IFT-UAM/CSIC-22-98

Collider Searches for Heavy Neutral Leptons:
beyond simplified scenarios

Asmaa<EMAIL_ADDRESS>Pablo
<EMAIL_ADDRESS>Xabier
<EMAIL_ADDRESS>and Gioacchino
<EMAIL_ADDRESS>
(a)Pôle Théorie, Laboratoire de Physique des 2 Infinis Irène Joliot Curie (UMR
9012)

CNRS/IN2P3, 15 Rue Georges Clemenceau, 91400 Orsay, France

(b) Instituto de Física Corpuscular (CSIC-Universitat de València),

Catedrático José Beltrán 2, E-46980 Paterna (Valencia), Spain

(c) Departamento de Física Teórica and Instituto de Física Teórica UAM/CSIC,

Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

With very few exceptions, the large amount of available experimental bounds on
heavy neutral leptons - HNL - have been derived relying on the assumption of
the existence of a single (usually Majorana) sterile fermion state that mixes
with only one lepton flavour. However, most of the extensions of the Standard
Model involving sterile fermions predict the existence of several HNLs, with
complex mixing patterns to all flavours. Consequently, most of the
experimental bounds for HNLs need to be recast before being applied to a
generic scenario. In this work, we focus on LHC searches of heavy neutral
leptons and discuss how to reinterpret the available bounds when it comes to
consider mixings to all active flavours, not only in the case with a single
HNL, but also in the case when more heavy neutral leptons are involved. In the
latter case, we also consider the possibility of interference effects and show
how the bounds on the parameter space should be recast.

###### Contents

  1. 1 Introduction
  2. 2 Status of HNL searches at high-energy colliders
  3. 3 Beyond the single mixing assumption
  4. 4 Beyond the single HNL
  5. 5 Conclusions
  6. A Decay process in presence of 2 HNL
    1. A.1 Same sign leptons: $W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{+}q^{\prime}\bar{q}$
    2. A.2 Different sign leptons: $W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}q^{\prime}\bar{q}$

##  1 Introduction

Generating neutrino masses and their mixing as observed in neutrino
oscillation phenomena requires to go for beyond the Standard Model of
Particles (BSM). Many options are presently explored as extensions of the
Higgs and/or gauge sectors, most of the time with new fields within the
particle content. In particular heavy neutral fermions, such as right-handed
neutrinos $\nu_{R}$, are often present as building blocks of several neutrino
mass generation mechanisms. For instance, at least two $\nu_{R}$ are required
to accommodate light neutrino masses via the type-I seesaw mechanism [1, 2, 3,
4, 5]. Moreover, in several variants of the type-I seesaw realized at low
scale, other sterile (from SM gauge interactions) fermions $\nu_{S}$ are
considered, as in the case for the Inverse [6, 7, 8] and Linear [9, 10] seesaw
mechanisms; these variants allow to have large neutrino Yukawa couplings with
a comparatively low seesaw scale, potentially within reach at colliders. From
now on, we will refer to these states ($\nu_{R,S}$) as Heavy Neutral Leptons
(HNL).

Due to the presence of HNLs, the charged and neutral currents are modified,
with the leptonic mixing matrix encoding now not only the PMNS mixing matrix
[11, 12], but also the active-HNL mixings $U_{\alpha N}$, $\alpha=e,\mu,\tau$.
With these modifications and depending on the mass scale of these new neutral
leptons, one expects an impact on numerous observables and thus to abundant
constraints on the plane ($M_{N},|U_{\alpha N}|^{2}$) (see Refs. [13, 14, 15]
and references therein).

In this work we will be interested in HNL searches at high-energy colliders,
mainly at the LHC due to its current extensive program dedicated to HNL
searches, see for instance [16], and the numerous dedicated works and analyses
[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58]. With very few exceptions, the large amount of available HNL
bounds have been derived relying on the assumption of the existence of a
single (usually Majorana) HNL that mixes with only one lepton flavour.
However, most of the BSM scenarios involving new neutral leptons address the
lepton mixing as a whole, as it impacts flavour physics studies and lepton
properties - Dirac or Majorana nature for neutrinos - with different lepton
number conserving/violating processes. The mixing pattern in these scenarios
is expected to be quite complex, so applying the bounds from negative searches
on the HNL space of parameters derived in the context of simplified hypotheses
(with only one HNL which mixes to only one active flavour) does not seem
adequate. Indeed, as we will see, using these limits directly will in general
overconstrain the parameter space. Consequently, most of the experimental
bounds for HNL need to be recast before being applied to a generic BSM
scenario.

The motivation for reinterpreting LHC bounds in general is a well-established
topic, see for instance Ref. [59] and references therein. In the context of
HNL searches, the reinterpretation of the obtained bounds on the HNL mixings
to active flavours has been addressed in previous works, as in for instance
Ref. [50] discussing searches for Heavy Neutral Leptons with Displaced
Vertices, and more recently in Ref. [58] focusing on HNLs decaying promptly to
a tri-lepton final state. In the latter, the single-flavour mixing results
obtained by ATLAS have been recast to a low-scale seesaw model with a pseudo-
degenerate pair of HNLs, the most minimal and simple extension in order to
accommodate neutrino oscillation data (the lightest neutrino being massless).
Due to the simplicity of this model, the active neutrino masses and mixings
determine the flavour pattern of the HNLs [60, 61], and it is possible to
define benchmark points beyond the single-flavour scenario [62]. While being a
very interesting scenario, as it is probing the parameter space connected to
light neutrino masses, this approach has the drawback of being model
dependent. For example, considering other sources for light neutrino
masses111This would actually be needed in order to relax the hypothesis of the
lightest active neutrino being massless and thus have in general 3 massive
active neutrinos., such as additional HNLs not necessarily within the LHC
range, could spoil the correlation between light and heavy sectors that
motivated the definition of these scenarios.

For this reason, in this work we will follow a different approach. We will
instead work with physical HNL states with independent mixings and masses,
with the motivation of covering every scenario that could be realized at
generic BSM models. We will also discuss how to go beyond the simplest single-
flavour mixing scenario, however we will not attempt to explain light neutrino
masses and mixings. This idea is actually the most straightforward extension
to what is usually assumed at LHC searches. In doing so, we will discuss what
would be the most relevant quantities to be bounded experimentally in order to
easily reinterpret the results.

Furthermore, we will also extend the study to the case where more than one HNL
is present and, in particular, taking into account possible interference
effect when at least two heavy neutral leptons are in the same mass regime
(nearly degenerate or possibly forming a pseudo-Dirac neutrino pair). This
possible interference effect can lead to different bounds on the active-
sterile mixings, see for instance [63, 64, 65].

This work is organised as follows: after having thoroughly discussed the
status of high-energy collider searches of HNL in Section 2, a first insight
of going beyond the single active-sterile mixing approximation is examined in
Section 3. In this section, we also discuss how one would reinterpret
experimental data from branching ratios to limits on the model parameter
space, i.e. HNL masses and mixings. In Section 4, we consider the generic case
of having more than one HNL and discuss possible changes in the interpretation
of the bounds on HNL, in particular when the interference effects can be
relevant. Final comments are collected in Section 5. Further details for
deriving the relevant amplitudes are collected in the Appendices.

##  2 Status of HNL searches at high-energy colliders

Heavy Neutral Leptons can be searched for in a wide variety of processes and
experiments, the HNL mass being the key ingredient to decide which is the
optimal one. HNLs lighter than the GeV scale can lead to signatures in nuclear
$\beta$ decays or in leptonic or semileptonic meson and tau decays, while
heavy HNL above the TeV are better explored indirectly by electroweak (EW)
precision observables or rare flavour processes. For a detailed review of all
these signals and experimental status, see for instance Refs. [13, 14, 15].

Here, we are interested in the intermediate regime, with HNL masses $M_{N}$
ranging from few to hundreds of GeVs. Such HNLs could be directly produced at
high-energy colliders and their lifetimes are usually short enough to decay
within the detectors, so we could discover them looking for their decay
products. As these are weak processes, the Standard Model boson masses
obviously define the relevant scale to be compared to. Along this work, we
will refer as light 222Light in the context of high-energy collider searches.
HNL to those with masses lighter than the $W$ boson, and as heavy HNL to the
ones with $M_{N}>M_{W}$. Extensive reviews about HNL searches at colliders can
be found for instance in Refs. [66, 67, 68]. Here we just summarize and update
the list of experimental analyses, introducing at the same time the most
relevant aspects that we will use in our discussion in the next sections.

As in any collider search looking for heavy unstable particles, we need to
consider both the production and the decay channels of the HNLs. At a hadronic
collider such as the LHC, the main production channel comes from Drell-Yan $W$
and $Z$ bosons,

$pp\to W^{(*)}\to N\ell^{\pm}\quad{\rm and}\quad pp\to Z^{(*)}\to N\nu\,,$ (1)

where the gauge bosons could be on- or off-shell, depending on whether the HNL
is lighter or heavier than the $W$ or $Z$ bosons. Additional production
channels could also arise from the Higgs ($H$) boson decays, which could be
motivated in several models providing large neutrino Yukawa couplings.
Unfortunately, Higgs bosons are produced less abundantly than weak bosons, so
they are usually neglected. Moreover the $W$ channel has the additional prompt
charged lepton that can help triggering the process and reducing backgrounds,
and thus experimental searches focused mostly on this channel. Nevertheless,
it is worth mentioning that for very heavy masses, at around the TeV scale,
vector boson fusion channels such as $W\gamma$ or $WW$ become important and
could even dominate the production of HNLs [25, 27, 57]. Indeed, the latest
CMS analysis [69] already included the $W\gamma$ channel in order to enhance
their sensitivity to high HNL masses.

After being produced, a HNL of several GeVs, but still lighter than $M_{W}$,
will decay dominantly via off-shell $W$ or $Z$ bosons to a 3-body final state

$\displaystyle N$ $\displaystyle\to\ell^{\pm}_{\alpha}jj\,,$ (2)
$\displaystyle N$
$\displaystyle\to\ell^{\pm}_{\alpha}\ell^{\mp}_{\beta}\nu_{\beta}\,,$ (3)
$\displaystyle N$ $\displaystyle\to\nu_{\alpha}jj\,,$ (4) $\displaystyle N$
$\displaystyle\to\nu_{\alpha}\ell_{\beta}^{\pm}\ell_{\beta}^{\mp}\,,$ (5)
$\displaystyle N$ $\displaystyle\to 3\nu\,.$ (6)

On the other hand, if $M_{N}$ is above the EW scale, the dominant decays will
be to on-shell $W,Z$ and $H$ bosons, i.e. $N\to\ell^{\pm}W^{\mp},\nu Z,\nu H$.
These 2-body decays will be followed by the decay of the heavy bosons, leading
at the end to the same final states as before. Nevertheless, it is important
to keep in mind that the kinematics in these two mass regimes will be
different.

Figure 1: Drell-Yann HNL production leading to a dilepton signature. The
thunder-shaped arrow indicates that the HNL could be of Dirac or of Majorana
nature, short- or long-lived.

Combining both production and decay channels, we get a full process such as
the example shown in Fig. 1. Depending on the relative size of $M_{N}$ and
$M_{W}$, either the first or the second $W$ boson will be on-/off-shell,
distinguishing the light and heavy HNL regimes. A full catalogue of HNL
signatures, combining the different production and decay processes, can be
found for instance in Ref. [70]. Here we focus only on those that have been
already searched for at the LHC, which we collect in Table 1. We notice that
similar searches considering the existence of right-handed currents have also
been performed [71, 72, 73, 74, 75, 76, 77, 78], and could in principle be
recast to our setup. Nevertheless, one would naively expect lower
sensitivities, as they are optimized for heavy right-handed gauge bosons.

Most of the LHC searches focused on the smoking gun signature for Majorana
neutrinos, the same sign (SS) dilepton final state:

$pp\to\ell_{\alpha}^{\pm}N\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}+nj\,.$ (7)

Here, the lepton pair is accompanied by at least two jets (see Fig. 1), unless
$M_{N}$ is much lighter or much heavier than $M_{W}$, which leads to boosted
objects and collimated jets that are reconstructed as a single one.

Being a LNV process, the SS dilepton does not suffer from severe SM
backgrounds. Unfortunately, current collider searches are sensitive only to
relatively large mixings between the HNL and the active neutrinos, too large
to explain the masses of the light neutrinos unless a symmetry protected
scenario is invoked. More specifically, this symmetry is an approximated
conservation of lepton number [79], which also suppressed the expected LNV
signal from HNLs (see however Refs. [80, 81, 82]).

From this point of view, searching for opposite sign (OS) dileptons, as done
by LHCb [83], seems more relevant to explore theoretically motivated
scenarios. The drawback is the large amount of background from
$Z\to\ell^{+}\ell^{-}$ decays, which reduces the sensitivity. A possible
alternative would be focusing on LFV channels to reduce backgrounds [32, 52].

Yet another alternative considers the fully leptonic process

$pp\to\ell_{\alpha}^{\pm}N\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}\ell_{\gamma}^{\mp}\nu\,.$
(8)

This channel has a trilepton signature, rather clean in a hadronic collider,
nevertheless it also has a source of MET, which might spoil the complete
reconstruction of $M_{N}$. The trilepton channel offers the possibility to
search for both LNV and LNC signals, however most of the experimental analyses
still focus only on the LNV channels as to reduce backgrounds, again from
$Z\to\ell^{+}\ell^{-}$. For example, ATLAS searched [84] for
$e^{\pm}e^{\pm}\mu^{\mp}$ and $\mu^{\pm}\mu^{\pm}e^{\mp}$ channels333Having an
undetected (anti)neutrino, it is not always possible to define a LNV or LNC
process unambiguously. Assuming the presence of a HNL that mixes only to
electrons or to muons, as ATLAS did, the $e^{\pm}e^{\pm}\mu^{\mp}$ and
$\mu^{\pm}\mu^{\pm}e^{\mp}$ channels are originated only from LNV processes.
However, this is not anymore true if the HNL mixes to both flavours [58]., but
not for $e^{\pm}e^{\mp}\mu^{\pm}$ and $\mu^{\pm}\mu^{\mp}e^{\pm}$. CMS did
something similar for light HNLs, although they also included channels with OS
but same flavour lepton pairs in the heavy HNL regime, removing only those
events with lepton pairs compatible with a decay of a $Z$ boson [69]. As
stated before for the SS dilepton channel, searching for HNLs without assuming
their Majorana nature will be helpful to probe scenarios compatible with
neutrino oscillation data, and thus with potentially suppressed LNV signals.

Finally, it is important to stress that improving the experimental
sensitivities to smaller values of mixings implies exploring HNL with longer
lifetimes, which can travel macroscopic distances before decaying. Such long-
lived HNL would avoid the searches mentioned so far, as they all assumed
prompt decaying HNLs, and therefore we need a dedicated search for this kind
of topologies. Recently, both ATLAS [85] and CMS [86] have searched for
displaced vertex signatures for light HNLs, setting the strongest constraints
for GeV masses up to 20 GeV. This can be seen in Fig. 2, where we summarize
all the relevant LHC constraints explained in this section.

Channel | Lepton Flavour | Experiment |  $\sqrt{s}$ [TeV] | $\mathcal{L}\leavevmode\nobreak\ [{\rm fb}^{-1}]$ |  $M_{N}$ [GeV]
---|---|---|---|---|---
prompt SS dilepton $pp\to\ell_{\alpha}^{\pm}N\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}+nj$ | $ee/\mu\mu$ |  CMS’12 [87] | 7 | 4.98 | (50, 210)
$\mu\mu$ |  CMS’15 [88] | 8 | 19.7 | (40, 500)
$ee/e\mu$ |  CMS’16 [89] | 8 | 19.7 | (40, 500)
$ee/\mu\mu$ |  ATLAS’15 [90] | 8 | 20.3 | (100, 500)
$ee/e\mu/\mu\mu$ |  CMS’18 [69] | 13 | 35.9 | (20, 1600)
$\mu\mu$ |  LHCb’20 [83] | 7-8 | 3.0 | (5, 50)
| prompt OS dilepton
---
$pp\to\ell_{\alpha}^{\pm}N\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}+nj$
$\mu\mu$ |  LHCb’20 [83] | 7-8 | 3.0 | (5, 50)
Prompt trilepton $pp\to\ell_{\alpha}^{\pm}N\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}\ell_{\gamma}^{\mp}\nu$ | $eee+ee\mu/\mu\mu\mu+\mu\mu e$ |  CMS’18 [91] | 13 | 35.9 | (1, 1200)
$ee\mu/\mu\mu e$ |  ATLAS’19 [84] | 13 | 36.1 | (5, 50)
Displaced trilepton $pp\to\ell_{\alpha}N,\leavevmode\nobreak\ N\to\ell_{\beta}\ell_{\gamma}\nu$ | $\mu-e\mu/\mu-\mu\mu$ |  ATLAS’19 [84] | 13 | 32.9 | (4.5, 10)
6 combinations of $e,\mu$ |  ATLAS’22 [85] | 13 | 139 | (3, 15)
6 combinations of $e,\mu$ |  CMS’22 [86] | 13 | 138 | (1, 20)
Table 1: HNL searches at the LHC, classified according to the type of signal
searched for. OS/SS are for opposite/same sign for the charges of final
leptons and $nj$ for a number $n$ of final jets.

Figure 2: Summary of direct HNL searches performed at the LHC so far, either
by CMS (dashed), ATLAS (dotted) or LHCb (dot-dashed), and group by colors for
different kind of searches as given in Table 1. In the upper (lower) panel a
single mixing scenario to electrons (muons) is assumed. Shadowed area cover
the area excluded by direct searches at LEP. Notice that below 2 GeV and above
(approx.) 100 GeV, bounds from meson decays and from non-unitarity of the
lepton mixing [92] dominate respectively over current LHC bounds, although we
do not show them explicitly for easier reading of the collider results.

### Searches at lepton colliders

Despite the fact that our current most powerful high-energy collider is a
hadronic one, it is important to stress that lepton colliders are extremely
relevant for HNL searches. Not only due to the impressive sensitivities
expected at future leptonic colliders such as the FCCee [93], but also because
HNL searches at LEP still provide the most relevant limits for some $M_{N}$
hypotheses.

The great advantage of a leptonic collider is its clean environment, in
contrast with the hadronic ones. In the case of LEP, they combined this
cleanliness with the huge amount of $Z$ bosons they collected to search for
HNLs produced in $Z\to\nu N$. Moreover, they considered both visibles and
semi-invisible HNL decays, such as monojet final states [94]:

$e^{+}e^{-}\to\nu N\to\nu\nu q\bar{q}\,,$ (9)

with the $q\bar{q}$ pair clustered as a single jet due to the large HNL boost
(efficient for $M_{N}\lesssim 30$ GeV). For heavier masses, $M_{N}\in(30,80)$
GeV, the signature was composed by two jets with or without a charged lepton.
Such a search would be very challenging at a hadronic collider, however it has
the advantage of being sensitive to all flavours, including the mixing to the
$\tau$ lepton, not explored so far by LHC searches. The DELPHI resutls [94],
derived for both long-lived and prompt light HNLs, were not improved (for
mixings to $e$ and $\mu$ flavours) by LHC until very recently, and still
dominate for some mass ranges (cf. Fig. 2).

Additionally, the L3 collaboration explored the heavy HNL regime by
considering their production via the t-channel $W$ diagram [95]. This process
dominates the heavy HNL production at a $e^{+}e^{-}$ collider running above
the $Z$ pole, however it is sensitive only to mixings to electrons. The
results by L3 still provide the strongest limits for masses between 100 and
200 GeV.

Despite the great effort in the search for HNLs by both LEP and LHC, it is
important to analyze their implications for realistic models introducing and
motivating the existence of HNLs. A common feature of all these searches is
the assumption of a simplified scenario, most of the time consisting on a
single HNL mixing to a single lepton flavour, which is not the standard
hypothesis one would use from the theory side. To our knowledge, the only
exceptions to these simplifications are provided by the recent ATLAS search
for long-lived HNLs that also considered a minimal but realistic 2HNL scenario
[85], and CMS searches for SS $e\mu$ final states [89, 69], although still
neglecting the mixing to taus. In the two following sections, we discuss the
importance of going beyond these simplified scenarios to explore more
realistic scenarios.

##  3 Beyond the single mixing assumption

As explained in the previous section, most of the LHC analyses are done
assuming the existence of just one HNL that mixes to a single flavour, which
we referred to as the single mixing scenario. In this section, we consider
deviations from this simplified hypothesis and discuss their implications for
reinterpreting the LHC bounds summarized in Fig. 2. In particular, we focus
only on prompt searches, while the implications for long-lived HNL were
discussed in, for instance, Ref. [50].

For simplicity, we still consider the presence of a single HNL (we will
discuss deviations from this hypothesis in the next section), however we open
the room for generic mixing patterns. Moreover, we will follow a bottom-up
approach where the SM is extended by ad-hoc masses for the 3 active neutrinos,
as required by oscillation phenomena, and by the presence of the additional
HNL $N$. This framework is useful to study, in particular, the collider
phenomenology of HNLs without assuming any specific underlying model or
mechanism of light neutrino mass and leptonic mixing generation444Notice
however that reproducing oscillation data in a given framework may introduce
relations between the HNL mass and mixings, shrinking the parameter space we
will consider..

In such a framework, the lepton mixing matrix is thus enlarged to a $4\times
4$ unitary matrix

$U_{\nu}=\left(\begin{array}[]{cc}U_{\nu\nu}^{3\times 3}&U_{\nu N}^{3\times
1}\\\ U_{N\nu}^{1\times 3}&U_{NN}^{1\times 1}\end{array}\right)\,,$ (10)

so the would-be-PMNS matrix $U_{\nu\nu}$ is no longer a unitary matrix, a
feature which is indeed used to constrain these models [92], and the fourth
column contains the HNL mixings to each flavour:

$U_{\nu N}^{T}=\big{(}U_{eN},U_{\mu N},U_{\tau N}\big{)}\,.$ (11)

For our discussion, it is interesting to parametrize this column as

$U_{\nu
N}^{T}=\sqrt{U^{2}}\,\big{(}\varepsilon_{e},\varepsilon_{\mu},\varepsilon_{\tau}\big{)}\,,$
(12)

where $U^{2}$ represents the total (squared) mixing of the HNL and the
$\varepsilon_{\alpha}$ its flavour strengths, with
$|\varepsilon_{e}|^{2}+|\varepsilon_{\mu}|^{2}+|\varepsilon_{\tau}|^{2}=1$.
Notice that this framework is precisely the one considered by most LHC
analyses, the only difference being that they simplify it by setting the a
priori non-relevant mixings to zero. Here, we are interested in knowing how
these bounds need to be modified in a generic mixing pattern scenario.

The reason why we expect the bounds to be modified is twofold. The first
reason is due to the importance of the HNL decay width, which depends on every
mixing $U_{\alpha N}$, and which plays a major role in the resonant searches
(on-shell produced HNL) we are interested in. This means that the final cross
sections will depend on all of the mixings, even on those flavours that are
not explicitly present in the charged leptons involved in the process. The
second reason is that for some channels, considering generic mixings could
open new contributing diagrams, which could modify the distributions and thus
the efficiencies of the searches as discussed thoroughly in Ref. [58].
Complete expressions for the computation of the total decay width $\Gamma_{N}$
can be found for instance in Ref. [96], however for our purposes we
parameterize it as

$\Gamma_{N}=|U_{eN}|^{2}\,\Gamma_{N}^{e}+|U_{\mu
N}|^{2}\,\Gamma_{N}^{\mu}+|U_{\tau N}|^{2}\,\Gamma_{N}^{\tau}\,,$ (13)

where $\Gamma_{N}^{\alpha}$ stands for the sum of partial decay widths
depending on the mixing $U_{\alpha N}$, after factorizing the $|U_{\alpha
N}|^{2}$ dependence itself. Thus, $\Gamma_{N}^{\alpha}$ are independent of the
mixings (at leading order) and depend only on the HNL mass. Moreover, when the
HNL is heavy enough so that we can neglect charged lepton masses, we get
$\Gamma_{N}^{e}\simeq\Gamma_{N}^{\mu}\simeq\Gamma_{N}^{\tau}$ and thus

$\Gamma_{N}\propto\sum_{\alpha}\big{|}U_{\alpha N}\big{|}^{2}=U^{2}\,.$ (14)

With this discussion in mind, we can now study how the different processes
displayed in Table 1 depend on the HNL mixings. Let us start focusing on the
dilepton channels, the most straighforward case as we only need to track the
effect of the HNL total decay width.

In the narrow width approximation, the processes with SS and same flavour
dileptons can be factorized in the production of the HNL together with a
charged lepton, times its subsequent decay to the same lepton plus jets. The
first part depends only on the mixing to the flavour of that lepton, however
the second one involves all the mixings due to the HNL decay width. More
explicitly, we have

$\sigma(pp\to\ell_{\alpha}^{\pm}N\to\ell_{\alpha}^{\pm}\ell_{\alpha}^{\pm}+nj)\propto|U_{\alpha
N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{\pm}jj)\propto\frac{|U_{\alpha
N}|^{4}}{\Gamma_{N}}\,,$ (15)

or, assuming a heavy enough HNL,

$\sigma(pp\to\ell_{\alpha}^{\pm}N\to\ell_{\alpha}^{\pm}\ell_{\alpha}^{\pm}+nj)\propto
U^{2}\,|\varepsilon_{\alpha}|^{4}\,.$ (16)

Then, we clearly see that those bounds obtained in the single mixing benchmark
($\varepsilon_{\alpha}=1$) will be relaxed in a general flavour scenario with
a fixed $U^{2}$, since in general we will have $|\varepsilon_{\alpha}|^{2}\leq
1$. This is actually the expected behaviour, as switching on other mixings
opens for new decay channels, so not every produced HNL will decay to the
final state we are searching for.

process (prompt) | Relevant parameters (Majorana HNL)
---|---
approx. | complete dependence
$pp\to\ell_{\alpha}^{\pm}\ell_{\alpha}^{\pm}+nj$ | $U^{2}\,|\varepsilon_{\alpha}|^{4}$ | $\big{|}U_{\alpha N}\big{|}^{2}\,{\rm BR}(N\to\ell_{\alpha}^{\pm}jj)$
$pp\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}+nj$ | $U^{2}\,|\varepsilon_{\alpha}|^{2}\,|\varepsilon_{\beta}|^{2}$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\beta}^{\pm}jj)+|U_{\beta N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{\pm}jj)$
|  |
$pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{+}\ell_{\alpha}^{-}+\not{E}_{T}$ | $U^{2}\,|\varepsilon_{\alpha}|^{4}$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{+}\ell_{\alpha}^{-}\bar{\nu}_{\alpha})+|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{-}\ell_{\alpha}^{+}\nu_{\alpha})$
$pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{+}\ell_{\beta}^{-}+\not{E}_{T}$ | $U^{2}\,|\varepsilon_{\alpha}|^{2}\,(|\varepsilon_{\alpha}|^{2}+|\varepsilon_{\beta}|^{2})$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{+}\ell_{\beta}^{-}\bar{\nu}_{\beta})+|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\beta}^{-}\ell_{\alpha}^{+}\nu_{\alpha})$
$pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{-}\ell_{\beta}^{+}+\not{E}_{T}$ | $U^{2}\,|\varepsilon_{\alpha}|^{2}\,(|\varepsilon_{\alpha}|^{2}+3|\varepsilon_{\beta}|^{2})$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{-}\ell_{\beta}^{+}\nu_{\phantom{\alpha}})+|U_{\beta N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{-}\ell_{\alpha}^{+}\nu_{\phantom{\alpha}})$
$pp\to\ell_{\alpha}^{+}\ell_{\beta}^{+}\ell_{\gamma}^{-}+\not{E}_{T}$ | $U^{2}\,{\scriptscriptstyle\displaystyle\sum_{\scriptstyle i=\alpha,\beta}}|\varepsilon_{i}|^{2}\,\big{(}1-|\varepsilon_{i}|^{2}\big{)}$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\beta}^{+}\ell_{\gamma}^{-}\nu_{\phantom{\alpha}})+|U_{\beta N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{+}\ell_{\gamma}^{-}\nu_{\phantom{\alpha}})$
Table 2: Summary table for generic flavour dependences of dilpeton and
trilepton channels at the LHC assuming a single Majorana HNL with generic
mixing patterns. Flavour indices are to be understood as different, i.e.
$\alpha\neq\beta\neq\gamma$. For trileptons, we are neglecting the effects of
differential distributions, as discussed in the text.

We can repeat the exercise for the different flavour SS dilepton processes.
Obviously, the minimal setup in this case requires to have two non-zero
mixings, which leads to two diagrams that in principle interfere.
Nevertheless, using the narrow width approximation, we can see that both
diagrams cannot resonate at the same time, so we can neglect the interference
and add both processes incoherently:

$\displaystyle\sigma(pp\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}+nj)$
$\displaystyle\propto\Big{(}|U_{\alpha N}|^{2}\,{\rm
BR}(N\to\ell_{\beta}^{\pm}jj)+|U_{\beta N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}^{\pm}jj)\Big{)}$ $\displaystyle\propto\frac{|U_{\alpha
N}|^{2}|U_{\beta N}|^{2}}{\Gamma_{N}}\propto
U^{2}\,|\varepsilon_{\alpha}|^{2}\,|\varepsilon_{\beta}|^{2}\,.$ (17)

The case of the trilepton channels can be more involved, mainly because we
cannot know the lepton number and the flavour carried by the missing
(anti)neutrino. Let us consider first the case of same flavour trileptons.
There are two contributing diagrams, one with a neutrino and one with an
antineutrino, which we can add incoherently555If light neutrinos are of
Majorana nature, then there is an interference term, which is however
negligible as it is proportional to light neutrino masses.. Then, considering
for simplicity a $W^{+}$ Drell-Yan channel, we have

$\sigma(pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{+}\ell_{\alpha}^{-}+\not{E}_{T})\propto\Big{(}|U_{\alpha
N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}^{+}\ell_{\alpha}^{-}\bar{\nu}_{\alpha})+|U_{\alpha
N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}^{-}\ell_{\alpha}^{+}\nu_{\alpha})\Big{)}\,.$ (18)

Notice that the first contribution is mediated by the Majorana nature of the
HNL, while the second one is of Dirac type. In principle, both contributions
are identical when integrated over the full phase space. However, different
spin-correlations induce different angular distributions, which might
translate into different acceptances under a given experimental analysis.
Still, both channels have the same flavour dependence, so we can write,

$\sigma(pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{+}\ell_{\alpha}^{-}+\not{E}_{T})\propto|U_{\alpha
N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}\ell_{\alpha}\nu_{\alpha})\propto\frac{|U_{\alpha
N}|^{4}}{\Gamma_{N}}\propto U^{2}\,|\varepsilon_{\alpha}|^{4}\,.$ (19)

When the trilepton signal involves leptons of two different flavours, we need
to consider two subcases: the same-sign same-flavour (SSSF) and opposite-sign
same-flavour (OSSF). These channels are trickier because in a generic flavour
pattern there are new diagrams not present in the single mixing scenario. For
instance, in the case of the SSSF, we have two types of contributions:

$\begin{array}[]{ll}\text{\rm Majorana-
like:}&pp\to\ell^{+}_{\alpha}N\,,N\to\ell^{+}_{\alpha}\ell^{-}_{\beta}\bar{\nu}_{\beta}\,,\\\
\text{\rm Dirac-
like:}&pp\to\ell_{\alpha}^{+}N\,,N\to\ell_{\beta}^{-}\ell_{\alpha}^{+}\nu_{\alpha}\,.\end{array}$
(20)

It is clear that the first process requires of a Majorana HNL, while the
second one needs mixings to both flavours. As before, the interference is
negligible, so we have

$\sigma(pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{+}\ell_{\beta}^{-}+\not{E}_{T})\propto\Big{(}|U_{\alpha
N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}^{+}\ell_{\beta}^{-}\bar{\nu}_{\beta})+|U_{\alpha
N}|^{2}\,{\rm
BR}(N\to\ell_{\beta}^{-}\ell_{\alpha}^{+}\nu_{\alpha})\Big{)}\,.$ (21)

Due to the missing (anti)neutrino, both processes are almost identical at the
LHC, with the only difference coming again from the different distributions
and acceptances of the experimental analysis. This was studied in detail in
Ref. [58] for the case of light HNLs. Nevertheless, in order to get a first
rough estimate, we can neglect these differences and write

$\sigma(pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{+}\ell_{\beta}^{-}+\not{E}_{T})\propto|U_{\alpha
N}|^{2}\,\frac{|U_{\alpha N}|^{2}+|U_{\beta N}|^{2}}{\Gamma_{N}}\propto
U^{2}\,|\varepsilon_{\alpha}|^{2}\,(|\varepsilon_{\alpha}|^{2}+|\varepsilon_{\beta}|^{2})\,.$
(22)

The case of OSSF is similar, although now the roles of Majorana and Dirac HNLs
are flipped:

$\begin{array}[]{ll}\text{\rm Majorana-
like:}&pp\to\ell^{+}_{\alpha}N\,,N\to\ell^{+}_{\beta}\ell^{-}_{\alpha}\bar{\nu}_{\alpha}\,,\\\
\text{\rm Dirac-
like:}&pp\to\ell_{\alpha}^{+}N\,,N\to\ell_{\alpha}^{-}\ell_{\beta}^{+}\nu_{\beta}\,.\end{array}$
(23)

Moreover, since we are working in the prompt HNL regime, we can also have
$pp\to\ell_{\beta}^{+}N,N\to\ell_{\alpha}^{+}\ell_{\alpha}^{-}\overset{\textbf{(--)}}{\nu}_{\alpha}$.
This means that a proper recasting of this kind of signals would require to
compute the efficiencies for all these diagrams. If, for the shake of this
discussion, we neglect these effects again, we get:

$\displaystyle\sigma(pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{-}\ell_{\beta}^{+}+\not{E}_{T})$
$\displaystyle\propto\Big{(}|U_{\alpha N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}^{-}\ell_{\beta}^{+}\nu_{\beta})+|U_{\alpha N}|^{2}\,{\rm
BR}(N\to\ell_{\beta}^{+}\ell_{\alpha}^{-}\bar{\nu}_{\alpha})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ +\,|U_{\beta
N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}^{-}\ell_{\alpha}^{+}\nu_{\alpha})+|U_{\beta
N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}^{+}\ell_{\alpha}^{-}\bar{\nu}_{\alpha})\Big{)}$
$\displaystyle\propto|U_{\alpha N}|^{2}\,\frac{|U_{\alpha N}|^{2}+3|U_{\beta
N}|^{2}}{\Gamma_{N}}\propto
U^{2}\,|\varepsilon_{\alpha}|^{2}\,(|\varepsilon_{\alpha}|^{2}+3|\varepsilon_{\beta}|^{2})\,.$
(24)

Finally, and even if no LHC searches have been performed so far, we can also
consider the case with 3 different flavours. Following the same steps, we get

$\displaystyle\sigma(pp\to\ell_{\alpha}^{+}\ell_{\beta}^{+}\ell_{\gamma}^{-}+\not{E}_{T})$
$\displaystyle\propto\Big{(}|U_{\alpha N}|^{2}\,{\rm
BR}(N\to\ell_{\beta}^{+}\ell_{\gamma}^{-}\bar{\nu}_{\gamma})+\,|U_{\alpha
N}|^{2}\,{\rm BR}(N\to\ell_{\gamma}^{-}\ell_{\beta}^{+}\nu_{\beta})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ +\,|U_{\beta
N}|^{2}\,{\rm
BR}(N\to\ell_{\alpha}^{+}\ell_{\gamma}^{-}\bar{\nu}_{\alpha})+|U_{\beta
N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{-}\ell_{\alpha}^{+}\nu_{\alpha})\Big{)}$
$\displaystyle\propto
U^{2}\,\Big{\\{}|\varepsilon_{\alpha}|^{2}\,\big{(}1-|\varepsilon_{\alpha}|^{2}\big{)}+|\varepsilon_{\beta}|^{2}\,\big{(}1-|\varepsilon_{\beta}|^{2}\big{)}\Big{\\}}\,.$
(25)

process (prompt) | Relevant parameters (Dirac HNL)
---|---
approx. | complete dependence
$pp\to\ell_{\alpha}^{\pm}\ell_{\alpha}^{\mp}+nj$ | $U^{2}\,|\varepsilon_{\alpha}|^{4}$ | $\big{|}U_{\alpha N}\big{|}^{2}\,{\rm BR}(N\to\ell_{\alpha}^{\pm}jj)$
$pp\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp}+nj$ | $U^{2}\,|\varepsilon_{\alpha}|^{2}\,|\varepsilon_{\beta}|^{2}$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\beta}^{\pm}jj)+|U_{\beta N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{\pm}jj)$
|  |
$pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{+}\ell_{\alpha}^{-}+\not{E}_{T}$ | $U^{2}\,|\varepsilon_{\alpha}|^{4}$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{-}\ell_{\alpha}^{+}\nu_{\alpha})$
$pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{+}\ell_{\beta}^{-}+\not{E}_{T}$ | $U^{2}\,|\varepsilon_{\alpha}|^{2}|\varepsilon_{\beta}|^{2}$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\beta}^{-}\ell_{\alpha}^{+}\nu_{\alpha})$
$pp\to\ell_{\alpha}^{+}\ell_{\alpha}^{-}\ell_{\beta}^{+}+\not{E}_{T}$ | $U^{2}\,|\varepsilon_{\alpha}|^{2}\,(|\varepsilon_{\alpha}|^{2}+|\varepsilon_{\beta}|^{2})$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{-}\ell_{\beta}^{+}\nu_{\beta})+|U_{\beta N}|^{2}\,{\rm BR}(N\to\ell_{\alpha}^{-}\ell_{\alpha}^{+}\nu_{\alpha})$
$pp\to\ell_{\alpha}^{+}\ell_{\beta}^{+}\ell_{\gamma}^{-}+\not{E}_{T}$ | $U^{2}\,|\varepsilon_{\gamma}|^{2}\,\big{(}1-|\varepsilon_{\gamma}|^{2}\big{)}$ | $|U_{\alpha N}|^{2}\,{\rm BR}(N\to\ell_{\gamma}^{-}\ell_{\beta}^{+}\nu_{\beta})+|U_{\beta N}|^{2}\,{\rm BR}(N\to\ell_{\gamma}^{-}\ell_{\alpha}^{+}\nu_{\alpha})$
Table 3: Same as Table 2, but for Dirac HNL. We give the OS dileptons in this
case, since the SS are not sensitive to Dirac HNLs.

We summarize our discussion in Table 2. Here, we assume a Majorana HNL,
although a similar table can be easily obtained for Dirac HNL by just
switching off the LNV channels we discussed above. This is done in Table 3.
Notice how some channels that were designed for Majorana HNLs are also
sensitive to Dirac HNLs in the case of having generic mixing patterns, as it
was already discussed in Ref. [58].

These tables are to be compared with the minimal mixing scenario where all of
the processes scale as $|U_{\alpha N}|^{2}$, or $|U_{\alpha N}|^{2}|U_{\beta
N}|^{2}/(|U_{\alpha N}|^{2}+|U_{\beta N}|^{2})$ for $\alpha\neq\beta$.
However, we see that in general each process is sensitive to a different
combinations of mixing strengths. This means that, in order to generalize the
bounds to a generic pattern, it is better to set bounds on the quantity on the
last columns of Tables 2 and 3, since then we only need to recompute the new
BRs for each mixing hypothesis.

While there are some experimental results also presenting (in the single
mixing assumption) bounds in the ($M_{N},|U_{\alpha N}|^{2}\times$BR) plane,
most of the results are given directly in the ($M_{N},|U_{\alpha N}|^{2})$
one. Translating the latter to the former is straightforward in most of the
cases, since the experimental collaborations usually assume a constant666This
is a well-justified approximation, but still an approximation that could be
avoided by setting bounds directly on $|U_{\alpha N}|^{2}\times$BR. BR for
channel under study, although not always specifying the precise value they
used. Nevertheless, this recasting to $|U_{\alpha N}|^{2}\times$BR is not
possible when the experimental results on $|U_{\alpha N}|^{2}$ are presented
after combining different channels. This is the case for instance for the
latest CMS searches for trileptons [91], where they combined channels like
$e^{\pm}e^{\pm}e^{\mp}$ and $e^{\pm}e^{\pm}\mu^{\mp}$ (see Table 1). While in
the single mixing scenario both channels depend only on $|U_{eN}|^{2}$, they
have a different dependence in the case of a generic mixing scenario (see
Table 2), and thus it is not easy to recast the obtained bounds without a
dedicated analysis. For this reason, together with the potential efficiency
differences discussed above, we will focus the rest of our discussion only on
the dilepton channels.

Figure 3: Rescaling of the bounds on $|U_{\mu N}|^{2}$ from CMS [69] (solid
red line) and LHCb [83] (solid blue line), fixing the active$-$sterile mixings
to the values corresponding to the green points $1-4$ in the ternary plot on
the left. The orange squared point represents the single mixing case.

As a case of study for the use of Table 2, let us consider the CMS [69] and
LHCb [83] searches for the LNV dimuon channel $\mu^{\pm}\mu^{\pm}$. For a
given total active-sterile mixing $U^{2}$, we can display the full flavour
mixing space in a ternary diagram, as in the left panel of Fig. 3. Then, the
single mixing scenario constraints by both CMS and LHCb lie in the top corner.
As we move along the ternary, we decrease the flavour strength to muons, so
the bounds are relaxed, as shown in the right panel (dashed lines). Here we
chose just few benchmark points for the light HNL mass regime, although the
same logic applies to the heavy one.

As discussed above, the physical reason for the relaxation of these bounds is
due to the new HNL decay channels in the generic mixing scenario. On the other
hand, this also implies that the experimental searches for the other channels
with different flavours might become relevant. In order to show the interplay
between different flavour channels, let us consider again a ternary diagram.
We can understand it as a subspace of the parameter space with fixed values of
$M_{N}$ and $U^{2}$, which is dissected in flavour space. Then, searches for
dimuon channels will cover the area close to the $\varepsilon_{\mu}=1$ corner,
becoming weaker as we move further away. Equivalently, dielectron searches
will cover the ternary from the $\varepsilon_{e}=1$ corner, while the
$e^{\pm}\mu^{\pm}$ channel will cover the area in between. This is depicted in
Fig. 4 for two benchmark points, one in the light regime ($M_{N}=30$ GeV) and
one in the heavy regime ($M_{N}=300$ GeV).

Figure 4 clearly shows the complementarity of the different dilepton searches,
to which we could supplementary add the bounds from trilepton channels if the
above mentioned concerns are solved. If the combination of every channel
covered all the area of the ternary, we could say that this $(M_{N},U^{2})$
point is excluded no matter which flavour mixing pattern we were considering.
This is not the case in any of the two examples in the figure, since the
bottom left corner is still allowed by LHC. Notice that this is always the
case at present, since there are no LHC searches for HNLs mixing to the tau
lepton, so the corner of $\varepsilon_{\tau}=1$ will always be allowed.
Although we already discussed that a single mixing scenario does not seem very
natural for a realistic model including HNLs, closing this gap still motivates
the need of performing dedicated searches in the tau sector.

On the other hand, it is important to stress that the benchmark points in Fig.
4 are chosen just for illustrative purposes, since they are already excluded
by LEP searches or global fit bounds. Indeed, this is actually the case in a
large part of the parameter space for the current LHC bounds discussed in Sec.
2. Nevertheless, it is worth emphasizing that LHC sensitivities are expected
to improve during the currently ongoing runs, pushing our knowledge about HNLs
beyond present limits. Therefore, combining the different LHC channels as we
discussed here will become crucial in order to determine whether a heavy
neutral lepton with a given mass and mixing is completely excluded or not.

Figure 4:  Combination of several bounds in a general mixing pattern for two
benchmark points of mass and total mixing $U^{2}$. Each bound correspond to
the latest SS dilepton searches by CMS (Table 1), which were derived within a
single mixing scenario (or assuming $U_{\tau N}$ for the $e^{\pm}\mu^{\pm}$
channel). The white area is still allowed by LHC searches.

##  4 Beyond the single HNL

Should HNL exist in Nature, there is a priori no bound on their number and in
general, BSM models involving HNLs do not introduce just one of them. For
example, in the standard type-I seesaw model at least two HNL are needed to
explain neutrino oscillation data. In this work, in order to explore
deviations from the single HNL hypothesis, we extend the framework considered
in the previous section to include more neutral fermions focusing here on the
minimal case of SM extension via two HNLs. In this case, the lepton mixing
matrix is now a $5\times 5$ unitary matrix

$U_{\nu}=\left(\begin{array}[]{cc}U_{\nu\nu}^{3\times 3}&U_{\nu N}^{3\times
2}\\\ U_{N\nu}^{2\times 3}&U_{NN}^{2\times 2}\end{array}\right)\,,$ (26)

with the fourth and fifth columns encoding the mixings of both HNLs to the
active leptons

$U_{\nu N}=\left(\begin{array}[]{cc}U_{eN_{1}}&U_{eN_{2}}\\\ U_{\mu
N_{1}}&U_{\mu N_{2}}\\\ U_{\tau N_{1}}&U_{\tau N_{2}}\end{array}\right)\,.$
(27)

For the sake of simplicity, we will assume in this section that the two HNLs
mix to just a single flavour. In a sense, it can be seen as extending the
interpretation of the bounds of Fig. 2 in horizontal in $U_{\nu N}$, to
additional columns, while in Section 3 we extended them in vertical, to
additional flavours. The most general case with several HNLs and arbitrary
mixing patterns can then be inferred as a combination of these two
discussions.

In the case where only one of these HNLs is within experimental reach or when
there are several HNLs but with well-separated mass regimes, our conclusions
derived following the single HNL scenario will apply to each of the HNLs.
Nevertheless, if two HNLs happen to be close in mass (as motivated by low-
scale seesaw models [61, 97, 98], resonant leptogenesis [99] or ARS
leptogenesis [100]), they could lead to interference effects and modify the
results and the bounds obtained in the single HNL hypothesis. Moreover, these
modifications might affect both LNV and LNC branching ratios, and thus
studying their correlation could shed light on the nature of the HNL (see for
instance [64] and references therein).

In this section we discuss how these effects could affect the LHC bounds
obtained in the single HNL scenario from searches for LNV and LNC channels,
and provide a recipe to combine both results (on LNV and LNC searches) in
order to bring forth more robust bounds on HNLs parameter space. The recipe is
also applicable to the case where HNLs interfere. We mostly focus on the LHCb
[83] results for the prompt dimuon channel, since this is the only available
analysis addressing both SS and OS dilepton channels (cf. Table 1). More
specifically, this search considers light HNLs which are dominantly produced
from on-shell DY $W$ bosons in Fig. 1, that is,

$pp\to
W^{+}\to\ell_{\alpha}^{+}N_{i}\to\ell_{\alpha}^{+}\ell_{\beta}^{+}\,q\,\overline{q}^{\prime}\,,\quad
pp\to
W^{+}\to\ell_{\alpha}^{+}N_{i}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}\,q\,\overline{q}^{\prime}\,.$
(28)

In presence of just a single Majorana HNL, although the angular distributions
will be different (see Appendix A for more details), the predictions for the
total rates of these two processes are of similar size: equal for channels
with $\alpha=\beta$ and twice as large for $\alpha\neq\beta$. The reason for
this difference is that the channel with crossed $\ell_{\alpha}$ and
$\ell_{\beta}$ is also contributing in the case of the LNV process. However,
it must be added incoherently to the process since the rate is dominated by
on-shell HNLs, fixing the momentum of the first lepton with the 2-body decay
kinematics, and thus the interference becomes subdominant. This means that for
channels with $\alpha\neq\beta$, the total rate for the LNV process is
enhanced by a factor of 2 with respect to the LNC one. On the other hand, in
channels with $\alpha=\beta$ there is an additional $1/2$ factor from having
two identical particles, and thus we obtain the same total rate for both LNV
and LNC.

When assuming the existence of two HNLs, we have two identical contributions
to the total amplitude of the processes, one for each $N_{i}$. The squared
amplitude is then given as sum of the individual contributions of each HNL,
plus a potential interference between $N_{1}$ and $N_{2}$ contributions. For
each individual contribution to the amplitudes, we find that they are
proportional to $U^{*}_{\alpha N_{i}}U^{*}_{\beta N_{i}}$ for the SS process
and to $U^{*}_{\alpha N_{i}}U_{\beta N_{i}}^{\phantom{*}}$ for the OS one,
proving convenient to define

$U_{\alpha N_{i}}=|U_{\alpha N_{i}}|\,e^{i\phi_{\alpha i}}\,,$ (29)

with $\phi_{\alpha i}\in[0,2\pi]$, $\alpha=e,\mu,\tau$ and $i=1,2$. In this
way, the interference term resulting from both amplitudes will be proportional
to

$\left|U_{\alpha N_{1}}\right|\left|U_{\alpha N_{2}}\right|\left|U_{\beta
N_{1}}\right|\left|U_{\beta N_{2}}\right|e^{i\delta\phi^{\pm}}\,,$ (30)

where we have defined $\delta\phi^{\pm}$ as follows:

$\displaystyle\delta\phi^{\pm}=\left(\phi_{\alpha 2}-\phi_{\alpha
1}\right)\pm\left(\phi_{\beta 2}-\phi_{\beta 1}\right)\ ,$ (31)

with $+/-$ for the SS/OS channel. Details with the complete analytical
amplitude and decay rates, which we have additionally checked with Whizard
[101], can be found in Appendix A, both for the case with just one HNL and
with two HNLs.

As already stated, when the mass difference of the two HNLs is too large
(compared to their decay width), the contribution of each HNL resonate
independently and the interference is negligible. In this case, the total
rates for the LNV and LNC rates are related as in the case of the single HNL
scenario. Therefore, in the following we will assume the scenario when the
interference effects can modify considerably the relative predictions between
LNV and LNC rates, which corresponds to the case where both HNLs are close in
mass. More specifically, we assume that the individual contribution of each
HNL is of similar size, $M_{N_{1}}\simeq M_{N_{2}}\equiv
M_{N},\Gamma_{N_{1}}\simeq\Gamma_{N_{2}}\equiv\Gamma_{N}$, and also that
$\left|U_{\alpha N_{2}}\right|\left|U_{\beta N_{2}}\right|=\left|U_{\alpha
N_{1}}\right|\left|U_{\beta N_{1}}\right|$. However, we consider that the two
HNL mass splitting $\Delta M_{N}\equiv M_{N_{2}}-M_{N_{1}}$ could be different
from zero as long as it is small compared to the decay width $\Gamma_{N}$.
Notice that this kind of scenario appears naturally in low-scale seesaws due
to the approximated lepton number conservation. The only difference is that in
these models the phases of the HNLs are fixed to be opposite, i.e.
$\delta\phi^{+}=\pi$ and $\delta\phi^{-}=0$ (the heavy neutrinos forming a
pseudo-Dirac neutrino pair), and that $\Delta M_{N}$ is somehow related to
light neutrino masses [81, 82], while in the following we will let them free,
in the spirit of the bottom-up approach described before.

Under these conditions, we can write the total decay rate driven by the two
HNLs in the case of $W^{+}$ channel as

$\left.\Gamma\left(W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{\pm}q\bar{q}^{\prime}\right)\right|_{N_{1}\&N_{2}}=2\,\mathcal{K}_{+}\left(y,\delta\phi^{\pm}\right)\,\left.\Gamma\left(W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{\pm}q\bar{q}^{\prime}\right)\right|_{N_{1}}\
,$ (32)

and equivalently for the $W^{-}$ channel with
$\mathcal{K}_{-}\left(y,\delta\phi^{\pm}\right)$ . Here, we have factorized
the total rate in presence of only one HNL, and defined the modulation
function

$\mathcal{K}_{\pm}\left(y,\delta\phi^{\pm}\right)\equiv\left(1+\cos\delta\phi^{\pm}\frac{1}{1+y^{2}}\mp\sin\delta\phi^{\pm}\frac{y}{1+y^{2}}\right),$
(33)

with

$y\simeq\frac{\Delta M_{N}}{\Gamma_{N}}\,.$ (34)

The function $\mathcal{K}_{\pm}$ codifies the role of the interference. In the
limit of $\Delta M_{N}\gg\Gamma_{N}$, the two HNL are too separated in mass,
coherence is lost and the total contribution is just twice the single HNL
contribution for both LNV and LNC. On the other hand, for $\Delta
M_{N}<\Gamma_{N}$, the modulation function can take values from 0 (maximally
destructive interference) to 2 (maximally constructive). Thus, we are
maximizing the effects of the interference between the two HNLs. Moreover,
these effects will be different for LNV and LNC, breaking the equal size
prediction in the single HNL scenario.

Figure 5: Left: rescaling of the bounds on $|U_{\mu N}|^{2}$ from LHCb [83] in
the presence of two HNLs. Dark blue line is the LHCb bound in the LNC
searches, while in lighter blue (lower curve) is the rescaled bound for
$y=\Delta M_{N}/\Gamma_{N}<1$. The dark green line is the bound in the LNV
searches, which can be relaxed (upper green region) if the $N_{1,2}$ form a
pseudo$-$Dirac pair ($\delta\phi^{+}=\pi$, $y\ll 1$), or strengthened (lower
green region) if $\delta\phi^{+}=0$. Right: $|U_{\mu N}|^{2}$ as a function of
$y$, with $M_{N}=30$ GeV and $\delta\phi^{+}=\pi$. Blue (green) region is for
LNC (LNV) channel with the thick lines corresponding to the (rescaled) LHCb
bounds to the case of a low-scale seesaw with pseudo-Dirac HNL pair.

These modulation function can be used to simply recast the bounds derived by
LHCb [83] under the assumption of a single HNL. Noticing that LHCb searched
only for the $W^{+}$ channel and given Eq. (15), to recast these bounds to our
scenario with two HNLs, we need to rescale the mixing as

$\left|U_{\mu N}\right|^{2}\rightarrow\left|U_{\mu N}\right|^{2}\times
2\mathcal{K}_{+}\left(y,\delta\phi^{\pm}\right)\,,$ (35)

with $\delta\phi^{-}=0$ in this channel with $\alpha=\beta$. Following this
modulation, we show in the left panel of Fig. 5 how the LHCb bounds might
vary, depending on the values of $y$ and the relative phases $\delta\phi^{+}$,
for both LNV and LNC searches, which defines the green and blue bands,
respectively. The vertical axis needs to be understood now as the (squared)
mixing of each of the HNLs to muons, which in absence of interference ($y\gg
1$) is just a factor of two stronger with respect to the single HNL scenario.
For $y\ll 1$, however, constructive interference can strengthen the bounds by
up to an additional factor of two, while destructive interference could relax
it, even avoid it completely in the case of LNV signals. The latter
corresponds to the case where $\delta\phi^{+}=\pi$, which is precisely when
the two HNLs have opposite phases (thus forming a pseudo-Dirac pair), as
required by low-scale seesaws with approximated lepton number conservation.
Interestingly, the same choice of parameters that maximizes the destructive
interference of LNV channels also maximizes the constructive interference for
the LNC ones, making the bounds from the latter stronger.

This interplay prompts us to consider both channels at the same time, using
their complementarity to set absolute bounds on the mixings that could not be
avoided even with ad-hoc values of the parameters that maximize the
interference. We show this in the right panel of Fig. 5 for a particular
example of $M_{N}=30$ GeV and opposite phases, $\delta\phi^{+}=\pi$, as
required by low-scale seesaws. When the contributions of the two HNLs is
maximally coherent ($y\ll 1$), the LNV searches are avoided at the price of
maximizing the LNC bounds. If coherence is lost ($y\gg 1$), then the stronger
LNV bounds always dominate over the LNC ones. We see then that the largest
possible mixing could be obtained in between the two cases, when the two
tendencies cross over, which we can consider as an absolute bound on the
mixing that cannot be avoided even with 2 interfering HNLs.

To summarize, searches for LNC and LNV processes are important and
complementary, since they can cover areas of the parameter space even in the
case where there is some interference (partially) cancelling any of the two
channels. This strongly motivates the need of searching for both LNC and LNV
channels in parallel, even if the latter is more challenging experimentally,
since combining both of them we could set more robust bounds on generic
scenarios including more than one heavy neutral lepton.

### CP violation

When a quasi-degenerated pair of HNLs is considered, new CP-violating phases
are introduced, which can induce differences in the decays of these particles
to leptons over antileptons. If HNLs were discovered in processes such as
those in Eq. (28), and provided that enough events were collected, one way of
measuring this potential CP asymmetry would be by defining the ratio [65]

$A_{\mathrm{CP}}^{\pm}=\frac{{\rm
BR}\left(W^{-}\rightarrow\ell_{\alpha}^{-}\ell_{\beta}^{\mp}\bar{q}q^{\prime}\right)-{\rm
BR}\left(W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{\pm}q\bar{q}^{\prime}\right)}{{\rm
BR}\left(W^{-}\rightarrow\ell_{\alpha}^{-}\ell_{\beta}^{\mp}\bar{q}q^{\prime}\right)+{\rm
BR}\left(W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{\pm}q\bar{q}^{\prime}\right)}\,.$
(36)

Using our previous results for the $W$ decays, see Eq. (32), it is
straightforward to see that $A_{\rm CP}^{\pm}$ takes the simple form

$A_{\rm
CP}^{\pm}=\frac{y\,\sin\delta\phi^{\pm}}{1+y^{2}+\cos\delta\phi^{\pm}}\,,$
(37)

where $+/-$ denotes again LNV/LNC processes.

Figure 6:  CP violating ratio $A_{\rm CP}^{\pm}$, defined in Eq. (37), as
function of the relevant ratio $y=\Delta M_{N}/\Gamma_{N}$ and for different
choices of the relative phases $\delta\phi^{\pm}$. Red dots were obtained with
Whizard as a cross-check of our analytical results.

As can be seen, this equation does not depend on the HNL masses, but on their
mass difference $\Delta M_{N}$ (through $y=\Delta M_{N}/\Gamma_{N}$). It
vanishes for the obvious case of $\delta\phi^{\pm}=0$, since then there is no
difference in both the $W$ decay and its CP equivalent; notice that it is also
the case when $y$ is close to zero or too large. This is shown in Fig. 6 where
we display the CP asymmetry for different values of $\delta\phi^{\pm}$.

Notice that a similar computation was performed in Ref. [65] with which our
results agree but with some slight differences. On the one hand, we find a
discrepancy when applying the narrow width approximation for the interference
term (see App. A for more details). On the other hand, we performed the
computation considering very prompt heavy neutral leptons, so we did not take
into account their time evolution and their possible oscillations before
decaying. Our approach is appropriate for heavy HNLs, while one should take
into account the evolution effect for very light HNLs, thus with longer decay
lifetimes, as it was done in Ref. [65]. Finally, we also point out that, in
order to measure $A_{\rm CP}$ at a proton-proton collider, we must take into
account that the production rates for $W^{+}$ and $W^{-}$ are not the same
[65].

##  5 Conclusions

In this work, we focused on LHC searches of heavy neutral leptons that decay
promptly (short-lived). In most of the searches, which we summarized in Sec.
2, the obtained bounds were derived under the hypothesis of the existence of a
single (usually Majorana) HNL that mixes with only one lepton flavour, while
most of the BSM scenarios involving new neutral fermions require more than one
HNL. Moreover, unless some specific symmetries are present, the mixing pattern
in these BSM scenarios is more complex, each HNL mixing in general with all
charged leptons, and thus, the bounds derived from negative searches on the
HNL parameter space have to be recast before being applied to a generic BSM
scenario.

In this study we discussed how to recast the present experimentally obtained
bounds on the parameter space, i.e. active-sterile mixings $U_{\alpha N}$,
$\alpha=e,\mu,\tau$, versus the mass of the HNL, to the case of generic mixing
to all active flavours as well as to the case with several HNLs. The former
was covered in Sec. 3, where we inspected the flavour dependencies of each of
the channels searched for by the LHC, and stressed the importance of setting
bounds not only on the mixings, but on the relevant combination of $|U_{\alpha
N}|^{2}\times{\rm BR}$ (see Tables 2 and 3). Considering the bounds on this
combination, we proposed a method to combine the results in flavour space,
using the ternary diagrams in Fig. 4 to conclude whether that area of
parameter space is fully excluded, regardless of the assumed mixing pattern.

In the case with several heavy neutral leptons, we focused on the scenario
when two HNLs are in the same mass regime, nearly degenerate or possibly
forming a pseudo-Dirac neutrino pair, paying special attention to the non-
trivial role of the interference between their contributions. To illustrate
its importance, we focused on dilepton channels and moreover considered both
channels with same and opposite charge of the final leptons, as it was done by
LHCb [83]. We showed the complementary of the LNC and LNV searches and the
importance of performing both of them in parallel. We stressed that by doing
this, we are not only taking into account the two possible nature of a single
HNL, Dirac or Majorana, but also covering the case when, for example, two
Majorana HNLs exist and however they interfere destructively, suppressing the
expected LNV signature.

To summarize, we have discussed the importance of going beyond simplified
scenarios such as the single mixing hypothesis. While they are useful for
simplifying experimental analyses, they are not directly applicable to BSM
models introducing HNLs. Unfortunately, recasting the bounds of each
experimental analysis to a given BSM scenario can be a tedious task. In this
work, we have proposed an alternative way of presenting the bounds on the
parameter space of the HNLs, which under some approximations can be both
directly constrained by experimental analyses and also easily recast to a
generic BSM scenario.

## Acknowledgements

PE is grateful for the hospitality of the Pôle de Physique Théorique of the
IJCLAB (Orsay) during the first stages of the project. This project has
received support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
860881 (HIDDe$\nu$ network). The work of PE is supported by the Spanish grants
PID2020-113775GB-I00 (AEI / 10.13039/501100011033), CIPROM/2021/054
(Generalitat Valenciana) and by the FPI grant PRE2018-084599. XM also
acknowledges partial support from the Spanish Research Agency (Agencia Estatal
de Investigación) through the Grant IFT Centro de Excelencia Severo Ochoa No
CEX2020-001007-S and Grant PID2019-108892RB-I00 funded by
MCIN/AEI/10.13039/501100011033.

##  Appendix A Decay process in presence of 2 HNL

In this appendix we collect the relevant details for the computation of the
$W$ boson decay into a HNL, followed by its semileptonic decay. This is the
relevant process for the searches performed by LHCb [83] and that we discussed
in Sec. 4 in the scenario with two heavy neutral leptons.

###  A.1 Same sign leptons:
$W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{+}q^{\prime}\bar{q}$

We start with the decay rate for the process
$W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{+}q^{\prime}\bar{q}$ mediated
by two HNLs $N_{1,2}$ almost degenerate in mass, similar to that shown in Fig.
1, but without the initial quarks. The amplitude reads

$\mathcal{M}^{+}=\sum_{i=1,2}\frac{g^{3}}{2\sqrt{2}}U_{{\alpha}N_{i}}^{*}U_{{\beta}N_{i}}^{*}\frac{M_{N_{i}}}{M_{W}^{2}}\epsilon_{\mu}^{*}\frac{\overline{\ell_{\alpha}^{c}}\gamma^{\mu}\gamma^{\nu}P_{L}\ell_{\beta}\bar{q}\gamma_{\nu}P_{L}q^{\prime}}{p_{N}^{2}-M_{N_{i}}^{2}+i\Gamma_{N_{i}}M_{N_{i}}}\,,$
(38)

with $p_{N}=p_{W}-p_{\ell_{\alpha}}$. The channel with crossed $\ell_{\alpha}$
and $\ell_{\beta}$ gives rise to the same amplitude, but it must be added
incoherently to our process since the rate is dominated by on-shell $N_{i}$,
and thus the momentum of the first lepton is fixed by the 2-body decay
kinematics. Therefore the two processes do not interfere and we can neglect
for the moment the crossed channel. The only modification results in a factor
of 2 in the rate.

Defining $U_{\alpha N_{i}}=|U_{\alpha N_{i}}|\,e^{i\phi_{\alpha i}}$,
$\delta\phi^{+}=\left(\phi_{\alpha 2}-\phi_{\alpha 1}\right)+\left(\phi_{\beta
2}-\phi_{\beta 1}\right)$ and using the narrow width approximation (NWA), the
squared matrix element becomes

$\displaystyle\overline{|\mathcal{M}^{+}|}^{2}=\left[\frac{g^{3}}{2\sqrt{2}M_{W}^{2}}\right]^{2}\pi\left(p_{\ell_{\beta}}\cdot
p_{q}\right)\left(2E_{\ell_{\alpha}}E_{q^{\prime}}+p_{\ell_{\alpha}}\cdot
p_{q^{\prime}}\right)\times$
$\displaystyle\Bigg{\\{}\left|U_{{\alpha}N_{1}}\right|^{2}\left|U_{{\beta}N_{1}}\right|^{2}\frac{M_{N_{1}}}{\Gamma_{N_{1}}}\delta\left(p_{N}^{2}-M_{N_{1}}^{2}\right)+\left|U_{{\alpha}N_{2}}\right|^{2}\left|U_{{\beta}N_{2}}\right|^{2}\frac{M_{N_{2}}}{\Gamma_{N_{2}}}\delta\left(p_{N}^{2}-M_{N_{2}}^{2}\right)$
$\displaystyle+2\left|U_{{\alpha}N_{1}}\right|\left|U_{{\alpha}N_{2}}\right|\left|U_{{\beta}N_{1}}\right|\left|U_{{\beta}N_{2}}\right|M_{N_{1}}M_{N_{2}}\left[\delta\left(p_{N}^{2}-M_{N_{1}}^{2}\right)+\delta\left(p_{N}^{2}-M_{N_{2}}^{2}\right)\right]$
$\displaystyle\left.\left[\cos\delta\phi^{+}\,\frac{\Gamma_{N_{1}}M_{N_{1}}+\Gamma_{N_{2}}M_{N_{2}}}{\left(\Delta
M_{N}^{2}\right)^{2}+\left(\Gamma_{N_{1}}M_{N_{1}}+\Gamma_{N_{2}}M_{N_{2}}\right)^{2}}-\sin\delta\phi^{+}\,\frac{\Delta
M_{N}^{2}}{\left(\Delta
M_{N}^{2}\right)^{2}+\left(\Gamma_{N_{1}}M_{N_{1}}+\Gamma_{N_{2}}M_{N_{2}}\right)^{2}}\right]\right\\}\,,$
(39)

where $\Delta M_{N}^{2}\equiv M_{N_{2}}^{2}-M_{N_{1}}^{2}$. Notice that, for
the interference term, we used the NWA as follows:

$\displaystyle\frac{1}{\left(p_{N}^{2}-M_{N_{1}}^{2}+i\Gamma_{N_{1}}M_{N_{1}}\right)\left(p_{N}^{2}-M_{N_{2}}^{2}-i\Gamma_{N_{2}}M_{N_{2}}\right)}=$
$\displaystyle\hskip
126.61467pt\frac{\pi\left(\Gamma_{N_{2}}M_{N_{2}}+\Gamma_{N_{1}}M_{N_{1}}\right)}{\left(\Delta
M_{N}^{2}\right)^{2}+\left(\Gamma_{N_{1}}M_{N_{1}}+\Gamma_{N_{2}}M_{N_{2}}\right)^{2}}\left[\delta\left(p_{N}^{2}-M_{N_{1}}^{2}\right)+\delta\left(p_{N}^{2}-M_{N_{2}}^{2}\right)\right]$
$\displaystyle\hskip 113.81102pt+\frac{i\pi\Delta M_{N}^{2}}{\left(\Delta
M_{N}^{2}\right)^{2}+\left(\Gamma_{N_{1}}M_{N_{1}}+\Gamma_{N_{2}}M_{N_{2}}\right)^{2}}\left[\delta\left(p_{N}^{2}-M_{N_{1}}^{2}\right)+\delta\left(p_{N}^{2}-M_{N_{2}}^{2}\right)\right],$
(40)

which differs from the expression in Ref. [65], as discussed in Section 4.
Assuming $M_{N_{1}}\simeq M_{N_{2}}\equiv
M_{N},\Gamma_{N_{1}}\simeq\Gamma_{N_{2}}\equiv\Gamma_{N}$ and $\Delta
M_{N}\equiv M_{N_{2}}-M_{N_{1}}\neq 0$, and considering that $\left|U_{\alpha
N_{1}}\right|\left|U_{\beta N_{1}}\right|=\left|U_{\alpha
N_{2}}\right|\left|U_{\beta N_{2}}\right|\equiv\left|U_{\alpha
N}\right|\left|U_{\beta N}\right|$, we get

$\displaystyle\overline{|\mathcal{M}^{+}|}^{2}\simeq$
$\displaystyle\left[\frac{g^{3}}{2\sqrt{2}M_{W}^{2}}\right]^{2}\pi\left(p_{\ell_{\beta}}\cdot
p_{q}\right)\left(2E_{{\alpha}}E_{q^{\prime}}+p_{{\alpha}}\cdot
p_{q^{\prime}}\right)\delta\left(p_{N}^{2}-M_{N}^{2}\right)\frac{M_{N}}{\Gamma_{N}}\left|U_{{\alpha}N}\right|^{2}\left|U_{{\beta}N}\right|^{2}$
$\displaystyle\times
2\Biggl{\\{}1+2\left[2\cos\delta\phi^{+}\frac{M_{N}^{2}\Gamma_{N}^{2}}{\left(\Delta
M_{N}^{2}\right)^{2}+4\Gamma_{N}^{2}M_{N}^{2}}-\sin\delta\phi^{+}\frac{M_{N}\Gamma_{N}\Delta
M_{N}^{2}}{\left(\Delta
M_{N}^{2}\right)^{2}+4\Gamma_{N}^{2}M_{N}^{2}}\right]\Biggr{\\}}\,.$ (41)

We observe that the squared amplitude in the case of only one sterile neutrino
factorizes out. Integrating over the phase space and after factorizing the
decay width in the single HNL framework, it is straightforward to obtain

$\Gamma\left(W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{+}q^{\prime}\bar{q}\right)=2\left(1+\cos\delta\phi^{+}\frac{1}{1+y^{2}}-\sin\delta\phi^{+}\frac{y}{1+y^{2}}\right)\left.\Gamma\left(W^{+}\rightarrow\ell_{\alpha}^{+}\ell_{\beta}^{+}q^{\prime}\bar{q}\right)\right|_{N_{1}}\,,$
(42)

where we have defined

$y\equiv\frac{\Delta
M_{N}^{2}}{2M_{N}\Gamma_{N}}\simeq\frac{M_{N_{2}}^{2}-M_{N_{1}}^{2}}{\left(M_{N_{1}}+M_{N_{2}}\right)\Gamma_{N}}=\frac{\Delta
M_{N}}{\Gamma_{N}}\,.$ (43)

Notice that we obtain the same result of Ref. [63]. This is because we are
using the NWA, which corresponds to consider on-shell HNL, leading to the same
result after integrating over $t$, the time evolution of the intermediate HNL,
from $0$ to $\infty$. In fact the factor $1/\Gamma$ coming from the NWA is
equivalent to the factor $\int^{\infty}_{0}\left|e^{-\Gamma
t/2}\right|^{2}=1/\Gamma$, and the interference part of Eq. (A.1) coincides
with the finding of Ref. [63].

###  A.2 Different sign leptons:
$W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}q^{\prime}\bar{q}$

In this appendix, the decay rate for the process
$W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}q^{\prime}\bar{q}$ mediated by the
two HNLs $N_{1,2}$ almost degenerate in mass is computed. The amplitude of
this process is given by

$\mathcal{M}^{-}=\sum_{i=1,2}\frac{g^{3}}{2\sqrt{2}M_{W}^{2}}U_{\alpha
N_{i}}^{*}U_{\beta
N_{i}}\epsilon_{\mu}^{*}\frac{\bar{\ell}_{\beta}\gamma^{\mu}\not{p}_{N}\gamma^{\nu}P_{L}\ell_{\alpha}^{+}\bar{q}\gamma_{\nu}P_{L}q^{\prime}}{p_{N}^{2}-M_{N_{i}}^{2}+i\Gamma_{N_{i}}M_{N_{i}}}\,,$
(44)

where, again, $p_{N}=p_{W}-p_{\ell_{\alpha}}$. With the help of FeynCalc
[102], we obtain for the squared amplitude,

$\displaystyle\left|\overline{\mathcal{M}^{-}}\right|^{2}=$
$\displaystyle\left(\frac{g^{3}}{2\sqrt{2}M_{W}^{2}}\right)^{2}\left\\{\left|U_{\alpha
N_{1}}\right|^{2}\left|U_{\beta
N_{1}}\right|^{2}\frac{K_{11}}{\left(p_{N}^{2}-M_{N_{1}}^{2}\right)^{2}+\Gamma_{N_{1}}^{2}\,M_{N_{1}}^{2}}\right.$
$\displaystyle+$ $\displaystyle\left|U_{\alpha N_{2}}\right|^{2}\left|U_{\beta
N_{2}}\right|^{2}\frac{K_{22}}{\left(p_{N}^{2}-M_{N_{2}}^{2}\right)^{2}+\Gamma_{N_{2}}^{2}\,M_{N_{2}}^{2}}$
$\displaystyle+$ $\displaystyle 2\left.\text{Re}\left[U_{\alpha
N_{1}}^{*}U_{\beta N_{1}}U_{\alpha N_{2}}U_{\beta
N_{2}}^{*}\frac{K_{12}}{\left(p_{N}^{2}-M_{N_{1}}^{2}+i\,\Gamma_{N_{1}}\,M_{N_{1}}\right)\left(p_{N}^{2}-M_{N_{2}}^{2}-i\,\Gamma_{N_{2}}\,M_{N_{2}}\right)}\right]\right\\}\,,$
(45)

with the $K_{ij}$ factors defined as

$\displaystyle K_{ij}=$ $\displaystyle-\frac{16}{M_{W}}\left(p_{\beta}\cdot
p_{q^{\prime}}\right)\biggl{\\{}M_{W}\,\left(p_{N_{i}}\cdot
p_{N_{j}}\right)\Bigl{[}\left(p_{\alpha}\cdot
p_{q}\right)+2\,E_{\alpha}\,E_{q}\Bigr{]}\biggr{.}-2\,E_{\alpha}\left[\left(p_{N_{i}}\cdot
p_{W}\right)\left(p_{N_{j}}\cdot p_{q}\right)\right.$
$\displaystyle\left.+\left(p_{N_{j}}\cdot p_{W}\right)\left(p_{N_{i}}\cdot
p_{q}\right)\right]\biggl{.}-M_{W}\left[\left(p_{N_{i}}\cdot
p_{q}\right)\left(p_{N_{j}}\cdot p_{\alpha}\right)+\left(p_{N_{j}}\cdot
p_{q}\right)\left(p_{N_{i}}\cdot p_{\alpha}\right)\right]\biggr{\\}}\,.$ (46)

Assuming as before that $M_{N_{1}}\approx M_{N_{2}}\equiv M_{N}$ and
$\Gamma_{N_{1}}\approx\Gamma_{N_{2}}\equiv\Gamma_{N}$, then
$K_{11}=K_{22}=K_{12}\equiv K$. Also, considering that $\left|U_{\alpha
N_{1}}\right|\left|U_{\beta N_{1}}\right|=\left|U_{\alpha
N_{2}}\right|\left|U_{\beta N_{2}}\right|\equiv\left|U_{\alpha
N}\right|\left|U_{\beta N}\right|$, we can simplify the expression notably

$\displaystyle\left|\overline{\mathcal{M}^{-}}\right|^{2}=$
$\displaystyle\left(\frac{g^{3}}{2\sqrt{2}M_{W}^{2}}\right)^{2}\left|U_{\alpha
N}\right|^{2}\left|U_{\beta
N}\right|^{2}\,2\,\left\\{\frac{K}{\left(p_{N}^{2}-M_{N}^{2}\right)^{2}+\Gamma_{N}^{2}\,M_{N}^{2}}\right.$
$\displaystyle+$
$\displaystyle\left.\text{Re}\left[\frac{K\,e^{i\delta\phi^{-}}}{\left(p_{N}^{2}-M_{N_{1}}^{2}+i\,\Gamma_{N_{1}}\,M_{N_{1}}\right)\left(p_{N}^{2}-M_{N_{2}}^{2}-i\,\Gamma_{N_{2}}\,M_{N_{2}}\right)}\right]\right\\}\,,$
(47)

with $\delta\phi^{-}=\left(\phi_{\alpha 2}-\phi_{\alpha
1}\right)-\left(\phi_{\beta 2}-\phi_{\beta 1}\right)$. And using the narrow
width approximation, we get

$\displaystyle\left|\overline{\mathcal{M}^{-}}\right|^{2}=$
$\displaystyle\,2\left(\frac{g^{3}}{2\sqrt{2}M_{W}^{2}}\right)^{2}\left|U_{\alpha
N}\right|^{2}\left|U_{\beta
N}\right|^{2}\,K\,\frac{\pi}{\Gamma_{N}\,M_{N}}\delta\left(p_{N}^{2}-M_{N}^{2}\right)$
$\displaystyle\left\\{1+\text{Re}\biggl{[}2\,\Gamma_{N}\,M_{N}\frac{2\,\Gamma_{N}\,M_{N}+i\,\Delta
M_{N}^{2}}{\left(\Delta
M_{N}^{2}\right)^{2}+4\,\Gamma_{N}^{2}\,M_{N}^{2}}e^{i\,\delta\phi^{-}}\biggr{]}\right\\}\,,$
(48)

Finally, it is straightforward to obtain the total decay width in terms of the
decay width mediated by just one HNL,

$\Gamma\left(W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}q^{\prime}\bar{q}\right)=2\left(1+\cos{\delta\phi^{-}}\frac{1}{1+y^{2}}-\sin{\delta\phi^{-}}\frac{y}{1+y^{2}}\right)\times\Gamma\left(W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{-}q^{\prime}\bar{q}\right)\Bigl{.}\Bigr{|}_{N_{1}}\,,$
(49)

with $y$ defined in Eq. (43).

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