aainstitutetext: Laboratório de Instrumentaçao e Física Experimental de
Partículas, Departamento de Física da Universidade do Minho, Campus de
Gualtar, 4710-057 Braga, Portugalbbinstitutetext: CAFPE and Departamento de
Física Teórica y del Cosmos, Universidad de Granada, E–18071 Granada, Spain

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

Guilherme Guedes<EMAIL_ADDRESS>b and José Santiago<EMAIL_ADDRESS>
###### Abstract

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.

##  1 Introduction

New fermions are a common occurrence in models of physics beyond the Standard
Model (SM). If they are vector-like delAguila:1982fs , 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 delAguila:2000rc ; delAguila:2008pw  and
single production (see however Aguilar-Saavedra:2013qpa  for strong
constraints in minimal models and Anastasiou:2009rv  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 deBlas:2013gla  put very stringent limits on
their mixing with the SM fermions, thus significantly reducing the possibility
of a sizeable single production at colliders. Pair-production via Drell-Yan is
quite model independent (see however Araque:2015cna ; Araque:2015cbe ;
Chala:2020odv ) but the smaller production cross-section than for vector-like
quarks makes the reach quite modest (see Altmannshofer:2013zba ;
Falkowski:2013jya ; Dermisek:2014qca ; Kumar:2015tna ; Bhattiprolu:2019vdu
for theoretical studies and Aaboud:2018jiw ; Aad:2015dha ; Sirunyan:2019ofn
for experimental searches). Furthermore, in all these cases, decays into only
SM particles are assumed. However, there are classes of models with 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. A prime example is T-parity in
Little Higgs models Cheng:2004yc ; Low:2004xc . 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 Dercks:2018hgz ).

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 Delaunay:2020vdb . 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 Chala:2017xgc  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.

##  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 $A_{H}^{\mu}$, 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
$\mathrm{BR}(E\to\ell H)$, $\mathrm{BR}(E\to\nu_{\ell}W)$,
$\mathrm{BR}(E\to\ell Z)$ and $\mathrm{BR}(E\to\ell A_{H})$, where $\ell$
stands for either electron or muon111Decays into tau leptons have been
considered, assuming SM decays only, in delAguila:2010es ; Kumar:2015tna ;
Bhattiprolu:2019vdu . and we assume the sum of these four branching ratios to
be equal to one but otherwise arbitrary.222The 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 Chala:2013ega . We focus on $E$ Drell-Yan pair production, with
subsequent decays governed by the corresponding branching ratios.333Studies in
which the production and/or decay of new vector-like fermions are dominated by
non-renormalizable interactions can be found in Criado:2019mvu ; Chala:2020odv
. Out of the four possible decay channels, the two that are easiest to detect
experimentally are $E\to\ell Z$ and $E\to\ell A_{H}$. The charged current one
into $\nu_{\ell}W$ is difficult to disentangle from the overwhelming
$W+\mbox{jets}$ background and the one into $\ell H$ 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 $\mathrm{BR}(E\to\ell
Z)$ and $\mathrm{BR}(E\to\ell A_{H})$. 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 $\ell Z$ and slepton searches.
Neither of them can be directly used in our more general scenario, except for
the former in the $\mathrm{BR}(E\to\ell A_{H})=0$ 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
$\mathrm{BR}(E\to\ell A_{H})\neq 1$ 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.

###  2.1 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 Alloul:2013bka  and leading order event
generation is done with MadGraph5_aMC@NLO Alwall:2014hca . 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 Sjostrand:2014zea  with the CMS CUETP8M1
Khachatryan:2015pea  underlying event tuning and the NNPDF23LO Ball:2012cx
parton distribution functions. The detector response is modeled with Delphes 3
deFavereau:2013fsa . We use the default CMS card for the LHC analysis and the
HL-LHC detector card for the $\sqrt{s}=27\,\mathrm{TeV}$ analysis. $95\%$ C.L.
limits are obtained using the $CL_{s}$ Read:2002hq  method by fitting the
relevant discriminant variables using OpTHyLic Busato:2015ola  which outputs
the upper limit on the signal strength, $\mu=\sigma_{up}/\sigma_{th}$, where
$\sigma_{up}$ is the upper limit on the cross-section and $\sigma_{th}$ is the
theoretical prediction obtained through the MadGraph simulation.

Figure 1: Pair production of a VLL singlet $E$ and decay channels our analysis
is most sensitive to.

####  2.1.1 Decays into SM particles

For the case in which the VLL decays exclusively to SM final states ($W\nu$,
$Z\ell$ and $H\ell$) we reproduce the analysis presented in Ref. Aad:2015dha ,
an ATLAS search performed at $\sqrt{s}=8\;\mathrm{TeV}$ and an integrated
luminosity of $\mathcal{L}=20.3\;\mathrm{fb}^{-1}$, looking for multi-lepton
signals coming from the $Z\ell$ 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 $\Delta R\equiv\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}<3$, with $\eta$ and
$\phi$ 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.

Selection Cuts | $ZZ$ | $WZ$ | $Z\gamma$
---|---|---|---
| OSSF lepton pair with
---
$|m_{\ell\ell}-m_{Z}|<10\;\mathrm{GeV}$
0.25 | 0.19 | 0.0024
$p^{\ell_{1}}_{T}>26\;\mathrm{GeV}$ | 0.25 | 0.19 | 0.0023
$\Delta R(Z,\mathrm{off-}Z\;\mathrm{lepton})<3$ | 0.17 | 0.11 | 0.0008
Table 1:  Cumulative efficiencies for the background events after applying the
selection cuts corresponding to the analysis performed at
$\sqrt{s}=8\;\mathrm{TeV}$. Efficiencies are presented as the number of events
which are selected over the number of initial events. We have included in the
generation only leptonic decays of the $Z$ and $W$ into electron, muons and
taus but apply our cuts only to final state electrons and muons.

The analysis searches for an excess in the distribution of the variable
$\Delta m=m_{3\ell}-m_{\ell\ell}$, where the mass of the reconstructed $Z$
boson, denoted by $m_{\ell\ell}$, is subtracted from the invariant mass of the
3-lepton system, $m_{3\ell}$. 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
$m_{W}-20\;\mathrm{GeV}<m_{jj}<150\;\mathrm{GeV}$ , with $m_{W}$ the $W$ boson
mass; and 3-lepton only region, in which exactly 3 leptons are identified with
no pairs of jets satisfying the previous condition on their invariant mass.

Figure 2: 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).

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 $ZZ$, $WZ$ and $Z\gamma$, 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 $M_{\mathrm{E}}\gtrsim
160\mbox{ GeV}$, which represents a difference of $\sim 7\%$ in comparison to
the expected limit obtained in original analysis.

$ZZ$ $WZ$ $Z\gamma~{}(\times 10^{-2})$ Selection cuts A B C A B C A B C OSSF
lepton pair with $|m_{\ell\ell}-m_{Z}|<10\;\mathrm{GeV}$ 0.25 0.25 0.25 0.18
0.18 0.18 0.48 0.48 0.48 $\Delta R(Z,\mathrm{off-}Z\;\mathrm{lepton})<3$ 0.16
0.16 0.16 0.10 0.10 0.10 0.14 0.14 0.14
$p_{T}^{\ell_{1}}>\\{80,\;100,\;120\\}$ 0.054 0.029 0.0098 0.029 0.015 0.0052
0.05 0.02 0.01 $p_{T}^{\ell_{2}}>\\{20,\;40,\;60\\}$ 0.054 0.025 0.0073 0.029
0.012 0.0035 0.05 0.02 0.01 $p_{T}^{\ell_{3}}>\\{0,\;0,\;20\\}$ 0.054 0.025
0.0067 0.029 0.012 0.0031 0.05 0.02 <0.01 off-$Z$ lepton = $e$ 0.029 0.013
0.0034 0.013 0.0053 0.0014 0.03 0.01 <0.01 $m_{T}<160\;\mathrm{GeV}$ ($3ljj$)
0.0079 0.0043 0.0014 0.0013 0.0009 0.0004 0.01 0.01 <0.01

Table 2:  Efficiencies for the background events after applying the selection
cuts corresponding to the new analysis performed at
$\sqrt{s}=13\;\mathrm{TeV}$. Efficiencies are presented as the number of
events which are selected over the number of initial events. The regions
represented by $\\{A,B,C\\}$ correspond to different values of the mass of the
VLL, $\\{M_{\mathrm{E}}<300,\;300\leq M_{\mathrm{E}}<400,\;M_{\mathrm{E}}\geq
400\\}$, in GeV. We have included in the generation only leptonic decays of
the $Z$ and $W$ into electron, muons and taus but apply our cuts only to final
state electrons and muons.

In order to see what the reach with the current recorded luminosity can be we
have repeated the same analysis at $\sqrt{s}=13\;\mathrm{TeV}$ and
$\mathcal{L}=139\;\mathrm{fb^{-1}}$. 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
$Z\gamma$ background, generating only events in which at least one lepton has
$p_{T}>62\,\mathrm{GeV}$.

Furthermore, at these higher energies, we can also remove almost the entirety
of the $WZ$ background by setting a cut on the transverse mass, $m_{T}$, of
the reconstructed $W$ boson. This cut is effective because in events from
$WZ\rightarrow\ell\nu\ell\ell$, in principle, the off-$Z$ lepton is coming
from the $W$ decay and the missing energy of the event, $\not{E}_{T}$,
originates from the neutrino. Therefore, for events from the $WZ$ background,
we have

$m_{T}=\sqrt{2\left(\not{E}_{T}p_{T\ell}-\mathbf{\not{E}_{T}}.\mathbf{p_{T\ell}}\right)}\leq
m_{W}\,,$ (1)

where $p_{T\ell}$ 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 $WZ$ 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
$410\;\mathrm{GeV}$ ($420\;\mathrm{GeV}$) 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.

Figure 3: Exclusion plot for a VLL singlet decaying into electrons (muons) on
the left (right) panel at the LHC for $\sqrt{s}=13\mbox{ TeV}$ and an
integrated luminosity of $139\;\mathrm{fb}^{-1}$ using the improved analysis.
See text for details.

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, $p\;p\rightarrow E^{+}E^{-}$, but
also the pair production of the neutral component, $p\;p\rightarrow NN$, and
the associated production of both, $p\;p\rightarrow E^{\pm}N$. For large
masses (we will consider both components degenerate in mass), the charged
component will decay equally to $\ell Z$ and $\ell H$, while the neutral
component decays solely to $\nu W$. 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 $\sim 720\;\mathrm{GeV}$ being excluded.

An analysis searching for VLL doublets of hypercharge $-1/2$ was performed by
the CMS collaboration in Ref. Sirunyan:2019ofn  with an integrated luminosity
$\mathcal{L}=77\;\mathrm{fb}^{-1}$. A bound $M_{L}\geq 790\;\mathrm{GeV}$ 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 $M_{L}^{\mathrm{expected}}\geq
690\;\mathrm{GeV}$. Rescaling our search to the same integrated luminosity we
find $M_{L}^{\mathcal{L}=77\;\mathrm{fb}^{-1}}\geq 730\;\mathrm{GeV}$,
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.

Figure 4: Exclusion plot for a VLL doublet decaying into electrons (or
electron neutrinos) at the LHC for $\sqrt{s}=13\mbox{ TeV}$ and an integrated
luminosity of $139~{}\mathrm{fb}^{-1}$ using our improved analysis.

####  2.1.2 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
Aaboud:2018jiw  at $\sqrt{s}=13\;\mathrm{TeV}$ and an integrated luminosity of
$\mathcal{L}=36.1\;\mathrm{fb}^{-1}$ searching for pair produced sleptons
decaying into a SM lepton and a neutralino, as represented in Figure 5. The
analysis selects events with 2 OSSF leptons ($e$, $\mu$), imposing a veto on
additional jets. It also rejects events with an invariant mass of the two
leptons $m_{\ell\ell}<40\;\mathrm{GeV}$. Several inclusive and exclusive
signal regions are defined in which different requirements are demanded for
$m_{\ell\ell}$ and the $m_{T2}$ variable Barr:2003rg ; Lester:1999tx , defined
by

$m_{T2}=\mathrm{min_{\mathbf{q_{T}}}}[\mathrm{max}(m_{T}(\mathbf{p_{T1},q_{T}}),m_{T}(\mathbf{p_{T2},\not{E}_{T}-q_{T}})]\,,$
(2)

where $\mathbf{p_{T1,2}}$ represent the transverse momentum of each of the
identified leptons and $\mathbf{q_{T}}$ is the vector that minimizes the
maximum of both transverse masses, defined as

$m_{T}(\mathbf{p_{T},q_{T}})=\sqrt{2(p_{T}q_{T}-\mathbf{p_{T}q_{T}})}\,.$ (3)

Figure 5: Pair production and decay of sleptons at hadron colliders.
$\tilde{\ell}$ represents a charged slepton and $\tilde{\chi}^{0}_{1}$ the
lightest neutralino.

The main backgrounds for this signal are diboson processes ($ZZ$, $WW$ and
$WZ$) and $t\bar{t}$. In order to maximize our statistics we always include 2
leptons (with an $m_{\ell\ell}>95\;\mathrm{GeV}$ partonic cut) in the final
state in the background generation. These 2 leptons can be of either family
for the diboson case and are restricted to the first two generations for the
$t\overline{t}$ 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 Aaboud:2018jiw . However, new
results are calculated considering only the signal region of
$m_{\ell\ell}>111\;\mathrm{GeV}$ and $m_{T2}>100\;\mathrm{GeV}$ as we find the
difference in regards to all signal regions not significant. Given that the
analysis in Ref. Aaboud:2018jiw  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 slepton-neutralino model. Fixing the neutralino
mass $M_{\tilde{\chi}^{0}_{1}}=1\;\mathrm{GeV}$ we obtain the results shown in
Figure 6, with a limit $M_{\tilde{\ell}}\geq 565\;\mathrm{GeV}$, very similar
to the expected limit obtained by the ATLAS collaboration, $\sim
570\;\mathrm{GeV}$.

Selection Cuts | $ZZ$ | $WZ$ | $WW$ | $t\overline{t}$
---|---|---|---|---
| OSSF lepton pair
---
0.33 | 0.33 | 0.23 | 0.53
$m_{\ell\ell}>40\;\mathrm{GeV}$ | 0.31 | 0.28 | 0.11 | 0.53
$p^{\ell_{1}}_{T}>25\;\mathrm{GeV}$ | 0.31 | 0.28 | 0.11 | 0.53
$p^{\ell_{2}}_{T}>20\;\mathrm{GeV}$ | 0.28 | 0.26 | 0.095 | 0.51
$p^{b-jet}_{T}<20\;\mathrm{GeV}$ | 0.24 | 0.24 | 0.093 | 0.13
$p^{jet}_{T}<60\;\mathrm{GeV}$ | 0.15 | 0.14 | 0.081 | 0.061
$m_{T2}>100\;\mathrm{GeV}$ | 0.0064 | 0.003 | 0.0002 | 0.0001
$m_{\ell\ell}>110\;\mathrm{GeV}$ | 0.0016 | 0.001 | 0.0002 | 0.0001
Table 3:  Cumulative efficiencies for the background events after applying the
selection cuts corresponding to the analysis performed at
$\sqrt{s}=13\;\mathrm{TeV}$ for slepton searches Aaboud:2018jiw . Efficiencies
are presented as the number of events which are selected over the number of
initial events (see text for details). Figure 6: Limit on the slepton mass,
$M_{\tilde{\ell}}$, for a neutralino mass
$M_{\tilde{\chi}^{0}_{1}}=1\;\mathrm{GeV}$ following the analysis in
Aaboud:2018jiw .

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 $\sqrt{s}=13\;\mathrm{TeV}$ and
an integrated luminosity $\mathcal{L}=139\;\mathrm{fb}^{-1}$ 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 $M_{E}\gtrsim 900\;\mathrm{GeV}$ the production cross-section is too
low and the analysis cannot constrain the signal regardless of $M_{A_{H}}$.

Figure 7: Expected (solid black) and 1-sigma band (green) $95\%$ C.L.
exclusion limit in the $M_{E}-M_{A_{H}}$ plane from the analysis in
Aaboud:2018jiw . The excluded region is the one below the curves.

###  2.2 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 $\mathrm{BR}_{g}(E\rightarrow
A_{H}\ell)=\mathrm{BR}_{g}(E\rightarrow Z\ell)=\mathrm{BR}_{g}(E\rightarrow
H\ell)=\mathrm{BR}_{g}(E\rightarrow W\nu)=0.25$, where the $g$ subscript
describes the generated sample. To probe a specific point with different BRs,
each event is weighted by $\mathrm{BR}_{p}^{i}/\mathrm{BR}_{g}^{i}$, 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 $\ell A_{H}$ and $\ell Z$, the results are presented in the
$\mathrm{BR}(E\rightarrow A_{H}\ell)\;\mathrm{vs}\;\mathrm{BR}(E\rightarrow
Z\ell)$ plane with the others branching ratios being fixed to

$\mathrm{BR}_{p}(E\rightarrow W\nu)=2\mathrm{BR}_{p}(E\rightarrow
H\ell)=\frac{2}{3}\left[1-\mathrm{BR}_{p}(E\rightarrow
A_{H}\ell)-\mathrm{BR}_{p}(E\rightarrow Z\ell)\right],$ (4)

which correspond to the relation between the different branching ratios in the
large $M_{E}$ limit for $\mathrm{BR}(E\to A_{H}\ell)=0$. 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 $\mu$ as a function of
the branching ratio into $W\nu$ 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 $M_{E}$.

Figure 8: Dependence of the signal strength, $\mu$, on $\mathrm{BR}(E\to
W\nu)$ for different values of the remaining parameters. The masses are taken
to be $M_{E}=500\;\mathrm{GeV}$ and $M_{A_{H}}=98\;\mathrm{GeV}$
($M_{E}=400\;\mathrm{GeV}$ and $M_{A_{H}}=98\;\mathrm{GeV}$) for the left
(right) panels. Values of $\mu$ smaller than 1 are considered excluded.

Our final result, that combines the two analyses discussed in the previous
section for arbitrary values of $\mathrm{BR}(E\to A_{H}\ell)$ and
$\mathrm{BR}(E\to Z\ell)$ are shown in Figure 9 for two different values of
$M_{A_{H}}=1\;\mathrm{GeV}$ (left panel) and $M_{A_{H}}=98\;\mathrm{GeV}$
(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 $M_{A_{H}}=1\;\mathrm{GeV}$. 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 $\mathrm{BR}(E\to A_{H}\ell)=0$, at the
relevant (mass dependent) $\mathrm{BR}(E\to Z\ell)$. The most stringent bounds
are along both axes, when the branching ratios into the channels we are most
sensitive to are maximized. The numerical value of the limits in these three
interesting cases are

$M_{E}\gtrsim\left\\{\begin{array}[]{l}405\;\mathrm{GeV},\quad[\mbox{VLL
singlet}],\\\ 630\;\mathrm{GeV},\quad[\mathrm{BR}(E\to\ell Z)=1],\\\
895\;\mathrm{GeV},\quad[\mathrm{BR}(E\to\ell
A_{H})=1],\end{array}\right.\quad[\sqrt{s}=13\;\mathrm{TeV},\mathcal{L}=139\;\mathrm{fb}^{-1}].$
(5)

Figure 9: $95\%$ C.L. lower bound on the VLL mass $M_{E}$ as a function of
$\mathrm{BR}(E\to A_{H}\ell)$ and $\mathrm{BR}(E\to Z\ell)$ for
$M_{A_{H}}=1\;\mathrm{GeV}$ (left panel) and $M_{A_{H}}=98\;\mathrm{GeV}$
(right panel). The limits are given as contour plots for fixed values of
$M_{E}$ in which the region above and to the right of the curves is excluded
and they are computed from a combination of the analyses described in the
previous section with $\sqrt{s}=13\,\mathrm{TeV}$ and an integrated luminosity
of $\mathcal{L}=139\;\mathrm{fb}^{-1}$. The contours correspond to masses that
grow from the $(0,0)$ vertex outwards. Figure 10: Projected limits on the mass
of the VLL, $M_{E}$ for arbitrary branching fractions for the HL-LHC. (See
text and Figure 9 for details.)

###  2.3 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 $\sqrt{s}=13\;\mathrm{TeV}$ and an
integrated luminosity of $\mathcal{L}=3\;\mathrm{ab}^{-1}$) 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. The result is shown, for $M_{A_{H}}=1\;\mathrm{GeV}$, in Figure
10. The correspond final reach of the HL-LHC in the limiting cases of a VLL
singlet, $\mathrm{BR}(E\to\ell Z)=1$ and $\mathrm{BR}(E\to\ell A_{H})=1$ is,
respectively,

$M_{E}\gtrsim\left\\{\begin{array}[]{l}785\;\mathrm{GeV},\quad[\mbox{VLL
singlet}],\\\ 1090\;\mathrm{GeV},\quad[\mathrm{BR}(E\to\ell Z)=1],\\\
1450\;\mathrm{GeV},\quad[\mathrm{BR}(E\to\ell
A_{H})=1],\end{array}\right.\quad[HL-LHC].$ (6)

When considering a higher energy collider, like the HE-LHC, for which we
consider $\sqrt{s}=27\;\mathrm{TeV}$ and $\mathcal{L}=3\;\mathrm{ab}^{-1}$, 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
$p_{T}>75\;\mathrm{GeV}$ of the leading lepton whereas for the analysis
focusing on the missing energy decay, backgrounds were generated with a
partonic cut of $p_{T}>100\;\mathrm{GeV}$ for the leading lepton. We were able
to use this cut since we updated the selection thresholds from Table 3 to
$p_{T}^{\ell_{1}}>120\;\mathrm{GeV}$ in the missing decay analysis. The
resulting reach, again for $M_{A_{H}}=1\;\mathrm{GeV}$, is reported for
arbitrary branching ratios in Figure 11. The estimated reach, in the limiting
cases is

$M_{E}\gtrsim\left\\{\begin{array}[]{l}1295\;\mathrm{GeV},\quad[\mbox{VLL
singlet}],\\\ 1770\;\mathrm{GeV},\quad[\mathrm{BR}(E\to\ell Z)=1],\\\
1965\;\mathrm{GeV},\quad[\mathrm{BR}(E\to\ell
A_{H})=1],\end{array}\right.\quad[\text{HE-LHC}].$ (7)

Figure 11: Projected limits on the mass of the VLL, $M_{E}$ for arbitrary
branching fractions for the HE-LHC. (See text and Figure 9 for details.)

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 colliderreach .
First we test the validity of this approach by extrapolating the current
luminosity results reported in Eq. (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 energy. The difference in the missing decays analysis drops to 14 %
when we extrapolate from the HL-LHC results. We can expect a similar effect
when extrapolating our results to the FCC. Assuming
$\sqrt{s}=100\;\mathrm{TeV}$ and $\mathcal{L}=3\;\mathrm{ab}^{-1}$ we obtain
the results shown in Figure 12 and the following limits in the VLL, pure $Z$
and pure $A_{H}$ decay cases

$M_{E}\gtrsim\left\\{\begin{array}[]{l}2525\;\mathrm{GeV},\quad[\mbox{VLL
singlet}],\\\ 3665\;\mathrm{GeV},\quad[\mathrm{BR}(E\to\ell Z)=1],\\\
3330\;\mathrm{GeV},\quad[\mathrm{BR}(E\to\ell
A_{H})=1],\end{array}\right.\quad[\text{hh-FCC (extrapolation)}].$ (8)

Figure 12: Projected limits on the mass of the VLL, $M_{E}$ for arbitrary
branching fractions for the 100 TeV hh-FCC. (See text and Figure 9 for
details.)

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.

Figure 13: VLL singlet pair production cross-section at ($pp$) hadron
colliders with $\sqrt{s}=13\,,27\,,100\,\mathrm{TeV}$.

##  3 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.

###  3.1 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) Cheng:2004yc ;
Low:2004xc , 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

$\displaystyle\mathcal{L}$
$\displaystyle=-c_{A_{H}h}g^{\prime\,2}\left(\sqrt{2}vhA_{H}^{\mu}A_{H_{\mu}}+\frac{1}{2}hhA_{H}^{\mu}A_{H_{\mu}}\right)+q_{H}g^{\prime}[\bar{E}_{L}\gamma_{\mu}\ell_{R}+\mathrm{h.c.}]A_{H}^{\mu}+\ldots\,,$
(9)

where $v\approx 174\;\mathrm{GeV}$, 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 $g^{\prime}$ to
make the connection with the LHT model more direct. In the LHT model
$c_{A_{H}h}=\frac{1}{8}$ and $q_{H}=\frac{1}{10}$ delAguila:2008zu .

The latest Planck results measured the relic density abundance to be $\Omega
h^{2}\sim 0.12$ Aghanim:2018eyx  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 Birkedal:2006fz . 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.

Figure 14: Relevant annihilation processes for the dark photon in the standard
freeze-out mechanism.

The calculation of the relic density is done using MadDM Ambrogi:2018jqj  by
inputting a UFO model Degrande:2011ua  which we generate through Feynrules
Alloul:2013bka . The results are presented in Figure 15 for a VLL mass
$M_{E}=500\;\mathrm{GeV}$ in the $M_{A_{H}}-c_{A_{H}h}$ plane for different
values of $q_{H}$. The curves represent the values for which the relic
abundance agrees with the observed value. The region below the curve is
excluded and the one above requires further sources of DM. We also show,
shaded in grey, the excluded region from direct detection using the latest
XENON1T data Aprile:2018dbl . For small values of $q_{H}$ ($<1$), the
s-channel annihilation through the Higgs 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 $q_{H}\sim 1.5$
allowing almost all 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 shown the excluded
region in Figure 15.

Figure 15: Contours of $\Omega h^{2}=0.12$ for fixed values of $q_{H}$. The
region below the curves gives too large DM density and is excluded. The shaded
region is excluded from direct detection experiments. See text for details.

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. Agrawal:2014ufa , 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
$A_{H_{\mu}}A_{H_{\nu}}F^{\mu\nu}$ 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.

Figure 16: Relevant direct detection process.

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 leptons the VLL couples to. The latest experimental results
are Aoyama:2012wj ; Muong-2:2021ojo

$\displaystyle\Delta a_{e}$
$\displaystyle=a_{e}^{\mathrm{Exp}}-a_{e}^{SM}=-1.06(0.82)\times 10^{-12}\;,$
(10) $\displaystyle\Delta a_{\mu}$
$\displaystyle=a_{\mu}^{\mathrm{Exp}}-a_{\mu}^{SM}=25.1(5.9)\times
10^{-10}\;,$ (11)

where the uncertainties include theoretical and experimental contributions.

The new contribution from $E$ and $A_{H}$ reads Agrawal:2014ufa ,

$a_{\ell}=-\frac{\epsilon^{2}}{48\pi^{2}r^{2}(1-r^{2})^{4}}q_{H}^{2}g^{\prime\,2}\left[5-14r^{2}+39r^{4}-38r^{6}+8r^{8}+18r^{4}\mathrm{ln}(r^{2})\right]+\mathcal{O}(\epsilon^{3})\;,$
(12)

where $\epsilon\equiv m_{\ell}/M_{E}$ and $r\equiv M_{A_{H}}/M_{E}$ and
$m_{\ell}$ 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.

Figure 17: Region that explains the electron $g-2$ anomaly for
$M_{E}=500\;\mathrm{GeV}$ (green) and $M_{E}=1\,\mathrm{TeV}$ (blue). The 2
sigma limits from the contribution to the muon $g-2$ are shown with solid
lines, for $M_{E}=500\;\mathrm{GeV}$ (yellow) and for $M_{E}=1\,\mathrm{TeV}$
(red). The region above the lines is excluded.

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 weekly 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 Baker:2015qna  increases the
annihilation cross-section since processes such as $A_{H}E\rightarrow SM\;SM$
and $A_{H}SM\rightarrow E\;SM$ 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 $M_{A_{H}}-M_{E}\sim T_{F}$. Knowing that
$M_{A_{H}}\sim 25\;T_{F}$, for cold DM, the splitting must be at most
$\Delta\sim 0.04$ where $\Delta\equiv(M_{E}-M_{A_{H}})/M_{A_{H}}$.

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 $\Omega h^{2}=0.12$ for
different values of $\Delta$. 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
$\Delta\lesssim 0.05$ a significant difference with respect to the standard
annihilation scenario is observed.

Figure 18: Contours of $\Omega h^{2}=0.12$ for fixed values of $\Delta$. The
region to the right of the curve gives too large DM density and is excluded.
The solid (dashed) curves correspond to $q_{H}=0.1$ ($q_{H}=0.2$). The shaded
region is excluded from direct detection experiments. See text for details.

In order to better understand the dependence on $q_{H}$ in this co-
annihilation regime we show, in Figure 19, the $\Omega h^{2}=0.12$ contours in
the $M_{A_{H}}-q_{H}$ plane, again for different values of $\Delta$ 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
$\Delta$ 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 $\Delta\gtrsim 0.05$, the annihilation process
mediated by the heavy lepton is important for low mass differences between the
VLL and $A_{H}$ and large $q_{H}$ and as such the annihilation cross-section
is influenced by changes in $\Delta$ even outside the co-annihilation regime.

Figure 19: Contours of $\Omega h^{2}=0.12$ for fixed values of $\Delta$ for
$c_{A_{H}h}=1$. The region below each curve gives too large DM density and is
excluded. The shaded region is excluded from direct detection experiments. See
text for details.

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 ATLAS:2019lng  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.

###  3.2 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 Hall:2009bx . 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
Delaunay:2020vdb  with emphasis on the DM phenomenology, thus setting the VLL
mass to a conservative $M_{E}=1\;\mathrm{TeV}$ 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 Delaunay:2020vdb :

$\Omega h^{2}\approx 0.12\times
10^{-9}\frac{M_{E}}{M_{A_{H}}}\left(\frac{g_{H}s}{5.3\times
10^{-17}}\right)^{2},$ (13)

where $g_{H}$ and $s$ are defined in Eqs. (17) and (20), respectively.

For each value of $M_{E}$ and $M_{A_{H}}$ the condition that $A_{H}$
corresponds to all of DM, i.e. Eq. (13) $=0.12$, fixes the product $g_{H}s$.
Choosing then a value for $s$ fixes all branching ratios of the VLL. As such,
by choosing a particular $s$ we can get the collider bound from our previous
analysis.

We present these results in Figures 20 and 21 for the LHC analysis at
$\sqrt{s}=13\;\mathrm{TeV}$ and $\mathcal{L}=139\;\mathrm{fb}^{-1}$ and
$\mathcal{L}=3\;\mathrm{ab}^{-1}$ 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 curve, we display it as a solid or dashed curve depending on whether the
most constraining analysis is the one which is most sensitive to the decays
into SM particles or into missing energy, respectively. This is relevant
since, in order to use the collider bounds we obtained, $E\rightarrow
A_{H}\ell$ must be a prompt decay for the missing energy search whereas
$E\rightarrow Z\ell$ is the most important channel to be prompt in the SM
decays analysis.

In Eq. (38) we show the minimum value $s$ must take so that $E\rightarrow
Z\ell$ is prompt. For $E\rightarrow A_{H}\ell$ the value of $g_{H}$ (fixed for
each mass point) is going to determine whether it is a prompt decay mode. In
Figures 20 and 21 we show, in shaded gray, the regions in which the
$E\rightarrow A_{H}\ell$ decay length is $\geq 1\;\mathrm{cm}$ (light gray) or
$\geq 5\;\mathrm{cm}$ (dark gray). This is relevant for the dashed part of the
curves and shows that, for smaller values of $s$, the limits on $M_{E}$ might
be significantly weaker. A more detailed analysis, 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.

Figure 20: Contours of collider limits, for the LHC with
$\sqrt{s}=13\;\mathrm{TeV}$ and $\mathcal{L}=139\;\mathrm{fb}^{-1}$, on models
that generate the observed DM relic abundance via the freeze-in mechanism for
different values of the fermion mixing parameter $s$. The region below the
curves is excluded by collider searches, either targeting SM decays (solid) or
decays into missing energy (dashed). The light (dark) shaded region correspond
to an $E\rightarrow A_{H}\ell$ decay length $\geq 1\;\mathrm{cm}$ ($\geq
5\;\mathrm{cm}$). See text and Appendix A for details.  Figure 21: Contours of
collider limits, for the LHC with $\sqrt{s}=13\;\mathrm{TeV}$ and
$\mathcal{L}=3\;\mathrm{ab}^{-1}$, on models that generate the observed DM
relic abundance via the freeze-in mechanism for different values of the
fermion mixing parameter $s$. The region below the curves is excluded by
collider searches, either targeting SM decays (solid) or decays into missing
energy (dashed). The light (dark) shaded region correspond to an $E\rightarrow
A_{H}\ell$ decay length $\geq 1\;\mathrm{cm}$ ($\geq 5\;\mathrm{cm}$). See
text and Appendix A for details.

##  4 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 $\mathrm{BR}(E\to
A_{H}\ell)$ and $\mathrm{BR}(E\to Z\ell)$ 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 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.

## Acknowledgments

We are grateful to N. Castro, M. Chala, M. Ramos and T. Vale for useful
comments. This work has been supported in part by the Ministry of Science,
Innovation and Universities (PID2019-106087GB-C22) and and by the Junta de
Andalucía grants FQM 101, SOMM17/6104/UGR, A-FQM-211-UGR18 and P18-FR-4314
(FEDER). GG acknowledges support by LIP (FCT, COMPETE2020-Portugal2020, FEDER,
POCI-01- 0145-FEDER-007334) as well as by FCT under project CERN/FIS-
PAR/0024/2019 and under the grant SFRH/BD/144244/2019.

##  Appendix A Explicit realization

We describe in this appendix an explicit realization of the framework used in
this work. Rather than aiming at full generality we focus on a minimal model
capable of generating the range of branching ratios we can be sensitive to at
the LHC and future colliders. The explicit realization we describe here is
well motivated as a good candidate for feebly interacting DM Delaunay:2020vdb
. 444Indeed our model corresponds to the one in Delaunay:2020vdb  with the
following replacements: $M_{0}\to M_{E},\Lambda_{1}\to
x_{E},V\to\omega/\sqrt{2},s\to\theta_{R}$. The model has an $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\times U(1)_{H}$ gauge symmetry. The matter fields
consist of the SM particles, which are all neutral under $U(1)_{H}$, a new
vector-like lepton with the following quantum numbers, with notation
$(SU(3)_{C},SU(2)_{L})_{U(1)_{Y},U(1)_{H}}$,

$E^{(0)}_{L,R}\sim(1,1)_{-1,1},$ (14)

and a complex scalar

$\Phi\sim(1,1)_{0,1}.$ (15)

At the renormalizable level we can write the following Lagrangian

$\mathcal{L}=\mathcal{L}_{\mathrm{SM}}-\frac{1}{4}F_{H}^{\mu\nu}F_{H\,\mu\nu}+|D_{\mu}\Phi|^{2}-V(\Phi)+\bar{E}^{(0)}(\mathrm{i}\cancel{D}-M_{0})E^{(0)}-\Lambda_{1}(\bar{E}^{(0)}_{L}\Phi
e^{(0)}_{R}+\mathrm{h.c.})+\ldots,$ (16)

where $V(\Phi)$ 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 555The order of magnitude expectation for kinetic mixing
Holdom:1985ag  is small enough to be negligible for most of the parameter
space and also well within the experimental limits Chun:2010ve . For values of
$s$ on the smaller side a small extra suppression might be needed
Gherghetta:2019coi . and that the VLL only couples to one of the SM RH charged
leptons, taken to be the electron here, denoted by $e^{(0)}_{R}$ but it could
equally well be the muon or tau, in the basis of diagonal charged lepton
Yukawa couplings. Hereafter we suppress all terms in the Lagrangian that are
not relevant for our discussion. The covariant derivative for the new fields
reads

$D_{\mu}=\partial_{\mu}-\mathrm{i}g_{H}A_{H\,\mu},$ (17)

where we have used $Q_{H}=1$.

Once $U(1)_{H}$ is spontaneously broken, the corresponding gauge boson,
$A_{H}$, acquires a mass

$M_{A_{H}}=\sqrt{2}g_{H}V,$ (18)

where we have denoted $V\equiv\langle\Phi\rangle$ the vacuum expectation value
(vev) of $\Phi$, and $e^{(0)}_{R}$ and $E^{(0)}_{R}$ mix

$\mathcal{L}=-\bar{E}^{(0)}_{L}(\Lambda_{1}Ve^{(0)}_{R}+M_{0}E^{(0)}_{R})+\mathrm{h.c.}+\ldots~{}.$
(19)

This mixing can be rotated away (thus defining the SM RH charged lepton) via
the following unitary rotation

$\begin{pmatrix}e_{R}^{(0)}\\\ E^{(0)}_{R}\end{pmatrix}=\begin{pmatrix}c&s\\\
-s&c\end{pmatrix}\begin{pmatrix}e_{R}\\\ E_{R}\end{pmatrix},$ (20)

where

$s\equiv\frac{\Lambda_{1}V}{M},\quad c\equiv\frac{M_{0}}{M},\quad
M\equiv\sqrt{M_{0}^{2}+\Lambda_{1}^{2}V^{2}}.$ (21)

Denoting $E^{(0)}_{R}\equiv E_{L}$ we have the SM extended with a VLL singlet
with hypercharge $-1$ and the following mass Lagrangian for the charged
leptons

$\mathcal{L}=\begin{pmatrix}\bar{e}_{L}&\bar{E}_{L}\end{pmatrix}\begin{pmatrix}m&m^{\prime}\\\
0&M\end{pmatrix}\begin{pmatrix}e_{R}\\\ E_{R}\end{pmatrix}+\ldots,$ (22)

where $m$ and $m^{\prime}$ are generated after EWSB and satisfy

$\frac{m^{\prime}}{m}=\frac{s}{c},$ (23)

and a new neutral heavy gauge boson with couplings

$\mathcal{L}=g_{H}A_{H}^{\mu}\begin{pmatrix}\bar{e}&\bar{E}\end{pmatrix}\gamma_{\mu}\left[\begin{pmatrix}0&0\\\
0&1\end{pmatrix}P_{L}+\begin{pmatrix}s^{2}&-sc\\\
-sc&c^{2}\end{pmatrix}P_{R}\right]\begin{pmatrix}e\\\
E\end{pmatrix}+\ldots~{}.$ (24)

The effect of mixing with extra vector-like fermions is well known
delAguila:1982fs . The physical basis is obtained by diagonalizing the mass
matrix in (22) via a bi-unitary rotation

$\begin{pmatrix}e_{\chi}\\\
E_{\chi}\end{pmatrix}\longrightarrow\begin{pmatrix}c_{\chi}&s_{\chi}\\\
-s_{\chi}&c_{\chi}\end{pmatrix}\begin{pmatrix}e_{\chi}\\\
E_{\chi}\end{pmatrix},$ (25)

where $\chi=L,R$ denotes the chirality and, in the $m^{\prime}\ll M$ limit
that we will be interested in we have

$s_{L}=\frac{m^{\prime}}{M}+\ldots,\quad
s_{R}=\frac{mm^{\prime}}{M^{2}}+\ldots,$ (26)

where the dots denote higher orders in $m^{\prime}/M\ll 1$. The corresponding
masses are

$m_{e}=m\left(1-\frac{m^{\prime\,2}}{2M^{2}}+\ldots\right)\approx m,\quad
M_{E}=M\left(1+\frac{m^{\prime\,2}}{2M^{2}}+\ldots\right)\approx M.$ (27)

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

$\displaystyle\mathcal{L}^{Z}=$
$\displaystyle\frac{g}{2c_{W}}Z_{\mu}\bar{\psi}^{i}_{Q}\gamma^{\mu}[X^{QL}_{ij}P_{L}+X^{QR}_{ij}P_{R}-2s_{W}^{2}Q\delta_{ij}]\psi^{j}_{Q},$
$\displaystyle\mathcal{L}^{W}=$
$\displaystyle\frac{g}{\sqrt{2}}W^{+}_{\mu}\bar{\psi}^{i}_{Q}\gamma^{\mu}[V^{QL}_{ij}P_{L}+V^{QR}_{ij}P_{R}]\psi^{j}_{Q-1}+\mathrm{h.c.},$
$\displaystyle\mathcal{L}^{H}=$
$\displaystyle-\frac{H}{\sqrt{2}}\bar{\psi}^{i}_{Q}Y^{Q}_{ij}P_{R}\psi^{j}_{Q}+\mathrm{h.c.},$
$\displaystyle\mathcal{L}^{A_{H}}=$ $\displaystyle
g_{H}A_{H\,\mu}\bar{\psi}^{i}_{Q}\gamma^{\mu}[Z^{QL}_{ij}P_{L}+Z^{QR}_{ij}P_{R}]\psi^{j}_{Q},$
(28)

where $\psi^{i}_{Q}$ is a fermion of electric charge $Q$, $g$ is the
$SU(2)_{L}$ gauge coupling, $c_{W}$ is the cosine of the weak angle and $i,j$
are flavor indices. The relevant couplings are, to leading order in the small
$m^{\prime}/M$ expansion parameter,

$\displaystyle X^{-1}_{L}\approx$
$\displaystyle\begin{pmatrix}-1&-\frac{m^{\prime}}{M}\\\
-\frac{m^{\prime}}{M}&-\frac{m^{\prime\,2}}{M^{2}}\end{pmatrix},\quad
X^{-1}_{R}=(0),$ $\displaystyle W^{0}_{L}\approx$
$\displaystyle\begin{pmatrix}U_{i1}&U_{i,1}\frac{m^{\prime}}{M}\end{pmatrix},\quad
W^{0}_{R}\approx(0),$ $\displaystyle vY^{-1}\approx$
$\displaystyle\begin{pmatrix}m&m^{\prime}\\\
m\frac{m^{\prime}}{M}&\frac{m^{\prime\,2}}{M^{2}}\end{pmatrix},$
$\displaystyle Z^{-1}_{L}\approx$
$\displaystyle\begin{pmatrix}\frac{m^{\prime\,2}}{M^{2}}&-\frac{m^{\prime}}{M}\\\
-\frac{m^{\prime}}{M}&1\end{pmatrix},\quad
Z^{-1}_{R}\approx\begin{pmatrix}s^{2}+2sc\frac{mm^{\prime}}{M^{2}}&-sc+(s^{2}-c^{2})\frac{mm^{\prime}}{M^{2}}\\\
-sc+(s^{2}-c^{2})\frac{mm^{\prime}}{M^{2}}&c^{2}-2sc\frac{mm^{\prime}}{M^{2}}\end{pmatrix},$
(29)

where $v\approx 174$ is the Higgs vev, $i$ denotes the neutrino flavor and we
have suppressed all input that is not directly relevant for our purposes.

In order to realize our scenario we consider the limit $M_{A_{H}}\ll M$, so
that $E$ can decay into $Ze$, $He$, $W\nu$ and $A_{H}e$ and $A_{H}$ can decay
into $\bar{e}e$ provided $M_{A_{H}}>2m_{e}$. The corresponding decay widths
are

$\displaystyle\sum_{i=1}^{3}\Gamma(E\to
W\nu_{i})\approx\sum_{i}\frac{g^{2}}{64\pi}\Big{[}(V^{0}_{L})^{2}_{iE}+(V^{0}_{R})^{2}_{iE}\Big{]}\frac{M_{E}^{3}}{m_{W}^{2}}\approx\frac{g^{2}s^{2}}{64\pi
c^{2}}\frac{m_{e}^{2}M_{E}}{m_{W}^{2}},$ (30) $\displaystyle\Gamma(E\to
Ze)\approx\frac{g^{2}}{128\pi
c_{W}^{2}}\Big{[}(X^{-1}_{L})^{2}_{eE}+(X^{-1}_{R})^{2}_{eE}\Big{]}\frac{M_{E}^{3}}{m_{Z}^{2}}\approx\frac{g^{2}s^{2}}{128\pi
c_{W}^{2}c^{2}}\frac{m_{e}^{2}M_{E}}{m_{Z}^{2}},$ (31)
$\displaystyle\Gamma(E\to
He)\approx\frac{1}{64\pi}\Big{[}|(Y^{-1})_{eE}|^{2}+|(Y^{-1})_{Ee}|^{2}\Big{]}M_{E}\left(1-2\frac{m_{H}^{2}}{M_{E}^{2}}\right)$
$\displaystyle\phantom{\Gamma(E\to He)}\approx\frac{s^{2}}{64\pi
c^{2}}\frac{m_{e}^{2}M_{E}}{v^{2}}\left(1-2\frac{m_{H}^{2}}{M_{E}^{2}}\right),$
(32) $\displaystyle\Gamma(E\to
A_{H}e)\approx\frac{g_{H}^{2}}{32\pi}\Big{[}(Z^{-1}_{L})^{2}_{eE}+(Z^{-1}_{R})^{2}_{eE}\Big{]}\frac{M_{E}^{3}}{M_{A_{H}}^{2}}$
$\displaystyle\phantom{\Gamma(E\to
A_{H}e)}\approx\frac{g_{H}^{2}}{32\pi}\left[\frac{m^{\prime\,2}}{M_{E}^{2}}+\left(-sc+(s^{2}-c^{2})\frac{m_{e}m^{\prime}}{M_{E}^{2}}\right)^{2}\right]\frac{M_{E}^{3}}{M_{A_{H}}^{2}}\approx\frac{g_{H}^{2}s^{2}c^{2}}{32\pi}\frac{M_{E}^{3}}{M_{A_{H}}^{2}},$
(33)

where we have shown the leading terms in the $x/M_{E}$, with
$x=m_{e},m_{Z},m_{W},m_{H}$, except for $m_{H}$, for which the subleading term
is relevant for low values of $M_{E}$. Using the properties

$m_{W}=\frac{gv}{\sqrt{2}}=c_{W}m_{Z},$ (34)

we recover the standard $2:1:1$ decay pattern into $W$, $Z$ and $H$ for large
values of $M_{E}$. Finally, assuming $M_{AH}\gg 2m_{e}$ we have

$\displaystyle\Gamma(A_{H}\to e^{+}e^{-})\approx$
$\displaystyle\frac{g_{H}^{2}}{24\pi}\left[(Z^{-1}_{L})_{ee}^{2}+(Z^{-1}_{R})_{ee}^{2}\right]M_{A_{H}}$
$\displaystyle\approx$
$\displaystyle\frac{g_{H}^{2}}{24\pi}\left[\left(\frac{m^{\prime\,2}}{M_{E}^{2}}\right)^{2}+\left(s^{2}+2sc\frac{m_{e}m^{\prime}}{M_{E}^{2}}\right)^{2}\right]M_{A_{H}}\approx\frac{g_{H}^{2}s^{4}}{24\pi}M_{A_{H}}.$
(35)

In order to realize our framework we need $A_{H}$ to be stable at detector
scales, $E$ to decay promptly, and the branching ratios of $E$ decaying into
$A_{H}$ and the SM bosons to be of similar order. Assuming a decay length
larger than $\sim 10\;\mathrm{m}$ for $A_{H}$ and smaller than
$10^{-2}\;\mathrm{m}$ for $E$, these conditions translate into

$\displaystyle\Gamma(A_{H}\to e^{+}e^{-})\lesssim 2\times
10^{-19}~{}\mathrm{GeV},\quad\mbox{(Invisible $A_{H}$)},$ (36)
$\displaystyle\Gamma(E\to Ze,A_{H}e)\gtrsim 2\times
10^{-16}~{}\mathrm{GeV},\quad\mbox{(Prompt $E$ decays)}.$ (37)

Using the expressions above we can find, for each value of $M_{E}$ and
$M_{A_{H}}$, the values of $g_{H}$ and $s$ that satisfy these conditions.
Indeed, requiring prompt $E\to Ze$ decays gives lower bound on $s$

$\frac{s}{c}\gtrsim\left[\frac{128\pi
c_{W}^{2}}{g^{2}}\frac{m_{Z}^{2}}{m_{e}^{2}M_{E}}2\times 10^{-16}\mbox{
GeV}\right]^{\frac{1}{2}}=\left\\{\begin{array}[]{l}3.1\times
10^{-3}\sqrt{\frac{500\mbox{ GeV}}{M_{E}}},\mbox{ electron},\\\ 1.5\times
10^{-5}\sqrt{\frac{500\mbox{ GeV}}{M_{E}}},\mbox{ muon}.\end{array}\right.$
(38)

Requiring that $A_{H}$ decays invisibly and the decay $E\to A_{H}e$ is prompt
provides in turn an upper limit on $s$

$\frac{\Gamma(A_{H}\to e^{+}e^{-})}{\Gamma(E\to A_{H}e)}\leq
10^{-3}\Rightarrow\frac{s}{c}\lesssim 2.7\times
10^{-2}\left(\frac{M_{E}}{M_{A_{H}}}\right)^{\frac{3}{2}}.$ (39)

This upper limit is of the same order of magnitude as the one from electroweak
precision data deBlas:2013gla . Note that for the values of $M_{E}$ we are
sensitive to, unless $M_{A_{H}}$ is very close to $M_{E}$, the two limits are
always compatible. Provided $s$ is fixed in the allowed range, we can fix a
minimum value of $g_{H}$ by requiring $E\to A_{H}e$ to be prompt

$g_{H}\gtrsim\left[\frac{32\pi}{s^{2}c^{2}}\frac{M_{A_{H}}^{2}}{M_{E}^{3}}2\times
10^{-16}GeV\right]^{\frac{1}{2}}\approx\frac{1.3\times
10^{-9}}{sc}\frac{M_{A_{H}}}{100\mbox{ GeV}}\left[\frac{500\mbox{
GeV}}{M_{E}}\right]^{\frac{3}{2}},$ (40)

and a maximum one by requiring $A_{H}$ to be stable at detector scales

$g_{H}\lesssim\left[\frac{24\pi}{s^{4}}\frac{2\times
10^{-19}}{M_{A_{H}}}\right]^{\frac{1}{2}}\approx\frac{4\times
10^{-10}}{s^{2}}\left[\frac{100\mbox{ GeV}}{M_{A_{H}}}\right]^{\frac{1}{2}}.$
(41)

Once $s$ and $g_{H}$ are fixed within the allowed values we have fixed the
relative decay of $E$ into $A_{H}$ and $Z$

$\mathcal{R}\equiv\frac{\Gamma(E\to Ze)}{\Gamma(E\to
A_{H}e)}\approx\frac{g^{2}}{4c_{W}^{2}c^{4}g_{H}^{2}}\frac{M_{A_{H}}^{2}}{m_{Z}^{2}}\frac{m_{e}^{2}}{M_{E}^{2}}\approx\frac{1.65\times
10^{-13}}{g_{H}^{2}c^{4}}\left(\frac{M_{A_{H}}}{100\mbox{
GeV}}\right)^{2}\left(\frac{500\mbox{ GeV}}{M_{E}}\right)^{2}.$ (42)

Using the minimum and maximum values of $g_{H}$ we get

$\displaystyle 10^{6}\frac{s^{4}}{c^{4}}\left(\frac{M_{A_{H}}}{100\mbox{
GeV}}\right)^{3}\left(\frac{500\mbox{GeV}}{M_{E}}\right)^{2}\lesssim\mathcal{R}\lesssim
10^{5}\frac{s^{2}}{c^{2}}\left(\frac{M_{E}}{500\mbox{ GeV}}\right).$ (43)

As an example, fixing $M_{E}=500\mbox{ GeV}$ and $M_{A_{H}}=100\mbox{ GeV}$ we
have

$3.1\times 10^{-3}\lesssim\frac{s}{c}\lesssim 0.094.$ (44)

Fixing for instance $s=0.01$ we now have

$1.3\times 10^{-7}\lesssim g_{H}\lesssim 4\times 10^{-6},$ (45)

and

$10^{-2}\lesssim\mathcal{R}\lesssim 10.$ (46)

Considering the muon instead of the electron increases $\mathcal{R}$ by a
factor $(m_{\mu}/m_{e})^{2}\approx 4.4\times 10^{4}$ and reduces the lower
limit of $s/c$ by a factor $m_{e}/m_{\mu}\approx 4.8\times 10^{-3}$.

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