# Regular black holes have stable cores

Alfio Bonanno  INAF, Osservatorio Astrofisico di Catania, via S.Sofia 78 and
INFN, Sezione di Catania, via S. Sofia 64, I-95123,Catania, Italy  Amir-Pouyan
Khosravi  Frank Saueressig  Institute for Mathematics, Astrophysics and
Particle Physics (IMAPP),
Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

###### Abstract

A common argument suggests that non-singular geometries may not describe black
holes observed in nature since they are unstable due to a mass inflation
effect. We analyze the dynamics associated with spherically symmetric, regular
black holes taking the full backreaction between the infalling matter and
geometry into account. We show that this results in a bounded mass function
and a stable spacetime. If Hawking radiation is taken into account, the
evaporation process ends with a non-singular remnant of Planckian size.

###### pacs:

04.60.-m, 11.10.Hi

The detection of gravitational waves emitted from the merger of two black
holes (Abbott  _et al._ , 2016a, b, 2017) and the first picture of an event
horizon published by the event horizon telescope (Akiyama  _et al._ , 2019)
have moved black holes into the focus of observational science. This raises
the intriguing question whether the black holes observed in nature are indeed
the black holes known from general relativity or black hole “mimickers”
resembling these objects. From the viewpoint of classical general relativity
the theoretical description of black holes is very simple: a Schwarzschild
black hole consists of a spacetime singularity, creating the curvature of
spacetime, which is hidden behind an event horizon. The latter realizes the
cosmic censorship hypothesis (Hawking and Ellis, 2011), stating that spacetime
singularities can not be observed by exterior observers. Moreover, the no-hair
theorem Israel (1967, 1968); Carter (1971) states that black holes in general
relativity are characterized completely by their mass, charge, and angular
momentum. Any perturbation created, e.g., during the formation of a black hole
in a stellar collapse, dies off quickly, ensuring that the configuration
settles to the simple, static solution. Mutatis mutandis, this simple picture
also holds for realistic Kerr-Newman black holes carrying angular momentum.

The occurrence of singularities is often taken as an indicator that general
relativity is incomplete and should be generalized to a quantum theory
expelling this feature. In anticipation of such a theory, various ad hoc
modifications of the Schwarzschild solution have been proposed (Bardeen, 1968;
Borde, 1997; Dymnikova, 1992, 2002; Hayward, 2006; Frolov and Zelnikov, 2017),
also see (Frolov, 2014; Ansoldi, ) for a reviews. Commonly, these
modifications replace the black hole singularity by a regular piece of de
Sitter space. As a consequence, regular black hole spacetimes must come with
two spacetime horizons, the (outer) event horizon and an (inner) Cauchy
horizon. The consistency of these models is challenged by the observation that
the Cauchy horizon is generally unstable to external perturbations (Penrose,
1968). In particular, the self-consistency arguments forwarded in (Carballo-
Rubio  _et al._ , 2018) suggest that regular black hole geometries are not
viable models for describing the black holes observed in nature since the
effect of mass inflation renders the regular core exponentially unstable
against perturbations. In this work, we perform the first complete dynamical
analysis of this instability. In conclusion, we show that in regular black
holes the divergence of the mass function is suppressed before the Cauchy
horizon is reached at very large advanced times. Thus regular black hole
geometries are viable candidates for describing black holes. Taking into
account the energy loss due to Hawking radiation, the black hole evaporation
terminates in a extremal black hole with zero Hawking temperature and
Planckian mass.

A generic spherically symmetric line element can always be written as
$ds^{2}=g_{ab}dx^{a}dx^{b}+r^{2}d\Omega^{2}$, $a,b=0,1$, where $r=r(x^{a})$ is
the radius of the 2-spheres with $x^{a}$ being constant. It is convenient to
introduce a generalized Schwarzschild mass function $M(x^{a})$ by means of the
gradient of $r(x^{a})$,111In the sequel, we set Newton’s constant $G=1$,
implying that all dimensionful quantities are measured in Planck units.

$g^{ab}\partial_{a}r\partial_{b}r=f(x^{a})=1-\frac{2M(x^{a})}{r}\,.$ (1)

The taxonomy of different static black hole geometries is encoded in the
radial behavior of $M(r)$. For a Schwarzschild black hole $M=m$ where $m$ is
the mass of the object at large distances $r\gg m$. In the physical picture
proposed by Sakharov (Sakharov, 1966), a phase transition to a false vacuum
occurs at Planckian distances from the center so that a de Sitter core
eventually develops (Dymnikova, 1992; Frolov  _et al._ , 1990) and $M(r)\sim
r^{3}$ for small $r$. An explicit model realizing such a phase transition is
the Hayward model (Hayward, 2006) where

$M(r)=\frac{mr^{3}}{r^{3}+2ml^{2}}\,.$ (2)

Here $l$ is a free parameter whose value should be fixed by the underlying
quantum gravity model. Other suggestions for regular black hole geometries
include the Bardeen black holes (Bardeen, 1968; Ayon-Beato and Garcia, 2000),
the asymptotically safe black holes (Bonanno and Reuter, 2000; Koch and
Saueressig, 2014), Plack stars (Rovelli and Vidotto, 2014; De Lorenzo  _et
al._ , 2015; Saueressig  _et al._ , 2016), and the loop black hole described
in Brown  _et al._ (2011), also see (Frolov, 2016) for additional references
and discussion.

$r=r_{CH}$$r=r_{CH}$$r=r_{EH}$$\Sigma$$v=const$Star$r=0$${\cal J}^{+}$Outer
potential barrier${\cal M}_{-}$${\cal M}_{+}$$H$

Figure 1: The Ori model of the interior of a regular black hole. The $r=0$
hypersurface is not singular in these models. Radiation enters the black hole
backscattered from the outer potential barrier as ingoing null rays. Outgoing
radiation is schematically represented by a thin shell $\Sigma$ which crosses
the Cauchy horizon. The “corner” $H$ is a singular point of the conformal
diagram. At variance with the classical Ori model, no singularity develops in
the future sector of the shell.

The occurrence of a de Sitter core induces a non-trivial topology change in
the black hole geometry. In variance with the Schwarzschild case, the horizon
condition $f(r)=0$ now has two solutions: beside the event horizon (EH)
located at $r_{+}$ there is also an inner, Cauchy horizon (CH) at a smaller
radius $r_{-}<r_{+}$. As a consequence, the spacetime is no longer globally
hyperbolic. We define he surface gravity $\kappa_{\pm}$ at the horizons by

$\kappa_{\pm}\equiv\pm\,\frac{1}{2}\,\left.\frac{\partial f(r)}{\partial
r}\right|_{r=r_{\pm}}\,,$ (3)

where the sign ensures $\kappa_{\pm}>0$. The event horizon emits Hawking
radiation with temperature $T_{\rm EH}=\kappa_{+}/(2\pi)$. Owed to the
resulting energy loss, the event and Cauchy horizons approach each other. A
complete evaporation is attained in an infinite time and the final, asymptotic
configuration is a regular, extremal black hole with vanishing surface
gravity. The mass $m_{\rm cr}$ of the remnant follows form the condition that
the position of $r_{\pm}$ coincide. For the Hayward model, $m_{\rm
cr}=\frac{3\sqrt{3}}{4}l$.

A classical analysis reveals that the Cauchy horizon is generally unstable to
external perturbations (Penrose, 1968). For this reason there have been doubts
about the consistency of regular black hole models (Frolov, 2018). The
physical picture underlying the analysis is as follows: the collapse of a
mass-distribution to form a black hole will lead to the emission of a stream
of gravitational waves when the black hole settles into its final “hairless”
state. A part of this wave tail will be reflected by the gravitational
potential at $r>r_{+}$, creating an ingoing flux of positive energy crossing
the event horizon. In the vicinity of the Cauchy horizon a part of this flux
will again be back-scattered by the gravitational potential in the black hole
interior, creating an outgoing positive energy flux. A consistent calculation
which takes into account the combined effect of the ingoing radation from the
collapsing star and the backscattered gravitational radiation near the CH
shows that a genuine scalar singularity develops at the Cauchy horizon Poisson
and Israel (1989); *poisson90. This situation is illustrated in Fig. 1.

Technically, it is convenient to formulate this analysis in terms of the
coordinates $(v,r)$ where time is parameterized by the ingoing Eddington-
Finkelstein coordinate $v$, defined by the condition that radially ingoing
light rays follow curves with $v=\text{const}$. One then introduces a time-
dependent perturbation in the mass function $M(r)\rightarrow M(r,v)$
describing a shell of outgoing lightlike dust on the interior geometry of the
black hole. The corresponding energy flux is encoded in the Isaacson effective
energy-momentum tensor Isaacson (1968a); *isaacson68b and its decay follows
Price’s law (Price, 1972a, b), $1/v^{p}$, where $p\geq 12$ is an integer
connected with the multi-pole moments of the gravitational radiation. The
behavior of the curvature invariants near the Cauchy horizon is rather
insensitive to the details of the local fields trapped inside the event
horizon. The rate of divergence of the Coulomb component of the Weyl
curvature222Here $C_{\mu\nu\rho\sigma}$ denotes the Weyl tensor and
$\\{l^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\\}$ is a complex null-tetraed with
$l^{\mu}n_{\mu}=-1$, $m^{\mu}\bar{m}_{\mu}=1$. $\Psi_{2}$ is the only non-zero
scalar for a Petrov type-D spacetime. Novikov and Frolov (1989),
$\Psi_{2}\equiv C_{\mu\nu\rho\sigma}l^{\mu}m^{\nu}\bar{m}^{\rho}n^{\sigma}$,
can then be characterized by the anomalous dimension of the instability

$\nu=\frac{d\ln|\Psi_{2}|}{d\ln v}\,.$ (4)

In the limit $v\rightarrow\infty$, $\Psi_{2}\sim v^{-p}\,{\rm
e}^{\kappa_{-}\,v}$, entailing that the anomalous dimension behaves as
$\nu\sim\kappa_{-}\,v-p$. Since $\Psi_{2}\propto M(r,v)$ this effect has has
been dubbed mass-inflation. It is difficult to imagine an efficient self-
regulator mechanism at work before the curvature has curbed the core at
Planckian levels (Anderson  _et al._ , 1993; Bonanno and Reuter, 1999): at
variance with Petrov type-N, type-D curvatures cannot be confined in thin
layers, and it is reasonable to expect that the complete past evolution of the
3-dimensional hyper-cylinder $r=r_{-}$ is singular.

This conclusion is less compelling, however, if the backreaction of the
Hawking radiation in the interior is taken into account. In this case, the
Cauchy horizon also receives a blue shifted influx of negative energy density
originating from the event horizon. This flux could halt the contraction of
the generators of the Cauchy horizon well before the curvature has reached
Planckian levels, at least for mini black holes (Balbinot  _et al._ , 1991).
Drawing a definite conclusion therefore requires comparing the characteristic
time scales of the evaporation process and the growth time of the mass-
inflation instability.

Such an analysis has recently been presented in Carballo-Rubio  _et al._
(2018), describing the evaporation process of the regular black hole through a
sequence of quasi-adiabatic states for which

$\nu_{E}\equiv-\frac{d\ln\kappa_{+}}{d\ln v}\ll\nu\,,$ (5)

always. This led to the conclusion that the characteristic time scale
associated with the evaporation process is too small to efficiently act as a
self-regulator for the interior geometry.

The estimation of the exponent $\nu$, controlling the time scale on which the
instability builds up, is however more delicate than in the original Poisson-
Israel model because of the non-linear functional form of the quasi-local mass
function $M(r,v)$ in these models. A simple self-consistent estimation of the
mass dynamics based on the DTR relations Dray and ’t Hooft (1985); *redmount
might therefore not be sufficient to consistently describe the contraction of
the generators spanning the Cauchy horizon and the energy influx near the
“corner” region $H$ in Fig. 1. In fact, the Cauchy horizon singularity of the
classical Poisson-Israel model is rather weak: the metric coefficient are
still regular and the integrated tidal forces are bounded. This necessitates a
consistent calculation of the back-reaction of the radiation on the Cauchy
horizon of a regular black hole, taking both the positive energy influx due to
classical perturbations and the backreaction of the regular geometry into
account. This work presents the results of this analysis, describing the
dynamics of a realistic gravitational collapse based on the Ori model (Ori,
1991) of the classical Poisson-Israel model Israel (1967, 1968).

Figure 2: Left: the evolution of the mass function $m_{+}$ in the future
sector of the shell for various values of the initial radius $r_{0}$, with
$p=12$, $\beta=1$, $m=4$, and $l=1$. Right: the evolution of the Kretschmann
scalar $K=R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}$ as a function of $v$ in a
logarithmic plot. The black-dashed lines represents the standard mass
inflation of the Poisson-Israel model.

In the Ori model the outgoing energy flux is modelled by a thin pressureless
null shell $\Sigma$ which divides the spacetime in two regions ${\cal
M}_{\pm}$, inside $(+)$ and outside ($-$) the shell. This shell acts as a
catalyst to trigger the divergence of the mass function. Assuming spherical
symmetry, the metric in each sector of spacetime can be written as

$ds^{2}=-f_{\pm}(r,v_{\pm})dv_{\pm}^{2}+2drdv_{\pm}+r^{2}d\Omega^{2}\,,$ (6)

where $f_{\pm}=1-2M(r,v_{\pm})/r$. The equality of the induced metric on
$\Sigma$ forces $r$ to be the same on ${\cal M}_{\pm}$. For this reason it is
convenient to choose $r$ as a parameter (not necessarily affine) along the
null generators $s_{\pm}^{\mu}=dx_{\pm}^{\mu}/dr=(2/f_{\pm},1,0,0)$ of
$\Sigma$. Einstein’s equations on each sector of the spacetime can then be
expressed in terms of the mass function $M(r,v)$ as

$\frac{\partial M}{\partial r}=-4\pi r^{2}T_{v}{}^{v}\,,\qquad\frac{\partial
M}{\partial v}=4\pi r^{2}T_{v}{}^{r}\,.$ (7)

Continuity of the flux across $\Sigma$ requires (Barrabes and Israel, 1991)

$[T_{\mu\nu}s^{\mu}s^{\nu}]=0\,.,$ (8)

where the square brackets indicate the “jump” of a scalar quantity across the
shell. This condition is consistent with the assumption of $\Sigma$ being
pressureless. In terms of the lapse and mass-function, eq. (8) then reads

$\frac{1}{f_{+}^{2}}\frac{\partial M_{+}}{\partial
v_{+}}=\frac{1}{f_{-}^{2}}\frac{\partial M_{-}}{\partial v_{-}}\,,$ (9)

evaluated at $\Sigma$. In general $v_{+}$ and $v_{-}$ have to be distinguished
because $v_{+}$ is not just the continuation of $v_{-}$ to $\mathcal{M}_{+}$.
The functional dependence of the two is fixed by noting that the position of
the null hypersurface $\Sigma$ in the two coordinate systems is

$f_{+}dv_{+}=f_{-}dv_{-}\;\quad{\rm along}\;\Sigma\,.$ (10)

In the following we shall use (10) to express all physical quantities in terms
of $v\equiv v_{-}$.

We now specify this general framework to the mass function $M(r,v)$ given in
(2), setting

$f_{\pm}=1-\frac{2m_{\pm}(v_{\pm})r^{2}}{r^{3}+2m_{\pm}(v_{\pm})l^{2}}\,.$
(11)

Since the shell moves light-like, we have $f_{-}dv_{-}=2dr$, implying

$\dot{r}(v)=\frac{1}{2}-\frac{m_{-}\,r^{2}}{r^{3}+2\,l^{2}\,m_{-}}\,.$ (12)

Here the dot stands for derivative with respect to $v$ and it is understood
that $r$ and $m_{\pm}$ are $v$-dependent functions specifying the position of
$\Sigma$ and the mass density. Combining eqs. (9) and (10) furthermore gives a
differential equation for the mass function

$\displaystyle\frac{r^{4}\,\dot{m}_{+}+6\,l^{2}\,m_{+}^{2}\,\dot{r}}{(r^{3}+2l^{2}m_{+})(r^{3}-2m_{+}(r^{2}-l^{2}))}$
$\displaystyle=\frac{r^{4}\,\dot{m}_{-}+6\,l^{2}\,m_{-}^{2}\,\dot{r}}{(r^{3}+2\,l^{2}\,m_{-})(r^{3}-2m_{-}(r^{2}-l^{2}))}\,.$
(13)

The system (Regular black holes have stable cores) and (12) is a coupled
system of non-linear differential equations encoding the evolution of $r(v)$
and $m_{+}(v)$.

The boundary condition at the event horizon is determined by the Price’s tail
behavior so that in the past sector of the shell the mass function $m_{-}(v)$
can be written as

$m_{-}(v)=m_{0}-\frac{\beta}{(v/v_{0})^{p}}$ (14)

where $\beta>0$ is a quantity with the dimension of a mass and $v_{0}$ is the
initial value of $v$. It is possible to show that the right-hand side of
(Regular black holes have stable cores) tends to a finite positive limit,
$c>0$, as $v\rightarrow\infty$. As a consequence

$m_{+}(v)\sim\frac{r_{-}^{3}}{2l^{2}}\,\frac{e^{2cr_{-}v}\,l^{2}-1}{e^{2cr_{-}v}(r_{-}^{2}-l^{2})+1}$
(15)

which is finite and positive in the $v\rightarrow\infty$ limit. Notably, (15)
develops a singularity if the parameter $l$ regularizing the geometry is sent
to zero. The observation that $m_{+}(v)$ asymptotically approaches a constant,
positive value actually extends to a number of other regular black hole
geometries, cf. Table 1. This suggests that the de Sitter core of the geometry
actually also guarantees the absence of the mass-inflation effect. We
highlight that the crucial feature discriminating the regular black hole case
from the classical Ori model is _the sign of the constant $c$_. The change
from $c<0$ (classical Ori model) to $c>0$ (regular black hole) ensures that
the latter geometry does not suffer from a mass-inflation effect. Moreover, we
observe that in our analysis the lapse function in the past sector of the
shell approaches zero as a power law, at variance with what has been assumed
in (Carballo-Rubio  _et al._ , 2018) when discussing the DTR relations.

| $f(r)$ | $\lim_{v\rightarrow\infty}m_{+}(v)$
---|---|---
(Hayward, 2006) | $1-\frac{2mr^{2}}{r^{3}+2ml^{2}}$ | $\frac{r_{-}^{3}}{2(r_{-}^{2}-l^{2})}$
(Bonanno and Reuter, 2000) |  $1-\frac{2mr^{2}}{r^{3}+\omega(2r+9m)}$ |  $\frac{r_{-}(r_{-}^{2}+2\omega)}{2r_{-}^{2}-9\omega}$
(Bardeen, 1968; Ayon-Beato and Garcia, 2000) | $1-\frac{2mr^{2}}{(r^{2}+a^{2})^{3/2}}$ | $\frac{(r_{-}^{2}+a^{2})^{3/2}}{2r_{-}^{2}}$
Table 1:  Asymptotic values of the mass function $m_{+}(v)$ for various
regular black hole geometries. Here $l,\omega$, and $a$ are parameters which
should be determined from quantum gravity.

In order to study the complete evolution of the shell we have numerically
integrated eqs. (Regular black holes have stable cores) and (12) for various
initial conditions. The results are depicted in Fig. 2. This establishes that
the initial exponential growth is suppressed at late times and both $m_{+}$
and the Kretschmann curvature $K=R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}$ are
finite in the $v\rightarrow\infty$ limit. As a consequence, the anomalous
dimension in eq. (4) vanishes asymptotically

$\nu\ll 1\,.$ (16)

Most remarkably, the non-singular black hole geometry (2) respects the
limiting curvature hypothesis (Markov, 1982, 1984; Polchinski, 1989; Frolov,
2016), even at the dynamical level, once the perturbations of the classical
Ori model are included.

We now compare the result (16) and the rate of evaporation $\nu_{E}$, cf. eq.
(5). The massloss due to Hawking radiation follows from the Stefan-Boltzmann
law,

$-\frac{dm_{-}(v)}{dv}=\sigma_{\rm
SB}\left(\frac{\kappa_{+}(v)}{2\pi}\right)^{4}\,A_{\rm EH}(v)\,,$ (17)

where $A_{\rm EH}$ and $\kappa_{+}$ are the area and the surface gravity of
the event horizon. Eq. (17) can be explicitly solved in the large $v$ limit,

$m_{-}(v)=m_{\rm cr}+const/v+O(v^{-2})\,,$ (18)

where $m_{\rm cr}$ is the mass of the asymptotic configuration. Substituting
eq. (18) into eq. (3) shows that the outer surface gravity $\kappa_{+}(v)\sim
v^{-1/2}$ so that the exponent controlling the evaporation time asymptotes to
$\nu_{\rm E}\sim 1/2$. In combination with (16) this implies $\nu_{\rm
E}\gg\nu$. Therefore we conclude that in regular black holes of the Hayward,
Bardeen, and asymptotic safety type, the backreaction of the Hawking radiation
acts on time scales much shorter than the slow growth of the mass function. In
these cases the final state of the geometry at $v\rightarrow\infty$ is a cold
extremal black hole which is not destabilized by the mass-inflation effect.

_Outlook._ It would be interesting to see whether the regularization mechanism
established in this work is also operative in other types of regular black
hole solutions including the black holes discussed by Dymnikova (Dymnikova,
1992, 2002; Platania, 2019). We hope to come back to this point in the future.

###### Acknowledgements.

_Acknowledgements_ A.-P. K. thanks the Osservatorio Astrofisico di Catania for
hospitality during the initial stages of this project. A.-P. K. acknowledges
financial support by an ACRI Young Investigators Training Program.

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# Probing the heavy neutrino hypothesis

Xabier Marcano  Departamento de Física Teórica and Instituto de Física Teórica
UAM/CSIC,
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

###### Abstract

There is a strong experimental program searching for massive sterile
neutrinos. Here we focus on the heavy regime above the GeV scale, where their
existence can be probed in either high-energy colliders or in high-precision
facilities. We first review the current experimental status at colliders,
showing that the LHC already improves LEP results for mixings to electrons and
muons, while mixings to taus remain challenging at a hadron collider. We also
discuss the importance of exploring both lepton number violating and
conserving signals, as well as different flavor channels. Finally, we present
the latest global analysis of electroweak precision and flavor observables,
showing that the intensity frontier provides the strongest constraints for
heavy neutrino masses above the electroweak scale.

##  1 Introduction

Right-handed neutrinos, also known as sterile neutrinos or heavy neutral
leptons (HNLs), are among the simplest yet best motivated new particles to
extend the particle content of the Standard Model (SM). They are introduced by
a plethora of models explaining neutrino mass generation, often in connection
with other open problems in the SM. Unfortunately, their mass scale is usually
unknown, motivating a strong experimental program covering all possible
ranges.

Here, we focus on the heavy neutrino hypothesis, when they are above the few
GeV scale. This energy scale represents the threshold above which the new
neutrinos cannot be produced in light meson or $\tau$ decays, so we enter the
high-energy colliders territory. Colliders such LEP and LHC can efficiently
produce heavy neutrinos below the electroweak (EW) scale, but the rates fall
down for heavier masses, until they become too heavy to be produced.
Nevertheless, their existence induces deviations from unitarity of the
leptonic mixing matrix, so they can still be probed for at the intensity
frontier via EW precision observables, universality ratios or charged lepton
flavor violating (cLFV) processes. In fact, and as we summarize here, direct
searches at colliders provide the strongest bounds for heavy neutrinos from
few GeV to the EW scale [1], above which indirect bounds from precision
physics [2] dominate the current heavy neutrino landscape.

##  2 Current LHC bounds for heavy neutrinos

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

Figure 2: Summary of LHC searches for Majorana heavy neutrinos. Updated from
Abada et al.[1] with recent LHC results . Included also the latest bounds from
non-unitarity [2], which dominate above the EW scale.

We focus first on heavy neutrino masses $M_{N}$ ranging from few to hundreds
of GeVs, which 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.

At a hadronic collider such as the LHC, the main production channel comes from
Drell-Yan W or Z bosons and, for heavy masses, also from vector boson fusion
channels such as $W\gamma$. Notice however that current searches usually
consider only the charged channel since the associated charged lepton helps
triggering the event. Then, the $N$ will decay to charged leptons, neutrinos
and quarks, leading to different signal topologies that we can classify
according to the number of charged leptons as dilepton (as in Fig. 1) or
tripletons. Moreover, depending on their masses and couplings, the heavy
neutrino could be short or long-lived, so we can further split the signatures
in prompt or displaced vertex signals.

We collect and classify in Figure 2 all the heavy neutrino searches done at
the LHC so far, shown in the usual mass-mixing plane. This is an updated
version of the figure by Abada et al.[1], where we have included the latest
results from CMS [3, 4, 5]. We did not include other recent results, such as
the CMS long-lived search for very light masses [6] or the vector boson fusion
searches [7, 8]. The former gives the most relevant bounds for $M_{N}\lesssim
2$ GeV, where there are stronger bounds from tau and light meson decays. The
latter are t-channel processes, so the heavy neutrinos are not directly
produced as in the other searches, which consider on-shell neutrinos.
Moreover, while this allows for sensitivities to very heavy masses, current
bounds are several orders of magnitude weaker than those from precision data,
as we will see later.

Figure 2 reveals several interesting aspects. On the one hand, it makes
manifest that the LHC is immersed in an intense program for heavy neutrino
searches. On the other hand, it shows the complementarity between the
different signal topologies to cover all the parameter space, especially
between prompt and long-lived regimes. As a result, the LHC is already able to
improve LEP results for all heavy neutrino masses and for mixings to electrons
or muons. This is a non-trivial statement, since lepton colliders are
extremely relevant for heavy neutrino searches and not easy to improve by a
hadron collider beyond the fact that they can probe heavier masses. The reason
is that LEP, and similarly future leptonic colliders such as the FCCee,
combined a clean environment with a huge amount of Z bosons collection to
search for HNLs produced in $Z\to\nu N$. This kind of search, which is
challenging to perform at the LHC 111Although proposals as the inclusive
displaced vertex searches [9] could help., has the advantage of being equally
sensitive to all neutrino mixings, explaining why the LHC is not able to
improve LEP results for $U_{\tau N}$.

Another important remark about Figure 2 is that it only shows the results
assuming that the heavy neutrino is a Majorana particle. This is in fact the
most frequent assumption, as it leads to lepton number violating (LNV)
signatures that help to keep backgrounds under control. Unfortunately, current
collider searches are sensitive only to relatively large mixings, 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, which also leads to the naive suppression of the expected
LNV signal. From this point of view, searching Dirac neutrinos via lepton
number conserving (LNC) signals, such as the opposite sign dilepton search by
LHCb [10], or some of the latests analyses for long-lived or triplepton
channels, seems more relevant to explore theoretically motivated scenarios.
Nevertheless, a closer look to the cancelation of LNV signals in symmetry
protected scenarios reveals that it is not as straightforward, and that it
depends on the details and parameter space of the model. In fact, studying
both LNV and LNC signals at the same time could provide us information not
only about the nature of the heavy neutrinos, but also about the neutrino mass
generation, being able even to disentangle between different low-scale seesaw
realizations [11]. Therefore, we stress the importance of keep exploring both
Majorana and Dirac hypotheses at colliders.

##  3 Beyond minimal assumptions in LHC searches

Figure 3:  Recast of same-sign dilepton searches by CMS to a general mixing
pattern for two benchmark points of mass and total mixing
$U^{2}\equiv|U_{eN}|^{2}+|U_{\mu N}|^{2}+|U_{\tau N}|^{2}$. The white area is
still allowed by LHC searches.

With very few exceptions, the large amount of available heavy neutrino bounds
have been derived relying on the assumption of the existence of a single
(usually Majorana) neutrino that mixes with only one lepton flavor. However,
most of the BSM scenarios involving new neutral leptons address the lepton
mixing as a whole. The mixing pattern in these scenarios is expected to be
quite complex, so applying the bounds derived in the context of simplified
hypotheses does not seem adequate. Indeed, using these limits directly will in
general overconstrain the parameter space [1]. Consequently, most of the
experimental bounds for heavy neutrinos need to be recast before being applied
to a generic BSM scenario.

The main reason why we expect the bounds to be modified 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 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 generic flavor dependences of dilepton
and trilepton channels at the LHC, assuming still a single heavy neutrino but
with generic mixing patterns, was given by Abada et al.[1]. The main
conclusion of that inspection of the different processes is that LHC results
could be easily recast to generic mixing patterns if the bounds are provided
for each of the flavor channels individually, besides the usually provided
combination of channels that is performed to increase the final sensitivity.

As an example, let us consider the same-sign prompt dilepton searches at CMS
[12]. This analysis provided individual bounds for the $ee$, $\mu\mu$ and
$e\mu$ channels, considering each time the corresponding simplified mixing
pattern. Nevertheless, given that we know how each of this individual channels
depend on the different mixings [1], we can easily combine these results to
cover all possible mixing patterns, as in the ternary in Figure 3. On the
other hand, trilepton searches, which currently give the best limits for most
of the masses, are derived combining different flavor channels, since they all
depend share the same $|U_{\alpha N}|^{2}$ dependency in the simplified
scenarios (e.g. both $\mu\mu\bar{\mu}$ and $\mu\mu\bar{e}$ channels scale as
$|U_{\mu N}|^{2}$ for a single Majorana neutrino that mixes only to muons).
Nevertheless, as soon as we turn to a more complex scenario, each of these
flavor channels will scale differently, and thus we cannot recast the bounds
of the simplified scenario. Consequently, in order to constraint any generic
BSM scenario, it is important to have access to the constraints imposed by
each of the flavor channels individually (or even better to the correlations),
not just the final combination.

##  4 Current precision bounds for heavy neutrinos

G-SS | LFC Bound | LFV Bound
---|---|---
|  $68\%$CL |  $95\%$CL |  $68\%$CL |  $95\%$CL
$\eta_{ee}$ | $[0.33,1.0]\cdot 10^{-3}$ | $[0.081,1.4]\cdot 10^{-3}$ | - | -
$\eta_{\mu\mu}$ | $1.5\cdot 10^{-5}$ | $1.4\cdot 10^{-4}$ | - | -
$\eta_{\tau\tau}$ | $1.6\cdot 10^{-4}$ | $8.9\cdot 10^{-4}$ | - | -
$\text{Tr}\left[\eta\right]$ | $[0.28,1.2]\cdot 10^{-3}$ | $2.1\cdot 10^{-3}$ | - | -
$\left\lvert\eta_{e\mu}\right\rvert$ | $1.4\cdot 10^{-4}$ | $3.4\cdot 10^{-4}$ | $\mathbf{8.4\cdot 10^{-6}}$ | $\mathbf{1.2\cdot 10^{-5}}$
$\left\lvert\eta_{e\tau}\right\rvert$ | $\mathbf{4.2\cdot 10^{-4}}$ | $\mathbf{8.8\cdot 10^{-4}}$ | $5.7\cdot 10^{-3}$ | $8.1\cdot 10^{-3}$
$\left\lvert\eta_{\mu\tau}\right\rvert$ | $\mathbf{9.4\cdot 10^{-6}}$ | $\mathbf{1.8\cdot 10^{-4}}$ | $6.6\cdot 10^{-3}$ | $9.4\cdot 10^{-3}$
Table 1: Upper bounds (or preferred intervals) from non-unitarity effects in a
general seesaw scenario. The LFC bounds are obtained from the global fit
analysis to EW flavor and precision observables. For off-diagonal
$\eta_{\alpha\beta}$ elements, we also derive limits from cLFV transitions and
highlight the strongest bound for each flavor sector. See Blennow et al.[2]
for details and other scenarios.

Heavy neutrinos can also be probed at the intensity frontier since, even if
they cannot be directly produced, their existence induces deviations from
unitarity of the leptonic mixing matrix. These deviations can be parametrized
in general through a small Hermitian matrix $\eta$:

$N=(\mathbb{I}-\eta)\,U\,,$ (1)

where $U$ is the unitary matrix that diagonalizes the Weinberg operator. This
parametrization is convenient as it encodes non-unitarity effects regardless
of the UV completion that originates the deviations and how many fields it
contains. In the particular case of heavy neutrinos, $\eta$ corresponds to
(half of) the coefficient of the dim-6 operator obtained upon integrating them
out, and therefore it is directly related to their mixings. Thus, setting
bounds on $\eta$ implies setting bounds on the sum of the mixings of all the
heavy neutrinos in the model.

Given the fact that the non-unitarity effects affect many low-energy
processes, we have performed a global analysis to the most important EW
precision and flavor observables. Our new analysis improves previous ones with
a better handle of the correlations and the statistical treatment, as well as
updating the list of observables. In particular, our fit includes: four
determinations of the $W$ boson mass; two determinations of the effective weak
mixing angle; five LEP observables at the $Z$-pole; a CMS measurement of the
$Z$ invisible width; five weak decay ratios of lepton flavor universality; ten
weak decays constraining the CKM unitarity; and cLFV observables. The results
for the case of a general seesaw model (G-SS) are given in Table 1. All the
details of the analysis, as well as the results for other well-motivated
scenarios, can be found in Blennow et al.[2].

These upper bounds can be understood as bounds on the total squared mixing
(the sum) of all heavy neutrinos to a given flavor. In the particular case
considered before for LHC searches, with a single heavy neutrino or a single
pair of pseudo-Dirac neutrinos, we have

$\eta_{\alpha\alpha}=\frac{1}{2}\big{|}U_{\alpha N}\big{|}^{2}\,,$ (2)

so we can directly compare the limits from colliders and from precision
observables, as done in Figure 2 (and neglecting the mild preference for a
non-zero $\eta_{ee}$). We see that non-unitarity effects provide the strongest
bounds for heavy neutrinos heavier than the EW scale.

##  5 Conclusions

Heavy neutrinos can be searched for at both energy- and intensity-frontiers.
In the former, the strong LHC program searching for heavy neutrinos is already
providing the strongest bounds for electron and muon sectors, improving
previous LEP limits thanks to the complementarity of the different kind of
searches (mainly prompt and long-lived). Exploring the tau sector at the LHC
is more challenging, although the latest searches already provide the first
LHC limits.

We have also discussed the potential of collider searches in going beyond the
usually considered simplified scenarios, stressing the importance of exploring
both LNV and LNC signals to learn about the model behind the neutrino mass
generation, and highlighting also the role of the different flavor channels in
exploring the whole parameter space. In particular, we emphasized the fact
that providing the exclusion limits from each of the flavor channels
individually, or even better with correlations, would facilitate recasting the
experimental results to generic and more realistic scenarios.

Finally, we presented an updated and improved global analysis of EW flavor and
precision observables, deriving the strongest bounds for heavy neutrinos with
masses above the EW scale, even beyond the direct ones obtained by LHC.
Therefore, we can summarize the current heavy neutrino landscape as being
dominated by the high-energy frontier from the few GeV to the EW scale, above
which we enter the intensity frontier territory.

## Acknowledgments

This project has received support from the European Union’s Horizon Europe
Programme under the Marie Skłodowska-Curie grant agreement no.
101066105-PheNUmenal, and from the Spanish Research Agency (Agencia Estatal de
Investigación) through the Grant IFT Centro de Excelencia Severo Ochoa No
CEX2020-001007-S.

## References

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