compat=1.1.0 ††institutetext: Instituut-Lorentz, Leiden University, Niels
Bohrweg 2, 2333 CA Leiden, The Netherlands

# Quasi-Dirac Heavy Neutral Leptons in the Left-Right Symmetric Model

Oleksii Mikulenko<EMAIL_ADDRESS>
###### Abstract

We discuss the phenomenology of a pair of degenerate GeV-scale Heavy Neutral
Leptons within the Left-Right Symmetric Model (LRSM) framework, with the third
fermion serving as a dark matter candidate. We highlight the potential of the
recently approved SHiP experiment to test the existence of the light DM
species, and the signatures of lepton number violation as a possible
experimental probe of the model in various experiments. Our findings include
concrete predictions, in some part of the model’s parameter space, for the
effective right-handed couplings
$(V^{R}_{e})^{2}:(V^{R}_{\mu})^{2}:(V^{R}_{\tau})^{2}=0.16:0.47:0.38$ (normal
neutrino hierarchy), $0.489:0.22:0.30$ (inverted hierarchy) of the degenerate
pair.

##  1 Introduction.

The Standard Model (SM) has achieved remarkable success in describing physics
at energies up to the electroweak scale, both in collider experiments and in
the universe. A few observed phenomena, however, still lack an explanation in
the SM: the existence of nonluminous, nonbaryonic dark matter that permeates
the Universe; masses of neutrinos, manifesting in the oscillations between
different flavors; and the striking asymmetry between the matter and
antimatter in our Universe. A minimalistic and natural scenario, capable of
handling all the aforementioned beyond the SM (BSM) problems, consists of
introducing right-handed counterparts of the SM neutrinos, i.e. sterile
neutrinos Minkowski:1977sc ; Mohapatra:1979ia ; Mohapatra:1980yp ;
Schechter:1980gr . As the name indicates, these new particles are not charged
under any of the SM gauge groups, may have Majorana mass, and are an example
of more general Heavy Neutral Leptons (HNL), see Mohapatra:2005wg ;
Abdullahi:2022jlv  for a review. Despite being sterile, the new particles
possess a feeble weak-like interaction as a consequence of mixing with active
neutrinos. The interaction is suppressed by the small mixing angle $U\ll 1$
and opens up the opportunity to search for these elusive particles at future
accelerator experiments, such as SHiP SHiP:2015vad ; SHiP:2018xqw , DUNE
DUNE:2015lol ; Ballett:2019bgd ; Krasnov:2019kdc , LHC-based experiments
Kling:2018wct ; Boyarsky:2021moj ; Curtin:2018mvb ; Dercks:2018wum ;
Hirsch:2020klk ; Aielli:2019ivi ; Feng:2022inv ; Ovchynnikov:2022its , and
future colliders Antusch:2016ejd ; Blondel:2022qqo ; Chrzaszcz:2020emg ;
Mekala:2022cmm ; Boyarsky:2022epg ; Li:2023tbx ; Kwok:2023dck . In addition to
this, HNLs can be constrained with astrophysics and cosmology Dolgov:2000jw ;
Sabti:2020yrt ; Boyarsky:2020dzc ; Vincent:2014rja ; Poulin:2016anj ;
Syvolap:2019dat ; Rembiasz:2018lok ; Suliga:2019bsq ; Coloma:2017ppo ;
Gustafson:2022rsz ; Fischer:2022zwu  and indirect searches such as electroweak
precision measurements Antusch:2014woa ; Fernandez-Martinez:2016lgt ;
Chrzaszcz:2019inj  and charge lepton number violation Bernstein:2013hba ;
Calibbi:2017uvl ; Urquia-Calderon:2022ufc .

The future experiments envisioned can observe HNLs with couplings orders of
magnitude beyond the current limits. Consequently, there is a potential for
abundant signal yield, which would facilitate scrutinizing the properties of
the newly seen particles Mikulenko:2023iqq ; Mikulenko:2023olf . A natural
question then arises: what is the landscape of models that could, in
principle, lead to such an observed signal, and how can we differentiate
between these models? This question is the main motivation of the presented
work.

The neutrino Minimal Standard Model ($\nu$MSM) Asaka:2005an ; Asaka:2005pn
introduces one keV-scale sterile neutrino $N_{1}$ and two heavier HNLs
$N_{2}$, $N_{3}$ with masses at or above the GeV scale. For the naming
convention, we will denote the heavy species that can be probed by direct
searches as HNLs ($N_{2}$, $N_{3}$ in this case). The sterile neutrino $N_{1}$
serves as a warm dark matter candidate Drewes:2016upu ; Boyarsky:2018tvu  and
has a somewhat narrow range of possible masses around the keV scale, bounded
from below by the Tremaine-Gunn bound Tremaine:1979we ; Boyarsky:2008ju  and
from above by the X-ray emission constraints from DM-dominated objects
Abazajian:2001vt . The two HNLs lead to two massive neutrino states via the
so-called type-I seesaw mechanism, leaving the lightest active neutrino
effectively massless.111Neglecting the contribution of the dark matter
$N_{1}$, which is tiny due to the constraints on the particle’s lifetime. The
naive scale of the HNL mixing angles is then defined by the seesaw limit
$U^{2}_{\text{seesaw}}=\sqrt{\Delta m^{2}_{\text{atm.}}}/m_{N}$, with $\Delta
m^{2}_{\text{atm.}}=(50\,\mathrm{meV})^{2}$ being the mass scale of the active
neutrinos. This scale is too small to be reached by the mentioned future
experiments. An attractive theoretical scenario is to consider the HNL pair as
a quasi-Dirac fermion, which alleviates some problems. First, an approximate
Dirac nature translates into a cancellation between the two seesaw
contributions: the mixing angles of HNLs may be large, within the reach of the
near-future experiments, without spoiling active neutrino masses. Second,
there is limited freedom in the relations between the coupling constants to
different lepton flavors, making the model less generic and more easily
verifiable. Finally, oscillations between the two GeV degenerate HNLs may
enhance the efficiency of leptogenesis Akhmedov:1998qx ; Canetti:2010aw ;
Canetti:2012vf ; Klaric:2020phc ; Klaric:2021cpi , a possible mechanism for
the generation of the baryon asymmetry in the Universe Fukugita:1986hr ;
Davidson:2008bu  which is otherwise viable only for HNL masses (much) above
the electroweak scale. Ultimately, the model deals with three BSM phenomena
and is a very attractive target for searches.

Beyond the minimal model, HNLs can possess additional interactions, which may
affect searches for such particles Magill:2018jla ; Ovchynnikov:2023wgg ;
Ovchynnikov:2022rqj ; deGiorgi:2022oks ; Abdullahi:2023gdj ; Fernandez-
Martinez:2023phj . One model that naturally leads to such nonminimal
interactions is the Left-Right Symmetric Model (LRSM) Pati:1974yy ;
Mohapatra:1979ia ; Mohapatra:1980yp , attempting to restore the symmetry
between the left and right particles. The model extends the SM with a new
$SU(2)_{R}$ gauge symmetry and equips HNLs, the right-handed neutrino siblings
in this case, with interactions through the right-handed charged current. The
new symmetry is spontaneously broken at a scale higher than the electroweak
scale, and the processes with the right-handed analog $W_{R}$ of the $W$-boson
are suppressed by the large mass of the mediator $m_{W_{R}}\gg m_{W}$. The
constraints on the scale of new physics come from direct searches CMS:2018hff
; ATLAS:2019lsy ; CMS:2021dzb ; ATLAS:2023cjo ; Cottin:2019drg , meson
precision measurements Bertolini:2014sua , and neutrinoless double beta decay
Tello:2010am ; Gluza:2016qqv . Current limits on the mass of the new boson lie
generally around or below $5\,\mathrm{TeV}$.

In this work, we embed the $\nu$MSM-motivated case of one sterile neutrino and
two HNLs in the quasi-Dirac limit into the minimal Left-Right Symmetric Model.
We study the flavor structure of the HNL couplings and analyze the additional
signatures arising from the new interactions, based solely on the seesaw
relation between the couplings and the active neutrino masses. Specifically,
we found the analytic form of the leptonic analog $V^{R}$ of the CKM matrix,
in the limit of type-I seesaw, depending on two unknown parameters. In this
case and for some experimental setups, the HNL pair can be described as a
single particle with effective coupling uniquely fixed by the experimentally
measured properties of active neutrinos. In addition, we compute the
corrections to the matrices of left and right-handed couplings arising from
the full type-II seesaw relation.

It must be stressed that the considered embedding has a few problems. From the
phenomenological perspective, we treat the $\nu$MSM as a limit of the LRSM, in
which the energy scale of new gauge bosons is too high to affect experimental
searches. The DM candidate $N_{1}$ can still remain cosmologically stable
independent of the new interaction if it is lighter than the electron, i.e.
the decay through right-handed current is kinematically forbidden. However,
the inclusion of gauge interactions with the new scale in the
$10\,\mathrm{TeV}-100\,\mathrm{TeV}$ range (within the reach of collider
experiments) introduces an additional means of thermalization of particles in
the early Universe. This imposes severe obstacles to the model’s capability to
account for both baryon asymmetry Frere:2008ct ; Bezrukov:2012as ;
BhupalDev:2014hro ; Dhuria:2015cfa  and dark matter Bezrukov:2009th . Thus,
some additional mechanisms are needed for the model to be viable. The thermal
overproduction of dark matter may be avoided with a period of early matter
domination by some heavy particles, diluting the abundance of hot relics
Nemevsek:2012cd ; Borah:2017hgt , although the realization of such a scenario
within the experimentally interesting region is challenging Nemevsek:2022anh ;
Nemevsek:2023yjl .

The structure of the paper is as follows. Section 2 describes the
phenomenology of HNLs with combined left and right-current interactions. In
Section 3, we constrain the flavor structure of the left and right coupling
constants from the initial LRSM lagrangian. Section 4 discusses various
interesting experimental signatures of the considered model, specifically the
lepton number violation signatures and dark matter search at the SHiP
experiment. We conclude in Section 5.

##  2 Phenomenology of HNLs below electroweak scale

In this section, we summarize the phenomenology of a unified left-plus-right
interaction of HNLs that will be relevant to us. Without going into details,
we assume that the gauge symmetry contains two subgroups $SU_{L}(2)$ and
$SU_{R}(2)$ that are coupled correspondingly to left and right fermions and
get spontaneously broken at different scales. The right-handed scale is
higher, making the associated gauge bosons much heavier than the SM $W$,
$Z$-bosons. Right-handed HNLs are connected to the light SM fermions via the
charged boson $W_{R}$ and through the Yukawa coupling to neutrinos (mixing).
For a right-handed Weyl spinor $N_{I}$, the Lagrangian contains the following
interactions through left (LH) and right-handed (RH) currents

$\displaystyle\mathcal{L}_{L}\supset\;$
$\displaystyle\frac{g}{\sqrt{2}}\bar{l}\,N_{I}^{c}W+\frac{g}{2\cos\theta_{W}}\bar{\nu}\,N_{I}Z$
$\displaystyle\mathcal{L}_{R}\supset\;$
$\displaystyle\frac{g}{\sqrt{2}}\bar{l}\,N_{I}W_{R}$

Here, $W$, $Z$ are the SM gauge bosons, $W_{R}$ is the charged boson of the
$SU_{R}(2)$, and $\alpha=e,\mu,\tau$ are leptonic flavors. In principle, left
and right bosons can mix with each other: thus, $W$ may mediate the RH
interactions as well. We do not consider such interactions, see Sec. 3.4 for
further details. The mixing between $Z$ and its heavier counterpart $Z_{R}$ is
relevant in the $Z$-pole experiments and is considered separately in Sec. 4.1.
The interactions mediated by the scalar sector of the model are not considered
here: we expect them to be subdominant because of an additional suppression
from the Yukawa couplings of particles in the considered GeV-scale. A review
of all interactions in LRSM can be found in Roitgrund:2014zka .

For HNLs below the electroweak scale, the interactions unify into a
generalized Fermi interaction with both left and right currents:

$\displaystyle\mathcal{L}\supset\quad$ $\displaystyle\theta^{L}_{\alpha
I}\frac{G_{F}}{\sqrt{2}}\bar{l}_{\alpha}N^{c}_{I}\times[\bar{\nu}_{\beta}l_{\beta}+V^{\text{CKM}}_{ij}\bar{u}_{i,L}d_{j,L}]+\theta^{L}_{\alpha
I}\frac{G_{F}}{\sqrt{2}}\bar{\nu}_{\alpha}N^{c}_{I}J_{Z}$
$\displaystyle\bigg{|}\,\,\text{LH}$ $\displaystyle+$
$\displaystyle\theta^{R}_{\alpha
I}\frac{G_{F}}{\sqrt{2}}\bar{l}_{\alpha}N_{I}\times\bigg{[}\tilde{V}^{R}_{J\beta}\bar{N}_{J}l_{\beta}+V^{R,\text{CKM}}_{ij}u_{i,R}d_{i,R}\bigg{]}$
$\displaystyle\bigg{|}\,\,\text{RH}$ $\displaystyle+$ h.c. (1)

containing some phenomenological couplings, with $G_{F}$ being the Fermi
constant and $J_{Z}$ being the SM neutral current. Both types of interactions
are suppressed by small parameters222We use the upper and lower-index notation
$\theta^{L}=\theta_{L}$, $V^{R}=V_{R}$, etc., interchangeably $\theta_{L,R}\ll
1$. To separate the suppression scale and the flavor structure, it is useful
to reparametrize them in the form

$\theta^{L}_{\alpha I}\equiv U^{L}_{I}V^{L}_{\alpha
I},\qquad\theta^{R}_{\alpha I}\equiv U^{R}_{I}V^{R}_{\alpha
I},\qquad\sum_{\alpha}|V^{R/L}_{\alpha I}|^{2}=1.$

The Lagrangian (1) may be viewed as an effective theory with a set of
arbitrary couplings. For some experimental probes, the constraints on the
$\theta_{L}$ can be recasted into the equivalent bounds for $\theta_{R}$. For
example, this can be applied to the $0\nu\beta\beta$ bounds Mitra:2011qr ; one
has to be careful, however, with the possible interference effects
Gluza:2016qqv . The Left-Right symmetric model imposes the following
constraints. First, there are 3 HNL species, and the right-handed suppression
scale $U^{R}_{I}=m^{2}_{W}/m^{2}_{W_{R}}$333Or, generally,
$g_{R}m_{W}^{2}/g_{L}m^{2}_{W_{R}}$, if one does not impose the equality of
the left and right gauge couplings $g_{L}\neq g_{R}$. is equal for all
species. Second, the unspecified two-HNL interaction couplings
$\tilde{V}^{R}_{J\beta}$ must be equal to $(V^{R}_{\beta J})^{*}$. Finally,
the right-handed CKM matrix $V^{R,\text{CKM}}$ is approximately equal to the
standard CKM matrix, up to the corrections that are bounded from above by
$2m_{b}/m_{t}\approx 0.05$ Senjanovic:2015yea . In this work, we assume that
these two CKM matrices are equal.

$N_{I}$$\theta^{L}_{\alpha
I}$$l_{\alpha}$$\nu_{\beta}/u_{i}$$\bar{l}_{\beta}/\bar{d}_{j}$$W$$N_{I}$$\theta^{L}_{\alpha
I}$$\nu_{\alpha}$$f$$\bar{f}$$Z$

$N_{I}$$V^{R}_{\alpha
I}$$l_{\alpha}$$u_{i}$$\bar{d}_{j}$$W_{R}$$N_{I}$$V^{R}_{\alpha
I}$$l_{\alpha}$$V^{*R}_{\beta J}$$N_{J}$$\bar{l}_{\beta}$$W_{R}$

Figure 1: Diagrams of the decay of an HNL. Left: decays through mixing with
active neutrinos. The suppression comes from the small mixing angle
$\theta^{L}_{\alpha I}$. Right: decays involving the right-handed interaction.
These diagrams are suppressed by $\theta^{R}_{\alpha I}=V^{R}_{\alpha
I}(m_{W}/m_{W_{R}})^{2}$ at energies below the $W_{R}$ mass. The bottom-right
diagrams have two HNLs while being suppressed by $\theta^{R}$ only once.

The phenomenology of HNLs with only mixing interactions, $U^{R}=0$, has been
reviewed in Bondarenko:2018ptm . The main qualitative difference that arises
from the inclusion of right-handed interactions is the appearance of the
interaction between different HNL species. In the minimal case, these would be
suppressed by the small left-handed couplings twice, once per each HNL
involved. This opens up an opportunity to probe HNLs which would otherwise be
inaccessible.

Below, we list the relevant production and decay channels for the model.

  1. Production.

For beam-dump experiments, the main HNL production channel is decays of
copiously produced mesons. Since strong interactions respect parity, the meson
form-factors remain the same regardless of whether the decay happens through
the left or the right quark current. The branching ratio for decay of a meson
$X$ has the following scalling

$\text{Br}(X\to Yl_{\alpha}N_{I})=(|\theta^{L}_{\alpha
I}|^{2}+|\theta^{R}_{\alpha I}|^{2})\text{Br}_{\text{norm.}}(X\to
Yl_{\alpha}N_{\alpha})$

where $(\text{norm.})$ with $N_{\alpha}$ stands for a quantity, normalized to
the $\theta^{L}_{\beta}=\delta_{\alpha\beta}$ with all the other couplings set
to zero. These can be found in Bondarenko:2018ptm .

At LHC and future high-energy colliders Blondel:2022qqo ; Curtin:2018mvb ;
Aielli:2019ivi ; Hirsch:2020klk , decays of $W$, $Z$ (and $H$) become
relevant. These happen only through the left current

$\text{Br}(W,Z,H\to\nu_{\alpha}N_{I})=|\theta^{L}_{\alpha
I}|^{2}\text{Br}_{\text{norm.}}(W,Z,H\to\nu_{\alpha}N_{\alpha})$

For even higher-energy searches, with HNL mass above the electroweak scale, a
more case-to-case basis is needed.

  2. Decays.

    1. (a)

Hadronic decays through charge current are mediated by both interactions and
have equal scaling:

$\Gamma(N_{I}\to l_{\alpha}h^{+})=(|\theta^{L}_{\alpha
I}|^{2}+|\theta^{R}_{\alpha I}|^{2})\Gamma_{\text{norm.}}(N_{\alpha}\to
l_{\alpha}h^{+}),$

under our assumption of the equality of left and right-handed CKM matrices.
These decays result in a fully detectable state of a charged lepton and
hadrons.

    2. (b)

Decays into neutral hadrons and neutrino is only mediated by left interaction

$\Gamma(N_{I}\to\nu_{\alpha}h^{0})=|\theta^{L}_{\alpha
I}|^{2}\Gamma_{\text{norm.}}(N_{\alpha}\to\nu_{\alpha}h^{0})$

    3. (c)

Purely leptonic decays $\Gamma(N_{I}\to\nu l\bar{l}\text{ or
}3\nu)\propto|\theta^{L}|^{2}$ are mediated by the left mixing only as well

    4. (d)

Finally, the additional channel of decay into a lighter HNLs $N_{I}\to
N_{J}l_{\alpha}\bar{l}_{\beta}$ is suppressed only once despite involving two
HNL species. There are two diagrams for this process (Fig. 2) that, in
general, interfere with each. However, if the lighter species is substantially
lighter than the decaying HNL, the two diagrams decouple and start to
correspond to the emission of $N_{J}$ with specific helicity: positive
(negative) when $N_{J}$ acts as a particle (antiparticle). In this limit, the
total decay width reads

$\Gamma(N_{I}\to N_{J}l_{\alpha}\bar{l}_{\beta})=U_{R}^{2}\left(|V^{R}_{\alpha
I}|^{2}|V^{R}_{\beta J}|^{2}+|V^{R}_{\beta I}|^{2}|V^{R}_{\alpha
J}|^{2}\right)\Gamma_{\text{norm.}}(N_{\alpha}\overset{W}{\to}l_{\alpha}\nu_{\beta}\bar{l}_{\beta})$

The decay is mediated only by $W$. The full expression for the decay width
with nonvanishing $m_{N_{J}}$ is given in App. A.

$N_{I}$$V^{R}_{\alpha I}$$l_{\alpha}$$V^{*R}_{\beta
J}$$N_{J}$$\bar{l}_{\beta}$$W_{R}$

$N_{I}$$V^{*R}_{\beta I}$$\bar{l}_{\beta}$$V^{R}_{\alpha
J}$$N_{J}$$l_{\alpha}$$W_{R}$

Figure 2: Two interfering diagrams, contributing to the $N_{I}\to
N_{J}l_{\alpha}\bar{l}_{\beta}$ process. When $N_{J}$ is produced being
ultrarelativistic, the diagrams decouple and correspond to the emission of a
specific helicity state.

##  3 Flavor structure from the seesaw relation.

In this section, we derive the numerical values of the $V^{L}$, $V^{R}$
matrices from the seesaw relation, which determines the properties of active
neutrinos.

###  3.1 LRSM Lagrangian

The leptonic Yukawa sector of the LRSM model is Senjanovic:2016vxw :

$\mathcal{L}\supset\bar{L}_{\alpha}([Y_{e}]_{\alpha\beta}\Phi-[Y_{\nu}]_{\alpha\beta}\sigma_{2}\Phi^{*}\sigma_{2})R_{\beta}+\bar{L}_{\alpha}^{c}[Y_{1}]_{\alpha\beta}i\sigma_{2}\Delta_{L}L_{\beta}+\bar{R}_{\alpha}^{c}[Y_{2}]_{\alpha\beta}i\sigma_{2}\Delta_{R}R_{\beta}+\text{h.c.}$

where $L_{\alpha}=(\nu,e_{L})_{\alpha}$, $R_{\alpha}=(N,e_{R})_{\alpha}$ are
left and right lepton doublets, and $\Phi$, $\Delta_{L}$, $\Delta_{R}$ are
scalar fields that acquire the following vacuum expectation values:

$\Phi\to v\,\text{diag}(\cos b,-\sin
b\,e^{-ia})\qquad\Delta_{L,R}\to\begin{pmatrix}0&0\\\ v_{L,R}&0\end{pmatrix}$

The generalized parity symmetry $\mathcal{P}$ that exchanges the left and
right fields $L\leftrightarrow R$, $\Delta_{L}\leftrightarrow\Delta_{R}$
implies the following relations:

$Y^{\dagger}_{e}=Y_{e},\quad Y^{\dagger}_{\nu}=Y_{\nu},\quad Y_{1}=Y_{2}$

Choosing the initial flavor basis that diagonalizes the charge lepton mass
matrix $m_{l}$ after the symmetry breaking, the Lagrangian takes the form

$\mathcal{L}\supset(m^{\text{diag}}_{l})_{\alpha\beta}\bar{l}_{\alpha}l_{\beta}-v[\tilde{Y}_{\nu}]_{\alpha\beta}\bar{\nu}_{\alpha}N_{\beta}+\frac{v_{L}}{v_{R}}M_{\alpha\beta}\bar{\nu}^{c}_{\alpha}\nu_{\beta}+M_{\alpha\beta}\bar{N}^{c}_{\alpha}N_{\beta}$

with the Dirac mass matrix:

$v\tilde{Y}_{\nu}\approx vY_{\nu}\frac{\cos^{2}b-e^{2ia}\sin^{2}b}{\cos
b}-m^{\text{diag}}_{l}e^{ia}\tan b$

The neutrino mass matrix becomes a sum of type-I and type-II seesaw
contributions:

$m_{\nu}=-v^{2}\tilde{Y}_{\nu}M^{-1}\tilde{Y}_{\nu}^{T}+\frac{v_{L}}{v_{R}}M$

The diagonalization of the HNL mass matrix $M_{\alpha\beta}$ is achieved via a
$V_{R}$ unitary rotation:

$N_{\alpha}=[V_{R}]_{\alpha i}\tilde{N}_{I},\qquad M\equiv
V^{*}_{R}m^{\text{diag}}_{N}V_{R}^{\dagger},\qquad\tilde{Y}_{\nu}\equiv
V_{R}YV_{R}^{\dagger}$

and the resulting seesaw relation reads:

$U_{\text{PMNS}}^{*}m_{\nu}^{\text{diag}}U_{\text{PMNS}}^{\dagger}=-v^{2}V_{R}Y[{m^{\text{diag}}_{N}}]^{-1}Y^{T}V_{R}^{T}+\frac{v_{L}}{v_{R}}V^{*}_{R}m^{\text{diag}}_{N}V^{\dagger}_{R}$
(2)

where the neutrino mass matrix is written explicitly diagonalized with the
PMNS matrix.

The identification of the effective couplings that enter Eq. (1) is
straightforward and given by

$\theta^{R}_{\alpha I}=\frac{m_{W_{L}}^{2}}{m_{W_{R}}^{2}}[V^{R}]_{\alpha
I},\qquad\theta^{L}_{\alpha I}=\frac{iv}{m_{N_{I}}}[V^{R}Y]_{\alpha I}$

###  3.2 Approximate symmetry limit in type-I seesaw, $b=0$

We start with the simplest scenario of the type-I seesaw, with no CP-violation
$b=0$ and no type-II contribution $v_{L}/v_{R}=0$. To remind the reader, we
are seeking a solution that involves only two degenerate HNLs whose mixing
angles are allowed to exceed the standard seesaw bound significantly. For
this, one can typically employ the Casas-Ibarra parametrization Casas:2001sr .
However, this approach faces the problem of ensuring that the Yukawa matrix
$Y_{\nu}$ is hermitian (with $b=0$), a requirement that is not automatically
imposed by the parametrization. An analysis discussing the reconstruction of
new physics properties from the seesaw relation can be found in
Nemevsek:2012iq ; Senjanovic:2016vxw ; Senjanovic:2018xtu ; Senjanovic:2019moe
. In this work, we employ a different approach to emphasize the quasi-Dirac
nature of the HNL pair.

We start by realizing an exact lepton symmetry that makes the mixing angles
completely independent from the neutrino masses. The exact lepton symmetry
that yields vanishing $m_{\nu}$ and hermitian $Y_{\nu}$ is achieved by
choosing:

$m_{N}^{\text{diag}}=m_{N}\begin{pmatrix}0&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\qquad Y=y\begin{pmatrix}0&0&0\\\ 0&1&-i\\\
0&i&1\end{pmatrix}$ (3)

The other choice with the matrix $Y\to Y^{T}$ simply corresponds to the
redefinition $N_{2}\leftrightarrow N_{3}$.

The left-mixing matrix takes the form

$\theta_{\alpha I}=\frac{v}{m_{N}}(V_{R}Y)_{\alpha
I}=\frac{\sqrt{2}yv}{m_{N}}V_{L}$
$V_{L}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&\quad
V^{R}_{12}+iV^{R}_{13}&\quad-i(V^{R}_{12}+iV^{R}_{13})\\\ 0&\quad
V^{R}_{22}+iV^{R}_{23}&\quad-i(V^{R}_{22}+iV^{R}_{23})\\\ 0&\quad
V^{R}_{32}+iV^{R}_{33}&\quad-i(V^{R}_{32}+iV^{R}_{33})\end{pmatrix}$

Its structure is completely determined by the right-handed matrix $V^{R}$.

Now, let us try to violate the exact symmetry in the type-I seesaw terms by
introducing small deviations that still respect the hermitian form of $Y$:

$m_{N}^{\text{diag}}=m_{N}\begin{pmatrix}0&0&0\\\ 0&1-\frac{\mu}{2}&0\\\
0&0&1+\frac{\mu}{2}\end{pmatrix},\qquad Y=y\begin{pmatrix}0&0&0\\\
0&1+\epsilon_{3}&-i(1+\epsilon_{2})+\epsilon_{1}\\\
0&i(1+\epsilon_{2})+\epsilon_{1}&1-\epsilon_{3}\end{pmatrix}$

where $|\epsilon_{i}|$, $|\mu|\ll 1$. The parameter $\mu$ controls the
relative mass splitting between the two species
$\mu=(m_{N_{3}}-m_{N_{2}})/m_{N}$ and can be either positive of negative.

In the linear order in terms of the small parameters, Eq. (2) takes the form

$U^{*}_{\text{PMNS}}m_{\nu}^{\text{diag}}U^{\dagger}_{\text{PMNS}}=-V_{R}XV_{R}^{T}$
(4)

with

$X=\frac{y^{2}v^{2}}{m_{N}}\begin{pmatrix}0&0_{1\times 2}\\\ 0_{2\times
1}&X_{2\times 2}\end{pmatrix},\qquad X_{2\times
2}=-2\epsilon_{2}+(-2i\epsilon_{1}+2\epsilon_{3}+\mu)(\sigma_{3}+i\sigma_{1})$
(5)

where $\sigma_{i}$ are Pauli matrices.

We adopt the definition in which
$m^{\text{diag}}_{\nu}=\text{diag}(0,m_{2},m_{3})$ is the mass-ordered
diagonal matrix of active neutrinos with masses:

$\displaystyle m_{2}=\sqrt{\Delta m^{2}_{\text{21}}}\approx 9\,\mathrm{meV}$
$\displaystyle\quad m_{3}=\sqrt{\Delta m^{2}_{\text{3l}}}\approx
50\,\mathrm{meV},$ (NH) $\displaystyle m_{2}=\sqrt{|\Delta
m^{2}_{\text{3l}}|}\approx 50\,\mathrm{meV}$ $\displaystyle\quad
m_{3}=\sqrt{|\Delta m^{2}_{\text{3l}}|+\Delta m^{2}_{\text{21}}}\approx
51\,\mathrm{meV},$ (IH) (6)

Moreover, for convenience, we define

$U_{\text{PMNS}}=\tilde{U}_{\text{PMNS}}P\begin{pmatrix}1&0&0\\\
0&e^{i\eta}&0\\\ 0&0&1\end{pmatrix},\qquad P=\begin{pmatrix}0&1&0\\\ 0&0&1\\\
1&0&0\end{pmatrix}\text{ for IH;}\quad P=1_{3\times 3}\text{ for NH}$ (7)

where $\tilde{U}_{\text{PMNS}}$ is an experimentally measured matrix as given
explicitly in Gonzalez-Garcia:2012hef . In this parameterization, $\eta$ is
the single Majorana phase associated with $m_{2}$ and the matrix $P$ reorders
the massive neutrino states to the standard convention.

The perturbation matrix $X$ can be rewritten as

$X=O^{\dagger}\begin{pmatrix}0&0&0\\\ 0&m_{2}&0\\\
0&0&m_{3}\end{pmatrix}O^{*}$

where the explicit form of the rotational matrix $O$ is given by

$O=\frac{1}{\sqrt{2(m_{2}+m_{3})}}\times\\\
\times\begin{pmatrix}\sqrt{2(m_{2}+m_{3})}&0&0\\\
0&-i(\sqrt{m_{3}}e^{-i\beta}\pm\sqrt{m_{2}}e^{i\beta})&\sqrt{m_{3}}e^{-i\beta}\mp\sqrt{m_{2}}e^{i\beta}\\\
0&-(\sqrt{m_{3}}e^{i\beta}\mp\sqrt{m_{2}}e^{-i\beta})&i(\sqrt{m_{3}}e^{i\beta}\pm\sqrt{m_{2}}e^{-i\beta})\end{pmatrix}$
(8)

and the initial perturbation parameters satisfy the relations

$\displaystyle\epsilon_{1}m_{s}$ $\displaystyle=\frac{m_{3}-m_{2}}{4}\sin
2\beta$ $\displaystyle\epsilon_{2}m_{s}$
$\displaystyle=\pm\frac{\sqrt{m_{2}m_{3}}}{2}$ (9)
$\displaystyle\left(\epsilon_{3}+\frac{\mu}{2}\right)m_{s}$
$\displaystyle=\frac{m_{3}-m_{2}}{4}\cos 2\beta$

where $m_{2}$, $m_{3}$ \- two active neutrino masses, $\beta$ is a free
parameter, and $m_{s}=\frac{y^{2}v^{2}}{m_{N}}$ is the neutrino mass scale
which one would expect without the approximate lepton symmetry. The last
constraint implies that the mass splitting $\mu$ remains a free parameter.
Apart from these parameters, the HNLs are parameterized by two sets of
couplings

$\displaystyle V^{R}=iU^{*}_{\text{PMNS}}O,$ $\displaystyle\qquad
U^{R}_{I}=\frac{m_{W_{L}}^{2}}{m_{W_{R}}^{2}}$ $\displaystyle V^{L}_{\alpha
2}=-iV^{L}_{\alpha
3}=\frac{e^{i\beta}}{\sqrt{m_{2}+m_{3}}}\tilde{U}^{*}_{\text{PMNS}}P\times\begin{pmatrix}0\\\
\mp e^{-i\eta}\sqrt{m_{2}}\\\ \sqrt{m_{3}}\end{pmatrix},$ $\displaystyle\qquad
U^{L}_{2,3}=\frac{\sqrt{2}yv}{m_{N}}$

which are functions of two free parameters: $\eta$ and $\beta$. The sign
change can be compensated for by an appropriate redefinition of the phases
$\eta\to\eta+\pi$, $\beta\to\beta+\pi/2$, which would result in an overall
phase of $V^{R}$, $V^{L}$. Hereafter, we choose the plus sign, whenever the
phase redefinition does not affect the conclusions. The model is fully
parametrized with 4 general parameters $m_{\text{DM}}$, $m_{N}$, $U_{R}$,
$U_{L}$, two internal parameters $\eta$, $\beta$ which fix the flavor matrices
$V^{L}$, $V^{R}$, and the mass splitting parameter $\mu=\Delta m_{N}/m_{N}$.

The approach and results presented here may seem contradictory to the
conclusions of Nemevsek:2012iq  that the Dirac masses cannot exceed the seesaw
limit. The conclusion was based on a naive application of the square root
operation to the Dirac mass matrix $M_{D}\sim\sqrt{m_{\nu}/M_{N}}$. It is
known that matrix square root can be singular, see App. B for a concrete
example. In the recent work Kriewald:2024cgr , closed analytic expressions for
the square root of $3\times 3$ matrix have been derived showing the same
potential singularity. The situation described here requires neutrino mass
matrix of a specific form, which may be considered as fine-tuning in the
bottom-up approach when one reconstructs the properties of heavy neutrinos. In
our top-bottom approach, however, this is a perfectly reasonable assumption.

It is worth to mention several relations in the derived formulae. First, the
parameter $\beta$ enters $V^{L}$ only as an overall phase and does not affect
the ratios between the LH couplings. The obtained results coincide with the
Casas-Ibarra parameterization in the approximate symmetry limit.

Second, the RH couplings of the DM candidate $V^{R}_{\alpha 1}$ do not depend
on the internal parameters, and their magnitudes are

$|V^{R}_{\alpha 1}|^{2}=|\tilde{U}^{\text{PMNS}}_{\alpha
1}|^{2}=\begin{cases}\begin{pmatrix}0.68^{+0.03}_{-0.04},&0.07^{+0.18}_{-0.02},&0.24^{+0.03}_{-0.18}\end{pmatrix},&\quad\text{NH}\\\
\begin{pmatrix}0.022^{+0.002}_{-0.002},&0.57^{+0.04}_{-0.17},&0.41^{+0.17}_{-0.04}\end{pmatrix},&\quad\text{IH}\end{cases}$
(10)

within the $3\sigma$ confidence intervals Esteban:2020cvm  (excluding the
Super-Kamiokande data).

The magnitudes of $V^{R}_{\alpha 2}$, $V^{R}_{\alpha 3}$ depend on both
$\beta$ and $\eta$. However, thanks to the unitarity of the matrix $V^{R}$,
the following combination is fixed:

$\frac{|V^{R}_{\alpha 2}|^{2}+|V^{R}_{\alpha
3}|^{2}}{2}=\frac{1-|V^{R}_{\alpha 1}|^{2}}{2}\equiv\\\ \equiv\langle
V^{R}_{\alpha}\rangle^{2}=\begin{cases}\begin{pmatrix}0.16^{+0.02}_{-0.02},&0.47^{+0.01}_{-0.09},&0.38^{+0.09}_{-0.02}\end{pmatrix},&\text{NH}\\\
\begin{pmatrix}0.489^{+0.001}_{-0.001},&0.22^{+0.09}_{-0.02},&0.30^{+0.02}_{-0.09}\end{pmatrix},&\text{IH}\end{cases}$
(11)

The summarizing ternary plots of the values of $V_{L}$, $V_{R}$ are shown in
Fig. 3.

Figure 3: Ternary plots for the couplings $|V^{L}|^{2}$, $|V^{R}|^{2}$ in the
type-I seesaw with approximate lepton symmetry, for the normal (left) and
inverse (right) neutrino hierarchies. The red lines show the range of possible
values of the LH couplings $|V^{L}_{\alpha 2}|^{2}=|V^{L}_{\alpha 3}|^{2}$.
The green point shows a fixed value of the RH couplings $|V^{R}_{\alpha 1}|$
of the DM candidate, given by Eq. (10). The dashed blue region represents the
range of the values for $|V^{R}_{\alpha 2}|^{2}$, $|V^{R}_{\alpha 3}|^{2}$.
The corresponding pairs of points are located symmetrically around the blue
circle, depicting the fixed value $\langle
V^{R}_{\alpha}\rangle^{2}\equiv(|V^{R}_{\alpha 2}|^{2}+|V^{R}_{\alpha
3}|^{2})/2$, see Eq. (11). The plot uses the best-fit neutrino oscillation
parameters Esteban:2020cvm  (without the SK data) and does not include the
corresponding experimental uncertainties.

###  3.3 Mixed type-I and II seesaw

With the type-II contribution, the seesaw equation becomes

$U_{\text{PMNS}}^{*}m^{\text{diag}}_{\nu}U_{\text{PMNS}}^{\dagger}=-V_{R}XV_{R}^{T}+\kappa(m_{2}+m_{3})V_{R}^{*}\begin{pmatrix}0&0&0\\\
0&1&0\\\ 0&0&1\end{pmatrix}V_{R}^{\dagger}$ (12)

where the introduced a dimensionless parameter

$\kappa=\frac{v_{L}m_{N}}{v_{R}(m_{2}+m_{3})}$ (13)

The parameters $m_{2}$, $m_{3}$ keep their original definition as given in Eq.
(6).

This adjustment leads to a few qualitative changes: first, the mass of the
lightest neutrino in $m_{\nu}^{\text{diag}}$ no longer has to be equal zero,
despite the fact that both terms in the r.h.s. only have rank two. The reason
is that, in general, these two terms are not simultaneously diagonalizable,
and one can easily check that a sum of such two matrices yields a rank three
matrix. Second, the neutrino mass matrix must be specified with three Majorana
phases, rather than only one as in Eq. (7). One additional phase is obviously
attributed to the new neutrino mass eigenvalue. The appearance of the second
phase can be explained in the following way: the previous type-I seesaw
relation (4) has a phase rotation symmetry $U_{\text{PMNS}}\to
e^{i\phi}U_{\text{PMNS}}$, $V_{R}\to e^{-i\phi}V_{R}$ that has been ignored by
choosing a specific form of $U_{\text{PMNS}}$. With two terms involving both
$V_{R}$ and its complex conjugated $V_{R}^{*}$, this symmetry does not hold,
and a change in the overall phase in $U_{\text{PMNS}}$ does modify the
magnitudes of the entries of the matrix $V_{R}$.444One can compare this to the
equation of the form $z^{2}=x^{2}-a(x^{*})^{2}$. For $b=0$, phase rotation in
$z$ can be trivially translated into the change in $x$, but this is no longer
the case for $a\neq 0$.

This is a fourth-order equation in the desired $V_{R}$ matrix, which
complicates the analytic solution. Therefore, we solve the equation
numerically in the following form:

$V_{R}=iU_{\text{PMNS}}e^{iC}O(\beta)$

where the matrix $O$ is given by (8) and $C$ is a hermitian correction matrix,
being zero for $\kappa=0$. Multiplication $V_{R}\to
V_{R}\cdot\text{diag}(e^{i\phi},1,1)$ by a phase leaves the equation
invariant, meaning that one free parameter remains. We add the constraint
$C_{11}=0$ to fix the solution uniquely.

The type-I mass matrix $X$ has three independent parameters $\epsilon_{1}$,
$\epsilon_{2}$, $\epsilon_{3}+\mu/2$. To use the same convenient
parametrization (9), we introduce two adjustment parameters $\delta\mu_{1}$,
$\delta\mu_{2}$:

$\displaystyle\epsilon_{1}m_{s}$
$\displaystyle=\left(\frac{m_{3}-m_{2}}{4}+\delta\mu_{1}(m_{2}+m_{3})\right)\sin
2\beta$ $\displaystyle\epsilon_{2}m_{s}$
$\displaystyle=\frac{\sqrt{m_{3}m_{2}}}{2}+\delta\mu_{2}(m_{2}+m_{3}),$
$\displaystyle\left(\epsilon_{3}+\frac{\mu}{2}\right)m_{s}$
$\displaystyle=\left(\frac{m_{3}-m_{2}}{4}+\delta\mu_{1}(m_{2}+m_{3})\right)\cos
2\beta$

With this choice or parametrization, the $\beta$ parameter drops out from
equation (12) identically.

In summary, the matrix equation (12) has 12 real equations (taking into
account that the l.h.s. matrix is symmetric) with eight entries of the
hermitian matrix $C$, two parameters $\delta\mu_{1}$, $\delta\mu_{2}$, and
four additional parameters: mass of the lightest neutrino, its associated
phase, and two other phases entering a modified version of Eq. (7):

$U_{\text{PMNS}}=\tilde{U}_{\text{PMNS}}P\begin{pmatrix}1&0&0\\\
0&e^{i\eta_{1}}&0\\\ 0&0&e^{i\eta_{2}}\end{pmatrix}$

To match the number of equations to the number of free parameters, we treat
$\eta_{1}$, $\eta_{2}$ as fixed parameters, and search for the solution in the
remaining twelve parameters. We build the solution iteratively: starting from
$\kappa=0$ and the exact solution, we increase $\kappa$ by a small step and
find the values of the unknown parameters by minimizing the total squared
error between the entries on the left and right sides. Once found, $\kappa$ is
increased once again and the process is repeated, producing a unique one-
parametric solution for the matrix $C(\kappa)$ and hence $V_{R}$ for each
fixed $\eta_{1}$, $\eta_{2}$.

The strictest bounds on the total neutrino mass come from cosmological
observations. We adopt the Planck limit $\sum_{\nu}m_{\nu}<0.12\,\mathrm{eV}$
Planck:2018vyg , although this constraint is tentative and could actually be
relaxed DiValentino:2021imh . Our numerical solutions that violate the bound
are excluded. Notably, for most of the combinations of $\eta_{1}$, $\eta_{2}$,
the minimal neutrino mass stays at a level below the bound even for the
largest $\kappa$ we considered.

We performed the scan in the range $\kappa\in(0,20)$ and, for each step, found
the minimum of the total squared error with the tolerance of $10^{-8}$. The
possible values of the left and right-handed couplings are shown in Fig. 4.

Figure 4: Numerical scan for the possible values of $|V^{L}_{\alpha
2}|^{2}=|V^{L}_{\alpha 3}|^{2}$ (top), $|V^{R}_{\alpha 1}|^{2}$ (middle), and
$|V^{R}_{\alpha 2,3}|^{2}$ (bottom) for the included type-II seesaw
contribution, controlled by the $\kappa=v_{L}m_{N}/v_{R}(m_{2}+m_{3})$
parameter. The right-handed couplings averaged over the heavy species $\langle
V^{R}_{\alpha}\rangle^{2}=(|V_{\alpha 2}^{R}|^{2}+|V_{\alpha 3}^{R}|^{2})/2$
can be obtained from the middle plot, using the unitarity of the coupling
matrix. The left (right) column corresponds to the normal (inverse) hierarchy
of neutrino masses. The colors match the legend of Fig. 3. The scan is
performed in the parameter space of $\kappa\in(0,20)$ and two Majorana phases
$\eta_{1}$, $\eta_{2}$. The numerical results indicate that for $\kappa<0.2$
the corrections to $|V^{R,L}|^{2}$ are limited to $\lesssim 0.1$. Higher
values of $\kappa$ effectively allow for arbitrary couplings but also imply
some amount of fine-tuning, to ensure cancellation between the type-I and
type-II contributions.

###  3.4 Type-I with $b\neq 0$

To this point, we neglected the contribution of the $b$ angle in the $\Phi$
vacuum expectation value. The two changes the angle introduces are a
correction to the seesaw relation and mixing between the charged bosons,
leading to RH interactions of HNLs through the Standard Model $W$-boson. We
first argue that the seesaw contribution naturally requires $b$ to be small.
For $b\ll 1$, the neutrino Dirac mass matrix is modified to

$vY_{\nu}\to vY_{\nu}-m^{\text{diag}}_{l}be^{ia}$

The second term will result in the contribution to the neutrino mass matrix at
the scale of $m_{\nu}\sim b^{2}m_{\tau}^{2}/m_{N}$ where the charge lepton
mass matrix is dominated by the tau-lepton mass. This relation can be used to
put a crude constraint

$b\lesssim 10^{-5}\sqrt{\frac{m_{N}}{\mathrm{GeV}}}$ (14)

which are much stricter for HNLs below the TeV scale than the current limits
from the amount of CP violation in the quark sector, $b\sin a\lesssim
m_{b}/m_{t}$ Maiezza:2010ic ; Senjanovic:2014pva ; Senjanovic:2015yea . The
effective coupling for the $W$-mediated coupling has the order of
Senjanovic:1978ev :

$\xi_{W}\sim\frac{m_{W}^{2}}{m^{2}_{W_{R}}}\,b=U_{R}\,b$ (15)

which is further suppressed compared to the effective coupling $U_{R}$ of the
$W_{R}$-mediated interaction.

The constraint (14) is crude because one can hypothesize an adjustment of the
$Y_{\nu}$ matrix to cancel the contribution of the charge lepton mass matrix.
While one cannot exclude such a possibility, this adjustment faces two
problems. First, a change in a hermitian matrix cannot cancel the CP-violating
part, proportional to $b\sin a$. Second, the misalignment between the heavy
states in the charged lepton and sterile neutrino sectors implies that such an
adjustment would necessarily include a contribution of the lightest $N_{1}$
neutrino species, which we forbid in the considered model. In any case, values
of $b$ larger than given by the constraint do require some amount of fine-
tuning on top of the assumed approximate lepton symmetry of the HNL pair.

In this section, we assume that $b$ is small to be treated as a correction and
repeat the same procedure as described in Sec. 3.2. The linearized equation
reads:

$U_{\text{PMNS}}^{*}m^{\text{diag}}_{\nu}U_{\text{PMNS}}^{\dagger}=-V_{R}XV_{R}^{T}+\frac{\tilde{b}e^{ia}}{\sqrt{2}}(m_{2}+m_{3})\\\
\times\left[\begin{pmatrix}0&0&0\\\ 0&m_{\mu}/m_{\tau}&0\\\
0&0&1\end{pmatrix}V_{R}\begin{pmatrix}0&0&0\\\ 0&1&-i\\\
0&i&1\end{pmatrix}V_{R}^{T}+(\text{transpose})\right]$ (16)

where we have kept muon mass $m_{\mu}$ but neglected electron mass. The
dimensionless parameter, equivalent to $\kappa$ in the case of type-II
correction, is given by

$\tilde{b}=b\frac{\sqrt{2}yvm_{\tau}}{m_{2}+m_{3}}=b\frac{m_{\tau}U}{m_{2}+m_{3}},\qquad\tilde{b}\ll\sqrt{\frac{m_{N}U^{2}}{m_{2}+m_{3}}}$
(17)

The inequality represents the necessary condition for the validity of the
linear expansion, which simply requires that the quadratic in $b$ term is
negligible. Namely, $b$ must be sufficiently below the constraint Eq. (14).

We employ the same parametrization as for the type-II correction. The sole
difference is the drop of the redundant Majorana phase $\eta_{2}$, which in
this case can actually be absorbed with a trivial phase rotation of both
$U_{\text{PMNS}}$ and $V_{R}$. The range of the numerical scan is limited to
$\tilde{b}<10$. The plots with the possible values of $|V^{R}|^{2}$,
$|V^{L}|^{2}$ are given in Fig. 5.

Figure 5: Numerical scan for the possible values of $|V^{L}_{\alpha
2}|^{2}=|V^{L}_{\alpha 3}|^{2}$ (top), $|V^{R}_{\alpha 1}|^{2}$ (middle), and
$|V^{R}_{\alpha 2,3}|^{2}$ (bottom) for various $\tilde{b}$, defined in Eq.
(17). The left (right) column corresponds to the normal (inverse) hierarchy of
neutrino masses.

##  4 Experimental probes

In this section, we analyze various experimental probes which may be used to
identify the model. For simplicity, we consider only the minimal case of Sec.
3.2, assuming that the corrections due to the type-II seesaw and nonzero $b$
are negligible. Possible non-minimality of the model, as well as experimental
details related to the precision of such probes, can and do affect the
conclusions of this session. We deliberately omit these details for the sake
of generality.

###  4.1 Sensitivity of FCC-ee in the $Z$-pole mode

The best sensitivity to minimal heavy neutral leptons in the $\sim
10-100\,\mathrm{GeV}$ mass range is offered by the FCC-ee, operating at the
$Z$-pole mass Blondel:2014bra ; Blondel:2022qqo , thanks to the abundant
sample reaching more than $10^{12}$ $Z$-bosons. The production in the decay of
$Z$-boson occurs through either the $Z\to N\nu$ process, mediated by the left-
handed interaction, or the yet-unaccounted mixing between $Z$ and a possible
heavy state $Z_{R}$, which would enable the $Z\to NN$ decay. Our analysis
focuses on the properties of fermions and has been largely independent of the
details of the bosonic sector of the model, only relying on the existence of
$W_{R}$. The process $Z\to NN$, however, does require model-dependent details.
Let us, for a moment, ignore this production channel. The decays of the HNLs
can still be mediated by both left and right-handed interaction, thus offering
a way to probe nonminimal HNL interactions. Additional interactions may
enhance the signal rate to the observable level in parts of the parameter
space otherwise unreachable. This refers specifically to the light HNLs with
small coupling constant $U^{2}_{L}$, for which, in LH-interaction case only,
the decay length reaches macroscopic values and suppresses the probability of
the decay within the detector system. With the new interaction, the HNL decay
width is increased, forcing the HNL to decay faster and avoiding the mentioned
obstacle.

To estimate the sensitivity of an experiment, hosted by FCC-ee, let us provide
simple scaling arguments. At the lower bound of sensitivity Bondarenko:2019yob
, the number of events is proportional to:

$N_{\text{ev}}\propto\underbrace{\left[U^{2}_{L}\text{Br}_{Z\to\nu
N}+c_{Z}U_{R}^{2}\text{Br}_{Z\to
NN}\right]}_{\text{production}}\quad\times\quad\underbrace{[U_{L}^{2}\cdot\text{Br}_{\text{vis.},L}+U_{R}^{2}\cdot\text{Br}_{\text{vis.},R}]}_{\text{decay
width}}$ (18)

where the production branchings

$\displaystyle\text{Br}_{Z\to\nu N}$
$\displaystyle=\left(1-\frac{m_{N}^{2}}{m_{Z}^{2}}\right)^{2}\left(1+\frac{m_{N}^{2}}{2m_{Z}^{2}}\right)$
$\displaystyle\text{Br}_{Z\to NN}$
$\displaystyle=\left(1-\frac{m_{N}^{2}}{m_{Z}^{2}}\right)\sqrt{1-\frac{4m_{N}^{2}}{m_{Z}^{2}}}$

carry the kinematic dependence on the HNL mass, while
$\text{Br}_{\text{vis},L(R)}$ stands for the visible branching ratio for the
pure left(right)-handed interactions. The parameter $c_{Z}$ contains the $ZNN$
couplings, except for the overall suppression scale $U^{2}_{R}$, and in the
minimal LRSM is given by Duka:1999uc :

$c_{Z}=\frac{1}{4}\frac{\cos^{4}2\theta_{W}}{\cos^{4}\theta_{W}}\approx 0.04$

This parameter is notably small. If the production is dominated by the $Z\to
NN$ decay, the decay width is also dominated by $U^{2}_{R}$ and the LH
interaction can be neglected. If $U_{R}\lesssim U_{L}$, both production and
decay are dominated by LH interactions. Nevertheless, there exists a region in
the parameter space between:

$\frac{U^{2}_{L}}{c_{Z}}\frac{\text{Br}_{Z\to\nu N}}{\text{Br}_{Z\to
NN}}\gtrsim U^{2}_{R}\gtrsim U^{L}_{2}$

in which production and decay become parametrically independent.

For a more rigorous analysis of the FCC-ee sensitivity to heavy neutral
leptons, we use the setup described in Blondel:2022qqo , with the expected
$5\cdot 10^{12}$ $Z$-bosons. Specifically, we assume the CLD design, with the
geometry of a cylinder with length $8.6\,\mathrm{m}$ and radius
$5\,\mathrm{m}$. The decay length of the HNL should exceed $400\,\mathrm{\mu
m}$. The number of required decay events is four.

For the sake of generality, we neglect masses of the charged leptons and
$u,d,s,c$ quarks, as neglect the decay of HNLs into a $b$-quark. The last
assumptions can safely done for $N\to lq_{u}b$ decays thanks to the smallness
of $ub$, $sb$ entries of CKM matrix. For the neutral current decays $N\to\nu
bb$, however, this can lead to an error of the order at most $5\%$ level in
the decay width. Finally, we assume that cross-HNL decays $N_{I}\to
N_{J}l_{\alpha}l_{\beta}$ are absent and only one HNL species is produced.
These assumptions, having a minor effect on the results, allow us to eliminate
the flavor-specific details and build general bounds which valid for any
combinations of $V^{L/R}_{\alpha I}$.

Figure 6: Left: sensitivity contours for $U^{2}_{L}$ (red-shaded lines),
assuming HNL production only in $Z\to\nu N$ decays. The filled red region is
the sensitivity of the standard model with only left-handed interactions.
Below the light-red line denoted as Z prod limit in the $U^{2}_{L}$ parameter
space, the number of HNLs produced is below the required limit of four
particles. With $W_{R}$ masses around $20-40\,\mathrm{TeV}$, the sensitivity
to small $U_{L}^{2}$ in the HNL mass range $20-30\,\mathrm{GeV}$ improves at
the lower bound. For larger $W_{R}$ mass, RH starts to dominate, as
demonstrated by the proper sensitivity reach to $U_{R}^{2}$ (blue line) from
Urquia-Calderon:2023dkf  (one displaced vertex, PMNS mixing), which includes
all production channels. The light-blue line corresponds to the production
limit from $Z\to NN$ in the $U^{2}_{R}$ space. Right: estimated sensitivity
contours in the $(U^{2}_{L},U^{2}_{R})$ space for various choices of the HNL
mass, from $7\,\mathrm{GeV}$ to $70\,\mathrm{GeV}$, including both $Z\to\nu N$
and $Z\to NN$ decay channels with $c_{Z}=0.04$.

Under these assumptions, the full decay width can be approximated as

$\Gamma_{L/R}=2U^{2}_{L,R}\times
C_{L/R}\times\frac{G_{F}^{2}m_{N}^{5}}{192\pi^{3}},$

where prefactor 2 counts the charge-conjugated decay modes for a Majorana
fermion. The coefficients are

$C_{R}=6,\qquad
C_{L}=\frac{1}{12}(153-84\sin^{2}\theta_{W}+152\sin^{4}\theta_{W})=11.8$

The decay width of the only invisible mode is $\Gamma(N\to
3\nu)=2U^{2}_{L}\gamma_{0}$. The final note is the account of different HNL
momentum for the two production modes, which determine the decay length.

The sensitivity of FCC-ee with to HNLs is shown in Fig. 6. For smaller mixing
angles and masses below $30\,\mathrm{GeV}$, the LH sensitivity drops due to
the large lifetime of HNLs. The RH interactions mediated by $W_{R}$ with mass
$20-40\,\mathrm{GeV}$ improve the sensitivity to $U_{L}$ in this region by
forcing the particles to decay more promptly. For small $U^{2}_{L}$ and masses
above $30\,\mathrm{GeV}$, the particle lifetime is sufficiently small, and the
sensitivity is saturated by the production limit: the LH interaction is too
tiny to produce any HNLs below the bound.

###  4.2 (De)coherence of the quasi-Dirac pair

Observation of lepton number violating (LNV) processes in collider experiments
would be a smoking gun for new physics, as illustrated by the famous Keung-
Senjanovic process Keung:1983uu  for searching for the LRSM right-handed
neutrinos. Lepton number violation is closely related to leptogenesis and the
nature of active neutrino masses, which, if Majorana, should exhibit
themselves in $0\nu\beta\beta$ decays Vergados:2012xy . In the scenario of
quasi-Dirac HNLs, LNV happens due to oscillations between the lepton and
antilepton-like parts of the pair. The analysis of the properties of these
oscillations and various collider probes can be found in Antusch:2017ebe ;
Antusch:2017pkq ; Drewes:2019byd ; Schubert:2022lcp . If one of the two SM
leptons, required to establish LNV directly, escapes detection, e.g., in the
form of neutrino at lepton colliders Antusch:2023jsa ; Mikulenko:2023ezx  or
is absorbed in the target in case of beam dump experiments Tastet:2019nqj ,
kinematic information may still be used to probe the nature of HNLs. Although
oscillations are believed to be directly defined by the innate properties of
the new particles (namely, mass splitting), Ref. Antusch:2023nqd  claims that
dynamical effects, arising from the uncertainty of particle energy, may lead
to decoherence between the two species and spoil the connection between the
observed LNV and particle parameters. In what follows, we study the properties
of HNL interactions in two limits, assuming that the oscillations are either
very slow or very fast and treating this choice as an independent model
parameter.

At the moment of HNL creation, the two species are produced in a
superposition. If the oscillations between them become too rapid at the moment
of decay, the superposition becomes decoherent: the two particles can be
treated as independent, and any observable signal may be computed as simply a
sum of two separate contributions. If the oscillations are sufficiently slow,
the HNLs remain coherent and interfere with each other, affecting the decay
pattern of such a superposition. In the pure left-handed case, the final
states that violate lepton number conservation cancel each other as a
manifestation of the quasi-Dirac nature of the HNL pair. The situation becomes
more complicated for the right-handed interactions, as they do not respect the
imposed approximate lepton symmetry, which is attributed only to the specific
choice of the Dirac mass matrix without any particular symmetry-driven
restrictions on $V^{R}$.

Below, we analyze the coherent and decoherent cases. For simplicity, we assume
a hierarchical relation between the small couplings $U^{2}_{L}$, $U^{2}_{R}$,
such that the two interactions do not mix in production/decay. We assume that
an HNL is produced together with a lepton of flavor $\alpha$, and interacts
with a $\beta$-flavored (anti)lepton.

  1. (a)

Rapid decoherence.

If the two HNLs are treated as independent particles, the total number $n_{I}$
of HNLs created with the $\alpha$ lepton flavor is simply proportional to
$|V_{\alpha I}|^{2}$. with $V$ standing for either $V^{R}$ or $V^{L}$,
depending on the type of interactions. Each of the HNLs then decays into
$\beta$-flavored leptons and antileptons independently with equal
probabilities.

The number of events with $l_{\alpha}$ accompanying the HNL production and
$l_{\beta}$ or $\bar{l}_{\beta}$ in the final state becomes, up to the overall
normalization, is given by the matrix:

$n_{\alpha\beta}=n_{\alpha\bar{\beta}}\propto P_{\alpha}D_{\beta}(|V_{\alpha
2}|^{2}|V_{\beta 2}|^{2}+|V_{\alpha 3}|^{2}|V_{\beta 3}|^{2})$ (19)

where $P_{\alpha}$ ($D_{\beta}$) is the kinematic factor, carrying the
dependence on the mass of the charged lepton. The kinematic matrix
$P_{\alpha}D_{\beta}$ is therefore determined by the particle mass (measured
by some other means) and the experimental setup, which may have different
acceptance and detection efficiencies for different leptons. In the case of LH
interactions, there is no difference between $I=2,3$, resulting in the
standard product $|V^{L}_{\alpha 2}|^{2}\cdot|V^{L}_{\beta 2}|^{2}$. In the RH
case, the event matrix depends explicitly on both internal model parameters —
phases $\beta$ and $\eta$.

  2. (b)

Coherent pair.

In the opposite scenario, the HNLs form a Dirac fermion. In a sequential
process of HNL production and decay, the intermediate wave function is a
mixture of $N_{2}$ and $N_{3}$. Despite each of them being Majorana particles
and having equal probabilities of transforming into either a lepton or an
antilepton, the same conclusion does not apply to their superposition because
of interference. The combined wave function created after the interaction is
determined by the corresponding currents:

$\displaystyle J_{L,\alpha}$ $\displaystyle\propto V^{L}_{\alpha
2}\bar{l}(N^{c}_{2}-iN^{c}_{3})+\text{h.c.}$ $\displaystyle J_{R,\alpha}$
$\displaystyle\propto V^{R}_{\alpha 2}\bar{l}N_{2}+V^{R}_{\alpha
3}\bar{l}N_{3}+\text{h.c.}$

The matrix of the number of events with different lepton flavors gets the
following form:

$\displaystyle n_{\alpha\bar{\beta}}$ $\displaystyle\propto
P_{\alpha}D_{\beta}|V_{\alpha 2}V^{*}_{\beta 2}+V_{\alpha 3}V^{*}_{\beta
3}|^{2}$ (LNC) $\displaystyle n_{\alpha\beta}$ $\displaystyle\propto
P_{\alpha}D_{\beta}|V_{\alpha 2}V_{\beta 2}+V_{\alpha 3}V_{\beta 3}|^{2}$
(LNV)

For LH interactions $V^{L}_{\alpha 2}=-iV^{L}_{\alpha 3}$, the standard result
is the absence of a lepton number violation in the final state. For RH
interactions, both matrices are always nonzero.

In the case of right-handed production and decay, may be simplified to:

$\displaystyle n_{\alpha\bar{\beta}}\propto$ $\displaystyle
P_{\alpha}D_{\beta}|\delta_{\alpha\beta}-[U_{\text{PMNS}}]_{\alpha
1}[U_{\text{PMNS}}]^{*}_{\beta 1}|^{2}$ (20) $\displaystyle
n_{\alpha\beta}\propto$
$\displaystyle\frac{P_{\alpha}D_{\beta}}{(m_{2}+m_{3})^{2}}\left|\left(U^{*}_{\text{PMNS}}\left(\begin{smallmatrix}0&\,\,0&\,\,0\\\
0&\,\,\mp i\sqrt{m_{2}m_{3}}&\,\,m_{3}-m_{2}\\\ 0&\,\,m_{3}-m_{2}&\,\,\mp
i\sqrt{m_{2}m_{3}}\end{smallmatrix}\right)U^{\dagger}_{\text{PMNS}}\right)_{\alpha\beta}\right|^{2}$
(21)

with the PMNS matrix given by (7) and the plus-minus sign corresponding to
that of Eq. (9). The Majorana phase $\eta$ only affects the matrix of LNV
events, while the LNC event matrix is fixed.

If the pair remains coherent, the dependence of the event matrices on the
parameter $\beta$ vanishes as a result of the underlying symmetry between the
two HNLs.

In some experimental setups, the initial lepton $l_{\alpha}$ may escape
detection. This applies to FCC-ee in the $Z$-pole mode, where the HNL can be
produced in $Z\to\nu N$ decays with an unobservable neutrino, and for beam-
dump experiments, in which all production-related information gets lost in the
target. In this case, the event matrices must be summed up over
$\alpha+\bar{\alpha}$. If the kinematics factors $P_{\alpha}$ are equal all
lepton flavors, most of the event matrices are reduced to the same parametric
dependence thanks to the unitarity of $V^{R}$:

$\sum_{\alpha}n_{\alpha\beta}\propto D_{\beta}(|V^{R}_{\alpha
2}|^{2}+|V^{R}_{\alpha 3}|^{2}),\qquad\text{ if }P_{\alpha}=\text{const}$ (22)

proportional to the averaged couplings $\langle V^{R}_{\alpha}\rangle^{2}$ of
Eq. (11), which are uniquely fixed up to neutrino hierarchy. The list of
matrices that reduce to this relation is given in Table. 1. In other words,
all the internal information of the models vanishes, and the apparent
experimental signal is equivalent to the one from a single HNL with a very
specific set of couplings, determined only by neutrino mass hierarchy. This
can be used a perfect benchmark model of quasi-Dirac HNLs within the minimal
seesaw scenario of Sec. 3.2. For the nonminimal seesaw scenarios, the same
result (22) holds with the appropriate values of $(V^{R}_{\alpha})^{2}$, which
are no longer fixed.

| Decoherent case | Coherent case
---|---|---
$R$ prod., $R$ decay | LNC, LNV | LNC, LNV
$L$ prod., $R$ decay | LNC, LNV | LNV + LNC
Table 1: Event matrices that reduce to the same parametric form of Eq. (22)
after summation over the initial lepton flavor $\alpha$. For LH production and
RH decay, the relation only holds for the sum of the lepton number violating
and conserving decays; in all other cases, the relation holds for both types
separately.

Let us now discuss the implications of the event matrix expressions for
differentiating between the coherent and decoherent cases. First, let us start
with the mentioned FCC-ee in the $Z$-pole regime, specifically the $Z\to\nu N$
channel. Production from $ee\to Z\to NN$ will be discussed below. The neutrino
in the $Z\to\nu N$ decay cannot be detected, therefore, neither the associated
lepton flavor or charge can be identified. One has to rely only on the
distribution of different lepton flavors in the final state, compared with Eq.
22. On the one hand, a deviation from this strict prediction would immediately
exclude the considered minimal model. On the other hand, if the experimental
results do agree with the model, it becomes significantly more challenging to
establish the nature of the new fermions. A possible solution is to employ the
nonzero polarisation of the $Z$-bosons and the resulting features in the
kinematic distribution of the final lepton Blondel:2021mss . Another
possibility consists of observing oscillations directly as time-dependent
variance in the properties of the HNLs Antusch:2023jsa .

At SHiP, oscillations can lead to potentially observable position-dependent
features in the kinematic distributions of decay products Tastet:2019nqj , if
their length is of the $1\,\mathrm{m}$ scale, comparable to the size of the
experiment. Otherwise, one has to rely directly on the event matrices in order
to probe the nature of HNLs. New physics particles are produced inside the
target, propagate to the decay vessel, and decay into a charged lepton and
hadrons. The measurement of particle mass is possible with the kinematics of
the decay products, while the absence of leptonic decays would indicate that
the right-handed interactions are dominant. The summation over the initial
lepton flavors $\alpha$ loses some information, but not as drastically as (22)
suggests. The reason is the kinematic difference between charged leptons in
the range of HNL masses relevant for beam-dump experiments, namely, around a
few GeV. For a concrete example, we consider HNLs with a mass of $\sim
1\,\mathrm{GeV}$: the main production channel is decays $D^{\pm}_{(s)}\to
Xl^{\pm}N$ of $D$-mesons, produced in the proton scatterings. For this mass,
tau-leptons are completely absent, $P_{\tau}=D_{\tau}=0$, while electrons and
muons may treated as equivalent: $P_{e}=P_{\mu}$, $D_{e}=D_{\mu}$ up to small
corrections due to muon mass. Assuming that these leptons are detected with a
unit efficiency, the theoretical prediction of the ratio
$N_{e}/(N_{e}+N_{\mu})$ between the number of events with electron/muon in the
final state can be computed directly. For the decoherent case, the range of
values the ratio can take is somewhat larger than that of the coherent case.
Fig. 7 shows the results for $N_{e}/(N_{e}+N_{\mu})$; if the experimental
measurement lies outside of the range defined by the blue bands (outside the
dashed region), the coherent case in the minimal scenario is excluded.

Figure 7: The fraction $N_{e}/(N_{e}+N_{\mu})$ of observed decays into $e$
versus $e+\mu$ as a function of $\eta$, $\beta$ for normal (left) and inverse
(right) hierarchy. HNLs with a mass around $1\,\mathrm{GeV}$ and only right-
handed interactions are considered. Corrections due to nonzero muon mass are
neglected, assuming $m_{\mu}\ll m_{N},m_{D}-m_{N}$. The absence of
$\tau$-coupled interactions allows the extraction of the flavor structure even
without observing the initial lepton in the $D^{\pm}\to Xl^{\pm}_{\alpha}N$
decay. The colored horizontal band on top shows the dependence on $\eta$
(bottom axis) for the coherent case, with the range of possible values (upper
axis) limited by the blue bands in the colormap. The main plot shows the
dependence of the ratio on the parameters $\beta$, $\eta$ for the decoherent
case, with the dashed regions depicting the values that lie between the blue
band. The coherent case can be excluded for points outside the dashed region.
The plots are periodic in $\beta$, $\eta$, and the plus-minus sign corresponds
to the sign in Eq. (9).

At hadron colliders, the Keung-Senjanović process $pp\to Xl_{\alpha}N\to
Xl_{\alpha}(Yl_{\beta})$ is a powerful probe since both leptons can be traced,
and the event matrix can be fully reconstructed. Misidentification of the
initial and final lepton, which might be an issue for short-lived HNLs, does
not lose information, provided that the event matrices are symmetric. With the
twelve independent entries of $n_{\alpha\beta}$ and $n_{\alpha\bar{\beta}}$, a
full fit of the model can be performed in the standard way. We focus on the
relation between the lepton number violating same-sign (SS) versus conserving
opposite-sign (OS) events. We assume that lepton masses can be neglected, and
consider only the coherent case since, for the decoherent case, the lepton
number conserving/violating events are trivially expected to occur with equal
probability. The relative fractions of SS and OS to the total number of events
as functions of the Majorana phase $\eta$ for different combinations of lepton
flavors are shown in Fig. 8. The fractions of OS events are fixed and can be
used to check the model’s consistency. The SS events provide information on
$\eta$.

Figure 8: The relative fractions of opposite-sign (OS, solid lines) and same-
sign (SS, dashed lines) events for various combinations of lepton flavors in
the final state and normal (left) or inverse (right) neutrino mass hierarchy.
For some values of the Majorana phase $\eta$, a conspicuous feature is the
peculiar combination of a significant fraction of lepton number violating
events with a much smaller amount of lepton number conserving decays. The
feature is most prominent in the $\mu\tau$, $e\mu$ channel for NH, and the
$e\mu$, $e\tau$ channels for IH.

At lepton colliders, the unified low-energy phenomenology described in Sec. 2
is not valid. The analysis of LRSM at lepton colliders can be found in Ref.
Urquia-Calderon:2023dkf . Two main differences that have to be taken into
account are HNL production through the neutral $Z_{R}$-boson of the extended
gauge group and the breakdown of the equivalence between the left- and right-
handed couplings — the adopted definition of $U^{R}$ loses its physical
interpretation. For a concrete example, the HNL production cross-section for
the left Mikulenko:2023ezx  and right-handed interactions Urquia-
Calderon:2023dkf  have the following scalings:

$\sigma^{L}\sim\frac{U_{L}^{2}}{m_{W}^{2}},\qquad\qquad\sigma^{R}(s<m_{W_{R}}^{2})\sim\frac{s}{m_{W_{R}}^{4}}\sim\frac{U_{R}^{2}}{m_{W}^{2}}\times\left(\frac{s}{m_{W_{R}}^{2}}\right)$

where the center-of-mass energy $\sqrt{s}$ is conservatively assumed to be
below the $W_{R}$ mass. It might be possible that the LH-production dominates
the RH one for the parameter space where, technically, $U_{R}>U_{L}$ for the
associated couplings.

The main process for HNLs with RH-interactions is a pair production and decay
$l^{+}l^{-}\to N_{I}N_{J}\to(l_{1}X)(l_{2}Y)$. The flavor and charge
composition of the final leptons provides a window into the properties of
HNLs. In comparison with the previous setups, the initial lepton flavor and
charge are fixed and two HNLs are present, making the relation of experimental
measurements to the couplings $V^{R}$ more convoluted. Therefore, we do not
present concrete numerical results but only a qualitative analysis. The
same/opposite sign signature, which refers to the charges of $l_{1}l_{2}$,
still serves as a powerful probe of nontrivial dynamics in the HNL sector. The
opposite-sign events occur through double LNV or double LNC decays of the two
$N$, while the same-sign events require one LNV and one LNC decay. Deviation
of the SS/OS ratios from the trivial unit (Majorana/decoherent case) or zero
(Dirac) values would be an indication of the coherent pair. The exact
procedure for computing these ratios for the coherent case has to account for
two interferences: one coming from the production of HNLs and one coming from
their decays as a single tied state. The schematic formula for the amplitude
is:

$\displaystyle\langle l^{+}_{\beta}l^{\pm}_{\gamma}|$
$\displaystyle(NN\text{-mediated})|l^{+}_{\alpha}l^{-}_{\alpha}\rangle$
$\displaystyle\Longrightarrow\sum_{IJ}\langle
l^{+}_{\beta}l^{\pm}_{\gamma}|(\text{decay})|N_{I}N_{J}\rangle(C_{W}V_{\alpha
I}V^{*}_{\alpha J}+C_{W}V^{*}_{\alpha I}V_{\alpha J}+C_{Z}\delta_{IJ})$
$\displaystyle\Longrightarrow\sum_{IJ}(\tilde{C}_{W}V_{\beta I}V^{(*)}_{\gamma
J}+\tilde{C}_{W}V^{(*)}_{\gamma I}V_{\beta J})(C_{W}V_{\alpha I}V^{*}_{\alpha
J}+C_{W}V^{*}_{\alpha I}V_{\alpha J}+C_{Z}\delta_{IJ})$

where $C_{W}$, and $C_{Z}$ are kinematic factors for production through the
charged and neutral current, respectively, and $\tilde{C}_{W}$ is the
kinematic factor for HNL decay. These factors are independent of the lepton
flavor by construction, once charge lepton masses are neglected. The notation
$V^{(*)}$ refers to $V$ for the $l^{+}_{\gamma}$ and $V^{*}$ for the
$l^{-}_{\gamma}$ choice. For a given model, the kinematic factors can be
computed explicitly and substituted into the formula to give the probability
of various combinations of final lepton states.

###  4.3 Dark matter at SHiP

Figure 9: Schematic depiction of new physics detection at SHiP. HNLs are
produced in proton-nucleus collisions in the target, move into the decay
volume, and decay via one of the three main decay channels: leptonic
$l\bar{l}+$invisible neutrino of light $N_{1}$, $l$CC into a charged lepton
and hadrons, and NC into hadrons with total charge zero plus invisible
neutrino.  Figure 10: Branching ratios for different theoretical setups:
dominant left-handed interactions (left), right-handed interactions (middle),
and right-handed interaction with light $N_{1}$.

This section aims to estimate the potential of the SHiP experiment to measure
non-minimal interactions of HNLs and ultimately probe the DM candidate
$N_{1}$. The SHiP experiment offers excellent sensitivity to long-living HNLs
in both the minimal scenario SHiP:2018xqw  and the LRSM Castillo-
Felisola:2015bha . For the dark matter searches, the situation is different:
the conventional approach SHiP:2020noy  relies on the scattering of new
physics in the Scattering and Neutrino Detector and has limited to no
potential when considering the sterile neutrino. The reason is that $N_{1}$
interacts even weaker than the active neutrino, for which less than $10^{7}$
events are expected Pastore:2020dgg . In addition, the active neutrino events
constitute a background for the new physics search, further suppressing the
sensitivity. In the minimal case of left-handed interactions, the search for
DM in the decays of long-living HNLs is not feasible as well: the production
of the light DM is suppressed by the small mixing angle squared, one for every
new species. In contrast, the right-handed interactions do not have such a
problem, for the equivalent small coupling originates from the propagator of
the heavy $W_{R}$, and thus, the probability is suppressed only once.

The analysis is based on the branching ratios of visible decay modes,
following the methodology of Mikulenko:2023iqq ; Mikulenko:2023olf . The main
idea is the following: $N_{1}$ is expected to be much lighter than the
detectable HNL pair $N_{2/3}$ and would give rise to the decay:

$N_{2/3}\to l_{\alpha}\bar{l}_{\beta}+\text{invisible}$

which mimics a similar process with a neutrino. The way this decay adjusts the
observed branching ratios is, however, non-trivial. The overall scheme of the
experiment is shown in Fig. 9 with the decay modes of interest: $l\bar{l}$,
$l$CC (charged lepton and hadrons) and NC (neutrino and neutral hadrons). The
$l$CC mode provides a precise measurement of the HNL mass. Simpler models
without a light $N_{1}$, such as the commonly accepted phenomenology of HNL
with left-handed interactions, would be insufficient and fail to fit the
observed set of final states.

Three basic scenarios serving as starting points are: LH-dominated case, RH-
dominated case, and RH-dominated with a light $N_{1}$ species. The qualitative
picture highlighting the differences between the three cases is shown in Fig
10. The main feature of both RH cases is the absence of the NC mode.
Furthermore, HNLs without the $N_{1}$ decay only semileptonically into $l$CC,
making this case the most easily identifiable. The LH and RH+$N_{1}$ are more
difficult to distinguish since both have the $l_{\alpha}l_{\beta}$ mode. To
establish the difference, one has to rely on the NC mode, as well as on the
relative ratios between $l\bar{l}$ and $l$CC. These three scenarios can
readily be distinguished with as few as tens of events.

Any observed experimental result must be tested compared to all possible
theoretical scenarios, including the case of left and right-handed interaction
strengths being of the same order. The main result of this section is a
demonstration that the DM candidate can still be probed in this scenario with
a sufficiently high amount of statistics. For the demonstration, we adopt the
following benchmark model: the HNLs have the mass of $1.5\,\mathrm{GeV}$,
decay decoherently, and have couplings

$\displaystyle|V^{L}_{e}|^{2}:|V^{L}_{\mu}|^{2}:|V^{L}_{\tau}|^{2}$
$\displaystyle=0.11:0.22:0.67$ $\displaystyle|V^{R}_{e2}|^{2}:|V^{R}_{\mu
2}|^{2}:|V^{R}_{\tau 2}|^{2}$ $\displaystyle=0.16:0.46:0.38$
$\displaystyle|V^{R}_{e3}|^{2}:|V^{R}_{\mu 3}|^{2}:|V^{R}_{\tau 3}|^{2}$
$\displaystyle=0.16:0.46:0.38$ $\displaystyle|V^{R}_{e1}|^{2}:|V^{R}_{\mu
3}|^{2}:|V^{R}_{\tau 3}|^{2}$ $\displaystyle=0.49:0.22:0.30$
$\displaystyle\mathcal{R}\equiv$
$\displaystyle\frac{U^{2}_{R}}{U^{2}_{L}+U^{2}_{R}}$

chosen to be equal for $N_{2}$ and $N_{3}$, and consistent with the parameter
space of type-I seesaw of Sec. 3.2 (normal hierarchy). The parameter
$\mathcal{R}$ is a free parameter that quantifies the contribution of RH
interactions.

For a given $\mathcal{R}<1$, we want to distinguish

  1. 1.

the benchmark model without light $N_{1}$ versus a single LH-interacting HNL
with arbitrary couplings $V^{L}$,

  2. 2.

the benchmark model with light $N_{1}$ versus a single LH-interacting HNL with
arbitrary couplings $V^{L}$,

  3. 3.

the benchmark model with the light $N_{1}$ versus a single HNL with both left
and right interactions and arbitrary couplings $V^{L}$, $V^{R}$, and
$\mathcal{R}$, but without $N_{1}$.

We compute the branching ratios for each case and use the package from
Mikulenko:2023iqq  to estimate the required number of events rejecting the
tested models at 90% confidence level with 90% probability. We assume the most
optimistic scenario of zero background and equal detection efficiency for all
decay channels. The results are shown in Fig. 11. For the dominant RH case
$\mathcal{R}=1$ and an existing light DM candidate, less than a hundred events
would suffice to exclude an HNL with arbitrary left and right couplings but no
DM. These estimates should be treated as general lower limits and must be
corrected in the proper analysis to account for the actual limitations of the
detector systems: particle misidentification, different reconstruction
efficiencies, and theoretical uncertainties in the branching ratios, such as
those arising from the imprecise HNL mass measurement.

It must be stressed that the third case is the most general test, assuming an
HNL with arbitrary left and right couplings. In this sense, the data analysis
does not have to rely in any way on the quasi-Dirac setup. Therefore, the
conclusions about the possibility of probing the light candidate are not
restricted to our specific model. The same analysis can be performed for an
arbitrary set of couplings of the light and heavier species; the inconsistency
of the signal with the model without $N_{1}$ will be more or less pronounced
but still present, unless for some fine-tuned cases, such as complete DM-tau
lepton coupling, $|V_{1\tau}^{R}|=1$.

Figure 11: Left: Number of observed events in the benchmark model, necessary
to exclude the tested model versus the benchmark model at 90% confidence level
with 90% probability. The solid lines correspond to the differentiation
between the benchmark models and the tested models. The filled regions
represent the variation of the results for some randomly picked combinations
of couplings, consistent with Type-I seesaw case of Sec. 3.2, in place of the
benchmark model. Right: an illustration of the branching ratios for the
benchmark model with a DM candidate and their incompatibility with LH+RH (top)
and LH (bottom) models.

In the case of the discovery of new physics, the observation may indicate some
kind of non-minimality that requires combined left and right interactions.
This would correspond to the region above the green line of Fig. 11. In such a
scenario, it would be imperative to investigate further whether the signal
contains a contribution of $N_{1}$ decays. The blue line quantifies the
minimal required sensitivity of a follow-up experiment, needed to give the
decisive answer. This estimate depends on the unknown underlying theory;
however, one can obtain a general reference point: In the case of a slight
deviation (say, $2\sigma$) of the observation from the pure LH model, the
number of required events should increase by roughly an order of magnitude.

##  5 Conclusions.

In this work, we have investigated the attractive scenario of a quasi-Dirac
pair of Heavy Neutral Leptons within the Left-Right Symmetric Model. A
degenerate pair can facilitate the generation of baryon asymmetry and, at the
same time, be within the reach of future experimental searches without leading
to too large masses of active neutrinos. Compared to the minimal case, the
HNLs with the Left-Right Symmetric Model possess additional interactions
through the right-handed current and have new phenomenological implications.

We analyzed the flavor structure of the right-handed HNL couplings from the
full seesaw relation in the limit. In the minimal type-I case analytically,
the analytical solution for the parameter space of the couplings is provided.
A notable result is concrete predictions for effective right-handed couplings.
These predictions serve as tangible targets for experimental validation.

Moreover, we have explored the potential impact of our findings on various
experimental probes in the considered model. Our assessment of the sensitivity
of FCC-ee to detect HNLs in the combined model, along with the provision of
simple scaling arguments, shows that right-handed interactions can improve the
sensitivity to the left-handed couplings.

We examined the properties of the quasi-Dirac HNL pair in the coherent and
decoherent state. Each case offers rich phenomenology. We discussed the
experimental signatures that can be employed for SHiP, FCC-ee, and future
hadron and lepton colliders.

Finally, we demonstrated for the first time the potential of the recently
approved SHiP experiment to probe the existence of a dark matter candidate in
decays of HNLs. Our analysis suggests that, in the best-case scenario, SHiP
could provide compelling evidence for dark matter with fewer than a hundred
events. Our findings have become possible thanks to leveraging, on the one
hand, ways to extend the new physics model by accounting for possible
nonminimal interactions, with the attempt to keep the model as predictive as
possible on the other hand. This highlights the necessity for further research
to explore the full potential of next-generation experiments.

## Acknowledgements

I thank Kevin Alberto Urquía Calderón, Marco Drewes, Miha Nemevšek, and Alexey
Boyarsky for their comments and suggestions. This project has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (GA 694896), from the NWO Physics Vrij
Programme “The Hidden Universe of Weakly Interacting Particles”, No.
680.92.18.03, which is partly financed by the Dutch Research Council NWO.
Feynman diagrams have been computed with the use of FeynCalc
Shtabovenko:2020gxv .

##  Appendix A Decay $N_{I}\to N_{J}l_{\alpha}l_{\beta}$

The decay width is given by

$\Gamma(N_{I}\to
N_{J}l_{\alpha}\bar{l}_{\beta})=U_{R}^{2}\frac{G_{F}^{2}m_{N_{I}}^{5}}{16\pi^{2}}\\\
\times\int^{(1-x_{J})^{2}}_{(x_{\alpha}+x_{\beta})^{2}}\frac{dx}{x}\sqrt{\lambda(1,x,x_{J}^{2})\lambda(x,x_{\alpha}^{2},x_{\beta}^{2})}(x-x_{\alpha}^{2}-x_{\beta}^{2})\\\
\bigg{(}(1+x_{J}^{2}-x)(|V^{R}_{\alpha I}|^{2}|V^{R}_{\beta
J}|^{2}+|V^{R}_{\beta I}|^{2}|V^{R}_{\alpha
J}|^{2})-2x_{J}\text{Re}[V^{R}_{\alpha I}V^{R}_{\alpha J}V^{*R}_{\beta
I}V^{*R}_{\beta J}]\bigg{)}.$

Here, $x_{J}=m_{N_{J}}/m_{N_{I}}$, $x_{\alpha}=m_{l_{\alpha}}/m_{N_{I}}$,
$x_{\beta}=m_{l_{\beta}}/m_{N_{I}}$, and

$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$

##  Appendix B Square root of $2\times 2$ matrix.

Two over two matrices are sufficiently simple and an explicit formula for the
square roots exists. For a complex $2\times 2$ matrix $X$, there are two pairs
of square roots (except for special cases) given by Levinger:1980 :

$R_{\pm}=\frac{X+s_{\pm}I}{\sqrt{t+2s_{\pm}}},\qquad t=\text{Tr}\,X,\quad
s_{\pm}=\pm\sqrt{\det X}$

where $I$ is a unit matrix, and each $R_{\pm}$ can be further multiplied by
$\pm 1$.

If the matrix $X$ is chosen to have $t\to 2|s|$, the denominator approaches
zero for one of the two pairs of roots. The numerator does not vanish and the
whole square root can increase indefinitely. To provide an example, a
symmetric $X$ that satisfies this condition is:

$X=\begin{pmatrix}a+b&ia\\\ ia&b-a\end{pmatrix}+O\left(\frac{1}{y}\right)$

where $y\gg a,b\sim 1$ is some large scale that introduces deviations. A
straightforward substitution gives a rather expected result for the large
square root:555This would be the choice in the Eq. (3) with
$\tilde{Y}_{\nu}=V^{*}YV^{\dagger}$, had we decided to follow Nemevsek:2012iq
and adopt generalized $\mathcal{C}$ symmetry.

$R_{-}=O(\sqrt{y})\begin{pmatrix}1&i\\\
i&-1\end{pmatrix}+O\left(\frac{1}{\sqrt{y}}\right)$

It is crucial to note that the eigenvalues of $X$ are equal, up to the
$1/\sqrt{y}$ correction, as is the case as well for the matrix of Eq. (5). It
might be tempting to identify the eigenvalues with the neutrino masses, which
are known to be different, and thus exclude the possibility that $y$ may be
arbitrarily large. However, the seesaw relation is written for symmetric
matrices, and the diagonalization of the matrix should be done by an
orthogonal rotation $X\to OXO^{T}$ rather than by a unitary transformation.
This operation does not leave eigenvalues invariant. Instead, the components
of diagonalized $X$

$|\lambda_{\pm}|^{2}=(\sqrt{|a|^{2}+|b|^{2}}\pm|a|^{2})^{2}$

have enough freedom to be chosen independently in the limit of $y\to\infty$.

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