is Z_Q → e⁺ e⁻ and hence only experiments measuring leptonic signatures (blue) can constrain the parameter space in the g_B − m_B plane. In [226, 240] it has been shown that the meson decays B → KZ_B and K^± → π^± Z_B as well as the Z boson decay Z → γZ_B are enhanced due to non-current conservation.

Zoom In Zoom Out Reset image size

The excluded blue regions were obtained using data from different kinds of experiments, such as the proton beam dumps NuCAL [125, 222], CHARM [223] and LSND [144, 177], where the Z_B boson is produced via meson decays or the proton bremsstrahlung process as well as LHCb constraints with a vector mediator decaying into muons. For a more detailed description of the limits, we refer to [216]. We also show the predicted sensitivities for the FASER (dashed pink) and FASER2 experiments (dashed purple), that were obtained using FORESEE code [73] together with the hadronic branching ratios from reference [216]. We can see from the figure that FASER will reach lower g_B couplings and will provide constraints in a new region where the current experiments lack efficiency and sensitivity. This is further outreached by FASER2 which covers an even larger region of the parameter due to the sensitivity to three pion and kaons decays of the vector boson.

Including DM. In [89], the U(1)_B model has been considered in context of DM. In particular, a complex scalar with a Lagrangian

$$\begin{equation}\mathcal{L}\supset \vert {\partial }_{\mu }\chi {\vert }^{2}-{m}_{\chi }^{2}\vert \chi {\vert }^{2}\end{equation} \tag{ 4.17 }$$

and a dark current

$$\begin{equation}{J}_{{Z}_{B}\bar{\chi }\chi }={g}_{\chi }\text{i}({\partial }_{\mu }{\chi }^{\ast }\chi -{\chi }^{\ast }{\partial }_{\mu }\chi )\end{equation} \tag{ 4.18 }$$

has been presented. This model evades constraints arising from precision measurements of the cosmic microwave background (CMB) anisotropies [241, 242] due to velocity-suppressed P-wave annihilations. The full model parameter space is specified by four parameters, m_B , g_B , m_χ and Q_χ . Two choices of Q_χ values have been discussed in [89], one where DM and SM particles have comparable interactions strengths, i.e. Q_χ = 1, and one where ${\alpha }_{\chi }={g}_{B}^{2}{Q}_{\chi }^{2}/(4\pi )$ was fixed. In the following, we will only present the case of Q_χ = 1 while the other case will be discussed in section 5.1.2 in the context of scattering signatures. If we further adopt the common convention m_B = 3m_χ , the analysis can be reduced to the two parameters g_B and m_B again.

Compared to the U(1)_B model without DM, already mentioned searches are modified with changed branching ratios and are complemented by dark matter DD, DM DIS, and DM elastic scattering limits. The DD bounds have to be taken with care since they do not apply for inelastic scalar DM if the mass splitting between the dark species is large enough to suppress upscattering of non-relativistic DM particles. In figure 74, the combined results for CRESST-III [243], DarkSide-50 [244], and Xenon 1T [245, 246] are shown as a very light gray shaded region assuming that Ω_χ h² ≃ 0.12 [247]. Note that for parameter points that are below the thermal target line (solid black), the DM abundance has to be explained by a non-standard cosmological scenario. DM scattering events can occur when the copiously produced hadrophilic mediator decays to dark matter particles Z_B → χχ. Particles in this DM beam can then scatter in detectors and can be searched for in experiments like in the downstream neutrino detector of MiniBooNE [248, 249], or as recently with the CCM LAr detector [250]. All existing constraints are shown in dark gray.

Zoom In Zoom Out Reset image size

Several of the proposed detectors under consideration for FPF have discovery prospects for the U(1)_B mediators coupling to DM. The projected sensitivity is shown in figure 74. Both FASER and FASER2 can detect decays of the long-lived vector mediator into visible final states, as shown by the blue lines. Note that this LLP signature covers a smaller but still significant part of the parameter space compared to the case of mediators that are not coupling to DM since a bigger share of the total decay width is going into invisible DM states. In addition, the proposed neutrino detectors at the FPF, FLArE-10 and FASERν2, have the capability to probe this scenario by searching for the scattering of the produced DM particles inside the detectors. The sensitivity has been estimated for DM elastic scatterings with nucleons χp → χp, which lead to visible single proton tracks in the detectors (dark red), and DM DIS off nuclei χN → χX, leaving a hadronic recoil with multiple charged tracks (light red) [89]. More details on the scattering signature will be discussed in section 5.1.2. Overall, all proposed signatures in the light of hadrophilic models highlight the rich phenomenology of FPF searches being able to both constrain couplings in the range 10⁻⁸ ≲ g_B ≲ 10⁻⁵ from LLP decays as well as above 10⁻⁴ ≲ g_B in DIS and elastic scattering events where DM can explain the relic abundance.

Experiments at HL facilities utilizing proton beams provide impressive sensitivity to new physics in the form of light weakly coupled degrees of freedom, namely dark sectors. The FPF at the LHC promises to provide an important new facility of this type, given its energy reach up to 14 TeV. The dominant production channels for dark sector degrees of freedom at proton beam facilities depend on the beam energy and the mass of the dark mediator. Among a variety of channels, forward bremsstrahlung of dark vectors and scalars is particularly important in the mass range from 500 MeV to a GeV, due to the possibility of enhancement via mixing with hadronic resonances with the same quantum numbers. However, computing this production rate in the forward region is a difficult task as it involves nonperturbative QCD physics associated with the forward pp cross section.

In recent work [251] we have revisited the calculation for the rate for bremsstrahlung production of light dark vector (A') and dark scalars (S), coupled through the corresponding renormalizable portal interactions, $\mathcal{L}\supset -\frac{1}{2}{\epsilon}{F}^{\mu \nu }{F}_{\mu \nu }^{\prime }-AS{H}^{{\dagger}}H$. For sub-GeV mass dark sector states, the dark sector state couples with the proton coherently, and one considers the induced coupling to protons which follows directly from the portal interactions,

$$\begin{equation}{\mathcal{L}}_{\text{eff}}\supset -{\epsilon}e{A}_{\mu }^{\prime }\bar{p}{\gamma }^{\mu }p-{g}_{\text{SNN}}\theta S\bar{p}p+\cdots \end{equation} \tag{ 4.19 }$$

where higher multipole couplings for ${A}_{\mu }^{\prime }$ have been ignored, the h − S mixing angle $\theta \simeq Av/{m}_{h}^{2}\ll 1$ for low mass scalars, and g_SNN = 1.2 × 10⁻³ follows from QCD low energy theorems. Based on these couplings, we considered a number of different approximation schemes in calculating the production rate,

• ISR and FSR in quasi-elastic scattering
• ISR in non-single diffractive scattering via the quasi-real approximation
• Hadronic generalization of the WW approximation
• Modified WW approximation

This approach allows for an assessment of relative precision in the calculations, and for consideration of more direct approaches that fully model initial state radiation (ISR) and final state radiation (FSR) within Pomeron-mediated forward diffractive and non-diffractive proton–proton scattering, with variants of the Weizäcker–Williams (WW) approximation, developed in the 1970's and generalizing the approach used for electron beams. In particular, the modified WW approximation of [179] has been used widely to model bremsstrahlung production of dark sectors within proton beams.

Our final results for production rates are shown in figure 76 for the 14 TeV LHC indicating various comparisons between different modes and calculational schemes. These plots show the rate within an angular region that matches the expected scale of experiments within the FPF. In all cases, a timelike form-factor for coupling to the proton provides resonant enhancements.

One of the interesting conclusions from this analysis was the recognition that the combination of t-channel ISR plus FSR for quasielastic Pomeron-mediated scattering (shown in figure 75) is suppressed via interference. This is seen explicitly in the green contour in figure 76. Since this cancellation is not expected to persist as scattering becomes non-single diffractive with more complex final states, ISR from the beam proton is predicted to provide a better approximation to the total rate. This rate was independently estimated using a quasi-real approximation for the incoming proton wit consistent results, and indeed leads to a rate that as shown in figure 76 is similar in magnitude to rate obtained by evaluating a splitting function using the modified WW approximation in [179].

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We are planning to model dark photon production modes in meson decays, hadronization, ISR, FSR in Herwig [252, 253] to study its relevance in various experimental setups. So far, it is assumed that dark photons A' would mainly be produced in meson decays, e.g. π⁰ → γA', Drell–Yan and in bremsstrahlung processes in proton–proton collisions and beam-dump experiments [117]. Meson decays are described in the framework of VMD [254–256] to determine decays like V → PA', P → γA' or via mixing V → A' where V is a vector meson, P a pseudoscalar meson and γ the SM photon [195]. For dark photons heavier than the meson masses, i.e. m_A' > m_π,η,..., bremsstrahlung and Drell–Yan are so far the dominant production mode. The bremsstrahlung process is typically modelled by using a Fermi–Weizäcker–Williams approximation [179, 181, 182] which uses proton form factors including an off-shell mixing with vector mesons. This results in a relatively sharp enhancement in the production around the ρ and ω mass $\sim 775$ MeV [117]. The calculations for proton bremsstrahlung have recently been revised in [251] as discussed in the previous section. Above these masses, the production rate through bremsstrahlung drops sharply down to unobservably small values, and hence, additional production modes could increase the potential of dark photon searches.

In their EOI [24], the CODEX-b collaboration has studied several production modes for ALPs with couplings to gluons [25] by using Pythia 8 [114]. In particular, they include (i) hadron decays via mixing with the light pseudoscalar π⁰, η, η' mesons: for any hadron decay into neutral pseudoscalar mesons, there is the possibility that this hadron decays into an ALP. (ii) Additionally, an ALP can be produced in flavor-changing neutral-current (FCNC) bottom and charm hadron decays as for example via b → sa. (iii) Axions can be radiated of any gluon in the parton shower. (iv) Finally, axions can be produced during hadronization if a $q\bar{q}$ pair has pseudoscalar quantum numbers like π⁰, η, η' and hence, forms an axion through mixing into it. For the case of CODEX-b, the production in the parton shower turns out to play an important role and boosts the overall production rate by orders of magnitude. Their results could possibly also be applied to MATHUSLA [26, 27, 257], ATLAS, CMS, and LHCb.

In the context of the FPF, the focus shifts more towards additional production modes in the forward direction. Let us take the dark photon as a model example. In figure 77, we show several ways of producing a dark photon in proton–proton collisions, for example in the beam remnants, ISR, in the hard process, FSR, or in hadronization.

Zoom In Zoom Out Reset image size

To accurately describe these production modes, we can take advantage of the existing infrastructure available in MC simulators. In particular, we would like to use Herwig to study the following dark photon production mechanisms:

• ISR and FSR for parton splittings q → qA',
• hadronization where a quark pair $q\bar{q}$ hadronizes into A',
• hadron decay to A'.

For the implementation of ISR and FSR in Herwig, we can make use of the previous work [258, 259] where an angularly-ordered EW parton shower has been implemented into Herwig. The general form for the q → q'V splitting function as given in [258] and shown in figure 78 can be easily translated into a splitting dark photon since its already been taken care of that all engaged particles have non-zero masses, especially the vector particle V, and additional longitudinal polarization states might be present.

Zoom In Zoom Out Reset image size

In fact, the dark photon with same couplings to left- and right-handed helicities a_L = a_R = 1, represents a simplified version of the EW vector case. The explicit expression for the dark photon splitting function ${P}_{q\to qA}(z,\tilde{q})$ is then given by

$$\begin{equation}{P}_{q\to qA}(z,\tilde{q})=\frac{{a}^{2}}{1-z}\left[1+{z}^{2}-\frac{2{m}_{q}^{2}+{m}_{A}^{2}}{z{\tilde{q}}^{2}}\right]\end{equation} \tag{ 4.20 }$$

with a light-cone momentum fraction z, evolution scale $\tilde{q}$ and quark and dark photon masses, m_q and m_A', respectively. The implementation of dark photon production in and after hadronization will probably follow the approach as done by the CODEX-b collaboration in [25]. A big advantage of using Herwig for studying vector mediators in collision experiments is that we could not only describe the production of those vector bosons but also the decay using earlier results implemented into Herwig [260]. Therein, in particular the hadronic decay of a vector particle with arbitrary couplings to quarks is described in the context of indirect detection [261, 262]. But the implemented channels could easily be used in the context of dark photons in other experimental setups.

It is yet unclear how these mechanisms affect the overall production rate and as a consequence predictions for various experiments. For detectors in the forward direction of a beam-collision such as FASER, we expect that ISR of dark photons and the production within the hadronization of beam remnants could provide a sizable contribution.

Our attempt to add dark photon production processes is of course not limited to vector particles produced in the forward direction. In experiments that are placed in a more transverse direction, dark photon radiation in the parton shower might yield a higher number of dark photon events. The results can be applied to various existing/proposed dark photon searches and experiments. We expect that many proton beam dump experiments, for example, would be sensitive to dark photons being produced in the shower, and hence their bounds would increase. Besides, not only vector particles could be produced. Since the project inherits the EW parton shower implementation of reference [258] including Higgs radiations, our study can, in principle, be extended to dark scalars as well.

The majority of vector mediator models consider leptonic decays when searching for signatures in experiments made or used for the detection of light vector particles. For the dark photon model, the coupling strength to the SM can be constrained for a wide range of masses from a few MeV to $\mathcal{O}(100)$ GeV. In this specific model, all possible decays, and in particular the total decay width, can be extracted from e⁺ e⁻ annihilations into leptons, quarks and hadrons since the dark photon kinetically mixes with the SM photon and hence, its couplings are proportional to the SM photon-fermion couplings. This yields to very accurate descriptions of dark photon decays and consequently predictions for experimental searches. Nevertheless, for the case of generic U(1) gauge group models this strategy cannot be conducted anymore due to different decay structures that will arise from differences in the mediator-meson couplings compared to the SM photon case. Especially in the low-energy range from MeV to $\sim 2$ GeV, hadronic decays might deviate largely from the dark photon case. In this range, the vector mediator ${Z}_{Q}^{\mu }$ directly mixes with the vector mesons ρ, ω, and ϕ through

$$\begin{equation}{\mathcal{L}}_{V{Z}_{Q}}=2{g}_{Q}{Z}_{Q}^{\mu }\,\mathrm{Tr}\left[{V}_{\mu }{Q}^{f}\right],\end{equation} \tag{ 4.21 }$$

with V^μ = T^a V^a,μ , where T^a are U(3) generators, V^a,μ the vector mesons with

$$\begin{equation}\begin{aligned}\hfill \rho :\,{\rho }^{\mu }{T}_{\rho }& ={\rho }^{\mu }\frac{1}{2}\,\mathrm{diag}(1,-1,0),\,\omega \!:{\!\omega }^{\mu }{T}_{\omega }\hfill & \hfill ={\omega }^{\mu }\frac{1}{2}\,\mathrm{diag}(1,1,0),\,\phi :\!{\phi }^{\mu }{T}_{\phi }={\phi }^{\mu }\frac{1}{\sqrt{2}}\,\mathrm{diag}(0,0,1).\!\end{aligned}\end{equation} \tag{ 4.22 }$$

and g_Q is the coupling constant. Whereas the dark photon with Q^f = diag(2/3, −1/3, −1/3) couples to all of the mesons, most baryophilic U(1) gauge bosons with Q^f = diag(1/3, 1/3, 1/3) only couple to ω and ϕ. The vector mesons further couple to other vector and pseudoscalar mesons which yields to a wealth of final state configurations. For example, the 2 and 4 pion channels for vector decays are purely driven by the mixing with the ρ meson whereas the πγ and 3 pion final states mostly come from the mixing with ω and ϕ. A comprehensive list of final states can be found in reference [216]. A change in the hadronic decays will even change the results for leptonic signatures, since limits and predictions do not depend on the partial decay width into leptons that can be calculated perturbatively, but on its relative contribution compared to other decays. Hence, to set robust constraints on these models, it is inevitable to precisely determine the branching ratio into all sorts of final states as well as the total decay width to determine the lifetime. That is the only way we can accurately predict where and how the vector boson will appear in the experimental setup.

As part the DarkCast tool presented in reference [195], a first attempt has been made to describe hadronic channels individually, especially its splitting into ρ, ω and ϕ contributions, to not only describe dark photon decays, but general vector mediator models with arbitrary couplings to quarks. In this section, based on reference [216], we describe how we improve the description of several channels and extend the number of final state configurations compared to reference [195] by extracting the hadronic currents obtained in low-energy e⁺ e⁻ fits. Some of the fits are taken from a previous study [260], and some are introduced for the first time. Decay widths, branching ratios, and other related decay quantities calculated in [216] are available in the python package DeLiVeR ²⁴⁰ .

As described in detail in reference [216], channels that have already been used in reference [195] are improved by considering contributions from more than one vector meson mixing with the vector mediator for each channel, by including more recent and complete data sets as well as rigorously following the VMD model [254–256] with only very little theoretical assumptions. As seen in the left panel of figure 79, major differences arise in the ω and ϕ contributions when describing e⁺ e⁻ annihilations to hadrons. In case of the ω meson (pink curve), this mainly comes from a different description of the πγ channel for energies below the ω mass [263], and because of including more processes that include the ω resonance or excited states thereof above 1.4 GeV. The big difference in the ϕ contribution (light green curve) between just above the ϕ resonance and 1.6 GeV can be traced back to a different handling of the KK and KKπ channels.

Zoom In Zoom Out Reset image size

In DarkCast, it is assumed that KK = 2 K⁺ K⁻ which leads to an overestimation of the KK contribution. Only attributing this channel then to the ϕ meson consequently results in an overestimation of the ϕ contribution. We fit both the charged e⁺ e⁻ → K⁺ K⁻ and ${e}^{+}{e}^{-}\to {K}^{0}{\bar{K}}^{0}$ components separately and consider contributions from ρ-like, ω-like and ϕ-like mesons. For the KKπ channel, DarkCast only considers the isoscalar subprocess e⁺ e⁻ → K*(892)K that is responsible for e⁺ e⁻ → K*(892)K → KKπ. Nevertheless, besides not accurately describing the three-body phase space of KKπ by K*K, the isovector contribution of K*K is missing. As before in the case of KK, the KKπ channel is assumed to be determined by the ϕ contribution only. We take into account the three components K⁰ K⁰ π⁰, K⁺ K⁻ π⁰ and K^± K⁰ π^∓ for the description of the KKπ channel. In all the three processes, we have a isovector contribution that is assigned to the ρ meson, and an isoscalar contribution that is described by the ϕ meson.

As an example on how the different treatment of hadronic decays affects the vector mediator branching ratios, we discuss the case of the B − L model. As seen from equation (4.21), in this model we only have a ω and ϕ contribution. In the right panel of figure 79, we can see that slightly above the ϕ mass, our branching ratios are smaller due to the lower ϕ contribution. For mediator masses above 1.5 GeV instead, the bigger ω contribution of our calculations results in a bigger branching ratio into hadrons. Above 1.75 GeV, we do not trust VMD anymore and assume perturbative mediator decays into quarks. Overall, the difference between our branching ratios and the ones from DarkCast is below the 10% level for hadrons. This difference is shared between electrons, muons and neutrinos. As a consequence, we observe that limits coming from ${Z}_{Q}\to {e}^{+}{e}^{-},{\mu }^{+}{\mu }^{-},\nu \bar{\nu }$ searches that reach over many order of magnitudes in the vector coupling will only change very little with differences around a few percent in the hadronic decays. Only for the case where one hadronic channel is dominating, as for example below the two-pion threshold in the U(1)_B model, we can observe visible differences in the limits [216].

Nevertheless, if we focus on signatures of a Z_Q decaying into hadrons as suggested by reference [89], the limits might change drastically. As an example, we take the hadrophilic U(1)_B model where the gauge boson couples directly to quarks but only through kinetic mixing with leptons. Hence, the mediator decays nearly a 100% into hadrons above the πγ channel threshold. Major differences in the individual hadronic channels are coming from πγ and KK which changes the relative contribution of all dominant channels as seen in the left panel of figure 80. With these branching ratios, we can now study the expected sensitivity for future experiments as for example FASER2. Whereas the overall sensitivity to all hadrons only changes mildly, the sensitivity to the KK channel reduces only to the region around the ϕ mass instead of reaching from 1.0–1.4 GeV as shown in the right panel of figure 80. We can clearly see that the individual branching ratio differences of up to 30% directly translates to different sensitivities. For deviations that cannot be explained by the branching ratios it is likely that they come from the difference of the vector mediator lifetime of both calculations.

Zoom In Zoom Out Reset image size

In this section, we presented improvements in the calculation of hadronic decays of vector particles in vector mediator models. This includes providing an almost complete set of all possible hadronic and leptonic partial decay widths, branching ratios, and related vector mediator decay quantities in the python package DeLiVeR. Especially in decay channels related to vector mediators mixing with the ω and ϕ mesons, we observe significant differences between our calculations and DarkCast. Limits that range over several orders of magnitude in the vector coupling and mass and are based on Z_Q decays into electrons, muons or neutrinos only change very little when having only a few percent differences in the corresponding branching ratios. Nevertheless, for regions that are dominated by one hadronic channel, the limits might differ. Moreover, if we study future sensitivities for hadronic decay signatures of vector mediators, we observe that, for example for the case of FASER2, the different treatment of hadronic decays will have a significant effect on the sensitivities.

Before proceeding to particular physics realization of the dark Higgs and other light scalars, let us pause and review the existing constraints on the parameter space. The landscape of the dark Higgs S, spanned by its mass m_S and mixing sin θ, is shown in the left panel of figure 82. As we can see, FASER2 at the FPF can detect light long-lived scalars produced in the high-energy pp collisions up to a few GeV scalar masses, as shown by the FASER2 target line [9, 37]. For comparison, we also show the existing limits (shaded regions) from FCNC meson decay searches and beam-dump experiments.

Zoom In Zoom Out Reset image size

The scalar S can obtain FCNC couplings to the SM quarks at one-loop level via its mixing with the SM Higgs, and thus contribute to rare FCNC decays of mesons, such as K → π + X, B → K + X and B → π + X with X = e⁺ e⁻, μ⁺ μ⁻, γγ or missing energy if X decays outside the detector. There are stringent limits from K⁺ → π⁺ e⁺ e⁻ and π⁺ μ⁺ μ⁻ in NA48/2 [271, 272]; K⁺ → π⁺ γγ and ${\pi }^{+}\nu \bar{\nu }$ at NA62 [273, 274] and E949 [275]; K_L → π⁰ e⁺ e⁻, π⁰ μ⁺ μ⁻ and π⁰ γγ in KTeV [276–279]; ${K}_{L}\to {\pi }^{0}\nu \bar{\nu }$ and π⁰ X (with X being a LLP) at KOTO [280]; and B → K + X with $X={e}^{+}{e}^{-},{\mu }^{+}{\mu }^{-},\,\nu \bar{\nu }$ at LHCb [281], BaBar [282, 283] and Belle [284]. Similarly, the MicroBooNE collaboration has recently performed a search for light monoenergetic scalars from kaon decay at rest K⁺ → π⁺ + S with S → e⁺ e⁻ [285]. The most stringent limits coming from NA62, E949, KOTO, MicroBooNE and LHCb experiments are shown as the shaded blue, light green, orange, dark green and pink regions respectively. For the detailed calculations of the FCNC decays and the derivation of these limits, see, e.g., references [286–288].

The light scalar S can also be produced in the high-intensity beam-dump experiments. The current most stringent limits come from kaon decays K → π + S in the CHARM experiment [223], as shown by the magenta shaded region. At the LSND experiment, the scalar S is predominately produced by the proton bremsstrahlung process, and the current LSND electron and muon data [289, 290] have excluded the brown shaded region [291]. The data from the fixed target experiment PS191 have also be reinterpreted in reference [292] to set limits on the light scalar from kaon decays K → π + S, as shown by the red shaded region. It is expected that the future high-intensity beam-dump experiments will significantly improve these limits. For instance, the Fermilab SBN experiment, which is a combination of using the SBND and ICARUS experiments with the BNB and NuMI beams respectively, can probe a mixing angle down to $\sim 1{0}^{-7}$ [293], as shown by the purple curve. The near detector at DUNE can further improve the prospects by over an order of magnitude [294], as shown by the green curve.

On the astrophysical side, the scalar S can be produced abundantly in the supernova cores with temperature T ∼ 30 MeV via the nucleon bremsstrahlung process. If it escapes the supernova core, it will contribute an additional channel for energy loss; therefore, the observed neutrino luminosity ${\mathcal{L}}_{\nu }$ of SN1987A can be used to set limits on the mixing angle sin θ of S with the SM Higgs [295, 296, 298–301]. Setting conservatively ${\mathcal{L}}_{\nu }=3\times 1{0}^{53}$ erg/sec and 5 × 10⁵³ erg/sec, the corresponding supernova limits are shown in figure 82 respectively as the lighter and darker black shaded regions. One can also derive similar limits [302] from energy loss criteria in the Sun, white dwarfs, red giants and horizontal-branch stars; however, their core temperature is only $\mathcal{O}(\mathrm{k}\mathrm{e}\mathrm{V})$, and therefore, the corresponding limits are not relevant for the parameter space shown in figure 82.

On the cosmology front, if the light S stays in thermal equilibrium with the SM particles and has a lifetime ≳1 s, it will contribute an extra degree of freedom and spoil the success of Big Bang nucleosynthesis (BBN) [303, 304]. However, it turns out that there is no parameter space of m_S and sin θ that satisfies both the conditions above, so the generic BBN limit is not applicable here [296]. The actual BBN constraint will depend on the particular thermal history of the scalar S in the early Universe and can in principle be evaded, without affecting the late-time scalar phenomenology we are interested in here. Therefore, we do not show the BBN limit in figure 82. However, if the scalar has an additional trilinear coupling to Higgs, it can be abundantly produced via rare Higgs decays h → SS and reach thermal equilibrium. The corresponding constraints are shown in figure 81, where we note that the exact constraint only mildly depends on the additional Higgs-scalar coupling [304].

Another motivation for a singlet scalar is provided by the relaxion mechanism which was introduced as a solution to the (little) hierarchy problem of the SM [305]. An initially large Higgs boson mass M ≫ v is reduced to the observed mass ${m}_{h}=\mathcal{O}(v)$ through the dynamical evolution of the relaxion ϕ in the early Universe. The Higgs-relaxion potential reads

$$\begin{equation}V=({M}^{2}-{g}_{1}M\phi ){h}^{2}-{g}_{2}{M}^{3}\phi -{{\Lambda}}^{2}{h}^{2}\,\mathrm{cos}\left(\frac{\phi }{f}\right)+\lambda {h}^{4},\end{equation} \tag{ 4.25 }$$

where g_1,2 are dimensionless couplings of the same order and h is the neutral component of the Higgs doublet. The cosine-term is generated by the coupling of the relaxion $(\phi /f){G}_{\mu \nu }^{\prime }{\tilde{{G}^{\prime }}}^{\mu \nu }$ to the field strength of a new strongly coupled gauge group QCD' whose strong dynamics set the scale Λ.

During cosmic inflation, the relaxion rolls down its potential, before triggering electroweak symmetry breaking at ϕ ∼ M/g₁. The Higgs field gets displaced, turns on the periodic potential of the relaxion, and the Hubble friction then stops the relaxion in one of the resulting minima. The relaxion mass and the Higgs-relaxion mixing angle sin θ in the minimum can be approximated as [270, 306]

$$\begin{equation}{m}_{\phi }^{2}\simeq \frac{{{\Lambda}}^{2}{v}^{2}}{2{f}^{2}}\left[\mathrm{cos}\left(\frac{{v}_{\phi }}{f}\right)-\frac{2{{\Lambda}}^{2}}{{m}_{h}^{2}}\,{\mathrm{sin}}^{2}\left(\frac{{v}_{\phi }}{f}\right)\right]\quad \text{and}\quad \mathrm{sin}\,\theta \simeq \frac{{{\Lambda}}^{2}v}{f{m}_{h}^{2}}\,\mathrm{sin}\left(\frac{{v}_{\phi }}{f}\right),\end{equation} \tag{ 4.26 }$$

where v_ϕ denotes the final relaxion field value. For the Hubble scale $H\,\gtrsim \,\sqrt{{g}_{2}{M}^{3}/f}$, the relaxion immediately stops in one of its first minima after electroweak symmetry breaking [270] and one finds $\mathrm{sin}({v}_{\phi }/f)\sim \mathrm{cos}({v}_{\phi }/f)\sim 1/\sqrt{2}$ such that,

$$\begin{equation}\mathrm{sin}\,\theta \simeq \frac{{2}^{1/4}{m}_{\phi }{\Lambda}}{{m}_{h}^{2}}.\end{equation} \tag{ 4.27 }$$

Validity of the effective theory equation (4.25), furthermore requires ²⁴¹ f ≳ v ≳ Λ which implies m_ϕ ≲ Λ. On the other hand, Λ below the GeV-scale is cosmologically unfavorable as the relaxion mechanism requires an enormous number of inflationary e-foldings in this case [307]. The FPF will be able to probe a large part of the relaxion parameter space. In figure 81 we used equation (4.27) with Λ = 2 GeV to provide an explicit FPF target line, which is labeled as relaxion.

In reference [308] it was noted that generically the relaxion would stop at the first minimum, which is characterized by a shallower potential compared with the one described by the single cosine term (of a generic axion-like models), while the mixing angle with the Higgs remains unmodified. It results in an 'unnaturally' large value for the mixing angle, basically, as the relaxion is generically relaxed to an unnatural point in its parameter space, and was dubbed as the 'relaxed-relaxion' with a mass-mixing-angle relation of

$$\begin{equation}{m}_{\phi }^{2}\simeq n\delta \frac{{{\Lambda}}_{\mathrm{b}\mathrm{r}}^{4}}{{f}^{2}}\quad \text{and}\quad \mathrm{sin}\,\theta \simeq \frac{{\epsilon}}{\sqrt{n}\delta }\frac{{m}_{\phi }^{2}}{{m}_{h}^{2}},\end{equation} \tag{ 4.28 }$$

where n denotes the nth minimum, ${\epsilon}={\left({{\Lambda}}_{\mathrm{b}\mathrm{r}}/4\right)}^{4}$ and δ = Λ_br/(vΛ). The upper bound of the relaxion mixing angle for a relaxion stopping in the first or a generic minimum is given by

$$\begin{equation}\mathrm{sin}\,\theta \leqslant {({m}_{\phi }/v)}^{2/3}.\end{equation} \tag{ 4.29 }$$

In addition, there is a lower bound on the mixing angle, that obtained from requiring f ≳ Λ ≳ 4πv:

$$\begin{equation}\mathrm{sin}\,\theta \geqslant {\left(\frac{{m}_{\phi }}{{n}^{\frac{1}{4}}v}\right)}^{\frac{4}{3}}.\end{equation} \tag{ 4.30 }$$

This gives rise to a band of the natural parameter space of the relaxion. In figure 81, we show the lower limit as target line labeled as relaxed relaxion. The upper limit would be found at larger couplings beyond those shown in the figure. As we can see, the FPF will be able to probe the relaxion band in the mass range above 300 MeV. The phenomenology and collider bounds of the long-lived relaxion with a mass of several GeV have been analyzed in reference [309].

If the dark Higgs/singlet scalar S is sufficiently light, it can be readily produced in the hot and dense nuclear matter in the cores of stars, supernovae and NSs via the nuclear bremsstrahlung process NN → NNS, where the S can couple to any of the four external nucleon legs or to the intermediate pion mediator. Note that a crucial difference between the production of dark Higgs and that of axion/ALP is that the latter being a CP-odd particle can only be emitted from the nucleon legs but not from the mediator pions at leading order. If the scalar coupling with the SM particles (induced via its mixing θ with the SM Higgs) is sufficiently weak, it will free-stream out of the astrophysical bodies, thus providing an additional energy loss mechanism that would affect their evolution. On the other hand, if the coupling is sufficiently strong, the scalar particle is trapped, thereby contributing to the transport properties of the astrophysical core.

An interestingly new astrophysical probe of the light scalar parameter space of interest comes from NS mergers [297]. Depending on the thermodynamic conditions in the merger, the scalar can have a wide range of MFP. Even after taking into account all the laboratory and supernova limits on the scalar, there is still some overlapping parameter space with FPF where the scalar MFP is small enough for it to be trapped inside the merger. In this case, the scalar S will form a Bose gas and contribute to thermal conductivity of the merger remnant. In the absence of any physics BSM, the neutrinos dominate the thermal transport inside the merger remnant [310]. The thermal conductivities due to the trapped scalars (κ_S ) and trapped neutrinos (κ_ν ) are compared in the right panel of figure 82 for a particular choice of the baryon density n_B equal to the nuclear saturation density n₀ and for a merger temperature T = 40 MeV (thermodynamic conditions which simulations reliably indicate that mergers reach [311]), as shown by the black dashed contours. The results for other choices of density and temperature can be found in reference [297]. In places where the ratio κ_S /κ_ν is larger than one, fast thermal equilibration is a signature of BSM physics, as no SM mechanism yields such a high thermal conductivity. This effect can potentially lead to observable signals in future NS merger events, but this needs to be further studied with the aid of updated merger simulations including the transport of trapped scalars. It is remarkable that some of the regions with κ_S /κ_ν > 1 can be directly probed at the future laboratory experiments such as SBN, DUNE and FASER2, thus providing a nice complementarity between the laboratory and astrophysical probes of dark Higgs.

We consider the non-minimal quartic inflation in a classically conformal U(1)_X extended SM [312]. By imposing the conformal invariance at the classical level on the minimal U(1)_X extended SM [313], all mass terms in the Higgs potential are forbidden. As a result, the U(1)_X gauge symmetry is radiatively broken by the Coleman–Weinberg (CW) mechanism [314], which subsequently drives the electroweak symmetry breaking through a mixing quartic coupling between the U(1)_X Higgs Φ and the SM Higgs H fields [315]. We identify the U(1)_X Higgs field with a non-minimal gravitational coupling as inflaton. Because of the classically conformal invariance, this scenario not only leads to the inflationary predictions consistent with the Planck 2018 results [316], but also provides a direct connection between the inflationary predictions and the LHC search for the U(1)_X gauge boson (Z') resonance. We will show that the inflaton mass and its mixing angle with the SM Higgs field lie in a suitable range, the inflaton can be searched by FASER2 with a direct connection to the inflationary predictions. Therefore, three independent experiments, namely, the inflaton search at FASER2, the Z' boson resonance search at the HL-LHC and the precision measurement of the inflationary predictions, are complementary to test our inflation scenario.

The particle content of the model is listed in the left panel of of figure 83, where the U(1)_X charge of a particle is defined as a linear combination of its SM hypercharge and its B − L (baryon minus lepton) number. The U(1)_X charges are determined by a real parameter, x_H , and the well-known minimal U(1)_B−L model is realized as the limit of x_H → 0. In the presence of the three right-hand neutrinos (RHNs), ${N}_{\text{R}}^{1,2,3}$, this model is free from all the gauge and the mixed gauge-gravitational anomalies. Once the U(1)_X Higgs field (Φ) develops a VEV, $\langle {\Phi}\rangle ={v}_{X}/\sqrt{2}$, the U(1)_X gauge symmetry is broken and the Z' boson becomes massive, m_Z' = 2g_X v_X , where g_X is the U(1)_X gauge coupling. The symmetry breaking also generates Majorana masses for the RHNs which play a crucial role in the type-I seesaw mechanism for generating the light neutrino masses.

• (a)  
  Radiative U(1)_X symmetry breaking. Imposing the classical conformal invariance, the Higgs potential of our model at the tree level is given by

  $$\begin{equation}V={\lambda }_{H}\!{\left({H}^{{\dagger}}H\right)}^{2}+{\lambda }_{{\Phi}}\!{\left({{\Phi}}^{{\dagger}}{\Phi}\right)}^{2}-{\lambda }_{\text{mix}}\!\left({H}^{{\dagger}}H\right)\!\left({{\Phi}}^{{\dagger}}{\Phi}\right),\qquad \end{equation} \tag{ 4.31 }$$

  where we set λ_H,Φ,mix > 0. Assuming λ_mix ≪ 1, we can separately analyze the Higgs potential for Φ and H. The CW potential for the Higgs field Φ at the one-loop level is given by [314]

  $$\begin{equation}V(\phi )=\frac{{\lambda }_{{\Phi}}}{4}{\phi }^{4}+\frac{{\beta }_{{\Phi}}}{8}{\phi }^{4}\left(\mathrm{ln}\left[\frac{{\phi }^{2}}{{v}_{X}^{2}}\right]-\frac{25}{6}\right),\end{equation} \tag{ 4.32 }$$

  where $\phi =\sqrt{2}\,\mathrm{Re}[{\Phi}]$, v_X is chosen as a renormalization scale, and the coefficient of the one-loop corrections is approximately given by $16{\pi }^{2}{\beta }_{{\Phi}}\simeq 96{g}_{X}^{4}-3{Y}_{M}^{4}$. The stationary condition, ${\left.\mathrm{d}V/\mathrm{d}\phi \right\vert }_{\phi ={v}_{X}}=0$, leads to

  $$\begin{equation}\bar{{\lambda }_{{\Phi}}}=\frac{11}{6}\bar{{\beta }_{{\Phi}}},\end{equation} \tag{ 4.33 }$$

  where the barred quantities are evaluated at ⟨ϕ⟩ = v_X . The mass of ϕ is given by

  $$\begin{equation}{m}_{\phi }^{2}={\left.\frac{{\mathrm{d}}^{2}V}{\mathrm{d}{\phi }^{2}}\right\vert }_{\phi ={v}_{X}}=\bar{{\beta }_{{\Phi}}}{v}_{X}^{2}\simeq \frac{6}{\pi }\bar{{\alpha }_{X}}{m}_{{Z}^{\prime }}^{2},\end{equation} \tag{ 4.34 }$$

  where ${\alpha }_{X}={g}_{X}^{2}/(4\pi )$, and we have neglected RHN contributions. The U(1)_X gauge symmetry breaking by $\langle {\Phi}\rangle ={v}_{X}/\sqrt{2}$ induces a negative mass squared for the SM Higgs doublet in equation (4.31) and triggers the electroweak symmetry breaking [315]. The SM(-like) Higgs boson mass (m_h = 125 GeV) is described as ${m}_{h}^{2}={\lambda }_{\text{mix}}{v}_{X}^{2}=2{\lambda }_{H}{v}_{h}^{2}$, where v_h = 246 GeV is the Higgs doublet VEV. The mass matrix for the Higgs bosons, ϕ and h, is given by

  $$\begin{equation}\mathcal{L}\supset -\frac{1}{2}\left[\begin{matrix}\hfill h\hfill & \hfill \phi \hfill \end{matrix}\right]\left[\begin{matrix}\hfill {m}_{h}^{2}\hfill & \hfill {\lambda }_{\text{mix}}{v}_{X}{v}_{h}\hfill \\ \hfill {\lambda }_{\text{mix}}{v}_{X}{v}_{h}\hfill & \hfill {m}_{\phi }^{2}\hfill \end{matrix}\right]\left[\begin{matrix}\hfill h\hfill \\ \hfill \phi \hfill \end{matrix}\right].\end{equation} \tag{ 4.35 }$$

  We diagonalize the mass matrix by

  $$\begin{equation}\left[\begin{matrix}\hfill h\hfill \\ \hfill \phi \hfill \end{matrix}\right]=\left[\begin{matrix}\hfill \mathrm{cos}\,\theta \hfill & \hfill \mathrm{sin}\,\theta \hfill \\ \hfill -\mathrm{sin}\,\theta \hfill & \hfill \mathrm{cos}\,\theta \hfill \end{matrix}\right]\left[\begin{matrix}\hfill \tilde{h}\hfill \\ \hfill \tilde{\phi }\hfill \end{matrix}\right],\end{equation} \tag{ 4.36 }$$

  where $\tilde{h}$ and $\tilde{\phi }$ are the mass eigenstates. For the parameter region which will be searched by FASER, the mixing angle θ ≃ v_h /v_X ≪ 1 and the mass eigenvalues ${m}_{\tilde{\phi },\tilde{h}}\simeq {m}_{\phi ,h}$ with ${m}_{\phi }^{2}\ll {m}_{h}^{2}$.
• (b)  
  Non-minimal U(1)_X Higgs inflaton. Our inflation scenario is defined by the action in the Jordan frame:

  $$\begin{equation}{\mathcal{S}}_{\text{J}}=\int {\mathrm{d}}^{4}x\sqrt{-g}\left[-\frac{1}{2}(1+2\xi {{\Phi}}^{{\dagger}}{\Phi})\mathcal{R}+{g}^{\mu \nu }{\left({D}_{\mu }{\Phi}\right)}^{{\dagger}}\left({D}_{\nu }{\Phi}\right)-V\right],\end{equation} \tag{ 4.37 }$$

  where the non-minimal gravitational coupling constant ξ > 0, and the reduced Planck mass of M_P = 2.44 × 10¹⁸ GeV is set to be 1 (Planck unit). In our analysis, we fix the number of e-folds N₀ = 50 to solve the horizon and flatness problems. In this non-minimal quartic inflation, all the inflationary predictions are determined as a function of ξ. Once ξ is fixed, the quartic coupling λ_Φ at the inflationary scale is uniquely determined. Extrapolating the λ_Φ to its value at v_X according to the renormalization group (RG) equations and using equation (4.33), we obtain a one-to-one correspondence between the set of two free parameters, {ξ, v_X } and {m_ϕ , θ}.
• (c)  
  Searching for the inflaton at the FPF. In the right panel of figure 83, we show our results in (m_ϕ , θ)-plane, together with FASER search reach, the search reach of other planned/proposed experiments (contours with the names of experiments indicated), and the current excluded region (gray shaded) from CHARM [223], Belle [284] and LHCb [317] experiments, as shown in reference [9]. Here, to ensure the readability of the figure, we have not shown the search reach of other experiments such as SHiP [29], MATHUSLA [318] and CODEX-b [24] presented in reference [9]. After our analysis, each point in FASER parameter space has a one-to-one correspondence with inflationary predictions and Z' boson search parameters. The diagonal dashed lines correspond to ξ = 9.8 × 10⁻³ (r = 0.064) and ξ = 0.12 (r = 0.01), respectively, from left to right. The light red shaded region (r > 0.064) is excluded by the Planck 2018 results. We find that the parameter region corresponding to the inflationary prediction r ≳ 0.01 can be searched by FASER2 in the future, a part of which is already excluded by the Planck 2018 result. The diagonal solid lines correspond to m_Z' (TeV) = 0.3, 0.4, 0.5, 0.7, 1.0, 1.3, 2.6, 5.0, and 10, from top to bottom. A point on a solid line corresponds to a fixed value of ξ, or equivalently, r. Along each line, the ξ (r) value increases (decreases) from left to right. For example, the left (right) diagonal dashed lines denote r = 0.0064 and r = 0.01.

Zoom In Zoom Out Reset image size

In conclusion, we have considered the non-minimal quartic inflation scenario in the minimal U(1)_X model with classical conformal invariance, where the inflaton is identified with the U(1)_X Higgs field. FASER can search for the inflaton when its mass and mixing angle with the SM Higgs field are in the range of 0.1 ≲ m_ϕ  (GeV) ≲ 4 and 10⁻⁵ ≲ θ ≲ 10⁻³. By virtue of the classical conformal invariance and the radiative U(1)_X symmetry breaking via the Coleman–Weinberg mechanism, the inflaton search by FASER, the Z' boson resonance search at the LHC, and the future measurement of r are complementary to test our inflationary scenario. For all the details of analysis presented here, see the original paper [319].

The minimal coupling of a singlet scalar S to the SM through renormalizable interactions with the Higgs doublet, S⁽²⁾ H², leads to the scalar acquiring couplings proportional to those of the Higgs boson. Notably, the Yukawa coupling of the scalar to fermions is proportional to the fermion mass. In the presence of higher dimensional effective interactions involving the scalar and SM fields, however, there are additional possibilities for the scalar-fermion couplings. We consider the Lagrangian

$$\begin{equation}\mathcal{L}=\frac{1}{2}{\partial }^{\mu }S{\partial }_{\mu }S-\frac{1}{2}{m}_{S}^{2}{S}^{2}-\left(\frac{{c}_{S}}{M}S{\bar{Q}}_{\text{L}}{u}_{\text{R}}{H}_{c}+\mathrm{h}.\mathrm{c}.\right)\end{equation} \tag{ 4.38 }$$

where c_S is a matrix containing the family structure of the S coupling to the up-type quarks and M is the mass scale of the new physics underlying the effective operator. While we restrict ourselves to scalar couplings to up-type quarks for simplicity, analogous theories can be written for down-type quarks and charged leptons. In particular, FPF searches for a muon-philic scalar could probe regions of parameter space consistent that could potentially explain the muon g − 2 anomaly [320].

Generally non-standard couplings such as those in equation (4.38) result in flavor changing neutral-currents, but certain choices of coupling structures lead to viable phenomenology. Specifically, suppose we work in the mass basis for the up quarks. If c_S ∝ diag(1, 0, 0), S couples to only the 1st generation up quark; we term this a flavor-specific hypothesis [320], and it may be thought of as a generalization of minimal flavor violation. With such a form for c_S , it can be shown that the radiatively generated flavor-changing couplings of the S are small due to symmetry considerations. After electroweak symmetry breaking, the scalar acquires a coupling to the up quark

$$\begin{equation}\mathcal{L}\supset -{g}_{u}S\bar{u}u,\ {g}_{u}=\frac{{c}_{S}v}{\sqrt{2}M}\end{equation} \tag{ 4.39 }$$

This effective coupling is sufficient to determine the production and decay of the S at the FPF [321], which can be used to calculate the projected sensitivity of LLP detectors such as FASER2 to up-philic scalars.

Because the scalar-quark coupling is not proportional to mass as in the Higgs portal model, the experimental landscape for flavor-specific scalars is markedly different from that for minimally coupled scalars. Up-philic scalars are primarily produced at the FPF through light meson decays, notably those of the η⁽'⁾ and kaons; this may be compared to Higgs portal scalars, which are dominantly produced at the FPF from B meson decays. Similarly, as up-philic scalars do not couple to leptons, the only possible decay mode of the S is to γγ for m_S < 2m_π . Above the pion threshold, S decays hadronically. We use the estimated meson spectra at the LHC in conjunction with the S decay widths in reference [321] to calculate the reach of FASER(2) and SHiP LLP searches as a function of the scalar mass and effective coupling g_u . We also translate the limits on S from current and future rare meson decay searches (MAMI, KLOE, BESII, REDTOP), fixed target experiments (CHARM), supernova data (SN1987A), and BBN. These limits are shown in figure 84.

Zoom In Zoom Out Reset image size

Now, the model described above may be augmented in several ways. From an effective field theory (EFT) perspective, perhaps the first question is the character of the UV completion of equation (4.38) at the heavy scale M. Two natural completions of the c_S interaction come from vector-like quarks and additional scalars, respectively [322]. It is of interest to ask whether direct and indirect searches for these particles would be complementary to FPF searches for S, and we explore the interplay between these searches below. In addition, the S could serve as a mediator between the SM and dark matter. Depending on the relative masses of the mediator and dark matter, it may be possible for S to decay invisibly. As our focus here is on LLP searches for the S assuming its visible decay, we will not consider the potential couplings of S to dark matter further.

Turning to the question of the UV physics underlying the effective theory in equation (4.38), we first write a completion with a vector-like quark U' that has the same charges under SU(3) × SU(2) × U(1) as the right-handed up quark:

$$\begin{equation}\mathcal{L}=\frac{1}{2}{\partial }^{\mu }S{\partial }_{\mu }S-\frac{1}{2}{m}_{S}^{2}{S}^{2}+{\bar{U}}^{\prime }(\text{i}\!\!{\hskip-3pt \!D}-M){U}^{\prime }-\left(y{\bar{Q}}_{\text{L}}{U}_{\text{R}}^{\prime }{H}_{c}+\lambda {\bar{U}}_{L}^{\prime }{u}_{\text{R}}S+\mathrm{h}.\mathrm{c}.\right)\end{equation} \tag{ 4.40 }$$

The new couplings y and λ are related to the effective coupling of the scalar through c_S = −yλ, and we have identified the heavy scale in equation (4.38) with the U' mass.

The presence of the U' leads to additional constraints beyond those in the effective theory. First, naturalness of radiative corrections to the S masses suggest that λ ≪ y, and we assume this hierarchy of couplings. Then, the U' mixes with the SM u_R. This affects the unitarity of the effective SM CKM matrix and electroweak precision tests (EWPT). Furthermore, the U' mediates box diagrams contributing to kaon mixing (FCNC). We show these limits, with future projections when available, in the left panel of figure 84. In addition, if there is a non-trivial phase among the couplings y and λ, the SM up Yukawa y_u , and M, a contribution to the neutron EDM arises. We show this constraint assuming an order-1 phase. Lastly, there are LHC constraints from direct U' searches, but these are subdominant in the parameter space of the left panel of figure 84 and are not shown.

Instead of being completed with a vector-like quark, the effective theory of an up-philic scalar may arise from integrating out a heavy new Higgs doublet H' with the Lagrangian

$$\begin{align}\hfill \mathcal{L}& =\frac{1}{2}{\partial }^{\mu }S{\partial }_{\mu }S-\frac{1}{2}{m}_{S}^{2}{S}^{2}+\vert {D}_{\mu }{H}^{\prime }{\vert }^{2}-{M}^{2}{H}^{\prime {\dagger}}{H}^{\prime }\hfill \\ \hfill & \quad -\left({y}^{\prime }{\bar{Q}}_{\text{L}}{u}_{\text{R}}{H}_{c}^{\prime }+\kappa MS{H}^{{\dagger}}{H}^{\prime }+\mathrm{h}.\mathrm{c}.\right)+\ \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\ \mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}\hfill \end{align} \tag{ 4.41 }$$

Similarly to the vector-like quark completion, the mass of the new Higgs doublet is the heavy scale M, and the effective coupling above can be thought of as c_S = −κy'.

In an analogous fashion to the U' theory, we may consider additional constraints on an up-philic scalar that is completed by an additional Higgs doublet. Once again, requiring the stability of the S mass under radiative corrections results in tight constraints on one of the couplings, and we will work in the regime κ ≪ y'. In addition, the H' contributes to neutral kaon mixing (FCNC). Then, the H' and S mediate one-loop diagrams involving the Z couplings to quarks, which affect EWPT. Also, since the H' couples to quarks, it is subject to dijet searches at the LHC. As before, we show all of these limits and their future projections in the right panel of figure 84. There is again a neutron EDM constraint if there is a phase among the new physics couplings and the SM up Yukawa, which is shown assuming that the phase is large.

From figure 84, it is clear that searches for long-lived up-philic scalars at FASER2 can provide additional sensitivity beyond existing limits. These results would be independent of the UV completion, and complementary to direct searches for new heavy particles mediating the effective coupling of a light scalar to up quarks. Beyond the minimal renormalizable portals of hidden sectors, flavor-philic scalar theories provide novel targets for long-lived searches at the FPF.

Another example of a theory that could provide a scalar particle that could be probed at the FPF is the type-I 2HDM. In the following, we will consider 2HDMs in which one of the neutral BSM scalars is very light and long lived, while the others are heavy.

The Higgs sector of the 2HDMs consists of two SU(2)_L scalar doublets Φ_i  (i = 1, 2) with hyper-charge Y = 1/2. After the electroweak symmetry breaking with the neutral components of Φ_1,2 develop VEVs of v_1,2 and tan β = v₂/v₁, the scalar sector of the 2HDMs [323, 324] consists of five physical scalars: h, H, A, H^±, with the CP-even Higgs mixing angle being α. In this work, h will be identified as the SM Higgs with m_h = 125 GeV, while H and A are the BSM neutral Higgses. The alignment limit under this convention is therefore cos(β − α) = 0. The effective Lagrangian for a (light) CP-even scalar H and CP-odd pseudoscalar A interacting with SM particles can be written as

$$\begin{align}\hfill \mathcal{L}& =-\sum\limits _{f}{\xi }_{H}^{f}\frac{{m}_{f}}{v}H\bar{f}f-{\xi }_{H}^{W}\frac{2{m}_{\text{W}}^{2}}{v}\phi {W}^{\mu +}{W}_{\mu }^{-}-{\xi }_{H}^{Z}\frac{{m}_{Z}^{2}}{v}H{Z}^{\mu }{Z}_{\mu }\hfill \\ \hfill & \quad +{\xi }_{H}^{g}\frac{{\alpha }_{s}}{12\pi v}H{G}_{\mu \nu }^{a}{G}^{a\mu \nu }+{\xi }_{H}^{\gamma }\frac{\alpha }{4\pi v}H{F}_{\mu \nu }{F}^{\mu \nu }+\sum\limits _{f=u,d,e}{\xi }_{A}^{f}\frac{\text{i}{m}_{f}}{v}\bar{f}{\gamma }_{5}fA\hfill \\ \hfill & \quad +{\xi }_{A}^{g}\frac{{\alpha }_{s}}{4\pi v}A{G}_{\mu \nu ,a}{\tilde{G}}^{\mu \nu ,a}+{\xi }_{A}^{\gamma }\frac{\alpha }{4\pi v}A{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }.\hfill \end{align} \tag{ 4.42 }$$

Expressions for the various coupling modifiers ${\xi }_{H,A}^{f,W,Z}$ as well as the loop induced couplings ${\xi }_{H,A}^{g,\gamma }$ can be found in references [325–328]. There are four different types of 2HDMs, depending on the couplings of Φ_1,2 with the quark and lepton sectors. Consequently, the resulting effective couplings of H and A with fermions, namely ${\xi }_{H,A}^{f}$, exhibit different tan β dependence.

After taking into account the theoretical considerations such as unitarity, perturbativity, and vacuum stability, as well as electroweak precision measurements (EWPM), the viable parameter space with a light H or A are:

$$\begin{equation}{m}_{H}\sim 0:\ {m}_{A}\sim {m}_{{H}^{\pm }}\lesssim 600\ \mathrm{G}\mathrm{e}\mathrm{V},\end{equation} \tag{ 4.43 }$$

$$\begin{equation}{m}_{A}\sim 0:{m}_{{H}^{\pm }}\sim {m}_{H}\lesssim {m}_{h},\end{equation} \tag{ 4.44 }$$

with $\lambda {v}^{2}\equiv {m}_{H}^{2}-{m}_{12}^{2}/(\mathrm{sin}\,\beta \,\mathrm{cos}\,\beta )\approx 0$ to accommodate a large range of tan β.

The flavor constraints such as ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$, K⁺ → π⁺ e⁺ e⁻, B → X_s γ, B_s,d → μ⁺ μ⁻, $B-\bar{B}$ mixing, decays of B and D baryons, impose strong constraints on the charged Higgs mass as well as the value of tan β. In type-II, type-L, and type-F, at least one of the Higgs Yukawa couplings to quarks or lepton are unsuppressed at all regions of tan β, which makes it hard to accommodate a very weakly coupled light neutral Higgs that has relatively long decay lifetime. Therefore, the only viable model is the type-I 2HDM, in which all the Higgs Yukawa couplings are suppressed at large tan β.

To accommodate the current SM-like Higgs coupling measurements [329, 330], we take the close to the alignment limit of cos(β − α) ∼ 0. For a light H/A with long enough lifetime, h → HH/AA is constrained by the invisible Higgs decay. Under the exact alignment limit of cos(β − α) = 0, unlike g_hAA , which is suppressed to be only a few percent of the value of the SM trilinear Higgs coupling, g_hHH recovers the SM value of ${m}_{h}^{2}/2v$, leading to an invisible decay branching fraction too big to accommodate the current bound of Br(h → invisible) < 0.24 [331–333]. In our study below for the CP-even light Higgs, we take the parameter choice of large tan β = 1/cos(β − α), which leads to simultaneous suppressions of the g_hHH coupling as well as the effective Yukawa couplings ${\xi }_{H,A}^{f}$.

The productions of a light CP even Higgs H are mostly via the flavor changing decays of kaons and B mesons. Depending on the mass of the CP-even scalar H, it can decay into pair of photons, leptons, and multiple hadrons. However, for larger values of tan β > 10, the decay into photons dominate. This is due to our parameter choice cos(β − α) = 1/tan β, which leads to suppressed couplings to fermions ${\xi }_{H}^{f}\sim 1/{\mathrm{tan}}^{3}\,\beta$. In contrast, the decay mode into photons arises through a W-boson or charged Higgs with corresponding coupling ${\xi }_{H}^{\gamma }\sim 1/\mathrm{tan}\,\beta$. The decay length cτ takes values between 10³ m and 10⁻³ m for light scalar masses m_H in the range between 100 MeV and 10 GeV for tan β = 100.

A light CP odd scalar A typically mixes with pseudo-Goldstone bosons π⁰, η and η'. Any process that produces those mesons would also produce the new pseudoscalar A. In addition, the scalar can be produced in the weak decays of SM mesons, in particular K → πA and b → X_s A. At low mass, it mainly decays to diphotons and dileptons. Once the hadronic channels open up, it could decay into multiple hadrons, as well as radiative hadronic decay. Decay length cτ typically reaches between 100 m and 10⁻⁷ m for m_A in the range of 100 MeV to 10 GeV and tan β = 100.

In figure 85, we show the FASER2 reach for the light CP even Higgs H (left panel) and CP odd Higgs A (right panel) in the parameter space of m_H/A vs tan β plane. Experimental constraints on the light scalars from various experiments are also shown in shaded gray regions. The benchmarks we chose are: light H with tan β = 1/cos(β − α), ${m}_{A}={m}_{{H}^{\pm }}=600\enspace \text{GeV}$; light A with cos(β − α) = 0, m_H = 90 GeV, ${m}_{{H}^{\pm }}=110\enspace \text{GeV}$. The reach for the light CP odd A has higher value for tan β since ${\xi }_{A}^{f}=1/\mathrm{tan}\,\beta$, while ${\xi }_{H}^{f}\sim 1/{\mathrm{tan}}^{3}\,\beta$ for the light H case. We can see that FASER2 will have the potential to probe a large region of this currently unconstrained region of 2HDM parameter space.

Zoom In Zoom Out Reset image size

Supersymmetry, if it exists in nature, should be spontaneously broken in a hidden sector. This breaking may occur at relatively low energy scale and it can be achieved in models with no-scale supergravity [336, 337] or gauge-mediation [338, 339]. Although the details of the hidden sector are model dependent, there should be a sector, where spontaneous supersymmetry breaking takes place, and this sector contains the Goldstone supermultiplet. Fermionic component of this supermultiplet is goldstino which becomes gravitino in supergravity extensions of the model. In the case of chiral Goldstone supermultiplet, the superpartners of goldstino are scalar S and pseudoscalar P sgoldstinos. The scale of sgoldstino masses depends on details of the hidden sector and it is phenomenologically acceptable to have it below 10 GeV. Here we discuss FASER2 reach for the light sgoldstinos [340] and assume, in accordance with results from LHC experiments, that all the superpartners of the SM particles and particles from the hidden sector (except for gravitino which is lightest supersymmetric particle (LSP) in the present scenario), to be much heavier neglecting their possible impact on sgoldstino phenomenology. In the case of light sgoldstinos their interactions with gravitino can also be safely neglected.

Sgoldstino interactions with SM fields are determined by the soft supersymmetry breaking parameters. Corresponding low energy interactions (see, e.g., references [341, 342]) include couplings to photons and gluons

$$\begin{align}\hfill \mathcal{L}& =-\frac{{M}_{\gamma \gamma }}{2\sqrt{2}F}S{F}_{\mu \nu }{F}^{\mu \nu }+\frac{{M}_{\gamma \gamma }}{4\sqrt{2}F}P{F}_{\mu \nu }{F}_{\lambda \rho }{{\epsilon}}^{\mu \nu \lambda \rho }-\frac{{M}_{3}}{2\sqrt{2}F}S{G}_{\mu \nu }^{a}{G}^{\mu \nu \,a}\hfill \\ \hfill & \quad +\frac{{M}_{3}}{4\sqrt{2}F}P{G}_{\mu \nu }^{a}{G}_{\lambda \rho }^{a}{{\epsilon}}^{\mu \nu \lambda \rho }\hfill \end{align} \tag{ 4.45 }$$

as well as to quarks q_i , i = 1, ..., 6 and charged leptons l_i , i = 1, 2, 3,

$$\begin{equation}\mathcal{L}=-S\frac{v{A}_{Q}{y}_{ij}^{q}}{\sqrt{2}F}{\bar{q}}_{i}{q}_{j}-P\frac{v{A}_{Q}{y}_{ij}^{q}}{\sqrt{2}F}{\bar{q}}_{i}{\gamma }_{5}{q}_{j}-S\frac{v{A}_{l}{y}_{ij}^{l}}{\sqrt{2}F}{\bar{l}\!}_{i}{l}_{j}-P\frac{v{A}_{l}{y}_{ij}^{l}}{\sqrt{2}F}{\bar{l}\!}_{i}{\gamma }_{5}{l}_{j}.\end{equation} \tag{ 4.46 }$$

Here M_γγ ≡ M₁ cos²  θ_W + M₂ sin²  θ_W with M_i , i = 1, 2, 3 being the gaugino masses. The combinations ${A}_{Q,l}{y}_{ij}^{q,l}$ play the role of trilinear coupling constants. For flavor conserving couplings we assume $v{y}_{ii}^{q,l}={m}_{i}^{q,l}$ for simplicity. The parameter $\sqrt{F}$ has the meaning of supersymmetry breaking scale. The model should be considered as a low energy EFT (LEFT) valid for energies $E\ll \sqrt{F}$ and in the weak coupling regime for ${m}_{\text{soft}}< \sqrt{F}$.

The interactions of scalar and pseudoscalar sgoldstino to the SM fields induce their decays to the lighter SM particles. The decay pattern of sgoldstino depends on the hierarchy of the soft supersymmetry breaking parameters; see, e.g., references [342–346]. In case of soft parameters of similar size, M₃ = M_γγ = A₀, the decays of scalar sgoldstino into hadrons dominate, if allowed, while decays into photons dominate, otherwise. Let us note that for pseudoscalar sgoldstinos the hadronic modes are suppressed for m_P ≲ 1 GeV, because they are three-body decays.

Sgoldstino couplings are naturally bounded from limits on soft supersymmetry breaking terms inferred from direct searches for squarks, gluinos, etc and observations of rare processes where the superpartners might contribute to. At the same time, the models with light sgoldstinos are also constrained from testing their predictions, e.g., direct searches for light sgoldstinos, performed at kaon experiments [347–349], e⁺ e⁻ [350, 351] and hadronic [352, 353] colliders, and at beam-dump facilities [346].

Production mechanisms of scalar and pseudoscalar sgoldstinos in proton collisions relevant for FASER2 include direct production via gluon fusion gg → S(P) and production in decays of mesons. The latter involve flavor-conserving and flavor-violating sgoldstino couplings to quarks as well as scalar sgoldstino mixing with the Higgs boson [354]. Examples of sgoldstino production in decays include decays of flavorless vector mesons V → S(P)γ and semihadronic decays of light and heavy pseudoscalar mesons K → πS, η → πS, B → KS(P). Expressions for various partial decay widths with scalar and pseudoscalar sgoldstinos in the final states can be found in reference [340].

In figures 86–88, we show several examples of FASER2 sensitivity regions, which depend on the signal signature (pair of photons, leptons or mesons) and sgoldstino production mechanism. We estimate the FASER2 sensitivities assuming dominance of a particular decay mode. In all cases we investigate only the regions in the model parameter space which are consistent with existing bounds on light sgoldstinos.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The model contains several parameters and it is difficult to specify precisely the total parameter space which can be probed by FASER2. We refer to reference [340] for detailed discussion of FASER2 sensitivity to models with light sgoldstino. To get an impression we can broadly say that FASER can probe the models with light sgoldstinos having supersymmetry breaking scale $\sqrt{F}\lesssim 1500\text{\mbox{--}}30\,000$ TeV at the first stage of experiment and about 1–2 orders of magnitude higher at the second one.

In recent years, there has been interest in cosmological solutions to the hierarchy problem of the SM, wherein novel dynamics in the early Universe selects a small Higgs mass [305, 355–361]. In contrast to traditional symmetry-based solutions to the hierarchy problem (e.g. weak-scale SUSY and composite Higgs), quadratically divergent corrections to the Higgs mass from new physics are not suppressed. These models generically require a new light, weakly-coupled particle that interacts with the Higgs [361].

Here we focus on the 'crunching dilaton' model of reference [362], which combines aspects of symmetry-based and cosmological approaches to address Higgs naturalness. The main phenomenological prediction is a new scalar (a dilaton) in the 0.1–10 GeV range which mixes with the Higgs. This dilaton could potentially be observed at FASER2. It differs from a dark Higgs in that it also has a tree-level coupling to the photon, which alters its phenomenology.

We will summarize the essential points of the model, and refer the reader to reference [362] for additional details. We postulate a multiverse of causally disconnected Hubble patches. Some scanning sector sets the value of the Higgs VEV in each patch, up to a UV cutoff scale. We introduce dynamics such that all patches rapidly undergo a cosmological crunch unless the Higgs VEV lies in a finite range, v_min < v < v_crit. Then the only patches that allow for a large observable Universe are those in which the Higgs VEV is in this range.

This can be explicitly realized through a new CFT sector in which the conformal symmetry is spontaneously broken; the Higgs couples to the dilaton, the Goldstone boson of the broken symmetry. We employ the Goldberger–Wise mechanism [363] to generate a stable minimum in the dilaton potential. The total vacuum energy in this minimum is always large and negative, so that any patches where the dilaton rolls down to the Goldberger–Wise minimum rapidly undergo a cosmological crunch. Interactions between the Higgs and dilaton generate a second, metastable minimum in the dilaton potential, but only when the Higgs VEV is smaller than some critical value v_crit. Crucially, the conformal symmetry allows v_crit to be $\mathcal{O}(\mathrm{T}\mathrm{e}\mathrm{V})$, hierarchically smaller than the UV cutoff, without fine-tuning. Thus we dynamically select a small Higgs VEV in the multiverse, generating an apparent naturalness problem.

Explicitly, the dilaton potential is given by

$$\begin{equation} V(\chi ,v)=-\lambda {\chi }^{4}+{\lambda }_{\text{GW}}\frac{{\chi }^{4+\delta }}{{k}^{\delta }}+{\lambda }_{2}{v}^{2}\frac{{\chi }^{2+\alpha }}{{k}^{\alpha }}-{\lambda }_{H{\epsilon}}{v}^{2}\frac{{\chi }^{2+\alpha +{\epsilon}}}{{k}^{\alpha +{\epsilon}}}-{\lambda }_{4}{v}^{4}\frac{{\chi }^{2\alpha }}{{k}^{2\alpha }}\end{equation} \tag{ 4.47 }$$

where χ is the dilaton, k is the UV cutoff of the CFT sector, the λ's are dimensionless couplings, and δ, α, and are related to anomalous dimensions of certain CFT operators. The first two terms correspond to the Goldberger–Wise potential, while the remaining terms depend on the value of the Higgs VEV and generate a metastable minimum when v is sufficiently small. In the 5D dual description of the CFT, where the dilaton is identified with the IR brane in a slice of AdS₅ space, these Higgs-dependent terms can be understood as arising from IR brane-localized terms in the potential. We omit a detailed discussion of the potential here because it is not necessary to understand the phenomenological features of the model.

For the Higgs VEV to drive the dilaton potential, the dilaton must be light relative to the Higgs, typically around m_χ = 1 GeV. The dilaton mixes with the Higgs and therefore inherits all the SM Higgs couplings suppressed by sin θ, where θ is the mixing angle. One can derive approximate analytical expressions for m_χ and sin θ in terms of the potential parameters, from which it follows

$$\begin{equation}\mathrm{sin}\,\theta \propto \frac{{m}_{\chi }^{2}}{{m}_{h}^{2}}.\end{equation} \tag{ 4.48 }$$

The dilaton has negligible tree-level couplings to the SM fermions. This is most easily understood in the 5D holographic description of the CFT: the dilaton is localized towards the IR brane, while the SM fermions are localized on the UV brane, and so they do not interact significantly with the dilaton at tree level. Conversely, the Higgs propagates in the AdS₅ bulk, and so the electroweak gauge bosons must live in the bulk as well. Therefore the dilaton has couplings to the electroweak gauge bosons of the form [370]

$$\begin{equation}\frac{1}{4{{\Lambda}}_{\text{EW}}}\chi \left({F}_{\mu \nu }^{2}+{Z}_{\mu \nu }^{2}+2{W}_{\mu \nu }^{2}\right),\end{equation} \tag{ 4.49 }$$

where Λ_EW is $\mathcal{O}(\mathrm{T}\mathrm{e}\mathrm{V})$. The coupling to the W and Z turns out to be negligible relative to the coupling obtained through mixing with the Higgs, but the coupling to the photon is phenomenologically important.

In summary, there is a dilaton with mass ${m}_{\chi }=\mathcal{O}(\mathrm{G}\mathrm{e}\mathrm{V})$ which mixes with the Higgs and also has a direct coupling to the photon. In the left panel of figure 89 we show the results of a random scan of the parameter space in the m_χ -sin²  θ plane. We take k = 10⁸ TeV, δ = 0.01, λ_GW = 2 × 10⁻⁶, λ = 1.1λ_GW, and α = 0.1, and sample the other parameters from ∈ (0.03, 0.1), λ₂ ∈ (0.5, 1) × 10⁻², λ_H ∈ (2, 4) × λ₂, and λ₄ ∈ (2, 3), where the reader is referred to reference [362] for a discussion of the naturalness of these parameter choices. The experimental bounds are similar to those on a dark Higgs, but must be rescaled to account for the photon coupling. The existing constraints on the parameter space come from searches for rare B meson decays at LHCb. From figure 89, we see that the unexplored parameter space below m_χ ≃ 1 GeV could be probed at the FPF with FASER2.

Zoom In Zoom Out Reset image size

One can also consider probing the photon coupling directly. The bounds on the photon coupling are shown in the right panel of figure 89. The model is not in tension with the existing experimental constraints. However, we also see that FASER2 would lack the sensitivity to probe the region of parameter space relevant to the model. Although it is not depicted in figure 89, we remark that this relevant parameter space could be fully explored at a future lepton collider such as the FCC-ee [371, 372].

A sterile neutrino is a neutral lepton which is a gauge singlet with respect to the SM gauge symmetry SU(3)_C ⊗ SU(2)_L ⊗ U(1)_Y . Several extensions of the SM consider a right-chiral sterile neutrino ν_R, which are CP conjugates of ${\nu }_{\text{L}}^{c}$, where ν_L is a left-chiral active neutrino and the superscript denotes a charge conjugate spinor: ${\psi }^{c}=C{\bar{\psi }\,\!}^{\text{T}}$ with the charge conjugation defined as C = iγ₂ γ₀. We consider three active and n > 0 sterile neutrinos, i.e. '3 + n' neutrinos.

From neutrino oscillation experiments it is clear that at least two of three active neutrinos are massive. Case n = 1 is insufficient as it yields only one massive active neutrino. Therefore the case n = 2 is minimal, while n = 3 is often used. Such a case complements the number of generations in the SM.

Sterile neutrinos may be either Dirac or Majorana fermions. Majorana mass term can be written directly into the Lagrangian without any need to generate it via a spontaneous symmetry breaking (SSB) mechanism. This allows practically any mass for the sterile neutrinos. In addition, sterile neutrinos may have Yukawa interactions with the SM Higgs boson: ${\Delta}{\mathcal{L}}_{Y}=-{Y}_{\text{D}}{\varepsilon }_{\alpha \beta }{L}_{\text{L}\alpha }{\phi }_{\beta }\bar{{\nu }_{\text{R}}}$. After SSB, they produce also a Dirac mass term. When both Dirac and Majorana mass terms are present the mass term can be written as

$$\begin{equation}{\Delta}\mathcal{L}=-\frac{1}{2}{\left(\genfrac{}{}{0.0pt}{}{{\nu }_{\text{L}}}{{\nu }_{\text{R}}^{c}}\right)}^{\text{T}}C\left(\begin{matrix}\hfill 0\hfill & \hfill {m}_{\text{D}}\hfill \\ \hfill {m}_{\text{D}}^{\text{T}}\hfill & \hfill {M}_{\text{R}}\hfill \end{matrix}\right)\left(\genfrac{}{}{0.0pt}{}{{\nu }_{\text{L}}}{{\nu }_{\text{R}}^{c}}\right)+\text{h.c.},\end{equation} \tag{ 4.50 }$$

where the neutrino mass matrix must be diagonalized with a unitary matrix U to obtain the physical mass eigenstates. Here ${m}_{\text{D}}={Y}_{\text{D}}v/\sqrt{2}$, where v ≈ 246 GeV is the VEV of the SM Higgs boson and $\mathrm{R}\mathrm{e}[{Y}_{\text{D}}]\in \,]0,\sqrt{4\pi }[$ is the Yukawa coupling. The case m_D = 0 (corresponding to the pure Majorana neutrino case) produces no mixing between active and sterile neutrinos. We consider instead the type-I seesaw mechanism [373–376], which requires 0 < m_D ≪ M_R. It gives an explanation for the lightness of active neutrino masses by requiring the sterile neutrinos to be very heavy. The light neutrino mass matrix is given by seesaw formula

$$\begin{equation}{m}_{\nu }\approx -{m}_{D}{M}_{\text{R}}^{-1}{m}_{\text{D}}^{\text{T}}=-\frac{{v}^{2}}{2}{Y}_{\text{D}}{M}_{\text{R}}^{-1}{Y}_{\text{D}}^{\text{T}},\end{equation} \tag{ 4.51 }$$

which is a valid approximation if the sterile neutrinos are heavier than $\sim 1$ eV and at most $\sim 1{0}^{15}$ GeV. Even higher seesaw scale produces too small m_ν . The unitary matrix U diagonalizes the neutrino mass matrix as