The present and future status of heavy neutral leptons

As neutrinos are neutral, they can be of two different types, Dirac or Majorana particles, corresponding to the conservation or not of the leptonic number symmetry. Differently from the case of Dirac particles, common to all other SM fermions, Majorana neutrinos satisfy the Majorana condition

$$\begin{eqnarray}&&\nu ={\nu }^{c}\equiv C{\bar{\nu }}^{T}\,,\end{eqnarray} \tag{ 8 }$$

where C is the charge-conjugation matrix and ν^c is the charge-conjugate of ν. Effectively neutrinos and antineutrinos are not distinguishable. Neutrino masses can be of different types, leading to Dirac or Majorana neutrinos.

A Dirac mass term, $\overline{\nu }{m}_{{\rm{D}}}\nu ={\overline{\nu }}_{L}{m}_{{\rm{D}}}{\nu }_{R}+{\rm{h}}.{\rm{c}}.$, requires the introduction of ν_R and is analogous to that of all the SM-charged fermions. This term conserves lepton number: giving both chiral components the same lepton number so that under a $U{(1)}_{{ \mathcal L }}$ transformation ν_L,R → e^{i η} ν_L,R , the mass term remains invariant.

Majorana masses require only one Weyl spinor ν_L as $-{{ \mathcal L }}_{\mathrm{Majorana}}\,=\tfrac{1}{2}\bar{{\nu }^{c}}{m}_{{\rm{M}}}\nu \,=-\tfrac{1}{2}{\nu }_{L}^{T}{C}^{\dagger }{m}_{{\rm{M}}}{\nu }_{L}+{\rm{h}}.{\rm{c}}.$ This term breaks lepton number by two units. Notice that analogously it is also possible to have Majorana masses for ν_R . If the theory contains also ν_R , it is possible to have Dirac mass term and both Majorana ones for ν_L and ν_R . In this case, the resulting massive neutrinos are of Majorana type, as can be understood by the fact that overall the Lagrangian does not conserve lepton number. In a full theory with 3 ν_L and N ν_R , there will be 3 light Majorana neutrinos and a number N of massive neutrinos that we will denote as HNLs, in the case in which their masses are much larger than the eV-scale. Typically, the latter will be nearly-sterile as only a small admixture of active neutrinos is allowed.

The mass terms discussed above are forbidden in the SM as there are no right-handed neutrinos and a Majorana mass term using only the ν_L fields violates the SM gauge invariance. Consequently, the SM, in its minimal form, does not allow for neutrino masses. How can one extend the SM in order to account for neutrino masses in a consistent framework? A vast number of models have been proposed and most of them require sterile neutrinos and the corresponding HNL. We briefly review the key features and we discuss more in detail the Seesaw type I mechanism and HNLs in section 2.3.

The SM simplest extension can be constructed by adding SM gauge singlets ν_R , typically called sterile neutrinos. The SM gauge group allows for the Yukawa interaction between ν_R , the leptonic doublet $L\equiv {\left({\nu }_{L}^{T},{{\ell }}^{T}\right)}^{T}$ and the Higgs doublet H:

$$\begin{eqnarray}&&-{{ \mathcal L }}_{y}=\overline{L}F\cdot \widetilde{{\rm{\Phi }}}\,{\nu }_{R}+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ 9 }$$

where $\widetilde{{\rm{\Phi }}}=i{\sigma }_{2}{{\rm{\Phi }}}^{* }$. After electroweak symmetry breaking, $\langle \tilde{H}\rangle ={\left({v}_{H}/\sqrt{2},0\right)}^{T}$, a Dirac mass term emerges

$$\begin{eqnarray}&&-{{ \mathcal L }}_{y}\to \langle \tilde{H} \rangle \ne 0-{{ \mathcal L }}_{\mathrm{Dirac}}=\frac{{v}_{H}}{\sqrt{2}}{\bar{\nu }}_{L}{y}_{\nu }{\nu }_{R}+{\rm{h}}.{\rm{c}}.\end{eqnarray} \tag{ 10 }$$

This Yukawa interaction and the resulting Dirac mass conserve lepton number. Indeed, a Majorana mass term for ν_R is not forbidden by gauge invariance and its absence must be imposed by requiring lepton number conservation. In this case, this symmetry goes from being an accidental symmetry of the SM to a fundamental ingredient of the theory of particle interactions. In this sense, this is a major departure from the SM.

Although a viable explanation for neutrino masses, alternatives have been sought as this has several shortcomings: apart from the issue of lepton number discussed above, the Yukawa coupling needs to be exceedingly small, many orders of magnitude smaller than the charged lepton ones, and moreover, one would expect the masses and mixing angles to have a similar hierarchy as the quark ones, a fact that is contradicted by experimental results.

Going beyond dimension-four operators, the SM admits a dimension-five one which is gauge invariant. The so-called Weinberg operator [45]

$$\begin{eqnarray}&&{{ \mathcal L }}_{M,\mathrm{BSM}}=\frac{{c}^{[5]}}{{\rm{\Lambda }}}{L}^{T}\cdot {\tilde{H}}^{* }{C}^{\dagger }{\tilde{H}}^{\dagger }\cdot L+{\rm{h}}.{\rm{c}}..\end{eqnarray} \tag{ 11 }$$

indeed leads to Majorana masses

$$\begin{eqnarray}&&{{ \mathcal L }}_{M,\mathrm{BSM}}\to \langle \tilde{H} \rangle \ne 0\frac{{c}^{[5]}{v}_{H}^{2}}{2{\rm{\Lambda }}}{\nu }_{L}^{T}{C}^{\dagger }{\nu }_{L}+{\rm{h}}.{\rm{c}}..\end{eqnarray} \tag{ 12 }$$

after electroweak symmetry breaking. It violates lepton number by two units and consequently the massive neutrinos are of Majorana type. This operator suggests the existence of a new theory at a scale Λ. The hunt for the new particles and interactions involved is at the center of much of current research in theoretical neutrino physics.

As the Weinberg operator is a low-energy effective term, the key question concerns the full theory which is completed above Λ, including new particles with mass M ≈ Λ. Conserving Lorentz and gauge symmetries, at tree level, there are three possible options for the origin of this operator:

• Seesaw type I [7–11] for a singlet fermion.
• Seesaw type II [11, 46] using heavy triplet scalars.
• Seesaw type III [47, 48] for triplet fermions,

as schematically shown in figure 1.

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These operators can be further embedded in a full theory such as GUT models, in which right-handed neutrinos emerge as a completion of the fermionic representations. These terms are not the only way to generate neutrino masses. Models in which they emerge at loop-level have been extensively studied [49–58].

A typical, albeit by no means ubiquitous, feature of most models behind neutrino masses is the presence of some new neutral fermionic states. Unless a new symmetry forbids a Yukawa interaction between these fermions and the leptonic doublet, generically mixing will arise with the active neutrinos, and potentially other neutral fermions as well. For mass scales in the MeV to multi-TeV, these states will be characterised as HNL with a distinct phenomenology and potentially the possibility to test them.

The smallness of neutrino masses relates to the scale of the new physics but can be ascribed to different approaches (for broad reviews and summaries, see [59–65]).

• Large Λ. From the Weinberg operator, we see that neutrino masses are suppressed (relative to the charged-fermion ones) by v/Λ. In many extensions, Λ is taken to be very large, even close to the Grand Unification scale, ∼10¹⁴ GeV. In this case, couplings are not required to be small and the suppression of neutrino masses is achieved thanks to the heavy masses of the mediator. In this formulation, the Seesaw mechanisms can be embedded in GUT theories: for instance SO(10) models include right-handed neutrinos required to implement a Seesaw type I mechanism. Leptogenesis can also be realised in this framework to explain the baryon asymmetry of the Universe [66–72]. Despite these many advantages, a significant drawback is that these models cannot be directly tested. Some indirect hints could be present in the case of GUT models, such as proton decay and a possible link to gravitational waves due to cosmic strings [73–76]. Moreover, if a stabilising mechanism for the electroweak symmetry breaking scale, e.g. supersymmetry, is absent, the Higgs mass receives corrections which are too large unless Λ < 10⁷ GeV [77–80] for Seesaw type I or smaller than the TeV scale for Seesaw type II models. In contrast, the corrections are too large in type III models if Λ < 10⁵ [81].
• Small couplings. This approach advocates c^[5] ≪ 1. Although generically the naive expectation is for couplings to be of order 1, the SM itself presents many examples of small Yukawa couplings, as small as 10⁻⁶. It is therefore possible that the couplings entering in c^[5] make it a small parameter, and the scale Λ is much lower. The advantage is that the new particles are in principle accessible directly at experiments as far as they couple sufficiently strongly to the SM. A lot of interest has been devoted recently to the TeV model as a prime target for searches at LHC searches. For instance, TeV HNL would induce same-sign dileptons plus jets with no missing energy and lepton flavour violating signals, e.g. [63, 82–85], see section 3. The fermion mass scale can be lowered even further for smaller couplings, with HNL with GeV, MeV and even eV masses. These low-energy Seesaw models have distinct signatures which depend on their mass and flavour mixing, as discussed in the other sections. Key signatures are energy peaks in meson decays, decays into visible SM particles, kinks and possible contributions to neutrinoless double beta decays and other lepton number violating processes, e.g. [12], as we will discuss in sections 4 and 5. The baryon asymmetry of the Universe could be explained also in this context via the ARS mechanism and resonant leptogenesis at low scales, see section 7 and references therein including [86–93]. Small couplings and light sterile neutrinos may naturally be realized in the setting of a warped extra dimension as in [94].
• Quasi-conserved symmetries. As discussed earlier, if a lepton symmetry is preserved, Majorana neutrino masses are zero. Therefore, a small breaking of such symmetries would explain the smallness of masses. This is the principle the inverse [95–98] and linear [99, 100], as well as extended, Seesaws. This approach is considered 'natural' as setting the small parameters to zero restores the lepton symmetry. The smallness of neutrino masses derives from the partial cancellation of the contributions of multiple HNL to it. As the lepton number violating parameters are small, the heavy neutral lepton states are pseudo-Dirac, i.e. are Majorana particles with nearly degenerate masses and opposite CP-parity [101, 102]. They can have low scales, including the MeV–TeV one, and large mixing, making them accessible to colliders and other terrestrial experiments. As the lepton symmetry is midly broken, lepton number violating processes in this type of framework are typically suppressed for Majorana masses above the electroweak scale while they remain potentially observable for long lives particles.

As neutrino experiments do not provide any indication of the scale of the new physics and theoretical considerations do not provide a unique preference for a given scale, both high energy, Λ ≫ TeV, and low energy Λ < multi TeV models can be considered. Above we discussed the advantages of high-scale Seesaw models, mainly their simple explanation of the smallness of neutrino masses, their possible embedding in GUT theories and successful leptogenesis, but at the same time their drawbacks, namely their untestability and the possible destabilisation of the Higgs mass.

Low-scale Seesaw models, and consequently the existence of HNLs, have strong support from various points of view (for an in-depth discussion see [64]). First of all, they are directly testable at experiments and therefore should be a prime choice for experimental searches in order to unveil the origin of neutrino masses. Secondly, in certain mass ranges, they also offer a framework to explain the baryon asymmetry and they can have connections with dark sectors, such as dark photons, dark matter, dark scalars, as discussed in section 2.5. In the minimal formulation of the νMSM, they can even explain evidence BSM, namely neutrino masses, dark matter and the baryon asymmetry, with a very small number of new particles and parameters. They are technically natural, meaning that small parameters are justified if the symmetry of the system is augmented by setting them to zero. In this sense, both options of low Λ and quasi-preserved symmetries are natural as they related to the small breaking of a lepton symmetry. Finally, it is non-trivial that the SM might be valid up to the Planck scale [103]. In this scenario, there is no intermediate scale above the electroweak one and neutrino masses must be generated by a low-energy mechanism, calling for Λ below the electroweak scale. The current lack of hints of new physics at the LHC might provide hints in this direction [63, 84].

This picture might be challenged by the possible SM vacuum meta-stability which may indicate the presence of the intermediate scale between the Planck and Fermi scales [103–107]. For a certain domain of the masses of the Higgs boson and the top-quark the effective potential for the Higgs field has an extra, deeper minimum, see figure 2. Given the latest measurements of the top quark mass [108] and the Higgs boson mass, the evidence for the meta-stability is rather slim and is not exceeding 2σ. If the SM vacuum is indeed meta-stable, cosmology may require modification of the SM at energies ≳10¹⁰ GeV, though even this conclusion can be evaded [106].

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Vice versa, a sufficiently long-lived or stable vacuum could be interpreted as tentative evidence for the absence of new physics between the electroweak and Planck scales [103], which can be interpreted as motivation for low-scale Seesaw scenarios [64]. The vacuum stability can also be affected by the impact that the HNLs themselves have on the Higgs potential [109], which imposes an upper bound on the mixing angle that is shown in figure 35.

The minimal LR symmetric model based on the SU(3)_c × SU(2)_L ×SU(2)_R × U(1)_B−L gauge group [293] may explain parity violation of weak interaction by restoring it and high energy scales and breaking SU(2)_R × U(1)_B−L to U(1)_Y spontaneously [294]. In the minimal model [7, 9] with scalar triplets Δ_LR , the masses of Majorana neutrinos N is tied to the LR breaking scale. Moreover, one has two sources of neutrino mass

$$\begin{eqnarray}&&{M}_{\nu }=-{M}_{D}^{T}{M}_{N}^{-1}{M}_{D}+\displaystyle \frac{{v}_{L}}{{v}_{R}}{M}_{N},\end{eqnarray} \tag{ 18 }$$

where the first term is usually referred to as type I and the second as type II contribution [295]. In contrast to the mostly sterile HNLs in type I Seesaw, the HNLs now also carry gauge interactions, mediated by heavy W_R , Z_LR gauge bosons. The Majorana masses m_N and LNV originate from a Yukawa interaction with the triplet Higgses, and thus N's may also couple to the SM sector via such interactions and scalar mixing. This leads to couplings such as hNN and ggΔ_R , which makes it possible to produce Ns at colliders in the Higgs sector as well.

The desire to restore parity leads to imposition of near-hermiticity or symmetricity on all the Yukawa matrices, effectively leading to V_R = V_L = V_ckm. With charged gauge currents flavor structure fixed, flavor constraints coming from K and B meson mixing come about. Early studies [296] had the lower limit ${M}_{{W}_{R}}\gt 2\,\mathrm{TeV}$, which has been revisited [297, 298] and updated to the current lower limit of about a few TeV, depending on the choice of LR parity [299], see [300] for recent reappraisal. While the early bound did not motivate the W_R searches at the Tevatron, the LHC has indeed surpassed the flavor constraints and is in the position to search for Ns even in the minimal LR model, as discussed in section 3. Finally, the Dirac mass matrices M_D are not arbitrary in the LR model, once parity is imposed. This is in contrast to the type I Seesaw, where there is significant freedom, encoded in the Casas–Ibarra parametrization [132], in the couplings that control the ν − N mixing angle. This connection between Majorana and Dirac mass matrices is particularly clean in the case of ${ \mathcal C }$-parity [301] and a bit more involved in the case of ${ \mathcal P }$ [302].

Apart from the collider signatures, low scale LR symmetry with ${M}_{{W}_{R}}\sim \mathrm{TeV}$ can manifest itself in many other phenomenological observables, related to lepton number and flavor violation. To focus on the most fundamental question of lepton number violation, one should point out the interplay between the LHC and neutrinoless double beta decay searches [63, 298, 303]. As pointed out time ago [295], LNV can manifest itself also in 0ν2β not only through the exchange of light neutrinos, but also via parity inverted diagrams, where W_L , ν_L are swapped for their right-handed counterparts. This interplay was exemplified in the case of type II dominance [303] and in the presence of additional interactions [304]. The upcoming 0ν2β decay searches will thus complement the LNV searches at the LHC, as seen on the left of figure 15.

In minimal models, the HNL production and decay are controlled by SM interactions and the mixing between HNLs and the active neutrino and typically result in relatively long lifetimes if the masses are in the MeV–GeV range. This is the basis of searches, e.g. at colliders and beam dump experiments. In models with additional interactions, HNL have new sources of production and decay channels. In the dark sector, interactions can even be strong and lead to enhanced scattering cross sections and fast decays, changing radically the HNL phenomenology. For instance, fast decays into neutrinos or other 'invisible' particles can weaken collider and beam dump limits, as the HNL would have not reached the detector or their signals are suppressed by the branching ratio into visible channels. Moreover, new signatures can emerge and have been advocated to explain the low-energy excess of electron-like (LEE) events at the MiniBooNE experiment [327–332], see figure 4.

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In this picture, HNLs, produced in the neutrino beam [367, 368] or directly inside the MiniBooNE detector via scattering [338, 340, 358, 369–374], subsequently decay to missing energy and an e⁺ e⁻ pair, e.g. ${\nu }_{h}\to {\nu }_{{h}^{{\prime} }}\,{e}^{+}{e}^{-}$. If the e⁺ e⁻ pair is sufficiently collinear or the energy of one of the particles falls below the detector energy threshold, the electromagnetic (EM) shower is reconstructed as a single electron or photon. In view of the recent results from the MicroBooNE experiment showing that the leading SM explanation of the excess is strongly disfavoured [375], and the tension of the most popular BSM explanation of the excess, 3 + 1 neutrino oscillations, with other oscillation searches and cosmology [43, 376–381], models of e⁺ e⁻ have become all the more intriguing.

New opportunities for their search are being offered by neutrino facilities and in particular by short-baseline accelerator neutrino experiments and the Near Detector (ND) complex of long baseline ones, such T2HK and DUNE. A host of new generation liquid argon (LAr) experiments within the Fermilab SBN program [382], including the MicroBooNE experiment [383] and the currently running ICARUS one [], are ideally suited for these searches. The hodoscopic nature of LAr detectors allows for unprecedented calorimetric and spatial resolution, potentially distinguishing the e⁻, e⁺ e⁻, γ, γ γ EM final states, as well as final state hadron multiplicity. For models in which the HNLs are produced in scattering, typical signatures include a primary neutrino interaction and a HNL decay vertex that can be displaced from it due to the finite HNL lifetimes, and EM showers, in some cases reconstructed as two separate electrons, that do not point back to the original vertex, due to missing energy. For incoherent scattering, there is also a proton track at the initial vertex. There are four distinct topologies in LAr that can be searched for—see figure 5.

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Beyond the SBN facilities, these models can leave signatures at e⁺ e⁻ colliders, e.g. BaBar [384, 385], Belle [386], and the next-generation experiments Belle II [387, 388] and BESIII [389]. In particular, monophoton searches at these experiments can test the semi-visible dark photon explanation of the muon g_μ − 2 []. Fixed-target experiments are another testbed for these models. HNLs can be produced in kaon decays at NA62 [390], leading to cascade decays with displaced vertices. Searches for inelastic dark matter produced in electron bremsstrahlung at NA64 are also sensitive to models of this kind [391–393].

Phenomenological type I Seesaw. At the LHC, HNLs can be produced in the GeV mass range through the decays of heavy meson, τ leptons, W bosons, H bosons, and even top quarks. If new interactions or gauge symmetries exist in nature, or if masses are above the electroweak (EW) scale, then HNLs can be produced in many more ways. For the minimal model, high-mass channels include the Drell–Yan process [396, 397], gluon fusion [398–400], W γ fusion [84, 401–403], and same-sign WW fusion [404, 405]. In the case that HNLs are Majorana particles, decays are allowed into both lepton number conserving and lepton number violating channels; if they are Dirac, only lepton number conserving processes are allowed. Both can mediate lepton flavor violating processes. Moreover, depending on the HNL's mass and mixing angle, the decays may be either prompt (short lifetime) or displaced (large lifetime) from the production vertex.

We start with current results from ATLAS and CMS with the case for displaced and prompt decays of HNLs in the minimal extension of the SM. Results that have been released so far at 13 TeV include an ATLAS displaced signature search, that sets constraints for mixing with muon-neutrinos at the level ∣U_μ4∣² ∼ 10⁻⁶ for HNL masses between 4 and 10 GeV based on a data sample of 33–36 fb⁻¹ [406]. The results are shown in figure 6. Reinterpretation of these results in terms of models with more than one HNL was performed by [407]. Similarly, CMS has a search with 35.9 fb⁻¹ of integrated luminosity at 13 TeV, which obtained the first direct bounds on HNLs up to and above 500 GeV, and giving ∣U_e4∣², ∣U_μ4∣² < 1 up to TeV HNL masses [408]. The results are shown in figure 7.

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A newly released CMS result of a low mass HNL search [409] uses the full dataset of Run 2 corresponding to 138 fb⁻¹, using the final state where the HNL decays in opposite charged leptons and a neutrino, hence the final state signature is three charged leptons of which two are displaced. The results are shown in figure 8. Both the Majorana (top) and Dirac (bottom) HNL decay channels are probed, using the fact that the HNL production cross section is the same but the HNL lifetime is twice as large for a Dirac HNL due to the missing LNV decay channels. The results of this study set constraints for mixing with muon-neutrinos at the level ∣U_μ4∣² ∼ 4 · 10⁻⁷ for HNL masses between 8 and 14 GeV. The sensitivity in the electron neutrino channels is around ∣U_e4∣² ∼ 10⁻⁶ for HNL masses in the same region.

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Both ATLAS and CMS have reported results at lower center-of-mass energies [410–413].

Extended gauge symmetries and effective field theories. HNLs may arise from an extended gauge symmetry, such as U(1)_B−L , which promotes baryon and lepton number conservation to a gauge symmetry, or LR symmetric theories, which restore parity at high-energy scales. For details on the construction of such models, see section 2.5. In contrast to HNLs in minimal type I models, heavy Majorana neutrinos N in extended gauge scenarios couple directly to new gauge bosons. Gauge bosons may have couplings that are naturally un-suppressed, as in LR and B−L theories, or masses that are naturally light, as in B−L theories. In these cases, HNLs can be produced copiously at the LHC, as long as the mediators are not too heavy and couplings are not too small.

The paradigmatic example of HNL production through new gauge bosons is the Keung-Senjanović process [396] within the minimal LR model, where a right-handed (RH) W_R can be produced in the s-channel and decays into a heavy Majorana N and a RH-charged lepton of various flavors. The N, in turn, decays into another charged lepton and two partons (quarks and antiquarks). Depending on the relative sizes of the W_R and N masses, one can obtain four qualitatively distinct signatures: (i) when ${M}_{{W}_{R}}\gtrsim {m}_{N}$, all final state particles are well separated and one can search for the classic collider signature pp → W_R → ℓ_i ℓ_j jj [396]. The leptons ℓ _i and ℓ _j may have the same or different electric charges and flavors, and all partons typically have momenta that scale like ${M}_{{W}_{R}}$. (ii) When ${M}_{{W}_{R}}\gg {m}_{N}$, N is boosted and its decay products are collimated. The resulting signature is pp → W_R → ℓ_i J, where J is a multi-prong jet and can be identified as the boosted N [414–416]. The momentum of ℓ and J are comparable to ${M}_{{W}_{R}}$. Importantly, J can be displaced [417]. (iii) When m_N is within the LHC's kinematic reach but ${M}_{{W}_{R}}$ is not, then the signature pp → ℓ_i ℓ_j jj is still possible through a far-off-shell W_R [418]. In this case, the kinematics of final state particles are comparable to m_N and strongly resemble the minimal type I scenario but result in different angular distributions [417, 418]. This limit also has connections to the so-called νSMEFT framework [419–421]. (iv) When m_N ≳ W_R , the kinematics and search methodology are analogous to a high-mass version of searches for HNLs in the minimal HNL scenario. The production HNLs through Z_R gauge bosons can similarly be categorized. Due to the heaviness of right-handed gauge bosons, vector boson fusion and associated production channels typically feature cross sections too small to be produced at the LHC.

Beyond production mechanisms through gauge bosons, HNLs in extended gauge theories can also be produced via Higgs sector couplings and Yukawa interactions. For example: the SM-like Higgs state at 125 GeV can mix with another scalar that breaks the extended gauge symmetry and is the source of heavy neutrino mass origin (such as the neutral component of an SU (2)_R triplet ${{\rm{\Delta }}}_{R}^{0}$ in the minimal LR model). In the mass basis, one of them has an exotic Higgs decay h → NN [422] that again results in two same-sign leptons and four jets, all possibly displaced if m_N is low enough. Likewise, one can produce the extra scalar resonance ${{\rm{\Delta }}}_{R}^{0}$ through gluon fusion or associated W/Z production [423], which leads to an even larger potential signal, as long as the WW, ZZ final states are suppressed, say for m_Δ ≲ 160 GeV. The final states are again LNV with soft ℓ^± ℓ^± and additional jets and may be displaced, thus forming one or more displaced vertices. It is also possible for $h\to {{\rm{\Delta }}}_{R}^{0}{{\rm{\Delta }}}_{R}^{0}$ to occur, which would subsequently produce 4 Ns and a striking ΔL = 4 final state with essentially zero background. It is important to stress that the production of NN pairs via a Seesaw Higgs occurs not only in LR scenarios but also in pure U(1)_B−L [424–426].

For HNLs with ${M}_{{W}_{R}}\gt {m}_{N}\gt { \mathcal O }(100)$ GeV, the final state products can be resolved and one gets a ΔL = 2 final state with ℓ^± ℓ^± jj. Current results at 13 TeV in the resolved channel exclude W_R masses up to about ${M}_{{W}_{R}}\approx 5$ TeV [427–429]. The precise bound depends on the amount of data recorded, with limits using 138 fb⁻¹ summarized by the green line in figure 9 (left). Weaker limits are reported by the two experiments at lower center-of-mass energies [413, 430–432]. For the boosted channel, experimental searches by CMS [427] and ATLAS [433] are more sensitive to the parameter space m_N ≲ 100 GeV and ${M}_{{W}_{R}}\gtrsim 1$ TeV, and exclude ${M}_{{W}_{R}}\lt 5.3$ TeV. This is illustrated in the blue line in figure 9 (left) for CMS and figure 9 (right) for ATLAS. For even lighter masses with m_N ≲ 50 GeV, a significant number of N's decay outside of the detector and one can recast [417] the prompt search for $W^{\prime} \to \ell \,+\,\mathrm{MET}$ [434] (superseded by [435]), which results in a stringent bound of about 5–6 TeV in the electron and muon channels, as shown in the orange contour on the right of figure 9. Using a reinterpretation of Run 1 data, searches for N via far-off-shell W_R only exclude ${M}_{{W}_{R}}\lt 1$ TeV for m_N < 500 GeV [418].

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Type III Seesaw. It may be the case that HNLs exist but that they are not singlets of the SM's gauge symmetries. In particular, they may appear as the neutral component of a fermionic EW triplet. In this case, not only do HNLs exist but also electrically charged partners E^± (these states are sometimes also labeled T⁰, T^±, Σ⁰, Σ^±, or L⁰, L^±). Furthermore, N and E^± carry SU (2)_L gauge charges, which allows them to be produced in pairs without active-sterile mixing suppression. At the LHC, NE^± pairs can be produced through the Drell–Yan and vector boson fusion processes. E⁺ E⁻ and E^± ℓ^∓ pairs can be produced through the Drell–Yan, photon fusion, gluon fusion, and vector boson fusion; the channels have a non-trivial interplay between mixing, PDF enhancements, and coupling enhancement/suppression [63, 399, 436]. Decays of heavy, vector leptons typically proceed to EW gauge bosons. In principle, N and E^± have the same masses due to gauge symmetry (they belong to the same multiplet). However, small, QED-included mass splittings between E^± and N open the E^± → N π^± decay mode [437]. Notably, the production of NE^± and E⁺ E⁻ pairs can result in final states with multiple leptons and high-p_T jets as well as final states with only many-lepton [82, 437–439]. In these cases, all high-p_T objects carry momentum comparable to the masses of N and E^±. For particularly heavy triplets, one also anticipates final states with boosted topologies [440].

Presently, searches by the ATLAS and CMS experiments for NE^± pairs in few- and many-lepton final states stringently constrain Type III leptons. More specifically, with 139 fb⁻¹, the ATLAS experiment has excluded masses below m_L < 790 GeV at 95% CL in the same- and opposite-sign dilepton, MET, and jets configuration [441]. The experiment has also excluded cross sections larger than σ = 22 (7.5) fb for representative benchmarks m_L = 400 (700) GeV in multi-lepton channels [442, 443]. These results are summarized in figure 10. In events with at least three charged leptons, the CMS experiment has excluded NE^± with masses below m_L = 880 GeV with 137 fb⁻¹ at 13 TeV under specific mixing assumptions [444, 445]. At 13 TeV with 36 fb⁻¹ of data, CMS has excluded triplet leptons with masses in the range m_L = 370–900 GeV in the multi-lepton final state under various mixing and branching rate assumptions, including the contribution from τ leptons [446]. At 8 TeV with 20 fb⁻¹ of data, ATLAS has excluded NE^± pairs with masses below m_L ≲ 100–468 GeV in the 3- and 4-lepton channels [447]. Less constraining results at 7 TeV have also been reported [448]. The precise bound depends on the specific lepton flavor configuration assumed. It is important to stress that while some of these searches have included signal processes with τ leptons, the flavor category remains understudied.

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Reinterpretation of LHC results. Experiments searching for HNLs typically report their results within simplified models consisting of a single HNL coupled to a single lepton flavor, as in figures 6 and 7. However, the type I Seesaw—which aims to describe neutrino oscillations—necessarily features at least two HNLs coupled to several flavors (see section 2.3). Reference [407] has performed a detailed reinterpretation of the latest ATLAS search [406] for prompt HNLs in W decays within a minimal type I Seesaw with two HNLs. It has shown that the exclusion limits can differ by several orders of magnitude from the limits obtained under the single HNL assumption. Hence comparing the mixing angles from a realistic model to the reported limits could lead to wrongly excluding large regions of the parameter space. Some results of [407] are shown in figure 11. The colored regions contain all the possible exclusion limits when scanning over the allowed combinations of mixing angles (see figure 3). As one can see, many allowed parameter sets lead to exclusion limits that are much weaker than the single-flavor ones. This drastic change in the experimental sensitivity is related to the opening of new channels (to which the search may not be sensitive) if more than one mixing angle is allowed.

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To facilitate reinterpretation, [407] proposed a reweighting method which allows—using only a handful of constants that can be easily computed and published by experiments—to exactly extrapolate the expected signal to any combination of mixing angles. This method is applicable to other models of feebly-interacting particles. In order to perform an accurate reinterpretation, modeling the background and the systematic uncertainties is also critical. To this end, [407] recommends that experiments release (preferably) their likelihood or (at least) the covariance matrix between the background counts in all bins from all channels. This suggestion is in line with the recommendations from the LHC Reinterpretation Forum [449].

Phenomenological type I Seesaw. New results will be released by both collaborations in 2022 based on the full Run 2 data and including more final states, for example, the final states where the HNL decays in a charged lepton and hadrons. These final states have larger branching ratios than the full leptonic ones, but tougher background conditions to fight. Nevertheless with the help machine learning (ML) techniques one expects that the resulting limits can be competitive with, potentially even by-pass, those from the 3-lepton final states analyses. Combining the overall ATLAS+CMS results one can anticipate to probe the coupling region down to ∣U_{μ N} ∣² ∼ 10⁻⁷ for HNL masses in the range of 5–15 GeV in the muon channel, which is the most sensitive channel.

Other channels that could be explored include ultralight HNLs from decays of bottom and charm mesons, as already searched for in the LHCb experiment. The large heavy meson decay samples that can potentially be collected at the experiments, e.g. via so-called 'data parking' techniques as exercised by the ATLAS and CMS experiments in 2018, may allow for additional sensitivity in the mass range below 5 GeV. CMS demonstrated in Run 2 that B meson enriched data samples containing more than 10¹⁰ B hadrons per year can be collected, leading to a potentially large source of neutrinos from leptonic B decays.

In 2022 the LHC will start a new run, called Run 3, which will continue till the end of 2025, and should more than double the luminosity accumulated so far, allowing to accumulate a total data sample of as much as perhaps 400–450 fb⁻¹. The increase in statistics will in particular allow to probe a mass range towards about 20 GeV for small couplings. The next phase will be the high-luminosity LHC option, i.e. the HL-LHC. For this era, machine and detector upgrades have been designed and are being constructed. These will replace present detector components that will retire after Run 3. The HL-LHC operation should start before the end of this decade, and collect a total data sample corresponding to about 3 ab⁻¹ per experiment in the following O(10) years. A simple scaling of the present results to the higher luminosity sample compared to the present one, and assuming the experimental upgrades will allow LHC central detectors to collect data samples of the same quality as the present ones despite higher pile-up levels, these data could allow ATLAS and CMS to probe the parameter space region towards mixing of ∣V_{ℓ N} ∣² ≈ 10⁻⁸ in the mass range of 10–20 GeV, i.e. complementing to the sensitivity by fixed-target beam-dump (and neutrino) experiments in a higher mass range. For lower masses, the increasing lifetime of the HNLs that is associated with smaller mixing reduces the potential gain with the present central detectors due to the reduced acceptance for the HNL decays in the detectors. To have more firm predictions of the future potential sensitivity of ATLAS and CMS, the full Run 2 analyses need to be completed first, and detailed simulation studies are required by the experimental groups to map the sensitivity contours for the HL-LHC's full potential. We strongly encourage such studies to be conducted in the forthcoming years.

In addition one can further certainly envisage that the future data selection efficiency can be enhanced by tools that are now being explored, but are still in an early phase, such as advanced ML techniques or any new generation of such advanced analysis algorithms that may become available in some years from now, resulting from the present ongoing wave of new developments. Early studies on applying ML techniques to HNL searches appear encouraging [450]. Furthermore, an important improvement of the new upgraded detectors that will become available as an additional handle for LLP analyses is the more global precise timing capabilities of the detectors, with an aimed precision of 30 pico-seconds. Hence, the hunt for HNL signals at ATLAS and CMS will certainly continue to improve over the lifetime of the operation of the LHC and should increasingly cover the mass region above O(1) GeV.

Several phenomenological studies have been conducted for HNL searches at the LHC. Focusing first on light HNLs, in [451] a phenomenological study was presented on the potential sensitivity to low-mass HNLs at the HL-LHC. The aim of that work was to investigate what HNL parameter space could be in reach in principle, based on present detectors without significant modifications to detector coverage but assuming optimizations to obtain similar to present reconstruction efficiencies, understanding of backgrounds etc. Specifically, it was assumed: to have a minimal displacement 5 mm to reduce B meson background; have a displaced vertex invariant mass of 2 GeV to suppress backgrounds from interactions with the material; a displaced vertex has to have at least two tracks with ΔR ≥ 0.1; a primary vertex lepton p_T (e, μ, τ) = 30, 25, 140 GeV for ATLAS and CMS; a primary vertex lepton p_T (e, μ, τ) = 10, 15, 50 GeV for LHCb; all spatial momenta ∣p∣ > 5 GeV to make it out of the magnetic field; muon chambers at CMS can be used for displaced vertex reconstruction. SM backgrounds were included in the study. These, somewhat optimistic, results shown in figure 12 give a clear target for the experiments for the parameter phase space that could in principle be covered under ideal conditions by future LHC data.

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As mentioned before, dedicated analyses at even lower masses based on searches in charm and bottom decays is also an interesting channel to pursue these experiments. Such searches will be discussed in the next section for LHCb; for ATLAS and CMS has no results to date been released that allow to evaluate the power of these analyses for the future.

For masses above a 100 GeV, the prospect of discovering HNLs is enhanced by the opening of several production mechanisms with competitive cross sections, as shown for example in figure 13. Some of these production channels involve vector boson fusion, which enhance sensitivity to HNL parameter space and to new models [85, 402, 405, 452, 453]. With such large masses, HNLs can decay to on-shell EW bosons that are potentially boosted. In addition to ML techniques [450], the development of novel jet-veto techniques shows promise to greatly improve signal-to-noise ratio in searches for heavy HNLs [85, 454]. As shown in figure 14, the multi-lepton and MET channel, active-sterile mixing as small as ∣V_{ℓ N} ∣² ∼ 10⁻⁵ − 10⁻² in the electron, muon, and tau sectors can be probed at 95% CL for heavy Dirac neutrinos masses in the range m_N = 200–1200 GeV with 3 ab⁻¹ of data at $\sqrt{s}=14$ TeV [85, 454]. Results are shown for the Dirac channel but comparable sensitivity is expected for the Majorana channel [85, 454].

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Extended gauge symmetries and effective field theories. Due to the significantly larger datasets, upgraded detector experiments, and improved Monte Carlo capabilities, the sensitivity to HNLs in extended gauge models is very promising at the HL-LHC. As shown in figure 15 (left), direct searches for HNLs and W_R in the standard and displaced channels are sensitive to ${M}_{{W}_{R}}$ well above 5–6 TeV with the full Run 3 dataset [417, 455]. Higher masses can be probed with the full HL-LHC dataset [415, 456, 457]. Searches for N via far-off shell W_R can probe ${M}_{{W}_{R}}\sim { \mathcal O }(10)$ TeV for m_N < 1 TeV with 1 ab⁻¹ of data [418]. More generally, searches for short- and long-lived HNLs with masses m_N > 100 GeV via dimension six operators show that EFT scales as large as ${\rm{\Lambda }}\sim { \mathcal O }(10)$ TeV can be probed [418, 458]. For smaller masses, active-sterile mixing below ∣V_{ℓ N} ∣² ∼ 10⁻¹⁰ can be probed for a fixed ${\rm{\Lambda }}={ \mathcal O }(10)$ TeV [459–461]. As for the Higgs-mediated channels, in the case of ${gg}\to {{\rm{\Delta }}}_{R}^{0}\to \,{NN}$ production, the LHC is able to probe significantly higher LR scales with ${M}_{{W}_{R}}\sim { \mathcal O }(10\,\mathrm{TeV})$, depending on the size of the Higgs-triplet mixing. With such high ${M}_{{W}_{R}}$, the N decay rates are further suppressed and final states may feature pairs of significantly displaced NN vertices. For further details, see figure 8 in [423].

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Finally, in minimal U(1)_B−L models, the LHC detector experiments with ${ \mathcal O }(100)$ fb⁻¹ of data are sensitive to $Z^{\prime}$ bosons as heavy as 4–5 TeV for effective B−L couplings of g_B−L = 0.05 [462, 463]. With more data, larger masses and smaller couplings can be probed. Due to its contribution to the Drell–Yan continuum, it is also possible to constrain very heavy $Z^{\prime}$ from B−L models in precision measurements of low-p_T dilepton pairs; the precise sensitivity at Run 3 and the HL-LHC depends on the $Z^{\prime}$ mass and coupling [425, 464, 465]. A particular novelty of U(1)_B−L models is that NN pair production is possible through coupling to the Z_B−L gauge boson and B−L Higgs mixing. Among other channels, this implies the potential production of displaced, long-lived NN pairs [423, 426, 466].

Type III Seesaw. If HNLs are the neutral components of a leptonic, EW triplet, as in the type III Seesaw, then the HL-LHC will greatly improve present sensitivity. In particular, the increase in integrated luminosity will increase sensitivity to heavy lepton pairs up to m_L = 1.8–2 TeV with 3 ab⁻¹ of data [63, 436]. This assumes the baseline analysis of [439], and varies depending on specific flavor-mixing assumptions. Such sensitivity also holds for many-lepton final states as well as multi-lepton finale states with fat jets, i.e. boosted, hadronically decay W/Z/H bosons [440, 467]. However, further studies are needed to investigate the ultimate sensitivity achievable when fully leptonic, semi-leptonic, and fully hadronic channels are combined.

The current LHCb results involving HNLs cover a wide mass region. The mass region can be divided into a low mass (m_N < m_W ) region and a high mass region (m_N > m_W ). In the low mass region LHCb can be competitive with other experiments. For the high mass region this is more challenging.

In the low mass region, the published result by LHCb so far covers the B⁺ → μ⁺ N → (π⁺ μ⁻) channel [468] via off-shell W decays and is shown on the left-hand side of figure 16. The published LHCb result was later corrected due to the initial use of incorrect decay rates for B^± and N [469]. In the high mass region the published result covers the W⁺ → μ⁺ μ^± jet channel via on-shell W decays and is shown on the right-hand side of figure 16 [470]. The searches via off-shell W decays suffer from a stringent cut on the B⁺ meson invariant mass. This cut has been removed for future studies where contributions from B⁰ and B⁰ _c will also play a central role.

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New results will be released in 2022 based on the full Run 2 data. The new results will cover both on- and off-shell W productions of HNLs. The first results to be published will most likely be with HNLs decaying in π μ final states. For off-shell W productions of HNLs the decay chain will therefore include, but not be limited to B⁺ → μ⁺ N → (π⁺ μ⁻). Further extensions, especially using B_c initial states should be expected. The increase of the delivered luminosity during Run 3 will also be accompanied by an essentially new detector [471]. For LHCb this translates to a total integrated luminosity of recorded data for Run 3 of about 50 fb⁻¹. Crucially, the yield for leptonic channels will be increased by a factor 10 and for hadronic channels by a factor 20. This is due not only to the increase in luminosity but also to a full-detector readout at 30 MHz. This, together with the removal of the hardware trigger stage will allow for an output of 2–5 GB s⁻¹ to storage. The LHCb experiment during Run 1 and 2 was able to cover low region of phase space thanks to its low p_T reach. This will increase even further for Run 3 thanks to a new tracker which will allow good efficiency down to particles with a p_T of about 0.5 GeV. The greater p_T resolution will be accompanied by a drastic reduction in ghost rate and a large gain in reconstruction time.

Further development should be expected with the increase in luminosity due to the HL-LHC conditions. A second upgrade has been already proposed and is currently under consideration by the funding agencies [472]. The upgrade, labeled Upgrade II, will potentially make LHCb the only general purpose flavor physics facility in the world on this timescale and a general purpose experiment in the forward direction. This will be crucial for exploring HNLs coupling to leptons in all generations. Nevertheless, LHCb was not built to work under HL-LHC conditions and therefore some of its key elements will have to be replaced. For example the whole tracking apparatus and some particle identification. If this will happen, LHCb will be able to explore parameter space in regions which will remain unexplored at the LHC. This will be particularly true for the semi-displaced regions just above the charm threshold.

The FASER experiment was proposed to be installed in the very forward region. FASER, which stands for ForwArd Search ExpeRiment will be in an excellent position to search for low mass and extremely weakly interacting particles at the LHC [476]. FASER uses the same interaction region as ATLAS and is located in the line-of-sight, behind 100 m of rock shielding, 480 m downstream from the ATLAS interaction point, in the unused service tunnel TI12. The LHC magnets sweep away charged particle backgrounds hence the experimental area experiences low radiation and low beam backgrounds. The FASER experiment (Phase1) has been approved on March 5th 2019. FASER consists of a tracker system, veto layers, electromagnetic calorimeter modules, a decay volume, and a permanent magnet. The detector is about 5 m long; the decay volume has a diameter of 20 cm and a length of 1.5 m. It makes use of spare ATLAS silicon tracking modules and LHCb electromagnetic calorimeter modules. FASER is ready to take data in 2022. An extension of the experimental equipment for neutrino detection based on emulsion technology is the FASERν experiment [477].

FASER will be able to place constraints on GeV scale HNLs for all flavors of mixing, ∣U_eN ∣², ∣U_{μ N} ∣², ∣U_{τ N} ∣² [478]. The sensitivity reach for FASER for HNLs is shown in figures 17 and 18, compared to other experiments. New leading results can be produced on ∣U_{τ N} ∣² as seen from the figure. An upgraded version of FASER, called FASER2 will also be able to produce leading results on the other HNL couplings as well. FASER2, not yet approved, is a larger acceptance version of FASER and proposed for the HL-LHC operation of the LHC. Recently FASER2 has been included in the FPF facility discussed below.

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The Scattering and Neutrino Detector (SND) at the LHC (SND@LHC) experiment is first and foremost a high efficiency neutrino type identification experiment [481]. It is installed in the TI18 tunnel at a distance of 480 m from the ATLAS interaction (on the opposite site of ATLAS compared to FASER) with a pseudorapidity range coverage of 7.2 < η < 8.6. The installation happens in a symmetrically and opposite location with respect to FASER and therefore SND@LHC will also benefit from the same rock shielding and LHC sweeping magnets.

The target region of the detector is equipped with both offline and online subdetectors. The offline detection consists of emulsion cloud chambers. These chambers are stacked with scintillating fibers; a muon identification system is placed downstream. Different particle identification techniques are used to identify different charged particles. Micrometric resolutions of the vertexes in combination with a timing resolution of around 100 ps make this target an ideal place to study displaced particles.

The SND@LHC collaboration has outlined different techniques that will be used to search for physics BSM. Light dark matter can be searched for using scattering off protons with both elastic and NC/CC (inelastic) signatures. When displaced particles decay to a pair of charged particles inside the target region, SND@LHC can also be sensitive to portals, among which HNLs play an important role. In the case of HNLs mixing with a ν_τ, the calorimeter property of the target are used to reconstruct the decay N → π⁰ ν. Figure 17 [479] shows what was previously reported in the literature. It can be seen that during Run 3 SND@LHC can explore a new parameter space region if HNLs mix exclusively with ν_τ . The explorable mass region sits below 1 GeV and that should close the existing gap between the CHARM and the DELPHI experiments. For a potential future upgrade SND@LHC will also be able to probe unexplored parameter space regions if HNLs mix exclusively with ν_μ .

The MATHUSLA project [482] proposes a very large surface detector based on an affordable technology to detect decays of ultra long-lived neutral weakly interacting particles, with proper c τ lifetimes up to 10⁷–10⁸ meter. This would allow coverage of lifetimes that range from the start of the early Universe to the Big Bang Nucleosynthesis (BBN) epoch. The presence of remaining heavy long-lived particles (LLPs) in that era could significantly disturb the abundance of the elements, which have strong constraints from data. Hence it means that early Universe LLPs should not have lived significantly beyond the start of the BBN phase. MATHUSLA is positioned physically at a significant distance from the LHC detectors interaction point, i.e. more than 100 m, and needs to be huge in size in order to have a suff × 100 × 30 m, build up from individual smaller modules so that a phased installation over years will be possible. Decays of LLP neutral particles will be detected in multiple planes of extruded scintillator, and perhaps complemented with resistive plate chambers.

MATHUSLA will be sensitive down to 10⁻⁹ for mixing matrix elements involving the e and μ flavors, and down to 10⁻⁸ for τ flavored mixing [482]. The sensitivity curves for MATHUSLA for HNLs are shown in figures 18 and 19(a).

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CODEX-b has similar new particle search goals as MATHUSLA but it proposes a detector at a shorter distance to the interaction point, needing a smaller detector volume to detect the decays. CODEX-b is proposed to be installed in the LHCb cavern, about 20 meters away from the interaction point in a 10 × 10 × 10 m³ area to the side of the LHCb experiment. A lead shield is added between the interaction point and the new detector, in order to stop hadronic activity coming from the interaction point. The detector volume will consist of multi-layered planes of resistive plate chambers, detecting the decays of neutral weakly interacting LLPs. Initial test measurements in the anticipated future CODEX-b area have been performed and it was found that the backgrounds were indeed as anticipated. CODEX-b can be sensitive down to 10⁻⁸ for mixing matrix elements involving the e and μ flavors, and down to 10⁻⁶ for τ flavored mixing [480]. The expected sensitivity is shown in figures 18 and 19(a).

The AL3X [488] project, an idea which is mentioned for completeness, proposes to use large parts of the present ALICE detector and turn it into a LLP experiment, based on a similar strategy as for CODEX-b. The idea is to move the interaction point from the center of ALICE to some 10 meter upstream, put a thick shield with an active veto between the collision point and AL3X, and re-use much of the present ALICE inner tracker, and the original L3 magnet that comes with the cavern, as a new LLP detector. It would require some revision and rebuilding of accelerator components, and potentially also a more optimized tracking detector for the physics, so this proposal would in any case not be a cheap endeavor. Obviously, the ALICE collaboration has ideas and plans of its own for the experiment. No follow-up activity on the study of this experimental opportunity has been reported during the last few years. The sensitivity to HNLs has been studied for AL3X in [489] and is shown in figure 19(a).

Recently in [490] ANUBIS (AN Underground Belayed In-Shaft experiment) was proposed, an auxiliary detector to be installed in one of the access shafts above the ATLAS or CMS interaction point, as a tool to search for LLPs. ANUBIS is proposed to have four (removable) tracking stations (TS) spaced 18.5 m apart from each other, providing a uniform coverage of the detector volume. This setup results in a cross-sectional area of 230 m² per TS. The tracking stations will be based on resistive plate chamber technology. A demonstrator test with a module of 2 × 1 × 1 m³ is proposed to be installed in the ATLAS shaft during Run 3. In [483] the sensitivity of this proposal is discussed for long-lived heavy neutral leptons (HNLs) in both minimal and extended scenarios. Both the minimal HNL model where both production and decay of the HNLs are mediated by active-sterile neutrino mixing, and the case of right-handed neutrinos in a LR symmetric model are studied.

Next, a U(1)_B−L extension of the SM is considered. In this model, HNLs are produced from decays of mostly SM-like Higgs boson, via mixing in the scalar sector of the theory. In all cases, ANUBIS is shown to have a sensitivity reach comparable to the proposed MATHUSLA detector. For the minimal HNL scenario, the contributions from Ws decaying to HNLs are more important at ANUBIS than at MATHUSLA, extending the sensitivity to slightly larger HNL masses for ANUBIS. The sensitivity curves for ANUBIS for HNLs are shown in figure 19(a).

MAPP stands for the MoEDAL Apparatus for Penetrating Particles [491]. The MoEDAL collaboration is proposing MAPP, an upgrade detector program for LLPs in a gallery near IP8 shielded from cosmic rays by an overburden of approximately 100 m of limestone. A first-stage new detector of MAPP, consisting of scintillator bars and slabs to hunt for milli-charged particles has recently been approved and will be installed for taking data during LHC Run 3. The next stage of the proposal, labeled MAPP-LLP, targets to deploy later three nested boxes of scintillator hodoscope detectors, in a 'ussian doll' configuration, following as far as possible the contours of the cavern. It is designed to be sensitive to long-lived neutral particles from new physics scenarios via their interaction or decay in flight in a decay zone of size approximately 5 m (wide) × 10 m (deep) × 3 m (high).

In [492] the pair production of right-handed neutrinos from the decay of an additional neutral $Z^{\prime}$ boson in the gauged B−L model is investigated. For this study, the sensitivity for an extended version of the MAPP detector proposal covering the whole available tunnel cavern area was used. The reachable sensitivity is shown in figure 20.

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The FPF [475] is a proposal to create a new experimental cavern with the space and infrastructure to support a suite of far-forward experiments at the Large Hadron Collider (LHC) during the High Luminosity era. Located along the beam collision axis and shielded from the interaction point by at least 100 m of concrete and rock, the FPF will house several experiments that will detect particles outside the acceptance of the existing large LHC experiments around the interaction points, and will observe rare and exotic processes in an extremely low-background environment. Experiments include a larger acceptance version of FASER, labelled FASER2, a liquid argon TPC, and an extended version of SND@LHC named AdvSND all of which can be used for HNL hunting. A recent discussion on the status of this project, experiments considered and initial physics potential studies can be found in [485].

FASER2 will have larger dimensions than FASER, namely a magnet aperture of 2 m diameter which will in particular open the acceptance for decays from D and B mesons, and a decay length volume of 10 m. For the sensitivity studies shown in this document a decay volume with a length of 5 is used, which corresponds to the original FASER2 proposal in pre-FPF study days. The angular acceptance of neutral pions will increases from 0.6% in FASER to 10% in FASER2. There is a significant improvement in sensitivities to LLPs produced in decays of heavy mesons, due to the additional acceptance of B-meson production. The larger decay volume also improves sensitivity to larger LLP masses and longer LLP lifetimes. The combined effect of all these factors, as well as the increased luminosity expected for the HL-LHC over LHC Run 3, is an improvement in reach of several orders of magnitude for certain new physics scenarios and will constitute a major step forward in sensitivity.

Likewise the Advanced SND project is meant to extend the physics case of the SND@LHC experiment [494]. It will consist of two detectors of which the one placed in the η region i.e. 7.2 < η < 8.4 can be placed in the FPF cavern. The detectors consist of three elements. Upstream is the target region for the vertex reconstruction and the electromagnetic calorimetric energy measurement. The target will be made of thin sensitive layers interleaved with tungsten plates, for a total mass of ∼5 tons. Next are a hadronic calorimeter and a muon identification detector. The final most downstream element will be a magnet for measurements of the muon charge and momentum, allowing for neutrino/anti-neutrino separation for muon neutrinos and for tau neutrinos in the muonic decay channel of the τ lepton.

FLArE is a LArTPC with transverse dimensions of 1.5 × 1.5 m² and a lenght of 7 m. The total LAr mass is 50 tonnes, the fiducial mass will be 10–20 tonnes. New ideas are being studied to base this detector on liquid krypton. In any case the active material will be dense which means that it will not be very favorable for the signal to background ratio for searches for rare decays. However in [495, 496] it is shown that an interesting sensitivity region can be covered to the search for neutrino magnetic moments via upscattering of a neutrino to a new sterile neutrino state, for heavy neutrino masses below 1 GeV.

At the FPF, HNL decays to hadrons and/or charged leptons will be visible in FASER2. LHC production of HNLs mixing with all three neutrino flavors have been studied in [475, 478, 484, 497, 498]. Although a HNL giving neutrino masses through the type I Seesaw would decay to all flavors, given the mixing angles required by neutrino oscillations, in less minimal models, such as the linear Seesaw or inverse Seesaw, there is more freedom to fit the neutrino data. In particular, scenarios can be built where the decay of the HNL is predominantly to taus. This benchmark was considered in figure 19(b) showing the expected FASER2 reach. Because the ν_τ has significant production from heavy meson decays, the large rates for forward D and B production at the LHC benefit the FPF, allowing for sensitivity to HNL masses up to several GeV. In AdvSND the same scattering technique as discussed for SND@LHC can be used for HNL detection.

FACET [493] is a proposal to add a set of very forward detectors to CMS, integrated within a new purposely-built wide beam-pipe, and proposed to be deployed during the high-luminosity era of the LHC in order to search for BSM LLPs, such as dark photons, dark Higgs, axion-like particles, as well as heavy neutral leptons. FACET stands for Forward-Aperture CMS ExTension, and will be sensitive to LLPs that can penetrate up to 50 m of magnetised iron and decay in an 18 m long, 1 m diameter wide vacuum pipe. The decay products will be measured in tracking and calorimeter detectors using identical technologies as for the the planned upgrades of CMS for the high-luminosity LHC running.

FACET is not proposed to become a new stand alone experiment, but rather as a new subsystem or extension of the CMS experiment that, while overlapping in the parameter space with other searches, will cover a different and unique region for LLPs. FACET will be sensitive to particles produced with polar angle 1 < θ < 4 mrad (7.6 > η > 6.2). It is closer to the interaction region (IP5) than e.g. FASER2 at IP1; it has an long decay volume spanning from 101 < z < 119 m from the IP; it is followed by an 8 m long region instrumented with various particle detectors. FACET covers a range of (unboosted) lifetimes c τ from 0.1 to 100 m. A unique feature among the proposed new LHC experiments is that the decay volume is at LHC vacuum quality, thus virtually eliminating any background from secondary interactions with e.g. air in the decay volume.

The reach for HNLs within the U(1)_B−L model with FACET has been studied [492]. In particular decays of $Z^{\prime}$ bosons to muon flavored neutrinos were studied. The $Z^{\prime}$ can be light (e.g. 20 GeV) and yet have escaped detection in earlier experiments due to the smallness of the coupling to SM particles. In this scenario, FACET has a unique sensitivity at high HNL masses, above about 10 GeV for lifetimes between 10 cm and 100 m. Figure 20 shows the coverage in the coupling ∣V_{μ N} ∣ − M(N) plane in the model, taken from [493].

At the electron-positron stage of the FCC, high-precision measurements will be performed; but FCC-ee also offers powerful opportunities for the discovery of new phenomena. Direct and indirect evidence for BSM physics, can be obtained via a combination of high precision measurements and direct searches. Among the later, searches for HNLs will have an extraordinary potential at the high-luminosity Z pole run [503, 504].

The main production mode will then be Z → ν N, with subsequent decays of the N into off-shell W or Z bosons. The FCC-ee will have good sensitivity above the charm mass all the way up to the W boson mass for different values of the mixing angle with the existing neutrinos. Sensitivity down to heavy-light mixing of ∣V_{ℓ N} ∣² ∼ 10⁻⁸ − 10⁻¹² was obtained in [505, 506], covering a large parameter space for heavy neutrino masses between 5 and 80 GeV. This sensitivity includes the regime where Z → N decays result in boosted HNLs [506].

For low values of the neutrino mixing angles, the decay length of the HNLs can be significant, leading to long-lived signatures [128, 504, 507]. At the FCC-ee, HNLs could be detected for decays up to about 1 m away from the collision point, generating a secondary vertex in the middle of the tracking system. These searches will suffer from very low backgrounds. In case of an observation and with enough luminosity it could even be possible to study kinematic properties that differ between Dirac and Majorana neutrinos [504, 508] using asymmetry observables as defined in [119, 128, 143, 509]. For illustration, figure 35 in section 7 shows an updated estimation of different sensitivities for current and proposed detectors including FCC-ee displaced vertex analysis [504].

Following the plans and ideas to add additional LLP experiments at the HL-LHC it is possible to also envision a similar concept at new future colliders. This is the motivation behind e.g. HECATE [510], a LLP detector concept for the FCC-ee or CEPC facility. The civil engineering of the FCC-ee will have much bigger detector caverns than needed for a lepton collider experiments since they are intended to be used later for a hadron collider facility, which requires larger volumes. This would allow to install extra instrumentation on the large cavern walls to further boost the sensitivity for HNLs. Studies for LLPs with detectors at displaced locations w.r.t the central detector for future lepton colliders have also been reported in [511]. Figure 20 (right) shows a phenomenological study on the reach for HNLs at the FCC-ee (and CEPC) and compare the results for a standard detector (i.e. central tracker and outer muon system) with detector scenarios where the additional space in the large cavern is equipped with muon detector layers. The quantities l₀ and l₁ denote the minimal and maximal distance from the interaction point where the cavern is instrumented to detect an HNL decay into charged particles. The study shown is made for the channels with HNLs decaying to muon final states.

Indirect constrains from precision SM measurements are also sensitive to HNLs: the potential sensitivity of the FCC-ee for the HNL reach has been discussed in [503]. Leptonic colliders are also well suited to look for light m_N ∼ 10 GeV produced in the Higgs sector of extended gauge theories, LR and U(1)_B−L . These may be produced in the associated $Z{{\rm{\Delta }}}_{R}^{0}$ or $W{{\rm{\Delta }}}_{R}^{0}$ or via WW fusion. In both cases, ee colliders have the advantage of reduced fake rates and absence of triggers and can thus access lighter (and displaced) m_N .

If any HNLs with masses at or below the EW scale are discovered, FCC-ee would provide an powerful tool to falsify or support the hypothesis that they are responsible for the neutrino masses and/or baryon asymmetry of the Universe, see section 7.2.1. FCC-ee could measure the flavor-mixing pattern displayed in figure 3 at the percent-level [138]. Leptogenesis imposes further constraints on the flavor-mixing pattern in addition to the experimental fits shown in figure 3, which can be tested by comparing flavored branching ratios in displaced decays. Finally, determing the angular distribution of decay products can help address the question whether HNLs are Dirac or Majorana or the potential existence of additional intermediate states [418, 457, 504, 508, 509, 512], see section 2.3, while its dependence on the displacement is sensitive to the HNL mass splitting [138], a crucial parameter for leptogenesis.

The FCC-hh is the second stage of the FCC and will be sensitive to higher masses and smaller mixing than accessible at the LHC for both prompt and displaced searches. In 100 TeV hadron collision, HNLs can be produced in a variety of channels just as they can be at the LHC. Notably, the gigantic increase in the gluon parton distribution function has a significant impact on production cross section [85, 403], particularly production channels involving intermediate the Z and Higgs bosons [85, 400]. The signatures and backgrounds at 100 TeV are largely the same as at 13–14 TeV. However, at 100 TeV, the hadronic environment is significantly more energetic and (fake) background rates categorically grow much faster than HNL signal rates [85]. This means analysis strategies designed for 14 TeV must be adapted for a qualitatively different environment.

In this stage, the environment would be much more complex than FCC-ee, and pile-up, backgrounds characterization, and triggers would be all crucial elements in the success of these searches. As shown in figure 14, assuming a comparable acceptance to the ATLAS and CMS detectors, it could be possible to test heavy, Dirac-like neutrinos with masses up to 10 TeV (100 GeV) to the level of ∣V_{ℓ N} ∣² ∼ 10⁻³ (10⁻⁴) in the trilepton and MET channel involving final state τ leptons (in their leptonic and hadronic decay modes) [85]. Comparable sensitivity holds for the Majorana case in the trilepton channel. Much smaller mixing can potentially be probed at lower masses and with lighter lepton flavors [85, 507]. It is possible that even higher mass HNL scales can be probed indirectly with the same-sign W^± W^± scattering channel [405, 513]. Finally, it is important to stress that the FCC-hh conditions also allow for characterization both in flavor and charge of the produced neutrino, thus information of the flavor-sensitive mixing angles and a test of the fermion violating nature of the intermediate (Majorana) particle.

Similarly to the case of the LHC, it would be possible to search for HNLs at FCC-hh-based experiments—by producing them at the main FCC-hh experiment and detecting their decays inside a large displaced decay volume. Given a huge number of produced B mesons and W bosons (a factors of ≃30 and ≃120 increase as compared to the LHC), FCC-hh may significantly push the lower bound of the probed HNL parameter space. This is especially important for masses m_N ≪ m_Z , where current and future searches at the main LHC experiments, as well as proposed experiments such as FCC-ee, have limited potential because of a small number of mother particles (Z, W bosons) and parametric suppression of the HNL decay probability, ${P}_{\mathrm{decay}}\propto {{\rm{\Gamma }}}_{N}\propto {m}_{N}^{5}$.

In order to reduce the expenses on the installation and construction of the experiment, it would be attractive to place the decay volume with detectors within the infrastructure already planned for FCC-hh. One of the possible placements is right around the main detectors, which are located inside the 60 m long cavern. The two configurations of the setup, the one with detectors covering ∣η∣ < 1.5 (the mid-η₁ experiment) and another one 1.5 < ∣η∣ < 2.6 (the mid-η₂ experiment), are shown in figure 21. Due to the large solid angle coverage and small mean γ factors of HNLs flying in the direction of these detectors, such configurations may extend the exclusion region by an order of magnitude and larger beyond that of SHiP and LHC-based experiments, which is also well below the FCC-ee reach for m_N < m_B [514].

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Recently, a study was reported on HNL production in the U(1)_B−L model discussed earlier, showing a potential sensitivity reach at the FCC-hh [517]. The study, which considered final states with muons, shows couplings can be probed down to ∣V_{ℓ N} ∣² ∼ (10⁻¹¹) for high HNL mass values around 1 TeV. Decays of a heavy Z' boson resulting in boosted decays of light HNLs in μ jj final states, boosted fatjet topologies or track bases can be used. While this is difficult for HL-LHC, for FCC-hh searches involving at least one displaced fatjet in the inner detector or displaced tracks in muon system can probe Seesaw region for light neutrino mass as low as 30 GeV [466].

The FCC-hh program would also greatly improve sensitivity to non-minimal HNL models. In LR models for instance, direct searches for W_R and N are sensitive to ${M}_{{W}_{R}}\sim \,$ 35–40 TeV with 10 ab⁻¹ of data [415]. Indirect searches can readily exceed this benchmark [418]. For HNLs in the Type III Seesaw, masses up to 8 TeV can be probed via E^± N⁰ and E⁺ E⁻ pair production and decay with 3 ab⁻¹ of data [63, 436]. However, these analyses can be improved in light of new techniques, e.g. ML, that have been developed over recent years.

The clean environment and low pileup at ep colliders makes it also an excellent place to search for LLP using the long lifetime and displaced vertex signature [518]. Electron-proton colliders can be used to search for heavy neutrinos, N, e.g. in the process ep → Nj; Nljj [507]. In particular, in low-scale see-saw models with two sterile neutrinos, as discussed before the mixing with SM neutrinos is expected to be very weak, leading to a displaced vertex from the decay of the heavy neutrino. Searches [519] for ep → μ + 3j based on BDT optimization, assuming a tracking resolution of 8 μm and a sensitivity of 40 μm for the displacement of the vertex, shows that the constraint on the mixing angles down to a few times 10⁻⁹ can be reached for an HNL mass between approximately 10 and 60 GeV.

The NA62 experiment at CERN [390] is focused on the measurement of the ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ decay rate to a 10% precision using the decay in flight technique, and pursues a wide rare kaon decay program. The experiment is served by the P42/K12 beamline from the SPS. Slowly extracted 400 GeV protons, delivered in 4.8 s spills at a nominal intensity of 3.3 × 10¹² protons per pulse (ppp), impinge over a 400 mm long beryllium target to obtain a monochromatic (75 GeV) secondary mixed π⁺, K⁺ and p beam. The NA62 Run 1 dataset collected in 2016–2018, corresponding to 2.2 × 10¹⁸ protons on target (pot), has led to the first observation of the ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ decay based on 20 candidates [586]. NA62 Run 2 started in 2021, and is approved until the CERN Long Shutdown 3 (LS3). In the longer term, a plan for a high-intensity K⁺ decay experiment (NA62-upgrade) to be performed at CERN between possibly LS3 and LS4 is taking shape [582]. The primary beam intensity in the high-intensity experiment is expected to be four times that of NA62, and collection of about 5 × 10¹⁹ pot is expected by 2038 or so.

The NA62 experiment has conducted a search for HNL production in the K⁺ → e⁺ N and K⁺ → μ⁺ N decays with the Run 1 dataset [28, 29]. The former search uses the main trigger line designed for the ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ decay, while the latter search is based on the control trigger downscaled by a factor of 400. The analysis represents a peak search in the reconstructed squared missing-mass, ${m}_{\mathrm{miss}}^{2}={\left({P}_{K}-{P}_{{\ell }}\right)}^{2}$, where P_K and P_ℓ are the measured kaon and lepton 4-momenta. The HNL mass ranges covered are 144–462 MeV in the electron case (complementing PIENU results for m_N < 135 MeV, see section 5.5), and 200–384 MeV in the muon case (complementing the K⁺ → μ⁺ N peak search in the 178–300 MeV range with stopped kaons at BNL E949 [23]). No HNL signals have been observed in either decay mode, and the corresponding upper limits of the decay branching fractions and the mixing parameters ∣U_ℓ4∣² have been established. The sensitivity is limited by backgrounds in both cases; in particular, the K⁺ → μ⁺ ν decay followed by the ${\mu }^{+}\to {e}^{+}\nu \bar{\nu }$ decay in flight represents an irreducible background to the K⁺ → e⁺ N process.The exclusion limits at 90% CL obtained with NA62 Run 1 data are shown in figures 27 and 28 for electron and muon couplings, respectively. The NA62 bounds on HNL production are comparable to those obtained from HNL decay searches at T2K with almost its full dataset [36].

The sensitivity to the mixing parameters ∣U_ℓ4∣² is expected to improve in future as the inverse square root of the integrated kaon flux. The projected sensitivities of the K⁺ → ℓ⁺ N searches at NA62-upgrade [582] are displayed in figures 27 and 28. A search for π⁺ → e⁺ N decay of the beam pions at NA62-upgrade is expected to improve the PIENU limits on ∣U_e4∣² [24] in the range m_N < m_π ; the corresponding sensitivity is yet to be evaluated and is not shown. In the electron case, the gap at m_N ≈ m_π between searches in π⁺ and K⁺ decays can be covered by the search for K⁺ → π⁰ e⁺ N decay, although with a limited sensitivity [587]. The P42/K12 beamline can also be operated in beam dump mode. A switch to beam dump mode is performed by removing the Beryllium target from the proton beamline. The primary proton beam is then dumped into a pair of 1.6m thick movable Cu-Fe collimators (TAX) placed between two pairs of bending dipole magnets. During the standard data taking in kaon mode, these bending magnets are used to select the momentum of the secondary beam passing through suitably aligned holes. In beam dump operation, the TAX collimator is used as hadron absorber, while the pair of magnets downstream the TAX is used to provide sweeping of the secondary muons produced in the TAX. A sample of 1.5×10¹⁷ pot was collected by NA62 in beam dump mode during a 1-week long run in November 2021 at a typical beam intensity of 6×10¹² ppp, using trigger streams for decays of exotic particles into charged and neutral particles. NA62 operation in beam dump mode with the goal of accumulating 10¹⁸pot by LS3 has been approved by the CERN scientific committees [588]; the corresponding sensitivity curves assuming electron, muon, and tau flavor dominance are displayed in figures 27, 28, 29. In the longer term, a collection of 10¹⁹ pot in beam dump mode by LS4, and of additional 4×10¹⁹ pot after LS4, is foreseen [583]. However, the exact schedule and beam sharing between kaon and dump mode will be a matter of scientific policy, which is why the end-date for collecting the required data samples varies between 2032 and 2038 in table 1.

A novel experiment, SHADOWS (Search for Hidden And Dark Objects With the SPS), has been recently proposed within the Physics Beyond Colliders CERN study group ⁴⁶ and the Expression of Interest submitted to the CERN PS and SPS Committee [583]. SHADOWS aims to use the 400 GeV primary proton beam (P42/K12 beamline) extracted from the CERN SPS currently serving the NA62 experiment, and will be located in the same experimental hall (ECN3) in the CERN North Area. SHADOWS will take data off-axis concurrently with NA62-upgrade when the beamline is operated in the dump mode. In fact HNLs (and in general feebly-interacting particles) emerging from the decays of charmed and beauty hadrons have a non-negligible transverse momentum and therefore can be detected by an experiment placed off-axis with respect to the direction of the impinging beam on the dump.

The off-axis position allows SHADOWS to be placed very close to the dump, at about 10 m with respect to the proton interaction point, with the twofold advantage of: (i) maximizing the signal acceptance due to the larger flux intercepted by the relatively small transverse dimensions; (ii) minimizing the background that is mostly concentrated in the forward direction. The SHADOWS current plan, synchronized with that of the NA62-upgrade (section 5.2.1), involves running at 4–6 times the nominal P42/K12 intensity (at 2 × 10¹³ pot per pulse), and collection of 10¹⁹ pot in a year of data taking between LS3 and LS4, and additional 4 × 10¹⁹ pot in 3–4 years of data taking after LS4. However, as in the NA62 case, the exact schedule and beam sharing between kaon and dump mode is still to be defined. Therefore in table 1 the end-date for collecting the required data sample (∼5 · 10¹⁹ pot) varies also in this case between 2032 and 2038.

The current detector baseline consists of a standard spectrometer of 2.5 × 2.5 m² transverse dimensions with a ∼20 m long, in-vacuum decay volume placed about 1 m off-axis with respect to the beam. The spectrometer comprises a tracking system with a dipole magnet of 1 Tm bending power, a timing detector, an electromagnetic and (possibly) a hadron calorimeter, and a muon detector. The upstream face of the decay vessel is instrumented with an active veto. A second spectrometer of 3 × 4 m² transverse dimensions and 30 m long decay volume following the first one is still compatible with the NA62-upgrade operation, and is being envisaged for a future upgrade.

The proton interactions with the dump, along with feebly-interacting particles, give rise to a copious direct production of short-lived resonances, pions and kaons. While the dump thickness (22 λ_I) is sufficient to absorb the hadrons and the electromagnetic radiation produced in the proton interactions, the decays of pions, kaons and short-lived resonances result in a large flux of muons and neutrinos. Muons and neutrinos from the dump are the two major sources of background for FIP searches. For an off-axis detector, however, the background originated by neutrino interactions with the residual air in the in-vacuum decay volume or the detector material is negligible as the large majority (those emerging from the decays of pions and kaons) is intrinsically produced in the forward direction. The most relevant backgrounds for SHADOWS are those originated by the beam-induced muon flux. Leveraging on the fact that off-axis muons have a much lower momentum than those on-axis (typically below 15 GeV for off-axis muons against up to 400 GeV for on-axis muons), a magnetized iron block placed in front of the SHADOWS decay vessel should be able to reduce this background to a manageable level. Preliminary studies are very promising.

The projections for the 90% CL exclusion limits on HNL couplings corresponding to 5 × 10¹⁹ pot assuming electron, muon and tau flavor dominance are shown in figures 27, 28, 29. The addition of a second spectrometer is equivalent to the increase of the proton intensity by a factor 2–5, depending on the value and type of coupling and the considered mass range. These results do not include trigger, reconstruction, and selection efficiencies. However, realistic estimates obtained in a similar spectrometer as NA62 show that the inclusion of such efficiencies would reduce the available dataset by at most a factor of two.

SHADOWS is currently under review by the SPS Committee at CERN. It aims to be installed during the LS3 (2026–2028) and take data in Run 4. It is fully compatible with the NA62-upgrade operation, and is complementary to NA62 in the HNL physics reach: NA62 is mostly sensitive below the kaon mass, while SHADOWS is sensitive mainly between the kaon and the beauty masses. The relatively small dimensions and the use of existing beamline, experimental area and related infrastructure make SHADOWS a relatively cheap, compelling, and competitive project for HNL searches in this decade.

The Search for Hidden Particles (SHiP) [589] is an experiment that would be placed at a proposed general purpose intensity-frontier facility, called the BDF. SHiP is designed to probe a variety of new physics models containing LLPs with masses in ${ \mathcal O }(10\,\mathrm{GeV})$ range and has unique potential for the HNL searches.

SHiP will be served by the SPS accelerator, exploiting the additional 4 × 10¹⁹ protons at 400 GeV per year that is not exploited by any experiment since the decommissioning of the CNGS facility. The experiment consists of a large density target [590–592], followed by a hadron stopper, and an active muon shield based on the magnetic deflection [593]. Downstream the muon shield, there is the SND, designed for studying scatterings of neutrinos and dark sector particles. Next to the SND there is the Hidden sector decay volume, which is a pyramidal frustum with a length of 50 m and upstream dimensions of 2.2 × 4.5 m² located on-axis approximately 45 meters downstream the center of the target. The decay volume ends by the Hidden Sector Decay Spectrometer (HSDS), consisting of straw trackers, spectrometer magnet, timing detector, EM calorimeter and the muon system. HSDS allows studying decays of hidden sector particles. It reconstructs the decay vertex of hidden sector particles, measures their invariant mass, and identifies decay products.

The background for decays, including the muon combinatorial background, muons and neutrinos DIS events in the detector material, is efficiently suppressed by the combination of the target configuration, the hadron stopper, the muon shield, HSDS surrounding background tagger, and HSDS timing detector. Full Monte Carlo numeric simulations performed with the help of the full MC suite called FairShip (based on the FairRoot software framework [594]), supplemented by the measurements done in test beam on the detector prototypes have shown that the background yield expected during the working time of the SHiP experiment is <0.1 for fully reconstructed signals and 6.8 for partially reconstructed signals. The latter may be easily eliminated by requiring the invariant mass of the reconstructed tracks to exceed 100 MeV [589].

The SHiP experiment has a perfect potential to probe HNLs with masses above the kaon mass, being complementary to the experiments optimized to search for HNLs produced in decays of long-lived kaons, such as DUNE and NA62 in the kaon mode. Indeed, first, kaons produced at SHiP would get mostly absorbed by the hadron stopper (see however [595] discussing the sensitivity of SHiP to HNLs from kaons). This allows for background-free searches for partially reconstructed signals from decays of HNLs. Second, the main production channels of HNLs in the mass range m_K < m_N < m_B are decays of D and B mesons. Due to the large intensity of the protons at SHiP, around 3 × 10¹⁷ of D mesons and 5 × 10¹³ of B mesons would be produced [596]. Their angular distribution is peaked in the far-forward region, and hence most of them fly within the geometrical acceptance of HSDS, providing the maximal geometric acceptance for HNLs.

The analysis of the SHiP sensitivity to HNLs has been performed in [515]. The signal, decays of HNLs into at least two charged particles, has been modeled by full MC simulations with the help of FairShip, with the detector response simulated by GEANT4 [597]. The 90% CL sensitivity regions for the setup with 2 × 10²⁰ pot for the case of the pure e, μ, τ mixings are shown in figures 27, 28 and 29. The sensitivity to the HNLs with arbitrary mixing pattern may be calculated using a script [598]. To summarize, SHiP may probe the HNL mixing angles down to U² ∼ 10⁻¹⁰ in the region ${m}_{N}\lt {m}_{{D}_{s}}$, and U² ∼ 10⁻⁸ in the region ${m}_{{D}_{s}}\lt {m}_{N}\lt {m}_{B}$.

A recent updated study for a possible location of SHIP and the BDF at CERN is reported in [599].

T2K (Tokai-to-Kamioka) [607] is a long baseline neutrino experiment located in Japan with the primary goal of measuring muon (anti-)neutrino oscillations using Super-Kamiokande as its far detector. The T2K neutrino beam is produced at the J-PARC center (Tokai, Ibaraki) by colliding 30 GeV protons on a graphite target. The produced pions and kaons are focused and selected by charge with magnetic horns and subsequently decay in flight to neutrinos. Depending on the polarity of the current in the horns, the experiment can be run either in neutrino or anti-neutrino mode. The ND ND280, located 280 meters from the proton target, is composed of several sub-detectors, in particular three time projection chambers (TPCs) filled with Argon gas [608] and two scintillator-based fine-grained detectors, surrounded by an electromagnetic calorimeter and a 0.2 T magnet.

The HNL search was performed by considering their production from kaon parents (${K}^{\pm }\to {{\ell }}_{\alpha }^{\pm }N,\alpha =e,\mu$) and subsequent decay in ND280 [36]. T2K data from November 2010 to May 2017 were considered for a total of 1.86 × 10²¹ pot (∼2:1 of neutrino to anti-neutrino mode). All the possible decay modes N → ℓ^± π^∓ and N → ℓ^± ℓ^∓ ν were considered, including the neutral current decay modes N → e⁺ e⁻ ν_τ and N → μ⁺ μ⁻ ν_τ that are directly sensitive to the mixing element ${U}_{\tau }^{2}$. Production and decay branching ratios were extracted from [609], and effects related to heavy neutrino polarization [610] and delayed arrival time (with respect to light neutrinos) are also taken into account.

In order to significantly improve the signal to background ratio, only events occurring in the TPC gas volume are considered for this analysis. Events are selected by identifying two tracks of opposite charge originating from a vertex in a TPC, and kinematic cuts are applied. Upstream detector activity is also vetoed. Five signal channels are then identified based on particle identification (obtained from TPC energy loss): μ^± π^∓, e⁻ π⁺, e⁺ π⁻, e⁺ e⁻, μ⁺ μ⁻. The remaining background events are mainly originating from neutrino-induced coherent pion production on argon nuclei in the TPC gas, as well as misreconstructed interactions outside the gas e.g. photon conversion.

No events are observed in the defined signal regions, which is consistent with the background-only hypothesis. Several approaches are applied to obtain final limits on the U_α mixing elements. On the one hand, each production/decay mode is considered independently and the corresponding analysis channel is used to put limits on the associated mixing elements. On the other hand, a combined method has been developed to fit simultaneously all the heavy neutrino production and decay modes and the ten different analysis/signal channels (five for each beam mode). The Bayesian framework is using a Markov Chain Monte Carlo to marginalize over nuisance parameters and flat priors on the mixing elements ${U}_{\alpha }^{2}$ are assumed. The limits on U_e ² and ${U}_{\mu }^{2}$ obtained with profiling (setting ${U}_{\mu }^{2}={U}_{\tau }^{2}=0$ and ${U}_{e}^{2}={U}_{\tau }^{2}=0$, respectively) are shown in figures 27 and 28, while [36] presents as well marginalized limits (including on ${U}_{\tau }^{2}$) and constraints in the plane ${U}_{e}^{2}-{U}_{\mu }^{2}$.

These limits have been recently reinterpreted to extend to lower masses (20 ≲ m_N ≲ 200 MeV) in [578] by considering N → e⁺ e⁻ ν_μ decays. The T2K group is currently performing the corresponding analysis including as well using pion production of heavy neutrinos, and searching for other decay modes, as well as taking benefit of the increased statistics (a total of 3.8 × 10²¹ POT has been collected as for February 2022).

ArgoNeuT was a LArTPC that operated in the Neutrinos from the Main Injector (NuMI) beam at Fermilab. The NuMI beam is produced colliding 120 GeV protons on a graphite target, producing mesons that are focused by a two-horn system and allowed to decay over a 675 m long decay pipe. ArgoNeuT collected data between 2009 and 2010, during which time the NuMI beam was operating in antineutrino mode for a total exposure of 1.25 × 10²⁰ POT. The detector had a target mass of 0.24 ton and an active volume of 40 × 47 × 90 cm³. It was situated a distance of 1033 m downstream of the proton target and 318 m from the NuMI absorber. Finally, it was located immediately upstream of the MINOS ND. The MINOS ND, being significantly larger and also magnetized, can act as a muon spectrometer (and a muon/pion separator) for particles that exit the ArgoNeuT LArTPC before stopping.

The ArgoNeuT collaboration [581] carried out a search for heavy neutral leptons with nonzero ∣U_τ4∣² where the N are produced in a chain of D meson decays–${D}_{(s)}^{\pm }\to {\tau }^{\pm }{\nu }_{\tau }$, τ^± → NX^± (where X^± is one or more SM particles). These are produced in the NuMI beam and absorber, and then can decay either inside or in front of the ArgoNeuT detector via N → ν_τ μ⁺ μ⁻. With no observed signal, [581] places a constraint improving on existing searches in this mass range for tau-coupled N from CHARM [576] and DELPHI [611]. However, in the time since, [33] demonstrated that CHARM has sensitivity beyond the reach of [576], yielding a more stringent constraint than ArgoNeuT.

The NOνA ND at FNAL has a potential for HNL studies and is briefly assessed here. There have been comparative studies, before 2018, of the sensitivity to HNL detection with the ND of the NOνA experiment hosted by Fermilab and falling under the general concept of Lightweight Dark Matter [612] The 990 m distance of the ND to the NuMI target could provide a search for micro-second HNL lifetimes. The acceptance efficiency into the 300 ton NOνA ND has been estimated to be on the order of 10⁻⁴. The main mass range for the HNL that can be produced by the Fermilab's NuMI beam extends primarily to 500 MeV from the decay of the light mesons that make up the secondary beam of NOνA. There is also a possibility for accessing masses up to 2 GeV from the decay of heavier mesons, like the charmed meson D_s , which is produced with 1 μ b within the NuMI beam [613, 614].

The preliminary studies with NOνA ND focused on the first few 10²⁰ POT of exposure and the dilepton decay channel for the HNL. The advantage of NOνA ND to leptons is based on its segmented design, the low-Z plastic, and the advanced PID efficiency used for measuring the NC neutrino interaction signatures. The current upper limit of the mixing between the electron neutrino and the HNL in the mass range 100–500 MeV to the SM is no more than 10⁻⁷ for an expected HNL yield of O(100) events [615–617]. There are much stronger HNL hadronic and semi-leptonic decays, still unexplored, within the NOνA existing dataset. There can be e-showers surrounded by pion-showers or purely multi-pion signatures, or combinations of a muon track of high energy with an e-shower. In particular, a muon with a jet combination is believed to be the strongest channel yet. However, the preliminary NOνA ND studies, before 2018, did not show any competitive sensitivity for these channels. This is mainly due to the lack of magnetic field in the fiducial volume which compromises any high energy reconstruction efficiency. Also, to a lesser degree, it was due to the hadronic PID analysis still being in the early development stage.

At the time of Snowmass 2021, though, the NOνA analysis-production of training samples for PID has been substantially increased and extended. The experience with the use of neural networks along with the maturity of analysis selections, the understanding of the reconstruction resolution and selections, has matured sufficiently to warrant a new investment in the sensitivity studies of HNL detection in the NOνA ND. The non-magnetic fiducial region may still restrict the resolution at high energy signatures, but this is somewhat compensated by the already existing large dataset of detected events, several years ahead of any new detector developments. Analysis of the existing dataset of NOνA can improve the understanding of backgrounds to HNL, further constrain model rates about the coupling to SM, and strengthen the case for such exotic studies of New Physics studies within the neutrino experiments.

The reported T2K results [36] are still statistically limited. The experiment will continue taking data up to 2026 (T2K-II phase), with an upgrade of the ND ND280 [618] complex as well as an increase of the accelerator beam power, with the aim to collect additional 10 × 10²¹ POT. The new ND280 detector will in particular embed two additional TPCs, that will approximately double the active volume available for HNL search. The operation will then continue in the context of the Hyper-Kamiokande (Hyper-K) experiment [619].

Ten years of Hyper-K operation with a usage of (upgraded) ND280 as a ND, correspond to statistics enhanced by a factor of ∼30 with respect to current T2K results on HNLs. Scaling the latter based on this factor, we obtain a sensitivity shown in figures 27 and 28. This is assuming the measurement would still be limited by the collected statistics, which is reasonable given the very low background levels reported in [36]. Furthermore, additional data will also allow one to better constrain the backgrounds by using more populated control regions, and improvements in event reconstruction are also expected that can potentially lead to an increase in the HNL signal selection efficiency and purity.

The SBN Program at Fermilab consists of three liquid argon time projection chamber detectors situated along the Booster Neutrino Beam (BNB), a neutrino beam produced colliding 8 GeV protons on a Be target, resulting in predominantly pions and kaons that are subsequently focused using a single horn and allowed to decay over a 50 m decay pipe. The Short-Baseline Near Detector (SBND) is located at 110 m from the neutrino beam origin and has 112 ton of active mass. The SBND detector is currently being installed, with beam data taking expected in 2023. MicroBooNE is placed at 463 m downstream the neutrino beam origin and has 85 ton of active mass. MicroBooNE started beam data taking on October 2015 and finished on October 2021 after accumulating ∼10²¹ POT. ICARUS is located at 600 m from the neutrino beam origin and has 476 ton of active mass, and is finalizing commissioning.

The MicroBooNE collaboration published its first results from searching for kaon decay production of N and its subsequent decay into a charged pion/muon pair in [579]. Due to large backgrounds from neutrino scattering (CC 1μ1π), the time structure of the neutrino beam was leveraged to reduce backgrounds and optimize signal searches. This (null) result leads to a constraint which is relatively weak compared to T2K and other existing constraints. In addition, MicroBooNE is exposed to the off-axis component of the NuMI beam. MicroBooNE is located at 680 m and 8° from the NuMI target and 100 m and 120° from the NuMI absorber. MicroBooNE has also searched for Higgs-Portal Scalars [620] where the new-physics particle is produced in the absorber of the NuMI neutrino beam. Reference [580] reinterpreted this result in the neutrino-portal context and found that it improves on existing constraints in the 30–150 MeV region, comparable to the results demonstrated for T2K in [578].

The ICARUS collaboration has recently started to study the physics case and technical feasibility for a dedicated HNL search using the NuMI beam. The ICARUS detector is located off-axis to the NuMI beamline with an angle of 5.7^◦, about 800 m from the NuMI target. The production of HNL mostly comes from charged mesons (i. e. pion/kaon) decay, with contributions from both decay-at-rest and decay in-flight. It is expected that the HNL flux will peak at off-axis angles (due to the mass of the HNLs), whereas the SM neutrino flux drops off significantly at these angles. This allows for HNL searches to benefit from an increased signal-to-background ratio.

The SBND collaboration is carrying out a sensitivity study for HNLs in the BNB beam. Due to the closer location of SBND to the beam origin (compared to both MicroBooNE and ICARUS), a search using a delayed trigger window exploiting the longer time of flight of the HNL, as employed by MicroBooNE [579], is expected to be less effective. Different strategies must be applied to search for HNL signatures and reduce the neutrino-scattering background, like the double ionization signature used by ArgoNeuT for dimuon searches [581] or exploit the internal structure of the proton beam as MiniBooNE did [621]. In order to obtain a realistic estimation of the signal efficiency and background rejection, SBND is using a full-detector simulation to include the reconstruction effects. SBND is projected to accumulate between 10 × 10²⁰ and 16 × 10²⁰ POT during its run.

The DUNE is a long baseline neutrino experiment using the neutrino beam delivered by the new LNBF accelerator facility under design and construction at FNAL. This facility will use 120 GeV protons and provide 1.1 × 10²¹ POT per year, delivering a neutrino beam with 1.2 MW of beam power. After a running period of approximately 6 years an upgrade of the beam facility power to 2.4 MW is planned, doubling the number of POT/year. Both neutrino and antineutrino beams will be provided roughly with equal running times. The neutrinos are sent to a 40 kton fiducial volume far detector located 1.5 km underground, based in the Sanford Underground Research Facility in South Dakota, located 1300 km northwest from FNAL.

Closer to the neutrino source will be a ND complex which, among many other tasks, will monitor the neutrino flux produced by the facility. Apart from being a neutrino source, the high-intensity beam interactions with the target constitute also a potential source of HNLs and other new exotic particles, which are produced in particular in decays of light mesons. With the LBNF beam energy and intensity, the DUNE ND will cover an interesting and novel search region for decay in-flight HNL particles with masses up to 2 GeV.

The dominant decay channels into visible final states for HNLs in the low-mass region are: HNL → e π, ν e μ, ν ee, ν μ μ, ν π⁰, and μ π. The mass range for HNLs up to 2 GeV can be explored in all flavor-mixing channels. Decay channels for masses above 500 MeV include channels containing heavier mesons and tau leptons (see, e.g. table 1 in [600]).

The expected main background to this search comes from SM neutrino—ucleon scattering events in which the hadronic activity at the vertex is below threshold, see e.g. [600, 606]. Charged current quasi-elastic events with pion emission from resonances are background to the semi-leptonic decay channels. Moreover, misidentification of long pion tracks as muons can constitute a background to three-body leptonic decays. Neutral pions are often emitted in neutrino scattering events and can be a challenging background for HNL decays that include a neutral meson or channels with electrons in the final state.

The DUNE ND [601] will be built in a shallow underground hall located 574 m downstream from the neutrino beam origin. Currently, it is expected that it will consist of three different components: a 150 tonne LArTPC (ND-LAr); a magnetized, gaseous argon time-projection chamber surrounded by an electromagnetic calorimeter (ND-GAr); and a smaller magnetized beam monitor consisting of straw tube chambers and an electromagnetic calorimeter (SAND). They will serve important individual and overlapping functions in the primary mission of the ND: the precise characterization of the neutrino beam energy and composition, as well as vastly improving knowledge of cross sections and particle yields for neutrino scattering processes in the few GeV energy region. The two argon TPCs (ND-LAr and ND-GAr) will be moved laterally off the beam axis for some of the running, as to sample the beam at off-axis angles to better constrain flux uncertainties. Details on this DUNE-PRISM project are given in [601].

The optimal detector component in this ND complex for HNL studies is the ND-GAr as it has a larger (decay) volume than SAND and has a gaseous rather than liquid argon medium [602, 606]. For detectors with a dense medium a much larger background from neutrino interactions with a similar topology as the signal decays is expected, reducing its discovery potential. The studies indicate that one can expect about a factor two better sensitivity for HNLs searches in the gas-based TPC. The LBNF and ND facility are expected to become operational at the start of the next decade.

Several studies have recently examined the sensitivity for a DUNE-like experimental set-up, all showing the interesting potential for HNL searches below the kaon mass limit, where the anticipated high-intensity proton beams may outmatch those of high-energy beam dump experiments [600–602, 604–606]. In [604] only the signal is studied, based on meson decays generated by PYTHIA. It creates its own neutrino flux from pp collisions. All the other studies make use of officially released neutrino fluxes delivered by the FNAL machine-experiment interface group [622]—albeit for different versions, as the proton accelerator and neutrino beamline have been still further optimized during the last few years. These flux files have been modified in [602, 605, 606] to replace the quasi-massless SM neutrinos by massive HNL and change correspondingly the kinematics of these new particles. In [605] effective Lagrangians are obtained for both the Dirac and Majorana scenarios, and made available as FeynRules models, so that fully differential event distributions can be easily simulated.

Rererence [602] highlighted the prospects of ND-GAr for such a study, given both the lower detector density (higher signal-to-background ratio) and the magnetization, allowing for charge-identification of the HNL decay products. If an HNL is discovered in a channel where it decays fully visibly, e.g. N → μ^± π^∓, then measurements of the respective rates of these two charges, μ⁺ π⁻ and μ⁻ π⁺ can allow for discrimination between Majorana- and Dirac-fermion hypotheses for the HNLs and SM neutrinos. With the magnetized ND-GAr and a relatively pure neutrino beam from LBNF, such discrimination is possible for many HNL masses in ten years of data collection [602].

For the sensitivity studies in this document we follow [606], which includes background estimates and a simple parametrized detector simulation. The studies have been performed for different scenarios of the total amount of protons on target (POT), implementation of the background, and selection of the NDs (ND-LAr + ND-GAr or ND-LAr only). Compared to the original study in [606] an improved background rejection method has been used. The study in [606] also includes the effect of the DUNE-PRISM running scheme, and shows that during the off-axis operation the experiment still has excellent prospects for discovering HNLs. For that study the detectors were kept on axis for 50% of the time and the rest of the time the detectors were moved laterally and with equal exposure at 5 different positions: 6 m, 12 m, 18 m, 24 m, 30 m. Figure 23 shows the sensitivity for a scenario of 6 years of LBNF with 1.2 MW followed by 6 years with 2.4 MW, corresponding to a total of 2 × 10²² POT for ND-GAr and 1.4 × 10²² POT for ND-GAr, which should have been collected by approximately ∼2040, with the ND-GAr staging scenario. The results show that DUNE will be very competitive for masses in the region of a few 100 MeV, and can provide interesting limits also in the higher mass region.

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For the summary figures 27, 28 and 29, the scenario of 12 years operation, and including signal plus background, was chosen, corresponding to the orange solid line in figure 23.

A new DUNE study is forthcoming, using the most recent neutrino flux calculations, including the flux corrections and decays as implemented in [605], for the signal and backgrounds, and including a parameterized response of the detectors for smearing of the kinematical variables and particle detection efficiency. Also the neutrino–nucleus background could possibly be further suppressed exploiting the timing difference between SM neutrinos and HNLs, if e.g. very fast timing detectors can be included. The use of ML is another direction which can be explored, utilizing the very detailed event information recorded by the detectors to select HNL decays. To quantify these background levels, a detailed simulation study is required to estimate the impact on the detector requirements.

At sufficiently small mixing angles, HNLs will be essentially stable on detector length scales, making their direct experimental observation difficult. However a recent proposal [623] suggests looking for a deficit in the muon or electron neutrino charged current events in a neutrino detector near the production point. This method may in principle allow one to probe mixing angles as low as ${ \mathcal O }({10}^{-5})$ for low masses (MeV region), it relies on a precise knowledge of the active neutrino flux coming from the target which has not yet been demonstrated.

The dense and hot early universe provides a natural avenue for HNL production and possible decay. However, these decays may also negatively impact the successes of standard BBN [151, 645, 661–663].

In this context, let us first discuss the case of HNLs that are heavy enough to decay into long-lived mesons such as π^± and K^0,± (depending on the mixing pattern, the threshold for π production ranges from m_N = m_π + m_e to m_N = m_η ).

For small lifetimes of these particles, τ_N ∼ (10⁻²–10)s, the main effect on BBN is the modification of the p ↔ n conversion processes via the inclusion of additional meson-mediated reactions like π⁻ + p⁺ ↔ n + π⁰/γ. Since the cross sections of these additional reactions are large compared to the weak p ↔ n conversion processes of the SM, even a small fraction of mesons that gets injected into the plasma can keep the neutron-to-proton ratio (n/p) close to unity (n/p) ≃ 1, which then also influences the subsequently produced helium-4 abundance ${Y}_{p}\simeq 2{\left(n/p\right)}_{{T}_{\mathrm{BBN}}}/[1+{\left(n/p\right)}_{{T}_{\mathrm{BBN}}}]$ at some characteristic BBN temperature T_BBN. Hence, if the neutron-to-proton ratio gets modified by the decay but cannot relax back to its SM value of (n/p) ≈ 1/7 once the HNL decay has concluded—and before the weak reactions freeze-out –, meson injection will lead to an overproduction of helium-4 and other light elements, and thus to conflicts with observations for lifetimes already as small as τ_N = 0.02 s [645]. For larger lifetimes, HNLs survive until the onset of nuclear synthesis reactions, and the injected mesons also start to induce hadrodisintegration reactions like π⁻ + ⁴He → p + 3n (see e.g. [664, 665] for more general bounds from hadrodisintegration). However, together with destroying nuclei, mesons still keep (n/p) ≃ 1. Therefore, as long as HNLs disappear before the decoupling of nuclear synthesis reactions, i.e. for τ_N ≲ 10⁴ s, these sizable amounts of free protons and neutrons get bounded in nuclei, thus still increasing the nuclear abundances compared to their SM values [638, 666].

The BBN constraints on ${ \mathcal O }(\mathrm{GeV})$ HNLs in models with negligible lepton asymmetry ⁴⁸ in the neutrino sector, i.e. η_ν ≪ 1, are summarized in figure 31.

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Lighter HNLs that cannot decay into mesons, mainly affect BBN via their decay into high-energy neutrinos or electromagnetic states and different effects can be dominant, namely (i) an altered n ↔ p conversion rates due to the presence of additional neutrinos (if the neutrinos are injected after neutrino decoupling), (ii) a modification of the expansion dynamics of the Universe due to modifications to the Hubble rate or the time-temperature relation, and/or (iii) the late-time modification of the light-element abundances after BBN via the process of photodisintegration [267, 667–669]. The first two effects can cause an increase in the deuterium and helium-4 abundances for the whole range of lifetimes from ${\tau }_{N}\gtrsim 0.05\,{\rm{s}}-{ \mathcal O }(1){\rm{s}}$, with a dependence on the HNL mass m_N ≳ 10 MeV [663, 666]. The latter effect instead concerns HNLs with lifetimes >10⁴ s, and mainly leads to deuterium underproduction and helium-3 overproduction via the dissociation of deuterium (γ + D → n + p) and helium-4 (γ + ⁴He → D + D,...), respectively. Regarding photodisintegration, the corresponding bounds can be obtained via the public code ACROPOLIS [669–671], which can be used to model the process of photodisintegration for a wide range of scenarios. The corresponding BBN constraints for the type-I Seesaw model with three right-handed neutrinos and Majorana masses below the pion mass, are summarized in figure 32.

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Similarly to BBN, HNLs may affect CMB if surviving until temperatures T ≃ few MeV. The impact of HNLs with masses ${m}_{N}\gtrsim { \mathcal O }(1\mathrm{MeV})$ and lifetimes below the Seesaw bound is indirect—via a change of the helium abundance and the effective number of ultrarelativistic degrees of freedom N_eff. The effect of these parameters on CMB that cannot be mimicked by a change of ΛCDM parameters within their error bars is the impact on the damping tail of the CMB spectrum [676].

The impact of short-lived HNLs on N_eff has been recently studied in several works [677, 678], where it has been argued that HNLs with masses m_N ≳ 50–70 MeV lifetimes τ_N ≲ 1s decrease N_eff even if decaying mostly into neutrinos (see figure 33).

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The current CMB constraints on HNLs with the pure τ mixing from [677] are shown in figure 33 (see also [672]); the case of the pure e, μ mixings is qualitatively similar. They are weaker than the current BBN bounds for masses m_N ≳ 30 MeV. Nevertheless, a number of upcoming and proposed CMB missions, such as the Simons Observatory [679] and CMB-S4 [145], could provide a determination of N_eff around the percent-level, which will improve the CMB bounds.

More broadly, it has long been known HNLs produced in the Early Universe can lead to an overabundance of matter, directly affecting the expansion rate [680, 681]. For sufficiently long-lived (lifetime ≳ t_eq ) HNLs with a relic abundance larger than the observed dark matter abundance, this can directly affect the measured value of the Hubble parameter H₀. This is illustrated as a green region in figure 34. If these HNLs are abundantly produced but decay before matter-radiation equality, they do not leave an imprint on H₀. However, they may lead to an early period of matter domination. This can alter the sound horizon at recombination, affecting the location of the CMB peaks, as well as the BAO peak as observed in the large scale matter power spectrum [672]. Such scenarios can be ruled out to be CMB and LSS data, as shown in the purple region of figure 34.

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Finally, heavier HNLs decaying into charged final states after recombination can lead to extra ionization and heating [682] during the dark ages. Rescattered CMB photons lead to a suppression of the angular power spectrum at high multipoles, as well as increased correlation in the EE power spectrum for lower ℓ. Including data from Planck and Lyman-α limits on heating on the intergalactic medium, this leads to a lifetime bound of τ ≳ 10²⁶ s [674]. Translated to a mixing angle, for HNLs heavier than twice the electron mass, this yields [672]:

$$\begin{eqnarray}&&{\sin }^{2}\theta \gtrsim \times {10}^{-5}\sqrt{\displaystyle \frac{192{\pi }^{3}}{{b}_{e}}{\left(\displaystyle \frac{{\mathrm{GeV}}^{-2}}{{G}_{F}}\right)}^{2}{\left(\displaystyle \frac{\mathrm{eV}}{{m}_{N}}\right)}^{7}},\end{eqnarray} \tag{ 22 }$$

where b_e is the fraction of decays into e⁺ e⁻ pairs. Equation (22) is valid for lifetimes longer than ∼300 000 years, corresponding to the time of recombination.

Such exotic energy constraints are close to cosmic-variance limited and could be marginally improved by future CMB experiments such as CORE. However, a precise measurement of the baryon temperature at redshifts 20–25 to (δ T_b ∼ 5–10 mK) may reliably improve such constraints by around an order of magnitude [673]. Energy injection before recombination could also lead to μ-type distortions of the CMB spectrum. A future spectral distortion mission such as PIXIE [683] could help set leading constraints on lifetimes between 10⁶ and 10¹³ s [673].

In addition, HNLs can be produced in a variety of astrophysical environments, e.g. supernovae [684–686].

Supernovae (SN) may be efficient ${ \mathcal O }\,(\mathrm{keV})$ HNL factories. This is thanks to matter-driven enhancement of mixing angle [687], allowing it even to reach the resonant value ${\theta }_{\mathrm{res}}=\pi /4$ inside particular regions of the SN. Leaving the resonance region, the produced HNLs freely escape SN, providing an additional mechanism of its cooling which may lead to shortening of the duration of the active neutrino burst (the main cooling stage). Comparing the results of observations and theoretical predictions, this might be used to impose constraints for HNL parameters.

The only supernova event occurred sufficiently close to observe the neutrino flux was the explosion of SN1987A. Three neutrino detectors have reported simultaneous detection of a series of events with electron anti-neutrinos: IMB [688], Kamiokande II [689] and Baksan scintillator telescope [690]. They have shown that the emission duration was Δt ∼ 10 s, with characteristic neutrino energy E_ν ∼ 10 MeV. This data allows to make an estimate on the total energy emitted by neutrinos as ${E}_{{\bar{\nu }}_{e}\mathrm{tot}}\sim 3\cdot {10}^{53}$ erg, assuming the neutrino flavour equipartition [691]. It fits the estimate of the gravitational energy of a neutron star as a potential remnant of SN1987a, which defines the total energy budget of the explosion.

Considering this data, in order to maintain the observed duration of the active neutrino emission, the energy loss driven by HNLs even during short period of time must not exceed ⁴⁹

$$\begin{eqnarray}&&{E}_{N}\lesssim {10}^{53}\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 23 }$$

The constraint was numerically justified for SN explosion in models with axions [692] and may be applied to the case of HNLs, although there are differences of their production mechanism.

The HNL production inside the SN has been studied in the works [686, 693–703]. However, any possible constraints are subject of either theoretical uncertainties related to the SN explosion model or lack of experimental data:

• –  
  SN neutrinos observation did not provide the information about the temperature of the SN withing the neutrino-sphere, where neutrinos are trapped, while HNLs are mostly produced there, see [700].
• –  
  There was no direct observation of the SN1987a remnant yet [704, 705]. ⁵⁰ Despite the neutron star is commonly believed to be the one, there is a probability to have a different remnant which may appear in a different explosion model [706].
• –  
  Most models are either built on toy models of SN explosion or used the results of SM-based numeric simulations (Core-collapse supernovae model [707, 708]) where the back-reaction from the HNL production on the SN explosion is either omitted or simplified [709].
• –  
  Assuming non-typical supernovae explosion with a various neutrino spectra, limits for total emitted energy might be extended (see [710]), spoiling the energy loss argument.

To summarize, the SN explosion data does not allow to set robust bound on HNL parameters from study of SN explosion at the current status of the problem.

As already mentioned in section 2, the observed BAU is one of the strongest hints pointing at existence of physics BSM, see [712]. Leptogenesis is an attractive solution to the question of the origin of matter as it connects the observed BAU with the origin of the light neutrino masses. In this mechanism, the same HNLs that are responsible for the origin of the light neutrino masses, can produce the matter-antimatter asymmetry via their CP violating decays in the early Universe [66]. Among the different realizations, leptogenesis in the type-I Seesaw is by far the most studied one [713], where the SM only needs to be extended by two or more HNLs. Leptogenesis can also be realised with scalar [714] or fermionic [715] triplets, see [716] for a review.

While initial calculations suggested that the mass scale of HNLs needed for leptogenesis is quite high—M ∼ 10⁹ GeV [280], it was soon realized that the mass of HNLs can be as low as the TeV scale in resonant leptogenesis [246], or even at the GeV scale in the case of leptogenesis via oscillations [86, 87]. While developed independently, these mechanisms rely on a similar enhancement, and can be described by the same equations (see e.g. [717]). Together they are often known by the term low-scale leptogenesis.

Due to the discovery potential associated with such light HNLs, this mechanism has received significant attention in the past decade. Below we discuss the constraints on the HNL properties of the imposed by the baryogenesis. In the case with two HNLs, their masses should be nearly degenerate, and we denote the common mass as M. Phenomenologically, one is interested in the maximal size of the HNL mixing angles U² as a function of M [88, 92, 136, 138, 205, 208–210, 248, 718]. The most recent results [92, 93, 210], accounting for the significant theoretical progress [254, 256, 259, 262, 263] (see also [717, 719]), are shown in figure 35.

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In addition to the total mixing angles U² and masses M various properties of the HNLs (e.g. their mass splittings, mixing with individual SM flavours, and the amount of LNV) can be probed in direct searches (see sections 3 and 5) or indirect probes (see e.g. [144]) to constrain the viable parameter space. In particular in the minimal model with two HNLs the dimensionality of the parameter space is small enough that the combined information from direct search experiments, neutrino oscillation data, and neutrinoless double β-decay provides a powerful probe of leptogenesis [88, 136, 138, 164, 209, 638], allowing for a full testability at least in part of the parameter space [88, 136], see section 2.3. Future energy and intensity-frontier experiments hence have the potential to unveil the common origin of neutrino masses and matter in the Universe.

Cosmological and astrophysical measurements indicate that about 25% of the total energy budget of the Universe does not interact with known particles or emit any light. This phenomenon is known as dark matter [730]. The nature of dark matter remains an unsolved puzzle. The central hypothesis of today is that dark matter (DM for short) is made of particles. The only SM particles that could play role of DM are neutrinos. This would require the sum of neutrino masses to be ∑_i m_i ∼ 11.5 eV [731] (clearly violating laboratory [41] and cosmological [732] constraints). Moreover, the neutrinos would be relativistic during cosmic structure formation, leading to clustering properties inconsistent with observation [733]. Finally, neutrino DM would violate the 'Tremaine-Gunn bound' [734] on the phase space density of DM. Summarizing drawback of neutrino dark matter, one can say that they are 'too light' and 'too abundant'. HNLs cure all of these drawbacks, see e.g. [735] for a review, and provide a testable dark matter candidate with reach astrophysical phenomenology.

HNL DM production models. First, the HNLs' feeble interactions imply a smaller abundance than SM neutrinos. This is true for DM produced via mixing with neutrinos [87, 211–215, 736, 737] in the minimal models like νMSM (section 2.4). Another production mechanism that does not require the introduction of new particles is provided by Einstein–Cartan theory [289]. Einstein–Cartan theory is a description of gravity which is obtained by gauging the Poincaré group [738], see e.g. [739] for a review. This theory is equivalent to general relativity augmented by a universal four-fermion interaction. Dark matter N is produced via the annihilation of the SM particles $X+\bar{X}\to N+\bar{N}$ at very high temperatures. The momentum distribution of DM produced in this way has a unique form and differs from both non-resonantly and resonantly produced sterile neutrinos. In this mechanism, the mixing of sterile neutrinos is not related to their abundance. Moreover, the mass of N can vary in a wide range of values, not being limited by the keV range and thus avoiding many of the structure formation constraints (see below).

Beyond the minimal HNL models, the particle content of the theory is broader, and thus more production mechanisms exist. HNL DM can be produced in decays of heavier particles: neutral scalars [740–743], charged scalars [744, 745], an additional Higgs doublet [745, 746], vector bosons [747] or fermions [748]. HNLs can also be charged with respect to new gauge interactions (as in LR symmetric models) [115, 292, 294, 749], see [750] for the recent summary. In these cases, the abundance is generically 'too high'. However, owing to the presence of extra species, these models can produce extra relativistic species ('generate entropy') thus bringing the total HNL abundance to the dark matter level [751–755].

Probing HNL DM. HNL DM particles can be constrained in a number of ways. Although none of them are specific for HNL DM, their combination and cross-correlation with the results of ground-based searches may provide a convincing argument for the detection of the HNL DM.

The phase space density analysis (similar to that of [734]) leads a lower limit on the HNL DM mass (as on any other fermionic DM). There are model-independent and model-dependent bounds (see [756] for a general discussion). The recent updates, based on the improved Jeans analysis of dwarf galaxies can be found in [757].

Structure formation constraints. HNL DM is produced relativistic both in the minimal and non-minimal models. Therefore, it free-streams out of overdense regions, erasers (or at least modifies) the power spectrum of primordial inhomogeneities beyond the free-streaming horizon scale [758] and affects the formation of small-scale structures. Observables in which this can be visible include the Lyman-α forest, weak lensing constraints on the matter power spectrum, see e.g. [759] or [760] for the recent results, cosmic void properties [761–763], the number of high-redshift galaxies [764–769], counts of various objects on comparably small scales (galaxy halos, Milky Way satellites) [764, 770–775].

The widely used tool is the Lyman-α data [758, 776–787]. The most crucial result of this analysis is that at small scales the data show a cut-off as predicted by realistic DM models [779, 780, 788, 789]. However, current data do not allow to identify the nature of this signal. Indeed, astrophysical effects can strongly intervene at scales of interest [779, 780, 789]. Most prominently, gas pressure prevents small lines from forming and thermal broadening also smooths lines [790, 791]. To derive conclusions about the nature of DM this degeneracy must be resolved [780, 789]. Significant improvement in the data quality, better modeling of intergalactic media physics at higher numerical resolution and the use of novel statistical observables [792] will allow to detect this effect, remove the degeneracy and obtain a definite result about the nature of DM.

Gravitational lensing (and microlensing) is a promising way to probe the presence of substructures directly (see [793–805]). The detection of low mass halos within the main lensing halo [see e.g. [806]] or along the line-of-sight [see e.g. [807]] relies on the detailed information contained within the multiple lensed images. A large sample of lenses together with a robust way to predict the average distribution of sub-halos is required in order to make a statistically robust prediction.

A promising way to the detection of dark substructures relies on gaps in the stellar streams. Stellar streams are thin bands of stars lying on trajectories of dissolved dwarf galaxies or globular clusters in the Milky Way halo [808, 809]. Dark subhalos can perturb these streams, leading to the gaps and other features in the streams. Several works have attempted to use these gaps to infer the number of dark satellites (and thus, constrain DM) [810–814]. However, gaps can also be created by many astrophysical reasons, including time-varying bar potential or giant molecular clouds [815–820] making robust constrains challenging. The ambiguity will somewhat be reduced for distant streams, potentially observable with the next-generation of photometric surveys.

Future potentially interesting observations include 21 cm (see e.g. [821, 822]). The recent EDGES result demonstrated a lot of potential power in these observables for the structure formation constraints [823, 824] as well as uncertainties, associated with unknown reionization history.

The wealth of new cosmological data expected in the coming years as well as methodological breakthroughs will eventually allow to distinguish 'nuisance' astrophysical processes from (potential) effects of DM free-streaming and thus to infer the nature of DM from cosmological observations. Joint analyses (see e.g. [812, 814, 825]) will help to reduce systematic uncertainties of improper astrophysical modeling in each specific.

DM decay line. Weak-like interactions of HNLs together with their mass allow them to decay [537]. The most interesting channel from a point of view of DM detection is the loop mediated radiative process N → ν γ [826, 827], which predicts a sharp photon emission line from DM regions [737, 828, 829]. The phase space constraints limit HNL DM mass to be above ${ \mathcal O }(0.1)$ keV which means that this line can be detected in x-ray or γ-ray telescopes [830]. The searches for DM decay lines have been extensively done in the last 20 years, with a generic bound for the lifetime being 10²⁷–10²⁸ s, see [830] for review.

In 2014 an unidentified feature at 3.5 keV in the x-ray spectra of galaxy clusters [831, 832] as well as Andromeda [832] and the Milky Way galaxies [833] was reported. The DM interpretation of this signal has been the subject of an active discussion within the community ever since, see [735] for a summary. The central point of the discussion is related to the background modelling [834–837] and statistical interpretation of the data [838, 839]. Similar uncertainties exist in observations with the NuStar [840–842], for which the 3.5 keV lies at the edge of the sensitivity interval, making a quantitative estimate of the errors difficult. The preliminary analysis with Hitomi data [843] demonstrates the power of high-resolution x-ray instruments. It has also excluded astrophysical origin of the 3.5 keV line [844–847]. Depending on cosmology and theoretical model, the presumed 3.5 keV x-ray signal could correspond to a sterile neutrino with a mixing testable in upcoming laboratory experiments [848, 849]. The final word in this discussion will be put by the XRISM instrument [850] that will either detect the line. Additionally, microcalorimeters on sounding rockets [851] looking into the direction of Galactic Centre may confirm the origin of the line.

In summary: models with HNLs can accommodate dark matter particles in a number of ways. The range of masses of HNL DM is from sub-keV range to MeV or even GeV (depending on the model/production mechanism). These models have many observational signatures that allow to probe them with existing and forthcoming astronomical and cosmological data.

Measurement of the Hubble constant H₀, which describes the expansion rate of the Universe, supplies vital information about cosmology as well as potential insights about new physics [852]. Discrepancy between the measurements of Hubble constant in the local universe (e.g. Cepheids, Type-Ia supernovae) of H₀ = 74.03 ± 1.42 km s⁻¹ Mpc⁻¹ [853] and Planck satellite measurements of CMB and BAO data of H₀ = 67.66 ± 0.42 km s⁻¹ Mpc⁻¹ [43] is at the ∼5σ level [854].

Consider a minimal scenario with a heavy sterile neutrino that primarily couples to the SM through a mixing $\sin \theta$ with an active neutrino ν_a (a = e, μ, τ). [701] demonstrated that such sterile neutrino with mass ${m}_{s}\simeq { \mathcal O }(30)$ MeV decaying just before BBN, predominantly into active neutrinos, can increase the effective number of relativistic neutrino species ΔN_eff and thus alleviate the H₀ discrepancy. Intriguingly, the necessary masses and couplings of the sterile neutrino, assuming it mixes primarily with ν_τ and/or ν_μ neutrinos, are within reach of the Super-Kamiokande as well as upcoming laboratory experiments such as NA62 and DUNE. More so, improved upcoming measurements of ΔN_eff by CMB-S4 [855] will further explore this scenario. As demonstrated by [662], in the presence of non-negligible lepton asymmetry, decays of even heavier sterile neutrinos with mass m_s ≃ 150 to 450 MeV can assist with alleviating the Hubble tension.