New leptons with exotic decays: collider limits and dark matter complementarity

We describe current and future hadron collider limits on new vector-like leptons with exotic decays. We consider the possibility that, besides standard decays, the new leptons can also decay into a Standard Model charged lepton and a stable particle like a dark photon. To increase their applicability, our results are given in terms of arbitrary branching ratios in the different decay channels. In the case that the dark photon is stable at cosmological scales we discuss the interplay between the dark photon and the vector-like lepton in generating the observed dark matter relic abundance and the complementarity of collider searches and dark matter phenomenology.


Introduction
New fermions are a common occurrence in models of physics beyond the Standard Model (SM). If they are vector-like [1], namely both chiralities have the same quantum numbers, their mass term is gauge invariant and therefore it is not tied to the electroweak scale. As a result, they do not contribute to anomalies and all their physical effects decouple as inverse powers of their mass. Their phenomenological implications have been extensively studied, in particular in the case of vector-like quarks (triplets under color SU(3) C ), as they are strongly pair-produced at hadron colliders. Furthermore, the fact that electroweak top couplings have been measured with less accuracy than for lighter fermions leaves more room for relatively large indirect effects [2,3] and single production (see however [4] for strong constraints in minimal models and [5] for ways to evade them in more realistic ones).
Vector-like leptons (VLL), neutral under SU(3) C , have received much less attention. Indirect constraints [6,7] put very stringent limits on their mixing with the SM fermions, thus significantly reducing the possibility of a sizeable single production at colliders. Pairproduction via Drell-Yan is quite model independent (see however [8][9][10]) but the smaller production cross-section than for vector-like quarks makes the reach quite modest (see [11][12][13][14][15][16][17][18][19][20] for theoretical studies and [21][22][23] for experimental searches). Furthermore, decays into only SM particles are assumed in all these works except for [17,18] in which SM decays mediated by extra scalars are also considered. However, there are classes of models with JHEP01(2022)111 new VLL that incorporate a discrete symmetry under which SM particles are even and new particles are odd, thus preventing the decay of the VLL into only SM particles. They typically decay into the lightest odd-symmetric particle, which is often a dark matter (DM) candidate (see ref. [24]). A prime example is T-parity in Little Higgs models [25,26]. The lightest (and therefore easiest to produce) VLL usually decays into a SM lepton and a stable particle that results in missing energy at colliders. Such a decay has not been considered by experimental collaborations in the context of VLL searches. The production and decay pattern is very similar to the one of slepton pair production but due to the different spin of the particles involved, the interpretation of the experimental results in terms of VLL searches requires a recast of the analysis by theorists (see for instance [27]).
Even more interestingly, the possibility of simultaneously having both types of decays, into a SM lepton plus a W , Z or Higgs boson and into a SM lepton and missing energy, has never been considered in the past, despite the fact that this possibility is easy to realize and is even well motivated in the context of feebly interacting DM [28]. In this article we consider the possibility that the new VLL can simultaneously decay into the usual SM final states as well as into a SM lepton and missing energy. We will leave the decay pattern completely general so that our results apply to a large number of phenomenological models involving VLL. (See [29] for a similar study for the case of vector-like quarks.) Inspired by the case of Little Higgs models with T-parity and by feebly interacting dark photon models we will consider the missing energy particle to be a dark photon, a massive vector that is stable at detector scales. However, this dark photon could be stable at much longer scales, of the order of the lifetime of the Universe and therefore be a good DM candidate. We will also explore this possibility and we will show that the VLL can play a crucial role in this regard. Indeed, it can open a large region of the allowed parameter space by either contributing to the relic abundance via co-annihilation with the dark photon or via the freeze-in mechanism. We will analyze these two possibilities and we will show that they can give complementary information in the former case and benefit from the collider searches in the latter one.
The rest of this article is organized as follows. We describe in section 2 the most relevant current experimental searches for a new VLL with general decays. We then optimize these searches and obtain the expected LHC bounds on new VLL with arbitrary branching ratios with the current recorded luminosity. This is one of the main results of this article and it allows us to immediately get the constraints on new VLL with arbitrary decay patterns. We then explore the reach of the high-luminosity (HL-LHC) and high-energy (HE-LHC) configurations of the LHC together with an estimation of the final hh-FCC reach. Section 3 is devoted to the case in which the missing energy particle is stable and can act as a good DM candidate, first assuming the standard freeze-out mechanism and then the freeze-in one. We will see that in both cases the interplay with the VLL is crucial for a successful model. We then present our conclusions in section 4. We present in appendix A an explicit realization of the scenario we consider in the main text.

JHEP01(2022)111 2 New vector-like leptons with general decays
The goal of this article is to study the current and future reach of hadron colliders on new VLL that can decay not only into SM particles but also into a SM charged lepton and a neutral particle that is stable at detector scales and therefore appears as missing energy. We will present our results in a model-independent way whenever possible, as a function of arbitrary branching ratios in the different channels. To show actual limits we will however focus on a new VLL singlet with electric charge -1, E L,R , and mass M E , and a massive vector boson A µ H , as the stable (at detector scales) particle, with mass M A H < M E so that E can decay into A H and a SM lepton.
An explicit realization of our model is given in appendix A but the details are not needed for the moment. The only relevant information is that the dominant E decays are given by the following branching ratios BR(E → H), BR(E → ν W ), BR(E → Z) and BR(E → A H ), where stands for either electron or muon 1 and we assume the sum of these four branching ratios to be equal to one but otherwise arbitrary. 2 We focus on E Drell-Yan pair production, with subsequent decays governed by the corresponding branching ratios. 3 Out of the four possible decay channels, the two that are easiest to detect experimentally are E → Z and E → A H . The charged current one into ν W is difficult to disentangle from the overwhelming W + jets background and the one into H is either also difficult to disentangle from the relevant background or suffers from small branching fractions into easier to detect channels. Thus, in the following we will focus on the cleaner channels and give our results in terms of BR(E → Z) and BR(E → A H ). We will show that the results are mostly insensitive to the value of the two extra branching ratios.
There are currently two experimental analyses that are most sensitive to these discovery channels, searches for VLL into Z and slepton searches. Neither of them can be directly used in our more general scenario, except for the former in the BR(E → A H ) = 0 limit. The slepton searches have to be completely recast because of the different spin of the intermediate particle and also because of the contamination of other channels in the BR(E → A H ) = 1 limit.
We will begin this section by recasting the two relevant experimental analyses. We will first compare our results with the ones published by the experimental collaborations and then extend the analyses by considering arbitrary decays into the different channels. We will also update the analyses to take full advantage of higher luminosity and/or center of mass energy.

Recasting existing analyses
Since our goal is to interpolate between the limiting cases in which the branching ratio of the VLL to the missing energy channel goes from 0 to 1, we start by reproducing searches that probe these two limiting cases. The VLL model is implemented in Feynrules [35] and 1 Decays into tau leptons have been considered, assuming SM decays only, in [14,15,30]. 2 The decays into SM particles are usually fixed by the quantum numbers of the VLL but in realistic models with a rich spectrum, the mixing between heavy states can lead to arbitrary decay patterns [31]. 3 Studies in which the production and/or decay of new vector-like fermions are dominated by nonrenormalizable interactions can be found in [10,[32][33][34]. Figure 1. Pair production of a VLL singlet E and decay channels our analysis is most sensitive to.
leading order event generation is done with MadGraph5_aMC@NLO [36] with the NNPDF23LO [37] parton distribution functions. For the background simulation generator level cuts are applied which are specified in the text. All of these were tested to verify that their influence was minimal to the final yield of events after the analysis. Showering and hadronization are performed by Pythia8 [38]. The detector response is modeled with Delphes 3 [39]. We use the default CMS detector card for the LHC analysis and the HL-LHC detector card for the √ s = 27 TeV analysis. 95% C.L. limits are obtained using the CL s [40] method by fitting the relevant discriminant variables using OpTHyLic [41] which outputs the upper limit on the signal strength, µ = σ up /σ th , where σ up is the upper limit on the cross-section and σ th is the theoretical prediction obtained through the MadGraph simulation.

Decays into SM particles
For the case in which the VLL decays exclusively to SM final states (W ν, Z and H ) we reproduce the analysis presented in ref. [22], an ATLAS search performed at √ s = 8 TeV and an integrated luminosity of L = 20.3 fb −1 , looking for multi-lepton signals coming from the Z decay of a singlet VLL. The main production and decay channels are depicted in figure 1. This analysis selects 2 opposite sign same flavour (OSSF) leptons to reconstruct a Z boson and a third lepton with a ∆R ≡ ∆η 2 + ∆φ 2 < 3, with η and φ the pseudo rapidity and azimuthal angle, respectively, from the reconstructed Z boson, which is defined as the off-Z lepton. The definition of the full cuts and the corresponding efficiencies are presented in table 1. 4 The analysis searches for an excess in the distribution of the variable ∆m = m 3 − m , where the mass of the reconstructed Z boson, denoted by m , is subtracted from the invariant mass of the 3-lepton system, m 3 . Furthermore, 3 exclusive signal regions are defined, depending on the number of identified leptons and hadronically decaying W : 4-lepton region, in which at least 4 leptons are identified; 3-lepton + jj region, in which precisely 3 leptons are identified together with 2 jets whose invariant mass must be in the range m W − 20 GeV < m jj < 150 GeV , with m W the W boson mass; and 3-lepton JHEP01(2022)111   Comparison of our recast of the VLL search with the ATLAS collaboration results. We show the case in which the off-Z lepton is an electron, with the 1 (green) and 2 (yellow) sigma exclusion region from our simulation together with the expected limit as reported in the ATLAS search (solid blue) and the theoretical pair production cross section (dashed red). The branching fractions are fixed to those of a VLL singlet with SM only decays (as a function of its mass). only region, in which exactly 3 leptons are identified with no pairs of jets satisfying the previous condition on their invariant mass.
The analysis is performed separately for the case in which the off-Z lepton is an electron or a muon, corresponding to the VLL coupling only to first or second generation leptons, respectively. The main backgrounds for this analysis are ZZ, W Z and Zγ, for which our simulation very accurately reproduces the shape. We normalize these backgrounds to the values reported in the experimental publication, which amounts to a factor between 1.4 and 3.5, depending on the signal region, including the corresponding K-factor. We show in figure 2 the comparison of our 1-and 2-sigma exclusion plot (Brazilian plot) with the expected limit reported in the experimental search, together with the theoretical pair production of the VLL, for the case in which the off-Z lepton is an electron. The case in which it is a muon shows a similar level of agreement. In this analysis the VLL branching fractions are fixed to those of an electroweak singlet (as a function of its mass) and the resulting limit on the VLL mass is M E 160 GeV, which represents a difference of ∼ 7% in comparison to the expected limit obtained in original analysis.
In order to see what the reach with the current recorded luminosity can be we have repeated the same analysis at √ s = 13 TeV and L = 139 fb −1 . However, we can take advantage of the higher center of mass energy to impose more stringent cuts, in particular on the transverse momentum of the leading leptons. Since the p T of the observed leptons in signal events increases with the increase in the VLL mass, we have defined 3 clusters of masses in which the selection threshold for p T of observed leptons varies. We present in table 2 the definition of these clusters and the efficiencies of all selection cuts. With these more stringent selections, we were able to apply generation level cuts on the Zγ background, generating only events in which at least one lepton has p T > 62 GeV.
Furthermore, at these higher energies, we can also remove almost the entirety of the W Z background by setting a cut on the transverse mass, m T , of the reconstructed W boson. This cut is effective because in events from W Z → ν , in principle, the off-Z lepton is coming from the W decay and the missing energy of the event, / E T , originates from the neutrino. Therefore, for events from the W Z background, we have where p T is the transverse momentum of the off-Z lepton. As such, this quantity should, in principle, be at most the mass of the W boson. In order to keep as many signal events as possible, this cut is only performed on the 3-lepton signal region, which contains most of the W Z background. Figure 3 shows the limits obtained with this improved analysis.
Assuming that the observed data corresponds to the expected background, a mass of the VLL up to 410 GeV (420 GeV) could be excluded by this analysis for the case in which the off-Z lepton is an electron (muon). Given the similarity between the limits obtained when the VLL couples to first or second generation of SM leptons, we will only explore the case in which it couples to electrons hereafter.
Despite being tailored for the case in which the VLL is a singlet of SU (2), this analysis can also be applied for a VLL doublet, L, of hypercharge −1/2. In this case we need to take into account not only the pair production of the charged component of the VLL doublet, p p → E + E − , but also the pair production of the neutral component, p p → N N , and the associated production of both, p p → E ± N . For large masses (we will consider both components degenerate in mass), the charged component will decay equally to Z and H, while the neutral component decays solely to νW . Therefore, our background remains the same, and as such we can recast the previous analysis to the doublet case, obtaining the limits shown in figure 4, where we considered the off-Z lepton to be an electron. As expected, we obtain much more stringent bounds than on the singlet case, with masses up to ∼ 820 GeV being excluded.
An analysis searching for VLL doublets of hypercharge −1/2 was performed by the CMS collaboration in ref. [23] with an integrated luminosity L = 77 fb −1 . A bound M L ≥ 790 GeV was obtained in this analysis due to a statistical fluctuation in the observed data. The expected limit in that analysis, which is the fair comparison to the bound we can compute, corresponded to M expected L ≥ 690 GeV. Rescaling our search to the same integrated luminosity we find M L=77 fb −1 L ≥ 730 GeV, remarkably close to the expected limit in the CMS search, despite the fact that the CMS analysis targets decays into tau leptons and therefore a direct comparison is not straight-forward.

Decays with missing energy
To explore the case in which the VLL decays predominantly into a SM lepton and missing energy (A H in our case), we consider an ATLAS analysis [21] at where p T1,2 represent the transverse momentum of each of the identified leptons and q T is the vector that minimizes the maximum of both transverse masses, defined as The main backgrounds for this signal are diboson processes (ZZ, W W and W Z) and tt. In order to maximize our statistics we always include 2 leptons (with an m > 95 GeV partonic cut) in the final state in the background generation. These 2 leptons can be of  Table 3. Cumulative efficiencies for the background events after applying the selection cuts corresponding to the analysis performed at √ s = 13 TeV for slepton searches [21]. Efficiencies are presented as the number of events which are selected over the number of initial events (see text for details). either family for the diboson case and are restricted to the first two generations for the tt one. The selection cuts and corresponding efficiencies are presented in table 3. We have validated the analysis including all signal regions proposed in the original analysis [21]. However, new results are calculated considering only the signal region of m > 111 GeV and m T 2 > 100 GeV as we find the difference in regards to all signal regions not significant. Given that the analysis in ref. [21] applies to sleptons and neutralinos, which are scalars and fermions, respectively, as opposed to our case with a VLL and a dark photon (fermion and vector, respectively), in order to validate our analysis we have implemented a sleptonneutralino model. Fixing the neutralino mass Mχ0 1 = 1 GeV we obtain the results shown in figure 6, with a limit M˜ ≥ 565 GeV, very similar to the expected limit obtained by the ATLAS collaboration, ∼ 570 GeV.  [21]. The excluded region is the one below the curves.

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Once we have validated the analysis we can apply it to the VLL model. Contrary to the case of purely SM decays, in this case we have a new degree of freedom in our analysis, the mass of the other new particle, M A H . Some models predict this mass to be close to the electroweak scale, such as the Littlest Higgs model with T-parity, but in other cases, we can have sub-GeV masses as is the case in feebly interacting massive particles (FIMP) in which this new particle plays the role of DM as we will see below. As such for each mass point of the VLL, we vary M A H from 1 GeV up to the mass of the VLL in question. We present the limits obtained for √ s = 13 TeV and an integrated luminosity L = 139 fb −1 in figure 7. As expected, the analysis is more constraining for lighter A H . As the mass difference between A H and the VLL decreases, the leptons from signal events become softer and more difficult to identify and pass the selection cuts. For M E 900 GeV the production cross-section is too low and the analysis cannot constrain the signal regardless of M A H .

Constraints on vector-like leptons with general decays
Once we have ensured the accuracy of our simulations to recast the experimental searches in the limiting cases in which the VLL decays only through SM or missing energy channels, we are in a position to interpolate between them and therefore consider the case of arbitrary branching fractions in the different channels. To do this, we have scanned different possible branching ratios for each of the decay channels. In order to not need to generate every signal corresponding to different BRs, we apply a weight to each signal event according to its decay. To do this, we generate a signal of Drell-Yan pair-produced VLLs with where the g subscript describes the generated sample. To probe a specific point with different BRs, each event is weighted by BR i p /BR i g , where p subscript represents the probed branching ratio and i superscript corresponds the specific decay -the decay of each event is determined at generator level. Which analysis is more constraining depends on the particular value of the branching ratios but since they specifically target the final states with either A H and Z, the results are presented in the BR(E → A H ) vs BR(E → Z ) plane with the others branching ratios being fixed to which correspond to the relation between the different branching ratios in the large M E limit for BR(E → A H ) = 0. We have checked that the corresponding bounds are quite insensitive to this latter choice. The residual dependence is due to cross-contamination between different channels into our signal regions. However, this effect is small as shown in figure 8, where we represent the change in the signal strength µ as a function of the branching ratio into W ν for two different values of the remaining parameters. The signal strength that represent our discriminating variable changes by 20% at most, which results in a very mild dependence of the final limit on M E . Our final result, that combines the two analyses discussed in the previous section for arbitrary values of BR(E → A H ) and BR(E → Z ) are shown in figure 9 for two different values of M A H = 1 GeV (left panel) and M A H = 98 GeV (right panel). As expected, the effect of the A H mass is more relevant in the region in which the missing energy signal dominates and for lighter values of the VLL mass, since the smaller mass difference results in a softer lepton. Still, except for very low branching ratios into the decay channels targeted by our analysis, the differences are minimal. Thus, from now on we will only report our results for M A H = 1 GeV. 5 We show the results as contours for fixed value of M E with the region above and to the right of each contour line being excluded for that mass at the 95% CL. The limit for the VLL singlet case with SM decays can be easily obtained by considering the vertical axis, which corresponds to BR(E → A H ) = 0, at the relevant 5 Larger masses of AH would give rise to softer final-state leptons and therefore worsen the obtained limits on the VLL mass. In particular, for the case near degenerate particles, very little can be taken from this collider analysis, which is why we study the complementarity with DM in the next section. The small difference between the bound on the VLL singlet reported here and the one found in figure 3 is due to the fact that here we are fixing the branching ratio to Z to 25% whereas before the branching ratios were set by the couplings and masses of the model (the branching ratios tend to 25% in the large mass limit).

Future projections
The constraints presented in figure 9 represent the current constraints on a new charged VLL with general decays. In this section we explore the potential of the LHC to probe new VLLs in its high-luminosity (HL-LHC) and high-energy (HE-LHC) configurations. We will also explore the potential reach of the 100 TeV hh-FCC.
Starting with the HL-LHC (for which we take √ s = 13 TeV and an integrated luminosity of L = 3 ab −1 ) we use the same improved analysis described in the previous section and in table 2, making sure that we generate enough statistics for the required integrated luminosity. When considering a higher energy collider, like the HE-LHC, for which we consider √ s = 27 TeV and L = 3 ab −1 , we can again afford to impose more stringent cuts on the different variables involved in the analysis, in particular in the lepton p T . In the SM decays analysis, we impose a partonic cut on all backgrounds of p T > 75 GeV of the leading lepton whereas for the analysis focusing on the missing energy decay, backgrounds were generated with a partonic cut of p T > 100 GeV for the leading lepton. We were able to use this cut since we updated the selection thresholds from table 3 to p 1 T > 120 GeV in the missing decay analysis. The resulting reach, again for M A H = 1 GeV, is reported for arbitrary branching ratios in figure 11. The estimated reach, in the limiting cases is (2.7) A detailed study of the reach of future circular colliders on VLLs with general decays is beyond the scope of the present work, however, we can use a crude estimate of the corresponding reach at the hh-FCC by considering the instantaneous luminosity as used in the Collider Reach tool [44]. First we test the validity of this approach by extrapolating the current luminosity results reported in eq. (2.5) to the HL-LHC and to the HE-LHC. We find that the extrapolation agrees with our detailed simulation within 6%(14%) in the case of the HL-LHC for the SM decays analysis (missing decays analysis) and within 6%(35%) for the HE-LHC for the SM decays analysis (missing decays analysis). The latter case shows the differences that arise not only from the increased production cross sections of signal and backgrounds but also from the more stringent cuts that we can imposed with higher (2.8) The results reported in figures 9-12 are completely general except for the fact that we are using the production cross-section of a VLL singlet with hypercharge -1 to obtain the mass limits. For the sake of generality, we provide in figure 13 the cross-sections we have used for the LHC, HE-LHC and hh-FCC so that our limits can be applied to more general VLLs by rescaling the corresponding pair production cross-section.

Dark photon as a dark matter candidate
So far we have just assumed that the lifetime of A H is large enough to appear as missing energy at detector scales. However, if A H has a lifetime larger than the age of the universe, it becomes a suitable candidate for DM. As such, we can use the observed relic density and direct detection experiments to further constrain these models. In this section we will focus on two possible production mechanisms for DM. We will first consider the case in which A H has a mass around the electroweak scale and its abundance is fixed through the  freeze-out mechanism. Then we will consider the possibility that A H is light and has a very weak coupling to the SM so that its production follows the freeze-in mechanism.

Standard freeze-out
For the case of a heavy DM candidate -with a mass around the eletroweak scale -we will consider that it is stabilized through a symmetry. An example of this arises in the Littlest Higgs model with T-parity (LHT) [25,26], in which A H is T-odd, as is the vector-like lepton, while the SM particles are T-even. Therefore, the VLL decays exclusively through the missing energy channel. Since A H is a singlet of the SM, we can write the following operators where v ≈ 174 GeV, the dots represent other couplings that are irrelevant for the viability of A H as a DM candidate and we have included explicit factors of the U(1) Y gauge coupling g to make the connection with the LHT model more direct. In the LHT model c A H h = 1 8 and q H = 1 10 [45]. The latest Planck results measured the relic density abundance to be Ωh 2 ∼ 0.12 [46] and therefore the model must predict a relic density equal to (A H accounts for all of DM) or smaller than (A H is only part of DM content) that number. The most relevant processes for the annihilation of A H are to b-quarks, W + W − or Z bosons or top quarks (depending on the mass of the DM candidate) through the s-channel exchange of a Higgs [47]. Furthermore the annihilation into leptons through the exchange of the VLL is also important -the corresponding diagrams are shown in figure 14. Therefore, as mentioned above, the relic density calculation will be controlled by the couplings of A H to the Higgs and the coupling to the VLL and SM lepton and thus we will scan different values for these couplings. Given that the VLL mediates one of these channels, when the s-channel annihilation is subdominant, the mass difference between A H and the VLL will also play an important role.
The calculation of the relic density and direct detection bounds are done using MadDM [48] by inputting a UFO model [49] which we generate through Feynrules [35]. The results are presented in figure 15  dominates; however, as we increase q H , the channel mediated by the VLL becomes more important and we get a significant rise in the annihilation cross-section, with a q H ∼ 1.7 allowing for a significant part of the depicted parameter space. As expected, we can also see (particularly for high enough values of q H ) that, as the mass difference between the VLL and the DM candidate decreases, the impact of the VLL-mediated channel increases. The coupling to the Higgs boson is also important for the spin-independent scattering cross-section with nucleons, as the dominant diagrams are the Higgs exchange with quarks or with gluons through a loop of heavy quarks as represented in figure 16. We have computed the corresponding scattering cross-section with MadDM and show, shaded in grey, the excluded region in figure 15 using the XENON1T data [50], considering that A H makes up all the DM.
Varying q H also affects direct detection constraints. In principle q H could be responsible for a 1-loop DM nucleon scattering amplitude, mediated by a photon. However, as noted in ref. [51], for the case of a real DM vector candidate, the coupling between 2 DM particles and a photon will be described by a dimension-6 operator, since the dimension-4 A Hµ A Hν F µν does not exist due to the antisymmetry of the field strength tensor. Moreover, the resulting amplitude will be further suppressed when one takes the non-relativistic limit. As such, we will neglect contributions from this process to direct detection bounds in this work.
Another experimental observable which may be affected by changing q H is the anomalous magnetic moment of both the electron and the muon, depending on which of these SM where the uncertainties include theoretical and experimental contributions.

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The new contribution from E and A H reads [51], where ≡ m /M E and r ≡ M A H /M E and m is the mass of the SM lepton for which the contribution is being calculated. This result is always negative and as such, it contributes in the direction of explaining the (g − 2) anomaly of the electron, whereas it goes in the wrong direction for the muon anomaly. Figure 17 shows the parameter space that is constrained by these measurements. For the case in which the VLL couples to electrons, we show the region which explains the observed anomalous magnetic moment. For the muon case, as this model increases the tension with the experimental result we constrain this contribution to be smaller than the combination of the experimental and theoretical uncertainties. The region above the curves is excluded for the muon case. The results shown in figure 15 reflect the well known tension between the production of the correct relic abundance and direct detection experiments for a standard weakly interacting massive particle. Such tension can be relaxed if the masses of the new particles are nearly degenerate (with the VLL being slightly heavier). This regime of co-annihilation [54] increases the annihilation cross-section since processes such as A H E → SM SM and A H SM → E SM can now contribute significantly. The importance of these contributions will be a function not only of this degeneracy in mass, but also of the coupling q H . An estimation of the needed mass splitting to have a significant contribution to the annihilation process can be obtained by considering that, at the freeze-out temperature, T F , both particles are still in equilibrium. For co-annihiliation to be important, one would have  In figure 18 we show the relic density abundance for cases in which co-annihilation can be important. We consider two values of q H = 0.1 (solid) and q H = 0.2 (dashed) and plot the contours of Ωh 2 = 0.12 for different values of ∆. The region to the right of the different curves is excluded (as it gives too large relic abundance). Again we show in shaded grey the region excluded by direct detection experiments. We see that only for ∆ 0.05 a significant difference with respect to the standard annihilation scenario is observed.
In order to better understand the dependence on q H in this co-annihilation regime we show, in figure 19, the Ωh 2 = 0.12 contours in the M A H − q H plane, again for different values of ∆ for c A H h = 1. The region excluded by direct detection experiments is, as usual, shaded in grey. While for fixed q H we observed that increasing ∆ collapses the relic density line into the non co-annihilation regime, this does not happen in this plot. In this case, even though co-annihilation effects can be negligible for ∆ 0.05, the annihilation process mediated by the heavy lepton is important for low mass differences between the VLL and A H for large q H and as such, the annihilation cross-section is influenced by changes in ∆ even outside the co-annihilation regime.
Note that this co-annihilation case is complementary to what was studied in the previous section at colliders. In this case, given the small mass difference, the final state leptons at colliders will be very soft and therefore are very difficult to identify. There is an ongoing effort to search for this cases of compressed mass states at colliders [55] namely in the context of sleptons. Our results show that the interpretation of such a search in the context of VLLs is very well motivated.

Freeze-in in feebly interacting dark matter
In the case that DM is very light and couples very weakly to other particles, its relic density can be set by the freeze-in mechanism [56]. In this case, the DM candidate is not in equilibrium with the thermal bath but is actually produced through the decay of other heavy particles, in our case, the decay of the VLL. This possibility has been recently explored in [28] with emphasis on the DM phenomenology, thus setting the VLL mass to a conservative M E = 1 TeV in order to avoid any collider constraint. In this subsection we aim to show the complementarity between DM experiments and the collider results we presented before for a FIMP. This scenario is realized by the explicit model that we present in appendix A, to which we refer the reader for the details. The relic density can be calculated as [28]: We present these results in figures 20 and 21 for the LHC analysis at √ s = 13 TeV and L = 139 fb −1 and L = 3 ab −1 respectively. The region below the curves can be excluded by collider searches. In the region above the curve, for that fixed value of s, all the values of M E and M A H are experimentally allowed and can provide the correct DM relic abundance. For each value of s we display the collider bound as a solid or dotted curve, for the constraint coming from the analysis focusing on the SM decays or the missing energy channel, respectively. This is relevant since, in order to use the collider bounds we obtained, E → A H must be a prompt decay for the missing energy search whereas E → Z is the most important channel to be prompt in the SM decays analysis.
In eq. (A.25) we show the minimum value s must take so that E → Z is prompt (note that for the plotted values of s, only a VLL that couples to the 2nd generation of SM leptons would decay promptly). For E → A H the value of g H (fixed for each mass point) is going to determine whether it is a prompt decay mode. For the parameter space probed in figures 20 and 21, no points would represent a prompt decay through the missing energy channel and as such, we cannot consider directly the bounds obtained by the analysis which focuses on this decay channel. A more detailed study, which is beyond the scope of the present work, targeting displaced vertices has the potential to significantly probe the allowed region of parameter space in this class of models. Ref. [28] also calculates the constraints from direct and indirect DM detection on this model and we can see in the reference that when we set A H to reproduce all of the DM relic abundance through the freeze-in mechanism the model avoids direct detection constraints and is only probed by indirect ones -CMB anisotropies and diffuse gamma rays -near M A H = 1 MeV.

Conclusions
New vector-like leptons are quite common in extensions of the Standard Model. In minimal extensions, with no further new particles or anomalous couplings, their decays are governed by their mixing with the Standard Model leptons, which is strongly constrained by electroweak precision data. These constraints eliminate the possibility of substantial single production, leaving Drell-Yan pair production as their dominant production mechanism. Realistic new physics models are, however, usually far from minimal and the new particles present in the spectrum can have a significant impact on the phenomenology of these new leptons. New stable particles allow the possibility of a decay of the vector-like lepton into a Standard Model charged lepton and missing energy. Such a signature has been only experimentally searched for in the context of supersymmetric models with slepton pair production decaying into leptons and neutralinos. From the information given in the experimental analyses it is difficult to directly translate the corresponding bounds to the vector-like lepton case, despite the fact that this signature is well motivated by natural models like the Little Higgs models with T parity. Furthermore, the case in which the new lepton can simultaneously decay into Standard Model particles and into a Standard Model charged lepton and missing energy has been never considered before. This possibility is however also well motivated as it naturally appears in models of feebly interacting dark matter models in which the dark matter relic abundance is generated via the freeze-in mechanism.
In order to fill this gap we have considered the possibility of a new charge −1 vector-like lepton that can decay, with arbitrary branching ratios into a Standard Model lepton together with a Z, H, W or missing energy, represented by a dark photon A H , which is assumed to be stable at detector scales. We have then considered the most relevant LHC analyses probing such a model and, after carefully validating our implementation of the analyses, we have computed the current and future constraints that hadron colliders can place on new vector-like leptons with these exotic decays. Our results, represented as mass limits as functions of BR(E → A H ) and BR(E → Z ) are provided in figures 9-12 for current data at the LHC, the HL-LHC, the HE-LHC and the 100 TeV hh-FCC, respectively. This is JHEP01(2022)111 one of the main results of our work, as it provides the experimental limits from current and future hadron colliders on a large number of models of vector-like leptons with exotic decays.
We have also considered the interesting possibility that the dark photon, A H , is not only stable at detector scales but also at cosmological scales. It can then be a good dark matter candidate and we have explored the interplay between the dark photon and the vector-like lepton to provide a successful explanation for the observed dark matter relic abundance. After showing that the standard freeze-out mechanism presents tension between the generation of the dark matter relic abundance and limits from direct detection experiments, leaving only a relatively small region of viable parameter space, we consider the case of near degeneracy between the vector-like lepton and the dark photon. This leads to a successful generation of dark matter via co-annihilation, compatible with all current experimental limits. The relevant region of parameter space is complementary to collider searches, as the compressed spectrum significantly deteriorates the collider reach. The possibility of specific searches that target these compressed spectra models becomes a very interesting probe of the model in this regime.
Finally, we have considered the case in which the dark photon is very light and feebly interacting, realizing the freeze-in mechanism. We have shown that in this case collider searches are very complementary to dark matter probes and we have found that models compatible with current dark matter phenomenology can be easily tested in current or future hadron colliders.

(A.2)
At the renormalizable level we can write the following Lagrangian is a suitable potential to spontaneously break U(1) H and to make the physical Higgs scalar of such breaking much heavier than all the other fields in the spectrum so that we can effectively neglect it. For simplicity we have assumed that kinetic mixing between the two abelian groups is negligible 7 and that the VLL only couples to one of the SM RH charged leptons, taken to be the electron here, denoted by e The effect of mixing with extra vector-like fermions is well known [1]. The physical basis is obtained by diagonalizing the mass matrix in (A.9) via a bi-unitary rotation where χ = L, R denotes the chirality and, in the m M limit that we will be interested in we have In this physical basis, the coupling of fermions to the electroweak gauge bosons, Z, W , the Higgs boson, H, and the heavy photon, A H , can be written as follows

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In order to realize our scenario we consider the limit M A H M , so that E can decay into Ze, He, W ν and A H e and A H can decay intoēe provided M A H > 2m e . The corresponding decay widths are